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+[    {        "title": "2401.11044v1.The_Significance_of_Data_Abstraction_Methods_in_Machine_Learning_Classification_Processes_for_Critical_Decision_Making.pdf",        "content": "Highlights\nThe Significance of Data Abstraction Methods in Machine Learning Classifi-\ncation Processes for Critical Decision-Making\nKarol Capa la, Paulina Tworek, Jose Sousa\n•Classification is characterised by the attributes of explicability and uncertainty\nmanagement\n•Medical, behavioural and financial datasets, among others, are often small and full\nof missing data\n•SaNDA was proposed as an explainable method tweaked for such datasets\n•This paper analyses how different abstraction methods influence SaNDA’s perfor-\nmance\n•Modified SaNDA with new abstraction methods is compared against first version\nof SaNDA and Random ForestarXiv:2401.11044v1  [cs.LG]  19 Jan 2024The Significance of Data Abstraction Methods in Machine\nLearning Classification Processes for Critical Decision-Making\nKarol Capa laa,b,∗, Paulina Tworeka, Jose Sousaa\naPersonal Health Data Science group, Sano – Centre for Computational Medicine, Czarnowiejska 36\nbuilding C5, 30-054 Krak´ ow, Poland\nbInstitute of Computer Science, AGH University of Krakow, Kawiory 21, 30-059 Krak´ ow, Poland\nAbstract\nThe applicability of widely adopted machine learning (ML) methods to classification\nis circumscribed by the imperatives of explicability and uncertainty, particularly evident\nin domains such as healthcare, behavioural sciences, and finances, wherein accountability\nassumes priority. Recently, Small and Incomplete Dataset Analyser (SaNDA) has been\nproposed to enhance the ability to perform classification in such domains, by develop-\ning a data abstraction protocol using a ROC curve-based method. This paper focuses\non column-wise data transformations called abstractions, which are crucial for SaNDA’s\nclassification process and explores alternative abstractions protocols, such as constant\nbinning and quantiles. The best-performing methods have been compared against Ran-\ndom Forest as a baseline for explainable methods. The results suggests that SaNDA can\nbe a viable substitute for Random Forest when data is incomplete, even with minimal\nmissing values. It consistently maintains high accuracy even when half of the dataset is\nmissing, unlike Random Forest which experiences a significant decline in accuracy under\nsimilar conditions.\nKeywords: Missing Data, Small Datasets, Explainable Models, Machine Learning,\nData Science, Uncertainty\n1. Introduction\nThe recent upswing in the adoption and advancement of Artificial Intelligence (AI)\napplied to classification for Critical Decision-Making (CDM) has been propelled by the\nimplementation of deep learning, grounded in the unique amalgamation of substantial\ndatasets (big data) and the availability of computational power [20]. This adoption of\ndeep learning is centred on the comparative analysis of outcomes between human decision-\nmaking and machine-driven processes. However, this prompts question on the degree of\ncomparability between these outcomes [24].\nClassification for CDM processes encompasses two fundamental attributes: explain-\nability [24] and uncertainty [34], characteristics notably absent in outcomes generated by\ndeep learning [6, 30]. The integration of deep learning into CDM processes has encoun-\ntered obstacles due to its opaque nature [19, 12], leading to the emergence of research\n∗Corresponding author\nEmail addresses: k.capala@sanoscience.org (Karol Capa la), p.komorek@sanoscience.org\n(Paulina Tworek), j.sousa@sanoscience.org (Jose Sousa)\nPreprint submitted to Information Sciences Journal January 23, 2024trends such as eXplainable Artificial Intelligence (XAI) [18, 27] and one-shot learning\n[17].\nExplainability holds particular and pivotal significance in the technological assimi-\nlation within CDM, primarily due to accountability concerns [6]. When ML informs\nCDM, a comprehensive explanation is imperative to elucidate the classification rationale,\nfacilitating a deeper understanding of the decision-making process.\nNowadays, among various ML methods, deep learning is commonly used, as it has a\nhuge advantage in finding patterns in the data. However, the application of deep learning\nrequires the collection of substantial datasets, which is not achievable in many areas.\nPresence of small and missing data, characteristic for many fields, causes uncertainty [30]\nto be an inherent property of CDM scenarios [28].\nDespite the advancements on data access [25], federated learning [37] and encrypted\nlearning techniques [33], these challenges persist among others across diverse domains,\nincluding healthcare, behavioural sciences, and finances [5, 8, 21]. Moreover, challenges\nsuch as privacy concerns, limited accessibility, ethical considerations, and inherent data\nnature impend the acquisition of significant datasets [26, 14, 35, 31]. Therefore, it is\nnecessary to focus on creating solutions for small and incomplete data.\nRandom Forest was originally presented as an approach for improving the accuracy\nof single decision tree which sometimes has problem with over-fitting and therefore with\ngeneralisation [3]. Random Forest [4] is based on an ensemble of individual decision tree\ntrained on random samples of the training data, thereby achieving different characteristics\nof the data distribution. Once the decision trees have been constructed, the random forest\nalgorithm makes a prediction by averaging the predictions of all of the trees reducing the\nvariance of the model. This produces more accurate predictions than any individual\ntree could make, while remaining explainable. Moreover, in case of medium size, tabular\ndatasets, Random Forest may outperform deep learning on classification task [10, 11].\nThese two properties make them suitable for use in many sensitive areas.\nHowever, similarly to the majority of the widely used ML algorithms, they are unable\nto properly proceed with missing values. Therefore, they require removal of incomplete\nentries, or, when it is not possible or undesired, filling them with the help of imputation.\nAn alternative explainable ML method, SaNDA, built on the use of incomplete data\nand with the classification explainability provided by knowledge graphs (KG), has been\nrecently proposed by us [15]. The solution eliminates the need for imputers, which can\nintroduce erroneous information into the model, especially when the number of missing\nvalues is high or there is a bias in their occurrence. In addition to KG, another impor-\ntant element of SaNDA are abstractions, which is the transformation of data values into\na smaller categorical distribution. More specifically, for each column of data, SaNDA\nassigns the values it contains to one of two states (categories) called UPorDOWN .\nHowever, this binary abstraction choice is not the only possible one. This paper in-\nvestigates the impact of other abstraction methods on the classification performance of\nSaNDA. The aim of the experiments is to improve SaNDA’s performance in classifica-\ntion processes for CDM, while maintaining functionality, even with very large amounts\nof missing data.\nSec. 2 provides a more detailed description of the classification process of SaNDA, as\nwell as the alternative abstraction methods proposed in this paper and the metrics used\nto evaluate model performance. Research questions and descriptions of the experiments\nundertaken to answer them are described in Sec. 3. Results of the numerical experiments\n2are given in Sec. 4. Finally, the paper is closed with a summary and conclusions (Sec. 5).\n2. Model\n2.1. Verification metrics\nSeveral metrics are used as a valuable tool for comparing the performance of ML\nclassification models. They provide a comprehensive assessment of a model’s performance\nacross different aspects. To define chosen metrics the following basic concepts as results\nof the classification task should be introduced:\n•True positives (TP) - number of correctly classified instances of the positive class,\n•False positives (FP) - number of incorrectly classified instances of the negative class,\n•True negatives (TN) - number of correctly classified instances of the negative class,\n•False negatives (FN) - number of incorrectly classified instances of the positive class.\nBased on the concepts several metrics of classification tasks are defined. Among these,\nbalanced accuracy (BA) provides a comprehensive assessment of model performance.\nBA is the primary metrics chosen for evaluation the performance of selected abstraction\nmethods and comparing them with Random Forest. It is one of the simplest metrics\nfor evaluating a classification model’s ability to accurately predict classes in the context\nof imbalanced datasets, which are common, among others, for medical and financial\nproblems. BA is a extension of standard accuracy metrics, it is an average accuracy from\nboth the minority and majority classes, i.e.,\nBA =TP\nTP+FN+TN\nTN+FP\n2. (1)\nBeyond a simple BA, there are other metrics which offer different insights into how\nwell a classification model is performing [13]. Recall, defined as\nrecall =TP\nTP + FN, (2)\nmeasures the ability for the correct identification of instances of the positive class from\nall the actual positive samples in the dataset. This metrics is widely employed in various\ndomains, particularly in application classification algorithms used, e.g. in medical and\nfinancial scenarios. For instance, in healthcare it is crucial especially in medical screening\nand diagnostic testing as high value of recall suggests that the test performs well in\ndetecting TP cases, thereby reducing the chances of missing instances (FN) [13]. In\nfinancial area, recall is frequently measured in the context of risk management and fraud\ndetection to ensure that potential risks or fraudulent activities are detected effectively\n[22].\nHowever, there are also scenarios, where the accuracy of positive predictions made by\na classification model is critical, e.g. consequences of carrying out medical or financial\ninterventions or procedures that are not actually required, stemming from false positive\nresults, can have a substantial impact. In such case precision should be monitored.\nPrecision provides information about the quality of positive predictions made by the\n3model. It is defined as the number of positive instances divided by the total number of\npositive predictions, both correctly and incorrectly classified as positive class:\nprecision =TP\nTP + FP(3)\nAchieving a careful equilibrium between recall and precision holds paramount impor-\ntance in many applications. However, the optimal metrics for evaluation ML models\nshould be chosen based on the specific scope and nature of the problem at hand.\n2.2. SaNDA classification method\nSaNDA consists of two main parts, which interacts with each other to provide compre-\nhensive method of data analysis: classification, and KG [15]. While classification module\ncan separate data into classes and provide information how well model captures properties\nof the data, KG enhance explainability of the results and provide deeper insight into inter-\ndependence between features. Since the primary objective of this paper is to investigate\nthe potential for enhancing SaNDA’s data representation capability, which is primar-\nily evaluated through the classification aspect, this section focuses on the explanation\nof classification method using SaNDA. The most important concepts for a classification\nproblem using SaNDA are abstractions and the classification algorithm itself.\nData abstraction involves simplifying a specific set of data by condensing it into a\nmore streamlined representation of the entire dataset. It allows for eliminating specific\ntraits from an object or concept to distill it down to a collection of fundamental elements.\nTo this end, data abstraction creates a simplified representation of the underlying data,\nwhile hiding its complexities and associated operations. In addition, abstraction method\nused in SaNDA ensures the anonymisation of the data. There are various approaches to\ncreating abstractions.\nThe abstraction is formally defined as [15]:\nDefinition 1. Given AandBtwo sets of numerical values, with |A| ≥ |B|, an abstraction\nis defined as a function F:A→Bthat maps the values of Ato values of B.\nIn other world, process of abstraction maps original values from the data into smaller\nset of values. It is important to mention that abstraction process is performed on column-\nby-column basis. In the founding paper, original data was transformed into two classes,\nusing ROC curve method. However, other methods resulting in the different number\nof abstractions (categories), i.e., cardinality of set |B| ̸= 2, may also be an appropriate\nsolution. All of the methods using ROC curves and the new proposed alternative methods\nare presented in the subsection 2.3. For the need of SaNDA, elements of the new set\nare represent as B=B1, ..., B |B|and can be treated as a set of categorical variables.\nTherefore, if every of nfeatures in the dataset is abstracted to the same amount of\nabstractions, each input is now described by a set of categorical values X={x1,···, xn}.\nThe input constructed in this way can be combined with the natively categorical data,\nenabling easy integration of different types of data [9].\nThen based on obtained data abstractions, we generate an explainable KG represen-\ntation, which is created as it was previously described [16, 15]. Through building the\ngraphs, a representation for each class based on available features is prepared. The sig-\nnificance of each vertex of the KG is represented by the probability of each feature being\nin given category of the given class.\n4Following paper focuses on binary classification, however SaNDA algorithm can be\ndescribed and performed also for arbitrary number of classes [9, 15]. Let Cjbe the jth\nclass. Then for given Cjthe following probability can be computed as:\nP(X|Cj) =nY\ni=1P(xi|Cj) (4)\nThe class with the highest P(X|Cj) is chosen, i.e.\nargmaxC{P(X|Cj)}. (5)\nTherefore, SaNDA assigns a given set of values to the class for which its occurrence is\nmost likely. If the given feature xiis empty (missing value) or is represented by null\nvalues, it is skipped in the calculation of the probability P(X|Cj).\nContrary to the majority of commonly used methods, SaNDA does not divide feature\nspace base on optimisation of some classification function. Instead values of every feature\nare individually divided based on abstraction methods, which does not need (but can)\ntake into account class distribution. From the perspective of the entire feature space, it\ncreates division into grind, which every “cell” is label as one of the classes based on the\nconditions given by Eqs. (4) and (5). These conditions do not consider every individual\n“cell” but rather try to approximate it base on every dimension separately. On the one\nhand, this approach may diminish the classification accuracy of the method by failing to\nincorporate more intricate, nonlinear relationships, particularly in the context of a sparse\ngrid (where the number of abstractions is low). On the other hand it allows to complete\nclassification test even in the present of missing values by considering lower-dimensional\nspace.\n2.3. Abstraction methods\nAbstraction as a procedure for reducing the number of values that a variable can take\nis defined in Sec. 2.2. This section presents methods of abstracting data used in this\npaper.\n2.3.1. ROC Curves\nROC curve abstractions split the values of a feature into two categories in a way that\nmaximizes the separation between the classes in the feature. To be more precise, the\nvalues of the feature in the abstracted column are split into smaller and larger than the\nvalue that maximizes BA, as given by Eq (1).\n2.3.2. constant binning\nAbstractions though constant binning divide the values of a feature into categories of\nequal range. Let Abe a set of the values taken by the feature. Then L= max A−minA\nis a range of values. constant binning divide range of the data L, into nequal intervals\nof length L/n, assigning the feature a number corresponding to the bin number in which\nits original value falls.\n52.3.3. Quantiles\nWhile constant binning divides data to keep the range of every bin equal, abstractions\nbased on quantiles divide data into categories of the equal size. Cut-off values are chosen\nbased on the values of quantiles. Quantile of the order nis defined based on the probability\ndistribution of the random variable ranging over the set Xis such a way that the locations\nyqof the quantile q(q=k/n, where k≤n) given by\nq=Zyq\n−∞p(x)dx. (6)\nIn other words, quantile yqis such value that cumulative distribution function takes at\nthis point value qor alternatively qth part of the data takes values smaller or equal yq.\nQuantile-based abstractions therefore assign the feature the largest kfor which x < y q,\ni.e., corresponding to the highest quantile larger than initial value.\nThe most commonly used quantiles are ones of the order n= 2 (median), n= 4\n(quartiles) and n= 10 (deciles).\n3. Experiments\nThe main goal of this research is to improve the efficacy of the SaNDA algorithm by\nexamining the abstraction phase. To achieve this, it is necessary to first explore how\ndifferent abstraction methods influence the metrics that describe SaNDA performance.\nBased on the results, determine whether there are criteria for choosing an abstraction\nmethod that can be applied in model design, and if so, how to choose the best abstraction\nmethod for a given use of the SaNDA algorithm. Finally, verify how the best-performing\nabstraction methods influences SaNDA performance in the presence of missing values in\nthe dataset. For the sake of comparison with the original research, we used the 5 datasets\nfrom the paper proposing SaNDA [15] in the following experiments, supplemented by 10\nsynthetic DIGEN datasets (8 4426, 10 8322, 17 6949, 22 2433, 23 5191, 24 2433, 32 5191,\n354426, 36 466, 39 5578). A selection of datasets is briefly described below.\nThe Ionosphere dataset [7, 32], a collection of 351 radar measures of the ionosphere in\nGoose Bay, Labrador, is used to classify its structure. The dataset includes 34 numeric\nfeatures that measure the number of free electrons and other electromagnetic signals in\nthe ionosphere.\nThe Sonar dataset [7, 1] consists of 60 numerical features that measure the shape and\ncharacteristics of the sonar signal. The classification task is to distinguish underwater\nsurfaces as rock or metal. It is the smallest number of records equal to 208.\nThe Wisconsin Breast Cancer dataset [7, 36] uses 30 numerical features that measure\nthe shape and composition of a breast mass to describe 569 fine needle aspirates. The\ntask is to distinguish between cancerous and non-cancerous samples. It consists of 569\nentries.\nThe Accelerometer dataset [7, 29] was generated for prediction of motor failure time\nwith application of an artificial neural network. It uses 4 numerical features; 3 of them\nrepresents the values of x, y and z axes, while the fourth is cooler maximum speed per-\ncentage ranging from 20% to 100% with 5% intervals. The fifth attribute was used as\ntarget class, where normal configuration was admitted as negative class, while perpen-\ndicular and opposite configuration as positive class. It has the largest number of records\nequal to 102000.\n6The HIGGS dataset [7, 2] is produced from Monte Carlo simulations of particle decays\nand contains 1 .1×107entrances. Each process is described by 28 features, of which 21\nare kinematic properties measured by the particle detectors in the accelerator and the\nremaining 7 are quantities derived from them. The classification task is to distinguish\nbetween measurements of background noise and those connected to the observation of the\nHiggs particle. 105randomly selected records were taken for the experiments presented\nin this work.\nDIGEN [23] datasets were designed to differentiate the performance of some of the\nleading classification methods. It is the collection of 40 synthetic datasets created from\neach of the generative mathematical functions for testing binary classification task. Every\ndataset from DIGEN contains 10 features of 1000 normally distributed values.\nIn summary, the experimental setup consists of 13 small and 2 medium datasets. This\ndisproportion is caused by the target domain for SaNDA, which is related to the nature\nof data occurring in personal health care or finances.\nTo achieve the goals of this research, we conducted two numerical experiments using\nthe datasets described above. The first experiment focuses on exploring the different\nabstraction methods presented in the Sec 2.3 and their impact on SaNDA performance\nwith an indication of classifciation accuracy. For this purpose, each of the studied datasets\nwas transformed using one of the following abstraction methods:\n•constant binning into 10 and 20 bins\n•quantiles of order 10 (deciles) and 20\n•using ROC curve method as a control group\nThen, using the method used in SaNDA, models were created for classification purposes.\nIt is important to note that in this experiment we do not create any missing data in the\ndatasets.\nFollowing the initial experiment, the abstraction methods yielding the highest BA\nwas selected for each dataset with the selected percentage of missing data. Subsequently,\nthe SaNDA classification algorithm was modified to incorporate this chosen abstraction\nmethods and applied to datasets containing 1%, 5%, 10%, and 50% of missing data.\nDatasets with n% missing data were created from the original datasets by randomly\nremoving n% of data from each column (feature). The results were compared to those\nobtained from the original, ROC curve-based SaNDA and Random Forest algorithms.\nFor Random Forest, missing data were replaced with zeros to be able to process the\ndatasets. Subsequently, for all of the methods the BA, recall and precision values were\ncompared.\n4. Results\nFigure 1 and Table 1 show the BA for experiment 1. For each dataset detailed in\nSection 3, it contrasts the BA acquired from SaNDA across various abstraction types:\nROC, deciles (quantiles 10), quantiles 20, 10 bins, and 20 bins. For the accelerometer,\nHiggs, WDBC, and DIGEN datasets, with the exception of DIGENs: 22 2433, 24 2433,\nand 17 6949, the use of quantiles 20 results in superior BA for classification. On the other\nhand, for the remaining DIGENs, Sonar and Ionosphere datasets, the application of 20\nbins abstraction yields higher BA compared to other abstraction methods.\n7Based on this experiment, it can be observed that overall quantiles 20 offers better\nclassification performance compared to 20 bins abstraction method. In the case of the\nopposite being true, the discrepancy is significantly narrower. Similar but weaker trend\ncan be also observed between deciles and 10 bins. In light of the results, quantile-\nbased abstractions can be regarded as a preferable choice of abstraction for SaNDA\nclassification.\nHowever, the number of categories into which data is transformed as a result of ab-\nstraction process is of far greater significance than the type of abstraction employed. This\ncan be explained by the fact that a higher number of abstractions (categories) enhances\nthe resolution of the grid created in the feature space. As a result, a denser grid enables\na more homogeneous distribution of data across its constituent parts, leading to more\naccurate classification performance.\nOne might naively believe that increasing the number of abstractions will enhance\nclassification accuracy. However, this approach harbours potential pitfalls that warrant\ncareful consideration. First, increasing the number of abstractions can result in a de-\ncrease in the average number of data points per category, potentially compromising the\naccurate representation of statistical properties. Furthermore, an increase in the number\nof abstractions can lead to an expansion in the number of nodes within the KG gener-\nated by SaNDA, potentially diminishing its explainability. Finally, an extensive number\nof abstractions can exert a detrimental impact on the computational time required for\ngenerating classification and KG, a factor that may become particularly crucial when\ncomputational resources are constrained or the analysed problem demands expediency.\nGiven these considerations, the maximum number of abstractions employed in this re-\nsearch was limited to 20.\nBased on the results of Experiment 1, Experiment 2 employed SaNDA classification\nusing 20 bins and quantiles 20 for comparison with the original ROC curves abstractions\nand Random Forest was performed. The results of this comparison for increasing per-\ncentage of missing data are presented in Figs 2–4 and in Tables 1–15. Fig. 2 shows BA\nas a function of missing data for each dataset separately. In all of the cases, 20 categories\nabstractions, bin- or quantile-based, outperform both ROC-based abstractions. Random\nForest achieves higher BA in only 4 of 15 datasets with non-zero amount of missing\ndata: Accelerometer, HIGGS and DIGENs: 39 5578, 23 5191. However, for HIGGS and\nDIGEN23 5191 the difference between 20 quantiles abstractions SaNDA and Random\nForest in the BA is slight. When the proportion of missing data increases the advan-\ntage of SaNDA with 20 categories abstraction methods increases, as it achieves higher\nBA for majority of tested datasets. A notable exception is observed in the Accelerom-\neter dataset, where Random Forest surpasses both proposed new abstraction methods\nregardless of amount of missing values. This behaviour can be attributed to the limited\nnumber of features, specifically 4, characterising this dataset. Additionally, the size of\nthe dataset, which can be considered average with approximately 105inputs, favors the\nefficiency of Random Forest.\nBA is not the only important metrics from the point of view of classification models.\nThe proposed system, due to its ability to anonymize data and dealing with missing\nvalues may be particularly relevant, among others, in areas of medicine, social sciences\nand finance where the high level of recall and precision is also very important. Therefore,\nwe decided to check whether increasing the number of abstractions (categories) would\naffect mentioned metrics and compare them with the values obtained for Random Forest.\n8The results presented in Figure 3 showed that the recall was higher for quantiles 20\nthan for ROC for all of the chosen datasets. Comparison of quantiles 20 and Random\nForest shows that Random Forest wins in the case of 6 datasets (Accelerometer, HIGGS,\nWisconsin Breast Cancer and DIGENs: 8 4426, 39 5578, 23 5191, when there aren’t any\nmissing values. Quantiles 20 or 20 bins achieves better results for 9 datasets (Ionosphere,\nSonar and DIGENs: 32 5191, 36 466, 35 4426, 10 8322, 22 2433, 24 2433, 17 6949) with\nan advantage for quantiles. The only dataset where Random Forest performance is much\nbetter is Accelerometer, but the same observation was noticed for BA and it may be\nrelated to low number of features.\nSimilar conclusions were reached for precision, see Fig. 4. For all datasets, the pro-\nposed new abstraction methods performed significantly better than ROC. In the ab-\nsence of missing data Random Forest achieves highest precision values for 6 (HIGGS,\nIonosphere, Wisconsin Breast Cancer and DIGENs: 39 5578, 23 5191, 22 2433) of 15\ndatasets, in the case of 2 of these datasets the difference was marginal (DIGENs: 23 5191\nand 22 2433). For the rest of the datasets, classification using either quantiles 20 or 20\nbins yields higher precision (Accelerometer, Sonar and DIGENs: 8 442, 32 5191, 36 466,\n354426, 10 8322, 24 2433, 17 6949). With 10% of the missing data quantiles 20 or 20 bins\noutperform Random Forest in 12 out of 15 cases for recall (HIGGS, Ionosphere, Sonar and\nDIGENs: 8 442, 32 5191, 39 5578, 36 466, 35 4426, 10 8322, 23 5191, 24 2433, 17 6949)\nand in 12 cases for precision (Accelerometer, HIGGS, Sonar and DIGENs: 32 5191,\n395578, 36 466, 35 4426, 10 8322, 23 5191, 24 2433, 17 6949). A greater proportion of\nmissing data works against Random Forest, which is particularly noticeable when 50%\nof data is removed - in such conditions Random Forest prevails by a minimal margin for\nonly 1 dataset (for recall - Accelerometer, while for precision - DIGEN39 5578).\nThe results clearly demonstrate that the proposed new data abstraction methods\nsignificantly outperform abstractions using ROC. Moreover, the values obtained from se-\nlected metrics indicate that classification using SaNDA with a quantiles 20 abstraction\nmethod can compete with Random Forest for complete datasets. Additionally, classifica-\ntion with missing data clearly shows the advantage of the new data abstraction methods\nover classical approach.\n5. Summary and Conclusions\nThe increasing research on data anonymisation, federated learning and encrypted\ncomputing aims at increasing the data size and availability for deep learning, the driving\nforce behind the recent developments and impact of ML. However, there are areas such\nas healthcare, behavioural sciences and finances, where data, to support CDM remains\nvery limited. Recently, the Small and Incomplete Dataset Analyser (SaNDA) [15] has\nemerged as a promising solution to facilitate the adoption of ML in such areas due to its\ncapability to use small and noisy datasets, while explaining the outcomes.\nSaNDA can effectively tackle classification tasks even with substantial amounts of\nmissing data, eliminating the need for data imputation or modelling, which can be chal-\nlenging, costly and time-consuming in extremely low-quality datasets. One of its funda-\nmental aspects is the use of abstractions, which involve column-wise transformations of\nnumerical data into categorical representations.\nIn this paper we describe experiments to compare constant width constant binning\nand quantiles-based abstraction protocols as alternatives to the previously employed ROC\n9curve-based method. Their influence on classification performance has been extensively\nexamined and measured.\nAs a general result, increasing the number of data abstractions consistently enhances\nthe performance of the SaNDA algorithm, however, this results into some potential lim-\nitations. Firstly, it was identified that splitting the data into many categories, results\nin a poor statistical representation of the real population. Secondly, it reduces the ex-\nplainability of the model due to the increase of the complexity of the KG representation,\nbecoming useless for Critical Decision-Making. Lastly, a substantial increase on the time\nrequired to create the classification model makes its usefulness questionable for the pro-\nposed classification environments.\nDue to these limitations, we decided to cap the number of abstractions to 20, regard-\nless of the chosen abstraction method. This limit proved to be effective in significantly\nimproving the balanced accuracy of SaNDA.\nIn conclusion, the SaNDA classification process, built through the meticulous ex-\nperiments of a chosen abstraction methodology, holds as a prospective alternative to\nconventional explainable ML paradigms such as Random Forest. This assertion gains\nparticular relevance in situations necessitating CDM within constraints of small datasets\nand missing data. The proposed SaNDA enhanced classification processes demonstrates\nits potential to yield robust and explainable outcomes, thereby presenting a viable option\nin contexts where deep learning methodologies may encounter limitations.\nA future direction of research may involve the choice of an appropriate method and\nlevel of abstraction. The abstraction method can be chosen to maximise accuracy while\nmaintaining computation time efficiency for a specific dataset. The abstraction level\ndictates how accurately it mirrors the intricacies of the problem domain and the extent to\nwhich it simplifies and generalizes domain concepts. It can range from being very detailed\nand specific to more generalized and fundamental, depending on the intended scope and\npurpose of the domain problems. Adopting a dynamic approach to selecting methods\nand determining abstraction levels can enhance the model’s versatility. This adaptability\nwould allow for easy customization to meet specific problem and user requirements.\nAcknowledgements\nThis research has been supported by the European Union’s Horizon 2020 research\nand innovation programme under grant agreement Sano No 857533. This publication\nis supported by Sano project carried out within the International Research Agendas\nprogramme of the Foundation for Polish Science MAB PLUS/2019/13, co-financed by\nthe European Union under the European Regional Development Fund. 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A survey on federated learning.\nKnowledge-Based Systems , 216:106775, 2021.\n12dataset quantiles 20 deciles ROC 10 bins 20 bins Random Forest\nDIGEN23 5191 0.681 0.633 0.564 0.601 0.657 0.686\nDIGEN24 2433 0.682 0.597 0.597 0.635 0.690 0.594\nDIGEN39 5578 0.674 0.622 0.601 0.603 0.661 0.757\nDIGEN17 6949 0.623 0.602 0.593 0.615 0.633 0.483\nDIGEN32 5191 0.665 0.602 0.568 0.605 0.658 0.447\nDIGEN10 8322 0.682 0.611 0.577 0.620 0.657 0.627\nDIGEN22 2433 0.668 0.603 0.563 0.611 0.672 0.645\nDIGEN8 4426 0.675 0.623 0.589 0.648 0.687 0.656\nDIGEN35 4426 0.664 0.627 0.560 0.611 0.663 0.640\nDIGEN36 466 0.687 0.629 0.584 0.630 0.660 0.628\nAccelerometer 0.718 0.713 0.676 0.638 0.678 0.889\nIonosphere 0.940 0.924 0.911 0.950 0.957 0.936\nWisconsin Breast Cancer 0.969 0.952 0.948 0.948 0.953 0.958\nSonar 0.981 0.906 0.847 0.923 0.986 0.832\nHIGGS 0.663 0.658 0.624 0.609 0.640 0.666\nTable 1: Balanced accuracy (BA) for selected SaNDA classification methods and Random Forest for all\nchosen datasets with 0% missing values\n13dataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.678 0.58 0.664 0.647\nDIGEN24 2433 0.692 0.593 0.686 0.611\nDIGEN39 5578 0.679 0.601 0.651 0.657\nDIGEN17 6949 0.648 0.584 0.632 0.553\nDIGEN32 5191 0.669 0.564 0.658 0.555\nDIGEN10 8322 0.685 0.574 0.663 0.654\nDIGEN22 2433 0.658 0.581 0.673 0.625\nDIGEN8 4426 0.683 0.592 0.683 0.663\nDIGEN35 4426 0.667 0.57 0.664 0.596\nDIGEN36 466 0.685 0.574 0.667 0.571\nAccelerometer 0.718 0.676 0.678 0.885\nIonosphere 0.940 0.911 0.959 0.941\nWisconsin Breast Cancer 0.970 0.946 0.953 0.958\nSonar 0.981 0.866 0.896 0.793\nHIGGS 0.663 0.624 0.639 0.661\nTable 2: Balanced accuracy (BA) for selected SaNDA classification methods and Random Forest for all\nchosen datasets with 1% missing values\ndataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.682 0.566 0.670 0.541\nDIGEN24 2433 0.678 0.580 0.666 0.687\nDIGEN39 5578 0.690 0.589 0.680 0.610\nDIGEN17 6949 0.661 0.588 0.677 0.618\nDIGEN32 5191 0.680 0.593 0.667 0.689\nDIGEN10 8322 0.669 0.588 0.658 0.572\nDIGEN22 2433 0.646 0.578 0.655 0.572\nDIGEN8 4426 0.682 0.588 0.657 0.693\nDIGEN35 4426 0.672 0.575 0.653 0.594\nDIGEN36 466 0.684 0.580 0.679 0.626\nAccelerometer 0.716 0.673 0.693 0.869\nIonosphere 0.936 0.901 0.967 0.941\nWisconsin Breast Cancer 0.969 0.952 0.953 0.942\nSonar 0.981 0.853 0.981 0.818\nHIGGS 0.661 0.620 0.637 0.614\nTable 3: Balanced accuracy (BA) for selected SaNDA classification methods and Random Forest for all\nchosen datasets with 5% missing values\n14dataset 20deciles roc static 20Random Forest\nDIGEN23 5191 0.678 0.593 0.652 0.633\nDIGEN24 2433 0.694 0.587 0.677 0.562\nDIGEN39 5578 0.665 0.591 0.679 0.637\nDIGEN17 6949 0.654 0.591 0.657 0.657\nDIGEN32 5191 0.659 0.563 0.655 0.508\nDIGEN10 8322 0.675 0.583 0.669 0.594\nDIGEN22 2433 0.674 0.577 0.667 0.594\nDIGEN8 4426 0.667 0.575 0.682 0.694\nDIGEN35 4426 0.678 0.563 0.665 0.660\nDIGEN36 466 0.678 0.565 0.657 0.556\nAccelerometer 0.715 0.668 0.671 0.848\nIonosphere 0.942 0.892 0.947 0.936\nWisconsin Breast Cancer 0.964 0.954 0.958 0.962\nSonar 0.975 0.871 0.991 0.700\nHIGGS 0.657 0.618 0.635 0.622\nTable 4: Balanced accuracy (BA) for selected SaNDA classification methods and Random Forest for all\nchosen datasets with 10% missing values\ndataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.656 0.555 0.646 0.490\nDIGEN24 2433 0.671 0.618 0.656 0.541\nDIGEN39 5578 0.648 0.569 0.665 0.493\nDIGEN17 6949 0.680 0.617 0.689 0.600\nDIGEN32 5191 0.687 0.607 0.679 0.526\nDIGEN10 8322 0.648 0.589 0.669 0.491\nDIGEN22 2433 0.679 0.575 0.640 0.513\nDIGEN8 4426 0.662 0.579 0.679 0.524\nDIGEN35 4426 0.684 0.569 0.669 0.496\nDIGEN36 466 0.679 0.622 0.640 0.520\nAccelerometer 0.678 0.615 0.607 0.711\nIonosphere 0.944 0.900 0.947 0.796\nWisconsin Breast Cancer 0.959 0.943 0.952 0.909\nSonar 0.990 0.911 0.995 0.718\nHIGGS 0.627 0.593 0.606 0.517\nTable 5: Balanced accuracy (BA) for selected SaNDA classification methods and Random Forest for all\nchosen datasets with 50% missing values\n15dataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.681 0.567 0.659 0.684\nDIGEN24 2433 0.683 0.600 0.689 0.569\nDIGEN39 5578 0.673 0.599 0.656 0.760\nDIGEN17 6949 0.621 0.600 0.639 0.530\nDIGEN32 5191 0.662 0.570 0.657 0.473\nDIGEN10 8322 0.680 0.575 0.661 0.639\nDIGEN22 2433 0.671 0.566 0.663 0.679\nDIGEN8 4426 0.672 0.590 0.691 0.654\nDIGEN35 4426 0.663 0.561 0.664 0.658\nDIGEN36 466 0.683 0.588 0.659 0.652\nAccelerometer 0.820 0.749 0.940 0.890\nIonosphere 0.944 0.938 0.969 1.000\nWisconsin Breast Cancer 0.958 0.922 0.961 0.972\nSonar 0.970 0.901 0.980 0.821\nHIGGS 0.636 0.615 0.630 0.683\nTable 6: Precision for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 0% missing values\ndataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.679 0.586 0.671 0.664\nDIGEN24 2433 0.690 0.594 0.683 0.584\nDIGEN39 5578 0.677 0.602 0.645 0.652\nDIGEN17 6949 0.649 0.585 0.640 0.601\nDIGEN32 5191 0.665 0.560 0.654 0.595\nDIGEN10 8322 0.687 0.571 0.668 0.671\nDIGEN22 2433 0.659 0.581 0.669 0.652\nDIGEN8 4426 0.680 0.593 0.686 0.660\nDIGEN35 4426 0.662 0.568 0.657 0.602\nDIGEN36 466 0.681 0.577 0.666 0.593\nAccelerometer 0.819 0.750 0.940 0.885\nIonosphere 0.944 0.938 0.969 0.972\nWisconsin Breast Cancer 0.962 0.917 0.961 0.972\nSonar 0.970 0.907 0.980 0.824\nHIGGS 0.637 0.615 0.630 0.673\nTable 7: Precision for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 1% missing values\n16dataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.665 0.570 0.650 0.609\nDIGEN24 2433 0.676 0.580 0.677 0.610\nDIGEN39 5578 0.687 0.587 0.658 0.686\nDIGEN17 6949 0.643 0.579 0.659 0.532\nDIGEN32 5191 0.665 0.562 0.661 0.564\nDIGEN10 8322 0.697 0.593 0.683 0.625\nDIGEN22 2433 0.663 0.596 0.681 0.647\nDIGEN8 4426 0.618 0.589 0.665 0.688\nDIGEN35 4426 0.684 0.585 0.674 0.700\nDIGEN36 466 0.665 0.588 0.656 0.600\nAccelerometer 0.816 0.754 0.833 0.870\nIonosphere 0.937 0.925 0.978 0.972\nWisconsin Breast Cancer 0.958 0.926 0.952 0.954\nSonar 0.970 0.876 0.970 0.897\nHIGGS 0.635 0.607 0.627 0.642\nTable 8: Precision for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 5% missing values\ndataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.651 0.549 0.643 0.497\nDIGEN24 2433 0.656 0.605 0.648 0.516\nDIGEN39 5578 0.632 0.553 0.642 0.645\nDIGEN17 6949 0.667 0.606 0.674 0.469\nDIGEN32 5191 0.690 0.625 0.694 0.546\nDIGEN10 8322 0.651 0.599 0.671 0.529\nDIGEN22 2433 0.704 0.604 0.661 0.513\nDIGEN8 4426 0.646 0.565 0.656 0.517\nDIGEN35 4426 0.692 0.583 0.680 0.539\nDIGEN36 466 0.657 0.595 0.616 0.555\nAccelerometer 0.740 0.689 0.698 0.678\nIonosphere 0.945 0.932 0.956 0.839\nWisconsin Breast Cancer 0.927 0.901 0.935 0.927\nSonar 0.990 0.934 0.990 0.725\nHIGGS 0.602 0.575 0.589 0.310\nTable 9: Precision for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 10% missing values\n17dataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.651 0.549 0.643 0.497\nDIGEN24 2433 0.656 0.605 0.648 0.516\nDIGEN39 5578 0.632 0.553 0.642 0.645\nDIGEN17 6949 0.667 0.606 0.674 0.469\nDIGEN32 5191 0.690 0.625 0.694 0.546\nDIGEN10 8322 0.651 0.599 0.671 0.529\nDIGEN22 2433 0.704 0.604 0.661 0.513\nDIGEN8 4426 0.646 0.565 0.656 0.517\nDIGEN35 4426 0.692 0.583 0.680 0.539\nDIGEN36 466 0.657 0.595 0.616 0.555\nAccelerometer 0.740 0.689 0.698 0.678\nIonosphere 0.945 0.932 0.956 0.839\nWisconsin Breast Cancer 0.927 0.901 0.935 0.927\nSonar 0.990 0.934 0.990 0.725\nHIGGS 0.602 0.575 0.589 0.310\nTable 10: Precision for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 50% missing values\ndataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.680 0.544 0.652 0.711\nDIGEN24 2433 0.678 0.584 0.692 0.608\nDIGEN39 5578 0.676 0.612 0.678 0.755\nDIGEN17 6949 0.630 0.560 0.612 0.488\nDIGEN32 5191 0.674 0.556 0.662 0.449\nDIGEN10 8322 0.688 0.590 0.644 0.614\nDIGEN22 2433 0.658 0.538 0.698 0.587\nDIGEN8 4426 0.684 0.582 0.676 0.693\nDIGEN35 4426 0.666 0.554 0.660 0.623\nDIGEN36 466 0.698 0.560 0.662 0.594\nAccelerometer 0.558 0.530 0.380 0.888\nIonosphere 0.982 0.933 0.969 0.872\nWisconsin Breast Cancer 0.962 0.943 0.929 0.963\nSonar 0.990 0.784 0.990 0.914\nHIGGS 0.664 0.564 0.586 0.697\nTable 11: Recall for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 0% missing values\n18dataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.674 0.546 0.644 0.612\nDIGEN24 2433 0.696 0.586 0.694 0.629\nDIGEN39 5578 0.684 0.594 0.670 0.682\nDIGEN17 6949 0.644 0.576 0.602 0.524\nDIGEN32 5191 0.682 0.594 0.670 0.456\nDIGEN10 8322 0.680 0.598 0.648 0.627\nDIGEN22 2433 0.654 0.578 0.686 0.581\nDIGEN8 4426 0.690 0.586 0.674 0.699\nDIGEN35 4426 0.684 0.582 0.686 0.630\nDIGEN36 466 0.696 0.554 0.670 0.535\nAccelerometer 0.558 0.527 0.379 0.884\nIonosphere 0.982 0.933 0.973 0.897\nWisconsin Breast Cancer 0.962 0.943 0.929 0.963\nSonar 0.990 0.804 0.990 0.800\nHIGGS 0.663 0.564 0.586 0.693\nTable 12: Recall for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 1% missing values\ndataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.692 0.608 0.664 0.553\nDIGEN24 2433 0.706 0.580 0.686 0.601\nDIGEN39 5578 0.668 0.596 0.654 0.722\nDIGEN17 6949 0.658 0.570 0.642 0.451\nDIGEN32 5191 0.690 0.596 0.698 0.582\nDIGEN10 8322 0.672 0.568 0.672 0.588\nDIGEN22 2433 0.656 0.544 0.666 0.568\nDIGEN8 4426 0.704 0.618 0.674 0.719\nDIGEN35 4426 0.662 0.548 0.644 0.682\nDIGEN36 466 0.682 0.586 0.666 0.503\nAccelerometer 0.558 0.513 0.484 0.868\nIonosphere 0.991 0.938 0.973 0.897\nWisconsin Breast Cancer 0.962 0.948 0.934 0.963\nSonar 0.981 0.804 0.990 0.743\nHIGGS 0.663 0.568 0.586 0.650\nTable 13: Recall for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 5% missing values\n19dataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.688 0.598 0.654 0.645\nDIGEN24 2433 0.706 0.600 0.696 0.524\nDIGEN39 5578 0.660 0.578 0.696 0.636\nDIGEN17 6949 0.622 0.510 0.600 0.457\nDIGEN32 5191 0.668 0.538 0.632 0.500\nDIGEN10 8322 0.704 0.572 0.664 0.575\nDIGEN22 2433 0.664 0.548 0.642 0.677\nDIGEN8 4426 0.668 0.608 0.718 0.634\nDIGEN35 4426 0.660 0.542 0.646 0.656\nDIGEN36 466 0.666 0.572 0.668 0.581\nAccelerometer 0.559 0.498 0.367 0.845\nIonosphere 0.996 0.902 0.964 0.872\nWisconsin Breast Cancer 0.953 0.957 0.939 0.972\nSonar 0.959 0.825 1.000 0.686\nHIGGS 0.659 0.569 0.586 0.597\nTable 14: Recall for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 10% missing values\ndataset quantiles 20 ROC 20 bins Random Forest\nDIGEN23 5191 0.674 0.612 0.658 0.480\nDIGEN24 2433 0.720 0.682 0.682 0.552\nDIGEN39 5578 0.710 0.720 0.744 0.562\nDIGEN17 6949 0.718 0.668 0.732 0.573\nDIGEN32 5191 0.678 0.536 0.640 0.409\nDIGEN10 8322 0.638 0.540 0.664 0.536\nDIGEN22 2433 0.618 0.434 0.574 0.510\nDIGEN8 4426 0.718 0.686 0.754 0.497\nDIGEN35 4426 0.662 0.484 0.638 0.494\nDIGEN36 466 0.748 0.766 0.742 0.541\nAccelerometer 0.548 0.418 0.377 0.804\nIonosphere 0.991 0.920 0.973 0.667\nWisconsin Breast Cancer 0.962 0.948 0.943 0.944\nSonar 0.990 0.876 0.990 0.829\nHIGGS 0.600 0.547 0.559 0.161\nTable 15: Recall for selected SaNDA classification methods and Random Forest for all chosen datasets\nwith 50% missing values\n20Figure 1: Balanced accuracy (BA) of SaNDA classification method as a function the different abstraction\nmethods.\n21Figure 2: Balanced accuracy (BA) as a function of percentage of missing data for SaDNA with two best\nperforming abstraction methods (order quantiles 20 and 20 bins), SaNDA with ROC curve and Random\nForest.\n22Figure 3: Recall as a function of percentage of missing data for SaDNA with two best performing\nabstraction methods (order quantiles 20 and 20 bins), SaNDA with ROC curve and Random Forest.\n23Figure 4: Precision as a function of percentage of missing data for SaDNA with two best performing\nabstraction methods (order quantiles 20 and 20 bins), SaNDA with ROC curve and Random Forest.\n24"    },    {        "title": "2401.11047v1.Resonances_for_vector_valued_Jacobi_operators_on_half_lattice.pdf",        "content": "arXiv:2401.11047v1  [math.SP]  19 Jan 2024RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON\nHALF-LATTICE\nEVGENY KOROTYAEV\nAbstract. We study resonances for Jacobi operators on the half lattice with matrix val-\nued coefficient and finitely supported perturbations. We describe a forbidden domain, the\ngeometry of resonances and their asymptotics when the main coeffi cient of the perturbation\n(determining its length of support) goes to zero. Moreover, we sh ow that\n1) Any sequence of points on the complex plane can be resonances f or some Jacobi operators.\nIn particular, the multiplicity of a resonance can be any number.\n2) The Jost determinant coincides with the Fredholm determinant up to the constant.\n3) The S-matrix on the a.c spectrum determines the perturbation u niquely.\n5) The value of the Jost matrix at any finite sequence of points on th e a.c spectrum determine\nthe Jacobi matrix uniquely. The length of this sequence is equals to t he upper point of the\nsupport perturbation.\nTable of Contents\n1. Introduction and main results 1\n2. Preliminaries 6\n3. Properties of the Jost solutions 13\n4. Proof of main theorems 17\n5. Fredholm determinants 24\n6. Appendix: Inverse problems for finite Jacobi operators 31\n1.Introduction and main results\n1.1.Introduction. We consider Jacobi operators Hwith matrix-valued coefficients acting\nony= (yx)∞\n1∈H:=ℓ2(N,Cd),d/greaterorequalslant1 and given by\n(Hy)x=ax−1yx−1+axyx+1+bxyx, x∈N={1,2,3,...}, (1.1)\nwhere formally y0= 0. We assume that ax>0,bx=b∗,x∈Nare sequences of complex d×d\nmatrices. We discuss resonances for finitely supported sequence sax−1 1d,bx, where 1 1 = 1 1 d\nis the identity d×dmatrix. There are a lot of results about resonances for the scalar case\nd= 1, see [7], [14], [24]–[30] [37], [49], [60] and the references therein. Th ere are results about\ndirect and inverse scattering theory for this class of operators w ithd >1, see for example,\nAptekarev–Nikishin [1], Geronimo [19], Nikishin [64, 65] and Serebryako v [68, 70] and recent\npapers [3], [4]. Different types of Jacobi operators were discussed in [12], [20], [13], [47], [46],\n[57], [58], [59], [71].\nWe consider also non-selfadjoint Jacobi operators Hc=Ho+Vcacting on H, where the\nunperturbed operator Hoand the perturbation Vchave the form\n(Hoy)x=yx−1+yx+1,(Vcy)x= (ax−1−1 1d)yx−1+bxyx+(sx−1 1d)yx+1, yx∈Cd,(1.2)\nDate: January 23, 2024.\n2020Mathematics Subject Classification. 34F15( 47E05).\nKey words and phrases. inverse problem, Jacobi operator, resonances.\n12 EVGENY KOROTYAEV\nwhere the operator Hois the discrete Laplacian on N. We assume that ax,sx,bx,x∈Nare\nsequences of complex d×dmatrices. We assume that Vc∈Jc\nqdefined below. We define the\nspaceM=Mdof alld×dcomplex matrices and its subsets\nMs=Ms\nd={b∈Md:b=b∗},Mo=Mo\nd={b∈Md: detb/\\e}atio\\slash= 0}.(1.3)\nWe study resonances for following classes of perturbations.\nDefinition V. Introduce the class Jc\nq,q∈ {2p−1,2p},p∈Nof all operators Vcgiven by (1.2),\nwhere sequences (bx,ax−1 1d,s1−1 1d,x∈N)are finitely supported and satisfy\n/braceleftigg\nax,sx∈Mo\nd, bx∈Md, x/greaterorequalslant1,\nax=sx= 1 1d, bx= 0,∀x>p,/braceleftigg\nap=sp= 1 1d,bp∈Mo\ndifq= 2p−1\n1 1−spap∈Mo\nd, ifq= 2p,(1.4)\nand its subset of self-adjoint perturbations V∈Jqgiven by\nJq={V∈Jc\nq:ax=sx>0, bx=b∗\nx,∀x/greaterorequalslant1}. (1.5)\nOur main goal is to study self-adjoint operators H=Ho+V, whereVis given by\n(Vy)x= (ax−1−1 1d)yx−1+bxyx+(a∗\nx−1 1d)yx+1, yx∈Cd. (1.6)\nWe consider resonances for two natural classes:\n•all matrices ax>0,\n•all matrices axare upper triangular with positive elements on the diagonal.\nThe last case corresponds precisely Jacobi matrices with 2 d+1 non-vanishing diagonals\nwith the extreme diagonals strictly positive. Without loss of the gene rality we assume that\nV∈Jq, since there are exact relations between these two classes, see S ect. 6. The class\nJc\nqcorresponds to non-self-adjoint operators Hcand the class Jqcorresponds to self-adjoint\noperatorsH.\nAnon-self-adjointoperator Hchasanessential spectrum [ −2,2]plusafinitenumber n•of\neigenvalues λj(with algebraicmultiplicity dj) onthe set C\\[−2,2]. Recall themain properties\nof the self-adjoint operator Hfrom [1]. It has purely absolutely continuous spectrum [ −2,2]\nplus a finite number nof eigenvalues λj(with multiplicity dj/lessorequalslantd) on the set R\\[−2,2]. These\neigenvalues of Hare denoted by\nλ−1−<···<λ−n−<−2<2<λn+<···<λ1,n=n++n−, (1.7)\nfor some n±/greaterorequalslant0. Thus the spectrum of Hhas the form (see Fig. 1)\nσ(H) =σac(H)∪σd(H), σ ac(H) = [−2,2], σ d(H)⊂R\\[−2,2].\nThe operators HandHcwith a perturbation V∈JqandVc∈Jc\nq, withq∈ {2p−1,2p},\n×λ−1\n×λ−2×λ−3\n-2σac(H)\n2× ×λ2λ1\nFigure 1. The spectrum of HRESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 3\nrepresent semi-infinite matrices of the form\nH=\nb1a10 0...0 0 0\na1b2a20...0 0 0\n0a2b3a3...0 0 0\n... ... ... ... ... ... ... ...\n0...0ap−1bpap0 0\n0...0 0ap0 1 1 0\n0...0 0 0 1 1 0 1 1\n... ... ... ... ... ... ... ...\n, Hc=\nb1s10 0...0 0 0\na1b2s20...0 0 0\n0a2b3s3...0 0 0\n... ... ... ... ... ... ... ...\n0...0ap−1bpsp0 0\n0...0 0ap0 1 1 0\n0...0 0 0 1 1 0 1 1\n... ... ... ... ... ... ... ...\n.\nDefine a standard spectral variable k=k(λ) by\nλ=λ(k) =k+1\nk, k∈D, λ∈Λ :=C\\[−2,2], (1.8)\nwhereDr={k∈C:|k|< r},r >0 is the disc, and D=D1. Hereλ(k) is a conformal\nmapping from Donto the cut domain Λ and its inverse function k(λ) is\nk(λ) =λ\n2−/radicalig\nλ2\n4−1, k(λ) =1+o(1)\nλasλ→ ∞. (1.9)\nFor an operator Hcwe introduce a Jost solution f(k) = (fx(k))∞\n0,fx(k)∈Mdto the equation\nax−1fx−1+sxfx+1+bxfx= (k+k−1)fx,|k|>0, x∈Z+={0,1,2,3,...},\nwherefx(k) =kx1 1d∀x>pandq={2p−1,2p}.(1.10)\nFor convenience, in order to define f0(k) we seta0:= 1 1d. For the operator Hcwe define\nthed×dJost matrix by ψ=f0(k), which is our basic function. The Jost matrix ψ(k) is a\nMd-valued polynomial in k∈Cof orderq. Define the Jost determinant f(k) = detψ(k). The\nfunction f(k) is a polynomial of order qdwith zerosr1,....,rqdlabeled by\n0<|r1|/lessorequalslant|r2|/lessorequalslant···/lessorequalslant|rqd|.\nEach zerorj∈Dis an eigenvalue of Hc. The function fcan also have a finite number of zeros\ninC\\Dwhich are called resonances, plus possible zeros at ±1. The points ±1 are called\nvirtual or resonances. In the self-adjoint case V∈Jqthe function fhasn•=n++n−/greaterorequalslant0 real\nzeros inDgiven by\nkj=k(λj), λj=kj+1\nkj, j∈n•:={−n−,...,−1,1,...,n+},\n−1<k−n−<...<k −1<0<k1<...<k n+<1.(1.11)\nEachkjhas a multiplicity dj/lessorequalslantd. The resonances of Hcan also be defined as the poles of its\nresolvent meromorphically continued in C, see Proposition 5.2.\nFor the operator Hcgiven by (1.2), we define matrices\ntx=a−1\nx,Tx=t1t2···tx, cx= 1−sxax,C0=Tp−1cptp,\nCo\n0=Tp−1bptp+T1\np−1cptp,C1\n0=T1\np−1bp+Tp−1cp−1, Bx=x/summationdisplay\n1bj,\nT1\nx=b1Tx+t1b2t2...tx+t1t2b3t3...tx+...+Tx−1bxtx, x∈Np={1,...,p}.(1.12)\nOur first results are devoted to the operator Hcwith a finitely supported perturbation, in-\ncluding a selfadjoint case, when cp= 1 1−a2\np. Define the ring of all zeros\nDze={r−/lessorequalslant|rj|/lessorequalslantr+}, r+= max|rj|, r−= min|rj|.4 EVGENY KOROTYAEV\nWe describe the basic properties of Jost matrices of the operator Hcand the forbidden domain\nfor resonances defined by Df={|k|>r+}.\nTheorem 1.1. LetVc∈Jc\nqfor someq/greaterorequalslant1. Then\ni) The Jost matrix ψ=f0(k)has asymptotics\nψ(k) =/braceleftigg\nC0kq−Co\n0kq−1+O(kq−2)if q= 2p\n−Tp−1bpkq+C1\n0kq−1+O(kq−2)if q= 2p−1ask→ ∞,(1.13)\nand\nψ(k) =Tp−T1\npk+O(k2)\nf(k) = (1−kTrBp+O(k2))detTp,ask→0. (1.14)\nii) The Jost determinant f= detψis a polynomial and has the form\nf(k) =CVqd/productdisplay\n1(k−rn), CV= (−1)dqhdetTp,h=/braceleftigg\ndetcpifq= 2p\ndetbpifq= 2p−1,(1.15)\nqd/productdisplay\n1rn=1\nh,qd/summationdisplay\n11\nrn=p/summationdisplay\nx=1Trbx. (1.16)\niii) If the Jost determinant fdoes not have zeros in Dforq>2, then\n1/lessorequalslantr−/lessorequalslant|h|−1\nqd/lessorequalslantr+/lessorequalslant|h|−1. (1.17)\niv) If|h|>1, thenHchasmeigenvalues, counted with multiplicities, and rm\n−<1/|h|.\nDefine the resolvent Ro(λ) = (Ho−λ)−1and the Fredholm determinant Dc(k) by\nDc(k) = det(I+VcRo(λ)), λ=k+1\nk, k∈D. (1.18)\nThis determinant is well defined since Vcis finitely supported. The zeros of Dc(k) in the disc\nDare the zeros of f. We will to show that Dc=Cffor some constant C/\\e}atio\\slash= 0. It is well known\nthat eachkj∈Dis a zero of Dcand has a multiplicity dj/lessorequalslantd. The S-matrix for His defined\nby\nS(k) =ψ−1(k)ψ(k), k∈S:={|z|= 1}. (1.19)\nThe function Shas a meromorphic continuation from the circle Sto the whole complex plane.\nThen the poles of S(k) are exactly the resonances of H.\nTheorem 1.2. LetVc∈Jc\nqfor someq/greaterorequalslant1. Then the Fredholm determinant Dc(k),k∈D\ndefined by (1.18) satisfies\nf(k) =Dc(k)detTp∀k∈D, (1.20)\nand then for all k∈C. Moreover, if V∈Jq, then the function fis real on the real line and\nthe resonances are symmetric with respect to the real line.\nRemarks. The proof of the identity f=DdetTpford= 1 is sufficiently simple. In the\nmatrix case it is not enough and we need to use additional arguments , see Theorem 5.9.\nNow we consider the resonances for the case ap=εa′\npwhere the coupling constant ε→0\nwhile all other coefficients are fixed, and a′\np,bp∈Mo\nd. Thus we discuss the case when the\nsupport of the perturbation Vbecomes smaller.RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 5\nTheorem 1.3. LetV∈Jq,q∈Nsatisfy\nap=εa′\np, a′\np,bp∈Mo\ndifq= 2p and b p=εb′\np, b′\np,cp−1∈Mo\nd,ifq= 2p−1,(1.21)\nandε→0. Letψ(k,ε)be the corresponding Jost matrix. Let ko\nj,j∈Nmbe zeros of f(k,0) =\ndetψ(k,0)with the multiplicity do\njfor someN. Then\ni) The function ψ(k,ε)converges to ψ(k,0)uniformly in each disk as ε→0.\nii) For any ε >0small enough there exist do\njzeros(counted with multiplicity) of f(k,ε) =\ndetψ(k,ε)in each disc Drε(ko\nj) :={|k−ko\nj|<rε}for somerε>0small enough.\niii) Moreover, f(k,ε)/\\e}atio\\slash= 0in the domain DRε\\∪Drε(ko\nj)andf(k,ε)hasdzeroskn(ε)∈ {|k|>\nRε}(counted with multiplicity) for some Rε>1big enough and they have asymptotics\nkn(ε) = [τn+O(ε1\nd)]/ε, n= 1,...,d, (1.22)\nwhereτn,n∈Ndare zeros of det(A+τB),τ∈C, andA= 2a′\np,B=bpifq= 2pand\nA=−b′\np,B=cp−1ifq= 2p−1, these zeros are real and /\\e}atio\\slash= 0.\nRemark. Similar results for d= 1 were obtained in [62], when all an−1,bn→0. Our case\nford= 1 was discussed in [49].\n1.2.Inverse problems. We discuss inverse problems for selfadjoint operators H. Firstly we\nconsider possible positions of eigenvalues and resonances.\nTheorem 1.4. i) Let finite sequence of numbers kj∈(−1,0)∪(0,1),j∈Nmwith the mul-\ntiplicitydj/lessorequalslantdfor somem/greaterorequalslant0be given. Then there exists a finite perturbation V∈Jq\nfor someq/greaterorequalslant1such that each kj,j∈Nmis an eigenvalue (with the multiplicity dj) of the\noperatorH.\nii) Let a sequence rn∈C\\D,n= 1,...,qdbe given for some q/greaterorequalslant1and a number of the points\nrn= 1(orrn=−1) have multiplicity /lessorequalslantd. Then there exists a finite perturbation V∈Jq\nsuch that each point rnis a resonance for the operator H.\niii) The multiplicity of a resonance can be any number for som e specific perturbation V∈Jq.\nRemark. The proof of the theorem is based on results from [49], see Section 4 .\nWe formulate our second result about inverse problems for Jacobi operators with finitely\nsupported perturbations on the half lattice. We show that the ope ratorHis uniquely deter-\nmined by the Jost matrix at a finite number of points.\nTheorem 1.5. Let an operator V∈Jqfor someq∈N. Then\ni) The S-matrix S(k)for allk∈SdetermineV∈Jq,q∈Nuniquely.\nii) Letζj∈C, j∈Nq+1be a sequence of distinct numbers. Then the sequence of the Jo st\nmatrixψ(ζj),j∈Nq+1determineVuniquely.\nRemark. The proof of Theorem 1.5 i) is based on results from [40], see Section 6 .\n1.3.Historical review. There are a lot papers devoted to resonances for various operat ors\nsee articles [16, 23, 33, 72, 77] concerning to one-dimensional Sch r¨ odinger operator with com-\npactly supported potentials and the book [14] and the reference s therein. In particular, the\ninverse resonances problem (uniqueness, reconstruction and ch aracterization) were solved by\nKorotyaev for a Schr¨ odinger operator with a compactly support ed real-valued potential on\nthe half-line [33] and on the real line [35]. See also Zworski [76] and Br own-Knowles-Weikard\n[8] concerning the uniqueness problem. The resonance stability was considered in papers\n[34, 61, 5]. Resonances are discussed for various perturbations o f one-dimensional Schr¨ odinger6 EVGENY KOROTYAEV\noperator. For example, Sch¨ odinger operator with periodic plus a c ompactly supported poten-\ntial were considered by Firsova [15], Korotyaev [37] and Korotyae v-Schmidt [55]. Resonances\nfor Schr¨ odinger operator with a step-like potential were discuss ed by Christiansen [11]. Stark\noperator with compactly supported potentials was studied in [17, 39 ]. Resonances were also\nconsidered for three and fourth order differential operator with compactly supported coeffi-\ncients on the line, see [40, 2]. Note that the inverse resonance scat tering for Dirac operators\nwas discussed in [28, 29, 50, 51, 63].\nNow we discuss the case of Schr¨ odinger operators on the cubic lat ticeZd,d/greaterorequalslant2 for\nself-adjoint potentials. For Schr¨ odinger operators with decayin g potentials on the lattice\nZd, Boutet de Monvel and Sahbani [6] used Mourre’s method to prove completeness of the\nwave operators, absence of singular continuous spectrum and loc al finiteness of eigenvalues\naway from threshold energies. Isozaki and Korotyaev [30] studie d the direct and the inverse\nscattering problem as well as trace formulas. Korotyaev and Moller [52] discussed the spectral\ntheory for potentials V∈ℓp,p >1. An upper bound on the number of discrete eigenvalues\nin terms of potentials was given by Korotyaev and Sloushch [56], Roz enblum and Solomyak\n[67]. Different types of trace formulas for Schr¨ odinger operator s on discrete periodic graphs\nare discussed in [45], [48], [53], [54].\nFinally, we briefly comment on background literature for Jacobi ope rators. There are\na lot of articles about resonances and inverse resonance theory f or scalar Jacobi operators\nwith finitely supported perturbation is well understood, see [7], [24 ], [25], [26], [27], [37],\n[49], [57], [60], [62], and see references therein. Inverse inverse res onance theory for scalar\nJacobi operators with finitely supported perturbation are discus sed in [7], [37], [49] and the\nstability problem are are discussed in [62], [49]. The forbidden domain f or resonances and the\nlocalization of complex and real resonances are considered in [49].\nThe corresponding theory for Jacobi matrices with matrix-valued coefficients is still\nlargely a wide open field. The scattering theory for Jacobi matrices with matrix-valued coef-\nficients is considered in [1], [19], [64], [65], [69], [70]. The inverse scatterin g problems for such\noperators are discussed in [68], [70]. The scattering matrix for Schr ¨ odinger operators with\nmatrix-valued potential on the lattice are discussed in [3], [4]. Some pa rticular results were\nobtained in [20], [12], [46], [47], [42], [13] and see references therein.\n2.Preliminaries\n2.1.Determinants and Traces. Let| · |be the operator norm in the space Cd. For a\nsetY⊂Z(belowY=N,Z+=N∪ {0}orZ) define the spaces ℓ2(Y) andℓ∞(Y) as the\nspaces of sequences ( hx)x∈Yequipped with the norms given by /bardblh/bardbl2\n2=/summationtext\nx∈Y|hx|2/greaterorequalslant0 and\n/bardblh/bardbl∞= supx∈Y|hx|respectively.\nLet us recall some well-known facts from [21]. Define the space Bof bounded operators\nacting on the Hilbert space Kequipped with the norm /bardbl · /bardbl. LetB2(K) andB1(K) be the\nHilbert-Schmidt class and the trace of compact operators on the H ilbert space Kequipped\nwith the norms /bardblA/bardbl2\nB2= Tr(A∗A)<∞and/bardbl·/bardblB1respectively.\n•LetA,B∈BandAB,BA ∈B1. Then\nTrAB= TrBA, (2.1)\ndet(I+AB) = det(I+BA). (2.2)\n•LetA∈ B1. Then\n|det(I+A)|/lessorequalslante/ba∇dblA/ba∇dblB1. (2.3)RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 7\n•Let a function A(·) :D→B1be analytic in a domain D⊂C, and the operator ( I+A(k))−1\nis bounded for any k∈D. Then the function F(k) = det(I+A(k)) is analytic in Dand\nsatisfies\nF′(k) =F(k)Tr(I+A(k))−1A′(k)∀k∈D. (2.4)\n•LetA∈B1. Then the series/summationtext∞\nj=1Tr(−A)j\njconverges and\ndet(I+A) = exp/bracketleftbigg\n−∞/summationdisplay\nj=1Tr(−A)j\nj/bracketrightbigg\n, (2.5)\nand, in particular, if /bardblA/bardblB1is small enough, then\nlndet(I+A) = Tr/bracketleftig\nA−A2\n2+O(/bardblA/bardbl3\nB2)/bracketrightig\n. (2.6)\n•LetAadjbe the adjugate matrix of a d×dmatrixA. IfAis an invertible matrix, then\nA−1=1\ndetAAadj. (2.7)\n•For ad×dmatrixAthe determinants has a decomposition\ndet(A−τ1 1d) =d/summationdisplay\n0Tjτd−j, (2.8)\nwhereT0= 1 and the functions Tjare given by (see p.331-333 in [66])\nT1=−T1,T2=−1\n2(T2+T1T1), .....,Tj=−1\nmj−1/summationdisplay\n0Tj−sTs,.., T j= TrAj, j∈N.(2.9)\n2.2.Jost functions. Recall that for operators Hcgiven by (1.2) we have defined matrices\ntx=a−1\nx,Tx=t1t2···tx,Tx,p=txt2···tp, cx= 1−sxax,C0=Tp−1cptp,\nCo\n0=Tp−1bptp+T1\np−1cptp,C1\n0=T1\np−1bp+Tp−1cp−1, Bx=x/summationdisplay\n1bj,\nT1\nx=b1Tx+t1b2t2...tx+t1t2b3t3...tx+...+Tx−1bxtx, x∈Np={1,...,p}.(2.10)\nWe discuss asymptotics of Jost solutions f= (fx) for non-self-adjoint operators, in general.\nLemma 2.1. Let an operator Vc∈Jc\nqfor someq/greaterorequalslant1. Then each function fx(k)is a matrix-\nvalued polynomial in k∈Chaving the form\nfp(k) =tpkp, f j(k) =kjFj(k), j= 0,1...,p−1, (2.11)\nwhere Fj(k)is a polynomial of a degree q−2j, andf0(k),f1(k)have asymptotics as k→ ∞:\nf0(k) =C0kq−Co\n0kq−1+O(kq−2)\nf1(k) =C0kq−1−(Co\n0−b1C0)kq−2+O(kq−3)\nfx(k) =tx...tp−1cptpkq−x+O(kq−x−1), x= 2,...p−1,, q= 2p, (2.12)8 EVGENY KOROTYAEV\nand\nf0(k) =−Tp−1bpkq+C1\n0kq−1+O(kq−2)\nf1(k) =−Tp−1bpkq−1+(C1\n0−b1Tp−1bp)kq−2+O(kq−2)\nfx(k) =−tx...tp−1bpkq−x+O(kq−x−1), x= 2,...p−1,, q= 2p−1.(2.13)\nProof.Letq= 2p. Equations axfx= (λ−bx+1)fx+1−sx+1fx+2andfx=kx1 1, x>p, give:\n•ifx=p, then\napfp= (λ��bp+1)fp+1−fp+2= (k+k−1)kp+11 1−kp+21 1 =kp1 1⇒fp=kptp,(2.14)\n•ifx=p−1, then\nap−1fp−1= (λ−bp)fp−spfp+1= ((k+k−1−bp)kp−spapkp+1)tp= (kp+1cp−bpkp+kp−1)tp,\nfp−1=tp−1(kp+1cp−bpkp+kp−1)tp=tp−1kp−1(k2cp−bpk+1)tp,\n•ifx=p−2, then\nap−2fp−2= (λ−bp−1)fp−1−sp−1fp=/bracketleftig\n(k+k−1−bp−1)tp−1(kp+1cp−bpkp+kp−1)−sp−1kp/bracketrightig\ntp\n=/bracketleftig\n(kp+2+kp)tp−1cp−kp+1bp−1tp−1cp−(kp+1+kp−1)tp−1bp+bp−1tp−1bpkp\n+tp−1(kp+kp−2)−bp−1tp−1kp−1−sp−1kp/bracketrightig\ntp.\nThus we have\nfp−2=tp−2/bracketleftig\nkp+2tp−1cp−kp+1/parenleftig\nbp−1tp−1cp+tp−1bp/parenrightig\n+kp/parenleftig\ntp−1cp+bp−1tp−1bp+tp−1−sp−1/parenrightig\n−kp−1/parenleftig\ntp−1bp+bp−1tp−1/parenrightig\n+tp−1kp−2/bracketrightig\ntp,\nand defining Γ p−j=tp−j···tp−1we get\nfp−2=kp+2Γp−2cptp−kp+1Co\np−2+O(kp),Co\np−2=tp−2tp−1bptp+tp−2bp−1tp−1cptp.\n•Ifx=p−3, then\nap−3fp−3= (λ−bp−2)fp−2−sp−2fp−1= (k−bp−2)fp−2+O(kp+1)\n= (k−bp−2)/bracketleftig\nkp+2Γp−2cptp−kp+1Co\np−2/bracketrightig\n+O(kp+1)\n=kp+3Γp−2cptp−kp+2bp−2tp−2tp−1cptp−kp+2Co\np−2+O(kp+1).\nThus we have\nfp−3=kp+3Γp−3cptp−Co\np−3kp+2+O(kp+1),\nCo\np−3=tp−3tp−2tp−1bptp+tp−3[bp−2tp−2tp−1+tp−2bp−1tp−1]cptp.\nRepeating this procedure we obtain (2.11) and (2.12) for f0. We show (2.12) for f1. Above\nconsiderations give as k→ ∞:\nf0(k) =C0kq−Co\n0kq−1+O(kq−2), f1(k) =Cokq−1−Akq−2+O(kq−3), f2(k) =O(kq−3)\nfor some matrix A. Then substituting these asymptotics into the equation for f0,f1,f2we\nhave\nf0= (k−b1)f1+(1\nkf1−s1f2) = (k−b1)f1+O(kq−2), (2.15)\nwhich yields A=Co\n0−b1C0.RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 9\nLetk= 2p−1 andap=sp= 1 1. Then we have tp= 1 1 andfp=kp1 1. Letx=p−1, then\nap−1fp−1= (λ−bp)fp−fp+1= (k+k−1−bp)kp−kp+1=−bpkp+kp−1,\nfp−1=tp−1(−bpkp+kp−1),\nNote that if q= 2pthenfp−1∼kp+1and but ifq= 2p−1 thenfp−1∼kp.\n•ifx=p−2, then\nap−2fp−2= (λ−bp−1)fp−1−sp−1fp= (k−bp−1)tp−1(−bpkp+kp−1)−sp−1kp+O(kp−1)\n=−kp+1tp−1bp+kp(bp−1tp−1bp+tp−1−sp−1)+O(kp−1).\nThus due to tx−sx=txcxwe have\nfp−2=−Γp−2bpkp+1+C1\np−2kp+O(kp−1),C1\np−2=tp−2/bracketleftig\nbp−1tp−1bp+tp−1cp−1/bracketrightig\n.\n•Ifx=p−3, then due to fp−1=O(kp) we have\nap−3fp−3= (λ−bp−2)fp−2+O(kp) = (k−bp−2)/bracketleftig\n−kp+1Γp−2bp+C1\np−2kp/bracketrightig\n+O(kp)\n=−kp+2Γp−2bp+kp+1/bracketleftig\nbp−2Γp−2bp+C1\np−2/bracketrightig\n+O(kp).\nThus we have\nfp−3=−kp+2Γp−3bp+C1\np−3kp+1+O(kp),C1\np−3=tp−3bp−2Γp−2bp+tp−3C,\np−2\nRepeating this procedure we obtain (2.11) and (2.13) for f0.\nWe show (2.13) for f1. Above considerations give as k→ ∞:\nf0(k) =−αkq+αokq−1+O(kq−2), f1(k) =−αkq−1+Bkq−2+O(kq−3), f2(k) =O(kq−3)\nfor some matrix Bandα=Tp−1bp, αo=C1\n0. Then the substitution of these asymptotics into\nthe equation for f0,f1,f2implies\nf0= (k−b1)f1+(1\nkf1−s1f2) = (k−b1)f1+O(kq−2),\nwhich yields B=αo−b1α.\nWe discuss the asymptotics of Jost solutions for non-selfadjoint o peratorsHcask→0\nand properties of the Jost function f(k) = detf0(k).\nLemma 2.2. LetV∈Jc\nqfor someq/greaterorequalslant1. Then the polynomial f0(k)and Jost function\nf(k) = detf0(k)have the following asymptotics as k→0:\nf0(k) =Tp−T1\npk+O(k2),\nf1(k) =Tpk−(T1\np−b1Tp)k2+O(k3),\nf2(k) =T2,pk2+O(k3),\nf(k) = (1−kTrBp+O(k2))detTp,(2.16)\nwhereTp=t1t2···tpandT1\np=b1Tp+t1b2t2...tp+...+Tp−1bptpand\nTrBx=x/summationdisplay\n1Trbj= TrT−1\nxT1\nx, x∈N. (2.17)10 EVGENY KOROTYAEV\nii) Moreover, the determinant f= detf0and its zeros rjsatisfy\nf(k) =CVqd/productdisplay\n1(k−rn), CV= (−1)dqhdetTp,h=/braceleftigg\ndetcpifq= 2p\ndetbpifq= 2p−1,(2.18)\nf(k) =CV(kq−kq−1A+O(kq−2)), A= Tr/braceleftigg\nc−1\npbp+/summationtextp−1\n1bjif q= 2p\nb−1\npcp−1+/summationtextp−1\n2bj, if q= 2p−1,(2.19)\nqd/productdisplay\n1rj=1\nh,qd/summationdisplay\n11\nrj= TrBpqd/summationdisplay\n1rj=A. (2.20)\nProof.i) Using the equation axfx= (λ−bx+1)fx+1−sx+1fx+2andfx=kx, x>p, we obtain\nfp=kptpand the following cases:\n•ifx=p−1, then\nap−1fp−1= (λ−bp)fp−spfp+1= ((k+k−1−bp)kp−spapkp+1)tp= (kp+1cp−bpkp+kp−1)tp,\nfp−1=tp−1(kp−1−bpkp+O(kp+1))tp;\n•ifx=p−2, then\nap−2fp−2= (λ−bp−1)fp−1−sp−1fp\n=/bracketleftig\n(k+k−1−bp−1)tp−1(kp−1−bpkp+O(kp+1))+O(kp)/bracketrightig\ntp\n=/bracketleftig\ntp−1kp−2−kp−1bp−1tp−1−kp−1tp−1bp+O(kp)/bracketrightig\ntp.\nThus we have\nfp−2=kp−2tp−2tp−1tp−Kp−2tpkp−1+O(kp),Kp−2=tp−2bp−1tp−1+tp−2tp−1bp.\n•Ifx=p−3, then\nfp−3=tp−3/braceleftig\n(λ−bp−2)fp−2−sp−2fp−1/bracerightig\n=tp−3/braceleftig\n(k+k−1−bp−2)/bracketleftig\nkp−2tp−2tp−1−Kp−2kp−1+O(kp)/bracketrightig\n+O(kp−1)/bracerightig\ntp\n=kp−3tp−3tp−2tp−1tp−Kp−3tpkp−2+O(kp−1),\nwhere\nKp−3=tp−3Kp−2+tp−3bp−2tp−2tp−1=tp−3bp−2tp−2tp−1+tp−3tp−2bp−1tp−1+tp−3tp−2tp−1bp.\nRepeating this procedure we obtain asymptotics of f0in (2.16) and using (2.6) we obtain\nasymptotics of det f0in (2.16), since the identity (2.17) holds true:\nTrT1\nxT−1\nx= Tr(b1Tx+t1b2t2...tx+...+Tx−1bxtx)T−1\nx=x/summationdisplay\n1Trbj= TrBx.\nWe show (2.16) for f1. Above considerations give as k→0:\nf0(k) =Tp−T1\npk+O(k2), f1(k) =Tpk−Bk2+O(k3), f2(k) =T2,pk2+O(k3)\nfor some matrix B. Then the substitution of these asymptotics into the equation for f0,f1,f2\nimplies\nf0= (k+1\nk−b1)f1−s1f2= (1\nk−b1)f1+O(k2) =Tp−(B+b1Tp)k+O(k2),(2.21)RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 11\nwhich yields (2.16) for f1.\nii) Asymptotics (2.12), (2.13) give (2.18).\nLetm=p−1. Ifq= 2p, then from (2.12), (2.18) we obtain A= TrC−1\n0Co\n0and\nA= Tr[Tmcptp]−1/parenleftbigg\nTmbptp+T1\nmcptp/parenrightbigg\n= Tr[c−1\npbp+T−1\nmT1\nm] = Tr[c−1\npbp+p−1/summationdisplay\n1bj]\nLetq= 2p−1. Then from (2.13), (2.18) we obtain A= Tr(Tp−1bp)−1(C1\n0−b1Tp−1bp) and\nA= Tr/parenleftbig\n(Tmbp)−1[T1\nmbp+Tmcm]−b1/parenrightbig\n= Tr[T−1\nmT1\nm+b−1\npcm−b1] = Tr[b−1\npcp−1+p−1/summationdisplay\n2bj].\nCollecting last identities we obtain (2.19).\nWe show (2.20). Let P(k) =/producttextqd\n1(k−rj). From (2.16) and (2.18) we obtain\nf(k) =CVP(k),f(0) = det Tp=CVqd/productdisplay\n1(−rj),\nwhereCV= (−1)dqhdetTp, which yields/producttextqd\n1rj= 1/handP(0) = (−1)dq/h.\nRecall Vieta’s formulas for our polynomial P(k) =/producttextdq\n1(k−rj):\nP(k) =kdq−kdq−1C1+.....−C2k+P(0), C1=dq/summationdisplay\n1rj, C 2=dq/summationdisplay\n1P(0)\nrj.(2.22)\nThen using det Tp=CVP(0) and Vieta’s formulas and asymptotics (2.16) we obtain\nf(k) =CVP(k) =CV(P(0)−kC2+....),\nf(k) = (detTp)(1−kTrBp+O(k2)) =CV(P(0)−kP(0)TrBp+O(k2)),\nwhich yields C2=/summationtextqd\n1P(0)\nrj=P(0)TrBpand thus/summationtextqd\n11\nrn= TrBp, sinceP(0)/\\e}atio\\slash= 0.\nFrom (2.19) and Vieta’s formulas we obtain\nf(k) =CV(kq−kq−1A+O(kq−2) =CV(kq−kq−1C1+O(kq−2)\nwhich yields C1=/summationtextqd\n1rj=A.\nFor Schr¨ odiger operators q= 2p−1, allax=tx=Tx1 1 and our matrices have the form\ncx= 0 =C0,Co\n0=bp,T1\nx=Bx,C1\n0=Bp−1bp,C1\n0−b1Tp−1bp=p−1/summationdisplay\n2bjbp.(2.23)\nNowwe collect properties oftheJost functionsforSchr¨ odinger o perators, i.e., when all ax= 1 1.\nFrom Lemma 2.1 and Lemma 2.2 we obtain the following .\nCorollary 2.3. (Schr¨ odinger operators) Letq= 2p−1and allax=sx= 1 1. Then\nf0(k) =−bpkq+Bp−1bpkq−1+O(kq−2)\nf1(k) =−bpkq−1+(Bp−1−b1)bpkq−2+O(kq−2), as k → ∞, (2.24)\nf0(k) = 1 1−Bpk+O(k2)as k→0, (2.25)12 EVGENY KOROTYAEV\nwhereBx=/summationtextx\n1bjand\nf(k) =CVq/productdisplay\nj=1(k−rj), C V= (−1)ddet(bp), (2.26)\nf(k) =CV(kq−kq−1Bp−1+O(kq−2)),ask→ ∞\nf(k) = (1−kTrBp+O(k2))detTp,ask→0.(2.27)\n2.3.Regular solutions. Define a polynomial\nβx(λ) =sx(λ−bx)sx−1(λ−bx−1)....s1(λ−b1), (2.28)\nwheresx=s−1\nx. The direct computation gives that the polynomial βx(λ) has asymptotics\nβx(λ) =Sxλx−λx−1So\nx+O(λx−2),\nSx=sxsx−1...s1,So\nx=sxbxsx−1...s1+sxsx−1bx−1...s1+.....+sxsx−1...s1b1.(2.29)\nDefine the regular solutions ϕ,ϑof the equation under the initial conditions:\nax−1yx−1+sxyx+1+bxyx=λyx,\nϕ0=ϑ1= 0, ϕ 1=ϑ0= 1 1.(2.30)\nLemma 2.4. LetV∈Jc\nqfor someq/greaterorequalslant1. Then the regular solutions ϕx+1(λ),x∈Npsatisfy\nϕx+1(λ) =βx(λ)+O(λx−2) =Sxλx−λx−1So\nx+O(λx−2) (2.31)\nasλ→+∞, whereSx=sxsx−1...s1andSo\nxare defined by (2.29).\nProof.Using the equation sx−1ϕx= (λ−bx−1)ϕx−1−ax−2ϕx−2we obtain for x= 2,3:\nϕ2=s1[(λ−b1)ϕ1] =s1(λ−b1), ϕ3=s2[(λ−b2)ϕ2−a1] =β2(λ)−s2a1.\nsinceϕ0= 0, ϕ1= 1 1. Ifx= 4,5, then we have\nϕ4=s3[(λ−b3)ϕ3−a2ϕ2] =β3(λ)+O(λ),\nϕ5=s4[(λ−b4)ϕ4−a3ϕ3] =β4(λ)+O(λ2).\nRepeating this procedure and using (2.29), we obtain (2.31).\n2.4.Examples. We discuss two simplest examples at q= 1 andq= 2.\nExample 1, q= 1.Consider the case q=p= 1,a1= 1 1,detb1/\\e}atio\\slash= 0. Consider the zeros\nkj,j∈Ndof the Jost function f= det(1 1−b1k). Then we have\nψ= 1 1−b1k, k−1\nj,j∈Ndare eigenvalues of b1. (2.32)\nIf|kj|>1, thenkjis a resonance of H.\nIf|kj|<1, thenkjis an eigenvalue. If |kj|= 1, thenkjis a virtual state.\nIf the eigenvalues of b1are±1. Then the operator Hhas only virtual states ±1, there are no\nresonances and eigenvalues.\nExample 2, q= 2.Consider the case q= 2,p= 1. In this case det c1/\\e}atio\\slash= 0, where c1= 1 1−a2\n1.\nThen from the proof of Lemma 2.1 at x=p−1 we obtain\nψ= (k2c1−b1k+1 1)a−1\n1,f= detψ= det(k2c1−b1k+1 1)deta−1\n1.(2.33)\n•Letb1= 0 and let ±kj,j∈Ndbe zeros of the Jost function f= det(k2c1+ 1 1)det(a1)−1,\nsince the function fis even. Then each k−2\njis real eigenvalues of −c1andk−2\nj∈(−1,∞).\nThen the zeros of fsatisfyRESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 13\nIfk2\nj∈(0,1), then±kjis an eigenvalue of H.\nIfk2\nj∈(1,∞), then±kjis a resonance of H.\nIfk2\nj∈(−∞,−1), then±kj∈iR\\[−1,1] is a resonance of H.\nIfk2\nj= 1, then ±kj∈ {±1}is a resonance of H.\n1•Ifb1/\\e}atio\\slash= 0, then we obtain\nf(k) = det(k2c1−b1k+1 1) =k2d(1 1+O(1/k))detc1 (2.34)\nThen the Jost function f(k) has 2d zeros on the plane C. Even in this simple case very difficult\nto determine eigenvalues and resonances of the operator Hfor larged.\n2•Ifb1= 0 anda1=√\n21 1d, then we have c1=−1 1 andf= det[(1 1 −k2)1 1]2−d/2. Thus all zeros\noffare±1 only. Then the operator Hdoes no have resonances and eigenvalues, all zeros of f\nare virtual states ±1 only.\n3.Properties of the Jost solutions\n3.1.Jost solutions and their Wronskian. Inthis section we discuss Jost functions for self-\nadjoint perturbations V∈Jq. Define two mapping for sequences of matrix-valued polynomials\ny(k) = (yx(k))∞\n0by\n/tildewidey(k) = (/tildewideyx(k))∞\n0,/tildewideyx(k) :=y∗\nx(k),\ny/squaresmallsolid(k) = (y/squaresmallsolid\nx(k))∞\n0, y/squaresmallsolid\nx(k) :=yx(k),k= 1/k.\nFor a Jacobi operator Hand for two sequences of matrix-valued polynomials y(k) = (yx(k))∞\n0,\nη= (ηx(k))∞\n0we introduce the Wronskian\n{y,η}x(k) =/tildewideyx(k)axηx+1(k)−/tildewideyx+1(k)axηx(k), x= 0,1,2,... (3.1)\nThe Wronskian has the property\n/tildewider{y,ξ}=−{ξ,y}. (3.2)\nIf bothyandηare solutions of the Jacobi equation (1.10), then the Wronskian {y,η}x(k)\ndoes not depend on x. Recall that a0= 1 1 by our convention. In particular, we obtain\n{ϑ,ϑ}x= 0,{ϕ,ϕ}x= 0,{ϑ,ϕ}x= 1 1, x/greaterorequalslant0, (3.3)\nand\n{f,f}x=/tildewidefxaxfx+1−/tildewidefx+1axfx= 0,\n{f/squaresmallsolid,f}x=/tildewidef/squaresmallsolid\nxaxfx+1−/tildewidef/squaresmallsolid\nx+1axfx=k−k,(3.4)\nwheref/squaresmallsolid\nx(k) =fx(k) and\n{f,ϕ}x=/tildewidefxaxϕx+1−/tildewidefx+1axϕx=/tildewideψ,\n{f/squaresmallsolid,ϕ}x=/tildewidef/squaresmallsolid\nxaxϕx+1−/tildewidef/squaresmallsolid\nx+1axϕx=/tildewideψ/squaresmallsolid,(3.5)\nwhere we omit the variables λandkfor shortness.\nDefine the Weyl-Titchmarsh functions\nF=ϑ+ϕm+, m +=f1f−1\n0. (3.6)14 EVGENY KOROTYAEV\nLemma 3.1. Let a perturbation V∈Jq, q∈N. ThenF,m+satisfy\nf=ϑψ+ϕf1=Ff0, (3.7)\n{F,F}={F,f}=m+−/tildewidem+= 0,\n{F,ϕ}= 1 1.(3.8)\nProof.Identities (3.5) follow from the definitions of solutions f,ϕ. We have f=ϑA+ϕB\nfor some matrices A,B. Then (3.3) imply\n−ψ={ϕ,f}={ϕ,ϑ}A+{ϕ,ϕ}B=−A.\nSimilar arguments imply B=f1. From the properties of ϕ,ϑwe obtain {F,F}=m+−/tildewidem+\nand the identity (3.4) gives (3.8). This and the identity F=ff−1\n0give{F,f}= 0.\nNow we present the regular solutions ϕxin terms of the Jost solutions.\nLemma 3.2. Let a perturbation V∈Jq, q∈ {2p−1,2p}andk= 1/k. Then\nϕ= (k−k)−1(f/tildewideψ/squaresmallsolid−f/squaresmallsolid/tildewideψ), (3.9)\nψ/tildewideψ/squaresmallsolid=ψ/squaresmallsolid/tildewideψ, (3.10)\nϕp=1\nk−ka−1\np(kp/tildewideψ/squaresmallsolid−kp/tildewideψ), (3.11)\nϕx=1\nk−k(kx/tildewideψ/squaresmallsolid−kx/tildewideψ), x>p, (3.12)\nProof.We haveϕx=fxC1+f/squaresmallsolid\nxC2for some matrices C1,C2. Then (3.4) imply\n/tildewideψ={f,ϕ}x={f,f}xC1+{f,f/squaresmallsolid}xC2={f,f}xC1+{f,f/squaresmallsolid}xC2= (k−k)C2,\nwhich yields C2=−/tildewideψ(k)/(k−k). Similar arguments imply C1=/tildewideψ/squaresmallsolid/(k−k). Identity (3.9)\ngives (3.10)-(3.12).\nNow we present the regular solutions ϑxin terms of the Jost solutions.\nLemma 3.3. Let a perturbation V∈Jq, q∈ {2p−1,2p}andk= 1/z. Then\nϑ=1\nk−k(f/tildewidef1−f/squaresmallsolid/tildewidef/squaresmallsolid\n1), (3.13)\nϑp(λ) =1\nk−ka−1\np(kp/tildewidef1−kp/tildewidef/squaresmallsolid\n1). (3.14)\nProof.We haveϑ(λ) =fC1+f/squaresmallsolidC2, wheref/squaresmallsolid=f(1/k),k∈CandC1,C2are some matrices.\nThe Wronskian gives at x= 0:\n{f,ϑ}x=/tildewidefxaxϑx+1−/tildewidefx+1axϑx=−/tildewidef1,\n{f/squaresmallsolid,ϑ}x=/tildewidef/squaresmallsolid\nxaxϑx+1−/tildewidef/squaresmallsolid\nx+1axϑx=−/tildewidef/squaresmallsolid\n1,\nsinceϑ/squaresmallsolid=ϑand then (3.4) gives\n−/tildewidef1={f,ϑ}x={f,fC1+f/squaresmallsolidC2}x={f,f}xC1+{f,f/squaresmallsolid}xC2= (k−k)C2,\nwhich yields C2=/tildewidef1/(k−k). Similar arguments imply C1=/tildewidef/squaresmallsolid\n1/(k−k). From (3.13) we have\n(3.14).\nNow we write down the Jost function in terms of the regular solutions .RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 15\nLemma 3.4. Let a perturbation V∈Jq, q∈ {2p−1,2p}andλ=k+1\nk. Then\n/tildewideψ(k) =kpϕp+1(λ)−apkp+1ϕp(λ), (3.15)\nf1(k) =−kpϑp+1(λ)+apkp+1ϑp(λ). (3.16)\nProof.From (3.5) at x=pwe have (3.15). From Lemma 3.3 we have (3.16).\n3.2.Spectral parameters. We discuss spectral parameters of the operator Hassociated its\nwith eigenvalues.\nDefinition of Spectral data. For each eigenvalue kj∈(−1,1),j∈n•with the multiplicity\ndj/lessorequalslantd, we define the spectral data:\n•the subspaces\nEj= Kerfo(kj)⊂Cd, E•\nj= Kerf∗\no(kj)⊂Cd,dimEj= dimE•\nj=dj/lessorequalslantd;(3.17)\n•the orthogonal projectors Pj:Cd→EjandP•\nj:Cd→E•\nj,\nPjCd=Ej, P•\njCd=E•\nj;\n•the positive norming matrix PjKjPj:Ej→Ej, where the matrix Kjis given by\nKj=/summationdisplay\nx/greaterorequalslant1fx(kj)∗fx(kj)>0, j∈n•.(3.18)\nWe define also the matrix-valued functions of k∈C:\nuj(k) =P•\njψ(k)Pj:Ej→E•\nj, v j(k) =P•\njf1(k)Pj:Ej→E•\nj.(3.19)\nLemma 3.5. LetV∈Jq, q∈ {2p,2p−1}and let{kj,j∈n•}be all zeros of detfo(k)on\n(−1,1)and letkj=k−1\nj. Then for any j∈n•we have\nKj=/summationdisplay\nx/greaterorequalslant1f∗\nx(kj)fx(kj) =1\n˙λ(kj)/parenleftig\n˙f∗\no(kj)f1(kj)−˙f∗\n1(kj)fo(kj)/parenrightig\n>0,(3.20)\nKj/greaterorequalslantCj1 1d, C j=k2p+2\nj\n1−k2\nj, (3.21)\nf1(kj)Pj⊂E•\nj, u j(kj) =f1(kj)Pj, (3.22)\nPjKjPj=1\n˙λ(kj)Pj˙ψ∗(kj)f1(kj)Pj=1\n˙λ(kj)˙u∗\nj(kj)vj(kj)/greaterorequalslantCjPj. (3.23)\nProof.Let ˙u=∂\n∂zuandλ=k+k−1,k∈R. From (1.10) we have\nax−1fx−1+axψx+1+bxfx=λfx, x∈Z+,\nax−1˙fx−1+ax˙fx+1+bx˙fx=λ˙fx+˙λfx,\n˙f∗\nx−1ax−1+˙f∗\nx+1ax+˙f∗\nxbx=λ˙f∗\nx+˙λf∗\nx,\nmultiplying the first equation by ˙f∗\nxand the third equation by fxand taking the difference\n˙f∗\nxax−1fx−1+˙f∗\nxaxfx+1+˙f∗\nxbxfx=λ˙f∗\nxfx, x∈Z+,\n˙f∗\nx−1ax−1fx+˙f∗\nx+1axfx+˙f∗\nxbxfx=λ˙f∗\nxfx+˙λf∗\nxfx,(3.24)\nwe obtain\n˙λf∗\nxfx=˙f∗\nx−1ax−1fx+˙f∗\nx+1axfx−˙f∗\nxax−1fx−1−˙f∗\nxaxψx+1.16 EVGENY KOROTYAEV\nSumming we have\n˙λ/summationdisplay\nx/greaterorequalslant1f∗\nxfx=/summationdisplay\nx/greaterorequalslant1/parenleftbigg\n[˙f∗\nx−1ax−1fx−˙f∗\nxaxψx+1]+[˙f∗\nx+1axfx−˙f∗\nxax−1fx−1]/parenrightbigg\n=˙f∗\n0a0f1−˙f∗\n1a0f0=˙f∗\n0f1−˙f∗\n1f0,\nwhich yields (3.20). We show (3.20). From (1.10) we have\n/summationdisplay\nx/greaterorequalslantp+1f∗\nx(kj)fx(kj) =k2p+2\nj/summationdisplay\nn/greaterorequalslant0k2n\nj1 1d=k2p+2\nj\n1−k2\nj1 1d,\nwhich yields (3.21) and (3.23). We show (3.22). From (3.4) we obtain\nψ∗(kj)f1(kj)Pj=f∗\n1(kj)f0(kj)Pj= 0,\nthis givesf1(kj)Pj⊂E•\nj, i.e.,P•\njf1(kj)Pj=f1(kj)Pj, which yields Pj˙f∗\n0(kj)f1(kj)Pj=\n˙u∗\nj(kj)vj(kj). Substituting it into (3.20) and using (3.21) we obtain (3.23).\nWe describe the poles of the matrix ψ(k)−1.\nLemma 3.6. LetV∈Jq, q∈Nand letkj∈(−1,1),j∈n•be a zero of f. Then\ni) The matrix ˙uj(kj) =P•\nj˙ψ(kj)PjandP•\njψ1(kj)PjfromEjtoE•\njare invertible.\nii) The following asymptotics holds true:\nψ(k)−1=Pj˙uj(kj)−1P•\nj\nk−kj+O(1)as k→kj. (3.25)\nProof.i) Using (3.21) and (3.22), we obtain\nCjPj/lessorequalslantPjKjPj=1\n˙λ(kj)˙u∗\nj(kj)vj(kj).\nSince the left hand side is positive definite on Ej, then the matrices vj(kj) =P•\njf1(kj)Pjand\n˙uj(kj) =P•\nj˙ψ(kj) acting from EjtoE•\njare invertible.\nii) Lett=k−kj→0. Below we omit jfor shortness. Let ψ=/parenleftbigg\nα β\nγ u/parenrightbigg\n, where\nαt= (P•\nj)⊥ψ(kj+t)P⊥\nj:E⊥\nj→(E•\nj)⊥, β t= (P•\nj)⊥ψ(kj+t)Pj:Ej→(E•\nj)⊥,\nγt=P•\njψ(kj+t)P⊥\nj:E⊥\nj→E•\nj, u t=P•\njψ(kj+t)Pj:Ej→E•\nj.\nDue toψ(kj)Pj= 0 andP•\njψ(kj) = 0, we have\nαt=αo+O(t), β t=t˙βo+O(t2), γt=t˙γo+O(t2), ut=t˙uo+O(t2),\nast→0, where\n˙uo=P•\nj˙ψ(kj)Pj= ˙u(kj), α o= (P•\nj)⊥ψ(kj)P⊥\nj=ψ(kj)P⊥\nj:E⊥\nj→(E•\nj)⊥.\nThe operator αois invertible, since ψh= 0 implies h∈Ej. Due to i) the operator ˙ uois\ninvertible. Define w=u−γα−1βfrom the properties of α,β,γ,u we deduce that\nwt=t(˙u(kj)+O(t)).RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 17\nTherefore, the standard formula for the inverse matrix\n/parenleftbigg\nα β\nγ u/parenrightbigg−1\n=/parenleftbigg\nα−1+α−1βw−1γα−1−α−1βw−1\n−w−1γα−1w−1/parenrightbigg\n, w=u−γα−1β,\ndet/parenleftbigg\nα β\nγ u/parenrightbigg\n= detαdet(u−γα−1β) = detudet(α−βu−1γ) (3.26)\ngives\nψ(k)−1=/parenleftbigg\nα−1\no+O(t)O(1)\nO(1)t−1˙u−1\no+O(1)/parenrightbigg\nast→0.\nIn particular, this implies res\nk=kj(ψ(k))−1=Pj˙u−1\noP•\nj.\nWe describe properties of eigenvalues and the corresponding proj ectors.\nLemma 3.7. LetV∈Jq, q∈Nandkj∈(−1,1),j∈n•be a zero of fandkj= 1/kj. Then\ni) The matrix P•\njψ(kj)satisfies\nψ∗(kj)P•\nj⊂Ej, P jψ∗(kj)P•\nj=u∗\nj(kj) =ψ∗(kj)P•\nj. (3.27)\nii) For the eigenvalue kjthe corresponding solution has the form\nϕ(λj)P•\nj=cjf(kj)uj(kj)∗,cj=1\nkj−kj, (3.28)\n˙uj(kj)uj(kj)∗=uj(kj)˙uj(kj)∗. (3.29)\nProof.i) From (3.10) we have ψ(kj)ψ∗(kj)P•\nj=ψ(kj)ψ∗(kj)P•\nj= 0 which yields ψ∗(kj)P•\nj⊂\nEjand (3.27).\nii) From (3.9) we obtain\nϕ(λj) =cj(f(kj)ψ∗(kj)−f(kj)/tildewideψ(kj)) =cj(f(kj)ψ∗(kj)−f(kj)ψ∗(kj)). (3.30)\nThen from the identity (3.27) ψ∗(kj)P•\nj= 0 we obtain (3.28). From (3.10), (3.27) we have\nP•\nj˙ψ(kj)ψ∗(kj)P•\nj=P•\njψ(kj)˙ψ∗(kj)P•\nj= ˙uj(kj)uj(kj)∗=uj(kj)˙uj(kj)∗.\n4.Proof of inverse problems\n4.1.S-matrix. We discuss properties of S-matrix S(k) =ψ−1(k)ψ(k), k∈S:={|z|= 1}.\nWe determine asymptotics of S-matrix.\nCorollary 4.1. LetV∈Jq. Then S-matrix S(k),k∈Shas a meromorphic extension on the\nwhole complex plane and satisfies\nS(k) =ψ−1(k)ψ(1/k) =/tildewideψ(1/k)/tildewideψ(k)−1, k∈C, (4.1)\nS∗(k) =S(k) =S(k)−1, k∈S. (4.2)\nMoreover, it has asymptotics:\nS(k) = (−k)−qC−1/parenleftbigg\n1 1+S1k−1+O(k−2)/parenrightbigg\nTpask→ ∞, (4.3)\nS(k) = (−k)−qT−1\np/parenleftbigg\n1 1−kS1+O(k2/parenrightbigg\nCask→0, (4.4)18 EVGENY KOROTYAEV\nC=/braceleftigg\nC0\nTp−1bp, S1=/braceleftigg\nCo\n0C−1−T1\npT−1\np, if q = 2p\nC1\n0C−1−T1\npT−1\np, if q = 2p−1(4.5)\nwhereT1\np,Tp,Co\n0,C1\n0,C0are given by (1.12).\nProof.Letk= 1/k. From the definition S(k) =ψ−1(k)ψ(k) andk= 1/konSand from\n(3.10) we obtain (4.1). From (4.1) we get\nS∗(k) =ψ−1(k)ψ(k) =S(k), S∗(k)S(k) =ψ−1(k)ψ(k)ψ−1(k)ψ(k) = 1 1.\n•Letq= 2pandC=C0. Ifk→ ∞, then from S(k) =ψ(k)−1ψ(k) and (2.12), (2.16) we have\n:\nS(k) =ψ−1(k)ψ(k) =k−qC−1[1 1−Co\n0C−1k+O(k2)]−1[Tp−kT1\np+O(k2)]\n=k−qC−1[(1 1+Co\n0C−1k+O(k2)][1 1−kT1\npT−1\np+O(k2)]Tp\n=k−qC−1/parenleftbigg\n1 1+k(Co\n0C−1−T1\npT−1\np)+O(k2)/parenrightbigg\nTp.\nIfk→0, then similar arguments give\nS(k) =ψ(k)−1ψ(k) =k−qT−1\np[1 1−T1\npT−1\npk+O(k2)]−1[1 1−Co\n0C−1k+O(k2)]C\n=k2pT−1\np[1 1+T1\npT−1\npk+O(k2)][1 1−Co\n0C−1k+O(k2)]C\n=kqT−1\np/parenleftbigg\n1 1+k(T1\npT−1\np−Co\n0C−1)+O(k2)/parenrightbigg\nC,\n•Letq= 2p−1 andC=Tp−1bp. Ifk→ ∞, then from (2.13), (2.16) we have\nS(k) =ψ(k)−1ψ(k) =−k−qC−1[1 1−C1\n0C−1k+O(k2)]−1[(1 1−kT1\npT−1\np+O(k2)]Tp\n=−k−qC−1[1 1+C1\n0C−1k+O(k2)][1 1−kT1\npT−1\np+O(k2)]Tp\n=−k−qC−1/parenleftbigg\n1 1+k(C1\n0C−1−T1\npT−1\np)+O(k2)/parenrightbigg\nTp.\nIfk→0, then similar arguments give\nS(k) =ψ−1(k)ψ(k) =−k−qT−1\np[1 1−kT1\npT−1\np+O(k2)]−1[1 1−C1\n0C−1k+O(k2)]C\n=−k−qT−1\np[1 1+k(T1\npT−1\np−C1\n0C−1)+O(k2)]C.\nWe rewrite the S-matrix in terms of the Weyl function Mj=ϕjϕ−1\nj+1,j=p,p+ 1 and\nconversly.\nLemma 4.2. LetV∈JqandMp(λ) =ϕp(λ)ϕp+1(λ)−1andλ(k) =k+1\nk,k∈S\\{±1}and\nletϕj(λ(k))/\\e}atio\\slash= 0,j=p,p+1,p+2,. Then\ni) We have det(ajMj(λ(k))−ζ)/\\e}atio\\slash= 0, wherej=p,p+1,andζ=k,kand the identities hold\ntrue:\nS(k) =1\nk2p·1 1−kapMp(λ(k))\n1 1−kapMp(λ(k))=1\nk2p+2·1 1−kMp+1(λ(k))\n1 1−kMp+1(λ). (4.6)\nii) The matrix k2jS(k)−1 1,j=p,p+1,p+2is invertible and\nMp(λ(k)) =a−1\npk2pS(k)−1 1\nk(2p+1)S(k)−k, M p+1(λ(k)) =k2(p+1)S(k)−1 1\nk(2p+3)S(k)−k.(4.7)RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 19\nProof.i) Letk∈S. From (3.15) we have\n/tildewideψ(k) =kp[ϕp+1(λ)−kapϕp(λ)] =kp[(1 1−kapMp(λ)]ϕp+1(λ),\nwhich implies det( Xj(λ)−ζ)/\\e}atio\\slash= 0. Moreover, due to the identity (4.1) we obtain\nS(k) =/tildewideψ(1/k)/tildewideψ(k)−1=k−2p[1 1−kapMp(λ)][1 1−kapMp(λ)]−1.\nii) From (4.6) we have k2pS(k)[1 1−kapMp(λ)] = 1 1−kapMp(λ), which yields (4.7).\nWe describe the poles of the matrix S(k) associated with eigenvalues of H.\nLemma 4.3. Letkj∈(−1,1)andkj= 1/kj,j∈n•. Then the matrix-valued function S(k)\nhas a simple pole at kjsuch that\nS(k) =Xj\nk−kj+O(1),\nXj=Pj˙uj(kj)−1P•\njψ(kj) =Pj˙uj(kj)−1uj(kj)>0 inEj,(4.8)\nwhere˙uj(kj) =P•\nj˙ψ(kj)Pj:Ej→E•\njis invertible for all j∈n•.\nProof.From (3.25), (3.27) we obtain as k→kj:\nS(k) =ψ−1(k)ψ(1/k) =/parenleftbiggPj˙uj(kj)−1P•\nj\nk−kj+O(1)/parenrightbigg\nψ(1/k)\n=Xj\nk−kj+O(1), X j=Pj˙uj(kj)−1P•\njψ(kj) =Pj˙uj(kj)−1uj(kj),(4.9)\nwhere due to Lemma 3.6 the mapping ˙ uj(kj) =P•\nj˙ψ(kj)Pj:Ej→E•\njis invertible for all\nj∈Nm. From (3.23) we have\nCjPj/lessorequalslantPjKjPj=1\n˙λ(kj)˙u∗\nj(kj)vj(kj) =1\n˙λ(kj)v∗\nj(kj)˙uj(kj). (4.10)\nFrom (3.4) and u(kj)Pj= 0 we obtain\nψ∗(kj)f1(kj)−f∗\n1(kj)ψ(kj) =kj−kj,\nPjψ∗(kj)f1(kj)Pj=u∗\nj(kj)vj(kj) =v∗\nj(kj)uj(kj) = (kj−kj)Pj.\nThenv∗\nj(kj) = (kj−kj)Pjuj(kj)−1and substituting it into (4.10) we obtain\nPjKjPj=1\n˙λ(kj)v∗\nj(kj)˙uj(kj) =1\n˙λ(kj)(kj−kj)Pjuj(kj)−1˙uj(kj) =kjPjuj(kj)−1˙uj(kj).\n4.2.Proof of main theorems. We begin to prove main theorems.\nProof of Theorem 1.1. i) From (2.12) and (2.16) we have (1.13), (1.14) respectively.\nii) Letk= 1/kandq= 2p. We show (1.15). From (1.13), (2.6) we obtain\nf(k) = det[C0kq+O(kq−1)] =kdqdetC0det[1 1+O(1/k)] =kdqdetC0(1 1+O(1/k)),(4.11)\nand detC0= detcpdetTp. This yields f(k) = detcpdetTp/producttextqd\n1(k−rn). The proof of (1.15) for\nq= 2p−1 is similar.\nThe asymptotics (1.13) yields f(0) = det Tp/\\e}atio\\slash= 0, thus/producttextqd\n1rn=1\nh. From (1.14) we have\nf′(0)\nf(0)=−/summationtextk\n11\nrn=−TrBp,which yields (1.16).20 EVGENY KOROTYAEV\niii) Recall that h= detcpifq= 2pandh= detbpifq= 2p−1. If the operator Hdoes\nnot have eigenvalues for some q >2, then all |rj|/greaterorequalslant1 (and some of them >1). From (1.15)\nwe get/producttextqd\n1|rj|= 1/|h|and we deduce that |h|<1 andr+<1\n|h|andrdq\n+/lessorequalslant1\n|h|, which yields\n1\n|h|1\nqd/lessorequalslantr+<1\n|h|. Similar arguments give rqd\n−/lessorequalslant1\n|h|. Collecting these estimates we obtain (1.17).\niv) If|h|>1, then the identity/producttextqd\n1|rj|=1\n|h|<1 gives that some |rj|<1 and the operator H\nhasmeigenvalues counted with multiplicities eigenvalues. This yields rm\n−<1/|h|.\nStatement v) is proved in Theorem 5.9.\nRecall results about the perturbation theory for polynomials, see e.g. p.5, from [66].\nTheorem 4.4. LetF(t,ε) =td+v1(ε)td−1+..+vd(ε)be an d-th degree polynomial in tand\nwhose coefficients are all analytic functions of ε. Supposet=tois a root of multiplicity mof\nF(t,0). Then for εnearεo= 0, there are exactly doroots (counting multiplicity) of F(t,ε)\nneartoand these roots are the branches of one or more multi-valued a nalytic functions with\nat worst algebraic branch points at ε=εo. Explicitly, there are positive integers d1,...,dkwith\nd1+...+dk=doand multi-valued analytic functions t1,...,tk(not necessary distinct) with\nconvergent Puiseux series (Taylor series in ε1\ndo)\nts(ε) =to+∞/summationdisplay\nj=1αs,jεj\nds, (4.12)\nso that the doroots neartoare given by the d1values oft1, thed2values oft2, etc.\nProof of Theorem 1.3. ConsiderH, whereV∈JqandVsatisfies (1.4), (1.21) for some\nq= 2p/greaterorequalslant2. The proof for the case q= 2p−1 is similar. The condition (1.21) gives\ncp= 1 1−a2\np=−2εa′\np−ε2a′\np2, and b p,a′\np∈Mo\nd.\nThe corresponding Jost matrix we denote by f0(k,ε). Let the Jost polynomial f(k,ε) =\ndetψ(k,ε) andψ(k,0) = det f(k,0). Here f(k,ε) is a polynomial in kof the order qdand\nf(k,0) is a polynomial in kof the order ( q−1)d.\ni) Letf(k,ε) = detψ(k,ε) andf(k,0) = detψ(k,0). Here f(k,ε),ε/\\e}atio\\slash= 0 is a polynomial in\nkof the order qdandf(k,0) is a polynomial in kof the order ( q−1)d. Letε→0. Then the\npolynomial f(k,ε) converges to the polynomial f(k,0) uniformly in each disk DR,R>0.\nLetko\nj,j∈NMbe zeros of f(k,0) with the multiplicity do\njfor someM. Consider the discs\nDrε(ko\nj) for somerε>0 small enough. Then due to by Rouch´ e’s Theorem, f(k,ε) has as many\nroots, counted with multiplicities, as f(k,0) in each disk Drve(ko\nj) and the remaining domain\nDRε\\∪Drε(ko\nj) for someRε>0 big enough.\nii) Consider the last dzeros in the domain C\\DRεand show that they satisfy (1.22). In order\nto compute these zeros we introduce new variable z= 1/kand the corresponding functions\nφ(z,ε) =zqψ(1/z,ε) =C0−Co\n0z+z2φ1(z,ε),\nC0=−εA−ε2B, A=Tp−12a′\nptp, B=Tp−1a′\np2tp,\nCo\n0=α−εβ−ε2β1, α=Tp−1bptp∈Mo\nd, β= 2T1\np−1a′\nptp, β1= 2T1\np−1a′\np2tp,\nwhere\n|φ1(z,ε)|/lessorequalslantCφ, (4.13)RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 21\nuniformly in z,ε∈D, for some constant Cφ. The large zeros kj(ε)∈C\\DRε,j∈Ndmap to\nzeroszj(ε) = 1/kj(ε)∈Drε,j∈Nd, whererε= 1/Rε.\nWe do a next transformation. Let z=εt. Then we have φ(εt,ε) =εg(t,ε) where\ng(t,ε) =−A−εB−(α−εβ−ε2β1)t+εt2φ1(εt,ε) =−go(t)+εg1(εt,ε),\ngo(t) =A+tα, g 1(εt,ε) =−B+tβ+εtβ1+t2φ1(εt,ε).(4.14)\nConsider the zeros of the determinant fo(z) = detgo(z). We have\nfo(t) = detTp−1[2a′\np+tbp]tp= detTpdet[2a′\np+tbp],\nwhere det[2 a′\np+tbp],t∈Cis the real polynomial of the order d. This polynomial has only real\nzeros since the matrix a′\np,bpare self-adjoint. Let to\nj,j∈Ndbe zeros of fowith the multiplicity\ndo\njfor somemand/summationtextm\n1do\nj=d. Then the determinant det g(t,ε) has the form\nf(t,ε) = detg(t,ε) = (−1)dfo(t)+εf1(t,ε),\nwheref1(t,ε) is an d-th degree polynomial in tand whose coefficients are all polynomial in\nofε. We apply Theorem 4.4 to the function f(t,ε). Due to Theorem 4.4 for any ε >0 and\neachj∈Nmthere exist do\njzerostj,s(ε) off(t,ε) in each disc Drε(to\nj) for some rε>0 small\nenough. The zeros tj,s(ε) are multivalued analytic functions in ε(not necessary distinct) with\nconvergent Puiseux series (4.12) (Taylor series in ε1\nd). Thus we have tj,s(ε) =to\nj+O(ε1\ndj).\nThen using the transformations k=1\nzandz=εt, we obtain\nkj,s(ε) = 1/(εtj,s(ε)) = (ε)−1(to\nj+O(ε1\ndj))−1= (ε)−1(τo\nj+O(ε1\ndj)),whereτo\nj= 1/to\nj.\n4.3.1-dim case. We consider Jacobi operators Hacting ony= (yx)∞\n1∈ℓ2(N) and given by\n(Hy)x=ax−1yx−1+axyx+1+bxyx, x∈N, (4.15)\nwhereax>0,bx∈Rand formally y0= 0. In the case d= 1 the set Jqconsists of finitely\nsupported sequences ax−1,bx,x∈Nand the condition (1.4) has the form\nax−p>0, ax= 1, bx= 0,∀x>p/braceleftigg\nap= 1,bp/\\e}atio\\slash= 0 ifq= 2p−1\nap/\\e}atio\\slash= 1, ifq= 2p.(4.16)\nIn order to describe eigenvalues and resonances of Hfor this case we define the set of all\npossible locations of the eigenvalues and resonances from [49].\nDefinition R .LetRq⊂Cq\\{0},q∈Nbe a set of vectors r= (rn)q\n1such that:\n1)0<|r1|/lessorequalslant|r2|/lessorequalslant...../lessorequalslant|rq|and if|rj|=|rj+1|for somej, then0/lessorequalslantargrj/lessorequalslantargrj+1<2π.\nA polynomial F(k) =/producttextq\n1(k−rn)is real on the real line.\n2) All zeros of FinDare real and simple. Denote the zeros of FinDaskj, j∈n•for some\nn−,n+/greaterorequalslant0arranged by\n−1<k−n−<..<k −1<0<k1<...<k n+<1,\nand they satisfy\nF(1/kj)/\\e}atio\\slash= 0∀j∈n•={−n−,...,−1,1,...,n+}. (4.17)\n3)Fhas an odd number /greaterorequalslant1of zeros on each interval (k−1\nj,k−1\nj−1), j=n•\\{−1}and an even\nnumber/greaterorequalslant0of zeros on each of intervals (1,k−1\nn+)and(k−1\n−n−,−1).22 EVGENY KOROTYAEV\n4) Ifq= 2p/greaterorequalslant2, then1\nF(0)<1or it is equivalent real F(0)/∈[0,1].\nHere the condition1\nf(0)<1 in 4) is equivalent to ap/\\e}atio\\slash= 1 atq= 2p.\nWe sometimes write ψn(V),rn(V),.. instead of ψn,rn,.. when several potentials are\nbeing dealt with. Now we define the mapping r:Jq→ Rq, q∈Nby\nV→r(V) = (rn(V))q\n1∈ Rq, (4.18)\nwhere (rn)q\n1is the sequence of zeros of the corresponding Jost function ψ(k,V). Recall results\nabout inverse resonance scattering for Jacobi operators (see e.g., [49]) with d= 1.\nTheorem 4.5. i) Each mapping r:Jq→ Rq, q∈Ngiven by (4.18) is a bijection between Jq\nandRq. There is an algorithm to recover the perturbation Vfromr(V).\nii) LetSjbe the S-matrix for two Jacobi operators Hjwith perturbations Vj∈Jq, q/greaterorequalslant1for\nj= 1,2. Let(ζj)q\n1be a sequence of distinct points on C−/intersectiontext{|z|= 1}such thatS1(ζj) =S2(ζj)\nfor allj∈Nq. Then we have V1=V2.\nRemark. This theorem solves the inverse resonance problem for Jacobi ope rators on the\nhalf-lattice with finitely-supported perturbations.\nFrom Theorem 4.5 we obtain the following results.\nCorollary 4.6. i) Let finite sequence rj∈(−1,0)∪(0,1),j∈Nmfor somem/greaterorequalslant0. Then there\nexists a finite perturbation V∈Jqfor someq/greaterorequalslant1such that each rj,j∈Nmis an eigenvalue\nfor the operator H.\nii) Let a vector r= (rn)q\n1∈ Rqfor someq >1. Then there exists a finite perturbation V∈Jq\nfor someq/greaterorequalslant1such that each rj∈(−1,1)is an eigenvalue for the operator Hand other\npointsrjare resonances for the operator H.\niii) The multiplicity of a resonance can be any number for som e specific perturbation V∈Jq.\nExample, separation of variables. Consider Jacobi operators HonH=ℓ2(N,Cd) given\nby (1.6). Let all matrices ax,bxare diagonal and have the from:\nax= diag{a1,x,a2,x,...,ad,x}, b x= diag{b1,x,b2,x,...,bd,x} ∀x∈N,\nfor some (aτ,x,bτ,x)∈R+×R,(τ,x)∈Nd×N. Then the operator HonH=ℓ2(N,Cd) is an\northogonal sum of a scalar Jacobi operator Hτacting onℓ2(N) and given by\nH=⊕d\n1Hτ,(Hτy)x=aτ,xyx+1+aτ,x−1yx−1+bτ,xyx, x∈N, y0= 0,(4.19)\nwherey= (yx)x∈N∈ℓ2(N). In this case each fτ,x(k) is the standard Jost solution for Hj,\nandψτ(k) =fτ,0(k) is the corresponding Jost function. The properties of ψτis well known,\nsee e.g, [49] and [7], [14], [24]–[30], [37], [49], [60]. In the scalar case the co ndition (1.4) has\nthe form (4.16). We assume that each sequence ( aτ,x,bτ,x,x∈N),τ∈Ndsatisfies condition\n(4.16). Note that the Jost function for the operator Hhas the for ψ(k) =/producttextd\n1ψτ(k).\nProof of Theorem 1.4. We consider Jacobi operators Honℓ2(N) given by (4.19), i.e.,\nH=⊕d\n1Hτ, where (Hτy)x=aτ,xyx+1+aτ,x−1yx−1+bτ,xyx,x∈N, andthenumbers ( aτ,x,bτ,x)∈\nR+×R,(τ,x)∈Nd×Nsatisfy (4.16). Thus the corresponding operator V∈Jq.\ni) Let a finite sequence kj∈(−1,0)∪(0,1),j∈Nmfor somem/greaterorequalslant0, where each rjhas the\nmultiplicity dj/lessorequalslantd. We take the Jacobi operator Hacting onℓ2(N,Cd) given by (4.19). Due\nto Theorem 4.5 we can assume that H1has simple eigenvalues kj∈(−1,0)∪(0,1),j∈Nm\nfor some operator V1acting onℓ2(N) and given by\n(V1y)x= (a1,x−1−1)yx−1+b1,xyx+(a1,x−1)yx+1 (4.20)RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 23\nfromJqfor anyq/greaterorequalslant2m−2. After this remaining numbers kjhave multiplicity dj−1. From\nthis remaining sequence we again due to Theorem 4.5 we assume that H2has these numbers\nas simple eigenvalues for some operator V2acting onℓ2(N) and having the form (4.20) from\nJqfor anyq/greaterorequalslant2m−2. Repeating this procedure we construct the operators H1,...,H dand\na finite perturbation V=⊕d\n1Hτ∈Jqfor someq/greaterorequalslant2m−2 such that each kj,j∈Nmis an\neigenvalue for the operator H.\nii) Let a vector r= (rn)m\n1∈ Rmfor somem >0. Then there exists a finite perturbation\nV∈Jqfor someq/greaterorequalslant1 such that each rj∈(−1,1) is an eigenvalue for the operator Hand\nother points rjare resonances for the operator H.\nLet we have a sequence R={rn∈C\\D,n∈Nqd}for someq/greaterorequalslant1, where a number of\nthe pointsrn= 1 (orrn=−1)/lessorequalslantd. From the main sequence Rwe take the first sequence\nR1={̺n,n∈Nq}such that̺1=−1 (if there exists such point in R) and̺q= 1 (if there\nexists such point in R) and̺n,n= 2,...,q−1 are any points from R. For this sequence\nR1due to Theorem 4.5 there exists a scalar Jacobi operator H1with a perturbation V1∈Jq\nhaving the form (4.20) such that each point from this small sequenc eR1is a resonance for\nthe scalar Jacobi operator H1. From the remaining sequence R\\R1due to Theorem 4.5 we\nagain take a new sequence R2={̺n,n∈Nq,̺n∈R\\R1}such that there exists a scalar\nJacobi operator H2with a perturbation V2∈Jqhaving the form (4.20). Here each point from\nthis second small sequence R2is a resonance for the scalar Jacobi operator H2. We repeat\nthis procedure and we obtain H=⊕d\n1Hτ, where each Hτis the scalar Jacobi operator with\na perturbation Vτ∈Jqand each point of the sequence R={rn,n∈Nqd}is a resonance\nfor some operator Hτ. Here the perturbation V=⊕d\n1Vτbelongs to Jqand we have used the\nidentityψ(k) =/producttextd\n1ψτ(k). The statement iii) follows from ii).\nProof of Theorem 1.5. i) Assume that Jacobi operators Hs=Ho+Vs,s= 1,2 with\ndifferent perturbations Vs∈Jqhas the same S-matrix S(k) for allk∈S. Then the identity\n(4.7) gives that V1,V2have the same Weyl-Titchmarsh matrix-valued function Mp+1:\nMp+1(λ) =S(k)−1 1k2p+2\nkS(k)−k(2p+3)k∈S,\nwhich hasa meromorphic extension onthe whole plane. Then Theorem 6.3 gives that V1=V2.\nii) Letψ=ψ(k) be theJost matrix forthe Jacobi operator Hs=Ho+Vwiththe perturbation\nV∈Jq. Letζj∈C, j∈Nq+1be a sequence of distinct numbers. Then due to (1.13) the\nmatrix-valued polynomial ψ=ψ(k) has asymptotics\nψ(k) =Ckq+O(kq−1) ask→ ∞,detC/\\e}atio\\slash= 0,\nfor some matrix C. Thus each component of ψ(k) is a polynomial of degree /lessorequalslantqand is given at\nof distinct numbers ζj∈C, j∈Nq+1. Then we recover it uniquely and then the Jost matrix\nψ(k) is also recovered uniquely. Thus the statement i) and the identity S(k) =ψ(k)−1ψ(1/k)\nyields the Jost function ψ(k) determine the perturbation Vuniquely.\nWe determine the derivative of the phase shift ξ.\nLemma 4.7. Let(rj)q\n1be zeros of the Jost function f(k)for some perturbation V∈Jq,where\nq∈N. Then for k∈Sits phase shift function ξ(k)satisfies\nξ′(k) =1\n2ikq/summationdisplay\nj=12−Re(krj)\n|k−rj|2, k∈S. (4.21)24 EVGENY KOROTYAEV\nProof.Taking the derivative of the both parts of the identity for the S-ma trix\ne−2iξ(k)= detS(k) =f(k)−1f(k),|k|= 1,Imk<0, (4.22)\nwheref(k) = detψ(k) andk=k−1, we obtain:\n−2ie−2iξ(k)ξ′(k) =−f′(k)\nk2f(k)−f(k)f′(k)\nf2(k)=−detS(k)/parenleftbiggf′(k)\nk2f(k)+f′(k)\nf(k)/parenrightbigg\n, (4.23)\nand\n2iξ′(k) =f′(k)\nk2f(k)+f′(k)\nf(k). (4.24)\nHere we have used that ψ(k) is a polynomial with real coefficients, for every complex root\nrjthere is also a root ¯ rj. From Theorem 1.1 we have ψ(k) =CΠq\nj=1(k−rj),C/\\e}atio\\slash= 0, which\nimplies\nf′(k)\nf(k)=q/summationdisplay\nj=11\nk−rj=q/summationdisplay\nj=1¯k−¯rj\n|k−rj|2,f′(k)\nf(k)=q/summationdisplay\nj=11\n1\nk−rj=q/summationdisplay\nj=11\nk−rj\nand\n2iξ′(k) =f′(1/k)\nk2f(1/k)+f′(k)\nf(k)=q/summationdisplay\nj=1¯k−¯rj+¯k2(k−rj)\n|k−rj|2=q/summationdisplay\nj=12−Re(krj)\nk|k−rj|2.\n5.Fredholm determinants\n5.1.Laplacians on Z.Consider the Laplacian ∆ on the lattice Zgiven by (∆ y)x=yx+1+\nyx−1,x∈Z. Let us introduce the shift operator on the lattice Z(andN) by\n(∂y)x=yx+1,(∂∗y)x=yx−1. (5.1)\nThe spectral properties of ∆ is easier to describe by passing to its u nitary transformation by\nthe Fourier series. Let T=R/(2πZ) = [−π,π] be the flat torus and Ube the unitary operator\nfromℓ2(Z) toL2(T),given by\n/hatwidey(t) = (Uy)(t) =/summationtext\nx∈Zyxe−it·x\n(2π)1\n2, t∈T.\nThe shift operator and the Laplacian on L2(T) are rewritten as /hatwide∂:=U∂U∗=eitand\n/hatwide∆ =U∆U∗= 2cost, t∈T. (5.2)\nLetR(k) = (∆−λ(k))−1and letR(x−x′,k),(x,x′,k)∈Z2×Dbe its kernel. Then the kernel\nR(x,k) has the following representation\nR(x,k) =k|x|\nk−1/k=−k|x|\n√\nλ2−4,for (x,k)∈Z×D, (5.3)\nandR(x,k) has a meromorphic continuation from DintoC, see e.g. [30]. The proof is very\nsimple. Indeed, residue calculus imply\nR(x,k) =1\n2π/integraldisplay2π\n0e−ixtdt\n2cost−k−1\nk=1\n2πi/integraldisplay\n|w|=1w−xdw\n(w−k)(w−1\nk)=k|x|\nk−1\nk.\nBy (1.9), we have1\nk=λ+√\nλ2−4\n2andk−1\nk=−√\nλ2−4, which proves (5.3).RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 25\nLemma 5.1. i) The kernel Ro(x,x′,k),(x,x′,k)∈N2×Dof the resolvent Ro(k) = (Ho−\nλ(k))−1onℓ2(N)is a polynomial and has the form\nRo(x,x′,k) =k|x−x′|−kx+x′\nk−1/k=k|x−x′|1−k2w\nk−1/k, (5.4)\nRo(x,x′,k) =−k1+|x−x′|(1+k2+k4+...+k2(w−1)), w=/braceleftigg\nx′, x/greaterorequalslantx′\nx, x′>x,(5.5)\nand it has an analytic continuation from DintoC.\nii) The following estimates hold true for x/lessorequalslantx′:\n|Ro(x,x′,k)|/lessorequalslantC|k||x−x′|+1, C=x or C =2\n|k2−1|,|k|/lessorequalslant1, (5.6)\n|Ro(x,x′,k)|/lessorequalslantC2|k|x′+x−1, C2=/braceleftigg\nx, if |k|/lessorequalslant2\n4, if |k|/greaterorequalslant2. (5.7)\nProof.i)LetL2(T) =L2\nev(T)⊕L2\nod(T), whereL2\nod(T)isthesetofoddfunctions f(−t) =−f(t)\nandL2\nev(T) is the set of even functions f(−t) =f(t) for allt∈[−π,π]. Introduce the similar\ndecomposition ℓ2(Z) =ℓ2\nev(Z)⊕ℓ2\nod(Z), where is ℓ2\nev(Z) the set of odd functions y−x=−yx\nandℓ2\nod(Z) is the set of even functions y−x=yxfor allx∈Z. From (5.2) we obtain\n∆ℓ2\ns(Z)⊂ℓ2\ns(Z), s∈ {od,ev},\n∆y=Hoy∀y∈ℓ2(N), y0= 0.(5.8)\nFor eachh∈ℓ2(N) we define an odd function hod∈ℓ2\nod(Z) by\nhod\nx=/braceleftigg\nhxif x∈N\n−hxif−x∈Nand/tildewideh0= 0.\nLetα=1\nk−1/k. We have\ngx=α/summationdisplay\nj∈N(k|x−j|−k|x+j|)hj=α/summationdisplay\nj∈Zk|x−j|hod\nj= (R(λ(k))hod)x=god\nx,∀x∈N,\nand (∆−λ)god=hod, which yields ((∆ −λ)god)x= ((Ho−λ)g)x=hod\nx=hxforx∈N, since\nhod\n0=god\no= 0. From here we obtain (5.4), which yields (5.5).\nii) Using (5.4) we obtain\nRo(x,x′,k) =k|x−x′|+1\nk2−1(1−k2w),2w=x+x′−|x−x′|=/braceleftigg\n2x′if x/greaterorequalslantx′\n2x if x /lessorequalslantx′,(5.9)\nwhich yields (5.4). Let x/lessorequalslantx′andε=k2. Then we get\n|1−k2x|/lessorequalslant|1−ε||1+ε+....+εx−1|/lessorequalslant|1−k2|x∀k∈C. (5.10)\nLet|k|/lessorequalslant1. The estimate for the case C=2\n|k2−1|follows from (5.4). For the case C=x,\n(5.10) implies |1−k2x|/lessorequalslant|1−k2|xand thus (5.9) yields\n|Ro(x,x′,k)|/lessorequalslant|k||x−x′|+1\n|k2−1||1−k2w|/lessorequalslantx|k||x−x′|+1. (5.11)26 EVGENY KOROTYAEV\nIf|k|/lessorequalslant2, then (5.10) implies |1−k2x|/lessorequalslant|1−k2|xk2(x−1)and thus (5.9) yields\n|Ro(x,x′,k)|/lessorequalslant|k||x′−x|+1\n|k2−1||1−kβ|/lessorequalslantx|k|x′−x+1+2(x−1)=x|k|x′+x−1.(5.12)\nIf|k|/greaterorequalslant2, then (5.4) yields\n|Ro(x,x′,k)|/lessorequalslant2|k|x+x′+1\n|k2−1|/lessorequalslant4|k|x+x′−1.\nCollecting the last two estimates we obtain (5.7).\nWe determine the kernel Rx,x′(k),x,x′∈Nof the resolvent ( H−λ)−1,λ=k+1\nk,k∈D,\nsee e.,g., p. 152 from [12].\nProposition 5.2. LetV∈Jqandλ=k+1\nk,k∈D. Then\ni) The resolvent R(k) = (H−λ)−1,λ∈Λ1ong∈ℓ2(N,Cd)is given by\ny=R(k)g=F(k)u(k)+ϕ(k)v(k),\nux(k) =x/summationdisplay\nt=1/tildewideϕt(k)gt, vx(k) =∞/summationdisplay\nt=x+1/tildewideFt(k)gt,(5.13)\nii) LetRx,t(k),x,t∈Nbe the kernel of the resolvent R(k),k∈D. For each x,t∈Nthe\nmatrix-valued function Rx,t(k)has a meromorphic extension from the unit disc in the whole\ncomplex plane with poles at zeros of the Jost function f(k) = detψ(k).\nProof.i) Consider the equation ( H−λ)y=g,λ∈Λ. We construct the solution\nyx=Fxux+ϕxvx, u0= 0,\nin terms of some unknown matrix-functions ux,vx. Assume that components ζx=ux+1−\nux, γx=vx+1−vxsatisfy\nFxζx+ϕxγx= 0,\nax/bracketleftig\nFx+1ζx+ϕx+1γx/bracketrightig\n=gx,(5.14)\nforx∈N. We compute ( Hy)x=ax−1yx−1+axyx+1+bxykforyx=Fxux+ϕxvx.The\ncomponents yx−1,yx+1in terms of ζx,γxhave the forms\nyx−1=Fx−1ux−1+ϕx−1vx−1= (Fx−1ux+ϕx−1vx)+(Fx−1ζx−1+ϕx−1γx−1),\nyx+1=Fx+1ux+1+ϕx+1vx+1= (Fx+1ux+ϕx+1vx)+(Fx+1ζx+ϕx+1γx).(5.15)\nSubstituting (5.15) into ( Hy)xwe compute\n(Hy)x=ax−1/bracketleftig\nFx−1ux+ϕx−1vx/bracketrightig\n+ax/bracketleftig\nFx+1ux+ϕx+1vx/bracketrightig\n+gx+bxyk=λyx+gx.\nWe need to show that the system (5.14) has a unique solution. From ( 5.14) we obtain\nγx=−ϕ−1\nxFxζx⇒Gxζx=gx, G x=axFx+1−axϕx+1ϕ−1\nxFx.\nThen from this solution and (3.3), (3.8) we get\nGx=axFx+1−/tildewideϕ−1\nx/tildewideϕx+1axFx=/tildewideϕ−1\nx/bracketleftig\n/tildewideϕxaxFx+1−/tildewideϕx+1axFx/bracketrightig\n=/tildewideϕ−1\nx{ϕ,F}=−/tildewideϕ−1\nx,\nwhich yields ζx=−/tildewideϕxgx. We need to determine γx. From (5.14) we obtain\nζx=−F−1\nxϕxγx⇒ax/bracketleftig\nϕx+1−Fx+1F−1\nxϕx/bracketrightig\nγx=gx.RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 27\nThen from (5.14), (3.8), we get\naxϕx+1−axFx+1F−1\nxϕx=axϕx+1−/tildewideF−1\nx/tildewideFx+1axϕx=/tildewideF−1\nx/bracketleftig\n/tildewideFxaxϕx+1−/tildewideFx+1axϕx/bracketrightig\n=/tildewideF−1\nx,\nwhich yields γx=/tildewideFxgx. We compute ux,vx. We have\nux=ζx−1+ux−1=ζx−1+ζx−2+ux−3=....=ζx−1+ζx−2+...+ζ1.\nLetrx(n) =−/summationtextn−1\nxγj. Similar arguments give\nvx=−γx+vx+1=−γx−γx+1+vx+2=−γx−γx+1−γx+2+vx+2=rx(n)+vn+2.\nThe sum/summationtext∞\nxγjis absolutely convergence since γj=Fjgj. These identities and ζx=−/tildewideϕxgx\nandγx=/tildewideFxgximply (5.13).\nii) From i) and the properties of the functions ϕ,F=ϑ+ϕm+,m+=f1f−1\n0we deduce that\nfor eachx,t∈Nthe matrix-valued function Rx,t(k) has a meromorphic extension from the\nunit disc in the plane Cwith poles at zeros of the Jost function f(k) = detψ(k).\nLemma 5.3. Let/bardbl·/bardblB2be the Hilbert-Schmidt norm on ℓ2(N)andk∈D. Then we have\n/bardblfRo(k)g/bardblB2/lessorequalslant|k|\n|k2−1|/bardblf/bardbl/bardblg/bardbl, f,g ∈ℓ2(N,Cd), (5.16)\n/bardblfRo(k)g/bardblB2/lessorequalslant√\n2|k|/bardblf/bardbl1/bardblg/bardbl1, f,g ∈ℓ2\n1(N,Cd), (5.17)\nwhere/bardblf/bardbl2=/summationtext\nx∈N|fx|2and/bardblf/bardbl2\n1=/summationtext\nx∈Nx2|fx|2.\nProof.From (5.6) we obtain\n/bardblfRo(k)g/bardbl2\nB2=/summationdisplay\nx,x′∈N|fx|2|k|x−x′|−k|x+x′||2\n|k−1/k|2|gx′|2/lessorequalslant1\n|k−1/k|2/summationdisplay\nx,x′∈N|fx|2|gx′|2,\nwhich yields (5.16). From (5.6) we obtain\n/bardblfRo(k)g/bardbl2\nB2=/summationdisplay\nx,x′∈N|fx|2|k|x−x′|−k|x+x′|2|\n|k−1/k|22\n|gx′|2\n/lessorequalslant|k|2/summationdisplay\nx>x′x′2|fx|2|gx′|2+|k|2/summationdisplay\nx<x′x2|fx|2|gx′|2/lessorequalslant2|k|2/bardblf/bardbl2\n1/bardblg/bardbl2\n1,\nwhich yields (5.17).\nRecall that we have defined ( ∂f)x=fx−1,(∂∗f)x=fx+1,x/greaterorequalslant1 acting on ℓ2(N) by\n∂f= (0,f1,f2,f3,...), ∂∗f= (f2,f3,f4,...).\nWe rewrite HcandVcin terms of ∂\nHc=∂a+b+s∂∗=Ho+Vc, V=∂(a−1 1)+b+(s−1 1)∂∗.\nWe need simple properties of the operator ∂:\n∂∗∂= 1 1, ∂∂∗f= (0,f2,f3,f4,...),\nTr∂nA= 0,∀(n,A)∈N×ℓ1(N,Md)(5.18)\nDefine Fredholm determinants Dc(λ),Dc(k) andD(λ),D(k),whereλ=k+1\nk∈Λ by\nDc(λ) = det(I+Yc(λ)), Dc(k) =Dc(λ(k)), Yc(λ) =VcRo(λ)χc,\nD(λ) = det(I+Y(λ)), D(k) =D(λ(k)), Y(λ) =VRo(λ)χ,(5.19)28 EVGENY KOROTYAEV\nwhereχc(andχ) is the characteristic function of the set supp( |ax−1 1|+|sx−1 1|+|bx|) (and\nsupp(|ax−1 1|+|bx|)), where |·|is a matrix norm in Cd. The determinants are well defined\nsinceVc,Vare finitely supported perturbations and thus Yc(λ),Y(λ)∈ B1.\nLemma 5.4. i) Let functions F,G∈ℓ2(N,Md)be finitely supported. Then operator-valued\nfunctions T(k) = (k2−1)FRo(k)G∈ B1for anyk∈D, the function T:D→ B1is analytic\ninDand is a polynomial in Cwith finite dimensional coefficients.\nii) LetV∈Jc\nq. Then the operator T(k) = (k2−1)Yc(k)∈B1for anyk∈Dand the operator-\nvalued function T:D→B1is analytic in Dand has an analytic extension onto the whole\nplane, which is a polynomial in Cwith finite dimensional coefficients.\niii) The Fredholm determinant Dc(k),k∈Dis analytic in Dand has a meromorphic contin-\nuation from Dinto the whole plane C, which is a polynomial in C.\niv) The set {k∈D:Dc(k) = 0}is finite and coincides with σd(Hc).\nProof.i) Letα=|x−x′|. From (5.4) we obtain\nRo(x,x′,k) =k1+2α1−k2w\nk2−1=k1+2α(1+k2+k4+...+k2(w−1)), (5.20)\nand eachRo(x,x′,k),x,x′∈Nhas a meromorphic continuation from DintoC.\nResults ii) and iii) follows from i). The statement iv) is well-known.\nWe need results from [30] about Fredholm determinants.\nLemma 5.5. Let an operator H=Ho+Vact on the Hilbert space K, where the operators\nHo=H∗\no,Vare bounded and the operator Vis trace class. Let D(λ) = det(I+VRo(λ)),\nwhereRo(λ) = (Ho−λ)−1,λ∈Λ :=C\\σ(Ho). Then\ni)D(λ)is analytic in Λ. Moreover\nD(λ) =D(λ), λ∈Λ,ifV∗=V, (5.21)\nD(λ) = 1+O(1/λ) as|λ| → ∞, (5.22)\nlogD(λ) =−∞/summationdisplay\nn=1(−1)n\nnTr (VRo(λ))n=−/summationdisplay\nn/greaterorequalslant1Fn\nn λn,\nF1= TrV, F 2= Tr(2VHo+V2), Fn= Tr(Hn−Hn\no), n/greaterorequalslant1,(5.23)\nwhere the series are absolutely convergent for |λ|>r0and the radius r0>0is large enough.\nii) The set {λ∈Λ;D(λ) = 0}is discrete and coincides with σd(H).\nWe discuss properties of the Fredholm determinant Dc(λ) = det(I+VcRo(λ)).\nLemma 5.6. Let an operator Hc=Ho+Vcact on H, whereVc∈Jc\nq. Then\ni) The determinant Dc(λ)satisfies i) and ii) from Lemma 5.5, where Λ =C\\σ(Ho).\nii) The function Dc(k) =Dc(λ(k)),λ=k+1\nkis a polynomial in k∈Cand satisfies\nlogDc(k) =−F1k−F2k2\n2+O(k3) ask→0, (5.24)\nF1= Tr/summationdisplay\nbx, F 2=∞/summationdisplay\nx=1Tr/parenleftbig\nb2\nx+2(axsx−1 1)/parenrightbig\n. (5.25)\niii) The set {λ∈Λ;Dc(λ) = 0}is finite and coincides with σd(Hc).RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 29\nProof. The statement i) follows from Lemma 5.5.\nii) Substituting1\nλ=k\n1+k2=k+O(k3) into (5.23) we obtain\n−logD(λ) =F1\nλ+F2\n2λ2+O(1)\nλ3=F1k+F2k2\n2+O(k3)\nWe compute F1,F2. Letα=a−1 1d,σ=a−1 1d,andVo=∂α+σ∂∗. From (5.23), (5.18) we\nobtain\nF1= TrVc= Tr(∂α+b+σ∂∗) = Trb, F 2= Tr(2VcH0+V2\nc).\nWe compute F2. From (5.18) we obtain the first term\nTr2VcHo= 2Tr(∂α+σ∂∗+b)(∂+∂∗) = 2Tr/bracketleftbig\n(∂α+σ∂∗+b)∂+(∂α+σ∂∗+b)∂∗/bracketrightbig\n= 2Tr/bracketleftbig\nσ∂∗∂+∂α∂∗/bracketrightbig\n= 2Tr∞/summationdisplay\n1/bracketleftbig\nαx+σx/bracketrightbig\n.\nLetVo:=∂α+σ∂∗. From (5.18) we obtain the second term\nTrV2\nc= Tr[Vo+b]2= Tr[b2+2Vob+V2\no] = Tr[b2+V2\no],\nTrV2\no= Tr(∂α+σ∂∗)2= Tr[∂α∂α+σ∂∗σ∂∗+∂ασ∂∗+σ∂∗∂α]\n= Tr[∂ασ∂∗+σ∂∗∂α] =∞/summationdisplay\nx=1Tr[αxσx+σxαx] = 2∞/summationdisplay\nx=1Trσxαx\nCollecting these identities and using Tr( αxσx+αx+σx=αxsx−1 1 we obtain (5.25).\nThe statement iii) follows from Lemma 5.5, ii), since Dc(k) is a polynomial.\n5.2.Determinants. Now we discuss the real Jacobi matrices. We prove the uniqueness . Let\nH2=H2(D) denote the Hardy class of functions fwhich are analytic in the unit disc D\nequipped with the norm\n/bardblf/bardbl2\nH2= sup\n1>r>01\n2π/integraldisplay2π\n0|f(reiφ)|2dφ<∞.\nLemma 5.7. Letfj,j= 1,2, be such that logfj∈ H2, and satisfy the following conditions:\ni)fj(k) = 1+o(1)ask→0,\nii)φj(t) := argfj(t)is continuous in t∈S\\{ko}for someko∈S,\niii)e−2iφ1(t)=e−2iφ2(t),for a.e.t∈S.\nThenf1=f2onD.\nProof. Recall that if φ1(t) =φ2(t),for a.e.t∈R. Then it is well known that f1(z) =f2(z)\nonD(see p.10 [32]). Hence we have only to show φ1(t) =φ2(t),for a.e.t∈S. Due to ii) and\niii) we deduce that φ1(t) =φ2(t) +πN,for a.e.t∈Sand for some N∈Z. Hence we have\nlogf1(k) = logf2(k)+iπNand i) yields that N= 0.\nWe discuss properties of f(k) = detf0(k) andD(k) with real coefficients.\nLemma 5.8. LetV∈Jq, qand let allax,bxare real. Then we have\nf(k) =f(k),∀k∈C. (5.26)\nLet in addition, D(1)/\\e}atio\\slash= 0orD(−1)/\\e}atio\\slash= 0. Then\nf=DdetTp. (5.27)30 EVGENY KOROTYAEV\nProof. Consider f= detψ, where the matrix-valued polynomial ψ(k) has matrix valued\ncoefficients, which are real matrices. Then it gives (5.26). In order to show (5.27) we define\nthe function F(k) =f(k)/detTp. The functions F(k),D(k) have the same zeros in Dand\nF(k) = 1+O(k), D(k) = 1+O(k) ask→0.\nFor the zeros kj∈(−1,1),j∈n•off(and simultaneously D(k)) we define the Blaschke\nproductB(k),k∈Cby: ifn•= 0, thenB= 1 and if n•/greaterorequalslant1, then\nB(k) =/productdisplay\nj∈n•|kj|\nkj(kj−k)\n(1−kjk). (5.28)\nDefine the functions FB=F/BandDB=D/B. From the Birman-Krein identity det S(k) =\nD(k)/D(k), and from S(k) =ψ(k)−1ψ(k),k∈Sand (5.26) we obtain\nD(k)\nD(k)=f(k)\nf(k)=f(k)\nf(k)=F(k)\nF(k).\nThis yieldsDB\nDB=FB\nFB. Thus due to Lemma 5.7 we deduce that DB=FBand thenD=F.\nNow we discuss Jacobi operators Hc=Ho+Vc, whereV∈Jc\nq. Consider the Jost\nfunctionfx(k,h) and the Jost determinant f(k,h) = detf0(k,h) for the operator Hc, where\nh= (hx)∞\n1,hx= (ax,sx,bx). Define determinants and functions\nf•\nx(k,h) = (detAx)fx(k,h),f•(k,h) = (detAx)detf0(k,h), Ax=a1a2....ax.(5.29)\nTheorem 5.9. LetVc∈Jc\nq, q∈ {2p,2p−1}. Then each f•\nx(k,h),x/greaterorequalslant0is a polynomial in\nk,a•\nx,a•\nj,bj,sj,j/greaterorequalslantx−1and satisfies\nf•\n0(k,h) =A•\np−kK•\n0+...,\nA•\np=a•\n1a•\n2···a•\np,K•\n0=b1a•\n1...a•\np+a•\n1b2a•\n2···a•\np+...+a•\n1···a•\np−1bpa•\np.(5.30)\nMoreover, the function detf•\n0(k,h)is a polynomial in k,a•\nx,sx,bx,x/greaterorequalslant1and satisfies\nf•(k,h) = 1+O(k),∀k→ ∞. (5.31)\nProof.i) Consider the matrix-valued function f•\nx(k) = det(Ax)fx(k,h),x/greaterorequalslant0. Recall that\nifaxis an invertible matrix, then tx=a−1\nx=1\ndetaxa•\nx, wherea•\nxbe the adjugate matrix\nof a matrix axand deta•\nx= 1. Using the equation axfx= (λ−bx+1)fx+1−sx+1fx+2and\nfx=kx, x>p, we obtain fp=kptpandf•\np=kpa•\npand the following cases:\n•ifx=p−1, then\nap−1fp−1= (λ−bp)fp−spfp+1= ((k+1\nk−bp)kp−spapkp+1)tp=kp−1(1 1−bpk+ηpk2)tp,\nf•\np−1=kp−1a•\np−1(1 1−bpk+ηpk2)a•\np, η x= 1 1−sxax;\n•ifx=p−2, then\nap−2fp−2= (λ−bp−1)fp−1−sp−1fp=/bracketleftig\n(k+1\nk−bp−1)tp−1(kp−1−bpkp+ηpkp+1)−sp−1kp/bracketrightig\ntp\n=/bracketleftig\ntp−1kp−2−kp−1(bp−1tp−1+tp−1bp)+..../bracketrightig\ntp,\nwhich yields\nf•\np−2=/bracketleftig\nkp−2a•\np−2a•\np−1a•\np−K•\np−2kp−1+..../bracketrightig\n,K•\np−2=a•\np−2bp−1a•\np−1a•\np+a•\np−2a•\np−1bpa•\np.RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 31\n•Ifx=p−3, then we have\nfp−3=tp−3/braceleftig\n(λ−bp−2)fp−2−sp−2fp−1/bracerightig\n=kp−3tp−3tp−2tp−1tp−Kp−3kp−2+....,\nwhere\nKp−3=tp−3bp−2tp−2tp−1tp+tp−3tp−2bp−1tp−1tp+tp−3tp−2tp−1bptp,\nwhich yields\nf•\np−3=kp−3a•\np−3a•\np−2a•\np−1a•\np−K•\np−3kp−2+....,\nK•\np−3=a•\np−3bp−2a•\np−2a•\np−1a•\np+a•\np−3a•\np−2bp−1a•\np−1a•\np+a•\np−3a•\np−2a•\np−1bpa•\np.\nRepeating this procedure we obtain (5.30). Moreover, we deduce t hatf•\nx(k,h),x/greaterorequalslant0 is a\npolynomial in k,ax,a•\nj,bj,sj,j/greaterorequalslantx. Furthermore, we obtain that the function det f•\n0(k,h) is\na polynomial in k,a•\nx,sx,bx,x/greaterorequalslant1 and satisfies (5.31).\nProof of Theorem 1.2. Consider the function Dc(k,h) = det(1 1+ VcRo(k)). The operator\nYc(k,h) =VcRo(k)χpisfinitedimensional matrixandthendue to(5.4)thefunction Dc(k,h)is\na polynomial of k,hx. The functions f•(k,h),Dc(k,h) are polynomials in k,hx. They coincide\nat realbx=b∗\nxandsx=ax>0. Then they coincide for all hand we have (1.20). If V∈Jq,\nthen (5.21) gives that the function fis real on the real line.\n6.Appendix: Inverse problems for finite Jacobi operators\n6.1.Classis of Jacobi operators. We define the class of Jacobi operators H=Ho+Vwith\ntriangular matrix coefficients:\nM∇\nd={a∈Mo\ndis a upper triangular matrix with diagonal positive entries },\nJ∇\nq={V∈Jc\nq:s∗\nx=ax∈M∇\nd, bx=b∗\nx,∀x/greaterorequalslant1},Mu\nd={c∈Mo\ndis unitary }.(6.1)\nWe discus presentations of Jacobi matrices.\nLemma 6.1. Consider a Jacobi operator Hc=Ho+Vc, wheres∗\nx=ax∈Mo\ndand a sequence\n(ax,bx)∈Mo\nd×Md,x/greaterorequalslant1satisfies supx/greaterorequalslant1(|ax|+|bx|)<∞, where| · |is the norm in\nCd. Introduce the unitary operator UonHby(Uf)x=uxfx,x∈N. Then the operator\nT=U∗HU=Ho+V, where the operator Vand\n(Vf)x= (ax−1−1 1)yx−1+(a∗\nx−1 1)yx+1+bxyx,\nax=u∗\nx+1a+\nxux,bx=uxbxu∗\nx, x∈N.(6.2)\ni) If the sequence uxhas the form\nux+1=ωxu1, ωx=vx−1···v2v1, x∈N (6.3)\nwherevx∈Muis uniquely defined by the identity ax=vxa+\nx,a+\nx>0, then\nax=u∗\nx+1a+\nxux>0. (6.4)\nii) Let the sequence ux∈Mube defined by the rule:\n•u1∈Muis any matrix and\n•ifux,x/greaterorequalslant1is given, then ux+1is a unique matrix under the condition u∗\nx+1axux∈M∇.\nThen\nax=u∗\nx+1axux∈M∇\nd (6.5)\niii) In the both cases, we have that Vc∈JciffV∈Jc.32 EVGENY KOROTYAEV\nProof. We consider the case b= (bx) = 0, the proof for b/\\e}atio\\slash= 0 is similar. Introduce\nthe operator Ubyyx= (Uf)x=uxfxwheref= (fx)∞\n1. For finitely supported sequences\ny= (yx)∞\n1we have the scalar products\n(Ty,y) =∞/summationdisplay\n1/parenleftbig\n(axyx,yx+1)+(a∗\nxyx+1,yx)/parenrightbig\n=∞/summationdisplay\n1/parenleftbig\n(axfx,fx+1)+(a∗\nxfx+1,fx)/parenrightbig\n,(6.6)\nwhereax=u∗\nx+1axux.\ni) Due to the identity ax=a+\nxvx, wherevxis uniquely defined, we have ax=u∗\nx+1a+\nxvxux.\nAssume that ux,x/greaterorequalslant1 satisfy\nux+1=vxux, x/greaterorequalslant1, (6.7)\nthenax=u∗\nx+1a+\nxvxux=u∗\nx+1a+\nxux+1>0.We show that the equation (6.7) has the solution.\nLetωx=vx···v2v1for anyx/greaterorequalslant2 andu1. Then we have:\nu2=v1u1, u3=v2u2=v2v1u1=ω2u1, ...,u x+1=ωxu1,....\nThus this sequence uxsatisfies the equation (6.7) for any u1and in particular for u1= 1 1. The\nuniqueness of uxfollows from the uniqueness in ax=a+\nxvx.\nii) We need the well known fact:\n•Let a matrix A∈Mo\nd. Then there exist unique matrices L∈M∇\ndandU ∈Mu\ndsuch that\nUA=L. (6.8)\nWe take any u1∈Mu. Due to (6.8) we have a1=u∗\n2a1u1∈M∇for some unique u2∈Mu.\nAgain due to (6.8) (2) we have a2=u∗\n3a2u2∈M∇for some unique u3∈Muand so on. The\nuniqueness of uxfollows from the uniqueness in (6.8).\niii) We show that Vc∈Jc\nqiffV∈Jc\nqfor the case i). The proof for ii) is similar. Thus ax= 1 1,vx\nfor allx > pandup+1=up+2=.... This yields ap+j=u∗\np+j+1ap+jup+j=u∗\np+1ap+jup+j= 1 1\nfor allj/greaterorequalslant1. Forx=pwe have\ndet[1 1−a2\np] = det[1 1 −u∗\np+1apup(u∗\npa∗\npup+1)] = det[1 1 −apa∗\np],detbp= det(upbpu∗\np) = detbp,\nwhich yields that Vc∈Jc\nqiffV∈Jc\nq.\nThe unitary operator Uin (6.2) has the matrix u1∈Mu\ndas a free parameter. All other\nmatrices are defined uniquely. If u1= 1 1, then the unitary operator Uis uniquely defined.\n6.2.Definitions. Recall results from [42]. We discuss a finite Jacobi operator Jpacting on\ny= (yx)p\n1∈Hp= (Cd)pwith the Dirichlet boundary condition and defined by\n/braceleftigg\n(Jpy)x=axyx+1+bxyx+ax−1yx−1, x∈Np={1,2,...,p},\na0=ap= 1 1d, y 0=yp+1= 0,,(6.9)\nwherebx=b∗\nx,ax>0 for allx. We can consider Jpas ap×pself-adjoint matrix given by\nJp=\nb1a10 0...0\na1b2a20...0\n0a2b3a3...0\n... ... ... ... ... ...\n0...0ap−2bp−1ap−1\n0...0 0ap−1bp\n, b x=b∗\nx, a x>0.(6.10)RESONANCES FOR VECTOR-VALUED JACOBI OPERATORS ON HALF-LAT TICE 33\nThe spectrum of Jpis a finite sequence of real eigenvalues µj,j∈Nmwith multiplicity dj/lessorequalslantd,\nwhich satisfy\nµ1<µ2<···<µm, d j/lessorequalslantd,m/summationdisplay\n1dj=dp.\nDefine the Md-valued solutions ϕ(λ) = (ϕx(λ))p+1\n0andχ(λ) = (χx(λ))p+1\n0of the equation\naxyx+1+bxyx+ax−1yx−1=λyx, x = 0,1,..,p+1,\nϕ0=χp+1= 0, ϕ1=χp= 1 1d, λ∈C,(6.11)\nRecall that ϕx(λ) andχx(λ),x∈Npare matrix-valued polynomials such that deg ϕx(λ) =x\nand degχx(λ) =p+1−x.IfJpy=µjy, then the eigenvector y∈Hpcorresponding to the\neigenvalueµjhas the form y= (yx)p\n1, whereyx=ϕx(µj)vfor somev∈Cd. Hence, the eigen-\nvalues of Jpare zeros of the polynomial det ϕp+1(·). We sometimes write ϕx(λ,Jp),µj(Jp),...\ninstead ofϕx(λ),µj,..., when several Jacobi matrices are being dealt with.\nDefinition of Spectral data. For each eigenvalue µj,j∈Nm, we define the subspace\nEj= Kerϕp+1(µj),dimEj=dj/lessorequalslantd. (6.12)\n•the orthogonal projector Pj:Cd→ EjontoEj⊂Cd,\n•the positive self-adjoint operator gj:Ej→ Ejgiven by\ngj=Gj|Ej, where G j=Pjp/summationdisplay\nn=1ϕ∗\nx(µj)ϕx(µj)Pj. (6.13)\nWe define the matrix-valued Weyl-Titchmarsh function M(λ) by\nM(λ) =−χ1(λ)χ0(λ)−1, λ∈C. (6.14)\nDefine the reflection operator ♭pacting ony= (yx)p\n1∈(Cp)mby\n(♭py)j=yp+1−j,∀j∈Np, (6.15)\nand the Jacobi matrix J♭\np:=♭pJp♭punder the reflection. Consider the relationship between\nour spectral data and Weyl-Titchmarsh function.\nProposition 6.2. i) The fundamental solutions χ0,χ1satisfy\nχ0(λ,Jp) =ϕ∗\np+1(λ,Jp) =ϕp+1(λ,J♭\np),\nχ1(λ,Jp) =−ϑ∗\np+1(λ,Jp) =ϕp(λ,J♭\np).(6.16)\nii) The function M(λ) =M∗(λ)is analytic in C\\σ(Jp)and has the form\nM(λ) =−m/summationdisplay\nj=1Bj\nλ−µj,m/summationdisplay\nj=1Bj= 1 1d, (6.17)\nwhere the matrices Bj= res\nλ=µjM(λ)are self-adjoint and are given by\nBj/vextendsingle/vextendsingle\nEj=g−1\nj, B j/vextendsingle/vextendsingle\nCq⊖Ej= 0, j∈Nm. (6.18)34 EVGENY KOROTYAEV\niii) An operator Ton(Cd)pgiven by (6.19) is positive, i.e.,\nT=\nP0P1...Pp−1\nP1P2...Pp\n... ... ... ...\nPp−1Pp...P2p−2\n>0,wherePs=m/summationdisplay\nj=1µs\njPj, s= 0,..,2p−2.(6.19)\nIn order to formulate our main result we need a restriction on spect ral data.\nExample. 1) Letp=manddj= dimEj= rankPj=dfor allj∈Nm, i.e.,Pj= 1 1 for all j.\nThen for each distinct values µ1<···<µmthe system ( µj,1 1)m\n1satisfies (6.19).\n2) Letp= 2,m= 3,d1+d2=d3=d. Then the system {(µ1,P1),(µ2,P2),(µ3,1 1)}satisfies\n(6.19) iff Ker P1∩KerP2={0}. Thus, in the continuous case terminology, Ker P2is the\n”forbidden subspace” (see [10]) for the projector P1and vice versa.\nFor eachd/greaterorequalslant2,p/greaterorequalslant2 we introduce the set of spectral data\nDp=\n\n(µj,Pj,gj)m\n1:p/lessorequalslantm/lessorequalslantpd, µ1<···<µmare real numbers;\n(µj,Pj)m\n1satisfies (6.19)\ngj: RanPj→RanPjare linear operators such\nthatgj=g∗\nj>0 and/summationtextm\nj=1Pjg−1\njPj= 1 1d\n\n.(6.20)\nWe formulate our main result for the mapping Ψ given by Ψ :/parenleftbig\n(ax)p−1\n1;(bx)p\n1/parenrightbig\n/ma√sto→(µj,Pj,gj)m\n1.\nTheorem 6.3. The mapping Ψ : (M+\nd)p−1×(Ms\nd)p→Dpis one-to-one and onto.\nAcknowledgments. Our study was supported by the RSF grant No 19-71-30002.\nReferences\n[1] A. Aptekarev, E. Nikishin, The scattering problem for a discrete Sturm–Liouville oper ator,(Russian)\nMat. Sb. (N.S.) 121(163) (1983), no. 3, 327–358.\n[2] A. Badanin; E. Korotyaev, Resonances of 4-th order differential operators, Asymp. 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Mekh. 1985, no. 5, 10–15.\n[70] Serebryakov,V. P. Properties of scattering data of a discrete Sturm-Liouvill e equation , (Russian) Trudy\nMoskov. Mat. Obshch. 49(1986), 130–140.\n[71] H. Schulz-Baldes, Rotation numbers for Jacobi matrices with matrix entries , Math. Phys. Electron. J.\n13(2007), 40 pp.\n[72] B. Simon, Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178(2000), no.\n2, 396–420.\n[73] Simon, B. Orthogonal polynomials on the unit circle. Part 1. Classica l theory. American Mathematical\nSociety Colloquium Publications, 54, Part 1. American Mathematical S ociety, Providence, RI, 2005.\n[74] Simon, B. Orthogonal polynomials on the unit circle. Part 2. Spectral theory. American Mathematical\nSociety Colloquium Publications, 54, Part 2. American Mathematical S ociety, Providence, RI, 2005.\n[75] M. Toda, Theory of nonlinear lattices, 2nd ed., SpringerSeries in Solid-State Sciences, vol. 20, Springer-\nVerlag, Berlin, 1989.\n[76] M. Zworski, A remark on isopolar potentials, SIAM J. Math. Anal. 32(2001), 1324–1326.\n[77] M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73(1987), 277–296.\nSaint-Petersburg State University, Universitetskaya nab . 7/9, St. Petersburg, 199034,\nRussia and HSE University, 3A Kantemirovskaya ulitsa, St. P etersburg, 194100, Russia,\nkorotyaev@gmail.com, e.korotyaev@spbu.ru"    },    {        "title": "2401.11168v1.Fermionic_signal_of_vacuum_polarization_in_strong_laser_fields.pdf",        "content": "arXiv:2401.11168v1  [hep-th]  20 Jan 2024Fermionic signal of vacuum polarization in strong laser fiel ds\nYa-Nan Dai,1Karen Z. Hatsagortsyan,2,∗Christoph H. Keitel,2and Yue-Yue Chen1,†\n1Department of Physics, Shanghai Normal University, Shangh ai 200234, China\n2Max-Planck-Institut f¨ ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany\n(Dated: January 23, 2024)\nVacuum polarization (VP) is investigated for the interacti on of a polarizedγ-ray beam of GeV photons with\na counterpropagating ultraintense laser pulse. In a conven tional setup of a vacuum birefringence measurement,\na VP signal is the emerging small circular (linear) polariza tion of the initially linearly (circularly) polarized\nprobe photons. The pair production via the nonlinear Breit- Wheeler process in such a high-energy environment\neliminates part of the γ-photons in the outgoing γ-beam, increasing the statistical error and decreasing the\naccuracy of this VP signal. In contrast, we investigate the c onversion of the emerging circular polarization of\nγ-photons into longitudinal polarization of the created pos itrons, considering the latter as the main VP signal.\nTo study the VP effects in the highly nonlinear regime, where the Euler-Heisen berg effective Lagrangian method\nbreaks down, we have developed a Monte-Carlo simulation met hod, incorporating vacuum birefringence and\ndichroism via the one-loop QED probabilities in the locally constant field approximation. Our Monte Carlo\nmethod will enable the study of VP e ffects in strong fields of arbitrary configuration. With 10 PW la ser systems,\nwe demonstrate the feasibility of detecting the fermionic s ignal of the VP effect at the 5σconfidence level with\na few hours of measurement time.\nQuantum electrodynamics (QED) predicts virtual electron-\npositron pair creation by a photon in vacuum, resulting in va c-\nuum polarization (VP) in strong electromagnetic fields and\nthe quantum vacuum behaving as a birefringent medium [1–\n3]. This intriguing phenomenon has not been directly imple-\nmented in an experiment despite continuous attempts [4–7].\nThis is important not only as a proof of nonlinear QED but\nalso because it may point towards new physics beyond the\nstandard model [8–11].\nThe vacuum birefringence (VB) signal is enhanced using\nstronger background fields, longer interaction distances, and\na higher probe frequency, and the main hindering factor is\nthe background noise. The long interaction distance has bee n\nimplemented in PVLAS [12, 13] and BMV [5] experiments,\nwhich aim to measure the ellipticity acquired by a linearly p o-\nlarized (LP) optical light propagating through a strong sta tic\nmagnetic field (8.8 T) of a long extension (1 m), however,\nwithout conclusive results so far [4].\nThe advent of high-intensity optical [14, 15] and x-ray\nfree-electron lasers (XFEL) [16], coupled with rapid advan ce-\nments in x-ray polarimetry (with achievable precision of 8 ×\n10−11[17]), has opened new perspectives for measuring VB\nwith the use of ultrastrong laser fields (with magnetic fields\nreaching 106T), and keV photons of XFELs [18–22]. Using a\n10-Petawatt class laser, the induced ellipticity signal ca n reach\nup to∼10−4for the XFEL probe [22]. The HIBEF consortium\nis developing the flagship experiment in this regime [23].\nFurther enhancement of the VB signal is envisaged for a\ncombination ofγ-ray sources [24] and PW laser facilities [25–\n28]. The ultrastrong laser fields can also be replaced by the\nfields of an aligned crystal [29]. The common VB signal dis-\ncussed in this setup is the polarization of the γ-ray beam after\nthe interaction, which relies on the feasibility of sensiti veγ-\nray polarimetry, which is a challenging task [27]. In the VB\nsetup via laser-γ-beam collisions copious real pairs are pro-\nduced due to nonlinear Breit-Wheeler process, which is thesource of vacuum dichroism (VD) [28]. This e ffect is espe-\ncially dramatic when the quantum nonlinearity parameter is\nlargeχγ/greaterorsimilar1 [30]. The pair production (PP) decreases the\nnumber ofγ-photons in the final state, increasing the statisti-\ncal error of the VB signal measurement, thus playing the role\nof undesirable noise.\nWhile in the case of optical and x-ray probes, the treat-\nment of VB is valid within the Euler-Heisenberg e ffective La-\ngrangian method, as the probe photon energy is negligible\nwith respect to the electron rest mass, the QED photon po-\nlarization operator in the strong background field should be\nemployed in the case of a γ-probe. The QED polarization op-\nerator within one-loop approximation has been investigate d in\nRefs. [30–34], which has been applied to the VP problem\n[28, 35]. In particular, in Ref. [28] the feasibility of dete cting\nVB and VD with 10-PW laser systems and GeV γ-photons on\nthe time scale of a few days was demonstrated. For VB in a\ncrystal, circular polarization (CP) of ∼18% is obtained with\nincident photons in the energy range of 180 GeV [29]. Re-\ncently, it has been proposed to use helicity flips to detect VB\n[35], however, the obtained signature is of high-order ( α2) in\nthe fine structure constant α, with a suppressed probability.\nIn this letter, we put forward a method for observing VB via\nthe created positron longitudinal polarization during the inter-\naction of LPγ-photons with a LP ultraintense laser pulse in\nthe highly nonlinear regime with χγ/greaterorsimilar1. The LPγ-photons\nas assumed to be produced by linear Compton scattering, and\nto propagate in a PW laser pulse, acquiring CP due to VP. The\nhelicity of the photons is subsequently transferred to the p ro-\nduced pairs during the nonlinear Breit-Wheeler process, ge n-\nerating longitudinally polarized positrons with polariza tion\nup to∼70%. Therefore, rather than the conventional pho-\ntonic signal of VP, we find a strong signature of VB in the\npositron polarization, see the scheme of the interaction in Fig.\n1. In contrast to previous schemes where the PP is undesir-\nable, increasing the statistical error of the VB measuremen t,2\ne−\ne+γ\nFIG. 1. Measurement scheme for VP: γ-photons of LP penetrate a\nstrong counterpropagating laser pulse, with LP aligned at 4 5 degrees\nwith respect to the γ-polarization. The γ-photons develop CP due to\nVB and align along the electric field due to VD. Subsequently, the\nCP ofγ-photons is transformed into the longitudinal polarizatio n of\nelectrons and positrons as generated in the nonlinear Breit -Wheeler\nprocess, yielding a discernible fermionic signal of VP.\nwe employ the pairs as a source for a valuable VB signal. To\ncarry out the investigation, we have developed a Monte Carlo\nmethod for the simulation of VB and VD of a γ-ray beam in\na highly nonlinear regime, which applies to an arbitrary con -\nfiguration of a background strong field. We demonstrate the\nexperimental feasibility of measuring VB with an average st a-\ntistical significance of 5 σon the measurement time scale of a\nfew hours in upcoming 10-PW laser facilities.\nLet us first introduce our Monte Carlo method, which al-\nlows us to treat the γ-photon polarization dynamics induced\nby the strong-field VB and VD, for details see Supplemen-\ntal Materials (SM) [36]. Until now, the QED Monte Carlo\nmethod is known for the simulation of the photon emission\nand PP processes [37–44], which employ the polarization re-\nsolved probabilities of the photon emission and PP in strong\nfields via the tree diagrams in the locally constant field ap-\nproximation, see overview in Ref. [45]. The loop diagram\ncontribution of the order of αvia the interference of the one-\nloop self-interaction with the forward scattered one, is al so\nincluded for the electron, describing the so-called “no-ph oton\nemission” probabilities for the electron polarization cha nge\n[34, 46]. However, the similar loop diagram contributions f or\na photon polarization change were missing in the present QED\nMonte Carlo codes, and have been implemented in this work.\nThe impact of radiative corrections to photon polarization\nincludes: a polarization generation of ξ3associated with VD,\nand a rotation ofξ⊥=(ξ1,ξ2) induced by VB [ ξi=(ξ1,ξ2,ξ3)\nare the Stokes parameters of incident photons. The former co r-\nresponds to the imaginary part of polarization operator, wh ich\nis related to PP probability via optical theorem, and the lat -\nter corresponds to the real part of polarization operator. I n\nour Monte Carlo code, at each simulation step ∆t, the PP is\ndetermined by the total PP probability, and the positron en-\nergy and polarization by the spin-resolved spectral probab ility\n[45], using the common algorithms [37–44]. If the PP event\nis rejected, the photon polarization state is determined by thephoton-polarization dependent loop probability wNP[36]:\nwNP(ξi)=1−/integraldisplaydWP\ndεdtdε∆t (1)\ndWP\ndεdt=C0/integraldisplay∞\nzpdxK1\n3(x)+ε2\n++ε2\nεε+K2\n3/parenleftig\nzp/parenrightig\n−ξi·ˆe3K2\n3/parenleftig\nzp/parenrightig,\nwhere C0=αm2√\n3πω2,zq=2\n31\nδ(1−δ)χγ,ωis the energy of probe\nphoton,εandε+the energy of the produced electron and\npositron, respectively. This loop contribution can be inte r-\npreted as a selection e ffect of initial photon polarization [47]\ndue to the PP dependence on the photon polarization. Due\nto this loop contribution, the photon polarization changes to a\nmixed state during the no-PP processes, which corresponds t o\nVD:\nξNP=ξi1−/integraldisplay\ndεC0/integraldisplay∞\nzpdxK1\n3(x)+ε2\n++ε2\nεε+K2\n3/parenleftig\nzp/parenrightig∆t\n+/integraldisplay\nC0dεˆe3K2\n3/parenleftig\nzp/parenrightig\n∆t/bracerightigg1\nwNP(ξi). (2)\nFurthermore, the VB e ffect arises due to the net contribution\nof theα-order loop process and the PP tree process (with par-\ntial cancellation). In our simulation, the VB is realized by\nrotation of the photon polarization vector in ( ξ1,ξ2) plane at\neach step [22, 28, 34]:\n/parenleftiggξf\n1\nξf\n2/parenrightigg\n=/parenleftiggcosϕsinϕ\n−sinϕcosϕ/parenrightigg/parenleftiggξ1\nξ2/parenrightigg\n, (3)\nwhereϕ=αm2\nω2∆t/integraltext\ndεGi′(ξ)\nξ, withξ=1/[δ(1−δ)χγ]2/3,δ=\nε/ω, and Gi′(x) the Scorer prime function.\nWe assume that the probe LP γ-photons are produced by\nlinear Compton scattering of a LP laser pulse, see SM [36].\nThe generatedγ-photons withinθmax=0.05 mrad are highly\npolarized withξi=(−0.91,0,0), and have an average energy\nofωγ=1.1 GeV , with the energy spread ∆ε0/ε0=0.55. The\nphoton yield within θ≤θmaxisNγ=1×106≈0.5N0\ne−, see SM\n[36]. Afterwards, these photons collide with a 10 PW laser\nbeam for the high-energy VB and VD experiment. Here we\nuse a focused Gaussian LP laser pulse, with the peak intensit y\nI∼1023W/cm2(a0=150), wavelengthλ0=800 nm, pulse\ndurationτp=50 fs, and the focal radius w0=5λ0[49, 50].\nThe simulation results for the final photons are shown in\nFigs. 2 and 3. The outgoing photon beam consists of the\nprobe photons, survived after PP ( ∼105), and a substantial\namount of new born photons from radiation of produced pairs\n(∼108). The remaining probe photons are still confined within\nθ≤θmaxas the off-forward scattering ( ∼α2) is negligible.\nAfter propagating through the laser field, the average polar iza-\ntion of probe photons changes to ξ=(−0.53,−0.60,0.37)[Fig.\n2 (c)-(e)], while the larger-angle photons exhibit a distin ct LP:\nξ′=(0,0,0.59) [Fig. 2 (b)].\nTo analyze the simulation results, we use simplified estima-\ntions. The VD is described by the following system of equa-3\n(a)\nθy(rad)θx(rad)(b)\nθy(rad)\n(c)θx(mrad)(d) (e)\n(f)\nθy(mrad)θx(mrad)(g)\nθy(mrad)(h)\nθy(mrad)\nFIG. 2. (Top row) The photon angular distribution after the i nterac-\ntion: (a) for the density d2Nγ/dθxdθy, (b) for the photon polarization\nξ3, withθx,yin mrad. (Middle row) The angular distribution of photon\npolarization within |θx,y|∈[0,θmax] for: (c) degree of linear polariza-\ntion at±45◦with respect to polarization basis PLP\n1=ξ1, (d) degree of\ncircular polarization with PCP=ξ2, (e) degree of linear polarization\nalong polarization basis with PLP\n3=ξ3. (Bottom row) Same as the\nmiddle row, but without VP e ffects.\ntions [36, 47]:\ndξ1,2\ndt=−/integraltext\ndεC0ξ3ξ1,2K2\n3/parenleftig\nzp/parenrightig\n,\ndξ3\ndt=/integraltext\ndεC0/parenleftig\n1−ξ2\n3/parenrightig\nK2\n3/parenleftig\nzp/parenrightig\n. (4)\nIn the case of the photon initial polarization ξ1≈1 andξ3≈0,\nthe VD acts as a polarization damper to reduce ξ1but to in-\ncreaseξ3. Meanwhile, the VB induces a polarization rotation\nfromξ1toξ2according to Eq. (3), resulting in a decrease of ξ1\nand an increase of ξ2. With these equations, we estimate the\naverage polarization for a 1 GeV photon after the interactio n\nξ=(0.53,0.65,0.39), which is in a qualitative accordance\nwith Fig. 2 (c)-(e).\nIn the highly nonlinear regime χγ/greaterorsimilar1, considerable\namounts of pairs are produced. The photons emitted by the\ngenerated electrons and positrons are mixed with probe pho-\ntons that carry photonic signals of VP. To clarify the impact\nof secondary photons and reveal the pure VP e ffects, we ar-\ntificially turn offthe polarization variation during the no-PP\nprocess. The average polarization of photons at small angle\nbecomesξ′=(−0.87,0.0,0.06), see Fig. 2 (f)-(h). The CP ξ′\n2\ndisappears without VP regardless of photon emissions. How-\never, the radiation of pairs a ffects the LP of final photons. The\naverage polarization of the emitted photons by unpolarizedelectrons (positrons), we estimate using the result of Ref. [43]:\nξ′\n1=ξ′\n2=0,ξ′\n3=K2\n3/parenleftig\nzq/parenrightigε2+ε′2\nε′εK2\n3/parenleftig\nzq/parenrightig\n−/integraldisplay∞\nzqdxK1\n3(x)−1\n,\n(5)\nwhere zq=2\n3ω\nχeε′withεandε′being the electron (positron)\nenergy before and after emission, respectively. Since ξ′\n3is\ninversely proportional to the emitted photon energy ω′, the\naverage polarization at a small angle is reduced by ∼1% be-\ncause of the mixing of the emitted hard photons. For soft pho-\nton emissions in the large angle region [Fig. 2 (b)], we have\nξ′\n1=ξ′\n2=0 andξ′\n3≈0.5 according to Eq. (5), resulting in an\naverage polarization of the entire beam as ξ=(0.0,0.0,0.59).\nIn the high-energy regime, photon emission of produced pair s\nsignificantly broadens the angular distribution [Fig. 2 (a) ,(b)],\nand changes the average polarization of detected photons.\nTherefore, accounting for the photon emissions is necessar y\nfor accurately distinguishing the VP e ffect.\nThe full spectrum including all photons is shown in Fig. 3.\nThe spectrum and polarization exhibit distinct behavior in the\ntwo regions divided by ωc=0.6 GeV . The density distribution\nin the low-energy region has a feature of synchrotron radiat ion\nas it mostly consists of emitted photons, while the high-ene rgy\nregion exhibits a flat-top structure just as for the probe pho tons\n[Fig. 3 (a)]. We find an increase of ξ2,3and decrease ofξ1\nin the high-energy region due to VP [Fig. 3(b)] because the\npolarization of probe photons is significantly a ffected by VB\nand VD. Interestingly, the photons emitted in the low-energ y\nregion also present a sizeable CP ξ2, indicating that the created\ne+e−pairs obtain longitudinal polarization when taking into\naccount VP.\nThe polarization features of the created positrons are show n\nin Fig. 4. The positrons are longitudinally polarized with\naverage polarization of ∼13% and highest polarization up\nto∼70% [Fig. 4(b) and (f)]. The yield of positrons are\nNe+e−≈7.5×105∼0.75Nγ[Fig. 4(a) and (e)]. In the high-\nenergy region, most of the probe photons are converted to\n0 0.5 1 1.5-1-0.500.51\n0 0.5 1 1.510-2100102104106108\n(a)\nωmdNγ/dωVPE\nw/o VPE(b)\nω¯ξi\nξ1VPE\nξ2VPE\nξ3VPE\nξ1w/o VPE\nξ2w/o VPE\nξ3w/o VPE\nFIG. 3. (a) Photon spectra with (solid line) and without (red dashed\nline) VP effect (VPE). (b) The average photon polarization vs photon\nenergyω(GeV):ξ1(blue solid line),ξ2(red dot-dashed line), ξ3\n(green dotted line), with VP e ffect (thick line) and without VP e ffect\n(thin line).4\n(a)\nθy(mrad)θx(mrad)(b)\nθy(mrad)\n(c)\nθy(mrad)θx(mrad)(d)\nθy(mrad)\n(e)\nε+(Gev)mdN e+/dε+ VPE\nw/o VPE(f)\nε+(Gev)P/bardbl\nVPE\nw/o VPE\nFIG. 4. (Top row) The positron angular distribution: (a) for the\nnumber density d2Ne+/dθxdθy(mrad−2), (b) for the longitudinal po-\nlarization P/bardbl, whenθx,yare in mrad. (Middle row) Same as top\nrow but without VP e ffects. (Bottom row) Positron number density\nmdN e+/dε+(e), and the longitudinal polarization (f) vs positron en-\nergyε+(GeV), with (blue solid line) VP, and without (red dashed\nline) VP effect.\npairs via nonlinear Breit-Wheeler process. The longitudin al\npolarization of positrons stems from the helicity transfer of\nCP from the probe photons, that is induced by VB at the early\nstage of interaction. The emitted photons, detrimental to t he\nhigh-precision measurement of photonic signals, have a neg -\nligible impact on the fermionic signal, as secondary PP from\nsoft radiation is minimal ( ∼10−2Ne+e−). Thus, the emergence\nof longitudinal polarization is essentially a pure signatu re of\nVB. As can be seen from Fig. 4(d), the longitudinal polariza-\ntion vanished without VP.\nFor experimental feasibility, we estimated the impact of\nprobe photons energy on fermionic signals of VB, see Fig. 5.\nAs the energy of the probe photon increases, the strength of\nVB signal also increases due to the larger χγ, because the high\nphoton energy could accelerate the rotation from ξ1toξ2for\na fixed laser duration. Therefore, the longitudinal polariz ation\nof positrons increases with photons energy within some limi ts,\nbefore reaching∼15% atω=0.75 GeV in the case of param-\neters of Fig. 5. Afterwards, the polarization saturates wit hin\nsome photon energy range, and further decreases with higher\nω. This is because with higher ω, and higherχγ, the probe\nphoton undergoes PP before attaining a significant CP due to\nVB. As a result, unlike the scaling law of positron density\nthat monotonously increases with photon energy, the polari za-tion purity has an optimal energy range within the interval o f\nω/GeV∈[0.75,1].\nLet us discuss the feasibility of VB detection taking advan-\ntage of the positron polarization. There are conventional t ech-\nniques for measuring longitudinal polarization of positro ns\n(electrons), such as Compton [51–53] and Møller polarime-\ntries [54–56]. For the discussed parameter regime the Mølle r\npolarimetry is more advantageous, where the longitudinal p o-\nlarization is deduced via the measured asymmetry /angbracketleftR/angbracketright=\nN+−N−\nN++N−, where N±are the number of scattered positrons when\nthe positron helicity is parallel or anti-parallel to the ta rget po-\nlarization. For positrons distributed in a large energy ran ge\n[Fig. 4(e)], the asymmetry reads [57]:\n/angbracketleftR/angbracketright≈/summationtext\niσ0iζz\niA′\nzziNe+\ni/summationtext\niσ0iNe+\niPz\nT, (6)\nwhereσ0iis the unpolarized cross-section, A′\nzzian angle-\ndependent asymmetry, ζziandNe+\niare the average longitudi-\nnal polarization and positron number for positrons with en-\nergyγi=εi/m, and Pz\nTis the polarization of the foil target.\nIn order to detect VB at the 5 σconfidence level with target\nfoil polarization Pz\nT=8.52%, target thickness 25 µm, elec-\ntron density nz∼1024cm−3, the required positron number is\n˜Ne+=2.4×108. The measurement time is 3 hours considering\nthe repetition rate of the PW laser is 1 /60 Hz, see [57].\nNote that, the photon polarization state also changes due\nto VP. Even though the γ-ray polarimetry for CP poses chal-\nlenges, the decrease of ξ1and increase ofξ3can be regarded\nas suitable photonic signals for detecting VB and VD, respec -\ntively. For the remaining probe photons, a single-shot mea-\nsurement could reach a 8 σconfidence level [57]. However,\nas for previous schemes [28], the feasibility relies on the c a-\npacity of post-selection techniques to reduce the substant ial\nbackground noise from radiation and cascaded detectors to e n-\nhance the conversion e fficiency [58–60] and suppress multiple\nCoulomb scattering [61]. The current experimental detecti on\n0.5 0.75 1 1.25 1.5 1.75 21.55.59.513.5104\n0.080.110.140.17\nωNe+P/bardbl\nNe+\nP/bardbl\nFIG. 5. The scaling law of positron number Ne+(solid black line)\nand longitudinal polarization P/bardbl(red dashed line) versus energy of\nprobe photonω(GeV). The probe photons’ number is Nγ=2×105\nandξ=(1,0,0). The laser parameters are same as in Figs. 2-4.5\ncapacity for gamma polarization (typically /greaterorsimilar10% [62]) is sig-\nnificantly lower than that for positrons (typically ∼0.5% [63]).\nConcluding, we analyzed a setup for high-energy VP mea-\nsurement using a 10 PW laser system with 1 GeV LP γprobe\nphotons. Deviating from the conventional photonic signal o f\nVP, we identified the fermionic signal of VB in the positron po -\nlarization that is free from disturbances caused by seconda ry\nemissions, and more feasible for VB detection. In our scheme ,\npreviously avoided real pairs are used as a better source for\ndetecting VB. 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Rokujo, Demonstration of polarization\nsensitivity of emulsion-based pair conversion telescope f or cos-\nmic gamma-ray polarimetry, Nuclear Instruments and Method s\nin Physics Research Section A: Accelerators, Spectrometer s,\nDetectors and Associated Equipment 833, 165 (2016).\n[63] A. Narayan, D. Jones, J. Cornejo, M. Dalton, W. Deconinc k,\nD. Dutta, D. Gaskell, J. Martin, K. Paschke, V . Tvaskis, et al. ,\nPrecision electron-beam polarimetry at 1 gev using diamond mi-\ncrostrip detectors, Physical Review X 6, 011013 (2016)."    },    {        "title": "2401.11186v1._π___and__K__meson_properties_for_large__N_f__and__N_c_.pdf",        "content": "π- and K-meson properties for large NfandNc\nAftab Ahmad∗and Mumtaz Khan†\nInstitute of Physics, Gomal University, 29220, D.I. Khan, Khyber Pakhtunkhaw, Pakistan.\nDynamical chiral symmetry restoration for higher number of light quark flavors Nfand breaking\nfor higher number of colors Ncimplies the suppression and enhancement of the dynamically\ngenerated quark mass. The study of various larger values of NfandNcmay have greater impact\non the internal structure of light hadrons. In this work, we study the properties of the π- (pion)\nandK-meson (kaon), such as mass, condensate, and leptonic decay constant, for various Nfand\nNc. We use the symmetry-preserving vector-vector flavor-dependent contact interaction model of\nquark. The dynamical quark masses are calculated by using the Schwinger-Dyson equation (SDE).\nThe masses of the pion ( mπ) and kaon ( mK) for different values of NfandNcare determined\nusing the homogeneous Bethe-Salpeter equation. For fixed Nf= 2 and Ncis increased, the\ndynamically generated quark mass Mu/d( mass of up and down quarks), strange quark mass\n(Ms), meson in-condensate κ1/3\n(π,K ), and decay constant f(π,K )all increases. The pion mass mπ\nremains approximately constant until Ncreaches around 6 .5, after which it grows rapidly. On the\nother hand, the kaon mass mKincreases slowly with increasing Ncuntil it reaches approximately\nNc≈7.5, beyond which it rises quickly. When Nc= 3 is fixed at and various values of Nfare\nconsidered, all the parameter values decrease as a function of Nf, except for the pion and kaon\nmass m(π,K ), which increase above a critical value of Nfaround 8. This is the region where chiral\nsymmetry is restored, and the pion and kaon behave as free particles, similar to their behavior in\nthe presence of a heat bath. The results obtained for fixed Nf= 2 and Nc= 3 are fairly in decent\nagreement with experimentally calculated statistics and previous model calculations based on the\nSchwinger-Dyson equation (SDE) and Bethe-Salpeter equation (BSE).\nPACS numbers: 14.40.Aq, 24.85.1p, 11.10.St, 12.38.-t, 12.38.Lg,12.38.Aw, 11.15.Pg\nI. INTRODUCTION\nQuantum Chromodynamics (QCD) is a well-\nestablished theory that explores the fascinating world\nof strong color interaction between quarks and gluons.\nThese interaction formed the bound states known\nas hadrons. QCD has two major regimes known as\nasymptotic freedom and quark confinement. Asymptotic\nfreedom, which was discovered by Gross and Wilczek\n[1, 2], describes the weak interaction between quarks\nat high energies. On the other hand, quark confine-\nment, as proposed by Wilson [3], explains the strong\ninteraction between quarks at low energies, preventing\nthem from existing in isolation. In addition to the color\nconfinement, the dynamical chiral symmetry breaking\nis another significant characteristic of low-energy QCD.\nThis concept is closely linked to the generation of\nconstituent quark masses. It’s widely recognized that\nQCD displays both confinement and chiral symmetry\nbreaking when considering a small number of light quark\nflavors, denoted as Nf. However, for larger values of\nNf, it is believed that there exists a critical value Nc\nf,\nbeyond which the chiral symmetry is restored and the\nquark becomes deconfined [4–6]. This critical value Nc\nf\nmust be smaller than the upper limit of the critical\nvalue, denoted as NAf,c\nf, where asymptotic freedom\n∗aftabahmad@gu.edu.pk\n†mkmarwat700@gmail.comis believed to exist. According to the research by\nPolitzer in 1973 [2], for a gauge group SU(3), the critical\nnumber is NAf,c\nf= 16 .5. Therefore, the QCD theory\nis considered to be conformal in the infrared regime,\ndriven by the presence of an infrared fixed point (the\nfixed point corresponds to a specific point where the\nβ-functions for the QCD couplings tends to zero) [7–13]\n. The range of fermion flavors Nc\nf≲Nf< NA,c\nf, it is\ncommonly referred to as the “conformal region” [4, 14].\nAs we approach the upper limit ( Nf≲NA,c\nf) of this\nregion, the infrared fixed point is situated in the weakly\ninteracting region, However, at the lower end ( Nf∼Nc\nf),\nthe infrared fixed point undergoes a shift towards the\nstrongly interacting region. In this region, the coupling\nbecomes increasingly strong as Nfdecreases. Conse-\nquently, the system enters a phase characterized by\nthe breaking of chiral symmetry and the confinement\nof quarks. Lattice QCD simulations [6, 15–18], as well\nas continuum methods of QCD [5, 19–23] and some\neffective models of QCD [24] in the fundamental SU(3)\nrepresentation have all highlighted the significance of the\nchiral symmetry restoration and deconfinement phases\noccurring as Nfenters the conformal zone 8 ≲Nc\nf<12\nand approximated the value of critical number of flavors\nNc\nf≈8. The low energy QCD theory, characterized by a\nlarger number of colors Ncin the fundamental SU(Nc)\nrepresentation, has a notable impact in the infrared\nregime as well. In Ref. [23, 24], it has been observed that\nthe dynamically chiral symmetry breaks beyond a critical\nvalue of Nc\nc≈2.2 and hence the dynamically generatedarXiv:2401.11186v1  [hep-ph]  20 Jan 20242\nquark mass increases with the increase of Nc. In recent\ncomprehensive analyses[24], the QCD phase diagram at\nfinite temperature and density has been explored, with\na particular focus on the influence of higher NfandNc.\nThe findings indicate that the critical line separating the\nhadron phase from the Quark gluon plasma experiences\nsuppression as the number of flavors increases, while\nit becomes more pronounced with higher values of Nc.\nAdditionally, the rate of Schwinger quark-antiquark\nproduction exhibits a swifter growth as the number of\nflavors, Nf, increases, whereas it demonstrates a slower\nincrease with rising values of Nc[25]. Exploring the\nimpact of higher values of NcandNfon light hadron\nbound states, especially the properties of the π- and K-\nmeson, would yield intriguing and meaningful results.\nThis investigation holds great importance as it allows us\nto gain a deeper understanding of how these fundamental\nparticles behave under varying conditions. Our primary\nobjective and motivation for this endeavor and research\nundertaking is to explore various parameters associated\nwith light hadrons, particularly the masses, condensate\nand leptonic decay constant of π- and K-meson, when\nconsidering a larger number of quark colors Ncand\nflavors Nf. The π- and K-meson are considered the\nlightest hadrons and serve as mediators for the long-\nrange interaction between hadrons. These mesons are\neasily generated in collisions involving electrons and\nnucleons, making them ideal for investigating models\nof hadronic structure and sub-nucleonic degrees of\nfreedom within nuclei. Being mesons, they consist of\na simple quark-antiquark valence-quark composition,\nmaking them the simplest light-quark systems to study\nas bound states influenced by strong interactions. This\nanalysis is necessary for developing a comprehensive\nunderstanding of their properties and interactions based\non the elementary degrees of freedom in QCD. There\nis a strong motivation in electron-ion-collider (EIC)\nin BNL beside HERA to probe into the structure of\nπ- and K-meson, see for example Ref. [26] for detail\nanalysis. In our present investigation, we employ the\nSchwinger-Dyson equation to calculate the dynamical\nquark mass, and we utilize the homogeneous Bethe-\nSalpeter equation (BSE) [27] to determine the mass\nof the π- and K- meson for higher values of Ncand\nNf. The homogeneous BSE can be understood as an\neigenvalue problem, where the eigenvalue ( P2=−m2\nH)\nrepresents the square of the mass of the bound state [28].\nThe eigenvector in this case corresponds to the bound\nstate amplitude. This bound state amplitude, also\nknown as the Bethe-Salpeter amplitude, plays a vital\nrole in the computation of production and scattering\nprocesses that involve mesons. By delving into these\ncalculations, we aim to gain a deeper understanding of\nthe intricacies and characteristics of light hadrons. Our\ngap equation and the BSE kernel will be formulated\nusing the symmetry-preserving, confining vector-vector\nflavor-dependent contact interaction (FCI) model of\nquarks [23–25]. We will be working in the Landau gaugeand employing an optimal Schwinger-proper time regu-\nlarization scheme within the rainbow-ladder truncation.\nThis approach allows us to maintain the desired symme-\ntries while accurately capturing the dynamics of quarks\nin our calculations. By utilizing these specific methods,\nwe aim to obtain comprehensive and reliable results in\nour research. The article is organized as follows: In\nSection 2, we provide a comprehensive overview of the\nflavor-dependent contact interaction model and the SDE\ngap equation. In Sec. 3, we present the BSE approach to\nmesons and dynamical quark masses. Additionally, we\ndiscuss the meson bound state quark interaction model,\nincluding the flavor-dependent CI model approach and\nthe Bethe-Salpeter amplitu (BSA) for mesons. Moving\non to Section 4, we present our desired findings and\nresults, focusing on both a higher number of flavors and\ncolors. Lastly, in Section 5, we summarize our research\nand highlight the outcomes we have achieved.\nII. FLAVOR-DEPENDENT CONTACT\nINTERACTION MODEL AND THE GAP\nEQUATION\nWe start our discussion from the Schwinger-Dyson\nequations (SDE) [29], where the dressed-quark propaga-\ntorSfcan be written as:\nS−1\nf(p) =S−1\nf,0(p) + Σ( p). (1)\nHere, Sf,0(p) = (/p+mf+iϵ)−1, is the bare quark propa-\ngator. The Sf(p) = (/p+Mf+iϵ)−1represents the dressed\nquark propagator. Where Σ( p) is the self-energy and is\ngiven by:\nΣ(p) =Zd4q\n(2π)4g2Dµν(q)λa\n2γµSf(q)λa\n2Γν(p, q),(2)\nhereg2is the QCD coupling constant and Γ ν(p, q) is the\ndressed quark-gluon vertex. Dµν(q) is the gluon propaga-\ntor and λa’s are the Gell-Mann matrices. In the SU(N c)\nrepresentation the Gell-Mann matrices satisfy the follow-\ning identity :\nN2\nc−1X\na=1λa\n2λa\n2=1\n2\u0012\nNc−1\nNc\u0013\nI, (3)\nwhere, Iis the identity matrix. In the present work,\nwe use a symmetry-preserving confining flavor-dependent\ncontact interaction (FCI) model [23–25] for the gluon\npropagator in the Landau gauge, in the infrared region\nwhere the gluons acquires a dynamically generated mass\nmg[30–34]. The model is given by\ng2Dµν(p−q) =4παir\nm2gs\n1−(Nf−2)\nNc\nfδµν=δµναeff(Nf),(4)\nwhere the parameter αir= 0.93πis the infrared-enhanced\ninteraction strength and mg= 800 MeV is the gluon mass3\nscale [35]. The Nc\nf=Nc\nf+ηrepresents some guessed\nvalues of the critical number of flavors [23, 24], where\nthe value of the parameter ηwas so adjusted to be in\nthe range 1 .8−2.3, to obtain the desired critical number\nNc\nf≈8, above which dynamical chiral symmetry is re-\nstored and deconfinement occurs. It has been explained\nand justified in Ref. [23] that the appearance of the pa-\nrameter ηis because of the factor ( Nf−2) in Eq. (4).\nWith a particular choice of the flavor-dependent model\nEq. (4), and after performing the trace over the Dirac\nmatrices, the dynamical quark mass function Mfis given\nby:\nMf=mf+αNc\neff(Nf)\n2Zd4q\n(2π)4Tr[Sf(k)].(5)\nHere Mfis the dynamical quark mass mfis current\nquark mass and\nαNc\neff(Nf) =1\n2s\n1−(Nf−2)\nNc\nf\u0012\nNc−1\nNc\u0013\n.(6)\nOn simplifying Eq. (5), we get\nMf=mf+ 2αNc\neff(Nf)Zd4q\n(2π)4Mf\nq2+M2\nf. (7)\nThe quark-anitquark condensate in this truncation is de-\nfined as:\n−⟨¯qq⟩=Mf−mf\n2αNc\neff(Nf). (8)\nAfter setting the variable s=q2, and using d4q=\n(1/2)q2dq2sin2θdθsinϕdϕdψ in Eq. (7) and then per-\nforming the integration, we get\nMf=mf+αNc\neff(Nf)\n8π2Z∞\n0dss\ns+M2\nf. (9)\nThe integral mentioned in equation Eq. (9) is not con-\nvergent and requires regulation. In this study, we utilize\nthe Schwinger proper-time regularization scheme [29] to\naddress this issue. This procedure involves exponentia-\ntion the denominator of the integrand and introducing an\nadditional infrared cut-off τir= 1/Λir, alongside the con-\nventional ultraviolet cut off τuv= 1/Λuv. The integrand\nof Eq. (9) can be written as\n1\ns+M2\nf=Z∞\n0dτe−τ(s+M2\nf)→Zτ2\nir\nτ2uvdτe−τ(s+M2\nf)\n=e−τ2\nuv(s+M2\nf)−e−τ2\nir(s+M2\nf)\ns+M2\nf. (10)\nThe infrared cutoff τiris employed to implement confine-\nment by ensuring the absence of quark production thresh-\nolds [36–39]. However, it is important to note that the CImodel, as described in Eq. (9), is a non-renormalizable\ntheory. This means that the parameter τuv, cannot be\neliminated and instead becomes an integral part of the\nmodel. It sets the scale for all dimensional quantities\nwithin the model. Furthermore, the ultraviolet cutoff\nΛuv, also plays a significant role in the study of heavier\nquarks. By increasing Λ uv, we can simulate the short-\ndistance effects that occur as the quark mass increases.\nThe adoption of this regularization procedure is signif-\nicant as it eliminates quadratic and logarithmic diver-\ngences and ensures satisfaction of the axial-vector Ward-\nTakahashi identity [40, 41]. From Eq. (9) and Eq. (10)\nand after performing the integration over ‘s’, the gap\nequation is reduced to:\nMf=mf+MfαNc\neff(Nf)\n8π2A01(M2\nf, τ2\nuv, τ2\nir),(11)\nwith\nAδζ(M2;τ2\nuv, τ2\nir) =(M2\nf)ϵ\nΓ(ζ)Γ(ζ−2, τ2\nuvM2, τ2\nirM2),(12)\nwhere ϵ=δ−(ζ−2) and Γ( a, y1, y2) = Γ( a, y1)−\nΓ(a, y2) with Γ( a, y) =R∞\nytα−1e−tdtbeing the incom-\nplete Gamma function. In this FCI model, the confine-\nment for higher NcandNfcan be triggered from the\nconfining length scale[23, 25, 42–44]:\n˜τir=τirM(3,2)\nM(Nc, Nf), (13)\nhere M(3,2) is the dynamical mass for fixed Nc= 3\nandNf= 2. M(Nc, Nf) is the generalized NcandNf\ndependent dynamical mass. The model introduces the\nparameter τirto simulate confinement by ensuring the\nabsence of quark production thresholds. In the presence\nofNfandNc,τirneeds to vary slightly with both Nfand\nNc. Therefore, the connection between dynamical chiral\nsymmetry breaking and confinement is expressed through\na regulator, denoted as ˜ τir, which explicitly depends on\nNfandNc. The confinement length scale, as described\nin Eq. (13), becomes infinite when chiral symmetry is re-\nstored in the chiral limit. This leads to deconfinement at\na certain critical value Nc\nf, where the production thresh-\nolds reappear [23, 25]. In the next section we discuss the\ngeneral formalism fo BSE with FCI Model.\nIII. BETHE-SALPETER EQUATION FOR\nPSEUDOSCALAR MESON AND THE FCI\nMODEL\nA comprehensive relativistic approach to meson bound\nstates is facilitated by the Bethe-Salpeter equation\n(BSE). The BSE reveals poles in the four-point function,\nwhich correspond to the existence of meson-bound states.\nThe homogeneous BSE establishes the conditions neces-\nsary for these poles to emerge in a specific JPCchannel.4\nSince quarks cannot be directly observed in high-energy\nexperiments, the study of bound states becomes crucial\nfor testing QCD. Specifically, the determination of a me-\nson bound-state problem in a specific JPCchannel relies\non its homogeneous counterpart and is given by:\n[ΓH(k;P)]tu=Zd4q\n(2π)4Ktu;rs(k, q;P)χ(q;P)sr,(14)\nwith\nχ(q;P) =Sf(q+)ΓH(q;P)Sg(q−), (15)\nis the Bethe-Salpeter wave-function and q+=q+ηP,\nq−=q−(1−η)P;η∈[0,1] is a momentum-sharing pa-\nrameter. Here kis the relative and Pis the total momen-\ntum of the quark-antiquark system. Γ H(k;P) represents\nthe meson Bethe-Salpeter amplitude (BSA). The symbol\nH=f¯gdenotes the quantum numbers and flavor compo-\nsition of the meson, and r,s,t, and urepresent color, fla-\nvor, and spinor indices respectively. As previously stated,\nSfis the dressed-quark propagator and K(k, q;P) is the\nquark-antiquark scattering kernel. The study of the pseu-\ndoscalar meson (i.e., pions, kanos etc. ) observables lacks\nsignificance unless it explicitly ensures the satisfaction\nof the vector and axial-vector Ward-Takahashi identities\n(axWTI) [45]. In the chiral limit ( mf= 0), the axWTI\ncan be written as:\n−PµΓ5µ(k;P) =S−1(k+)γ5+γ5S−1(k−). (16)\nWhere k+=k+P,k−=kand Γ 5µ(k, P) is the axial\nvector vertex, which is determined by\nΓ5µ(k;P) =γ5γµ+Zd4q\n(2π)4K(k, q;P)χ5µ(q+, q).(17)\nEq. (16), which capture the phenomenological aspects of\ndynamical chiral symmetry breaking (DCSM) in QCD, it\nis essential to establish a connection between the axial-\nvector vertex, denoted as Γ 5µ(k;P), and the quark prop-\nagator, represented as Sf(k). This connection implies\na correlation between the kernel in the BSE Eq. (16)\nand the kernel in the quark SDE Eq. (1). Any trun-\ncation scheme applied to the SDE-BSE coupled system\nmust preserve this relationship, thereby influencing the\ncharacteristics of the quark-antiquark scattering kernel\nK(p, q;P). In the context of the contact interaction (CI)\nand rainbow-ladder truncation (bare-vertex approxima-\ntion) [38, 45, 46], the K(p, q;P) is given as:\nK(p, q;P) =−g2Dµν(p−q)\u0014λa\n2γµ\u0015\n⊗\u0014λa\n2γν\u0015\n,(18)\nUpon inserting Eq. (3) and Eq. (4) in Eq. (18) and then\nusing in Eq. (14), we have the homogeneous BSE ( η= 1),\nin the FCI model Eq. (6), for ground state pseudoscalar\nmeson ( JPC= 0−+) can be expressed as:\nΓ0−+(k;P) =−4αNc\neff(Nf)Zd4q\n(2π)4(19)\nγµSf(q+P)Γ0−+(q;P)S¯g(q)γµ. (20)In the FCI symmetry preserving regularization scheme\nand in the rainbow-ladder truncation the Bethe-Salpeter\namplitude cannot depend on relative momentum ( see for\nexample for the detail analysis [38, 46]). A general de-\ncomposition for pseudoscalar meson in the FCI scenario\nhas the following form:\nΓ0−+(P) =iγ5E0−+(P) +1\nMRγ5γ.PF 0−+(P),(21)\nwhere MR=MfM¯g/(Mf+M¯g),E0−+(P) and F0−+(P)\nare the Bethe-Salpeter amplitudes (BSA). Substituting\nEq. (21) in Eq. (20), and by adopting the same procedure\ndescribed in Refs. [38, 46–49] allows for the easy deriva-\ntion of the explicit expression for the BSE of the pseu-\ndoscalar mesons. To ensure the preservation of Eq. (16),\nit is necessary to incorporate a regularization technique.\nThis necessity leads us to the following form of the\naxWTI [38, 46–49]:\n0 =Z1\n0dαZd4q\n(2π)41\n2q2+W\n[q2+W]2, (22)\nwhere αis a Feynman parameter and W =\nW(Mg, Mf, α, P2) =M2\ng(1−α) +M2\nfα+α(1−α)P2.\nEq. (22) can be further simplified as\n0 =Z1\n0dα{A01(W) +A02(W)}, (23)\nwithA02(x) =−x(d/dx)A01(x) and ¯A02(x) =A02(x)/x.\nNow we can write the explicit form of the Bethe-Salpeter\nequation Eq. (20) in the FCI model as:\n\u0012\nE0−+(P)\nF0−+(P)\u0013\n=αNc\neff(Nf)\nπ2 \nK0−+\nEEK0−+\nEF\nK0−+\nFEK0−+\nFF!\u0012\nE0−+(P)\nF0−+(P)\u0013\n,\n(24)\nEq. (24) is an Eigenvalue equation which has a solution at\nP2=−m2\n0−+, with m0−+is the mass of pseudoscalar me-\nson. The explicit form of the Kernels of the BSE Eq. (24)\nfor the pseudoscalar meson (See for example for detail\nAppendix A of Ref. [47] and Ref. [49]) are:\nK0−+\nEE=Z1\n0dα\b\nA01(W)−[MgMf+α(1−α)P2− W]¯A02(W)\t\n,\nK0−+\nEF=P2\n2MRZ1\n0dα[Mg(1−α) +Mfα]¯A02(W),\nK0−+\nFF=−1\n2m2Z1\n0dα[MfMg+M2\ng(1−α)+M2\ngα]¯A02(W),\nK0−+\nFE=−2MR\nP2K0−+\nEF. (25)5\nFor the purpose of computation of the observable, the\nBSA has to be normalized. In the rainbow ladder trunca-\ntion of the BSE, the normalization condition (we assume\nη= 1) is of the form:\nPµ=αNcZd4q\n(2π)4Tr[Γ0−+(−Q)\n×∂\n∂PµSf(q+P)Γ0−+(Q)S¯g(q)], (26)\nwhere αNcis the color factor in the SU(Nc) representa-\ntion as discussed in Eq. (3). The condition Eq. (26) can\nbe further simplified as:\n1 =d\ndP22αNcZd4q\n(2π)4\n×Tr[Γ0−+(−Q)Sf(q+P)Γ0−+(Q)S¯g(q)],(27)\nwith P2=−m2\n0−+, where m0−+represents the mass of\npseudoscalar meson and at P=Q. By utilizing the\namplitudes and propagators, it is possible to calculate\nvarious characteristics of the pseudoscalar mesons (the\nπ-meson and K-meson in the present case) within the\nrainbow ladder truncation. This includes the leptonic de-\ncay constants and the condensates for the mesons. The\ndecay constant in terms of canonically normalized ampli-\ntudes can be expressed as:\nf0−+Pµ=αNcZd4q\n(2π)4Tr[γ5γµS(q+)Γ0−+(P)S(q−)],\n(28)\nor on simplifying we have (see for example [50]):\nf0−+=1\nMRαNc\n4π2[E0−+K0−+\nFE+F0−+K0−+\nFF], (29)\nand the in-meson condensate (see for an instance for de-\ntail discussion in Refs. [28, 46, 51]) is given by:\nκ0−+=f0−+1\nMRαNc\n4π2[E0−+K0−+\nEE+F0−+K0−+\nEF].(30)\nIn the next section, we discuss the numerical solution of\nthe SDE-BSE to calculate the properties of π-meson and\nK-meson for higher number of light quark flavors Nfand\nhigher number of color Ncin the fundamental SU(Nc)\nrepresentation from FCI model.\nIV. NUMERICAL RESULTS\nIn this section, we present our numerical results and\ndiscussion. We will be utilizing the technique of flavor de-\npendent contact interaction with renormalized SDE for\nthe dynamical masses of up ( u), down ( d) and strange\n(s) quarks and homogeneous BSE for the masses of π\nandK-meson . It is important to note that in this work,\nwe will be discussing different possibilities for the massesTABLE I. Parameters of the FCI model used as inputs for the\nSDE and BSE. These parameters were determined to repro-\nduced the properties of ground state π- and ρ-mesons (for\nNf= 2 and Nc= 3) in Ref. [46] and for K-meson in [50],\nexcept an additional parameter in FCI model which was de-\ntermined in [23]. All the enlisted dimensioned quantities are\nin GeV.\nmu=dms Λir Λuv αir mgNc\nf\n0.007 0.17 0.24 0.905 0.93 π0.8 9.98\nTABLE II. In the following table we presents the solution\nof the SDE and the BSE and calculated all these physical\nquantities for fixed Nf= 2 and Nc= 3. All the computed\nquantities are in the units GeV.\nMu/d Ms mπ mk fπ fk κ1/3\nπ κ1/3\nK\n0.367 0.533 0.139 0.499 0.101 0.111 0.243 0.247\nof light hadrons. Firstly, we will vary the number of fla-\nvorsNf, while keeping Nc= 3 fixed. Then, we will keep\nthe value of Nf= 2 fixed and vary Nc. We will consider\nboth scenarios and compute the dynamical quark masses\nofu-,d-, and s-quarks, the mass of π-meson, the mass\nofK-meson, the pion decay constant fπ, the kaon decay\nconstant fK, the pion condensate κ1/3\nπand the kaon con-\ndensate κ1/3\nK. All the model parameters to be used in the\nmodel are enlisted in Table I. To check the consistency\nof our results we first fixed Nf= 2 and Nc= 3 and ob-\ntained the mentioned properties of the π-and K-meson\nwith the FCI Model from SDE-BSE equations and en-\nlisted the results in Table II . Our results in this scenario\nfrom FCI model are consistent with results obtained from\nCI model [46, 50, 52–54]. In the next subsection, we fixed\nNf= 2 and increase the number of color Ncand will dis-\ncuss the results.\nA. Properties of π-and K-meson for a various Nc\nIn this subsection, we present the numerical results\nof the SDE and BSE for a higher number of Ncwhile\nkeeping Nf= 2 fixed. The dynamical quark mass Mu/d\nwith a bare quark mass of m= 0.007 GeV is plotted in\nFig. 1(a), and the dynamical strange quark mass Mswith\na bare strange quark mass of ms= 0.17 GeV is shown in\nFig. 1(b). These plots indicate that the dynamical mass\ngrows rapidly above a critical value of Nc\nc≈2.2, beyond\nwhich the dynamical chiral symmetry is broken. The\ncritical value can be calculated from the color-gradient\nof the mass function ∂NcM[24, 44]. The in-meson con-\ndensate Eq. (30), for pion represented by κ1/3\nπ, is shown\nin Fig. 2(a) as a function of Nc. Similarly, the conden-\nsate for kaon κ1/3\nKis presented in Fig. 2(b), both plots\ndemonstrate the evolution of the condensates as Ncin-\ncreases. The decay constant given in Eq. (29) for the6\npion fπ, is plotted in the Fig. 3(a) as a function of Nc,\nwhile Fig. 3(b) displays the kaon decay constant fKfor\nvarious Nc. Both plots exhibit a smooth growth as Nc\nincreases. Lastly, the mass of the pion mπis plotted as\na function of Ncin Fig. 4(a), where it remains nearly\nconstant until Nc≈6.5 and then rapidly increases. The\nmass of the kaon mKis plotted in Fig. 4(b), also slightly\nincreases as Ncincreases until it reaches Nc≈7.5 where\nit rises up rapidly. One of the differences in the behavior\nof the pion mass mπand the kaon mass mKis that the\nstrange quark mass has a higher bare quark mass com-\npared to the light up (or down) bare quark mass. We\nhave compiled a Table IV with all the parameter values\nfor different values of Ncand a fixed Nf= 2. In the\nfollowing subsection, we will keep Nc= 3 fixed and vary\nNf.\nB. Properties of π- and K- meson for various Nf\nIn this subsection, we present the numerical solution of\nthe SDE and the BSE for a higher number of flavors, de-\nnoted as Nf, while keeping the number of colors Nc= 3\nfixed. The dynamical quark mass, Mu/d, with a bare\nquark mass of m= 0.007 GeV, is plotted as a function of\nNfin Fig. 5(a). This plot illustrates that the dynamical\nmass for the light quarks, Mu/d, decreases monotonically\nasNfincreases, until it reaches a critical value of approx-\nimately Nc\nf≈8. Beyond this critical value, the dynami-\ncal chiral symmetry is restored, and the bare quark mass\nremains. The critical value can be determined from the\nflavor-gradient of the mass function, ∂NfMu/d[24, 25].\nThe dynamical strange quark mass Mswith bare strange\nquark mass ms= 0.17 GeV is shown in the Fig. 5(b),\nthis plot shows that the Msmonotonically decreases as\nwe increase Nftill the critical Nc\nf≈11.97, where the dy-\nnamical mass vanishes and the bare strange quark mass\nsurvives. The critical number in the case of s-quarks is\nhigher then that of the u- ord-quark is because of its\nheavier bare mass. The pion condensate κ1/3\nπas a func-\ntion of Nfis depicted in the Fig. 6(a) whereas κ1/3\nKis\npresented in the Fig. 6(b), both plots shows the mono-\ntonically decrease of both the condensate as we increase\nNf. The decay constant fπas a function of the number\nofNfis depicted in Fig. 7(a) and in the Fig. 7(b) we\nshow the plot of fKas a function of higher Nf. Both\nplots shows the monotonic decrease with an increase of\nNf. We plotted the mass of pion mπas function of Nf\nin Fig. 8(a), it remains almost constant until Nf≈8 and\nabove it rises rapidly in the chiral symmetry restoration\nregion. In Fig. 8(b), we show the evolution of mass of\nkaon mK,it slightly increase as we increase Nfuntil it\ngrows up to higher values. The situation is quite similar\nto that of finite temperature where the masses of pion or\nkaon smoothly rises up at the chiral symmetry restora-\ntion temperature [55, 56]. We have tabulated all the\nparameters values for various Nfand for fixed Nc= 3 in\nTable IV.V. SUMMARY AND PERSPECTIVES\nIn this research paper, we have conducted a study on\nthe properties of the π- and K-meson, including their\nmass, condensate, and leptonic decay constant, con-\nsidering various values of NfandNc. To accomplish\nthis, we have utilized the symmetry-preserving vector-\nvector flavor-dependent contact interaction (FCI) model\nof quarks. By employing the Schwinger-Dyson equation\n(SDE), we have determined the dynamical quark masses\nMu/dfor the up or down quark (with mu=md= 0.007\nGeV to maintain isospin symmetry) and the dynamical\nmass of the strange quark Ms(with ms= 0.17 GeV),\nconsidering different values of NfandNc. Addition-\nally, we have computed the masses of the pion mπand\nkaon mKfor various values of NfandNcusing the ho-\nmogeneous Bethe-Salpeter equation. Initially, we con-\nducted calculations on the properties of the πandK-\nmeson with fixed values of Nf= 2 and Nc= 3. Our\nfindings align with previous model calculations based on\nthe Schwinger-Dyson equation (SDE) and Bethe-Salpeter\nequation (BSE). Subsequently, we kept Nf= 2 constant\nand varied Nc, leading us to observe that the dynam-\nically generated quark mass Mu/d,Ms, the condensate\nκ1/3\n(π,K), the decay constant f(π,K), all increase as Ncis\nraised. This can be attributed to the fact that a larger Nc\ntends to enhance the dynamical chiral symmetry. Con-\nversely, the pion mass mπremains relatively stable un-\ntilNcreaches approximately 6.5, after which it expe-\nriences a rapid increase. On the other hand, the kaon\nmass mKexhibits a slight rise with increasing Ncuntil\nit reaches around Nc≈7.5, beyond which it increases\nrapidly. Next, we kept Nc= 3 fixed and varied Nf. We\nobserved that all parameter values decrease as a func-\ntion of Nf, except for the pion and kaon masses m(π,K),\nwhich rise near the chiral symmetry restoration region\n(near and above a critical value of Nf≈8). This region\nis where chiral symmetry is restored, and the pion and\nkaon behave as free particles, similar to their behavior in\nthe presence of a heat bath. Therefore, according to the\nFCI Model prediction, the masses of the π- and Kmeson\nincrease with larger NcandNf. However, the dynamical\nquark masses of u,d, and squarks, decay constant fπ,K,\nand meson in-condensate κ1/3\n(π,K)increase as we increase\nNc, while they decrease when we increase Nf. In the\nnear future, we are interested in extending this work to\ncalculate the electromagnetic form factors for higher Nc\nandNf.\nACKNOWLEDGMENTS\nWe acknowledge A. Bashir, B.Masud and M.A. Bedolla\nfor their guidance and suggestion. We also grateful to the\ncolleagues of the Institute of Physics, Gomal University\nfor their hospitality and support.7\n2 4 6 80.00.20.40.60.81.01.2\nNcMu/d[GeV]\n(a)\n2 4 6 80.00.20.40.60.81.01.2\nNcMs[GeV] (b)\nFIG. 1. (a) Behavior of the dynamical quark mass Mu/das a function of the number of colors Nc, for fixed number of flavors\nNf= 2, and with bare quark mass mu/d= 0.007 GeV. The dynamical quark mass increases as Ncincreases at and above some\ncritical value Nc\nf≥2.2, the dynamical chiral symmetry is broken. (b) Behavior of the dynamical strange quark mass Mswith\nms= 0.17 GeV as a function of Nc.\n2 4 6 8 100.00.10.20.30.40.5\nNcκπ1/3[GeV]\n(a)\n2 4 6 8 100.00.10.20.30.40.50.6\nNcκK1/3[GeV] (b)\nFIG. 2. (a) Behavior of pion condensate as a function of the number of colors Ncfor fixed Nf= 2. (b) Behavior of the kaon\ncondensate as a function of Nc, while keeping Nf= 2 fixed.\n[1] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343\n(1973).\n[2] H. D. Politzer, Phys. Rev. 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(a) The variation of the pion decay constant for various Ncat fixed Nf= 2. (b) The variation of the kaon decay\nconstant for various Ncat fixed Nf= 2.\n3 4 5 6 7 8 9 100.00.20.40.60.8\nNcmπ[GeV]\n(a)\n3 4 5 6 7 8 9 100.30.40.50.60.70.80.9\nNcmK[GeV] (b)\nFIG. 4. (a) The Behavior of pion mass mπwith Nc, while keeping Nf= 2. The pion mass approximately constant untill\nNc≈6.5 while above it rises quickly. (b) Behavior of the kaon mass mKas a function of Nc, while keeping Nf= 2 fixed. It\nslowly rises till Nc≈7.5 where above it grows faster.\n[21] A. Doff and A. A. Natale, Phys. Rev. D 94, 076005\n(2016), arXiv:1610.02564 [hep-ph].\n[22] D. Binosi, C. D. Roberts, and J. Rodriguez-Quintero,\nPhys. Rev. D 95, 114009 (2017), arXiv:1611.03523 [nucl-\nth].\n[23] A. Ahmad, A. Bashir, M. A. Bedolla, and J. J.\nCobos-Mart´ ınez, J. Phys. G 48, 075002 (2021),\narXiv:2008.03847 [hep-ph].\n[24] A. Ahmad and A. Murad, Chin. Phys. C 46, 083109\n(2022), arXiv:2201.09980 [hep-ph].\n[25] A. Ahmad and A. 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(a) The dynamical quark mass Mu/das a function of number of flavors Nffor fixed Nc= 3 with mu/d−0.007 GeV.\nThe mass decreases monotonically as we increase Nfwhich clearly indicates that the chiral symmetry restored above some\ncritical number of flavors Nc\nf= 8. (b) The behavior of the dynamical strange quark mass Msas a function of Nf, while keeping\nNc= 3 fixed with bare strange quark mass ms= 0.17 GeV. The Msdecreases as we increase Nf.\n0 2 4 6 8 100.100.150.200.25\nNfκπ1/3[GeV]\n(a)\n0 2 4 6 8 100.100.150.200.250.30\nNfκK1/3[GeV] (b)\nFIG. 6. (a) The pion condensate κ1/3\nπas a function of number of flavors Nffor fixed Nc= 3. It decrease as we increase Nf.\n(b) Behavior of the kaon condensate κ1/3\nKas a function of Nf, for fixed Nc= 3. The κ1/3\nKdecreases as we increase Nf.\n2 4 6 8 100.000.020.040.060.080.100.12\nNffπ[GeV]\n(a)\n2 4 6 8 100.050.060.070.080.090.100.110.12\nNffK[GeV] (b)\nFIG. 7. (a) The evolution of pion decay constant fπas a function of Nffor fixed Nc= 3. The Pion decay constant decreases\nas we increase Nf. (b) The evolution of the kaon decay constant fKas a function of Nf, while keeping Nc= 3 fixed. The fK\ndecreases as we increase Nf.10\nTABLE III. Dynamical generated quark mass of up( u), down( d) and strange ( s) quarks, masses of π-and K-mesons, the pion\ndecay constant fπ, the kaon decay constant fkand their condensates ( κ1/3\nπ) and ( κ1/3\nπ) for higher number of colors Ncfor fixed\nNf= 2.\nNc Mu/d Ms mπ mk fπ fk κ1/3\nπ κ1/3\nk\n3 0.367 0.533 0.139 0.499 0.101 0.111 0.243 0.247\n4 0.520 0.637 0.137 0.515 0.144 0.149 0.278 0.277\n5 0.618 0.713 0.139 0.530 0.177 0.180 0.300 0.298\n6 0.689 0.722 0.140 0.542 0.204 0.207 0.316 0.313\n7 0.745 0.820 0.385 0.552 2.837 0.231 0.329 0.326\n8 0.792 0.860 0.576 0.646 0.260 2.052 0.340 0.336\n9 0.831 0.895 0.703 0.747 0.287 0.283 0.349 0.346\n10 0.866 0.926 0.800 0.829 0,313 0.308 0.358 0.354\n2 4 6 8 100.00.10.20.30.40.5\nNfmπ[GeV]\n(a)\n2 4 6 8 100.400.450.500.550.600.65\nNfmK[GeV] (b)\nFIG. 8. (a) The mass of pion mπas a function of number of flavors Nf, for fixed Nc= 3. The mπgrows faster near and above\nNc\nf≈8 where the chiral symmetry restoration occurs. (b) The behavior of the Kaon mass mKas a function of Nf, while\nkeeping Nc= 3. The mKSlightly increases as as we increase Nfand near and above Nc\nf≈8, it grows faster.\nTABLE IV. Dynamically generated quark mass of up( u), down( d) and strange ( s) quarks, masses of π-and K-mesons, the pion\ndecay constant fπ, the kaon decay constant fkand their in-condensates ( κ1/3\nπ) and ( κ1/3\nπ) for higher number of flavors Nffor\nfixed Nc= 3.\nNf Mu/d Ms mπ mk fπ fk κ1/3\nπ κ1/3\nk\n2 0.367 0.533 0.139 0.499 0.101 0.111 0.243 0.247\n3 0.340 0.517 0.140 0.498 0.099 0.111 0.240 0.245\n4 0.309 0.499 0.142 0.498 0.095 0.111 0.236 0.243\n5 0.271 0.479 0.146 0.499 0.090 0.110 0.230 0.240\n6 0.223 0.457 0.153 0.505 0.081 0.108 0.221 0.235\n7 0.158 0.432 0.172 0.519 0.063 0.104 0.202 0.223\n8 0.077 0.402 0.235 0.551 0.034 0.095 0.163 0.192\n9 0.035 0.366 0.352 0.589 0.015 0.085 0.126 0.154\n10 0.020 0.323 0.481 0.634 0.009 0.074 0.105 0.13011\n[43] A. Ahmad, Chin. Phys. 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Suppl. 4, 021124 (2023)."    },    {        "title": "2401.11191v1.Angular_velocity_and_linear_acceleration_measurement_bias_estimators_for_the_rigid_body_system_with_global_exponential_convergence.pdf",        "content": "arXiv:2401.11191v1  [eess.SY]  20 Jan 2024Angular velocity and linear acceleration measurement bias\nestimators for the rigid body system with global exponentia l\nconvergence\nSoham Shanbhag∗and Dong Eui Chang†\nSchool of Electrical Engineering, Korea Advanced Institute of Sc ience and Technology,\nDaejeon, Republic of Korea\nAbstract\nRigid body systems usually consider measurements of the pos e of the body using onboard cam-\neras/LiDAR systems, that of linear acceleration using an ac celerometer and of angular velocity using\nan IMU. However, the measurements of the linear acceleratio n and angular velocity are usually biased\nwith an unknown constant or slowly varying bias. We propose a measurement bias estimator for such\nsystems under assumption of boundedness of angular velocit y. We also provide continuous estimates to\nthe state of the system, i.e. the pose, linear velocity, and p osition of the body. These estimates are\nglobally exponentially convergent to the state of the rigid body system. We propose two bias estimators\ndesigned with the estimate of the pose in the ambient Euclide an space of the Special Euclidean group\nand show global exponential convergence of the proposed obs ervers to the state of the system. The first\nobserver assumes knowledge of bounds of the angular velocit y, while the second observer uses a Riccati\nobserver to overcome this limitation. We show the convergen ce with an example of a rigid body rotation\nand translation system on the special Euclidean group. We sh ow that the observer is able to estimate\nthe bias using data collected from an Intel Realsense camera .\n1 Introduction\nResearch in quadcopter design and control has been increasing du e to their use in military and civilian\napplications[ 2]. For effective use of these systems, the study of control of the se systems is paramount.\nHowever, for feedback based control, one important design cons ideration is the integration of sensors in\nthose systems for providing the state of the system. Sensors pr oposed for this quadcopter system usually\nmeasure the position of various landmarks in the environment, given by the equation\nyi=RT(pi−p),\nwhereRis the matrix denoting the rotation of the quadcopter frame with re spect to the ground frame, p\nis the position of the quadcopter in the ground frame, and pis andyis are the position of the landmarks\nin the ground and quadcopter frame, respectively. Moreover, me asurements concerning angular velocity\nare collected using an IMU and linear velocity measurements are collec ted using GPS sensors or Doppler\nvelocity sensors[ 5,15,14]. These problems are known as “Attitude and Heading Reference Sy stem” problems\nin the literature[ 8]. However, the measurement of linear velocity is difficult if velocity mea surements are not\navailable, like in GPS denied environments. Alternatively, acceleromet er measurements have been also\nexploited as an alternative to linear velocity measurements due to th eir ease of use.\nUsually, IMU and accelerometer sensor measurements are corrup ted with a constant bias. Deterministic\nobservers for state and bias estimation have been designed to est imate this bias in an online fashion. The\n∗sshanbhag@kaist.ac.kr\n†Corresponding author, dechang@kaist.ac.kr\n1authors in [ 9] design an almost semi-globally uniformly ultimately bounded observer for this system. The\nauthors in [ 7] and [10] propose locally exponentially stable observers using the continuou s Riccati equation.\nThe authors in [ 1] propose an asymptotically stable observer using invariant extend ed Kalman filters. The\nauthors in [ 3] propose a globally convergent observer for the bias in the acceler ometer readings, but assume\nunbiased angular velocity measurements.\nAs can be seen, the observers proposed are unable to achieve glob al convergence in the presence of IMU\nand accelerometer bias. This is due to a topological obstruction on S O(3) not allowing global observers[ 4].\nTwo methods are used to overcome this obstruction. The authors in [16] design a globally exponentially\nstable observer for the rigid body system. They design a hybrid (co ntinuous + discrete) observer for the\nrigid body system with the state consisting of the rotational pose, translation pose, and the linear velocity.\nTo the best of our knowledge, this is the first observer proposing g lobal exponential stability. However, the\nhybrid nature of the observer leads to discontinuities in the observ er dynamics, specifically the pose. Since\nthe estimate of the pose is used in classical control methods, a disc ontinuity in the estimate of the pose is\nundesirable. Another alternative is the extension of the dynamics t o the ambient Euclidean space. This\napproach has been used to solve similar problems, for example in [ 6,12]. We use this approach to design\nglobally convergent continuous observers for the rigid body syste m. An observer using a similar approach\nhas been designed for the system considering linear velocity measur ements instead of linear acceleration\nmeasurements in [ 13].\nTheambientspaceextensionallowsustodesigndeterministicobserv erswhicharecontinuousandglobally\nconvergent. We propose two observers for this purpose. The fir st observer assumes knowledge of bounds of\nthe angular velocity and proposes a constant gain observer. The s econd observer is a Riccati based variable\ngain observer, which is computationally expensive but assumes only t he existence of a bound on the angular\nvelocity, and no knowledge of the said bound. To our knowledge, the se are the first observers able to provide\nglobal exponential convergence without discontinuities.\n2 Preliminaries\nWe denote the estimate of the state Aby¯A. Denote by SO(3) the set of orthogonal matrices of dimension 3\nwith determinant 1, and by so(3) the corresponding Lie algebra. The elements of so(3) are skew symmetric\nmatrices in R3×3. The matrix representation of the cross product with a vector vis denoted by v×:R3→\nso(3) such that for all w∈R3,v×w=v×w. The inverse of the ( ·)×operatoris denoted by ( ·)∨:so(3)→R3,\nwhere for all a∈R3,(a×)∨=aand for all A∈so(3),(A∨)×=A. The Euclidean inner product of two\nmatrices in Rm×nis denoted by /an}bracketleft·,·/an}bracketri}ht:Rm×n×Rm×n→Rsuch that /an}bracketleftA,B/an}bracketri}ht= trace(ATB). The Euclidean\nnorm of a matrix A∈Rm×nis defined as /bardblA/bardbl=/radicalbig\n/an}bracketleftA,A/an}bracketri}ht. The orthogonal projection of A∈R3×3to\nso(3) is given by πso(3)(A) = (A−AT)/2. LetA⊗Bdenote the Kronecker product of AandBinRn×n.\nThe matrix consisting of all zeros in Rm×nis denoted by 0 m×n. Sometimes, we write 0 mto mean 0 m×1\nfor the sake of compactness. The identity matrix in Rn×nis denoted by In. We use the property that for\nanyC∈Rm×n,CCT≥0. We denote the smallest and largest eigenvalues of a matrix Abyλmin(A) and\nλmax(A), respectively. All vectors are considered as column vectors.\nConsider the system evolving on SO(3) ×R3×R3⊂R3×3×R3×R3as\n˙R=RΩ×, (1a)\n˙p=v, (1b)\n˙v=g+Ra. (1c)\nMeasurementsofΩand aareavailablewithaconstantadditivebiasintheformΩ m= Ω+bΩandam=a+ba,\nrespectively. This additive bias is a random turn-on bias, present du e to thermal, physical, mechanical, and\nelectrical properties of the sensor, and varies every time the sen sors are started. In this paper, we assume\nthe measurements of Randpbeing available. Usually, measurements are available as a function of Rand\np. In this case, the measurements may be constructed using availab le measurements. Note that we do not\nassume availability of measurement of the linear velocity, v, of the system.\nThe following is assumed about the angular velocity of the system:\nAssumption 1. The angular velocity of the system Ωis bounded.\n2We assume measurements are available continuously. Since we design deterministic observers, the mea-\nsurements of the states are assumed to have no noise. In practic e, the measurements of the states will be\nnoisy. Also, the biases considered may contain noise which can be mod elled as a Gauss-Markov process.\nHowever, as numerical and experimental simulations show, the pro posed observers perform satisfactorily in\nthe presence of some noise in the measurements.\n3 Proposed observers\nWe present two observers in this section. The first observer assu mes knowledge of the bounds of Ω, and\nprovides a constant gain observer for the system. The second ob server assumes no such knowledge of the\nbounds, but only that a bound on Ω exists. However, it uses a Riccat i equation based observer, which is\ncomputationally expensive as compared to the first observer.\n3.1 Constant Gain Observer\nChoose the observer equations as:\n˙¯R=R(Ωm−¯bΩ)×+k1(R−¯R), (2a)\n˙¯bΩ=k2πso(3)(RT¯R)∨, (2b)\n˙¯p= ¯v+k3(p−¯p), (2c)\n˙¯v=g+R(am−¯ba)+k4(p−¯p), (2d)\n˙¯ba=−k5RT(p−¯p), (2e)\nwhere (¯R,¯bΩ,¯p,¯v,¯ba)∈R3×3×R3×R3×R3×R3, andk1,k2,k3,k4,k5∈R.\nLemma 1. Givenc >0, there exist k3,k4,k5∈Rwhich satisfy the inequalities\nY:=\n2k2\n3−2k4\n−k2\n5k3k4−k3k2\n5−k3k5\nk3k4−k3k2\n52k2\n4−2k3k5\n−k2\n3k2\n5−k4k5\n−k3k5 −k4k52k2\n5−c2\n>0, Z:=\nk3 k4 −k5\nk4(k3k4−k5)−k3k5\n−k5−k3k5k4k5\n>0.(3)\nProof.Note that Ycan be simplified as\nY=\nk3\nk4\n−k5\n/bracketleftbigk3k4−k5/bracketrightbig\n+\nk5\n−k3k5\n0\n/bracketleftbigk5−k3k50/bracketrightbig\n+\nk2\n3−2k4−2k2\n5 0 0\n0 k2\n4−2k3k5−2k2\n3k2\n50\n0 0 k2\n5−c2\n.\nIfk3,k4andk5are chosen such that k5> c,k2\n3−2k4−2k2\n5>0, andk2\n4−2k3k5−2k2\n3k2\n5>0, thenY >0.\nThe principal minors of Zcan be written as k3,k2\n3k4−k3k5−k2\n4, andk5(k2\n3k2\n4−k3\n3k5−k3\n4+k2\n5). To\nensure that Zis positive definite, it suffices to choose k3,k4andk5such that k3>0,k2\n3k4−k3k5−k2\n4>0,\nandk2\n3k2\n4−k3\n3k5−k3\n4>0 ifk5>0.\nWe define the set K(c) containing the permissible values of k3,k4andk5as\nK(c) =/braceleftbig\n(k3,k4,k5)|k5> c,k3>0,k2\n4−2k3k5−2k2\n3k2\n5>0,k2\n3k4−k3k5−k2\n4>0,\nk2\n3k2\n4−k3\n3k5−k3\n4>0,k2\n3−2k4−2k2\n5>0} (4)\nThe setK(c) is an open set. Note that these conditions are one example of the p ossible conditions on k3,k4\nandk5for positive definiteness of YandZ, and hence these conditions do not define an exhaustive set.\nIfc≤1, (k3,k4,k5) = (10,40,2)∈K(c). SinceK(c) is an open set, there are infinitely many values in the\nneighbourhood of (10 ,40,2) which also belong to K(c). Consider ( k′\n3,k′\n4,k′\n5)∈K(1). Let c >1, and choose\n3k3=ck′\n3,k4=c2k′\n4andk5=ck′\n5. Then,\nk5=ck′\n5> c, k 3=ck′\n3>0, k2\n3−2k4−2k2\n5=c2k′2\n3−2c2k′\n4−2c2k′2\n5>0,\nk2\n3k4−k3k5−k2\n4=c4k′2\n3k′\n4−c2k′\n3k′\n5−c4k′2\n4> c4k′2\n3k′\n4−c4k′\n3k′\n5−c4k′2\n4>0,\nk2\n3k2\n4−k3\n3k5−k3\n4=c6k′2\n3k′2\n4−c4k′3\n3k′\n5−c6k′3\n4> c6k′2\n3k′2\n4−c6k′3\n3k′\n5−c6k′3\n4>0,\nk2\n4−2k3k5−2k2\n3k2\n5=c4k′2\n4−2c2k′\n3k′\n5−2c4k′2\n3k′2\n5> c4k′2\n4−2c4k′\n3k′\n5−2c4k′2\n3k′2\n5>0.\nHence, (k3,k4,k5) = (ck′\n3,c2k′\n4,ck′\n5)∈K(c), and hence satisfy inequalities ( 3). Since K(c) is an open set,\nthere exist infinite solutions in the neighbourhood of ( ck′\n3,c2k′\n4,ck′\n5) which satisfy inequalities ( 3). This\nconcludes the proof.\nDefine the observer error terms\nER=R−¯R, e p=p−¯p, ev=v−¯v, (5a)\neΩ=bΩ−¯bΩ, ea=ba−¯ba. (5b)\nTheorem 1. Choosek1,k2>0andk3,k4,k5satisfying inequalities (3)withc= supt≥0/bardblΩ(t)/bardbl. Then the\nproposed observer (2)converges exponentially fast to the system (1)under Assumption 1for all(¯R(0),¯bΩ(0),¯p(0),¯v(0),¯ba(0))∈\nR3×3×R3×R3×R3×R3.\nProof.Differentiating the errors along the system ( 1) and (2), the error system is\n˙ER=−ReΩ×−k1ER, (6a)\n˙eΩ=k2πso(3)(RTER)∨, (6b)\n˙ep=ev−k3ep, (6c)\n˙ev=−Rea−k4ep, (6d)\n˙ea=k5RTep. (6e)\nDefinex= (ep,ev,ea). Then the update equation for xcan be written as\n˙x=\n−k3I3I303×3\n−k4I303×3−R\nk5RT03×303×3\nx=A(t)x.\nSince the dynamics of ER-eΩsystem are independent of the xsystem, consider the Lyapunov functions\nV1=k2\n2/an}bracketleftER,ER/an}bracketri}ht+/an}bracketlefteΩ,eΩ/an}bracketri}ht,\nV2=xTP−1x,\nwherePis defined as\nP=\nk3I3 k4I3 −k5R\nk4I3(k3k4−k5)I3−k3k5R\n−k5RT−k3k5RTk4k5I3\n.\nDifferentiating V1along system ( 6), we get\n˙V1=k2/an}bracketleftER,−ReΩ×−k1ER/an}bracketri}ht+2/an}bracketlefteΩ,k2πso(3)(RTER)∨/an}bracketri}ht=−k1k2/an}bracketleftER,ER/an}bracketri}ht,\nwhich is negative semi-definite. Hence, ERandeΩare bounded since V1is bounded. Moreover, since V1is\nbounded below by 0 and non-increasing, V1(t) has a finite limit as t→ ∞. Differentiating ˙V1along system\n(6), we have ¨V1=−k1k2/an}bracketleftER,−ReΩ×−k1ER/an}bracketri}htwhich is bounded since ERandeΩare bounded, R∈SO(3) is\nbounded, and Ω is bounded due to Assumption 1. Hence, ˙V1is uniformly continuous. By Barbalat’s lemma,\nsinceV1(t) has a finite limit as t→ ∞,˙V1(t)→0 ast→ ∞. Hence, lim t→∞ER(t) = 0.\n4Differentiating equation ( 6a), we get\n¨ER=k2\n1ER+k1ReΩ×−RΩ×eΩ×−k3Rπso(3)(RTER),\nwhich is bounded since ER,RandeΩare bounded, and Ω is bounded by Assumption 1. Hence, ˙ERis\nuniformly continuous. Using Barbalat’s Lemma, since lim t→∞ER(t) = 0, lim t→∞˙ER(t) = 0. Hence,\neΩ(t) =−(R(t)T(˙ER(t) +k1ER(t)))∨→0 ast→ ∞. Since the system under consideration is linear, the\nerror system is exponentially stable (Theorem 4.11 [ 11]).\nSince\nP=/bracketleftbiggI606×3\n03×6RT/bracketrightbigg\n(Z⊗I3)/bracketleftbiggI606×3\n03×6R/bracketrightbigg\n,\nwhereZis as defined in inequalities ( 3),PandZhave the same eigenvalues. Consequently, defining\nW1(x) =/bardblx/bardbl2\nλmax(Z), (7)\nW2(x) =/bardblx/bardbl2\nλmin(Z), (8)\nwe have that W1(x)≤V2(t,x)≤W2(x) sinceλmin(Z)>0. Using c= supt≥0/bardblΩ(t)/bardbl, define the matrices\nC,DandQas\nC=/bracketleftbig\nk5I3k3k5I3−RΩ×/bracketrightbig\n, D=/bracketleftbigg\n06×606×3\n03×6c2I3−ΩT\n×Ω×/bracketrightbigg\n,andQ=/bracketleftbigg\nI606×3\n03×6RT/bracketrightbigg\n(Y⊗I3)/bracketleftbigg\nI606×3\n03×6R/bracketrightbigg\n,\nwhereYis as defined in inequalities ( 3). Differentiating V2along system ( 6),\n˙V2=xT(A(t)TP−1+P−1A(t)−P−1˙PP−1)x=xTP−1(A(t)P+PAT(t)−˙P)P−1x,\n=−xTP−1/parenleftbig\nCTC+D+Q/parenrightbig\nP−1x.\nNote that CTC≥0 for allC∈R3×9,D≥0 sincec2≥λmax(ΩT\n×Ω×), andQ >0 sinceY >0. Hence, ˙V2(x)\nis negative definite.\nNote that\nP−1QP−1=/bracketleftbiggI606×3\n03×6RT/bracketrightbigg\n(Z−1YZ−1⊗I3)/bracketleftbigg\nI606×3\n03×6R/bracketrightbigg\n,\nwhere we have used the property ( A⊗B)(C⊗D) =AC⊗BDand (A⊗B)−1=A−1⊗B−1. Hence, the\neigenvalues of P−1QP−1are the same as the eigenvalues of Z−1YZ−1. Define\nW3(x) =λmin(Z−1YZ−1)/bardblx/bardbl2,\nsuch that W3(x)≤ −˙V2(t,x), i.e.˙V2(t,x)≤ −W3(x). From equations ( 7) and (8) we have W1(x) andW2(x)\nsuch that W1(x)≤V2(t,x)≤W2(x). From Theorem 4.10 [ 11], the error system is exponentially stable.\nHence, global exponential convergence of the constant gain obs erver to the state is shown.\nAs can be seen from the required conditions, this observer require s the knowledge of the value of the\nbound supt/bardblΩ(t)/bardbl. Since this bound may not always be available, we propose the following variable gain\nobserver.\n3.2 Variable Gain Observer\nWe first consider the following linear system\n˙x(t) =A(t)x(t),\ny(t) =Cx(t),(9)\nwhereA∈Rn×nandC∈Rm×n.\n5Lemma 2. The bounded system (9)is uniformly completely observable if there exists a µ >0andp∈N\nsuch that for all t,M(t)MT(t)≥µIn>0, whereM(t)is defined as M(t) =/bracketleftbig\nM0(t)... M p(t)/bracketrightbig\nwith\nM0(t) =CT(t)andMi+1(t) =˙Mi(t)+AT(t)Mi(t).\nConsider the continuous Riccati equation(hereafter abbreviate d as CRE)\n˙P=A(t)P+PAT(t)−PCTQ(t)CP+V(t), (10)\nwhereQ∈Rm×mandV∈Rn×nare symmetric positive definite matrices. Let P(t)∈Rn×nbe the solution\nof equation ( 10) withP(0)∈Rn×na symmetric positive definite matrix.\nLemma 3. If system (9)is uniformly completely observable then the solution P(t)is well defined on R+.\nAlso, there exist constants 0< qm≤qM<∞such that qmIn≤P(t)≤qMIn.\nChoose the observer equations as:\n˙¯R=R(Ωm−¯bΩ)×+k1(R−¯R), (11a)\n˙¯bΩ=k2πso(3)(RT¯R)∨, (11b)\n˙¯p= ¯v+K3(p−¯p), (11c)\n˙¯v=g+R(am−¯ba)+K4(p−¯p), (11d)\n˙¯ba=K5(p−¯p). (11e)\nwhere (¯R,¯bΩ,¯p,¯v,¯ba)∈R3×3×R3×R3×R3×R3,k1,k2∈R>0andK3,K4,K5∈R3×3are given by/bracketleftbigKT\n3KT\n4KT\n5/bracketrightbigT=P(t)/bracketleftbigI303×303×3/bracketrightbigTQ(t), where P(t)∈R9×9,t≥0 is the solution of the CRE\nequation with P(0) =I9withQ(t)>0 andA(t) andCdefined as\nA(t) =\n03×3I303×3\n03×303×3−R(t)\n03×303×303×3\n, C=/bracketleftbigI303×303×3/bracketrightbig\n. (12)\nLemma 4. There exists a solution for the CRE given in equation (10)for the above choice of A(t)andC.\nProof.Consider the linear time varying system ( 9). From Lemma 2, withAandCdefined as in 12, we have\nthat\nM0(t) =\nI3\n03×3\n03×3\n, M1(t) =\n03×3\nI3\n03×3\n, M2(t) =\n03×3\n03×3\n−RT\n.\nHence,M(t)MT(t) =I9, implying that the system ( 9) is uniformly completely observable. Hence, the\nsolution P(t) of the CRE given in equation ( 10) starting at a symmetric positive definite matrix exists and\nis well defined with the chosen A(t) andCfrom Lemma 3.\nDefine the observer error terms as in equation ( 5). The following theorem proposes a variable gain\nobserver.\nTheorem 2. WithQandVdefined as constant symmetric positive definite matrices in t he CRE equation\n(10), the observer system (11)converges to the system (1)exponentially fast under Assumption 1for all\n(¯R(0),¯bΩ(0),¯p(0),¯v(0),¯ba(0))∈R3×3×R3×R3×R3×R3.\nProof.Differentiating the errors, the error system is\n˙ER=−ReΩ×−k1ER, (13a)\n˙eΩ=k2πso(3)(RTER)∨, (13b)\n˙ep=ev−K3ep, (13c)\n˙ev=−Rea−K4ep, (13d)\n˙ea=−K5ep. (13e)\n6Definex= (ep,ev,ea). Then the update equation for xcan be written as\n˙x=A(t)x−P(t)CTQCx,\nwithA(t) andCas defined in equation ( 12).\nSince the dynamics of R-Ω system are independent of the xsystem, consider the two Lyapunov functions\nV1=k2\n2/an}bracketleftER,ER/an}bracketri}ht+/an}bracketlefteΩ,eΩ/an}bracketri}ht,\nV2=xTP−1x.\nThe exponential convergence of the ER-eΩsystem follows similar to that of Theorem 1.\nNote that P(t) is bounded above by qMI9and below by qmI9. Defining\nW1(x) =/bardblx/bardbl2/qM, (14)\nW2(x) =/bardblx/bardbl2/qm, (15)\nwe have that W1(x)≤V2(t,x)≤W2(x). Differentiating V2along the trajectory of the system ( 13),\n˙V2=xT((A−PCTQC)TP−1+P−1(A−PCTQC)−P−1˙PP−1)x\n=xT(ATP−1+P−1A−2CTQC−P−1(AP+PAT−PCTQCP+V)P−1)x\n=−xT(CTQC+P−1VP−1)x\nNote that CTQC≥0 sinceQ >0, andP−1VP−1>0 sinceP,V >0. Defining\nW3(x) =λmin(V)/bardblx/bardbl2\nq2\nM,\nweseethat W3(x)≤λmin(V)/bardblP(t)−1x/bardbl2since1/qMisthesmallestsingularvalueof P−1duetosymmetricity\nofP. Hence, W3(x)≤ −˙V2(t,x), i.e.˙V2(t,x)≤ −W3(x). Since W1(x)≤V2(t,x)≤W2(x) from equations\n(14) and (15), the system is exponentially stable (Theorem 4.10 [ 11]). Hence, the proposed variable gain\nobserver is globally exponentially convergent.\n0 2 4 6 8 10 12 14012345\nTime/bardblR−¯R/bardblObserver I\nObserver II\n(a)0 2 4 6 8 10 12 1400.511.522.533.5\nTime/bardblp−¯p/bardblObserver I\nObserver II\n(b)0 2 4 6 8 10 12 140123456\nTime/bardblbv−¯bv/bardblObserver I\nObserver II\n(c)\n0 2 4 6 8 10 12 140123456\nTime/bardblbΩ−¯bΩ/bardblObserver I\nObserver II\n(d)0 2 4 6 8 10 12 140123456\nTime/bardblba−¯ba/bardblObserver I\nObserver II\n(e)\nFigure 1: Simulation of observer in presence of noise: observer I: c onstant gain observer, observer II: variable\ngain observer\n70 10 20 30 40-0.8-0.6-0.4-0.200.20.40.60.8\nTime(s)bΩi\n(a)0 10 20 30 40-10-5051015\nTime(s)bai\n(b)\n0 10 20 30 40024681012\nTime(s)bΩi\n(c)0 10 20 30 40-50510152025\nTime(s)bai\n(d)\nFigure 2: Experimental simulation of observer with data collected fr om a Intel Realsense T265: observer I:\nconstant gain observer (solid line), observer II: variable gain obse rver (dashed line). Legend: square: b(Ω,a)x,\nplus:b(Ω,a)y, circle:b(Ω,a)z. Figures (a) and (b) correspond to no added bias, Figures (c) and (d) correspond\nto added bias of 10 units.\n4 Numerical simulation\nTo simulate the observers, the system is initialised as ( R(0),p(0),v(0)) = (exp( −πe3×/3),03,03). The\ntrue angular velocity and linear acceleration of the system are chos en as (−sin(10t),cos(10t),0.6sin(5t)) and\n(cos(0.5t),sin(0.5t),cos(t)), respectively. The measurement ofthese values arecorrupte d with Gaussian noise\nof amplitude 0 .01 and constant biases bΩ= (−1,1,5) andba= (1,−5,1), respectively. For the measurement\nof the pose, we assume availability of lmarkers(expressed as homogeneousvectors) b= [b1,b2,...,b l]∈R4×l\nin the inertial frame and their corresponding measurements in the b ody frame r= [r1,r2,...,r l]∈R4×l,\nwhich are corrupted by Gaussian noise of amplitude 0 .01. The measurement of the state of the system is\narrived at using the landmark measurements as\n/bracketleftbiggRmpm\n0 1/bracketrightbigg\n=πSE(3)(rb†),\nwhereπSE(3)(A) represents projection of A∈R4×4onto the special Euclidean group of three dimensions,\nandb†represents the pseudo-inverse of b.\nThe observer state ( ¯R(0),¯p(0),¯v(0),¯bΩ(0),¯ba(0)) = (I3,03,03,03,03) is chosen as the initial state of both\nthe observers. The simulation is run for 15 seconds. We choose k1= 1,k2= 1,k3= 3.4,k4= 5.5 and\nk5= 1.3 for the constant gain observer and k1= 1,k2= 1,P(0) =I9,V= 0.1I9andQ=I3for the variable\n8Table 1: Estimated Biases on data collected using Intel Realsense T2 65. Observer I: constant gain observer,\nobserver II: variable gain observer\nObserver I Observer II\nbΩ ba bΩ ba\nNo added bias (0.05, -0.08, 0.03) (-2.28, 9.58, -7.69) (0.05, -0.08, 0.03) (-2.29, 9.72, -7.76)\nAdded bias of 10 units (9.90, 9.79, 9.88) (7.26, 19.11, 1.85) (9.9, 9.79, 9.88) (7.29, 19.3, 1.82)\ngain observer. The results of the simulation is shown in Figure 1. From the simulation, it can be seen that\nthe observers converge to the system state fairly quickly. Moreo ver, the bias value is estimated correctly\neven in the presence of noise in the system.\n5 Experiments\nWe perform experimental simulations to check the feasibility of the o bserver to estimate the bias in the\nangular velocity and the linear acceleration. We collect data of the po se using an Intel Realsense sensor.\nWe use an IMU for the measurement of the angular velocity of the sy stem, and an accelerometer for the\nmeasurement of the linear acceleration. We use the proposed obse rvers for the estimation of the bias in the\nmeasurements. Since the noise in the measurement devices may be c omparable to the bias, we also run the\nobservers with the same data by adding a large(as compared to the noise) constant bias. In this experiment,\nwe add (10 ,10,10) to the measurements of the linear acceleration and angular velo city. The convergence of\nthe bias to a constant value ensures that the bias has been accura tely estimated.\nThe gains are chosen as k1= 1,k2= 1,k3= 3.4,k4= 5.5 andk5= 1.3 for the constant gain observer\nand ask1= 1,k2= 1,P(0) =I9,V= 0.1I9, andQ=I3for the variable gain observer. The simulation\nresults can be seen in Figure 2. As expected, the biases converge to a constant value. The estim ated biases\nare shown in Table 1. We see that the observer estimates the estimated biases well and converges to an\nalmost constant value fairly quickly. We also see that the two experim ents arrive at a similar value of bias\n(after removing the added bias of 10 units), and hence are consist ent. The final estimate in the bias(taken\nas mean of the four biases arrived at) in the angular velocity is ( −0.025,−0.145,−0.045) and in the linear\nacceleration is ( −2.505,9.428,−7.945).\n6 Conclusion\nWe have designed two observers which show globally exponentially sta ble convergence to the rigid body\nrotation and translation system under assumptions of boundedne ss of angular velocity. These observers\nestimate the unknown bias in the linear acceleration and angular veloc ity measurements exponentially fast.\nThe first observer assumes knowledge of bounds of the angular ve locity, and proposes a constant gain\nobserver. The second observer assumes no such knowledge, but provides a Riccati like observer, which is\ncomputationally expensive as compared to the first one. A possible f uture research topic would be proposing\na constant gain observer without the knowledge of the bounds of a ngular velocity. An observer for discrete\nmeasurements with a continuous time system can also be designed by using the results of the proposed\nobserver. Also, another possible future research topic would be u sing the ambient space extension technique\nto propose a machine learning based observer for the rigid body sys tem.\nReferences\n[1] Axel Barrau and Silv` ere Bonnabel. The invariant extended Kalma n filter as a stable observer. IEEE\nTransactions on Automatic Control , 62(4):1797–1812, April 2017.\n[2] Omar Ibrahim Dallal Bashi, Wan Zuha Wan Hasan, Norhafiz Azis, Suh aidi Shafie, and Hiroaki Wa-\ngatsuma. Unmanned aerial vehicle quadcopter: A review. Journal of Computational and Theoretical\nNanoscience , 14(12):5663–5675, December 2017.\n9[3] Pedro Batista, Carlos Silvestre, and Paulo Oliveira. On the observ ability of linear motion quantities in\nnavigation systems. Systems & Control Letters , 60(2):101–110, February 2011.\n[4] Sanjay P. Bhat and Dennis S. Bernstein. A topological obstruct ion to global asymptotic stabilization\nof rotational motion and the unwinding phenomenon. In Proceedings of the 1998 American Control\nConference. ACC (IEEE Cat. No.98CH36207) , volume 5, pages 2785–2789, Philadelphia, PA, USA,\nJune 1998. IEEE.\n[5] S´ ergio Br´ as, Maziar Izadi, Carlos Silvestre, Amit Sanyal, and Pa ulo Oliveira. Nonlinear observer for 3D\nrigid body motion estimation using Doppler measurements. IEEE Transactions on Automatic Control ,\n61(11):3580–3585, November 2016.\n[6] Dong Eui Chang. Globally exponentially convergent continuous ob servers for velocity bias and state\nfor invariant kinematic systems on matrix Lie groups. IEEE Transactions on Automatic Control ,\n66(7):3363–3369, July 2021.\n[7] Simone de Marco, Minh-Duc Hua, Tarek Hamel, and Claude Samson. Position, velocity, attitude\nand accelerometer-bias estimation from IMU and bearing measurem ents. In 2020 European Control\nConference (ECC) , pages 1003–1008, Saint Petersburg, Russia, May 2020. IEEE.\n[8] Jay A. Farrell. Aided Navigation: GPS with High Rate Sensors . McGraw Hill Professional, April 2008.\n[9] Hashim A. Hashim. GPS-denied navigation: Attitude, position, linea r velocity, and gravity estimation\nwith nonlinear stochastic observer. In 2021 American Control Conference (ACC) , pages 1149–1154,\nMay 2021.\n[10] Minh-Duc Hua and Guillaume Allibert. Riccati observer design for p ose, linear velocity and gravity\ndirection estimation using landmark position and IMU measurements. In2018 IEEE Conference on\nControl Technology and Applications (CCTA) , pages 1313–1318, August 2018.\n[11] Hassan Khalil. Nonlinear Systems . Prentice Hall, Upper Saddle River, New Jersey, 2002.\n[12] Jae Hyeon Park, Karmvir Singh Phogat, Whimin Kim, and Dong Eui C hang. Transversely stable\nextended Kalman filters for systems on manifolds in Euclidean spaces .Journal of Dynamic Systems,\nMeasurement, and Control , 143(6), February 2021.\n[13] Soham Shanbhag and Dong Eui Chang. Globally exponentially conv ergent observer for the rigid body\nsystem on SE(3). In 2022 IEEE 61st Conference on Decision and Control (CDC) , pages 1257–1262,\nDecember 2022.\n[14] Jos´ e Fernandes Vasconcelos, Rita Cunha, Carlos Silvestre, a nd Paulo Oliveira. Landmark based nonlin-\near observerfor rigid body attitude and position estimation. In 2007 46th IEEE Conference on Decision\nand Control , pages 1033–1038, December 2007.\n[15] Jos´ e Fernandes Vasconcelos, Rita Cunha, Carlos Silvestre, a nd Paulo Oliveira. A nonlinear position and\nattitude observer on SE(3) using landmark measurements. Systems & Control Letters , 59(3):155–166,\nMarch 2010.\n[16] Miaomiao Wang and Abdelhamid Tayebi. Hybrid nonlinear observers for inertial navigation using\nlandmark measurements. IEEE Transactions on Automatic Control , 65(12):5173–5188,December 2020.\n10"    },    {        "title": "2401.11247v1.Applying_stiff_integrators_for_ODEs_and_DDEs_to_problems_with_distributed_delays.pdf",        "content": "APPLYING STIFF INTEGRATORS FOR ODES AND DDES TO\nPROBLEMS WITH DISTRIBUTED DELAYS\nNICOLA GUGLIELMI∗AND ERNST HAIRER†\nAbstract. There exist excellent codes for an efficient numerical treatment of stiff and differential-\nalgebraic problems. Let us mention Radau5 which is based on the 3-stage Radau IIA collocation\nmethod, and its extension to problems with discrete delays Radar5 . The aim of the present work is\nto present a technique that permits a direct application of these codes to problems having a right-\nhand side with an additional distributed delay term (which is a special case of an integro-differential\nequation). Models with distributed delays are of increasing importance in pharmacodynamics and\npharmacokinetics for the study of the interaction between drugs and the body.\nThe main idea is to approximate the distribution kernel of the integral term by a sum of exponen-\ntial functions or by a quasi-polynomial expansion, and then to transform the distributed (integral)\ndelay term into a set of ordinary differential equations. This set is typically stiff and, for some\ndistribution kernels (e.g., Pareto distribution), it contains discrete delay terms with constant delay.\nThe original equations augmented by this set of ordinary differential equations can have a very large\ndimension, and a careful treatment of the solution of the arising linear systems is necessary.\nThe use of the codes Radau5 andRadar5 is illustrated at three examples (two test equations and\none problem taken from pharmacodynamics). The driver programs for these examples are publicly\navailable from the homepages of the authors.\nKey words. Stiff systems, differential-algebraic equations, delay equations, integro-differential\nequations, distributed delays, Runge-Kutta methods, approximation by sum of exponentials, gamma\ndistribution, Pareto distribution, Radau5 ,Radar5 .\nAMS subject classifications. 65L06, 45D05, 65F05\n1. Introduction. The motivation of the present work is in dealing numerically\nwith mathematical models consisting of delay integro-differential equations, as those\narising in pharmacokinetics and pharmacodynamics, where the interaction between\ndrugs and the body is studied (e.g., [16, 11, 5]). The problem is usually modeled by or-\ndinary or delay, differential or differential-algebraic equations, which typically contain\nintegral terms of convolution type (also termed distributed delay). Differential equa-\ntions with a distributed delay term are a special case of Volterra integro-differential\nequations. There is a large literature on the numerical discretization of such problems,\nand their accuracy and stability is well investigated. Let us mention the monographs\nby H. Brunner [3, 4]. A different approach for problems with weakly singular kernel\nis the use of discretized fractional calculus by C. Lubich [13] (see also [8]), and the\noblivious convolution quadrature [14, 12] for more general kernels. Available codes\nmostly make use of constant step size.\nImplementing an integration method for the considered class of problems is more\nor less straight-forward, if constant time steps are used. However, in realistic situa-\ntions one is often confronted with initial layers (like the problem of Section 6.3) and/or\nfast transitions between different states, so that flexibility in the choice of step size is\nan essential ingredient for efficiency. Although there exist excellent codes for solving\nnonstiff and stiff ordinary differential equations, codes for differential-algebraic equa-\ntions, and codes for delay differential equations equipped with sophisticated step size\n∗Division of Mathematics, Gran Sasso Science Institute, Via Crispi 7, I-67100 L’Aquila, Italy.\nEmail: nicola.guglielmi@gssi.it\n†Section de Math´ ematiques, Universit´ e de Gen` eve, CH-1211 Gen` eve 24, Switzerland.\nEmail: ernst.hairer@unige.ch\n1arXiv:2401.11247v1  [math.NA]  20 Jan 20242 N. GUGLIELMI, E. HAIRER\nstrategies, we are not aware of a code based on a variable step size time integrator\nthat can efficiently treat problems with distributed delays.\nInstead of extending a numerical integrator for Volterra integro-differential equa-\ntions to a variable step size code, we propose to change the problem in such a way that\nany code for stiff and differential-algebraic (delay) equations can be applied. The idea\nis to approximate the distribution kernel of the integral term by a sum of exponential\nfunctions (multiplied by a polynomial) and then to transform the integral delay term\ninto a set of differential equations. The use of an approximation by a sum of expo-\nnential functions is not new in numerical analysis. In [2] it is applied for an efficient\ncomputation of high-dimensional integrals. In [12] it is used for developing a variable\nstep size integrator that is applied to interesting problems like a blow-up problem for\na nonlinear Abel integral equation and a fractional diffusion-reaction system. New in\nthe present work is that we do not propose another time integrator, but we show that\na large class of problems with distributed delay can be solved by standard software\nfor stiff and differential-algebraic (delay) equations. This is illustrated at the hand of\nRadau5 [9] for stiff and differential-algebraic equations, and its extension Radar5\n[6, 7] for problems including delay arguments. New is also an application to problems\nin pharmacokinetics and pharmacodynamics.\nOutline of the paper. Section 2 introduces the class of problems that can be treated\nwith the codes presented in this work. It comprises nonstiff, stiff, differential-algebraic,\nand delay differential equations, with the additional feature that distributed delay\nterms are included. In approximating the kernel of a distributed delay term by a sum\nof exponential functions multiplied by a polvnomial, the problem is transformed into\na system without distributed delay terms. Since the resulting system is typically of\na much larger dimension than the original problem, Section 3 presents an algorithm\nthat reduces considerably the complexity of the required solution of linear systems.\nFurthermore, the stability of the proposed algorithm and the determination of the\naccuracy parameters are discussed. Section 4 describes the approach by Beylkin and\nMonz´ on [1] for approximating the factor t−α(typically present in weakly singular\nkernels) by a sum of exponential functions. Parameters in the approximation are\nselected to keep the approximation error under a given level. Section 5 presents\nimportant examples of distributed delays (the gamma distribution and the Pareto\ndistribution) commonly used in pharmacodynamics and pharmacokinetics. Finally,\nSection 6 provides numerical evidence of the efficiency of the proposed approach by\napplying the codes Radau5 andRadar5 to two test examples and to an example\nfrom chemotherapy-induced myelosuppression. The codes together with drivers for\nproblems with distributed delays are made publicly available.\n2. Differential equations with distributed delay. We consider differential\nequations of the form\nM˙y(t) =f\u0000\nt, y(t), y(t−τ), I(y)(t)\u0001\n, y (0) = y0, y(t) =η(t) for t <0,(2.1)\nwhere y∈Rd,Mis a constant d×dmatrix, the delay satisfies τ≥0, and\nI(y)(t) =Zt\n0k(t−s)g\u0000\ns, y(s)\u0001\nds (2.2)\nis a distributed delay term. The vector functions η(t),f(t, y, v, I ), and the scalar\nfunctions k(t),g(t, y) are assumed to be smooth. The kernel k(t) is allowed to have\nan integrable singularity at the origin.Applying codes for ODEs and DDEs to problems with distributed delays 3\nThis formulation contains ordinary differential equations, differential-algebraic\nequations (for singular M), delay differential equations, and integro-differential equa-\ntions as special cases. All considerations of the present work extend straight-forwardly\nto the case, where the delay τ=τ(t, y(t))≥0 is time and state dependent, and where\nseveral delay terms and several integral terms Ii(t)(y) (with different kernels ki(t)\nand different functions gi(t, y)) are present. For notational convenience we restrict\nour study to the formulation (2.1). We focus on the situation, where at least one\nintegral term (2.2) is present in (2.1).\nAssume that we have at our disposal a reliable code for solving differential equa-\ntions of the form (2.1), where no integral term is present. Let us mention the code\nRadau5 of [9] for stiff and differential-algebraic equations and its extension Radar5\nof [6] for delay differential-algebraic equations. Our aim is to demonstrate how such\ncodes can be applied to the solution of (2.1) with integral terms included, without\nchanging them. The main idea is to approximate the kernel k(t) by a finite sum of\nexponential functions multiplied by a polynomial. We thus assume that\nk(t) =nX\ni=1pi(t) e−γit, p i(t) =miX\nj=0ci,jtj(2.3)\nwith real coefficients ci,jandγi, so that the distributed delay term (2.2) can be written\nas\nI(y)(t) =nX\ni=1miX\nj=0ci,jzi,j(t), z i,j(t) =Zt\n0(t−s)je−γi(t−s)g\u0000\ns, y(s)\u0001\nds. (2.4)\nDifferentiation of zi,j(t) with respect to time yields, for i= 1, . . . , n ,\n˙zi,j(t) =−γizi,j(t) +(\ng\u0000\nt, y(t)\u0001\nj= 0\nj zi,j−1(t)j= 1, . . . , m i.(2.5)\nWe insert (2.4) into the differential equation (2.1), we denote zi=\u0000\nzi,0, . . . , z i,mi\u0001⊤,\nand we add these differential equations for zi,j(t) to the original equations. This gives\nthe augmented system of differential equations\nM˙y(t) = f\u0010\nt, y(t), y(t−τ),nX\ni=1miX\nj=0ci,jzi,j(t)\u0011\n,y(0) = y0\ny(t) =η(t) for t <0\n˙z1(t) = J1z1(t) +e1g\u0000\nt, y(t)\u0001\nz1(0) = 0\n··· ···\n˙zn(t) = Jnzn(t) +eng\u0000\nt, y(t)\u0001\nzn(0) = 0(2.6)\nfor the super vector\u0000\ny, z1, . . . , z n\u0001\nof dimension d+ (m1+ 1) + . . .+ (mn+ 1). Here,\nwe use the notation\nJi=\n−γi0··· ··· 0\n1−γi......\n0 2 −γi..................0\n0··· 0mi−γi\n∈Rmi+1,mi+1, e i=\n1\n0......\n0\n∈Rmi+1\nfor the lower bidiagonal matrix and the first unit vector, both of dimension mi+ 1.\nThis gives a formulation, where also those codes can be applied, that are not adapted\nto the treatment of distributed delays (2.1).4 N. GUGLIELMI, E. HAIRER\n3. Solving distributed delay equations via the augmented system. For\nthe case that the original system (2.1) is nonstiff and the exponents γiin (2.3) are of\nmoderate size, any code for nonstiff ODEs or DDEs can be applied to the augmented\nsystem (2.6). However, in important applications (see Section 4) some of the γican\nbe very large, so that the augmented system is stiff independent of the behaviour of\n(2.1). Therefore, a stiff solver (e.g., Radau5 for the case of ODEs, and Radar5 in\nthe presence of retarded arguments) has to be applied to (2.6). Stiff solvers are in\ngeneral implicit and require an efficient solution of linear systems of the form (see [9,\nSection IV.8])\n\u0010\n(γh)−1M − J\u0011\nu=a, (3.1)\nwhere a= (a0, a1, . . . , a n)⊤is a given vector and u= (u0, u1, . . . , u n)⊤is the solution\nof the system. Here, γis a real or complex parameter of the stiff integrator, his the\ntime step size, M= Diag\u0000\nM, I 1,···, In\u0001\nis a block diagonal matrix with Iibeing\nthe identity matrix of dimension mi+ 1, and the Jacobian matrix of the augmented\nsystem is\nJ=\nJ C 1C2··· Cn\nB1J10··· 0\nB20J2......\n............0\nBn0··· 0Jn\n, (3.2)\nwhere J= (∂f/∂y )(t, y, v, I ) is a square matrix of dimension d, and the rank-one\nmatrices BiandCiare given as follows: Bi=eig⊤\nywith gy= (∂g/∂y )(t, y), and\nCi=fIc⊤\niwith fI= (∂f/∂I )(t, y, v, I ) and ci= (ci,0, ci,1, . . . , c i,mi)⊤. All derivatives\nare evaluated at the current integration point.\n3.1. Efficient solution of the linear system (3.1).The linear system (3.1)\nis of dimension d+N, where N= (m1+ 1) + . . .+ (mn+ 1) can be much larger than\nd, the dimension of the original problem. Without exploiting the special structure of\n(3.1) its solution costs O\u0000\n(d+N)3\u0001\nflops. We propose to solve the linear system in\nthe following way.\nThe first block of (3.1) is equivalent to\n\u0010\n(γh)−1M−J\u0011\nu0=a0+nX\ni=1Ciui, (3.3)\nand the other blocks give\n\u0000\n(γh)−1Ii−Ji\u0001\nui=ai+Biu0. (3.4)\nThis equation permits to express uiin terms of u0. Inserted into (3.3) yields\n\u0010\n(γh)−1M−J\u0011\nu0=a0+nX\ni=1Ci\u0010\n(γh)−1Ii−Ji\u0011−1\u0000\nai+Biu0\u0001\n, (3.5)Applying codes for ODEs and DDEs to problems with distributed delays 5\nwhich can be written as\n\u0010\n(γh)−1M−bJ\u0011\nu0=a0+fInX\ni=1c⊤\ni\u0010\n(γh)−1Ii−Ji\u0011−1\nai,\nbJ=J+fIg⊤\nynX\ni=1c⊤\ni\u0010\n(γh)−1Ii−Ji\u0011−1\nei. (3.6)\nNote that bJis a rank-one perturbation of J. Since\u0000\n(γh)−1Ii−Ji\u0001\nis a lower bidi-\nagonal matrix of dimension mi+ 1, the computation of the constant in the rank-one\nperturbation and the computation of the right-hand side of the linear system (3.6)\nrequire not more than O(N) flops. Having computed u0, the uiare obtained from\n(3.4). All in all, this gives an algorithm that has a cost of O(d3) +O(N) flops.\n3.2. Stable approximation of the distributed delay term: exponential\ncase. It may happen that several γnin the kernel approximation (2.3) are very large\nand positive, so that the summands are not well scaled. Consider, for example,\nthe kernel (5.1) defined by a gamma distribution (see Section 5.1 below), which is\napproximated by\nk(t)≈chN−1X\nn=Meαnhe−(enh+κ)t. (3.7)\nFor large positive n(nhcan be of size 50 or more), the exponent γn= enh+κas well\nas the coefficient eαnhare very large, and the computation has to be done with care.\nFor the kernel approximation above, the integral term (2.2) is approximated as\nI(y)(t) =Zt\n0k(t−s)g\u0000\ns, y(s)\u0001\nds≈chN−1X\nn=Meαnhzn(t),\nzn(t)≈Zt\n0e−γn(t−s)g\u0000\ns, y(s)\u0001\nds, (3.8)\nwhere zn(t) is the solution of the differential equation (see Section 2)\n˙zn(t) =−γnzn(t) +g\u0000\nt, y(t)\u0001\n, z n(0) = 0 .\nIt is discretized numerically by a time integrator, typically a Runge–Kutta method.\nFor example, the θ-method, gives the approximation zk\nn≈zn(tk) via\nzk+1\nn=zk\nn−γn∆t\u0010\nθzk+1\nn+(1−θ)zk\nn\u0011\n+∆tgk+1\nn, gk+1\nn=g\u0010\ntk+θ∆t, θyk+1\nn+(1−θ)yk\nn\u0011\n,\nwhich can also be written as\nzk+1\nn=R(−γn∆t)zk\nn+ ∆tS(−γn∆t)gk+1\nn, R (µ) =1 + (1 −θ)µ\n1−θµ, S(µ) =1\n1−θµ.\nSolving this recursion and using z0\nn= 0 yields\nzk+1\nn= ∆tkX\nj=0R(−γn∆t)jS(−γn∆t)gk+1−j\nn . (3.9)6 N. GUGLIELMI, E. HAIRER\nAssuming |g\u0000\nt, y(t)\u0001\n| ≤Gto be bounded, and |R(−γn∆t)|<1 we obtain (with\nµ=−γn∆t)\n|zk+1\nn| ≤∆t\f\fS(µ)\f\f1− |R(µ)|k+1\n1− |R(µ)|G≤∆t\f\fS(µ)\f\fG\n1− |R(µ)|. (3.10)\nAny other Runge–Kutta method will lead to a similar recursion, where R(µ) is its\nstability function, and S(µ) is a rational function bounded by O(µ−1). This shows\nthat for a stable approximation of the integral term (3.8), A0-stability (i.e., |R(µ)|<1\nforµ <0) is a necessary condition, and |R(∞)|<1 is recommended. These conditions\nimply that |zk+1\nn| ≤ O (∆t|µ|−1|) =O(γ−1\nn).\nLet us go back to the question of large coefficients in the approximation (3.8).\nIn the exact solution, a large coefficient eαnhis compensated by the small factor\ne−(enh+κ)tin (3.7). In the numerical approximation, it is compensated by γ−1\nn≤e−nh.\n3.3. Quasipolynomial case. Consider now the situation, where some of the\npolynomials in (2.3) have a degree at least 1. In this case zn(t) is a vector and\nsatisfies the differential equation\n˙zn(t) =Jnzn(t) +eng\u0000\nt, y(t)\u0001\n, z n(0) = 0 ∈Rmn+1.\nA Runge-Kutta discretization yields\nzk+1\nn =R(∆tJn)zk\nn+ ∆tS(∆tJn)engk+1\nn and\nzk+1\nn = ∆ tkX\nj=0R(∆tJn)jS(∆tJn)engk+1−j\nn (3.11)\nwith R(∆tJn) and S(∆tJn) now rational matrix functions. They are of the form\n(with γ=γn,m=mn, and µ=−γn∆t)\nR(∆tJn)=\nR(µ) 0 0 ··· 0\n\u00001\n1\u0001\n∆tR′(µ) R(µ) 0...\n\u00002\n2\u0001\n∆t2R′′(µ)\u00002\n1\u0001\n∆tR′(µ) R(µ)......\n............ 0\u0000m\nm\u0001\n∆tmR(m)(µ)\u0000m\nm−1\u0001\n∆tm−1R(m−1)(µ)···\u0000m\n1\u0001\n∆tR′(µ)R(µ)\n\nThis is a consequence of R(−∆tγI+ ∆N) =Pm−1\nl=0∆tl\nl!R(l)(µ)Nl, where Nis the\nnilpotent matrix whose only non-zero elements are in the first subdiagonal. For the\nmatrix S(∆tJn) we get the same formulas with R(µ) replaced by S(µ). We are inter-\nested to bound the norm of the vector zk+1\nnin (3.11). The first component zk+1\nn,0is given\nby the relation (3.9) and therefore satisfies the estimate (3.10). To estimate the sec-\nond component zk+1\nn,1we have to compute the elements of the second row of R(∆tJn)j.\nThe diagonal element is R(µ)jand the element left to it is j∆tR′(µ)R(µ)j−1. From\n(3.11) we thus get\nzk+1\nn,1= ∆tkX\nj=0\u0010\nj∆tR′(µ)R(µ)j−1S(µ) +R(µ)j∆tS′(µ)\u0011\ngk+1−j\nn .Applying codes for ODEs and DDEs to problems with distributed delays 7\nApplying the triangle inequality, using the bound |g\u0000\nt, y(t)\u0001\n| ≤Gand the estimates\n(for 0 ≤r <1)\nkX\nj=0rj=1−rk+1\n1−r≤1\n1−r,kX\nj=0j rj−1≤1\n(1−r)2,kX\nj=0\u0012j\nl\u0013\nrj−l≤1\n(1−r)l+1,\n(3.12)\n(the second and last inequalities are obtained by differentiating the infinite geometric\nseries and by truncating higher order terms, which are positive) we get\n|zk+1\nn,1| ≤∆t2\u0012\f\fR′(µ)S(µ)\f\f\n(1− |R(µ)|)2+\f\fS′(µ)\f\f\n1− |R(µ)|\u0013\nG. (3.13)\nApplying the fact that |S(µ)|=O(µ−1),|S′(µ)|=O(µ−2), and |R′(µ)|=O(µ−2), we\nfind that |zk\nn,1|=O(γ−2\nn), and the same conclusion as in Section 3.2 can be drawn.\nFor general mwe use the multinomial theorem (the indices ilare non-negative)\n\u0012m−1X\nl=0∆tl\nl!R(l)(µ)Nl\u0013j\n=X\ni1+...+im=j\u0012j\ni1, . . . , i m\u0013m−1Y\nl=0\u0010∆tl\nl!R(l)(µ)Nl\u0011il+1(3.14)\nand the inequalities (3.12) for values of lup to m. Since the nilpotent matrix satisfies\nNm= 0, only the terms with\ni2+ 2i3+. . .+ (m−1)im≤m−1 (3.15)\ngive raise to non-vanishing expressions in (3.14). Therefore, only i1can be large. It\ncan take only the values j, j−1, . . . , j −m+ 1, because otherwise i2+. . .+imwould\nhave to be larger than m−1, contradicting the restriction (3.15).\nWe start with i1=j. In this case all other ilare zero and there is only one term\nin (3.14). The sum over jfrom 0 to kis bounded by the truncated geometric series\n(see (3.10)).\nFori1=j−1 only one index among {i2, . . . , i m}can be non-zero (equal to 1, if\nm≥2). In each case the multinomial coefficient in (3.14) equals j, and the second\nsum in (3.12) provides the desired bound (see (3.13)).\nFori1=j−2 we have two possibilities. Either one among the indices {i2, . . . , i m}\nequals 2 (if m≥3), or two of them equal 1 (provided that m≥4). In each case the\nmultinomial coefficient in (3.14) equals j(j−1)/2 or j(j−1), and the estimate of\n(3.12) with l= 2 can be applied. Using S′′(µ) =O(µ−3) and similar estimates for\nthe stability function R(µ), we obtain |zk\nn,2|=O(γ−3\nn).\nThis procedure can be continued until i1=j−m+ 1. We obtain the bounds\n|zk\nn,l|=O(γ−l−1\nn) for the ( l+ 1)th component of the vector zk\nn.\n3.4. Determination of the accuracy parameters. The variables zi,j(t) are\nauxiliary variables in the system (2.6) and the question arises, how accurate they\nshould be. For the original problem (2.1) the user has to specify the desired accuracy\nof the solution. This is typically done with help of the parameters Atol andRtol, with\nthe aim of having a local error for y(t) that is bounded by Atol +|y|Rtol (for more\ndetails see [9, p. 124]). Of course, such a requirement can be applied component-wise.\nFor solution components with a very small modulus that have a strong impact on the\nother solution components it is in general advisable to use a parameter Atol that is\nmuch smaller than Rtol.8 N. GUGLIELMI, E. HAIRER\nIn the situation of the present work, it is not necessary that all variables zi,j(t)\nare computed very accurately. Since the solution y(t) only depends on a linear com-\nbination of them (with possibly very small coefficients), it is sufficient that this linear\ncombination is sufficiently accurate. This can be achieved as follows: we augment the\ndimension of y(t) by one and introduce the new variable yd+1(t) by\n0 =nX\ni=1miX\nj=0ci,jzi,j(t)−yd+1(t). (3.16)\nWe replace the double sum in (2.6) by yd+1(t) and we add the relation (3.16) to (2.6).\nThis gives a differential-algebraic system where the number of algebraic variables is\naugmented by one. The advantage of this is that we can require different accuracies\nfor the zi,jand for their sum (3.16). For a given tolerance Tolwe propose (in general)\nto put Atol =Rtol =Tolfor all d+ 1 components of the augmented vector y, and\nwe put Atol =Rtol =ωTol(with ω≥1) for all variables zi,j. To increase efficiency\nwithout spoiling accuracy we propose to take ω= 100 or even larger (see the second\nexperiment in Section 6.1 and Table 6.2).\n4. Approximating the kernel with a sum of exponentials. Important ker-\nnels, such as the gamma distribution and the Pareto distribution, contain the function\nt−αas factor. This section is devoted to approximate it by a sum of exponential func-\ntions.\n4.1. Approach of Beylkin and Monz´ on. Since the Laplace transform of the\nfunction zα−1is Γ(α)t−αforα >0, we have the integral representation\nt−α=1\nΓ(α)Z∞\n0e−tzzα−1dz=1\nΓ(α)Z∞\n−∞e−teseαsds, α > 0. (4.1)\nTo express this function as a sum of exponentials, [1] proposes to approximate the\nintegral to the right by the trapezoidal rule. With a step size h >0 this yields\nT(t, h) =h\nΓ(α)∞X\nn=−∞eαnhe−enht. (4.2)\n•Error of the trapezoidal rule. It follows from [15, Theorem 5.1] that the error due\nto this approximation can be bounded by\n\f\ft−α−T(t, h)\f\f≤2C\ne2πa/h−1, C = sup\nb∈(−a,a)1\nΓ(α)Z∞\n−∞e−tescosbeαsds= (tcosa)−α\nfor any h >0 and any 0 < a < π/ 2. This gives the estimate\n\f\ft−α−T(t, h)\f\f≤cαt−α, c α=2(cos a)−α\ne2πa/h−1. (4.3)\nTo achieve cα≤ε, the step size hhas to satisfy the inequality in\nh≤2πa\nln\u0000\n1 +2\nε(cosa)−α\u0001, a =π\n2\u0010\n1−α\n(α+ 1) ln ε−1\u0011\n. (4.4)\nThe value of ais chosen so that, for a given ε >0, the estimate for hin (4.4) turns\nout to be close to maximal.Applying codes for ODEs and DDEs to problems with distributed delays 9\n•Error from truncating the series (4.2) at−∞.Since the function s7→e−teseαsis\nmonotonically increasing for s≤ln(α/t) we have, for Mh≤ln(α/t),\nEM(t, h) =h\nΓ(α)M−1X\nn=−∞eαnhe−enht≤1\nΓ(α)ZMh\n−∞e−teseαsds\n=t−α\nΓ(α)ZteMh\n0e−σσα−1dσ=t−α\u0012\n1−Γ(α, teMh)\nΓ(α)\u0013\n,(4.5)\nwhere Γ( α, x) =R∞\nxe−σσα−1dσis the incomplete Gamma function.\n•Error from truncating the series (4.2) at+∞.The function s7→e−teseαsis\nmonotonically decreasing for s≥ln(α/t). The same computation as before therefore\nshows that for Nh≥ln(α/t)\nEN(t, h) =h\nΓ(α)∞X\nn=Neαnhe−enht≤1\nΓ(α)Z∞\nNhe−teseαsds\n=t−α\nΓ(α)Z∞\nteNhe−σσα−1dσ=t−αΓ(α, teNh)\nΓ(α).(4.6)\n•Choice of the step size h.For a fixed accuracy requirement ε, we choose haccording\nto (4.4). To study the quality of this approximation, we plot in Figure 4.1 for several\nvalues of αthe relative error of the trapezoidal rule as a function of the step size h.\nWe see that for α= 1/2 the error of the trapezoidal rule is bounded by ε= 10−5\nwhenever h≤0.78. The estimate (4.4) gives h≤0.70, which is a reasonably good\napproximation.\n.4 .6 .8 1.0 1.210−1010−810−610−4\nerror of trapezoidal rule\nα= 0.1α= 0.9\nstep size h\nFig. 4.1 .Relative error\f\ft−α−T(t, h)\f\f\u000e\nt−αas a function of the step size h. The different\ncurves correspond to α= 0.1,0.2, . . . , 0.9(bold curve for α= 0.5).\n•Choice of the truncation indices MandN.For a treatment of the two truncation\nerrors, we restrict tto the interval 0 < δ≤t≤T <∞, and we use the estimates\n(4.5) and (4.6). The incomplete Gamma function is monotonically decreasing and\nsatisfies Γ( α,0) = Γ( α), Γ(α,∞) = 0. We let x∗be such that Γ( α, x∗)≥Γ(α)(1−ε),\nandx∗be such that Γ( α, x∗)≤Γ(α)ε. Both truncation errors are bounded by ε t−α\nfort∈[δ, T] ifMandNare chosen according to TeMh≤x∗andδeNh≥x∗. Since10 N. GUGLIELMI, E. HAIRER\nΓ(α, x)≥Γ(α)−xα/α, we can approximate x∗by the relation xα\n∗= Γ( α+ 1)ε.\nFrom Γ( α, x)≤xα−1e−xa suitable x∗is given by ( x∗)α−1e−x∗≤Γ(α)ε, which is\napproximately x∗=−ln\u0000\nΓ(α)ε\u0001\n.\n−60 −50 −40 −30 −20 −1010−810−610−410−2100\n−30−20−10 0 10 20 30 40 50 6010−810−610−410−2100\ntruncation error at −∞\nMT= 10T= 1010truncation error at + ∞\nNδ= 10−10\nδ= 102\nFig. 4.2 .Relative error of the the truncation errors as a function of MandN, respectively.\nTo get an impression of the size of the truncation indices MandN, we fix the\nvalues α= 0.5 and h= 0.78, which correspond to ε= 10−5(see Figure 4.1). In\nthe left picture of Figure 4.2 we plot max\b\nEM(t, h)/t−α\f\ft∈[10−10, T]\t\n(the relative\ntruncation error) as a function of M, for T= 10,102, . . . , 1010. In the right picture\nof Figure 4.2 we plot max\b\nEN(t, h)/t−α\f\ft∈[δ,500]\t\nas a function of N, for δ=\n102,10,1,10−1, . . . , 10−10. Replacing the right end of the interval [ δ,500] by a larger\nvalue does not change the picture. We have included the curves for large values of δ,\nbecause they are of interest for a treatment of the Pareto distribution (see Section 5.2).\nAs a conclusion we see that for an accuracy requirement of ε= 10−5and for an\ninterval [ δ, T] with δ=ε= 10−5andT= 100, we can choose the step size h= 0.78,\nand the truncation indices M=−35 and N= 18 or (using hfrom (4.4)) h= 0.70,\nM=−40 and N= 20.\n•Total approximation error. The error of approximating the function t−αby a sum\nof exponentials is composed by the error of the trapezoidal rule and by those due to\ntruncation. We have, for δ≤t≤T,\n\f\ft−α−TN\nM(t, h)\f\f≤3ε t−α, TN\nM(t, h) =h\nΓ(α)N−1X\nn=Meαnhe−enht, (4.7)\nprovided that h,M, and Nare chosen as discussed above.\n4.2. Approach of Braess and Hackbusch. An efficient computation of high-\ndimensional integrals is another motivation for studying the approximation of t−α\n(α= 1/2 or 1) by exponential sums (see [2]). The differences to the approach\nof Beylkin and Monz´ on are that the coefficients ciandγiin the approximation\nun(t) =Pn\ni=1ciexp (−γit) are computed for minimal nnumerically with a Remez-like\nalgorithm, and the absolute error\f\ft−α−un(t)\f\f(not the relative error as in (4.7)) is\nkept below a tolerance ε.\n5. Important examples of distributed delays. Because of their application\nin pharmacodynamics we are mainly interested in the situation where the kernel\nof the integral delay term (2.2) is a distribution, i.e., a non-negative function withR∞\n0k(s) ds= 1. The following distributions are taken from the list given in [16].Applying codes for ODEs and DDEs to problems with distributed delays 11\n5.1. Gamma distribution. For positive parameters κ >0 and 0 < α < 1, the\ngamma distribution is given by\nk(t) =κ1−α\nΓ(1−α)t−αe−κt, t > 0. (5.1)\nWe choose δ > 0 such thatRδ\n0k(t) dt≤ε, and T > 0 such thatR∞\nTk(t) dt≤ε.\nRoughly speaking, the small parameter δand the large parameter Thave to be such\nthat\n(κ δ)1−α\nΓ(2−α)≤ε and(κ T)−αe−κT\nΓ(1−α)≤ε. (5.2)\nWe use (4.7) for approximating the gamma distribution by a sum of exponential func-\ntions. For a given accuracy requirement ε, Algorithm 1 summarizes the computation\nof the parameters δ, Tas well as the parameters h, M, N needed in (4.7).\nAlgorithm 1: Choice of the parameters for the Gamma distribution.\nData: α,κ,ε,δmin,tf\nResult: h, T, δ, M, N\nbegin\n1 Seta=π\n2\u0010\n1−α\n(α+ 1) ln ε−1\u0011\n2 Seth=2πa\nln\u0000\n1 +2\nε(cosa)−α\u0001\n3 Compute Tsuch that ( κ T)−αe−κT/Γ(1−α) =ε\n4 SetT= min {tf, T}\n5 Setx∗= (Γ( α+ 1)ε)1/α\n6 Setx∗=−ln\u0000\nΓ(α)ε\u0001\n7 Compute Msuch that TeMh=x∗\nCompute δsuch that ( κ δ)1−α/Γ(2−α) =ε\n8 Setδ= max {δ, δmin}\n9 Compute Nsuch that δeNh=x∗\nreturn\n•Negative values of α.For negative values of αthe function k(t) vanishes at the\norigin, but the sum of exponentials (4.2) does not. Let αsatisfy −1< α < 0. The\nidea is to split t−α=t·t−α−1, so that α+ 1∈(0,1), and the formula (4.2) can be\napplied to the factor t−α−1. This results in a linear combination of functions te−γt,\nwhich can be treated as explained in Section 2.\nFor values of αsatisfying −2< α < −1, we split t−α=t2·t−α−2. The above\nprocedure then leads to kernels of the form (2.3) with mi= 2. An extension to other\nnon-integer negative values of αis straight-forward.\n5.2. Pareto distribution. The so-caled type I Pareto distribution [16] is given\nforα >0 and β >0 by\nk(t) =(\n0 0 ≤t < β\nα βαt−α−1t≥β.(5.3)\nNote that there is no singularity at the origin. For the approximation (4.7) of t−α−1,\nwe can choose δ=βindependent of the accuracy ε, so that the condition δeNh≥x∗12 N. GUGLIELMI, E. HAIRER\nis less restrictive for the truncation index N. The conditionR∞\nTk(t) dt≤εforT\nreduces to Tα≥βα/ε. The truncation index Mis determined by TeMh≤x∗, where\nTis the minimum of β/ε1/αand of the length of the integration interval Tend(see\nAlgorithm 2).\nAlgorithm 2: Choice of the parameters for the Pareto distribution.\nData: α,β,ε,tf\nResult: h, T, M, N\nbegin\n1 SetT=β ε−1/α\n2 SetT= min {tf, T}\n3 Seta=π\n2\u0010\n1−(α+ 1)\n(α+ 2) ln ε−1\u0011\n4 Seth=2πa\nln\u0000\n1 + 2 ε−1(cosa)−(α+1)\u0001\n5 Setx∗= Γ(α+ 2)ε\n6 Setx∗=−ln\u0000\nΓ(α+ 1)ε\u0001\n7 Compute Msuch that TeMh=x∗\nCompute Nsuch that βeNh=x∗\nreturn\nIn this case the integral in (2.2) is from 0 to t−β. When replacing the kernel (on\nthe interval [ β, T]) by a sum of exponential functions, we are thus lead to\nM˙y(t) =f\u0000\nt, y(t), y(t−τ), IP(t−β)\u0001\n(5.4)\nwhere the function IP(t) is given by (with γn= enh)\nIP(t) =\n\n0 if t≤0\nαβαh\nΓ(α)N−1X\nn=Me(α+1)nhe−γnβzn(t) if t≥0(5.5)\nand zn(t) =Zt\n0e−γn(t−s)g\u0000\ns, y(s)\u0001\ndsgives raise to the differential equation\n˙zn(t) =−γnzn(t) +g\u0000\nt, y(t)\u0001\n, z n(0) = 0 . (5.6)\nAs we have done in Section 3.4 we consider Ip(t) as a new variable of the system,\nand we add the algebraic relation (5.5) to the augmented system (5.4)-(5.6). In this\nway we can monitor the choice of the accuracy parameters Atol andRtol as explained\nin Section 3.4. The introduction of the variable IP(t) in the system has the further\nadvantage that during the numerical integration only the back values of the scalar\nfunction IP(t) have to be stored. Note that the augmented system has additional\nbreaking points at iτ+jβwith non-negative integers i, j.\n•Remark. For small positive α(e.g., α= 0.15 and smaller, see [16, Table 1]) the\ncondition Tα≥βα/εcan lead to a very large T, which then results in a large negative\ntruncation index M. To increase efficiency, we propose to introduce the new variable\nyd+1(t) =αβαZt−β\n0(t−s)−α−1g\u0000\ns, y(s)\u0001\nds,Applying codes for ODEs and DDEs to problems with distributed delays 13\nwhich is the argument I(y)(t) in (2.1), and to differentiate it with respect to t. This\nyields the delay differential equation\n˙yd+1(t) =αβ−1g\u0000\nt−β, y(t−β)\u0001\n−α(α+ 1)βαZt−β\n0(t−s)−α−2g\u0000\ns, y(s)\u0001\nds,\nwhich can be added to the system for y(t). The integral in the right-hand side is\nagain of Pareto type, but with αreplaced by α+ 1. Replacing the new kernel by a\nsum of exponentials gives a more efficient algorithm. Of course, this procedure can\nbe repeated to increase the parameter αeven more.\n6. Numerical experiments. This section provides numerical evidence of the\nproposed algorithm. The first two examples are scalar test equations (one ordinary\ndifferential equation with a Gamma-distributed delay term, the other a delay differen-\ntial equation with a Pareto-distributed delay term). The third example is taken from\napplications in chemotherapy-induced myelo-suppression. For the time integration\nwe make use of the code Radau5 (see[9]), if the augmented system is an ordinary\ndifferential equation, and of the code Radar5 (see [6, 7]), if the augmented system is\na delay differential equation.\n6.1. Example 1: ordinary differential equation with a Gamma-distri-\nbuted delay term. As a first test example we consider a linear differential equation\n˙y(t) =\u0000\n1−y(t)\u0001\nerf\u0012√\nt\n2\u0013\n−e−t/4√\nt√π+I\u0000\nt, y(t)\u0001\n+1\n2, y (0) = 0 , (6.1)\nwhere y(t) is a real-valued function, erf denotes the error function, and the distributed\ndelay term is\nI(y)(t) =Zt\n0k(t−s)y(s) ds, k (t) =e−t/4\n2√\nπt(6.2)\nHere, k(t) is the Gamma-distribution (5.1) with parameters κ= 1/4 and α= 1/2.\nNote that the kernel is weakly singular, but that there is no singularity in the aug-\nmented system (2.6). The inhomogeneity of the equation is chosen such that\ny(t) =t/2\nis the exact solution. This follows from the fact that with y(t) =t/2 we have\nI(y)(t) =t−2\n2erf\u0012√\nt\n2\u0013\n+e−t/4√\nt√π.\nFirst experiment (connection between Tol and ε).We fix the tolerance Tol= 10−8\nfor the time integrator, and we vary the value of ε, which determines the accuracy of\nthe approximation by the sum of exponentials. We consider the integration interval\n[0, tf] with tf= 50, so that the exact solution at the final point is y(tf) = 25.\nAccording to Algorithm 1 of Section 5.1 we compute, for given ε >0 and for\nδmin= 0, the parameters h,T,M, and N. For ε=Tol= 10−8, we compute\nT= min {50,65.79}= 50 and δ=π·10−16. The suggested parameter values are\nh= 0.4638, M=−89,N= 84 which are indicated in bold in Table 6.1. There, the\nvalues of h, T, M, N are also given for further values of ε. The value of δisδ=πε2.14 N. GUGLIELMI, E. HAIRER\nε h T M N err\n10−40.84 30.49−27 24 2.45·10−4\n10−50.70 39.20−39 35 2.75·10−5\n10−60.60 48.00−54 49 2.35·10−6\n10−70.52 50.00−70 65 2.40·10−7\n10−80.46 50.00−89 84 1.71·10−8\n10−90.42 50.00−110 104 4.72·10−10\n10−100.38 50.00−133 127 2.14·10−9\n10−110.35 50.00−158 152 2.08·10−9\nTable 6.1\nError behavior obtained by applying Radau5 to the test problem (6.1) with final point tf= 50\nand Tol = 10−8for different computed values of h,M, and N(corresponding to ε).\nFor the numerical integration we apply the code Radau5 (the code Radar5\nwould give similar results) with accuracy requirement Atol =Rtol =Tol= 10−8\nto the augmented system (2.6) with parameters h, M, N , determined by different\nvalues of ε. The initial step size is h=ε. The last column of Table 6.1 reports\nthe relative error err := |y(tf)−¯y|/|y(tf)|at the final point. Here, ¯ ydenotes the\nnumerical approximation computed at time tf. We can observe that for ε≤Tolthe\nerror is essentially proportional to ε, which corresponds to the error of the kernel\napproximation. For ε≥Tolthe error remains close to Tol, which shows the error of\nthe time integration.\nω= 1 ω= 10 ω= 100\nε err #fe cpu err #fe cpu err #fe cpu\n10−42.5e-4 81 2.5e-4 2.5e-4 66 2.2e-4 2.5e-4 66 2.2e-4\n10−62.4e-6 162 7.9e-4 2.3e-6 132 6.6e-4 2.3e-6 117 6.0e-4\n10−81.8e-8 365 2.7e-3 1.6e-8 279 2.1e-3 1.5e-8 243 1.8e-3\n10−105.8e-11 773 7.6e-3 1.1e-11 587 6.2e-3 1.2e-10 482 5.1e-3\nTable 6.2\nError behavior of Radau5 applied to the test problem (6.1) with tf= 50 , initial step size\nh= 0.1, and Atol =Rtol =εforyandyd+1(see (3.16) ), and Atol =Rtol =ωεfor the auxiliary\nvariables zi,j. Here, err,#fe,cpuindicate the relative error of y(t), the number of function evalu-\nations and the cpu time.\nSecond experiment (choice of the accuracy parameters). As proposed in Section 3.4 we\nadd a new variable y1+1(t) to the equation (6.1), and we apply the code Radau5 to the\naugmented system with accuracy parameters Atol =Rtol =εfor the y-components\nandAtol =Rtol =ωεfor the z-components. We first put ω= 1, and then study\nthe behaviour for increasing ω. The relative error of y(t), the number of function\nevaluations, and the cpu time are given in Table 6.2. We observe that increasing ω\ndoes not affect the precision of the numerical approximation for y(t), but significantly\nreduces the number of function evaluations and the cpu time. This is due to the fact\nthat the numerical integrator can take larger step sizes. For every εthere seems to\nexist a critical value of ω, such that the numerical result does not change any more\nwhen ωis further increased. Numerical computations show that this happens for\nε= 10−4when ω= 10, for ε= 10−6when ω= 100, for ε= 10−8when ω= 1000, etc.\nWe thus propose to use for this example ω= 0.1ε−1/2, i.e., Rtol =√ε/10.Applying codes for ODEs and DDEs to problems with distributed delays 15\n6.2. Example 2: delay differential equation with a Pareto-distributed\ndelay term. We next consider a test example with Pareto-distributed delay term.\nSince such a kernel leads in any case to a delay equation, we add a discrete delay term\nand consider\n˙y(t) =−5I(y)(t)−y\u0000\nt−τ\u0001\n−2\ny(t) + 1, y (t) =tfort≤0, (6.3)\nwhere y(t) is a scalar real function, the discrete delay term is τ=π/4, and the\ndistributed delay term is\nI(y)(t) =√β\n2Zt−β\n0(t−s)−3/2y(s) ds, (6.4)\nwhere the kernel is a Pareto distribution (5.3) with α= 1/2. Since t−s≥β >0,\nthere is no singularity in the integrand. Note that the system (5.4) depends on y(t−τ)\nand also IP(t−β), which gives rise to breaking points at integral multiples of τand\nβand also their integer linear combinations.\nε h M N err\n10−11.662 −3 1 7.69·10−2\n10−21.116 −6 2 8.97·10−4\n10−30.851 −11 3 2.31·10−4\n10−40.692 −17 4 2.81·10−5\n10−50.586 −24 5 1.37·10−6\n10−60.509 −32 6 3.46·10−7\n10−70.451 −41 7 1.90·10−7\n10−80.405 −51 89.83·10−8\n10−90.368 −62 9 5.95·10−8\n10−100.337 −75 10 1.75·10−7\n10−110.311 −88 11 2.40·10−7\nTable 6.3\nError behavior obtained by applying Radar5 to the test problem (6.3). for different values of\nh,MandN, computed according Algorithm 2, with Tol = 10−8\nIn our experiments we set β= 1 and we consider the integration interval [0 , tf]\nwith tf= 10. With this choice we compute a reference solution to high precision,\ny(tf)≈0.570525788119 .\nAs explained in Section 5.2, an approximation of the kernel by a sum of ex-\nponentials gives rise to a delay differential equation. For an approximation error ε\nAlgorithm 2 yields the parameters h,M, and Nthat are needed for the description\nof the augmented system.\nForε= 10−8, the values are h= 0.405,M=−51 and N= 8, which are indicated\nin bold in the right side of Table 6.3. In contrast to the situation of Example 1, the\nresulting augmented system is not very stiff.\nFor the numerical solution of this problem we use the code Radar5 . We consider\na fixed tolerance Tol= 10−8, and we apply the code with Atol =Rtol =Tol. We\ninclude in the mesh the first 10 breaking points,\nτ, β, 2τ, τ+β,2β,3τ,2τ+β,2β+τ,3β,4τ,16 N. GUGLIELMI, E. HAIRER\nand let the code possibly compute further breaking points.\nAs initial step size we choose h=Atol. Similar to the first experiment for\nExample 1, we report the relative error in Table 6.3 at the final point. We get 120\nsteps (with no rejections), 854 function evaluations, 72 Jacobian evaluations, 99 LU\ndecomposition of the matrix in the linear system and 244 solutions of triangular\nsystems.\nThe result is very similar to that of Table 6.1. For a fixed tolerance Tol, the error\nis proportional to εas long as ε >Tol. It stagnates at Tolfor values of εthat are\nsmaller than Tol.\n6.3. Example 3: a distributed delay model of chemotherapy-induced\nmyelosuppression. We consider a model system that is proposed and studied in\n[10, p. 56]. It uses ideas of the publication [5], where transit compartments in a\nsemi-physiological model by Friberg are replaced by a convolution integral with a\nGamma-distribution (see also [11]). The equations of the model are\n˙y(t) =\u0010\nκ\u0010w0\nw(t)\u0011γ\n−ksC(t)−κ\u0011\ny(t)\n˙w(t) = −κw(t) +κI\u0000\nt, y(t)\u0001\n˙A(t) = −VmaxA(t)\nKm+C(t), C (t) =A(t)\nV,(6.5)\nwhere y(t) gives the proliferating precursor cells for granulocytes, w(t) the circulating\ngranulocytes, and A(t) is the amount of drug in the plasma. The distributed delay\nterm, with a Gamma-distribution kernel, is given by\nI(y)(t) =Zt\n0k(t−s)y(s) ds, k (t) =κ1−α\nΓ(1−α)t−αe−κt. (6.6)\nTable 6.4 gives two sets of parameters, which are taken from [10, Table 2]. Concerning\nthe units of the parameters we refer to [10]. Time is measured in hours.\nν= 1−α κ w0 γ ks Vmax Km V\n0.964 ν/47.514.4 0.664 0.0328 77.2 16.9 1.35\n1.46 ν/55.614.4 0.507 0.0213 100 22 1.03\nTable 6.4\nValues of the parameters for the system (6.5)\nWe note that the third equation of (6.5) is a scalar differential equation for A(t),\nwhich is independent of the other two variables of the system. It can be solved\nanalytically by separation of variables. In this way we find that A(t) is solution of\nthe nonlinear equation\nKm\nVmax\u0010\nlnA(t)−lnA0\u0011\n+A(t)−A0\nVmaxV=−(t−t0).\nTo get a better conditioning for its numerical treatment we write it as\nA(t) =A0exp\u0010\n−1\nKmV\u0000\nA(t)−A0\u0001\n−Vmax\nKm(t−t0)\u0011\n. (6.7)Applying codes for ODEs and DDEs to problems with distributed delays 17\n0 100 200 300 400 500 600 7000102030\n0 100 200 300 400 500 600 7000102030\nty(t)\nw(t)problem 1\nty(t)w(t)problem 2\nFig. 6.1 .Solution components y(t)andw(t)of the system (6.5) for the parameter values of\nTable 6.4: upper row (problem 1) and lower row (problem 2).\nWithout changing the solution of the problem we can replace in (6.5) the differential\nequation for A(t) by the algebraic equation (6.7). This then leads to a differential-\nalgebraic system of index 1, which can be solved efficiently by Radau5 and by\nRadar5 . The formulation as a differential-algebraic system has the advantage that\nduring the numerical integration there is no accumulation of errors in the variable\nA(t). In our experience the DAE formulation is slightly more efficient than the ODE\nformulation (see Table 6.7).\nAs proposed in Section 3.4 we add a new variable y3+1(t) to the equation (6.5),\nand we apply the code Radau5 to the augmented system. For various values of εwe\nuseAtol =Rtol =εfor the three components of (6.5), Atol =Rtol = 10−2εfor the\ny3+1-component corresponding to the integral delay, and Atol =Rtol = 102εfor the\nz-components. As initial step size we take max( ε,10−5).\nFirst experiment. We consider the problem (6.5) with parameters taken from the\nupper row of Table 6.4. Initial values are\ny(0) = w0, w (0) = w0, A (0) = A0= 127 . (6.8)\nThe solution components y(t) and w(t) are plotted in the left picture of Figure 6.1.\nWe do not include the graph of A(t), because it monotonically decreases, rapidly\napproaches zero, and stays there for all t. We observe that the function y(t) has an\ninitial layer. It rapidly decreases from y(0) = 14 .4 to a small value and then behaves\nsmoothly.\nThe challenge of this problem is the fact that the value of α= 0.036 is very small.\nAccording to Algorithm 1 this then leads to a very small x∗, and consequently to a\nlarge negative value of M, so that the dimension of the augmented system becomes\nlarge. The values of M,N(as well as h) are computed by Algorithm 1 of Section 5.1.\nThey are presented for different values of εin Table 6.5.\nWe apply the code Radau5 for values of εranging between 10−3and 10−10on\nthe interval [0 ,100]. Since the Jacobian of the vector field is not banded, we apply\nthe code in the standard way, where the linear algebra considers the Jacobian as a\nfull matrix. The cpu time (measured in seconds) is given in the row of Table 6.5\nmarked “cpu (full)”. We also have implemented the treatment of the arising linear\nsystems according to the algorithm of Section 3.1, and the cpu times are listed in the\nrow of Table 6.5 marked “cpu (sumexp)”. The improvement is amazing. Already for\nε= 10−3the cpu time is decreased by a factor of 100 and for ε= 10−10, where the18 N. GUGLIELMI, E. HAIRER\nε 10−310−410−610−710−910−10\nM −157 −268 −582 −783 −1276 −1567\nN 4 8 20 27 45 56\ncpu (full) 1.9·10−11.3·1002.6·1018.3·1015.1·1021.2·103\ncpu (sumexp) 9.6·10−42.1·10−38.9·10−31.6·10−25.3·10−29.5·10−2\nTable 6.5\nComparison of different linear algebra solvers for the system (6.5) with parameters of the upper\nrow of Table 6.4.\ndimension of the augmented system is very large, the cpu time is even decreased by\na factor of 104.\nSecond experiment. We next consider the set of parameters given in the lower row\nof Table 6.4. The solution is plotted in the right picture of Figure 6.1. Here, the\nvalue of α=−0.46 is negative, and the trick explained in Section 5.1 has to be\napplied. This means that the factor t−αin (6.6) is written as t−α=t·t−(α+1)with\nα+ 1 = 0 .54∈(0,1), and only the term t−(α+1)is replaced by a sum of exponentials.\nThis yields an approximation of the form (2.3) with mi= 1.\nAs before we apply the code Radau5 with several different values of ε, and initial\nvalues (6.8). We only use the option (sumexp). This time we study the error of the\nnumerical approximation. A reference solution is computed with a very high accuracy\nrequirement. The exact solution of A(100) is far below round-off. The row, indicated\nas “err” in Table 6.6 shows the relative error of y(t) and w(t) at time t= 100. The\nvalues of MandNare computed by Algorithm 1. They are of moderate size, even\nfor very small ε. We realize that the error is nicely proportional to the accuracy\nrequirement Tol.\nε 10−310−510−710−910−11\nM −17 −38 −67 −105 −150\nN 13 35 67 108 156\ncpu (sumexp) 4.7·10−41.7·10−35.1·10−31.5·10−23.9·10−2\nerr 5.3·10−41.4·10−51.5·10−73.3·10−91.0·10−10\nTable 6.6\nStudy of accuracy of Radau5 for the system (6.5) with parameters of the lower row of Table 6.4.\nThird experiment. The equations (6.5) are an ODE formulation of the problem of\nSection 6.3. When the differential equation for A(t) is replaced by the algebraic\nequation (6.7) we get a DAE formulation with two differential equations and one\nalgebraic relation. The code Radau5 can be applied to each of these formulations,\nand it is interesting to know which one performs better. In Table 6.7 we present for\nvarious εthe error at the final point tf= 100, as well as the required number of steps\nand function evaluations. We observe that, for the DAE formulation (the final three\ncolumns), not only the error is smaller (excepting ε= 10−3), but also the number of\nsteps and function evaluations (and consequently the cpu time) are smaller.\n6.4. Implementation of the approach of Section 3.1 for solving the lin-\near system. The code Radau5 [9] (and similarly the code Radar5 [6]) consists of\nthree files: Radau5.f (Radar5.f ) contains the time integrator, decsol.f the lin-\near algebra subroutines, and dcdecsol.f (dcdecdel.f ) contains the subroutinesApplying codes for ODEs and DDEs to problems with distributed delays 19\nε h M N err1 #st1#fe1 err2 #st2#fe2\n10−31.04−17 13 5.34·10−425 161 9.54·10−323 154\n10−50.69−38 35 1.37·10−547 287 9.06·10−638 262\n10−70.52−67 67 1.52·10−780 507 5.47·10−868 483\n10−90.42−105 108 3.33·10−9152 985 4.98·10−10126 945\n10−110.35−150 156 1.05·10−10309 2036 2.41·10−11256 1933\nTable 6.7\nPerformance of Radau5 for the problem (6.5) (coefficients from the lower row of Table 6.4)\nwithtf= 100 andεranging from 10−3to10−11. Here, err1,#st1,#fe1indicate the relative error,\nthe number of steps and the number of function evaluations for an implementation as ODE, while\nerr2,#st2,#fe2are the numbers corresponding to an implementation as DAE.\nthat link the time integrator with the linear algebra subroutines. When using the\nlinear algebra approach of Section 3.1 one only has to replace the file dcdecsol.f\n(ordcdecdel.f ) bydcsumexp.f (ordcsumexpdel.f )1. The other files need not\nbe touched. In the driver the Jacobian has to be defined as follows: first of all, if d\nis the dimension of yin the original system and nthe number of summands in (2.3),\nwe have to define the dimension of the system as\nN=\u001ad+ 1 + n+ 2 if all mi= 0 in (2.3)\nd+ 1 + 2 n+ 2 if all mi= 1 in (2.3).\nThe first dcomponents correspond to those of the original system (2.1), the ( d+ 1)th\ncomponent is for the integral delay (3.16), and the next n(or 2n) components are for\nthe variables zi,j. The final 2 components are not used in the right-hand side function\nof the problem. They are used only for the Jacobian.\nThe Jacobian has to be declared as banded with upper bandwidth mujac =d\nand lower bandwith mljac = max(2 −d,0). The array fjac then has Ncolumns\nand max( d+ 1,3) rows. In the left upper ( d+ 1)×(d+ 1) matrix, the derivative of\nthe vector field f(augmented by (3.16)) with respect to ( y, yd+1) has to be stored, in\ncolumn d+ 2 the derivative of fwith respect to the integral term I, and in column\nd+ 3 the gradient of g(t, y) with respect to y. For the case that all mi= 0, the\ncoefficients {ci0|i= 1, . . . n}are stored in the first row starting at position d+ 4, the\nexponents {γi|i= 1, . . . n}in the second row, and zeros in the third row. Finally, for\nthe case that all mi= 1, the coefficients {(ci0, ci1)|i= 1, . . . n}are stored in the first\nrow starting at position d+ 4, the exponents {(γi, γi)|i= 1, . . . n}in the second row,\nand the subdiagonal (1 ,0,1,0, . . . , 1,0) of (3.2) in the third row.\nThe codes Radau5 andRadar5 together with the files decsol.f ,dcdecsol.f ,\ndcsumexp.f together with drivers for the examples of this article are available at\nthe address http://www.unige.ch/ ~hairer/software.html .\nConclusions. In this article we have described an efficient methodology to deal\nwith important classes of distributed delays, by using Radau5 orRadar5 , which are\ncodes for the numerical integration of a large class of stiff and differential-algebraic\nequations, and with discrete delays (for Radar5 ).\nThe methodology is based on a suitable expansion of the kernels in terms of\nexponential functions, which allows to transform the distributed delay into a set of\nODEs or DDEs. Our numerical experiments confirm the efficiency of this methodology\n1The subroutines dcdecdel.f anddcsumexpdel.f are restricted to the case where Mis a\n(possibly singular) diagonal matrix.20 N. GUGLIELMI, E. HAIRER\nwhich allows to get a much more versatile code, able to deal with both discrete and\ndistributed memory effects, a feature that makes it very appealing in disciplines where\nthese kind of delays naturally arise, as pharmacodynamics and pharmacokinetics.\nAcknowledgments. Nicola Guglielmi acknowledges that his research was sup-\nported by funds from the Italian MUR (Ministero dell’Universit` a e della Ricerca)\nwithin the ‘PRIN 2022 Project “Advanced numerical methods for time dependent\nparametric partial differential equations with applications” and the Pro3 joint project\nentitled “Calcolo scientifico per le scienze naturali, sociali e applicazioni: sviluppo\nmetodologico e tecnologico”. He is also affiliated to the INdAM-GNCS (Gruppo\nNazionale di Calcolo Scientifico).\nErnst Hairer acknowledges the support of the Swiss National Science Foundation,\ngrant No.200020 192129.\nREFERENCES\n[1] G. Beylkin and L. Monz´ on. Approximation by exponential sums revisited. Appl. Comput.\nHarmon. Anal. , 28:131–149, 2010.\n[2] D. Braess and W. Hackbusch. On the efficient computation of high-dimensional integrals and\nthe approximation by exponential sums. In A. Kunoth R. DeVore, editor, Multiscale,\nNonlinear and Adaptive Approximation , volume 56, pages 39 – 74. Springer Verlag, 2009.\n[3] H. Brunner. Collocation methods for Volterra integral and related functional differential equa-\ntions , volume 15 of Cambridge Monographs on Applied and Computational Mathematics .\nCambridge University Press, Cambridge, 2004.\n[4] H. Brunner and P. J. van der Houwen. The numerical solution of Volterra equations , volume 3\nofCWI Monographs . North-Holland Publishing Co., Amsterdam, 1986.\n[5] D. Cˆ amara De Souza, M. Craig, T. Cassidy, J. Li, F. Nekka, J. B´ elair, and A.R. Humphries.\nTransit and lifespan in neutrophil production: implications for drug intervention. Journal\nof Pharmacokinetics and Pharmacodynamics , 45(1):59 – 77, 2018.\n[6] N. Guglielmi and E. Hairer. Implementing Radau IIA methods for stiff delay differential equa-\ntions. Computing , 67(1):1–12, 2001.\n[7] N. Guglielmi and E. Hairer. Computing breaking points in implicit delay differential equations.\nAdv. Comput. Math. , 29(3):229–247, 2008.\n[8] E. Hairer and P. Maass. Numerical methods for singular nonlinear integro-differential equations.\nAppl. Numer. Math. , 3(3):243–256, 1987.\n[9] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-\nAlgebraic Problems . Springer Series in Computational Mathematics 14. Springer-Verlag,\nBerlin, 2nd edition, 1996.\n[10] W. Krzyzanski. Ordinary differential equation approximation of gamma distributed delay\nmodel. Journal of Pharmacokinetics and Pharmacodynamics , 46(1):53 – 63, 2019.\n[11] W. Krzyzanski, S. Hu, and M. Dunlavey. Evaluation of performance of distributed delay model\nfor chemotherapy-induced myelosuppression. Journal of Pharmacokinetics and Pharma-\ncodynamics , 45(2):329 – 337, 2018.\n[12] M. L´ opez-Fern´ andez, C. Lubich, and A. Sch¨ adle. Adaptive, fast, and oblivious convolution in\nevolution equations with memory. SIAM J. Sci. Comput. , 30(2):1015–1037, 2008.\n[13] C. Lubich. Discretized fractional calculus. SIAM J. Math. Anal. , 17(3):704–719, 1986.\n[14] A. Sch¨ adle, M. L´ opez-Fern´ andez, and C. Lubich. Fast and oblivious convolution quadrature.\nSIAM J. Sci. Comput. , 28(2):421–438, 2006.\n[15] L.N. Trefethen and J.A.C. Weideman. The Exponentially Convergent Trapezoidal Rule. SIAM\nReview , 56(3):385–458, 2014.\n[16] C.A. Wesolowski, S.N. Wanasundara, P.S. Babyn, and J. Alcorn. Comparison of the gamma-\npareto convolution with conventional methods of characterising metformin pharmacokinet-\nics in dogs. J. Pharmacokinet. Pharmacodyn. , 47:19–45, 2020."    },    {        "title": "2401.11273v1.Einstein_Gauss_Bonnet_Cosmological_Theories_at_Reheating_and_at_the_End_of_the_Inflationary_Era.pdf",        "content": "arXiv:2401.11273v1  [gr-qc]  20 Jan 2024Einstein-Gauss-Bonnet Cosmological Theories at Reheatin g and at the End of the\nInflationary Era\nV.K. Oikonomou1,2,∗Pyotr Tsyba2,†and Olga Razina2‡\n1)Department of Physics, Aristotle University of Thessaloni ki, Thessaloniki 54124, Greece and\n2)L.N. Gumilyov Eurasian National University - Astana, 01000 8, Kazakhstan\nIn this work we study the GW170817-compatible Einstein-Gau ss-Bonnet theories during the re-\nheating and the end of inflationary era. Given the scalar field potential V(φ) which can have some\nintrinsic importance for the theory, determining the scala r coupling function ξ(φ) can be cumber-\nsome due to lack of analyticity. The GW170817 observation co nstrains the scalar coupling function\nand the scalar field potential to have some interdependence, thus during the slow-roll era one can\ncalculate the scalar coupling function. However, when the s low-roll era ends, it is expected that the\nscalar coupling function should have a different form and the same applies for the reheating era,\nassuming that the scalar potential of the theory does not cha nge. In this work we exactly aim to\nhighlight this feature of Einstein-Gauss-Bonnet theories , as the Universe evolves through distinct\nsequential evolution eras, and we focus on how to determine t he scalar coupling function during\nthe various evolutionary eras, from inflation to the reheati ng era. Regarding both the end of the\ninflationary era and the reheating era, it is found that the Hu bble rate obeys a constant-roll-like\ncondition of the form ˙H=δH2, thus the determination of the scalar Gauss-Bonnet functio nξ(φ)\nis reduced to solving a differential equation. A mentionable feature of the era exactly at the end\nof inflation is that the Klein-Gordon equation is decoupled f rom the field equations, because the\nGauss-Bonnet invariant is zero. We provide several example s of interest to support our arguments.\nPACS numbers: 04.50.Kd, 95.36.+x, 98.80.-k, 98.80.Cq,11. 25.-w\nI. INTRODUCTION\nUndoubtedly, during the next decade inflationary theories [1–4] will be in the mainstream of interest since they will\nbe directly tested either by stage four Cosmic MicrowaveBackgrou ndexperiments [5, 6], or indirectly via gravitational\nwaveexperiments[7–15]whichwillcapturethestochasticimprintof inflation. Amongthecandidatetheories, Einstein-\nGauss-Bonnet theories [16–53] serve as an important class of the ories, which in some cases predict a blue-tilted\ntensor spectrum and also are theoretically motivated since these c ontain string corrections. The striking GW170817\nobservation back in 2017 altered our perception for several cand idate inflationary theories, including the Einstein-\nGauss-Bonnet theories. The reason was the fact that the kilonov a GW170817 event indicated that the gravitational\nwave speed is almost equal to light speed. Einstein-Gauss-Bonnet t heories, and other similar string corrected theories,\npredict a propagation speed for tensor perturbations that are n ot equal to that of light, thus these classes of theories\nwere put into existential peril, see [54–57] for a discussion on which t heories contradict GW170817. However, if\none requires that the propagation speed of tensor perturbation s is equal to that of light, the theoretical issues of\nEinstein-Gauss-Bonnet gravity may be refined, and this was done in a series of articles [58–60]. In these theories, the\nGauss-Bonnet scalar coupling function ξ(φ) satisfies a differential equation ¨ξ=H˙ξand in effect the scalar potential\nand the scalar Gauss-Bonnet coupling are connected and are not f ree to choose. The formalism developed in [58–60]\nleads to GW170817-compatibleand viable inflationary theories. Howe ver, given the scalar potential, the scalarGauss-\nBonnet coupling function can be found only semi-analytically, and only for the evolutionary patch of the Universe\nwhich generates a specific Hubble rate. Thus in principle, one cannot know the exact form of the scalar Gauss-Bonnet\ncoupling function, and only approximate forms of it may be found. In this article we aim to highlight this important\nfeature in Einstein-Gauss-Bonnet theories of gravity, focusing o n two eras that follow the inflationary era, the end of\ninflation era and the reheating era. In both these cases, the Hubb le rate and its time derivative satisfy a constant-\nroll-like relation ˙H=δH2, thus this constrains and simplifies the field equations. In effect, th e scalar Gauss-Bonnet\nfunction obeys a set of differential equations and may be obtained a nalytically for each evolutionary era, given the\n∗Electronic address: Corresponding author: voikonomou@ga pps.auth.gr;v.k.oikonomou1979@gmail.com\n†Electronic address: pyotrtsyba@gmail.com\n‡Electronic address: olvikraz@mail.ru2\nscalar potential. We use several illustrative examples, and we demon strate that indeed the scalar Gauss-Bonnet\ncoupling acquires distinct functional forms during the various distin ct evolutionary eras of our Universe. Also for a\nspecific viable inflationary case, we examine the behavior of the Gaus s-Bonnet term after inflation and during the\ninflationary era, and we show that for this interesting example, the effect of the Gauss-Bonnet coupling is minor, thus\nthe reheating era is mainly controlled by the scalar potential.\nII. A BRIEF OVERVIEW OF GW170817-COMPATIBLE EINSTEIN-GAUS S-BONNET FRAMEWORK\nLet us start our analysis by recalling how the Gauss-Bonnet scalar c oupling function is obtained during the infla-\ntionary era, by using the conditions required so that the gravitatio nal wave speed is equal to unity in natural units,\nand also the slow-roll conditions. We consider the Einstein-Gauss-B onnet gravitational action,\nS=/integraldisplay\nd4x√−g/parenleftbiggR\n2κ2−1\n2∂µφ∂µφ−V(φ)−1\n2ξ(φ)G/parenrightbigg\n, (1)\nwithRstanding for the Ricci scalar, κ=1\nMpwith being Mpthe reduced Planck mass, and also Gdenotes the Gauss-\nBonnet invariant in four dimensions, which is G=R2−4RαβRαβ+RαβγδRαβγδwhereRαβandRαβγδdenote the\nRicci and Riemann tensors respectively. For the rest of the article we shall assume that the geometric background is\na flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, with line element,\nds2=−dt2+a(t)23/summationdisplay\ni=1(dxi)2, (2)\nwherea(t) denotes as usual the scale factor, and for the FLRW backgroun d, the Gauss-Bonnet invariant becomes\nG= 24H2(˙H+H2), withHbeing the Hubble rate H=˙a\na. Notice that one requires that an inflationary era occurs,\none must have ˙H≪H2, however when inflation ends, at this time instance, we have that th e Gauss-Bonnet invariant\nis exactly zero, since ˙H=−H2. We shall consider this scenario later on in this article. Now, assuming that the scalar\nfield has only a time-dependence, upon varying the gravitational ac tion (1) with respect to the metric and the scalar\nfield, we obtain the field equations,\n3H2\nκ2=1\n2˙φ2+V+12˙ξH3, (3)\n2˙H\nκ2=−˙φ2+4¨ξH2+8˙ξH˙H−4˙ξH3, (4)\n¨φ+3H˙φ+V′+12ξ′H2(˙H+H2) = 0. (5)\nThe inflationary era is realized when ˙H≪H2and also we assume that the slow-roll conditions hold true for the\nscalar field, so in total we have the following conditions holding true du ring the inflationary era,\n˙H≪H2,˙φ2\n2≪V,¨φ≪3H˙φ. (6)\nNow the GW170817 event imposed serious constraints on the gravit ational wave speed of inflationary theories. The\npropagation speed of the tensor perturbations of a flat FLRW met ric has the form,\nc2\nT= 1−Qf\n2Qt, (7)\nwith the functions Qf,FandQbdefined above for Einstein-Gauss-Bonnet theories being equal to Qf= 8(¨ξ−H˙ξ),\nQt=F+Qb\n2,F=1\nκ2andQb=−8˙ξH. Hence in order to have exactly c2\nT= 1, we must have Qf= 0, which\nconstrains the Gauss-Bonnet scalar coupling function to obey the differential equation ¨ξ=H˙ξ. We can express this\ndifferential equation in terms of the scalar field as follows,\nξ′′˙φ2+ξ′¨φ=Hξ′˙φ, (8)3\nwhere the “prime” denotes differentiation with respect to the scala r field, and also we used ˙ξ=ξ′˙φandd\ndt=˙φd\ndφ.\nBy making the assumption,\nξ′¨φ≪ξ′′˙φ2, (9)\nwhich is actually motivated by the slow-roll conditions obeyed by the s calar field, Eq. (8) becomes,\n˙φ≃Hξ′\nξ′′. (10)\nThen Eqs. (5) and (10) yield,\nξ′\nξ′′≃ −1\n3H2/parenleftbig\nV′+12ξ′H4/parenrightbig\n. (11)\nClearly this is a differential equation that relates the scalarfield pote ntialV(φ) with the scalar Gauss-Bonnet function\nξ(φ), used in the previous literature [58–60]. This can be further simplifie d, by assuming that,\nκξ′\nξ′′≪1, (12)\nand furthermore the following assumptions,\n12˙ξH3= 12ξ′2H4\nξ′′≪V , (13)\nwhich are related to the constraint of Eq. (12). By combining Eqs. ( 6), (10) and (13), we get the simplified form of\nthe field equations,\nH2≃κ2V\n3, (14)\n˙H≃ −1\n2κ2˙φ2, (15)\n˙φ≃Hξ′\nξ′′. (16)\nAlso, due to Eq. (14), the condition (13) acquires the simpler form,\n4κ4ξ′2V\n3ξ′′≪1. (17)\nFurthermore, the differential equation (11), takes the simpler fo rm,\nV′\nV2+4κ4\n3ξ′≃0, (18)\nand constrains basically the scalar potential and the scalar Gauss- Bonnet coupling during the inflationary era. It is\nvery important to further stress that the differential equation ( 18) holds true only during the inflationary era and\nalso by assuming a slow-rolling scalar field, so this is not a general relat ion between the scalar field potential and\nthe scalar Gauss-Bonnet coupling function. It is the aim of this work to find the relation of the scalar Gauss-Bonnet\ncoupling and the scalar potential, at the end of inflation and during th e reheating era. Note that, this difficulty arises\ndue to the difficulty to solve the field equations analytically, given the p otential, and perhaps the Hubble rate. But\neven if these last two were given, one would still have to determine th e scalar Gauss-Bonnet coupling function during\nthe various stages of evolution, for example during the inflationary era for a quasi-de Sitter evolution, during the\nreheating and so on, with only the differential equation ¨ξ=H˙ξholding true. The lack of analyticity forces us to\nbreak the problem to smaller problems, and thus this shows the motiv ation of this work. Now, having the differential\nequation (18) at hand, we can determine the scalar potential and fi nd the scalar Gauss-Bonnet coupling function. We4\nshall consider three potentials that may lead to analytic results in th e case of reheating era and the end of inflation\nera, two of which we present here. Consider the following scalar pot ential,\nV(φ) =βexp(κλφ), (19)\nwhereβis a constant with mass dimensions [ M]4. By solving the differential equation (18), we obtain the following\nscalar Gauss-Bonnet function,\nξ(φ) =3e−κλφ\n4βκ4. (20)\nAlso consider the following scalar potential,\nV(φ) =λκφn, (21)\nwhereλin this case is a constant with mass dimensions [ M]5−n. By solving the differential equation (18), we obtain\nthe following scalar Gauss-Bonnet function,\nξ(φ) =3φ−n\n4κ5λ. (22)\nWe shall compare these solutions for the Gauss-Bonnet scalar cou pling function ξ(φ) to the solutions that will be\nobtained from the end of inflation era and from the reheating era.\nIII. EINSTEIN-GAUSS-BONNET GRAVITY DURING THE REHEATING E RA AND AT THE END OF\nINFLATION\nHaving discussed the inflationary era of the Einstein-Gauss-Bonne t gravity, let us now consider the reheating era\nand the end of inflation era. In the reheating era, the scale factor of the Universe is a(t)∼t1/2, thus, the Hubble rate\nand its time derivative satisfy a constant-roll like relation of the for m˙H=βH2, and specifically,\n˙H=−2H2, (23)\nso by taking this into account and also the fact that ¨ξ=H˙ξ, the field equations (3) and (4) yield the following\nconstraint for the scalar field,\n˙φ2= 2V . (24)\nNow, in ordinary scalar field theory in the absence of Gauss-Bonnet corrections, the field equation (5) combined with\nthe constraint (24) would yield an exponential potential for the sc alar field. However, due to the presence of the\nGauss-Bonnet corrections, the constraint (24) can be inserted and used in the field equation for the scalar field, thus\none obtains the following differential equation for the Gauss-Bonne t scalar coupling function,\n2V′+3H˙φ−12ξ′H4= 0. (25)\nFrom the Friedmann equation, by using the constraint (24), we get ,\nH˙φ==3H2\nκ2−2V\n12ξ′H2, (26)\nthus combining Eq. (25) and the Friedmann equation we get,\n3H3\nκ2−7HV=−V′√\n2V , (27)\nwhere we used that ˙φ=−√\n2Vbecause the scalar field is expected to decrease as the time evolves . The algebraic\nequation (27) can be solved with respect to the Hubble rate and yield s the following real solution,\nH(V,V′) =73√\n2κ2V(φ)\n33/radicalBig√\n2/radicalbig\n81κ4V(φ)V′(φ)2−686κ6V(φ)3−9√\n2κ2/radicalbig\nV(φ)V′(φ)(28)5\n0.0 0.5 1.0 1.5 2.0-800-600-400-2000\nϕ\nMpξ'(ϕ)\n1010\nFIG. 1: The behavior of the scalar Gauss-Bonnet coupling fun ctionξ′(φ) during the reheating era for the scalar potential (37).\n+3/radicalBig√\n2/radicalbig\n81κ4V(φ)V′(φ)2−686κ6V(φ)3−9√\n2κ2/radicalbig\nV(φ)V′(φ)\n33√\n2.\nUsing Eqs. (28), (25) and also the fact that ˙φ=−√\n2V, one obtains a differential equation for the Gauss-Bonnet\nscalar coupling function ξ(φ), which is the following,\n2V′−3H(V,V′)√\n2V−12ξ′H(V,V′)4= 0, (29)\nso given the scalar potential, the above can yield the function ξ(φ) during the reheating era by solving the differential\nequation, or simply find ξ′(φ) by solving the algebraic equation (29) with respect to ξ′(φ). Now let us investigate\nthe solutions for the function ξ(φ), for the potentials used in the previous section, and we compare t he new solutions\nforξ(φ) during the reheating era and the inflationary era. Let us start wit h the exponential potential of Eq. (19) in\nwhich case the scalar Gauss-Bonnet function for a slow-roll inflatio nary era yields the solution (20). If we plug the\nscalar potential (19) in Eq. (28) we get,\nH(φ) =Aeκλφ\n2, (30)\nwhereAstands for,\nA=22/3/parenleftBig/radicalbig\nβ3κ6(81λ2−686)−9β3/2κ3λ/parenrightBig2/3\n+14βκ2\n3 25/63/radicalBig/radicalbig\nβ3κ6(81λ2−686)−9β3/2κ3λ. (31)\nUsing the solution (30) and plugging it in the differential equation (29) we get,\nξ(φ) =e−κλφ/parenleftbig\n3√\n2A√β−2βκλ/parenrightbig\n12A4κλ. (32)\nApparently, this is a similar solution to that of Eq. (20) but this is an ac cidental feature because of the simplicity\nof the model. Thus in this case, the scalar coupling function during th e inflationary era is a rescaled function of the\nreheating era scalar coupling function. This feature is model depen dent though, as we now show. Let us assume that\nthe scalar potential has the form (21), and in the inflationary era t he scalar Gauss-Bonnet coupling function is given\nby relation (22). Now during the reheating era, by plugging the pote ntial (21) in Eq. (28) we get,\nH(φ) =14κ3λφn+22/3S(φ)2/3\n3 25/63/radicalbig\nS(φ), (33)6\nwhere the function S(φ) is defined as follows,\nS(φ) =φ/radicalbig\nκ7λ3φ3n−2(81n2−686κ2φ2)−9κ2n(κλφn)3/2\nφ. (34)\nUsing the solution (33) and plugging it in the differential equation (29) we get,\nξ′(φ) =a1(φ)S(φ)2/3+a2(φ)3/radicalbig\nS(φ)+a5(φ)\na3(φ)S(φ)2/3+a4(φ)3/radicalbig\nS(φ)+a6(φ), (35)\nwhere the functions Q(φ),a1(φ),a2(φ),a3(φ),a4(φ),a5(φ) anda6(φ) are defined as follows,\nQ(φ) =/radicalbig\n−κ7λ3φ3n−2(686κ2φ2−81n2), (36)\na1(φ) = 9 22/3√κ√\nλφ2−n\n2Q(φ)−81 22/3κ4λ2nφn+1,\na2(φ) = 1623√\n2κ9/2λ5/2n2φ3n\n2−183√\n2κλnφQ(φ),\na3(φ) =−108 22/3κ7λ3n2φ2n−1372 22/3κ9λ3φ2n+2+12 22/3κ7/2λ3/2nφn\n2+1Q(φ),\na4(φ) = 35283√\n2κ9λ3nφ2n+1−3923√\n2κ6λ2φn+2Q(φ),\na5(φ) = 126κ7/2λ3/2φn\n2+2Q(φ)−1134κ7λ3nφ2n+1,\na6(φ) =−6048κ10λ4n2φ3n+19208κ12λ4φ3n+2+672κ13/2λ5/2nφ3n\n2+1Q(φ).\nHence it is obvious that the scalar Gauss-Bonnet function during th e reheating and the slow-roll era for the scalar\npotential (21) are quite distinct. The two models we presented are just demonstrational potentials and will not be\nfurther studied. However, an interesting question is the following: suppose we started with a specific potential and\nthe corresponding inflationary scalar Gauss-Bonnet function whic h both generate a viable inflationary era, like for\nexample the following two which were developed in Ref. [58–60],\nV(φ) =3\n4βκν+4φν+3γκ4, (37)\nwhereβandγare two dimensionless constants. By solving the differential equatio n (18), we obtain the following\nscalar Gauss-Bonnet function,\nξ(φ) =βκνφν, (38)\nwhat isthe behaviorofthe Gauss-Bonnetscalarcouplingfunction d uringthe reheatingera? This will indicatewhether\nthe Gauss-Bonnet string corrections affect the reheating era, a t least in this case. This can easily be examined by\nusing the potential (37) and plugging it in Eq. (28) and we get,\nH(φ) =7κ2Q(φ)+/parenleftbig\n18√\n2βνκν+6φν−1Q(φ)5/2+S(φ)/parenrightbig2/3\n√\n33/radicalBig\n18√\n2βνκν+6φν−1Q(φ)5/2+S(φ), (39)\nwhere,\nS(φ) =/radicalBigg\n648β2ν2κ2ν+6φ2ν−2−343κ8(4βκνφν+3γ)2\nκ14(4βκνφν+3γ)5, (40)\nQ(φ) =1\nκ4(4βκνφν+3γ).\nUsing the solution (39) and plugging it in the differential equation (29) we get,\nξ′(φ) =−3R(φ)/parenleftbig\n18√\n2βνκν+6φν−1Q(φ)5/2+S(φ)/parenrightbig4/3\n4/parenleftBig\n7κ2Q(φ)+/parenleftbig\n18√\n2βνκν+6φν−1Q(φ)5/2+S(φ)/parenrightbig2/3/parenrightBig4, (41)7\n-2 -1 0 1 2-6-4-2024\nϕ\nMpξ(ϕ)\n1010\nFIG. 2: The behavior of the scalar Gauss-Bonnet coupling fun ctionξ(φ) at exactly the end of the inflationary era for the scalar\npotential (37).\nwhereR(φ) is defined to be equal to,\nR(φ) = 24βνκν+4φν−1Q(φ)2+3√\n2/radicalbig\nQ(φ)/parenleftBig\n7κ2Q(φ)+/parenleftbig\n18√\n2βνκν+6φν−1Q(φ)5/2+S(φ)/parenrightbig2/3/parenrightBig\n3/radicalBig\n18√\n2βνκν+6φν−1Q(φ)5/2+S(φ).(42)\nApparently, the scalar Gauss-Bonnet function during the reheat ing and the slow-roll era for the scalar potential (37)\nare quite distinct, so let us see how the scalar Gauss-Bonnet funct ion behaves during the reheating era. A set of\nthe parameter values that yield a viable inflationary era are ( β,γ,ν) = (1.2×1010,1000,15), and for these values\nin Fig. 1 we plot ξ′(φ) as a function of the scalar field. As time evolves the scalar field value s decrease, thus the\nreheating era is approached as the scalar field values tend to zero. As it can be seen, the derivative of the scalar\ncoupling function tends to zero, and therefore this means that ξ(φ)∼const. Therefore, the Gauss-Bonnet term in\nthe Lagrangian is effectively ∼ Gthus the action integral of this term is just a surface term, and th erefore for this\nmodel, the Gauss-Bonnet correction terms play no role in the rehea ting era, and therefore the reheating era is solely\ndetermined by the behavior of the scalar field potential.\nHaving discussed the behavior of the scalar Gauss-Bonnet coupling function during the reheating era, now we shall\nconsider the end of inflation era, that is, the exact time instance th at inflation ends. In this case, the total effective\nequation of state parameter is w=−1\n3thus we have ˙H=−H2which is again a constant-roll behavior. In ordinary\nscalar field theory, this would result to an exponential potential fo r the scalar field, but in the Einstein-Gauss-Bonnet\ncase, this constraints the scalar Gauss-Bonnet function as we no w evince. By taking into account the fact that\n˙H=−H2and also the fact that ¨ξ=H˙ξ, the field equations (3) and (4) yield the following constraint for the scalar\nfield,\n˙φ2=V . (43)\nAs we mentioned, in ordinary scalar field theory in the absence of Gau ss-Bonnet corrections, the field equation (5)\ncombined with the constraint (43) would yield an exponential potent ial for the scalar field but in our case, it will\nsimply constrain the scalar Gauss-Bonnet function. From Eq. (43) we have ¨φ=V′\n2. Now, due to the fact that\n˙H=−H2, the scalar field equation of motion becomes,\n¨φ+3H˙φ+V′= 0, (44)\nso by substituting ¨φ=V′\n2and Eq. (43) and taking the square, we have,\nH2=/parenleftbiggV′\n2V/parenrightbigg2\n, (45)8\nso by substituting this in the Friedmann equation (3) we finally have,\n3\nκ2/parenleftbiggV′\n2V/parenrightbigg2\n=3V\n2−6ξ′/parenleftbiggV′\n2V/parenrightbigg2\nV′, (46)\nwhich is a differential equation that, given the scalar potential, can b e solved with respect to ξ(φ), or it can be viewed\nas an algebraic equation that can yield ξ′(φ). Now let us consider the examples we gave in the case of the reheat ing\nera, and we compare the solutions for the scalar Gauss-Bonnet co uplings. Consider first the scalar potential of Eq.\n(19), so plugging this exponential potential in the differential equa tion (46), we easily get the following solution for\nξ(φ),\nξ(φ) =λe−κλφ\nβκ+2φ\n2κ3λ3, (47)\nwhich is quite different from the expression of Eq. (20). Accordingly , for the potential of Eq. (21), the scalar\nGauss-Bonnet coupling function at the end of the inflationary era is ,\nξ(φ) =φ4\n4n3−φ2−n\n2κ3λ(2−n)n, (48)\nwhich is quite distinct from that of Eq. (22). Finally, for the potentia l of Eq. (37) which recall that it generates a\nviable inflationary era, the solution of the differential equation (46) is,\nξ(φ) =φ2κ−3ν/parenleftBig\n128β4ν2κ4ν+2φν\nν+2+162βγ2κνφ2−2ν\nν−2+81γ3φ2−3ν\n3ν−4/parenrightBig\n192β3ν3(49)\n+φ2\n192β3ν3κ−3ν/parenleftBigg\n−72β2γκ2νφ−ν/parenleftbig\nγκ2(ν−4)ν2−6(ν−2)φ2/parenrightbig\n(ν−4)(ν−2)−48β3κ3ν/parenleftbig\nφ2−2γκ2ν2/parenrightbig/parenrightBigg\n,\nwhich is of course quite different from that of Eq. (38). Now for this last solution it is worth to investigate the\nbehavior of the scalar coupling function as a function of the scalar fi eld. To this end, for the same set of parameter\nvalues used in the reheating era case, that is, for ( β,γ,ν) = (1.2×1010,1000,15), which recall that these yield a\nviable inflationary era, in Fig. 2 we plot ξ(φ) as a function of the scalar field. In this case too, as time evolves, t he\nscalar field values decrease, hence we focus on values of the scalar field in the sub-Planck region. As it can be seen,\nthe scalar Gauss-Bonnet coupling function is nearly zero for sub-P lanck values of the scalar field, but for nearly zero\nvalues, the scalar Gauss-Bonnet coupling function blows up. Howev er, it is not certain that the scalar field actually\ntook such values at the end of inflation, so considering that the sca lar field values at the end of inflation are of the\norder of the Planck mass, this means that the scalar Gauss-Bonne t coupling function is nearly zero, thus it does not\naffect the evolution at the end of inflation. This is an interesting feat ure, combined with the one we obtained in the\nreheating era case developed previously.\nIn conclusion, in this section we demonstrated that given the scalar potential, it is actually difficult to have an\nanalytic form of the scalar coupling Gauss-Bonnet function, if this is assumed that it realizes the various evolutionary\npatches of our Universe. Thus we presented the formalism on how t o find the analytic form of ξ(φ) or its derivative\nξ(φ), during the inflationary era, the reheating era and the end of infla tion era. This may yield important information\nfor the reheating era, and to which extent it affects the reheating and subsequent radiation domination era. This\nformalism can also be applied in other modified gravities, so we defer th is task for future works.\nConclusions\nIn this work we aimed to demonstrate that the scalar Gauss-Bonne t coupling function can be found analytically in\ndistinct evolutionary epochs of the Universe, and that in general it has distinct approximate forms for these epochs.\nWe considered Einstein-Gauss-Bonnet gravities, which are compat ible with the GW170817 event. We demonstrated\nthat during the inflationary era, the scalar Gauss-Bonnet coupling and the scalar potential are connected and satisfy\na specific differential equation, which yields the approximate form of the scalar Gauss-Bonnet coupling during the\nslow-rollera. During the end ofinflation era, the Hubble rateand its time derivativesatisfy a constant-rollevolutionof\nthe form ˙H=−H2and thus the scalar potential and the scalar Gauss-Bonnet couplin g satisfy a different differential\nequation which provides us with a different solution for the scalar Gau ss-Bonnet coupling. Furthermore, during9\nthe reheating era, the Hubble rate and its time derivative satisfy an alternative constant-roll evolution of the form\n˙H=−2H2and therefore the scalar potential and the scalar Gauss-Bonnet coupling satisfy another distinct from\nthe other two differential equation, which provides us with a differen t solution for the scalar Gauss-Bonnet coupling.\nWe presented several illustrative and easy to study analytically exa mples, and we showed explicitly that the scalar\nGauss-Bonnet coupling indeed is different during the various evolutio nary epochs we considered. We also presented a\nmodel which generates a viable inflationary era, and we showed that the Gauss-Bonnet coupling tends to zero at the\nend of inflation and during the reheating era. Thus for this example, the Gauss-Bonnet term has a minimal effect on\nthe reheating era, and therefore the latter is affected solely by th e scalar potential. The formalism we presented in\nthis paper can also be used in other modified gravity frameworks and we aim to study such scenarios in future works.\nFinally, the inflationary framework of theories at hand can produce a blue-tilted tensor spectrum, and this could\nbe valuable in view of the latest NANOGrav observation in 2023 [61], alth ough an extremely high blue-tilted tensor\nspectral index would be needed [62]. 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In three-dimensional topo-\nlogical insulators, for instance, the axion coupling is of the order of the fine structure constant, and hence a\nperturbative analysis of the field equations is plenty justified. In this paper we use Green’s function techniques\nto obtain time-dependent solutions to the axion field equations in the presence of a planar domain-wall separat-\ning two media with different topological order. We apply our results to investigate the radiation of a short linear\nantenna near the domain-wall.\nI. INTRODUCTION\nTopological materials are not distinguished by local ordered parameters, but by topological quantum numbers which could be\ndescribed and computed using topological field theories in the low energy limit. The best studied of these are the topological\ninsulators (TIs), where the corresponding bulk band gap is similar to an ordinary insulator, but the boundary of the material hosts\ngapless states which are protected by the topological numbers of the bulk geometry [1]. The experimental discovery of TIs in\nHgTe quantum wells [2] and stoichiometric crystals [3–5] encourages further investigations regarding its exotic properties.\nApart from their spectroscopic distinguishing features, TIs have unconventional electromagnetic response described by ax-\nion electrodynamics, which is defined by the standard Maxwell Lagrangian plus the additional θ-term (in SI units) Lθ=\n(α/π)p\nϵ0/µ0θE·B, where EandBare the electromagnetic fields, αis the fine structure constant, and θis an angular\nvariable known in particle physics as the axion field or axion angle [6]. The axion field was originally proposed to solve the\nstrong-CP problem in QCD [7–9], but its weak interaction strength with standard model particles and an elegant production\nmechanism in the early universe make it a promising candidate to explain dark matter as well [10–12]. In the context of TIs,\ntime-reversal (TR) symmetry indicates that θ= 0, π(mod 2π), and hence the θ-term has no effect on Maxwell equations in the\nbulk. The nontrivial topological property, a surface half-integer quantum Hall effect, manifests only at the boundary between\ntwo manifolds with different topologies, in which case θis not globally constant. Therefore, from the macroscopic point of view,\nthe axion angle θcan be viewed as an effective parameter characterizing the media, in the same footing as its permittivity ϵand\npermeability µ.\nTheθ-term encodes the most salient feature of the electromagnetic response of TIs: the topological magnetoelectric effect,\nwhere an electric field can induce a magnetic polarization and a magnetic field can induce an electric polarization [13]. This\nis why, in the condensed matter literature, the axion angle is termed the topological magnetoelectric polarization (TMEP). A\nnumber of topological magnetoelectric effects have been predicted on the basis of this theory, namely, the appearance of charge\nand current densities of magnetic monopoles [14–21], the magneto-optical Faraday and Kerr rotation [22–26], the Casimir effect\n[27–33], as well as the influence of TIs on atomic systems nearby [34–39].\nTime-independent solutions to the axion field equations have been elaborated by using the electric-magnetic duality of Z2\nTIs [15] as well as Green’s function techniques [17, 19]. However, general solutions to the time-dependent case are still under\ninvestigation [40–42]. In this work we aim to fill in this gap by computing, in a perturbative fashion, solutions to the time-\ndependent axion field equations in the presence of a planar domain wall separating two media with different topologies. The\nvalidity of a perturbative scheme for the solution of the field equations is as follows. In the high-energy context the validity is\nplenty justified: the axion-to-two- photon coupling gθγγwhich converts photons into axions is expected to be very small [43]. In\nthe condensed matter framework, the smallness of the θ-term is controlled by the coupling αθ, which is small since α≈1/137\nandθ=πfor a TI with a single Dirac fermion on its surface. In the hypothetical situation of a TI with a large number of Dirac\nfermions on its surface, our perturbative approach ceases to be valid as αθapproaches unity. In view that exact solutions to the\nstatic axion field equations have been reported in the literature, we shall first validate our perturbative scheme by reproducing\nwell-known results in this case. Next we will safely extend our analysis to the time-dependent case. Finally, we apply our results\nto study the radiation of a short linear antenna near the domain-wall.\nThe paper is organized as follows. In Sec. II we briefly review the basics of axion electrodynamics. In Sec. III we solve,\nin a perturbative fashion, the static field equations and verify its consistency with previously reported results for the Green’s\nfunction for two ponderable topological insulators separated by a planar interface. Section IV is devoted to the evaluation of the\n∗martin i@ciencias.unam.mx\n†alberto.martin@nucleares.unam.mxarXiv:2401.11285v1  [cond-mat.mes-hall]  20 Jan 20242\ntime-dependent Green’s function by assuming equal optical properties in the two media. This configuration allow us to isolate\nthe topological effects. The details of technical calculations are relegated to the appendices. Finally, in Sec. V we apply our\nresult to investigate the radiation of a short linear antenna near a θdomain-wall.\nII. TOPOLOGICAL FIELD THEORY\nThe effective field theory governing the electromagnetic response of topological insulators, independently of the microscopic\ndetails, is defined by the action (in SI units) [44]:\nS=Z\nd4x\u00141\n2\u0012\nϵE2−1\nµB2\u0013\n+α\nπrϵ0\nµ0θE·B\u0015\n, (1)\nwhere α=e2/(4πϵ0ℏc)≃1/137is the fine structure constant, ϵandµare the permittivity and permeability of the material,\nrespectively, and θis the topological magnetoelectric polarizability (axion field). Under periodic boundary conditions, the\npartition function and all physical quantities are invariant under shifts of θby any multiple of 2π. Since E·Bis odd under\nTR, the only symmetry allowed values of θare0orπ(modulo 2π). This leads to the Z2classification of three-dimensional TR\ninvariant TIs. Clearly, the θ-term in Eq. (1) has no effect on Maxwell equations in the bulk, since it is a total derivative term.\nThe nontrivial topological property, a surface half-integer quantum Hall effect, manifests only when a TR-breaking perturbation\nis induced on the surface to gap the surface states, thereby converting the material into a full insulator. This can be achieved by\nintroducing magnetic dopants to the surface [45] or by the application of an external static magnetic field [46]. In this situation,\nθis quantized in odd integer values of π, where the magnitude and sign of the multiple is controlled by the strength and direction\nof the TR-breaking perturbation.\nThe electromagnetic response of a system in the presence of the axionic term is still described by the ordinary Maxwell\nequations, but the constituent relations which define the electric displacement Dand the magnetic field Hacquire an extra term\nproportional to θ, i.e.D=ϵE+α(θ/π)p\nϵ0/µ0BandH=µ−1B−α(θ/π)p\nϵ0/µ0E. Here, the θ-dependent terms,\nwhich arise from the axionic term in Eq. (1), describe the topological magnetoelectric effect. The frequency components of\nthe electromagnetic fields (introduced as usual: by making a Fourier decomposition in time of the sources and fields) obey the\ninhomogeneous equations\n∇ ·[ϵ(r, ω)E(r, ω)] =ρ(r, ω)−α\nπrϵ0\nµ0∇θ(r)·B(r, ω), (2)\n∇ ×\u0002\nµ−1(r, ω)B(r, ω)\u0003\n+iωϵ(r, ω)E(r, ω) =J(r, ω) +α\nπrϵ0\nµ0∇θ(r)×E(r, ω), (3)\nwhile the homogeneous equations are untouched due to gauge invariance of the action (1):\n∇ ·B(r, ω) = 0 , (4)\n∇ ×E(r, ω) =iωB(r, ω). (5)\nOne can further verify that the divergence equations (2) and (4) are subsumed within the curl equations (3) and (5), which can\nbe verified by taking the divergence of the latters. We observe that equations (2)-(3) depend only on the space gradient of\nthe axion field. So, the θ-term does not modify the field equations in the bulk of topological insulators. However, for a TI in\ncontact with a conventional insulator, as shown in Fig. 1, the axion field is piecewise constant across the interface Σand then\n∇θ(r) = ( θ1−θ2)δ(Σ)ˆn, where ˆnis the outward unit normal to Σ. These are the equations we aim to solve in a perturbative\nfashion along this paper. Our general approach will be to first Taylor expand the electromagnetic potentials in powers of the fine\nstructure constant and then solve recursively the resulting equations. Before embarking to the solution of the time-dependent\nfield equations, we first tackle the time-independent case and demonstrate consistency of our results with previously reported\nones.\nIII. SOLUTION TO THE STATIC AXION FIELD EQUATIONS\nLet us consider the geometry shown in Fig. 1 with the interface Σbeing the plane z= 0. The half-space z <0is occupied by\na topological medium with dielectric constant ϵ2, magnetic permeability µ2and axion angle θ2, whereas the half-space z >0is\noccupied by a medium with dielectric constant ϵ1, magnetic permeability µ1and axion angle θ1. Then the inhomogeneity in the\npermittivity, permeability and the axion field are limited to a finite discontinuity across the planar surface z= 0.3\nFIG. 1. Geometry of two semi-infinite topologically insulating media separated by the surface Σ. The media have different magnetoelectric\npolarizabilities, permittivities and permeabilities, as depicted.\nTo proceed further we write the field equations (2) and (3) in terms of the electrostatic potential ϕ(r)and vector potential\nA(r). In the Coulomb gauge, the static electromagnetic potentials satisfy\n−∇ · [ϵ(r)∇ϕ(r)] =ρ(r)−˜αrϵ0\nµ0δ(z)ˆez· ∇ × A(r), (6)\n∇ ×\u0002\nµ−1(r)∇ ×A(r)\u0003\n=J(r)−˜αrϵ0\nµ0δ(z)ˆez× ∇ϕ(r), (7)\nwhere ˜α=α(θ1−θ2)/π. To obtain a perturbative solution for the electromagnetic potentials in the presence of arbitrary external\nsources, we introduce the expansions ϕ(r) =ϕ(0)(r) +P∞\nn=1ϕ(n)(r)andA(r) =A(0)(r) +P∞\nn=1A(n)(r), where ϕ(n)and\nA(n)(r)are understood to vanish as ˜αn. We now substitute the power expansion into equations (6) and (7), and comparing\ncoefficients of each power of ˜αresults in an infinite series of simultaneous equations. The zeroth-order equations are simply the\nMaxwell equations in the absence of the θ-term,\n−∇ ·h\nϵ(r)∇ϕ(0)(r)i\n=ρ(r), ∇ ×h\nµ−1(r)∇ ×A(0)(r)i\n=J(r). (8)\nThen-order equations, which are the main concern in this work, are\n∇ ·h\nϵ(r)∇ϕ(n)(r)i\n= ˜αrϵ0\nµ0δ(z)ˆez· ∇ × A(n−1)(r),∇ ×h\nµ−1(r)∇ ×A(n)(r)i\n=−˜αrϵ0\nµ0δ(z)ˆez× ∇ϕ(n−1)(r).\n(9)\nThe process of solving these equations begins with the solution to the zeroth-order equations (8), which involves the well-\nknown electro- and magnetostatic Green’s functions. We now adopt the following notation. Since translational invariance is\nbroken in the zdirection due to the presence of the interface, it is natural decompose any vector Cinto components parallel and\nperpendicular to the normal direction ˆez, i.e.Cz=CzˆezandC⊥=Cxˆex+Cyˆey, respectively. So, the general solution for\nthe electrostatic potential is\nϕ(0)(r) =Z\nGϵ(r,r′)ρ(r′)d3r′, (10)\nwhere the electric Green’s function is given by\nGϵ(r,r′) =1\n4πϵ1\"\n1p\n|r⊥−r′⊥|2+ (z−z′)2+ϵ1−ϵ2\nϵ1+ϵ21p\n|r⊥−r′⊥|2+ (|z|+z′)2#\n(11)\nin which r⊥=xˆex+yˆeyandr′⊥=x′ˆex+y′ˆey[47]. The solution to the magnetostatic problem is a bit more subtle.\nDecomposing the current density and the vector potential into their components parallel and perpendicular to ˆez, i.e.J=J⊥+Jz\nandA=A⊥+Az, respectively, the general solution for the vector potential, in the half-space z >0, can be written as\nA(0)\nξ(r) =Z\nG(ξ)\nµ(r,r′)Jξ(r′)d3r′, ξ =⊥, z (12)4\nwhile for the region z < 0the solution is simply A(0)(r) =R\nG(⊥)\nµ(r,r′)J(r′)d3r′. The corresponding magnetic Green’s\nfunction is given by [47]\nG(ξ)\nµ(r,r′) =µ1\n4π\"\n1p\n|r⊥−r′⊥|2+ (z−z′)2+χµ2−µ1\nµ2+µ11p\n|r⊥−r′⊥|2+ (|z|+z′)2#\n, (13)\nwhere χ= +1 ( −1)forξ=⊥(z).\nWith all of the above we are ready to solve equations (9). This is a simple but not straightforward task, so we relegate the\ndetails to Appendix A. The final results, up to second order in the parameter ˜α, are\nϕ(r) =ϕ(0)(r) +Zh\nG(1)\n0i(r,r′)Ji(r′) +G(2)\n00(r,r′)ρ(r′)i\nd3r′, (14)\nAi(r) =A(0)\ni(r) +Zh\nG(1)\n0i(r,r′)ρ(r′) +G(2)\nij(r,r′)Jj(r′)i\nd3r′, (15)\nwhere the indices i, jrange over the transverse coordinates x, y, and\nG(1)\n0i(r,r′) =˜αp\nϵ0/µ0\n2π(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji(r⊥−r′\n⊥)j\n|r⊥−r′\n⊥|2\"\n1−|z|+z′\np\n|r⊥−r′\n⊥|2+ (|z|+z′)2#\n, (16)\nG(2)\n00(r,r′) =(˜αp\nϵ0/µ0)2\n2π(ϵ1+ϵ2)2(1/µ1+ 1/µ2)1p\n|r⊥−r′⊥|2+ (|z|+z′)2, (17)\nG(2)\nij(r,r′) =(˜αp\nϵ0/µ0)2\n2π(ϵ1+ϵ2)(1/µ1+ 1/µ2)2\"\nδijp\n|r⊥−r′⊥|2+ (|z|+z′)2+∂iKj(r,r′)#\n, (18)\nwith the definition\nKj(r,r′) =(r⊥−r′\n⊥)j\n|r⊥−r′\n⊥|2\u0012q\n|r⊥−r′\n⊥|2+ (|z|+z′)2−(|z|+z′)\u0013\n. (19)\nClearly, the electric (vector) potential is even (odd) under time-reversal operation, as expected.\nImage magnetic monopole effect\nAs a consistency check of our results, we shall verify the image magnetic monopole effect, which has been extensively\ndiscussed in the literature by using different approaches, for example, the method of images [14], the electric-magnetic duality\nof TIs [15] and Green’s function techniques [19, 20]. This elusive effect consists in the appearance of an image magnetic\nmonopole when an electric charge is placed at a certain distance from the TI’s surface.\nWe now consider for the charge density a pointlike electric charge qatz=z0. Equations (14) and (15), together with the\nGreen’s functions (16)-(18), give the electric and vector potentials. For z >0, the electric potential can be interpreted as due to\ntwo pointlike electric charges, one of strength qatz=z0, and the other, the image charge, of strength\nq′=q\"\nϵ1−ϵ2\nϵ1+ϵ2+2ϵ1(˜αp\nϵ0/µ0)2\n(ϵ1+ϵ2)2(1/µ1+ 1/µ2)#\n(20)\nat−z0. Forz <0only one pointlike electric charge appears, of strength q′+qlocated at z0. Unremarkably, the above expression\nreduce to the standard textbook one in the nontopological limit ( ˜α= 0) [47]. Due to the magnetoelectricy of TIs, a magnetic\nfield is induced. The corresponding vector potential is found to be\nA(r) =g\n4πxˆey−yˆex\nr2\n⊥\"\n1−|z|+z0p\nr2\n⊥+ (|z|+z0)2#\n, (21)\nwhich we identify as the vector potential of a straight vortex line or Dirac string over the zaxis. For z >0(z <0) the string is\nover the negative (positive) zaxis ending at a magnetic monopole at −z0(z0). In other words, the magnetic field for z >0can\nbe interpreted as that of a magnetic monopole of strength\ng=2q˜αp\nϵ0/µ0\n(ϵ1+ϵ2)(1/µ1+ 1/µ2)(22)\nat the image point −z0. Similarly, for z <0the magnetic field can be understood as originated by a monopole of strength −g\nlocated at z0. Our results are in a agreement with the ones reported in Refs. [14, 15, 19, 20]. This can be further verified by\npower expanding the image electric and magnetic charges up to order ˜α2, thus yielding our results (20) and (22), respectively.5\nIV . TIME-DEPENDENT SOLUTIONS TO THE AXION FIELD EQUATIONS\nIn this Section we consider the same configuration as in Sec. III, but we shall investigate now time-dependent solutions to the\nfield equations (2)-(5). In order to isolate the contribution arising from the surface Hall effect, we restrict ourselves to the case in\nwhich the permittivities are the same, i.e. ϵ1=ϵ2≡ϵ, but the topological magnetoelectric polarizability is a piecewise constant\nfunction: θ1forz >0andθ2forz <0. We assume the sources located in the region z >0.\nWe now introduce the field-potential relationships in the frequency domain, B(r, ω) =∇ ×A(r, ω)andE(r, ω) =\n−∇ϕ(r, ω) +iωA(r, ω). In the Lorentz gauge, ∇ ·A(r, ω) =iω\nv2ϕ(r, ω)withv= 1/√ϵµ, the electromagnetic potentials\nsatisfy\n−\u0000\n∇2+k2\n0\u0001\nϕ(r, ω) =ρ(r, ω)\nϵ−˜α\nϵrϵ0\nµ0δ(z)ˆez· ∇ × A(r, ω), (23)\n−\u0000\n∇2+k2\n0\u0001\nA(r, ω) =µJ(r, ω)−˜αµrϵ0\nµ0δ(z)ˆez×[∇ϕ(r, ω)−iωA(r, ω)], (24)\nwhere k0=ω/v. To solve this system of equations we follow the same perturbative program than before: we introduce the\nexpansions ϕ(r, ω) =ϕ(0)(r, ω) +P∞\nn=1ϕ(n)(r, ω)andA(r, ω) =A(0)(r, ω) +P∞\nn=1A(n)(r, ω), where ϕ(n)(r, ω)and\nA(n)(r, ω)vanish as ˜αn, and solve the system recursively. The zeroth-order equations are exactly Maxwell equations for ˜α= 0,\ni.e.\n−\u0000\n∇2+k2\n0\u0001\nϕ(0)(r, ω) =ρ(r, ω)\nϵ, (25)\n−\u0000\n∇2+k2\n0\u0001\nA(0)(r, ω) =µJ(r, ω), (26)\nwhile the n-order potentials are determined by\n−\u0000\n∇2+k2\n0\u0001\nϕ(n)(r, ω) =−˜α\nϵrϵ0\nµ0δ(z)ˆez· ∇ × A(n−1)(r, ω), (27)\n−\u0000\n∇2+k2\n0\u0001\nA(n)(r, ω) =−˜αµrϵ0\nµ0δ(z)ˆez×h\n∇ϕ(n−1)(r, ω)−iωA(n−1)(r, ω)i\n. (28)\nThe first step towards solving the higher-order equations (27)-(28) is to solve the zeroth-order equations (25)-(26). This requires\nthe well-known frequency-dependent Green’s function of electromagnetism [47]:\nGω(r,r′) =eik0|r−r′|\n4π|r−r′|, (29)\nsuch that\nϕ(0)(r, ω) =1\nϵZ\nGω(r,r′)ρ(r′, ω)d3r′,A(0)(r, ω) =µZ\nGω(r,r′)J(r′, ω)d3r′. (30)\nWith these results we can evaluate now equations (27) and (28). In this case we solve the system up to first order in the parameter\n˜α. As shown in the Appendix B 1, for the vector potential one gets:\nAi(r, ω) =A(0)\ni(r, ω) +Z\nG(1)\nij(r,r′)Jj(r′, ω)d3r′, (31)\nand for the scalar potential one spot the identity iω\nv2ϕ(r, ω) =∇ ·A(r, ω). Here,\nG(1)\nij(r,r′) =i1\nω\u0002\n(ˆez× ∇)i∂j−k2\n0ϵijz\u0003\nGω(r,r′), (32)\nwhere\nGω(r,r′) =−˜α\n8πµ\nϵrϵ0\nµ0Z∞\n0dk⊥ei√\nk2\n0−k2\n⊥(|z|+|z′|)\nk2\n0−k2\n⊥k⊥J0(k⊥|r⊥−r′\n⊥|). (33)\nA detailed derivation of this expression can be found in the Appendix B 2. It is worth to mention that the exact electromagnetic\npotentials (and hence the fields) must be calculated using Eq. (33), which unfortunately cannot be evaluated in an analytical6\nFIG. 2. Short linear antenna (of length 2dand frequency ω) at a distance z0from the planar interface between two topologically insulating\nmedia.\nfashion. However, for many practical calculations, one can obtain an approximated expression valid for |r⊥−r′\n⊥| ≫1and\n|z|+|z′| ≫1. In the Appendix B 3 we derive, by two different methods, the aforementioned expression. The approximation\nrelays in the fact that the integrand in Eq. (33) is a rapidly oscillating function for large |r⊥−r′\n⊥|and|z|+|z′|, and hence the\nleading contribution can be obtained by using, for example, the stationary phase approximation. The final result is\nGω(r,r′)≈i˜α\n8πµ\nϵrϵ0\nµ0eik0√\n|r⊥−r′\n⊥|2+(|z|+|z′|)2\nk0(|z|+|z′|). (34)\nOnce we have found the indexed Green’s function, we are ready to tackle problems involving different time-dependent sources\nnear the domain wall.\nV . RADIATION OF A SHORT LINEAR ANTENNA NEAR A θDOMAIN-WALL\nThe problem of a short linear antenna near a dielectric half-space has attracted great attention for over the years due to\nits importance to many practical applications, in particular, in the context of THz applications [48], near-field optics [49],\nplasmonics [50] and nanophotonics [51]. The exact solution to this problem was given first by Sommerfeld in 1909 [52]. Since\nthen numerous investigations have been carried out in this subject. The main goal of this Section is to address the problem of the\nradiation pattern produced by a short linear antenna placed near the boundary between two θ-half spaces. To this end, we shall\nemploy the theory developed in the previous Section.\nThe geometry of the problem is shown in Fig. 2. The left half-space z <0is occupied by a topological insulator with axion\nangle θ1, while the right half-space z >0is occupied by a TI with axion angle θ2. We consider a simple model for a transmitting\nantenna oriented parallel to the interface, say along the the x-axis and placed at a distance z0from the interface, with uniform\ncurrent flowing along a conductor of length 2d. We assume that this current varies sinusoidally with time as\nJ(r, t) =I0sin(ω0t)δ(y)δ(z−z0)Θ(d− |x|)ˆex, (35)\nwhere I0andω0denotes the current amplitude and angular frequency, respectively, the δ’s are Dirac deltas and Θis the Heaviside\nstep function. Because the current is discontinuous at the ends, charge must be deposited there, being of opposite sign at each\nend, thus forming an electric dipole. The exact zeroth-order scalar and vector potentials can be calculated in a simple fashion\nand can be found in a variety of textbooks on electrodynamics [47]. The only nonzero component of the vector potential (which\nis in the same direction as the current) in the time-domain is\nA(0)\nx(r, t) =µI0\n4πZ+d\n−ddx′1\n|r−r′|sin\u0014\nω0\u0012\nt−|r−r′|\nc\u0013\u0015\n, (36)\nwhere r′(x′) =x′ˆex+z0ˆez. The result for the scalar potential in the time-domain is\nϕ(0)(r, t) =−I0\n4πϵω0\ncosh\nω0\u0010\nt−|r−r+|\nc\u0011i\n|r−r+|−cosh\nω0\u0010\nt−|r−r−|\nc\u0011i\n|r−r−|\n, (37)\nwhere r±=r′(±d). One can easily verify that these potentials satisfy the Lorentz gauge condition.7\nFIG. 3. Illustration of cross-sections of the normalized electric field [panels (a) and (b)] and magnetic field [panels (c) and (d)] generated by a\nshort linear antenna in vacuum at z0.\nFIG. 4. Illustration of cross-sections of the normalized electric field [panels (a) and (b)] and magnetic field [panels (c) and (d)] generated by a\nshort linear antenna at a distance z0from a θ-domain wall at z= 0.\nIn order to evince the impact of the axion field, we first evaluate numerically the electromagnetic potentials and fields at first\norder in θby using Eq. (31) together with the exact Green’s function (32) with Gωgiven by Eq. (33). To this end we introduce the\ndimensionless quantities X=x/z0,Y=y/z0andZ=z/z0. Figure 3 shows the electric and magnetic fields in the case θ= 0,\ni.e. in the absence of the topological magnetoelectricity. Panels (a) and (b) show the electric field lines (in units of E0=µIv\n4πz0)\non the planes Y= 0 andX=d/Z0(where the end-points of the antenna are located) for ωt=π/4, respectively. They\nexhibit the expected behavior: dipole-like in panel (a) and almost point-like in panel (b).Panels (c) and (d) show the magnetic\nfield lines (in units of B0=E0/v) on the planes X=d/z0andY= 0 forωt=π/4, respectively. The former exhibits the\nusual current-encircling behavior dictated by the Amp `ere law, while the latter exhibits the expected results, whose filed lines\ninvert their direction periodically. In order to show the full electromagnetic fields, including the effects of the magnetoelectric\ncoupling, we have to introduce further dimensionless quantities besides the coordinates (X, Y, Z ), namely, T=t/τ,D=d/z0,\nΛ =ω0z0/candτ=z0/c. This is so because the appearance of an additional length scale in the system: the antenna-surface\ndistance z0. In Fig. 4 we show the total electromagnetic fields for θ=π, which is a typical value for topological insulators.\nWe observe that the cross-sections X= 0 exhibit the same behavior as in the case θ= 0, for both, the electric (panel (a)) and\nmagnetic (panel (c)) fields; however, as we move from X= 0 the field lines deviates from the nontopological case. Panels (b)\nand (d) show the Y= 0cross-section, in which case we observe a considerably different behavior, which cannot be interpreted in\na simple manner. Indeed, the charge and current densities sourcing these fields are far from trivial, but they can be understood in\nterms of the half-integer Hall current at the interface. The normalized Hall current jHall(in units of σHallE0, with σHall=e2/2h)\nis\njHall(r⊥, t) =ˆex∂YF(X, Y) +ˆey\u0002\n∂XF(X, Y)−Λ2G(X, Y)\u0003\n, (38)\nwhere\nF(X, Y) =cos [Λ( T−R+)]\nR+−cos [Λ( T−R−)]\nR−, G (X, Y) =Z+D\n−DdX′cos [Λ( T−S(X′))]\nS(X′), (39)\nwithS(X′) =p\n(X−X′)2+Y2+ 1andR±=p\n(X∓D)2+Y2+ 1.\nNow we are interested in the fields that survive at large distances from the source, i.e. in the radiation zone, which requires\napproximating the potentials for r≫λandr≫d, where λ= 2πc/ω is the wavelength. To this end we use the expansions8\n|r−r′| ≈r−n·r′and|r−r±| ≈r−zz0\nr∓xd\nr, where n=r/r. All in all, the zeroth order potentials (up to order 1/r) become\nA(0)\nx(r, t)≈µI0\n2πsin\u0014\nω0\u0012\nt−r2−z0z\nvr\u0013\u0015\nsin\u0012ω0d\nvx\nr\u0013v\nω0x, ϕ(0)(r, t) =vx\nrA(0)\nx(r, t). (40)\nUsing these potentials we find the electromagnetic fields in the radiation zone:\nB(0)=µI0\n2πcos\u0014\nω0\u0012\nt−r2−z0z\nvr\u0013\u0015\nsin\u0012ω0d\nvx\nr\u00131\nxˆex׈n,E(0)=−vˆn×B(0). (41)\nFollowing the same program one can derive expressions for the electromagnetic potentials at first-order. The details are relegated\nto the Appendix C. Using Eq. (31) one finds that the nonzero components of the vector potential in the time-domain at first-order\nbecome\nA(1)\nx(r, t) =˜α\n8πµ\nϵrϵ0\nµ0I0\nω0k0(|z|+z0)∂2\n∂x∂yZ+d\n−ddx′sin\u0014\nω0\u0012\nt−|r∗−r′\ni|\nv\u0013\u0015\n, (42)\nA(1)\ny(r, t) =−˜α\n8πµ\nϵrϵ0\nµ0I0\nω0k0(|z|+z0)\u0012∂2\n∂x2+k2\n0\u0013Z+d\n−ddx′sin\u0014\nω0\u0012\nt−|r∗−r′\ni|\nv\u0013\u0015\n, (43)\nwhere r′\ni(x′) =x′ˆex−z0ˆezis the image point of r′(x′)andr∗=xˆex+yˆey+|z|ˆez. The first-order scalar potential in the\ntime-domain is\nϕ(1)(r, t) =−I0˜α\n8πµ\nϵrϵ0\nµ0y\n|z|+z0Z+d\n−ddx′sinh\nω0\u0010\nt−|r∗−r′\ni|\nv\u0011i\n|r∗−r′\ni|. (44)\nIn the radiation zone these potentials can be further simplified. Indeed, one realizes than only the component A(1)\nyof the vector\npotential contributes to radiation (to be precise, the term proportional to k2\n0in Eq. (43)), while the others decay faster than 1/r.\nUsing the expansion |r∗−r′\ni| ≈r∗−xx′−z0|z|\nr∗ withr∗=r, we get\nA(1)\ny(r, t)≈ −˜αI0\n4πϵω0rϵ0\nµ0µ\nϵr/x\n(|z|+z0)1\nvsin\u0014\nω0\u0012\nt−r2+z0|z|\nv r\u0013\u0015\nsin\u0012ω0d\nvx\nr\u0013\n, ϕ(1)(r, t) =vy\nrA(1)\ny(r, t).(45)\nKeeping terms of the order 1/rthe electromagnetic fields become\nB(1)=−˜αµI0\n4πrϵ0\nµ0µ\nϵcos\u0014\nω0\u0012\nt−r2+z0|z|\nv r\u0013\u0015\nsin\u0012ω0d\nvx\nr\u0013r/x\n|z|+z0ˆey׈n,E(1)=−vˆn×B(1). (46)\nAs shown in Ref. [17], energy conservation is unchanged in the presence of the θterm, i.e. given the Poynting vector Sand\nenergy density udefined as usual S=E×Handu=1\n2(E·D+B·H), respectively, one realizes that ∂tu+∇ ·S= 0.\nMomentum conservation is modified by a surface term due to the self-induced charge and current densities arising there, but\nit does not concern us in this paper. The angular distribution of the radiated power takes the usual formdP\ndΩ=v\nµr2|B|2,\nwith however a total magnetic field which acquires a contribution arising from the θ-term. Keeping the lowest order in the\nmagnetoelectric polarizability ˜αand averaging (in time) over a complete cycle, the angular distribution of the radiated power by\nthe linear antenna becomes\n\u001cdP\ndΩ\u001d\n≈vµI2\n0\n8π2sin2\u0012ω0d\nvx\nr\u0013\u00141−(x/r)2\n(x/r)2−2(θ/π)σHallrµ\nϵy\nxr\n|z|cos\u0012z0ω\nvz+|z|\nr\u0013\u0015\n. (47)\nThe first term corresponds to the usual radiated power by a linear antenna embedded in an infinite material media described\nby the permittivity ϵand permeability µ. The second term, which is proportional to the surface Hall conductivity, exhibits an\ninteresting divergent behavior near the interface, as is depicted in Fig. 5. The physical origin of this term is understood from the\nsurface Hall current and charge densities at the interface, which generates electromagnetic field which contribution to radiations\nin the near zone to the interface.\nVI. SUMMARY AND DISCUSSION\nTopological phases are an emerging class of materials which have attracted much attention in condensed matter physics.\nThe remarkable microscopic properties encoded in the band structure of topological phases produces exceptional macroscopic9\nFIG. 5. Angular distribution of the radiated power by a thin linear antenna near a θdomain-wall. The position of the interface is evident.\nelectromagnetic response, which is described by a topological field theory akin to axion electrodynamics. In general, a linear\nmagnetoelectric media can be described by adding the term ∼θijEiBjto the free enthalpy of the system, where θijis the\nmagnetoelectric tensor. In this frame, axion electrodynamics is a particular case when the ME tensor is of the form θij=θδij.\nIn this paper we derived, in a perturbative fashion, solutions to the time-dependent axion electrodynamics field equations for\na picewise constant axion angle. The validity of the perturbative program relies in the fact that the axion coupling, in the case\nof topological insulators, is proportional to the fine structure constant. In the particle physics context this condition is vastly\nfulfilled, since it is expected to be very small. The difference is that in the former case the presence of boundaries is a natural\ntask, while in the latter the axion is assumed to be distributed over the whole space.\nTo validate our perturbative scheme, we first tackle the static axion field equations and derive the components of the Green’s\nmatrix up to second order in the axion coupling. We show that our results are consistent with the previously reported exact\nGreen’s matrix when Taylor expanded up to second order in the coupling. We discuss in passing the image magnetic monopole\neffect, which is a salient manifestation of the unconventional magnetoelectric response of topological insulators. In the time-\ndependent case, the axion field equations become more involved and no analytical results have been reported yet, although there\nare some first attempts towards this direction. Here we aim to fill in this gap by applying the same perturbative program to the\naxion field equations. For the sake of simplicity, in this paper we choose that the bulks have the same optical properties, namely,\nϵ1=ϵ2≡ϵandµ1=µ2≡µ, which is made to highlight and isolate the effects of the axion coupling. In a forthcoming paper\nwe will relax this condition and generalize our results to the case of two topologically insulating media characterized by different\nnontrivial optical parameters. This generalization is of course far from trivial. To illustrate the importance of the axion angle in\nthe time-dependent case considered here, we investigate the radiation of a short linear antenna above a topologically insulating\ndomain-wall with different values of θ.\nWe first computed numerically the electromagnetic fields due to the linear antenna by using the exact Green’s function and\nverify that they exhibit the expected behavior. We interpret the induced fields in terms of the Hall current at the interface between\nthe two half-spaces characterized by different θ-angles. Next, to investigate the effects of the θ-term in the power radiated, we\nuse the asymptotic expression for the Green’s function and evaluate the fields in the radiation zone. These field allow us to\ncompute the time-averaged angular distribution of the radiated power, which as expected, is dominated by the usual Maxwellian\nterm, but becomes corrected due to the θ-term. Such correction is quite interesting, since exhibits a divergent-type behavior\nnear the interface, which arises from the field produced by the surface charge and current densities there. This is a distinctive\nbehavior with respect to the usual radiation pattern.\nIn summary, a general solution to the static axion field equations has been previously elaborated in terms of Green’s func-\ntion techniques; however, the time-dependent case has received less attention mainly due to its complexity. In this paper we\nintroduced a perturbative scheme which was successfully validated with the static theory and then we extended the analysis\nto the time-dependent case. We obtained general expressions for the corresponding matrix of Green’s function which we also\napproximated in the far-field regime. We expect our method and results will be of considerable relevance and that they may\nconstitute the basis for numerous other researches which include more intricate model of antennas and the radiation of particles\nnear the θ-interface.10\nACKNOWLEDGMENTS\nM. I.-M. was supported by the CONAHCyT fellowship No. 843426. A. M.-R. and M. I.-M. has been partially supported\nby DGAPA-UNAM Project No. IA102722 and by Project CONAHCyT (M ´exico) No. 428214. Useful discussions with L. F.\nUrrutia, M. Cambiaso and L. Medel are warmly appreciated.\nDATA A V AILABILITY STATEMENT\nNo datasets were generated or analyzed during the current study.\nDECLARATIONS\nConflict of interest The authors declare no conflict of interest.\nAppendix A: Static Green’s function\n1. First-order Green’s functions\nLet us consider the first-order corrections. The solution to Eq. (9) are\nϕ(1)(r) =Z\nG(1)\n0i(r,r′)Ji(r′)d3r′, A(1)\ni(r) =Z\nG(1)\ni0(r,r′)ρ(r′)d3r′, (A1)\nwhere\nG(1)\n0i(r,r′) =−˜αrϵ0\nµ0Z\nGϵ(r,r′′)δ(z′′)ϵzji∂′′\njG(⊥)\nµ(r′′,r′)d3r′′, (A2)\nG(1)\ni0(r,r′) =−˜αrϵ0\nµ0Z\nG(⊥)\nµ(r,r′′)δ(z′′)ϵzji∂′′\njGϵ(r′′,r′)d3r′′. (A3)\nNoting that partial derivatives apply only in the transverse coordinates x′′andy′′, the properties of the Dirac delta allow us to\ncompute the integrals with respect to the z′′-coordinate in a simple fashion. Now, if we use ∂′′\njG(⊥)\nµ(r′′,r′) =−∂′\njG(⊥)\nµ(r′′,r′)\nand∂′′\njGϵ(r′′,r′) =−∂′\njGϵ(r′′,r′)forj=x, y, the double primed derivatives can be get out of the integrals as a minus primed\nderivative. So we are left with\nG(1)\n0i(r,r′) = ˜αrϵ0\nµ0ϵzji∂′\njZ\nΣG(⊥)\nµ(r,r′′\n⊥)Gϵ(r′′\n⊥,r′)d2r′′\n⊥, (A4)\nG(1)\ni0(r,r′) = ˜αrϵ0\nµ0ϵzji∂′\njZ\nΣGϵ(r,r′′\n⊥)G(⊥)\nµ(r′′\n⊥,r′)d2r′′\n⊥. (A5)\nUsing the electric (11) and magnetic (13) Green functions we realize that\nG(1)\n0i(r,r′) =G(1)\ni0(r,r′) =˜αp\nϵ0/µ0\n(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji∂′\njZ\nΣ1p\n|r⊥−r′′⊥|2+z21p\n|r′′\n⊥−r′⊥|2+z′2d2r′′\n⊥\n(2π)2. (A6)\nTo carry out integration, we use the following integral representation:\n1\n|r−r′|= 4πZ\nΣk⊥d2k⊥\n(2π)2eik⊥·(r⊥−r′\n⊥)e−k⊥|z−z′|\n2k⊥, (A7)\nwhere Σk⊥is the two-dimensional space kx-ky. Using this representation twice in Eq. (A6) we get\nG(1)\n0i(r,r′) =G(1)\ni0(r,r′) =˜αp\nϵ0/µ0\n(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji\n×∂′\njZ\nΣk⊥d2k⊥\n2πe−k⊥|z|\nk⊥Z\nΣk′\n⊥d2k′\n⊥\n2πe−k′\n⊥z′\nk′\n⊥Z\nΣeik⊥·(r⊥−r′′\n⊥)eik′\n⊥·(r′′\n⊥−r′\n⊥)d2r′′\n⊥\n(2π)2.(A8)11\nIntegration with respect to the coordinates r′′\n⊥produces a Dirac delta on the momenta k⊥andk′\n⊥, i.e.\nG(1)\n0i(r,r′) =G(1)\ni0(r,r′) =˜αp\nϵ0/µ0\n(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji\n×∂′\njZ\nΣk⊥d2k⊥\n2πe−k⊥|z|\nk⊥Z\nΣk′\n⊥d2k′\n⊥\n2πe−k′\n⊥z′\nk′\n⊥eik⊥·r⊥−ik′\n⊥·r′\n⊥δ(k′\n⊥−k⊥)\n=˜αp\nϵ0/µ0\n(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji∂′\njZ\nΣk⊥d2k⊥\n(2π)2eik⊥·(r⊥−r′\n⊥)e−k⊥(|z|+z′)\nk2\n⊥, (A9)\nwhere we used the properties of the Dirac delta to perform integration over k′\n⊥. The resulting double integral becomes easier to\nperform if we express the area element in polar coordinates, i.e. d2k⊥=k⊥dk⊥dφ, and choose the kxaxis in the direction of\nthe vector r⊥−r′\n⊥. Noting that k⊥·(r⊥−r′\n⊥) =|r⊥−r′\n⊥|k⊥cosφ, we can write\nG(1)\n0i(r,r′) =G(1)\ni0(r,r′) =˜αp\nϵ0/µ0\n2π(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji∂′\njZ∞\n0dk⊥e−k⊥(|z|+z′)\nk⊥\u00121\n2πZ2π\n0dφ ei|r⊥−r′\n⊥|k⊥cosφ\u0013\n.\n(A10)\nThe parentheses in this equation enclose an integral representation of the Bessel function J0. The resulting integral is\nG(1)\n0i(r,r′) =G(1)\ni0(r,r′) =˜αp\nϵ0/µ0\n2π(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji∂′\njZ∞\n0dk⊥J0(|r⊥−r′\n⊥|k⊥)e−k⊥(|z|+z′)\nk⊥, (A11)\nand taking the primed derivative\nG(1)\n0i(r,r′) =G(1)\ni0(r,r′) =˜αp\nϵ0/µ0\n2π(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji(r⊥−r′\n⊥)j\n|r⊥−r′\n⊥|Z∞\n0dk⊥J1(|r⊥−r′\n⊥|k⊥)e−k⊥(|z|+z′).(A12)\nThis integral is well known [53]. The final result is then\nG(1)\n0i(r,r′) =G(1)\ni0(r,r′) =˜αp\nϵ0/µ0\n2π(ϵ1+ϵ2)(1/µ1+ 1/µ2)ϵzji(r⊥−r′\n⊥)j\n|r⊥−r′\n⊥|2\"\n1−|z|+z′\np\n|r⊥−r′\n⊥|2+ (|z|+z′)2#\n. (A13)\n2. Second-order Green’s functions\nWe now turn to the second-order corrections to the GFs. Integration of Eq. (9) produces\nϕ(2)(r) =Z\nG(2)\n00(r,r′)ρ(r′)d3r′, A(2)\ni(r) =Z\nG(2)\nij(r,r′)Jj(r′)d3r′, (A14)\nwhere\nG(2)\n00(r,r′) =−˜αrϵ0\nµ0Z\nGϵ(r,r′′)δ(z′′)ϵzji∂′′\njG(1)\ni0(r′′,r′)d3r′′, (A15)\nG(2)\nij(r,r′) =−˜αrϵ0\nµ0Z\nG(⊥)\nµ(r,r′′)δ(z′′)ϵzki∂′′\nkG(1)\n0j(r′′,r′)d3r′′. (A16)\nUsing the first-order Green’s function given by Eq. (A13), we verify that ∂′′\njG(1)\ni0(r′′,r′) =−∂′\njG(1)\ni0(r′′,r′)forj= 1,2,\nand hence the double primed derivatives can be get out of the integrals as a minus primed derivative, as before. Performing the\nintegration with respect to z′′we get\nG(2)\n00(r,r′) = ˜αrϵ0\nµ0ϵzji∂′\njZ\nΣGϵ(r,r′′\n⊥)G(1)\ni0(r′′\n⊥,r′)d2r′′\n⊥, (A17)\nG(2)\nij(r,r′) = ˜αrϵ0\nµ0ϵzki∂′\nkZ\nΣG(⊥)\nµ(r,r′′\n⊥)G(1)\n0j(r′′\n⊥,r′)d2r′′\n⊥. (A18)12\nUsing the integral representations (A4) for the 0i-components and (A5) we find\nG(2)\n00(r,r′) = (˜αp\nϵ0/µ0)2ϵzjiϵzki∂′\nj∂′\nkZ\nΣZ\nΣGϵ(r,r′′\n⊥)Gϵ(r′′\n⊥,r′′′\n⊥)G(⊥)\nµ(r′′′\n⊥,r′)d2r′′′\n⊥d2r′′\n⊥, (A19)\nG(2)\nij(r,r′) = (˜αp\nϵ0/µ0)2ϵzkiϵzlj∂′\nl∂′\nkZ\nΣZ\nΣG(⊥)\nµ(r,r′′\n⊥)Gϵ(r′′\n⊥,r′′′\n⊥)G(⊥)\nµ(r′′′\n⊥,r′)d2r′′′\n⊥d2r′′\n⊥, (A20)\nand substituting the electric (11) and magnetic (13) Green functions we get\nG(2)\n00(r,r′) =(˜αp\nϵ0/µ0)2\n2π(ϵ1+ϵ2)2(1/µ1+ 1/µ2)tr[I(r,r′)], (A21)\nG(2)\nij(r,r′) =(˜αp\nϵ0/µ0)2\n2π(ϵ1+ϵ2)(1/µ1+ 1/µ2)2Iij(r,r′), (A22)\nwhere we have defined the symmetric matrix I= (Iij)∈R2×2, with components\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl∂′\nmZ\nΣZ\nΣ1p\n|r⊥−r′′⊥|2+z21p\n|r′′\n⊥−r′′′⊥|2+ 021p\n|r′′′\n⊥−r′⊥|2+z′2d2r′′′\n⊥\n2πd2r′′\n⊥\n2π\n(A23)\nTo evaluate this expression, we use thrice the integral representation (A7), i.e.\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl∂′\nmZ\nΣZ\nΣ Z\nΣk⊥d2k⊥\n2πeik⊥·(r⊥−r′′\n⊥)e−k⊥|z|\nk⊥!  Z\nΣk⊥d2k′\n⊥\n2πeik′\n⊥·(r′′\n⊥−r′′′\n⊥)e−k′\n⊥(0)\nk′\n⊥!\n Z\nΣk⊥d2k′′\n⊥\n2πeik′′\n⊥·(r′′′\n⊥−r′\n⊥)e−k′′\n⊥z′\nk′′\n⊥!\nd2r′′′\n⊥\n2πd2r′′\n⊥\n2π. (A24)\nIntegrating with respect to r′′\n⊥andr′′′\n⊥produces two Dirac deltas in momentum:\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl∂′\nmZ\nΣk⊥Z\nΣk⊥Z\nΣk⊥d2k⊥d2k′\n⊥d2k′′\n⊥\n2πe−k⊥|z|e−k′′\n⊥z′\nk⊥k′\n⊥k′′\n⊥eik⊥·r⊥−ik′′\n⊥·r′\n⊥δ(k′\n⊥−k⊥)δ(k′′\n⊥−k′\n⊥)\n= (δijδlm−δilδjm)∂′\nl∂′\nmZ\nΣk⊥d2k⊥\n2πe−k⊥(|z|+z′)\nk3\n⊥eik⊥·(r⊥−r′\n⊥)(A25)\nwhere we have used the properties of the Dirac delta. We now introduce the polar coordinates (kx, ky) =k⊥(cosφ,sinφ)and\nchoose the vector r⊥−r′\n⊥along the kxdirection. So, the angular integral yields\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl∂′\nmZ∞\n0dk⊥e−k⊥(|z|+z′)\nk2\n⊥J0(|r⊥−r′\n⊥|k⊥). (A26)\nThis integral does not converge as is, but its derivatives does. Taking one of the derivatives we find\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl\"\n(r⊥−r′\n⊥)m\n|r⊥−r′\n⊥|Z∞\n0dk⊥e−k⊥(|z|+z′)\nk⊥J1(|r⊥−r′\n⊥|k⊥)#\n. (A27)\nTo perform this integral in a convergent manner, we use the following trick:R∞\nΛe−k⊥λdλ=e−k⊥Λ/k⊥, and therefore\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl\"\n(r⊥−r′\n⊥)m\n|r⊥−r′\n⊥|Z∞\n|z|+z′dλZ∞\n0dk⊥e−k⊥λJ1(|r⊥−r′\n⊥|k⊥)#\n, (A28)\nand this integral is well-known; see, for example, Ref. [53]. The result is\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl\"\n(r⊥−r′\n⊥)m\n|r⊥−r′\n⊥|2Z∞\n|z|+z′dλ \n1−λp\n|r⊥−r′\n⊥|2+λ2!#\n, (A29)13\nfrom which we finally obtain\nIij(r,r′) = (δijδlm−δilδjm)∂′\nl\u0014(r⊥−r′\n⊥)m\n|r⊥−r′\n⊥|2\u0012q\n|r⊥−r′\n⊥|2+ (|z|+z′)2−(|z|+z′)\u0013\u0015\n. (A30)\nFrom this expression we find\ntr[I(r,r′)] =1p\n|r⊥−r′⊥|2+ (|z|+z′)2, (A31)\nand\nIij(r,r′) =δijtr[I(r,r′)] +∂iKj(r,r′), (A32)\nwhere\nKj(r,r′) =(r⊥−r′\n⊥)j\n|r⊥−r′\n⊥|2\u0012q\n|r⊥−r′\n⊥|2+ (|z|+z′)2−(|z|+z′)\u0013\n. (A33)\nAppendix B: Frequency-dependent Green’s function\n1. Derivation of the Green’s functions\nSubstituting the zeroth-order solutions (30) into Eqs. (27) and (28) one finds that the first order solution are given by\nϕ(1)(r, ω) =−ϵzij∂iZ\nGω(r,r′)Jj(r′, ω)d3r′(B1)\nA(1)\ni(r, ω) =−ϵizj∂jZ\nGω(r,r′)ρ(r′)d3r′+iω\nv2ϵizjZ\nGω(r,r′)Jj(r′, ω)d3r′, (B2)\nwhere we have defined\nGω(r,r′) = ˜αµ\nϵrϵ0\nµ0Z\nGω(r,r′′\n⊥)Gω(r′′\n⊥,r′)d2r′′\n⊥. (B3)\nWe now perform son manipulations in Eqs. (B1) and (B2). Taking the divergence of Eq. (B2) we obtain\n∂iA(1)\ni(r, ω) =iω\nv2ϵizj∂iZ\nGω(r,r′)Jj(r′, ω)d3r′=iω\nv2ϕ(1)(r, ω), (B4)\nwhere we used that ϵzij∂i∂j= 0. So, one we have computed the vector potential, the scalar potential becomes fully determined.\nIn order to manipulate Eq. (B2) we use the local charge conservation: ∇ ·J=iωρ. Substituting the charge density in Eq. (B2)\nproduces\nA(1)\ni(r, ω) =−1\niωϵizj∂jZ\nGω(r,r′)[∂′\nkJk(r′, ω)]d3r′+iω\nv2ϵizjZ\nGω(r,r′)Jj(r′, ω)d3r′. (B5)\nIntegrating by parts and using the identity ∂′\niGω(r,r′) =−∂iGω(r,r′)we obtain\nA(1)\ni(r, ω) =1\niωϵzij∂j∂kZ\nGω(r,r′)Jk(r′, ω)d3r′+iω\nv2ϵizkZ\nGω(r,r′)Jk(r′, ω)d3r′\n=i\nω\u0012\n−ϵzij∂j∂k+ω2\nv2ϵizk\u0013Z\nGω(r,r′)Jk(r′, ω)d3r′, (B6)\nwhich can be written as\nA(1)\ni(r, ω) =Z\nGij(r,r′)Jj(r′, ω)d3r′, (B7)\nwhere\nGij(r,r′) =i\nω\u0012\n−ϵzij∂j∂k+ω2\nv2ϵizk\u0013\nGω(r,r′). (B8)\nThis expression corresponds to Eq. (32) of the main text.\nIn the following we derive an exact expression for the function Gω(r,r′), which cannot be expressed in term of known\nfunctions. We also obtain an analytical expression in the radiation zone (far field approximation) by using two different methods.14\n2. Exact expression for the function Gω(r,r′)\nThe integral we have to evaluate is given by Eq. (B3). Substituting the frequency-dependent Green’s function (29) the integral\nbecomes:\nGω(r,r′) = ˜αµ\nϵrϵ0\nµ0Z\nd2r′′\n⊥eik0|r−r′′\n⊥|\n4π|r−r′′\n⊥|eik0|r′′\n⊥−r′|\n4π|r′′\n⊥−r′|. (B9)\nTo evaluate this integral we now use the Sommerfeld identity:\neik0|r|\n4π|r|=Z\nΣk⊥d2k⊥\n(2π)2eik⊥·r⊥iei√\nk2\n0−k2\n⊥|z|\n2p\nk2\n0−k2\n⊥, (B10)\nwhere Σk⊥is the two-dimensional space kx-kyandr=r⊥+|z|ˆez. Using this representation twice in Eq. (B9) we get\nGω(r,r′) = ˜αµ\nϵrϵ0\nµ0Z\nΣk⊥d2k⊥\n(2π)2Z\nΣk′\n⊥d2k′\n⊥\n(2π)2ei(k⊥·r⊥−k′\n⊥·r′\n⊥)iei√\nk2\n0−k2\n⊥|z|\n2p\nk2\n0−k2\n⊥iei√\nk2\n0−k′2\n⊥|z′|\n2p\nk2\n0−k′2\n⊥Z\nd2r′′\n⊥ei(k′\n⊥−k⊥)·r′′\n⊥(B11)\nIntegration over r′′\n⊥produces a Dirac delta in momentum, which allow us to evaluate one of the momentum integrals. We obtain:\nGω(r,r′) =−˜α\n4µ\nϵrϵ0\nµ0Z\nΣk⊥d2k⊥\n(2π)2eik⊥·(r⊥−r′\n⊥)ei√\nk2\n0−k2\n⊥(|z|+|z′|)\nk2\n0−k2\n⊥, (B12)\nAs before, we now introduce the polar coordinates (kx, ky) =k⊥(cosφ,sinφ)and choose the vector r⊥−r′\n⊥along the kx\ndirection. So, the angular integral yields\nGω(r,r′) =−˜α\n8πµ\nϵrϵ0\nµ0Z∞\n0dk⊥ei√\nk2\n0−k2\n⊥(|z|+|z′|)\nk2\n0−k2\n⊥k⊥J0(k⊥|r⊥−r′\n⊥|). (B13)\nThis is an exact expression for the required function. It corresponds to Eq. (33) of the main text.\n3. Far-field approximation\nHere we shall approximate the integral (B13) in the radiation zone, i.e. for ρ,|z| ≫ 1. For crosscheking our result, we\ncompute the integral by two different methods. One of them used the Sommerfeld integral ( Method 1 ) and the other is the\nstationary phase approximation ( Method 2 ).\nMethod 1 . We start by using the identity q−1=R∞\n0dξ e−qξin Eq. (B13), for q=p\nk2\n0−k2\n⊥to obtain\nGω(r,r′) =−˜α\n8πµ\nϵrϵ0\nµ0Z∞\n0dξZ∞\n0dk⊥k⊥p\nk2\n0−k2\n⊥ei√\nk2\n0−k2\n⊥(|z|+|z′|+iξ)J0(k⊥|r⊥−r′\n⊥|). (B14)\nUsing the Sommerfeld integral (B10) this expression can be written in the simple form:\nGω(r,r′) =i˜α\n8πµ\nϵrϵ0\nµ0Z∞\n0dξeik0Rξ\nRξ, (B15)\nwhere Rξ≡Rξ(r,r′) =p\n|r⊥−r′\n⊥|2+ (|z|+|z′|+iξ)2, such that R0=p\n|r⊥−r′\n⊥|2+ (|z|+|z′|)2. Next we shall\napproximate this integral in the far-field region. To this end we use the fact that Rξ=R0q\n1 +2iξ(|z|+|z′|)−ξ2\nR2\n0to rewritte the\nintegral (B15) as\nGω(r,r′) =i˜α\n8πµ\nϵrϵ0\nµ0Z∞\n0dχeiζ√\n1+2iχ˜z−χ2\np\n1 + 2 iχ˜z−χ2, (B16)15\nwhere ζ=k0R0and˜z= (|z|+|z′|)/R0. Numerical evaluation of this integral for ζ≫1and˜z≪1reveals that only small\nvalues of χcontribute to the integral. So we can expand the root and retain solely the leading order, i.e.\nGω(r,r′)≈i˜α\n8πµ\nϵrϵ0\nµ0eiζZ∞\n0dχe−(ζ˜z)χ\n1 +iχζ=i˜α\n8πµ\nϵrϵ0\nµ01\n˜z[π+iEi(iζ)], (B17)\nwhose approximation for ζ≫1is\nGω(r,r′)≈i˜α\n8πµ\nϵrϵ0\nµ0eik0R0\nk0(|z|+|z′|). (B18)\nThis result corresponds to Eq. (34) of the main text.\nMethod 2 . Since the Bessel function and the complex exponential in the integrand of Eq. (B13) are rapidly oscillating\nfunctions in the far-field region, one can use the stationary phase approximation to evaluate the integral. To this end we first use\nthe identity J0(x) =1\n2[H(1)\n0(x)+H(2)\n0(x)]together with the reflection formula H(1)\n0(eiπx) =−H(2)\n0(x), where H(1)\n0andH(2)\n0\nare the Hankel functions. In this case, the integral (B13) can be written as\nGω(r,r′) =−˜α\n16πµ\nϵrϵ0\nµ0Z∞\n−∞dk⊥k⊥\nk2\n0−k2\n⊥eikz(|z|+|z′|)H(1)\n0(k⊥|r⊥−r′\n⊥|). (B19)\nWe now implement the far-field approximation into the Hankel function. For x≫1we have H(1)\n0(x)≈q\n2\nπxeix−iπ\n4, and the\nintegral (B19) becomes\nGω(r,r′)≈ −˜α\n16πµ\nϵrϵ0\nµ0Z∞\n−∞dk⊥k⊥\nk2\n0−k2\n⊥s\n2\nπk⊥|r⊥−r′\n⊥|exp\u0014\ni|r⊥−r′\n⊥|\u0012\nk⊥+q\nk2\n0−k2\n⊥|z|+|z′|\n|r⊥−r′\n⊥|\u0013\n−iπ\n4\u0015\n.\n(B20)\nWe observe that, in the far-field, the exponential factor is a rapidly oscillating function of k⊥and hence we can apply the\nstationary phase approximation to evaluate the dominant contribution. A simple calculation reveals that the stationary phase\npoint locates at\nk∗\n⊥=k0r\n1 +\u0010\n|z|+|z′|\n|r⊥−r′\n⊥|\u00112. (B21)\nFinally, we use the well-known asymptotic formula\nZ\nRg(x)eiλf(x)dx≈g(x0)eiλf(x0)+iπ\n4sgn(f′′(x0))s\n2π\nλ|f′′(x0)|, (B22)\nwhere λ≫1andf′(x0) = 0 , to obtain\nGω(r,r′)≈i˜α\n8πµ\nϵrϵ0\nµ0eik0R0\nk0(|z|+|z′|), (B23)\nwhich is again the result of Eq. (B18). This validates our expression for the far-field approximation.\nAppendix C: Evaluation of the first-order potentials for a short linear antenna\nOur starting point here is Eq. (31) for the components of the vector potential. The first order contribution in the frequency-\ndomain is read off as\nA(1)\ni(r, ω) =i1\nω\u0002\n(ˆez× ∇)i∂j−k2\n0ϵijz\u0003Z\nGω(r,r′)Jj(r′, ω)d3r′, (C1)16\nwhere Gω(r,r′)is the corresponding Green’s function. The potential in the time-domain is computed by inverse Fourier trans-\nforming the above expression. We obtain:\nA(1)\ni(r, t) =i1\n2π\u0014\n(ˆez× ∇)i∂j+1\nv2∂2\n∂t2ϵijz\u0015Z\ndω e−iωt1\nωZ\nGω(r,r′)Jj(r′, ω)d3r′. (C2)\nThe current source (35) in the frequency domain is\nJ(r, ω) =Z\neiωtJ(r, t)dt=I δ(y)δ(z−z0)Θ(d− |x|)ˆexπ\ni[δ(ω+ω0)−δ(ω−ω0)]. (C3)\nSubstituting (C3) into Eq. (C2) and integrating the frequency one obtains\nA(1)\ni(r, t) =−1\n2π\u0014\n(ˆez× ∇)i∂x+1\nv2∂2\n∂t2ϵixz\u0015Iπ\nω0Z+d\n−ddx′\u0002\neiω0tG−ω0(r,r′′) +e−iω0tGω0(r,r′′)\u0003\n, (C4)\nwhere r′′=x′ˆex+z0ˆez. Since we are interested in the far-field domain, we use the Green’s function (34). In such case the\nvector potential becomes\nA(1)\ni(r, t) =−˜α\n8πµ\nϵrϵ0\nµ0I\nω0k0(|z|+z0)\u0014\n(ˆez× ∇)i∂x−ω2\n0\nv2ϵixz\u0015Z+d\n−ddx′sin\u0014\nω0\u0012\nt−|r∗−r′\ni|\nv\u0013\u0015\n, (C5)\nwhere r′\ni(x′) =x′ˆex−z0ˆezis the image point of r′(x′)andr∗=xˆex+yˆey+|z|ˆez. Equations (42) and (43) of the main text\nare derived from this result by taking i=xandi=y, respectively. Now, to derive the scalar potential, we appeal to the Lorentz\ngauge condition. Taking the divergence of the vector potential we obtain\n∂iA(1)\ni(r, ω) =−˜α\n8πµ\nϵrϵ0\nµ0I\nω0k0(|z|+z0)ω2\n0\nv2∂\n∂yZ+d\n−ddx′sin\u0014\nω0\u0012\nt−|r∗−r′\ni|\nv\u0013\u0015\n=−1\nv2∂\n∂tϕ(1)(r, t), (C6)\nwherefrom we find\nϕ(1)(r, t) =−˜α\n8πµ\nϵrϵ0\nµ0I\nk0(|z|+z0)∂\n∂yZ+d\n−ddx′cos\u0014\nω0\u0012\nt−|r∗−r′\n0i|\nv\u0013\u0015\n, (C7)\nwhich, after taking the y-derivative, corresponds to Eq. (44) of the main text.\nAppendix D: Derivation of Eq. (47)\nThe angular distribution of the radiated power is given bydP\ndΩ=v\nµr2|B|2. Keeping linear order in the magnetic field, i.e.\n|B|2≈ |B(0)|2+ 2B(0)·B(1), one findsdP\ndΩ≈v\nµr2\u0002\n|B(0)|2+ 2B(0)·B(1)\u0003\n. Substituting the zeroth (41) and first (46) order\nmagnetic fields we obtain\ndP\ndΩ≈v\nµ\u0012µI0\n2π\u00132\ncos\u0014\nω0\u0012\nt−r2−z0z\nvr\u0013\u0015\nsin2\u0012ω0d\nvx\nr\u0013\u001a\ncos\u0014\nω0\u0012\nt−r2−z0z\nvr\u0013\u00151−(x/r)2\n(x/r)2\n+2˜αrϵ0\nµ0µ\nϵcos\u0014\nω0\u0012\nt−r2+z0|z|\nv r\u0013\u0015y/x\n|z|+z0r\u001b\n, (D1)\nwhere we used that (ˆex׈n)·(ˆex׈n) = 1−(x/r)2and(ˆex׈n)·(ˆey׈n) =−xy/r2.\nFinally, averaging in time over a complete cycle yields\n\u001cdP\ndΩ\u001d\n≈v\n2µ\u0012µI0\n2π\u00132\nsin2\u0012ω0d\nvx\nr\u0013\u001a1−(x/r)2\n(x/r)2+ 2˜αrϵ0\nµ0µ\nϵcos\u0014(z+|z|)z0ω\nrv\u0015y/x\n|z|+z0r\u001b\n, (D2)\nwhere we have used:\nlim\nT→∞1\n2TZ+T\n−Tcos2\u0014\nω0\u0012\nt−r2−z0z\nvr\u0013\u0015\ndt=1\n2, (D3)\nlim\nT→∞1\n2TZ+T\n−Tcos\u0014\nω0\u0012\nt−r2−z0z\nvr\u0013\u0015\ncos\u0014\nω0\u0012\nt−r2+z0|z|\nvr\u0013\u0015\ndt=1\n2cos\u0014(z+|z|)z0ω\nrv\u0015\n. (D4)17\nThese results stablish Eq. (5).\n[1] X.-L. Qi and S.-C. 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We also discuss the Harbourne-Huneke Containment and the a forementioned\nDemailly’s Conjecture for general points and show the resul ts for sufficiently many general points\nand general points in projective spaces with low dimensions .\n1.Introduction\nGiven a set of sreduced points X={P1,...,P s}inPN/CZ, over an algebraically closed field\nwith coordinate ring R= /CZ[PN/CZ], one of the fundamental questions asks about the least degr ee\nof a homogeneous polynomial in Rvanishing at Xwith multiplicity at least m. IfIXdenotes\nthe ideal defining X, then by Zariski-Nagata theorem, the m-th symbolic power of IX, defined as\nI(m)=∩s\ni=1pm\ni, wherepiis the ideal defining the point Pi, consists of all homogeneous polynomials\nvanishing at Xwith multiplicity at least m. Letα(I) be the least degree of elements in I(or initial\ndegree of I). The Waldschmidt constant of Iis defined as /hatwideα(I) = lim\nm→0α(I(m))\nm. Demailly proposed\nthe following lower bound for the Waldschmidt constant of th e defining ideal of points Iwhich\nsimultaneously gives lower bounds for all α(I(m)) (see Definition 2.2).\nConjecture 1.1. [Dem82] IfIXis the defining ideal of X={P1,...,P s} ⊂PN\nC, then\n/hatwideα(IX)/greaterorequalslantα(I(m)\nX)+N−1\nm+N−1,for allm/greaterorequalslant1.\nTheaboveinequality issharp: itbecomesequalitywhen,for instance, Iisacompleteintersection.\nIt is also asymptotically an equality (when m→ ∞). Form= 1, Conjecture 1.1is known as the\nChudnovsky’s conjecture [ Chu81] and is well studied, see, [ Chu81,HH13,Dum12,Dum15,BH10,\nGHM13,FMX18,DTG17,FMX18,BGHN22b ,BN22].\nEsnault and Viehweg [ EV83] verified Conjecture 1.1forN= 2. Bocci, Cooper, and Harbourne\n[BCH14] provided a more elementary proof using ideal containment f ors=n2general points,\nand for at least ( m+ 1)2−1 general points for any fixed minP2. In dimesion N/greaterorequalslant3, the\nconjecture is wide open. Malara, Szemberg, and Szpond [ MSS18] proved that for any fixed integer\nm, any set of at least ( m+1)Nvery general points satisfies Demailly’s conjectural inequ ality. This\nwas later improved to at least mNvery general points by the work of Dumnicki, Szemberg, and\nSzpond [ DSS24]. Trok and Nagel [ NT19] proved that Conjecture 1.1holds for at most N+2 and\nspecific cases of N+ 3 general points. Chang and Jow [ CJ20] verified the conjecture for s=kN\ngeneral points, where k∈N. The authors along with Grifo and H` a [ BGHN22a ] proved that for\nany fixed m∈N, any set of at least (2 m+ 2)Ngeneral points verifies Demailly’s conjecture. It\nwas also shown to be true for special configurations of points including the star configurations\n2010Mathematics Subject Classification. 14N20, 13F20, 14C20.\nKey words and phrases. Demailly’s Conjecture, Cremona Transformation, Waldschm idt Constant, Ideals of Points,\nSymbolic Powers, Containment Problem, Stable Harbourne-H uneke Conjecture.\n1of points [ MSS18,BGHN22a ], (a more elementary proof for) the Fermat configurations an d other\nconfigurations of points in P2[Ngu23b], and even beyond the points setting including determinant al\nideals [BGHN22a ], star configurations of hypersurfaces [ BGHN22a ] (also known as the uniform\na−fold product ideal, see, Lin and Shen [ LS22]), and special configurations of lines [ Ngu23a].\nBesides classical applications in complex analysis, see fo r instance [ Wal77,Mor80], Demailly’s\nbound was shown to be beneficial for the study of the ideal cont ainment problem [ TX20], as well as\ncomputing the values α(I(m)\nX) for special configurations [ Ngu23a,Ngu23b]. Demailly’s bound also\ngives rise to upper bounds for the regularity values of a quot ient by ideals generated by powers of\nlinear forms [ DNS23], which is useful to study dimensions of certain spline spac es [DV21].\nWe briefly recall the descriptions of very general and genera l points. We follow the exposition\nfrom the work of Fouli, Mantero, and Xie [ FMX18, Page 4]. A set of points X={P1,...,P s}is an\nelement of the Chow variety of algebraic 0-cycles of degree sinPN(or an element in the Hilbert\nscheme of spoints in PN). A property Pholds for svery general set of points in PN/CZ, if there exist\ncountable infinite open dense subsets Ut,t∈N, of the Chow variety of algebraic 0-cycles of degree\nssuch that the property Pholds for all X∈/intersectiontext∞\nt=1Ut. If there exists one open dense subset Usuch\nthat the property Pholds for all X∈U, then the property Pholds for sgeneralsets of points.\nIn this manuscript, we focus on a specific case of Conjecture 1.1whenm= 2 and N/greaterorequalslant3.\nConjecture 1.2. IfIXis the defining ideal of X={P1,...,P s} ⊂PN\nC, then\n/hatwideα(I)/greaterorequalslantα(I(2))+N−1\nN+1.\nAs mentioned before, Conjecture 1.2was proved for at least 2Nvery general points [ DSS24], for\nat mostN+2 and some cases of N+3 general points [ NT19], and for at least 63general points in\nP3, at least 54general points in P4, and at least 4Ngeneral points in PNwhereN/greaterorequalslant5 [BGHN22a ].\nOur main result is to complete the proof of Conjecture 1.2for any set of very general points. In\naddition, we verify the conjecture for at least 2Ngeneral points, and general points in P3,P4, and\nP5except for four cases in P5. This is not a routine checking as it involves finding better b ounds\nfor the Waldschmidt constants in given cases. In general, pr oving the results for general points is\nmuch harder than that for very general points as we have to mov e from countably infinite open\nconditions to one open condition.\nMain Result. Conjecture 1.2holds for the points in the following list (*).\n•(Theorem 3.2)svery general points where N+3/lessorequalslants/lessorequalslant2N.\n•(Corollary 4.6) at least 2Ngeneral points.\n•(Corollary 5.4)sgeneral points in PN, for the following cases:\n(i)6/lessorequalslants/lessorequalslant63, inP3\n(ii)8/lessorequalslants/lessorequalslant54, inP4\n(iii)s= 8,9and14/lessorequalslants/lessorequalslant45, inP5.\nCoupled with results in [BGHN22a ,DSS24,NT19], Demailly’s Conjecture when m= 2now holds\nfor any set of very general points, any set of at least 2Ngeneral points, and any set of general points\ninP3,P4,andP5, except for 10/lessorequalslants/lessorequalslant13general points in P5.\nThe result for 10 /lessorequalslants/lessorequalslant13 general points in P5is still not known. We believe a more clever\ntreatment for the Waldschmidt constant of 10 points would le ad to the affirmative answer to these\nfour cases. We now briefly discuss our strategies. This invol ves deriving and using regularity and\nthe initial degree of the second symbolic power of the definin g ideal of generic points from its\nHilbert function. Let pibe the defining ideal of the point Pi∈PN, then the fat point scheme\nY=m1P1+···+msPsis defined by the ideal IY=pm1∩···∩pms. The Hilbert function of Yis\n2defined as follows\nHFR/IY(t) = dim/CZ(R/IY)t,for allt∈Z+.\nThe Alexander-Hirschowitz theorem provides the Hilbert fu nction of Y, whenY= 2P1+···+2Ps.\nFrom the definition, we obtain dim/CZ[IY]t=/parenleftbigN+t\nN/parenrightbig\n−HFR/IY(t). Thus the least t >0 for which\nHFR/IY(t)</parenleftbigN+t\nN/parenrightbig\nwill give the initial degree α(IY).\nForthevery general case, wecontrol α(I(2))usingtheAlexander-Hirschowitz Theoremandderive\nin Lemma 3.1the following: if Iis the ideal defining sgeneric points then to show Conjecture 1.2\nit is enough to prove\n(1) /hatwideα(I)/greaterorequalslantN+ℓ\nN+1,where/parenleftbiggN+ℓ\nN/parenrightbigg\n/lessorequalslant(N+1)s </parenleftbiggN+ℓ+1\nN/parenrightbigg\n.\nNext, to handle /hatwideα(I), in Theorem 3.2, we split the number of points into two adequate parts to\nuse induction on N, reduction methods and results from [ DSS24,BN22] to obtain the proper lower\nbound for /hatwideα(I) to prove inequality 1and thus prove Conjecture 1.2for very general points.\nForgeneralpoints, we need an additional tool. To prove Conjecture 1.2for general sets of\npoints in the List (*), we prove the containment I(r(N+1)−N+1)⊆mr(N−1)/parenleftbig\nI(2)/parenrightbigr, for some r∈\nN, for ideals defining generic points, as in [ BGHN22a , Theorem 2.9]. Following the strategy in\n[BCH14,BGHN22a ]), we show that a specific inequality between the Waldschmid t constant and\nthe regularity of the second symbolic power would imply the a forementioned containment.\nTheorem 4.1LetI⊂R= /CZ[PN] be the defining ideal of sgeneric points. If\n/hatwideα(I)/greaterorequalslantN−1+reg(I(2))\nN+1,\nthen there exists rsuch that I(r(N+1)−N+1)⊆mr(N−1)/parenleftbig\nI(2)/parenrightbigr.\nThus, we can focus on proving the hypothesis inequality of th e above theorem for each case of\ngeneral points. Firstly, upper bounds for the regularity of the second symbolic power of the ideal\ndefining generic points can be derived from the Alexander-Hi rschowitz theorem. Recall that the\nregularity of Yis the least integer t+1 such that HF R/IY(t) =/summationtexts\ni=1/parenleftbigN+mi−1\nN/parenrightbig\n. In Lemma 4.2, we\nderive the upper bounds and prove that: if Iis the defining ideal of sgeneric points then to prove\nthe hypothesis of Theorem 4.1it is sufficient to prove\n(2) /hatwideα(I)/greaterorequalslantN+ℓ+1\nN+1,where/parenleftbiggN+ℓ\nN/parenrightbigg\n<(N+1)s/lessorequalslant/parenleftbiggN+ℓ+1\nN/parenrightbigg\n.\nSecondly, we need the proper lower bounds for the Waldschmid t constant of the defining ideal\nof generic points for proving inequality 2. Using reduction methods from [ BN22], we establish\nreasonable lower bounds for the Waldschmidt constant in The orem4.4, which leads to Corollary\n4.6. Again, using reduction methods and splitting the points, a s in [DSS24,BN22], we establish\nreasonable lower bounds for the Waldschmidt constant in The orems5.1,5.2, and5.3, which lead to\nCorollary 5.4. Thus combining we prove Conjecture 1.2forsmany general points in the List (*).\nWe believe that our strategies can be used to prove Demailly’ s Conjecture 1.1for higher values of\nm. Following our method, onecan derive similar inequalities as (1) and(2). After that computation\nof adequate lower bounds for the Waldschmidt constant using methods in [ DSS24,BN22] or other\nmethods would lead to the desired conjectural bound 1.1.\nThis paper is outlined as follows. In Section 2, we recall all results needed for later sections.\nIn Section 3, we prove Conjecture 1.2for any set of very general points. In Section 4we prove\nConjecture 1.2for at least 2Nmany general points in PNwithN/greaterorequalslant6. In Section 5we prove\nthat any set of general points in PN,N= 3,4,5 satisfies Conjecture 1.2, except for the case\n(N= 5,10/lessorequalslants/lessorequalslant13).\n3Acknowledgments. The authors thank Huy T` ai H` a for his comments and suggestio ns. The\nsecondauthorispartially supportedbytheNAFOSTED(Vietn am) underthegrant number101.04-\n2023.07.\n2.Preliminaries\nIn this section, we collect important definitions and notati ons used in this paper. For unex-\nplained terminology, we refer the interested reader to the t ext [CHHVT20 ]. We will only work over\nalgebraically closed field of charateristic zero.\nDefinition 2.1. LetRbe a commutative ring and let I⊆Rbe an ideal. For m∈N, we define\nthem-th symbolic power ofIas\nI(m)=/intersectiondisplay\np∈Ass(I)(ImRp∩R).\nAlthough symbolic powers are notoriously hard to compute in general, symbolic powers of ideals\nof points have a nice description. It is well-known that if Xis the set {P1,...,P s} ⊆PN/CZofsmany\ndistinct points and pi⊆ /CZ[PN/CZ] is the defining ideal of Pi, thenI=p1∩···∩psis the ideal defining\nX, and the m-th symbolic power of Iis given by\nI(m)=pm\n1∩···∩pm\ns.\nTo study lower bounds for the initial degree of symbolic powe rs of ideals, the Waldschmidt\nconstant turned out to be very useful by the following definition-lemm a.\nDefinition 2.2. LetI⊆ /CZ[PN/CZ] be a homogeneous ideal and α(J) denote the initial degree of a\nhomogeneous ideal J. TheWaldschmidt constant ofIis defined as\n/hatwideα(I) := lim\nm→∞α(I(m))\nm= inf\nm∈Nα(I(m))\nm.\nSee, for example, [ BH10, Lemma 2.3.1].\nWe recall the statement of Demailly’s Conjecture in terms of the Waldschmidt constant.\nConjecture 2.3 (Demailly) .IfI⊆ /CZ[PN/CZ] is the defining ideal of a set of (reduced) points in PN/CZ,\nthen, for each m∈N,\n/hatwideα(I)/greaterorequalslantα(I(m))+N−1\nm+N−1.\nA direct approach to tackle this conjecture is to treat the tw o sides of the inequality separately.\nWe will see that the right-hand side, when m= 2 and the points are general, can be computed by\nthe Alexander-Hirschowitz Theorem, while the left-hand si de can be bounded by a combination of\nreduction techniques. We also define the following notation for later use.\nDefinition 2.4. Fori= 1,...s, letpibe the defining ideal of a point Pi∈X={P1,...,P s} ⊂\nPN/CZandm= (m1,...,m s) be a tuple of positive integers. The fat point scheme denote d by\nm1P1+m2P2+···+msPsis the scheme with the defining ideal\nI(m) =I(m1,...,m s) =pm1\n1∩pm2\n2∩···∩pmss.\nWhenmj/lessorequalslant0, we take pmj\nj= /CZ[PN/CZ]. We will also use the following notation:\nm×s= (m,m,...,m )/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nstimes.\nIf we want to emphasize the dimension of the projective space where the points live in, we also use\nthe notation I(PN,m) orI(PN,m×s).\n4It is known that the set of all collections of s(not necessarily distinct) points in PN/CZis parameter-\nized by the Chow variety G(1,s,N+1) of 0-cycles of degree sinPN/CZ(see [GKZ94]) (or the Hilbert\nscheme of spoints in PN). A property Pis said to hold for a generalset orvery general set ofs\npoints in PN/CZif it holds for all configuration of spoints in an open dense subset U⊆G(1,s,N+1)\nor in a countable intersection of such open dense sets, respe ctively.\nThe paper [ FMX18] offers an alternative way to describe general and very genera l points that\nprovides a practically useful way to approach Demailly’s co njecture and containment problem for\ngeneral and very general sets of points. Let ( zij)1/lessorequalslanti/lessorequalslants,0/lessorequalslantj/lessorequalslantNbes(N+1) new indeterminates. We\nusezandato denote the collections ( zij)1/lessorequalslanti/lessorequalslants,0/lessorequalslantj/lessorequalslantNand (aij)1/lessorequalslanti/lessorequalslants,0/lessorequalslantj/lessorequalslantN, respectively. Let /CZ(z)\nbe the field extension adjoining all zij,\nPi(z) := [zi0:···:ziN]∈PN/CZ(z),andX(z) :={P1(z),...,P s(z)}.\nWe call the set X(z) the set of sgenericpoints in PN/CZ(z). For any a∈As(N+1)/CZ, letPi(a) andX(a)\nbe obtained from Pi(z) andX(z), respectively, by setting zij=aijfor alli,j. There exists an open\ndense subset W0⊆As(N+1)/CZsuch that X(a) is a set of distinct points in PN/CZfor alla∈W0, and all\nsubsets of spoints in PN/CZarise in this way. It is shown that if a property Pholds for X(a) whenever\nain an open dense set U⊆As(N+1)/CZ(or a countable union of open dense subsets), the property P\nholds for a general (or very general, respectively) set of spoints in PN/CZ, see [FMX18, Lemma 2.3].\nMoreover, this construction behaves well and compatible wi th thespecialization process given in\nthe work of Krull [ Kru48], see also [ BGHN22b , Definition 2.8, Remarks 2.9, 2.10]. From now on,\nwe will use the terms general and very general as in this const ruction. We also collect the following\nwell-known facts about the Waldschmidt constant of a set of g eneric points for later use.\nLemma 2.5. If/hatwideα(s) :=/hatwideα(I(1×s))is the Waldschmidt constant the defining ideal of sgeneric\npoints in PN, then the followings are true\n(i)/hatwideα(s)/greaterorequalslant/hatwideα(k)whenever s/greaterorequalslantk.\n(ii)α(I(m×s))/greaterorequalslantm/hatwideα(s).\n(iii)/hatwideα/parenleftbig\nI/parenleftbig\n1×kN/parenrightbig/parenrightbig\n=k.[Eva05,DTG17]\n(iv)/hatwideα(s)/greaterorequalslant/hatwideα(J)for anyJdefining ideal of spoints.\nOur proof of Demailly’s Conjecture when m= 2 relies on the lower bound for the Waldschmidt\nconstant of a set of generic points and the upper bound of the C astelnouvo-Mumford regularity\nand the initial degree of the second symbolic power of the defi ning ideal of the set of generic points.\nTo study the former, we utilize the Cremona reduction process, see for instance, [ Dum12,Dum15,\nBN22], and gluing results from the Waldschmidt decomposition te chnique given in [ DSS24]. For\nthe latter, the celebrated Alexander-Hirschowitz Theorem provides bounds for the regularity value\nand the initial degree of the second symbolic power. We recal l these results here.\nDefinition 2.6. The Cremona transformation is the following birational map\nΦ :PN/CZ→PN/CZ,defined by Φ( x0:···:xN)/ma√sto→(x−1\n0:···:x−1\nN).\nWe recall some useful results from Cremona reductions.\nLemma 2.7. [BN22, Lemma3.1] ,[Dum12, Proposition8] LetI(m1,...,m s)be the ideal of sgeneric\npoints or general points with multiplicities m1,...,m s. If\nI(m1,...,m s)d/ne}ationslash= 0,thenI(m1+k,...,m N+1+k,mN+2,...,m s)d+k/ne}ationslash= 0,\nwherek= (N−1)d−/summationtextN+1\nj=1mj.\nLemma 2.8. [Dum15,Proposition10] Letm1,...,m r,m′\n1,...,m′\ns,t,kbe integers. If I(m1,...,m r)k=\n0andI(m′\n1,...,m′\ns,k+1)t= 0,thenI(m1,...,m r,m′\n1,...,m′\ns)t= 0.\n5We also have the following useful result whose proof using th e Waldschmidt decomposition.\nLemma 2.9 (Theorem 4.1 [ DSS24]).Denote/hatwideα(PN,r)the Waldschmidt constant of the ideal of r\nvery general points in PN. LetN/greaterorequalslant2andk/greaterorequalslant1. Assume that for some integers r1,...,rk+1and\nrational numbers a1,...,ak+1we have\n/hatwideα(PN−1,rj)/greaterorequalslantaj,for allj= 1,...,k+1,\nk/lessorequalslantaj/lessorequalslantk+1,forj= 1,...,k, a 1> k, a k+1/lessorequalslantk+1.\nThen,\n/hatwideα(PN,r1+...+rk+1)/greaterorequalslant\n1−k/summationdisplay\nj=11\naj\nak+1+k.\nLastly, we recall thecelebrated Alexander-Hirschowitz Th eorem proven ina series of work [ Ale88,\nAH92b,AH92a,AH95], and explain how it gives the regularity value and the initi al degree of the\nsecond symbolic powers.\nTheorem 2.10. LetI⊂R= /CZ[PN]be the defining ideal of sgeneral points in PN, then\nHFR/I(2)(d) = min/braceleftbigg/parenleftbiggd+N\nN/parenrightbigg\n,s(N+1)/bracerightbigg\n,\nwhereHFAdenotes the Hilbert function of a graded algebra A, except the following cases\n(i)d= 2,2/lessorequalslants/lessorequalslantN,\n(ii)d= 3,N= 4,s= 7,\n(iii)d= 4,2/lessorequalslantN/lessorequalslant4,s=/parenleftbigN+2\n2/parenrightbig\n−1.\nIn our context, the Castelnouvo-Mumford regularity of the d efining ideal of fat point schemes\ncan be interpreted as follows. First, for an ideal J, we define the postulation number ofR/Jto be\npost(R/J) = max{d∈Z|HFR/J(d)/ne}ationslash=HPR/J(d)}.\nInotherwords, post( R/J) is thelast valuewheretheHilbertfunctionandHilbert fun ctiondisagree.\nBy the following Serre’s formula,\nHFR/J(i)−HPR/J(i) =dim(R/J)/summationdisplay\nj=depth( R/J)(−1)jdimk(Hj\nm(R/J))i,∀i∈Z,\nwhenR/Jis Cohen-Macaulay, as depth( R/J) = dim( R/J), we have the following definition.\nDefinition 2.11. LetJ⊂R= /CZ[PN] be a Cohen-Macaulay homogeneous ideal. The regularity of\nR/Jis defined by reg( R/J) = post( R/J)+dim(R/J).\nLemma 2.12. LetI⊂R= /CZ[PN]be the defining ideal of sgeneral points, we have\nreg(R/I(2)) = post( R/I(2))+1 = min/braceleftBig\nd∈Z|HFR/I(2)(d) = HPR/I(2)(d) =s(N+1)/bracerightBig\n.\nIn particular, for each dsuch that HFR/I(2)(d) =s(n+1), we have reg(I(2))/lessorequalslantd+1.\nProof.The results follow since dim( R/I(2)) = 1 and reg( I(2)) = reg(R/I(2))+1. /square\nSimilarly, for the initial degree α(I(2)), it is clear that\nα(I(2)) = min/braceleftbigg\nd∈N|HFR/I(2)(d)/ne}ationslash=/parenleftbiggd+N\nN/parenrightbigg/bracerightbigg\n/lessorequalslantmin/braceleftbigg\nd∈N|s(N+1)</parenleftbiggd+N\nN/parenrightbigg/bracerightbigg\n.\nNote that equality holds except for some cases arising from t he exceptional cases in Theorem 2.10.\n63.Demailly’s Conjecture when m= 2for Very General Points\nInthissection, weshowthatDemailly’s Conjecturewhen m= 2holdsforanysetof sverygeneral\npoints when N/greaterorequalslant3. Demailly’s Conjecture was proved completely in P2, and when m= 2,N/greaterorequalslant3,\nit was proved for s/lessorequalslantN+2, see [NT19], ands/greaterorequalslant2Nnumberof very general points, see [ DSS24]. We\nwill show theconjecture for all remainingcases of very gene ral points, that is, when N+3/lessorequalslants/lessorequalslant2N.\nWefirstbeginwiththefollowinglemmagivingawaytocontrol theright-handsideoftheconjectural\ninequality.\nLemma 3.1. For a fix ℓ∈N, letIbe the defining ideal of a set of sgeneric points or very\ngeneral points such that/parenleftbiggN+ℓ\nN/parenrightbigg\n/lessorequalslant(N+1)s </parenleftbiggN+ℓ+1\nN/parenrightbigg\n. If/hatwideα(I)/greaterorequalslantN+ℓ\nN+1,then Demailly’s\nConjecture when m= 2holds for I, that is\n/hatwideα(I)/greaterorequalslantα(I(2))+N−1\nN+1.\nProof.If/parenleftbigN+ℓ\nN/parenrightbig\n/lessorequalslant(N+1)s </parenleftbigN+ℓ+1\nN/parenrightbig\nthen by Theorem 2.10,α/parenleftbig\nI(2)/parenrightbig\n/lessorequalslantℓ+1. Thus the inequality\n/hatwideα(I)/greaterorequalslantN+ℓ\nN+1implies Demailly’s inequality for each ℓ. /square\nTheorem 3.2. LetN/greaterorequalslant3andI⊂R= /CZ[PN]be the defining ideal of N+ 3/lessorequalslants/lessorequalslant2Ngeneric\npoints or very general points. Then, the Demailly’s Conjectur e holds for m= 2, that is,\n/hatwideα(I)/greaterorequalslantN−1+α(I(2))\nN+1.\nProof.The result for very general points follows from specializat ion the result for generic points,\nsee [FMX18]. Hence, we will assume that Iis the defining ideal of sgeneric points. Moreover, we\nwill prove the (stronger) results for N= 3,4 in Section 5, hence, we will assume here that N/greaterorequalslant5.\nBy Lemma 3.1, it suffices to show that\n/hatwideα(I)/greaterorequalslantN+ℓ\nN+1,\nwhere/parenleftbiggN+ℓ\nN/parenrightbigg\n/lessorequalslant(N+1)s </parenleftbiggN+ℓ+1\nN/parenrightbigg\nandℓ/greaterorequalslant2 as/parenleftbiggN+2\nN/parenrightbigg\n<(N+3)(N+1)/lessorequalslants(N+1).\nFrom [DHSTG14 , Proposition B.1.1], we have /hatwideα(I(1×N+3))/greaterorequalslantN+2\nN. Thus when, N+3/lessorequalslants, and/parenleftbigN+2\nN/parenrightbig\n/lessorequalslants(N+1)</parenleftbigN+3\nN/parenrightbig\nthen by Lemma 2.5, we have\n/hatwideα(I(1×s))/greaterorequalslant/hatwideα(I(1×N+3))/greaterorequalslantN+2\nN>N+2\nN+1.\nSimilarly, when N+3/lessorequalslants, and/parenleftbiggN+3\nN/parenrightbigg\n/lessorequalslant(N+1)s </parenleftbiggN+4\nN/parenrightbigg\n, we have\n/hatwideα(I(1×s))/greaterorequalslant/hatwideα(I(1×N+3))/greaterorequalslantN+2\nN>N+3\nN+1,\nas desired. On the other hand, we have 2N(N+1)/lessorequalslant/parenleftbigg2N\nN/parenrightbigg\nfor allN/greaterorequalslant5. Therefore, it suffices\nnow to show that /hatwideα(I)/greaterorequalslantN+ℓ\nN+1,where/parenleftbiggN+ℓ\nN/parenrightbigg\n/lessorequalslant(N+1)s </parenleftbiggN+ℓ+1\nN/parenrightbigg\nand 4/lessorequalslantℓ/lessorequalslantN−1.\nWe will use induction on N. The base cases N= 3 and 4 will be proved in Section 5as mentioned\nearlier. We consider the following two cases.\n7Case 1: Whenℓ/lessorequalslant3+√\n4N+5\n2. We first claim that for such ℓ, we have\n/parenleftbiggN+ℓ\nN/parenrightbigg\n/greaterorequalslant(N+1)/parenleftbiggN+ℓ−2\nN/parenrightbigg\n.\nIn fact, the above inequality is equivalent to ( N+ℓ)(N+ℓ−1)/greaterorequalslant(N+1)ℓ(ℓ−1), which is equivalent\nto\nℓ2−3ℓ−(N−1)/lessorequalslant0,\nhence, is equivalent to3−√\n4N+5\n2/lessorequalslantℓ/lessorequalslant3+√\n4N+5\n2. Thus, the claim follows since we assume\nℓ/greaterorequalslant2>3−√4N+5\n2.\nNow, for ℓ/lessorequalslant3+√\n4N+5\n2, if/parenleftbiggN+ℓ\nN/parenrightbigg\n/lessorequalslant(N+1)s, we have that s/greaterorequalslant/parenleftbiggN+ℓ−2\nN/parenrightbigg\n. Sinceℓ/greaterorequalslant4,\nwe have ℓ−2/greaterorequalslant2, thus, by the same proof as that of [ BN22, Lemma 4.5, Lemma 4.9, Theorem\n4.10], we have that\n/hatwideα(I)/greaterorequalslantN+ℓ−1\nN>N+ℓ\nN+1,\nas desired.\nCase 2: Whenℓ >3+√\n4N+5\n2. We first claim that for such ℓ, we have\n/parenleftbigN+ℓ\nN/parenrightbig\nN+1/greaterorequalslant/parenleftbigN−1+ℓ\nN−1/parenrightbig\nN+/parenleftbigN−1+ℓ−1\nN−1/parenrightbig\nN+1.\nThe inequality is equivalent to\n(N+ℓ−2)!\nℓ!(N+1)!(ℓ2−3ℓ−N+1)/greaterorequalslant1.\nNote that ℓ2−3ℓ−(N−1)>0 if and only if ℓ >3+√\n4N+5\n2since3−√\n4N+5\n2<0 when\nN/greaterorequalslant5. Suppose that ℓ=ℓ1=⌊ℓ0⌋+1, where ℓ0=3+√\n4N+5\n2. Then\n(N+ℓ1−2)!\nℓ1!(N+1)!(ℓ2\n1−3ℓ1−N+1) =(N+ℓ1−2)···(N+2)(ℓ2\n1−3ℓ1−N+1)\nℓ1(ℓ1−1)(ℓ1−2)···3·2·1\n=(N+ℓ1−2)\nℓ1·(N+ℓ1−3)\nℓ1−1···(N+4)\n6·(N+3)\n4·3·(N+2)\n5·2·(ℓ2\n1−3ℓ1−N+1).\nSinceℓ2\n1−3ℓ1−N+1>0anditisaninteger, wehave ℓ2\n1−3ℓ1−N+1/greaterorequalslant1. IfN/greaterorequalslant9theneach term\nin the product is at least 1, and thus(N+ℓ1−2)!\nℓ1!(N+1)!(ℓ2\n1−3ℓ1−N+1)/greaterorequalslant1. IfN= 5,6,7 or 8, we have\nthatℓ1= 5, and one can check directly that for these values,(N+ℓ1−2)!\nℓ1!(N+1)!(ℓ2\n1−3ℓ1−N+1)/greaterorequalslant1.\nNow, when ℓ/greaterorequalslant⌊ℓ0⌋+2, we have ℓ/greaterorequalslantℓ0+1 and by writing\n(N+ℓ−2)!\nℓ!(N+1)!(ℓ2−3ℓ−N+1) =(N+ℓ−2)\nℓ...(N+3)\n5·(N+2)\n4·(ℓ+ℓ0)\n6·(ℓ−ℓ0),\nwe can see that all the factors of the right hand side are great er than 1, as ℓ+ℓ0/greaterorequalslant2ℓ0/greaterorequalslant8 when\nN/greaterorequalslant5. Hence, the claim is proved.\n8Thus, for ℓ >3+√\n4N+5\n2, if/parenleftbiggN+ℓ\nN/parenrightbigg\n/lessorequalslant(N+1)s, since\n/parenleftbigN+ℓ\nN/parenrightbig\nN+1/greaterorequalslant/parenleftbigN−1+ℓ\nN−1/parenrightbig\nN+/parenleftbigN−1+ℓ−1\nN−1/parenrightbig\nN+1,\nwe have that\ns/greaterorequalslant/ceilingleftBigg/parenleftbigN+ℓ\nN/parenrightbig\nN+1/ceilingrightBigg\n/greaterorequalslant/ceilingleftBigg/parenleftbigN−1+ℓ\nN−1/parenrightbig\nN+/parenleftbigN−1+ℓ−1\nN−1/parenrightbig\nN/ceilingrightBigg\n+1/greaterorequalslant/ceilingleftBigg/parenleftbigN−1+ℓ\nN−1/parenrightbig\nN/ceilingrightBigg\n+/ceilingleftBigg/parenleftbigN−1+ℓ−1\nN−1/parenrightbig\nN/ceilingrightBigg\n.\nTherefore, we can split sintor1/greaterorequalslant/ceilingleftBigg/parenleftbigN−1+ℓ\nN−1/parenrightbig\nN/ceilingrightBigg\nandr2/greaterorequalslant/ceilingleftBigg/parenleftbigN−1+ℓ−1\nN−1/parenrightbig\nN/ceilingrightBigg\n. Considering r1andr2very\ngeneral points in PN−1, by the induction hypothesis, we have\n/hatwideα(PN−1,r1)/greaterorequalslantN−1+ℓ\nN=a1and/hatwideα(PN−1,r2)/greaterorequalslantN−2+ℓ\nN=a2.\nNote that a1>1 anda2/lessorequalslant2 asℓ/lessorequalslantN−1, hence, by Lemma 2.9, we get\n/hatwideα(PN,s)/greaterorequalslant/parenleftbigg\n1−N\nN−1+ℓ/parenrightbiggN−2+ℓ\nN+1/greaterorequalslantN+ℓ\nN+1,\nwhere the last inequality is equivalent to ℓ2−3ℓ+ 2/greaterorequalslant0, which is true for all ℓ. This gives the\ndesired inequality for generic points as well by Lemma 2.5(iv). /square\n4.Ideal Containment and Demailly’s Conjecture for Sufficient ly Many General\nPoints\nIn this section, we will show that Demailly’s Conjecture whe nm= 2 holds for any set of at least\n2Nmany general points when N/greaterorequalslant6. Our method is via ideal containment. Harbourne-Huneke\nproposed the following containment for the defining ideal Iof a set of points in PN.\nI(r(m+N−1))⊂mr(N−1)/parenleftBig\nI(m)/parenrightBig\n,for allrandm/greaterorequalslant1.\nThe aforementioned containment implies Demailly’s Conjec tural bound. Our aim is to prove that\nfor the defining ideal Iofsgeneric points, there exists an integer rsuch that we have the following\nstronger containment\nI(r(N+1)−N+1)⊆mr(N−1)/parenleftBig\nI(2)/parenrightBigr\n,\nthen follow the strategy in [ BGHN22a , Theorem 2.9]. To prove this containment, using the same\nidea as in [ BCH14, Lemma 2.6], it suffices to show that α(I(r(N+1)−N+1))/greaterorequalslantrreg(I(2))+r(N−1).\nTheorem 4.1. LetI⊂R= /CZ[PN]be the defining ideal of sgeneric points. If\n/hatwideα(I)/greaterorequalslantreg(I(2))+N−1\nN+1,\nthen there exists an integer rsuch that\nI(r(N+1)−N+1)⊆mr(N−1)/parenleftBig\nI(2)/parenrightBigr\n.\nProof.We have\nlim\nr→∞α(I(r(N+1)−N+1))\nr(N+1)= lim\nr→∞α(I(r(N+1)−N+1))\nr(N+1)−N+1=/hatwideα(I)/greaterorequalslantreg(I(2))+N−1\nN+1.\n9Thus, there exists rsuch that,\nα(I(r(N+1)−N+1))\nr(N+1)/greaterorequalslantreg(I(2))+N−1\nN+1,\nhence,α(I(r(N+1)−N+1))/greaterorequalslantrreg(I(2))+r(N−1).This implies I(r(N+1)−N+1)⊆mr(N−1)/parenleftbig\nI(2)/parenrightbigr./square\nLemma 4.2. For a fix ℓ∈N, letIbe the defining ideal of a set of sgeneric points or very general\npoints such that/parenleftbiggN+ℓ\nN/parenrightbigg\n<(N+1)s/lessorequalslant/parenleftbiggN+ℓ+1\nN/parenrightbigg\n. If/hatwideα(I)/greaterorequalslantN+ℓ+1\nN+1,then the inequality,\n/hatwideα(I)/greaterorequalslantreg(I(2))+N−1\nN+1,\nholds. Moreover, Demailly’s Conjecture when m= 2holds for a set of sgeneral points in PN.\nProof.If/parenleftbigN+ℓ\nN/parenrightbig\n<(N+1)s/lessorequalslant/parenleftbigN+ℓ+1\nN/parenrightbig\nthen by Lemma 2.12, reg/parenleftbig\nI(2)/parenrightbig\n/lessorequalslantℓ+2. Thus the inequality\n/hatwideα(I)/greaterorequalslantN+ℓ+1\nN+1implies the inequality /hatwideα(I)/greaterorequalslantreg(I(2))+N−1\nN+1for allℓ. By Theorem 4.1, the stable\ncontainment I(r(N+1)−N+1)⊆mr(N−1)/parenleftbig\nI(2)/parenrightbigrholds for sgeneric points, hence, by the proof of\n[BGHN22a , Theorem 2.9], the Demailly’s Conjecture when m= 2 holds for a set of sgeneral\npoints in PN. /square\nLemma 4.3. The following inequalities hold:\n(a) IfN/greaterorequalslant5, then4N(N+1)/lessorequalslant/parenleftbigg3N+2\nN/parenrightbigg\n.\n(b) IfN/greaterorequalslant11, then3N(N+1)/lessorequalslant/parenleftbigg2N+2\nN/parenrightbigg\n.\n(c) IfN/greaterorequalslant7, then3N(N+1)/lessorequalslant/parenleftbigg2N+3\nN/parenrightbigg\n.\nProof.(a).ForN= 5, we can check that 45·6 = 6144, and/parenleftbig3·5+2\n5/parenrightbig\n= 6188. Now assume that\n4N(N+1)/lessorequalslant/parenleftbig3N+2\nN/parenrightbig\nholds for an arbitrary N/greaterorequalslant5. Now\n/parenleftbigg3(N+1)+2\nN+1/parenrightbigg\n=(3N+5)(3N+4)(3N+3)\n(N+1)(2N+4)(2N+3)/parenleftbigg3N+2\nN/parenrightbigg\n/greaterorequalslant(3N+5)(3N+4)(3N+3)\n(N+1)(2N+4)(2N+3)·4N(N+1)\n=(3N+5)(3N+4)(3N+3)\n(2N+4)(2N+3)4N\n/greaterorequalslant4N+1(N+2).\nThus the result follows by induction. The proofs for parts ( b) and (c) are essentially the same. /square\nTheorem 4.4. LetN/greaterorequalslant6, andIbe the defining ideal of sgeneric points in PN, where2N/lessorequalslants/lessorequalslant4N.\nThen the following holds\n/hatwideα(I)/greaterorequalslantreg(I(2))+N−1\nN+1.\nProof.If the number of points satisfies 3N/lessorequalslants/lessorequalslant4N, then by Lemma 4.3(a) we have s(N+1)/lessorequalslant/parenleftbiggN+2N+2\nN/parenrightbigg\n. Hence, by Lemma 2.12, we have reg( I(2))/lessorequalslant2N+3. The inequality follows as in\nthis case, by Lemma 2.5,/hatwideα(I)/greaterorequalslant3>N−1+2N+3\nN+1.\n10Now we consider the case where 2N/lessorequalslants/lessorequalslant3N. By Lemma 4.2, we only need to prove that\n(3)/hatwideα(I)/greaterorequalslantN−1+ℓ+2\nN+1=N+ℓ+1\nN+1whenever/parenleftbiggN+ℓ\nN/parenrightbigg\n<(N+1)s/lessorequalslant/parenleftbiggN+ℓ+1\nN/parenrightbigg\n.\nNote that 3N(N+1)/lessorequalslant/parenleftbig2N+2\nN/parenrightbig\n, for allN/greaterorequalslant11by Lemma 4.3(b). Thuswehave s(N+1)/lessorequalslant/parenleftbigN+N+2\nN/parenrightbig\nwhenever N/greaterorequalslant11, and by Lemma 2.12, we get reg( I(2))/lessorequalslantN+3. Hence, by Lemma 2.5, we get\n/hatwideα(I)/greaterorequalslant2 =N−1+N+3\nN+1,for allN/greaterorequalslant11.\nNow we consider the cases when 6 /lessorequalslantN/lessorequalslant10.\nCase 1:WhenN= 6, it is enough to check inequality ( 3) for 4/lessorequalslantℓ/lessorequalslant9 (and 26/lessorequalslants/lessorequalslant36).\n(i) When ℓ= 6 or 7, since s/greaterorequalslant26, then by Lemma 2.5, we have /hatwideα(I(1×s))/greaterorequalslant2/greaterorequalslant6+ℓ+1\n6+1.\n(ii) When ℓ= 8, then s/greaterorequalslant429 = 26·6 + 45. We first show that /hatwideα/parenleftbig\nI/parenleftbig\n2×6,1×45/parenrightbig/parenrightbig\n/greaterorequalslant22\n10.\nAssume that I/parenleftbig\n20m×6,10m×45/parenrightbig\n22m−1/ne}ationslash= 0. Let k= 5(22m−1)−(6·20m+ 10m) =\n−20m−5 and applying Lemma 2.7, we get I/parenleftbig\n10m×44/parenrightbig\n2m−6/ne}ationslash= 0, which is a contradiction.\nThus/hatwideα/parenleftbig\nI/parenleftbig\n2×6,1×45/parenrightbig/parenrightbig\n/greaterorequalslant22\n10. Hence, by [ BN22, Proposition 3.6] we have /hatwideα(I(1×s))/greaterorequalslant\n/hatwideα/parenleftBig\nI/parenleftBig\n1×6·26,1×45/parenrightBig/parenrightBig\n/greaterorequalslant/hatwideα/parenleftbig\nI/parenleftbig\n2×6,1×45/parenrightbig/parenrightbig\n/greaterorequalslant22\n10>6+8+1\n6+1.\n(iii) When ℓ= 9, then s/greaterorequalslant715/greaterorequalslant7×26. Thus by [ BN22, Theorem 3.2]\n/hatwideα/parenleftbig\nI(1×s)/parenrightbig\n/greaterorequalslant/hatwideα/parenleftBig\nI(1×7·26)/parenrightBig\n/greaterorequalslant2/hatwideα(I(1×7)) = 2·7\n6>6+9+1\n6+1.\nCase 2:When 7/lessorequalslantN/lessorequalslant10, it is enough to check inequality ( 3) for 2N/lessorequalslants/lessorequalslant3Nandℓ/lessorequalslantN+2 as\n3N(N+1)/lessorequalslant/parenleftbig2N+3\nN/parenrightbig\nfor allN/greaterorequalslant7 by Lemma 4.3(c).\n(i) When ℓ/lessorequalslantN+1, and s/greaterorequalslant2Nwe have /hatwideα(I(1×s))/greaterorequalslant2/greaterorequalslantN+ℓ+1\nN+1.\n(ii) When ℓ=N+2, thenwecancheck directlythatthenumberofpointssatis fiess/greaterorequalslant2N(N+1)\nwith these values of N. Thus by [ BN22, Theorem 3.2]\n/hatwideα/parenleftbig\nI(1×s)/parenrightbig\n/greaterorequalslant/hatwideα/parenleftBig\nI(1×2N(N+1))/parenrightBig\n/greaterorequalslant2/hatwideα/parenleftBig\nI/parenleftBig\n1×(N+1)/parenrightBig/parenrightBig\n= 2·N+1\nN>N+N+2+1\nN+1.\n/square\nCorollary 4.5. Suppose that N/greaterorequalslant3, and let Ibe the defining ideal of sgeneric points in PN,\nwheres/greaterorequalslant2N. Then\n/hatwideα(I)/greaterorequalslantreg(I(2))+N−1\nN+1.\nProof.The case where N= 3,4 and 5 will be proved in Section 5. The case N/greaterorequalslant6 follows directly\nfrom Theorem 4.4and by the proof of [ BGHN22a , Lemma 2.8] and [ BGHN22a , Remark 2.10]. /square\nCorollary 4.6. Suppose that N/greaterorequalslant3, and let Ibe the defining ideal of sgeneral points in PN,\nwheres/greaterorequalslant2N. ThenIverifies Demailly’s Conjecture when m= 2, that is,\n/hatwideα(I)/greaterorequalslantα(I(2))+N−1\nN+1.\nProof.The proof goes along the same line as that of [ BGHN22a , Theorem 2.9]. /square\n115.Ideal Containment and Demailly’s Conjecture for General Poi nts in Low\nDimension\nIn this section, we shall show Demailly’s Conjecture when m= 2 for any set of general points in\nP3,P4andP5, except four numbers of points in P5where the result is still not known. Note that\nthe results in this section are used for the base cases in the p roof of Theorem 3.2. In the following,\nas the conjecture was shown to be true in several cases: small number of general points in [ NT19]\nand sufficiently many general points in [ BGHN22a ], we will only consider remaining cases.\nTheorem 5.1. LetIbe the defining ideal of sgeneric points in P3. If6/lessorequalslants/lessorequalslant63= 216, then\n/hatwideα(I)/greaterorequalslant2+reg(I(2))\n4.\nProof.Since 20 =/parenleftbig3+3\n3/parenrightbig\n<4·6</parenleftbig3+4\n3/parenrightbig\n= 35, and 816 =/parenleftbig3+15\n3/parenrightbig\n<4·63</parenleftbig3+16\n3/parenrightbig\n= 969, then by\nLemma4.2, it suffices to show that for 3 /lessorequalslantℓ/lessorequalslant15,\n/hatwideα(I)/greaterorequalslantN−1+ℓ+2\nN+1=4+ℓ\n4,\nwhereIis the defining ideal of sgeneric points with/parenleftbig3+ℓ\n3/parenrightbig\n<4s/lessorequalslant/parenleftbig3+ℓ+1\n3/parenrightbig\n. We consider case by\ncase as follows.\nCase 5.1.1: ℓ= 3,s= 6,7,8.By [DHSTG14 , Proposition B.1.1], we have /hatwideα(I)/greaterorequalslant/hatwideα(6)/greaterorequalslant5\n3>4+3\n5.\nCase 5.1.2: ℓ= 4,9/lessorequalslants/lessorequalslant14.Sinces >23, we have /hatwideα(I)/greaterorequalslant2, thus,/hatwideα(I)/greaterorequalslant2 =4+4\n4.\nCase 5.1.3: ℓ= 5,15/lessorequalslants/lessorequalslant21.By [Dum15, Proposition 11], /hatwideα(I)/greaterorequalslant/hatwideα(14)/greaterorequalslant7\n3>4+5\n4.\nCase 5.1.4: ℓ= 6,22/lessorequalslants/lessorequalslant30.By [Dum15, Proposition 11], /hatwideα(I)/greaterorequalslant/hatwideα(21)/greaterorequalslant8\n3>4+6\n4.\nCase 5.1.5: ℓ= 7,8,30/lessorequalslants/lessorequalslant55.Sinces >33, we have /hatwideα(I)/greaterorequalslant3, thus/hatwideα(I)/greaterorequalslant3 =4+8\n4.\nCase 5.1.6: ℓ= 9,56/lessorequalslants/lessorequalslant71.The result holds as\n/hatwideα(I)/greaterorequalslant/hatwideα(56)/greaterorequalslant2/hatwideα(7) =56\n15>4+9\n4.\nwhere/hatwideα(7)/greaterorequalslant28\n15holds by [ Dum15, Proposition 11] and /hatwideα(56)/greaterorequalslant2/hatwideα(7) holds by [ BN22, Theorem\n3.2].\nCase 5.1.7: 10/lessorequalslantℓ/lessorequalslant12,72/lessorequalslants/lessorequalslant140.Sinces >43, we have /hatwideα(I)/greaterorequalslant4, thus,/hatwideα(I)/greaterorequalslant4 =4+12\n4.\nCase 5.1.8: 13/lessorequalslantℓ/lessorequalslant15,141/lessorequalslants/lessorequalslant216.Sinces >53, we have /hatwideα(I)/greaterorequalslant5, thus,/hatwideα(I)/greaterorequalslant5>4+15\n4.\n/square\nTheorem 5.2. LetIbe the defining ideal of sgeneric points in P4. If8/lessorequalslants/lessorequalslant54= 625, then\n/hatwideα(I)/greaterorequalslant3+reg(I(2))\n5.\nProof.Since 35 =/parenleftbig4+3\n4/parenrightbig\n<5·8<70 =/parenleftbig4+4\n4/parenrightbig\nand 3060 =/parenleftbig4+14\n4/parenrightbig\n<5·625</parenleftbig4+15\n4/parenrightbig\n= 3876, then by\nLemma4.2it suffices to show that for 3 /lessorequalslantℓ/lessorequalslant14,\n/hatwideα(I)/greaterorequalslantN−1+ℓ+2\nN+1=5+ℓ\n5,\n12whereIis the defining ideal of sgeneric points with/parenleftbig4+ℓ\n4/parenrightbig\n<5s/lessorequalslant/parenleftbig4+ℓ+1\n4/parenrightbig\n. We again consider case\nby case as follows.\nCase 5.2.1: ℓ= 3,8/lessorequalslants/lessorequalslant14.By [BN22, Lemma 3.9], /hatwideα(I)/greaterorequalslant/hatwideα(8)/greaterorequalslant8\n5=5+3\n5.\nCase 5.2.2: ℓ= 4,15/lessorequalslants/lessorequalslant25.For finding the right lower bound for /hatwideα(I), we will use the similar\ntrick to what was used in [ BN22, Lemma 4.5]. Since /hatwideα(1×6)/greaterorequalslant6\n4=30\n20, we have\nI((20m)×6)30m−1= 0,for allm.\nWe claim that\nI/parenleftbig\n(20m)×9,(30m)×1/parenrightbig\n36m−1= 0,∀m.\nApplying Lemma 2.7for three times with largest possible multiplicities prove s the claim. Indeed,\neach time the reductor k= 3d−/summationtext5\ni=1miaffects the coefficients of minmianddsignificantly. In\nthe following table we capture the coefficient of mink, and change of coefficients of minmiand\nd. The change in the constant term does not affect so we omit them i n the table.\ndm1m2m3m4m5m6m7m8m9m10k\n36302020202020 20 20 20 20 −2\n342818 18 18 18 20 20202020−6\n282218181818 14 14 14 14 20 −12\n161066618 14 14 14 14 8\nFrom the last row, since, the coefficient of min the degree dis less than that of the fifth point, we\nconclude that I/parenleftbig\n(20m)×9,(30m)×1/parenrightbig\n36m−1= 0,∀m. Now, by applying the Lemma 2.8one time, we\ngetI((20m)×15)36m−1= 0, for all m, therefore, /hatwideα(I)/greaterorequalslant/hatwideα(15)/greaterorequalslant36\n20=5+4\n5.\nCase 5.2.3: ℓ= 5,26/lessorequalslants/lessorequalslant42.Sinces/greaterorequalslant24, we have /hatwideα(I)/greaterorequalslant2, thus/hatwideα(I)/greaterorequalslant2 =5+5\n5.\nCase 5.2.4: ℓ= 6,43/lessorequalslants/lessorequalslant66.Since/hatwideα(1×6)/greaterorequalslant6\n4=30\n20, we have\nI((20m)×6)30m−1= 0,for allm.\nWe claim that I/parenleftbig\n(20m)×1,(30m)×7/parenrightbig\n44m−1= 0,∀m.In fact, if it is not the case for some m, by\napplying Lemma 2.7withk= 3(44m−1)−5·30m=−18m−3, we can reduce to\nI/parenleftbig\n(20m)×1,(30m)×2,(12m−3)×5/parenrightbig\n26m−4/ne}ationslash= 0,\nwhich is a contradiction as 26 m−4<30m. Now, by applying the Lemma 2.8seven times, we get\nI((20m)×43)44m−1= 0, for all m, therefore, /hatwideα(I)/greaterorequalslant/hatwideα(43)/greaterorequalslant44\n20=5+6\n5.\nCase 5.2.5: ℓ= 7,67/lessorequalslants/lessorequalslant99.Again, since /hatwideα(1×6)/greaterorequalslant6\n4=30\n20, we have\nI((20m)×6)30m−1= 0,for allm.\nWe claim that I/parenleftbig\n(20m)×1,(30m)×11/parenrightbig\n48m−1= 0,∀m.Similarly, we apply Lemma 2.7for two times\nwith largest possible multiplicities proves the claim. The following table we capture the coefficient\nofmink, and change of coefficients of minmiandd. Again, the change in the constant term\ndoes not affect so we omit thoses in the table.\n13dm1m2m3m4m5m6m7m8m9m10m11m12k\n48303030303030 30 30 30 30 30 20 −6\n4224 24 24 24 24 30 3030303030 20 −24\n1824 24 24 24 24 6 6 6 6 6 30 20 −12\nFrom the last row, since, the coefficient of min the degree dis less than that of the first point,\nwe conclude that I/parenleftbig\n(20m)×1,(30m)×11/parenrightbig\n48m−1= 0,∀m.Now, by applying the Lemma 2.8eleven\ntimes, we get I((20m)×67)48m−1= 0, for all m, therefore, /hatwideα(I)/greaterorequalslant/hatwideα(67)/greaterorequalslant48\n20=5+7\n5.\nCase 5.2.6: 8/lessorequalslantℓ/lessorequalslant10,100/lessorequalslants/lessorequalslant273.Sinces >34, we have /hatwideα(I)/greaterorequalslant3, thus/hatwideα(I)/greaterorequalslant3 =5+10\n5.\nCase 5.2.7: 11/lessorequalslantℓ/lessorequalslant14,274/lessorequalslants/lessorequalslant625.Sinces >44, we have /hatwideα(I)/greaterorequalslant4, thus/hatwideα(I)/greaterorequalslant4>5+14\n5.\n/square\nTheorem 5.3. LetIbe the defining ideal of sgeneric points in P5. If8/lessorequalslants/lessorequalslant45= 1024, except\n10/lessorequalslants/lessorequalslant13then\n/hatwideα(I)/greaterorequalslant4+reg(I(2))\n6.\nThe result for 10/lessorequalslants/lessorequalslant13is not known.\nProof.Since 21 =/parenleftbig5+2\n5/parenrightbig\n<6·8<56 =/parenleftbig5+3\n5/parenrightbig\nand 4368 =/parenleftbig5+11\n5/parenrightbig\n<6·1024</parenleftbig5+12\n5/parenrightbig\n= 6188, then\nby Lemma 4.2it suffices to show that for 2 /lessorequalslantℓ/lessorequalslant11,\n/hatwideα(I)/greaterorequalslantN−1+ℓ+2\nN+1=6+ℓ\n6,\nwhereIis the defining ideal of sgeneric points with/parenleftbig5+ℓ\n5/parenrightbig\n<6s/lessorequalslant/parenleftbig5+ℓ+1\n5/parenrightbig\n. We consider case by\ncase as follows.\nCase 5.3.1: ℓ= 2,s= 8,9.By [DHSTG14 , Proposition B.1.1], we have /hatwideα(P5,8)/greaterorequalslant5+2\n5>6+2\n6.\nCase 5.3.2: ℓ= 3and14/lessorequalslants/lessorequalslant21.Applying Lemma 2.9using Theorem 5.2as follows:\npickr1= 8,since/parenleftbigg4+3\n4/parenrightbigg\n<5r1/lessorequalslant/parenleftbigg4+4\n4/parenrightbigg\n,we can pick a1=5+3\n5=8\n5,and\nr2= 6,using the fact that /hatwideα(PN,N+2)/greaterorequalslantN+2\nN,we can pick a2=4+2\n4=6\n4,thus, we have\n/hatwideα(P5,14)/greaterorequalslant(1−1\na1)a2+1 = (1 −5\n8)6\n4+1 = 1.5625>1.5 =6+3\n6.\nCase 5.3.3: ℓ= 4and22/lessorequalslants/lessorequalslant42.Similarly, applying Lemma 2.9using Theorem 5.2as follows:\npickr1= 15,since 70 =/parenleftbigg4+4\n4/parenrightbigg\n<5r1/lessorequalslant/parenleftbigg4+5\n4/parenrightbigg\n,we can pick a1=5+4\n5=9\n5,and\nr2= 7,using the fact that /hatwideα(PN,N+2)/greaterorequalslantN+2\nN,we can pick a2=4+2\n4=6\n4,thus, we have\n/hatwideα(P5,22)/greaterorequalslant(1−1\na1)a2+1 = (1 −5\n9)6\n4+1 =6+4\n6.\nCase 5.3.4: ℓ= 5,6and43/lessorequalslants/lessorequalslant132.Sinces/greaterorequalslant42>25we have, /hatwideα(P5,s)/greaterorequalslant2>6+6\n6.\nCase 5.3.5: ℓ= 7and133/lessorequalslants/lessorequalslant214.Again, applying Lemma 2.9using Theorem 5.2as follows:\npickr1= 67,since 330 =/parenleftbigg4+7\n4/parenrightbigg\n<5r1/lessorequalslant/parenleftbigg4+8\n4/parenrightbigg\n,we can pick a1=5+7\n5=12\n5,\n14r2= 43,since 210 =/parenleftbigg4+6\n4/parenrightbigg\n<5r2/lessorequalslant/parenleftbigg4+5\n4/parenrightbigg\n,we can pick a2=5+6\n5=11\n5,and\nr3= 15,since 70 =/parenleftbigg4+4\n4/parenrightbigg\n<5r3/lessorequalslant/parenleftbigg4+5\n4/parenrightbigg\n,we can pick a3=5+4\n5=9\n5,thus, we have\n/hatwideα(P5,133)/greaterorequalslant/hatwideα(P5,125)/greaterorequalslant/parenleftbigg\n1−/parenleftbigg1\na1+1\na2/parenrightbigg/parenrightbigg\na3+1 =/parenleftbigg\n1−/parenleftbigg5\n11+5\n12/parenrightbigg/parenrightbigg9\n5+2>2.23>6+7\n6.\nCase 5.3.6: ℓ= 8and215/lessorequalslants/lessorequalslant333.By [BN22, Theorem 3.2], we have\n/hatwideα(P5,215)/greaterorequalslant/hatwideα(P5,6·32)/greaterorequalslant2/hatwideα(6)/greaterorequalslant2·6\n5>6+8\n6.\nCase 5.3.8: 9/lessorequalslantℓ/lessorequalslant11and334/lessorequalslants/lessorequalslant1024.Sinces/greaterorequalslant334>35, we have /hatwideα(P5,s)/greaterorequalslant3>6+11\n6.\n/square\nAs a consequence of Theorems 4.1,5.1,5.2, and5.3, we deduce the following corollary regarding\ngeneral points in PNforN= 3,4, and 5.\nCorollary 5.4. LetIbe the defining ideal of sgeneral points in PN. Ifssatisfies the following\nconditions\n(i)6/lessorequalslants/lessorequalslant63= 216, whenN= 3;\n(ii)8/lessorequalslants/lessorequalslant54= 624, whenN= 4;\n(iii)s= 8,9and14/lessorequalslants/lessorequalslant45= 1024, whenN= 5;\nthen\n/hatwideα(I)/greaterorequalslantα(I(2))+N−1\nN+1.\nThe case 10/lessorequalslants/lessorequalslant13inP5is not known.\nProof.The proof has the same structure as [ BGHN22a , Theorem 2.9]. /square\nRemark 5.5. Combined results in [ BGHN22a ,DSS24,NT19], Demailly’s Conjecture when m= 2\nnow holds for any set of very general points, any set of at leas t 2Ngeneral points, and any set of\npoints in P3,P4, andP5, except for 10 /lessorequalslants/lessorequalslant13 inP5.\nReferences\n[Ale88] J. Alexander. Singularit´ es imposables en positio ns g´ en´ erale ` a une hypersurface projective. Compositio\nMath., 68:305–354, 1988. 6\n[AH92b] J. Alexander and A. Hirschowitz. 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The initial degree of symbolic powers of Fermat-like id eals of planes and lines arrange-\nments.Comm. Algebra , 51(1):29–45, 2023. 2\n[Ngu23b] T. T. Nguy ˜ˆ en. Initial degree of symbolic powers of ideals of Fermat co nfigurations of points. Rocky\nMountain J. Math. , 53(3):859–874, 2023. 2\n[NT19] U. Nagel and B. Trok. Interpolation and the weak Lefsc hetz property. Trans. Amer. Math. Soc. ,\n372(12):8849–8870, 2019. 1,2,7,12,15\n[TX20] S. Tohaneanu and Y. Xie. On the containment problem fo r fat points ideals. Proc. Amer. Math. Soc. ,\n148(6):2411–2419, 2020. 2\n[Wal77] M. Waldschmidt. Propri´ et´ es arithm´ etiques de fo nctions de plusieurs variables. II. In S´ eminaire Pierre\nLelong (Analyse) ann´ ee 1975/76 , pages 108–135. Lecture Notes in Math., Vol. 578. 1977. 2\n16Arizona State University, School of Mathematical and Stati stical Sciences, Tempe, AZ 85287-1804,\nUSA\nEmail address :sankhaneel.bisui@asu.edu, sbisui@tulane.edu\nMcMaster University, Department of Mathematics and Statis tics, 1280 Main Street West, Hamil-\nton, Ontario, Canada, and University of Education, Hue Univ ersity, 34 Le Loi St., Hue, Viet Nam\nEmail address :nguyt161@mcmaster.ca\n17"    },    {        "title": "2401.11301v1.Nonequilibrium_dynamics_of_the_Hubbard_dimer.pdf",        "content": "Nonequilibrium dynamics of the Hubbard dimer\nYaroslav Pavlyukh\nInstitute of Theoretical Physics, Faculty of Fundamental Problems of Technology,\nWroclaw University of Science and Technology, 50-370 Wroclaw, Poland\nyaroslav.pavlyukh@pwr.edu.pl\nKeywords: Nonequilibrium Green’s function theory, generalized Kadanoff-Baym Ansatz, excited states\nElectron dynamics in a two-sites Hubbard model is studied using the nonequilibrium Green’s function approach. The study is moti-\nvated by the empirical observation that a full solution of the integro-differential Kadanoff-Baym equation (KBE) is more stable and\noften accompanied by artificial damping [Marc Puig von Friesen, C. Verdozzi, and C.-O. Almbladh (2009)] than its time-linear refor-\nmulations relying on the generalized Kadanoff-Baym ansatz (GKBA). Additionally, for conserving theories, numerical simulations\nsuggest that KBE produces natural occupations bounded by one and zero in agreement with the Pauli exclusion principle, whereas,\nin some regimes, GKBA-based theories violate this principle. As the first step for understanding these issues, the electron dynam-\nics arising in the adiabatic switching scenario is studied. Many-body approximations are classified according to the channel of the\nBethe-Salpeter equation in which electronic correlations are explicitly treated. They give rise to the so-called second Born, T-matrix\nandGW approximations. In each of these cases, the model is reduced to a system of ordinary differential equations, which resem-\nble equations of motion for a driven harmonic oscillator with time-dependent frequencies. A more complete treatment of electronic\ncorrelations is achieved by combining different correlation channels, with parquet theory serving as a starting point.\n1 Introduction\nCoherent electron dynamics in correlated materials attracts considerable attention due to the possibil-\nity of creating and controlling new quantum states by time-dependent perturbations [ 1]. Lattice sys-\ntems such as the Hubbard model and its numerous extensions represent a versatile playground for test-\ning many-body theories [ 2,3,4] and provide insight into the properties of correlated materials such as\ncuprates, transition metal oxides [5] and dichalcogenides [6].\nCoherent dynamics in such systems can launched by periodic field driving, quenching of system parame-\nters with electric fields and in pump-probe experiments exploiting various combinations of phase-locked\ninfrared (IR) and extreme ultraviolet (XUV) pulses. Coherent dynamics can be understood on the basis\nof the nonequilibrium Green’s function (NEGF) approach, which can be applied perturbatively or com-\nbined with dynamical mean field theory. In a recent sequence of works [ 7,8,9] it has been demonstrated\nthat numerical NEGF approach can be significantly accelerated by the use of the so-called generalized\nKadanoff-Baym ansatz (GKBA) [ 10] which leads to the theory formulation in the form of coupled ordi-\nnary differential equations (GKBA+ODE).\nThe strength of GKBA+ODE lays in the possibility of various extensions [ 11,12], such as the inclusion\nof electron-phonon interactions [ 13], multiparticle correlations [ 9], or transport [ 14]. It is also known that\nvery systematic and balanced treatment of electronic correlations can be achieved by working not with\nself-energies, but rather with vertex functions. This gives rise to the so-called parquet method [ 15,16],\nwhich combines correlations in the three channels: particle-particle ( pp) and 2 particle-hole ( ph,ph).\nIn this work, I present the unexpected finding of exact analytic solutions for the Hubbard dimer sep-\narately in all three correlation channels within the NEGF+GKBA approximation and demonstrate\nthe simplest possible way to combine the channels using the so-called fluctuating-exchange approxi-\nmation (FLEX). Despite its simplicity, this model attracts recurrent attention in the pure electronic\ncase [ 17,18,19,20,21], linearly [ 22,23,13] and quadratically [ 24] coupled with phonons, and can be\nstudied experimentally as ultracold atoms in optical lattice systems [25].\nThe outline of the work is as follows. The GKBA+ODE approach is overviewed, presenting a uniform\nformulation for the three correlation channels. Next, it is demonstrated that the respective collision\nintegrals can be combined together such that the double counting of Feynman diagrams can be avoided.\nFinally, analytic solutions for the driven Hubbard model using all aforementioned approximations are\ndiscussed.\n1arXiv:2401.11301v1  [cond-mat.str-el]  20 Jan 20242 Compendium of the GKBA+ODE scheme\nConsider first a general form of the electronic Hamiltonian\nˆH(t) =X\nijhij(t)ˆd†\niˆdj+1\n2X\nijmnvijmn(t)ˆd†\niˆd†\njˆdmˆdn, (1)\nexpressed in terms of fermionic operators ˆd†\ni,ˆdj, where imay stand for spatial degrees of freedom and\nspin. In the NEGF formalism the fundamental unknowns are the electronic lesser/greater single-particle\nGreen’s functions\nG<\nij(t, t′) =i⟨ˆd†\nj(t′)ˆdi(t)⟩, G>\nij(t, t′) =−i⟨ˆdi(t)ˆd†\nj(t′)⟩, (2)\nThey satisfy the Kadanoff-Baym equations of motion, which are mathematically integro-differential equa-\ntions:\n[i∂t−he(t)]G≶(t, t′) =\u0002\nΣ≶·GA+ ΣR·G≶\u0003\n(t, t′), (3)\nwhere [A·B](t, t′)≡R\nd¯t A(t,¯t)B(¯t, t′),is a real-time convolution and XR/A(t, t′) are the retarded/advanced\nfunctions, and Σ is the correlation part of the self-energy. We work in zero-temperature formalism, there-\nfore contributions due to vertical track of the Keldysh contour are not included in Eq. (3). The time-\nlocal mean-field part is incorporated into the Hartree-Fock Hamiltonian hHF,ij(t) =hij(t)+P\nmnwimnjρ<\nnm(t)\nwith wimnj=vimnj−vimjn, and we also introduced densities according\nρ≶\nij(t) =−iG≶\nij(t, t). (4)\nBy combining Eq. (3) with its adjunct and going to the equal times limit one obtains:\nd\ndtρ<(t) =−i\u0002\nhHF(t), ρ<(t)\u0003\n−\u0000\nI(t) +I†(t)\u0001\n. (5)\nThe collision term can be expressed in terms of the two-particle Green’s function (2-GF)\nIlj(t) =−iX\nimnvlnmi(t)Gimjn(t). (6)\nEq.(5)is not closed because the Gimjn(t) can be expressed as a functional of the two-times Green’s func-\ntionG<\nij(t, t′). The complicated time-dependence in G<\nij(t, t′) is decoupled with the help of GKBA\nG≶(t, t′) =−GR(t, t′)ρ≶(t′) +ρ≶(t)GA(t, t′), (7)\nwhereby using a simpler form of the retarded propagator:\nGR(t, t′) =−iθ(t−t′)Tn\ne−iRt\nt′dτ hHF(τ)o\n. (8)\nLet us consider now a large class of approximations in which 2-GF is given as a solution of the one-\nchannel Bethe-Salpeter equation. Depending on the channel in which electronic correlations are treated,\npp,phorph, they are known as T-matrix and GW approximations. For Hubbard models, the cancella-\ntion of direct and exchange diagrams for electrons with equal spin also allows to formulate the second\nBorn approximation as the common leading term of all three methods, and it can be treated on equal\nfooting.\nA particularly concise formulation is achieved by introducing matrix notations [ 9] for rank-4 tensors like\n2-GF G, Coulomb interaction v, and the response-functions: the full χand the noninteracting χ0ones.\nThe index order is different for each approximation as summarized in Tab. 1. The one-time G(t) can be\nrepresented then as\nG(t) =−iZt\n0dt′n\nχ>(t, t′)v(t′)χ0,<(t′, t)−(>↔<)o\n, (9)\n2Table 1: Definitions of electronic two-particle tensors. The vertically grouped indices are combined into one super-index.\nQuantity 2B and GW TppTph\niχ0,≶\n13\n24(t, t′)G≶\n13(t, t′)G≷\n42(t′, t)−G≶\n13(t, t′)G≶\n24(t, t′)−G≶\n13(t, t′)G≷\n42(t′, t)\nG13\n24(t) G4132(t) G1234(t) G1432(t)\nh13\n24(t) h13δ42−δ13h42 h13δ24+δ13h24 h13δ42−δ13h42\nv13\n24(t) v1432(t) v1243(t) v1423(t)\nρ<\n13\n24(t) ρ<\n13(t)ρ>\n42(t) ρ<\n13(t)ρ<\n24(t) ρ<\n13(t)ρ>\n42(t)\nHereχis the response function describing the pure electronic screening in the respective channel. In\nFig. 1, the representation is illustrated for the GW case in application to the Hubbard model. By differ-\nentiating Eq. (9) with respect to time one obtains the following equation of motion\nid\ndtG(t) =−Ψ(t) +\u0002\nh(t) +aρ∆(t)v(t)\u0003\nG(t)−G(t)\u0002\nh(t) +av(t)ρ∆(t)\u0003\n, (10)\nwhere his effective two-particle Hamiltonian. Other ingredients are defined as\nρ∆(t)≡ρ>(t)−ρ<(t), (11)\nΨ(t)≡ρ>(t)v(t)ρ<(t)−ρ<(t)v(t)ρ>(t), (12)\nwhere ρ≶and the lesser/greater two-particle densities. Constant ais method-dependent and is equal to\n0,−1 and 1 for second Born, GW andT-matrix approximations, respectively.\n3 Application to the Hubbard dimer\nSpecifying Eq. (1)to the site-spin basis i≡(i, σi), restricting to the nearest neighbors ( <i,j>) hop-\nping hiσijσj=hδ<i,j>δσiσj, and setting\nviσijσjkσklσl=Uδijδklδjkδσiσlδσjσk, (13)\none obtains the Hubbard Hamiltonian\nˆHH=ˆT+ˆHU=hX\nσX\n<i,j>ˆd†\niσˆdjσ+UX\niˆni↑ˆni↓, (14)\nwhere his the hopping parameter, and Uis the on-site repulsion and ˆniσ=ˆd†\niσˆdiσ. In what follows, I\nwill focus on the case where a lot of progress can be done analytically: the half-filled two-site Hubbard\nmodel ( N= 2 andP\ni⟨ˆniσ⟩= 1). This system has a large number of symmetries ( D∞hpoint group).\nFrom the translational invariance follows that all physical operators are given by Toeplitz matrices. In\nparticular, the one-body Hamiltonian in the site basis is represented as a matrix\nT=\u0012\n0h\nh0\u0013\n,or for brevity T=\u00020h\u0003\n. (15)\nOther symmetries (reflections) impose further restrictions on the density matrix: it is not only Hermi-\ntian, but also symmetric and can be written in terms of just one parameter (site basis, using shorthand\nnotation for Toeplitz matrices)\nρσ=\u00021/2a\u0003\n. (16)\nThe Hartree-Fock Hamiltonian with this density reads\nhHF=\u0002U/2h\u0003\n, (17)\n3a)/uni2191+/uni2193 /uni2191+/uni2193 /uni2191+/uni2193 /uni2191+/uni2193j/uni2191\ni/uni2191m,σ\nn,σ /u1D4A2lmjn =v v v v v\nb)/uni2191 /uni2193 /uni2191 /uni2193j/uni2191\ni/uni2191m,/uni2193\nn,/uni2193/u1D4A2lmjn =\nv´ v´ v´ v´ v´Figure 1: Diagrammatic representation of the 2-GF ( GW approximation) for two forms of the Coulomb interaction in the\nHubbard model.\nwith the eigenvalues\nϵHF\ni=U/2±h. (18)\nDensity matrix based on the lowest energy eigenvector of Hamiltonian (17) then reads\nρHF\nα=\u00021/2−1/2\u0003\n. (19)\nConsider now the driving protocol in which U=U(t) and h= 1, and the density matrix possesses\nthe full symmetry of the system at every time instance. This discards the possibility of spontaneous\ndimerization. In such a scenario the density matrix in site basis can always be written in the form (16)\nwith a time-dependent parameter a(t). This means that HF Hamiltonian has always the same eigenvec-\ntors, but time-dependent eigenvalues (18)(via parametric dependence of Uon time). Thus, HF basis\nis fixed. HF-Hamiltonian is time-dependent and diagonal in this basis. Likewise, the density matrix is\ntime-dependent and diagonal. Since diagonal matrices commute [ hHF(t), ρσ(t)] = 0, in our driving proto-\ncol the density matrix is driven exclusively by the collision term\n˙a(t) =−2U(t)Φ2(t), (20)\nwhere Φ m(t) =−iG11m1(t) is introduced, and the initial condition according to Eq. (19) reads\na(ti) =−1/2. (21)\nNow the equations of motion for matrix elements of 2-GF for various approximations will be formulated\nand analysed.\nSecond Born approximation The EOM for Φ 2(t) follows from Eq. (10). After long but trivial calcula-\ntions one obtains that 2-GF fulfills a driven oscillator equation\n¨Φ2(t) + 16Φ 2(t) =˙L(t). (22)\nwith the time-dependent driving\nL(t) =U(t)\u0000\na(t)3+1\n4a(t)\u0001\n≡U(t)λ(t). (23)\n2BsGW\nTppTph\n-10 -8 -6 -4 -2 2 4t-0.495-0.490-0.485-0.480-0.475ρ12(t)U=1\n2BsGW\nTppTph\n-10 -8 -6 -4 -2 2 4t-0.4-0.3-0.2-0.1ρ12(t)U=10\n2BsGW\nTppTph\n-10 -8 -6 -4 -2 2t-0.4-0.3-0.2-0.10.0ρ12(t)U=20\n2BsGW\nTppTph\n-200 -150 -100 -50t-0.4-0.3-0.2-0.10.0ρ12(t)U=20\nFigure 2: The GKBA thermalization of the Hubbard dimer computed using exact equations for different correlated methods. It is\nquite surprising that Tphand s GW produce very different results for small- U, whereas the two methods asymptotically converge to\nthe same value for U→ ∞ . Exact thermalized values determined by Eq. (33) are shown as dashed lines.\n4T-matrix approximations A marked feature of these correlated methods is that the oscillator eigenfre-\nquency in Eq. (22) becomes modulated by the single-particle density. Introducing\nK(t) =U(t)a(t), (24)\none obtains two coupled oscillators equations\n¨Φ2(t) + 16(1 ±K(t))Φ2(t) =˙L(t)±2˙K(t)Φ1(t), (25a)\n˙Φ1(t) =−4Φ2(t), (25b)\nwhere the upper and lower signs correspond to the phandppchannels, respectively. The initial condi-\ntions are\nΦm(ti) =˙Φm(ti) = 0 . (26)\nGWapproximation Eq.(13)yields the Hubbard interaction Uni↑ni↓only in conjunction with the Pauli\nprinciple. However, it is not guaranteed when working perturbatively that the only scattering processes\nare those between the particles of opposite spin, Fig. 1(a). In this example the intermediate fermionic\nbubbles may carry arbitrary spin leading to overscreening. Same-spin scattering can be eliminated by\nintroducing a spin-dependent interaction [3, 26]\nv′\niσijσjkσklσl=viσijσjkσklσl(1−δσiσj). (27)\nleading to 2-GF in Fig. 1(b). This method will be called spin- GW orsGW . Corresponding 2-GF equa-\ntion(10)becomes more involved due to the necessity of considering correlators with two different spin-\norders that for brevity are denoted as a=↑↓↑↓ andb=↑↑↑↑:\n¨Φa\n2(t) + 16Φa\n2(t) =−2˙K(t)Φb\n1+ 16K(t)Φb\n2+˙L(t), (28a)\n¨Φb\n2(t) + 16Φb\n2(t) =−2˙K(t)Φa\n1(t) + 16 K(t)Φa\n2, (28b)\n˙Φa\n1(t) =−4Φa\n2(t), (28c)\n˙Φb\n1(t) =−4Φb\n2(t). (28d)\nNotice that the driving term is present only for the mixed spin component, which now enters the density\nequation (20).\nTo summarize, the EOM for the density matrix (5)is reduced to a single differential equation (20)for\na single off-diagonal element. The EOM for 2-GF (10)takes different form depending on the approx-\nimation: Eq. (22)(2B), Eq. (25)(T-matrix), and Eq. (28)(sGW ) (see Supplemental Information for\ndetailed derivations). Numerical solutions of these equations for the adiabatic switching protocol\nU(t) =(\nUsin2\u0010\nπ\n2(t+τ)\nτ\u0011\nτ < t < 0,\nU t ≥0;(29)\nand different final Hubbard- Uand switching times ( ti=τ) are shown in Fig. 2. There are two observa-\ntions in comparison with earlier works using the full KBE propagation. First of all, a crucial difference\nbetween the GKBA method and the full solution of KBE is that the former does not lead to the artifi-\ncial damping observed in the paper of Marc Puig von Friesen, C. Verdozzi, and C.-O. Almbladh [ 2]. Be-\nsides the numerical evidences, this observation is supported by the structure of the obtained equations,\nwhich have a driven oscillator form without a damping term. The damping was discussed in another\npapers by the same authors [ 3,27], and a plausible explanation is that it results from self-consistent\ntreatments, leading to an infinite number of poles in the electron Green’s function. In contrast, an exact\nsolution should only have a finite number of poles for finite systems. Another interesting effect is the\nlack multiple steady states [ 2,3]. This is also an artefact of exact calculation not observed in the GKBA\nscheme. As comparison of Fig. 2(c) and (d) shows, fast switching of the interaction leads to oscillations,\nbut there is never a transition to a state different from the one obtained by slow adiabatic switching.\nThis holds true even when artificial damping term is added. However, it is not possible to exclude that\nunder some combination of parameters an artificial steady state can be reached.\n53.1 Asymptotic analysis\n3.1 Asymptotic analysis\nStarting from the zero-temperature adiabatic assumption the correlated density matrix of the ground\nstate of the system can be determined analytically for 2B and T-matrix methods. To this end equations\nof motion are written in the form common to all methods, the ¨Φ2(t) is neglected assuming infinitesimally\nslow switching, parametrix dependence of physical quantities on Uis introduced, i.e., a(U)≡a(U(t)),\nandf(u)≡Φ1(U(t)). We obtain\n2DU[a(U)] =UD U[f(U)], (30a)\n−4DU[f(U)](1−a(U)ηU) =DU[uλ(a(U))]−2ηf(U)DU[Ua(U)], (30b)\na(0) =−1/2, f(0) = 0 . (30c)\nwhere η= 1,−1,0 for Tpp,Tph, 2B approximations, respectively. DUdenotes the derivative with respect\ntoUandλ(a) =a3+1\n4a. Surprisingly, these equations can be analytically integrated (see Supplemental\nInformation for detailed derivations) leading to\na2h\n16 + 16\u0000\n1 + 4 a2\u0001\n+U2\u0000\n1 + 4 a2\u00012i\n= 12, 2B; (31a)\na2(Ua+ 2)2=Ua+ 1, Tph; (31b)\na\u0000\na(Ua−2)2+U\u0001\n= 1, Tpp. (31c)\nThe derivation for the spin GW method proceeds along the same line, except there are more equations\nto solve. As the consequence, it is not possible to integrate all of them. However, one can reduce (Sup-\nplemental Information) them to a single nonlinear ordinary differential equation\np(a, U)a′(U)2+q(a, U)a′(U) +r(a, U) = 0 , (32)\nwhere\np(a, U) = 3 U2a2\u0000\n36U4a4+ 3\u0000\nU2−48\u0001\nU2a2−U4+ 8U2+ 192\u0001\n−\u0000\nU2+ 16\u00012,\nq(a, U) = 2 Ua\u0000\n9U4a4−3U2\u0000\nU2−4\u0001\na2−(U2+ 16)\u0001\n,\nr(a, U) =U2a2\u0000\n3U2a2\u0000\n3a2−1\u0001\n−1\u0001\n.\nWhile we were not able to solve Eq. (32)analytically, it is possible to perform series expansions and to\ncompare with the exact solution [28] that has much simpler form\na(U) =−2√\nU2+ 16. (33)\nFrom the implicit equations (31), differential equation (32)and algebraic equation (33)we obtain the\nseries expansions for U→0\na=−1\n2+U2\n64−3U4\n2048+O\u0000\nU6\u0001\n, 2B; (34a)\na=−1\n2+U2\n64+9U4\n4096+O\u0000\nU6\u0001\nsGW ; (34b)\na=−1\n2+U2\n64+U3\n128+9U4\n4096+O\u0000\nU6\u0001\n, Tph; (34c)\na=−1\n2+U2\n64−U3\n128+9U4\n4096+O\u0000\nU6\u0001\n, Tpp; (34d)\na=−1\n2+U2\n64−3U4\n4096+O\u0000\nU6\u0001\n, exact . (34e)\nAll of them are exact of to the second order. This is expected because the second order diagrams are\nfully taken into account by all the methods. T-matrix methods contain terms of the thirds order cor-\nresponding to the ladder diagrams with three interaction lines. Exact result does not have these terms.\n63.1 Asymptotic analysis\nsGW1.024/u\n0.5 100.010.10110\nU-a(U)\n0 5 10 15 20-0.5-0.4-0.3-0.2-0.10.0\nUa(U)2BsGWTphTppexacta u+10\nFigure 3: Asymtpotics of the off-diagonal density matrix element a(U) for the spin- GW method (left) and comparison of\na(U) for different methods (right).\nThus, these diagrams must be compensated by other third order diagrams not accessible with our method.\nThe spin GW method contains only the even order terms. This is expected from the diagrammatic con-\nstruction illustrated in Fig. 1. 2B method also contains only even order terms. Terms of fourth order\nand higher arise from the self-consistent solution of the Dyson equation.\nFor large values of Uthe off-diagonal density approaches zero as\na=−2√\n3\nU+O\u0000\nU−3\u0001\n, 2B; (35a)\na=−1.024\nU+O\u0000\nU−3\u0001\n, sGW ; (35b)\na=−1\nU+O\u0000\nU−3\u0001\n, Tph; (35c)\na=−1\nU1/3+1\nU+O\u0000\nU−3\u0001\n, Tpp; (35d)\na=−2\nU+O\u0000\nU−3\u0001\n, exact . (35e)\nIt is quite unexpected that Tppapproximation fails to provide even the −1/Uasymptotic dependence of\nthe exact solution. It should also be noticed that spin- GW is numerically very close to the Tphmethod,\nhowever, the asymptotic coefficient (determined numerically) is slightly different [ −1.024, see Fig. 3(left)].\nThis indicates that physically the two methods are physically different, and the observed similarity is\nrather a coincidence.\nAnalysing further the a(U) dependence based on Eqs. (31)allows to conclude that physical (fulfilling\nthe initial condition) branch of the considered theories always satisfies −1/2< a(U)≤0 in equilibrium.\nThis indicates that natural occupations ni(U) are always in the range 0 ≤ni≤1, as stipulated by the\nPauli exclusion principle. This is a non-trivial finding as apart from the Hartree-Fock approximation\nthere is no proof that the natural occupations should adhere to physical limits in correlated theories\ntreated within GKBA. In fact, there are electronic [ 21] and electron-phonon [ 12] systems, where under\nsome conditions violations of these limits occur.\nIt is also interesting to note that the structure of time-dependent equations (25)is very similar to the\nstructure of electron self-energies derived for the same system in the ground state [ 29]. For instance\nforTphandTpp, the self-energy poles (see Eqs.(53, 43) therein) are expressed in terms of the effective\nHamiltonian that reduces to the double effective frequency in Eq. (25)(Ω2= 1±aU) when Hartree-Fock\na=−1/2 is used therein. It also explains why the Tphtheory of Ref. [ 29] is unstable when Uapproaches\n7=F +/uni039B /uni03A6ph/uni03A6pp/uni03A6ph/uni0336 + +\n/uni039B ≈a)\nb)Figure 4: (a) Vertex function as a sum of irreducible and two-particle reducible correlators, Eq. (36). (b) Approximation of the\nlowest-order irreducible vertex by the direct Coulomb interaction (exchange is not included).\n2, and why such an instability is absent in our case. As can be see from Fig. 3(right), in our approach a\ndepends on Uand therefore the effective frequency 1 + aU(dashed line) never becomes equal to zero.\n4 Combining channels\nNumerical examples shown above indicate that sGW ,TppandTphapproximations, despite containing\ninfinite sequence of Feynman diagrams via the solution of the Bethe-Salpeter equation in the respec-\ntive channels, are describing physically very different scenarios. Can one exploit the advantages of each\napproximation? In the following, I derive such an approximation starting from the exact parquet equa-\ntions [ 16]. This derivation is complementary to the self-energy based derivation in Ref. [ 30]. The moti-\nvation for this starting point is two-fold: on one side this derivation establishes that there is no double\ncounting of Feynman diagrams. On the other side, it demonstrates difficulties of going beyond. They are\nassociated with multiple time-arguments of constituent vertex functions [ 31]. In particular, working in\none channel allows one to express the one-time 2-GF in terms of two-times response functions, see for\ninstance Eq. (28) of Ref. [ 9], and to close the equation of motion for it. Combining the channels requires\nin general to deal with 4-times quantities for which GKBA is not known.\nConsider the full (reducible) vertex F, which is just 2-GF with amputated fermionic lines, i. e., iG=\nχ0Fχ0. This equation is written schematically as it does not reflect the time arguments of the ingredi-\nents. Reducing Gto two-times form is the main goal here. Quite generally, the vertex function Fcan be\nwritten as a sum of the fully irreducible vertex Λand reducible vertices Φi\nF=Λ+X\niΦi, (36)\nwhere i=pp, ph, phdenotes the channels, in which Φiaretwo-particle reducible . Eq. (36)is, therefore,\nonly a topological statement (Fig. 4). It is important because it rules out any double counting of the\nresulting Feynman diagrams.\nThe reducible vertices Φifulfill a set of interrelated equations (see first line of Fig. 5 for the exact equa-\ntion in the phchannel). Two approximations can be introduced allowing to transit from four to two\ntime-arguments: (i) The leading term for each equation is written as ( Fχ0Λ)i(second line of Fig. 5) in\nall channels. Here, the subscript iindicates that quantities in the brackets are given in the index order\npertinent to the channel, other terms are discarded; (ii) Approximating therein\nF≈Λ+Φi, (37)\n(third line) and using Λ≈v[Fig. 4(b)] leads to the Bethe-Salpeter equation for the two-particle re-\nducible functions (forth line of Fig. 5)\nΦi(z, z′)≈v(z)χ0\ni(z, z′)v(z′) +Z\nCd¯zΦi(z,¯z)χ0\ni(¯z, z′)v(z′), (38)\nwhere z,z′are times on the Keldysh contour C.χ0\niis written explicitly for each channel in Tab. 1, where\nGW approximation corresponds to the treatment of phchannel, Tppapproximation — ppchannel, and\nTph—phchannel. Notice that while exact Bethe-Salpeter equations are formulated for 4-times correla-\ntors, the final approximated equation is closed for Φidependent on two-times.\n8/uni03A6ph=F/uni039B + + /uni03A6pp F /uni03A6ph/uni0336 F\n≈F/uni039B\n≈ /uni039B /uni03A6ph /uni039B/uni039B +\n≈ /uni03A6ph+ v/uni03C70v v/uni03C70Figure 5: Derivation of the Bethe-Salpeter equation (38) for the two-particle phreducible correlator. phandppchannels are treated\nin the same way. Notice that the first approximation means setting the irreducible vertex in the phchannel to the fully irreducible\nvertex (and similarly in other channels). This is exactly what defines the FLEX approximation [32].\nDecorating Φiwithχ0\nion both sides of Eq. (38)one obtains the RPA equations for the full response\nfunctions χi(z, z′)v(z′) = [χ0\ni·(v+Φi)](z, z′):\nχi(z, z′) =χ0\ni(z, z′) +Z\nCd¯zχi(z,¯z)v(¯z)χ0\ni(¯z, z′). (39)\nLet us now decorate the vertex functions in Eq. (37)with four fermionic lines and go to the equal time-\nlimit ( t) for the external time-arguments\niGi(t) = [χ0\ni·v·χ0\ni+χ0\ni·Φi·χ0\ni](t) = [χi·v·χ0\ni](t). (40)\nIn this way Eq. (9)was re-derived for each channel. However, one can do better by using the fullvertex\nas in Eq. (36)instead of the partial vertex as in Eq. (37). To this end, let us decorate Eq. (36)(Λ≈v)\nwith four fermionic lines with equal external time-arguments ( t). It follows then\nG(t) =−2G2B(t) +X\niGi(t). (41)\nNotice that the matrix notations (Tab. 1) are not used here because each approximation has own order\nof indices. Instead, Gs are interpreted here as usual rank-four tensors, like in Eq. (6). A crucial point of\nthis derivation is to realize that\niG2B(t) = [χ0\ni·v·χ0\ni](t) (42)\nis independent on the channel iand is given simply by the second Born approximation for 2-GF. This\ncan be verified by using definitions in Tab. 1 ( provided that lhs of this equation is written with index\norder pertinent to the rhs). Therefore Eq. (41)can be written explicitly for the Hubbard model with\nspin-dependent interactions (27) as\nGFLEX(t) =−2G2B(t) +GsGW(t) +GTpp(t) +GTph(t). (43)\nThe resulting approximation is denoted as FLEX —the fluctuating-exchange approximation — which\nseems to be the common term [ 33,34,8]. It may be viewed as a first-order element in a hierarchy of\nsuccessive approximations to the full parquet solution [35, 32].\nThermalization is possible only for small values of Uas depicted in Fig. 6 (left). There is, however, a\nsimple procedure that stabilizes the GKBA+ODE methods. It was already mentioned that EOMs for\nthe matrix elements of 2-GF have a driven harmonic oscillator form, viz. Eq. (22, 25, 28). By introduc-\ning a “velocity damping” with coefficient k(see Supporting Information for the exact form of equations)\nconverged solutions can be obtained even for strongly correlated cases, Fig. 6 (right). If for a given\nmethod the adiabatic switching procedure can be performed, then the final density matrix of the sys-\ntem is characterised by the a(U) value independent of a small damping k. The term “small” stipulates\nthat the magnitude of damping should not exceed other energy scales in the system (see Supporting\nInformation for the investigation of the role of k). While the “damping” approach work excellently for\n92BsGW\nTppTph\nFLEX\nU=0.75\n-10 -8 -6 -4 -2 2 4t-0.498-0.496-0.494-0.492-0.490-0.488ρ12(t)\n2BsGW\nTppTph\nFLEX\n2B(k=0)U=0.75\nk=1.0\n-10 -8 -6 -4 -2 2 4t-0.498-0.496-0.494-0.492-0.490-0.488ρ12(t)\n2BsGW\nTppTphFLEX\n2B(k=0) U=10\nk=2.5\n-10 -8 -6 -4 -2 2 4t-0.4-0.3-0.2-0.1ρ12(t)Figure 6: The GKBA thermalization of the Hubbard dimer computed using exact equations for different correlated methods.\nDashed horizontal line depicts exact thermalized value of a(t) given by Eq. (33).\nthe Hubbard dimer, further work is needed in order to generalize this approach to more complicated\nsystems.\nFinally, we remark that our approach is different from the dynamically-screened-ladder approximation in\nRef. [8] in that four ingredient 2-GFs in Eq. (43) are independently propagated.\n5 Conclusions and outlook\nExact many-body electron dynamics in a finite basis representation is described by the linear differen-\ntial equation for the wave-function. Reformulations of this dynamics on the language of reduced quan-\ntities, such as nonequilibrium Green’s functions, inevitably lead to the equations of motion with much\nmore complicated mathematical structure. One of the first attempts to analyze the integro-differential\nKadanoff-Baym equations in this perspective have been made by Marc Puig von Friesen, C. Verdozzi,\nand C.-O. Almbladh [ 3]. They have made a number of interesting observations concerning non-unique\nsteady states and the role of the correlation-induced damping. These findings are revisited here in view\nof the rapid development of the time-linear methods that can be formulated as a system of coupled or-\ndinary differential equations. In this work, the first steps are taken in the investigation of the nonlinear\nform of the GKBA+ODE scheme by applying it to the two-sites Hubbard model. This involves deriving\noscillator-like equations for the components of the two-particle GF and analyzing the adiabatic switching\nscenario and the steady-state limit. This has led to the observation of similarities between equations in\ndifferent channels, systematic under- and over-estimations of the correlational effects pertinent to dif-\nferent schemes and very close but distinct asymptotic limits for the TphandsGW methods. It was also\npossible to prove analytically that second Born, T-matrix and spin- GW approximations never violate\nthe Pauli exclusion principle in the steady state limit. Nonetheless, such violations may take place under\nstrongly nonequilibrium conditions. Further investigations of these effects are needed. The structure of\nthe GKBE+ODE equations in the dimer case hints that this is a challenging mathematical problem.\nSupporting Information\nSupporting Information is available from the Wiley Online Library or from the author.\nAcknowledgements\nI would like to thank Enrico Perfetto and Gianluca Stefanucci for insightful discussions, and Hugo U.R.\nStrand, Claudio Verdozzi, Michael Bonitz for the excellent organization of the PNGF8 conference. This\nresearch was part of project no. 2021/43/P/ST3/03293 cofunded by the National Science Centre and the\nEuropean Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie\ngrant agreement no. 945339.\n10REFERENCES\nReferences\n[1] D. N. Basov, R. D. Averitt, D. Hsieh, Nat. Mater. 2017 ,16, 11 1077.\n[2] M. P. von Friesen, C. Verdozzi, C.-O. Almbladh, Phys. Rev. Lett. 2009 ,103, 17 176404.\n[3] M. Puig von Friesen, C. Verdozzi, C.-O. Almbladh, Phys. Rev. B 2010 ,82, 15 155108.\n[4] D. Karlsson, A. Privitera, C. 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Drchal, V. Janiˇ s, J. Kudrnovsk´ y, V. S. Oudovenko, X. Dai, K. Haule, G. Kotliar, J. Phys. Con-\ndens. Matter 2005 ,17, 1 61.\n[35] N. E. Bickers, S. R. White, Phys. Rev. B 1991 ,43, 10 8044.\n12"    },    {        "title": "2401.11333v1.Error_bounds_of_constant_gain_least_mean_squares_algorithms.pdf",        "content": "arXiv:2401.11333v1  [eess.SP]  20 Jan 2024Error bounds of constant gain least-mean-squares algorith ms\nChang Liu1,†and Antwan D. Clark2Senior Member, IEEE\nAbstract —Constant gain least-mean-squares (LMS)\nalgorithms have a wide range of applications in trajec-\ntory tracking problems, but the formal convergence of\nLMS in mean square is not yet fully established. This\nwork provides an upper bound on the constant gain\nthat guarantees a bounded mean-squared error of LMS\nfor a general design vector. These results highlight the\nrole of the fourth-order moment of the design vector.\nNumerical examples demonstrate the applicability of\nthis upper bound in setting a constant gain in LMS,\nwhile existing criteria may fail. We also provide the\nassociated error bound, which can be applied to design\nvectors with linearly dependent elements.\nI. Introduction\nLEAST mean-squares (LMS) algorithms have a wide\nrange of applications from signal processing and\nadaptive control [1] to geophysics [2]. Particularly, un-\nderstanding the impact of the constant gain in these\nalgorithms is important in trajectory tracking and the\nestimation of temporal parameters where some applica-\ntions include the works of [3], [4], [5], [6], [7] due to\nthe fact that non-vanishing gains ensures the fidelity of\nadaptations to the time-dependent true parameter. Some\nadvancements include the works of [8], [9], [10], [11] where\nstochastic approximation theory has been used to find\nformal convergence criterion of recursive LMS algorithms\nin which the main requirement is that the indexed gain\nak→0 as k→ ∞ [12]. Exploiting the concept of\nweak convergence [13], [14], researchers have found the\nconnections between the constant gain and stochastic\nbehavior [15], [16] as well as sensitivity to certain data\nproperties [17]. Practical convergence benchmarks were\nalso developed, which include the earlier works of [18], [19 ],\nand [20] who provided informal convergence arguments.\nLater, more rules of thumb were proposed such as those\nrelated to mean-squared stability [21] and stochastic gra-\ndient methods with constant or non-decaying gains [22],\n[23], [24], [25].\nHowever, finding the bounds on these algorithms is still\nan open research problem. Although recent developments\nprovided more measurable criterion, this is limited to\nsufficiently small gains and is not applicable for linearly\ndependent stochastic design vectors [21]. Here, we provide\ntheoretical upper bounds on the constant gain of LMS\nalgorithms that guarantees the bounded mean-squared er-\nror for a general design vector. Furthermore, these results\nprovide upper bounds on the estimation errors as well as\nillustrate the correlations between the fourth-order mo-\nment of general design vectors. Our results, via synthetic\n1School of Mechanical, Aerospace, and Manufacturing Engine er-\ning, University of Connecticut, CT 06269, USA.2Department of\nApplied Mathematics and Statistics, Johns Hopkins Univers ity, MD\n21218, USA. †chang liu@uconn.eduand practical data, illustrate the promise of our proposed\nrule of thumb as well as demonstrate improvement over\nprevious results for both general and linearly dependent\nstochastic design vectors [18], [21].\nThe rest of this paper is presented in the following\nmanner. In Section II, we provide the upper bound on the\ngain to guarantee a bounded mean-squared error of LMS\nand the associated error bound. Section III provides ex-\namples demonstrating their fidelity. Section IV concludes\nthis paper, where we highlight some future directions.\nII. Error Bounds of LMS\nOur analysis considers the simple linear regression\nmodel based on the scalar output measurement given by\nzk=hT\nkθ∗+ǫk, k= 1,2, ..., n, (1)\nwhere θ∗is the m×1 dynamic response parameter process\nof interest, zkis the scalar output measurement, and\nǫk=N/parenleftbig\n0, σ2\nǫ/parenrightbig\nis the sequence of independent normally\ndistributed random noise. Furthermore, hkis a m×1\nstochastic design vector that is independent of ǫkconsist-\ning of a finite-magnitude covariance matrix E[hkhT\nk]/{ollowsequal0\n(∀k∈N). The supposition in our analysis is that the\ncovariance matrix is independent of kand is represented\nby the Cholesky decompositions\nE[hkhT\nk] =NNT. (2)\nAdditionally, we consider the minimization of the loss\nfunction ˆLk= (zk−hT\nkθ∗)2/2 in which tracking the\ndynamic parameter is represented as\nˆθk=ˆθk−1−ahk(hT\nkˆθk−1−zk), (3)\nwhere a >0 is the positive constant gain. Hence, we have\nthe following results.\nTheorem 1. If∃P≻0and0< χ < 2/asuch that:\naE[hkhT\nkP h khT\nk]−PE[hkhT\nk]−E[hkhT\nk]P/precedesequal −χP,\n(4)\n∀k∈N+, then the LMS algorithm with the constant gain a\nin equation (3) will provide estimation ˆθof true value θ∗\nsuch that E[||ˆθk−θ∗||2\n2]≤Πk<∞is bounded ∀k∈N+.\nFurthermore, the asymptotic limit of the upper bound is\nlim\nk→∞Πk=a\nχλmin(P)σ2\nǫtr/bracketleftBig\nPE/bracketleftbig\nhkhT\nk/bracketrightbig/bracketrightBig\n. (5)\nProof:\nConsidering equations (1) and (3) the LMS estimation\nerror ˜θk:=ˆθk−θ∗is given by\n˜θk= (I−ahkhT\nk)˜θk−1+ahkǫk, (6)Iis the identity matrix. Since P≻0 and noting\nthat V(˜θk) = ˜θT\nkP˜θkis a scalar quantity such that\ntr/bracketleftbig\nV/parenleftbig˜θk/parenrightbig/bracketrightbig\n= tr/bracketleftBig\n˜θT\nkP˜θk/bracketrightBig\n=V/parenleftbig˜θk/parenrightbig\nyields\nV/parenleftbig˜θk/parenrightbig\n= tr/bracketleftBig\n˜θT\nkP˜θk/bracketrightBig\n. (7)\nNoting the right hand side of (7) with ˜θkdefined by (6)\nwhile also noting the following property\ntr/bracketleftBig\n˜θT\nkP˜θk/bracketrightBig\n= tr/bracketleftBig\nP˜θk˜θT\nk/bracketrightBig\n, (8)\nthe right hand side of (8) yields\ntr/braceleftBig\nP˜θk−1˜θT\nk−1+/parenleftBig\n−ahkhT\nk/parenrightBig\nP/parenleftBig\n−ahkhT\nk/parenrightBig\n˜θk−1˜θT\nk−1\n+P/parenleftBig\n−ahkhT\nk/parenrightBig\n˜θk−1˜θT\nk−1+/parenleftBig\n−ahkhT\nk/parenrightBig\nP˜θk−1˜θT\nk−1\n+aǫkP/parenleftBig\nI−ahkhT\nk/parenrightBig\n˜θk−1hT\nk\n+aǫkP h k˜θT\nk−1/parenleftBig\nI−ahkhT\nk/parenrightBig\n+a2ǫ2\nkP h khT\nk/bracerightBig\n.\n(9)\nNoting that ˜θkdepends on the history ( Hk,Ψk) where\nHk= (hk,hk−1, ...,h1),Ψk= (ǫk, ǫk−1, ..., ǫ 1) and that\nhkandǫkare measurable with respect to ( Hk,Ψk), tak-\ning the conditional expectation of V/parenleftbig˜θk/parenrightbig\ngiven ( Hk,Ψk)\nwhile considering (9) yields\nE/bracketleftBig\nV/parenleftbig˜θk/parenrightbig/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracketrightBig\n=E/braceleftbigg\ntr/bracketleftBig\nP˜θk−1˜θT\nk−1+/parenleftBig\n−ahkhT\nk/parenrightBig\nP/parenleftBig\n−ahkhT\nk/parenrightBig\n˜θk−1˜θT\nk−1\n+P/parenleftBig\n−ahkhT\nk/parenrightBig\n˜θk−1˜θT\nk−1+/parenleftBig\n−ahkhT\nk/parenrightBig\nP˜θk−1˜θT\nk−1\n+aǫkP/parenleftBig\nI−ahkhT\nk/parenrightBig\n˜θk−1hk\n+aǫkP h k/parenleftBig\nI−ahkhT\nk/parenrightBig\n˜θk−1\n+a2ǫ2\nkP h khT\nk/bracketrightBig/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracerightbigg\n(10a)\n=tr/braceleftbigg\nPE/bracketleftBig\n˜θk−1˜θT\nk−1/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracketrightBig\n+a2hkhT\nkP h khT\nkE/bracketleftBig\n˜θk−1˜θT\nk−1/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracketrightBig\n−aP h khT\nkE/bracketleftBig\n˜θk−1˜θT\nk−1/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracketrightBig\n−ahkhT\nkPE/bracketleftBig\n˜θk−1˜θT\nk−1/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracketrightBig\n+aǫkP/parenleftBig\nI−ahkhT\nk/parenrightBig\nE/bracketleftBig\n˜θk−1/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracketrightBig\nhT\nk\n+aǫkP h kE/bracketleftBig\n˜θT\nk−1/vextendsingle/vextendsingle/vextendsingleHk,Ψk/bracketrightBig /parenleftBig\nI−ahkhT\nk/parenrightBig\n+a2ǫ2\nkP h khT\nk/bracerightbigg\n. (10b)\nApplying iterative conditional expectation while also not -\ning that ǫk=N/parenleftbig\n0, σ2\nǫ/parenrightbig\nis independent and identically\ndistributed (IID) gives\nE/bracketleftbig\nV/parenleftbig˜θk/parenrightbig/bracketrightbig\n=tr/braceleftbigg/parenleftBig\nP+a2E/bracketleftBig\nhkhT\nkP h khT\nk/bracketrightBig−aPE/bracketleftBig\nhkhT\nk/bracketrightBig\n−aE/bracketleftBig\nhkhT\nk/bracketrightBig\nP/parenrightBig\nE/bracketleftBig\n˜θk−1˜θT\nk−1/bracketrightBig\n+ 0 + 0 + a2σ2\nǫPE/bracketleftBig\nhkhT\nk/bracketrightBig/bracerightbigg\n(11a)\n=tr/bracketleftBig\nPE/bracketleftBig\n˜θk−1˜θT\nk−1/bracketrightBig/bracketrightBig\n+ tr/bracketleftBig\naQkE/bracketleftBig\n˜θk−1˜θT\nk−1/bracketrightBig/bracketrightBig\n+a2σ2\nǫtr/bracketleftBig\nPE/bracketleftBig\nhkhT\nk/bracketrightBig/bracketrightBig\n, (11b)\nwhere Qk=aE[hkhT\nkP h khT\nk]−PE/bracketleftBig\nhkhT\nk/bracketrightBig\n−E/bracketleftBig\nhkhT\nk/bracketrightBig\nP\nis the left-hand side of (4). Noting that E/bracketleftbig˜θk−1˜θT\nk−1/bracketrightbig\n=\nMMTby Cholesky decomposition while also considering\nthe following trace property\ntr/bracketleftBig\naQkE/bracketleftbig˜θk−1˜θT\nk−1/bracketrightbig/bracketrightBig\n= tr/bracketleftBig\naQkMMT/bracketrightBig\n=atr/bracketleftBig\nMTQkM/bracketrightBig\n. (12)\nHence, employing condition (4), equation (12) becomes\ntr/bracketleftBig\naQkE/bracketleftBig\n˜θk−1˜θT\nk−1/bracketrightBig/bracketrightBig\n≤ −aχtr/bracketleftBig\nMTP M/bracketrightBig\n=−aχtr/bracketleftBig\nP MMT/bracketrightBig\n=−aχtr/bracketleftBig\nPE/bracketleftBig\n˜θk−1˜θT\nk−1/bracketrightBig/bracketrightBig\n=−aχE/bracketleftbig\nV(˜θk−1)/bracketrightbig\n. (13)\nCombining equations (11) and (13), we have\nE/bracketleftbig\nV(˜θk)/bracketrightbig\n≤(1−aχ)E/bracketleftbig\nV(˜θk−1)/bracketrightbig\n+a2σ2\nǫtr/bracketleftBig\nPE/bracketleftbig\nhkhT\nk/bracketrightbig/bracketrightBig\n(14a)\n≤(1−aχ)kE/bracketleftbig\nV(˜θ0)/bracketrightbig\n+ [1 + (1 −aχ) +...(1−aχ)k−1]\n×a2σ2\nǫtr/bracketleftBig\nPE/bracketleftbig\nhkhT\nk/bracketrightbig/bracketrightBig\n(14b)\n=(1−aχ)kE/bracketleftbig\nV(˜θ0)/bracketrightbig\n+1−(1−aχ)k\n1−(1−aχ)\n×a2σ2\nǫtr/bracketleftBig\nPE/bracketleftbig\nhkhT\nk/bracketrightbig/bracketrightBig\n, (14c)\nwhere the inequality (14b) is obtained by applying in-\nequality (14a) iteratively and the inequality (14c) is\nobtained via geometric series. Therefore, E/bracketleftbig\nV(˜θk)/bracketrightbig\nis\nbounded ∀k∈N+as|1−aχ|<1 based on the fact that\n0< χ < 2/a. Using the inequality λmin(P)/bardblˆθ−θ∗/bardbl2\n2≤\nV(˜θk) asP≻0, we have\nE/bracketleftBig\n/bardblˆθk−θ∗/bardbl2\n2/bracketrightBig\n≤E/bracketleftbig\nV(˜θk)/bracketrightbig\n/λmin(P), (15)\nwhich leads to the bounded mean-squared error:\nE/bracketleftBig\n/bardblˆθk−θ∗/bardbl2\n2/bracketrightBig\n≤Πk:=(1−aχ)k\nλmin(P)E/bracketleftbig\nV(˜θ0)/bracketrightbig\n+1−(1−aχ)k\nχλmin(P)aσ2\nǫtr/bracketleftBig\nPE/bracketleftbig\nhkhT\nk/bracketrightbig/bracketrightBig\n.\n(16)\nTaking lim\nk→∞Πk, we have the asymptotic limit of the\nupper bound as\nlim\nk→∞Πk=a\nχλmin(P)σ2\nǫtr/bracketleftBig\nPE/bracketleftbig\nhkhT\nk/bracketrightbig/bracketrightBig\n. (17)/square\nThe case where P=Iproduces the following corollary.\nCorollary 2. If∃0< χ < 2/asuch that\naE[hkhT\nkhkhT\nk]−2E[hkhT\nk]/precedesequal −χI, (18)\nthen the LMS algorithm with the constant gain ain\nequation (3) will provide estimation ˆθof true value θ∗\nsuch that E[||ˆθk−θ∗||2\n2]≤Πk<∞is bounded ∀k∈N+.\nFurthermore, the asymptotic limit of the upper bound is\ngiven by\nlim\nk→∞Πk=a\nχσ2\nǫtr/bracketleftBig\nE/bracketleftbig\nhkhT\nk/bracketrightbig/bracketrightBig\n. (19)\nIt is also important to note the following remarks.\nRemark 1. If we consider additional assumptions, we\nhave condition (18) in Corollary 2 the same as conditions\nreported in the literature as summarized in Table I.\nTABLE I\nAssumption Convergence Condition\n(with Reference)\nE/bracketleftBig\nhkhT\nkhkhT\nk/bracketrightBig\n=E/bracketleftBig\nhkhT\nk/bracketrightBig\nE/bracketleftBig\nhkhT\nk/bracketrightBig\na <2/slashBig /parenleftBig\nλmax/braceleftBig\nE/bracketleftBig\nhkhT\nk/bracketrightBig/bracerightBig/parenrightBig\n([18, p. 102])\ntr/braceleftBig\nE/bracketleftBig\nhkhT\nkhkhT\nk/bracketrightBig/bracerightBig\n=/parenleftBig\ntr/braceleftBig\nE/bracketleftBig\nhkhT\nk/bracketrightBig/bracerightBig /parenrightBig 2\na <2/slashBig\ntr/parenleftBig\nE/bracketleftBig\nhkhT\nk/bracketrightBig/parenrightBig\n([18, p. 103])\nE/bracketleftBig\nhkhT\nkhkhT\nk/bracketrightBig\na </parenleftBig\n2λmin/braceleftBig\nE/bracketleftBig\nhkhT\nk/bracketrightBig/bracerightBig/parenrightBig\n/precedesequal/parenleftBig\nλmax/braceleftBig\nE/bracketleftBig\nhkhT\nk/bracketrightBig/bracerightBig /parenrightBig 2\nI ÷/parenleftBig\nλmax/braceleftBig\nE/bracketleftBig\nhkhT\nk/bracketrightBig/bracerightBig/parenrightBig 2\n([21, Eq. (9)])\nRemark 2. If the fourth-order moment term in (18) in\nCorollary 2 vanishes (i.e., E[hkhT\nkhkhT\nk] =0), then\nχ= 2λmin/braceleftBig\nE[hkhT\nk]/bracerightBig\n. (20)\nEquation (19) in Corollary 2 becomes\nlim\nk→∞Πk=aσ2\nǫtr/braceleftBig\nE[hkhT\nk]/bracerightBig\n2λmin/braceleftBig\nE[hkhT\nk]/bracerightBig, (21)\nwhich is the same as the upper bound in [21, equation (11)].\nThe motivation of these results stem from analyzing the\nLyapunov stability of dynamical systems [26]. Specifically ,\nTheorem 1 provides sufficient conditions for convergenceof the LMS mean-squared error, where V(˜θk) is an analo-\ngous Lyapunov function and the dynamical system is the\ndynamic behavior that is dependent on k. The case where\nP=I(as presented in Corollary 2) provides not only a\nconservative estimate but also a connection to the energy\nstability [27]. Furthermore, the remarks in this paper\nconnect Theorem 1 and Corollary 2 to existing results in\n[18], [21] via considering the additional assumptions on hk.\nHowever, these assumptions are in general not guaranteed\nand thus the associated convergence conditions may fail\nwhen applied in practical applications.\nIII. Numerical examples\nIn this section, we consider two numerical examples\nto obtain the upper bound of constant gain aof LMS\nthrough applying Theorem 1 and Corollary 2, which will\nbe compared with the upper bound in Remark 1 according\nto [18], [21]. Next, we will compare the asymptotic limit\nof the upper bound of mean-squared error lim\nk→∞Πkin The-\norem 1 and Corollary 2 against that in [21] as mentioned\nin Remark 2. Although the synthetic Example 1 is not\nconsidering a E[hkhT\nk] varying over k, this example can\nstill demonstrate the advantage of proposed Theorem 1\nand Corollary 2 compared with the literature [18], [21].\nExample 1. We consider hk∈R2×1as a two-dimensional\nrandom variables with zero mean and respective standard\ndeviation σ1and σ2, and the correlation coefficient ρ.\nThus, we have covariance matrix\nE[hkhT\nk] =/bracketleftbiggσ2\n1 ρσ1σ2\nρσ1σ2 σ2\n2/bracketrightbigg\n. (22)\nWe denote P=/bracketleftbigg\np11p12\np21p22/bracketrightbigg\nleading to each component of\nE[hkhT\nkP h khT\nk] =:/bracketleftbigg\nf11f12\nf21f22/bracketrightbigg\nas\nf11=3σ4\n1p11+k2σ2\n1σ2\n2p22+k3σ3\n1σ2(p12+p21), (23a)\nf12=σ2\n1σ2\n2(p12+p21) +k3σ3\n1σ2p11+k3σ1σ3\n2p22,(23b)\nf21=k2σ2\n1σ2\n2(p12+p21) +k3σ3\n1σ2p11+k3σ1σ3\n2p22,(23c)\nf22=3σ4\n2p22+k2σ2\n1σ2\n2p11+k3σ1σ3\n2(p12+p21), (23d)\nwhere k2= 1+2 ρ2andk3= 3ρ. We consider ǫk∼ N(0, σ2\nǫ)\nand fix σǫ= 0.1. We consider four different sets of\nparameters σ1,σ2, and ρspecified in Table II.\nσ1σ2ρ\nExample 1A 1 1 0\nExample 1B 1 2 0\nExample 1C 1 1 0.5\nExample 1D 1 1 1\nTABLE II: Parameters σ1,σ2andρof Examples 1A-1D.\nExample 2. We use the oboe reed data\n(https://www.jhuapl.edu/ISSO/PDF-txt/reeddata-fit.pr n)\nas discussed in Section 3.4 and Exercise 3.16 of [12]. Here,\nwe consider the curvilinear model the same as equation(3.26) in [12]. We use the sample mean to approximate\nE[hkhT\nk]andE[hkhT\nkhkhT\nk]leading to\nE[hkhT\nk]\n=\n1.740 3 .850 2 .347 2 .122\n3.850 9 .770 5 .206 4 .910\n2.347 5 .206 3 .666 2 .936\n2.121 4 .910 2 .936 3 .585\n, (24)\nE[hkhT\nkhkhT\nk]\n=\n35.274 83 .016 51 .333 48 .512\n83.016 216 .007 120 .071 118 .083\n51.333 120 .071 83 .068 71 .289\n48.512 118 .083 71 .289 91 .267\n. (25)\nFor each example, we apply Theorem 1 and Corollary\n2 to obtain the lowest upper bounds of the constant gain\nsupaof LMS. For implementing equation (4) in Theorem\n1, we employ YALMIP [28] of version R20190425 with the\nsemi-definite programming (SDP) solver Mosek of version\n9.0 (www.mosek.com). The lowest upper bound of the\nconstant gain ais then obtained through a bisection search\nsolving\nmax a,s.t. Theorem 1 is satisfied . (26)\nSimilarly, the lowest upper bound of constant gain a\naccording to Corollary 2 is obtained by solving\nmax a,s.t. Corollary 2 is satisfied . (27)\nThis can be solved by the same toolbox as Theorem 1,\nbut Corollary 2 can be also solved by directly computingeigenvalues without any semi-definite programming (SDP)\nsolver.\nTable III compares the lowest upper bound of\nconstant gain sup aobtained through Theorem 1\nand Corollary 2 against informal arguments for\nconvergence of the constant gain LMS [18] as well\nas criteria in [21]. Here, the lowest upper bounds of\nthe constant gain sup aobtained from both Theorem\n1 and Corollary 2 are typically smaller than those\nobtained from informal arguments as 2/slashBig\nλmax{E[hkhT\nk]}\nand 2/slashBig\ntr{E[hkhT\nk]}[18]. This corresponds to the\nobservation that/parenleftBig\nE[hkhT\nk]/parenrightBig2\n/ne}ationslash=E[hkhT\nkhkhT\nk]\nin general, which provides a possible explanation\nfor why the informal criteria [18] typically fail to\nguide practical applications. Regarding Example 1A,\nthe criteria 2 λmin{E[hkhT\nk]}/slashBig /parenleftBig\nλmax{E[hkhT\nk]}/parenrightBig2\n[21] is the same as 2/slashBig\nλmax{E[hkhT\nk]}and thus\nalso larger than those obtained by Theorem 1\nand Corollary 2. Regarding Examples 1B-1C,\n2λmin{E[hkhT\nk]}/slashBig /parenleftBig\nλmax{E[hkhT\nk]}/parenrightBig2\n[21] gives the\nupper bound on ain a similar order to Theorem 1 and\nCorollary 2. This 2 λmin{E[hkhT\nk]}/slashBig /parenleftBig\nλmax{E[hkhT\nk]}/parenrightBig2\nis not applicable when E[hkhT\nk] has a zero eigenvalue,\nwhich appears when some elements of hkare linearly\ndependent [21] (see Example 1D).\nEx. 1A Ex. 1B Ex. 1C Ex. 1D Ex. 2\nTheorem 1 0.5000 (C) 0.1610 (C) 0.4227 (C) 0.3333 (C) 0.0875\nCorollary 2 0.5000 (C) 0.1538 (C) 0.4000 (C) 0.3333 (C) 0.071 6\n2/slashBig\nλmax{E[hkhT\nk]}[18] 2.0000 (N) 0.5000 (N) 1.3333 (N) 1.0000 (N) 0.1169\n2/slashBig\ntr{E[hkhT\nk]}[18] 1.0000 (N) 0.4000 (N) 1.0000 (N) 1.0000 (N) 0.1066\n2λmin{E[hkhT\nk]}/slashBig /parenleftBig\nλmax{E[hkhT\nk]}/parenrightBig2\n[21] 2.0000 (N) 0.1250 (C) 0.4443 (C) 0.0000 (-) 0.0007\nTABLE III: The lowest upper bound of constant gain sup aof LMS obtained using Theorem 1, Corollary 2, and\narguments in [18] and [21] for Examples 1A-1D and Example 2, r espectively. Inside of the brackets, ‘C’ denotes a\nbounded terminal error E[/bardblˆθk−θ∗/bardbl2\n2]<10, while ‘N’ represents unbounded terminal error E[/bardblˆθk−θ∗/bardbl2\n2]>108for\nnumerical Examples 1A-1D after k= 104iteration using LMS with a= sup a−10−4. ‘-’ means LMS is not applicable\ndue to zero gain. Here, E[/bardblˆθk−θ∗/bardbl2\n2] is averaged over 103replication.\nTable III shows that the upper bound of constant\ngain aobtained from Corollary 2 is smaller than that\nderived from Theorem 1, but Corollary 2 is much easier to\nsolve, which can provide an initial guidance on practical\napplications. Here, we also note that the upper bounds\nobtained from Theorem 1 and Corollary 2 are the same for\nExample 1, where the design vectors are uncorrelated ran-\ndom variables and having the same means and variances.\nIn Example 1D with perfect correlated elements in thedesign vector, the upper bound of the constant gain sup a\nobtained from Theorem 1 and Corollary 2 also overlap.\nHowever, Theorem 1 has the potential to provide a less\nconservative upper bound on acompared to Corollary 2\n(see Examples 1B and 1C).\nNext, we run LMS using the constant gain a= sup a−ξ\nwith ξ= 10−4according to a range of criteria in Table III.\nWe set the true value of parameter θ∗as/bracketleftbig\n1 1/bracketrightbigTand the\ninitial estimation ˆθ0as a random vector ∼ N(0,I2). Here,we run LMS with iteration k= 104and 103replications.\nThe convergence based on the expectation of the terminal\nerrorE[/bardblˆθk−θ∗/bardbl2\n2] (k= 104) is also indicated in the\nbracket of Table III. Here, we can see that the criteria\n2/slashBig\nλmax{E[hkhT\nk]}and 2/slashBig\ntr{E[hkhT\nk]}[18] lead to the\nunbounded estimation error for Examples 1A-1D, and the\ncriteria 2 λmin{E[hkhT\nk]}/slashBig /parenleftBig\nλmax{E[hkhT\nk]}/parenrightBig2\n[21] is also\nnot sufficient to provide a guarantee of a bounded mean-\nsquared error; see Example 1A. However, the upper bound\non the constant gain sup aobtained from Theorem 1 and\nCorollary 2 are able to provide a bounded E[/bardblˆθk−θ∗/bardbl2\n2]\nfor Examples 1A-1D.\nEx. 1A Ex. 1B Ex. 1C Ex. 1D\nGain a 0.4999 0.1537 0.3999 0.3332\nEq. (5) in Thm. 1 25.0 0.049 0.11 327\nEq. (19) in Cor. 2 25.0 1.01 10.66 Inf\nEq. (21) 0 .0050 0.0038 0.0080 Inf\nSimulation 0.072 0.0071 0.025 0.11\nTABLE IV: The selected gain a= sup a−10−4according\nto Corollary 2 and associated asymptotic upper bound\nofE[/bardblˆθk−θ∗/bardbl2\n2] based on equation (5) in Theorem 1,\nequation (19) in Corollary 2 and equation (21) based on\n[21, equation (11)]. These are compared with E[/bardblˆθk−θ∗/bardbl2\n2]\nobtained from numerical simulations after k= 104itera-\ntions averaged over 103replications.\nWe further analyze the asymptotic limit of the upper\nbound Π kas provided in equation (5) in Theorem 1,\nequation (19) in Corollary 2 and equation (21) according\nto Ref. [21]. We firstly select a constant gain athat is\nthe upper bound according to Corollary 2 minus ξ= 10−4\nsuch that (18) can be satisfied leading to a positive χusing\nCorollary 2. We then solve the following optimization\nproblem\nmax χ,s.t. Theorem 1 is satisfied . (28)\nThe resulting χandPare then used to obtain the error\nbound in equation (5) of Theorem 1. Similarly, we solve\nmax χ,s.t. Corollary 2 is satisfied , (29)\nwhere χcan be used to obtain error bound in (19) of\nCorollary 2. We also compute the error bound as (21)\naccording to [21]. The selected constant gain aand re-\nsulting asymptotic upper bounds lim\nk→∞Πkare summarized\nin Table IV. We also perform simulations and evaluate\nE/bracketleftBig\n/bardblˆθk−θ∗/bardbl2\n2/bracketrightBig\natk= 104averaged over 103replications,\nwhich is reported also in Table IV. Here, we can see that\nthe upper bounds provided by Theorem 1 and Corollary\n2 are larger than the simulation results, while the upper\nbound according to [21] is violated in Examples 1A-1C.\nIn the work of [21] the fourth-order term is neglected\nassuming ais small enough (see Remark 2, which is likely\nto result in the failure of the error bound [21] as shown\nhere). Moreover, Theorem 1 provides an error bound that\nis significantly tighter than that based on Corollary 2 inExamples 1B-1C. In Example 1D, the correlated elements\nof the design vector lead to λmin{E[hkhT\nk]}= 0 and thus\nequation (21) leads to infinity upper bound, which is also\nmentioned as one limitation [21]. The upper bound in\nCorollary 2 also similarly leads to an infinity upper bound.\nHowever, Theorem 1 is able to provide a finite upper\nbound, which benefited from a general matrix P.\nθF θAE θTF θTF\nTheorem 1 0.330 0.086 0.012 0.010\nCorollary 2 0.275 0.081 0.041 0.027\n2/slashBig\nλmax{E[hkhT\nk]}[18] 302 3275 3049 -3446\n2/slashBig\ntr{E[hkhT\nk]}[18] 533 426 278 -193\n2λmin{E[hkhT\nk]}\n(λmax{E[hkhT\nk]})2[21] 0.082 0.184 0.112 0.106\nBatch LS [12] 0.584 0.101 0.078 0.034\nRecursive LS [12] 0.557 0.106 0.086 0.036\nTABLE V: Terminal estimation ˆθ161using constant gain\na= sup a−10−4of LMS using sup aobtained from\nTheorem 1, Corollary 2, and informal arguments [18] and\ncriterion [21] for Example 2 as reported in Table III.\nResults are compared with those obtained from batch\nleast-squares (LS) and recursive LS using the same data\nset [12].\nFor Example 2 we further consider\nthe data from practical measurements\n(https://www.jhuapl.edu/ISSO/PDF-txt/reeddata-fit.pr n)\nas discussed in Section 3.4 and Exercise 3.16 in [12]. The\nobtained upper bounds are also compared with those\nobtained from informal arguments [18] as shown in Table\nIII. In Table V, we report the terminal estimation ˆθ161\nof Example 2 after using all available measurements.\nThe initial estimation ˆθ0=0is employed, which is the\nsame as that used in Exercise 3.16 for LMS in [12]. In\nTable V, we also compare the results obtained from the\nbatch least-squares method and recursive least-squares\nas reported in [12, Table 3.2]. Here, we can see that the\nupper bound of the constant gain proposed in this paper\nis also applicable in practical data, while the informal\nargument in [18] is likely to have a large estimation error.\nThe criterion 2 λmin{E[hkhT\nk]}/slashBig\n(λmax{E[hkhT\nk]})2[21]\nleads to a relatively small upper bound of aas shown in\nTable III, which may lead to a slow converging rate of\nLMS as shown in Table V. This suggests that Theorem 1\nand Corollary 2 are also applicable to practical data.\nIV. Conclusion and future work\nTheorem 1 and Corollary 2 propose upper bounds on\nthe constant gain of LMS that guarantees a bounded\nmean-squared error. These sufficient conditions highlight\nthe role of fourth-order moments in the design vector.\nWe demonstrate the applicability of these upper bounds\nnumerically based on synthetic and practical data, while\nexisting criteria [18], [21] introduce additional assump-\ntions on the design vector and may lead to unboundedestimation results within constant gain LMS. The main\nresult in Theorem 1 is computed based on semi-definite\nprogramming (SDP), while Corollary 2 does not require\nan SDP solver which is able to provide initial guidance\nof the upper bound on the gain. However, Theorem 1\nhas the potential to provide a tighter upper bound on\nconstant gain athan Corollary 2 which may lead to a\nfaster convergence rate.\nThese theoretical results also provide upper bounds\nof estimation error that demonstrate consistency with\nnumerical examples, which is an improvement of [21].\nAdditionally, the error bound of estimation error in The-\norem 1 is applicable when elements of the design vector\nare linearly dependent, while both Corollary 2 and Ref.\n[21] fail to provide a finite upper bound (Example 1D).\nMoreover, the numerical examples further demonstrate\nthat Theorem 1 and Corollary 2 provide the same lowest\nupper bound on the constant gain and estimation error\nwhen elements of the design vector are independent with\nthe same variance (Example 1A).\nThis work can be naturally extended to consider the\ninherent dependence between {hk}and{ǫk}, where the\nassociated fourth-order moment E[ǫ2\nkhkhT\nk] is also likely to\nbe important; see the last term in (10b). 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SIAM, 1994.\n[27] C. Liu and D. F. Gayme, “Input-output inspired method\nfor permissible perturbation amplitude of transitional wa ll-\nbounded shear flows,” Physical Review E , vol. 102, p. 063108,\n2020.\n[28] J. 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Here, for the first time, we develop an analytic model of an axisymmetric switchback\nwith uniform magnetic field strength. In this model, three parameters control the geometry of the\nswitchback: height (length along the background magnetic field), width (thickness along radial direc-\ntion perpendicular to the background field), and the radial distance from the center of switchback\nto the central axis, which is a proxy of the size of the switchback along the third dimension. We\ncarry out three-dimensional magnetohydrodynamic simulations to investigate the dynamic evolution\nof the switchback. Comparing simulations conducted with compressible and incompressible codes, we\nverify that compressibility, i.e. parametric decay instability, is necessary for destabilizing the switch-\nback. Our simulations also reveal that the geometry of the switchback significantly affects how fast\nthe switchback destabilizes. The most stable switchbacks are 2D-like (planar) structures with large\naspect ratios (length to width), consistent with the observations. We show that when plasma beta ( β)\nis smaller than one, the switchback is more stable as βincreases. However, when βis greater than one,\nthe switchback becomes very unstable as the pattern of the growing compressive fluctuations changes.\nOur results may explain some of the observational features of switchbacks, including the large aspect\nratios and nearly constant occurrence rates in the inner heliosphere.\n1.INTRODUCTION\nOne of the most striking findings made by Parker Solar Probe (PSP) is that the nascent solar wind is almost filled\nwith magnetic switchbacks (“switchbacks” hereinafter) (Kasper et al. 2019; Bale et al. 2019). Switchbacks are local\npolarity reversals of the radial magnetic field and are typically Alfv´ enic structures, with highly correlated velocity and\nmagnetic field fluctuations and nearly constant magnetic field strength (McManus et al. 2020; Woolley et al. 2020;\nLarosa et al. 2021). For a comprehensive description of switchbacks and their properties, please refer to the review\npaper (Raouafi et al. 2023a) and references therein.\nPSP observations have revealed that switchbacks significantly modify the properties of solar wind turbulence, high-\nlighting their influential role in the solar wind dynamics (Dudok de Wit et al. 2020; Bourouaine et al. 2020; Martinovi´ c\net al. 2021; Shi et al. 2022b). Besides, as large-amplitude fluctuations, switchbacks are believed to be an important\nenergy source for the heating and acceleration of the solar wind (Halekas et al. 2023). Indeed, observations show that\nthe plasma properties, including proton temperature and temperature anisotropy (Farrell et al. 2020; Larosa et al.\n2021; Woolley et al. 2021; Woodham et al. 2021; Luo et al. 2023; Huang et al. 2023; Laker et al. 2024), are different\ninside and outside the switchbacks, with a possible enhancement of proton temperature inside the switchbacks.\nCorresponding author: Chen Shi\ncshi1993@ucla.eduarXiv:2401.11334v3  [physics.space-ph]  31 Mar 20242 Shi et al.\nAlthough there is no doubt that understanding switchbacks is crucial for a complete understanding of the solar\nwind, how are the switchbacks generated and how do they propagate in the solar wind is still under debate. There\nare three widely accepted theories for the generation mechanisms of switchbacks, namely the interchange reconnection\nhappening in the solar corona (Yamauchi et al. 2002; Fisk & Kasper 2020; Zank et al. 2020; Drake et al. 2021; He\net al. 2021; Telloni et al. 2022; Upendran & Tripathi 2022; Hou et al. 2023), the velocity shear (Landi et al. 2006;\nToth et al. 2023) that may arise due to coronal jets (Sterling & Moore 2020; Magyar et al. 2021; Raouafi et al.\n2023b) or motion of the magnetic footpoint between source regions of fast and slow streams (Schwadron & McComas\n2021), and the natural evolution of Alfv´ en waves in the expanding solar wind (Belcher 1971; Hollweg 1974; Squire\net al. 2020; Shoda et al. 2021; Mallet et al. 2021). However, which mechanism is dominant is still under debate.\nAs different generation mechanisms take place at different radial distances to the Sun, it is thus important to study\nthe radial evolution of the occurrence rate of switchbacks that may suggest which generation mechanism is most\neffective. Unfortunately, no solid conclusion has been made on this topic either. Mozer et al. (2020) suggest that the\nswitchback occurrence rate does not change much with radial distance, while other studies indicate a dependence of\nthe occurrence rate on the duration of the switchback, i.e. the occurrence rate of larger switchbacks increases with the\nradial distance and the occurrence rate of shorter-duration switchbacks shows a decreasing trend (Tenerani et al. 2021;\nJagarlamudi et al. 2023). Moreover, there is evidence showing that the occurrence rate of switchbacks is smaller inside\nthe Alfv´ en radius than outside (Pecora et al. 2022; Bandyopadhyay et al. 2022). Hence, it is likely that a subset of the\nswitchbacks observed in-situ are generated in the lower corona while the others are generated locally (Tenerani et al.\n2021). Another important question is how does a switchback evolve in the solar wind, because the dynamic evolution\nof switchbacks inevitably affects the characteristics of switchbacks observed in-situ, including their occurrence rates\nas well as their topology (Laker et al. 2021). Magnetohydrodynamic (MHD) simulations (Tenerani et al. 2020) show\nthat a 2D Alfv´ enic switchback maintains stable for quite a long time (tens of Alfv´ en crossing time) and is eventually\ndestabilized by the growth of density fluctuations, i.e. parametric decay instability (PDI). In addition, the Parker\nspiral (Johnston et al. 2022; Squire et al. 2022) and Hall effect (Tenerani et al. 2023) also modify the evolution and\ndissipation of the switchbacks.\nThe objective of this study is to numerically investigate the dynamic evolution of 3D switchbacks. The 3D effect has\nnot been fully investigated in previous studies because of the difficulty to establish a 3D analytic model that can satisfy\nboth the divergence-free condition and a uniform magnetic field strength. Squire & Mallet (2022) developed a time-\nintegration method that grows the amplitude of the initially weak spherically-polarized magnetic field fluctuations (with\nnormalized magnetic field fluctuation δB/|B| ≈0.2) while keeping the uniform-magnitude constraint and divergence-\nfree condition. Here, we construct an analytic solution of a single 3D switchback, so that we are able control the\ngeometry of the switchback and investigate the effect of the switchback’s geometry on its evolution. In addition, we\nanalyze how the plasma beta affects the destabilization process of the switchbacks.\nThe paper is organized as follows: In Section 2, we describe in detail the analytic model of the 3D switchback. In\nSection 3, we show the results of a series of 3D MHD simulations conducted based on the analytic model. In Section\n4, we summarize the study and discuss possible future investigation.\n2.ANALYTIC MODEL OF A 3D AXISYMMETRIC MAGNETIC SWITCHBACK\nWe consider a cylindrical coordinate system ( r, ϕ, z ) and assume axisymmetric ( ∂/∂ϕ≡0). In this case, the magnetic\nfield can be expressed by a scalar flux function ψ(r, z) plus a ϕ-component:\nB=1\nr∇ψ׈eϕ+Bϕˆeϕ=−1\nr∂ψ\n∂zˆer+1\nr∂ψ\n∂rˆez+Bϕˆeϕ, (1)\nwhich automatically satisfies the ∇ ·B= 0 condition. We further decompose ψinto a uniform background field\nψ0=1\n2B0r2, which corresponds to B0=B0ˆez, and a fluctuation part ψ1(r, z). In order to have a switchback, we\nassume that ψ1has the form\nψ1(r, z) = ∆ Bexp\u0012\n−z2\nH2\u0013\nf(r) (2)\nwhere ∆ Bis the strength of the switchback, His the scale length of the switchback along zdirection, i.e. its “height”.\nTo confine the switchback within a region in radial direction, we further require\n1\nrd f\ndr= exp\u0014\n−(r−R0)2\nR2\n1\u0015\n(3)3D magnetic switchbacks 3\nFigure 1. Visualization of the analytic model of the magnetic field described in Section 2. The model parameters are:\n∆B/B 0=−2,R0= 2, R1= 0.5,H= 3, Rm= 2, and Rd= 1.5. (a) Radial profiles of the r(blue), ϕ(orange), z(green)\ncomponents, and magnitude (black) of the magnetic field. (b) Two magnetic field lines (green and white) over-plotted on three\northogonal surfaces which are color-coded with Bz. The background field is along zdirection.\nwhere R0is the radial location of the switchback’s center, and R1is the radial size of the switchback, i.e. its “width”.\nThe above model of magnetic field is 2.5D instead of fully 3D as we are assuming symmetry in ϕ. Nonetheless, the\nparameter R0serves as a proxy of the size of the switchback along the third dimension: The larger R0is, the more\n2D-like, i.e. planar, the structure is. In Appendix A, we present an alternative model which consists of trigonometric\nfunctions instead of Gaussian functions.\nThe solution of f(r) to Equation (3) is\nf(r) =−1\n2R2\n1\"\ne−(r−R0)2\nR2\n1+√πR0\nR1erf\u0012R0−r\nR1\u0013\n+C#\n(4)\nwhere erf( x) is the error function and the integral constant Cis chosen such that f(0) = 0, i.e.\nC=−\"\ne−R2\n0\nR2\n1+√πR0\nR1erf\u0012R0\nR1\u0013#\n(5)\nThus, the analytic form of BrandBzare\nBr(r, z) =−∆BzR2\n1\nrH2e−z2\nH2\"\ne−(r−R0)2\nR2\n1+√πR0\nR1erf\u0012R0−r\nR1\u0013\n+C#\n(6a)\nBz(r, z) =B0+ ∆Bexp\u0014\n−(r−R0)2\nR2\n1−z2\nH2\u0015\n(6b)\nEquation (6b) indicates that the switchback is centered at ( r=R0, z= 0) and has a size ( R1, H) in the r−zplane.\nIdeally, Bϕcan be used to maintain the constant- |B|condition. However, Equation (6) does not ensure B2\nr+B2\nz≤B2\n0,\nand there are certain regions where the constant- |B|condition cannot be preserved. The reason is that in the above\nmodel, Br→1/rfor large r, which is not fast enough to maintain B2\nr+B2\nz≤B2\n0. This also leads to a difficulty in\nsetting up the boundary condition because we cannot directly implement periodic boundary conditions if Brdoes not\ndecay to zero at the boundaries. To overcome this difficulty, we modify the flux function with a predefined function\np(r) such that Brdecays exponentially outside the switchback region and Bzdoes not change in the z=0 plane:\n˜ψ(r, z) =p(r)ψ(r, z) + [1−p(r)]ψ(r,0). (7)4 Shi et al.\nForp(r), we choose the following function\np(r) =\n\nexp\u0014\n−\u0010\nr−Rm\nRd\u00112\u0015\nr≥Rm\n1 r < R m(8)\nwhere Rmis the radial location at which the modulation of the magnetic field starts, and Rdis the scale length of\nthe modulation. With the modified flux function, one can calculate the modified magnetic field which results in the\nmodified BrandBz:\n˜Br=p(r)Br(r, z) (9a)\n˜Bz=p(r)Bz(r, z) + [1−p(r)]Bz(r,0) +1\nrdp\ndr[ψ(r, z)−ψ(r,0)] (9b)\nthat satisfy ˜B2\nr+˜B2\nz≤B2\n0and hence Bϕ=\u0010\nB2\n0−˜B2\nr−˜B2\nz\u00111/2\n. We note that this modulation indeed does not change\nthe magnetic field at the z= 0 plane. A different but equivalent approach to do the modulation is to assume Bris\nmodified by a given function p(r) (Equation (9a)) and Bzis modified by an unknown function q(r, z), and then solve\nq(r, z) such that the modified magnetic field preserves the divergence-free condition. This approach is described in\nAppendix B. We neglect the tildes above BrandBzhereinafter for convenience.\nIn Figure 1, we visualize the above analytic model with parameters ∆ B/B 0= 2,R0= 2,R1= 0.5,H= 3,Rm= 2,\nandRd= 1.5. Panel (a) shows the radial profiles of the three components (colored) and magnitude (black) of the\nmagnetic field. Solid lines are z= 0, i.e. through the center of the structure, and dashed lines are z=H/2. In\npanel (b), the three orthogonal surfaces are color-coded with Bz, and the green and white lines are two magnetic field\nlines. Here, the green line is connected to the background magnetic field, while the white line is confined within the\nswitchback region. That is to say, in our model, not all of the field lines are connected to the Sun. In contrast, in-situ\nobservations show that the magnetic field inside switchbacks is connected to the Sun (Balogh et al. 1999; Kasper et al.\n2019). Thus, this model is a combination of switchbacks and flux ropes. However, we note that magnetic flux ropes\nare occasionally observed near the switchbacks by PSP (Choi et al. 2024). This phenomenon implies the possibility\nthat reconnection takes effect either during the generation of the switchbacks (Drake et al. 2021; He et al. 2021) or\nduring the propagation of the switchbacks in the solar wind. In Appendix C, we briefly discuss how the reconnection\nbetween two switchbacks may generate a flux rope.\nOne limitation of the model is that, in the region |z|> H andRm≤r≲Rm+Rd,Bzis smaller than B0due to\nthe introduced modification (see panel (b) of Figure 1 and Figure 2). Consequently, there is a finite Bϕin this region,\ni.e. the background field is twisted azimuthally. This low- Bzregion extends infinitely in z. Actually, the ∇ ·B= 0\ncondition means it is impossible to find a switchback model that has finite extents in both r(thex−yplane) and z.\nThe proof is simple: Suppose that the fluctuation part of the field totally vanishes outside a finite radial distance Rb\nin the x−yplane and consider a cylinder with r=Rband 0 ≤z≤Hb. Clearly there is no magnetic flux across the\nside surface r=Rb, hence the divergence-free condition writes as\nZZ\nz=Hb,r≤RbBz=ZZ\nz=0,r≤RbBz. (10)\nAs the perturbation in Bzat the z= 0 surface is negative definite, we immediately see that Bz(z=Hb)≡B0cannot\nsatisfy the above equation, i.e. for any horizontal plane z=Hb, there must be some regions where Bz< B 0.\n3.NUMERICAL SIMULATIONS\n3.1. Simulation setup\nBased on the analytic model described in Section 2, we conduct a series of 3D MHD simulations using a pseudo-\nspectral MHD code (Shi et al. 2019, 2020, 2022a; Tenerani et al. 2023; Shi et al. 2023). The code evolves the MHD\nequation set in conservation form on a Cartesian grid. The initial condition comprises a uniform density ρ0≡1, a\nuniform thermal pressure P0, magnetic field as described in Section 2 with B0= 1, and velocity that is anti-correlated\nwith the magnetic field ( V=−B/√ρ0). We note that the code uses normalized quantities such that permeability µ0\nis eliminated.\nThe parameters for the conducted simulations are summarized in Table 1. In all the runs listed in Table 1, we\nuse the same spatial resolution ∆ = 0 .047 along all dimensions. In addition, several fixed parameters include the3D magnetic switchbacks 5\nRun ∆B/B 0R0R1H RmRdP0Lx, Ly, Lz nx, ny, nz\n0, 0IC -2 20.5 3 21.5 1.0 24,24,24 512,512,512\n1 -2 40.5 3 41.5 1.0 48,48,24 1024,1024,512\n2 -2 20.5 6 21.5 1.0 24,24,48 512,512,1024\n3 -2 20.5 1.5 21.5 1.0 24,24,12 512,512,256\n4 -2 20.5 3 21.5 0.75 24,24,24 512,512,512\n5 -2 20.5 3 21.5 0.5 24,24,24 512,512,512\n6 -2 20.5 3 21.5 0.2 24,24,24 512,512,512\n7 -2 20.5 3 21.5 0.1 24,24,24 512,512,512\n8 -2 20.5 3 21.5 0.05 24,24,24 512,512,512\nTable 1. Simulation parameters. Additional low-resolution runs to verify the grid-convergence are not listed here (see the\ntext).\nswitchback strength ∆ B/B 0≡2, the width of the switchback R1≡0.5, and the parameters for modification of the\nmagnetic field outside the switchback region: Rm≡R0,Rd≡1.5. We do not implement explicit resistivity and\nviscosity. Instead, we apply a numerical filter in the wave-vector space, which is similar to the filters usually adopted\nin compact-finite-difference schemes (Lele 1992), to all quantities after each time step to maintain numerical stability.\nFor a detailed description of the filter, please refer to equations (16,17) of Shi et al. (2020). To verify grid convergence\nof the simulations, several low-resolution runs (not listed in Table 1) based on Run 0 are conducted, including one run\nwith nz= 256, one run with nx=ny= 256, one run with nx=ny=nz= 256, and one run with nx=ny=nz= 256\nand weaker numerical filter. We have verified that the linear growth rate of the instability is not affected by the\nresolution and the filter strength.\nCompared with Run 0, Run 1 has doubled R0, Run 2 has doubled H, and Run 3 has one half of H, so that we can\ninvestigate the effect of the geometry on the switchback evolution. We also carry out one run, labeled as Run 0IC,\nwith exactly the same initial condition with Run 0, but using incompressible version of the simulation code. For Runs\n0-3,P0= 1.0 so that the plasma beta is β= 2.0. To investigate how βinfluences the evolution, we conduct Runs 4-8\nwhich have the same initial configuration with Run 0 but different values of P0so that βvaries in the range 0.1-2.0.\n3.2. Results\nIn Figure 2, we show the snapshots of Runs 0 (top), 0IC (middle), & 8 (bottom) color-coded with Bz. The plots\nare 2D slices in x−zplane, across the central axis ( y= 0). The black lines are magnetic field lines projected on\nthe 2D plane. In Run 0IC, the switchback remains stable until the end of the simulation ( t= 300), while in Run 0\nand Run 8, the switchback remains stable for a certain time but eventually destabilizes. These results indicate that\ncompressibility is a necessary condition for the destabilization of the switchbacks, verifying the conclusion of Tenerani\net al. (2020), i.e. parametric decay instability is the major mechanism in the destabilization of the switchbacks. .\nIn Figure 3, we plot the time evolution of the normalized density fluctuation ( δρ/⟨ρ⟩) (top) and minimum Bzin the\nsimulation domain (bottom) in different runs. Here δρis the root-mean-square and ⟨ρ⟩is the average of density. Let us\nfirst focus on the left column, which shows results of Runs 0 (blue), 1 (orange), 2 (green), and 3 (red). Apparently, the\ndensity fluctuations grow exponentially over certain time intervals in all the Runs before they saturate. As the density\nfluctuations grow, minimum Bzgradually decreases below −1. After the density fluctuations saturate, minimum Bz\nstarts to rise rapidly and eventually becomes positive. That is to say, the magnetic field is first compressed as the PDI\ngrows and then destabilizes rapidly after the PDI saturates. We note that, due to the numerical diffusivity, min( Bz)\ngradually increases with time if there is no compressibility or the growth rate of PDI is small. Comparing Runs 0,\n2, and 3, which have different values of H(length/height of the switchback), we can see that the growth rate of the\nPDI is largest in Run 3 and smallest in Run 2, indicating that longer switchbacks are more stable than the shorter\nones, consistent with the PSP observations which show that the switchbacks in the solar wind are typically very long\nstructures (Horbury et al. 2020; Laker et al. 2021). Comparing Run 0 and Run 1, we see that the switchback is much\nmore stable in Run 1, in which R0, the size of the structure along the third dimension, is larger. That is to say, the\nswitchback is more stable if it is more 2D-like or planar. In conclusion, Runs 0-3 show that the most stable switchbacks\nhave 2D-like structures and large aspect ratios (length to width).6 Shi et al.\nFigure 2. Snapshots of the x−zplane across the central axis ( y= 0) color-coded with Bzfor Runs 0 (a,top), 0IC (b,middle),\nand 8 (c,bottom). Black curves are magnetic field lines projected on the 2D plane.\nThen we investigate the effect of plasma beta by inspecting the right column of Figure 3. From the time profiles of\nδρ/⟨ρ⟩, we see that the evolution in runs with β >1 is quite different from the evolution in runs with β≤1. The\ngrowth rates of the instability are almost identical in Run 0 ( β= 2.0) and Run 4 ( β= 1.5). When βdrops to unity\n(Run 5), the growth rate decreases significantly. As we further decrease βto 0.4 (Run 6), the growth rate does not\nchange much compared with Run 5. Eventually, as βcontinues to drop (Run 8, β= 0.1), the growth rate starts to\nincrease and becomes comparable to β >1. Although the growth rates of the instability are comparable in Runs 0,\n4, and 8, the disturbance of the magnetic field is much slower in Run 8 than in Runs 0 & 4 (bottom right panel),\nimplying that the characteristics of the density fluctuation may be very different depending on plasma beta.3D magnetic switchbacks 7\nFigure 3. Time evolution of (top) the normalized density fluctuations δρ/⟨ρ⟩where δρis the root-mean-square of the density\nand⟨ρ⟩is the average density and (bottom) the minimum value of Bzin the simulation domain. In the left column, we show\nthe results for Runs 0, 1, 2, 3. In the right column we show the results for Runs 0, 4, 5, 6, 7, 8.\nIt is well known that the growth rate of PDI depends on β(Derby Jr 1978; Jayanti & Hollweg 1993). For a\nmonochromatic, circularly polarized Alfv´ en wave, the dispersion relation of PDI is described by the equation (Derby Jr\n1978; Tenerani & Velli 2013; R´ eville et al. 2018)\n(ω+k+ 2) ( ω+k−2) (ω−k)\u0010\nω2−˜βk2\u0011\n−A2k2\u0000\nω3+kω2−3ω+k\u0001\n= 0 (11)\nwhere ωandkare frequency and wavenumber of the density fluctuation normalized to the frequency and wavenumber\nof the mother wave, A=δB/B 0is the amplitude of the mother wave normalized to the background magnetic field, and\n˜β=C2\ns/V2\nAwhere CsandVAare the sound speed and Alfv´ en speed of the background plasma. Note that we use the\nsymbol ˜βinstead of βsince ˜β= 5/6×βassuming an adiabatic index of 5 /3. In panels (d)-(f) of Figure 4, we plot the\nsolved dispersion relation based on Equation (11) for (d) A= 0.1, (e) A= 1.0, and (f) A= 2.0 respectively. In each\npanel, the solid lines are growth rate and the dashed lines are frequency. Colors of the lines correspond to different\nvalues of ˜β. Clearly, regardless of the wave amplitude A, the growth rate decreases with ˜βand the wavenumber\ncorresponding to the maximum growth rate decreases toward k0. As pointed out by (Jayanti & Hollweg 1993), for\nβ >1, the instability is more like a beat instability as the wavenumber and frequency of the generated sound wave\nare close to those of the mother wave. In this case, the secondary transverse fluctuation is dominated by the forward\npropagating Alfv´ en wave with wavenumber close to 2 k0. In panels (a1)-(c2), we show the snapshots color-coded with\nρ1in Runs 0 (a, β= 2.0), 5 (b, β= 1.0), and 8 (c, β= 0.1) with the magnetic field lines. Here ρ1=ρ− ⟨ρ⟩is\nthe density fluctuation. We can see that, for β= 2.0, during the linear stage (panel (a1)), the density fluctuation is\nconcentrated within the switchback region. For small β(= 0.1) (panels (c1,c2)), the density fluctuation has shorter8 Shi et al.\nFigure 4. Panels (a1)-(c2): Snapshots of the x−zplane across the central axis ( y= 0) color-coded with ρ1for Runs 0 (a,top),\n5 (b,middle), and 8 (c,bottom) where ρ1=ρ− ⟨ρ⟩is the density fluctuation. Black curves are magnetic field lines projected\non the 2D plane. Panels (d)-(f): Linear growth rate (solid) and frequency (dashed) of the parametric decay instability for a\nmonochromatic circularly-polarized Alfv´ en wave with amplitude (d) A= 0.1, (e) A= 1.0, and (f) A= 2.0. In each panel,\ndifferent curves correspond to different values of ˜β=C2\ns/V2\nA, where CsandVAare the sound speed and Alfv´ en speed of the\nbackground field. ω0,k0are the frequency and wavenumber of the mother Alfv´ en wave. A=δB/B 0is the amplitude of the\nmother wave normalized to the background magnetic field.\nwavelength and is distributed along the z-direction, similar to that observed in 2D simulations conducted by Tenerani\net al. (2020) (see their Figure 6). For intermediate β(Run 5, β= 1.0), we see growth of density fluctuations both\ninside and outside the switchback, and the wavelength of the growing mode is larger than Run 8. These results are\nqualitatively consistent with the linear theory. However, a major discrepancy is that, in our simulations, the growth\nrate of the instability for β >1 is larger than β <1 except for very small β(β= 0.1), while in the linear theory, the\nmaximum growth rate monotonically decreases with β. We point out that, the simulated switchback is not a simple\nmonochromatic, circularly-polarized, one-dimensional Alfv´ en wave, so the stability of this structure is likely to deviate\nfrom linear theory described by Equation (11) (Malara & Velli 1996).3D magnetic switchbacks 9\n4.SUMMARY\nIn this study, we develop an analytic model of an axisymmetric 3D magnetic switchback with uniform magnetic field\nstrength. A series of 3D MHD simulations are conducted to investigate the evolution of the switchback. We focus on\nhow the geometry and plasma beta modify the switchback evolution. The major conclusion of the simulations is\n•Compressibility is a necessary condition for the destabilization of the switchbacks (see also Tenerani et al. 2020).\nThe switchback is stable in the incompressible-MHD simulation.\n•The most stable switchbacks are 2D-like structures with large aspect ratios, i.e. planar structures elongated\nalong the background magnetic field direction.\n•The plasma beta plays an important role in the destabilization of switchbacks. For β≪1, the structure is\nunstable due to the classic parametric decay instability. β≲1 is the most stable regime for the structure. For\nβ >1, the structure becomes unstable again due to the beat instability (Jayanti & Hollweg 1993). Destabilization\nof the magnetic field in this case is much faster than β <1.\nOur results may explain why the switchbacks observed by PSP are mostly long structures (Horbury et al. 2020; Laker\net al. 2021). Moreover, the β-dependence of the stability shown by the simulations suggests a scenario where the\noccurrence rate of switchbacks in the inner heliosphere ( β≲1) does not change significantly with radial distance to\nthe Sun (Mozer et al. 2020; Pecora et al. 2022), but drops significantly when βgrows to greater than unity during the\npropagation of the solar wind. This may explain why much less switchbacks are observed beyond 1 AU, e.g. Ulysses\nobservation (e.g. Balogh et al. 1999), than inside 1 AU. An investigation of the relation between the occurrence rate of\nswitchbacks and the plasma beta will be necessary to verify this point. Last, we note that our simulations start with a\nswitchback that is already generated. An important question is whether and how different generation mechanisms lead\nto different structures of the switchbacks. However, this is beyond the scope of the current study. Future numerical\nand observational studies are necessary to answer this question.\nThis work is supported by NSF SHINE #2229566 and NASA ECIP #80NSSC23K1064. The numerical simulations\nare conducted on Extreme Science and Engineering Discovery Environment (XSEDE) EXPANSE (allocation No. TG-\nAST200031.) at San Diego Supercomputer Center (SDSC), which is supported by National Science Foundation grant\nnumber ACI-1548562 (Towns et al. 2014), and the Derecho: HPE Cray EX System (https://doi.org/10.5065/qx9a-\npg09) of Computational and Information Systems Laboratory (CISL), National Center for Atmospheric Research\n(NCAR) (allocation No. UCLA0063). We thank Dr. Jaye L Verniero for very inspiring discussions.1\n2\n3\n4\n5\n6\nSoftware: Matplotlib (Hunter 2007), NumPy (Harris et al. 2020)\nAPPENDIX\nA.AN ALTERNATIVE ANALYTIC MODEL OF THE SWITCHBACK\nIn Section 2, we introduced an analytic model of the switchback (Equation (6)) based on Gaussian functions. Here,\nwe show an alternative model which is based on simpler trigonometric functions and is more compact spatially. The\ndefinition of the parameters ∆ B,R0,R1,H,Rm,Rdremains the same with the main text.\nThe background field is still ψ0=1\n2B0r2. For the fluctuation part, we write\nψ1(r, z) = ∆ B×g(z)f(r) (A1)\nwhere\ng(z) =(\ncos2\u0000π\n2z\nH\u0001\n|z| ≤H\n0 |z|> H(A2)\nWe define s=R1/π,x= (r−R0)/2s, and write f(r) as\nf(r) =s2\u0002\nx2+xsin(2x) + cos(2 x)/2\u0003\n+sR0[x+ sin(2 x)/2] +C,|r−R0| ≤R1 (A3)10 Shi et al.\nThe constant Cis such that f(R0−R1) = 0. We require f(r > R 0+R1)≡f(R0+R1) and f(r < R 0−R1)≡0. One\ncan easily show that the above f(r) and g(z) lead to\nBz(r, z) =B0+ ∆Bcos2\u0010π\n2z\nH\u0011\ncos2\u0012π\n2·r−R0\nR1\u0013\n,|z| ≤H,|r−R0| ≤R1 (A4)\nSimilar to the model described in Section 2, we need to modify the magnetic field so that Brvanishes outside a certain\nradius. Here, we set the modification function p(r) as\np(r) =\n\n1 r < R m\ncos4\u0010\nπ\n2·r−Rm\nRd\u0011\nRm≤r≤Rm+Rd\n0 r > R m+Rd(A5)\nThe finalized flux function is then described by Equation (7). Based on experiments, we find that Rm≤R0is necessary\nif ∆B=−2B0, and Rm< R 0+R1for ∆ B >−2B0.\nB.AN ALTERNATIVE APPROACH TO MODULATE THE MAGNETIC FIELD OUTSIDE THE\nSWITCHBACK REGION\nIn this section, we describe an approach that is equivalent to Equation (7) but directly modifies the magnetic field\ninstead of the flux function. Suppose we have a predefined function p(r) so that the modified Bris\n˜Br(r, z) =p(r)Br(r, z). (B6)\nObviously, Bzalso needs to be modified by a function q(r, z)\n˜Bz(r, z) =q(r, z)Bz(r, z) (B7)\nin order to preserve the divergence-free condition ∇ ·˜B= 0, which leads to an equation for q(z) at a fixed r:\ndq(z)\ndz+P(z)q(z) =Q(z) (B8)\nwhere\nP(z) =1\nBz∂Bz\n∂z, Q (z) =p\nBz∂Bz\n∂z−Br\nBz∂p\n∂r.\nWe note that ∇·B= 0 is used to simplify the equation. Equation (B8) is a typical first-order linear ordinary differential\nequation, and has a solution of the following form\nq(z) =1\nI(z)\u0014\nCq+Zz\n0I(z′)Q(z′)dz′\u0015\n(B9)\nwhere Cqis an integral constant and\nI(z) = exp\u0014Zz\n0P(z′)dz′\u0015\n=Bz(z)\nBz(0)\nWe further require q(z= 0)≡1 so that Bzis not modified at the central plane z= 0, which then leads to Cq= 1.\nHence, the final form of q(z) is\nq(z) =Bz(0)\nBz(z)\u001a\n1 +1\nBz(0)Zz\n0\u0014\np∂Bz\n∂z−Br∂p\n∂r\u0015\ndz′\u001b\n(B10)\nSince we assume p(r) is not a function of z, the integral on the right-hand-side of the above equation can be calculated\neasily:Zz\n0p∂Bz\n∂zdz′=p[Bz(z)−Bz(0)] = p×∆Bexp\u0014\n−(r−R0)2\nR2\n1\u0015\u0010\ne−z2\nH2−1\u0011\n(B11)\nandZz\n0Br∂p\n∂rdz′=dp\ndr×∆BR2\n1\nr\"\ne−(r−R0)2\nR2\n1+√πR0\nR1erf\u0012R0−r\nR1\u0013\n+C#\n×1\n2\u0010\ne−z2\nH2−1\u0011\n(B12)\nOne can verify that the result acquired using the above procedure is exactly the same with Equation (9).3D magnetic switchbacks 11\nFigure 5. Illustration of two interacting switchbacks. (a) Reconnection happens at two different locations marked by the red\ndots. (b) Two new switchbacks and a flux rope are generated.\nC.A POSSIBLE MECHANISM THAT GENERATES FLUX ROPES NEAR SWITCHBACKS\nIn all the analytic models of switchbacks (current study and previous study by Tenerani et al. (2020)), closed field\nlines, i.e. flux ropes, are inevitably included as there is no straightforward way to construct a model with open\nmagnetic field lines only. The existence of flux ropes around switchbacks indicates prior reconnection events. However,\nwe emphasize that switchbacks accompanied by flux ropes are not necessarily generated by interchange reconnection\n(Drake et al. 2021; He et al. 2021). One possible mechanism is the interaction between two switchbacks during the\npropagation of the solar wind, as illustrated by Figure 5. When the two switchbacks collide, reconnection happens\nat two different locations (a). At each reconnection site, the inverse part of one field line reconnects with the normal\npart of the other filed line. After the reconnection (b), two new switchbacks and a flux rope are generated. We note\nthat, in practice, it may not be easy to trigger reconnection in switchbacks as they are highly Alfv´ enic. Nonetheless,\nreconnection is occasionally observed near the switchback boundaries (Froment et al. 2021). Thus, the mechanism\nshown by Figure 5 should not be excluded.\nREFERENCES\nBale, S., Badman, S., Bonnell, J., et al. 2019, Nature, 576,\n237\nBalogh, A., Forsyth, R., Lucek, E., Horbury, T., & Smith,\nE. 1999, Geophysical research letters, 26, 631\nBandyopadhyay, R., Matthaeus, W., McComas, D., et al.\n2022, The Astrophysical journal letters, 926, L1\nBelcher, J. 1971, Astrophysical Journal, vol. 168, p. 509,\n168, 509\nBourouaine, S., Perez, J. C., Klein, K. 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D., et al. 2021,\nMonthly Notices of the Royal Astronomical Society, 508,\n236\nYamauchi, Y., Suess, S. T., & Sakurai, T. 2002,\nGeophysical research letters, 29, 21\nZank, G., Nakanotani, M., Zhao, L.-L., Adhikari, L., &\nKasper, J. 2020, The Astrophysical Journal, 903, 1"    },    {        "title": "2401.11341v1.Sensory_adaptation_in_a_continuum_model_of_bacterial_chemotaxis____working_range__cost_accuracy_relation__and_coupled_systems.pdf",        "content": "Sensory adaptation in a continuum model of bacterial chemotaxis - working range,\ncost-accuracy relation, and coupled systems\nVansh Kharbanda1, 2and Benedikt Sabass1, 2\n1Faculty of Physics and Center for NanoScience, Ludwig-Maximilians-Universit¨ at M¨ unchen\n2Institute for Infectious Diseases and Zoonoses, Department of Veterinary Sciences,\nLudwig-Maximilians-Universit¨ at M¨ unchen\n(Dated: January 23, 2024)\nSensory adaptation enables organisms to adjust their perception in a changing environment. A\nparadigm is bacterial chemotaxis, where the output activity of chemoreceptors is adapted to differ-\nent baseline concentrations via receptor methylation. The range of internal receptor states limits\nthe stimulus magnitude to which these systems can adapt. Here, we employ a highly idealized,\nLangevin-equation based model to study how the finite range of state variables affects the adap-\ntation accuracy and the energy dissipation in individual and coupled systems. Maintaining an\nadaptive state requires constant energy dissipation. We show that the steady-state dissipation rate\nincreases approximately linearly with the adaptation accuracy for varying stimulus magnitudes in\nthe so-called perfect adaptation limit. This result complements the well-known logarithmic cost-\naccuracy relationship for varying chemical driving. Next, we study linearly coupled pairs of sensory\nunits. We find that the interaction reduces the dissipation rate per unit and affects the overall\ncost-accuracy relationship. A coupling of the slow methylation variables results in a better accuracy\nthan a coupling of activities. Overall, the findings highlight the significance of both the working\nrange and collective operation mode as crucial design factors that impact the accuracy and energy\nexpenditure of molecular adaptation networks.\nI. INTRODUCTION\nThe surroundings of all living organisms constantly\nchange and diverse mechanical, electrical, and chemical\nsignaling pathways enable organisms to continuously de-\ntect these changes [1–4]. Reliable detection of changes in\ndifferent signal backgrounds requires the adjustment of\nthe sensitivity, which is called sensory adaptation [5]. A\nparadigm of sensory adaptation is chemotaxis of the bac-\nterium Escherichia coli (E. coli ) [6]. Chemotaxis allows\nE. coli to navigate up or down chemical concentration\ngradients, e.g., to find an environment with higher sugar\nconcentrations. Chemotactic movement is a result of a\nstochastic control of the rotation direction of the flagella\nthat drive bacterial swimming [3]. E. coli swims in a pat-\ntern that comprises of successive straight runs and tum-\nble phases, during which it reorients. As the bacterium\nmoves towards higher concentrations of a chemoattrac-\ntant, the tumbling frequency is reduced, which leads to\nincreasingly long runs that take the cell in the direction\nof higher concentration. Conversely, the tumbling fre-\nquency is increased as the bacterium swims away from a\nrepellent.\nThe presence of ligands is sensed by transmembrane re-\nceptors that are called methyl-accepting chemotaxis pro-\nteins. Depending on ligand binding, receptors can either\nbe in an active or an inactive state. Chemoattractant\nbinding inhibits receptor activity, which suppresses tum-\nbling by lowering the phosphorylation state of a response\nregulator CheY [7]. Simultaneously, an adaptation of the\nreceptors to the ligand concentration occurs via methy-\nlation or demethylation of their four to five glutamate\nresidues with the help of the methyltransferase CheR and\nthe methylesterase CheB, respectively [8, 9]. CheR pref-erentially methylates inactive receptors and increases re-\nceptor activity through methylation if receptor activity is\nreduced by an attractant. CheB preferentially demethy-\nlates active receptors and thereby lowers activity upon re-\nmoval of attractants. Consequently, adaptation restores\nthe receptor activities to their original value after an ini-\ntial response to a ligand concentration change [5]. This\nfeedback mechanism can in principle ensure a precise and\nrobust detection of concentration gradients, regardless of\nthe ambient background concentration of ligands. How-\never, experiments also revealed a limited working range\nof adaptation with imprecise adaptation to very high con-\ncentrations of attractants [10–12]. This limitation most\nlikely results from a saturation of available methylation\nsites [13]. While the impact of a limited working range\non the sensory performance have been studied theoreti-\ncally [11, 13], it is less well-understood how the energetic\ncost of adaptation depends on the working range of the\nmechanisms. Chemosensory adaptation that is realized\nby a finite number of receptor methylation sites can be\nidealized as a dynamical process with a bounded range\nof internal variables. How such a bounded internal phase\nspace affects the energetics of nonlinear biological pro-\ncesses is generally an open question. This issue motivates\nour study of sensory adaptation with a highly idealized\nmodel where the interplay of phase-space boundaries of\ninternal variables, energetics, and adaptive performance\ncan be studied.\nChemotaxis receptor molecules form clusters at the cell\nmembrane [14, 15], where individual molecules form ho-\nmodimers and three such homodimers form a trimer [16–\n18]. A receptor core complex, the minimal unit of signal-\ning, contains two trimers and shows a non-linear dose-\nresponse [19]. Assemblies of multiple chemoreceptorsarXiv:2401.11341v1  [q-bio.CB]  20 Jan 20242\nwork in a cooperative manner to enhance the sensitivity\nof the response to subtle concentration gradients [20–22].\nOn a molecular level, cooperativity can result from en-\nzymes that change the methylation state of the dimer\nthey are bound to as well as the methylation state\nof neighboring dimers, e.g., CheR and CheB [23, 24].\nThe molecular mechanisms of receptor cooperativity, as\nwell as its functional advantages, e.g., to achieve opti-\nmal signal-to-noise ratios, have been studied extensively.\nHowever, the impact of receptor cooperativity on the en-\nergy consumption required for adaptation is hardly ex-\nplored and warrants further theoretical research.\nA variety of models for sensory adaptation in chemo-\ntaxis have been proposed over the past decades [25–31].\nIdealized feedback systems can be represented by only\na few coupled differential equations [32, 33]. However,\nchemosensory adaptation is usually modeled as a stochas-\ntic process since the number of molecules constituting a\nsensory apparatus in individual cells is typically quite\nsmall and thermal noise and active processes result in\nnoisy signals. This noise affects the reliability of infor-\nmation transmission between an input signal and the\noutput [34–36] and thereby determines the precision of\nthe cellular response. However, fluctuations are also inti-\nmately connected with the energetic cost of adaptation.\nFrom an information-theoretic viewpoint, sensory adap-\ntation always entails a thermodynamic cost since it im-\nplies an irreversible erasure of stored information along\nwith a measurement [37]. Moreover, the two fundamental\nmodels of noisy cellular adaptation, namely the negative\nfeedback mechanism in chemotaxis and the incoherent\ntype-1 feed-forward mechanism, only show adaptation if\nthe systems’ states are held in non-equilibrium [38, 39].\nThe energetic cost of maintaining a non-equilibrium state\nrequired for adaptation can be related to sensory perfor-\nmance by an energy-speed-accuracy relation [38], which\nstates that the dissipation rate is proportional to the neg-\native logarithm of the adaptation error, both measured\nin steady state after application of a constant stimulus.\nMost work on the energy-speed-accuracy relationship\nis based on a discrete description of receptor states using\nmaster equations [38], for which the original relationship\ncan be generalized [40]. Here, one assumes two discrete\nstates for the fast receptor activity and four to five dis-\ncrete states that represent the slowly changing methyla-\ntion level that acts as a a negative control element in E.\ncolichemotaxis. Alternatively, the receptor states can\nbe represented in a continuum picture using Langevin\nequations for both the response and the control variables.\nBased on simulation results for such a model, the general-\nity of the energy-speed-accuracy relation has been ques-\ntioned [41]. The Langevin-equation based model is highly\nidealized and not as directly applicable to chemotaxis as\nthe discrete model. However, it has the advantage that\nthe adaptation process and the associated fluxes of proba-\nbility can be clearly visualized in a two-dimensional phase\nspace. This facilitates an intuitive understanding of the\nadaptation process. In principle, one may even studya hybrid model where the states of the control variable\nof the receptor are discrete and the response variable is\ncontinuous, although the technical advantage of such an\napproach is not obvious.\nIn this article, we take advantage of the clear rep-\nresentation offered by the Langevin-equation approach\nto study the relationships between adaptation, working\nrange, and energetic cost for individual and coupled sys-\ntems. Unlike earlier studies [38, 40], we directly simulate\nthe Langevin dynamics, taking into account the phase-\nspace boundaries. Plotting the adaptation error against\nthe dissipation rate for varying input signals, we find that\nthe finite working range results in a linear cost-accuracy\nrelation, which highlights the importance of the finite\nworking range of the internal variables on the perfor-\nmance and the cost of adaptation. This result comple-\nments the well-known logarithmic energy-speed-accuracy\nrelation [38]. Also, to underpin our numerical findings,\nwe derive an analytical expression for the energetic cost of\nadaptation as a function of input stimulus strength. Fur-\nthermore, motivated by the discovery that receptor clus-\nters consist of minimal sensory units with a non-linear\nresponse [19], we consider pairs of receptors in the spirit\nof Ref. [42] and ask how a linear coupling of the sensor\nstates affects the overall cost-accuracy relationship. To\nthis end, we study two examples of coupled systems and\nfind that, in general, a linear coupling reduces the cost\nof adaptation without affecting its accuracy.\nII. A SINGLE SENSORY SYSTEM\nA. Model Introduction\nAn idealized model of bacterial adaptation can be built\nfrom a three-node network with a negative feedback. We\nfocus on a Langevin-type model suggested in Ref. [38],\nwhere all variables are assumed to have non-negative,\nreal values. The model comprises of a variable afor the\nrapidly changing receptor activity and a slowly changing\ncontrol variable m, representing the receptor methylation\nstate. The activity aresponds to changes in an input\nstimulus sand also represents the output of the system.\nFollowing a stimulus, a change in the activity adrives\na slow compensatory change in the control variable\nm, which ultimately brings aback to the stimulus-\nindependent setpoint a0, see FIGs. 1a and 1b. The dy-\nnamics of the activity and the control (methylation) vari-\nables is described by a pair of coupled Langevin equations\nas\n˙a=Fa(a, m, s ) +ηa(t); ˙m=Fm(a, m, s ) +ηm(t).(1)\nHere, ηa(t) and ηm(t) both represent white noise. De-\nnoting expectation values as ⟨. . .⟩, the noise satisfies\n⟨ηi(t)ηj(t′)⟩= 2∆ iδijδ(t′−t) where the indices iand\njrepresent aorm. The functions FaandFmprovide\na lumped description of the nonlinear biochemical reac-3\ntions that govern the dynamics of fast response and slow\nadaptation, respectively [38].\nThe Fokker-Planck equation (FPE) that corresponds\nto the Langevin equations (1) describes the evolution of\nthe phase-space probability distribution function (PDF)\nP≡P(a, m, t ) as\n∂tP=−∂aJa−∂mJm, (2)\nwhere Ja≡FaP−∆a∂aPandJm≡FmP−∆m∂mPare\nthe components of the probability current J.\nIt can be shown that the system must obey the follow-\ning condition to satisfy detailed-balance\n∆m∂mFa= ∆ a∂aFm, (3)\nwhich is also derived in the Supplementary Material.\nThis detailed-balance condition for the negative feedback\ncontrol mechanism implies that adaptation cannot be re-\nalized in equilibrium and is inherently dissipative [43].\nWe specify the functional form of the forces driving the\ndynamics of aandmas\nFa(a, m, s ) =−ωa[a−G(s, m)] (4)\nand\nFm(a, m, s ) =−ωm(a−a0)[β−(1−β)C∂mG(s, m)],(5)\nwhere\nC≡∆m/ωm\n∆a/ωa(6)\nis assumed to be a constant and β∈[0,1] is a param-\neter that determines whether the system is in equilib-\nrium, β= 0, or out-of-equilibrium, β > 0. The rate of\nthe response and adaptation processes are determined in\nEqs. (4, 5) by the constants ωaandωm, respectively. We\nassume that\nωa≫ωm, (7)\nsuch that the time scales of response and adaptation are\nwell-separated.\nThe function G(s, m) is chosen to be a Hill equation\nin the Michaelis-Menten limit, i.e., with Hill coefficient\nH= 1 as\nG(s, m) = (1 + s/(K0e2m)H)−1, (8)\nwhere we set K0= 1. The functional form chosen in\nEq. (8) ensures that ∂mG > 0 and ∂sG < 0, which is\nrequired for the negative feedback mechanism [38].\nIn a deterministic setting, for any β, there exists a\nfixed point where the two nullclines of the deterministic\nsystem intersect. Consequently, there is a value of the\ncontrol variable m=m∗that satisfies\nG(s, m∗) =a0. (9)Moreover, there exists a βcsuch that the fixed point is\nstable only if β > β c, where 0 < βc<1 [38]. Thus, stable\nadaptation is only achieved if β > β c. Phase portraits of\nthe system with stable fixed points for β= 1 and unsta-\nble fixed points for β= 0 are shown in FIGs. 1c and 1e,\nrespectively. The corresponding probability distributions\nare presented in FIGs. 1d and 1f. Throughout this arti-\ncle, we focus for simplicity on the case of β= 1, which\nis called the fully adaptative state which represents the\nperfect adaptation limit.\nB. Assumptions & Definitions\n1. Phase space & Adaptation Error\nFor the model described in Eqs. (1), the activity aand\ncontrol state mare stochastic variables that lie in a phase\nspace denoted as ¯Ω≡[0,1]×[0, m0], with m0correspond-\ning, e.g., to the maximum number of methylation sites\nin a receptor. To solve the FPE, Eq. (2), on the interior\nof the phase space, Ω ≡¯Ω\\∂Ω, a set of well-defined\nboundary conditions is required. For this model, we im-\npose reflective boundary conditions, which state that the\ncurrent vector obeys\nn·J= 0 for ( a, m)T∈∂Ω. (10)\nwhere nis the outward-pointing vector normal to the\nboundary ∂Ω.\nThe setpoint for the activity a0is assumed to be a con-\nstant. Since the function Fmis designed to be propor-\ntional to ( a−a0), the generalized force driving methyla-\ntion changes vanishes at the setpoint. The integral feed-\nback mechanism ensures that the statistical average of\nthe activity\n⟨a(t)⟩=ZZ\nΩa P(a, m, t ) dadm (11)\napproaches a0in the steady state [38]. Denoting the\nsteady-state average as ⟨. . .⟩ss, the adaptation error\nϵ≡ |1− ⟨a⟩ss/a0| (12)\nquantifies the accuracy of adaptation in the stationary\nlimit after the decay of transient responses to a signal.\n2. Effective Temperature & Energy Dissipation Rate\nIn equilibrium, the fluctuation-dissipation theorem\n(FDT) can be used to relate the temperature of the sys-\ntem to the noise amplitude and the damping in the sys-\ntem [44]. In the following, we set the Boltzmann constant\nto unity, kB≡1, and define the effective temperatures\npertaining to the stochastic variables of the system in\nEq. (1) as follows\nTeff(a)≡∆a\nωa, T eff(m)≡∆m\nωm. (13)4\nFIG. 1. (a)A minimal three-node network with a negative feedback loop. (b)Following a change of the stimulus s, the\ninstantaneous response of the activity ais cancelled by a gradual change in the control variable m. The adaptation error is\ndefined as the deviation of the expectation value of afrom its setpoint a0in the stationary limit. (c)Phase portrait of the\ndynamics in the absence of noise. For β= 1, the fixed point of the deterministic dynamics ( a0, m∗) is stable and represents the\nadaptive state of the system. (d)Steady-state probability distribution of the system variables in the presence of noise. The\ndeterministic fixed point lies at the center of the probability distribution, that is, ( ⟨a⟩ss,⟨m⟩ss) = (a0, m∗).(e)Forβ= 0, the\nchemical driving vanishes and the fixed point ( a0, m∗) is an unstable saddle node. (f)In the presence of noise, the state of the\nnon-adaptive system is driven to the corners of the phase space. Simulation parameters: s= 20 ,ωa= 50, ωm= 5, T= 0.01.\nFor (f), we chose s= 3to illustrate that the state localizes in both corners of the phase space.\nWe assume that the response and adaptation processes\ndescribed in Eq. (1) are coupled to the same heat bath.\nTherefore, the effective temperatures are equal Teff(m) =\nTeff(a) =T, which entails that C= 1.\nSince the product of temperature and the entropy pro-\nduction rate equals the dissipation rate in a stationary,\nisothermal setting, see Ref. [45], the stationary dissipa-\ntion rate can be expressed as\n˙Wdiss=TZZ\nΩ\u0014J2\na\n∆aP+J2\nm\n∆mP\u0015\ndadm. (14)\nThis equation for the dissipation rate is only valid for a\nsystem with single heat bath and temperature T. For a\nsystem with multiple heat baths, additional terms must\nbe included in Eq. (14) to account for the heat flux be-\ntween reservoirs [46, 47].\nAn analytical approximation for the dissipation rate\nwas given in Ref. [38], where Eq. (7) and Laplace’s in-\ntegral method were used to estimate the integrals inEq. (14), leading to the following expression\n˙Wdiss≈˙Wa\ndiss=ωmT\nC\u0000\n1 +C ∂mG(s, m)\u00012\f\f\nm=m∗\n=ωmT(1 +∂mG(s, m))2\f\f\nm=m∗,(15)\nwhere C= 1, the superscript “a” stands for the analytical\napproximation proposed in Ref. [38], and m∗is the m-\ncoordinate of the fixed point of the deterministic system\ni.e., it satisfies Eq. (9).\nAlso, employing the framework of stochastic thermo-\ndynamics [45, 46], the dissipation rate for an individual\ntrajectory “t” of the system can be expressed as\n˙Wt=T\u0000\n∆−1\naFa◦˙a+ ∆−1\nmFm◦˙m\u0001\n, (16)\nwhere ◦indicates a product in the Stratonovich\nsense [46]. For an ensemble of trajectories, we take the\nstatistical average of Eq. (16) to obtain an expression for\nthe dissipation rate in stationary state as\n˙Wdiss=\nT\u0000\n∆−1\naFa◦˙a+ ∆−1\nmFm◦˙m\u0001\u000b\nss. (17)5\nIn contrast to Eq. (14), Eq. (17) can be straight-forwardly\nevaluated using the trajectories generated by simulations\nof the stochastic dynamics.\nC. Simulation Methods\nTo simulate the system of Langevin equations (1),\nwe employ an explicit first-order method as described\nin Ref. [48], where the boundary conditions can be ac-\ncounted for by adding a term that ensures normal re-\nflections at the boundary [49]. However, to make the\neffect of the boundaries explicit in analytical calculations\nof the dissipation rate, we mostly employ smooth poten-\ntials that approximate the no-flux boundary conditions\nof the system. The potentials are\nV(a, m)≡1\nωaVb(a, amax) +1\nωmVb(m, m max),(18)\nwhere ( a, m)∈Ω and the function Vbwith arguments a\normis given by\nVb(r, rmax) =V0\u0002\nexp(−r\nσ) + exp(r−rmax\nσ)\u0003\n.(19)\nThe potential Vbis a superposition of exponential func-\ntions which approximate hard, reflecting walls in the limit\nof small σ >0. Non-zero values of σensure that the po-\ntential is smooth and bounded for r∈(0, rmax).To test if\nthe finite range of the wall potentials affects the simula-\ntion results, we also perform simulations using “reflection\noperators”, the details of which can be found in the Sup-\nplementary Material. The results of computations using\nthe reflection operators are also presented in the Supple-\nmentary Material (FIG. S2) and agree with results that\nwere obtained from computations that make use of the\nboundary potentials.\nD. Analytical and Numerical Results\nIn this section, we study how the finite variable-range\nin a fully adaptive system leads to a non-zero adaptation\nerror that increases with the magnitude of the stimulus.\nConcomitantly, the boundaries reduce dissipation, which\nresults in a novel functional form of the cost-accuracy\ntrade-off.\n1. Simulation Results: Dependence of Adaptation Error on\nSystem Variables\nSince the system consists of fluctuating state variables,\nunderstanding its adaptive performance requires a con-\nsideration of the probability distribution of the variables.\nThe probability distribution is affected by the presence of\nphase-space boundaries that determine the range of thesystem variables. We define the bulk of the phase space,\ndenoted by B ⊆ Ω⊂¯Ω, as the largest subset of the\nphase space in which the boundary-induced changes of\nthe probability distribution do not result in measurable\nchanges in adaptation error.\nThe capacity of a system to adapt after a change in\nstimulus sis limited by the phase-space boundaries that\ndetermine the maximum and minimum values of system\nvariables [50, 51]. In the present model, this claim can\nbe proven by separating the adaptation error, Eq. (12),\ninto three contributions,\nϵ=|ϵ1−ϵ2+ϵ3|, (20)\nwith\nϵ1=1\nβωma0Z1\n0da∆mP(a,0)da, (21a)\nϵ2=1\nβωma0Z1\n0da∆mP(a, m 0)da, (21b)\nϵ3=1−β\nβa0ZZ\nΩdadm P(a, m)∂mG(s, m)(a−a0),(21c)\nwhich are obtained using the time-scale separation of\nadaptation and response. In Eq. (20), ϵ1andϵ2are con-\ntributions due to the phase-space boundaries, while ϵ3\naccounts for the error in the bulk. For perfect adapta-\ntion, β= 1, the bulk contribution vanishes [38]. There-\nfore, in a fully adaptive state, any error in the stationary\nvalue of the adapted state results from the limited range\nof the internal variables.\nSince in a perfectly adaptive state the adaptation error\nis only non-zero near the boundaries, the adaptation er-\nror for a large stimulus sdepends on the maximum value\nthat the methylation variable can assume. This maxi-\nmum value is determined by m0. Results from numerical\ncomputations displayed in FIG. 2a show that the depen-\ndence of the adaptation error on m0can be fitted by a\nFermi-Dirac-like function as\n˜ϵ(s, m 0) :=1\nc1+ec2(m0−c3)=1\nc1+e˜c2ˆk(m0−c3)/s,(22)\nwhere c1, c2= ˜c2ˆk\ns,and c3are fit parameters and ˆk= 104\nis a constant scale factor. In FIG. 2a, the values of m0\nare shown at which the analytical solution of the de-\nterministic system predicts an error of 5% for different\ns. These values indicate the points where the boundary\neffects start to become significant for a given stimulus.\nFigure 2b illustrates that the adaptation error also de-\npends on the setpoint a0. A larger setpoint results in an\nincreased adaptation error.\n2. Simulation Results: Dissipation Rate\nIf the stimulus magnitude sexceeds a threshold s0,\nthe solution of Eq. (9) yields a fixed point m∗> m 0that\nlies outside the variable domain, m∗̸∈¯Ω. Then, the6\nFIG. 2. (a)Dependence of the adaptation error on the\nrange of the control (methylation) variable m∈[0, m0], with\nˆk= 104. For large m0, the error vanishes for a fully adap-\ntive system since the state rarely encounters the boundary\natm0. Simulation results can be fitted with an exponential\nform, see Eq. (22). Vertical lines show values of m0at which\nthe corresponding deterministic systems produce an error of\n5%.(b)Adaptation error as a function of stimulus strength\nat fixed system size with m0= 4. (c)The dissipation rate\n˙Wdissof the adaptive system decreases with reduced system\nsize if the state encounters the phase-space boundary. For\nsmall stimuli, s= 102, the state remains in the center of the\nspace and the dissipation is is independent of system size. (d,\ne)The function ∂mG(s, m)|m=⟨m⟩ssand the dissipation rate\nare roughly quadratic functions of the setpoint distance from\nthe neutral state a0= 0.5, which lies in between the active\nstate ( a= 1) and the inactive state ( a= 0) of the system.\n(f)Dissipation rate as a function of the stimulus strength\ns. Simulation results ˙Wdissare compared with an analytical\napproximation ˙Wa\ndissand an empirical formula W(s, λ). The\nthree quantities agree for small stimuli s < s 0. Otherwise,\n˙Wa\ndissdiffers from the simulation results due to the break-\ndown of the analytical approximation at the boundaries.steady-state expectation value ⟨m⟩ssdeviates from m∗.\nTo compute the dissipation rate ˙Wa\ndiss, see Eq. (15), out-\nside the bulk B, we replace m∗with⟨m⟩ss. A comparison\nofm∗and⟨m⟩ssas a function of the stimulus scan be\nfound in the Supplementary Material, FIG. S3.\nThe interaction of state trajectories with phase-space\nboundaries reduces the system’s energy dissipation rate\nin the stationary state. As shown in FIG. 2c, for a small\nvalue of the input stimulus, s= 102, the dissipation rate\nof the system remains constant for varying m0. How-\never, for larger sthe dissipation of the system starts to\ndecrease when the probability distribution touches the\nboundary at m0. Furthermore, when sis fixed, changing\nthe setpoint a0of the activity variable also affects the en-\nergy cost, as moving the distribution along the a−axis in-\ntroduces effects of the boundaries at a= 0 or a= 1. The\nplot in FIG. 2e shows that the dissipation rate decreases\nsymmetrically as a0is moved away from the center at\na0= 0.5.\n3. Breakdown of Approximation for the Dissipation Rate\nand Empirical Formula\nComparison of the dissipation rate in simulations,\n˙Wdiss, with the approximate expression ˙Wa\ndiss, Eq. (15),\nshows that the latter only holds if the setpoint is close\nto the center of the phase space, a0= 0.5, see FIG. 2e.\nThus, the analytical approximation holds only if the ef-\nfect of the boundaries is very small whereas ˙Wdiss≤˙Wa\ndiss\notherwise.\nThe behavior of the dissipation rate ˙Wdisscan be fur-\nther analyzed by defining a threshold value s0for the\ninput stimulus. Above this threshold, the probability\ndistribution is sufficiently altered by the boundary at\nm=m0to change the dissipation rate by more than\nsome fixed percentage. For a0∈(0.3,0.7), see FIG. 2e,\nand a large enough value of ssuch that m∗̸≈0, we con-\njecture the following expression for the dissipation rate\nWdiss:=(˙Wa\ndiss s < s 0\nK0e−λ(s−s0)+ϵ0(1−e−λ(s−s0))s≥s0\n(23)\nwhere K0≡˙Wa\ndiss\f\f\ns=s0is the dissipation coefficient and\nϵ0≡min\ns∈(0,∞)˙Wdiss. Results from the empirical expression\nWdissand ˙Wdissare shown in FIG. 2f. The expression\nin Eq. (23) describes the numerical data well, while the\napproximation ˙Wa\ndissfails.\n4. Cost-Accuracy Relation\nWhat is the energetic cost of accurate adaptation?\nA well-known energy-speed-accuracy relation, proven in\nRef. [38], asserts a monotonous increase of the dissipa-\ntion rate with adaptation accuracy for varying chemical7\ndriving force. More precisely, the energy-speed-accuracy\nrelationship states that the dissipation rate is propor-\ntional to the negative logarithm of the error. See the\nSupplementary Material (FIG. S4) for a recapitulation\nof this result in our simulations where the chemical driv-\ning is controlled by the value of β. Optimal accuracy\nis reached in the limit of perfect adaptation where the\ndissipation is maximal.\nTo understand how the finite range of internal variables\naffects the relationship between dissipation and accuracy,\nwe vary here the magnitude of the external stimulus s.\nWe focus on the limit of a fully adaptive model, i.e.,\nβ= 1. As shown in FIG. 3a, the adaptation error ob-\ntained in simulations decreases linearly with a growing\ndissipation rate. In other words, the accuracy increases\nwith the energetic cost also if the error solely results from\na finite variable range. This relationship is thus qualita-\ntively similar to the energy-speed-accuracy relation, but\nthe functional forms and implications are different.\nTo buttress our numerical result for the cost-accuracy\nrelation, we next perform analytical calculations of the\ndissipation rate. As illustrated in FIGs. 2e and 2f, the\nvalidity of the approximate expression ˙Wa\ndissbreaks down\nwhen significant adaptation errors occur close to phase-\nspace boundaries. Thus, we derive a new equation for\nthe dissipation rate by explicitly taking into account the\nboundaries of the variable domain. We consider the sys-\ntem of Langevin equations with any smooth respulsive\nboundary potential Vas\n˙a=Fa(a, m, s )−ωa∂aV+ηa, (24a)\n˙m=Fm(a, m, s )−ωm∂mV+ηm. (24b)\nThese equations are linearized around any point near\nthe phase-space boundary ( ac, mc) of their determinis-\ntic counterparts. Using Eqs. (4, 5) and β= 1, we obtain\n˙a=−ωa(a−ac)\u0002\n1 +∂2\naV\f\f\na=ac\u0003\n+ωa∂mG(s, m)\f\f\nm=mc(m−mc) +ηa,(25a)\n˙m=−ωm(a−ac)−ωm∂2\nmV\f\f\nm=mc(m−mc)+ηm,(25b)\nwhere the higher order terms of the expansion are ignored\nand the leading terms in the Langevin equations drop\nout, as we define the fixed point ( ac, mc) to satisfy,\nG(s, mc)−∂aV\f\f\na=ac−ac= 0, (26a)\n(ac−a0) +∂mV\f\f\nm=mc= 0. (26b)\nThe above two equations represent a modified version\nof Eq. (9). The boundary potentials ensure that the\nsolution mcexists for all values of the input and that\nmc=m∗fors < s 0.\nFIG. 3. (a)Cost-accuracy relation for varying stimulus\nstrength. The adaptation error ϵis linearly related to the\nthe dissipation rate ˙Wdissat different values of s. Ana-\nlytical results ˙We\ndissestimating the dissipation rate at weak\nand strong stimuli, s < s 0ands≫s0, agree with the\nsimulation data. The steepest decent approximation used\nfor˙Wa\ndissbreaks down for strong stimuli. (b)Steady-state\nprobability distribution after application of a weak stimu-\nlus,s= 103< s 0. The distribution resembles a Gaussian\nand the averages co-localize with the solution of Eq. (26),\n(ac, mc).(c)State probability distribution after application\nof a strong stimulus, s= 5·104> s 0. The distribution is\nskewed and the averages do not co-localize with the max-\nimum of the distribution ( ac, mc).Simulation parameters:\nV0= 1/2, ωm= 5, ωa= 50, σ= 0.01, T= 0.01.\nThe fixed point ( ac, mc) corresponds to the point of\nmaximum likelihood - the point in the phase space where\nthe probability density function reaches its maximum\nvalue, see FIGs. 3b and 3c. The FPE corresponding\nto the linearized equations (25) is solved by a Gaussian\nansatz. Substitution of the solution into the expression\nfor the dissipation rate, Eq. (14), yields\n˙We\ndiss=Teff(1 +∂mG(s, m))2ωaωm\nωa+ωm∂2mV+ωa∂2aV\f\f\f\fa=ac\nm=mc. (27)\nHere, ∂2\naVand∂2\nmVare both non-negative due to the\nchoice of boundary potentials. The superscript “e” in\n˙We\ndissindicates that this expression arises from a linear\nexpansion of the governing equations of the system.\nThe approximation in Eq. (27) is only valid when ei-\nther s < s 0ors→ ∞ because the state probability\ndistribution can be approximated by a Gaussian distri-\nbution only for such values of s, see FIG. 3b. For other\nvalues of s, the distribution is skewed and an expansion8\naround the point of maximum likelihood results in larger\nerrors.\nAs illustrated in the Supplementary Material, simula-\ntions with the chosen exponential boundary potentials\nyield very similar results to simulations employing a re-\nflection operator with very small timesteps. Thus, the\nresults are expected to also hold for reflective bound-\naries. Also, a master-equation based chemotaxis recep-\ntor model with discrete molecule states yields qualita-\ntively the same cost-accuracy relation for varying stim-\nulus strength as shown for the Langevin-equation based\nmodel in this section, see Supplementary Material.\nE. Discussion of Results for a Single Sensory\nSystem\nUsing a minimal Langevin-equation-based model for\nadaptation [38], we analyze how a limited range of inter-\nnal variables affects adaptation accuracy and energetic\ncost. The model consists of a stochastic variable acalled\nactivity, a slow, stochastic control variable m, and an\nexternally controlled stimulus s. In this work, the adap-\ntation error ϵis defined as the systematic, relative devi-\nation of the activity afrom its setpoint a0, see Eq. (12).\nThe adaptation process can be visualized by following\nthe motion of the system state ( a, m) in the rectangular\nvariable domain ¯Ω with a∈[0,1] and m∈[0, m0]. For\nthe fully adaptive model considered here, the adaptation\nerror solely results from the limited range of the inter-\nnal variables aandm. To explain the onset of the error\nϵ, consider the state to be initially at its setpoint with\n⟨a⟩=a0. After a sudden increase of the stimulus s,⟨m⟩\nslowly shifts to higher values to bring ⟨a⟩back to its set-\npoint. However, if sbecomes larger than a certain input-\nthreshold s0, such that the control variable approaches\nits upper limit m0, the reflective boundaries alter the\nprobability distribution, see FIGs. 3c and 1d. Then the\ncontrol variable cannot increase sufficiently to counteract\nthe effect of the external stimulus on the activity aand\nthe perfect adaptability of the system is compromised.\nUnlike the error due to the limited range of the control\nvariable, which results from the deterministic dynamics\nof the system, the error resulting from finite domain of\nais a consequence of the finite, non-zero variance of the\nnoise in the system. Also, as shown in FIG. 2a, the adap-\ntation error saturates for very large stimulus sto a max-\nimum value lim s→∞ϵ≡ϵmax<1.ϵmaxis independent\nof both the setpoint and the system size as long as m0is\nfinite. The limit on the maximum error results from the\nskewness of the probability distribution.\nThe considered energetic cost of adaptation is the\nsteady-state dissipation rate after a change of the input\nsignal. Perpetual dissipation in the adaptive system re-\nsults from the non-equilibrium forces that maintain the\nstate ( a, m) close to a fixed point in the presence of fluc-\ntuations. The fluctuations drive the state away from its\nfixed point and work is required to bring it back. Uponstrong stimulation, the probability distribution for ( a, m)\nmoves to the vicinity of the boundaries. Intuitively, the\neffect of boundaries can be interpreted here as passive\nforces that also maintain the state at a fixed point. More\nformally, Eq. (10) ensures that the normal component\nof the fluxes at the boundary vanishes, thereby reduc-\ning the contribution of these fluxes to the integrand in\nEq. (14). Consequently, the dissipation rate decreases at\nthe boundary, see FIG. 2f.\nA plot of adaptation error against dissipation rate as a\nfunction of stimulus strength yields a linear cost-accuracy\nrelationship as a result of the limited range of internal\nvariables. Thus, our result pertain to the behavior of a\nsensory system in a regime where its internal variables\nreach their physical limits. In that case, we see that the\nsystem is unable to adapt, but it also reduces the energy\nit uses for adaptation in a linear fashion. By contrast,\nvariation of the chemical driving force yields a logarith-\nmic cost-accuracy relationship, called the energy-speed-\naccuracy relation [38]. Thus, the functional form of cost-\naccuracy relations depends on the nature of the inter-\nnal constraints imposed on the adaptation process. We\nemphasize, however, that the energy-speed-accuracy re-\nlation can directly imply a design trade-off for cells since\nthe adaptive performance can be improved by investing\nmore energy. This is not the case for the cost-accuracy\nrelationship studied here, which depends on the range of\ninternal variables and the external stimulus. In order to\nturn our results into a proper trade-off relationship, one\nwould need to determine how the cost of enlarging the\nstate space, i.e., to generate molecules with more methy-\nlation sites in the case of chemotaxis, improves chemo-\ntaxis.\nIII. INTERACTING SYSTEMS\nWe now shift the focus from single sensory units\nto pairs of interacting sensory units as illustrated in\nFIGs. 4a.(i) and 4b.(i). The state of a sensory unit with\nindex i∈ {1,2}is described by the variables ( ai, mi) that\nrepresent the activity and the control (methylation), re-\nspectively. The four-dimensional phase space is denoted\nas¯Ω = [0 ,1]2×[0, m0]2.\nA. Overall Adaptation Error\nBefore specifying how the units interact, we introduce\nthe adaptation accuracy of a combined system generi-\ncally. For the coupled pair of systems, the overall activity\naand overall error ϵare defined as\na≡(a1+a2)/2, (28)\nϵ≡ |1− ⟨a⟩ss/a0|, (29)\nwhere ⟨a⟩ssrepresents the stationary-state average of the\noverall activity and a0is the setpoint value, which is as-9\nFIG. 4. (a.(i), b.(i)) Composite systems consisting of two sensory units, where each realizes an adaptive feedback loop. The\nunits are linearly coupled either via their activity or their control (methylation) variables. The overall activity aof the system\nis the average of the individual activities a1anda2.(a.(ii) - a(v)) Steady-state probability distribution Pvisualized on\ntwo-dimensional planes in the phase space for coupled activities. A large coupling constant ω12> ωa{1,2}leads to a strong\ncorrelation of the rapidly changing activities a1anda2. The control variables m1andm2are also strongly correlated. As\na result, the distributions on the ai−miplanes for the individual units are also altered. (b.(ii) - b.(v)) State probability\ndistribution Pvisualized on two-dimensional planes in the phase space for coupled control variables. Due to the slow dynamics\nof the control variables in the individual units determined by ωa{1,2}≫ωm{1,2}, the variances of the m{1,2}are smaller than\nthe variances of the a{1,2}. As a result, even if a large coupling constant enforces a strong correlation of m1andm2, the\nactivities are only weakly correlated as seen by the almost radially symmetric distribution in the a1−a2plane. Consequently,\nthe probability distributions for the individual units shown in (b.(ii) - b.(iii)) have largely the same shape as in absence of\ncoupling, see FIG. 1d. Simulation parameters: k= 2000 ,s= 100 ,ωa{1,2}= 50, ωm{1,2}= 5, T= 0.01, m0= 4.\nsumed be the same for both units. By employing the defi-\nnition of overall activity, Eq. (28), and the sub-additivity\nof the norm, we can establish an upper bound on the\noverall adaptation error for the interacting system\nϵ=\f\f\f\f\f⟨a1⟩ss−a0\n2a0+⟨a2⟩ss−a0\n2a0\f\f\f\f\f\n≤1\n2\f\f\f\f\f⟨a1⟩ss−a0\na0\f\f\f\f\f+1\n2\f\f\f\f\f⟨a2⟩ss−a0\na0\f\f\f\f\f=1\n2(ϵ1+ϵ2).(30)Here, ϵ1andϵ2are the errors of the individual units,\nwhich are calculated with respect to the joint probabil-\nity distribution of the interacting system. The upper\nbound (30) can be generalized to a system comprising an\narbitrary finite number of sensory units.\nB. Coupled Activities\nThe first interacting system we investigate involves\ncoupling of the activities of two units. Coupling is as-10\nsumed to be linear in the variables with symmetric coef-\nficients.\n1. Model Description\nThe system is described by a set of two Langevin equa-\ntions for each unit, denoted with the indices i, j∈ {1,2}.\nFor unit iand with neighboring unit j̸=ithe equations\nare given by\n˙ai=˜Fai(ai, aj, mi, s) +ηai,\n˙mi=Fmi(ai, mi, s) +ηmi,(31)\nwhere η{mi,ai}is Gaussian white noise as in Eq. (1) and\nallη...with subindices {a1, a2, m1, m2}, are statistically\nindependent of each other. The generalized force ˜Faithat\nappears in Eqs. (31) is defined as\n˜Fai(ai, aj, mi, s)≡Fai(ai, mi, s)−ωij(ai−aj).(32)\nAs the interaction between the two units is assumed to\nbe symmetric, we have ω12=ω21. In Eq. (32) we set\nFai=−ωai[ai−Gi(s, mi)], (33)\nwhich represents the driving forces that depend on the\ninput signal sthrough Gi(s, mi) = (1 + s/e2mi)−1.\nThe time evolution of the joint probability distribu-\ntion function P≡P(a1, m1, a2, m2, t) is governed by\na Fokker-Planck equation. Conditions for detailed bal-\nance in absence of driving can be derived analogously\nto the non-interacting case, see Supplementary Material.\nThese conditions are consistent with the assumption of\none effective temperature for the two units with identical\nfriction and noise strength coefficients. To control the\ninteraction strength, we introduce a tuning parameter k\nsuch that\nω12=k(ωa1T) =ω21=k(ωa2T), (34)\nwhere 0 ≤k <∞andωa1=ωa2.\nFurthermore, the detailed-balance conditions require\nthatFmiis independent of ajfori̸=jandi, j∈ {1,2}.\nIntegration of the detailed-balance conditions allows one\nto systematically construct an expression for Fmias\nFmi=ωmi(ai−a0)[β+\n(1−β)∆mi\n∆aiωai\nωmi∂miGi(s, mi)](35)\nwhere 0 ≤β≤1 and∆mi\n∆aiωai\nωmi= 1 since the effective\ntemperatures are assumed to be equal. We take β= 1\nunless specified otherwise.\nThe boundary conditions for the system in Eq. (31) are\nsimilar to Eq. (10), but the probability flux is now a four-\ndimensional vector denoted by J= (Ja1, Ja2, Jm1, Jm2)T.\nThe boundary potential is re-defined accordingly.The dissipation rate for the interacting system can be\nobtained from the dissipation rate integral of the system,\nanalogous to Eq. (14). For system consisting of a pair of\nsensory units we obtain\n˙W=TX\ni∈{1,2}\n\u0000\n∆−1\nai˜Fai◦˙ai+ ∆−1\nmiFmi◦˙mi\u0001\u000b\nss.(36)\n2. Numerical Results for Coupled Activities\nThe set of Langevin equations (31) are solved numeri-\ncally with the Euler-Maruyama method, as for the non-\ninteracting system. Results are shown in FIGs. 4a.(ii)-\n4a.(v) and FIGs. 5a.\nThe probability distribution function Phas four di-\nmensions and plotting the distribution requires a selec-\ntion of two-dimensional planes in the phase space as\nshown in FIGs. 4a.(ii)-4a.(v). Figures 4a.(ii) and 4a.(iii)\ndisplay distributions for two sensory units with strong\ncoupling, i.e., k≫1. Comparison with results for a sin-\ngle unit, see FIG. 1d, shows that the coupling distorts\nthe probability distributions since it introduces a corre-\nlation between the two activity variables. Consequently,\nthe probability distribution on the a1−a2plane aligns\nalong the line a1=a2and the distributions in the two\na−mplanes also exhibit a stretched variance along the\nm−axis.\nFigure 5a.(i) illustrates the relationship between the\noverall adaptation error and the dissipation rate for dif-\nferent stimulus magnitudes s. As for single sensory units,\nsee FIG. 3a, the cost-accuracy relation resulting from the\nnumerical solution of Eq. (31) is linear. For vanishing in-\nteraction strength, k= 0, the pair of units together have\nthe same adaptation accuracy as an individual system,\nalbeit with twice the dissipation rate. As kis increased,\nthe slope of the linear relation becomes more negative,\nimplying that interaction reduces the dissipation rate.\nFigure 5a.(iii) illustrates that a linear coupling of the ac-\ntivities reduces accuracy of the system. However, the\noverall adaptation error increases only slightly while the\ndissipation is strongly affected by coupling.\nC. Coupled Methylation\nAfter studying coupled activities, we next consider a\ncoupling the control (methylation) variables. Again, the\ninteraction is assumed to be linear and symmetric.\n1. Model Description\nIn analogy to Eq. (31), the dynamics of a pair of sen-\nsory units labeled with the indices i, j∈ {1,2}and with\ncoupled variables m1andm2is described by Langevin\nequations. For unit iand with neighboring unit j̸=i11\nFIG. 5. Cost-accuracy relation for composite systems consisting of pairs of linearly coupled sensor units. (a.(i)) Adaptation\nerror vs. dissipation rate for varying stimulus magnitude sin systems with linearly coupled activities. As for a single, non-\ninteracting unit, the cost-accuracy relation is linear. The parameter kcontrols the coupling strength. When k= 0, the\nunits do not interact and the overall dissipation rate is twice the dissipation rate of a single unit. As the coupling strength k\nincreases, less work is required to achieve the same accuracy as without coupling. (a.(ii), a.(iii)) Dissipation rate and overall\nadaptation error as a function of the stimulus magnitude for different coupling strengths k. With increasing coupling strengths,\nthe dissipation rate decreases while the adaptation error increases slightly when activities are coupled. (b.(i)) Adaptation\nerror vs. dissipation rate for varying stimulus strength sin systems with linearly coupled control (methylation) variables.\nFor strong coupling, k≥20k0, the overall dissipation rate nearly equals that of a single non-interacting system. (b.(ii),\nb.(iii)) Dissipation rate and overall adaptation error as a function of the stimulus magnitude for different coupling strengths\nk. With increasing coupling strength, the dissipation rate decreases while the adaptation error remains unchanged. Linear\ncoupling of the slow control variables thus reduces the energetic cost of adaptation per unit without compromising the adaptive\nperformance. (c.(i), c(ii)) The adaptation error is consistently lower if control variables miare coupled, rather than activity\nvariables ai, for otherwise identical systems. Dots are simulation results, dashed lines show data trend. Simulation parameters:\nk0= 100 ,ωa{1,2}= 50, ωm{1,2}= 5, T= 0.01, σ= 0.01, V0= 1/2.12\nthe equations are given by\n˙ai=Fai(ai, mi, s) +ηai,\n˙mi=Fmi(ai, mi, s)−ωij(mi−mj) +ηmi,(37)\nwhere the white noises η{ai,mi}are again pairwise inde-\npendent of each other. The generalized forces Faiare\nthe same as in Eq. (33). To guarantee detailed balance\nin equilibrium, the system must satisfy six constraints as\nshown in the Supplementary Material. These constraints\nare used to construct Fmi, resulting in the same expres-\nsion as given in Eq. (35). The coupling constants are\nscaled with the effective temperature Tas\nω12=k(ωm1T) =ω21=k(ωm2T), (38)\nwhere the parameter 0 ≤k <∞is again used to tune the\ninteraction strength. Lastly, as the interaction strength\nis symmetric and linear, the dissipation rate can be again\ncomputed using Eq. (36).\n2. Numerical Results for Coupled Methylation\nResults from simulations of the system described by\nEq. (37) are shown in FIGs. 4b(ii)-4b.(v) and 5b. The\nformer displays the histogram of trajectories for different\nvariable combinations, representing the probability dis-\ntribution on different two-dimensional planes. The latter\nillustrates the cost-accuracy relation of the interacting\nsystem for varying interaction strengths.\nTypically, the range of the control variable is much\nlarger than the amplitude of the noise, m0≫∆m. There-\nfore, the probability distribution in the m1−m2plane\nappears highly localized, see FIG. 4b.(v). Coupling the\ncontrol elements reduces the spread of this distribution\non the m1−m2plane even more. The two slow variables\nare strongly correlated. However, the two fast activities\nare almost independent of each other and their joint dis-\ntribution is only slightly elongated along the a1=a2axis,\nsee FIG. 4b.(iv). Overall, the probability distribution on\nthea1−m1anda2−m2planes are very similar to the\ndistributions for a non-interacting system, FIGs. (4b(ii),\n4b.(iii)). Therefore, the distributions of the coupled units\nare distorted after application of a stimulus almost the\nsame way as for a non-interacting system. Adaptation\nerrors of the fully adaptive, interacting system are conse-\nquently almost the same as for a single sensory system.\nA coupling of the control variables reduces the over-\nall dissipation rate both for trajectories that stay in the\nphase space interior and for trajectories that interact\nwith the boundaries. For a given coupling strength k,\nthe dissipation rate is slightly higher if the miare cou-\npled, compared to coupled activities ai. Figure 5b.(i)\nshows a linear relationship between the adaptation error\nand the dissipation rate required for adaptation, similar\nto FIG. 5a.(i). Overall, coupling of the control variables\nreduces the dissipation rate and the adaptation error ap-\npears to be independent of interaction strength for this\ncoupling, see FIG. 5b.(iii).D. Discussion of Interacting Systems\nTo study the adaptive performance for an idealized sys-\ntem consisting of two sensory units, we define an over-\nall adaptation error in terms of the overall activity, see\nEqs. (29, 28). These definitions result in an upper bound\nfor the overall adaptation error of the system in terms of\nthe individual errors of the interacting sensory units. If\none measures the error of individual adaptive units, the\noverall error must be smaller than or equal to the mean\nof the individual errors. This inequality is generic as it is\nrobust to the type of interaction between the units and is\ntrue even if the systems are not identical, provided they\nhave the same setpoint a0.\nThe first interacting system that we consider is one in\nwhich the activities aiare coupled, see Eq (31). For this\nsystem, we find that increasing the interaction strength\nleads to a reduction of the dissipation rate and a slightly\nworse adaptation accuracy. This change of the system\nbehavior is due to a stretch of the probability distri-\nbutions for the ( ai, mi) along the mi−axes. After an\nincrease of the stimulus s, the probability distributions\nshift to higher values of the control variables mi. Dur-\ning this shift, the distributions interact earlier with the\nboundaries of the phase space if they are stretched. Thus,\nboundary conditions of the variables influence the behav-\nior of activity-coupled units at smaller stimuli sthan for\nthe corresponding non-interacting unit. Instead of cou-\npling the fast activities, we next couple the slow control\nvariables, see Eq. (37). Similarly to the previous case,\nthe linear, symmetric coupling of the control variables\nreduces the dissipation rate. However, the state proba-\nbility distributions for the individual units in the system\nclosely resemble the distribution for a single, uncoupled\nunit. Therefore, the adaptation accuracy is not compro-\nmised by this coupling.\nFor both types of interacting systems, the finite range\nof internal variables is irrelevant for small stimulus val-\nuessif the states ( ai, mi) rarely assume their extreme\nvalues. In absence of such boundary effects, an increase\nof the coupling constant reduces the overall dissipation\nfrom originally twice the dissipation rate of a single unit\ndown to almost the dissipation rate of only one unit.\nThus, a linear interaction of sensory units in the fully\nadaptive limit reduces the number of wasteful cycles and\neffectively lowers the energetic cost of adaptation.\nIV. CONCLUSION\nFrom bacterial chemosensors to yeast osmotic pressure\nsensors and mammalian light sensors, a wide variety of\nsystems in biology rely on biochemical feedback networks\nto enable sensory adaptation [2, 52, 53]. In the case of\nE. coli chemotaxis, transmembrane receptors sense and\nadapt to the concentration of extracellular ligands. This\nprocess is characterized by a rapid response in sensor\nactivity following a change in ligand concentration and a13\nslowly varying methylation that counteracts the effect of\nthe stimulus and returns the activity to a set point [32].\nWe employ an idealized model of bacterial adaptation\nto visualize how the working range affects the accuracy\nand energetics of adaptation. First, we study the three-\nnode model for adaptation of a single sensory unit. Large\nstimuli drive the state of the modelled system to its phys-\nical limits, represented by boundary conditions of the\nphase space of internal variables. We find that the adap-\ntation error is a linear function of the dissipation rate.\nAs the strength of a stimulus is increased, the error in-\ncreases while the dissipation rate decreases. The results\nfrom simulations are supported with analytical approxi-\nmations. Next, we investigate the the same relationship\nfor interacting systems. For pairs of sensory units, the\noverall adaptation error is determined by the average ac-\ntivity of the two units. An interaction between the units\nis modelled in two ways. Either by coupling the output\nactivities of the two units or by coupling their control\nvariables representing the methylation state. We find\nthat both types of interactions reduce the dissipation rate\nand therefore affect the cost-accuracy relationship. Fur-\nthermore, the simulation results suggest that a coupling\nof the methylation variables may be more advantageous\nthan a coupling of the activities because, in this case,\nthe interaction reduces the energetic cost cost of pow-\nering multiple units without compromising the accuracy\nof adaptation. Overall, a cooperative operation of many\nsensors amplifies sensitivity [21] and our models predictthat the amplification can be energy efficient since the\ndissipation rate per sensor is reduced by coupling.\nLooking ahead, one can consider to study the energet-\nics of larger systems of coupled units, as well as how they\nbehave in the regime of imperfect adaptation. Clearly,\nlinear receptor arrays or randomly clustered receptors\nwith next-neighbor couplings may exhibit more complex\ncollective energetics. From a more formal point of view,\nit would also be interesting to study how different lev-\nels of noise in the different components of the system\nwould affect the adaptive performance while at the same\ntime driving energy flows. Finally, further theoretical\nand experimental work is needed to assess the biologi-\ncal relevance of a trade-off between the benefit of a large\nworking range and the energetic cost of generating suf-\nficiently complex sensory systems. 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Cell Biol. 5, 1024\n(2005)."    },    {        "title": "2401.11367v1.Classifying_representations_of_finite_classical_groups_of_Lie_type_of_dimension_up_to___ell_4_.pdf",        "content": "arXiv:2401.11367v1  [math.RT]  21 Jan 2024CLASSIFYING REPRESENTATIONS OF FINITE CLASSICAL GROUPS OF LIE\nTYPE OF DIMENSION UP TO ℓ4\nLUIS GUTI ´ERREZ FREZ AND ADRI ´AN ZENTENO\nAbstract. In this article we classify the irreducible representation s of finite classical groups Gof Lie type\nof rank ℓwith dimension at most a constant proportional to ℓ4according to whether Gis of type Aℓor\nnot. The dimensions of those representations are explicitl y computed by hand. We conclude the work by\naddressing certain cases of the inverse Galois problem via t he foregoing classification.\n2020Mathematics Subject Classification . 20C33, 20G40, 11F80.\nContents\n1. Introduction 1\n2. Preliminaries on representations of finite simple groups of Lie type 3\n2.1. Root systems 3\n2.2. Representations 4\n2.3. Multiplicity and dimension 4\n3. Multiplicities and dimensions for groups of type Aℓ 4\n4. Representations of finite groups of type Aℓ 8\n5. Multiplicities and dimension in finite classical groups 11\n5.1. Type Bℓ 11\n5.2. Type CℓandDℓ 22\n6. Some applications 24\n7. Apendix 25\n7.1. Type Cℓ 25\n7.2. Type Dℓ 26\nReferences 27\n1.Introduction\nIt is well-known the multiples bridges among representation theory w ith other theories in mathematics.\nWe are specially concerned about the irreducible representations o f finite classical groups of Lie type and\nits connections with the large image conjecture for automorphic Ga lois representations and the inverse\nGalois problem. More precisely, let GQ:= Gal(Q/Q) be the absolute Galois group of QandAQthe ring\nof adeles of Qandπbe a cuspidal algebraic regular automorphic representations of GL n(AQ). According\nto global Langlands correspondence [ 5], for each prime pthere should exist a p-adic Galois representation\nρπ,p:GQ→GLn(Qp) associated to π. A folklore conjecture (usually known as the large image conjectur e)\npredicts that the image of the proyectivization ρproj\nπ,p:GQ→PGLn(Fp) of the residual representation\nρπ,p:GQ→GLn(Fp) ofρπ,pshould be as large as possible for almost all primes p(i.e., all but finitely\nmany), except when an automorphic reason impede it. For instance , whenπis polarizable [ 2,§2.1], this\nmeans that the image of ρproj\nπ,pshould contain either PSpn(Fps) or PΩ±\nn(Fps) (for some positive integer s)\nfor almost all p. This conjecture is, in general, known in very few cases ( n≤6) and the proofs in all known\ncases when πis polarizable are done by considering all possible images of ρproj\nπ,pgiven by the classification\nof the maximal subgroups of PSpn(Fps) or PΩ±\nn(Fps) -see [15] for more details. As we will see at the end\nof this paper, this conjecture has important consequences on th e Inverse Galois problem for finite groups of\nLie type.\n1In the eighties, Aschbacher classified the maximal subgroups of alm ost all of the finite almost simple clas-\nsical groups [ 1]. Essentially, this classification divides these subgroups into two typ es. Roughly speaking, the\nfirst of these consists of subgroups that preserve some kind of g eometric structure which are well understood.\nOn the other hand, the second class consists of those absolutely ir reducible subgroups that can be obtained\n(at least for psufficiently large) as representations of finite groups of Lie type. H owever, this classification\nis not refined enough, particularly for subgroups in the second clas s. Thereby, a best knowledge of the\nclassification of irreducible representations of finite simple groups o f Lie type would be a significant input to\nembark on the large image conjecture (for automorphic Galois repr esentations). It is worth recalling that\na finite simple group of Lie type P Gin characteristic pmay be attached to a connected reductive simple\nalgebraic group GoverFpof simply connected type and a Frobenius endomorphism F:G(Fp)→G(Fp) such\nthat PG∼=GF/Z, whereGF:={γ∈G(Fp) :F(γ) =γ}is the group of fixed points of FandZis the center\nofGF. Then, projective representations of P Gin characteristic pare the same as linear representations of\nGFin characteristic p(see [13], p. 49, items (ix) and (x)), which can be constructed by restrictin g highest\nweight representations of GtoGF. Thus we are interested in classifying the irreducible representatio ns ofG\nwhen it is a finite classical group of rank ℓwith dimension at most a constant proportional to ℓ4, extending\nresults in [ 10] and [11], where constants proportional to ℓ2andℓ3are studied respectively. Subsequently,\nby using our classification and the philosophy in [ 14], we deduce some results concerning to the large image\nconjecture for automorphic Galois representations and the inver se Galois problem.\nWe would like to point out that we are assuming Gof simply connected type, which ensures a corre-\nspondence between representations of the group Gand representations of its Lie algebra gfor almost all p.\nTherefore, through this paper, we will consider psufficiently large in order to account with such a correspon-\ndence. The main result of this paper splits into three theorems which tackle the classification of irreducible\nrepresentations of Gof dimension less or equal to ℓ4/16 orℓ4according to whether the group is of type Aℓ\nor not. Thanks to the work of Chevalley, we know that each irreduc ibleG(Fp)-module Vis determined (up\nto isomorphism) by its highest weight and for each dominant weight λthere is an irreducible G(Fp)-module\nV(λ) with highest weight λ. Thereby, classifying those irreducible representations is equivale nt to determine\nall dominant weights λsuch that the dimension of the irreducible G(Fp)-module V(λ) of highest weight λ\nsatisfies the foregoing condition-we will call them admissible weights . Determining this class of weights will\nimply to compute diverse dimensions and consequently to handle a con siderable number of multiplicities.\nWe then will present certain examples with these calculations first to illustrate admissible and not admissible\nweights. Nevertheless, one of the key motivations to give these ex amples is to reminisce about the computing\nof multiplicities and its recursive nature since we will have to compute lo nger multiplicities and dimensions\nalong the work. In addition, a second good motivation is that those e xamples are placed in such a manner\nto contribute in the proofs of our main results, helping to be more re adable such demonstrations.\nThe classification of admissible weights splits into three stages accor ding toGis of type Aℓ,Bℓor of type\nCℓandDℓ. Our strategy to this end will be to analyze the behavior of diverse m ultiplicities of weights µ/pre⌋edesequalλ\nto study the dimension of V(λ) whereby we obtain a good list of admissible weights in order to show that this\nlist exhaust the admissible ones. We therefore commence by applying successfully the mentioned strategy to\nGof typeAℓas starting case. Later on, we tackle the casewhen Gis oftype Bℓ, where we may point out that\ndetermining the admissible weights certainly presents additional cha llenges. Indeed, computing the required\nmultiplicities to study admissibility of weights implies much more efforts th an the case Aℓ. In particular,\nthese calculations turn out to be very laborious with extensive comp utations to use the Freudenthal formula\nto determine multiplicities at hand. We are able to carry out all those c omputations and find a good list of\nadmissible weights. Once we have ended that, obtaining good lists for typesCℓ,Dℓwill follow more natural.\nThe second stage of the strategy also will be a subtle task in the sen se that from an arbitrary admissible\nweight we will have to prove this one can be reduced to a one of our go od list,i.e., the list exhausts the\nadmissible ones which gives rise to the desired classification. We here c ontinue case-by-case tackling first\nthe type Aℓand subsequently the case of type Bℓ. This work takes a significant time and whence the cases\nCℓandDℓwill be deduced directly. Hence we only outline the proof by refining th e arguments according\nto the case and by proving the admissibility of the new weight that app ears when Gis of type CℓandDℓ.\nAt this point, we would like to observe that the required multiplicities in t hese cases are listed in two tables\nof the Appendix, where we bring all information to compute them, th us avoiding repetitive arguments and\ncalculations.\n2We conclude the Introduction by giving the structure of the paper .\nThis article begins with a brief background on representation theor y of finite groups of Lie type in §2.\nHere we introduce the root systems and fix the orders that we will b e used through the work. Also we recall\nthe Freudenthal multiplicity formula and the expression to compute dimension of the G(Fp)-module V(λ)\nwith highest weight λ. In§3we introduce the concept of admissible weight λwhenGis of type Aℓand\nobtain a good list -Proposition 3.6- which is the first part toward the classification of admissible weights of\nG. Later on, §4is dedicated completely to Theorem 4.1, where the classification of the admissible weights is\nobtained when Gis of type Aℓ. In§5we tackle our aim for groups Gof typeBℓ,CℓandDℓ. In the first case\nwe prove a series of lemmas on required multiplicities to obtain a list of ad missible weights -Proposition 5.6.\nThus, we may show such a classification for Gof typeBℓin Theorem 5.9. While Theorem 5.10determines\nall the admissible weights for the remainder two types by appealing to Appendix -Tables 1 and 2- for the\nrequired multiplicities. We present the last two results of the article in §6by providing applications of our\nclassification to large image conjecture for automorphic Galois repr esentation in Theorem 6.2and on the\ninverse Galois problem in Corollary 6.3. We conclude our work with an appendix §7where we give the tables\nwith the all required multiplicities when Gis of type CℓorDℓ.\n2.Preliminaries on representations of finite simple groups o f Lie type\nIn this section we will recall some results about algebraic groups and their defining characteristic repre-\nsentations that we will need in this paper.\n2.1.Root systems. LetGbe a connected reductive simple algebraic group over Fpof simply connected\ntype and rank ℓ. LetTbe a maximal torus of G,X∼=Zℓits character group and Y∼=Zℓits co-character\ngroup. Let {α1,...,α ℓ} ⊂Xbe a set of simple roots for Gwith respect to Tand ∆ its associated root\nsystem. Let α∨\ni∈Ythe coroot corresponding to αifor each 1 ≤i≤ℓ. Viewing X⊗Ras Euclidean space\nwe can define the fundamental weights λ1...,λℓ∈X⊗Ras the dual basis of α∨\n1,...,α∨\nℓ.\nFor our purposes in accordance with [ 11] we shall use the opposite order from the one given in [ 3,8].\nThus, the Dynkin diagrams for classical groups are given as follows:\nAℓ:1 2 3 4 ℓ\n... Bℓ:1 2 3 4 ℓ\n...\nCℓ:1 2 3 4 ℓ\n...Dℓ:1\n23 4 5 ℓ\n...\nWe also recall that each α∈∆ determines a reflection σαby means of\nσα(β) = (α,β), β∈E,\nwhere (E,(−,−)) is the underlying inner product real vector space of finite dimens ional associated to ∆.\nThe Weyl group Wof ∆ is the subgroup of the orthogonal group on Egenerated by the reflections σαi.\nThus one observes that this group acts on ∆ by permutation. In fa ct,Wis generated by the reflections σαi\nwith 1≤i≤ℓby appealing to [ 9, Prop. 2.62].\nFurthermore, one obtains a partial ordering in Xby declaring that a weight λis smaller than a weight λ′\n(denoted by λ≺λ′) if and only if λ′−λis a non-negative linear combination of simple roots. We shall also\nuse/pre⌋edesequalin an obvious way. In addition, a weight λ∈Xis dominant if it is a non-negative linear combination\nof the fundamental weights. The group Wacts on the set of the dominant weights by means of its action\non the simple roots α1,···,αℓ. Given a dominant weight λ, we will denote by Orb W(λ) and Stab W(λ) the\norbit and the stabilizer of λunder the W-action, respectively.\nItisworthrecallingthatthestabilizersubgroupStab W(λ) ofadominantweight λ=a1λ1+a2λ2+···+aℓλℓ\nis the parabolic subgroup of Ggenerated by the reflections along the simple roots αisuch that ai= 0.\n32.2.Representations. LetVbe a finite-dimensional G(Fp)-module. Considering this as T-module, we\nhave a decomposition V=/circleplustext\nλ∈XVλinto weight spaces Vλ={v∈V:vt=λ(t)vfor allt∈T}. Then, the\nset of weights λ∈XwithVλ/ne}ationslash={0}is called the set of weights ofV. This set is a union of W-orbits.\nAs we pointed out in the Introduction, from a result of Chevalley (se e Section 31.3 of [ 8]) we have that,\nifVis irreducible, then the set of weights of Vcontains a unique element λsuch that for all weight λ′ofV\nwe have λ′≺λ. Thisλis called the highest weight ofV. This is dominant and dim( Vλ) = 1. Moreover, we\nhave that each irreducible G(Fp)-module Vis determined (up to isomorphism) by its highest weight and for\neach dominant weight λ∈Xthere is an irreducible G(Fp)-module V(λ) with highest weight λ.\nFor example, when Gis ofAℓtype, the standard representation V=V(λ) has highest weight λ=λℓand\ndimension ℓ+ 1. Moreover, the anti-symmetric tensor product/logicalandtextk(V) =V(λ) is irreducible with highest\nweightλ=λℓ−k+1+···+λℓand dimension/parenleftbigℓ+1\nk/parenrightbig\n. More generally, the irreducible representation V(λ), with\nhighest weight λ=a1λ1+(a1+a2)λ2+···+(a1+···+aℓ)λℓ, is contained in\nSyma1(ΛℓV)⊗···⊗Symaℓ−1(Λ2V)⊗Symaℓ(V)\nby Weyl’s Construction [ 7, Lecture 15]. In particular, taking aℓ/ne}ationslash= 0 andai= 0,i= 1,···,ℓ−1, we have that\nV(aℓλℓ) is contained in the aℓ-symmetric power Symaℓ(V) ofV, which has dimension/parenleftbigℓ+aℓ\naℓ/parenrightbig\n. In fact, we will\nprove that Symaℓ(V) is irreducible by computing the dimension of V(aℓλℓ). In general, when Gis of type\nBℓ,CℓandDℓ, we can only guarantee that V(kλℓ)⊆Symk(V) andV(λℓ−k+1+···+λℓ)⊆Λk(V). Through\nthis paper, we also compute the exact dimensions of these irreducib le representations V(λ) fork= 2,3,4 by\ncomputing multiplicities as follows.\n2.3.Multiplicity and dimension. First, we introduce a special weight which appears in computing of\nmultiplicities as we will see here below. Specifically, the strongly dominan t weight δis defined by\nδ:=1\n2/summationdisplay\nα≻0α=ℓ/summationdisplay\ni=1λi,\nfollowing [ 8,§10.2 and §13.3].\nNow, let V(λ) be a finite-dimensional G(Fp)-module with highest weight λandµ/pre⌋edesequalλbe a dominant\nweight. We define the multiplicity of µinV(λ) as\nmλ(µ) = dimV(λ)µ\nwhich can be calculated by using the recursive Freudenthal multiplicit y formula [ 8,§22.3]:\nmλ(µ) =2/summationtext\nα∈Φ+/summationtext∞\ni=1(µ+iα,α)mλ(µ+iα)\n/⌊ard⌊lλ+δ/⌊ard⌊l2+/⌊ard⌊lµ+δ/⌊ard⌊l2=2/summationtext\nα∈Φ+/summationtext∞\ni=1(µ+iα,α)mλ(µ+iα)\n(λ,λ)−(µ,µ)+2(λ−µ,δ)\nwhere Φ +denotes the set of positive roots of ∆ and mλ(λ) = 1.\nIt follows from the proof of Theorem in [ 8,§21.2] that the dimension of V(λ) is\n(2.1) dim V(λ) =/summationdisplay\nµ∈Π(λ)mλ(µ)l(µ)\nwhere Π( λ) :={µ:µis a dominant weight such that µ/pre⌋edesequalλ}andl(µ) denotes the length of the W-orbit\nOrbW(µ).\n3.Multiplicities and dimensions for groups of type Aℓ\nIn order to be ready for computing certain multiplicities required alon g the section, let us start by giving\na system of simple roots of type Aℓand the fundamental weights associated.\nLet{α1,α2,···,αℓ}be the simple roots system of type Aℓgiven by\nαi:=ei−ei+1,1≤i≤ℓ.\nWe can normalize as follows: ( αi,αi) = 1, ( αi,αi+1) =−1/2 and (αi,αi+k) = 0,k >1. Then, the\nfundamental weights are given by\nλi=1\nℓ+1[(ℓ−i+1)α1+2(ℓ−i+1)α2+i(ℓ−i+1)αi+i(ℓ−i)αi+1+···+iαℓ].\n4The following result will useful for computing the dimensions at hand.\nLemma 3.1. For any 1≤i,j≤ℓwe have that\n(ℓ+1)2(λi,λj) =i(j−1)(ℓ−j+1)2·(−1/2)+ijℓ−j+1/summationdisplay\nk=1k.\nProof.Without loss of generality suppose that i≤j, namely j=i+rwithr≥0. Thus, we have\nλi= (ℓ−i+1)α1+···+i(ℓ−i+1)αi+···+i(ℓ−i−r+1)αj+i(ℓ−i−r)αj+1+···+iαℓ\nλj= (ℓ−j+1)α1+···+(ℓ−j+1)αi+···+j(ℓ−j+1)αj+j(ℓ−j)αj+1+···+jαj.\nThen we can calculate\n(ℓ+1)2(λi,λj)\n= (ℓ−i+1)(ℓ−j+1)+2( ℓ−i+1)(ℓ−j+1)·(−1/2)\n+2(ℓ−i+1)(ℓ−j+1)·(−1/2)+4(ℓ−i+1)(ℓ−j+1)+6( ℓ−i+1)(ℓ−j+1)·(−1/2)\n+6(ℓ−j+1)(ℓ−i+1)·(−1/2)+9(ℓ−i+1)(ℓ−j+1)+12( ℓ−j+1)(ℓ−i+1)·(−1/2)\n...\n+i(j−2)(ℓ−j+1)(ℓ−i−r+2)·(−1/2)+i(j−1)(ℓ−i−r+2)(ℓ−j+1)+ij(ℓ−j+1)(ℓ−i−r+2)·(−1/2)\n+i(j−1)(ℓ−i−r+1)(ℓ−j+1)·(−1/2)+ij(ℓ−i−r+1)(ℓ−j+1)+ij(ℓ−i−r+1)(ℓ−j)·(−1/2)\n+ij(ℓ−i−r)(ℓ−j+1)·(−1/2)+ij(ℓ−i−r)(ℓ−j)+ij(ℓ−i−r)(ℓ−j−1)·(−1/2)\n+ij(ℓ−i−r−1)(ℓ−j)·(−1/2)+ij(ℓ−j−1)2+ij(ℓ−i−r−1)(ℓ−j−2)·(−1/2)\n...\n+6ij·(−1/2)+4ij+2ij·(−1/2)\n+2ij·(−1/2)+ij\n=i(j−1)(ℓ−j+1)2·(−1/2)+ij(ℓ−j+1)2+ij(ℓ−j+1)(ℓ−j)·(−1/2)\n+ij(ℓ−j)(ℓ−j+1)·(−1/2)+ij(ℓ−j)2+ij(ℓ−j)(ℓ−j−1)·(−1/2)\n+ij(ℓ−j−1)(ℓ−j)·(−1/2)+ij(ℓ−j−1)2+ij(ℓ−j−1)(ℓ−j−2)·(−1/2)\n...\n+6ij·(−1/2)+4ij+2ij·(−1/2)\n+2ij·(−1/2)+ij\n=i(j−1)(ℓ−j+1)2·(−1/2)+ijℓ−j+1/summationdisplay\nk=1k2−ℓ−j+1/summationdisplay\nk=1k(k−1)\n=i(j−1)(ℓ−j+1)2·(−1/2)+ijℓ−j+1/summationdisplay\nk=1[k2−k(k−1)]\n=i(j−1)(ℓ−j+1)2·(−1/2)+ijℓ−j+1/summationdisplay\nk=1k.\n/square\nNow we introduce a pivotal concept, specially in the next two section s.\nDefinition 3.2. LetGbe a connected reductive simple algebraic group of type AℓoverFpof simply connected\ntype and λbe a dominant weight. We will say that λis admissible if the dimension of the G(Fp)-module\nV(λ)is at most ℓ4/16.\n5We see for instance.\nExample 3.3. WhenGis as above with ℓ >2, the dominant weight λ1+λ2is admissible.\nWe need to compute the dimension of V(λ1+λ2)to verify the admissibility of λ=λ1+λ2. Observe\nfirst that λ3=λ−(α1+α2), we then have to compute mλ(λ3+α1)and is idem for mλ(λ3+α2); we set\nµ=λ3+α2=λ−α1. Thus we have that\nmλ(µ) = 2·mλ(µ+α2)(µ+α2,α2)\n(λ+δ,λ+δ)−(µ+δ,µ+δ).\nImmediately we see that mλ(µ+α2) =mλ(λ) = 1and(µ+α2,α2) = 1/2. Thanks to Lemma 3.1we\ndeduce that\n(λ,λ) =5ℓ−4\n2(ℓ+1)and(µ,µ) =5ℓ−4\n2(ℓ+1).\nWe then have\nmλ(µ2) = 2·mλ(µ+α2)(µ+α2,α2)\n(λ,λ)−(µ,µ)+2(λ−µ,δ)\n= 2·1·1\n2\n5ℓ−4\n2(ℓ+1)−5ℓ−4\n2(ℓ+1)+1\n= 1.\nHence we compute the multiplicity of µinV(λ)as follows\nmλ(µ) =(µ+α1,α1)mλ(µ+α1)+(µ+α2,α2)mλ(µ+α2)+(µ+α1+α2,α1+α2)mλ(µ+α1+α2)\n(λ+δ,λ+δ)−(µ+δ,µ+δ)\n= 2·1·1+1·1+1·1\n5ℓ−4\n2(ℓ+1)−3(ℓ−2)\n2(ℓ+1)+2\n= 2.\nThus, we conclude that\ndimV(λ1+λ2) =mλ(λ)l(λ)+mλ(λ3)l(λ3) =ℓ(ℓ+1)+2 ·(ℓ−1)ℓ(ℓ+1)\n6\n=ℓ(ℓ+1)[1+( ℓ−1)/3]\n=ℓ(ℓ+1)(ℓ+2)/3,\nas desired.\nIn contrast with example above we observe.\nExample 3.4. For allℓ≥12, the weight λ=λℓ−2+λℓis not admissible.\nIt suffices to show that the multiplicity of λℓ−3/pre⌋edesequalλis3by resorting as follows\ndimV(λℓ−2+λℓ)≥mλ(λℓ−3)l(λℓ−3) = 3·(ℓ−2)(ℓ−1)ℓ(ℓ+1)\n24\n=ℓ4(1−2/ℓ)(1−1/ℓ)(1+1/ℓ)\n8\n>ℓ4\n16,\nby using (1−2/ℓ)(1−1/ℓ)≥(1−2/12)(1−1/12)>1/2for any integer ℓ≥12and(1+1/ℓ)≥1. Thus,λ\ncannot be admissible.\nThen we shall compute mλ(λℓ−3). We first note\nµ:=λℓ−3=λ−(αℓ+αℓ−1+αℓ−2)\nThus, we have to compute\nmλ(µ) =(∗)\n(λ,λ)−(µ,µ)+2(λ−µ,δ),\n6where\n(∗) = 2/summationdisplay\nα∈Φ+∞/summationdisplay\ni=1(µ+iα,α)mλ(µ+iα)\n= (µ+(αℓ+αℓ−1+αℓ−2),αℓ+αℓ−1+αℓ−2)mλ(µ+(αℓ+αℓ−1+αℓ−2))\n+(µ+(αℓ+αℓ−1),αℓ+αℓ−1)mλ(µ+(αℓ+αℓ−1))\n+(µ+(αℓ−1+αℓ−2),αℓ−1+αℓ−2)mλ(µ+(αℓ−1+αℓ−2))\n+(µ+αℓ,αℓ)mλ(µ+αℓ)+(µ+αℓ−1,αℓ−1)mλ(µ+αℓ−1)\n+(µ+αℓ−2,αℓ−2)mλ(µ+αℓ−2).\nBy computing -term by term- as in Example 3.3we may infer that\nmλ(µ+αℓ)(µ+αℓ,αℓ) = 1·2\nmλ(µ+αℓ−1)(µ+αℓ−1,αℓ−1) = 1·2\nmλ(µ+αℓ−2)(µ+αℓ−2,αℓ−2) = 1·2\nmλ(µ+αℓ−1+αℓ)(µ+αℓ−1+αℓ,αℓ−1+αℓ) = 1·2\nmλ(µ+αℓ−1+αℓ−2)(µ+αℓ−1+αℓ−2,αℓ−1+αℓ−2) = 1·2\nmλ(µ+αℓ−2+αℓ−1+αℓ)(µ+αℓ−2+αℓ−1+αℓ,+αℓ−2+αℓ−1+αℓ) = 1·2.\nFinally, since\n(λ,λ)−(µ,µ)+2(λ−µ,δ) = (λ,λ)−(µ,µ)+2(αℓ+αℓ−1+αℓ−2,δ) = 6−4+2·3 = 8,\nwe have:\nmλ(µ) = 2·2·1+2·1+2·1+2·1+2·1+2·1\n8= 2·12\n8= 3.\nHenceλ=λℓ−2+λℓis not admissible.\nRemark 3.5. It is worth to emphasizing the obvious inequality\ndimV(λ)≥mλ(µ)l(µ)≥l(µ)\nthat we will frequently use for any dominant weight µ≺λ.\nProposition 3.6. LetGbe as above with ℓ≥15. Then, all weights in the following table are admissible.\nλ dimension of V(λ)\n0 1\nλ1,λℓ ℓ+1\nλ2,λℓ−1 ℓ(ℓ+1)/2\nλ3,λℓ−2 (ℓ−1)ℓ(ℓ+1)/6\nλ4,λℓ−3 (ℓ−2)(ℓ−1)ℓ(ℓ+1)/24\n2λ1,2λℓ (ℓ+2)(ℓ+1)/2\n3λ1,3λℓ (ℓ+3)(ℓ+2)(ℓ+1)/6\nλ1+λℓ ℓ(ℓ+2)\nλ1+λℓ−1,λ2+λℓ(ℓ+2)(ℓ+1)(ℓ−1)/2\nλ1+2λℓ,2λ1+λℓ (ℓ+1)ℓ(ℓ+3)/2\nλ1+λ2,λℓ−1+λℓ ℓ(ℓ+1)(ℓ+2)/3\nIn addition for ℓ≥16, the weights 4λ1and4λℓare also admissible whose associated irreducible represen -\ntations have dimension (ℓ+4)(ℓ+3)(ℓ+2)(ℓ+1)/24.\nProof.Firstly, we shall prove that, for ℓ≥15, the following weights\n3λ1,3λℓ, λ3, λℓ−2, λ4, λℓ−3, λ2+λℓ, λ1+λℓ−1, λ1+2λℓ,2λ1+λℓ, λ1+λ2, λℓ−1+λℓ.\nare admissible. We start remarkingthat V(kλℓ) is irreducible by definition and is contained in the symmetric\npower tensor product Symk(V), where Vis the standard irreducible representation of dimension ℓ+1. Now,\n7let us start by deducing the dimension of the weight space V(3λℓ). Note that the dominant weights µ/pre⌋edesequal3λℓ\nare: 3λℓ,λℓ−1+λℓ, andλℓ−2. In addition, their stabilizers are contained in the reflection groups of type\nAℓ−1,Aℓ−2andAℓ−3×A2, respectively. Since its multiplicities are at least 1, by formula ( 2.1) we can\nobserve\ndimV(3λℓ)≥l(3λℓ)+l(λℓ−1+λℓ)+l(λℓ−2)\n= (ℓ+1)+ℓ(ℓ+1)+ℓ(ℓ−1)(ℓ+1)/6\n= (ℓ+1)(ℓ+2)(ℓ+3)/6\n= dimSym3(V).\nThis then implies that V(3λℓ) = Sym3(V) and dim V(3λℓ) = (ℓ+1)(ℓ+2)(ℓ+3)/6≤ℓ4/6. By duality\nwe infer the dimensions of V(3λ1). Similarly, we can deduce that the anti-symmetric power tensor pr oduct\nΛi+1(V) =V(λℓ−i) fori= 2,3 by observing that V(λℓ−i)⊆Λi+1(V) and\ndimV(λℓ−i)≥l(λℓ−i) =(ℓ+1)!\n(ℓ−i)!(i+1)!=/parenleftbiggℓ+1\ni+1/parenrightbigg\n= dimΛi+1(V).\nAgain by duality we obtain the dimensions of V(λ3) andV(λ4). It remains to compute the dimensions\nof the weight spaces associated to λ2+λℓ, λ1+λℓ−1, λ1+2λℓ,2λ1+λℓ, λ1+λ2, λℓ−1+λℓ. In fact, we\nshall only need to compute the dimensions of V(λ1+λ2), V(λ1+λℓ−1) andV(λ1+2λℓ) by duality. The\ndimension of the former G-module has been calculated in Example 3.3and the other two dimensions can be\nobtained in the same way.\nNow, when ℓ≥16 we may check that λ= 4λℓis admissible by verifying first that the dominant weights\nµ/pre⌋edesequal4λℓare: 4λℓ,λℓ−1+2λℓ, 2λℓ−1,λℓ−2+λℓ−1andλℓ−3, and later to check\ndimV(4λℓ)≥l(4λℓ)+l(λℓ−1+2λℓ)+l(2λℓ−1)+l(λℓ−2+λℓ)+l(λℓ−3) = dimSym4(V).\nThis implies that V(4λℓ) = Sym4(V) and dim V(4λℓ) = (ℓ+1)(ℓ+2)(ℓ+3)(ℓ+4)/24≤ℓ4/16. Then, by\nduality, we deduce the dimension of V(4λ1). Finally, we may note that the remaining weights were studied\nby L¨ ubeck in [ 11,§5]. /square\nRemark 3.7. Notice that the argument used to prove Λi+1(V) =V(λℓ−i)fori= 2,3(resp.V(iλℓ) =\nSymi(V)fori= 3,4) works for any integer isuch that 1≤i≤ ⌊ℓ/2⌋+ 1. This tell us in particular that\nΛi+1(V)(resp.Symi(V)) is irreducible. Moreover, by duality we may observe that\ndimV(λi+1) = dimΛi+1(V),(resp.dimV(iλ1) = dimSymi(V)).\n4.Representations of finite groups of type Aℓ\nIn order to determine all admissible weights when Gis a finite group of Lie type AℓoverFpwe shall\nproceed by a series of steps to reduce the possibilities according to whether the coefficients are zeroes\nor not. Let λ=a1λ1+a2λ2+···+aℓλℓbe a dominant weight. The first step, nullity of coefficients,\nconsists of identifying all indexes isuch that ai= 0. Each of the other reductions will be associated to a\nnonzero coefficient ai, where we determine all the possibilities to the other nonzero coeffic ientsajreducing\nus exclusively to admissible weights. Explicitly, we start with.\nNullity of coefficients −Claim:ai/ne}ationslash= 0 only if i≤4 ori≥ℓ−3.\nSuppose ai/ne}ationslash= 0 for some isatisfying 5 ≤i≤ℓ−4. This implies that the W-stabilizer of λis contained\nin a reflection group of type Ai−1×Aℓ−i. Thus,\ndimV(λ)≥l(λ) =(ℓ+1)!\ni!(ℓ+1−i)!.\nWe shall prove that ( ℓ+1)!/i!(ℓ+1−i)!> ℓ4/16, for any 5 ≤i≤ℓ−4. To do this, assume firstly that\ni≤ℓ+1−i. In this case, we need to show that\n(ℓ+2−i)(ℓ+3−i)···(ℓ−3)(ℓ−2)(ℓ−1)ℓ(ℓ+1)\ni!>ℓ4\n16.\n8We shall first verify the inequality for i= 5, that is,\n(ℓ−3)(ℓ−2)(ℓ−1)ℓ(ℓ+1)\n5!>ℓ4\n16⇐⇒(ℓ−3)(ℓ−2)(ℓ−1)(ℓ+1)\n3·5>ℓ3\n2.\nIn fact, since ℓ≥15, we can infer that ( ℓ+1)/15>1 and 1−j/ℓ≥1−j/15 for 1 ≤j≤3. Then, by\ncombining these facts we can deduce\n(ℓ−3)(ℓ−2)(ℓ−1)(ℓ+1)\n3·5>(ℓ−3)(ℓ−2)(ℓ−1)\n≥ℓ3(1−3/ℓ)(1−2/ℓ)(1−1/ℓ)\n≥ℓ3(1−3/15)(1−2/15)(1−1/15)\n≥ℓ312\n15·13\n15·14\n15\n>ℓ3\n2.\nThus, the case i= 5 follows by multiplying by ℓ/8.\nThe general case i≤ℓ+1−iis then deduced from the base case i= 5 by noting that\n(ℓ+2−i)(ℓ+3−i)···(ℓ−4)(ℓ−3)(ℓ−2)(ℓ−1)ℓ(ℓ+1)\ni(i−1)···6·5!\n=/parenleftbigg(ℓ+2−i)(ℓ+2−(i+1))···(ℓ+2−6)\ni(i−1)···6/parenrightbigg/parenleftbigg(ℓ−3)(ℓ−2)(ℓ−1)ℓ(ℓ+1)\n5!/parenrightbigg\nand the first parenthesis is greater than 1 by using the condition ℓ≥15. Finally, the case ℓ+1−i < i≤ℓ\nfollows by using a symmetric argument. Therefore, we reduces us t o\nλ=a1λ1+a2λ2+a3λ3+a4λ4+aℓ−3λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ.\nLiberty of coefficients −Assuming that aℓ−3/ne}ationslash= 0 we shall show that aℓ−3= 1 and ai= 0 for any i≤4\nandi≥ℓ−2. In fact, if aℓ−j/ne}ationslash= 0 for all j= 0,1,2 then the W-stabilizer of λis contained in a reflection\ngroup of type Aℓ−4. We then see that\n(4.1) dim V(λ)≥l(λ) =(ℓ+1)!\n(ℓ−3)!= (ℓ−2)(ℓ−1)ℓ(ℓ+1)>ℓ4\n16.\nOn the other hand, when exactly two coefficients aℓ−jare nonzero, the W-stabilizer of λis contained in\na reflection group of type Aℓ−3×A2. An argument as above shows that dim V(λ)> ℓ4/16. Next, if just one\ncoefficient aℓ−jis nonzero then the W-stabilizer of λis contained in a reflection group of type Aℓ−4×A1×A1\norAℓ−4×A2according to j= 1 or not. Thus, we may rule out this possibility of λ. It follows that aℓ−j= 0\nfor allj= 0,1,2 and by a symmetry argument one checks that ai= 0 for all i= 1,2,3. Thus, we reduce to\nλ=aℓ−3λℓ−3.\nWe now will prove that aℓ−3= 1 by supposing the contrary i.e.,aℓ−3≥2. Then, we observe that the\n(ℓ−4)-coefficient of\nλ−αℓ−3=aℓ−3λℓ−3−(−λℓ−4+2λℓ−3−λℓ−2) =λℓ−4+(aℓ−3−2)λℓ−3+λℓ−2\nis one, which is not possible by Nullity of Coefficients. Whence aℓ−3= 1 and then λ=λℓ−3is admissible by\nProposition 3.6.\nNow assume that aℓ−3= 0 and aℓ−2/ne}ationslash= 0, that is,\nλ=a1λ1+a2λ2+a3λ3+a4λ4+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ.\nWe shall prove that ai= 0 for 1 ≤i≤4. Suppose first a1/ne}ationslash= 0 and a2=a3=a4= 0. We note that\ntheW-stabilizer of λis contained in a reflection group of type Aℓ−4orAℓ−4×A1orAℓ−4×A2according\ntoaℓ−1/ne}ationslash= 0/ne}ationslash=aℓor not. Then, reasoning as above, we can deduce that dim V(λ)> ℓ4/16. Similarly each\npossibility of some ai/ne}ationslash= 0 for 1 ≤i≤4 discards λas admissible weight, and thus\nλ=aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ.\n9We now show that aℓ−2≤1. Otherwise, assuming that aℓ−2≥2 one observes that the coefficients at\n(ℓ−3) and (ℓ−1) of the dominant weight λ−aℓ−2are nonzero, but this is not possible by reduction above.\nHence, we reduce our analysis to study\nλ=λℓ−2+aℓ−1λℓ−1+aℓλℓorλ=aℓ−1λℓ−1+aℓλℓ.\nWhenaℓ−2= 1 andaℓ−1≥2 then the term 2 λℓ−2appears in λ−αℓ−1, which is impossible by applying the\nreasoning above. So, any λof the form λℓ−2+aℓ−1λℓ−1+aℓλℓwithaℓ−1≥2 is not admissible and we reduce\nus toλof the form λ=λℓ−2+λℓ−1+aℓλℓorλ=λℓ−2+aℓ−1λℓ−1. For the case λ=λℓ−2+λℓ−1+aℓλℓ,\nifaℓ≥2 we may observe similarly that the term 2 λℓ−1appears in the dominant weight λ−αℓ−1which is a\ncontradiction by iterating the previous reasoning. This reduces ou r analysis to study the dominant weights\nλℓ−2+λℓ−1+λℓandλℓ−2+λℓ−1. In the same vein, we discard the possibility of a dominant weight of\nthe form λℓ−2+aℓλℓwithaℓ≥4. The cases aℓ= 2,3 are reduced again to analyze the dominant weights\nλℓ−2+λℓ−1+λℓandλℓ−2+λℓ−1. Finally, when λ=λℓ−2+aℓλℓwithaℓ≥2 cannot be admissible by\nusing similar arguments as above. Therefore, we only need to deter minate the dimensions of V(λ) for the\nfollowing dominants weights\nλℓ−2+λℓ−1, λℓ−2+λℓ−1+λℓ, λℓ−2+λℓ, λℓ−2.\nIn the first case we see that λℓ−3+λℓ≺λℓ−2+λℓ−1. Thus, we can deduce as before that\ndimV(λℓ−2+λℓ−1)≥l(λℓ−3+λℓ) =ℓ4(1−2/ℓ)(1−1/ℓ)(1+1/ℓ)\n6>ℓ4(1−2/ℓ)(1−1/ℓ)\n6> ℓ4/12> ℓ4/16,\nsince (1−2/ℓ)(1−1/ℓ)≥((1−2/15)(1−1/15)>1/2 for any ℓ≥15. In the same way, we can rule out the\nsecond case by using that λℓ−3+2λℓ≺λℓ−2+λℓ−1+λℓ. Note that λℓ−2+λℓis not admissible by appealing\nto Example 3.4andλℓ−2is admissible by Proposition 3.6.\nNext, suppose aℓ−3=aℓ−2= 0 and aℓ−1/ne}ationslash= 0:\nλ=a1λ1+a2λ2+a3λ3+a4λ4+aℓ−1λℓ−1+aℓλℓ.\nIfaℓ−1≥2 we note again that λℓ−2andλℓappear in the writing of the dominant weight λ−αℓ−1, which\nis impossible as was showed previously. Thus, we reduce to aℓ−1= 1,i.e.,\nλ=a1λ1+a2λ2+a3λ3+a4λ4+λℓ−1+aℓλℓ.\nIn this case we may prove that ai= 0 for 2 ≤i≤4. Here, many cases arise for the W-stabilizer of λ\naccording to the vanished for some of these ai(2≤i≤4). For instance, if a2/ne}ationslash= 0 and a3=a4= 0 we\nsee that W-stabilizer of λis contained in a reflection group of type A1×Aℓ−4orAℓ−4according to a1= 0\nor not. Thus, we may verify that dim V(λ)> ℓ4/16,i.e.,λis not admissible. Indeed, in any case we may\ndeduce that the dimension V(λ) is bigger than the bound ℓ4/16. This implies\nλ=a1λ1+λℓ−1+aℓλℓ.\nReasoning as above one verifies that a1andaℓcannot be simultaneously nonzero. In fact, assume first\nthata1≥2 we obtain the dominant weight\nλ−α1= (a1−2)λ1+λ2+λℓ−1+aℓλℓ,\nwhich is ruled out by computations as above. On the other hand, if aℓ≥2 then 2λℓ−1appears in the writing\nofλ−αℓ, which is impossible too. Moreover, we may discard λ=λ1+λℓ−1+λℓby arguing as in Example\n3.4. This lefts us the dominant weights λ1+λℓ−1,λℓ−1+λℓandλℓ−1which are admissible by Proposition\n3.6.\nFinally suppose that aℓ−3=aℓ−2=aℓ−1= 0 and aℓ/ne}ationslash= 0,i.e., we will deal with\nλ=a1λ1+a2λ2+a3λ3+a4λ4+aℓλℓ\nWe may verify, similarly as above, that a3=a4= 0, reducing our analysis to\nλ=a1λ1+a2λ2+aℓλℓ.\nObserve that if a2/ne}ationslash= 0 then a1= 0 by using a dual reasoning to the previous case. This implies that\nλ=λ2+aℓλℓ, which gives just one new admissible weight λ=λ2+λℓ. Finally, if a2= 0 then we have that\nλ=a1λ1+aℓλℓand we get the new admissible weights λ1+λℓ, 2λ1+λℓ,λ1+2λℓandaℓλℓforaℓ= 1,2,3,4.\n10In summary, we have proved the first of our main results:\nTheorem 4.1. LetGbe a finite group of Lie type AℓoverFp. Ifℓ≥15then the list giving in Proposition\n3.6exhausts the admissible weights.\n5.Multiplicities and dimension in finite classical groups\nIn this section we deal with the rest of finite classical groups. Let Gbe a connected reductive simple\nalgebraic group of type Bℓ,CℓorDℓof simply connected type. By analogy to Definition 3.2we present the\nnotion of admissibility for these types.\nDefinition 5.1. LetGbe a connected reductive simple algebraic group of type Bℓ,CℓorDℓoverFpof\nsimply connected type and λbe a dominant weight. We will say that λis admissible if the dimension of the\nG(Fp)-module V(λ)is at most ℓ4.\nWe shall begin the desired classification of admissible weights by consid eringGof typeBℓ.\n5.1.TypeBℓ.In order to compute the multiplicities involved in this section, we start by considering\n{α1,α2,···,αℓ}the simple roots system (of type Bℓ) given by\nα1:=eℓandαi:=eℓ−i+1−eℓ−i+2,2≤i≤ℓ,\nwhere{e1,e2,···,eℓ}is an orthogonal basis for E∼=Rℓ. We then note that ( α1,α1) = 1, (α1,α2) =−1,\nand fori,j >1 withi/ne}ationslash=jwe see that\n(5.1) ( αi,αj) =\n\n2,ifi=j\n−1,ifi=j±1\n0,otherwise .\nThen the fundamental weights λi(1≤i≤ℓ), such that ( λi,αj) =δij, are given by\nλ1= (1/2)(αℓ+2αℓ−1+···+ℓα1)\nλi=αℓ+2αℓ−1+···+(ℓ−i)αi+1+(ℓ−i+1)(αi+αi−1+···+α2+α1),1< i < ℓ\nλℓ=αℓ+αℓ−1+···+α2+α1.\nIn particular, we have that ( λi,λi) =ℓ−i+1.In addition, the half sum of the positive roots is\nδ=1\n2ℓ/summationdisplay\ni=1(ℓ−i+1)(ℓ+i−1)αi.\nIt is easy to verify\n(αi,δ) =/braceleftBigg\n1\n2,ifi= 1\n1,otherwise .\nLet us start with the first example of admissibility for this type.\nExample 5.2. For allℓ≥12, the weight λ= 4λℓis admissible. To see that, it suffices to show that\ndimSym4(V)is less or equal to ℓ4sinceV(4λℓ)⊆Sym4(V). In fact, we observe that\ndimSym4(V) =(2ℓ+1)(2ℓ+2)(2ℓ+3)(2ℓ+4)\n24\n=(2ℓ+1)(ℓ+1)(2ℓ+3)(ℓ+2)\n6\n=ℓ4·(2+1/ℓ)(1+1/ℓ)(2+3/ℓ)(1+2/ℓ)\n6\n≤ℓ4,\nby verifying the inequality\n(2+1/ℓ)(1+1/ℓ)(2+3/ℓ)(1+2/ℓ)≤(2+1/12)(1+1 /12)(2+3 /12)(1+2 /12)≤6,\nfor anyℓ≥12. Thusλ= 4λℓis an admissible weight.\n11To find all the admissible weights λand compute dimensions of certain G-modules V(λ), we will need the\nfollowing lemmas.\nLemma 5.3. For allℓ≥12, the multiplicities of all dominant weights µ/pre⌋edesequaliλℓ,i= 3,4, are given in the\nfollowing table:\nhighest weight dominant weight µ/pre⌋edesequalλmultiplicity of µ\n3λℓ 3λℓ 1\nλℓ−1+λℓ 1\nλℓ−2 1\n2λℓ 1\nλℓ−1 1\nλℓ ℓ\n0 ℓ\n4λℓ 4λℓ 1\nλℓ−1+2λℓ 1\n2λℓ−1 1\nλℓ−2+λℓ 1\nλℓ−3 1\n3λℓ 1\nλℓ−1+λℓ 1\nλℓ−2 1\n2λℓ ℓ\nλℓ−1 ℓ\nλℓ ℓ\n0 ℓ(ℓ−1)/2\nProof.Letλ= 3λℓ. We shall compute those multiplicities mλ(µ) where 2 λℓ≺µ/pre⌋edesequal3λℓ. Sinceµ:=λℓ−1+\nλℓ=λ−αℓwe have that ( λ,λ) = 9 and ( µ,µ) = 5. In addition as ( αℓ,δ) = 1 and ( λ,αℓ) = 3(λℓ,αℓ) = 1,\nwe have\nmλ(µ) = 2·mλ(µ+αℓ)(µ+αℓ,αℓ)\n(λ,λ)−(µ,µ)+2(αℓ,δ)= 2·1·3\n9−5+2·1= 1.\nNext, we compute the multiplicity of λℓ−2=λ−(αℓ−1+2αℓ). To do this, we observe that the positive\nroots involved in Fruedenthal formula are αℓ−1,αℓ, 2αℓandαℓ−1+αℓ. Notice, at this point, that λis not\nincluded in the formula since µ+αℓ−1+2αℓis not of the form µ+iαfor some positive root αby appealing\nto [3, Plate II].\nSetµ:=λ−(αℓ−1+αℓ).\n(1) Setµℓ−1:=µ+αℓ−1=λ−αℓ. We first see that mλ(µℓ−1+αℓ) = 1 and ( µℓ+αℓ,αℓ) = 9. In\naddition, we already have ( λ,λ) = 9, and we verify that ( µℓ−1,µℓ−1) = 5. Thus, we conclude\nmλ(µℓ−1) = 2·mλ(µℓ−1+αℓ)(µℓ−1+αℓ,αℓ)\n(λ,λ)−(µℓ−1,µℓ−1)+2(αℓ,δ)= 2·1·3\n9−5+2= 1.\n(2) Forµℓ:=µ+αℓ=λ−αℓ−1, we can conclude that mλ(µℓ) = 0 since ( µℓ+αℓ−1,αℓ−1) = 0.\nHence, by verifying that ( µ+(αℓ−1+αℓ),αℓ−1+αℓ) = 3, (µ,µ) = 5 and ( αℓ−1+αℓ,δ) = 2\nmλ(µ) = 2·mλ(µℓ−1+αℓ−1)(µℓ−1+αℓ−1,αℓ−1)+mλ(µℓ+αℓ)(µℓ+αℓ,αℓ)+mλ(λ)(λℓ,αℓ−1+αℓ)\n(λ,λ)−(λℓ−2,λℓ−2)+2(αℓ−1+αℓ,δ)= 1\n= 2·1·1+0+1 ·3\n9−5+4= 1.\nWe continue with the computing of mλ(λ−2αℓ). To this end we may deduce as above that mλ(λ−αℓ) = 1\nand next we shall compute the multiplicity mλ(µ+2αℓ). We can verifythat ( µ+αℓ,αℓ) = 1, (µ+2αℓ,αℓ) = 3\n12and (µ,µ) = 5. Thus, we have\nmλ(λ−2αℓ) = 2·mλ(λ−αℓ)(λ−αℓ,αℓ)+mλ(λ)(λ,αℓ)\n(λ,λ)−(µ,µ)+2(2αℓ,δ)\n= 2·1·1+1·3\n9−5+4= 1.\nGoing back to computing of mλ(µ) withµ=λℓ−2=λ−(αℓ−1+2αℓ) we have\nmλ(µ) = 2·(∗)\n(λ,λ)−(µ,µ)+2(αℓ−1+2αℓ,δ)\nwhere\n(∗) = (µ+αℓ,αℓ)mλ(µ+αℓ)+(µ+αℓ−1)mλ(µ+αℓ−1,αℓ−1)\n+(µ+2αℓ,αℓ)mλ(µ+2αℓ)+(µ+αℓ−1+αℓ,αℓ−1+αℓ)mλ(µ+αℓ−1+αℓ)\n= (µ+αℓ,αℓ)mλ(λ−(αℓ−1+αℓ))+(µ+αℓ−1,αℓ−1)mλ(λ−2αℓ)\n+(µ+2αℓ,αℓ)mλ(λ−αℓ−1)+(µ+αℓ−1+αℓ,αℓ−1+αℓ)mλ(λ−αℓ).\nComputing as above we see that\n(λ,λ)−(µ,µ)+2(αℓ−1+2αℓ,δ) = 9−3+6 = 12 .\nNow, by using the multiplicities calculated above and computing the inne r products, we have\nmλ(µ) =2[1·2+1·2+0·4+1·2]\n12= 1.\nThe remaining multiplicities mλ(µ), withµ/pre⌋edesequal2λwere compute in [ 11].\nFinally, the case λ= 4λℓislongerbut completelysimilarthan the oneabovebycomputing the mu ltiplicity\nofλℓ−1+2λℓas first step. /square\nLemma 5.4. For allℓ≥12, the multiplicities of all dominant weights µ/pre⌋edesequalλℓ−i,i= 2,3, are given in the\nfollowing table:\nhighest weight dominant weight µ/pre⌋edesequalλmultiplicity of µ\nλℓ−2 λℓ−2 1\nλℓ−1 1\nλℓ ℓ−1\n0 ℓ\nλℓ−3 λℓ−3 1\nλℓ−2 1\nλℓ−1 ℓ−1\nλℓ ℓ\n0 ℓ(ℓ+1)/2\nProof.We shall present the computations for i= 2 and we may remark that the case i= 3 follows very\nsimilarly by resorting to the analogous arguments. So, we put λ=λℓ−2andµ=λℓ−1. Since µ=\nλ−(α1+α2+···+αℓ−2) we have to consider in the Freudenthal formula the positive roots of the form\nαi+αi+1+···+αj,1≤i≤j≤ℓ−2.\nIndeed, the formula for mλ(µ) involves the multiplicities of µ+(αi+···+αj) for any 1 ≤i≤j≤ℓ−2. We\nthen observe\nmλ(µ) = 2·/summationdisplay\n1≤i≤j≤ℓ−2mλ(µ+(αi+···+αj))(µ+(αi+···+αj),αi+···+αj)\n(λ,λ)−(µ,µ)+2(/summationtextℓ−2\nk=1αk,δ).\n13First, by using ( λℓ−2,αk) =δ(ℓ−2)k, (α1,α1) = 1 and ( αk,αk+1) =−1 we deduce that\n(λ,λ)−(µ,µ) = (λℓ−2,λℓ−2)−/parenleftBigg\nλℓ−2−ℓ−2/summationdisplay\nk=1αk,λℓ−2−ℓ−2/summationdisplay\nk=1αk/parenrightBigg\n= 2/parenleftBigg\nλℓ−2,ℓ−2/summationdisplay\nk=1αk/parenrightBigg\n−/parenleftBiggℓ−2/summationdisplay\nk=1αk,ℓ−2/summationdisplay\nk=1αk/parenrightBigg\n= 2(λℓ−2,αℓ−2)−(α1,α1)−ℓ−2/summationdisplay\nk=2(αk,αk)−2ℓ−3/summationdisplay\nk=1(αk,αk+1)\n= 2−1−2(ℓ−3)+2(ℓ−3) = 1.\nThus\n(λ,λ)−(µ,µ)+2(ℓ−2/summationdisplay\nk=1αk,δ) = 1+2(1 /2+(ℓ−3)) = 2(ℓ−2).\nNow, we claim that the positive roots α1+α2+···+αk, with 1 ≤k≤ℓ−2, are the only ones that\ncontribute in the computation of mλ(µ). Indeed, we can show that mλ(µ+ (αi+···+αj)) = 0, for any\n2≤i≤j≤ℓ−2. To see that, we directly verify that mλ(λ−αk) = 0 for any 1 ≤k≤ℓ−3 which implies\ninductively that mλ(λ−(α1+···+αi−1)) = 0,i.e,we prove the case j=ℓ−2. Now, by assuming now that\nj < ℓ−2 we have µ+(αi+···+αj) =λ−(α1+···+αi−1)+(αj+1+···+αℓ−2).\nThen we can note that the kind of positive roots that contribute in t he Fruedental formula to compute\nthe foregoing multiplicity are of the form αi′+···+αj′andαj, withi≤i′≤j′≤j, so we may deduce\nthatmλ(λ−(α1+···+αi−1+αj)) = 0 for any i+1≤j≤ℓ−2. By combining these multiplicities we can\nconclude that mλ(λ−(α1+···+αi−1)−(αj+1+···+αt)) = 0, for j+1≤t≤ℓ−2. In summary, we have\nproved that\n(5.2) mλ(µ+(αi+···+αj)) = 0,2≤i≤j < ℓ−2.\nNext by proving\n(5.3) mλ(µ+(α1+···+αk)) = 1,1≤k < ℓ−2,\nwe can deduce\nmλ(µ) = 2·/summationdisplay\n1≤i≤j≤ℓ−2mλ(µ+(αi+···+αj))(µ+(αi+···+αj),αi+···+αj)\n(λ,λ)−(µ,µ)+2(/summationtextℓ−2\nk=1αk,δ)\n= 2·ℓ−2/summationdisplay\nk=0mλ(λℓ−2−(αk+1+···+αℓ−2))(λ−(αk+1+···+αℓ−2),α1+···+αk)\n2(ℓ−2)\n= 2·ℓ−2\n2(ℓ−2)= 1.\nRegarding the proof of the equation ( 5.3) we proceed by induction. We begin by verifying\nmλ(µ+(α1+···+αℓ−4)) =mλ(λ−(αℓ−3+αℓ−2)) = 1\nthanks to mλ(λ−αℓ−2) = 1,mλ(λ−αℓ−3) = 0 and mλ(λ) = 1. Now we assume by induction that\nmλ(µ+(α1+···+αk−1)) =mλ(λ−(αk+···+αℓ−2)) = 1,1≤k < ℓ−2,\nand we write µk=λ−(αk−1+αk+···+αℓ−2).\nBy arguing as above and thanks to ( 5.2) only the weights of the form µk+(αk−1+αk+···+αj) will\ncontribute in the Fruedenthal formula of mλ(λ−(αk−1+αk+···+αℓ−2)). So, we observe\n(µk+(αk−1+···+αj),αk−1+···+αj) = (λ−(αj+1+···+αℓ−2),αk−1+···+αj) =−(αj+1,αj) = 1.\n14Thus,\nmλ(µj) = 2·ℓ−2/summationdisplay\nk=jmλ(µj+(αj+···+αk))(µj+(αj+···+αk),αj+···+αk)\n(λ,λ)−(µj,µj)−2(/summationtextℓ−2\nk=jαk,δ)\n= 2·ℓ−2/summationdisplay\nk=jmλ(λ−(αk+1+···+αℓ−2))(λ−(αk+1+···+αℓ−2),αj+···+αk)\n(λ,λ)−(µj,µj)−2(/summationtextℓ−2\nk=jαk,δ)\n= 2·ℓ−j−1\n2(ℓ−j−1)= 1.\nThis completes the computation of mλ(λℓ−1) = 1.\nNext we shall compute mλ(λℓ). By observing\nµ:=λℓ=λℓ−2−ℓ−2/summationdisplay\ni=1αi−ℓ−1/summationdisplay\ni=1αi\nwe can infer that the only terms nonzero in the Freudenthal formu la depend on the following positive roots:\nα=ℓ−2/summationdisplay\ni=1αi+ℓ−1/summationdisplay\ni=1αi,\nα=α1+···+αk,1≤k≤ℓ−1,\nα=αk+···+αℓ−1,2≤k≤ℓ−1,\nαk,j= 2(α1+···+αk)+(αk+1+···+αj),1≤k≤ℓ−3 andℓ−2≤j≤ℓ−1.\nAt this point, we need to remark that mλ(µ+2(α1+···+αk))(µ+2(α1+···+αk),(α1+···+αk)) also\ncontributes in the Freudenthal formula of mλ(µ), when 1 ≤k≤ℓ−3. According to the terms associated to\nthe above positive roots in the foregoing formula, we first compute that\nmλ/parenleftBigg\nµ+ℓ−2/summationdisplay\ni=1αi+ℓ−1/summationdisplay\ni=1αi/parenrightBigg\n·/parenleftBigg\nµ+ℓ−2/summationdisplay\ni=1αi+ℓ−1/summationdisplay\ni=1αi,ℓ−2/summationdisplay\ni=1αi+ℓ−1/summationdisplay\ni=1αi/parenrightBigg\n=mλ(λ)/parenleftBigg\nλ,ℓ−2/summationdisplay\ni=1αi+ℓ−1/summationdisplay\ni=1αi/parenrightBigg\n= 1·2.\nSupposing that αk=α1+···+αk, where 1 ≤k≤ℓ−1, we directly verify that ( µ+αk,αk) = 1. Now\nwe shall prove that mλ(µ+αk) = 1. Let us start by noticing that the multiplicities for k=ℓ−i, i= 1,2\nwere computed in the above case, thus we have to compute mλ(µ+αk) fork≤ℓ−3. For the general case,\nthe Freudenthal formula involves computing many terms of the for mmλ(µk+α))(µk+α,α) for appropriate\npositive roots αwe start by setting µk=µ+αk. Each of them implies in its turn to compute many other\nmultiplicities. Nevertheless, we introduce a case sufficiently illustrativ e of these computations, say mλ(µℓ−3).\nFirstly, we see that\nµℓ−3=λℓ−2−(α1+···+αℓ−3+2αℓ−2+αℓ−1).\nImmediately, we calculate\n(λ,λ)−(µℓ−3,µℓ−3)+2(αℓ−1,δ) = 3−3+2ℓ= 2ℓ.\nNow by appealing to similar arguments and strategies as in the comput ing ofmλℓ−2(λℓ−1) we may deduce\nthe only positive roots contributing to the formula to compute mλ(µℓ−3) are\nα1+···+αj,1≤j≤ℓ−1.\nHere we also argue as in mλ(µℓ−1) we may obtain\nmλ(µ+α1+···+αk)(µ+α1+···+αℓ−1,α1+···+αk) = 1·1,1≤k≤ℓ−1, k/ne}ationslash=ℓ−3,\nmλ(µ+α1+···+αℓ−3)(µ+α1+···+αℓ−1,α1+···+αℓ−3) = 1·2.\n15Therefore, we get\nmλ(µ) =2[(ℓ−2)·1+1·2]\n2ℓ= 1.\nRegarding the term mλ(µ+2αk)(µ+2αk,αk), for 1≤k≤ℓ−3, we can proceed similarly as above to\ndeduce each multiplicity mλ(µ+2αk). At this point, routine computations allow to calculate the remaining\ncomponents to obtain the multiplicities.\nFinally, suppose that αk,j= 2(α1+···+αk)+(αk+1+···+αj), where 1 ≤k≤ℓ−3 andℓ−2≤j≤ℓ−1.\nWe compute ( µ+αk,j,αk,j) as follows:\n•Forj=ℓ−1 we have\n(µ+αk,ℓ−1,αk,ℓ−1) = (λℓ−2−(αk+1+···+αℓ−2),2(α1+···+αk)+(αk+1+···+αℓ−1))\n= (λℓ−2,αℓ−2)−2(αk+1,αk)−(αk+1,αk+1)−(αℓ−2,αℓ−1)\n= 2.\nLikewise we verify that ( µ+αk,ℓ−2,αk,ℓ−2) = 2.\n•Whenk≤ℓ−3 we get\n(µ+αk,ℓ−1,αk,ℓ−1) = (λℓ−2−(αk+1+···+αℓ−2),2(α1+···+αk)+(αk+1+···+αℓ−1))\n= (λℓ−2,αℓ−2)−2(αk+1,αk)−(αk+1,αk+1)−(αℓ−2,αℓ−1)\n= 2,\nand identically we get that ( µ+αk,ℓ−2,αk,ℓ−2) = 2.\nOn the other hand, we may resort as above to determine the only po sitive roots αinvolving in the formula\nofmλ(µ+αk,j) we recursively deduce that mλ(λℓ−2−(αk+1+···+αℓ−2)) = 1 as in the previous case. Since\n1≤k≤ℓ−3 andℓ−2≤j≤ℓ−1 it follows that the number of positive roots αk,jis\n1·2+2·2+···+2·(ℓ−3) = 2·(ℓ−2)(ℓ−3)/2 = (ℓ−2)(ℓ−3).\nIn summary, the computations from the cases above provide resp ectively the following summands to the\nformula of mλ(µ)\n1·2,(ℓ−1)·1·1,(ℓ−3)·1·2,(ℓ−2)·1·2,1·2(ℓ−2)(ℓ−3)\nThen we can compute\nmλ(µ) =2[2·1+(ℓ−1)·1+2(ℓ−2)+2(ℓ−3)+2(ℓ−2)(ℓ−3)]\n2(2ℓ−3)\n=2(ℓ−1)(2ℓ−3)\n2(2ℓ−3)\n=ℓ−1.\nTo obtain the multiplicity mλ(0), we observe that\nµ= 0 =λ−/parenleftBiggℓ−2/summationdisplay\ni=2αℓ−i+ℓ−1/summationdisplay\ni=1αℓ−i+ℓ/summationdisplay\ni=0αℓ−i/parenrightBigg\n.\nThis writing allows to apply the same strategy as above to determine a ll terms involved in the formula of\nmλ(0) and thereby to conclude the desired multiplicity.\nThe case λ= 4λℓis only longer, but completely similar than the ones considered above. The natural first\nstep is to compute the multiplicity of λℓ−1+2λℓ. /square\nLemma 5.5. For allℓ≥12, the multiplicities of all dominant weights µ/pre⌋edesequalλℓ−1+λℓare given in the\nfollowing table:\n16highest weight dominant weight µ/pre⌋edesequalλmultiplicity of µ\nλ=λℓ−1+λℓλℓ−1+λℓ 1\nλℓ−2 2\n2λℓ 1\nλℓ−1 2\nλℓ 2ℓ−1\n0 2 ℓ−1\nProof.We shall start by computing the multiplicity of λℓ−2=λ−(αℓ−1+αℓ). To do this, we can verify as\nabove that\nmλ(λ−αℓ) = 2·mλ((λ−αℓ)+αℓ)((λ−αℓ)+αℓ,α��)\n(λ,λ)−(µℓ,µℓ)+2(αℓ,δ)= 2·1·1\n5−5+2·1= 1.\nand similarly we check that mλ(λ−αℓ−1) = 1. Thus, we may deduce that the multiplicity mλ(λℓ−2) = 2 by\ncomputing as follows\nmλ(λℓ−2) = 2·mλ(µℓ−1+αℓ−1)(µℓ−1+αℓ−1,αℓ−1)+mλ(µℓ−1+αℓ)(µℓ−1+αℓ,αℓ)+mλ(λ)(λ,αℓ−1+αℓ)\n(λ,λ)−(λℓ−2,λℓ−2)+2(αℓ−1+αℓ,δ)\n= 2·1·2+1·2+1·2\n5−3+4= 2.\nIn order to compute mλ(2λℓ) we see that\nµ:= 2λℓ=λ−ℓ−1/summationdisplay\nk=1αk.\nA direct computation shows\n(λ,λ)−(µ,µ)+2(ℓ−1/summationdisplay\nk=1αk,δ) = 5−4+2(1/2+(ℓ−2)) = 2(ℓ−1).\nBy arguing as in proof of Lemma 5.3, we may infer that the only required multiplicities to compute\nmλ(2λℓ) aremλ(µ+(α1+···+αk)), for 1 ≤k≤ℓ−2, andmλ(λ) itself. Inductively, as Lemma 5.4, we\nmay check that mλ(µ+(α1+···+αk)) = 1 for any 1 ≤k≤ℓ−2. In addition, we can also verify in this\ncase that\n(µ+(α1+···+αk),α1+···+αk) = (λ−(αk+1+···+αℓ−1),α1+···+αk) =−(αk+1,αk) = 1.\nfork≤ℓ−2. Thus, we deduce\nmλ(µ) = 2·ℓ−1/summationdisplay\nk=1mλ(µ+(α1+···+αk))(µ+(α1+···+αk),α1+···+αk)\n(λ,λ)−(µ,µ)−2(/summationtextℓ−1\nk=1αk,δ)\n= 2·ℓ−1/summationdisplay\nk=1mλ(λ−(αk+1+···+αℓ−2))(λ−(αk+1+···+αℓ−2),α1+···+αk)\n2(ℓ−1)\n= 2·ℓ−1\n2(ℓ−1)= 1.\nWe now start the computations of mλ(λℓ−1). We first note that\nµ:=λℓ−1=λ−ℓ/summationdisplay\nk=1αk.\nWe have\n(λ,λ)−(µ,µ)+2(ℓ/summationdisplay\nk=1αk,δ) = 5−2+2(1/2+(ℓ−1)) = 2(ℓ+1).\n17In addition we already know that mλ(λ−(αℓ−1+αℓ)) = 2 and mλ(λ−αℓ) = 1 - by calculations above.\nThus, we see that mλ(λ)(λ,α1+···+αℓ) = 1·2. Indeed, we may argue as in Lemma 5.3to deduce\nmλ(λ−(αk+1+···+αℓ))(λ−(αk+1+···+αℓ),α1+···+αk) = 2·1,\nfor any 1 ≤k≤ℓ−2, andmλ(λ)(λ,α1+···+αℓ) = 1·2. In the same way, we can verify\nmλ(λ−(α1+···+αℓ−1))(λ−(α1+···+αℓ−1),αℓ) = 1·2.\nTherefore, by noting that the foregoing terms are the only nonze ro summands in the Fruedenthal formula\nas in Lemma 5.4we conclude that mλ(λℓ−1) = 2.\nWhile the multiplicity mλ(λℓ) may be computed by means of the arguments used in Lemma 5.4observing\nthat\nµ:=λℓ=λ−ℓ−1/summationdisplay\nk=1αk−ℓ/summationdisplay\nk=1αk=λ−2ℓ−1/summationdisplay\nk=1αk−αℓ\nand\n(λ,λ)−(µ,µ)+2(2ℓ−1/summationdisplay\nk=1αk+αℓ,δ) = 5−1+4(ℓ−1) = 4ℓ.\nIdem for mλ(0) taking in mind\nµ:= 0 =λ−ℓ−1/summationdisplay\nk=1αk−2ℓ/summationdisplay\nk=1αk= 3(2ℓ−1).\nand\n(λ,λ)−(µ,µ)+2(ℓ−1/summationdisplay\nk=1αk+2ℓ/summationdisplay\nk=1αk,δ) = 5−0+3(2ℓ−1)) = 2(3 ℓ+1).\n/square\nThus we can prove.\nProposition 5.6. LetGbe as above of type Bℓwithℓ≥12. Then, all weights in the following table are\nadmissible.\nλ dimension of V(λ)\n0 1\nλℓ 2ℓ+1\n2λℓ ℓ(2ℓ+3)−1\n3λℓ ℓ(2ℓ+1)(2ℓ+5)/3\n4λℓ ℓ(ℓ+1)(2ℓ+1)(2ℓ+7)/6\nλℓ−1 2ℓ2+ℓ\nλℓ−2 (2ℓ−1)2ℓ(2ℓ+1)/6\nλℓ−3(2ℓ−2)(2ℓ−1)2ℓ(2ℓ+1)/24\nλℓ+λℓ−1(2ℓ+3)(2ℓ+1)(2ℓ−1)/3\nIn addition, when 12≤ℓ≤15the table has to include the weight λ1whose associated irreducible repre-\nsentation V(λ1)has dimension 2ℓ.\nProof.Suppose that ℓ≥12. The dimensions of the weight spaces associated to the highest w eights 0, λℓ,\nand 2λℓ, are computed in Table 2 of [ 11]. We next shall compute the dimension of V(3λℓ), the dimension of\nV(4λℓ) follows by means of the same computations. To this end and accord ing to Proposition 2.1we have to\ncompute l(µ), for each dominant weight µ/pre⌋edesequal3λℓgiven in Lemma 5.3. We first observe that the stabilizers\nof the weights µi=iλℓ, fori= 1,2,3, are reflection groups of type Bℓ−1, and hence l(µi) = 2ℓ. On the\nother hand, we also see that the stabilizers of λℓ−1+λℓ,λℓ−1andλℓ−2are reflection groups of type Bℓ−2,\nBℓ−2×A1andBℓ−3×A2, so the length of its orbits are 4 ℓ(ℓ−1), 2ℓ(ℓ−1), 4ℓ(ℓ−1)(ℓ−2)/3 respectively\nand clearly l(0) = 1. Then, by resorting to the multiplicities given in Lemma 5.3and Proposition 2.1we get\ndimV(3λℓ) = 2ℓ+4ℓ(ℓ−1)+4ℓ(ℓ−1)(ℓ−2)/3+2ℓ+2ℓ(ℓ−1)+2ℓ2+ℓ\n=ℓ(2ℓ+1)(2ℓ+5)/3.\n18Nowwe compute the dimension of V(λℓ−3) since the case of V(λℓ−2) followsin the sameway, evenshorter,\nandV(λℓ−1) was computed in [ 11]. Certainly dim V(λℓ−3) may be calculated as above, but let us give an\nalternative way to deduced it. We first observe that\ndimV(λℓ−3)≥mλℓ−3(λℓ−3)l(λℓ−3)+mλℓ−3(λℓ−2)l(λℓ−2)+mλℓ−3(λℓ−1)l(λℓ−1)+mλℓ−3(λℓ)l(λℓ)+mλℓ−3(0)l(0).\nAs above we see that the stabilizer of λℓ−iis a reflection group of type Bℓ−i−1×Ai, fori= 1,2,3\nand of type Bℓ−2forλℓ. Then, l(µ) is 2ℓ(ℓ−1)(ℓ−2)(ℓ−3)/3,4ℓ(ℓ−1)(ℓ−2)/3,2ℓ(ℓ−1),2ℓ,1 for\nµ=λℓ−3, µ=λℓ−2, µ=λℓ−1, µ=λℓ, µ= 0, respectively. Thus, by using the multiplicities given by\nLemma5.3we see that\ndimV(λℓ−3)≥2ℓ(ℓ−1)(ℓ−2)(ℓ−3)/3+4ℓ(ℓ−1)(ℓ−2)/3+2ℓ(ℓ−1)(ℓ−2)+2ℓ(ℓ−1)+ℓ(ℓ−1)/2\n= (2ℓ−2)(2ℓ−1)2ℓ(2ℓ+1)\n= dimΛ4(V).\nBy combining the last inequality with V(λℓ−3)⊆Λ4(V) we conclude that V(λℓ−3) is equal to Λ4(V).\nOn the other hand, we can deduce the dimension of V(λℓ−1+λℓ) by appealing Lemma 5.5and arguing\nas inV(3λℓ).\nFinally, for λ=λ1and 12≤ℓ≤15 we infer that there not exist µ/pre⌋edesequalλexceptλitself. In addition, we\ndirectly verify that Orb W(λ) = 2ℓ. This then says that dim V(λ) = 2ℓ< ℓ4forℓin the mentioned rank, i.e.,\nλis admissible. /square\nRemark 5.7. LetGbe as above of type BℓandVbe the irreducible G-module of dimension 2ℓ+1. Then,\nthe anti-symmetric tensor product Λi(V)is irreducible, for i= 1,2,3,4.\nIn order to prove that the weights in the previous proposition are t he only admissible weights of type\nBℓ, we will prove the following combinatorial lemma which will be very usefu l below. This result is perhaps\nwell-known to the experts, but we give a proof for lack of a referen ce.\nLemma 5.8. The coefficients of\n(2+x)n=n/summationdisplay\nk=0/parenleftbiggn\nk/parenrightbigg\n2n−kxk\nforn≥12are strictly increasing until k=⌊n\n3⌋and then they are strictly decreasing.\nProof.Firstly we put tk:= 2n−k/parenleftbign\nk/parenrightbig\nfor each integer 0 ≤k≤n. Then\ntk:= 2n−k/parenleftbiggn\nk/parenrightbigg\n= 2·2n−(k+1)n(n−1)···(n−(k+1))\nk!.\nWe begin by observing t0= 2n−1·2<2n−1·n=t1because 2 < n. We also note that\nt1= 2n−1n= 2n−2n·2<2n−2n·n−1\n2=t2,\nsince 2<(n−1)/2. Thus, t0< t1< t2. In general, if k <⌊n\n3⌋we may infer that 2 k <(n−(k+ 1)). It\nworks for n≥12 and proceed as above to show tk< tk+1. We see that the worst case is when k=⌊n\n3⌋−1.\nIn such case, the inequality follows from\nn−/parenleftBig/floorleftBign\n3/floorrightBig\n−1/parenrightBig\n≥n−n\n3+1 = 2n\n3+1≥2/floorleftBign\n3/floorrightBig\n.\n19We then deduce\nt⌊n\n3⌋−1= 2n−⌊n\n3⌋+1/parenleftbiggn\n⌊n\n3⌋−1/parenrightbigg\n= 2n−⌊n\n3⌋+1 n!\n(⌊n\n3⌋−1)!(n−⌊n\n3⌋+1)!\n= 2n−⌊n\n3⌋+1 n!\n(⌊n\n3⌋−1)!(n−⌊n\n3⌋)!(n−⌊n\n3⌋+1)\n= 2n−⌊n\n3⌋ n!\n(⌊n\n3⌋−1)!(n−⌊n\n3⌋)!·2\n(n−⌊n\n3⌋+1)\n<2n−⌊n\n3⌋ n!\n(⌊n\n3⌋−1)!(n−⌊n\n3⌋)!·1\n⌊n\n3⌋\n=t⌊n\n3⌋,\nthanks to the inequality 2 /(n−⌊n\n3⌋+1)<1/⌊n\n3⌋. This proves that the coefficients of (2+ x)nare strictly\nincreasing until k=⌊n\n3⌋.\nOn the other hand, by similar arguments, when k >⌊n\n3⌋we have that 2 ≥n−(k−1)\nk. Then, by observing,\n2n−(k+1)n(n−1)···(n−(k−2))\n(k−1)!·2≥2n−kn(n−1)···(n−(k−2))\n(k−1)!·(n−(k−1))\nk\nwe show that the coefficients of (2+ x)nare strictly decreasing when k >⌊n\n3⌋. This finishes the proof. /square\nNow we are ready to state and give a proof of our main result of this s ection.\nTheorem 5.9. LetGbe a finite group of Lie type BℓoverFp. Ifℓ≥12then the list giving in Proposition\n5.6exhausts the admissible weights.\nProof.We shall proceed case-by-case by arguing as in Theorem 4.1. Start by setting\nλ=a1λ1+a2λ2+···+aℓλℓ.\nNullity of coefficients −λis admissible implies ai= 0 for 1 ≤i≤ℓ−4.\nSuppose first that ai/ne}ationslash= 0, for some 2 ≤i≤ℓ−4. Then, the stabilizer of λis contained in a reflection\ngroup of type Bi−1×Aℓ−i- it is worth mentioning that we are using the convention B1=A1as in [6]. This\nimplies\ndimV(λ)≥ti:=2ℓℓ!\n2i−1(i−1)!(ℓ−i+1)!= 2ℓ−(i−1)/parenleftbiggℓ\ni−1/parenrightbigg\n.\nBy appealing to Lemma 5.8we have that the ( i−1)th-coefficient tiof (2 +x)ℓare strictly increasing for\n2≤i≤ ⌊ℓ\n3⌋+1. Thus, for ℓ≥16 we use the fact that 2ℓ≥ℓ4to deduce\ndim(V(λ))≥ti> t0= 2ℓ≥ℓ4.\nNow, by numerical inspection we also may verify for each 12 ≤ℓ≤15 that\ndimV(λ)≥2ℓ−(i−1)/parenleftbiggℓ\ni−1/parenrightbigg\n> ℓ4,for 2≤i≤ ⌊ℓ\n3⌋+1.\n20On the other hand, we will demonstrate that dim V(λ)> ℓ4by using the sequence ( ti) is decreasing for\ni >⌊ℓ\n3⌋+1. Indeed, we shall show that tℓ−4> ℓ4forℓ≥12 by observing\ntℓ−4> ℓ4⇐⇒25ℓ(ℓ−1)(ℓ−2)(ℓ−3)(ℓ−4)\n5!> ℓ4\n⇐⇒25(ℓ−1)(ℓ−2)(ℓ−3)(ℓ−4)\n5!> ℓ3\n⇐⇒25ℓ3(1−1/ℓ)(1−2/ℓ)(1−3/ℓ)(ℓ−4)\n2·3·4·5> ℓ3\n⇐⇒22(1−1/ℓ)(1−2/ℓ)(1−3/ℓ)(ℓ−4)\n3·5>1.\nSinceℓ≥12 we can infer that 1 −j/ℓ≥(12−j)/12 for 1≤j≤3, and (ℓ−4)/15≥8/15. By combining\nthese facts we can infer\n22(1−1/ℓ)(1−2/ℓ)(1−3/ℓ)(ℓ−4)\n3·5= 22(1−1/ℓ)(1−2/ℓ)(1−3/ℓ)(ℓ−4)/15\n≥22·11\n12·10\n12·9\n12·8\n15>1.\nWith the foregoingequivalence and inequality above, we can conclude thattℓ−4> ℓ4forℓ≥12. This implies\nthatti≥tℓ−4> ℓ4, fori≥ ⌊ℓ\n3⌋+ 1, when ℓ≥12, by appealing to the sequence ( ti) is decreasing in this\nrank. We then conclude\ndim(V(λ))≥ti≥tℓ−4> ℓ4,\nTo deal the case i= 1 we proceed as follows. If a1≥2 we deduce that λ′:= (λ−αℓ−1)≺λcan be\nwritten as nonnegative linear combination of fundamental weights c ontains the term λ2. Resorting as above\nwe have that\ndim(V(λ))≥|W|\n|StabW(λ′)|> ℓ4,\ni.e.,λis not admissible when a1≥2.\nBy appealing to the same arguments, a1= 1 implies ai= 0 for any 2 ≤i≤ℓ. Hence we reduce to\nλ=λ1orλ=aℓ−3λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ.\nBy Proposition 5.6we have that λ=λ1is admissible when 12 ≤ℓ≤15 and not otherwise. Next we consider\nλ=aℓ−3λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ. Ifaℓ−3≥2, we produce a dominant weight λ′:=λ−αℓ−3, whose\nwriting as nonnegative linear combination of fundamental weights co ntains the term λℓ−4, but it cannot be\npossible due to the arguments above. Thus, we reduce the analysis to\nλ=λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓorλ=aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ.\nFor the former case, i.e., assuming aℓ−3= 1, we claim that aℓ−j= 0 forj= 0,1,2. In fact, if some of\nthese coefficient were nonzero then the stabilizer Stab W(λ) ofλwere contained in a reflection group of type\nBℓ−4×A1,Bℓ−4×A1×A1orBℓ−4×A2. In any of those cases we can verify that\ndimV(λ)≥|W|\n|StabW(λ)|> ℓ4.\nby arguing as cases above.\nOn the other hand, when λ=aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓwe can prove that aℓ−2<2. Otherwise, we\nobserve that λℓ−3andλℓ−1appear in the writing of λ−αℓ−2, but this case was already covered previously.\nThus, we get\nλ=λℓ−2+aℓ−1λℓ−1+aℓλℓorλ=aℓ−1λℓ−1+aℓλℓ.\nIn both cases we may deduce that λis not admissible if aℓ−i>1 withi= 0 or 1 by arguing as above.\nFor instance, if λ=λℓ−2+aℓ−1λℓ−1+aℓλℓ. withaℓ−1≥2 we can observe that the term 2 λℓ−2appears in\nλ′:=λ−αℓ−1, but such a weight λ′is not admissible by the previous case. It follows that λ=λℓ−2+aℓλℓ\n21orλ=λℓ−2+λℓ−1+aℓλℓ. Moreover, we may proceed very similarly for aℓwhich implies to analyze the\ndimensions of the representations associated to the weights\nλℓ−2+λℓ−1, λℓ−2+λℓandλℓ−2+λℓ−1+λℓ\nForλ:=λℓ−2+λℓ−1, by using the identity,\nαℓ−2+αℓ−1=−λℓ−3+λℓ−2+λℓ−1−λℓ\nwe deduce that µ:=λℓ−3+λℓ≺λℓ−2+λℓ−1. The same calculations as above show that the length of orbit\nofµis greater to ℓ4, and hence dim V(λ)> ℓ4,i.e.,λis not admissible.\nOn the other hand, λ:=λℓ−2+λℓis not admissible by Example 3.4. While for λ:=λℓ−2+λℓ−1+λℓone\nnotices that\n(λℓ−2+λℓ−1+λℓ)−(λℓ−3+2λℓ) =−λℓ−3+λℓ−2+λℓ−1−λℓ=αℓ−2+αℓ−1\nwhich says λℓ−3+2λℓ≺λ. Thus, we have\ndimV(λ)≥l(λℓ−3+2λℓ)\n= 23ℓ(ℓ−1)(ℓ−2)(ℓ−3)/3\n=ℓ4·23(1−1/ℓ)(1−2/ℓ)(1−3/ℓ)/3\n> ℓ4,\nby observing 23(1−1/ℓ)(1−2/ℓ)(1−3/ℓ)/3>1 for any ℓ≥12. This implies that λ=λℓ−2+λℓ−1+λℓis\nnot admissible either.\nFinally, we consider the case λ=aℓλℓ. By appealing to Lemma 5.6, the weights of the form λ=aℓλℓ\nare admissible for aℓ= 1,2,3,4. To complete this case we observe that none λwithaℓ>4 provides a new\nadmissible weight. To see that, it suffices to prove\nλℓ−4≺λℓ−3+λℓ≺5λℓ/pre⌋edesequalλ.\nforcing then dim V(λ)≥l(λℓ−4)> ℓ4.\nTherefore, the Table in Proposition 5.6exhausts the admissible weights which conclude the proof. /square\nFinally, we will deal with the rest of the classic groups Lie type in the ne xt subsection.\n5.2.TypeCℓandDℓ.\nTheorem 5.10. LetGbe a group of type CℓorDℓoverFp. Ifℓ≥12then the lists giving in the table here\nbelow exhaust the admissible weights in each case.\n22typeλ dimension\nCℓ\n0 1\nλℓ 2ℓ\n2λℓ (2ℓ+1)(2ℓ)/2\n3λℓ (2ℓ+2)(2ℓ+1)(2ℓ)/6\n4λℓ (2ℓ+3)(2ℓ+2)(2ℓ+1)(2ℓ)/24\nλℓ−1 2ℓ2−ℓ−1\nλℓ−2 2ℓ(2ℓ−1)(2ℓ−2)/6−2ℓ\nλℓ−3 ((2ℓ−3)(2ℓ−2)(2ℓ−1)2ℓ/24)−((2ℓ+1)(2ℓ)/2)\nλℓ+λℓ−1 (2ℓ+2)(2ℓ)(2ℓ−2)/3\nDℓ\n0 1\nλℓ 2ℓ\n2λℓ (2ℓ+1)(2ℓ)/2−1\n3λℓ (2ℓ+2)(2ℓ+1)(2ℓ)/6−2ℓ\n4λℓ ((2ℓ+3)(2ℓ+2)(2ℓ+1)(2ℓ)/24)−((2ℓ+1)(2ℓ)/2)\nλℓ−1 2ℓ2−ℓ\nλℓ−2 2ℓ(2ℓ−1)(2ℓ−2)/6\nλℓ−3 (2ℓ−3)(2ℓ−2)(2ℓ−1)2ℓ/24\nλℓ+λℓ−1 (2ℓ+2)(2ℓ)(2ℓ−2)/3\nIn addition, when 12≤ℓ≤17andGis of type Dℓ, the second table has to include the weights λ1andλ2\nwhose associated irreducible representation has dimensio n2ℓ−1.\nProof.The tables in Appendix contain all the information to calculate directly the dimensions of the weights\nspacesV(λ). Thus, we may deduce that the weights in the tables above are effe ctively admissible. Next we\nshall prove that the list is exhausted respect to admissible ones. To this end, let\nλ=a1λ1+a2λ2+···+aℓλℓ\nbe a dominant weight and assume that Gis of type Cℓ. By arguing as in proof of Theorem 5.9(nullity of\ncoefficients) we obtain that λadmissible implies\nλ=λ1orλ=aℓ−3λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ\nOne here finds a little difference in determining the admissible weights wit h the case Bℓ, forℓ=\n12,13,14,15. Explicitly, λ=λ1is not admissible by noticing first λ3≺λand that\ndim(V(λ)≥ |OrbW(λ3)|= 2ℓ−3ℓ(ℓ−1)(ℓ−2)> ℓ4.\nThus\nλ=aℓ−3λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+a1λℓ.\nFrom this point, the rest of the route of Bℓworks perfectly for the current case Cℓby resorting to parallel\narguments and Appendix -Table 1.\nNow, assume that Gis of type Dℓ. By arguing as in proof of Theorem 5.9too, we obtain that λadmissible\nimplies\nλ=λ1, λ=λ2orλ=aℓ−3λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓ\nContrary to the previous case, here, the weights λ=λ1,λ=λ2are admissible for 12 ≤ℓ≤17 since\nthe dimension of V(λi) is 2ℓ−1fori= 1,2. On the other hand, the dominant weights of the form λ=\naℓ−3λℓ−3+aℓ−2λℓ−2+aℓ−1λℓ−1+aℓλℓcan be addressed as above by using Appendix -Table 2. /square\nRemark 5.11. It is worth mentioning that:\n•IfGis of type CℓandVthe irreducible G-module of dimension 2ℓ. Then, the i-symmetric power\nSymi(V)is irreducible and the anti-symmetric tensor product Λi(V)is reducible for i= 2,3,4.\n•In contrast with the case Cℓ, whenGis of type Dℓwe may observe that Symi(V)is reducible and\nΛi(V)is irreducible for i= 2,3,4.\n236.Some applications\nLetρp:GQ→GOn(Qp)⊆GLn(Qp) be ap-adic orthogonal Galois representation. Examples of this\nkind of Galois representations can be obtained from the so-called re gular algebraic, polarizable, cuspidal\nautomorphic representations of GL n(AQ), via the global Langlands correspondence [ 2,§2.1].\nIn this case, the largeimageconjecture predictsthat, foralmost allprimes p, the imageofthe proyectiviza-\ntionsρproj\np:GQ→PGOn(Fp) of the residual representations ρp:GQ→GOn(Fp) should contains PΩ±\nn(Fps)\nfor some positive integer s. As we pointed out in the introduction, this conjecture is known in ve ry few\ncases. However, if we impose certain conditions on the local compon ents ofπ, at least we can bound the\npossible images of ρp. For example, in [ 14, Proposition 3.4], the second author proved the following criterion\nconcerning to the image of certain families of orthogonal Galois repr esentations.\nProposition 6.1. LetR={ρp}pbe a family of p-adic orthogonal Galois representations ρp:GQ→\nGOn(Qp)as in[14, Theorem 2] . Then, for almost all primes p, the image of ρproj\npis contained in an almost\nsimple group with socle a finite simple group of Lie type in cha racteristic p.\nAs observed in [ 14, Section 5], if we only had a best knowledge of the classification of irre ducible represen-\ntations of finite simple groups of Lie type, we could improve Propositio n6.1. In particular, the main result\nof [14] (Theorem 1.1), states that for almost all primes pandn= 4̟(where̟is a prime number such that\n17≤̟≤73), the image of ρproj\npis equal to one of the following groups:\n(6.1) PΩ±\nn(Fps),PSO±\nn(Fps),PO±\nn(Fps),PGO±\nn(Fps)\nfor some positive integer s. The idea of the proof is just to look at the classification of project ive irreducible\nrepresentations of finite simple groups of Lie type P Gof dimension n≤292, given in [ 11], and prove that\nforpsufficiently large and n= 4̟there are not irreducible representations of dimension nother than those\ncoming from a finite simple group of type D2̟. This improving had an interesting consequence concerning\nto the inverse Galois problem. Explicitly, [ 14, Corollary 1.2] states that at least one of the groups in ( 6.1)\ncan be realized as a Galois group over Qfor infinitely many primes pand infinitely many positive integers s.\nIn the same line of thought, using the strategy in Section 5 of [ 14] combined with Theorems 4.1,5.9and\n5.10, we may prove the following improvement of Proposition 6.1forn= 4̟with̟a prime number such\nthat 79≤̟≤787.\nTheorem 6.2. LetR={ρp}pbe a compatible system of n-dimensional orthogonal Galois representations\nρp:GQ→GOn(Qp)as in[14, Theorem 2] . Ifn= 4̟, where̟is a prime number such that 79≤̟≤787,\nthen for almost all pthe image of ρproj\npis equal to\n(6.2) PΩ±\nn(Fps),PSO±\nn(Fps),PO±\nn(Fps),PGO±\nn(Fps)\nor to a finite group PG(Fps), whereGis of type Aℓ(ℓ≤7),Bℓ,Cℓ,Dℓ(ℓ≤5) orG2, for some positive\nintegers.\nProof.The proof closely follows the proof of [ 14, Theorem 1.1]. Nevertheless, we will write a brief sketch\nof this for completeness. As we pointed out at the Introduction, p rojective representations of P Gin char-\nacteristic pare the same as linear representations of GFin characteristic p, which can be constructed by\nrestricting highest weight representations of GtoGF. Thus, according to the discussion in [ 14,§5] preceding\nthe proof of Theorem 1.1 (which works for any dimension), it is enoug h to prove that the only irreducible\n4̟-dimensional representations ρp:GF→Ω±\n4̟(Fqǫ) that could occur, appear when Gis of type Aℓ(ℓ≤7),\nBℓ,Cℓ,Dℓ(ℓ≤5),G2orDℓ.\nAssume that Gis of a Lie type different from Aℓ(ℓ≤7),Bℓ,Cℓ,Dℓ(ℓ≤5) andG2. In order to\ndetermine all irreducible representations in defining characteristic for groups of a fixed Lie type, we first\nneed to compute all factorizations of ninto factors greater than 1. Such factorizations are 4 ̟= 4·̟=\n2·2·̟= 2·2̟. Appealing to Tables 6.6-6.53 of [ 11] and the assumptions on G, there are no irreducible\nrepresentations of dimension 2 or 4. Thus, there are no tensor de composable representations of dimension\n4̟. Moreover, according to Theorems 4.1,5.9,5.10and Tables 6.6-6.53 of [ 11], the only irreducible tensor\nindecomposable self-dual representations of dimension 4 ̟are the natural representations of C2̟andD2̟.\nHowever,thenaturalrepresentationof C2̟issymplecticthentheonlyirreducible4 ̟-dimensionalorthogonal\nrepresentation occurs when Gis of type D2̟. /square\n24Then, we have the following result:\nCorollary 6.3. For almost all prime pand each nof the form 4̟, where̟is a prime number such that\n79≤̟≤787, we have that at least one of the following groups:\n(6.3) PSL ℓ+1(Fps),PGLℓ+1(Fps),PSUℓ+1(Fps),PGUℓ+1(Fps) 2≤ℓ≤7.\n(6.4) G2(Fps),PΩ2ℓ+1(Fps),PSO2ℓ+1(Fps),PSp2ℓ(Fps),PGSp2ℓ(Fps),2≤ℓ≤5.\n(6.5) PΩ±\n2ℓ(Fps),PSO±\n2ℓ(Fps),PO±\n2ℓ(Fps),PGO±\n2ℓ(Fps),2≤ℓ≤5orℓ= 2̟.\noccurs as a Galois group over Qfor infinitely many primes pand infinitely many positive integers s.\nRemark 6.4. In the previous result, we include the possible occurrence o f (6.3), (6.4), and (6.5), for\n2≤ℓ≤5, as Galois groups of Q. In the future, computational tools could be useful to compu te all the\nrepresentations of these groups for dimensions n≤3148. Moreover, it is worth that the result is new for\n̟ >181while for ̟≤181the foregoing result follows from [11].\nRemark 6.5. Finally we also emphasize that if we assume the computations on the web page of Frack L¨ ubek1\nthe Theorem 6.2works for 2477≤̟ <10000and for3037≤̟ <12155if we include groups of type C6.\nIn this case, the result is new for ̟ >283and for̟≤283follows from L¨ ubek’s computations.\n7.Apendix\nNext we shall present the multiplicities and lengths of the dominants w eightsµ/pre⌋edesequalλ, for each highest\nweightλlisted in the table in Theorem 5.10. We may remark that these tables provide all the information\nto compute the dimensions of V(λ) which implies admissibility of the foregoing weights λ.\nIn what follows we consider orthogonal basis {e1,e2,···,eℓ}ofE∼=Rℓ.\n7.1.TypeCℓ.Start by considering {α1,α2,···,αℓ}the simple root system of type Cℓgiven by\nα1:= 2eℓandαi:=eℓ−i+1−eℓ−i+2,2≤i≤ℓ,\nWe then note that ( α1,α1) = 4, (α1,α2) =−2 and\n(7.1) ( αi,αj) =\n\n2,ifi=j\n−1,ifi=j±1\n0,otherwise ,\nfori,j >1. The fundamental weights λi(1≤i≤ℓ), such that ( λi,αj) =δij, are given by\nλ1=αℓ+2αℓ−1+···+(ℓ−1)α2+ℓ(1/2)α1\nλi=αℓ+2αℓ−1+···+(ℓ−i)αi+1+(ℓ−i+1)(αi+···+(1/2)α1),2< i < ℓ,\nλℓ=αℓ+αℓ−1+···α2+(1/2)α1.\nWe note that ( λi,λi) =ℓ−i+1. The half sum of the positive roots is\nδ=ℓ/summationdisplay\ni=1λi=ℓ/parenleftbiggℓ+1\n2/parenrightbigg\n·1\n2α1+ℓ/summationdisplay\ni=2(ℓ−i+1)/parenleftbiggℓ+i\n2/parenrightbigg\nαi.\nIt is easy to verify that\n(αi,δ) =/braceleftBigg\n2,ifi= 1\n1,otherwise .\nWith the notation above in mind, the table below may be deduced by rou ting each step of the case Bℓ.\n1http://www.math.rwth-aachen.de/ Frank.Luebeck/chev/W MSmall/index.html\n25highest weight dominant weight µ/pre⌋edesequalλmultiplicity of µ length of W-orbit\n3λℓ 3λℓ 1 2 ℓ\nλℓ−1+λℓ 1 4 ℓ(ℓ−1)\nλℓ−2 1 22ℓ(ℓ−1)(ℓ−2)/3\nλℓ ℓ 2ℓ\n4λℓ 4λℓ 1 2 ℓ\nλℓ−1+2λℓ 1 4 ℓ(ℓ−1)\n2λℓ 1 2 ℓ\nλℓ−2+λℓ 1 22ℓ(ℓ−1)(ℓ−2)\nλℓ−3 1 24ℓ(ℓ−1)(ℓ−2)(ℓ−3)/24\n2λℓ ℓ 2ℓ\nλℓ−1 ℓ 2ℓ(ℓ−1)\n0 ℓ(ℓ+1)/2 1\nλℓ−2 λℓ−2 1 23ℓ(ℓ−1)(ℓ−2)/6\nλℓ ℓ−2 2 ℓ\nλℓ−3 λℓ−3 1 24ℓ(ℓ−1)(ℓ−2)(ℓ−3)/24\nλℓ−1 ℓ−3 2 ℓ(ℓ−1)/6\n0 ℓ(ℓ−3)/2 1\nλℓ−1+λℓλℓ−1+λℓ 1 22ℓ(ℓ−1)\nλℓ−2 2 22ℓ(ℓ−1)(ℓ−2)/3\nλℓ 2(ℓ−1) 2 ℓ\nTable 1.\n7.2.TypeDℓ.Start by considering {α1,α2,···,αℓ}the simple root system of type Dℓgiven by\nα1:=eℓ−1+eℓandαi:=eℓ−i+1−eℓ−i+2,2≤i≤ℓ.\nWe then note that ( α1,α1) = 2, (α1,α2) = 0, (α1,α3) =−1 and\n(7.2) ( αi,��j) =\n\n2,ifi=j\n−1,ifi=j±1\n0,otherwise ,\nfori,j >1. The fundamental weights λi(1≤i≤ℓ), such that ( λi,αj) =δij, are given by\nλ1= (1/2)(αℓ+2αℓ−1+···+(ℓ−2)α3+(1/2)(ℓ−2)α2+(1/2)ℓα1)\nλ2= (1/2)(αℓ+2αℓ−1+···+(ℓ−2)α3+(1/2)ℓα2+(1/2)(ℓ−2)α1),\nλi=αℓ+2αℓ−1+···+(ℓ−i)αi+1+(ℓ−i+1)(αi+αi−1+···+α3)+(1/2)(ℓ−i+1)(α2+α1),2< i < ℓ,\nλℓ=αℓ+αℓ−1+···+α3+(1/2)(α2+α1).\nWe observe that ( λi,λi) =ℓ−i+ 1 for 3 ≤i≤ℓand (λ1,λ1) = (λ2,λ2) =ℓ/4. The half sum of the\npositive roots is\nδ=1\n4ℓ(ℓ−1)(α1+α2)+1\n2ℓ/summationdisplay\ni=3(ℓ−i+1)(ℓ+i−2)αi.\nIt is easy to verify that\n(αi,δ) =/braceleftBigg\n2 ifi= 1,2\n1 otherwise .\nWith the notation above in mind, the table below may be deduced as in th e previous subsection.\n26highest weight dominant weight µ/pre⌋edesequalλmultiplicity of µ length of W-orbit\n3λℓ 3λℓ 1 2 ℓ\nλℓ−1+λℓ 1 4 ℓ(ℓ−1)\nλℓ−2 1 22ℓ(ℓ−1)(ℓ−2)/3\nλℓ ℓ−1 2 ℓ\n4λℓ 4λℓ 1 2 ℓ\nλℓ−1+2λℓ 1 4 ℓ(ℓ−1)\n2λℓ−1 1 2 ℓ(ℓ−1)\nλℓ−2+λℓ 1 22ℓ(ℓ−1)(ℓ−2)\nλℓ−3 1 22ℓ(ℓ−1)(ℓ−2)(ℓ−3)/24\n2λℓ ℓ 2ℓ\nλℓ−1 ℓ 2ℓ(ℓ−1)\n0 ℓ(ℓ−1)/2 1\nλℓ−2 λℓ−2 1 23ℓ(ℓ−1)(ℓ−2)/6\nλℓ ℓ−1 2 ℓ\nλℓ−3 λℓ−3 1 24ℓ(ℓ−1)(ℓ−2)(ℓ−3)/24\nλℓ−1 ℓ−2 2 ℓ(ℓ−1)\n0 ℓ(ℓ−1)/2 1\nλℓ−1+λℓλℓ−1+λℓ 1 22ℓ(ℓ−1)\nλℓ−2 2 23ℓ(ℓ−1)(ℓ−2)/6\nλℓ 2(ℓ−1) 2 ℓ\nTable 2.\nReferences\n[1] M. Aschbacher. On the maximal subgroups of the finite classical groups . Invent. Math. 76(1984), 469-514.\n[2] T. Barnet-Lamb, T. Gee, D. Geraghty and R. Taylor, Potential automorphy and change of weight. Ann. of Math. 179\n(2014), 501-609.\n[3] N. Bourbaki. Lie groups and Lie algebras. Chapters 4-6 . Elements of Mathematics. Springer-Verlag, Berlin 2002.\n[4] J. N. Bray, D. F. Holt and C. M. Roney-Douglas. The maximal subgroups of the low-dimensional finite classic al groups .\nWith a foreword by Martin Liebeck. London Mathematical Soci ety Lecture Note Series, 407. Cambridge University Press,\nCambridge, 2013.\n[5] K. Buzzard and T. Gee. The conjectural connections between automorphic represen tations and Galois representations ,\nAutomorphic forms and Galois representations. Vol. 1, Lond on Math. Soc. Lecture Note Ser., 414Cambridge Univ. Press,\nCambridge, 2014, pp. 135-187.\n[6] Conway. Atlas. Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wi lson, R. A. Atlas of finite groups. Maximal\nsubgroups and ordinary characters for simple groups. With comput. assist. from J. G. Thackray. Oxford: Clarendon Press.\nXXXIII, 252 p.(1985).\n[7] W. Fulton and J. Harris. Representation Theory: A First Course. Graduate Texts in Mathematics, 129. Springer, 2004.\n[8] J. E. Humphreys. Linear Algebraic Groups . Graduate Texts in Mathematics, 21. Springer-Verlag, Berlin 1975.\n[9] J. A. W. Knapp. Lie groups beyond an introduction -second edition. Progress in Mathematics, Birkhauser, Boston 2002.\n[10] M. Liebeck, On the orders of maximal subgroups of the finite classical gro ups.Proc. Lond. Math. Soc., III. Ser. 50(1985),\n426-446.\n[11] F. L¨ ubeck. Small degree representations in defining characteristic. LMS J. Comput. Math. 4(2001), 135-169.\n[12] R. Steinberg. Representations of algebraic groups . Nagoya Math. J. 22(1963), 33-56.\n[13] R. Steinberg. Lectures on Chevalley groups . Notes prepared by John Faulkner and Robert Wilson. Revised and corrected\nedition of the 1968 original. With a foreword by Robert R. Sna pp. University Lecture Series, 66. American Mathematical\nSociety, Providence, RI, 2016.\n[14] A. Zenteno. L¨ ubeck’s classification of representations of finite simpl e groups of Lie type and the inverse Galois problem\nfor some orthogonal groups. J. Number Theory 206(2020), 182-193.\n[15] A. Zenteno. Galois representations with large image in global Langland s correspondence . arXiv:2306.02493.\n27Luis Guti´ errez Frez ( Corresponding author )\nInstituto de Ciencias F ´ısicas y Matem ´aticas\nUniversidad Austral de Chile\nCampus Isla Teja s/n, Valdivia\nRegi´on de los R ´ıos, Chile\nE-mail address: luis.gutierrez@uach.cl\nAdri´ an Zenteno\nCentro de Investigaci ´on en Matem ´aticas A.C.\nJalisco s/n. Col. Valenciana, CP 36023\nGuanajuato, Mexico.\nE-mail address: adrian.zenteno@cimat.mx\n28"    },    {        "title": "2401.11449v1.Energy_Consumption_Analysis_for_Continuous_Phase_Modulation_in_Smart_Grid_Internet_of_Things_of_beyond_5G.pdf",        "content": "1\nEnergy Consumption Analysis for Continuous\nPhase Modulation in Smart-Grid Internet of Things\nof beyond 5G\nHongjian Gao, Yang Lu, Shaoshi Yang, Jingsheng Tan, Longlong Nie and Xinyi Qu\nAbstract —Wireless sensor network (WSN) underpinning the\nsmart-grid Internet of Things (SG-IoT) has been a popular\nresearch topic in recent years due to its great potential for\nenabling a wide range of important applications. However, the\nenergy consumption (EC) characteristic of sensor nodes is a key\nfactor that affects the operational performance (e.g., lifetime of\nsensors) and the total cost of ownership of WSNs. In this paper, to\nfind the modulation techniques suitable for WSNs, we investigate\nthe EC characteristic of continuous phase modulation (CPM),\nwhich is an attractive modulation scheme candidate for WSNs\nbecause of its constant envelope property. We first develop an EC\nmodel for the sensor nodes of WSNs by considering the circuits\nand a typical communication protocol that relies on automatic\nrepeat request (ARQ)-based retransmissions to ensure successful\ndata delivery. Then, we use this model to analyze the EC char-\nacteristic of CPM under various configurations of modulation\nparameters. Furthermore, we compare the EC characteristic of\nCPM with that of other representative modulation schemes, such\nas offset quadrature phase-shift keying (OQPSK) and quadrature\namplitude modulation (QAM), which are commonly used in\ncommunication protocols of WSNs. Our analysis and simulation\nresults provide insights into the EC characteristics of multiple\nmodulation schemes in the context of WSNs; thus, they are\nbeneficial for designing energy-efficient SG-IoT in the beyond-5G\n(B5G) and the 6G era.\nIndex Terms —continuous phase modulation (CPM), wireless\nsensor network (WSN), energy efficient, modulation optimization,\nsmart grid, Internet of Things (IoT), B5G, 6G.\nI. I NTRODUCTION\nSMART grid is the energy infrastructure for smart cities,\ntelecommunications, networks, and the computing indus-\ntry. It upgrades traditional power grid systems with state-of-\nthe-art information and communication technologies, such as\nwireless sensor network (WSN) techniques and the Internet\nof Things particularly designed for the smart grid industry\n(SG-IoT). In fact, SG-IoT heavily relies on WSN, which is\ncharacterized by a variety of distinct performance metrics,\nsuch as transmission rate, signal coverage, energy consumption\n(EC), and network lifetime [1]. (defined as the number of\njoules consumed per successfully transmitted bit) of wireless\nsensors, because the energy supply requirements of sensors\nCorresponding author: S. Yang (E-mail: shaoshi.yang@bupt.edu.cn)\nH. Gao and Y . Lu are with the State Grid Smart Grid Research Institute\nCo., Ltd., Beijing 102209, China.\nS. Yang, J. Tan, L. Nie, and X. Qu are with the School of Infor-\nmation and Communication Engineering, Beijing University of Posts and\nTelecommunications, and also with the Key Laboratory of Universal Wireless\nCommunications, Ministry of Education, Beijing 100876, China.\nPublished on Sensors , vol. 24, no. 2, article number 533, Jan. 2024,\nhttps://doi.org/10.3390/s24020533are stringent in WSNs (i.e., very limited energy supply) and\nall the other performance metrics can be affected by the EC\ncharacteristic of sensors. To elaborate a little further, in many\napplication scenarios of the smart grid industry, it is often\ninconvenient or unsafe for humans to work in the deployment\nsite and the lifetime of sensors is often expected to be over\nseveral years. Therefore, low EC is of vital importance for\nthese scenarios.\nBy contrast, in wireless communication systems that operate\nwith the support of power grid infrastructure, it is more\nappropriate to invoke energy efficiency (EE), which is typically\ndefined as the number of bits successfully transmitted per\njoule. This concept is at the heart of green communications,\na vision globally recognized for reducing the Carbon foot-\nprint produced by the networking sector, especially in the\nera of 5G, 5G-Advanced, and 6G [2]. Obviously, EE is the\nreciprocal of EC. Extensive studies have been devoted to\noptimizing the EE of wireless networks in the past decade.\nFor instance, in [3], based on the fractional programming\nframework, the joint power and subcarrier allocation prob-\nlem was solved for maximizing the EE of a multi-user,\nmulti-relay, single-cell orthogonal frequency-division multi-\nple access (OFDMA) cellular network composed of single-\nantenna nodes. Afterwards, system models that are more\ncomplicated were considered: the joint transmit and receive\nbeamforming-based multi-user, multi-relay, multi-input multi-\noutput (MIMO)-OFDMA cellular networks [4], [5]; the multi-\ncell single-antenna OFDMA networks [6]; the partial/full\ninterference alignment-based multi-user, multi-relay, multi-\ncell MIMO-OFDMA networks [7]; the massive MIMO-aided,\nmulti-pair, one-way decode-and-forward relay system [8]; and\nthe fully connected K-user interference channel with each user\nhaving either a single antenna or multiple antennas [9]. Addi-\ntionally, the EE of wireless networks that are delay-sensitive\nwas also studied by maintaining statistical quality-of-service\nQoS guarantees in OFDMA networks [10] and by considering\nthe uplink ultra-reliable low-latency communication (URLLC)\ntraffic in the MIMO-aided grant-free access [11] of 5G and its\nbeyond. In [12], secrecy-energy efficient hybrid beamforming\nschemes were designed for a satellite-terrestrial integrated\nnetwork in order to maximize the achievable secrecy-EE while\nsatisfying the signal-to-interference-plus-noise ratio (SINR)\nconstraints of both the earth stations and the cellular users;\nfurther, in [13], the secrecy-energy efficient beamforming in\nmultibeam satellite systems was investigated with the metric\nof signal-to-leakage-plus-noise ratio (SLNR).arXiv:2401.11449v1  [eess.SP]  21 Jan 20242\nSince wireless sensors are typically powered by batteries,\nit is more appropriate to use EC in the context of wireless\nsensors. EC is closely related to the selected modulation\nscheme. Firstly, this selection may influence the type of\nelectronic components utilized, such as a power amplifier\n(PA) or analog-to-digital converter (ADC), because different\nmodulation schemes may require different circuit designs and\nimplementations. Secondly, the specific choice of modulation\nschemes also affects the number of bits transmitted in a single\nsymbol duration that consumes a certain amount of energy.\nThirdly, different modulation schemes may incur different\npacket error rates (PERs), which influence the number of\nretransmissions that also consume energy and are necessary\nfor successful packet delivery between any pair of wireless\nsensors. Therefore, it is important to investigate the impact of\ndifferent modulation schemes on the EC and identify the most\nappropriate scheme for WSNs of SG-IoT.\nPrior research mainly focused on studying the EC of\nmodulation schemes that are sensitive to the nonlinearity of\nPAs. More specifically, in [14], the EC minimization problems\ncorresponding to M-ary quadrature amplitude modulation\n(MQAM) and multiple frequency-shift keying (MFSK) were\nstudied. In [15], [16], the authors studied the relationship be-\ntween the total EC per successfully transmitted information bit\nand the transmission distance while assuming different mod-\nulation methods, such as binary phase-shift keying (BPSK),\nquadrature phase-shift keying (QPSK), and 16QAM. They also\nstudied the average signal-to-noise ratio (SNR) values required\nto achieve the optimal EC. In [17], the transmission power of\nMQAM was optimized by using a particular model to achieve\nthe minimum EC. In [18], the EC per successfully transmitted\nbit for modulation techniques including binary frequency-\nshift keying (BFSK), BPSK, QPSK, 16QAM, and 64QAM\nwas studied under various channel conditions. However, these\nmodulation techniques require the use of linear PAs, which\nresults in lower energy utilization efficiency. In contrast,\nconstant envelope modulation techniques are insensitive to the\nnonlinearity of PAs; thus, they constitute a promising solution\nto improving the energy utilization efficiency. However, there\nis a scarcity of research focusing on the impact of constant\nenvelope modulation techniques on the achievable EC in the\ncontext of WSNs underpinning SG-IoT.\nAgainst the above backdrop, in this paper, we endeavor\nto investigate the impact of a constant envelope modulation\ntechnique, i.e., continuous phase modulation (CPM), on the\nEC of sensor nodes in WSNs suitable for SG-IoT. Our novel\ncontributions are summarized as follows.\n•We establish a realistic power consumption model\nthrough the analysis of circuit power consumption, trans-\nmission power consumption, and reception power con-\nsumption on a point-to-point communication link; in par-\nticular, we consider three operation modes of the sen-\nsors, including sleeping mode, transient mode, and active\nmode.\n•Based on the above power consumption model and a\ntypical automatic repeat request (ARQ)-based wireless\ntransmission protocol, the EC incurred by successfully\nsending a single information bit is numerically evaluated\nPreamble Header Payload LP LH LL\nLFig. 1: Physical layer packet structure.\nunder different configurations of CPM parameter values.\nIn particular, we consider different waveform pulses of\nthe CPM, including the rectangular pulse, rising cosine\npulse, and GMSK pulse, for comprehensive coverage. We\nalso investigate the impact of the distance between the\ntransmitter and the receiver, the impact of the received\nSNR, the impact of the modulation order, and the average\nnumber of transmissions required for sending a single\npacket, under various modulation schemes considered.\n•We compare the EC per successfully transmitted bit of\nthe CPM with that of conventional non-constant envelope\nmodulation methods, such as offset quadrature phase-\nshift keying (OQPSK) used in the Zigbee standard and\nQAM modulation supported by the current 5G standard.\nOur simulation results and analysis demonstrate that\nCPM enjoys a significantly lower EC than OQPSK and\n16QAM in the scenario considered, which is valuable\nfor the standard evolution of beyond 5G tailored for the\nimportant use case of low-power SG-IoT.\nII. T HEEC M ODEL\nTo analytically determine the amount of energy consumed\nwhen a single bit is transmitted without error, an EC model\nneeds to be established. We make the assumption that each\npacket transmitted in the forward direction induces an error-\nfree feedback packet in the reverse direction, which acknowl-\nedges the successful reception of the data packet or requests\nfor retransmission.\nA. Packet Structure\nIn wireless communication systems, the general format of\nthe physical layer packet structure is shown in Figure 1 and\nconsists of three parts: a pilot code for clock synchronization,\na packet header specifying the configuration of transmission\nparameters, and a data payload carrying the transmitted data.\nWe assume that the entire packet uses the same modulation\nand that the symbol error probability (SEP) is determined\nby the received SNR γand the modulation scheme adopted.\nWe also assume that symbol errors are independently and\nidentically distributed (i.i.d.) and no channel coding is used,\ni.e., no redundant bits are added (redundant bits increase\nEC). If a symbol is erroneously detected at the receiver,\nthe entire packet will be retransmitted until all the symbols\nin the packet are correctly detected at the receiver. The packet\nerror probability (PEP) can be expressed as a function of the\nSEP, the packet length L, and the number of bits per symbol\nm= log2M, i.e.,\nPEP = 1−(1−SEP)L/m, (1)3\nwhere Mdenotes the size of the modulation constellation.\nHence, by utilizing the packet length Land the probability\nthat a packet is successfully transmitted, i.e., 1−PEP , the av-\nerage amount of data successfully delivered per transmission\nduration of a packet is formulated as\nNc=L(1−SEP)L/m. (2)\nThen, upon assuming that the feedback signal indicating\nwhether a retransmission is needed or not is reliably transmit-\nted on the reverse link, the average number of transmissions\nrequired for successfully delivering a packet is given by\nNre=L\nNc=1\n1−PEP=1\n(1−SEP)L/m. (3)\nB. Basics of CPM\nCPM is an attractive modulation scheme for WSNs un-\nderpinning SG-IoT because its carrier phase is modulated in\na continuous manner and it is typically implemented as a\nconstant-envelope waveform, i.e., the transmitted carrier power\nis constant. The phase continuity requires a relatively small\npercentage of the power to occur outside of the intended\nband (e.g., low fractional out-of-band power), leading to high\nspectral efficiency. Meanwhile, the constant envelope yields\nexcellent power/energy efficiency. However, the primary draw-\nback of CPM is the high implementation complexity required\nfor an optimal receiver.\nFor systems that employ CPM, the transmitted signal at time\ninstant tcan be expressed as[19]\ns(t,I) =r\n2E\nTcos(2 πfct+ϕ(t,I) +ϕ0), (4)\nwhere Eis the symbol energy, Tis the symbol interval, fc\nis the carrier frequency, and ϕ0is an arbitrary constant initial\nphase shift that can be set to zero without loss of generality\nwhen coherent transmission is considered. In addition, ϕ(t,I)\nis the time-varying information-carrying phase formulated as\nϕ(t,I) = 2 πKX\nk=−∞hkIkq(t−kT), KT≤t≤(K+1)T,(5)\nwhere I={Ik|k∈(−∞,···,−1,0,+1,···, K)}is an\ninfinitely long sequence of uncorrelated M-ary data sym-\nbols, each having one of the values from the alphabet\nA={±1,±3,···,±(M−1)}with equal probability 1/M;\n{hk}is a sequence of modulation indices defined as hk=\n2fd,kT, with fd,kbeing the peak frequency deviation. When\nhk=hfor all k, the modulation index remains fixed for\nall symbols. When the modulation index changes from one\nsymbol to another, the signal is called multi- hCPM, with hk\nvarying in a cyclic manner. q(t)is some normalized waveform\nshape that represents the baseband phase response (i.e., phase\npulse) and is obtained from the frequency pulse g(t)by\nq(t) =Zt\n−∞g(τ)dτ. (6)\nIf the duration of g(t)is equal to the symbol interval T,\nnamely, g(t) = 0 fort > T , the modulated signal is calledfull-response CPM. If the duration of g(t)is larger than the\nsymbol interval T, namely, g(t)̸= 0fort > T , the modulated\nsignal is called partial-response CPM.\nSuppose the length of the frequency pulse g(t)in terms of\nthe number of symbol intervals is N. Thus, N= 1 yields\nfull-response CPM. If g(t)is selected as a rectangular pulse,\nnamely,\ng(t) =(\n1\n2NT,0≤t≤NT,\n0, otherwise ,(7)\nthen for a full-response CPM, we have\nq(t) =\n\n0, t≤0,\nt\n2T,0≤t≤T,\n1\n2, t≥T.(8)\nIt is evident that the performance of CPM is influenced by\ncertain parameters, including but not limited to M,hk,N,\nand the frequency pulse g(t). Note that by choosing different\npulse shapes g(t)and varying M,hk, and N, an infinite\nvariety of CPM signals may be generated, each with its unique\ncharacteristics and performance.\nFor a CPM signal, the error rate performance can be derived\nbased on the maximum-likelihood sequence detection (MLSD)\nreceiver, which is conventionally computed using the Viterbi\nAlgorithm (V A). Specifically, for a given CPM scheme, we\nhave\nSEP =KminQ(q\nd2\nminγ). (9)\nAccording to Anderson’s seminal work on digital phase\nmodulation [20], Kmindenotes the total number of feasible\npaths that satisfy the constraint of the minimum Euclidean\ndistance dminwithin the observation interval on the CPM\nphase grid. The value of Kminincreases with the modulation\norder M. Both Kminanddmindepend on critical parameters\nincluding M,hk,N, and the pulse shaping function g(t).\nC. Circuit Power Consumption\nIn wireless communication systems, a significant portion of\nenergy is dedicated to signal transmission and reception cir-\ncuits, which are mainly composed of the baseband (BB) digital\nsignal processing unit and the radio frequency (RF) signal\nprocessing unit, as shown in Figure ??. To elaborate a little\nfurther, the BB signal processing unit mainly includes source\ncoding/decoding, pulse shaping, channel coding/decoding, dig-\nital modulation/demodulation, channel estimation, synchro-\nnization, and so on. For a wireless sensor, since the data rate\nrequirement is usually low, the BB symbol rate is also low.\nMeanwhile, typically, no computation-intensive signal pro-\ncessing techniques, such as multi-user detection and iterative\ndecoding, are used in an energy-constrained wireless sensor;\nhence, the BB power consumption is significantly smaller than\nthe RF circuit power consumption.\nA typical model of an RF signal processing unit, also known\nas an RF chain, is shown in Figure 3 [14], [16], [21], [22],\n[23]. Specifically, on the transmitter side, the BB signal is first\nconverted to an analog signal by the digital-to-analog converter\n(DAC). Then, the analog signal is filtered by the low-pass filter4\n濕濴瀆濸濵濴瀁濷澳 \n濗濼濺濼瀇濴濿澳濦濼濺瀁濴濿澳 \n濣瀅瀂濶濸瀆瀆濼瀁濺澳 \n濖濼瀅濶瀈濼瀇 Tx RF \nCircuit\nRx RF \nCircuitTx/Rx Switch \nPTBB /PRBB PTRF \nPRRF \nFig. 2: The communication modules and the corresponding\npower consumption model of a point-to-point wireless com-\nmunication system.\nLO DAC \nADC IFA Filter\nFilterFilter\nFilter FilterMixer\nLNA MixerPA Duplexer\nPDAC Pfilt\nPLo PPA Pfilt\nPfiltPfilt PfiltPdup \nPLNA Pmix Pmix \nPIFA \nPADC PRF \nFig. 3: A typical power consumption model of the RF signal\nprocessing unit of a point-to-point wireless communication\nsystem.\nand upconverted by the mixer, whose output is then filtered\nagain, amplified by the power amplifier (PA), passed through\nthe duplexer, and finally transmitted to the wireless channel.\nOn the receiver side, the RF signal is sequentially filtered,\namplified by the low-noise amplifier (LNA), cleaned by the\nanti-aliasing filter, downconverted by the mixer, filtered again\nbefore passing through the intermediate frequency amplifier\n(IFA) that has an adjustable gain, and finally converted back\nto a digital signal by the analog-to-digital converter (ADC).\nNote that the mixers operate with the aid of the local oscillator\n(LO) and, among all the RF components, the PA and LNA\nusually have much higher power consumption than the others.\nAccordingly, the total power consumption during transmis-\nsion can be expressed as\nPTx(d) =PTBB+PTRF+PPA(d)|{z}\nPTRF=PT0+ξ\nηPT(d),(10)\nwhere PTBB andPTRFrepresent the power consumption of\nthe transmitter’s BB processing unit and RF processing unit\nexcluding the PA, respectively, and both of them can be\nregarded as constant values that are collectively denoted by\nPT0;PPA(d)andPT(d)are defined as the power consumption\nof the PA and the transmit power, respectively, both of which\nare functions of the transmission distance dupon assuming\nadaptive power control and are related via PPA(d) =ξ\nηPT(d);\nandηandξrepresent the drain efficiency and the peak-to-\naverage power ratio (PAPR) of the PA, respectively [18].Similarly, the total power consumption of the receiver is\nexpressed as\nPRx=PRBB+PRRF+PLNA|{z}\nPRRF=PR0, (11)\nwhere PRBB andPRRFrepresent the power consumption of\nthe receiver’s BB processing unit and RF processing unit\nexcluding LNA, respectively, and both of them can be regarded\nas constant values; PLNA represents the power consumption\nof the LNA, which is also constant upon assuming that the\nLNA is appropriately designed and biased, so that necessary\nsensitivity is provided for reliably receiving, demodulating,\nand decoding a minimum power signal. Hence, PRBB,PRRF,\nandPLNAare collectively denoted by the constant PR0.\nD. Transmission Power Consumption\nDue to the path-loss, scattering, reflection, and other phe-\nnomena in the wireless channel, a certain amount of energy\nis inevitably lost during the transmission of electromagnetic\nwaves that carry data symbols, thus resulting in transmission\nenergy dissipation. The path-loss depends on several factors,\nsuch as the distance between the transmitter and receiver,\nthe frequency of the signal, the type of antennas used, and the\nenvironment through which the signal propagates.\nWhen the signal with a transmit power PT(d)is propagated\nthrough the wireless channel, the received power PR(d)at the\nreceiver can be formulated as\nPR(d) =GTGRPT(d)\u0010λ\n4πd\u0011α\n=PT(d)\nA0dα, (12)\naccording to Friis’ transmission equation. This expression\ncharacterizes the dependency between the received power\nwith respect to several parameters. Specifically, the constant\nA0depends on the transmit antenna gain GT, the receive\nantenna gain GR, and the carrier wavelength λ. Additionally,\nthe path-loss exponent is α. It is noted that the received\npower is inversely proportional to the distance draised to the\npower of α, and the received SNR γis proportional to 1/A0\ndivided by dα. These relationships collectively describe how\nthe aforementioned parameters influence the received power\nand the received SNR [18]:\nγ=PR(d)\nN0WN fMl, (13)\nwhere Wrepresents the transmission bandwidth, N0denotes\nthe power spectral density of the baseband-equivalent additive\nwhite Gaussian noise (AWGN), and they are primary determi-\nnants of γ. Furthermore, the noise figure of the RF front-end\nof the receiver, denoted as Nf, and any additional noise or\ninterference, represented by the link margin term Ml, can also\nimpact the received SNR γ.\nAccordingly, by substituting Equation (12) into Equa-\ntion (13), the relationship between the transmit power PT(d),\nthe communication distance d, the received SNR γ, and other\nparameters can be quantitatively expressed as\nPT(d) =A0dαN0WN fMlγ=Adαγ, (14)\nwhere we have A=A0N0WN fMl.5\nE. EC per Successfully Transmitted Bit\nAs mentioned in Section II, we assume that each packet\ntransmitted in the forward direction is matched by an error-\nfree feedback packet in the reverse direction in order to guar-\nantee reliable transmission. Both directions of transmissions\nconsume energy. The above transmission process, usually\nincorporating retransmissions, continues until the entire packet\nof the forward direction is correctly decoded at the receiver.\nAdditionally, we assume that the sensor node transceiver\ncircuitry works in a multi-mode manner: (1) when there are\ndata to transmit or receive, all circuits of the sensor work in the\nactive mode; (2) when neither transmission nor reception are\nneeded, the circuits of the sensor enter sleep mode by default,\nwhich uses the minimum possible power (small enough to be\nnegligible) to ensure that the circuits can be activated when\nnecessary; (3) when the sensor is in the period of switching\nfrom sleep mode to active mode, it is in a transient mode that\nalso consumes non-negligible energy. Note that the transient\nduration from active mode to sleep mode is sufficiently short\nto be neglected but the start-up process from sleep mode to\nactive mode can be slow.\nTherefore, for a single round-trip transmission (forward\ndirection transmission and reverse direction feedback) on a\npoint-to-point communication link, the total EC of the system\ncan be divided into two parts: the EC of the forward direction\ntransmission and the EC of the reverse direction feedback.\nFor the forward direction transmission, the EC is composed\nof the start-up energy consumption 2ESTin transient mode\n(both the transmitting node and the receiving node may be\nin sleep mode initially), the transmitter energy consumption\nETx(d)in active mode, and the receiver energy consumption\nERxin active mode. Therefore, the EC of the forward direction\ntransmission is expressed as\nEFW= 2EST+ETx(d)+ERx= 2EST+PTx(d)TDTA+PRxTDRA,\n(15)\nwhere TDTA is the transmission duration for sending a data\npacket in the forward direction and TDRAis the corresponding\nduration of signal processing at the receiver of the forward di-\nrection.\nAdditionally, the EC of the reverse direction transmission\nis expressed as\nERV=PTx(d)TFTA+PRxTFRA, (16)\nwhere PTx(d)is the total power consumption for transmission\nof a feedback packet in the reverse direction by the receiving\nnode of the forward direction and TFTAis the corresponding\ntime duration. Note that PTx(d)may be different from PTx(d)\nof the forward direction because a feedback packet may have\na different transmission rate and reliability requirements com-\npared with a data packet. TFRAis the duration of processing\nthe feedback packet at its receiver (i.e., the transmitting node\nof the forward direction) in active mode.\nBased on the above analysis, we obtain the EC per suc-\ncessfully transmitted bit as (17), where PT(d)is the transmit\npower for the feedback packet transmission and Nreis the\naverage number of retransmissions.TABLE I: The value of d2\nminwhen we set M= 2,4,8,16,\nh= 0.75, andN= 3. For GMSK, the time-bandwidth product\nBT is set to 0.3, where Bis the −3dB bandwidth of the\nGaussian pulse.\nWaveform M= 2 M= 4 M= 8 M= 16\nREC 2.31648 1.41550 2.12325 2.831\nRC 2.96059 5.30037 6.12447 8.16596\nGMSK 2.89955 4.69275 5.95011 7.93348\nIII. CPM P ARAMETER SELECTION\nAs mentioned in Section II-B, the main parameters that af-\nfect the performance of CPM are M,hk,N, and the frequency\npulse shaping function g(t), whilst the minimum squared\nEuclidean distance d2\nminalso depends on these parameters.\nTherefore, it is necessary to determine how these parameters\ninfluence the EC per successfully transmitted bit.\nIn principle, it is possible to implement an infinite number\nof different CPM signals using various combinations of design\nparameters. However, for practical implementation, we must\nconsider the trade-off between the achievable performance\nand the cost incurred. It is well known that partial-response\nCPM signals usually have better spectral efficiency than full-\nresponse CPM signals. However, the computational complex-\nity of the optimal MLSD receiver exponentially increases\nwithN, which is the length of g(t)in terms of the number\nof symbol intervals. In this study, we focus on the partial-\nresponse CPM with a moderate value of N= 3. The modu-\nlation order Malso significantly influences the computational\ncomplexity and the requirements for demodulation devices\n[24], which can lead to high EC. Therefore, small values, such\nasM= 2,4,8,16, etc., are generally chosen. We assume\nthe modulation index hk=h, which is also an important\nparameter and has a complex functional relationship with the\nminimum squared Euclidean distance d2\nmin[24], [25]. Smaller\nvalues of hresult in narrower bandwidth, more concentrated\nsignal energy, and narrower transition bands. However, they\nalso make the phase variations less obvious and can increase\nthe complexity of demodulation decisions. For single index\nmodulation, h= 0.5or0.75is commonly used. Finally, the\npulse shaping function g(t)usually utilizes rectangular pulse\n(REC), rising cosine pulse (RC), and Gauss minimum-phase\nshift-keying pulse (GMSK) [26].\nGiven the imperative to minimize EC for sensors,\nthe present study focuses on CPM signals that can be imple-\nmented with simple devices and require low computational\ncomplexity. Accordingly, CPM signals with three different\ng(t)functions are selected for investigation, assuming M=\n2,4,8,16,h= 0.75, and N= 3. The values of d2\nminunder\nthese parameter configurations are calculated with the methods\ngiven in [24], [25] and listed in Table I.\nIV. E VALUATION THE EC OFCPM\nA. Identification of Major Performance Influencing Factors\nFrom Equation (17), it can be observed that the EC per\nsuccessfully transmitted bit, Eb, is predominantly determined\nby four parameters: the forward-link transmit power PT(d), the\nreverse-link transmit power PT(d), the specific CPM scheme6\nEb=Nre(EFW+ERV)\nL\n=EFW+ERV\nNc\n=2EST+PTx(d)TDTA+PRxTDRA+PTx(d)TFTA+PRxTFRA\nL(1−SEP)L/m\n=2EST+ (PT0+ξ\nηPT(d))TDTA+PRxTDRA+ (PT0+ξ\nηPT(d))TFTA+PRxTFRA\nL(1−KminQ(p\nd2\nminγ))L/m.(17)\nTABLE II: A summary of the pertinent simulation parameters.\nParameters Values\nSymbol rate for transmitted signals 20 ksps\nLP,LH, and LL 4, 3, and 30 bytes\nPower spectral density of AWGN at the receiver ( N0) −174 dBm/Hz\nNoise figure of the RF front-end of the receiver ( Nf) 10 dB\nEquivalent antenna gain ( A0) 30 dB\nBandwidth ( W) 20 kHz\nAdditional noise ( Ml) 10 dB\nPT0 15.9 mW\nPR0 58.2 mW\nM 2, 4, 8, 16\nh 0.75\nN 3\nPath-loss exponent ( α) 3.5\nDrain efficiency ( η) 0.7 for CPM; 0.35 for OQPSK and 16QAM\nPeak-to-average power ratio ( ξ) 0 dB for CPM; 3.5 dB for OQPSK; 6.7 dB for 16QAM\nadopted, and the received SNR γ. From another perspective,\nwe can also say that Ebis mainly determined by PT(d),\nPT(d), the achievable SEP, and the modulation efficiency m.\nUpon inspection of Equations (13) and (9), it becomes clear\nthat the SEP on the forward link can be expressed as\nSEP =KminQ\ns\nd2\nminPT(d)\nA0dαN0WN fMl\n. (18)\nAlthough a high transmit power is desirable for achieving\na low SEP and thus reducing the number of retransmissions\nNre, it may also result in excessively large transmission power\nfor the sensor and increase the EC of each single-direction\ntransmission. Hence, an optimal transmit power (or in turn\nthe received SNR γ) exists and is yet to be found for\nminimizing the EC per successfully transmitted bit. Similarly,\nthe relationship between the EC per successfully transmitted\nbit and other major parameters mentioned above needs to be\nstudied.\nB. Simulation Results and Discussions\nIn the following numerical simulations, we consider the\nradio links of a WSN designed for smart grid and operating\nin the industrial–science–medical (ISM)-oriented 2.4 GHz\nfrequency band. Table II provides a summary of the pertinent\nsimulation parameters, including circuit-related parameter set-\ntings as well. Due to the constant envelope characteristic of the\nCPM signal, a nonlinear PA with a high ηvalue is employed,\nin contrast to the general radio architecture.First, let us compare the EC performance of different CPM\nschemes and of other representative modulation schemes,\nsuch as OQPSK and 16QAM, by observing how Ebvaries\nwith the received SNR γ. As shown in Figure 4, for all\nmodulation schemes, the achievable Ebvalues firstly descend\nand then increase with the received SNR. This is because\nin the low-SNR region, the number of retransmissions plays\nan important role, while the number of retransmissions is\nreduced to its minimum under a sufficiently high SNR, and\nthen an even higher SNR means unnecessary energy wastage.\nIn addition, for most SNRs, the achievable Ebvalues of REC\n(d2\nmin= 2.831,h= 0.75andN= 3), RC ( d2\nmin= 8.16596 ,\nh= 0.75andN= 3), and GMSK ( d2\nmin= 7.93348 ,h= 0.75\nandN= 3) are lower than those of OQPSK and 16QAM,\nwhich verifies the advantages of CPM in terms of energy\nsaving. The reason why a higher order CPM scheme has a\nlower Ebcan be explained by referring to Equation (17) as\nfollows: (1) a larger Mcauses a smaller number of symbols\nper packet (i.e., the smaller exponent L/m ) and a smaller SEP\n(i.e., the larger base number 1−SEP), thus the denominator\nL(1−SEP)L/mincreases with M; (2) the variable TDTAin\nthe numerator becomes smaller when Mbecomes larger, while\nall the other terms can be regarded as constants. Furthermore,\nwe can see that the lowest values of Ebachieved by the four\nmodulation schemes having M= 16 (i.e., REC, RC, GMSK,\nand 16QAM) are almost the same, while OQPSK having\nM= 4achieves the largest Ebin the high-SNR region and the\nsecond-largest Ebin the low-SNR region. It is also observed\nthat when the SNR is above a specific threshold, the three\ndifferent CPM waveforms exhibit the same EC performance.7\n0 5 10 15 20 25 30 35\nSNR (dB)11.21.41.61.822.22.42.62.83Eb (J/bit)10-6\nREC\nRC\nGMSK\nOQPSK\n16QAM\nFig. 4: The relationship between the EC per successfully\ntransmitted bit ( Eb) and the received SNR ( γ) for OQPSK,\n16QAM, and the CPM signals with different pulse shaping\nfunctions (REC, RC, and GMSK), while assuming M= 16\nandN= 3 for the three CPM waveforms, as well as d= 10\nm and the AWGN channel for all the modulation schemes\nconsidered.\nFigure 5 characterizes the relationship between Eband the\ncommunication distance dfor OQPSK, 16QAM, and the CPM\nsignals with different pulse shaping functions. We can see that\nEbincreases with dfor all the modulation schemes considered.\nThis is because large distance reduces the received SNR\nvalue under a given transmit power, thus degrading the SEP\nperformance and increasing the number of retransmissions. We\nalso see that all the three CPM waveforms considered have\nthe same EC performance curves, which are consistently and\nsignificantly better than those of OQPSK and 16QAM when d\nis sufficiently large. Here, we assume γ= 8 dB for the three\nCPM waveforms and γ= 15 dB for OQPSK and 16QAM. The\ntwo SNR values are selected according to the results shown\nin Figure 4, where the three CPM waveforms achieve their\noptimal Ebat about γ= 8 dB, while OQPSK and 16QAM\nachieve their near-optimal Ebat about γ= 15 dB.\nFigure 6 shows how the Ebvalues of the three CPM\nwaveforms vary with the modulation efficiency m, while\nassuming γ= 8 dB and d= 10 m. It is observed that for\neach given CPM waveform, the Ebvalue becomes smaller as\nmincreases. Although the RC-based CPM signaling exhibits\nthe highest Ebwhen m= 1,2,3, all the three CPM waveforms\nachieve almost the same Ebwhen m= 4(i.e.,M= 16 ). This\nis because the three CPM waveforms achieve almost the same\nSEP performance when m= 4,γ= 8 dB, and d= 10 m.\nThese observations are consistent with the results shown in\nFigure 4.\nFigure 7 demonstrates the relationship between the average\nnumber of transmissions Nrerequired for successfully sending\na single packet and the received SNR γover the AWGN\nchannel employing different modulation schemes, including\nOQPSK, 16QAM, and the three CPM waveforms. Obviously,\n10 20 30 40 50\nd (m)11.21.41.61.822.22.42.62.8Eb (J/bit)10-6\nREC\nRC\nGMSK\nOQPSK\n16QAMFig. 5: The relationship between the EC per successfully\ntransmitted bit ( Eb) and the communication distance ( d) for\nOQPSK, 16QAM, and the CPM signals with different pulse\nshaping functions (REC, RC, and GMSK) over the AWGN\nchannel, while assuming M= 16 ,N= 3 andγ= 8 dB for\nthe three CPM waveforms, as well as γ= 15 dB for OQPSK\nand 16QAM signals.\n16QAM incurs the largest Nre, while the REC- and GMSK-\nbased CPM schemes require the smallest Nre, under all the\nthree SNR values of 6dB,8dB, and 10dB. In addition, the\nOQPSK scheme requires a smaller and a larger Nrethan the\nRC-based CPM scheme under γ= 6 dB and γ= 8 dB,\nrespectively. However, when the received SNR is sufficiently\nhigh, e.g., γ= 10 dB, all the modulation schemes, except\n16QAM, require only a single transmission on average for\nsuccessfully sending a packet.\nV. C ONCLUSIONS\nIn this paper, the EC characteristics of various CPM\nschemes are compared with those of OQPSK and 16QAM\nin the context of WSN-based SG-IoT of beyond 5G. We\nfirst propose an EC model for the sensor nodes of WSNs by\nconsidering the circuits and a typical communication protocol\nthat relies on ARQ-based retransmissions. Our analytical and\nsimulation results demonstrate that all the CPM schemes based\non the pulse shaping functions of REC, RC, and GMSK sig-\nnificantly outperform OQPSK used in the Zigbee standard and\n16QAM used in the current 5G standard, in terms of the EC\nper successfully transmitted bit, Eb. We also show that for all\nthe modulation schemes considered, the individual optimum\nvalues of Ebare achieved with the received SNR that is neither\ntoo small nor too large. In addition, we show that Ebincreases\nwith the communication distance dfor all the modulation\nschemes considered, and decreases with the modulation order\nMfor the three CPM schemes. Overall, it is observed that the\nREC- and GMSK-based CPM schemes achieve the best EC\nperformance of all the modulation schemes considered.8\n1 2 3 4\nm01234567Eb (J/bit)10-6\nREC\nRC\nGMSK\nFig. 6: The relationship between the EC per successfully\ntransmitted bit ( Eb) and the modulation efficiency mfor the\nCPM signals with different pulse shaping functions (REC, RC,\nand GMSK) over the AWGN channel, while assuming N= 3,\nγ= 8 dB and d= 10 m.\n6 8 10\nSNR (dB)0.811.21.41.61.822.22.4The average number of transmissionsREC\nRC\nGMSK\nOQPSK\n16QAM\nFig. 7: The relationship between the average number of trans-\nmissions required for successfully sending a single packet and\nthe received SNR ( γ), while considering OQPSK, 16QAM,\nand the CPM signals with different pulse shaping functions\n(REC, RC, and GMSK) over the AWGN channel. Assume\nthatM= 16 andN= 3 for the three CPM waveforms, and\nd= 10 m for all the modulation schemes compared.REFERENCES\n[1] Z. Fei, B. Li, S. Yang, C. Xing, H. 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Sundberg, “Continuous phase mod-\nulation – Part II: Partial response signaling,” IEEE Transactions on\nCommunications , vol. 29, no. 3, pp. 210–225, Mar. 1981.\n[25] K. Kassan, H. Far `es, D. C. Glattli, and Y . Lou ¨et, “Performance vs.\nspectral properties for single-sideband continuous phase modulation,”\nIEEE Transactions on Communications , vol. 69, no. 7, pp. 4402–4416,\nJul. 2021.\n[26] M. Foruhandeh, M. Uysal, I. Altunbas, T. Guven, and A. Gercek,\n“Optimal choice of transmission parameters for LDPC-coded CPM,”\ninProc. IEEE Military Communications Conference (MILCOM’14) ,\nBaltimore, USA, Oct. 2014, pp. 368–371."    },    {        "title": "2401.11518v3.An_asymmetrical_body__example_of_analytical_solution_for_the_rotation_matrix_in_elementary_functions_and_Dzhanibekov_effect.pdf",        "content": "An asymmetrical body: example of analytical solution for the rotation matrix in\nelementary functions and Dzhanibekov effect.\nAlexei A. Deriglazov1,∗\n1Depto. de Matem´ atica, ICE, Universidade Federal de Juiz de Fora, MG, Brazil\n(Dated: 1 de fevereiro de 2024)\nWe solved the Poisson equations, obtaining their exact solution in elementary functions for the\nrotation matrix of a free asymmetrical body with angular velocity vector lying on separatrices.\nThis allows us to discuss the temporal evolution of Dzhanibekov’s nut directly in the Laboratory\nsystem, where it is observed. The rotation matrix depends on two parameters with clear physical\ninterpretation as a frequency and a damping factor of the solution. Qualitative analysis of the\nsolution shows that it properly describe a single-jump Dzhanibekov effect.\nI. INTRODUCTION.\nDzhanibekov’s nut is an asymmetrical body rotating along the axis very close to the intermediate axis of inertia of\nthe body. Being in the state of an apparently stationary rotation, at regular intervals such a body makes a sudden\njump, abruptly changing the direction of rotation vector by 180 degrees. This paradoxical behavior was first observed\nby V. A. Dzhanibekov during his trip in outer space, and bears his name. A similar behavior of a body on the surface\nof the Earth (that is in a gravitational field) is called the tennis racket effect. The clue for a theoretical explanation for\nthis phenomenon is contained in Euler equations, which show that stationary rotation around the intermediate axis\nis non stable [1–6]. Various aspects of the behavior of such a body, predicted from Euler equations, are intensively\ndiscussed in the literature [7–15].\nHowever, solutions to the Euler equations describe the movement of instantaneous axis of rotation (that is the\nangular velocity) with respect to a body-fixed frame, and thus contain only indirect information about the character\nof the movement in Laboratory system, in which the effect is observed. To understand the movement of a body in\nspace, we need to find its rotation matrix, the latter is guided by Poisson equations. That is, in addition to the Euler\nequations, we also need to analyse the Poisson equations.\nPoisson equations contain the angular velocity, which must first be found from Euler equations. Unfortunately,\nsolutions to Euler equations generally can be written only in the form of elliptic integrals, which are not very il-\nluminative. However, under special initial conditions there are two (non trivial) particular solutions in elementary\nfunctions [4], and it is they that turn out to be closely related to the Dzhanibekov effect. These two trajectories of\nangular velocity vector are called separatrices. Then the natural question is whether it is also possible to solve the\nPoisson equations in elementary functions for this case? In this work we give an affirmative answer to this question,\nobtaining the explicit form of rotation matrix of an asymmetrical body with angular velocity vector moving along the\nseparatrices.\nTemporal evolution of the rotation axis along a separatrix is not a periodic function [4]. As we will see, the same\nis true for the rotation matrix. In this case an ideal body experiences a sudden jump only once: let it be launched\nwith the rotation axis lying on a separatrix, and almost antiparallel to the conserved angular momentum. Then after\nsome time it will make a sudden jump so that the rotation axis becomes almost parallel to the angular momentum,\nafter which it will asymptotically approach it1. This is the physical content of the rotation matrix that we found.\nThe trajectories lying near separatrix are time-periodic functions [4]. So it is expected that an ideal body along such\na trajectory will experiences multiple jumps at regular intervals, that is the Dzhanibekov effect.\nThe work is organised as follows. In Sect. II we describe the necessary notation and present the basic equations\nto be analysed. In Sect. III we solve the Poisson equations obtaining the rotation matrix in elementary functions,\nsee Eq. (29). In concluding section IV we present qualitative analysis of the solution, showing that it properly\ndescribe a single-jump Dzhanibekov effect. The rotation matrix will be constructed without assuming any kind of its\nparametrization like Euler angles. To avoid possible confusion, in the Appendix we discuss some peculiarities, that\nshould be taken into account when working with the Euler angles.\n∗Electronic address: alexei.deriglazov@ufjf.br\n1The body launched with the rotation axis almost parallel to the angular momentum, will approach it without any jump.arXiv:2401.11518v3  [nlin.SI]  30 Jan 20242\nII. NOTATION AND BASIC EQUATIONS OF THE PROBLEM.\nConsider a free asymmetrical body with the inertia moments I3> I2> I1. To describe its rotational movement,\nwe place the origin of Laboratory frame in the center of mass. Besides, at the initial instant of time, say t= 0, the\nLaboratory axes (with unit vectors ei) should be chosen in the directions of axes of inertia (with unit vectors Ri(t)),\nthe latter we assume to be the body-fixed frame. Then the temporal evolution of the body can be obtained solving\nthe Euler-Poisson equations2\nI˙Ω= [IΩ,Ω], (1)\n˙Rij=−ϵjkmΩkRim, (2)\nwith the inertia tensor of a diagonal form: Iij=diagonal (I1, I2, I3). The basic variables in these 3 + 9 equations are\nthe rotation matrix Rij(t) and the components Ω i(t) of instantaneous angular velocity vector ω(t) with respect to\nbody-fixed frame: ω(t)≡ωi(t)ei= Ω i(t)Ri(t). The initial conditions for the problem are Rij(0) = δij, Ωi(0) = ˜Ωi,\n˜Ωi∈R. Any solution Rij(t) of the system (1), (2) with these initial conditions is an orthogonal matrix at any t.\nBesides, columns of the rotation matrix are just basis vectors of the body-fixed frame: R(t) = (R1(t),R2(t),R3(t)).\nFor the latter use, we recall that velocity of rotation can be described using four different quantities. They are\ninstantaneous angular velocity ωi, its components in body-fixed frame Ω i, conserved angular momentum miand its\ncomponents in body-fixed frame Mi. They are related as follows [17]:\nm=RM=RIRTω=RIΩ. (3)\nAtt= 0 we get\nm=M(0) = Iω(0) = IΩ(0). (4)\nTwo particular solutions to Euler equations in elementary functions. Euler equations (1) have two\nintegrals of motion, which are the conserved energy and square of angular momentum\nI1Ω2\n1+I2Ω2\n2+I3Ω2\n3= 2E, I2\n1Ω2\n1+I2\n2Ω2\n2+I2\n3Ω2\n3=m2. (5)\nHence for any solution Ω(t) of Euler equations, the end of this vector at each tlies on the trajectory that is an\nintersection of two ellipsoids (5). If we consider movements with fixed energy E, the intersections are possible when\n2EI1≤m2≤2EI3. When\nm2= 2EI2, (6)\nthe two resulting trajectories (called separatrices) pass through vertices of intermediate axis of E-ellipsoid, see Figure 1.\nGeometrically, they turn out to be two ellipses [4]. All trajectories (5) intersect the plane Ω 2= 0 twice, so without loss\nof generality we can choose the initial conditions for the problem (1) on a half-plane as follows: Ω(0) = ( ˜Ω1,0,˜Ω3>0)T.\nThen Eqs. (5) at t= 0 give the initial conditions through Eandm2as follows\n˜Ω2\n1=−m2+ 2EI3\nI1I2I3−1, ˜Ω2\n3=m2−2EI1\nI3I2I3−1,where I3−1≡I3−I1\nI2, I 2−1≡I2−I1\nI3,and so on . (7)\nUsing Eqs. (5) at t= 0 together with (6), we get the initial conditions for selecting the separatrices\nΩ(0) = ( ϵa˜Ω3,0,˜Ω3>0)T, ϵ =±1, a ≡s\nI3−2\nI2−1. (8)\nTo solve the Euler equations with these initial conditions, we use Eqs. (5)-(8) to represent Ω 1and Ω 2through Ω 3\nas follows:\nΩ1=ϵaΩ3, Ω2=±bq\n˜Ω2\n3−Ω2\n3, where b≡s\nI3−1\nI2−1. (9)\n2We use the notation from [16, 17]. In particular, by ( a,b) and [ a,b]i=ϵijkajbkwe denote the scalar and vector products of the vectors\naandb.ϵijkis Levi-Civita symbol in three dimensions, with ϵ123= 1.3\nmR\nR2(t)(0)(0)\neee1\n3\n2 2=\n-m\nΩ\nFigura 1: Two separatrices are drawn with bold lines. The line denoted R2(t) shows the trajectory of intermediate axis for a\nsingle-jump Janibekov effect.\nSubstituting these expressions into third Euler equation: ˙Ω3=I1−2Ω1Ω2, we get\n˙Ω3=±p\nI3−2I3−1Ω3q\n˜Ω2\n3−Ω2\n3. (10)\nThis equation with + and −signs has the same solutions, so we will continue with the sign +. Separating the vari-\nablesdΩ3\nΩ3√˜Ω2\n3−Ω2\n3=p\nI3−2I3−1dt, it is immediately integrated−1\n2˜Ω3\u0012\nln\u0014\n˜Ω3+q\n˜Ω2\n3−Ω2\n3\u0015\n−ln\u0014\n˜Ω3−q\n˜Ω2\n3−Ω2\n3\u0015\u0013\n=\np\nI3−2I3−1t+c. Using the initial condition Ω 3(0) = ˜Ω3we conclude c= 0, then\n˜Ω3+q\n˜Ω2\n3−Ω2\n3\n˜Ω3−q\n˜Ω2\n3−Ω2\n3=e−ωt, ω ≡p\nI3−2I3−1˜Ω3. (11)\nSeparating Ω 3from this equality we get\nΩ3(t) =˜Ω3\ncoshωt. (12)\nPutting this expression into (9), we obtain the remaining components up to a sign. The vectors Ω(t), for which Ω 1(t)\nand Ω 2(t) have the same sign, turn out to be solutions to Euler equations (1). So the final expression for the time\nevolution of angular velocity along the separatrices is\nΩ(t) = \nϵa˜Ω3\ncoshωt, ϵb ˜Ω3tanhωt,˜Ω3\ncoshωt!T\n, ϵ =±1, a≡s\nI3−2\nI2−1, b≡s\nI3−1\nI2−1, ω≡p\nI3−2I3−1˜Ω3.(13)\nThe evolution is not a periodic function of time: Ω(±∞) = (0 ,±ϵb˜Ω3,0)T. This equality means that the vector\nΩ(t) in the body-fixed frame Riasymptomatically approaches to the intermediate axis R2.\nThe following identities (for ϵ= +1):\nΩ1=aΩ3,d\ndt\u00121\nΩ3\u0013\n=−I1−2aΩ2\nΩ3, bω =I3−1a˜Ω3, (14)\nwill be repeatedly used in subsequent calculations.\nIII. SOLUTION TO POISSON EQUATIONS IN ELEMENTARY FUNCTIONS.\nDenoting the rows of rotation matrix RijasG1,G2andG3, the original system of 9 Poisson equations (2) splits\ninto three systems: ˙Gi= [Gi,Ω]. So, denoting any one row Gibyγ, we need to solve the following equations:4\n˙γ= [γ,Ω(t)], or, in components\n˙γ1=γ2Ω3−γ3Ω2, ˙γ2=γ3Ω1−γ1Ω3, ˙γ3=γ1Ω2−γ2Ω1, (15)\nwith known functions Ω i(t) written in (13). To be specific, we work with only one separatrix, with ϵ= +1. These\nequations have two integrals of motion. The first is γ2= const. The second follows from conservation of angular\nmomentum: mk= (RIΩ)k=P\niIiΩi(Gk)i, and reads as follows:\nI1Ω1γ1+I2Ω2γ2+I3Ω3γ3=c. (16)\nThe whole rotation matrix will be restored, choosing three particular solutions to the system (15) and (16). G1(t) is\nthe solution γ(t) that obeys the initial condition γ(0) = (1 ,0,0), and taking c=m1= (IΩ(0))1=I1a˜Ω3according to\n(16), (4) and (13). G2(t) is the solution γ(t) that obeys the initial condition γ(0) = (0 ,1,0) and taking c=m2= 0.\nAt last, G3(t) is the solution γ(t) that obeys the initial condition γ(0) = (0 ,0,1) and taking c=m3=I3˜Ω3.\nUsing (13) in (15) and (16) we get\n˙γ1=γ2Ω3−γ3Ω2, (17)\n˙γ2=aγ3Ω3−γ1Ω3, (18)\n˙γ3=γ1Ω2−aγ2Ω3, (19)\nI1aγ1+I3γ3=1\nΩ3(c−I2Ω2γ2). (20)\nSolving the equations (18) and (20) with respect to γ1andγ3\nγ1=1\nI2b2Ω3[ac−aI2Ω2γ2−I3˙γ2], γ 3=1\nI2b2Ω3[c−I2Ω2γ2+I1a˙γ2], (21)\nwe substitute them into (17). After tedious calculations we get the following closed equation for determining γ2(t):\nd2\ndt2\u0012γ2\nΩ3\u0013\n+\u0014\nΩ2\n2+I3−1\nI2−1Ω2\n3\u0015γ2\nΩ3=I3−2+ 1\nI3cΩ2\nΩ3. (22)\nRemarcably, substituting Ω 2and Ω 3from (13), we obtain equation of a harmonic oscillator for the quantityγ2\nΩ3, with\ntime-independent proper frequency k, and under influence of an external time-dependent force as follows:\nd2\ndt2\u0012γ2\nΩ3\u0013\n+k2γ2\nΩ3=I3−I2+I1\nI1I3cbsinhωt, k ≡b˜Ω3. (23)\nIts general solution is\nγ2(t) = Ω 3(t)\u0014\nAcos(kt+k0) +cb\nI2k2sinhωt\u0015\n. (24)\nUsing this in Eq. (21), and presenting the resulting expressions in terms of initial value ˜Ω3, the general solution to\nequations (17)-(20) with integration constants Aandk0reads as follows\nγ1(t) =1\nI2b2\u0014\nb˜Ω3I3Asin(kt+k0)−b˜Ω3aI1Acos(kt+k0) tanh ωt+I1ac\nI2˜Ω31\ncoshωt\u0015\n,\nγ2(t) =˜Ω3\ncoshωtAcos(kt+k0) +c\nI2b˜Ω3tanhωt,\nγ3(t) =1\nI2b2\u0014\n−b˜Ω3aI1Asin(kt+k0)−b˜Ω3I3Acos(kt+k0) tanh ωt+I3c\nI2˜Ω31\ncoshωt\u0015\n. (25)\nAt the initial instant t= 0 we get\nγ1(0) =1\nI2b2\u0014\nb˜Ω3I3Asink0+I1ac\nI2˜Ω3\u0015\n, γ 2(0) = ˜Ω3Acosk0, γ 3(0) =1\nI2b2\u0014\n−b˜Ω3aI1Asink0+I3c\nI2˜Ω3\u0015\n.(26)\nAs we saw above, the three rows of the matrix Rijare obtained from Eqs. (25) if we adjust the integration constants\nAandk0with initial dates as indicated below Eq. (16). Solving the equations (26) with these dates, we get, in each\ncase\nA=I3\nI2b˜Ω3, k 0=π\n2; A=1\n˜Ω3, k 0= 0; A=−I1a\nI2b˜Ω3, k 0=π\n2. (27)5\nSubstituting these values into Eqs. (25), we get the rotation matrix R. The most compact expression for it appears\nif instead of the parameters aandbwe introduce unit vector ˆmin the direction of conserved angular momentum m.\nAccording to Eqs. (4) and (8), it is expressed in terms of aandbas follows: ˆm= ( ˆm1,0,ˆm3)T≡m\n|m|=IΩ(0)\n|IΩ(0)|=\n(I1a\nI2b,0,I3\nI2b)T. In terms of the moments of inertia it is\nˆm= \nI1\nI2s\nI3−2\nI3−1,0,I3\nI2s\nI2−1\nI3−1!T\n. (28)\nWith these notation, the rotation matrix R+for the movement along the separatrix reads as follows\n\nˆm2\n1\ncoshωt+ ˆm2\n3coskt+ ˆm1ˆm3sinkttanhωt−ˆm3sinkt\ncoshωt+ ˆm1tanhωtˆm1ˆm3(1\ncoshωt−coskt) + ˆm2\n3sinkttanhωt\nˆm3sinkt−ˆm1coskttanhωtcoskt\ncoshωt−ˆm1sinkt−ˆm3coskttanhωt\nˆm1ˆm3(1\ncoshωt−coskt)−ˆm2\n1sinkttanhωtˆm1sinkt\ncoshωt+ ˆm3tanhωtˆm2\n3\ncoshωt+ ˆm2\n1coskt−ˆm1ˆm3sinkttanhωt\n(29)\nRotation matrix R−for the separatrix with ϵ=−1 is obtained from this expression replacing ˆ m1on−ˆm1. As it\nshould be, the obtained matrix is orthogonal at any t, in particular Rij(0) = δij. The final expression admits the\nlimit of a symmetrical body with I3=I2> I1. This could be a fairly prolated cylinder of uniform density. In this\nlimit Eq. (29) turn into the expected expression\n\ncos˜Ω3t−sin˜Ω3t0\nsin˜Ω3tcos˜Ω3t0\n0 0 1\n, (30)\nthat describe a rotation of the cylinder around its transverse axis ˆm= (0,0,1).\nThe movement (29) is determined by the frequency\nk≡s\nI3−1\nI2−1˜Ω3, (31)\nand by the damping factor\nω≡p\nI3−2I3−1˜Ω3. (32)\nTo estimate the latter, let us consider the homogeneous rectangular plate of sides a= 7 cm., b= 4 cm., c= 2 cm.,\nand with ˜Ω3= 2π×5 rad/sec. Its inertia moments are [4]: I3=µ\n12(a2+b2) =µ\n1265,I2=µ\n12(a2+c2) =µ\n1253 and\nI1=µ\n12(b2+c2) =µ\n1220. Then cosh−1(ωt) = cosh−1(22.423t) and tanh( ωt) = tanh(22 .423t). In particular, at t= 0.3\nsec. we get cosh−1(22.423×0.3) = 0 ,0023 and tanh(22 .423×0.3) = 0 .999997. The graphs of these functions are\npresented in Figure 2. The function cosh−1(ωt) makes a sharp jump on the interval t∈[−0.3,0.3], and is practically\nequal to zero outside it. The function tanh( ωt) also makes a sharp jump approximately on the same interval, and is\npractically equal to ±1 outside it.\nIn the limit t→ ±∞ the rotation matrix (29) acquires the form\nR+\nij(±∞) =\nˆm2\n3coskt±ˆm1ˆm3sinkt±ˆm1−ˆm1ˆm3coskt±ˆm2\n3sinkt\nˆm3sinkt∓ˆm1coskt 0 −ˆm1sinkt∓ˆm3coskt\n−ˆm1ˆm3coskt∓ˆm2\n1sinkt±ˆm3ˆm2\n1coskt∓ˆm1ˆm3sinkt\n. (33)\nThis is the rotation of the frequency karound the vector of conserved angular momentum ±m. That is in this limit\nthe intermediate axis R2(t) occupies the position of ±m, while other two axes rotate around it.\nIV. CONCLUSION. QUALITATIVE DISCUSSION OF THE ROTATION MATRIX AND\nDZHANIBEKOV EFFECT.\nWe solved Poisson equations (2) for an asymmetrical body with angular velocity vector lying on a separatrix,\nobtaining the expression (29) for the rotation matrix in elementary functions. The resulting matrix depends on two6\n-0.8-0.8-0.6 -0.6-0.4 -0.4-0.2 -0.20.2 0.20.4 0.40.6 0.60.8 0.81 11.2 1.2-2\n-2-1.5-1.5-1-1-0.5-0.50.50.5111.51.50\n0ggf\nf\nFigura 2: Graphs of the damping functions cosh−1(22.423t) and tanh(22 .423t).\nparameters which determine the frequency and asymptotic damping of the solution at t→ ±∞ . The motion guided\nby this matrix has a simple physical meaning: the body in the process of motion tends to superpose its axis of\ninstantaneous rotation with the direction of conserved vector of angular momentum. To see this we consider, for\ndefiniteness, the homogeneous rectangular plate described below Eq. (32). Consider the plate that at t= 0 has\nangular velocity Ω(0) = ( ϵa˜Ω3,0,˜Ω3>0)T. According to (4), the conserved angular momentum of the plate is\nm= (I1a˜Ω3,0, I3˜Ω3). Both vectors lie in the plane of Laboratory vectors e1,e3, see Figure 1. The intermediate\naxisR2(t) of the plate is the second column of the rotation matrix (29). Let us analyse its time evolution during the\ninterval t∈[−t1, t1] where t1>>1. During the interval t∈[−t1,−0.3] we can approximate the axis as follows\nR2(t) =\u0012\nˆm1tanhωt−ˆm3sinkt\ncoshωt,coskt\ncoshωt,ˆm3tanhωt+ˆm1sinkt\ncoshωt\u0013T\n≈\n(−ˆm1−0.002 ˆm3sinkt,0.002 cos kt,−ˆm3+ 0.002 ˆm1sinkt)T. (34)\nThis means that the intermediate axis moves in a vicinity of the vector −m= (−ˆm1,0,−ˆm3) in an expanding spiral\nwith a very small step. Next, during the interval t∈[−0.3,0.3] of 0 .6 sec., the intermediate axis abruptly changes\nits position (see Figure 2), ending up in the vicinity of the vector m= ( ˆm1,0,ˆm3). During the interval t∈[0.3, t1]\nwe can approximate the axis as follows\nR2(t) =\u0012\nˆm1tanhωt−ˆm3sinkt\ncoshωt,coskt\ncoshωt,ˆm3tanhωt+ˆm1sinkt\ncoshωt\u0013T\n≈\n( ˆm1−0.002 ˆm3sinkt,0.002 cos kt, ˆm3+ 0.002 ˆm1sinkt)T. (35)\nThis means that the intermediate axis asymptotically approaches to the conserved angular momentum m= ( ˆm1,0,ˆm3)\nin a tapering spiral.\nIn resume, our solution (29) for the rotation matrix in elementary functions, that corresponds to the angular velocity\nvector moving along a separatrix, describe a single-jump Dzhanibekov effect. The trajectories lying near the separatrix\nare time-periodic functions [4]. So it is expected that an ideal body along such a trajectory will experiences multiple\njumps at regular intervals, that is the Dzhanibekov effect.\nAcknowledgments\nThe work has been supported by the Brazilian foundation CNPq (Conselho Nacional de Desenvolvimento Cient´ ıfico\ne Tecnol´ ogico - Brasil).7\nV. APPENDIX\nThe rotation matrix has been constructed above without assuming any kind of its parametrization like Euler angles.\nThe rotation matrix in terms of Euler angles reads as follows:\nR=\ncosψcosφ−sinψcosθsinφ−sinψcosφ−cosψcosθsinφsinθsinφ\ncosψsinφ+ sin ψcosθcosφ−sinψsinφ+ cos ψcosθcosφ−sinθcosφ\nsinψsinθ cosψsinθ cosθ\n, (36)\nand then the problem reduces to the search for their dependence on time. To avoid possible confusion, here we discuss\nsome peculiarities, that should be taken into account when working with the Euler angles.\nLet us assume that we solved the Euler equations for some given initial conditions\nω(0) = Ω(0) = ( ˜Ω1,˜Ω2,˜Ω3). (37)\nAccording to Eq. (3), we can write the following vector equality with known left hand side: Ω(t) =\nI−1RT(θ(t), φ(t), ψ(t))m. In components we get\nΩ1(t) =m1\nI1[cosψcosφ−sinψcosθsinφ] +m2\nI1[cosψsinφ+ sin ψcosθcosφ] +m3\nI1sinψsinθ,\nΩ2(t) =m1\nI2[−sinψcosφ−cosψcosθsinφ] +m2\nI2[−sinψsinφ+ cos ψcosθcosφ] +m3\nI2cosψsinθ,\nΩ3(t) =m1\nI3sinθsinφ−m2\nI3sinθcosφ+m3\nI3cosθ. (38)\nSo it is remain to resolve these algebraical equalities with respect to the Euler angles. Let us try to do this in accordance\nwith the following widely used reasoning. Before solving the equations (38), we simplify them as follows. We take the\nLaboratory axis e3in the direction of constant in space vector of angular momentum m. Then m= (0,0, M), where\nM=|m|, and Eqs. (38) acquire the form\nMsinψsinθ=I1Ω1(t)≡M1, M cosψsinθ=I2Ω2(t)≡M2, M cosθ=I3Ω3(t)≡M3, (39)\nwhere Mi(t) are components of angular momentum in the body-fixed frame3. Using the previously obtained\nsolutions Ω i(t) to Euler equations, we immediately get two of the desired quantities: cos θ(t) = I3Ω3(t)/M,\ntanψ(t) =I1Ω1(t)/I2Ω2(t).\nTo understand, what is wrong here, note the following. Since we are working with equations in which the inertia\ntensor is diagonal, then by choosing the third axis of Laboratory in the direction of m, we also have the third axis of\ninertia R3(0) in the same direction. Besides, from Rij(0) = δijtogether with Eq. (4) we get\nω(0) = I−1m= (0,0, M/I 3)∥R3(0). (40)\nThat is, under the assumptions made above, the initial angular velocity of the body turns out to be directed along\nthe third axis of inertia. But in this case, the subsequent evolution of an ideal body is just a stationary rotation\naround this axis. It is clear, that calculations made using the equations (39) have nothing to do with this movement.\nComparing (37) with (40), we conclude that this occurred because the Euler equations were solved in a coordinate\nsystem different from that chosen to simplify the Poisson equations (38). In other words, writing the equations (39),\nthe components of two vectors taken in two different coordinate systems were equalized. More detailed discussion\naround these points can be found in [17] and [18].\n[1] E. T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies , (Cambridge: at the University press,\n1917).\n[2] W. D. MacMillan, Dynamics of rigid bodies , (Dover Publications Inc., New-York, 1936).\n[3] E. Leimanis, The general problem of the motion of coupled rigid bodies about a fixed point , (Springer-Verlag, 1965).\n[4] L. D. Landau and E. M. Lifshitz, Mechanics , Volume 1, third edition, (Elsevier, 1976).\n3These equalities should be compared with Eqs. (37.13) in [4].8\n[5] F. L. Chernousko, L. D. Akulenko, D. D. Leshchenko, Evolution of motions of a rigid body about its center of mass ,\n(Springer, 2017).\n[6] H. M. Yehia, Rigid body dynamics. A Lagrangian approach , Advances in Mechanics and Mathematics, V. 45, Birkh¨ auser,\n2022.\n[7] M. S. Ashbaugh, C. C. Chicone, R. H. Cushman, The twisting tennis racket , Dyn. Diff. Equat. 3(1991) 67-85.\n[8] H. Kinoshita, Analytical expansions of torque-free motions for short and long axis modes , Celestial Mech. Dyn. Astron. 53\n(1992) 365-375.\n[9] A. G. Petrov, S. E. Volodin, Janibekov’s effect and the laws of mechanics , Dokl. Phys., 58, N 8 (2013) 349-353.\n[10] V. F. Zhuravlev, G. M. Rozenblat, Estimates of solutions during motion of the Euler-Poinsot top and explanation of the\nexperiment with Dzhanibekov’s nut , Rus. J. Nonlin. Dyn., 16N 3, (2020) 517-525.\n[11] C. Murakami, Analytical solution of the Euler-Poinsot problem , J. Geom. Symmetry Phys. 60 (2021) 25-46;\narXiv:2103.16056.\n[12] S. Sailer, S. R. Eugster and R. I. Leine, The tippedisk: a tippetop without rotational symmetry , Regular and Chaotic\nDynamics, 25N 6 (2020) 553.\n[13] C. Peterson, and W. Schwalm, Euler’s rigid rotators, Jacobi elliptic functions, and the Dzhanibekov or tennis racket effect ,\nAmerican Journal of Physics 89(2021) 349.\n[14] V. V. Koryanov and A. S. Kukharenko, Influence of the inertia resultant moments inequality on the aerial vehicle rotational\nmotion , Engineering Journal: Science and Innovation, 2(2023) 134.\n[15] P. Trivailo and H. Kojima, Enhancement of the attitude dynamics capabilities of the spinning spacecraft using inertial\nmorphing , Aeronautica Journal, 1241276, (2020) 838-871.\n[16] A. A. Deriglazov, Euler-Poisson equations of a dancing spinning top, integrability and examples of analytical solutions ,\nCommun. Nonlinear Sci. Numer. Simulat. 127(2023) 107579; arXiv:2307.12201.\n[17] A. A. Deriglazov, Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system ,\nEur. J. Phys. 44(2023) 065001; arXiv:2301.10741.\n[18] A. A. Deriglazov, Comment on the Letter ”Geometric Origin of the Tennis Racket Effect” by P. Mardesic, et al, Phys.\nRev. Lett. 125, 064301 (2020) , arXiv:2302.04190."    },    {        "title": "2401.11536v1.Nonlinear_Model_Predictive_Detumbling_of_Small_Satellites_with_a_Single_axis_Magnetorquer.pdf",        "content": "NonlinearModelPredictiveDetumblingofSmallSatelliteswitha\nSingle-axis Magnetorquer\nKota Kondo∗\nKyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan\nIlya Kolmanovsky†\nUniversity of Michigan, Ann Arbor, Michigan 48109\nYasuhiro Yoshimura‡\nKyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan\nMai Bando§\nKyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan\nShuji Nagasaki¶\nKyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan\nToshiya Hanada‖\nKyushu University, Fukuoka-shi, Fukuoka, 819-0395, Japan\nNomenclature\n𝑩,𝑩0= Earth’s magnetic field vector in body and orbital frame, respectively\n𝐻 = Hamiltonian\n𝑖 = inclination\n𝐽cost = cost function\n𝑱 = moment of inertia\n𝒎 = magnetic dipole moment\n𝑚max = maximum control input of magnetic torquer\n𝑚𝑥 = magnetic moment of the single magnetic torquer along the 𝑥-axis\n𝑁 = discretized step number on prediction horizon\n𝑸,𝑅1,𝑅2= weight matrices\n𝑸t = terminal cost\n𝑟 = distance between the satellite and the center of the Earth\nRelated work, \"Model Predictive Approach for Detumbling an Underactuated Satellite,\" presented as AIAA 2020-1433 at AIAA Scitech 2020\nForum, Orlando, FL, 6-10 January 2020\n∗Student, Department of Mechanical and Aerospace Engineering; kongon1004@gmail.com, AIAA Student Member\n†Professor, Department of Aerospace Engineering; ilya@umich.edu, AIAA Associate Fellow.\n‡Assistant Professor, Department of Aeronautics and Astronautics.\n§Associate Professor, Department of Aeronautics and Astronautics.\n¶Assistant Professor, Department of Aeronautics and Astronautics.\n‖Professor, Department of Aeronautics and Astronautics.arXiv:2401.11536v1  [eess.SY]  21 Jan 2024𝑻 = control torque vector\n𝑇𝑠 = prediction horizon\n𝑼 = optimal input matrix\n𝑉 = Lyapunov function\n𝑣 = dummy input\n𝜆 = Lagrange multiplier\n𝝎 = angular velocity vector\n𝜔𝑒 = argument of perigee\n𝜇 = Lagrange multiplier for equality constraint\n𝜃 = true anomaly\nSubscripts\n𝑖 =𝑖-th time step on prediction horizon\n∗ = conditions on prediction horizon\nI. Introduction\nVariousactuators are used in spacecraft to achieve attitude stabilization, including thrusters, momentum wheels,\nand control moment gyros. [ 1–3]. Small satellites, however, have stringent size, weight, and cost constraints,\nwhich makes many actuator choices prohibitive. Consequently, magnetic torquers have commonly been applied to\nspacecrafttoattenuateangularrates[ 4,5]. Approachesfordealingwithunder-actuationduetomagneticcontroltorques\ndependency on the magnetic field and required high magnetic flux densities have been previously considered in [6, 7].\nGenerallyspeaking,controlofasatellitethatbecomesunder-actuatedasaresultofon-boardfailureshasbeena\nrecurrentthemeintheliterature,seee.g.,[ 8,9]andreferencestherein. Methodsforcontrollingspacecraftwithfewer\nactuatorsthandegrees offreedomareincreasinglyin demandduetotheincreased numberofsmallsatellitelaunches\n[10].\nMagnetictorquershavebeenextensivelyinvestigatedformomentummanagementofspacecraftwithmomentum\nwheels [11] and for nutation damping of spin satellites [ 12], momentum-biased [ 13], and dual-spin satellites [ 14].\nNonetheless, severely under-actuated small spacecraft that carry only a single-axis magnetic torquer have not been\npreviously treated.\nThisnoteconsidersthedetumblingofasmallspacecraftusingonlyasingle-axismagnetictorquer. Evenwitha\nthree-axis magnetictorquer, the spacecraftis under-actuated,while, in thecase ofonlya singleaxis magnetic torquer,\ntheproblemisconsiderablymoredemanding. Ournoteexaminesthefeasibilityofspacecraftattitudecontrolwitha\nsingle-axis magnetic torquer and possible control methods that can be used.\n2Ourspecific contributionsareasfollows. We demonstrate,throughanalysis andsimulations,that theconventional\nB-dot algorithm for spacecraft detumbling fails with a single-axis magnetic torquer. Also, there has not been any\nprevious analysis of a satellite’s controllability and stabilizability with a single magnetic actuator. We discuss these\nproperties; this discussion motivates consideration of more advanced control approaches such as Nonlinear Model\nPredictive Control (NMPC) [ 15,16]. Closed-loop simulation results with NMPC are reported, which illustrate the\npotential of NMPC to perform spacecraft detumbling with a single-axis magnetic torquer. These developments, which\nshow the improved capability to detumble the spacecraft with NMPC as compared to the classical B-dot law in the case\nof a single magnetic torquer, contribute to advancements in small satellite technology.\nII. Spacecraft Rotational Dynamics\nThe body-fixed frame of a rigid spacecraft is assumed to be located at the center of mass and to be aligned with the\nprincipal axes of inertia. The evolution of body frame components of spacecraft angular velocity vector is described by\nthe classical Euler’s equations [17]:\n𝑱¤𝝎+𝝎×𝑱𝝎=𝑻 (1)\nAsingle-axismagnetictorquerinteractswiththeEarth’slocalmagneticfieldandgeneratescontroltorqueaccording\nto [18]:\n𝑻=𝒎×𝑩 (2)\nGiven a single-axis magnetic torquer with the coil along the 𝑥-axis, the control torque with the single-axis\nmagnetorquer is written as 𝑻=[0,−𝐵𝑧𝑚𝑥, 𝐵𝑦𝑚𝑥]𝑇. Thus, no torque is generated about the 𝑥-axis, which makes\nangular rate stabilization challenging. By aggregating the above equations, we obtain\n¤𝜔𝑥\n¤𝜔𝑦\n¤𝜔𝑧=1\n𝐽𝑥(𝐽𝑦−𝐽𝑧)𝜔𝑦𝜔𝑧\n1\n𝐽𝑦(𝐽𝑧−𝐽𝑥)𝜔𝑧𝜔𝑥−𝐵𝑧𝑚𝑥\n𝐽𝑦\n1\n𝐽𝑧(𝐽𝑥−𝐽𝑦)𝜔𝑥𝜔𝑦+𝐵𝑦𝑚𝑥\n𝐽𝑧(3)\nwhere𝑚𝑥is the control input.\nIII. Detumbling Control Law\nA. B-dot Algorithm\nThe conventional approach to detumbling small satellites with magnetic torquers exploits the B-dot algorithm [ 19].\nTheB-dotalgorithm’sprincipleistoadddampingthroughcontrolmoments,whichleadstoareductioninspacecraft\nangular velocities. This section analyzes closed-loop stability with the B-dot law in the case of a single-axis magnetic\n3torquer. Considering the Euler’s equations for spacecraft dynamics given is Eq. (1), define a Lyapunov function [ 19] as\n𝑉(𝝎)=1\n2𝝎𝑇𝑱𝝎 (4)\nThisLyapunovfunctionispositiveeverywhereexceptwhen 𝝎=0(i.e.,theequilibriumpoint). Withtheuseof\nEq. (1), its time derivative along trajectories of the system is found as\n¤𝑉(𝝎)=𝝎𝑇𝑱¤𝝎 (5)\n=𝝎𝑇(−𝝎×𝑱𝝎+𝑻) (6)\n=𝝎𝑇𝑻 (7)\nThe conventional B-dot feedback law in [ 20,21] generates each axis magnetic dipole moment as follows. For\ninstance, for the 𝑥-axis,\n𝑚𝑥=−𝑚max¤𝐵𝑥\n||¤𝐵𝑥||(8)\nwhere𝑚maxis the maximum magnitude of the magnetic dipole moment.\nThe time derivative of Earth’s magnetic field vector 𝑩with respect to an inertial frame is given by\n𝐼¤𝑩=¤𝑩+𝝎×𝑩 (9)\nwhere the left superscript 𝐼on𝑩indicates ”with respect to an inertial frame.” Assuming sufficiently large angular\nvelocity, 𝝎,sothat𝐼¤𝑩issmallenoughinmagnitudecomparedto ¤𝑩,wherethelatteristhederivativeofEarth’smagnetic\nfield vector with respect to a body fixed frame, Eq. (9) can be approximated as [4, 21].\n0≈¤𝑩+𝝎×𝑩 (10)\n⇒¤𝑩≈ − 𝝎×𝑩 (11)\nFollowing [ 22], which treated the case of three single-axis magnetic torquers, suppose we proceed with closed-loop\nstabilizability analysis by computing the time derivative of 𝑉along closed-loop system trajectories. In the case of a\n4single-axis magnetic actuation, we obtain\n¤𝑉(𝝎)=−(𝝎×𝑩)𝑇𝒎 (12)\n=−𝑚max\n||¤𝑩||(𝝎×𝑩)𝑇[𝜔𝑦𝐵𝑧−𝜔𝑧𝐵𝑦,0,0]𝑇(13)\n=−𝑚max\n||¤𝑩||(𝜔𝑦𝐵𝑧−𝜔𝑧𝐵𝑦)2(14)\nItisclearthat,although ¤𝑉(𝝎)≤0,theexpressionfor ¤𝑉doesnotdependon 𝜔𝑥. Furthermore,since 𝑉(𝑡)=𝑉(𝝎(𝑡))\nisanon-increasingfunctionof 𝑡,𝝎isbounded. Moreover ¥𝑉isacontinuousfunctionof 𝑩and𝝎,whicharebounded.\nHence,¤𝑉(𝑡)is uniformly continuous in time [ 23]. Therefore, by Barbalat’s lemma, we conclude lim𝑡→∞¤𝑉(𝑡)=0,\nwhichindicatesthatinthelimitas 𝑡→∞either𝜔𝑦𝐵𝑧=𝜔𝑧𝐵𝑦=0or𝜔𝑦=𝜔𝑧=0. Intheeithercase, since 𝜔𝑥canbe\narbitrary, B-dot algorithm appears to be incapable of detumbling a satellite with a single-axis magnetic actuation. This\nis confirmed by our subsequent numerical simulations.\nB. Controllability Analysis\nThissectiondiscussesthecontrollabilitypropertiesofthesatelliteangularvelocitydynamicswiththesingle-axis\nmagneticactuation. Notethatboththreeandtwo-axismagneticallyactuatedsatelliteshavebeenshowntobecontrollable\n[24,25]. Thesingle-axismagneticactuationcaseismorechallengingsincethereisonlyasinglecontrolinput;thiscase\nhas not previously been addressed in the literature.\nForthecaseofsingle-axismagneticactuation,localweakcontrollabilityinthesenseof[ 26,27]canbedemonstrated.\nThe weak local controllability is necessary for local controllability. It implies that the set of reachable states at a given\ntime from a given state starting at another given time instant contains an open neighborhood of the state space. Clearly,\nweak local controllability is necessary but not sufficient for local controllability.\nNote that equations of motion (3) can be written as\n¤𝝎=𝑓0(𝝎,𝑡)+𝑓1(𝝎,𝑡)𝑚𝑥 (15)\n=1\n𝐽𝑥(𝐽𝑦−𝐽𝑧)𝜔𝑦𝜔𝑧\n1\n𝐽𝑦(𝐽𝑧−𝐽𝑥)𝜔𝑧𝜔𝑥\n1\n𝐽𝑧(𝐽𝑥−𝐽𝑦)𝜔𝑥𝜔𝑦+0\n−𝐵𝑧\n𝐽𝑦\n𝐵𝑦\n𝐽𝑧𝑚𝑥. (16)\nThe system is time-varyingas 𝐵𝑦and𝐵𝑧depend on the spacecraftposition in orbit and hence on time. The necessary\nand sufficient conditions for local weak controllability of a time-varying nonlinear system can be obtained by extending\nthe state vector with the time, 𝑡, as an extra state with the dynamics ¤𝑡=1, and analyzing the resulting autonomous\nsystem. This leads to conditions such as Theorem 4 in [28] which we adopt here.\n5Define the operators ⟨𝜉,𝜂⟩and[𝜉,𝜂]for two time-varying vector fields 𝜉an𝜂as\n⟨𝜉,𝜂⟩≡[𝜉,𝜂]−𝜕𝜉\n𝜕𝑡, (17)\n[𝜉,𝜂]=𝜕𝜂\n𝜕𝜔𝜉−𝜕𝜉\n𝜕𝜔𝜂. (18)\nNote that[𝜉,𝜂]is the conventional Lie Bracket. The controllability distribution △can now be defined for our\ntime-varying nonlinear system based on Algorithm 1.\nAlgorithm 1: Controllability distribution for time-variant nonlinear systems\nSet△0=span{𝑓1}and𝑘=0;\nwhile dim(△𝑘⊕⟨△𝑘, 𝑓0⟩⊕[△𝑘, 𝑓1])>dim(△𝑘)do\nSet△𝑘+1=△𝑘⊕⟨△𝑘, 𝑓0⟩⊕[△𝑘, 𝑓1];\n𝑘=𝑘+1;\nend\n⊕sums up the span of two vector fields.\nThenecessaryandsufficientconditionsforlocalweakcontrollabilityin[ 28]leadtothefollowingresult: Algorithm1\nconverges in at most 2steps for the system (15) and △2is nonsingular and has rank 3at a given 𝝎0and𝑡0if and only if\nthe system is locally weakly controllable from ( 𝝎0,𝑡0).\nBy examining the form of △2(see calculations in the Appendix), we observe that 𝐽𝑦≠𝐽𝑧(unequal moments of\ninertia about the two principal axes which are orthogonal to the axis along which the magnetic actuator is aligned) is a\nnecessary condition for weak local controllability (and hence for local controllability). This is also apparent from (15)\nas the angular velocity component about 𝑥-axis becomes decoupled from the rest of the dynamics if 𝐽𝑦=𝐽𝑧.\nFurthermore, the numerical evaluation of △2along the orbits used in our NMPC simulations with IGRF model (36)\nfor𝐵𝑦and𝐵𝑧confirms that the rank of △2is equal to 3and hence the system is locally weakly controllable.\nWe note that we cannot establish a stronger property of (small-time) local controllability of the satellite with\na single-axis magnetic actuation with the above analysis. While it appears to hold in our numerical simulations,\nsuch property is much harder to demonstrate and is left to future research. Nevertheless, conditions for local weak\ncontrollability are necessary for local controllability and hence are useful.\nC. Stabilizability Analysis\nExisting approaches [ 17,24,29,30] to stabilizing the spacecraft with magnetic actuation typically rely on the\napplicationofthetheoryofaveraging[ 31]. Followingthisrouteinthecaseofsingle-axismagneticactuationencounters\ntechnical difficulties as we now illustrate.\n6Consider the equations of motion given by Eq. (15) and suppose a control law of the form,\n𝑚𝑥=𝜖2¯𝑚𝑥(𝝎\n𝜖,𝑡) (19)\nhas been specified, where 𝜖 >0is a small parameter. Let\n𝒉=𝝎\n𝜖=\u0014\nℎ𝑥ℎ𝑦ℎ𝑧\u0015T\n,\nthen\n¤ℎ𝑥=𝜖1\n𝐽𝑥(𝐽𝑦−𝐽𝑧)ℎ𝑦ℎ𝑧\n¤ℎ𝑦=𝜖1\n𝐽𝑦(𝐽𝑧−𝐽𝑥)ℎ𝑧ℎ𝑥−𝜖\n𝐽𝑦𝐵𝑧(𝑡)¯𝑚𝑥(𝒉,𝑡) (20)\n¤ℎ𝑧=𝜖1\n𝐽𝑧(𝐽𝑥−𝐽𝑦)ℎ𝑥ℎ𝑦+𝜖\n𝐽𝑧𝐵𝑦(𝑡)¯𝑚𝑥(𝒉,𝑡)\nis in the form to which the theory of averaging [31] can be applied. Letting\n𝑢𝑦(𝒉)=−𝜖\n𝐽𝑦lim\n𝑇→∞1\n𝑇∫𝑇\n0𝐵𝑧(𝑡)¯𝑚𝑥(𝒉,𝑡)𝑑𝑡\n𝑢𝑧(𝒉)=𝜖\n𝐽𝑧lim\n𝑇→∞1\n𝑇∫𝑇\n0𝐵𝑦(𝑡)¯𝑚𝑥(𝒉,𝑡)𝑑𝑡\nand assuming that the limits exist, the averaged version of Eq. (20) has the form,\n¤¯ℎ𝑥=𝜖1\n𝐽𝑥(𝐽𝑦−𝐽𝑧)¯ℎ𝑦¯ℎ𝑧\n¤¯ℎ𝑦=𝜖1\n𝐽𝑦(𝐽𝑧−𝐽𝑥)¯ℎ𝑧¯ℎ𝑥+𝑢𝑦(¯𝒉) (21)\n¤¯ℎ𝑧=𝜖1\n𝐽𝑧(𝐽𝑥−𝐽𝑦)¯ℎ𝑥¯ℎ𝑦+𝑢𝑧(¯𝒉)\nBy the straightforward application of Brockett’s necessary condition [ 32], the averaged system given by Eq. (21)is\nnotsmoothlyorevencontinuouslystabilizable,i.e.,astabilizingtime-invariantcontrollawgivenby 𝑢𝑦(¯𝒉),𝑢𝑧(¯𝒉),if\nexists, would have to be discontinuous. This conclusion is consistent with the interpretation of the averaged system\ndynamicsassimilartoarigidbodycontrolledbytwocontroltorques;suchasystemisknowntobenotsmoothlyoreven\ncontinuously stabilizable by time-invariant feedback laws. Consequently, ¯𝑚𝑥(𝝎\n𝜖,𝑡)would have to be discontinuous as a\nfunction of 𝝎. Unfortunately, the theory of averaging [ 31,33] assumes smoothness (twice continuous differentiability)\nin [31] of the right hand side of ordinary differential equations being averaged.\n7Hence, there is a complication in using the classical averaging theory to develop conventional control laws for the\ncaseofsingle-axismagneticactuation. Ontheotherhand,NMPC,ifsuitablyformulated,isabletostabilizesystems\nthat are not smoothly or even continuously stabilizable, including underactuated spacecraft [34].\nD. NMPC Formulation\nThis section formulates an NMPC approach to detumbling the satellite with the nonlinear dynamics represented by\nEq.(3) based on the following receding horizon optimal control problem,\nminimize𝐽cost=1\n2𝝎T(𝑡+𝑇𝑠)𝑸t𝝎(𝑡+𝑇𝑠)+∫𝑡+𝑇𝑠\n𝑡{1\n2(𝝎T(𝜏)𝑸𝝎(𝜏)+𝑅1𝑚2\n𝑥(𝜏))−𝑅2𝑣(𝜏)}𝑑𝜏\nsubject to 𝑱¤𝝎+𝝎×𝑱𝝎=𝑻\n𝑻=𝒎×𝑩,where 𝒎=[𝑚𝑥,0,0]T\n𝑚2\n𝑥+𝑣2−𝑚2\nmax=0(22)\nwhere𝑡is the current time instant, 𝑇𝑠is the prediction horizon, 𝑸,𝑸t,𝑅1, and𝑅2are positive-definite weight matrices,\nand𝑸tisterminalcost. Theauxiliaryinput, 𝑣,isintroducedfollowing[ 35]toenforcethecontrolconstraintsbyrecasting\nthem as equality constraints in Eq. (22). The negative sign preceding 𝑅2in the cost function being minimized promotes\nkeeping𝑣positive and control constraints strictly satisfied. This receding horizon optimal control problem is chosen as\nit is synergistic with the continuation/generalized minimal residual method (C/GMRES) method [ 36]. Reference [ 37]\nprovides a comparison of different strategies to handle inequality constraints in such a setting.\nE. NMPC with C/GMRES Algorithm\nThe C/GMRES method [ 36] is applied to design NMPC based on Eq. (22). As C/GMRES method has small\ncomputational footprint, its use is advantageous in small satellites with limited computational and electric power.\nFollowing C/GMRES method, the problem is first discretized as follows:\n𝝎∗\n𝑖+1(𝑡)=𝝎∗\n𝑖(𝑡)+𝑓(𝝎∗\n𝑖(𝑡),𝒖∗\n𝑖(𝑡))Δ𝜏 (23)\n𝝎∗\n0(𝑡)=𝝎(𝑡) (24)\n𝐶(𝝎∗\n𝑖(𝑡),𝒖∗\n𝑖(𝑡))=0 (25)\n𝐽cost=𝜓(𝝎∗\n𝑁(𝑡))+𝑁−1∑︁\n𝑖=0𝐿(𝝎∗\n𝑖(𝑡),𝒖∗\n𝑖(𝑡))Δ𝜏 (26)\nwhere𝑓(𝝎,𝒖)istherighthandsideofequationsofmotioninEq. (22),𝐶(𝝎,𝒖)istheequalityconstraintinEq. (22),\n𝐿(𝝎,𝒖)=1\n2(𝝎𝑇𝑸𝝎+𝑅1𝑚2\n𝑥)−𝑅2𝑣,andΔ𝜏=𝑇𝑠/𝑁. Setting the initial state of the discretized problem to the current\n8angularvelocityvectoras 𝝎∗\n0(𝑡)=𝝎(𝑡),asequenceofcontrolinputs {𝒖∗\n𝑖(𝑡)}𝑁−1\n𝑖=0isfoundateachtimeinstant 𝑡;then\nthe control given to the system is based on the first element of this sequence and is defined as 𝒖(𝑡)=𝒖∗\n0(𝑡).\nThe solution of the discretized problem is based on introducing the Hamiltonian, 𝐻, as\n𝐻(𝝎,𝝀,𝒖,𝝁)=𝐿(𝝎,𝒖)+𝝀𝑇𝑓(𝝎,𝒖)+𝝁𝑇𝐶(𝝎,𝒖) (27)\nwhere 𝝀is the vector of co-states and 𝝁is the Lagrange multiplier associated with the equality constraint. The\nfirst-order necessary conditions for optimality dictate [ 38] that{𝒖∗\n𝑖(𝑡)}𝑁−1\n𝑖=0,{𝝁∗\n𝑖(𝑡)}𝑁−1\n𝑖=0,{𝝀∗\n𝑖(𝑡)}𝑁−1\n𝑖=0, satisfy the\nfollowing conditions:\n𝐻𝒖(𝝎∗\n𝑖(𝑡),𝝀∗\n𝑖+1(𝑡),𝒖∗\n𝑖(𝑡),𝝁∗\n𝑖(𝑡))=0 (28)\n𝝀∗\n𝑖(𝑡)=𝝀∗\n𝑖+1(𝑡)+𝐻𝑇\n𝝎(𝝎∗\n𝑖(𝑡),𝝀∗\n𝑖+1(𝑡),𝒖∗\n𝑖(𝑡),𝝁∗\n𝑖(𝑡))Δ𝜏 (29)\n𝝀∗\n𝑁(𝑡)=𝜓𝑇\n𝝎(𝝎∗\n𝑵(𝑡)) (30)\nTo determine{𝒖∗\n𝑖(𝑡)}𝑁−1\n𝑖=0and{𝝁∗\n𝑖(𝑡)}𝑁−1\n𝑖=0, which satisfy Eqs.(23–25) and (28–30), we define a vector of the inputs\nand multipliers in Eq. (31) as\n𝑼(𝑡)=[𝑚𝑥∗\n0(𝑡), 𝑣∗\n0(𝑡), 𝜇∗\n0(𝑡), ..., 𝑚𝑥∗\n𝑁−1(𝑡), 𝑣∗\n𝑁−1(𝑡), 𝜇∗\n𝑁−1(𝑡)]𝑇(31)\nThis vector has to satisfy the equation,\n𝑭(𝑼(𝑡),𝝎(𝑡),𝑡)≡(𝜕𝐻\n𝜕𝑢)𝑇(𝝎∗\n0(𝑡),𝑢∗\n0(𝑡),𝜆∗\n1(𝑡),𝑡)\n𝑚2\n𝑥0+𝑣2\n0−𝑚2\nmax=0\n...\n(𝜕𝐻\n𝜕𝑢)𝑇(𝝎∗\n𝑵−1(𝑡),𝑢∗\n𝑁−1(𝑡),𝜆∗\n𝑁(𝑡),𝑡+𝑇)\n𝑚2\n𝑥𝑁−1+𝑣2\n𝑁−1−𝑚2\nmax=0, (32)\nwhere𝐻=1\n2(𝝎𝑇𝑸𝝎+𝑅1𝑚2\n𝑥)−𝑅2𝑣+𝜆𝑥{1\n𝐽𝑥(𝐽𝑦−𝐽𝑧)𝜔𝑦𝜔𝑧}+𝜆𝑦{1\n𝐽𝑦(𝐽𝑧−𝐽𝑥)𝜔𝑧𝜔𝑥−𝐵𝑧𝑚𝑥}\n+𝜆𝑧{1\n𝐽𝑧(𝐽𝑥−𝐽𝑦)𝜔𝑥𝜔𝑦+𝐵𝑦𝑚𝑥}+𝜇{𝑚2\n𝑥+𝑣2−𝑚2\nmax}(33)\nIn C/GMRES [36, 39], Eq. (32), which has to haold at each time instant, 𝑡, is replaced by a stabilized version,\n𝑑\n𝑑𝑡𝑭(𝑼(𝑡),𝝎(𝑡),𝑡)=−𝜁𝑭(𝑼(𝑡),𝝎(𝑡),𝑡) (𝜁 >0) (34)\n9and then by\n𝜕𝑭\n𝜕𝑼¤𝑼(𝑡)=−𝜁𝑭−𝜕𝑭\n𝜕𝑼¤𝝎(𝑡)−𝜕𝑭\n𝜕𝑡(35)\nwhere arguments are omitted.\nFinally,¤𝑈can be determined from Eq. (35) with C/GMRES resulting in a form of a predictor-corrector strategy.\nIV. Simulation Results\nThissectionpresentssimulationresultsofboththeB-dotalgorithmandNMPCforaspacecraftinasun-synchronous\norbit with orbital elements given in Table 1.\nA. Orbital Elements\nTable 1 Six elements of Aeolus (sun synchronous orbit)\nSemi-major axis 6691.6 [km]\nEccentricity 0.00046440\nInclination 96.700[deg]\nRight Ascension of Ascending Node 100.90 [deg]\nArgument of perigee 119.70 [deg]\nMean anomaly 240.49 [deg]\nB. Earth’s Magnetic Field Model\nWeemployInternationalGeomagneticReferenceField(IGRF)[ 40]asanEarth’smagneticmodelinoursimulations.\nHowever, to reduce the onboard computational complexity, NMPC uses a simpler magnetic dipole model described in\nEq. (36) [41].\n𝐵0𝑥\n𝐵0𝑦\n𝐵0𝑧=𝐷𝑚3\n2sin𝑖sin 2𝜂\n−3\n2sin𝑖\u0010\ncos 2𝜂−1\n3\u0011\n−cos𝑖(36)\nwhere𝜂=𝜃+𝜔𝑒,𝐷𝑚=−𝑀𝑒\n𝑟3,𝑀𝑒=8.1×1025[gauss·cm3],𝑟is a distance between the satellite and the center of the\nEarth,𝜃istrueanomaly,and 𝜔𝑒isargumentofperigee. Figure1showsthatthereisaslightdiscrepancybetweenthe\ntwomodelsonthesun-synchronousorbit,which,aswewillseefromthesimulationresults,willnotprecludeNMPC\ncontroller from achieving detumbling.\n10Fig. 1 Magnetic field in dipole and IGRF on the sun-synchronous orbit.\nC. Comparison of B-dot and NMPC on Asymmetric Satellite\nThis section demonstrates NMPC’s advantages over the B-dot algorithm, using a general satellite model whose\nmoments of inertia are given in Table 2. The maximum magnetic moment is set to 1.0 [A ·m2].\nTable 2 Moment of inertia of the asymmetric satellite\nMoments of inertia 𝐽𝑥𝐽𝑦𝐽𝑧\nValue [kg·m2] 0.020 0.030 0.040\n1. The B-dot algorithm\nAs shown in Eq.8, the B-dot law finds control inputs depending the Earth’s magnetic field. Following the common\npractice, to avoid the B-dot law generating unnecessary control inputs, its implementation is based on\n𝑚𝑥= \n0, if¤𝐵𝑥<10−7\n−𝑚max¤𝐵𝑥\n||¤𝐵𝑥||,otherwise(37)\n112. NMPC\nWe here demonstrate NMPC’s capability of detumbling the spacecraft. Simulations are conducted with the NMPC\nparameters listed in Table 3, which were determined by trial and error.\nTable 3 NMPC properties for the asymmetric satellite\n𝑇𝑠 𝑄 𝑄 t 𝑅1𝑅2𝑁Δ𝜏\n10 [sec] diag([ 104,102,5×10]) diag([ 104,102,5×10])10−110−1101.0 [sec]\nwhere𝑇𝑠ispredictionhorizon, 𝑄,𝑄t𝑅1,and𝑅2areweightmatrices, 𝑁isdiscretizedstepnumberalongprediction\nhorizon, and Δ𝜏=𝑇𝑠/𝑁.\nThe initial conditions for the four study cases for which simulation results are reported below are chosen randomly\nwith angular velocity components between -3.0 and 3.0 [deg/s]. The initial angular velocities and the results are all\ngiven in Table 4. These four simulations are representatives of a larger number of simulation case studies that we have\nperformed.\nTable 4 Initial conditions and results\n𝜔𝑥[deg/s]𝜔𝑦[deg/s]𝜔𝑧[deg/s] B-dot detumbled? NMPC detumbled?\nCase 1 2.429286 2.878490 -0.366780 No Yes\nCase 2 -1.576299 -0.246907 2.778531 No Yes\nCase 3 0.047150 -2.486905 -1.425107 Yes Yes\nCase 4 -0.626909 -0.795380 2.927892 No No\nFigure 2 to 5 present the simulation results from the four case studies. When the magnitude of all the angular\nvelocities are less than 0.10 [deg/s], the simulations are set to be terminated in both B-dot and NMPC simulations. The\nsimulations are run for the maximum time of 150 [min].\n12Fig. 2 Case 1: time history of angular velocities.\nAs can be seen in Fig. 2, although 𝜔𝑦and𝜔𝑧are well attenuated, the B-dot algorithm is not able to sufficiently\nreduce𝜔𝑥. NMPC, in contrast, is able to achieve detumbling.\nFig. 3 Case 2: time history of angular velocities.\n13Figure3alsoreportsanexamplewheretheB-dotlawdoesnotachieveallaxesdetumbling,buttheNMPCalgorithm\ndoes. Note that in the beginning of the trajectory, NMPC increases 𝜔𝑧to be able to reduce 𝜔𝑥, which is a challenging\nvariable to control.\nFig. 4 Case 3: time history of angular velocities.\nFigure4isacasewhereboththeB-dotmethodandtheNMPCapproachachievedetumbling. Thisrequiresabout\n150 [min] for the B-dot algorithm, while NMPC is able to achieve this within a much shorter time period.\nFinally,Fig.5showcasesaplotwhereneithertheB-dotnorNMPCareabletodetumblethespacecraftwithinthe\nallocatedtimeof150[min]. InthecaseoftheB-dotalgorithm, 𝜔𝑥persistsatanonzerovalue. NMPC,ontheother\nhand, is able to gradually attenuate all angular velocity components but is not able to fully detumble the spacecraft\nwithin the given time.\n14Fig. 5 Case 4: time history of angular velocities.\nFig. 6 Control input comparison.\nFigs. 6 shows the time histories of control inputs in all the four cases. As can be seen, the NMPC controller requires\nmuch smaller control inputs. This can be advantageous in terms of reduced energy consumption and reduced electrical\n15disturbance.\nV. Conclusions\nThe satellite’s detumbling with only on a single-axis magnetic actuator is a challenging problem, in particular,\nrequiringadifferentapproachtostabilizationthanforthree-axisandtwo-axismagneticactuationsystems. Thenecessary\nconditionsforthecontrollabilityofspacecraftangularvelocitiesinvolve: (1)spacecraftnothavingequalmomentsof\ninertia about the two principal axes that are orthogonal to the axis along which the magnetic actuator is aligned and (2)\nsatisfyingthespecificrankcontrollabilityconditionsderivedinthenote. Thelatterhavebeenshowntoholdnumerically\nfor the spacecraft and the orbit considered in the simulations. The classical B-dot law appears to be incapable of\neliminating spacecraft rotational motion in the simulations, which has also been predicted from the theoretical analysis.\nThe Nonlinear Model Predictive Control (NMPC) strategy based on the continuation/generalized minimal residual\n(C/GMRES) method has been shown to achieve detumbling within the allocated time through simulations in most\ncases. Thepossibilityofdetumblingspacecraftwithonlythesingle-axismagneticcoilopensthepossibilityforsmall\nspacecraft missions with stringent cost and packaging constraints.\nAppendix\nBelow are the controllability distributions in Algorithm 1:\n△0=span{𝑓1} (38)\n=span{[0,−𝐵𝑧\n𝐽𝑦,𝐵𝑦\n𝐽𝑧]𝑇} (39)\n△1=△0⊕⟨△ 0, 𝑓0⟩⊕[△ 0, 𝑓1] (40)\n=span{𝑓1}⊕1\n𝐽𝑥𝐽𝑧𝐵𝑦𝜔𝑦(𝐽𝑦−𝐽𝑧)−1\n𝐽𝑥𝐽𝑦𝐵𝑧𝜔𝑧(𝐽𝑦−𝐽𝑧)\n1\n𝐽𝑦𝜕𝐵𝑧\n𝜕𝑡−1\n𝐽𝑦𝐽𝑧𝐵𝑦𝜔𝑥(𝐽𝑥−𝐽𝑧)\n−1\n𝐽𝑧𝜕𝐵𝑦\n𝜕𝑡−1\n𝐽𝑦𝐽𝑧𝐵𝑧𝜔𝑥(𝐽𝑥−𝐽𝑦)(41)\n=span{𝑓1,𝑔1} (42)\n16where𝑔1is the second term in Eq. (41).\n△2=△1⊕⟨△ 1, 𝑓0⟩⊕[△ 1, 𝑓1] (43)\n=span{𝑓1,𝑔1}⊕span{⟨𝑓1, 𝑓0⟩,⟨𝑔1, 𝑓0⟩}⊕ span{[𝑓1, 𝑓1],[𝑔1, 𝑓1]} (44)\n=span{𝑓1,𝑔1}⊕span{𝑔1,𝑔2}⊕span{0,𝑔3} (45)\n=span{𝑓1,𝑔1,𝑔2,𝑔3} (46)\nwhere𝑔2=⟨𝑔1, 𝑓0⟩and𝑔3=[𝑔1, 𝑓1]. The explicit calculations of 𝑔2and𝑔3give:\n𝑔2=−2\n𝐽𝑥𝐽𝑦𝐽𝑧(𝐽𝑦−𝐽𝑧)(𝐽𝑦𝜕𝐵𝑦\n𝜕𝑡𝜔𝑦−𝐽𝑧𝜕𝐵𝑧\n𝜕𝑡𝜔𝑧)\n−1\n𝐽𝑥𝐽2𝑦𝐽𝑧(𝐽𝑦(𝐵𝑧𝐽2\n𝑥𝜔2\n𝑥−2𝜕𝐵𝑦\n𝜕𝑡𝐽2\n𝑥𝜔𝑥−𝐵𝑧𝐽𝑥𝐽𝑧𝜔2\n𝑥+2𝜕𝐵𝑦\n𝜕𝑡𝐽𝑥𝐽𝑧𝜔𝑥−𝐵𝑧𝐽𝑥𝐽𝑧𝜔2\n𝑧\n+𝜕2𝐵𝑧\n𝜕𝑡2𝐽𝑥𝐽𝑧+𝐵𝑧𝐽2\n𝑧𝜔2\n𝑧)−𝐵𝑧𝐽3\n𝑥𝜔2\n𝑥−𝐵𝑧𝐽3\n𝑧𝜔2\n𝑧+𝐵𝑧𝐽2\n𝑥𝐽𝑧𝜔2\n𝑥+𝐵𝑧𝐽𝑥𝐽2\n𝑧𝜔2\n𝑧)\n1\n𝐽𝑥𝐽𝑦𝐽2𝑧(𝐽𝑧(𝐵𝑦𝐽2\n𝑥𝜔2\n𝑥+2𝜕𝐵𝑧\n𝜕𝑡𝐽2\n𝑥𝜔𝑥−𝐵𝑦𝐽𝑥𝐽𝑦𝜔2\n𝑥−2𝜕𝐵𝑧\n𝜕𝑡𝐽𝑥𝐽𝑦𝜔𝑥−𝐵𝑦𝐽𝑥𝐽𝑦𝜔2\n𝑦\n+𝜕2𝐵𝑦\n𝜕𝑡2𝐽𝑥𝐽𝑦+𝐵𝑦𝐽2\n𝑦𝜔2\n𝑦)−𝐵𝑦𝐽3\n𝑥𝜔2\n𝑥−𝐵𝑦𝐽3\n𝑦𝜔2\n𝑦+𝐵𝑦𝐽2\n𝑥𝐽𝑦𝜔2\n𝑥+𝐵𝑦𝐽𝑥𝐽2\n𝑦𝜔2\n𝑦)(47)\n(48)\n𝑔3=2\n𝐽𝑥𝐽𝑦𝐽𝑧𝐵𝑦𝐵𝑧(𝐽𝑦−𝐽𝑧)\n0\n0(49)\nReferences\n[1]Jin, J., Park, B., Park, Y., and Tahk, M.-J., “Attitude control of a satellite with redundant thrusters,” Aerospace Science\nand Technology , Vol. 10, No. 7, 2006, pp. 644 – 651. https://doi.org/https://doi.org/10.1016/j.ast.2006.04.005, URL\nhttp://www.sciencedirect.com/science/article/pii/S1270963806000514.\n[2]ZhangFan,ShangHua,MuChundi,andLuYuchang,“Anoptimalattitudecontrolofsmallsatellitewithmomentumwheel\nand magnetic torquerods,” Proceedings of the 4th World Congress on Intelligent Control and Automation (Cat. 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Wedeterminethegradientflowscale 𝑡0usingpionandkaonmasses\nanddecayconstantsintheisospinsymmetriclimitofQCDasexternalphysicalinput. Weemploy\nmodel variation techniques to explore the systematic uncertainties in the extraction of the ground\nstate signal of lattice observables, as well as for the continuum-chiral extrapolations. We observe\nthat the combined analysis of the mixed action data with that based on 𝑂(𝑎)-improved Wilson\nvalence quarks, provides an improved control of the extrapolation of 𝑡0to the physical point.\nThe 40th International Symposium on Lattice Field Theory (Lattice 2023)\nJuly 31st - August 4th, 2023\nFermi National Accelerator Laboratory\n∗Speaker\n©Copyright owned by the author(s) under the terms of the Creative Commons\nAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2401.11546v2  [hep-lat]  24 Jan 2024Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\n1. Introduction\nThe validity of the Standard Model of particle physics has been verified in multiple ways\nover the years, through the consistency between its theoretical predictions and the corresponding\nexperimental measurements. However, it is expected that the Standard Model is only an effective\ntheory valid up to some energy scale. Searches for physical phenomena that deviate from its\npredictions are one of the main objectives in contemporary fundamental physics. One of the areas\nwhereNewPhysicsisexpectedtoappearisthequark-flavorsectoroftheStandardModel. Aprecise\ntheoretical determination of the strong interaction effects is needed in order to reduce the oretical\nuncertainties on flavour observables, and to uncover possible inconsistencies between theory and\nexperiment. In this context, Lattice Field Theory is the key tool for calculating non-perturbative\nQCD contributions from first principles.\nWeconsideralatticesetup[1–5]aimedtoaddresstheleadingsystematicuncertaintiesaffecting\ncharm-quarkobservables. Itisbasedonamixed-actionregularisationconsistingofWilsontwisted\nmass valence quarks combined with CLS ensembles with 𝑂(𝑎)-improved Wilson sea quarks [11,\n15]. Furthermore, the slowing down of the sampling of topological sectors at fine lattice spacings\ncan be tamed by the use of open boundary conditions in the time direction [10].\nAfeatureofthislatticeregularisationisthatwhenvalencetwistedmassfermionsaretunedto\nmaximaltwist,anautomatic 𝑂(𝑎)improvement–uptoresidualmasseffectsfrom 𝑢,𝑑,𝑠seaquarks\n–canbeachieved[2,12]. Thisisofparticularrelevancewhenworkingwithheavyquarks. Inthis\nwork we present an update of the use of this lattice formulation in the light (up/down) and strange\nquark sectors. This is a necessary step before studying heavy quark physics, since a matching\nbetween the valence quark masses and the 𝑁𝑓=2+1flavors in the sea is needed. Finally, we can\ncarry out an independent computation of light-quark observables such as the pion and kaon decay\nconstants and perform the scale setting using the gradient flow scale 𝑡0.\n2. Sea-quark sector: 𝑂(𝑎)-improved Wilson fermions\nIn the sea sector, we employ the set of CLS ensembles with open boundary conditions in the\ntime direction collected in Table 1. They employ the Lüscher-Weisz gauge action with 𝑁𝑓=2+1\nflavours of non-perturbatively 𝑂(𝑎)-improved Wilson quarks.\nThese ensembles follow a chiral trajectory defined by a constant trace of the bare sea quark\nmass matrix\nTr\u0000Mq\u0001=mu+md+ms=constant. (1)\nThis ensures that the improved bare coupling ˜𝑔0is kept constant up to 𝑂(𝑎2)effects when varying\nthe quark masses at a fixed coupling 𝛽. However, since this condition is defined for the bare quark\nmasses, we instead opt to follow a renormalised chiral trajectory by imposing that\n𝜙4=8𝑡0\u0012\n𝑚2\n𝐾+1\n2𝑚2\n𝜋\u0013\n=8𝑡0𝑚2\n𝐾+1\n2𝜙2, (2)\nis kept fixed to its physical value. This choice corresponds at LO in 𝜒PT to fixing the sum of the\nrenormalized quark masses, since\n𝜙4∝mR\nu+mR\nd+mR\ns. (3)\n2Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\n𝛽 𝑎[fm] id 𝑚𝜋[MeV]𝑚𝐾[MeV]\n3.40 0.086 H101 420 420\nH102 350 440\nH105 280 460\n3.46 0.076 H400 420 420\n3.55 0.064 N202 420 420\nN203 340 440\nN200 280 460\nD200 200 480\n3.70 0.050 N300 420 420\nN302 340 440\nJ303 260 470\nTable 1:𝑁𝑓=2+1CLSensembles[15]usedintheseasector. Theseensemblesemploynon-perturbatively\n𝑂(𝑎)-improved Wilson fermions and open boundary conditions in the time direction.\nThe target chiral trajectory can be reached through a mass-shift of the simulated quark masses by\nTaylor expanding an observable 𝑂in the bare quark masses [16] as follows\n\n𝑂(𝑚′\n𝑢,𝑚′\n𝑑,𝑚′\n𝑠)\u000b\n=⟨𝑂(𝑚𝑢,𝑚𝑑,𝑚𝑠)⟩+∑︁\n𝑞(𝑚′\n𝑞−𝑚𝑞)𝑑⟨𝑂⟩\n𝑑𝑚𝑞. (4)\nFollowing [21], the sum in eq. (4) is restricted to 𝑞=𝑠since shifting only the strange quark mass\nleads to an improved precision in the target observables. An educated guess for the physical value\n𝑡ph\n0,√︃\n𝑡iter\n0=0.1445(6)fm. (5)\nisselectedasinputofthefinaliterationleadingtothevalueof 𝜙4towhichallobservablesarethen\nmass-shifted. Thisspecificvalueineq.(5),istheoutcomeofthefirststepsofan iterativeprocedure\nstarting from an initial guess of 𝑡0, which is chosen free of uncertainties, and iterates the complete\nscale setting analysis, including correlations, until convergence to the value of 𝑡0is reached. The\nsteps of this procedure are illustrated in the right panel of Fig. 4. Using the isospin symmetric\n(isoQCD) values of the pion and kaon meson masses recommended in Ref. [25],\n𝑚isoQCD\n𝜋=𝑚exp\n𝜋0=134.9768(5)MeV, (6)\n𝑚isoQCD\n𝐾=𝑚exp\n𝐾0=497.611(13)MeV, (7)\nleads to the value for 𝜙∗\n4=1.101(9).\n3. Valence-quark sector: Wilson twisted mass fermions\nIn the valence sector we use the Wilson twisted mass (Wtm) fermion action at maximal twist.\nThis regularisation adds a chirally rotated mass term to the massless Wilson Dirac operator 𝐷W\n3Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\nincluding the Sheikholeslami-Wohlert term,\n𝐷Wtm=𝐷W+𝑚val\n0+𝑖𝛾5𝜇val\n𝑞, (8)\nwhere the𝑚val\n0mass enters in the standard subtracted quark mass, 𝑚val\n𝑞=𝑚val\n0−𝑚cr, and𝜇val\n𝑞is\nthe twisted mass parameter. Physical observables constructed from the Wtm Dirac operator are\n𝑂(𝑎)-improved–saveforresiduallatticeartefactsof 𝑂(𝑎𝑔4\n0Tr\u0000Mq\u0001)arisingfromseaquarkmasses\n–oncethemaximaltwistconditionisfixedbytuningthehoppingparameter, 𝜅val=(2𝑎𝑚val\n0+8)−1,\nso that the light ( 𝑢,𝑑) valence PCAC quark mass vanishes, 𝑚val\n12=0, on each ensemble.\n4. Matching sea and valence sectors\nSinceweareusingamixedactionsetup,torecovertheunitarityofthetheoryinthecontinuum\nlimit, we require the matching of the sea and valence quark masses. This matching can also be\nimposed in terms of the observables 𝜙2,𝜙4defined in eq. (2), depending on the kaon and pion\nmasses,\n𝜙val\n2≡𝜙sea\n2, 𝜙val\n4≡𝜙sea\n4. (9)\nMore specifically, we employ a set of measurements in the valence hyperplane (𝜅val,𝑎𝜇val\n𝑙,𝑎𝜇val\n𝑠)\nthatallowstoperformsmallinterpolationstothevalenceparametersthatsatisfyeq. (9),inaddition\nto simultaneously imposing the maximal twist condition, 𝑚val\n12=0. The interpolation fit functions\ncan be based on LO 𝜒PT. The matching procedure is illustrated in Fig. 1.\nHaving determined the valence matching parameters(𝜅val,∗, 𝑎𝜇val,∗\n𝑙, 𝑎𝜇val,∗\n𝑠)at which the\nmaximaltwistandthematchingconditionsineq. (9)aresimultaneouslysatisfied,itisthenpossible\nto also interpolate the valence pion and kaon decay constants to the same matching point.\nWe note that, throughout the analysis, finite volume effects are corrected using LO 𝜒PT\n[23]. These corrections are found to be smaller than the size of the statistical uncertainties for all\nobservablesandensembles. Theextractionofthegroundstatesignaloflatticeobservablesisbased\non a model variation over the Euclidean time fit intervals.\n5. Scale setting and determination of 𝑡ph\n0\nThe scale setting is carried out in terms of a linear combination of the pion and kaon decay\nconstants in units of 𝑡0[16],\n√︁\n8𝑡0𝑓𝜋𝐾=√︁\n8𝑡0×2\n3\u0012\n𝑓𝐾+1\n2𝑓𝜋\u0013\n. (10)\nUp to logarithmic corrections, this quantity remains constant in SU(3) 𝜒PT at NLO along the\nrenormalized chiral trajectory defined by 𝜙4≡𝜙∗\n4.\nWe will consider three sets of data in the scale setting analysis: (i) the unitarysetup where\nthe same𝑂(𝑎)-improved Wilson Dirac operator is used in the sea and valence sectors, which are\nreferred to as the “Wilson” case; (ii) the mixed action setup after the matching procedure, referred\nto as “Wtm”; and (iii) the Combined set of data, in which the two previous sets are analysed\nsimultaneously by imposing the universality of continuum-limit result. More specifically, the\n4Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\n0.0002\n0.00000.00020.00040.00060.0008am(val)\n12\n7.282 7.283 7.284 7.285\n1/(v)\n0.330.340.350.36(val)\n2\n0.0028 0.0029 0.0030\n(v)\nl\nFigure 1: Toprow: Light( 𝑢,𝑑)valencePCACquarkmassfromthevalence gridofpoints–inthehyperplane\nofinputparameters (𝜅val,𝑎𝜇val\n𝑙,𝑎𝜇val\n𝑠)–interpolatedtothemaximaltwistcondition, 𝑚val\n12=0. Bottomrow:\n𝜙val\n2alongthevalencegrid,interpolatedtotheseavalue 𝜙sea\n2. Theseasectorparameterscorrespondtothose\nof Table 1 for ensemble H105. The orange band in both figures represents the interpolation along the grid\nof valence parameters, while the horizontal grey line and band represent the target value to which we want\nto interpolate both observables. In the case of 𝑎𝑚val\n12, it is set to zero, and for 𝜙val\n2to𝜙sea\n2. The interpolation\nof𝜙val\n4is carried out in a similar way to that of 𝜙val\n2.\n“Wilson”and“Wtm”datasetscanbeanalysedindependently,leadingtotwodeterminationsofthe\nphysicalvalueof 𝑡ph\n0. Forthe Combined case,asimultaneouslyfitofbothdatasetswithindependent\nsetsofparameterscharacterisingcutoffeffectsisperformed. Thelatticedatacanbeparameterised\nas follows\u0010√︁\n8𝑡0𝑓𝜋𝐾\u0011latt\n=\u0010√︁\n8𝑡0𝑓𝜋𝐾\u0011cont\n+c(𝑎,𝜙 2), (11)\nwith c(𝑎,𝜙 2)a function describing cutoff effects. Several possible choices for the continuum\nbehaviour and the cutoff effects are explored. For the continuum mass-dependence we consider\nSU(3)𝜒PT expressions at NLO\n\u0010√︁\n8𝑡0𝑓𝜋𝐾\u0011cont\n=𝑝1\n8𝜋√\n2\"\n1−7\n6𝐿 \n𝜙2\n𝑝2\n1!\n−4\n3𝐿 \n𝜙4−𝜙2/2\n𝑝2\n1!\n−1\n2𝐿 \n4𝜙4/3−𝜙2\n𝑝2\n1!\n+𝑝2𝜙4#\n,(12)\nwhere,𝐿(𝑥)=𝑥log(𝑥). An alternative is to employ a Taylor expansion around the symmetric\npoint𝜙sym\n2,\u0010√︁\n8𝑡0𝑓𝜋𝐾\u0011cont\n=𝑝1+𝑝2(𝜙2−𝜙sym\n2)2. (13)\nTo characterise the lattice spacing dependence, we consider\n𝑐(𝑎,𝜙 2)=𝑐1𝑎2\n𝑡0+𝑐2𝜙2𝑎2\n𝑡0+𝑐3𝛼ˆΓ\n𝑠𝑎2\n𝑡0, (14)\nandweexaminetheeffectofswitchingon/offthedifferent 𝑐𝑖’s. Anexploratorystudyoftheimpact\nof including logarithmic corrections of 𝑂(𝑎2𝛼ˆΓ\n𝑠)considering the smallest value of the anomalous\ndimensions ˆΓreported in [26] is also incorporated in the analysis by including the 𝑐3term while\n5Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\n0.2 0.4 0.6 0.8\n2\n0.2850.2900.2950.3000.3050.3100.3150.3200.3258t0fK\n0.00 0.01 0.02 0.03 0.04\na2/8t0=3.40\n=3.46\n=3.55\n=3.70\nFigure 2: Left: Chiral and continuum extrapolations of√8𝑡0𝑓𝜋𝐾. We show the measurements for Wilson\n(empty points) and Wtm (filled points). The fit form in eq. (12) is used for mass dependence together with\neq. (14)with 𝑐2=𝑐3=0toparameterisethelatticespacingdependence. Nocutsareappliedinthisspecific\nfit. Pointswiththesamecolourrefertoacommonvalueofthelatticespacing. Thegreybandrepresentsthe\ncontinuumlimitdependenceforthe Combined datasetanalysis,whilethecolouredbandsrepresentthechiral\nfits at each lattice spacing for the Wilson data, and, similarly, the dashed lines (without showing uncertainty\nin the fits) correspond to the Wtm case. Right: Continuum limit of symmetric point ensembles, 𝑚𝜋=𝑚𝐾,\nwith𝜙2=0.740(9). A common continuum limit is not imposed in this fit.\nsetting𝑐1=0. We observe that the quality of the fit with cutoff effects of 𝑂(𝜙2𝑎2)is poor for the\ncurrent set of Wilson data, while a good description can be obtained for the Wtm data.\nInadditiontoavariationofthefunctionalforms,weexplorethepossibilityofperformingcuts\nin the data, by removing the coarsest lattice spacing 𝛽=3.40, or by cutting the symmetric point\nensembles with 𝑚𝜋=420MeV.\nThe correlations present in the Monte Carlo data are retained throughout the analysis. In\nthe chiral-continuum fit, the 𝜒2function includes the correlations among the√𝑡0𝑓𝜋𝐾data points\nwhile the residual cross-correlations between√𝑡0𝑓𝜋𝐾and𝜙2are neglected in the fit. We observe,\nhowever, that this leads to a tiny deviation of the expectation value of the chi-squared [19],\n𝜒2\u000b\n,\nawayfromthenumberofdegree-of-freedom,i.e. theexpectedvaluewhenconsideringacorrelated\nfit.\nTo study the systematic effects associated with model variation in the chiral and continuum\nextrapolations,weusethemodelaveragingmethodintroducedin[17]withtheinformationcriterion\nproposedin[18]totakeintoaccountfitsthatarenotfullycorrelated. Accordingtothisinformation\ncriterion, each fit model is assigned a weight\n𝑊∝exp\u0012\n−1\n2\u0010\n𝜒2−\n𝜒2\u000b\u0011\u0013\n, (15)\n6Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\n0.1440.146t0[fm]\n0.00.5p-value\n[][a2][a2][]\n[][a2][a2][>3.40]\n[][a2][a2][m<420MeV]\n[][a2][a2][mL>3.9]\n[][a2][a2][mL>4.1]\n[][a2\ns][a2\ns][]\n[][a2\ns][a2\ns][>3.40]\n[][a2\ns][a2\ns][m<420MeV]\n[][a2\ns][a2\ns][mL>3.9]\n[][a2\ns][a2\ns][mL>4.1]\n[][a2][a22][]\n[][a2][a22][>3.40]\n[][a2][a22][m<420MeV]\n[][a2][a22][mL>3.9]\n[][a2][a22][mL>4.1]\n[][a2][a2+a22][]\n[][a2][a2+a22][>3.40]\n[][a2][a2+a22][m<420MeV]\n[][a2][a2+a22][mL>3.9]\n[][a2][a2+a22][mL>4.1]\n[Tay][a2][a2][]\n[Tay][a2][a2][>3.40]\n[Tay][a2][a2][m<420MeV]\n[Tay][a2][a2][mL>3.9]\n[Tay][a2][a2][mL>4.1]\n[Tay][a2\ns][a2\ns][]\n[Tay][a2\ns][a2\ns][>3.40]\n[Tay][a2\ns][a2\ns][m<420MeV]\n[Tay][a2\ns][a2\ns][mL>3.9]\n[Tay][a2\ns][a2\ns][mL>4.1]\n[Tay][a2][a22][]\n[Tay][a2][a22][>3.40]\n[Tay][a2][a22][m<420MeV]\n[Tay][a2][a22][mL>3.9]\n[Tay][a2][a22][mL>4.1]\n[Tay][a2][a2+a22][]\n[Tay][a2][a2+a22][>3.40]\n[Tay][a2][a2+a22][m<420MeV]\n[Tay][a2][a2+a22][mL>3.9]\n[Tay][a2][a2+a22][mL>4.1]\n0.00.1W\nFigure 3: Results for√︃\n𝑡ph\n0for the Combined data set and for each model considered. The p-value and\n𝜒2\u000b\nare computed following [19]. The blue band represents the model average following [17] where the\nsystematic error contribution from the model variation has been included. The horizontal axis refers to the\nconsideredmodels: thefirsttag([ 𝜒]or[Tay])referstothecontinuumparameterisationaccordingtoeq. (12)\nor eq. (13), respectively; the second and third tags label the cutoff effects of the Wilson and Wtm subsets of\ndata,respectively([ 𝑎2]ifonly𝑐1ineq. (14)isswitchedon,[ 𝑎2𝛼ˆΓ\n𝑠]ifonly𝑐3isused,[𝜙2𝑎2]if𝑐2isused,or\n[𝑎2+𝜙2𝑎2] if both𝑐1and𝑐2are considered); and the fourth tag corresponds to the cuts performed in both\nWilson and Wtm data for the fit ([ 𝛽 >3.40] if only ensembles with 𝛽 >3.40are kept or [𝑚𝜋<420MeV]\nif symmetric point ensembles are discarded). Although only a subset of the models do have a significant\nweight, we observe that the error band from the model averaging covers the results of all the individual\nmodels. The first model in the 𝑥-axis corresponds to the one illustrated in Fig. 2.\nwhichallowstocomputeaweightedaveragefor√𝑡0𝑓𝜋𝐾overtheexploredmodels. Themethodalso\nassignsasystematicuncertaintycomingfromthemodelvariation. 1Themodelvariationprocedure\nandthecorrespondingmodelaverageareillustratedinFig. 3. Thespecificmodelbasedoneq. (12)\nfor the continuum behaviour and 𝑐2=𝑐3=0term in eq. (14) is shown in Fig. 2.\nOnce√8𝑡0𝑓𝜋𝐾is determined at the physical point (i.e. continuum limit result with physical\npion and kaon masses), using a prescription for the isoQCD values of the pion and kaon decay\nconstants [25]\n𝑓isoQCD\n𝜋=130.56(13)MeV, 𝑓isoQCD\n𝐾=157.2(5)MeV, (16)\nas physical inputs, it is possible to determine the physical value of the gradient flow scale 𝑡ph\n0. Our\nresults for the three types of analysis (Wilson, Wtm and Combined), together with a comparison\nwithotherstudiesalsobasedon 𝑁𝑓=2+1CLSensembles,arepresentedinFig. 4. Weobservea\nshift of the central value of 𝑡0with respect to the result of Bruno et al. [16] — which is based on a\nsimilar set of ensembles than the one employed in this work — depending on whether the physical\ninputs of FLAG21 [25] or FLAG16 [24] are considered.\n1We have also considered this model averaging technique to extract the ground state signal of the relevant lattice\nobservables by scanning over Euclidean-time fit intervals.\n7Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\n0.144 0.146 0.148\nt0[fm]\nThis work, Combined, FLAG21This work, Wtm, FLAG21This work, Wilson, FLAG21Bruno et al. '16 [16]\nFLAG Review 2021 [25]\nThis work, Wilson, FLAG16Bali et. al '22 [22]\nStrassberger '23 [21]\n0 1 2 3\niter0.14300.14350.14400.14450.14500.1455t0[fm]\ncombined\nWtm\nWilson\nFigure 4: Left: Results for the gradient flow scale√𝑡0in physical units based on the model average (pink\npoints) for our three types of analysis (Wilson, Wtm and Combined), compared with the 𝑁𝑓=2+1FLAG\naverage[25]andwithotherdeterminationsalsobasedon 𝑁𝑓=2+1CLSensembles(bluepoints). Sincein\nBrunoetal. [16]theFLAG16[24]prescriptionwasusedtosetinthephysicalinput,wealsoshowtheresult\nofouranalysisusingtheWilsonregularisationwithFLAG16physicalinput(\"Thiswork,Wilson,FLAG16\")\ninsteadofFLAG21[25](\"Thiswork,Wilson,FLAG21\"). TheCLSresults,Balietal. [22]andStrassberger\n[21],useFLAG21[25]physicalinput. Right: resultsof√𝑡0forthethreetypesofanalysisateachiterationof\nthescalesettinganalysis. Theinitialiterationstartsfroma 𝑡0valuewithouterrorandthe 𝑖-thiterationstarts\nwith the value of 𝑡ph\n0of the(𝑖−1)-th iteration. After four iterations of the scale setting analysis we observe\nsigns of convergence for the value of 𝑡0.\n6. Conclusions\nWe have presented an update of a scale setting procedure based on physical inputs for the\npion and kaon decay constants, using a mixed action consisting of twisted mass valence quarks on\nCLS𝑂(𝑎)-improved sea quarks [27]. We have explored the systematic effects associated to model\nvariation in the chiral and continuum limits, and demonstrated the effectiveness of combining the\nWilson and mixed action (Wtm) calculations to improve the precision of 𝑡0. In the near future\nwe plan to extend this study by adding ensembles at finer lattice spacings and with physical pion\nmasses, which will allow to consider an analysis where physical input from only the pion decay\nconstantisemployedinthedeterminationof 𝑡0. Furthermore,weplantodeterminethelightquark\nmasses and to pursue our charm physics project based on this mixed action approach [7–9].\nAcknowledgements\nWe are grateful to our colleagues in the Coordinated Lattice Simulations (CLS) initiative for the\ngeneration of the gauge field configuration ensembles employed in this study. We acknowledge\nPRACE for awarding us access to MareNostrum at Barcelona Supercomputing Center (BSC),\nSpain and to HAWK at GCS@HLRS, Germany. The authors thankfully acknowledge the com-\nputerresourcesatMareNostrumandthetechnicalsupportprovidedbyBarcelonaSupercomputing\nCenter (FI-2020-3-0026). We thank CESGA for granting access to Finis Terrae II. This work\nis partially supported by grants PGC2018-094857-B-I00 and PID2021-127526NB-I00, funded by\nMCIN/AEI/10.13039/501100011033andby“ERDFAwayofmakingEurope”,andbytheSpanish\nResearchAgency(AgenciaEstataldeInvestigación)throughgrantsIFTCentrodeExcelenciaSevero\n8Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\nOchoaSEV-2016-0597andNoCEX2020-001007-S,fundedbyMCIN/AEI/10.13039/501100011033.\nWe also acknowledge support from the project H2020-MSCAITN-2018-813942 (EuroPLEx), un-\nder grant agreement No. 813942, and the EU Horizon 2020 research and innovation programme,\nSTRONG-2020 project, under grant agreement No 824093.\nReferences\n[1] G.Herdoíza,C.Pena,D.Preti,J.Á.RomeroandJ.Ugarrio,“AtmQCDmixed-actionapproach\nto flavour physics”, EPJ Web Conf. 175 (2018), 13018, [arXiv:1711.06017 [hep-lat]].\n[2] A. Bussone et al. [ALPHA], “Heavy-quark physics with a tmQCD valence action”, PoS\nLATTICE2018 (2019), 270, [arXiv:1812.01474 [hep-lat]].\n[3] J. Ugarrio et al. [Alpha], “First results for charm physics with a tmQCD valence action”, PoS\nLATTICE2018 (2018), 271, [arXiv:1812.05458 [hep-lat]].\n[4] A. Bussone et al. [ALPHA], “Matching of 𝑁𝑓=2+1CLS ensembles to a tmQCD valence\nsector”, PoS LATTICE2018 (2019), 318, [arXiv:1903.00286 [hep-lat]].\n[5] J. Frison et al., “Heavy semileptonics with a fully relativistic mixed action”, PoS LAT-\nTICE2019 (2019), 234, [arXiv:1911.02412 [hep-lat]].\n[6] G. Herdoiza et al., PoS LATTICE2022 (2022), 268.\n[7] A. Bussone, A. Conigli, J. Frison, G. Herdoiza, C. Pena, D. Preti, A. Saez, J. Ugarrio,\n“Hadronic physics from a Wilson fermion mixed-action approacg: Charm quark mass and\n𝐷(𝑠)meson decay constants”, [arXiv:2309.14154].\n[8] A. Conigli et al., PoS LATTICE2022 (2022), 235.\n[9] J. Frison et al., PoS LATTICE2022 (2022), 378.\n[10] M.LüscherandS.Schaefer,“LatticeQCDwithouttopologybarriers”,JHEP07(2011),036,\n[arXiv:1105.4749 [hep-lat]].\n[11] M.Brunoetal.,“SimulationofQCDwith 𝑁𝑓=2+1flavorsofnon-perturbativelyimproved\nWilson fermions”, JHEP 02 (2015), 043, [arXiv:1411.3982 [hep-lat]].\n[12] R. Frezzotti and G. C. Rossi, JHEP 08 (2004), 007, doi:10.1088/1126-6708/2004/08/007,\n[arXiv:hep-lat/0306014 [hep-lat]].\n[13] R.Frezzottietal.[Alpha],“LatticeQCDwithachirallytwistedmassterm”,JHEP08(2001),\n058, [arXiv:hep-lat/0101001 [hep-lat]].\n[14] C. Pena, S. Sint and A. Vladikas, “Twisted mass QCD and lattice approaches to the Delta\nI=1/2 rule”, JHEP 09 (2004), 069, [arXiv:hep-lat/0405028 [hep-lat]].\n[15] D. Mohler, S. Schaefer and J. Simeth, “CLS 2+1 flavor simulations at physical light- and\nstrange-quark masses”, EPJ Web Conf. 175 (2018) 02010, [1712.04884].\n[16] M. Bruno, T. Korzec and S. Schaefer, “Setting the scale for the CLS 2+1flavor ensembles,”\nPhys.Rev.D 95(2017)no.7,074504,doi:10.1103/PhysRevD.95.074504,[arXiv:1608.08900\n[hep-lat]].\n[17] William I. Jay and Ethan T. Neil, “Bayesian model averaging for analysis of lattice field\ntheory results”, Phys. Rev. D 103, 114502 (2021), doi:10.1103/PhysRevD.103.114502,\n[arXiv:2008.01069].\n[18] J. Frison, “Towards fully bayesian analyses in Lattice QCD”, [arXiv:2302.06550].\n9Determination of 𝑡0from a Mixed Action with Wilson Twisted Mass Valence Quarks Alejandro Saez\n[19] Mattia Bruno and Rainer Sommer, “On fits to correlated and auto-correlated data”,\n[arXiv:2209.14188].\n[20] Jochen Heitger, Fabian Joswig, Simon Kuberski, “Determination of the charm quark mass in\nlatticeQCDwith2+1flavoursonfinelattices”,JHEP2021,288(2021),[arXiv:2101.02694].\n[21] B.Straßberger,“TowardsHigherPrecisionLatticeQCDResults: ImprovedScaleSettingand\nDomain Decomposition Solvers,” doi:10.18452/26517\n[22] G.S.Bali et al.[RQCD],“ScalesettingandthelightbaryonspectruminN 𝑓=2+1QCDwith\nWilson fermions,” JHEP 05(2023), 035 doi:10.1007/JHEP05(2023)035 [arXiv:2211.03744\n[hep-lat]].\n[23] GilbertoColangeloandStephanDurrandChristophHaefeli,“Finitevolumeeffectsformeson\nmasses and decay constants”, [arXiv:hep-lat/0503014].\n[24] S. Aoki et. al, “Review of lattice results concerning low-energy particle physics”,\n[arXiv:1607.00299].\n[25] Y. Aoki et. al, “FLAG Review 2021”, Eur.Phys.J.C 82 (2022) 10, 869,\ndoi:10.1140/epjc/s10052-022-10536-1, [arXiv:hep-lat/2111.09849].\n[26] Nikolai Husung, “Logarithmic corrections to 𝑂(𝑎)and𝑂(𝑎2)effects in lattice QCD with\nWilson or Ginsparg-Wilson quarks”, [arXiv:hep-lat/2206.03536].\n[27] A. Saez, A. Conigli, J. Frison, G. Herdoíza, C. Pena and J. Ugarrio, PoS LATTICE2022\n(2023), 357 doi:10.22323/1.430.0357\n10"    },    {        "title": "2401.11559v1.Scale_dependent_cosmology_from_effective_quantum_gravity_in_the_invariant_framework.pdf",        "content": "Scale-dependent cosmology from effective quantum gravity\nin the invariant framework\nNicolas R. Bertini1,Davi C. Rodrigues2,3,4andIlya L. Shapiro1\n1Departamento de Física, ICE, Universidade Federal de Juiz de Fora\nCampus Universitário - Juiz de Fora, 36036-900, MG, Brazil\n2Departamento de Física & Cosmo-Ufes, Universidade Federal do Espírito Santo,\n29075-910, Vitória, ES, Brazil\n3Institut für Theoretische Physik, Universität Heidelberg,\nPhilosophenweg 16, 69120. Heidelberg, Germany\n4Instituto de Física, Universidade Federal do Rio de Janeiro,\n21941-972, Rio de Janeiro, RJ, Brazil\nAbstract\nWe explore the possibility of a consistent cosmology based on the gauge-fixing\nindependent running of the gravitational and cosmological constants ( Gand\nΛ) in the framework of effective quantum gravity. In particular, their running\nin this framework was found to satisfy G∝Λ4. In the cosmological setting,\nthe covariance of the theory provides energy conservation relations, which are\nimpossible to satisfy with the unique scale parameter. However, by introducing\nthe second sub-dominant scale corresponding to the higher-loop corrections and\nhigher-derivative terms, one can close the system of equations for the running of\nparameters and arrive at the consistent cosmological solutions. This approach\nyields a change in the cosmological expansion history that affects the ratio of\nthe Hubble parameter today to the Hubble parameter at high redshift.\nKeywords: Cosmology, quantum gravity, running couplings, multi-scale run-\nning\nMSC:83C45, 81T17, 81T15, 83F05\n1 Introduction\nThe idea to apply the renormalization group running to the gravitational physics has a long\nhistory. One can mention, e.g., the works in this direction developed in [1, 2, 3, 4, 5], based\non very different approaches and frameworks. The reason is that it cannot be ruled outarXiv:2401.11559v1  [gr-qc]  21 Jan 2024that the equation of state for the elusive Dark Energy may be observationally confirmed\ndifferent from −1. From the quantum field theory (QFT) viewpoint, the main candidate\nfor the Dark Energy is the cosmological constant (CC), and in this case, the deviation from\nthecanonical −1canbeaconsequenceoftheIRrunningoftheCCortheNewtonconstant,\nor both. Taking this into account, it is important to explore the theoretical possibility of\nsuch a running in different possible QFT or quantum gravity frameworks,\nDepending on the theoretical background of renormalization group running, the appli-\ncations to cosmology can be, in particular, separated into the ones based on semiclassical\nor perturbative quantum gravity effects. In the semiclassical case, the phenomenologically\nreasonable running demands to assume the quadratic decoupling of massive fields effects\nin the IR [6, 4]. It is important to note that there is no consistent way to derive this\ndecoupling in the QFT framework, as it was discussed in detail in [7]). All the attempts\nin this direction which emerged since then are completely speculative in the sense that\nthe result depends on the ad hocchoice of the scheme of quantum calculations. On the\nother hand, the quadratic decoupling looks a phenomenologically reasonable assumption,\ntaking into account that the same behavior can be directly obtained in other sectors of the\nvacuum action (see, e.g., [8]). The same relations can be also obtained from the covariance\narguments [9]. Starting from this point, one can develop cosmological models based on\na running cosmological and Newton constants, as pioneered in [10] and [11]. There were\nmany subsequent works in very different frameworks. One can see, e.g., [12], [13], or [14],\nand references therein.\nOn the other hand, trying to apply the perturbative quantum gravity running to cos-\nmology or to the black-hole physics leads to another kind of problem. The first works in\nthis direction, such as [1] and [2], were based on some versions of renormalizable quantum\ngravity [15] and the corresponding running [16, 17]. However, there are two problems in\napplying the higher derivative models to the real world and, in particular, to cosmology.\nFirst of all, the renormalizability of these theories is owing to the presence of massive\nghosts (may be, also, tachyonic ghosts) with the masses of the Planck order of magnitude.\nCorrespondingly, instabilities of vacuum state and of the classical solutions is expected.\nSecond, even assuming that there can be a satisfactory solution to the problem of ghosts,\nthere remains another obstacle for the cosmological applications. The massive degrees of\nfreedom contributing to the running, should decouple in the IR. This feature was always\nexpected to hold, and was also proved recently in [18] for a particular higher derivative\nmodel. This means, the running in the renormalizable models of quantum gravity can\nbe applied only at the energy scales much higher than the typical masses, i.e., above the\nPlanck scale. This does not look appropriate for the applications to cosmology since even\ninflationoccursmanyordersbelowthePlanckscale. Inthissituation, itisnecessarytolook\n2for an effective theory of quantum gravity and try to explore the running in the framework\nappropriate for the relatively low energies.\nIn the effective approach, because of the decoupling of the massive degrees of freedom,\nwe are left with the quantum effects of gravitons. Thus, the general relativity (GR) can\nbe considered a universal theory of effective IR quantum gravity [19]. Therefore, it is\ninteresting to apply the effective approach [20] (see, e.g., [21] for general introduction and\n[22] for a review), using quantum gravity based on GR as a basis for the cosmologically\nrelevantrunning. However, usingthetraditionalapproachtoderivingquantumcorrections,\none meets strong gauge-fixing dependence of the beta functions, such that only the on shell\nversion of renormalization group can be invariant [16]. The unique known way out is to\nuse the special version of quantum field theory, called Vilkovisky-DeWitt unique effective\naction [23, 24] (see also [25] for the first practical calculation in quantum gravity and [26]\nfor important general discussions). The necessary calculations were performed by Taylor\nand Veneziano [27] and recently extended in [28] (see also [29] for further attempt of the\nphysical analysis of the output), to include the beta functions for higher derivative terms\nwhich have to be renormalized beyond the one-loop approximation, regardless these terms\ndo not affect the quantum rules. What is relevant for us is that these terms have a much\nweaker running in the IR, simply because the higher derivative terms are suppressed in\nthe IR. To implement this in the Minimal Subtraction scheme, the running of these terms\nshould may correspond to the scale parameter different from the one of the running of\ncosmological and Newton constants.\nDifferent from the flat-space QFT, in curved spacetime, one needs a special effort to link\nthe results obtained within the minimal subtraction scheme of renormalization with a given\nphysical situation (see, e.g., [30] for the discussion and further references). This means,\nan application of the IR running for the cosmological (and other gravitational) problems\nrequires identification of the artificial renormalization group scale µwith a certain physical\nquantity. These identifications are not a trivial issue. For instance, in the paper on a\nsemiclassical running of the CC [31] this scale was identified with the temperature of the\ncosmic background radiation. From the QFT viewpoint, this is not a consistent choice.\nThe reason is we expect that the scale is associated to the energy of the gravitational modes\nwhich appear in the external lines of the Feynman diagrams [4, 6]. It is clear that µshould\nbe associates with a dimensional parameter characterizing the energy of the gravitational\nfield. Forinstance, inthedescriptionofexpandingUniverseitcanbetheHubbleparameter\nH, or its time derivative ˙H. It is interesting that the QFT-based arguments provide the\nresult which is equivalent to the specially designed scale-setting procedure in cosmology\n[32] and in astrophysics [33, 34] (see also [14] for the covariant identifications and further\nreferences on the subject).\n3A pertinent question is whether there should be only one typical scale, or several scales.\nThe multi-scale option has fair justification from the phenomenological side. For instance,\nin cosmology one can consider (and should do so, in principle) different energy scales for\nthe conformal factor of the metric and for the cosmic perturbations, because the typical\nlengths in these cases may be quite different. From the QFT and running viewpoints, this\nmeans that the energies in external lines of the Feynman diagrams should be also different.\nIt is clear that the multi-scale extension may lead to some changes in the bound on the\nrunning derived in the two types of the cosmological models in Refs. [35] and [36].\nComingbacktotheeffectivequantumgravityrunning,theMinimal-Subtractionscheme-\nbased renormalization based on the unique effective action [28] revealed the usual logarith-\nmic running for the second, fourth and six-order terms. Thus, even in this, relatively\nsimple, situation, the natural way to show that the higher derivative terms are suppressed,\nis to introduce the second, e.g., sub-leading, scale parameter.\nIn the next sections, we construct the identification of scale for the effective quantum\ngravity beta functions [27, 28]. In the semiclassical case applied to cosmology, the relation\nbetween µ-dependence of the cosmological and Newton constants may be established by us-\ning the conservation law. In the case of universal quantum gravity running obtained in [28],\nthis is impossible to do because both beta-functions are well-defined by the “one-fourth”\nrule. In what follows we show that this rule becomes compatible with the conservation law\nonly if we include the running of the higher derivative terms with the second, numerically\nsuppressed scale.\nThe rest of the article is organized as follows. In the next Sec. 2, we introduce the action\nof our model and construct the framework for describing the running. The considerations\nare based on the simplified form of the higher derivative terms. Furthermore, we consider\nthe conservation law and the process of identifying the scale. Some technical details are\nseparated in the Appendix. As an example, Sec. 3 describes the evolution of the Universe\ndominated by dust and CC. Finally, in Sec. 4 we draw our conclusions and discuss the\nfurther possible developments concerning the effective quantum gravity - based running.\nThe notations include the signature (−,+,+,+), the definition of the Riemann tensor\nRα\n. βµν= Γα\nβν, µ−Γα\nβµ, ν+.... The Ricci tensor is defined as Rαβ=Rµ\n. αµβand the scalar\ncurvature as R=Rα\nα.\n2 Running couplings and conservation law in cosmology\nThe effective approach to quantum gravity assumes that the Feynman rules are constructed\nstarting with the Einstein-Hilbert action. Then, the unique physical degree of freedom is\nthe graviton and there is a duality between UV and IR contributions. The derivation\n4of the IR effects requires only the non-local terms in the effective action, but those can\nbe recovered from the UV divergences. The same concerns the beta functions, and not\nonly the ones for the Newton and cosmological constants, but also for the higher derivative\nterms, which can be included into the action just to be renormalized [28]. As we shall see in\nwhat follows, these terms are important for closing the conservation equations, so we have\nto include some expression for these terms. On the other hand, since quantum general\nrelativity is a non-renormalizable theory, there is an unbounded quantity of such terms\nand, for practical purposes, one has to perform a truncation. Our purpose is to explore\nthe dynamics of the conformal factor in the lowest possible approximation. This means,\nin the fourth-derivative sector we can ignore the square of the Weyl tensor, topological\nand surface terms, such that the general form of the action boils down to the R2. This\nreduction has an additional sense because this term is also required for inflation [37, 38].\nIn the next order, we have many relevant six-derivative terms. However, since these terms\nare relevant only from the second-loop order, we can choose a particular simplest version\nfor these term.\nTaking these arguments into account, we arrive at the Einstein-Hilbert action with an\nadditional f(R)function\nS[g,Ψ] =1\n16πZ\nd4x√−gn1\nG(R−2Λ) + f(R)o\n+Sm[g,Ψ], (1)\nwhere GandΛare constants, f(R)is a polynomial on Rwith minimum order 2 and Ψ\nrepresents matter fields. The metric field equations are\nGαβ+GH αβ= 8πG(Tαβ−ρΛgαβ), (2)\nwhere ρΛ=Λ\n8πGand\nHαβ=fRRαβ−1\n2gαβf+ (gαβ2− ∇ α∇β)fR, (3)\nTαβ=−2√−gδSm\nδgαβ, (4)\nwith fR=∂f\n∂R.\nWe consider scale-dependent effects on GandΛ, such that G=G(µ1)andΛ = Λ( µ1),\nin agreement with the results of [27, 28]. As commented in Sec. 1, a second scale µ2is\nboth possible and necessary for the consistency of the field equations. A third scale will\nnot be necessary (similarly to ref. [14]). For the higher derivative terms, we assume the\npolynomial dependence\nf(R) =ξ(µ1)R2+ξ3(µ1, µ2)R3+ξ4(µ1, µ2)R4. (5)\n5This dependence on two scales is essential to ensure that the conservation law expressed\nby the continuity equation is satisfied. We justify this from here, beginning with the\nderivation of the continuity equation.\nFrom the definition of Hαβ, it can be computed that\n∇αHαβ=−1\n2X\nb∂f\n∂µb∇βµb. (6)\nHence, the divergence of Tα\nβ, using Eq. (2), can be written as\n∇αTα\nβ=∇βρΛ−G−1(Tα\nβ−δα\nβρΛ)∇αG+1\n8πGHα\nβ∇αG−1\n16πX\nb∂f\n∂µb∇αµb.(7)\nLet us additionally assume that Tαβhas the form of the energy-momentum tensor of a\nperfect fluid, i.e.\nTαβ= (ρ+p)uαuβ+gαβp, (8)\nwhere pis the pressure, ρis the energy density, and uαis the four-velocity of the fluid.\nThe usual conservation law for such a fluid is\nuβ∇αTα\nβ=−uα∇αρ−(ρ+p)∇αuα= 0. (9)\nContracting (7) with uβ, we obtain\nuβ∇αTα\nβ=uα\u0000\n∇αµ1\u0001hdρΛ\ndµ1+1\nG(ρ+ρΛ)dG\ndµ1i\n+1\n8πGuβ(∇αµ1)Hα\nβdG\ndµ1−1\n16πuαX\nb\u0000\n∇αµb\u0001∂f\n∂µb.(10)\nFrom (2) we get\nρ+ρΛ=1\n8πGuαuβ\u0000\nGαβ+GH αβ\u0001\n. (11)\nThus Eq. (10) becomes\nuβ∇αTα\nβ=uα\u0000\n∇αµ1\u0001hdρΛ\ndµ1+1\n8πG2dG\ndµ1uσuλGσλi\n−1\n16πuα\u0000\n∇αµb\u0001∂f\n∂µb\n+\u0000\n∇αµ1\u0001h\nuαuσuλHσλ+uβHα\nβi1\n8πGdG\ndµ1. (12)\nA well-motivated and widely used cosmological scale identification is with the Hubble\nparameter [4, 11, 36]. Thus, we set\nµ1=H. (13)\n6The running of G,Λandξ, was calculated in [28] and found to be\nΛ(µ1) = Λ 0\u0010\n1 +νlnµ1\nµ1\u0011−1/5\n, (14)\nG(µ1) = G0\u0010\n1 +νlnµ1\nµ1\u0011−4/5\n, (15)\nξ(µ1) = ξ0\u0010\n1 +κlnµ1\nµ1\u0011\n. (16)\nIn these formulas the values µ1correspond to the fiducial scale, and the quantities νand\nκare constants. These solutions of the renormalization group equations were derived in\n[28] using a combination of the effective approach to quantum gravity and the Vilkovisky-\nDeWitt unique effective action.\nThe above relations imply that G∝Λ4, a relation which was not explored in previous\ncosmological frameworks1. Indeed, for instance, in ref. [11] and other works based on it, Λ\ndepends on µ2, while Gdepends on ln µ; hence they do not have a proportionality relation.\nIn refs. [42, 43, 14], on the other hand, one finds Λ∝G−1, which also clearly deviates from\nthe picture described in this framework.\nAdopting the FLRW metric\nds=−dt2+a2(t)δijdxidxj(17)\nin the comoving frame, the r.h.s.of (10) gives non-zero result unless fdepends on more\nthan one scale. Since the non-vanishing terms depend on ˙H, we set\nµ2=˙H. (18)\nOn the other hand, for the full set of third and fourth-order terms there are no quantum\ncalculations of the running. Furthermore, there are indications that the corresponding beta\nfunctions at higher than the one-loop orders may be dependent on the renormalization\nscheme [39]. Therefore, the scale dependencies of the corresponding parameters can be\nfound by using the conservation law, that means solving Eqs. (12) up to the first order\ninν. The details of this solution can be found in Appendix 4 and here we give the final\n1This relation reflects the running of the two parameters, while their absolute values have very different\nmagnitudes.\n7results,\nξ3(µ1, µ2) = ξ30\u001a\n1−16πξ0κ\n3ξ30(2µ2\n1+µ2)lnµ1\nµ1\n+ν\n45ξ30G0(2µ2\n1+µ2)3h\n−2(µ2\n1−µ2\n1) + Λ 0lnµ1\nµ1i\u001b\n, (19)\nξ4(µ1, µ2) = ξ40\u001a\n1 +4πξ0κ\n9ξ40(2µ2\n1+µ2)2lnµ1\nµ1\n+ν\n360ξ40G0(2µ2\n1+µ2)4h\n2(µ2\n1−µ2\n1)−Λ0lnµ1\nµ1i\u001b\n. (20)\nThe most important qualitative detail is that without the higher derivative terms, i.e.,\nwithout f(R), the solutions (14) and (15) are incompatible with the conservation law.\nFrom the physical side, this means if the energy exchange between gravity and matter is\nforbidden (see the discussions and further references in [11], [13] and [30]) fixing both Λ(µ)\nandG(µ)does not leave enough space to fit the conservation law.\n3 Application to late-time cosmology\nAs a first phenomenological investigation, we consider late-time cosmology. In such sce-\nnario, the field equations are significantly simpler since Hαβis suppressed with respect to\nthe other terms in Eq. (2) and thus there is no explicit dependence on µ2. Adopting the\nmetric (17), the 0-0 component of Eq. (2) is\n3H2−Λ(H) = 8 πG(H)ρ , (21)\nwhere Λ(H)andG(H)are given by Eqs. (14) and (15), respectively. We assume that the\nusual continuity equation remains satisfied, i.e.,\n˙ρ+H(ρ+p) = 0 . (22)\nFrom the physical side, this means we do not allow for the energy exchange between\nvacuum and matter, as this was the case in [11]. The reason is that such an exchange is\nphysically impossible at low energies [40] and can take place only in the early Universe\n[13].\nBefore starting the phenomenological analysis of the previous dynamical equations, one\nneeds to specify the value of the reference scale µ1. The µ1value is such that the renor-\nmalization group corrections become relevant only when µ1̸≈µ1. Since such corrections\nare expected to be stronger in the primordial universe, we take the future de Sitter phase\n8as the one without renormalization group corrections. Being µ1the Hubble value at the\nasymptotic de Sitter phase, we set\nµ1=r\nΛ0\n3. (23)\nFor a universe dominated by dust, Eq. (22) provides ρ=ρ0(1 + z)3, where ρ0is a\nconstant. Therefore Eq. (21) can be written as\nH2\nH2\n0= Ω Λ+ Ω m(1 +z)3, (24)\nwhere H0is the Hubble parameter at z= 0. The expressions for ΩΛandΩmare, up to the\nfirst order in ν, given by\nΩΛ= Ω Λ0\u0012\n1−ν\n5lnH\nµ1\u0013\n, Ωm= Ω m0\u0012\n1−4ν\n5lnH\nµ1\u0013\n, (25)\nbeing ΩΛ0≡Λ0/3H2\n0andΩm0≡8πG0ρ0/3H2\n0the density parameters of the CC and\nmatter, respectively, at z= 0. With these definitions µ1can be rewritten in the form\nµ1=H0p\nΩΛ0. (26)\nν=0.1\nν=0\nν= - 0.1\n0.1 0.5 1 5 10 50 1001101001000\nzH/H0\nν=0.1\nν=0.01\nν= - 0.1\nν= - 0.01\n0.1 0.5 1 5 10 50 100-0.2-0.10.00.10.2\nzHν\nFigure 1: Left:Hubble parameter versus redshift for different νvalues. Right:Same as\nthe previous plot, but for the relative deviation Hν.\nInFig.1wedisplaynumericalresultsfortheHubbleparameterassumingsmall νvalues.\nWhen H=H0, Eq. (24) poses a relation between ΩΛ0,Ωm0andν. For small ν, the two\nformer ones should not be far from the ΛCDM values. To illustrate this relation, let Ωm0\nbe about its corresponding value in ΛCDM: Ωm0= 0.3. Then, the standard cosmological\ncase is found with ν= 0, implying ΩΛ0= 0.7. For ν= 0.1, one finds a quite small change\ninΩΛ0, namely: ΩΛ0= 0.705. Instead of specific phenomenological results, we henceforth\n9use this picture above and fix Ωm0= 0.3. Further results with observational data will be\nconsidered in a future work.\nIn the left panel of Fig. 1, we compare the ν-corrected Hubble parameter H(z;ν)with\nthe conventional one H(z;ν= 0). One sees that as zbecomes larger, H(z;ν)becomes\nsmaller if ν > 0. The right-hand panel shows the results in the form of the relative\ndeviation\nHν(z)≡H(z;ν)−H(z; 0)\nH(z; 0). (27)\nThe RG correction with ν > 0may alleviate the tension that exists between local\nestimates of H0and measurements derived from the cosmic microwave background, since\nlocal measurements favor larger H0values (see ref. [41] for a recent review). Nonetheless,\nwe stress that this is at the moment a possible application. A robust evaluation of this issue\nrequires both theoretical and phenomenological developments. Considering the theoretical\npart, it is necessary to extend this approach towards high redshift, when νln(H/µ1)needs\nnot to be small. And the proper setting of the RG scale considering perturbations needs\nto be done.\n4 Conclusions and discussions\nWe extended the framework initially proposed in [28], applying it for the first time to\ncosmology. Within this framework, the gravitational constant Gand the cosmological\nconstant Λexhibit dependency on the energy scale µ1, as shown in eqs. (14) and (15). The\nspecific running found in ref. [28] satisfies G∝Λ4, being clearly differently from others\n(e.g., [11, 42, 43]). We have shown that, in this context, it is not consistent to simply use\nthe scale µ1∝Htogether with energy-momentum conservatio. However, introducing a\nsecond scale ( µ2) solves the issue, as we have shown here.\nFollowing ref. [28], the evolution of G,Λ, and the R2coupling constant ξshould de-\npend solely on µ1. It is conceivable that higher order R-corrections exist, necessitating\nadditional coupling constants. These additional coupling constants can dependent on a\nsecondary scale µ2. Our analysis introduces R3andR4corrections with the coupling\nconstants ξ3(µ1, µ2)andξ4(µ1, µ2)respectively, achieving theoretical consistency.\nIn this context, we have analyzed the background cosmological evolution for late-time\ncosmology. For ν >0, our results show that the ratio of the current Hubble parameter to\nthat at high redshift can be substantially lower than that predicted by standard cosmology\n(where ν= 0). ThesefindingsholdpotentialimplicationsforaddressingtheHubbletension\n(e.g., [41]). However, further theoretical investigation is required, particularly during the\n10inflationary period, when quantum corrections may not be negligible. We plan to explore\nthese aspects in future work.\nAcknowledgements\nNRB acknowledges the support from Conselho Nacional de Desenvolvimento Científico\ne Tecnológico - CNPq, for the post-doctoral fellowship. DCR thanks Heidelberg Uni-\nversity for hospitality and support. He also acknowledges partial support from Con-\nselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and Fundação\nde Amparo à Pesquisa e Inovação do Espírito Santo (FAPES-Brazil) (TO 1020/2022,\n976/2022, 1081/2022). I.Sh. acknowledges the partial support from CNPq under the\ngrant 303635/2018-5.\nAppendix A. Derivation of ξ3andξ4\nLet us describe the procedure for obtaining the functions (19) and (20) that maintain\nthe conservation of the energy-momentum tensor in the cosmological context. For the\nmetric (17) and in a comoving frame, the last term in Eq. (12) is zero and, on top of\nthis, uαuβGαβ= 3H2. Using the identifications (13), (18) and the function (5), Eq. (12)\nbecomes\nuβ∇αTα\nβ=˙H\u0014dρΛ\ndH+3H2\n8πG2dG\ndH−1\n16π\u0010dξ\ndHR2+∂ξ3\n∂HR3+∂ξ4\n∂HR4\u0011\u0015\n−¨H\n16π\u0012∂ξ3\n∂˙HR3+∂ξ4\n∂˙HR4\u0011\n. (28)\nForuβ∇αTα\nβ= 0, the coefficients of ˙Hand ¨Hmust simultaneously be zero. Therefore,\nusing (14), (15) and (16) and denoting x=Handy=˙Hto simplify the notations, the\ncoefficient of ˙His\nα\nx+βx+σ\nxR2−∂ξ3\n∂xR3−∂ξ4\n∂xR4= 0. (29)\nThe coefficient of ¨His\n∂ξ3\n∂yR3+∂ξ4\n∂yR4= 0, (30)\nwhere\nα=6Λ0ν\n5G0, β =−24ν\n5G0, σ =−16πξ0κ, R = 6(2 x2+y). (31)\n11The system of partial differential equations presented above can be solved analytically. It\nis convenient to find the equations involving only ξ3, and only ξ4. In the second case, this\ncan be done as follows. By differentiating (29) with respect to y, we get\n2σ\nxR∂R\n∂y−R3∂2ξ3\n∂y∂x−3R2∂ξ3\n∂x∂R\n∂y−∂2ξ4\n∂y∂xR4−4R3∂ξ4\n∂x∂R\n∂y= 0, (32)\nDifferentiating the Eq. (30) with respect to xand using this same equation to eliminate\n∂ξ3/∂y, results in\n−∂2ξ3\n∂x∂yR3=R3∂ξ4\n∂y∂R\n∂x+R4∂2ξ4\n∂x∂y. (33)\nOn the other hand, Eq. (29) can be rewritten as\n−∂ξ3\n∂xR2=∂ξ4\n∂xR3−σ\nxR−βx\nR−α\nxR. (34)\nUsing (33) and (34) in (32) with the conditions\n∂2ξ3\n∂x∂y=∂2ξ3\n∂y∂x,∂2ξ4\n∂x∂y=∂2ξ4\n∂y∂x, (35)\nwe get\nR3\u0012∂ξ4\n∂y∂R\n∂x−∂ξ4\n∂x∂R\n∂y\u0013\n−\u0012σR\nx+3βx\nR+3α\nxR\u0013∂R\n∂y= 0. (36)\nSolving this equation by requiring ξ4to be a constant, ξ40, corresponding to x=x0(i.e.,\nto the fiducial scale µ0), we arrive at\nξ4(x, y) =ξ40−β(x2−x2\n0)\n864(2 x2+y)2−α−12(2x2+y)σ\n432(2 x2+y)4ln\u0010x\nx0\u0011\n. (37)\nTaking this function in (30) and integrating over yresults in\nξ3(x, y) =β(x2−x2\n0)\n108(2 x2+y)3+α+ 18σ(2x2+y)2\n54(2x2+y)3ln\u0010x\nx0\u0011\n+ξ30(x), (38)\nwhere ξ30(x)is an arbitrary function. Inserting this result into (29) gives the condition\ndξ30/dx= 0, then ξ30must be constant. 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Piattella, Scalar-Tensor gravity with\nsystem-dependent potential and its relation with Renormalization Group extended Gen-\neral Relativity , JCAP 09(2015), 009 [arXiv:1504.05119 [gr-qc]].\n16"    },    {        "title": "2401.11580v1.Age_of_Gossip_in_Random_and_Bipartite_Networks.pdf",        "content": "Age of Gossip in Random and Bipartite\nNetworks\nThomas J. Maranzatto\nDepartment of Mathematics, Statistics, and Computer Science\nUniversity of Illinois Chicago\nChicago, IL, USA\nEmail: tmaran2@uic.edu\nJanuary 23, 2024\nAbstract\nIn this paper we study gossip networks where a source observing a\nprocess sends updates to an underlying graph. Nodes in the graph com-\nmunicate to their neighbors by randomly sending updates. Our interest is\nstudying the version age of information (vAoI) metric over various classes\nof networks. It is known that the version age of Knis logarithmic, and\nthe version age of Knis linear. We study the question ‘how does the\nvAoI evolve as we interpolate between KnandKn’ by studying Erdős-\nReyni random graphs, random d-regular graphs, and bipartite networks.\nOur main results are proving the existence of a threshold in G(n, p)from\nrational to logarithmic average version age, and showing G(n, d)almost\nsurely has logarithmic version age for constant d. We also characterize\nthe version age of complete bipartite graphs KL,R, when we let Lvary\nfrom O(1)toO(n).\n1 Introduction\nWe live in a world of connectivity, and randomized communication networks\nare becoming more ubiquitous. From IoT applications [1] to federated machine\nlearning topologies [2] to connectivity in drone swarms [3], it is clear that anal-\nysis of information flow in random networks is needed.\nIn this paper we consider the version age of information (vAoI) metric over\nvarious gossip network topologies. Herein, a source node observing a process\nrandomly shares its status with a network of Nnodes, and the nodes in the\nnetwork randomly share (or ‘gossip’) their statuses amongst each other. A\ncentral question in gossip networks is to characterize how out of date the status\nof the network is compared to the source, and the vAoI is just one among many\nmetrics used in this analysis. Others ways to measure ‘freshness’ include the age\nofinformation[4],ageofincorrectinformation[5],andageofsynchronization[6].\n1arXiv:2401.11580v1  [cs.IT]  21 Jan 2024The vAoI metric was initially used to show that the age of gossip in the\nclique Knscales as Θ(log n), and the age of gossip in the disconnected graph\nKnscales as Θ(n)[7]. A conjecture was also given that the cycle Cnshould\nhave version age O(√n)which was subsequently proven [8]. Other works have\nstudied the 2-d lattice [9] and generalizations of cycles [10].\nThis paper investigates the question, ‘how does the vAoI evolve as we inter-\npolatebetween KnandKn’? Thisisanaturalquestiontoask, andwasstatedin\nother works [9] and partially answered in the context of generalized cycles [10].\nWe choose to focus on complete bipartite graphs KL,Rand random graph mod-\nels. For bipartite graphs, we investigate how version age increases when the left\npartition increases from 0ton\n2. For random graphs, we show a phase transition\ntakes place where the graph goes from having rational version age to logarith-\nmic version age. We also study random d-regular graphs and show under the\nuniform distribution, these almost surely always have logarithmic version age.\nOur main theorems are presented in section 3.\n2 System Model and Background\n2.1 Version Age of Information\nWe consider a source node n0sending updates to a network G= (N, E)over\nnnodes. We let N= [1, .., n]. The source updates itself via a Poisson process\nwith rate λe. The source also sends updates to each v∈Gas separate Poisson\nprocesses with ratesλ\nn. In all graphs we consider, an undirected edge ij∈E\nfacilitates two-way communication between nodes iandj, with λi(j) =λ\ndeg(i)\ndenoting the Poisson rate from itoj, and λj(i) =λ\ndeg(j)denoting the Poisson\nrate from jtoi. In general λi(j)̸=λj(i). If node ihas no neighbors then for\nallj,λi(j) = 0.\nThe source and every node in the network have internal counters; when a\nnode i∈ N ∪ { n0}communicates to a neighbor j(because i’s Poisson process\nupdated) isends its current counter value. The counter for n0increments if\nand only if the process for n0updates. Contrast this with j∈ Nwhose counter\nincrements if and only if jreceives a newer version from one of its neighbors.\nLetXj(t)be the number of versions node jis behind n0at time t, and for any\nsubset Sof vertices let XS(t) = min i∈SXi(t). Then the limiting average version\nage of SisvG(S) = lim t→∞XS(t).\nThe information flow from iinto set Sis denoted by λi(S) =P\nj∈Sλi(j) =\nλ|N(i)∩S|\nn. Similarly λ0(S) =λ|S|\nn. Then the main result of Yates [7] is that the\nstochastic quantity vG(S)can be computed combinatorially by unrolling the\nfollowing recursion:\nvG(S) =λe+P\ni̸∈Sλi(S)vG(S∪ {i})\nλ0(S) +P\ni̸∈Sλi(S)(1)\nIn any network where λandλeare constant, λ0(S) =λ|S|\nn, and λi(j) =\n2λ\ndeg(i), the version age is monotonic and bounded as Ω(log n) =vG(S) =O(n)\nas shown in Appendix A. We use this fact numerous times throughout and will\nattempt to be explicit when it is applied.\n2.2 Random Graphs\nA graph is d-regular if all vertices have degree d. We let G(n, d)denote the\nuniform probability distribution over all n-node dregular graphs. The Erdős-\nReyni model G(n, p)denotes the distribution on n-node graphs where each\nedge is chosen i.i.d. with probability p. We say a graph property Qholds\nasymptotically almost surely (a.a.s.) under G(n, d)ifP[G(n, d)∈Q]→1as\nn→ ∞, and likewise for G(n, p).\nA graph property is called monotone if adding edges to a graph does not\nchange the property and the property is invariant to permuting the labels of\nvertices (e.g. connectivity is monotone, 3-colorability is not). Recall the defini-\ntion of a random graph threshold:\nDefinition 1. A function p∗=p∗(n)is a threshold for graph property Rin\nG(n, p)if\nlim\nn→∞P[G(n, p)∈R] =(\n0ifp/p∗→0\n1ifp/p∗→ ∞\nAsharp threshold is defined similarly, where the limits to 0 and ∞can be\nbounded by 1−ϵand1 +ϵrespectively, for every ϵ >0. (See eg. [11] for a more\nformal definition of sharp thresholds.)\n3 Notation and Summary of Results\nThroughout we assume λ0andλare constants and let n→ ∞. All inequalities\nare meant to hold for nsufficiently large. We say a graph property Qholds\nasymptotically almost surely (a.a.s.) under G(n, d)ifP[G(n, d)∈Q]→1as\nn→ ∞, and likewise for G(n, p). We use vG(S)to denote the version age\nofS⊆ N, and when there is no ambiguity about the graph the subscript G\nwill be dropped. The graph KL,Ris the complete bipartite graph with vertex\nbipartition N=L∪RandE(G) ={uv:u∈L, v∈R}. For any S⊂ N, the\nneighborhood of SisN(S) ={v∈ N : dist G(v, S) = 1}and its edge boundary\nis∂S={ij∈E:i∈S, j̸∈S}. Finally we use the notation ˜Θ(·)to suppress\nlogarithmic factors. Our main results are presented below.\nTheorem 1. LetL:=L(n)andR:=R(n)be non-decreasing functions such\nthat|L|+|R|=nand for all n,|L|<|R|. For KL,R, ifj∈Rthen,\n1.L= Θ(1) = ⇒v({j}) = Θ( n)\n2.L=f(n)andf(n) =o(n) =⇒v({j}) = Ω( n/f(n))\n3.L=nαforα∈(0,1) = ⇒v({j}) =˜Θ(n1−α)\n34.L= Θ( n) = ⇒v({j}) = Θ(log n)\nTheorem 2. For any fixed d≥3, a.a.s the worst-case version age of any vertex\ninG(n, d)isΘ(log n).\nTheorem 3. Letϵ >0. Ifp=(1−ϵ) logn\nnthen a.a.s. the average version age of\na vertex in G(n, p)isΩ(nϵ). Ifp=(100+ ϵ) logn\nnthen the average version age of\na vertex is Θ(log n). Furthermore there are constants α >0and1≤c∗≤100\nsuch that p=c∗logn\nnis a threshold for the graph property “ Ghas average version\nage less than αlogn”\nTo summarize in words, the worst-case version age in complete bipartite\ngraphs is inversely related to the size of the smaller component. This intuitively\nmakes sense, as K0,n=Knis the empty graph and has linear version age, and\nthe balanced bipartite graph Kn/2,n/2is very dense so information should flow\nreadily (and we will show has the same version age scaling as Knup to constant\nfactors). Theorem 1 makes this intuition precise. Our concern with rational\npartition sizes in KL,Ris to compare these graphs to generalized rings with\nrational degree as studied in Srivastava and Ulukus [10]: in both cases rational\ndegree distribution leads to rational version age scaling.\nPerhaps surprising is the contrast between Theorem 2 and Theorem 3, where\nd-regular graphs have logarithmic version age and G(n, p)only achieves loga-\nrithmic version age for expected degree ∼logn. An explanation for this is\nthat for any fixed d,G(n, d)is a very good expander, but G(n, p)only achieves\nconnectivity once p=logn\nnand so below this point is a worst-case expander.\nExpansion is a good measure for long-range connectivity in graphs, and connec-\ntivity should be a condition for the quick spread of gossip. In fact the proof for\nTheorem 2 relies heavily on the expansion properties of G(n, d), and likewise\nthe lower bound of Theorem 3 relies on counting isolated vertices in G(n, p).\n4 Bipartite graphs\nWe prove theorem 1. By symmetry of the complete bipartite graph KL,R, the\nversion age of a subset only depends on the number of nodes in the left and\nright parts. Therefore for any subset S⊂Vwith|S∩L|=i,|S∩R|=jdefine\nv(i, j) := vKL,R(S)and likewise for λw(i, j). Finally define uKL,R(i, j) =\nλ\nλevKL,R(i, j).\nLemma 1. LetKL,Rbe a complete bipartite graph on nvertices. Then for any\nS⊂VwithS∩L=i,S∩R=j,\nuKL,R(i,j)=1 +(|L|−i)j\n|R|u(i+ 1, j) +(|R|−j)i\n|L|u(i, j+ 1)\ni+j\nn+(n−i)j\n|R|+(n−j)i\n|L|.\n4Proof.Rearranging equation 1 and using v(N) =λe\nλ,\nv(N)λ=λ0(S)v(S) +X\ni̸∈Sλi(S)(v(S∪ {i}))\nNote that u(S) =v(S)\nv(N), and λ0(S) =λ|S|\nn. By symmetry of the network we can\nsplit the sum and simplify,\n1 =i+j\nnu(i, j) +X\n(S∩L)cj\n|R|(u(i, j)−u(i+ 1, j))\n+X\n(S∩R)ci\n|L|(u(i, j)−u(i, j+ 1))\n=u(i, j)\u0012i+j\nn+(|L| −i)j\n|R|+(|R| −j)i\n|L|\u0013\n−(|L| −i)j\n|R|u(i+i, j)−(|R| −j)\n|L|u(i, j+ 1)\nSolving for u(i, j)completes the proof\nProof.(Theorem 1) We split the proof into four parts corresponding to the four\nregimesintheTheorem. Since u(S)isaconstantmultipleof v(S),wearecontent\nto bound the former. If L= Θ(1), then there are constants C1<lim inf L(n)\nandC2>lim sup L(n). Then for large enough n,\nλ\nλev(0,1) = u(0,1)≤1 +C2\nn−C2u(1,1)\n1\nn+C2\nn−C2≤n−C2\nC2= Θ( n)\nWhere the second inequality uses the fact that for all graphs uG(S)>0. The\nlower bound is almost identical, save for switching the roles of C1andC2.\nFor the second case that L=f(n)\nnandf(n) =o(n),\nu(0,1)≥1 +f(n)\nn−f(n)u(1,1)\n1\nn+f(n)\nn−f(n)= Ω\u0012n−f(n)\nf(n)\u0013\n= Ω\u0012n\nf(n)\u0013\nwhere the last equality follows from the upper bound on f.\nFor the third case when L=nα, we have the lower bound Ω(n1−α)from the\nabove observation. We also have\nu(0,1) =1 +nαu(1,1)\nn(1−nα)\n1\nn+nα\nn(1−nα)=O\u0000\nn1−α\u0001\n+O(u(1,1))\nso it is enough to show u(1,1) = ˜O\u0000\nn1−α\u0001\n. To that end we state and prove the\nfollowing lemma.\n5Lemma 2. For any complete bipartite graph KL,R,\nuKL,R(1,1)≤min{|R|log(|L|),|L|log(|R|)}.\nProof.Forclaritywewrite LandRinsteadof |L|and|R|anddropthesubscript\nonu(·,·). By Lemma 1\nu(k,1)≤1 +L−k\nRu(k+ 1,1) +k(R−1)\nLu(k,2)\nL−k\nR+k(R−1)\nL\n≤1 +L−k\nRu(k+ 1,1) +k(R−1)\nLu(k,1)\nL−k\nR+k(R−1)\nL\n=RL+L(L−k)u(k+ 1,1) +kR(R−1)u(k,1)\nL(L−k) +kR(R−1)\nWhere the second inequality follows since increasing the size of a set can only\ndecrease its version age. Letting D=L(L−k) +kR(R−1), we have\nu(k,1)\u0012L(L−k)\nD\u0013\n≤RL\nD+L(L−k)\nDu(k+ 1,1)\n=⇒u(k,1)≤R\nL−k+u(k+ 1,1)\nStartingat u(1,1)andapplyingthisinequalityrecursively Ltimesyields u(1,1)≤PL\ni=1R\nL−i≤Rlog(L). An analogous argument holds for the quantity Llog(R)\nby instead expanding u(1, k).\nApplying Lemma 2 completes the proof for the third case.\nFor the fourth case when L= Θ( n), similar to case one we could find con-\nstants C1, C2so that the tail of L(n)is bounded as nC1≤L(n)≤nC2. For\nclarity we are content to let L=cn, and K(n) :=Kcn,(1−c)nfor some absolute\nconstant c. We also drop floors and ceilings on cnwhen this isn’t an integer;\nthis obviously doesn’t change the asymptotics. We briefly prove the following\nfact.\nFact 1. uK(n)(1,2)≤uK(2n)(1,2)anduK(n)(2,1)≤uK(2n)(2,1)\nProof.When we move from Kcn,(1−c)ntoK2cn,2(1−c)n, every arc in the larger\nnetwork has half the capacity of a corresponding arc in the smaller network.\nTherefore by symmetry of the network any subset (A, B)⊂K2cn,2(1−c)nwill\nhave the same version age scaling as (A/2, B/2)⊂Kcn,(1−c)n.\n6Now there is a δ=δ(c)>0, and for any ϵ >0andnlarge enough,\nuK(n)(1,1) =1 +nc−1\nn(1−c)uK(n)(2,1) +n(1−c)−1\nncuK(n)(1,2)\n2\nn+nc−1\nn(1−c)+n(1−c)−1\nnc\n≤1\n1\nn+nc−1/2\n(1−c)n+n(1−c)−1/2\nnc×\n\u0012\n1 +nc−1\nn(1−c)uK(2n)(2,1)\n+n(1−c)−1\nncuK(2n)(1,2)\u0013\n+ϵ\n≤1\n1\nn+nc−1/2\n(1−c)n+n(1−c)−1/2\nnc×\n\u0012\n1 +nc−1/2\nn(1−c)uK(2n)(2,1)\n+n(1−c)−1/2\nncuK(2n)(1,2)\u0013\n+ϵ−δ\n=uK(2n)(1,1) +ϵ−δ\nThe equality is by Lemma 1. In the first inequality we applied Fact 1 and noted\nthe limits are the same. In the second inequality we used the fact that the\nversion age of a subset in any network is at most linear. Therefore uK(n)(1,1)≤\nuK(2n)(1,1) + ϵ−δ, setting ϵ < δimplies uK(n)(1,1) = O(logn). The lower\nbound follows since the version age of the clique KnisΘ(log n).\n5 Random Regular Graphs\nThe proof for Theorem 2 uses similar techniques to [9] for bounding the age of\ngossip on a grid. Therein a key part of the argument is understanding expansion\nproperties of the network; how many edges exist in any cut of G. We recall the\nedge expansion number for a graph (also known as the Cheeger constant, or\nisoperimetric number).\nDefinition 2. For any graph G, the edge expansion number h(G)is given by\nh(G) := min\n|S|≤n/2|∂S|\n|S|\nwhere ∂Sis the set of edges in the cut spanning SandSc.\nRecall that G(n, d)is the uniform probability distribution over all d-regular\ngraphs. A result by Bollobás [12] shows for constant d,G(n, d)generates good\nedge expanders. We need a weaker version of their result,\n7Theorem 4. Bollobás [12] For every fixed d≥3. Then there is a constant\ncd<1\n2such that\nP[h(G(n, d))≥dcd]→1asn→ ∞\nProof.(Theorem 2) Recalling identity (4) from [9] which is just a rearrangement\nof equation (1),\nλe=λ0(S)v(S) +X\ni̸∈Sλi(S)(v(S)−V(S∪ {i})) (2)\nSince G(n, d)is regular, we can partition ∂Sinto sets A1, ..., A dwhere Aj=\n{v̸∈S:|N(v)∩S|=j}. Then,\nλe=λ0(S)v(S) +dX\ni=1X\nj∈Aiλj(S)(v(S)−v(S∪j))\n≥λ0(S)v(S) +dX\ni=1|Ai|λi\ndmin\nj∈Ai(v(S)−v(S∪j))\n≥λ0(S)v(S) + \nλ\nddX\ni=1i|Ai|!\n(v(S)−max\ni∈N(S)(v(S∪i)))\nSincePd\ni=1i|Ai|=∂S, by Theorem 4, when |S|< n/ 2a.a.s. G(n, d)satisfies:\nλe≥λ|S|\nnv(S) +λ\ndcdd|S|(v(S)−max\ni∈N(S)v(S∪i))\n=⇒v(S)≤λe\nλ+cd|S|max i∈N(S)vS∪i\n|S|\nn+cd|S|(3)\nBy an analogous argument for when |S|> n/ 2:\nv(S)≤λe\nλ+cd(n− |S|) max i∈N(S)v(S∪i)\n|S|\nn+cd(n− |S|)(4)\nTherefore when unrolling recursion (2) for v({i}), ifS < n/ 2we use inequal-\nity (3), otherwise we use (4). To that end let Xbe the sum corresponding to\nsmall subset size and letting j:=|S|,\nX≤λe\nλ\u00121\ncd+1\nn\u0013\n1 +n/2X\ni=1iY\nj=1cdj\nj+1\nn+cd(j+ 1)\n\n≤λe\ncdλ\n1 +n/2X\ni=1iY\nj=1cdj\nj+1\nn+cd(j+ 1)\n\n≤λe\ncdλ\n1 +n/2X\ni=1iY\nj=1j\nj+ 1\n\n8=λe\ncdλ\n1 +n/2X\ni=11\ni+ 1\n=O(logn)\nLetting Ybe the terms corresponding to |S|> n/ 2, it can be shown that\nY≤λe\nλ+λe\nλ\n1 +n−2X\ni=n/2iY\nj=n/2cd(n−j)\nj\nn+cd(n−j−1)\n\n×n/2−1Y\nj=1cdj\nj\nn+cd(j+ 1)×1\n1/2 +cdn/2\n≤C′λe\nλlogn=O(logn)\nWhere we omitted computations that are analogous to those found in the pre-\nvious step and in [9]. Finally noting that for any network v({i}) = Ω(log n)\ncompletes the proof.\n6 Erdős-Reyni Random Graphs\nWe need the following lemmas on properties of G(n, p),\nLemma 3. The following holds a.a.s.\n1. If d <1andp=d/n, then G(n, p)has no component of size larger than\nO(logn)\n2. If1\n2< d < 1andp=dlogn\nnthen G(n, p)hasΩ(n1−d)isolated vertices\n3. If d >100andp=dlogn\nnthen G(n, p)is connected and for any ϵ >0,\nevery v∈G(n, p)satisfies deg(v)∈((1−ϵ)np,(1 +ϵ)np)\nProof.Items one and three are easily verified ( [13], [14]). We leave Item two\nfor Appendix B.\nLemma 4. Let0< δ < 1andp≥30 log n\nδ2n. Then a.a.s. any subset SofG(n, p)\nwith|S|<n\n2satisfies\nP(|∂S| ̸∈(1±δ)E[|∂S|]) =o(1)\nProof.For any k=|S| ⊂ Nandα=δp\nk(n−k)p, define AS:=P\u0002\n|∂S| ̸∈\n(1±δ)E[|∂S|]\u0003\n. Since in G(n, p)edges are included i.i.d, the Chernoff bound\nstates\nP[AS]≤3 exp\u0012−α2\n8\u0013\n.\nthen by the choice of p, fornlarge,\n9 log\u0000\nn\u0000n\nk\u0001\u0001\nδ2k(n−k)≤9 log(ne\nk)\nδ2(n−k)+9 log( n)\nδ2(n−1)≤9(1 + log n)\nδ2n\n2+10 log n\nδ2n≤p\n9So that by the definition of α,\nP\n[\nS⊂N\n|S|≤n/2AS\n≤3n/2X\nk=1\u0012n\nk\u0013\nexp\u0012−δ2k(n−k)p\n8\u0013\n≤3n/2X\nk=1o\u00121\nn\u0013\n=o(1)\nProof.(Theorem 3) Let ϵ >0and suppose p≤(1−ϵ) logn\nn. Then by Lemma 3.2,\nthere are O(nϵ)isolated vertices. By equation (1) any isolated vertex ihas\nversion age v({i}) =λen\nλ= Θ( n). For any ithat is not isolated,\nvG(n,p)({i}) = Ω(log n).\nThen the average version age of a vertex is greater than\n1\nn(Θ\u0000\nn1+ϵ\u0001\n+ Ω (( n−nϵ)) log n) = Ω( nϵ).\nNow let p≥100 log n\nn. Define Aj={v∈V(G) :|N(v)∩S|=j}and\nu(S) =v(S)\nv(N). Applying Lemma 3.3 with ϵ= 1/2and Lemma 4 with δ= 1/3,\na.a.s. for every k=|S| ⊂ Nwe have\nv(N)λ=λ0(S)v(S) +X\ni̸∈Sλi(S)(v(S∪ {i}))\n=λk\nnv(S) +X\ni̸∈S|N(i)∩S|\ndeg(i)(v(S)−V(S∪ {i}))\nWhere the first line is identity (2). Then a.a.s.\nn≥ku(S) +2n\n3 lognX\nℓ<3 lognAℓ(u(S)−u(S∪ {i})) (5)\n≥ku(S) +n\n2 logn×k(n−k)2\n3logn\nn×(u(S)−max\ni∈N(S)u(s∪i))(6)\n≥ku(S) +1\n3k(n−k)(u(S)−max\ni∈N(S)u(s∪i))\nWhere(5)usesLemma3andconnectivity, and(6)usesLemma4andsymmetry.\nRearranging,\nu(S)≤n+1\n3k(n−k) max u(S∪ {i})\nk+1\n3k(n−k)(7)\nLetPi=Qn−1\nj=i1\n3j(n−j)\nj+1\n3j(n−j). Expanding a single vertex ℓ∈ N,\nu({ℓ})≤n\n1 +n−1\n3+1\n3(n−1)\n1 +1\n3(n−1)+nX\ni=1n\n(i+ 1) +1\n3(i+ 1)( n−i−1)Pi\n10Observing that Pi=Qn−1\nj=in−j\nn−j+3≤Qn−1\nj=in−j\nn−j+1=1\nn−i+1,\nu({ℓ})≤C1+ 3nnX\ni=11\ni(n−i+1\n3)(8)\nWhere the1\n3is present to avoid diving by zero and facilitate the next step.\nBriefly recall some properties of the Digamma function [15]: for every complex\nzthat is not a negative integer, the Digamma function has series representation\nψ(1 +z) =−γ+P∞\nk=1z\nk(k+z), where γis the Euler–Mascheroni constant. Also\nψ(1−x) =ψ(x) +πcot(πx)andψ(1 +x)∼logxifx∈(0,∞). Therefore (8)\nis bounded as, Therefore (8) is bounded as,\nu({ℓ})≤C1+∞X\ni=13n\ni(n−i+1\n3)+∞X\ni=n+13n\ni(i−n−1\n3)\n≤C2+ 3∞X\ni=1−(n+1\n3)\ni(i−(n+1\n3))+ 3∞X\ni=1n\ni(n+i)\n≤C3+ψ(1−(n+1\n3)) +ψ(n+ 1)\n=O(logn)\nWeconcludebyremarkingontheexistenceofathresholdforsuper-logarithmic\nto logarithmic version age in G(n, p). Fix ϵ > 0,δ > 0and the sequences\np0(n) :=(1−ϵ) logn\nnandp1(n) :=(100+ ϵ) logn\nn. Consider the graph property\nP=‘Ghasaverageversionagelessthan αlogn’. Wejustprovedthat P[G(n, p0)∈\nP]≤δ, and P[G(n, p1)∈P]≥1−δ, when αis suitably large. Since adding\nedges to any Gcan only decrease the average version age (Appendix A), Pis\na monotone graph property invariant under vertex permutations. Therefore for\nalln, there exists a p∗(n)∈(p0(n), p1(n))where P[G(n, p∗)∈P] =1\n2. By the\nshrinking critical window, limn→∞p1(n)−p0(n)\npc(n)exists, so definition 1 is satisfied\nand a threshold exists.\nWe briefly note that part 1 of Lemma 3 implies for d <1andp < d/n, the\nexpected version age of a vertex in G(n, p)isΩ\u0010\nn\nlogn\u0011\n. For pmuch greater than\nin part 3, the version age still scales logarithmically, but the constant factors\ndecrease.\n7 Remarks\nWhile we have roughly characterized the behavior of vAoI in G(n, p), the ques-\ntion of identifying the precise scaling of the threshold is still open. We haven’t\nmade much effort to optimize the constants in Theorem 3. More work is needed\nto show the existence of a sharp threshold, and the author believes this will\nrequire novel structural results for G(n, p)at or slightly above the sharp con-\nnectivity threshold.\n11There are many open problems left for the study of randomness in gossip\nnetworks. For instance, the 2-d lattice has version age scaling as n1/3[9]. What\nwould happen if we considered bond percolation on the lattice instead? How\nwould the version age scale with p? There also exist random spatial graph\nmodels, such as preferential attachment and nearest neighbor attachment which\ncould model real-world networks in a more realistic way. The combinatorial\nnature of the vAoI may make studying these models tractable.\nAcknowledgements\nThis research was funding in part by NSF grant ECCS-2217023. I would like\nto thank Sennur Ulukus and Arunabh Srivastava for helpful discussions on sec-\ntion 5, and Xing Gao for help with details in Theorem 2. Thanks to Lev Reyzin\nfor pointing out a need for Theorem 5.\n12References\n[1] M. Mohamed, H. ElSawy, and W. Mesbah, “Optimized degree-aware ran-\ndom patching for thwarting iot botnets,” IEEE Networking Letters , vol. 5,\nno. 1, pp. 59–63, 2023.\n[2] B. Isik, F. Pase, D. Gunduz, T. Weissman, and Z. Michele, “Sparse\nrandom networks for communication-efficient federated learning,” in The\nEleventh International Conference on Learning Representations , 2023.\n[Online]. Available: https://openreview.net/forum?id=k1FHgri5y3-\n[3] D. Liu, Y. Xu, J. Wang, Y. Xu, A. Anpalagan, Q. Wu, H. Wang, and\nL. Shen, “Self-organizing relay selection in uav communication networks:\nA matching game perspective,” IEEE Wireless Communications ,\nvol. 26, no. 6, p. 102–110, Dec. 2019. [Online]. Available: http:\n//dx.doi.org/10.1109/MWC.001.1700434\n[4] Y. Sun, I. Kadota, R. Talak, and E. H. Modiano, “Age of\ninformation: A new metric for information freshness,” Synthesis\nLectures on Communication Networks , 2019. [Online]. Available: https:\n//api.semanticscholar.org/CorpusID:213862731\n[5] A. Maatouk, S. Kriouile, M. Assaad, and A. Ephremides, “The age of\nincorrect information: A new performance metric for status updates,”\nIEEE/ACM Transactions on Networking , vol. 28, no. 5, p. 2215–2228, Oct.\n2020. [Online]. Available: http://dx.doi.org/10.1109/TNET.2020.3005549\n[6] J. Zhong, R. D. Yates, and E. Soljanin, “Two freshness metrics\nfor local cache refresh,” 2018 IEEE International Symposium on\nInformation Theory (ISIT) , pp. 1924–1928, 2018. [Online]. Available:\nhttps://api.semanticscholar.org/CorpusID:52020072\n[7] R. D. Yates, “The age of gossip in networks,” in 2021 IEEE International\nSymposium on Information Theory (ISIT) . IEEE, Jul. 2021. [Online].\nAvailable: http://dx.doi.org/10.1109/ISIT45174.2021.9517796\n[8] A. Srivastava and S. Ulukus, “Age of gossip on generalized rings,”\ninMILCOM 2023 - 2023 IEEE Military Communications Conference\n(MILCOM) . IEEE, Oct. 2023. [Online]. Available: http://dx.doi.org/10.\n1109/MILCOM58377.2023.10356227\n[9] ——, “Age of gossip on a grid,” in 2023 59th Annual Allerton Conference\non Communication, Control, and Computing (Allerton) . IEEE, Sep.\n2023. [Online]. Available: http://dx.doi.org/10.1109/Allerton58177.2023.\n10313495\n[10] ——, “Age of gossip on generalized rings,” in MILCOM 2023 - 2023\nIEEE Military Communications Conference (MILCOM) . IEEE, Oct.\n2023. [Online]. Available: http://dx.doi.org/10.1109/MILCOM58377.2023.\n10356227\n13[11] E. Friedgut and G. Kalai, “Every monotone graph property has a sharp\nthreshold,” 1996. [Online]. Available: https://api.semanticscholar.org/\nCorpusID:9721337\n[12] B. Bollobás, “The isoperimetric number of random regular graphs,”\nEuropean Journal of Combinatorics , vol. 9, no. 3, pp. 241–244, 1988.\n[Online]. Available: https://www.sciencedirect.com/science/article/pii/\nS0195669888800143\n[13] A. Blum, J. Hopcroft, and R. Kannan, Random Graphs . Cambridge Uni-\nversity Press, 2020, p. 215–273.\n[14] M. Krivelevich and B. Sudakov, “The phase transition in random graphs:\nA simple proof,” Random Structures and Algorithms , vol. 43, no. 2, p.\n131–138, Sep. 2012. [Online]. Available: http://dx.doi.org/10.1002/rsa.\n20470\n[15] H. Srivastava and J. Choi, “Zeta and q-zeta functions and associated\nseries and integrals,” in Zeta and q-Zeta Functions and Associated Series\nand Integrals , H. Srivastava and J. Choi, Eds. London: Elsevier, 2012,\npp. 1–140. [Online]. Available: https://www.sciencedirect.com/science/\narticle/pii/B9780123852182000013\n14Appendix A\nWe show the vAoI function is monotone. We assume λ, λeare constants, and\nλi(j) =λ\ndeg(i).\nLemma 5. For any graph G, letG′be obtained from Gby adding an edge (uv).\nThen for any S⊂G,\nvG′(S)≤vG(S)\nProof.Recalling equality 1, note that if u∈Sandv∈S, then the version age\nofSinGandG′are identical. Therefore we only need to consider the cases\nwhen one of u, vare in S, or neither are in S. We prove these cases separately\nvia reverse induction on S⊂G. To that end, the base case is S=Nand\nthe statement is true by the observation above. For the inductive hypothesis,\nsuppose the claim is true for |S′|=k >1. We now consider sets |S|=k−1.\nCase 1 is that u∈Sbutv̸∈S. Then,\nvG′(S)=λe\nλ+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)vG′(S∪ {i})\u0011\n+|S∩NG′(v)|\ndegG′(v)vG(S∪ {v})\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)\n≤λe\nλ+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)vG(S∪ {i})\u0011\n+|S∩NG′(v)|\ndegG′(v)vG(S∪ {v})\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)\n≤\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)\nvG(S)\n+\n|S∩NG′(v)|\ndegG′(v)\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)\nvG(S∪ {v})\n≤\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)\nvG(S)\n+\n|S∩NG′(v)|\ndegG′(v)\n|S|\nn+P\ni̸∈S∪{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)\nvG(S)\n=vG(S)\n15Where the first inequality is from the inductive hypothesis, the second is multi-\nplying by 1 and applying equation 1, the third is the fact that adding a vertex\nto a set cannot increase the version age, and the fourth is just rearranging the\ncoefficients in the large brackets.\nCase 2 is when neither uorvare in S. Notice that the only terms obtained\nby unrolling equation 1 that are different in GandG′are those when either u\norvare neighbors of the S; otherwise the extra edge uvdoesn’t play a role in\nthe expansion, or the edge is present in Sso the terms are identical. Therefore\nwe only need to consider those sets which are distance 1 from {u, v}. For\ncompleteness we can then perform analogous computations to above.\nvG′(S) =\nλe\nλ+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)vG′(S∪ {i})\u0011\n+|S∩NG′(v)|\ndegG′(v)vG′(S∪ {v}) +|S∩NG′(u)|\ndegG′(u)vG′(S∪ {u})\n|S|\nn+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)+|S∩NG′(u)|\ndegG′(u)\n≤λe\nλ+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)vG(S∪ {i})\u0011\n+|S∩NG′(v)|\ndegG′(v)vG(S∪ {v}) +|S∩NG′(u)|\ndegG′(u)vG(S∪ {u})\n|S|\nn+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)+|S∩NG′(u)|\ndegG′(u)\n≤\n|S|\nn+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n|S|\nn+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)+|S∩NG′(u)|\ndegG′(u)\nvG(S)\n+\n|S∩NG′(v)|\ndegG′(v)\n|S|\nn+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)+|S∩NG′(u)|\ndegG′(u)\nvG(S∪ {v})\n+\n|S∩NG′(u)|\ndegG′(u)\n|S|\nn+P\ni∈NG′(S)\\{u}\\{v}\u0010\n|S∩NG(i)|\ndegG′(i)\u0011\n+|S∩NG′(v)|\ndegG′(v)+|S∩NG′(u)|\ndegG′(u)\nvG(S∪ {u})\n≤vG(S)\nWe have the easy corollary, which follows from above and results by [7].\nCorollary 1. For any graph Gand any i∈G,\nΩ(log n)≤vG({i})≤O(n)\nAppendix B\nWe prove Lemma 3.2\n16Proof.By linearity of expectation, the expected number of isolated vertices µ\nis\nn(1−p)n−1≈n\u0012\n1−dlogn\nn\u0013n\n≈nexp(−dlogn) =n\nnd=n1−d.\nLetIibe the indicator r.v. that vertex iis isolated. Then,\nE\nX\ni∈NIiX\nj∈NIj\n=E\nX\ni,j∈NIiIj\n\n=E\nX\ni̸=jIij\n+E\nX\ni=jIij\n\n=µ+ 2\u0012n\n2\u0013\n(1−p2n−3)\n≈µ+ 2\u0012n\n2\u0013\nexp\u0012−dlogn\nn(2n−3)\u0013\n=µ+n(n−1)n−2d+3d\nn\n=µ+ (n−1)n1−2d+3d\nn\n≤µ+n\n2\nWhere the last inequality is for nlarge enough, since d > 1/2. Let Xbe\nthe r.v. counting the number of isolated vertices. We have asymptotically\nVar(X)≤n\n2+n1−d−n2(1−d)≤n. Then by Chebyshev’s inequality,\nP\u0002\f\fn1−d−X\f\f≥nd\u0003\n≤Var(X)\nn2d≤n\nn2d=o(1)\nSo,\nP\u0002\nX≤n1−d−nd\u0003\n=o(1)\nTherefore a.a.s. for our choice of p, there are at least n1−d−nd=O\u0000\nn1−d\u0001\nisolated vertices in G(n, p).\n17"    },    {        "title": "2401.11589v1.Uniform_bounds_for_the_density_in_Artin_s_conjecture_on_primitive_roots.pdf",        "content": "arXiv:2401.11589v1  [math.NT]  21 Jan 2024UNIFORM BOUNDS FOR THE DENSITY\nIN ARTIN’S CONJECTURE ON PRIMITIVE ROOTS\nANTONELLA PERUCCA AND IGOR E. SHPARLINSKI\nAbstract. We consider Artin’s conjecture on primitive roots over a number field K,\nreducing an algebraic number αPKˆ. Under the Generalised Riemann Hypothesis,\nthere is a density dens pαqcounting the proportion of the primes of Kfor which αis a\nprimitive root. This density dens pαqis a rational multiple of an Artin constant Apτqthat\ndepends on the largest integer τě1 such that αP pKˆqτ. The aim of this paper is\nbounding the ratio dens pαq{Apτq, under the assumption that dens pαq ‰0. Over Q, this\nratio is between 2 {3 and 2, these bounds being optimal. For a general number field Kwe\nprovide upper and lower bounds that only depend on K.\n1.Introduction\nWe consider Artin’s conjecture on primitive rootsover number fields , pointing the reader\nto Moree’s survey [ 6] for an extensive introduction to this topic and a very rich list of ref -\nerences. The classical conjecture concerns the field Qand the reductions of a rational\nnumberαR t0,˘1uwhich is not a square. Then, under the Generalised Riemann Hypoth-\nesis (GRH), the following set of prime numbers phas a positive Dirichlet density dens pαq:\nthe primes psuch that αis a primitive root modulo p, which means that vppαq “0 and\nthat pαmodpqgenerates pZ{pZqˆ. Letting τě1 be the largest integer such that αis a\nτ-th power in Qˆ, the heuristic density is the Artin constant\nApτq:“ź\nℓ|τˆ\n1´1\nℓ´1˙ź\nℓ∤τˆ\n1´1\nℓpℓ´1q˙\n,\nwhereℓvaries over the prime numbers. It is known that Ap1q „0.37 andApτq ďAp1qis\na positive rational multiple of Ap1q. The density dens pαqis a positive rational multiple of\nApτq, and the aim of this work is bounding the ratio dens pαq{Apτq.\nIfτ“1, then we have\n84\n85ďdens pαq\nAp1qď6\n5\nwhere the bounds are attained for α“ ´15 andα“ ´3, respectively. In general, we have\nthe following bounds (attained for α“ p´15q15andα“ p´3q3, respectively):\n2\n3ďdens pαq\nApτqď2.\n2010Mathematics Subject Classification. 11A07, 11M26, 11R45.\nKey words and phrases. primitive root, Artin’s conjecture.\n12 A. PERUCCA AND I.E. SHPARLINSKI\nWe deduce that the largest Artin density over Qis\ndens p´3q “6\n5Ap1q „0.45.\nWe also consider this problem for a general number field K. We again assume the GRH\n(more precisely, we assume the Extended Riemann Hypothesis for t he zeta function of\nnumber fields). We suppose that αPKˆis not a root of unity and it is not a square in\nKˆ, and we exclude the finitely many primes pofKsuch that vppαq ‰0, which ensures\nthat pαmodpqis well-defined and non-zero. Then dens pαqis the density of the primes p\nofKsuch that pαmodpqgenerates the cyclic group pO{pqˆ, whereOdenotes the ring of\nintegers of K.\nFor every positive integer nletζn“expp2πi{nq. Denote by Bthe square-free part of the\nsmallest even integer Ω ě1 such that the largest abelian subextension of K{Qis contained\nin the cyclotomic field QpζΩq(in particular, Bdepends only on K). We restrict to the\ncase dens pαq ‰0 (see Lenstra’s work [ 5] for a study of this condition) and in particular\nthe largest integer τsuch that αP pKˆqτis odd. By Theorem 5.1we have\ndens pαq\nApτqďź\nℓ|B,ℓ|τℓ´1\nℓ´2ź\nℓ|B,ℓ∤τℓ2´ℓ\nℓ2´ℓ´1ď2ź\nℓ|pB{2qℓ´1\nℓ´2.\nSo we have an upper bound that only depends on B: sinceApτqcan be arbitrarily small\nby varying τ, our uniform bound has an advantage with respect to the trivial bo und\ndens pαq ď1.\nFinally, we prove that there exists an explicit positive constant cBthat only depends on\nB(in particular, it only depends on K) such that\ndens pαq\nApτqěcB.\nOur results for Qrely on the explicit formulas for dens pαqprovided by Hooley [ 2]. For\ngeneralK, our upper bound stems from the following observation (where by ‘in dex’ we\nmean the index of pαmodpq, namely the index of the group xpαmodpqyinside pO{pqˆ):\nthe set of primes pofKsuch that the index is 1 is contained in the set of primes p\nofKsuch that all prime divisors of the index divide B. Finally, our lower bound is\nbased on insights on the so-called entanglements , namely, the interdependencies among the\ncyclotomic-Kummer extensions Kpζℓ,ℓ?αq{K, whereℓvaries in the set of prime numbers.\nWe may replace αby a finitely generated and torsion-free subgroup GofKˆof positive\nrank, and then compare the Artin density dens pGqto a suitable Artin constant AR(see\nDefinition 1). We are able to provide upper and lower bounds for dens pGq{ARthat only\ndepend on K, see Corollary 5.3.\nFinally remark that Section 4contains Kummer theory results describing the entangle-\nments among the field extensions Kpζℓ,ℓ?\nGq{K, whereℓvaries in the set of prime numbers.\nThose results are unconditional and are of independent interest.UNIFORM BOUNDS FOR THE DENSITY IN ARTIN’S CONJECTURE 3\nAcknowledgements. The first author got curious about the fact that ´3 maximizes the\nArtin density over the rational numbers. Then discussions betwee n the two authors in 2022\nat the Max-Planck-Institut f¨ ur Mathematik in Bonn and the Univer sity of Luxembourg\nled to the uniform bounds over Q. The hospitality of both institutions is gratefully ac-\nknowledged. Finally, the first author generalised the bounds to num ber fields in 2024. We\nthank Hendrik Lenstra, Pieter Moree, Pietro Sgobba and Kate Sta nge for encouragement,\nand Olli J¨ arviniemi for confirming the plausibility of our main result. La st but not least,\nwe are grateful to Fritz H¨ ormann for providing the very general proof of Proposition 4.4.\nDuring the preparation of this work, I.S. was supported in part by A ustralian Research\nCouncil Grants DP230100530 and DP230100534.\n2.The Artin density for the rational numbers\nIn this section we work over Q, keeping the notation of the introduction and assuming\nthe GRH. Remark that we use the notation ℓto denote prime numbers, and µis the\nMoebius function.\nLetαPQzt0,˘1ube not a square in Qˆ, and call δthe square-free integer such that\nα{δis a square in Qˆ. Denote by τthe largest integer for which αP pQˆqτand remark\nthatτis odd. By Hooley’s work [ 2], the Artin density can then be written as\n(2.1) dens pαq “Apτq#\n1 if δı1pmod 4 q,\n1´µp|δ|qfτpδqotherwise ,\nwhere\nfτpδq “ź\nℓ|δ, ℓ|τ1\nℓ´2ź\nℓ|δ, ℓ∤τ1\nℓ2´ℓ´1.\nAs the above quantities µp|δ|qandfτpδqare only needed for the case δ”1pmod4 qwe may\ndefineδ, as done by Moree in [ 6], as the discriminant of Qp?αq.\nTheorem 2.1. Forτ“1, we havedens pαq\nAp1qď6\n5and for arbitrary τ\n2ědens pαq\nApτqě$\n’’’’’’’’&\n’’’’’’’’%84\n85ifτ“1,\n18\n19ifτis power of 3,\n2\n3if3|τandt“5,\n14\n15if3∤τandt“5,\n1´1\nmint19,t´2uif3|τandtě7,\n1´1\n5min t19,t´2uif3∤τandtě7,\nwheretis the smallest prime which exceeds 3and divides τ, if it exists, and t“0otherwise,\nand each of the lower and upper bounds is attained.\nProof.We rely on the explicit formula ( 2.1). Ifδı1pmod4 qwe have dens pαq{Apτq “1,\nso we may restrict to the case δ”1pmod4 q. We need to maximize4 A. PERUCCA AND I.E. SHPARLINSKI\nfτpδq “ź\nℓ|δ, ℓ|τFpℓqź\nℓ|δ, ℓ∤τGpℓq,\nwhere\nFpℓq:“1\nℓ´2, G pℓq:“1\nℓ2´ℓ´1.\nTogetanupper (respectively, lower) boundwehavetochoose δwithnegative(respectively,\npositive) Moebius function µp|δ|q. We can take |δ|prime (respectively, the product of two\nprimes) because fτpδqdoes not increase by adding prime factors to δ. Remark that\nFp3q “1ąGp3q “1{5ěmaxtFpℓq,Gpℓq:ℓě5,prime u;\nFp5q “1{3ąGp5q “1{19ěmaxtFpℓq,Gpℓq:ℓě23,prime u.(2.2)\nWe have in particular\nf1pδq “ź\nℓ|δGpℓq ďmin\nℓ|δGpℓq ďGp3q “1\n5“f1p´3q.\nSo, supposing τ“1, forα“ ´3 we get the largest value dens pαq{Ap1q “6{5. Now\nconsider a general τě1. We have\nfτpδq “ź\nℓ|δ, ℓ|τFpℓqź\nℓ|δ, ℓ∤τGpℓq !max\"\nmax\nℓ|δ, ℓ|τFpℓq,max\nℓ|δ, ℓ∤τGpℓq*\n.\nThus, considering ( 2.2), we get fτpδq ďf3p´3q “1. So for α“ p´3q3we get the largest\nvalue dens pαq{Apτq “2.\nTo get a lower bound, we have to find the optimal choice for two distin ct odd primes\nℓ1,ℓ2and take δP t˘ℓ1ℓ2u, recalling that δ”1 mod 4. By ( 2.2) we can take ℓ1“3. In\nthe first four cases of the lower bound we may take ℓ2“5 and hence δ“ ´15, so we easily\nconclude those cases by computing dens pp´15qτq{Apτq.\nIn the penultimate case (where 5 ∤τ) by (2.2) we take ℓ2“5 iftě23 andℓ2“t\notherwise and obtain\nfτpδq “Fp3qmaxtGp5q,Fptqu “1\nmint19,t´2u.\nIn the last case, a similar argument leads to the same choice of δand hence\nfτpδq “Gp3qmaxtGp5q,Fptqu “1\n5min t19,t´2u,\nwhich concludes the proof. /square\nExample 2.2. By Hooley’s formula (2.1)we clearly obtain the largest value for dens pαq\nwithτ“1. Then the largest Artin density over Qisdens p´3q “6\n5Ap1q. In fact, ´3\nmaximizes the Artin density because if the index of p´3 modpqis odd, then it is automat-\nically coprime to 3(indeed, we are considering primes pthat do not split completely in\nQp?´3q “Qpζ3q).UNIFORM BOUNDS FOR THE DENSITY IN ARTIN’S CONJECTURE 5\n3.Setup for the general case\n3.1.Notation. We assume the Extended Riemann Hypothesis for the zeta funct ion of\nnumber fields.\nLetKbe a number field, and work within a fixed algebraic closure of K. Writeζnfor a\nprimitive n-th root of unity.\nLetαPKˆbe not a root of unity, while Gis a finitely generated and torsion-free\nsubgroup of Kˆof positive rank r. While considering the reductions of α(respectively, G)\nmodulo primes of Kwe tacitly exclude the finitely many primes psuch that the reduction\nmodulopis not well-defined or it is not contained in the multiplicative group of the residue\nfieldatp. Thankstothisrestrictionwehaveawell-definedmultiplicativeindexo fαmodulo\np(respectively, of Gmodulop) in the multiplicative group of the residue field at p.\nWe call dens pαqthe density of primes pofKfor which ind pαmodpq “1. We similarly\ndefine dens pGqrequiring ind pGmodpq “1.\n3.2.Artin constants. By the very general framework developed by the first author with\nJ¨ arviniemi and Sgobba [ 3,4], we know in particular that the Artin density dens pGqis a\nrational multiple of the following Artin constant (which is at least Ap1qand it is strictly\nless than 1):\nź\nℓprimeˆ\n1´1\nℓrpℓ´1q˙\n.\nWe now define a rational multiple of the above constant with the aim of producing an\nArtin constant that better fits dens pGq:\nDefinition 1. For every prime ℓletrℓbe an integer in the range from 0tor, such that\nrℓ“rholds for all but finitely many ℓ. Then we define the Artin constant\nAR:“ź\nℓprimeˆ\n1´1\nℓrℓpℓ´1q˙\n.\nGivenGas above, for every prime ℓwe define rℓas the rank of the group G{pGXKˆℓq.\nThen we have ARą0 if we suppose that r2ą0 (in fact, r2“0 implies that Gconsists of\nsquares and hence dens pGq “0).\nExample 3.1. ForG“ xαywe have rℓP t0,1u. Suppose that αis not a square in Kˆ,\nand callτthe largest (odd) integer such that αP pKˆqτ. Then we have rℓ“0if and only\nifℓ|τ, which implies that AR“Apτq.\n4.Entanglements\nThe results in this section are unconditional. We denote by Qthe product of the prime\nnumbers qsuch that ζqPK. Moreover, we denote by Bthe square-free part of the smallest\neveninteger Ω ě1such thatthelargestabeliansubextension of K{Qiscontainedin QpζΩq.\nIn particular, we have Q|B. Remark that QandBare constants that only depend on K\n(more precisely, they only depend on the largest abelian subextens ion ofK{Q). We remark6 A. PERUCCA AND I.E. SHPARLINSKI\nthat for the results in this section it is possible to replace Bby any positive multiple of it\nwhich is again square-free.\nRemark 4.1. Because of our choice of Bthe following holds. For all primes ℓ∤B\nthe cyclotomic fields Kpζℓqare linearly disjoint and have maximal degree ℓ´1overK.\nMoreover, for any fixed prime ℓ∤B, and denoting any prime by rℓ, we have\nKpζℓq Xź\nrℓ‰ℓKpζrℓq “K.\nConsequently, we have\nKpζBq Xź\nℓ∤BKpζℓq “K.\nProposition 4.2. For every prime ℓ∤Bthe degree of the cyclotomic-Kummer extension\nKpζℓ,ℓ?\nGq{Kequalsℓrℓpℓ´1q. For every prime ℓsuch degree divides ℓrℓpℓ´1q.\nProof.Thesecondassertionisstraight-forwardbyclassicalKummerthe ory. ByRemark 4.1\nwe are left to prove that for ℓ∤Bthe degree of Kpζℓ,ℓ?\nGq{Kpζℓqequalsℓrℓ. We claim that\nrℓis the rank of G{pGXKpζℓqˆℓq, so the assertion follows from classical Kummer theory.\nThe claim holds because ℓ‰2 so with the language of [ 1] the parameters for ℓ-divisibility\nofGare the same over Kand over Kpζℓq. /square\nProposition 4.3. For every prime ℓ∤Bwe haveKpζℓ,ℓ?\nGq XKpζ8q “Kpζℓq. Moreover,\n(for any choice of the ℓ-th roots) we have Kpℓ?\nGq XKpζ8q “K. For all primes ℓ∤Bthe\ncyclotomic-Kummer extensions Kpζℓ,ℓ?\nGq{Kare linearly disjoint over K.\nProof.The field Kpζℓ,ℓ?\nGq XKpζ8qis an abelian extension of Kso by Schinzel’s result\non abelian radical extensions [ 7, Theorem 2] it must be contained in Kpζℓqbecauseℓ∤Q.\nThe second assertion follows because the extensions Kpζℓq{KandKpℓ?\nGq{Khave coprime\ndegrees.\nFor the last assertion it suffices to observe that for every prime ℓ∤Band by varying rℓ\nin the prime numbers we have\nKpζℓ,ℓ?\nGq Xź\nrℓ∤ℓBKpζrℓ,rℓ?\nGq ĎKpζℓq Xź\nrℓ∤ℓBKpζrℓ,rℓ?\nGq “Kpζℓq Xź\nrℓ∤ℓBKpζrℓq “K.\nThe inclusion is because Kpℓ?\nGq{KandKprℓ?\nGq{Khave coprime degrees and then we may\napply the first part of the statement. The former equality again fo llows from the first part\nof the statement, while the latter equality is a consequence of Rema rk4.1. /square\nLetBGbe the squarefree integer which is the smallest positive multiple of Bsuch that\nfor any fixed prime ℓ∤BGand denoting any prime by rℓwe have\nKpζℓ,ℓ?\nGq Xź\nrℓ‰ℓKpζrℓ,rℓ?\nGqq “KUNIFORM BOUNDS FOR THE DENSITY IN ARTIN’S CONJECTURE 7\n(the existence of BGfollows from Kummer theory, see [ 3, Section 3] by the first author and\nJ¨ arviniemi). We remark that in the following results we could replace BGby a positive\nmultiple of it which is again square-free.\nA proof of our next result has been communicated to the authors b y Fritz H¨ ormann.\nProposition 4.4. LetKbe a field, let L1,L2,Mbe finite abelian extensions of Kand\nconsider the field E:“ML1XL2. Assuming that L1XL2“K, the Galois group GalpE{Kq\nofE{Kis a quotient of a subgroup of the Galois group GalpM{KqofM{K.\nProof.The compositum F:“L1L2Mis again a finite abelian extension of K. We write\nG:“GalpF{Kqand forXP tL1,L2,Muwe write GX:“GalpF{Xq. By our assumption,\nwe haveG“GL1GL2. First observe that we have\n(4.1) Gal pE{Kq –GL1GL2\nGL2pGL1XGMq–GL1\npGL1XGMqpGL1XGL2q,\nwhere the second isomorphism is obtained by observing that GL2X pGLXGMq “1 (by\nthe definition of F) and hence\nGL1GL2\nGL2pGL1XGMq–GL1GL2\nGL2\nGL1XGM–GL1\nGL1XGL2\nGL1XGM–GL1\npGL1XGMqpGL1XGL2q.\nThe last group in ( 4.1) is a quotient of\nGL1\nGL1XGM–GL1GM\nGM.\nFinally, this last group is a subgroup of\nGL1GL2\nGM–GalpM{Kq\nand the result follows. /square\nWe recall that the exponent of an abelian extension is defined as the exponent of its\nGalois group.\nWe instantly derive from Proposition 4.4one of our main tools.\nCorollary 4.5. LetL1,L2andMbe finite abelian extensions of Ksuch that L1XL2“K.\nThen the field ML1XL2ĎL2is an abelian extension of Kof degree dividing rM:Ksand\nof exponent dividing the exponent of M{K.\nWe also need the following result.\nProposition 4.6. For every prime ℓletGℓbe a subgroup of G. Consider the field\nF:“ź\nℓ|BKpζℓ,ℓa\nGℓq Xź\nℓ∤BKpζℓ,ℓa\nGℓq.\nThen we have\nFĎ´\nKpζBqź\nq|QKpqa\nGqq¯\nXKpζBG{Bq8 A. PERUCCA AND I.E. SHPARLINSKI\nandF{Kis a Kummer extension of exponent dividing Qand degree dividing QrQ, where\nrQ:“maxq|QrankGq.\nProof.The second assertion follows fromthe first as a consequence of Co rollary4.5(setting\nM“ś\nq|QKpqa\nGqq,L1“KpζBqandL2“KpζBG{Bq) because M{Kis a Kummer\nextension of exponent dividing Qand degree dividing QrQ. Also remark that by the\ndefinition of BGwe have\nF“ź\nℓ|BKpζℓ,ℓa\nGℓq Xź\nℓ|pBG{BqKpζℓ,ℓa\nGℓq.\nFor any prime rℓ, we consider the rℓ-partFrℓofF, namely the largest subextension of F\nwhose degree is a power of rℓ. For a prime q|Qwe clearly have\nFqĎ´\nKpqa\nGqqź\nℓ|pB{qqKpζℓq¯\nXź\nℓ|pBG{BqKpζℓq.\nSo we conclude by proving that for rℓ∤Qwe haveFrℓ“K.\nForrℓ∤Bwe have\nFrℓĎź\nℓ|BKpζℓq X´\nKpζrℓ,rℓa\nGrℓqź\nℓ|pBG{Bq,ℓ‰rℓKpζℓq¯\n.\nConsidering the former field on the right-hand-side, Frℓ{Kis abelian. So we may replace\nthe latter field by its largest abelian subextension. Since ζrℓRK, by Schinzel’s Theorem on\nabelian radical extensions [ 7, Theorem 2] we then have\nFrℓĎź\nℓ|BKpζℓq Xź\nℓ|prℓBG{BqKpζℓq “K,\nwhere the equality holds by Remark 4.1.\nForrℓ|Bwith rℓ∤Qwe may reason analogously because we have\nFrℓĎ´\nKpζrℓ,rℓa\nGrℓqź\nℓ|pB{rℓqKpζℓq¯\nXź\nℓ|pBG{BqKpζℓq,\nwhich concludes the proof. /square\nAs remarked above, BandBGin Proposition 4.6can be replaced by positive square-free\nmultiples that still satisfy B|BG.\nRemark 4.7. As a special case of Proposition 4.6we have\nź\nℓ|BKpζℓ,ℓ?\nGq Xź\nℓ∤BKpζℓ,ℓ?\nGq ĎKpζB,Q?\nGq XKpζBG{Bq.UNIFORM BOUNDS FOR THE DENSITY IN ARTIN’S CONJECTURE 9\n5.General bounds for the Artin density\nAll densities mentioned in this section exist thanks to the results in [ 3] by the first\nauthor and J¨ arviniemi (which are conditional under the Extended Riemann Hypothesis).\nLetBbe as in Section 4. We denote by dens BpGqthe density of the primes pofKsuch\nthat for every prime divisor ℓ|Bwe have ℓ∤indpGmodpq. For any prime ℓ, we write\ndensℓpGqfor the density of the primes pofKsuch that ℓ∤indpGmodpq. Finally, we\ndenote by ĆdensBpGqthe density of the primes pofKsuch that for every ℓ∤Bwe have\nℓ∤indpGmodpq.\nTheorem 5.1. Suppose that dens pGq ‰0(in particular, r2ą0). Then we have\ndens pGq\nARďź\nℓ|Bˆ\n1´1\nℓrℓpℓ´1q˙´1\nď2ź\nℓ|pB{2qℓ´1\nℓ´2.\nProof.The second inequality is clear because we can estimate the factor ℓ“2 usingr2“1\nand each factor ℓ‰2 usingrℓ“0.\nFor every prime ℓ∤Bthe degree of Kpζℓ,ℓ?\nGq{Kequalsℓrℓpℓ´1qby Proposition 4.2.\nMoreover, such fields are linearly disjoint by Proposition 4.3. Remark that by Hooley’s\nmethod (see Moree’s survey [ 6]) the condition ℓ∤indpGmodpqcan be replaced by pnot\nsplitting completely in Kpζℓ,ℓ?\nGq. We deduce that\nĆdensBpGq “ź\nℓ∤BdensℓpGq “ź\nℓ∤Bˆ\n1´1\nℓrℓpℓ´1q˙\n.\nSince\ndens pGq\nARďĆdensBpGq\nAR“ź\nℓ|Bˆ\n1´1\nℓrℓpℓ´1q˙´1\n,\nwe conclude the proof. /square\nWe recall the definition of BGfrom Section 4.\nTheorem 5.2. There exists a positive constant cBthat depends only on Bsuch that if\ndens pGq ‰0then we have\ndens pGq\nARěcB.\nProof.Wereplace Bby the smallest square-free positive multiple Bso that forevery prime\nℓ∤Bthe integer ℓ´1 is at least 2 Q, where, as before, Qdenotes the product of the prime\nnumbers qsuch that ζqPK. Observing that QďBwe see that Bonly depends on\nthe original value for B. We call BGthe least common multiple between BandBGand\nremark that BGis squarefree. We point out that our proof strategy involves conc entrating\non the prime divisors of BG, distinguishing the primes that divide Bby those who don’t.10 A. PERUCCA AND I.E. SHPARLINSKI\nBy the independence and maximality of the cyclotomic-Kummer exten sions at primes\nnot dividing BG(as described in Section 4) we have\ndens pGq “dens BGpGqź\nℓ∤BGdensℓpGq “dens BGpGqź\nℓ∤BGˆ\n1´1\nℓrℓpℓ´1q˙\n.\nThus we have\ndens pGq\nAR“dens BGpGq\nś\nℓ|BG´\n1´1\nℓrℓpℓ´1q¯ědens BGpGq\nś\nℓ|pBG{Bq´\n1´1\nℓrℓpℓ´1q¯.\nSince dens pGq ‰0 we must have dens BpGq ‰0 so there exists a Galois automorphism τ\non ź\nℓ|BKpζℓ,ℓ?\nGq,\nwhich is not the identity on any of the fields Kpζℓ,ℓ?\nGq. We can fix a set of generators of\nGand for any ℓ|Bwe choose one of those generators, call it αℓ, which is not fixed by τ,\nsettingαℓ“1 if no such generator exists. We then call τ|Lthe restriction of τto the field\nL:“ź\nℓ|BKpζℓ,ℓ?αℓq.\nFor every positive integer Nwe write KN:“ś\nℓ|NKpζℓ,ℓ?\nGq. We then define fBG,τpGq\nas the number of automorphisms in Gal pKBG{Kqwhose restriction to Lisτ|L, and such\nthat for every ℓ| pBG{Bqthe restriction to Kpζℓ,ℓ?\nGqis not the identity. By our choice\nofτwe clearly have\nfBG,τpGq\nrKBG:Ksďdens BGpGq\nso we are left to prove that there is a constant c‹\nB(that depends only on B) such that\n(5.1)fBG,τpGq\nrKBG:Ksěc‹\nBź\nℓ|pBG{Bqˆ\n1´1\nℓrℓpℓ´1q˙\n.\nWe consider the field\nE:“LXKBG{B.\nWe can write\n(5.2)fBG,τpGq\nrKBG:Ks“1\nrL:Es¨fBG,τ|EpGq\nrKBG{B:Ks,\nwherefBG,τ|EpGqcounts the automorphisms in Gal pKBG{B{Kqwhose restriction to E\nequalsτ|Eand such that for every ℓ| pBG{Bqthe restriction to Kpζℓ,ℓ?\nGqis not the\nidentity. Indeed, once we have fixed a suitable automorphism in Gal pKBG{B{Kq, then we\nonly need to extend it in a prescribed way to LKBG{B, and we have rLKBG{B:KBG{Bs “\nrL:Es. Since rL:Es ď rL:Ks ďś\nℓ|Bℓpℓ´1q, the inverse of rL:Escan be bounded\nfrom below only in terms of B.UNIFORM BOUNDS FOR THE DENSITY IN ARTIN’S CONJECTURE 11\nIfBG“B, then the last ratio in ( 5.2) is 1 and we may conclude, so in what follows we\nwill suppose that BGhas more prime divisors with respect to B.\nRecall that by our choice of Bwe have\nrKBG{B:Ks “ź\nℓ|pBG{Bq1\nℓrℓpℓ´1q\nso, by (5.1), we are left to prove that there exists a constant c7\nBdepending only on Bsuch\nthat\n(5.3) fBG,τ|EpGq ěc7\nBź\nℓ|pBG{Bqpℓrℓpℓ´1q ´1q.\nBy Proposition 4.6(settingGℓ“ xαℓyforℓ|BandGℓ“Gotherwise, so in particular\nrQ“1) we deduce that E{Kis a Kummer extension of degree dividing Qcontained in\nKpζBG{Bq.\nFix a prime qdividing rE:Ks, so in particular we have q|Q. Theq-part of the\nextension E{Kis a cyclic extension Cq{Kof degree qcontained inś\nℓPSqKpζℓq, whereSq\nis a subset of the primes dividing BG, not dividing B, and such that q| rKpζℓq:Ks. We\nmay assume that Sqis minimal.\nThen (on the q-part of the considered Galois groups) the condition of extending t he au-\ntomorphism τ|Cqis a compatibility of q-characters related to the subextensions of degree q\nofKpζℓqforℓPSq. And this condition is satisfied if we fix ℓqPSqand select an appropriate\nautomorphism on the degree qsubextension of Kpζℓqq{K, such choice depending on the\nfree choices that we can make for Kpζℓq{Kby varying ℓPSqwithℓ‰ℓq.\nFor every ℓ| pBG{BqcallQℓthe product of the primes qas above such that ℓ“ℓq\n(settingQℓ“1 for an empty product). So, up to considering fewer automorphis ms, we\nare able to separate the conditions at the different primes, namely\n(5.4) fBG,τ|EpGq ěź\nℓ|pBG{Bqgrℓ,Qℓ,\nwheregrℓ,Qℓis the amount of automorphisms in Kpζℓ,ℓ?\nGqthat are not the identity and\nthat are the fewest by varying a prescribed restriction to the cyc lic subextension of degree\nQℓofKpζℓq. We have (recalling that ℓ´1ě2Qě2Qℓ)\n(5.5)grℓ,QℓěrKpζℓ,ℓ?\nGq:Ks\nQℓ´1“ℓrℓpℓ´1q\nQℓ´1ě1\nmint2,QℓuQℓpℓrℓpℓ´1q ´1q.\nRemark thatź\nℓ|pBG{BqQℓ|Q\nand that the number of primes ℓ| pBG{Bqsuch that Qℓ‰1 is bounded by the number\nof prime divisors of Q, which is some constant CKthat only depends on the field K.12 A. PERUCCA AND I.E. SHPARLINSKI\nSubstituting ( 5.5) in (5.4) we then obtain\nfBG,τ|EpGq ě1\n2CKQź\nℓ|pBG{Bqℓrℓpℓ´1q ´1,\nwhich implies ( 5.3) and concludes the proof. /square\nFinally, since Bdepends only on K, we observe that Theorems 5.1and5.2imply that\ndens pGq{ARcan be bounded only in terms of K, which gives an analogues of Theorem 2.1\nfor arbitrary number fields Kand finitely generated groups of Kˆ, albeit less explicit.\nCorollary 5.3. There exist two positive constants c0pKqandC0pKqthat depend only on\nKsuch that if dens pGq ‰0then we have\nc0pKq ďdens pGq\nARďC0pKq.\nReferences\n[1] C. Debry and A. Perucca, Reductions of algebraic integers , J. Number Theory 167(2016), 259–283.\n6\n[2] C. Hooley, Artin’s conjecture for primitive roots , J. Reine Angew. Math. 225(1967), 209–220. 2,3\n[3] O. J¨ arviniemi and A. Perucca, Unified treatment of Artin-type problems , Res. Number Theory 9\n(2023), Art. 10. 5,7,9\n[4] O. J¨ arviniemi, A. Perucca and P. Sgobba, Unified treatment of Artin-type problems II , submitted for\npublication (status: minor revisions), arXiv:2211.15614 .5\n[5] H. W. Jr. Lenstra, On Artin’s conjecture and Euclid’s algorithm in global field s, Invent. Math. 42\n(1977), 201–224. 2\n[6] P. Moree, Artin’s primitive root conjecture – a survey , Integers 12(2012), 1305–1416. 1,3,9\n[7] A. Schinzel, Abelian binomials, power residues and exponential congrue nces, Acta Arithm. 32(1977),\n245–274. 6,8\nA.P.: Department of Mathematics, University of Luxembourg , Esch-sur-Alzette, L-\n4364, Luxembourg\nEmail address :antonella.perucca@uni.lu\nI.S.: School of Mathematics and Statistics, University of N ew South Wales. Sydney,\nNSW 2052, Australia\nEmail address :igor.shparlinski@unsw.edu.au"    },    {        "title": "2401.11659v1.Dynamical_invariant_based_shortcut_to_equilibration.pdf",        "content": "Dynamical invariant based shortcut to equilibration\nMohamed Boubakour,1,∗Shimpei Endo,2, 3Thom´ as Fogarty,1and Thomas Busch1\n1Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa 904-0495, Japan\n2Department of Engineering Science, The University of Electro-Communications, Tokyo 182-8585, Japan\n3Department of Physics, Tohoku University, Sendai 980-8578, Japan\n(Dated: January 23, 2024)\nWe propose using the dynamical invariants, also known as the Lewis–Riesenfeld invariants, to\nspeed-up the equilibration of a driven open quantum system. This allows us to reverse engineer the\ntime-dependent master equation that describes the dynamics of the open quantum system and sys-\ntematically derive a protocol that realizes a shortcut to equilibration. The method does not require\nadditional constraints on the timescale of the dynamics beside the Born-Markov approximation and\ncan be generically applied to boost single particle quantum engines significantly. We demonstrate it\nwith the damped harmonic oscillator, and show that our protocol can achieve a high-fidelity control\nin shorter timescales than simple non-optimized protocols. We find that the system is heated during\nthe dynamics to speed-up the equilibration, which can be considered as an analogue of the Mpemba\neffect in quantum control.\nI. INTRODUCTION\nUnderstanding and controlling open quantum systems\nis a major challenge for the exploration of quantum phe-\nnomena in the presence of dissipative effects, the de-\nterministic preparation of quantum states, and the de-\nvelopment of quantum devices [1–4]. A particular in-\nteresting and relevant question is how to accelerate the\nequilibration of open quantum systems, which, for ex-\nample, has applications in enhancing the performance of\nquantum heat engines [5–8]. Recently this question has\nattracted some attention and various techniques based\non approaches such as optimal control [9–12], linear re-\nsponse theory [13], techniques inspired by shortcuts to\nadiabaticity [14, 15] and reverse engineering [16–18] have\nbeen developed. Some of these have also been success-\nfully applied to design optimized quantum Carnot cycles\n[19] or an optimized quantum Otto cycle [20, 21], both\nof which show that a potential quantum advantage can\nbe achieved.\nThe challenge of accelerating the equilibration of an\nopen quantum system comes with the challenge of de-\nscribing its dynamics in a practical way for control. Even\nin the well-established framework of using master equa-\ntions to describe open quantum systems [22], their deriva-\ntion can be a very difficult task for an underlying general\ntime-dependent Hamiltonian. For example, a shortcut\nto equilibration in Refs. [16, 18] was realized by deriving\na non-adiabatic time-dependent master equation in the\ninertial limit [23, 24], which assumes small variations of\nthe adiabatic parameter of the system. While this allows\nto obtain the Lindblad operators explicitly, the resulting\ndriving protocol can be restricted.\nIn this work, we propose a shortcut to equilibration re-\nalized by using the dynamical invariant (DI) to describe\nthe dynamics of driven open quantum systems. Also\n∗mohamed.boubakour@oist.jpknown as the Lewis-Riesenfeld invariant, it was originally\nintroduced by the latter to solve the time-dependent dy-\nnamics of closed quantum systems [25]. However, they\ncan also be used to derive a time-dependent master equa-\ntion in a comprehensive way, without restrictions on the\ndriving protocol, and which also gives a clear picture of\nthe influence of the driving protocol on the dissipative\neffects [26]. Using the master equation, we present a\ngeneral formulation of the shortcut between the equilib-\nrium states of the initial and final Hamiltonian through\nreverse engineering.\nTo show the power of this approach, we apply this\ntechnique to the damped harmonic oscillator. This is\na well-known model that can describe atomic and opto-\nmechanical systems, and is commonly used to explore the\noperation of quantum heat engines [27–29]. Moreover,\ndissipative harmonic oscillator systems have recently at-\ntracted additional attention as an interesting framework\nto describe Bose polaron systems [30, 31]. These systems\nare promising platforms to explore quantum thermody-\nnamics phenomena like heat transport between meso-\nscopic quantum gases [32] or thermometry [33]. We find\nthat our protocol outperforms non-optimized protocols,\nshowing much higher fidelity and shorter time to reach\nthe target equilibrium state. Notably, the effective tem-\nperature of the system in the middle of the dynamics\nneeds to increase to achieve faster equilibration.\nII. DYNAMICAL INVARIANT BASED\nTIME-DEPENDENT MASTER EQUATION\nA driven, open quantum system is in general described\nby a Hamiltonian of the form\nH(t) =HS(t) +HB+HI, (1)\nwhere HS(t) is the time-dependent Hamiltonian of the\nsystem of interest and HBis the Hamiltonian of the en-\nvironment. In our case, the environment correspondsarXiv:2401.11659v1  [quant-ph]  22 Jan 20242\nto a thermal bath at a given temperature Tand thus\nits state is given by ρB=Z−1\nBexp\u0010\n−HB\nkBT\u0011\n. The final\nterm describes the interaction between the system and\nthe environment, HI=P\nkAk⊗Bk, where the Akact\non the system and the Bkact on the bath. In order to\nderive a tractable master equation, we use the Born ap-\nproximation that assumes that the coupling strength is\nsufficiently weak to neglect correlations between the sys-\ntem and the bath during the dynamics. The state of the\nsystem and the bath can then be written as a product\nof the reduced states, ρ(t)≈ρS(t)⊗ρB. We also apply\nthe Markov approximation and assume that the correla-\ntions inside the bath decay much faster than any other\ntimescale of the system, which allows one to derive a local\nmaster equation. After tracing out the bath, we obtain\na Redfield master equation in the interaction picture of\nthe form\nd˜ρS(t)\ndt=−1\nℏ2X\nk,lZ∞\n0Bkl(τ)h\n˜Ak(t),˜Al(t−τ)˜ρS(t)i\n−Blk(−τ)h\n˜Ak(t),˜ρS(t)˜Al(t−τ)i\ndτ,\n(2)\nwhere Bkl(τ) = Tr B\u0010\n˜Bk(τ)BlρB\u0011\nis the two-point cor-\nrelation function of the bath and the tilde indicates op-\nerators in the interaction picture. In the presence of a\ntime-dependent system Hamiltonian, the evaluation of\n˜Ak(t) can be challenging since there is no general proce-\ndure that allows to calculate the time evolution operator\nof the system US(t) =T←exp\u0010\n−i\nℏRt\n0HS(τ)dτ\u0011\n(where\nT←is the time-ordering operator).\nHowever, this problem can be solved by using the\nDI. Let us consider the closed dynamics of the system\niℏ∂t|ψ(t)⟩=HS(t)|ψ(t)⟩. A DI of the Hamiltonian\nHS(t) is a Hermitian operator Iobeying [25]\ndI(t)\ndt=∂I(t)\n∂t+1\niℏ[I(t), HS(t)] = 0 . (3)\nThis means that the solution of the closed dynamics\ncan be written as a linear combination of the instanta-\nneous eigenstates of the DI |ψ(t)⟩=P\nncneiαn(t)|ϕn(t)⟩\nwhere the dynamical phases are given by αn(t) =\n1\nℏRt\n0⟨ϕn(τ)|iℏ∂\n∂τ−H(τ)|ϕn(τ)⟩dτ. The time-evolution\noperator of the system can therefore also be obtained\nfrom the eigenstates of the DI as\nUS(t) =X\nneiαn(t)|ϕn(t)⟩⟨ϕn(0)|, (4)\nwhich in turn allows one to calculate the operators acting\non the system in the interaction picture as [26]\n˜Ak(t) =X\nm,nei(αn(t)−αm(t))⟨ϕm(t)|Ak|ϕn(t)⟩Fmn,(5)\nwith Fmn=|ϕm(0)⟩⟨ϕn(0)|. The operators can therefore\nbe written as products of time-dependent scalar functionsthat contain the information on the driving protocol,\nand time-independent operators Fmn. Those operators\nare jump operators constructed with the DI eigenstate\nsuggesting that the dissipative part of the open dynam-\nics will involve transitions of the system between those\nstates.\nFocusing on the case of the time-dependent harmonic\noscillator, the Hamiltonian of the system is given by\nHS(t) =p2\n2m+1\n2mω(t)2x2. A DI of this Hamiltonian\nis [25, 34]\nI(t) =\u0010\nb(t)p−m˙b(t)x\u00112\n2m+1\n2mω2\n0\u0012x\nb(t)\u00132\n,(6)\nwhere ω0=ω(0) and b(t) corresponds to a dimensionless\nscaling function that satisfies the Ermakov equation\n¨b(t) +ω2(t)b(t) =ω2\n0\nb3(t). (7)\nWe notice that the invariant has the structure of a har-\nmonic oscillator with a constant frequency ω0, a position\nx/b(t) and momentum b(t)p−m˙b(t)x. Thus the eigen-\nstates can be obtained by using the standard ladder op-\nerators.\nIII. SHORTCUT TO EQUILIBRATION FOR\nTHE DAMPED HARMONIC OSCILLATOR\nA. Time-dependent master equation\nIn what follows, we design a shortcut to equilibra-\ntion (STE) protocol for the time-dependent damped har-\nmonic oscillator (DHO). The Hamiltonian of the bath\nis given by HB=P\nnℏωn\u0000\nb†\nnbn+1\n2\u0001\nand the interac-\ntion is described in the rotating wave approximation by\nHI=P\nngn\u0000\na†bn+ab†\nn\u0001\n, where the gnare constant\ncoupling strengths between the particle and the n-th\nmode of the bath. By using the DI of Eq. (6), we describe\nthe dynamics of the driven system by deriving the follow-\ning Lindblad master equation in the interaction picture\n(see Appendix A for details)\nd˜ρS(t)\ndt=−i\nℏh\n˜HLS(t),˜ρS(t)i\n+|D(t)|2\n2ℏ2γ+(˜ω(t))\u0012\naI˜ρS(t)a†\nI−1\n2{a†\nIaI,˜ρS(t)}\u0013\n+|D(t)|2\n2ℏ2γ(˜ω(t))\u0012\na†\nI˜ρS(t)aI−1\n2{aIa†\nI,˜ρS(t)}\u0013\n,(8)\nwhere ˜HLS(t) is the time-dependent Lamb shift in\nthe interaction picture and D(t) = b(t) + 1 /b(t) +\ni˙b(t)/ω0. The time-dependent decay rates characteriz-\ning the emission and absorption are given by γ+(˜ω(t)) =\nπJ(˜ω(t)) (1 + n(˜ω(t))) and γ(˜ω(t)) = γ+(˜ω(t))e−ℏ˜ω(t)\nkBT,3\nwithJ(ω) =P\nng2\nnδ(ω−ωn) being the bath spectral den-\nsity function and n(ω) = (eℏω\nkBT−1)−1the Planck distri-\nbution. The dissipative part of the dynamics occurs at a\ntime-dependent Bohr frequency given by ˜ ω(t) =ω0/b(t)2\nwhich gives a new physical interpretation of the scaling\nfunction b(t) in the context of open quantum systems\ntheory. The Lindblad operators aIanda†\nIcorrespond\nrespectively to the annihilation and creation operator of\nthe invariant at t= 0. However, if one considers a driving\nprotocol with a continuous start from the initial Hamil-\ntonian [ HS(0), I(0)] = 0, we recover the creation and\nannihilation operators of the particle, aI=a(a†\nI=a†).\nThe approximations used to derive this master equation\nwith regards to the different time scales involved in the\nsystem are detailed in Appendix A.\nB. Formulation of the shortcut protocol and\nreverse engineering\nThe STE protocol we want to design is the follow-\ning: the particle is initially at equilibrium with the\nthermal bath with an initial frequency ω(0) = ω0and\nthus the state is given by the Gibbs state ρS(0) =\nZ−1\n0e−HS(0)/kBTwhere Z0is the partition function. We\nwant to find a protocol such that at the end of it, the par-\nticle is in a new equilibrium state with the bath at the fi-\nnal frequency ω(tf) =ωfi.eρS(tf) =Z−1\nfe−HS(tf)/kBT,\nwhere tfis the duration of the protocol. We focus on the\ncompression stroke ωf> ω 0but the expansion can also\nbe done in the same way.\nWe know that the state of the particle will be Gaussian\nduring the dynamics and thus can be fully determined\nby the expectation values ⟨a†a⟩(t) = Tr\u0000\na†aρS(t)\u0001\nand\n⟨a2⟩(t) = Tr\u0000\na2ρS(t)\u0001\ncharacterizing the excitation and\nthe squeezing of the particle. The equations describing\ntheir evolution during the driving protocol are obtained\nin the interaction picture from the master equation (8)\nas\nd⟨˜a†˜a⟩\ndt=π\n2ℏ2|D(t)|2J(˜ω(t))\u0000\nn(˜ω(t))− ⟨˜a†˜a⟩\u0001\n,(9)\nd⟨˜a2⟩\ndt=−π\n2ℏ2|D(t)|2J(˜ω(t))⟨˜a2⟩. (10)\nSince the particle is initially in a Gibbs state with\n⟨˜a2⟩(0) = 0, it follows from Eq. (10) that ⟨˜a2⟩= 0\nat any time during the dynamics. Thus, the proto-\ncol is described by the differential equation (9) alone.\nFurthermore, the initial and target state are equilib-\nrium states which implies ⟨˜a†˜a⟩(0) = ( eℏω0\nkBT−1)−1,\n⟨˜a†˜a⟩(tf) = ( eℏωf\nkBT−1)−1andd⟨˜a†˜a⟩(0)\ndt=d⟨˜a†˜a⟩(tf)\ndt= 0.\nIn addition to these boundary conditions, we impose\nd2⟨˜a†˜a⟩(0)\ndt2=d2⟨˜a†˜a⟩(tf)\ndt2 = 0 to ensure a smooth evolution\nof the system between the initial state and the target\nstate.\nFigure 1. (a) Fidelity between the final state and the target\nstate as a function of tf. (b) Profile of the trap frequency\nfor the STE protocol as a function of time for different pro-\ntocol durations. The black dashed line shows the reference\nramp. (c) Coherence generated during the dynamics of the\nSTE as a function of time. (d) Effective temperature of the\nparticle during the STE protocol as a function of time. The\nfinal compression is ωf= 3ω0and the bath temperature is\nT=ℏω0/kB. The spectral density function is an Ohmic dis-\ntribution with an abrupt cut-off J(ω) =γωΘ(Λ−ω) with\nγ=ℏ2/500 and Λ = 100 ω0, and the number of particles in\nthe bath is N= 600.\nThe protocol can now be found by reverse-engineering\nEq. (9) to obtain the scaling function b(t), which in turn\nallows one to obtain the trap frequency from the Ermakov\nequation (7). The boundary conditions on ⟨˜a†˜a⟩com-\nbined with Eq. (9) imply b(0) = 1, b(tf) =p\nω0/ωfand\n˙b(0) = ˙b(tf) = 0. We also obtain additional boundary\nconditions from the Ermakov equation ¨b(0) = ¨b(tf) = 0.\nWe can thus consider a 6-th order polynomial ansatz for\nthe scaling function b(t) =P6\nn=0an(t/tf)n, in which the\nfirst 6 coefficients allow to satisfy the above boundary\nconditions. The 6-th order term can then ensure that the\nscaling function connects the initial state and the target\nstate through Eq. (9). The coefficient a6is simply found\nby maximizing the fidelity between the target state and\nthe state of the particle at the end of the protocol.\nC. Properties of the shortcut\nTo quantify the performance of the shortcut, we cal-\nculate the fidelity between the target state ρTand the\nstate of the particle at the end of the protocol\nF(ρS(tf), ρT) = Tr\u0012q√ρTρS(tf)√ρT\u00132\n. (11)\nSince the total Hamiltonian (1) is quadratic for the DHO,\nthe states of the particle and the bath remain Gaus-4\nsian. We can thus use an efficient numerical method\nto solve the dynamics of the total Hamiltonian (1) dur-\ning the protocol. In fact, this can be done by time-\nevolving the covariance matrix of the system plus the\nbath, which corresponds to a (2 N+2)×(2N+2) matrix\n(where Nis the number of particles in the bath), with\nthe Heisenberg equations of motion [35]. This allows us\nto see when the validity of the master equation actually\nbreaks down for a given set of bath parameters (num-\nber of particles, spectral density function) and when the\nshortcut therefore does not work anymore. To demon-\nstrate the benefit of our shortcut, we compare it with\ntwo simple protocols: the sudden quench ωq(t >0) = ωf\nand a reference ramp described by a polynomial function\nωr(t) =ω0+ 10∆ ω(t/tf)3−15∆ω(t/tf)4+ 6∆ ω(t/tf)5\nwith ∆ ω=ωf−ω0. For the DHO, the quench protocol is\ncharacterized by an asymptotic exponential convergence\nof the fidelity to one as F≈1−e−tf/τqwhere τqis a\ncharacteristic time related to the decay rates [16].\nThe fidelity as a function of tfis shown in Fig. 1(a).\nThe STE outperforms the sudden quench and the ref-\nerence ramp that also shows an asymptotic exponential\nbehavior but with a longer characteristic time than the\nsudden quench. However, one can see that for short dura-\ntions, the fidelity for the STE protocol collapses because\nat shorter times the description provided by the master\nequation deviates from the exact dynamics. The fidelity\nreaches approximately the value 0.999 around tf≈16/ω0\nand then keeps increasing to one.\nIn order to obtain physical insights on the STE pro-\ntocol the profile of the trap frequency ω(t) is shown in\nFig. 1(b) for different protocol durations tfand compared\nto the reference ramp (black dashed line). We see that\nthe profiles are quite different: while for the reference\nramp the frequency increases monotonically toward ωf,\nin the STE protocol the particle is driven to large trap\nfrequency values at intermediate times before decreasing\nto reach the final frequency at t=tf. Notably, for faster\nprotocols the particle is driven to higher frequency val-\nues. Inversely, for larger tf, the amplitude of the trap\nfrequency decreases and we observe that the STE proto-\ncol gets closer to the reference ramp when tfapproaches\nthe adiabatic limit.\nSuch a trap frequency profile necessarily implies non-\nequilibrium features in the dynamics. To quantify them,\nwe calculate the coherence in the system during the STE,\nwhich has been suggested to play a key role in the control\nof open quantum systems [19]. We define it as the change\nof entropy between the diagonal part of the state and the\nfull density matrix in the instantaneous eigenenergy basis\n[36]\nC(t) =S(ρdiag(t))−S(ρS(t)), (12)\nwhere S(ρ) =−Tr(ρlog(ρ)) is the von Neumann en-\ntropy. The coherence is shown in Fig. 1(c). One can\nsee that its profile reflects the results we have shown be-\nfore: coherence is generated in the system when the trap\nfrequency is changing. It vanishes when the maximumtrap frequency is reached, and increases again when the\ntrap frequency decreases toward the final value. There\nis no coherence remaining in the system at the end of\nthe protocol and the amount generated during the dy-\nnamics increases for faster protocols. This also allows\nto explain why the quench protocol works better than\nthe reference ramp. Indeed, coherence causes transitions\nof the particle between its eigenstates and can be used\nas a catalysis that helps to accelerate the thermalization\nof an open quantum system. A controlled manipulation\nof coherence allows to reach the new equilibrium state\nand this is what the STE achieves. Even though the\nnon-optimal protocols do not reach the target state, the\nsudden quench will always perform better than a smooth\nramp that generates much less non-adiabatic excitations.\nNotably, this is in contrast with shortcuts to adiabaticity\nprotocols which require the suppression of such excita-\ntions to reach the desired target state [37].\nFinally we characterize the dynamics of the STE\nby calculating the effective temperature of the sys-\ntem. During the dynamics, the state of the particle\ncan be written in the interaction picture as ˜ ρS(t) =\nZ(t)−1P\nne−ϵ(t)n|ϕn(0)⟩⟨ϕn(0)|with\n⟨˜a†˜a⟩(t) = (eϵ(t)−1)−1, (13)\nand the effective partition function is given by\nZ(t) = (1 −e−ϵ(t))−1. Thus, back to the\nSchr¨ odinger picture the Hamiltonian is always diago-\nnal in the instantaneous eigenbasis of the DI ρS(t) =\nZ(t)−1P\nne−ϵ(t)n|ϕn(t)⟩⟨ϕn(t)|, which shows that sim-\nilarly to the closed dynamics, the eigenstates of the DI\ngive us the states that the system will explore for a given\nprotocol. We then simply define the effective tempera-\nture of the system as\nTeff(t) =ℏω(t)\nkBϵ(t). (14)\nThe effective temperature of the particle during the STE\nis shown in Fig. 1(d). It deviates significantly from the\nbath temperature before returning to it at the end of\nthe protocol. More interestingly, the particle is driven to\nstates that are effectively hotter, and the faster the short-\ncut is, the hotter the state of the particle is. While the\nshortcut is designed for an isothermal compression, which\ncorresponds to a cooling process, the strategy adopted by\nthe STE actually consists of warming up the particle in\norder to cool it down faster. This is reminiscent of the\nMpemba effect [38], an empirical phenomenon where a\nhot liquid can freeze faster than a cold liquid. Recently,\nthe Mpemba effect has been discussed and predicted for\na quantum dot coupled to two reservoirs [39]. Here, we\nobserve a similar feature to the thermal Mpemba effect\nin the context of driven open quantum systems.5\nIV. CONCLUSION\nIn the present work, we have presented results that\npave the way for improving the control of driven open\nquantum system and the performance of quantum heat\nengines in the future. Dynamical invariants have been\nshown to be a powerful tool for describing and acceler-\nating the equilibration for a well-known model, without\nadditional restrictions on the timescale of the dynam-\nics beside the Born-Markov approximation. Our work\nalso brings new physical interpretations of the dynami-\ncal invariant. Indeed the scaling function b(t), that fully\ncharacterizes the invariant, sets both the driving protocol\nand the decay rates. This allows to derive protocols that\nmodify both the unitary part and the dissipative part of\nthe dynamics. The STE protocol is characterized by a\nmanipulation of the coherence that drives the particle to\nhotter states in the case of the isothermal compression.\nOur observation resonates with the thermal Mpemba ef-\nfect and a rigorous formulation of this phenomenon in\nthe context of driven open quantum systems would be\nan interesting direction to take in the future.\nIt is also an interesting avenue to design shortcuts and\naddress the question whether the geometric bound [40–\n42] can be reached by minimizing the dissipated work.\nWhile we have considered the isothermal stroke in this\nwork, one can consider different strokes, or start from\nnon-equilibrium states. It would be also possible to de-\nsign shortcuts for applications other than ones thermody-namics. For example, our method could be used to design\nfast and robust protocols for quantum gates [3, 43]. Also\nrecently, a similar approach has been used to quickly gen-\nerate entangled states in a double two-level system [44].\nFinally, an important extension would be to go beyond\nthe single particle problem and optimize the equilibra-\ntion of interacting many-body states. While not an easy\ntask, two possible directions are within reach: two parti-\ncles with short-range interactions [45] and the hardcore\nTonks–Girardeau limit [46]. Beside having well-known\nanalytical results, both systems have showed enhanced\nperformances compared to non-interacting quantum en-\ngines [47–49], paving the way for fully-optimized many-\nbody quantum heat engines.\nACKNOWLEDGEMENTS\nThis work was supported by the Okinawa Institute of\nScience and Technology Graduate University, and used\nthe computing resources of the Scientific Computing and\nData Analysis section at OIST. The authors also ac-\nknowledge support from a SHINKA grant from OIST and\nTohoku University. SE is supported by JSPS KAKENHI\nGrant Numbers JP21H00116 and JP22K03492. TF ac-\nknowledges support from JSPS KAKENHI Grant Num-\nber JP23K03290. TF and TB are also supported by JST\nGrant Number JPMJPF2221.\n[1] C. Brif, R. Chakrabarti, and H. Rabitz, Control of quan-\ntum phenomena: past, present and future, New J. Phys.\n12, 075008 (2010).\n[2] C. P. 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Wu, Dynamics of a driven\nopen double two-level system and its entanglement gen-\neration, Phys. Rev. A 107, 032409 (2023).\n[45] T. Busch, B.-G. Englert, K. Rza˙ zewski, and M. Wilkens,\nTwo Cold Atoms in a Harmonic Trap, Foundations of\nPhysics 28, 549 (1998).\n[46] M. Girardeau, Relationship between Systems of Impene-\ntrable Bosons and Fermions in One Dimension, J. Math.\nPhys. 1, 516 (1960).\n[47] J. Jaramillo, M. Beau, and A. del Campo, Quantum\nsupremacy of many-particle thermal machines, New J.\nPhys. 18, 075019 (2016).\n[48] T. Fogarty and T. Busch, A many-body heat engine at\ncriticality, Quantum Science and Technology 6, 015003\n(2020).\n[49] M. Boubakour, T. Fogarty, and T. Busch, Interaction-\nenhanced quantum heat engine, Phys. Rev. Res. 5,\n013088 (2023).\nAppendix A: Time-dependent master equation of\nthe damped harmonic oscillator\nWe start from the Redfield equation given by Eq. (2)\napplied to the damped harmonic oscillator\nd˜ρS(t)\ndt=−1\nℏ2Z∞\n0B12(τ)\u0002\n˜a†(t),˜a(t−τ)˜ρS(t)\u0003\n−B21(−τ)\u0002\n˜a†(t),˜ρS(t)˜a(t−τ)\u0003\n+B21(τ)\u0002\n˜a(t),˜a†(t−τ)˜ρS(t)\u0003\n−B12(−τ)\u0002\n˜a(t),˜ρS(t)˜a†(t−τ)\u0003\ndτ,(A1)7\nwhere the bath two-point correlation functions are given\nby\nB12(τ) =X\nng2\nnTrB(˜bn(τ)b†\nnρB) =X\nne−iωnτg2\nn(1 +n(ωn))\n=Z∞\n0e−iωτJ(ω)(1 + n(ω))dω,\nB21(τ) =X\nng2\nnTrB(˜b†\nn(τ)bnρB) =X\nneiωnτg2\nnn(ωn)\n=Z∞\n0eiωτJ(ω)n(ω)dω.\n(A2)\nNow we need to evaluate the ladder operators of the\nparticle in the interaction picture. Since they are con-\njugate, we can just focus on the annihilation operator,\nwhich we can write with the position and momentum\noperators in the interaction picture as\n˜a(t) =rmω0\n2ℏ\u0012\n˜x(t) +i˜p(t)\nmω0\u0013\n. (A3)\nWe use the invariant (6) to evaluate the operator in\nthe interaction picture. Since the invariant is a har-\nmonic oscillator with a position x/b(t) and momentum\nΠ = b(t)p−m˙b(t)x, we can express the position and\nmomentum of the particle in terms of the instantaneous\nladder operators of the invariant that we denote aItand\na†\nIt\nx=b(t)r\nℏ\n2mω0(aIt+a†\nIt),\np=Π\nb(t)+m˙b(t)x\nb(t)=r\nℏmω0\n2\u0010\nC(t)aIt+C∗(t)a†\nIt\u0011\n,\n(A4)\nwith the complex function\nC(t) =˙b(t)\nω0−i\nb(t). (A5)\nWe deduce that the annihilation operator of the har-\nmonic oscillator in the Schr¨ odinger picture is related tothe ladder operators of the invariant through the follow-\ning Bogoliubov transformation\na=1\n2\u0010\nD1(t)aIt+D∗\n2(t)a†\nIt\u0011\n, (A6)\nwhere\nD1,2(t) =b(t)±1\nb(t)±i˙b(t)\nω0. (A7)\nWe now need to calculate the ladder operators of the\ninvariant in the interaction picture. This is easily done\nby using the time-evolution operator written with the\neigenstates of the invariant (Eq. (4)) and using the ex-\npression of the dynamical phase for the harmonic oscil-\nlator αn(t) =−ω0(n+ 1/2)Rt\n01/b(τ)2dτ\n˜aIt(t) =X\nn,mei(αm(t)−αn(t))|ϕn(0)⟩⟨ϕn(t)|aIt|ϕm(t)⟩⟨ϕm(0)|\n=X\nn,mei(αm(t)−αn(t))√mδn,m−1|ϕn(0)⟩⟨ϕm(0)|\n=e−iφ(t)X\nn√\nn+ 1|ϕn(0)⟩⟨ϕn+1(0)|=e−iφ(t)aI0,\n(A8)\nwhere the phase φis given by\nφ(t) =Zt\n0ω0\nb(τ)2dτ=Zt\n0˜ω(τ)dτ. (A9)\nFor the next we will use aI(a†\nI) instead of aI0(a†\nI0)\nto denote the ladder operators of the invariant at t= 0.\nNow we can obtain an explicit expression of the annihi-\nlation operator of the particle in the interaction picture\n˜a(t) =1\n2\u0010\nD1(t)e−iφ(t)aI+D∗\n2(t)eiφ(t)a†\nI\u0011\n.(A10)\nWe can insert Eq. (A10) in the Redfield equation (A1).\nWe only explicitly write it down for the first commutator\nin the right-hand side of Eq.(A1) since the same treat-\nment can be straightforwardly done for the other terms.\nAfter expanding the commutator, we obtain\n[˜a†(t),˜a(t−τ)˜ρS(t)] =1\n4\u0010\ne−i(φ(t)−φ(t−τ))D2(t)D∗\n2(t−τ)[aI, a†\nI˜ρS(t)] +e−i(φ(t)+φ(t−τ))D2(t)D1(t−τ)[aI, aI˜ρS(t)]\n+ei(φ(t)+φ(t−τ))D∗\n1(t)D∗\n2(t−τ)[a†\nI, a†\nI˜ρS(t)] +ei(φ(t)−φ(t−τ))D∗\n1(t)D1(t−τ)[a†\nI, aI˜ρS(t)]\u0011\n.\n(A11)\nThe integral in the Redfield equation is dominated\nby the bath two-point correlation function that rapidly\ndecays with a characteristic time τB. The decay timeis given by the cut-off of the bath τB∼Λ−1. Based\non the Markov approximation, the decay time must be\nmuch smaller than the typical timescale of the system8\ngiven by ω(t)−1i.eτB≪ω(t)−1. We can thus use\nthe first order approximation of the phase in the inte-\ngralφ(t−τ)≈φ(t)−˜ω(t)τ. We also make the zero-th\norder approximation Di(t−τ)≈Di(t) meaning that the\nvariations of the scaling function b(t) and its derivative\nare negligible in the time window [0 , τB]. Formally, it im-\nplies τB≪\f\f\fDi(t)\n˙Di(t)\f\f\f. This approximation can be reformu-\nlated as τB≪τDwhere we introduce a driving timescale\nτD= min i,t\f\f\fDi(t)\n˙Di(t)\f\f\f[23]. Let us remark that this approx-\nimation is not necessary to derive the master equation,\nhowever it allows to simplify the reverse-engineering for\nthe shortcut.The last approximation we will use is the secular ap-\nproximation. We neglect the non-secular terms, to derive\na master equation in Lindblad form and ensure that the\nstate of the system remains physical. This means that the\nnon-secular contributions contain fast oscillating terms\nthat average to zero. This implies φ(t) +φ(t−τ)≫\nφ(t)−φ(t−τ). By using a first order expansion and\nthe Markov approximation, we obtain φ(t)≫˜ω(t)τBi.eRt\n0˜ω(τ)dτ≫˜ω(t)τB.\nTaking account of the different approximations in\nEq. (A11), we obtain\nZ∞\n0B12(τ)\u0002\n˜a†(t),˜a(t−τ)˜ρS(t)\u0003\ndτ≈|D2(t)|2\n4Z∞\n0B12(τ)e−i˜ω(t)τdτ[aI, a†\nI˜ρS(t)]+|D1(t)|2\n4Z∞\n0B12(τ)ei˜ω(t)τdτ[a†\nI, aI˜ρS(t)].\n(A12)\nThe integrals can be calculated by using the well-\nknown resultR∞\n0eiωτdτ=πδ(ω) +iP(1/ω) where P\ndenotes the principal value. After combining the differ-\nent terms and few lines of algebra, we obtain the time-\ndependent master equation for the damped harmonic os-\ncillator given by Eq. (8), where we replaced D1(t) by\nD(t). The time-dependent Lamb shift is given by\n˜HLS(t) =ℏ\n4\u0012\n|D1(t)|2PZ∞\n0J(ω)\n˜ω(t)−ωdω\n−|D2(t)|2PZ∞\n0J(ω)\n˜ω(t) +ωdω\u0013\n.(A13)"    },    {        "title": "2401.11696v1.Self_interaction_corrected_SCAN_functional_for_molecules_and_solids_in_the_numeric_atom_center_orbital_framework.pdf",        "content": "arXiv:2401.11696v1  [cond-mat.mtrl-sci]  22 Jan 2024Self-interaction corrected SCAN functional for molecules and solids in the numeric\natom-center orbital framework\nSheng Bi,1,2,3Christian Carbogno,1,∗Igor Ying Zhang,2,4,†and Matthias Scheffler1\n1The NOMAD Laboratory at the FHI of the Max-Planck-Gesellsch aft and IRIS-Adlershof\nof the Humboldt-Universit¨ at zu Berlin, Faradayweg 4-6, D- 14195 Berlin-Dahlem Germany\n2Department of Chemistry, Fudan University, Shanghai 20043 3, People’s Republic of China\n3Research Center for Intelligent Supercomputing, Zhejiang L ab, Hangzhou 311100, P. R. China\n4MOE Key Laboratory of Computational Physical Sciences, Sha nghai Key Laboratory of Bioactive Small Molecules\n(Dated: January 23, 2024)\nSemilocal density-functional approximations (DFAs), inc luding the state-of-the-art SCAN func-\ntional, are plagued by the self-interaction error (SIE). Wh ile this error is explicitly defined only\nfor one-electron systems, it has inspired the self-interac tion correction method proposed by Perdew\nand Zunger (PZ-SIC), which has shown promise in mitigating t he many-electron SIE. However, the\nPZ-SIC method is known for its significant numerical instabi lity. In this study, we introduce a novel\nconstraint that facilitates self-consistent localizatio n of the SIC orbitals in the spirit of Edmiston-\nRuedenberg orbitals [Rev. Mod. Phys. 35, 457 (1963)]. Our practical implementation within the\nall-electron numeric atom-centered orbitals code FHI-aims guarantees efficient and stable conver-\ngence of the self-consistent PZ-SIC equations for both mole cules and solids. We further demonstrate\nthat our PZ-SIC approach effectively mitigates the SIE in the meta-GGA SCAN functional, signifi-\ncantly improving the accuracy for ionization potentials, c harge-transfer energies, and band gaps for\na diverse selection of molecules and solids. However, our PZ -SIC method does have its limitations.\nIt can not improve the already accurate SCANresults for prop erties such as cohesive energies, lattice\nconstants, and bulk modulus in our test sets. This highlight s the need for new-generation DFAs\nwith more comprehensive applicability.\nI. INTRODUCTION\nKohn-Sham (KS) density-functional theory (DFT) [1,\n2] is currently the most widely used electronic-structure\nmethod across various scientific disciplines [3–5]. To a\nlarge extent, this is due to the favorable balance be-\ntween computational accuracy and efficiency. The key\ncontribution to this favorable balance is the existence\nof density-functional approximations (DFAs) to the ex-\nact exchange-correlation (XC) functional [2, 5]. In this\ncontext, the non-empirical SCAN functional proposed by\nSun, Ruzsinszky, and Perdew in 2015 [6] is particularly\npromising. Compared to earlier semi-local DFAs, includ-\ning the local-density approximations (LDAs), the gener-\nalized gradient approximations(GGAs), and other meta-\nGGAs [7–9], the SCAN meta-GGA functional yields con-\nsistent and notable improvements in describing many\nchemical and physical properties of both molecules and\nsolids [10–12]. This can be attributed to its compli-\nance with 17 constraints, which the exact XC functional\nmust satisfy. However,like all othersemi-localDFAs, the\nSCANfunctional isnotimmune tothe self-interactioner-\nror (SIE) [13–15]. This limitation accounts for its inabil-\nity to reliably describe the ionization potential, specifi-\ncallytheenergyofthehighestoccupiedmolecularorbital,\nas well as charge-transferproperties and band gaps in in-\nsulators and semiconductors [16, 17].\n∗christian.carbogno@fhi-berlin.mpg.de\n†igorzhangying@fudan.edu.cnFormally, the SIE arises from the ground of the\nXC functional within the KS-DFT framework, i.e., the\nHartree term [18]:\nEH[n] :=1\n2/integraldisplay/integraldisplayn(r)n(r′)\n|r−r′|d3r′d3r,(1)\nwhich captures the most part of electron-electron\nCoulomb interaction in the system. However, since the\ndensityn(r) encompasses all electrons, the Hartee term\nalso introduces non-physical Coulomb interactions where\nindividual electrons interact with themselves. In one-\nelectron systems, this spurious interaction is precisely\ncounterbalanced by the Fock term within the Hartree-\nFock (HF) theory. In other words, the HF method is\nfree from one-electron self-interactions. However, it is\nwell-documented that the HF approach is not enough to\ncancel out these non-physical self-interactions in multi-\nelectron systems [19]. Within the KS-DFT framework,\nthese non-physical interactions are expected to be pre-\ncisely canceled out by the exact XC functional.\nUnfortunately, this cancellation is insufficiently\nachieved for all existing semi-local DFAs. The residual\nself-interaction implies the SIE of a given DFA, leading\nto incorrect electron delocalization [13, 20]. Obviously, a\nsingle orbital’s XC energy must cancel its self-interacted\nCoulombinteraction. Thisanalysishasmotivatedthede-\nvelopment of so-called self-interaction corrections (SIC),\ninitially proposed by Perdew and Zunger in the early\n1980s (PZ-SIC) [15]:\nEPZ-SIC[{ni}] :=−Ne/summationdisplay\ni/parenleftbig\nEDFA\nxc[ni]+EH[ni]/parenrightbig\n.(2)2\nHere,Nerepresents the total number of electrons in\nthe system, and {ni(r) =|φi(r)|2}denotes the densi-\nties of occupied single-electron orbitals, referred to as\nSIC orbitals {φi(r)}henceforth (for further details, see\nSectionII). Accordingly, EDFA\nxc[ni] is the DFA-specific\nsingle-particle XC energy evaluated for the SIC orbital\ndensityni(r). Meanwhile, EH[ni] is the Coulomb self-\ninteraction energy of ni(r), calculated using the Hartree\nequation (Eq. ( 1)). By incorporating these one-electron\ncorrection terms into the semi-local DFA XC energy, the\nPZ-SIC scheme effectively eliminates the SIE of a given\nDFA in one-electron systems. Meanwhile, it has been\nshownthatthePZ-SICschemecansubstantiallymitigate\nmany-electronSIE when the appropriateSIC orbitalsare\nemployed [21–23].\nWhile SIC orbitals coincide with KS orbitals in\none-electron systems, they generally diverge in many-\nelectron systems. Theoretically, the optimal SIC orbitals\nshould be determined by variationallyminimizing the to-\ntal energy of the self-interaction corrected DFA (SIC-\nDFA) [15]. This minimization must adhere to the total-\ndensity constraint , which stipulates that the sum of all\noccupied SIC orbital densities should yield the total elec-\ntron density n(r) =/summationtextNe\nini(r) (for further details, see\nSectionII). Moreover, as initially proposed by Peder-\nson, Heaton, and Lin, the variational minimization of\nthe SIC-DFA total energy not only produces the PZ-SIC\nequations for KS orbitals but also imposes the so-called\norbital potential constraint on SIC orbitals [24]. Similar\nto the standard KS equations, PZ-SIC equations should\nbe solved iteratively in a self-consistent manner because\nthey depend on both KS and SIC orbitals (also see Sec-\ntionIIfor more details).\nIn practice, the PZ-SIC equations often yield multi-\nple self-consistent solutions for SIC orbitals, all satisfy-\ning the total density and orbital potential constraints.\nAs a result, while the correct SIC orbitals that mini-\nmize the SIC-DFA total energy are among these multiple\nself-consistent solutions, the specific solution obtained is\nhighly sensitive to the choice of initial guess.\nIn the seminal work of PZ-SIC, Perdew and Zunger\nobserved that localized orbitals often provide lower SIC-\nDFAtotalenergiescomparedtotheirdelocalized(canoni-\ncal)counterparts, suchasstandardKSorbitals[15]. This\nobservation led to frequently characterizing SIC orbitals\nas “localized orbitals”. This insight has inspired sev-\neral empirical guidelines for initializing SIC orbitals [25].\nA variety of localization methods have been used to\ngenerate localized initializations [26], including Pipek-\nMezey [27], Edmiston-Ruedenberg [28], von Niessen [29],\nFoster-Boys [30], and the fourth-moment [31] methods.\nLehtola, Head-Gordon, and J´ onsson [26] systematically\nexamined various initial SIC orbitals for atoms in the\nfirst three rows of the periodic table. Remarkably, the\nSIC total energies associated with these different sets\nof orbitals can diverge by more than 0.5 eV. Conse-\nquently, identifying the correct SIC orbitals remains a\nsignificant challenge, even for atoms and simple diatomicmolecules [24, 25].\nMore recently, Pederson, Ruzsinszky, and Perdew\nnoted the topological similarity between Fermi orbitals\nwith L¨ owdin orthogonalization (FLOs) and previously\nemployed localized orbitals. Therefore, they suggested\nconstructing the SIC orbitals using FLOs [32–35], giving\nrise to a modified PZ-SIC algorithm known as FLO-SIC.\nThis innovative approach effectively filters out the FLO-\nlike SIC orbitals from the multiple self-consistent solu-\ntions, mitigating the issue of multiple solutions. FLO-\nSIC has demonstrated considerable promise for atoms\nand molecules [36–38], with particular success observed\nin systems containing πbonds [36, 39].\nForextendedmaterials,findingthecorrectSICorbitals\nbecomes even more serious [40]. To capture the localized\nnature of SIC orbitals, Heaton, Harrison, and Lin sug-\ngested representing them in terms of real-space Wannier\norbitals[22] under periodic boundary conditions (PBCs).\nHowever, it does not thoroughly solve the issue of multi-\nple solutions. Different sets of Wannier orbitals can sat-\nisfy the total density and orbital-potential constraints.\nConsequently, discrepancies arise; for instance, the SIC-\nLDAbandgapoftheArcrystalreportedbyHeaton, Har-\nrison, and Lin differs from that of Stengel and Spaldin as\nmuch as 0.7 eV [22, 41]. Large uncertainties are reported\nfor the band gaps of bulk Silicon [23, 41] and transition-\nmetal mono-oxides (MnO, FeO, CoO, and NiO) [42, 43].\nThe SIC orbitalsofthose results were all filtered by satis-\nfyingthe constraintsoftotaldensityandorbitalpotential\nbut using different localization definitions for the Wan-\nnier orbitals.\nMore recently, in 2020, Shinde et al.introduced the\nuse of conceptually related FLOs for filtering out the\nWannier orbitals (WFLO) [44]. As a result, the WFLO-\nSIC approach identifies proper SIC orbitals for solids,\nleading to a significant improvement in the band gap of\nthe semi-local PBE functional [45]. Additionally, alter-\nnative localization strategies have an enormous poten-\ntial to establish the SIC orbitals [23, 26], including the\nmaximally localized Wanner orbitals (MLWOs) [46] or\nthe Edmiston-Ruedenberg (E-R) type orbitals [28]. The\nE-R orbitals are similar to MLWOs in molecular sys-\ntems [47] and are essentially identical to MLWOs in ex-\ntended periodic systems [48]. However, these works eval-\nuate the SIC-DFA total energies in a non-self-consistent\nmanner. To our knowledge, there is no report on the\nself-consistent SIC-DFA implementation that explicitly\nincorporates MLWOs or E-R localization.\nBesides the SIC problems discussed above, there is an-\nother potential source of inaccuracy in most of the dis-\ncussed PZ-SIC studies, i.e., the usage of semi-local DFAs\nat the level of LDA or GGA [14, 22, 35, 39, 49, 50]. Since\nthe SIC is inherently linked to the employed DFA via\nEq. (2), it is yet unclear to which extent the observed\ntrends are related to the SIC or to the DFA. This is fur-\nther substantiated by the fact that the combination of\nPZ-SIC and the state-of-the-art meta-GGA SCAN has\nshown some promising results [32, 51–53]. Consequently,3\nthere is an urgent need for practical, self-consistent SIC\nimplementations compatible with the SCAN functional\nfor both molecules and solids.\nIn this work, we overcome these challenges by building\nupon the total-density constraint of Perdew and Zunger\nand theorbital potential constraint of Pederson, Heaton,\nand Lin. We introduce a novel third constraint, the or-\nbital density-potential constraint , to PZ-SIC equations.\nInspiredby the E-Rorbitals[28], ourproposedconstraint\naims to maximize spatial localization of the SIC orbitals\nunder the orbital potential constraint . As demonstrated\nin the subsequent sections, this approach facilitates ef-\nficient convergence of the PZ-SIC equations for both\nmolecular and extended systems.\nWe demonstrate the efficacy of our proposed method-\nology through its implementation in the all-electron nu-\nmeric atom-center orbital (NAO) package FHI-aims [54,\n55]. BesidesapplyingourPZ-SICapproachforsemi-local\nDFAs such as LDA and GGA, we primarily focus on PZ-\nSIC calculations with the meta-GGA SCAN functional\n(SIC-SCAN). For the latter, we observe significant im-\nprovementsin the highest occupied orbital energies when\ncompared to the ionization potential of atoms from H to\nAr, as well as for 18 molecules taken from reference [56].\nFurthermore, the band gaps of various solids, including\nSi, MnO,diamond-C,MgO,andLiF,arecalculatedusing\nSIC-SCAN, yielding comparable results to GWcalcula-\ntions. The combination of PZ-SIC and SCAN offers en-\nhancedaccuracyinelectronicstructurecalculationswhile\nmaintaining a moderate computational burden, particu-\nlarly for periodic solids.\nThe paper is organized as follows: Section IIprovides\na derivation of the standard PZ-SIC formula and intro-\nduces the three constraints used to solve the PZ-SIC\nequations, both for finite and periodic systems. In par-\nticular, we analyze and discuss the physical motivation\nbehind the orbital density-potential constraint by using\nthe carbon atom, the Helium dimer (He 2), the Helium\ncrystal, and the periodic polyethylene chain. In section\nIII, we assess the performance of PZ-SIC in conjunction\nwith SCAN by examining various metrics such as SIC\nenergy, ionization potential, band gap, and cohesive en-\nergyfor a groupofmolecules and solids. Conclusions and\noutlook are presented in section IV.\nII. THEORY\nWith the PZ-SIC concept (Eq. ( 2)), the total energy\nof a self-interaction corrected DFA, termed SIC-DFA, is\nwritten as [15]\nESIC-DFA[n,{ni}] :=EDFA[n]+EPZ-SIC[{ni}].(3)\nHereEDFA[n] is the total energy of a given DFA in the\nstandard KS framework [2]:\nEDFA[n] :=Ts[n]+Eext[n]+EH[n]+EDFA\nxc[n],(4)whereTsis the kinetic energy of the KS non-interacting\nelectrons. Eext[n] andEH[n] are the external poten-\ntial energy and the Hartree energy, respectively, both\nof which are explicit functionals of the electron density\nn(r).EDFA\nxc[n] is the XC energy of the DFA.\nIn the standard KS-DFT scheme, the KS equations\nare obtained via a variational minimization of the DFA\nenergyEDFA[n] (Eq. (4)), whereby the conservation of\nthe number of electrons Nehas to be ensured. This is\nequivalent to enforcing orthonormalizationon the canon-\nical occupied KS orbitals {ψl}[15], leading to the Euler\nequation\nδ\nEDFA−Ne/summationdisplay\nijǫij(/an}bracketle{tψi|ψj/an}bracketri}ht−δij)\n= 0.(5)\nBecause the DFA energy EDFA[n] remains invariant un-\nder the unitary transformations of the canonical occu-\npied KSorbitals {ψl}, anyorthonormalizedsetsofsingle-\nparticleorbitals {φi}, derivedthroughaunitarytransfor-\nmationfrom {ψl}, satisfiesEq.( 5)andyieldstheminimal\nDFA energy EDFA[n]. It is worth pointing out that while\nthe unitary transformation preserves the total density\nNe/summationdisplay\ni=1|φi(r)|2=Ne/summationdisplay\ni=1|ψi(r)|2=n(r),(6)\nit does not necessarily maintain the individual orbital\ndensities {ni(r)}\nni(r) =|φi(r)|2/ne}ationslash=|ψi(r)|2. (7)\nThe variational minimization of the SIC-DFA en-\nergyESIC-DFA(Eq. (3)) is more complicated because\nthe PZ-SIC contribution EPZ-SIC[{ni}]—the second part\nof Eq. (3)—is not invariant under unitary transforma-\ntions of the occupied KS orbitals, {ψl}. Consequently,\nthere exists an alternative orthonormalized set of single-\nelectron orbitals {φi}, which preserves the total density\n(Eq. (6)). This set can further minimize the PZ-SIC con-\ntribution and thus the overall SIC-DFA energy. These\norbitals, {φi}, referred to as SIC orbitals, are derived\nfrom unitary transformations of the KS orbitals {ψl},\nwhicharenot necessarilyidentical tothe KSorbitals[40].\nTherefore, the effort to minimize ESIC-DFA(Eq. (3)) ne-\ncessitates an additional orthonormalizationconstrainton\nthe occupied SIC orbitals {φi}[25, 26], leading to the\ngeneralized Euler equation\nδ\nESIC-DFA−Ne/summationdisplay\nijǫij(/an}bracketle{tψi|ψj/an}bracketri}ht−δij)\n−Ne/summationdisplay\nabλab(/an}bracketle{tφa|φb/an}bracketri}ht−δab)/bracketrightBigg\n= 0.(8)4\nA. PZ-SIC One-Electron Equations\nUnfortunately, the fact that two different sets of or-\nbitals, i.e., {φi}and{ψi}, enter Eq. ( 8) and that these\norbitals are inherently coupled complicates the solution\nof this equation. In practice, the problem is approached\nby solving two distinct problems: one to determine the\nKS orbitals {ψl}and one for the SIC orbitals {φi}. To\nensure self-consistency, these two sets of equations are\nsolved iteratively.\nThe PZ-SIC one-electron equations for the KS non-\ninteracting systems are\nˆhSIC-DFAψl=ǫlψl. (9)\nHere,ˆhSIC-DFAis the KS non-interactingHamiltonian for\nthe SIC-DFA\nˆhSIC-DFA=ˆts+ ˆvext+ ˆvH+ ˆvxc+ ˆvSIC,(10)\nwhich includes the kinetic energy operator ˆts, the exter-\nnal potential ˆ vext, the Hartree potential ˆ vH, the XC po-\ntential ˆvxc, and the SIC operator ˆ vSIC. Compared to the\nstandard KS Hamiltonian, Eq. ( 10) additionally features\nthe SIC operator\nˆvSICψl=δEPZ-SIC[{ni}]\nδn·ψl=Ne/summationdisplay\niˆv1eSIC\ni|φi/an}bracketri}ht/an}bracketle{tφi|ψl/an}bracketri}ht.\n(11)\nHere, the one-electron SIC potential {ˆv1eSIC\ni}in the SIC\noperator is the negative of summing the single-particle\nHartree potentials ˆ ves[ni] and the DFA-specific single-\nparticle XC potentials ˆ vxc[ni], evaluated for the SIC or-\nbital densities ni(r):\nˆv1eSIC\ni=−(ˆvH[ni]+ ˆvxc[ni]) (12)\nIt is important to note that the SIC operator ˆ vSICde-\npends on the SIC orbitals {φi}. Consequently, the com-\nputations of the SIC operator in this work occur within\nthe framework of the generalized Kohn-Sham (gKS) the-\nory [57–59] in analogy to the gKS derivation of the meta-\nGGASCANpotential[6]. (PleaserefertoRef.[14,15,24]\nfor more detailed derivations). Given a proper initial\nguess for {φi}, the KS orbitals {ψi}can be obtained by\nsolving the PZ-SIC equations (Eq. ( 9)). From this, the\nground-state density is derived as Eq. ( 6), which in turn\nyields the minimal SIC-DFA total energy (Eq. ( 3)) cor-\nresponding to the initial guess {φi}[26]. In practical\nscenarios, the inherent challenge of multiple solutions in\nthe standard PZ-SIC scheme manifests as a sensitivity\nof the self-consistent SIC orbitals to this initial guess.\nThis phenomenon is further exemplified by the cases of\nthe Carbon atom, Helium dimer, and Helium crystal in\nSec.IIBand Sec. IIC.B. PZ-SIC Constraints\nIn the pursuit of minimizing the SIC-DFA total energy\nusing the self-consistent PZ-SIC scheme, it is imperative\nto iteratively refine the SIC orbitals {φi}based on the\nKS orbitals {ψl}from the prior iteration (Eq. ( 9)). This\nupdating process for the SIC orbitals must adhere to\nspecific physical constraints.\nConstraint 1. Total density constraint (TDC):\nThe SIC orbitals {φi}are enforced to produce the same\ndensityn(r) as the canonical KS orbitals {ψl}:\nNe/summationdisplay\ni|φi(r)|2=n(r) =Ne/summationdisplay\nl|ψl(r)|2.(13)\nTechnically speaking, this implies that the occupied SIC\norbitals must be related to the occupied KS orbitals via\na unitary transformation [15]:\nφi=Ne/summationdisplay\nlTilψl,\nδml=Ne/summationdisplay\niT∗\nimTil.(14)\nUnder this constraint and under the assumption that\nthe KS orbitals {ψl}and the total density n(r) from\nthe preceding iteration remain unchanged, the general-\nized Euler equation (Eq. ( 8)) yields the one-electron SIC\npotential equations:\nˆv1eSIC\niφi=/summationdisplay\naλaiφa,(15)\nwhich lead to the Pederson constraint that follows in\nEq. (16) (Please refer to Ref. [24] for more detailed\nderivations).\nConstraint 2. Orbital potential constraint (OPC):\n/an}bracketle{tφm|ˆv1eSIC\nm|φn/an}bracketri}ht=/an}bracketle{tφm|ˆv1eSIC\nn|φn/an}bracketri}ht.(16)\nFor the KS orbitals {ψl}from the preceding iteration,\nthe minimal SIC-DFA total energy (Eq. ( 3)) is provided\nby the SIC orbitals that satisfies this orbital potential\nconstraint. The resultingSICorbitalsarefrequently(but\nnot always) localized [25].\nIn summary, to find the minimal SIC-DFA energy\nvia the variational minimization scheme, we must solve\nthe PZ-SIC one-electron equations self-consistently un-\nder Constraints 1 and 2 (TDC and OPC, Eqs. ( 13) and\n(16)). The well-documented challenge of multiple solu-\ntions arises from the fact that the ESIC-DFAfunctional\n(Eq. (3)) possesses many local minima. As a result, PZ-\nSIC outcomes are highly sensitive to the initial guess of\nSIC orbitals [26].5\nAs discussed in the introduction, the benefits of lo-\ncalization procedures have been established within the\nframework of PZ-SIC [15]. This is a logical outcome\nsince more localized densities generally result in stronger\nself-interaction, leading to a most significant reduction in\nthe total energy within the SIC-DFA scheme. Moreover,\nlocalization procedures, as exemplified by FLO, can re-\nduce the number of unitary transformations (Eq. ( 14)),\nthereby facilitating the self-consistent PZ-SIC conver-\ngence. Inspired by these localized methods, we pro-\npose the construction of SIC orbitals by self-consistently\nenforcing additional strong localization. Note that dif-\nferent procedures and approaches exist for this pur-\npose [23, 25, 26]. In this work, we choose the Edmiston-\nRuedenberg (E-R) restriction [28], which, by definition,\nmaximizes the integrals over the squared densities of the\nindividual SIC orbitals\nmax/integraldisplay\ndr/summationdisplay\nin2\ni(r)⇔min/integraldisplay\ndr/summationdisplay\ni/negationslash=jni(r)nj(r),(17)\nwhich is equivalent to minimizing the overlap of different\nSIC orbital densities [29], and thus maximizing the sum\nof the single-particle Hartree energy/summationtextNe\niEH[ni] [28].\nFor this reason, Perdew and Zunger suggested in the\noriginal PZ-SIC paper that E-R localized orbitals are an\nappropriatechoicefor the PZ-SICframeworkas alsocon-\nfirmed by Pederson and Lin [25].\nInordertoincorporatetheE-Rrestriction(Eq.( 17))as\na constraint into the self-consistent PZ-SIC framework,\nwe follow the spirit and the derivation of the OPC to\nobtain the self-consistently localized SIC Euler equation\nδ/bracketleftBigg/integraldisplay\nd3r/summationdisplay\nin2\ni(r)−η(δESIC-DFA)/bracketrightBigg\n= 0.(18)\nThis variationleadsto anew constraint, seethe following\nEq. (19).\nConstraint 3. Orbital density-potential constraint\n(ODPC):\n/an}bracketle{tφm|nmˆv1eSIC\nn|φn/an}bracketri}ht=/an}bracketle{tφm|ˆv1eSIC\nmnn|φn/an}bracketri}ht,(19)\nwhich canbe formulatedasa set oflocalizationequations\n(PleaseseeSupplementarySec. Iforadetailedderivation\nof these relationships). The herein-introduced constraint\nbased on the E-R restriction imposes an even stronger\nemphasis on localization than the orbital-potential con-\nstraint. As a consequence, it helps to steer the solution\nof the PZ-SIC equations towards the correct global min-\nimumESIC-DFA, especially when multiple solutions as-\nsociated with nonlocal minima are possible if only the\ntwo first constraints are enforced. First, we explain and\nillustrate this for two simple systems, He 2and C. Be-\nlow, we will then further substantiate this by extended\nbenchmarks for solids, see Sec. IIC.\nFIG. 1. Depictions of the KS orbital densities for the He 2\nmolecule in (a) and their corresponding SIC orbital densiti es\nin (b). For the C atom, the KS orbital densities are presented\nin (c), and the SIC densities in (d). The (SIC-)SCAN calcu-\nlations in FHI-aims were used to assess both sets of orbitals,\nKS and SIC, employing tight tier -1 NAO basis sets and de-\nfault output settings. The results were visualized using th e\nopen-source tool Jmol[60] with its predefined settings.\nAs a first example, we consider the closed-shell dihe-\nlium molecule He 2with a bond length of 1.1 ˚A. Here,\n1σgand 1σuare the fully occupied KS orbitals featur-\ning two electrons each. These occupied KS orbitals are\ndelocalized over the atoms, cf. Fig. 1(a). As derived in\nSupplementary Sec. II, there are two sets of orbitals that\nfulfill both TDC and OPC. These are the canonical KS\norbitals{1σg;u}and the more localized orbitals {φA;B},\nsee Fig.1(b), that can be obtained via the unitary trans-\nformation:\nφA;B=1√\n2[1σg±1σu]. (20)\nMorespecifically,Constraint1(TDC,Eq.( 13))isfulfilled6\nvia\nn(r) = 2|1σg(r)|2+2|1σu(r)|2\n= 2|φA(r)|2+2|φB(r)|2,(21)\nand Constraint 2 (OPC, Eq. ( 16)) via:\n/an}bracketle{t1σg|ˆv1eSIC\ng|1σu/an}bracketri}ht=/an}bracketle{t1σg|ˆv1eSIC\nu|1σu/an}bracketri}ht,\n/an}bracketle{tφA|ˆv1eSIC\nA|φB/an}bracketri}ht=/an}bracketle{tφA|ˆv1eSIC\nB|φB/an}bracketri}ht.(22)\nThis means that the SIC orbitals cannot be uniquely de-\ntermined via Constraints 1 and 2 (TDC and OPC), even\nin this relatively simple case, in which {φA;B}are the\nobviously better, more localized choice.\nThis hurdle is overcome by Constraint 3 (ODPC,\nEq. (19)), which is only fulfilled by the set of {φA;B},\nbut not by {1σg;u}:\n/an}bracketle{t1σg|ngˆv1eSIC\nu|1σu/an}bracketri}ht /ne}ationslash=/an}bracketle{t1σg|ˆv1eSIC\ngnu|1σu/an}bracketri}ht,\n/an}bracketle{tφA|nAˆv1eSIC\nB|φB/an}bracketri}ht=/an}bracketle{tφA|ˆv1eSIC\nAnB|φB/an}bracketri}ht.(23)\nAs expected, the {φA;B}orbitals then also yield a SIC-\nSCAN total energy that is about 5.29 eV lower than that\nof the{1σg;u}orbitals [61]. This is in line with the origi-\nnal intention of the constraint to enforce localization and\nshowcases how ODPC is a useful and often necessary\ncomplement to PZ-SIC theory.\nSecond, we showcase a similar but less trivial exam-\nple, the SIC-SCAN total energy calculations for the car-\nbon atom with the electronic ground state configuration\n1s2s(2p)2. The atomic orbitals {1s,2s,2px,2py}shown\nin Fig.1(c) satisfy both TDC and OPC [25] but not\nODPC. However, all three constraints can be simultane-\nously met by the following set of orbitals obtained from\na unitary transformation of the atomic orbitals:\nφ1= 1s\nφ20=1√\n32s−√\n2√\n32px\nφ21=1√\n32s+1√\n62px+1√\n22py\nφ22=1√\n32s+1√\n62px−1√\n22py.(24)\nAs shown in Fig. 1, these hybridized orbital densities\n(|φ20|2and|φ21|2) break the spherical harmonic sym-\nmetry and are thus more localized, i.e., they exhibit a\nsmaller overlap compared to the atomic orbital densities\n(|2s|2and|2px|2). Accordingly, these hybridized orbitals\ndefined in Eq. ( 24) result in a SIC-SCAN energy that\nis about 0.86 eV lower than that obtained using atomic\norbitals.\nC. Periodic Systems\nIn periodic boundary conditions (PBCs), the PZ-SIC\none-electron equations have to be solved for multiple k-\npoints in the first Brillouin zone\nˆhSIC-DFAψl,k=ǫl,kψl,k. (25)The KS orbitals {ψl,k(r)}are the generalized Bloch or-\nbitalslocated in the firstBrillouinzone. Due to the local-\nizednatureoftheSICorbitals,Heaton,Harrison,andLin\nproposedthatit ismoreconvenienttoexpresstheSICor-\nbitalsφi,I(r) in terms of Wannier orbitals {Ψl,L(r)}[22].\nAccordingly, the SIC orbital φi,I(r) centered in the Ith\nunit cell is expressed as:\nφi,I(r) =Ne/summationdisplay\nlNL/summationdisplay\nLTiI,lLΨl,L(r). (26)\nHere,Neis the number of electrons in a unit cell, and NL\nis the unit cell number (also the k-point number) in the\nsupercell. The Wannier orbitals {Ψl,L(r)}can be gen-\nerated via a Fourier transformation of the KS orbitals\n{ψl,k(r)}and thus span the same space as the KS or-\nbitals,\nΨl,L(r) = Ψl(r−RL)\n=1√NLNL/summationdisplay\nkexp[−ik·RL]ψl,k(r).(27)\nHere,{RL}are the lattice vectors, with the subscript\nL= [L1,L2,L3] being the index of a given lattice vector.\nIn such a Wannier representation, TDC reads\n1\nNLNeNL/summationdisplay\niI|φi,I(r)|2=n(r) =1\nNLNeNL/summationdisplay\nl,k|ψl,k(r)|2.(28)\nSimilarly, OPC and ODPC are expressed as\n/an}bracketle{tφm,M|ˆv1eSIC\nm,M|φn,N/an}bracketri}ht=/an}bracketle{tφm,M|ˆv1eSIC\nn,N|φn,N/an}bracketri}ht(29)\nand\n/an}bracketle{tφm,M|nm,Mˆv1eSIC\nn,N|φn,N/an}bracketri}ht=/an}bracketle{tφm,M|ˆv1eSIC\nm,Mnn,N|φn,N/an}bracketri}ht.\n(30)\nIn this case, ˆ v1eSIC\ni,I=δEPZ-SIC[{ni,I}]/δni,Iis the one-\nelectron SIC potential under PBCs, and ni,I=|φi,I|2\nis the associated SIC orbital density (Please refer to\nRef. [22, 44] and Supplementary Sec. I for more detailed\nderivations).\nAs mentioned in the introduction, the SIC multiple-\nsolution problem is known to be more serious for\nsolids [22, 41–43, 63]. To illustrate this and to showcase\nhow the introduction of ODPC helps to alleviate this is-\nsue, we here discuss crystal He, a prototypical case with\na multiple-solution problem in standard SIC approaches.\nAs there are two He atoms in each unit cell, see Fig. 2(a),\nthe occupied KS orbitals in the Wannier representation\ncan be marked as 1Σ g,Iand 1Σ u,I. Naturally, these or-\nbitals have the same shapes as the occupied molecular\norbitals in the closed-shell dihelium He 2molecule, see\nFig.1-(a). All orbitals {(Φ1,I,Φ2,I)}that satisfy TDC\ncanbe expressedviaunitary transformationsofthe Wan-\nnier KS orbitals (1Σ g,I,1Σu,I). In this particular simple\ncase, two distinct angles θI=1,2∈[0,π/2] can be used7\nFIG. 2. (a) Schematic representation of the hcp Helium cryst al’s unit cell on the left; while the depiction of two unit cel ls\nincorporating periodic boundary conditions on the right. G eometry was taken from Ref. [62]. (b) The energy profile of EPZ-SIC\n(Eq. (2)) with respect to the rotation angles ( θ1,θ2). (c) The absolute errors (AE) defined in Eq. (31) pertaining to OPC\n(Constraint 2) and (d) ODPC (Constraint 3), respectively. ( e) Combined AE of both OPC and ODPC for all possible orbitals\nin the 1×1×2 Helium unit cell. These orbitals are unitarily transforme d from the 1Σ g,I,1Σu,Iorbitals using rotation angles θ1\nandθ2. In the subfigures (c-e), the regions in white indicate the so lutions for OPC, ODPC, and their combinations, respectivel y.\nBoth SCAN and SIC-SCAN calculations in FHI-aims employed PBCs and tight tier -1 NAO basis sets.\nto characterize this unitary transformation, as derived in\ndetail in Supplementary Sec. III. Accordingly, the orig-\ninal Wannier KS orbitals (1Σ g,I,1Σu,I) are obtained by\nchoosing the angles to be ( θ1= 0,θ2=π/4).\nTo visualize the influence of the actual choice of\n{(Φ1,I,Φ2,I)}, we here show EPZ-SICas an energy surface\nas function of θI=1,2in Fig.2-(b). The global EPZ-SIC\nminimum of 0.32 eV is obtained for the SIC orbitals\nΦ1;2,Iwith (θ1=π/4,θ2=π/2). Respective SIC or-\nbitals at these angles give1√\n2[1Σg,I±1Σu,I], which, not\ntoo surprisingly, localize at He atoms and have similar\nshapes as in the case of the non-periodic, closed-shell\nHe2molecule, see Fig. 1-(b). At variance with Fig. 2-(b),\nwhich is produced using all three constraints, Fig. 2-(c)\nand Fig. 2-(d) show what happens if only TDC and OPC\nviz. TDC and ODPC are used, respectively. Addition-\nally, Fig. 2-(e) shows, once more, the results of using all\nthree constraints. To better visualize the differences in\nthese cases, we here locate the solutions of constraints in\nthe individual plots by using the absolute error (AE) ofOPC or ODPC:\nAE =\n\nmax{|/an}bracketle{tφm,M|ˆv1eSIC\nm,M−ˆv1eSIC\nn,N|φn,N/an}bracketri}ht|}\nmax{\n|/an}bracketle{tφm,M|nm,Mˆv1eSIC\nn,N−ˆv1eSIC\nm,Mnn,N|φn,N/an}bracketri}ht|}.\n(31)\nThis reveals that the standard SIC method, which only\naccounts for TDC and OPC, is multi-valued since the\nthree solutions at (0, π/4), (π/2,π/4), and (π/4,π/2)\nsatisfy TDC and OPC, as shown in Fig. 2-(c). Similarly,\nusingonly TDC andODPC alsoyields multi-valued solu-\ntions at (π/4, 0.0) and ( π/4,π/2), as shown in Fig. 2-(d).\nThe solution becomes uniquely single-valued only when\nall three constraintsare employed, as shown in Fig. 2-(e).\nIn this case, only one solution, i.e., ( π/4,π/2), fulfills all\nconstraints. This position also corresponds to the mini-\nmal SIC energy shown in Fig. 2-(a).\nAs a last example, we turn our attention to the\npolyethylene chain, (C 2H4)n. Its geometry is depicted\nin Fig.3-(a) and (b). In such a complex system, the8\nFIG. 3. (a) Primitive unit cell of the periodic polyethylene\nchain, (C 2H4)n. (b) Schematic view of the periodic polyethy-\nlene chain with two unit cells along the chain direction. Ge-\nometry was taken from Ref. [64]. (c) PZ-SIC energies based\non the SCAN functional (Eq. (2)). Different initial guesses o f\nSIC orbitals were generated with respect to the mixing facto r\nadefined in Eq. (32). All calculations were performed using\ntight tier -1 NAO basis sets in FHI-aims .\nSIC orbitalscannotbe exhaustivelyexploredusing alow-\ndimensional parametric model, unlike the simpler model\nsystemspreviouslydiscussed. Nonetheless, wehereinves-\ntigate the dependence of the SIC procedure by employ-\ning a set of initial guesses for the SIC orbitals, which are\ngrounded in physical reasoning. These initial guesses are\nformulated by mixing the atomic orbitals ϕicentered at\nthe atomic position Rac\niand the Wannier orbitals ΨSCAN\ni,L\nobtained from SCAN calculations without including SIC:\nφinit\ni,I=1\nAi[aϕi(r−Rac\ni−RI)\n+(1−a)ΨSCAN\ni(r−RI) ].(32)\nHere,Aidenotes a normalization factor, and ais a mix-\ning factor that ranges from 0 to 1. Accordingly, turning\nupafrom 0 to 1, the initial guess φinit\ni,Ican be varied\nfrom having the pure atomic-orbital character to being a\nWannier orbital, i.e., a KS molecular orbitals associated\nwith the unit cell Iin real space. This allows tuning the\nlocalization of the initial guess, since, as discussed in the\ncase of He 2, the atom orbitals that surround the atomic\ncenters can be expected to be more localized than the\nWannier orbitals [63]. As shown in Fig. 3-(c), different\nsolutions are found for different values of a, with energies\nvarying by about 0.3 eV at most, if only TDC and OPC\nare taken into account. This indicates that the problem\nhas multiple local minima. Let us emphasize that even\nwhen starting from fully localized orbitals a= 0, TDC\nand OPC are not converging to the lowest energy solu-\ntion. However, this doesn’t apply when ODPC is addi-\ntionally imposed because, in this case, all initial guessesconverge to the same lowest-energy solution.\nIII. RESULTS\nWe have implemented the aforementioned PZ-SIC\nmethod within the FHI-aims package [54]. Detailed as-\npects of the implementation can be found in Supplemen-\ntary Sec. IV. We here present benchmark results for a\nrangeoffinite and periodicsystems, including fundamen-\ntal atoms, molecules, and bulk solids. It is worth noting\nthat there are two distinct approaches for representing\nSIC orbitals in finite systems [65]. One can either use a\nreal-valuedunitarytransformationmatrix Toracomplex\none (Eq. ( 14)). While both realandcomplex transforma-\ntion matrices yield identical total densities, the resulting\nSIC energies can vary [66]. This section will present re-\nsultsforboth realandcomplex SIC orbitalswhen dealing\nwith finite systems. For periodic systems, where KS or-\nbitals are inherently complex, we will exclusivelyconsider\ncomplex SIC orbitals.\nA. Finite Systems\n1. Ionization Potential\nThe ionization potential (IP) measures the capability\nof an element to participate in chemical reactions that\nnecessitate ion formation or electron transfer. IP is typ-\nically calculated via the total energy difference between\nthe neutral molecule and the corresponding ion. Within\nthe framework of (g)KS approach, as applied in specific\nDFAs, the IP is theoretically equivalent to the negative\nof the energy of the highest occupied molecular orbital\n(HOMO), asdelineated in Ref. [67, 68]. However, this re-\nlationship often encounters deviations due to the errors,\nmost notably the many-electron SIE. While gKS outper-\nforms KS in orbital energy calculations [69], these errors\nstill lead to significant discrepancies in the IP and pre-\ndicted HOMO energies using (g)KS approaches [68]. In\nother words, this inconsistency affects the accuracy of IP\nestimationsderivedfrom HOMO energies. Consequently,\nthe precision of calculated IPs based on HOMO energies\nremains a crucial metric for evaluating the effectiveness\nof various electronic-structure theory methodologies, in-\ncluding SIC [56].\nThe HOMO energies for atoms ranging from H to Ar,\nas calculated by LDA, PBE, SCAN, and SIC-SCAN, are\npresented in Fig. 4, with experimental values [70] pro-\nvided for reference. The pronounced SIE in LDA and\nPBE [71] leads to a notable underestimation of the IPs.\nThe performance of SCAN offers a marginal improve-\nment. The mean absolute percentage errors (MAPE) are\n42.03% for LDA, 41.75% for PBE, and 38.5% for SCAN,\nsubstantiating that SCAN also suffers under SIEs. How-\never, with SIC-SCAN, there is a notable improvement in9\nthe predictions. As shown in Fig. 4-top, the MAPE is re-\nduced to 5.02% when using realSIC orbitals and further\ndrops to 3.95% for complex SIC orbitals. This is in line\nwith the findings in Ref. [65], where the use of complex\nSIC orbitals yielded superior results compared to their\nrealcounterparts.\nFIG. 4. Performance of various DFAs in describing the\nionization potentials for the atoms ranging from H to Ar.\nSIC-SCAN calculations using realandcomplex SIC orbitals\nare marked as SIC-SCAN(real) and SIC-SCAN(complex), re-\nspectively. (top) Relative errors of −ǫHOMOenergies to IPs,\nwhile the −ǫHOMOvalues are given in (bottom). All calcula-\ntions were performed using tight tier -1 NAO basis sets. The\nexperimental IPs were taken from the NIST database [70].\nFig.5showcases the performance of various DFAs in\npredicting negative HOMO energies ( −ǫHOMO) for a set\nof 18 molecules. As observed, LDA, PBE, and SCAN\ncalculations tend to underestimate the −ǫHOMOenergies\nsignificantly, whereas the SIC approach considerably im-\nproves the results. Specifically, all absolute percentage\nerrors of SIC-SCAN(complex) are under 23%, and its\nMAPE is 12 .8%, which is substantially lower than the\nSCAN error (33 .5%). We note, however, that the PZ-\nSIC method tends to slightly over-correct the SCAN cal-\nculations, resulting in the −ǫHOMOenergies consistently\nhigher than the experimental IPs. To contextualize these\nresults,G0W0@PBE, as reported in Ref. [56], offers IP\npredictions with a mere 3% MAPE.\n2. Energy Curve and Broken Symmetry\nAs the SIE tends to delocalize electrons, semi-local\nDFAs typically struggle to describe charge transfer pro-\ncesses. A typical example is that semi-local DFAs tend\nFIG. 5. (top) Performance of various DFAs in describing\nthe ionization potentials of 18 molecules. The mean absolut e\npercentage errors (MAPEs) of HOMO energies are given in\n(bottom). Calculations of LDA, PBE, SCAN, and SIC-SCAN\nmethods were carried out in FHI-aims usingtight tier -1 NAO\nbasis sets. G0W0@PBE results were taken from Ref. [56],\nwhich were also calculated in FHI-aims but utilized def2-\nQZVP basis sets. The experimental vertical ionization en-\nergies, taken from the NIST database [70], are reported for\ncomparison.\nto overly preserve the ground state symmetry of disso-\nciating neutral heterodimers by producing fractionally\ncharged fragments. For instance, SCAN incorrectly pre-\ndicts a positive fractional charge on the H atom in the\ndissociation of the molecule H-F (Fig. 6), hence giv-\ning too high energies in the dissociation limit. While\nKimet al.[72] highlighted that a meticulous selection\nof orbital occupations can address this issue, the hybrid\nfunctional PBE0 [73] outperforms both PBE and SCAN\nwith self-consistent calculations. This superiority is at-\ntributed to incorporating a fraction (0.25) of Fock ex-\nchange in PBE0, which reduces the SIE in PBE. Never-\ntheless, PBE0 cannot correctly predict the change distri-\nbution at distances between 1.5-2.5 ˚A, as illustrated in\nFig.6-bottom. In contrast, SIC reproduces comparable\ncharge analyses of references and successfully maintains\nelectron-neutral atoms in the dissociation limit. Here,\naccurate results from coupled cluster theory with single,\ndouble, and perturbative triple excitations (CCSD(T))\nare used as references. As shown, the energy curve\nof SIC-SCAN(complex) matches the CCSD(T) curve in\nboth equilibrium (around r≈0.8˚A) and stretched-bond\n(r >0.8˚A) regions. For comparison, the energy curve10\nof SIC-SCAN(real) deviates from the one of CCSD(T) in\nthe equilibrium region.\nFIG. 6. (top) HF dissociation energy curves of various\nmethods with the zero-energy level set to the total energy\nof isolated atoms/ions. (bottom) Mulliken charge analyses\non the H atom along the dissociation. All calculations were\nperformed in FHI-aims .Tight tier -1 NAO basis sets were\nemployed for PBE, PBE0, SCAN and SIC-SCAN incorporat-\ningrealandcomplex SIC orbitals, while the CCSD(T) results\nutilize cc-pVTZ Gaussian-type basis sets.\nFig.7illustrates how the SIE in SCAN calculations\naffects open-shell configurations, e.g., H+\n2, He+\n2and Ne+\n2.\nFor H+\n2, which contains only one electron, SIC exactly\noffsets the SIE in the SCAN calculation. However, for\nthe He+\n2and Ne+\n2dissociations with more electrons, this\nexactcancellationdoesnotoccur. Asthe dissociationbe-\nhaviorsof He+\n2and Ne+\n2aresimilar, we take the Ne+\n2sys-\ntem with more electrons as an example to analyze in the\nfollowing. The conclusions drawn for Ne+\n2are applicable\nto He+\n2as well. Nevertheless, SIC, whether using realor\ncomplex SIC orbitals, significantly improves the SCAN\ncurve. The resulting SIC-SCAN curves align closely\nwith CCSD(T) calculations, indicating that the PZ-SIC\nscheme successfully rectifies the majority of the many-\nelectron SIEs in the SCAN calculations of the Ne+\n2disso-\nciation. As shown in Fig. 7, both SCAN and SIC-SCAN,\nwithout symmetry breaking, predict the same chargedis-\ntribution, with each Ne atom carryinga fractionalcharge\nof 0.5 (Ne0.5). It is also worth noting the combined total\nenergy of Ne and Ne+1, which is taken as the reference\nforthe Ne+\n2dissociationcurvein Fig. 7. Regardlessofthe\nchosen SIC orbitals (either realorcomplex), SIC-SCAN\nclosely reproducesthe degeneracybetween the configura-\ntions of Ne+···Ne and Ne0.5···Ne0.5, with a deviation\nof less than 0.3 eV. This observation suggests that SIC-\nFIG. 7. Dissociation curves of H+\n2(a), He+\n2(b), and Ne+\n2\n(c), with the zero-energy reference set to the total energy o f\nisolated atoms/ions. The mirror symmetry ensures no charge\ntransfer in these two dissociations, as demonstrated by the\nMulliken charge analyses. All calculations were performed\ninFHI-aims .Tight tier -1 NAO basis sets were employed for\nSCANandSIC-SCANincorporating realandcomplex SICor-\nbitals, while the CCSD(T) results utilize cc-pVTZ Gaussian -\ntype basis sets. The Hartree-Fock (HF) method, being exact\nfor one-electron systems, provided the H+\n2dissociation refer-\nence curve.11\nSCAN can provide an accurate description of fractional\ncharged systems, which has been widely used to discuss\nthe SIE in the conventional DFAs [74].\nB. Periodic Systems\n1. Performance for Crystal Properties\nGiven that our SIC-SCAN implementation effectively\naddresses the multiple solution issue in both finite and\nperiod systems, we have extended our attention to the\ncrystal properties, including cohesive energy, bulk mod-\nulus, andlatticeconstant. Weevaluatethecohesiveener-\nFIG. 8. (a) Lattice constants at equilibrium ( ˚A), bulk mod-\nulus (GPa) and cohesive energies (eV) of the diamond Si, dia-\nmond C, rocksalt LiH, fcc Ne, and zincblende AlP, calculated\nusing SCAN and SIC-SCAN calculations. (b) Absolute errors\nin the calculated lattice constants ( ˚A), bulk modulus (GPa),\nand cohesive energies (eV) for SCAN and SIC-SCAN. All cal-\nculations were carried outin FHI-aims usingtight tier -1 NAO\nbasis sets and the k-grid setting of 4 ×4×4. The experimental\nreference values were collected from the main-group test se t\n[75, 76].\ngies of a fundamental set of materials, including diamond\nSi, diamond C, rocksalt LiH, fcc Ne, and zincblende AlP,\nas presented in Fig. 8-(a). Additionally, we computed\nthe lattice constant and bulk modulus for this test set,\nwith results also displayed in Fig. 8-(a). The relevant\nabsolute errors (AEs) are plotted in Fig. 8-(b) for com-\nparative analysis.Consistent with findings in previous studies [52, 77],\nour calculations suggest that the SIC contribution tends\ntodeterioratethe accuracyofSCANpredictionsforcohe-\nsive energies and bulk modulus. However, the impact of\nSIC on SCAN’s lattice constant predictions is relatively\nminor. AsatypicalsystemofvanderWaals(vdW) inter-\naction, the fcc Ne poses a challenge for standard DFAs.\nOurresultssuggestthat theSCANmethodtendstoover-\nestimate the vdW interaction, with its calculated cohe-\nsive energy being double the experimental value and its\nlattice constant being 0.37 ˚A shorter. Given that the\nPZ-SIC scheme is irrelevant to addressing the SCAN er-\nror for vdW interactions, SIC-SCAN does not improve\nthe terrible performance of SCAN in predicting the lat-\ntice constant of fcc Ne. The observed improvements in\nthecohesiveenergyand bulkmodulusofthefcc Nemight\nbe due to a fortuitous offset between the inappropriate\nSIC contribution and SCAN’s overestimation of vdW in-\nteractions.\n2. Band Gaps\nThe fundamental band gap is a ground-state property,\nwhich is defined as the difference between the ioniza-\ntion energy and the electron affinity [67]. For periodic\nsolids,ithasbeendemonstratedthat, foranyDFAwithin\nthe KS or gKS framework, the fundamental band gap is\nequivalenttotheenergydifferencebetweenthelowestun-\noccupiedorbital(LUMO)andHOMO,providingthatthe\ngiven DFA potential is continuous upon introducing an\nelectron or hole [67, 69]. This finding is consistent across\nboth the KS and gKS methods. The well-documented\nunderestimation of the band gap in semi-local DFAs, in-\ncluding LDA, PBE, and SCAN, is largely attributed to\nsignificant SIEs present in these approximations [2, 67].\nIn this work, we evaluate the performance of various\nmethods in predicting band gaps for a selection of 10\nfundamental solids. This includes covalent crystals (such\nas Si, C), ionic crystals (such as NaCl, LiF), and Mott\ninsulators composed of transition-metal oxides (MnO).\nThe experimental band gaps are taken as the reference.\nIt’sworthemphasizingthat althoughelectron-phononin-\nteractions can influence experimental band gaps [78–80],\nthe remarkable agreement between the band gaps deter-\nmined by the G0W0method and experimental data sug-\ngests that, at least for the solids investigated here, the\nimpactofthisinteractionisprobablylesssignificantcom-\npared to the SIE in DFAs. As illustrated in Fig. 9, the\nmean absolute percentage errors (MAPEs) of LDA and\nPBE are 45.4 % and 43.2 %, respectively, substantially\nhigher than the G0W0value (4.7 %). SCAN outperforms\nLDA and PBE with a lower MAPE (31.5 %), as the step\noutside KS to gKS is to enhance orbital energies. Unfor-\ntunately, However, Figure 9reveals that SCAN still has\npoor accuracy in predicting band gaps, consistent with\nprevious findings [17]. Our results suggest that applying\nSICtoSCANcanreducethe bandgaperror,loweringthe12\nMAPE to 18.5 %. We acknowledge that SIE may not be\nthe sole source of error for SCAN, but it significantly\ncontributes to inaccuracies in band gap calculations.\nMnO is classified as a Mott insulator characterized by\nstrong electron interactions [81, 82]. Accurately predict-\ning the band gap of MnO is a challenge for conventional\nsemi-local DFAs [83]. Our results confirm that LDA,\nPBE,andSCANmethodspredictthebandgapsforMnO\naround 0.60 eV, a significant underestimation compared\nto the experimental value of 3.70 eV. A common remedy\nis the DFT+ Umethod, which introduces an empirical\nelectron repulsion “ U” [84, 85]. For instance, GGA+ U\ngives a band gap of approximately 4 eV when apply-\ning a “U” of 5.25 eV via the linear-response approach\n[86]. Moreover, the self-consistent GW0approach, based\non the LDA+ Uwave functions, produces a band gap of\n3.32 eV [87]. In contrast, our research indicates that the\nPZ-SIC scheme can also effectively improve the SCAN\ndescriptionforMnO’s bandgap. The SIC-SCANmethod\ncalculates a band gap of 3.73 eV, which is impressively\nclose to the experimental value, deviating by a mere 0.03\neV. This result aligns with previous studies, suggesting\nthat PZ-SIC captures the strong electron interactions of-\nten overlooked in standard semi-local DFAs [15, 83].\nTofurtherimprovethecalculatedbandgapsin theKS-\nDFT framework, a scaled-down PZ-SIC (sd-SIC) variant\nhas been proposed [44, 90, 91]. Compared to the stan-\ndard PZ-SIC scheme (Eq. 2), the pivotal adjustment in\nsd-SIC is to introduce a set of orbital-density specific\nscaling factors, denoted as {Xi}, into the self-interaction\ncorrection\nEsd-SIC-DFA=EDFA+Esd-SIC\nEsd-SIC[{ni}] =−Ne/summationdisplay\niXi/parenleftbig\nEDFA\nxc[ni]+EH[ni]/parenrightbig\n.(33)\nIn addition to determining the scaling factor for each or-\nbital, adopting a uniform scaling factor across all orbital\ndensities (Xi=X) presents a straightforward and effec-\ntive strategy, albeit it takes on empirical impacts. For\ninstance, it has been observed that a scaling factor of\n0.4 enhances thermochemistry [90], while a factor of 0 .2\nyields desirable band gaps in both silicon and diamond\n[92]. Furthermore, the coefficient X= 2/3, as proposed\nby Shinde et al.[44] in 2020, to enhanced the SIC-PBE\nresults for fundamental solids. Building upon this, our\nstudy investigates the impact of this coefficient on the\nSIC-SCAN band gaps. We discovered that a scaling fac-\ntor of 0.60 yields optimal results for SIC-SCAN. These\nfindings are comprehensively discussed in Supplemen-\ntary Section V. The outcomes are illustrated in Fig. 9,\nwhere they are denoted as scaled-down SIC-SCAN (sd-\nSIC-SCAN). In agreement with the previous research by\nShindeet al.[44], we find that the scaled-down algorithm\nwith a constant scaling factor enhances the accuracy of\nthe calculated band gaps. The sd-SIC-SCAN approach\noutperforms the SIC-SCAN method, bringing down the\nMAPE to 12.9%. However, it is crucial to highlight that\nFIG. 9. Calculated band gaps using LDA, PBE, SCAN, (sd-\n)SIC-SCAN, and GWagainst experimental counterparters.\nThe graph features a black diagonal line representing the pe r-\nfect agreement to the experiment. Mean absolute percentage\nerrors (MAPEs) are collected below the graph. Calculations\nof LDA, PBE, SCAN, and (sd-)SIC-SCAN were performed in\nFHI-aims . Ak-grid setting of 5 ×5×5 was used together with\ntight tier -1 NAO basis sets. Geometries were taken from the\nexperiments. Results for G0W0@PBE and experimental data\nwere collected from Refs. [88, 89].\nthe standard PZ-SIC method is exact for one-electron\nsystems, which, unfortunately, is not the case in the sd-\nSIC method.\n3. Crystalline trans-Polyacetylene\nWe then turn to a challenging problem for periodic\nsystems, the broken symmetry in crystalline polyacety-\nlene (CH) x. The polyacetylene chain has two configu-\nrations, named cisandtrans. In 1957, Ooshika found\nthat the transconfiguration is a stable configuration for\nthe long-chain polyacetylene (TPA) [95–97], where a se-\nquence of alternating single and double carbon bonds is\nformed along the chain, as illustrated in Fig. 10-(a-d).\nThe dimerization parameter ∆ zdescribes the length dif-\nference of these bonds along the chain’s direction ( zsand\nzd):\n∆z=zs−zd. (34)13\nFIG. 10. (a) Experimental unit cell of the TPA structure [93] . Carbon atoms are depicted in gray, while hydrogen atoms are\nin white. (b) and (c) are the right and bottom side views, resp ectively. (d) The schematic view of a single (CH) xchain with\nthe dimerization parameter ∆ z, representing the difference between the distance of single -bond and double-bond carbon atoms\nalong the chain direction zsandzd. (e) Total energy curves as a function of the dimerization pa rameter ∆ z, computed using\nthe SCAN, (sd-)SIC-SCAN, PBE0, and RPA@PBE methods. All cal culations were executed in FHI-aims . Vertical red dashed\nlines in the subfigure (e) mark the experimental dimerizatio n parameter (∆ z≈0.047˚A) [94]. The calculated total energies for\n∆z= 0 were set as the zero-energy reference. Markers pinpointe d the optimized dimerization for each method. A dense k-grid\nsetting of 14 ×12×16 was used together with tight tier -1 NAO basis sets for these calculations. As the dimerizatio n factor\nvaries, the combined length of zs+zdremained equal to the lattice vector length |c|. A vertical height hof 0.70˚A–from the\nmiddle carbon atom to the connection of the other two carbon a toms–was maintained.\nHere,zsandzdare vertical distances from the central\ncarbon atom on both sides. X-ray studies on crystalline\nTPA identified a dimerization of ∆ z≈0.047˚A [94]. The\nunderlying mechanism of symmetry break is yet to be\nclarified. On the one hand, the formation of alternating\nbonds and charge transfer between carbons has been ex-\nplained by using the celebrated 1D Su-Schrieffer-Heeger\nmodel, emphasizing the phonon/electron-phonon inter-\naction [98]. On the other hand, the electron-electron in-\nteraction could be another possible driving force for the\ninduced dimerization [73, 99, 100].\nFig.10-(e) displays the TPA energy curves as a func-\ntion of the dimerization factor ∆ z, which are calculated\nusing the SCAN, SIC-SCAN, PBE0, and RPA@PBE\nmethods. In this context, “RPA@PBE” denotes the ran-\ndom phase approximation based on PBE density and\norbitals[101]. The standard hybrid functional PBE0 [73],\nwhich incorporates a portion of the Hartree-Fock ex-\nchange with α= 0.25, predicts a dimerization of 0.044 ˚A.\nThis aligns closely with the experimental findings (0.047\n˚A) and the RPA@PBE calculation (0.045 ˚A) [101]. Our\nanalysis indicates that the calculated dimerization is sen-\nsitive to the αvalue. When αis increased to 0.50, the\nresulting dimerization parameter rises to 0.059 ˚A, over-\nshooting the experimental value by 0.012 ˚A.\nWhile all the methods investigated here predict a\nsymmetry-broken transconfiguration with ∆ z >0, thedimerization factor ∆ zdetermined by the SCAN method\nis 0.026 ˚A, significantly shorter than the experimental\nmeasurement of 0.047 ˚A. In agreement with a previous\nstudy [102], our results confirm that the PZ-SIC scheme\nenhances the accuracy of conventional semi-local DFAs\nfor this system. SIC-SCAN predicts a more extended\ndimerizationparameterofaround0.036 ˚A. While the cal-\nculateddimerizationparameterremainsunchangedwhen\napplyingauniformscalingfactor X= 0.60viathescaled-\ndown SIC formula (Eq. ( 33)), the sd-SIC-SCAN method\nconsiderably diminishes the energy well compared to the\nSIC-SCAN method.\nFinally, we turn our attention to the band gap of crys-\ntalline TPA, using its experimental geometry from ref-\nerence [94]. Direct and indirect adsorption gaps of TPA\nhave been identified by Fincher et al., which are 1.4 eV\nand 1.1 eV, respectively [94]. Fig. 11presents the direct\nand indirect band gaps as calculated by various methods.\nThe standard hybrid PBE0 with α= 0.25 slightly under-\nestimates the indirect band gap at 0.95 eV. In contrast,\nusingα= 0.50 with PBE0 leads to a notable overesti-\nmation. As expected, due to the serious SIE, SCAN sig-\nnificantly underestimates the band gap, erroneously pre-\ndicting identical values of 0.12 eV for both direct and in-\ndirect gaps. The SIC-SCAN method, on the other hand,\nrectifies the difference between direct and indirect gaps\nbut greatly overestimates the indirect band gap to 4.5014\nFIG. 11. Calculated band gaps of crystalline trans-\npolyacetylene using the SCAN, PBE0 and SIC-SCAN meth-\nods. All calculations were performed in FHI-aims . The ex-\nperimental geometry was taken from Ref. [94]. A dense k-grid\nsetting of 14 ×12×16 was used together with the tight tier -1\nNAO basis sets. The blue dash line and orange line presented\nthe direct and indirect experimental band gaps, respective ly,\nwhich were taken from Ref. [94].\neV. Introducing a uniform scaling factor, X, alleviates\nthe overcorrection of PZ-SIC, but the calculated indirect\nband gap of sd-SIC-SCAN remains considerably larger\nthan the experimental findings.\nCrystalline TPA represents a challenging periodic sys-\ntem to study. Numerous factors, including lattice con-\nstants [103], can influence its calculated properties. As\nillustrated in Fig. 10and Fig. 11, significant discrepan-\ncies are evident in energy and band gap values across\nvarious computational methods. The widely used hy-\nbrid functional PBE0, with a default alpha of 0 .25, pre-\ndicts dimerization that closely aligns with the results of\nRPA@PBE. However, increasing the alpha value to 0.5\nadversely affects PBE0’s performance. It’s worth not-\ning that PBE0 outperforms the (SIC-)SCAN method in\npredicting these TPA properties, highlighting that while\nmitigating the SIE is crucial, it is not the only factor to\nconsider when analyzing crystalline TPA using density\nfunctional theory.\nIV. CONCLUSION\nThe state-of-the-art, non-empirical meta-GGA SCAN\nfunctional provides a significant and consistent improve-\nment over the non-empirical GGA PBE across a range of\nchemical interactions in both molecules and solids. How-\never, it is not immune to the pervasive self-interaction\nerror. In this study, we implement and generalize the\nself-interaction correction algorithm proposed by Perdew\nand Zunger (PZ-SIC), to alleviate this error within theSCAN functional. Specifically, we introduce the orbital\ndensity-potential constraint, which, in conjunction with\nthe total density constraint and the orbital potential con-\nstraint, facilitates the self-consistent localization of the\nSIC orbitals, leading to an accurate PZ-SIC energy. Our\nfindings indicate that the orbital density-potential con-\nstraint can narrow down multiple potential SIC orbitals\nto a singular optimal choice in systems like C, He 2, hcp-\nHelium, and the CH-chain.\nOur PZ-SIC approach markedly enhances the per-\nformance of SCAN, as evidenced by the highest occu-\npied molecular orbital (HOMO) energies ǫHOMOof H-Ar\natoms and a set of 18 molecules. Built upon the accu-\nrate description of orbital energies, our PZ-SIC approach\nyields commendable energy curves and corrected charge\ntransfers in the dissociations of H-F and Ne-Ne+. When\ntested on 10 foundational solids, SIC-SCAN consistently\noutperforms SCAN in band gap predictions. However,\nfor other crystalline properties, such as cohesive energy,\nbulk modulus, and lattice constants, SIC-SCAN does not\nfare as well as SCAN for our test set, which includes dia-\nmondSi, diamondC,rocksaltLiH,fccNe,andzincblende\nAlP. We then delve into the ground state of crystalline\npolyacetylene, a renowned challenge for periodic systems\nthat remains elusive for semi-local DFT calculations like\nthe SCAN functional. Our findings indicate that the\nPZ-SIC methodology adeptly identifies the symmetry-\nbreaking in crystalline polyacetylene, emphasizing the\nrole of electron-electron interactions.\nSUPPLEMENTARY INFORMATION\nThe following files are available free of charge:\n•supplementary.pdf, containing additional informa-\ntion needed for detailed discussions.\n•data.zip, containing geometries and electronic-\nstructure energy results produced in this project\nand also available on the NOMAD repository with\nthe DOI: 10.17172/NOMAD/2023.11.28-1 .\nACKNOWLEDGMENTS\nSheng thanks Zhenkun Yuan, Hagen-Henrik Kowal-\nski and Florian Knoop for inspiring discussions. This\nwork received funding from the European Union’s Hori-\nzon 2020 Research and Innovation Programme (grant\nagreement No. 951786, the NOMAD CoE), and the\nERC Advanced Grant TEC1P (No. 740233). 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Guo2,4\n1Jiangxi Normal University, Nanchang 330022, China\ne-mail: frlin@jxnu.edu.cn\n2Sino-French Joint Laboratory for Astrometry, Dynamics and Space Science, Jinan University, Guangzhou 510632, China\ne-mail: tpengqy@jnu.edu.cn\n3Guangdong Ocean University, Zhanjiang 524000, China\n4Department of Computer Science, Jinan University, Guangzhou 510632, China\nFebruary 14, 2024\nABSTRACT\nContext. Geometric distortion (GD) critically constrains the precision of astrometry. Its correction required GD calibration observa-\ntions, which can only be obtained using a special dithering strategy during the observation period. Some telescopes lack GD calibration\nobservations, making it impossible to accurately determine the GD e ffect using well-established methods. This limits the value of the\ntelescope observations in certain astrometric scenarios, such as using historical observations of moving targets in the solar system to\nimprove their orbit.\nAims. We investigated a method for handling GD that does not rely on the calibration observations. With this advantage, it can be\nused to solve the GD models of telescopes which were intractable in the past. Consequently, astrometric results of the historical\nobservations obtained from these telescopes can be significantly improved.\nMethods. We used the weighted average of the plate constants to derive the GD model. The method was implemented in Python and\nreleased on GitHub. It was then applied to solve GD in the observations taken with the 1-m and 2.4-m telescopes at Yunnan Obser-\nvatory. The resulting GD models were compared with those obtained using well-established methods to demonstrate the accuracy.\nFurthermore, the method was applied in the reduction of observations for two targets, the moon of Jupiter (Himalia) and the binary\nGSC2038-0293, to show its e ffectiveness.\nResults. The GD of the 60-cm telescope at Yunnan Observatory without calibration observations was accurately solved by our method.\nAfter GD correction, the astrometric results for both targets show improvements. Notably, the mean residual between observed and\ncomputed position ( O−C) for the binary GSC2038-0293 decreased from 36 mas to 5 mas.\nKey words. astrometry – methods: data analysis – techniques: image processing\n1. Introduction\nBoth space and ground-based observations in the field of op-\ntical astrometry are inevitably a ffected by geometric distortion\n(GD). Various factors contribute to GD, such as flaws in the\nshape of optical components, installation deviations, irregulari-\nties in filters, and manufacturing imperfections in CCD detectors\n(Bernard et al. 2018).\nIn the majority of scenarios, GD serves as the main e ffect\nlimiting high-precision astrometry (Peng et al. 2017; Wang et al.\n2017; McKay & Kozhurina-Platais 2018; Wang et al. 2019;\nCasetti-Dinescu et al. 2021). For example, observations captured\nby the UVIS channel of the Wide Field Camera 3 on the Hubble\nSpace Telescope (HST) would be a ffected by GD, causing shifts\nof up to 35 pixels (McKay & Kozhurina-Platais 2018); For the\nobservations taken with the 2.4-m telescope at Yunnan Observa-\ntory, the astrometric precision nearly improved by a factor of 2\nafter GD correction (Peng et al. 2017).\nBesides its e ffect on astrometric precision, GD correction is\nalso necessary for some applications to improve computational\nspeed or meet fundamental requirements. For instance, in real-\ntime applications that involve handling substantial data and com-\nputational loads for detecting and tracking near-Earth objects,\nZhai et al. (2018) employed GD correction when converting the\npixel coordinate to the equatorial coordinate, so that this processwas accelerated while achieving high-precision results (Zheng\net al. 2021).\nAdopting high-order plate constants for reduction can miti-\ngate the e ffect of GD. However, it requires su fficient reference\nstars exist in the field of view (FOV), which is not met in spe-\ncific high-precision applications. For some ground-based obser-\nvations of moving targets, instrument limitations and other fac-\ntors may result in a lack of observed reference stars available\nfrom the observations. Consequently, the GD solution derived\nfrom calibration observations is essential for obtaining high-\nprecision positions of these moving targets. In addition, in HST\nextremely deep field observations where the AB magnitude of\nstars can reach up to 30 mag (Illingworth et al. 2013), there\nis no star catalogue that provides accurate positions for refer-\nence stars. The infeasibility of applying the high-order plate con-\nstants, as a result, makes GD correction necessary.\nThe most notable research on addressing GD is the study\nconducted by Anderson & King (2003), which focuses on the\nGD e ffect of the Wide Field Planetary Camera 2 (WFPC2) in-\nstalled on the HST. The GD e ffect at the edges of the WFPC2\nFOV reached up to 5 pixels, severely degrading the astromet-\nric precision of the HST. For this reason, Anderson & King\n(2003) developed a self-calibration technique using observations\nacquired at various orientations. After GD correction, the aver-\nage positional residuals decreased to 0.02 pixels and 0.01 pixels\nArticle number, page 1 of 8arXiv:2401.11703v2  [astro-ph.IM]  13 Feb 2024manuscript no. new_GDC_arXiv\nfor the planetary camera and the wide-field camera, respectively.\nFrom then on, the HST started to fully demonstrate its astromet-\nric potential. This self-calibration technique, which does not rely\non a star catalogue, has been applied to solve GD for multiple\ncameras on the HST (Bellini & Bedin 2009) as well as several\nground-based telescopes (Anderson et al. 2006; Bellini & Bedin\n2010), achieving high-precision astrometry.\nAn alternative approach for solving GD is based on the\nknown reference star positions that are not a ffected by GD. These\npositions can be calculated by an external astrometric reference\nsystem, such as a high-precision astrometric catalogue, HST ob-\nservation positions with GD correction (Service et al. 2016), or\neven star positions obtained from non-optical wavelength bands\n(Reid & Menten 2007). By comparing the positional deviations\nbetween observed and reference positions, the influence of GD\non astrometry can be determined. This method requires fewer\nobservations for GD solution than the self-calibration technique,\nbut the accuracy may be a ffected by the errors present in the ex-\nternal reference system (Bernard et al. 2018). Peng et al. (2012)\nproposed a novel method for mitigating the influence of cata-\nlogue errors on GD solution. This method involves algebraically\nmanipulating the measurement errors of identical stars across\ndifferent frames, thereby reducing the accuracy requirements of\nthe external reference catalogue. Furthermore, it has been im-\nproved by Wang et al. (2019) to handle the GD in observations\ncaptured by the 2.3-m Bok telescope at Kitt Peak (Peng et al.\n2023).\nThe above methods can e ffectively solve the GD model and\nachieve positional measurements with precision up to the 0.01\npixel level. Nonetheless, these methods necessitate well-planned\nobservational strategies, obtaining optimally dithered and over-\nlapping frames (Anderson & King 2003; Peng et al. 2012), to\noffset the e ffects of GD or catalogue errors (Zheng et al. 2021).\nThese GD calibration observations require additional telescope\ntime. It is sometimes di fficult to meet due to observation con-\nditions. More importantly, some telescopes may not have had\nsuch observation plans for a significant period of time. Due to\nthe dynamic changes in GD, a prolonged absence of GD calibra-\ntion observations can substantially degrade the accuracy of the\nastrometry.\nTo address this issue, a method is proposed to derive an an-\nalytical GD model without the requirement of GD calibration\nobservations. We demonstrate the performance of this method\nby comparing it with the well-established GD correction method\n(Peng et al. 2012; Wang et al. 2019). Additionally, we perform\nthe reduction of Himalia (J6) observations captured by the 2.4-m\ntelescope at Yunnan Observatory and the 60-cm telescope obser-\nvations for a low-mass close binary system GSC02038-00293\n(Dal et al. 2012). These observations are initially reduced using\nplate constants without GD correction, but the results are un-\nsatisfactory. Like many telescopes primarily employed for pho-\ntometric purposes, the 60-cm telescope at Yunnan Observatory\nhas never performed any GD calibration observations. However,\na novel research focus of the binary GSC02038-00293 highlights\nthe importance of astrometry. Since our method is e ffective and\neasy to implement, it is especially applicable at this time. Ap-\nplying our method to correct GD significantly improves the as-\ntrometry of the targets, leading to more reliable results than any\nother reduction methods. The results indicate that our method is\nparticularly useful for reducing historical observations that lack\nof GD calibration observations. The advantages of our method\nmake it valuable in studying the long-term changes in the posi-tions of the targets, as well as their formation and evolution. A\nPython implementation of this method is available on GitHub1.\nThis paper is structured as follows: Section 2 provides de-\ntailed information about the observations and the corresponding\ninstruments used to capture them, while Section 3 describes the\nGD correction method based on the Gaia DR3 (Gaia Collabora-\ntion et al. 2023). Section 4 presents the performance comparison\nbetween our method and the well-established method, as well as\nthe advantages of the new method in reducing the observations\nof the targets J6 and GSC02038-00293. Finally, Section 5 con-\ncludes the paper with some closing remarks.\n2. Observations\nObservations obtained from multiple telescopes are used in\nthis paper. These telescopes include the 60-cm telescope (Zang\net al. 2022), the 1-m telescope (IAU code 286, longitude–\nE102◦47′18′′, latitude–N25◦1′46′′, and height–2000m above sea\nlevel) at Yunnan Observatory, and the 2.4-m telescope (IAU code\nO44, longitude–E100◦1′51′′, latitude–N26◦42′32′′, and height–\n3193m above sea level) at Yunnan Observatory (YNO 60-cm,\nYNO 1-m, and YNO 2.4-m). More instrumental details of the\nreflectors and CCD detectors are listed in Table 1. The patterns\nand magnitudes of GD experienced by these instruments are dif-\nferent.\nObservation sets 1 and 2 were captured using the dithering\nstrategy, which takes multiple dithered exposures of the same\nsky field with di fferent o ffsets (Peng et al. 2012). They are used\nto demonstrate that the proposed method in this paper achieves\nthe same accuracy as other well-established GD correction meth-\nods. Observation sets 3 and 4 are significantly a ffected by higher-\norder GD, but only a dozen or so bright stars can be used to solve\nthe plate constants, thus the GD solution is very important for\nhigh-precision astrometry of the targets. The detailed informa-\ntion of these observations is provided in Table 2. In this paper,\nbright stars refer to the stars with su fficient signal-to-noise ra-\ntio, whose astrometric precision is not dominated by centering\nerrors.\n3. Methods\nThe method investigated in this paper derives an analytical GD\nmodel, which is characterized by a high-order polynomial, using\nthe distortionless positions of stars provided by the Gaia cata-\nlogue. The analytical GD model can e ffectively describe the GD\neffect in ground-based observations, because the majority of GD\ncomponents can be characterized by polynomials. The remain-\ning components, which are typically described using a lookup\ntable (Wang et al. 2019), generally only account for a minor por-\ntion of GD. This is confirmed in the subsequent section through\nexperimentation.\nThis method extracts the GD e ffect present in each frame of\nobservations, and then derives the GD model based on the GD\neffect extracted from these multiple frames. The implementation\ndetails of this method are as follows. A two-dimensional Gaus-\nsian fitting is used to determine the pixel positions of the ob-\nserved stars. These observed stars are then cross-matched with\nthe stars given in the Gaia catalogue (Gaia Collaboration et al.\n2023) to obtain their reference positions. Specifically, the refer-\nence positions are topocentric astrometric positions of the stars\ncalculated from their catalogue positions. To ensure the accu-\nracy of the GD solution, we also account for additional factors\n1https://github.com/JxnuLin/GDSolver\nArticle number, page 2 of 8Lin et al.: A GD solution and its implementation.\nTable 1. Specifications of the 1-m and 2.4-m telescopes and the corresponding CCD detectors.\nParameter 1-m telescope 2.4-m telescope 60-cm telescope\nApproximate focal length 1330 cm 1920 cm 750 cm\nF-Ratio 13.1 8.0 12.5\nDiameter of primary mirror 101.6 cm 240.0 cm 60.0 cm\nApproximate scale factor 0.234 arcsec pixel−10.283 arcsec pixel−10.383 arcsec pixel−1\nSize of CCD array (e ffective) 4096×4112 1900 ×1900 2048 ×2048\nSize of pixel 15.0 µm×15.0µm 13.5µm×13.5µm 13.5µm×13.5µm\nTable 2. Details of the observations. The first column is the identification of the observation set. Column (2) and Column (3) list the target and the\nobservational date. Column (4) gives the number of CCD frames in each observation set. Column (5) is the range of zenith distance. Column (6)\nis the telescope used. Column (7) provides the number of stars with su fficient signal-to-noise ratio, whose astrometric precision is not dominated\nby centering errors. Column (8) gives the exposure time.\nID Target Obs Date Frames Zenith Distance Telescope Bright Star Exposure\n(y-m-d) (No.) (degree) (No.) (second)\n(1) (2) (3) (4) (5) (6) (7) (8)\n1 NGC2168 2018-11-13 38 1 ∼17 YNO 1-m 350 ∼400 60\n2 NGC1664 2015-02-10 44 19 ∼21 YNO 2.4-m 150 ∼180 40\n3 Himalia 2017-04-08 15 35 ∼37 YNO 2.4-m ∼10 30\n4 GSC02038-00293 2011-02-28 187 1 ∼30 YNO 60-cm ∼12 40\nthat may cause deterioration to its accuracy. These factors in-\nclude di fferential colour refraction and charge transfer e fficiency\nissues, which can be e ffectively addressed using the method pre-\nsented in Lin et al. (2020). Consequently, we can obtain the pixel\ncoordinate ( xi,yi) and the equatorial coordinate ( αi,δi) of each\nstari. The standard coordinate ( ξi,ηi) can be converted from the\nequatorial coordinate via the central projection, which is pre-\nsented in Green (1985).\nTo extract the GD e ffect on pixel positions, we need to solve\na six-parameter linear transformation to obtain the approximate\npixel positions ( xL\ni,yL\ni) of the reference stars. The linear transfor-\nmation is expressed as:\nxi=eaξi+ebηi+ec,\nyi=edξi+eeηi+ef,(1)\nwhere the coe fficientsea∼ef(denoted as CL\nstdhereafter) can be es-\ntimated through the least-squares fitting. Using the linear trans-\nformation, the standard coordinates ( ξi,ηi) can be converted to\nthe approximate pixel positions ( xL\ni,yL\ni). The coe fficients of the\nlinear transformation CL\nstdare initially inaccurate because they\nare affected by the GD. During the iterative solving process of\nGD, the pixel positions ( xi,yi) in Equation 1 will be replaced by\nthe positions after GD correction in each new iteration. As a re-\nsult, the approximate pixel positions ( xL\ni,yL\ni) would converge to\nthe distortionless pixel positions.\nBased on the pattern and magnitude of GD experienced by\nthe optical system of each telescope, we select a polynomial of\nappropriate order Nto characterize its analytical GD model. The\ngeneral formula of the polynomial is given as:\nU=X\nm,n,0⩽m+n⩽NkmnXmYn,\nV=X\nm,n,0⩽m+n⩽NjmnXmYn,(2)\nwhere kmnandjmnare the parameters to be fitted. Setting ( X,Y)\nas coordinates ( xL\ni,yL\ni) and ( U,V) as coordinates ( xi,yi), an Nth-\norder polynomial that characterizes the GD e ffect can be fitted.We denote the coe fficients of this polynomial as Cpix. By solving\nfor the coe fficients Cpixof each frame in an observation set and\napplying a weighted average based on image quality, an average\nGD solution Cpixcan be obtained. Most of the random errors are\noffset in the weighted average of the information from multiple\nframes, leaving only the GD e ffect.\nNow we can determine the GD e ffect at any given pixel po-\nsition using a polynomial with coe fficients Cpix. However, when\nthe GD e ffect changes dramatically within a small image range,\nthere would be a significant di fference in the GD e ffect between\nthe distortionless pixel position of the star and its actual observed\npixel position. In order to handle this issue, we determine the\ntransformation from the pixel positions ( xi,yi) to the distortion-\nless positions to correct GD. Specifically, we construct a 16 ×16\ngrid uniformly distributed across the pixel coordinates of the\nimage (as shown in Figure 2). Then the grid positions ( xg,yg)\nare transformed via a polynomial using the coe fficients Cpix, re-\nsulting in their distorted positions ( xGDg,yGDg). Finally, the inverse\ntransformation coe fficients Cinvare determined by fitting from\n(xGDg,yGDg) to ( xg,yg). The pixel position with GD correction can\nbe calculated by setting ( X,Y) as the coordinate ( xi,yi) and us-\ningCinvas the coe fficients in Equation 2. Figure 1 describes the\nsolving process for these coe fficients and the transformations be-\ntween di fferent positions.\nConsidering that the pixel positions of stars are contaminated\nby di fferent levels of random noise, weights have been intro-\nduced into all fitting procedures related to the pixel positions\n(xi,yi). The GD model is derived through an iterative procedure,\nwith the weights initially set to be uniform. After the first itera-\ntion of the GD solution, the weight for each star is determined\nas the inverse of the variance in positional measurements. The\nvariance can be calculated by fitting a sigmoidal function to the\nMag-SD data (as shown in Figure 3). The sigmoidal function is\nexpressed as:\nσ(m)=(A1−A2)/(1+e(m−m0)/dm)+A2, (3)\nwhere mis the magnitude of the star, σ(m) is the positional SD of\nthe star in the previous iteration, A1,A2,m0anddmare the free\nArticle number, page 3 of 8manuscript no. new_GDC_arXiv\nFig. 1. The transformations between di fferent positions and the solving process of the corresponding coe fficients in one iteration of our geometric\ndistortion (GD) solution. The arrows in the figure indicate the processes of coordinate transformations through the given coe fficients, as well as\nother operational processes denoted in bold. The target in the figure represents the fitting targets of the input positions, while the coeffdenotes the\nfitted coe fficients obtained from the fitting process.\nparameters. Detailed calculation procedure for the weights can\nbe found in Lin et al. (2019). The weights and the coe fficients\nCL\nstdare updated in each iteration, so that a more accurate GD\nmodel can be solved. The final analytical GD model is obtained\nthrough two to four iterations of the aforementioned procedure.\nFor comparison, a classical method of the plate constants re-\nduction is also applied in this paper. The method can be simply\ndescribed as solving a polynomial transformation from pixel po-\nsitions of reference stars to their standard positions, and then\nusing the transformation to calculate the astrometric position of\nthe target. Using a Nth-order polynomial for reduction, GD not\nhigher than Nth-order can be handled if there are enough refer-\nence stars (Green 1985; Peng & Fan 2010).\n4. Results\nCompared with the GD model determined by the well-\nestablished method (Peng et al. 2012; Wang et al. 2019), the\naccuracy of the GD model obtained through our method is veri-\nfied. Furthermore, our method is applied to reduce observations\nof Himalia (J6) and GSC02038-00293 to demonstrate its advan-\ntages.\n4.1. Comparison with the well-established method\nSince observation sets 1 and 2 were acquired by the dithering\nstrategy, the well-established methods can also be used to solve\nGD. The GD models for these observation sets were solved by\nthe methods described in Wang et al. (2019) and Section 3, re-\nspectively. Figure 2 presents the results, which include the GD\nmodels for the YNO 1-m and 2.4-m telescopes solved by each\nmethod. The di fferences of the GD models solved by these two\nmethods for each telescope are also presented in the right panels\nof the figure. The analytical GD model for the YNO 1-m tele-\nscope is characterized using a 4th-order polynomial, while the\nmodel for the YNO 2.4-m telescope using a 5th-order polyno-\nmial.\nAfter GD correction, the six-parameter plate constants are\nused for the reduction of observations to obtain astrometric re-\nsults. Hereafter, the astrometric results corrected by the method\nof Wang et al. (2019) are used as a reference. Figure 3 showsthe positional standard deviation (SD) of each star, which is cal-\nculated asσsum=√\nσ2\nαcosδ+σ2\nδ. In addition, the di fference in the\nmean ( O−C) (i.e., the residual between the observed and com-\nputed position) between our results and the reference results is\nshown in Figure 4. As can be seen from Figure 3 and 4, the as-\ntrometric results corrected using our GD solution are consistent\nwith the reference results. For the YNO 1-m telescope observa-\ntions, which is less a ffected by GD, the mean ( O−C) difference\nbetween our results and the reference results is merely 1 mas.\nThe di fference is only 2 mas for the observations captured by the\nYNO 2.4-m telescope. In addition, the SDs of the astrometric re-\nsults obtained by the two methods are equivalent. That is to say,\nthe method proposed in this paper can e fficiently correct GD and\nobtain reliable astrometric results.\n4.2. Application of the GD solution\nAs stated in Section 1, our method is particularly useful in sce-\nnarios where only a limited number of bright reference stars\n(typically a dozen or so) can be used in the reduction. This usu-\nally happens when observing a sparse FOV , where the moving\ntarget may pass through. In this section, we processed and an-\nalyzed the observations of two targets that satisfy the scenario.\nThese observations were not taken with a dithered FOV . Hence,\nthe aforementioned well-established GD solutions are not appli-\ncable.\nFigure 5 shows the astrometric results of the J6 observations\ncaptured by the YNO 2.4-m telescope, the upper panel shows\nan obviously greater SD for the target than other bright stars.\nThis is because there are insu fficient reference stars available for\nreduction, leading to overfitting of the 3rd-order plate constants.\nTo address this issue, we corrected the GD corresponding\nto the 3rd-order polynomial using the method proposed in this\nwork, and then used the six-parameter plate constants for reduc-\ntion. The astrometric precision of the target J6 is improved after\nGD correction, with the positional SD decreased from 20 mas to\n17 mas. The result is shown in the lower panel of Figure 5.\nThe improvement is more significant for the observations\ntaken with the YNO 60-cm telescope. Figure 6 shows the GD\nmodel of the telescope solved with observation set 4. To obtain\nreliable astrometric results, this GD solution was applied in the\nArticle number, page 4 of 8Lin et al.: A GD solution and its implementation.\nFig. 2. The GD models of the YNO 1-m and 2.4-m telescopes solved by our method and the method of Wang et al. (2019), respectively. The\ndifferences of these two GD solutions for each telescope are presented in the right panels. The upper panels display the GD models of the YNO\n1-m telescope, while the lower panels show the GD models of the YNO 2.4-m telescope. The statistics of the GD vectors are presented at the top\nof each panel. The vectors are suitably magnified to clearly visualize the shape of the GD.\nreduction of observation set 4. For comparison, Figure 7 also\npresents the results obtained by using plate constants of di fferent\norder for reduction. The left panel in the figure gives the posi-\ntional SD and the right panel the corresponding mean ( O−C)\ncalculated by⟨O−C⟩sum=√\n⟨O−C⟩2\nαcosδ+⟨O−C⟩2\nδ.\nDue to the observation set points to a fixed FOV , using low-\norder plate constants for reduction is possible to achieve precise\npositional measurement for the target (see 25.0 mas in panel (a)\nof Figure 7). However, despite the good fit of the plate constants\nat this time, the results for all stars displayed in the right panel\n(b) show large mean ( O−C), suggesting the presence of signifi-\ncant GD e ffect and rendering these astrometric results unreliable.\nAs the order of the plate constants increased, the positional SD\nof the target given in panels (c) and (e) becomes greater than that\nof other bright stars. That is to say, even if only 2nd-order plate\nconstants are used, overfitting will occur and become more pro-\nnounced as the order increases. This is consistent with the pre-\nvious astrometric results of J6 observations. Additionally, pan-\nels (d) and (f) show that the mean ( O−C) values of the refer-\nence stars are decreased. This is due to overfitting resulting in\nthe residual being absorbed in the reduction process. The target\nshould not be involved in the fitting of the plate constants, so\nits mean ( O−C) values remain large, indicating unsatisfactory\nastrometric results.The bottom two panels in Figure 7 provide the results of ap-\nplying our GD correction method first, followed by reduction us-\ning the six-parameter plate constants. As evident from the pan-\nels, these results show significant advantages compared to the\nresults of other methods. On the one hand, the astrometric preci-\nsion of the target is comparable to that achieved using the low-\norder plate constants. On the other hand, panel (h) reveals that\nthe mean ( O−C) values for both the target and the reference\nstars are significantly smaller than those obtained using the 2nd\nand 3rd-order plate constants. This demonstrates that the system\nerror caused by GD is significantly decreased after applying the\nGD correction.\nEmpirically, when using the weighted least squares method\nto determine the plate constants for data reduction, the number\nof bright reference stars should be approximately 1.5 times the\nnumber of fitting parameters. This is crucial when determining\nthe necessity of a GD solution.\n5. Conclusions\nWe investigated a GD correction method based on the high-\nprecision Gaia catalogue. The major advantage of this method\nis easy to implement. It was found that no more than fifteen\nframes, with only a dozen bright stars per frame, are enough\nto derive the accurate GD solution. As fitting errors are elim-\nArticle number, page 5 of 8manuscript no. new_GDC_arXiv\n/s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s32/s32\n/s32/s84/s104/s105/s115/s32/s119/s111/s114/s107/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s48/s46/s48\n/s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s32/s32\n/s32/s87/s97/s110/s103/s32/s101/s116/s32/s97/s108/s46/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s48/s46/s48\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50/s48/s53/s48/s49/s48/s48/s83/s68/s32/s40/s109/s97/s115/s41\n/s32/s32\n/s32/s84/s104/s105/s115/s32/s119/s111/s114/s107\n/s71 /s32/s109/s97/s103/s49/s51/s46/s53\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48 /s50/s50/s48/s53/s48/s49/s48/s48\n/s32/s32\n/s32/s87/s97/s110/s103/s32/s101/s116/s32/s97/s108/s46/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s51/s46/s51\nFig. 3. Comparison of the positional standard deviation (SD) obtained after GD correction using our method versus the reference method (Wang\net al. 2019). The upper panels present the results for observation set 1 captured by the YNO 1-m telescope, and the bottom panels show the results\nfor observation set 2 captured by the YNO 2.4-m telescope. The horizontal axis is Gaia G-mag and the vertical axis is the positional SD calculated\nbyσsum=√\nσ2\nαcosδ+σ2\nδ. The horizontal dashed line marks the median of the positional SDs for stars brighter than the magnitude indicated by the\nvertical line, the number on the right is the median.\n/s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s45/s49/s48/s48/s49/s48\n/s32/s32\n/s32/s89/s78/s79/s32/s49/s45/s109/s32/s116/s101/s108/s101/s115/s99/s111/s112/s101/s99/s111/s115 /s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s32/s61 /s32/s48/s46/s54 /s32/s109/s97/s115\n/s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s45/s49/s48/s48/s49/s48\n/s32/s32\n/s32/s89/s78/s79/s32/s49/s45/s109/s32/s116/s101/s108/s101/s115/s99/s111/s112/s101/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s32/s61 /s32/s48/s46/s54 /s32/s109/s97/s115\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48\n/s32/s32\n/s32/s89/s78/s79/s32/s50/s46/s52/s45/s109/s32/s116/s101/s108/s101/s115/s99/s111/s112/s101/s99/s111/s115 /s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s32/s61 /s32/s49/s46/s54 /s32/s109/s97/s115\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48\n/s32/s32\n/s32/s89/s78/s79/s32/s50/s46/s52/s45/s109/s32/s116/s101/s108/s101/s115/s99/s111/s112/s101/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s32/s61 /s32/s49/s46/s55 /s32/s109/s97/s115\nFig. 4. The di fferences of the mean ( O−C) in R.A. and Decl directions between our results and the reference results. The upper panels are the\nresults for observation set 1 and the bottom panels are for observation set 2. Stars with low signal-to-noise ratios are excluded. The standard\ndeviation of these di fferencesσis shown in each panel.\ninated by averaging the coe fficients from multiple frames, the\nfinal GD solution does not have overfitting issues even if only a\ndozen bright stars can be used to solve the high-order polyno-\nmial. To evaluate the accuracy of our GD solution, observations\nof open clusters taken with the 1-m and 2.4-m telescopes at Yun-\nnan Observatory were reduced. In the reduction, our GD solu-\ntion and a well-established GD solution (Peng et al. 2012; Wang\net al. 2019) were used for GD correction, respectively. The re-\nsults demonstrate that both methods achieve the same precision.\nIn addition, the mean ( O−C) difference between our results andthe reference results is only 1 mas for the YNO 1-m telescope ob-\nservations and 2 mas for the YNO 2.4-m telescope observations.\nIt is evident that our GD solution is convenient and e ffective.\nThis method presents significant value for some observa-\ntions, such as those in which GD correction is unattainable due\nto the absence of relevant calibration data. This situation often\noccurs when dealing with historical observations, which are cap-\ntured by the telescopes employed for photometric purposes. Be-\ncause of the new research focus, astrometric analysis for these\nobservations may become important. The observations of the bi-\nArticle number, page 6 of 8Lin et al.: A GD solution and its implementation.\n/s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s48/s49/s55/s45/s48/s52/s45/s48/s56\n/s50/s48/s46/s49\n/s32/s32\n/s32/s32 /s72/s105/s109/s97/s108/s105/s97/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s49/s46/s49/s51/s114/s100/s32/s79/s114/s100/s101/s114\n/s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s48/s49/s55/s45/s48/s52/s45/s48/s56\n/s49/s55/s46/s52\n/s32/s32\n/s32/s32 /s72/s105/s109/s97/s108/s105/s97/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s56/s46/s51/s71/s68/s67/s32/s43/s32/s49/s115/s116/s32/s79/s114/s100/s101/s114\nFig. 5. The positional SD of Himalia (J6) before and after the GD cor-\nrection.\nFig. 6. The GD model of YNO 60-cm telescope.\nnary GSC02038-00293 taken with the YNO 60-cm telescope are\nan example that satisfies the situation. By applying our GD so-\nlution in the reduction, the astrometric results of this target were\nsignificantly improved, with the mean ( O−C) decreased from 36\nmas to 5 mas. The J6 observations in a sparse FOV taken with\nthe YNO 2.4-m telescope were also corrected by the method, and\nthe astrometric results of the target has been improved to some\nextent. Performance of our method in the processing of observa-\ntions a ffected by large magnitude of GD, such as the Bok tele-\nscope observations (Zheng et al. 2022; Peng et al. 2023), will be\ndiscussed in the future.\nAcknowledgements\nThis work was supported by the National Natural Science\nFoundation of China (Grant Nos. 12203019), by the National\nKey R&D Program of China (Grant No. 2022YFE0116800),by the National Natural Science Foundation of China (Grant\nNos. 11873026, 11273014) by the China Manned Space Project\n(Grant No. CMS-CSST-2021-B08) and the Joint Research Fund\nin Astronomy (Grant No. U1431227). The authors would like\nto thank the chief scientist Qian S. B. of the 1-m telescope and\nhis working group for their kindly support and help. And thank\nthem for sharing the observations of the binary GSC02038-\n00293. 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F. 2022, A&A, 661,\nA75, doi: 10.1051/0004-6361/202141725\nArticle number, page 7 of 8manuscript no. new_GDC_arXiv\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s50/s53/s46/s48\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s50/s51/s46/s50/s49/s115/s116/s32/s111/s114/s100/s101/s114/s40/s97/s41\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s40/s98/s41\n/s51/s53/s46/s56\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51/s79/s45/s67\n/s115/s117/s109 /s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s51/s46/s55\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s50/s56/s46/s48\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s53/s46/s57/s50/s110/s100/s32/s111/s114/s100/s101/s114/s40/s99/s41\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s40/s100/s41/s79/s45/s67\n/s115/s117/s109 /s32/s40/s109/s97/s115/s41\n/s50/s56/s46/s54\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51\n/s71 /s32/s109/s97/s103/s54/s46/s52\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s40/s101/s41\n/s52/s50/s46/s50\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s49/s52/s46/s54/s51/s114/s100/s32/s111/s114/s100/s101/s114\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s40/s102/s41\n/s49/s56/s46/s48\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51/s79/s45/s67\n/s115/s117/s109 /s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s51/s46/s49\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s40/s103/s41\n/s50/s54/s46/s51\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51/s83/s68/s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s50/s48/s46/s48/s71/s68/s67/s32/s43/s32/s49/s115/s116/s32/s111/s114/s100/s101/s114\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s40/s104/s41\n/s53/s46/s50\n/s32/s32\n/s32/s71/s83/s67/s48/s50/s48/s51/s56/s45/s48/s50/s57/s51/s79/s45/s67\n/s115/s117/s109 /s32/s40/s109/s97/s115/s41\n/s71 /s32/s109/s97/s103/s53/s46/s48\nFig. 7. Comparison of the positional SD and mean ( O−C) obtained by using di fferent methods for reduction. From top to bottom in this figure\nare the results of reduction using 1st, 2nd and 3rd-order plate constants, as well as the results obtained by reduction using the six-parameter plate\nconstants after GD correction. The left panels show the mean ( O−C) for each star, while the right panels present the corresponding positional SD.\nThe numbers in the panels are the statistics of the target GSC02038-00293. The horizontal dashed line marks the median of the positional SD or\nmean ( O−C) in each panel for stars brighter than 15 magnitude.\nArticle number, page 8 of 8"    },    {        "title": "2401.11706v1.Nambu_Goldstone_Gravity_and_New_Cosmology.pdf",        "content": "arXiv:2401.11706v1  [physics.gen-ph]  22 Jan 2024Nambu-Goldstone Gravity and New Cosmology\nKimihide Nishimura∗\nNihon-Uniform, 1-4-1 Juso-Motoimazato, Yodogawa-ku Osak a 532-0028 Japan\n(Dated: January 23, 2024)\nThe reformulation of gravity as the Nambu-Goldstone mode as sociated with broken gauge sym-\nmetry is given and applied to cosmology. We find that; the new g ravity consists of a scalar mode\nand a tensor mode, of which the tensor mode corresponds to the Einstein gravity; the universe\npulsates without spacial curvature; the gravitational con stant increases from zero as the universe\nexpands and decreases to zero as the universe contracts; the dark energy is identified with the sum\nof the gravitational energies of scalar and tensor modes, wh ile the dark matter is the mirage of the\ngravitational energy caused by the projection into the ΛCDM spectrum; the gravitational equation\ndictates the cosmological constant to be zero; the universe begins with inflation, and the present\nage of the universe is approximately 550Gy.\nI. INTRODUCTION\nThe recent discoveries in cosmology by the advance of observation al technologies; 1. the necessity of dark energy,\ndark matter, and inflation for understanding in particular the acce lerating expansion of the universe[1, 2] and the\nanisotropies of cosmic microwave background[3–5]; 2. the abundan ce of matured galaxies with high redshifts found\nby the James Webb Space Telescope (JWST), which possibly contrad icts the present age of the universe[6, 7], suggest\nthat the Einstein theory of gravity has already been inadequate fo r describing our universe. In order to accommodate\nthese new entities, it seems necessary to revise the theory of gra vity.\nFrom theoretical point of view, one of the unsatisfactory points w ith Einstein gravity is that it is not based on\nquantum theory. The equivalence principle, on which Einstein gravity is based, alone is not adequate to determine\nthe dynamics of gravity, though the Einstein-Hilbert action is the sim plest one.\nAnother distinction of gravity from other gauge interactions is the weakness of the force. Dirac [8] once tried\nto understand the small gravitational constant by supposing tha t it decreases as the universe expands. However,\na quantum theoretical reconsideration on this problem suggests t he opposite possibility: the gravitational constant\nincreasing from zero as the universe expands. This possibility is realiz ed when gravity is considered as the Nambu-\nGoldstone (NG) mode associated with broken symmetry [9–12]. If th e universe begins with spontaneous symmetry\nbreakdown, there would be no NG bosons before the breakdown, a nd therefore no gravity. Further, if the breakdown\nis not yet complete, gravity would be weak. Then, the hypothesis of NG gravity transfers the problem to finding out\nthe symmetry which breaks down at the beginning of the universe, w hich will in turn provide a missing principle to\ndetermine the dynamics of gravity. Here we emphasize that what is r equired to be broken is not spacetime symmetry.\nThe hypothetical necessity of spacetime symmetry breakdown se ems to stem from the desire to understand spinning\nmassless particles as the NG bosons, for example, photons and gra vitons [13]. Of those, the idea of NG photon was\nnot so attractive, since there already exists gauge theory. On th e other hand, the NG graviton is far more important,\nsince it enable us to understand gravity, which is otherwise based on the utterly different principle, in the context of\ngauge theory.\nSpontaneous symmetry breakdown, which plays the essential role for the unified picture of elementary particles and\nforces[14, 15], postulates that some quantum field develops the va cuum expectation value, which is supposed to be\nthe same at all spacetime points. Considering this phenomenon in the presence of gravity, the vacuum expectation\nvalue independent of spacetime points seems too restrictive.\nHowever, according to the equivalence principle, we can find at ever y spacetime point a local inertial frame where\ngravity is absent, and where spacetime symmetry holds locally. This o bservation suggests that, even in the presence\nof gravity, we may still assume a common vacuum expectation value a t every local Lorentz frame. In addition,\nit makes aware us an important fact that the local Lorentz frame t ransforms spacetime symmetry (global Lorentz\nsymmetry) into gauge symmetry (local Lorentz symmetry), which is not accompanied by any spacetime coordinate\ntransformations.\nHitherto in elementary particle physics the fields which break gauge in variance are supposed to be Lorentz scalars\nin order not to break Lorentz invariance. The reasoning above will r emove this restriction, if we consider symmetry\n∗Electronic address: kimihiden@dune.ocn.ne.jp2\nbreaking in a local Lorentz frame. Then, any local Lorentz tensor s with gauge indices become allowable as the\nsymmetry breaking fields, which do not break the general coordina te invariance nor the equivalence principle.\nWhen the gauge-breaking field is a local Lorentz tensor, its vacuum expectation value will take the form of a\nconstant mixed tensor with both local Lorentz indices and the gaug e indices. If we want to preserve local Lorentz\ninvariance even after symmetry breakdown, the constant mixed t ensor should have the form of some multiple of the\nMinkowski metric. This requirement implies that the gauge group to b e broken should have as a subgroup Lorentz\ngroup of four dimensions, or it should be some extension of Lorentz group, for example, a Lorentz group of higher\ndimensions. We soon recognize that five dimensional Lorentz symme try is adequate for generating gravity as NG\nmode. In this case, no extra symmetry other than the ordinary Lo rentz symmetry survives. Actually, as presently\nshown, the vierbein, which is introduced as an auxiliary field to set up a local Lorentz frame, acquires dynamics by\nspontaneous symmetry breakdown in a local Lorentz frame.\nThe simplest choice of the gauge-breaking tensor field is to take the gauge field itself. We construct in Sec.II the\naction of NG gravity explicitly based on the Yang-Mills gauge field with fiv e dimensional local Lorentz symmetry on\nfour dimensional Minkowski spacetime. The resulting action contain s a scalar field and the quadratic term of the\nRiemann curvature, which are both absent in the Einstein theory, w here one of the interaction terms of the scalar\nfield becomes the germ of the Einstein action of gravity.\nSection III derives the gravitational equations for the scalar and the tensor modes. The gravitational constant is\ndetermined by the vacuum expectation value of the scalar field, as o nce supposed by Brans and Dicke [16]. The new\ngravitational equation of tensor mode is expressible as the total e nergy-momentum tensor to be zero, which implies\nthat there remains no role for the cosmological constant except c alibrating the zero points of the constituent energy\ndensities.\nSec.IV applies the new gravity to the expanding universe to obtain th e Friedmann equations.\nSec.V shows that the Friedmann equations thus derived admit one pa rameter family of exact solutions, where the\nscalar gravity determines the time evolution of the universe, while th e tensor gravity determines the energy spectrum\nin the universe. The solutions show that the universe pulsates witho ut spacial curvature, and begins with the power\nlaw inflation.\nSec.VI shows that the dark energy and the cold dark matter in the B ig-Bang cosmology are both originated from\nthe same gravitationalenergy obeying different equations of stat e, though the cold dark matter is simply the reflection\nof the gravitational energy converted into the state of cold matt er.\nSec.VII clarifies that the cause of the phase transition of the unive rse from inflationary expansion to decelerating\nexpansion is the change of the nature of gravity from repulsive into attractive.\nSec.VIII calculates the temperatures of thermal radiation and no rmal matter to reconsider the cosmic microwave\nbackground (CMB), the Big-Bang nucleosynthesis (BBN), and the baryon asymmetry (baryogenesis) in the context\nof new cosmology.\nSec. IX determines the free parameter remaining until the end in ou r cosmological model, and estimates the present\nage of the universe, by comparing the rate of change of the gravit ational constant with observational data.\nSec.X represents the results obtained in graphics.\nII. NEW LAGRANGIAN OF GRAVITY\nWe first construct the Yang-Mills action with five dimensional local Lo rentz gauge invariance in four dimensional\nMinkowski spacetime. The simplest presentation is to consider the s calar multiplet VAwith gauge index A= (α,5) =\n(0,1,2,3,5), which transforms as a local Lorentz vector in the gauge space ,\nδVA=θABVB, (1)\nwhereθAB=−θBAare the gauge parameters for the extended local Lorentz trans formations. A subscript or a\nsuperscript is interchangeable by contracting it with the five dimens ional Minkowski metric ηAB=ηBAof signature\n(+,−,−,−,−). The gauge connection ωµAB=−ωµBAis introduced to define the covariant derivative,\nDµVA=∂µVA+ωµABVB, (2)\nwhereµis the coordinate index of four dimensional Minkowski spacetime: µ= (0,1,2,3). The gauge covariance\nrequires that ωµABtransforms as\nδωµAB=θACωµCB+θBCωµAC−∂µθAB. (3)\nThe curvature tensor, or the field strength Ω µνABis obtained from the relation:\n[Dµ,Dν]VA= ΩµνABVB,ΩµνAB=∂µωνAB−∂νωµAB+ωµACωνCB−ωνACωµCB. (4)3\nThen, the Yang-Mills Lagrangian in the general coordinate system h as the form\nLω=−1\n8g2gµρgνσΩµνABΩρσAB, (5)\nwhereµ,ν, etc. are, from now on, the general coordinate indices, while the o riginal Minkowski coordinate index µis\ntransferred to the local Lorentz index αby introducing the vierbein eµαas an auxiliary field. The metric tensor gµν\nis then expressed as gµν=ηαβeµαeνβ, andgis a dimensionless gauge coupling constant.\nWe next decompose the five dimensional local Lorentz tensor into t he four dimensional local Lorentz components\nand the others by introducing notations: ωµα:=ωµα5=−ωµα5, and Ω µνα:= Ωµνα5=−Ωµνα5to obtain\nLω=1\n4g2gµρgνσΩµναΩρσα−1\n8g2gµρgνσΩµναβΩρσαβ, (6)\nΩµνα=∇µωνα−∇νωµα,∇µωνα:=∂µωνα−Γρµνωρα+ωµαβωνβ, (7)\nΩµναβ=∂µωναβ−∂νωµαβ+ωµαγωνγβ−ωναγωµγβ+ωµαωνβ−ωναωµβ. (8)\nSince the Lagrangian has local Lorentz symmetry of four dimension s as the subgroup, the component ωµαβinωµAB\nis identical with the ordinary local Lorentz connection, and theref ore, the derivative ∇µin (7) is covariant under\nboth the general coordinate transformations and the four dimen sional local Lorentz transformations. The Christoffel\nsymbol Γρµνis understood to be calculated for the metric gµν, though it is cancelled in Ω µνα.\nWe then break the five dimensional local Lorentz symmetry down to four dimensions in a local Lorentz frame by\nassuming that ωµαdevelops the vacuum expectation value of the form:\n∝angb∇acketlefteµαωµβ∝angb∇acket∇ight=g√\n3∝angb∇acketleftφ∝angb∇acket∇ightηαβ, (9)\nwhereφis a real scalar field of mass dimension one, and ηαβis the ordinary four dimensional Minkowski metric in\nthe local Lorentz frame. The normalization factor has been chose n for later convenience. The vacuum expectation\nvalue (9) preserves the ordinary local Lorentz invariance and the general coordinate invariance. What are broken are\nthe extra four generators of the five dimensional local Lorentz g roup.\nFrom the above observations, the following parametrization of ωµαis possible [12],\nωµα=g√\n3φeµα, (10)\nsince the vierbein has been introduced as an auxiliary field. This form s atisfies automatically the condition for the\nvacuum expectation value (9). The desired symmetry breakdown is realized by introducing to the Lagrangian (6) the\nbreaking term:\nLsb=η2\n2gµνωµαωνα−Λ0,Λ0:=g2η4\n3, (11)\nwhereηis a constant with mass dimension one, which gives the scale of symmet ry breaking. The constant term Λ 0,\nwhich is unnecessary for symmetry breaking, has been introduced tentatively as the standard cosmological constant.\nThe implication of the cosmological constant in the new gravity forma lism, which is one of the main themes of this\npaper, will be discussed later.\nIf the torsion constraint: ∇µeνα−∇νeµα= 0 is imposed on the vierbein, the total Lagrangian Lg=Lω+Lsbis\nreduced to the form:\nLg=1\n2gµν∂µφ∂νφ−V(φ)−φ2\n6R−1\n8g2RµνρσRµνρσ, V(φ) =g2\n3(φ2−η2)2, (12)\nwhereRµνρσis the Riemann curvature, and R=eµαeνβRµναβis the scalar curvature. Owing to the standard\ncosmological constant Λ 0, the minimum of the potential V(φ) becomes zero at φ=±η. We adopt Lagrangian Lgfor\nthe action of “Nambu-Goldstone gravity”.\nIII. GRAVITATIONAL EQUATIONS\nThe gravitational equation in the presence of matter is derived by a dding to Lgthe normal matter Lagrangian Ln\ncorresponding to leptons, quarks, gauge bosons, the Higgs boso n, and the others which may include yet unknown4\nparticles and sources of energies besides gravity, and varying the total action with respect to φandgµν. Then, we\nobtain the new gravitational equations:\n/parenleftbigg\n∇ρ∇ρ+R\n3/parenrightbigg\nφ+∂V\n∂φ=jn, (13)\nRµαβγRναβγ−gµν\n4RαβγδRαβγδ+2∇ρ∇σRµρνσ= 2g2(Tµν\nφ+Tµν\nn), (14)\nwhere\njn:=δLn\nδφ, (15)\nTµν\nn:=−2/radicalbig\n−|g|δ(/radicalbig\n−|g|Ln)\nδgµν, (16)\nTµν\nφ:=∇µφ∇νφ−gµν/parenleftbigg1\n2∇ρφ∇ρφ−V/parenrightbigg\n+1\n3/parenleftbig\n∇µ∇νφ2−gµν∇ρ∇ρφ2/parenrightbig\n−φ2\n3/parenleftbigg\nRµν−gµν\n2R/parenrightbigg\n.(17)\nThe new gravitational equation for the tensor mode (14) is the fou rth oder partial differential equation with the\ndimensionless gauge coupling constant g. We can verify that\n∇µTµν\nφ=jn∇νφ, (18)\n∇µTµν\nn=−jn∇νφ, (19)\n∇µTµν\nt= 0, Tµν\nt:=1\n2g2/bracketleftbigg\n−RµαβγRναβγ+gµν\n4RαβγδRαβγδ−2∇ρ∇σRµρνσ/bracketrightbigg\n. (20)\nThe equation (18) follows from (13) and (17), while (19) follows from (18) and the gravitational equation (14), since\nit requires ∇µ(Tµν\nφ+Tµν\nn) = 0. The conservation of the energy-momentum tensor for tens or gravity (20) follows from\nthe Bianchi identity alone. If the φ-dependence of Lnis negligible ( jn= 0), then ∇µTµν\nφ=∇µTµν\nn= 0.\nOwing to the notation Tµν\nt, the tensor equation is expressible as the total energy-momentu m tensor to be zero,\nTµν=Tµν\nφ+Tµν\nt+Tµν\nn= 0. (21)\nThe Einstein equation is obtained by decomposing Tµν\nφintoTµν\nsandTµν\nc, where\nTµν\nφ=Tµν\ns−Tµν\nc, Tµν\nc:=φ2/parenleftbigg\nRµν−gµν\n2R/parenrightbigg\n. (22)\nThen, we have\nRµν−gµν\n2R= 8πG(Tµν\ng+Tµν\nn), Tµν\ng:=Tµν\ns+Tµν\nt, (23)\nwhere\nG=1\n8πφ2, (24)\nis the Newton “constant”. If jn= 0, we have\n∇µTµν\nn= 0,∇µTµν\ng=/parenleftbigg\nRµν−gµν\n2R/parenrightbigg\n∇µφ2,∇µG(Tµν\ng+Tµν\nn) = 0. (25)\nWe consider here the cosmological constant in the new gravitationa l equations. While the scalar equation (13) does\nnot depend on the cosmological constant Λ 0, the tensor equation (21) requires the total cosmological const ant to be\nzero. Therefore, Tµν\nnshould have the cosmological constant −Λ0gµνto cancel the cosmological constant in Tµν\nφ, since\nTµν\ntdoes not have the cosmological constant term by definition. This ob servation shows that new gravity essentially\nsettles the cosmological constant problem [17].\nThe cosmological constant becomes a serious problem if it originates from the vacuum energy or the zero-point\nenergy in quantum field theory, since it is infinite. In this view, quantu m corrections may induce for each constituent\nenergy-momentum tensor: Tµν\nφ,Tµν\nt, andTµν\nnin (21) the cosmological constant terms of their own, Λ φgµν, Λtgµν,\nand Λ ngµν, respectively. Then, the gravitational equation (21) dictates th at the total cosmological constant should\nbe zero: Λ = Λ φ+Λt+Λn= 0. If we adopt the convention that Λ c= 0, the cosmological constant for the normal\nmatter Λ nshould be cancelled by that for gravity Λ g= Λs+Λt, where Λ gmay differ from Λ 0.5\nIV. FRIEDMANN EQUATIONS\nWe apply new gravitational equations (13) and (23) to the expandin g universe and clarify the difference between\nthe Einstein gravity and the new gravity. We assume here the Friedm ann-Robertson-Walker metric with zero spacial\ncurvature:\nds2=gµνdxµdxν=dt2−a(t)2δijdxidxj, (26)\nand that the energy-momentum tensor for normal matter Tµν\nnhas the form of a perfect fluid:\nT00\nn=ρn, Tij\nn=a(t)−2δijpn. (27)\nIn this metric, the scalar equation (13) takes the form\n¨φ+3H˙φ−a−2∂2\niφ−2φ(˙H+2H2)+∂V(φ)\n∂φ= 0, H=˙a\na,˙a=da\ndt, (28)\nwhile the tensor equation (23) is expressible in the form\n/bracketleftbigg\nρc\npc/bracketrightbigg\n=/bracketleftbigg\nρg\npg/bracketrightbigg\n+/bracketleftbigg\nρn\npn/bracketrightbigg\n, (29)\nwhere\n/bracketleftbigg\nρc\npc/bracketrightbigg\n=/bracketleftbigg3H2φ2\n−(2˙H+3H2)φ2/bracketrightbigg\n,/bracketleftbigg\nρg\npg/bracketrightbigg\n:=/bracketleftbigg\nρs\nps/bracketrightbigg\n+/bracketleftbigg\nρt\npt/bracketrightbigg\n, (30)\n/bracketleftbigg\nρs\nps/bracketrightbigg\n=\n1\n2/parenleftBig\n˙φ−2Hφ/parenrightBig2\n+V(φ)\n1\n2˙φ2−V(φ)+2\n3/parenleftBig\nφ¨φ+˙φ2+2Hφ˙φ/parenrightBig\n−2\n3/parenleftBig\n2˙H+3H2/parenrightBig\nφ2\n, (31)\n/bracketleftbigg\nρt\npt/bracketrightbigg\n=\n3g−2/parenleftBig\nH¨H+3H2˙H−1\n2˙H2/parenrightBig\n−g−2/parenleftBig...\nH+6H¨H+9H2˙H+9\n2˙H2/parenrightBig\n. (32)\nV. EXACT SOLUTIONS\nIfjn= 0, the scalar equation (28) admits the exact solutions of the form :\nφ(t) =βη\nsinθ, a(t) = sinγθ, θ=αt, (33)\nwhereαis a constant of mass dimension one, while βandγare dimensionless numbers. Putting (33) into (28), we\nfind\nα2=4g2η2\n3(4γ2+3γ−1), β2=4γ2+γ−2\n4γ2+3γ−1. (34)\nSince these are positive numbers, the value of γis in the range\nγ >√\n33−1\n8= 0.593···,orγ <−1. (35)\nThe first equation in (33) implies that φstarts from the infinity: φ(0) =∞, which is certainly another symmetric\npoint of the potential V(φ) other than φ= 0. It will be a natural condition rather than assuming that φstarts from\nthe top of a small hill of the potential: V(0), if the early universe was very hot and dense.\nThe energy conservation ˙φ2/2+Veff(φ) = 0 gives the effective potential:\nVeff(φ) =−2g2\n3bφ2(φ2−β2η2), b:= 4γ2+γ−2, (36)6\nwhich has a fairly gentle slope for a large γ. In this effective potential, the scalar φrolls up from Veff(+∞) =−∞to\nVeff(βη) = 0, and then returns to the infinity (Fig.1).\nThe second equation of (33) implies that the evolution of the univers e can be periodic in time, if γis an integer.\nForγ >0, it begins from a point at t= 0, and expands to the maximum. Then it returns to a point again (Fig .2).\nForγ <0, on the other hand, it begins from the infinity and contracts to th e minimum, and then blows up again to\nthe infinity in a finite time.\nThe Hubble parameter H, and the deceleration parameter qare calculated as\nH=αγcotθ, q=−1−˙H\nH2=−1+1\nγcos2θ. (37)\nFrom the second equation, we notice that 1+ q >0 forγ >0, and 1+ q <0 forγ <0 at all times. According to the\nrecent observational data, the present value of the deceleratio n parameter is q0=−0.51 [5], and therefore 1+ q0>0.\nConsequently, the solution with γ <0 will not correspond to our universe. We consider in the following only the case\nwith a positive γ.\nFurther, if γ >1 the deceleration parameter qvanishes at the phase transition points\nθx±= arccos( ±1/√γ). (38)\nThen, the universe experiences accelerating expansion during 0 < θ < θ x+, while decelerating expansion during\nθx+< θ <90◦. From the first equation, we see that H= 0 atθ= 90◦, where the universe stops expanding and then\nbegins contracting to a point again.\nThere are several reasons to expect that γis large, for example, γ >10, orγ >100. One reason is that a large\nγwill induce an inflation [18–20], which seems necessary for understan ding the anisotropy of the cosmic microwave\nbackground (CMB). The second is, as we presently see, that γgrows large as the present dark energy component Ω0\nΛ\nin the ΛCDM cosmology approaches 2 /3, which approximates the observational value Ω0\nΛ= 0.683 [3]. Finally, the\nobservational constraint on the time variation of the gravitationa l constant also requires a large γ[21].\nThe energy density ρaand the pressure pain the previous section are expressible in the form:\n/braceleftbigg\nρc=e/parenleftbig\nc1sin−4θ+c2sin−2θ/parenrightbig\n,\npc=e/parenleftbig\nc3sin−4θ+c4sin−2θ/parenrightbig\n,(39)\n/braceleftbigg\nρs=e/parenleftbig\ns1sin−4θ+s2sin−2θ+λs/parenrightbig\n,\nps=e/parenleftbig\ns3sin−4θ+s4sin−2θ−λs/parenrightbig\n,(40)\n/braceleftbigg\nρt=e/parenleftbig\nt1sin−4θ+t2sin−2θ+λt/parenrightbig\n,\npt=e/parenleftbig\nt3sin−4θ+t4sin−2θ−λt/parenrightbig\n,(41)\n/braceleftbigg\nρg=e/parenleftbig\ng1sin−4θ+g2sin−2θ+λg/parenrightbig\n,\npg=e/parenleftbig\ng3sin−4θ+g4sin−2θ−λg/parenrightbig\n,(42)\n/braceleftbigg\nρn=e/parenleftbig\nn1sin−4θ+n2sin−2θ+λn/parenrightbig\n,\npn=e/parenleftbig\nn3sin−4θ+n4sin−2θ−λn/parenrightbig\n,(43)\n(44)\nwhere we have introduced the partial cosmological constants: λs,λt,λgandλn, in view of the argument in Sec.III.\nThese can be specified at will under the conditions: λg=λs+λtandλg+λn= 0, without affecting the gravitational\nequations. The standard cosmological constant Λ 0corresponds to λs=λ0, where\nλ0= Λ0/e=(b+2γ+1)2\n12bγ2, e= 3(αβγη)2. (45)\nThe numerical coefficients ci,si, andtiare given by\n\n\nc1= 1,\nc2=−1,\nc3=2\n3γ−1,\nc4= 1,\n\ns1= 1+3\n4γ,\ns2=−4\n3−7\n6γ,\ns3=−1−1\n12γ+1\nγ2,\ns4=4\n3+17\n18γ−7\n9γ2,\n\nt1=−4γ\nb/parenleftBig\n1−1\n2γ/parenrightBig\n,\nt2=4γ\nb/parenleftBig\n1−2\n3γ/parenrightBig\n,\nt3=4γ\nb/parenleftBig\n1−1\n2γ/parenrightBig/parenleftBig\n1−4\n3γ/parenrightBig\n,\nt4=−4γ\nb/parenleftBig\n1−2\n3γ/parenrightBig2\n,(46)7\nwhilegiandniare given by gi:=si+ti,ni=ci−gi, (i= 1,2,3,4). In particular, the following expressions are\nuseful in later considerations:\n\n\nn1=−3\n4γ+4γ\nb/parenleftBig\n1−1\n2γ/parenrightBig\n=1\n4γ/parenleftBig\n1−3\nγ+···/parenrightBig\n,\nn2=1\n3+7\n6γ−4γ\nb/parenleftBig\n1−2\n3γ/parenrightBig\n=1\n3/parenleftBig\n1+1\n2γ+···/parenrightBig\n,\nn3=/parenleftBig\n4\n3γ−1/parenrightBig\nn1=−1\n4γ/parenleftBig\n1−13\n3γ+···/parenrightBig\n,\nn4=/parenleftBig\n2\n3γ−1/parenrightBig\nn2=−1\n3/parenleftBig\n1−1\n6γ+···/parenrightBig\n,(47)\nwhere the second equalities are the asymptotic expansions with res pect toγ−1.\nWe remarkthat the critical density ρc=esin−4θcos2θbecomes zerowhen the universe stops expanding at θ= 90◦.\nAccordingly, it will be convenient to calibrate the energy densities ρs,ρt,ρg, andρnso that each component also\nbecomes zero at θ= 90◦. This is allowable, and realizes if we take λa=−a1−a2fora=s,t,g,n. In this case, λgor\n−λnis found very close to λ0for a large γ:\nλg=−λn=λ0+1\n12γ2+O(γ−3). (48)\nVI. DARK ENERGY AND DARK MATTER\nWe first introduce the notion of the “cosmological gauge transfor mation” for the energy momentum tensor Tµν\na:\nTµν\na→Tµν\na+Λa(t)gµν,or,/bracketleftbigg\nρa\npa/bracketrightbigg\n→/bracketleftbigg\nρa+Λa(t)\nps−Λa(t)/bracketrightbigg\n,(a=s,t,g,n), (49)\nwhere the gauge parameter Λ a(t) is a function of the cosmological time t. When Λ a(t) is constant, it becomes an\nordinarycosmologicalconstant. We do not consider the gaugetra nsformation of the critical density ρc, nor the critical\npressure pc, since these are regarded as fiducial quantities.\nWe consider again the cosmological constant problem [17] in the cont ext of our new formalism. If the gravitational\nequation is invariant under the above gauge transformation, then the cosmological constant problem will disappear,\nsince the problem becomes simply a matter of gauge fixing.\nWhile the scalar equation (28) is invariant under any change of gauge parameter Λ s, the tensor equation (21), or\n(29) is invariant under the condition Λ g+Λn= Λs+Λt+Λn= 0. However, as is well known, since the Friedmann\nequations for the energy density component and the pressure co mponent (29) are equivalent, the two equations can\nbe cast into the gauge invariant form in terms of the “invariant dens ities” ˜ρa:=ρa+pa:\n˜ρc= ˜ρg+ ˜ρn,or,1 =˜Ωg+˜Ωn,˜Ωg:=˜ρg\n˜ρc,˜Ωn:=˜ρn\n˜ρc, (50)\nwhich contrasts to the ordinary (gauge dependent) energy spec trum of the universe:\nρc=ρg+ρn,or,1 = Ωg+Ωn,Ωg:=ρg\nρc,Ωn:=ρn\nρc. (51)\nConsequently, the equation (50) for the tensor gravity also beco mes completely independent of the gauge parameters.\nWe next consider the physical implication of the invariant densities ˜ ρa. Since the new gravitational equations (28)\nand (50) are “gauge invariant”, we can take use of this gauge free dom to convert each specific energy-momentum\ntensor into the form of “cold matter” with zero pressure:\n/bracketleftbigg\nρa\npa/bracketrightbigg\n→/bracketleftbigg\nρa+Λa(t)\npa−Λa(t)/bracketrightbigg\n=/bracketleftbigg\nρa+pa\n0/bracketrightbigg\n,Λa(t) =pa,(a=g,n). (52)\nWe find from (52) that the invariant density of a-fluid: ˜ρa=ρa+pais the effective energy density as cold matter.\nThen, the Friedman equation in the gauge invariant form (50) implies t he energy spectrum of the cold matter.\nIn order to find out the correspondence of the quantities Ω g, Ωn,˜Ωg, and˜Ωnto those in the Big-Bang cosmology,\nwe cast the Friedmann equations (29) into the ΛCDM spectrum:\n/bracketleftbigg\nρc\npc/bracketrightbigg\n=/bracketleftbigg\nΛd\n−Λd/bracketrightbigg\n+/bracketleftbigg\nρm\n0/bracketrightbigg\n, (53)8\nwhere Λ dis the cosmological constant as the “dark energy”, and ρmis the cold matter density. Then, we find the\nexpressions of Λ dandρmin the new formalism:\nΛd=−pc, ρm=ρc+pc= ˜ρc, (54)\nfrom which we have\nΩΛ:=Λd\nρc= 1−Ωm,Ωm:=ρm\nρc=˜ρc\nρc= Ωgcm+Ωncm,Ωgcm:=˜ρg\nρc= Ωm˜Ωg,Ωncm:=˜ρn\nρc= Ωm˜Ωn,(55)\nwhere Ω gcmis the component for the gravitational cold matter, while Ω ncmis that for the normal cold matter.\nTherefore, the following identifications are natural:\nΩd= Ωgcm,Ωb= Ωncm, (56)\nwhere Ω dis for the “cold dark matter”, while Ω bis for the (cold) baryonic matter, since the dominant component of\nthe normal cold matter will be baryons.\nWe next examine the correspondences (55) and (56) quantitative ly, in particular, for the present values. In order\nto do that, it is convenient to specify the “standard present time” t0byq0=−1/2. Since\nΩΛ=−pc\nρc= 1+2˙H\n3H2=1−2q\n3,Ωm=˜ρc\nρc=2\n3γcos2θ, (57)\nwe obtain the relations:\ncosθ0=/radicalbigg\n2\nγ, θ=αt=/radicalbigγ\n2−1\nγH0t, t0=γ/radicalbig\nγ/2−1H−1\n0arccos/radicalbigg\n2\nγ, (58)\nwhereH0is the Hubble constant at the standard present time. Then, we hav e by definition (Ω0\nΛ,Ω0\nm) = (2/3,1/3),\nwhich approximate well those obtained by the astronomical observ ations: Ω0\nΛ= 0.683 [3], and q0=−0.51 [5]. Owing\nto (47), we find the expressions\nΩn= 1−Ωg=n1+(n1+n2)sin2θ, (59)\n˜Ωn= 1−˜Ωg= 2n1+n2sin2θ, (60)\nwhere Ω nand˜Ωnincrease as the universe expands, and decrease as the universe c ontracts. These normal matter\ncomponents, or the gravitational components, take nearly the s ame value for a large γ:\n˜Ωn−Ωn= Ωg−˜Ωg=n1cos2θ≃cos2θ\n4γ. (61)\nFurther, we obtain as the present values\nΩ0\nn= 1−Ω0\ng=1\n3−17\n12γ2+···, (62)\n˜Ω0\nn= 1−˜Ω0\ng=1\n3−11\n12γ2+···, (63)\nand therefore,\nΩ0\ng≃2/3,Ω0\nd≃2/9,Ω0\nb≃1/9, (64)\nwhich are compared to the observational values: (Ω0\nΛ,Ω0\nd,Ω0\nb) = (0.683,0.268,0.049) [3]. Then, we confirm from\n(Ω0\ng,Ω0\nn)≃(Ω0\nΛ,Ω0\nm) that the true nature of dark energy is the gravitational energy , though they behave differently\nas the universe expands, since the dark energy Λ dand the gravitational energy ρgobey different equations of state.\nThe same caution applies for the cold matter ρmin the ΛCDM cosmology and the normal matter ρnin the new\ncosmology.\nWe find further that the distinction between the cold dark matter a nd the baryons (normal cold matter) in the\nΛCDM cosmology is simply the reflection of the distinction between the gravitational energy and the normal matter\nenergy caused by casting them into the ΛCDM spectrum (Fig.3).\nThe value of Ω0\nbin (64) is a little large compared to the observational data, possibly s ince we assumed tentatively\nthat the normal matter consists only of baryons.9\nVII. VARYING EQUATION OF STATE\nWhenγis large, the pulsating universe (33) begins with inflation. The era of in flation continues till the universe\nreaches the phase transition point θ=θx+(38), and then enters the era of decelerating expansion. Astron omical\nobservations reveal that our universe is still in the phase of accele rating expansion [1, 2, 5], and therefore θ0< θx+.\nThe cause of the phase transition at θx+in the pulsating universe is inferred from the equations of state for the normal\nmatter and gravitational fluids: pa=waρa,(a=n,g).\nWe first consider the case of normal matter wn=pn/ρn(43). Owing to the relations (47), we have\nwn(θ) =−1+2\n3γcos2θ/bracketleftbigg\n1+n1cos2θ\nn1+(n1+n2)sin2θ/bracketrightbigg\n. (65)\nThen,wnincreases monotonically from a negative value wn(0◦) to the positive infinity wn(90◦) = +∞, where\nwn(0◦) =−1+4\n3γ. (66)\nAccordingly, the normal matter fluid behaves like a cosmological con stant (the dark energy) w=−1 at the beginning\nof the universe, and then becomes to behave successively like a cold matter, a radiation, and finally an ideal gas\nw=∞(sinceρ∝T1+1/wfor a perfect fluid) at θ= 90◦. In particular, when the universe is at the phase transition\npointθx+,wntakes the value\nwn(θx+) =−1\n3+O(γ−2). (67)\nOn the other hand, we have for the gravitational fluid wg=pg/ρg(42):\nwg(θ) =−1+2\n3γcos2θ/bracketleftbigg\n1−n1cos2θ\n1−n1−(n1+n2)sin2θ/bracketrightbigg\n, (68)\nwhich shows that wgincreases monotonically from a negative value wg(0◦) to the positive infinity wg(90◦) = +∞,\nwhere\nwg(0◦) =−1+2\n3γ+O(γ−2). (69)\nWe have also in this case,\nwg(θx+) =−1\n3+O(γ−2), (70)\nwhich is approximately the same as wn(θx+), ifγis large. We see that the two fluids obey almost the same equation\nof statep=−ρ/3 at the phase transition point θx+, which looks like a kind of thermal radiation with the opposite\npressure (Fig.4).\nWe next show that the appearance of this “thermal anti-radiation ” is the symptom that the gravitational force\nchanges its sign. The linear gravity approximation gives as the gravit ational potential VGbetween small portions of\nthe static fluid with w1=p1/ρ1atx=x1, and that with w2=p2/ρ2atx=x2:\nVG=−G/integraldisplayd3x1d3x2\n|x1−x2|/bracketleftbigg\nTµν(x1)Tµν(x2)−1\n2Tρρ(x1)Tσσ(x2)/bracketrightbigg\n=−Gζ(w1,w2)m1m2\nr12, ζ(w1,w2) := 1+3 w1+3w2−3w1w2. (71)\nwheremi:=ρid3xi. The gravitational force becomes repulsive for ζ(w1,w2)<0. In the case of two portions of the\nnormal matter fluid: w1=w2=wn, for example, we have\nζ(wn,wn) = 1+6 wn−3w2\nn<0 forwn<1−2√\n3=−0.1547···, wn>1+2√\n3= 2.1547···.(72)\nIf one of the fluids is a cold matter: w2= 0, we have instead,\nζ(wn,0) = 1+3 wn<0 forwn<−1\n3. (73)10\nIn either case, the normal matter exerts repulsive force when wn<−1/3. If one of the fluids is the thermal radiation\nw= 1/3, on the other hand, the gravity returns to attractive:\nζ(wn,1/3) = 2+2 wn>0 forwn>−1. (74)\nAccordingly, the phase transition point θx+corresponds to the time at which wnandwgbegin to exceed −1/3, and\ntherefore, at which the both fluids cease to repel, and begin to att ract the cold matter. In this view, the cause of the\naccelerating expansion of the universe is attributed to the effect o f repulsive gravity, and the subsequent decelerating\nexpansion is due to the change of the nature of gravity from repuls ive into attractive.\nVIII. THERMAL PROPERTY OF THE UNIVERSE\nThe Big-Bang cosmology assumes that the cosmic microwave backgr ound (CMB) at present originates from the\nadiabatic cooling of radiation, which was decoupled from thermal equ ilibrium with matter at redshift z≃103. This\nhypothesis is, however, no longer maintainable in the pulsating univer se. In fact, if the adiabatic cooling of thermal\nradiation is assumed, we have the relation for the energy density of radiation ρR:\nρR\nρ0\nR= (1+z)4, (75)\nwhile the critical density ρc(39) behaves as\nρc\nρ0c=γ\n2/bracketleftbigg\n(1+z)4/γ−/parenleftbigg\n1−2\nγ/parenrightbigg\n(1+z)2/γ/bracketrightbigg\n. (76)\nThen, if Ω0\nR=ρ0\nR/ρ0\nc= 5×10−5forT0= 2.725K is assumed, ρRwould surpass ρcatz≃21 forγ= 10, or at z≃16\nforγ= 100, which implies that the actual temperature of CMB at z≃103was rather low, and it has experienced\nreheating after the decoupling from matter at T≃3000K.\nThe energy density of radiation ρRmay be calculated by recasting the Friedmann equation (29) into the sum of\nthe cosmological parameter, the cold matter, and the thermal ra diation:\n/bracketleftbigg\nρc\npc/bracketrightbigg\n=/bracketleftbigg\nΛd\n−Λd/bracketrightbigg\n+/bracketleftbigg\nρm\n0/bracketrightbigg\n+/bracketleftbigg\nρR\nρR/3/bracketrightbigg\n, (77)\nfrom which we have\nρm= (1−r)˜ρc,˜ρR=4\n3ρR=r˜ρc,(0≤r≤1). (78)\nThe ratio ris determined as follows. We may assume that r(t= 0) = 1 at the beginning of the universe, since all the\nknown elementary particles, including the scalar graviton and the te nsor graviton, would be in thermal equilibrium\nwith the photon. Due to the periodical nature of time evolution of th e universe, rwill be expressible as the sum of the\nLegendre polynomials, or the power series of cos θ. Accordingly, we may assume as the most dominant contribution\nthe simplest form r= cosνθ, where the number νis not necessarily an integer. Then, we have\nρR=e\n2γsin−4θcosνθ (79)\nfrom which cosνθ0= 4Ω0\nR= 2×10−4follows. We next define the “reduced temperature” TRby\nρR=π2\n15T4\nR, TR:=N1/4\nγT, (80)\nwhereNγ(1≤N1/4\nγ<3) is the effective multiplicity of the photon, which is half the total deg rees of freedom of\nparticles in thermal equilibrium with the photon. Then, we have\nTR\nT0=sinθ0\nsinθ/parenleftbiggcosθ\ncosθ0/parenrightbiggν/4\n(81)11\nWe can approximate the formula (81) by\nTR\nT0= (4Ω0\nR)−1/4sinθ0\nsinθ= (4Ω0\nR)−1/4(1+z)1/γ,/parenleftBigg\nsinθ <sinθǫ:=/radicalBigg\nǫln(γ/2)\n−ln(4Ω0\nR)/parenrightBigg\n, (82)\nwithin the error ǫ= 1−cosν/4θ. For CMB, we have sin θcmb≃7.64×10−3with sinθǫ≃4.35×10−2forǫ= 10−2\nandγ= 10. Then, the formula (82) is correct within the error of 1% for TR>3000K and γ >10, and we find that\nthe redshift at T=TR= 3000K is z≃102.1γin the new cosmology.\nFurther, when θ < θcmb, sinθis well approximated by αt, and the formula (82) is further reduced to the form\nt= (Ω0\nR)−1/4H−1\n0√γT0\nTR= 43.4√γy/bracketleftbigg1010K\nTR/bracketrightbigg\n, (83)\nwhere the second equality is for H0= 73Kms−1Mpc−1= (13.4Gy)−1. We find that tcmb>0.46Gy for γ >10, which\nis compared to tcmb≃0.38My in the Big-Bang cosmology.\nIf CMB at the period of T=TR≃3000K was cooled down by adiabatic expansion, then the present te mperature\nof CMB would be less than 3000 /(1+z)≃3×10−18K despite our assumption T0= 2.725K. This result confirms that\nCMB has been reheated after the decoupling. In fact, we have fro m (79)\n˙ρR\nρR+4H=α[4(γ−1)cotθ−νtanθ]∝negationslash= 0. (84)\nSinceα∝H0, however, the total heat added during ∆ tis proportional to H0∆t, and the adiabatic hypothesis is still\nwell applied for the processes accomplished within a very short time r elative to the age of the universe. The similar\nargument applies also for the cosmological nucleosynthesis [22, 23] and baryogenesis [24, 25].\nThe period of Big-Bang nucleosynthesis (BBN) [22] ends when the fr ee neutrons decay away, which is characterized\nby the temperature T= 1.5×1010K≃1.29MeV, or TR=N1/4\nγT= 2.3×1010K forNγ= 43/8 [26]. Then, in the\nnew cosmology, the time available for the cosmological nucleosynthe sis will be tcns= 19√γ >60y forγ >10, and\naccordingly all the chemical elements in the universe could be synthe sized by that time, whereas in the Big-Bang\ncosmology only a few light elements can be synthesized within a few minu tes.\nHowever, the formula (82) shows that the redshift correspondin g to that time is zcns≃109γ, while the normal cold\nmatter density ˜ ρnat that time is\n˜ρn≃ρ0\nc\n6γ(1+zcns)4/γ. (85)\nIf ˜ρnis identified with the baryonic energy density, the contribution of th e cosmological nucleosynthesis to the present\nbaryon density will be\n∆Ω0\nb=1\n6γ(1+zcns)4/γ−3<10−9(3γ−4). (86)\nWe further imagine the epoch when the entire universe was filled with n ucleons. Due to the saturation property of\nnucleons, the energy density at that time would be (180MeV)4. If this is identified with ˜ ρnat that time, we have\n1+znuc=/bracketleftbigg\n6γ˜ρn\nρ0c/bracketrightbiggγ/4\n>/bracketleftbigg˜ρn\nρ0c/bracketrightbiggγ/4\n≃1011γ. (87)\nThen, the contribution of baryons at that time to the present bar yon density would be\n∆Ω0\nb=1\n6γ(1+znuc)4/γ−3<10−11(3γ−4). (88)\nBoth cases show that these contributions are completely negligible. Nevertheless, we have already seen that the\npresent proportion of baryons is roughly 10% in the critical density (64). Consequently, either case shows that most\nof the baryons in the present universe have been (re-)generate d after the cosmological (Big-Bang) nucleosynthesis.\nHow this can be realized is inferred from the following observation.\nThe intrinsic state of baryons in the pulsating universe is the normal matter satisfying the equation of state (65),\nwhere the cold baryons are related to the normal matter by the ga uge transformation. We have seen in Sec.VII that\nthe state of normal matter changes from the dark energy like sta te:wn≃ −1 to the cold matter like state: wn≃012\nas the universe expands. When the normal matter is in the “dark st ate”:−1< wn<0, the photons and the baryons\nincluded: the “dark photons” and the “dark baryons” will have the tendency to proliferate in the expanding universe.\nIn fact, the normal matter satisfies the conservation law (25), a nd therefore experiences adiabatic expansion. For\na short time during which wncan be regarded as constant, the thermodynamics gives the relat ions for an adiabatic\nprocess:\nρn∝V−1−wn, Un=ρnV∝V−wn, (89)\nwhereVis some volume, for which we may take the intrinsic volume a3. When the normal matter, including the\nphoton gas and the baryonic fluid, is in the dark state: −1< wn<0,ρndiverges, while the internal energy Un\nvanishes as the universe shrinks to a point. Then, there would be no photons nor baryons at the beginning of the\nuniverse, which implies that the universe would be too small to accomm odate even a single elementary particle.\nContrary, ρndecreases, while Unincreases as the universe expands, which will explain the reheating o f the thermal\nradiation and the increase of the baryon number in the expanding un iverse.\nThe above observation suggests further that the temperature of the universe at the beginning would be very low,\ncontrary to the Big-Bang hypothesis, which can be confirmed by co nsidering the temperature of the normal matter\nitself.\nActually, if we assume jn= 0 and the adiabatic expansion for the normal matter, the equatio n of thermodynamics\ngives\ndS=ρ′\nnV\nTndTn+ρn+pn\nTndV, (90)\nwhere the prime implies the derivative with respect to the temperatu reTn. The integrability condition for the entropy\ngives\ndTn\nTn=dpn\nρn+pn=−ds2\ns22n3+n4s2\nn1+n3+(n2+n4)s2, (91)\nwheres= sinθ. Using the relations (47), we can integrate it to obtain\nTn∝/parenleftbigg2n1\nn2+sin2θ/parenrightbigg\nsin3γ−4θ. (92)\nIn terms of the redshift, we have for a large γandθ∼90◦:\nTn\nT0n=/bracketleftbigg\n1+2\nγln(1+z)/bracketrightbigg\n(1+z)−3≤e3/parenleftbigg\n1−1\nγ/parenrightbigg\n,/parenleftbig\nz≥e−1−1/parenrightbig\n, (93)\nwhereT0\nnis the temperatureofnormalmatter at present, and eis here the baseof naturallogarithm. The temperature\nof the normal matter reaches its maximum Tmax\nn≃20T0atz=e−1−1, orθ= 90◦, though elsewhere it is almost\nzero (Fig.5). If the temperature of the normal matter and that o f CMB are the same at present, then Tmax\nn≃55K.\nIn particular, the temperature of the normal matter at the begin ning of the universe is precisely zero, as expected.\nConsequently, the baryogenesis will continue as long as baryons ar e in the dark state, and it will end when the dark\nbaryons upheave to the cold baryons. The genesis of matter in the new cosmology does not date back to the very\nearly universe, but still now going on, since wn<0 yet at present.\nIX. GRAVITATIONAL CONSTANT VARYING WITH TIME\nFrom (24) and (33), we have the gravitational constant varying w ith time:\nG=sin2αt\n8πβ2η2,˙G\nG=2\nγH. (94)\nThen, the value of γwill be directly obtained by knowing the present values ( ˙G/G)0andH0. The most stringent\nevaluation of the value ( ˙G/G)0is provided by the Lunar Laser Ranging (LLR) [27]:\n/parenleftBigg˙G\nG/parenrightBigg\n0= (4±9)×10−13y−1. (95)13\nThis result suggests ˙G0>0: an increasing gravitational constant, which favors our new gra vity theory, though\ncontrary to the expectations by Dirac and Brans-Dicke.\nWith (˙G/G)0= 4×10−13y−1andH0= 73kms−1Mpc−1= (13.4Gy)−1, we have\nγ= 2H0/parenleftBigg˙G\nG/parenrightBigg−1\n0≃370. (96)\nAccordingly, the variation of the gravitational constant with resp ect to the redshift zis fairly small:\nGz\nG0= (1+z)−2/γ≃1−2\nγln(1+z), G10= 0.987G0, G20= 0.984G0. (97)\nThe relation in (58) gives\ndt\ndθ=α−1= 27.3H−1\n0= 6.38Gy/degree, (98)\nfrom which the standard age of the present universe for θ0= arccos/radicalbig\n2/γ= 85.8◦is\nt0= 40.9H−1\n0= 548Gy . (99)\nFurthermore, we find that the phase transition will occur at tx+= 555Gy, and the universe will stop expanding at\ntmax= 574Gy. Accordingly, the lifetime of the universe will be 1150Gy. The value of the symmetry braking scale η\nand the gauge coupling constant gare obtained from (24), (34) and (37):\nη≃0.2MP= 2.43×1018GeV, g=/radicalBigg\n3πb\nγG0H2\n0≃1.5×10−59, (100)\nwhereMPis the Planck mass. The coupling constant gis very small, and we confirm the validity of the assumption\njn= 0, since due to (10), jnwill be proportional to g.\nX. GRAPHICS\nAll the graphs in this section assume γ= 370, except for the first one.\nV(/u1D719)\nVeff /uni0028.9/u1D719 /uni0029.10/u1D719//u1D702 !!0 /u1D737V\nFIG. 1: The scalar field φin Sec.V, rolling down and up in the potential V(φ) (a bold line), while rolling up and down in the\neffective potential Veff(φ) (a grey line). The vertical scale is arbitrary, and the slop e ofVeff(φ) is exaggerated. The turning\npointφ/η=βcorresponds to the maximum of the scale factor a(t).14\nPulsating Universe ●\n/u1D7030/u1D703x+ \n!/u1D703x- \n!\n●a(/u1D703)\n/u1D703=/u1D6FCt /u1D703B\nFIG. 2: The comparison between the pulsating universe (a bol d line) and the Big-Bang universe (a grey thinner line) in Sec .V,\nwhereθ0corresponds to the “standard present time”. The Big-Bang wo uld occur at θB, while the pulsating universe experiences\nthe phase transition from inflationary expansion to deceler ating expansion at θx+, and the phase transition from accelerating\ncontraction to deflationary contraction at θx−. The portion of the bold line under the horizontal dashed lin e corresponds to\nthe era of repulsive gravity dominance, while that above to t he era of attractive gravity dominance.\n1\n01/3 Ωn=1-Ω g\n/u1D703=0 /uni309C.644 /u1D703=180 /uni309C.644 /u1D703=/u1D7030Ωm= 1-Ω Λ ~\nΩbΩn=1-Ω g~Ω\nFIG. 3: Thetime evolutionoftheenergyspectruminthepulsa tinguniverseinSec.VI,andthatcasted intotheΛCDM spectr um.\nThe normal matter component Ω n(θ) and the normal cold matter component ˜Ωn(θ) (bold lines) are regular functions of θ,\nthough graphically not distinguishable, while Ω m= 1−ΩΛ(a grey line) is singular at θ= 90◦. Nevertheless, the graph shows\nthat Ω mand Ω ntake almost the same value at the standard present time t0=α−1θ0. The region sandwiched between Ω mand\nΩbcorresponds to the cold dark matter Ω d, though it is the mirage of the distinction between the gravi tational energy and the\nnormal matter energy in the new cosmology.15\nw=-1 w=-1/3 w=0 \n/u1D7030 /u1D703=90 /uni309C.644wn(/u1D703), wg(/u1D703)\n/u1D703x /u1D703=/u1D6FCt\nFIG. 4: The varying equations of state: wn=pn/ρn, andwg=pg/ρgin Sec.VII. The both are graphically not distinguishable.\nWe see that w≃ −1/3 at the phase transition point θ=θx+, where the era of repulsive gravity ends and the era of attrac tive\ngravity begins. We are at present still in the era of repulsiv e gravity: w0\nn<−1/3 andw0\ng<−1/3.\nTn TR\nT0\n/u1D7030 /u1D703T\n/u1D703=90 /uni309C.644e (1-1/ /u1D6FE) T 03\nFIG. 5: The time evolution of the reduced temperature of ther mal radiation TRand that of normal matter Tnin Sec.VIII,\nwhen the both are the same at the standard present time. Tnis almost zero for θ < θ0. The maximum value of Tnatθ= 90◦\nise3(1−1/γ)T0.\n[1] Adam G. Riess et al., Observational Evidence from Supern ovae for an Accelerating Universe and a Cosmological Consta nt,\nAstron. J. 116, 1009, (1998); arXiv:astro-ph/9805201.\n[2] S. Perlmutter et al., Measurements of Ω and Λ from 42 High- Redshift Supernovae, Astron. J. 517, 565 (1999);\narXiv:astro-ph/981213316\n[3] Planck Collaboration: N. Aghanim, et al., Planck 2018 re sults. VI. Cosmological parameters, Astron. Astrophys. 641, A6\n(2020).\n[4] G. Hinshaw, et al., Nine-Year Wilkinson Microwave Aniso tropy Probe (WMAP) Observations: Cosmological Parameter\nResults, ApJS, 208, 19 (2013); arXiv:1212.5226.\n[5] Adam G. Riess, et al., A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc\nUncertainty from the Hubble Space Telescope and the SH0ES Te am; arXiv:2112.04510 [astro-ph.CO]\n[6] Ivo Labbe, et al., A population of red candidate massive g alaxies 600 Myr after the Big Bang, Nature 616, 266 (2023).\n[7] A. C. Carnall et al., A massive quiescent galaxy at redshi ft 4.658, Nature 619, 716 (2023).\n[8] P. A. M. Dirac, The cosmological constants, Nature, 139, 323 (1937); A new basis for cosmology, Proc. Roy. Soc. A165,199\n(1938).\n[9] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345;124, 246 (1961).\n[10] J. Goldstone, Field theories with ≪Superconductor ≫solutions, Nuovo. Cimento 19, 154 (1961).\n[11] J. Goldstone, A. Salam and S. Weinberg, Broken symmetri es, Phys. Rev. 127, 965 (1962).\n[12] Kimihide Nishimura, Extra equation of gravity induced by spontaneous local Lorentz symmetry breakdown, Phys. Rev . D\n103, 064056 (2021); arXiv:1812.02557.\n[13] J. D. Bjorken, A dynamical origin for the electromagnet ic field, Ann. Phys. (N.Y.) 24, 174 (1963).\n[14] Peter W. Higgs, Broken Symmetries and the Masses of Gaug e Bosons, Phys. Rev. Lett. 13, 508 (1964).\n[15] Steven Weinberg, A Model of Leptons, Phys. Rev. Lett. 19, 1264 (1967).\n[16] C. Brans and R. H. Dicke, Mach’s Principle and a Relativi stic Theory of Gravitation, Phys. Rev. 124, 925 (1961).\n[17] Steven Weinberg, The cosmological constant problem, R ev. Mod. Phys. 61, 1 (1989).\n[18] K. Sato, First-order phase transition of a vacuum and th e expansion of the Universe, Mon. Not. Roy. Astron. Soc. 195,\n467 (1981).\n[19] Alan. H. Guth, Inflationary universe: A possible soluti on to the horizon and flatness problems, Phys. Rev. D23, 347 (1 981);\nFluctuations in the New Inflationary Universe, Phys. Rev. Le tt.49, 1110 (1982).\n[20] F. Lucchin and S. Matarrese, Power-law inflation, Phys. Rev. D32, 1316 (1985).\n[21] Steven Weinberg, Gravitation and Cosmology, Chap. 16, Sec. 4, p.619, John Wiley & Sons, Inc. (1972).\n[22] R. A. Alpher, H. Bethe, and G. Gamow, The Origin of Chemic al Elements, Phys. Rev. 73, 803 (1948).\n[23] G. Steigman, Primordial Nucleosynthesis in the Precis ion Cosmology Era, Annu. Rev. Nucl. Part. Sci. 57, 463 (2007).\n[24] A. D. Sakhalov, Violation of CP Invariance, C asymmetry , and baryon asymmetry of the universe, JETP Lett. 5, 24 (1967).\n[25] Mark Trodden, Electroweak baryogenesis, Rev. Mod. Phy s.71, 1463 (1999).\n[26] Steven Weinberg, Cosmology, Chap. 3, Oxford Universit y Press Publishing (2008).\n[27] James G. Williams, Salva G. Turyshev and Dale H. Boggs, P rogress in Lunar Laser Ranging Tests of Relativistic Gravit y,\nPhys. Rev. Lett. 93, 261101 (2004), arXiv:gr-qc/0411113."    },    {        "title": "2401.11722v1.Spatial_discretization_effects_in_spanwise_forcing_for_turbulent_drag_reduction.pdf",        "content": ".1\nBanner appropriate to article type will appear here in typeset article\nSpatial discretization effects in spanwise forcing for\nturbulent drag reduction\nEmanuele Gallorini and Maurizio Quadrio\nDipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano,\nItaly\n(Received xx; revised xx; accepted xx)\nWall-based spanwise forcing has been experimentally used with success by Auteri et al.\n(Phys. Fluids vol. 22, 2010, 115103) to obtain large reductions of turbulent skin-friction\ndrag and considerable energy savings in a pipe flow. The spatial distribution of the azimuthal\nwall velocity used in the experiment was not continuous, but piecewise-constant. The present\nstudy is a numerical replica of the experiment, based on a set of direct numerical simulations\n(DNS); its goal is the identification of the effects of spatially discrete forcing, as opposed to\nthe idealized sinusoidal forcing considered in the majority of numerical studies.\nRegardless of the discretization, with DNS the maximum drag reduction is found to\nbe larger: the flow easily reaches complete relaminarization, whereas the experiment was\ncapped at 33% drag reduction. However, the key result stems from the observation that,\nfor the piecewise-constant forcing, the apparent irregularities of the experimental data\nappear in the simulation data too. They derive from the rich harmonic content of the\ndiscontinuous travelling wave, which alters the drag reduction of the sinusoidal forcing.\nA detailed understanding of the contribution of each harmonic reveals that, whenever e.g.\ntechnological limitations constrain one to work far from the optimal forcing parameters, a\ndiscrete forcing may perform very differently from the corresponding ideal sinusoid, and in\nprinciple can outperform it. However, care should be exercised in comparison, as discrete\nand continuous forcing have different energy requirements.\n1. Introduction\nTurbulent flow control for drag reduction is an active discipline, pursuing a technological\ngoal of steadily increasing economic and environmental importance. A subset of flow control\nstudies aims at reducing the drag of wall-bounded turbulent flows. This is a particularly\ndifficult challenge: once geometry is simplified to consider a parallel flow, drag is only due\nto friction, and high levels of turbulent friction stem from the interaction between turbulence\nand the wall. Altering this interaction to improve the overall energetic efficiency, by following\neither active or passive approaches, is as fundamentally appealing as practically difficult.\nAn active, open-loop strategy to reduce friction drag which gained popularity in recent\nyears is made by spanwise forcing (a thorough review has been recently provided by Ricco\net al. 2021), notably in its spatially distributed version made by the streamwise-travelling\nwaves (StTW), where the spanwise velocity component 𝑊at the wall is prescribed according\nto:\n𝑊(𝑥,𝑡)=𝐴sin(𝜅𝑥𝑥−𝜔𝑡) (1.1)\nAbstract must not spill onto p.2arXiv:2401.11722v1  [physics.flu-dyn]  22 Jan 20242 E. Gallorini, M. Quadrio\nin which𝑥,𝑡are the streamwise coordinate and time, and 𝜅𝑥and𝜔are the streamwise\nwavenumber and frequency of the oscillating wave, which is thus characterized by a phase\nspeed𝑐=𝜔/𝜅𝑥. StTW, introduced by Quadrio et al. (2009), are interesting owing mainly\nto their good energetic performance, which is preserved at high Reynolds numbers (Gatti\n& Quadrio 2016). StTW are effective even in compressible and supersonic flows (Ruby &\nFoysi (2022); Gattere et al. (2024)), and have been demonstrated to affect favorably the\naerodynamic drag of a three-dimensional body (Banchetti et al. 2020), up to the point that\nan actuation over a limited portion of the wing of an airplane in transonic flight reduces the\ntotal drag of the aircraft by nearly 10% at a negligible energy cost (Quadrio et al. 2022).\nThe major obstacle to the deployment of StTW (and of spanwise forcing in general) in\npractical applications is the lack of suitable actuators. Very few laboratory implementations\nof StTW are available, and typically the idealized sinusoidal waveform of the wall velocity\ncannot be achieved. Bird et al. (2018) describe a planar actuator for StTW, formed by a\ntensioned membrane skin mounted on a kagome lattice; they discuss why the measured drag\nreduction turns out to be less than expected, and attribute it to the out-of-plane velocity\ncomponent created by the actual forcing. Benard et al. (2021) report preliminary results\non the implementation of steady longitudinal waves of spanwise forcing via plasma DBD\nactuators, based on electrodes designed with a suitable shape, which are affected by the\nnon-ideal response of the actuator. For a long time, the sole available laboratory experiment\nwith StTW (and the most successful one) was the water pipe flow experiment carried out\nby Auteri et al. (2010) (referred to as ABBCQ in this paper), who set up a low-Reynolds\nturbulent pipe flow modified by StTW and reported up to 33% of drag reduction. StTW were\nimplemented with an original device, recently replicated by Marusic et al. (2021) in the plane\ngeometry, in which the circular pipe is subdivided into axial slabs that independently oscillate\nin the azimuthal direction. Different rotational speeds of nearby slabs provided the desired\nstreamwise variation of transverse velocity, realizing a discrete travelling wave (DTW). A\nsketch of the DTW concept is provided in figure 1 (left). The experimental setup of ABBCQ\nconsisted of 10 sets of 6 independently movable segments and allowed to test waves made\nby a variable number 𝑠=2,3,6 of segments discretizing each sinusoid. As emphasized in\nfigure 1, the actual streamwise distribution of the forcing spanwise velocity with DTW is far\nfrom the sinusoidal one (red line) and is instead well approximated by a staircase function\n(blue line).\nThe design of the ABBCQ pipe flow experiment was guided by the original numerical\nstudy by Quadrio et al. (2009), carried out for the plane channel flow; however, the two flows\nare not identical. The drag reduction datasets, while in broad agreement, do present a few\ndifferences which have not been examined in depth so far. Figure 1 (right) compares the\nexperimental drag reduction data measured by ABBCQ for 𝑠=3 to the corresponding subset\nof DNS data, at the same wavelength of the travelling wave, with a similar forcing amplitude\nand Reynolds number. The largest drag reduction rate measured in the experiment is 33%\ninstead of more than 50% of the DNS. No drag increase is found in the experiment, and only\na drop of drag reduction to nearly zero level is observed, albeit at the very same frequency\n𝜔+≈0.1 where the DNS study predicts drag reduction to become negative. Moreover, the\nexperimental data present a rather irregular, wiggling dependence upon 𝜔+compared to the\nsmooth evolution of the DNS plane channel: especially at negative frequencies, some of the\nexperimentally measured drag reduction rates are comparable and even marginally larger\nthan the simulations, but other measurement points lie well below the numerical curve.\nABBCQ discuss some possible reasons behind these differences. The geometry (plane\nchannel vs circular pipe) certainly does play a role; the value of the Reynolds number is not\nidentical between the two studies, being 𝑅𝑒𝜏=200 in the plane channel DNS and 𝑅𝑒𝜏≈175\nin the circular pipe experiment; the same applies to the forcing amplitude, 𝐴+=12 in DNS3\nFlow\nWall velocity\n-0.2 -0.1 0 0.2 0.1-1001020304050\nω+100Rexp\nDNS\nFigure 1: Left: sketch of StTW actuation (continuous wave, red line) and DTW actuation\nwith𝑠=3 (discrete wave, blue line), as applied to the cylindrical pipe geometry by\nABBCQ. Right: percentage drag reduction Rexperimentally measured for DTW by\nABBCQ in the pipe flow (symbols), compared at the same value of 𝜅+to available DNS\ninformation for StTW from the DNS study of Quadrio et al. (2009) in the planar geometry\n(continuous black line). Besides the different geometry, the numerical study has a slightly\nsmaller forcing amplitude and a slightly larger Reynolds number.\nand𝐴+=13.8 in the experiment; the periodic boundary conditions employed in the DNS are\nnot fully equivalent to the actual flow conditions at the inflow and outflow sections of a finite\nlength of a pipe, implying that a temporal transient in the former can easily be discarded,\nwhereas the equivalent spatial transient in the experiment cannot be eliminated and affects\nthe measurement. However, the major difference — and the one the present work sets out\nto investigate — consists in the nature of the spatially distributed forcing applied at the\nwall: it is a sinusoidal function in the DNS study, where Eq.(1.1) is enforced as a boundary\ncondition, whereas in the experiment it consists of a piecewise-constant and time-periodic\nfunction. The importance of waveform discretization was already discussed by ABBCQ.\nThey mentioned how the wave realized with two segments only, i.e. 𝑠=2, is a limiting\ncase of the discrete travelling wave (DTW), in which there can be no travelling direction\nfor the wave, which becomes standing and works in a region of the parameter space where\nperformance is suboptimal, especially in terms of energy saving. Moreover, the harmonic\ncontent of DTW was suggested to potentially explain some features of the experimental\nresults shown in figure 1.\nThe importance of an in-depth understanding of discretization effects descends from the\nnecessity of any experimental realization of such forcing to be, to an extent, spatially discrete.\nFor temporal discretization (which is not particularly critical in experiments), these effects\nwere systematically studied by Cimarelli et al. (2013), who considered various temporal\nwaveforms to implement the spatially uniform spanwise-oscillating walls, and concluded\nthat the sinusoidal waveform is the best overall in terms of energy savings. Spatially discrete\nforcing, instead, has been rarely used so far and never discussed in terms of discretization\neffects. Kiesow & Plesniak (2003) experimentally generated a localized crossflow in a three-\ndimensional turbulent boundary layer using a transverse running belt. In a numerical study,\nMishra & Skote (2015) used only the positive cycle of steady square waves to enforce\ndrag reduction in a turbulent boundary layer, under the rationale that similar performance\ncompared to the complete control could be obtained with lower energy consumption. Straub\net al. (2017) studied the spanwise oscillating wall technique in a low-aspect-ratio duct, and\nconsidered how it performs when only a fraction of the available surface is actuated. Benard4 E. Gallorini, M. Quadrio\net al. (2021) dealt with the issue that several DBD actuators must be placed side by side to\nachieve a near-wall distribution of velocity that should be as spanwise uniform as possible.\nThe present paper describes the replica (including, first of all, the spatially discrete forcing)\nof the ABBCQ pipe flow experiment, based on direct numerical simulations (DNS), with\na focus on drag reduction and energy efficiency. Not much information is available for\nspanwise forcing applied to cylindrical geometries. Besides the few early works (Orlandi &\nFatica 1997; Quadrio & Sibilla 2000; Nikitin 2000) which proved with DNS the validity of\nspanwise (azimuthal) forcing in a pipe flow, only recently Liu et al. (2022) tested spatially\nnon-uniform spanwise forcing (in the form of stationary waves) in a pipe flow. Strictly\nspeaking, thus, StTW for drag reduction have never been tested numerically in a pipe. For\nthat purpose, in our DNS study we employ an original DNS solver, designed for efficient\nsimulations of high-Reynolds number turbulent pipe flows, which addresses the problem of\nexcess azimuthal resolution near the pipe axis: it exploits a mixed discretization (Fourier in\nthe homogeneous directions and compact finite differences in the radial direction) to decrease\nthe azimuthal resolution as the pipe axis is approached.\nThe structure of the paper is as follows. In §2 the DNS code will be presented and\nvalidated, the DTW forcing will be introduced, and the computational procedures adopted\nfor the present study will be described. Results of the simulation campaign will be presented\nin§3, touching upon drag reduction, power budget, and flow statistics. Discussion of the\ndiscretization effects and a critical evaluation of the present results against the ABBCQ\nexperiment are contained in §4; summarizing conclusions are provided in §5.\n2. Methods\n2.1. The DNS code\nWe solve by direct numerical simulation (DNS) the incompressible Navier–Stokes equations,\nwritten in non-dimensional form and cylindrical coordinates, for the primitive variables\npressure𝑝and velocity 𝒖. The axial, radial, and azimuthal directions are indicated with\n𝑥,𝑟, and𝜃; the respective velocity components are 𝑢,𝑣and𝑤. The axial length of the\ncomputational domain is 𝐿, the pipe radius is 𝑅; the complete azimuthal extent of 2 𝜋is\nconsidered.\nTemporal discretization of the equations follows the usual partially-implicit approach in\nDNS of wall-bounded flows: the code implements a combination of the implicit Crank-\nNicholson scheme for the viscous terms with a library of explicit schemes for the non-\nlinear convective terms. In this work, the three-substeps, low-storage Runge–Kutta scheme\ndescribed in Rai & Moin (1991) is used.\nSpatial discretization, instead, deserves a specific discussion. The discretization is mixed,\nas in the Cartesian DNS solver (Luchini & Quadrio 2006) which has inspired the present\ncode. The homogeneous directions 𝑥and𝜃call for a spectral discretization, naturally\nenforcing the required periodic boundary conditions with the computational efficiency of\nthe pseudospectral approach. Compact, fourth-order accurate finite differences are used to\ndiscretize differential operators in the radial direction. The mixed discretization is at the root\nof an interesting feature of the present code, which addresses a fundamental problem with\nthe DNS of the turbulent pipe flow. Once the number of azimuthal Fourier modes is set\naccording to physics-based considerations to yield the adequate azimuthal spatial resolution\nat the pipe wall, the azimuthal resolution increases above the required level as the pipe axis\nis approached. Such excess resolution not only is useless and thus a waste by itself, but also\ncauses a steep rise in the computational cost of the simulation, since a vanishing spanwise cell\nsize implies a vanishing time step, as the stability of the explicit temporal scheme requires\nFocus on Fluids articles must not exceed this page length5\nthat the Courant–Friedrichs–Lewy (CFL) number remains below the threshold dictated by\nthe time integration scheme. Thus, while in general the computational cost of a DNS quickly\ngrows with𝑅𝑒, for a pipe flow with a Fourier azimuthal discretization the cost has an even\nsteeper rise because of the rapidly shrinking time step.\nTo handle this problem, short of accepting the vanishingly small time step, recent high- 𝑅𝑒\nsimulations of pipe flow (e.g. Pirozzoli et al. 2021) resort to an implicit treatment of the\nazimuthal convection terms. The alternative approach that is followed here derives from and\nextends the one introduced two decades ago by Quadrio & Luchini (2002), who developed a\nDNS solver for the incompressible Navier–Stokes equations written in velocity-vorticity form,\nand used it for the DNS of the turbulent flow in an annular pipe. Although the present code\nsolves the Navier–Stokes equations in their primitive-variables formulation with a monolithic\napproach, in both cases the truncation of the azimuthal Fourier series changes with the\nradial coordinate, in such a way that the actual azimuthal resolution remains approximately\nconstant across the pipe. This can be achieved smoothly and without interpolation, provided\nthe spanwise Fourier series is combined, as in the present case, with a collocated method for\nthe discretization of the radial direction.\nThe radially varying number of azimuthal modes implies that some modes exist at the\npipe wall at 𝑟=𝑅but do not reach the pipe axis at 𝑟=0: hence, these modes end in\nthe bulk of the flow, where a suitable boundary condition has to be provided for them.\nOn the ground that a well-designed DNS neglects, i.e. puts to zero, all the modes above\nits maximum wavenumber, in their original formulation Quadrio & Luchini (2002) used a\nsimple homogeneous boundary condition also for these modes terminating in the fluid. Here,\nwe take one step further and use as a boundary condition the same regularity conditions that\nare employed at the pipe axis. There, as shown for example by Lewis & Bellan (1990), scalar\nand vector quantities require a different treatment. Let 𝛽be the spanwise wavenumber, and\nlet the superscript ˆ·indicate the Fourier coefficient of a variable. For pressure, ˆ 𝑝(𝑟)∼𝑟|𝛽|\nas𝑟→0. For the velocity vector, the equivalent condition becomes:\n \nˆ𝑢(𝑟)∼𝑟|𝛽|\nˆ𝑣(𝑟)∼𝑟|𝛽|−1\nˆ𝑤(𝑟)∼𝑟|𝛽|−1when𝛽≠0 \nˆ𝑢(𝑟)∼𝑟0\nˆ𝑣(𝑟)∼𝑟\nˆ𝑤(𝑟)∼𝑟when𝛽=0. (2.1)\nThese regularity conditions are enforced not only for those standard modes that span the\nfull radial extent of the pipe 0 < 𝑟 < 𝑅 but also for those terminating in the bulk; this\nprovides them with a smooth decay in the radial direction. Of course, as long as the baseline\nresolution is well chosen, no differences are expected in flow statistics between the present\napproach and the one employed by Quadrio & Luchini (2002).\nThe variable-modes approach can be easily programmed in a simple and general way. In the\ncode, written in the CPL language (Luchini 2020, 2021), a two-dimensional (for generality)\narray of pointers is used to reference into a variable-sized one-dimensional array, storing all\nthe nonzero coefficients as a function of 𝑟. Although this exceeds the scope of this work,\nsuch a programming approach makes it straightforward to extend the use of variable modes:\nfor example, in the plane channel flow it could be advantageous to have the number of both\nstreamwise and spanwise modes become a function of the wall-normal coordinate.\nAs its Cartesian counterpart, the code is capable of parallel computing. It sports at the\nsame time a shared-memory and a distributed-memory algorithm where a low-level message-\npassing strategy is employed on a computational domain subdivided into wall-parallel slices\nso that the two-dimensional inverse/direct FFTs can be computed locally to each machine. A\nstandard message-passing version based on the MPI library is also available.6 E. Gallorini, M. Quadrio\n0 0.2 0.4 0.6 0.8 1-101\nx/λW/ANo filter\nFilter\n0 0.2 0.4 0.6 0.8 1-101\nx/λW/ASinusoid\ns= 3\ni=0\ni=1i=2\nFigure 2: Schematics of the spatial waveform for the control: the red line is the sinusoidal\nwave, and the blue line is its discrete approximation by three segments per period, i.e.,\n𝑠=3. The corresponding representation after the Fourier transform without (empty\nsymbols) and with (filled symbols) spatial filtering is also depicted.\n2.2. The discrete travelling waves (DTW)\nThe piecewise-continuous wall forcing shown with a blue line in figure 1 does not lend\nitself to an immediate description within a Fourier discretization. An equivalent problem\nwas faced by e.g. Ricco & Hahn (2013) and Wise & Ricco (2014), who described spatially\ndiscontinuous forcing distributions with a Fourier discretization. Mishra & Skote (2015),\nwho dealt with the same issue, reported that oscillations introduced at the discontinuities\n(the so-called Gibbs phenomenon) produce instabilities and large numerical errors, and\ntherefore need to be adequately treated.\nThe present DTW forcing, shown in figure 2, is defined by a number 𝑠of segments\ndiscretizing the continuous sinusoidal counterpart. One wavelength of the piecewise-constant\ntravelling wave is written analytically as:\n𝑤(𝑥,𝑡;𝑠)=𝐴sin\u0010\n𝜔𝑡−2𝜋𝑖\n𝑠\u0011𝑖\n𝑠𝜆⩽𝑥 <𝑖+1\n𝑠𝜆 (2.2)\nwhere𝜆=2𝜋/𝜅𝑥stands for the wavelength of the wave, and 𝑖is an integer spanning the\nrange 0 ⩽𝑖 <𝑠−1 (note that this definition is non-unique, as the phase difference between\nthe DTW and the sinusoid could be chosen differently without altering the results discussed\nin the following). The waveform is then periodically extended to the whole axial length of\nthe pipe, which always fits an integer number of wavelengths.\nTo avoid the appearance of spurious oscillations, the discontinuities in the DTW are\nregularized via a smoothing Gaussian filter. The expression of the filter in physical space is:\n𝐺(𝑥)=\u00126\n𝜋Δ2\u00131/2\nexp\u0010\n−6𝑥2\nΔ2\u0011\n(2.3)\nwithΔthe filter width; the filtered wall forcing is obtained via convolution of the original\none with the kernel 𝐺(𝑥)in physical space, and then Fourier-transformed. The filter width\nwas carefully determined after a parametric study in the preliminary work carried out by\nBiggi (2012); the value of Δis chosen as to strike a compromise between the range of\nscales/positions affected by the filter and the magnitude of the residual oscillations. The\nemployed filtering is represented in figure 2, where it can be appreciated that oscillations are\nreduced to a very small level, while the step function remains relatively sharp.7\n0 Nθ,max10\nNθrNθ,min\nr0\nFigure 3: Schematic representation of the radially-varying number of azimuthal nodes\n(only the positive ones are shown). The maximum value 𝑁𝜃,𝑚𝑎𝑥 is constant from the pipe\nwall at𝑟=1 to the radial position 𝑟=𝑟0=0.8𝑅, then decreases linearly to the value\n𝑁𝜃,𝑚𝑖𝑛 at the pipe axis. In this plot, as in the main simulations, 𝑁𝑟=100 and every other\nradial point is omitted for clarity.\n2.3. Validation\nBefore delving into the actual study, we preliminarily assess, for a canonical turbulent pipe\nflow, to what extent the radially varying number of azimuthal modes affects the solution\nand the computational efficiency of the code. For a turbulent pipe flow at a bulk Reynolds\nnumber of𝑅𝑒𝑏=𝑈𝑏𝐷/𝜈=4900, which is the value used for the rest of the study, two\nconfigurations are considered: one in which the spanwise truncation of azimuthal modes\nis simply kept constant at the value 𝑁𝜃,𝑚𝑎𝑥 , and the other in which the azimuthal modes\ndecrease with 𝑟from𝑁𝜃,𝑚𝑎𝑥 to𝑁𝜃,𝑚𝑖𝑛 . As shown schematically in figure 3, based on\nprevious experience, we have set up the azimuthal discretization such that the maximum\nresolution with 𝑁𝜃,𝑚𝑎𝑥=96 (i.e. 193 azimuthal modes, from −𝑁𝜃,𝑚𝑎𝑥 to+𝑁𝜃,𝑚𝑎𝑥 ) holds\nin a near-wall layer 𝑟0⩽𝑟⩽𝑅, with𝑟0=0.8𝑅; for larger wall distances, the number of\nazimuthal modes decreases linearly with 𝑟so that𝑁𝜃,𝑚𝑖𝑛=4 (i.e. 9 modes, from −𝑁𝜃,𝑚𝑖𝑛 to\n+𝑁𝜃,𝑚𝑖𝑛 ) at the centerline. The discretization also employs 384 streamwise modes and 100\nradial points, for a pipe with a length of 𝐿=22𝑅.\nIn figure 4 we compare the spectral energy density Φ+\n𝑢𝑢(𝛽)of the streamwise velocity\nfluctuations in the 𝑟−𝛽plane, as computed from the two simulations. As expected, the\nsimulation with the standard discretization has an energy density (black contour lines) that\npeaks near the wall; toward the axis of the pipe, it decreases to very low values already\nat a rather small azimuthal wavenumber. For the considered spanwise resolution, the near-\nwall maximum captured at the largest 𝛽is about 10−4; as the pipe axis is approached, the\nenergy levels decrease, and at the axis, the energy levels drop below 10−30for the largest\n𝛽. Once a radially varying number of azimuthal modes is employed (colored contour), this\nwaste is avoided, and densities below 10−7are not computed in the core region of the pipe.\nImportantly, the white dashed contour lines of the variable-modes simulation demonstrate\nthat the two datasets overlap perfectly in the region of the 𝑟−𝛽plane where they are both\ndefined.\nThe two cases produce a virtually identical value of the friction coefficient, namely 𝐶𝑓=\n9.52·10−3, which is very near to the value of 9 .45·10−3predicted by the Blasius power-law8 E. Gallorini, M. Quadrio\n10 20 30 40 50 60 70 80 9010.80.60.40.20\nβrΦ+uu\n100\n10−2\n10−4\n10−6\n10−8\nFigure 4: Spectral energy density Φ+𝑢𝑢of the axial velocity fluctuations in the 𝑟−𝛽plane.\nBlack contour lines are from the simulation with constant azimuthal resolution, and the\ncolormap with white dashed contours is from the the simulation where the number of\nazimuthal modes varies with 𝑟. Contour levels start from 100, spaced by one order of\nmagnitude.\n𝐶𝑓=0.0791𝑅𝑒−1/4\n𝑏(Schlichting & Gersten 2000): a more than satisfactory agreement, given\nthe known difficulties for such correlations at low values of the Reynolds number.\nAn additional validation step is carried out in the presence of the spanwise forcing. The\navailability of the recent work by Liu et al. (2022), who numerically tested the drag reduction\ncapabilities of standing waves, allows us to repeat a representative set of their simulations for\ncomparison. As in that work, a fixed 𝑅𝑒𝜏=180 is enforced, and the pipe length is 𝐿=6𝜋𝑅.\nWe keep the same spatial resolution employed above since it is nearly identical to the one\nemployed in their study. We replicate one of their cases at 𝐴+=12 and𝜆+=424 with 40.4%\ndrag reduction, obtaining 40.6% drag reduction. Another case with 𝐴+=6 and𝜆+=1695 is\nreported to yield 28.1% drag reduction; here it yields 28.5%. Finally, the case with 𝐴+=30\nand𝜆+=1695 is confirmed to lead to a complete relaminarization of the flow.\nIn terms of computational efficiency, the code has been tested on an Intel Cascade Lake\n8260 processor. The single-core solution of one Runge–Kutta time step (i.e. the sum of the\nthree substeps) requires ≃20 seconds when the full azimuthal nodes are retained, and ≃15\nseconds with the variable modes set up as described above, with a 25% savings in computing\ntime. However, as previously discussed, the true advantage of employing a radially varying\nnumber of azimuthal modes lies in the larger time step size allowed by the stability condition.\nIn these tests, the same value of (unitary) CFL corresponds in the first case to a time step\nofΔ𝑡+≃0.008, which becomes in the second Δ𝑡+≃0.08, demonstrating one order of\nmagnitude larger time steps when the azimuthal modes are truncated. Finally, the usage\nof variable modes is beneficial also for the memory occupation of the code, which in this\nconfiguration (i.e. with a Runge–Kutta method that stores the solution at one previous time\nlevel) amounts to 331 MB of RAM instead of 434 MB when 𝑁𝜃is kept constant at 𝑁𝜃,𝑚𝑎𝑥\nacross the pipe.\n2.4. Computational procedures\nThe numerical study described in the remainder of the paper is carried out at the same\nReynolds number of the ABBCQ experiment, namely 𝑅𝑒𝑏≡𝑈𝑏𝐷/𝜈=4900, where 𝑈𝑏9\nis the bulk velocity, and 𝐷=2𝑅is the pipe diameter. Every simulation is performed by\nadjusting at every time step the axial homogeneous pressure gradient in such a way that the\nflow rate does not vary in time, i.e. by following the Constant Flow Rate (CFR) strategy, as\ndefined by Quadrio et al. (2016). An uncontrolled simulation serves as the reference case and\nestablishes the corresponding nominal values for the drag and the friction-based Reynolds\nnumber,𝑅𝑒𝜏≈170.\nThe length of the computational domain is 22 𝑅; 384×192 Fourier modes are used to\ndiscretize the streamwise and azimuthal directions respectively, while 𝑁𝑟=100 nodes\ndiscretize the radial direction. Once the additional modes used to remove the aliasing error\nare accounted for, the spatial resolution of the reference case is Δ𝑥+=4.8 and𝑟Δ𝜃+=2.8 at\nthe pipe wall. The radial resolution varies from Δ𝑟+=0.7 near the wall to Δ𝑟+=2.4 at the\ncenterline. The radially-varying modes are set up as described above in §2.3, with a linear\nvariation from 𝑁𝜃,𝑚𝑖𝑛=4 to𝑁𝜃,𝑚𝑎𝑥=96, and𝑟0=0.8𝑅. The time step is dynamically\nadjusted to satisfy the constraint CFL=1.\nThe reference experimental dataset by ABBCQ, already shown in figure 1, consists of data\ntaken at various oscillation frequencies, ranging from 𝜔+=−0.25 to𝜔+=0.2 (note that\nquantities indicated with the + superscript are defined in terms of the friction velocity of the\nuncontrolled flow). The forcing amplitude is also fixed at 𝐴/𝑈𝑏=1 (or𝐴+≈14). Owing\nto the fixed size of the device producing the DTW, the wavelength could not be changed\ncontinuously in the experiment. For the DTW produced with three moving segments, 𝑠=3, it\nwas𝜆/𝑅=4.38 (equivalently, the streamwise wavenumber was 𝜅𝑥𝑅=1.43 or𝜅+\n𝑥=0.0082).\nThe present numerical experiments include three sets of simulations. In one, labeled SIN,\nthe idealized sinusoidal boundary condition (1.1) is applied, while the second and third sets\nconsider DTW, realized with a relatively fine ( 𝑠=6, case S6) and coarse ( 𝑠=3, case\nS3) discretization of the waveform, according to Eq.(2.2). In particular, case S3 has the\nclosest correspondence to the experiment. The wavelength and amplitude of the forcing are\nnominally identical to those of the experiments, whereas the oscillation frequency is varied\nwith a relatively fine step, and covers a slightly larger range.\nTwenty-nine simulations are performed for each forcing type, for a total of 88 cases\nincluding the reference one. Understanding the harmonic content of the forcing led us to run\n45 additional simulations where only some harmonics of the DTW are included. Each case\nis run for a total of 1000 convective time units; the first half of the time history is discarded to\navoid the influence of the initial transient, which can be particularly long at times, especially\nfor the cases with the highest drag reduction. In a few selected cases, a database is saved\nfor later statistical analysis; for these, 50 oscillation cycles are sampled, and four phases are\nstored for each cycle. Most of the simulations are run serially (i.e. on one core each) on a\nsingle machine equipped with an Intel Xeon Phi processor and 68 cores, which was kept\nbusy for approximately two months.\n3. Results\n3.1. Drag reduction\nWe start by looking first at the raw changes in skin-friction drag. The drag reduction rate is:\nR=1−𝐶𝑓\n𝐶𝑓 ,0=1−𝜏𝑤\n𝜏𝑤,0(3.1)10 E. Gallorini, M. Quadrio\n-0.3 -0.2 -0.1 0 0.3 0.2 0.1-0.100.10.20.30.40.50.6Laminar\nω+RS6\nS3\nSIN\nEXP\nFigure 5: Drag reduction rate Rcomputed with DNS against experimental measurements\nfrom ABBCQ (blue triangles, blue line). The plot shows the laminar limit (upper dashed\nline), and quantifies with bars the error deriving from the finite averaging time (see text).\nwhere the subscript 0 indicates quantities evaluated for the uncontrolled flow, 𝐶𝑓is the\nfriction coefficient, defined in the usual way as:\n𝐶𝑓=2𝜏𝑤,𝑥\n𝜌𝑈2\n𝑏,\nand the last equality in Eq.(3.1) only holds for CFR simulations. Note that, in cylindrical\ncoordinates, the longitudinal and azimuthal components of the mean wall shear stress are:\n𝜏𝑤,𝑥=−𝜇𝜕𝑢\n𝜕𝑟\f\f\f\n𝑟=𝑅, 𝜏 𝑤,𝜃=−𝜇\u0012𝜕𝑤\n𝜕𝑟−𝑤\n𝑅\u0013\f\f\f\n𝑟=𝑅.\nwhere the overbar indicates temporal average.\nFigure 5 compares the output of our simulations with the ABBCQ measurements for 𝑠=3.\nThe first striking observation is that the experimental data, labelled as EXP, are quite far\nfrom the results obtained with the ideal sinusoidal forcing, case SIN. The qualitative look\nof the curves is similar. The sudden shift from drag reduction to drag increase for waves\ntravelling at a phase speed comparable to the convection speed of the near-wall turbulent\nstructures known to take place in the planar caase (Quadrio et al. 2009) is confirmed. However,\nsignificant quantitative differences do exist. The maximum drag reduction obtained with SIN\nis about twice the experimental one, and peaks at 66%, which at this 𝑅𝑒corresponds to full\nrelaminarization of the flow. Moreover, the SIN data do not present the evident wiggles of the\nexperimentalR=R(𝜔+)curve. The error bars plotted in figure 5 refer to the finite averaging\ntime, and are computed according to the procedure introduced by Russo & Luchini (2017).\nThey are generally small, and confirm the deterministic nature of the wiggles, which are not\nan artifact of the measurement procedure. An evident exception are the points at 𝜔+=0.04,\nwhere the flow is on the verge of relaminarisation, and alternately visits a turbulent state and\nRapids articles must not exceed this page length11\na nearly laminar state, switching between them over a long time scale. This observation also\nexplains the apparent disagreement of the curves SIN, S3 and S6 at this very specific control\npoint.\nSuch differences between the experimental data and the expected drag reduction from\nplane channel DNS were already noticed in the original paper by Auteri et al. (2010), in\na comparison with plane-channel information at 𝑅𝑒𝜏=200, but their identification and\ninterpretation are now easier. The one-to-one comparison made possible by the present data\nrules out several alternate explanations for these discrepancies, as for example the difference\nbetween circular and planar geometries (which, in fact, acts to actually reduce differences),\nor the slightly different parameters of the forcing.\nCase S3 (green curve) is the one that should most closely correspond to EXP (blue curve);\nindeed, experiments and simulations, while not overlapping, do show much better agreement.\nBoth datasets present, especially at negative frequencies, the large wiggles that are missing\nin the SIN curve. Notably, at certain frequencies, the discrete wave realized by S3 achieves\na distinctly largerRthan SIN, a feature that was not previously observed. Relaminarization\nis only partial for S3, with a maximum Rof 57%, obtained at a frequency slightly larger\nthan the optimum for SIN; as for EXP, the region of drag increase at 𝜔+≈0.1 is correctly\nidentified, butRremains positive and no drag increase is measured.\nCase S6 implements the same travelling wave tested in experiments, but yields a better\napproximation of the sinusoidal waveform. Drag reduction data from S6 resemble very\nmuch those from SIN, and full laminarization is achieved at the smallest positive frequency,\nindirectly confirming the key role of control discretization. Still, a slightly diminished drag\nincrease and the presence of wiggles (albeit of smaller amplitude) indicate that discretization\neffects remain at work even in the S6 case. Reasons explaining the observed discrepancies\namong these data sets will be discussed later in §4.\n3.2. Power budget\nWith active control, it is important to complement the information regarding drag reduction\nwith the cost of the input power, conveyed by the power ratio P𝑖𝑛between the power\nrequired to create the control action and the power 𝑃0per unit wall area required to drive the\nuncontrolled pipe flow. Hence, if saving energy is the ultimate interest, more than the drag\nreduction rateRitself, the most informative quantity is the net power saving S, defined as:\nS=R−P 𝑖𝑛. (3.2)\nIn the present application, the control acts in the azimuthal direction only, and the input\npower that an ideal control system transfers to the viscous fluid, normalized with the pumping\npower𝑃0, is:\nP𝑖𝑛=1\n𝑃01\n2𝜋𝑅𝐿𝑇∫𝑇\n0∫2𝜋\n0∫𝐿\n0−𝜇𝑤\u0012𝜕𝑤\n𝜕𝑟−𝑤\n𝑅\u0013\f\f\f\n𝑟=𝑅𝑅𝑑𝑥𝑑𝜃𝑑𝑡. (3.3)\nNote that the term 𝑤/𝑅in the integrand of Eq.(3.3) is sometimes incorrectly omitted in\nexisting studies, but its present in the correct expression of the wall shear stress in cylindrical\ncoordinates. Figure 6 (top) plots how P𝑖𝑛changes with the control frequency, in comparative\nform between SIN and cases S3, and S6. The outcome is not obvious. The input power\nvaries significantly with the control parameters; as in the planar case, it is minimum in\nthe region (small positive frequencies) where the drag reduction is maximum. Of primary\ninterest here, however, are the relative differences between the sinusoidal forcing and DTW,\nhighlighted in the right panel. It is seen that at not-too-small negative frequencies (say\n𝜔+<−0.02) and large positive ones ( 𝜔+>0.14), DTW are less expensive than their12 E. Gallorini, M. Quadrio\n-0.2 -0.1 0 0.2 0.13.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\nω+Pin\nS3\nS6\nSIN\n-0.2 -0.1 0 0.2 0.10.811.21.41.61.82\nω+Pin,DTW /Pin,SIN\n-0.2 -0.1 0 0.2 0.10.5\n0\n-0.5\n-1.0\n-1.5\n-2.0\nω+SS3\nS6\nSIN\n-0.2 -0.1 0 0.2 0.1-0.5-0.4-0.3-0.2-0.100.10.20.3\nω+SDTW−SSIN\nFigure 6: Input power ratio P𝑖𝑛(top) and net savings S(bottom). The right panels\nemphasize variations from the SIN case.\nsinusoidal counterpart, with larger differences, of the order of 10%, observed for S3 and\nbackward-travelling waves. However, the opposite scenario is observed at small positive\nfrequencies, i.e. in the most interesting region where the control is supposed to work, owing\nto the lower absolute energetic costs. For example, at 𝜔+=0.02 the powerP𝑖𝑛required by\nS3 is twice that for SIN.\nThe combined dependence of both RandP𝑖𝑛on the control parameters determines the\nchanges to the net savings S, plotted in figure 6 (bottom). The large differences in terms of\nrelaminarization (or lack thereof) observed at the smallest frequencies for the various types\nof forcing blurs the picture further; moreover, the ratio of Sobtained with DTW and SIN\nbecomes a delicate indicator whenever Sapproaches zero. The most expensive S3 forcing\nshows significant extra savings compared to SIN; however, this happens for frequencies\nwhereSis large and negative, which makes the improved performance pointless. Wherever\nSis positive (or, as in this case at a relatively large forcing amplitude, wherever it approaches\nzero), S3 presents a significant efficiency gap compared to SIN. At any rate, in correspondence\nof the optimum parameters, SIN remains the best forcing type, in terms of both P𝑖𝑛andS.13\n100101102051015\n102025\ny∗u∗ω+=−0.08\nRef\nSIN\nS3\n100101102051015\n102025\ny∗ω+= 0.11\nRef\nSIN\nS3\nFigure 7: Mean streamwise velocity profile for two cases with drag reduction\n(𝜔+=−0.08, left) and drag increase or minimal drag reduction ( 𝜔+=0.11, right). Cases\nSIN and S3 are compared to the uncontrolled flow.\n3.3. Flow statistics\nTurbulence statistics are inspected here for the sole purpose of verifying whether or not\nthe discrete form of the forcing alters the flow significantly, besides the already quantified\ndifferent level ofR. To this aim, we focus on SIN and S3 and pick two cases: one is at\n𝜔+=−0.08 and consistently yields a large positive R, whereas the second is at 𝜔+=0.11\nand yields drag increase — more precisely, a negative Rfor SIN and a very small positive\nvalue ofRfor S3. In figure 7, the mean streamwise velocity profile 𝑢is plotted against the\nwall distance 𝑦=1−𝑟in the law-of-the-wall form by using the actual friction velocity\nas a reference velocity. This is the so-called true viscous scaling (Quadrio 2011), indicated\nwith an asterisk superscript. It can be appreciated that the profiles of 𝑢∗differ essentially\nonly because of the different value of R, which translates into a different vertical shift of the\nlogarithmic portion of the profile. Indeed, owing to the scaling employed, all profiles collapse\nin the viscous sublayer; the vertical shift Δ𝐵∗appears in the logarithmic region (which is not\nparticularly wide, since 𝑅𝑒is low and further lowered by drag reduction). As a proxy for the\nshift, we consider the value of 𝑢∗at𝑦∗=100, and obtain a Δ𝑢∗of 5.1 for SIN and 6.9 for\nS3 when𝜔+=−0.08, in agreement with the larger drag reduction of the latter case. For the\npositive frequency, instead, Δ𝑢∗is−0.8 for SIN and+0.9 for S3, once again in agreement\nwith the small drag increase of the former and the small drag reduction of the latter. Besides\nthe different levels of Rand the consequent different vertical shifts, no other difference can\nbe appreciated in the profiles.\nA similar picture emerges by looking at second-order statistics, i.e. for example the variance\nof velocity fluctuations. In the present case, the fluctuations should be defined by accounting\nnot only for the mean flow but also for the control-induced coherent velocity field with zero\naverage. To this purpose, we use a classic triple decomposition, where a generic quantity 𝑎\nis decomposed as 𝑎=𝑎+˜𝑎+𝑎′′, i.e. into its mean, coherent, and stochastic components.\nThe sum of the coherent and stochastic components is indicated as 𝑎′=˜𝑎+𝑎′′. The mean\ncomponent is obtained by averaging each quantity in time and along the azimuthal and\nstreamwise directions; the coherent part, instead, derives from averaging together points at\nthe same phase 𝜙=𝜅𝑥𝑥−𝜔𝑡after removal of the mean value. The difference between the14 E. Gallorini, M. Quadrio\n100101102-1012345678\ny∗ω+=−0.08\nRef\nSIN\nS3\nu′′u′′∗\nv′′v′′∗w′′w′′∗\nu′′v′′∗\n100101102-1012345678\ny∗ω+= 0.11\nRef\nSIN\nS3u′′u′′∗\nv′′v′′∗w′′w′′∗\nu′′v′′∗\nFigure 8: Wall-normal profiles of the stochastic turbulent stresses. Colors as in figure 7.\ninstantaneous field and the corresponding mean and coherent parts defines the stochastic\nfield.\nIn the present work, we do not consider the coherent components, since their magnitude\nis negligible compared to the stochastic ones. An exception is obviously the Stokes layer\ncontributions ˜ 𝑤andg𝑤𝑤, and a small streamwise modulation of the ˜ 𝑢component, which\nbecomes apparent in its wall derivative. Figure 8 depicts the wall-normal profiles of the\n(stochastic) Reynolds stresses normalized with the actual friction velocity, i.e. 𝑢′′𝑢′′∗,\n𝑣′′𝑣′′∗,𝑤′′𝑤′′∗, and𝑢′′𝑣′′∗. The streamwise fluctuations always decrease with control, either\ncontinuous or discrete, for both drag reduction and drag increase. This can be attributed to the\nstrengthened redistribution action of the pressure-strain term, which moves energy towards\nthe spanwise and wall-normal fluctuations. The redistribution is enhanced by the tilting of\nthe structures (Yakeno et al. 2014; Gallorini et al. 2022). For example, the maximum tilt\nangle, defined as in Yakeno et al. (2014), here is 14◦for𝜔+=−0.08, and becomes 34◦for\n𝜔+=0.11 with SIN and≈27◦at the same frequency with S3.\nFor the negative drag-reducing frequency 𝜔+=−0.08, the reduction in intensity is\naccompanied by a slight shift of the near-wall peak further from the wall. The shift is\nlarger, and the decrease of the peak value is much larger, for S3 than for SIN. Since the\ndifferentRis already accounted for by the * scaling, the extra reduction of the peak value\nprovided by S3 is attributed to the highly subcritical turbulent state (described later in §4.2)\nreached by the turbulent flow in this case. The positive drag-increasing frequency 𝜔+=0.11\nhas the location of the wall-normal peak approximately unchanged, but the profiles after the\npeak show a linear (in semi-logarithmic scale) region that is absent in the reference profile\n(at these𝑅𝑒). Such linear regions also appear in the profile for 𝑤′′𝑤′′∗, and resemble those\nobserved by Lee & Moser (2015) in canonical channel flows, but at much higher Reynolds\nnumbers (𝑅𝑒𝜏>550). It is worth mentioning that the shift of 𝑢′′𝑢′′∗in the wall-normal\ndirection qualitatively agrees with the observation of Gallorini et al. (2022), in which the\nauthors pointed out a movement away from the wall of the quasi-streamwise vortices related\ntoRin case of drag reduction and minor modifications in case of drag increase. The spanwise\nand wall-normal diagonal components of the Reynolds stress tensor show minor changes for\nthe negative frequency, but very large increases for the positive frequency, suggesting lesser\nchanges in the drag-reduced turbulent flow, and minor effects of the discrete forcing as long\nas it works to reduce drag, whereas drag increase implies important modifications also far15\n0 π 2π050100y∗SIN\n0 π 2π050100S3\n01234567/angbracketleftu/prime/primeu/prime/prime/angbracketright∗\nθ,t\n0 π 2π050100150SIN\n0 π 2π050100150S3\n01234567/angbracketleftu/prime/primeu/prime/prime/angbracketright∗\nθ,t\n0 π 2π050100y∗\n0 π 2π050100\n00.20.40.6/angbracketleftv/prime/primev/prime/prime/angbracketright∗\nθ,t\n0 π 2π050100150\n0 π 2π050100150\n00.20.40.60.81.01.21.4/angbracketleftv/prime/primev/prime/prime/angbracketright∗\nθ,t\n0 π 2π050100y∗\n0 π 2π050100\n00.51/angbracketleftw/prime/primew/prime/prime/angbracketright∗\nθ,t\n0 π 2π050100150\n0 π 2π050100150\n01234/angbracketleftw/prime/primew/prime/prime/angbracketright∗\nθ,t\n0 π 2π050100\nφy∗\n0 π 2π050100\nφ0.6\n0.4\n0.2\n0-/angbracketleftu/prime/primev/prime/prime/angbracketright∗\nθ,t\n0 π 2π050100150\nφ\n0 π 2π050100150\nφ0.8\n0.6\n0.4\n0.2\n0-/angbracketleftu/prime/primev/prime/prime/angbracketright∗\nθ,tω+=−0.08 ω+= 0.11\nFigure 9: Components of the stochastic Reynolds stress tensor as a function of the radial\ncoordinate𝑟and the phase 𝜙, for two cases with drag reduction ( 𝜔+=−0.08, left) and\ndrag increase or minimal drag reduction ( 𝜔+=0.11, right).\nfrom the wall. The off-diagonal component 𝑢′′𝑣′′∗presents changes that are expected from\nthe FIK identity Fukagata et al. (2002) (although contributions from additional terms are\npresent due to streamwise inhomogeneity) and decreases whenever drag reduction is present.\nWith a streamwise-varying forcing, the streamwise direction is not homogeneous anymore,\nand flow statistics may vary also with the phase 𝜙=𝜅𝑥𝑥−𝜔𝑡of the forcing. Figure 9 plots\nthe stochastic Reynolds stresses as a function of 𝑦and𝜙, after averaging in time and over the\nazimuthal direction. For the cases with drag reduction, as originally observed by Quadrio\net al. (2009), no streamwise modulation of the statistics is observed, if not for some residual\nstatistical noise: the flow does not get directly altered along the forcing wavelength, and its\nmain change is the reduced level of wall friction. The cases with drag increase, instead, do\nshow a strong modulation over the forcing wavelength, which extends quite far from the wall.\nAs in the planar case, the periodic modulation possesses a structure in the radial direction that\nis influenced by the specific spatial shape of the coherent generalized Stokes layer (Quadrio\n& Ricco 2011). The modulation is interpreted as a sort of resonance between the convection\nspeed of the near-wall turbulence structures and the phase speed 𝜔/𝜅𝑥of the travelling wave.16 E. Gallorini, M. Quadrio\n0 1−101\nx/λt=0\n4T\n0 1\nx/λt=1\n4T\n0 1\nx/λt=2\n4T\n0 1\nx/λt=3\n4T\nFigure 10: SIN (red line) and S3 (blue line) wall forcing along one wavelength, at four\ndifferent times during the period 𝑇. At each time, the dots mark the position at which SIN\nis maximum.\nIt is worth noting that such modulation is visible for S3 too, although Ris slightly positive.\nThe coexistence of (small) drag reduction and a streamwise-modulated flow can be attributed\nto the complex interactions taking place with DTW: as shown later in §4.1, the superposition\nof different harmonics is such that, in this case, features of drag-reducing and drag-increasing\nflows are observed simultaneously.\n4. Discussion\nReplicating the ABBCQ experiment by DNS has confirmed that some features of the\nexperimental data derive from the different nature of continuous and discrete forcing. In\nthis Section, we discuss the reason for the two major differences: the wiggles in the curve\nR=R(𝜔+)(which are always present for discrete forcing, regardless of the nature of the\nmeasurement), and the larger maximum drag reduction observed in the simulations.\n4.1. The role of different harmonics\nAlthough in both the continuous and discrete case the spanwise wall velocity 𝑊varies in\ntime between±𝐴, a meaningful comparison between DTW and SIN at the same amplitude\nis not obvious.\nIt is easy to show that the staircase function sketched in figure 1 and expressed analytically\nby Eq.(2.2) has a lower amplitude compared to the sinusoidal forcing once averaged over\nthe forcing period. Figure 10 plots SIN and S3 at four different instants during one period.\nAs time progresses, the sinusoidal wave is simply shifted in space, whereas DTW sees its\nwaveform modified during the period: by focusing on a specific phase 𝜙=𝜅𝑥𝑥−𝜔𝑡(for\nexample, as in the figure, where SIN has its maximum), it can be seen that DTW assumes\ndifferent values at different times. For the generic phase 𝜙, the average intensity of DTW can\nbe quantified analytically. By defining Δ𝑠=2𝜋/𝑠as the width of the each constant piece of\nthe DTW,𝑊(𝜙)can assume any value from 𝐴sin(𝜙−Δ𝑠/2)to𝐴sin(𝜙+Δ𝑠/2). Taking the\naverage of the sinusoidal function over the interval Δ𝑠, one obtains:\n𝑊(𝜙)=1\nΔ𝑠∫𝜙+Δ𝑠/2\n𝜙−Δ𝑠/2𝐴sin(𝜙′)𝑑𝜙′=−𝐴\nΔ𝑠h\ncos(𝜙+Δ𝑠\n2)−cos(𝜙−Δ𝑠\n2)i\n. (4.1)\nThe previous equation can be rearranged, with the aid of the prosthaphaeresis formulae,17\n0.20.40.60.81\n−11\nx/λW/AS3\n0.20.40.60.81\n−11\nx/λS6\n1SD\n1OD\n2SD\n2OD\nFigure 11: Sketch of the first two pairs ( 𝑚=0 and𝑚=1) of harmonics for S3 (left) and\nS6 (right). Each pair is made by two waves: one travelling in the same direction (1SD,\n2SD) and the other in the opposite direction (1OD, 2OD) of the nominal DTW.\ninto the following expression:\n𝑊(𝜙)=2𝐴\nΔ𝑠sin\u0012Δ𝑠\n2\u0013\nsin(𝜙)=𝑠\n𝜋sin\u0010𝜋\n𝑠\u0011\n|     {z     }\n𝛾(𝑠)𝐴sin(𝜙). (4.2)\nHence, the averaged DTW can be written as 𝛾(𝑠)𝐴sin(𝜙), i.e. proportional to 𝐴sin(𝜙),\nbut its effective amplitude is equal to 𝐴only for the ideal sinusoidal forcing described by\nEq.(1.1), for which 𝑠→∞ and lim 𝑠→∞𝛾(𝑠)=1. Compared to the nominal amplitude\n𝐴, the amplitude of DTW is decreased by a factor 𝛾(𝑠), which depends on the number of\nslabs discretizing the sinusoid. For the S3 case, 𝛾(3)=0.83. This is consistent with the S3\nforcing not reaching relaminarization, as shown in figure 5, because of its smaller effective\namplitude. In fact, we have verified with an additional simulation, run with sinusoidal forcing\nfor𝜔+=0.02 and a forcing intensity reduced by a factor 0.83, that the flow does not reach\nthe fully laminar state, and yields R=0.43.\nHowever, this is only part of the whole picture. Once the importance of discretization is\nrecognized, it is necessary to proceed further to properly describe DTW: in fact, the reduced\namplitude discussed above cannot explain, for example, other results reported in figure 5,\nwhere in the range −0.12⩽𝜔+⩽−0.06 drag reduction due to the discrete S3 forcing is\nlarger than that achieved by SIN, while the reduced effective amplitude discussed above\nwould suggest otherwise.\nThe DTW described by Eq.(2.2) can be expanded into an infinite Fourier series. Following\nABBCQ, a DTW characterized by its three parameters 𝐴,𝜅𝑥and𝜔, and discretized into 𝑠\nsegments, is rewritten as the following sum:\n𝑊(𝑥,𝑡;𝑠)=𝐴∞∑︁\n𝑚=0sin[(𝑚𝑠+1)𝜋/𝑠]\n(𝑚𝑠+1)𝜋/𝑠sin[𝜔𝑡−𝜅𝑥(𝑚𝑠+1)𝑥]+\n+sin[((𝑚+1)𝑠−1)𝜋/𝑠]\n((𝑚+1)𝑠−1)𝜋/𝑠sin[𝜔𝑡+𝜅𝑥((𝑚+1)𝑠−1)𝑥].(4.3)\nThe DTW written as in (4.3) is made by two families of sinusoidal waves: one family has\na phase speed of the same sign of the nominal wave (and thus travels in the same direction,\nSD), while the other has the opposite sign and travels in the opposite direction (OD). The\ntemporal frequency of the waves is unchanged (in absolute value) and equals the nominal\none, but the effective amplitude (always ⩽𝐴) and streamwise wavenumber (always ⩾𝜅𝑥) of\neach wave depend on the number of slabs 𝑠and on the index 𝑚of the series. The amplitude\nof the harmonics decreases with 𝑚, and the wavenumber correspondingly increases. The\naveraged DTW derived above in Eq.(4.2) is recognized as the first SD wave with 𝑚=0.\nFigure 11 plots as an example the first ( 𝑚=0) pair (1SD and 1OD) and the second ( 𝑚=1)\npair (2SD and 2OD), for a visual appreciation of the relative amplitude and wavelength.18 E. Gallorini, M. Quadrio\nCase SIN S3 1SD 1OD 1p 1p+2p S6 1SD 1OD 1p 1p+2p\n𝜔+=−0.2 0.231 0.143 0.203−0.203 0.075 0.115 0.254 0.221 0.188 0.253 0.263\n𝜔+=−0.08 0.324 0.381 0.305 0.319 0.364 0.375 0.311 0.312 0.118 0.312 0.318\n𝜔+=0.11−0.097 0.041−0.094 0.146−0.022 0.029−0.060−0.094 0.033−0.081−0.066\nTable 1:Rfor different harmonics of the Fourier series (4.3), cases S3 and S6. Columns\nmarked with 1SD (1OD) correspond to simulations where only the first same-direction\n(opposite-direction) wave is present, whereas 1p means the first harmonic pair combined,\nand 1p+2p the first two harmonic pairs combined.\nCase SIN S3 1SD 1OD 1p 1p+2p S6 1SD 1OD 1p 1p+2p\n𝜔+=−0.22.222.09 1.52 0.33 1.86 2.00 2.14 2.02 0.05 2.07 2.13\n𝜔+=−0.081.461.29 1.00 0.12 1.12 1.21 1.42 1.33 0.04 1.37 1.41\n𝜔+=0.11 1.421.46 0.97 0.31 1.29 1.38 1.43 1.30 0.08 1.38 1.42\nTable 2:P𝑖𝑛for different harmonics of the Fourier series (4.3), cases S3 and S6. Columns\nas in Table 1.\nIt remains to be established to what extent the (linear) superposition of the DTW harmonics\ncan be useful in understanding the pattern of drag reduction and the differences between SIN\nand S3. To this aim, additional simulations (for cases S3 and S6) are run at three frequencies:\n𝜔+=0.11 (drag increase), 𝜔+=−0.08 (drag reduction, where S3 performs better than SIN\nand S6 equals SIN), and 𝜔+=−0.2 (drag reduction, where S3 performs worse than SIN,\nand S6 performs better). Instead of the actual DTW given by Eq.2.2, the employed forcing\nis sinusoidal, and contains one or more of the harmonics of the series 4.3. In particular,\nwe consider the first two pairs of harmonics, isolated or in combination, for a total of eight\nadditional simulations for each case.\nHence, in these numerical experiments, the wall forcing is given by various combinations\nof the following sinusoids:\n𝑊(𝑥,𝑡; 3)=3√\n3\n2𝜋𝐴h\nsin(𝜔𝑡−𝜅𝑥𝑥)+1\n2sin(𝜔𝑡+2𝜅𝑥𝑥)−1\n4sin(𝜔𝑡−4𝜅𝑥𝑥)−1\n5sin(𝜔𝑡+5𝜅𝑥𝑥)i\n(4.4)\nfor case S3, and\n𝑊(𝑥,𝑡; 6)=3\n𝜋𝐴h\nsin(𝜔𝑡−𝜅𝑥𝑥)+1\n5sin(𝜔𝑡+5𝜅𝑥𝑥)−1\n7sin(𝜔𝑡−7𝜅𝑥𝑥)−1\n11sin(𝜔𝑡+11𝜅𝑥𝑥)i\n(4.5)\nfor case S6.\nResults from these experiments are shown in tables 1 and 2, in terms of RandP𝑖𝑛\nrespectively. By focusing first on R, the first and relatively trivial observation is that the\nseries (4.3) converges rather quickly in terms of drag reduction: for example at 𝜔+=−0.2,\nthe first pair of harmonics of S3 yields a drag reduction of 7.5%, adding the second pair\nyields 11.5% and the actual DTW at 𝑚→∞ yields 14.3%. A less obvious observation is\nthe very different role played by the two harmonics at a given 𝑚, and their highly nonlinear\ncombination. As an example, for the drag-reducing case at 𝜔+=−0.2, the wave 1SD alone\nyields a significant 20.3% drag reduction, whereas 1OD produces precisely the same amount\nof drag increase. However, the non-linear interaction between the two is such that, when19\nA/W = 0.83\nR=−0.09A/W = 0.41\nR= 0.15A/W = 0.21\nR= 0.18A/W = 0.17\nR= 0.02\nA/W = 0.83\nR= 0.31A/W = 0.41\nR= 0.32A/W = 0.21\nR= 0.05A/W = 0.17\nR= 0.09\n−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.800.010.020.030.04\nω+κ+x\n−0.2−0.10.00.10.20.30.4\n1SD1OD2SD2OD\nR\nFigure 12: Position on the drag reduction map of the first two harmonic pairs of the DTW\nwith𝑠=3, for a drag-reducing case (white dots: 𝜔+=−0.08,𝜅+𝑥=0.0082 andR=0.38)\nand a nearly drag-increasing one (black dots: 𝜔+=0.11,𝜅+𝑥=0.0082 andR=0.04).\nNote how each pair of corresponding harmonics of the same case (e.g. white dots for\n𝜔+=0.08) have the same frequency with opposite sign. The drag reduction map is\nadapted from Gatti & Quadrio (2016), for a plane channel flow at 𝑅𝑒𝜏=200 and\n𝐴+=14.2. The legend below each dot quantifies the amplitude 𝐴/𝑊of the harmonic, and\nthe drag reduction achieved by that harmonic alone.\nthe two harmonics are at play together, drag reduction prevails over drag increase with an\noutcome ofR=0.075.\nConsidering just the first pair of harmonics explains some of the features observed in figure\n5. The amplitude of the 1SD component is closer to 𝐴for S6 than S3, and the amplitude\nof 1OD decreases more rapidly for S6 than S3; the opposite happens for the wavenumber.\nBy noting that, apart from the amplitude and a phase-shift, the 1SD wave for S6 correspond\nto the 2OD wave for S3, one understands why the S3 data are further away from SIN\nthan S6 data, show stronger wiggles, and never achieve relaminarization. For the two cases\ncorresponding to the second and third row of table 1 (i.e. drag reduction and drag increase),\nthe position of the first S3 harmonics in the plane of the control parameters is shown in\nfigure 12. The drag reduction map is recomputed from the information provided by Gatti\n& Quadrio (2016), hence at 𝑅𝑒𝜏=200 and for the planar geometry, and is adapted to\nthe forcing intensity of the present case to convey a qualitative information of the behavior\nexpected for the harmonic. Results at 𝜔+=0.11 are interesting to understand the effects\nof the drag-increasing components. The dominant wave travels at the same phase speed as\nthe nominal wave and produces the same drag-increasing effect, quantitatively not too far\nfrom the one by SIN. However, as clearly shown by figure 12, higher harmonics possess a\ndifferent phase speed and a different wavenumber: they are all located in drag-reducing areas\nof the𝜔+−𝜅+\n𝑥plane. Hence, they act to weaken the drag-increasing effect of the nominal\nwave, resulting in a small drag reduction. The different 1OD wave between cases S3 and S6\nexplains why S6 instead achieves drag increase.\nThe case at𝜔+=−0.08 demonstrates an interesting interplay between the waves of the\nfirst pair. They lie both in the drag-reducing regime, but their combination results in an\nenhancement ofRfor S3, while for S6 the drag variation is essentially unchanged. Once\nagain, this happens because 1OD sits in a more effective region of the parameter space (in this\ncase, nearly at the optimal position). Lastly, at 𝜔+=−0.2, the 1OD waves have an opposite20 E. Gallorini, M. Quadrio\n-0.1 0 0.1-0.100.10.20.30.40.50.6\nω+R1SD\n1OD\n1p\nS3\nFigure 13: Drag reduction rate Rcomputed with DNS for the first pair of harmonics (1SD\nand 1OD) and their combination (indicated with 1p) compared against S3.\neffect: drag increase for S3, and drag reduction for S6. This affects the combined wave and\nreducesRfor S3, but increases it for S6.\nTable 2 reportsP𝑖𝑛for the same cases considered in table 1. Again, the variation of the\nDTW compared to SIN can be ascribed to the different harmonic components, which differ\nin their amplitude and their position in the 𝜔+−𝜅+\n𝑥plane. In terms of power, however, the\nvarious harmonics are almost perfectly additive, in sharp contrast with drag reduction data.\nThis is reasonable, as the input power for the spanwise forcing has little to do with the\nsuperimposed turbulent flow, and can be described quite well by the laminar transverse flow\nalone (Quadrio & Ricco 2011).\nA global view of the effect of the 1SD and 1OD waves, as well as their combination\n(indicated with 1p) is provided in figure 13, where results from an additional set of numerical\nexperiments are plotted. For 15 points with frequency −0.10⩽𝜔+⩽0.16, the drag reduction\nrate from case S3 is compared with the ones obtained with forcing by the sinusoidal wave\n1SD, 1OD and their combination 1p, all with the amplitude prescribed by (4.3). First of\nall, one notices that the first pair of harmonics is responsible for most (but not all) of the\neffects of the DTW, especially at negative frequencies. The exception at 𝜔=+0.04, where\nS3 nearly achieves laminarization but the 1p harmonics do not, has been already discussed.\nThe figure also provides detailed information on the non-trivial contribution of the isolated\nwaves 1SD and 1OD. At first approximation, the 1SD curve resembles SIN, but for the\nreduced amplitude of the forcing; the 1OD curve, instead, acts at a doubled wavenumber, as\npreviously exemplified in figure 12, so that its horizontal axis is enlarged by a factor 2 and\nalso reversed by virtue of the phase speed reversal.\nOverall, the general picture emerging from this analysis is that the sinusoidal forcing is\nalways best, as long as one can work with the optimal control parameters that yield maximum\ndrag reduction. However, as soon as the forcing parameters do not assume their optimal values\n(something that is not inconceivable e.g. because of technological limitations), the higher\nharmonics of a non-sinusoidal forcing affect the outcome in a way that depend on their\nlocation in the parameter space, so that the discrete forcing may outperform the sinusoidal\nforcing. This is exactly the conclusion reached one decade ago by Cimarelli et al. (2013),\nwho experimented with the temporal waveform for the spanwise oscillating wall. In both the\nspatial and temporal cases, regions of the parameters space exist where the sinusoidal forcing21\nRef\nω+= 0\nω+=−0.2\nFigure 14: Instantaneous snapshot of the reference and sinusoidally forced pipe flow, with\nparameters yielding drag reduction above ( 𝜔+=0) and below ( 𝜔+=−0.2) the apparent\nexperimental threshold for R. The background colormap (from blu to red) encodes the\nazimuthal velocity of the wall; the grey isosurfaces are drawn for 𝜆+\n2=−0.03.\ncan be outperformed, in terms of both RandS. However, such regions are far from the global\noptimum, in correspondence of which the sinusoidal forcing remains the best choice.\nThe final message is that, whenever the forcing is spatially distributed, its spatial dis-\ncretization is an essential ingredient to evaluate drag reduction and its success in terms of\nenergy savings. Experimental studies where a discrete form of spanwise forcing is employed\ncan hardly be compared to results from a sinusoidal forcing, unless discretization is properly\naccounted for.\n4.2. Localized turbulence\nOnce it is recognized that S3 and EXP data present the very same wiggles, which should not\nbe attributed to experimental errors, but are a direct consequence of forcing discretization,\none remaining difference becomes even more evident. In figure 5, EXP data do not reach\nabove an apparent threshold of maximum R≈ 0.33, while S3, S6 and SIN achieve much\nhigher values ofR, up to the complete relaminarization of the flow.\nEven though the only experimental information provided by ABBCQ is the drag reduction\nvalue, the additional insight allowed by the present numerical study highlights a peculiar\nfeature of the solution, which could explain the discrepancy. Instantaneous snapshots of the\nsinusoidally forced flow for two SIN cases with drag reduction above and below the apparent\nexperimental threshold (namely at 𝜔+=0 and𝜔+=−0.2, withR=0.42 andR=0.23\nrespectively) are compared to a snapshot from the reference flow in figure 14. The background\ncolormap shows the azimuthal velocity at the pipe wall; isosurfaces for 𝜆2(Jeong et al. 1997)\nare plotted at the level 𝜆+\n2=−0.03, and provide a qualitative idea of the turbulent structures at\nplay within the flow. As expected, the control action reduces the presence of structures at both\nfrequencies, with a more evident effect when drag reduction is higher. However, the most\nstriking feature of the plot is that turbulent activity for 𝜔+=0, where drag reduction is larger22 E. Gallorini, M. Quadrio\nand goes above the apparent experimental threshold, appears to be spatially intermittent,\nwhereas in the reference flow and also for the lower drag reduction case, it assumes a more\nconventional, spatially uniform look. Turbulent structures at 𝜔+=0 appear clustered, and\nsurrounded by a refractory region where turbulence is absent or nearly so. One can envisage\nthe arrangement of the flow structures in what is usually called a puff (Barkley 2016; Avila\net al. 2023), which is typical of transitional pipe flow; a notable difference here is that the\nflow is turbulent, the Reynolds number is relatively high, and the structures also undergo a\nswirling motion due to the spanwise flow induced by the forcing.\nHence, we link the state of turbulent localization observed in DNS-computed flow fields\nto the ability to reach Rlevels above the plateau of approximately 30 – 35% found in the\nexperiment. This suggestion is supported by the observation that the local friction oscillates\naround the laminar value far from the turbulent regions, whereas in correspondence to\nthem it becomes comparable to the experimentally measured value. Several reasons might\nhave prevented the ABBCQ pipe flow from reaching such a higher drag reduction state.\nExamples include the periodic streamwise boundary conditions as opposed to a fixed level\nof disturbance entering the experimental pipe via the inlet, or the idealized setting of the\nsimulation, which is free from environmental disturbances.\n5. Conclusions\nIn this work, direct numerical simulations have been used to replicate the successful turbulent\npipe flow experiment by Auteri et al. (2010) for skin-friction drag reduction based on\nspanwise forcing. The experiment was carried out at a rather low value of the Reynolds\nnumber, which is replicated precisely here. We are aware that this is a marginally low 𝑅𝑒,\nand in fact several control configurations lead to total of partial relaminarization of the flow.\nHowever, the study is not designed to address the important question of the 𝑅𝑒dependence\non drag reduction (in general, and with spanwise forcing in particular), for which the discrete\nnature of the forcing should not play any crucial role. Rather, the goal here is to exploit the\ndetailed comparison between experimental and simulation data obtained under nominally\nidentical conditions to understand how a spatially discrete implementation of streamwise-\ntravelling waves of azimuthal wall velocity affects the outcome of the flow control technique.\nIn fact, any experimental implementation of the forcing unavoidably differs from the ideal\nsinusoidal function typically considered in numerical studies; accounting for the effect of\nspatial waveform is required for a proper assessment of the forcing performance.\nThe DNS simulations have been carried out with an original code that uses Fourier\ndiscretization for the homogeneous directions and resorts to compact finite differences for\nthe radial one. Such mixed discretization is particularly suited for the turbulent pipe flow,\nand provides an efficient strategy to solve the issue of excess azimuthal resolution near the\npipe axis, by allowing the number of azimuthal spanwise modes to gradually decrease as\nthe axis is approached. Thanks to the efficiency of the code in terms of CPU and memory\nrequirements, most of the present study has been carried out on a single Xeon Phi processor.\nThe experimental conditions have been replicated by implementing the spanwise forcing\nas a piecewise-continuous, streamwise- and time-periodic function. Differences between\nthe drag reduction rate Rcomputed by DNS and measured in the experiment have been\nidentified and explained. The much higher drag reduction measured in simulations for both\ncontinuous and discrete forcing is suggested to derive from low- 𝑅𝑒effects, which enable the\nnumerically simulated flow to reach a state of spatially localized turbulence and eventually\nfully relaminarize. The apparently irregular behaviour of experimental Ris observed in\nthe numerical results too, and is shown to derive from the discretization of the sinusoidal\nwaveform, locally resulting into either larger of smaller Rdepending on the combination of23\ncontrol parameters. By expanding the piecewise-continuous function into a Fourier series,\nthe discrete forcing is shown to be equivalent to two families of sinusoidal waves, each with\ndecreasing amplitude and wavelength as the order of the harmonics is increased: one family\nis made of waves travelling in the same direction of the discrete wave, whereas the other\ncontains waves travelling in the opposite direction. Additional simulations have been run to\nobserve the role of the various harmonics, concluding that discretization has a predictable\nbut non-trivial effect depending not only on the degree of discretization, but also on the\nposition of the nominal (sinusoidal) wave in the plane of the control parameters. In the end,\nthe drag reduction performance of the discrete wave can be predicted from its harmonic\ncontent, provided full information for the sinusoidal waves is available\nTo progress from the idealized setting of a DNS towards experimental or real-world\napplications, where a discrete spatial waveform is unavoidable, it is essential to fully\nunderstand the differences between continuous and discrete waveforms, and to exercise\ncare when comparing DNS data computed for sinusoidal forcing with experimental, non-\nsinusoidal ones. This applies not only to raw drag reduction data, but also to the energetic\nrequirements of the forcing. Provided a comparison is carried out properly, a sinusoidal\nforcing remains the best option in correspondence of the optimal forcing parameters. Yet,\nwhenever e.g. technological limitations prevent the actuator to operate in correspondence of\nthe best forcing conditions, the discrete forcing can in principle outperform the sinusoidal\none.\nDeclaration of Interests\nThe authors report no conflict of interest.\nAcknowledgments\nThe authors gratefully acknowledge the preliminary work carried out by Martina Biggi during\nher Master thesis work.\nREFERENCES\nAuteri, F., Baron, A., Belan, M., Campanardi, G. & Quadrio, M. 2010 Experimental assessment of\ndrag reduction by traveling waves in a turbulent pipe flow. Phys. Fluids 22(11), 115103/14.\nAvila, M., Barkley, D. & Hof, B. 2023 Transition to Turbulence in Pipe Flow. Annual Review of Fluid\nMechanics 55(1), 575–602.\nBanchetti, J., Luchini, P. & Quadrio, M. 2020 Turbulent drag reduction over curved walls. J. 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A 369(1940), 1428–1442.\nQuadrio, M., Chiarini, A., Banchetti, J., Gatti, D., Memmolo, A. & Pirozzoli, S. 2022 Drag reduction\non a transonic airfoil. J. Fluid. Mech. 942, R2, 1–10.\nQuadrio, M., Frohnapfel, B. & Hasegawa, Y. 2016 Does the choice of the forcing term affect flow\nstatistics in DNS of turbulent channel flow? Eur. J. Mech. B / Fluids 55, 286–293.\nQuadrio, M. & Luchini, P. 2002 Direct numerical simulation of the turbulent flow in a pipe with annular\ncross-section. Eur. J. Mech. B / Fluids 21, 413–427.\nQuadrio, M. & Ricco, P. 2011 The laminar generalized Stokes layer and turbulent drag reduction. J. Fluid\nMech. 667, 135–157.\nQuadrio, M., Ricco, P. & Viotti, C. 2009 Streamwise-traveling waves of spanwise wall velocity for\nturbulent drag reduction. J. Fluid Mech. 627, 161–178.\nQuadrio, M. & Sibilla, S. 2000 Numerical simulation of turbulent flow in a pipe oscillating around its\naxis. J. 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L¨ u1,3\n1Center for Joint Quantum Studies and Department of Physics,\nSchool of Science, Tianjin University,\nYaguan Road 135, Jinnan District, Tianjin 300350, China\n2Department of Physics, Shahid Beheshti University, 1983969411, Tehran, Iran\n3Joint School of National University of Singapore and Tianjin University,\nInternational Campus of Tianjin University, Binhai New City, Fuzhou 350207, China\nABSTRACT\nThis paper analyzes the possibility of bouncing and non-bouncing universes in the frame-\nwork of four-dimensional Einstein-Gauss-Bonnet (4D-EGB) gravity, corresponding respectively\nto negative and positive coupling constants λof the Gauss-Bonnet term. We also use the\nHorndeski-type scalar-tensor theory to assess the role of a scalar charge Cas a geometrical\ncontribution to the radiation in the Universe. We modify the expansion history of the universe\nto allow for modifications induced by the 4D-EGB gravity. Using Planck measurements of\nthe cosmic microwave background anisotropies as well as various datasets of baryonic acoustic\noscillations, we set the upper bounds λ≤10−16(km/s/Mpc)−2andλ≤10−30(km/s/Mpc)−2\nfor the non-bouncing and bouncing scenarios. The upper limit in the latter case is mainly\ndriven by the requirement to conservatively respect the thermal history at energy scales of\nthe standard model of particle physics. We also find that the contribution of the geometrical\nradiation-like term of the model cannot exceed 10% of the current radiation in the Universe.\nThis study shows the feasibility of a bouncing universe, even with a normal matter sector,\nin the 4D-EGB gravity. More theoretical investigation is required to further explore possi-\nble observational predictions of the model that can distinguish between general relativity and\n4D-EGB gravity.\nhkhodabakhshi@tju.edu.cn m farhang@sbu.ac.ir (*Corresponding author) mrhonglu@gmail.comarXiv:2401.11732v1  [astro-ph.CO]  22 Jan 2024Contents\n1 Introduction 2\n2 Model 3\n3 Analysis and Datasets 6\n4 Results 7\n4.1 Non-bouncing universe ( k= 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n4.2 Bouncing universe ( k=−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n5 Conclusions 12\n1 Introduction\nGeneral relativity (GR), a main pillar of the ΛCDM scenario, has passed many observational\ntests in a wide range of scales and plays a crucial role in understanding large-scale universe.\nHowever, given the conceptual challenges and observational tensions of ΛCDM, serious ef-\nforts have been put to exploring alternative gravities. Among these is modification to the\nEinstein-Hilbert Lagrangian by integrating Gauss-Bonnet terms [1]. These terms are impor-\ntant in low-energy gravity theories, especially in heterotic string theory [2, 3] and studying\nquantum fields in curved spacetimes [4]. A key feature of Gauss-Bonnet terms is their ability\nto produce ghost-free, second-order field equations [1, 5]. Gauss-Bonnet terms do not alter\ngravitational field equations in 4-dimensional spacetime. However, there exists a D→4 limit\nof Gauss-Bonnet gravity by rescaling the coupling constant λ→λ/(D−4) [6], as well as\nmodifications through the Kaluza-Klein reduction [7–11] that would change the field equations\nin four dimensions. These approaches have led to novel insights into gravitational dynamics\nand important discoveries in cosmology [11–20].\nSeveral studies of 4-dimensional Einstein-Gauss-Bonnet (4D-EGB) gravity have led to\nvarious constraints on the parameter λ. According to [15], inflation in the early Universe\nrestricts negative values of λ, with an upper limit suggested at 10−49(km/s/Mpc)−2. On\nthe other hand, [16] notes that 4D-EGB gravity, with no cosmological constant, is chal-\nlenged by current cosmological and gravitational-wave data. This study estimates λto be\nbetween −10−20(km/s/Mpc)−2and 10−19(km/s/Mpc)−2. In addition, employing the Affine\nInvariant Markov chain Monte Carlo (MCMC) Ensemble sampler, EMCEE [21], and vari-\nous datasets, [18] constrains the upper bound of λto be approximately 10−22(km/s/Mpc)−2.\n2Moreover, based on the need for 4D-EGB theory to be able to reproduce the lightest observed\nblack holes, [22] suggests that the maximum value of minus λin 4D-EGB gravity should be\nnear 10−28(km/s/Mpc)−2. In these works, although positive and negative values for λwere\nconsidered, their cosmological consequences have not been explored.\nIn [20] we explored the cosmological aspects of 4D-EGB gravity, minimally coupled to\na perfect fluid, by considering both positive and negative values of λ. It has been shown\nthat selecting a negative value for λresults in a universe that begins with a bounce rather\nthan the traditional Big Bang. This bouncing scenario notably does not violate the null\nenergy condition, contrasting with many existing models. Furthermore, using the scalar-tensor\nreformulation and the Kaluza-Klein reduction [6,7,9–11], a scalar charge Cis introduced [8],\ncontributing a radiation-like component λC4/H2\n0to the model. In [23], using CMB data\nfrom the Atacama Cosmology Telescope (ACT) [24], it is shown that at the small- λlimit,\nλC4/H2\n0= (−9±6)×10−6. This result leads to an imaginary field for positive values of λ.\nWe will discuss how we can avoid an imaginary field in a bouncing universe.\nIn this paper, based on our results for the 4D-EGB model presented in [20], we aim to\ninvestigate observational tests for both bouncing and non-bouncing scenarios, examining the\nimpact of the radiation-like component. To do so, we employ data from the temperature and\npolarization anisotropy spectra of the cosmic microwave background (CMB) radiation and its\nlensing, as measured by Planck [25], as well as the measurements of the Baryonic Acoustic\nOscillations [26].\nThe structure of the paper is as follows: In Section 2 we present the equations that describe\nthe expansion history in the 4D-EGB model. Section 3 briefly introduces the methodology for\ndata analysis as well as the datasets used in this work. The results are presented in Section 4\nand we discuss the results and conclude in Section 5.\n2 Model\nThe 4D-EGB gravity action in the framework of the four-dimensional Horndeski-type scalar-\ntensor theory is described by [7,10]\nA=Z\nd4x√−g\u0012\nR+kλ\u0000\nϕLGB+ 4Gab∇aϕ∇bϕ−4□ϕ(∇ϕ)2+ 2(∇ϕ)4\u0001\u0013\n, (1)\nwhere k=±1, and λ≥0 represents the coupling constant which, depending on the choice of\nk, would lead to positive or negative coupling. LGB=R2−4RabRab+RabcdRabcdis the Gauss-\nBonnet term and Gabis the Einstein tensor and the curvature of the internal space is considered\nzero [6,8,9,11]. Considering a minimally-coupled perfect fluid matter with energy-momentum\n3tensor Ta\nb= diag( ρ, p, p, p ), in the context of FLRW metric, the time-time component of the\nEinstein field equations can be expressed as [6,8,20]\n\u0012˙a\na\u00132\n+kλ\"\u0012˙a\na\u00134\n−C4\na4#\n=8πG\n3ρ+Λc2\n3. (2)\nThe constant Cis the scalar charge associated with the shift symmetry ϕ→ϕ+const in the\nframework of four-dimensional Horndeski-type scalar-tensor theory. The cosmological constant\nis denoted as Λ, and ρ=ρm+ρris the density of dark and baryonic matter and radiation in\nour universe. The Friedmann equation (2) can be reformulated as\nH2+kλH4=Λc2\n3+8πG\n3(ρm+ρr) +kλC4\na4. (3)\nThe eq. (3) yields two positive solutions for H2,\nH2(z, λ) =−1 +ϵp\n1 + 4 kλ[H2\nE(z) +kλC4(1 +z)4]\n2kλ, ϵ =±1, (4)\nwhere the Einstein Hubble HEis defined as\nH2\nE=H2\n0\u0002\nΩr(1 +z)4+ Ω m(1 +z)3+ Ω Λ\u0003\n, (5)\nandH0is the Hubble constant and z=−1 + 1 /arepresents redshift. The density parameters\nfor Λ, matter and radiation are defined as\nΩΛ=c2Λ\n3H2\n0, Ωm=8πG\n3H2\n0ρ0m, Ωr=8πG\n3H2\n0ρ0r, (6)\nwhere ρ0mandρ0rare the current density of matter and radiation. In eq. (4), the radiation-\nlike term with the coefficient λC4can be absorbed into the radiation term in H2\nEthrough the\nredefinition\nΩ′\nr= Ω r+kλC4\nH2\n0, (7)\nwhere scalar charge Cis treated as a free parameter and is allowed to take both positive and\nnegative values. For simplicity, in this section we will consider C= 0. However, in the rest\nof the paper and for comparison with data, we will aslo consider C̸= 0 case. The universe\nwould experience very different histories depending on the choice of k. For a comprehensive\nexplanation of these solutions and their properties, including how they impact the dynamics\nof the universe, a detailed study is provided in [20]. Here we briefly review some of the results.\nThe solution with ϵ= 1 does not converge to Einstein limit when the dimensionless parameter\nλH2\nEgoes to zero. We therefor focus only on scenarios with ϵ=−1. The k= 1 case corresponds\nto a universe starting from a singular point, the so called Big Bang, and expanding over time.\nOn the other hand, the solution for k=−1 represents a bouncing universe, suggesting that\n4Figure 1: The Hubble parameter Has a function of redshift zin the very early universe (108< z < 109),\nplotted for Einstein gravity (red dotted-line) and the 4D-EGB gravity in a non-bouncing (blue solid-\nline) and bouncing universe (green dashed-line). In this plot we consider λ= 10−35(km/s/Mpc)−2for\nboth non-bouncing and bouncing universes. The bounce redshift can be obtained from eq. (8). Smaller\nλ’s would lead to bounce at higher redshifts. The ΛCDM standard parameter values used in this plot\nare based on the bestfit measurements by Planck 2018 [26].\nthe universe avoids the initial singularity by going through a bounce. The bounce is shown to\nhappen when H2(z0) = 1 /(2λ), or, equivalently when\n1−4λH2\nE(z0) = 0 , (8)\nwhere z=z0is the redshift of the bouncing point (see eq. (4)). Using this equation, the value\nofλcan be determined if the bounce redshift is given. This point is illustrated in Fig. 1.\nThe Hubble parameter as given by eq. (4), with ϵ=−1, would converge to the Einstein\nlimit as λH2\nEapproaches zero\nH2=H2\nE−kλ(H2\nE)2+. . . . (9)\nThe difference between this new Hfor both k=±1 and HEwould increase as one goes to\nhigher redshifts (see Fig. 1).\nGiven the observational achievements of the Einstein gravity, one may require that the\ndeviation of 4D-EGB gravity from Einstein gravity stays small even in the early universe.\nThis can be ensured by choosing small enough values for λ. In Fig. 2 for the case k= 1, we\nillustrate the maximum relative deviation of 4D-EGB gravity from Einstein gravity as\n∆H(λ;z1, z2) = max\u0012\f\f\f\fH2(z, λ)−H2\nE(z)\nH2\nE(z)\f\f\f\f\u0013\f\f\f\fz2\nz1, (10)\nwhere the max is taken over zin an assumed redshift range z1toz2. The dimensionless\nparameter λHin Fig. 2 is defined as λH≡λH2\n0. The figure shows the high sensitivity of the\n5Figure 2: The maximum relative deviation ∆ Hforz1= 103,z2= 109(red solid-line), and for z1= 0,\nz2= 103(blue dashed-line) are plotted for k= 1 and various log( λH) values. Here, λH=λH2\n0\nparameterizes the dimensionless coupling constant. The deviation from the ΛCDM model is significant\nforλ≈10−10(km/s/Mpc)−2atz2= 103, while considering higher redshifts at z2= 109constrains the\nupper limit of λto approximately 10−37(km/s/Mpc)−2. The ΛCDM standard parameter values used\nin this plot correspond to the bestfit measurements by Planck 2018 [26].\n4D-EGB model to the choice of λ. The root of this sharp sensitivity to lambda is related to\nthe different behavior of H2/(1 +z)4→0 and H2\nE/(1 +z)4→H2\n0Ωrasz→ ∞ .\nThe overall behavior of ∆ Hfor the k=−1 case is quite similar to that of the k= 1 case, but\nwith different values for λ, which would in turn determine the redshift of the bouncing point\nin the early universe, as shown in eq. (8). Caution should be taken when choosing λif the new\nscenario is required not to violate the thermal history of the Universe. In this work we choose\nto respect the thermal history and therefore assume that the bounce, if any, has occurred\nbefore, zbounce≳1010. In the following sections we investigate observational constraints on λ\nfor both the bouncing and non-bouncing universes. We also explore the implications for the\nscalar field ϕ, or equivalently for the λC4term —as a correction to radiation.\n3 Analysis and Datasets\nIn our implementation of the 4D-EGB model, we modified the public Cosmological Monte\nCarlo (CosmoMc) package, originally designed for the ΛCDM model, to include the expansion\nhistory in the 4D-EGB scenario. There are two new parameters in this model: the coupling\nconstant parameterized by λHand the new relativistic degree of freedom that characterizes\nthe scalar charge (see eq. (7)). We perform the search for λHlogarithmically to cover an\nexhaustively broad range in the parameter space. One should note that for the bouncing\n6universe, corresponding to the k=−1 case, H2becomes imaginary at high redshifts if λis not\nsufficiently small. We discard these non-physical λ’s in the analysis by associating to them an\na priori zero probability. The highest allowed value for λin each scenario would also set the\nlowest possible redshift for the bounce. We request this bounce to happen before the highest\nredshift used in the analysis, i.e., z <1010, which would correspond to λH≲10−30, If the\nvalues for the other parameters are set based on the bestfits from Planck 2018 [26]. The degree\nof freedom corresponding to the scalar charge is included in the analysis by letting the effective\nnumber of relativistic species, Neff, be a free parameter. Changing Neffwould effectively alter\nthe radiation density as\nΩr= \n1 +7\n8\u00124\n11\u00134/3\nNeff!\nΩγ, Ωγ=8πG\n3H2\n0ργ, (11)\nwhere ργis the energy density of photons. The fiducial value for the effective number of\nrelativistic species, Nfid\neff= 3.046, corresponds to the standard scenario with three flavors of\nmassless neutrinos. Thus, in this parametrization, any deviation ∆ Nefffrom the fiducial value\nwould imply non-zero contribution of the radiation-like term λC4to the radiation density,\nλC4\nH2\n0≈0.2 Ωγ|∆Neff|. (12)\nTo constrain the parameters of the 4D-EGB model along with the standard cosmological\nparameters, the predictions of the model need to be compared with observations. As the\ndeparture from Einstein gravity grows with redshift, we expect the tightest constraint to come\nfrom the early universe data. We therefor use the temperature and polarization power spectra\nof CMB anisotropies (TT+TE+EE) and CMB lensing as measured by Planck 2018 [26], as well\nas various measurements of the baryonic acoustic oscillations (BAO). We note that we carried\ntwo sets of parameter estimation, with CMB data alone, and with the combined CMB+BAO.\nSince the implications for the 4D-EGB model were found to be consistent and indeed quite\nsimilar, we only report the results with the combined data. We refer to this combined dataset\nas P18+BAO.\n4 Results\nIn this section we use observations of CMB anisotropies and BAO to constrain the parameters\nin the cosmological scenario based on the 4D-EGB gravity. We separately explore two cases\nof non-bouncing ( k= 1) and bouncing ( k=−1) universes. In both scenarios, the parameter\nλcharacterizes the deviation from GR, and we also investigate the impact of the scalar charge\nby comparing the results for C= 0 and a varying Ccase. Table 1 and Figures 3 and 4\n7ΛCDM 4D-EGB, k= 1 4D-EGB, k=−1\nC= 0 C̸= 0 C= 0 C̸= 0\nΩbh20.02242±0.00014 0 .02240±0.00028 0 .02236±0.00035 0 .02240±0.00026 0 .02235±0.00035\nΩch20.11933±0.00091 0 .11960±0.00185 0 .11840±0.00560 0 .11960±0.00180 0 .11820±0.00540\nH0 67.66±0.42 67 .55±0.86 67 .10±2.10 67 .55±0.80 67 .10±2.10\nλ − <10−17<10−16<10−3110−30\nτ 0.0561±0.0071 0 .0570±0.0150 0 .0560±0.0150 0 .0570±0.0135 0 .0560±0.0135\nNeff − − 2.97±0.315 − 2.96±0.325\nln(1010As) 3.047±0.014 3 .049±0.029 3 .044±0.031 3 .049±0.027 3 .044±0.030\nns 0.9665±0.0038 0 .9659±0.0074 0 .9630±0.0130 0 .9657±0.0067 0 .9630±0.0130\nlog(λH) − <−13 <−12 <−27 <−26\nσ8 0.810±0.006 0 .811±0.012 0 .807±0.019 0 .812±0.012 0 .807±0.018\nTable 1: Constraints on the parameters of the 4D-EGB gravity for non-bouncing ( k= 1) and bouncing\nuniverses ( k=−1) measured by P18+BAO. In both scenarios, two cases of C= 0 and free Care\nexplored which respectively correspond to no scalar charge and with a scalar charge associated to the\nshift symmetry of the scalar field in the theory. This radiation-like degree of freedom is parametrized\nthrough Neff. The ΛCDM results are also presented for comparison.\nsummarize these results. The second column of the table presents the ΛCDM measurements\nof the cosmological parameters, as the baseline for comparison. It should be noted that any\nmodification to Neffin this work is mainly associated to the scalar charge, and is thus expected\nto be absent in the Einstein gravity. This should be contrasted to various extensions to ΛCDM\nwhere variation in Neffis associated to non-standard radiation-like contribution to the energy-\nmomentum tensor of the Universe. In the table, we report the measurements for both λand\nlog(λH), while in the plots we only present log( λH). The former is more conveniently used for\ncomparison with the quoted constraints in the literature while the latter is numerically easier\nto work with.\n4.1 Non-bouncing universe ( k= 1)\nThek= 1 columns in Table 1 present the results of parameter estimation with P18+BAO\nof a non-bouncing universe with a 4D-EGB gravity without a scalar charge (third column),\nand with the possibility of a scalar charge with its amplitude set as a free parameter to be\nmeasured by data (forth column). The ΛCDM scenario would correspond to the λ→0 limit\nof the 4D-EGB model. The parameter log( λH) is found to be consistent with zero, with tight\nupper bounds, which slightly relaxes as the scalar charge degree of freedom is included in the\n8analysis. The measured mean value for Neffis lower than expected from the standard model\nof particle physics ( Neff= 3.046). This is however slightly tricky to analyze. Firstly, the\nsignificance of this deviation is statistically low. Moreover, the reported Nefffrom P18+BAO\nin the ΛCDM scenario is also low, i.e., Neff= 2.99±0.33 [25], again with low significance,\nyet with tighter errors compared to our results, as expected. It is therefore not clear that this\nsmall deviation can be associated to the new degrees of freedom allowed in the 4D-EGB model.\nBased on these results and using eq. (12) we can estimate the order of magnitude of the upper\nbound on λC4/H2\n0, i.e., the radiation-like contribution to H2from this scenario (see eq. (3)).\nWe find λC4/H2\n0≲10−6(1σ), where |∆Neff| ∼10−1.\n4.2 Bouncing universe ( k=−1)\nThek=−1 columns in Table 1 present the results of parameter estimation with P18+BAO\nfor the bouncing universe within the 4D-EGB gravity, corresponding to the C= 0 and varying\nCcases. The parameter λis tightly constrained in both cases, with λ≤10−31(km/s/Mpc)−2\nforC= 0 and λ≤10−30(km/s/Mpc)−2when marginalized over C. These constraints are\nsignificantly tighter compared to those in the non-bouncing scenario. It should be noted that\nthe bounce redshift can be calculated for a given λfrom eq. (8). For example, if we require\nthe bounce to happen before the lepton era, with z∼1015, and assuming that the ΛCDM\nparameters are set to their Planck bestfits, we find λ≈10−60(km/s/Mpc)−2. In our work\nwe require zbounce≳1010to guarantee the bounce does not happen earlier than the highest\nredshift directly used in the analysis. This would correspond to λ≲10−35(km/s/Mpc)−2if\nΛCDM parameter values are set to their Planck bestfits. Marginalizing over these parameters\nwould relax this bound to the reported values in Table 1.\nThe order of magnitude of the upper bound on λC4/H2\n0for the k=−1 case is also similar\nto that found in Section 4.1. It should however be noted that, unlike the k= 1 scenario, the\nmeasured Neffin this case should be lower than the fiducial Neffto avoid an imaginary C.\nUnlike the Big Bang model, which starts from a singularity, a bouncing universe begins\nexpansion from a non-singular state characterized by certain density and temperature which\ncan be calculated from the bounce redshift. Moreover, the very early Universe is widely\nbelieved to have undergone the inflationary phase of rapid accelerated expansion. In [20]\nthe exact solution of t(a) was calculated in the radiation-dominated era using eq. (4). The\nbehaviour of a(t) indicates that a radiation-filled Universe with the 4D-EGB gravity does not\nexperience an inflationary phase, in nor bouncing neither non-bouncing scenarios. This should\nbe contrasted to some other models, such as the ‘matter bounce scenario’, where that conditions\nat the bounce are shown to naturally lead to inflation [27]. Furthermore, assuming inflation\n9Figure 3: Marginalized 1D and 2D likelihoods of the parameters in the 4D-EGB cosmology, as measured\nby P18+BAO, for a non-bouncing universe ( k= 1), with C= 0 and a free C, parametrized through\nNeffdue to the radiation-like behavior of this degree of freedom.\n10Figure 4: Similar to Figure 3 but for a bouncing universe ( k=−1).\n11is driven by a single canonical scalar field Φ with potential energy V(Φ), it is demonstrated\nin [16] that achieving inflation in the 4D-EGB model with k= 1 is challenging. We leave a\ndetailed analysis of the initial conditions and inflationary dynamics to future work.\n5 Conclusions\nThis study provided a comprehensive study of the observational tests of the 4D-EGB cosmology.\nIt is shown [20] that the 4D-EGB gravity model leads to two different scenarios: a bouncing\nand a non-bouncing universe, depending on the sign of the coupling constant (in our work\nrepresented by k= 1 and k=−1 respectively). In the Horndeski-type scalar-tensor theory\nframework [6, 7, 10, 11], the 4D-EGB model includes a scalar charge C, which contributes to\nthe radiation density [8]. We employed data from Planck 2018 (TT+TE+EE+lensing) and\nBAO [25, 26] to constrain both λandλC4/H2\n0contribution to radiation and explored the\ncosmological aspects of the 4D-EGB model in its validation and predictions. In this paper, we\nworked with the unit of λas (km/s/Mpc)−2and the unit of Cas equivalent to km/s/Mpc.\nFor a universe without a bounce, i.e. with a positive coupling constant, we find λ≤\n10−17(km/s/Mpc)−2with no scalar charge (i.e, with C= 0) and λ≤10−16(km/s/Mpc)−2if\nmarginalized over C. The detailed results are listed in Table 1 and illustrated in Figure 3. In\na bouncing universe [20], where the initial singularity can be avoided, our analysis suggests\ntighter bounds of λ≤10−31(km/s/Mpc)−2andλ≤10−30(km/s/Mpc)−2ifC= 0 and if Cis\nmarginalized over, respectively. See Table 1 and Figure 4. The bounds on λin the latter case\nare mainly determined by the requirement of an early enough bounce, if it exists, occurring\nbefore the highest redshift used in our analysis ( zbounce≳1010). This is a conservative choice,\nas a higher bounce redshift (and thus a lower λ) is required to respect the thermal history of\nthe universe with an energy scale corresponding to the standard model of particle physics. We\nalso found that the contribution of the scalar charge to the radiation density is on the order\nof 10−1, indicating a significant influence in the overall radiation density of the universe.\nIn brief, the results are found to be consistent with general relativity, with no observational\nhints for deviations as suggested by a 4D-EGB gravity. However, the theoretical appeal of\nthis theory, especially in the bouncing scenario where the initial singularity can be avoided,\nis encouraging for further investigation. In particular, the possibility of an early inflationary\nscenario in this model requires further exploration. We also note that the current work only\nsearched for observational imprints of the model through its impact on the expansion history\nof the Universe and did not investigate the modified perturbation equations [23]. The detailed\nstudy of these are left to future work.\n12The calculated bounds for λare roughly consistent with previous works [15–19, 23]. The\nfeasibility of a bouncing universe within the framework of the 4D-EGB model provides new\ninsights into the early Universe and demonstrates how modifications to gravity can alter our\nunderstanding of the Universe. Further theoretical investigation is required to fully understand\nthe cosmological implications of the 4D-EGB model and the bouncing universe hypothesis.\nAcknowledgement\nThis work is supported in part by the National Natural Science Foundation of China (NSFC)\ngrants No. 11935009 and No. 12375052\nReferences\n[1] D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. 12 498 (1971)\ndoi:10.1063/1.1665613.\n[2] B. Zwiebach, “Curvature Squared Terms and String Theories,” Physics Letters B 156,\n315 (1985).\n[3] R.I. Nepomechie, “On the Low-energy Limit of Strings,” Physical Review D 32, 3201\n(1985).\n[4] N.D. Birrell, N.D. Birrell, P. Davies, “Quantum fields in curved space,” 7 (Cambridge\nuniversity press, 1984).\n[5] H. Khodabakhshi, H. L¨ u, “Holographic relations and alternative boundary conditions for\nLovelock gravity,” Phys. Rev. D 95, [arXiv:2203.08839 [hep-th]].\n[6] D. Glavan, C. 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In the design phase, this\nis achieved through co-simulation of component mod-\nels. However, complex co-simulation environments are\nrarely (re-)used in the verification and validation phases,\nin which mixed real-virtual prototypes (e.g. Hardware-\nin-the-Loop) are already available. One reason for this\nare coupling errors such as time-delays, which inevitably\noccur in co-simulation of virtual and real-time systems,\nand which influence system behavior in an unknown and\ngenerally detrimental way. This contribution introduces\na novel, adaptive method to compensate for constant\ntime-delays in potentially highly nonlinear, spatially dis-\ntributed mixed real-virtual prototypes, using small feed-\nforward neural networks. Their optimal initialization\nwith respect to defined frequency domain features results\nfrom a-priori frequency domain analysis of the entire cou-\npled system, including coupling faults and compensation\nmethods. A linear and a nonlinear example demonstrate\nthemethodandemphasizeitssuitabilityfornonlinearsys-\ntems due to online training and adaptation. As the com-\npensation method requires knowledge only of the band-\nwidths, the proposed method is applicable to distributed\nmixed real-virtual prototypes in general.\nINTRODUCTION\nCurrent trends in the automotive sector like connected\nand autonomous driving functions are leading to a closer\ncoupling of different vehicle domains. This is for example\nthe case in the development of an emergency brake assis-\ntants through the interaction of longitudinal control and\nbrake management. In order to enable a time- and cost-\nefficient development of such cross-domain vehicle func-\ntions, the interactions of the domains must be considered\nat an early stage of the development process. Simula-\ntion experts interconnect the different simulation mod-\nels, by coupling various tools using the methods of co-simulation [9]. This way complex cross-domain model-\nin-loop (MiL) co-simulations are implemented. Since\nthe modeling and integration effort to set up such co-\nsimulations is high, the demand is coming up to use the\nsame co-simulation environment not only during the de-\nsign phase, but also in the verification and validation\nphase of the development process. In those phases, first\ncomponents of the vehicle are available as real hard-\nware on test benches. Coupled to the existing MiL co-\nsimulation, detailed mixed real-virtual prototypes are re-\nalizedtotestthehardwareorsoftwareinopencontextun-\nder realistic (e.g. traffic) conditions. But, due to the real-\ntime requirement, couplings between simulation models\nand hardware (HiL) always come with coupling faults\n(e.g. time-delay or measurement noise). If, in addition,\nthe unchanged models from the MiL co-simulation are to\nbe used, the coupling faults even increase, since the mod-\nels can generallynot be compiled on a real-time operating\nsystem,butrunonastandardWindowsPC.Furthermore,\nthere are use cases for cross-company collaboration using\nmixed real-virtual prototypes, since the complexity of the\nsystems is increasing and their handling requires a wide\nrange of different competencies, which most companies\ndo not have in house. The additional distance between\nthe coupled systems in those use cases further increases\nthe coupling faults and their negative effect on the overall\ncoupled system.\nThis contribution addressesthe time-delays in distributed\nmixedreal-virtualprototypes,byproposinganovel,adap-\ntive compensation method combined with adetailed anal-\nysis of the dynamic effects a simulator distribution has\non the overall system. After covering related work from\nthe fields of distributed mixed real-virtual prototypes and\ncoupling fault compensation, the compensation method\nbased on a feedforward neural network is presented. The\nresults of the following analysis of the overall system in\nfrequency domain are then used for an optimal compen-\nsation method design. Finally, the applicability of the\nmethodology on a nonlinear example is shown.\nRELATED WORK\nRecently the development of mixed real-virtual proto-\ntypes in the automotive sector gained some momentum\nthrough the release ofthe Distributed Co-Simulation Pro-\ntocol(DCP) [12]. TheDCP isdesigned tostandardizethecoupling of real-time or non real-time simulators and thus\nreducingtheintegrationeffortofspatiallydistributedpro-\ntotypes.\nIn literature many examples for the transition from vir-\ntual to real testing using mixed real-virtual prototypes\ncan be found. A toolchain for a seamless transition from\na MiL co-simulation to heterogeneous HiL testing is pre-\nsented [11]. Using this toolchain, an engine test bench\nis coupled to real-time vehicle dynamics and environment\nmodels. Mixed real-virtual prototypes can also be used to\nincorporate the driver in the testing process, e.g. by in-\nvestigating the interaction between humans and an auto-\nmatic transmission in a driving simulator [15]. Since test\nbenches usually do not stand side by side, it is reasonable\nto use the internet when coupling them. Examples for\nmixed real-virtual prototype coupled via the internet can\nbe found in [7] and [16]. In [2] additionally the DCP is\nused for the integration.\nIn all mentioned examples, the focus is on implement-\ning the coupling itself. Coupling faults as delays and\ndropouts, specified in [16], are known but the use of com-\npensation methods is not yet widespread. In the sci-\nentific fields of telerobotics [13] and networked control\nsystems [1] many methods are developed to compensate\nfor these coupling faults, but the compensation is imple-\nmented in the controls themselves. For mixed real-virtual\nprototypes, however, the physical models should not be\nmodified, which is why the compensation must be im-\nplemented in the coupling signals [17]. In [18] a model-\nbased-coupling method is presented which is meant for\nthe usage for mixed real-virtual prototypes and is tested\non an engine test bench. The parameters of two second\norder linear systems are identified online to compensate\nfor the delay. A linear fourth order FIR filter is intro-\nduced in [17]. Together with a recursive least squares\nalgorithm as identification method, the filter is designed\nto cope with communication time delays, data-losses and\nnoisy measurements.\nHowever,thereisnocompensationmethodcapableofrep-\nresentingnonlinearsignalbehavior. Inaddition, the influ-\nence of the compensation on the overall system should be\npredictable and verifiable and its parameterizationshould\nideally be based on the dynamics of the coupled system\nitself.\nCOMPENSATION METHOD\nThis paper considers the coupling process between two\ndistributively coupled systems (e.g. vehicle components)\n“A” and “B”, at least one of which is a real-time system.\nThe communication ormacrostep size ∆ Tis the constant\nrate at which data is exchanged between the two systems.\nThe coupling faults that occur in a distributed coupling\nofmixed real-virtualprototypesare attributable to effects\nlike communication time-delay, jitter, determinism and\nmessageloss[16]. Since the time-delay is usually the most\ndominant fault in a distributed system, the others are\nneglected in this paper. This simplifies the faults to a\nconstant time-delay τwhich represents the time betweensending a message from “A” and receiving it at “B” and\nvice versa. The macro step size fixes its resolution, it\nholdsτ=k·∆Twithk∈N.\nTo compensate for the time-delay, an algorithm is needed\nwhich is placed at each input uof each system participat-\ning in the distributed real-time co-simulation. The com-\npensation method extrapolates the delayed input ut−τto\nget a predicted input value ˆ utat timet. In order to make\nthe method applicable for as many simulation tools and\nreal-time systems as possible, the extrapolation is signal-\nbased and only the current as well as past values (and no\nderivatives) of the inputs are used for the extrapolation.\nFigure 1 givesan overviewof the coupled system, the sub-\nscript of y,uand ˆustand for the point in time at which\nthe signal is evaluated.\nSys Atime-delay\nSys Byt\ntime-delayneural network\nneural networkˆut ut−τ\nFigure 1: Overview of the Considered Coupled System\nFor extrapolation, small feedforward neural networks are\nused here. If the activation function of all network nodes\nis linear and a vector /vector uconsisting of ppast signalvalues is\nthe input to the network, the output of the network reads\nas\nˆut=/vector aT/vector u+bwith/vector a=\na1\na2\n...\nap\nand/vector u=\nut−τ\nut−τ−∆T\n...\nut−τ−(p−1)∆T\n.\n(1)\nThe vector /vector aand the scalar bare calculated from the\nweights of the network. As already stated by [6], this\nequals a linear autoregressive model, which is a gen-\neralization of commonly used extrapolation methods in\nco-simulation as zero-order-hold (constant extrapolation,\nZOH) and first-order-hold (linear extrapolation, FOH).\nThe recursive FIR-Filter for time-delay compensation\nin [17] is based on the same function.\nThe main advantage of using a feed forward neural net-\nwork for the extrapolation is the fact that it can be ex-\ntendedtorepresentnonlinearsignalbehavior,bychoosing\na nonlinear activation function in the hidden layer(s). If\nRectified Linear Units (ReLU) or, to prevent parts of the\nnetwork from “dying”, leaky ReLUs are chosen as activa-\ntion function, the network is enabled to switch between\ndifferent configurationsof the parameters /vector aandb. For ex-\nample, a hidden layer with nneurons with (leaky) ReLU\nactivationimplements n2differentparametersetsof /vector aand\nbdepending on which hidden node is active and which is\nnot. But between the switching points the network re-\ntains its linear behavior from equation (1), which allows\na detailed analysis of the compensation behavior, even in\nthe nonlinearcase. Furthermore, the structureofthe neu-\nral network allows using efficient algorithms for the initial\ntraining aswell as an online adaptionof the weightsofthe\nnetwork.ANALYSIS OF THE COUPLING PROCESS\nThe couplingfaults influence the behaviorofadistributed\nmixed real-virtual prototype significantly. It is not un-\nlikely that an originally stable system gets unstable due\nto the coupling and the associated coupling faults. In\norder to increase the confidence in mixed real-virtual pro-\ntotypes by making statements on robustness regarding\nthe coupling faults, this section aims to analyze the entire\nclosed-loop system including coupling faults and compen-\nsation method in frequency domain. First, the transfer\nfunction of the coupling process including coupling faults\nand compensation method is calculated and second it is\nshown using an example how this transfer function can\nbe utilized to analyze the influence of the coupling faults\nand of the compensation method on the closed-loop sys-\ntem. Furthermore, it is shown in the next section that the\ncompensation method can be initialized optimally based\non this analysis.\nAs already mentioned in [3], a detailed analysis of a co-\nsimulation is also possible in time domain via a multi-rate\napproach, but due to the variable step size solvers usually\nused for physical models, the analysis would become very\ncomplex. Instead, in this paper, the analysis is carried\nout in frequency domain which leads to the following as-\nsumptions:\n1. All sub-systems, their inputs and their outputs are\nassumed to be ideally time continuous. It follows\nthat the numerical errors made by a solver due to\nevaluating the system equations at discrete points in\ntime are neglected. The measurement error that oc-\ncurs when reading out values of real-time systems\n(e.g. test benches) is also neglected.\n2. Themacrostep size∆ Tisconstantandsmallenough\nto avoid aliasing in all coupling signals. This is veri-\nfied via the Nyquist-Shannon theorem with the con-\ndition ¯ω∆T << πfor the ratio of the maximal band-\nwidth ¯ωof the coupling signal and ∆ T[4].\n3. The coupling faults are simplified to a constant time-\ndelayτ(see previous section).\nSince all inputs and outputs of the sub-systems are time\ncontinuous, it is reasonable to assume that also the cou-\npling process is a time continuous element which includes\nthe overall correlation between the continuous output sig-\nnal of a sub-system and the continuous input signal of\nanother sub-system. The coupling process includes two\ndifferent effects. On the one hand, the disturbing effects\nof sampling and delay due to data exchange and commu-\nnication and, on the other hand, the added methods of\ncompensation and reconstruction to compensate for these\neffects. Figure2displaysanoverviewoftwodistributively\ncoupled systems “A” and “B” and the coupling element.\nTo enable a closed-loop analysis of the coupled system\nin frequency domain, the Laplace transform is applied to\neach component separately. The transfer function of the\ndisturbing effects Gf(s) describes the correlation between\nan output signal y(t) and the ideally reconstructed input\nus(t) of another sub-system in frequency domain. Hence,Sys Asampling time-delay compensation reconstruction\nSys B\nsampling time-delay compensation reconstructiony(t) us(t) ˆ u(t)\nGp(s)Gc(s) Gf(s)\nFigure 2: Components of the Real-Virtual Prototype\nunder assumption 2) and 3) holds\nU(s) =1\n∆T·e−sτ\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nGf(s)·Y(s). (2)\nA detailed derivation on why the influence of the sam-\npling can be representedby a scalingwith1\n∆Tunder these\ncircumstances can be found in [4]. The authors carry\nout the Laplace transformation of a coupling process of\na non-iterative offline co-simulation (without delay and\ncompensation).\nThe transfer function Gc(s) represents the frequency do-\nmain correlation of the ideally reconstructed input us(t)\nand the applied input ˆ u(t). It holds\nˆU(s) =Gc(s)·U(s). (3)\nGc(s) is now derived for a feedforward neural network\nwith linear activation function as compensation and ZOH\nas the reconstruction method. Starting point to get Gc(s)\nisequation(1)whichcontainsthesampledbehaviorofthe\ncompensationmethodunderconsiderationofthe constant\ntime-delay τ. Including the ZOH reconstruction leads to\na piece wise constant function in time domain\nˆu(t) =/vector aT/vector u+bwithn∆T≤t <(n+1)∆T.(4)\nThe Laplace transform of ˆ u(t) is defined as\nL{ˆu(t)}(s) =∞/summationdisplay\nn=0/dispvarint(n+1)∆T\nn∆Tˆu(t)e−s∆Tdt.(5)\nThe linearity property of the Laplace transform allows a\nseparatetransformationofeachsummand of ˆ u(t). Forthe\nfirst summand ˆ u1(t) =a1·ut−τequation (5) simplifies to\nL{ˆu1(t)}(s) =ˆU1(s) =∞/summationdisplay\nn=0/dispvarint(n+1)∆T\nn∆Tˆu1(t)e−s∆Tdt.(6)\nSince ˆu1(t) is piece wise constant in, it holds\nˆU1(s) =∞/summationdisplay\nn=0ˆu1(t)e−sn∆T−e−s(n+1)∆T\ns(7)\n=a11−e−s∆T\ns/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nGc1(s)∞/summationdisplay\nn=0ut−τe−sn∆T\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nU1(s).(8)\nThisresultsintheLaplacetransformofthefirstsummand\nGc1(s) ofGc(s), which is equal to a ZOH extrapolationscaled by a1. The transform for all other summands of\nˆu(t) works accordingly under consideration of an addi-\ntional time shift. Therefore it holds\nGc(s) =p−1/summationdisplay\nn=0an+1e−ns∆T·1−e−s∆T\ns+b\nse−s∆T(9)\nfor a compensation taking ppast signal values into ac-\ncount. Finally, using equations (2) and (3), the overall\ncoupling process in frequency domain Ge(s) can be writ-\nten as\nGp(s) =Gf(s)·Gc(s) (10)\n=p−1/summationdisplay\nn=0an+1e−(τ+n∆T)s·1−e−s∆T\ns∆T+b\ns∆Te−(τ+∆T)s.\n(11)\nRemarks:\n•The methodology also allows using other methods to\nreconstruct the input between macro time steps (e.g.\nFOH) by changing equation (4) accordingly [4].\n•When using a piece wise linear activation function\nin the neural network (e.g. ReLU or leaky ReLU)\nthe behavior is nonlinear. Of course in this case the\nLaplacetransformisnotvalidanymore,butsincethe\nnetwork will still maintain a piece wise linear struc-\nture, each linear component can be transformed sep-\narately, to examine the different possibilities of the\ntransmission behavior of the neural network.\nLinear Two-Mass Oscillator\nUsing the example of a two-mass oscillator, it is shown\nhow the transfer function of the coupling process Gp(s)\ncan be used to estimate the overall system behavior of a\ndistributively coupled system. Here, this is done purely\nin simulation with synthetic coupling faults between the\nsub-systems.\nTwo via spring and damper connected models of single\nmass oscillators form the two-mass oscillator as can be\nseen in figure 3. At the dashed line, the oscillator is di-\nm1 m2c1 cc c2\nd1 dc d2x1 x2\nFigure 3: Linear Two-Mass Oscillator\nvided into two sub-models each containing one mass. The\nmodels are coupled using the so called force/displacement\ncoupling [14] approach, where one model calculates the\ncoupling force and receives position and velocity of the\nmass of the other model. Table 1 contains the parame-\nters of the slightly damped coupled system.\nA Laplace transform of the system equations of the two\nsingle mass oscillators [14] yields their transfer functionsTable 1: Parameters of the Two-Mass Oscillator (SI)\nParameter Value\nm1,m2100,1\nc1,c2,cc10\nd1,d2,dc0.01\nGmass1(s) andGmass2(s). Together with the transfer\nfunction of the coupling process Gp(s) the coupled sys-\ntem can be interpreted as a control circuit, which allows\nstability analysis with Bode and Nyquist plots. Figure 4\nsketches the idea.\nGp(s) Gmass1 Gp(s) Gmass2\nFigure 4: Interpretation of the System as Control\nCircuit.\nIn order to determine Gp(s) explicitly, numerical values\nmust be assigned to the parameters of the distributed\nsimulation and the compensation method. The macro\nstep size is set to ∆ T= 0.001sand the delay per coupling\ndirection to τ= 0.003s. The resulting round-trip-time of\nRTT= 0.006sis much lower than e.g. in a coupling\nvia the internet, but since the two-mass oscillator is a\nstrongly coupled system, even this short round-trip-time\nhas an effect on the system behavior.\nThe neural network which is used for compensating the\ndelay is implemented in Python using the Keras pack-\nage[5]. The sizeofthe networkand especiallythe number\nof inputs (considered past signal values) is a trade-off be-\ntween computation resourcesneeded and the capability of\nthe network to represent nonlinear signal behavior. With\na linear activation function of course, the capability ofthe\nnetwork does not increase with the number of neurons [6],\nbut for the time-delay compensation of a nonlinear signal\na single neuron is not sufficient. The chosen network size\noffour input neurons, two hidden neuronsand one output\nneuron is the smallest, which gives good results also for\nnonlinear signals (see nonlinear example).\nFor this example, the weights of the network are obtained\nbytrainingthenetworkbeforesimulationonaself-created\ndata set which is based on simulation results of the two-\nmass oscillator with different initial conditions (without\nfaults). Since this is cumbersome, section presents how\nthe information from the analysis shown here can be used\ntooptimallyparameterizethenetworkinadvancewithout\ntraining. After the training the parameters (equation (1))\ncalculated from the weights of the network read as\n/vector a=\n2.4748\n−0.6470\n−0.1664\n−0.6664\n,b= 0. (12)\nThus, the coupling process including compensation for\nthis example can be determined by equation (11). Multi-plying the transfer functions, as shown in figure 4, results\nin the open-loop transfer function representation of the\noverall distributively coupled system\nGsys(s) =Gmass2(s)·Gp(s)·Gmass1(s)·Gp(s).(13)\nFigure 5 shows the open loop Bode plot of Gsys(s).\nThe blue curve is the reference without coupling effects\n(Gp(s) = 1), red curve is with included faults and triv-\nial (ZOH) compensation and the yellow curve is with in-\ncluded faults and neural network compensation. First of\nall, the two resonance frequencies of the two-mass oscil-\nlator are clearly visible and, more importantly, they do\nnot change by adding faults or the compensation method.\nThis means the qualitative system behavior stays the\nsame. The Bode plot is also useful to verify, that the\nfrequency bandwidth of the system (areas with high fre-\nquency amplification) is below 6rad /sand therefore small\nenough to avoid aliasing (assumption 2). It is further ob-\nserved, that the magnitude for high frequencies is larger\nwhen the compensation method is included in the con-\ntrol circuit. However, this amplification is not critical,\nsince the two-mass oscillator itself dampens very strongly\nin this frequency range. In the bode phase diagram a\nrapidly decreasing phase for high frequencies is visible,\nwhich is typical for systems with delays. It is notice-\n10−110010110210310410−810−3102\nω[rad/s]Magnitude [-]reference\nfaults\nfaults and compensation\n10−1100101102103104−300−200−1000\nω[rad/s]Phase [degree]\nFigure 5: Bode Plot of Gsys(s)\nable that without compensation the phase deviates away\nfrom the reference within the bandwidth, whereas with\ncompensation the phase follows the reference in a larger\nfrequency range.\nHow this affects stability can be investigated using the\nNyquist plot in figure 6. A close look at the area around\nthe critical point Pc= (−1,0j) reveals a deviation of the\nNyquist locus in the case without compensation. The en-\ncirclementoftheNyquist locusaroundthe criticalpoint is\ndifferentfromthereferencewhenthefaultsareintroduced\nand changesback to the reference when the compensation\nis added (Nyquist Stability Criterion). This behavior is\nconfirmedwith simulation results: The system with faults\nis unstable without the compensation and can be stabi-\nlized by adding the compensation method. The referencesystem is of course stable, since a two-mass oscillator is a\nmechanically stable system.\n−5 −4 −3 −2 −1 0−0.0100.01\nReal AxisImaginary Axisreference\nfaults\nfaults and compensation\ncritical point Pc\nFigure 6: Enlargement of Nyquist Plot of Gsys(s) with a\npre-trained Neural Network as Compensation Method\nThis section revealed two important things: Firstly, an\nanalysis of the overall system behavior of a distributed\nreal-virtual prototypes in the frequency domain is possi-\nble. Thisallows,forexample, tocheckinadvancewhether\nasystem canbe stabilized under acertaintime-delaywith\na certain compensation and reconstruction method. Of\ncourse these statements are subject to some simplifica-\ntions and assumptions and can therefore not be trans-\nferred one-to-one to reality, but the basic system behavior\nunder certain coupling errors can be demonstrated with\nthis analysis approach. Secondly, it is shown that the\ntrained neural network is able to compensate for the cou-\npling faults.\nDESIGN OF THE COMPENSATION METHOD\nThe idea is to use the analysis method from the last sec-\ntion not only for checking the system behavior under the\ninfluence of delays and a compensation method, but also\nfor an optimal design of the compensation method itself.\nThis eliminates the need for training the neural network\nin beforehand.\nThe compensation method is very flexible due to the pa-\nrameters /vector aandb. Therefore, requirements for the be-\nhavior of the compensation in the frequency domain are\nmade first, before on that basis an optimization problem\nis defined, by whose solution the optimal parameters are\nfound. Since the compensation depends strongly on the\nconfiguration and properties of the coupled systems, the\nrequirements are defined for the transfer function of the\noverall coupling process Gp(s,/vector a,b).\nThe parameters /vector aandbmust be chosen such that\n•/summationtext\npap+b= 1, whichleadsto lim w→0Gp(s=jω) = 1\nand thus a correct extrapolation of constant signals.\n•the magnitude of Gp(s) within the bandwidth of the\ncoupled system is neutral ( |Gp(jω)|= 1).\n•the phase shift of Gp(s) within the bandwidth of the\ncoupled system is neutral ( ∠Gp(jω) = 0◦).\n•outsidethe bandwidth frequenciesofthe coupled sys-\ntem, the combined magnitude of all coupling pro-cessesGp(s) does not increase faster than the magni-\ntude of the coupled system decreases. This guaran-\ntees that additional amplifications of Gsys(s), which\nare introduced by Gp(s), will be damped by the dy-\nnamics of the coupled system itself.\nThe second and third requirement ensure the compensa-\ntion of the coupling faults in frequency ranges where the\ncoupled system is dynamically active.\nAll requirements are weighted and combined into a sin-\ngle objective function, but a multi-criteria optimization\nwould also be possible. The combined objective function\nof the optimization problem with s=jωreads as\nJ(a,b) =αJa+βJp+γJr, with (14)\nJa=/dispvarintωbw,max\nωbw,min1−|Gp(jω)|\nωbw,max−ωbw,mindω (15)\nJp=/dispvarintωbw,max\nωbw,min∠Gp(jω)\nωbw,max−ωbw,mindω (16)\nJr=/dispvarintωbw,min\n0max(|Gp(jω),1)dω−ωbw,min (17)\n+/dispvarint2π\n∆T\nωbw,maxmax/parenleftbigg\n|Gp(jω)|−/parenleftbiggω\nωbw,max/parenrightbiggv\n,0/parenrightbigg\ndω.\n(18)\nThe bandwidth frequencies of the coupled system are\nwithin the interval [ ωbw,min,ωbw,max] and the remaining\nfrequencies (up to the sampling frequency) in [0 ,ωbw,min)\nand (ωbw,max,2π\n∆T]. The optimization problem results in\nmin\na,bJ(a,b) such that/summationdisplay\npap+b= 1.(19)\nRemarks:\n1. It shall apply γ >> α,β , to ensure that the coupling\nprocess stays inside the magnitude boundary for fre-\nquenciesoutsidethe bandwidthfrequencies. Further-\nmore holds α= 100βto punish a phase difference of\n1◦equally as a magnitude error of 1% [4].\n2. Parameter vdepends on the relative degree rof the\ncoupled system. It holds v=1\n2r\n3. The calculation of the objective function is possible\nwith very little system information. Transfer func-\ntions of the coupled subsystems are not necessary.\nThe bandwidth of the coupling signals could instead\nbe estimated by Fourier transformations of the cou-\npling signals and the relative degree can be set con-\nservatively to one in case of doubt.\nThe bandwidth of the two mass oscillator is in the range\n[1rad\ns,6rad\ns] and the relative degree is r= 2. Thus, the\nnumerically found solution of equation (19) is\n/vector aopt=\n6.5103\n−1.5509\n−9.9296\n5.9702\n,bopt= 0. (20)\nFigure 7 and 8 show the bode plot and the enlarged\nnyquist plot with the neural network compensation, ini-\ntialized with the optimized /vector aoptandbopt. Similar resultsas in the trained version of the neural network are ob-\ntained. The deviation of the nyquist curve between the\nreference and the compensated version is even smaller\nthan with the trained neural network.\n10−110010110210310410−810−3102\nω[rad/s]Magnitude [-]reference\nfaults\nfaults and compensation\n10−1100101102103104−300−200−1000\nω[rad/s]Phase [degree]\nFigure 7: Bode Plot of Gsyswith an Optimally\nInitialized Neural Network as Compensation\n−5 −4 −3 −2 −1 0−0.0100.01\nReal AxisImaginary Axisreference\nfaults\nfaults and compensation\ncritical point Pc\nFigure 8: Enlargement of Nyquist Plot of Gsyswith an\nOptimally Initialized Neural Network as Compensation\nNONLINEAR EXAMPLE\nIn the lastsection, anoptimal time-delaycompensationof\nthe form of equation (1) was designed on the basis of very\nlittle system information. Now the question arises why\nneural networks should be used as compensation method\natall, sincethey aremorecomplexto calculateand imple-\nment than equation (1) (and with linear activation func-\ntion they behave exactly alike). The advantages of the\nneural network is that it can be adapted online by train-\ning in parallel and is also capable of represent nonlinear\nsignal behavior.\nTo show this, a nonlinearity extends the two-mass oscil-\nlator system from figure 3. A mechanical stop prevents\nthe first mass from positioning beyond x1,stop=−0.1m.\nThis is implemented by reversing the velocity of the first\nmassv′=−e·vatx1,stop[8]. The coefficient of resti-tution is e= 0.7, which leads to a partly inelastic col-\nlision. For x1> x1,stopthe system behaves linear and\ncorresponds exactly to the two-mass oscillator from sec-\ntion (all system parameters are identical), whereby the\noptimized compensation parameters from equation (20)\nare also optimal in the nonlinear system.\nFigure 9 shows simulation results with the nonlinear two-\nmassoscillator. The oscillationstartsfrom the initial con-\nditionx1,0=x2,0= 1 and no external forces act on the\nsystem. The bouncing of the first mass is visible and all\ncurves lie on top of each other, which means that the cou-\npling faults haveno significanteffect on x1during the first\n50sof the simulation.\n0 10 20 30 40 5000.51\nTime [s]x1 [m]reference\nfaults\nfaults and compensation\nFigure 9: Position of the First Mass for t∈[0s,50s]\nFigure 10, which shows the same simulations 400 slater,\nrevealsa large deviation of the simulation results and also\ndifferent convergence properties of the coupled system.\nThe slight damping reduces the oscillation amplitude of\n450 460 470 480 490 500−0.100.10.2\nTime [s]x1 [m]reference\nfaults\nfaults and compensation\nFigure 10: Position of the First Mass for t∈[450s,500s]\nx1in each period in the reference simulation as well as\nin the simulation with faults and compensation. In the\nend all the energy will be dissipated and the system will\nend up in its equilibrium. In the simulation with faults\nand without compensation, the oscillation amplitude of\nx1stays at 0 .1mand the mass hits the mechanical stop\nin each period. The energy fed into the system each pe-\nriod due to the instability caused by the faults (figure 6)\nis dissipated by the partly elastic collision with the me-\nchanical stop.Figure 11 shows, however, the problem of the compensa-\ntion method designed on the linear system. The velocity\nreversal at the mechanical stop causes jumps in the cou-\npling signal ˙ x1, which leads to an overshoot by factor\na1,opt= 6.5103 of the compensation. In this particular\ncase, the large compensation errors at the jumps do not\neffect the overall system behavior significantly (system is\nstill stable, see figure 10), but in general such large errors\nshould be avoided.\n0 10 20 30 40 50−4−2024\nTime [s]˙x1[m/s]reference\nfaults and compensation\nFigure 11: Velocity of the First Mass with Linear Neural\nNetwork as Compensation\nHere the advantages of the neural network can be fully\nutilized. Instead of a linear activation function, leaky Re-\nLUs are now used in the two neurons of the hidden layer,\nwhich enables the network to switch between four differ-\nent behaviors. The network is initialized the same way as\nthe linear neural network from the previous example, but\nisnowadaptedduring the simulation. The online training\nis performed in parallel in an external process, in order\nnot to influence the computation time of the compensa-\ntion. For this purpose, the values of the sampled coupling\nsignal are stored during the simulation and sent to the\ntraining process together with the current configuration\nof the neural network. There, training data is created\nfrom the data points. Each training sample consist of an\ninput vector /vector xwithpconsecutive data points and a re-\nsponsey, which contains the data point k=τ\n∆Tsteps\nafter the latest value of the input vector. The neural net-\nwork from the compensation process is then duplicated\nand optimized in the training process, to minimize the\ncost\nC(w) =1\nNN/summationdisplay\ni=1L(f(xi,yi,w)) (21)\nforNtraining samples. The function fstands for the\nneural network with the weights w. The loss function\nLimplements the mean-squared-error algorithm and the\noptimizer Adam [10], a gradient-descent algorithm with\nadaptive learning rate, solves the optimization problem.\nOnce the optimization is complete, the updated weights\nare sent to the compensation process, where the neural\nnetwork is updated. This online adaption process can be\nrepeated several times during a simulation.\nFigure 12 shows the simulation results of ˙ x1again, but\nthis time the online training was active. After the firstjump at 4 .5s, the first online training cycle starts and fin-\nishes a few seconds later. This way, the network is able\nto switch its behavior at the second jump and the pre-\ndicted signal does not overshoot anymore. If the simula-\ntion would now be carried out using the adapted neural\nnetwork as compensation method right from the start,\nalso no overshoot would occur at the first jump.\n0 10 20 30 40 50−4−2024\nStart of first online training\nEnd of first online training\nTime [s]˙x1[m/s]reference\nfaults and compensation\nFigure 12: Velocity of the First Mass with Nonlinear and\nOnline Adapting Neural Network as Compensation\nCONCLUSION AND FUTURE WORK\nCurrent trends in the automotive sector require dis-\ntributed mixed real-virtual test approaches. In this work\nafeedforwardneuralnetworkisusedasageneralizedcom-\npensation method for the coupling faults (constant time-\ndelay) of mixed real-virtual prototypes. In order to be\nable to make statements in advance on how the com-\npensation method will behave in a spatially distributed\nco-simulation, an analysis method in frequency domain\nis proposed that describes the overall coupling process.\nEven with very little information on the coupled subsys-\ntems, it is possible to use the analysis to determine the\nparametersofthe neural network via optimization in such\na waythat the coupling faults are compensated optimally.\nFurthermore, it is shown that the neural network based\ncompensation is also able to adapt to nonlinear signal be-\nhavior using an online learning strategy.\nFuture work will on the one hand focus on the consider-\nation of different fault effects to be able to analyze the\ncoupling process in a more realistic way. On the other\nhand the entire methodology will be tested on a real, in-\ndustrial test bench coupled to a MiL simulation.\nREFERENCES\n[1] John Baillieul and Panos J. Antsaklis. Control and com-\nmunication challenges in networked real-time systems.\nProceedings of the IEEE , 95(1):9–28, 2007.\n[2] Peter Baumann, Martin Krammer, Mario Driussi, Lars\nMikelsons, Josef Zehetner, Werner Mair, and Dieter\nSchramm. Using the distributed co-simulation protocol\nfor a mixed real-virtual prototype. In Proceedings of 2019\nIEEE International Conference on Mechatronics , Ilme-\nnau, Germany, 2019. IEEE Industrial Electronics Society.[3] Martin Benedikt and Edo Drenth. Relaxing stiff system\nintegration by smoothing techniques for non-iterative co-\nsimulation. In IUTAM Symposium on Solver-Coupling\nand Co-Simulation , pages 1–25. Springer, 2019.\n[4] Martin Benedikt, Daniel Watzenig, and Anton Hofer.\nModelling and analysis of the non-iterative coupling pro-\ncess for co-simulation. Mathematical and Computer Mod-\nelling of Dynamical Systems , 19(5):451–470, 2013.\n[5] Fran¸ cois Chollet et al. Keras. https://keras.io , 2015.\n[6] Georg Dorffner, editor. Neural networks for time series\nprocessing . Citeseer, 1996.\n[7] Tulga Ersal, Mark Brudnak, Ashwin Salvi, Jeffrey L.\nStein, Zoran Filipi, and Hosam K. Fathy. Development of\nan internet-distributed hardware-in-the-loop simulatio n\nplatform for an automotive application. American So-\nciety of Mechanical Engineers, 2009.\n[8] Heinz A. Gall. Zur simulation von haftreibung und mech-\nanischen anschl¨ agen. ASIM Nachrichten , 1:4–9, 2001.\n[9] Cl´ audio Gomes, Casper Thule, David Broman, Pe-\nter Gorm Larsen, and Hans Vangheluwe. Co-simulation:\na survey. ACM Computing Surveys (CSUR) , 51(3):49,\n2018.\n[10] Diederik P. Kingma and Jimmy Ba. Adam: A method for\nstochastic optimization. arXiv preprint arXiv:1412.6980 ,\n2014.\n[11] Serge Klein, Rene Savelsberg, Feihong Xia, Daniel Guse ,\nJakob Andert, Torsten Blochwitz, Claudia Bellanger, Ste-\nfan Walter, Steffen Beringer, Janek Jochheim, and Nico-\nlas Amringer. Engine in the loop: Closed loop test\nbench control with real-time simulation. SAE Interna-\ntional Journal of Commercial Vehicles , 2017.\n[12] Martin Krammer, Martin Benedikt, Torsten Blochwitz,\nKhaled Alekeish, Nicolas Amringer, Christian Kater, Ste-\nfan Materne, Roberto Ruvalcaba, Klaus Schuch, Josef Ze-\nhetner, Micha Damm-Norwig, Viktor Schreiber, Natara-\njan Nagarajan, Isidro Corral, Tommy Sparber, Serge\nKlein, and Jakob Andert. The Distributed Co-Simulation\nProtocol for the Integration of Real-Time Systems and\nSimulation Environments. 2018.\n[13] Dale A. Lawrence. Stability and transparency in bilate ral\nteleoperation. IEEE transactions on robotics and automa-\ntion, 9(5):624–637, 1993.\n[14] PuLi. On the Numerical Stability of Co-Simulation Meth-\nods. PhDthesis, Technische Universit¨ at Darmstadt, 2017.\n[15] Niko Maas, Benjamin Hesse, Martin Koppers, and Di-\neter Schramm. Simulator setup according to use case\nscenarios-a human-oriented method for virtual develop-\nment. IEEE, 2014.\n[16] Viktor Schreiber, Valentin Ivanov, Klaus Augsburg,\nMatti Noack, Barys Shyrokau, Corina Sandu, and\nPieter Schalk Els. Shared and distributed x-in-the-loop\ntests for automotive systems: Feasibility study. IEEE\nAccess, 6:4017–4026, 2018.\n[17] Georg Stettinger, Martin Benedikt, Markus Tranninger ,\nMartin Horn, and Josef Zehetner, editors. Recursive FIR-\nFilter design for fault-tolerant real-time co-simulation ,\n2017.\n[18] Georg Stettinger, Martin Horn, Martin Benedikt, and\nJosef Zehetner. A model-based approach for prediction-\nbased interconnection of dynamic systems. IEEE, 2014."    },    {        "title": "2401.11822v1.Axions_and_Primordial_Magnetogenesis__the_Role_of_Initial_Axion_Inhomogeneities.pdf",        "content": "Prepared for submission to JCAP\nAxions and Primordial\nMagnetogenesis: the Role of Initial\nAxion Inhomogeneities\nFilippo Anzuinia,b, Angelo Maggic\naSchool of Physics, The University of Melbourne, Parkville, Victoria 3010, Australia\nbAustralian Research Council Centre of Excellence for Gravitational Wave Discovery (Oz-\nGrav), The University of Melbourne, Parkville, Victoria 3010, Australia\ncTheoretical Particle Physics and Cosmology, King’s College London, Strand, London WC2R\n2LS, UK\nE-mail: filippo.anzuini@gmail.com, angelo.maggi@kcl.ac.uk\nAbstract. The relic density of dark matter in the ΛCDM model restricts the parameter space\nfor a cosmological axion field, constraining the axion decay constant, the initial amplitude of\nthe axion field and the axion mass. It is shown via lattice simulations how the relic density of\naxion-like particles with masses close to the one of the QCD axion is affected by axion-gauge\nfield interactions and by initial axion inhomogeneities. For pre-inflationary axions, once the\nHubble parameter becomes smaller than the axion mass, the latter starts to oscillate, and part\nof its energy density is spent producing gauge fields via parametric resonance. If the gauge\nfields are dark photons and Standard Model photons, the energy density of dark photons\nbecomes higher than the one of the axion, while the high conductivity of the primordial\nplasma damps the oscillations of the photon field. Such a scenario allows for the production\nof small-scale, primordial magnetic fields, and it is found that the relic density of axions\nwith a low decay constant are within the bounds set by the ΛCDM model, while GUT-\nscale axions are far too abundant. It is also shown that initial inhomogeneities of the axion\nfield can change substantially the gauge field production, boosting or suppressing (depending\non the axion parameters and couplings) the magnetogenesis mechanism with respect to an\nhomogeneous axion field. It is found that when the axion mass is far lighter than the QCD\naxion model and the initial axion field is inhomogeneous, weak but cosmologically relevant\nmagnetic field seeds can be generated on scales of the order of 0.1kpc.arXiv:2401.11822v1  [hep-ph]  22 Jan 2024Contents\n1 Introduction 1\n2 Axion and gauge field dynamics 2\n2.1 Homogeneous axion field 2\n2.2 Inhomogeneous axion field 4\n3 Lattice simulations 5\n3.1 Magnetogenesis when TR< f 6\n3.2 Magnetogenesis when TR> f 9\n4 Conclusion 12\nA Additional results for inhomogenous axions 14\nB Higher resolution runs 15\n1 Introduction\nAxions were first introduced as a dynamical solution to the strong-CP problem (QCD axions)\n[1–6]. They are pseudo Nambu-Goldstone bosons emerging from the breaking of the U(1)\nPeccei-Quinn symmetry, with masses and couplings to Standard Model particles which follow\na strict, inverse proportionality to the energy scale at which the Peccei-Quinn symmetry is\nbroken. However, several theoretical and experimental efforts [7] are devoted to the search of\nlight and ultra-light generalizations of QCD axions (generally called axion-like particles, or\nALPs), whose masses and couplings differ from the QCD axion model. From a cosmological\npoint of view, ALPs have been proposed as candidates for the inflaton field driving the cosmic\nexpansion during inflation [8, 9], they may constitute the totality of dark matter [10], and lead\nto a rich phenomenology when coupled to other Beyond Standard Model and Standard Model\nparticles [11, 12], such as magnetogenesis [13] and the production of primordial gravitational\nwaves [14–17].\nCurrent constraints on the relic density of dark matter [18] restrict the viable parameter\nspace for axions, if they constitute the entirety of dark matter. This translates on bounds on\nthe axion decay constant f, the initial amplitude of the axion field and its mass. For the QCD\naxion, the decay constant is constrained to f≲1012GeV (for an initial amplitude of the axion\nfield of order ≈f), while higher values of fare compatible with the ΛCDM model only if the\ninitial amplitude of the axion field is ≪f. One way to relax such constraints and deplete\nthe axion abundance is allowing for the coupling of axions (either QCD axions or ALPs) with\ngauge fields in the Early Universe, leading to the sourcing of U(1)Abelian gauge fields (such\nas dark photons) [11–13], or SU(2)non-Abelian gauge fields (see for example [19], where the\naxion plays the role of the inflaton field). If the axion-gauge field couplings are sufficiently\nlarge, axion oscillations lead to a resonant production of the gauge fields [11, 12, 20, 21], which\nterminates when the gauge field abundance becomes comparable to the one of the axion, while\nthe latter suffers a sharp drop due to the backreaction of the sourced gauge fields. For the\nspecific case of the axion-photon and axion-photon-dark photon coupling, axions may lead to\n– 1 –the generation of primordial magnetic fields. The magnetic fields produced in the primordial\nuniverse may constitute the seeds for the large-scale, feeble magnetic fields that permeate\ngalaxies, galaxy clusters and cosmic voids [13, 22, 23].\nIn this work we focus on the interaction of ALPs with Abelian gauge fields, and in-\nvestigate via lattice simulations how the production of primordial, standard and dark elec-\ntromagnetic fields at the expense of the axion field leads to a depletion of the axion abun-\ndance. We first simulate axions with decay constants above the inflationary energy scale (with\nf∈[1014,1016]GeV), and assume that the Peccei-Quinn symmetry is not restored after the\nend of inflation, i.e. that the reheating temperature TRsatisfies TR< f. In these conditions\nthe axion field is initially homogeneous, and we find that while the ALP-photon coupling\ndoes not lead to a significant magnetic field production, large couplings among axions, dark\nphotons and photons [24, 25] can overcome the damping of photon oscillations due to the high\nconductivity of the primordial plasma [26], producing early seeds of small-scale primordial\nmagnetic fields. Such a mechanism was exploited in [13] for ultra-light ALPs after the Big\nBang Nucleosynthesis epoch. Here we extend those results and apply them to earlier times,\nwhen the temperature of the universe Tis in the range T≳1MeV, and to heavier ALPs\n(with masses in the range 10−10-10−8eV) compared to the case studied in [13], where masses\nof the order of 10−17eV were considered. We find that while axions with f≲1015GeV have\nan abundance lower than ΩDM(where ΩDMis the dark matter relic density [18]), axions with\nfclose to the GUT scale remain overabundant. We also perform simulations where the axion\nfield is initially inhomogeneous, as it would be the case for example for axion field configu-\nrations emerging from the decay of a network of strings. In particular, we show that in the\nlatter case the magnetogenesis mechanism can be far less efficient than in the pre-inflationary\nscenario. However, for certain coupling strengths and axion parameters, the intensity of the\nsourced magnetic fields increases with respect to scenarios where magnetogenesis is triggered\nby an initially homogeneous axion field. We also show that ultra-light ALPs can produce\nweak, standard magnetic fields on scales of the order of 0.1kpc.\nThis work is structured as follows. In Section 2 we outline the model lagrangian density\nemployed throughout this work. We also describe how initial inhomogeneities in the axion\nfield can arise. The results of our lattice simulations are presented in Section 3.\n2 Axion and gauge field dynamics\nIn this section we discuss the microphysical model that we input in our lattice simulations.\nWe first describe in Section 2.1 the prototype model and introduce the relevant couplings,\nassuming that the axion field interacts with dark photons and photons. We highlight the\nrole played by the primordial plasma and of the source terms appearing in the equations of\nmotion. We then turn to inhomogeneities in the initial axion field configuration in Section\n2.2.\n2.1 Homogeneous axion field\nThe lagrangian density for a system composed by ALPs, dark photons and photons reads\n[11, 13]\n– 2 –L=1\n2∂µϕ∂µϕ−V(ϕ)−1\n4FµνFµν−gϕA\n4fϕFµν˜Fµν−1\n4XµνXµν\n−gϕX\n4fϕXµν˜Xµν−gϕAX\n2fϕFµν˜Xµν+JµAµ. (2.1)\nIn Eq. (2.1), ϕis the ALP field, and the ALP potential reads\nV(ϕ) =m2\nϕf2\u0014\n1−cos\u0012ϕ\nf\u0013\u0015\n, (2.2)\nwhere mϕis the ALP mass. The field strength tensor Fµνis given by Fµν=∂µAν−∂νAµ\n(where Aµisthephotonfield), while Xµν=∂µXν−∂νXµisthecorrespondingoneforthedark\nphoton field ( Xµ). The ˜Fµνand ˜Xµνtensors correspond to the duals of FµνandXµνrespec-\ntively. The coupling constants for the axion-photon, axion-dark photon and axion-photon-\ndark photon interactions are denoted by gϕA, gϕXandgϕAXrespectively [11–13, 24, 25], which\nin principle can be large [27], while Jµis the standard four-dimensional electromagnetic cur-\nrent. Note that in Eq. (2.1) we neglect a possible kinetic mixing between dark photons\nand photons, i.e. a lagrangian density contribution of the form ϵFµνXµν[28]. Due to the\nsmallness of ϵ(typically ϵ≲10−3), we expect a negligible correction with respect to the other\ninteraction terms proportional to the spacetime gradients of the axion field. For simplicity\nwe will consider the kinetic mixing term in a separate work.\nUsing the FRW background metric\nds2=a2(τ)\u0000\ndτ2−dx2\u0001\n(2.3)\n(where a(τ)is the scale factor and τis the conformal time), the equations of motion read\n¨ϕ+2H˙ϕ−∇2ϕ+a2∂V\n∂ϕ=−1\na2\u0014gϕA\nf˙A·∇×A+gϕX\nf˙X·∇×X+gϕAX\nf\u0012\n˙X·∇×A+˙A·∇×X\u0013\u0015\n,(2.4)\n¨A+σ\u0014\n˙A+v×(∇×A)\u0015\n+∇×(∇×A) =gϕA\nf\u0012\n˙ϕ∇×A−∇ϕ×˙A\u0013\n+gϕAX\nf\u0012\n˙ϕ∇×X−∇ϕ×˙X\u0013\n,(2.5)\n¨X+∇ ×(∇ ×X) =gϕX\nf\u0012\n˙ϕ∇ ×X− ∇ϕ×˙X\u0013\n+gϕAX\nf\u0012\n˙ϕ∇ ×A− ∇ϕ×˙A\u0013\n. (2.6)\nIn the equations above, we imposed the temporal gauge A0= 0andX0= 0, and\noverdotsrepresentderivativeswithrespecttoconformaltime, while HistheconformalHubble\nparameter(reading H=aH,where HistheHubbleparameter)and σistheconformalelectric\nconductivity of the primordial plasma. Note that the term including σis proportional to the\nvelocity of the plasma v. A self-consistent treatment of such term requires to obtain vby\nsolving the appropriate fluid equations, which must be coupled with the equations of motion\nfor the axion-gauge field system. We defer such task to a future publication, and below we\nfollow [13] and neglect such term. The conformal conductivity σis related to the physical\nconductivity σphviaσph=σ/a. We use the expression in [26] for TQGP≪T≪MW(where\n– 3 –TQGPis the temperature of the quark-gluon plasma and MWis the mass of the Wboson)\ngiven by\nσph=J\nE=Nl\nNl+ 3PNq\nqQ2q3ζ(3)\nln(2)αln(1/αN l)T , (2.7)\nwhere Jis the electric current density, Ethe electric field, NlandNqcount, respectively, the\nlepton and quark species with masses below T,Qqis the quark charge in units of the electric\ncharge e, and α=e2/4π. At lower temperatures, we use the expression [26]\nσph=3ζ(3)\nln(2)αln(1/α)T . (2.8)\nThe temperature is evolved in time using the relation in [29], reading\nTMeV=\u00122.4√g∗t\u00131\n2\n, (2.9)\nwhere TMeVistheuniversetemperatureinMeV, g∗denotestheenergydensityeffectivedegrees\nof freedom, and tis the cosmic time.\nThe system of coupled equations of motion leads to a complex, non-linear dynamics\nthat requires lattice simulations. However, it is possible to get some insight by inspecting the\nsource terms in the equations of motion. If fis larger than the inflationary Hubble parameter\nHIandTR< f, the initial axion field is homogeneous (pre-inflationary axions). If the axion is\nfarfromtheminimumofitspotential(misalignementmechanism), itsenergydensityisseveral\norders of magnitude larger than the initial electromagnetic energy density of the AandX\nfields, and its oscillations are damped by the expansion rate of the universe (given by the term\n2H˙ϕ). When the Hubble parameter becomes smaller than the axion mass, the axion begins\nto oscillate. The time derivative of the axion field acts as a source term for the gauge fields\n(see the terms on the right-hand side of the corresponding equations of motion), producing\nthe gauge fields via parametric resonance [11–13, 20, 21, 30], and leading to a “dynamo” effect\nthat generates electric and magnetic fields1. While the energy density of the dark-photon field\ngrows resonantly, the plasma conductivity curbs the growth of the photon field due to axion\noscillations, acting as a damping term similarly to the effect due to the expansion rate of the\nuniverse for the axion. For example, the term proportional to gϕA˙ϕ∇×Adoes not lead to an\nefficient dynamo mechanism, since it depends on the weak, standard magnetic field ∇ ×A.\nHowever, as shown below, it is possible to partially overcome the plasma-damping effect via\nthe axion-photon-dark photon coupling terms, such as gϕAX˙ϕ∇×X. The latter term depends\non the magnetic field associated with X, rather than the standard, suppressed magnetic field,\nand allows for a more efficient production of standard electromagnetic fields. In general we\nfind that while the dark electromagnetic fields are large, the standard electromagnetic fields\nremain subdominant due to the high plasma conductivity.\n2.2 Inhomogeneous axion field\nInhomogeneities develop naturally for post-inflationary axions, i.e. when f≲HI. When\nthe temperature of the universe is of the order of T∼f, topological defects such as do-\nmain walls and networks of strings form via the Kibble mechanism [32–41], which release\n1We note that if the axion is initially close to the minimum of its potential, its oscillations can induce\nperturbative particle production, cf. for example [31] and references therein.\n– 4 –relativistic axions. Numerically, there are two possible approaches to study the evolution of\nthe axion field in this scenario [41]. The first consists in solving the equation of motion for\nthe Peccei-Quinn field, simulating the formation of topological defects and the production of\nrelativistic axions, which later scale as non-relativistic matter. The second approach relies on\nthe construction of initial conditions for axions from the string network spectrum, and the\nsimulations start without the presence of topological defects, but with an axion field whose\ndegree of inhomogeneity depends on the properties of the topological defects.\nHowever, even for pre-inflationary axions there can be conditions in which the axion field\nbecomes inhomogeneous. For example, if the reheating temperature TR(with TR≲√HIMPl,\nwhere MPlis the Planck mass) after the inflationary period is larger than the axion decay\nconstant, the global symmetry associated with the axion is restored after inflation. At later\ntimes, the breaking of such symmetry leads to the formation of a cosmic network of strings,\nand the resulting axion field is inhomogeneous.\nTo study how initial inhomogeneities affect the ability of the axion to source gauge fields,\nwe perform simulations of axions with decay constants in the range f∈[1011,1015]GeV. Such\na range encompasses scenarios in which inhomogeneities of the initial axion field develop when\nf≲HI, or when TR≳f. We use simplified initial conditions, which are meant to reproduce\n(at least qualitatively) the complex axion field configurations due to a network of strings and\ndomain walls [33–35, 39–42]. We initialize the axion field drawing a random value of the\nfield for each Hubble patch from a gaussian distribution [43]. We assume that the field is\nhomogeneous on sub-Hubble scales, which corresponds to considering only the misalignement\ncontribution for the axion field [42, 43]. We stress that such initial conditions are employed in\norder to study qualitatively how the gauge field production mechanism responds to an initial,\nhighly inhomogeneous field configuration. Moreover, the typical resolution employed in the\nliterature for simulations with post-inflationary axions (without gauge fields) is >10003grid\npoints, which are necessary to resolve properly the formation of small-scale axion structures.\nRunning such simulations demands a significant computational power, which increases further\nif the axion interacts with multiple gauge fields, which is our primary goal. To the best of\nour knowledge, there are no numerical campaigns in the literature where such high-resolution\nsimulations are employed to study inhomogeneous axions interacting with two gauge fields.\nThe capabilities of the numerical code that we developed do not allow us to perform runs in\na reasonable time including axions and gauge fields with such resolutions, and rather allow\nus to use the standard resolution employed in the literature of axions interacting with dark\nphotons only, typically employing 1283and2563grid points. While the limited resolution\nof our runs makes our results for initially inhomogeneous axion fields qualitative, we aim at\nperforming higher-resolution simulations which will be presented in future work.\n3 Lattice simulations\nWe now report the details of the lattice simulations performed in this work. We consider\nthe ALP in our model to be slightly lighter than the QCD axion at zero temperature, i.e.\nmϕ= 0.8mQCD(f). In all of the following simulations, the gauge fields are initialized in\nmomentum space as in [12] (albeit dark photons are massless in our model), with root-mean\nsquare amplitude in Fourier space\np\n⟨|Ak|2⟩=1p\n2|k|/a, (3.1)\n– 5 –Figure 1 : Volume-averaged energy densities for axions (with f= 1014GeV), dark photons\nand photons ( ρϕ, ρXandρArespectively) versus conformal time. Left panel: axion and dark\nphoton energy densities for gϕX=gϕAX = 60andgϕX=gϕAX = 80. The inset, smaller\npanel displays (in linear scale for mϕτ) that the orange solid line scales as a−3at later times\nwith respect to the teal curve due to the stronger coupling. Right panel: photon energy\ndensity for the same couplings. In all of the panels, the axion-photon coupling is fixed to\ngϕA=−2.2×10−3, and the axion mass is mϕ= 4.7×10−8eV. The initial value of the axion\nfield is ϕ=f.\nwhere kdenotes the comoving momentum. The same initialization procedure is applied to\nthe dark photon field. The simulations are performed in a cubic volume with linear dimension\nL=π/(4mϕ)using 1283grid points, applying periodic boundary conditions at the edges of\nthe simulation box and starting at the conformal time τi= 0.1L. We follow [12] and evolve\nthe system of equations in time using a leap-frog algorithm and an implicit scheme where the\nterms multiplying the updated values of the field time-derivatives are inserted in a matrix\nMlm. We solve the system MlmVm(τ+ ∆τ) =Zl, where Vmis a vector containing the\nupdated values of the time-derivatives of the fields, Zlis a vector containing the values of the\ntime-derivatives at the previous time step, and ∆τis the conformal time-step. We also check\nthat the Gauss constraint [11, 16] is satisfied in all of the runs.\n3.1 Magnetogenesis when TR< f\nWe show results of our simulations with f= 1014GeV in Figure 1, assuming that TR< fand\nhence that the axion field is initially homogeneous with initial amplitude ϕ=f. In the left\npanel of Figure 1 we show the volume-averaged energy densities of ALPs and dark photons\n(denotedwith ρϕandρXrespectively)normalizedby m2\nϕf2andmultipliedby (a/aosc)3, where\nwe define aoscas the value of the scale factor when H=mϕ. The axion-photon coupling is\nfixed throughout this paper to gϕA=−2.2×10−3(as for the KSVZ axion model [5]), while we\nvary the remaining couplings and use gϕX=gϕAX= 60(teal curves) and gϕX=gϕAX= 80\n(orange curves). At the beginning of the simulation, the axion is pinned by the Hubble term.\nLater, the axion starts to oscillate and its energy density ρϕscales as a−3until ρXbecomes\ncomparable to ρϕ. Then, the latter drops sharply due to the backreaction of the sourced\ndark photon field, and the axion field develops significant inhomogeneities. After a transient\nperiod in which axions and dark photons keep interacting, at the end of the simulations the\n– 6 –101102\nk/mφ10−710−610−510−410−3d(a3ρφ)/dlnk[a3\noscm2\nφf2]\nmφτ= 10\nmφτ= 100\nmφτ= 160\n101102\nk/mφ10−510−410−310−210−1d(a4ρX)/dlnk[a4\noscm2\nφf2]\nmφτ= 10\nmφτ= 100\nmφτ= 160Figure 2 : Spectra of axions (left panel) and dark photons (right panel) for f= 1014GeV,\nmϕ= 4.7×10−8eV and gϕX=gϕAX= 60.\naxion energy density decreases as a−3(i.e. axions behave as cold dark matter), while the\nenergy density of dark photons scales as a−4(i.e. the gauge fields behave as dark radiation).\nThe stronger is the coupling, the shorter is the conformal time interval required for the\nenergy density of dark photons to become higher than the one of the axion. The right panel\nin Figure 1 displays the evolution of the volume-averaged, standard electromagnetic energy\ndensity ρAfor the coupling values employed in the other panel. We find that ρAis far smaller\nthan ρϕandρX, attaining a peak of ρA(a/aosc)3/m2\nϕf2≈6×10−34forgϕX=gϕAX= 80.\nWhile the growth of the dark photon field takes place via the conversion of axion energy\ndensity into dark photons (with efficiency regulated by the magnitude of the coupling gϕX),\nthe ALP field does not source photons efficiently. As described in the previous section, the\nphoton oscillations are damped by the highly conductive nature of the primordial plasma, the\nstandard magnetic field ∇ ×Aremains small, and the gϕA˙ϕ∇ ×Asource term is inefficient\neven for gϕA≈ O(100). However, when the dark photon abundance increases due to the\naxion-driven dynamo, the coupling gϕAXallows for a secondary dynamo effect that sources\nstandard electric and magnetic fields. This secondary dynamo is more efficient than the\none due to the axion-photon coupling because the corresponding source term in the photon\nequation of motion (Eq. (2.5)) is proportional to gϕAX˙ϕ∇ ×X, with |∇ ×X| ≫ |∇ × A|.\nAfter the exponential growth of ρA, the latter scales as a−4. We note that the coupling gϕAX\nin our model is larger than the one employed in [13]. This is because the constraints on the\nproduct of the axion-photon-dark photon constant and the sourced magnetic for ultra-light\naxions (with masses ≲10−14eV) are stricter than for heavier axions [23].\nIn Figure 2 we show the axion spectra (left panel), as well as the dark photon spectra\n(right panel) for the runs in Figure 1. In both panels, the spectra show a single peak which\nis first located at k/m ϕ≈30and later moves to higher momenta ( k/m ϕ≳100) [16]. As\nthe conformal time increases, one can see how the power transfers to higher momenta due\nto the scattering of low-momentum axions with high-momentum dark photons, leading to\nthe production of axions and dark photons with higher momenta and hence to the growth of\nthe spectra for k/m ϕ≳80. The spectrum of the standard electromagnetic energy density is\n– 7 –101102\nk/mφ10−3810−3710−3610−35d(a4ρA)/dlnk[a4\noscm2\nφf2]\nmφτ= 10\nmφτ= 160Figure 3 : Spectrum of the photon energy density. The axion parameters and couplings are\nas in Figure 2.\n50 60 70 80\ngφX10−410−310−210−1100Ωah2\nf= 1014GeV\nf= 1015GeV\nf= 1016GeV\n40 50 60 70 80\ngφX10−2710−2610−25B0(G)\nf= 1014GeV\nf= 1015GeV\nf= 1016GeV\nFigure 4 : Relic axion energy density versus the coupling gϕX(left panel) and standard\nmagnetic field strength redshifted to today (right panel). The mass of the axion is fixed to\nmϕ= 4.7×10−8eV (for f= 1014GeV), mϕ= 4.7×10−9eV (for f= 1015GeV) and\nmϕ= 4.7×10−10eV (for f= 1016GeV). The results are obtained with ϕ=fat the start of\nthe simulations, and with gϕX=gϕAX.\nreported in Figure 3. The spectral features are similar to the ones of the dark photon spectra,\nand display a transfer of power from smaller to larger momenta at late times.\nIn Figure 4 we show the axion relic density Ωah2(left panel) and the strength of the\nstandard magnetic field redshifted to today B0(right panel) versus the axion-dark photon\ncoupling (we use gϕX=gϕAX). In the left panel, the axion relic density Ωah2approaches\nthe value ΩDMh2= 0.12(horizontal black line) for f= 1015GeV, while it is lower or higher\nforf= 1014GeV and f= 1016GeV respectively. The right panel shows that, as expected,\n– 8 –the higher is the coupling, the more intense is the relic magnetic field strength. Note that\nwhile B0is maximal for f= 1016GeV, it corresponds to having Ωah2≫0.12; hence, only\nforf≤1015GeV we obtain that Ωah2is compatible with ΩDMh2, and at the same time\nwe find a sizable B0. We stress however that due to the typical axion masses studied in this\nsection, the correlation length of the magnetic fields in our simulations (of the order of 1-10\npc) is far shorter than the typical correlation lengths of large-scale magnetic fields. In the\nnext section however we consider lighter axions and hence larger correlation lengths for the\nsourced magnetic fields.\n3.2 Magnetogenesis when TR> f\nWe now study the production of gauge fields in the presence of an initially inhomogeneous\naxion field, fixing f= 1014GeV and assuming TR≳f, as outlined in Section 2.2. We\nintroduce a parameter ζthat quantifies the ratio between the initialtotal energy density of\nthe axion (i.e. gradient plus potential energy densities) and the magnitude of the potential\nenergy of the axion given by m2\nϕf2, i.e. ζ=ρϕ(τi)/(m2\nϕf2). As ζincreases, so does the\ninitial average value of the axion field, leading to a higher axion relic density2. For ζ≈106,\nthe average value of the axion field is ⟨ϕ2/f2⟩ ≈π2/3[10]. As in Section 3.1, we start the\nsimulations at τi= 0.1L, which in the scenario considered in this section corresponds to the\nmoment when relativistic axions are emitted from the topological defect network and start\ninteracting with the gauge fields.\nFigure 5 reports the energy densities of axions, dark photons and photons in the top\nleft, top right and bottom panels respectively for three different values of ζand for gϕX=\ngϕAX = 60. At the beginning of the simulations, the top left panel shows that ρϕscales\nasa−4, i.e. axions behave as radiation. This is because the initial gradient energy of the\naxion is far larger than the potential term, and the axion behaves as a nearly massless field.\nWhile ρϕis initially far larger than in the scenario studied in Section 3.1 ( ζ≫1), it quickly\napproaches the values attained in Figure 1. Once the gradient energy density of the axion\nbecomes comparable or smaller than its potential energy density, ρϕscales as a−3, behaving\nlike dark matter. The higher is ζ, the longer it takes for the axion field to scale as a−3(cf. the\nblack and teal curves with the red and orange ones). The top right panel of Figure 5 shows\nthat the production of dark photons is far less efficient when the axion field is inhomogeneous\nand, for ζ≈103,ρXremains well below ρϕ. The high degree of inhomogeneity of the axion\nfield prevents the latter from acting as a coherently oscillating condensate, limiting the energy\ntransfer from the axion to the gauge sector compared to when TR< ffor the same couplings\n(cf. Figure 1). The situation changes when ζ≳104, allowing ρXto become larger than ρϕ.\nThe bottom panel shows the growth of ρA. The latter becomes a few orders of magnitude\nlarger than in Figure 1 when ζ≳104, signalling that an inhomogeneous axion field has a\nbeneficial effect on the magnitude of the produced standard magnetic field. Thanks to the\nlarge axion gradients, the source terms in the gauge field equations of motion lead to a more\nefficient magnetogenesis despite the suppression due to the plasma conductivity.\nIn Figure 6 we compare ρAobtained for pre-inflationary axions (black curves, with the\ninitial condition of the homogeneous axion field given by ϕ=f) with results obtained for\nTR> f. For both the gϕX=gϕAX = 40andgϕX=gϕAX = 80cases, when ζ≳105\nthe production of standard electromagnetic fields is boosted, exceeding by a few orders of\n2We note that simulations of post-inflationary axions report relic densities which can be far larger than the\none obtained from the standard misalignement mechanism for pre-inflationary axions (cf. [10] and references\ntherein).\n– 9 –Figure 5 : Production of gauge fields with f= 1014GeV, mϕ= 4.7×10−8eV,gϕX=gϕAX=\n60and assuming that TR> f. The top-left panel shows that the axions scale as radiation\nuntil mϕτ≈100, i.e. until the gradient energy is much larger than the potential term. The\ntop-right panel shows that ρX> ρϕonly for ζ≳104. Similarly, the bottom panel shows that\nthere is no production of standard electromagnetic fields for ζ≲103.\nmagnitude the results obtained with pre-inflationary axions. In particular, we find that ρA\nobtained for ζ≈106is four orders of magnitude larger than the values attained by the black\ncurves. We report additional results for initially inhomogeneous axions in Appendix A. We\nalso confirm the magnetogenesis boost due to inhomogeneous axions by performing a set of\nshort simulations using 2563and3203grid points and f= 1014GeV, which are reported in\nAppendix B along with the spectra of ρX. The spectra of ρAobtained from such runs are\nreported in Figure 7, where we fix the couplings to gϕX=gϕAX= 80and vary ζ. The teal and\nred curves peak at around k/m ϕ≈40, while the orange curve shows a peak at k/m ϕ≈80.\nThe latter curve also shows a bump at relatively low momenta ( k/m ϕ≈20) compared to the\nflatter slope of the spectra for ζ≈104andζ≈105. We emphasize that the results in this\nsection should be interpreted qualitatively, and higher-resolution runs are required to resolve\nproperly the growth of ρX,ρAand their spectral features.\nFigure 8 reports the relic axion density (left panel) and the standard magnetic field\nstrength redshifted to today (right panel) versus gϕX(we use gϕX=gϕAX). We find that for\nζ≲105,Ωah2is well below ΩDMh2(given by the horizontal black line), and attains values\nsimilar to the ones found in the left panel of Figure 4 for f= 1014GeV. On the other hand,\n– 10 –10−1100101102\nmφτ10−4510−3910−3310−27ρAa3/(a3\noscm2\nφf2)gφX,gφAX= 40\nζ≈104\nζ≈105ζ≈106\nPre\n10−1100101102\nmφτ10−4510−3910−3310−27ρAa3/(a3\noscm2\nφf2)gφX,gφAX= 80\nζ≈104\nζ≈105ζ≈106\nPreFigure 6 : Comparison of the standard electromagnetic energy produced by pre-inflationary\naxions (black curves, obtained with the initial condition ϕ=f) and an initially inhomoge-\nneous axion field for f= 1014GeV and mϕ= 4.7×10−8eV. The left panel is obtained for\ngϕX=gϕAX= 40, while the right panel for gϕX=gϕAX= 80.\n101102\nk/mφ10−3610−3410−3210−30d(a4ρA)/dlnk[a4\noscm2\nφf2]\nζ≈104\nζ≈105\nζ≈106\nFigure 7 : Spectrum of the standard electromagnetic energy density for an initially inhomo-\ngeneous axion field with parameters fixed to f= 1014GeV , mϕ= 4.7×10−8eV and with\ngϕX=gϕAX= 80.\nforζ≈106we find that Ωah2approaches ΩDMh2. The right panel shows that, for ζ≈106,\nB0is two orders of magnitude larger than in the scenario studied in Figure 4 for f= 1014\nGeV, with a maximum strength of B0≈8×10−25G, which is close to the value obtained for\nthe case in which magnetogenesis takes place after the Big Bang Nucleosynthesis [13].\nIn general, the results in this section show that if the axion is initially inhomogeneous\n(TR> f), theproductionofgaugefieldsisseverelyhamperedfor ζ≲104andgϕX, gϕAX≲60.\nIn this case dark photons remain far less abundant than the axion, in contrast to the results in\nSection 3.1, which correspond to the scenario thoroughly studied in the literature [11, 12, 16].\nHowever, inhomogeneous axions can lead to stronger magnetic fields, which in the case of\n– 11 –40 50 60 70 80\ngφX10−410−310−210−1100Ωah2\nζ≈104\nζ≈105\nζ≈106\n40 50 60 70 80\ngφX10−2910−2710−2510−23B0(G)\nζ≈104\nζ≈105\nζ≈106Figure 8 : Relic axion density versus the coupling gϕX(left panel) and standard magnetic\nfield strength redshifted to today (right panel) for axions with f= 1014GeV, mϕ= 4.7×10−8\neV and assuming that TR> f. We also use gϕX=gϕAX.\nphotons are roughly two orders of magnitude more intense than in the scenario TR< f, at\nleast for high values of ζand relatively large couplings.\nIn the results above the typical magnetic field correlation length is smaller than the\none characteristic of large-scale, cosmological magnetic fields due to our choice of axion pa-\nrameters. It is possible to obtain larger correlation lengths by considering lighter axions.\nFigure 9 shows the standard relic magnetic field strength for axions with f= 1015GeV,\nmϕ≈5.9×10−12eV3andmϕ≈2.4×10−11eV, and ζ≈106(assuming TR> fand hence\nthat the axion field is initially inhomogeneous). With such parameters, the magnetic field has\na correlation length of order O(0.1)kpc, and attains B0≈10−25G for the relatively large\ncouplings gϕX=gϕAX= 150. Note that the sourced magnetic field is weaker with respect to\nthe intensities reported in the right panel of Figure 8. This is mostly due to the numerical\ninitialization procedure employed in our code (cf. Appendix A).\n4 Conclusion\nThe parameter space of axions is restricted by the ΛCDM model, which gives a relic dark\nmatter density of ΩDMh2= 0.12[18]. This translates into bounds on the initial axion field\namplitude, mass and on the axion decay constant.\nWhen the axion field is initially homogeneous, we find that the production of dark\nphotons and photons via axion oscillations alleviates the tension between the relic density\nof axions with large decay constants and ΩDM, and allows for the production of small-scale,\nstandard magnetic fields with strengths redshifted to today of the order of 10−27-10−26G and\ncorrelation lengths of the order of 1-10pc. We also perform the first simulations where the\ninitial axion field is highly inhomogeneous, which is typical for post-inflationary axions (i.e.\n3Note that the axion parameters mϕ≈5.9×10−12eV and f= 1015GeV are in principle excluded by the\nrecent bounds on fandmϕobtained in [44], where the constraints are obtained from compactifications of IIB\nstring theory (cf. also [45] for current bounds on mϕandf). Here we consider such a combination of mϕand\nffor illustration purposes only.\n– 12 –80 100 120 140\ngφX10−2610−2510−24B0(G)\nmφ≈5.9×10−12eV\nmφ≈2.4×10−11eVFigure 9 : Standard relic magnetic field strength for ultra-light axions with f= 1015GeV\nand for mϕ= 5.9×10−12eV (teal curve) and mϕ= 2.4×10−11eV (orange curve). The relic\nmagnetic field strength is obtained using gϕX=gϕAXandζ≈106.\naxions with f < H I), while for axions with large fthese conditions are possible if TR> f.\nIn such a scenario, the axion relic density is close to ΩDMh2forf= 1014GeV and ζ≈106.\nWe also show that a cosmologically relevant abundance of dark photons is more challenging\nto achieve, unless the gϕXcoupling is large (with gϕX≳60) and/or ζ≳104. On the other\nhand, an initially inhomogeneous axion field has the beneficial effect of boosting the strength\nof the sourced standard magnetic field by two orders of magnitude, if the axion gradients\nand the axion-photon-dark photon coupling are relatively large. In this case, the strength\nof the standard magnetic field redshifted to today attains ≈10−25G, both in the case of\nALPs with masses close to the QCD axion, and for ultra-light ALPs with masses of the order\nof10−12−10−11eV. For the latter, the typical correlation length is of the order of ≈0.1\nkpc, showing that it is possible to obtain large-scale magnetic field seeds that are relevant for\ngalactic scales.\nThe present work can be expanded in several ways. Among them, it is desirable to\nimprove the initial conditions for inhomogeneous axions employed in this work, for example\nalong the lines of [41]. Moreover, we aim at performing higher-resolution runs with 5123grid\npoints or more to resolve properly the axion-gauge field dynamics and their spectra when the\naxion field is initially inhomogeneous.\nAcknowledgments\nIt is a pleasure to thank Liina Jukko for providing us with the code to compute the spectra\nand David J. E. Marsh, Wen-Yuan Ai and Andrew Melatos for many useful comments and\ndiscussions. FA thanks Jun’ichi Yokoyama, Ryusuke Jinno and the RESCEU group at the\nUniversity of Tokyo, and Shuichiro Yokoyama, Kiyotomo Ichiki and the Kobayashi-Maskawa\nInstitute for hospitality while part of this research was completed. FA is supported by the\nAustralian Research Council (ARC) Centre of Excellence for Gravitational Wave Discovery\n(OzGrav), through project number CE170100004. The numerical simulations presented in\n– 13 –this work were performed on the Ngarrgu Tindebeek supercomputing cluster at Swinburne\nUniversity.\nFigure 10 : Production of gauge fields by post-inflationary axions with f= 1011GeV,\nmϕ= 4.7×10−5eV and using gϕX=gϕAX= 60. Axions scale as radiation until mϕτ≈100.\nThe top-right panel shows that the production of dark photons is efficient only for ζ≳104,\nandthebottompanelshowsthatstandardelectromagneticfieldsarenotproducedfor ζ≲103.\nAppendix\nA Additional results for inhomogenous axions\nIn Section 3.2 we report results for an initially inhomogeneous axion field using f= 1014\nGeV, which requires to have TR> f. Here we study the case of axions with f= 1011GeV,\nwhich can be labeled as post-inflationary axions if f < H I. Figure 10 displays ρϕ,ρXand\nρA, which show a similar behaviour to Figure 5 in Section 3.2. The main difference can be\nseen in the bottom panel: ρAattains a larger maximum value than in Figure 5, which is due\n– 14 –to the different initial conditions for the gauge fields employed for runs with f= 1011GeV\nandf= 1014GeV. Numerically, our initialization procedure follows the one in the publicly\navailable code LATTICEEASY [46], so the initial magnitude of the gauge fields depends on\nthe axion parameters (besides the size of the simulation box and the grid-spacing).\nB Higher resolution runs\n10−1100\nmφτ10−410−1102105ρXa3/(a3\noscm2\nφf2)\nζ≈104\nζ≈105ζ≈106\n10−1100\nmφτ10−4010−3510−3010−25ρAa3/(a3\noscm2\nφf2)\nζ≈104\nζ≈105ζ≈106\nFigure11 : Electromagneticenergydensitiesobtainedfromsimulationswith 2563gridpoints.\nρXandρAshow a consistent behaviour with the results shown in Figure 5, which are obtained\nwith 1283grid points. The couplings are fixed to gϕX=gϕAX= 80. We also use f= 1014\nGeV and mϕ= 4.7×10−8eV, and assume that TR> f.\nThe results presented in this paper are mostly obtained performing simulations with\n1283grid points. While in the literature post-inflationary axions are simulated on lattices\nwith far higher resolutions [39–42] (but without couplings to gauge fields), we check that the\ngrowth of ρAdisplayed in Figures 5 and 6 occurs also by increasing substantially the number\nof grid points, i.e. on grids with 2563and3203points. Figure 11 shows indeed the same rapid\ngrowth of ρXandρAin runs employing 2563grid points, confirming the results presented in\nthe main text (the same behaviour is observed in runs with 3203grid points, not reported in\nthis paper). We notice that the maximum value of ρAin Figure 11 is higher than the values\nobtained in the main text. 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Tkachev, LATTICEEASY: A program for lattice simulations of scalar fields in\nan expanding universe ,Computer Physics Communications 178(2008) 929 [ hep-ph/0011159 ].\n– 18 –"    },    {        "title": "2401.11824v2.Rethinking_Centered_Kernel_Alignment_in_Knowledge_Distillation.pdf",        "content": "Rethinking Centered Kernel Alignment in Knowledge Distillation\nZikai Zhou1,Yunhang Shen2,Shitong Shao3,Linrui Gong4,Shaohui Lin1∗\n{choukai003,shenyunhang01,shaohuilin007 }@gmail.com,\nshaoshitong@seu.edu.cn,\nttrr2021@sjtu.edu.cn\nAbstract\nKnowledge distillation has emerged as a highly\neffective method for bridging the representa-\ntion discrepancy between large-scale models and\nlightweight models. Prevalent approaches involve\nleveraging appropriate metrics to minimize the di-\nvergence or distance between the knowledge ex-\ntracted from the teacher model and the knowl-\nedge learned by the student model. Centered\nKernel Alignment (CKA) is widely used to mea-\nsure representation similarity and has been applied\nin several knowledge distillation methods. How-\never, these methods are complex and fail to un-\ncover the essence of CKA, thus not answering\nthe question of how to use CKA to achieve sim-\nple and effective distillation properly. This paper\nfirst provides a theoretical perspective to illustrate\nthe effectiveness of CKA, which decouples CKA\nto the upper bound of Maximum Mean Discrep-\nancy (MMD) and a constant term. Drawing from\nthis, we propose a novel Relation-Centered Ker-\nnel Alignment (RCKA) framework, which practi-\ncally establishes a connection between CKA and\nMMD. Furthermore, we dynamically customize the\napplication of CKA based on the characteristics\nof each task, with less computational source yet\ncomparable performance than the previous meth-\nods. The extensive experiments on the CIFAR-100,\nImageNet-1k, and MS-COCO demonstrate that our\nmethod achieves state-of-the-art performance on\nalmost all teacher-student pairs for image classifi-\ncation and object detection, validating the effective-\nness of our approaches.\n1 Introduction\nTremendous efforts have been made in compressing large-\nscale models into lightweight models. Representative meth-\nods include network pruning [Frankle and Carbin, 2019 ],\nmodel quantization [Wu et al. , 2016 ], neural architec-\nture search [Wan et al. , 2020 ]and knowledge distilla-\ntion (KD) [Hinton et al. , 2015 ]. Among them, KD has re-\ncently emerged as one of the most flourishing topics due\nto its effectiveness [Liuet al. , 2021b; Huang et al. , 2022;Gong et al. , 2023; Shao et al. , 2023a ]and wide applica-\ntions [Chong et al. , 2022; Chen et al. , 2023; Shao et al. ,\n2023b ]. Particularly, the core idea of KD is to transfer\nthe acquired representations from a large-scale and high-\nperforming model to a lightweight model by distilling the\nlearning representations in a compact form, achieving pre-\ncision and reliable knowledge transfer. Out of the consen-\nsus of researchers, there are two mainstream approaches to\ndistilling knowledge from the teacher to the student. The\nfirst approach is the logit-based distillation, which aims to\nminimize the probabilistic prediction (response) scores be-\ntween the teacher and student by leveraging appropriate met-\nrics[Zhao et al. , 2022; Hinton et al. , 2015 ]. The other is\nfeature-based distillation, which investigates the knowledge\nwithin intermediate representations to further boost the dis-\ntillation performance [Yang et al. , 2022b; Liu et al. , 2021a;\nChen et al. , 2021; Ahn et al. , 2019 ]. Among them, the design\nof metrics is essential in knowledge transfer and has been\nattractive to scholars. Kornblith at al [2019 ]proposes the\nCentered Kernel Alignment (CKA) for the quantitative under-\nstanding of representations between neural networks. CKA\nnot only focuses on model predictions but also emphasizes\nhigh-order feature representations within the models, provid-\ning a comprehensive and enriched knowledge transfer.\nRecent studies [Qiuet al. , 2022; Saha et al. , 2022 ]intro-\nduce CKA to quantitatively narrow the gap of learned repre-\nsentations between the teacher model and the student model.\nUndeniably, these methods have achieved significant success.\nHowever, their designs are excessively complex and need a\nlarge amount of computational resources, making it challeng-\ning to achieve fine-grained knowledge transfer and leading\nto low scalability. Moreover, these methods fail to uncover\nthe essence of CKA, lacking an in-depth analysis of CKA in\nknowledge distillation. The reason why CKA is effective has\nnot been explored. Therefore, we focus on the theoretical\nanalysis of CKA and rethink a more reasonable architecture\ndesign that ensures simplicity and effectiveness while gener-\nalizing well across various tasks.\nIn this paper, we provide a novel perspective to illustrate\nthe effectiveness of CKA, where CKA is regarded as the up-\nper bound of Maximum Mean Discrepancy (MMD) with a\nconstant term, specifically. Drawing from this, we propose\na Relation-Centered Kernel Alignment (RCKA) framework,\nwhich practically establishes a connection between CKA andarXiv:2401.11824v2  [cs.CV]  28 Feb 2024Figure 1: The overall framework of the proposed Relation-based Centered Kernel Alignment (RCKA). We first transform the feature map\nfrom the shape of (B, C, HW )into (B, CHW )and then compute the CKA similarity of feature maps between the teacher and the student.\nBesides, we compute the inter-class and intra-class CKA similarity of logits between teacher and student. Here, Nrefers to the number of\nsamples, and Prefers to the corresponding probability of class to which this sample belongs.\nMMD. Besides, we dynamically customize the application\nof CKA on instance-level tasks, and introduce Patch-based\nCentered Kernel Alignment (PCKA), with less computational\nsource yet competitive performance when compared to pre-\nvious methods. Our method is directly applied not only to\nlogit-based distillation but also to feature-based distillation,\nwhich exhibits superior scalability and expansion. We uti-\nlize CKA to compute high-order representation information\nboth between and within categories, which better motivates\nthe alleviation of the performance gap between the teacher\nand student.\nTo validate the effectiveness of our approaches, we con-\nduct extensive experiments on image classification (CIFAR-\n100[Krizhevsky and Hinton, 2009 ]and ImageNet-1k [Rus-\nsakovsky et al. , 2015 ]), and object detection (MS-COCO [Lin\net al. , 2014 ]) tasks. As a result, our methods achieve state-of-\nthe-art (SOTA) performance in almost all quantitative com-\nparison experiments with fair comparison. Moreover, fol-\nlowing our processing architecture, the performance of the\nprevious distillation methods is further boosted in the object\ndetection task.\nOur contribution can be summarized as follows:\n•We rethink CKA in knowledge distillation from a novel\nperspective, providing a theoretical reason for why CKA\nis effective in knowledge distillation.\n•We propose a Relation Centered Kernel Alignment\n(RCKA) framework to construct the relationship be-\ntween CKA and MMD, with less computational source\nyet comparable performance than previous methods,\nwhich verifies our theoretical analysis correctly.\n•We further dynamically customize the application of\nCKA for instance-level tasks and propose a Patch-based\nCentered Kernel Alignment (PCKA) architecture for\nknowledge distillation in object detection, which furtherboosts the performance of previous distillation methods.\n•We conduct plenty of ablation studies to verify the ef-\nfectiveness of our method, which achieves SOTA perfor-\nmance on a range of vision tasks. Besides, we visualize\nthe characteristic information of CKA and discover new\npatterns in it.\n2 Related Work\nVanilla Knowledge Distillation [Hinton et al. , 2015 ]proposes\naligning the output distributions of classifiers between the\nteacher and student by minimizing the KL-divergence, dur-\ning training the emphasis on negative logits can be fine-tuned\nthrough a temperature coefficient, which serves as a form of\nnormalization during the training process of a smaller stu-\ndent network. Tremendous efforts [Tung and Mori, 2019;\nHuang et al. , 2022; Qiu et al. , 2022; Zagoruyko and Ko-\nmodakis, 2016a; Park et al. , 2019 ]have been made on how\nto design a good metric to align the distribution between the\nteacher and student.\nDesign an suitable alignment method for KD can start\nfrom two typical types: Drawing on representations, numer-\nous methods have made significant strides by aligning the in-\ntermediate features [Zagoruyko and Komodakis, 2016a ], the\nsamples’ correlation matrices [Tung and Mori, 2019 ], and the\noutput logits between the teacher and student [Huang et al. ,\n2022 ]. From a mathematical standpoint, some measure the-\nories are introduced to illustrate the similarity between the\nteacher and student, such as mutual information [Ahn et al. ,\n2019 ]. Among these, Centered Kernel Alignment (CKA) is a\nvaluable function for measuring similarity. It simultaneously\nconsiders various properties during similarity measures, such\nas invariance to orthogonal transformations. While the ef-\nfectiveness of CKA in KD has been demonstrated in someworks [Qiuet al. , 2022; Saha et al. , 2022 ], the essence of\nCKA has not been thoroughly explored, and the unavoid-\nable additional computational costs also limit its application\nprospect.\nIn this paper, we will revisit CKA in KD and provide a\nnovel theoretical perspective to prove its effectiveness and an-\nalyze how it functions across various distillation settings.\n3 Methodolgy\nIn this section, we first revisit the paradigm of knowledge\ndistillation and then introduce the formula of Centered Ker-\nnel Alignment (CKA). Specifically, we derive the formula of\nthe relationship between CKA and Maximum Mean Discrep-\nancy (MMD), where CKA can be decoupled as the upper\nbound of MMD with a constant term. In light of the above\ndeduction, we outline the methodology of our paper. We ap-\nply the proposed methods in image classification and object\ndetection, dynamically customizing CKA for each task.\n3.1 The Paradigm of Knowledge Distillation\nThe existing KD methods can be categorized into two groups.\nParticularly, the logits-based KD methods narrow the gap be-\ntween the teacher and student models by aligning the soft\ntargets between them, which is formulated as following loss\nterm:\nLlogits =Dlogits(Ts(σ(zs;τ)),Tt(σ(zt;τ))), (1)\nwhere zsandztare the logits from students and teachers, re-\nspectively. And σ(·)is the softmax function that produces\nthe category probabilities from the logits, and τis a non-\nnegative temperature hyper-parameter to scale the smooth-\nness of the predictive distribution. Specifically, we have\nσi(z;τ) = softmax(exp( zi/τ)).Dlogits is a loss function to\ncapture the discrepancy distributions, e.g. Kullback-Leibler\ndivergence. And TsandTtdenote the transformation func-\ntions in students and teachers, respectively, which usually\nrefer to the identity mapping in Vanilla KD [Hinton et al. ,\n2015 ].\nSimilarly, the feature-based KD methods, which aim to\nmimic the feature representations between teachers and stu-\ndents, are also represented as a loss item:\nLfeat=Dfeat(Ts(Fs),Tt(Ft)), (2)\nwhere FsandFtdenote feature maps from students and\nteachers, respectively. Transformation modules TsandTt\nalign the dimensions of FsandFt.Dfeatcomputes the dis-\ntance between two feature maps, such as ℓ1- orℓ2norm.\nTherefore, the KD methods can be represented by a generic\nparadigm. The final loss is the weighted sum of the cross-\nentropy loss Lce, the logits distillation loss, and the feature\ndistillation loss:\nL=Lce+αLlogits +βLfeat, (3)\nwhere αandβare hyper-parameters controlling the trade-off\nbetween these three losses.3.2 Distilling with the Upper Bound\nCentered Kernel Alignment (CKA) has been proposed as a\nrobust way to measure representational similarity between\nneural networks. We first prove that CKA measures the co-\nsine similarity of the gram matrix between teachers and stu-\ndents.\nTheorem 1 (Proof in Appendix C.1) .LetXandYbeN×P\nmatrices. The CKA similarity ∥Y⊤X∥2\nFis equivalent to the\ncosine similarity of XX⊤andY Y⊤, which denote the gram\nmatrix of XandY, respectively. In other words,\nSCKA(X, Y ) =∥Y⊤X∥2\nF\n∥XTX∥F∥YTY∥F\n=vec(XX⊤)⊤vec(Y Y⊤)\n∥vec(XX⊤)∥2∥vec(Y Y⊤)∥2,\nwhere vecoperator represents reshaping the matrix to a vec-\ntor.\nWe then derive the formula of the relationship between\nCKA and MMD, where CKA can be regarded as the upper\nbound of MMD with a constant term.\nTheorem 2 (Proof in Appendix C.2) .Maximizing CKA sim-\nilarity is equivalent to minimizing the upper bound of MMD\ndistance:\n∥Y⊤X∥2\nF\n∥XTX∥F∥YTY∥F=−NEi,j[⟨xi, xj⟩ − ⟨yi, yj⟩]2+ 2\n≤ −N(Ei,j[⟨xi, xj⟩]−Ei,j[⟨yi, yj⟩])2+ 2,\nwhere the inequality is given by Jesen’s inequality.\nAccording to Jesen’s inequality, CKA can be decoupled as\nthe upper bound of MMD with a constant term. The first\nterm corresponds to minimizing the upper bound of MMD\ndistance with the RKHS kernel. In contrast, the latter constant\nterm acts as a weight regularizer, enhancing the influence of\nMMD, where it promotes the similarity between features of\nthe same batch, not only instances in the same class but also\nin different classes. On one hand, optimizing the upper bound\nof MMD, which has additional stronger constraints, allows it\nto converge to the optimal solution more quickly and stably.\nOn the other hand, the latter term serves as a weight scaling\nmechanism, effectively avoiding the challenges of optimiza-\ntion caused by excessively small MMD values, which result\nin small gradients.\nAccording to the deduction, we successfully transformed\nour optimization objective from maximizing CKA to mini-\nmizing the upper bound of MMD, which makes our method\nmore intuitive and concise. Building upon these findings, we\npropose our methods, which are more effective than previous\nmethods.\n3.3 Relation-based Centered Kernel Alignment\nAs illustrated in Fig. 1, we propose a Relation Centered\nKernel Alignment (RCKA) framework in image classifica-\ntion. In this framework, we leverage CKA as a loss func-\ntion to ensure that the centered similarity matrix is distilled\nrather than forcing the student to mimic the teacher‘s simi-\nlarity matrix with a different scale. This is very importantFigure 2: The overall framework of PCKA. We dynamically customize the framework of proposed method based on the characteristics of\nobject detection. In this framework, we first patch the featuremap of the teacher and student with the patchsize (PH, PW), then transform\nthe featuremap in order to get the gram matrix between each patch. Finally, we calculate the loss LPCKA , and get average from dimension\nC. Here, B, C, H, W refer to the batchsize, channels, height and width of the featuremap, respectively. NPH, NPWdenote the number of\npatches cutting along the height and width, respectively.\nbecause a model‘s discriminative capability is dependent on\nthe distribution of its features rather than its scale, which is\ninconsequential for class separation [Nguyen et al. , 2020;\nOrhan and Pitkow, 2017 ].\nAssume we have a large-scale teacher model tand a\nlightweight student model s. The activation map from layer\nlof the teacher is denoted as F(l)\nt∈Rb×c×h×w, whereas\nthe activation map of layer l′of the student is denoted as\nF(l′)\ns∈Rb×c′×h′×w′.c,h, andwdenote the channel, height,\nand width of the teacher, whereas c′, h′andw′denote that of\nthe student. The mini-batch size is denoted by b. The logits\nof the teacher and student are denoted as zt∈RN×Pand\nzs∈RN×P′, where NandP(orP′) refer to the number\nof samples and the corresponding probability for which class\nthis sample belongs to. Therefore, the formula of our method,\nsimilar to Eqn. 3, is represented as:\nLRCKA =LCE+αLFCKA (4)\n+β(LIntra−LCKA +LInter−LCKA ) (5)\nwhere αandβare hyper-parameters controlling the trade-\noff between the features loss LFCKA and logits loss\n(LIntra−LCKA +LInter−LCKA ).\nTheLFCKA ,LIntra−LCKA andLInter−LCKA are repre-\nsented as:\nLFCKA =SCKA(T(Ft),T(Fs)), (6)\nLIntra−LCKA =SCKA(zt, zs), (7)\nLInter−LCKA =SCKA(zT\nt, zT\ns), (8)\nwhere Tin Eqn. 6 refers to the transformation module\nRb×c×h×w→Rb×chw.\nCompared with the previous methods, our method has su-\nperior scalability and expansion and can be directly applied toboth feature and logits distillation. We calculate the gram ma-\ntrix to collect high-order inter-class and intra-class represen-\ntations, encouraging the student to learn more useful knowl-\nedge. Also, we provide the relationship between CKA and\nMMD in Appendix C.2 to better demonstrate the theoretical\nsupport of our method.\nBecause the value of CKA ranges from [0,1], at the begin-\nning of the training process, LCEplays a more important role\nthan all CKA losses to drive the optimization of the student,\nwhich helps the student avoid matching extremely complex\nrepresentations.\n3.4 Patch-based Centered Kernel Alignment\nIn this subsection, we further adapt the proposed RCKA to\ninstance-level tasks such as object detection. However, di-\nrectly applying RCKA to instance-level tasks may deteriorate\nperformance, as the above tasks are usually trained with a\nsmall size of mini-batches ( e.g. 2or4per GPU), causing\nthe failure of the gram matrix to collect enough knowledge.\nBesides, increasing the mini-batch size requires a significant\namount of computational resources, making it infeasible in\npractice. Thus, we dynamically customize our RCKA method\nfor object detection.\nRecent works [Shuet al. , 2021; Heo et al. , 2019 ]find that\ndistilling the representations of intermediate layers is more\neffective than distilling the logits in object detection. There-\nfore, we adjust our method to only target intermediate lay-\ners. Still, we follow our core idea in the classification task,\nwhich calculates the similarities between different instances\nby using CKA. So, we divide the image feature maps into\nseveral patches and compute the similarities between differ-\nent patches.\nOur redesigned method is illustrated in Fig. 2. In thisTable 1: Results on the CIFAR-100 test set. “Same” and “Different” in the first row refer to whether the model architecture is the same for\nteachers and students. Combined distillation type means that this method transfers the knowledge both on the features and logits. Our methods\nsurpass almost all algorithms with the same distillation type. “RN”, “WRN”, “SN”, and “MN” denote ResNet, Wide ResNet, ShuffleNet, and\nMobileNet, respectively.\nArchitecture Same Different\nDistillation\nTypeTeacherRN-110 RN-110 WRN-40-2 WRN-40-2 RN-32 ×4 VGG-13 WRN-40-2 RN-32 ×4 VGG-13\n74.31 74.31 75.61 75.61 79.42 74.64 75.61 79.42 74.64\nStudentRN-20 RN-32 WRN-40-1 WRN-16-2 RN-8 ×4 VGG-8 SN–V1 SN–V1 MN–V2\n69.06 71.14 71.98 73.26 72.50 70.36 70.50 70.50 64.60\nFeature-basedFitNet [2014 ] 68.99 71.06 72.24 73.58 73.50 71.02 73.73 73.59 64.14\nATKD [2016a ] 70.22 70.55 72.77 74.08 73.44 71.43 73.32 72.73 59.40\nSPKD [2019 ] 70.04 72.69 72.43 73.83 72.94 72.68 74.52 73.48 66.30\nCCKD [2019 ] 69.48 71.48 72.21 73.56 72.97 70.71 71.38 71.14 64.86\nRKD [2019 ] 69.25 71.82 72.22 73.35 71.90 71.48 72.21 72.28 64.52\nVID [2019 ] 70.16 70.38 73.30 74.11 73.09 71.23 73.61 73.38 65.56\nCRD [2020 ] 71.46 73.48 74.14 75.48 75.51 73.94 76.05 75.11 69.73\nOFD [2019 ] - 73.23 74.33 75.24 74.95 73.95 75.85 75.98 69.48\nReviewKD [2021 ] - 71.89 75.09 76.12 75.63 74.84 77.14 76.93 70.37\nICKD-C [2021a ] 71.91 74.11 74.63 75.57 75.48 73.88 75.19 74.34 67.55\nDPK [2022 ] 72.44 74.89 75.27 76.42 - 74.96 74.43 76.00 68.63\nFCKA(ours) 71.49 73.64 74.70 75.53 74.93 74.35 75.98 75.67 68.97\nLogit-basedKD[2015 ] 70.67 73.08 73.54 74.92 73.33 72.98 74.83 74.07 67.37\nDKD [2022 ] - 74.11 74.81 76.24 76.32 74.68 76.70 76.45 69.71\nDIST [2022 ] 69.94 73.55 74.42 75.29 75.79 73.74 75.23 75.23 68.48\nIKL-KD [2023 ] - 74.26 74.98 76.45 76.59 74.88 77.19 76.64 70.40\nNKD [2023 ] 71.26 73.79 75.23 76.37 76.35 74.86 76.59 76.90 70.22\nLCKA(ours) 70.87 73.64 74.63 75.78 75.12 74.35 76.12 76.43 69.37\nCombinedSRRL [2021 ] 71.51 73.80 74.75 75.96 75.92 74.40 76.61 75.66 69.14\nRCKA(ours) 72.26 74.31 75.34 76.51 76.11 74.97 77.21 76.97 70.12\nTable 2: Results on the ImageNet validation set. We use ResNet-34 and ResNet-50 released by Torchvision [Marcel and Rodriguez, 2010 ]as\nour teacher’s pre-training weight.\nArchitecture Accuracy Feature-based Logit-based Combined\nTeacher Student Teacher Student OFD CRD ReviewKD ICKD-C MGD [2022b ]KD RKD DKD DIST SRRL Ours\nResNet-34 ResNet-18Top-1 73.31 69.76 71.08 71.17 71.61 72.19 71.80 70.66 70.34 71.70 72.07 71.73 72.34\nTop-5 91.42 89.08 90.07 90.13 90.51 90.72 90.40 89.88 90.37 90.41 90.42 90.60 90.68\nResNet-50 MobileNet-V1Top-1 76.16 70.13 71.25 71.37 72.56 - 72.59 70.68 - 72.05 73.24 72.49 72.79\nTop-5 92.86 89.49 90.34 90.41 91.00 - 90.74 90.30 - 91.05 91.12 90.92 91.01\nframework, we first patch the feature maps of the teacher and\nstudent with a patch size of (PH, PW), then transform the fea-\nture maps to get the gram matrix between each patch. Finally,\nwe calculate the loss LPCKA and get the average from dimen-\nsionC. Here, NPHandNPWdenote the number of patches\ncutting along the height and width, respectively. Therefore,\nthe Patch-based CKA loss is represented as:\nLPCKA =γSCKA(Ns\nPH·Ns\nPW, Nt\nPH·Nt\nPW), (9)\nwhere Ns\nPH·Ns\nPWandNt\nPH, Nt\nPWare denoted as the num-\nber of the student patches and the teacher patches, respec-\ntively. Usually, Ns\nPH·Ns\nPW=Nt\nPH, Nt\nPW.γrefers to the\nloss weight factor.\n4 Experiments\nWe conduct extensive experiments on image classification\nandobject detection benchmarks. The image classification\ndatasets include CIFAR-100 [Krizhevsky and Hinton, 2009 ]\nand ImageNet-1k [Russakovsky et al. , 2015 ]and the ob-\nject detection dataset includes MS-COCO [Linet al. , 2014 ].\nMoreover, we present various ablations and analyses for the\nproposed methods. More details about these datasets are inAppendix A. We apply a batch size of 128 and an initial\nlearning rate of 0.1for the SGD optimizer on CIFAR-100.\nAnd we follow the settings in [Huang et al. , 2022 ]for the\nResNet34-ResNet18 pair and the ResNet50-MobileNet pair\non ImageNet-1k. The settings of other classification and de-\ntection tasks are in Appendix B. Our code will be publicly\navailable for reproducibility.\n4.1 Image Classification\nClassification on CIFAR-100. We compare state-of-the-\nart (SOTA) feature-based and logit-based distillation algo-\nrithms on 9student-teacher pairs. Among them, 6pairs have\nthe same structure for teachers and students, and the rest of\nthem have different architectures. The results are presented\nin Tab. 1. Our proposed method outperforms all other al-\ngorithms on 4student-teacher pairs and achieves compara-\nble performance on the rest of them, meanwhile requiring\nextremely less computational resources and time consump-\ntion than the SOTA methods DPK [Qiuet al. , 2022 ]and Re-\nviewKD [Chen et al. , 2021 ]. The comparisons of computa-\ntional cost are in Appendix D.Table 3: Results on the COCO validation set (T →S refers to the distillation from T to S). Here, the content in brackets to the right of “Ours”\nrefers to the methods applied in the distillation process. In addition, CM RCNN-X101 stands for Cascade Mask RCNN-X101.\nT→S CM RCNN-X101 [2017 ]→Faster RCNN-R50 [2015 ] RetinaNet-X101 →RetinaNet-R50 [2017 ] T→S FCOS-R101 →FCOS-R50 [2019 ]\nType Two-stage detectors One-stage detectors Type Anchor-free detectors\nMethod AP AP 50 AP75 APSAPM APL AP AP 50 AP75 APSAPM APL Method AP AP 50 AP75 APSAPM APL\nTeacher 45.6 64.1 49.7 26.2 49.6 60.0 41.0 60.9 44.0 23.9 45.2 54.0 Teacher 40.8 60.0 44.0 24.2 44.3 52.4\nStudent 38.4 59.0 42.0 21.5 42.1 50.3 37.4 56.7 39.6 20.0 40.7 49.7 Student 38.5 57.7 41.0 21.9 42.8 48.6\nKD[2015 ] 39.7 61.2 43.0 23.2 43.3 51.7 37.2 56.5 39.3 20.4 40.4 49.5 KD [2015 ] 39.9 58.4 42.8 23.6 44.0 51.1\nCOFD [2019 ] 38.9 60.1 42.6 21.8 42.7 50.7 37.8 58.3 41.1 21.6 41.2 48.3 FitNet [2014 ]39.9 58.6 43.1 23.1 43.4 52.2\nFKD [2021 ] 41.5 62.2 45.1 23.5 45.0 55.3 39.6 58.8 42.1 22.7 43.3 52.5 GID [2021 ] 42.0 60.4 45.5 25.6 45.8 54.2\nDIST [2022 ] 40.4 61.7 43.8 23.9 44.6 52.6 39.8 59.5 42.5 22.0 43.7 53.0 FRS [2021 ] 40.9 60.3 43.6 25.7 45.2 51.2\nDIST+mimic [2022 ]41.8 62.4 45.6 23.4 46.1 55.0 40.1 59.4 43.0 23.2 44.0 53.6 FGD [2022a ]42.1 - - 27.0 46.0 54.6\nOurs 41.4 62.1 45.2 23.5 45.6 54.9 40.3 59.9 43.0 23.3 44.2 54.9 Ours 39.8 59.0 42.4 22.2 43.6 52.5\nOurs + mimic 42.4 63.3 46.1 24.3 46.7 56.1 40.7 60.4 43.4 23.9 44.7 55.1 Ours + mimic 40.7 60.5 43.1 23.4 44.8 53.1\nTable 4: Results on the COCO validation set (T →S refers to the\ndistillation from T to S). Here, the content in brackets to the right\nof “Ours” refers to the methods applied in the distillation process.\nIn addition, “Inside GT Box” means we use the GT boxes with the\nsame stride on the FPN layers as the feature imitation regions. “Main\nRegion” means we imitate the features within the main distillation\nregion.\nT→S GFL-R101 →GFL-R50 [2020 ]\nMethod AP AP 50 AP75 APSAPM APL\nTeacher 44.9 63.1 49.0 28.0 49.1 57.2\nStudent 40.2 58.4 43.3 23.3 44.0 52.2\nFT[2014 ] 40.7 58.6 44.0 23.7 44.4 53.2\nInside GT Box 40.7 58.6 44.2 23.1 44.5 53.5\nDeFeat 40.8 58.6 44.2 24.3 44.6 53.7\nMain Region 41.1 58.7 44.4 24.1 44.6 53.6\nFGFI [2019 ] 41.1 58.8 44.8 23.3 45.4 53.1\nFGD [2022a ] 41.3 58.8 44.8 24.5 45.6 53.0\nGID [2021 ] 41.5 59.6 45.2 24.3 45.7 53.6\nSKD [2022 ] 42.3 60.2 45.9 24.4 46.7 55.6\nOur 42.8 61.2 46.3 24.8 47.1 55.4\nClassification on ImageNet-1k. We also conduct exper-\niments on the large-scale ImageNet to evaluate our meth-\nods. Our RCKA achieves comparable results with other algo-\nrithms, even outperforms them, as shown in Tab. 2. We find\nthat with the increasing of categories and instances, it is more\nchallenging for the student to mimic the high-order distribu-\ntion of the teacher. Moreover, in Appendix 8, we We explore\nthe feature distillation for ViT-based models on ImageNet-1k.\nIt is noted that our method outperforms other methods, which\nmeans that our method has good scalability and good perfor-\nmance.\n4.2 Object Detection\nDetection on MS-COCO. Comparison experiments are\nrun on three kinds of different detectors, i.e., tow-stage de-\ntectors, one-stage detectors, and anchor-free detectors. As\nshown in Tab. 3, PCKA outperforms the precious meth-\nods almost on all three kinds of metrics, by aligning the\nhigh-order patch-wise presentations. We believe that align-\ning feature maps of the student and teacher in low-order\ncould also improve the performance of PCKA, driven by\nmimicking low-order representations in the early stage and\nthen learning high-order and complex representations gradu-\nally. Thus, we follow [Huang et al. , 2022 ]by adding aux-\niliary mimic loss, i.e., translating the student feature maps\nfrom the teacher feature map by a convolution layer and su-pervising them utilizing LMSE , to the detection distillation\ntask. Ultimately, we conclude from Tab. 3 that PCKA-based\nmimic loss achieves the best performance on Cascade RCNN-\nX101-Cascade RCNN-R50 and RetinaNet-X101-RetinaNet-\nR50 pairs. We also conduct experiments on the other four\narchitectures, as shown in Tab. 4 and Tab. 9. These results\nfurther validate the effectiveness of our proposed method.\n4.3 Ablations and Visualizations\nWe conduct ablation studies in three aspects: (a)the effect of\nhyper-parameters. (b)effectiveness of the proposed modules.\n(c)unexplored phenomenon during training.\nAblation studies on hyperparameters. As shown in\nTab. 12, Tab. 11 and Tab. 13 in Appendix, we conduct the\nablation studies on the size of mini-batch, loss scaling fac-\ntorγonLPCKA and the number of intermediate layers for\ndistilling. We find the local optima values are the mini-batch\nsize12, loss scaling factor 10, and 3layers of distillation.\nThe upper bound of MMD. In Theorem 2, we derive the\nrelationship between CKA and MMD, where CKA is the up-\nper bound of MMD with a constant term. To validate this,\nWe conduct the experiment, which is shown in Tab. 6, we no-\ntice that CKA, which is the upper bound of MMD, has addi-\ntional stronger constraints. Because of this, CKA converges\nto the optimal solution more quickly and stably, compared\nwith MMD.\nThe dimension to average. In PCKA framework, we cut\nthe activations of the teacher and student in the shape of\n(C, N PH·NPW, B·PH·PW). We also carry out the experi-\nments of averaging on different dimensions, shown in Tab. 5.\nWe find averaging on channel dimension is the optima.\nPatch distillation. We explore the effectiveness of cutting\nactivations into patches. As shown in Tab. 7, several standard\ndistillation methods [Hinton et al. , 2015; Zagoruyko and Ko-\nmodakis, 2016a; Huang et al. , 2022 ]all perform well with\npatch cutting, validating the effectiveness of cutting patches.\nWith the smaller representation distribution in patches, it is\neasier to align the teacher and student. Thus, the proposed\nPCKA architecture amazingly boosts the previous methods.\nVisualize the CKA value. We present some visualizations\nto show that our method does bridge the teacher-student gap\nin logit-level. In particular, we visualize the logit similarity\nfor6teacher-student pairs in Appendix E. We find that our\nmethod significantly improves the logit-similarity.Table 5: The ablation study on the COCO validation set (T →S refers\nto the distillation from T to S). Here, we explore which dimension\nwe should choose to get better results. ”Mix-up” means the 1st dis-\ntilling layer uses Batch avg. method, the 2nd distilling layer uses\nSpatial avg. method and the final distilling layer uses Channel avg.\nmethod\nT→S RetinaNet-X101 →RetinaNet-R50\nMethod AP AP 50 AP75 APSAPM APL\nTeacher 41.0 60.9 44.0 23.9 45.2 54.0\nStudent 37.4 56.7 39.6 20.0 40.7 49.7\nBatch avg. 38.5 57.9 40.8 20.7 41.5 52.5\nSpatial avg. 39.3 58.7 41.9 21.4 41.3 50.9\nMix-up avg. 38.2 58.1 40.4 21.3 41.9 50.9\nChannel avg. 40.3 59.9 43.0 23.3 44.2 54.9\nVisualize the training process. We further visualize the\ntraining process of different detectors and the patch effect on\nthe RetinaNet-X101-RetinaNet-R50 pair. The results are in\nTab. 4 and Tab. 3 in the Appendix.\nTable 6: Experiments on the upper bound of MMD. We derive the\nformula that CKA is the upper bound of MMD with a constant term.\nFrom these experiments, we can prove that optimizing the upper\nbound of MMD can better improve the performance, compared with\nMMD.\nT→S RetinaNet-X101 →RetinaNet-R50\nMethod AP AP 50 AP75 APSAPM APL\nTeacher 41.0 60.9 44.0 23.9 45.2 54.0\nStudent 37.4 56.7 39.6 20.0 40.7 49.7\nMMD w/ patch 38.5 57.7 40.9 22.2 42.8 51.3\nPCKA 40.3 59.9 43.0 23.3 44.2 54.9\nVisualize the inference outputs. We first visualize the con-\nfusion matrix of the proposed method in Fig. 6, and then vi-\nsualize the annotated images of training with/without patches\naveraging different dimensions in Fig. 7, respectively. These\nfigures reveal that our method can collect the similarities be-\ntween different classes, and also show the effectiveness of our\nmethod on the object detection task.\n5 Discussion\nPCKA in image classification. We apply PCKA to the\nclassification task, and it also outperforms well on the meth-\nods with the same distillation type, as shown in Tab. 14. How-\never, PCKA performs very badly on the teacher-student pairs\nwith different architectures. As cutting activations of differ-\nent architectures contain more dissimilar and harmful rep-\nresentations, bringing difficulty in transferring knowledge to\nthe student.\nAverage on the channel, boosting the performance. The\nresults in Tab. 7 reveal an interesting phenomenon, where the\nperformance of the previous distillation methods is boosted\nby averaging the loss on channel dimension after the acti-\nvations are cut into patches. Instead of directly matching\nthe whole representation distribution in the activations, cut-\nting patches makes the alignment between the teacher and\nstudent easier with a smaller representation distribution inTable 7: Ablation study of distillation methods with(w/) or with-\nout(w/o) patch on the COCO validation set (T →S refers to the dis-\ntillation from T to S). Here, we surprisingly notice that previous dis-\ntillation method performance can be improved by image patching.\nT→S RetinaNet-X101 →RetinaNet-R50\nMethod AP AP 50 AP75 APSAPM APL\nTeacher 41.0 60.9 44.0 23.9 45.2 54.0\nStudent 37.4 56.7 39.6 20.0 40.7 49.7\nKD 37.2 56.5 39.3 20.4 40.4 49.5\nKD w/ patch 39.3 58.7 41.9 21.4 41.3 50.9\nAT 34.4 52.3 36.4 17.7 37.2 47.8\nAT w/ patch 37.4 56.6 39.9 20.8 40.6 49.8\nDIST 39.8 59.5 42.5 22.0 43.7 53.0\nDIST w/ patch 40.2 59.6 43.2 22.7 44.8 53.9\nPCKA w/o patch 36.4 55.8 38.7 20.6 39.8 48.7\nPCKA(ours) 40.3 59.9 43.0 23.3 44.2 54.9\npatches. Besides, cutting into patches follows the idea in the\nframework of classification proposed by us, thus PCKA cal-\nculates the inter-class similarities and intra-class similarities\nin patches. Moreover, due to the superiority of cosine sim-\nilarity over distance-based losses [Boudiaf et al. , 2020 ]and\nhigh-order distribution representations collected by the gram\nmatrix, PCKA outperforms DIST and AT.\nPositional information loss. In PCKA, we cut the activa-\ntion of the teacher and student into patches, and then flatten\nthem into a vector. Although this operation damages the orig-\ninal positional information, performance does not deteriorate.\nWe suppose that CKA ensures the focus of the optimization\nis on the shape of the distribution, rather than the raw values\nin the Gram matrix, which is vital because a model‘s discrim-\ninative capability is dependent on the distribution of its fea-\ntures rather than its scale. Besides, at the beginning, the effect\nbrought by PCKA is smaller, compared with CE loss. There-\nfore, CE loss motivates the optimization of the student model\nsteadily, and starting from a certain moment, PCKA drives\nthe student model to align complex and high-order represen-\ntations, improving the generalization ability.\n6 Conclusion\nIn this paper, we provide a novel theoretical perspective of\nCKA in knowledge distillation, which can be simplified as\nthe upper bound of MMD with a constant term. Besides, we\ndynamically customize the application of CKA based on the\ncharacteristics of each task, with less computational source\nyet comparable performance than previous methods. Further-\nmore, we propose a novel processing architecture for knowl-\nedge distillation in object detection task, which can further\nboost the performance of previous distillation methods. 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OpenReview.net, 2021.\n[Zhao et al. , 2022 ]Borui Zhao, Quan Cui, Renjie Song,\nYiyu Qiu, and Jiajun Liang. Decoupled knowledge distil-\nlation. In CVPR , pages 11953–11962, New Orleans, LA,\nUSA, Jun. 2022. IEEE.\n[Zhixing et al. , 2021 ]Du Zhixing, Rui Zhang, Ming Chang,\nShaoli Liu, Tianshi Chen, Yunji Chen, et al. Distilling\nobject detectors with feature richness. NeurIPS , 34:5213–\n5224, Dec. 2021.\n[Zhou and Chen, 2023 ]Zikai Zhou and Huanran Chen. Afn:\nAdaptive fusion normalization via an encoder-decoder\nframework. In ArXiv preprint , 2023.A Datasets\nCIFAR-100. Dataset CIFAR-100 [Krizhevsky and Hinton,\n2009 ]is the subsets of the tiny image dataset and consists of\n60,000 images with the size 32 ×32. Specifically, the train-\ning set contains 50,000 images, and the testing set contains\n10,000 images.\nImageNet-1k. Dataset ImageNet-1k [Russakovsky et al. ,\n2015 ], also commonly referred to as ILSVRC 2012, has 1000\nclasses, and the benchmark is trained using the training set\nand tested using the validation set. Its training and validation\nsets contain 1281,167 and 50,000 images, respectively.\nMS-COCO. Dataset MS-COCO [Lin et al. , 2014 ]is a\nlarge-scale object detection dataset. The benchmark is the\nsame as ImageNet-1k, using the training set for training and\nthe validation set for testing. The training/validation split was\nchanged from 83K/41K to 118K/5K in 2017. Researchers\ncommonly apply the 2017 version for experiments.\nB Hyperparameter Settings\nClassification. For the classification experiments on\nCIFAR-100, the batch size is 128, the total number of epochs\nis 240, and the learning rate is initialized to 0.1 and scheduled\nby ALRS [Chen et al. , 2022 ]. In addition, we employ an\nSGD optimizer for training and set the weight decay and\nmomentum as 5e-4 and 0.9, respectively. For the classifica-\ntion experiments on ImageNet-1k (ResNet34-ResNet18 pair\nand ResNet50-MobileNet pair), the total batch size is 512,\nthe total number of epochs is 240, the batch size in every\nGPU is 128, the number of GPUs is 4 and the learning rate\nis initialized to 0.1 and scheduled by ALRS. Besides, we\nemploy an SGD optimizer for training and set the weight\ndecay and momentum as 1e-4 and 0.9, respectively. The loss\nweight scaling factors α, β are both 5.\nDetection. For the detection experiments on MS-COCO,\nwe utilize mmdetection [Chen and Wang, 2019 ]and mm-\nrazor [Contributors, 2021 ]for both training and testing.\nFollowing [Shu et al. , 2021; Park et al. , 2019; Zhou and\nChen, 2023 ], we use the same standard training strategies on\nthe Cascade RCNN-X101-Faster RCNN-R50 and RetinaNet-\nX101-RetinaNet-R50 pairs. To be specific, the total batch\nsize is 12, the total number of epochs is 24, the batch size in\nevery GPU is 3, the number of GPUs is 4 and the learning\nrate is divided by 10 at 16 and 22 epochs. The initial learning\nrate is set as 0.02 and 0.01 on Cascade RCNN-X101-Faster\nRCNN-R50 and RetinaNet-X101-RetinaNet-R50 pairs, re-\nspectively. Besides, the setting on the FCOS-R101-FCOS-\nR50 pair is following [Yang et al. , 2022a ]. Compared with the\nRetinaNet-X101-RetinaNet-R50 pair, the only difference is\nwe apply a warm-up learning rate on the FCOS-R101-FCOS-\nR50 pair. Furthermore, the settings on GFL-R101-GFL-R50,\nRepPoints-X101-RepPoints-R5, RetinaNet-R101-RetinaNet-\nR50 and Faster RCNN-R101-Faster RCNN-R50 pairs are\nthe same with the setting on RetinaNet-X101-RetinaNet-R50\npair.C Derivations\nC.1 Revisit the formula of CKA\nWe prove CKA measures the cosine similarity between\nteacher‘s gram matrix and student gram matrix.\nProof.\n∥Y⊤X∥2\nF\n∥XTX∥F∥YTY∥F=tr((Y⊤X)⊤Y⊤X)\n∥XTX∥F∥YTY∥F\n=tr(X⊤Y Y⊤X)\n∥XTX∥F∥YTY∥F=tr(XX⊤Y Y⊤)\n∥XTX∥F∥YTY∥F\n=vec(XX⊤)⊤vec(Y Y⊤)\n∥vec(XX⊤)∥2∥vec(Y Y⊤)∥2.\nC.2 Connection between CKA and MMD\nIn Theorem 1, we already prove the relationship between\nCKA distance and cosine similarity. Denote the i-th row of\nmatrixX√\n||XXT||Fasxiand i-th row of matrixY√\n||Y YT||Fas\nyi, we can get:\n∥Y⊤X∥2\nF\n∥XTX∥F∥YTY∥F\n=vec(XX⊤)⊤vec(Y Y⊤)\n∥vec(XX⊤)∥2∥vec(Y Y⊤)∥2\n=− ∥vec(XX⊤)\n∥vec(XX⊤)∥2−vec(Y Y⊤)\n∥vec(Y Y⊤)∥2∥2\n2\n+∥vec(XX⊤)\n∥vec(XX⊤)∥2∥2\n2+∥vec(Y Y⊤)\n∥vec(Y Y⊤)∥2∥2\n2\n=− ∥vec(XX⊤)\n∥vec(XX⊤)∥2−vec(Y Y⊤)\n∥vec(Y Y⊤)∥2∥2\n2+ 2\n=−\nX\ni,j[⟨xi, xj⟩ − ⟨yi, yj⟩]2\n+ 2\n=−NEi,j[⟨xi, xj⟩ − ⟨yi, yj⟩]2+ 2\n≤ −N(Ei,j[⟨xi, xj⟩]−Ei,j[⟨yi, yj⟩])2+ 2\nConsequently, maximizing the Centered Kernel Alignment\n(CKA) similarity is tantamount to minimizing the upper\nbound of Maximum Mean Discrepancy (MMD) distance.\nD Compare the average running time\nWe show the superior average training time cost than other\nmethods.\nE CKA curve in training.\nTo qualitatively analyze the proposed method, we visualize\nthe CKA similarities in the training phase. As shown in\nFig. 5, we can find the CKA values increase with training,\nwhich demonstrates that our method does narrow the gap of\nteacher-student models at the logit level.Table 8: Results on the ImageNet validation set. ∗indicates the teacher is pre-trained on ImageNet-21K. We evaluate our method on DeiT\narchitectures.\nArchitecture Accuracy Methods\nTeacher Student Teacher Student KD VIT-KD [2022c ]VIT-KD+RCKA\nDeiT III-Small* [2020 ]DeiT-TinyTop-1 82.76 74.42 76.01 76.06 77.23\nTop-5 - 92.29 93.26 93.16 93.67\nDeiT III-Base* DeiT-SmallTop-1 85.48 80.55 82.52 81.95 83.12\nTop-5 - 95.12 96.30 95.64 96.41\nTable 9: Results on the COCO validation set (T →S refers to the distillation from T to S). Here, the content in brackets to the right of “Ours”\nrefers to the methods applied in the distillation process.\nT→S Faster RCNN-R101 →Faster RCNN-R50 RetinaNet-R101 →RetinaNet-R50 T →S RepPoints-X101 →RepPoints-R50 [2019 ]\nMethod AP AP 50 AP75 APSAPM APL AP AP 50 AP75 APSAPM APL Method AP AP 50 AP75 APSAPM APL\nTeacher 39.9 60.1 43.3 23.5 44.2 51.5 38.9 58.0 41.5 21.0 42.8 52.4 Teacher 44.2 65.5 47.8 26.2 48.4 58.5\nStudent 38.4 59.0 42.0 21.5 42.1 50.3 37.4 56.7 39.6 20.0 40.7 49.7 Student 38.6 59.6 41.6 22.5 42.2 50.4\nFGFI [2019 ]39.3 59.8 42.9 22.5 42.3 52.2 38.6 58.7 41.3 21.4 42.5 51.5 FKD [2021 ]40.6 - - 23.4 44.6 55.3\nGID [2021 ] 40.2 60.7 43.8 22.7 44.0 53.2 39.1 59.0 42.3 22.8 43.1 52.3 Our 41.0 62.2 44.0 23.6 44.6 55.6\nFGD [2022a ]40.4 - - 22.8 44.5 53.5 39.6 - - 22.9 43.7 53.6\nOur 40.1 60.7 43.8 22.7 44.1 52.6 38.6 57.8 41.3 20.7 42.3 51.9\nTable 10: Average throught (batches / second) of training ResNet-\n18 student with ResNet-34 teacher on ImageNet. The speed is tested\nbased on our implementations on 8 NVIDIA V100 GPUs.\nMethod KD RKD SRRL CRD DIST RCKA\nThroughout 14.28 11.11 12.98 8.33 14.19 11.31\nFigure 3: Training process visualization of all experimented detec-\ntors.\nTable 11: Ablation study of loss weight γ.\nT→S RetinaNet-X101 →RetinaNet-R50\nLoss Weight AP AP 50 AP75 APSAPM APL\nTeacher 41.0 60.9 44.0 23.9 45.2 54.0\nStudent 37.4 56.7 39.6 20.0 40.7 49.7\nγ= 5 39.8 59.4 42.5 22.3 43.5 53.9\nγ= 10 40.3 59.9 43.0 23.3 44.2 54.9\nγ= 15 38.5 58.0 41.0 20.7 41.8 51.3\nFigure 4: The effect of patch on different dimensions. This experi-\nment is conducted on the RetinaNet-X101-RetinaNet-R50 pair\nTable 12: Ablation study of Batchsize. Noted that ”4 ×1” means 4\nGPUs, each batchsize is 1, totally batchsize is 4×1 = 4\nT→S RetinaNet-X101 →RetinaNet-R50\nAP AP 50 AP75 APSAPM APL\nTeacher 41.0 60.9 44.0 23.9 45.2 54.0\nStudent 37.4 56.7 39.6 20.0 40.7 49.7\nBatchsize 4 ×139.9 59.5 42.8 21.7 43.9 53.3\nBatchsize 4 ×240.3 59.8 43.1 22.5 44.2 54.4\nBatchsize 4 ×340.3 59.9 43.0 23.3 44.2 54.9\nBatchsize 4 ×440.0 59.8 42.6 22.9 43.8 54.1\nF PCKA in image classification\nWe apply PCKA to the classification task, and it also outper-\nforms well on the methods with the same distillation type,\nas shown in Table 14. However, PCKA performs very badly\non the teacher and student pairs with different architectures.\nThe possible reason is that the cutting activations of differ-\nent architectures contain more dissimilar and harmful repre-Table 13: Ablation study of number of distillation layers. Noted that\n”Layer=1” means choosing 1 layer to distill.\nT→S RetinaNet-X101 →RetinaNet-R50\nAP AP 50 AP75 APSAPM APL\nTeacher 41.0 60.9 44.0 23.9 45.2 54.0\nStudent 37.4 56.7 39.6 20.0 40.7 49.7\nLayer=1 39.9 59.7 42.8 21.9 43.5 54.1\nLayer=2 40.2 59.9 42.6 21.9 44.0 54.6\nLayer=3 40.3 59.9 43.0 23.3 44.2 54.9\nTable 14: Results on the CIFAR-100 test set. We apply PCKA to\nthe classification task, and it also outforms some of the previous\nmethods.\nDistillation\nTypeTeacherResNet110 ResNet110 WRN-40-2 [2016b ]VGG13\n74.31 74.31 75.61 74.64\nStudentResNet20 [2016 ] ResNet32 WRN-16-2 VGG8\n69.06 71.14 73.26 72.50\nFitNet [2014 ] 68.99 71.06 73.58 71.02\nATKD [2016a ] 70.22 70.55 74.08 71.43\nSPKD [2019 ] 70.04 72.69 73.83 72.68\nCCKD [2019 ] 69.48 71.48 73.56 70.71\nRKD [2019 ] 69.25 71.82 73.35 71.48\nVID [2019 ] 70.16 70.38 74.11 71.23\nKD[2015 ] 70.67 73.08 74.92 73.33\nDKD [2022 ] - 74.11 76.24 76.32\nDIST [2022 ] 69.94 73.55 75.29 75.79\nPCKA(ours) 70.97 74.11 75.61 74.98\nsentations, causing difficulty in transferring knowledge to the\nstudent.Figure 5: CKA curve in training phase on CIFAR-100. We visualize the CKA similarities in the training for six teacher-student pairs.Figure 6: The confusion matrix is from the RetinaNet-R50, distilled by PCKA with teacher model RetinaNet-X101, averaging on channel.Figure 7: Top 5 inference results with RetinaNet-R50, distilled by PCKA without patch or with patch."    },    {        "title": "2401.11894v1.Exact_Normal_Modes_of_Quantum_Plasmas.pdf",        "content": "Exact Normal Modes of Quantum Plasmas\nTian-Xing Hu,1Dong Wu,1,a)Z. M. Sheng,2and J. Zhang1, 3, b)\n1)Key Laboratory for Laser Plasmas and School of Physics and Astronomy, and Collaborative Innovation Center of IFSA,\nShanghai Jiao Tong University, Shanghai, 200240, China\n2)Institute for Fusion Theory and Simulation, Department of Physics, Zhejiang University, Hangzhou 310027,\nChina.\n3)Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.\nThe normal modes, i.e., the eigen solutions to the dispersion relation equation, are the most fundamental properties\nof a plasma, which also of key importance to many nonlinear effects such as parametric and two-plasmon decay,\nand Raman scattering. The real part indicates the intrinsic oscillation frequency while the imaginary part the Landau\ndamping rate. In most of the literatures, the normal modes of quantum plasmas are obtained by means of small damping\napproximation (SDA), which is invalid for high- kmodes. In this paper, we solve the exact dispersion relations via the\nanalytical continuation (AC) scheme, and, due to the multi-value nature of the Fermi-Dirac distribution, reformation\nof the complex Riemann surface is required. It is found that the change of the topological shape of the root locus\nin quantum plasmas is quite different from classical plasmas, in which both real and imaginary frequencies of high- k\nmodes increase with kin a steeper way than the typical linear behaviour as appears in classical plasmas. As a result,\nthe temporal evolution of a high- kperturbation in quantum plasmas is dominated by the ballistic modes.\nI. INTRODUCTION\nThe study of quantum plasmas has important applications in\nthe fields of, e.g., astrophysics1, nano-physics2,3, warm dense\nmatter4, and inertial confinement fusion5–7, and thus received\nwidespread attention in recent years. However, some of its\nfundamental properties have rarely been seriously discussed.\nFor example, the exact eigen solution of a degenerate quan-\ntum plasma. In this paper, we solve the Landau damping rate\nby means of analytical continuation (AC) method, and avoid\nthe branch cut discontinuity by extending the Riemann sur-\nface of the 1d Fermi-Dirac distribution function (1dFDDF).\nThe AC method is a crucial step to comprehend the damping\nnormal modes of plasmas, proposed by Landau a long time\nago, which is widely used in the field of classical plasmas, but\nrarely discussed for quantum plasmas. The AC method is able\nto solve the exact normal modes with arbitrarily high wave\nnumbers, which are important when mirco-scale quantum ki-\nnetic effects are involved. And, the exact linear dispersion\nrelation of real frequency helps us better understand nonlin-\near phenomena, namely, the coupling between different linear\nmodes, and the exact imaginary frequency determines the life-\ntime of a plasmon.\nThe theoretical basis of this paper is the so-called collision-\nless quantum kinetic theory (QKT), which is, mathematically\nspeaking, a Wigner-Poisson system (WPS) of equations. The\nlinearized WPS is equivalent to the famous Random Phase\nApproximation (RPA), or the time-dependent Hartree approx-\nimation, which is well-proved to be a very successful model\nfor quantum plasmas when the density is higher than the typ-\nical solid density ( ∼1024cm−3). Such a dense, degenerate\nelectron environment is ubiquitous in the universe. For ex-\nample, the core of the sun is partially degenerate, the core\na)Electronic mail: dwu.phys@sjtu.edu.cn\nb)Electronic mail: jzhang1@sjtu.edu.cnof a white dwarf is extremely degenerate, and, the progeni-\ntors of most of energetic astrophysics events such as X-/ γ-ray\nburst and supernova explosion, are assumed to be related to\nextreme environments8,9. In laboratories, some of the inertial\nconfinement fusion experiment also produce degenerate quan-\ntum plasmas, e.g., the double-cone ignition5(DCI) scheme.\nHence, this work may has a wide range of applications.\nThis paper is organized as follow. In Sec. II, we introduced the\ncomplex structure of the Fermi-Dirac distribution function,\nthen briefly reviewed the collisionless QKT and the corre-\nsponding linear response theory, which is the theoretical basis\nof this paper. In Sec. III, we discussed the complex structure\nof the dielectric function of quantum plasmas. And thus the\nexact linear dispersion relation of quantum Langmuir wave is\nsolved by means of the aforementioned methods. Numerical\nsimulations are also presented to verify the results. Further\ndiscussion and main conclusion are presented in Sec. IV.\nII. THEORIES AND METHODS\nWe define a quantum parameter\nˆ¯h≡¯hωp\n2EF, (1)\nsometime is refereed to as the normalized Planck’s constant.\nHere, ωp=p\n4πe2n/meis the plasma frequency, and the\nFermi energy EF=¯h2\u0000\n3π2n\u0001\n/2me. It seems strange that ˆ¯h\nis proportional to n−1/2, since we expect that quantum effects\nshould be stronger for higher density. However, quantum ef-\nfects are also stronger for lower temperature, while ˆ¯hsolely\ndependent on n. Hence, to measure the importance of quan-\ntum wave effects, a more appropriate choice would be\n˜¯h≡¯hωp\n2kBT, (2)arXiv:2401.11894v1  [physics.plasm-ph]  22 Jan 20242\nand once the degeneracy Θ=kBT/EF(the inverse of which\nmeasures the importance degeneracy) is fixed, decreasing ˆ¯h\nmeans increasing ˜¯h.\nIn this paper, we adopt the natural unit system, where ¯h=\nme=e=kB=1. And the frequency, number density, velocity,\nlength, and energy, are normalized to ωp,ne(electron density),\nvF(Fermi velocity), λF=vF/ωp, and EFrespectively.\nA. Analytical structure of the 1dFDDF\nThe equilibrium state of electron obeys the 3d Fermi-Dirac\ndistribution (3dFDDF)\nf3d(v) =3\n4π1\ne(v2−µ)/Θ+1\n=−3\n4πLi0h\n−e(µ−v2)/Θi\n,(3)\nwhere Li nis the n-th order polylogarithm, and the value\nof the chemical potential µ=µ(Θ)is chosen such thatRf3d(v)d3v=1.\nAs a fundamental study, we only consider the interaction of\nelectrons to a plane wave field, hence we can integrate over the\ndimensions perpendicular to the wave, namely, we only care\nthe about the 1d Fermi-Dirac distribution (1dFDDF) f1d(v∥) =RRdv⊥f3d(v∥,v⊥). It is easy to prove that,\nf1d(v) =−3\n4ΘLi1\n2h\n−e(µ−v2)/Θi\n=3\n4Θlnh\ne(µ−v2)/Θ+1i\n.(4)\nLetvbe a complex variant v=vr+ivi, then the 1dFDDF is a\nmulti-value function, the branch cuts of which are\nvrvi=±π\n2ℓΘ, ℓ∈2Z, (5)\nand the branch points are located at the hyperbola:\nv2\nr−v2\ni=µ. (6)\nThe structure of the 1dFDDF is also thoroughly discussed in\nRef. 10. The density of the branch points on the hyperbola in-\ncreases with decreasing Θ. In Fig. 1, we plotted the 1dFDDF\nin complex- vplane, one can see that its imaginary part is dis-\ncontinued at the branch cuts. These discontinuities are equal\nto the height of a Riemann leaf, which is 3 πΘ/2 here, and\nthey exist because we considered only a single leaf the Rie-\nmann surface, see Fig. 2 (a). However, the Riemann surface\nof 1dFDDF has an infinite leaves because of the logarithm in\nEq. (4), see Fig. 2 (b). Discontinuity may cause unphysi-\ncal effects, as we shall see in the next subsection. Hence, we\nextend the area enclosed by the first branch cut and the hyper-\nbola to the whole Riemann surface, as is shown in Fig. 2 (c),\nwe then obtain a smooth surface between the two branches of\nthe hyperbola. Noticing that, after this operation, the discon-\ntinuities do not vanish, but are moved from the branch cuts to\nthe hyperbola.\nFigure 1. (a) Real (b) Imaginary part of the 1dFDDF ( Θ=0.1). The\ndash-lines are the hyperbolae of Eq. (6), and the white dotted-lines\nare the branch cuts of Eq. (5).\nB. The Linearized WPS\nThe electrons in a collisionless quantum plasma obey the\nWigner equation11\n∂tf+v·∂xf+iϑiˆ¯h∂p[φ(x)]f=0. (7)\nHere f=f(x,v,t)is the Wigner quasi-distribution function,\nwhich is the quantum counter part of the classical distribution\nfunction, and the pseudo-differential operator ϑis defined by\nϑy[O(x)]≡O\u0010\nx+y\n2\u0011\n−O\u0010\nx−y\n2\u0011\n. (8)\nNoticing that when y≪x,ϑy[O(x)]≃y·∇O, then Eq. (7)\nreduces to the Vlasov equation, thus the Wigner equation is\nalso called the quantum Vlasov equation.\nAnd we need the Poisson equation\n−∇2φ=nb−eZ\nfdv, (9)\nwhere nbstands for the background ion density, to close the\nsystem. Let f=f0+δf,φ=φ0+δφ, and consider only the\ndirection parallel to the wave vector, the linear evolution of\nthe perturbed field can be formally written as12\nδφ(t,k) =i\nk2ZZ\ndv∥dω\n2πδfk0\u0000\nv∥\u0001\ne−iωt\n\u0000\nω−kv∥\u0001\nε(ω,k), (10)\nwhere δfk0(v)is the initial perturbation, and the dielectric\nfunction (DF) is defined as\nε(ω,k) =1+1\nk2W\u0000ˆ¯hω,ˆ¯hk\u0001\n, (11)3\nFigure 2. (a) A single leaf (b) Seven leaves (c) Extended single leaf of the 1dFDDF ( Θ=0.1) Riemann surface.\nwhere\nW(ω,k) =Z\ndvϑk[f(v)]\nω−kv(12)\nis the Lindhard response function13(LRF). As Θ→∞,f(v)\ntends to Maxwellian, namely, in the classical limit, it reduces\nto\nW(ω,k) =2\nΘ1√\n2πZ\ndvve−v2\n2\nv−ω/kΘ=−2\nΘZ′\u0012ω\nkΘ\u0013\n,(13)\nwhere kΘ=kp\nΘ/2, and Zis the famous plasma dispersion\nfunction. It is evident that in quantum plasma, the linear re-\nsponse is dependent on both ωandk, but in classical plasma,\nit depends only on the ratio of ωtok. The response function\nthen does not depend on ˆ¯h, which means that when Θis large\nenough, the system naturally returns to classical, no matter the\nvalue of ˆ¯h.\nThe roots of the eigen-equation\nε(ω,k) =0 (14)\nyield the dispersion relation of the normal modes. Here,\nω=ωr+iωiis a complex number. To calculate the DF (11)\nexactly with negative value of ωiand solve Eq. (14), analyt-\nical continuation (AC) is needed for ωr=0 is a branch cut,\nnamely, when ωi<0, the response function (12) should be\nmodified to\nW(ω,k) =Z\ndvϑk[f(v)]\nω−kv−2πiϑkh\nf\u0010ω\nk\u0011i\n, (15)\nwhere the 2 πi term stems from the residue of the integrand of\nEq. (12). In the classical limit, it is\nΘ\n2W(η) =1−√\n2ηF\u0012η√\n2\u0013\n+ir\nπ\n2ηe−η2\n2, (16)\nwhere η=ω/kΘ, and\nF(x) =e−x2Zx\n0et2dt (17)\nis the Dawson integral.III. QUANTUM LANGMUIR WAVE\nA. Solving the Normal Modes\nThe Landau damping rate is the negative imaginary part of the\neigen-frequency. To calculation the DF with negative imagi-\nnary frequency, analytical continuation is needed. Noticing\nthat for extremely degenerate plasmas, analytical continuation\nis not needed when k<kF=vF=ωpλ−1\nF, since the dielectric\nfunction has an analytic solution when Θ→0:\nRe[ε] =1+3\n2k2\u0014\n1+1\n2ˆ¯hk\u0000\n1−b2\n−\u0001\nln\f\f\f\f1+b−\n1−b−\f\f\f\f\n−1\n2ˆ¯hk\u0000\n1−b2\n+\u0001\nln\f\f\f\f1+b+\n1−b+\f\f\f\f\u0015\n,(18)\nIm[ε] =3π2\n4ˆ¯hk3ln1+exp\u0002\u0000\nb2\n+−µ\u0001\n/Θ\u0003\n1+exp\u0002\u0000\nb2−−µ\u0001\n/Θ\u0003, (19)\nwhere\nb±=ω\nk±ˆ¯hk\n2. (20)\nThus one can see than only when |b±|>1, namely,\n−k±ˆ¯hk2\n2<ω<k±ˆ¯hk2\n2, (21)\nthe DF has finite imaginary part. The area enclosed by (21) is\nreferred as to the electron-hole excitation continuum (EHEC),\nin which a plasmon delay into an electron and a hole. The\nabove formula is obtained based on the small damping ap-\nproximation (SDA):\nlim\ny→01\nx±iy=P1\nx∓iπδ(x), (22)\nwhich is incorrect when ωiis finite. In the following, we di-\nrectly solve the Eq. (15) without any approximation scheme.4\nFigure 3. (a) (b): Real and imaginary part of the dispersion function\nwithout analytical continuation, and (c) (d): with analytical contin-\nuation. The dashed-lines are mappings of the branch cuts. The red\nlines stand for Re [ε(ω,k)] = 0, and blue lines for Im [ε(ω,k)] = 0.\nIn Fig. 3, the numerical results of dispersion function with\nor without AC are presented ( Θ=0.1,ˆ¯h=0.6), where the\nwave number k=0.3. The upper two panels (a) and (b) are\nthe real and imaginary parts of the DF without AC, while the\nlower two are with. It is shown that the discontinuity of the\n1DFDDF is explicitly mapped from the complex vplane to the\ncomplex ωplane by the ϑoperator, since in the calculation,\nwe used the non-extended 1DFDDF, i. e., the single leaf Rie-\nmann surface of Fig. 2 (a). The series of pairs of black dots\nin the lower half plane are located at two hyperbolas, whose\nvertices are µk±ˆ¯hk2/2 respectively.\nIn the rest of this paper, the imaginary part of the DF would\nnot be presented, but we still keep the Im [ε(ω,k)] = 0 lines\nin the real DF diagrams. And the colorbar will also be ne-\nglected since the absolute values of the DF are irrelevant in\nthe context.\nTake an example of a degenerate plasma, say, with density\nn=3×1026cm−3, and temperature T=300eV. Then we\nhave the quantum parameter ˆ¯h=0.2 andΘ=0.2. The real DF\nof such parameters in complex frequency plane are plotted In\nFig. 4, where the real root loci are indicated by red lines and\nthe imaginary loci by blue lines. Noticing that the real and\nimaginary loci have multiple intersections, which stand for\nmultiple solutions of the normal modes. Generally, we only\ncare about the least damping mode, i.e., the intersection point\nwith maximum ωi, which we indicated by black crosses in\nFig. 4 and labeled its value. In panels (a) to (e), we used the\nsingle leaf 1DFDDF, the discontinuities occur at the branch\ncuts. One can see that there are no intersections in panel (e),\nso we have to use the extended 1DFDDF to move the discon-\ntinuities to the hyperbolas, as is seen in panel (f), in which a\ndamping mode is obtained.For another example of extremely degenerate plasma, we\nchoose density n=3×1029cm−3, and temperature T=\n103eV, which could be the interior of a typical small white\ndwarf. Then we have the ˆ¯h=0.06 and Θ=0.005. The DF\nresults are presented in Fig. 5. One finds that, at such small\nΘ, the distance among each branch cuts are extremely small.\nAll the modes with finite damping rate are located between\nthe two hyperbolas (a gap opened by quantum wave effect).\nHence the extend 1DFDDF is indispensable in this case. From\nFig. 5, one also finds that when ωiis finite,\nωr≃µk+ˆ¯hk2\n2, (23)\nand since µ→1 when Θ→0, this is upper bound of the\nEHEC.\nNow we briefly discuss the topological features of the root\nloci. Generally, the imaginary locus starting from ∞+0i, pass-\ning through every pair of branch point, while the real locus\nstarting from one of the branch points and end with another. If\nthe starting and ending points are a pair, and the locus is above\nthe line joining the two branch points, we refer to this topolog-\nical shape as “classical”, for it is topologically identical to the\nclassical solution, see Fig. 6 (The classical thumb-like figure\nis very common in the field of classical plasma instabilities,\ne.g., see Ref. 14). Otherwise it is a “quantum” shape. For\nexample, the (a) ~ (d) panels in Fig. 4 are classical, while (d),\n(e) and (f) are quantum. And in extremely degenerate case\nlike Fig. 5, the classical shape does not exist at all. Further-\nmore, a quantum shape means that the least damping mode is\nmost likely located within the two hyperbolas.\nIn Fig. 6, the classical sub-figure is calculated via Eq. (16),\nand the other two are quantum results with Θ=4, which is\nalmost Maxwellian, it is shown that the normal modes in these\nthere sub-figure are fairly close to one another. As Θ→∞, all\nthe branch points going to −i∞, and we know that the value\nofˆ¯h(or˜¯h) only effects the distance between a pair of branch\npoints. This diagrammatically demonstrated that the larger Θ\nis, the less important the value of ˆ¯h(or˜¯h) is.\nB. Exact Solution of the Normal Modes\nIt is worth mentioning that, ε(ω,k) =0 has multiple roots in\na finite region of the complex ωplane, as one can see from\nFig. 3, but we only care about the lowest damping mode.\nHence, to solve the full dispersion relation of the QLW, we can\nstarting from k≃0+, and calculate a small region centered at\nω0=1+0+i to find the first root ω1and record the difference\n∆ω1=ω1−ω0. Then increasing kwith a small value and cal-\nculating a next region centered at ω1+∆ω1to find the second\nroot. Repeating this procedure, then a continuous curve of the\ncomplex frequency of the QLW as a function of kis obtained.\nThe full solutions of normal modes ω(k)with different den-\nsities are plotted in Fig. 7. In panel (a), where ˆ¯h=0.4 is\ncorresponding to electron number density n=4×1024cm−3,\nandΘ=0.2, 0.4, and 0 .8, are corresponding to the temper-\nature T=19, 37 and 74eV . In panel (b), where ˆ¯h=0.2 is5\nFigure 4. Complex frequency plane of dispersion function, with Θ=0.2,ˆ¯h=0.2.\nFigure 5. Complex frequency plane of dispersion function, with Θ=0.006,ˆ¯h=0.06.\nFigure 6. The DF complex frequency plane of classical and almost-\nclassical plasmas.\ncorresponding to electron number density n=3×1026cm−3,\nandΘ=0.1, 0.2, and 0 .4, are corresponding to the tempera-\ntureT=170, 340 and 680eV . In both the two cases, the eigen\nfrequency of lower temperature plasmas surpass the higher\ntemperature when kis large enough, which is the result of\nthe quantum topological shape. In panel (c), all the curvesare calculated with Θ=0.02, since the shape of the curves\nhardly change when Θfurther decreases, we annotated they as\nΘ<0.02, or, one can simply treat them as zero-temperature\nresults. The dashed-lines attached to each solid line are the\ncorresponding upper bounds of the EHEC. One can see that,\nthe solid lines are slightly lower than the dashed-lines only\nin a very small region, which means that the EHEC predicted\nby Eq. (21) is incorrect when kis large. This is because the\nderivation of Eq. (21) is based on the SDA, which is incorrect\nfor finite damping rate.\nC. Numerical Benchmark of the Normal Modes\nThe WPS, i. e., Eq. (7) and (9) can be solved numerically15,16\nas a initial value problem. We set the initial perturbation as\nf(x,v,0) = [ 1+Acos(k0x)]f1d(v), (24)\nand let it evolve, then measure the frequency and damping rate\nof the perturbation. The factor Ais chosen to be very small to\navoid nonlinear effects. When the initial perturbation is set,6\nFigure 7. Dispersion relation calculated via AC method, where the solid lines stand for real frequency and dashed-lines for imaginary, and the\ncolored markers are simulation results (square for real and triangle for imaginary part).\nthe time-dependent behavior of the perturbed field δφconsist\nof a ballistic mode and an infinity number of normal mode,\nand this can be shown by rewrite Eq. (10) as\nδφ(t,k) =1\nk2Z\ndvδfk0(v)\n×\u0014e−ikvt\nε(kv,k)+∑\nne−iωnt\n(ωn−kv)∂ωnε\u0015\n,(25)\nwhere the first term in the bracket stands for the ballistic mode\nwhile the second the normal modes.\nSome results are presented also in Fig. 7, where the colored\ncircles stand for real frequency measured from simulation re-\nsult, and the triangles for damping rate. The simulation re-\nsults and the numerical results of normal modes match up per-\nfectly. However, the simulation results of high- kmodes are\nnot shown, especially in panel (c), where the damping rates\nof the simulation points with the largest kare one order lower\nthan they real frequency. We do not present the result of high k\nbecause it is found that those high- kmodes have uncertain fre-\nquency, or, do not damp exponentially, it is impossible to mea-\nsure the complex frequency nor the damping rate. The reason\nfor this non-exponential behavior is, as we have mentioned\npreviously, in degenerate plasmas, when kis large enough,\nthe topological shape of the real locus become non-classical,\nwhich results in the frequency and the damping rate of nor-\nmal modes increase faster than linear with k. Hence, beyond\na critical value of k, the damping rate of the normal mode will\nsurpass the ballistic mode, the time-dependent behavior then\ndominated by the ballistic mode. This phenomenon does not\noccur in a classical plasma since its dispersion relation is al-\nways linear as kincreases (when kλD≳1).\nThe proof of the above statement is represented in Fig. 8,\nwhere the red lines are the pure ballistic evolution obtained by\nintegrate the first term in Eq. (25). In panel (a), where Θ=0.2\nand ˆ¯h=0.6, when k=1.3, the ballistic mode damps faster\nthan the normal mode, hence the simulation result shows aclear normal mode with ωi=−0.185 (the dotted line stands\nfor e−0.185t). However, when k=2.1, the normal mode should\ngiveωi=−0.944, while the simulation curve damps slower\nthan e−0.944tand is almost identical to the ballistic mode, and\nit confirms our conclusion. Also noticing that, the simulation\ncurve does not have a clear real frequency. Similarly, in panel\n(b), where Θ=0.02 and ˆ¯h=0.05, a clear normal mode with\nωi=−0.0957i is measured for k=2.1, but for k=2.9 the\nlong time behavior is interfered by ballistic mode, hence no\nclear normal mode can be seen.\nIV. DISCUSSION AND CONCLUSION\nIn this paper, we adopt the analytical continuation scheme to\nsolve the dispersion relation of degenerate plasma. Compared\nto SDA scheme, the AC scheme can solve the normal modes\nof a dielectric system with arbitrarily high k, which is crucial\nwhen small-wavelength quantum kinetic effects are encoun-\ntered. We find that in degenerate plasmas, the normal mode\nfrequency and its damping rate increase with ksteeper than\nlinear, which is related to the “quantum” topological shape of\nthe root locus. As a result, the temporal evolution of a high-\nkperturbation in quantum plasmas is dominated by ballistic\nmode. Especially, the exact solution of linear dispersion rela-\ntion is the basis of the quantitative analysis of nonlinear effects\nsuch as parametric decay, two-plasmon decay, Raman scatter-\ning, and etc., in dense plasmas.\nV. ACKNOWLEDGEMENTS\nThis work was supported by the Strategic Priority Re-\nsearch Program of Chinese Academy of Sciences (Grant No.\nXDA250050500), the National Natural Science Foundation\nof China (Grant No. 12075204), and Shanghai Municipal7\nFigure 8. Temporal evolution of linear perturbations calculated by\nnumerical simulation (the blue lines), and the ballistic evolution (the\nred lines): (a) Θ=0.2,ˆ¯h=0.6, (b) Θ=0.02,ˆ¯h=0.05. The dotted-\nlines indicate the damping rate calculated by the eigen equation (14).\nScience and Technology Key Project (No. 22JC1401500).\nDong Wu thanks the sponsorship from Yangyang Develop-\nment Fund.REFERENCES\n1Dmitri A Uzdensky and Shane Rightley. Plasma physics of extreme astro-\nphysical environments. 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Physical Review E , 105(6):065203, 2022."    },    {        "title": "2401.11954v1.RUMBoost__Gradient_Boosted_Random_Utility_Models.pdf",        "content": "RUMBoost: Gradient Boosted Random Utility Models\nNicolas Salvad ´ea,∗, Tim Hillela\naDepartment of Civil, Environmental and Geomatic Engineering, University College London, Gower Street, London, WC1E 6BT, United Kingdom\nAbstract\nThis paper introduces the RUMBoost model, a novel discrete choice modelling approach that combines the inter-\npretability and behavioural robustness of Random Utility Models (RUMs) with the generalisation and predictive abil-\nity of deep learning methods. We obtain the full functional form of non-linear utility specifications by replacing each\nlinear parameter in the utility functions of a RUM with an ensemble of gradient boosted regression trees. This enables\npiece-wise constant utility values to be imputed for all alternatives directly from the data for any possible combina-\ntion of input variables. We introduce additional constraints on the ensembles to ensure three crucial features of the\nutility specifications: (i) dependency of the utilities of each alternative on only the attributes of that alternative, (ii)\nmonotonicity of marginal utilities, and (iii) an intrinsically interpretable functional form, where the exact response of\nthe model is known throughout the entire input space. Furthermore, we introduce an optimisation-based smoothing\ntechnique that replaces the piece-wise constant utility values of alternative attributes with monotonic piece-wise cu-\nbic splines to identify non-linear parameters with defined gradient. We demonstrate the potential of the RUMBoost\nmodel compared to various ML and Random Utility benchmark models for revealed preference mode choice data from\nLondon. The results highlight both the great predictive performance and the direct interpretability of our proposed\napproach. Furthermore, by analysing the non-linear utility functions, we can identify complex behaviours associated\nwith di fferent transportation modes which would not have been possible with conventional approaches. The smoothed\nattribute utility functions allow for the calculation of various behavioural indicators such as the Value of Time (V oT)\nand marginal utilities. Finally, we demonstrate the flexibility of our methodology by showing how the RUMBoost\nmodel can be extended to complex model specifications, including attribute interactions, correlation within alternative\nerror terms (Nested Logit model) and heterogeneity within the population (Mixed Logit model).\nKeywords: Discrete Choice, Mode Choice, Machine Learning, Random Utility, Ensemble Learning\n1. Introduction and literature review\nDiscrete choice models (DCMs), based on Random Utility theory, have been used extensively to model choices\nover the last 50 years (Ben-Akiva and Lerman, 1985; Train, 2009), including the choice of travel mode. DCMs have\nmany desirable qualities: most crucially, their parametric form is directly interpretable and allows for the integration\nof expert knowledge consistent with behavioural theory. For example, with a DCM, it is possible to: (i) ensure\nmarginal utilities in the model are monotonic (e.g. that increasing the cost of an alternative will always decrease\nits utility); (ii) restrict the utilities of each alternative to be dependent only on the attributes (or features) of that\nalternative; and (iii) define arbitrary interactions of socio-economic characteristics and alternative-specific attributes.\nThese three points are essential to derive key behavioural indicators, such as elasticities and Value of Time (V oT),\nused to inform transport policies and investment decisions. However, the parametric form of DCMs is not without\ndisadvantages. Crucially, the linear-in-parameters utility functions are relatively inflexible and must be specified in\nadvance by the modeller. As such, these models may fail to capture complex phenomena and non-linear e ffects in\nhuman behaviour. Among DCMs, one of the best-known and most widely used models is the Multinomial Logit\nmodel (MNL) derived by McFadden et al. (1973). The model assumes that the error term is a type 1 extreme value\n∗Corresponding author\nEmail addresses: nicolas.salvade.22@ucl.ac.uk (Nicolas Salvad ´e),tim.hillel@ucl.ac.uk (Tim Hillel)\nPreprint submitted to Transportation Research Part C: Emerging Technologies January 23, 2024arXiv:2401.11954v1  [cs.LG]  22 Jan 2024(EV) random variable, which is independent and identically distributed ( i.i.d.) across all alternatives and observations.\nThis formulation results in a closed-form expression of the probability, allowing for easier estimation of the model\nparameters. However, more complex model specifications exist, for example to capture: (i) di fferent distributions of\nthe error term; (ii) nesting (joint consideration) of alternatives; (iii) behavioural heterogeneity across individuals in\nthe population; and (iv) behavioural heterogeneity across sequential choices.\nThere have been numerous attempts to apply machine learning (ML) probabilistic classification algorithms, such\nas neural networks and ensemble learning, to investigate choice behaviour. These models exhibit high predictive\nperformance and, thanks to their data-driven nature, do not require any utility functions to be specified in advance of\nmodel estimation. However, they lack an underlying behavioural model and so it is not possible to guarantee con-\nsistency of forecasts or derive behavioural indicators such as Value of Time (V oT) or willingness-to- pay from the\nmodel parameters. Initial approaches for analysing these models from a behavioural perspective rely on approximat-\ning the partial derivatives of the output probabilities of unconstrained ML classifiers in order to define elasticities (and\ntherefore marginal rates of substitution /V oT) for variables of interest. Wang et al. (2020) use this approach to extract\neconomic indicators from neural networks for mode choice problems, whilst ´Angel Mart ´ın-Baos et al. (2023) extend\nthis to several other classification algorithms, including Gradient Boosting Decision Trees, the Random Forest, and\nSupport Vector Machines. Unlike marginal utilities from a DCM, which have a parametric functional form, the prob-\nability derivatives of ML classifiers provide only a numeric estimate of the point elasticities at observed data points.\nFurthermore, as the underlying models are unconstrained, they exhibit several qualities that are inconsistent with\nrandom utility theory, including: (i) non-monotonic elasticities, leading to unwanted behaviours, such as a negative\nV oT; and (ii) including all features for all alternatives uniformly, therefore violating the independence of irrelevant\nalternatives assumption. As such, these techniques have seen limited real-world use and practitioners continue to\nrely predominantly on parametric DCMs. That being said, the ability of ML models to capture complex non-linear\nrelationships as well as their improved predictive accuracy makes them an attractive proposition.\nIn response to these limitations, there has been an emergence of hybrid data-driven utility models in recent years,\nthat attempt to combine the benefits of ML and DCMs. These can largely be grouped into two di fferent approaches:\n1. adding additional constraints to machine learning models (e.g. monotonicity, alternative-specific attributes, etc)\nso that their output can mimic DCM utility values; and\n2. using data-driven approaches to automate or assist with identifying suitable parametric utility functions.\nThere have been several studies which attempt to incorporate key components from DCMs through making appro-\npriate constraints on the structure of ML models. This analysis stems from the fact that probabilistic ML classifiers,\nsuch as neural networks and GBDTs, make use of the same logistic function (commonly referred to as softmax in ML)\nas the MNL to generate choice probabilities over each alternative; replacing the linear-in-parameter utility functions\nof the MNL with a complex network of neurons with non-linear activation functions (in the case of the neural net-\nwork) or ensemble of regression trees (in the case of GBDTs). As such, with suitable constraints on the model, the\npre-softmax regression values for each alternative can be considered as pseudo-utilities . The most important of these\nconstraints is that the pseudo-utility of each alternative is a function of only the attributes of that alternative (alongside\nthe socio-economic characteristics of the decision-maker), therefore obeying the independence of irrelevant alter-\nnatives assumption. Without this assumption, it is not possible to perform a utility-based behavioural analysis as\nmodifying one attribute would a ffect the utility values of all alternatives. Wang et al. (2021) build on their prior work\nof modelling choices with DNNs by defining a separate sub-network per alternative, dependent only on the attributes\nof that alternative, thus allowing for more robust behavioural indicators to be extracted. Similarly, Mart ´ın-Baos et al.\n(2021) make use of kernel Logistic Regression (KLR) (where the non-linearity is achieved through transforming the\ninput data with a kernel) to learn the deterministic part of each alternative utility by having one kernel per alternative.\nThis allows for behavioural indicators to be defined in the pseudo-utility space, rather than the probability space. Other\nstudies instead use a neural network to complement a conventional utility model. Sifringer et al. (2020) use a Con-\nvolutional Neural Network (CNN) to learn a linear-in-parameter utility function for the alternative-specific attributes\nalongside a DNN to get socio-economic interactions, adding them to the pseudo-utility output in a second stage. In\na similar approach, Wong and Farooq (2021) use a Residual Neural Network to add a non-linear, fully-connected\nresidual on top of the linear pseudo-utility at each layer of the network.\n2However, these studies still lack of a key component of the interpretability and extrapolation ability of DCMs:\nmonotonicity of marginal pseudo-utilities. Whilst not relying on traditional deep learning models, there are several\nstudies that allow for monotonicity of marginal utilities in ML models. Kim and Bansal (2023) use a lattice network\nwith input and output calibration layer to account for non-linearities and partial monotonicity. Krueger and Daziano\n(2022) replace the sum operator in a traditional Mixed Logit model with the Choquet integral, accounting for attribute\ninteraction under monotonicity. Lastly, Aboutaleb (2022) take advantage of the sum of squares of polynomials to\nlearn data-driven utility functions under monotonicity and shape constraints.\nFinally, some researchers retain the traditional structure of the utility function but add data-driven components\nsuch as learned parameters or assisted utility specification. Han et al. (2022), with the TasteNet model, use a Deep\nNeural Network to learn individual-specific parameters from socio-economic characteristics, constraining the sign\nof the parameters with the appropriate activation function. Instead of replacing the deterministic utility by a data\ndriven model, Hillel et al. (2019) and Ortelli et al. (2021) use data-driven models to assist the modeller with utility\nspecification. The first paper suggests potential non-linear interactions of attributes from the gain of split points in a\nGBDT algorithm. The second study proposes an algorithm for automatically evaluating multiple model specifications\nby drawing the Pareto frontier, which is useful in assessing the quality of model specification under multiple objectives.\nHowever, the modeller still needs to pre-specify the input space and identify non-linearities.\nThe summary of state-of-the-art practices in hybrid utility models is in Table 1. The features that we consider\nin the table are if the models: (i) include alternative-specific attributes; (ii) incorporate monotonicity constraints;\n(iii) have intrinsically interpretable pseudo-utilities, as in DCMs; (iv) provide automatic identification of non-linearity\nin the pseudo-utilities, i.e the complete non-linear transformations from the input variable to the pseudo-utilities are\nobservable; and (v) include non-i.i.d. error terms, therefore allowing for complex model specifications such as the\nNested Logit or the Mixed Logit models; In this paper, we provide an algorithm to directly learn the non-linear full\nfunctional form of the utility, while ensuring alternative-specific attributes and monotonicity on key attributes.\nTable 1: Summary of state-of-the-art practice in hybrid utility models. A cross (X) means that the model implements the feature and a tilde ( ∼)\nmeans that the model partly incorporates the feature. Complete explanations on the features that we consider for each column are in the text.\nPapersAlternative\nspecific\nattributesMonotonicity\nconstraintIntrinsically\ninterpretable\nutility\nfunctionAutomatic\nidenfication of\nnon-linearitiesNon-i.i.d\nerror\nterms\nWang et al. (2021) X\nMart ´ın-Baos et al. (2021) X\nWong and Farooq (2021) X ∼ X\nSifringer et al. (2020) X ∼ X\nHan et al. (2022) X X X\nKim and Bansal (2023) X X ∼ ∼\nKrueger and Daziano (2022) X X ∼ X\nAboutaleb (2022) X X X X\nOrtelli et al. (2021) X X X X\nWhilst the above review evaluates the literature in hybrid data-driven utility modelling, it is also worth considering\nthe more general field of eXplainable Artificial Intelligence (XAI), where researchers aim to add explanations to the\npredictions of general purpose ML algorithms. The most well known XAI techniques, in use across multiple fields,\nare: LIME (Ribeiro et al., 2016) and SHAP (Lundberg and Lee, 2017). Several researchers use them to provide\ninterpretability of ML classifiers in a mode choice context (Tamim Kashifi et al., 2022; Ren et al., 2023; Dahmen\net al., 2023; ´Angel Mart ´ın-Baos et al., 2023). However these methods are local approximations of the model relying\non data points and, therefore, have no guarantee of validity outside of the input space. On the other hand, Explainable\n3Boosting Machine (EBM) from Caruana et al. (2015) is an example of interpretable Gradient Boosting Decision Trees\nalgorithm. They achieve interpretability by growing shallow regression trees. However, the model does not allow the\nmodeller to specify alternative-specific variables and monotonicity is only applied as a post-processing tool on features\nwithout interaction. In addition, a feature can interact with several other features simultaneously, which is inconsistent\nwith random utility theory. We also observe that the majority of research on ML models for choice modelling focuses\non neural networks. Although neural networks have numerous applications in many fields and show great results, other\nalgorithms exhibit better performance on classification tasks such as choice modelling. In particular, GBDT models,\nincluding XGBoost (Chen and Guestrin, 2016) and LightGBM (Ke et al., 2017) have demonstrated state-of-the-art\npredictive performance for a variety of classification tasks (Hillel et al., 2018; ´Angel Mart ´ın-Baos et al., 2023). These\nmodels, despite lacking of behaviour interpretability, consistency and robustness (like other ML models) show great\ninterpretability potential due to their additive nature. Furthermore, existing ML approaches have continued to rely on\nthe logit formulation, with i.i.d. EV error terms. As such, they cannot capture complex behavioural phenomenons\n(e.g. panel e ffects, dynamic or sequential choice situations, nesting of alternatives, etc).\nIn this paper, we present Random Utility Models with Boosting (RUMBoost) which aims to combine the predictive\npower of GBDTs with the interpretability and behavioural consistency of DCMs. At a high level, RUMBoost replaces\neach parameter in the utility specifications of traditional DCMs with an ensemble of regression trees, allowing for non-\nlinear parameters to be extracted directly from data. Algorithmically, RUMBoost consists of two parts: (i) Gradient\nBoosted Utility Values (GBUV), where ensembles of regression trees are used to impute piece-wise constant values\nfor each parameter in each pseudo-utility specification; and (ii) Piece-wise Cubic Utility Functions (PCUF), where\nmonotonic piece-wise cubic splines are optimised to fit the GBUV outputs, to allow for a defined gradient for each\nparameter where elasticities are needed.\nThe GBUV output e ffectively defines piece-wise constant pseudo-utility specifications that are consistent with\na DCM, with the ability to constrain the model to have: (i) pseudo-utility functions depending only on their al-\nternative-specific attributes and socio-economic characteristics; (ii) monotonicity of marginal pseudo-utilities; and\n(iii) arbitrary interactions between variables as defined by the modeller. Furthermore, the GBUV output is intrinsi-\ncally interpretable, with the exact functional form observable to the modeller over the full input space of each variable.\nHereafter, we no longer distinguish between utility and pseudo-utility as we believe that, with the constraints afore-\nmentionned, the GBUV output can be interpreted as utility values.\nAs the GBUV output is piece-wise constant, it does not have a defined gradient, with the gradient being either zero\nor infinite at any given point. As such, behavioural indicators such as elasticities and V oT cannot be extracted directly.\nThis is addressed through the second element of RUMBoost, PCUF. PCUF only needs to be applied for variables for\nwhich elasticities are needed (typically the alternative-specific attributes), which optimises a set of piece-wise cubic\nsplines (polynomials) to obtain a defined gradient over the full range of the input space. This allows the modeller to\ncompute behavioural indicators such as the Value of Time (V oT).\nThe flexibility of our approach allows for estimating complex model structures with correlated error terms such as\nthe Nested Logit model or accounting for population heterogeneity such as the Mixed Logit model. The model is im-\nplemented with Python and the code is freely available on GitHub ( https://github.com/NicoSlvd/RUMBoost ).\nThe rest of the paper is organised as follows: Section 2 presents the methodology of this paper, the results of a case\nstudy are presented in Section 3, Section 4 provides some model extensions, and Section 5 concludes and suggests\nareas of improvement for future work.\n2. Methodology\n2.1. Theoretical background\nWe summarise here key theory of both RUMs and GBDTs under a unified notation in order to help the reader\nunderstand our approach. For a more detailed background on each approach, we direct the reader to the following\ntexts: Ben-Akiva and Lerman (1985); Train (2009); Friedman (2001); Chen and Guestrin (2016); Ke et al. (2017).\n2.1.1. RUMs\nDiscrete Choice Models based on Random Utility theory assume that individuals choose the alternative that max-\nimises their utility, a latent measure of the attractivity of an alternative. The utility is usually defined as follows:\n4Uin=Vin+εin (1)\nwhere Vinis the observable utility and εinthe error term capturing unobserved variables for alternative iand individual\nn. In a DCM, Vinis a manually specified linear-in-parameters function of the variables, such that:\nVin=βi,0+KX\nkβi,kxikn (2)\nwhere K+1 parameters associated with Kvariables xiknhave to be estimated from the data. For simplicity, we\nconsider here the MNL model, where the error term is assumed to be i.i.d, following an extreme value distribution of\ntype 1 (Gumbel). We will later demonstrate more complex model specifications where the error terms are allowed to\nbe correlated across alternatives. For the MNL, in a classification problem with Jclasses, the probabilities are given\nby the multi-class logistic function:\nPin=eVin\nPJ\njeVjn(3)\nThe qualities of the MNL model are directly derived from the parametric form of Vin. In particular, it is intrinsically\ninterpretable because there are no variable interactions.\nIn addition, the modeller can easily incorporate domain knowledge through alternative-specific attributes and\nmonotonicity of marginal utilities. For a multi-class classification problem with Jclasses, and Kexplanatory variables,\nwe define:\nxi=ai∪s (4)\nwhere xiis the set of variables for alternative iand\n(i)|xi|≤K, i.e. the cardinality of xishould be smaller or equal than the number of attributes K\n(ii) aiis the set of alternative-specific attributes for alternative i, such that ai∈x,∀iandai∩aj=∅,∀i,j\n(iii) sis the set of socio-economic characteristics such that s∈xands∩ai=∅,∀i\nDiscrete choice models also guarantee the monotonicity of marginal utilities. A positive (resp. negative) mono-\ntonic relationship implies that an increasing attribute xkwill increase (resp. decrease) the value of the utility function\nVin. The signs of parameters in DCMs allow the modeller to easily verify the monotonicity of an attribute and, if\nneeded, to constrain it with parameter bounds.\nFinally, the gradient is always defined and easy to compute, which allows for the derivation of behavioural indi-\ncators. Among them, the Value of Time (V oT) is an important measure of the perceived cost of travelling time by\nindividuals. The V oT of an alternative iis defined as follows:\nVoT i=∂Vi\n∂xi,time·1\n∂Vi\n∂xi,cost=βi,time\nβi,cost(5)\nwhereβi,timeandβi,costare the parameters associated with the time xi,timeand cost xi,costattributes of the utility function\nVi.\n2.1.2. Gradient boosting\nThe following notation combines that of Friedman (2001) and Chen and Guestrin (2016), summarising the GBDT\nalgorithm commonly used in popular libraries such as XGBoost and LightGBM. Deep learning models, including\nDNNs and GBDT use the same multi-class logistic function (typically referred to as the softmax function) as the MNL,\nwhich means that they are based on the same assumptions on the error term. In other words, these models first estimate\na latent regression value for each class which is then used to generate choice probabilities. The advantage of these\nmodels over a linear-in-parameter RUM is that they are inherently non-linear, and so can capture complex relationships\n5between the input features and the choice probabilities. However, unlike in RUMs, these regression models: (i) are\na function of all explanatory variables /features (i.e. attributes of each alternative); (ii) do not contain any constraints\nto represent behavioural assumptions; (iii) allow for complex feature interactions that cannot be constrained (or in\nmany cases even observed) by the modeller; and (iv) have an unknown functional form that is not observable by the\nmodeller. Formally, the ML predictive function Fi(x) is used to replace the deterministic part of the utility function\nVi(x). In GBDT algorithms, the predictive function is an additive function of the form:\nVi(x)=Fi,m(x)=X\nmfi,m(x) (6)\nwhere fi,m(x) is the output of a single regression tree. At each boosting round m, assuming Jclasses, Jregression\ntrees are induced to directly minimise the following objective:\nL=NX\nn=1JX\ni=1ℓ(yin,ˆyin,m−1+fi,m(xin)) (7)\nwhere:\n•Nis the number of observations in the dataset;\n•yin=1 if the choice inof the individual nisi, and 0 otherwise; and\n•ˆyin,m−1is the predicted probability of class iand observation nwith attributes xinat iteration m−1,i.e.ˆyin,m−1=\nFi,m−1(xin).\nThis optimisation problem has no closed-form solution. It is generally solved by taking the Taylor second-order\napproximation of the loss function:\nL=NX\nn=1JX\ni=1ℓ(yin,ˆyin,m−1)+ginfi,m(xin)+1\n2hinfi,m(xin)2(8)\nwhere gin=∂ℓ(yin,ˆyin,m−1)/∂ˆyin,m−1andhin=∂2ℓ(yin,ˆyin,m−1)/∂2ˆyin,m−1are the first and second derivative of the loss\nfunction with respect to the prediction for observation nand class i. Here, the Hessian matrix is replaced by a diagonal\napproximation following Friedman (2001), which means that each class prediction is assumed to be independent. This\napproximation is crucial for computational e fficiency and allows for e fficient optimisation of the loss function. In\naddition, we can ignore the constant term ℓ(yin,ˆyin,m−1) for the minimisation task.\nAssuming that we have Lterminal nodes (i.e. the bottom nodes of the regression tree) resulting in Lregions Lim\nfor a tree of class iat iteration m, each observation will belong uniquely to one of the region Lim, such that Equation\n8 becomes:\nL=JX\ni=1X\nl∈Lim(X\nn∈lgin)γl,im+1\n2(X\nn∈lhin)γ2\nl,im (9)\nwithγl,imbeing the leaf value at region lfor class i. Therefore, by taking the first derivative with respect to γand\nsetting it to 0, we obtain the optimal leaf value:\nγl,im=−(P\nn∈lgin)\n(P\nn∈lhin)(10)\nBy substituting Equation 10 into 9, and assuming a split point that would lead to a left Lle f t,imand right Lright,im\nregions, we obtain the loss reduction for splitting:\nLsplit=1\n2JX\ni=1X\nlle f t∈Lle f t,im(P\nn∈lle f tgin)2\nP\nn∈lle f thin+X\nlright∈Lright,im(P\nn∈lrightgin)2\nP\nn∈lrighthin−X\nl∈Lim(P\nn∈lgin)2\nP\nn∈lhin(11)\n6Therefore, we can choose the split point that maximises the loss reduction. For a classification problem, the loss\nfunction is usually defined as:\nℓ=yin·log(pi(xin)), (12)\nwhere pi(xin)=ˆyin. This equation is typically referred to as the cross-entropy loss in ML contexts, though is equivalent\nto the log likelihood in RUMs. As in the MNL, we can obtain the predicted probabilities of each class using the\nsoftmax function:\npi(xin)=eVi(xin)\nPJ\nj=1eVj(xjn), (13)\nPlugging Equations 12 and 13 into Equation 10 gives:\nγl,im=J−1\nJP\nxin∈Rl,imyin−pi(xin)\nP\nxin∈Rl,impi(xin)(yin−pi(xin))(14)\nwhereJ−1\nJis a factor accounting for redundancy.\nThe utilities of each ensemble for each individual can then be updated at each iteration:\nVi,m(xin)=Vi,m−1(xin)+LX\nl=1γl,im·1(xin∈Lim) (15)\nwhere 1(·) is the indicator function, i.e. equals 1 if the argument is true, 0 otherwise.\nMonotonicity is easily implemented in regression trees. If we assume a positive (resp. negative) monotonic\nrelationship for an explanatory variable, a split point that partitions the data in two such that xle f t<xrightis only\nconsidered if:\nγle f t<γ right (resp.γle f t>γ right) (16)\nNote that, in order to satisfy the monotonicity over the full tree, the subsequent left and right leaf values are bounded\nby(γle f t+γright)\n2. The side of the tree and the nature (positive or negative) of the constraint determine if it is a lower or\nupper bound.\n2.2. The RUMBoost model\nWe first explain here how we adapt the general GBDT model to output Gradient Boosted Utility Values (GBUV)\nto emulate parametric RUMs. We then present the Piecewise-Cubic Utility Functions (PCUF) algorithm, that outputs\nsmoothed monotonic non-linear parameters.\n2.2.1. Gradient Boosted Utility Values (GBUV)\nIn RUMBoost-GBUV , we replace each parameter in the utility functions of a RUM with an ensemble of regression\ntrees, where the leaves in the regression trees represent the partial utility contribution for the corresponding value of\nthat variable. These can then be added over each tree in the ensemble to find the contribution of each variable to the\nutility. The overall utility for each alternative can therefore be found by summing the ensembles for each variable over\nall variables in the utility function. For K parameters applied to K variables, we have:\nVin=ASC i+KiX\nkMikX\nmfimk(xink) (17)\nwhere ASC iis an Alternative-Specific Constant for alternative iandMikis the number of regression trees in the\nensemble for parameter kfor alternative i. In other words, by ordering the split points for each variable in ascending\norder, and adding leaf values for the appropriate values, we can interpret the sum as raw utility values for each variable.\nThis allows us to impute the full functional form of the ”non-linear parameters” with piece-wise constants. In this\nform, the utility function is non-linear (and non-continuous) and known over the full input space. Probabilities for\n7each alternative can then be calculated with the appropriate transformation (e.g. softmax /logistic function for the\nMNL model).\nFor simplicity of notation, the above formulation shows each parameter being applied to a single variable. How-\never, it is possible to define arbitrary feature interactions by allowing the regression trees in each ensemble to split on\nmultiple variables e.g. for an interaction of two variables k1andk2we would havePMikmfimk(xink1,xink2). This allows\nthe modeller to specify any desired feature interactions as required.\nWe modify the standard GBDT boosting algorithm in order to identify optimal RUMBoost-GBUV ensembles for\neach variable (or combination of variables). For a J-class problem, we introduce J trees at each boosting iteration\nduring model training, one for the utility function of each alternative. At each iteration, the model computes choice\nprobabilities for the previous estimates of the utility functions, derives the gradient and hessian of the log-loss, and\nuses them for boosting the next set of Jtrees. Each regression tree is added to the ensemble for a single variable,\nautomatically selected by the algorithm in an exhaustive search (for computational e fficiency the search is typically\nlimited to a finite number of possible split points) in order to minimise the loss function.1\nOnce the model has been fully trained, we can extract normalised ASCs. We define the non normalised ASCs as:\nASC′\ni=KiX\nkMikX\nmfimk(0) (18)\nSince in RUM only the di fference in utilities matter, one of the ASCs can be normalised to 0. Assuming that the\nASC of the alternative jis normalised, we obtain the following set of ASCs:\nASC i=ASC′\ni−ASC′\nj,∀i=1,...,J (19)\nWe build our algorithm on top of LightGBM (Ke et al., 2017), such that each ensemble is a LightGBM Booster ob-\nject. Therefore, we can make use of the already implemented monotonicity constraint feature to impose monotonicity\non utilities as required.\nThe RUMBoost-GBUV model training algorithm is described formally in Algorithm 1. Note that the algorithm is\nindependent of the assumption on the error terms. This allows RUMBoost-GBUV to be used to emulate any arbitrary\nmodel formulation for which the gradient and Hessian of the loss function is defined, including Nested Logit and\nMixed Logit models. Furthermore, as the code has been re-implemented from scratch, any ML regressor that can\nsatisfy the constraints described above can be used in place of GBDT.\nAlgorithm 1 : RUMBoost-GBUV\nxi=ai+s,∀i=1,...,J\nPositive monotonic set of attributes x+\ni⊆xi,∀i=1,...,J\nNegative monotonic set of attributes x−\ni⊆xi,∀i=1,...,J\nx+\ni∩x−\ni=∅,∀i=1,...,J\nSpecify variables interactions\nVin(xin)=0,∀i=1,...,J\nform=1 toMdo\nCompute pin(xin)\nfori=1 toJdo\nCompute the first and second derivatives of the loss function\nChoose the split point that maximises the loss reduction of any variable k\nAdd the regression tree fimk(xink) to its corresponding ensemble\nend for\nend for\n1Note that for the practical implementation of RUMBoost-GBUV , there is actually a single ensemble per alternative utility function, with each\ntree in the ensemble restricted to only split on the corresponding variable(s). The trees for each variable are then grouped into a separate ensemble\nonce the model has been fully trained. However, this is equivalent to having a separate ensemble for each variable during model training, which we\nbelieve to be a more intuitive interpretation of the underlying functionality, and so present the algorithm in that way here.\n82.2.2. Piece-wise Cubic Utility Function (PCUF)\nThe GBUV ensembles for each parameter in Section 2.2.1 are non-continuous, and so have a gradient of either\nzero or infinite at any point. However, many behavioural indicators require the utility function to have defined gradient\nto be computed. Therefore, we interpolate the utility values into a smooth function using piece-wise cubic Hermite\nsplines. Using splines ensure that the underlying cubic polynomials have equal values and derivatives at the knots\n(i.e. boundary points). Given the number of knots and their positions, only the derivative at each knot needs to\nbe computed, making Hermite splines very attractive for e fficient computations. Using the approach introduced by\nFritsch and Carlson (1980), it is possible to guarantee monotonic splines, where the gradient is always negative or\npositive (or zero) as required. The interpolation must satisfy two conflicting objectives: (i) fitting the data as well\nas possible to maintain good predictive power on out-of-sample data; and (ii) being as smooth as possible to obtain\nrelevant behavioural indicators.\nThe first objective favours a higher number of knots, while the second aims for a lower number so that the deriva-\ntive is well defined. A natural objective function to capture the trade-o ffof both these objectives is the Bayesian\nInformation Criterion (BIC), which takes the following form:\nBIC=−2N·L+d f·ln(N) (20)\nwhere Lis the loss function described in Equation 12, d fis the degree of freedom of the model, and Nis the number\nof observations. The first part of the function aims for a better fit of the data, while the second part penalises the model\nfor its complexity.\nRUMBoost-PCUF, therefore, has two parameters to tune: (i) the number of knots; and (ii) their positions. Given\na sequence of Q+1 knots ak=t0,k<t1,k<...< tQ,k=bkfor an attribute kwhere akandbkare the domain where that\nattribute is defined, the optimal positions and numbers of knots are determined by the following optimisation problem:\nmin\ntq,k−2N·L+d f·ln(N)\ns.t. tq+1,k−tq,k>0∀q=0,...,Q,∀k\nt0,k=ak ∀k\ntQ,k=bk ∀k(21)\nGiven the number of knots, there is an optimal position of knots that minimises the loss function. Therefore, the\ntwo hyperparameters can be tuned sequentially: the number of knots is selected first and their optimal positions are\nfound with a constrained optimisation solver afterwards. However, this can be a complex optimisation problem if the\nnumber of attributes is high, and it has been shown in the literature that it is acceptable to fix the positions of knots\nover the range of the attribute (e.g., quantile) and optimise only their numbers (Hastie and Tibshirani, 1990). The\nRUMBoost-PCUF algorithm is summarised in Algorithm 2.\nAlgorithm 2 : RUMBoost-PCUF\nxi=ai+s,∀i=1,...,J\nGradient boosted utility values Vik∀i=1,...,J,k=1,...,K\nNumber of iterations for hyperparameter search Niter\nforn=1 toNiterdo\nfori=1 toJdo\nfork=1 toKdo\nifxk∈aithen\nChoose a number of knots Qk\nDefine the initial position of knots tq,k=Qk-quantile( xk,q)∀q=0,...,Qk\nend if\nend for\nend for\nOptimise the knots position to minimise the BIC\nend for\n92.3. Code and implementation\nWe implement the model in Python, making use of the library LightGBM for the utility regression ensembles (Ke\net al., 2017). Our implementation creates a regression ensemble for each alternative in which the input attributes (or\nfeatures) can be specified independently. We set up the logistic /softmax function so that in each round of boosting,\nthe trees (and split points in each tree) are selected to directly minimise the log-loss (cross-entropy loss), therefore\nemulating maximum likelihood estimation (MLE). Separate ensembles for each parameter are obtained by restricting\nthe possible set of feature interactions in each tree. The code makes use of the existing monotonic constraints func-\ntionality to guarantee monotonicity of the marginal utilities as required. We have therefore implemented an interface\nwhich allows the modeller to specify:\n•which attributes should be included in each utility function\n•control attribute interactions\n•specify which attributes should have monotonic marginal utilities.\nEarly stopping is used to determine the appropriate number of trees in each ensemble, with boosting terminated\nonce the log-loss does not improve on out-of-sample data for a given number of iterations (e.g. 100). Furthermore,\nwe have written a script that converts model files from the popular choice modelling software Biogeme (Bierlaire,\n2023) to be used directly within RUMBoost, therefore allowing modellers to easily replicate any MNL model in\nRUMBoost. This conversion works using the utility specification to define alternative-specific attributes and control\nattribute interactions, and using bounds on the parameters to define monotonic constraint. The code is freely available\non Github ( https://github.com/NicoSlvd/RUMBoost )\n3. Detailed case study\nWe apply our methodology on a case study, where we benchmark RUMBoost against a MNL model and three ML\nclassifiers: Neural Network (NN), Deep Neural Network (DNN) and LightGBM. These models are re-implemented\nfrom ´Angel Mart ´ın-Baos et al. (2023)2. In addition, we show the non-linear utility functions and compute behavioural\nindicators. To ensure a fair comparison, RUMBoost is built using the same model specification as the MNL model.\n3.1. Case study specifications\nWe use the London Passenger Mode Choice (LPMC) (Hillel et al., 2018) dataset for our case study, a publicly\navailable dataset providing details of more than 80000 trips in London, alongside their associated mode choice deci-\nsions. It is an augmented version of the London Travel Demand Survey (LTDS) trip diary dataset, to include the travel\ntime and cost of alternatives. The dataset contains observations from 17615 households over a three-year period,\nand there are four possible alternatives: walking, cycling, public transport and driving. The MNL model, also used\nto create RUMBoost, is a 62-parameter model with alternative-specific constants (ASC). When estimating the MNL\nmodel, we normalise the ASC, the generic attributes and the socio-economic characteristics of the walking alternative\nto zero. The model specification is summarised in Table 2. Lastly, the RUMBoost and ML models are trained on the\nfirst two years of the dataset with a 5-fold cross validation scheme designed in such a way that trips performed by the\nsame household members cannot be in di fferent folds, to avoid data leakage. We include an early stopping criterion of\n100 iterations, i.e. we stop the training if the performance on the validation set is not improving during 100 iterations.\nFor the ML classifiers, we also include a hyperparameter search. This search is done with the python library Hyperopt\n(Bergstra et al., 2013), and the search space and results are summarised in Appendix A.\n3.2. RUMBoost model specification\nThe MNL model is directly used to specify the constraints of the RUMBoost model. The alternative-specific\nattributes constraint is directly satisfied by the MNL utility specification. Interactions between attributes are restricted,\nsuch that each tree corresponds to a single parameter. Finally, monotonicity constraints are obtained from the bounds\n10Table 2: Variables used in the LPMC RUMBoost and MNL models. For the MNL estimation, the socio-economic characteristics and generic\nattributes are normalised to 0 for the walking alternative. Purpose and Fuel type are dummy variables where one category is normalised. The\nconstants are not included in the RUMBoost training, but are reconstructed afterwards, following Equation 19.\nWalking Cycling Public Transport Driving\nAlternative-specific attributes\nConstant ✓ ✓ ✓\nTravel time ✓ ✓ ✓ ✓\nAccess time ✓\nTransfer time ✓\nWaiting time ✓\nNum. of PT changes ✓\nCost ✓ ✓\nCongestion charge ✓\nSocio-economic characteristics and generic attributes\nStraight-line distance ✓ ✓ ✓ ✓\nStarting time ✓ ✓ ✓ ✓\nDay of the week ✓ ✓ ✓ ✓\nGender ✓ ✓ ✓ ✓\nAge ✓ ✓ ✓ ✓\nDriving license ✓ ✓ ✓ ✓\nCar ownership ✓ ✓ ✓ ✓\nPurpose: home-based work ✓ ✓ ✓ ✓\nPurpose: home-based education ✓ ✓ ✓ ✓\nPurpose: home-based other ✓ ✓ ✓ ✓\nPurpose: employers business ✓ ✓ ✓ ✓\nPurpose: non-home-based other ✓ ✓ ✓ ✓\nFuel type: diesel ✓ ✓ ✓ ✓\nFuel type: hybrid ✓ ✓ ✓ ✓\nFuel type: petrol ✓ ✓ ✓ ✓\nFuel type: average ✓ ✓ ✓ ✓\nthat would be applied to the MNL beta parameters (see Table 3), and so applied negatively on travel time, headway,\ncost and distance, and positively on car ownership and driving license (when applicable).\nOne big advantage of RUMBoost over the more flexible GBDT model is that the additional constraints help to\nregularise the model and therefore has a lower propensity to overfit compared to the unconstrained GBDT model.\nWe find that the modelling results are less dependent on hyperparameter values, including regularisation parameters.\nThus, we use the LightGBM default parameters except for the learning rate, the maximum depth of trees and the\nnumber of trees. The number of boosting rounds is obtained with the cross-validation early stopping criterion, where\nwe average the best number of trees of each fold to obtain the final number of trees. As such, we make use of the\nfollowing parameters for each regression ensemble:\n•learning rate: 0.1\n•max depth: 1 (following attribute interaction constraints)\n•minimum data and sum of hessian in leaf: 20 (default) and 0.001 (default)\n•maximum number of bins and minimum of data in bins: 255 (default) and 3 (default)\n2The code is freely available at https://github.com/JoseAngelMartinB/prediction-behavioural-analysis-ml-travel-mode-choice\n11Table 3: Attributes constrained to monotonicity. Travel time, cost and distance are monotonic negative, i.e. an increase of these attributes will\ndecrease the utility function value. Car ownership and driving license are positive monotonic and applied only on for the driving alternative.\nMonotonic negative Travel time, cost, distance\nMonotonic positive Car ownership*, driving license*\n*Only for the driving alternative\n•monotone constraint method: advanced\n•number of boosting rounds (trees): 1300\n3.3. Comparisons with other ML models and MNL\nThe results of the RUMBoost model are benchmarked against the MNL and ML models presented in Section 3.1.\nWe compare the models with their cross-entropy loss (CLE) on the test set (lower is better) and their computational\ntime per cross-validation iteration. The results are shown in Table 4.\nTable 4: Benchmark of Classification on LPMC Dataset. The models are compared with their CEL (negative Cross-Entropy Loss, lower the\nbetter) on the test set and their computational time for one CV iteration. The training results of RUMBoost-PCUF are the results of the\noptimisation problem described in Section 2.2.2.\nModels MetricsLPMC\n5 fold CV Holdout test set\nMNLCEL 0.6913 0.7085\nComp. Time [s] 242.14 -\nNNCEL 0.6516 0.6667\nComp. Time [s] 7.85 -\nDNNCEL 0.6613 0.6735\nComp. Time [s] 3.89 -\nLightGBMCEL 0.6381 0.6537\nComp. Time [s] 4.64 -\nRUMBoost-GBUVCEL 0.6570 0.6737\nComp. Time [s] 6.48 -\nRUMBoost-PCUFCEL 0.6479* 0.6730\nComp. Time [s] 712.48* -\n*Not with CV\nOverall, the RUMBoost model outperforms the MNL model on both training and testing validations, whilst still\nensuring a directly interpretable functional form. Whilst there is a marginal performance sacrifice of the RUMBoost\nmodels compared to the unconstrained GBDT and NN models, it is important to note that the functional form of the\nRUMBoost utilities is directly interpretable over the full input space, allowing for guarantees of behavioural consis-\ntency of forecasts, and in the case of RUMBoost-PCUF, extraction of behavioural indicators, which is not possible\nfor the GBDT and NN models. Note that further enhancements of the RUMBoost model, including the functional\neffects model (see Section 4.3) further narrow this gap. Interestingly, the loss of information due to smoothing is\nminimal, and the CE loss even improves on the LPMC dataset, even with an objective function penalising complex\nmodels. Therefore, we deduce that the piece-wise splines act as further regularisation of the RUMBoost model when\nthere is a su fficient amount of data and splitting points. Computationally, the GBUV algorithm shows similar results\nthan the ML classifiers, which is much better than the MNL model. The PCUF algorithm on the other hand has a big\ncomputational time, illustrating the complexity of the optimisation problem.\n123.4. Gradient boosted utility values\nThe primary advantage of using RUMBoost over other unconstrained ML algorithms is that non-linear utility\nfunctions can be observed directly. We have access to the ensemble for each parameter thanks to the custom training\nfunction. Therefore, we can delve into each tree of each ensemble to retrieve leaf values and splitting points. More\nspecifically, each split point in a regression tree represents a step in the GBUV output for the corresponding parameter.\nThe utility contribution at point xis the cumulative sum of all corresponding leaf values of the trees in the ensemble.\nThese functions are presented for the travel time, cost, departure time and age parameter in Figures 1a, 1b, 2a and 2b.\nFigure 1a shows the impact of travel times on parameter values for the LPMC dataset, with the public transport\nalternative including both bus and train travel times. From the graph, it can be noted that the walking and driving\ntime parameters have a convex shape, indicating that the increase in a short trip has a greater influence on the utility\nfunction than a longer trip. This contrasts with the two parameter values linked to public transport, where bus travel\ntime has an approximately linear curve, and train travel time impact results in a concave parameter values. Finally,\nthe parameter values of cycling is initially convex, then reaches a plateau between 0.5 and 1 hour of travel time, and\ndecreases rapidly after 1 hour. Figure 1b depicts the constant utility contributions of travel cost parameter for driving\nand PT. Interestingly, both parameter values exhibit a similar behaviour. There is first a sharp decrease representing\nthe disutility of travelling, then a plateau, and a final drop around 2 £.\nFigures 2a and 2b show the constant utility values of the age and departure time parameters for each alternative.\nFollowing behavioural theory, there are no monotonicity constraints on these variables, but we still observe that\nincreasing age mainly reduces the utility of walking and cycling, with a mostly convex shape. For public transport,\nthe parameter values is the lowest at older ages, and peaks around the age of 20. Finally, for the driving alternative,\nthe utility is higher with younger and older ages, but the lowest point is at the age of 20. Also note that passengers\nare included in the driving alternatives, explaining the high values for younger and older individuals. Finally, the\ndeparture time utilities are stable during the day. The walking and cycling utilities exhibit a sharp increase around\nmidnight. The public transport utility has a higher value on the morning, corresponding to the morning commuting\ntime. Finally, the driving utility decreases until 7h, and increases until the end of the day after that.\n(a) Travel time (LPMC)\n (b) Cost (LPMC)\nFigure 1: Utility contributions of a) travel time and b) cost on the LPMC dataset, both under a negative monotonic constraint. Each step represents\na split point of a regression tree in the corresponding ensemble.\n13(a) Age (LPMC)\n (b) Departure time (LPMC)\nFigure 2: Utility contributions of a) age and b) departure time on the LPMC dataset. Both variables are non-monotonic. Each step represents a\nsplit point of a regression tree in the corresponding ensemble.\n3.4.1. GBUV robustness\nTo demonstrate the robustness of the GBUV outputs, we perform bootstrap sampling for 100 iterations. We plot\nthe resulting parameter values with their average and show the results for the travel time parameters in Figure 3. To\nvisualise the distribution of the input space, we plot a histogram of the data distribution on top of each figure.\nFigure 3 show that, even with sampling with replacement, i.e. changing the distribution of the population, the\nutility values are robust. While their values may vary slightly, their shape is similar in all utilities, especially when the\ndensity of observations is high. The walking and driving travel time (Figures 3a and 3d) exhibit the best robustness,\nwhile cycling travel time (Figure 3b) and the rail travel time (Figure 3c) are the one showing the most scale variability.\nHowever, the cycling alternative is the least chosen alternative, and a high number of individuals have no travel time\nfor the PT alternative, which can explain this finding. These bootstrapped utilities could be further used to calculate\nconfidence interval.\n14(a) Walking travel time\n (b) Cycling travel time\n(c) PT - Rail travel time\n (d) Driving travel time\nFigure 3: Utility contributions of the travel time on the LPMC dataset with bootstrapping for a) walking, b) cycling, c) PT and d) driving\nalternative. Each line with transparency corresponds to a bootstrap sampling iteration. The mean is highlighted, and the distribution of data is\nshown on top of each figure. The figures are cropped at 2 hours of travel time.\n153.5. Piece-wise cubic utility functions\nWe make use of the SciPy (Virtanen et al., 2020) implementation of monotonic cubic splines (Fritsch and Carlson,\n1980; Fritsch and Butland, 1984) to smooth the GBUV outputs to produce piece-wise cubic utility functions. We\ntreat finding the number and location of the knots as a heurisitic optimisation problem where we minimise the BIC of\nthe model likelihood, following the methodology introduced in Section 2.2.2. We make use of the Hyperopt Python\nlibrary to identify optimal solutions. The case-study model has 15 variables for which we wish to extract marginal\nutilities. Each search in Hyperopt involves selecting a di fferent number of knots, constrained to be an integer value\nbetween a minimum of 3 and up to 8. In total, 25 searches are conducted (i.e. 25 di fferent combinations of numbers\nof knots for each variable).\nThe inner optimisation loop then identifies optimal knot locations, given a fixed number of knots for each variable,\nusing the SLSQP (Sequential Least Squares Quadratic Programming) algorithm, implemented in SciPy. We constrain\nthe first and last knots to be at the location of the first and last observations for each variable. For the initial solution\nfor the remaining knots, we follow Wang et al. (2023), where we set the position of K+1 knots at the k/(K+1)th\nquantile.\nThe optimised number of knots for each variable are shown in Table 5. The use of the BIC as the objective\nfunction results in parsimonious solutions, where the number of splines is limited for simple transformations. The\nstraight-line distance is not included for public transport as there were no regression trees in the parameter ensemble.\nThe smoothed PCUF outputs for the travel time and cost parameters are shown in Figure 4. From the graph, it is clear\nthat PCUF output has successfully smoothed the GBUV functions, rather than overfitting with too many splines.\nTable 5: Optimal number of knots for PCUF. The number of knots is chosen with a hyperparameter search of 25 iterations\nAttributes Number of knots\nWalking\nTravel time 6\nDistance 6\nCycling\nTravel time 6\nDistance 6\nPublic transport\nRail travel time 3\nBus travel time 4\nAccess travel time 4\nInterchange waiting time 8\nInterchange walking time 7\nCost 3\nDriving\nTravel time 5\nDistance 4\nCost 3\nUsing the gradients from the PCUF functions, the Value of Time (V oT) can be computed for the alternatives that\ninclude both travel time and cost parameters. We define the plot only in the area where the cost derivative is not zero,\nand we cap the maximum value at 100 £ /h and minimum value at 0.1 £ /h. We also exclude the flat areas at low and\nhigh values of both attributes. As both the travel time and cost attributes are non-linear, the V oT is unique to each\ncombination of travel time and cost. As such, it is represented as a 3D distribution, though note that this distribution\nis homogeneous across the population. Figure 5 displays the V oT for the PT and driving alternatives on the LPMC\ndataset on a logarithmic (base 10) scale. We use a logarithmic scale to better visualise the di fferences among lower\n16(a) Travel time\n (b) Cost\nFigure 4: Piece-wise monotonic cubic spline interpolation of a) travel time and b) cost on the LPMC dataset. The knots are drawn in black and the\nfirst and last knots are omitted for clarity. The GBUV used for interpolation are plotted as a scatter plot.\nV oTs. The value of time of rail (Figure 5a) ranges from 2 to 5 £ /h for trips lasting less than 0.6 hour and increases to\n10 to 20 £ /h for travel times of 0.6 to 1 hour. It decreases again after one hour. This suggests that between 0.6 and 1\nhour of travel time, individuals are willing to pay more to reduce their travel times. Regarding the value of time for\ndriving (Figure 5b), we observe overall a decrease of the travel time with increasing travel time, and an increasing of\nV oT with increasing cost. The lowest value is around 2 £ /h for no cost and 1.3 hours of travel time, and the highest\nvalue is capped at 100 £ /h for 4£ cost and no travel time. We deduce that individuals with high cost and low travel\ntimes are willing to pay more to reduce their travel time.\n(a) Rail (LPMC dataset)\n (b) Driving alternative (LPMC dataset)\nFigure 5: Value of Time (V oT) for a) rail, b) driving. The V oT is capped at 100£ /h, and displayed only where the utility functions derivatives are\nnon zero.\nIn addition to the contour plot V oT, we also compute the V oT across the population. To do so, we remove\nobservations that have a zero travel time for the rail alternative, and exclude the 0.1% highest values (from 99.9%\nto 100%). Then, we calculate the V oT of all remaining individuals with their respective costs and travel times. The\n17results are shown in Figure 6. For the PT alternative (Figure 6a), the distribution of V oT peaks below 1 £ /h and then\ncontinuously decreases until 5£ /h. For driving V oT (Figure 6b), there is a sharp peak at 17.5 £ /h. These values are\nboth lower than the V oT extracted from linear-in-parameter RUM models for the same dataset, of 8.73 £ /h and 40 £ /h\nrespectively (see Hillel (2019), p.133), showing the impact of the non-linear utility specification.\n(a) PT alternative\n (b) Driving alternative\nFigure 6: Histogram of the population Value of Time (V oT) for a) rail, b) driving. The observations with zero travel times, as well as the highest\n0.1% V oT values are excluded. The solid line represents the kernel density estimates.\n4. Extensions of RUMBoost\nIn this section, we present three extensions of RUMBoost, highlighting how the approach can be generalised to\ndifferent modelling scenarios: a) incorporating attribute interactions (Section 4.1); b) assuming alternative correlation\nwithin the error term (Section 4.2); and c) accounting for correlation within trips made by the same individual (Section\n4.3).\n4.1. Second-order attribute interactions\nBy allowing two continuous attributes to interact, we can consider attribute interactions. In doing so, it is still\npossible to interpret the ensemble output for these two attributes on a contour plot. As an example, we arbitrarily\nallow age and travel time to interact. We allow for a max depth of two in each tree to allow for feature interactions\nand, through early stopping, perform 680 total boosting rounds. Figure 7 shows the resulting contour plot for all\nalternatives. The contour plots are only monotonic with respect to travel time, as specified in the model. For the\nwalking alternative (Figure 7a), longer travel times lead to increased disutility for both younger and older ages, while\nshorter travel times result in relatively uniform disutility. For cycling (Figure 7b), we observe a similar phenomenon\nfor longer travel times but the disutility is still pronounced for older individuals with shorter travel times, which aligns\nwith the expectation that cycling may be less feasible for older individuals. In the case of public transport (Figure\n7c), disutility is more significant for younger ages and less pronounced for adults and shorter travel times. Lastly,\nthe disutility associated with an increase in age or travel time for the car alternative (Figure 7d) remains relatively\nconstant, but it is mitigated for ages below 10 and travel times shorter than 0.5 hour.\n4.2. Nested RUMBoost\nUntil now, we assumed the error terms for each alternative to be distributed i.i.d., leading to the MNL model\nformulation. We show that we can relax this assumption with our approach by assuming an error term correlated\nwithin alternatives. In RUM, this error term leads to the Nested Logit (NL) model. We update the probability formula\nand update the gradient and hessian (second derivative) computations accordingly, giving:\nP(i)=P(i|m)P(m) (22)\nwhere the probability of choosing iknowing the nest mis:\n18(a) Walking alternative\n (b) Cycling alternative\n(c) Public transport alternative\n (d) Driving alternative\nFigure 7: Utility function in the form of a contour plot for the travel time (x axis) with age (y axis) interaction on the LPMC dataset for a) walking,\nb) cycling, c) PT and d) driving alternative. Only the travel time is subject to monotonicity constraint.\nP(i|m)=eµmVi\nP\nj∈meµmVj, (23)\nwhile the probability to choose the nest mis:\nP(m)=e˜Vm\nPM\np=1e˜Vp, (24)\nwhere:\n•˜Vm=1\nµmln\u0010P\ni∈meµmVi\u0011\n•Mthe number of nest\n19•µmthe scaling parameter of nest m\nWe treatµas a hyperparameter, where we search its optimal value with a 5-fold cross validation scheme on the\ntraining test. More details are given in the appendix Appendix A. We train the Nested RUMBoost model on the LPMC\ndataset, and compare its performance against a NL model. The nest is assumed to be within the motorised modes (PT\nand driving) while walking and cycling are in their own nests. The optimal value of µfound with hyperparameter\ntuning is 1.16, while the one estimated with the Nested Logit model is 1 .35 (see Appendix B). The di fference of\nflexibility of these two models explains this di fference. The results are shown in Table 6.\nTable 6: Benchmark of classification on the LPMC Dataset for the Nested RUMBoost with the RUMs and RUMBoost. The models are compared\nwith their CEL (negative Cross-Entropy Loss, lower the better) on the test set and their computational time for one CV iteration.\nMNL Nested Logit RUMBoost-GBUV Nested RUMBoost\nCE Time [s] CE Time [s] CE Time [s] CE Time [s]\n5 fold CV 0.6913 242.14 0.6921 1067.04 0.6570 8.9 0.6568 48.53\nHoldout test set 0.7085 - 0.7091 - 0.6737 - 0.6731 -\nWhilst the Nested Logit model does not show a performance improvement over the MNL mode in terms of out-of-\nsample validation, the nested RUMBoost model marginally outperforms the base RUMBoost model. Note that these\nresults are without PCUF smoothing, but the Nested RUMBoost could be smoothed, as in Section 3.5.\n4.3. Functional E ffect RUMBoost\nLastly, we propose the Functional E ffect RUMBoost (FE-RUMBoost), a model accounting for observations cor-\nrelation. This model draws some parallels with the Mixed E ffect model, where the fixed e ffect part is RUMBoost\nwithout attribute interaction and the random e ffect part includes all socio-economic characteristics with full interac-\ntion. By doing so, we keep the full utility interpretability on the trip attributes, and we learn an individual- specific\nconstant for each alternative with the second part of the model. In other words, the second part of the model is a latent\nrepresentation of each individual from their socio-economic characteristics, which can capture correlation for panel\ndata or other observation correlated situations. We apply the model on the LPMC dataset and we compare it with the\nbenchmarks of Section 3.3 in Table 7.\nTable 7: Benchmark of classification on the LPMC Dataset for the FE-RUMBoost with the ML classifiers. The models are compared with their\nCEL (negative Cross-Entropy Loss, lower the better) on the test set and their computational time for one CV iteration.\nLightGBM NN DNN FE-RUMBoost\nCE Time [s] CE Time [s] CE Time [s] CE Time [s]\n5 fold CV 0.6381 4.64 0.6516 7.85 0.6613 3.89 0.6447 10.9\nHoldout test set 0.6537 - 0.6667 - 0.6735 - 0.6626 -\nThis extension of the model substantially improves the prediction performance on the test set. Our model outper-\nforms both the NN and DNN classifiers and narrows the gap vs the unconstrained LightGBM model, while keeping\nthe full interpretability on key alternative-specific attributes. The computational cost induced by the greater complex-\nity is minimal compared to the initial RUMBoost. Again, the model presented here is without smoothing but, because\nall the trip attributes that were previously smoothed are in the first part of the model, we can apply smoothing as in\nSection 3.5. In addition, we can visualise the individual specific constant per alternative. We show the distribution of\nthe constants per alternative in Figure 8 in the form of histograms. These histograms show clearly that for cycling, PT\nand driving the distribution of individual-specific constants are bi-modal.\nThese three extensions demonstrate the RUMBoost model’s ability to incorporate attribute interactions, account\nfor correlated error terms within alternatives, and learn an individual-specific constant for observation correlated data.\nThe extensions can also be combined, as they are applied to di fferent parts of the model. This showcases the flexibility\nand generalisability of the RUMBoost framework. In future work, any complex choice situations that lead to a defined\ngradient and hessian could be applied to the model.\n20Figure 8: Histograms of the individual-specific constant learnt with the Functional E ffect RUMBoost of a) walking, b) cycling, c) PT and d)\ndriving alternative. For each histogram, we plot the kernel density estimate as a solid line. Overall, cycling, PT and driving individual constants\nexhibit a bi-modal distribution, and walking alternative a uni-modal distribution centered at zero.\n5. Conclusion and further work\nThe methodology presented in this paper allows for a fully interpretable ML model (RUMBoost) based on GBDT\nand inspired by random utility models. In short, RUMBoost replaces every parameter of a RUM by an ensemble\nof regression trees. By re-implementing classification for GBDT, we can provide specific attributes for alternative\nutilities, control attribute interactions in the ensemble, and apply monotonic constraints on key attributes based on\ndomain knowledge. These constraints considerably improve the predictions of traditional RUMs and enable the\nderivation of non-linear utility functions. Furthermore, we apply piece-wise monotonic cubic splines to interpolate\nthe utility functions and obtain a smooth utility function. We find that the smoothing acts as further regularization\nand enables us to compute the Value of Time (V oT). We also show that the modularity of our approach allow for\nthe estimation of complex model specification such as error term accounting for correlation within alternatives or\ncorrelation within observations. Our approach o ffer to observe the full functional form of the utility function with\ndefined gradient, just like in DCMs. The key di fference is that the utility function is directly learnt from the data.\nWhilst applied here to choice models, this methodology could be used in place of any linear-in-parameters models,\n21for regression, classification, or any task for which the gradient and hessian of the cost function are well defined. Fur-\nther work includes applying the model to various problems to demonstrate this statement. Further work also includes\napplying the RUMBoost model to other complex model specifications such as the Cross-Nested Logit (CNL) model.\nThe PCUF algorithm could be improved by applying B-splines, which would provide a C2monotonic interpolation\nof the data, where shape constraint could be incorporated. Finally, the GBUV could be computed directly with linear\ntrees, quadratic trees or splines, to obtain directly piece-wise utility functions with defined gradient.\nCRediT authorship contribution statement\nNicolas Salvad ´e: Conceptualization, Methodology, Software, Writing - Original Draft, Visualization. Tim Hillel :\nConceptualization, Methodology, Writing - Review and Editing, Supervision.\nDeclaration of competing interest\nNone\nAcknowledgements\nWe sincerely thank Prof. Michel Bierlaire for his guidance on the smoothing process. We also would like to\nexpress our sincere gratitude to Dr. Jos ´e´Angel Mart ´ın-Baos for his assistance with the benchmarks on the LPMC.\nAppendix A. Hyperparameter search\nTable A.8: Hyperparameter search and optimal value for RUMBoost, Nested RUMBoost, and FE-RUMBoost on the LPMC dataset\nRUMBoost Nested RUMBoost FE-RUMBoost\nNumber of searches 1 25 100\nTime [s] 44.5 5378 5366\nHyperparameter Distribution Search space\nMean of num iterations early stopping 100 1300 1256 1099\nbagging fraction uniform [0.5, 1] - - 0.700\nbagging freq choice (0, 1, 5, 10) - - 10\nfeature fraction uniform [0.5, 10] - - 0.867\nlambda l1 log uniform [0.0001, 10] - - 6.592\nlambda l2 log uniform [0.0001, 10] - - 1.028\nlearning rate fixed 0.1 - - 0.1\nmax bin uniform [100, 500] - - 131\nmin data inleaf uniform [1, 200] - - 27\nmin gain tosplit log uniform [0.0001, 5] - - 0.800\nmin sum hessian inleaf log uniform [1,100] - - 1.783\nnum leaves uniform [2, 100] - - 74\nmu uniform [1, 2] - 1.167 -\n22Table A.9: Hyperparameter search and optimal value for LightGBM, NN, and DNN on the LPMC dataset.\nLightGBM NN DNN\nNumber of searches 1000 1000 1000\nTime [s] 24882 28736 26343\nHyperparameter Distribution Search space\nMean of CV num iterations early stopping 100 1493 - -\nbagging fraction uniform [0.5, 1] 0.7204 - 0.700\nbagging freq choice (0, 1, 5, 10) 1 - 10\nfeature fraction uniform [0.5, 10] 0.6007 - 0.867\nlambda l1 log uniform [0.0001, 10] 0.0242 - 6.592\nlambda l2 log uniform [0.0001, 10] 0.0001 - 1.028\nlearning rate fixed - 0.1 adaptive 0.1\nmax bin discrete uniform [100, 500] 237 - 131\nmin data inleaf discrete uniform [1, 200] 156 - 27\nmin gain tosplit log uniform [0.0001, 5] 0.0007 - 0.800\nmin sum hessian inleaf log uniform [1,100] 1.4136 - 1.783\nnum leaves discrete uniform [2, 100] 16 - 74\nactivation fixed - - tanh(x) relu(x)\nbatch size choice (128, 256, 512, 1024) - 1024 1024\nhidden layer size discrete uniform [10, 500] - - 20\nlearning rateinit uniform [0.0001, 1] - 0.0907 -\nsolver choice [lbfgs, sgd, adam] - sgd -\ndepth discrete uniform [2,10] - - 2\ndrop choice (0.5, 0.3, 0.1) - - 0.3\nepochs discrete uniform [50, 200] - - 90\nwidth choice (25, 50, 100, 150, 200) - - 50\nAppendix B. Estimation of the MNL models\nTable B.10: Parameter estimates of the LPMC MNL. Out of the 62 parameters, 9 are not significant at a 95% confidence interval.\nLPMC - MNL\nValue Active bound Rob. p-value\nASC Bike -3.380 0.000 0.000\nASC Car -2.592 0.000 0.000\nASC Public Transport -1.908 0.000 0.000\nBageBike -0.004 0.000 0.032\nBageCar 0.005 0.000 0.000\nBagePublic Transport 0.011 0.000 0.000\nBcarownership Bike 0.036 0.000 0.590\nBcarownership Car 0.694 0.000 0.000\nBcarownership Public Transport -0.213 0.000 0.000\nBconcharge Car -1.147 0.000 0.000\nBcost driving fuel Car 0.000 1.000 1.000\nBcost transit Public Transport -0.115 0.000 0.000\nBdayofweek Bike -0.020 0.000 0.201\nBdayofweek Car 0.030 0.000 0.000\nBdayofweek Public Transport -0.044 0.000 0.000\n23Table B.10: Parameter estimates of the LPMC MNL. Out of the 62 parameters, 9 are not significant at a 95% confidence interval.\nLPMC - MNL\nValue Active bound Rob. p-value\nBdistance Bike -0.232 0.000 0.040\nBdistance Car 0.000 1.000 1.000\nBdistance Public Transport 0.000 1.000 1.000\nBdriving license Bike 0.678 0.000 0.000\nBdriving license Car 0.663 0.000 0.000\nBdriving license Public Transport -0.526 0.000 0.000\nBdurcycling Bike -2.670 0.000 0.000\nBdurdriving Car -4.777 0.000 0.000\nBdurptaccess Public Transport -4.608 0.000 0.000\nBdurptbusPublic Transport -1.911 0.000 0.000\nBdurptintwaiting Public Transport -4.284 0.000 0.000\nBdurptintwalking Public Transport -2.335 0.000 0.027\nBdurptrailPublic Transport -1.338 0.000 0.000\nBdurwalking Walk -8.596 0.000 0.000\nBfemale Bike -0.834 0.000 0.000\nBfemale Car 0.100 0.000 0.002\nBfemale Public Transport 0.160 0.000 0.000\nBfueltype Avrg Bike -0.691 0.000 0.000\nBfueltype Avrg Car -1.400 0.000 0.000\nBfueltype Avrg Public Transport -0.221 0.000 0.000\nBfueltype Diesel Bike -0.822 0.000 0.000\nBfueltype Diesel Car -0.228 0.000 0.000\nBfueltype Diesel Public Transport -0.419 0.000 0.000\nBfueltype Hybrid Bike -1.000 0.000 0.000\nBfueltype Hybrid Car -0.721 0.000 0.000\nBfueltype Hybrid Public Transport -0.945 0.000 0.000\nBfueltype Petrol Bike -0.867 0.000 0.000\nBfueltype Petrol Car -0.242 0.000 0.000\nBfueltype Petrol Public Transport -0.323 0.000 0.000\nBptninterchanges Public Transport -0.101 0.000 0.154\nBpurpose BBike -0.029 0.000 0.775\nBpurpose BCar -0.043 0.000 0.543\nBpurpose BPublic Transport -0.012 0.000 0.874\nBpurpose HBE Bike -1.054 0.000 0.000\nBpurpose HBE Car -0.756 0.000 0.000\nBpurpose HBE Public Transport -0.237 0.000 0.000\nBpurpose HBO Bike -0.773 0.000 0.000\nBpurpose HBO Car -0.352 0.000 0.000\nBpurpose HBO Public Transport -0.442 0.000 0.000\nBpurpose HBW Bike -0.291 0.000 0.000\nBpurpose HBW Car -1.062 0.000 0.000\nBpurpose HBW Public Transport -0.502 0.000 0.000\nBpurpose NHBO Bike -1.233 0.000 0.000\nBpurpose NHBO Car -0.379 0.000 0.000\nBpurpose NHBO Public Transport -0.715 0.000 0.000\nBstart time linear Bike 0.017 0.000 0.015\nBstart time linear Car 0.027 0.000 0.000\n24Table B.10: Parameter estimates of the LPMC MNL. Out of the 62 parameters, 9 are not significant at a 95% confidence interval.\nLPMC - MNL\nValue Active bound Rob. p-value\nBstart time linear Public Transport 0.010 0.000 0.016\nBtrafficperc Car -2.404 0.000 0.000\nTable B.11: Parameter estimates of the LPMC NL. Out of the 63 parameters, 10 are not significant at a 95% confidence interval.\nLPMC - NL\nValue Active bound Rob. p-value\nASC Bike -3.346 0.000 0.000\nASC Car -2.439 0.000 0.000\nASC Public Transport -1.969 0.000 0.000\nBageBike -0.004 0.000 0.026\nBageCar 0.007 0.000 0.000\nBagePublic Transport 0.011 0.000 0.000\nBcarownership Bike 0.062 0.000 0.351\nBcarownership Car 0.628 0.000 0.000\nBcarownership Public Transport -0.037 0.000 0.369\nBconcharge Car -0.816 0.000 0.000\nBcost driving fuel Car 0.000 1.000 1.000\nBcost transit Public Transport -0.077 0.000 0.000\nBdayofweek Bike -0.023 0.000 0.155\nBdayofweek Car 0.022 0.000 0.007\nBdayofweek Public Transport -0.033 0.000 0.000\nBdistance Bike -0.225 0.000 0.042\nBdistance Car -0.005 0.000 0.960\nBdistance Public Transport 0.000 1.000 1.000\nBdriving license Bike 0.705 0.000 0.000\nBdriving license Car 0.484 0.000 0.000\nBdriving license Public Transport -0.396 0.000 0.000\nBdurcycling Bike -1.839 0.000 0.003\nBdurdriving Car -3.409 0.000 0.000\nBdurptaccess Public Transport -3.410 0.000 0.000\nBdurptbusPublic Transport -1.445 0.000 0.000\nBdurptintwaiting Public Transport -3.036 0.000 0.000\nBdurptintwalking Public Transport -1.876 0.000 0.017\nBdurptrailPublic Transport -1.085 0.000 0.000\nBdurwalking Walk -8.171 0.000 0.000\nBfemale Bike -0.831 0.000 0.000\nBfemale Car 0.112 0.000 0.000\nBfemale Public Transport 0.156 0.000 0.000\nBfueltype Avrg Bike -0.680 0.000 0.000\nBfueltype Avrg Car -1.175 0.000 0.000\nBfueltype Avrg Public Transport -0.322 0.000 0.000\nBfueltype Diesel Bike -0.817 0.000 0.000\nBfueltype Diesel Car -0.251 0.000 0.000\n25Table B.11: Parameter estimates of the LPMC NL. Out of the 63 parameters, 10 are not significant at a 95% confidence interval.\nLPMC - NL\nValue Active bound Rob. p-value\nBfueltype Diesel Public Transport -0.391 0.000 0.000\nBfueltype Hybrid Bike -0.985 0.000 0.000\nBfueltype Hybrid Car -0.755 0.000 0.000\nBfueltype Hybrid Public Transport -0.941 0.000 0.000\nBfueltype Petrol Bike -0.864 0.000 0.000\nBfueltype Petrol Car -0.259 0.000 0.000\nBfueltype Petrol Public Transport -0.315 0.000 0.000\nBptninterchanges Public Transport -0.088 0.000 0.093\nBpurpose BBike -0.032 0.000 0.748\nBpurpose BCar -0.066 0.000 0.337\nBpurpose BPublic Transport -0.053 0.000 0.445\nBpurpose HBE Bike -1.036 0.000 0.000\nBpurpose HBE Car -0.665 0.000 0.000\nBpurpose HBE Public Transport -0.271 0.000 0.000\nBpurpose HBO Bike -0.771 0.000 0.000\nBpurpose HBO Car -0.317 0.000 0.000\nBpurpose HBO Public Transport -0.397 0.000 0.000\nBpurpose HBW Bike -0.272 0.000 0.000\nBpurpose HBW Car -0.976 0.000 0.000\nBpurpose HBW Public Transport -0.570 0.000 0.000\nBpurpose NHBO Bike -1.235 0.000 0.000\nBpurpose NHBO Car -0.415 0.000 0.000\nBpurpose NHBO Public Transport -0.678 0.000 0.000\nBstart time linear Bike 0.016 0.000 0.016\nBstart time linear Car 0.024 0.000 0.000\nBstart time linear Public Transport 0.012 0.000 0.002\nBtrafficperc Car -1.947 0.000 0.000\nMU m 1.391 0.000 0.000\nReferences\nAboutaleb, Y .M., 2022. 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URL: https://linkinghub.elsevier.com/retrieve/pii/S0968090X21000802 , doi: 10.1016/\nj.trc.2021.103050 .\n27"    },    {        "title": "2401.11978v1._Λ__b__rightarrow_Pl__factorization_in_QCD.pdf",        "content": "Λb→Plfactorization in QCD\nLei-Yi Lia,b1, Cai-Dian Lüa,b2, Jin Wanga,b3, Yan-Bing Weic4\naInstitute of High Energy Physics, CAS, P.O. Box 918(4) Beijing 100049, China\nbSchool of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049,\nChina\ncSchool of Physics and Optoelectronic Engineering, Beijing University of Technology, Beijing\n100124, P.R. China\nAbstract\nWe calculate the form factors for the baryon number violation process of a\nheavy-flavor baryon decaying into a pseudoscalar meson and a lepton. In the\nframework of the Standard Model effective field theory, the leptoquark operators\nat the bottom quark scale, whose matrix elements define the form factors, are\nderived by integrating out the high energy physics. Under the QCD factorization\napproach, the form factors of the baryon number violation process at leading power\ncan be factorized into the convolution of the long-distance hadron wave function\nas well as the short-distance hard and jet functions representing the hard scale\nand hard-collinear scale effects, separately. Based on measurements of the baryon\nnumberviolationprocessbyLHCb,wefurtherimposeconstraintsonthenewphysics\nconstants of leptoquark operators.\n1Corresponding author: lily@ihep.ac.cn\n2Corresponding author: lucd@ihep.edu.cn\n3Corresponding author: wangjin99@ihep.ac.cn\n4Corresponding author: yanbing.wei@bjut.edu.cnarXiv:2401.11978v1  [hep-ph]  22 Jan 20241 Introduction\nIntheStandardModel,thebaryonnumber( B)andtheleptonnumber( L)arestrictlyconserved.\nSakharov’s conditions [1] state that the violation of the baryon number will provide a new\nsource of CP violation to explain the matter-antimatter asymmetry in the universe. Therefore,\nthe study of baryon number violation is an important topic in the search for new physics\nsignal. In new physics Beyond Standard Model, such as the Grand Unified Theory [2, 3] and\nSupersymmetry Theory [4], there do exist baryon number violation couplings. Since the new\nphysics energy scale is generally far above the top quark mass, we usually utilize the framework\nof Standard Model effective field theory (SMEFT), in which heavy fields are integrated out to\nconstruct the model-independent leptoquark operator [5–8].\nExperimentally, thesebaryonnumberviolationcouplingsaresearchedthroughprotondecay.\nIn addition, experimental researchers have also started studying baryon number violation\nprocesses in heavy flavor physics at colliders [9–14], thanks to the improvement of experimental\nprecision. The investigation of leptoquark operators has attracted much attention in heavy\nflavor physics. In Ref. [15–19], the introduction of leptoquark operators aims to address the\nanomalies in R(D)andR(D∗). In the Baryogenesis model [20–23], the form factors of Bmeson\ndecaying into proton and dark antibaryon have been calculated using light-cone sum rules\n[22]. This calculation seeks to explain baryon number violation and the matter-antimatter\nasymmetry in the universe. Recently, BaBar experiment gives the upper limit for this process\n[24]. For processes involving the violation of baryon and lepton numbers, Ref. [25] calculated\nthe form factors for the decay process D0→¯pe+using light-cone sum rules, with the intention\nof providing theoretical input for BESIII measurements [26].\nIn Ref. [13], the authors measured the baryon and lepton number violation process Λb→\nKµat the LHCb experiment and provided an upper limit. To constrain the new physics\nparameters in conjunction with the experimental data, the theoretical exploration of these\ndecays is indispensable. For the Λb→Kµdecay, there exist ∆(B−L) = 0process [5] Λb→\nK−l+and∆(B−L) = 2process [27] Λb→K+l−. Our work computes the form factors of\nΛb→Plspectator processes for both ∆(B−L) = 0and(B−L) = 2, with P=π, Kdenoting a\nlight pseudoscalar meson. In the framework of the SMEFT, effective operators of leptoquark at\nthembscaleareobtainedbyintegratingouttheheavyparticlesatnewphysicsscale. Thematrix\nelements of the leptoquark operators are calculated with the QCD factorization approach [28–\n31], which is widely used in the semileptonic and nonleptonic decays of Bmesons. At leading\npower, our calculations indicate that the form factors of the Λb→Plprocess are factorized into\nthe convolution of hard function, jet function, and wave function without endpoint divergence.\nThe short-distance hard function and jet function, which respectively correspond to the hard\nscaleandhard-collinearscalecontributions,couldbecalculatedperturbatively. Theformfactors\nof baryon number violation processes can be combined with experimental data to constrain the\n1new physics parameters of leptoquark operators. These form factors can also serve as input for\ncalculations in other new physics theories.\nThe framework of this paper is as follows: In the next section, we will introduce the effective\nHamiltonian and the leptoquark operators. In Section 3, the form factor of Λb→Pwill be\ncalculated in the QCD factorization approach. In section 4, the numerical result will be given.\nWe present our conclusions in the last section.\n2 leptoquark operator\nIn the theory of baryon number violation, the introduction of leptoquark can be traced back to\nGrand Unification Theories [2, 3], and the leptoquark operator theory [5–8]. In the framework\nof the SMEFT, we integrate out the new physics scale, to obtain the effective Hamiltonian for\ntheΛb→Plspectator processes as\nHnew=7X\nα=1Gnew,αOα, (1)\nwhere Gnew,αrepresents the effective new physics coupling constants. The corresponding\nleptoquark operators of Λb→Plspectator processes are\nO1=ϵijk(¯dc\niΓbj)(¯lΓsk), O 2=ϵijk(¯sc\niΓbj)(¯lΓdk),\nO3=ϵijk(¯dc\niΓsj)(¯lΓbk), O 4=ϵijk(¯dc\niΓbj)(¯lΓdk),\nO5=ϵijk(¯dc\niΓdj)(¯lΓbk), O 6=ϵijk(¯uc\niΓbj)(¯uc\nkΓl),\nO7=ϵijk(¯uc\niΓuj)(¯bc\nkΓl),(2)\nwhere l=e, µrepresent the leptons. ϵijkis a fully antisymmetric tensor and the Latin letters\nof superscript represent color indices. The violation of fermion flow often occurs in the process\nof baryon number violation. Typically, people introduce eigenstates of charge conjugation to\nconstruct leptoquark operators, with the specific form\nψc=C¯ψT, ¯ψc=ψTC, (3)\nwhere Cis the matrix of charge conjugation. In Fig. 1, the bold line represents a heavy quark\nfield, the thin lines represent light quark fields, and the dashed line represents the lepton field.\nThe shaded blocks in gray represent the Γmatrices connecting two fermion fields. The Lorentz\nstructures of leptoquark operators are represented by\nΓ ={1, γ5, γµ, γµγ5, σµν}, (4)\n2bk\n(2) (3)bk l−\n¯sm¯dj¯dj ¯sml−\n(1)\n¯dj¯dml−\n(4)\n¯ui ¯uml+\n(7)bk\nbk(5) (6)\n(8) (9)¯djbk\nl−¯dm¯djbk\nl−¯sm\nbk l−\n¯dm¯dj\n¯uibk\nl+¯um bk\n¯um ¯uil+Figure 1: Leptoquark operators of Λb→Plspectator processes below the mbscale, with the\nbold line representing a heavy quark field, the thin lines representing light quark fields, and the\ndashed line representing the lepton field. The shaded blocks in gray represent the Γmatrices\nconnecting two fermion fields.\nwhere σµν=i\n2[γµ, γν]. For the leptoquark operator O4andO6, there exist two quark with same\nfalvor, when we chose different situation quark as initial state, the diffenrent channel will come\nout. For the convenience of distinguishing different decay channels, we label the color of u, d, b\nquark in Λbbaryon as i, j, kandmrepresent the color in meson to distinguish the diffenent\nchannel. The diagrams of leptoquark operators are shown in Fig. 1, where diagrams (1)∼(3)\nrepresent operators O1∼O3respectively. Diagrams (4)and(5)represent the s-channel and\nt-channel of O4operator, diagram (6)represent O5operator. Simlarily, diagrams (7)and(8)\nrepresent the s-channel and t-channel of O6operator, diagram (9)represent O7operator. The\nleptoquark operator O1∼O3contribute to the process of Λb→K+l−;O4andO5contribute\nto the process of Λb→π+l−;O6andO7contribute to the process of Λb→π−l+. It is easy\nto see that the operator O1∼O5violates B−Lnumber; while operator O6andO7conserves\nB−Lnumber.\n3 QCD calculation\nIn the last section, the leptoquark operators are obtained at the bottom quark mass scale\nafter integrating out the high-energy physics. For the processes below the mbscale, the QCD\n3factorization approach [28–31] is employed to investigate the hadron decay. This approach\nprovides a factorization scheme for decay amplitudes to be expanded in the power of λ∼\nΛQCD/mb, which is widely employed in Bmeson decay. Under the QCD factorization approach,\nthe form factor of heavy-to-light current at leading power can be factorized as [30]\nfi(q2) =Ci·ξP(E) +ϕB⊗Ti⊗ϕP. (5)\nIn Eq. (5), the first term on the right-hand side of the equation is the hard function Citimes the\nsoftformfactor ξP(E)whichabsorbsthenon-perturbativeeffectsbelowthehard-collinearscale.\nTrying to further factorize ξP(E)into a convolution of a jet function and light-cone distribution\namplitudes (LCDAs), one will encounter the well known endpoint divergence which destroy the\nfactorization. The second term on the right-hand side of the equation represents the hard-\nscattering kernel Ticonvoluted with the LCDAs of the Bmeson and the light pseudoscalar\nmeson. We will show in the following that for the baryon decay the form factor could be\nfactorized, thus the factorization formula of the form factor only retain the second term in\nEq. (5) but with the Bmeson LCDA replaced by the baryon one.\n3.1 Kinematics\nSince the Λbbaryon is very heavy, the massless final-state meson and lepton in the baryon\nnumber violation process can be approximated to be on the light-cone, with momenta pand\nq, respectively. In the light-cone coordinate system, we will introduce two light-cone vectors n\nand¯n\nnµ= (1,0,0,1), ¯nµ= (1,0,0,−1), (6)\nwhich satisfy n2= ¯n2= 0andn·¯n= 2. We choose the large momentum component of the\npseudoscalarmesoninthecollineardirectionas ¯n·p=mΛbandthelargemomentumcomponent\nof the lepton in the anti-collinear direction as n·q=mΛb. In the light-cone coordinate, the\nmomentum can be expanded as\nkµ=n·k¯nµ\n2+ ¯n·knµ\n2+kµ\n⊥, (7)\nwhere we chose the convention k∼(n·k,¯n·k, k⊥). Since the Λbbaryon involves a heavy\nbottom quark, we consider it within the framework of Heavy Quark Effective Theory (HQET)\n[32], defining the velocity of the heavy baryon\nvµ= (1,0,0,0), (8)\n4satisfying v2= 1. For the heavy b-quark decay processes in QCD, there exist five different\nmomentum regions\nhard : ( 1 ,1,1 )mb,\nhard−collinear : ( λ,1,√\nλ)mb,\ncollinear : ( λ2,1, λ)mb,\nanti−collinear : ( 1 , λ2, λ)mb,\nsoft : ( λ, λ, λ )mb.(9)\nExcept for the heavy quark, the rest light degree of freedom in the baryon has soft momentum.\nThe final-state pseudoscalar meson and lepton represent the collinear mode and anti-collinear\nmode respectively. The interaction between the collinear field and the soft field is mediated by\nthe hard-collinear field.\n3.2 Light-cone distubation amplitude\nTo calculate the decay amplitude at leading power, we need to introduce the definition of the\nLCDAs for the light pseudoscalar meson [30, 31]\n⟨P(p)|[¯q(t¯n)]A[t¯n,0] [q(0)]B|0⟩=ifP\n4¯n·p\u0014/n\n2γ5\u0015\nBAZ1\n0dx eixt¯n·pϕP(x, µ),(10)\nwhere xand¯xis the momentum fraction of the ¯qquark and the qquark respectively in the\nmeson, and ¯x= 1−x. Leading-twist pseudoscalar meson LCDAs can be expanded in terms of\nGegenbauer polynomials as\nϕP(x, µ) = 6 x(1−x)\u0014\n1 +∞X\nn=1aP\nn(µ)C(3/2)\nn(2x−1)\u0015\n. (11)\nThe Gegenbauer moments expansion of the pseudoscalar meson is presented up to the first two\norders in Table 1. The definition of gauge link [33] is\n[t¯n,0] =Pexp\u0014\nigZt\n0dx¯n·A(x¯n)\u0015\n, (12)\nconnecting along the light-cone direction from 0tot¯nat the coordinate space.\nThe definition of the LCDA for the Λbbaryon is as follows: [35–38]\n⟨0|[ui(t1n)]A[0, t1n] [dj(t2n)]B[0, t2n] [hv,k(0)]C|Λb(v)⟩\n=ϵijk\n4Nc!f(2)\nΛb(µ) [uΛb(v)]C\u0014/¯n\n2γ5CT\u0015\nBAZ∞\n0dω ωZ1\n0dy e−iω(t1y+it2¯y)ψ2(y, ω),(13)\nwhere ωis the sum of the two light quark n-direction momenta. yand¯ycorrespond to the\n5Table 1: Imput parameter at µ0= 1 GeV.\nmΛb= 5.6196 GeV [34] τΛb= 1.471 ps [34]\nω0= 0.280+0.047\n−0.038GeV [35] f(2)\nΛb(µ0) = 0 .030±0.005 GeV3[35]\nfπ= 0.1304±0.0002 GeV [31] fK= 0.1562±0.0007 GeV [31]\naπ\n1(µ0) = 0 [31] aπ\n2(µ0) = 0 .29±0.08 [31]\naK\n1(µ0) =−0.07±0.04[31] aK\n2(µ0) = 0 .24±0.08 [31]\nmomentum fraction of uanddquarks, respectively with ¯y= 1−y.uΛb(v)is the Dirac spinor\nin HQET, satisfying\n/vuΛb(v) =uΛb(v). (14)\nThe definition of gauge link [35] is\n[0, tin] =Pexp\u0014\n−igZti\n0dx n·A(xn)\u0015\n, (15)\nwhere i= 1,2. The specific form of the wave function at leading twist [35–38] is as follows\nψ2(y, ω) =y(1−y)ω21\nω4\n0e−ω/ω 0, (16)\nwhere ω0is a non-perturbative input parameter, whose numerical value given in Table 1.\n3.3 Λb→PlDecay amplitude\nTo calculate the form factors of the Λb→Plspectator processes, we need to construct\ncorrelation functions associated with the leptoquark operators. The leading order Feynman\ndiagrams of quark field for the baryon number violation process are shown in Fig. 2. Each\ninteraction vertex in the diagram can be replaced by one of the leptoquark operators from\nEq. (2). In QCD, the interaction vertex between quark and gluon is\nLint(x) =¯ψ(x)g/A(x)ψ(x). (17)\nSince the operators in Eq. (2) involve eigenstates of charge conjugation, we need to introduce\ninteraction vertices with charge conjugation\nLc\nint(x) =−ψT(x)g/AT(x)¯ψT(x). (18)\n6hard\nhard−collinear\ncollinear\nsoft\nΛb\nPΛb\nPΛb\nP\n(a) ( b) ( c)Figure 2: Leading order factorization diagram of Λb→Pform factor.\nAccording to discrete symmetry, the Lagrangian is invariant under conjugate transformations\nLint(x) =Lc\nint(x). The corresponding correlation function in Fig. 2 is:\nTα=Z\nd4x d4y T{Oα(0), iLint(x), iLint(y)}. (19)\nIntheframeworkofQCDfactorization, wecanwritedownthedecayamplitudefortheFig.2(a)\nbased on the correlation function:\nAa\nα(Λb→P+l−) =Gnew,α ζa\nΛb→P+,αׯul(q)Ma\nαuΛb(v), α = 1∼5,\nAa\nα(Λb→P−l+) =Gnew,α ζa\nΛb→P−,α×uT\nΛb(v)Ma\nαvl(q), α = 6∼7.(20)\nThe form factor ζa\nΛb→P+,αandζa\nΛb→P−,αatO(αs)can be factored into the decay constants fP\nandfΛbtimestheconvolutionofthehardfunctions Ca\n±α(x, y, ω, µ ), jetfunctions Ja\n±α(x, y, ω, µ ),\nand non-perturbative wave functions ϕP(x, µ0)andψ2(y, ω)at leading power\nζa\nΛb→P±,α=fPf(2)\nΛbZ1\n0dxZ∞\n0dω ωZ1\n0dy Ca\n±α(x, y, ω, µ )Ja\n±(x, y, ω, µ )ϕP(x, µ0)ψ2(y, ω).(21)\nThe perturbative hard and jet function receives contributions from the hard mode and the\nhard-collinear mode, respectively. At the tree-level, the hard function Ca\n±α(x, y, ω, µ ) = 1, and\nthe jet functions are\nJa\n+(x, y, ω, µ ) =π αs(µ)Ta\nc,α\n4 ¯n·p1\nx y ω2,Ja\n−(x, y, ω, µ ) =π αs(µ)Ta\nc,α\n4 ¯n·p1\nx¯y ω2,(22)\nwhere Ta\nc,αrepresents the color factors of different leptoquark operators. The numerical result\nof the color factors are Ta\nc,1=Ta\nc,7=−Ta\nc,2=−Ta\nc,3=−Ta\nc,4=−Ta\nc,5=−Ta\nc,6=−2/9.\nFor the decay amplitude of the baryon number violation processes, there exist spinor\nstructures formed by the lepton spinor and the heavy baryon spinor. The difference of spinor\n7structure between Λb→P+l−andΛb→P−l+process arises from whether the fermion flow of\nthe leptoquark operators in Eq. (2) is broken. The fermion flow in the lepton part of O1∼O5\noperators is not broken, hence the spin structure is the bilinear form ¯ul(q)Ma\nαuΛb(v). On the\nother hand, the fermion flow in the lepton part of operators O6andO7is broken, resulting in\na spin structure in the form of uT\nΛb(v)Ma\nαvl(q).Ma\nαrepresents the matrix element between the\nheavy baryon spinor field and the lepton spinor field\nMa\n1=−Ma\n2=−Ma\n4/2 =−2 ¯n·p m1/2\nΛbΓ/n\n2Γ,\nMa\n6= 4 ¯n·p m1/2\nΛbΓTC/n\n2Γ,Ma\n3=Ma\n5= 2 ¯n·p m1/2\nΛbTr\u001a\nΓ/n\n2\u001b\nΓ,\nMa\n7= 2 ¯n·p m1/2\nΛbTr\u001a\nΓ/n\n2\u001b\nCΓ,(23)\nwhere m1/2\nΛbisfromthedifferenceofthespinorbetweenQCDandHQET uΛb(pΛb) =m1/2\nΛbuΛb(v).\nFrom Eq. (2) and Fig. 1, one can see that for operators O3, O5andO7, the Dirac fermion flow\nis ab-quark going to a lepton, while one of the light quark in Λbbaryon going to the final\nstate pseudoscalar meson. In our calculation, the light quark in the four-fermion operator is\ncombined with the spectator quark making a scalar di-quark forming the pseudoscalar final\nstate meson, which is similar to the semileptonic Bdecay. In this case the calculation will give\na contribution as a trace of the Dirac matrices shown in the second line of Eq. (23). Because of\nthis trace in Eq. (23), only Γ =γµkind of operators can contribute in our calculation, which is\nsimilar as the Bdecay case. For the other kinds of operators, shown in the first line of Eq. (23),\nno trace required, such that all kinds of operators contribute except that for Γ = σµν. It is\neasy to see from Eq. (23) that these kinds of operators will not contribute to any of the decays\nconsidered here, due to the anti-symmetric nature of σµν.\nKinematically, the above calculation can be described by the factorization diagram for the\nform factor of the baryon number violation process, as depicted in Fig. 2(a). The b-quark\ndecays through new physics into two antiquark fields and a lepton field, with one of the hard-\ncollinear antiquark fields annihilating with the soft quark from the initial state to produce a\nhard-collinear gluon, which converting the soft spectator quark into a collinear one. The jet\nfunction in Eq. (22) is obtained when the hard-collinear fields are integrated out, which gives a\nleading power contribution.\nComparing Fig. 2(a) and (b), the light quark propagator is replaced by a heavy quark\npropagator in the perturbative calculation, thus the contribution from Fig. 2(b) is ΛQCD/mb\nsuppressedcomparedwithFig.2(a). Wewillnotconsiderthisnext-to-leadingpowercontribution\nin the present paper.\nBased on the correlation function in Eq. (19), we can similarly express the amplitude of\n8Λbπ\nΛbK\n0.15 0.20 0.25 0.30 0.350.00050.00100.00150.00200.00250.00300.0035\nω0(GeV )|ζΛb→P,α|(GeV )Figure 3: The parameter dependence of Λb→Pform factor.\nFig. 2(c) as\nAc\nα(Λb→P+l−) =Gnew,α ζc\nΛb→P+,αׯul(q)Mc\nαuΛb(v), α = 1∼5,\nAc\nα(Λb→P−l+) =Gnew,α ζc\nΛb→P−,α×uT\nΛb(v)Mc\nαvl(q), α = 6∼7.(24)\nIt is easy to show that at leading power the matrix element Mc\nαbetween spinors for any of the\neffective operators for this Feynman diagram is proportional to\nMc\nα∼/n\n2γµ/n\n2γµ/¯n\n2=/¯n\n2γµ/n\n2γµ/n\n2= 0. (25)\nIn fact, this diagram is similar to the one in semi-leptonic Bdecay, which has the same power\nwith Fig. 2(b). Remarkably, the amplitude of Fig. 2(b) and (c) are the major contribution of\ntheBmeson heavy-to-light form factor at leading power [30, 39, 40], which are suppressed in\nour baryon decay process. As a result, only Fig. 2(a) will contribute to the decay amplitude\nat leading power. While the endpoint divergence of the jet function at small values of x, yand\nωwill not show up in the baryon decay because the convolution of the jet function with the\nLCDAs in Eq. (21) is convergent, as we can see employing in Eq. (11) and (16)\n4 Numerical result\nIn the previous section, we provided the factorized formula of the form factor for the baryon\nnumber violation process. Taking into account the input parameters from Table 1, we can\nchoose the strong coupling constant at hard-collinear scale αs(µ= 2GeV) ≃0.3and calculate\nthe form factors for the Λb→P+l−andΛb→P−l+spectator processes. The numerical result\n9Table 2: Constraints of the new physics effective couplings from LHCb experiments.\n|Gnew,α|2[GeV−4] Γ = 1 orγ5 Γ =γµ Γ =γµγ5 Γ =σµν\nα= 1,2 <2.6×10−17<6.4×10−18<6.4×10−18-\nα= 3 - <1.6×10−18- -\nofζa\nΛb→K+,αform factor is calculated when α= 1,2,3as\n|ζa\nΛb→K+,α|= 1.09+0.36\n−0.42×10−3GeV. (26)\nThe numerical result of form factors ζa\nΛb→π+,αwhen α= 4,5andζa\nΛb→π−,αwhen α= 6,7are\n|ζa\nΛb→π+,α|=|ζa\nΛb→π−,α|= 9.00+2.92\n−3.42×10−4GeV. (27)\nIt is easy to see that there is only one independent form factor at the heavy quark limit\nfor each kind of decays, just like the Bmeson decays [32]. During the calculation of the form\nfactor,theprimarysourceoftheoreticaluncertaintiesoriginatesfromthenon-perturbativeinput\nparameter ω0of the Λbwave function. The dependence of form factors |ζa\nΛb→π,α|and|ζa\nΛb→K,α|\non this non-perturbative input is shown in Fig. 3. It can be observed that the influence of the\nnon-perturbative input ω0on the form factor |ζa\nΛb→P,α|is monotonically decreasing.\nThe LHCb experiment has ever searched for the Λb→K+µ−andΛb→K−µ+processes\n[13], with the joint upper limit on the branching ratios of\n[B(Λb→K−µ+) +B(Λb→K+µ−)]×3.1×10−6\nB(Λb→pK−)<1.95×10−9at CL = 90 % .(28)\nCombining the branching ratios B(Λb→pK−) = (5 .4±1.0)×10−6provided by the particle\ndategroup[34], wecanobtaintheconstraintsofnewphysicscouplingsforleptoquarkoperators.\nAs stated in section 2, the leptoquark operator O1∼O3contribute to the process of Λb→\nK+l−. Assumingthatonlyonekindofoperatorscontributetothecorrespondingdecaychannel,\nconstraints on new physics couplings Gnew,αfor different Lorentz structures are presented in\nTable 2. As discussed in the previous section, only vector current contributes to the effective\noperators O3,O5andO7, so the constraint to operator O3only occur for the vector current\ncase. The tensor does not contribute to any kinds of effective operators, so there is no constraint\nfrom the current experiment. As for operator O1andO2, we have two kinds of constraints: one\nis for 1orγ5operators, the other is for γµorγµγ5currents.\n105 Conclusion\nIn this work, we introduced factorization theorems for baryon and lepton number violation\nprocesses in heavy baryon decay. Within the framework of the SMEFT, we integrated out the\nnew physics particles and absorbed them into effective new physics couplings Gnew,α, obtaining\nleptoquark operators at the mbscale. In the QCD factorization approach, we factorized the\nform factor for the process Λb→Plinto the convolution of the hard function, jet function, and\nwave function. The combination of the effective field theory of the Standard Model and QCD\nfactorization allows us to understand the factorization behavior of baryon number violation\nprocesses even in a low-energy situation. We computed the numerical results of the form factor\nfor these effective operators, which can also be applied to similar baryon number violation\nprocesses or serve as input for other leptoquark new physics theories.\nFinally, as an example of application, utilizing measurements of Λb→K±µ∓decays from\nLHCb experiments, we derive constraints on some of the effective new physics couplings of the\nleptoquark operators. With the upgrade and renovation of the High-Luminosity Large Hadron\nCollider, future experiments will be able to conduct more precise measurements of baryon\nnumber violation processes.\nAcknowledgements\nWe are grateful to Dong-Hao Li for helpful discussions. The work is partly supported by\nthe National Natural Science Foundation of China with Grant No.12275277 and the National\nKey Research and Development Program of China under Contract No.2020YFA0406400 and\n2023YFA1606000.\nReferences\n[1] A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967).\n[2] J. C. 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Catania, I-95123, Catania, Italy and\nCNR-IMM, Via S. Sofia 64, I-95123, Catania, Italy\nPertti J. Hakonen\nLow Temperature Laboratory, Department of Applied Physics, Aalto University,\nPuumiehenkuja 2B Otaniemi, P.O. Box 15100, FI-00067 AALTO, Finland and\nInstituteQ - The Finnish Quantum Institute, Otakaari 1, FI-02150 Espoo, Finland\nE. Paladino\nDipartimento di Fisica e Astronomia Ettore Majorana,\nUniversit´ a di Catania, Via S. Sofia 64, I-95123, Catania, Italy\nINFN, Sez. Catania, I-95123, Catania, Italy and\nCNR-IMM, Via S. Sofia 64, I-95123, Catania, Italy\nFundamental issues of 1 /fnoise in quantum nanoscience are reviewed starting from basic statisti-\ncal noise processes. Fundamental noise models based on two-level systems (TLS) are described. We\nemphasize the importance of TLSs in materials parameter fluctuations, such as dielectric constant.\nThe present understanding of 1 /fnoise in superconducting quantum interferometers and in single\nelectron devices is summarized. For coherent quantum nanoscience, we introduce superconducting\nqubits and the relation between decoherence and 1 /fnoise using the filter function formulation. We\nalso clarify the qubit noise spectroscopy and emphasize the importance of materials with reduced\n1/fnoise for future quantum coherent nanodevices.\nKey Points\n•Basic characteristics of 1 /fnoise in condensed mat-\nter systems\n•Theoretical issues concerning stationarity, ergodic-\nity, and TLS with uniform distribution for tunnel-\ning parameter\n•Specific features of 1 /fnoise in nanoscale systems\nand quantum materials\n•Significance of TLS for dissipation in superconduct-\ning materials for low-loss quantum circuits\n•1/fnoise in quantum circuits and devices, both in\nthe flux and charge fluctuation regimes\n•Influence of 1 /fnoise on coherence properties of\nqubits\n•Role of low-frequency fluctuations in quantum sens-\ning\nI. INTRODUCTION\nFluctuations with power spectral density S(f) increas-\ning with decreasing frequency have been observed in a\ngreat variety of systems from physics, electronic engi-\nneering, and biology to music, economic and social sci-\nences, even medicine. Solid-state materials and devices\nhave formed the supreme arena where the command ofthe physics of low-frequency fluctuations has been deeply\nexploited. Strong motivation comes both from funda-\nmental research and from the quest for electronic de-\nvices with improved performance. The advent of modern\nnanoscience has opened a new era in which this subject\nhas entered the realm of quantum coherence.\nMeasurements of fluctuations in a vast number of phys-\nical systems yield a noise spectral density of the form\nS(f) =C f−α(1)\nwith αranging from 0 to 2, down to the lowest measured\nfrequencies ∼µHz. Since values of α≈1 are typically\nobserved, the phenomenon is called 1 /fnoise. The al-\nternative terms ”flicker noise” or ”excess noise” are less\nfrequently used.\nIn solid-state materials and devices fluctuations of\ntransport and dielectric properties exhibit a ∼1/fspec-\ntrum in addition to intrinsic thermal noise and quantum\n(shot) noise (see Fig. 1). They share the same distinc-\ntive features as the frequency dependence of the noise,\nthe time dependence of correlations, weak stationarity\nand other statistical properties.\nWith the advent of nanofabrication technologies, the\nnumber of devices and electronic components showing\nexcess low-frequency noise has increased, and they all\nshare the features of the ubiquitous 1 /fnoise. Charge\nfluctuations with 1 /fspectrum in substrate materi-\nals have been found to influence devices operating in\nthe Coulomb blockade regime where quantized charge\nin a small, weakly-coupled conducting island is acting\nas the quantum mechanical observable. In the regimearXiv:2401.11989v1  [cond-mat.mes-hall]  22 Jan 20242\nof superconducting coherence, flux noise generates un-\nwanted currents with 1 /fcharacter, which is detrimen-\ntal in many magnetic field sensors and superconduct-\ning qubits. Also, various heterostructures (including e.g.\nfield effect (FET) and high-electron-mobility transistor\n(HEMT) structures), either based on materials deposi-\ntion, modulation doping, or stacking of atomically lay-\nered two-dimensional (2D) materials, display 1 /fnoise\nin the number of charge carriers owing to charge traps\nin the vicinity of the active device channel. Beyond elec-\ntronic and physical systems, 1 /fnoise is found even in\nhealthy human heartbeat rates.\nThe appearance of 1 /f-like behavior in many phenom-\nena of diverse nature and its scale invariance, i.e. the\nfact that each decade of frequencies contains the same\nintegrated power, suggested that a common, deeper level\ndescription may exist, such as for the universality of the\nexponents in critical phenomena, i.e. that 1 /f-like be-\nhavior is an inherent property of all systems and that a\nuniversal minimum frequency exists. However, the large\namount of experimental data in solid-state systems indi-\ncates that there is no universal spectrum of low-frequency\nnoise.\nIn modern electronic devices with small active vol-\nume bistable switching of the resistance called random\ntelegraph noise (RTN) is observed. The phenomenon is\nattributed to capture and emission processes from de-\nfects suggesting a possible microscopic mechanism of 1 /f\nnoise. In fact, in many solid-state systems experiments\nare satisfactorily explained by the combined action of\nmany bistable defects or traps. The distribution of the\nobserved physical characteristics accounts for the rather\nlarge range of time constants generating 1 /fnoise.\nBesides universality, other fundamental problems have\nbeen posed as the necessary existence of low-frequency\ncutoff to avoid unphysical divergencies and the related\nquestions of stationarity. On the experimental side, it\nis often difficult to give an unambiguous interpretation\nof data as the ensemble-averaged power spectra provide\nincomplete information on individual fluctuators.\nWhile for metals and spin glasses, some understand-\ning of the mechanism of 1 /fnoise has been developed\nthe problem is still unsolved for the majority of physical\nsystems. Advances in the study of the microscopic na-\nture are of practical importance since 1 /fnoise limits the\nperformance of high-end electronic devices and sensors at\nlow frequencies. Devices for metrological applications are\nalso affected since their high operating frequencies show\nfluctuations with 1 /fspectrum.\nThis chapter focuses on 1 /fnoise in quantum\nnanoscience . Quantum nanoscience comprises of natural\nor engineered nanoscale systems whose functionality and\nstructure can only be explained through quantum mech-\nanisms such as discretisation, superposition and entan-\nglement (definition from Ref. Milburn and Woolley [1]).\nThese quantum properties facilitate, among other things,\nthe enormous power of quantum information processing\nas well as novel quantum sensors. However, superposi-\nFIG. 1. Qualitative representation of noise spectrum in\ncoherent nanodevices on a log-log scale. Lines describe the\nexpected behavior in different frequency ranges. Dots rep-\nresent typical observations in frequency windows depending\non the noise detection method (different symbols and dots),\nsee discussion in Section IV. Long-dashed lines indicate the\n1/fdependence between the intrinsic low- and high-frequency\ncutoffs γ1andγ2, which in general are not known. Thick lines\nindicate the 1 /fαdependence, with α≈0.9. Ω denotes the\ntypical qubit splitting, of the order of 1010Hz in supercon-\nducting qubits. In this range, quantum noise is commonly\nwhite or Ohmic (dotted and thin lines). The dash-dotted line\nindicates the linearly increasing Johnson-Nyquist noise.\ntion states are fragile and quickly destroyed by 1 /fnoise,\nwhich needs to be considered for their functional appli-\ncations. Our scope here is to discuss first the influence of\n1/fnoise on ”traditionally-behaving” nanodevices and\nsensors, and then describe specific features encountered\nin the quantum coherent case.\nNew fundamental problems related to 1 /fnoise arise\nin quantum nanoscience and quantum information since\nthe flicker noise may be the main source of decoherence\npresent. For coherent quantum devices such as qubits,\nthis has led to original ways for quantifying the impact\nof 1/fnoise on the qubit, and to strategies for recovering\nthe coherence using well-chosen pulse sequences.\nProgress in quantum nanoscience has also provided\nnew tools for the characterization of 1 /fnoise sources.\nNanoelectronic devices can be employed for example to\ndetect charge noise at the micro electron charge level,\nthereby allowing to chart substrate characteristics in\nterms of trap states or fluctuators modelled by TLSs.\nCoherent TLS, qubits, on the other hand, can be em-\nployed as spectrometers detecting electromagnetic noise\nover a large span of frequencies. Qubits as spectrometers\nprovide an excellent means to address the dynamics of\nimpurity spins, which often form the culprit for coher-\nence in quantum nanosystems.3\nII. THEORETICAL OVERVIEW\nFor a stochastic process x(t) gated in a time window T,\nx(t)̸= 0 for |t|< T/ 2, the power spectrum Sx(f), repre-\nsenting the spectral resolved ensemble-averaged power of\nthe signal, is independent on Tif and only if the stochas-\ntic process is stationary. For 1 /f-like noise the statisti-\ncal properties change slightly over time scales typically\nslow relative to the observation times. Thus it is con-\nsidered a wide-sense (or weakly) stationary process, i.e.\nthe correlation functions up to second order are (nearly)\ninvariant for time translation. Then Wiener-Khintchine\ntheorem applies, stating that Sx(f) is given by the uni-\nlateral Fourier transform of the autocorrelation function\nϕx(τ) :=⟨δx(t+τ)δx(t)⟩=Z∞\n0dω\n2πSx(ω) cos( ωτ).(2)\nThe integrated power in the spectrum between two fre-\nquencies [ f1, f2] is given by\nPx(f1, f2) =Zf2\nf1dω\n2πSx(ω) =Cxln\u0010f2\nf1\u0011\n(3)\nimplying that the integrated power per decade of frequen-\ncies is constant, a property known as scale invariance.\nThe autocorrelation function for τ= 0 yields the vari-\nance⟨(δx)2⟩of the stochastic process which according to\nEq.(3) diverges logarithmically both at low and at high-\nfrequencies. This divergence is known as a ”paradox” of\n1/fnoise. Of course, the experimentally observed vari-\nance is finite since the frequency band for measurements\nis limited. Indeed it is bounded from below due to the\nfinite duration of the realization of x(t) being measured\nwhose maximum value is set by the overall measurement\ntime tm. On the other hand, at large frequencies 1 /f\nnoise becomes much smaller than other contributions to\nfluctuations and it is not measurable, thus high frequen-\ncies are also cut off. This may happen at frequencies\nranging from ∼102to∼107Hz in different systems.\nIt is worth stressing that there is no experimental evi-\ndence for the existence of a low-frequency cutoff fc, there-\nfore strictly speaking the stationarity of the process is\nstill an open question. Difficulties are due to the fact\nthat measuring the low-frequency part of the spectrum,\nsay down to the sub- µHz range would require a duration\ntm∼1/fcof weeks. Therefore experimental statistical\nquantities depend on tm. In particular this applies to\nthe variance, ⟨(δx)2⟩=Cxln(f2tm), which increases log-\narithmically with tm. This is another peculiar property\nof 1/fnoise of experimental relevance. In particular,\nit may have a large impact on applications in coherent\nnanodevices where large fluctuations produce strong de-\nphasing. From the mathematical point of view, this be-\nhavior of statistical quantities implies that 1 /f-like noise\nis not strictly stationary thus it is also non-ergodic. On\nthe other hand, the logarithmic dependence on tmis weak\nenough to determine weak stationarity allowing the use of\nthe Wiener-Khintchine theorem which relates the power\nFIG. 2. Power spectral density of individual bistable fluc-\ntuators (coloured solid curves) whose combined effect (black\nsolid curve) yields close to 1 /fspectrum (indicated by dashed\nblack line). If one of the fluctuators is more strongly coupled\n(dashed orange) a bump appears in the whole power spectrum\n(solid orange curve).\nspectra to correlation functions accessed experimentally.\nThese latter show strongly nonexponential behavior de-\ncaying logarithmically with time in the important range\nf−1\n2< t < t m< f−1\nc, another feature observed in several\nexperiments.\nUnderstanding the microscopic nature of 1 /fnoise is a\nkey problem both from the fundamental point of view and\nfor applications aiming at fabricating improved electronic\nand coherent devices for modern nanoscience. Typically,\nmodels for 1 /fnoise in the solid state are based on a col-\nlection of TLSs describing bistable defects or trap states\nwith a wide distribution tunnelling rates [2–6]. A large\ncollection of such states with broadly distributed param-\neters generates wide band 1 /fnoise and in what follows\nwe will draw this connection.\nFluctuations of a bistable defect are described by a\nstochastic process xi(t) =±1 characterized by a single\nrelaxation rate γi. Its kinetics leads to RTN where corre-\nlations decay exponentially, ϕi(τ) = exp( −γiτ), yielding\nthe Lorentzian spectral density\nSi(f) = 4Z∞\n0dτ ϕ i(τ) cos(2 πfτ) =4γi\n(2πf)2+γ2\ni\nas given by the Wiener-Khintchine theorem. The overall\nnoise is constructed as a linear superposition of these\nrelaxation processes, x(t) =P\nipixi(t), and its power\nspectrum is given by\nSx(f) =X\nipiSi(f)→Z∞\n0dγ p(γ)4γ\n(2πf)2+γ2\nwhere we have allowed for a continuous distribution\nof relaxation rates. Since ⟨(δx)2⟩=R∞\n0d f Sx(f) =4\nR∞\n0dγ p(γ), the quantity p(γ)dγis the contribution\nto the variance of processes with rates in the interval\n[γ, γ+dγ]. For a weighting function p(γ) =Cx/γin the\ninterval [ γ1, γ2] and vanishing elsewhere we obtain\nSx(f) =γ2Z\nγ14Cxdγ\n(2πf)2+γ2=\n\nconst f≪γ1\n2π\nCx/fγ1\n2π≪f≪γ2\n2π\nCx(γ2−γ1)\n(πf)2γ2\n2π≪f\n(4)\nreproducing the 1 /fbehavior for intermediate frequen-\ncies, γ1≪2πf≪γ2(see Fig. 2). Notice that the ex-\ntremal rates γ1andγ2provide soft frequency cutoffs to\nSx(f), avoiding the divergence of ⟨(δx)2⟩. They are easily\nidentified for a finite system of discrete fluctuators. We\nhave still to justify the form of p(γ) which is the other key\nassumption determining the 1 /fbehavior of the power\nspectrum. In the Mc Worther model [7] for semicon-\nductors, the kinetics is modelled as tunneling processes.\nThe rates have the form γ(z) =γ0exp(−z/C) depend-\ning on the parameter zsummarizing the exponential de-\npendence on the width and on the height of the tunnel\nbarriers. Now, if zis distributed uniformly in the interval\n[z2, z1] then p(γ) =C/γin [γ(z1), γ(z2)]. The same result\nis obtained if the kinetics stems from thermal activation,\nγ(E) =γ0exp(−E/k BT), assuming that the activation\nenergies Eare uniformly distributed as originally pro-\nposed by Du Pr´ e and by Van der Ziel [8, 9]. Notice that\nthis picture naturally accounts for the weak stationarity\n(implying non-ergodicity) of 1 /fnoise since during the\nmeasurement, the TLSs with relaxation rates γ <1/tm\nare frozen in some non-equilibrium configuration.\nSpecific models for 1 /fnoise based on quantum TLSs\ninteracting with phonons (in insulating solids) or elec-\ntrons (in metals) yielding physically motivated distribu-\ntions of parameters have been used to explain observed\nproperties of 1 /fnoise. These models apply in rater sim-\nple situations of non-interacting impurities while in gen-\neral, the physical properties of disordered systems depend\non their many-body nature. Still, the 1 /fbehavior in the\nsolid-state appears to be a consequence of the universal\nproperties of the kinetics rather than the universality of\nlow-frequency fluctuations [3].\nFinally, we remark that noise from a collection of TLSs\nis not Gaussian. The Gaussian limit is approached when\nthe noise is generated by a sufficiently large number of\nindependent fluctuators each of which contributes negli-\ngibly to the variance. Non-Gaussianity in 1 /fnoise has\nbeen observed in different nanosystems, although it poses\nexperimental challenges since it requires finite-bandwidth\nmeasurement with large observation times tm[4]. On the\nother hand, resolving the contribution of individual fluc-\ntuators is needed in view of miniaturization and of the\nstudy of the impact of 1 /ffluctuators on the reliabil-\nity of quantum architectures. Non-Gaussian features of\n1/fnoise and their impact on phase-coherent quantum\nsystems are discussed in depth in Ref. Paladino etal.\n[10].III. 1/F NOISE IN NANOSCIENCE\nThe relative 1 /fresistance fluctuations δR/R have\nbeen found to decrease as 1 /Volume in bulk materials.\nSimilarly, the relative 1 /fcurrent noise in FETs has been\nobserved to scale as ∝1/Area of the gate electrode. So\ninevitably, the role of 1 /fnoise will grow when going\nto nanoscale quantum devices. Lately, nano-scale FETs\nhave been fabricated using heterostructures of atomically\nthin 2D materials. In such components, the surface-to-\nvolume ratio becomes the largest, which may lead to ma-\njor issues with the quality of surfaces or interfaces in\nsuch nanodevices. As Wolfgang Pauli said ”God made\nthe bulk; surfaces were invented by the devil”, this state-\nment becomes more and more relevant when entering the\nnanoscale regime.\nQuantized conductance in nanoscience (mostly 1D or\n2D transport) is typically described using the Landauer-\nB¨ uttiker formalism [11], in which current is given as a\nsum over transmission channels with transmission τik:\nI= ΣτikG0where G0= 2e2/hand the indices kandire-\nfer to subbands and quantized states within them, respec-\ntively. If we take only one channel with transmission τ,\nthe conductance can be written as G−1= (1−τ)/τ+G−1\n0\nwhere the first term is the four-terminal resistance and\nthe second term describes the contact resistance for per-\nfect contacts. Since contact resistance is constant, fluc-\ntuations in τyield δR/R =−δτ/τ both for the two- and\nfour-lead resistance R. If the fluctuations in transmission\nchannels are independent, we may add fluctuations of\nthe channels incoherently, which yields δG2= Σδτ2\nikG2\n0.\nEach fluctuating channel can have its own frequency de-\npendence, which in the simplest case is Lorentzian with a\ncut-off given by a characteristic two-level fluctuator fre-\nquency fc\nik(see below). If the cut-off frequencies fc\nikare\ndistributed uniformly on log-scale, summation over all\nδτik(f) yields a 1 /fnoise spectrum in transmission. In\npractice, contacts are not ideal and may also have fluctu-\nations. In the nearly ballistic limit, the nonidealities of\ncontacts can be integrated into the transmission values\nτik, and no distinction between scattering in the contact\nvs. the channel needs to be made.\nThe conductance in diffusive electrical transport can\nbe written as G=eNµ where Nis the total number of\ncharge carriers and µis their mobility; eis the electron\ncharge. Therefore, the conductance variation is either\ndue to mobility or carrier number fluctuations:\nδG=eNδµ +eµδN (5)\nIn bulk metallic conductors, Ncan be made very large,\nand the relative fluctuations coming from carrier number\ncan be made to vanish as 1 /√\nN. However, when the\nsample size is brought towards the nano regime, the dom-\ninance of mobility fluctuations is suppressed and both\ncontributions have to be taken into account. Carrier\nnumber is varied by tunneling into trap states that reside\nin the nearby dielectric, for example in the gate dielec-\ntric of a MOSFET transistor. In a small quantum device,5\nsuch as a quantum point contact (QPC), the trap state\nmay result in distinct RTN signature in the resistance\n(see Sec. II). The RTN noise in a QPC can be described\nusing Machlup’s formula for a single two-level fluctuator\n[12]:\nSG(f) = [Gτ1−Gτ2]2w1w2τ\n1 + (2 πfτ)2(6)\nwhere w1andw2= 1−w1denote the probabilities for\nfinding the QPC in conductance states Gτ1andGτ2, re-\nspectively, while 1 /τdenotes the total transition rate\nbetween the conductance states (sum of back and forth\nrates).\nEven though nanoscience devices are small, more\nthan one fluctuator is typically acting on them. The\nMcWhorter model is obtained by considering a collection\nof single fluctuators which have an equal strength and a\nuniform distribution of corner frequencies fc= 1/(2πτ)\non logarithmic scale. The resulting 1 /fspectrum is given\nin Eq. (4) where 1 /τcorresponds to γ.\nThis 1 /fspectrum will cover a large frequency span\nprovided that the trap lifetimes [ τ1. . . τ 2] are distributed\nover a wide range. In the standard McWhorter model,\nthe trapped charge is set to one electron charge e, but\nwith separate TLSs, the charge change in the device in-\nduced by the fluctuators may be a fraction of e, which\nreduces the prefactor Cxin Eq. (4). The McWhorter\nmodel has been successfully applied to semiconductor de-\nvices and, indeed, good agreement is obtained between\nexperimental results in MOSFETs and the model [6, 13].\nTypically, the noise is given as relative fluctuations\n\u0000δG\nG\u00012=\u0000δR\nR\u00012=\u0000δN\nN\u00012+\u0010\nδµ\nµ\u00112\n, i.e. scaled by the\ntotal conductance of the system. Ideally, the scaling con-\nductance should be governed by those processes which\nare governing the fluctuations. Unfortunately, this is not\nalways the case and then the relevant scattering processes\nshould be separated from the data before scaling. The-\nories can also incorporate correlations between carrier\nnumber and mobility fluctuations. The fluctuations are\ncorrelated because an electron in the trap state can also\nmodify the scattering governing the mobility of the sam-\nple. This is common in 2D material FETs, for example,\nfabricated using graphene [14].\nGood electrical contacting is difficult to achieve in\nnanosamples. Consequently, the low-frequency noise for-\nmula in Eq. (5) has to be supplemented by the contact\nnoise term, which adds incoherently with the noise aris-\ning from the nanodevice. For the relative fluctuations in\nthe total conductance GTone obtains then\u0010\nδGT\nGT\u00112\n=\nR2\nc\n(R+Rc)2\u0010\nδRc\nRc\u00112\n+R2\n(R+Rc)2\u0000δR\nR\u00012in which R= 1/Gand\nRcdenote the sample and contact resistance, respec-\ntively, and the weighting factor of the relative contact\nnoiseδRc\nRcgrows with the ratioRc\n(R+Rc).\nBoth δµandδNin Eq. (5) may depend on tempera-\nture, which can lead to low-frequency conductance fluc-\ntuations owing to temperature fluctuations δT[3]. Theinfluence of δTfluctuations is expected to increase in\nsmall nanodevices, because\nδT2\u000b\n∝kBT2/Cwhere C\nis the heat capacity of the system. These temperature\nfluctuations can, for example, limit the ultimate energy\nresolution in nanosized calorimeters, as discussed in Ref.\nKarimi etal. [15].\nTLSs discussed in Sec. II lead to noise and dissipation\nboth in acoustic and electric systems. The acoustic cou-\npling is due to strain fields (elastic dipole moment) that\nevery TLS possesses when embedded into elastic mate-\nrial [16]. Some of the TLSs possess also an electric dipole\nmoment, which makes them a source of 1 /fnoise in elec-\ntrical circuits, both for the conductance noise as well\nas for fluctuations in dielectric properties of materials\n[16, 17]. Therefore, TLSs make a contribution to the real\nand imaginary parts of the dielectric constant εTLS(f, T).\nIn weak electric fields, the loss tangent δTLS(f, T) of the\ndielectric constant is dominated by TLS energies close to\nhf:\nδTLS(f, T) =Im{εTLS(f, T)}\nRe{ε}=δ0tanh\u0012hf\n2kBT\u0013\n(7)\nwhere the hyperbolic tangent describes the equilibrium\npopulation difference of the two levels and δ0is the intrin-\nsic loss parameter proportional to the density of TLS per\nunit volume and energy. If the asymmetry and tunneling\nparameters of two-level states are uniformly distributed\nin energy as assumed in the standard tunneling model\n[16], the low-temperature loss tangent becomes indepen-\ndent of frequency. In the limit kBT≫hf, the upper\nstate will become populated, leading to δTLS(f, T)∝\nhf/k BT. Clearly, the role of TLSs in dissipation van-\nishes when f→0. According to Kramers-Kronig rela-\ntions, Eq. (7) can be utilized to obtain the reactive part\nofεTLS(f, T), and the equations for fluctuations of reac-\ntance (capacitance) can be derived [17].\nThe capacitance modulation due to TLSs provides an\nefficient means to investigate two-level fluctuators and\nthe ensuing 1 /fnoise using superconducting microwave\ncavities [16–18]. Furthermore, understanding the perfor-\nmance limits of such cavities is vital, because high-quality\nsuperconducting microwave cavities are critical elements\nfor efficient qubit readout [19] and kinetic inductance de-\ntectors for sensitive bolometry or single photon detection\n[20]. A summary of recent results obtained for the loss\ntangent are given in Ref. [18] for several dielectric and su-\nperconducting materials which are relevant for coherent\nquantum nanoscience. Most recently, promising results\nhave been obtained for superconducting tantalum [21];\nin comparison to Nb, this is assigned to more kinetically\nlimited and chemically robust oxides in Ta [22].\nMeasurement power is known to modify the influence\nof TLSs [18]. Dissipation in a superconducting mate-\nrial is first governed by TLSs, but increasing power leads\nto the creation of quasiparticles that dominate losses at\nhigh rf power. On the other hand, when there are spa-\ntial variations in the superconducting order parameter,\ni.e. the superconducting gap, it is possible to have quasi-6\nparticles trapped into local minima of this gap variation.\nIn fact, TLSs due to trapped quasiparticles on the sur-\nface of Al superconductor have been observed to produce\nlow energy TLSs up to energies of h ×20 GHz [23]. The\nobserved level separation agrees with theoretical values\nobtained using estimates of the second derivative of the\nspatial variation of the gap [23].\nIn standard TLS models, the number of two-level\nstates Nis evenly distributed in energy, i.e. the den-\nsity of states N(E) =dN/dE = const .Plenty of experi-\nments have been performed to test this hypothesis [2]. In\nquantum nanodevices, there are recent experiments that\nindicate an energy-dependent density of states N(E) [16].\nEnergy dependence can arise for example from inter-\nactions between TLSs, which yields an Efros-Shklovskii\ntype of pseudogap at small energies. There are indica-\ntions that N(E)∝Eα/2with α∼0.2−0.7; the same\nexponent appears also in the temperature dependence of\n1/fnoise SV∝1/Tα[16, 24], which leads to increase of\n1/fnoise with decreasing T.\nSurface spins, such as physisorbed atomic hydrogen,\nhave been found to behave as TLSs that cause significant\ndielectric and flux noise. Removal of surface adsorbants\nby suitable physical or chemical treatment has led to a\nfactor of 10 reduction in frequency fluctuations of a mi-\ncrowave cavity at ∼5 GHz frequency [25]. The scaling of\n1/fas∝Area is quite conspicuous in superconducting\nquantum nanoscience, for example in the superconduct-\ning loops SQUIDs (Superconducting Quantum Interfer-\nence Devices, see Sec. III A). It is also becoming apparent\nthat surfaces and interfaces impose large noise contribu-\ntions in superconducting microwave circuits, which may\neven dominate the defect/impurity contributions scaling\nwith volume.\nBesides TLSs, non-equilibrium quasiparticles may\ncause excess noise and losses in superconductors. In gen-\neral, the number of quasiparticles observed in supercon-\nducting films at T≲150 mK amounts to 10−8. . .10−6\nper Cooper pair, which is clearly larger than predicted\nfrom the BCS theory for a superconductor in equilibrium\nwith the environment at T= 10 mK. The reason for ex-\ncess quasiparticles is not fully resolved yet, but recent ex-\nperiments have indicated that non-equilibrium phonons,\ncreated by e.g. cosmic rays, do play an essential role in\nthe process [26].\nA. Noise in quantum circuits\nQuantum circuits are based on two canonically con-\njugate variables, charge Qand flux Φ (see Sec. IV A).\nThe strength of quantum fluctuations in these variables\ndepends on the impedance of the quantum circuit Z0. If\nZ0is large (fluctuations for the impedance Z0=p\nL/C\nare derived from corresponding LC oscillator [27]), the\nfluctuations of charge are given at the quantum limit by\nδQ2\u000b\n=ℏ\n2Z0, while the fluctuations for flux amount to\nδΦ2\u000b\n=ℏZ0\n2. Hence, quantum circuits at large Z0aresusceptible to external charge noise while external flux\nnoise is detrimental in the case of small Z0.\nFlux noise is produced by resistance fluctuations when\ncurrent is passed through a normal conductor. In metal-\nlic conductors, the commonly accepted view is that 1 /f\nnoise is determined entirely by fluctuations in charge car-\nrier mobility [28]. In addition to models based on local-\nized states and fluctuating scattering cross sections [29],\nthere are more refined models based on a modification of\nelectronic wave interference induced by a defect displace-\nment by a hopping event. These models, the universal\nconductance fluctuation noise and local interference noise\nmodel, have been quite successful in accounting for many\nexperimental observations in metals with defects as sum-\nmarized in Ref. Giordano [30].\nSuperconducting Quantum Interference Devices\nClassical SQUIDs are inherently low impedance devices,\nwhich is highlighted by the small shunt resistance in-\nserted in them for stabilization of their phase dynam-\nics. Hence, their ultimate sensitivity is limited by the\nlow-frequency flux noise, or alternatively by internal fluc-\ntuations of the Josephson energy of the tunnel barriers.\nLow-frequency noise in SQUIDs displays distinct 1 /fα\nbehavior [31, 32]. In spite of 50 years of investigations,\nthe origin of flux noise in SQUIDs is still poorly under-\nstood. It has been found that its power law exponent α\nvaries substantially around α= 1, and the exponent dis-\nplays strong temperature dependence [33]. An increase\nof noise linear with the area of the superconductor in the\nSQUID loop has been found [33], but no clear correlations\nhave been observed neither on the selection of materials\nnor on the details of the device geometry. Even though\nthe fundamental mechanism for flux noise in SQUIDs re-\nmains unclear, there are plenty of experiments indicating\nthat impurity atoms with a spin magnetic moment are\ninvolved in the noise phenomena.\nFlux noise in a SQUID may originate from paramag-\nnetic impurities residing on the superconducting material\nof the SQUID loop. Paramagnetic impurities produce\nflux according to paramagnetic susceptibility χ(T)∝\n1/T, the fluctuation of which produces flux noise. Such\nfluctuating paramagnetic impurities have been observed\nin Al, Nb, Au, Re, and Ag among others [34, 35]. The\nfluctuating spins are believed to reside on the metal sur-\nface as the flux noise scales with the surface area of the\nSQUID loop [33]. In order to obtain a correct order of\nmagnitude of flux noise, one has to assume a magnetic\nimpurity concentration of ∼1×1018/m2with an impu-\nrity moment of one Bohr magneton, given by electron\nrest mass meand electron charge easµB=eℏ\n2me=\n9.274×10−24J/T, i.e. a distance of 1 nm between the\nimpurities [34]. There is also experimental evidence for\nadsorbed oxygen molecules producing a magnetic signa-\nture and flux noise [36]. With its paramagnetism, oxygen\ncan meet the above-stated requirement for strength and\ndensity of impurity moments.7\nHydrogen atoms provide an alternate source for flux\nnoise [37, 38]. Electron spin resonance measurements on\nsapphire have revealed a resonance with a 1.42 GHz line\nsplitting which matches free H atoms. In addition, flux\nnoise measurements using the qubit relaxation rate indi-\ncate a peak at 1.4 GHz. Further support for the presence\nof H is provided by DFT calculations of interstitial hy-\ndrogen atoms embedded in bulk sapphire, which yield an\nESR splitting of 1 .28−1.36 GHz [39]. Recently, complex\nradicals have also been found to contribute to the noise\nand decoherence phenomena [40].\nSingle electron tunneling devices\nThe most sensitive charge detectors in the solid state are\nsingle electron transistors (SET) and charge qubits. Both\ndevices suffer strongly from background charge fluctua-\ntions which often have 1 /fcharacter with some over-\nlying Lorentzian spectra due to strongly coupled TLSs\n[10]. Using Al/AlO x/Al tunnel barriers on top of com-\nmon dielectric such as Si, SiO 2, and Si 3N4, the charge\nnoise amounts to 10−4−10−3e/√\nHz at 10 Hz, both in\nsuperconducting and normal metal regimes. The best\nAl/AlO x/Al devices using stacked design have reached\n8×10−6e/√\nHz at 10 Hz [41]. Charge noise in SETs\narises typically from charged two levels states capaci-\ntively coupled to the island [10]. Such TLSs are present\nin all back-gated devices at the interface of the gate di-\nelectric and the conducting materials. The best charge\nsensitivities in Al-based devices have been obtained in\nstructures where the contact area to dielectric substrate\nmaterial has been eliminated with a stacked design [41].\nThe achieved levels of charge noise have not been ade-\nquate for fabricating well-functioning charge qubits, i.e.\nCooper pair boxes, using Al-technology [10, 42].\n1/fcharge noise is a very important characteristic for\nindustrially fabricated silicon quantum dot MOSFET de-\nvices which are being developed for spin qubits [43]. Typ-\nical noise values in multielectron quantum dots at 1 Hz\nare in the range 1 −10µV referred to the gate elec-\ntrode [44, 45] (corresponding to 80 −800µe/√\nHz using\na gate capacitance 10−17F). Small charge noise in these\nquantum dots is important both for the coherence and\nread-out of spin qubits [46].\nIn 1D and 2D nanodevices, the density of states can\nbe strongly energy dependent. This causes fluctuations\nin the chemical potential µFto generate carrier num-\nber fluctuations, which leads to a one-to-one correspon-\ndence between the conductance noise spectrum and the\nfrequency spectrum of µFfluctuations. The fluctuations\ninµFmay be induced, for example, by chemical dop-\ning or background charge fluctuations [47]. Such fluctu-\nations in 1D and 2D conductors are similar to Coulomb\neffects, where the shift of chemical potential is given in\nterms of variation in the island charge Q=Qi. The\ninfluence of Qion the current Ican be described using\ntransconductance gm=∂I/∂V g=Ceff∂I/∂Q iwhere\nCeffspecifies the connection of the island charge to theshift of electrochemical potential. Typical charge noise\nvalues for carbon nanotube [48, 49] and 2D material de-\nvices [50] with good transconductance are in the range\n4×10−5. . .10−3e/√\nHz at 10 Hz.\nBesides charged TLSs, background fluctuations can\narise from charge localized at interfaces where it can be\ntransported by impurities/defects easier than within the\nbulk. This calls for extreme care when stacking 2D het-\nerostructures in order to keep their 1 /fnoise as small\n[14, 50]. In addition, there has been a substantial ef-\nfort in studies on suspended 2D materials, in particu-\nlar, graphene, which has been employed as a low-noise\nplatform for investigations of noise induced by diffusion\nof mobile scattering centers on the membrane [51]. As\nan interpretation of the observed noise results, intrinsic\nlong-term correlations due to clustering and declustering\nof adsorbed atoms has been proposed [51]. Interactions\nbetween TLSs and non-Gaussian fluctuations can also be\ninvestigated by the second spectrum [52, 53].\nIV. 1/F NOISE IN QUANTUM-COHERENT\nNANODEVICES\nNanoscale-based quantum technologies build on har-\nnessing fundamental properties of quantum states, such\nas superposition, quantum coherence and entanglement\nto enable novel functionalities [54]. The extraordinary\nimpetus in quantum-technology research, from academia\nto spin-offs and industries, since the last few years of the\n20thcentury, has led to a number of breakthroughs. At\nthe same time, it has pointed out that fluctuations char-\nacterized by power spectral density 1 /fαwith α≈1 in\na setup-dependent low-frequency range, are paramount\nin most of the coherent nanodevices. On one side, they\nlimit the ability to control the quantum-coherent behav-\nior of engineered nanostructures, leading to a gradual\nloss of phase coherent evolution or decoherence . On the\nother side, the systematic analysis of decoherence un-\nder different control sequences allows for reconstructing\nthe frequency spectrum of stochastic signals, a task com-\nmonly referred to as noise spectroscopy . Here, we present\nan overview on how fluctuations with 1 /fspectrum in-\nfluence the phase coherent evolution of artificial atoms\nplaying the twofold roles of elementary quantum bits and\nof quantum sensors, in particular of noise spectrometers.\nWe will refer to selected paradigmatic examples of coher-\nent nanodevices.\nA. Artificial atoms\nNowadays micro and nano solid-state devices can be\nfabricated. Tehy are called artificial atoms (AA) [55]\nsince they show functional analogies with natural atomic\nor molecular systems.\nElectrons in semiconductors can be confined in a ”box”\ncalled quantum dot, small enough that they do not be-8\nhave as in bulk solids but rather as electrons in indi-\nvidual atoms. Indeed they have a discrete spectrum of\nenergy and at low temperatures, electrons can occupy\nonly the lowest-energy levels. A qubit can be encoded in\nthe charge or the spin of electronic configurations. Spin\nqubits couple weakly to their environment and leveraging\non the available fabrication technologies they are inter-\nesting candidates for building large-scale quantum com-\nputers.\nSuperconducting artificial atoms are devices made of\ntunnel junctions displaying the Josephson effect. In a\nsuperconductor, the many-body condensate is described\nby a single-particle wavefunction, and in coherent de-\nvices, its phase behaves as a quantum degree of freedom\nanalogous to the coordinate of a fictitious particle in a\nconfining potential. Superconducting AAs may encode a\nqubit in the eigenstates of the potential (phase qubits),\nin the flux enclosed in a superconducting quantum in-\nterference device (SQUID) ring (flux qubit), in the extra\ncharge stored in a superconducting ”box” (charge qubits)\nto mention the simplest instances (see Fig. 3).\nAs an example, the rf-SQUID illustrated in Fig.3a is\nmodelled by an equivalent circuit with concentrated ele-\nments (Fig. 3b). The circuit’s dynamics emulates a par-\nticle moving in the potential\nU(X) =−EJcos2πX\nΦ0+EL\n2\u0012X−Φx\nΦ0\u00132\n(8)\nwhere the ”coordinate” Xis the total magnetic flux\nconcatenated with the ring Φ Tand the ”momentum”\nPis the charge Qaccumulated at the surface of the\njunction’s electrodes. The Hamiltonian describing the\nquantum-coherent regime is obtained by quantizing the\ncanonical coordinates which become non-commuting op-\nerators, [ ˆQ,ˆΦT] =iℏ. Here Φ xis the flux of an exter-\nnal bias field concatenated to the ring, Φ 0=h/(2e)≈\n2.068833 ×10−15Wb is a universal constant named flux\nquantum and other circuit parameters are defined in\nFig. 3).\nIn principle, an rf-SQUID biased by an external flux\nΦx≈Φ0/2 may function as a qubit. In this case U(X)\nis a nearly symmetric double-well potential (see Fig. 3c)\nwith a low-energy spectrum consisting of a doublet. The\nqubit Hamiltonian is obtained by truncating the system\nto these two levels. In the spin language, it is written as\nˆH=−ε(Φx)\n2σz−∆\n2σx (9)\nwhere εis the asymmetry of the potential, modulable\nby Φ xand ∆ is a tunneling matrix element between its\nminima. The two eigenvalues of σzcorrespond to states\nlocalized in one of the minima of the potential, the total\nconcatenated flux being Φ T= 0,Φ0. These are the basic\nprinciples of the so-called flux-qubit whose actual design\nis a multijunction version of the rf-SQUID [56].\nClassical noise induces fluctuation of the parameters.\nFor instance, flux noise is described by adding a stochas-tic external flux, Φ x→Φx+˜Φx(t), which induces fluc-\ntuations ˜ ε(t) of the potential asymmetry, as illustrated\nin Fig. 3c. Instead, fluctuations of EJdue for instance\nto the impurities in the Josephson junction induce fluc-\ntuations of the barrier height ˜U0of the potential and in\nturn fluctuations ˜∆(t) of the tunnel splitting. In general,\nthe Hamiltonian acquires a fluctuating term δˆH(t) =\n−1\n2\u0002\n˜ϵ(t)σz+˜∆(t)σx\u0003\nwhere ˜ ϵ(t) and ˜∆(t) are classical\nstochastic processes. Experiments show that all of them\nexhibit ∼1/fbehavior at low frequencies [10, 57]. The\napproach outlined for classical noise may be extended\nto describe phenomenologically quantum noise. In this\ncase, the stochastic processes are substituted by opera-\ntors describing physical fields exerted by the microscopic\nquantum degrees of freedom of one or more physical en-\nvironments [10].\nThe other prominent family of superconducting qubits,\nincluding the transmon [58] which provides the fore-\nfront device for present superconducting quantum pro-\ncessors [59–61], may be introduced following the same\nroadmap. The simplest instance is the so-called charge\nqubit, Fig. 3(d) [62] where the conjugate variables are\nthe discrete charge ˆQinto the superconducting island\nand the cyclic coordinate ˆΦ = ℏˆϕ/(2e) related to the\nquantized phase difference ϕbetween the macroscopic\nwave functions of the electrodes of the Josephson junc-\ntion. Again the Hamiltonian can be written in the form\nEq.(9) where now fluctuations ˜ εare related to charge\nnoise whereas ˜∆(t) originates from fluctuations of EJor\nfrom flux noise.\nThe main effect of low-frequency noise on the dynam-\nics of the qubit is dephasing. For instance, the popula-\ntions of the eigenstates of an isolated qubit with Hamil-\ntonian Eq.(9) undergo coherent oscillations with the an-\ngular frequency given by the qubit energy splitting (we\ntake ℏ= 1) Ω =√\nϵ2+ ∆2. Due to the coupling to the\nenvironment, which induces fluctuations of Ω, the am-\nplitude of coherent oscillations decays as A(t) = e−Γ(t).\nDephasing strongly depends on how the qubit is biased\nor manipulated. This extreme sensitivity is exploited to\nmitigate dephasing or, on the contrary, for quantum sens-\ning of environmental noise. For instance, let’s set the flux\nbias such that ε(Φx) = 0 and consider the effect of fluc-\ntuations ˜∆ (called ”longitudinal”). Then the decay of\noscillations is given by\nΓ(t) =t2\n2Z∞\nt−1\nmd fF0(ft)S∆(f) (10)\nbeing determined by the power spectrum of the fluc-\ntuations of ˜∆(t) ”filtered” by the function F0(x) =\nsin2(πx)/(πx)2. Longitudinal 1 /fnoise may determine\na very strong decay Γ( t)≈1\n2⟨(δ˜∆)2⟩t2where ⟨(δ˜∆)2⟩ ∼Rf2\n1/tmd f C ∆/f≈C∆log(f2tm). This strong decay of co-\nherences determines the complete phase randomization\nin a generic solid-state system at equilibrium. Instead,\n˜εnoise (called ”transverse”) contributes at second order9\nFIG. 3. (a) The rf-SQUID is a superconducting ring with inductance L, and associated energy EL= Φ2\n0/L, interrupted by a\nJosephson junction with critical current Ic, Josephson energy EJ= 2e Ic/ℏand capacitance C; several noise sources may affect\nits dynamics, many of them displaying a 1 /flow-frequency behavior. (b) The equivalent circuit of the rf-SQUID, the flux Φ x\nbeing induced by a current Ixflowing in a nearby circuit. (c) The rf-SQUID potential U(X) Eq.(8): the device is analogous\nto a particle of ”mass” Cmoving in this potential, modulable by varying Φ x. (d) Equivalent circuit of the Cooper-pair box,\nwhere a discrete-valued charge ˆQis stored in a superconducting island (red) connected to the circuit by a Josephson junction\nand modulated by the external voltage Vxvia a gate capacitance Cx.\non the fluctuations of the qubit splitting ∆ thus their\nimpact is more limited resulting in a power-law decay\nΓ(t) =1\n4lnh\n1 +\u0010⟨(δ˜ε)2⟩t\n∆\u00112i\n(11)\nTherefore, to reduce the impact of low-frequency noise,\nan optimal bias strategy has been used, where the qubit\nis biased in such a way that the main source of noise is\ntransverse. This way, the impact of flux (charge) noise\nin flux (Cooper-pair box-based) qubits is minimized.\nIn the last generation of qubits, significant coherence\ntimes are achieved by designing the device in a way that\nprotection from noise is stronger for a larger range of\nparameters. For instance, dynamical sweet spots based\non the application of periodic drives have been identi-\nfied [63], demonstrating reduced susceptibility to low-\nfrequency noise and predicting an enhancement of pure\ndephasing by 3 orders of magnitude in the fluxonium [64].\nFor a recent review on noise-protected superconducting\nquantum circuits, we refer the reader to Ref. Gyenis etal.\n[65]. Here we mention that, in addition to design strate-\ngies, considerable advances in materials science and en-\ngineering materials provided significant noise protection,\nas discussed in Ref. [22].\nIn general, amorphous or glassy materials with plenty\nof TLSs may be disadvantageous in coherent quantum\ndevices. Besides residing in the tunnel barrier material,\nTLSs may influence the dielectric constant of substrates\nas well as insulating coatings between wiring layers. The\ninverse time constants of the two-level states have been\nfound to range from mHz to GHz frequencies. Hence, in\naddition to limiting the quality factors of qubits, TLSs\ncan influence readout resonators for qubits at GHz fre-quencies as well as low-frequency SQUID magnetometers,\nas discussed in the next paragraph.\nDefects in oxides and interfaces also represent the main\nlimitation of gate fidelities in semiconductor-based quan-\ntum dot charge qubits, where they cause voltage fluctu-\nations on the control electrodes. Analogously to super-\nconducting qubits, dephasing due to 1 /f-charge noise is\nminimized by working at sweet spots [66–71]. Spin-based\nquantum dot qubits [72] in semiconductors, are sensitive\nto magnetic field noise due to the surrounding nuclear\nspin bath and to charge noise. Recent progress in this\nfield has been discussed in Refs. Harvey [43], Laucht\netal. [73].\nB. Quantum Sensing\nCoherent nanosystems are at the core of investigation\nalso in the recent field of quantum sensing. For a recent\nreview, we refer the reader to Ref. Degen etal. [74] and\nreferences therein. The idea behind is to turn into an ex-\nploitable property the sensitivity to external disturbances\nrepresenting an intrinsic weakness for quantum comput-\ning purposes. Quantized energy levels, quantum coherent\nevolution or quantum entanglement are used to measure\na physical quantity or to improve sensitivity beyond clas-\nsical measurements. To function as quantum sensors,\nphysical systems need to satisfy the DiVincenzo crite-\nria of efficient initialization and coherent manipulation\n[75]. In addition, they have to interact with the physical\nquantity to be detected V(t) which leads to shifts of the\nquantized energy levels or splittings, E(V). The relevant\nquantity is the transduction parameter γ=∂qE/∂Vq,10\nFIG. 4. (a) In the spin language a pure state of a qubit is a unit vector of the Bloch sphere and the qubit Hamiltonian Eq.(9)\nis represented by a vector ⃗H. Coherent oscillations correspond to a precession of the state vector about ⃗H, whose direction\nand modulus fluctuate due to noise, affecting the qubit dynamics. (b) Coherent oscillations of the qubit populations in an\nisolated system (gray line) are strongly suppressed by longitudinal low-frequency noise (blue line), while the effect of transverse\nnoise (red line) is much weaker. (c) Frequencies of the 1 /fspectrum (grey lines, the thick one for a finite low-frequency cutoff)\nentering dephasing are filtered by the qubit dynamics: the blue curves represent F0(f) entering Eq.(10) at different times, larger\ntyielding smaller bandwidths. The orange, magenta and green curves represent the action of the filter Eq.(12) for periodic\ndynamical decoupling operated by N π-pulses, N= 2 representing spin-echo; the bandwidth migrates to larger frequencies\nsuppressing the effect of low-frequency noise.\nwhere usually q= 1 or q= 2. In any physical im-\nplementation, the sensitivity to external signals carries\nalso coupling with unwanted external noise sources. The\ntrade-off between strong coupling to the quantity to be\ndetected and weak disruptive noise effects are quantified\nby the sensitivity which scales as ∝1/(γ√\nT), where Tis\nthe relevant decoherence or relaxation time [74]. Quan-\ntum sensors require maximal transduction and long de-\ncoherence time. In general, the role of 1 /fnoise in these\nschemes is that of limiting the coherence time which gov-\nerns the measurement time available for quantum sens-\ning. Thus, lowering the amount of 1 /fnoise will increase\nthe sensitivity and consequently also the dynamic range\nof quantum sensing (the largest signal is set by the small-\nest measurement time.)\nThe effect of low-frequency fluctuations can be illus-\ntrated in nitrogen-vacancy (NV) centers in diamond [76],\none of the most advanced platforms for quantum sens-\ning. The use of electron spin states of NV-centers in\ndiamond for highly sensitive magnetometry is an estab-\nlished fact with relevant applications. For an in depth\ndescription of this topic we refer to References Rondin\netal. [77], Schirhagl etal. [78], Rembold etal. [79]. Key\nproperties of these colour centers are the possibility to\noptically initialize, read out and coherently manipulate\nat ambient conditions. The sensing protocol is essen-\ntially the measurement of the Zeeman splitting induced\nby an external magnetic field. For advanced sensing ap-\nplications, the transition frequency is detected via inter-\nferometric techniques. The basis of these techniques is\nRamsey interference, the spin equivalent of an opticalMach–Zehnder interferometer. The qubit is prepared in\nthe superposition |0⟩+eiϕ0|1⟩and Ramsey interference is\nbased on the phase accumulated by the Larmor preces-\nsion during the interrogation time, This phase is related\nto the addressed transition frequency, which depends on\nthe external field to be measured. This allows the de-\ntection of static, slowly varying, or broadband near-DC\nsignals. The main limitation of DC magnetometry with\nNV centers is dephasing due to slowly varying inhomo-\ngeneities of the dipolar fields due to unpolarized spin im-\npurities in the diamond crystal, mostly13C and14N.\nDynamical decoupling (DD) schemes inspired by NMR\n[80], have been used to reduce the effect of low-frequency\nnoise components [10]. These techniques are by now rou-\ntinely employed both as active control tools for increas-\ning the quality factors of quantum gates and in quantum\nsensors. The simplest instance is Hanh’s echo, consisting\nof aπ-pulse applied halfway of the detection in such a\nway as to reverse the qubit evolution so that the phases\naccumulated in the two-time intervals cancel out.\nDD refers to sequences of multiple π-pulses which allow\nshaping of the frequency response of the quantum sensor,\nin particular attaining higher sensitivity when measuring\nAC magnetic fields and longer coherence times for quan-\ntum bits and gates.\nThe DD control protocol is conveniently formulated\nin terms of filter functions entering the qubit coherent\noscillations, analogously to Eqs. 10, 11. For Gaussian\nnoise coupled longitudinally, the filter function takes the11\nform [81]\nFN(f, t) =1\n(2πft)2\f\f1 + (−1)N+1ei2πft\n+ 2NX\nj=1(−1)jei2πtj\f\f2.(12)\nNis the number of pulses applied at time tjwithin\nthe measurement time, t, perpendicularly with respect\nto the qubit quantization axis. A variety of pulse se-\nquences have been investigated, differentiated by pulse\ndirections and timings. The resulting filters effectively\neliminate from the qubit evolution spectral noise compo-\nnents to different degrees (see Fig. 4c). For instance, the\nCarr–Purcell sequence [82], which consists of nequidis-\ntantπpulses, acts as a narrow quasi-monochromatic and\ntunable filter. The first pulse time, t1selects the pass-\nband frequency, while ndetermines the filter width. The\nCP sequences allow the detection of monochromatic AC\nfields with periodicity commensurate with the 2 t1, the in-\nterpulse timing. In NV sensor settings, periodic protocols\nhave been exploited to detect and characterize individual\n13C nuclear spins placed a few nanometers apart from the\nNV center [83, 84].\nDespite the extraordinary progress, the sensitivities\ncurrently achieved are lower than the theoretical limits\nestablished by the qubits’ lifetimes. Material synthesis\ntechniques have been developed to reduce paramagnetic\nimpurities and strain effects in the host crystal. In addi-\ntion, quantum optimal control (QOC) methods have been\napplied to quantum sensors [79, 85, 86]. These techniques\nrepresent an alternative route to enhance the coherence\ntime when noise at high-frequencies dominates.\nNoise Spectroscopy\nNoise spectroscopy is an important tool in quantum\nsensing. The term refers to methods for reconstructing\nthe frequency spectrum of stochastic signals. The most\ncommon tools are based on Ramsey interferometry or\nmore involved rf pulse sequences [74]. Noise spectroscopy\ncan provide relevant information both on the signal to be\ndetected and on the intrinsic noise of the sensor. It can\nbe formulated again in terms of filter functions. In fact,\nin addition to the decoupling regime, there exist spectral\nregions of the filters of positive gain. The effect of the\ncorresponding spectral components of noise is amplified\n[10].\nNoise spectroscopy is routinely used as a powerful tool\nto characterize 1 /fcharge and magnetic flux noise in\nsuperconducting qubits. Here we only mention some\nrecent results in this platform. Exploiting the narrow-\nband filtering properties of the Carr Purcell Meibum\nGill (CPMG) sequence [87], authors of Ref. Bylander\netal. [57] measured flux noise in a persistent current\nqubit away from the optimal point in the spectral re-\ngion 0 .2–20 MHz, where 1 /f0.9noise was detected. Thedecay of Rabi oscillations provided an independent mea-\nsurement of the noise spectrum at frequencies of the or-\nder of Ω Rabi≈MHz. Remarkably, the same power law\nbehavior has been detected by Ramsey interferometry in\nthe much lower frequency range 0 .01−100 Hz in Ref.Yan\netal. [88]. By studying the decay of Rabi oscillations un-\nder strong driving conditions authors of Ref. Yoshihara\netal. [89] inferred flux noise up to 1 .7 GHz. Around\n300 MHz a spectrum close to 1 /fwas detected where ev-\nidence of a Lorentzian component, possibly due to a more\nstrongly coupled fluctuator was pointed out, see Figure\n2. The result of these types of measurements is sketched\nin Figure 1. Most noise spectroscopy protocols have been\ndeveloped tailored to Gaussian noise models. Norris and\nViola in Ref. [90] introduced a noise spectroscopy proto-\ncol allowing for higher-order spectral estimation of non-\nGaussian noise, as demonstrated in a flux-tunable super-\nconducting qubit sensor [91]. This represents a valuable\ntool for characterizing discrete microscopic sources with\na 1/fspectrum. More recently, multilevel noise spec-\ntroscopy has been realized via a spin-locking-based pro-\ntocol [92]. The additional information gained from prob-\ning the higher-excited levels enabled the identification of\ncontributions from different underlying noise mechanisms\nin a flux-tunable transmon qubit. Noise spectroscopy\ntracking multilevel fluctuations represents also an effi-\ncient tool for materials characterization, as it has been\ndemonstrated in a transmon fabricated with tantalum\nmetal [93].\nSilicon quantum dot spin qubits have also been em-\nployed as metrological devices to study the noise envi-\nronment experienced by the qubit. Noise spectroscopy\nhas been done for an implanted phosphorus donor qubit\nin silicon (Si:P) [94] and a SiGe quantum dot spin qubit\n[95]. Charge noise spectroscopy in an isotopically pu-\nrified electron spin qubit in a28Si/SiGe quantum dot\nhas been realized by an independent estimate of S(f)\naround 0 .01–1 Hz by Ramsey interferometry, and around\ntens of kilohertz by CPMG sequence. Remarkably, the\ntwo estimates are consistent with 1 /f1.01spectrum ex-\ntending over seven decades of frequencies [95]. In this\nexperiment, rapid spin rotations are applied by using a\nproximal micromagnet, which induces spin-electric cou-\npling fields. In Ref. Chan etal. [96] the environment\nof a silicon metal–oxide–semiconductor (SiMOS) quan-\ntum dot spin qubit has been characterized via CPMG dy-\nnamical decoupling pulse sequences. The detected charge\nnoise is 1 /fαwith α≈0.8−1 for frequencies between 2\nand 20 kHz. Noise in the energy splitting of an electron\nspin qubit in a highly isotopically purified28Si/SiGe de-\nvice has been detected via Ramsey spectroscopy down to\n10−5Hz and by Hanh-echo at higher frequencies in Ref.\nStruck etal. [97]. Measurements revealed a 1 /f2be-\nhavior below 5 ×10−3Hz followed by 1 /ftrend. These\ndependencies have been found to be consistent over 8\ndecades with charge noise independently detected in the\nsame setup via measurement the current noise of a nearby\nSET sensor.12\nV. CONCLUSION\nIt is widely recognized that the road to avoiding 1 /f\nnoise in nanodevices is to improve the characteristics of\nthe employed materials. Silicon crystal growth technol-\nogy has been perfected in the past, and nowadays ex-\ntremely pure silicon single crystals are available. Similar\nquality of materials would be required for various sub-\nstances that are employed in quantum technology [22].\nOne possible route for achieving high-quality materials\nis to tailor devices atom by atom [98]. The era of such\na “designer matter” approach has recently begun, but\nplenty of work lies ahead before the quality and through-\nput of these materials will become relevant for alleviating\n1/fnoise in mainstream quantum science and technol-\nogy.\nProgress in the more regular “engineered matter” is\nbeing made using composite, hybrid materials, for exam-\nple, encapsulated graphene with superconducting con-\ntacts [99]. Such hybrid junctions have been employed\nin superconducting qubits, but it appears that there is\nstill substantial 1 /fnoise present at the interfaces [100].\nIn addition, different mechanisms may originate critical\ncurrent fluctuations, for instance, charge carrier density\nfluctuations might also play a role [101]. The resistance\nnoise properties of regular AlO xtunnel junctions are still\nsuperior, and the influence of their EJ(or critical current,\nIc) noise on qubit coherence time is small. It will become\nessential when the coherence time will be in the ms range\n[102].\nOne of the emerging, striking fields of 2D materials is\nthe magic angle twisted bilayer graphene (MATBG), in\nwhich superconductivity [103] is assigned to flat energy\nbands [104–106]. The MATBG samples can be tuned be-\ntween superconductivity and Mott insulating states with\na rather small change in the charge carrier density. How-\never, owing to variations in local charge doping or in\ntwist angle, the superconducting properties of these sam-\nples may fluctuate in space or time, and thus display 1 /f\nnoise in critical current. A study of such noise will es-\ntablish the quality of this material for certain sensitive\ndetector applications, for example, their use in kinetic\ninductance detectors for radiation.\n2D materials offer a platform for the miniaturization\nof components for coherent quantum circuits [107, 108].\nLarge, low-loss capacitance per unit area can be achieved\nusing a few layer hBN as an insulating layer. Further-\nmore, by utilizing h-BN as a gate dielectric, strong gate\ncoupling for FETs can be obtained. Compared with\namorphous hafnium oxide, the TLS density of h-BN di-\nelectric is definitely smaller, which promises smaller 1 /f\nnoise for components in such 2D material technology.With the increasing miniaturization of electronic quan-\ntum components, the relative magnitude of 1 /fnoise has\nbeen constantly growing over the years as the active part\nof the component is influenced strongly by one single im-\npurity. Similar extreme purity as for single silicon crys-\ntals would be needed for other technologically relevant\nmaterials in the future [22]. An alternative solution to\nthe 1 /fnoise dilemma is to harness independent single\natoms or molecules [109] as building blocks for quantum\ncomponents and sensors. This has already been achieved\nquite well in quantum sensors based on NV centers in\ndiamond or in optically read magnetometers based on\nrubidium gas for example [110].\nBesides dynamical decoupling schemes, the use of lim-\nited quantum resources can be bolstered by using phase\nestimation algorithms [111]. These algorithms provide a\nsensitivity improvement directly proportional to the mea-\nsurement time, i.e. they allow to achieve the Heisenberg\nlimit in measurement accuracy. Combined with Bayesian\nlearning algorithms, phase estimation schemes may pro-\nvide ultimate speed for high-sensitivity sensing, e.g. mag-\nnetometry [112], in spite of the influence of 1 /fnoise\nembedded in the coherence time of the sensing qubit.\nQuantum Optimal Control is another promising\nmethodology both for sensing applications and for quan-\ntum computation applications which require high-fidelity\ngate operation. One instance where QOC can provide a\nsolution is in reducing the cumulative errors due to hard\npulses (i.e. rectangular pulses) employed with DD se-\nquences.\nFluctuations can provide important additional infor-\nmation in sensing, for example in gas recognition. Low-\nfrequency noise due to adsorbant molecules and atoms\non graphene has been demonstrated to yield distinct\n1/fspectra [113]. In combination with supervised-\nlearning algorithms, based on statistical learning theory,\nfluctuation-enhanced gas recognition has already been in-\nvestigated [114]. 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Email: gyzhang@iphy.ac.cn ; wei.yang@iphy.ac.cn   \n \nAbstract  \nDouble  layer  quantum  system s are promising platform s for realizing novel quantum phases.  Here, w e \nreport a study of quantum oscillations  (QOs) in a weakly coupled double layer system, composed  of a \nlarge angle twisted double bilayer graphene  (TDBG) . We observe two different QOs at low temperature , \none with a periodicity in carrier density ( n), i.e. Shubnikov –de Haas  oscillation  (SdHO)  due to Landau \nquantization, and the other one in displacement field ( D), resulting a grid pattern . We quantify  the \ninterlayer coupling  strength  by measuring  the interlayer capacitance from  the grid pattern with a \ncapacitance model, revealing an electron -hole asymmetry.  At high temperature when SdHO are \nthermal smeared , we observe resistance peaks  when LLs from two minivalleys  in the moiré Brillion \nzone  are aligned , regardless of carrier density ; eventually, it results in  a two-fold increase of oscillating \nfrequency in D, serving as a smoking -gun evidence of  the magneto -intersubband oscillations (MISO) \nin a double layer system . The temperature dependence of MISO suggests electron -electron interaction \nbetween two minivalleys play a crucial rule in the scattering , and the scattering times obtained from \nMISO thermal damping are  found to be  correlated w ith the interlayer coupling strength.  Our study \nreveals an intriguing interplay among Landau quantization, moiré band structure, and scatterings.  \n \n  2 \n Main  Text \nDouble  layer  quantum  systems are ideal  platforms for realizing novel quantum phases1–9, thanks \nto the rich interplay among charge, spin, and layer degree of freedom . Recently, t wisted graphene \nmultilayers have been demonstrated as a promising paradigm  for developing novel states of matter10–\n18. A key ingredient lies in the twist angle, which determines not only the lateral size of moiré \nsuperlattice in real space, i.e. moiré Brillouin zone in reciprocal space, but also the coupling between \nthe layers . At some optimal  angle, strong interlayer coupling will lead to flat moiré bands with the \nhybridized electronic states indistinguishable between the two layers, which favors correlated states19–\n23. At a larger angle , in the so -called weak coupling regime, the moiré  superlattice reduces in size  and \nthe moiré bands are usually dispersive with a reduced fermi velocity, and  importantly  the wave function \nof minivalleys is layer polarized24–28. Essentially, the interlay er coupling defines the electronic \ncorrelation between two layers, and to tune this coupling  is crucial  for realiz ing collective quantum \nphases  in various double layer  systems . The ability to quantify the coupling  in double layer systems  is \nimportant yet rarely explored.   \nHere, we focus on a weak coupled double layer moiré system, which is composed of a large angle \ntwisted double bilayer graphene (TDBG) , by measuring  the quantum oscillations (QOs) due to Landau \nquantization . The QOs, in principle,  are determined by the  combined  band structure  of the two layers , \nand thus could be serve as a probe for the interlayer coupling.  Displacement field  (D) is a powerful \nknob in tuning electronic structure in TDBG. In the strong coupling regime, it helps to reveal  spin \npolarized correlated insulators and valley polarized insulators at different D12–14,29 –31, and even leads \nto insulating QOs in band inversion condition32. In the weak coupling regime, D drives a charge \nredistribution between the two layers, or equivalently between two minivalleys24,26,27. The scatterings \nbetween layers are then reflected on the interminivalley scatterings , as shown in Fig. 1 (b) and 1 (e). For \na twist angle of around 2°, due to the small size of the moiré Brillion zone  (Fig. 1 (c)), scatterings \nbetween minivalleys could be significant25,27. In the following, we will first quantify the interlayer \ncoupling strength from the periodicity of the QOs  at low temperature in a capacitance model  of D-field \ntuned SdHO from two minivalleys , yielding an electron -hole asymmetry. Then , we reveal a two -fold \nincrease of oscillating frequency in D for the QOs at elevated temperature when SdHO are thermal \nsmeared, and we attribute them as the enhanced interminivalley scatterings when the LLs at k and k’ \nminivalleys are aligned. Lastly, we demonstrate a n intima te connection  between interlayer coupling \nstrength and the interminivalley scatterings, by  quantify ing the scattering  lifetime  from  the high \ntemperature oscillat ing magnetoresistance . \nWe fabricate ultraclean TDBG device s by the cut and stack  methods33–35. In the following, w e \nfocus on device D 1 with a twist angle of 2.25 ° that reveals high-quality oscillation data  with low \nresistance of few tens of ohms . The device is in a dual gate geometry, allowing an independent tuning \nof total carrier density n and displacement  field D, i.e. 𝑛=(𝐶𝑡𝑉𝑡+𝐶𝑏𝑉𝑏)/𝑒 and 𝐷/𝜀0=(𝐶𝑡𝑉𝑡−\n𝐶𝑏𝑉𝑏)/2/𝜀0, where 𝐶𝑏 (𝐶𝑡) is the geometrical capacitance per area for bottom (top) gate, e is the 3 \n electron charge , and ε0 is vacuum permittivity . Fig. 2(a) is a color mapping of measured longitudinal \nresistance  Rxx as a function of  n and D/ε0 at B = 2 T. The oscillations due to Landau quantization are \nvividly revealed in grid pattern , indicated by the alternative change s of the color between blue and \nwhite . The oscillation data are plotted in Fig. 2(b), where the  resistance  is minimum at LL filling factor \nvLL = 8N for D/ε0 = 0, and at vLL = 8(N+1/2) for D/ε0 = -0.04 V/nm,  with N a nonzero integer . The data \nis of high quality, with a dip value of ~15 Ω and oscillation amplitude of ~60 Ω at around vLL = -50.  \nvLL = n0/B is the LL filling factor and 0 is magnetic flux quantum.   \nTo begin with, we consider Landau level crossing s of the two minivalleys  that are  approximated \nas two layer -polarized parabolic energy bands (Fig.  2(d)). They are quantized into a series of Landau \nlevels (LLs) at a given B, described by  \nEt/b = (Nt/b + 1/2) ℏ ∓ eV/2.  (1) \nHere t/b corresponds to the top /bottom layer, N is the Landau level index,  ℏ is the reduced Planck \nconstant,   = eB/m* is the cyclotron frequency , and V is the interlayer potential difference  that \ndepends on  D. With the change of V, the two sets of LLs will intersect each other as shown in Fig. \n2(e). By taking m* = 0.08 me and LLs broadening of 0.35  ℏ, the calculated Landau level DOS \nmapping (details are shown in Methods ) is shown in Fig. 2(c). The calculations  agree  well with  the \nexperimental data in Fig. 2(a), indicating a linear relationship between V and D in the moiré valence \nband.   \nNext, we quantitatively analyze the QOs  to reveal the intimate link between V and D. The TDBG  \ndevice  follows  a capacitive model36–38 (Fig. 1(e) and details in supporting information ), and it gives   \nV = (D + en/2)/Cm,  (2) \nwhere Cm is measured interlayer capacitance  and n is the carrier density difference between the top \nand bottom bilayer graphene . The magnitude of interlayer capacitance  Cm reflect s the degree of layer \npolarization39,40, and the larger the Cm the stronger the coupling between  two twisted layers . \nExperimentally , we can extract the  interlayer  capacitance Cm according to the equation  (2). At the \nintersection point of two sets of LLs in Fig. 2 (a) and 2 (c), one could obtain ne/h = ±(Nt – Nb) × 4eB/h \nand Ve/h = ∓(Nt – Nb) × ℏB/m*, where the subscript e/h represent s the electron/hole  and m* is 0.08  \nme for holes and 0.06  me for electrons extracted from the thermal damping  effect  of QOs at D/ε0 = 0. \nThe obtained Cm is plotted against vLL in Fig. 2 (f). For valence band,  Cm ~ 6.3  μF/cm2 is constant, \nregardless of vLL; this suggests the valance band is  a well -defined decoupled system36,40. For \nconduction band, on the contrary, it shows a strong carrier density dependence . Cm reaches up to 19 \nμF/cm2 near the CNP , and it decreases and tends to saturate at a small value of ~7 μF/cm2 at high \ndensity in Fig. 2(c). Such a large capacitance indicat es a strong coupling in the conduction band , and \nits strong doping dependence suggests the interlayer coupling strength is highly sensitive to the change \nof fermi surface . Note that similar results are reproduced in device D2 (Supplementary Fig. 1 6). It is \nalso worth noted that  the observed electron -hole asymmetry in capacitance model match es well with 4 \n our continuum model calculations, where the degree of layer polarization for holes is higher than that \nfor electrons  (Supplementary Fig. 4). \nTo investigate the scatterings in the weakly coupled double layer system,  we measure the QOs at \nhigh temperature . Fig. 3(a) shows a color mapping of longitudinal resistance Rxx (n, D/ε0) at B = 2 T \nand T = 20 K , and it reveals a striped pattern, unlike the previous grid pattern at low temperature. The \ndisappearance of oscillating Rxx(n) at a fixed D indicates that conventional SdHO s are thermal smeared.  \nThe striped patterns depend only by D, irrespective of total carrier density n; in other words, the strip \npatterns are determined only by carrier density difference between top and bottom layer ( k and k’ \nminivalley) . The observation agrees perfectly with resonant condition  when LLs from two minivalleys \nare aligned , and are the main features of the magneto -intersubband oscillations (MISO)24,41 –46. Take \nthe dashed line in Fig. 3(a) for example, comparisons of Rxx(D/ε0) at different T are shown in Fig. 3(c). \nThe transition from SdHO to MISO is captured in the doubling of D-periodicity as T is increased from \n1.8 K to ≥10 K.  Such transition is also vividly  captured in the temperature evolution of the peak width \nand peak amplitude at D/ε0 = 0 in Fig. 3 (d); it decreases rapidly from ~26  mV/nm  at T < 4K to ~16  \nmV/nm at 10  K, and then it barely changes at T ≥10 K.  Note that, by considering the spacing between \npeaks at T ≥10 K as ℏ, a peak width of ~ 16 mV/nm is corresponding to an energy linewidth of ~ 1.85 \nmeV .  In addition, the high temperature MISO are found emanating from D/ε0 = 0 at B = 0 T  in Fig. \n3(b), which resembles the conventional Landau fan diagram when n is varied. The resemblance \nindicates that the carrier density difference between two minivalleys are linearly generated by D. And \nmost importantly , the resistance peak at D/ε0 = 0 indicates additional scattering processes for the \nresonant condition , as shown in the schematics of Fig. 3(e) , which is exactly the predicted \ninterminivalley  scatterings in the MISO model. Similar observations are obtained for electrons in \nSupplementary Note 6 and 7 . \nTo further establish a close connection between the interlayer coupling obtained at low \ntemperature and the interminivalley scatterings at high temperature , we measure the resistance  as a \nfunction of B at high temperature. At a fixed carrier density  of -2.16 × 1012 /cm2 in the valence band, \nthe temperature  (T) dependent QOs are plotted  against 1/ B for D/ε0 = 0 and -0.08 V/nm in Fig. 4(a) \nand 4(b), respectively. For D/ε0 = -0.08 V/nm , it evolves from a beating pattern at low T to a single \nfrequency oscillation at high T. The changes are better  revealed in the fast Fourier transform (FFT) \nspectrum in the inset of Fig. 4(b), where the oscillation frequency (Bf) changes from  three peaks at T \n= 1.8 K to one peak at T = 15 K. By contrast, for D/ε0 = 0 V/nm in Fig. 4(a), Bf ~ 11 T keeps unchanged  \nwhile  its amplitude quickly fades, as T is increased.  Note that t he frequency Bf is linearly correlated \nwith n, defined as Bf = nh/ge. Considering the degeneracy g = 4 for each minivalley, the total \ndegeneracy is 8  at D/ε0 = 0 V/nm and Bf ~11 T is corresponding to n = 2.13 × 1012 /cm2 that matches \nwell with the gate induced carrier density , which is then assigned as SdHO . At nonzero D/ε0 = -0.08 \nV/nm, minivalleys degeneracy is lifted,  and the  three oscillation modes at Bf ~4, 8.2, and 12.2  T in Fig.  \n4(b) are corresponding to n = 0.38 7, 0.79 4, and 1.18 1 × 1012 /cm2, respectively . The two  high f requency  5 \n modes in Fig.  4(b) share the same T dependence as the one in Fig. 4(a), and are attributed to the SdHO \nin two minivalley k and k’. What’s more, the lowest frequency mode at Bf = 4 T is equal to the difference \nBf (k) - Bf (k’); in other words, the slow oscillation mode is determined by minivalley carrier density \ndifference  n. The observation of the  low frequency mode could persist at high T resembles the high \ntemperature QOs in weak coupled TBG, is another characteristic  of MISO24,41 –46 in TDBG .  \nThe details of the scattering lifetime are quantitatively revealed in the thermal damping of the \nQOs. The thermal  damping  of SdHO  is described by Lifshitz -Kosevich ( L-K) formula , i.e. R(T) ~ \n(2π2kBT/ℏ)/sinh(2 π2kBT/ℏ). As shown in Fig. 4(c), the amplitude of SdHO is barely visible  at T = \n10 K, and similar T dependence for B1 and B2 indicate identical effective mass for the minivalleys . By \ncontrast , MISO observed  in Fig.  4(c) are thermal damped  with a  slow  temperature dependence  \ndescribed as exp (-T2) (Fig. 4(c)). In a clean limit (  >>1,  is the total scattering time)41,42,44,47,48with \nthe assumption of identical scattering time for two minivalley , MISO in TDBG could be  described  as: \n = 2( inter) exp(-2π/(q)) cos(2 π(Et-Eb)/ℏ).     (3) \nHere  is the Drude resistivity ;  and inter are the total and interminivalley scattering time, \nrespectively;   exp(-2π/(q)) is the Dingle factor , with q the quantum scattering time. The observed \ndecay of exp (-T2) indicates a scattering rate of  ~ T2, which f urther suggests  an important role played \nby electron -electron interaction44. We attribute the prominent quadratic law to the enhanced Umklapp  \nelectron scatterings in moiré system, which also agrees with the quadratic power law of zero-field \nresistivity at low temperatures  in Supplementary Fig.1 4 when phonon contributions are negligible  in \nTDBG49,50. At a fixed T = 18 K, the log () vs 1/ B gives quantum scattering time  τq ~ 1.04 ps, which \nis of the same order of magnitude compared to the mean free time ; by changing the T, we could obtain \nτq vs T, see Supplementary Fig. 12. It is worth mentioning that the low resistance, tens of ohm in device  \nD1, as well repeatable results in device D2 , enables us to reveal the intrinsic MISO a nd to rule out \nother extrinsic impurities or phonon scatterings mechanisms, distinct from previous report in TDBG . \nIn addition , we can quantitively extract the ratio of  inter by fitting the  high temperature  \nmagnetoresistance curve with the equation  (3). Fig.4 (d) shows the obtained  inter for both the \nelectrons and holes. For holes, the ratio is small, ranging from 0.048 to 0.2 , indicating a small \nintervalley scatting rate  Г ~1/inter; while, for electrons, it is much larger, ranging from ~0.4 to 0.8 , \nindicating a comparable contribution between intravalley and intervalley scatterings . For instance, at \na fixed n = ±2.0 × 1012 /cm2 and T = 18 K (Supplementary Fig. 11 and 1 3), one could obtain a n \nintervalley scatter time of ~120 ps and ~6 ps for holes and electrons, respectively. It is worthy to note \nthat t he obtained   inter also agrees well with the electron -hole asymmetric interlayer coupling from \nCm in Fig. 2 (c), thus establishing an intimate connection between intervalley scatterings and interlayer \ncouplings .  \nIn conclusion, we systematically investigate the interlayer coupling and interminivalley \nscattering s from quantum oscillations in a weakly coupled moiré superlattice  of TDBG . The results \nshow that the valance band is a well -defined decoupled system , whereas  the interlayer coupling 6 \n strength of conduction band highly depends on the doping levels , indicating strong electron -hole \nasymmetry . In addition , we demonstrate strong interminivalley scattering in this system according to \nthe observed MISO for both electron s and hole s. The MISO can be effectively tuned by D, linearly  via \nE(D) ~ D; the temperature dependence of MISO suggests electron -electron interaction s and interlayer \ncouplings play a crucial rule in the scattering . Being a highly tunable  moiré  system, TDBG  provides \nan excellent platform to investigate the interplay among Landau quantization, moiré band structure, \nand scatterings . \n \nMethods \n1. Device fabrication  and characterization . \nThe twisted double bilayer graphene (TDBG) devices are  fabricated by the cut and stack technique33. \nBilayer graphene, few-layer graphite (FLG), and hexagonal boron nitride (h -BN) (15 –40 nm thick) \nwere firstly exfoliated on SiO 2. As shown in Supplementary Fig. 1, the layer number of graphene is \nidentified by the reflection contrast from the optical microscope images51 as well as the Raman \nspectrum52. Then we used an atomic force microscope (AFM) needle to cut a large bilayer graphene \nflake into two pieces. Next, we used poly (bisphenol A carbonate) (PC) supported by \npolydimethylsiloxane (PDMS) on a glass slide to pick up the top h -BN, two pieces of b ilayer graphite \nwith a twist angle of 60° + , bottom h -BN, and bottom graphite gate in turn.  \nThe fabrication of the metal top gate and electrodes followed a standard electron -beam lithography \nprocess and electron -beam metal evaporation. The devices were designed as a Hall bar structure and \nshaped by traditional reactive ion etching with a CHF 3 and O 2 gas mixture . Finally, Cr/Au electrodes \nare evaporated on silicon wafer as 1D edge contacts to connect with Hall bars.  The combination of two \ngate voltages  allows independent tuning of the carrier density n and D, i.e. 𝑛=(𝐶𝑡𝑉𝑡+𝐶𝑏𝑉𝑏)/𝑒 and \n𝐷=(𝐶𝑡𝑉𝑡−𝐶𝑏𝑉𝑏)/2. \nThe transport behaviors are measured at the base temperature (~1.8 K) of our helium -4 cryostat . The \ntwisted angle of the TDBG devices are determined by the carrier density difference between CNP and \ngap opening at full filling, i.e. ns = 4/A ≈ 8θ/(√3a2), where A is the area of a moiré unit cell, θ is \ntwisted angle, and a is the lattice constant of graphene.  \n2. Continuum model calculations.  \nThe band structure of TDBG is calculated by continuum model19,53. In calculation, we use the \nparameters from reference27,53  \n \n(0, 1, 3, 4, wAA, wAB) = (3100,  400, 320, 44, 100,100 ) meV  \n \nHere 0 is the intralayer nearest -neighbor hopping term; 1 is the interlayer hopping term in AB -stacked \nbilayer graphene, from bottom layer A sublattice to top layer B sublattice;  is the interlayer hopping 7 \n term from bottom layer B sublattice to top layer A sublattice;  is the interlayer hopping term from \nbottom layer A(B) sublattice to top layer A(B) sublattice.  and  will bring electron -hole asymmetry \nin twisted double bilayer graphene systems. The electric displacement field changes the on -site energy \nof every layer in twisted graphene systems, described by the following matrix,  \n \nV=\n(     U\n21̂000\n0U\n61̂00\n00−U\n61̂0\n000−U\n21̂)     \n \nwhere U is electrostatic  interlayer  potential energy, 1̂ is a 2×2 unit matrix.  \n3. Landau levels spectrum calculations.  \nThe band structure of k and k’ minivalley is approximately equivalent to two parabolic bands localized \nin top and bottom layers, respectively. Now we focus on the case of hole -type valance band. The \nLandau levels energy spectrum can be described by the following formulas.  \nEtn = (n + 1/2) h - eV0/2         \nEbn = (n + 1/2) h + eV0/2        \n = eB/m* \nNext , we calculate the density of states (DOS) of Landau levels based on the  Gauss distribution by \nconsidering the disorder induced Landau levels broaden.  \n𝐷𝑛(𝐸)=𝑔𝑛𝐿√2\n𝜋𝛤2exp (−2(𝐸−𝐸𝑛)2\n𝛤2)     \nHere 𝛤 corresponds to  the broaden of Landau levels,  and we use a n estimation of 𝛤/ℎ𝑤 = 0.35  to \nfit our experimental data.  𝑔 is the degeneracy of energy band and 𝑛𝐿 is the carrier density to full fill \na Landau level. At a fixed interlayer voltage V, DOS can be expressed as  \n𝐷𝑉(𝐸)=∑(𝐷𝑛𝑢𝑝(𝐸)𝑛 +𝐷𝑛𝑑𝑜𝑤𝑛(𝐸))     \nThe filling factor v is calculated by integrating DOS with E. \n𝑣=𝑛/𝑛𝐿=1\n𝑛𝐿∫𝐷𝑈(𝐸)𝑑𝐸     \nAcknowledgements  \nWe acknowledge supports from the National Key Research and Development Program (Grant No. \n2020YFA0309600), National Science Foundation of China (NSFC, Grant Nos. 12074413 , 61888102, \n11834017), the Strategic Priority Research Program of CAS (Grant Nos. XDB33000000) and the Key -\nArea Research and Development Program of Guangdong Province (Grant No. 2020B0101340001). \nK.W. and T.T. acknowledge support from the JSPS KAKENHI (Grant Numbers 20H00354, 21H05233 \nand 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan.  \n 8 \n Author contributions  \nW.Y . and G.Z. supervised the project. W.Y ., L.L. designed the experiments. Y.Y., L.L. fabricated the \ndevices. Y.L., L.L. performed the magneto -transport measurement. 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(a) shows  the \ntwo Brillouin zone s of top bilayer (TBL , red) and bottom bilayer (BBL , blue ), with a twist angle  () (b) \nis the zoomed -in of the resulted moiré Brillouin zone , with the moiré k (red) and k’ (blue) from top \nand bottom layer , respectively . The curved arrow (purple) denotes  the intervalley scatterings. (c) The \ncalculated m oiré periodicity (black) and the effective mass (red) as a function of twist angle in TDBG.  \n(d) Sideview schematics of the stacked TDBG device structure . (e) Schematics of the interlayer \ncapacitance ( Cm) between the top bilayer (TBL) and bottom bilayer (BBL) in TDBG , with t he curved \narrow (purple) for the interlayer scatterings.  (f) A typical transfer curve of Rxx (n) at T = 1.8  K for \ndevice D1, and a twist  angle  of ~2.25o is obtained from carrier density at full fillings of moiré \nsuperlattice ( ns). The inset shows the Hall bar structure of D1.  \n \n  \n13 \n  \nFIG. 2. Quantifying the interlayer coupling strength from the quantum oscillation spectra in a \ncapacitance model. (a) A color mappings of Rxx(vLL, D/ε0) at T = 1.8 K and  B = 2 T. (b) plots of Rxx at \nD/ε0 = 0 and -0.04 V/nm,  corresponding to the blue and red dotted lines in (d), respectively.  The red \nline is offset by 100 Ω for clarity.  (c) Calculated DOS as a function of Landau level (LL)  filling factor \nvLL and interlayer potential difference V at B = 2 T . (d) shows the schematic of  the parabolic \napproximation for the minivalley  k (red) and k’ (blue)  and the  LLs structure in magnetic field  for DOS \ncalculation, and  (e) is the evolution of the LLs spectr ums of k and k’ as a function of energy E and the \ninterlayer potential difference V at B = 2 T. (f) The measured  Cm versus n, where Cm are extracted \nfrom the intersection pattern in ( a) according to the capacitance model .  \n  \n14 \n  \nFIG. 3. The displacement field dependence of the carrier density insensitive quantum oscillations. (a) \nRxx as a function of  n and D/ε0 at B = 2 T and T = 20 K. (b) Rxx as a function of D/ε0 and B at n = -2.18 \n× 1012 /cm2 and T = 20 K. (c) Rxx versus D/ε0 at T = 1.8  K, 10  K and 20  K. Here B = 2 T, n = -2.18 × \n1012 /cm2 corresponding to the blue dotted line in ( a). (d) ∆D/ε0 versus T, where ∆ D/ε0 is the FWHM \nof the resistance peak at  D/ε0 = 0 in (c). Inset: Amplitude of resistance peaks ∆ Rxx versus  T at D/ε0 = 0 \n(red) and D/ε0 = 0.043 V/nm (blue).  (e). Schematics of the enhanced intervalley scatterings at the \nresonant condition when LLs from k and k’ are aligned, corresponding to the yellow stars in (b).  The \ngreen dashed lines denote the fermi level.  \n  \n15 \n  \nFIG. 4. Temperature dependence of the quantum oscillations. (a) and ( b) are temperature dependent \nRxx(1/B) for D/ε0 = 0 and -0.08 V/nm, respectively, where n is fixed at -2.16 × 1012 /cm2 and the inserts \nare FFT spectrums. (c) Normalized FFT amplitude verse T for the mode B0, B1 and B2 in (b). B0 mode \nis fitted with exp( -T2) decay , while B1 and B2 are fitted with L -K formula. (d) The ratio of  inter \nversus n at T = 18 K, where  inter are extracted  by fitting the high temperature magnetoresistance \ncurve with the equation (3).  \n \n"    },    {        "title": "2401.12008v1.Four_Gluon_Vertex_from_Lattice_QCD.pdf",        "content": "Four Gluon Vertex from Lattice QCD\nManuel Colaço,∗Orlando Oliveira,†and Paulo J. Silva‡\nCFisUC, Departamento de Física, Faculdade de Ciências e Tecnologia,\nUniversidade de Coimbra, 3004-516 Coimbra, Portugal\nA lattice QCD calculation for the four gluon one-particle irreducible Green function in the Landau\ngauge is discussed. Results for some of the associated form factors are reported for kinematical\nconfigurations with a single momentum scale. Our results show that the computation of this Green\nfunction requires large statistical ensembles with 10K or larger number of gauge configurations. The\nsimulations considered herein have a clear Monte Carlo signal for momenta up to ∼1GeV. The\nform factors show an hierarchy, with the form factor associated with the tree level Feynman rule\nbeing dominant and essentially constant for the range of momenta accessed. The remaining form\nfactors seem to increase as the momentum decreases, suggesting that a possible logdivergence may\noccur. The computed form factors are, at least, in qualitative agreement with the results obtained\nwith continuum approaches to this vertex, when available.\nContents\nI. Introduction 1\nII. The Four Gluon Vertex 2\nA. An orthogonal set of operators 6\nB. Full Green function and lattice form factors 6\nC. Lattice form factors F(i)and 1-PI form factors 7\nIII. Lattice setup and choice of momenta 7\nIV. Lattice form factors 8\nV. Summary and Conclusions 11\nAcknowledgments 14\nA. Three gluon contributions for proportional momenta 14\nB. Implementation of color-Lorentz projectors to measure F(i)14\nReferences 17\nI. INTRODUCTION\nThedynamicalcontentoftheQCDGreenfunctionsisofupmostimportancetoallhadronicphysics,speciallyitsnon-\nperturbative contents. Indeed, among other phenomena, the interpretation of the hadron spectra, the comprehension\nof dynamical chiral symmetry breaking and of the quark and gluon confinement mechanisms are outside the scope of\nthe perturbative solution of the theory.\nThe analysis of the QCD Green functions started looking at those functions with smaller number of external legs,\nnamely the propagator or two point correlation functions. Their tensorial structure is simpler, as it requires smaller\ntensor basis, and can be guided with the help of the Slavnov-Taylor identities. All the two point QCD correlation\nfunctionshavebeenstudiedindetailwithnon-perturbativemethodsandafairlygoodunderstandingofthesefunctions\nhas been achieved [1–7]. A good and coherent picture of the two point QCD correlation functions, obtained with\n∗Electronic address: manuel.sc.colaco@gmail.com\n†Electronic address: orlando@uc.pt\n‡Electronic address: psilva@uc.ptarXiv:2401.12008v1  [hep-lat]  22 Jan 20242\ndifferent non-perturbative methods, has been achieved but there are issues that are still under debate. An example\nis their analytical structure, required in the computation of meson and baryon properties with continuum methods,\nthat remains to be understood [8–12].\nIn QCD the three point one-particle irreducible (1-PI) Green functions have also been object of study. Indeed, the\nghost-gluon vertex [13–16], the three gluon vertex [17–35] and the quark-gluon vertex [36–58] have been investigated\nwith some degree of detail and its main features are now understood. If for some of the vertices a qualitative picture\nof their dependence for different kinematics can be claimed, a complete and accurate description is still to be done.\nThe next level of QCD one-particle irreducible Green functions to be studied with non-perturbative methods are\nthose with four external legs. These include the four quark and the four gluon correlation functions. The four quark\nGreen function is required to understand, for example, the meson spectra. The four gluon correlation function has\na similar role as the four quark function but for the glueball spectra. Moreover, it also allows for the definition of a\nrenormalization group invariant QCD charge, it contributes to the Dyson-Schwinger equation for the gluon propagator\nand impacts on the gluon propagator itself [59]. Herein, our focus of attention is the four gluon one-particle irreducible\nGreen function.\nThenon-perturbativecontentofthefourgluon1-PIGreenfunctionhasbeenstudiedwiththecontinuumformulation\nof QCD using the Dyson-Schwinger equations [60–62]. A full description of this four leg function requires a tensor\nbasis with a large number of elements [61, 63, 64], that makes the analysis of this four point correlation function\ndifficult. Continuum studies of this function have to make simplifications to arrive at a manageable computational\nscenario and the calculations performed have to consider a basis with a reduced number of elements. The continuum\nbased Dyson-Schwinger calculations identify a number of form factors that display very mild dependence with the\nmomentum, as seems to be the case of the form factor associated with the tree level tensor structure that, nevertheless,\nseems to be suppressed at low momenta, and others that show logdivergences in the IR region. These divergences are\nassociated with the unprotected log’s coming from the massless ghost loops in the Dyson-Schwinger equations [61].\nThe continuum QCD methods based on the Dyson-Schwinger equation for this 1-PI Green function seem to favour\nsuch type of scenario.\nThe picture that emerges from the various continuum approaches to the four gluon 1-PI Green function, although\nbeing consistent at the qualitative level, also show differences that probably come from the different truncations\nand/or the required modelling to solve the underlying equations. The differences between the various studies provide\nan extra motivation to a lattice calculation of the four gluon vertex. Indeed, the interplay of the various non-\nperturbative methods is important to achieve a proper and sufficiently accurate description of the QCD one-particle\nirreducible Green function.\nIn this work we address the computation of the four gluon one-particle irreducible Green function within pure\nYang-Mills lattice QCD simulations in the Landau gauge. Preliminary results can be found in [65]. The manuscript\nis organized as follows. In Sec. II it is given a description of the four gluon full Green function, accessed in the lattice\nQCD simulations, and how it relates to the corresponding 1-PI function. Furthermore, details on the extraction of\nthe 1-PI contribution in a simulation in the Landau gauge are also discussed there, together with the tensor basis\nthat will be used for the calculation. The definitions of the form factors to be measured can also be found in this\nsection. The lattice setup, the choice of the kinematic configurations to be used in the measurements and some details\nof the lattice calculation are given in Sec. III. The lattice form factors, including those associated with the amputated\nGreen functions, are reported and discussed in Sec. IV. Finally in Sec. V we summarize and conclude. Some of the\ndetails of the calculation, namely a proof of the decoupling of the three gluon contribution to the four point function\nfor a certain class of kinematical configuration and the lattice version of the continuum operators, are given in the\nappendices.\nII. THE FOUR GLUON VERTEX\nLet us start by discussing the computation of the four gluon 1-PI Green function using the continuum formulation\nof QCD in Minkowski spacetime. By doing so we aim to arrive at a better understanding of this complex Green\nfunction before plugging into the measurement of the Green function with lattice QCD simulations.\nIn the numerical Monte Carlo simulations of QCD on a lattice only the full Green functions are accessed. In\nreal space, these Green functions are defined as the vacuum expectation values of the time order product of the\nfundamental fields. In QCD, the gluon N-point full Green function reads\nGa1···aNµ1···µN(x1,···, xN) =⟨Aa1\nµ1(x1)Aa2\nµ2(x2)···AaNµN(xN)⟩, (1)\nwhere ⟨···⟩represent the vaccuum expectation value, realised as an ensemble average over gauge configurations\nsampled with an appropriate action. Any other quantity, like form factors, have to be computed a posteriori by\nhandling a set of full Green functions Ga1···aNµ1···µN. The computation of the 1-PI four gluon Green function is no exception3\nFIG. 1: Diagramatic representation of the four gluon full Green function defined in Eq (4). The filled vertex in the left is the\ncomplete four point Green function, the blob filled in back is the connected Green function and the blobs in grey stand for one\nparticle-irreducible diagrams. Color, Lorentz and momentum indices are omitted.\nand the first step to perform is to decompose the 4-point full Green function in terms of QCD 1-PI Green functions.\nFor pure Yang-Mills theory the functional generator of the full Green functions Z[J]is\nZ[J] =ei W[J]=Z\nDA ei S[A]+i(J,A), (2)\nwhere S[A]is the effective action associated with the Yang-Mills theory, that has contributions coming from the gauge\nand ghost fields,\n(J, A) =Z\nd4x Aa\nµ(x)Ja µ(x), (3)\nW[J]is the generating functional of the connected Green functions and Jare classical external sources. It follows\nfrom the definition of the full Green function that the four gluon full Green function is given by\nGa1···a4\nµ1···µ4(x1,···, x4) =δ4Z[J]\ni δJa1µ1(x1)···i δJa4µ4(x4)\f\f\f\f\nJ=0\n=iδ4W[J]\nδJa1µ1(x1)···δJa4µ4(x4)\f\f\f\f\nJ=0\n−δ2W[J]\nδJa1µ1(x1)δJa2µ2(x2)\f\f\f\f\nJ=0δ2W[J]\nδJa3µ3(x3)δJa4µ4(x4)\f\f\f\f\nJ=0\n−δ2W[J]\nδJa1µ1(x1)δJa3µ3(x3)\f\f\f\f\nJ=0δ2W[J]\nδJa2µ2(x2)δJa4µ4(x4)\f\f\f\f\nJ=0\n−δ2W[J]\nδJa1µ1(x1)δJa4µ4(x4)\f\f\f\f\nJ=0δ2W[J]\nδJa2µ2(x2)δJa3µ3(x3)\f\f\f\f\nJ=0. (4)\nThe diagrammatic representation of Ga1···a4µ1···µ4(x1,···, x4)in terms of connected Green functions is given in Fig. 1. The\nfour gluon full Green function is a sum of a four-gluon connected Green function, represented by the full black blob\nin Fig. 1, and disconnected parts that are functions of the two gluon Green function, i.e. of the gluon propagator,\nthe full blobs in grey. The four gluon connected Green function itself can be written in terms of 1-PI functions. In\norder to workout this decomposition let us consider the classical Yang-Mills field given by\nAa\ncl,µ(x) =δW[J]\nδJa µ(x)(5)\nand introduce the generating functional for the one-particle irreducible correlation functions\nΓ[Acl] =W[J]−(J, Acl). (6)\nThe one-particle irreducible Green functions are the functional derivatives of the Γ[Acl]at vanishing classical fields.\nIt follows from the above definitions that\nδΓ\nδAa\ncl,µ(x)=−Ja µ(x) (7)\nand, from this relation combined with Eqs (5) and (6), the orthogonality relation\nZ\nd4zδ2W[J]\nδJaµ(x)δJcν(z)δ2Γ[Acl]\nδAc\nclν(z)δAb\nclζ(y)=−δabgµζδ(x−y) (8)4\nFIG. 2: Diagramatic representation of the four gluon connected Green function (full black blob) appearing in Fig. 1 in terms\nof one particle-irreducible functions (full grey blobs).\nis derived. By taking functional derivatives of Eq (7), using the orthogonality relation (8), and after some straight-\nforward algebra, one arrives at the equality\nδ4W[J]\nδJaµ(x)δJbν(y)δJcι(z)δJe\nζ(t)\f\f\f\f\f\nJ=0=Z\nd4w1d4w2d4w3(\nDbb′\nνν′(y−w1)Dee′\nζζ′(t−w2)δ3W\nδJaµ(x)δJcι(z)δJc′\nι′(w3)δ3Γ\nδAb′\ncl ν′(w1)δAc′\ncl ι′(w3)δAe′\ncl ζ′(w2)\n+Dcc′\nιι′(z−w1)Dee′\nζζ′(t−w2)δ3W\nδJaµ(x)δJbν(y)δJb′\nν′(w3)δ3Γ\nδAb′\ncl ν′(w3)δAc′\ncl ι′(w1)δAe′\ncl ζ′(w2)\n+Dcc′\nιι′(z−w1)Dee′\nνν′(y−w2)δ3W\nδJaµ(x)δJb\nζ(t)δJb′\nζ′(w3)δ3Γ\nδAb′\ncl ζ′(w3)δAc′\ncl ι′(w1)δAe′\ncl ν′(w2))\n+Z\nd4w1d4w2d4w3d4w4Daa′\nµµ′(x−w1)Dbb′\nνν′(y−w2)Dcc′\nιι′(z−w3)Dee′\nζζ′(t−w4)×\n×δ4Γ\nδAa′\ncl µ′(w1)δAb′\ncl ν′(w2)δAc′\ncl ι′(w3)δAe′\ncl ζ′(w4)(9)\nwhose diagrammatic representation is given in Fig. 2. The connected four gluon Green function is a sum of terms that\ninclude the four gluon 1-PI Green function and contributions proportional to the three gluon 1-PI Green functions.\nOn a lattice simulation it is the full Green function Ga1···a4µ1···µ4, see Eq. (4) and Fig. 1, that is measured. Its decompo-\nsition in terms of connected Green functions calls for the four-gluon connected Green function and disconnected terms\nthat are proportional to products of gluon propagators. In the lattice formulation of QCD, the Green functions are\nproportional to the lattice volume and, therefore, the terms represented in Fig. 1 are such that the four gluon con-\nnected part is proportional to the lattice volume V, while the terms proportional to the propagators are proportional\ntoV2. The lattice volume dependence of the terms that define Ga1···a4µ1···µ4make it almost impossible to have a good\nMonte Carlo signal for the connected four gluon diagram, unless the contribution of the disconnected parts vanish.\nThis can be achieved considering only those kinematical configurations such that the momenta piandpjassociated,\nrespectively, with the external legs iandjare such that pi+pj̸= 0for all iandj. From now on, it will be assumed\nthat these conditions on the momenta of the external legs are fulfilled and, therefore, the contributions due to the\ndisconnected diagrams can be discarded.\nThe four gluon connected Green function, represented diagrammatically in Fig. 2, is given by a term that includes\nthe four gluon 1-PI Green function and diagrams that include three gluon 1-PI Green functions and gluon propagators.\nThen, to access the four gluon 1-PI Green function from the connected four gluon Green function, the three gluon\ncontributionsneedstobeknown. However, suchinformationonthethreegluoncontributionsiscurrentlynotavailable.\nOne can either introduce some modelling for the three gluon 1-PI Green function, in such a way that its contribution\ncan be removed, or further constrain the kinematical configuration of the external legs to suppress the contributions\nthat call for three-gluon one-particle irreducible diagrams. For example, as discussed in [61], this is case when the\nincoming momenta are p1=p,p2=p,p3=pandp4=−3p. A first and preliminary analysis of this kinematical\nconfiguration using lattice QCD simulations can be found in [66].5\nIngeneral, fortheLandaugauge, duetotheorthogonalityofthegluonpropagator, forthekinematicalconfigurations\nwhere the momenta of the external legs are all proportional, i.e. for the class of momenta such that\np1=p, p 2=η p, p 3=λ p, p 4=−(1 +η+λ)p , (10)\nwhere ηandλare real numbers, the terms in the decomposition of the four gluon connected Green function that\nrequire the contribution of the three-gluon vertex vanish. The proof of this statement can be found in App. A. Herein,\nin order to avoid any type of extra modelling, we choose to consider a subset of the kinematical configurations that\nbelong to the class of momenta defined in Eq. (10). Although the modelling of the three gluon 1-PI is avoided, our\nchoice based on the kinematics constraint the type of information that can be accessed in the simulation.\nThe choice of momenta is important to access the four gluon 1-PI Green function without extra assumptions.\nHowever, the color-Lorentz structure of this Green function is complex. For a general kinematical configuration, the\nnumber of tensors required to define a tensor basis to fully describe the four gluon 1-PI is large. For example, for\nthe symmetric point, the four gluon vertex requires more than one hundred different tensors for its full description\n[63]. General discussions of the tensor basis for this Green function can be found in [61, 64]. Ideally, one would\naccess each of the form factors that multiply each tensor of the complete basis for the Green function with the help\nof a suitable projection operator. However, not only the lattice simulations convolute the 1-PI functions with the\ngluon propagators but also the quality of the signal-to-noise ratio in a Monte Carlo simulation favor certain types of\nprojecting operators. In principle, the signal-to-noise ratio can always be improved by generating larger ensembles of\ngauge configurations, but a practical computation has always a limited statistical ensemble.\nAcompletetensorbasisdescribingthefourgluon1-PIincludesseveraltypesofoperatorsthatareproportionaltothe\nmetric tensor and/or momenta. However, given that in a lattice simulation the measured function convolutes the four\ngluon 1-PI with gluon propagators, for the class of kinematical configurations considered here that is characterized\nby a single momentum, see Eq. (10), due to the orthogonality of the gluon propagator in the Landau gauge, the\ntensors proportional to momentum do not contribute to the full Green function that is measured in a simulation.\nThis property simplifies considerably the tensor analysis the Green function and the tensors that contribute can only\ninclude the metric tensor for its Lorentz component and fabcordabc, the fully anti-symmetric and fully symmetric\nstructure constants, and/or δabin the description of the color part of the Green function. Then, by imposing Bose\nsymmetry it is possible to build the possible tensor structures that, for the chosen kinematics, contribute to the Green\nmeasured on a simulation. A particular tensor that belongs to the class of allowed operators is the tensor structure\nthat appears in the perturbative tree level four gluon Feynman rule\neΓ(0)abcd\nµνηζ=fabrfcdr(gµηgνζ−gµζgνη) +facrfbdr(gµνgηζ−gµζgνη) +fadrfbcr(gµνgηζ−gµηgνζ).(11)\nFrom this tensor another operator that also contributes to the lattice Green function can be build replacing the fully\nanti-symmetric SU(3) structure constants fabcby the fully symmetric dabcand symmetrizing the Lorentz structure\npart. The corresponding tensor reads\neΓ(1)abcd\nµνηζ=dabrdcdr(gµηgνζ+gµζgνη) +dacrdbdr(gµζgνη+gµνgηζ) +dadrdbcr(gµνgηζ+gµηgνζ),(12)\nBesides the two above tensors, one may also consider the tensor operator\neΓ(2)abcd\nµνηζ=\u0010\nδabδcd+δacδbd+δadδbc\u0011 \u0010\ngµνgηζ+gµηgνζ+gµζgνη\u0011\n(13)\nthat was mentioned in [61]. The three above tensors are not orthogonal to each other and their color-Lorentz scalars\nare\neΓ(0)·eΓ(0)= 108 N2\u0010\nN2−1\u0011\n= 7776 , (14)\neΓ(0)·eΓ(1)=−36\u0010\nN4−5N2+ 4\u0011\n=−1440 , (15)\neΓ(0)·eΓ(2)= 0 (16)\neΓ(1)·eΓ(1)= 12\u0012\n17N4−209N2−496\nN2+ 688\u0013\n=4640\n3, (17)\neΓ(1)·eΓ(2)=288\u0000\nN4−5N2+ 4\u0001\nN= 3840 , (18)\neΓ(2)·eΓ(2)= 216\u0010\nN4−1\u0011\n= 17280 , (19)\nwhere the numbers in the r.h.s. refer to the particular case where N= 3. In general, the above set of tensors do\nnot define a complete tensor basis for the Green function that is measured on a simulation and others should be\nconsidered. However, in the current work only their contribution will be measured.6\nA. An orthogonal set of operators\nAlthough the operators eΓ(0),eΓ(1)andeΓ(2)are not orthogonal in the color-Lorentz space, by a linear combination\nof these operators it is possible to build three orthogonal operators from eΓ(i). Indeed, a straightforward calculation\nshows that the tensor operators\nt(1)abcd\nµνηζ =eΓ(0)abcd\nµνηζ, (20)\nt(2)abcd\nµνηζ =N2−4\n3N2eΓ(0)abcd\nµνηζ +eΓ(1)abcd\nµνηζ−4\u0000\nN2−4\u0001\n3N(N2+ 1)eΓ(2)abcd\nµνηζ\n=5\n27eΓ(0)abcd\nµνηζ +eΓ(1)abcd\nµνηζ−2\n9eΓ(2)abcd\nµνηζ, (21)\nt(3)abcd\nµνηζ =eΓ(2)abcd\nµνηζ (22)\nare orthogonal in the color-Lorentz space and, therefore,\nt(i)·t(j)=Njδij, (23)\nwhere the normalization factors Njread\nN1= 108 N2\u0010\nN2−1\u0011\n= 7776 , (24)\nN2=1280\n3, (25)\nN3= 216\u0010\nN4−1\u0011\n= 17280 ; (26)\n(27)\nonce more the numbers in the r.h.s. are for N= 3. For a general kinematics, the three tensors t(1),t(2)andt(3)do\nnot define a complete basis neither for the full Green function nor for the one-particle function. The exception being\nthe configuration of momenta such that p1=p2=p3=pandp4=−3pwhere t(1),t(2)andt(3)describe completely\nthe correlation function [61].\nB. Full Green function and lattice form factors\nFor the special kinematics under discussion herein, the full Green functions measured in lattice simulations gets\nonly contributions from the connected four gluon full Green function. In the Landau gauge, the full Green function\ncan be written as\nGabcd\nµνηζ(p1, p2, p3, p4) =\u0010\nP⊥(p1)\u0011\nµµ′\u0010\nP⊥(p2)\u0011\nνν′\u0010\nP⊥(p3)\u0011\nηη′\u0010\nP⊥(p4)\u0011\nζζ′D(p2\n1)D(p2\n2)D(p2\n3)D(p2\n4)\n \nF(p2\n1, . . .)t(1)abcd\nµ′ν′η′ζ′+G(p2\n1, . . .)t(2)abcd\nµ′ν′η′ζ′+H(p2\n1, . . .)t(3)abcd\nµ′ν′η′ζ′+···!\n(28)\nwhere F(p2\n1, . . .),G(p2\n1, . . .),H(p2\n1, . . .), etc, are Lorentz scalar form factors, ···represent the contribution of the\nremaining components of the tensor basis, supposedly but not necessarilly orthogonal to the space spanned by t(1)to\nt(3), and\n\u0010\nP⊥(p)\u0011\nµν=gµν−pµpν\np2(29)\nis the orthogonal projector in momentum space that appears in the definition of the Landau gauge gluon propagator\nDab\nµν(p) =δab\u0010\nP⊥(p)\u0011\nµνD(p2). (30)\nThe 1-PI form factors to be measured from the full Green function are\nF(0)=−eΓ(0)abcd\nµνηζGabcd\nµ′ν′η′ζ′(p1, p2, p3, p4)gµµ′gνν′gηη′gζζ′, (31)\nF(1)=−eΓ(1)abcd\nµνηζGabcd\nµ′ν′η′ζ′(p1, p2, p3, p4)gµµ′gνν′gηη′gζζ′, (32)\nF(2)=−eΓ(2)abcd\nµνηζGabcd\nµ′ν′η′ζ′(p1, p2, p3, p4)gµµ′gνν′gηη′gζζ′. (33)7\nIn general, the F(i)are linear combinations of the form factors F,G,H···that describe the 1-PI four gluon function.\nIt follows from the definitions (31) – (33) that their r.h.s. are symmetric under interchange of any pair of momenta\nand Bose symmetry demands that the F(i)can only depend on the momenta through the combinations\nF(i)≡F(i)\u0010\np2\n1+p2\n2+p2\n3+p2\n4,(p1p2) + (p1p3) + (p1p4) + (p2p3) + (p2p4) + (p3p4)\u0011\n. (34)\nFor the kinematical configurations investigated in the current work, where all momenta are proportional to each other\nand there is a unique momentum scale, one can write that F(i)≡F(i)(p2)to simplify the notation.\nC. Lattice form factors F(i)and 1-PI form factors\nOur primer concern in this work is to compute the F(i). These functions are not the form factors that describe the\n1-PI four gluon vertex, see the definition in Eq. (28), but are given by linear combinations of F,G,H,. . .Assuming\nthat the functions F,GandHgive the dominant contributions to the form factors lattice form factors F(i), then for\nthe kinematical configurations (p1, p2, p3, p4) = (0 , p, p, −2p)and(p1, p2, p3, p4) = (0 , p, 2p,−3p), a\nstraightforward calculation gives\nF(0)(p2) = 7776 F(p2), (35)\nF(1)(p2) =−1440F(p2) +1280\n3G(p2) + 3840 H(p2), (36)\nF(2)(p2) = 17280 H(p2) (37)\nwhile for (p1, p2, p3, p4) = (p, p, p, −3p)one has\nF(0)(p2) = 3888 F(p2), (38)\nF(1)(p2) =−720F(p2) +640\n3G(p2) + 2400 H(p2), (39)\nF(2)(p2) = 10800 H(p2). (40)\nIfFandHare directly related to F(0)andF(2), the form factor G(p2)can be accessed by the linear combination of\nall the lattice form factors\n5\n27F(0)(p2) +F(1)(p2)−2\n9F(2)(p2) =\n\n1280\n3G(p2),all kinematics except (p1, p2, p3, p4) = (p, p, p, −3p)\n640\n3G(p2),for(p1, p2, p3, p4) = (p, p, p, −3p).\n(41)\nWe close this section by recalling the reader that for the kinematics (p1, p 2, p 3, p 4) = ( p, p, p, −3p)the three\ntensor structures considered define a complete basis [61].\nIII. LATTICE SETUP AND CHOICE OF MOMENTA\nThe main goal of this work is to measure the form factors that describe the four gluon one-particle irreducible\ncorrelation function using lattice simulations. The results described below refer to simulations that use the Wilson\naction for β= 6.0. For this bare coupling constant the corresponding lattice spacing, measured from the string\ntension, is a= 0.1016(25) fm or, equivalently, 1/a= 1.943(48)GeV. In the conversion from lattice to physical units\nwe use the central values just quoted1. The computer simulations were performed using the Chroma [68] and PFFT\n[69] libraries and run on the Navigator supercomputer at the University of Coimbra.\nThe form factors F(i)defined in Eqs (31) to (33) were computed with several ensembles of gauge configurations.\nOur first try was to consider the ensembles used in [5, 70, 71] that have 2000 gauge configurations for the 644\nlattice and 1801 gauge configurations for the 804lattice, both rotated to the Landau gauge2. For these ensembles\n1Note that there is an uncertainty of ∼2.5% in the scale setting towards physical momentum. However, in the following, the uncertainty\nassociated with 1/awill be ignored and only the central value will be used. See [7, 67] for discussions.\n2The details on the sampling and the rotation of the gauge configurations towards the Landau gauge can be found in the above cited\nworks.8\nof configurations the signal-to-noise ratios are quite poor, preventing from achieving any reliable measurement for\ntheF(i). To overcome the problem of the signal-to-noise ratio we have generated 9038 gauge configurations, in the\nLandau gauge, for a 324lattice with the same βvalue, together with 4560 gauge configurations for a 484lattice and\nβ= 6.0. Our choice for βrelies on the study of the gluon propagator [5] that suggests that for this βthe lattice\nspacing effects are under control with possible mild volume effects in the IR that show up only in the smallest lattice.\nFrom now on, we will refer only to the measurements with these two latter lattices.\nOur choice of momenta to measure the form factors F(i)considers only a subset of the class of momenta defined\nin (10). The subset of momenta was chosen with the aim to minimize the effects due to the breaking of rotational\nsymmetry and our choice was to use momenta pias close to each other as possible. The momentum configurations\nconsidered are\np1= 0, p 2=p, p 3=p, p 4=−2p referred as (0, p, p, −2p),\np1= 0, p 2=p, p 3= 2p, p 4=−3p referred as (0, p, 2p,−3p),\np1=p, p 2=p, p 3=p, p 4=−3p referred as (p, p, p, −3p).(42)\nIn a simulation that uses hypercubic lattices, the breaking of rotational symmetry implies that the measured form\nfactors F(i)are not only functions of p2but also of the H4 invariants p[n][24, 72–74]. If for the gluon propagator the\nbuilding of these invariants and the evaluation of the effects due to the breaking of rotational symmetry is relatively\nstraightforward, a similar analysis of Green functions with higher number of external legs becomes rather involved.\nFor the three point functions, the effects due to the breaking of rotational symmetry were discussed in [24] and for\nan hypercubic lattice, the breaking of rotational symmetry is minimized when one of the momenta is proportional to\n(1,1,1,1). Similar conclusions have been observed for the two point functions [73–75] At least for the two point and\nthree point gluon functions it was observed that the replacement of the naive momentum by an improved momentum,\nsee Eq. (43) in Sec. IV, seem to handle the effects due to the breaking of rotation symmetry.\nFor the kinematical configurations under consideration, the form factors F(i)are functions of a single momentum\nscale. In order to improve the signal-to-noise ratio, given that the form factors are functions of p2, in all cases\nwe average over equivalent momenta, including negative momenta, before performing the ensemble averages. For\nexample, for each gauge configuration the form factors associated with the momenta\n( 1,1,1,1 ) ( −1,1,1,1 ) ( −1,−1,1,1 ) ( −1,−1,−1,1 ) ( −1,−1,−1,−1 )\n( 1 ,−1,1,1 ) ( −1,1,−1,1 ) ( −1,−1,1,−1 )\n( 1 ,1,−1,1 ) ( 1 ,−1,−1,1 ) ( −1,1,−1,−1 )\n( 1 ,1,1,−1 ) ( −1,1,1,−1 ) ( 1 ,−1,−1,−1 )\n( 1 ,−1,1,−1 )\n( 1 ,1,−1,−1 )\nare averaged before performing the ensemble averages. By doing so we are assuming that the dependence of the form\nfactors on the H4 invariants p[n]is small or negligible.\nIV. LATTICE FORM FACTORS\nThe continuum form factors F(i)defined in Eqs (31) to (33) were measured in the simulations using their lattice\nimplementations as given in App. B. For the ensembles referred previously, it turns out that only the simulations using\nthe324lattice have a good signal-to-noise ratio on a range of momenta. On the other hand, the results coming from\nthe simulation with the 484lattice show a good signal-to-noise ratio only for F(0)but not for the amputated functions.\nAnyway, despite the relative large errors associated with this latter lattice, the outcome of the simulation confirms\nthe tendency observed in the 324results and the two sets of data are compatible within one standard deviation. In\nall cases, the statistical errors were computed with the bootstrap method with a confidence level of 67.5%. The bare\nlattice data for the form factors will be described in terms of the improved lattice momenta\npµ= (2/a) sin( π nµ/L)with nµ=−L/2,−L/2 + 1 ,···,0,1,···, L/2−1, (43)\nwhere Lis the length of the lattice. At least for the gluon propagator the use of the improved momenta handles part\nof the lattice effects.\nThe bare form factors F(0),F(1)andF(2)associated with the full Green functions, i.e. that incorporate the\ncontributions of the gluon propagators, are shown in Fig. 4 for all the kinematics and for all the momenta accessed\nin the simulation. The black circles are the data from simulations using the 324lattice volume, while the red squares\nis the data measured on the 484lattice volume. Note the different vertical scales that are associated with each of the9\nFIG. 3: Bare gluon propagator in the Landau gauge using the data from the simulations with the 324and484lattices. The\ndata reported is a subset of all the lattice momenta. Indeed, to handle the effects associated with the breaking of rotational\nsymmetry we follow the recipe of [75] and perform the so-called cylindrical and conical cuts in momentum space, while for\np <0.7GeV all data is included.\nF(i). Their order of magnitude can be understood looking at the bare gluon propagator function D(p2), see Fig. 3,\nthat is included in the data of Fig. 4.\nThe data reported shows there is a range of momenta where there is a clear sign for all the form factors that goes\nup to p∼1GeV. For higher momenta the statistical errors prevent the access to the behaviour of the various F(i).\nReduced errors can be accessed by using larger statistical ensembles of gauge configurations. However, the data also\nshows that the measurement of the four gluon Green functions is possible for gauge ensembles with, at least, about\n10k configurations.\nLet us now consider the same form factors but associated with the amputated Green functions. The form factors\ndescribing the 1-PI Green function, i.e. the amputated Green function, can be computed from the F(i)dividing the\nlater functions by the gluon propagators that are associated with each of the external legs of the full Green function.\nFor the size of each gauge ensemble, the statistical errors associated with the two point function are tiny, see the\nresults of Fig. 3, and will be ignored when performing the division. Then, the statistical errors for the amputated\nGreen function form factors come only from the statistical errors on the F(i).\nOn a lattice simulation the (bare) computed functions relevant for our purpose are, using now a simplified notation,\na4⟨A(1)···A(4)⟩for the four point function and a2⟨A(1)A(2)⟩for the propagator. From the division of the bare F(i)\nby the bare the gluon propagator, the would be amputated Green functions are\na4⟨A(1)···A(4)⟩\n\u0010\na2⟨A(1)A(2)⟩\u00114=⟨A(1)···A(4)⟩\na4\u0010\n⟨A(1)A(2)⟩\u00114. (44)\nTo arrive at the corresponding dimensionless amputated Green functions, these ratios have to be multiplied by a4.\nThen, the bare lattice amputated Green function reads\na4a4⟨A(1)···A(4)⟩\n\u0010\na2⟨A(1)A(2)⟩\u00114(45)\nand can be seen in Fig. 5 as a function of the momenta. The data reported shows that there is a clear Monte Carlo\nsignal for momenta up to ∼1 GeV for all the amputated form factors. For larger momenta the statistical errors\nbecome quite large and it is hard to comment on the behaviour of the amputated F(i). The dependence on pshows an\nhierarchy between the various amputated form factors, with the form factor associated with tree level tensor structure\nF(0)being the largest of the three F(i). In general, for the same form factor, the statistical error associated with\nkinematical configuration (p, p, p, −3p)is the largest. This can be understood looking at the lattice operators given\nin App. B. Indeed, for this kinematical configuration the number of averages performed for each configurations is10\nFIG. 4: Dimensionless bare lattice form factors F(i)for the various kinematics. Black dots are the data from the simulations\non the 324lattice, while the red squares refer to the simulation on 484. Due to the large statistical errors the data from the\nlarger volume is omitted in some of the plots. Note the different vertical scales used in each Fig. See main text for details.\nsubstantially smaller than for the remaining ones and, therefore, the data associated with (p, p, p, −3p)should have\nlarger fluctuations, when compared with the other two kinematics investigated here.\nIn what concerns the dependence of the amputated form factors with momentum, the curves suggest that F(0)is\nessentially constant for p≳0.6GeV (although for p≳1GeV the large statistical errors prevent any firm conclusion)\nand decreases for smaller momenta. The systematics for F(1)andF(2)are harder to understand from Fig. 5. As\ndiscussed below, by taking into account the Bose symmetry and its implication on the amputated form factors gives\na clear understanding on their dependence with momentum scales.\nThe gluon field is a bosonic field and, therefore, the form factors should depend on momenta as described in Eq.\n(34). In this sense, the form factors should be a function not of pbut of\ns=\u0000\np2\n1+p2\n2+p2\n3+p2\n4\u0001\n/4 (46)\nand of the combination of scalar products of the various momenta mentioned in Eq. (34). Let us ignore this latter\ndependence and redo the plots, looking at the amputated form factors as a function of sand combining the various\nkinematics in the same plot. The lattice data for the three amputated form factors is reported in Fig. 6. In general,\nthe lattice data for the amputated form factors is compatible with a dependence on s. For the level of statistical\nprecision achieved in the simulation, the description of the lattice does not seem to require any further variable than\nsitself. The data of the amputated F(0)is compatible with a constant for s≳1GeV. However, F(0)is slightly11\nFIG. 5: Dimensionless bare lattice amputated form factors F(i)for the simulation on 324lattice. The data from the 484\nsimulation was omitted as it has a two large statistical error to add any useful information. For the lower momenta the data\nfrom the simulation with a 484, that have smaller statistical errors, follow the trend observed in the 324simulation and the two\nsets of data are compatible within one standard deviation.\nsuppressed at lower momenta. On the other hand F(1)andF(2)seem to be constant for s≳1GeV but increase when\nsbecomes smaller. Moreover, the lattice data for the amputated F(1)suggests that this form factor should change\nsign at s∼0.3GeV. No change of sign is seen or suggested for F(2). The relative strong increase in the F(1)andF(2)\nobserved when sapproaches zero, seem to suggest that these form factors may have logarithmic divergences in the IR\nregion. However, given the limited access to the low momenta region, the question of the logarithmic divergences is\nnot solved within the current simulation. To answer this question one should have access to the deep infrared region\nthat require simulations using larger lattices or with different lattice spacing.\nV. SUMMARY AND CONCLUSIONS\nHerein, the calculation of the four gluon one-particle irreducible Green function, in the Landau gauge, using lattice\nQCD simulations is addressed. The determination of the associated form factors from the full Green function is\ndiscussed and some of the 1-PI Green function form factors are computed. Our study shows that lattice simulations\nto access the four gluon 1-PI Green function require large ensembles of gauge configurations, meaning 10k or more\ngauge configurations per ensemble. Moreover, to resolve the 1-PI contribution to the full four gluon Green function,\nthat is the Green function directly measured in the simulation, the kinematics of the external legs have to be carefully\nchosen. In particular, for the kinematics characterized by a unique momentum, it is possible to disentangle the four12\nFIG. 6: Dimensionless bare lattice amputated form factors F(i)as a function of sfor the simulation on 324lattice. The plots\ninclude the data for all the kinematical configurations. In the right plots the data for kinematical configuration with the large\nstatistical error was omitted for a better reading on the dependence of the form factors with s.\ngluon 1-PI contribution from the disconnected parts and from the diagrams associated with the three gluon 1-PI in\nthe full Green function.\nIn order to arrive at such large ensembles of configurations, the simulations reported use either a hypercubic lattice\nsize of 324, that has a physical volume of (∼3.3fm)4, or a hypercubic lattice size of 484, whose physical volume is\nabout (∼4.9fm)4. For the smaller lattice our ensemble has about 10K configurations, for the larger lattice the size of\nthe ensemble is of the order of 4K. This prevents the larger lattice volume simulation to achieve a good signal-to-noise\nratio and the statistical errors for the 1-PI form factors for the 484lattice are quite large. Nevertheless, they confirm\nthe tendency observed in the simulation with the 324lattice. We are currently working on the production of larger\nstatistical ensembles to improve our results, i.e. to achieve a good signal for larger range of momenta and to explore\nother kinematical configurations. The results of such simulations will be made public as soon as possible.13\nFIG. 7: Dimensionless bare lattice amputated form factors F(0)as a function of sfor the simulation on 324lattice as in Fig.\n6 replacing the various data points for the same momentum with an weighted average using the statistical error squared as\nweight.\nThe complete description of the four gluon 1-PI irreducible Green function calls for a tensor basis with more than\none hundred independent operators. However, for the kinematical configurations under consideration, the number\nof relevant operators that contribute to full Green functions is relatively small and, in some cases, can be handled\nexactly. Of the possible tensor operators that define the basis for the 1-PI, we have considered the three operators\n(11), (12) and (13), measured the associated form factors F(0),F(1),F(2)defined in (31), (32), (33), respectively, and\ntheir bare amputated versions that appear in the 1-PI four gluon irreducible Green function.\nThe tensor operator used to define the amputated form factor F(0)is the operator that appears in the tree level\nFeynman rule for the four gluon vertex. Our simulation gives an amputated form factor that is essentially constant\nfor momentum above ∼0.5GeV (see Fig. 6), with the data suggesting a suppression at small momenta. Despite the\nlarge ensemble, for momentum above ∼1GeV it is difficult to disentangle its functional form. For momenta in the\nrange ∼1−1.5GeV, despite the size of the statistical errors, the data seems to be compatible with a constant value;\nthis can be better seen by replacing the various data points for the same momentum by their weighted averages using\nthe statistical error squared as a weight as reported in Fig. 7 for F(0). This result is in qualitative agreement with\nthe outcome of recent Dyson-Schwinger calculations [61, 62, 65] but not with the calculation described in [60].\nThe analysis of F(1)shows a negative form factor whose absolute value is about a quarter of the absolute value of\nF(0), it has the opposite sign of this latter form factor and seems to approach zero as sis decreased. This behaviour\nsuggests a change of sign for F(1)ats∼0.15GeV2and suggests a divergent behaviour for F(1)in the deep infrared\nregion. Again, the observed F(1)is qualitatively in agreement with the Dyson-Schwinger result reported in [65].\nThe bare amputated F(2)form factor has, apart the possible change of sign, a similar behaviour as observed for\nF(1), i.e. F(2)∼F(0)/4, with both form factors having the same sign and with the F(2)data suggesting a possible\nlogdivergence in the deep infrared region. The observed lattice data for F(2)is qualitatively in agreement with the\nrecent Dyson-Schwinger calculation reported in [65]. Note that the remaining Dyson-Schwinger studies do not access\neither F(1)orF(2).14\nAcknowledgments\nThe authors acknowledge Arlene C. Aguilar, Mauricio N. Ferreira, Joannis Papavassiliou and Leonardo R. San-\ntos for helpful discussions and careful reading of the manuscript. This work was partly supported by the FCT\n– Fundação para a Ciência e a Tecnologia, I.P., under Projects Nos. UIDB/04564/2020 ( https://doi.org/10.\n54499/UIDB/04564/2020 ), UIDP/04564/2020 ( https://doi.org/10.54499/UIDP/04564/2020 ). and CERN/FIS-\nPAR/0023/2021. P. J. S. acknowledges financial support from FCT contract CEECIND/00488/2017 ( https:\n//doi.org/10.54499/CEECIND/00488/2017/CP1460/CT0030 ). The authors acknowledge the Laboratory for Ad-\nvanced Computing at the University of Coimbra ( http://www.uc.pt/lca ) for providing access to the HPC resources\nthat have contributed to the research within this paper. Access to Navigator was partly supported by the FCT\nAdvanced Computing Project 2021.09759.CPCA.\nAppendix A: Three gluon contributions for proportional momenta\nThe three-gluon one-particle diagram can be analysed using the Ball-Chiu tensor basis introduced in [76]; see also\n[77]. For the momenta configuration of type p1=p,p2=�� pandp3=−(1 + η)p, only the so called longitudinal\ncomponent contributes. Following the notation of Ball-Chiu, the one-particle longitudinal component reads\n(1−η)A(p2\n1, p2\n2;p2\n3)gµ1µ2pµ3+ (1 + 2 η)A(p2\n2, p2\n3;p2\n1)gµ2µ3pµ1−(2 + η)A(p2\n3, p2\n1;p2\n2)gµ3µ1pµ2\n+ (1 + η)B(p2\n1, p2\n2;p2\n3)gµ1µ2pµ3−(1 + 2 η)B(p2\n2, p2\n3;p2\n1)gµ2µ3pµ1−η B(p2\n3, p2\n1;p2\n2)gµ3µ1pµ2\n−η(1−η)C(p2\n1, p2\n2;p2\n3)p2P⊥\nµ1µ2(p)pµ3+η(1 + η)C(p2\n2, p2\n3;p2\n1)p2P⊥\nµ2µ3(p)pµ1\n−(1 +η)(2 + η)C(p2\n3, p2\n1;p2\n2)p2P⊥\nµ1µ3(p)pµ2−2η(1 +η)S(p2\n1, p2\n2, p2\n3)pµ1pµ2pµ3 (A1)\nwhere A,B,C,Sare the Ball-Chiu form factors and\nP⊥\nµν(p) =gµν−pµpν\np2with pµP⊥\nµν(p) = 0 (A2)\nis the orthogonal projector for vector fields. The external legs are proportional to pat least for one of the Lorentz\nindices, given that in the contribution to the four-gluon connected Green function all the external legs are contracted\nwith a gluon propagator that, in the Landau gauge, is proportional to P⊥\nµν(p), given that P⊥\nµν(κp) =P⊥\nµν(p), where\nκis a constant, then it follows that for this class of kinematical configuration the terms in Fig. 2 that contain the\nthree-gluon irreducible diagram vanish. Note that this result is valid even if one of the momenta vanishes.\nAppendix B: Implementation of color-Lorentz projectors to measure F(i)\nThe tensor operators that define the lattice form factors F(i), see Eqs (31) to (33), require the SU(3) totally\nantisymmetric group structure constants fabc, the totally symmetric structure constants dabcor color diagonal δabop-\nerators. The first two types of operators can be computed from color traces of the generators given in the fundamental\nrepresentation taas\nTr\u0000\ntatbtc\u0001\n=1\n4\u0000\ndabc+i fabc\u0001\n(B1)\nor, alternatively,\n−2iTr\u0000\u0002\nta, tb\u0003\ntc\u0001\n=fabcand 2Tr\u0000\b\nta, tb\t\ntc\u0001\n=dabc. (B2)\nThe Euclidean version of the tensors in Eqs (11) to (13) is\neΓ(0)abcd\nµνηζ =fabrfcdr\u0010\nδµηδνζ−δµζδνη\u0011\n+facrfbdr\u0010\nδµνδηζ−δµζδνη\u0011\n+fadrfbcr\u0010\nδµνδηζ−δµηδνζ\u0011\n,(B3)\neΓ(1)abcd\nµνηζ =dabrdcdr\u0010\nδµηδνζ+δµζδνη\u0011\n+dacrdbdr\u0010\nδµζδνη+δµνδηζ\u0011\n+dadrdbcr\u0010\nδµνδηζ+δµηδνζ\u0011\n,(B4)\neΓ(2)abcd\nµνηζ =\u0010\nδabδcd+δacδbd+δadδbc\u0011 \u0010\nδµνδηζ+δµηδνζ+δµζδνη\u0011\n(B5)\nand\nF(i)=eΓ(i)abcd\nµνηζ ⟨Aa\nµ(p1)Ab\nν(p2)Ac\nη(p3)Ad\nζ(p4)⟩. (B6)\nThen, plugging in these definitions it turns out that for the (K. Conf. 1) kinematics15\np1= 0, p2=p3=p, p 4=−2p\nF(0)(p2) =−8X\nr(\nTr\u0010\n[Aµ(0), Aν(p)]tr\u0011\nTr\u0010\n[Aµ(p), Aν(−2p)]tr\u0011\n−Tr\u0010\n[Aµ(0), Aν(p)]tr\u0011\nTr\u0010\n[Aν(p), Aµ(−2p)]tr\u0011\n+Tr\u0010\n[Aµ(0), Aν(−2p)]tr\u0011\nTr\u0010\n[Aµ(p), Aν(p)]tr\u0011)\n, (B7)\nF(1)(p2) = 8X\nr(\nTr\u0010\n{Aµ(0), Aν(p)}tr\u0011\nTr\u0010\n{Aµ(p), Aν(−2p)}tr\u0011\n+Tr\u0010\n{Aµ(0), Aν(p)}tr\u0011\nTr\u0010\n{Aν(p), Aµ(−2p)}tr\u0011\n+Tr\u0010\n{Aµ(0), Aν(−2p)}tr\u0011\nTr\u0010\n{Aµ(p), Aν(p)}tr\u0011)\n,(B8)\nF(2)(p2) = 4(\n2Tr\u0010\nAµ(0)Aµ(p)\u0011\nTr\u0010\nAν(p)Aν(−2p)\u0011\n+2Tr\u0010\nAµ(0)Aν(p)\u0011\nTr\u0010\nAµ(p)Aν(−2p)\u0011\n+ 2Tr\u0010\nAµ(0)Aν(p)\u0011\nTr\u0010\nAν(p)Aµ(−2p)\u0011\n+ 2Tr\u0010\nAµ(0)Aν(−2p)\u0011\nTr\u0010\nAµ(p)Aν(p)\u0011\n+Tr\u0010\nAµ(0)Aµ(−2p)\u0011\nTr\u0010\nAν(p)Aν(p)\u0011)\n, (B9)\nfor the (K. Conf. 2) kinematics\np1= 0, p2=p, p 3= 2p, p 4=−3p16\nF(0)(p2) =−4X\nr(\nTr\u0010\n[Aµ(0), Aν(p)]tr\u0011\nTr\u0010\n[Aµ(2p), Aν(−3p)]tr\u0011\n−Tr\u0010\n[Aµ(0), Aν(p)]tr\u0011\nTr\u0010\n[Aν(2p), Aµ(−3p)]tr\u0011\n+Tr\u0010\n[Aµ(0), Aν(2p)]tr\u0011\nTr\u0010\n[Aµ(p), Aν(−3p)]tr\u0011\n−Tr\u0010\n[Aµ(0), Aν(2p)]tr\u0011\nTr\u0010\n[Aν(p), Aµ(−3p)]tr\u0011\n+Tr\u0010\n[Aµ(0), Aν(−3p)]tr\u0011\nTr\u0010\n[Aµ(p), Aν(2p)]tr\u0011\n−Tr\u0010\n[Aµ(0), Aν(−3p)]tr\u0011\nTr\u0010\n[Aν(p), Aµ(2p)]tr\u0011)\n,(B10)\nF(1)(p2) = 4X\nr(\nTr\u0010\n{Aµ(0), Aν(p)}tr\u0011\nTr\u0010\n{Aµ(2p), Aν(−3p)}tr\u0011\n+Tr\u0010\n{Aµ(0), Aν(p)}tr\u0011\nTr\u0010\n{Aν(2p), Aµ(−3p)}tr\u0011\n+Tr\u0010\n{Aµ(0), Aν(2p)}tr\u0011\nTr\u0010\n{Aµ(p), Aν(−3p)}tr\u0011\n+Tr\u0010\n{Aµ(0), Aν(2p)}tr\u0011\nTr\u0010\n{Aν(p), Aµ(−3p)}tr\u0011\n+Tr\u0010\n{Aµ(0), Aν(−3p)}tr\u0011\nTr\u0010\n{Aµ(p), Aν(2p)}tr\u0011\n+Tr\u0010\n{Aµ(0), Aν(−3p)}tr\u0011\nTr\u0010\n{Aν(p), Aµ(2p)}tr\u0011)\n,(B11)\nF(2)(p2) = 4(\nTr\u0010\nAµ(0)Aµ(p)\u0011\nTr\u0010\nAν(2p)Aν(−3p)\u0011\n+Tr\u0010\nAµ(0)Aµ(2p)\u0011\nTr\u0010\nAν(p)Aν(−3p)\u0011\n+Tr\u0010\nAµ(0)Aµ(−3p)\u0011\nTr\u0010\nAν(p)Aν(2p)\u0011\n+Tr\u0010\nAµ(0)Aν(p)\u0011\nTr\u0010\nAµ(2p)Aν(−3p)\u0011\n+Tr\u0010\nAµ(0)Aν(p)\u0011\nTr\u0010\nAν(2p)Aµ(−3p)\u0011\n+Tr\u0010\nAµ(0)Aν(2p)\u0011\nTr\u0010\nAµ(p)Aν(−3p)\u0011\n+Tr\u0010\nAµ(0)Aν(2p)\u0011\nTr\u0010\nAν(p)Aµ(−3p)\u0011\n+Tr\u0010\nAµ(0)Aν(−3p)\u0011\nTr\u0010\nAµ(p)Aν(2p)\u0011\n+Tr\u0010\nAµ(0)Aν(−3p)\u0011\nTr\u0010\nAν(p)Aµ(2p)\u0011)\n(B12)\nand for the (K. Conf. 3) kinematics\np1=p2=p3=p, p 4=−3p17\nF(0)(p2) = 24X\nrTr\u0010\n[Aµ(p), Aν(p)]tr\u0011\nTr\u0010\n[Aν(p), Aµ(−3p)]tr\u0011\n, (B13)\nF(1)(p2) = 24X\nrTr\u0010\n{Aµ(p), Aν(p)}tr\u0011\nTr\u0010\n{Aµ(p), Aν(−3p)}tr\u0011\n, (B14)\nF(2)(p2) = 12(\nTr\u0010\nAµ(p)Aµ(p)\u0011\nTr\u0010\nAν(p)Aν(−3p)\u0011\n+ 2Tr\u0010\nAµ(p)Aν(p)\u0011\nTr\u0010\nAµ(p)Aν(−3p)\u0011)\n. (B15)\nThe expressions in Eqs (B7) to (B15) will be used in the simulations to measure each of the lattice form factors F(i).\n[1] A. Cucchieri and T. Mendes, PoS LATTICE2007 , 297 (2007), arXiv:0710.0412 [hep-lat] .\n[2] A. C. Aguilar, D. Binosi, and J. Papavassiliou, Phys. Rev. D 78, 025010 (2008), arXiv:0802.1870 [hep-ph] .\n[3] C. S. Fischer, A. Maas, and J. M. Pawlowski, Annals Phys. 324, 2408 (2009), arXiv:0810.1987 [hep-ph] .\n[4] I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker, and A. Sternbeck, Phys. Lett. B 676, 69 (2009), arXiv:0901.0736\n[hep-lat] .\n[5] O. Oliveira and P. J. Silva, Phys. Rev. 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B 869, 417 (2013), arXiv:1210.2331 [hep-ph] ."    },    {        "title": "2401.12009v1.Experimental_investigation_and_scale_analysis_on_melting_of_salty_ice_in_a_3D_printed_cavity_filled_with_porous_media.pdf",        "content": "Experimental investigation and scale analysis on melting of \nsalty ice in a 3D-printed cavity filled with porous media \nLi Xiaotian1, WangYuming2, Yang Wei2, Yao wei1, * \n1. Department of Space Exploration, Qianxuesen Laborotary of Space science and \nTechnology, China Academy of Space science and Technology, Beijing 100094, China \n2. Department o f Energy and Dynamics , College of Engineering, Peking University, Beijing \n100871, China \nAbstract \nWhile significant interests have been devoted to the double diffusive phase change \nheat transfer of binary solutions, the understanding on the melting heat transfer of salty \nice in porous media is still imcomplete. This work aims to explore the melting heat \ntransfer characteristics of salty ice in a square cavity filled with homogeneous porous \nmedia subjected to lateral heating. In order to facilitate the visualization of melting \ndynamics in a uniform porous media, a three-dimensional (3D) printed transparent \ncavity filled with porous matrix was manufactured, featuring an open upper end. \nAqueous solutions of sodium chloride at concentration both higher and lower than the \neutectic concentration were investigated. Lateral heating experiments with constant \nheat flux were conducted in a subzero temperature environment. The effect of heating \npower, initial concentrations and different sizes of porous matrix was investigated. \nDistinct phenomena were observed based on the concentration of the aqueous solution. \nSpecifically, the concentration lower than the eutectic concentration results in an \nupward bending melting interface. Whereas the concentration surpassing the eutectic \nconcentration resulted in a downward bending melting interface. In the presence of salt, \na melting convex appears at the bottom of the melting interface and manifests during \nthe whole melting process. Notably, smaller matrix enables faster melting rate. \nFurthermore, three distinct stages were revealed in our experiments, characterized as \nconduction, mixture of conduction and ultimately convection stage. The correlations for Nu and the average propagation rate of mushy/liquid interface were established by \norder of magnitude analysis and fit well with the experiments. \n \n*. Corresponding author \nYao Wei, 1. Department of Space exploration, Qianxuesen Laborotary of Space \nscience and Technology, Chinese Academy of Space science and Technology, Beijing \n100094, China, Mail.yaowei1972@hotmail.com \nIntroduction \nDouble diffusive convective heat transfter occurs in various nature processes and \nfinds considerable applications in engineering contexts. The complication of heat \ntransfer mechnisms during double diffusive convection are governed by a combination \nof temperature and concentration gradients within fluids. Researches and advancements \non double diffusive convective heat transfer have been the interests of scholary \ncommunities for decacdes (Bennacer et al., 2001; Bourich et al., 2004; Mohamad and \nBennacer, 2001; Trevisan and Bejan, 1990, 1985) . Notably, a particular facet namely \nthe phase-change double diffusive convective heat transfer has attracted significant \nattentions recently. The phase-change double diffusive heat transfer holds importance \nin diverse fields such as environmental science (e.g., solar ponds), oceanography (e.g., \nsea ice melting), geology (e.g., permafrost melting), astrophysics (e.g., extraterritial \nplanets climate), semiconductor processing (e.g., crystal crystallisation), energy storage \n(e.g., phase change material). Extensive investigations have been devoted to both the \nfundamental aspects and application-oriented problems prevail to the double diffusion \nconvective phase-change heat transfer. \nDouble-diffusion convection arising from melting or solidification of binary \naqueous solutions without porous media has been extensively studied. Chen et al. (1971) \nstudied the behavior of the flow field induced by lateral heating in a stably stratified \nfluid of constant gradient, and showed that the celluar convection was induced by the \ninstability of fluid. Choi et al. (1980) experimentally studied the solidification of Na2CO3 in various configurations, and found the occurrence of finger instability when \ncooling from bottom. Additionally, the opposing gradients of solutions resulted in the \nemergence of small double-diffusive layering. Thompson and Szekely (1988) explored \nthe solidifacation of solutions from the lateral wall. A compositional and density \nstratification was discoverd during solidification. Christenson and Incropera (1989) \nexperimentally investigated the lateral solidifications of solutions, revealing that the \nsolidification rates, localized remelting, and macroscopic solute redistribution were \ninfluenced by the solutally driven flows. Marcoscopic mathmatical model related to the \ntransport process was reviewed by Viskanta (1990) while Voller and Brent (1990) \nmodelled the mushy region of a binary solidification system. Nishimura and Imoto \n(1993) reported the solidification of NH 4Cl-H2O in a cavity cooled laterally. Double \ndiffusive cells with nearly constant concentration were identified within the liquid \nregion during solidification and controlled by the buoyancy ratio and thermal Rayleigh \nnumber.  \nCao et al. (1989) reported the solidification of aqueous sodium chloride solutions \nwithin a densely packed bed of glass spheres. Remelting phenomena were observed in \nthe mush zone. They found that the enriched and stratified fluid layers appeared at the \nbottom of the fluid. The intensity of natural convection within this system was governed \nby the Darcy number and modified Rayleigh number. Matsumoto et al. (1993) studied \nthe solidification of porous media saturated with aqueous solutions in a rectangular cell \nand characterized the solidification process both experimentally and numerically. The \nsolidification rate is slow at the top and bottom due to the combined effect of thermal \nand salt buoyancy. The permeability coefficient in the mushy zone could be \napproximately expressed by the exponential formula for the liquid phase rate. Choi and \nViskanta (1993) studied the solidification of binary alloys (water and ammonium \nchloride) in the saturated state of a spherical packed bed. Remelting was observed at \nboth solid/mushy and mushy/liquid interfaces. Comparison of the experimental results \nwith a simple analytical model that ignores the presence of convection reveals \nreasonable success in the solid and slurry regions. Song et al. (1993) experimentally \nand theoretically investigated the upward solidification of a porous medium saturated with a binary solution. An analytical model was proposed and predictions were \ncompared with experimental data with good agreement. An \"over-enrichment\" \nphenomenon was also observed in the viscous region away from the liquid phase. Song \net al. (2001) studied the experimental and theoretical investigation of lateral freezing of \nnon-uniform porous media saturated with saline solution, which deepened the \nunderstanding of solute redistribution during solid/liquid phase transition. The presence \nof a porous structure significantly alters the distribution and growth kinetics of the \nprimary solid phase. When the permeability of the porous matrix phase decreases, the \npredicted macroscopic devitrification decreases. \n However, the melting heat transfer of binary mixture has received less attention. \nBeckermann and Viskanta (1990) experimentally investigated the melting of vertical \nice into NH 4Cl-H2O solution and the melting of vertical NH 4Cl-H2O eutectic layer into \nwater by laterally heating. It was found that in both cases the melting process results in \nthe formation of a double diffusive layer in an initially homogeneous liquid. The \nexperiments showed that relatively small changes in the thermal boundary conditions \nof melting could lead to large differences in the layering process and double diffusion \nin the liquid also had a significant effect on the melting process. Yin et al. (2000) \ninvestigated the development of layered structures, convective mixing of indium and \ngallium, bidirectional flow patterns, and solid-liquid interfaces. The presence of \nlayering and multiple double convective layers in the binary Ga-In5 melt was observed \nin real time by X-ray visualization under transient horizontal temperature gradients. \nMultiple double diffusive layers were interspersed at the top of the melt layers. A close \nrelationship between the double diffusive layers and the solid/liquid interface was \nidentified. The interface is divided into several parts by characteristic morphological \nprotrusion features.  \n The melting process of binary eutectics in porous media is also very important. For \nexample, the melting of salty ice occurs widely in thawing of frozen soil, Mars \nexploration (Mellerowicz et al., 2022), thermal storage , etc. However, only one study \non the melting of binary eutectics in porous media was identified through literature \ninvestigation. Ouestlai et al. (2008) used numerical methods to study the effect of density change on double diffusive convection of meltwater in a side-heated porous \nmedium considering a nonlinear density expression (Oueslati et al., 2008). The author \nchose an initial temperature of 0°C and used initial concentrations of 5 and 10 wt%, at \nwhich point the solution did not freeze due to the lowering of the freezing point, so the \nconclusions of the article are questionable.  \nIn order to better understand the mechanism of double diffusive convective heat \ntransfer in the melting in low temperature environments, we conducted experiments \nwith 3D printing technology to faciliate visualization. The natural convection and heat \ntransfer characteristics of the melting process in transparent cavity with saturated \nporous matrix were investigated. The correlations of Nu number and average \npropogation speed of melting interface were established by the order of magnitude \nmethod for the hypoeutetic region.  \nExperimental equipments and procedures \nAs shown in Fig.1, a transparent square container was fabricated with 3D printing \nmethod to facilitate direct visualization of the melting process. A desktop-scale \nexperimental setup in the laboratory was built to simulate the subzero temperature \nenvironment of Mars. The transient evolution of melting interface was filmed by a \nCanonTM Legria HFR 86 video camera and the temperature data was measured by \nthermocouple and recorded by a digital data acquisition unit during the melting process. \nThe cavity is 40 mm wide, 40 mm high, 2 mm thick, and has a 2 mm base for fixation. \nA rectangular array of small black cylinders was printed inside the square cavity with \nthe integrated 3D printing technology to simulate the solid matrix in a porous medium. \nA cylindrical cavity with a diameter of 6 mm, a height of 40 mm, and a wall thickness \nof 1 mm is attached to the left side of the square cavity for mounting a cylindrical heater. \n During the experiment, a metal heating rod with a diameter of 6 mm and a height \nof 40 mm was placed into the cavity, and the inside of the cavity was filled with \nthermally conductive silicone grease to reduce the contact thermal resistance between \nthe heating rod and the wall. A schematic diagram of the test cell is shown in Figure 2. Twenty-five type K thermocouples (wire diameter 0.21 mm) were taped to the outside \nof the square cavity to measure the transient temperature distribution of the square \ncavity shell.  \n   \nFigure 1 Schematic of the assembled cavity with soluiton saturated porous \nmatrix  \nThe entire cavity was fixed to an acrylic base by screws to eliminate the effect of \ntilting of the square chamber on convection. Deionized water was mixed with the \ndesired NaCl in different ratios to obtain the desired salt concentration. The solution \nwas heated to 80 °C to remove the bubbles from the water, and the degassed aqueous \nNaCl solution was carefully injected into the interior of the square cavity using a \nhypodermic syringe (5 mm diameter of the needle) to minimize the gas bubbles in the \nsolution. Prior to the test, the solution-filled square chamber was frozen in a low-\ntemperature refrigerator. Since the addition of NaCl lowers the freezing point of the \naqueous solution, the refrigerator temperature was set to -40oC during freezing to allow \nthe solution to freeze completely and rapidly, which lasted for 10 minutes, to keep the \nsolute as evenly as possible.  \nA Qingsheng™ QS2000 thermal cycling chamber was used to simulate the low \ntemperature environment. The lowed temperature of the chamber is -40 oC. A square \nchamber filled with the frozen solution was placed inside the chamber and the left wall \nof the chamber was heated by an electric cylinder heater controlled at a constant power \nlevel. Before starting the test, the hot/cold circulator was turned on for refrigeration, \nand after confirming that the temperature inside had stabilized to the desired low \ntemperature by observing the display, the door of the circulator was opened. Then the \nsquare chamber, which had been fixed on the base, was taken out of the refrigerator and \nquickly put into the hot/cold circulator. Then the door of the circulator was closed. For \nthe visualization test, the outer walls of the square chambers used were free of \nthermocouples to avoid interference with the filming. For temperature measurement \ntests, K-type thermocouples were uniformly attached to the outer wall of the square \nchamber. The thermocouple tail leads were routed through a small hole on the left side \nof the recirculation box and the hole was plugged with a rubber stopper. All type K \nthermocouples were calibrated with ice water at 0 oC, and a temperature data acquisition \nsystem was used to record temperature changes at predetermined time intervals (30 s). \nDuring the course of the experiments, when a melting liquid appeared, a trace amount \nof dry methyl blue powder was pushed into the cavity via a hypodermic syringe \nsuspended above the left wall surface in the square cavity to stain the melting zone in \norder to better distinguish the solid-liquid interface. To perform the visualization \nexperiments, a white LED planar light source was used to illuminate from behind the \nsquare cavity. During the visualization experiments, the front and back of the square \nchamber were covered with a heat shield, while the top and bottom of the chamber were \nin direct contact with cold air. Photographs were taken at 5 min intervals, and the heat \nshield was removed when taking the photographs, and then the front and back of the \ncavity were covered after taking the photographs in order to minimize the heat loss \nduring the photographs. For temperature measurement experiments, the heat shield was \nkept over the square cavity. Figure 2 shows the optical picture taken without the heat \nshield. \n  \nFigure 2 Schematic diagram of the test setup \n \nFigure 3 Schematic of thermocouple arrangement with a spatial separation of 0.8 centimeters \nbetween each consecutive thermocouple. \n \nFigure 4 Phtograph of forzen sample  \nSix sets of experiments were carried out and the conditions are shown in Table 1. \nExperiments 1-3 are to study the effect of different heating powers on the melting \nprocess. Experiments 4-5 are to investigate the effect on the melting process by \nmeasuring the change in initial concentration, and Experiment 6 is to explore the effect \nof different porous matrix diameters on the melting process. Each set of experiments \nwas performed with three repetitions of visualization tests and three repetitions of \ntemperature measurements, respectively. In this case, Experiment 2 was used as a \ncontrol group with 5 wt.% initial salt concentration, 5 W heating power and 2 mm \ncylinders. The eutectic point of the aqueous NaCl solution used was -22.4 °C and the \neutectic concentration was 23.1 wt.%. In all experimental conditions, the ambient \ntemperature was -30oC. \n Table 1 Summary of Ice Melt Experimental Conditions  \nWorking condition Initial concentration Heating power Cylinder diameter  \n1 5 wt.% 3 W 2 mm \n2 5 wt.% 5 W 2 mm \n3 5 wt.% 7 W 2 mm \n4 10 wt.% 5 W 2 mm \n5 25 wt.% 5 W 2 mm \n6 5 wt.% 5 W 1 mm \n  \nExperimental results and discussion \n The photographs of the melting process of the binary mixture in the cavity is shown \nin Figure 5. The lighter part is the liquid region and the darker part is the mushy region, \nwhile the darkest part is the frozen region. The mushy/solid interface was identified as \nmelting interface in this article. In the initial stage, the melting interface propogate \nfaster at the center. This is due to that the top and bottom of the square cavity are in \nconctact with the cold environment exchanging heat with it, thus the melting rate is \nlower. At 5 minutes, the melting interface is approximately parallel to the left wall, \nmainly because the heat transfer mechanism is conduction. At 10 minutes, the melting \nregion at the top of the square cavity grows faster, indicating that thermal convection \nhas developed in the upper part of the melting region. The fluid ajacent to the left wall \nrises along the heated wall and accumulates at the top of the square cavity, thus \naccelerating the melting in the upper part. At the bottom of the square cavity, the melting \nalso starts to accelerate, forming a nose protruding to the right, which is mainly due to \nthe redistribution of the solutes in the liquid zone adjoining the melting interaface. \nWhen the eutectic mixture melts into solution, the solute depletion occurs in this region \ndue to the difference in solubility of NaCl in the liquid and solid phase. Solutes \nprecipitate out of the ice into the mushy zone (Brimblecombe et al., 1987; Kumar et al., \n2007; Terwilliger and Dizon, 1970) , thus increasing the concentration and resulted iin \nthe denser liquid near the melting interface. The denser liquid sinks to the bottom and \nlowers the freezing point, thus accelerating the melting of the frozen solution at the \nbottom. A stagant zone was build at bottom and double diffusion layer appeared \nbetween warmer, less salty liquid at the upper section and colder, more salty liquid at \nthe bottom. The mechnism is different from Yin et al. (2001). In their melting \nexperiments, the unmelted In elements snow down the bottom from the interface and \nincrease the thermal conductivity at the bottom, which leads to an accelerated melting \nrate. Unlike Beckerman et al. (1990) as well, their experiments involved eutectics \ndissolving into fresh water, increasing the concentration of neighboring fresh water, \nwhich increased the density of the water and snow down to the bottom, thus increasing the melt rate at the bottom.  \n  \nFig. 5 Evolution of the melting interface for the heating power of 5 W, the initial salt \ncontent of 5 wt.%, the porous matrix with small cylinders of 1 mm diameter and 0.5 mm \nspacing, and an ambient temperature of -30 oC. (In the figure, the light-colored part is the \nmelting region, the dark-colored part is the frozen region, and the melting interface is the \nintersection line between the light-colored and dark-colored parts, and the subsequent \nphotographs are the same)  \nAs shown in Fig. 6, the melting interface evolution is plotted for the heating power \nof 5 W, the initial salt content of 5 wt.%, and the small cylindrical diameter of 2 mm, at \nthe ambient temperature of -30 oC. The melting is slower in the initial moments in the \ncase of the 2 mm matrix than in the case of a 1mm matrix, and is still predominantly by \nthermal conduction at 5 minutes. Although the spacing is the same, the larger diameter \nresults in a smaller number of matrices per unit volume, and therefore allows for a larger \npore volume and greater porosity for the development of convection. However, in terms \nof heat transfer, the thermal conductivity of the matrix decreases due to the increase in \nmatrix porosity, therefore its heat transfer capacity will decrease and the melting rate \nwill be slower. The increase in porosity increases the convective strength, but the \nincrease in thermal convection is relatively more pronounced. The effect of the \nenhancement of thermal convection and decrease in thermal conductivity on the melting \nrate, on the other hand, needs to be analyzed quantitatively and will be discussed further \nin the subsequent parts of the article.  \n  \nFig. 6 Melting interface evolution for the heating power of 5 W, the initial salt content of \n5 wt.%, and the porous matrix with small cylinders with diameter of 2 mm at the ambient \ntemperature of -30 oC \nAs shown in Fig. 7, the evolution of the melting interface is photographed for a \nheating power of 5 W, an initial salt content of 10 wt.%, and a porous matrix cylinder \nwith a diameter of 2 mm and a spacing of 0.5 mm. Compared with the 5 wt.% \nconcentration, the melting rate increases significantly, with a more pronounced \ninterfacial bending observed at nearly 10 min. This phenomenon is attributed to the \ninverse relationship between concentration and freezing points. The higher \nconcentrations lowered freezing points and accelerated the melting rate. With the heat \nconduction mechanism, the melting interface propogated to the right for a longer \ndistance due to the faster melting rate. During this stage, heat convection mechanism is \nnot yet fully developed. The upper section of the melting interface bends to right rather \nthan to left. The melting rate is higher at a concentration of 10 wt.% as opposed to the \n5 wt.%. This augmented melting area intensified the thermal convection against the \nheat loss between the upper section of the square cavity and its surroundings. Thus, the \nmelting rate at the upper scetion was promoted. For the 5 wt.% case, on the other hand, \nthe cumulative effect of heat at the top due to thermal convection is weaker than the \nheat loss by convective heat exchange between the top and the surrounding, so the \ntemperature at the top decreases and the temperature in the middle of the interface is \nhigher. Correspondingly, the rate of motion of the top interface is slower than that of \nthe middle, which is manifested by the melting interface bending to the left at the top.  \n \nFig. 7 Evolution of the melting interface for the heating power of 5 W, the initial salt content \nof 10 wt.%, small cylinders of porous matrix with a diameter of 2 mm and a spacing of 0.5 mm, \nand the ambient temperature of -30 oC. \nAs shown in Figure 8, the photographs of the melting process for Test 5 are \npresented. In contrast to the 5 wt.% and 10 wt.% cases, the melting interface bends \ndownward to the right. It is simply because at this concentration the solubility already \nexceeds the eutectic concentration, salt precipitated and settled to the bottom of the \nsquare cavity once the metling occured. When the melting started, the salt concentration \nat the bottom was higher, which greatly reduced the melting point, so the melting rate \nin the lower part was much higher than in the upper part. This is similar to Beckerman \net al. (1990). With the increase of the melting region, the thermal convection is \nenhanced. While the effect of salt concentration gradient on the flow pattern is much \nlarger than that of thermal convection. Due to the influence of the salt concentration \ngradient in the vertical direction, the melting interface continues to bend to the lower \nright, while the area influenced by thermal convection is small. So, the temperature at \nthe top is very low, resulting in lower melting rate. With the enhancement of thermal \nconvection, the effect caused by the salt concentration gradient is partially canceled at \nthe upper section, where the degree of bending gradually decreases and an inflectioin \nappears as time elpased. \n \nFig. 8 Evolution of the melting interface for the power of 5 W, the initial salt concentration of \n25 wt.%, small cylinders of porous matrix with a diameter of 2 mm and the spacing of 0.5 mm, \nand an ambient temperature of -30 oC. \nIn order to further quantitatively analyze the melting rate for different working \nconditions, the liquid fraction as a function of time in the cavity under different working \nconditions was analysed using matlab and ImageJ software. The melting rates were \ncompared for different initial salt concentrations, porous media matrix sizes, and \nheating powers, respectively. \n As shown in Fig. 9, the variation of liquid fraction with time for different initial \nsalt concentrations is shown. Overall, the melting becomes faster as the salt \nconcentration increases. The reason for this phenomenon is that the melting point \ndecreases as the concentration increases, boosting the melting rate. In addition, the \nmelting rate gradually slows down with time. The first turning point of the curves \noccurs after 5 min, implying the development of convection. A second turning point \noccurs after 15min, indicating heat equilibrium with ambient is gradually built at this \npoint. Between 15min and 30min, the liquid fraction changes little, indicating that a \nsteady state is almost reached at this point. Comparatively speaking, the melting curves \nof 5wt.% and 10wt.% mixture have similar trends, which can be almost repeatible by \nmerely curve translation, because their flow patterns are dominated by heat convection \nand the melting interface is deflected to the upper right. The flow pattern of melting of \n25 wt.% mixture is different from the previous two. The melting interface is deflected \ndownward to the right, and thermal convection is limited due to the solutal gradient, so \nthe shape of the liquid fraction curves is also different. \n \nFig. 9 Variation of liquid phase rate with time in the square cavity for different initial salt \nconcentrations \nAs shown in Fig. 10, the liquid fraction curves are shown for different diameters \nof porous matrices. There lies an inverse relationship between the diameter of the \nporous matrix and liquid fraction. As the diameter decreases, the volume of each pore \ndecreases, resulting in a higher quantity of porous matrices per unit volume and leading \nto a reduction in porosity. However, the thermal conductivity is increased. The decrease \nin porosity constrains the developement of thermal convection, while the consequential \nincrease in thermal conductivity exerts a more crucial influence on the melting rate. \nConsequently, the net effect is an enhancement in the overall liquid fraction, due to the \nreduction of the diameter of the porous matrix. This complicate interplay between \nporosity, thermal conductivity, and melting rate unfolds the dynamics governing heat \ntransfer mechnism within the porous matrix. \n \n \nFigure 10 Melting rate for different porous matrix diameters \nThe liquid fraction profiles for various heating powers are shown in Fig. 11. As the \nheating power increases, the melting rate grows. The higher the power, the faster to \nreach thermal equilibrium. This is because a faster melting rate leads to an earlier \ninitiation of the thermal convection. After the fully developement of thermal convection, \nthe thermal equilibrium is built (Jany and Bejan, 1988). In our experiments, due to the \nlow ambient temperature and heating power level, after reaching thermal equilibrium, \nthe melting rate decreases gradually and the melting interface couldn’t make contact \nwith the right-side wall. \n \n \n Fig. 11 Variation of liquid phase rate with time for different heating powers \nOrder of magnitude analysis \nThe visualized experimental results indicate that the melting process could be \ndevided into three stages, as shown in Fig. 12, which differs from four stages of Jany \net al. (1988) due to the melting interface couldn’t make the contact with the right wall \nin our experiments. Heat conduction dominates the melting process in the early stage \nof melting. During conduction stage, heat flow is perpendicular to the hot wall and the \nmelting front is almost parallel to the heated wall. Then the thickness of the liquid \nregion increased and natural convection was developed. As a result, the heat flux along \nthe heated wall becomes inhomogeneous, which deforms the melt interface, indicating \nthe mixture heat transfer stage starts. In the final stage, natural convection develops \nover time and dominates the phase change process. In total, natural convection is \naffected by both the thermal and solutal gradients, which influences the heat transfer \nmechanism and thus the melting process. In order to further clarify the heat transfer \nmechanisms, a theoretical analysis based on order of magnitude analysis is performed \non the hypoeutectic side. \n \nFigure 12 Schematic diagram of solid-liquid interfacial evolution, with the liquid phase \non the left of the green line  \n(1) Heat conduction stage  \nAs shown in Fig. 12 (a), the melt front is almost parallel the heated wall. The order \nof s is comparable with H. By applying the law of conservation of energy to the liquid-\nphase region, the density of heat flow introduced through the left wall is of the same \norder of magnitude as the enthalpy change of melting at the interface. According to the \nanalysis, it is obtained  \n(்ೢି்)\n௦ᇣᇧᇤᇧᇥ\n୦ୣୟ୲ ୡ୭୬ୢ୳ୡ୲୧୭୬𝐻~ఘ௦\n௧ถ𝐻\n୦ୣୟ୲ ୭ ୳ୱ୧୭୬  \nwhere ℎ  is the enthalpy of melting, k is the thermal conductivity, ℎ  is the \nforced convective heat transfer coefficient with the environment, and 𝑇  is the \ntemperature of the ambient. The order of magnitude of the average distance of the \nmelting interface propogated from the heated wall can be derived as:  \n𝑠~𝐻(𝜃)ଵ/ଶ   \nwhere 𝜃= 𝑆𝑡𝑒 𝐹𝑜 , \n 𝑁𝑢=ℎ0𝐿\n𝑘  \n 𝑆𝑡𝑒=(்ೢି்)\n  \n 𝐹𝑜=ఈ௧\nுమ   \nFrom the eqation, it is known that the distance of the melting interface from the left \nheating wall shows a relationship with the height of the square cavity, the heating time \nand the specific heat capacity of the fluid itself, the thermal diffusivity and the enthalpy \nof melting. The solute affects the thermophysical properties of water to some extent, \nand therefore indirectly affects the rate of motion of the melting interface at this stage. \nUnder constant heat flow density conditions, 𝑇௪−𝑇 is unknown. (𝑞ᇱᇱ𝑠)/𝑐~(𝑇௪−\n𝑇), where ��ᇱᇱ is the heat flow density. \nThe average Nu number of the heated wall is: \n𝑁𝑢=−1\n𝑇௪−𝑇න\n∂𝑇\n∂𝑥ቤ\n௫ୀ𝑑𝑦 \nSubstituting  𝑑𝑇~𝑇௪−𝑇,dx~s, gives  \n𝑁𝑢~𝐻\n𝑠~𝐻𝜃ିଵ\nଶ \n(2) Mixed heat transfer stage \nAs the melting continues, thermal convection was activated and intensitified in the \nupper section of the cavity as the liquid region increases, accelerating the melting \nprocess. As a result, the liquid region in the upper section becomes wider with a height \nof a, whereas in the lower section with a height of H-a, is still dominated by thermal \nconduction, including the stagnant region at the bottom. At the mixed heat transfer stage, \nthe heat flux is introduced along the heated wall and is in equilibrium with the fluid \nheat transfer. The fluid heat transfer can be divided into two sections: the upper section \ndominated by thermal convection heat transfer and the lower section dominated by heat \nconduction. \nWithin the convective region, which can be divided into the main flow region and \nthe boundary layer, the changes of temperature and concentration mainly occur in the \nboundary layer, so s ~𝛿்,𝐻−𝑎~𝐻. \nIn the initial stage, heat conduction is predominant, so according to the \nconservation of energy: \n∂𝑇\n∂𝑡∼𝛼∂𝑇\n∂𝑥ଶ \nApplying substitution yields \n𝛥𝑇\n𝑡∼𝛼𝛥𝑇\n𝛿்ଶ \nThe thickness of thermal boundary layer can be found as 𝛿்∼(𝛼𝑡)ଵ\nଶ \nAccording to mass conservation  \n𝜕𝑢\n𝜕𝑥+𝜕𝑣\n𝜕𝑦= 0 \nwhere u is the velocity in x-direction, v is the velocity in y-direction. Applying \nthe order of magnitude analysis yields \n𝑢\n𝛿்~𝑣\n𝐻 \nThe x-direction omentum equation  is \n0=−𝜕𝑝\n𝜕𝑥−𝑢\n𝐷𝑎 \nThe y-direction momentum equation can be rewritten as \n0=−𝜕𝑃\n𝜕𝑦−𝑣\n𝐷𝑎+𝑔𝛽𝛥𝑇𝜃𝐿\n𝜈𝑣+𝑔𝛽்𝛥𝐶𝜙𝐿\n𝜈𝑣 \nTaking the partial differential ∂/∂y for the x-direction momentum equation, and the \npartial differential ∂/∂x for the y-direction momentum equation, and subtracting them \nyields \n𝜕𝑣\n𝜕𝑥−𝜕𝑢\n𝜕𝑦=𝐷𝑎(𝑔𝛽𝛥𝑇𝜃𝐿\n𝜈𝑣𝜕𝜃\n𝜕𝑥 +𝑔𝛽்𝛥𝐶𝜙𝐿\n𝜈𝑣 𝜕𝜙\n𝜕𝑥 ) \nAccording to the phase diagram of NaCl solution, in the hypoeutectic side, the \nliquidus line is approximated as a slash and the dimensionless melting point can be \nexpressed by  𝜃~−𝑚 𝜙,thus, leading to 𝜃~𝜙 , where 𝑚 is the slope of the liquidus \nline. The above equation can be rewritten as \n𝑣~𝐷𝑎(𝑔𝛽𝛥𝑇𝜃𝐿\n𝜈−𝑔𝛽்𝛥𝐶𝜙𝐿\n𝜈  )c \nWhen steady state is achieved, the buoyancy force is balanced by the Darcy \nfrictional resistance. At this equilibrium point, the velocity of the fluid becomes \nconstant and can be expressed as  \n𝑣~𝑔𝛽𝐾(1−𝑁)𝛥𝑇\n𝜈𝛿் \nThe buoyancy force is determined by the density difference casused by the gradient \nof temperature and concentration. Upon entering the convection dominated stage, the \nheat conduction term in the energy equation can be neglected, thus 𝜎Δ𝑇\n𝑡~𝑣Δ𝑇\n𝐻 \nFrom this we can find \n𝑡∼𝜎𝐻\n𝑣 \nAt this point, the thermal boundary layer maintains nearly constant, so  \n𝛿்=𝐻[(1−𝑁)𝑅𝑎]ିଵ\nଶ \nwhere N is the buoyancy ratio and 𝑅𝑎 is the Darcy Rayleigh number, the \nheight of the convection layer is 𝑎, which leads to  \n𝛿்\n𝑍~ቀ(1−𝑁)𝑅𝑎,௭ቁିଵ\nଶ \nwhere 𝑅𝑎,௭=(𝑅𝑎𝑧)/𝐻. At the bottom, the width 𝑠 and height are of the \nsame order, so  \n𝑧(1−𝑁)𝑅𝑎,௭~𝐻( Ste 𝐹𝑜)ଵ\nଶ \nTherefore, the order of z is  \n \n𝑧~(1−𝑁)𝐻𝑅𝑎 Ste 𝐹𝑜 \nThe mixing heat transfer stage ends at when z equals H. Therefore, the mixing heat \ntransfer phase ends at \n𝜃ଵ~[(1−𝑁)  𝐻𝑅𝑎]ିଵ \nAccording to the above analysis, the heat transfer mechanism consists of two parts: \nconvection and conduction. The average Nu number of the heated wall is \n𝑁𝑢=𝐻−𝑧\n𝑠+𝑧\n𝛿் \nSubstituting into the above equation yields: \n𝑁𝑢~(𝑆𝑡𝑒𝐹𝑜)ିଵ\nଶ+(1−𝑁)𝑅𝑎Ste 𝐹𝑜ଵ\nଶ \n(3) Convection stage \nWhen the bottom of the thermal convection region touches the top of the salt \nconvection region, we consider that convection has entered the fully developed stage \nafter which  𝑧 =  𝐻, thus \n𝑁𝑢~𝐻\n𝛿் At the stage of full convective development, both the thermal boundary layer and \nthe convective height remain constant, so the 𝑁𝑢 number remains constant \n𝑁𝑢~(1−𝑁)𝑅𝑎 \nCompared with Jany et al.'s study, we introduce the effect of salt and porous media \non the melting process, which is reflected in the correlation equation by increasing the \nbuoyancy ratio  𝑁 and the Darcy number 𝐷𝑎, where 𝐷𝑎 can be coupled into the Darcy \nRayleigh number  𝑅𝑎. \nComparison with tests \nIn order to verify the results of the theoretical analysis, we analyzed the average \n𝑁𝑢  number of the heated wall and the average propogation speed of the melting \ninterface under different working conditions. The formulas for data processing are as \nfollows. The average heat transfer coefficient is  \nℎି\n=𝑄௧\n𝐴(𝑇௪−𝑇) \nwhere 𝑇௪ is the average wall temperature, 𝑇 is the melting temperature, A is \nthe heat transfer area, 𝑄௧ is the net heating power. Then the 𝑁𝑢ି\n caculated from \nthe data is  \n𝑁𝑢ି\n= ష\n\n \nwhere 𝐿 is the characteristic length and 𝑘 is the thermal conductivity \n𝑞௧=𝜌௦ℎ௦𝑑𝑉\n𝑑𝑡 \nwhere 𝑞௧ is the heat transfer rate at mushy/re,  𝜌௦ is the solution density, ℎ௦ \nis the latent heat of melting, and 𝑉 is the liquid phase volume. Stefan number can be \nexpressed as \n𝑆𝑡𝑒=𝐶ቀ𝑞𝐿\n𝑘ቁ\nℎ௦   \nwhere 𝐶 is the specific heat capacity, 𝑞 is the input heat flux, 𝐿 is the \ncharacteristic length, and  ℎ௦ is the latent heat of phase transition.  \nFig. 13 Average Nu number of heated wall with dimensionless time \nTable 2 Fitting Functional of 𝑁𝑢 \nPower \n(W) Conduction stage Mixed heat transfer \nmechanism stage Convection fully \ndeveloped stage \n7 𝑦=0.126∗𝑆𝑡𝑒𝐹𝑜ି.ହ \n(0<𝑆𝑡𝑒𝐹𝑜<0.007)  y =0.126∗𝑆𝑡𝑒𝐹𝑜ି.ହ \n−2.83∗𝑆𝑡𝑒𝐹.ହ \n(0.007<𝑆𝑡𝑒𝐹𝑜<0.026) 𝑦=0.10∗𝑅𝑎ି.ଶହ  \n(0.026<𝑆𝑡𝑒𝐹𝑜\n<0.04) \n5 𝑦=0.086∗(𝑆𝑡𝑒𝐹𝑜)ି.ହ \n(0<𝑆𝑡𝑒𝐹𝑜<0.006) 𝑦=0.085∗(𝑆𝑡𝑒𝐹𝑜)ି.ହ \n+3.88∗𝑆𝑡𝑒𝐹.ହ \n(0.006<𝑆𝑡𝑒𝐹𝑜<0.029) 𝑦=0.080∗𝑅𝑎ି.ଶହ  \n(0.029<𝑆𝑡𝑒𝐹𝑜\n<0.04) \n3 𝑦=0.063∗(𝑆𝑡𝑒𝐹𝑜)ି.ହ \n(0<𝑆𝑡𝑒𝐹𝑜<0.006) 𝑦=0.012∗(𝑆𝑡𝑒𝐹𝑜)ି.ହ \n−5.55∗𝑆𝑡𝑒𝐹.ହ \n(0.006<𝑆𝑡𝑒𝐹𝑜<\n0.028) 𝑦=0.078∗𝑅𝑎ି.ଶହ  \n(0.028<𝑆𝑡𝑒𝐹𝑜\n<0.04) \nIn order to verify the results of the order-of-magnitude analysis, we carried out \nexperiments with input powers of 3W, 5W, and 7W, measured the temperature data and \nprocessed the relevant data according to the above equations. As is shown in Fig.13, \nthe 𝑁𝑢  number increases as the input power increase. The three different power \nconditions share the same tendency, indicating that the flow pattern in the liquid phase \nregion is similar. The change of  𝑁𝑢 can be roughly divided into three successive stages. \nAt the first stage 𝑁𝑢  decreases rapidly, indicating the predominance of the heat \nconduction. The propogation of the melting interface to the right increases the heat \ntransfer thermal resistance.  \nThe second stage occurs when notable inflection point appears, indicating a \ntransition in the heat transfer mechanism. Convective heat transfer is activated and \nevolves into a mixed convection mechanism alongside conduction. In the third stage, \nthe 𝑁𝑢 number curve approaches to be parallel with the horizontal coordinate axis, \nexhibiting a nearly constant value independent of time. This implies the dominance of \nconvective heat transfer approaching to a fully developed state, or a quasi-steady state. \nStages of transformation were initially determined by comparing interfacial evolution \ndiagrams during the melting process, followed by fitting each stage using the order-of-\nmagnitude correlations. Notably, the average Nu number under each input power fits \nwell with the experimental results, affirming the validity of the order-of-magnitude \nanalysis. The expressions of the fitted function are detailed in Table 2.  \n \nFigure 14 Plot of average melt interface travel distance against dimensionless time. \nAs shown in Fig. 14, the propogation distance of the melting interface is plotted as \na function of SteFo for different concentrations and matrix diameters. Under identical \nporous media parameters, the increase in concentration leads to the increase of \nmovement speed of the melting interface. While under identical concentration, the \nreduction in the diameter of the porous medium matrix results in the increased average \nmovement rate of the melting interface. The variation of the average displaced distance \nof the melting interface can be divided into two stages. The first phase lasts for a about \n𝐿(𝑆𝑡𝑒𝐹𝑜)ିభ\nమ until the end of the heat conduction. The correlations of the second phase \nis approximately  𝐿𝑆𝑡𝑒𝐹𝑜𝑅𝑎ିభ\nర .Both of the expressions show alliance with the \nexperimental data. \nConclusion \nAn experimental visualization of the melting process of a frozen binary solution \nwithin a square cavity subjected to lateral heating is performed. The experiments \nemploy a 3D-printed transparent square cavity filled containing porou matrix, allowing \nfor the observation and order-of-magnitude analysis of the fluid dynamics associated \nwith phase change heat transfer governed by double diffusion. Thermal and solutal \ngradients both affected the heat transfer and melting process. Thermal convection \ninduced by thermal gradient promoted the melting of the upper section, while salt \nperticipated enhanced the melting of the lower section. The following conclusions were \nobtained from the study: \n (1) During melting in the hypoeutectic side, a protruding melting region at the \nlower part of the melting interface, attributed to concentration stratification and the \nconcentration-induced reduction in melting point. When melting in the hypereutectic \nside, the entire melting interface is downward curved due to the fact that salt \nstratification affects the flow pattern to a greater extent than thermal convection. \n (2) The melting process is discernibly delineated into three distinct stages, with soil \nmatrix influencing flow patterns, thereby impacting heat transfer and melting rates. By employing dimensionless analysis, a correlation expression for the Nusselt (Nu) \nnumber has been derived. The correlation equation reveals that the melting process is \npredominantly influenced by several key parameters, including the buoyancy ratio, \nDarcy number, sample height, thermal diffusivity, and Rayleigh number. These factors \ncollectively govern the convective heat transfer characteristics during the melting \nprocess and provide valuable insights into the underlying physics of the system. \n(3) Through order of magnitude analysis, in the first stage of melting, heat \nconduction is dominant and the 𝑁𝑢  number is mainly affected by the temperature \ndifference, the thermal diffusivity of the solution and the height of the square cavity. In \nthe second stage, 𝑁𝑢  number are positively related to 𝑅𝑎  , incorprating the \ninfluence of permeability, buoyancy ratio and heating power. Larger 𝑅𝑎indicates a \nhigher intensity of the thermal natural convection. Also, the relationship between \nthe 𝑁𝑢  number and SteFo becomes complex. In the thrid stage, the Nu number is \nindependent of SteFo and depends only on the permeability of the porous medium, the \ndensity of heat flow and the buoyancy ratio. \nBased on the expressions obtained from the order-of-magnitude analysis, it is \npossible to estimate the time scale required to melt a given amount of salty ice for \ndifferent input powers and using different heights of heater, as well as for different \nthermal conductivities, permeabilities, and salinities, which may provide insights for \nthe related applications. \nReferences \nBennacer, R., Mohamad, A.A., Akrour, D., 2001. Transient natural convection in an \nenclosure with horizontal temperature and vertical solutal gradients. Int. J. \nTherm. Sci. 40, 899–910. https://doi.org/10.1016/S1290-0729(01)01276-5 \nBourich, M., Hasnaoui, M., Amahmid, A., 2004. A scale analysis of thermosolutal \nconvection in a saturated porous enclosure submitted to vertical temperature and \nhorizontal concentration gradients. Energy Convers. Manag. 45, 2795–2811. \nhttps://doi.org/10.1016/j.enconman.2004.01.010 Brimblecombe, P., Clegg, S.L., Davies, T.D., Shooter, D., Tranter, M., 1987. \nObservations of the preferential loss of major ions from melting snow and \nlaboratory ice. Water Res. 21, 1279–1286. https://doi.org/10.1016/0043-\n1354(87)90181-3 \nJany, P., Bejan, A., 1988. Scaling theory of melting with natural convection in an \nenclosure. Int. J. Heat Mass Transf. 31, 1221–1235. \nhttps://doi.org/10.1016/0017-9310(88)90065-8 \nKumar, A., Dutta, P., Sundarraj, S., Walker, M.J., 2007. Remelting of solid and its \neffect on macrosegregation during solidification. Numer. Heat Transf. Part A \nAppl. 51, 59–83. https://doi.org/10.1080/10407780600710391 \nMellerowicz, B., Zacny, K., Palmowski, J., Bradley, B., Stolov, L., Vogel, B., Ware, \nL., Yen, B., Sabahi, D., Ridilla, A., Nguyen, H., Faris, D., Van Susante, P., \nJohnson, G., Putzig, N.E., Hecht, M., 2022. RedWater: Water Mining System for \nMars. New Sp. 10, 166–186. https://doi.org/10.1089/space.2021.0057 \nMohamad, A.A., Bennacer, R., 2001. Natural convection in a confined saturated \nporous medium with horizontal temperature and vertical solutal gradients. Int. J. \nTherm. Sci. 40, 82–93. https://doi.org/10.1016/S1290-0729(00)01182-0 \nOueslati, F.S., Bennacer, R., Sammouda, H., Belghith, A., 2008. Thermosolutal \nconvection during melting in a porous medium saturated with aqueous solution. \nNumer. Heat Transf. Part A Appl. 54, 315–330. \nhttps://doi.org/10.1080/10407780802084637 \nTerwilliger, J.P., Dizon, S.F., 1970. Salt rejection phenomena in the freezing of saline \nsolutions. Chem. Eng. Sci. 25, 1331–1349. https://doi.org/10.1016/0009-\n2509(70)80010-0 \nTrevisan, O. V., Bejan, A., 1990. Combined Heat and Mass Transfer by Natural \nConvection in a Porous Medium. Adv. Heat Transf. 20, 315–352. \nhttps://doi.org/10.1016/S0065-2717(08)70029-7 \nTrevisan, O. V., Bejan, A., 1985. Natural convection with combined heat and mass \ntransfer buoyancy effects in a porous medium. Int. J. Heat Mass Transf. 28, \n1597–1611. https://doi.org/10.1016/0017-9310(85)90261-3  "    },    {        "title": "2401.12022v1.Damping_Enhanced_Magnon_Transmission.pdf",        "content": "Damping-Enhanced Magnon Transmission\nXiyin Ye,1Ke Xia,2Gerrit E. W. Bauer,3, 4and Tao Yu1,∗\n1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China\n2School of Physics, Southeast University, Jiangsu 211189, China\n3WPI-AIMR and Institute for Materials Research and CSRN, Tohoku University, Sendai 980-8577, Japan\n4Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing 100190, China\n(Dated: January 23, 2024)\nThe inevitable Gilbert damping in magnetization dynamics is usually regarded as detrimental\nto spin transport. Here we demonstrate in a ferromagnetic-insulator–normal-metal heterostructure\nthat the strong momentum dependence and chirality of the eddy-current-induced damping causes\nalso beneficial scattering properties. Here we show that a potential barrier that reflects magnon wave\npackets becomes transparent in the presence of a metallic cap layer, but only in one direction. We\nformulate the unidirectional transmission in terms of a generalized group velocity with an imaginary\ncomponent and the magnon skin effect. This trick to turn presumably harmful dissipation into useful\nfunctionalities should be useful for future quantum magnonic devices.\nIntroduction .—Magnonic devices save power by ex-\nploiting the collective excitations of the magnetic or-\nder, i.e., spin waves or their quanta, magnons, for non-\nreciprocal communication, reprogrammable logics, and\nnon-volatile memory functionalities [1–10]. The possibil-\nity to modulate magnon states and their transport in fer-\nromagnets by normal metals or superconductors brings\nfunctionalities to spintronics [11–14], quantum informa-\ntion [15–21], and topological materials [22, 23]. The pre-\ndiction of inductive magnon frequency shifts by supercon-\nducting gates on magnetic insulators [24–30] have been\nexperimentally confirmed [31]. Normal metals are not\nequally efficient in gating magnons [32–35], but the stray\nfields of magnetically driven “eddy currents” [36–43] sig-\nnificantly brake the magnetization dynamics [36].\nThe intrinsic Gilbert damping seems to be detrimental\nto transport since it suppresses the magnon propagation\nlength. However, in high-quality magnets such as yt-\ntrium iron garnet (YIG) films, this is not such an issue\nsince the magnon mobility is often limited by other scat-\ntering processes such as two-magnon scattering by disor-\nder, and measurements can be carried out in far smaller\nlength scales.\nNatural and artificial potential barriers are impor-\ntant instruments in electronics and magnonics by confin-\ning and controlling the information carriers. They may\nguide magnon transport [31, 44], act as magnonic logic\ngate [45], induce magnon entanglement [18, 46], and help\ndetecting exotic magnon properties [47–50]. In the lin-\near transport regime, the transmission of electrons and\nmagnons through an obstacle has always been assumed\nto be symmetric, i.e., the same for a wave or particle\ncoming from either side.\nIn this Letter, we address the counter-intuitive ef-\nfect that the strong momentum-dependent eddy-current-\ninduced damping by a normal metal overlayer as shown\nin Fig. 1 may help surmount obstacles such as mag-\nnetic inhomogeneities [51], artificial potential barriers\nformed by surface scratches [52], or dc-current carryingwires [46]. Here we focus on the band edges of magnetic\nfilms that are much thinner than the extinction length of\nthe Damon-Eshbach surface states in thick slabs and are\ntherefore not chiral. Instead, the effect therefore origi-\nnates from the Oersted fields generated by the eddy cur-\nrents in the overlayer that act in only half of the recip-\nrocal space [7] and causes magnon accumulations at the\nsample edges or magnon skin effect [8, 9]. The trans-\nmission through a barrier that is small and symmetric\nfor magnons with opposite wave numbers in an uncov-\nered sample becomes unidirectional with the assistance\nof dissipative eddy currents.\nFIG. 1. Ferromagnetic insulator-normal metal heterostruc-\nture. An in-plane external magnetic field H0orients the\nmagnetization at an angle θwith the ˆz-direction. The yellow\nsheet between the normal metal and ferromagnetic insulator\nindicates suppression of the exchange interaction and conven-\ntional spin pumping.\nModel and non-perturbation theory .—We consider the\nferromagnetic insulator (FI)-normal metal (NM) het-\nerostructure with thickness 2 dFanddMand an in-plane\nmagnetic field H0in Fig. 1. The saturated equilib-\nrium magnetization Msmakes an angle θwith the ˆz-\ndirection such that the torques exerted by the external\nand anisotropy fields cancel. For convenience, we set\nθ= 0 in the following discussion and defer results forarXiv:2401.12022v1  [cond-mat.mes-hall]  22 Jan 20242\nfinite θto the Supplemental Material (SM) [53]. We gen-\neralize a previous adiabatic theory [7, 36] to the full elec-\ntrodynamics of the system by self-consistently solving the\nMaxwell equations coupled with the linearized Landau-\nLifshitz (LL) equations and Ohm’s Law. This treatment\nbecomes exact in the limit of an instantaneous response\nof the metal electrons and high-quality ultrathin mag-\nnetic films.\nThe driving force is an externally generated spatiotem-\nporal magnetization dynamics M(r, t) =M(r, ω)e−iωtat\nfrequency ω. According to Maxwell’s theory, the electric\nfieldEobeys the wave equation ∇2E(r, ω)+k2\n0E(r, ω) =\n−iωµ0JM, where the wave number k0=ω√µ0ε0,µ0(ε0)\nis the vacuum permeability (permittivity), and JM=\n∇×Mis the “magnetization current” [54]. Disregarding\nthe intrinsic Gilbert damping, the LL equation\niωM=−µ0γM×Heff[M] (1)\ngoverns the magnetization dynamics in the FI, where γ\nis the gyromagnetic ratio. The effective magnetic field\nHeff[M] =−δF[M]/δM(r), where the free energy Fis\na functional of the magnetization. It includes the static\nfieldH0, the dipolar field Hd, and (in the FI) the ex-\nchange field Hex=αex∇2Mthat depends on the spin-\nwave stiffness αex. In the presence of the NM layer,\nHeff[M] also contains the Oersted magnetic fields gen-\nerated by the “eddy” currents J=σE, where the elec-\ntrical conductivity σis real. This defines a closed self-\nconsistency problem that we solve numerically.\nWe consider a thin FI film with constant Ms=\n(0,0, Ms). The transverse fluctuations M(r, ω) =\n(Mx(k, ω), My(k, ω),0)eik·rwith in-plane wave vectors\nk= (0, ky, kz) are small precessions with iMx(k, ω) =\nakMy(k, ω), where the complex ellipticity akbecomes\nunity for circular motion.\nThe electric-field modes outside the magnet are plane\nwaves with wave numbers km=p\nω2µ0ε0+iωµ0σ,\nwhere σ= 0 in the absence of an NM layer. The continu-\nity of electric and magnetic fields provides the interface\nboundary conditions. The field in the FI\nEη={x,y,z}(−dF⩽x⩽dF)\n=E(0)\nη(−dF⩽x⩽dF) +RkE(0)\nη(x=dF)e−iAk(x−dF)\nis now modified by the reflection coefficient\nRk=\u0000\nA2\nk−B2\nk\u0001\neiBkdM−\u0000\nA2\nk−B2\nk\u0001\ne−iBkdM\n(Ak−Bk)2eiBkdM−(Ak+Bk)2e−iBkdM,(2)\nwhere E(0)is the solution of Eq. (1) inside the FI without\nthe NM cap [53], Ak=p\nk2\n0−k2, and Bk=p\nk2m−k2.\nThe reflection is isotropic and strongly depends on the\nwave vector. Naturally, Rk= 0 when dM= 0. On the\nother hand, when |k|= 0, the electric field cannot escape\nthe FI, since the reflection is total with Rk=−1.A corollary of Maxwell’s equation—Faraday’s Law—\nreads in frequency space iωµ0[Hd(r, t) +M(r, t)] =∇ ×\nE(r, t). When the magnetization of sufficiently thin mag-\nnetic films is uniform, the Zeeman interaction is propor-\ntional to the spatial average Hdover the film thickness.\nReferring to SM for details [53], we find\nHd,x=\u0014\n−Rk\n4A2\nkdFak(e2iAkdF−1)2(−iAkak+ky)\n+i\n2AkdF(e2iAkdF−1)\u0015\nMx≡ζx(k)Mx,\nHd,y=\"\n−Rk\n4iAkdF(e2iAkdF−1)2 \n−ky\niAkak+k2\ny\nA2\nk+ 2!\n+k2\ny\nA2\nk−k2\ny\nA2\nk1\n2iAkdF(e2iAkdF−1)#\nMy≡ζy(k)My.\nBy substitution into the LL equation (1), the spin wave\neigenfrequencies and ellipticities become\nω(k) =µ0γq\n(˜H0−ζx(k)Ms)(˜H0−ζy(k)Ms),(3a)\nak=q\n(˜H0−ζy(k)Ms)/(˜H0−ζx(k)Ms), (3b)\nwhere ˜H0=H0+αexk2Ms. Imω(k)̸= 0 because of the\nJoule heating due to the eddy currents in the cap layer.\nChiral damping and frequency shifts .—The stray elec-\ntric fields of spin waves propagating perpendicular to the\nmagnetization are chiral, i.e., they depend on their prop-\nagation direction by a hand rule. When kz= 0,E=Ezˆz\nis along the equilibrium magnetization and Ez∝My\nis complex only for positive ky. We illustrate the re-\nsults of the self-consistent calculations for dF= 100 nm,\ndM= 500 nm, conductivity σ= 6.0×107(Ω·m)−1\nfor copper at room temperature [55], applied magnetic\nfield µ0H0= 0.02 T, µ0Ms= 0.178 T, the exchange\nstiffness αex= 3×10−16m2for YIG [56], and γ=\n1.77×1011(s·T)−1. The presence of the NM cap lay-\ners shifts the relative phases between the stray electric\nfields and that of the generating spin waves. We focus\nhere on the wave numbers ky=±1µm−1in Fig. 2(a)\n[Fig. 2(b)] at which the electric field is in-phase (out-\nof-phase) with the transverse magnetization Myˆy. The\nresponse to an in-phase (out-of-phase) electric field is dis-\nsipative (reactive). Both components decay in the FI and\nthe vacuum as ∝1/|k|. In the NM, the in-phase compo-\nnent is screened only in the metal region on the scale of\na skin depth λ=p\n2/(ωµ0σ)∼1.5µm at ω= 11 GHz.\nThe out-of-phase electric field, on the other hand, cre-\nates only a reactive response and is therefore symmetric\nabove and below the metallic film. Also in this case the\ndamping is modulated for constant Gilbert damping by\nthe associated spin wave frequency shift in Fig. 2(b), an\neffect that cannot be captured by the adiabatic approxi-\nmation [7, 36].3\nFIG. 2. The system responds strongly to a phase difference\nbetween the spin waves and their wave vector-dependent ac\nelectric stray fields E. ReEcauses damping [(a)] and Im E\na frequency shift [(b)]. Im Ezgoverns the spin wave vector\ndependence of the chiral damping [(c)]. (d) illustrates the\nstrong ky-dependence of the damping of the lowest standing\nspin wave for Cu thicknesses dM={50,100,200,500}nm. (e)\nshows the real and imaginary parts of the reflection coefficient\nRkthat causes the frequency shifts plotted in (f).\nThe chirality of the radiated electric field controls the\nbackaction of the NM layer that modifies the magnon\ndispersion in a chiral fashion. Figure 2(c) illustrates\nthe strong wave vector-dependent damping coefficient\nαeff(k) =|Imωk|/Reωk. Spin waves propagating in the\npositive ˆy-direction decay much faster than those along\nthe negative direction, while the damping for positive\nand negative kzis the same. According to Fig. 2(d), the\ncalculated damping for kz= 0 in Fig. 2(c) increases (de-\ncreases) with the thickness of the Cu (YIG) film. The\nenhancement of the damping saturates for NM thick-\nnesses dN>1/p\nk2+ 1/λ2, depending on the skin depth\n(λ∼1.5µm) and the wave number 1 /kof the electric\nfield. Moreover, the Kittel mode at k= 0 in Fig. 2(e)\nis not affected by the metal at all because the reflection\ncoefficient Rk=−1, which implies that the dynamics\nof the FI and metal fully decouple. Indeed, recent ex-\nperiments do not find a frequency shift of the FMR by a\nsuperconducting overlayer [57, 58]. The additional damp-\ning by eddy currents reported by Ref. [39] is caused bythe width of the exciting coplanar waveguide, a finite-size\neffect that we do not address here.\nThe real part of Rkin Fig. 2(e) causes an in-phase\nOersted magnetic field that chirally shifts the spin wave\nfrequencies by as much as ∼1 GHz, see Fig. 2(f). Refer-\nence [59] indeed reports a frequency shift of perpendicular\nstanding spin wave modes in Bi-YIG films in the presence\nof thin metallic overlayers.\nThe predicted effects differ strongly from those caused\nby spin pumping due to the interface exchange coupling\nαsp= (ℏγ/M sdF)Reg↑↓, where g↑↓is the interfacial spin\nmixing conductance [60]. αspdoes not depend on the\nthickness of the metal and vanishes like 1 /dF. The fre-\nquency shift scales like Im g↑↓/dFand is very small even\nfor very thin magnetic layers. In contrast, the eddy\ncurrent-induced damping is non-monotonic, scaling like\n∝dFwhen 2 kdF≪1, vanishing for much thicker mag-\nnetic layers, and reaching a maximum at dF∼2λ.\nUnidirectional transmission of wave packets through a\npotential barrier .—The transmission of a wave packet im-\npinging from the left or right at a conventional potential\nbarrier is the same [61]. In the presence of a metal cap,\nthis does not hold for magnons in thin magnetic films.\nBefore turning to the potential scattering in this\nmodel, we have to address the effect of the edges. When\nmagnons propagate in the negative direction without\ndamping but decay quickly when propagating in the op-\nposite one, those reflected at the left boundary of the\nsample accumulate, which is a non-Hermitian skin ef-\nfect [62–65]. We substantiate this conclusion by nu-\nmerical calculations for a two-dimensional square lat-\ntice model with ˆ mi= (1 /√\nN)P\nkˆmkeik·ri, where ˆ mk\nis the annihilation operator of magnons with frequency\nωkfrom Eq. (3a) and ilabels the sites and Nis the\nnumber of sites. The Hamiltonian in the real space\nˆH0=P\nijtjiˆm†\njˆmi, where tji= (1/N)P\nkℏωkeik·(rj−ri)is\na hopping amplitude between possibly distant sites iand\njand the summation is over the first Brillouin zone. With\na coarse-grained lattice constant of ay=az= 0.1µm the\nreciprocal lattice vector 2 π/ay,zis much larger than the\nmagnon modes of interest (refer to the SM [53] for de-\ntails). When the frequencies ωkare complex, the Hamil-\ntonian is non-Hermitian, i.e.,tji̸=t∗\nij.\nFigure 3(a) shows the winding path of the real and\nimaginary eigenfrequencies with wave number. In the\ninterval ky= [−25,25]µm−1and an applied magnetic\nfield parallel to the boundary with θ= 0, the complex\ncomponent is hysteretic, indicating localization of modes\nat opposite boundaries. Figure 3(b)-(c) show the average\nspatial distributions W(r) = (1 /Nm)PNm\nl=1|ϕl(r)|2ofNm\nlowest-frequency eigenstates ϕl(r) for ky∈[−1,1]µm−1\nandkz∈[−1,1]µm−1. When the static magnetic field\naligns with the sample boundary z-axis, i.e.θ= 0 in\nFig. 3(b), the magnons tend to accumulate at the left4\nedge. In the antiparallel configurations θ=π[Fig. 3(c)],\nthe magnons aggregate at the right. In the noncollinear\nconfiguration with θ=π/4 [Fig. 3(d)], the maxima shifts\nto the upper-left corner. While Wis an average, we\nalso illustrate the localization of individual low-frequency\nmodes in SM [53].\nFIG. 3. The magnon skin effect caused by chiral damping.\n(a) Complex spectral winding under periodic boundary con-\nditions when kyevolves from −25 to 25 µm−1forθ= 0. (b)-\n(d) corresponds to the edge or corner aggregations of magnon\neigenstates for other magnetic configurations θ∈ {0, π, π/ 4}.\nWe now illustrate the effect of square potential barri-\ners of width dand height u0,ˆV(y) =u0[Θ(y+d/2)−\nΘ(y−d/2)], where Θ( x) is the Heaviside step function,\non the magnon transmission along ˆy(⊥Ms). With in-\ncoming ⟨y|k0⟩=eik0y, the scattered states |ψs⟩obey the\nLippmann-Schwinger formula [66]\n|ψs⟩=|k0⟩+1\niℏ∂t−ˆH0+i0+ˆV|ψs⟩. (4)\nwhere ˆH0=P\nkℏωkˆm†\nkˆmkis the magnon Hamiltonian for\nan extended film. The transmitted waves read\n⟨y|ψs⟩=\u001aT+(k0)eik0y,{y, k0}>0\nT−(k0)eik0y,{y, k0}<0. (5)\nIn the weak scattering limit |u0d| ≪ | ℏvk0|,\nT±(k0) = 1±\u0012iℏvk0\nu0d−vk0\n2|vk0|\u0013−1\n≈1∓iu0d\nℏvk0,(6)\nwhere vk0=∂ωk/∂k|k=k0ˆyis a generalized group ve-\nlocity that dissipation renders complex. The imaginary\npart of the group velocity and transmission amplitudes\ndepend on the direction of the incoming wave:\nD±(k0) =|T±(k0)|2≈1±2Im\u0012u0d\nℏvk0\u0013\n. (7)For example, with u0/ℏ= 30.5 GHz, d= 0.1µm,k0=\n±0.8µm−1,vk0>0= (2.32 + 0 .52i) km/s and vk0<0=\n−(2.64 + 0 .16i) km/s lead to T+(k0>0)≈0.6 while\nT−(k0<0)≈0.9, so even in the weak scattering limit the\nNM cap layer significantly and asymmetrically reduces\nthe transmission probability.\nWe can assess the strong scattering regime with |u0d|≳\n|ℏvk0|by numerical calculations but find dramatic ef-\nfects on the time evolution of a real-space spin-wave\npacket as launched, e.g., by a current pulse in a mi-\ncrowave stripline. We adopt a Gaussian shape Ψ( r,0) =\ne−(r−r0)2/(2η2)eiq0·rcentered at r0with a width η≫ay,z\nthat envelopes a plane wave with wave vector q0and\nˆV(r) = u0f(r) with either f(|y−˜y0|< d) = 1 or\nf(|z−˜z0|< d) = 1, where ˜ y0and ˜z0are the center of the\nbarriers. According to Schr¨ odinger’s equation Ψ( r, t) =\neiˆHt/ℏΨ(r, t= 0) with ˆH=ˆH0+ˆV(r). Numerical results\nin Fig. 4(a) and (b) u0d≪ |ℏvk0|agree with perturbation\ntheory (7) in the weak scattering regime. However, when\n|ℏvk0|≲u0dand|Im(v−k0)| ≪ | Im(vk0)|≲|Re(v±k0)|\nthe transmission and unidirectionality becomes almost\nperfect. Figure 4(c) and (d) show a nearly unidirectional\ntransmission of the wave packet through the potential\nbarrier for the Damon-Eshbach configuration q0⊥Ms;\nit is transparent for spin waves impinging from the left,\nbut opaque for those from the right. In the calculations,\nq0=q(0)\nyˆywith q(0)\ny=±5µm−1andη= 3µm≫d.\nThe potential barrier is peaked with d=ay,z= 0.1µm\nand its height u0/ℏ= 15 GHz is relatively weak (the\nregular on-site energy ∼13 GHz). Also, dM= 50 nm\nanddF= 20 nm. The results are insensitive to the de-\ntailed parameter values (see SM [53]). The red and blue\ncurves are the incident and reflected wave packets, re-\nspectively. When q(0)\ny<0, the barrier does not affect the\nwave packet that propagates freely through the poten-\ntial barrier and accumulates on the left edge [Fig. 4(c)].\nWhen q(0)\ny>0, as shown in Fig. 4(d), the barrier reflects\nthe wave packet nearly completely, which we associate\nagain with the skin effect since these magnons cannot ac-\ncumulate on the right side. The unidirectional transmis-\nsion is therefore a non-local phase-coherent phenomenon\nthat involves the wave function of the entire sample.\nSince we find the skin effect to be crucial, its absence\nin waves propagating in the ˆz-direction must affect the\ntransport over the barrier. Indeed, our calculations in\nFig. 4(e) and (f) find strong reflection for both propaga-\ntion directions, even when reducing the barrier height by\nan order of magnitude to u0= 1.5 GHz (see SM [53]).\nDiscussion and conclusion .—In conclusion, we cal-\nculate the chiral damping, chiral frequency shift, and\nanomalous transport of magnonic modes in ferromag-\nnetic films with NM cap layers beyond the adiabatic ap-\nproximations. We predict anomalous unidirectional spin\ntransport over potential barriers. This effect is rooted\nin the non-Hermitian magnon skin effect and reflects the5\nFIG. 4. Calculated transmissions [(a) and (b)] and time evolu-\ntion of spin-wave packets in the presence of a potential barrier\nat the origin when q0⊥Ms[(c) and (d)] and q0∥Ms[(e)\nand (f)], where Msand the applied magnetic field are parallel\nto the sample edge with θ= 0. The red and blue curves rep-\nresent, respectively, the incident and scattered wave packets\nwith propagation directions indicated by arrows.\nglobal response of the entire system to a local perturba-\ntion. 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Using eight cluster lens\nmodels for supernova Refsdal, we infer H0= 64.2+4.4\n−4.3km s−1Mpc−1and using two cluster models most\nconsistent with the observations, we infer H0= 66.3+3.8\n−3.6km s−1Mpc−1. Our estimations of H0are in\n1σagreement with the results assuming a flat ΛCDM model and the uncertainties are comparable.\nOur constraint results on H0from the eight lens models and the two lens models indicate 2 σand 1 .8σ\ntension with that estimated by SH0ES, respectively. However, our most probable values of H0from\nthe two sets of lens models show good consistency with H0inferred from Planck CMB observations\nassuming ΛCDM model within 1 σ. We also find that our results for H0indicate 2 σdeviations and\n1.7σdeviations from the constraint results of H0using six time-delay quasars by H0LiCOW with the\nsame analysis method.\nKeywords: Hubble Constant — gravitational lensing — supernovae\n1.INTRODUCTION\nThe Hubble constant ( H0) describes the present ex-\npansion rate of the Universe, defined as H0≡˙a/a, when\nthe scale factor a= 1 or z= 0, where ˙ a= da/dt.\nCurrently, there is a significant tension about Hubble\nconstant arising from the discrepancy between the local\nmeasurement of H0bythe Supernova H0 for the Equa-\ntion of State (SH0ES) collaboration (Riess et al. 2016,\n2018, 2019, 2022), and the estimation of H0using Planck\ncosmic microwave background (CMB) and other cosmo-\nlogical observations assuming ΛCDM model (Ade et al.\n2016; Aghanim et al. 2018). The tension has reached\n>5σlevel, indicating a potential problem with current\nstandard cosmology. Either there exists unknown sys-\ntematic errors in the observations or there are cosmo-\nlogical models beyond standard ΛCDM model.\nThe time delays from strong gravitational lensing sys-\ntems provide a part-directly method to determine H0\nrelying on well know physics (i.e., gravity) (Millon et al.\nCorresponding author: Kai Liao\nliaokai@whu.edu.cn2020). Time-delay distance D∆tcould be obtained with\nadditional information of lensing potential and line-of-\nsight environment measurements for a certain lensing\nsystem. D∆tis a combination of three angular diame-\nter distances and depends on H0and other cosmologi-\ncal parameters. Thus time-delay lensing could be used\nto determine H0. Currently, time-delay cosmography\nof lensed quasars has achieved percent level precision on\nthe measurements of H0(Millon et al. 2020; Birrer et al.\n2020).\nStrongly Lensed supernavae (SLSNe) could also pro-\nvide time-delay measurements with higher precision and\nrequiring shorter observation periods ∼months (Oguri\n& Kawano 2003; Oguri 2019; Dobler & Keeton 2006;\nGoobar et al. 2017; Suyu et al. 2023; Liao et al. 2022).\nIn 2014, Hubble Space Telescope (HST) observed four\nimages of a single SN forming an Einstein cross config-\nuration around an elliptical galaxy at redshift z= 0.54\nin the MACS J1149.6+2223 cluster. The cluster’s grav-\nitational potential also creates multiple images of the\nspiral SN host galaxy at z= 1.49. Lens models of the\ngalaxy cluster predicted that an additional image of SN\nwould be observed in 2015 (Kelly et al. 2015), which\nwas detected by HST on 2015 December 11 UT (KellyarXiv:2401.12052v1  [astro-ph.CO]  22 Jan 20242\net al. 2016). The reappearance of SN was used to esti-\nmate the relative time delays and performed a blinded\nmeasurement of H0(Kelly et al. 2023). The authors in-\nferred H0= 64.8+4.4\n−4.3km/s/Mpc using eight cluster lens\nmodels and H0= 66.6+4.1\n−3.3km/s/Mpc using two cluster\nlens models most consistent with observations.\nHowever, in the work of Kelly et al. (2023), the au-\nthors fixed the matter density Ω M= 0.3 assuming the\nflat ΛCDM model to infer the value of H0with time\ndelay measurements from the lensed SN. They claimed\nthat the effect of the uncertainty from fixing Ω Mand\nΩΛis smaller than ∼0.7 km s−1Mpc−1given the weak\ndependence of H0on Ω Mand Ω Λ. Note that the key\nproblem on H0is related with whether the flat ΛCDM\nmodel is the correct description. In a non-ΛCDM model\nframework, the bias could be unforeseeable. Another\npoint to be concerned is that, their estimation of H0is\nconsistent with H0= 67.36±0.54 km s−1Mpc−1which\nwas obtained from Planck CMB observations assuming\nΛCDM model (Aghanim et al. 2018) while in tension\nwith the latest local measurements from SH0ES which\ngave H0= 73 .04±1.04 km s−1Mpc−1(Riess et al.\n2022). Therefore, it is worth double-check the inferred\nH0value in different viewpoints.\nIn this work, we are going to constrain the value of H0\nmodel-independently based on Gaussian process (GP)\nregression. By using GP regression, we generate pos-\nterior samples of unanchored SN distance cosmological-\nindependently and anchor them with the lensed SN sys-\ntem.\nThis paper is organized as follows: in Section 2, we\nbriefly introduce the data and the analysis method. Our\nresults and discussions are presented in Section 3. Fi-\nnally, we conclude in Section 4.\n2.DATA AND METHOD\nIn our analysis, we use the time-delay measurements\nfrom the reapearance of SN Refsdal in combination with\nSN data set from Pantheon+ (Scolnic et al. 2022).\nThe reappearance of SN Refsdal, designed as SX was\ndetected in HST observations taken on 2015 Decemple\n11 UT and the observation was used to estimate the rel-\native time delay between SX and the previous image, S1,\nas 320-380 days. Given the matter distribution in the\nforeground MASCJ1149 cluster lens, one can constrain\nthe value of H0with time-delay cosmography.\nIn our work, we are going to use time-delay measure-\nments of SN Refsdal considering two sets of lens models\nto make constraints on H0following the work in Kelly\net al. (2023), in which the authors first used a set of pre-\ndictions from eight models, e.g., the Diego-a free-form\nmodel (Treu et al. 2016), the Zitrin-c* light-trace-mass(LTM) model (Treu et al. 2016), and the Grillo-g(Treu\net al. 2016), Oguri-a* and Oguri-g*(Treu et al. 2016),\nJauzas-15.2* (Jauzac et al. 2016) and Sharon-a* and\nSharon-g* (Treu et al. 2016) simply parameterized mod-\nels. The second, the authors only used the Grillo-g and\nOguri-a* models. During our analysis, we use the time-\ndelay results from the eight cluster models as well as the\ntwo cluster models to estimate the value of H0model-\nindependently based on GP regression.\nFor the SN data, we use the so far largest data set,\nthe Pantheon+ sample (Scolnic et al. 2022), to provide\nan unachored cosmological distance.\nWith Pantheon SN data sets, we generate 1000 pos-\nterior samples of DLH0independently of cosmological\nmodel using GP regression (Holsclaw et al. 2010a,b;\nShafieloo et al. 2012; Hwang et al. 2023). The GP re-\ngression used here is based on the GPhist code (Kirkby\n& Keeley 2021) first used in Joudaki et al. (2018). GP\nregression works by generating a random set of cosmo-\nlogical functions whose statics are characterized by a\ncovariance function, in which the kernel plays an im-\nportant role. Moreover, the mean function also plays\nan important role in GP regression and the final recon-\nstruction results are not quite independent of the mean\nfunction, however, it has a modest effect on the final re-\nconstruction results because the values of hyperparame-\nters help to trace the deviations from the mean function\n(Shafieloo et al. 2012; Shafieloo et al. 2013). In our work,\nwe use a squared-exponential kernel for the covariance\nfunction with\n< φ(si)φ(sj)>=σ2\nfexp\u0012\n−|si−sj|2\n2ℓ2\u0013\n(1)\nwhere si= ln(1 + zi)/ln(1 + zmax) and zmax= 2.261 is\nthe maximum redshift of the SN Ia sample. σfandℓare\ntwo hyperparameters which are marginalized over. φis\njust a random function drawn from the distribution de-\nfined by the covariance function of Equation (1) and we\ntake this function as φ(z) = ln\u0000\nHmf(z)/H(z)\u0001\n, where\nHmf(z) is the mean function, which we choose to be the\nbest-fit ΛCDM model from Pantheon+ dataset.\nFor the details of the reconstruction with GP, we refer\nthe readers to (Rasmussen & Williams 2006; Holsclaw\net al. 2010a,b; Shafieloo et al. 2012; Hwang et al. 2023)\nand reference therein.\nThen, we convert the 1000 posterior samples of DLH0\nto unanchored angular diameter distance DAH0with\nDAH0=DLH0\n(1 +z)2. (2)\nAfter that, we evaluate the values of 1000 DAH0at\nthe redshift of the lens galaxy, zd= 0.54 and the SN3\nRefsdal’s host galaxy, zs= 1.49 to get 1000 values of\nD∆tH0with\nH0D∆t(zd, zs) = (1 + zd)(H0Dd)(H0Ds)\n(H0Dds).(3)\nIn the end, we compute the likelihood of H0from each\nof the 1000 realizations in combination with the time-\ndelay measurements, D∆tand then, marginalize over the\nrealizations to form the posterior distribution of H0.\nWe need to note that, throughout this work, a spatial\nflat universe is assumed. Thus, the angular diameter\ndistance between the lens and the source is calculated\nvia\nDds=Ds−1 +zd\n1 +zsDd. (4)\n3.RESULTS AND DISCUSSIONS\nWith unanchored cosmological distance from SNe Ia\nanchored by time-delay distance from SN Refsdal us-\ning GP regression, the posteriors of H0are shown in\nFigure 1. For the cluster models of SN Refsdal, two\nsets of models are considered. The left plot is ob-\ntained with the full set of eight models constructed be-\nfore the reappearance while the right plot is obtained\nwith two pre-appearance models: Grilla-g and Oguri-\na* models using the reapppearance’s position. Our\ncosmological-model-independent constraints give H0=\n64.2+4.4\n−4.3km s−1Mpc−1(best-fit values and the 1 σun-\ncertainties) for eight cluster models of lensed SN Refs-\ndal and H0= 66 .3+3.8\n−3.6km s−1Mpc−1for the two pre-\nappearance models,respectively.\nOur two estimations of H0are within 1 σagreement\nwith the results in Kelly et al. (2023), in which the au-\nthors assumed flat ΛCDM model with fixed Ω M. This\nconsistency confirmed that, the effect by fixing a spe-\ncific cosmological model is smaller given the weak de-\npendence of H0on Ω Mand Ω Λ. Our constraint results\nonH0from the eight lens models and two lens mod-\nels indicate 2 σand 1 .8σtension with that estimated\nby SH0ES in Riess et al. (2022), respectively. How-\never, our most probable values of H0from the two sets\nof lens models show good consistency with H0inferred\nfrom Planck CMB observation assuming ΛCDM model\nwithin 1 σ(Aghanim et al. 2018). Furthermore, we re-\nalize that, the uncertainties for our cosmological-model-\nindependent analysis are comparable to the results as-\nsuming a specific cosmological model, while reducing\npossible bias. The H0values from this lensed SN are\nalso in tension with that obtained from six or seven\ntime-delay measurements of lensed quasars, which give\nH0= 73 .3+1.7\n−1.8km s−1Mpc−1in Wong et al. (2020),\nH0= 74.0+1.7\n−1.8km s−1Mpc−1in Millon et al. (2020) and\nH0= 74.6+5.6\n−6.1km s−1Mpc−1in Birrer et al. (2020).Comparing our constraint results to the time-delay\ncosmography using six time-delay quasars in combina-\ntion with SNe Ia with GP regression in Liao et al. (2020),\nwe find our best-fit results for H0are smaller than H0es-\ntimation from time-delay quasars by 8 .6 km s−1Mpc−1\nfrom the eight cluster models and 6 .5 km s−1Mpc−1\nfrom the two cluster models, which correspond to 2 σ\ndeviations and 1 .7σdeviations, respectively. The domi-\nnant source of uncertainty in H0estimation is the clus-\nter lens model. If the lens cluster is perfectly modeled,\nit will provide a much more precise constraints on the\nvalue of H0.\nIn the end, we show the values of H0from time-delay\nlensing measurements in Figure 2 for direct compari-\nson. The values of H0are adapted from Wong et al.\n(2020); Millon et al. (2020); Birrer et al. (2020); Liao\net al. (2020); Kelly et al. (2023); Aghanim et al. (2018);\nRiess et al. (2022).\n4.CONCLUSION\nWe provide a cosmological-model-independent\nmethod using GP regression to estimate Hubble con-\nstant H0by anchoring the SNe Ia from Pantheon+\nsample with time-delay measurements of SN Refsdal\nassuming a spatial flat universe. Using two sets of\nlens models, we obtained the constraints on H0of\n64.2+4.4\n−4.3km s−1Mpc−1and 66 .3+3.8\n−3.6km s−1Mpc−1.\nOur results are within 1 σconsistent with the results\nfrom Kelly et al. (2023) assuming a specific cosmologi-\ncal model. Despite not assuming a specific cosmological\nmodel, the uncertainties in our constraints are compa-\nrable to the results assuming ΛCDM model, while re-\nducing possible bias towards the ΛCDM model. The\nresults are in tension with that estimated by SH0ES\nin Riess et al. (2022) but agrees well with H0inferred\nfrom Planck CMB observations assuming ΛCDM model\n(Aghanim et al. 2018).\nWe further compare our results to the H0estimation\nusing six time-delay quasars using a same method de-\nscribed in our work (Liao et al. 2020). The results show\nabout 2 σdeviations and 1 .7σdeviations for the two sets\nof lens models, respectively. If we can model the lens\ncluster better in the future, we could obtain a much\nmore precise estimation on the value of H0.\nFor future estimations of H0, current surveys such as\nVera C. Rubin Observatory’s Legacy Survey of Space\nand Time (LSST), Euclid, and Nancy Grace Roman\nSpace Telescope (ROMAN) (Spergel et al. 2015; Ake-\nson et al. 2019; Oguri & Marshall 2010; Collett 2015),\nwill bring us over one hundred lensed SN Ia with well-\nmeasured time delays. Moreover, further analysis will4\n45 50 55 60 65 70 75 80 85\nH0 [km/s/Mpc]PDFH0=64.2+4.4\n4.3\n45 50 55 60 65 70 75 80 85\nH0 [km/s/Mpc]PDFH0=66.3+3.8\n3.6\nFigure 1. The posteriors of H0from Pantheon+ sample in combination with SN Refsdal. The left plot is obtained with the full\nset of eight models constructed before the reappearance while the right is obtained with two pre-appearance models: Grilla-g\nand Oguri-a* models using the reapppearance’s position.\nhelp to improve the cluster model and provide more pre-\ncise constraints on H0.\nSN data will also be improved with ongoing and future\nsurveys. Not only the sample of the SNe will become\nlarger with better precision, but also the redshift cover-\nage will become wider. With future time-delay distance\nmeasurements together with a larger and more precise\nSN Ia data set, one can obtain H0more precisely with\nthe method described in this work and understand the\nH0tension better.ACKNOWLEDGEMENTS\nWe thank Patrick Kelly for providing the posteriors in\ntheir paper. This work was supported by National Natu-\nral Science Foundation of China (NSFC) Nos. 12222302,\n12003006.\nREFERENCES\nAde, P. 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Lett., 819, L8,\ndoi: 10.3847/2041-8205/819/1/L8\n—. 2023, Science, 380, abh1322,\ndoi: 10.1126/science.abh1322\nKirkby, D., & Keeley, R. 2021, gphist: Cosmological\nexpansion history inference using Gaussian processes,\nAstrophysics Source Code Library, record ascl:2109.023.\nhttp://ascl.net/2109.0235\n55 60 65 70 75 80\nH0 [km/s/Mpc]Lensed SNLensed \n Quasars\nthis work\n+SN, model-independently\neight cluster modelstwo cluster modelsKelly et al. 2023depending on flat-CDM\neight cluster modelstwo cluster modelsLiao et al. 2020\n6 time delay lenses+SN,\n model-independently\nH0LiCOWBirrer et al. 2020\n7 time-delay lenses(+33 SLACS lenses), depending on flat-CDM\nTDCOSMO+SLACSSDSS+IFUTDCOSMO+SLACSSDSSTDCOSMO+SLACSIFUTDCOSMO onlyMillon et al. 2020\n6 time-delay lenses(5 H0LiCOW+1STRIDES), depending on flat-CDM\nTDCOSMO(NFW+stars/constant M/L)\nTDCOSMO (power law)Wong et al. 2020\n6 time-delay lenses, depending on flat-CDM\nH0LiCOW (average of PL and NFW + stars/constant M/L)\nFigure 2. 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H., Goobar, A., Collett, T., More, A., &\nVernardos, G. 2023, arXiv e-prints, arXiv:2301.07729,\ndoi: 10.48550/arXiv.2301.07729\nTreu, T., et al. 2016, Astrophys. J., 817, 60,\ndoi: 10.3847/0004-637X/817/1/60\nWong, K. C., et al. 2020, Mon. Not. Roy. Astron. Soc., 498,\n1420, doi: 10.1093/mnras/stz3094"    },    {        "title": "2401.12054v3.Twisting_asymptotic_symmetries_and_algebraically_special_vacuum_solutions.pdf",        "content": "arXiv:2401.12054v3  [gr-qc]  29 Mar 2024Twisting asymptotic symmetries and algebraically special vacuum\nsolutions\nPujian Mao and Weicheng Zhao\nCenter for Joint Quantum Studies and Department of Physics,\nSchool of Science, Tianjin University, 135 Yaguan Road, Tia njin 300350, China\nEmails: pjmao@tju.edu.cn, zhaoweichengok@tju.edu.cn\nABSTRACT : In this paper, we study asymptotic symmetries and algebrai cally\nspecial exact solutions in the Newman-Penrose formalism. R emoving the hyper-\nsurface orthogonal condition in the well studied Newman-Un ti gauge, we obtain\na generic asymptotic solution space which includes all poss ible origins of prop-\nagating degree of freedom. The asymptotic symmetry of the ge neralized system\nextends the Weyl-BMS symmetry by two independent local Lore ntz transforma-\ntions with non-trivial boundary charges, which reveals new boundary degrees\nof freedom. The generalized Newman-Unti gauge includes alg ebraically special\ncondition in its most convenient form. Remarkably, the gene ric solutions satis-\nfying the algebraically special condition truncate in the i nverse power of radial\nexpansions and the non-radial Newman-Penrose equations ar e explicitly solved\nat any order. Hence, we provide the most general algebraical ly special solution\nspace and the derivation is self-contained in the Newman-Pe nrose formalism.\nThe asymptotic symmetry with respect to the algebraically s pecial condition is\nthe standard Weyl-BMS symmetry and the symmetry parameters consist only\nthe integration constant order. We present the Kerr solutio n and Taub-NUT so-\nlution in the generalized Newman-Unti gauge in a simple form .Contents\n1 Introduction 2\n2 NP formalism with twist 5\n3 Twisting solution space in NP formalism 8\n4 Twisting asymptotic symmetries and charges 10\n5 Algebraically special solutions 15\n5.1 Solution space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n5.2 Asymptotic symmetries and transformation law . . . . . . . . . . . . . . 18\n6 Exact solutions with twist 19\n6.1 Kerr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n6.2 Taub-NUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n7 Applications and future directions 21\nA Finite twisting asymptotic symmetry transformations 22\n1 Introduction\nExact solutions are of fundamental importance in General re lativity and other theories of\ngravity. Normally, the equations of motion of gravity theor y are very complicated due\nto their highly non-linear property. Certain simplificatio ns or approximations are essen-\ntial in obtaining solutions. The Petrov classification [ 1] provides a very powerful tool\nto classify exact solutions. Algebraically special condit ion will significantly reduce the\nsystem and exact solutions in the reduced system are much mor e tractable [ 2]. While, if\none cares more about the asymptotic behavior of the solution , Bondi and collaborators\nestablished an elegant framework to formulate the Einstein equation as a characteristic\ninitial value problem [ 3,4]. Meanwhile, Newman and Penrose (NP) construct a remark-\nable formalism to understand gravitational theory by means of four null tetrad bases [ 5].\nUnder the Newman-Unti (NU) gauge [ 6], the results in Bondi-Sachs (BS) system [ 3,4]\ncan be equivalently obtained in the NP formalism. The NU syst em is connected to the BS\nsystem by radial coordinate transformation [ 7]. In principle, any asymptotically flat solu-\ntion can be put into the BS or NU system in series expansion. Ho wever they are in some\n2sense not always convenient for exact asymptotically flat so lutions, nor some generalized\nsystems [ 8–12]. The reason of the inconvenience is from the choice of diffe rent null di-\nrections. The NU system is built on a hypersurface orthogona l null direction. While to\ncharacterize algebraically special solution, the princip le null direction (PND) is the best\nchoice for representing exact solutions. In this paper, we w ill present an asymptotic anal-\nysis in the NP formalism by removing the hypersurface orthog onal condition, which we\nrefer to as generalized NU system. The main motivation is to i nclude exact solutions,\nsuch as Kerr solution [ 13] and Taub-NUT solution [ 14,15] in a simple form.\nThe advantage of the NP formalism is that it manifests the hyp ersurface orthogonal\ncondition and other geometric meanings of certain gauge cho ices. In other words, one can\napply the gauge choice to reduce the system in a very direct wa y. The other advantage in\nparticular for asymptotic analysis in the NP formalism is en coded in its first order nature.\nFirst order differential conditions in metric formalism be come algebraic conditions in\nfirst order formalism and one needs, in principle, one less or der of symmetry parameters\nand solutions for computing the charges. In the NP formalism , the gauge transformation\nconsists of diffeomorphism and local Lorentz transformati ons. Removing the hypersur-\nface orthogonal condition leads to independent Lorentz tra nsformations, which extends\nthe Weyl-BMS symmetry [ 16–18]. We refer to the new residual Lorentz symmetry as\ntwisting asymptotic symmetry. The full asymptotic symmetr y is complete in the sense\nthat all the radial dependence of the symmetry parameters ar e fixed. There are non-trivial\nfinite charges associated to the twisting asymptotic symmet ries which indicates that they\npresent new boundary degrees of freedom.\nThe obtained solution space includes all possible origins o f the propagating degree of\nfreedom, which consists of three parts encoded in the evolut ion of the asymptotic shear\nσ0, the tetrad field M0which determines the twist, and the conformal factor of the t wo\nsurfaceP.1The reduction of our results to the NU system is transparent, which is simply\ngiven by turning off the tetrad field M0. The transformation laws of those fields indicate\nthat one can turn off two of them without losing any propagati ng degree of freedom. A\nmore appreciated choice is to turn off the twist and keep the a symptotic shear [ 3,4,6].\nIn the second part of this paper, we consider the alternative choice. The main motiva-\ntion is precisely from exact solutions. In the other choice, one has the possibility to set\nthe full shear tensor σto zero which incorporates with algebraically special cond ition.\nAccording to the Goldberg-Sachs theorem [ 20], for any algebraically special spacetime,\none can turn off the NP variables κandσ, where the repeated PND is chosen as one\nof the real NP tetrad basis. This null direction is normally n ot hypersurface orthogonal\nfor exact solutions such as Kerr or Taub-NUT solution. We imp lement the simplification\nfrom the algebraically special condition and perform a self -contained asymptotic analysis\n1The degree of freedom from Pis relevantly independent. A common choice is first to fix it as a round\nsphere or punctured complex plane [ 16,19].\n3in the NP formalism. We find that the solution space has a remar kable simplification in\nthe radial direction. In terms of series expansion in the com plex expansion ρandρ, the\nsolution space is given in a truncated form. We exactly verif y that the non-radial NP equa-\ntions only provide constraints at their leading order which was proposed by Newman and\nUnti [ 6]. To the best of our knowledge, this has never been proven or v erified elsewhere\nfor a generic algebraically special solution space in the NP formalism. This truncation is\nnew in the NP formalism, it is known in metric formalism, see e .g., in Chapter 27 and 29\nof [2], and the original references therein. The motivation of ou r derivation in the NP for-\nmalism is that, apart from verifying the NU conjecture for al gebraically special solution,\nsome geometric conditions and relations are better appreci ated from a self-contained NP\nanalysis than translation from the metric formalism. Note a lso that the translation itself\nmay not be straightforward. Our NP derivation is coordinate independent in the sense\nthat we do not need any specific arrangement of the coordinate s system. The only needed\ningredient is the affine parameter of the null geodesic assoc iated to the repeated PND.\nOur derivation has one technical difference than that in [ 2]. We choose a different gauge\ncondition by setting α0`β0“0, while the choice of [ 2] isτ0“0in the NP convention.\nThough the choice τ0“0may be simpler in part of the computation, our choice is for\napplying the extremely valuable ðoperator. Due to the existence of twist, it is more con-\nvenient to generalize the ðoperator by including a time derivative piece. We denote the\ngeneralized opertator as ðn. The commutator of the generalized operator ðnincludes also\na twist piece. We perform the reduction of the asymptotic sym metry from the generic case\nto the algebraically special case. Remarkably, we recover t he standard Weyl-BMS sym-\nmetry [ 16–18] and all the symmetry parameters consist of only the integra tion constant\npiece. This is another realization of the BMS symmetry, whic h is much more tractable\nconsidering the significant simplifications in the radial di rection. We also present the Kerr\nsolution and Taub-NUT solution in the generalized NU system . They are given in a very\nsimple form.\nThe organization of this paper is as follows. In the next sect ion, we review the NP\nformalism and specify the conditions with a twist. In sectio n3, we derive the asymp-\ntotic solution space with a twist. In section 4, we compute the asymptotic symmetry of\nthe generalized NU system and the associated twisting charg es. In section 5, we apply\nthe algebraically special condition in the asymptotic anal ysis. The solution space and\nasymptotic symmetries in closed form are obtained. Section 6presents the Kerr solution\nand Taub-NUT solution in the generalized NU system. We close with some comments\non future directions in the last section. There is one Append ix which provides a parallel\nderivation of asymptotic symmetries in terms of finite trans formations.\n42 NP formalism with twist\nThe NP formalism [ 5] is a tetrad formalism with two real null basis, denoted as e1“l“\ne2, e2“n“e1, and two complex null basis e3“m“ ´e4, e4“¯m“ ´e3. The two\ncomplex basis are conjugate of each other to guarantee that t he metric constructed from\nthe tetrad is real. The null basis vectors are set to have the f ollowing orthogonality and\nnormalization conditions,\nl¨m“l¨¯m“n¨m“n¨¯m“0, l ¨n“1, m ¨¯m“ ´1. (1)\nThe spin connection is labelled by 12 complex scalars denote d by Greek symbols,\nκ“Γ311“lνmµ∇νlµ, π“ ´Γ421“ ´lν¯mµ∇νnµ,\nǫ“1\n2pΓ211´Γ431q “1\n2plνnµ∇νlµ´lν¯mµ∇νmµq,\nτ“Γ312“nνmµ∇νlµ, ν“ ´Γ422“ ´nν¯mµ∇νnµ,\nγ“1\n2pΓ212´Γ432q “1\n2pnνnµ∇νlµ´nν¯mµ∇νmµq,\nσ“Γ313“mνmµ∇νlµ, µ“ ´Γ423“ ´mν¯mµ∇νnµ,\nβ“1\n2pΓ213´Γ433q “1\n2pmνnµ∇νlµ´mν¯mµ∇νmµq,\nρ“Γ314“¯mνmµ∇νlµ, λ“ ´Γ424“ ´¯mν¯mµ∇νnµ,\nα“1\n2pΓ214´Γ434q “1\n2p¯mνnµ∇νlµ´¯mν¯mµ∇νmµq.(2)\nThe Weyl tensor is represented by five complex scalars, denot ed asΨ0,Ψ1,Ψ2,Ψ3,Ψ4,\nΨ0“ ´C1313,Ψ1“ ´C1213,Ψ2“ ´C1342,Ψ3“ ´C1242,Ψ4“ ´C2324.\nWe will deal with asymptotically flat vacuum solution in this work. So the Ricci tensor\nvanishes. We would refer to [ 21] for the other notations.\nOne can obtain the following relations from the orthogonali ty conditions and normal-\nization conditions in ( 1),\nlν∇νlµ“ pǫ`¯ǫqlµ´κ¯mµ´¯κmµ,\nlν∇νnµ“ ´pǫ`¯ǫqnµ`¯π¯mµ`πmµ,\nlν∇νmµ“ pǫ´¯ǫqmµ´κnµ`¯πlµ,\nnν∇νlµ“ pγ`¯γqlµ´τ¯mµ´¯τmµ,\nmν∇νlµ“ pβ`¯αqlµ´σ¯mµ´¯ρmµ.(3)\nIn the asymptotic analysis in the NP formalism near null infin ity, the NU gauge [ 6] is\nwell appreciated, which is based on two assumptions that lis tangent to null geodesics\nand is a hypersurface orthogonal null direction. Those two c onditions have the following\n5consequences to the NP variables, κ“0andρ“ρ. Then, one can use third and first\nclasses of tetrad rotations to set ǫ“0“πandτ“α`β. Correspondingly, lis tangent\nto null geodesics with affine parameter and is the gradient of a scalar field. The rest three\nnull basis are parallely transported along l. One normally choose this affine parameter\nas radial coordinate rand choose the scalar field as time coordinate u, hencel“B\nBrand\nl“du.\nWhen we consider algebraically special solution, the repea ted PNDs are usually not\nhypersurface orthogonal. Hence, exact solutions, such as K err solution, are not in a simple\nform when transformed into the NU gauge. If one adopts the rep eated PND to construct\nthe null basis by choosing it as l, the Goldberg-Sachs theorem [ 20] yields that σ“0“κ.\nThen, one can turn off ǫby a third class of tetrad rotation, which will not touch the\nconditions σ“0“κ. Hence, the PND lis tangent to a null geodesic with affine\nparameter. A first class of null rotation can turn off πwithout changing σ“κ“ǫ“0.\nIn this work, we will implement an asymptotic study which can incorporate both\nhypersurface orthogonal null direction and PND. We only imp ose gauge conditions π“\nκ“ǫ“0, hencel“B\nBr. Then the tetrad system can be constructed as\nn“WB\nBu`UB\nBr`XAB\nBxA,\nl“B\nBr, m “MB\nBu`ωB\nBr`LAB\nBxA.(4)\nThough we specify a coordinate system pu,r,xAq, it is important to point out that we\ndo not have any delicate arrangement for the choice of the non -radial coordinates. The\nfall-off conditions are chosen as follows\nW“1`Opr´1q, XA“Opr´1q, U “Oprq, M “Opr´1q,\nLz“Opr´2q, L¯z“Opr´1q, ω “Opr´1q, ρ “ ´1\nr`Opr´2q,\nRepρq “ ´1\nr`Opr´3q, σ “Opr´2q, α “Opr´1q, β “Opr´1q\nα`β“Opr´2q, λ “Opr´1q, µ “Opr´1q, τ “Opr´1q.(5)\nThose conditions can be reached for solution space by three c lasses of tetrad rotations and\nthe combined coordinates transformation which preserves t he gauge conditions κ“ǫ“\nπ“0,2see, e.g., the arrangement for some of fall-off conditions i n [6]. As directional\nderivatives, the basis vectors are designated by special sy mbols,\nD“lµBµ,∆“nµBµ, δ “mµBµ. (6)\nThe full vacuum NP equations with respect to conditions κ“ǫ“π“0are arranged as\n2σ“0will be also preserved for algebraically special case. Henc e, any algebraically special solution\ncan be set into those fall-off forms as well.\n6Radial equations\nDρ“ρ2`σσ, (7)\nDσ“ pρ`ρqσ`Ψ0, (8)\nDα“ρα`βσ, (9)\nDβ“ασ`ρβ`Ψ1, (10)\nDτ“τρ`τσ`Ψ1, (11)\nDλ“ρλ`σµ, (12)\nDµ“ρµ`σλ`Ψ2, (13)\nDγ“τα`τβ`Ψ2, (14)\nDν“τµ`τλ`Ψ3, (15)\nD∆“ ´pγ`γqD`τδ`τδ, (16)\nDδ“ ´pα`βqD`ρδ`σδ, (17)\nDΨ1´δΨ0“4ρΨ1´4αΨ0, (18)\nDΨ2´δΨ1“3ρΨ2´2αΨ1´λΨ0, (19)\nDΨ3´δΨ2“2ρΨ3´2λΨ1, (20)\nDΨ4´δΨ3“ρΨ4`2αΨ3´3λΨ2. (21)\nNon-radial equations\n∆λ“δν´ pµ`µqλ´ p3γ´γqλ`νp3α`β´τq ´Ψ4, (22)\n∆ρ“δτ´ρµ´σλ´τpτ`α´βq ` pγ`γqρ´Ψ2, (23)\n∆α“δγ`ρν´ pτ`βqλ` pγ´µqα`γpβ´τq ´Ψ3, (24)\n∆µ“δν´µ2´λλ´ pγ`γqµ`νpα`3β´τq, (25)\n∆β“δγ´µτ`σν`βpγ´γ´µq ´αλ`γpα`β´τq, (26)\n∆σ“δτ´σµ´ρλ´τpτ´α`βq ` p3γ´γqσ, (27)\n∆δ“δ∆`νD` pα`β´τq∆` pγ´γ´µqδ´λδ, (28)\nδρ´δσ“ρpα`βq ´σp3α´βq `τpρ´ρq ´Ψ1, (29)\nδα´δβ“µρ´λσ`αα`ββ´2αβ`γpρ´ρq ´Ψ2, (30)\nδλ´δµ“νpρ´ρq ´µpα`βq `λpα´3βq ´Ψ3, (31)\nδδ´δδ“ pµ´µqD` pρ´ρq∆` pα´βqδ` pβ´αqδ, (32)\n∆Ψ0´δΨ1“ p4γ´µqΨ0´ p4τ`2βqΨ1`3σΨ2, (33)\n∆Ψ1´δΨ2“νΨ0` p2γ´2µqΨ1´3τΨ2`2σΨ3, (34)\n∆Ψ2´δΨ3“2νΨ1´3µΨ2` p2β´2τqΨ3`σΨ4, (35)\n∆Ψ3´δΨ4“3νΨ2´ p2γ`4µqΨ3` p4β´τqΨ4. (36)\n73 Twisting solution space in NP formalism\nThe NP variables satisfying the NP equations in asymptotic e xpansions are given by\nΨ0“Ψ0\n0pu,z,¯zq\nr5`Opr´6q,Ψ1“Ψ0\n1pu,z,¯zq\nr4´ðnΨ0\n0\nr5`Opr´6q,\nΨ2“Ψ0\n2pu,z,¯zq\nr3´ðnΨ0\n1\nr4`Opr´5q,Ψ3“Ψ0\n3pu,z,¯zq\nr2´ðnΨ0\n2\nr3`Opr´4q,\nΨ4“Ψ0\n4pu,z,¯zq\nr´ðnΨ0\n3\nr2`Opr´3q,\nρ“ ´1\nr`iΣpu,z,¯zq\nr2`Σ2´σ0σ0\nr3`Opr´5q,\nσ“σ0pu,z,¯zq\nr2´pΣ´σ0σ0qσ0`1\n2Ψ0\n0\nr4`Opr´5q,\nα“α0pu,z,¯zq\nr`α0σ0´iΣα0\nr2\n`pα0σ0`iΣα0qσ0´ pΣ2´σ0σ0qα0´iΣpα0σ0´iΣα0q\n2r3`Opr´4q,\nβ“ ´α0pu,z,¯zq\nr´α0σ0`iΣα0\nr2\n´pα0σ0´iΣα0qσ0´ pΣ2´σ0σ0qα0`iΣpα0σ0`iΣα0q `Ψ0\n1\n2r3`Opr´4q,\nτ“τ0pu,z,¯zq\nr´τ0σ0`iΣτ0\nr2`Opr´3q,\nλ“λ0pu,z,¯zq\nr´µ0σ0`iΣλ0\nr2`Opr´3q,\nγ“γ0pu,z,¯zq `τ0α0´τ0α0\nr`iΣpα0τ0´α0τ0q ´1\n2Ψ0\n2\nr2`Opr´3q, (37)\nµ“µ0pu,z,¯zq\nr´σ0λ0`Ψ0\n2´iΣµ0\nr2`Opr´3q,\nν“ν0pu,z,¯zq ´τ0µ0`τ0λ0`Ψ0\n3\nr`Opr´2q,\nLz“ ´¯Ppu,z,¯zqσ0\nr2`Opr´3q, L¯z“Ppu,z,¯zq\nr`iΣP\nr2`Opr´3q,\n¯L¯z“ ´Pσ0\nr2`Opr´3q,¯Lz“¯P\nr´iΣ¯P\nr2`Opr´3q,\nM“M0pu,z,¯zq\nr´σ0M0´iΣM0\nr2`Opr´3q,\nω“ω0pu,z,¯zq\nr´σ0ω0`1\n2Ψ0\n1´iΣω0\nr2`Opr´3q,\nXz“ ´τ0¯P\nr`Opr´2q, W “1´τ0M0`τ0M0\nr`Opr´2q,\n8U“ ´pγ0`γ0qr`U0pu,z,¯zq\n´1\n2pΨ0\n2`Ψ0\n2q ` pτ0ω0`τ0ω0q ´2iΣpα0τ0´α0τ0q\nr`Opr´2q,\nwhere\nα0“1\n2pM0BulnP`¯PBlnPq, γ0“ ´1\n2Buln¯P, τ0“ ´BuM0´2M0γ0,\nµ0“2iΣγ0`µ0\nr, µ0\nr“ ´ðnα0´ðnα0,2iΣ“ðnM0´ðnM0,\nω0“2iΣτ0´iðnΣ`ðnσ0, λ0“ Buσ0` p3γ0´γ0qσ0` pτ0q2´ðnτ0\nν0“ðnpγ0`γ0q ´τ0pγ0`γ0q,Ψ0\n3“2iΣν0`ðnµ0´ðnλ0,\nΨ0\n4“ðnν0´4γ0λ0´τ0ν0´ Buλ0,\nU0“ðnτ0´iΣpγ0`3γ0q ´iBuΣ`µ0\nr´τ0τ0,\nΨ0\n2´Ψ0\n2“iΣp2U0`µ0`µ0q `ðnω0´ðnω0,\nand\nBuΨ0\n0` pγ0`5γ0qΨ0\n0“ðnΨ0\n1`3σ0Ψ0\n2´4τ0Ψ0\n1,\nBuΨ0\n1`2pγ0`2γ0qΨ0\n1“ðnΨ0\n2`2σ0Ψ0\n3´3τ0Ψ0\n2,\nBuΨ0\n2`3pγ0`γ0qΨ0\n2“ðnΨ0\n3`σ0Ψ0\n4´2τ0Ψ0\n3,\nas well as the identities\nBuα0“ ´ðnγ0`γ0τ0´2γ0α0,\nBuµ0“ðnν0´2µ0pγ0`γ0q ´τ0ν0,\nBuΨ0\n3`2p2γ0`γ0qΨ0\n3“ðnΨ0\n4´τ0Ψ0\n4,\nwhich are satisfied automatically when the precise expressi ons of the NP variables are\ninserted. Note that U0is proven to be real. We define an operator ðnwhich generalizes\nthe extremely useful ðoperator at null infinity [ 22] for a field of spin sas,\nðnηpsq“ pP¯B `M0Bu`2sα0qηpsq\n“P¯P´s¯Bp¯Psηpsqq `M0¯P´sBup¯Psηpsqq,(38)\nand\nðnηpsq“ p¯PB `M0Bu´2sα0qηpsq\n“¯PPsBpP´sηpsqq `M0PsBupP´sηpsqq,(39)\nwhere psqdenotes the spin weight of the field ηpsq. The generalized operators ðnandðn\nhave the same effect for raising and lowering the spin weight asðandð. They are the\n9Table 1: Spin weights\nðnτ0α0γ0ν0µ0σ0λ0Ψ0\n4Ψ0\n3Ψ0\n2Ψ0\n1Ψ0\n0\ns11´10´102´2´2´1012\nspecial case of the very generic eth operator defined in [ 23] with restriction to the null\ninfinity. The commutator of the generalized operators is\nrðn,ðnsηpsq“ ´p2sµ0\nr`2iΣBuqηpsq. (40)\nThe spin weights of relevant fields are listed in Table 1.\nTo summarize, the solution space of the system is fully chara cterized by three arbitrary\nfunctions σ0P, andM0of three variables pu,z,¯zqand initial data Ψ0,Ψ0\n1, the real part\nofΨ0\n2of two variables pu0,z,¯zq.\n4 Twisting asymptotic symmetries and charges\nAs a tetrad system, the gauge transformation of the NP formal ism is a combination of\na diffeomorphism and a local Lorentz transformation. Let ξµandωab“ ´ωbadenote\nparameters for the two types of infinitesimal transformatio ns, the gauge transformation of\nthe NP variables are [ 17]\nδξ,ωeaµ“ξνBνeaµ´ Bνξµeaν`ωabebµ,\nδξ,ωΓabc“ξνBνΓabc´eµ\ncBµωab`ωadΓdbc`ωbdΓadc`ωcdΓabd,\nδξ,ωCabcd“ξνBνCabcd`ωafCfbcd`ωbfCafcd`ωcfCabfd`ωdfCabcf.(41)\nThe ten gauge conditions yield\n•0“δξ,ωeu\n1“ ´Brξuù ñξu“fpu,z,¯zq.\n•0“δξ,ωΓ311“ ´Brω24`σω23`ρω24ù ñω24“ω24\n0pu,z,¯zq\nr´ω24\n0iΣ`ω23\n0σ0\nr2 `Opr´3q.\n•0“δξ,ωΓ411“ ´Brω23`ρω23`σω24ù ñω23“ω23\n0pu,z,¯zq\nr`ω23\n0iΣ´ω24\n0σ0\nr2 `Opr´3q.\n•0“δξ,ωΓ211“ ´Brω12`pα`βqω24`pα`βqω23ù ñω12“ω12\n0pu,z,¯zq`Opr´3q.\n•0“δξ,ωer\n1“ ´Brξr`ω2aer\naù ñξr“ ´ω12\n0r`Zpu,z,¯zq´ω23\n0ω`ω24\n0ω0\nr2 `Opr´2q.\n•0“δξ,ωeA\n1“ ´B rξA`ω2aeA\naù ñξz“Yzpu,z,¯zq ´ω24\n0¯P\nr`Opr´2qand\nξ¯z“Y¯zpu,z,¯zq ´ω23\n0P\nr`Opr´2q.\n10•0“δξ,ωΓ321“ ´Brω14´λω23´µω24ù ñω14“ω14\n0pu,z,¯zq `ω24\n0µ0`ω23\n0λ0\nr`\nOpr´2q.\n•0“δξ,ωΓ421“ ´Brω13´µω23´λω24ù ñω13“ω13\n0pu,z,¯zq `ω23\n0µ0`ω24\n0λ0\nr`\nOpr´2q.\n•0“δξ,ωΓ431“ ´Brω34` pα´βqω23´ pα´βqω24ù ñω34“ω34\n0pu,z,¯zq `\n2α0ω24\n0´2α0ω23\n0\nr`Opr´1q.\nNote that all the omitted subleading orders of those symmetr y parameters are completely\nfix once a solution of the NP system is given. The fall-off cond itions yield\n•δξ,ωeu\n2“Opr´1q ù ñω12\n0“ Buf.\n•δξ,ωeA\n2“Opr´1q ù ñ B uYA“0.\n•δξ,ωez\n3“Opr´2q ù ñ¯BYz“0, henceYz“Ypzq.\n•δξ,ωe¯z\n4“Opr´2q ù ñ BY¯z“0, henceY¯z“¯Yp¯zq.\n•δξ,ωpΓ314`Γ413q “Opr´3q ù ñZ“1\n2pðnω23\n0`ðnω24\n0´ω23\n0τ0´ω24\n0τ0q.\n•δξ,ωΓ213“Opr´2q ù ñω14\n0“ pγ0`¯γ0qω24\n0´ðnω12\n0.\n•δξ,ωΓ214“Opr´2q ù ñω13\n0“ pγ0`¯γ0qω23\n0´ðnω12\n0.\nThe parameters of residual gauge transformations are chara cterized by six functions\nfpu,z,¯zq, Yz“Ypzq, Y¯z“¯Yp¯zq,\nω23\n0pu,r,z q, ω24\n0pu,r,z q, ω34\n0pu,r,z q.(42)\nAs was done in [ 17], it is useful to reparametrize residual gauge symmetries b y trading\nthe real function Bufpu,z,¯zqand the imaginary ω34\n0pu,z,¯zqfor a complex parameter\nΩpu,z,¯zqaccording to\nBuf“1\n2r¯B¯Y´¯Y¯BlnpP¯Pq ` BY´YBlnpP¯Pqs `fpγ0`γ0q `1\n2pΩ`¯Ωq,\nω34\n0“1\n2r¯B¯Y´¯Y¯BlnP`¯Y¯Bln¯P´ BY`YBln¯P´YBlnPs\n`fpγ0´γ0q `1\n2pΩ´¯Ωq.(43)\nThis redefinition of parameters simplifies the transformati on onPand¯Pas\nδP“ΩP, δ¯P“¯Ω¯P, (44)\n11which we refer to as a complex Weyl rescaling. It then determi nes forfin terms of an\nintegration function Tpz,¯zqwhich characterizes the supertranslation,\nfpu,z,¯zq “1?\nP¯PrTpz,¯zq `˜u\n2pBY`¯B¯Yq ´YB˜u´¯Y¯B˜u`1\n2p˜Ω`˜¯Ωqs, (45)\nwhere\n˜u“żu\nu0du1a\nP¯P,˜Ω“żu\nu0du1a\nP¯PΩ. (46)\nThe new symmetries represented by ω23\n0pu,r,z qandω24\n0pu,r,z qare explicitly from re-\nmoving the hypersurface orthogonal condition. The asympto tic symmetry characterized\nby the parameters pξ,ωqrealize a symmetry algebra at ull infinity as\nrδξ1,ω1,δξ2,ω2sφα“δˆξ,ˆωφα,\nˆξµ“ rξ1,ξ2sµ,\npˆωqab“ξ1ρBρω2ab`ω1acω2cb´ p1Ø2q,(47)\nwhereφαdenotes an arbitrary field. In particular,\nˆf“YA\n1BAf2`f1Buf2´ p1Ø2q,\nˆYA“YB\n1BBYA\n2´YB\n2BBYA\n1,\nˆΥ“f1BuΥ2`YA\n1BAΥ2´ Buf1Υ2´Υ1Λ2´ p1Ø2q,\nˆ¯Υ“f1Bu¯Υ2`YA\n1BA¯Υ2´ Buf1¯Υ2`¯Υ1Λ2´ p1Ø2q,\nˆΛ“f1BuΛ2`YA\n1BAΛ2´ p1Ø2q,(48)\nwhere we define Υ“ω23\n0andΛ“ω34\n0for notational brevity. The asymptotic symmetries\nhave the following actions on the solution space\nδM0“fBuM0`YABAM0`M0Buf`ΛM0`Υ´ðnf,\nδΣ“fBuΣ`YABAΣ´i\n2pðnΥ´ðnΥ`Υτ0´Υτ0q,\nδτ0“fBuτ0`YABAτ0`τ0Buf`Λτ0´ BuΥ´2γ0Υ`ðnBuf,\nδσ0“fBuσ0`YABAσ0`σ0Buf`2Λσ0`Υτ0´ðnΥ,\nδΨ0\n0“fBuΨ0\n0`YABAΨ0\n0`3Ψ0\n0Buf`2ΛΨ0\n0`4ΥΨ0\n1,\nδΨ0\n1“fBuΨ0\n1`YABAΨ0\n1`3Ψ0\n1Buf`ΛΨ0\n0`3ΥΨ0\n2,\nδΨ0\n2“fBuΨ0\n2`YABAΨ0\n2`3Ψ0\n2Buf`2ΥΨ0\n3,\nIt is clear from the transformation law that one can turn off M0by the residual Lorentz\ntransformation characterized by Υ. Correspondingly, the twist vanishes. Indeed, this can\nalways be done near the null infinity. Namely, one can always h ave a hypersurface or-\nthogonal null direction to construct the tetrad system. Thi s is because the null infinity has\na very universal structure for any asymptotically flat space time. The whole framework\n12is based on the existence of a null infinity and the solution ca n be considered as flowing\nfrom null infinity to the bulk in the form of series expansion. However, for certain inter-\nesting exact solutions, it is very hard to do so. The main diffi culty is from the proof of\nthe integrability condition for transforming into a hypers urface orthogonal null direction.\nWhile, series expansion in1\nrsignificantly simplifies the radial integration.\nThe three arbitrary functions σ0P, andM0can all represent propagating degree of\nfreedom. One can turn off two of them by residual gauge transf ormations which still\nkeeps the full propagating degree of freedom. Regarding to t he transformation laws,\none can find that turning off both σ0andM0reduces the supertranslation to ordinary\ntranslations [ 24]. Hence, the common choice at null infinity is to turn off M0and fixP\nto be a round sphere [ 4,25].\nNormally, the next step is to compute the charges associated to the asymptotic sym-\nmetries, which will characterize the boundary degrees of fr eedom. As was stated in the\nintroduction, our main motivation of the present work for ge neralizing the NU system\nis to incorporate with exact solutions and clarify the inter play between hypersurface or-\nthogonal null direction and PND. We will not demonstrate the full charge analysis here\nbut only compute the charges associated to the symmetries as sociated to the independent\nLorentz transformation, including the twisting asymptoti c symmetries and the imaginary\npart of the Weyl rescaling since they are the additions to the well studied Weyl-BMS sym-\nmetry [ 17,18].3This will confirm that removing the hypersurface orthogonal arises new\nboundary degrees of freedom.\nIt is very efficient to use the form language to compute bounda ry charges for the NP\nformalism [ 26,27]. The exact formulas of the forms eaare given by the co-tetrad system.\nIn asymptotic form, they are\nl“du´M0\n¯Pdz´M0\nPd¯z`Opr´2q,\nn““\npγ0`γ0qr´U0‰\ndu`dr´«\nM0pγ0`γ0qr\n¯P`ω0´M0U0\n¯Pff\ndz\n´„M0pγ0`γ0qr\nP`ω0´M0U0\nP\nd¯z`Opr´1q, (49)\nm“ ´τ0du`˜\n´r\n¯P`M0τ0´iΣ\n¯P¸\ndz`M0τ0´σ0\nPd¯z`Opr´1q,\n¯m“ ´τ0du`M0τ0´σ0\n¯Pdz`ˆ\n´r\nP`M0τ0`iΣ\nP˙\nd¯z`Opr´1q.\n3Note that the imaginary part of the Weyl rescaling was reveal ed in [ 17], the charge associated to this\nsymmetry has not been computed. While the charge of the real p art of the Weyl rescaling was computed\nin [18] for au-indepdent Weyl factor of the boundary metric.\n13The connection one forms Γabare given as\nΓ12“ ´pγ`γql`τm`τ¯m, (50)\nΓ13“ ´τl`ρm`σ¯m, (51)\nΓ23“νl´µm´λ¯m, (52)\nΓ34“ pγ´γql´ pα´βqm` pα´βq¯m. (53)\nThe boundary charge from the Palatini action is defined by [ 26–30]\nδ{HPa“1\n32πGǫabcdż\nBΣrδpiξΓabec^edq ´iξpδΓab^ec^edq ´δpωabec^edqs,(54)\nfor field dependent symmetry parameters where Scan be any constant- utwo surface at\nthe null infinity to evaluate the boundary charge. The imagin ary part of the Weyl rescaling\ncharge is integrable and is given by\nQΛ“1\n8πGż\nSΛ\nP¯PpM0ω0´M0ω0q. (55)\nWe have computed the full expression of the twisting charge. There is a divergent piece\nat orderr,\nδ{Qprq\nΥ,¯Υ“r\n8πGż\nSZδ1\nP¯P, Z “1\n2pðnΥ`ðnΥ´Υτ0´Υτ0q. (56)\nOne can in general regularize the divergent boundary charge s, see, e.g., relevant treat-\nments in [ 12,18,31–33]. Here, we will simply remove the divergence by choosing a\nreduction of the solution space, as the main motivation here is to manifest the degrees of\nfreedom associated to the twist. We choose a fixed Weyl factor of the boundary metric.\nFor simplicity, we set P¯P“1, which yields the null infinity with topology RˆC. Hence,\nP“eiP0pu,z,¯zq,¯P“e´iP0pu,z,¯zq. (57)\nThis choice will turn off the real part of the Weyl rescaling a nd keep the imaginary part\nΛ. The twisting charge in the reduced case is obtained as\nδ{Qp0q\nΥ,¯Υ“1\n8πGż\nSδ”\npΥM0`ΥM0qpM0ν0`M0ν0qı\n`Z„\n2M0δα0`2M0δα0`2pM0γ0δM0`M0γ0δM0q\n`2ˆ\nΣ´ipγ0´γ0qM0M0`ipM0α0´M0α0q `i\n2pM0τ0´M0τ0q˙\nδP0\n.(58)\nThe charge is not integrable in this form. One may consider to include a field-dependent\nredefinition of the symmetry parameters [ 10,32,34–37] to make the charge integrable.\nBut there are three dynamical fields in the charge M0,M0, andP0. While there are only\n14two symmetry parameters ΥandΥ. The system is overdetermined. Note also that the\nΛcharge involves σ0. So there are more dynamical fields than symmetry parameters in\nthis system. One can further reduce the solution space by cho osing a non-dynamical P0.\nThen it is possible to have the twist charge in an integrable f orm. Here, we sketch a naive\nway for achieving that. The charge can be written as\nδ{Qp0q\nΥ,¯Υ“1\n8πGż\nSˆΥδM0`ˆΥδM0, (59)\nwhereˆΥ,ˆΥcan be derived from the algebraic equations by identifying ( 59) with ( 58).\nHowever, it is not completely clear to us if one can inverse th e relation to solve out\nΥ,¯Υas there are derivatives of Υ,¯Υin the parameter Z. Suppose that one can solve\noutΥ,¯Υin terms of ˆΥ,ˆΥand the dynamical fields M0,M0. Then, considering ˆΥ,ˆΥ\nas field independent parameters will make the twist charge in tegrable. An interesting\nfeature of the twist and Λcharges is that they consist of only news functions. The char ges\nare not conserved and there is no flux-balance law constraini ng them. Once we reduce\nthe solution space by choosing a non-dynamical and u-independent P0, the twist charge\nvanishes and the Λsymmetry is turned off. We find a similar type of charge in a thr ee\ndimensional setup [ 35] where the charge is related to the Weyl anomaly coefficient [ 38].\nClearly, the charge expressions ( 55) and ( 58) indicate that the new boundary degrees\nof freedom are indeed associated to the relaxation of removi ng hypersurface orthogonal\ncondition. Setting M0“0“M0turns off both charges ( 55) and ( 58). Then one can turn\noffP0by trivial gauge transformation in the reduced solution spa ce. A normal choice for\na punctured complex plane null infinity is to set directly P“¯P“1[16,19]. Here we\nshow that the existence of twist arises interesting boundar y degree of freedom associated\nto the imaginary part of the Weyl rescaling. Hence, the choic eP“¯P“1for a punctured\ncomplex plane null infinity with twist will lose physical deg ree of freedom.\n5 Algebraically special solutions\nIn this section, we will manifest the algebraically special condition in the asymptotic\nanalysis in the NP formalism. We will choose the repeated PND as the null basis l.\nConsequently, we will have extra gauge conditions σ“Ψ0“Ψ1“0.4We will show\nthat there is a remarkable simplification in the solution spa ce and asymptotic symmetry,\ngiving the r-dependence of all NP variables and symmetry parameters in t erms of simple\nrational functions of r.\n4Note that those conditions can not be reached by residual gau ge transformation. They are extra con-\nditions which can decrease propagating degrees of freedom. By default, those conditions exclude all alge-\nbraically general solutions.\n155.1 Solution space\nA generic solution of ρfor the radial NP equation is [ 39]\nρ“ ´1\nr`iΣ. (60)\nThen the twist parameter Σwill be propagated to everywhere of the solution space. But\nwith this expression, the full solution space is given in a cl osed form. The solutions to the\nradial equations are given by\nρ“ ´1\nr`iΣ, α “α0\nr`iΣ, β “ ´α, τ “τ0\nr`iΣ, λ “λ0\nr`iΣ,\nM“M0\nr´iΣ, ω “ω0\nr´iΣ, L¯z“P\nr´iΣ,\nW“1´τ0M0\nr´iΣ´τ0M0\nr`iΣ, Xz“ ´τ0¯P\nr`iΣ,\nΨ2“Ψ0\n2\npr`iΣq3, µ “µ0\nr´iΣ´rΨ0\n2\npr`iΣq2pr´iΣq,\nγ“γ0`τ0α0\nr´iΣ´τ0α0\nr`iΣ´Ψ0\n2\n2pr`iΣq2,\nU“ ´pγ0`γ0qr`U0´Ψ0\n2\n2pr`iΣq´Ψ0\n2\n2pr´iΣq´τ0ω0\nr`iΣ´τ0ω0\nr´iΣ, (61)\nΨ3“Ψ0\n3\npr`iΣq2`Ψ1\n3\npr`iΣq3`Ψ2\n3\npr`iΣq4,\nΨ1\n3“ ´ðnΨ0\n2,Ψ2\n3“3\n2Ψ0\n2piðnΣ`ω0q,\nΨ4“Ψ0\n4\nr`iΣ`Ψ1\n4\npr`iΣq2`Ψ2\n4\npr`iΣq3`Ψ3\n4\npr`iΣq4`Ψ4\n4\npr`iΣq5,\nΨ1\n4“ ´ðnΨ0\n3,Ψ2\n4“3\n2λ0Ψ0\n2`Ψ0\n3piðnΣ`ω0q ´1\n2ðnΨ1\n3,\nΨ3\n4“Ψ1\n3piðnΣ`ω0q ´1\n3ðnΨ2\n3,Ψ4\n4“Ψ2\n3piðnΣ`ω0q,\nν“ν0´Ψ0\n3`τ0λ0\nr`iΣ´τ0µ0\nr´iΣ`τ0Ψ0\n2\n2pr2`Σ2q´Ψ1\n3\n2pr`iΣq2´Ψ2\n3\n3pr`iΣq3.\nCorrespondingly, the co-tetrad system is given by\nl“du´M0\n¯Pdz´M0\nPd¯z,\nn“„\npγ0`γ0qr´U0`rK`JΣ\nr2`Σ2\ndu`dr\n´«\nM0\n¯Pˆ\npγ0`γ0qr´U0`rK`JΣ\nr2`Σ2˙\n`ω0\n¯Pff\ndz\n16´„M0\nPˆ\npγ0`γ0qr´U0`rK`JΣ\nr2`Σ2˙\n`ω0\nP\nd¯z, (62)\nm��� ´τ0du`˜\n´r\n¯P`M0τ0´iΣ\n¯P¸\ndz`M0τ0\nPd¯z,\n¯m“ ´τ0du`M0τ0\n¯Pdz`ˆ\n´r\nP`M0τ0`iΣ\nP˙\nd¯z,\nwhereKandJare the real and imaginary parts of Ψ0\n2,\nK“1\n2pΨ0\n2`Ψ0\n2q, J “ ´i\n2pΨ0\n2´Ψ0\n2q.\nAll the NP variables are in closed form in the expansion of ρ“ ´1\nr`iΣandρ“ ´1\nr´iΣ.\nThen, it is possible to solve all the non-radial equations to every order, which will be\nimplemented explicitly. Starting from ( 29), we obtain that\nω0“2iΣτ0´iðnΣ. (63)\nFrom ( 27), we obtain\nλ0“ pτ0q2´ðnτ0. (64)\nA constraint of Ψ0\n2is yielded by ( 34) as\nðnΨ0\n2“3τ0Ψ0\n2. (65)\nFrom ( 30), we get\nµ0“2iΣγ0`µ0\nr, (66)\nwhereµ0\nr“ ´ðnα0´ðnα0is the real part. Eq. ( 31) yields\nΨ0\n3“2iΣν0`ðnµ0´ðnλ0. (67)\nFrom the u-component of ( 32), one can derive that\n2iΣ“ðnM0´ðnM0“ ´”\nPp¯PW ´M0Xzq ´M0¯PX¯zı2\nP¯PlruBzl¯zs. (68)\nIt is clear that lis hypersurface orthogonal once Σ“0. Note that lruBrlzs“0from the\nsolution of the radial equations. From ¯z-component of ( 32), it is obtained that\nα0“1\n2pM0BulnP`¯PBlnPq. (69)\nTher-component of ( 32) fixed the imaginary part of Ψ0\n2as\nΨ0\n2´Ψ0\n2“iΣp2U0`µ0`µ0q `ðnω0´ðnω0. (70)\n17We continue with the u-component of ( 28), which yields\nτ0“ ´BuM0´2M0γ0. (71)\nThe¯z-component and r-component of ( 28) lead to\nγ0“ ´1\n2Buln¯P. (72)\nand\nν0“ðnpγ0`γ0q ´τ0pγ0`γ0q. (73)\nFrom ( 23), we obtain\nU0“ðnτ0´iΣpγ0`3γ0q ´iBuΣ`µ0\nr´τ0τ0. (74)\nOne can prove that U0is real using the expression of τ0and the commutator of general-\nized operator ðnin (40). From ( 35), there is another constraint of Ψ0\n2,\nBuΨ0\n2`3pγ0`γ0qΨ0\n2“ðnΨ0\n3´2τ0Ψ0\n3. (75)\nEq. ( 22) yields\nΨ0\n4“ðnν0´4γ0λ0´τ0ν0´ Buλ0. (76)\nFinally, Equations ( 24)-(26) and ( 36) are identities and do not lead to new relation. In\nparticular,\nBuα0“ ´ðnγ0`γ0τ0´2γ0α0, (77)\nand\nBuµ0“ðnν0´2µ0pγ0`γ0q ´τ0ν0. (78)\nThe solutions of the non-radial equations recover the leadi ng results of previous section\nwithσ0“0. Remarkably, we exactly prove that there is no new constrain t from any\nsubleading order. This has been suggested by Newman and Unti [6]. Here we give a\ndirect verification for any algebraically special solution . To summarize, for any given M0\nandP, an exact algebraically special solution of vacuum Einstei n equation is specified\nonce a solution of the constraints for Ψ0\n2, namely Eqs. ( 65), (70), (75), is obtained.\n5.2 Asymptotic symmetries and transformation law\nThe extra conditions Ψ1“Ψ0“σ“0yield that ω23\n0“0“ω24\n0. The asymptotic\nsymmetries are in closed form as\nξu“fpu,z,¯zq, Yz“Ypzq, Y¯z“¯Yp¯zq, ξr“ ´Bufr,\nω12“ Buf, ω13“ ´ðnBuf, ω14“ ´ðnBuf, ω34“Λpu,z,¯zq.(79)\n18The symmetry algebra recovers precisely the Weyl-BMS algeb ra [17]. The asymptotic\nsymmetries have the following transformation law on the sol ution space,\nδM0“fBuM0`YABAM0`M0Buf`ΛM0´ðnf,\nδΣ“fBuΣ`YABAΣ,\nδΨ0\n2“fBuΨ0\n2`YABAΨ0\n2`3Ψ0\n2Buf.\nNote that the residual gauge transformation can not turn off Σ. Hence, the twist in this\nsystem must not be pure gauge from the point of view of both pro pagating and boundary\ndegrees of freedom. It is amusing to notice that the inhomoge neous piece in the trans-\nformation of M0is only supertranslation relevant as ´ðnfwhich is in a similar form as\nthe transformation law of σ0in the twist-free case [ 17,24]. The difference of them is\nonly from their different spin weight. The shear of the gener ator of the null infinity λ0is\ncompletely fixed by τ0in (64). Hence, in the algebraically special case, τ0is effectively\nthe news tensor of this system, which indicates the propagat ing degree of freedom. The\nNP variables M0andτ0in the algebraically spacial case are the analogue of σ0andλ0of\nthe twist-free case.\nThe next step will be the computation of the asymptotic charg es. As we have shown\nthat the solution space and asymptotic symmetry parameters are all in truncated forms.\nWe expect that the charges are in closed form which indicates that the BMS symmetry\nbecomes symplectic symmetry similar to three dimensional t heories [ 40], see also [ 41].\nSurprisingly, there are propagating degree of freedom in th e present case. This will be the\nfirst example of enhancement from asymptotic symmetry or nea r horizon symmetry to\nsymplectic symmetry in the system with propagating degree o f freedom. We will present\nthe full charge analysis elsewhere.\n6 Exact solutions with twist\nIt is very important for the applications of asymptotic symm etry and the associated charges\nto have exact solutions in the proposed system. In this secti on, we will present the exact\nforms of Kerr and Taub-NUT solution in the generalized NU sys tem. Both solutions are\ngiven in a very simple forms in the generalized NU gauge.\n6.1 Kerr\nThe Kerr line-element in the Kerr-Schild coordinate pu,r,θ,φ qis\nds2“△\n̺¯̺pdu´asin2θdφq2´sin2θ\n̺¯̺radu´ pr2`a2qdφs2\n`2drpdu´asin2θdφq ´̺¯̺pdθq2,(80)\n19where we define\n△“r2´2Mr`a2, ̺ “r`iacosθ,¯̺“r´iacosθ. (81)\nThe inverse metric is given by\ngµν“1\n̺¯̺¨\n˚˚˚˝´a2sin2θ r2`a20 ´a\nr2`a2´△0a\n0 0 ´1 0\n´a a 0´1{sin2θ˛\n‹‹‹‚(82)\nWe choose the tetrad system as\nl“B\nBr, n “B\nBu` p´1\n2`Mr\n̺̺qB\nBr,\nm“eiφ\n?\n2¯̺ˆ\niasinθB\nBu´iasinθB\nBr´B\nBθ`icscθB\nBφ˙\n.(83)\nThe non-vanishing spin coefficients and Weyl scalars are\nρ“ ´1\n̺, α “e´iφcotθ\n2\n2?\n2̺, β “ ´α,\nµ“ ´1\n2̺`Mr\n¯̺̺2, γ “M\n2̺2, ν “iaMe´iφsinθ?\n2̺3,\nΨ2“ ´M\np̺q3,Ψ3“ ´3iaMe´iφsinθ?\n2̺4,Ψ4“3a2Me´2iφsin2θ\n̺5.(84)\nIn the complex stereographic coordinates pz,¯zq, whereθ“2arccot?z¯zandφ“ ´i\n2lnz\n¯z,\nthe tetrad system becomes\nl“B\nBr, n “B\nBu` p´1\n2`Mr\n̺̺qB\nBr,\nm“1?\n2¯̺ˆ\niaeiφsinθB\nBu´iaeiφsinθB\nBr` p1`z¯zqB\nB¯z˙\n.(85)\nThe non-vanishing spin coefficients and Weyl scalars are now\nρ“ ´1\n̺, α “¯z\n2?\n2̺, β “ ´α,\nµ“ ´1\n2̺`Mr\n¯̺̺2, γ “M\n2̺2, ν “iaMe´iφsinθ?\n2̺3,\nΨ2“ ´M\np̺q3,Ψ3“ ´3iaMe´iφsinθ?\n2̺4,Ψ4“3a2Me´2iφsin2θ\n̺5.(86)\n206.2 Taub-NUT\nThe Taub-NUT solution in the complex stereographic coordin ates adapted to the NP con-\nventions is [ 42]\nds2“fprqˆ\ndt`2iNzd¯z´¯zdz\n1`z¯z˙2\n´dr2\nfprq´ pr2`N2q4dzd¯z\np1`z¯zq2, (87)\nwherefprq “r2´2Mr´N2\nr2`N2. Here,Nis the NUT parameter and Mis the mass parameter.\nDefining\n˜u“t´r`2?\n´M2´N2arctanr´M\n´M2´N2`Mlnpr2´2Mr´N2q, (88)\nas the new time coordinate, we choose the null bases for the me tric ( 87) as\nl“ Br, n “ B˜u´f\n2Br,\nm“i?\n2Nz\nr´iNB˜u´1`z¯z?\n2pr´iNqB¯z,\n¯m“ ´i?\n2N¯z\nr`iNB˜u´1`z¯z?\n2pr`iNqBz.(89)\nThe non-zero NP variables are\nρ“ ´1\nr`iN, α “ ´¯z\n2?\n2pr`iNq, β “ ´α, γ “M`iN\n2pr`iNq2,\nµ“ ´1\n2pr´iNq`pM`iNqr\npr`iNq2pr´iNq,Ψ2“ ´M`iN\npr`iNq3.(90)\nBoth null bases landnare repeated principal null directions without shear but wi th twist\n[42].\n7 Applications and future directions\nThere are a number of interesting applications and open ques tions that should be ad-\ndressed in the future. We list some of them here.\n• The most direct one is to find new exact solutions of vacuum Ei nstein gravity from\nthe three constraints of Ψ0\n2for particular choice of M0andP.\n• It will be very meaningful to check if the investigation of t he algebraically spacial\nsolution can be extended to the case with a cosmological cons tant.\n• Another extension is to include electromagnetic fields or o ther types of matter\nfields.\n21• It is interesting to see if one can proof some kind of uniquen ess theorem from the\ntruncated algebraically special solution space when furth er constraint is required,\nsuch as stationary or axial symmetric, see, e.g., [ 43] for relevant investigation from\nasymptotic point of view.\n• Another interesting direction is to find the most general as ymptotic symmetry al-\ngebra at null infinity. The present work focuses on the relaxa tion with a twist. The\nresulting asymptotic Killing vector has a fixed r-component. Another interesting\nrelaxation was proposed in [ 11,12] by imposing the differential NU gauge condi-\ntion where there is new symmetry arise from the r-component of the asymptotic\nKilling vector. Constructing the metric from our tetrad ass umption ( 4) indicates\nthat the only gauge condition is grr“0. It seems that this gauge choice is more\nrelaxed than [ 11,12]. However the rest conditions are hidden in the tetrad and sp in\ncoefficient relations which represent the existence of spec ial null directions that can\nnot been seen directly from the gauge conditions on the metri c.\nAcknowledgments\nThe authors thank Zhoujian Cao, Marc Geiller, and Xiaoning W u for useful discussions.\nP.M. would like to thank Glenn Barnich for long term collabor ations and supports in\nrelevant research topics. This work is supported in part by t he National Natural Science\nFoundation of China (NSFC) under Grants No. 11935009 and No. 11905156.\nA Finite twisting asymptotic symmetry transformations\nIn this Appendix, we work out the finite version of the asympto tic symmetry. In the NU\ngauge, this was explicitly derived by Barnich and Troessaer t [16]. Here, we will follow\nthe same procedure with relaxed gauge and fall-off conditio ns, see also [ 44] for other\nbut relevant generalizations. In the NP formalism, the loca l Lorentz transformation is\ndescribed in three classes of rotation of the tetrad system [ 21]. A combined form of the\nthree rotations is given by\n˜l“ p1`¯ABqp1`A¯BqeERl`B¯Be´ERn`¯Bp1`¯ABqeiEIm`Bp1`A¯Bqe´iEI¯m,\n˜n“A¯AeERl`e´ERn`¯AeiEIm`Ae´iEI¯m, (91)\n˜m“Ap1`¯ABqeERl`Be´ERn` p1`¯ABqeiEIm`ABe´iEI¯m.\nA combined change of coordinates is in the form\nu“upu1,r1,z1,¯z1q, r“rpu1,r1,z1,¯z1q, xA“xApu1,r1,z1,¯z1q. (92)\n22The gauge conditions of limply that\nBxµ\nBr1“˜lµ. (93)\nHence,\nBu\nBr1���B¯Be´ERW`¯Bp1`¯ABqeiEIM`Bp1`A¯Bqe´iEIM,\nBz\nBr1“XzB¯Be´ER`Lz¯Bp1`¯ABqeiEI`¯LzBp1`A¯Bqe´iEI, (94)\nBr\nBr1“ p1`¯ABqp1`A¯BqeER`UB¯Be´ER`ω¯Bp1`¯ABqeiEI`ωBp1`A¯Bqe´iEI,\nwhich generalizes Eq. (6.41) of [ 16]. The unprimed coordinates will be fixed up to four\nintegration constants of r1, for which we will denote with a subscript “0”. To implement\nthe gauge condition κ“π“ǫ“0, one needs to check the transformed spin coefficients,\nκ1“ ´˜lν˜lµ∇ν˜mµ,π1“ ´˜lν˜mµ∇ν˜nµ, ǫ1“˜lν˜nµ∇ν˜lµ, (95)\nwhich eventually lead to\n˜DB “eE“\nB¯Be´ERτ`¯Bp1`¯ABqeiEIσ`Bp1`A¯Bqe´iEIρ‰\n,\n˜D¯A“¯A2eE“\nB¯Be´ERτ`¯Bp1`¯ABqeiEIσ`Bp1`A¯Bqe´iEIρ‰\n´e´E“\nB¯Be´ERν`¯Bp1`¯ABqeiEIµ`Bp1`A¯Bqe´iEIλ‰\n, (96)\n˜DE “ ´2¯AeE“\nB¯Be´ERτ`¯Bp1`¯ABqeiEIσ`Bp1`A¯Bqe´iEIρ‰\n´2“\nB¯Be´ERγ`¯Bp1`¯ABqeiEIβ`Bp1`A¯Bqe´iEIα‰\n.\nwhere˜D“B\nBr1because of the gauge conditions on l. The following fall-off conditions of\nthe symmetry parameters are assumed for asymptotically flat case [ 16]\nr“eER0r1`Op1q, u “Op1q, xA“Op1q,\nA“Op1q, E “Op1q, B “Opr´1q.(97)\nHence, the three classes of tetrad rotation will be fixed up to six integration constants of\nr1at their leading order. To proceed, we use the fall-off condi tions of the NP variables to\nfix the integration constants. We start from the transformed tetradn1,\nn1u1“e´ER0Buu1\n0`Opr´1q, n1r1“Oprq, n1z1“e´ER0Buz1\n0`Opr´1q, (98)\nwhich implies\nER0“ Buu1\n0, z1\n0“Ypz,¯zq. (99)\nThe next one is m1,\nm1u1“Opr´1q, m1r1“Opr´1q, m1z1“eiEI0PB¯zz1`Opr´2q, m1¯z1“Opr´1q.\n(100)\n23This yields z1\n0“Ypzq. The condition Γ213“Opr´2qyields that\nA0“ pγ0`γ0qe´2ER0B0`e´ER0ð1\nnER0. (101)\nThe condition Repρq “Opr´3qfixesr1\n0as\nr1\n0“1\n2A0B0`1\n2A0B0`1\n2e´ER0pð1\nnB0`ð1\nnB0q\n´1\n2e´ER0´\nB0ð1\nnE0`B0ð1\nnE0`eEI0τ0B0`e´EI0τ0B0¯\n.(102)\nThere is no condition to fix B0. 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D 99no. 10, (2019) 104024 ,arXiv:1901.04010 [gr-qc] .\n27"    },    {        "title": "2401.12058v1.The_Dimension_Strikes_Back_with_Gradients__Generalization_of_Gradient_Methods_in_Stochastic_Convex_Optimization.pdf",        "content": "arXiv:2401.12058v1  [cs.LG]  22 Jan 2024The Dimension Strikes Back with Gradients: Generalization of\nGradient Methods in Stochastic Convex Optimization\nMatan Schliserman Uri Sherman Tomer Koren\nJanuary 23, 2024\nAbstract\nWe study the generalization performance of gradient method s in the fundamental stochas-\ntic convex optimization setting, focusing on its dimension dependence. First, for full-batch\ngradient descent (GD) we give a construction of a learning pr oblem in dimension d=O(n2),\nwhere the canonical version of GD (tuned for optimal perform ance of the empirical risk) trained\nwithntraining examples converges, with constant probability, t o an approximate empirical\nrisk minimizer with Ω(1) population excess risk. Our bound t ranslates to a lower bound of\nΩ(√\nd) on the number of training examples required for standard GD to reach a non-trivial\ntest error, answering an open question raised by Feldman (20 16) and Amir, Koren, and Livni\n(2021b) and showing that a non-trivial dimension dependenc e is unavoidable. Furthermore,\nfor standard one-pass stochastic gradient descent (SGD), w e show that an application of the\nsame construction technique provides a similar Ω(√\nd) lower bound for the sample complexity\nof SGD to reach a non-trivial empirical error, despite achie ving optimal test performance. This\nagain provides an exponential improvement in the dimension dependence compared to previous\nwork (Koren, Livni, Mansour, and Sherman, 2022), resolving an open question left therein.\n1 Introduction\nThe study of generalization properties of stochastic optim ization algorithms has been at the heart\nof contemporary machine learning research. While in the mor e classical frameworks studies largely\nfocused on the learning problem (e.g., Alon et al., 1997; Blumer et al., 1989), in the past dec ade it\nhas become clear that in modern scenarios the particular alg orithm used to learn the model plays\na vital role in its generalization performance. As a promine nt example, heavily over-parameterized\ndeep neural networks trained by first order methods output mo dels that generalize well, despite the\nfact that an arbitrarily chosen Empirical Risk Minimizer (E RM) may perform poorly (Zhang et al.,\n2017; Neyshabur et al., 2014, 2017). The present paper aims a t understanding the generalization\nbehavior of gradient methods, specifically in connection wi th the problem dimension, in the funda-\nmental Stochastic Convex Optimization (SCO) learning setu p; a well studied, theoretical framework\nwidely used to study stochastic optimization algorithms.\nThe seminal work of Shalev-Shwartz et al. (2010) was the first to show that uniform con-\nvergence, the canonical condition for generalization in st atistical learning (e.g., Vapnik, 1971;\nBartlett and Mendelson, 2002) may not hold in high-dimensio nal SCO: they demonstrated learn-\ning problems where there exist certain ERMs that overfit the t raining data (i.e., exhibit large\npopulation risk), while models produced by e.g., Stochasti c Gradient Descent (SGD) or regularized\nempirical risk minimization generalize well. The construc tion presented by Shalev-Shwartz et al.\n(2010), however, featured a learning problem with dimensio n exponential in the number of training\n1examples, which only served to prove an Ω(log d) lower bound on the sample complexity for reach-\ning non-trivial population risk performance, where dis the problem dimension. In a followup work,\nFeldman (2016) showed how to dramatically improve the dimen sion dependence and established an\nΩ(d) sample complexity lower bound, matching (in terms of d) the well-known upper bound ob-\ntained from standard covering number arguments (see e.g., S halev-Shwartz and Ben-David, 2014).\nDespite settling the dimension dependence of uniform conve rgence in SCO, it remained unclear\nfrom Shalev-Shwartz et al. (2010); Feldman (2016) whether t he sample complexity lower bounds\nfor uniform convergence actually transfer to natural learn ing algorithms in this framework, and in\nparticular, to common gradient-based optimization method s. Indeed, it is well-known that in SCO\nthere exist simple algorithms, such as SGD, that the models t hey produce actually generalize well\nwith high probability (see e.g., Shalev-Shwartz and Ben-Da vid, 2014), despite these lower bounds.\nMore technically, the construction of Feldman (2016) relie d heavily on the existence of a “peculiar”\nERM which does not seem reachable by gradient steps from a dat a-independent initialization, and\nit was not at all clear (and in fact, stated as an open problem i n Feldman, 2016) how to adapt the\nconstruction so as to pertain to ERMs that could be found by gr adient methods.\nIn an attempt to address this issue, Amir et al. (2021b) recen tly studied the population per-\nformance of batch Gradient Descent (GD) in SCO, and demonstr ated problem instances where it\nleads (with constant probability) to an approximate ERM tha t generalizes poorly, unless the num-\nber of training examples is dimension-dependent.1Subsequently, Amir et al. (2021a) generalized\nthis result to the more general class of batch first-order alg orithms. However, due to technical\ncomplications, the constructions in these papers were base d in part on the earlier arguments of\nShalev-Shwartz et al. (2010) rather than the developments b y Feldman (2016), and therefore fell\nshort of establishing their results in dimension polynomia l in the number of training examples. As\na consequence, their results are unable to rule out a sample c omplexity upper bound for GD that\ndepends only (poly-)logarithmically on problem dimension .\nIn this work, we resolve the open questions posed in both Feld man (2016) and Amir et al.\n(2021b): Our first main result demonstrates a convex learnin g problem where GD, unless trained\nwith at least Ω(√\nd) training examples, outputs a bad ERM with constant probabi lity. This bridges\nthe gap between the results of Feldman (2016) and actual, con crete learning algorithms (albeit with\na slightly weaker rate of Ω(√\nd), compared to the Ω( d) of the latter paper) and greatly improves on\nthe previous Ω(log d) lower bound of Amir et al. (2021b), establishing that the sa mple complexity\nof batch GD in SCO has a significant, polynomial dependence on the problem dimension.\nFurthermore, in our second main result we show how an applica tion of the same construction\ntechnique provides a similar improvement in the dimension d ependence of the empirical risk lower\nbound presented in the recent work of Koren et al. (2022), thu s also resolving the open question left\nin their work. This work demonstrated that in SCO, well-tune d SGD may underfit the training data\ndespite achieving optimal population risk performance. At a deeper level, the overfitting of GD and\nunderfitting of SGD both stem from a combination of two condit ions: lack of algorithmic stability,\nand failure of uniform convergence; as it turns out, this com bination allows for the output models to\nexhibit a large generalization gap , defined as the difference in absolute value between the empir ical\nand population risks. Our work presents a construction tech nique for such generalization gap\nlower bounds that achieves small polynomial dimension depe ndence, providing for an exponential\nimprovement over previous works.\n1Here we refer to GD as performing T=niterations with stepsize η= Θ(1 /√n), where ndenotes the size of the\ntraining set, but our results hold more generally; see below for a more detailed discussion of the various regimes.\n21.1 Our contributions\nIn some more detail, our main contributions are as follows:\n(i) We present a construction of a learning problem in dimens iond=O(nT+n2+η2T2) where\nrunning GD for Titerations with step ηover a training set of ni.i.d.-sampled examples leads,\nwith constant probability, to a solution with population er ror Ω(η√\nT+1/ηT).2In particular,\nfor the canonical configuration of T=nandη= Θ(1/√n), the lower bound becomes Ω(1) and\ndemonstrates that GD suffers from catastrophic overfitting a lready in dimension d=O(n2).\nPut differently, this translates to an/tildewideΩ(√\nd) lower bound the number of training examples\nrequired for GD to reach nontrivial test error. See Theorem 1 below for a formal statement\nand further implications of this result.\n(ii) Furthermore, we give a construction of dimension d=/tildewideO(n2) where the empirical error of one-\npass SGD trained over T=ntraining examples is Ω( η√n+ 1/ηn). Assuming the standard\nsetting ofη= Θ(1/√n), chosen for optimal test performance, the empirical error lower\nbound becomes Ω(1), showing that the “benign underfitting” p henomena of one-pass SGD\nis exhibited already in dimension polynomial in the number o f training samples. Rephrasing\nthis lower bound in terms of the number of training examples r equired to reach nontrivial\nempirical risk, we again obtain an/tildewideΩ(√\nd) sample complexity lower bound. See Theorem 2 for\nthe formal statement and further implications.\nBoth of the results above are tight (up to logarithmic factor s) in view of existing matching\nupper bounds of Bassily et al. (2020). We remark that the cons tructions leading to the results\nfeature differentiable Lipschitz and convex loss functions, whereas the lower boun ds in previous\nworks concerned with gradient methods (Amir et al., 2021b,a ; Koren et al., 2022) crucially applied\nonly to the class of non-differentiable loss functions. From the perspective of general non-smooth\nconvex optimization, this feature of the results imply that our lower bounds remain valid under any\nchoice of a subgradient oracle of the loss function (as oppos ed to only claiming that there exists a\nsubgradient oracle under which they apply, like prior resul ts do).\n1.2 Main ideas and techniques\nOur work builds primarily on two basic ideas. The first is due t o Feldman (2016), whereby an\nexponential number (in n) of approximately orthogonal directions, that represent t he potential\ncandidates for a “bad ERM,” are embedded in a Θ( n)-dimensional space. The second idea, under-\nlying Bassily et al. (2020); Amir et al. (2021b,a); Koren et a l. (2022) is to augment the loss function\nwith a highly non-smooth component, that is capable of gener ating large (sub-)gradients around\ninitialization directed at all candidate directions, that could steer GD towards a bad ERM that\noverfits the training set.\nThe major challenge is in making these two components play in tandem: since the candidate\ndirections of Feldman (2016) are only nearly orthogonal, the progress of GD towards one specific\ndirection gets hampered by its movement in other, irrelevan t directions. And indeed, previous\nwork in this context fell short of resolving this incompatib ility and instead, opted for a simpler\nconstruction with a perfectly-orthogonal set of candidate directions, that was used in the earlier\nwork of Shalev-Shwartz et al. (2010). Unfortunately though , this latter construction requires the\nambient dimensionality to be exponential in the number of sa mplesn, which is precisely what we\naim to avoid.\n2By population error (or test error) we mean the population ex cess risk, namely the gap in population risk between\nthe returned solution and the optimal solution.\n3Our solution for overcoming this obstacle, which we describ e in length in Section 3, is based\non several novel ideas. Firstly, we employ multiple copies o f the original construction of Feldman\n(2016) in orthogonal subspaces, in a way that it suffices for GD to make a single step within\neach copy so as to reach, across all copies, a bad ERM solution ; this serves to circumvent the\n“collisions” between consecutive GD steps alluded to above . Secondly, we carefully design a convex\nloss term that, when augmented to the loss function, forces s uccessive gradient steps to be taken\nin a round-robin fashion between the different copies, so tha t each subspace indeed sees a single\nupdate step through the GD execution. Lastly, we introduce a novel technique that memorizes the\nfull training set by “encoding” it into the iterates in a conv ex and differentiable manner, so that\nthe GD iterate itself (to which the subgradient oracle has ac cess) contains the information required\nto “decode” the right movement direction towards a bad ERM. W e further show how all of these\nadded loss components can be made differentiable, so as to all ow for a differentiable construction\noverall. A detailed overview of these construction techniq ues and a virtually complete description\nof our construction are provided in Section 3.\n1.3 Additional related work\nLearnability and generalization in the SCO model. Our work belongs to the body of litera-\nture on stability and generalization in modern statistical learning theory, pioneered by Shalev-Shwartz et al.\n(2010) and the earlier foundational work of Bousquet and Eli sseeff (2002). In this line of research,\nHardt et al. (2016); Bassily et al. (2020) study algorithmic stability of SGD and GD in the smooth\nand non-smooth (convex) cases, respectively. In the genera l non-smooth case which we study here,\nBassily et al. (2020) gave an iteration complexity upper bou nd ofO(η√\nT+ 1/ηT+ηT/n ) test\nerror forTiterations with step size ηover a training set of size n. The more recent work of\nAmir et al. (2021b) showed this to be tight up to log-factors i n the dimension independent regime,\nand Amir et al. (2021a) further extends this result to any opt imization algorithm making use of\nonly batch gradients (i.e., gradients of the empirical risk ). Even more recently, Kale et al. (2021)\nconsiders (multi-pass) SGD and GD in a more general SCO model where individual losses may\nbe non-convex (but still convex on average), and prove a samp le complexity lower bound for GD\nshowing it learns in a suboptimal rate with any step size and a ny number of iterations.\nSample complexity of ERMs. With relation to the sample complexity of an (arbitrary) ERM\nin SCO, Feldman (2016) showed that reaching ǫ-test error requires Ω( d/ǫ+ 1/ǫ2) training samples,\nbut did not establish optimality of this bound. In a recent wo rk, Carmon et al. (2023) show this\nto be nearly tight and presents a/tildewideO(d/ǫ+ 1/ǫ2) upper bound for any ERM, improving over the\nO(d/ε2) upper bound that can be derived from standard covering numb er arguments. Another\nrecent work related to ours is that of Magen and Shamir (2023) , who provided another example for\na setting in which learnability can be achieved without unif orm convergence, showing that uniform\nconvergence may not hold in the class of vector-valued linea r (multi-class) predictors. However, the\ndimension of their problem instance was exponential in the n umber of training examples.\nImplicit regularization and benign overfitting. Another relevant body of research focuses\non understanding the effective generalization of over-para meterized models trained to achieve zero\ntraining error through gradient methods (see e.g., Bartlet t et al., 2020, 2021; Belkin, 2021). This\nphenomenon appears to challenge conventional statistical wisdom, which emphasizes the importance\nof balancing data fit and model complexity, and motivated the study of implicit regularization (or\nbias) as a notion for explaining generalization in the over- parameterized regimes. Our findings in\nthis paper could be viewed as an indication that, at least in S CO, generalization does not stem\n4from some form of an implicit bias or regularization; see Ami r et al. (2021b); Koren et al. (2022)\nfor a more detailed discussion.\n2 Problem setup and main results\nWe consider the standard setting of Stochastic Convex Optim ization (SCO). The problem is char-\nacterized by a population distribution Dover an instance set Z, and loss function f:W×Z��R\ndefined over convex domain W⊆Rdind-dimensional Euclidean space. We assume that, for any\nfixed instance z∈Z, the function f(w,z) is both convex and L-Lipschitz with respect to its first\nargumentw. In this setting, the learner is interested in minimizing th epopulation loss (orrisk)\nwhich corresponds to the expected value of the loss function over D, defined as\nF(w) =Ez∼D[f(w,z)], (population risk/loss)\nnamely, finding a model w∈Wthat achieves an ε-optimal population loss, namely such that\nF(w)≤F(w∗) +ε,wherew∗∈arg minw∈WF(w) is a population minimizer.\nTo find such a model w, the learner uses a set of ntraining examples S={z1,...,zn}, drawn\ni.i.d. from the unknown distribution D. Given the sample S, the corresponding empirical loss (or\nrisk), denoted/hatwideF(w), is defined as the average loss over samples in S:\n/hatwideF(w) =1\nnn/summationdisplay\ni=1f(w,zi). (empirical risk/loss)\nWe let/hatwidew∗∈arg minw∈W/hatwideF(w) denote a minimizer of the empirical risk, refered to as an em pirical\nrisk minimizer (ERM). Moreover, for every w∈W, we define the generalization gap atwas the\nabsolute value of the difference between the population loss and the empirical loss, i.e., |F(w)−\n/hatwideF(w)|.\nOptimization algorithms. We consider several canonical first-order optimization alg orithms\nin the context of SCO. First-order algorithms make use of a (d eterministic) subgradient oracle\nthat takes as input a pair ( w,z) and returns a subgradient g(w,z)∈∂wf(w,z) of the convex loss\nfunctionf(w,z) with respect to w. If|∂wf(w,z)|= 1, the loss f(·,z) is differentiable at wand the\nsubgradient oracle simply returns the gradient at w; otherwise, the subgradient oracle is allowed\nto emit any subgradient in the subdifferential set ∂wf(w,z).\nFirst, we consider standard gradient descent (GD) with a fixe d step size η >0 applied to the\nempirical risk/hatwideF. We allow for a potentially projected, m-suffix averaged version of the algorithm\nthat takes the following form:\ninitialize at w1∈W;\nupdate wt+1= ΠW/bracketleftigg\nwt−η\nnn/summationdisplay\ni=1g(wt,zi)/bracketrightigg\n, ∀1≤t<T ;\nreturn wT,m:=1\nmm/summationdisplay\ni=1wT−i+1.(1)\nHere ΠW:Rd→Wdenotes the Euclidean projection onto the set W; whenWis the entire space\nRd, this becomes simply unprojected GD. The algorithm returns either the final iterate, the average\nof the iterates, or more generally, any m-suffix average (1 ≤m≤T) of iterates.\n5The second method that we analyze is Stochastic Gradient Des cent (SGD), which is again\npotentially projected and/or suffix averaged. This method us es a fixed stepsize η >0 and takes\nthe following form:\ninitialize at w1∈W;\nupdate wt+1= ΠW/bracketleftbigwt−ηg(wt,zt)/bracketrightbig, ∀1≤t<T ;\nreturn wT,m:=1\nmm/summationdisplay\ni=1wT−i+1.(2)\nMain results. Our main contributions in the context of SCO are tight lower b ounds for the\npopulation loss of GD and for the empirical loss of SGD, where the problem dimension is polynomial\nin the number of samples nand stepsT. First, for the population risk performance of GD, we\nprove the following:\nTheorem 1. Fixn>0,T >32002and0≤η≤1\n5√\nTand letd= 178nT+ 2n2+ max {1,25η2T2}.\nThere exists a distribution Dover instance set Zand a convex, differentiable and 1-Lipschitz loss\nfunctionf:Rd×Z→Rsuch that for GD (either projected or unprojected; cf. Eq. (1)withW=Bd\norW=Rdrespectively) initialized at w1= 0with step size η, for allm= 1,...,T , them-suffix\naveraged iterate has, with probability at least1\n6over the choice of the training sample,\nF(wT,m)−F(w∗) = Ω/parenleftig\nmin/braceleftig\nη√\nT+ 1/ηT, 1/bracerightig/parenrightig\n. (3)\nFor SGD, we prove the following theorem concerning its conve rgence on the empirical risk:\nTheorem 2. Fixn > 2048 and0≤η≤1\n5√nand letd= 712nlogn+ 2n2+ max {1,25η2n2}.\nThere exists a distribution Dover instance set Zand a convex, 1-Lipschitz and differentiable loss\nfunctionf:Rd×Z→Rsuch that for one-pass SGD (either projected or unprojected ; cf. Eq. (2)\nwithW=BdorW=Rdrespectively) over T=nsteps initialized at w1= 0with step size η, for\nallm= 1,...,T , them-suffix averaged iterate has, with probability at least1\n2over the choice of the\ntraining sample,\n/hatwideF(wT,m)−/hatwideF(/hatwidew∗) = Ω/parenleftig\nmin/braceleftig\nη√\nT+ 1/ηT, 1/bracerightig/parenrightig\n. (4)\nDiscussion. As noted in the introduction, both of the bounds above are tig ht up to logarithmic\nfactors in view of matching upper bounds due to Bassily et al. (2020). For GD tuned for optimal\nconvergence on the empirical risk, where T=nandη= Θ(1/√n), Theorem 1 gives an Ω(1)\nlower bound for the population error, which precludes any sa mple complexity upper bound for\nthis algorithm of the form O(dp/ǫq) unlessp≥1\n2. In particular, this implies an Ω(√\nd) lower\nbound the number of training examples required for GD to reac h a nontrivial population risk. In\ncontrast, lower bounds in previous work (Amir et al., 2021b) only implies an exponentially weaker\nΩ(logd) dimension dependence in the sample complexity. We note how ever that there is still a\nsmall polynomial gap between our sample complexity lower bo unds to the known (nearly tight)\nbounds for generic ERMs (Feldman, 2016; Carmon et al., 2023) ; we leave narrowing this gap as an\nopen problem for future investigation.\nMore generally, with GD fixed to perform T=nα,α> 0 steps, and setting ηso as to optimize\nthe lower bound, the right-hand side in Eq. (3) becomes Θ( n−α/4), which rules out any sample\n6complexity upper bound of the form O(dp/ǫq) unless it satisfies max {2,α+ 1}p+1\n4αq≥1.3\nSpecifically, we see that any dimension-free upper bound wit hT=nmust have at least an 1 /ǫ4\ndependence on ǫ; and that for matching the statistically optimal sample com plexity rate of 1 /ǫ2,\none must either run GD for T=n2steps or suffer a polynomial dimension dependence in the rate\n(e.g., forT=nthis dependence is at least d1/4).\nSimilar lower bounds (up to a logarithmic factor) are obtain ed for SGD through Theorem 2,\nbut for the empirical risk of the algorithm when tuned for opt imal performance on the population\nrisk withT=n. In this case, the bounds provide an exponential improvemen t in the dimension\ndependence over the recent results of Koren et al. (2022), sh owing that the “benign underfitting”\nphenomena they revealed for one-pass SGD is exhibited alrea dy in dimension polynomial in the\nnumber of training samples.\nFinally, we remark that our restriction on ηis only meant for placing focus on the more common\nand interesting range of stepsizes in the context of stochas tic optimization. It is not hard to extend\nthe result of Theorems 1 and 2 to larger values of η(in this case the lower bounds are Ω(1), the\nsame rate the theorems give for η= Θ(1/√\nT)), in the same way this is done in previous work\n(e.g., Amir et al., 2021b; Koren et al., 2022).\n3 Overview of constructions and proof ideas\nIn this section we outline the main ideas leading to our main r esults and give an overview of the\nlower bound constructions. As discussed above, the main tec hnical contribution of this paper is in\nestablishing the first Ω( η√\nT) term in Eqs. (3) and (4) using a loss function in dimension po lynomial\ninnandT, and this is also the focus of our presentation in this sectio n. In Sections 3.1 to 3.4 we\nfocus on GD and describe the main ideas and technical steps to wards proving our first main result;\nin Section 3.6 we survey the additional steps and adjustment s needed to obtain our second main\nresult concerning SGD.\nStarting with GD and Theorem 1, recall that our goal is to esta blish a learning scenario where\nGD is likely to converge to a “bad ERM”, namely a minimizer of t he empirical risk whose population\nloss is large. We will do that in four steps: we will first estab lish that such a “bad ERM” actually\nexists; then, we will show how to make such a solution reachab le by gradient steps from the origin;\nwe next describe how the information required to identify th is solution can be “memorized” by GD\ninto its iterates; and finally, we show how to combine these co mponents and actually drive GD\ntowards a bad ERM.\n3.1 A preliminary: existence of bad ERMs\nOur starting point is the work of Feldman (2016) that demonst rated that in SCO, an empirical risk\nminimizer might fail to generalize, already in dimension li near in the number of training samples.\nMore concretely, they showed that for any sample size n, there exists a distribution Dover convex\nloss functions in dimension d= Θ(n) such that, with constant probability, there exists a “bad\nERM”: one that overfits the training sample and admits a large generalization gap.\nTheir approach was based on a construction of a set of unit vec tors of size 2Ω(n), denotedU, that\nare “nearly orthogonal”: the dot product between any two dis tinctu,v∈Usatisfies |/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht| ≤1\n8.4\n3To see this, let r= max {2, α+ 1}and note that for our construction d=O(nT+n2) =O(nr) and ǫ= Ω( n−α/4);\nthe sample complexity upper bound O(dp/ǫq) can be therefore rewritten in terms of nasO(nrp+αq/4), and since this\nshould asymptotically upper bound the number of samples n, one must have that rp+1\n4αq≥1.\n4The original construction by Feldman (2016) satisfied sligh tly different conditions, which we adjust here for our\nanalysis.\n7Then, they take the power set Z=P(U) ofUas the sample space (namely, identifying samples\nwith subsets of U), the distribution Dto be uniform over Z, and the (convex, Lipschitz) loss\nhF16:RΘ(n)×Z→Rto be defined as follows:\nhF16(w,V) = max/braceleftig\n1\n2,max\nu∈V/a\\}b∇acketle{tu,w/a\\}b∇acket∇i}ht/bracerightig\n. (5)\nFor this problem instance, they show that with constant prob ability over the choice of a sample\nS={V1,...,Vn}iid∼ Dnof sizen=O(d), at least one of the vectors in U, sayu0∈U, will not be\nobserved in any of the sets V1,...,Vn, namelyu0∈U\\/uniontextn\ni=1Vi. Finally, they prove that such a\nvectoru0is in fact an Ω(1)-bad ERM (for which the generalization gap i s Ω(1)).\nTo see why this is the case, note that, since every vector u∈Uis in every training example\nViwith probability1\n2, the setU(whose size is exponential in n) is large enough to guarantee the\nexistence of a vector u0/∈/uniontextn\ni=1Viwith constant probability. Consequently, the empirical lo ss of\nsuchu0equals1\n2(since /a\\}b∇acketle{tu0,v/a\\}b∇acket∇i}ht ≤1\n8for anyv∈Viand therefore hF16(u0,Vi) =1\n2for alli). However,\nfor a fresh example V∼ D, with probability1\n2it holds that u0∈V, and therefore hF16(u0,V) = 1,\nin which case the population risk of u0is at least =1\n2·1\n2+1\n2·1\n2=3\n4and the generalization gap is\ntherefore at least1\n4.\n3.2 Ensuring that bad ERMs are reachable by GD\nAs Feldman (2016) explains in their work, although there exi sts an ERM with a large generalization\ngap, it is not guaranteed that such a minimizer is at all reach able by gradient methods, within a\nreasonable (say, polynomial in n) number of steps. This is because in their construction, the loss\nfunctionhF16remains flat (and equals1\n2, see Eq. (5)) inside a ball of radius Ω(1) around the origin,\nwhere GD is initialized; within this ball, all models are ess entially “good ERMs” that admit zero\ngeneralization gap. It remains unclear how to steer GD, with stepsize of order η=O(1/√\nT) over\nTsteps, away from this flat region of the loss towards a bad ERM, such as the u0identified above.\nTo address this challenge, we modify the construction of Fel dman (2016) in a fundamental\nway. The key idea is increase dimensionality and replicate F eldman’s construction in Torthogonal\nsubspaces; this would allow us to decrease, in each of the sub spaces, the distance to a bad ERM to\nonlyO(η), rather than Θ(1) as before. Then, while each of these subsp ace ERMs is only Ω( η)-bad,\nwhen taken together, they constitute an Ω(1)-bad ERM in the l ifted space whose distance from the\norigin is roughly η√\nT= Θ(1), yet is still reachable by Tsteps of GD.\nMore concretely, we introduce a loss function h:Rd′×P(U)→R(ford′= Θ(n)) that resembles\nFeldman’s function from Eq. (5) up to a minor adjustment:\nh(w′,V) = max/braceleftig\n3\n32η,max\nu∈V/angbracketleftbigu,w′/angbracketrightbig/bracerightig\n. (6)\nAs in the original construction by Feldman, Vhere ranges over subsets of a set U⊆Rd′of\nsize 2Ω(d′), the elements of which are nearly-orthogonal unit vectors. Then, we construct a loss\nfunction in dimension d=Td′by applying hinTorthogonal subspaces of dimension d′, denoted\nW(1),...,W(T), as follows:5\nℓ1(w,V) =/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2/parenleftig\nh(w(k),V)/parenrightig2\n. (7)\nHere and throughout, w(k)refers to the k’th orthogonal component of the vector w, that resides\nin the subspace W(k). Finally, the distribution Dis again taken to be uniform over Z=P(U),\n5The summation starts at k= 2 due to technical reasons that will become apparent later i n this proof sketch.\n8and a training set is formed by sampling S={V1,...,Vn} ∼ Dn. As before, we know that with\nprobability at least1\n2, there exists a vector u0such thatu0∈U\\/uniontextn\ni=1Vi.\nWith this setup, it can be shown that ℓ1is indeed convex and O(1)-Lipschitz, and further,\nthat any vector wsatisfyingw(k)=cηu0for a sufficiently large constant c >0 and Ω(T)-many\ncomponents k, is an Ω(1)-bad ERM with respect to ℓ1. The important point is that, unlike in\nFeldman’s original construction, such bad ERMs are potenti ally reachable by GD: it is sufficient\nto guide the algorithm to make a single, small step (with step sizeη) towardsu0in each subspace\nW(k).\n3.3 Memorizing the dataset in the iterate\nThere is one notable obstacle to the plan we just described: t he vectoru0is determined in a rather\ncomplex way by the full description of the training set and it is unclear how to reproduce such a\nvector through subgradients of the loss function. Indeed, r ecall that the only access GD has to\nthe training set is through subgradients of individual func tionsg(w,V 1),...,g (w,Vn) (and linear\ncombinations thereof), and none of these has direct access t o the full training set that could allow\nfor determining a vector u0∈U\\/uniontextn\ni=1Vi.\nTo circumvent this difficulty, another key aspect of our const ruction involves a mechanism that\neffectively memorizes the full training set in the iterate witself, using the first few steps of GD. For\nthis memorization, we can, for example, further increase th e dimension of the domain Wand create\nan “encoding subspace,” denoted as W(0)(and the corresponding component of a vector w∈Wis\nindicated by w(0)), which is orthogonal to W(1),...,W(T). In this subspace, each step taken with\nrespect to (a linear combination of) individual gradients g(wt,Vi) encodes the information on the\nsetsV1,...,Vninto the iterate wt. Then, since the subgradient oracle receives wtas input, it can\nreconstruct the training set encoded in wt(0)and recover u0, in every subsequent step.\nOn its own, the task of memorizing the training set is not part icularly challenging and can\nbe addressed in a rather straightforward manner.6What turns out to be more challenging is to\ndesign the encoding in such a way that u0is realized as the unique subgradient (i.e., the gradient)\nof the loss function. This would be crucial for establishing that our lower bound is valid for any\nsubgradient oracle, and not only for an adversarially chose n one (as well for making the construction\ndifferentiable; we discuss this later on, in Section 3.5).\nLet us describe an encoding mechanism where u0acts as the unique subgradient at w1= 0.\nWe will employ an encoding subspace W(0)of dimension Θ( n2), and augment samples with a\nnumberj∈[n2], drawn uniformly at random; namely, each sample in the trai ning set is now a pair\n(Vi,ji)∈P(U)×[n2], fori= 1,...,n . We then create an encoding function φ:P(U)×[n2]→W(0)\nsuch thatφ(V,j) maps the set Vinto thej’th (2-dimensional) subspace of the encoding space. The\nrole ofjis to ensure that, with constant probability, sets in the tra ining sample are mapped to\ndistinct subspaces of the encoding space W(0), and thus can be uniquely inferred given the encoding.\nTo implement the encoding within the optimization process, we introduce the following term into\nthe loss:\nℓ2(w,(V,j)):=/a\\}b∇acketle{t−φ(V,j),w(0)/a\\}b∇acket∇i}ht. (8)\nFollowing a single step of GD, the iterate becomes w2(0)=η\nn/summationtextn\ni=1φ(Vi,ji), and by the properties\nof the encoding φit is then possible, with constant probability, to fully rec over the sets V1,...,Vn\nin the training set, given the iterate w2.\n6One simple approach is to utilize a one-dimensional encodin g space to encode every individual training example\nV∈P(U) in the least significant bits of wt(0), in a way that guarantees there are no collisions between diff erent\npossible values of such sets. This allows the subgradient or acle to calculate the specific u0from the training set\nencoded in the least significant bits of wt(0)and return it as a subgradient.\n9Next, we introduce an additional term into the loss function , whose role is to “decode” the\ntraining set V1,...,Vnfromwtand produce a vector u0∈U\\/uniontextn\ni=1Vias a subgradient. For this,\nwe represent every potential training set using a vector ψ∈Ψ⊆B2n2, and define a mapping\nα:R2n2→Uthat, for every ψ∈Ψ, provides a vector α(ψ)∈Uthat does not appear in any of\nthe setsViin the training sample associated with ψ(if such a vector exists). Finally, we add the\nfollowing term to the loss function,\nℓ3(w):= max/braceleftbigg\nδ1,max\nψ∈Ψ/braceleftig\n/a\\}b∇acketle{tψ,w(0)/a\\}b∇acket∇i}ht −β/a\\}b∇acketle{tα(ψ),w(1)/a\\}b∇acket∇i}ht/bracerightig/bracerightbigg\n, (9)\nwhereβ,δ1>0 are small predefined constants. We can show, assuming that i n the first step, the\ntraining set was encoded to the iterate ( w2(0)=η\nn/summationtextn\ni=1φ(Vi,ji)), that for a suitable choice of the\nencoder (φ) and decoder ( ψandα), in the following iteration, the vector ψ∗∈Ψ that represents\nthe actual training set V1,...,Vnis realized as a unique maximizer in Eq. (9), which in turn tri ggers\na gradient step along u0:=α(ψ) in the subspace W(1).\n3.4 Making GD converge to a bad ERM\nOur final task is to finally make GD converge to a “bad ERM,” name ly to a model wsuch that\nw(k)=cηu0for a sufficiently large constant c >0 and Ω(T)-many values of k, assuming it was\nsuccessfully initialized at wwithw(1)=c1u0(andw(k)= 0 fork > 1) as we just detailed. We\nwill accomplish this by forcing GD into making a single step t owardsu0in Ω(T) of the subspaces\nW(1),...,W(T).\nTo this end, we employ a variation of a technique used in previ ous lower bound construc-\ntions (Bassily et al., 2020; Amir et al., 2021b; Koren et al., 2022) to induce gradient instability\naround the origin. In these prior instances, however, the po tential directions of progress—analogous\nto vectors in our set U—were perfectly orthogonal (and thus, the dimension of spac e was required\nto be exponential in n). In contrast, in our scenario the vectors in Uare only approximately or-\nthogonal, and directly applying this approach could lead to situations where gradient steps from\nconsecutive iterations may interfere with progress made in correlated directions in previous itera-\ntions.\nTo address this, we introduce a careful variation on this tec hnique, based on augmenting the\nloss function with the following convex term:\nℓ4(w) = max/braceleftbigg\nδ2,max\nu∈U,k<T/braceleftig\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w(k+1)/a\\}b∇acket∇i}ht/bracerightig/bracerightbigg\n, (10)\nwhereδ2>0 is a small constant (that will be set later). The key idea her e is that following the\ninitialization stage, the inner maximization above is alwa ys attained at the same vector u=u0, and\nfor values of kthat increase by 1 in every iteration of GD. Consequently, su bgradient steps with\nrespect to this term will result in making a step towards u0in each of the components w(1),w(2),...\none by one, avoiding interference between consecutive step s. At the end of this process, there are\nΩ(T) values ofksuch thatw(k)=1\n8ηu0, which is what we set to achieve.\nIn some more detail, assuming GD is successfully initialize d at a vector wwithw(1)=c1u0and\nw(k)= 0 fork >1 (c1>0 is a small constant), note that the maximum in Eq. (10) is uni quely\nattained at k= 1 andu=u0. Consequently, the subgradient of ℓ4at initialization is a vector g\nsuch thatg(1)=3\n8u0,g(2)=−1\n2u0(andg(k)= 0 fork/\\e}atio\\slash= 1,2), and taking a subgradient step with\nstepsizeηresults inw(1)= (ηβ−3η\n8)u0andw(2)=η\n2u0(fork/\\e}atio\\slash= 1,2,w(k)remains as is). In each\nsubsequent iteration, the maximization in Eq. (10) is attai ned at an index kfor whichw(k)=η\n2u0\n10and atu=u0.7Subsequently, every gradient step adds −3η\n8u0tow(k)andη\n2u0tow(k+1)and\nresults inw(2)=w(3)=...=w(k)=η\n8u0andw(k+1)=η\n2u0(whereas for all s>k +1,w(s)remains\nzero).\nFinally, we note that the GD dynamics we described ensure tha t the iterates w1,...,wTremain\nstrictly within the unit ball Bd, even when the algorithm does not employ any projections. As\na consequence, the construction we described applies equal ly to a projected version of GD, with\nprojections to the unit ball, and the resulting lower bound w ill apply to both versions of the\nalgorithm.\n3.5 Putting things together\nWe can now integrate the ideas described in Sections 3.1 to 3. 4 into a construction of a learning\nproblem where GD overfits the training data (with constant pr obability), that would serve to prove\nour lower bound for gradient decent. To summarize this const ruction:\n• The examples in the learning problem are parameterized by p airs (V,j)∈Z:=P(U)×[n2],\nwhereUis the set of nearly-orthogonal vectors described in Sectio n 3.1, andP(U) is its power\nset;\n• The population distribution Dis uniform over pairs ( V,j)∈Z, namely such that V∼\nUnif(P(U)) (i.e.,Vis formed by including every element u∈Uindependently with proba-\nbility1\n2) andj∼Unif([n2]);\n• The loss function in this construction, f:W×(P(U)×[n2])→R, is then given by:\n∀(V,j)∈Z, f (w,(V,j)):=ℓ1(w,V) +ℓ2(w,(V,j)) +ℓ3(w) +ℓ4(w), (11)\nwith the terms ℓ1,ℓ2,ℓ3,ℓ4as defined in Eqs. (7) to (10) respectively.\nWith a suitable choice of parameters, this construction ser ves to proving Theorem 1. We remark\nthat, while fin this construction is convex and O(1)-Lipschitz, it is evidently non-differentiable.\nFor obtaining a construction with a differentiable objectiv e that maintains the same lower bound\nand establish the full claim of Theorem 1, we add one final step of randomized smoothing of the\nobjective. This argument hinges on the fact that the subgrad ients offare unique along any possible\ntrajectory of GD , so that smoothing in a sufficiently small neighborhood would preserve gradients\nalong any such trajectory (and thus does not affect the the dyn amics of GD), while making the\nobjective differentiable everywhere. The full proof of Theo rem 1 is deferred to Appendix A.\n3.6 Additional adjustments for SGD\nMoving on to discuss our second main result for SGD, we provid e here a brief overview of the\nnecessary modifications upon the construction for GD to esta blish the lower bound for SGD in\nTheorem 2; further details can be found in Section 5. In the ca se of SGD, our goal is to establish\nunderfitting: namely, to show that the algorithm may converg e to a solution with an excessively\nlarge empirical risk despite successfully converging on th e population risk.\nThe main ideas leading to our construction for SGD are simila r to what we discussed above, but\nthere are several necessary modifications that arise from th e fact that, whereas in GD the entire\n7For this value of k, it holds that max u∈U/braceleftbig3\n8/an}bracketle{tu, w(k)/an}bracketri}ht −1\n2/an}bracketle{tu, w(k+1)/an}bracketri}ht/bracerightbig\n=3\n16η(attained at u=u0), whereas for\nother values of kthis quantity is at most ≈1\n8·3\n8η+1\n8·/parenleftbig1\n2/parenrightbig2η <1\n8ηdue to the near-orthogonality of vectors in U. It\nfollows that the subgradient is a vector gsuch that g(k)=3\n8u0,g(k+1)=−1\n2u0(and zeros elsewhere).\n11training set is revealed already in the first iteration, in SG D it is revealed sequentially, one training\nsample at a time. In particular, unlike in the case of GD where it is possible to identify a bad\nERMu0at the few first steps of the algorithm and steer the algorithm in this direction in every\nsubspaceW(1),W(2),..., for SGD the required progress direction in W(t),represented as a “bad\nsolution”ut, can be only determined in the t’th step based on the encoded training set up to that\npoint,V1,...,Vt−1. As a result, it is crucial to modify the loss function such th at the process of\ndecoding such utfromV1,...,Vt−1occurs in every iteration t.\nAnother essential adjustment involves identifying a solut ion with a large generalization gap\n(namely large empirical risk, low population risk) and guid ing the SGD iterates to converge to\nsuch a solution. Considering the function ℓ1defined in Eq. (7), such a solution is represented by\na vectoru∈Uthat appears in all of the sets V1,...,Vnin the training sample. However, since ut\ncannot depend on future examples, our goal within every subs paceW(t)is to take a single gradient\nstep towards a vector utpresent only in sets up to that point, namely in/intersectiontextt−1\ni=1Vi(note that such\nutmaximizes the corresponding loss functions ℓ1(w,V 1)...ℓ 1(w,Vt−1)). Additionally, to ensure\nthat gradients for future loss functions remain zero and do n ot affect the algorithm’s dynamics,\nit is necessary to to guarantee that ut∈/intersectiontextn\ni=tVi; in other words, we are looking for a solution\nut∈/intersectiontextt−1\ni=1Vi∩/intersectiontextn\ni=tVi. For ensuring that such a vector actually exists (with const ant probability),\nwe lift the dimension of the set Uand the subspaces {W(k)}n\nk=1tod= Θ(nlogn) (instead of Θ( n)\nas before) and modify the distribution Dso as to have that Vis sampled such that every element\nu∈Uis included in Vindependently with probability 1 /4n2.\nWith these adaptations in place, we can obtain Theorem 2; for more details we refer to Section 5.\n4 Overfitting of GD: Proof of Theorem 1\nIn this section, we provide a formal proof of our main result f or GD. We establish a lower bound of\nΩ(η√\nT) for the population loss of GD, where the hard loss function i s defined in a d-dimensional\nEuclidean space, where the dimension dis polynomial in the number of examples n. In Appendix A\nwe complete the proof of Theorem 1, by showing a lower bound of min{1/ηT, 1}, and a construction\nof a differentiable objective that holds the lower bound stat ed in Theorem 1.\nFull construction. For the first step, for a dimension d′that will be set later, we use a set of\napproximately orthogonal vectors in Rd′with size (at least) exponential in d′, the existence of which\nis given by the following lemma, adapted from Feldman (2016) .\nLemma 1. For anyd′≥256, there exists a set Ud′⊆Rd′, with |Ud′| ≥2d′/178, such that for all\nu,v∈Ud′,u/\\e}atio\\slash=v, it holds that | /a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht | ≤1\n8.\nNow, letnbe the number of examples in the training set. We define the set U:=Ud′to be a\nset as specified by Lemma 1 for d′= 178n. Then, as outlined in Section 3, we define the sample\nspaceZ:={(V,j) :V⊆U,j∈[n2]}and the hard distribution Das the uniform distribution.\nMoreover, we consider the loss function f:Rd→R(defined in Eq. (11) for d:=Td′+ 2n2=\n178nT+ 2n2. This loss function is convex and 5-Lipschitz over Rd, as established in the following\nlemma:\nLemma 2. For every (V,j)∈Z, the loss function f(w,(V,j))is convex and 5-Lipschitz over Rd\nwith respect to its first argument.\nFor this construction of distribution and loss function, we obtain the following theorem.\n12Theorem 3. Assume that n > 0,T > 32002andη≤1√\nT. Consider the distribution Dand the\nloss function fthat defined in Section 3.5 for d= 178nT+ 2n2,ε=1\nn2(1−cos(2π\n|P(U)|)),β=ǫ\n4T2,\nδ1=η\n2nandδ2=3ηβ\n16. Then, for Unprojected GD (cf. Eq. (1)withW=Rd) on/hatwideF, initialized at\nw1= 0with step size η, we have, with probability at least1\n6over the choice of the training sample:\n(i) The iterates of GD remain within the unit ball, namely wt∈Bdfor allt= 1,...,T ;\n(ii) For all m= 1,...,T , them-suffix averaged iterate has:\nF(wT,m)−F(w∗) = Ω/parenleftbigη√\nT/parenrightbig.\nAlgorithm’s dynamics. We next give a key lemma that characterizes the trajectory of GD\nwhen applied to the empirical risk/hatwideFformed by the loss function fand the training sample S=\n{(Vi,ji)}n\ni=1. The characterization holds under a certain “good event”, g iven as follows:\nE:=/braceleftbig/uniondisplayn\ni=1Vi/\\e}atio\\slash=U/bracerightbig∩/braceleftbigjk/\\e}atio\\slash=jl,∀k/\\e}atio\\slash=l/bracerightbig. (12)\nIn words, under the event Ethere exists at least one “bad direction” (which is a vector i n the set\nU\\/uniontextn\ni=1Vi) and there is no collision between the indices j1,...,jn. In the following lemma we\nshow that Eholds with a constant probability. The proof is deferred to A ppendix B.2.\nLemma 3. For the event Edefined in Eq. (12), it holds that Pr(E)≥1\n6.\nUnder this event, the dynamics of GD are characterized as fol lows.\nLemma 4. Assume the conditions of Theorem 3, and consider the iterate s of unprojected GD on\n/hatwideF, with step size η≤1/√\nTinitialized at w1= 0. Under the event E, we have for all t≥5that\nwt(k)=\n\nη\nn/summationtextn\ni=1φ(Vi,ji)k=0;/parenleftbig−3\n8+t−2\n4ǫ\nT2/parenrightbigηu0k= 1;\n1\n8ηu0 2≤k≤t−3;\n1\n2ηu0 k=t−2;\n0 t−1≤k≤T,(13)\nwhereu0is a vector such that u0∈U\\/uniontextn\ni=1Vi.\nTo prove Lemma 4, we break down to the different components of t he loss and analyze how the\ntermsℓ1,ℓ3andℓ4affects the dynamics of GDunder the event E. For each of these components,\nwhich involve maximum over linear functions, we show which t erm achieves the maximum value\nfor eachwtand derive the expressions for the gradients at those points by the maximizing terms.\nFirst, we show that under this event, the gradients of ℓ1do not affect the dynamics since in any\niterationtthe gradient of ℓ1is zero, as stated in the following lemma. The proof is deferr ed to\nAppendix B.\nLemma 5. Assume the conditions of Theorem 3 and the event E. Letw∈Rdbe such that for\nevery 2≤k≤T,w(k)=cηu0forc≤1\n2andu0∈U\\/uniontextn\ni=1Vi. Then, for every i, it holds that\n(i) for every k≥2, it holds maxu∈Vi/a\\}b∇acketle{tw(k),u0/a\\}b∇acket∇i}ht ≤η\n16;\n(ii)ℓ1is differentiable at wand for all i∈[n], we have ∇ℓ1(w,Vi) = 0.\n13Next, for the term ℓ3, as outlined in Section 3.3, it is used for identifying the ac tual training\nsetS={(Vi,ji)}n\ni=1given an encoding ψ∗=1\nn/summationtextn\ni=1φ(Vi,ji) in the iterate wt(0)and ensuring a\nperformance of gradient step in W(1)towards a corresponding vector u0∈U\\/uniontextn\ni=1Viin the following\niteration. It is done by getting ψ∗as a maximum of linear functions (with positive constant mar gin)\nover the set Ψ which contains all possible encoded datasets. This idea is formalized in the following\nlemma.\nLemma 6. Assume the conditions of Theorem 3 and the event E. Letψ∗=1\nn/summationtextn\ni=1φ(Vi,ji)and\nw∈Rdbe suchw(0)=ηψ∗, and letw(1)=cηu0for|c| ≤1andu0∈U\\/uniontextn\ni=1Vi. Then\n(i) For every ψ∈Ψ,ψ/\\e}atio\\slash=ψ∗:\n/a\\}b∇acketle{tw(0),ψ∗/a\\}b∇acket∇i}ht −ǫ\n4T2/a\\}b∇acketle{tα(ψ∗),w(1)/a\\}b∇acket∇i}ht>/a\\}b∇acketle{tw(0),ψ/a\\}b∇acket∇i}ht −ǫ\n4T2/a\\}b∇acketle{tα(ψ),w(1)/a\\}b∇acket∇i}ht+ηǫ\n4;\n(ii) Forψ=ψ∗, it holds that\n/a\\}b∇acketle{tw(0),ψ∗/a\\}b∇acket∇i}ht −ǫ\n4T2/a\\}b∇acketle{tα(ψ∗),w(1)/a\\}b∇acket∇i}ht>δ1+η\n16n;\n(iii)ℓ3is differentiable at wand the gradient is given as follows:\n(∇ℓ3(w))(k)=\n\nψ∗k= 0;\n−ǫ\n4T2u0k= 1;\n0 otherwise.\nFinally, for ℓ4, as detailed in Section 3.4, the role of this term is to make th e last iterate wT\nholdwT(k)=η\n8ηu0for Ω(T) many sub-spaces W(k). In the following lemma, we show that in every\niterationt, every gradient step increases the amount of such ks by 1, namely, in every iteration t,\nthe maximum of ℓ4is attained at u=u0and indexkt= arg max {k:wt(k)/\\e}atio\\slash= 0}, which increases\nby 1 in every iteration, making the wt+1(kt)=η\n8ηu0.\nLemma 7. Assume the conditions of Theorem 3 and the event E. Letw∈Rd,u0∈U\\/uniontextn\ni=1Vi\nand3≤m < T be such that w(1)=cηu0for−3\n8≤c≤0,w(k)=η\n8u0for every 2≤k≤m−1,\nw(k)=η\n2u0andw(k)= 0for everyk≥m. Then, it holds that,\n(i) For every pair u∈Uandk<T such thatk/\\e}atio\\slash=moru/\\e}atio\\slash=u0,\n3\n8/a\\}b∇acketle{tu0,w(m)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu0,w(m+1)/a\\}b∇acket∇i}ht>3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w(k+1)/a\\}b∇acket∇i}ht+η\n64\n(ii)\n3\n8/a\\}b∇acketle{tu0,w(m)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu0,w(m+1)/a\\}b∇acket∇i}ht>δ2+η\n64.\n(iii)ℓ4is differentiable at wand the gradient is given as follows:\n/parenleftbig∇ℓ4(w)/parenrightbig(k)=\n\n3\n8u0k=m;\n−1\n2u0k=m+ 1;\n0 otherwise.\n14Proof of Lemma 4. We prove the lemma by induction on t; the base case, for t= 5, is proved in\nLemma 21 in Appendix B and here we focus on the induction step. For this, fix any t≥5 and\nassume the that the lemma holds for wt; we will prove the claim for wt+1.\nFirst, forℓ1, note that, by the hypothesis of the induction, for every 2 ≤k≤T,wt(k)=cηu0\nforc≤1\n2, thus, by Lemma 5, for every i,∇ℓ1(wt,Vi) = 0.\nForℓ2, we know that, for every i,\n/parenleftbig∇ℓ2(wt,(Vi,ji))/parenrightbig(k)=/braceleftigg\n−φ(Vi,ji)k= 0;\n0 otherwise .\nForℓ3, using the hypothesis of the induction, which implies that wt(1)=cηu0for|c| ≤1 and\nwt(0)=η\nn/summationtextn\ni=1φ(Vi,ji), by Lemma 6, we get that,\n/parenleftbig∇ℓ3(wt)/parenrightbig(k)=\n\n1\nn/summationtextn\ni=1φ(Vi,ji)k= 0;\n−ǫ\n4T2u0 k= 1;\n0 otherwise .\nForℓ4, by the hypothesis of the induction we know that wt(1)=/parenleftbig−3\n8+(t−2)\n4ǫ\nT2/parenrightbigηu0, thus,wt(1)=\ncηu0for−3\n8c≤0. Then the conditions of Lemma 7 hold for m=t−2, thus, it holds that,\n/parenleftbig∇ℓ4(wt)/parenrightbig(k)=\n\n3\n8u0k=t−2;\n−1\n2u0k=t−1;\n0 otherwise .\nCombining all together, we get that,\n/parenleftbig∇/hatwideF(wt)/parenrightbig(k)=\n\n−ǫ\n4T2u0k= 1;\n3\n8u0k=t−2;\n−1\n2u0k=t−1;\n0 otherwise ,\nwhereu0∈U\\/uniontextn\ni=1Vi, and the lemma follows. /square\nProof of Lower Bound. Now we can turn to prove Theorem 3. Here we prove the lower boun d\nfor the case of suffix averaging with m= 1, namely, when the output solution is the final iterate\nwTof GD; the full proof for the more general case can be found in A ppendix B.3.\nProof of Theorem 3 ( m= 1case). We prove the theorem under the condition that Eoccurs. First,\nin Lemma 22 in appendix Appendix B we know that for every t, we have that /ba∇dblwt/ba∇dbl ≤1.\nNext,wTis as in Eq. (13). Now, we notice that if a vector v∈Uis in a setV⊆U, it holds\nthat max u∈V/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht= 1. However, if v /∈V, it holds that max u∈V/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht=1\n8. As a result, by the fact\nthat every vector for a fresh pair ( V,j)∼D,u0∈Uis inVwith probability1\n2, the following holds:\nEV/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/braceleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT(k)/a\\}b∇acket∇i}ht/bracerightbigg2\n≥EV/radicaltp/radicalvertex/radicalvertex/radicalbtT−3/summationdisplay\nk=2max/braceleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT(k)/a\\}b∇acket∇i}ht/bracerightbigg2\n=EV/radicaligg\n(T−4) max/braceleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,η\n8u0/a\\}b∇acket∇i}ht/bracerightbigg2\n15=η√\nT−4\n8EVmax/braceleftbigg3\n4,max\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/bracerightbigg\n≥η√\nT−4\n8/parenleftbigg3\n4Pr(u0/∈V) + Pr(u0∈V)/parenrightbigg\n=7η\n64√\nT−4.\nMoreover, we notice that for every t,V⊆Uandj∈[n2],ℓ2(wt,(V,j))≥ −/ba∇dblwt(0)/ba∇dbl ≥ −η,\nℓ3(wt)≥δ1andℓ4(wt)≥δ2, thus, it holds that\nF(wT)≥7η\n64√\nT+δ1+δ2−η≥η/parenleftbigg7\n64√\nT−1/parenrightbigg\n;\nF(w∗)≤F(0)≤3η\n32√\nT+η.\nThen, since Tis assumed large enough so that 2 ≤1\n128√\nT, we conclude\nF(wT)−F(w∗)≥η/parenleftbigg1\n64√\nT−2/parenrightbigg\n≥η\n128√\nT. /square\n5 Underfitting of SGD: Proof of Theorem 2\nIn this section we show a formal proof of our main result for SG D. As inGD, we construct a hard\nloss function, which is defined in a d-dimensional Euclidean space such that dis polynomial in the\nnumber of examples n. Using this construction, we establish a lower bound of Ω( η√\nT) for the\nempirical loss of SGD with T=niterations. We complete the proof of Theorem 2 in Appendix A.\nFull construction. For the first step of the construction, we use Lemma 1 (see Sect ion 4), which\nshows for every dimension d′an existence of a set of approximately orthogonal vectors in U∈Rd′\nwith size exponential in d′. We define the set Uto beU:=Ud′ford′= 712nlognand the sample\nspace to be ZSGD:={V:V⊆U}. Moreover, we define the hard distribution DSGDto be such\nthat everyu∈Uis included in V⊆Uindependently with probability δ=1\n4n2.\nFor the hard loss function, we continue referring to every ve ctorw∈Rdas a concatenation of\nvectors,w= (w(0),w(1),w(2),...,w(n)), where for 1 ≤k≤n,w(k)∈R712nlognandw(0)∈R2n2.\nIn this construction, w(0)is also a concatenation of nvectorsw(0,1),...,w(0,n)such that each for\neveryr∈[n],w(0,r)∈R2n\nOur approach is, as in GD, in every iteration t, to encode the set Vt, sampled from DSGDinto\nthe iteratew(0)\nt+1. For this, we construct an encoder, φ:P(U)×[n]→R2n, a decoder α:R2n→U,\na real number ǫ >0 andnsets denoted as Ψ 1,..., Ψn. Here, the idea behind the construction is\nsuch setψkrepresents all of the possible training sets with kexamples, {V1...,Vk}, and in every\niterationt, it is possible to get the vector ψ∗\nt−1∈Ψt−1that is recognized with the actual sets\nV1,...,Vt−1that are sampled before this iteration, as a maximizer of a li near function with margin\nǫ. Then, as outlined in Section 3.6, we aim to output a vector ut∈/intersectiontextn\ni=tVi. The exact construction\nof suchǫ,φ,α, Ψ1,..., Ψnis detailed in Lemma 23 in Appendix C.\nThen, forǫ,φ,α, Ψ1,..., Ψnandd:=nd′+ 2n2= 712n2logn+ 2n2we define the loss function\nin our construction. The loss function fSGDis composed of three terms: ℓSGD\n1,ℓSGD\n2,ℓSGD\n3, and is\n16defined as follows,\nfSGD(w,V):=/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nℓSGD\n1(w,V):=(14)\n+ max/parenleftbigg\nδ1, max\nk∈[n−1],u∈U,ψ∈Ψk/parenleftbigg3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht\n− /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg\n+/a\\}b∇acketle{tw(0,1),−1\n4n2φ(V,1)/a\\}b∇acket∇i}ht − /a\\}b∇acketle{t1\nn3u1,w(1)/a\\}b∇acket∇i}ht\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nℓSGD\n3(w,V):=,\nwhere the second term is denoted ℓSGD\n2(w,V) andu1is an arbitrary vector in U. In the following\nlemma, we establish that the above loss function in indeed co nvex and Lipschitz over Rd. The proof\nappears in Appendix C.\nLemma 8. For everyV∈Z, the loss function fSGD(w,V)is convex and 4-Lipschitz over Rdwith\nrespect to its first argument.\nFor this construction of distribution and loss function, we show the following theorem,\nTheorem 4. Assume that n > 2048 andη≤1√\nT. Consider the distribution DSGDand the loss\nfunctionfSGDwithd= 712n2logn+ 2n2,ε=1\nn2(1−cos(2π\n|P(U)|))andδ1=η\n8n3. Then, for\nUnprojected SGD (cf. Eq. (2)withW=Rd) withT=niterations, initialized at w1= 0with step\nsizeη, we have, with probability at least1\n2over the choice of the training sample,\n(i) The iterates of SGD remain within the unit ball, namely wt∈Bdfor allt= 1,...,T ;\n(ii) For all m= 1,...,T , them-suffix averaged iterate has:\n/hatwideFSGD(wT,m)−/hatwideFSGD(/hatwidew∗) = Ω/parenleftbigη√\nT/parenrightbig.\nAlgorithm’s dynamics. As in GD, we provide a key lemma that characterizes the trajec tory\nof SGD under a certain ”good event” . For this good event, give n a random training set sample\nS={Vi}n\ni=1, we denote Pt=/intersectiontextt−1\ni=1ViandSt=/intersectiontextt=n\ni=tVi. Moreover, if Pt/\\e}atio\\slash=∅, we denote rt=\narg min {r:Vt∈Pt}andJt=vrt∈U. The good event is given as follows,\nE′={∀t≤T Pt/\\e}atio\\slash=∅andJt∈St} (15)\nIn the following lemma we show that E′occurs with a constant probability. The proof appears in\nAppendix C.\nLemma 9. ForT=nand the event E′defined in Eq. (15), it holds that Pr(E′)≥1\n2.\nUnder this event, the dynamics of SGD is characterized as follows,\n17Lemma 10. Assume the conditions of Theorem 4, and consider the iterate s ofunprojected SGD ,\nwith step size η≤1√\nTinitialized at w1= 0. Under the event E′, we have for t≥4ands/\\e}atio\\slash= 0,\nwt(k)=\n\n−3\n8ηu1+ (t−1)η\nn3u1k= 1\n1\n8ηuk 2≤k≤t−2\n1\n2ηut−1 k=t−1\n0 t≤k≤n,\nand fors= 0,\nwt(0,k)=\n\nη\n4n2/summationtextt−1\ni=2φ(Vi,1)k= 1\nη\n4n2/summationtextt−1\ni=1φ(Vi,i)k=t−1\n0 k /∈ {1,t−1}.\nwhereu1∈Uand every another vector ukholdsuk∈/intersectiontextk−1\ni=1Vi∩/intersectiontextn\ni=kVi.\nFor proving this key lemma, we analyze how the terms ℓSGD\n1,ℓSGD\n2affects the dynamics of SGD\nunder the event E′. First, we show that the gradients of ℓSGD\n1does not affect the dynamics of SGD ,\nas the gradient of this term in any iterate wtis zero. The idea is formalized in the following lemma.\nThe proof is deferred to Appendix C.\nLemma 11. Assume the conditions of Theorem 4 and the event E′. Letw∈Rdandtbe such that\nfor every 2≤k≤t−1,w(k)=cηukforc≤1\n2and every such ukholdsuk∈/intersectiontextk−1\ni=1Vi∩/intersectiontextn\ni=kVi, and\nfor everyt≤k≤T,w(k)= 0. Then, for every t, it holds that, ℓSGD\n1is differentiable at (w,V)and\nfor alli∈[n], we have ∇ℓSGD\n1(w,Vt) = 0.\nNow, we analyze the gradient of ℓSGD\n2. The role of this component is to decode the next ”bad\nsolution”α/parenleftig\n1\nn/summationtextt−1\ni=1φ(Vi,i)/parenrightig\nfrom the sets V1,...,Vt−1, and make a progress in this direction in\nsome subspace W(t−1). In the following lemma, we show that the gradient of ℓSGD\n2, serves this goal.\nLemma 12. Assume the conditions of Theorem 4 and the event E′. For every k, letψ∗\nk=\n1\nn/summationtextk\nt=1φ(Vt,t). Moreover, let m≥3andw∈Rdsuch thatw(1)=cηu1for−3\n8≤c≤0and\nu1∈U, for every 2≤k≤m−1,w(k)=1\n8ηuksuch that every ukholdsuk∈/intersectiontextk−1\nt=1Vt∩/intersectiontextn\nt=kVt,\nw(m)=1\n2ηumwhereumholdsum∈/intersectiontextm−1\nt=1Vt∩/intersectiontextn\ni=mVtand for every m+ 1≤k≤T,w(k)= 0.\nMoreover, assume that wholdsw(0,m)=η\n4nψ∗\nm,/ba∇dblw(0,1)/ba∇dbl ≤ηand for every k /∈ {m,1},w(0,k)= 0.\nThen, for every V⊆U,ℓSGD\n2is differentiable at (w,V)and, we have for k/\\e}atio\\slash= 0,\n∇ℓSGD\n2(w,V)(k)=\n\n3\n8um k=m\n−1\n2α(ψ∗\nm)k=m+ 1\n0 k /∈ {m,m + 1}\nand,\n∇ℓSGD\n2(w,V)(0,k)=\n\n1\n4n2/summationtextm\nt=1φ(Vt,i) k=m\n−1\n4n2/summationtextm\nt=1φ(Vt,i)−1\n4n2φ(V,i)k=m+ 1\n0 k /∈ {m,m + 1}.\n18Now we can prove Lemma 10.\nProof of Lemma 10. We assume that E′holds and prove the lemma by induction on t. We begin\nfrom the basis of the induction, t= 4, which is proved in Lemma 26 in Appendix C. Now, we\nassume the hypothesis of the induction, that the lemma holds for iteration tand turn to show the\nrequired for iteration t+ 1.\nFirst, we notice that for every 2 ≤k≤t−1,wt(k)=cηukforc≤1\n2and every such ukholds\nuk∈/intersectiontextk−1\ni=1Vi∩/intersectiontextn\ni=kVi, and for every t≤k≤T,wt(k)= 0. Then, by Lemma 11, we have that\n∇ℓSGD\n1(wt,Vt) = 0.\nSecond,ℓSGD\n3is a linear function, thus,\n∇ℓSGD\n3(wt,Vt)(s)=\n\n−1\nn3u1s= 1\n−1\n4n2φ(Vt,1)s= 0,1\n0 otherwise .\nThird, For ℓSGD\n2(wt,Vt), we notice for m=t−1≥3 it holds that wt(1)=cηu1for−3\n8≤c≤0 and\nu1∈U, for every 2 ≤k≤m−1,wt(k)=1\n8ηuksuch that every ukholdsuk∈/intersectiontextk−1\nt=1Vt∩/intersectiontextn\nt=kVt,\nwt(m)=1\n2ηumwhereumholdsum∈/intersectiontextm−1\nt=1Vt∩/intersectiontextn\ni=mVt, and for every m+ 1≤k≤T,w(k)= 0.\nMoreover,wtholdsw(0,m)=η\n4nψ∗\nm,/ba∇dblwt(0,1)/ba∇dbl ≤ηand for every k /∈ {m,1},wt(0,k)= 0. Then, by\nLemma 12, we get that, we have for k/\\e}atio\\slash= 0,\n∇ℓSGD\n2(wt,Vt)(k)=\n\n3\n8ut−1k=t−1\n−1\n2α(ψ∗\nt−1)k=t\n0 k /∈ {t−1,t}\nand,\n∇ℓSGD\n2(wt,Vt)(0,k)=\n\n1\n4n2/summationtextt−1\ni=1φ(Vi,i)k=m\n−1\n4n2/summationtextt\ni=1φ(Vi,i)k=m+ 1\n0 k /∈ {m,m + 1}.\nNow, by Lemma 23, for j= arg mini{i:vi∈/intersectiontextt−1\ni=1Vi}, we get that\nα(ψ∗\nt−1) =vj∈t−1/intersectiondisplay\ni=1Vi.\nWe notice that/intersectiontextt−1\ni=1Vi=Ptand thusα(ψ∗\nt−1) =Jt. Then, by E′,α(ψ∗\nt−1) also holds α(ψ∗\nt−1)∈St.\nCombining the above together, we get, for ut=α(ψ∗\nt−1)∈Pt∩St,\n∇f(wt,Vt)(k)=\n\n−1\nn3u1k= 1\n3\n8ut−1k=t−1\n−1\n2utk=t\n0k /∈ {1,t−1,t},\nand,\n∇f(wt,Vt)(0,k)=\n\n−1\n4n2φ(V3,1) k= 1\n1\n4n2/summationtextt−1\ni=1φ(Vi,i)k=t−1\n−1\n4n2/summationtextt\ni=1φ(Vi,i)k=t\n0 k /∈ {1,t−1,t},\n19and the lemma follows. /square\nThe proof of Theorem 4 is similar to Theorem 3, using Lemma 10 i nstead of Lemma 4, and is\ndeferred to Appendix C.\nAcknowledgments\nThis project has received funding from the European Researc h Council (ERC) under the European\nUnion’s Horizon 2020 research and innovation program (gran t agreements No. 101078075; 882396).\nViews and opinions expressed are however those of the author (s) only and do not necessarily reflect\nthose of the European Union or the European Research Council . Neither the European Union nor\nthe granting authority can be held responsible for them. 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Cambridge University\nPress, 2014. ISBN 9781107057135.\nS. Shalev-Shwartz, O. Shamir, N. Srebro, and K. Sridharan. L earnability, stability and uniform\nconvergence. The Journal of Machine Learning Research , 11:2635–2670, 2010.\nV. Vapnik. On the uniform convergence of relative frequenci es of events to their probabilities.\nTheory of Probability and its Applications , 16(2):264–281, 1971.\nC. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals. Under standing deep learning requires\nrethinking generalization. In 5th International Conference on Learning Representations , ICLR\n2017, 2017.\nA Differentiability and Proofs of Theorems 1 and 2\nIn this section, we complete the proof of Theorems 1 and 2, by s howing a construction of a differen-\ntiable objective that maintains the same lower bounds given in Theorems 3 and 4 and Lemma 33.\nOur general approach is to use a randomized smoothing of the o riginal objectives. Then, we use\nthe fact that the subgradients are unique along any possible trajectory of GD, to show that when\nsmoothing is applied within a sufficiently small neighborhoo d, gradients along any such trajectory\nare preserved. Consequently, this approach does not impact the dynamics of the optimization\nalgorithm, while simultaneously ensuring the objectives b ecome differentiable everywhere.\n21A.1 Proof of Theorem 1\nFull construction. The hard distribution Dis defined to be as in Section 4. The hard loss\nfunction is a smoothing of f(Eq. (11)), and is defined as\n˜f(w,(V,j)):=Ev∈B[f(w+δv,(V,j))], (16)\nfor a sufficiently small δ>0 and thed-dimensional unit ball B. Analogously, we denote the empir-\nical loss and the population loss with respect to the loss fun ction ˜fas/hatwide˜F(w) =1\nn/summationtextn\ni=1˜f(w,(Vi,ji))\nand ˜F(w) =E(V,j)∼D˜f(w,(V,j)), respectively. The loss function ˜fis differentiable, 5-Lipschitz\nwith respect to its first argument and convex over Rd, as stated in the following lemma.\nLemma 13. For every (V,j)∈Z, the loss function ˜fis differentiable, convex and 5-Lipschitz with\nrespect to its first argument and over Rd.\nWe first prove the following theorem,\nTheorem 5. Assume that n>0,T >32002andη≤1√\nT. Consider the distribution Dand the loss\nfunction ˜fford= 178nT+ 2n2,ε=1\nn2(1−cos(2π\n|P(U)|)),β=ǫ\n4T2,δ=ηβ\n32,δ1=η\n2nandδ2=3ηβ\n16.\nThen, for Unprojected GD (cf. Eq. (1)withW=Rd) on/hatwideF, initialized at w1= 0with step size η,\nwe have, with probability at least1\n6over the choice of the training sample:\n(i) The iterates of GD remain within the unit ball, namely wt∈Bdfor allt= 1,...,T ;\n(ii) For all m= 1,...,T , them-suffix averaged iterate has:\n˜F(wT,m)−˜F(w∗) = Ω/parenleftbigη√\nT/parenrightbig.\nAlgorithm dynamics. Now we show that the dynamics of GD when is applied on/hatwide˜Fis identical\nto dynamics of the algorithm on/hatwideF, as stated in the following lemma.\nLemma 14. Under the conditions of Theorems 3 and 5, let wt,˜wtbe the iterates of Unprojected\nGD with step size η≤1√\nTandw1= 0, on/hatwideFand/hatwide˜Frespectively. Then, if Eoccurs, then for every\nt∈[T], it holds that wt= ˜wt.\nProof of Theorem 5. Next, we set out to establish the proof for Theorem 5.\nProof of Theorem 5. LetwT,mbe them-suffix average of GDwhen is applied on/hatwideF. Letw∗=\narg minwF(w). By Lemma 14, we know that, with probability of at least1\n6,Eoccurs and wT,m=\nwT,m. Then, by Theorem 3 and Lemma 32,\nη\n3200√\nT≤F(wT,m)−F(w∗)\n=F(wT,m)−F(w∗)\n≤˜F(wT,m) + 5δ−˜F(w∗) + 5δ\n≤˜F(wT,m) + 5δ−˜F(w∗) + 5δ,\nand,\n˜F(wT,m)−˜F(w∗)≥η\n3200√\nT−10ηǫ\n128T2\n22≥η\n3200√\nT−η\n10T2\n≥η\n6400√\nT. (T≥30)\n/square\nNow we can finally prove Theorem 1. The proof is an immediate co rollary from Theorem 5\nand the lower bound of Ω/parenleftig\nmin/parenleftig\n1\nηT,1/parenrightig/parenrightig\ngiven in Lemma 35 in Appendix E. It’s important to\nhighlight that we offer a rigorous proof for a modified version of Theorem 1, where the loss function\nfpossesses a Lipschitz constant of only 5. By scaling down thi s loss function by a factor of1\n5and\nsimultaneously adjusting the step size ηby a factor of 5, we can employ the same proof to establish\nthe validity of Theorem 1.\nProof of Theorem 1. We know that η≤1\n5√\nT. First, by Theorem 5, we know that for Unprojected\nGDandd1= 178nT+ 2n2, there exist a distribution Dover a probability space Z, a constant C1\nand a loss function ˜f:Rd1×Z→Rsuch that, with probability of at least1\n6,\n˜F(wT,m)−˜F(w∗)≥C1η√\nT.\nSecond, by Lemma 35, we know that for Unprojected GDandd2= max(25η2T2,1), there exist a\nconstantC2and a deterministic loss function ˜fOPT:Rd2→Rsuch that\n˜fOPT(wT,m)−˜fOPT(w∗)≥C2min/parenleftbigg\n1,1\nηT/parenrightbigg\nNow, letC=1\n2min (C1,C2). Ifη≥T−3\n4, then,η√\nT≥min(1,1\nηT), and we get,\n˜F(wT,m)−˜F(w∗)≥C/parenleftbigg\nη√\nT+ min/parenleftbigg\n1,1\nηT/parenrightbigg/parenrightbigg\n≥C/parenleftbigg\nmin/parenleftbigg\n1,η√\nT+1\nηT/parenrightbigg/parenrightbigg\n.\nOtherwise, we get that,\n˜fOPT(wT,m)−˜fOPT(w∗)≥C/parenleftbigg\nη√\nT+ min/parenleftbigg\n1,1\nηT/parenrightbigg/parenrightbigg\n≥C/parenleftbigg\nmin/parenleftbigg\n1,η√\nT+1\nηT/parenrightbigg/parenrightbigg\n.\nSince in both cases, by Lemma 35 and Theorem 5, wt∈Bdfor everyt∈[T], the theorem is\napplicable also for Projected GD. /square\nA.2 Proof of Theorem 2\nFull construction. The hard distribution DSGDis defined to be as in Section 5. The hard loss\nfunction is a smoothing of fSGD(Eq. (14)), and is defined as\n˜fSGD(w,V):=Ev∈B/bracketleftig\nfSGD(w+δv,V )/bracketrightig\n, (17)\nfor a sufficiently small δ > 0 and the d-dimensional unit ball B. Analogously, we denote the\nempirical loss and the population loss with respect to the lo ss function ˜fSGDas/hatwide˜FSGD\n(w) =\n1\nn/summationtextn\ni=1˜fSGD(w,Vi) and ˜FSGD(w) =EV∼D˜fSGD(w,V), respectively. The loss function ˜fSGDis\ndifferentiable, 4-Lipschitz with respect to its first argume nt and convex over Rd, as stated in the\nfollowing lemma.\n23Lemma 15. For everyV∈Z, the loss function ˜fSGDis differentiable, convex and 4-Lipschitz with\nrespect to its first argument and over Rd.\nWe first prove the following theorem,\nTheorem 6. Assume that n > 2048 andη≤1√n. Consider the distribution DSGDand the loss\nfunction ˜fSGDwithd= 712n2logn+ 2n2,ε=1\nn2(1−cos(2π\n|P(U)|)),δ=ηε\n32n3andδ1=η\n8n3. Then,\nfor Unprojected SGD (cf. Eq. (2)withW=Rd) withT=niterations, initialized at w1= 0with\nstep sizeη, we have, with probability at least1\n2over the choice of the training sample,\n(i) The iterates of SGD remain within the unit ball, namely wt∈Bdfor allt= 1,...,n ;\n(ii) For all m= 1,...,n , them-suffix averaged iterate has:\n/hatwide˜FSGD\n(wn,m)−/hatwide˜FSGD\n(/hatwidew∗) = Ω/parenleftbigη√n/parenrightbig.\nAlgorithm’s dynamics. Now, As in GD, the main step in proving Theorem 6 is to show that\ntaking expectation of fSGDfor every point win a ball with small enough radius does not change\nthe dynamics of SGD .\nLemma 16. Under the conditions of Theorems 4 and 6, let wt,˜wtbe the iterates of Unprojected\nSGD with step size η≤1√\nTandw1= 0, on/hatwideFSGDand/hatwide˜FSGD\nrespectively. Then, if E′occurs, then\nfor everyt∈[T], it holds that wt= ˜wt.\nProof of Theorem 6. Next, we set out to establish the proof for Theorem 6.\nProof of Theorem 6. Letwn,mbe them- suffix average of SGD when is applied on fSGDand let\n/hatwidew∗= arg minw/hatwideFSGD(w). By Lemma 16, we know that, with a probability1\n2,wn,m=wSGD\nn,m. Then,\nby Theorem 4 and Lemma 32,\nη\n64000√n≤/hatwideFSGD(wn,m)−/hatwideFSGD(/hatwidew∗)\n=/hatwideFSGD(wn,m)−/hatwideFSGD(/hatwidew∗)\n≤/hatwide˜FSGD\n(wn,m) + 4δ−/hatwide˜FSGD\n(/hatwidew∗) + 4δ\n≤/hatwide˜FSGD\n(wn,m) + 4δ−/hatwide˜FSGD\n(/hatwidew∗) + 4δ,\nand,\n/hatwide˜FSGD\n(wn,m)−/hatwide˜FSGD\n(/hatwidew∗)≥η\n64000√n−ηǫ\n4n3\n≥η\n64000√n−η\n4n3\n≥η\n128000√n. (n≥40)\n/square\nNow we can finally prove Theorem 2.\n24Proof proof of Theorem 2. We know that T=nandη≤1\n5√\nT. First, by Theorem 6, we know that\nfor Unprojected SGD andd1= 712n2logn+2n2, there exist a distribution DSGDover a probability\nspaceZ, a constant C1and a loss function ˜fSGD:Rd1×Z→Rsuch that, with probability of at\nleast1\n2,\n/hatwide˜FSGD\n(wT,m)−/hatwide˜FSGD\n(/hatwidew∗)≥C1η√\nT.\nSecond, by Lemma 35, we know that for Unprojected SGD andd2= max(25η2T2,1), there exist\na constantC2and a deterministic loss function ˜fOPT:Rd2→Rsuch that\n˜fOPT(wT,m)−˜fOPT(/hatwidew∗)≥C2min/parenleftbigg\n1,1\nηT/parenrightbigg\nNow, letC=1\n2min (C1,C2). Ifη≥T−3\n4, then,η√\nT≥min(1,1\nηT), and we get,\n/hatwide˜FSGD\n(wT,m)−/hatwide˜FSGD\n(/hatwidew∗)≥C/parenleftbigg\nη√\nT+ min/parenleftbigg\n1,1\nηT/parenrightbigg/parenrightbigg\n≥C/parenleftbigg\nmin/parenleftbigg\n1,η√\nT+1\nηT/parenrightbigg/parenrightbigg\n.\nOtherwise, we get that,\n˜fOPT(wT,m)−˜fOPT(/hatwidew∗)≥C/parenleftbigg\nη√\nT+ min/parenleftbigg\n1,1\nηT/parenrightbigg/parenrightbigg\n≥C/parenleftbigg\nmin/parenleftbigg\n1,η√\nT+1\nηT/parenrightbigg/parenrightbigg\n.\nSince in both cases, by Lemma 35 and Theorem 6, wt∈Bdfor everyt∈[T], the theorem is\napplicable also for Projected SGD. /square\nB Proofs of Section 4\nB.1 Proofs for the full construction\nProof of Lemma 1. Letr= 2−d′\n178. For every 1 ≤i≤rand 1 ≤j≤d′we define the random variable\nuj\nibe a random variable to be1√\nd′with probability1\n2and−1√\nd���with probability1\n2. Then, for every\n1≤i≤r, we define the vector uiwhich itsjth entry is uj\niand look at the set U={u1,u2,...ur}.\nThis set will hold the required property with positive proba bility. First, for every i/\\e}atio\\slash=k,/a\\}b∇acketle{tui,uk/a\\}b∇acket∇i}htare\nsums ofdrandom variables that taking values in [ −1\nd′,1\nd′] withE/a\\}b∇acketle{tui,uk/a\\}b∇acket∇i}ht= 0. Then by Hoeffding’s\ninequality,\nPr(|/a\\}b∇acketle{tui,uk/a\\}b∇acket∇i}ht| ≥1\n8)≤2e−2(1\n8)2\nd′·4\nd′2= 2e−d′\n128\nThen, by union bound on the/parenleftbigr\n2/parenrightbigpairs of vectors in U,\nPr(∃i,k|/a\\}b∇acketle{tui,uk/a\\}b∇acket∇i}ht| ≥1\n8)≤2e−d′\n128·/parenleftigg\nr\n2/parenrightigg\n<2e−d′\n128·1\n2r2≤1.\n/square\nLemma 17. Letn,d≥1and a setU⊆Bd. LetP(U)be the power set of U. Then, there exist a\nsetΨ⊆R2n2, a number 0<ǫ<1\nnand two mappings φ:P(U)×[n2]→R2n2,α:R2n2→Usuch\nthat,\n(i) For every j∈[n2]andV⊆U, it holds /ba∇dblφ(V,j)/ba∇dbl ≤1;\n25(ii) For every ψ∈Ψ, it holds /ba∇dblψ/ba∇dbl ≤1;\n(iii) LetV1,...,Vnbe arbitrary subsets of U. Ifj1,...,jnhold thatji/\\e}atio\\slash=jkfori/\\e}atio\\slash=k,ψ∗=\n1\nn/summationtextn\ni=1φ(Vi,ji)is that,\n•/angbracketleftig\nψ∗,1\nnn/summationdisplay\ni=1φ(Vi,ji)/angbracketrightig\n>7\n8n;\n•For everyψ∈Ψ,ψ/\\e}atio\\slash=ψ∗:\n/angbracketleftig\nψ∗,1\nnn/summationdisplay\ni=1φ(Vi,ji)/angbracketrightig\n≥/angbracketleftig\nψ,1\nnn/summationdisplay\ni=1φ(Vi,ji)/angbracketrightig\n+ǫ;\n•If/uniontextn\ni=1Vi/\\e}atio\\slash=U, then it holds that α(ψ∗) =vi∗∈U\\/uniontextn\ni=1Vifori∗= min {i:vi∈U\\/uniontextn\ni=1Vi}.\nProof. First, we consider an arbitrary enumeration of P(U) = {V1,...V|P(U)|}and define g:\nP(U)→R2,g(Vi) =/parenleftig\nsin/parenleftig\n2πi\n|P(U)|/parenrightig\n,cos/parenleftig\n2πi\n|P(U)|/parenrightig/parenrightig\n. Now, we refer to a vector a∈R2n2as a\nconcatenation of n2vectors in R2,a(1),...,a(n2). Then, we define δ= 1−cos/parenleftig\n2π\n|P(U)|/parenrightig\n,ǫ=δ\nn2\nand\nφ(V,j)(i)=/braceleftigg\ng(V)i=j\n0 otherwise\nAs a result, for every Vi,jit holds that\n/ba∇dblφ(Vi,j)/ba∇dbl=/ba∇dblg(Vi)/ba∇dbl=/radicaligg\nsin/parenleftbigg2πi\n|P(U)|/parenrightbigg2\n+ cos/parenleftbigg2πi\n|P(U)|/parenrightbigg2\n= 1\nMoreover, if j1/\\e}atio\\slash=j2,\n/a\\}b∇acketle{tφ(Vi,j1),φ(Vi,j2)/a\\}b∇acket∇i}ht= 0,\nand ifi>k ,\n/a\\}b∇acketle{tφ(Vi,j),φ(Vk,j)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tg(Vi),g(Vk)/a\\}b∇acket∇i}ht\n= sin/parenleftbigg2πi\n|P(U)|/parenrightbigg\nsin/parenleftbigg2πk\n|P(U)|/parenrightbigg\n+ cos/parenleftbigg2πi\n|P(U)|/parenrightbigg\ncos/parenleftbigg2πk\n|P(U)|/parenrightbigg\n= cos/parenleftbigg2π(i−k)\n|P(U)|/parenrightbigg\n≤cos/parenleftbigg2π\n|P(U)|/parenrightbigg\n(cos is monotonic decreasing in [0 ,π/2])\n= 1−δ\nWe notice that 0 < δ < 1. Now, we consider an arbitrary enumeration of U={v1,...v |U|}, and\ndefine the following set Ψ ⊆R2n2and the following two mappings σ:R2n2→P(U),α:R2n2→U,\nΨ = {1\nnn/summationdisplay\ni=1φ(Vi,ji) :∀i Vi⊆U, ji∈[n2] andi/\\e}atio\\slash=l=⇒ji/\\e}atio\\slash=jℓ}\n26Note that, for every ψ∈Ψ,\n/ba∇dblψ/ba∇dbl=/ba∇dbl1\nnn/summationdisplay\ni=1φ(Vi,ji)/ba∇dbl ≤1\nnn/summationdisplay\ni=1/ba∇dblφ(Vi,ji)/ba∇dbl= 1.\nThen, for every a∈R2n2andj∈[n2], we denote the index q(a,j)∈[|P(U)|] as\nq(a,j) = arg max\nr/a\\}b∇acketle{tg(Vr),a(j)/a\\}b∇acket∇i}ht,\nand define the following mapping σ:R2n2→P(U),\nσ(a) =n2/uniondisplay\nj=1,a(j)/negationslash=0Vq(a,j).\nMoreover, for every a∈R2n2, we denote the index p(a)∈[|U|] as\np(a) = arg min\ni{i:vi∈U\\σ(a)},\nand define the following mapping α:R2n2→U,\nα(a) =/braceleftigg\nv|U|σ(a) =U\nvp(a)σ(a)/\\e}atio\\slash=U.\nNow, LetV1,...,Vn⊆Uandj1,...jnthat are sampled uniformly from [ n2], We prove the last\npart of the lemma under the condition that ji/\\e}atio\\slash=jkfori/\\e}atio\\slash=k.ψ∗=1\nn/summationtextn\ni=1φ(Vi,ji) holds\n/a\\}b∇acketle{tψ∗,1\nnn/summationdisplay\ni=1φ(Vi,ji)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t1\nnn/summationdisplay\ni=1φ(Vi,ji),1\nnn/summationdisplay\ni=1φ(Vi,ji)/a\\}b∇acket∇i}ht\n=1\nn2n/summationdisplay\ni=1/a\\}b∇acketle{tφ(Vi,ji),φ(Vi,ji)/a\\}b∇acket∇i}ht\n=1\nn\n>7\n8n\nForψ=1\nn/summationtextn\nl=1φ(V′\nl,j′\nl) such thatψ/\\e}atio\\slash=ψ∗, there are at most npairsi,lsuch that /a\\}b∇acketle{tφ(V′\ni,j′\ni),φ(V′\nl,j′\nl)/a\\}b∇acket∇i}ht /\\e}atio\\slash=\n0. thus, there exists a pair ( V′\nr,j′\nr) that (V′\nr,j′\nr)/∈ {(Vi,ji) :i∈[n]}. and for every i,/a\\}b∇acketle{tφ(Vi,ji),φ(V′\nl,j′\nl)/a\\}b∇acket∇i}ht ≤\n1−δ. As a result,\n/a\\}b∇acketle{tψ,1\nnn/summationdisplay\nl=1φ(Vi,ji)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t1\nnn/summationdisplay\ni=1φ(V′\nl,j′\nl),1\nnn/summationdisplay\ni=1φ(Vi,ji)/a\\}b∇acket∇i}ht\n=1\nn2n/summationdisplay\ni=1n/summationdisplay\nl=1/a\\}b∇acketle{tφ(Vi,ji),φ(V′\nl,j′\nl)/a\\}b∇acket∇i}ht\n≤1\nn2\n1−δ+n/summationdisplay\ni=1,i/negationslash=r1\n\n≤1\nn2(1−δ+n−1)\n27=1\nn−δ\nn2\n=/a\\}b∇acketle{tψ∗,1\nnn/summationdisplay\ni=1φ(Vi,ji)/a\\}b∇acket∇i}ht −ǫ\nFurthermore, since if all jiare distinct, for every iit holds that,1\nn/summationtextn\ni=1φ= (Vi,i)(ji)=1\nng(Vi),\nthus,\nq/parenleftigg\n1\nnn/summationdisplay\ni=1φ(Vi,ji),ji/parenrightigg\n= arg max\nr/a\\}b∇acketle{tg(Vr),1\nnn/summationdisplay\ni=1φ(Vi,ji)(ji)\n/a\\}b∇acket∇i}ht\n= arg max\nr/a\\}b∇acketle{tg(Vr),1\nng(Vi)/a\\}b∇acket∇i}ht\n=i,\nand we get,\nσ(ψ∗) =σ/parenleftigg\n1\nnn/summationdisplay\ni=1φ(Vi,ji)/parenrightigg\n=n2/uniondisplay\nj=1,1\nn/summationtextn\ni=1φ(Vl,ji)(j)/negationslash=0Vq(1\nn/summationtextn\ni=1φ(Vi,ji),j)\n=n/uniondisplay\ni=1Vq(1\nn/summationtextn\ni=1φ(Vi,ji),ji)(The indices that are non-zero are {ji}n\ni=1})\n=n/uniondisplay\ni=1Vi\nFinally, assuming that/uniontextn\ni=1Vi/\\e}atio\\slash=U,\nα(ψ∗) =vp(a)∈U\\n/uniondisplay\ni=1Vi.\n/square\nProof of Lemma 2. We prove that ℓ1,ℓ2andℓ4are convex and 1-Lipschitz and ℓ3is convex and\n12-Lipschitz.\nFirst, by Lemmas 1 and 17 for every u∈UandV∈P(U),j∈[n2], it holds that /ba∇dblu/ba∇dbl= 1,\n/ba∇dblφ(V,j)/ba∇dbl= 1. Then, ℓ2is a 1-Lipschitz linear function, and ℓ4is a maximum over 1-Lipschitz linear\nfunctions, thus, both functions are convex and 1-Lipschitz . Moreover, for every possible ψ∈Ψ\n/ba∇dblψ/ba∇dbl=/ba∇dbl1\nnn/summationdisplay\nl=1φ(Vl)/ba∇dbl ≤1\nnn/summationdisplay\nl=1/ba∇dblφ(Vl)/ba∇dbl= 1.\nthus,ℓ3is a maximum over 2-Lipschitz linear functions, thus, it is c onvex and 2-Lipschitz. Now,\nforℓ1, for every set V⊆U, letαV(w)∈RT−1to be the vector which its k’th coordinate is\nαV(w)(k)= max/parenleftig3η\n32,maxu∈V/a\\}b∇acketle{tuw(k+1)/a\\}b∇acket∇i}ht/parenrightig\nand prove convexity and 1-Lipshitzness. For establishing\nconvexity, observe\n/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,(λx+ (1 −λ)y)(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n28=/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/parenleftbigλ/a\\}b∇acketle{tu,x(k)/a\\}b∇acket∇i}ht+ (1 −λ)/a\\}b∇acketle{tu,y(k)/a\\}b∇acket∇i}ht/parenrightbig/parenrightbigg2\n≤/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/parenleftbigλ/a\\}b∇acketle{tu,x(k)/a\\}b∇acket∇i}ht/parenrightbig+ max\nu∈V/parenleftbig(1−λ)/a\\}b∇acketle{tu,y(k)/a\\}b∇acket∇i}ht/parenrightbig/parenrightbigg2\n(convexity of max & monotonicity of square root)\n≤/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2/parenleftbigg\nλmax/parenleftbigg3η\n32,max\nu∈V/parenleftbig/a\\}b∇acketle{tu,x(k)/a\\}b∇acket∇i}ht/parenrightbig/parenrightbigg\n+ (1 −λ) max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,y(k)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n=/ba∇dblλαV(x) + (1 −λ)αV(y)/ba∇dbl2\n≤λ/ba∇dblαV(x)/ba∇dbl2+ (1 −λ)αV(y)/ba∇dbl2 (convexity of ℓ2norm)\n=λ/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tux(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n+ (1 −λ)/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tuy(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n.\nFor 1-Lipschitzness, for every w∈Rdand sub-gradient g(w,V)∈∂ℓ1(w,V), there exists a sub\ngradienth(w,V)∈∂/parenleftbigg/summationtextT\nk=2max/parenleftig3η\n32,maxu∈V/a\\}b∇acketle{tuw(k)/a\\}b∇acket∇i}ht/parenrightig2/parenrightbigg\nsuch that\n/ba∇dblg(w,V)/ba∇dbl=/ba∇dblh(w,V)/ba∇dbl\n2/radicalbigg\n/summationtextT\nk=2max/parenleftig\n3η\n32,maxu∈V/a\\}b∇acketle{tuw(k)/a\\}b∇acket∇i}ht/parenrightig2=/ba∇dblh(w,V)/ba∇dbl\n2/radicalig/summationtextT\nk=2αV(w)(k)2.\nMoreover, for every kand sub gradient bk,V(w)∈∂/parenleftig\nαV(w)(k)/parenrightig\nwe denote rk,V(w)∈Rdthe\nvector with rk,V(w)(k)=bk,V(w) and forj/\\e}atio\\slash=k,rk,V(w)(j)= 0. Then, for every sub gradient\nh(w,V)∈∂/parenleftbigg/summationtextT\nk=2max/parenleftig3η\n32,maxu∈V/a\\}b∇acketle{tuw(k)/a\\}b∇acket∇i}ht/parenrightig2/parenrightbigg\n, there exists T−1 such vectors rk,V(w)∈Rd\n(2≤k≤T) such that,\nh(w,V) = 2T/summationdisplay\nk=2rk,V(w)αV(w)(k)\nAs a result, by the fact that every sub gradient of bk,V(w)∈∂/parenleftig\nαV(w)(k)/parenrightig\nis either 0 or λ1u1+\nλ2u2+...+λpupfor/summationtext\niλi≤1, such that for all every j,k,uj∈UandαV(w)(k)=/a\\}b∇acketle{tuj,w()k/a\\}b∇acket∇i}ht,\ncombining by the facts that for distinct k,k′,rk,V,rk′,Vare orthogonal, it holds, for u2\nj,...uT\nj∈U\nsuch that for every k,αV(w)(k)=/a\\}b∇acketle{tuk\nj,w(k)/a\\}b∇acket∇i}ht,\n/ba∇dblh(w,V)/ba∇dbl=/ba∇dbl2T/summationdisplay\nk=2rk,V(w)αV(w)(k)/a\\}b∇acket∇i}ht/ba∇dbl\n=/ba∇dbl2T/summationdisplay\nk=2,bk,Vrk,V(w)αV(w)(k)/a\\}b∇acket∇i}ht/ba∇dbl\n= 2/ba∇dblT/summationdisplay\nk=2rk,V/a\\}b∇acketle{tuk\nj,w(k)/a\\}b∇acket∇i}ht/ba∇dbl.\n29Now, we denote ck\nj(w)∈Rdthe vector with ck\nj(w)(k)=uk\njand forj/\\e}atio\\slash=k,ck\nj(w)(j)= 0, and,\n/ba∇dblh(w,V)/ba∇dbl= 2/ba∇dblT/summationdisplay\nk=2rk,V/a\\}b∇acketle{tck\nj,w/a\\}b∇acket∇i}ht/ba∇dbl\n≤2/radicaltp/radicalvertex/radicalvertex/radicalbt/a\\}b∇acketle{tT/summationdisplay\nk=2rk,V/a\\}b∇acketle{tck\nj,w/a\\}b∇acket∇i}ht,T/summationdisplay\nl=2rl,V/a\\}b∇acketle{tcl\nj,w/a\\}b∇acket∇i}ht/a\\}b∇acket∇i}ht\n= 2/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2/ba∇dblrk,V/ba∇dbl2/a\\}b∇acketle{tck\nj,w/a\\}b∇acket∇i}ht2\n≤2/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2/a\\}b∇acketle{tuk\nj,w(k)/a\\}b∇acket∇i}ht2\n= 2/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2αV(w)(k)2.\nThe lemma follows. /square\nB.2 Proof of algorithm’s dynamics\nIn this section we describe the dynamics of GD when applied on/hatwideFfor training set Sthat is\nsampled from a distribution D. We begin with showing that the good event E(Eq. (12)) occurs\nwith a constant probability.\nProof of Lemma 3. By the fact that every Viandjiare independent, it is enough to show that\nPr/parenleftiggn/uniondisplay\ni=1Vi/\\e}atio\\slash=Ud′/parenrightigg\n≥1\n2,\nand,\nPr (for every k/\\e}atio\\slash=l,jk/\\e}atio\\slash=jl)≥1\n3.\nFor the former, for every u∈Ud′, sinceViare sampled independently and every vector u∈Ud′is\nin everyViwith probability1\n2,\nPr/parenleftigg\nu∈n/uniondisplay\ni=1Vi/parenrightigg\n= 1−Pr/parenleftigg\nu /∈n/uniondisplay\ni=1Vi/parenrightigg\n= 1−2−n,\nthus, since by Lemma 1, |Ud′| ≥d′\n178=n, it holds that,\nPr/parenleftiggn/uniondisplay\ni=1Vi=Ud′/parenrightigg\n= Pr/parenleftigg\n∀u∈Ud′u∈n/uniondisplay\ni=1Vi/parenrightigg\n=/parenleftbig1−2−n/parenrightbig|Ud′|\n≤/parenleftbig1−2−n/parenrightbig2d′\n178\n=/parenleftbig1−2−n/parenrightbig2n\n≤1\ne\n30<1\n2.\nWe conclude,\nPr/parenleftiggn/uniondisplay\ni=1Vi/\\e}atio\\slash=Ud′/parenrightigg\n≥1\n2.\nFor the latter, since all jis are sampled independently, for a single pair k/\\e}atio\\slash=l, it holds that\nPr(jk/\\e}atio\\slash=jl) = 1 −1\nn2\nAs a result,\nPr (for every k/\\e}atio\\slash=l,jk/\\e}atio\\slash=jl) =/parenleftbigg\n1−1\nn2/parenrightbiggn(n−1)\n2≥/parenleftbigg\n1−1\nn2/parenrightbiggn2\n2≥1√\n2e≥1\ne. /square\nFrom now on, we analyze the dynamics of the GD conditioned on E(Eq. (12)). We begin with\nseveral lemmas.\nProof of Lemma 5. For the first part, we know that, for every 2 ≤k≤T,w(k)=cηu0forc≤1\n2. In\naddition, by the facts that u0∈U\\/uniontextn\ni=1Viand that for every u/\\e}atio\\slash=v∈U, it holds that /a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht ≤1\n8,\nwe get for every i, maxu∈Vi/a\\}b∇acketle{tu0,u/a\\}b∇acket∇i}ht ≤1\n8, thus, for every iandk≥2,\nmax\nu∈Viuw(k)= max\nu∈Vi/a\\}b∇acketle{tu,cηu 0/a\\}b∇acket∇i}ht ≤1\n8·cη≤η\n16.\nFor the second part, for every sub-gradient g(w,Vi)∈∂ℓ1(w,Vi), there exists a sub gradient\nh(w,Vi)∈∂/parenleftbigg/summationtextT\nk=2max/parenleftig\n3η\n32,maxu∈V/a\\}b∇acketle{tuw(k)/a\\}b∇acket∇i}ht/parenrightig2/parenrightbigg\nsuch that\ng(w,Vi) =h(w,Vi)\n2/radicalbigg\n/summationtextT\nk=2max/parenleftig\n3η\n32,maxu∈Vi/a\\}b∇acketle{tuw(k)/a\\}b∇acket∇i}ht/parenrightig2.\nThen, since for every k, it holds that max u∈U/a\\}b∇acketle{tw(k),u0/a\\}b∇acket∇i}ht ≤η\n16, every such sub-gradient h(w,Vi) is\nzero, ∇ℓ1(w,Vi) = 0. /square\nProof of Lemma 6. First, for the first part, by Lemma 17, the fact that for every ψ,/ba∇dblα(ψ)/ba∇dbl ≤1,\nand by /ba∇dblw(1)/ba∇dbl ≤η, for everyψ∈Ψ,ψ∗=1\nn/summationtextn\ni=1φ(Vi,ji) holds,\n/a\\}b∇acketle{tw(0),ψ∗/a\\}b∇acket∇i}ht −1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ∗),w(1)/a\\}b∇acket∇i}ht ≥ /a\\}b∇acketle{tη\nnn/summationdisplay\ni=1φ(Vi,ji),ψ∗/a\\}b∇acket∇i}ht −ηǫ\n4\n≥η/a\\}b∇acketle{t1\nnn/summationdisplay\ni=1φ(Vi,ji),ψ∗/a\\}b∇acket∇i}ht −ηǫ\n4\n≥η/a\\}b∇acketle{t1\nnn/summationdisplay\ni=1φ(Vi,ji),ψ/a\\}b∇acket∇i}ht+ηǫ−ηǫ\n4(Lemma 17)\n=η/a\\}b∇acketle{t1\nnn/summationdisplay\ni=1φ(Vi,ji),ψ/a\\}b∇acket∇i}ht+ηǫ\n2+ηǫ\n4\n>/a\\}b∇acketle{tw(0),ψ/a\\}b∇acket∇i}ht −1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ),w(1)/a\\}b∇acket∇i}ht+ηǫ\n4,\n31thus,\narg max\nψ∈Ψ/parenleftbigg\n/a\\}b∇acketle{tw(0),ψ/a\\}b∇acket∇i}ht −1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ),w(1)/a\\}b∇acket∇i}ht/parenrightbigg\n=ψ∗=1\nnn/summationdisplay\ni=1φ(Vi,ji).\nFor the second part, by the fact that ǫ<1\nnand Lemma 17,\n/a\\}b∇acketle{tw(0),ψ∗/a\\}b∇acket∇i}ht −1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ∗),w(1)/a\\}b∇acket∇i}ht ≥7η\n8n−η\n4n>η\n2n+η\n16n=δ1+η\n16n.\nNow, by E, foru0=α(ψ∗), which is the uwith the minimal index in U\\/uniontextn\ni=1Vi,\nα(ψ∗) =u0∈U\\n/uniondisplay\ni=1Vi.\nAs a result, by the fact that the maximum is attained uniquely atψ∗, we derive that,\n∇ℓ3(w)(k)=\n\n1\nn/summationtextn\ni=1φ(Vi,ji)k= 0\n−1\n4ǫ\nT2u0 k= 1\n0 otherwise .\n/square\nProof of Lemma 7. We show that the maximum is attained uniquely at k=mandu=u0. For\nk= 1 and every u∈U,\n3\n8/a\\}b∇acketle{tu,wt(1)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,wt(2)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu,cηu 0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,η\n8u0/a\\}b∇acket∇i}ht ≤9η\n512+η\n128=13η\n512.\nMoreover, for every 2 ≤k≤m−2 and every u∈U,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w(k+1)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu,η\n8u0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,η\n8u0/a\\}b∇acket∇i}ht ≤3η\n64+η\n128=7η\n128.\nFork=m−1 and every u∈U,\n3\n8/a\\}b∇acketle{tu,w(m−1)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w(m)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu,η\n8u0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,η\n2u0/a\\}b∇acket∇i}ht ≤3η\n64+η\n32=5η\n64.\nFork=mandu=u0,\n3\n8/a\\}b∇acketle{tu,w(m)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w(m+1)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu0,η\n2u0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu0,0/a\\}b∇acket∇i}ht=3η\n16.\nFork=mandu/\\e}atio\\slash=u0,\n3\n8/a\\}b∇acketle{tu,w(m)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu′,w(m+1)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu,η\n2u0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu′,0/a\\}b∇acket∇i}ht ≤3η\n128.\nFor everym+ 1≤k<T −1 and every u∈U,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu′,w(k+1)/a\\}b∇acket∇i}ht= 0.\nMoreover, since T≥4,η < 1,ǫ< 1,δ1≤3η\n1024, and\n3\n8/a\\}b∇acketle{tu,w(m)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w(m+1)/a\\}b∇acket∇i}ht=3η\n16>δ2+η\n64.\n32We derive that,\n∇ℓ4(w)(k)=\n\n3\n8u0k=m\n−1\n2u0k=m+ 1\n0 otherwise .\n/square\nLemma 18. Under the conditions of Theorem 3, if Eoccurs and wtis the iterate of Unprojected\nGD on/hatwideF, with step size η≤1√\nTandw1= 0, then, fort= 2it holds that,\nw2(k)=/braceleftiggη\nn/summationtextn\ni=1φ(Vi,ji)k= 0\n0 otherwise.\nProof. Fort= 1,w1= 0. By Lemma 5 we know that for every i,∇ℓ1(w1,Vi) = 0. Moreover, by\nthe fact that δ1,δ2>0 the maximum in ℓ3andℓ4is attained in δ1andδ2, respectively, thus we\nget that\n∇ℓ3(w1) =∇ℓ4(w) = 0\nAs a result,\n∇/hatwideF(w1)(k)=1\nnn/summationdisplay\ni=1∇ℓ2(w1,(Vi,ji))(k)=/braceleftigg\n−1\nn/summationtextn\ni=1(Vi,ji)k= 0\n0 otherwise ,\nand,\nw2(k)=/braceleftiggη\nn/summationtextn\ni=1φ(Vi,ji)k= 0\n0 otherwise .\n/square\nLemma 19. Under the conditions of Theorem 3, if Eoccurs and wtis the iterate Unprojected GD\non/hatwideF, with step size η≤1√\nTandw1= 0, then, fort= 3it holds that,\nw3(k)=\n\nη\n4ǫ\nT2u0 k= 1\n0 2 ≤k≤T\nη\nn/summationtextn\ni=1φ(Vi,ji)k= 0,\nwhereu0∈U\\/uniontextn\ni=1Vi.\nProof. By Lemma 18, w2(1),...w 2(T)= 0, thus, by Lemma 5, we know that for every i,∇ℓ1(w1,Vi) =\n0. Moreover, by the fact that δ2>0, we get that ∇ℓ4(w2) = 0. Forℓ3(w2), by Lemma 6, using the\nfact thatw2(1)= 0 andw2(0)=η\nn/summationtextn\ni=1φ(Vi,ji), we get that\n∇ℓ3(w2)(k)=\n\n1\nn/summationtextn\ni=1φ(Vi,ji)k= 0\n−1\n4ǫ\nT2u0 k= 1\n0 otherwise .\n33Forℓ2(w2), for every i, the gradient is\n∇ℓ2(w2,(Vi,ji))(k)=/braceleftigg\n−φ(Vi,ji)k= 0\n0 otherwise .\nCombining all together, we conclude that, for u0∈U\\/uniontextn\ni=1Vi, it holds that,\n∇/hatwideF(w2)(k)=\n\n−1\n4ǫ\nT2u0k= 1\n0 2 ≤k≤T\n0 k= 0,\nand\nw3(k)=\n\nη\n4ǫ\nT2u0 k= 1\n0 2 ≤k≤T\nη\nn/summationtextn\ni=1φ(Vi,ji)k= 0.\n/square\nLemma 20. Under the conditions of Theorem 3, if Eoccurs and wtis the iterate Unprojected GD\non/hatwideF, with step size η≤1√\nTandw1= 0, then, fort= 4it holds that,\nw4(k)=\n\n−3η\n8u0+η\n2ǫ\nT2u0k= 1\nη\n2u0 k= 2\n0 3 ≤k≤T\nη\nn/summationtextn\ni=1φ(Vi,ji)k= 0,\nwhereu0∈U\\/uniontextn\ni=1Vi.\nProof. We start with ℓ1,ℓ2,ℓ3. Forℓ1, by Lemma 19, for every 2 ≤k≤T,w3(k)= 0, thus, by\nLemma 5, we know that for every i,∇ℓ1(w1,Vi) = 0. Forℓ2, we know that, for every i,\n∇ℓ2(w3,(Vi,ji))(k)=/braceleftigg\n−φ(Vi,ji)k= 0\n0 otherwise .\nForℓ3, by Lemma 6, using the fact that w3(1)=cηu0for|c| ≤1 andw3(0)=η\nn/summationtextn\ni=1φ(Vi,ji), we\nget that,\n∇ℓ3(w3)(k)=\n\n1\nn/summationtextn\ni=1φ(Vi,ji)k= 0\n−1\n4ǫ\nT2u0 k= 1\n0 otherwise .\nNow, forℓ4, we show that the maximum is attained uniquely in k= 1 andu=u0: Fork/\\e}atio\\slash= 1,\nfor everyu∈U\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w 3(k+1)/a\\}b∇acket∇i}ht= 0.\nFork= 1 andu/\\e}atio\\slash=u0,\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w 3(k+1)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu,w 3(1)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w 3(2)/a\\}b∇acket∇i}ht\n34=3\n8/a\\}b∇acketle{tu,η\n4ǫ\nT2u0/a\\}b∇acket∇i}ht\n≤3η\n256ǫ\nT2\nFork= 1 andu=u0,\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w 3(k+1)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu0,w3(1)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu0,w3(2)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu0,η\n4ǫ\nT2u0/a\\}b∇acket∇i}ht\n=3η\n32ǫ\nT2\n>δ2\nWe derive that,\n∇ℓ4(w3)(k)=\n\n3\n8u0k= 1\n−1\n2u0k= 2\n0 3 ≤k≤T\n0k= 0.\nCombining all together, we get that,\n∇/hatwideF(w3)(k)=\n\n3\n8u0−1\n4ǫ\nT2u0k= 1\n−1\n2u0 k= 2\n0 3 ≤k≤T\n0 k= 0,\nand\nw4(k)=\n\n−3η\n8u0+η\n2ǫ\nT2u0k= 1\nη\n2u0 k= 2\n0 3 ≤s≤T\nη\nn/summationtextn\ni=1φ(Vi,ji)k= 0,\nwhereu0∈U\\/uniontextn\ni=1Vi. /square\nLemma 21. Under the conditions of Theorem 3, if Eoccurs and wtis the iterate Unprojected GD\non/hatwideF, with step size η≤1√\nTandw1= 0, then, fort= 5it holds that,\nw5(k)=\n\n−3\n8ηu0+3η\n4ǫ\nT2u0k= 1\n1\n8ηu0 k= 2\n1\n2ηu0 k= 3\n0 4 ≤s≤T\n1\nn/summationtextn\ni=1φ(Vi,ji)k= 0,\nwhereu0∈U\\/uniontextn\ni=1Vi.\n35Proof. We begin with ℓ1,ℓ2,ℓ3. Note that, by Lemma 20, for every 2 ≤k≤T,w4(k)=cηu0for\nc≤1\n2, thus, by Lemma 5, for every i,∇ℓ1(w4,Vi) = 0. Forℓ2, we know that, for every i,\n∇ℓ2(w4,(Vi,ji))(k)=/braceleftigg\n−φ(Vi,ji)k= 0\n0 otherwise .\nForℓ3, by Lemma 6, using Lemma 20, where we showed that w4(1)=cηu0for|c| ≤ 1 and\nw4(0)=η\nn/summationtextn\ni=1φ(Vi,ji), we get that,\n∇ℓ3(w4)(k)=\n\n1\nn/summationtextn\ni=1φ(Vi,ji)k= 0\n−1\n4ǫ\nT2u0 k= 1\n0 otherwise .\nIt is left to calculate ∇ℓ4(w4). We show that the maximum is attained uniquely at k= 2 and\nu=u0. First,\n3\n8/a\\}b∇acketle{tu,η\n2ǫ\nT2u0/a\\}b∇acket∇i}ht=3η\n16ǫ\nT2/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht ≤3η\n16T2,\nthus, since T≥4,\n3\n8/a\\}b∇acketle{tu,w 4(1)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w 4(2)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu,−3η\n8u0+η\n2ǫ\nT2u0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,η\n2u0/a\\}b∇acket∇i}ht ≤9η\n512+η\n32+9η\n256=43η\n512<3η\n16.\nFork= 2 andu=u0,\n3\n8/a\\}b∇acketle{tu,w 4(2)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,w 4(3)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu0,η\n2u0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu0,0/a\\}b∇acket∇i}ht=3η\n16(>δ2).\nFork= 2 andu/\\e}atio\\slash=ut−2,\n3\n8/a\\}b∇acketle{tu,w 4(2)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu′,w3(3)/a\\}b∇acket∇i}ht=3\n8/a\\}b∇acketle{tu,η\n2u0/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,0/a\\}b∇acket∇i}ht ≤3η\n128.\nFor every 3 ≤k≤T−1,\n3\n8/a\\}b∇acketle{tu,w 4(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu′,w4(k+1)/a\\}b∇acket∇i}ht= 0.\nWe derive that,\n∇ℓ4(w4)(k)=\n\n3\n8u0k= 2\n−1\n2u0k= 3\n0 3 ≤k≤T\n0k= 0.\nCombining all together, we get that,\n∇/hatwideF(w4)(k)=\n\n−1\n4ǫ\nT2u0k= 1\n3\n8u0k= 2\n−1\n2u0k= 3\n0 4 ≤k≤T\n0k= 0\n36and\nw5(k)=\n\n−3\n8ηu0+3η\n4ǫ\nT2u0k= 1\n1\n8ηu0 k= 2\n1\n2ηu0 k= 3\n0 4 ≤s≤T\n1\nn/summationtextn\ni=1φ(Vi,ji)k= 0,\nwhereu0∈U\\/uniontextn\ni=1Vi.\n/square\nLemma 22. Assume the conditions of Theorem 3, and consider the iterate s of unprojected GD on\n/hatwideF, with step size η≤1/√\nTinitialized at w1= 0. Under the event E, we have for all t∈[T]that\n/ba∇dblwt/ba∇dbl ≤1.\nProof. IfEholds, by Lemmas 4 and 18 to 20, we know that for every t≥2,/ba∇dblwt(1)/ba∇dbl ≤η\n2,/ba∇dblwt(t−1)/ba∇dbl ≤\nη\n2and for every k∈ {2,...,t −2},/ba∇dblwt(t−1)/ba∇dbl ≤η\n8. As a result,\n/ba∇dblwt/ba∇dbl2≤d/summationdisplay\ni=1wt[i]2\n≤T/summationdisplay\nk=0/ba∇dblwt(k)/ba∇dbl2\n<2·/parenleftbiggη\n2/parenrightbigg2\n+ (T−3)/parenleftbiggη\n8/parenrightbigg2\n+/vextenddouble/vextenddouble/vextenddoubleη\nnn/summationdisplay\ni=1φ(Vi,ji)/vextenddouble/vextenddouble/vextenddouble2\n≤η2\n2+η2(T−3)\n64+η2\n≤1\n64+3\n2T(η≤1√\nT)\n≤1 ( T≥2)\n/square\nB.3 Proof of Theorem 3\nProof of Theorem 3. By Lemma 3, with probability of at least1\n6,Eoccurs and by Lemma 4, it\nholds for every 2 ≤k≤T−3 that,\nwT,m(k)=1\nmm/summationdisplay\ni=1wT−i+1(k)=/braceleftiggη\n8u0 k≤T−m−2\n1\nm/parenleftbigη\n2+η\n8(T−k−2)/parenrightbigu0k≥T−m−1(18)\n=/braceleftiggη\n8u0k≤T−m−2\nη(T−k+2)\n8mu0k≥T−m−1\nThen, we denote αV∈RT−4the vector which its kth entry is max/parenleftig\n3η\n32,maxu∈V/a\\}b∇acketle{tu,wT,m(k+1)/a\\}b∇acket∇i}ht/parenrightig\n. By\nthe fact that every vector u∈Uis inVwith probability1\n2, the following holds,\nEV/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n≥EV/radicaltp/radicalvertex/radicalvertex/radicalbtT−3/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n37=EV/radicaltp/radicalvertex/radicalvertex/radicalbtT−4/summationdisplay\nk=1max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k+1)/a\\}b∇acket∇i}ht/parenrightbigg2\n=EV/ba∇dblαV/ba∇dbl\n≥ /ba∇dblEVαV/ba∇dbl\n=/radicaltp/radicalvertex/radicalvertex/radicalbtT−3/summationdisplay\nk=2/parenleftbigg\nEVmax/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\nThen, by Eq. (18),\nEV/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n≥/radicaltp/radicalvertex/radicalvertex/radicalbtT−m−2/summationdisplay\nk=2/parenleftbigg\nEVmax/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n+T−3/summationdisplay\nk=T−m−1/parenleftbigg\nEVmax/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n≥/radicaltp/radicalvertex/radicalvertex/radicalbtT−m−2/summationdisplay\nk=2/parenleftbigg\nEVmax/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,η\n8u0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n+T−3/summationdisplay\nk=T−m−1/parenleftbigg\nEVmax/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,η(T−k+ 2)\n8mu0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtT−m−2/summationdisplay\nk=2/parenleftbigg\nEVmax/parenleftbigg3\n4,max\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n+T−3/summationdisplay\nk=T−m−1/parenleftbigg\nEVmax/parenleftbigg3\n4,T−k+ 2\nmmax\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtT−m−2/summationdisplay\nk=2/parenleftbigg\nEVmax/parenleftbigg3\n4,max\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n+T−3/summationdisplay\nk=T−m−1/parenleftbigg\nEVmax/parenleftbigg3\n4,T−k+ 2\nTmax\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtT−m−2/summationdisplay\nk=2/parenleftbigg\nEVmax/parenleftbigg3\n4,max\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n+m−1/summationdisplay\nk=1/parenleftbigg\nEVmax/parenleftbigg3\n4,k+ 4\nTmax\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\nNow, treating each of the term separately, with probability1\n2onV, maxu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht ≤1\n8(otherwise\nit is 1), thus,\nEVmax/parenleftbigg3\n4,max\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg\n=1\n2·3\n4+1\n2·1 =7\n8\nMoreover, if k≤3T\n4−4\nEVmax/parenleftbigg3\n4,k+ 4\nTmax\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg\n=3\n4,\notherwise,\nEVmax/parenleftbigg3\n4,k+ 4\nTmax\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg\n≥1\n2max/parenleftbigg3\n4,k+ 4\nT/parenrightbigg\n+1\n2·3\n4\n≥3\n8+k+ 4\n2T\nThen, we get, if m≥T−3, (note that it implies l−1≥3T\n4−4),\nEV/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n38≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtm−1/summationdisplay\nk=1/parenleftbigg\nEVmax/parenleftbigg3\n4,k+ 4\nTmax\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\nk:1≤k≤3T\n4−49\n16+/summationdisplay\nk:3T\n4−4<k≤m−1/parenleftbigg3\n8+k+ 4\n2T/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\nk:1≤k≤3T\n4−49\n16+/summationdisplay\nk:3T\n4<k≤T/parenleftbigg3\n8+k\n2T/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbt27T−144\n64+/summationdisplay\nk:3T\n4<k≤T/parenleftbigg9\n64+3k\n8T/parenrightbigg\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalvertex/radicalbt27T−144\n64+\n9T\n256+3\n8T/summationdisplay\nk:3T\n4<k≤Tk\n\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbt27T−144\n64+/parenleftigg\n9T\n256+3T2\n16T−3(3T\n4+ 1)2\n16T/parenrightigg\n(i2\n2≤/summationtextn\ni=1i2≤(i+1)2\n2)\n=η\n8/radicaligg\n27T−144\n64+/parenleftbigg9T\n256+3T\n16−27T\n256−3\n16T−9\n32/parenrightbigg\n=η\n8/radicaligg\n148T\n256−45\n32−3\n16T\n≥η\n8/radicaligg\n147T\n256(T≥512 = ⇒45\n32+3\n16T≤T\n256)\n≥3η\n32·101√\nT\n100.\nOtherwise, if m<T −4, by similar arguments,\nEV/radicaltp/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈V/a\\}b∇acketle{tu,wT,m(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtT−m−2/summationdisplay\nk=2/parenleftbigg\nEVmax/parenleftbigg3\n4,max\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n+T−3/summationdisplay\nk=T−m−1/parenleftbigg\nEVmax/parenleftbigg3\n4,T−k+ 2\nTmax\nu∈V/a\\}b∇acketle{tu,u 0/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtT−m−2/summationdisplay\nk=2/parenleftbigg7\n8/parenrightbigg2\n+/summationdisplay\nk:1≤k≤3T\n4−49\n16+/summationdisplay\nk:T\n2<k≤m+3/parenleftbigg3\n8+k\n2T/parenrightbigg2\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtT/summationdisplay\nk=m+4/parenleftbigg7\n8/parenrightbigg2\n+/summationdisplay\nk:1≤k≤3T\n4−49\n16+/summationdisplay\nk:T\n2<k≤m+3/parenleftbigg3\n8+k\n2T/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\nk:1≤k≤3T\n4−49\n16+/summationdisplay\nk:3T\n4<k≤T/parenleftbigg3\n8+k\n2T/parenrightbigg2\n(3\n8+k\n2T≤7\n8)\n39≥3η\n32·101√\nT\n100.\nMoreover, we notice that for every t,ℓ2(wt)≥ −/ba∇dblwt(0)/ba∇dbl ≥ −η,ℓ3(wt)≥δ1andℓ4(wt)≥δ2, thus,\nit holds that,\nF(wT,l)≥303η\n3200√\nT+δ1+δ2−η≥η/parenleftbigg303\n3200√\nT−1/parenrightbigg\nand\nF(w∗)≤3η\n32√\nT+η\nThen, with probability of at least1\n6,\nF(wT,l)−F(w∗)≥η(303\n3200√\nT−2−3\n32√\nT)\n≥η(303\n3200√\nT−302\n3200√\nT) ( T≥32002=⇒2≤2\n3200√\nT)\n=η\n3200√\nT.\n/square\nC Proofs of Section 5\nC.1 Proofs for the full construction\nLemma 23. Letn, a setU∈Rd. LetP(U)be the power set of U. Then, there exist sets\n{Ψ1,...Ψn} ⊆R2n, a number 0< ǫ <1\nnand two mappings φ:P(U)×[n]→R2n,α:R2n→U\nsuch that,\n1. For every j∈[n]andV⊆U,/ba∇dblφ(V,j)/ba∇dbl ≤1.\n2. For every k,ψ∈Ψk,/ba∇dblα(ψ)/ba∇dbl ≤1,/ba∇dblψ/ba∇dbl ≤1.\n3. LetV1,...,Vk⊆U. Then, for every k,ψ∗\nk=1\nn/summationtextk\ni=1φ(Vi,i)holds,\n•For everyψ∈Ψk,ψ/\\e}atio\\slash=ψ∗\nk:\n/a\\}b∇acketle{tψ∗\nk,1\nnk/summationdisplay\ni=1φ(Vi,i)/a\\}b∇acket∇i}ht ≥ /a\\}b∇acketle{tψ,1\nnk/summationdisplay\ni=1φ(Vi,i)/a\\}b∇acket∇i}ht+ǫ;\n•If/intersectiontextk\ni=1Vi/\\e}atio\\slash=∅andm= arg mini{i:vi∈/intersectiontextk\ni=1Vi}, thenα(ψ∗) =vm∈/intersectiontextk\ni=1Vi.\nProof of Lemma 23. The construction is similar to Lemma 17. First, we consider a n arbitrary enu-\nmeration of P(U) ={V1,...V|P(U)|}and defineg:P(U)→R2,g(Vi) =/parenleftig\nsin/parenleftig\n2πi\n|P(U)|/parenrightig\n,cos/parenleftig\n2πi\n|P(U)|/parenrightig/parenrightig\n.\nHere, we refer to a vector a∈R2nas a concatenation of nvectors in R2,a(1),...,a(n). Then, we\ndefineδ= 1−cos/parenleftig\n2π\n|P(U)|/parenrightig\n,ǫ=δ\nn2and\nφ(V,j)(i)=/braceleftigg\ng(V)i=j\n0 otherwise\n40As a result, for every Vi,jit holds that\n/ba∇dblφ(Vi,j)/ba∇dbl=/ba∇dblg(Vi)/ba∇dbl=/radicaligg\nsin/parenleftbigg2πi\n|P(U)|/parenrightbigg2\n+ cos/parenleftbigg2πi\n|P(U)|/parenrightbigg2\n= 1\nMoreover, if j1/\\e}atio\\slash=j2,\n/a\\}b∇acketle{tφ(Vi,j1),φ(Vi,j2)/a\\}b∇acket∇i}ht= 0,\nand ifi>k ,\n/a\\}b∇acketle{tφ(Vi,j),φ(Vk,j)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tg(Vi),g(Vk)/a\\}b∇acket∇i}ht\n= sin/parenleftbigg2πi\n|P(U)|/parenrightbigg\nsin/parenleftbigg2πk\n|P(U)|/parenrightbigg\n+ cos/parenleftbigg2πi\n|P(U)|/parenrightbigg\ncos/parenleftbigg2πk\n|P(U)|/parenrightbigg\n= cos/parenleftbigg2π(i−k)\n|P(U)|/parenrightbigg\n≤cos/parenleftbigg2π\n|P(U)|/parenrightbigg\n(cos is monotonic decreasing in [0 ,π/2])\n= 1−δ\nWe notice that 0 <δ< 1. Now, we consider an arbitrary enumeration of U={v1,...v |U|}, and define\nthe following sets Ψ 1,...Ψn⊆R2nand the following two mappings σ:R2n→P(U),α:R2n→U,\nΨk={1\nnk/summationdisplay\ni=1φ(Vi,i) :∀i Vi⊆U}\nNote that, for every ψ∈Ψ,\n/ba∇dblψ/ba∇dbl=/ba∇dbl1\nnk/summationdisplay\ni=1φ(Vi,ji)/ba∇dbl ≤1\nnk/summationdisplay\ni=1/ba∇dblφ(Vi,ji)/ba∇dbl ≤1.\nThen, for every a∈R2nandj∈[n], we denote the index q(a,j)∈[|P(U)|] as\nq(a,j) = arg max\nr/a\\}b∇acketle{tg(Vr),a(j)/a\\}b∇acket∇i}ht,\nand define the following mapping σ:R2n→P(U),\nσ(a) =n/intersectiondisplay\nj=1,a(j)/negationslash=0Vq(a,j).\nMoreover, for every a∈R2n, we denote the index p(a)∈[|U|] as\np(a) = arg min\ni{i:vi∈σ(a)},\nand define the following mapping α:R2n2→U,\nα(a) =/braceleftigg\nv|U|σ(a) =∅\nvp(a)σ(a)/\\e}atio\\slash=∅.\n41Note that for every a∈R2n,α(a)∈U, thus, /ba∇dblα(a)/ba∇dbl ≤1.\nNow, LetV1,...,Vn⊆U,k∈[n] andψ∗\nk=1\nn/summationtextk\ni=1φ(Vi,i). Then,\n/a\\}b∇acketle{tψ∗,1\nnk/summationdisplay\ni=1φ(Vi,i)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t1\nnk/summationdisplay\ni=1φ(Vi,i),1\nnk/summationdisplay\ni=1φ(Vi,i)/a\\}b∇acket∇i}ht\n=1\nn2k/summationdisplay\ni=1/a\\}b∇acketle{tφ(Vi,i),φ(Vi,i)/a\\}b∇acket∇i}ht\n=k\nn2\nForψ=1\nn/summationtextk\ni=1φ(V′\ni,i) such that ψ/\\e}atio\\slash=ψ∗, there exists a index rsuch thatV′\nr/\\e}atio\\slash=Vr,thus,\n/a\\}b∇acketle{tψ,1\nnk/summationdisplay\nl=1φ(Vi,i)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t1\nnk/summationdisplay\ni=1φ(V′\ni,i),1\nnk/summationdisplay\ni=1φ(Vi,i)/a\\}b∇acket∇i}ht\n=1\nn2k/summationdisplay\ni=1/a\\}b∇acketle{tφ(Vi,i),φ(V′\ni,i)/a\\}b∇acket∇i}ht\n≤1\nn2\n1−δ+k/summationdisplay\ni=1,i/negationslash=r1\n\n≤1\nn2(1−δ+k−1)\n=k\nn2−δ\nn2\n=/a\\}b∇acketle{tψ∗\nk,1\nnn/summationdisplay\ni=1φ(Vi,ji)/a\\}b∇acket∇i}ht −ǫ\nFurthermore, it holds that,1\nn/summationtextn\ni=1φ(Vi,i)(i)=1\nng(Vi), thus,\nq/parenleftigg\n1\nnk/summationdisplay\ni=1φ(Vi,i),i/parenrightigg\n= arg max\nr/a\\}b∇acketle{tg(Vr),1\nnk/summationdisplay\ni=1φ(Vi,i)(i)\n/a\\}b∇acket∇i}ht\n= arg max\nr/a\\}b∇acketle{tg(Vr),1\nng(Vi)/a\\}b∇acket∇i}ht\n=i,\nthus, we get,\nσ(ψ∗) =σ/parenleftigg\n1\nnk/summationdisplay\ni=1φ(Vi,i)/parenrightigg\n=n/intersectiondisplay\nj=1,1\nn/summationtextk\ni=1φ(Vi,i)(j)/negationslash=0V\nq(1\nn/summationtextk\ni=1φ(Vi,i)(j),j)\n=k/intersectiondisplay\nj=1Vq(1\nn/summationtextk\ni=1φ(Vi,i),j)(The indices that are non-zero are j= 1,...,k )\n=k/intersectiondisplay\ni=1Vi\n42Then, assuming that/intersectiontextk\ni=1Vi/\\e}atio\\slash=∅, and let and m= arg mini{i:vi∈/intersectiontextk\ni=1Vi},p(a) =mand,\nα(ψ∗) =vm∈k/intersectiondisplay\ni=1Vi.\n/square\nProof of Lemma 8. First,ℓSGD\n1is convex and 1-Lipschitz by the fact that ℓSGD\n1=ℓ1and Lemma 2.\nMoreover, by Lemma 23, ℓSGD\n2is a maximum over 1-Lipschitz linear functions, thus, ℓSGD\n2is convex\nand 1-Lipschitz. Finally, ℓSGD\n3 is a summation of two 1-Lipschitz linear functions, thus, ℓSGD\n3is\nconvex and 2-Lipschitz. Combining all together, we get the l emma. /square\nC.2 Proof of algorithm’s dynamics\nIn this section we describe the dynamics of SGD. We begin with showing that the good event E′\n(Eq. (15)) occurs with a constant probability.\nProof of Lemma 9. First, by union bound,\nPr (∀t∈[n]Pt/\\e}atio\\slash=∅andJt∈St)≥1\n2= 1−Pr(∃tPt=∅orJt/∈St)\n≥1−n/summationdisplay\nt=1Pr(Pt=∅or (Pt/\\e}atio\\slash=∅andJt/∈St))\n≥1−n/summationdisplay\nt=1Pr (Pt=∅)−n/summationdisplay\nt=1Pr (Pt/\\e}atio\\slash=∅andJt/∈St)\n= 1−n/summationdisplay\nt=1Pr (Pt=∅)−n/summationdisplay\nt=1Pr (Pt/\\e}atio\\slash=∅) Pr (Jt/∈St|Pt/\\e}atio\\slash=∅).\nNow, for every vl∈U,\nPr(vl/∈t−1/intersectiondisplay\ni=1Vi) = 1 −Pr(vl/∈t−1/intersectiondisplay\ni=1Vi) = 1 −δt−1,\nand,\nPr (vl/∈St) = 1 −(1−δ)n−t+1≤1−(1−δ)n.\nThen,\nPr (Pt=∅) = Pr/parenleftiggt/intersectiondisplay\ni=1Vi=∅/parenrightigg\n= Pr( ∀vl∈U w / ∈t/intersectiondisplay\ni=1Vi)\n= (1 −δt−1)|U|\n≤(1−δn)|U|.\nMoreover, by the fact that for every t,Ptis independent of Vt+1,...Vn,\nPr(Pt/\\e}atio\\slash=∅) Pr (Jt/∈St|Pt/\\e}atio\\slash=∅) =/summationdisplay\nl:vl∈UPr (Pt/\\e}atio\\slash=∅) Pr (vl/∈St|Pt/\\e}atio\\slash=∅) Pr(Jt=vl)\n43=/summationdisplay\nl:vl∈UPr (Pt/\\e}atio\\slash=∅) Pr (vl/∈St) Pr(Jt=vl)\n≤1−(1−δ)n\nCombining all of the above, we get that,\nPr(∀t∈[n]Pt/\\e}atio\\slash=∅andJt∈St)\n= 1−n/summationdisplay\nt=1Pr (Pt=∅)−n/summationdisplay\nt=1Pr (Pt/\\e}atio\\slash=∅) Pr (Jt/∈St|Pt/\\e}atio\\slash=∅)\n≥1−n(1−δn)|U|−n(1−(1−δ)n).\nForδ=1\n4n2, by the fact that |U| ≥2d′\n178= 24nlog(n)=n4n,\n|U|δn≥n4nn−2n4−n≥n2n4−n≥log(4n)\nPr (∀t∈[n]Pt/\\e}atio\\slash=∅andJt∈St)≥1\n2≥1−n(1−δn\n500)|U|−n(1−(1−δ)n)\n≥1−ne−|U|δn\n500−n(1−(1−nδ))\n≥1−ne−log(4n)−n2δ\n≥1−1\n4−1\n4\n=1\n2.\n/square\nProof of Lemma 11. First, by the fact that for every t≤k≤T,w(k)= 0, for every such k,\nmax\nu∈Vt/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht= 0<3η\n32,\nFor 2 ≤k≤t−1,w(k)=cηuk, wherec≤1\n2and everyuk∈/intersectiontextT\ni=kVi⊆Vt, thus,\nmax\nu∈Vt/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht ≤η\n2·1\n8<3η\n32.\nWe derive that ∇ℓSGD\n1(wt,Vt) = 0. /square\nProof of Lemma 12. First, we show that the maximum of ℓSGD\n2(w,V) is attained with k=mand\nu=um. Fork≥m+ 1, for every u∈Uandψ∈Ψk,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht= 0.\nFork= 1, for every u∈Uandψ∈Ψ1, by Lemma 23, we know that for every ψ,V,j ,\n/ba∇dblψ/ba∇dbl,/ba∇dblφ(V,j)/ba∇dbl ≤1, andα(ψ)∈U, thus,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3c\n8/a\\}b∇acketle{tu1,u/a\\}b∇acket∇i}ht −η\n16/a\\}b∇acketle{tu2,α(ψ)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht −0 + 0\n44≤9η\n512+η\n128+η\n4n\n<η\n8. (n≥4)\nFor 2 ≤k≤m−2, for every u∈Uandψ∈Ψk, by Lemma 23, we know that for every ψ,V,j ,\n/ba∇dblψ/ba∇dbl,/ba∇dblφ(V,j)/ba∇dbl ≤1, andα(ψ)∈U, thus,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n64/a\\}b∇acketle{tuk,u/a\\}b∇acket∇i}ht −η\n16/a\\}b∇acketle{tuk+1,α(ψ)/a\\}b∇acket∇i}ht+ 0−0 + 0\n≤3η\n64+η\n16\n<η\n8.\nFork=m−1, for every u∈Uandψ∈Ψk, by Lemma 23, we know that for every ψ,V,j ,\n/ba∇dblψ/ba∇dbl,/ba∇dblφ(V,j)/ba∇dbl ≤1, andα(ψ)∈U, thus,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n64/a\\}b∇acketle{tuk,u/a\\}b∇acket∇i}ht −η\n4/a\\}b∇acketle{tuk+1,α(ψ)/a\\}b∇acket∇i}ht+ 0− /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),1\n4n2φ(V,m)/a\\}b∇acket∇i}ht\n≤3η\n64+η\n32+1\n16n2+1\n16n3\n<η\n8. (n≥4)\nFork=m,u/\\e}atio\\slash=umand every ψ∈Ψm, by Lemma 23, we know that for every ψ,V,j ,\n/ba∇dblψ/ba∇dbl,/ba∇dblφ(V,j)/ba∇dbl ≤1, andα(ψ)∈U, thus,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w(0,k)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,η\n2um/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w(0,k)/a\\}b∇acket∇i}ht\n≤3η\n128+η\n16n2\n<η\n32. (n≥4)\nFork=m,u=umand everyψ∈Ψm, by Lemma 23, we know that for every ψ,V,j ,/ba∇dblψ/ba∇dbl,/ba∇dblφ(V,j)/ba∇dbl ≤\n1, andα(ψ)∈U, thus,\n3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,wt(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,wt(0,k)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,η\n2um/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,wt(0,k)/a\\}b∇acket∇i}ht\n≥3η\n16−η\n16n2\n45>5η\n32(n≥4)\n>δ1.\nSecond, we show that when k=mandu=um, the maximum among ψ∈Ψmis attained\nuniquely in ψ∗\nm=1\nn/summationtextm\nt=1φ(Vt,t). For any ψ∈Ψm, withψ/\\e}atio\\slash=ψ∗\nm, by Lemma 23, for k=m,\nu=um,\n3\n8/a\\}b∇acketle{tu,w(m)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ∗\nm),w(m+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,m),1\n4nψ∗\nm/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,m+1),1\n4nψ∗\nm/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,m+1),−1\n4n2φ(V,m + 1)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,w(m)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ∗\nm,w(0,m)/a\\}b∇acket∇i}ht\n=3η\n16+η\n16n2/a\\}b∇acketle{tψ∗\nm,1\nnm/summationdisplay\nt=1φ(Vt,t)/a\\}b∇acket∇i}ht\n≥3η\n16+η\n16n2/a\\}b∇acketle{tψ,1\nnm/summationdisplay\nt=1φ(Vt,t)/a\\}b∇acket∇i}ht+ηǫ\n16n2\n=3\n8/a\\}b∇acketle{tu,w(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w(0,m)/a\\}b∇acket∇i}ht+ηǫ\n16n2\n=3\n8/a\\}b∇acketle{tu,w(m)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w(m+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,m),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw(0,m+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,m+1),−1\n4n2φ(V,m + 1)/a\\}b∇acket∇i}ht+ηǫ\n16n2\nWe derive that,\n∇ℓSGD\n2(w,V)(k)=\n\n3\n8um k=m\n−1\n2α(ψ∗\nm)k=m+ 1\n0 k /∈ {m,m + 1}\n∇ℓSGD\n2(w,V)(0,k)=\n\n1\n4n2/summationtextm\nt=1φ(Vt,t) k=m\n−1\n4n2/summationtextm\nt=1φ(Vt,t)−1\n4n2φ(V,m + 1)k=m+ 1\n0 k /∈ {m,m + 1}.\n/square\nLemma 24. Under the conditions of Theorem 4, if E′occurs and wtis the iterate of Unprojected\nSGD with step size η≤1√nandw1= 0,\nw2(k)=/braceleftiggη\nn3u1k= 1\n0k≥2,\nand,\nw2(0,k)=/braceleftiggη\n4n2φ(V1,1)k= 1\n0 k/\\e}atio\\slash= 1.\nProof.w1= 0, thus, for every k,\nmax\nu∈V1/a\\}b∇acketle{tu,w 1(k)/a\\}b∇acket∇i}ht= 0<3η\n32,\n46and we derive that ∇ℓSGD\n1(w1,V1) = 0. By the same argument, ∇ℓSGD\n2(w1,V1) = 0 (where the\nmaximum is attained uniquely in δ2). Moreover, ℓSGD\n3is a linear function, then, we get that,\n∇ℓSGD\n3(w1,V1)(k)=/braceleftigg\n−1\nn3u1k= 1\n0k≥2,\nand,\n∇ℓSGD\n3(w1,V1)(0,k)=/braceleftigg\n−1\n4n2φ(V1,1)k= 1\n0 k/\\e}atio\\slash= 1,\nand the lemma follows. /square\nLemma 25. Under the conditions of Theorem 4, if E′occurs and wtis the iterate of Unprojected\nSGD with step size η≤1√nandw1= 0,\nw3(k)=\n\n2η\nn3u1−3η\n8u1k= 1\nη\n2u2 k= 2\n0 3 ≤k≤n\nw3(0,k)=\n\nη\n4n2φ(V2,1) k= 1\nη\n4n2φ(V1,1) +η\n4n2φ(V2,2)k= 2\n0 k≥3.\nwhereu1∈U, andu2holdsu2∈P2∩S2.\nProof. First, by the fact that for every 2 ≤k≤T,w2(k)= 0, for every such k,\nmax\nu∈V2/a\\}b∇acketle{tu,w 2(k)/a\\}b∇acket∇i}ht= 0<3η\n32,\nand we derive that ∇ℓSGD\n1(w2,V2) = 0.\nMoreover,ℓSGD\n3is a linear function, thus,\nℓSGD\n3(w2,V2)(k)=/braceleftiggη\nn3u1k= 1\n0k≥2,\nand,\nℓSGD\n3(w2,V2)(k)=/braceleftiggη\n4n2φ(V2,1)k= 1\n0 k/\\e}atio\\slash= 1.\nForℓSGD\n2(w2,V2), we get by the fact that for every k≥1,w2(k+1)=w2(0,k+1)= 0,\nℓSGD\n2(w2,V2) = max/parenleftigg\nδ2, max\nk∈[n−1],u∈U,ψ∈Ψk/parenleftbigg3\n8/a\\}b∇acketle{tu,w 2(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w 2(0,k)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg\nAs a first step, we show that the the maximum is attained with k= 1 andu=u1, Fork/\\e}atio\\slash= 1, for\neveryu∈Uandψ∈Ψk,\n3\n8/a\\}b∇acketle{tu,w 2(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w 2(0,k)/a\\}b∇acket∇i}ht= 0.\n47Fork= 1,u/\\e}atio\\slash=u1and everyψ∈Ψ1, by the fact that /ba∇dblψ/ba∇dbl,/ba∇dblφ(V1,1)/ba∇dbl ≤1,\n3\n8/a\\}b∇acketle{tu,w 2(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w 2(0,k)/a\\}b∇acket∇i}ht ≤3η\n64n3+η\n16n3=7η\n64n3<3η\n16n3.\nFork= 1,u=u1and everyψ∈Ψ1, by the fact that /ba∇dblψ/ba∇dbl,/ba∇dblφ(V1,1)/ba∇dbl ≤1,\n3\n8/a\\}b∇acketle{tu,w 2(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4n2ψ,w 2(0,k)/a\\}b∇acket∇i}ht ≥3η\n8n3−η\n16n3>3η\n16n3>δ1.\nAs a second step we show that the maximum among ψ∈Ψ1is attained uniquely in ψ∗\n1=\n1\nnφ(V1,1). For any ψ∈Ψ1, withψ/\\e}atio\\slash=ψ∗\n1. By Lemma 23, for k= 1,u=u1,\n3\n8/a\\}b∇acketle{tu,w 2(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ∗\n1,w2(0,k)/a\\}b∇acket∇i}ht=3η\n8n3+/a\\}b∇acketle{t1\n4nψ∗\n1,η\n4n2φ(V1,1)/a\\}b∇acket∇i}ht\n=3η\n8n3+η\n16n2/a\\}b∇acketle{tψ∗\n1,1\nnφ(V1,1)/a\\}b∇acket∇i}ht\n≥3η\n8n3+η\n16n2/a\\}b∇acketle{tψ,1\nnφ(V1,1)/a\\}b∇acket∇i}ht+ηǫ\n16n2\n=3\n8/a\\}b∇acketle{tu,w 2(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w 2(0,k)/a\\}b∇acket∇i}ht+ηǫ\n16n2\nWe got that the maximum is uniquely attained at k= 1,u=u1,ψ=1\nnφ(V1,1). Now, by Lemma 23,\nforj= arg mini{i:vi∈V1}, we get that\nα(ψ) =vj∈V1.\nWe notice that V1=P2and thusα(ψ) =J2. Then, by E′,α(ψ) also holds α(ψ)∈S2. Combining\nthe above together, we get, for u2=α(ψ)∈P2∩S2,\n∇f(w2,V2)(k)=\n\n3\n8u1−1\nn3u1k= 1\n−1\n2u2 k= 2\n0 k≥3\nand,\n∇f(w2,V2)(0,k)=\n\n1\n4n2φ(V1,1)−1\n4n2φ(V2,1)k= 1\n−1\n4n2φ(V1,1)−1\n4n2φ(V2,2)k= 2\n0 k≥3,\nand the lemma follows. /square\nLemma 26. Under the conditions of Theorem 4, if E′occurs and wtis the iterate of Unprojected\nSGD with step size η≤1√nandw1= 0,\nw4(k)=\n\n3η\nn3u1−3η\n8u1k= 1\nη\n8u2 k= 2\nη\n2u3 k= 3\n0 k≥4,\nand,\nw4(0,k)=\n\nη\n4n2φ(V2,1) +η\n4n2φ(V3,1) k= 1\nη\n4n2φ(V1,1) +η\n4n2φ(V2,2) +η\n4n2φ(V3,3)k= 3\n0 k /∈ {1,3}.\n48Proof. First, we notice that by Lemma 25, it holds that w3(2)=cηu2forc≤1\n2andu2holds\nu2∈V1∩/intersectiontextn\ni=2Vi, and for every 3 ≤k≤T,wt(k)= 0. Then, by Lemma 11, we have that\n∇ℓSGD\n1(w3,V3) = 0.Moreover,ℓSGD\n3is a linear function, thus,\nℓSGD\n3(w3,V3)(k)=/braceleftiggη\nn3u1k= 1\n0k≥2,\nand,\nℓSGD\n3(w3,V3)(k)=/braceleftiggη\n4n2φ(V3,1)k= 1\n0 k/\\e}atio\\slash= 1.\nForℓSGD\n2(w3,V3), we first show that the the maximum is attained with k= 2 andu=u2. For\nk≥3, for every u∈Uandψ∈Ψk,\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w3(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw3(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht= 0.\nFork= 1, for every u∈Uandψ∈Ψ1, by the fact that for every ψ,V,j ,/ba∇dblψ/ba∇dbl,/ba∇dblφ(V,j)/ba∇dbl ≤1,\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w3(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw3(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n8(2η\nn3−3η\n8)/a\\}b∇acketle{tu1,u/a\\}b∇acket∇i}ht −η\n4/a\\}b∇acketle{tu2,α(ψ)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4n2φ(V2,1),ψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tη\n4n2φ(V1,1) +η\n4n2φ(V2,2),ψ/a\\}b∇acket∇i}ht\n+/a\\}b∇acketle{tη\n4n2φ(V1,1) +η\n4n2φ(V2,2),1\n4n2φ(V3,2)/a\\}b∇acket∇i}ht\n≤9η\n512+η\n32+η\n4n2+η\n2n2+η\n8n4\n<29η\n256(n≥4)\n<η\n8.\nFork= 2,u/\\e}atio\\slash=u2and everyψ∈��2, , by the fact that for every ψ,V,j ,/ba∇dblψ/ba∇dbl,/ba∇dblφ(V,j)/ba∇dbl ≤1,\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w3(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw3(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w 3(0,k)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,η\n2u2/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,η\n4n2φ(V1,1) +η\n4n2φ(V2,2)/a\\}b∇acket∇i}ht\n≤3η\n128+η\n8n3\n<η\n32. (n≥4)\nFork= 2,u=u2and everyψ∈Ψ2, by the fact that /ba∇dblψ/ba∇dbl,/ba∇dblφ(V1,1)/ba∇dbl ≤1,\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w3(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw3(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w 3(0,k)/a\\}b∇acket∇i}ht\n49=3\n8/a\\}b∇acketle{tu,η\n2u2/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,η\n4n2φ(V1,1) +η\n4n2φ(V2,2)/a\\}b∇acket∇i}ht\n≥3η\n16−η\n8n3\n>5η\n32(n≥4)\n>δ1.\nSecond, we show that the maximum among ψ∈Ψ2is attained uniquely in ψ∗\n2=1\nnφ(V1,1) +\n1\nnφ(V2,2). For any ψ∈Ψ2, withψ/\\e}atio\\slash=ψ∗\n2, by Lemma 23, for k= 2,u=u2,\n3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ∗\n2),w3(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k),1\n4nψ∗\n2/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw3(0,k+1),1\n4nψ∗\n2/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht\n=3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ∗\n2,w3(0,k)/a\\}b∇acket∇i}ht\n=3η\n16+/a\\}b∇acketle{t1\n4nψ∗\n2,η\n4n2φ(V1,1) +η\n4n2φ(V2,2)/a\\}b∇acket∇i}ht\n=3η\n16+η\n16n2/a\\}b∇acketle{tψ∗\n2,1\nnφ(V1,1) +1\nnφ(V2,2)/a\\}b∇acket∇i}ht\n≥3η\n16+η\n16n2/a\\}b∇acketle{tψ,1\nnφ(V1,1) +1\nnφ(V2,2)/a\\}b∇acket∇i}ht+ηǫ\n16n2\n=3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t1\n4nψ,w 3(0,k)/a\\}b∇acket∇i}ht+ηǫ\n16n2\n=3\n8/a\\}b∇acketle{tu,w 3(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tα(ψ),w3(k+1)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k),1\n4nψ/a\\}b∇acket∇i}ht − /a\\}b∇acketle{tw3(0,k+1),1\n4nψ/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tw3(0,k+1),−1\n4n2φ(V,k+ 1)/a\\}b∇acket∇i}ht+ηǫ\n16n2\nWe got that the maximum is uniquely attained at k= 2,u=u2,ψ=ψ∗\n2. Now, by Lemma 23, for\nj= arg mini{i:vi∈V1∩V2}, we get that\nα(ψ) =vj∈V1∩V2.\nWe notice that V1∩V2=P3and thusα(ψ) =J3. Then, by E′,α(ψ) also holds α(ψ)∈S3.\nCombining the above together, we get, for u1∈U,u2∈P2∩S2andu3=α(ψ∗\n2)∈P3∩S3,\n∇f(w3,V3) =\n\n−1\nn3u1 s= 1\n3\n8u2 s= 2\n−1\n2u3 s= 3\n0 4 ≤s≤n\n−1\n4n2φ(V3,1) s= 0,1\n1\n4n2φ(V1,1) +1\n4n2φ(V2,2) s= 0,2\n−1\n4n2φ(V1,1)−1\n4n2φ(V2,2)−1\n4n2φ(V3,3)s= 0,3\n0 s= 0,kfork≥3,\nand the lemma follows. /square\nC.3 Proof of Theorem 4\nProof of Theorem 4. We show that the theorem holds if the event E′occurs. First, we prove that\nfor everyt,/ba∇dblwt/ba∇dbl ≤1. By Lemma 10,\n/ba∇dblwt/ba∇dbl ≤/radicaltp/radicalvertex/radicalvertex/radicalbtd/summationdisplay\ni=1wt[i]2\n50≤/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay\nk=1/ba∇dblwt(k)/ba∇dbl2+n/summationdisplay\nl=1/ba∇dblwt(0,l)/ba∇dbl2\n</radicaligg\n2·/parenleftbiggη\n2/parenrightbigg2\n+ (n−2)/parenleftbiggη\n8/parenrightbigg2\n+ 2·/parenleftbiggη\n4n/parenrightbigg2\n≤/radicaligg/parenleftbiggη2\n2/parenrightbigg\n+η2(n−2)\n64+ 2η2\n≤/radicalbigg1\n64+5\n2n(η≤1√n)\n≤1 ( n≥4)\nNow, denote αV∈Rn−3the vector which its kth entry is max/parenleftbigη\n16,maxu∈Vi/a\\}b∇acketle{tu,(n−k+ 2)η\n8uk+1/a\\}b∇acket∇i}ht/parenrightbig.\nForwn=wn,n, and any 2 ≤s≤n−2,\nwn(s)=η\n2nus+ (n−s−1)η\n8nus= (n−s+ 3)η\n8nus.\nThen,\n1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,wn(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n≥1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,wn(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n=1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,(n−k+ 3)η\n8nuk/a\\}b∇acket∇i}ht/parenrightbigg2\n=1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn−3/summationdisplay\nk=1max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,(n−k+ 2)η\n8nuk+1/a\\}b∇acket∇i}ht/parenrightbigg2\n=1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn−3/summationdisplay\nk=1max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,(n−k+ 2)η\n8nuk+1/a\\}b∇acket∇i}ht/parenrightbigg2\n=1\nnn/summationdisplay\ni=1/ba∇dblαVi/ba∇dbl\n≥ /ba∇dbl1\nnn/summationdisplay\ni=1αVi/ba∇dbl\n=/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,(n−k+ 3)η\n8nuk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,n−k+ 3\nnuk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,n−k+ 2\nnuk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\nNow, by the fact that if E′holds, by Lemma 10, for 2 ≤k≤n−2,uk∈Pk=/intersectiontextk−1\ni=1Vk,\n1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,wn(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n51≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2/parenleftigg\n1\nnk−1/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,n−k+ 2\nnuk/a\\}b∇acket∇i}ht/parenrightbigg\n+1\nnn/summationdisplay\ni=kmax/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,n−k+ 2\nnuk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2/parenleftbigg3(n−k+ 1)\n4n+k−1\nnmax/parenleftbigg3\n4,n−k+ 2\nn/parenrightbigg/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\n2≤k≤n\n4−2/parenleftbigg3(n−k+ 1)\n4n+(k−1)(n−k+ 1)\nn2/parenrightbigg2\n+/summationdisplay\nn\n4−3<k≤n−2/parenleftbigg3(n−k+ 1)\n4n+3(k−1)\n4n/parenrightbigg2\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\n2≤k≤n\n4−2/parenleftbigg(n−k+ 1)(3n+ 4(k−1))\n4n2/parenrightbigg2\n+27n\n64\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\n1≤k≤n\n4−3/parenleftbigg(n−k)(3n+ 4k)\n4n2/parenrightbigg2\n+27n\n64\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt/summationdisplay\n1≤k≤n\n4−3/parenleftbigg3\n4+k\n4n−k2\nn2/parenrightbigg2\n+27n\n64\nNow, the fact that forn\n8≤k≤n\n4,k\n4n≤k2\nn2and fork≤n\n8,k\n8n≤k2\nn2,\n1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,wn(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbt/summationdisplay\n1≤k≤n\n8/parenleftbigg3\n4+k\n8n/parenrightbigg2\n+9n\n128−27\n16+27n\n64\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt9n\n128+3\n64n⌊n\n8⌋/summationdisplay\nk=1k+9n\n128−27\n16+27n\n64\n≥η\n8/radicaligg\n1\n2/parenleftbiggn\n8−1/parenrightbigg2\n−27\n16+36n\n64\n≥η\n8/radicalbiggn\n512−27\n16+36n\n64(n≥16)\n≥η\n8/radicalbigg577n\n1024(n≥2048)\n≥3η\n32·2001\n2000\nNow, form<n and 2 ≤k≤n−2,\nwn,m(k)=/braceleftiggη\n8uk k≤n−m−1\n1\nm/parenleftbigη\n2uk+ (n−k−1)η\n8us/parenrightbigk≥n−m\n=/braceleftiggη\n8uk k≤n−m−1\nη(n−k+3)\n8musk≥n−m.\n52Then, by similar arguments, it holds that,\n1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,wn,m(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n≥1\nnn/summationdisplay\ni=1/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,wn,m(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n≥/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3η\n32,max\nu∈Vi/a\\}b∇acketle{tu,wn,m(k)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n=η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtn−m−1/summationdisplay\nk=2/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,uk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n+n−2/summationdisplay\nk=n−m/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,n−k+ 3\nmuk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtn−m−1/summationdisplay\nk=2/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,uk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n+n−2/summationdisplay\nk=n−m/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,n−k+ 2\nnuk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n≥η\n8/radicaltp/radicalvertex/radicalvertex/radicalbtn−2/summationdisplay\nk=2/parenleftigg\n1\nnn/summationdisplay\ni=1max/parenleftbigg3\n4,max\nu∈Vi/a\\}b∇acketle{tu,n−k+ 2\nnuk/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg2\n(k≥2 =⇒n−k+2\nn≤1)\n≥3η\n32·2001\n2000(calculation for wn,n)\nAs a result, we notice that for every t,ℓSGD\n2(wt)≥ −1\n4n2−1\nn3andℓ2(wt)≥δ1thus, it holds\nthat,\n/hatwideF(wn,m)≥3η√n\n32·2001\n2000−1\n4n2−1\nn3+δ1\n≥3η√n\n32·2001\n2000−η\n2n2\n≥3η√n\n32·2001\n2000−η√n\n80000(n≥256)\n≥3η√n\n32·/parenleftbigg2001\n2000−1\n4000/parenrightbigg\n≥3η√n\n32·4001\n4000\nand\n/hatwideF(/hatwidew∗)≤/hatwideF(0)≤3η\n32√n\nThen, if E′holds\n/hatwideF(wn,m)−/hatwideF(/hatwidew∗)≥3η√n\n32·2001\n2000−3η\n32√n\n=η√n\n64000\n/square\nD Proofs of Appendix A\nD.1 Proofs of Appendix A.1\nThe proof of Theorem 5 appears in Appendix A.1. Here we prove s ome auxiliary lemmas that are\nused for the proof.\n53Proof of Lemma 13. First, differentiability can be derived immediately from Le mma 27. Second,\nfor 5-Lipschitzness, for every ( V,j)∈Z, we define ˜fV,j:Rd→Ras˜fV,j(w):=˜f(w,(V,j)). By the\n5-Lipschitzness of fwith respect to its first argument and Jensen Inequality, for everyx,y∈Rd, it\nholds that\n|˜fV,j(x)−˜fV,j(y)|=|Ev∈δB(fV,j(y+v))−Ev∈δB(fV,j(w+v))|\n=|Ev∈δB(fV,j(x+v)−fV,j(y+v))|\n≤Ev∈δB|(fV,j(x+v)−fV,j(y+v))|\n≤5|x−y|.\nThird, for convexity, by the convexity of ffor everyx,y∈Rdandα∈[0,1],\n˜fV,j(αx+ (1 −α)y) =Ev∈δB(fV,j(αx+ (1 −α)y+v))\n=Ev∈δB(fV,j(α(x+v) + (1 −α)(y+v)))\n≤Ev∈δB(αfV,j(x+v) + (1 −α)fV,j(y+v)))\n=αEv∈δB(fV,j(x+v)) + (1 −α) (Ev∈δBfV,j(y+v))\n=α˜fV,j(x) + (1 −α)˜fV,j(y).\n/square\nLemma 27. (Lemma 1 in Flaxman et al. (2005)) Let dandδ >0,Bbe thed-dimensional unit\nball and Sbe thed-dimensional unit sphere. Moreover, let DBandDSbe the uniform distributions\nonB,Srespectively. If ˜f(x) =Ev∼DB[f(x+δv)], then,\n∇˜f(x) =d\nδEa∼DS[f(x+δa)a]\nLemma 28. (e.g., Muller (1959)) Let d. LetSbe thed-dimensional unit sphere and DSthe\nuniform distributions on S. Moreover, we define random variables Y1,...,Yd∈R,X1,...,Xd∈R\nandY∈Rdsuch thatXi∼N(0,1)(whereN(0,1)is the normal univariate distribution with\nexpectation 0 and variance 1),Yi=xi/radicalig/summationtextd\ni=1X2\niandY= (Y1,...,Yd). Then,Y∼ DS.\nLemma 29. Letd. LetBbe thed-dimensional unit ball and DBthe uniform distributions on\nB. Letζ1> ζ 2>0, a function g:R→Randa1,...al∈B. Moreover, let h:B→R,\nh(x) =g(max(ζ1,max 1≤r≤l/a\\}b∇acketle{tar,x/a\\}b∇acket∇i}ht)andx0∈Bsuch that max 1≤r≤l/a\\}b∇acketle{tar,x0/a\\}b∇acket∇i}ht ≤ζ2. We define\n˜h(x):=Ev∼DB[h(x+δv)]. Then, for any 0<δ<ζ 1−ζ2,\n∇˜h(x0) = 0,\n˜h(x0) =g(ζ1).\nProof. First, for every randv∈B, by Cauchy-Schwartz Inequality,\n/a\\}b∇acketle{tar,x0+δv/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tar,x0/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tar,δv/a\\}b∇acket∇i}ht ≤ζ2+δ<ζ 1\nThen,\nmax(ζ1,max\n1≤r≤l/a\\}b∇acketle{tar,x0+δv/a\\}b∇acket∇i}ht) =ζ1,\nand\nh(x0+δv) =g(max(ζ1,max\n1≤r≤l/a\\}b∇acketle{tar,x0+δv/a\\}b∇acket∇i}ht) =g(ζ1).\n54As a result,\n˜h(x0) =Ev∼DB[h(x0+δv)] =g(ζ1)\nand by Lemma 27,\n∇˜h(x0) =d\nδEv∼DS[h(x0+δv)v]\n=d\nδEv∼DS/bracketleftbigg\ng/parenleftbigg\nmax(ζ1,max\n1≤r≤l/a\\}b∇acketle{tar,x0+δv/a\\}b∇acket∇i}ht/parenrightbigg\nv/bracketrightbigg\n=d\nδEv∼DS[g(ζ1)v]\n=d\nδg(ζ1)Ev∼DS[v]\n= 0\n/square\nLemma 30. LetdandK. LetBbe thedK-dimensional unit ball and DBthe uniform distributions\nonB. Letζ1>ζ2>0anda1,...al∈Bd. Moreover, let g:Bd→R,g(x) = max(ζ1,max 1≤r≤l/a\\}b∇acketle{tar,x/a\\}b∇acket∇i}ht)\nandh:B→R,h(x) =/radicalig/summationtextK\nk=1g(x(k))2. Letx0∈Bsuch that for every k,max 1≤r≤l/a\\}b∇acketle{tar,x0(k)/a\\}b∇acket∇i}ht ≤ζ2.\nWe define ˜h(x):=Ev∼DB[h(x+δv)]. Then, for any 0<δ<ζ 1−ζ2,\n∇˜h(x0) = 0,\n˜h(x0) =ζ1√\nK.\nProof. First, for every k,randu∈Bd, by Cauchy-Schwartz Inequality,\n/a\\}b∇acketle{tar,x0(k)+δu/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tar,x0(k)/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tar,δu/a\\}b∇acket∇i}ht ≤ζ2+δ<ζ 1\nThen,\ng(x0(k)+δu) = max(ζ1,max\n1≤r≤l/a\\}b∇acketle{tar,x0(k)+δu/a\\}b∇acket∇i}ht) =ζ1,\nand for every v∈B,\nh(x0+δv) =/radicaltp/radicalvertex/radicalvertex/radicalbtK/summationdisplay\nk=1g(max(ζ1,max\n1≤r≤l/a\\}b∇acketle{tar,x0(k)+δv(k)/a\\}b∇acket∇i}ht) =ζ1√\nK.\nAs a result,\n˜h(x0) =Ev∼DB[h(x0+δv)] =ζ1√\nK.\nNow, by Lemma 27,\n∇˜h(x0) =d\nδEv∼DS[h(x0+δv)v]\n=d\nδEv∼DS\n\n/radicaltp/radicalvertex/radicalvertex/radicalbtK/summationdisplay\nk=1/parenleftbigg\nmax(ζ1,max\n1≤r≤l/a\\}b∇acketle{tar,x0(k)+δv(k)/a\\}b∇acket∇i}ht/parenrightbigg2\n·v\n\n=d\nδEv∼DS/bracketleftig\nζ1√\nKv/bracketrightig\n55=d\nδζ1√\nKEv∼DS[v]\n= 0.\n/square\nLemma 31. Letd. LetBbe thed-dimensional unit ball and DBthe uniform distributions on\nB. Letζ1> ζ 2,ζ3>0and vectors a1,...al∈Bd\nG(0). Moreover, let h:B→R,h(x) =\nmax (ζ3,max 1≤r≤l/a\\}b∇acketle{tar,x/a\\}b∇acket∇i}ht)andx0∈B,r0∈[l]such that /a\\}b∇acketle{tar0,x0/a\\}b∇acket∇i}ht=ζ1andmax 1≤r≤l,r/negationslash=r0/a\\}b∇acketle{tar,x0/a\\}b∇acket∇i}ht ≤\nζ2. We define ˜h(x):=Ev∼DB[h(x+δv)]. Then, for any 0<δ<1\n2G(ζ1−max (ζ2,ζ3)),\n˜h(x0) =/a\\}b∇acketle{tar0,x0/a\\}b∇acket∇i}ht\n∇˜h(x0) =ar0\nProof. First, by Cauchy-Schwartz Inequality,\nmax/parenleftbigg\nζ3,max\nr/negationslash=r0/a\\}b∇acketle{tar,x0+δv/a\\}b∇acket∇i}ht/parenrightbigg\n≤max/parenleftbigg\nζ3,max\nr/negationslash=r0/a\\}b∇acketle{tar,x0/a\\}b∇acket∇i}ht+ max\nr/negationslash=r0/a\\}b∇acketle{tδv,ar/a\\}b∇acket∇i}ht/parenrightbigg\n≤max (ζ3,ζ2+Gδ)\n≤max (ζ3,ζ2) +Gδ\n<1\n2(ζ1+ max (ζ3,ζ2)),\nand forr0,\n/a\\}b∇acketle{tar0,x0+δv/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tar0,x0/a\\}b∇acket∇i}ht+/a\\}b∇acketle{tδv,ar0/a\\}b∇acket∇i}ht ≥ζ1−Gδ>1\n2(ζ1+ max (ζ2,ζ3)).\nWe derive that for every v∈B,\nh(x0+δv) = max/parenleftbigg\nζ3,max\n1≤r≤l/a\\}b∇acketle{tar,x+δv/a\\}b∇acket∇i}ht/parenrightbigg\n) =/a\\}b∇acketle{tar0,x+δv/a\\}b∇acket∇i}ht.\nand that the maximum is attained in r0. Then,\n˜h(x0) =Ev∼DB[h(x0+δv)]\n=Ev∼DB[/a\\}b∇acketle{tar0,x0+δv/a\\}b∇acket∇i}ht]\n=/a\\}b∇acketle{tar0,x0+δEv∼DBv/a\\}b∇acket∇i}ht\n=/a\\}b∇acketle{tar0,x0/a\\}b∇acket∇i}ht\nand by Lemma 27,\n∇˜h(x0) =d\nδEv∼DS/bracketleftbigg\nmax/parenleftbigg\nζ3,/parenleftbigg\nmax\n1≤r≤l/a\\}b∇acketle{tar,x0+δv/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg\nv/bracketrightbigg\n=d\nδEv∼DS[(/a\\}b∇acketle{tar0,x0+δv/a\\}b∇acket∇i}ht)v]\n=/a\\}b∇acketle{tar0,x0/a\\}b∇acket∇i}htd\nδEv∼DS[v] +d\nδEv∼DS[/a\\}b∇acketle{tar0,δv/a\\}b∇acket∇i}htv]\n= 0 +daT\nr0Ev∼DS/bracketleftig\nvvT/bracketrightig\n=daT\nr0Ev∼DS/bracketleftig\nvvT/bracketrightig\n.\n56Now, we define random variables Y1,...,Yd∈R,X1,...,Xd∈RandY∈Rdsuch thatXi∼\nN(0,1) (whereN(0,1) is the normal univariate distribution with expectation 0 and variance 1),\nYi=xi/radicalig/summationtextd\ni=1X2\ni. By Lemma 28, we get, for the standard basis vectors e1...ed,\n∇˜h(x) =daT\nr0EY1,...,Y d/bracketleftiggd/summationdisplay\ni=1Y2\nieieT\ni/bracketrightigg\n=daT\nr0d/summationdisplay\ni=1EYi/bracketleftig\nY2\ni/bracketrightig\neieT\ni\n=daT\nr0d/summationdisplay\ni=1EXi/bracketleftigg\nX2\ni/summationtextd\nl=1X2\nl/bracketrightigg\neieT\ni\n=daT\nr0d/summationdisplay\ni=11\ndeieT\ni\n=ar0\n/square\nProof of Lemma 14. We assume that E(Eq. (12)) holds and show Lemma 14 under this event. We\nprove the claim by induction on t. Fort= 1, it is trivial. Now, we assume that wt= ˜wt.\nForℓ1, in everyt, by the proofs of Lemmas 18 to 21 and Lemma 4, it can be observed that for\neveryi∈[n],k≥2, maxtmaxu∈Vi/a\\}b∇acketle{tu,wt(k)/a\\}b∇acket∇i}ht ≤η\n16, thus, in every iteration the term that gets the\nmaximal value is3η\n32. Then, by Lemma 30 and the hypothesis of the induction, for ev eryi,\n∇˜ℓ1( ˜wt,Vi) =∇˜ℓ1(wt,Vi) = 0 = ∇ℓ1(wt,Vi).\nFor˜ℓ2and everyw∈Rd,V⊆Uandj∈[n2], by linearity of expectation,\n˜ℓ2(w,(V,j)) =Ev∈δB/parenleftig\n/a\\}b∇acketle{tw(0)+v(0),−φ(V,j)/a\\}b∇acket∇i}ht/parenrightig\n=ℓ2(w,(V,j)) +/parenleftig\n/a\\}b∇acketle{tEv∈δBv(0),−φ(V,j)/a\\}b∇acket∇i}ht/parenrightig\n=ℓ2(w,(V,j))\nThen, we derive that for every wandi,∇˜ℓ2(w,(Vi,ji)) = ∇ℓ2(w,(Vi,ji)).\nFor˜ℓ3, which is a 2-Lipschitz linear function, for t= 1, by the proof of Lemma 18, the term\nthat gets the maximal value is δ1. Moreover, it can be observed that for such t, and every ψ∈Ψ,\nmax\nψ∈Ψ/parenleftbigg\n/a\\}b∇acketle{tw(0)\n1,ψ/a\\}b∇acket∇i}ht −1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ),w1(1)/a\\}b∇acket∇i}ht/parenrightbigg\n= 0.\nThen, we can apply Lemma 29, and get by the hypothesis of the in duction,\n∇˜ℓ3( ˜w1) =∇˜ℓ3(w1) = 0 = ∇ℓ3(w1).\nIft≥2, it can be observed that\nℓ3(wt) = max/parenleftbigg\nδ2,max\nψ∈Ψ/parenleftbigg\n/a\\}b∇acketle{tw(0)\nt,ψ/a\\}b∇acket∇i}ht −1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ),wt(1)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightbigg\n= max\nψ∈Ψ/parenleftbigg\n/a\\}b∇acketle{tw(0)\nt,ψ/a\\}b∇acket∇i}ht −1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ),wt(1)/a\\}b∇acket∇i}ht/parenrightbigg\n57Then, by the proofs of Lemmas 19 to 21 and Lemma 4, the maximal v alue of /a\\}b∇acketle{tw(0),ψ/a\\}b∇acket∇i}ht−1\n4ǫ\nT2/a\\}b∇acketle{tα(ψ),w(1)/a\\}b∇acket∇i}ht\nis attained in ψ=ψ∗and the difference from the second maximal possible value of t his term is at\nmostǫ\n2T2. As a result, using the fact that this maximum is also larger t hanδ2by at leastη\n8n(which\nis also larger than δ), we can apply Lemma 31 and get by the hypothesis of the induct ion that,\n∇˜ℓ3( ˜wt)(k)=∇˜ℓ3(wt)(k)\n=\n\n1\nn/summationtextn\ni=1φ(Vi,ji) k= 0\n−ǫ\n4T2α(1\nn/summationtextn\ni=1φ(Vi,ji))k= 1\n0 otherwise\n=∇ℓ3(wt)(k).\nFor˜ℓ4, fort∈ {1,2}, by the proofs of Lemmas 18 and 19, the term that gets the maxim al value\nisδ2. Moreover, it can be observed that for every such t, and every k∈[T−1] andu∈U,\n3\n8/a\\}b∇acketle{tu,wt(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,wt(k+1)/a\\}b∇acket∇i}ht= 0.\nThen, we can apply Lemma 29, and get by the hypothesis of the in duction,\n∇˜ℓ4( ˜wt) =∇˜ℓ4(wt) = 0 = ∇ℓ4(wt).\nFort= 3, it can be observed by the proof of Lemma 20 that,\nℓ4(wt) = max/parenleftigg\nδ2,max\nk∈[T−1],u∈U/parenleftbigg3\n8/a\\}b∇acketle{tu,wt(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,wt(k+1)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg\n= max\nk∈[T−1],u∈U/parenleftbigg3\n8/a\\}b∇acketle{tu,wt(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,wt(k+1)/a\\}b∇acket∇i}ht/parenrightbigg\nMoreover, the maximal value is3ηǫ\n32T2and is attained in k0= 1,u=u0=α(ψ∗). The second\nmaximal possible value of this term is δ2=3ηǫ\n64T2, then, by the fact that δ <3ηǫ\n32T2−3ηǫ\n64T2=3ηǫ\n64T2,\nwe can apply Lemma 31 and get by the hypothesis of the inductio n that\n∇˜ℓ4( ˜wt)(k)=∇˜ℓ4(wt)(k)\n=\n\n3\n8u0k= 1\n−1\n2u0k= 2\n0 otherwise\n=∇ℓ4(wt)(k).\nFort≥4, it can be observed by the proofs of Lemmas 4 and 21 that,\nℓ4(wt) = max/parenleftigg\nδ2,max\nk∈[T−1],u∈U/parenleftbigg3\n8/a\\}b∇acketle{tu,wt(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,wt(k+1)/a\\}b∇acket∇i}ht/parenrightbigg/parenrightigg\n= max\nk∈[T−1],u∈U/parenleftbigg3\n8/a\\}b∇acketle{tu,wt(k)/a\\}b∇acket∇i}ht −1\n2/a\\}b∇acketle{tu,wt(k+1)/a\\}b∇acket∇i}ht/parenrightbigg\nMoreover, the maximal value is3η\n16and is attained in k0=t−2,u=u0=α(ψ∗). The second\nmaximal possible value of this term is smaller then5η\n64, then we can apply again Lemma 31 and get\nby the hypothesis of the induction that\n∇˜ℓ4( ˜wt)(k)=∇˜ℓ4(wt)(k)\n58=\n\n3\n8u0k=t−2\n−1\n2u0k=t−1\n0 otherwise\n=∇ℓ4(wt)(k).\nIn conclusion, we proved that ∇/hatwideF(wt) =∇/hatwide˜F( ˜wt), thus, by the hypothesis of the induction,\nwt+1=wt− ∇/hatwideF(wt) = ˜wt− ∇/hatwide˜F( ˜wt) = ˜wt+1\n/square\nLemma 32. Letdandδ>0. Letf:Rd→Rbe aG-Lipschitz function. Let Bbe thed-dimensional\nunit ball. Moreover, let DBbe the uniform distributions on B. If ˜f(x) =Ev∼DB[f(x+δv)], then\nfor everyx,\n|˜f(x)−f(x)| ≤Gδ\nProof. By the fact that fisG-Lipschitz,\n|˜f(x)−f(x)|=|Ev∼DB[f(x+δv)]−f(x)|\n≤ |Ev∼DB[f(x)] +GδEv∼DB+ [/ba∇dblv/ba∇dbl]−f(x)|\n=GδEv∼DB[/ba∇dblv/ba∇dbl]\n≤Gδ\n/square\nD.2 Proofs of Appendix A.2\nProof of Lemma 15. First, differentiability can be derived immediately from Le mma 27. Second,\nfor 4-Lipschitzness, for every V∈Z, we define ˜fSGD\nV:Rd→Ras˜fSGD\nV(w):=˜fSGD(w,V). By the\n5-Lipschitzness of fSGDwith respect to its first argument and Jensen Inequality, for everyx,y∈Rd,\nit holds that\n|˜fSGD\nV(x)−˜fSGD\nV(y)|=/vextendsingle/vextendsingle/vextendsingleEv∈δB/parenleftig\nfSGD\nV(y+v)/parenrightig\n−Ev∈δB/parenleftig\nfSGD\nV(w+v)/parenrightig/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingleEv∈δB/parenleftig\nfSGD\nV(x+v)−fSGD\nV(y+v)/parenrightig/vextendsingle/vextendsingle/vextendsingle\n≤Ev∈δB/vextendsingle/vextendsingle/vextendsingle/parenleftig\nfSGD\nV(x+v)−fSGD\nV(y+v)/parenrightig/vextendsingle/vextendsingle/vextendsingle\n≤4|x−y|.\nThird, for convexity, by the convexity of fSGDfor everyx,y∈Rdandα∈[0,1],\n˜fSGD\nV(αx+ (1 −α)y) =Ev∈δB/parenleftig\nfSGD\nV(αx+ (1 −α)y+v)/parenrightig\n=Ev∈δB/parenleftig\nfSGD\nV(α(x+v) + (1 −α)(y+v))/parenrightig\n≤Ev∈δB/parenleftig\nαfSGD\nV(x+v) + (1 −α)fV(y+v))/parenrightig\n=αEv∈δB/parenleftig\nfSGD\nV(x+v)/parenrightig\n+ (1 −α)/parenleftig\nEv∈δBfSGD\nV(y+v)/parenrightig\n=α˜fSGD\nV(x) + (1 −α)˜fSGD\nV,j(y).\n/square\n59Proof of Lemma 16. We assume that E′(Eq. (15)) holds and prove Lemma 16 under this event.\nWe prove the claim by induction on t. Fort= 1, it is trivial. Now, we assume that wt= ˜wt. First,\nfor˜ℓSGD\n3and everywandV, by linearity of expectation,\n˜ℓSGD\n3(w,V) =Ev∈δB/parenleftbigg\n/a\\}b∇acketle{tw(0,1)+v(0,1),−1\n4n2φ(V,1)/a\\}b∇acket∇i}ht − /a\\}b∇acketle{t1\nn3u1,w(1)+v(1)/a\\}b∇acket∇i}ht/parenrightbigg\n=ℓSGD\n3(w,V) +/parenleftbigg\n/a\\}b∇acketle{tEv∈δBv(0,1),−1\n4n2φ(V,1)/a\\}b∇acket∇i}ht − /a\\}b∇acketle{t1\nn3u1,Ev∈δBv(1)/a\\}b∇acket∇i}ht/parenrightbigg\n=ℓSGD\n3(w,V)\nThen, we derive that for every w,∇˜ℓSGD\n3(w,V) =∇ℓSGD\n3(w,V). Now, for r∈ {1,2}we show that\nin each term ˜ℓSGD\nr(wt,Vt), the argument that gives the maximum value is the same as ℓSGD\nr(wt,Vt).\nFor˜ℓSGD\n1(wt,Vt), in everyt, by the proofs of Lemma 24, Lemma 25 and Lemma 10, the maximal\nvalue is3η\n32. Moreover, it can be observed that for every k≥2, maxtmaxu∈Vt/a\\}b∇acketle{tu,wSGD\nt(k)/a\\}b∇acket∇i}ht ≤η\n16.\nThen, by Lemma 30, and the hypothesis of the induction,\n∇˜ℓSGD\n1( ˜wt,Vt) =∇˜ℓSGD\n1(wt,Vt) = 0 = ∇ℓSGD\n1(wt,Vt).\nNow, for ˜ℓSGD\n2, fort= 1, ∇ℓSGD\n2(w1,V1) = 0 and the maximum is attained uniquely in δ1=η\n8n3\n(the second maximal value is zero). Then, we can apply Lemma 2 9 and by the hypothesis of the\ninduction, it follows that,\n∇˜ℓSGD\n2( ˜w1,V1) =∇˜ℓSGD\n2(w1,V1) = 0 = ∇ℓSGD\n2(w1,V1).\nFor everyt≥2, the maximum is attained uniquely in the linear term of k=t−1,u=ut−1and\nψ=ψ∗\nt−1such that the difference between the maximal value the second largest value is larger\nthanηǫ\n16n2. Then, we can apply Lemma 31 and by the hypothesis of the induc tion, it follows that,\n∇˜ℓSGD\n2( ˜wt,Vt) =∇˜ℓSGD\n2(wt,Vt) =∇ℓSGD\nt(wt,Vt).\nIn conclusion, we proved that ∇˜fSGD( ˜wt,Vt) = ∇fSGD(wt,Vt), thus, by the hypothesis of the\ninduction,\nwt+1=wt− ∇fSGD(wt,Vt) = ˜wt− ∇˜fSGD( ˜wt,Vt) = ˜wt+1.\n/square\nE Lower bound of Ω/parenleftbigg\nmin/parenleftbigg\n1,1\nηT/parenrightbigg/parenrightbigg\nIn this section, we prove the Ω/parenleftig\nmin/parenleftig\n1,1\nηT/parenrightig/parenrightig\nlower bound. Since our hard construction for getting\nthis bound involves a deterministic loss function, GD is equ ivalent to SGD. For clarity, we refer\nin our proof to the performance of GD, however, the same resul t is applicable also for SGD with\nT=niterations.\nE.1 Construction of a non-differentiable loss function.\nFord= max(25η2T2,1), we define the hard loss function fOPT:Rd→R, as follows,\nfOPT(w) = max/parenleftigg\n0,max\ni∈[d]{1√\nd−w[i]−ηi\n4d}/parenrightigg\n. (19)\nFor this loss function, we prove the following lemma,\n60Lemma 33. Assumen,T > 0,η≤1\n5√\nT. Consider the loss function fOPTthat is defined in Eq. (19)\nford= max(25η2T2,1). Then, for Unprojected GD (cf. Eq. (1)withW=Rd) onfOPT, initialized\natw1= 0with step size η, we have,\n(i) The iterates of GD remain within the unit ball, namely wt∈Bdfor allt= 1,...,T ;\n(ii) For all m= 1,...,T , them-suffix averaged iterate has:\nfOPT(wT,m)−fOPT(w∗) = Ω/parenleftigg\nmin/parenleftbigg\n1,1\nηT/parenrightbigg/parenrightigg\n.\nAlgorithm’s dynamics We start by proving a lemma that characterizes the dynamics o f the\nalgorithm.\nLemma 34. Assume the conditions of Lemma 33, and consider the iterate o f Unprojected GD on\nfOPT, initialized at w1= 0with step size η≤1\n5√\nTLetwtbe the iterate of Then, it holds that,\n(i) For every i∈[d]and for every t∈[T],\nwt[i]≤1\n2√\nd\n(ii) For every t∈[T], there exists an index jt∈[d]such thatk/\\e}atio\\slash=jt,\n1√\nd−wt[jt]−ηj\n4d>1√\nd−wt[k]−ηk\n4d+η\n8d.\n(iii) For every t∈[T],jtalso holds\n1√\nd−wt[jt]−ηjt\n4d>η\n8d.\nProof. We prove by induction on t. Fort= 1,wt= 0, thus,\nw1[i] = 0 ≤1\n2√\nd.\nMoreover, the maximizer is j1= 1. Then, we notice that for both d= 1 andd= 25η2T2,\nη≤1\n5√\nT=⇒η≤1\n5√\nd. Then, it holds that,\n1√\nd−w1[j1]−ηj1\n4d≥1√\nd−w1[j1]−η\n4\n≥19\n20√\nd\n>η\n8d,\nand, for every k/\\e}atio\\slash=j1,\n1√\nd−w1[j1]−ηj1\n4d=1√\nd−w1[k]−ηk\n4d+η(k−j1)\n4d\n61≥1√\nd−w1[k]−ηk\n4d+η\n4d\n>1√\nd−w1[k]−ηk\n4d+η\n8d.\nIn the step of the induction we assume that the lemma holds for everys≤tand prove it for\ns=t+ 1. By the hypothesis of the induction, we know that for every iterations≤t,/ba∇dblwt/ba∇dbl2≤1\n2,\nas a result, we know that the projections does not affect the dy namics of the algorithm until the\niterationt. Moreover, we know that for every iteration s≤tthere exists an index js∈[d] such\nthat the term that achieve the maximum value in wsis1√\nd−ws[js]−ηj\n4d. This maximum is attained\nuniquely in jsby margin that is strictly larger thanη\n8d. As a result, we derive that, for every s≤t,\n∇f(ws) =−ejs. Now, for every index m∈[d], we define,\nnm\nt=|{s≤t:m= arg max\ni∈[d]{1√\nd−ws[i]−ηi\n4d}}|.\nWe get that, for every iit holds that,\nwt+1[i] =ηni\nt.\nThen,\n/ba∇dblwt+1/ba∇dbl1=/summationdisplay\niηni\nt≤ηt,\nand ,thus, there exists a entry k∈[d] withwt+1[k]≤ηt\nd. Now, we prove the first part of the lemma\nusing this observation and the step of the induction. For eve ryi/\\e}atio\\slash=jt,\nwt+1[i] =wt[i]≤1\n2√\nd.\nOtherwise, we know that, by the definition of jt\n1√\nd−wt[i]−ηi\n4d>1√\nd−wt[k]−ηk\n4d+η\n8d,\nwt[i]<wt[k] +η(k−i)\n4d−η\n8d\n≤ηt\nd+η\n4\n≤1\n25√\nd+1\n20√\nd\nand,\nwt+1[i]≤wt[i] +η\n≤1\n25√\nd+1\n20√\nd+1\n5√\nd\n≤1\n2√\nd,\nwhere we again used the fact that η≤1\n5√\nTimpliesη≤1\n5√\ndfor bothd= 1 andd= 25η2T2.\n62For the second part of the lemma, we define Jt⊆[d],Jt= arg minj{nj\nt}andjt+1= min {j∈Jt}\nand show that jt+1holds the required. We know, for every j/\\e}atio\\slash=i∈[d],\nwt+1[i]−wt+1[j] =η(ni\nt−nj\nt).\nFork/\\e}atio\\slash=jt+1withnk\nt>njt+1\nt,\n1√\nd−wt+1[jt+1]−ηjt+1\n4d≤1√\nd−wt+1[k]−η−ηjt+1\n4d\n=1√\nd−wt+1[k]−η−ηk\n4d+η(k−jt+1)\n4d\n≤1√\nd−wt+1[k]−η−ηk\n4d+η\n4\n<1√\nd−wt+1[k]−ηk\n4d−η\n2.\nin contradiction to the fact that jt+1gets the maximal value. For k/\\e}atio\\slash=jt+1withnk\nt>njt+1\nt, it holds\nthatwt+1[jt+1]≤wt+1[k]−η, and,\n1√\nd−wt+1[jt+1]−ηjt+1\n4d≥1√\nd−wt+1[k] +η−ηjt+1\n4d\n=1√\nd−wt+1[k] +η−ηk\n4d+η(k−jt+1)\n4d\n≥1√\nd−wt+1[k] +η−ηk\n4d−η\n4\n>1√\nd−wt+1[k]−ηk\n4d+η\n8d,\nas required. For the third part of the lemma, we know that for e veryi∈[d],\n1√\nd−wt+1[i]−ηi\n4d≥1\n2√\nd−η\n4\n=9\n20√\nd\n>η\n8d.\n/square\nProof of lower bound. Now we can prove Lemma 33.\nProof of Lemma 33. The first part of the theorem is an immediate corollary from Le mma 34.\nMoreover, by applying this lemma again, we know that, for eve ryi∈[d],\nwT,m[i]≤1\n2√\nd,\nthus,\nfOPT(wT,m)−fOPT(w∗)≥1\n2√\nd−η\n4−0\n63≥1\n2√\nd−η\n20√\nd\n>1\n4√\nd\n= min/parenleftbigg1\n4,1\n20ηT/parenrightbigg\n.\n/square\nE.2 Construction of a differentiable loss function.\nIn this section, we prove the lower bound for a smoothing of fOPT, defined as\n˜fOPT(w) =Ev∈Bdmax/parenleftigg\n0,max\ni∈[d]{1√\nd−w[i]−δv[i]−ηi\n4d}/parenrightigg\n, (20)\nnamely, we prove the following lemma,\nLemma 35. Assumen,T > 0,η≤1\n5√\nT. Consider the loss function ˜fOPTthat is defined in Eq. (20)\nford= max(25η2T2,1)andδ=η\n16d. Then, for Unprojected GD (cf. Eq. (1)withW=Rd) on\nfOPT, initialized at w1= 0with step size η, we have,\n(i) The iterates of GD remain within the unit ball, namely wt∈Bdfor allt= 1,...,T ;\n(ii) For all m= 1,...,T , them-suffix averaged iterate has:\n˜fOPT(wT,m)−˜fOPT(w∗) = Ω/parenleftigg\nmin/parenleftbigg\n1,1\nηT/parenrightbigg/parenrightigg\n.\nFirst, we prove that the smoothing of the loss function does n ot affect the dynamics of the\nalgorithm, as stated in the following lemma,\nLemma 36. Under the conditions of Lemmas 33 and 35, let wt,˜wtbe the iterates of Unprojected\nGD with step size η≤1\n5√\nTandw1= 0, onfOPTand ˜fOPTrespectively. Then, for every t∈[T],\nit holds that wt= ˜wt.\nProof. We proof the lemma by induction on t. Fort= 1, we know that w1= ˜w1= 0. Now, we\nassume that wt= ˜wt. By Lemma 34, we know that the maximum of the loss function is a ttained\nuniquely with the property that the difference between the ma ximal value and the second maximal\nvalue is larger thenη\n8d. As a result, by the facts that fis 1-Lipschitz and δ=η\n16d, we can use\nLemma 31 for/hatwideF(wt) and get that,\n∇/hatwideF(wt) =∇/hatwide˜F(wt) =∇/hatwide˜F( ˜wt).\nIt follows by the hypothesis of the induction that,\nwt+1=wt− ∇/hatwideF(wt) = ˜wt− ∇/hatwide˜F( ˜wt) = ˜wt+1.\n/square\nNow we can prove Lemma 35.\n64Proof of Lemma 35. LetwT,mbe them-suffix average of GD when is applied on fOPT. Let\nw∗= arg minwfOPT(w). By Lemma 36, we know that, wT,m=wT,m. Then, by Lemma 33 and\nLemma 32,\n1\n4√\nd≤fOPT(wT,m)−fOPT(w∗)\n=fOPT(wT,m)−fOPT(w∗)\n≤˜fOPT(wT,m) +δ−˜fOPT(w∗) +δ\n≤˜fOPT(wT,m) +δ−˜fOPT(w∗) +δ,\nand,\n˜fOPT(wT,m)−˜fOPT(w∗)≥1\n4√\nd−η\n8d\n≥1\n4√\nd−1\n8√\nd\n≥1\n8√\nd\n≥min(1\n8,1\n40ηT).\n/square\n65"    },    {        "title": "2401.12091v1.Quantum_Eigensolver_for_General_Matrices.pdf",        "content": "Quantum Eigensolver for General Matrices\nXiao-Ming Zhang,1, 2Yukun Zhang,2Wenhao He,3, 4and Xiao Yuan2,∗\n1School of Physics, South China Normal University, Guangzhou 510006, China\n2Center on Frontiers of Computing Studies, School of Computer Science, Peking University, Beijing 100871, China\n3School of Physics, Peking University, Beijing 100871, China\n4Center for Computational Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\nThe eigenvalue problem, a cornerstone in linear algebra, provides profound insights into studying matrix\nproperties. Quantum algorithms addressing this problem have hitherto been constrained to special normal\nmatrices assuming spectral decomposition, leaving the extension to general matrices an open challenge. In this\nwork, we present a novel family of quantum algorithms tailored for solving the eigenvalue problem for general\nmatrices, encompassing scenarios with complex eigenvalues or even defective matrices. Our approach begins\nby tackling the task of searching for an eigenvalue without additional constraints. For diagonalizable matrices,\nour algorithm has ˜𝑂(𝜀−1)complexity with an error 𝜀, achieving the nearly Heisenberg scaling. Subsequently,\nwe study the identification of eigenvalues closest to a specified point or line, extending the results for ground\nenergy and energy gap problems in Hermitian matrices. We achieve an accuracy scaling of ˜𝑂(𝜀−2)for general\ndiagonalizable matrices, further refining to ˜𝑂(𝜀−1)under the condition of real eigenvalues or constant distance\nfrom the reference point. The algorithm’s foundation lies in the synergy of three techniques: the relationship\nbetween eigenvalues of matrix 𝐴and the minimum singular value of 𝐴−𝜇𝐼, quantum singular value threshold\nsubroutine extended from quantum singular-value estimation, and problem-specific searching algorithms. Our\nalgorithms find applications in diverse domains, including estimating the relaxation time of Markov chains,\nsolving Liouvillian gaps in open quantum systems, and verifying PT-symmetry broken/unbroken phases. These\napplications underscore the significance of our quantum eigensolvers for problems across various disciplines.\nEigenvalue, a fundamental concept in linear algebra, repre-\nsents scalar factors by which a matrix transformation stretches\nor compresses space. Formally, complex value 𝜆𝑗and nor-\nmalized vector|𝑣𝑗⟩are considered as the eigenvalue and the\ncorresponding eigenvector of matrix 𝐴if\n𝐴|𝑣𝑗⟩=𝜆𝑗|𝑣𝑗⟩. (1)\nThe applications of quantum computing in solving Eq. (1) have\nmostly been restricted to the Hermitian cases 𝐴=𝐴†[1–3],\nwhere𝜆𝑗are real, and|𝑣𝑗⟩forms an orthonormal basis. For\na general matrix, however, we encounter complex and even\ndefective eigenvalues, and eigenvectors are not necessarily or-\nthogonal. While quantum computing naturally favours Her-\nmiticity, its applicability in solving Eq. (1) for non-Hermitian\nmatrices remains an outstanding question.\nRelating to the eigenvalues, one can perform Jordan decom-\nposition of𝐴as\n𝐴=𝑃Λ𝑃−1, (2)\nwhereΛis a matrix in the Jordan canonical form (JCF), whose\ndiagonal elements correspond to the eigenvalues, and 𝑃is an\ninvertible matrix. 𝐴is called diagonalizable ifΛis diagonal.\nNote that a diagonalizable matrix is not necessarily Hermitian\nsince𝑃is generally non-unitary. Most generally, Λis a block-\ndiagonal matrix as close to a diagonal matrix as possible\nΛ=Λ 1⊕Λ2⊕···⊕Λ𝑀with𝑀⩽𝑁. Each Jordan block Λ𝑗\nis in the form ofΛ𝑗=©\n«𝜆𝑗1\n𝜆𝑗...\n...1\n𝜆𝑗ª®®®®®\n¬. (3)\nIf there exists a nontrivial Jordan block with a dimension larger\nthan one, we call 𝐴adefective matrix. In this case, we have\nstrictly𝑀 < 𝑁 and there are less than 𝑁linearly independent\neigenvectors.\nIt is worth noting that eigenvalues are fundamentally differ-\nent from singular values. There is no direct correspondence\nbetween eigenvalues and singular values in general unless the\nmatrix is normal with spectral decomposition. Therefore, the\neigenvalue problem does not trivially fit into the framework of\nquantum singular value transformation [4, 5].\nHere, we propose quantum eigensolver algorithms for gen-\neral matrices. The complexity of the algorithms for solving\nEq. (1) depends on two properties of the matrix 𝐴. The first\none is how defective 𝐴is, which can be formally defined as the\nlargest dimension of the Jordan block 𝑚′\nmax≡max𝑗dim(Λ𝑗).\nThe second one is the condition number of the matrix 𝑃,\ndenoted as𝜅𝑃. Regarding the nonuniqueness of the Jordan\ndecomposition, we define 𝜅𝑃as the minimum value for all\npossible𝑃in the following text. In our discussion, we assume\nthat we have knowledge of the upper bound of both quantities,\ni.e.𝑚maxand𝐾satisfying𝑚max⩾𝑚′\nmaxand𝐾⩾𝜅𝑃.\nEigenvalue Problems. We first summarize the detailed def-\ninitions of the eigenvalue problems. The first problem we\nconsider is to output an estimation of an eigenvalue defined as\nfollows.arXiv:2401.12091v1  [quant-ph]  22 Jan 20242\nRe[\u0000]Im[\u0000]\n<latexit sha1_base64=\"KcfBQytsgMQ+Tm0cfGsMDXIMS/s=\">AAAB6XicdVDLSgMxFL3js9ZX1aWbYBFcDZlSq8uiG5dV7APaoWTSTBuayQxJRihD/8CNC0Xc+kfu/BszfYCKHgg5nHMv994TJIJrg/Gns7K6tr6xWdgqbu/s7u2XDg5bOk4VZU0ai1h1AqKZ4JI1DTeCdRLFSBQI1g7G17nffmBK81jem0nC/IgMJQ85JcZKd72sXypjt4ZzIOxWl6QyJ547+3EZFmj0Sx+9QUzTiElDBdG66+HE+BlRhlPBpsVeqllC6JgMWddSSSKm/Wy26RSdWmWAwljZJw2aqd87MhJpPYkCWxkRM9K/vVz8y+umJrz0My6T1DBJ54PCVCATo/xsNOCKUSMmlhCquN0V0RFRhBobTtGGsLwU/U9aFderuee31XL9ahFHAY7hBM7Agwuoww00oAkUQniEZ3hxxs6T8+q8zUtXnEXPEfyA8/4FsGSNew==</latexit>{<latexit sha1_base64=\"GeLjeJ3GeCC47QfLzsS1vptzGkI=\">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVB71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MHz7WM9Q==</latexit>g(a)<latexit sha1_base64=\"GeLjeJ3GeCC47QfLzsS1vptzGkI=\">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVB71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MHz7WM9Q==</latexit>g<latexit sha1_base64=\"KcfBQytsgMQ+Tm0cfGsMDXIMS/s=\">AAAB6XicdVDLSgMxFL3js9ZX1aWbYBFcDZlSq8uiG5dV7APaoWTSTBuayQxJRihD/8CNC0Xc+kfu/BszfYCKHgg5nHMv994TJIJrg/Gns7K6tr6xWdgqbu/s7u2XDg5bOk4VZU0ai1h1AqKZ4JI1DTeCdRLFSBQI1g7G17nffmBK81jem0nC/IgMJQ85JcZKd72sXypjt4ZzIOxWl6QyJ547+3EZFmj0Sx+9QUzTiElDBdG66+HE+BlRhlPBpsVeqllC6JgMWddSSSKm/Wy26RSdWmWAwljZJw2aqd87MhJpPYkCWxkRM9K/vVz8y+umJrz0My6T1DBJ54PCVCATo/xsNOCKUSMmlhCquN0V0RFRhBobTtGGsLwU/U9aFderuee31XL9ahFHAY7hBM7Agwuoww00oAkUQniEZ3hxxs6T8+q8zUtXnEXPEfyA8/4FsGSNew==</latexit>{Re[\u0000]Im[\u0000]\n<latexit sha1_base64=\"yn72uj8s/DTU/8kMNFSQDJp4LEM=\">AAAB6HicbVDLSgMxFL1TX7W+qi7dBIvgqsyIr2XRjcsW7APaQTLpnTY2kxmSjFCGfoEbF4q49ZPc+Tem7Sy09UDgcM655N4TJIJr47rfTmFldW19o7hZ2tre2d0r7x+0dJwqhk0Wi1h1AqpRcIlNw43ATqKQRoHAdjC6nfrtJ1Sax/LejBP0IzqQPOSMGis16g/lilt1ZyDLxMtJBXLY/FevH7M0QmmYoFp3PTcxfkaV4UzgpNRLNSaUjegAu5ZKGqH2s9miE3JilT4JY2WfNGSm/p7IaKT1OApsMqJmqBe9qfif101NeO1nXCapQcnmH4WpICYm06tJnytkRowtoUxxuythQ6ooM7abki3BWzx5mbTOqt5l9aJxXqnd5HUU4QiO4RQ8uIIa3EEdmsAA4Rle4c15dF6cd+djHi04+cwh/IHz+QOs2Yze</latexit>PRe[\u0000]Im[\u0000]\n<latexit sha1_base64=\"KcfBQytsgMQ+Tm0cfGsMDXIMS/s=\">AAAB6XicdVDLSgMxFL3js9ZX1aWbYBFcDZlSq8uiG5dV7APaoWTSTBuayQxJRihD/8CNC0Xc+kfu/BszfYCKHgg5nHMv994TJIJrg/Gns7K6tr6xWdgqbu/s7u2XDg5bOk4VZU0ai1h1AqKZ4JI1DTeCdRLFSBQI1g7G17nffmBK81jem0nC/IgMJQ85JcZKd72sXypjt4ZzIOxWl6QyJ547+3EZFmj0Sx+9QUzTiElDBdG66+HE+BlRhlPBpsVeqllC6JgMWddSSSKm/Wy26RSdWmWAwljZJw2aqd87MhJpPYkCWxkRM9K/vVz8y+umJrz0My6T1DBJ54PCVCATo/xsNOCKUSMmlhCquN0V0RFRhBobTtGGsLwU/U9aFderuee31XL9ahFHAY7hBM7Agwuoww00oAkUQniEZ3hxxs6T8+q8zUtXnEXPEfyA8/4FsGSNew==</latexit>{<latexit sha1_base64=\"GeLjeJ3GeCC47QfLzsS1vptzGkI=\">AAAB6HicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdQx68ZiAeUCyhNlJbzJmdnaZmRXCki/w4kERr36SN//GSbIHTSxoKKq66e4KEsG1cd1vZ2V1bX1js7BV3N7Z3dsvHRw2dZwqhg0Wi1i1A6pRcIkNw43AdqKQRoHAVjC6m/qtJ1Sax/LBjBP0IzqQPOSMGivVB71S2a24M5Bl4uWkDDlqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTTUmlI3oADuWShqh9rPZoRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjjZ1wmqUHJ5ovCVBATk+nXpM8VMiPGllCmuL2VsCFVlBmbTdGG4C2+vEya5xXvqnJZvyhXb/M4CnAMJ3AGHlxDFe6hBg1ggPAMr/DmPDovzrvzMW9dcfKZI/gD5/MHz7WM9Q==</latexit>g\n<latexit sha1_base64=\"ZulQnuAxajCQqeGX0rA170utLio=\">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKeyKr2PQiwcPCZgHJEuYnXSSMbOzy8ysEJZ8gRcPinj1k7z5N06SPWhiQUNR1U13VxALro3rfju5ldW19Y38ZmFre2d3r7h/0NBRohjWWSQi1QqoRsEl1g03AluxQhoGApvB6HbqN59QaR7JBzOO0Q/pQPI+Z9RYqXbfLZbcsjsDWSZeRkqQodotfnV6EUtClIYJqnXbc2Pjp1QZzgROCp1EY0zZiA6wbamkIWo/nR06ISdW6ZF+pGxJQ2bq74mUhlqPw8B2htQM9aI3Ff/z2onpX/spl3FiULL5on4iiInI9GvS4wqZEWNLKFPc3krYkCrKjM2mYEPwFl9eJo2zsndZvqidlyo3WRx5OIJjOAUPrqACd1CFOjBAeIZXeHMenRfn3fmYt+acbOYQ/sD5/AGmyYza</latexit>L(b)(c)\nFIG. 1: Sketch of (a) the energy gap for Hermitian matrices with reference point 𝑃=0, (b) the point gap for non-Hermitian matrices with\nreference point 𝑃=0, and (c) the line gap for non-Hermitian matrices with reference line 𝐿={𝑖𝑏,𝑏∈R}.\nProblem 1 (Eigenvalue searching) .Given a square matrix 𝐴\nwith∥𝐴∥⩽1, output an estimation of an eigenvalue ˆ𝜆, such\nthatmin|ˆ𝜆−𝜆|⩽𝜀for some error 𝜀, where𝜆is one of the\neigenvalue solutions to Eq. (1).\nHere,∥·∥ refers to the spectral norm of a matrix. Problem 1\nhas no restrictions on the eigenvalue. We may require that\nthe eigenvalue to be estimated have certain properties. Take\nthe Hermitian matrix as an example, there are two important\nquestions related to eigenvalues. The first one is the lowest\neigenvalue problem. For a quantum many-body system de-\nscribed by a Hermitian Hamiltonian, this corresponds to the\nground-state energy of the system [1–3]. The second one\nis the eigenvalue gap problem, which plays a critical role in\nmany-body physics phenomena, such as conductivity and su-\nperconductivity. Extending from Hermitian to non-Hermitian\nmatrices with complex eigenvalues, the generalization of both\nquestions is not unique [6, 7], which correspond to the eigen-\nvalue searching problems under different restrictions. In par-\nticular, we consider the following two problems.\nProblem 2. Given a square matrix ∥𝐴∥⩽1, a reference point\n𝑃∈D( 0,1)and accuracy 𝜀∈(0,1). Let𝑔≡min𝜆𝑗≠𝑃\f\f𝜆𝑗−\n𝑃\f\f,S ≡\b\n𝜆𝑗\f\f|𝑃−𝜆𝑗|∈[𝑔,𝑔+𝜀]\t\n. The goal is to output\nthe gap estimation 𝑔′and eigenvalue estimation 𝜆′, such that\n|𝑔′−𝑔|⩽𝜀and|𝜆′−𝜆𝑗|⩽𝜀for some𝜆𝑗∈S.\nProblem 3. Given a square matrix ∥𝐴∥⩽1, a reference\nline𝐿in the complex plain such that 𝐿ÐD(0,1)≠∅, and\naccuracy𝜀∈ (0,1). Let𝑔=min𝜆𝑗∉𝐿,𝑝∈𝐿|𝜆𝑗−𝑝|,S ≡\b\n𝜆𝑗\f\fmin𝑝∈𝐿|𝜆𝑗−𝑝|∈[𝑔,𝑔+𝜀]\t\n. The goal is to output the\ngap estimation 𝑔′and eigenvalue estimation 𝜆′, such that|𝑔′−\n𝑔|⩽���and|𝜆′−𝜆𝑗|⩽𝜀for some𝜆𝑗∈S.\nHere, we have defined the disk as D(𝜇,𝑟)≡\b\n𝑥\f\f|𝑥−𝜇|⩽𝑟\t\n.\nIn case𝑃or𝐿have no overlap with eigenvalues, Problem 2\nor 3 corresponds to finding an eigenvalue that is closest to the\nreference point 𝑃, or line𝐿(up to an accuracy 𝜀). Therefore,\nthey can be considered as two different ways of the generaliza-\ntion of the ground energy problem for Hermitian matrices. On\nthe other hand, when 𝑃or𝐿overlaps with at least one of theeigenvalues, Problems 2 and 3 become Point gap , orLine gap\nproblems [6–8]. As illustrated in Fig. 1, they can be considered\nas two different ways of the generalizations from the energy\ngap problem of the Hermitian case. We note that the point gap\nand line gap are different. In some cases, both point gaps and\nline gaps are non-vanishing. But there exists matrices with\nnon-vanishing point gap, but zero line gap. Physics systems\nwith Hamiltonian corresponding these two cases may emerge\nfrom different symmetries and topologies [6–8]. In particular,\nthe second case does not have a Hermitian counterpart, the\ncorresponding physics system is also named to be genuinely\nnon-Hermitian [7].\nFurthermore, eigenvalue and eigenvector are related in gen-\neral. We discuss in [9] that an accurate eigenvalue estimation\nimplies a good approximation to the corresponding eigenvec-\ntors. In the remaining of the main text, however, we focus on\nthe eigenvalue problems.\nResults. Here, we introduce our results for the three eigenvalue\nproblems. We only summarize the results in the main text and\nrefer to [9] for details.\nWe first discuss the assumptions of the algorithms. Given a\ngeneral square matrix 𝐴∈C𝑁×𝑁with𝑁=2𝑛, we consider its\nblock encoding that provides unitary access to the matrix. For\na unitary 𝒪𝐴, we say it is a block encoding of 𝐴if it encodes\nthe desirable matrix 𝐴such that𝐴=(⟨0𝑎|⊗𝐼)𝒪𝐴(|0𝑎⟩⊗𝐼),\nwith𝐼the𝑁-dimensional identity. Note that we have neglected\na scaling factor compared to the conventional definition as it\ncan be absorbed in matrix 𝐴. Block-encoding is a standard way\nof encoding the classical description of a matrix to quantum\noperations [4, 5, 10–12]. In practice, 𝒪𝐴may be constructed by\nsparse-access input model or linear combination of unitaries\n(LCU) [10], depending on the form of 𝐴being presented.\nFor Hermitian matrices, the eigenvalue problem is typically\nsolved by assuming the existence of an initial state that can\nbe prepared to have a reasonable lower-bounded overlap with\nthe targeted eigenstate [1, 2]. Otherwise, the problems are in\ngeneral QMA-compete [13]. Here, a similar assumption is\nalso made for general matrices. We introduce an oracle 𝒫𝐴,\nwhich given an input 𝜇satisfying|𝜇|⩽1, outputs a quantum3\nstate|𝜓ini\n𝜇⟩satisfying|⟨𝜓ini\n𝜇|𝑢0(𝜇)⟩|⩾𝛾. Here,|𝑢0(𝜇)⟩is the\nright singular vector of the matrix 𝐴−𝜇𝐼corresponding to its\nsmallest singular value. 𝒫𝐴may be constructed quantumly,\nwith methods like variational quantum algorithms [14–16] or\nadiabatic state preparation [1]. Alternatively, one may find an\napproximated model of 𝐴whose eigenvalue can be calculated\nefficiently on a classical computer.\nHere and after, we assume that 𝒪𝐴,𝒫𝐴, and their inverses\ncan be queried efficiently. For simplicity, we also count the\nquery to controlled 𝒪𝐴or𝒫𝐴as a single query to them.\nFor Problem 1, we have the following result.\nTheorem 1. With success probability at least 1−𝛿, Problem 1\ncan be solved with\n˜O\u0010\n𝐾3𝜀−3𝑚max+2𝛾−1\u0011\n(4)\nuses of the query to 𝒪𝐴,𝒫𝐴and their inverses, and extra\nsingle- and two-qubit gates.\nHere ˜O(·) omits the polylogarithmic dependence on 1 /𝛿, 1/𝜀,\n𝐾, and𝑁. We also clarify that for qubit systems, the extra\nsingle- and two-qubit gate number contains a dependency of\nqubit number 𝑂(𝑛), which is however neglected by ˜𝑂.\nWhen𝑚max=1, i.e.𝐴is diagonalizable, Eq. (4) reduces\nto˜O\u0000𝐾3𝜀−1𝛾−1\u0001, achieving a nearly-optimal dependency on\n𝜀, which is also called the Heisenberg scaling. Besides, the\ndependency on 𝐾can be reduced if we have further restrictions\non𝐴. For example, if the eigenvalue is promised to be real,\nthe complexity can be further reduced to ˜𝑂(𝐾2𝜀−1𝛾−1).\nWe achieve the following result for Problems 2 and 3.\nTheorem 2. Promised that 𝑔⩾𝜀. With success probability\nat least 1−𝛿, Problem 2 and 3 can be solved with\n˜𝑂(𝐾3𝜀−3𝑚max−1𝛾−1) (5)\nqueries to 𝒪𝐴,𝒫𝐴and their inverses, and extra single- and\ntwo-qubit gates.\nFor diagonalizable matrix, the accuracy dependency is\n˜𝑂(𝜀−2). It is open whether the above theorem is optimal\nor not. The complication is that different from Problem 1,\nwe should exclude the possibility for eigenvalues with a\ngap smaller than 𝑔. Besides, if more restrictions exist on\nthe eigenvalue distribution, the query complexity may be\nfurther reduced. For example, if we are promised that the\neigenvalues are real, or in a circle, the complexity can be\nreduced to ˜𝑂(𝐾2𝜀−1𝛾−1), achieving the Heisenberg scaling.\nThe former is consistent with results for Hermitian case [2],\nwhile the latter is consistent with the result of quantum phase\nestimation [17].\nBelow, we briefly introduce the main idea of our algorithms\nachieving Theorem 1 and 2 and refer to [9] for details. Tobegin with, we consider an equivalent form of Eq. (1) as\n(𝐴−𝜆𝑗𝐼)|𝑣𝑗⟩=0. (6)\nOur algorithm is based on the following key observation.\n𝜎0(𝐴−𝜇𝐼)=0if and only if 𝜇is the solution to Eq. (6),\nwhere𝜎0(·)is the minimum singular value of a matrix. Specif-\nically, eigenvalue problems can be transferred to the problem\nof searching for 𝜇, such that𝐴−𝜇𝐼has zero singular values.\nIn practice, however, we can only estimate the singular value\nup to a certain accuracy. To this end, we define a cost function\n𝐶(𝜇)≡𝜎0(𝐴−𝜇𝐼). (7)\nThe distance from 𝜇to an eigenvalue can be bounded by 𝐶(𝜇)\nin the following lemma.\nLemma 1. When𝐴is diagonalizable, we have\n𝐶(𝜇)⩽min\n𝑗|𝜇−𝜆𝑗|⩽𝜅𝑃𝐶(𝜇). (8)\nWhen𝐴is defective, we have\n𝐶(𝜇)⩽min\n𝑗|𝜇−𝜆𝑗|⩽3(𝜅𝑃𝐶(𝜇))1/𝑚max. (9)\nAs we have assumed that 𝜅𝑃⩽𝐾, Lemma. 1 indicates that for\ndiagonalizable matrix, it suffice to find 𝜇satisfying𝐶(𝜇)⩽\n𝜀𝐾−1to achieve accuracy min |𝜇−𝜆𝑗|⩽𝜀, and similar for\ndefective matrices.\nTo solve Problem 1, the remaining task is to search for\na𝜇with small𝐶(𝜇). Our searching method is based on\na subroutine, called the singular value threshold subroutine\n(SVTS). The SVTS gives us the following result.\nLemma 2 (SVTS) .Let𝜇∈ D( 0,1),˜𝜀,𝛿∈ (0,1)be the\ncentre, threshold and success probability of the SVTS. We\ndefine𝑂𝐶(𝜇,˜𝜀,𝛿)as the output of SVTS, which satisfies the\nfollowing\nPr\u0002\n𝑂𝐶(𝜇,˜𝜀,𝛿)=True\f\f𝐶(𝜇)⩽˜𝜀/2\u0003\n⩾1−𝛿 (10a)\nPr\u0002\n𝑂𝐶(𝜇,˜𝜀,𝛿)=False\f\f𝐶(𝜇)⩾˜𝜀\u0003\n⩾1−𝛿 (10b)\nThen, SVTS can be constructed with ˜O\u0000˜𝜀−1𝛾−1\u0001uses of the\nquery to 𝒪𝐴,𝒫𝐴and there inverses, and extra single- and\ntwo-qubit gates.\nIn [9], we prove Lemma. 2 by constructing 𝑆𝑉𝑇𝑆 with quantum\nsingular value transformation [4]. Note that there is a “blur”\nregion𝐶(𝜇)∈( ˜𝜀/2,˜𝜀), in which the output of the oracle is\narbitrary. The runtime of SVTS depends on the range of the\nblur region.\nBased on Lemma. 2, one can find the eigenvalue with brute\nforce search using SVTS, but it requires total 𝑂(1/𝜀2)queries\nto SVTS. We develop a divide-and-conquer method for eigen-\nvalue searching, which uses only 𝑂(polylog(𝜀−1))queries to\nSVTS, and achieve scaling claimed in Theorem 1.4\nProblem 2 and 3 are more challenging. Take Problem 2\nas an example, the complication is that to claim 𝑔′is a good\nestimation of 𝑔with accuracy 𝜀, we should ensure that there\nis no eigenvalue in the region D(𝑃,𝑔′−𝜀)/D(𝑃,𝜀). Our\niterative strategy is as follows. Suppose at the 𝑗th step, we are\nconfidence that 𝑔∈[𝑅min\n𝑗,𝑅max\n𝑗]. LetΔ𝑗=𝑅max\n𝑗−𝑅min\n𝑗, we\nreduceΔ𝑗by querying a set of SVTSs. The process is termi-\nnated untilΔ𝑗⩽𝜀. After that, we search for an eigenvalue\nnear the circle with radius 𝑔′.\nNow we discuss the applications of our results in different\nproblems.\nApplications 1: Relaxation time of Markov chain. Markov\nchains model systems transitioning between states with prob-\nabilities determined solely by their current state, with broad\napplications in both natural and social science [18–22]. A\nfinite Markov chain can be described by the stochastic matrix.\nIn this case, we haveÍ\n𝑖𝐴𝑖,𝑗=1 for each column 𝑗. For every\nfinite Markov chain, the largest eigenvalue is 1. So we can de-\nfine the absolute spectral gap of 𝐴as𝑔mar≡1−max𝜆𝑗≠1|𝜆𝑗|.\nThe relaxation time, defined as 𝑡rel≡1/𝑔mar, is a crucial pa-\nrameter in understanding the properties of the Markov chain.\nIn particular, for irreducible, time-reversible Markov chain,\n𝑡relcan be used to upper bound the mixing time, i.e. the time\nconverging to the stationary distribution [22].\nWith a similar strategy to solving Problem 2 and 3, we can\nestimate the relaxation time with complexity consistent with\nTheorem 2 as follows.\nTheorem 3. Suppose𝐴is a stochastic matrix describing a\nMarkov chain. Promised that the relaxation time 𝑡relis larger\nthan𝜀∈(0,1). Then,𝑡relcan be estimated to accuracy 𝜀with\n˜𝑂(𝐾3𝜀−3𝑚max+1𝛾−1)queries to 𝒪𝐴,𝒫𝐴and their inverses, and\nextra single- and two-qubit gates.\nApplications 2: Liouvillian gap for open quantum systems.\nThe Liouvillian gap (LG) is an important quantity character-\nizing the decaying behaviour and phase transitions of open\nquantum systems [23–28]. The dynamics of an open quantum\nsystem can be described by the Lindblad master equation ¤𝜌=\n−𝑖[𝐻,𝜌]+Í\n𝜇\u0010\n2𝐿𝜇𝜌𝐿†\n𝜇−n\n𝐿†\n𝜇𝐿𝜇,𝜌o\u0011\n≡L(𝜌)for the Hamil-\ntonian𝐻and dissipators 𝐿𝜇. Because the Liouvillian operator\nLis a linear superoperator, we can perform vectorization of\n¤𝜌=L(𝜌)as¤˜𝜌=˜L·˜𝜌, where ˜𝜌=Í\n𝑚,𝑛𝜌𝑚𝑛|𝑚⟩⊗|𝑛⟩and ˜L=\n−𝑖𝐻⊗𝐼+𝑖𝐼⊗𝐻𝑇+Í\n𝜇\u0010\n2𝐿𝜇⊗𝐿∗\n𝜇−𝐿†\n𝜇𝐿𝜇⊗𝐼−𝐼⊗𝐿𝑇\n𝜇𝐿∗\n𝜇\u0011\n.\nLet𝜆𝑗(˜L)be the eigenvalues of ˜Lordered according to the\nmagnitude of the real part, i.e. Re 𝜆0(˜L)⩾Re𝜆1(˜L)⩾···.\nFor a general Lindbladian, there exists at least one steady state\n𝜌nesssatisfying¤˜𝜌ness=˜L˜𝜌ness=0, resulting in 𝜆0=0. The\nLG is formally defined as\n𝑔L≡|Re𝜆1(˜L)|. (11)\n𝑔Lhas a close relation to the relaxation behaviour of the open\nquantum system. In most cases, the relaxation time 𝜏of an\nopen quantum system satisfies 𝜏≲1/𝑔L[26].For many-body systems, analytic solutions to LG only ex-\nist for some special cases, while numerical calculation with\nclassical computers suffers from the exponential increase of\nthe Hilbert space. On the other hand, LG can potentially\nbe solved with a quantum computer efficiently based on our\nquantum eigensolver. One may assume that the vectorized Li-\nouvillian has a Jordan decomposition ˜L=𝑃Λ𝑃−1, where the\ncondition number of satisfies 𝜅𝑝⩽𝐾. We may also assume\nthat ˜Lis diagonalizable. Compared to Problem 3, LG is a line\ngap problem with 𝐿={𝑖𝑏,𝑏∈R}. According to Theorem 2,\nwe have the following result.\nTheorem 4. Promised that 𝑔L⩾𝜀and ˜Lis diagonalizable\nand∥˜L∥⩽1. With success probability at least 1−𝛿,𝑔Lcan\nbe estimated to accuracy 𝜀with\n˜𝑂(𝐾3𝜀−2𝛾−1) (12)\nqueries to 𝒪˜L,𝒫˜Land their inverses, and extra single- and\ntwo-qubit gates.\nIn most open quantum system models, ˜Lcan be decomposed\ninto the linear combination of Pauli strings. Accordingly, the\nblock encoding of ˜L, up to a rescaling factor, can be con-\nstructed by the linear combination of unitaries technique [29].\nApplications 3: PT-broken/unbroken phase classification.\nIn quantum systems described by non-Hermitian operators,\nthe eigenvalue does not necessarily to be complex. A typi-\ncal example is the parity-time ( 𝑃𝑇) symmetry operators [30–\n34]. An operator is called 𝑃𝑇symmetry if it is invariant\nunder simultaneous application of parity-reversal operator P\nand time-reversal operator Trespectively. The eigenvalues of\nthe𝑃𝑇-symmetry operator can either be real only or appear as\ncomplex conjugate pairs. The former possesses 𝑃𝑇-symmetry\nand is therefore categorized as 𝑃𝑇-unbroken phase when the\nmatrix is diagonalizable [34, 35]. In the second case, 𝑃𝑇-\nsymmetry is simultaneously broken and therefore categorized\nas the𝑃𝑇-broken phase. The transition between these two\nphases is of broad interest with applications in quantum sens-\ning [36, 37].\nTo verify whether the quantum system is in the 𝑃𝑇-broken\nor𝑃𝑇-unbroken phase, it suffices to determine if it contains\ncomplex eigenvalues. In practice, we may allow a certain error\n𝜀. When all eigenvalues are at most 𝜀distance away from the\nreal axis, the matrix is categorized as 𝑃𝑇-unbroken. With a\nmild modification of the algorithms for solving Problem 3, one\ncan solve this problem with a similar complexity claimed in\nTheorem 2. More specifically, we have the following result.\nTheorem 5. Given a square, diagonalizable matrix ∥𝐴∥⩽1.\nWith success probability 1−𝛿, one can verify whether 𝐴has\neigenvalues satisfying\f\fIm[𝜆𝑗]\f\f⩾𝜀, or all eigenvalues are in\nthe real axis, with ˜𝑂(𝐾3𝜀−2𝛾−1)queries to 𝒪𝐴,𝒫𝐴and there\ninverses, and extra single- and two-qubit gates.\nBased on Theorem 5, we can characterize the 𝑃𝑇-\nbroken/unbroken phase readily under the diagonalizable as-\nsumption.5\nDiscussions. We have developed quantum algorithms for solv-\ning eigenvalue problems. The idea can also be generalized to\nthe study of the properties related to eigenvectors. Future\nworks include finding more applications in physics, data sci-\nence, and other related fields.\nAcknowledgement. We thank Seth Lloyd, Xiaogang Li and\nDong Yuan for their helpful discussions. 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Dong, L. Lin, and Y. Tong, Ground-state preparation and\nenergy estimation on early fault-tolerant quantum computers via\nquantum eigenvalue transformation of unitary matrices, PRX\nQuantum 3, 040305 (2022).\n[42] G. H. Low and I. L. Chuang, Optimal hamiltonian simulation by\nquantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).7\nSupplemental material (Partial)\nI. BOUND THE ACCURACY OF EIGENVALUE WITH 𝐶(𝜇): PROOF OF LEMMA. 1\n𝐶(𝜇)⩽min|𝜇−𝜆𝑗|follows straightforwardly from Weyl’s Theorem Below, we focus on the upper bound of min |𝜇−𝜆𝑗|. We\nbegin with the diagonalizable matrix. We should proof that min |𝜇−𝜆𝑗|⩽𝜅𝑃𝐶(𝜇). Let∥·∥=𝜎max(·)be the operator norm.\nAccording to definition, the cost function satisfies\n𝐶(𝜇)=0 𝜇=𝜆𝑗, (S-1)\n𝐶(𝜇)=\r\r(𝑀−𝜇𝐼)−1\r\r−1𝜇≠𝜆𝑗. (S-2)\nWhen𝜇=𝜆𝑗, Lemma. 1 holds obviously. We now consider the case when 𝜇≠𝜆𝑗. Because𝑃𝑃−1=𝑃−1𝑃=𝐼, we have\n𝑀−𝜇𝐼=𝑃(Λ−𝜇𝐼)𝑃−1, (S-3)\nand\n(𝑀−𝜇𝐼)−1=𝑃(Λ−𝜇𝐼)−1𝑃−1. (S-4)\nTherefore,\n∥(𝑀−𝜇𝐼)−1∥=∥𝑃(Λ−𝜇𝐼)−1𝑃−1∥\n⩽∥𝑃∥∥(Λ−𝜇𝐼)−1∥∥𝑃−1∥\n⩽∥𝑃∥∥𝑃−1∥∥(Λ−𝜇𝐼)−1∥ (S-5)\nBy definition, we have 𝜅𝑃≡𝜎max(𝑃)/𝜎min(𝑃), where𝜎min(𝑃)is the minimum singular value of 𝑃. Because 1/𝜎min(𝑃)=\n𝜎max(𝑃−1), we have\n𝜅𝑃=𝜎max(𝑃)𝜎max(𝑃−1)=∥𝑃∥∥𝑃−1∥. (S-6)\nMoreover, we have\n∥(Λ−𝜇𝐼)−1∥=1\nmin|𝜇−𝜆𝑗|. (S-7)\nCombining Eq. (S-5), (S-6) and (S-7), we have\n𝐶(𝜇)−1=𝜅𝑃\nmin|𝜇−𝜆𝑗|, (S-8)\nwhich is equivalent to Eq. (8).\nWe then consider the defective matrix case. We first consider the Jordan normal form of the matrix 𝑀−𝜇𝐼. It can be expressed\nas𝑀−𝜇𝐼=𝑃˜Λ𝑃−1, where ˜Λ≡Λ−𝜇𝐼≡˜Λ1⊕˜Λ2⊕···⊕ ˜Λ𝑀is a block-diagonal matrix, where each Jordan block is\n˜Λ𝑗=©\n«𝜆𝑗−𝜇 1\n𝜆𝑗−𝜇...\n... 1\n𝜆𝑗−𝜇ª®®®®®\n¬. (S-9)8\nAccording to Eq. (S-2), we have\n𝐶(𝜇)=\r\r(𝑀−𝜇𝐼)−1\r\r−1\n=\r\r\r𝑃diag\u0010\n˜Λ−1\n1,˜Λ−1\n2,···,˜Λ−1\n𝑁\u0011\n𝑃−1\r\r\r−1\n⩾𝜅−1\n𝑃\r\r\rdiag\u0010\n˜Λ−1\n1,˜Λ−1\n2,···,˜Λ−1\n𝑁\u0011\r\r\r−1\n⩾𝜅−1\n𝑃\r\r\r˜Λ−1\n𝑗\r\r\r−1\n=𝜎min(˜Λ𝑗)\n𝜅𝑃. (S-10)\nNote that Eq. (S-10) is applied for arbitrary 𝑗. According to Ref.[39] (see also Ref. [40]), let 𝛿𝑗=|𝜆𝑗−𝜇|, we have\n𝜎min(˜Λ𝑗)⩾𝛿𝑚𝑗\n𝑗\n(1+𝛿𝑗)𝑚𝑗−1. (S-11)\nBecause the operator norm of 𝐴is bounded by∥𝐴∥⩽1, we also have|𝜆𝑗|⩽1 for all eigenvalues. Our searching region is also\nrestricted by|𝜇|⩽1, so we have 𝛿𝑗⩽2. We can simplify Eq. (S-11) as\n𝜎min(˜Λ𝑗)⩾\u0012𝛿𝑗\n1+𝛿𝑗\u0013𝑚𝑗\n(1+𝛿𝑗)⩾(𝛿𝑗/3)𝑚𝑗. (S-12)\nCombining Eq. (S-10) with Eq. (S-12), we have\n𝜅𝑃𝐶(𝜇)⩾(𝛿𝑗/3)𝑚𝑗, (S-13)\nwhich gives\n𝛿𝑗⩽3(𝜅𝑃𝐶(𝜇))1/𝑚𝑗. (S-14)\nBecause min 𝑗|𝜇−𝜆𝑗|⩽𝛿𝑗, we have\nmin\n𝑗|𝜇−𝜆𝑗|⩽3(𝜅𝑃𝐶(𝜇))1/𝑚𝑗. (S-15)\nWhen𝐶(𝜇)⩽1/𝜅𝑃, we have𝜅𝑃𝐶(𝜇)⩽1, and the right hand side of Eq. (S-15) increases monotonically with 𝑚𝑗. So\nmin𝑗|𝜇−𝜆𝑗|⩽3(𝜅𝑃𝐶(𝜇))1/𝑚max. When𝐶(𝜇)>1/𝜅𝑃, we have𝜅𝑃𝐶(𝜇)>1, so the right hand side of Eq. (S-15) is larger than\n3. Because we always have min 𝑗|𝜇−𝜆𝑗|⩽2, so we also have min 𝑗|𝜇−𝜆𝑗|⩽3(𝜅𝑃𝐶(𝜇))1/𝑚max, and the proof is of Lemma. 1\ncompleted.\nII. SOLUTIONS TO PROBLEM 1\nIn this section, we discuss the solution to Problem 1, which is summarized as the pseudo-code in Algorithm. 1. Our protocol\nis based on the following lemma that can be straightforwardly verified from Lemma. 1 and the definition of 𝑂𝐶in Lemma. 2.\nLemma 3. For diagonalizable matrix, if min𝑗|𝜇−𝜆𝑗|⩽𝑟/(2𝐾), with probability at least 1−𝛿, the output of 𝑂𝐶(𝜇,𝑟/𝐾,𝛿)is\nTrue. If min𝑗|𝜇−𝜆𝑗|⩾𝑟, with probability at least 1−𝛿, the output of 𝑂𝐶(𝜇,𝑟/𝐾,𝛿)is False.\nFor defective matrix, we define\n𝜈(𝑟)=(𝑟/3)𝑚max(2𝐾)−1. (S-16)\nifmin𝑗|𝜇−𝜆𝑗|⩽𝜈(𝑟), with probability at least 1−𝛿, the output of 𝑂𝐶(𝜇,2𝜈(𝑟),𝛿)is True. If min𝑗|𝜇−𝜆𝑗|⩾𝑟, with probability\nat least 1−𝛿, the output of 𝑂𝐶(𝜇,2𝜈(𝑟),𝛿)is False.\nLemma. 3 is also illustrated in Fig. S1a. For each SVTS (i.e. 𝑂𝐶(𝜇,𝑟/𝐾,𝛿)for defective matrix or 𝑂𝐶(𝜇,2𝜈(𝜇),𝛿)), we\nmark the region of 𝜇in which the output of SVTS is “True” (with probability at least 1 −𝛿) using yellow color. We also mark9\nGray circle: Initial guess region\nUpdated guess regionGray ring: Initial guess region\nUpdated guess regionTrueFalse\nSVTSEigenvalues inside yellow region: output “True” is likely\nEigenvalues outside green region: output “False” is unlikely\n(a)(b)(c)\nSupplementary Figure S1: (a) Sketch of SVTS. (b) Sketch of Algorithm. 2 for shrinking the range of eigenvalue searching (Problem 1). The\ninitial and updated guess region is enclosed by grey circles. (c) Sketch of Algorithm. 4 for shrinking the range of point gap. The initial guess\nregion is a ring enclosed by two grey circles (Problem 2). The updated guess region is a ring enclosed by a grey circle and red (one of the\nSVTS has output “True”) or blue (all of the SVTS has output “False”) circles.\na larger region with a green color. If all eigenvalues are outside the green region, the output of SVTS is unlikely to be “True”\n(with probability at least 1 −𝛿). Note that the radius of yellow regions is 𝑟/2𝐾or𝜈(𝑟)for diagonalizable and defective matrices,\nthe radius of green regions is 𝑟.\nBecause∥𝐴∥⩽1, all eigenvalues are in D(0,1). Our strategy is to iteratively shrink the region in which there is at least one\neigenvalue in it (with high probability). Our method contains 𝐽=⌈log2(1/𝜀)⌉steps, and the process of each step is illustrated in\nFig. S1b (see also Algorithm. 2). Suppose that before the 𝑗th step, we are confidence that there is at least one eigenvalue in the\nregionD(𝜆gss,𝐷). At this step, we shrink the radius of such confidence region from 𝐷to𝐷/2. This is achieved by introducing\na set of SVTSs, whose yellow region covers D(𝜆gss,𝐷). This ensures that at least one of the SVTS has output “True”. Another\nrestriction is that the green region of each SVTS has a radius 𝐷/2. In this way, once we obtain an output “True”, we are confident\nthat at least one eigenvalue is in the region D(𝜆′\ngss,𝐷/2), where𝜆′\ngssis the center of such SVTS with output “True”.\nNote that in Algorithm. 2, we have introduced a set of points Nnet(𝜆gss,𝐷,𝑚 max). It represents the centers of all SVTSs\nsatisfying the criteria above. Equivalently, we have\nD(𝜆gss,𝐷)⊂Ø\n𝜇∈N net(𝜆gss,𝐷,1)D(𝜇,𝐷/4𝐾), (S-17)\nwhen𝑚max=1, or\nD(𝜆gss,𝐷)⊂Ø\n𝜇∈N net(𝜆gss,𝐷,𝑚 max)D(𝜇,𝜈(𝐷/2)). (S-18)\nwhen𝑚max>1.\nWe first estimate the complexity of diagonalizable matrices. According to Lemma. 3, the query to each SVTS has complexity\n˜𝑂(𝐾𝐷−1𝛾−1). Moreover, the area of D(𝜆gss,𝐷)and the yellow regions of SVTSs are 𝜋𝐷2and𝜋𝐷2/(4𝐾)2respectively. So it\nsuffices to use 𝑂(𝐾2)number of SVTSs to cover D(𝜆gss,𝐷). Therefore, the complexity at each step is ˜𝑂(𝐾𝐷−1𝛾−1)×𝑂(𝐾2)=\n˜𝑂(𝐾3𝐷−1𝛾−1). In Algorithm. 1, the total algorithm contains 𝐽=⌈log2(1/𝜀)⌉steps, and we have 𝐷⩾𝜀. So the total complexity\nis˜𝑂(𝐾3𝜀−1𝛾−1).\nFor defective matrix, the threshold of each SVTS is 2 𝜈(𝐷/2)=𝑂(𝐷𝑚max/𝐾). The complexity of each query to SVTS is\ntherefore ˜𝑂(𝐾𝐷−𝑚max𝛾−1). The yellow region of each SVTS has area 𝑂(𝐷2𝑚max/𝐾2), so totally𝑂(𝐾2𝐷−2𝑚max+2)number\nof SVTSs is required to cover D(𝜆gss,𝐷). Therefore, the complexity for each step is ˜𝑂(𝐾3𝐷−3𝑚max+2𝛾−1), while the total\ncomplexity of Algorithm. 1 is ˜𝑂(𝐾3𝜀−3𝑚max+2𝛾−1).10\nAlgorithm 1 Quantum eigenvalue searching for Problem 1.\n𝐷←1,𝛿′←𝛿/⌈log2(𝐷/𝜀)⌉\nwhile𝐷 >𝜀 :\n𝜆gss←ℛdiag\u0000𝜆gss,𝐷,𝛿′\u0001\n𝐷←𝐷/2:\nend while\nreturn𝜆gss\nAlgorithm 2 ℛ(𝜆gss,𝐷,𝛿)\n𝛿′←𝛿/|N net(𝜆gss,𝐷,𝑚 max)|\nfor all𝜇∈N net(𝜆gss,𝐷,𝑚 max):\nif𝑚max=1:\n𝐵←𝑂𝐶(𝜇,𝐷/4𝐾,𝛿′)\nelse if :𝑚max>1:\n𝐵←𝑂𝐶(𝜇,2𝜈(𝐷/2),𝛿′)\nend if\nif𝐵=True:\nbreak for\nend if\nend for\nreturn𝜇\nIII. SOLUTION TO PROBLEM 2\nIn below, we briefly introduce the main idea of our protocols for solving Problem 2. For Problem 3 and other related problems,\nthe processes are similar. We specify the reference point as the original point, i.e. 𝑃=0. The goal is then to find an eigenvalue\nthat is closest to, but not equal to 0. In case 𝑃≠0, we can always define a new matrix ˜𝐴=(𝐴−𝑃𝐼)/(1+|𝑃|), and the problem\nthen reduces to the point gap problem for ˜𝐴with the original point as the reference point.\nOur algorithm contains two stages, in the first stage, we output an estimation of the gap 𝑔≡min𝜆𝑗|𝑃−𝜆𝑗|to accuracy𝜀. In\nthe second stage, we search the estimation of a target eigenvalue to accuracy 𝜀.\nA. Stage 1\nAccording to definition in Problem 2, we initially have 𝑔∈[𝑅min\n0,𝑅max\n0]with𝑅min\n0=𝜀and𝑅max\n0=1. Our strategy is to shrink\nthe range of 𝑔iteratively. The full process is summarized in Algorithm. 3. Suppose at the 𝑗th step, we are confidence that\n𝑔∈[𝑅min\n𝑗−1,𝑅max\n𝑗−1]. (S-19)\nIn this step,𝑅min\n𝑗−1or𝑅max\n𝑗−1is updated by querying the eigenvalue range shrinking subroutine (ERSS)\n𝒮(𝑅min,𝑅max,𝑟,𝛿)→( ˜𝑅min,˜𝑅max) (S-20)\ndefined in Algorithm. 4 (4.1, and 4.2 corresponds to diagonalizable and defective matrix respectively). The ERSS contains four\ninput parameters. 𝑅minand𝑅maxare the recent confidence region in which the gap 𝑔is in. The third parameter 𝑟 >0 controls the\nstep size of updating. It is required that 𝑟⩽𝑅minand𝑟⩽𝑅max−𝑅min. The first requirement ensures that the output of ERSS will\nnot be affected by eigenvalue at the original point, if any. The second requirement ensures that the current gap Δ𝑗≡𝑅max\n𝑗−𝑅min\n𝑗\nis non-increasing. The last parameter 𝛿is the failure probability. The ERSS has the following property.\nLemma 4. Let𝐴be a square matrix satisfying ∥𝐴∥⩽1. Let\u0000˜𝑅min,˜𝑅max\u0001be the output of 𝒮(𝑅min,𝑅max,𝑟,𝛿)for some\n0< 𝑅min< 𝑅max⩽1, and 0< 𝑟⩽min(𝑅min,(𝑅max−𝑅min)/2), and𝛿∈ (0,1). Then, suppose the point gap satisfies\n𝑔∈[𝑅min,𝑅max], with probability 1−𝛿, we have𝑔∈[˜𝑅min,˜𝑅max]. Here, ˜𝑅minand ˜𝑅maxare defined in Algorithm. 4.\nMoreover, the complexity of ERSS is given by the following.11\nLemma 5. 𝒮(𝑅min,𝑅max,𝑟,𝛿)defined in Algorithm. 4 can be realized with ˜𝑂\u0000𝐾2𝑅min𝑟−2𝑚max𝛾−1\u0001queries to 𝒪𝐴,𝒫𝐴and there\ninverses, and single- and two-qubit gates.\nBase on Lemma 4, we are able to update the region of 𝑔by𝒮\u0010\n𝑅min\n𝑗−1,𝑅max\n𝑗−1,𝑟,𝛿′\u0011\n→ (𝑅min\n𝑗,𝑅max\n𝑗), where𝛿′is set as a\nsufficiently small value. From Algorithm. 4, it can also be verified that\n|Δ𝑗−1−Δ𝑗|⩾Ω(𝑟𝑚max/𝐾). (S-21)\nStage 1 is separated into two substages. In substage 1, we set 𝑟=𝑅min\n𝑗−1at each step, and this substage terminates when\n𝑅min\n𝑗⩾𝑅max\n𝑗/2. The complexity of each step is ˜𝑂\u0010\n𝐾2(𝑅min\n𝑗)−2𝑚max+1𝛾−1\u0011\n. From Eq. (S-21), it can be verified that this substage\nterminates with at most 𝑂(𝐾𝜀−𝑚max+1)steps. Because 𝑅min\n𝑗⩾𝜀, The total complexity for this substage is\n˜𝑂\u0010\n𝐾2𝜀−2𝑚max+1𝛾−1\u0011\n×𝑂(𝐾𝜀−𝑚max+1)=˜𝑂\u0010\n𝐾3𝜀−3𝑚max+2𝛾−1\u0011\n. (S-22)\nIn substage 2, we set 𝑟=\u0010\n𝑅max\n𝑗−1−𝑅min\n𝑗−1\u0011\n/2, and this substage terminates when Δ𝑗⩽𝜀. In this substage, the complexity of each\nstep is ˜𝑂\u0000𝐾2𝜀−2𝑚max𝛾−1\u0001. This substage contains ˜𝑂(𝐾𝜀−𝑚max+1)steps as can be verified from Eq. (S-21). So the total complexity\nof substage 2 is\n˜𝑂\u0010\n𝐾2𝜀−2𝑚max𝛾−1\u0011\n×𝑂(𝐾𝜀−𝑚max+1)=˜𝑂\u0010\n𝐾3𝜀−3𝑚max+1𝛾−1\u0011\n. (S-23)\nCombining Eq. (S-22) and Eq. (S-23), the total complexity of stage 1 is ˜𝑂\u0000𝐾3𝜀−3𝑚max+1𝛾−1\u0001.\nAlgorithm 3 Stage 1 for solving Problem 2\n𝑅min\n0←𝜀;𝑅max\n0←1;𝑗←1\nwhile𝑅min\n𝑗−1< 𝑅max\n𝑗−1/2: # Substage 1\n(𝑅min\n𝑗,𝑅max\n𝑗)←𝒮(𝑅min\n𝑗−1,𝑅max\n𝑗−1,𝑅min\n𝑗−1,𝛿′)\n𝑗←𝑗+1\nend while\nwhile𝑅max\n𝑗−1−𝑅min\n𝑗−1>𝜀: # Substage 2\n(𝑅min\n𝑗,𝑅max\n𝑗)←𝒮\u0010\n𝑅min\n𝑗−1,𝑅max\n𝑗−1,\u0010\n𝑅max\n𝑗−1−𝑅min\n𝑗−1\u0011\n/2,𝛿′\u0011\n𝑗←𝑗+1\nend while\nreturn\u0010\n𝑅min\n𝑗−1,𝑅max\n𝑗−1\u0011\nB. Stage 2\nAfter stage 1, we are confidence that 𝑔∈[𝑅min\n𝐽,𝑅max\n𝐽]for some𝑅max\n𝐽−𝑅min\n𝐽⩽𝜀. In other words, we are confident that\nthere exists at least one eigenvalue in the region ˜D=D(0,𝑅max\n𝐽)/D( 0,𝑅min\n𝐽), while there is no eigenvalue in the region\nD(0,𝑅min\n𝐽)/D( 0,𝜀).\nThe remaining task is then to find an eigenvalue estimation in ˜D. This is achievable with a similar strategy for Problem 1.\nWe introduce a set of SVTSs, whose green region has radius 𝜀, and the yellow region covers ˜𝐷. Accordingly, the complexity\nof implementing each SVTS is ˜𝑂(𝜈(𝜀))=˜𝑂(𝐾𝜀−𝑚max). The area of ˜Dis upper bounded by 2 𝜋𝜀, while the yellow region of\neach SVTS is 𝜈(𝜀)=˜𝑂(𝜀𝑚max/𝐾). So it suffices to use totally ˜𝑂(𝐾𝜀−𝑚max+1)number of SVTSs to cover ˜D. Therefore, the total\ncomplexity of this stage is ˜𝑂(𝐾𝜀−𝑚max)ט𝑂(𝐾𝜀−𝑚max+1)=˜𝑂(𝐾2𝜀−2𝑚max+1).\nCombining the complexity for stage 1 and stage 2, the total complexity for solving Problem 2 is ˜𝑂\u0000𝐾3𝜀−3𝑚max+1𝛾−1\u0001.\nIV. EIGENVALUE RANGE SHRINKING SUBROUTINE\nIn this section, we give detailed construction of the ERSS. We described the ERSS for diagonalizable matrix and defective\nmatrix separately in Algorithm. 4.1 and Algorithm. 4.2. They are denoted as SdiagandSdefrespectively.12\nAlgorithm 4 𝒮(𝑅min,𝑅max,𝑟,𝛿)\nif𝑚max=1\n(˜𝑅min,˜𝑅max)←𝒮diag(𝑅min,𝑅max,𝑟,𝛿)\nelseif𝑚max>1\n(˜𝑅min,˜𝑅max)←𝒮def(𝑅min,𝑅max,𝑟,𝛿)\nend if\nreturn(˜𝑅min,˜𝑅max)\nAlgorithm 4.1 𝒮diag(𝑅min,𝑅max,𝑟,𝛿)(i.e. eigenvalue range shrinking subrutine for diagonalizable matrix)\n𝛿′←𝛿/|N ring(𝑅min,𝑟/𝐾)|\nfor all𝑡∈N ring(𝑅min,𝑟/𝐾):\n𝐵←𝑂𝐶(𝑡,𝑟/𝐾,𝛿′)\nif𝐵=True:\nbreak for\nend if\nend for\nif𝐵=True:\n˜𝑅min←𝑅min\n˜𝑅max←𝑅min+𝑟\nelse if𝐵=False:\n˜𝑅min←𝑅min+𝑟/(4𝐾)\n˜𝑅max←𝑅max\nend if\nreturn\u0000˜𝑅min,˜𝑅max\u0001\nAlgorithm 4.2 𝒮def(𝑅min,𝑅max,𝑟,𝛿)(Eigenvalue range shrinking subrutine for defective matrix)\n𝛿′←𝛿/|N ring(𝑅min,𝜈(𝑟))|\nfor all𝑡∈N ring(𝑅min,𝜈(𝑟)):\n𝐵←𝑂𝐶(𝑡,𝜈(𝑟),𝛿′)\nif𝐵=True:\nbreak for\nend if\nend for\nif𝐵=True:\n˜𝑅min←𝑅min\n˜𝑅max←𝑅min+𝑟\nelse if𝐵=False:\n˜𝑅min←𝑅min+𝜈(𝑟)/4\n˜𝑅max←𝑅max\nend if\nreturn\u0000˜𝑅min,˜𝑅max\u0001\nHere, we have also defined\nNring(𝑅,𝑠)=n\n𝑅𝑒𝑖2𝜋𝑚/𝑀(𝑅,𝑠)\f\f𝑚∈{1,2···,𝑀(𝑅,𝑠)}o\n, (S-24)\nwhere\n𝑀(𝑅,𝑠)=2𝜋\narctan(𝑠/(2𝑅)). (S-25)\nNring(𝑅,𝑠)defines a set of points at the circle with radius 𝑅and center 0. The main idea of ERSS is illustrated in Fig. S1c. The13\nyellow region of all SVTSs covers the edge of the circle with radius 𝑅min. If either of the SVTS has output true, we have 𝐵=\nTrue. In this case, with confidence at least 1 −𝛿, there exists at least one eigenvalue in the region covered by the green circle. So\n𝑅maxis updated. Otherwise, we have 𝐵=False. In this case, with confidence at least 1 −𝛿, all of the eigenvalues are outside in\nthe region covered by the yellow circle. So 𝑅minis updated.\nV. SINGULAR VALUE THRESHOLD SUBROUTINE: PROOF OF LEMMA. 2\nGiven a general matrix 𝐴∈C𝑁×𝑁satisfying∥𝐴∥⩽1, we can always rewrite it in the form of singular value transformation\nform\n𝐴=𝑁−1∑︁\n𝑗=0𝜎𝑗|𝑤𝑗⟩⟨𝑢𝑗| (S-26)\nfor some singular value 0 ⩽𝜎0⩽𝜎1···, orthonormal left singular vectors {|𝑤𝑗⟩}and right singular vectors {|𝑢𝑗⟩}. The SVTS\naims to determine if there are singular values smaller than a threshold. Below, we show how this can be realized with the\nblock-encoding of matrix 𝐴.\nLet𝑁=2𝑛, we choose the simplest definition of block encoding, i.e. an (𝑛+𝑎)qubit unitary 𝒪𝐴is called the block encoding\nof𝐴if(⟨0𝑎|⊗𝐼)𝒪𝐴(|0𝑎⟩⊗𝐼)=𝐴. It is typically required that 𝑎=𝑂(poly(𝑛)). Let𝑃(·)be a real polynomial function, we\ndefine the singular value transformation of a matrix as\n𝑃(svt)(𝐴)=\u001a𝑃(𝜎𝑗)|𝑤𝑗⟩⟨𝑢𝑗|if the degree of 𝑃(·)is odd\n𝑃(𝜎𝑗)|𝑢𝑗⟩⟨𝑢𝑗|if the degree of 𝑃(·)is even(S-27)\nAccording to [4], QSVT can be effectively constructed with 𝒪𝐴and few extra elementary quantum gates, if 𝑃(·)satisfies some\nreasonable criteria. More specifically, we have the following.\nLemma 6 (QSVT for real polynomials with definite parity, adapted from Theorem 4 in [4]) .Let𝑃∈Rbe a polynomial function\nsatisfying (1) The degree of 𝑃is at most𝑑; (2)𝑃is either of even or odd parity; (3) For ∀𝑥∈[− 1,1],|𝑃(𝑥)|⩽1.\nThen there exists a block encoding of 𝑃(𝐴)using𝑑queries of 𝒪𝐴,𝒪†\n𝐴, one extra ancillary qubit, and 𝑂(𝑎+1)𝑑extra single-\nand two-qubit gates.\nAs mentioned earlier, the aim of Lemma. 2 is approximately determine where the targeted eigenvalue lies, a task similar to\nthe fuzzy bisection scheme proposed in Ref. [41]. To this end, we also utilize the Heavised function for the bisection. Yet, the\nmain difference between our construction and Dong et al. [41]’s construction is that to deal with complex eigenvalues, we take\nadvantage of the relationship between eigendecomposition and singular value decomposition as given by Lemma. 1. That is given\nthe construction of the shifted matrix 𝐴−𝜇𝐼, if the shifted value 𝜇is close enough to the targeted eigenvalue 𝜆𝑗,𝐶(𝜇)is then\nclose to zero. Therefore, we can decide whether there is an eigenvalue 𝜆𝑗that is close to the attempted shift 𝜇by determining\nthe existence of singular value signals close to zero by QSVT techniques [4]. As such, an approximated Heaviside function of\nthe matrix is constructed. The assumption on the initial state |𝜓𝐼⟩then dictates the threshold (lower-bound value) of the signal\nthat we are seeking. The main difference between our and Dong et al. [41]’s scenario is that the polynomial function we applied\nis for the singular values but not eigenvalues.\nThe next step is thus to approximate a shifted Heaviside function 𝐻(𝑥−𝜃)with a polynomial function 𝑃(svt)\n𝐻(·)using QSVT\nmethods. The shifted Heaviside function is given by\n𝐻(𝑥−𝜃)=(\n1, 𝑥≤𝜃\n0, 𝑥 >𝜃(S-28)\nRegarding the approximated block encoding, we say that a unitary 𝑈𝐴is the(𝛼,𝑎,𝜂)-block encoding of 𝐴if∥𝛼(⟨0𝑎|⊗𝐼)𝑈𝐴(|0𝑎⟩⊗\n𝐼)−𝐴∥⩽𝜂. From [42], we have the following lemma.\nLemma 7. LetΔ,𝜂∈(0,0.5). Given a matrix 𝐴with its(1,𝑎,0)block-encoding 𝒪𝐴, we can construct a (1,𝑎+1,𝜂)-block-\nencoding of𝑃svt\n𝐻(𝐴)satisfying|𝑃𝐻(𝑥)−1|≤𝜂,∀𝑥∈[− 1,Δ/2]and|𝑃𝐻(𝑥)|≤𝜂,∀𝑥∈[Δ,1]using𝑂\u0010\n1\nΔlog\u0010\n1\n𝜂\u0011\u0011\napplications\nof𝒪𝐴and𝒪†\n𝐴, and𝑂\u0010\n𝑎\nΔlog\u0010\n1\n𝜂\u0011\u0011\nextra one- and two-qubit gates.14\nHere, we have approximated the Heaviside function with a shift 𝜃=3Δ/4. It is worth noting that the function between interval\n𝑥∈[Δ/2,Δ]often takes values that smoothly interpolate the function value of the two endpoints of the interval. See Sec. V of\nRef. [41] for an example. For simplicity, we will denote the (1,𝑎+1,𝜂)-block encoding unitary of 𝑃svt\n𝐻(𝐴)as𝑈𝐻.\nAs discussed in the main text, we further assume that we have a unitary oracle 𝒫𝐴which prepares an initial state with nontrivial\noverlap to|𝑢0⟩. More specifically, we have 𝒫𝐴|0𝑛⟩=|𝜓⟩, for some|⟨𝑢0|𝜓⟩|⩾𝛾. Applying𝑈𝐻to the join state of ancillary\nqubits at state|+⟩|0𝑎+1⟩and data qubit at state |𝜓⟩, we obtain\n𝑈𝐻|+⟩|0𝑎+1⟩|𝜓⟩=|+⟩|0𝑎+1⟩∑︁\n𝑗𝑐𝑗𝑃𝐻(𝜎𝑗)|𝑢𝑗⟩+|garb⟩ (S-29)\nfor some|𝑐0|⩾𝛾, and\n(⟨+|⟨ 0𝑎+1|⊗𝐼)|garb⟩=0. (S-30)\nIf we project the ancillary qubits to |+⟩|0𝑎+1⟩, the success probability of the projection is given by\n𝑝suss≡\r\r(⟨+|⟨ 0𝑎+1|⊗𝐼)𝑈𝐻|+⟩|0𝑎+1⟩|𝜓⟩\r\r (S-31)\n=∑︁\n𝑗|𝑐𝑗|2|𝑃𝐻(𝜎𝑗)|2. (S-32)\nIf the smallest singular value of 𝐴satisfies𝜎0⩽Δ/2, we have\n𝑝suss⩾|𝑐0|2|𝑃𝐻(𝜎0)|2⩾|𝑐0|2(1−𝜂)2⩾𝛾/4. (S-33)\nIf𝜎0(𝐴)⩾Δ, we have\n𝑝suss⩽𝜂2. (S-34)\nWe note that 𝜂decays rapidly with order 𝑑for the polynomial function. For example, we may require that the probability in the\nsecond case is at most half of the probability in the first case, i.e.\n𝜂2⩽(𝛾/4)/2=𝛾/8. (S-35)\nThis can be achieved with 𝑑=𝑂\u0010\nlog(1/𝛾)\nΔ\u0011\n. To distinguish whether Eq. (S-33) or Eq. (S-34) are satisfied, we can use the Monte\nCarlo method by performing the projection process many times. To achieve a constant correct probability, this method requires\nsampling size 𝑂(𝛾−2), and each run of the quantum circuit requires a single query to 𝑈𝐻(Lemma 9 of Ref [41], see also Ref [2]).\nAlternatively, we can improve the dependency on 𝛾to𝑂(𝛾−1)with the amplitude amplification method.\nLemma 8 (Lemma.12 in Ref [41]) .Given a unitary 𝑊applied at𝑛𝑤+1qubits, and let\n𝜔=∥(⟨0|⊗𝐼2𝑛𝑤)𝑊|0⟩|0𝑛𝑤⟩∥, (S-36)\nwhere𝐼2𝑛𝑤is2𝑛𝑤-dimensional identity. It is further promised that either 𝜔⩽𝛾1or𝜔⩾𝛾2for some 0⩽𝛾1< 𝛾 2. These two\ncases can be distinguished with success probability at least 1- 𝛿with𝑂\u0000(𝛾2−𝛾1)−1log(𝛿−1)\u0001queries to𝑊and one additional\nancilla qubit.\nWe define 𝒫′\n𝐴=(𝐼2⊗Hard⊗𝐼2𝑎+1⊗𝒫𝐴), where Hard is Hardamard gate. Following the definition in Eq. (S-29), it can be\nverified that\n(𝐼2⊗𝑈𝐻)𝒫′\n𝐴|0⟩|0𝑛+𝑎+2⟩ (S-37)\n=|0⟩|+⟩|𝜓⟩ (S-38)\n=|0⟩ \n|+⟩|0𝑎+1⟩∑︁\n𝑗𝑐𝑗𝑃𝐻(𝜎𝑗)|𝑢𝑗⟩+|garb⟩!\n. (S-39)15\nWe define|𝜓′⟩=|+⟩|0𝑎+1⟩Í\n𝑗𝑐𝑗𝑃𝐻(𝜎𝑗)|𝑢𝑗⟩, and a controlled rotation 𝑅|𝜓′⟩≡𝐼2⊗|𝜓′⟩⟨𝜓′|+𝑋⊗(𝐼−|𝜓′⟩⟨𝜓′|). According\nto Eq. (S-30), we have\n𝑝suss=��(⟨0|⊗𝐼2𝑎+2)𝑅|𝜓′⟩|0⟩|0𝑎+2⟩∥. (S-40)\nHere,𝑝sussis the success probability of projection defined in Eq. (S-31). According to Eq. (S-33) and Eq. (S-34), we can set two\nthresholds of projection success probabilities to be 𝛾1=𝜂2⩽𝛾/8 and𝛾2=𝛾/4 respectively. According to Lemma. 8, we can\ndistinguish whether 𝑝suss⩽𝛾1or𝑝suss⩾𝛾2with𝑂\u0000(𝛾2−𝛾1)−1log(𝛿−1)\u0001=˜𝑂\u0000𝛾−1\u0001queries to𝑅|𝜓′⟩.\n𝑅|𝜓′⟩requires single query to 𝑈𝐻,𝒫′\n𝐴and𝑂(𝑛)extra single- and two-qubit quantum gates. Summing up the complexity for\n𝑈𝐻and𝒫′\n𝐴, we have\nLemma 9. LetΔ,𝜂∈(0,0.5), Given𝐴∈C𝑁×𝑁satisfying∥𝐴∥⩽1, and promised that either 𝜎0(𝐴)⩽Δ/2or𝜎0(𝐴)⩾Δ. We\ncan distinguish these two cases with probability at least 1−𝛿using ˜𝑂(Δ−1𝛾−1)queries to 𝒪𝐴,𝒫𝐴and their inverses, and extra\nsingle- and two-qubit gates.\nNote that ˜𝑂(), has neglected the dependency on 𝑛. The difference of Lemma. 2 from Lemma. 9 is that we should deal with the\nshifted matrix(𝐴−𝜇𝐼). This can be achieved by performing rescaling ˜𝐴=(𝐴−𝜇𝐼)/(1+|𝜇|), which ensures that ∥˜𝐴∥⩽1. As\nshown in Sec. VI, we can encode ˜𝐴with𝑂(1)extra single- and two-qubit gates and one extra ancillary qubit. Moreover, because\n|𝜇|<1, the accuracy scaling is identical for dealing with 𝐴and(𝐴−𝜇𝐼)/(1+|𝜇|). Then, Lemma. 2 can be verified readily.\nVI. BLOCK ENCODING OF 𝐴−𝜇𝐼\nIn the block encoding implementation, we typically require that the spectral norm of the matrix being encoded is bounded by\n1. However, we have ∥𝐴−𝜇𝐼∥>1 in some cases. This problem can be solved by encoding a rescaled matrix (𝐴−𝜇𝐼)/(1+|𝜇|)\ninstead, because∥(𝐴−𝜇𝐼)/(1+|𝜇|)∥⩽1. For brevity, we simply denote 𝒪𝐴,𝜇as the block encoding of matrix (𝐴−𝜇𝐼)/(1+|𝜇|).\nWe have the following result about its construction.\nLemma 10. 𝒪𝐴,𝜇can be constructed with one query of single-qubit controlled 𝒪𝐴, single ancillary qubit, and a constant number\nof extra single- and two-qubit gates.\nProof. Let𝜃=arccos\u0010√︃\n1\n1+|𝜇|\u0011\nand\n𝑅(𝜃)=\u0012cos𝜃−sin𝜃\nsin𝜃cos𝜃\u0013\n,Ph(𝜙)=\u00121 0\n0𝑒𝑖𝜙\u0013\n. (S-41)\nThe block encoding unitary is constructed as\n𝒪𝐴,𝜇=(𝑅(−𝜃)⊗𝐼)\u0010\n|0⟩⟨0|⊗𝒪𝐴−|1⟩⟨1|⊗𝑒𝑖arg(𝜇)𝐼\u0011\n(𝑅(𝜃)⊗𝐼), (S-42)\nwhich is equivalent to the following quantum circuit\n𝑅(𝜃) Ph(𝜋+arg(𝜇))𝑅(−𝜃)\n𝑂𝐴\nIt can be verified that.\n⟨0𝑎+1|𝒪𝐴,𝜇|0𝑎+1⟩=(cos𝜃⟨0|+sin𝜃⟨1|)⊗𝐼\u0010\n|0⟩⟨0|⊗𝒪𝐴−|1⟩⟨1|⊗𝑒𝑖arg(𝜇)𝐼\u0011\n(cos𝜃|0⟩+sin𝜃|1⟩)𝐼\n=cos2𝜃𝒪𝐴−sin2𝜃𝑒𝑖arg(𝜇)𝐼\n=𝒪𝐴−𝜇𝐼\n1+|𝜇|. (S-43)\n□16\nWe note that for algorithms related to QSVT, the runtime is affected by the rescale factor 1 +|𝜇|. However, because we always\nhave|𝜇|⩽1 in our application, the rescale factor is bounded by a constant and will affect the runtime significantly."    },    {        "title": "2401.12094v2.CLIQUE_as_an_AND_of_Polynomial_Sized_Monotone_Constant_Depth_Circuits.pdf",        "content": "arXiv:2401.12094v2  [cs.CC]  23 Jan 2024CLIQUE as an AND of Polynomial-Sized\nMonotone Constant-Depth Circuits\nLevente Bodn´ ar\nFebruary 10, 2022\nAbstract\nThis paper shows that calculating k-CLIQUE on nvertex graphs, re-\nquirestheANDofatleast2n/4kmonotone, constant-depth,andpolynomial-\nsized circuits, for sufficiently large values of k. The proof relies on a new,\nmonotone, one-sided switching lemma, designed for cliques .\n1 Introduction\nAlmost all Boolean functions require exponentially large circuits [Sha4 9] how-\never, a super-polynomial circuit lower bound on NP complete problem s appear\nout of reach. As lower bounds for general circuits seem difficult to p rove, there\nhas been more focus on restricted circuit models.\nA well understood restriction, where there are strong lower boun ds, is the\nclass of constant-depth circuits. Early results [Ajt83, FSS81] pr oved that PAR-\nITY requires super-polynomial size constant-depth circuits. This was subse-\nquently refined in [Yao85, Has86], developing the powerful switching method.\nFor monotone functions, another fruitful restriction is the class of mono-\ntone circuits. Razborov was the first to prove super-polynomial b ound on the\nNP complete problem CLIQUE, for monotone circuits using the metho d of\napproximations [Raz85]. This was improved by [And85, AB87, Ros10] a nd\nrecently [CKR20] with the current best truly exponential bound nΩ(k)when\nk≤n1/3−o(1).\nThe simplest circuit expressing CLIQUE, is an OR of monomials, one for\neach possible clique. The top gate has fan-in/parenleftbign\nk/parenrightbig\n. Alternatively, one can express\nCLIQUE as monotone AND of monomials, one for each maximal graph w ithout\nk-clique. The number of maximal graphs without triangles was investig ated in\n[BP14], where they show that their number is asymptotically 2n2/8+o(n). The\nsame paper (in remark 7) constructs 21−1/(k−1)+o(1)\n4n2many maximal graphs\nwithoutk-clique, but method only works for small k <log4/3(n). Note that any\nmaximal graph without k-clique can be extended to a maximal graph without\n1k′-clique, by simply adding k′−kpoints connected to everything. This gives\na 21−1/log4/3(n−k)+o(1)\n4(n−k)2lower bound on the number of maximal, k-clique\nfree graphs, when k >log4/3(n). Correspondingly, there is a monotone, depth-\n2 circuit, with top AND gate, calculating k-CLIQUE, with top fan-in lower\nbounded by the same value. This paper is a small progress towards a nswering\nthenaturalquestion,whetherCLIQUEcanbeexpressedastheA NDofasmaller\nnumber, monotone, constant-depth, and polynomial-sized circuit s.\nTheorem 1. For any cd,csconstants, large enough nandlog(n)cd+6< k,one\ncannot have less than 2n/4kmany monotone circuits {fj}on/parenleftbign\n2/parenrightbig\ninput bits, each\nwith depth and size bounded by cdandncsrespectively, such that/logicalandtext\njfjcomputes\nk-CLIQUE.\nThe bound looks odd (and surprisingly strong for small k) compared to\nclassical circuit bounds for CLIQUE due to the restricted setting. This shows\nthat OR gates are much better at expressing CLIQUE than AND gat es in this\nsense. The result follows from the main lemma of this paper, that mon otone\nconstant-depth polynomial-sized circuits with zero-error on YES in stances have\n2−n/4kcorrelationwith cliquesonNO instances. Thisgivesastrongasymmet ric\nresult; note that a single monomial, stating that a certain ksubset is a clique,\nhas zero error on NO instances, but on YES instances, has/parenleftbign\nk/parenrightbig−1correlation.\nThemainnoveltyoftheproofisanapproximationofamonotoneCNFc ircuit\nhaving larger fan-in, by a monotone DNF circuit having a small fan-in, where\nthis approximation is correct on most of the relevant cliques. Appro ximation\nof Boolean functions with depth 2 circuits was initially investigated by [O W07].\nContinuing this line of work, [BHST14] showed strong one-sided boun ds for all\nmonotone functions.\nSection 2 contains the main notation used in this paper, section 3 cov ers the\nmain definitions, lemmas and an overall view of the proof. Section 4 co ntains\nall the technical steps and proofs of lemmas and theorems. The pa per finishes\nwith a few concluding thoughts and open problems.\n2 Notation\nFor positive integer n, use [n] for the set {0,1,...,n−1}. For a set Uand integer\nmwrite/parenleftbigU\nm/parenrightbig\n={A⊆U:|A|=m}. Whenm= 2 write K(U) instead/parenleftbigU\n2/parenrightbig\n. For\nthe power set write [2]U={A⊆U}.\nUseA,Bfor subsets of [ n]. Similarly G,Hfor subsets of K([n]) and identify\nthem with graphs. kis a number that can depend on n, and use X,Y,Zfor\nsubsets of/parenleftbig[n]\nk/parenrightbig\n. The curly versions represent families, so A,Bare used for\nfamilies of subsets of [ n]. Similarly G,Hare families of subsets of K([n]). For a\nfamilyAwriteK(A) ={K(A):A∈ A}.\nMonotone circuits are represented using the letters f,g, andh. Given a\ncircuit with input set U, for any A⊆Uwritef(A) for the value of the circuit on\n2inputA. Thereforethey canbeusedasfunctions f:[2]U→[2]. Forconvenience\nusef=gto mean that fandgcompute the same function, not that the circuits\nare the same. The depth d(f) of a circuit is the longest path from the output\nnode to any of the input nodes. The size |f|is the number of nodes in the\ncircuit. For f,gon the same input set U, writef≤gif∀A⊆U f(A)≤g(A).\nFor given A ⊆[2]Uthe DNF with monomials defined by AisiA:[2]U→[2].\nThis gives the function\niA(B)/ma√sto→/braceleftBigg\n1 if∃A∈ A(A⊆B)\n0 otherwise .\nThe letters ρ,σare monotone restrictions. On an input set Uthey are\nfunctions ρ:U→ {1,⋆}. Given a restriction ρand a circuit fthe restricted\ncircuitfρ:[2]ρ−1(⋆)→[2] is the circuit where every input source from ρ−1(1) is\nsettoconstanttrue. Givenaninputset U,useRp\nUasadistributiononmonotone\nrestrictions on U, that maps ρ(u) to⋆with probability p(and therefore maps\nto 1 with probability 1 −p) independently for each u∈U.\n3 Outline\nThe result follows from the following proposition on a single bounded de pth and\nsize monotone circuit.\nProposition 2. For any cd,csconstants, large enough nand any monotone\ncircuitfon/parenleftbign\n2/parenrightbig\ninput bits, with depth d(f)≤cdand size |f| ≤ncssuch that\nf≥iK/parenleftBig\n([n]\nk)/parenrightBig,fmust also satisfy\nP\nG∼ER(n,1−p)(f(G))≥1−2−n/3k\nwhere\np=c(cs,cd)log(n)−cd−4.\nThe method is a switching of the bottom layers between small ( O(log(n)2))\nfan-in DNF and medium ( O(n/k)) fan-in CNF. The rationale behind the num-\nbers is the desire to ensure that the circuits remain ineffective at ap proximating\nCLIQUE. This holds true intuitively for CNFs even when larger fan-ins are al-\nlowed. However, for DNFs, a stronger fan-in restriction is necess ary. To express\nthis correlation with CLIQUE, the following approach will be employed:\nDefinition 3. For a monotone restriction ρ:K([n])→ {1,⋆}and a monotone\ncircuitf:[2]K([n])→[2]both on graphs, write Zρ(f)for the maximal collection\nof cliques implying the function. It is the largest Zρ(f)⊆/parenleftbig[n]\nk/parenrightbig\nwhere it holds\nthat/parenleftbigg\niK(Zρ(f))/parenrightbigg\nρ≤fρ.\n3ThemainLemmaofthepapershowsthattheCNFtoDNFswitchingred uces\nthe possible cliques forcing the circuit to true (the set Z(f)) by a negligible\namount.\nLemma 4 (CNF to DNF switching with small clique error) .For any f:\n[2]K([n])→[2]monotone s-CNF and a monotone restriction ρ:K([n])→ {1,⋆},\none can find gmonotone t-DNF, that satisfies gρ≤fρand\n/vextendsingle/vextendsingleZρ(f)\\Zρ(g)/vextendsingle/vextendsingle≤/parenleftbiggn\nk/parenrightbigg/parenleftbigg\nsk\nn/parenrightbigg√\nt/2\n.\nThe DNF to CNF switching requires a random restriction, but after t he\nrestriction the functions are the same\nLemma 5 (DNF to CNF switching) .For any gmonotone t-DNF after taking\nρ∼R1/(2t)\nUrestriction gcan be written as an equivalent s-CNF with probability\n≥1−2−s−1.\nThe end result is a O(log(n)2) fan-in DNF that is still true on a largenumber\nof cliques, therefore it holds that a random 0-1 assignment on the r emaining\nvariables satisfies the circuit with high probability. The probability tha t one of\nthe DNF to CNF switching fails or that the final random assignment do es not\nforce the circuit to true, will be exponentially small, allowing the large f an-in\nbound on the output gate.\n4 Proofs\nThe following is a list of simple observations, the proof is not included.\nObservation 6.\n1. IfG ⊆ HtheniG≤iH.\n2. Iff≤gthenZ(f)⊆Z(g).\n3. Ifρ∼Rp\nUandσ∼Rq\nρ−1(⋆)thenσ◦ρ∼Rpq\nU.\n4. IfG=ρ−1(1)whereρ∼Rp\nK([n])thenG∼ER(n,1−p).\nThe idea for Lemma 4 is to build a monotone decision tree both by cliques\nand by edges simultaneously based on the formula. This gives enough control\nover the cliques they imply. The extension of edge sets to cliques can only\ndecrease the function but the clique implications Z(f) will not change. The\ntree is pruned to include only clauses below the cut. This can be achiev ed with\na minimal loss of cliques.\n4Proof of Lemma 4. First simplify the function fto only include edges from\nρ−1(⋆). If any of the clauses become trivial, then fρ≡1 and can set g≡1.\nLet’s name the clauses fρ=/logicalandtextm\nj=1/parenleftBig/logicalortext\ne∈qje/parenrightBig\nwhere|qj| ≤s.\nCreate two trees T⊆T′.Thas its nodes labeled by subsets of [ n], whileT′\nhas the labels from subsets of K([n]). Furthermore associate to every non-leaf\na clause from fρboth inTandT′. Forv∈TdenoteA(v)⊆[n] the label\nandqT(v) the associated clause. Similarly denote G(w)⊆K([n]) the label and\nqT′(w) forw∈T′the clause. Construct the labels such that for v∈T⊆T′\nit holds that/uniontextG(v) =A(v) meaning in particular that G(v)⊆K(A(v)). Also\nforv∈Tnot a leaf in Tthe labels will have qT(v) =qT′(v).\nThe root r∈Tandr∈T′has∅=A(r) =G(r). A node v∈Tis a leaf if\nfρ(K(A(v))) = 1, similarly w∈T′is a leaf if fρ(G(w)). Ifv∈Tis not a leaf\n(therefore not a leaf in T′) find the clauses in fρnot satisfied by K(A(v)). Let\nthe first clause be qj, then assign qT(v) =qT′(v) =qj. Ifwis a leaf in Tbut\nnot inT′or is not a member of Tat all, then choose qjto be the first clause\nnot satisfied by G(w) and set qT′(w) =qj. Then for v∈Twith a clause label\nqT(v) =qT′(v) and for each edge e∈qT(v) addv′a child of vboth inTand\nT′with label A(v′) =A(v)∪eandG(v′) =G(v)∪ {e}. This preserves that/uniontextG(v′) =A(v′). Forw∈T′\\Tnot a leaf, only add the w′children and G(w′)\nlabels without any constrains from T.\nq1∅\nq3{1,2} q2{1,3}\n{1,2,4}\n{1,2,5}\n{1,2,4,5}q3{1,2,3}\n{1,3,4}\n{1,2,3,4}\n{1,2,3,5}\n{1,2,3,4,5}q1∅\nq3{12} q2{13}\nq3{13,12} q3{13,34}\n{12,14}\n{12,25}\n{12,45}\n{13,12,45}{13,12,25}{13,12,14}{13,34,14}\n{13,34,25}\n{13,34,45}\nThe trees on formula f=q1∧q2∧q3= (12∨13)∧(12∨34)∧(14∨25∨45)\nwith clauses numbered accordingly.\nCall a node’s depth its distance from root. Let Adbe the collection of A(v)\nwherevis a leaf with depth ≤dinT. LetBdbe the collection of A(v) wherevis\nany node in Twith depth exactly d(not necessarily a leaf). Similarly define Gd\nto be the collection of G(w) forw∈T′leaf node of depth ≤d. Letd(T),d(T′)\n5be the maximal depth of the trees.\nClaim 7 (Relations involving the Gd,Ad,Bdsets).\n1.iGd(T′)=fρ\n2.iK(Ad(T))≤fρ\n3.Zρ/parenleftBig\niK(Ad(T))/parenrightBig\n=Zρ(f)\n4.iK(Ad(T))≤iK(Ad∪Bd+1)\n5.|Bd| ≤sd\nProof of Claim 7.\n1. Working in U=ρ−1(⋆), for any u⊆Uit holds that fρ(u) iff for all qj\nit is true that u∩qj/\\e}atio\\slash=∅. Therefore every non leaf w∈T′has an edge\new∈qT′(w) such that ew∈u. Following the label set increments G(w)\ntoG(w)∪ {ew}, provides a path from root to a leaf wsuch that each\nlabel in that path is a subset of uresulting in G(w)⊆uand therefore\niGd(T′)(u) = 1. This gives iGd(T′)≥fρ. For the other direction take usuch\nthatiGd(T′)(u) = 1. Then there is a leaf w∈TwhereG(w)⊆u. By\ndefinition of a leaf, G(w) already satisfies fρgiving that by monotonicity\nfρ(u) = 1.\n2. Consider any u⊆Usuch that iK(Ad(T))(u) = 1, then there must be a leaf\nv∈Tand a corresponding A(v)∈ Ad(T)withK(A(v))⊆u. Then as vis\na leafK(A(v)) satisfies fρand by monotonicity so does u.\n3. Using observation6 and the previouspoint it holds that Zρ/parenleftbigg\niK(Ad(T))/parenrightbigg\n⊆\nZρ(f). For the other direction, take any B∈Zρ(f). ThenK(B) satisfies\nfρwhich by above agrees with iGd(T′). Take a leaf w∈T′whereG(w)⊆\nK(B). Then as T⊆T′the path from the root to wmust contain a\nleaf from T, say it is v. ThenG(v)⊆G(w)⊆K(B) therefore A(v) =/uniontextG(v)⊆/uniontextG(w)⊆BsoK(B) satisfies iK(Ad(T))as required.\n4. Suppose for u⊆Uit also holds that iK(A⌈(T))(u) = 1, then there must be\na leafv∈Tand a corresponding A(v)∈ Ad(T)withK(A(v))⊆u. If the\ndepth of vis≤dthenv∈ AdandiK(Ad)(u) = 1. Otherwise there must\nbe a path from root to vgoing through a node at depth d+1, giving that\niK(Bd+1)(u) = 1. Using observation 6, the claim follows.\n5. Every set in Bdcan be identified with a path from the root to a node at\nleveld. As the tree branches at each step to at most snew nodes, the\nmentioned bound follows.\n6Combining the above with observation 6 and by noting for all dit holds that\nAd⊆ Ad(T),\nZρ/parenleftBig\niK(Ad)/parenrightBig\n⊆Zρ(f)⊆Zρ/parenleftbigg\niK(Ad∪Bd+1)/parenrightbigg\nthis gives\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleZρ(f)\\Zρ/parenleftBig\niK(Ad)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleZρ/parenleftbigg\niK(Ad∪Bd+1)/parenrightbigg\n\\Zρ/parenleftBig\niK(Ad)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/summationdisplay\nB∈Bd+1/parenleftbiggn−|B|\nk−|B|/parenrightbigg\nnote that every B∈ Bd+1is constructed in d+ 1 steps, each increasing the\ncardinality by at least 1 but at most 2, therefore d+1≤ |B| ≤2d+2 meaning\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleZρ(f)\\Zρ/parenleftBig\niK(Ad)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sd+1/parenleftbiggn−(d+1)\nk−(d+1)/parenrightbigg\n≤/parenleftbiggn\nk/parenrightbigg/parenleftbigg\nsk\nn/parenrightbiggd+1\nas iniK(Ad)each clause has at most/parenleftbig2d\n2/parenrightbig\nedges, setting dsuch that t=/parenleftbig2d\n2/parenrightbig\ngives the result with g=iK(Ad).\nProof of Lemma 5. Build a tree T′the samewayasin Lemma 4 (without caring\nabout any T) using the clauses of g. WriteHdfor the collection of G(w) sets\nwherewis a node at depth exactly d. Using this tree, write gρas as-CNF if all\ntheHs+1ORclausesaresatisfiedby ρ. SimilartoClaim 7thereis |Hs+1| ≤ts+1\nand each H∈ Hs+1has exactly s+ 1 edges. Therefore a ρ∼R1/(2t)\nUsatisfies\nall the OR clauses in Hs+1with probability lower bounded by\n1−|Hs+1|/parenleftbigg1\n2t/parenrightbiggs+1\n≥1−2−s−1\nas needed.\nThen proposition 2 follows from the combination of the two switchings .\nProof of Proposition 2. First transform the circuit to having alternating AND\nand OR layers of gates and extend the bottom with dummy gates to h ave fanin\n1. This results in f′agreeing with fon all inputs, while still |f′| ≤ncs+1\nandd(f′)≤cd+1. Fix t= 2(cs+2)2log(n)2ands=n/2kthis is to ensure\nthat/parenleftBig\nsk\nn/parenrightBig√\nt/2\n=n−(cs+2)The proof relies on the changing of the bottom layer\nbetween s-CNFs and t-DNFs, after each switch, merging with the layer above\ntherefore reducing the depth.\n7Provided the bottom layer is a t-DNF, use Lemma 5 for each DNF, intro-\nducing at depth ia random ρj∼R1/(2t)\nρ−1\n(<i)(⋆)where write ρ(<i)=◦\nj<iρjfor the\ncomposition of previous restrictions. The change is unsuccessful with probabil-\nity≤2−s−1at each DNF, otherwise results in a s-CNF.\nProvided the bottom layer is a s-CNF use Lemma 4. Each gate gin the\nbottom CNF layer is replaced with g′at-DNF satisfying g′≤gand by the\nselection of parameters, it holds that\n/vextendsingle/vextendsingle/vextendsingleZρ(≤i)(g)\\Zρ(≤i)(g′)/vextendsingle/vextendsingle/vextendsingle≤/parenleftbiggn\nk/parenrightbigg/parenleftbigg\nsk\nn/parenrightbigg√\nt/2\n=/parenleftbiggn\nk/parenrightbigg\nn−cs−2.\nThe probability that any of the DNF to CNF switching fails is ≤ncs+12−s−1\nby the union bound. Write ρ=ρ(<cd+2). The total number of cliques lost from\nCNF to DNF switches is at most/parenleftbign\nk/parenrightbig\nn−1from the union bound. This gives that\nwith probability ≥1−ncs+12−s−1it is possible to find f′′at-DNF such that\n(f′′)ρ≤(f′)ρand by noting Z(f′) =/parenleftbig[n]\nk/parenrightbig\nit holds that/vextendsingle/vextendsingleZρ(f′′)/vextendsingle/vextendsingle≥/parenleftbign\nk/parenrightbig\n(1−n−1).\nSuppose f′′=/logicalortext\nj/parenleftBig/logicalandtext\ne∈qje/parenrightBig\n.To conclude the proof disjoint sets among qj\nwill be selected. Iteratively pick the sets rjfromqjsuch that they are disjoint.\nStart with r1=q1and suppose inductively that r1,...,rxare selected so far and\nthey are all disjoint. Consider\nX={A∈Zρ(f′′):∀j≤x, rj∩K(A) =∅}\nnotice that\n|X| ≥/parenleftbiggn\nk/parenrightbigg/parenleftbigg\n1−n−1−xsk(k−1)\nn(n−1)/parenrightbigg\nsince each |rj| ≤sand the possible cliques containing one particular edge has\nsize/parenleftbign−2\nk−2/parenrightbig\n. Therefore each edge in rjremoves this number of cliques at most.\nProvided |X|>0 it is possible to pick A∈Xand asf′′(K(A)∪ρ−1(1)) = 1\nthere must be one qjsatisfied by the inclusion of K(A) giving that qjis disjoint\nfrom{rj:j≤x}and it is possible to extend the collection. Therefore, at\nleastx=n(n−1)\n2sk(k−1)many disjoint clauses can be constructed, giving that after\nassigningeach remaining ρ−1(⋆) edges to ⋆with probability 1 /2tgiving an extra\nσ∼R1/2t\nρ−1(⋆)restriction, it holds that\nP/parenleftbigg\nf′′/parenleftBig\nσ−1(1)/parenrightBig/parenrightbigg\n≥1−x/productdisplay\nj=1/parenleftBigg\n1−/parenleftbigg\n1−1\nt/parenrightbigg|rj|/parenrightBigg\n≥1−/parenleftbigg\n1−1\ne2/parenrightbigg n(n−1)\n2log(n)k(k−1)\n.\nThis combined with the probability that any of the cdmany switching fails,\ngives the bound\nP/parenleftbigg\nf/parenleftBig\n(σ◦ρ)−1(1)/parenrightBig/parenrightbigg\n≥1−ncs+12−n\n2k−1−/parenleftbigg\n1−1\ne/parenrightbigg n(n−1)\n2log(n)k(k−1)\n≥1−2−n\n3k.\n8At each restriction a σ,ρj∼R1/(2t)\nUis used. Every second layer uses one such\nrestriction, and an additional restriction used at the last step, giv ing a total\nσ◦ρ∼Rp\nK([n])where\np≥/parenleftbigg1\n2t/parenrightbiggcd/2+2\n= (2cs+2)−cd−4log(n)−cd−4.\nProof of Theorem 1. Use proposition 2 with the same cdandcsconstants. This\ngives every AC0 circuit satisfying the constraints is true on a G∼Rp\nK([n])\ninput with probability 1 −2−n/3k, therefore all the fjis true on input Gwith\nprobability 1 −o(1) using the union bound, but note that for log( n)cd+6< k\nthe probability that a clique appears in Gis upper bounded by\n/parenleftbiggn\nk/parenrightbigg\n(1−p)(k\n2)≤2log(n)k−k2\nc′(cd,cs)log(n)cd+3=o(1)\nusingpfrom proposition 2. Therefore the circuit can not express clique as there\nis a nonzero probability that Gforces all the fjto true, but Gstill has no\nclique.\n5 Concluding remarks\nAs outlined in the introduction, one can write CLIQUE as an OR of/parenleftbign\nk/parenrightbig\nmany\nmonotone polynomial-sized circuits, with a top OR gate. This gives a/parenleftbign\nk/parenrightbig−1\none-sided correlation between monotone polynomial-sized circuits a nd clique,\nwhich is correct on all negative instances. One can ask the dual que stion:\nQuestion8. What isS(n,k), the maximum one-sided correlation betweenmono-\ntone polynomial-sized circuits and k-CLIQUE on nvertices, which is correct on\nall positive instances?\nHere, a partial answer is given, a S(n,k)≤2−n/4k(for large enough k), that\nis unfortunately far from the best known constructions. It would be interesting\nto know tighter bounds for S(n,k).\nConjecture 9. S(n,k) = 2−ω(n)forkin a suitable range (for example k=nα\nfor some 0< α <1).\nThis would also imply that monotone monadic second-order sentence s with\nuniversalmonadic predicatesarenot strongenoughto express k-CLIQUE,while\nmonotoneexistentialsecond-ordersentencescanexpressit. L ookingattheproof\nof Lemma 4, the inequality\n/summationdisplay\nB∈Bd+1/parenleftbiggn−|B|\nk−|B|/parenrightbigg\n≤sd+1/parenleftbiggn−(d+1)\nk−(d+1)/parenrightbigg\n9is used, following from d+ 1≤ |B| ≤2d+ 2. A better approximation of\nthe set sizes on average could provide a stronger result, in particu lar if all\n|B| ≈2d+ 2, then the resulting bound is S(n,k)≤2−O(n2/k). The following\nstronger conjecture captures this:\nConjecture 10. It holds that S(n,k)≤2−Θ(n2/k)forkin a suitable range.\nNote that while this paper discusses the monotone question, as the method\nheavily uses that condition, the general question is equally interest ing:\nQuestion 11. What is the maximum one-sided correlation between polynomi al-\nsized circuits and k-CLIQUE on nvertices, which is correct on all positive\ninstances?\nReferences\n[AB87] Noga Alon and Ravi B. Boppana. The monotone circuit complex ity of\nboolean functions. Combinatorica , 7(1):1–22, Mar 1987.\n[Ajt83] M. Ajtai./summationtext1\n1-formulae on finite structures. Ann. Pure Appl. Log. ,\n24:1–48, 1983.\n[And85] A. E. Andreev. On a method for obtaining lower bounds for t he com-\nplexity of individual monotone functions. Sov. 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In\nProceedings of the eighteenth annual ACM symposium on Theor y of\ncomputing - STOC ’86 , pages6–20,Berkeley,California,UnitedStates,\n1986. ACM Press.\n[OW07] Ryan O’Donnell and Karl Wimmer. Approximation by DNF: Exam-\nples and counterexamples. In Automata, Languages and Programming ,\npages 195–206, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg .\n10[Raz85] A. A. Razborov. Lower bounds on monotone complexity of t he logical\npermanent. Mathematical Notes of the Academy of Sciences of the\nUSSR, 37(6):485–493, June 1985.\n[Ros10] Benjamin Rossman. The monotone complexity of k-clique on r andom\ngraphs. In 2010 IEEE 51st Annual Symposium on Foundations of\nComputer Science , pages 193–201, 2010.\n[Sha49] Claude. E. Shannon. The Synthesis of Two-Terminal Switch ing Cir-\ncuits.Bell System Technical Journal , 28(1):59–98, January 1949.\n[Yao85] Andrew C-C. Yao. Separating the polynomial-time hierarchy by ora-\ncles. In Proc. 26th Annual Symposium on Foundations of Computer\nScience, page 1–10. IEEE Press, 1985.\n11"    },    {        "title": "2401.12100v1.Metrics_matter__a_Formal_comment_on_Ward_et_al_Plos_One_2016_paper___Is_decoupling_GDP_growth_from_environmental_impact_possible_.pdf",        "content": "1 \n Metrics matter : Formal  comment on Ward et al Plos-One paper (2016 ) “Is decoupling GDP growth \nfrom environmental impact possible? ” \nHervé Bercegola,b *, Paul E. Brockwayc \nAffiliation s: \na) SPEC, CEA, CNRS, Université Paris -Saclay, CEA Saclay, 91191 Gif -sur-Yvette, France.  \nb) Laboratoire Interdisciplinaire des Énergies de Demain , Université Paris Cité, CNRS, UMR  8236 -LIED, \n75013 Paris, France . \nc)  Sustainability Research Institute, School of Earth and Environment, University of Leeds, Leeds LS2 \n9JT, UK  \n* herve.bercegol@cea.fr  \nAbstract:  \nThe Ward et al . (2016) Plos-One paper is an important, heavily -cited paper in the \ndecoupling literature. The authors present evidence of 1990 -2015 growth in material and \nenergy consumption and GDP at a world level, and for selected countries. They find only \nrelative decoupling has occurred, leading to their  central claim that future absolute \ndecoupling is implausible. H owever, the authors  have made two key errors in th eir \ncollected data: GDP data is in current prices  which includes inflation , and their global \nmaterial use data is the total mass of fossil energy  material s. Strictly , GDP data should be \nin constant prices  to allow for its comparison over time , and material  inputs to an \neconomy should be the  sum of mineral  raw material s. Amending  for these errors, we find \nmuch smaller level s of energy -GDP relative decoupling, and n o materials -GDP decoupling  \nat all  at a global level.  We check these new results by adding data for 1900 -1990 to provide \na longer time  series , and find consistently low (and even no) levels of global relative \ndecoupling of material use. The central claim  for materials over the implausibility of future \nabsolute decoupling therefore not only remains valid  but is reinforced by the corrected \ndatasets . \nWard et al. (2016) paper: foundational importance in the decoupling debate  \nA circular economy in which material resources are kept in the economy as long as possible (to reduce \nthe need for ne w resources) is often presented as a difficult and necessary goal of future socio -\neconomic  transformations.  In addition, reducing energy use is one of the core climate change policy \ngoals  [1–3], in combination with a rapid switch to renewable energy. The decoupling between GDP \nand the use of material and energy resources is therefore a basic  indicator of progress towards those \ngoal s. Thus,  the study of decoupling is a hot subject for anyone interested in the sustainability of \nhuman civilization . Hence, Ward et al . [4] gathered a strong interest , illustrated by 545 citations to \ndate ( October  3rd 2023) according to the Google Scholar web site.  \nUnfortunately, two key mistakes in the presentation and use of data affect this work : namely in the \nhistorical datasets relating to GDP and minerals use. Due to the foundational importance of the Ward \net al.  [4] paper, it is important to correct the datasets, present and discuss the implications of the \nupdated results. We set out the errors and apply corrections following Ward et al.’s  [4] sections: their \nIntroduction, and their Australia case study.  We additionally provide extended historical  data to obtain \nlong -run energy and material decoupling levels from GDP, and finally conclude.  2 \n 1. Corrections to Ward et al.’s Introduc tion  \nWe introduce two key corrections to the data and results contained in the Introduction . The first \ncorrection is to the GDP data. Economic data is usually collected in the local currency that was used in \nthe economic exchanges. However, to compare rea l values of products and services, this common \ncurrency must be adjusted  for inflation, all data being converted  into a constant currency. Therefore, \nfor use in decoupling studies – i.e. to study the relation between the economic activity and the usage \nof physical  natural resources  – one has to use real GDP, expressed in constant currency , as noted by \nSemieniuk  [5], who provides an in -depth treatmen t of the GDP metric choice.  However, it appears in \nFigure 1 of Ward et al.  [4] that nominal GDP in current US$ was used  for the Introduction’s examples \nof the World, O ECD, China and Germany. Indeed, any confusion about their GDP metric choice is \nclarified by their online journal comment1 which states that their Figure 1 data uses current GDP in \nmarket prices.  \nThe second correction is to material use data , which Ward et al. [4] report they  take  from materials \ndata accessed in 2016 from Lutte r et al . [6]. We have access to Lutter et al.  [7] data published online \nin 20232, in which we find that the material use (in tonnes/yr  normalized to 1990 ) shown in their Figure \n1 (rising from index of 100 in 199 0 to ~150 in 2015  for the World case ) is much slower t han the sum of \nthe non -metallic and metallic minerals from Lutter et al.  [7], which is closer to 250 in 2015 (relative to \n100 in 199 0). Correcting the data means a much higher minerals growth rate is observed . \nWe present the original and corrected World plots in Figure 1:  \n  \n  \nFigure 1 : comparison of world level data in figure 1 of Ward et al.  [4] with a reproduction of the figure and a figure obtained \nwith the corrected and adequate data.  Scale is normalised to 100 in 1990. a) scan of original figure 1 of Ward et al.  [4] for the \nworld level.  b) reproduction of a) with data available today for materials [7], GDP [8], and energy [9]. c) corrected plot that \nshows the same energy consumption data as b), but with a series of GDP in constant international curren cy at Parity of \nPurchasing Power and corrected materials data of metals and non -metallic material consumption  (green triangles ), and all \n                                                           \n1 https://journals.plos.org/plosone/article/comment?id=10.1371/annotation/3a36e651 -ba77 -4c92 -b3a4 -\n5c2fdc14c34d  \n2 Downloading data in 2023 versus 2016 means there may be slight data differences for the same periods. \nHowever, we do not observe noticeable changes, as shown in Figure 1 a and 1b  comparison plots  \n3 \n material consumption (green crosses) . All data  for materials [7], GDP [8], and energy [9] are obtained  from the same sources \nas Ward et al.  [4] , and are normalized to 1990 values.  \nFig. 1a is a scan of the World plot from Ward et al.  [4], figure 1; Fig. 1b reproduces 1a with nominal \nGDP, fossil materials usage data and primary energy consumption series. Figure 1c shows the correct \nplot with GDP in constant internation al currency at Parity of Purchasing Power and corrected materials \ndata . Figure 1 uses the same sources  [7–9] as Ward et al.  [4], for GDP, materials and energy. As for \nmaterials data, we gathered from Lutter et al.  [7] metallic and non metallic raw material consumption \n(green triangles in Fig. 1c) and the total raw material consumption (green crosses in Fig. 1c). The total \nraw mate rials include fossil fuels and biomass added to metal and non -metallic minerals . Fig. 1c shows \nclearly that energy consumption grows slower than real GDP, which is described as relative decoupling  \n[10,11] . It also shows that  the metals and non-metallic  minerals  consumption  (green triangles  in Fig. \n1c) grow s slightly quicker than GDP at a global level . When fossil fuels and biomass are added to \nmineral materials, one obtains a curve intermediary between energy and GDP (green crosses in Fig.1c)  \nNext, in our Figure 2,  we also compare for China the original data of Ward et al.  [4], in its figure 1 (Fig. \n2a) and those reproduced (Fig. 2b) and corrected (Fig. 2c). Whereas the global ca se of Fig. 1 shows \nclearly that raw material use has been growing nearly linearly with global GDP on the period, the \nsituation is different for individual countries. In the case of China, raw material consumption (green \ntriangles  and crosses  in Fig. 2c) de couple from real GDP after 1997 . Then the total material use (crosses \nin Fig.2c) follows energy growth, whereas metallic and non -metallic minerals use ( triangles  in Fig.2c) \ngrows at an intermediary rate between energy and GDP. One also observes this latter  series cease \ngrowing after 2013. This peculiar behavior will be discussed further after broaching the case of \nAustralia (see below and Fig. 3).  \n  \n  \nFigure 2 : comparison of China data in figure 1 of Ward et al.  [4] with a reproduction of the figure.  Scale is normalised to 100 \nin 1990.  a) scan of original figure 1 of  Ward et al.  [4] for China.  b) reproduction of a) with available data today, using the \nsame sources as Ward et al.  [4] c) corrected plot with real GDP, metal and non -metallic minerals use and primary energy \nconsumption (the energy is the same as in b).  References indicate sources of the data.  \nIt appears that what is labelled materials in Ward et al.  [4] plots (Fig 1a and Fig 2a of this work) follows \nrather closely the primary energy consumption data. In our reproduction in  Fig. 1b, the so -called \n4 \n material series seems  limited to global fossil fuel consumption . In the case of China, Fig.2 b, the material \nseries of Ward et al.  [4] seems to  be the total material use, including energetic materials (fossil fuels \nand biomass), represented as crosses in Fig.  2b and 2c.  \nAt the world level, there is no indication of any deviation from a linear relation between GDP and \nmineral material consumption , as seen in Fig. 1c. Previous observations of partial decoupling of mineral \nuse and global GDP was due either to a mistake, as in Ward et al.  [4], or to a questionable  convention. \nThe last case happens when fossil fuels and biomass are mixed with other mineral extracts into a global \nextraction quantity, as in Krausmann et al . [12]. Since energy efficiency increases, the growth rates  of \nenergy materials (most of fossil fuels, and some of biomass) and other mineral resources is slower than \nGDP. In Fig 1c, we have used the sum of metallic and non -metallic raw material consumption, from the \nsame source  [7] as Ward et al.  [4], in the last 30 years, it grew at an average rate greater or equal to \nthat of GDP. If instead of mineral materials, one takes the series of all material usage, including fossil \nfuels and biomass, as in Fig.1c for the global material use (crosses), one g ets a growth rate  intermediary \nbetween that of energy and that of metals and non -metallic materials . \nOverall, f rom these revised Figures, we observe that constant prices based GDP growth rates are much \nlower : for the world ~ 3%/year (versus 5%/year for current prices), and for China ~10%/ye ar (versus \n14%/year for current prices)  in the period 1990 -2012. Second, we see that material use growth rates \nare higher than Ward et al.  [4]– we find growth rates for the world of ~3%/year (versus ~2%/year), and \nfor China 7%/year (versus 5%/year). The combined effect (lower GDP growth rate, higher material use \ngrowth rate)  is to narrow the coupling between resource use and GDP. We still observe (though \nsmaller) relative decoupling for energy -GDP. However, for materials the previous relative decoupling \nfinding has gone at the world level, where we are left with close 1 -1 cou pling. The tightening of the \ncoupling of global energy and material use with GDP is perhaps nuanced, but is important , as some \nauthors (e.g. [13,14] ) refer to  Ward et al.  [4] as a key study of observed  relative decoupling  in both \nenergy and materials . \n2. Reappraisal of Ward et al.’s Australia case study  \nHaving set their introductory framing ( i.e. only relative  decoupling  in selected  countries and at global \nlevel) , Ward et al.  [4] next use data from Hatfield -Dodds et al.  [15] on Australia’s historical energy, \nmaterials and GDP, in order to construct a simple model estimate of required material and energy \nreduction use rates to achieve a state of absolute decoupling  for Australia . Reviewing the Hatfield -\nDodds et al.  [15] data, we find the final energy use (in TJ) data appears correct, but  questions are raised  \nabout the choice of the material series. Moreover, we observe that the behavior of mineral extraction \nduring the last years, which occurred between the publication of Ward et al.  [4] and the present, \nevidence the inadequacy of the modelling of  Ward et al.  [4] and Hatfield -Dodds et al.  [15]. \nContrary to material data of the introduction, obtained from  Lutter et al.  [7], Australian data in Ward \net al.  [4] come from Scha ndl and West [16]. It is clear that the material data in Schandl and West [16] \nequates to all material consumption: fossil fuels, biomass, metal ores, industrial and construction \nminerals . We suggest that metal and non metallic minerals (industrial and construction related) should \nbe separated from energy carriers (fossil fuels and biomass). To operate this distinction, we use once \nagain the data of Lutter et al.  [7], since Hatfield -Dodds et al.  [15] and Schandl and West [16] do not \nprovide detailed numbers for separate series.   \nTo reproduce  Australian data of resource intensities  of the economy (figure 2  of Ward et al.  [4]) we \nused materia l and energy consumption series from  Lutter et al.  [7] and bp plc [9]. GDP series obtained \nfrom  the World Bank [8] were expressed in constan t Australian dollars  using the constant ratio \nbetween Australian dollars 2010 and US$ 2015 calculated by comparing World Bank [8] to Hatfield -5 \n Dodds et al.  [15] and Schandl and West [16] data . In our Fig. 3, we represent the ratio to real GDP of \nprimary energy  consumption in Fig. 3a , of all material extraction  series in Fig. 3b and separate material \nextraction series in Fig 3c.   \n  \n  \nFigure 3 : evolution of energy and raw materials intensity of the Australian economy. a) ratio of primary energy consumption \nto GDP, with exponential curve fitting. b) ratio of all materials extraction  to GDP , with data from Ward et al.  [4] (green circles) \nand from Lutter et al.  [7] (orange squares) . c) ratio of material extraction  to GDP , separating Fossil fuel and biomass (blue \ndiamonds) from metals and non mineral materials (green triangles ). Data  sources for materials [7], GDP [8], and energy [9] \nare used throughout , except green circles in b) taken from  Ward et al.  [4]. \nPrimary energy usage (Fig. 3a)  shows a clear, albeit partial decoupling from GDP, for Australia as for \nother countries or groups of country shown in figure 1 of Ward et al.  [4]. In Fig. 3b, we compare the \nAustralian material intensity data of Ward et al.  [4], with our own values calculate d using available \ndata for materials [7], GDP [8], and energy [9]. We observe that the series differ noticeably in the \nperiod 1980 -2000 but are similar in 2000 -2010. At this point we cannot explain th e discrepancy during \nthe older period (it might come from different conventions …) : we do not comment on these features. \nWhat seems more relevant to decoupling is the sudden jump of material intensity in the later period:  \nfrom 2012 to 2015, the ratio grows from 1.2 ton per 1000 Aus$ 2010 to 1.5 ton . This kind of behavior \ncannot be rendered by the exponential curve fitting used in Ward et al.  [4] and is not envisaged in \nscenarios of  Hatfield -Dodds et al.  [15]. Moreover, when ma terials with mainly an energetic use (fossil \nfuels and biomass) are separated from metal and non -metallic minerals as in Fig. 3c, one sees that the \nsudden increase of the material intensity is totally due to a similar increase (+0.3 ton/1000 Aus$ 2010) \nin the extraction of the second type. To our point of view  thus , a more relevant series to study the \nevolution of  the economic intensity of material use is that of the extraction of metals and non-metallic  \nmaterial, excluding energetic materials (fossil fuels  and biomass) . \nComing back to the case of material consumption in China (green triangles in Fig. 2c): the leveling of \nthe curve after 2013 should not be interpreted as a complete decoupling of Chinese GDP from raw \nmaterials consumption after 2013. Indeed, knowing the entanglement (imbrication) of national \noutputs in the global economy, such a statement is totally unwarranted, especially when one sees in \nFig 3c the concomitant increase of raw material intensity in Australia at the same time. Nevertheless, \n6 \n these interesting features are certainly worth some economic analyses, which are beyond the scope \nof this comment.  \n \n3. Extension of historical global data supports the no decoupling hypothesis  \nFinally, t o test the corrected Introduction results for Ward et al.  [4] but over a longer timeseries, we \nadd data for 1900 -1990. Fig 4 establishes more firmly the  linear relation between  global  GDP and usa ge \nof mineral material resources . Using data made available with  the publications of Krausmann et al .   \n[12,17]  these graphs show  that annual consumption mineral materials all follow  closely the pace of \nglobal GDP. It is both true for the last 30 years (Fig. 4a), and from the beginning of the 20th century (Fig. \n4b). When all national contributions add -up, one get the behavior of Fig. 4: the consumption of mineral \nraw materials grows proportionally to GDP and has been doing so for the last hundred years.  \n  \nFigure 4 : comparison of data for global GDP [8,18] , Primary Energy Consumption [9,19]  and raw material consumption  [7,17] . \na) from 1990, normalized to 1990 values. b) from 1900, normalized to 19 00 values , semi -log scale.  References indicate sources \nof the data.  \nAt a global level, GDP grew at an average 3% annual pace for the last hundred years, whereas energy \nconsumption showed an average 2% annual growth rate  [11], indicating thus a partial -only decoupling. \nThe so -called energy rebound effect(s) is a key candidate  to explain the inexistence of absolute \ndecoupling in available economic data  [10].  \n \n4. Conclusion  \nThe importance of using the correct data in decoupling studies is paramount , and researcher need to \npay close at tention to their datasets . Here we found in an influential paper that GDP and material use \ndata was incorrect. Overall, the corrected data means much lower levels of global relative decoupling \nof material and energy use – something we reinforce by adding d ata for 1900 -1990 to provide a longer \ntime  series. Indeed, our corrections and extensions add further weight to their central claim - of the \nimplausibility of future absolute decoupling  under historical GDP growth rate s. \n \n5. Declaration of competing interest  \nThe authors declare that they have no known competing financial interests or personal relationships \nthat could have appeared to influence the work reported in this paper.  \n \n6. Data Availability  \nAll relevant data are within the paper and its Supporting Information files.  \n7 \n  \n7. Acknowledgements  \nPaul Brockway’s time was funded by the UK Research and Innovation (UKRI) Council, supported under \nEngineering and Physical Sciences Research Council Fellowship award EP/R024254/1.  \n \n8. CRediT author  contributions  \nHB: Conce ptualization, Methodology, Validation, Formal analysis, Investigation, Data Curation, \nWriting - Original Draft, Writing - Review & Editing, Visualization . PB: Conceptualization, \nMethodology, Investigation, Writing - Review & Editing, Funding acquisition . \n \nReferences:  \n1.  Creutzig F, Roy J, Lamb WF, Azevedo IML, Bruine De Bruin W, Dalkmann H, et al. Towards \ndemand -side solutions for mitigating climate change. Nat Clim Chang. 2018;8(4):268 –71.  \n2.  Creutzig F, Niamir L, Bai X, Callaghan M, Cullen J, Díaz -José J, et al. Demand -side solutions to \nclimate change mitigation consistent with high levels of well -being. Nat Clim Chang. \n2022;12(1):36 –46.  \n3.  Grubler A, Wilson C, Bento N, Boza -kiss B, Krey V, McCollum DL, et al. A low energy demand \nscenario for meeting the 1.5 °C target and sustainable development goals without negative \nemission technologies. Nat Energy. 2018;3(June):515 –27.  \n4.  Ward JD, Sutton PC, Werner AD, Costanza R, Mohr SH, Simmons CT. I s Decoupling GDP \nGrowth from Environmental Impact Possible? PLoS One. 2016;11(10):1 –14.  \n5.  Semieniuk G. Inconsistent Definitions of GDP: Implications for Estimates of Decoupling. UMass \nEcon Work Pap Ser. 2022;20.  \n6.  Lutter S, Lieber M, Giljum S. Global  Material Flow database: Material extraction data. 2016;  \n7.  Lutter S, Lieber M, Giljum S. Global Material Flow database: 2023 [Internet]. 2023. Available \nfrom: http://www.materialflows.net  \n8.  World Bank. World Bank Open Data 2023 [Internet]. 2023. Avail able from: \nhttp://data.worldbank.org/  \n9.  bp plc. Primary energy consumption from bp -stats -review -2021 -all-data.xlsx [Internet]. 2021. \nAvailable from: https://www.bp.com  \n10.  Brockway PE, Sorrell S, Semieniuk G, Heun MK, Court V. Energy efficiency and econ omy -wide \nrebound effects: A review of the evidence and its implications. Renew Sustain Energy Rev. \n2021;141(August 2020):110781.  \n11.  Bercegol H, Benisty H. An energy -based macroeconomic model validated by global historical \nseries since 1820. Ecol Econ. 2 022;192(October 2021).  \n12.  Krausmann F, Gingrich S, Eisenmenger N, Erb KH, Haberl H, Fischer -Kowalski M. Growth in \nglobal materials use, GDP and population during the 20th century. Ecol Econ. \n2009;68(10):2696 –705.  \n13.  Kemp -Benedict E. Dematerialization , Decoupling, and Productivity Change. Ecol Econ. \n2018;150(March):204 –16.  8 \n 14.  Parrique T, Barth J, Briens F, Kerschner C, Kraus -Polk A, Kuokkanen A, et al. Decoupling \nDebunked! Evidence and arguments against green growth as a sole strategy for sustainabi lity. \n2019;11.  \n15.  Hatfield -Dodds S, Schandl H, Adams PD, Baynes TM, Brinsmead TS, Bryan BA, et al. Australia is \n“free to choose” economic growth and falling environmental pressures. Nature. \n2015;527(7576):49 –53.  \n16.  Schandl H, West J. Resource use and  resource efficiency in the Asia -Pacific region. Glob \nEnviron Chang. 2010;20(4):636 –47.  \n17.  Krausmann F, Wiedenhofer D, Lauk C, Haas W, Tanikawa H, Fishman T, et al. Global \nsocioeconomic material stocks rise 23 -fold over the 20th century and require half  of annual \nresource use. Proc Natl Acad Sci U S A. 2017;114(8):1880 –5.  \n18.  GapMinder. GapMinder World GDP Data [Internet]. 2023. Available from: \nhttps://www.gapminder.org/  \n19.  Malanima P. The limiting factor: energy, growth, and divergence, 1820 –1913. Econ Hist Rev. \n2020;73(2):486 –512.  \n "    },    {        "title": "2401.12119v1.Temperature_as_Joules_per_Bit.pdf",        "content": "arXiv:2401.12119v1  [quant-ph]  22 Jan 2024Temperature as Joules per Bit\nCharles Alexandre B´ edard1, Sophie Berthelette2,\nXavier Coiteux-Roy3, 4, and Stefan Wolf1, 5\n1Faculty of Informatics, Universit` a della Svizzera italia na, Lugano, Switzerland\n2D´ epartement de physique, Universit´ e de Montr´ eal, Montr´ e al, Canada\n3School of Computation, Information and Technology, Techni sche Universit¨ at M ¨ unchen\n4Munich Center for Quantum Science and Technology (MCQST), Muni ch, Germany\n5Facolt` a indipendente di Gandria, Gandria, Switzerland\nJanuary 2024\nAbstract\nBoltzmann’s constant reflects a historical misunderstandi ng of the concept of\nentropy, whose informational nature is obfuscated when exp ressed in J/K. We\nsuggest that the development of temperature and energy, his torically prior to\nthat of entropy, does not amount to their logical priority: T emperature should\nbe defined in terms of entropy, not vice versa. Following the pr ecepts of in-\nformation theory, entropy is measured in bits, and coincide s with information\ncapacity at thermodynamic equilibrium. Consequently, not only is the tem-\nperature of an equilibrated system expressed in J/bit, but it acquires an opera-\ntional meaning: It is the cost in energy to increase its infor mation capacity by\n1 bit. Our proposal also supports the notion of available capacity , analogous to\nfree energy. Finally, it simplifies Landauer’s cost and clari fies that it is a cost\nofdisplacement , not of erasure.\n1 A Historical Artifact\n“Careful, that’s hot!” Temperature is one of the most intuitive physical concepts,\nas we can vividly feel it through our skin. Arguably, temperature is more in-\ntuitive than energy ; and definitely more than entropy . Perhaps unsurprisingly,\nthe historical development of the three concepts has also fo llowed that order.\nThe first thermometers were built in the 17thcentury, by the end of which\nsome precursor of kinetic energy, the so-called vis viva , was introduced by\nLeibniz [1]. As Young [2] coined the term energy at the beginn ing of the 19thcentury, its existence in various forms is quickly realized . Notably, in 1824,\nCarnot [3] studied engines that could use energy of the caloric type to produce\nenergy of the work type. Clausius [4] then characterized the form of energy\nthat is inevitably lost during a thermodynamical cycle in su ch engines. Only\nthen did he introduce entropy as ∆S=Q/T , whereQis the amount of heat\nentering or escaping the system and Tis the absolute temperature of the sys-\ntem. This tortuous path to entropy explains why it carries th e units of joules\nper kelvin, which, still today, prevail as the units of entro py; they are the ones\nadvocated by the International System of Units.\nBoltzmann’s approach [5] to entropy, measured as the logari thm of the\nnumber of microstates in a given macroscopic description of a system, can\nbe thought of as the first proposal to identify entropy with an i nformation-\ntheoretic measure of capacity. If all the microstates compa tible with the macro-\nscopic description are assumed to be possible, and if the log arithm is taken in\nbase 2, Boltzmann’s entropy is a measure of capacity which is operationally\nidentified with the maximum number of bits that can be stored in the system.\nSimilar to the memory capacity of computing devices, it can b e measured in\nbits. Boltzmann’s entropy is a priori very remote from joule s per kelvin, yet,\nin order to make it consistent with Clausius’ definition, Plan ck rescaled it by\na constant kBthat carries the units of J/K. We suggest that the introduction\nof this constant is unnecessary and prevents a clear and cris p understanding\nof entropy, namely, one that is compatible not only with stea m engines but\nwith any manipulation of thermodynamic systems, including those that are\ninformation-theoretic in nature.\nTo think of entropy as joules per kelvin is thus merely a histo rical artifact,\nwhich introduces into our view of entropy all the parochial d ependencies en-\ncompassed in the development of the Kelvin scale. For instanc e, this includes\nthe aggregate behaviour of water and the number 100 — and with it, our\nphysiology: the number of our fingers. Furthermore, expressi ng entropy as\nsuch misleadingly signals that entropy must be defined in term s of tempera-\nture. But this is true only for the historically first notion of entropy, which,\nas groundbreaking as it was, cannot be expected to arise in it s final form —\nthe existence of atoms was still unknown, let alone quantum t heory and in-\nformation theory. The chronological discovery of ideas doe s not amount to\ntheir logical priority: We suggest seeing entropy as a more f undamental con-\ncept than temperature and, following the modern informatio n-theoretic view,\nmeasuring entropy in bits, i.e., inbasic information uni t.\nThe significant role of information in thermodynamics is emph asized in\nthe following section, which motivates the information-th eoretic account of\nentropy. As explained in Section 3, at thermodynamic equili brium, entropy\namounts to information capacity. In Section 4, we show how te mperature\narises as a bridge between energy and information capacity, and how it ac-\nquires an operational meaning. Not only does it yield a notio n of unused\ncapacity completely analogous to free energy, but it also sh eds light on Lan-\ndauer’s erasure.\n22 Information in Thermodynamics\nCrucial insights and results were borne by bringing informa tion-processing\nconsiderations into the picture of thermodynamics. For mod ern reviews, see\nRefs [6, 7]. A serious challenge to the second law was propose d by Maxwell [8,\n9, 10] with his famous demon who sorts a system via measuremen ts and ma-\nnipulations, thereby reducing its entropy. Szilard’s engi ne [11] has become the\ncanonical system on which a demon is imagined to perform its p uzzling ma-\nnipulations. It consists of a one-molecule gas container th at can be bisected\nby a partition, which thereupon acts as a bidirectional pist on. After insert-\ning the partition, the demon observes on which side the molec ule is trapped,\nand then expands adiabatically the piston in the opposite di rection. This ex-\ntractskBTln2 J of free energy from the molecule. After thermalization with\nthe environment, the process can be repeated, ifthe demon has kept his abil-\nity to measure and act on the system. Szilard suggested that t he second law\nshould be saved by the act of measurement of the demon, which, he thought,\nshould unavoidably create kBTln2 J of heat, namely, just enough to compen-\nsate. The idea to tie a thermodynamic cost to the act of measur ement per se has\nbeen followed by many [12, 13, 14], and was likely stimulated by the confusion\nthat arose with measurement in quantum theory.\nIn 1961, Landauer [15] correctly identified information eras ure — and not\nmeasurement — as the precise thermodynamically irreversib le step which needs\nto be compensated by heat dissipation. The erasure of one bit of information\nmust be accompanied by kBTln2 J of free energy lost to the environment, or\nto the non-information-bearing degrees of freedom of a comp uter. The rea-\nson for heat dissipation is purely physical: Information ca nnot be processed\nindependently of real devices, or as Landauer would have it, “information is\nphysical .”\nThe fact that erasure has a thermodynamic cost does not a priori preclude\nmeasurement from also having such a cost. This concern was pa rt of the\nlarger misconception which held that computing devices sho uld unavoidably\ninvolve logical irreversibility, and, with it, heat dissip ation. Upon demonstrat-\ning that universal computation can be done via logically rev ersible steps, Ben-\nnett [16] paved the way for developing thermodynamically re versible mod-\nels of computation; the most spectacular of which is perhaps Fredkin and\nToffoli’s ballistic computer [17]. In this light, the statu s of measurement has\nbeen clarified [18]: An apparatus initialized in a ready state can measure non-\ndissipatively. This yields a detailed and satisfactory res olution of Maxwell’s\nproblem: With an initialized memory, the demon canmeasure the system and\nact on it to reduce its entropy. But this is no paradox, as it me rely displaces\nthe entropy of the system onto the demon’s memory. To operate in a cycle, the\ndemon needs to get rid of the information stored — an erasure w hich, by Lan-\ndauer’s bound, dissipates a quantity of heat greater or equa l to the entropy\nreduction of the system times the environmental temperatur e.\nThe logical reversibility of dynamical laws applies to all p hysical systems,\n3including those with the ability to store information and ac t based on it. Information-\nprocessing agents cannot avoid the second law. Yet, the poin t of view of infor-\nmation processing has offered information-theoretic limi ts to what can or can-\nnot be done, thermodynamically. In particular, Landauer’s erasure cost can\nbe considered one of the many expressions of the second law, w hich, as elo-\nquently stated by Schumacher [19], mandates that “No physic al process has\nas its sole result the erasure of information.”\nThe resolution of Maxwell’s demon has been highly influential in promot-\ning the role of information theory in thermodynamics. The sc ientific literature\non the topic has been booming and ramifying in many ways. The a dvent of\nShannon’s mathematical theory of communication [20] gener ated insights [21,\n22, 23] (and debates [24, 25]). Algorithmic information the ory [26, 27, 28] per-\nmits a quantification of information based on individual obje cts, yielding more\nsophisticated notions of entropy [29, 30] and macrostates [ 31]. The significance\nof quantum information was realized [32, 33, 34, 35] and inco rporated into var-\nious thermodynamical analyses [36, 37, 38, 39, 40, 41]. And a xiomatic recon-\nstructions of thermodynamics have been suggested [42, 43, 4 4].\nThis work is part of the program of modernizing thermodynami cs — of\ncleaning the soot out of its steam pipe origins. We consider p hysics and in-\nformation theory to be both embodied in thermodynamics, rep resented by\nenergy and entropy, respectively. Temperature arises as th e bridge between\nthem. Our suggestion is simple: Entropy, not temperature, d eserves a unit of\nits own. Yet, it is potentially transformative, for it genui nely suggests a modi-\nfication to the International System of Units and a dissolutio n of Boltzmann’s\nconstant, which reduces to unity the countless “ kBln2”s found in thermody-\nnamics. That simplification arises not by mathematical conve nience, but rather\nby integrating information into physics.\n3 Thermodynamic Equilibrium: Entropy as In-\nformation Capacity\nEnergy is a fundamental physical quantity whose conservati on principle has\nboth permitted significant theoretical advancements and res isted to severe ex-\nperimental tests. In a conversation about thermodynamics, we shall be satis-\nfied with its SI unit, the joule ( J), itself expressed in terms of the second, the\nmeter, and the kilogram. However, in Section 1, we express ou r dissatisfaction\nwith the SI unit of entropy as measured in J/Kand advocate instead for the\ninformation-theoretic view of entropy, measured in bits. Ma ny such entropy\nmeasures have been proposed, most of which are operationall y tied to bits.\nFor instance, Shannon entropy H(X)is the expected number of bits required\nto communicate the outcome of a random variable Xin an optimal prefix code.\nMore convoluted measures were proposed, yet, for our purpose s, we step back\non what is perhaps the simplest: information capacity.\n4The information capacity S(E)of a system at a given energy Eis quantified\nby\nS(E) = log2Ω(E)bits,\nwhereΩ(E)is the number of possible configurations a system at energy Ecan\nadopt, and a bit is the capacity of a 2-level system. The logar ithm of a mi-\ncrostate count that corresponds to a given constraint is the gist of Boltzmann’s\nentropy. By using the base 2instead of the natural logarithm1, and prefactor-\ning it by1bit instead of kB, we obtain a measure of capacity in bits, correspond-\ning to a given energy E. Information capacity is a counterfactual property of\nthe system, namely, an assessment of all the configurations in which the sys-\ntem could be , and it does not depend on the system’s actual configuration. N or\ndoes it depend on any probabilistic assessment, thereby avo iding the question\nof subjectivity and agent dependency.\nLike a distance in space measured by the number of meters that can fit in\nthat space, the capacity of a system is measured by the number of bits that can\nfit in that system. Operationally, S(E)is the number of bits required to label\nall distinguishable states at energy E, or equivalently, the number of bits that\ncan be encoded in the system if it is seen as a storage resource .\nIn modern days, it has become second nature to quantify the st orage ca-\npacity of devices in bits. In a similar way, any physical syst em can be viewed\nas an information-storage system. Compared to the memory of a computing\ndevice that has been engineered to be stable in some relevant set of environ-\nments, a generic physical system has microstates which, for all practical pur-\nposes, cannot be prepared nor maintained in any chosen configu ration. De-\nspite this technological (but not fundamental) impractica lity for user interplay,\nit remains that information can be encoded in physical syste ms, a quantity\nwhich is bounded by its information capacity. The assumptio n of thermody-\nnamic equilibrium amounts to viewing a system’s entropy as m aximal and,\ntherefore, reaching its full capacity2.\n4 Temperature as Joules per Bit\nIn this Section, we develop the logical implication that tem perature is better\nunderstood as J/bit. The information capacity Sis a function of its internal\nenergyE, which can be inverted into a function E(S). As with the usual statis-\ntical mechanical definition of temperature, Tis obtained as the slope ∂E/∂S\nbetween energy and information capacity. Therefore, with e ntropy measured\n1A valid defence of the natural logarithm is its convenience w hen the machinery of differen-\ntial calculus is used to elaborate the theory. While true, th is is easily resolved by the conver-\nsionlog2Ω(E)bits= lnΩ(E)nat, where the nat is the natural unit of information.\n2The fact that a system at maximal entropy saturates its capac ity can be easily recognized when\nquantifying entropy in the probabilistic setting: In this c ase, the Gibbs–Shannon entropy S(ρ) =\n−/summationtextΩ(E)\ni=1pilog2piof a distribution ρover a configuration space of Ω(E)possible values is maximal\nfor the uniform distribution, and is then equal to the capaci ty.\n5in bits, temperature is in J/bit. Note that the insistence on having entropy in\nterms of temperature ( e.g.inJ/K) yields a blatant circularity for those relying\non a statistical mechanical temperature. For instance, whe ndefining tempera-\nture, Schroeder [45] justifies the survival of the kelvin by: “ Thanks to the factor\nof Boltzmann’s constant in the definition of entropy, the slop e∂S/∂E of a sys-\ntem’s entropy vs. energy graph has the units of (J/K)/J = 1/K. If we take the\nreciprocal of this slope, we get something with units of kelv ins, just what we\nwant for temperature.”\nNon-exotic systems are of positive temperature and positiv e heat capacity,\nyielding both positive first and positive second derivatives ofEwith respect\ntoS, as displayed in Figure 1.1. In thermodynamic contexts, whe re large sys-\ntems are concerned, 1bit is small compared to the relevant scale of the energy\nvs capacity graphs, and the operational meaning of the tempe rature of a sys-\ntem can be made clear.\nEnergy (E)\nInformation capacity ( S)∆S∆E\n(1)Systems usually have a concave up\nshape.\nEnergy (E)\nInformation capacity ( S)\n(2)A heat bath is a linear idealization.\nFigure 1: Graphs of energy vs capacity.\nOperational definition of temperature: it is the increase in internal energy (in J)\nrequired to increase the information capacity by 1bit.\nTemperature as such is independent of an a priori notion of heat baths , which\ninstead can be understood in terms of systems whose E(S)function is of con-\nstant slope, namely, of constant temperature, as displayed in Figure 1.2. Such\nsystems can be thought of as idealizations of very large syst ems or, in fact,\nclose-ups of some E(S)function, which then appears to be linear. The en-\nergy cost to increase the information capacity of a heat bath at temperature T\nbyxbits isxTJ. However, for systems that are not well approximated by\na heat bath, the energy cost for an additional xbits need not be linear: The\nenergy cost must be integrated over the interval of increase d information ca-\npacity.\n64.1 Available Energy and Capacity\nIn 1873, Gibbs [46] found out that systems whose entropy is no t maximal have\navailable energy — now known as free energy — as the system can be used to\nproduce work. This energy is extracted from the system by tra nsforming it\ninto a state of lower energy, while conserving its entropy, u ntil the capacity\ncurve is reached. It is thus the amount of energy that can be ex tracted from\nthe system with no need to store an excess of entropy. See Figu re 2.Energy (E)\nInformation capacity ( S)Available\nenergyAvailable\ncapacity\nFigure 2: The point’s coordinates are given by the system’s entropy an d\nenergy, respectively on the xandyaxes. The red dashed line represents\nthe available energy while the blue dashed line, the availab le capacity.\nIn the same breath, Gibbs presents what he calls the capacity for entropy , or\nas we like to view it, available capacity3. It is the number of bits that can be\nencoded in the system at no extra energy cost. Available capa city quantifies\nthe amount of structure in the system, like blanks on a tape, o r the empty\nregisters in the memory of Maxwell’s demon. The view advocate d here, in\nwhich energy and entropy are more fundamental than temperat ure, highlights\nwell the duality captured by Gibbs between available energy and available\ncapacity.\nMoreover, that duality is also captured by idealized systems that are effi-\ncient to store either energy or entropy. A battery is a system designed to keep\nthe energy available, namely, the internal energy of the sys tem can be changed\nwith no significant changes in entropy. As an example, conside r a weight in a\ngravitational field. The (potential) energy of the weight can be easily changed\nwithout affecting the information that is encoded in it. Whe n changing the\nposition of the weight, except for the energy, all the intrin sic properties of the\nsystem are unchanged. Given the interpretation of temperat ure as joules per\n3Available capacity is also known by the names negative entropy (Schr¨ odinger [47]) and negen-\ntropy (Brillouin [48]).\n7bit, a battery is hot: Its energy can be changed with minimal i nformation pro-\ncessing. On the other hand, some systems are efficient for stor ing information\n(or entropy) with no energy change — we call them tapes . Degenerate ground\nstates, or hard disk drives, are instances of those tapes. An empty tape is cold,\nsince information can be encoded in it at no energy cost.\n4.2 Landauer’s “Displacement”\nBe it classical or quantum, information processing follows the logical reversibil-\nity of the physical laws of motion. As a consequence, the info rmation (i.e.\nentropy) of an isolated system does not decrease spontaneou sly. When consid-\nering the entire universe as one isolated system, the conclu sion is straightfor-\nward: “No information is ever lost.” This statement has been recognized as a\nformulation of the second law of thermodynamics [49, 50].\nThe term “information erasure” is, therefore, a misnomer: I nformation is\nnever erased as if free energy had the power of fundamental erasure. Rathe r,\ninformation is displaced . If information leaves a system, it moves to its environ-\nment.\nThe environment, therefore, should be seen as a memory which is itself\ndescribed by an energy vs information capacity curve of the s ort shown in\nFigure 1. Landauer’s principle then reflects a clear operati onal application of\nTemperature as Joules per Bit: To accommodate one additiona l bit of informa-\ntion, the environment’s capacity must be expanded via a prec ise quantity of\nfree energy (in joules). By definition, this quantity is the te mperature Tof the\nenvironment times one bit. No unnecessary kB. No unnecessary ln2.\n5 Conclusion\nUpon arguing the primality of bits over kelvins, we offer an i nterpretation\nof thermodynamics centred on the duality between energy and information.\nTemperature arises as the nexus between them, linking joule s to bits.\nWhen a system’s entropy is maximal, its capacity is a useful p roxy for its en-\ntropy, as they coincide. The system’s temperature is then op erationally given\nby the energy cost to increase the capacity by one bit. When th e system’s en-\ntropy is not maximal, it has available energy, or available c apacity, like batter-\nies and tapes do. And viewing the environment as a storage of i nformation is\nwhat explains Landauer’s “displacement”; namely, since a b it is never erased,\nthe environment’s capacity needs to be increased by one bit t o accommodate\nit, which is achieved at the expense of TJ, whereTis the temperature’s value\nin joules per bit.\nLike energy, which takes very different forms across the man y domains of\nphysics, information might also be characterized in differ ent ways. Investigat-\ning them and interlinking information-theoretic notions w ith many of the tra-\nditional concepts of thermodynamics is part of the large res earch programme\n8briefly summarized in Section 2.\nWhile the authors are now adept of “temperature as joules/bi t,” they un-\nsurprisingly do not use these units to discuss the water temp erature when they\ngo swimming. In fact, most of them still use the Fahrenheit. N evertheless, as\n1Kis about 9.57×10−24J/bit, their preferred water temperature would be\naround2.87×10−21J/bit.\nAcknowledgements\nThe authors are grateful to Charles H. Bennett, David Deutsc h, Paul Erker,\nand Maria Violaris for fruitful discussions and thought-pro voking comments\nabout earlier versions of this work. The authors also thank J ude the Obscure\nfor its enlightening atmosphere.\nCAB’s work is supported by the Fonds de recherche du Qu´ ebec – Nature\net technologie, the Hasler Foundation, and the Swiss Nation al Science Foun-\ndation. XCR and SW acknowledge support from the Swiss Nation al Science\nFoundation (SNSF).\nReferences\n[1] Gottfried W Leibniz. Specimen dynamicum pro admirandis Naturae\nlegibus circa corporum vires et mutuas actiones detegendis , et ad suas\ncausas revocandis. Acta Eruditorum , 4:145–157, 1695.\n[2] Thomas Young. A course of lectures on natural philosophy and the mechanical\narts: in two volumes , volume 2. Johnson, 1807.\n[3] Sadi Carnot. 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Enoc1,2,\nE. Esparza-Borges1,2, A. Fukui16,1, S. Geraldía-González1,2, E. A. Gilbert17, A. P. Hatzes18, Y . Hayashi15,\nTh. Henning19, E. Herrero20, J. M. Jenkins21, J. Lillo-Box12, N. Lodieu1,2, M. B. Lund13, R. Luque22, D. Montes14,\nE. Nagel23, N. Narita16,24,1, H. Parviainen1,2, A. S. Polanski25, S. Re ffert26, M. Schlecker27, P. Schöfer3,\nR. P. Schwarz6, A. Schweitzer28, S. Seager29,10,30, K. G. Stassun31, H. M. Tabernero14, Y . Terada32,33,\nJ. D. Twicken34,21, S. Vanaverbeke35,36,37, J. N. Winn38, R. Zambelli39, P. J. Amado3, A. Quirrenbach26, A. Reiners23,\nand I. Ribas40,20\n(Affiliations can be found after the references)\nReceived 1 December 2023 /Accepted 21 January 2024\nABSTRACT\nPlanets with orbital periods shorter than 1 day are rare and have formation histories that are not completely understood. Small ( Rp<2R⊕) ultra-\nshort-period (USP) planets are highly irradiated, probably have rocky compositions with high bulk densities, and are often found in multi-planet\nsystems. Additionally, USP planets found around small stars are excellent candidates for characterization using present-day instrumentation. Of\nthe current full sample of approximately 5500 confirmed exoplanets, only 130 are USP planets and around 40 have mass and radius measurements.\nWolf 327 (TOI-5747) is an M dwarf ( R⋆=0.406±0.015 R⊙,M⋆=0.405±0.019 M⊙,Teff=3542±70 K, and V=13 mag) located at a\ndistance d=28.5 pc. NASA’s planet hunter satellite, TESS, detected transits in this star with a period of 0.573 d (13.7 h) and with a transit depth\nof 818 ppm. Ground-based follow-up photometry, high resolution imaging, and radial velocity (RV) measurements taken with the CARMENES\nspectrograph confirm the presence of this new USP planet. Wolf 327b is a super-Earth with a radius of Rp=1.24±0.06R⊕and a mass of\nMp=2.53±0.46M⊕, yielding a bulk density of 7 .24±1.66 g cm−3and thus suggesting a rocky composition. Owing to its close proximity to its\nhost star ( a=0.01 au), Wolf 327b has an equilibrium temperature of 996 ±22 K. This planet has a mass and radius similar to K2-229b, a planet\nwith an inferred Mercury-like internal composition. Planet interior models suggest that Wolf 327b has a large iron core, a small rocky mantle, and\na negligible (if any) H /He atmosphere.\nKey words. techniques: photometric – techniques: radial velocities – planets and satellites: detection – planets and satellites: terrestrial planets –\nstars: individual: Wolf 327 – stars: late-type\n1. Introduction\nOne of the most uncommon types of exoplanets discovered to\ndate are ultra-short-period (USP) planets. USP planets are de-\nfined arbitrarily as planets with orbital periods shorter than one\nday (Sahu et al. 2006; Sanchis-Ojeda et al. 2013). Owing to their\nshort planet-to-star distance, they experience levels of irradia-\ntion on the order of hundreds or thousands of times stronger\nthan Earth’s, and are thus subject to extreme temperatures. These\nplanets typically have small sizes, with planetary radii up to\nRp<2R⊕(Winn et al. 2018), and appear to have mostly Earth-\nlike compositions (Dai et al. 2019), some of them even exhibit-\ning iron-enhanced densities (e.g., Rappaport et al. 2013; San-\nterne et al. 2018; Price & Rogers 2020). USP planets have cer-\ntain characteristics that put them in a distinct category of exo-\nplanets. For example, unlike hot Jupiters, the presence of USP\nplanets does not strongly correlate with the metallicity of their\nparent star (Winn et al. 2017). Another characteristic that sep-\narates USP planets from hot Jupiters is that the former are of-\nten found in multi-planetary systems (Sanchis-Ojeda et al. 2014;\nWinn et al. 2018). Furthermore, the abundance of USP planets\nseems to depend on the spectral type of their host star; Sanchis-Ojeda et al. (2014) computed the USP planet occurrence rates\nfor di fferent stellar spectral types and found 0 .15±0.05% for\nF-type stars, 0 .51±0.07% for G-type stars, and 1 .1±0.4% for\nM-type stars. Although the measured occurrence of USP plan-\nets is higher for M dwarfs than for F dwarfs, that di fference is of\nmodest statistical significance (about 2 σ). More significant is the\nhigher occurrence around G dwarfs compared to F dwarfs (about\n4σ), making it seem reasonable that the occurrence decreases as\na function of stellar mass, in general.\nThe formation pathway of USP planets is still unknown. Be-\ncause of the high irradiation levels they experience, USP planets\ntypically lie within the region where dust is sublimated, which\nmeans that they probably had to form farther away from their\ncurrent orbit. Proposed formation mechanisms include photo-\nevaporation of sub-Neptune planets (e.g., Lundkvist et al. 2016;\nLee & Chiang 2017) and migration of rocky planets (e.g., Petro-\nvich et al. 2019).\nHighly irradiated rocky USP planets may o ffer a unique op-\nportunity to study the surface and internal composition of exo-\nplanets with current instrumentation. Planets with surface tem-\nperatures above∼1100 K will probably have large molten re-\ngions since this temperature range marks the melting point of\nArticle number, page 1 of 26arXiv:2401.12150v1  [astro-ph.EP]  22 Jan 2024A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\nrocks for Earth-like compositions (Katz et al. 2003). This sur-\nface temperature range can be achieved by USP planets owing\nto their short orbital periods and because they are most probably\ntidally locked to their star. Although USP rocky planets proba-\nbly lose their primordial atmospheres in the early stages of their\nformation owing to the e ffects of the stellar activity and winds of\ntheir host star, some volatile elements associated with the magma\noceans located in the planet’s dayside may be detected in emis-\nsion (Ito et al. 2015; Nguyen et al. 2020, 2022).\nCurrently, no confirmed detection of a USP rocky planet at-\nmosphere has been reported; however, the canonical USP rocky\nexoplanet 55 Cnc e ( P=0.73 d, Rp=1.87R⊕, and Mp=\n7.99M⊕; McArthur et al. 2004; Dawson & Fabrycky 2010;\nWinn et al. 2011; Demory et al. 2011) is a promising candi-\ndate for the detection of a secondary atmosphere with an origin\nassociated with its magma ocean. The depths of the secondary\neclipses of 55 Cnc e have shown evidence of variability at both\noptical (Meier Valdés et al. 2022; Demory et al. 2023; Meier\nValdés et al. 2023) and infrared wavelengths (Demory et al.\n2016; Tamburo et al. 2018), while the primary transit depths\ndo not exhibit any significant variability (Tamburo et al. 2018;\nMeier Valdés et al. 2023). Heng (2023) proposed an explanation\nfor 55 Cnc e observations in which a secondary atmosphere is\nproduced by geochemical outgassing events on the dayside of\nthe planet; this gas envelope is transient owing to atmospheric\nescape processes. In this scenario an eclipse with a depth con-\nsistent with zero is produced when there is no secondary atmo-\nsphere; when the outgassed atmosphere accumulates in the day-\nside, this then has the e ffect of increasing the eclipse depth at op-\ntical wavelengths owing to a combination of Rayleigh scattering\nand thermal emission; and finally fluctuations in temperature are\nresponsible for the variable infrared eclipse depth. Future JWST\nobservations may detect evidence for this proposed scenario for\n55 Cnc e; however, more observations are needed to explore\nwhether other rocky USP planets also have variable infrared sec-\nondary eclipse depths, such observations perhaps opening the\ndoor to the study of secondary atmospheres in exoplanets.\nTo further improve our understanding of the formation and\nevolution of USP planets, it is important to increase the sam-\nple of this type of object with accurately measured radii and\nmasses. In this paper we report the detection of a USP planet\naround the star Wolf 327. This star is an M2.5-type star located\natd=28.5 pc. The transit events were discovered by the Tran-\nsiting Exoplanet Survey Satellite (TESS) as part of its search for\ntransiting planets. This paper is organized as follows: in Sect. 2\nand 3 we describe TESS and the ground-based data of this tar-\nget; Sect. 4 presents the stellar properties of the host star; Sect. 5\ndescribes our data analysis methods and results; and Sect. 6 dis-\ncusses our results and puts Wolf 327b in the context of known\nUSP planets; and finally in Sect. 7 we present our conclusions.\n2. TESS observations\nThe transiting planet candidate was discovered by TESS (Ricker\net al. 2014). TESS takes photometric data continuously for ∼27 d\nand sends data to Earth every ∼13.7 d. Wolf 327 (TIC 4918918,\nTOI-5747) was observed at 2-minute cadence in sector 21 from\n2020 January 21 to 2020 February 18 and in sector 48 from\n2022 January 28 to 2022 February 26. Once the data are re-\nceived, the TESS Science Processing Operations Center (SPOC,\nJenkins et al. 2016) at the NASA Ames Research Center gener-\nates simple aperture-photometry (SAP, Morris et al. 2020) light\ncurves that are then processed to remove systematic e ffects using\nthe Presearch Data Conditioning (PDC) pipeline module (Smithet al. 2012; Stumpe et al. 2012, 2014). The light curves are\nthen scanned for transit-like signals with a wavelet-based adap-\ntive noise-compensating matched filter (Jenkins 2002; Jenkins\net al. 2010, 2020). A limb-darkened transiting planet model is\nfitted (Li et al. 2019), and a suite of false-positive detection tests\nare applied to rule out certain non-planetary scenarios (Twicken\net al. 2018). Once this transit search and model-fitting process is\ncompleted, the candidate reports are vetted by the TESS Science\nOffice (TSO) at Massachusetts Institute of Technology (MIT),\nand the planet candidates are announced to the community and\nassigned a TESS object of interest (TOI) number. On 2022\nSeptember 1, TSO announced the detection of a transiting planet\ncandidate around Wolf 327. A transit signature with an orbital\nperiod of 0.573 d ( ∼13.7 h) was detected in the SPOC search of\nthe combined light curve from the two sectors on 2022 May 26.\nThe transit signature passed all the diagnostic tests presented in\nthe SPOC Data Validation reports. The SPOC Data Validation\nprocess performs di fference imaging (out-of-transit baseline im-\nage minus in-transit image) and centroid o ffset analysis as de-\nscribed by Twicken et al. (2018) to establish the location of the\ntransit source with respect to the location of the target star. A\nmean di fference image is produced in each sector (with transits)\nfor a given threshold crossing event and then a centroid based on\nthe pixel response function is computed to determine the location\nof the transit source. Afterwards a centroid o ffset is determined\nby subtracting the location of the target star, corrected for proper\nmotion. Wolf 327 was observed in two sectors (sectors 21 and\n48); for targets observed in multiple sectors SPOC computes a\nmean centroid o ffset by robustly averaging over the centroid o ff-\nsets associated with the individual sectors. The mean o ffset lo-\ncated the source of the transit at 5 .13±4.20′′from the coordinates\nof Wolf 327. There are no other TESS Input Catalog (TIC) ob-\njects within 3 σof the mean centroid o ffset, hence all TIC objects\nare excluded as potential sources of the transit signal. After pass-\ning all the diagnostics tests performed by SPOC, the candidate\nwas assigned TOI number TOI-5747.01 by the TESS Science\nOffice.\nIn this study we have analyzed TESS PDCSAP photometry\nof Wolf 327 from sectors 21 and 48, both taken with a 2-minute\ncadence. Figure 1 shows the TESS target pixel file (TPF) pro-\nduced with tpfplotter ,1the figure shows the field around the\ntarget, highlighting in red the apertures used to compute the light\ncurves. TESS apertures were computed by SPOC’s compute op-\ntimal aperture, which selects the photometric aperture pixels for\neach target and sector. The code also calculates a “crowding met-\nric” for each photometric aperture; this is the fraction of light in\nthe photometric aperture due to the target star after background\nsubtraction. In sector 21 the crowding metric was 0.9259, there-\nfore 7.41% of the median flux level was subtracted on each ca-\ndence to account for light from other nearby stars in the PDC-\nSAP light curve. In sector 48 the crowding metric was 0.9060,\nhence 9.40% of the median flux level was subtracted to account\nfor dilution. Otherwise, the transit depths would be artificially\nsmall. This dilution correction is based on a scalar average of\neach sector and is not dynamic.\nFigure 2 shows the TESS light curves for sectors 21 and\n48; both time series clearly show the eclipses of the nearby star\nTIC 4918919 ( V=12.1 mag, J=10.7 mag). TIC 4918919 is\nlocated at a projected separation of r=55′′from Wolf 327. This\nnearby star is a known eclipsing binary classified by Gaia (Gaia\nDR3 796185373590623360), and TESS observes an eclipse ev-\nery∼3.89 d. This eclipsing binary is a background star ( Gaia\n1https://github.com/jlillo/tpfplotter\nArticle number, page 2 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nDR3 parallax π=1.53±0.02 mas) and is not bound to Wolf 327\n(Gaia DR3 parallax π=35.00±0.03 mas). Although this star\nis positioned outside TESS’s photometric aperture for both sec-\ntors (see Fig. 1) owing to its brightness, part of its point spread\nfunction (PSF) is included inside the aperture used to integrate\nthe flux of Wolf 327. As previously mentioned, the constant flux\ncontamination level of TIC 4918919 is taken into account by\nthe PDCSAP pipeline; however, this dilution correction is not\ndynamic and the eclipses are still seen in the final TESS light\ncurve. In Sect. 5.2 we describe the steps taken to account for\nthe contributions of the transiting candidate and eclipsing binary\npresent in both TESS time series.\n3. Ground-based observations\n3.1. Seeing-limited photometry\nThe TESS pixel scale is ∼21′′pixel−1and photometric apertures\ntypically extend out to roughly 1′, generally causing multiple\nstars to blend in the TESS photometric aperture. To determine\nthe true source of the TESS detection, we acquired ground-based\ntime-series follow-up photometry of the field around Wolf 327 as\npart of the validation process of the candidate.\n3.1.1. Long-term photometric monitoring\nWe searched for long-term photometric data of Wolf 327 in\narchival databases. We found a time series taken by the All-Sky\nAutomated Survey for Supernovae (ASAS-SN, Shappee et al.\n2014). ASAS-SN is a survey dedicated to finding bright super-\nnovae and uses an array of 24 robotic telescopes located in sev-\neral countries (USA, Chile, South Africa, China). The project\nprovides light curves for stars with magnitudes between 9 and 18\ningandVfilters through their Sky Patrol portal.2We searched\nthe ASAS-SN portal for Wolf 327 photometry and found ∼200\nepochs taken with the Vfilter, spanning a time baseline of ∼2060\nd. The median values of the photometric measurements are ¯V=\n12.98 mag with a median photometric uncertainty of σV=0.02\nmag; the standard deviation of the time series is σdev=0.016\nmag. During the period of time in which the star was visible, the\ncadence of the observations was roughly one visit every 3 d.\nA long term photometric campaign of Wolf 327 was carried\nout from April to December 2023 with the 0.8 m Joan Oró tele-\nscope (TJO, Colomé et al. 2010) at the Montsec Observatory in\nLleida, Spain. We obtained a total of 353 images on 61 di fferent\nnights with an exposure time of 60 s each using the Johnson Rfil-\nter of the LAIA imager, a 4k ×4k with a field of view of 30′and\na scale of 0′′.4 pixel−1. The images were calibrated with darks,\nbias, and flat fields with the ICAT pipeline (Colome & Ribas\n2006) of the TJO. The di fferential photometry was extracted with\nAstroImageJ (Collins et al. 2017) using the aperture size that\nminimized the root-mean-square (rms) of the resulting relative\nfluxes, and a selection of the ten brightest comparison stars in\nthe field that did not show variability. Then, we used our own\npipelines to remove outliers and measurements a ffected by poor\nobserving conditions or with a low signal-to-noise ratio (S /N).\nThe resulting rms of the di fferential photometry from the TJO in\ntheRfilter is∼11 ppt.\nOwing to the relatively large photometric uncertainties and\nthe cadence of the observations, it is di fficult to detect the transit\nevents in the ASAS-SN and TJO data; however, the long-term\nmonitoring of Wolf 327 provides insights into the level of stellar\n2https://asas-sn.osu.edu/photometryactivity of the star and helps to establish the rotation period of\nWolf 327 (see Sect. 4.2).\n3.1.2. MuSCAT2 photometry\nWolf 327 was observed on the night of 2022 November 6 with\nthe 1.5 m Telescopio Carlos Sánchez (TCS) at Teide Observa-\ntory, Spain. The data were acquired by the TCS MuSCAT2 im-\nager (Narita et al. 2019). This instrument is equipped with four\ncameras and is capable of obtaining simultaneous images in g′,\nr′,i′, and z-short with little overhead time.\nDuring the observations the exposure times were set to 10 s\nfor the four bands. Standard data reduction and calibrations were\nperformed by the MuSCAT2 pipeline (Parviainen et al. 2019); in\naddition to basic reduction, this pipeline also optimizes the pho-\ntometric aperture while fitting a transit model (including system-\natic e ffects) to the time series.\nSince the data covered an egress of the transit and not a full\nevent, they were not used in the final fit. However, the MuS-\nCAT2 observations were useful to detect hints of an egress on\nWolf 327, confirming the transit event on the target star and rul-\ning out the presence of hitherto unrecognized eclipsing binaries\nlocated inside TESS’s photometric aperture that could mimic a\nsignal similar to that caused by a planetary transit.\n3.1.3. LCOGT photometry\nWolf 327 was included as part of the TESS Follow-up Observing\nProgram (TFOP; Collins 2019)3. The on-target follow-up light\ncurves are also used to place constraints on the transit depth and\nthe TESS ephemeris. To schedule our transit observations we\nused the TESS Transit Finder , which is a customized ver-\nsion of the Tapir software package (Jensen 2013).\nWe observed three full transit windows of Wolf 327b on\n2022 November 19, 2022 December 2 and 2022 December 12\nin the Pan-STARRS z-short band and two in the Sloan i′band,\nrespectively, using the Las Cumbres Observatory Global Tele-\nscope (LCOGT; Brown et al. 2013) 1.0 m network nodes at Teide\nObservatory on the island of Tenerife and at McDonald Obser-\nvatory near Fort Davis, Texas, United States. The images were\ncalibrated with the standard LCOGT BANZAI pipeline (McCully\net al. 2018) and di fferential photometric data were extracted us-\ningAstroImageJ . We used circular photometric apertures with\na radius of 4′′.7 for the z-short band observations and 12′′.4 and\n5′′.8 for the two i′band observations, respectively, to extract\nthe di fferential photometry. The smaller target star apertures ex-\ncluded all of the flux from the nearest known neighbor in the\nGaia DR3 catalog ( Gaia DR3 796185407950360320), which is\n∼9′′southwest of Wolf 327. The observational mode on the\nnight of 2022 December 2 used greater defocusing and thus re-\nsulted in a larger photometric aperture that includes flux from\nthe nearest known neighbor. However the neighbor is more than\na hundred times fainter than Wolf 327 and insignificantly a ffects\nthe transit depth. The three light curves are included in the global\nmodeling described in Sect. 5.2.\nTwo full transit windows and two ingress windows of\nWolf 327b were observed simultaneously in Sloan g′,r′,i′, and\nPan-STARRS z-short bands on 2023 January 19, 2023 January\n23, 2023 March 25, and 2023 April 25 using the LCOGT 2 m\nFaulkes Telescope North at Haleakala Observatory on Maui,\nHawai’i. The telescope is equipped with the MuSCAT3 multi-\nband imager (Narita et al. 2020). The images were calibrated\n3https://tess.mit.edu/followup\nArticle number, page 3 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\n1662 1664 1666 1668 1670 1672\nPixel Column Number15241526152815301532Pixel Row Number\nE\nNm= -2\nm= 0\nm= 2\nm= 4\nm= 5\n123\n45\n67\n89101112TIC 4918918 - Sector 21\n0.51.01.52.02.5\nFlux ×103 (e/s)\n736 738 740 742 744 746\nPixel Column Number151215141516151815201522Pixel Row Number\nE\nNm= -2\nm= 0\nm= 2\nm= 4\nm= 5\n123\n45\n67\n8 9101112TIC 4918918 - Sector 48\n0.51.01.52.0\nFlux ×103 (e/s)\nFig. 1. Tpfplotter (Aller et al. 2020) target pixel file (TPF) images of Wolf 327 for sectors 21 and 48. The red squares correspond to the TESS\naperture used to compute the photometry, the size of the red circles represent the Gaia DR3 (Gaia Collaboration et al. 2023) magnitudes of the\nstars, and the gray arrows shows the Gaia DR3 proper motion directions of each star inside the field of view. Wolf 327b is positioned at the center\n(star 1), star 7 (TIC 4918919) is a known eclipsing binary.\nFig. 2. TESS PDCSAP photometry for sectors 21 and 48. The deep transit events occurring with a period of ∼3.89 d are from the known eclipsing\nbinary TIC 4918919. The solid black line shows the best-fitting model from the joint data fit described in Sect. 5.2. The vertical lines represent the\ncentral time of the transit of the USP Wolf 327b (orange) and the eclipsing times of the EB TIC 4918919 (blue).\nusing the standard LCOGT BANZAI pipeline (McCully et al.\n2018), and photometric data were extracted using AstroImageJ\n(Collins et al. 2017). We used circular apertures with a radius\nof∼5′′.0 to extract the di fferential photometry. The two ingress\nwindow observations on 2023 January 23 and 2025 March 25\nwere not included in the global modeling.\n3.2. High resolution imaging\nAs part of our standard process for validating transiting exoplan-\nets to assess the possible contamination by bound or unboundcompanions on the derived planetary radii (Ciardi et al. 2015),\nwe observed Wolf 327 with high-resolution near-infrared adap-\ntive optics (AO) imaging at Keck Observatory and with optical\nspeckle observations at Sternberg Astronomical Institute.\n3.2.1. Near-infrared AO at Keck\nObservations of Wolf 327 were taken with the NIRC2 instrument\non Keck-II behind the natural guide star AO system (Wizinowich\net al. 2000) on 2023 April 26 in the standard 3-point dither pat-\ntern that is used with NIRC2 to avoid the left lower quadrant\nArticle number, page 4 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nof the detector, which is typically noisier than the other three\nquadrants. The dither pattern step size was 3′′and was repeated\ntwice, with each dither o ffset from the previous dither by 0′′.5.\nNIRC2 was used in the narrow-angle mode with a full field of\nview of∼10′′and a pixel scale of approximately 0′′.0099442\nper pixel. The Keck observations were made in the Kcont filter\n(λo=2.2706; ∆λ=0.0296µm) with an integration time in each\nfilter of 20 s for a total of 180 s. Flat fields were generated from a\nmedian average of dark-subtracted dome flats. Sky frames were\ngenerated from the median average of the nine dithered science\nframes; each science image was then sky-subtracted and flat-\nfielded. The reduced science frames were combined into a single\ncombined image using an intra-pixel interpolation that conserves\nflux and shifts the individual dithered frames by the appropriate\nfractional pixels; the final resolution of the combined dithers was\ndetermined from the full-width half-maximum (FWHM) of the\npoint spread function (0′′.100). To within the limits of the AO\nobservations, no stellar companions were detected. The final 5 σ\nlimit at each separation was determined from the average of all\nof the determined limits at that separation and the uncertainty\non the limit was set by the root-mean-square dispersion of the\nazimuthal slices at a given radial distance (Figure 3, left panel).\n3.2.2. Optical speckle at SAI\nWolf 327 was observed on 2023 January 17 with the speckle po-\nlarimeter on the 2.5 m telescope at the Caucasian Observatory of\nSternberg Astronomical Institute (SAI) of Lomonosov Moscow\nState University. The speckle polarimeter uses a high-speed\nlow-noise CMOS detector Hamamatsu ORCA–quest (Strakhov\net al. 2023). The atmospheric dispersion compensator was ac-\ntive, which allowed use of the Icband. The respective angular\nresolution is 0′′.083, while the long-exposure atmospheric seeing\nwas 0′′.8. We did not detect any stellar companions brighter than\n∆Ic=3.9 and 6.2 at ρ=0′′.25 and 1′′.0, respectively, where ρis\nthe separation between the source and the potential companion.\nThe sensitivity curve obtained is shown in Fig. 3 (right panel).\n3.2.3. Gaia assessment\nIn addition to employing high-resolution imaging, we have uti-\nlized Gaia DR3 to identify possible wide companions bound\nto Wolf 327. Typically, these stars are already cataloged in the\nTESS Input Catalog, and their impact on the transit events has\nbeen factored into the analysis of the transits and the associated\nderived parameters. From a comparison of parallaxes and proper\nmotions (e.g., Mugrauer & Michel 2020, 2021; Mugrauer et al.\n2022, 2023), our investigation reveals no additional widely sep-\narated companions identified by Gaia .\nFurthermore, the astrometry from Gaia offers supplementary\ninsights into the potential existence of close companions that\nmight have evaded detection by both Gaia and high-resolution\nimaging. The Gaia Renormalized Unit Weight Error (RUWE)\nserves as a metric, akin to a reduced chi-square, with values\napproximately ≲1.4 indicating that the astrometric solution is\ncompatible with a solitary star. Conversely, RUWE values ≳1.4\nsuggest an excess of astrometric noise, potentially attributed to\nthe presence of an unseen companion (e.g., Ziegler et al. 2020).\nWolf 327, with a RUWE of 1 .3, aligns with this single-star\nmodel, suggesting no unseen companions causing excess astro-\nmetric noise.3.3. CARMENES radial velocity measurements\nWolf 327 was observed with the CARMENES spectrograph as\npart of an RV follow-up program of TESS candidates (22B-3.5-\n006; PI: E. Pallé). CARMENES is installed at the 3.5 m tele-\nscope at the Calar Alto Observatory in Almería, Spain. The in-\nstrument has two channels: the visible one (VIS, spectral range\n0.52–0.96µm) and the near-infrared one (NIR, spectral range\n0.96–1.71µm); with an average spectral resolution for both\nchannels of R=94 600 and R=80 400, respectively (Quir-\nrenbach et al. 2014, 2018).\nWe collected a total of 22 CARMENES spectra from 2023\nJanuary 6 until 2023 February 6, covering a time baseline of 31\nd. The exposure time used to acquire the spectra was 1800 s.\nThe data reduction was done with the CARACAL pipeline (Ca-\nballero et al. 2016b), which performs basic data reduction (bias,\nflat, and cosmic ray corrections) and extracts the spectra using\ntheFOXoptimal extraction algorithm (Zechmeister et al. 2014).\nThe wavelength calibration is done following the algorithms de-\nscribed in Bauer et al. (2015). The radial velocity measurements\nwere computed with SERVAL4(Zechmeister et al. 2018). SERVAL\nproduces a template spectrum by co-adding and shifting the ob-\nserved spectra and computes the RV shift relative to this tem-\nplate using a χ2minimization with the RV shift as a free param-\neter. The RV measurements were corrected for other e ffects such\nas barycentric motion, secular acceleration, instrumental drifts,\nand nightly zero-points (e.g., Trifonov et al. 2018; Tal-Or et al.\n2018).\nSERVAL also computes several activity indices, such as spec-\ntral line indices (the Na idoubletλλ589.0 nm, 589.6 nm; H α\nλ656.2 nm, and the Ca iiinfrared triplet λλλ849.8 nm, 854.2 nm,\n866.2 nm), and other indicators such as the RV chromatic index\n(CRX) and the di fferential line width (dLW). These indicators\nare useful for monitoring the stellar activity of the star and its\neffect on the RV measurements (cf. Zechmeister et al. 2018).\nThe uncertainties of the RV measurements derived by\nSERVAL have a median value of 1 .88 m s−1and a standard de-\nviation ofσdev=1.43 m s−1for the VIS channel. In the NIR\nchannel, the median RV uncertainty value is larger at 9 .69 m s−1,\nwith a standard deviation of σdev=4.95 m s−1. The median over-\nall signal-to-noise ratio for the spectra taken with the VIS chan-\nnel was 329, with a standard deviation of 99 (minimum S /N=\n88, maximum S /N=469). For the NIR spectra, the median S /N\nwas 446, with a standard deviation of 138 (min. S /N=111, max.\nS/N=602).\nTheSERVAL RV measurements for both channels are shown\nin Fig. 4. In this study, we used the RV data extracted using the\nVIS channel, as this dataset presented lower scatter than the NIR\nmeasurements (see Bauer et al. 2020 for a comparison between\nCARMENES VIS and NIR RVs). The CARMENES VIS chan-\nnel velocity measurements and activity indicators used in this\nstudy can be found in Tables A.1 and A.2, respectively.\n4. Stellar properties\n4.1. Stellar parameters\nThe coordinates, photometric magnitudes, astrometric values,\nand stellar parameters of Wolf 327 are listed in Table 1. The stel-\nlar parameters were computed using a combined CARMENES\nstellar spectrum. The e ffective temperature ( Teff), surface grav-\nity (log g), and iron abundance ([Fe /H]) of the star were derived\n4https://github.com/mzechmeister/serval\nArticle number, page 5 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\nFig. 3. Adaptive optics and speckle imaging of Wolf 327. Left panel: Keck NIR AO imaging and sensitivity curve for Wolf 327 taken with\ntheKcont filter. The black points represent the 5 σlimits and are separated in steps of 1 FWHM; the purple area represents the 1 σazimuthal\ndispersion of the contrast determinations (see text). The inset image of the primary target shows no additional close-in companions. Right panel:\nIcband speckle sensitivity curve of Wolf 327 taken with the SPeckle Polarimeter (SPP) at the Caucasian Observatory. No stellar companions are\ndetected with magnitudes brighter than ∆Ic=3.9 and 6.2 at ρ=0′′.25 and 1′′.0 respectively. The inset shows the speckle autocorrelation function.\n0 5 1060\n40\n20\n02040RV [m/s]\nVIS NIR\n29 30 31 32\nBJD - 2459950 [days]\nFig. 4. Wolf 327 CARMENES radial velocity measurements obtained\nusing SERVAL for the visible (orange circles) and near-infrared (blue\nsquares) channels.\nusing SteParSyn5code (Tabernero et al. 2022). SteParSyn\nmakes use of the line list (for the visible and near-infrared wave-\nlength ranges available in CARMENES data) and model grid de-\nscribed by Marfil et al. (2021). The error in Teffis the quadratic\nsum of the measurement error given by SteParSyn (8 K) and\nthe weighted average of the spreads of the di fferences to liter-\nature values (69 K) given in Marfil et al. (2021). The target’s\nspectral type was determined with ±0.5 dex accuracy from the\ncolor-, absolute magnitude-, and luminosity-spectral type rela-\ntions of Cifuentes et al. (2020). The stellar mass ( M⋆) and radius\n(R⋆) were determined following Schweitzer et al. (2019), and the\nstellar luminosity ( L) was computed following Cifuentes et al.\n(2020). We determine that Wolf 327 is an M2.5 V dwarf with an\neffective temperature of Teff=3542±70 K, and a stellar mass\n5https://github.com/hmtabernero/SteParSyn/and radius of M⋆=0.405±0.019M⊙andR⋆=0.406±0.015R⊙,\nrespectively.\nWe searched for common proper-motion companions to\nWolf 327 in the Gaia DR3 catalog up to a radius of 1◦. We im-\nposed a restriction on the parallax with a range of 35 ±5 mas,\nbracketing the parallax of Wolf 327 (see Table 1). The query re-\nturned only Wolf 327, which thus does not seem to share motion\nand distance with any other star within one degree down to the\ndepth of Gaia . At the distance of Wolf 327 ( d=28.57±1.02 pc),\nGaia is sensitive to all stellar companions and even the highest-\nmass brown dwarfs (Reylé et al. 2021; Lodieu et al. 2019).\nThe galactocentric velocities ( U,V,W) of the star presented\nin Table 1 were computed following Cortés Contreras (2017),\nusing Gaia DR3 astrometry and adopting a right-handed sys-\ntem with no solar motion correction. The derived Galactic ve-\nlocities indicate that Wolf 327 belongs to the thin disk (Montes\net al. 2001), suggesting an age >1 Gyr. According to BANYAN\nΣ6(Gagné et al. 2018), the U,V,Wvalues suggest that Wolf 327\ndoes not belong to any young moving group, a finding support-\ning a relatively old age for this star.\n4.2. Stellar rotation from seeing-limited photometry\nWe used the long-term photometric V-band data from ASAS-\nSN and the R-band data from TJO to search for periodic flux\nvariations attributable to the rotation period of the star. We at-\ntempted to fit the available datasets jointly using a unified model-\ning approach. However, when adopting a common modeling pro-\ncedure, the final model for the TJO photometry exhibited clear\nsigns of overfitting. Consequently, we decided to rely solely on\nthe ASAS-SN dataset to establish the rotation period of the star.\nIn this section, we describe our adopted results for the ASAS-SN\ndataset, while our results for TJO are presented in Section B.\nTo model the flux variations, we used a combination of a lin-\near function in addition to a periodic term using Gaussian pro-\n6https://www.exoplanetes.umontreal.ca/banyan/\nbanyansigma.php\nArticle number, page 6 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nTable 1. Stellar parameters of Wolf 327.\nParameter Value Reference\nIdentifiers\nWolf 327 Wolf (1919).\nLSPM J0953 +3534 Lépine & Shara (2005)\nKarmn J09535 +355 Caballero et al. (2016a)\nTOI 5747 TESS Alerts\nTIC 4918918 Stassun et al. (2018)\nGaia DR3 796185407950360192 Gaia Collaboration et al. (2023)\n2MASS J09533093 +3534171 Skrutskie et al. (2006)\nCoordinates and spectral type\nα(ICRS, J2000) 09:53:30.90 Gaia DR3\nδ(ICRS, J2000) +35:34:16.7 Gaia DR3\nSpectral type M2.5 V This work\nPhotometry\nB[mag] 14 .586±0.070 UCAC4\nGBP[mag] 13 .247±0.003 Gaia DR3\nV[mag] 13 .018±0.062 UCAC4\nG[mag] 11 .943±0.003 Gaia DR3\nGRP[mag] 10 .811±0.003 Gaia DR3\nJ[mag] 9 .308±0.022 2MASS\nH[mag] 8 .682±0.021 2MASS\nKs[mag] 8 .435±0.017 2MASS\nW1 [mag] 8 .320±0.023 AllWISE\nW2 [mag] 8 .211±0.020 AllWISE\nW3 [mag] 8 .104±0.021 AllWISE\nW4 [mag] 7 .838±0.165 AllWISE\nParallax and kinematics\nπ[mas] 35 .00±0.03 Gaia DR3\nd[pc] 28 .56+0.02\n−0.03Bailer-Jones et al. (2021)\nµαcosδ[mas yr−1]−87.76±0.02 Gaia DR3\nµδ[mas yr−1]−177.14±0.02 Gaia DR3\nγ[km s−1] 9 .39±0.42 Gaia DR3\nU[km s−1]−11.26±0.25 This work\nV[km s−1]−26.04±0.04 This work\nW[km s−1]−0.10±0.32 This work\nPhotospheric parameters\nTeff[K] 3542 ±70 This work\nlogg[cm/s2] 4 .87±0.05 This work\n[Fe/H] dex −0.17±0.04 This work\nvsini[km/s] <2 This work\nPhysical parameters\nM⋆[M⊙] 0 .405±0.019 This work\nR⋆[R⊙] 0 .406±0.015 This work\nL[10−4L⊙] 234 .5±1.2 This work\nProt[d] 44 .4±0.4 This work\nAge [Gyr] 4 .1+3.2\n−2.4This work\nReferences. Gaia DR3: Gaia Collaboration et al. (2023); UCAC: Zacharias et al. (2013); 2MASS: Skrutskie et al. (2006); AllWISE: Cutri et al.\n(2021).\ncesses (GPs; e.g., Rasmussen & Williams 2010; Gibson et al.\n2012) of the form\nL(t)=zpt+αt+ki jQP, (1)\nwhere zptandαare the zero point and slope of the linear func-\ntion, respectively, and ki jQPis a GP periodic kernel. For the pe-\nriodic term, we used the implementation provided by the GP\npackage George (Ambikasaran et al. 2015) and chose the quasi-\nperiodic kernel described by\nki jQP=Aexp\"−(|ti−tj|)2\n2l2−Γ2sin2 π|ti−tj|\nProt!#\n, (2)where|ti−tj|is the di fference between two epochs or observa-\ntions, Ais the variance, lis the evolution timescale, Γis the scale\nfactor and Protis the period of the sinusoidal variation, that is,\nthe rotational period of star (for a discussion of the physical in-\nterpretation of these terms see Nicholson & Aigrain 2022). We\nalso added a jitter term ( σphot jitter ) as a free parameter to fit the\nwhite noise component present in the time series.\nThe fitting procedure was performed in two steps: first we\noptimized a posterior probability function using PyDE7and then\nwe used the optimal set of parameters to start exploring the pa-\n7https://github.com/hpparvi/PyDE\nArticle number, page 7 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\nTable 2. Prior functions and fitted parameters values for the ASAS-SN\nV-band photometry of Wolf 327.\nParameter Prior Value\nzpt[mag] U(−20,20) 12.98±0.02\nα(×10−6) [mag /d]U(−50,50) 6.7+17.3\n−16.2\nA[mmag] U(10−6,15) 0.53+2.5\n−0.4\nl[d] U(1.0,1000) 841+115\n−200\nΓ U(10−6,100) 0.44+0.51\n−0.27\nProt[d]U(1.0,100) 44 .4±0.4\nσphot jitter [mmag]U(10−6,10) 1.3+1.4\n−0.9\nNotes.Urepresent a uniform prior function. The slope of the linear\nfunction (α) was computed relative to the mid-time of the time series\nTmid=2457420 HJD.\nrameter space using emcee (Foreman-Mackey et al. 2013). We\nletemcee run for 3 000 iterations as a burn-in period and then\nran the main Markov chain Monte Carlo (MCMC) procedure\nfor 10 000 iterations. The final values of the fitted parameters\nwere obtained by computing the median and 1 σpercentiles of\nthe posterior distributions. Table 2 presents the values and 1 σ\nuncertainties of our fit.\nFigure 5 shows the photometric ASAS-SN V-band time se-\nries and its periodogram. The left panel shows the median model\n(black line) computed using the posterior distributions of the fit-\nted parameters; the blue-shaded area shows 1 σuncertainty of\nthe model. The right panel shows the Generalized Lomb-Scargle\n(GLS) periodogram computed using the python implementation\nby Zechmeister & Kürster (2009).8The root-mean-square of the\nresiduals of the fit is rms=0.013 mag.\nThe periodogram for the ASAS-SN observations presents a\nsignificant peak around 44 d, which we attribute to the stellar\nrotation of the star. With our fitting procedure we find a stellar\nrotational period of Prot=44.4±0.4 d, which will be our final\nadopted value. This value is in agreement with the results found\nfor the TJO data set ( Prot=42.1+3.5\n−5.8d, see Sec. B). Finally, we\nused the age–rotation period relation described in Angus et al.\n(2015) to get an estimate of the age of the star. Using our ro-\ntation period value (including uncertainties) and equation 1 of\nAngus et al. (2015), we find an age of 4 .1+3.2\n−2.4Gyr. Considering\nthe uncertainties, this value is in agreement with the estimate\nobtained using the age–rotation relation for M dwarfs derived by\nEngle & Guinan (2018); with this relation we obtain an age of\n2.7±1.1 Gyr.\n4.3. Stellar activity indicators\nWe investigated the spectral line diagnostics of stellar activity\nacross both the VIS and NIR channels of CARMENES. In the\nCARMENES wavelength range we computed a total of sixteen\nactivity indices following the methods described in Schöfer et al.\n(2019) and Je ffers et al. (2018). Because the spectra of M dwarfs\ndo not exhibit extensive continuum regions, the spectral line ac-\ntivity indices were computed using the pseudo-equivalent width\n(pEW) of the lines by integrating the fluxes in regions centered\nat the position of the line and in two nearby regions (see Table 1\nof Schöfer et al. 2019). When dealing with the absorption bands\n8https://github.com/mzechmeister/GLSof TiO and VO present in the spectrum, the pEW proves inef-\nfective as a measure for quantifying strength. This inadequacy\narises due to the potential extension of these bands across mul-\ntiple spectral orders, where they can become blended with other\nspectral features. Instead of using pEWs, the TiO and VO ac-\ntivity levels were computed by measuring the mean fluxes in\ntwo regions and taking the ratio between these two quantities\n(see Table 3 of Schöfer et al. 2019). The same approach was em-\nployed for the iron hydride (FeH) Wing-Ford band (Wing & Ford\n1969; Schiavon et al. 1997; Schöfer 2021). The regions where\nthe mean fluxes were computed are defined by 989.8 – 990.7 nm\nand 988.7 – 989.6 nm (see Table 2.3 of Schöfer 2021). The pho-\ntospheric and chromospheric line activity indices studied here\nwere: log( LHα/Lbol), pEW(Hα), He D 3, Na D, Ca IRT-a,-b,-c,\nHe I 10830, Pa β, CaH 2, CaH 3, TiO 7050, TiO 8430, TiO 8860,\nVO 7436, VO 7942, and FeH Wing-Ford (see Table 3).\nIn addition to the absorption line indicators, we also inves-\ntigated the RV chromatic (CRX) and the di fferential line width\n(dLW) indices. These indicators serve the purpose of monitor-\ning the stellar activity of the star and assessing its impact on the\nRV measurements. The CRX index is the slope of the radial ve-\nlocity (computed in each echelle spectral order) as a function of\nthe logarithm of the wavelength (Zechmeister et al. 2018). The\ndLW index is used as a proxy to track di fferential changes in the\nabsorption line widths of the spectrum. This index is computed\nby comparing the second derivative of the high signal-to-noise\nspectrum template to the residuals from the fit for the best RVs.\nThis comparison is performed in each echelle spectral order and\nthen averaged taking into account the uncertainties of the spec-\ntra (Zechmeister et al. 2018). As is the case for the RVs, we use\nonly the CRX and dLW values derived using the CARMENES\nVIS channel. Considering all the stellar activity indices, we stud-\nied a total of twenty activity diagnostic parameters.\nOur results show that Wolf 327 is a relatively stable and H α\ninactive star. Stars with positive pEW(H α) values have H αlines\nin absorption in contrast to H α-active stars with H αin emission\nand negative pEW(H α) values. The low activity level is consis-\ntent with the∼44 day rotation period of Wolf 327.\nWe also investigated whether there was a statistical corre-\nlation between the twenty activity diagnostics and the RV val-\nues using Pearson’s r, as previously described by Je ffers et al.\n(2020). None of the twenty diagnostics had a statistically sig-\nnificant correlation with the RV which is defined as Pearson’s r\nhaving a value r>|0.7|. Given the low number of RV measure-\nments, any further statistical analysis would be misleading.\n5. Analysis and results\n5.1. Radial velocity and activity indices periodogram analysis\nWe computed the GLS periodogram of the CARMENES radial-\nvelocity and activity-indicator measurements for Wolf 327. Fig-\nure 6 shows the periodograms for the CARMENES RV measure-\nments (VIS channel), the RVs after subtracting a linear trend and\nthe planetary signal (see Sect. 5.2), and the activity indices pro-\nvided by SERVAL , the vertical black line marks the orbital period\nof the transiting candidate around Wolf 327.\nThe RV periodogram shows a broad peak consistent with the\nperiod of the transiting candidate detected by TESS. The peak\nis slightly above the 1% false alarm probability (FAP) level, al-\nthough this peak is not the one with the highest power in the\nperiodogram. The RV signature of the rotation period of the star\nis not present in any of the periodograms, probably because the\nRV measurements were taken with a time baseline shorter than\nArticle number, page 8 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\n12.92\n12.94\n12.96\n12.98\n13.00\n13.02\n13.04\n13.06V magASAS-SN\n0 500 1000 1500 2000\nHJD - 2456300 [days]0.04\n0.02\n0.000.020.04Residuals\n0.010 0.100 1.000Frequency [days1]\n1 10 100\nPeriod [days]0.000.050.100.150.20Power (ZK)Prot = 44.4 days\nFig. 5. Long-term photometric archival data of Wolf 327 from ASAS-SN ( Vband). Left panel: photometric time-series and best-fitted model\ncomputed using GP adopting a quasi-periodic kernel (black line). The 1 σuncertainty regions of the fit are shown in light blue. The residuals of the\nfit are shown in the bottom panel. Right panel: GLS periodogram (Zechmeister & Kürster 2009) for ASAS-SN V-band photometry. The vertical\ngreen line shows the fitted rotational period of the star. The horizontal lines represent the false alarm probability (FAP) levels of 10% (red dotted\nline) and 1% (blue dashed line).\nTable 3. Wolf 327 activity indicators computed from a coadded\nCARMENES VIS and NIR channel spectrum.\nIndicator Value\npEW(Hα) 0 .0256±0.0087\nlog(LHα/Lbol) Inactive\nHe D 3 0.0201±0.0083\nNa D −0.4883±0.0203\nCa IRT a 0 .0320±0.0062\nCa IRT b 0 .0302±0.0048\nCa IRT c 0 .0282±0.0031\nFe 8691 0 .0132±0.0031\nHe 10833 0 .0073±0.0006\nPaβ 0.0002±0.0007\nCaH 2 0.9197±0.0008\nCaH 3 0.7644±0.0004\nTiO 7050 0 .6506±0.0003\nTiO 8430 0 .8407±0.0007\nTiO 8860 0 .9125±0.0006\nVO 7436 0 .9237±0.0008\nVO 7942 0 .9612±0.0005\nFeH Wing-Ford 0 .9696±0.0021\nthe rotation period of the star (31 d versus Prot=44.4±0.4 d\nfrom photometric measurements) and the observation strategy,\nas the majority of the RV data acquisition was made during two\nfull nights of continuous observations.5.2. TESS, ground-based light curves, and CARMENES data\nTo derive the planetary orbital parameters we decided to use\nTESS, ground-based light curves, and CARMENES RV mea-\nsurements to perform a joint fit of the data. We use the same\nmethod as described in previous papers (e.g., Murgas et al. 2022,\n2023). To model the transits we use PyTransit9(Parviainen\n2015) and we adopted a quadratic limb darkening (LD) law. The\nfitted LD coe fficients were compared through a likelihood func-\ntion to the predicted values computed by LDTK10(Parviainen &\nAigrain 2015) using the stellar parameters presented in Table\n1. During the fitting process we used the LD parameterization\nof Kipping (2013). The CARMENES radial velocity time series\nwas modeled using RadVel11(Fulton et al. 2018).\nFor the analysis we selected only ground-based observations\nthat covered a full transit of the planet candidate and presented a\ndetection of the transit event (for details see Appendix C). In the\nend we used the observations taken on 2022 November 19, 2022\nDecember 2, and 2022 December 12 by LCOGT 1 m telescopes;\nand the 2023 January 19 and 2023 April 25 time series taken with\nLCOGT 2 m telescope using the MuSCAT3 instrument. We used\nground-based light curves that were corrected by observational\nand instrumental e ffects (i.e., detrended light curves) provided\nby the ExoFOP team.\nTo model the systematic e ffects of unknown origin present\nin TESS and ground-based photometry we used GPs. For all\nthe light curves analyzed here we adopted the commonly used\n9https://github.com/hpparvi/PyTransit\n10https://github.com/hpparvi/ldtk\n11https://github.com/California-Planet-Search/radvel\nArticle number, page 9 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\n0.10 1.00Frequency [days1]\n0.00.20.40.60.8 RV (VIS)\n0.00.20.40.60.8 RV Resi.\n0.00.20.40.60.8 Ca IRTa\n0.00.20.40.60.8 Ca IRTb\n0.00.20.40.60.8 Ca IRTc\n0.00.20.40.60.8 CRX\n0.00.20.40.60.8 dLW\n0.00.20.40.60.8 H\n0.00.20.40.60.8 Na D1\n0.00.20.40.60.8 Na D2\n0.5 1.0 10.0 50.0\nPeriod [days]0.00.20.40.60.8 WindowPower (ZK)\nFig. 6. GLS periodograms of the CARMENES RV measurements taken\nwith the optical channel, RV residuals (i.e., linear trend and planet sig-\nnal removed), spectral activity indices, and window function. The hor-\nizontal lines represent the FAP levels of 10% (purple dotted line) and\n1% (blue dash line). The vertical black line marks the orbital period of\nWolf 327b ( P=0.5734 d).\nMatérn 3 /2 kernel:\nki j=c2\nk1+√\n3|ti−tj|\nτkexp−√\n3|ti−tj|\nτk, (3)\nwhere|ti−tj|is the time between epochs in the series, and the\nhyperparameters, ckandτk, were allowed to be free, with kin-\ndicating the di fferent time series (i.e. individual TESS sectors,\nground-based observations of transit events).We performed an initial test, fitting TESS data and\nCARMENES measurements simultaneously using RV models\nwith and without GPs, and a zero-point o ffset for each observing\nnight. We found that all the methodologies tested deliver similar\nresults in the determination of the radial velocity semi-amplitude\n(see Appendix D, Fig. D.1). For the case in which we included\nGPs in the RVs to model the systematic noise present in the mea-\nsurements, we adopted a Gaussian kernel of the form\nki jRV=c2\nrvexp \n−(ti−tj)2\nτ2rv!\n, (4)\nwhere ti−tjis the time between epochs in the series, and the hy-\nperparameters, crvandτrv, were set free. We decided to use this\nkernel since it has few free parameters and assumes that points\ncloser in time will be more correlated than observations sepa-\nrated by a long time baseline. With this approach we expect to\nmodel the e ffect on the RVs of transient stellar activity events\non the star. The use of a more complex kernel, such as the pop-\nular quasi-periodic kernel, is not warranted since the RV peri-\nodogram did not show any evidence of a strong correlation with\nthe rotation period of the star.\nTo model the planetary transits and RV variations we set as\nfree parameters the planet-to-star radius ratio ( Rp/R⋆), the LD\ncoefficients ( q1,q2following Kipping 2013), the central time\nof the transit (T c), the planetary orbital period ( P), the stel-\nlar density ( ρ⋆), the transit impact parameter (b), the RV semi-\namplitude ( KRV), the host star systemic velocity ( γ0), the long-\nterm linear trend of the radial velocity variation (˙ γ) and the RV\njitter (σRV jitter ). Owing to the short orbital period of the candi-\ndate, we decided to assume a circular orbit and fix the eccen-\ntricity to 0. As shown in Fig. 2, TESS light curves show deep\neclipses caused by the nearby star TIC 4918919 (see Sect. 2); we\nmodel the contribution of this EB with its own light curve model\nby setting as free parameters the transit depth, period, central\ntime of transit, stellar density, and impact parameter. The LD co-\nefficients for TIC 4918919 were fixed to the values computed\nwith LDTK using the TESS Input Catalog (Stassun et al. 2018,\n2019) stellar parameter values for this star.\nThe fitting procedure started with the optimization of a pos-\nterior probability function with PyDE . After the optimization fin-\nished, we used the optimal parameter vector as a start of an\nMCMC procedure using emcee to explore the parameter space.\nWe ran the MCMC using a number of chains that were at least\n5.5 times the number of free parameters for 35 000 iterations\nas a burn-in stage to ensure convergence. We then ran the main\nMCMC for another 15 000 iterations. We computed the final val-\nues of the fitted parameters using the median and 1 σlimits of the\nposterior distributions.\nWe compared the results of di fferent modeling approaches\nfor the joint fit of all the datasets available (i.e. TESS, ground-\nbased photometry; CARMENES RVs) through the Bayesian in-\nformation criterion metric (BIC, Schwarz 1978). Because of the\nphotometric variability present in TESS and ground-based light\ncurves, we decided to keep the use of GPs for these datasets. In\nthe case of the RV measurements we tested models without and\nwith GPs. We computed the BIC metric for three circular orbit\nmodels using di fferent modeling approaches for the RVs: Model\n1) RVs without GPs, Model 2) RVs including GPs, and Model 3)\nRVs using the zero-point o ffset method (no GPs). The computed\nBIC di fferences between the models were ∆BIC M1−M2=−10.4\nand∆BIC M1−M3=−74.4, meaning that Model 1 presented the\nlowest BIC value. Therefore, according to the BIC metric, the\nbest model was the simplest one, i.e. transit analysis including\nArticle number, page 10 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nFig. 7. TESS phase-folded light curve of Wolf 327b after subtracting\nthe photometric variations from the time series (top panel). The best-fit\ntransit model is shown in black, the circles are TESS binned observa-\ntions. Bottom panel : residuals of the fit.\nGPs for TESS and ground-based observations and no GPs for\nthe RV measurements. Hence, we adopted the parameter values\nof Model 1 as our final results. Using this approach we had a\ntotal of 50 free parameters with which to model the planet can-\ndidate (transits and RVs), the eclipsing binary TIC 4918919 in\nTESS light curves, and systematic e ffects.\nTable 4 presents the median and 1 σuncertainty range for the\nfitted and derived parameters for Wolf 327b. In Appendix D, Fig-\nure D.2 shows the posterior distributions of the transit and RV\nmeasurements fitted parameters (without the GP hyperparame-\nters for easy viewing). Figures 7, 8, and 9 present the best-fitting\nmodel for TESS and ground-based light curves, and RV mea-\nsurements, respectively. Figure 10 shows the phase-folded TESS\ndata and best-fitting model for the eclipsing binary TIC 4918919.\nThe radial velocity residuals (see Fig. 9 middle panel) shows a\nlinear trend with time. This trend may be caused by another mas-\nsive object in the system; however, the relatively short baseline\nof the RV observations (31 d) does not allow us to put limits on\nthe mass of the object. The potential presence of other planets in\nWolf 327 system is in agreement with what is known of the USP\npopulation (Sanchis-Ojeda et al. 2014; Winn et al. 2018); how-\never, more RV observations are needed to confirm the presence\nof more objects in the system and rule out certain stellar activity\ninduced signal that may be the caused of this linear trend.\nUsing our fitted values and the stellar parameters presented\nin Table 1, we find that Wolf 327b has a radius of Rp=1.24±\n0.06R⊕and a mass of Mp=2.53±0.46M⊕. This planet orbits\nits host star with a period of P=0.57347 d (13.76 h) and has an\nequilibrium temperature of Teq=996±22 K.\n6. Discussion\n6.1. Comparison with other exoplanets\nOnce we confirmed the planetary nature of Wolf 327b, we stud-\nied how it compares to the known population of USP planets.\nFigure 11 shows a period versus radius diagram of known exo-\nplanets with short orbital periods ( P<1 d), similar to the figures\npresented in the studies of Adams et al. (2021) and Morello et al.(2023). The planets with radius and mass measurements are rep-\nresented with circles, the size and color of the symbols being\nproportional to the mass of the planet; planets with only radius\nmeasurements are marked by the green squares. The figure high-\nlights the range of sizes presented by USP planets: from Jupiter-\nsized planets, to a less populated sub-Jovian region, and with a\nmore populated region inhabited by small ( Rp<3R⊕) planets.\nThe radius of Wolf 327b ( Rp=1.24±0.06R⊕) places it well be-\nlow the sub-Jovian desert and in a region where the typical mass\nrange goes up to a few tens of Earth masses, in agreement with\nour measured planetary mass.\nFigure 12 shows the mass and radius values of known tran-\nsiting exoplanets (data taken from TEPcat12catalog; Southworth\n2011). In the figure we also show the composition models of\nZeng et al. (2016, 2019). As expected for small USP planets,\nWolf 327b’s position in this diagram indicates that this planet\nprobably has a rocky composition, with little to no atmosphere,\nand with a higher density than that corresponding to an Earth-\nlike composition for the planet’s range of mass and radius.\nIn terms of mass and radius, the USP planet with the most\nresemblance to Wolf 327b is K2-229b. This planet was discov-\nered by Santerne et al. (2018), who measured an orbital of pe-\nriod of P=0.58 d (13.9 h) and estimated a planetary radius\nofRp=1.17±0.07R⊕, a mass of Mp=2.59±0.43M⊕and\na bulk density of ρp=8.9±2.1 g cm−3. K2-229b is the in-\nner planet of a planetary system, with a second planet in a 8.32\nd orbit (K2-229c, Rp=2.12±0.11R⊕,Mp<21.3M⊕) and\na possible third transiting candidate in a 31 d orbit (K2-229d,\nRp=2.65±0.24R⊕,Mp<25.1M⊕). Incidentally, given the\ntrend observed in the CARMENES RV data, Wolf 327b might\nalso be part of a multiplanet system. Although K2-229b is sim-\nilar in terms of size and mass, this planet has an equilibrium\ntemperature of Teq=1960±40 K because it is orbiting a star\nhotter than Wolf 327, a K0 dwarf ( Teff=5185±32K). Santerne\net al. (2018) modeled the planetary interior composition of K2-\n229b with only a metallic core and a silicate mantle, and found a\ncore-mass fraction of 68%, making K2-229b a planet with an en-\nlarged core in comparison with its mantle, similar to Mercury. In\nthe following section we explore the composition of Wolf 327b\nusing theoretical planetary interior models.\n6.2. Planet interior modeling\nWe utilized ExoMDN13(Baumeister & Tosi 2023) to model the\ninterior of Wolf 327b, using as input our derived values for the\nplanetary mass, radius, and equilibrium temperature while tak-\ning into account their uncertainties. ExoMDN is an exoplanet inte-\nrior inference model that adopts a machine-learning approach.\nThe program uses a set of mixture density networks (MDN)\ntrained on 5.6 million synthetic planet models computed with\ntheTATOOINE code (Baumeister et al. 2020; MacKenzie et al.\n2023). The planets from the training set have masses below\n25M⊕, equilibrium temperatures in the range 100–1000 K, and\nhave compositions consisting of an iron core, a silicate man-\ntle, a water and high-pressure ice layer, and an H /He atmo-\nsphere. Since our derived equilibrium temperature for Wolf 327b\n(Teq=996±23 K) places the planet at the edge of tempera-\nture range of ExoMDN’s training dataset, we recommend that the\nresults obtained by this model should be considered just as an\napproximation.\n12https://www.astro.keele.ac.uk/jkt/tepcat/\n13https://github.com/philippbaumeister/ExoMDN\nArticle number, page 11 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\nFig. 8. Ground-based individual transit observations of Wolf 327b used in the joint fit, systematic e ffects were removed from the light curves. The\npoints show the individual observations, the circles represent binned data points, and the best-fit model is shown by the black line.\nFigures 13 and 14 present the thickness and mass fraction of\nthe interior layers of Wolf 327b computed with ExoMDN . The\ninterior model predicts that the planet is dominated in terms\nof layer size and mass fraction by a large planetary iron core;\nWolf 327b’s core could take up as much as 78% of the planet’s\nsize and 93% of its mass. For the other layers the model pre-\ndicts a relatively small mantle, and a negligible fraction of wa-\nter and gas layers. The predicted lack of an atmosphere is in\nagreement with what is expected for highly irradiated planets of\nrelatively small sizes. Interestingly, ExoMDN ’s prediction of a rel-\natively large planetary core is in agreement with the simulations\nmade for K2-229b in Santerne et al. (2018), thus confirming the\nsimilarities between these two planets.\n6.3. Prospects for atmospheric characterization\nStudying the prerequisites for habitability and the orbital lim-\nits of the habitable zone requires the characterization of both\ntemperate and hot planets (e.g., Turbet et al. 2019; Schlecker\net al. 2023). With Sp=233.9±18.3S⊕, Wolf 327b is well\nwithin the latter regime (Kopparapu et al. 2013, 2014). If the\nplanet accreted significant amounts of water or other volatiles\nduring its formation, it probably went through a runaway green-house phase with an inflated atmosphere (Ingersoll 1969; Kast-\ning 1988; Turbet et al. 2020; Mousis et al. 2020), in which\nwater is removed through photolysis and subsequent hydrody-\nnamic escape of the hydrogen (Wordsworth & Pierrehumbert\n2013, 2014). The expected past evolution of the planet’s instella-\ntion (Bara ffe et al. 2015) makes it likely that the planet never left\nthis phase before being completely desiccated (Luger & Barnes\n2015). Our density estimate of Wolf 327b (see Fig. 12) is consis-\ntent with a post-runaway greenhouse state without an extended\natmosphere (Turbet et al. 2020), which may have been further\ndepleted through intense irradiation at the orbital distance of the\nplanet (Owen & Wu 2013).\nTo evaluate the possibility of further detailed studies of\nthis planet we computed the transmission spectroscopy metric\n(TSM) and emission spectroscopy metric (ESM) proposed by\nKempton et al. (2018). Both metrics are used to evaluate the\nfeasibility of detecting atmospheric signals during a primary\ntransit (TSM) or detecting the eclipse (ESM) with JWST. For\nWolf 327b we find a TSM of 13, Kempton et al. (2018) rec-\nommends a threshold of TSM >10 for planets with radii of\nRp<1.5R⊕which marks this planet as a potential candidate\nfor future transmission spectroscopy studies. However, owing to\nthe high levels of irradiation su ffered by Wolf 327b, it is possi-\nble that this planet has lost its original gas envelope. As for the\nArticle number, page 12 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nTable 4. Prior functions, fitted, and derived parameters of Wolf 327b. The fitted GP hyperparameters were left out on purpose for easy viewing.\nParameter Prior Value\nFitted transit and orbital parameters\nρ∗[g cm−3] N(8.5,1.08) 8 .54±0.30\nRp/R∗ U(0.0015,0.10) 0 .0280±0.0007\nTc−2 457 000 [BJD] U(2252.8167,2253.1167) 2252 .9811±0.0003\nP[d] U(0.4734,0.6734) 0 .5734745±0.0000003\nb=a/R∗cos(i) U(0,1) 0 .47±0.03\ne 0 (fixed)\nKRV[m/s] U(0,60) 3 .54+0.61\n−0.62\nγ0[m/s] U(−22.2,27.8) 4 .15+0.65\n−0.68\n˙γ[m/s d−1] U(−100,100) −0.22+0.04\n−0.05\nσRV[m/s] U(0.01,12.0) 0 .69+0.71\n−0.48\nDerived orbital parameters\na/R∗ 5.30±0.06\ni[deg] 84 .89+0.41\n−0.38Derived planet parameters\nRp[R⊕] 1 .24±0.06\nMp[M⊕] 2 .53±0.46\nρp[g cm−3] 7 .24±1.66\ngp[m s−2] 16 .0±3.3\na[au] 0 .0100±0.0004\n⟨Fp⟩[103W m−2] 318 .3±25.0\nSp[S⊕] 233 .9±18.3\nTeq(ABond=0.3) [K] 996 ±22\nFitted LD coe fficients\nq1T ES S U(0,1) 0 .271±0.007\nq2T ES S U(0,1) 0 .260±0.007\nq1g U(0,1) 0 .598±0.025\nq2g U(0,1) 0 .309±0.013\nq1r U(0,1) 0 .605±0.034\nq2r U(0,1) 0 .276±0.022\nq1i U(0,1) 0 .344±0.013\nq2i U(0,1) 0 .258±0.009\nq1z U(0,1) 0 .223±0.006\nq2z U(0,1) 0 .254±0.007\nDerived LD coe fficients\nu1T ES S 0.270±0.005\nu2T ES S 0.250±0.010\nu1g 0.477±0.013\nu2g 0.296±0.026\nu1r 0.430±0.025\nu2r 0.349±0.040\nu1i 0.303±0.007\nu2i 0.284±0.015\nu1z 0.240±0.004\nu2z 0.233±0.010\nNotes.U,Nrepresent uniform and normal prior functions respectively. ABondis the Bond albedo. The term ˙ γwas computed relative to Tbase=\n2459966.0 BJD.\npossibility of eclipse detection, we find an ESM of 12 .5; the rec-\nommended ESM threshold for rocky planets is of 7 .5 which in-\ndicates that this planet is also a good candidate for eclipse spec-\ntroscopy with JWST. Both emission and transmission JWST ob-\nservations would be a good way to test the hypothesis that the\nplanet has little or no atmosphere.At the moment of submission, no JWST results on USP\nrocky planets had been reported. Early JWST transmission spec-\ntroscopy studies have focused on small planets with orbital peri-\nods longer than 1 d and equilibrium temperatures below 1000 K.\nThese planets are: TRAPPIST-1b ( P=1.51 d, Rp=1.11R⊕, and\nTeq≈400 K; Greene et al. 2023; Ih et al. 2023), TRAPPIST-1c\n(P=2.42 d, Rp=1.09R⊕, and Teq≈340 K; Zieba et al. 2023;\nArticle number, page 13 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\n55 60 65 70 75 8015\n10\n5\n051015202530RV [m s1]\nCARMENES2023.02 2023.04 2023.06 2023.08 2023.10Year\na)\n55 60 65 70 75 80\nBJD - 2459900.0 [days]16\n8\n0816Residuals\nb)\n0.4\n 0.2\n 0.0 0.2 0.4\nPhase10\n5\n0510RV [m s1]\nc) Pb = 0.57 days\nKb = 3.54 m s1\neb = 0.00 \nFig. 9. Wolf 327b radial velocity measurements taken with\nCARMENES. (a) RV time series and best-fitting model (blue line). The\nbest-fitting model was computed using the median values of the pos-\nterior distribution of the fitted parameters. (b) Residuals of the fit after\nsubtracting the single-planet Keplerian, the velocity linear trend seen\nhere was included in the RV modeling. (c) Phase-folded RV measure-\nments.\nFig. 10. Top panel : TESS phase-folded light curve of the eclipsing bi-\nnary TIC 4918919. The photometric variations modeled by the GPs\nwere removed from the time series. The best-fit transit model is shown\nin black, the circles are TESS binned observations. Bottom panel : resid-\nuals of the fit.\nLincowski et al. 2023), LHS 475b ( P=2.03 d, Rp=0.99R⊕,\nandTeq≈590 K; Lustig-Yaeger et al. 2023), and GJ 486b\n(P=1.46 d, Rp=1.30R⊕, and Teq≈700 K; Moran et al. 2023).\nThese planets are not considered to be USP planets and were\nexpected to have some gaseous envelope. The observations for\nTRAPPIST-1b and 1c are consistent with a thin atmosphere or a\nbare rock surface, for LHS 475b JWST found a featureless trans-mission spectrum, and for GJ 486b the transmission spectrum\ncan be explained by either the presence of water or by the e ffects\nof unocculted spots in the transmission spectrum. As discussed\nin Sect. 1, the USP rocky planet 55 Cnc e is a prime candidate\nfor JWST observations and has two approved Cycle 1 programs.\nBrandeker et al. (2021) will study whether the planet is in a 3:2\nspin-orbit resonance to explain the secondary eclipse depth vari-\nability, and Hu et al. (2021) will try to detect any volatiles in the\natmosphere of 5 Cnc e using the spectral features of H 2O, CO,\nCO 2, and SiO.\nDuring the reviewing process of this article, Zhang et al.\n(2024) announced the first JWST results on a USP planet. The\ntarget of the study is GJ 367b (Lam et al. 2021), a USP sub-Earth\nwith a radius and mass of Rp=0.699±0.024 R⊕andMp=\n0.633±0.050 R⊕(Goffo et al. 2023), respectively, and an or-\nbital period of 0.322 d (7.7 h). Zhang et al. (2024) measured the\nphase curve of GJ 367b in the mid-infrared wavelength range (5–\n12µm) using the Mid-Infrared Instrument (MIRI; Kendrew et al.\n2015) on board of JWST. The team measured an eclipse depth of\n79±4 ppm, with a lack of any detectable atmosphere, and featur-\ning a dark surface (i.e., low albedo) within the MIRI bandpass.\nThe measured emission spectrum is consistent with a blackbody\nspectrum. The measured low albedo value for GJ 367b is consis-\ntent with what is expected for lava worlds with molten surfaces\n(Essack et al. 2020).\n6.4. Planet-star interaction\nIn this section we study the prospects for detecting radio emis-\nsion from magnetic star-planet interaction between Wolf 327 and\nits newly discovered planet. The phenomenon behind this emis-\nsion is the electron cyclotron maser (ECM) instability (Melrose\n& Dulk 1982), which can generate auroral radio emission in both\nthe planet and its host star. The characteristic frequency of this\nemission is given by the electron gyrofrequency, νG=2.8B(in\nMHz), with Bbeing the local magnetic field (in G). The resulting\nnonthermal emission has a coherent nature, with a high degree\nof circular polarization (close to 100% in some instances), and\ncan have a large bandwidth ( ∆ν∼νG/2).\nWe note that any potential radio emission coming directly\nfrom the planet is most probably impossible to detect. Since the\nmagnetic field of a planet is merely a few gauss, any resulting\nemission would fall well below the ionosphere cut-o ff. However,\nif the planet is in the sub-Alfvénic regime, where the speed of\nthe plasma wind relative to the planet is smaller than the Alfvén\nspeed, then energy and momentum can be transferred back to the\nstar via Alfvén wings, and coherent electron-cyclotron emission\ncan originate close to the star. Since the global magnetic fields of\nM-dwarf stars are typically larger than 100 G, sometimes reach-\ning several kG, the electron gyrofrequency would be of a few\nhundred MHz, or even a few GHz, making the resulting auroral\nemission potentially detectable with current radio telescopes.\nWe estimated the stellar magnetic field of Wolf 327 using\nthe relations in Reiners et al. (2022), which depend on the value\nof the Rossby number, Ro. From the two di fferent Roestimates\nin Wright et al. (2018), we get a magnetic field of the star,\nB⋆≈292 G. To estimate the radio emission arising from star–\nplanet interaction, we used the models in Pérez-Torres et al.\n(2021), where we consider two di fferent magnetic field geome-\ntries: a closed dipole and an open Parker spiral. We assumed that\nthe stellar wind was isothermal, with T=3×106K, and that the\nsolid angle covered by the ECM emission was 1.6 steradians. We\nconsidered two di fferent emission models, Saur /Turnpenney’s\nand Zarka /Lanza’s, which are respectively shown as orange and\nArticle number, page 14 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\n0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\nPeriod [days]110Radius [R]\nUltra-Hot-Jupiters\nSub-Jovian Desert (Rp 3-10 R)\nRocky USP Planets\n0.00.51.01.52.02.53.03.5\nlog10(Mp [M])\nFig. 11. Orbital period versus radius dia-\ngram for ultra-short-period planets. Confirmed\nplanets with only radius measurements are\nshown with green squares, and planets with\nmass and radius values are shown with cir-\ncles. The color and size of the circles repre-\nsent the mass of the planet. The position of\nWolf 327b is marked by the red star. Data taken\nfrom NASA Exoplanet Archive ( https://\nexoplanetarchive.ipac.caltech.edu/ ).\n0.5 1.0 2.0 4.0 6.0 8.0\nMass [M]\n0.60.81.01.21.41.61.82.0Radius [R]\n100% Fe 100% MgSiO3 Earth-like\nFig. 12. Mass–radius diagram for Wolf 327b\n(red star) and known transiting planets with\nmass determinations with a precision better\nthan 30% (parameters taken from the TEPcat\ndatabase; Southworth 2011). Planets orbiting\nM-type stars ( Teff≤4000 K) are marked with\ncircles, orange indicate planets with periods\nP>1 d, and red circles indicate USP planets\n(P≤1 d) around M dwarfs. The lines in the\nmass–radius diagram represent the composition\nmodels of Zeng et al. (2016, 2019) for plan-\nets with pure iron cores (100% Fe, solid brown\nline), Earth-like rocky compositions (32.5% Fe\nplus 67.5% MgSiO 3, dashed green line) and\npure rock (100% MgSiO 3, dash-dotted purple\nline).\nblue in the plots. For each model we consider e fficiency values\nranging from 0.2 to 2%. Since the planet is very close to the star,\nthe interaction does indeed take place in the sub-Alfvénic regime\n(see top panels of Fig. 15).\nWe show in Fig. 15 (panels aandb) the radio flux density\nfrom sub-Alfvénic star–planet interaction as a function of the\nmass-loss rate of the star and find that the emission is strong and\nvery similar in both a close dipolar and an open Parker spiral ge-\nometry of the magnetic field. Assuming a conservative threshold\nvalue of 100 µJy for the rms obtained in a given observation, we\nshould be able to detect radio emission from star–planet interac-\ntion, regardless of the geometry, e fficiency, model, or magnetic\nfield of the planet.\n7. Conclusions\nWe report the discovery of Wolf 327b, an ultra-short-period\nsuper-Earth orbiting around an M2.5 V dwarf with a period of\n0.573 d. The detection of the transit events was led by NASA’s\nTESS mission, and ground-based observations confirmed that\nthe transit events occur on the star Wolf 327. Follow-up RV mea-\nsurements taken with the CARMENES spectrograph allowed usto establish the mass of the candidate and confirm its planetary\nnature.\nFrom a coadded CARMENES mean stellar spectrum we de-\ntermined stellar parameters for the host star. We estimate that\nWolf 327 has an e ffective temperature of Teff=3542±70 K, an\niron abundance of [Fe /H]=−0.17±0.04 dex, a derived stellar\nmass of M⋆=0.405±0.019 M⊙, and a derived stellar radius\nofR⋆=0.406±0.015R⊙. Using long-term ground-based pho-\ntometric observations, we estimate a stellar rotation period of\nProt=44.4±0.4 d.Gaia DR3 measurements indicate the host\nstar does not present any proper-motion companions, based on\nthe star’s Galactic velocity components it is likely that it does\nnot belong to any young moving group association.\nWe analyzed photometric data taken by TESS (sectors 21\nand 48), five ground-based individual full-transit observations,\nand 22 RV measurements taken with the CARMENES spec-\ntrograph. All the datasets were fitted simultaneously using an\nMCMC procedure that included the contribution of red noise\nsources for each time series modeled by Gaussian processes.\nWe note that after subtracting the signal of the USP planet from\nthe RV measurements, the residuals present a slope that might\nindicate the presence of an additional companion in the sys-\ntem. From our joint data fit we find that Wolf 327b is a super-\nArticle number, page 15 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\n0.00.20.40.60.81.0dMantle\n0.00.20.40.60.81.0dWater\n0.0 0.5 1.0\ndCore0.00.20.40.60.81.0dGas\n0.0 0.5 1.0\ndMantle\n0.0 0.5 1.0\ndWater0.0 0.5 1.0\ndGasdCore=0.78+0.11\n0.14\ndMantle=0.05+0.14\n0.05\ndWater=0.07+0.10\n0.05\ndGas=0.07+0.08\n0.02\nFig. 13. Predicted thickness of the interior layers of Wolf 327b com-\nputed with ExoMDN .\n0.00.20.40.60.81.0wMantle\n0.00.20.40.60.81.0wWater\n0.0 0.5 1.0\nwCore0.00.20.40.60.81.0wGas\n0.0 0.5 1.0\nwMantle\n0.0 0.5 1.0\nwWater0.0 0.5 1.0\nwGaswCore=0.93+0.06\n0.21\nwMantle=0.06+0.21\n0.05\nwWater=0.01+0.04\n0.01\nwGas=0.00+0.00\n0.00\nFig. 14. Predicted mass fraction of the interior layers of Wolf 327b com-\nputed with ExoMDN .\nEarth with a radius of Rp=1.24±0.06R⊕and mass of\nMp=2.53±0.46M⊕. These measurements imply that this planet\nhas an iron-like bulk density of ρp=7.24±1.66 g cm−3. Owing\nto its short orbital period ( P=0.57347 d), the planet has an equi-\nlibrium temperature of Teq=996±22 K. Given its density and\nequilibrium temperature, it is likely that Wolf 327b possesses a\nrocky composition with little to no atmosphere.In terms of its mass and radius Wolf 327b, is very similar to\nthe known transiting planet K2-229b. Previous studies of K2-\n229b indicate that this planet probably possesses a large iron\ncore, with an internal structure similar to that of Mercury. Planet\ninterior modeling applied to Wolf 327b arrives at a similar result:\nthe internal structure of this planet is probably characterized by\na substantial iron core, which dominates both in terms of layer\nsize and mass fraction. Surrounding this core is a comparatively\nsmall mantle layer, accompanied by a negligible atmosphere (if\nany).\nOverall, Wolf 327b is an interesting addition to the known\nUSP planets around M dwarfs. This planet presents an interest-\ning internal structure and favorable metrics for secondary tran-\nsit detection with the JWST. Furthermore, since USP planets are\nusually found in multi-planetary systems, Wolf 327 also presents\na good opportunity for additional RV follow-up to search for\nplanets in wider orbits (as the data so far seem to indicate).\nAcknowledgements. CARMENES is an instrument at the Centro Astronómico\nHispano en Andalucía (CAHA) at Calar Alto (Almería, Spain), operated jointly\nby the Junta de Andalucía and the Instituto de Astrofísica de Andalucía\n(CSIC). CARMENES was funded by the Max-Planck-Gesellschaft (MPG),\nthe Consejo Superior de Investigaciones Científicas (CSIC), the Ministerio de\nEconomía y Competitividad (MINECO) and the European Regional Develop-\nment Fund (ERDF) through projects FICTS-2011-02, ICTS-2017-07-CAHA-\n4, and CAHA16-CE-3978, and the members of the CARMENES Consortium\n(Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Lan-\ndessternwarte Königstuhl, Institut de Ciències de l’Espai, Institut für Astro-\nphysik Göttingen, Universidad Complutense de Madrid, Thüringer Landesstern-\nwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte,\nCentro de Astrobiología and Centro Astronómico Hispano-Alemán), with ad-\nditional contributions by the MINECO, the Deutsche Forschungsgemeinschaft\n(DFG) through the Major Research Instrumentation Programme and Research\nUnit FOR2544 “Blue Planets around Red Stars”, the Klaus Tschira Stiftung,\nthe states of Baden-Württemberg and Niedersachsen, and by the Junta de An-\ndalucía. This paper includes data collected by the TESS mission, which are pub-\nlicly available from the Mikulski Archive for Space Telescopes (MAST). Fund-\ning for the TESS mission is provided by NASA’s Science Mission directorate.\nResources supporting this work were provided by the NASA High-End Com-\nputing (HEC) Program through the NASA Advanced Supercomputing (NAS)\nDivision at Ames Research Center for the production of the SPOC data prod-\nucts. We acknowledge the use of public TESS data from pipelines at the TESS\nScience O ffice and at the TESS Science Processing Operations Center. This re-\nsearch has made use of the Exoplanet Follow-up Observation Program website,\nwhich is operated by the California Institute of Technology, under contract with\nthe National Aeronautics and Space Administration under the Exoplanet Explo-\nration Program. This work makes use of observations from the LCOGT network.\nPart of the LCOGT telescope time was granted by NOIRLab through the Mid-\nScale Innovations Program (MSIP). MSIP is funded by NSF. The Joan Oró Tele-\nscope (TJO) of the Montsec Observatory (OdM) is owned by the Catalan Gov-\nernment and operated by the Institute for Space Studies of Catalonia (IEEC).\nThis work is partly supported by JSPS KAKENHI Grant Number JP18H05439,\nJST CREST Grant Number JPMJCR1761. This article is based on observations\nmade with the MuSCAT2 instrument, developed by ABC, at Telescopio Car-\nlos Sánchez operated on the island of Tenerife by the IAC in the Spanish Ob-\nservatorio del Teide. This paper is based on observations made with the MuS-\nCAT3 instrument, developed by the Astrobiology Center and under financial\nsupports by JSPS KAKENHI (JP18H05439) and JST PRESTO (JPMJPR1775),\nat Faulkes Telescope North on Maui, HI, operated by the Las Cumbres Obser-\nvatory. Some of the data presented herein were obtained at Keck Observatory,\nwhich is a private 501(c)3 nonprofit organization operated as a scientific partner-\nship among the California Institute of Technology, the University of California,\nand the National Aeronautics and Space Administration. The Observatory was\nmade possible by the generous financial support of the W. M. Keck Founda-\ntion. The authors wish to recognize and acknowledge the very significant cul-\ntural role and reverence that the summit of Maunakea has always had within\nthe Native Hawaiian community. We are most fortunate to have the opportu-\nnity to conduct observations from this mountain. This research has made use\nof the Exoplanet Follow-up Observation Program (ExoFOP; DOI: 10.26134 /Ex-\noFOP5) website, which is operated by the California Institute of Technology,\nunder contract with the National Aeronautics and Space Administration under\nthe Exoplanet Exploration Program. We acknowledge financial support from the\nAgencia Estatal de Investigación (AEI /10.13039 /501100011033) of the Ministe-\nrio de Ciencia e Innovación and the ERDF “A way of making Europe” through\nprojects PID2022-137241NB-C4[1:4], PID2021-125627OB-C3[1:2], PID2019-\nArticle number, page 16 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\na)b)\nc)d)\nFig. 15. Flux density arising from star-planet interaction as a function of the stellar mass-loss rate (panels aandb) and the magnetic field of the\nplanet (panels candd).a): Expected flux density for a closed dipolar geometry. b): Expected flux density for an open Parker spiral geometry.\nc): Expected flux density for a closed dipolar geometry. d): Expected flux density for an open Parker spiral geometry. For c)andd)we assume\n˙M=0.1˙M⊙. Note that the resulting radio emission is una ffected by the magnetic field of the planet, unless the latter is very large ( ∼4 G for closed\ndipole or even >10 G for a Parker spiral). The emission expected from Saur /Turnpenney’s model is shown in orange, and the emission expected\nfrom Zarka /Lanza’s model is shown in blue. The overlap between both models is shown in brown. The dashed black line represents the assumed\ndetection threshold of 100 µJy.\nArticle number, page 17 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\n109522GB-C5[1:4], PID2019-107061GB-C61, and RYC2021-031640-I, and the\nCentre of Excellence “Severo Ochoa” and “María de Maeztu” awards to the Insti-\ntuto de Astrofísica de Canarias (CEX2019-000920-S), Instituto de Astrofísica de\nAndalucía (CEX2021-001131-S) and Institut de Ciències de l’Espai (CEX2020-\n001058-M). This work was also funded by the Generalitat de Catalunya /CERCA\nprogramme; the M. V . Lomonosov Moscow State University Program of Devel-\nopment; the Massachusetts Institute of Technology through the TESS mission\nvia subaward s3449; the University of La Laguna under the EU Next Gen-\neration funds through the Margarita Salas Fellowship from the Ministerio de\nUniversidades ref. UNI /551/2021-May 26; the DFG through the priority pro-\ngram SPP 1992 “Exploring the Diversity of Extrasolar Planets” (JE 701 /5-1);\nthe DFG through the Major Research Instrumentation Programme and Research\nUnit FOR2544 “Blue Planets around Red Stars”; the Jet Propulsion Labora-\ntory, California Institute of Technology, under contract 80NM0018D0004 with\nthe National Aeronautics and Space Administration; the Francqui Foundation\nand the F.R.S.-FNRS through grant T.0109.20; and IPAC through a Visiting\nResearch Fellowship. 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B. 2016, ApJ, 819, 127\nZhang, M., Hu, R., Inglis, J., et al. 2024, arXiv e-prints, arXiv:2401.01400\nZieba, S., Kreidberg, L., Ducrot, E., et al. 2023, Nature, 620, 746\nZiegler, C., Tokovinin, A., Briceño, C., et al. 2020, AJ, 159, 191Instituto de Astrofísica de Canarias (IAC), calle Vía Láctea s /n,\n38205 La Laguna, Tenerife, Spain\ne-mail: fmurgas@iac.es\n2Departamento de Astrofísica, Universidad de La Laguna (ULL),\n38206 La Laguna, Tenerife, Spain\n3Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la\nAstronomía s /n, 18008 Granada, Spain\n4Center for Astroparticles and High Energy Physics (CAPA), Univer-\nsidad de Zaragoza, E-50009 Zaragoza, Spain\n5School of Sciences, European University Cyprus, Diogenes street,\nEngomi, 1516 Nicosia, Cyprus\n6Center for Astrophysics |Harvard & Smithsonian, 60 Garden Street,\nCambridge, MA 02138, USA\n7Max Planck Institute for Solar System Research, Justus-von-Liebig-\nWeg 3, D-37077, Göttingen, Germany\n8Centro Astronónomico Hispano en Andalucía, Observatorio de\nCalar Alto, Sierra de los Filabres, 04550 Gérgal, Almería, Spain\n9Astrobiology Research Unit, Université de Liège, 19C Allée du 6\nAoût, 4000 Liège, Belgium\n10Department of Earth, Atmospheric and Planetary Science, Mas-\nsachusetts Institute of Technology, 77 Massachusetts Avenue, Cam-\nbridge, MA 02139, USA\n11Sternberg Astronomical Institute Lomonosov Moscow State Univer-\nsity 119992, Universitetskii prospekt, 13, Moscow, Russia\n12Centro de Astrobiología (CSIC-INTA), ESAC campus, 28692 Vil-\nlanueva de la Cañada, Madrid, Spain\n13NASA Exoplanet Science Institute, IPAC, California Institute of\nTechnology, Pasadena, CA 91125 USA\n14Departamento de Física de la Tierra y Astrofísica and IPARCOS-\nUCM (Instituto de Física de Partículas y del Cosmos de la UCM),\nFacultad de Ciencias Físicas, Universidad Complutense de Madrid,\n28040, Madrid, Spain\n15Department of Multi-Disciplinary Sciences, Graduate School of\nArts and Sciences, The University of Tokyo, 3-8-1 Komaba, Me-\nguro, Tokyo 153-8902, Japan\n16Komaba Institute for Science, The University of Tokyo, 3-8-1\nKomaba, Meguro, Tokyo 153-8902, Japan\n17Jet Propulsion Laboratory, California Institute of Technology, 4800\nOak Grove Drive, Pasadena, CA 91109, USA\n18Thüringer Landessternwarte Tautenburg, Sternwarte 5, D-07778\nTautenburg, Germany\n19Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Hei-\ndelberg, Germany\n20Institut d’Estudis Espacials de Catalunya (IEEC), c /Gran Capità 2-\n4, 08034 Barcelona, Spain\n21NASA Ames Research Center, Mo ffett Field, CA 94035, USA\n22Department of Astronomy and Astrophysics, University of Chicago,\nChicago, IL 60637, USA\n23Institut für Astrophysik und Geophysik, Georg-August-Universität,\nFriedrich-Hund-Platz 1, 37077 Göttingen, Germany\n24Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\n25Department of Physics and Astronomy, University of Kansas,\nLawrence, KS 66045, USA\n26Landessternwarte, Zentrum für Astronomie der Universität Heidel-\nberg, Königstuhl 12, 69117 Heidelberg, Germany\n27Steward Observatory and Department of Astronomy, The University\nof Arizona, Tucson, AZ 85721, USA\n28Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112,\n21029 Hamburg, Germany\n29Department of Physics and Kavli Institute for Astrophysics and\nSpace Research, Massachusetts Institute of Technology, Cambridge,\nMA 02139, USA\n30Department of Aeronautics and Astronautics, MIT, 77 Mas-\nsachusetts Avenue, Cambridge, MA 02139, USA\n31Department of Physics and Astronomy, Vanderbilt University,\nNashville, TN 37235, USA\n32Institute of Astronomy and Astrophysics, Academia Sinica, P.O.\nBox 23-141, Taipei 10617, Taiwan, R.O.C.\nArticle number, page 19 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\n33Department of Astrophysics, National Taiwan University, Taipei\n10617, Taiwan, R.O.C.\n34SETI Institute, Mountain View, CA 94043 USA\n35Vereniging V oor Sterrenkunde (VVS), Oostmeers 122 C, 8000\nBrugge, Belgium\n36Centre for Mathematical Plasma-Astrophysics, Department of\nMathematics, KU Leuven, Celestijnenlaan 200B, 3001 Heverlee,\nBelgium\n37Public Observatory ASTROLAB IRIS, Provinciaal Domein “De\nPalingbeek”, Verbrandemolenstraat 5, 8902 Zillebeke, Ieper, Bel-\ngium\n38Department of Astrophysical Sciences, Princeton University, 4 Ivy\nLane, Princeton, NJ 08544, USA\n39Società Astronomica Lunae, Castelnuovo Magra, Italy\n40Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, c /de Can\nMagrans s /n, 08193 Bellaterra, Barcelona, Spain\nArticle number, page 20 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nAppendix A: Radial velocity data\nTable A.1 present the RV measurements of Wolf 327 taken with\nthe visible channel of CARMENES. Table presents A.2 the line\nactivity indicators computed with SERVAL .\nAppendix B: Stellar rotation from TJO photometry\nIn Fig. B.1 we show the ground-based photometric observations\nof Wolf 327 taken with the 0.8 m Joan Oró telescope (TJO). On\nthe left panel we present the normalized R-band flux versus time,\nin the right side panel the we show the GLS periodogram of\nthe time-series. The periodogram presents a maximum peak at\nPmax=40.6 d. We attempted to fit the light curve using the\nquasi-periodic kernel described in Sec. 4.2, however we found\nthat by using this procedure the kernel was overfitting our data.\nWe decided to model the light curve using a combination of a\nlinear and a sinusoidal function:\nL(t)=zpt+αt+Asin 2πt\nP+ϕ!\n, (B.1)\nwhere tis the epoch of the measurements, the free parameters\nzptandαare the zero point and slope of the linear function, re-\nspectively. For the sinusoidal function the free parameters are the\namplitude ( A), the rotation period ( Prot), and a phase angle ( ϕ).\nTo model the red noise present in the data, we used the GP im-\nplementation by Celerite (Foreman-Mackey et al. 2017) and\nchose a simple exponential kernel of the form\nki jexp=a2exp−c(|ti−tj|), (B.2)\nwhere|ti−tj|is the time between epochs in the series, and the\nhyperparameters, aandc, were allowed to be free.\nThe fitting procedure was the same as the one described in\nSec. 4.2. In Table B.1 we present the median and 1 σuncertain-\nties of the fitted parameters. With this procedure we find a ro-\ntation period of Prot=42.1+3.5\n−5.8d, consistent with the rotation\nperiod found by fitting the ASAS-SN V-band data.\nAppendix C: Ground-based transit light curves\nIn this section we describe our initial fitting process and selec-\ntion for the ground-based light curves considered for the joint fit\nof the data. Figure C.1 shows all the ground-based transit obser-\nvations available in ExoFOP and a time series taken with MuS-\nCAT2, along with a transit model for each individual light curve\n(black line). The transit model was computed using normal pri-\nors in the central time of the transit and orbital period of the\ncandidate, the values for these parameters were taken from the\nresults of a test joint fit of the TESS and CARMENES datasets.\nTo model some of the systematics present in the data we use a\ntransit baseline taking into account the position of the star, the\nFWHM, and airmass during the observations:\nB=f0+cXXp+cYYp+cfFw+caAir, (C.1)\nwhere the free parameters are f0as a flux o ffset, the coe fficients\ncXandcYmodel the flux variations caused by changes in the\nstar’s position on the detector ( Xp,Yp),cfis the coe fficient as-\nsociated with the changes of the PSF’s FWHM ( Fw) during the\nobservations, and cais the free parameter used to model flux\nchanges depending on the airmass ( Air). For the individual tran-\nsit fits we did not use GPs to model the red noise component\npresent in each of the time series.From Figure C.1 we can see that the observations that cover\na full transit event are from 2022 November 19, 2022 Decem-\nber 2 and 12, 2023 January 19, and 2023 May 25. These full\ntransit observations were reanalyzed with AstroImageJ using\nsome transit parameter values as priors taken from a TESS and\nCARMENES joint fit. Then systematic instrumental e ffects were\nremoved from the data, which were used for the final joint fit.\nAppendix D: Joint fit results\nDuring our data fitting test phase we compared the results of a\njoint fit performed on TESS photometric data and CARMENES\nRV measurements but using di fferent approaches to deal with\nthe RVs. In one test we used GPs to model the red noise of\nboth datasets (TESS and CARMENES), for a second model we\nused GPs for TESS and no GPs for CARMENES, and for the\nthird model we used GPs only for TESS data and no GPs for\nCARMENES but assuming di fferent zero-point o ffsets for each\nobserved night. The zero-point o ffset method assumes that each\nnight has a di fferent radial velocity zero-point o ffset, meaning\nthat we added five free γparameters corresponding to each night\nthat the target was observed. Figure D.1 shows the posterior dis-\ntribution of the radial velocity semi-amplitude Kobtained from\nall the models we tested, from the distributions we can see that\neither using or not using GPs, or the zero-point o ffset method, we\nget the same posterior distribution for K; meaning the that deter-\nmination of the mass of the planet is una ffected by the use of\nGPs to model the systematics present in the RV measurements.\nFigure D.2 presents the posterior distributions for the fit-\nted orbital parameters of Wolf 327b using all the data avail-\nable (TESS, ground-based transit observations, and RV measure-\nments from CARMENES). The fitted parameters for the eclips-\ning binary TIC 4918919 and Gaussian processes components\nwere left out intentionally from the plot for easy viewing.\nArticle number, page 21 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\nTable A.1. CARMENES VIS channel corrected radial velocities, instrumental drift, and nightly zero-point values computed by SERVAL . The RVs\npresented here were corrected by instrumental drifts and nightly zero-point o ffsets.\nBJD – 2 459 000 RV [m s−1] Instrumental drift [m s−1] Nightly zero-point [m s−1]\n950.552461 4 .615±1.618−0.737±0.442−1.108±1.015\n952.431876 14 .547±3.801 2 .820±0.465−1.198±1.677\n952.657952 9 .342±8.446−1.546±0.847−1.198±1.677\n958.562133 4 .469±1.829 2 .570±0.341−1.544±1.229\n958.737953 5 .320±2.080 1 .565±0.398−1.544±1.229\n979.352322 2 .662±2.464−9.406±0.455−1.813±0.501\n979.393815 7 .136±2.065−12.834±0.512−1.813±0.501\n979.443206 6 .017±2.354−16.900±0.590−1.813±0.501\n979.492442 6 .279±2.489−21.756±0.688−1.813±0.501\n979.540070 6 .173±2.073−26.460±0.542−1.813±0.501\n979.586515−0.143±1.839−30.637±0.454−1.813±0.501\n979.633090 2 .549±2.081−33.864±0.502−1.813±0.501\n979.680690−2.050±1.891−37.903±0.577−1.813±0.501\n979.736000−1.475±2.532−40.312±0.445−1.813±0.501\n981.363716−1.984±1.620−8.574±0.645−1.941±0.589\n981.411369−3.989±1.758−9.762±0.657−1.941±0.589\n981.459496−2.455±1.404−14.079±0.484−1.941±0.589\n981.523068−2.870±1.526−18.812±0.456−1.941±0.589\n981.568274−1.682±1.792−21.967±0.652−1.941±0.589\n981.663415 1 .686±1.636−29.524±0.448−1.941±0.589\n981.697833 3 .666±1.436−32.412±0.456−1.941±0.589\n981.749403 2 .931±1.866−35.338±0.476−1.941±0.589\n0.970.980.991.001.011.02Norm. Flux\nTJO photometry\nR\n0 50 100 150 200 250\nBJD - 2460050 [days]0.04\n0.02\n0.000.020.04Residuals\n0.010 0.100 1.000Frequency [days1]\n100\n101\n102\nPeriod [days]0.00.10.20.30.4Power (ZK)Prot = 42.1 days\nFig. B.1. Long-term photometric observations of Wolf 327 taken with the Joan Oró telescope (TJO). Left panel: normalized R-band flux (orange\npoints) versus time. The best-fitted model is shown by the black line, the 1 σuncertainty regions of the fit are shown in light blue. Right panel:\nGLS periodogram (Zechmeister & Kürster 2009) for TJO R-band photometry. The vertical green line shows the best-fitted rotation period ( Prot=\n42.1+3.5\n−5.8d). The horizontal lines represent the false alarm probability (FAP) levels of 10% (red dotted line) and 1% (blue dashed line).\nArticle number, page 22 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nTable A.2. CARMENES spectroscopic line activity indicators for Wolf 327 computed by SERVAL .\nBJD – 2 459 000 CRX [m s−1N−1\np] dLW [m2s−2] Hα[m s−1] Na iD1[m s−1] Na iD2[m s−1] Ca iiIRTa [m s−1] Ca iiIRTb [m s−1] Ca iiIRTc [m s−1]\n950.552461 17 .6±12.3−3.2±1.3 0.836±0.002 0.213±0.005 0.207±0.005 0.590±0.002 0 .439±0.002 0 .412±0.002\n952.431876 −34.6±35.4 154.9±10.0 0.792±0.005 0.448±0.016 0.388±0.016 0.583±0.004 0 .427±0.005 0 .402±0.004\n952.657952 33 .5±88.5 200.1±11.5 0.755±0.013 0.320±0.046 0.280±0.048 0.583±0.010 0 .448±0.012 0 .409±0.011\n958.562133 −18.9±13.8−12.8±2.1 0.828±0.003 0.191±0.006 0.177±0.006 0.596±0.002 0 .440±0.002 0 .409±0.002\n958.737953 16 .7±17.6−5.2±2.1 0.823±0.003 0.342±0.009 0.262±0.009 0.580±0.003 0 .435±0.003 0 .408±0.003\n979.352322 −35.4±26.0 9.5±3.0 0.869±0.004 0.321±0.014 0.274±0.015 0.588±0.003 0 .449±0.003 0 .414±0.003\n979.393815 −39.8±20.3−3.9±3.1 0.845±0.004 0.285±0.014 0.202±0.015 0.597±0.003 0 .448±0.003 0 .419±0.003\n979.443206 26 .3±23.7 1.4±3.1 0.841±0.005 0.320±0.015 0.282±0.015 0.599±0.004 0 .446±0.004 0 .413±0.004\n979.492442 52 .3±23.6 2.1±2.4 0.844±0.004 0.319±0.013 0.267±0.013 0.583±0.003 0 .448±0.003 0 .407±0.003\n979.540070 10 .9±20.8−9.5±2.1 0.831±0.003 0.262±0.010 0.197±0.010 0.591±0.003 0 .431±0.003 0 .408±0.003\n979.586515 −24.2±18.1−15.2±2.6 0.836±0.004 0.201±0.010 0.166±0.010 0.598±0.003 0 .444±0.003 0 .413±0.003\n979.633090 −0.6±21.2−15.7±2.2 0.844±0.004 0.247±0.012 0.197±0.012 0.595±0.003 0 .436±0.003 0 .416±0.003\n979.680690 9 .9±18.6 3.0±2.7 0.831±0.004 0.372±0.013 0.304±0.013 0.592±0.004 0 .438±0.004 0 .413±0.003\n979.736000 −19.2±26.8 6.1±3.2 0.838±0.004 0.462±0.014 0.323±0.014 0.587±0.003 0 .450±0.004 0 .412±0.003\n981.363716 0 .9±15.1 2.0±1.8 0.829±0.003 0.261±0.009 0.223±0.009 0.592±0.002 0 .436±0.002 0 .417±0.002\n981.411369 −16.0±16.2 1.4±1.3 0.838±0.002 0.215±0.005 0.199±0.005 0.593±0.002 0 .439±0.002 0 .411±0.002\n981.459496 6 .8±12.5−5.3±1.2 0.821±0.002 0.171±0.005 0.170±0.005 0.582±0.002 0 .436±0.002 0 .413±0.002\n981.523068 3 .9±14.2−8.8±1.4 0.824±0.002 0.189±0.005 0.168±0.005 0.584±0.002 0 .439±0.002 0 .414±0.002\n981.568274 12 .1±16.6−6.6±2.0 0.816±0.003 0.192±0.006 0.179±0.006 0.589±0.002 0 .439±0.002 0 .404±0.002\n981.663415 −7.6±15.7−5.8±1.9 0.819±0.003 0.189±0.006 0.171±0.006 0.587±0.002 0 .440±0.002 0 .411±0.002\n981.697833 −29.8±12.6−0.3±1.4 0.820±0.002 0.214±0.006 0.190±0.006 0.589±0.002 0 .439±0.002 0 .416±0.002\n981.749403 30 .3±18.4 25.2±3.2 0.822±0.003 0.391±0.009 0.301±0.009 0.584±0.003 0 .434±0.003 0 .406±0.002\nArticle number, page 23 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\nFig. C.1. Ground-based transit light curves of Wolf 327b. The best fitted model (black line) includes systematic e ffects but no red noise component\n(i.e., without GPs).\nArticle number, page 24 of 26F. Murgas et al.: Wolf 327b: A new member of the pack of ultra-short-period super-Earths around M dwarfs\nTable B.1. Prior functions and fitted parameters values for the TJO R-\nband photometry of Wolf 327.\nParameter Prior Value\nzpt U(−2,2) 1.000+0.001\n−0.001\nα(×10−6) [day−1]U(−50,50) 36 .8+14.7\n−14.6\nA U(10−6,10) 0.0058+0.0017\n−0.0023\nProt[d]U(1.0,100) 42 .1+3.5\n−5.8\nϕ[rad] U(0,π) 1.87+0.65\n−1.09\nσphot jitter U(0,1) 0.0015+0.0001\n−0.0002\nlogaU(−13.8,0.0)−10.1±0.3\nlogc[day−1]U(−4.6,4.6)−0.5±0.5\nNotes.Urepresent a uniform prior function. The slope of the linear\nfunction (α) was computed relative to the mid-time of the time series\nTmid=2460187 BJD.\n1 2 3 4 5 6\nK [m/s]0.00.10.20.30.40.50.60.7Normalized frequencyRVs with GP\nRVs without GP\nZP offset\nFig. D.1. Posterior distribution of the radial velocity semi-amplitude\n(K) for a joint fit using only TESS and CARMENES RV observations.\nWe show the results for a fit using GPs to model the red noise of both\ndatasets (light yellow), GPs for TESS data and no GPs for CARMENES\nmeasurements (light blue), and a fit of TESS photometry including GPs\nand RVs without GPs but using a di fferent zero-point (ZP) o ffset for\neach night (light purple).\nArticle number, page 25 of 26A&A proofs: manuscript no. Wolf327b_CARMENES_Murgas\nFig. D.2. Correlation plot for the fitted transit and orbital parameter for Wolf 327b. The fitted orbital parameters of the eclipsing binary TIC 4918919\nand the free parameters used to model the systematic e ffects were intentionally left out for easy viewing. The blue lines mark the median values of\nthe distribution. For plotting purposes the distributions for the central time of the transit and orbital period were o ffset by T1 =2459252 d.\nArticle number, page 26 of 26"    },    {        "title": "2401.12191v1.Information_Problem_in_Black_Holes_and_Cosmology_and_Ghosts_in_Quadratic_Gravity.pdf",        "content": "Information Problem in Black Holes and\nCosmology and Ghosts in Quadratic Gravity\nIgor Volovich\nSteklov Mathematical Institute, Russian Academy of Sciences,\nGubkina str. 8, 119991, Moscow, Russia\nE-mail: volovich@mi-ras.ru\nAbstract: Black hole information problem is the question about unitarity of the\nevolution operator during the collapse and evaporation of the black hole. One can\nask the same question about unitarity of quantum and inflationary cosmology. In\nthis paper we argue that in both cases, for black holes and for cosmology, the answer\nis negative and we face non-unitarity.\nSuch a question can not be addressed by using the fixed classical gravitational\nbackground since one has to take into account the backreaction. To his end one\nuses the semi-classical gravity, which includes the expectation value of the energy -\nmomentum tensor operator of the matter fields. One has to renormalize the energy-\nmomentum tensor and one gets an effective action which contains quadratic terms\nin scalar curvature and Ricci tensor. Such quadratic gravity contains ghosts which\nin fact lead to violation of unitarity in black holes and cosmology. We discuss the\nquestion whether black holes will emit ghosts.\nOne can try to restrict ourselves to the f(R) gravity that seems is a good ap-\nproximation to the semi-classical gravity and widely used in cosmology. The black\nhole entropy in f(R) gravity is different from the Bekenstein-Hawking entropy and\nfrom entanglement island entropy. The black hole entropy in R+R2gravity goes\nto a constant during the evaporation process. This can be interpreted as another\nindication to the possible non-unitarity in black holes and cosmology.arXiv:2401.12191v1  [hep-th]  22 Jan 2024Contents\n1 Introduction 1\n1.1 Classical and semi-classical gravity 2\n2 Semi-classical gravity and quadratic gravity 2\n3 Effective f(R)gravity. 3\n4 Black Holes in f(R)gravity 4\n5 The entropy in f(R)gravity 5\n6 Conclusions 6\n7 Acknowledgements 7\n1 Introduction\nBlack hole information problem is the question about unitarity of the evolution op-\nerator during the collapse and evaporation of the black hole [1, 2]. Various proposals\nhave been discussed in this context, such as black hole explosion [3, 4], gravitational\ncollapse of quantum scalar field [5], complementarity [6], firewalls [7], islands [8–14],\nnear critical state equations [15, 16] and negative dimensions [17, 18].\nIf one considers a fixed classical gravitational background then we cannot justify\ndiscussion of the black hole evaporation and therefore there is no reason to speak\nabout information problem in such approximation. One can try to attack informa-\ntion loss problem if we would investigate the collapse of quantum field or discuss\nsemi-classical gravity equations. These equations include the expectation value of\nthe energy-momentum tensor operator of the matter fields. One has to renormal-\nize the energy-momentum tensor and one gets an effective action which contains\nhigher order terms in curvature including not only linear term in curvature, but also\nquadratic in scalar and Ricci tensor. Such quadratic gravity is renormalizable but\nit contains ghosts [19, 20]. These ghosts lead to violation of unitarity in black holes\nand cosmology. We discuss the question whether black holes will emit ghosts.\nIn this note, in the context of the black hole information problem, we consider\nalso an effective classical local f(R) gravity which seems is a rather good approxi-\nmation to the semi-classical gravity. In a simple case it describes an interaction of\ngravity with quantum conformal matter. f(R) gravity is wildly used in inflationary\ncosmology [21–23]. Black holes in f(r) gravity are considered in [24].\n– 1 –1.1 Classical and semi-classical gravity\nHawking radiation has been derived by considering a free quantum field in the clas-\nsical gravitational background. For example, the Schwarzschild black hole metric\nsatisfies the Einstein equation\nRµν−1\n2Rgµν= 0 (1.1)\nIn such an approach we cannot investigate the black hole information problem since\nduring the evaporation process we have to taken into account the back reaction of\nthe radiation to the metric. To this end usually one uses the semi-classical gravity\napproximation.\nSemi-classical gravity is the approximation to the theory of quantum gravity in\nwhich one treats matter fields as being quantum and the gravitational field as being\nclassical but including quantum corrections.\nThe curvature of the spacetime is given by the semi-classical Einstein equations,\nwhich relate the curvature of the spacetime to the expectation value of the energy-\nmomentum tensor operator Tµνof the matter fields:\nRµν−1\n2Rgµν= 8πGD\nˆTµνE\nψ, (1.2)\nwhere G is the gravitational constant and ψindicates the quantum state of the matter\nfields. Such equations are non-linear and non-local equations that are difficult to\nanalyze. Therefore we consider the effective gravity.\nIn cosmology with the Friedmann metric\nds2=−dt2+a(t)2dΩ (1.3)\nthe semi-classical gravity with conformal matter was considered on [21] and is called\nthe Starobinsky inflationary model. It is shown in [22] that this model in 1-loop\napproximation is equivalent to the effective R+R2gravity.\nBy analogy with cosmology we will consider the R+R2gravity as the approx-\nimation to the semi-classical gravity in theory of black holes and use it to describe\nthe evaporation process.\nThe black hole entropy in f(R) gravity is different from the Bekenstein-Hawking\nentropy and from the island entropy. It goes to a constant during the evapora-\ntion process. This can be interpreted as an indication to the possible none-unitary\nevolution of the system.\n2 Semi-classical gravity and quadratic gravity\nFor the semi-classical Einstein equation (1.2) we have to perform renormalizations\nin both parts. The renormalized version of the left hand site of Eq.(1.2) reads (see\n– 2 –Eq.(6.52) [26])\nRµν−1\n2Rgµν+α(1)Hµν+β(2)Hµν+γHµν, (2.1)\nwhere\n(1)Hµν= 2R;µν−2gµν□R−1\n2gµνR2+ 2RRµν, (2.2)\n(2)Hµν= 2Rα\nµ;να−□Rµν−1\n2gµν□R+ 2Rα\nµRαν−1\n2gµνRαβRαβ (2.3)\nHµν=−1\n2gµνRαβγδRαβγδ+ 2RµαβγRαβγ\nν−4□Rµν+ 2R;µν−4RµαRα\nν+ 4RαβRαµβν,\n(2.4)\nwhere α, β, γ are constant.\nOne has the relations\n(1)Hµν=−1√−gδ\nδgµν(x)Z\ndx√−gR2(2.5)\nand similar for the square of the scalar curvature and the Ricci tensor. It is known\nthat the divergences in the energy-momentum tensor can be remover by usin the\nregularization of the type ⟨ϕ(x)ϕ(y)⟩asx→y. So one get the renormalized action\nin the form\nSren=Z\nd4x√−g(16πG)−1(R−2Λ) + aR2+bRαβRαβ+cRαβγδRαβγδ (2.6)\nThis action describes quadratic gravity. It is known quadratic gravity is renormaliz-\nable field theory, but it contains ghosts [19, 27]. That this ghost lead to the violation\nof the unitarity theory. Therefore the semi-classical also should lead to the violation\nof unitarity.\nA natural question arises whether the black hole will emit ghost by mechanism\nsimilar to the usual Hawking radiation. Lets remand that in classical field theory with\nhigher derivative there are not with negative energy. Which after the quantization\ncan be represented as Sghost. In both cases, classical and quantum theories, this\nmeans a lack of stability. Whether we have ghostly radiation from a black hole is a\nquestion worthy of further investigation.\n3 Effective f(R)gravity.\nLet us consider the gravitational field with matter in the following form I=IEH+Imat\nIEH=1\n16πGZ\nd4x√−gR, (3.1)\nwhere gµνis a metric, g= det gµνandRis a Ricci scalar. If we consider a scalar\nfield, then the action for the matter field Imathas the following form\nImat=1\n2Z\nd4x√−g(ξφ2R−gµν∂µφ∂νφ), (3.2)\n– 3 –where ξis a constant.\nPartition function in quantum gravity is given by the Euclidean path integral\nZ=Z\nDgµνDφ e−I=Z\nDgµνe−Ieff(g)(3.3)\nOne can expand Ieff(g) as follows\nIeff(g) =f(R) +f1(RµνRµν) +f2(RµναβRµναβ) +f3(□R) +... (3.4)\nThe action of f(R) gravity is [28–32]\nI=1\n16πGZ\nd4x√−g f(R), (3.5)\nwhere f(R) is analytical or smooth function. Variation of the action with respect to\nthe metric gµνgives the field equations\nf′(R)Rµν−f(R)\n2gµν=∇µ∇νf′(R)−gµν□f′(R), (3.6)\nwhere a prime denotes differentiation with respect to R.\nVacuum equations of motions are\nRµν(f′(R))−1\n2gµν(f(R)) = 0 (3.7)\nThe constant scalar curvature solutions R=R0means:\n2f′(R0)R0−4f(R0) = 0 (3.8)\nFor this kind of solution an effective cosmological constant may be defined as Λeff\nD≡\nR0/D, for a recent derivation of the inflationary cosmological constant, see [33].\nThus, for any solution with constant curvature R=R0andf′(R0)̸= 0 holds:\nRµν=f(R0)\n2f′(R0)gµν (3.9)\n4 Black Holes in f(R)gravity\nThe general static spherically symmetric solution can be written as\nds2=−h(r)dt2+h−1(r)dr2+r2dΩ2, (4.1)\nwhere\nh(r) = 1 −2GM\nr+Q2\nr2. (4.2)\n– 4 –This solution corresponds to a Reissner-Nordstr¨ om solution, i.e. a charged mas-\nsive BH solution with mass Mand charge Q. Below we restrict ourselves to the\nSchwarzschild solution, i.e. Q= 0.\nProblem of ghosts in R2gravity is analyzed in [34, 35]. Note that there are\nghosts in R+R2gravity in the Schwarzschild background. Consider perturbations\naround the Schwarzschild background where gµν=gµν+hµνis the Schwarzschild\nsolution with R= 0. Then the R2action one can consider in the following form\nS=Z\nd4x√gR2. (4.3)\nThen the R2action is\nS2=bZ\nd4x[DµDνhµν−□h]2, (4.4)\nwhere Dµis covariant derivative with respect to the Schwarzschild metric. We see the\nappearance of ghost in this action, similarly to the appearance of ghosts in Minkowski\nbackground. If we add to the pure R2action other linear scalar term then we get\npropagating ghosts. Note that in the de-Sitter black hole there are no ghosts, see\n[36].\n5 The entropy in f(R)gravity\nThe entropy of the black hole in the f(r) gravity is [24, 32, 37, 38]\nS=f′(RH)A\n4G, (5.1)\nwhere A= 16πG2M2is the area of the horizon of the black hole with mass Mand\nRHis the curvature on the horizon. For the Schwarzschild black hole one has\nRH=2\nr2\nh=1\n2G2M2. (5.2)\nThis formula for entropy was obtained by the Noether charge method proposed by\nWald [25]\nS= 4πZ\nS2∂L\n∂Rd2x, (5.3)\nwhere Lis the Lagrangian density and the integral is over the horizon at r=rh. We\nconsider\nf(R) =R+λGR2, (5.4)\nwhere λis dimensionless number. Finally, for the entropy of the black hole we have\nS=f′(RH)A\n4G=(1 +λ\nGM2)16πG2M2\n4G= 4πGM2+ 4λ. (5.5)\n– 5 –So, the entropy for effective R+R2gravity is equal to the sum of the Bekenstein-\nHawking entropy\nSBH=A\n4G= 4πGM2(5.6)\nand the constant 4 λ. If we apply this formula to the evaporation process, when\nthe mass of the black hole Mdecreases, after the complete evaporation we have the\nnon-vanishing entropy S= 4λ.\n6 Conclusions\nIt is pointed out in this paper that the semi-classical Einstein equation after renor-\nmalization are reduced to the quadratic gravity that contains ghosts. The presence\nof ghosts violates unitarity of theory, in other words there is information lost in black\nholes in cosmology. In that situation arises question whether black holes will emit\nghosts by mechanism similar to the usual Hawking radiation.\nThe effective f(R) gravity is also considered in the context of the black hole\ninformation problem. It is noted that, since this problem must be studied taking\ninto account collapse and evaporation processes, one cannot limit oneself to the\nclassical fixed background of a black hole, but it is necessary to take into account the\neffect of the back reaction of radiation on the metric. It is argued that, by analogy\nwith inflationary cosmology, effective f(R) gravity can be used to take into account\nsuch a back reaction non-perturbatively. One considers the Friedmann equation in\nsemi-classical gravity when the expectation value Eq.(1.2) is computed by one-loop\napproximation. It is shown [22] that in this approximation, quasi-classical gravity\ncan be reduced to R+R2gravity, which in cosmology is called the Starobinsky\ninflationary model.\nIn this paper the conformal invariant gravity f(R) =R+R2has been considered\nand it is shown that at the end of evaporation the black hole still has non-zero\nconstant entropy. This raises the question of what complete evaporation of a black\nhole means even for the Schwarzschild metric. In the Schwarzschild coordinates one\ncan send the mass of the black hole to zero to get the Minkowski metric, but in the\nKruskal coordinates one has the singularity in this limit, see [4, 15]. It is important\nto study time-dependent spherically symmetric solutions of f(R) gravity.\nIn inflationary cosmology the justification of the effective R+R2gravity is based\non experimental results. We hope that applying this model to the black hole infor-\nmation problem will also be quite reasonable.\nNote that actually the information problem of black holes in cosmology is par-\nticular case of fundamental problem of irreversibility and entropy increasing is con-\nsidered in foundation of statistical mechanics and theory of open quantum systems,\nsee [41–43].\n– 6 –7 Acknowledgements\nI am very grateful to I.Aref’eva, V.Frolov, T.Rusalev and D.Stepanenko for fruitful\ndiscussions.\nReferences\n[1] S. W. Hawking, Breakdown of Predictability in Gravitational Collapse,” Phys. Rev.\nD14(1976), 2460-2473\n[2] V. Frolov, I. Novikov, Black Hole Physics: Basic Concepts and New Developments,\nSpringer, 2012.\n[3] S. W. Hawking, Black hole explosions,” Nature 248(1974), 30-31\n[4] I. Aref’eva and I. Volovich, Quantum Explosions of Black Holes and Thermal\nCoordinates,” Symmetry 14(2022) no.11, 2298, arXiv:2104.12724 [hep-th].\n[5] B. Berczi, P. M. Saffin and S. Y. Zhou, Gravitational collapse with quantum fields,”\nPhys. Rev. D 104(2021) no.4, L041703, arXiv:2010.10142 [gr-qc].\n[6] L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole\ncomplementarity,” Phys. Rev. D 48(1993), 3743-3761, arXiv:hep-th/9306069\n[hep-th].\n[7] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or\nFirewalls?,” JHEP 02(2013), 062, arXiv:1207.3123 [hep-th].\n[8] G. Penington, Entanglement Wedge Reconstruction and the Information Paradox,”\nJHEP 09(2020), 002, arXiv:1905.08255 [hep-th].\n[9] A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk\nquantum fields and the entanglement wedge of an evaporating black hole,” JHEP 12\n(2019), 063, arXiv:1905.08762 [hep-th].\n[10] A. Almheiri, R. Mahajan and J. 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Mathematical foundations of quantum information\nand computation and its applications to nano-and bio-systems. Springer Science &\nBusiness Media.\n– 9 –"    },    {        "title": "2401.12268v1.Generalized_Ordinal_Patterns_Allowing_for_Ties_and_Their_Applications_in_Hydrology.pdf",        "content": "arXiv:2401.12268v1  [stat.AP]  22 Jan 2024Generalized Ordinal Patterns Allowing for Ties and\nTheir Applications in Hydrology\nAlexander Schnurra, Svenja Fischerb,∗\naDepartment Mathematik, Universit¨ at Siegen, D-57072 Sieg en, Germany\nbSPATE research unit, Ruhr-University Bochum, D-44801 Boch um, Germany. Phone: +49\n234 - 32 28961\nAbstract\nWhen using ordinal patterns, which describe the ordinal structur e within a data\nvector, the problem of ties appeared permanently. So far, model classes were\nused which do not allow for ties; randomization has been another att empt to\novercomethis problem. Often, time periodswith constantvaluesev enhavebeen\ncounted as times of monotone increase. To overcome this, a new ap proach is\nproposed: it explicitly allows for ties and, hence, considers more pat terns than\nbefore. Ties are no longer seen as nuisance, but to carry valuable in formation.\nLimit theorems in the new frameworkare provided, both, for a single time series\nand for the dependence between two time series.\nThe methods are used on hydrologicaldata sets. It is common to dis tinguish\nfive flood classes (plus ‘absence of flood’). Considering data vector s of these\nclasses at a certain gauge in a river basin, one will usually encounter s everal\nties. Co-monotonic behavior between the data sets of two gauges (increasing,\nconstant, decreasing) can be detected by the method as well as s patial patterns.\nThus, it helps to analyze the strength of dependence between diffe rent gauges\nin an intuitive way. This knowledge can be used to asses risk and to plan future\nconstruction projects.\nKeywords: ordinal data analysis, discrete time series, limit theorems, flood\n∗Corresponding author\nEmail addresses: schnurr@mathematik-uni-siegen.de (Alexander Schnurr),\nsvenja.fischer@rub.de (Svenja Fischer)\nPreprint submitted to Computational Statistics & Data Anal ysis January 24, 2024events\n2010 MSC: 62M10, 62H20, 62F12, 60F05, 05A05, 62G30\n1. Introduction\nIn the paper [1], the authors have suggested to use so called ordina l patterns\nfor the analysis of hydrological time series. The origin of the concep t of ordinal\npattern lies in the theory of dynamical systems [cf. 2, 3]. Ordinal pa tterns have\nbeen used in order to analyze the entropy of data sets [4] and to es timate the\nHurst-parameter of long-range dependent time series [cf. 5, 6]. A dditionally,\nthese methods have proved to be useful in the context of EEG-da ta in medicine\n[cf. 7], index data in finance [cf. 8] and flood data in extreme value the ory [cf.\n9].\nThe classical approach in ordinal pattern analysis works as follows: One\ndecides for a small number n∈N, that is, for the length of the data windows\nunderconsideration. In eachwindow, onlythe ordinalinformationis considered.\nThere are various ways to encode the ordinal information of an n-dimensional\nvector in a pattern. The one that is used most often reads as follow s:\nLetSndenote the set of permutations of {1,...,n}, which we write as n-\ntuples containing each of the numbers 1 ,...,nexactly one time. By the ordinal\npattern of order nof the vector x= (x1,...,xn) we refer to the permutation\nΠ(x1,...,xn) := (π1,...,πn)∈Sn\nwhich satisfies\nxπ1≥...≥xπn.\nWhen using this definition, it is often assumed that the probability of c oin-\ncident values within the vector ( x1,...,xn) is zero.\nHowever, in high-frequency data (mathematical finance) and in ca tegorical\ndata (e.g. hydrology) one will encounter ties between the values. T he following\napproaches have been used in order to overcome this problem in est imation\n2procedures:\na) skip the respective data points (leave out these vectors)\nb) randomize, e.g. by adding a small noise to the data, in order to avo id ties\nc) add the following to the definition above:\n...andπj−1> πjifxπj−1=xπjforj∈ {2,...,n}.\nEachofthese‘solutions’hasitsowndrawbacks. Incasea)wemight loosealotof\ninformation. Consideringcategorialdata sets with a small number o fcategories,\nwe could even loose most of the information. In case b) we might unde restimate\nco-movement of two data sets (see below) and in case c) the vecto rs (4,4,4,4)\nand (1,10,100,1000) are mapped on the same pattern (4 ,3,2,1), hence, they\nare considered to be equal from the ordinal point-of-view.\nIn earlier papers on the subject, data sets were used which only co ntained\nvery few vectors with ties. Having categorial data in mind, one could consider\ncases where the length of patterns exceeds the number of categ ories. In this\ncase (and there are indeed applications, where this approach is can onical, see\nbelow), every single vector has to have ties(!).\nHere, we present a new approach on how to overcome the problem o f ties\nin a natural way. We explicitly allow for ties and assign a larger number of\npatterns with the data . Ties are not seen as being disruptive; they are rather\nconsidered to carry valuable information.\nThere are various ways on how to describe the patterns one finds in the\nclassical approach. We have already recalled the one which is used mo st often.\nHowever, a different one is much more suitable to be used in the gener alized\nform we have in mind.\nThis works as follows:\nConsider a vectorof nconsecutivevalues ( x1,...,xn). Write down the values,\nwhich are attained ( y1,...,ym) such that y1< y2< ... < ym. The number of\ndifferent values attained in x1, ...,xnism∈N. Now define the generalized\npattern of ( x1,...,xn) to be the vector t= (t1,...,tn)∈Nnsuch that\ntj=kif and only if xj=yk.\n3By this definition, we obtain that the vector (1 ,2,4,3)yields the pattern t=\n(1,2,4,3) which is very convenient. Using strictly monotone transformatio ns\nN→Rone obtains all elements in R4having this pattern. We write Ψ for the\nfunction which assigns the generalized pattern with each vector in c ontrast to\nΠ which has been used in the classical case. The set of all generalized patterns\nof order nis denoted by Tn(in contrast to the classical permutations Sn).\n(1,2,3) /Bullet/Bullet/Bullet\n(3,2,1)/Bullet\n/Bullet\n/Bullet\n(1,2,2) /Bullet/Bullet /Bullet\n(1,1,2)/Bullet /Bullet/Bullet\n(2,1,1)/Bullet\n/Bullet /Bullet\n(1,3,2) /Bullet/Bullet\n/Bullet(1,1,1)/Bullet /Bullet /Bullet\n(3,1,2)/Bullet\n/Bullet/Bullet\n(1,2,1) /Bullet/Bullet\n/Bullet(2,2,1)/Bullet /Bullet\n/Bullet(2,1,2)/Bullet\n/Bullet/Bullet\n(2,3,1)/Bullet/Bullet\n/Bullet (2,1,3)/Bullet\n/Bullet/Bullet\nFigure 1: The 13 generalized patterns of length 3\nAs in the case of classical ordinal patterns, one can think of the st rings (of\nnumbers) as an archetype structure (see Figure 1 for the case n= 3).\nIf ties do not appear, the question of how many patterns one has t o consider\nboils down to the classical case: in how many ways can one write down t he\nnumbers 1,2, ..., n. It is well known that this number is n! =n·(n−1)·...·1.\nFor the generalized patterns we obtain 3 for n= 2, 13 for n= 3 and 75 for\nn= 4.\nLet us consider the number of patterns under consideration for g eneraln:\nthis number is equivalent to the number of ways ncompetitors can rank in a\ncompetition if each competitor is identifiable and if we allow for the poss ibility\nof ties. This series in Nis known under the name Fubini numbers . In cal-\nculating these numbers one first decides how many first places, sec ond places,\n4third places and so on appear (e.g. (1,1,2,2,3)). For each of these po ssibilities\none then calculates the number of re-arrangements of vectors o f this kind (e.g.\n(1,2,1,3,2)). For the readers’ convenience we recall the first 7 Fub ini numbers\nin Figure 2.\n1234567\n131375541468347293\nFigure 2: The first Fubini numbers\nThe name comes from the fact that the numbers coincide with the nu mbers\nof possibilities on how to calculate n-dimensional integrals iteratively using Fu-\nbini’s theorem. More information on these numbers can be found at o eis.com.\nWhile writing this paper, we were informed that the authors of [10] h ave also\nconsidered patterns with ties. However, they do not prove any limit theorems\nor use them in order to analyze dependence structures. They do c alculate the\nnumber of patterns [cf. 10, Appendix].\nAnalyzing the patterns within one time series, we can describe period icities,\ntimes of monotonicity and detect instationarities in a convenient way , just to\nname a few applications. In the next section, we use ordinal patter ns in order\nto describe the dependence between two time series. Classically, th is is often\ndone by using correlation. However, our method works even if the m odels under\nconsideration do not have second moments.\nThe notation we are using is more or less standard. (Ω ,F,P) denotes a\nprobability space in the background. We write Efor the expected value w.r.t.\nP, whileNdenotes the positive integers starting with one.\nThe paper structure is as follows: first, the generalized ordinal pa tterns are\nincorporated in a novel measure for dependence and respective s ample estima-\ntorsare introduced (Section 2). For these measures, limit theore ms areprovided\nin Section 3. There, we differentiate between two cases: the gener alized ordi-\nnal pattern dependence between two time series (in this case for s hort range\ndependent data) and the spatial consideration, where the patte rn is considered\n5as shifting vector over time. With these two cases, the following app lication\nof flood classes in a temporal and spatial manner is supported theo retically\n(Section 4).\n2. Methodology\nIn this section, the above-introduced generalization of the classic al ordinal\npatterns is used to construct a new measure for dependence: the ordinal pattern\ndependence . This dependence measure makes use of the generalized ordinal\npatterns to obtain information of the co-movement of two time ser ies. It can be\nextended by a metrical approach that weakens the assumption of strict equality\nof patterns. Finally, sample estimators for the different measures are given.\n2.1. Generalized Ordinal Pattern Dependence\nBy now we have only considered a single time series, respectively a sing le\ndata set. Additionally, we would like to use our generalized patterns ( with ties)\nin order to analyze the dependence between time series, respectiv ely data sets\n[cf. 8, 11, 12]). The main idea is that a co-monotonicbehavior betwee n two data\nsets leads to an increased number of coincident patterns at the sa me instants in\ntime.\n/Bullet/Bullet /Bullet/Bullet\n/Bullet/Bullet\n/Bullet /Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet/Bullet /Bullet/Bullet/Bullet\n/Bullet\n/Bullet /Bullet/Bullet\n/Bullet/Bullet\n1 2 3 4 5 6 7 8 9 10 11123456\nx1y1\nx11y11\nFigure 3: Two categorial data sets with partially co-monoto nic behavior\n6LetX= (Xk)k≥1andY= (Yk)k≥1be two stationary time series; let us\nmention that, although stationarity is theclassical assumption in this context,\nour results hold under weaker conditions: stationary increments o r even ordinal\npattern stationary, that is, stationarity of the pattern probab ilities over time,\nare enough.\nIn orderto quantifythe co-monotonicbehavioroftwotime series, weanalyze\nthe probability of coincident patterns at the same instants in time. S ince the\ntime series are stationary (or have at least stationary pattern pr obabilities), it\nis enough to consider the time points 1 to n\np(X,Y):=P/parenleftig\nΨ(X1,X2,...,Xn) = Ψ(Y1,Y2,...,Yn)/parenrightig\nand compare this with the hypothetical case of independence\nq(X,Y):=/summationdisplay\nt∈TnP(Ψ(X1,X2,...,Xn) =t)·P(Ψ(Y1,Y2,...,Yn) =t).\nWe call q(X,Y)comparison value. If we want to consider an anti-monotonic\nbehavior instead, we consider r(X,Y):=p(X,−Y)ands(X,Y):=q(X,−Y). If the\ntime series under consideration are fixed, we omit the subscripts. W e say that\nthe two time series exhibit generalized positive ordinal pattern depe ndence if\np > q; and generalized negative ordinal pattern dependence if r > s. In analogy\nto the classical case [11, Formula (5)] we can define a standardized ordinal\npattern coefficient\nord(X,Y) :=/parenleftbiggp−q\n1−q/parenrightbigg+\n−/parenleftbiggr−s\n1−s/parenrightbigg+\n. (1)\nThis coefficient has been compared in the classical case with other co ncepts of\nmultivariate dependence in [12]. It admits values between -1 and 1, ze ro in\ncase of independence and 1 (resp. -1) in case of total positive (re sp. negative)\ndependence.\n2.2. Generalized Average Weighted Ordinal Pattern Depende nce\nWhen dealing with classical ordinal patterns (not allowing for ties), it has\nproved to be useful to consider not only identical patterns when a nalyzing co-\nmovement, but to also consider almost similar patterns . This generalization of\n7the method is particularly useful, if the patterns are long (big values ofn) and\nhence more sensitive to noise. Defining a metric on the patterns (an d a weight\nfunction) was straight forward in the case of classical patterns w ithout ties. On\nthe space of classical ordinal patterns, that is, permutations, t here exist several\nmetrics which have proved to be useful in different areas of statist ics (cf. [13]).\nWhen analyzing co-movement, the metric related to the L1-norm has proved\nto be most suitable. This is due to the fact that permutations are clo se to\neach other (measured with this metric) if and only if, high values of on e vector\ncorrespond to high values of the other vector; and the same holds true for low\nvalues. However, this metric which was proposed by [11] is not sufficie nt in our\nnew context. Hence, we have to develop a metric that takes into ac count the\nspecial characteristics of generalized patterns.\nLet us first motivate the new metric: Consider the data vectors v1=\n(5,5,5,5),v2= (5,5,5,6) andv3= (5,5,5,4) of length 4. Intuitively, the\ndifferences between the patterns of v1andv2respectively the patterns of v1\nandv3should coincide. We obtain the patterns: t1= (1,1,1,1),t2= (1,1,1,2)\nandt3= (2,2,2,1). Applying the L1-metric would lead to different distances:\n/bardblt2−t1/bardbl1= 1 and /bardblt3−t1/bardbl1= 3.\nIn order to overcome this, we consider each pattern as an equivale nce class\nof vectors. Loosely speaking, (1 ,1,2,2) should be equivalent to (2 ,2,3,3) and\nhence, it should be close to (1 ,2,3,3). Let us emphasize that (2 ,2,3,3) is not\na pattern any more. More precisely, we want to consider not only th e pattern\nitself, but any shift caused by an addition of k·(1,1,1,1) withk∈Z. In the\nexample given here this would mean that we treat the vector w= (2,2,2,2)\nequally to t1such that if we compare wandt3instead of t1andt3they have\nthe same distance of 1 as t1andt2.\nConsequently, we define a new metric for generalized ordinal patte rnsdfas\nfollows:\n8Definition 2.1. Lett,u∈Tn. Then\ndf(t,u) = min\nk∈Z(/bardblt+k·en−u/bardbl1),\nwhereenis the unit pattern (1 ,...,1) of length n.\nProposition 2.2. The mapping df:Tn×Tn→Ris a metric.\nProof.We have to verify the axioms of a metric. First, consider\ndf(t,t) = min\nk∈Z(/bardblt+k·en−t/bardbl1) = min\nk∈Z(/bardblk·en/bardbl1) = 0\nConversely, if df(t,u) = 0, there exists a k∈Zs.t. min\nk∈Z(/bardblt+k·en−u/bardbl1) = 0.\nEitherk= 0, then t=uand we are done or k/ne}ationslash= 0. In this second case, we\nobtain the following: t∈Tn, hence it takes values in {1,...,m}and, therefore,\nutakes values in {k,...,m+k}. This leads to the contradiction u/∈Tn.\nThe symmetry property is obtained as follows ( t,u∈Tn):\ndf(t,u) = min\nk∈Z(/bardblu−k·en−t/bardbl1) = min\nℓ∈Z(/bardblu+ℓ·en−t/bardbl1) =df(u,t)\nwhere we have multiplied the entries of the vector by -1. Finally, the t riangle\ninequality follows by ( t,u,v∈Tn):\ndf(t,u)+df(u,v) = min\nk∈Z(/bardblt+k·en−u/bardbl1)+min\nℓ∈Z(/bardblu+ℓ·en−v/bardbl1)\n≥min\nk∈Zmin\nℓ∈Z(/bardblt+k·en−u+u+ℓ·en−v/bardbl1)\n= min\nk∈Zmin\nℓ∈Z(/bardblt+(k+ℓ)·en−v/bardbl1)\n= min\nm∈Z(/bardblt+m·en−v/bardbl1)\nIn order to implement the new metric, the range of kcan be limited to the\nlength of the ordinal pattern such that\ndf(t,u) = min\n−n≤k≤n(/bardblt+k·en−u/bardbl1).\nThis simplification still covers all necessary comparisons of pattern s.\n9This metric measures how close patterns are. In case of coincident patterns,\nwe would like to assign the ‘score’ one. The further the patterns ar e away from\neach other, the lower the score should be. Following [11], instead of u singd\ndirectly, we use the following anti-monotone weight function on the v alues ofd\nfor patterns of length less than six\nw(x) := 1·1{x=0}+0.5·1{x=1}, (2)\nand for patterns of length greater than or equal to six\nw(x) := 1·1{x=0}+0.75·1{x=1}+0.5·1{x=2}+0.25·1{x=3},(3)\nwhere 1 {x=t}is the indicator function, being one, when x=t, and zero\nelsewhere. Too small steps in the weight function in the presence of small\npatterns could lead to a too high acceptance of deviations from the observed\npatterns and hence could increase ordinal pattern dependence a rtificially. For\nlargern, an appropriate weight function might be defined, for example by us ing\nsmaller steps of weights. Since we limit the applications to small values o fn,\nthese weight functions are sufficient for our purposes. Let us rem ark that the\nweight functions are different than in the classical case: Since the L1-distance\nbetween two permutations in Snis always an even number, the weight function\nis - in that case - defined on even numbers only.\nWe define the score function for t,u∈Tnas\ns(t,u) :=w◦df(t,u) =w(df(t,u)). (4)\nTo obtain the score for the dependence between the two time serie s, the\nso-called total score is used, defined as\nS=/summationdisplay\nt∈Tn/summationdisplay\nu∈Tns(t,u)·P(Ψ(X1,...,Xn) =t,Ψ(Y1,...,Yn) =u).\nThe higher the total score is, the stronger the co-movement bet ween the time\nseries (resp. data sets).\n10Of course, this score can also be applied to the non-weighted case, where\nsimply a sum of 1s is considered. In principle it is possible to calculate a\ncomparison value in the same spirit as above.\nAn approach via generalized ordinal patterns has (almost) the sam e advan-\ntages as the classical ordinal pattern approach. In addition, we a re now able\nto deal with ties and hence we can handle categorial data. Let us em phasize\nthat the whole analysis is stable under strictly monotone transform ations of the\nstate space. Structural breaks in a single time series do not affect the ordinal\npattern dependence very much. In hydrology, e.g. a shift in mean c an be caused\nby a new dam. The algorithms in order to analyze data sets by our met hod are\nvery quick and in the analysis of ‘co-monotonic behavior’ no second m oments\nare needed. These are inevitable when applying classical correlation .\nSmall noise might (in contrast to the classical approach) change th e ordinal\nstructures, since ties might become increasing or decreasing patt erns. However,\nkeep in mind that the new approach is designed in first place to deal wit h\ncategorial data, where a small noise does not make any sense.\n2.3. Natural Estimators for the Ordinal Pattern Coefficient a nd the Total Score\nThe above given theoretical characteristics for ordinal pattern s can be es-\ntimated in the most natural way by their sample analogues. Given the ob-\nservations of ( X1,...,X N) and (Y1,...,Y N), the following estimators can be\nderived.\nFor the probability of a coincident ordinal pattern p, the sample estimator\nis given by\nˆpN=1\nN−n+1N−n+1/summationdisplay\nj=11{Ψ(Xj,...,Xj+n−1)=Ψ(Yj,...,Yj+n−1)}.\nA similar estimator was used in [11] in the context of classical ordinal p at-\nterns.\nIn order to estimate the standardized ordinal pattern coefficient , additional\nparameters are required. Sample estimators for these paramete rsq,randsare\ngiven in the following\n11ˆqN=/summationdisplay\nt∈Tn/parenleftbigg1\nN−n+1/parenrightbigg2N−n+1/summationdisplay\ni=11{Ψ(Xi,...,Xi+n−1)=t}N−n+1/summationdisplay\ni=11{Ψ(Yi,...,Yi+n−1)=t}\nˆrN=1\nN−n+1N−n+1/summationdisplay\ni=11{Ψ(Xi,...,Xi+n−1)=Ψ(−Yi,...,−Yi+n−1)}\nˆsN=/summationdisplay\nt∈Tn/parenleftbigg1\nN−n+1/parenrightbigg2N−n+1/summationdisplay\ni=11{Ψ(Xi,...,Xi+n−1)=t}N−n+1/summationdisplay\ni=11{Ψ(−Yi,...,−Yi+n−1)=t}.\nWith these, the standardized ordinal pattern coefficient can be es timated by\n/hatwidestord(X,Y) =/parenleftbiggˆpN−ˆqN\n1−ˆqN/parenrightbigg+\n−/parenleftbiggˆrN−ˆsN\n1−ˆsN/parenrightbigg+\n.\nWhen considering the total score S, a sample estimator is given by\nˆSN=1\nN−n+1N−n+1/summationdisplay\ni=1ω(df(Ψ(Xi,...,X i+n−1),Ψ(Yi,...,Y i+n−1))).\nHere, the considered window for the pattern is shifted in time by one step\neach time. Alternatively, the window could also be shifted by the lengt h of the\nordinal pattern n. This way, an overlapping of the windows could be omitted.\nHowever, it is not always a priori known, which time steps should be pa rt of one\nwindow. If, e.g., monthly data is used, a natural window would be the c hoice of\none year, capturing the seasonality. However, for annual data t his is no longer\nobvious. Hence, a loss of information could be obtained when choosin g a step\nof length n.\n3. Limit Theorems\nIn this section, we provide limit theorems under short range depend ence for\nthe previously introduced estimators in the context of ordinal pat tern depen-\ndence. Moreover, a limit theorem for the multivariate case, i.e., when consid-\nering a pattern vector that is shifted in time, is given. These limit theo rems\nprepare the applications in our temporal respectively spatial appr oach in the\nsubsequent section.\n123.1. Pairwise comparison in an SRD regime\nThe first case that is handled is the ordinal pattern dependence. T his means\nlimit theorems are required for a bivariate vector of random variable s. We can-\nnot use the limit theorems as presented in [11] for the following reas on: Already\nthe fundamental first theorem in that paper needs the assumptio n that the\ndistribution functions of the underlying random variables are Lipsch itz continu-\nous. Our random variables might be discrete (cf. the application on fl ood alert\nclasses in Section 4) and hence a more generalized setting is required .\nLimit theorems are developed under a general setting of short-ra nge de-\npendence to consider auto-correlation in the data. Here, we cons ider absolute\nregularity.\nDefinition 3.1. LetA,B ⊂ Fbetwoσ-fieldsontheprobabilityspace(Ω ,F,P).\nThe absolute regularity coefficient of AandBis given by\nβ(A,B) =Esup\nA∈A/vextendsingle/vextendsingleP(A|B)−P(A)/vextendsingle/vextendsingle.\nIndependence of AandBis equivalent to β(A,B) = 0.\nIf (Xk)k∈Nis a stationary process, then the absolute regularity coefficients\nof (Xk)k∈Nare given by\nβ(ℓ) = sup\nk∈Nβ(Fk\n1,F∞\nk+ℓ).\n(Xk)k∈Nis called absolutely regular, if β(ℓ)→0 asl→ ∞.\nThis concept of short range dependence has been introduced by [1 4] (and\nthere attributed to Kolmogorov) and includes many well-known mode ls used in\napplicationslikecertainMarkovchainsorAR-processes. Gaussians equencesare\nalso absolute regular, as long as their spectral density is of a certa in form [15].\nMoreover, renewal processes and solutions of stochastic differe ntial equations\nwere shown to be absolutely regular random processes.\nUnder this dependency assumption, the following limit theorem holds.\n13Theorem 3.2. Let(Zk= (Xk,Yk))k∈Nbe an absolutely regular, stationary\nprocess on the probability space (Ω,F,P). Then,\nˆpN→pP-a.s.\nThe theorem is in line with [6] and [5], where similar results for the long-\nrange dependent and weak dependence cases are stated.\nProof.Since we assume that the underlying short-range dependent proc ess is\nabsolute regular ( β-mixing), it is also ergodic. This makes the ergodic theorem\nby Birkhoff applicable. We apply it to the partial sums\nN−n+1/summationdisplay\ni=1(f(Xi,...,X i+n−1,Yi,...,Y i+n−1)−Ef(X1,...,X n,Y1,...,Y n))\nwithf(Xi,...,X i+n−1,Yi,...,Y i+n−1) = 1 {Ψ(Xi,...,Xi+n−1)=Ψ(Yi,...,Yi+n−1)}\nto obtain\n1\nN−n+1N−n+1/summationdisplay\ni=1f(Xi,...,X i+n−1,Yi,...,Y i+n−1)→Ef(X1,...,X n,Y1,...,Y n) =p\nP-almost surely.\nBesides consistency of the estimator, also a limit distribution is helpfu l, e.g.\nto derive confidence bands. The respective theorem is given in the f ollowing,\ntogether with limit theorems for the remaining parameters q,rands, such that\nalso a limit distribution of the standardized ordinal pattern coefficien t can be\nobtained.\nTheorem 3.3. Let(Zk= (Xk,Yk))k∈Nbe an absolutely regular, stationary\nprocess on the probability space (Ω,F,P)with coefficients/summationtext∞\nℓ=1(β(ℓ))1/2<∞.\nThen,\n√\nN(ˆpN−p)→ N(0,σ2),\n14where\nσ2=Var(1 {Ψ(X1,...,Xn)=Ψ(Y1,...,Yn)})\n+2∞/summationdisplay\nm=2Cov(1 {Ψ(X1,...,Xn)=Ψ(Y1,...,Yn)},1{Ψ(Xm,...,Xm+n−1)=Ψ(Ym,...,Ym+n−1)}).\nThe concept of the bivariate mixing process used here is very similar t o the\ntheorem given in [16]. There, a limit theorem is proven for a strongly m ixing\nprocessbyusingtherespectiveinequalitiesof[17]. Wewillobtainasimila rresult\nby applying a theorem of [18], that explicitly omits any continuity assum ption\non the distribution.\nProof.First, define( fi)i∈Nwithfi= 1{Ψ(Xi,...,Xi+n−1)=Ψ(Yi,...,Yi+n−1)}−passta-\ntionary process derived from the random variable f(Z1,...,Z i−1,Zi,Zi+1,...).\nThough the stationary process may not automatically be absolutely regular,\nTheorem 2.1 of [18] gives a theorem with sufficient conditions to never the-\nless derive a limit theorem for ( fi)i∈N. It is E(f1) =P(Ψ(X1,...,X n) =\nΨ(Y1,...,Y n))−p= 0 and Ef2\n1<∞due to the boundedness of f1as and\nindicator function and hence assumption one of Theorem 2.1 of [18] f ollows\nfrom stationarity. The remaining assumptions of Theorem 2.1 follow f rom the\nassumptions made in our theorem. Therefore, we can apply Theore m 2.1 of [18]\nto (fi)i∈N. This gives us\nP/parenleftbiggf1+...+fN\nσ√\nN< x/parenrightbigg\n−→1√\n2π/integraldisplayx\n∞e−u2/2du\nforN→ ∞and thus\n1√\nNN/summationdisplay\ni=1/parenleftig\nE(Ψ(Xi,...,X i+n−1) = Ψ(Yi,...,Y i+n−1))−p/parenrightig\n=√\nN(ˆpN−p)→ N(0,σ2),\nforσ/ne}ationslash= 0.\n15The variance σ2is given by\nσ2=Ef2\n1+2∞/summationdisplay\nj=1E(f1fi)\n=Var(1 {Ψ(X1,...,Xn)=Ψ(Y1,...,Yn)})\n+2∞/summationdisplay\nm=2Cov(1 {Ψ(X1,...,Xn)=Ψ(Y1,...,Yn)},1{Ψ(Xm,...,Xm+n−1)=Ψ(Ym,...,Ym+n−1)})<∞,\nwhich completes the proof.\nRemark 3.4.For applications, it might be helpful to estimate σin the previous\ntheoremsince in generalthis parameteris unknown. Such an estima tor ˆσN, that\nis consistent, can be derived by applying the sample estimators for U-statistic\nvariances in [19]. Then,\nˆσ2\nN=1\nNN−n+1/summationdisplay\ni=1N−n+1/summationdisplay\nj=1k/parenleftbiggi−j\nbN/parenrightbigg/parenleftbig\n1{Ψ(Xi,...,Xi+n−1)=Ψ(Yi,...,Yi+n−1)}−ˆpN/parenrightbig\n/parenleftbig\n1{Ψ(Xj,...,Xj+n−1)=Ψ(Yj,...,Yj+n−1)}−ˆpN/parenrightbig\n.\nApplication of Slutsky’s Theorem gives\n√\nNˆpN−p\nˆσN→ N(0,1).\nunder the conditions of Theorem 3.3.\nAdditionally, limit theorems for the remaining parameters which have b een\nintroduced above can be obtained.\nTheorem 3.5. Under the same conditions as in Theorem 3.3,\n√\nN(ˆqN−q)→ N(0,δ2),\n16where\nδ2=/summationdisplay\nt1,t2∈TnP(Ψ(Y1,...,Y n) =t1)∞/summationdisplay\nk=−∞Cov(1 {Ψ(X1,...,Xn)=t1},1{Ψ(Xk,...,Xk+n−1=t2)})\nP(Ψ(Y1,...,Y n) =t2)\n+2/summationdisplay\nt1,t2∈TnP(Ψ(X1,...,X n) =t1)∞/summationdisplay\nk=−∞Cov(1 {Ψ(X1,...,Xn)=t1},1[Ψ(Yk,...,Yk+n−1=t2)})\nP(Ψ(Y1,...,Y n) =t2)\n+/summationdisplay\nt1,t2∈TnP(Ψ(X1,...,X n) =t1)∞/summationdisplay\nk=−∞Cov(1 [Ψ(Y1,...,Yn)=t1},1[Ψ(Yk,...,Yk+n−1=t2)})\nP(Ψ(X1,...,x n) =t2)\nProof.We will first treat the two random vectors ( Xk)k∈Nand (Yk)k∈Nsepa-\nrately. Define\nqX(t) =P(Ψ(X1,...,X n) =t)\nqY(t) =P(Ψ(Y1,...,Y n) =t)\nwitht∈Tnand the respective sample analogons\nˆqX(t) =1\nNN−n+1/summationdisplay\ni=11{Ψ(Xi,...,Xi+n−1)=t}\nqY(t) =1\nNN−n+1/summationdisplay\ni=11{Ψ(Yi,...,Yi+n−1)=t}.\nApplying again Theorem 2.1 of [18], where the conditions are fulfilled due to\nthe boundedness of the indicator function and the assumptions on the abso-\nlute regular coefficients, together with the Cram´ er-Wold device it f ollows in the\nmultivariate case that\n√\nN((ˆqX(t)−qX(t))T\nt∈Tn,(ˆqY(t)−qY(t))T\nt∈Tn)T→ N(0,Σ)\n17with\nΣ =\nΣXΣXY\nΣYXΣY\n\nand\nΣX=/parenleftigg∞/summationdisplay\nm=1Cov(1 {Ψ(X1,...,Xn)=t1},1{Ψ(Xm,...,Xm+n−1)=t2})/parenrightigg\nt1,t2∈Tn\nΣXY= ΣYX=/parenleftigg∞/summationdisplay\nm=1Cov(1 {Ψ(X1,...,Xn)=t1},1{Ψ(Ym,...,Ym+n−1)=t2})/parenrightigg\nt1,t2∈Tn\nΣY=/parenleftigg∞/summationdisplay\nm=1Cov(1 {Ψ(Y1,...,Yn)=t1},1{Ψ(Ym,...,Ym+n−1)=t2})/parenrightigg\nt1,t2∈Tn.\nInthenextstep, wewanttoapplythedeltamethod. Forthispurpo se, wede-\nfine the function f:N2n!→Rwithf(x,y) =/summationtextn!\ni=1xiyiwithx= (x1,...,x n!)\nandy= (y1,...,y n!)suchthat q=f(qX,qY)and ˆqN=f((ˆqX(t))t∈Tn,(ˆqY(t))t∈Tn).\nIt remains to show that fis differentiable. To show this, we first have a look\nat the partial derivatives\n∂\n∂xif(x,y) =yiand∂\n∂yif(x,y) =xi.\nHence, ∆ f= (x,y) =\nx\ny\nwhich make the delta method applicable. This\nresults in\n√\nN(ˆqN−q)→ N(0,δ2)\nwith\nδ2= (∆f(qX,qY))TΣ∆f(qX,qY) = (qT\nX,qT\nY)Σ\nx\ny\n\nwhich completes the proof.\n18Analogous results can be obtained for the sample estimators of randsusing\nthe same techniques. With these results, the limit theorem for /hatwidestord(X,Y) then\nfollows simply from the delta method using the assumptions from Theo rem 3.3.\nSecondly, limit theorems for the sample estimator of the total scor e are\nrequired. Under the same conditions as in the above theorems, it ca n be shown\nthat\n√\nN(ˆSN−S)→N(0,γ2)\nwith\nγ2=Var(ω(df(Ψ(X1,...,X n) = Ψ(Y1,...,Y n))))\n+2∞/summationdisplay\nm=2Cov(ω(df(Ψ(X1,...,X n) = Ψ(Y1,...,Y n))),\nω(df(Ψ(Xm,...,X m+n−1) = Ψ(Ym,...,Y m+n−1)))).\nbyapplicationoftheCentralLimitTheoremforfunctionalsofabsolu telyregular\nprocesses of [18] and the Cram´ er-Wold device.\n3.2. Limit Theorems in a Multivariate i.i.d. Setting\nDealingwith multivariatedata, wealsoconsiderthe patternsateach pointin\ntime. Let ( Xk)k∈Nbe ad-dimensional time series. The pattern Ψ(( X1\nk,...,Xd\nk))\ndescribes in a short way, where the highest value, second highest v alue... in the\nvectorXkis. While we have dependence within the vector Xk, we encounter\nsituationswherethetimeseries( Xk)k∈Nconsistsofindependentrandomvectors.\nHence, we can use classical limit theorems:\n1\nNN/summationdisplay\nk=11{Ψ(Xk)=t}→P(Ψ(X1) =t)P-a.s.\nholds by the (strong) law of large numbers. Writing Pt:=P(Ψ(X1) =t) we\nobtain in addition\n1√\nNN/summationdisplay\nk=1(1{Ψ(Xk)=t}−Pt)→N(0,P2\nt)\n19by the Central Limit Theorem for i.i.d. Bernoulli random variables. The se\nsimple convergence theorems are stated here only for the sake of completeness\nsince they are used in the spatial approach of the subsequent sec tion.\n4. Application and Simulation: Temporal and Spatial Patter ns in\nFlood Classification\nLet us consider the following general situation: we observe d-dimensional\ndata vectors at discrete time points k∈N. This is modeled as a d-dimensional\ntime series ( Xk)k∈N. We want to analyze the dependence between the one-\ndimensional time series ( Xa\nk)k∈Nand (Xb\nk)k∈Nfor 1≤a,b≤d. We will do\nthis using two different approaches which both make use of generaliz ed ordinal\npatterns.\nX1\n1X1\n2/vextendsingle/vextendsingleX1\n3X1\n4···\nX2\n1X2\n2X2\n3X2\n4···\nX3\n1X3\n2/vextendsingle/vextendsingleX3\n3X3\n4···\n...............\nXd\n1Xd\n2Xd\n3Xd\n4···\n\nX1\n1X1\n2X1\n3X1\n4···\nX2\n1X2\n2X2\n3X2\n4···\nX3\n1X3\n2X3\n3X3\n4···\n...............\nXd\n1Xd\n2Xd\n3Xd\n4···\n\n\nFigure 4: The temporal approach (left) and the spatial appro ach (right)\nI. Temporal approach: we compare the time series ( Xa\nk)k∈N(for 1≤a≤d\npairwise). To this end, we use the ordinal pattern dependence as d escribed\nabove. This allows us to analyze co-monotonic behavior over time.\nII. Spatial approach: we assign the patterns Ψ( Xk) with vectors Xkat each\nfixed time point k. If the probability\nP(Ψ(Xa\n1) = Ψ(Xb\n1))\nis high, this is an indicator for strong dependence between the ath and the bth\ncomponent of the vector. However, the spatial pattern contain s more informa-\ntion. The frequency of certain patterns can reveal structures in a network. If,\n20for example, the frequency of the unit pattern (1 ,1,...,1) is high, this implies\nan overall high dependence between all vector entries. In the follo wing, the\nexample of flood classes will be introduced. If a gauging network with in a river\nbasin is treated as vector, the application of spatial ordinal patte rns can deliver\ninformation on which gauges often react similarly, due to weather pa tterns or\nanthropological impacts.\nFlood events within a river basin almost always occur at several gaug es at\nthe same time. Due to spatially distributed rainfall, snowmelt and flood wave\npropagation, flood events show patterns within a basin. Especially w eather\npatterns like the Vbweather pattern, that is responsible for almost all large\nflood events in Germany over the last decades, are characterized by a certain\ncirculation[20]. Hence, spatial patternsofflood eventsareinvest igatedby many\nresearcherstounderstandthe extentandpossiblehazardofex tremefloodevents\n[21]. Nevertheless, it is not straightforward to compare the obser ved magnitude\nof the events at different gauges. The peak of the flood wave, whic h is mostly\nused to characterize a flood event and for flood statistics [22], dep ends much\non the catchment size and characteristics and can be influenced by overlaid\nwaves of tributaries. The same holds true for the flood volume. Ins tead, [23]\npropose a flood classification. This classification has the advantage that it is\ndistribution-free and ordinal but, since it is an extension of the Che bychev-\ninequality, still depends on probabilities, which is necessary for flood statistics.\nIn their research, they also detected certain patterns in the clas sification of\nsingle flood events within the river basin, but they have not been eva luated\nstatistically concerning significance. Hence, we will adopt this classifi cation and\ninvestigate the significance of the spatial patterns of flood classe s with ordinal\npatterns. Forthispurpose, accordingto[23], fivefloodclassesar edefined (Table\n1).\nOf course, there does not exist spatial coherence of flood event s only, but\nalso temporal patterns can be observed for floods. In hydrology , often so called\nflood-rich and flood-poor periods are observed. These are define d as significant\nclusters of floods above or below a certain threshold. Nevertheles s, significance\n21Flood Class Description Non-exceedance Probability pof Peak\n0 small flood p <0.5\n1 normal flood 0 .5≤p <0.8\n2 medium flood 0 .8≤p≤0.933\n3 large flood 0 .933≤p≤0.966\n4 very large flood 0 .966≤p\nTable 1: Five flood classes to characterize the magnitude of fl ood events.\nof these clusters is hardly detected with common techniques, espe cially for non-\nannual data [24]. Again, the use of flood classes instead of thresh olds in combi-\nnation with ordinal patterns shall help us to detect such significant patterns in\nthe temporal series.\nFor our analyses, 14 gauges of the Mulde river basin with an observa tion\nperiod starting in 1926 (due to uniformity) are available, including hea d as well\nas estuary catchments. For these gauges, 314 flood events hav e been separated\nand classified with the classification described above. As a referenc e gauge we\nuse the Golzern gauge, being the one at the estuary and hence inclu ding the\nwhole basin. For this gauge, the classification resulted in 9 flood even ts of class\n4 (including the extraordinary large floods in 2002 and 2013), 7 flood events\nof class 3, 39 events of class 2, 56 events of class 1 and 203 events of class 0.\nOf course, not all events occurred at all gauges. If a certain eve nt cannot be\ndetected at a gauge (meaning that no flood occurred), it is assigne d with class\n-1, since it is different from all flood classes. With this assumption, a c om-\nparison of all gauges and events is made possible. A map of the study area is\nprovided in Figure 5.\n4.1. The Temporal Approach\nTemporal coherence between the flood classes of the gauges is inv estigated\nsimilarly to [1], with shifting ordinal patterns of the flood classes of len gthn= 4\nby one event per step and compare the patterns between the gau ges. Here, the\nmetrical approach is used to include small variations in the patterns . All 13\n220 1000 2000 3000 4000Altitude [m. a. s. l.]\nStreckewaldeWechselburg\nZoeblitzZwickau\nHopfgartenLichtenwaldeNossen\nAueBerthelsdorfBorstendorfGolzern\nGoeritzhain\nFigure 5: Map of the Mulde River basin in Saxony, Germany and i ts location in Central\nEurope.\n23gauges are considered. The metrical coherence (in percent) for all combination\nof gauges is given in the supplement. The coherence between the ga uges in\ngeneral is high, indicating strong dependencies between the gauge s (Table 2).\nSpatial patterns become visible. An example for the time series of flo od classes\nfor the four main gauges is given in Figure 6a, where clear coherence especially\nfor extreme events of classes 3 and 4 becomes visible.\nThe comparison value was estimated using ˆ qNin combination with the con-\nsidered metric. The results are given in the supplement. All comparis on values\nare much smaller than the obtained ordinal pattern dependence. T his implies\nsignificant dependency between all gauges.\nAs second comparison, also two time series ( Zt)t∈Zof type INGARCH(p,q)\n[25] have been simulated. More precisely, we simulated\nZt|Ft−1∼Poi(νt),\nwhereFt−1denotesthehistoryoftheprocessuptotime t−1withconditional\ndistribution chosen as Poisson. With this definition, the linear predict or is the\nconditional mean\nνt=β0+β1Zt−i1+...+βpZt−ip+α1νt−j1+...+αqνt−jq+η1Xt,1+...+ηrXt,r.\nFor the simulation, we chose β0= 2,p= 1,β1= 0.3,q= 0 and r= 0 (the\nINAR(1) model) for sake of simplicity. For the two time series with t= 1000,\nthe ordinalpattern metrical coherencehas been assessedfor 1 000repetitions. In\nmean, a coherence of 22.2% has been obtained, which is far below all o btained\nvalues for the flood time series (Table 2). Obviously, even an auto-c orrelated\ntime series does not lead to such high coherence. If we increase the correlation\nby assuming β1= 0.6, the coherence stays at 20.4%. This can also be seen in\nFigure 6b.\nSo far, we have only statedthat our method is beneficial to the general\nhandling of ties in the context of ordinal patterns. In the following, we will\nshortly demonstrate this. For this purpose, two standard handlin g methods of\ntiesinordinalpatternsarecomparedwithgeneralizedordinalpatt ernintermsof\n24GolzernNossenLichtenwaldeWechselburg\n1940 1960 1980 2000−101234\n−101234\n−101234\n−101234\nY earFlood classa)\nTimeSeries4TimeSeries3TimeSeries2TimeSeries1\n0 50 100 150 200 250051015\n051015\n051015\n051015\nTimeINAR(1)b)\nFigure 6: a) Time series of flood classes of the four main gauge s in the Mulde River basin in\nthe years 1927-2015; b) four time series of sample size n= 250 generated from an INAR(1)-\nprocess with β1= 0.6 without cross-correlation.\nTable 2: Results of the ordinal pattern dependence for all ca tchments in the Mulde River\nbasin and simulated INAR(1)-processes. Values are given in %.\nTime Series mean min max\nMulde flood classes 58.2 48.3 75.2\nINAR(1) with β1= 0.3 18.6 26.7 22.2\nINAR(1) with β1= 0.6 16.5 23.9 20.4\n25Table 3: Results of the ordinal pattern dependence for all 13 gauges with different ties-\nhandling. Values are given in %.\nn= 4 n= 6\nApproach mean min max mean min max\nGeneralized Ordinal Pattern Dependence 58 48 75 49 39 67\nRandomized ties 21 13 32 9.2 4.5 18\nApproach c) 21 7 42 14 7.0 30\nordinal pattern dependence. More precisely, we will use cases b) ( randomization\nofties)andc)(firstappearance)fromabove,sincecasea)(the removalfromties)\nis not meaningful in this context with small patterns and many ties. A gain, all\n13gaugesfromabovearecomparedconcerningtheir ordinalpatt ern dependence\nusing the metrical approach. To exclude the possibility of dependen ce of the\nresults on the pattern length, the lengths n= 4 as well as n= 6 are used.\nSince the methodology differs from the one proposed here, the met ric has to be\nadjusted. For these approaches, only even numbers of differenc e are obtained,\nhence the metric should only consider these [11]. For the case of n= 4, the\nmetric\nw(x) := 1·1{x=0}+0.5·1{x=2}, (5)\nis applied, while for n= 6, we apply\nw(x) := 1·1{x=0}+0.75·1{x=2}+0.5·1{x=4}+0.25·1{x=6}.(6)\nThe resultsaresummarizedin Table3. Detailedresultsforeachcomb ination\nof gauges are given in the supplement.\nTheclassicalties-handlingapproachesunderestimatethedepend encepresent\nin the data by altering the ties structure. Moreover, the length of the pattern\naffects the performance of the handling of ties and becomes worse the larger\nthe pattern. This can be explained by the application of the metric. D ue to\nthe metrical structure, for small patterns changes in the patte rn, as done in\n26the presence of ties, the effect is reduced since these patterns a re still similar\nto other patterns and the metric delivers comparably high values. F or larger\npatterns, this effect decreases.\nThis application demonstrates the drawbacks of classical ties-han dling in\nordinal pattern dependence. Due to the changes of the pattern structure, the\nordinal pattern dependence is significantly reduced and below the c omparison\nvalue for significance, hence indicating independence of the time ser ies. With\nour approach, we overcome these drawbacks in a natural way.\n4.2. The Spatial Approach\nFor the spatial approach, we compare the flood classification of th e peak\ndischarges at the gauges for each event. Due to computational lim its, the maxi-\nmum of eight gauges (Aue, Zwickau, Wechselburg, Hopfgarten, Zo eblitz, Licht-\nenwalde, Nossen and Golzern) and the subset consisting only of the estuary\nGolzern and the three main tributaries Wechselburg, Lichtenwalde a nd Nossen\nare considered. This means we obtain 314 ordinal patterns of lengt hn= 8\nrespectively n= 4. For these patterns, a percental frequency is estimated. Of\ncourse, it is not only of interest, if there is a coherence between th e gauges.\nThis can be assumed for most flood events due to the non-local nat ure of pre-\ncipitation. More interestingly, we want to investigate, which patter n occurs\nmost. That is, are there certain gauges that often behave differe ntly than the\nremaining gauges. For this purpose, we compare the ordinal patte rn of each\nevent with all possible theoretical ordinal patterns. The results s how, which\nordinal pattern occurs most and if this occurrence is significant. T he patterns\ncan be assumedto be independent since eachpattern belongsto a d ifferent flood\nevent which have been separated such that it can be guaranteed t hat they do\nnot influence each other. Moreover, we have tested the auto-co rrelation of each\ntime series with the Cram´ er V approach suggested in [25], where no s ignificant\nauto-correlation occurred for any of the gauges (with mean value s between 0.1\nand 0.16 for lags up to k= 100).\nIf weonlyconsiderthe fourestuarialcatchmentsWechselburg, L ichtenwalde,\n27Nossen and Golzern, representing the three tributaries and the e stuary, we can\nobserve that the unit pattern (1 ,1,1,1) =e4representing the vectors k·e4\n(k=−1,...,4) occurs with highest frequency of all patterns, being significant ly\nhigher than the theoretical value (Table 4). This implies that in gener al we\nhave very similar flood classes for events in the tributaries and the e stuary and\nthat their occurrence is significantly higher than expected. The se cond highest\nfrequency can be observed for the pattern where the central t ributary has flood\nclass that differs by one from the remaining classes and third highest where the\nGolzern gauge and exactly one of the tributaries differ by one class f rom the\nremaining two gauges. Obviously, the magnitude of a flood at the est uary is\ndetermined by the largest flood class of the three tributaries. The fourth most\ntype of pattern seems to be surprising on the first sight: it is the pa ttern where\nthe estuary gauge has a flood class differing by one class from the re maining\ngauges. From the hydrological perspective this can be explained by the phenom\nof overlaid flood waves. Due to temporally similar occurrence of flood event\nin the tributaries caused by spatially extended rainfall, often flood w aves can\noverlap at the estuary leading to so called superpositioned waves [2 6]. This\nleads to much larger flood waves in the estuary and hence to much lar ger flood\nclasses. In theory, it should be a significantly lower coherence and h ence this\npattern is significant for flood classes. Non-observed patterns ( 0%) are those\nwhere two gauges show flood classes that differ by more than two cla sses from\nthe other tributaries. Obviously, the most extreme events indicat ed by large\nflood classes do not occur locally in one tributary. Instead, at least moderate\nfloods can be observed for the whole basin. We would expect these p atterns\nto occur in 1% of all patterns. Also non-observed are patterns wh ere only one\ngauge has the flood class zero (0%). This emphasizes the strong de pendence\nwithin the basin, leading to similar reactions for the whole basin.\nIf we include all gauges, i.e., also headgauges, the results remain simila r.\nDue to the increased possibilities of patterns, the overall proport ion of distinct\npatterns reduces, but still the same significant patterns can be o bserved. For\nexample, the unit pattern has a frequency of 23 .2%, where we would expect\n28Table 4: Frequency of the most-frequent ordinal patterns of flood classes for the four main\ngauges in the Mulde River basin and theoretical comparison v alues. Vectors of flood classes\nrefer to the order (Golzern, Wechselburg, Lichtenwalde, No ssen),kcan take integer values\nbetween -1 and 4, respectively 0 and 4 for Golzern gauge. Valu es are given in %.\nVector of flood classes Ordinal pattern observed theoretical\n(k,k,k,k) (1,1,1,1) 58.3 48.2\n(k,k,k+1,k) (1,1,2,1) 21.1 13.2\n(k+1,k+1,k,k) (2,2,1,1)\n\n8.1 3.9 (k+1,k,k+1,k) (2,1,2,1)\n(k+1,k,k,k+1) (2,1,1,2)\n(k+1,k,k,k) (2,1,1,1) 6.4 3.5\n7.3%. Nevertheless, also new structures within the basin become visib le. Inter-\nestingly, patterns where only a sub-basin has a flood class one large r than the\nremaining sub-basins, which is proceeding to the main gauges and the outlet\ngauge, has highest frequency of 40 .16%. This implies that the head gauges do\nnot havesuch that ahigh dependence as the gaugeslocated furth erdownstream.\nWeather patterns do not affect those gauges alike, but instead mo re frequent\naffect only the downstream parts, further away from the Ore mou ntains.\n5. Summary and Conclusions\nThe detection ofcoherencebetween time seriesin largedatasetsis an impor-\ntant and challenging task. There exist various methods to do this fo r different\nsettings. Here, we have proposed a method based on ordinal patt erns. Ordinal\npatterns offer a simple consideration where only the relative position s of sub-\nsequent data points are considered and not the values itself. They have been\napplied in manydisciplines like medicine, financeand hydrology. However ,what\nhas been overlooked so far was the handling of ties. Ties occur in man y time\nseries and are of particular importance when analyzing categorical data. In this\nwork, we have explicitly considered ties and hence: generalized patt erns. New\nlimit theorems under short range dependent conditions were develo ped. The\n29general concept of short range dependence (absolute regularit y) that was used\nmakes an application to many classical time series models possible. The gen-\neral structure of the proofs using U-Statistics allows for an exte nsion to further\ndependence concepts.\nThe methodology then was applied to flood classes in a river basin in Ger -\nmany. It was shown that there exists a highly significant dependenc e between\nthe flood classes of neighboring catchments. At the same time, cer tain spatial\npatterns where detected that occurred significantly frequent. These patterns\nwere linked to flood-generating processes like the Vb weather patt ern, which\nis typical for this region and often causes extreme floods, and floo d superpo-\nsition. Finally, a simulation study was performed that demonstrates how the\nproposed handling of ties is more efficient than the classical treatme nt of ties\n(randomization, removal) and does not lead to a loss of information.\nFuture extensions of ordinal patterns for data with ties could be t he con-\nsideration of different dependence structures like near-epoch de pendence, which\nwouldallowanapplicationtomoreclassesoftimeseriesmodelssuchasG ARCH.\nMoreover, long-range dependent processes could be considered which also may\nbe of interest in the case of hydrological data.\nAcknowledgments\nWe would like to thank the Saxon State Office for Environment, Agricu lture\nand Geology for providing the discharge data, which are available fro m the data\nportal iDA (www.umwelt.sachsen.de/umwelt/46037).\nFunding: This work was supported by the German Research Founda tion\n(DFG) [grant numbers FOR 2416; SCHN 1231-3/2].\nSUPPLEMENTARY MATERIAL\nTable S1: Generalized Ordinal Pattern Dependence for Tempo ral Approach with n=4:\nTable with all ordinal pattern dependencies (in percent) estimated with\n30the generalized ordinal pattern approach with pattern length n= 4.\nPlease note that the results are symmetric. (.csv file)\nTable S2: Generalized Ordinal Pattern Dependence for Tempo ral Approach with n=6:\nTable with all ordinal pattern dependencies (in percent) estimated with\nthe generalized ordinal pattern approach with pattern length n= 6.\nPlease note that the results are symmetric. (.csv file)\nTable S3: Comp. value of Gen. Ordinal Pattern Dependence for Temporal Approach with n=4:\nTable with all comparison values for ordinal pattern dependencies ( in per-\ncent) estimated with the generalized ordinal pattern approach wit h pat-\ntern length n= 4. Please note that the results are symmetric. (.csv\nfile)\nTable S4: Comp. value of Gen. Ordinal Pattern Dependence for Temporal Approach with n=6:\nTable with all comparison values for ordinal pattern dependencies ( in per-\ncent) estimated with the generalized ordinal pattern approach wit h pat-\ntern length n= 6. Please note that the results are symmetric. (.csv\nfile)\nTable S5: OPD with Ties handled by Randomization for Tempora l Approach with n=4:\nTable with ordinal pattern dependencies (in percent) estimated wit h the\nclassical ordinal pattern approach with pattern length n= 4 and ties\nhandled by randomization (case b). Please note that the results ar e sym-\nmetric. (.csv file)\nTable S6: OPD with Ties handled by Randomization for Tempora l Approach with n=6:\nTable with ordinal pattern dependencies (in percent) estimated wit h the\nclassical ordinal pattern approach with pattern length n= 6 and ties\nhandled by randomization (case b). Please note that the results ar e sym-\nmetric. (.csv file)\nTable S7: OPD with Ties handled by Case c for Temporal Approac h with n=4:\nTable with ordinal pattern dependencies (in percent) estimated wit h the\n31classical ordinal pattern approach with pattern length n= 4 and ties\nhandled by case c. Please note that the results are symmetric. (.cs v file)\nTable S8: OPD with Ties handled by Case c for Temporal Approac h with n=6:\nTable with ordinal pattern dependencies (in percent) estimated wit h the\nclassical ordinal pattern approach with pattern length n= 6 and ties\nhandled by case c. Please note that the results are symmetric. (.cs v file)\nReferences\n[1] S. Fischer, A. Schumann, A. Schnurr, Ordinal pattern depend ence between\nhydrological time series, Journal of Hydrology 548 (2017) 536–55 1.\n[2] C. Bandt, Ordinal time series analysis, Ecological Modelling 182 (20 05)\n229–238.\n[3] C. Bandt, F. Shiha, Order patterns in time series, J. Time Ser. An al. 28\n(2007) 646–665.\n[4] C. Bandt, B. Pompe, Permutation entropy: A natural complexit y measure\nfor time series, Phys. Rev. Lett. 88 (2002) 174102.\n[5] M. Sinn, K. Keller, Estimation of ordinal pattern probabilities in gau ssian\nprocesses with stationary increments, Comp. Stat. Data Anal. 55 (2011)\n1781–1790.\n[6] A. Betken, J. Buchsteiner, H. Dehling, I. 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A. P. Perdiga o, The june\n2013 flood in the upper danube basin, and comparisons with the 2002 , 1954\nand 1899 floods, Hydrol. Earth Syst. Sci. 17 (2013) 5197–5212.\n34"    },    {        "title": "2401.12277v1.How_informative_are_summaries_of_the_cosmic_21_cm_signal_.pdf",        "content": "Astronomy &Astrophysics manuscript no. main ©ESO 2024\nJanuary 24, 2024\nHow informative are summaries of the cosmic 21-cm signal?\nDavid Prelogovi ´c and Andrei Mesinger\nScuola Normale Superiore, Piazza dei Cavalieri 7, 56125 Pisa, Italy\ne-mail: david.prelogovic@sns.it\nReceived - - - /Accepted - - -\nABSTRACT\nThe cosmic 21-cm signal will bring data-driven advances to studies of the Cosmic Dawn (CD) and Epoch of Reionization (EoR).\nRadio telescopes such as the Square Kilometre Array (SKA) will eventually map the HI fluctuations over the first billion years – the\nmajority of our observable Universe. With such large data volumes, it becomes increasingly important to develop “optimal” summary\nstatistics, allowing us to learn as much as possible about the CD and EoR. In this work we compare the astrophysical parameter\nconstraining power of several 21-cm summary statistics, using the determinant of the Fisher information matrix, det F. Since we do\nnot have an established “fiducial” model for the astrophysics of the first galaxies, we compute for each summary the distribution\nof det Facross the prior volume. Using a large database of cosmic 21-cm lightcones that include realizations of telescope noise,\nwe compare the following summaries: (i) the spherically-averaged power spectrum (1DPS), (ii) the cylindrically-averaged power\nspectrum (2DPS), (iii) the 2D Wavelet scattering transform (WST), (iv) a recurrent neural network (RNN) trained as a regressor; (v)\nan information-maximizing neural network (IMNN); and (vi) the combination of 2DPS and IMNN. Our best performing individual\nsummary is the 2DPS, having relatively high Fisher information throughout parameter space. Although capable of achieving the\nhighest Fisher information for some parameter choices, the IMNN does not generalize well, resulting in a broad distribution across\nthe prior volume. Our best results are achieved with the concatenation of the 2DPS and IMNN. The combination of only these two\ncomplimentary summaries reduces the recovered parameter variances on average by factors of ∼6.5 – 9.5, compared with using each\nsummary independently. Finally, we point out that that the common assumption of a constant covariance matrix when doing Fisher\nforecasts using 21-cm summaries can significantly underestimate parameter constraints.\nKey words. Methods: data analysis – Methods: statistical – Cosmology: theory – dark ages, reionization, first stars\n1. Introduction\nThe cosmic 21-cm signal, corresponding to the spin flip transi-\ntion of the ground state of HI, provides a window on the first\nbillion years of the Universe’s evolution. This under-explored\nperiod witnessed fundamental cosmic milestones, including the\nCosmic Dawn (CD) of the first galaxies and the final phase\nchange of our Universe: the Epoch of Reionization (EoR). Cur-\nrent radio interferometers are putting increasingly tighter upper\nlimits on its power spectrum (PS; e.g. Trott et al. 2020; Mertens\net al. 2020; HERA Collaboration et al. 2023), while the up-\ncoming Square Kilometre Array (SKA)1is promising to provide\na 3D image of the EoR in the next decade(s) (e.g. Koopmans\net al. 2015; Mesinger 2020). The unknown properties of the first\ngalaxies and IGM structures are encoded in the timing and mor-\nphology of the cosmic 21-cm signal (e.g. Pritchard & Furlanetto\n2007; McQuinn & O’Leary 2012; Visbal et al. 2012; Pacucci\net al. 2014).\nHow do we infer these properties? The only statistically-\nrobust way is through Bayesian inference: sampling from the\ntheory and comparing forward models to the observation(s).\nHowever what should we use to compare the forward model to\nthe data; i.e. how should we construct the likelihood? The cos-\nmic 21-cm signal is intrinsically a 3D lightcone map . Performing\ninference directly on the lightcone would be incredibly challeng-\ning, due to the high-dimensionality of the data. E ffectively, at\neach frequency the SKA should obtain a sky map comparable to\ncurrent CMB maps, but with the full lightcone including thou-\n1https://www.skatelescope.orgsands of such frequency slices (e.g. Loeb & Zaldarriaga 2004).2\nInstead, the data is compressed into a summary statistic . Sum-\nmary statistics usually involve some form of averaging, which\nincreases the signal to noise (S /N) and can motivate assuming an\napproximately Gaussian likelihood when performing inference\n(e.g. Greig & Mesinger 2015; Gazagnes et al. 2021; Watkinson\net al. 2022).3\nThe most common choice of a 21-cm summary statistic is\nthe spherically-averaged power spectrum (1D PS). The 1D PS\nmaximizes the signal-to-noise (S /N) of an interferometric obser-\nvation – a primary concern in the run up to a first detection (e.g.\n2Inference on high-dimensional lightcones is still possible in some\ncases, for example galaxy large scale structure (LSS) maps (e.g. Ki-\ntaura & Enßlin 2008; Jasche & Kitaura 2010; Jasche & Wandelt 2012,\n2013; Leclercq et al. 2017; McAlpine et al. 2022; Dai & Seljak 2022;\nBayer et al. 2022). Although such studies still compress the full galaxy\nobservations, only treating them as biased tracers of the matter field,\nthey are able to recover the phase information of our Universe’s initial\nconditions (so-called “constrained realizations”). It remains to be seen\nif such studies can be extended to the Epoch of Reionization and Cos-\nmic Dawn (though see Zhao et al. 2023b for a recent proof-of-concept\nstudy).\n3The gaussianity of summaries constructed by averaging can be\nloosely motivated by appealing to the Central Limit Theorem. However,\nthe theorem holds only for independent, identically distributed random\nvariables. Unfortunately, we only have a single observable Universe;\nthus summaries of cosmological data sets (e.g. the power spectrum) of-\nten involve binning over correlated (i.e. not independent) and /or differ-\nently evolving fields (i.e. not identically distributed), resulting in non-\nGaussian likelihoods (e.g. Prelogovi ´c & Mesinger 2023).\nArticle number, page 1 of 13arXiv:2401.12277v1  [astro-ph.CO]  22 Jan 2024A&A proofs: manuscript no. main\nGreig & Mesinger 2018; Trott et al. 2020; Mertens et al. 2020;\nHERA Collaboration et al. 2023). Another common choice of a\nsummary is the cylindrically-averaged (2D) PS, which includes\nadditional information on the anisotropy of the cosmic 21-cm\nsignal (e.g. Bharadwaj & Ali 2005; Barkana & Loeb 2006; Datta\net al. 2012; Mao et al. 2012). Fourier space is also a natural basis\nto isolate the cosmic signal from instrumental e ffects and fore-\ngrounds, which primarily reside in a wedge region in the 2D\nPS (e.g. Morales et al. 2012; Vedantham et al. 2012; Trott et al.\n2012; Parsons et al. 2014; Liu et al. 2014a,b; Murray & Trott\n2018). However, the power spectrum ignores Fourier phases,\nwhich can encode significant information for a highly non-\nGaussian signal such as 21-cm from the EoR /CD (e.g. Bharad-\nwaj & Pandey 2005; Mellema et al. 2006; Shimabukuro et al.\n2016; Majumdar et al. 2018; Watkinson et al. 2019). Because\nof this, several studies have also explored non-Gaussian statis-\ntics, such as the bispectrum (Shimabukuro et al. 2016; Majumdar\net al. 2018; Watkinson et al. 2019, 2022), trispectrum (Cooray\net al. 2008; Flöss et al. 2022), morphological spectra (Gazagnes\net al. 2021), bubble size distribution (Lin et al. 2016; Giri et al.\n2018; Shimabukuro et al. 2022; Doussot & Semelin 2022; Lu\net al. 2023), one-point statistics (Harker et al. 2009; Watkinson\n& Pritchard 2015; Kittiwisit et al. 2022), genus topology (Hong\net al. 2014), Wavelet Scattering Transform (Greig et al. 2022,\n2023; Zhao et al. 2023a), Minkowski functionals and tensors\n(Gleser et al. 2006; Yoshiura et al. 2017; Chen et al. 2019; Ka-\npahtia et al. 2019; Spina et al. 2021) and Betti numbers (Giri &\nMellema 2021; Kapahtia et al. 2021).\nIs there an “optimal” choice of summary statistic? One way\nto define “optimal” is by how tightly we can recover cosmologi-\ncal/astrophysical parameters. This can be quantified through the\nFisher information matrix (Fisher 1935). The Fisher information\nis commonly used to make forecasts using pre-determined sum-\nmary statistics, or even to define the summary statistic itself. Op-\ntimal summaries /algorithms that are based on the Fisher matrix\ninclude Karhunen-Loève methods and Massively Optimised Pa-\nrameter Estimation and Data compression (MOPED) for linear\ntransformations (e.g. Tegmark et al. 1997; Heavens et al. 2000),\nas well as non-linear generalizations like Information Maximiz-\ning Neural Networks (IMNNs, Charnock et al. 2018) and Fish-\nnets (Makinen et al. 2023). However, these summary statistics\ncome at the cost of interpretability, making the compression\nmore of a “black box” compared to physically-motivated and\neasy to interpret summaries like the PS.4\nIt is also important to note that, unlike ΛCDM, there is\nno obvious or unique choice to parametrize the astrophysics\nof early galaxies whose radiation determines the cosmic 21-cm\nsignal. Neither do we have a good idea of a “fiducial” model.\nThus there is no guarantee that a summary statistic that maxi-\nmizes Fisher information at a single parameter value for a single\nmodel parametrization would generalize well to other parame-\nters/models.\nHere we systematically compare the information content of\nseveral common 21-cm summaries on the basis of their Fisher\ninformation. These include: (i) the spherically-averaged power\nspectrum (1DPS), (ii) the cylindrically-averaged power spectrum\n4An interesting question is why would we even expect a truly opti-\nmal (lossless) compression to exist? The answer to this can be moti-\nvated through the manifold hypothesis (Fefferman et al. 2016), which\nstates that physical probability distributions of the data often lie on a\nlow-dimensional manifold. Optimal (lossless) compression would then\n“extract” such manifolds from the original high-dimensional data. The\nabove-mentioned algorithms do not attempt this directly, but aim at\nmaking compression as optimal as possible, given certain assumptions.(2DPS), (iii) the 2D Wavelet scattering transform (WST), (iv) a\nrecurrent neural network (RNN) trained as a regressor; and (v)\nan IMNN. Several previous studies compared the constraining\npower of 21-cm summaries (e.g. Gazagnes et al. 2021; Watkin-\nson et al. 2022; Greig et al. 2022; Zhao et al. 2023b; Hothi\net al. 2023). Here we explore a broader range of summary statis-\ntics, introducing also IMNNs to the field. Unlike previous 21-\ncm Fisher studies, we do not make the simplifying assumption\nof a constant covariance matrix. Most importantly, we compare\nthe summary statistics across a broad range of parameter space,\ninstead of choosing only one or two fiducial values to make a\nmock 21-cm observation. This is very important to verify the ro-\nbustness of the summary, since we do not have a good idea of a\n“fiducial” astrophysical model for the first galaxies.\nThe paper is organized as follows. In section §2 we de-\nfine the information content of a summary through the Fisher\nmatrix and introduce the Fisher information distribution. In\nsection §3 we introduce our 21-cm simulator and explain\nthe summaries we consider in detail. In §4 we discuss our\ndatabase of prior samples for evaluating the Fisher informa-\ntion. In §5 we present our results, quantifying the most in-\nformative summaries as well as the error introduced by the\ncommon assumption of a constant covariance matrix. Finally,\nwe present concluding remarks and future prospects in §6.\nAll quantities are quoted in co-moving units, and we as-\nsume a standard ΛCDM cosmology: (ΩΛ,ΩM,Ωb,n,σ8,H0)=\u0010\n0.69,0.31,0.048,0.97,0.81,68 km s−1Mpc−1\u0011\n, consistent with\nthe results from Planck Collaboration et al. (2020).\n2. The Fisher matrix as a measure of information\ncontent\nHow do we measure the information content of a summary? If\nwe have a theoretical model with a given parametrization, we can\nsee how sensitive the summary is to changes in the astrophysi-\ncal/cosmological parameters around some fiducial value. Specif-\nically, we can calculate the Fisher matrix (e.g. Spall 2005):\nF(θ∗)mn=Z\nddP(d|θ∗)∂\n∂θmlnP(d|θ∗)·∂\n∂θnlnP(d|θ∗)\n=E\"∂\n∂θmlnP(d|θ)·∂\n∂θnlnP(d|θ)\f\f\f\f\fθ∗#\n, (1)\nwhere dis the data summary (c.f. Fig. 1), P(d|θ) is the likelihood,\nandEdenotes the expectation value over the likelihood (i.e. an\nempirical average over many realizations) evaluated at a given\npoint in parameter space θ∗. The Fisher information matrix pro-\nvides the maximum constraining power, known as the Cramer-\nRao bound (Rao 1992; Cramér 1999): Var( ˆθm)≥(F−1)mm, where\nˆθis an unbiased estimator of the parameters. In the multivariate\ncase, one can equivalently write:\nCov( ˆθ)≥F−1, (2)\nwhere the matrix inequality is interpreted as Cov( ˆθ)−F−1being\na positive semi-definite matrix. One can then prove the following\ninequality (see Appendix A):\ndet Cov( ˆθ)≥detF−1. (3)\nTherefore, the determinant of the Fisher matrix measures the\ntightest parameter volume that can be constrained by the data.\nFrom now on, we will refer to detF(θ∗)as simply the Fisher in-\nformation around θ∗- the higher its value, the more constraining\nthe summary is around θ∗.\nArticle number, page 2 of 13David Prelogovi ´c and Andrei Mesinger: How informative are summaries of the cosmic 21-cm signal?\n2.1. Gaussian approximation\nIn almost all cosmological scenarios, we do not explicitly know\nthe full likelihood function P(d|θ), nor its “score”∇θlnP(d|θ).\nInstead for performing quick forecasts, it is common to ap-\nproximate the likelihood as a Gaussian in data space P (d|θ)≈\nN(d|µ(θ),Σ(θ)), where the mean µ(θ) and covariance Σ(θ) are\ngeneral functions of the parameter space.\nAssuming a Gaussian likelihood, one can compute the inte-\ngral in Eq. 1 analytically (Appendix B; see also Tegmark et al.\n1997; V ogeley & Szalay 1996):\nFmn≡Fµ,mn+FΣ,mn\n=∂µT\n∂θmΣ−1∂µ\n∂θn+1\n2tr\"\nΣ−1∂Σ\n∂θmΣ−1∂Σ\n∂θn#\n. (4)\nHere the first (second) term measures how the mean (covari-\nance) of the data summary changes with respect to the param-\neters. Cosmological Fisher forecasts usually ignore one of the\ntwo terms (most often the second term FΣ), effectively fixing ei-\nther the mean or the covariance matrix to some fiducial value.\nHowever, we will see below that both terms are non-negligible\nfor some summaries (e.g. Carron 2013).\n2.2. Finite differencing\nIn order to numerically evaluate Eq. 4, we group the terms into\n“fixed” (µ,Σ,Σ−1) and “di fferentiated” (∇θµ,∇θΣ). We calcu-\nlate “fixed” terms from realizations of the summary around the\nsame parameter value θ∗. To compute the mean, we can use the\nmaximum likelihood estimator (MLE):\nµ(θ∗)=1\nNNX\ni=1d(θ∗)i, (5)\nwhere the index ilabels di fferent realizations of initial condi-\ntions and other sources of stochasticity (e.g. telescope noise). It\nis known that such an estimator of the mean is unbiased. How-\never for the covariance matrix the situation is more complex, as\nthe MLE\nΣMLE(θ∗)=1\nNNX\ni=1(d(θ∗)i−µ(θ∗))(d(θ∗)i−µ(θ∗))T(6)\nis biased. One can show that the estimators of the covariance and\nits inverse can be unbiased in the following way (e.g. Hartlap\net al. 2007):\nΣ(θ∗)=N\nN−1ΣMLE(θ∗), (7)\nΣ−1(θ∗)=N−D−2\nN−1Σ−1\nMLE(θ∗), (8)\nwhere Dis dimensionality of the summary d. The correction\nprefactor for the inverse of the covariance is particularly impor-\ntant when Dis close to the number of samples N. Without it,\nboth FµandFΣwould end up over-confident, with the e ffect on\nFΣbeing quadratic (see Eq. 4).\nA similar approach is used to estimate the “di fferentiated pa-\nrameters”. In the case in which the simulator is not di fferentiable,\nwe can use finite di fferencing to estimate gradients of the mean\nFig. 1. Schematic illustrating our pipeline for producing data summaries\nfor a given astrophysical parameter combination (adapted from Prelo-\ngovi´c & Mesinger 2023). Starting from a realization of the cosmological\nsignal simulated with 21cmFAST , we remove the mean at each redshift\nand add a telescope noise realization corresponding to a 1000h observa-\ntion with SKA1-Low. The resulting lightcone is compressed into di ffer-\nent summaries. Explicitly-defined summaries are marked in blue, while\nneural network summaries are marked in red. This realization is com-\nputed at the fiducial parameter set, which used only for IMMN training\nand visualizations of the summaries. See text for more details.\nand covariance:\nδd(θ∗)i\nδθm≡d(θ∗+δθm/2)i−d(θ∗−δθm/2)i\nδθm, (9)\n∂µ\n∂θm\f\f\f\f\fθ∗=1\nMMX\ni=1δd(θ∗)i\nδθm, (10)\n∂Σ\n∂θm\f\f\f\f\fθ∗=2\nL−1LX\ni=1 δd(θ∗)i\nδθm−∂µ\n∂θm\f\f\f\f\fθ∗!\n(d(θ∗)i−µ(θ∗))T.\n(11)\nHere,δθmrepresents a small change in the parameter over which\nthe derivative is computed and δd(θ∗)i/δθmcorresponds to the\nnumerical derivative of a given summary realization (i.e. fixing\nall sources of stochasticity).\n2.3. The distribution of the Fisher information over parameter\nspace\nBelow we will show examples of common 21-cm summaries\nevaluated at a “fiducial” set of parameters, θfid. Unfortunately,\nthe cosmic 21-cm signal does not yet have a well-defined fidu-\ncial model and our choice of θfidis only a “best guess”. Therefore\nin order to properly assess the quality of a particular summary,\nwe sample many points θ∗from the prior P(θ), calculating the\nFisher information det F(θ∗) at each sample. We use this prior-\nweighted distribution of Fisher information as the main metric\nfor assessing the quality of a given summary .\n3. The cosmic 21-cm signal and its summary\nstatistics\nFor a given sample of astrophysical parameters, θ∗, we compute\na realization of mock data using the following steps (illustrated\nin Fig. 1):\nArticle number, page 3 of 13A&A proofs: manuscript no. main\n1. Sampling cosmological initial conditions, we simulate a re-\nalization of the 21-cm lightcone using the 21cmFAST code.\n2. We subtract the mean from each frequency slice, mimicking\nan interferometric observation.\n3. We simulate a 1000h tracked scan with the SKA1-low tele-\nscope, including the corresponding uvcoverage and realiza-\ntion of the instrument noise.\n4. We compress the resulting lightcone into a given summary\nobservation.\nWe summarize these steps in more detail below.\nThe cosmic 21-cm signal is simulated using the semi-\nnumerical code, 21cmFAST v35(Mesinger & Furlanetto 2007;\nMesinger et al. 2011; Murray et al. 2020). In brief, we use the\nPark et al. (2019) galaxy model, varying five parameters that\ncharacterize the unknown UV and X-ray properties of high- z\ngalaxies through scaling relations:\n–f∗,10- the fraction of galactic gas in stars, normalized at the\nhalo mass of 1010M⊙. The stellar to halo mass relation of\nthe faint galaxies that drive reionization is well described by\na power law: M∗/Mh=f∗,10(Mh/1010M⊙)0.5(Ωb/Ωm) (see\nPark et al. 2019 and references therein).\n–fesc,10- the ionizing UV escape fraction, normalized at the\nhalo mass 1010M⊙. Again, the (mean) escape fraction is as-\nsumed to be a power law with halo mass, and here we fix its\nindex toαesc=−0.5 (e.g. Qin et al. in prep).\n–Mturn- the characteristic host halo mass below which galax-\nies are ine fficient at forming stars, due to slow gas accretion,\nsupernovae feedback, and /or photoheating.\n–E0- the characteristic X-ray energy below which photons\nare absorbed by the interstellar medium (ISM) of the host\ngalaxy.\n–LX<2keV/SFR - the X-ray soft band (in the energy range\nE0– 2 keV) luminosity per star formation rate (SFR), in\nunits of erg s−1keV−1M−1\n⊙yr. The SFR is taken to be: ˙M∗=\nM∗/(0.5H(z)−1), where the Hubble time, H(z)−1, also scales\nwith the dynamical time at the virial radius.\nOur fiducial, θfidparameter vector corresponds to the follow-\ning: log10f∗,10=−1.3, log10fesc=−1, log10Mturn[M⊙]=8.7,\nlog10LX<2keV/SFR[erg s−1keV−1M−1\n⊙yr]=40,E0[keV] =0.5.\nThis particular choice is only used when training the IMMN, and\nfor the visualizations of the summaries below. For more details\non the model and its motivations, see Park et al. 2019 and refer-\nences therein.\nTo simulate the observation we follow the procedure de-\nscribed in Prelogovi ´c & Mesinger (2023) (c.f. Fig. 1). In brief,\nwe calculate the 21-cm signal from redshifts 30 to 5, on a 300\nMpc coeval cube, with 1.5 Mpc resolution. The snapshots of\nthe evolved coeval cube are then stacked to produce the final\nlightcone of the signal. After the subtraction of the mean for ev-\nery sky-plane slice, we add thermal noise corresponding to a a\n6 h/day, 10 s integration time tracked scan with SKA1-Low for a\n1000 h observation in total. Before computing each summary, we\nadditionally smooth the lightone with a box-car filter obtaining\na final resolution of 6 Mpc. This final step does not impact our\nresults as it is comparable to the SKA1-Low beam, but is needed\nin order to minimize GPU memory usage for certain summaries\n(see below for more details). We also note that in this work we\nmake the optimistic assumption of perfect foreground removal,\nthus exploring the maximum future potential of our set of sum-\nmary statistics.\nThe resulting 3D lightcone is compressed into a given sum-\nmary statistic. We explore the following summaries in this work:\n5https://github.com/21cmfast/21cmFAST\n6 8 10 12 14 16\nz100101102103104δ¯T2\nb∆2\n21[mK2]k = 0.06Mpc−1\nk = 0.20Mpc−1\nk = 0.40Mpc−1\nwith noise\nwithout noiseFig. 2. 1D power spectrum for the fiducial lightcone shown in Figure 1.\nDifferent colors correspond to di fferent wavemodes while solid (dashed)\ncorrespond to with (without) noise. Comparing the solid and dashed\nlines, we see that the signal is noise dominated at higher kmodes and\nredshifts (see also Fig. 1). All power spectra are computed from co-\nmoving cubes extracted from the lightcone, centered on the redshifts\nindicated by the vertical dashed lines.\n1DPS - spherically-averaged 1D power spectrum,\n2DPS - cylindrically-averaged 2D power spectrum,\nWavelets - 2D wavelet scattering transform,\nRNN - recurrent neural network,\nIMNN - information maximizing neural network.\n2DPS +IMNN - concatenation of the 2DPS and IMNN sum-\nmaries.\nThe first four are explicitly defined while the following two are\n“learned” by neural networks. The last, 2DPS +IMNN, is a com-\nbination of the two individual summaries. We describe them in\ndetail below.\n3.1. Spherically-averaged (1D) Power Spectrum\nAs discussed in the introduction, the spherically-averaged (1D)\npower spectrum, 1DPS, is the most common choice of a sum-\nmary statistic. Despite the cosmic 21-cm signal being non-\nGaussian, the 1DPS is a natural choice when seeking to maxi-\nmize the S /N of an interferometric first detection.\nThe 1DPS of the 21-cm signal δTb(x,z), where xis the sky-\nplane coordinate and zthe redshift, is defined as:\nδ¯T2\nb∆2\n21(k,z′)≡k3\n2π2V\u001c\f\f\fδTb(k,z′)−δ¯Tb(z′)\f\f\f2\u001d\nk. (12)\nδTb(k,z′) is calculated from the Fourier transforms of mean-\nsubtracted lightcone segments centered around z′. Here we cal-\nculate the 1DPS in three log-spaced k-bins, computed at eight\ndifferent z′. In Fig. 2 we show the 1DPS of the fiducial lightcone\n(see Fig. 1). Solid curves correspond to the full signal: cosmic +\nnoise, while the dashed curves are only the cosmic signal. We see\nthe usual three peaked redshift evolution of the large scale power\natk∼1 Mpc−1, tracing fluctuations in the IGM ionized frac-\ntion, the IGM temperature and Lyman alpha background (e.g.\nPritchard & Furlanetto 2007; Mesinger et al. 2014). By compar-\ning the solid and dashed curves, we see that the signal is noise\ndominated at small scales and early times. Indeed, k≥0.4 Mpc−1\nremains noise dominated virtually at all redshifts ( z>7), for this\nfiducial model and choice of 1000h integration time.\nArticle number, page 4 of 13David Prelogovi ´c and Andrei Mesinger: How informative are summaries of the cosmic 21-cm signal?\n3.2. Cylindrical (2D) Power Spectrum\nThe cylindrically-averaged 2D PS distinguishes between sky-\nplane ( k⊥) and line-of-sight ( k∥) modes. Redshift space dis-\ntortions (e.g. Bharadwaj & Ali 2005; Barkana & Loeb 2005)\nand lightcone evolution (e.g. Greig & Mesinger 2018; Mondal\net al. 2018) result in an anisotropic cosmic signal. Therefore, the\n2DPS should encode more physical information compared to the\n1DPS. Moreover, the instrument /foregrounds are better charac-\nterized in the 2D PS (e.g. see the review in Liu et al. 2014b),\nmaking it a natural summary observable for preliminary, low S /N\nmeasurements.\nThe 2DPS is defined as:\nδ¯T2\nb∆2\n21(k⊥,k∥,z′)≡k2\n⊥k∥\n4π2V\u001c\f\f\fδTb(k,z′)−δ¯Tb(z′)\f\f\f2\u001d\nk⊥,k∥, (13)\nwhere the Fourier transform is performed on the same lightcone\nchunks as in the 1D case, with the di fference being that the ex-\npectation value is calculated over ( k⊥,k∥) bins. Here we pick 3 ×3\nlog-spaced bins, aligned to the 1D case as close as possible. The\n2DPS for the fiducial lightcone (see Fig. 1) is shown in Fig. 3.\nThe panels correspond to the mean redshifts of the lightcone\nchunks, coinciding with the ones of the 1DPS (see Fig. 2). The\ncolumns correspond to the cases with and without noise, as la-\nbeled at the top. We can verify that both the cosmic signal and\nthe noise are anisotropic. For the cosmic signal, this is most ev-\nident for z≲10 where the noise is subdominant. For the noise,\nthe same can be clearly seen for higher redshifts, where the noise\ndominates the signal at high k∥much more than in high k⊥.\n3.3. Wavelet Scattering Transform\nThe wavelet transform (Gabor 1946; Goupillaud et al. 1984;\nTrott et al. 2012) and wavelet scattering transform (WST) (Mal-\nlat 2011) are based on convolutional filters designed on a Fourier\nbasis with an additional Gaussian envelope along the frequency\ndirection. As such, they capture local features in both image and\nFourier domains, which is often relevant in audio and image pro-\ncessing. Higher-order convolutions combined with neural net-\nworks (wavelet-based CNNs) have been very successful for such\npurposes (Bruna & Mallat 2012; Sifre & Mallat 2013; Anden &\nMallat 2014). The WST has been extensively explored in cos-\nmological (e.g. Cheng et al. 2020) and 21-cm (e.g. Greig et al.\n2022, 2023; Zhao et al. 2023a; Hothi et al. 2023) analyses.\nHere we briefly introduce how the WST is defined. It repre-\nsents a convolution of an input image I(x) by a set of wavelet\n(here specifically Morlet) filters. They are characterized by the\nphysical scale jand rotation moment l. The actual physical scale\nof the filter ψj,lwill then be 2jpixels, and its orientation angle\nl·π/L. The wavelet coe fficients correspond to the average of the\nconvolved images, to which we refer as simply wavelets . Note\nthat in this work we restrict ourselves to the 2D WST (see e.g.\nZhao et al. 2023a for recent work using the 3D transform).\nThe first and second order wavelets can be written as:\nsj1,l1\n1=⟨|I(x)∗ψj1,l1|⟩, (14)\nsj1,l1,j2,l2\n2=⟨||I(x)∗ψj1,l1|∗ψj2,l2|⟩. (15)\nOne can see the similarity between first order wavelets and the\nPS by writing PS( k)=⟨|I(x)∗ψ′|2⟩, withψ′=e−ik·x. To extract\nonly cylindrically-averaged information imprinted in the 21-cm\nsignal (analogous to calculating 2DPS in a Fourier basis), onecan average coe fficients over the rotations:\nSj1\n1=⟨sj1,l1\n1⟩l1, (16)\nSj2\n2=⟨sj1,l1,j2,l2\n2⟩l1,l2. (17)\nFor further information about the WST and wavelet coe fficients\nwe refer the reader to Cheng et al. (2020); Greig et al. (2022,\n2023).\nOur setup for calculating wavelets is the following. We first\nselect a series of 8 images, centered at the same redshifts as\nfor the power spectra. For each image we calculate the first and\nsecond-order wavelets and combine them for the final summary.\nAs the WST scale 2J×2Jhas to be smaller than the sky-plane\nimages (50×50), we set J=4. Further, we fix L=4 as a com-\nmon choice (Cheng et al. 2020; Greig et al. 2022). For comput-\ning convolutions, we use the scattering_transform6pack-\nage (Cheng et al. 2020). For each redshift slice, we average the\nfirst and second order coe fficients computed on a rolling 32x32\nwindow. Our final WST summary then consists of J=4S1and\nJ(J+1)/2=10S2coefficients for each of the 8 redshift slices,\nfor a total of 112 numbers.\n3.4. Recurrent Neural Network\nA Recurrent Neural Network (RNN) is a type of Neural Net-\nwork (NN) architecture designed to e fficiently encode sequen-\ntial or time-evolving information. In our previous work (Prel-\nogovi ´c et al. 2022), we used RNNs along the redshift axis in\ncombination with Convolutional NNs (CNNs) along the sky-\nplane, to capture the anisotropic evolution of the cosmic 21-cm\nsignal along the lightcone. Specifically, we used a long short-\nterm memory (LSTM, Hochreiter & Schmidhuber (1997); Shi\net al. (2015)) RNN, which is more stable to train compared with\nolder RNN versions (see e.g. the review in Schmidt 2019). Using\nRNNs resulted in a more accurate regression of astrophysical pa-\nrameters, compared to only using CNNs (e.g. Gillet et al. 2019;\nLa Plante & Ntampaka 2019; Mangena et al. 2020; Kwon et al.\n2020; Zhao et al. 2022a; Heneka 2023).\nIn this work, we use the flagship model from Prelogovi ´c et al.\n(2022), SummaryRNN , together with the exact weights from that\nwork. The model has been trained in a supervised manner as a\nregressor. The database used for training had the same telescope\nsimulator, but used an older version of the 21cmFAST simulator\nwith a di fferent astrophysical parametrization.\nOur RNN summary in this work is the 32-dim output from\nthe first dense layer of the network (see Table B3 in Prelogovi ´c\net al. 2022, where convolutional and recurrent layers are fol-\nlowed by dense layers, ending with prediction of four parame-\nters). Using such a higher-dimensional layer instead of the final\nparameter regressor output (e.g. Zhao et al. 2022a), makes the\nsummary more sensitive to the general lightcone features instead\nof the specific model and /or parametrization used to create the\ntraining set. This is similar to so-called “transfer learning” (e.g.\nJiang et al. 2022), where one uses a NN that has been pre-trained\non a di fferent domain (i.e. database) and re-trains its last layers\non a new domain. As our original RNN takes as input 25 ×25\nsky-plane slices, we roll a 25 ×25 window over each redshift\nslice, averaging the corresponding dense layer outputs to obtain\nthe final RNN summary.\n6https://github.com/SihaoCheng/scattering_transform\nArticle number, page 5 of 13A&A proofs: manuscript no. main\n0.04 0.12 0.4\nk⊥[Mpc−1]0.040.10.31k/bardbl[Mpc−1]\n0.04 0.12 0.4\nk⊥[Mpc−1]\n0.040.10.31k/bardbl[Mpc−1]\n0.040.10.31k/bardbl[Mpc−1]\n0.040.10.31k/bardbl[Mpc−1]\n0.04 0.12 0.4\nk⊥[Mpc−1]0.040.10.31k/bardbl[Mpc−1]\n0.04 0.12 0.4\nk⊥[Mpc−1]\n0.040.10.31k/bardbl[Mpc−1]\n0.040.10.31k/bardbl[Mpc−1]\n0.040.10.31k/bardbl[Mpc−1]\n−2−10 1 2 3log10δ¯T2\nb∆2\n21/[mK2]z= 8.6z= 7.6z= 6.7z= 5.9\nz= 15.4z= 13.1z= 11.3z= 9.8without noise with noise without noise with noise\nFig. 3. 2D power spectrum of the fiducial model shown in Figure 1. Di fferent columns show the 2DPS for the pure cosmic signal and including\nthermal noise, as labeled on the top. Each pair corresponds to a di fferent central redshift, coinciding with those used in the 1DPS summary (see\nFig. 2). Furthermore, the bins between 1DPS and 2DPS coincide as close as possible. As in the 1DPS case, one can notice that high- kmodes and\nhigh redshifts are noise dominated.\n3.5. Information Maximizing Neural Network\nAn Information Maximizing Neural Network (IMNN; Charnock\net al. 2018)7is an unsupervised learning algorithm, where the\nNN is specifically trained to maximize the Fisher information\ndetFof the summary compression, at a single fixed point in\nparameter space θfid. IMNNs have been previously applied to\ngalaxy LSS maps (e.g. Makinen et al. 2022) and catalog-free\nmodeling of galaxy types (Livet et al. 2021). Here we apply them\nfor the first time to 21-cm maps of the EoR and CD. In what fol-\nlows, we outline the main ingredients of the algorithm and the\ntraining procedure.\nTaking some NN architecture (in our case a simple CNN),\none compresses the lightcone linto a summary vector d\nd=NN(l). (18)\nThe dimensionality of the summary is equal to the parameter\nspace dimensionality ( D). This is motivated by the fact that, for\na given model parametrization , the “score” of the real likelihood\nfunction∇θlnP(l|θ) is su fficient for optimal parameter recovery,\nand thus the dimensionality of the summary which maximizes\ninformation around one fiducial point can be the same as the\ndimensionality of the parameter space (e.g. Heavens et al. 2000;\nCharnock et al. 2018).\n7https://github.com/tomcharnock/IMNNWe train the IMMN using the following loss function (c.f.\nCharnock et al. 2018; Makinen et al. 2021):\nFIMNN (θfid)mn≡∂µT\n∂θm\f\f\f\f\f\fθfid∂µ\n∂θn\f\f\f\f\fθfid(19)\nLF(Σ(θfid))≡∥Σ(θfid)−1∥F (20)\nL2(WNN)≡∥WNN∥2(21)\nL=−ln det FIMNN\n+λLF(Σ) tanh LF(Σ) (22)\n+λWL2(WNN)\nwhere\nµ(θfid)=1\nNMX\ni=1d(θfid)i, (23)\nΣ(θfid)=1\nN−1NX\ni=1(d(θfid)i−µ(θfid)) (d(θfid)i−µ(θfid))T,\n(24)\n∂µ\n∂θm\f\f\f\f\fθfid=1\nMMX\ni=1d(θfid+δθm/2)i−d(θfid−δθm/2)i\nδθm, (25)\nArticle number, page 6 of 13David Prelogovi ´c and Andrei Mesinger: How informative are summaries of the cosmic 21-cm signal?\nLF(Σ(θfid)) is the Frobenius norm between the covariance and\nthe unit matrix 1,L2(WNN) is the l2norm of the networks’\nweights. The final loss is then the sum of the negative loga-\nrithm of the Fisher information, together with the covariance\nand weights regularizers. The tanh term in the covariance reg-\nularizer is used to turn-o ffthe regularization once the covari-\nance becomes close to the unity. The main goal of the IMNN is\ntherefore to maximize the Fisher information, while keeping the\ncovariance of the summary fixed to unity.8Note that the Fisher\nmatrix is calculated only using the first term and without covari-\nance inverse between the two derivatives (see Eq. 4). Both of\nthese simplifications were needed in order to stabilize the train-\ning of the IMMN, and they are correct as long as the covariance\nis kept at unity.9\nFor the IMNN training, we create a database around the fidu-\ncialθfid(see Figs. 1 and 4) for N=M=1024 (Eqs. 23, 24, 25),\nwith 11264 realizations in total. The training and validation sets\nare formed by splitting the database in half, following the origi-\nnal work of Charnock et al. (2018). We train the network using\nstochastic gradient descent with a batch size b=128, regular-\nization strengths λ=10,λW=10−9and the Adam optimizer.\nTo increase the e ffective database size and improve the training,\nthe telescope noise is added to the simulations “on-the-fly”. The\nfinal model has been trained on 8 V-100 GPUs in parallel, for\n≈120k epochs and ≈20k GPUh. The IMNN code developed\nfor the purposes of this work we make publicly available under\n21cmIMNN10package.\n3.6. 2DPS + IMNN\nFinally, we also compute the Fisher information of the concate-\nnation of the individual 2DPS and IMNN summaries. As we\nshall see in the following section, these are the two best per-\nforming individual summaries. Note that one can cleanly com-\nbine summaries when performing Simulation Based Inference\n(SBI), as the likelihood does not need to be specified explicitly.\nThe main restriction is the size of the required database and the\nstability of the SBI training (e.g. Cranmer et al. 2020; Lueck-\nmann et al. 2021). When computing the Fisher information in\nthis work, we are restricted to these two summaries due to the\nlimits imposed by our database size. In particular, the minimal\nnumber of samples needed to estimate the covariance matrix is\nproportional to the dimensionality of the total summary (e.g.\nHartlap et al. 2007). Furthermore, the dimensions of the final\nsummary should be uncorrelated, as otherwise the covariance is\nsingular and non-invertible. This can result in incorrect Fisher\nmatrix estimation (see however Tegmark et al. 1997 for possible\nalternative solutions).\n8For any summary dof dimensionality D, one can define a new sum-\nmary, d′=a+Bd, where ais a constant D-dim vector serving to nor-\nmalize d′to be zero mean, and Bis a non-singular D×Dmatrix, serving\nto normalize d′to be unit variance. One can then show that ln P(d|θ)=\nlnP(d′|θ)+ln det B, or equivalently ∂lnP(d|θ)/∂θm=∂lnP(d′|θ)/∂θm.\nThis means that one always has the freedom to transform the summary\ndto be zero-mean, unit-variance at a single point θfid, without changing\nthe Fisher information anywhere in the parameter space.\n9The original work by Charnock et al. (2018) had slightly di fferent\nloss function, by having the inverse of the covariance in the Fisher term\nand the additional Frobenius norm between Σ−1and1. We found that\nthese choices destabilized the training and are not crucial for the con-\nvergence (see also Makinen et al. 2021).\n10https://github.com/dprelogo/21cmIMNN\n−1.5−1.0−0.50.0fesc\n8.08.59.09.5Mturn\n3839404142LX\n−1.50\n−1.35\n−1.20\n−1.05\nf∗30060090012001500E0\n−1.5\n−1.0\n−0.50.0\nfesc\n8.08.59.09.5\nMturn\n3839404142\nLX30060090012001500E0Fig. 4. Prior volume of the astrophysical parameter space. Dashed lines\ndemarcate the fiducial parameter values to train the IMNN, 2D contours\nenclose 95% of the prior volume, and the points denote the 152 prior\nsamples at which we compute the Fisher information for our summaries.\nNote that all units and log10factors have been removed for clarity - see\nthe main text and Fig. 1 for reference on the units.\n4. Database\nSince we do not have a strongly-motivated fiducial choice for our\ngalaxy parameters, we want to compare the Fisher information\nof 21-cm summaries across our prior volume. In order to do so,\nwe construct a “Fisher database”, sampling 152 parameter com-\nbinations from our prior. We use a flat prior for the X-ray emis-\nsion parameters, LXandE0, over the ranges shown in Fig. 4 (for\nmore motivation of these choices see Fragos et al. 2013; Lehmer\net al. 2016; Das et al. 2017). The remaining parameters char-\nacterizing star formation and ionizing emission of galaxies are\nconstrained by observed UV luminosity functions (e.g. Bouwens\net al. 2015, 2017; Oesch et al. 2018) as well as estimates of the\nreionization history (e.g. McGreer et al. 2015; Planck Collabo-\nration et al. 2016). As our model is nested in the 6D Park et al.\n(2019) model, we use the posterior from that work, calculating\nthe corresponding conditional distributions over f∗,10,fesc,10and\nMturn, fixing the remaining parameters to their fiducial values,\nas discussed in the previous section. To compute the conditional\ndistributions, we use the conditional_kde11code.\nOur prior and samples are shown in Fig. 4. The diagonal pan-\nels illustrate the 1D prior PDFs, while the contours in the lower\nleft enclose 95% of the 2D distributions. The 152 samples are\nshown as points, while the dashed vertical and horizontal lines\ndemarcate the fiducial choice used to train our IMNN.\nWe simulate N=130 realizations for each of the 152 param-\neter samples, θ∗, in order to estimate the mean and the covariance\nof each summary (Eqs. 5, 6), plus an additional 2 ×5×M=150\nrealizations at θ∗±δθm/2 to estimate the derivative of the mean\n(Eq. 10) along each of the 5 parameter basis vectors. We note\nthat no new simulations are needed to estimate the derivative of\nthe covariance matrix, since we use the same L=M=15 real-\n11https://github.com/dprelogo/conditional_kde\nArticle number, page 7 of 13A&A proofs: manuscript no. main\n0 20000 40000 60000 80000 100000 120000\nEpoch01020304050log10(detF)IMNNθfidtrain\nIMNNθfidval\nIMNNP(θ) val\nFig. 5. Fisher information of the IMNN during training. Dark (light)\nblue curves show the Fisher information for the training (validation) set\nas the IMNN is trained at θfid. The violin plots show the distribution of\nthe Fisher information over the full prior volume excluding θfid,P(θ),\nat every interval of 10k training epochs. Black dots and error bars in-\nside each violin denote the median and 68% C.L. of the distributions.\nThe divergence between the Fisher information computed at θfidand its\nPDF over the remaining prior volume clearly indicates that the IMNN\nis over-specializing to the specific parameter combination, θfid, used in\nthe training.\nizations that are used to compute the first Fisher term mentioned\nabove (c.f. Eq. 11 and associated discussion).\n5. Results\nWe now present the main results of this work. We first discuss\nthe training of the IMNN in Section 5.1, then compare the per-\nformance of all of our 21-cm summaries in Section 5.2, adding a\nfinal discussion on the importance of including the second Fisher\nterm (c.f. Eq. 4) in Section 5.3. Throughout, our figure of merit\nwill be the PDF of the Fisher information (log10(detF)), over the\nprior volume (see previous section). We quote both the median\nand the variance of this distribution. A high value of the median\nmeans that the typical parameter value is well constrained by the\nsummary, while a narrow variance means that the constraining\npower of the summary is relatively constant over the prior vol-\nume. We note that the Fisher information is a measure of the\naverage constraining power over our five parameters; we do not\ndiscuss the relative constraints on individual parameters in this\nwork.\n5.1. IMNN training\nIn Fig. 5 we show the training results for the IMNN. Blue lines\nshow the training and validation Fisher information (see the first\nterm of the IMNN loss, Eq. 22) at the fiducial point θfid. We\nrecall that the training and validation sets each consist of 5632\nlightcone realizations, evaluated at θfidwith di fferent cosmic ini-\ntial conditions and telescope noise seeds. The Fisher information\nof the training and validation sets are largely overlapping mean-\ning the network does not overfit to the training set. We see that\nby the end of the training, the IMNN has learned to improve the\nFisher information by 40(!) orders of magnitude, corresponding\nto parameter constraints that are improved by an average factor\nof∼108in the variance!\nIs the compression learned by the IMNN comparably infor-\nmative at other parameter combinations? We can answer thissummary median+34%\n−34%σ2/σ2\n2DPS\nRNN 9 .0+1.6\n−2.14.17\nWavelets 9 .6+2.9\n−3.13.16\nIMNN 90k 11 .8+8.2\n−5.71.15\n1DPS 11 .8+3.1\n−3.41.15\n2DPS 12 .1+3.1\n−3.31.00\nIMNN 90k +2DPS 15 .9+5.2\n−4.90.17\nTable 1. Companion table to Fig. 6, summarizing the Fisher informa-\ntion probability distribution across the prior volume for each summary.\nIn particular, the first column represents values of log10(detF) for the\nmedian and 68% of probability volume around it. The second column\nshows the single parameter variance (Eq. 26), which serves as a guide\nto how much would the recovered variance of a parameter increase or\ndecrease if one uses that particular summary instead of the 2DPS.\nquestion by comparing the solid lines of Fig. 5 to the violin\ndistributions. The latter show the distribution of the Fisher in-\nformation over the full prior volume, P(θ), at every interval of\n10k training epochs, excluding θfid. We see that although the\nIMNN is not overfitting in the classical sense (the Fisher infor-\nmation of the validation set keeps increasing), it quickly “over-\nspecializes” to the specific parameter value used for training, θfid.\nThus the compression learned by the IMNN does not generalize\nwell across parameter space.12The PDF of the Fisher informa-\ntion over the prior volume does not change significantly after\nepoch 40k, with the median beginning to decrease beyond epoch\n90k. We take epoch 90k as the final state of the IMNN in the\nremainder of this work.\n5.2. Comparison of all summaries\nIn Fig. 6 we show the main result of this work: the distribution\nof Fisher information across the prior volume, for all of the sum-\nmary statistics we consider. On the top axis we denote the cor-\nresponding relative improvement in constraints on the single pa-\nrameter variance σ2. We define it as an e ffective variance in each\ndimension with no correlation, giving the same Fisher informa-\ntion. For a 5-dim parameter space we have:\nσ2=(detF)−1/5. (26)\nHere, one can ignore the di fference in the units (i.e. scaling) be-\ntween parameter dimensions, see Appendix C for more details.\nSpecifically, the top axis of Fig. 6 denotes the single parameter\nvariance normalized to its median value of the 2DPS summary.\nTo give an example, a factor of 10 times larger Fisher informa-\ntion for a 5-dim parameter space would result in a 37% smaller\nsingle parameter variance. The median values and 68% C.L. of\nthe Fisher information and single parameter variance are also\nlisted in Table 1.\n12One solution could be to train the IMNN around several di fferent\npoints. However, this is extremely expensive in our application, both in\nterms of database size and GPU time. Furthermore, it is not clear if such\nprocedure would result in a meaningful summary, as the concept of one\npoint training is inherently built into the IMNN. Alternatively, it might\nbe possible to construct a generalized IMNN, maximizing the Fisher\ninformation throughout parameter space; however, it is not obvious how\nthis could be done.\nArticle number, page 8 of 13David Prelogovi ´c and Andrei Mesinger: How informative are summaries of the cosmic 21-cm signal?\n10210110010−110−2σ2/σ2\n2DPS\n0 5 10 15 20 25\nlog10(detF)Probability1DPS\n2DPS\nWavelets\nRNN\nIMNN 90k\n2DPS + IMNN 90k\nFig. 6. Distribution of the Fisher information (det F) across the prior volume, for all summaries considered in this work. For the IMNN summary,\nwe use the weights that give the highest median value (corresponding to 90k epochs; see Fig. 5). On the top axis we denote the corresponding\nrelative improvement in the single parameter variance (Eq. 26), normalized to its median value of the 2DPS summary. The 2DPS has the largest\nmedian value of all of the individual summaries, while the combination of the 2DPS +IMNN dramatically outperforms every individual summary.\nSee Table 1 for quantitative values.\nAmong the individual summaries, the 2DPS is a clear win-\nner. In comparison, the 1DPS has roughly a factor of two smaller\nFisher information across the prior volume, which translates to\na∼15% larger single parameter variance (Eq. 26). This shows\nthat indeed the anisotropy of the cosmic 21-cm signal helps in\nparameter recovery, though not very significantly. The distribu-\ntion of the 2DPS Fisher information is also relatively narrow, in-\ndicating its constraining power does not vary enormously across\nparameter space, compared to most other summaries.\nThe median Fisher information of the IMNN is comparable\nwith that of the 1DPS. However, the IMNN results in the widest\ndistribution of all of the summaries. This means that the com-\npression that the IMNN learned at θfidcan be much more in-\nformative but also much less informative, depending on the pa-\nrameter choice. Therefore, if one has a good idea of where the\nmaximum likelihood value will be (e.g. from complementary ob-\nservations; Park et al. 2019; Abdurashidova et al. 2022; HERA\nCollaboration et al. 2023; Breitman et al. 2024), it could be ben-\neficial to train the IMNN at that “best guess” value and use the\nresulting summary in inference. An optimal strategy could be to\nperform inference using the 2DPS in order to obtain the maxi-\nmum likelihood, train the IMNN at this ML parameter combi-\nnation, then perform inference again using the resulting IMNN\nsummary.The wavelet summary results in a median Fisher information\nthat is a factor of∼300 smaller than that of the 2DPS at their me-\ndian values (factor of ∼3 increase in the single parameter vari-\nance (Eq. 26). Intuitively, one would expect more information\ncontent in wavelets compared with the PS, as they e ffectively\ninclude higher order correlations. However, our WST followed\nthe definition in Greig et al. (2022, 2023), in which the wavelets\nare only computed at one slice for each redshift chunk. Thus,\nthe physics-rich, line-of-sight modes inside each redshift bin are\nlost for this choice of WST. The weaker performance of the WST\ncompared to the power spectra implies that there is more infor-\nmation in the line-of-sight modes than there is in higher-order\ncorrelations of the transverse modes.\nWe note that recent works also using 2D wavelets achieved\nbetter results than we do here (Greig et al. 2022; Hothi et al.\n2023). However, their analysis di ffers from ours. In particular,\nthey use more redshift bins and they do not average the coe ffi-\ncients over the sky-plane (rolling the filter as described above).\nWe confirm that without the additional smoothing that we per-\nform in this work, the Gaussian approximation for the likeli-\nhood that is intrinsic to the Fisher estimate (c.f. Eq. 4) is notably\nworse, for both the wavelets and the RNN. Therefore, the higher\nwavelet information found by Greig et al. (2022); Hothi et al.\n(2023) could be partially due to a less appropriate application of\nthe Fisher estimate. Alternatively, the information content could\nArticle number, page 9 of 13A&A proofs: manuscript no. main\nbe improved by using 3D wavelets instead of 2D ones (see e.g.\nZhao et al. 2023a).\nOf the individual summaries, the RNN performs the worst\nwith a median Fisher information that is three orders of magni-\ntude smaller than that of the 2DPS. However, its distribution is\nthe most homogeneous across the prior volume (i.e. the narrow-\nest). This could be interpreted as sacrificing the overall informa-\ntion content for robustness throughout the parameter space. In-\ndeed the RNN was trained as a regressor over a large parameter\nspace, and so it is understandable that it is much more consistent\nwith respect to the IMNN which was trained on a single parame-\nter combination. The overall low median information content of\nthe RNN compression could be due to the fact that it was trained\non a di fferent parametrization of the cosmological signal than\nused in this work. We expect that retraining the RNN as a re-\ngressor on the same parametrization would improve the median\ninformation content. Indeed, Zhao et al. (2022a) find that a re-\ngressor CNN slightly outperforms the 1DPS, resulting in tighter\nparameter constraints when using two di fferent parameter com-\nbinations for the mock observation. However, it is likely that this\nincrease in the information content would come at the cost of ro-\nbustness (i.e. a broader PDF), as the RNN over-specializes to\nfeatures inherent in that model’s parametrization and its training\nset.\nFinally, our concatenation of the 2DPS and IMNN signifi-\ncantly outperforms all of the individual summaries (c.f. the red\ncurve in Fig. 6). As we saw above, these two summary statistics\nare highly complimentary: the 2DPS is robust (i.e. a narrow in-\nformation PDF), while the IMNN summary can be much more\nconstraining for some parameters but much less for others (i.e. it\nhas a very broad information PDF). The distribution of Fisher in-\nformation of 2DPS +IMNN retains the narrowness of the 2DPS,\nbut is shifted to the right, including the high-value tail from the\nIMNN.13The combined summary is a factor of ∼6.5–9.5 bet-\nter in the single parameter variance (Eq. 26), averaged over the\nprior samples. Combining multiple 21-cm statistics has already\nbeen proven to tighten parameter constraints (e.g. Gazagnes et al.\n2021; Watkinson et al. 2022). However, previous work using tra-\nditional inference was limited by the need to derive a tractable\nlikelihood which becomes impossible for non-trivial summaries.\nFortunately, recent advances in applying SBI to 21-cm (e.g. Zhao\net al. 2022a,b; Prelogovi ´c & Mesinger 2023; Saxena et al. 2023)\nwill allow us to cleanly combine multiple summary statistics\nwithout having to explicitly define the likelihood. Unfortunately,\nthe neural density estimation that is intrinsic to SBI can be un-\nstable and /or require large training sets, if the summary statis-\ntics become very high dimensional. This work illustrates how\ncombining only two complimentary statistics can result in much\ntighter, more robust parameter constraints than would be avail-\nable using either individually.\n5.3. One-term vs. full Fisher matrix\nThe full Fisher matrix of a Gaussian likelihood can be separated\nin two terms F=Fµ+FΣ(see Eq. 4), where the first term\nincludes the gradient of the mean and the second term the gra-\ndient of the covariance matrix (e.g. Appendix B; Tegmark et al.\n13We note that in the case in which some value of the IMNN summary\nwould be completely degenerate with some 2DPS bin, the resulting co-\nvariance matrix would be singular and therefore non-invertible. This\nwould make estimating the Fisher matrix unreliable. However, as the\nIMNN compression and 2DPS are very di fferent summaries, this is un-\nlikely to be a problem.\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nlog10(detF)Probability1DPSFµ\n2DPSFµ1DPSFµ+FΣ\n2DPSFµ+FΣFig. 7. Distributions of Fisher information for the 1DPS (red) and 2DPS\n(blue). Solid lines account for both terms in the Fisher matrix (c.f. Eq.\n4), while the dashed lines correspond to the common simplification of a\nconstant covariance (i.e. including only the first term in Eq. 4). The as-\nsumption of a constant covariance underpredicts the Fisher information\nby a factor of∼2. For a 5-parameter model, this translates to a ∼15%\nlarger single parameter variance (Eq. 26).\n0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0\nlog10(detF)ProbabilityIMNN 90k Fµ\nWaveletsFµ\nRNNFµIMNN 90k Fµ+FΣ\nWaveletsFµ+FΣ\nRNNFµ+FΣ\nFig. 8. Analogous to Fig. 7, but for the other 21-cm summary statistics.\n1997). All of our previous results were calculated using the full\nexpression, including both terms. However, it is common in the\nliterature to ignore one of the terms, by approximating the mean\nor the covariance matrix as a constant (e.g. Carron 2013). In\nmost of cosmological Fisher forecasts using the power spectrum\n(e.g. Sailer et al. 2021; Abazajian et al. 2022; Euclid Collabo-\nration et al. 2022; Mason et al. 2023; Bykov et al. 2023), the\nsecond term, FΣ, is neglected.14In some cases, the covariance\nmatrix can indeed be independent or only weakly dependent on\nthe parameters, justifying ignoring the second Fisher term. How-\never, the importance of the FΣterm for 21-cm Fisher forecasts\nhas not yet been investigated (e.g. Greig et al. 2022; Balu et al.\n2023; Mason et al. 2023; Hothi et al. 2023). Indeed in Prelo-\ngovi´c & Mesinger (2023) we showed that assuming a parameter-\nindependent covariance matrix of the 21-cm 1DPS likelihood\ncan bias the posterior estimation. In this section, we quantify\nthe importance of the FΣterm in determining the Fisher infor-\n14For examples of ignoring the first term, see e.g. Abramo (2012);\nd’Assignies D et al. (2023).\nArticle number, page 10 of 13David Prelogovi ´c and Andrei Mesinger: How informative are summaries of the cosmic 21-cm signal?\nmation15, for all of our 21-cm summaries. It is important to note\nthat adding this term will always increase the information con-\ntent, as det( Fµ+FΣ)≥detFµ. For more details, see the end of\nAppendix A.\nIn Fig. 7 we plot the Fisher information PDF of the\ncommonly-used 1DPS and 2DPS. The solid curves are the same\nas in the previous figure, corresponding to the full expression for\nthe Fisher information. The dashed curves only include the first\nterm, assuming a constant covariance matrix. For both 1D and\n2D power spectra, neglecting the second term underestimates the\nFisher information by a factor of ∼2.3 on average over the prior\nsamples. For our 5-parameter model, this translates into an un-\nderestimate of the single parameter variance (Eq. 26) by 15%.\nIn Fig. 8 we compare the Fisher information PDFs with and\nwithout the second Fisher term for the remaining individual sum-\nmaries. We see that for the other 21-cm summaries, the second\nFisher term is even more important than for the power spectra.\nFor example, assuming a constant covariance underestimates the\nmean Fisher information of the RNN for a factor of ∼4700 and a\nfactor of∼11100 for the wavelets, averaged over the prior sam-\nples. Note that although the IMNN is trained to normalize the\ncovariance to unity, thus ensuring a constant covariance matrix\nby construction, this is only done at θfid. For the other points in\nthe parameter space, there is no guarantee that this condition is\nmet.\n6. Conclusions\nThe cosmic 21-cm signal will revolutionazie our understand-\ning of the CD and Eo). \"Big Data\" analysis techniques will\nbe required to cope with radiotelescopes such as the Square\nKilometre Array (SKA). In this work we compare several ap-\nproaches of compressing 21-cm lightcone data, on the basis\nof their Fisher information. Specifically, we compare the de-\nterminant of the Fisher information for the following summary\nstatistics: (i) the spherically-averaged power spectrum (1DPS),\n(ii) the cylindrically-averaged power spectrum (2DPS), (iii) the\n2D Wavelet scattering transform (WST), (iv) a recurrent neu-\nral network (RNN) trained as a regressor; (v) an information-\nmaximizing neural network (IMNN); and (vi) the combination\nof 2DPS and IMNN. Importantly we compare their Fisher infor-\nmation across parameter space, since there is currently no stan-\ndard model for the astrophysics of the first galaxies, which de-\ntermine the cosmic 21-cm signal.\nThe 2DPS is the individual summary with the highest me-\ndian Fisher information (Fig. 6, Table 1). The distribution of\nits Fisher information is also relatively narrow across parame-\nter space. This means that compared to most other summaries,\nthe 2DPS can recover astrophysical parameters with relatively\nsimilar precision regardless of what are the \"true\" values of our\nUniverse. We also show that the 2DPS slightly outperforms the\n1DPS, confirming that the anisotropy of the 21-cm signal helps\nconstrain parameters. The information content of both RNN and\nwavelets is generally lower than either the 1DPS or the 2DPS.\nHowever, we caution that both of these summaries could be de-\nfined in several di fferent ways, potentially improving their per-\nformances (e.g. Greig et al. 2022, 2023; Hothi et al. 2023; Zhao\net al. 2023a).\n15Assuming a constant covariance matrix can result in other biases at\nthe level of the full Fisher matrix. For example, it could impact param-\neter degeneracies and /or improve constraints of some parameters at the\ncost of others. Here we only focus on the e ffect this simplification has\non the determinant of the Fisher matrix as this quantity measures the\naverage parameter constraining power of the summary statistics.In this work, we also introduce IMNNs to the field of 21-cm\ncosmology. Although capable of achieving the highest Fisher in-\nformation for some parameter choices, the IMNN does not gen-\neralize well, resulting in a broad distribution across the prior vol-\nume. Indeed, we find an enormous di fference between its Fisher\ninformation computed at the parameter set used for training, θfid,\nand its information computed at the remaining prior samples\n(Fig. 5). This means that the IMNN overfits to the information\ncontent of the fiducial. The regularization inherent to the IMNN\nloss (c.f. Eq. 22 and the discussion in Footnote 8) makes it dif-\nficult to avoid this overfitting by training over many di fferent\nchoices ofθfid. A simpler solution could be to first perform infer-\nence with a less optimal summary, in order to determine a maxi-\nmum likelihood point, and then train the IMNN at that point.\nCombining the IMNN with the 2DPS results in a summary\nwhose median Fisher information is almost ∼4 orders of magni-\ntude larger than the 2DPS alone. This results in a factor of ∼6.5–\n9.5 tighter single parameter variance averaged over the prior\nsamples. This means the IMNN extracts complementary infor-\nmation with respect to the 2DPS. Performing simulation-based\ninference using such a summary could yield extremely tight and\nrobust parameter constraints. Moreover, combining only these\ntwo complementary summaries still preserves a high level of\ncompression, with our lightcone of ∼1.6·106cells reduced to\njust 77 numbers.\nFinally, we stress the importance of not assuming a constant\ncovariance when performing Fisher forecasts using 21-cm ob-\nservables. This common assumption underestimates the Fisher\ninformation of the power spectrum on average by a factor of\n∼2.3, corresponding to a 15% increase in the single parameter\nvariance (Eq. 26). For other summaries, the assumption under-\nestimates the Fisher information by orders of magnitude.\nAcknowledgements. D.P. thanks T. Charnock for many discussions and in depth\nexplanations on how the IMNN function, as well as L. Makinen on help with sta-\nbilizing IMNN training. A.M. acknowledges support from the Ministry of Uni-\nversities and Research (MUR) through the PRIN project \"Optimal inference from\nradio images of the epoch of reionization\" as well as the PNRR project \"Centro\nNazionale di Ricerca in High Performance Computing, Big Data e Quantum\nComputing\". We gratefully acknowledge computational resources of the HPC\ncenter at SNS. We acknowledge the CINECA award under the ISCRA initiative,\nfor the availability of high performance computing resources and support (grant\nIMC21cm - HP10CWIEF7). This work used the Extreme Science and Engineer-\ning Discovery Environment (XSEDE), which is supported by National Science\nFoundation grant number ACI-1548562. 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(A.2)\nFrom this, it follows:\nxTCx≥xTF−1x (A.3)\nexp \n−1\n2xTCx!\n≤exp \n−1\n2xTF−1x!\n(A.4)\nZ\ndxexp \n−1\n2xTCx!\n≤Z\ndxexp \n−1\n2xTF−1x!\n(A.5)\n1√det(C/2π)≤1p\ndet(F−1/2π)(A.6)\ndetC≥detF−1(A.7)\nHere the trick was to push the inequality towards the multivariate\nGaussian integral. After integrating, what is left is only the in-\nverse of a normalization constant. Notice that if one wants to\ninterpret Cas a covariance matrix and Fas a Fisher matrix,\nthen it would make more sense to integrate over xTC−1xand\nxTFx. However this is irrelevant for the proof, as it would only\nresult in the change of the inequality direction in each row, end-\ning with det C−1≤detF. We usually assume the matrices are\nnon-singular and thus invertible, which is needed here.\nNote that the identical procedure can be used to prove that\nthe one-term Fisher is a lower bound of the two-term Fisher. For\nF=Fµ+FΣone can write xT(Fµ+FΣ)x≥xTFµx, which by\nfollowing the same steps translates into det( Fµ+FΣ)≥detFµ.\nAppendix B: Derivation of a Gaussian Fisher\ninformation matrix\nIn order to derive Eq. 4, i.e. the Fisher information of a multi-\nvariate Gaussian distribution, we start by writing the Fisher in-\nformation matrix equation (Eq. 1) in the equivalent form:\nF(θ∗)mn=−Eh\n∂2\nmnlnP(d|θ)\f\f\fθ∗i\n, (B.1)\nwhere we shorten the notation of derivatives as ∂/∂θm≡∂mand\n∂2/∂θm∂θn≡∂2\nmn. This form comes from the fact that\n−∂2\nmnlnP(d|θ)=∂mlnP(d|θ)·∂nlnP(d|θ)−1\nP(d|θ)∂2\nmnP(d|θ).\n(B.2)\nTaking the expectation value on both sides, together with\nE\"1\nP(d|θ)∂2\nmnP(d|θ)\f\f\f\f\fθ∗#\n=∂2\nmnZ\nddP(d|θ∗)=0, (B.3)\nproves the equality between Eqs. 1 and B.1.\nBefore diving into evaluating B.1 for a Gaussian likelihood\nN(d|µ(θ),Σ(θ)), we firstly list several useful relations:\n∂mΣ−1=−Σ−1∂mΣ Σ−1, (B.4)\n∂m|Σ|=|Σ|tr\u0010\nΣ−1∂mΣ\u0011\n, (B.5)\nE\u0002A(d−µ)\u0003=0, (B.6)\nEh\n(d−µ)TA(d−µ)i\n=tr(AΣ). (B.7)The first one follows from ΣΣ−1=1and calculating its deriva-\ntive, the second one is Jacobi’s relation for the derivative of the\ndeterminant. The last two are expectation relations for Gaussian\ndistributions for an arbitrary matrix A(where dim A=dimΣ).\nThe first of these follows from the anti-symmetry of the integral,\nand the second is left as an exercise for the conscientious reader.\nWith this at hand, we can attack the problem of calculating\nthe Fisher matrix for a Gaussian likelihood. By taking the loga-\nrithm\nlnP(d|θ)=const.−1\n2ln|Σ|−1\n2(d−µ)TΣ−1(d−µ) (B.8)\nand computing its second derivative, we get:\n∂2\nmnlnP(d|θ)=−∂mµTΣ−1∂nµ (B.9)\n+1\n2tr\u0010\nΣ−1∂mΣ Σ−1∂nΣ\u0011\n(B.10)\n−(d−µ)TΣ−1∂mΣ Σ−1∂nΣ Σ−1(d−µ) (B.11)\n−1\n2tr\u0010\nΣ−1∂2\nmnΣ\u0011\n(B.12)\n+1\n2(d−µ)TΣ−1∂2\nmnΣ Σ−1(d−µ) (B.13)\n+∂2\nmnµTΣ−1(d−µ) (B.14)\n−\u0010\n∂mµTΣ−1∂nΣ+∂nµTΣ−1∂mΣ\u0011\nΣ−1(d−µ)\n(B.15)\nTaking the negative expectation value of the final equation can\nnow be done by using relations B.6 and B.7. One can easily see\nthat only the first three terms contribute, with others canceling\neach other or being equal to zero. With this, we finally have:\nFmn=∂mµTΣ−1∂nµ+1\n2tr\u0010\nΣ−1∂mΣ Σ−1∂nΣ\u0011\n. (B.16)\nAppendix C: Fisher information under linear\ntransformations\nImagine we calculate the Fisher information matrix in parameter\nspaceθaroundθ∗:\nFθ(θ∗)mn=E\"∂\n∂θmlnP(d|θ)·∂\n∂θnlnP(d|θ)\f\f\f\f\fθ∗#\n. (C.1)\nIf we now take a linear transformation of the parameter space\nη=a⊙θ+b, where⊙is element-wise multiplication, we get:\nFη(η∗=a⊙θ∗+b)=A Fθ(θ∗)A, (C.2)\nwhere A=diag( a) is a diagonal matrix with elements on the\ndiagonal equal to the vector a. That further means:\ndetFη=detAFθA=(detA)2detFθ, (C.3)\nln det Fη=C+ln det Fθ. (C.4)\nThe final relation can be useful when thinking about Fisher infor-\nmation calculated in di fferent physical units (di fferent scalings).\nThe only consequence of such procedure is that the logarithm\nof the Fisher information shifts by a constant C, independently\non the position in the parameter space. This further means that\nthe distribution of the Fisher information shifts by the same con-\nstant.\nFor the single parameter variance definition (Eq. 26), one can\nsee the units (scaling) do not play a role, as we always express it\nwith respect to the 2DPS, therefore:\nσ2\nσ2\n2DPS= detFθ\ndetFθ,2DPS!−1/5\n= detFη\ndetFη,2DPS!−1/5\n. (C.5)\nArticle number, page 13 of 13"    },    {        "title": "2401.12342v1.Discretisations_of_mixed_dimensional_Thermo_Hydro_Mechanical_models_preserving_energy_estimates.pdf",        "content": "Discretisations of mixed-dimensional\nThermo-Hydro-Mechanical models preserving energy\nestimates\nJ´ erˆ ome Droniou∗1,3, Mohamed Laaziri†2, and Roland Masson‡2\n1IMAG, Universit´ e de Montpellier, CNRS, Montpellier, France\n2Universit´ e Cˆ ote d’Azur, Inria, CNRS, Laboratoire J.A. Dieudonn´ e, team Coffee, Nice, France\n3School of Mathematics, Monash University, Victoria 3800, Australia\nAbstract\nIn this study, we explore mixed-dimensional Thermo-Hydro-Mechanical (THM)\nmodels in fractured porous media accounting for Coulomb frictional contact at ma-\ntrix fracture interfaces. The simulation of such models plays an important role in many\napplications such as hydraulic stimulation in deep geothermal systems and assessing\ninduced seismic risks in CO 2storage. We first extend to the mixed-dimensional frame-\nwork the thermodynamically consistent THM models derived in [ 1616] based on first and\nsecond principles of thermodynamics. Two formulations of the energy equation will be\nconsidered based either on energy conservation or on the entropy balance, assuming\na vanishing thermo-poro-elastic dissipation. Our focus is on space time discretisations\npreserving energy estimates for both types of formulations and for a general single phase\nfluid thermodynamical model. This is achieved by a Finite Volume discretisation of the\nnon-isothermal flow based on coercive fluxes and a tailored discretisation of the non-\nconservative convective terms. It is combined with a mixed Finite Element formulation\nof the contact-mechanical model with face-wise constant Lagrange multipliers account-\ning for the surface tractions, which preserves the dissipative properties of the contact\nterms. The discretisations of both THM formulations are investigated and compared\nin terms of convergence, accuracy and robustness on 2D test cases. It includes a Dis-\ncrete Fracture Matrix model with a convection dominated thermal regime, and either\na weakly compressible liquid or a highly compressible gas thermodynamical model.\nKeywords: Thermo-Hydro-Mechanical (THM) model, Mixed-dimensional model, Dis-\ncrete Fracture Matrix model, Contact-mechanics, Thermodynamically consistent dis-\ncretisation, Finite Volume, Mixed Finite Element formulation.\n1 Introduction\nThermo-Hydro-Mechanical (THM) models in fractured/faulted porous rocks play an impor-\ntant role in addressing the challenges of a sustainable exploitation of subsurface resources.\nThis is for example typically the case of deep geothermal energy production and CO 2geo-\nlogical storage. THM models offer valuable insights into the complex interactions between\ntemperature changes, fluid flow, rock deformation and fracture/fault mechanical behavior\n∗jerome.droniou@umontpellier.frjerome.droniou@umontpellier.fr\n†mohamed.laaziri@univ-cotedazur.frmohamed.laaziri@univ-cotedazur.fr\n‡roland.masson@univ-cotedazur.frroland.masson@univ-cotedazur.fr\n1arXiv:2401.12342v1  [math.NA]  22 Jan 2024within the subsurface. They play a pivotal role in assessing potential risks, informing mit-\nigation strategies and managing hydraulic stimulation of geothermal systems. Similarly,\nin CO 2sequestration projects, THM models are instrumental in predicting and preventing\nissues like fault reactivation which can potentially induce CO 2leakage or seismicity.\nThe THM models considered in this work initially integrate a mixed-dimensional approach,\ncoupling a non-isothermal Poiseuille flow within the network of fractures/faults represented\nas co-dimension one surfaces, to the non-isothermal Darcy flow in the surrounding porous\nrock, known as the matrix. Let us refer to [ 11,2121,2727,2929,1010,1111,2222,3131,44] and the references\nthere-in for the derivation and for various discretisations of such mixed-dimensional models\nin the context of single phase Darcy flows. The second key component is the thermo-poro-\nmechanical model coupling the deformation of the rock with the non-isothermal Darcy\nflow within the matrix domain. Let us refer to the monograph [ 1616] as a key reference\ntextbook on a thermodynamically consistent derivation of such models. In this work, we\nwill follow this framework further assuming small strains and porosity variations as well as\na linear thermo-poro-elastic behavior of the porous rock. The third ingredient is related\nto the mechanical behavior of the fractures/faults typically based on contact-mechanics\ntaking into account frictional contact at matrix-fracture interfaces. This type of mixed-\ndimensional poromechanical models have been the object of many recent works both in the\nisothermal case [ 3030,2424,33,2323,55,66,77,88] and in the non-isothermal case [ 3232,2525,3333].\nBuilding upon existing research, our work focuses on discretisations of mixed-dimensional\nTHM models preserving energy estimates for a general single phase fluid thermodynamical\nmodel, an aspect not thoroughly explored in previous studies. We first extend to the\nmixed-dimensional framework the thermodynamically consistent THM models derived in\n[1616] based on first and second principles of thermodynamics. Two formulations of the energy\nequation will be considered based either on energy conservation or on the entropy balance,\nassuming a vanishing thermo-poro-elastic dissipation. The entropy balance formulation\nis frequently combined with a small Darcy velocity assumption and a linearisation of the\nFourier term based on a small temperature variation assumption [ 1212,99]. It leads to an\napproximate entropy equation which will be investigated in this work both theoretically and\nnumerically from the points of view of energy estimates and accuracy of the solution. One\ndifficulty that needs to be dealt with when starting from such entropy balance equation is\nthe ability to return back to the energy equation at the discrete level. This will be addressed\nin this work by preserving at the discrete level the links between the energy conservation\nand entropy balance equations.\nTo achieve these goals, the space and time discretisations of the non-isothermal flow and\nof the contact-mechanics must be selected carefully. Regarding the flow, our framework is\nbased on coercive fluxes and incorporate a possible upwind approximation of the convection\nterms in order to deal with convection dominated regimes. The discretisation of the non-\nconservative convective terms in the entropy balance formulation is designed in order to\npreserve the link with the energy formulation. Although, this setting accounts for a large\nclass of coercive Finite Volume schemes, we focus in the following on the Hybrid Finite\nVolume (HFV) discretisation [ 1919,1010] in order to simplify the presentation. The time inte-\ngration scheme is based on a forward Euler discretisation including a semi-implicit variation\nin the entropy balance approach. Regarding the discretisation of the mechanical model, it\nwill be based, for the sake of simplicity, on a conforming Finite Element approximation of\nthe displacement field accounting for its discontinuity at matrix fracture interfaces. The\ndiscretisation of the frictional contact along the fracture network is a key feature both for\n2the robustness of the scheme and for the derivation of energy estimates. Different formu-\nlations such as mixed or stabilized mixed formulations [ 2626,3434,2828], augmented Lagrangian\n[1313] and Nitsche methods [ 1414,1515,22] have been developed to discretise Coulomb frictional\ncontact. In line with [ 2020,2323,77], a mixed formulation with face-wise constant Lagrange\nmultipliers is selected. This choice maintains at the discrete level the dissipative properties\nof the contact terms and the lower bound on the fracture aperture. It enables the straight-\nforward handling of complex fracture networks, including corners, tips, and intersections,\nand lead to local expressions of the contact equations, making possible the use of efficient\nsemi-smooth Newton nonlinear solvers.\nThe rest of this article is organised as follows. Section 22presents the mixed-dimensional\nTHM models with energy equations based either on an energy conservation formulation\nor on an approximate entropy balance formulation. The energy estimates satisfied by\nboth models are formally derived in Section 2.32.3. Their discretisations are described in\nSection 33including the Finite Volume discretisations of the non-isothermal flow in Section\n3.23.2and the mixed formulation of the contact-mechanics in Section 3.33.3. The discrete\nenergy estimates are derived in Section 3.43.4for the discretisations of both models which\nare compared numerically in Section 44in terms of convergence, accuracy and robustness.\nWe first consider in Section 4.14.1a manufactured solution for an incompressible fluid and\ndifferent Peclet numbers on a 2D square domain without fractures. Then, a 2D Discrete\nFracture Matrix (DFM) model with a six fracture network is investigated in Section 4.24.2\nboth for the case of a weakly compressible liquid and for the case of a highly compressible\nperfect gas.\n2 Mixed-dimensional Thermo-Hydro-Mechanical models\n2.1 Mixed-dimensional geometry and function spaces\nIn what follows, scalar fields are represented by lightface letters, vector fields by boldface\nletters. We let Ω ĂRd,dPt2,3u, denote a bounded polytopal domain, partitioned into a\nfracture domain Γ and a matrix domain Ω zΓ. The network of fractures is defined by\nΓ“ď\niPIΓi,\nwhere each fracture Γ iĂΩ,iPI, is a planar and simply connected polygonal domain,\nwhich is relatively open in the hyperplane it spans. Without restriction of generality, we\nwill assume that the fractures may only intersect at their boundaries (Figure 11), that is,\nfor any i, jPI, i‰jit holds Γ iXΓj“H, but not necessarily ΓiXΓj“H.\nThe two sides of a given fracture of Γ are denoted by ˘in the matrix domain, with unit\nnormal vectors n˘oriented outward from the sides ˘. We denote by γathe trace operators\non the side aPt`,´uof Γ for functions in H1pΩzΓqand by γBΩthe trace operator for the\nsame functions on BΩ. The jump operator on Γ for functions uinpH1pΩzΓqqdis defined by\nJuK“γ`u´γ´u,\nand we denote by\nJuKn“JuK¨n`and JuKτ“JuK´JuKnn`\n3Figure 1. Illustration of the dimension reduction in the fracture aperture for a 2D domain\nΩ with three intersecting fractures Γ i,iPt1,2,3u, with the equi-dimensional geometry on\nthe left and the mixed-dimensional geometry on the right.\nits normal and tangential components. The notation JqmKnwill also be used to denote the\nnormal jump of flux functions qmPHdivpΩzΓqdefined by\nJqmKn“γ`\nnqm`γ´\nnqm\nwith γa\nnthe normal trace operator on the side aof Γ oriented outward to the side aPt`,´u.\nThe notation γa\nnwill also be applied to tensor fields. The tangential gradient and divergence\nalong the fractures are respectively denoted by ∇τand div τ. The symmetric gradient\noperator εis defined such that εpvq“1\n2p∇v`p∇vqtqfor a given vector field vPH1pΩzΓqd.\nTo fix ideas, the pressure pand temperature Tare assumed to be continuous at the matrix–\nfracture interface, that is, these functions belong to H1pΩq. As a consequence, letting γ:\nH1pΩqÑL2pΓqbe the trace operator on the fracture, the fracture pressure and temperature\nare given by pf“γpandTf“γT.\nThe space for the displacement field is defined by\nU0“tvPH1pΩzΓqd:γBΩv“0u.\n2.2 Mixed-dimensional models\nWe consider a Thermo-Hydro-Mechanical (THM) model under the hypothesis of small\nperturbations for the skeleton accounting for small strain, displacement and variations of\nporosity [ 1616]. Linear isotropic thermo-poro-elastic constitutive laws are considered for the\nskeleton assuming small variations of temperature around the reference temperature Tref.\nThe Darcy law is used for the fluid velocity and the Fourier law for the thermal conduction.\nThermal equilibrium is assumed between the fluid and the skeleton, and the mechanical\ninertial term is modelled using the frozen specific average fluid-rock density m0.\nThe fluid thermodynamical properties, depending on its pressure pand temperature T, are\n•ϱpp, Tq: specific density,\n•epp, Tq: specific internal energy,\n•hpp, Tq“epp, Tq`p\nϱpp,Tq: specific enthalpy, and\n•ηpp, Tq: dynamic viscosity.\nFor a fluid property Ξ “ϱ, e, h, η , we use the short notations Ξ m“Ξpp, Tqin the matrix\nand Ξ f“Ξppf, Tfq“γΞmalong the fracture network.\n4The primary unknowns of the model are the fluid pressure p, the fluid temperature Tand\nthe skeleton displacement field u. They are solutions of the nonlinear system of PDEs\ncoupling the mixed-dimensional fluid mass conservation equation, the mixed-dimensional\ntotal energy conservation equation, the skeleton momentum balance equation, and the\nfrictional contact conditions at matrix fracture interfaces. The model is detailed below\nstarting with the equations in the matrix followed by the equations along the fracture\nnetwork. Two different formulations of the energy equation will be considered.\n2.2.1 Matrix model\nIn the matrix, the model accounts for the mass and energy conservation equations coupled\nto the skeleton momentum balance equation. Letting tFbe the final simulation time, we\ntherefore consider\nBtpϱmϕq`divpϱmVmq“Gminp0, tFqˆΩ, (1a)\nTBtSs`pBtϕ`Btpϱmϕemq`divpϱmhmVm`qmq“Hminp0, tFqˆΩ, (1b)\nm0B2\ntu´divσ“F inp0, tFqˆΩ, (1c)\nwith the Darcy and Fourier laws governing respectively the matrix fluid velocity and the\nmatrix conductive thermal flux:\nVm“´Kpϕq\nηm∇p,qm“´Λmpϕq∇T, (1d)\nwhere Kand Λ mare respectively the rock permeability and the fluid rock average thermal\nconductivity, both possibly depending on the matrix porosity ϕ. The energy equation ( 1b1b)\nhas been obtained under the hypothesis of reversible mechanical deformation in the sense\nof zero thermo-poro-mechanical dissipation:\nTBtSs`pBtϕ`σ:Btεpuq´B tEs“0. (2)\nAssuming a linear isotropic thermo-poro-elastic behavior of the skeleton, this gives the\nfollowing constitutive laws\nBtϕ“bdivBtu´αϕBtT`1\nNBtp , (3a)\nBtSs“αsKsdivBtu´αϕBtp`Cs\nTrefBtT, (3b)\nσ“σepuq´b pI´αsKspT´TrefqI, (3c)\nσepuq“2µεpuq`λdivuI, (3d)\nwhich derive from the following volumetric skeleton internal energy [ 1616]:\nEs“µ|εpuq|2`λ\n2pdivpuqq2`1\n2”\np Tı\nM«\np\nTff\n`αsKsTrefdivu, (4)\nwhere the matrix M, defined below, is assumed to be definite positive:\nM:“«\n1\nN´αϕ\n´αϕCs\nTrefff\n.\n5In the equations above, Ssis the volumetric skeleton entropy, σthe total stress tensor and\nσethe effective stress tensor. The parameters µandλare the effective Lame coefficients,\nNis the Biot modulus, bthe Biot coefficient, Ksis the bulk modulus, αsis the volumetric\nskeleton thermal dilation coefficient, αϕis the volumetric thermal dilation coefficient related\nto the porosity, and Csis the skeleton volumetric heat capacity.\nConsidering the combination1\nTˆ(1b1b)´hm\nTˆ(1a1a) leads to the following alternative formu-\nlation of the energy equation\nBtSs`ϱmϕ\nTBtem`pϱm\nTϕBt1\nϱm`1\nTϱmVm¨∇hm`1\nTdivqm“Hm\nT´hm\nTGm.(5)\nAn approximation of ( 55) is obtained based on the classical assumptions of small Darcy ve-\nlocity Vmand small variations of temperature around Tref. Then, using the approximation\nϱmVm¨∇hm“ϱmVm¨∇em`pϱmVm¨∇1\nϱm`Vm¨∇p\n„ϱmVm¨∇em`pϱmVm¨∇1\nϱm,\nand the linearisation1\nTdivqm„1\nTrefdivqm, we obtain the following approximate equation:\nBtSs`ϱmϕ\nTBtem`pϱm\nTϕBt1\nϱm`1\nTϱmVm¨∇em\n`p\nTϱmVm¨∇1\nϱm`1\nTrefdivqm“Hm\nT´hm\nTGm.(6)\nNote that, using Tds“de`pd1\nϱwith sthe fluid specific entropy, equation ( 66) becomes\nBtSs`ϱmϕBtsm`ϱmVm¨∇sm`1\nTrefdivqm“Hm\nT´hm\nTGm.\nThis equation is a classical approximate non conservative formulation of the entropy equa-\ntion [ 1616], which motivates the terminology adopted in the following of approximate entropy\nequation for (66) and of entropy equation for (55).\n2.2.2 Fracture model\nThe mass and energy conservation equations of the reduced fracture model are obtained by\nintegration along the fracture width of the equi-dimensional equations taking into account\nthe mass and energy normal flux continuity at matrix fracture interfaces. This process leads\nto\nBtpϱfdfq`divτpϱfVfq´ JϱmVmKn“Gfinp0, tFqˆΓ, (7a)\npfBtdf`Btpϱfdfefq`divτpϱfhfVf`qfq´ JϱmhmVm`qmKn“Hfinp0, tFqˆΩ,\n(7b)\nwith the fluid tangential velocity and thermal conductive flux integrated along the fracture\nwidth defined by\nVf“´Cfpdfq\nηf∇τpf,qf“´Λfpdfq∇τTf, (7c)\nand where\ndf“d0´JuKn (7d)\n6is the fracture aperture with d 0the aperture at contact state as illustrated in Figure 22.\nIn (7c7c),Cfis the fracture hydraulic conductivity typically given by the Poiseuille law\nCf“pdfq3\n12, and Λ fis the fracture thermal conductivity also possibly depending on d f.\nAt matrix fracture interfaces, a contact Coulomb frictional model is considered defined as\nfollows $\n’’’’’’&\n’’’’’’%T``T´“0 onp0, tFqˆΓ,\nTnď0,JuKnď0,JuKnTn“0 onp0, tFqˆΓ,\n|Tτ|ď´ F Tn onp0, tFqˆΓ,\npBtJuKτq¨Tτ´F Tn|BtJuKτ|“0 onp0, tFqˆΓ,(7e)\nwhere the vectorial surface tractions and their normal and tangential components are de-\nfined by $\n’’’&\n’’’%Ta“γa\nnσpuq`pfnaonp0, tFqˆΓ,aPt`,´u,\nTn“T`¨n`onp0, tFqˆΓ,\nTτ“T`´pT`¨n`qn`onp0, tFqˆΓ.\nIn a similar way as in the matrix, we also consider the entropy equation defined by the\ncombination1\nTˆ(7b7b)´hm\nTˆ(7a7a) leading to\nϱfdf\nTfBtef`pfϱf\nTfdfBt1\nϱf`1\nTfϱfVf¨∇hf`1\nTfpdivqf´JqmKnq\n´1\nTf´\nJϱmhmVmKn´hfJϱmVmKn¯\n“Hf\nTf´hf\nTfGf.(8)\nBy continuity of the pressure and temperature (and thus of the fluid specific enthalpy),\nJϱmhmVmKn´hfJϱmVmKn“0 and the last term in the left-hand side of ( 88) therefore van-\nishes. However, at the discrete level, as a result of a possible upwinding this compensation\nmay not necessarily occur. To keep that in mind in the continuous model, and justify the\ndiscretisation chosen in Section 33, we therefore write this vanishing term in the form of a\n(fictitious) jump of the enthalpy between the matrix and the fracture.\nJϱmhmVmKn´hfJϱmVmKn“ÿ\naPt`,´uγa\nnpϱmVmqpγahm´hfq.\nTherefore, ( 88) can be recast as\nϱfdf\nTfBtef`pfϱf\nTfdfBt1\nϱf`1\nTfϱfVf¨∇hf`1\nTfpdivqf´JqmKnq\n´1\nTfÿ\naPt`,´uγa\nnpϱmVmqpγahm´hfq“Hf\nTf´hf\nTfGf.(9)\nIts approximation based on small Darcy velocity and temperature variations assumptions\nwrites\nϱfdf\nTfBtef`pfϱf\nTfdfBt1\nϱf`1\nTfϱfVf¨∇ef`pf\nTfϱfVf¨∇1\nϱf`1\nTrefpdivqf´JqmKnq\n´1\nTfÿ\naPt`,´uγa\nnpϱmVmq´\nγaem´ef`pfp1\nγaϱm´1\nϱfq¯\n“Hf\nTf´hf\nTfGf.(10)\n7Figure 2. Conceptual fracture model with contact at asperities, d0being the fracture\naperture at contact state.\nThe model is closed by considering no-flow and no-energy flux boundary conditions both\nfor the matrix and fractures. Let us also recall that the pressure and temperature are\nassumed to be continuous at matrix pressure interfaces. At fracture intersections, the\nfracture pressure pfand temperature Tfare classically also continuous, and mass and\nthermal flux conservation is imposed.\nIn the sequel, we use the following terminology to refer to the two mixed-dimensional THM\nsystems:\n(11)-(22)-(33)-(77): “enthalpy-based ” THM model, ( H-model)\n(11)-(22)-(33)-(77) with ( 1b1b) replaced by the approximation ( 66) and\n(7b7b) replaced by the approximation ( 1010): “entropy-based ” THM model.(S-model)\n2.3 Energy estimates\nLet us first consider the enthalpy-based THM model ( H-modelH-model ). Summing the integral\nover Ω of ( 1b1b)`Btu¨(1c1c) and the integral over Γ of ( 7b7b), taking into account the contact\nand boundary conditions, the following contact persistence condition (which results from\n(7e7e))ż\nΓTnJBtuKndσ“0,\nand the zero dissipation equation ( 22), we obtain formally the following energy estimate:\nBtż\nΩpEs`ϱmϕemqdx`Btż\nΓϱfdfefdσ`ż\nΓ´FTn|JBtuKτ|dσ\n“ż\nΩpHm`F¨Btuqdx`ż\nΓHfdσ.(11)\nLet us now consider the entropy-based THM model ( S-modelS-model ). By construction, the energy\nestimate for this model can be deduced from the previous estimate ( 1111) just by adding to\nthe left hand side of ( 1111) the term\nż\nΩ´\n´Vm¨∇p`pT\nTref´1qdivqm¯\ndx`ż\nΓ´\n´Vf¨∇τpf`pTf\nTref´1qpdivqf´JqmKnq¯\ndσ,\nwhich corresponds to the difference between the exact terms and their approximations.\nRecalling the definitions ( 1d1d) and ( 7c7c) ofVm,qm,Vfandqf, this leads to the following\n8stronger form of the energy estimate\nBtż\nΩpEs`ϱmϕemqdx`Btż\nΓϱfdfefdσ`ż\nΓ´FTn|JBtuKτ|dσ\n`ż\nΩ´K\nηm∇p¨∇p`Λm\nTref|∇T|2¯\ndx`ż\nΓ´Cf\nηf|∇τpf|2`Λf\nTref|∇τTf|2¯\ndσ\n“ż\nΩpHm`F¨Btuqdx`ż\nΓHfdσ.(12)\n3 Discretisation\nThe objective is to design a discretisation preserving the energy estimates ( 1111) and ( 1212)\nfor respectively the enthalpy-based and entropy-based THM models. Both types of for-\nmulations will be compared in the numerical section. The schemes must account for both\ndiffusive and convective dominated energy transport. This motivates the choice of a Fi-\nnite Volume formulation of the mass and energy equations with possible upwinding of the\nmobilities. To fix ideas, we will consider in this work the mixed-dimensional Hybrid Fi-\nnite Volume (HFV) discretisation introduced in [ 1010] accounting for the continuity of the\npressure and temperature at matrix fracture interfaces. It will be combined with a mixed\nvariational formulation of the contact-mechanics with a conforming Finite Element dis-\ncretisation of the displacement field and a facewise constant approximation of the Lagrange\nmultiplier introduced to represent the surface tractions along the fractures. This choice\nhandles fracture networks with corners and intersections, and leads to a local formulation\nof the contact conditions. It also ensures the discrete persistence property of the contact\nterm and a dissipative frictional term which are key conditions to obtain an energy estimate\nfor the coupled THM systems.\n3.1 Space and time discretisations\nLetMdenote the set of polytopal cells, and Fthe set of faces of the mesh, with internal\nfaces gathered in Fintand boundary faces in Fext. The subset FKĂFdenotes the set of\nfaces of the cell KPM. We denote by σ“K|Lthe internal face shared by the two cells\nK, L gathered in the subset Mσ“tK, Lu. The notations σ“K|¨andMσ“tKuare used\nfor a face σPFKXFext. The mesh is assumed conforming to the fracture network Γ in\nthe sense that there exists a subset FΓofFsuch that\nΓ“ď\nσPFΓσ.\nThe subset of edges of a face σis denoted by Eσand we define the set of edges of Γ by\nEΓ“Ť\nσPFΓEσ. For a given edge ζPEΓ, let us denote by FΓ,ζthe subset of fracture faces\nsharing the edge ζ. We denote by |K|thed-dimensional measure of the cell KPM, and\nby|σ|thepd´1q-dimensional measure of the face σPF.\nThe HFV method is a Gradient Discretisation (GD) defined by the vector space of discrete\nunknowns\nXD“tvD“ppvKqKPM,pvσqσPF,pvζqζPEΓqu“RMYFYEΓ,\nand the reconstruction operators Πm\nD,∇m\nDin the matrix and Πf\nD,∇f\nDalong the fractures\n(see [ 1010] for their detailed definition). We assume that Πm\nDand Πf\nDare piecewise constant\n9reconstructions [ 1717, Definition 2.12] and, more specifically, pΠm\nDvDq|K“vKfor all KPM,\nandpΠf\nDvDq|σ“vσfor all σPFΓ. From the gradient reconstruction operator ∇m\nDin the\nmatrix, and given any KPMand any symmetric positive definite dˆdmatrix DK, fluxes\nFK,σpDK;¨q:XDÑR(forσPFK) are defined such that, for all vD, wDPXDand all\nKPM, ż\nKDK∇m\nDvD¨∇m\nDwDdx“ÿ\nσPFKFK,σpDK;vDqpwK´wσq. (13)\nLikewise, from the fracture tangential gradient reconstruction operator ∇f\nD, for any σPFΓ\nand any symmetric positive definite pd´1qˆpd´1qmatrix Dσ, fracture fluxes Fσ,ζpDσ;¨q:\nXDÑRare defined (for ζPEσ) such that, for all vD, wDPXD,\nż\nσDσ∇f\nDvD¨∇f\nDwDdσ“ÿ\nζPEσFσ,ζpDσ;vDqpwσ´wζq. (14)\nThese fluxes are local to each cell KPM(resp. each fracture face σPFΓ) in the sense\nthat they only depend on pvK,pvσqσPFKq(resp.pvσ,pvζqζPEσq) [1010].\nIn order to account for convective dominated energy transport, an upwind approximation\nneeds to be introduced according to the sign of the Darcy flux FK,σpKK;pDqusing the\nfollowing notations. Let the index “ Kσ,`” denote either σif no upwinding is used, or an\nupwind choice between KandLifσ“K|LPFintzFΓ, or an upwind choice between K\nandσifσ“K|¨PFextYFΓ(note that, if σ“K|Lis an internal face and not a fracture\nface, this upwinding must not depend on KorL; for example, ϱKσ,`“ϱLσ,`).\nSimilarly, for a given fracture edge ζ, the index “ σζ,`” denote either ζif no upwinding is\nused, or an upwind value between σandσ1ifFΓ,ζ“tσ, σ1uandζis not a boundary edge,\nor an upwind value between σandζin the other cases. The choice of the upwind value is\ndone according to the sign of the Darcy flux Fσ,ζpCf,σ;pDq.\nFor any fluid thermodynamical property Ξ “h, e, η, ϱ , and pD, TDPXD, we define Ξ DPXD\ncomponent by component by setting Ξ ν“Ξppν, Tνqfor all νPMYFYEΓ.\nThe displacement field uis discretised using a finite dimensional subspace UDofU0,\ntypically given by a Finite Element Method.\nWe consider a time discretisation ptnqn“0,...,N of the time interval p0, tFqwith t0“0 and\ntN“tF, and denote by ∆ tn“tn´tn´1the time step n. Iff“pfnqn“0,...,Nis a family of\nfunctions or vectors of XD, the discrete time derivative of fis defined as\nδn\ntf“fn´fn´1\n∆tn.\nThe second time derivative of the discrete displacement field is defined by\nδn\nt9uD“29un\nD´ 9un´1\nD\ntn´tn´2with 9un\nD“δn\ntuD.\nIt corresponds to the generalisation to non-uniform time steps of the standard centered\ndiscretisation of B2\ntu. To alleviate notations, whenever wDPXN`1\nDand‚“t m, fu, we set\nδn\nt,‚wD“δn\ntpΠ‚\nDwDq. We also consider the discrete time derivative of products of functions\nand elements of XD, or elements of XD, by setting: δn\nt,‚pfvDwDq“δn\ntpfΠ‚\nDvDΠ‚\nDwDqfor\nall families of functions f“pfnqn“0,...,N and elements vD, wDPXN`1\nD.\n10The discrete porosities pϕn\nDqn“0,...,N and skeleton entropies pSn\ns,Dqn“0,...,N are families of\npiecewise constant functions Ω ÑRonMsuch that ϕ0\nD, S0\ns,Dare given (e.g., as projections\nof the continuous initial porosity and entropy) and, for all n“1, . . . , N ,\nδn\ntϕD“bπMpδn\ntdivuDq´αϕδn\nt,mTD`1\nNδn\nt,mpD, (15)\nδn\ntSs,D“αsKsπMpδn\ntdivuDq´αϕδn\nt,mpD`Cs\nTrefδn\nt,mTD, (16)\nwhere πMis the projection on piecewise constant functions on M(that is,pπMfq|K“\n1\n|K|ş\nKffor all KPM).\nIn the fracture, we define the discrete apertures as pdn\nf,Dqn“0,...,N, with dn\nf,D: ΓÑRgiven\nby\ndn\nf,D“πF,Γpd0´Jun\nDKnq. (17)\nHere, πF,Γis the L2-projection on facewise constant functions on Γ.\nWe also define, for n“1, . . . , N , the function pGn\nm: ΩÑRwhich is piecewise constant\nonMequal on KPMto the average of Gmonptn´1, tnqˆK. The piecewise constant\nfunctions pHn\nmandpFnonM, as well as pGn\nfandpHn\nfonFΓare similarly defined. In the\nfollowing, we use the notations\nVn\nK,σ“FK,σ´Kpϕn´1\nKq\nηn´1\nK;pn\nD¯\nand Vn\nσ,ζ“Fσ,ζ´Cfpdn´1\nf,σq\nηn´1σ;pn\nD¯\n, (18)\nfor the matrix and fracture Darcy flux, as well as\nQn\nK,σ“FK,σ´\nΛmpϕn´1\nKq;Tn\nD¯\nand Qn\nσ,ζ“Fσ,ζ´\nΛfpdn´1\nf,σq;Tn\nD¯\n(19)\nfor the matrix and fracture Fourier fluxes. Note that K, Λm,Cfand Λ fcould also explicitly\ndepend on x. This dependence is omitted for the sake of simplicity.\n3.2 Discretisation of the mass and energy equations\nLet us first consider the HFV discretisation of the mass and energy conservation equations\nof the enthalpy-based THM model ( H-modelH-model ). For all time step n“1, . . . , N , it couples\nthe discrete mass conservation equations\n|K|δn\ntpϱKϕKq`ÿ\nσPFKϱn\nKσ,`Vn\nK,σ“|K|pGn\nm,K@KPM, (20a)\n|σ|δn\ntpϱσdf,σq`ÿ\nζPEσϱn\nσζ,`Vn\nσ,ζ´ÿ\nKPMσϱn\nKσ,`Vn\nK,σ“|σ|pGn\nf,σ@σPFΓ,(20b)\nÿ\nKPMσVn\nK,σ“0@σPFzFΓ, (20c)\n´ÿ\nσPFΓ,ζϱn\nσζ,`Vn\nσ,ζ“0@ζPEΓ,(20d)\nto the discrete total energy conservation equations\n|K|´\nTn\nKδn\ntSs,K`pn\nKδn\ntϕK`δn\ntpϱKϕKeKq¯\n`ÿ\nσPFKpϱn\nKσ,`hn\nKσ,`Vn\nK,σ`Qn\nK,σq“|K|pHn\nm,K@KPM,(21a)\n11|σ|´\npn\nσδn\ntdf,σ`δn\ntpϱσdf,σeσq¯\n`ÿ\nζPEσpϱn\nσζ,`hn\nσζ,`Vn\nσ,ζ`Qn\nσ,ζq\n´ÿ\nKPMσpϱn\nKσ,`hn\nKσ,`Vn\nK,σ`Qn\nK,σq“|σ|pHn\nf,σ@σPFΓ,(21b)\nÿ\nKPMσQn\nK,σ“0@σPFzFΓ, (21c)\n´ÿ\nσPFΓ,ζpϱn\nσζ,`hn\nσζ,`Vn\nσ,ζ`Qn\nσ,ζq“0@ζPEΓ,(21d)\nNote that ( 20c20c), (21c21c) embed both the conservation of fluxes across internal faces, as well as\nthe zero-flux conditions on boundary faces. The same remark holds for the flux conservation\nequations ( 20d20d), (21d21d) at fracture edges but keeping the full nonlinear fluxes due to the\ndependence on σof the upwind density and enthalpy when FΓ,ζcontains more than two\nfaces (which happens at internal crossing lines between three or more fractures). Similar\nconsiderations hold for ( 23c23c) and ( 23d23d) below.\nTemporarily dropping the time index nfor simplicity, the HFV discretisation of the approx-\nimate entropy equations ( 66)-(1010) is based on the following discretisation of ϱmVm¨∇wm“\ndivpϱmwmVmq´wmdivpϱmVmqon a given cell K, which includes a possible upwinding:\nforwDPXD,\nÿ\nσPFKwKσ,`ϱKσ,`VK,σ´wKÿ\nσPFKϱKσ,`VK,σ“ÿ\nσPFKϱKσ,`VK,σpwKσ,`´wKq.\nThis is a key choice which preserves the link between the non conservative entropy and the\nconservative energy formulations in the sense that it is designed to satisfy the following\ndiscrete version of wmdivpϱmVmq`ϱmVm¨∇wm“divpϱmwmVmq:\nwKÿ\nσPFKϱKσ,`VK,σ`ÿ\nσPFKϱKσ,`VK,σpwKσ,`´wKq“ÿ\nσPFKwKσ,`ϱKσ,`VK,σ.\nLikewise, in the fractures, we use the following discretisations of ϱfVf¨∇τwfÿ\nζPEσϱσζ,`Vσ,ζpwσζ,`´wσq,\nonσPFΓ, and ÿ\nσPFΓ,ζ´ϱσζ,`Vσ,ζpwσζ,`´wζq,\nonζPEΓ. In the same spirit, the terms such as ϱmϕBtwmin the matrix are discretised by\nΠm\nDϱn´1ϕn´1\nDδn\nt,mwDto ensure that\nΠm\nDϱn´1\nDϕn´1\nDδn\nt,mwD`Πm\nDwn\nDˆδn\nt,mpϱDϕDq“δn\nt,mpϱDϕDwDq. (22)\nThe same discretisation is applied for the terms such as ϱfdfBtwfin the fractures. Following\nthis methodology, the approximate entropy equations are discretised for all time step n“\n1, . . . , N by\n|K|δn\ntSs,K`|K|\nTn\nKϱn´1\nKϕn´1\nK´\nδn\nteK`pn\nKδn\nt1\nϱK¯\n`1\nTrefÿ\nσPFKQn\nK,σ\n`1\nTn\nKÿ\nσPFKϱn\nKσ,`Vn\nK,σ´\nen\nKσ,`´en\nK`pn\nKp1\nϱn\nKσ,`´1\nϱn\nKq¯\n“|K|\nTn\nKppHn\nm,K´hn\nKpGn\nm,Kq @ KPM,(23a)\n12|σ|\nTnσϱn´1\nσdn´1\nf,σ´\nδn\nteσ`pn\nσδn\nt1\nϱσ¯\n`1\nTref´ÿ\nζPEσQn\nσ,ζ´ÿ\nKPMσQn\nK,σ¯\n`1\nTnσÿ\nζPEσϱn\nσζ,`Vn\nσ,ζ´\nen\nσζ,`´en\nσ`pn\nσp1\nϱn\nσζ,`´1\nϱnσq¯\n´1\nTnσÿ\nKPMσϱn\nKσ,`Vn\nK,σ´\nen\nKσ,`´en\nσ`pn\nσp1\nϱn\nKσ,`´1\nϱnσq¯\n“|σ|\nTnσppGn\nf,σ´hn\nσpGn\nf,σq @ σPFΓ,(23b)\nÿ\nKPMσQn\nK,σ“0@σPFzFΓ, (23c)\n´1\nTn\nζÿ\nσPFΓ,ζϱn\nσζ,`Vn\nσ,ζ´\nen\nσζ,`´en\nζ`pn\nζp1\nϱn\nσζ,`´1\nϱn\nζq¯\n´1\nTrefÿ\nσPFΓ,ζQn\nσ,ζ“0@ζPEΓ,\n(23d)\nBy construction, applying this space time discretisation to the entropy equations ( 55)-(99)\nwould lead to an equivalence between the entropy and energy formulations at the discrete\nlevel when combined with the discrete mass equations. It results that the following Lemma\nstates the equivalence up to the terms which have been neglected or linearised in the\napproximate entropy equations.\nLemma 3.1. For all cell KPM,Tn\nKˆ(23a23a)`hn\nKˆ(20a20a)is equivalent to (21a21a)upon\ncorrecting the left-hand side by adding\nˆTn\nK\nTref´1˙ÿ\nσPFKQn\nK,σ´ÿ\nσPFKVn\nK,σppn\nKσ,`´pn\nKq.\nFor all fracture face σPFΓ,Tn\nσˆ(23b23b)`hn\nσˆ(20b20b)is equivalent to (21b21b)upon correcting\nthe left-hand side by adding\nˆTn\nσ\nTref´1˙´ÿ\nζPEσQn\nσ,ζ´ÿ\nKPMσQn\nK,σ¯\n´ÿ\nζPEσVn\nσ,ζppn\nσζ,`´pn\nσq`ÿ\nKPMσVn\nK,σppn\nKσ,`´pn\nσq.\nFor all fracture edge ζPEΓ,Tn\nζˆ(23d23d)`hn\nζˆ(20d20d)is equivalent to (21d21d)upon correcting\nthe left-hand side by adding\n´ˆTn\nζ\nTref´1˙ÿ\nσPFΓ,ζQn\nσ,ζ`ÿ\nσPFΓ,ζVn\nσ,ζppn\nσζ,`´pn\nζq.\nProof. Let us detail only the proof for the cell terms since fracture face and edge terms are\nsimilar. Gathering the terms of Tn\nKˆ(23a23a)`hn\nKˆ(20a20a) leads to the equation:\n|K|„\nTn\nKδn\ntSs,K`hn\nKδn\ntpϱKϕKq`ϱn´1\nKϕn´1\nK´\nδn\nteK`pn\nKδn\nt1\nϱK¯ȷ\n`hn\nKÿ\nσPFKϱn\nKσ,`Vn\nK,σ`ÿ\nσPFKϱn\nKσ,`Vn\nK,σ´\nen\nKσ,`´en\nK`pn\nKp1\nϱn\nKσ,`´1\nϱn\nKq¯\n`Tn\nK\nTrefÿ\nσPFKQn\nK,σ“|K|pHn\nm,K.\n13Recalling that h“e`p{ϱ, we obtain\n|K|”\nTn\nKδn\ntSs,K`´\nen\nKδn\ntpϱKϕKq`ϱn´1\nKϕn´1\nKδn\nteK¯\n`pn\nK´\nϱn´1\nKϕn´1\nKδn\nt1\nϱK`1\nϱn\nKδn\ntpϱKϕKq¯ȷ\nÿ\nσPFKϱn\nKσ,`Vn\nK,σ´\nhn\nKσ,``ppn\nK´pn\nKσ,`q1\nϱn\nKσ,`¯\n`Tn\nK\nTrefÿ\nσPFKQn\nK,σ“|K|pHn\nm,K.\nInvoking ( 2222), this relation reduces to\n|K|´\nTn\nKδn\ntSs,K`pn\nKδn\ntϕK`δn\ntpϱKϕKeKq¯\n`ÿ\nσPFK´\nϱn\nKσ,`hn\nKσ,`Vn\nK,σ`Qn\nK,σ¯\n´ÿ\nσPFKVn\nK,σppn\nKσ,`´pn\nKq`ˆTn\nK\nTref´1˙ÿ\nσPFKQn\nK,σ“|K|pHn\nm,K,\nwhich is the expected result for the cell term.\n3.3 Mixed variational discretisation of the contact-mechanical model\nThe subspace MDĂL2pΓqdenotes the set of piecewise constant functions on the partition\nFΓand we set MD“pMDqd. ForλDinMD, we use the decomposition λD“pλD,n,λD,τq\nwith λD,n“λD¨n`,λD,τ“λD´λD,nn`. We denote by λσthe constant value of λDPMD\non the face σPFΓand by λn,σandλτ,σits normal and tangential components.\nWe consider the mixed formulation based on facewise constant Lagrange multipliers, which\nseamlessly deals with fracture intersections, corners and tips and leads to local expressions\nof the discrete contact conditions and efficient semi-smooth Newton solvers. On the other\nhand, this approach requires to assume the following uniform inf-sup condition between the\nspace UDof displacement fields and the space MDof Lagrange multipliers: there exists c‹\nindependent on the mesh such that\ninf\nµDPMDsup\nvDPUDż\nΓµD¨JvDKdσ\n}vD}U0}µD}H´1\n2pΓqděc‹ą0. (24)\nLet us define the discrete dual cone of normal Lagrange multipliers as\nΛD“tλD,nPMD|λD,ně0 on Γu,\nand the discrete dual cone of vectorial Lagrange multipliers given λD,nPΛDas\nΛDpλD,nq“tµD“pµD,n,µD,τqPMD|µD,ně0,|µD,τ|ďFλD,non Γu.\nNote that the friction coefficient Fis assumed to be facewise constant on the partition FΓ.\nThe mixed discretisation of the quasi static contact-mechanical model reads: find puD,λDqP\nUDˆΛDpλD,nqsuch that, for all pvD,µDqPUDˆΛDpλD,nq,\nż\nΩ´\nm0δn\nt9uD¨vD`σpun\nDq:εpvDq´rbΠm\nDpn\nD`αsKspΠm\nDTn\nD´TrefqsdivpvDq¯\ndx\n`ż\nΓλn\nD¨JvDKdσ`ż\nΓΠf\nDpn\nDJvDKndσ“ż\nΩpFn¨vDdx,(25a)\n14ż\nΓ´\npµD,n´λn\nD,nqJun\nDKn`pµD,τ´λn\nD,τq¨Jδn\ntuDKτ¯\ndσď0, (25b)\nThe variational inequality in ( 25b25b) is equivalent to the contact conditions between λσand\nthe face average of JuDKdenoted by JuDKσ(see e.g. Lemma 4.1 of [ 77]). This is a key\nproperty to obtain the following discrete persistence property of the normal contact term\nż\nΓλn\nD,nJδn\ntuDKndσě0, (26)\nleading to the dissipative property of the contact term\nż\nΓλn\nD¨Jδn\ntuDKdσěż\nΓFλn\nD,n|Jδn\ntuDKτ|dσě0. (27)\nWe refer the interested reader to [ 77] for the proof.\n3.4 Summary of the schemes and discrete energy estimates\nTo summarise, the discretisations of the enthalpy-based and entropy-based formulations are\nas follows:\n(2020)-(2121)-(2525)-(1515)-(1616)-(1717) for the enthalpy-based model ( H-modelH-model ). ( H-scheme)\n(2020)-(2323)-(2525)-(1515)-(1616)-(1717) for the entropy-based model ( S-modelS-model ). ( S-scheme)\nLet us define the discrete skeleton internal energy En\ns: ΩÑRby\nEn\ns“1\n2”\nΠm\nDpn\nDΠm\nDTn\nDı\nM«\nΠm\nDpn\nD\nΠm\nDTn\nDff\n`αsKsTrefdivun\nD`µ|εpun\nDq|2`λ\n2pdivun\nDq2.\nThen, the following propositions state the discrete energy estimates for both formulations\nof the energy equation.\nProposition 3.2. The scheme (H-schemeH-scheme )based on the conservative enthalpy formulation\nof the energy equation satisfies, for all n“1, . . . , N , the following energy estimate\nż\nΩm0\n2δn\nt|9uD|2dx`ż\nΩδn\ntEs`ż\nΩδn\nt,mpϱDϕDeDq`ż\nΓδn\nt,fpϱDdf,DeDq\n`ż\nΓFλn\nD,n|Jδn\ntuDKτ|dσďż\nΩ´\npHn\nm`pFn¨δn\ntuD¯\ndx`ż\nΓpHn\nfdσ.(28)\nProof. Adding ( 25a25a) with vD“δn\ntuD“ 9un\nDto the sums over Mof (21a21a), over FΓof (21b21b)\nand over EΓof (21d21d), and taking into account the flux conservativity and the homogeneous\nNeumann boundary conditions ( 20c20c)-(21c21c), together with the definitions of the discrete\nfracture aperture ( 1717), porosity ( 1515) and skeleton entropy ( 1616), we obtain the following\nequality\nż\nΩm0δn\nt9uD¨9un\nDdx`ż\nΩδn\nt,mpϱDϕDeDqdx`ż\nΓδn\nt,fpϱDdf,DeDqdσ\n`ż\nΩ´”\nΠm\nDpn\nDΠm\nDTn\nDı\nMδn\nt,m«\npD\nTDff\ndx`σepun\nDq:δn\ntεpuDqdx`αsKsTrefdivδn\ntuD¯\ndx\n`ż\nΓλn\nD¨Jδn\ntuDKdσ“ż\nΩ´\npHn\nm`pFn¨δn\ntuD¯\ndx`ż\nΓpHn\nfdσ.\n15Then, using the diffusive properties of the implicit Euler time stepping (that is e.g., δn\nt9uD¨\n9un\nDě1\n2|9un\nD|2´1\n2|9un´1\nD|2) and the dissipative property of the discrete contact term ( 2727), we\nobtain the energy estimate ( 2828).\nProposition 3.3. The scheme (S-schemeS-scheme )based on the non-conservative approximate en-\ntropy equation satisfies, for all n“1, . . . , N , the following energy estimate\nż\nΩm0\n2δn\nt|9uD|2dx`ż\nΩδn\ntEs`ż\nΩδn\nt,mpϱDϕDeDq`ż\nΓδn\nt,fpϱDdf,DeDq\n`ż\nΩ´Kpϕn´1\nDq\nΠm\nDηn´1\nD∇m\nDpn\nD¨∇m\nDpn\nD`Λmpϕn´1\nDq\nTref|∇m\nDTn\nD|2¯\ndx\n`ż\nΓ´Cfpdn´1\nf,Dq\nΠf\nDηn´1\nD|∇f\nDpn\nD|2`Λfpdn´1\nf,Dq\nTref|∇f\nDTn\nD|2¯\ndσ\n`ż\nΓFλn\nD,n|Jδn\ntuDKτ|dσďż\nΩ´\npHn\nm`pFn¨δn\ntuD¯\ndx`ż\nΓpHn\nfdσ.(29)\nProof. The estimate ( 2929) is a consequence of the proof of Proposition 3.23.2combined with\nLemma 3.13.1and the definitions of the Darcy ( 1818) and Fourier ( 1919) fluxes from the coercive\nmatrix ( 1313) and fracture ( 1414) fluxes.\n4 Numerical experiments\nTwo test cases are considered in this Section. The first is based on a manufactured solution\nfor both the entropy and enthalpy-based models on a square domain with no fracture. The\nobjective is to assess and compare the convergence of the schemes for both models and\nboth for centred and upwind approximations of the thermal convection. Two values of the\npermeability are tested to induce either convection dominated or equilibrated diffusion and\nconvection thermal regimes.\nThe second example considers a Discrete Fracture Matrix (DFM) model introduced in [ 33]\nincluding a six fracture network. The setting of the simulation follows the test case proposed\nin [3333] and the physical and numerical behavior of the discrete models are investigated both\nfor the case of a slightly compressible liquid and for the case of a perfect gas.\nIn all these numerical experiments, the HFV discretisations of the non-isothermal flow are\ncombined with the P2conforming Finite Element discretisation of the displacement field.\nThis choice ensures the inf-sup condition ( 2424) for the P2´P0mixed formulation of the\ncontact-mechanics ( 2525). It also satisfies the inf-sup condition between the displacement\nand pressure discrete spaces which prevents potential oscillations of the pressure field at\nshort times in the undrained regime.\nThe coupled nonlinear system is solved at each time step using a fixed-point method on the\nfunction:\ngp,T:pp, Tq ÝÑ\nContact Mechanics\nSolveuhÝÑ\nFlow\nSolvep˜p,˜Tq,\naccelerated by a Newton–Krylov algorithm – which have proven to be efficient within this\ncontext in the isothermal case [ 77]. The stopping criteria is set to 10´10on the relative\n16residual. We refer to [ 1212] for an investigation of various fixed-point algorithms for THM\nmodels.\nAt each iteration of the Newton–Krylov algorithm, the p, Tsub-system is solved using\na Newton–Raphson algorithm and the contact-mechanics is solved using a semi-smooth\nNewton method. In both cases, the stopping criteria is defined by a relative norm of the\nresidual set to 10´10or a scaled maximum Newton increment of 10´10. At each Newton\niteration, the linear system is solved using the sparse direct solver SuperLU version 4.3\nboth for the non-isothermal flow and for the contact-mechanics.\n4.1 Manufactured solution without fractures\nWe investigate in this section the numerical convergence of the schemes ( H-schemeH-scheme ) and\n(S-schemeS-scheme ) for respectively the ( H-modelH-model ) and ( S-modelS-model ) models using the following ana-\nlytical solution\nupx, tq“10´1e´t˜\nx2y2\n´x2y2¸\n,\nppx, tq“e´tsinpxqsinpyq, Tpx, tq“e´tp2´cospxqcospyqq,\non the domain Ω “ p0,1q2m2and time interval p0, tFqwith tF“1 s. The fluid is as-\nsumed incompressible and its specific internal energy is defined by epTq“T. Two values\nK“100Im2andK“Im2(with Ithe identity matrix) of the homogeneous isotropic\npermeability are considered. They are chosen such that the resulting Darcy velocity Vm\ncorresponds to a thermal convection dominated regime in the first case, and to similar orders\nof magnitude for thermal convection and diffusion in the second case. Dirichlet boundary\nconditions are imposed for p,Tanduonp0, tFqˆB Ω and the source terms Gm,Hmand\nFare computed from the analytical solution based on the data set defined in Table 11. The\ndomain Ω is discretised using the first family of triangular meshes from [ 1818] as illustrated\nin Figure 33. Each mesh indexed by mPt1,2,3,4uincludes # M“56ˆ4m´1triangles.\nWe consider a uniform time stepping of p0, tFqwith time step ∆ t“2.10´5s chosen small\nenough to reduce the error due to the time discretisation and focus on the convergence in\nspace.\nThe convergence of the L2space time errors for p, T,uand their gradients are exhibited for\nboth schemes and both permeabilities in Figures 44-55as functions of the mesh step. Both\nthe upwind and centred schemes are considered for the thermal convection. Note that the\ncentred schemes are not presented for K“100Isince they fail to provide a solution due\nto their instability in the convection dominated regime. From Figures 44-55, we first observe\nthat the discretisations of both models provide a very similar convergence behavior for a\ngiven choice of the permeability and of the centred or upwind approximation of the thermal\nconvection terms. Regarding the displacement field, second and first order convergence\nrates are observed in all cases for respectively uand∇u. This is in accordance with the\ncellwise constant reconstruction Πm\nDof the pressure and temperature in the displacement\nfield variational formulation ( 25a25a). The convergence rates for pand∇pare respectively\nroughly 2 and 1 in all cases as could be expected. On the other hand, the converge rates\nforTand∇Tdepend on the approximation of the convection term and on the convection–\ndiffusion regime. For equilibrated convection and diffusion, the centred scheme provides a\nhigher convergence rate of order roughly 2 for Tthan the upwind scheme of order between\n1 and 2. An order 1 is observed on ∇Tfor both the centred and upwind schemes. In the\n17convection dominated regime, the upwind scheme exhibits a convergence rate slightly better\nthan 1 for T, and an order between 0 .5 and 1 for ∇Tslightly better for the entropy-based\nthan for the enthalpy-based model.\nFigure 3. Square domain Ω with its triangular mesh m“2 using 56ˆ4 cells.\nTable 1. Material Properties\nSymbol Quantity Value Unit\nE Young modulus 2 .5 Pa\nν Poisson coefficient 0 .25´\nN Biot modulus 0 .25 Pa´1\nb Biot coefficient 1 .0´\nKs Bulk modulus 2 .0 Pa\nη Fluid viscosity 1 .0 Pa s\nϕ0Initial porosity 4 ´\nΛm Effective thermal conductivity 0 .1 W m´1K´1\nϱ The fluid specific density 1 Kg m´3\nαs The volumetric skeleton thermal dilation coefficient 1 K´1\nαϕ The volumetric thermal dilation coefficient related to the porosity 1 K´1\nTref Reference temperature 1 K\nm0 Average fluid skeleton specific density 0 Kg m´3\nCs The skeleton volumetric heat capacity 0 .5 J m´3K´1\n18p T u ∇p ∇T ∇u\n10´1.810´1.610´1.410´1.210´110´0.810´410´310´210´1\n12\nhL2Relative ErrorsS-based Upwind scheme - K“I\n10´1.810´1.610´1.410´1.210´110´0.810´510´410´310´210´1\n12\nhS-based Centred scheme\n10´1.810´1.610´1.410´1.210´110´0.810´410´310´210´1\n12\nhL2Relative ErrorsS-based Upwind scheme - K“100I\nFigure 4. Convergence of the relative L2errors for the temperature T, pressure p, and\ndisplacement uand their gradients for the discretisation of the ( S-modelS-model ) model and both\nthe centred (for K“I) and upwind (for K“I,100I) schemes.\n19p T u ∇p ∇T ∇u\n10´1.810´1.610´1.410´1.210´110´0.810´410´310´210´1\n12\nhL2Relative ErrorsH-based Upwind scheme - K“I\n10´1.810´1.610´1.410´1.210´110´0.810´510´410´310´210´1\n12\nhL2Relative ErrorH-based Centred scheme\n10´1.810´1.610´1.410´1.210´110´0.810´410´310´210´1100\n12\nhL2Relative ErrorsH-based Upwind scheme - K“100I\nFigure 5. Convergence of the relative L2errors for the temperature T, pressure p, and\ndisplacement uand their gradients for the discretisation of the ( H-modelH-model ) model and both\nthe centred (for K“I) and upwind (for K“I,100I) schemes.\n4.2 2D Discrete Fracture Matrix (DFM) model\nWe consider the DFM model exhibited in Figure 66and introduced in [ 33]. It is defined on\nthe domain Ω “p0,1qˆp0,2qm2and includes a network Γ of six fractures, among which\nfracture 1 made up of two sub-fractures forming a corner, nearly intersecting fractures\nand the non-immersed fracture 5 with one tip lying on the left boundary. The setting of\nthe simulation follows the test case proposed in [ 3333] and consists in three stages on the\ntime intervals I1“ p0, tp1qq,I2“ rtp1q, tp2qqandI3“ rtp2q, tFs. These three stages are\nmonitored by the boundary conditions triggering respectively mechanical, hydraulic and\nthermal driving forces. As exhibited in Figure 77, the displacement u“tp5,´2q10´4m is\nimposed at the top boundary for tą0. At the left boundary, a high pressure p“8.106Pa\nis set for tětp1q, and a low temperature T“285 K is prescribed for tětp2q.\nThe initial pressure and temperature are fixed to p0“105Pa and T0“300 K. The initial\ndisplacement field u0is computed by solving the discrete contact-mechanical problem ( 25a25a)\nat given p0, T0with a zero displacement u“0at the bottom and top boundaries and\n20a free boundary condition at the left and right sides. Throughout the simulation, the\ntop and bottom boundaries are assumed impervious with zero heat flux while the right\nboundary condition imposes a fixed pressure p“105Pa and temperature T“300 K. A\nzero displacement u“0is prescribed at the bottom boundary and the left and right sides\nare kept with homogeneous total stress conditions.\nIn order to compare the ( H-modelH-model ) and ( S-modelS-model ) models and their discretisations on\ndifferent fluid behaviors, we consider in the following two cases corresponding first to a\nslightly compressible liquid and second to a perfect gas. The simulation parameters, not\ndepending on the fluid thermodynamical model, are reported in Table 22together with\nαϕ“ pb´ϕ0qαs,Ks“λ`µ,N“Ks\npb´ϕ0qp1´bq. The times tp1q,tp2q,tFwill be fixed\naccording to the fluid thermodynamical model in such a way that a stationary state is\nroughly reached at the end of each stage.\nThe simulations are performed on a family of 4 uniformly refined meshes indexed by mP\nt0,¨¨¨,3uwith 2855ˆ4mtriangular cells and 88 ˆ2mfracture faces. The finest mesh m“3\nis used for the reference solution in the numerical convergence investigations. Due to the\nthermal convection dominated regime during stage 3, the convection terms are upwinded\nfor the discretisation of both models.\nFigure 6. Two-dimensional domain Ω “p0,1qˆp0,2qm2including the six fracture network\nΓ.\nTable 2. Material Properties common to the liquid and gas test cases.\nSymbol Quantity Value Unit\nE Young modulus 40 GPa\nν Poisson coefficient 0 .15 ´\nF Friction coefficient 0 .5 ´\nb Biot coefficient 0 .65 ´\nK Permeability coefficient˜\n1 0\n0 0.5¸\nˆ10´15m2\nϕ0Initial porosity 0 .1 ´\nd0 Contact aperture 5 .10´4m\nΛm Effective thermal conductivity 2 W m´1K´1\nαs The volumetric skeleton thermal dilation coefficient 1 .5 10´5K´1\nm0 Average fluid skeleton specific density 0 Kg m´3\nCs The skeleton volumetric heat capacity 2 MJ m´3K´1\n21(a) Stage 1 (b) Stage 2\n(c) Stage 3\nFigure 7. Set up of the simulation in terms of initial condition at time t“0 and of boundary\nconditions during each of the three stages corresponding to the time intervals I1“p0, tp1qq,\nI2“rtp1q, tp2qqandI3“rtp2q, tFs. Here 1 bar corresponds to 105Pa.\n4.2.1 Weakly compressible liquid case\nWe consider a weakly compressible liquid with thermodynamical constitutive laws deriving\nfrom a free enthalpy potential. It is characterised by its specific density ϱpp, Tqsuch that\nϱref\nϱpp, Tq“1´pp´prefq\nKf`αfpT´Trefq,\nand its specific internal energy\nepp, Tq“CfT´αf\nϱref´\npp´prefqTref`ppT´Trefq¯\n`pp2´p2\nrefq\n2ϱrefKf,\ngiven the parameters Tref“300 K, pref“105Pa,ϱref“103Kg.m´3,Kf“2.18 GPa,\nαf“2.07 10´4K´1, and Cf“4180 J.Kg´1.K´1. The fluid viscosity is set to η“10´3Pa s.\nThe time intervals for each of the three stages are given by tp1q“100 s, tp2q“200 s and\ntF“5 days. The time stepping is defined by a small initial time step of 0 .1 s in order to\ncapture the undrained regime time scale at the beginning of stage 1 and by the maximum\ntime steps of 5 s for stage 1 and 2 and of 0 .1 day for stage 3.\nFigure 99exhibits the evolution of the contact state (open, contact stick or contact slip)\nalong the fractures at different times during the three stages. At time tą0, due to the\nimposed displacement at the top, most of the fractures switch from open to contact. During\nthe undrained regime, at the very beginning of stage 1, the high increase of the pressure\n(see Figure 88(a)) induces a slip state for most of the fractures in contact, as a result of the\nreduction of the normal surface traction. Toward the end of stage 1, these fractures switch\nback to stick state due to the pressure relaxation (see the evolution of the mean pressure\nin Figure 1010(a)). During stage 2, fractures 1 and 3 switch back to slip state as a result\nof the high pressure front propagation (see Figure 88(b) and 1010(a)) while they partially\n22or totally open during the cold temperature front propagation in stage 3 (see Figures 88(c)\nand1010(b)) as a result of the matrix shrinkage.\n(a) Stage 1 (b) Stage 2\n(c) Stage 3\nFigure 8. Discrete solution obtained with the enthalpy-based model on the finest mesh\nm“3: (a) pressure pat the very beginning of stage 1 ( t“0.1 s), (b) pressure patt“116\ns during stage 2, (c) temperature Tatt“27715 s time during stage 3.\nFigure 1010compares the evolution in time of mean in space variables ( p,pf,T,Tf, and\nscaled df´d0,JuKτ, and ϕ´ϕ0) for the entropy-based and enthalpy-based discrete solutions\nobtained on the finest mesh m“3. A very good match is observed on all variables with\nonly small differences during stage 3 which has been checked to result from the neglected\nterms V¨∇pin the approximate entropy equations of the ( S-modelS-model ) model. When these\nterms are added back, the discrete solutions of both models fully match. Note that the\nlinearisation of the Fourier term is not a significant source of discrepancy for this test case\ndue to the thermal convection dominated regime.\nFigure 1111compares, for the discrete solutions of both models, the convergence of the L2\nerrors in time of mean in space variables as functions of the mesh step h(1/total number\nof fracture faces). The errors are computed w.r.t. the fine mesh m“3 reference solutions\nobtained using the same time stepping and the same model. The observed convergence\nbehaviors of the discretisations of the two models are very similar, with a rate of convergence\nroughly equal to 1 .5.\n23(a) Initial state pp“1 bar , T“300 Kq\n(b) Beginning of Stage 1\n (c) End of stage 1\n(d) Stage 2\n (e) Stage 3\nFigure 9. Contact state (open, contact stick, contact slip) along the fractures at different\ntimes for the discrete solution on the finest mesh m“3 of the enthalpy-based model.\np T ϕ df JuKτpf Tf\n10´2.610´2.410´2.210´210´310´2\n1.5\nh (1/total number of fracture faces)L2ErrorConvergence Rates\n10´2.610´2.410´2.210´210´310´2\n1.5\nh (1/total number of fracture faces)Convergence Rates\nFigure 11. Relative L2in time errors vs. the mesh step h(1/total number of fracture faces)\nof mean in space p,pf,T,Tfdf,JuKτ,ϕdiscrete solutions for both enthalpy-based (full\nlines) and entropy-based models (dash lines). The errors are computed using the fine mesh\nm“3 reference solutions with the same time stepping and model.\n2410´11001011021031041051060123¨106\nTime (s)mean PressureH-p\nH-pf\nS-p\nS-pf\n(a)10´1100101102103104105106285290295300\nTime (s)mean TemperaturesH-T\nH-Tf\nS-T\nS-Tf\n(b)\n10´1100101102103104105106´1´0.500.51¨10´3\nTime (s)Scaled mean variationsH-df\nH-JuKτ\nH-ϕ\nS-df\nS-JuKτ\nS-ϕ\n(c)\nFigure 10. (a) Matrix and fracture mean pressures, (b) matrix and fracture mean tempera-\ntures, and (c) scaled mean JuKτ,df´d0andϕ´ϕ0as functions of time for the solutions on\nthe finest mesh m“3, for the discretisations of the enthalpy-based (H) and entropy-based\n(S) models.\n4.2.2 Perfect gas case\nWe now consider the case of a perfect gas characterised by the following specific enthalpy\nand density\nhpp, Tq“CfT, ϱpp, Tq“Mfp\nRT,\nwith gas specific heat capacity Cf“1000 J.Kg´1.K´1, molar mass Mf“28.9645 10´3\nKg.mol´1and the perfect gas constant R“8.3149 J.mol´1.K´1. The gas viscosity is fixed\ntoη“1.72 10´5Pa s.\nThe time intervals for each of the three stages are given by tp1q“100 s, tp2q“500 s and\ntF“3.005 105s. The time stepping is defined with a single time step during stage 1 since\nthe coupling of the displacement with the pressure is very small during this stage due to the\nhigh gas compressibility (see the pressure solution during stage 1 in Figure 1212(a)). Stage\n2 is initialised with a time step of 10 s and the maximum time steps are set to 100 s for\n25stage 2 and 8000 s for stage 3.\nIn Figure 1313we present the L2-errors in time, vs. the mesh size, for the spacial means of\nrelevant variables. Both schemes provide roughly a convergence order of 1 .5. It can also be\nnoticed that the enthalpy-based discrete solution is not very well captured by the coarser\nmesh, which explains the strong decrease of the errors between the first two meshes. The\nsignificant differences between the convergence plots of the two models result from the high\ndiscrepancy between the enthalpy-based and entropy-based discrete solutions.\nFigure 1414better illustrates the difference between the solutions of both models; we see in\nparticular that the entropy-based model exhibits a temperature variation far below ´15\nK during stage 3 while it should physically be close to 15 K as it is the case for the\nenthalpy-based model. We demonstrate in Figure 1515that this discrepancy is due to the\nneglected terms Vm¨∇pin the approximate entropy equations of the ( S-modelS-model ) model.\nOnce these terms are added, the discrete solutions of both models fully match. This can be\neasily explained by comparing during stage 3 the order of magnitude of the neglected term\nVm¨∇pwith that of ϱmVm¨∇hm. The ratio between both terms is of the order of roughly\n10 for the gas test case while it is roughly 0 .1 for the liquid test case. This is explained by\nthe rather high pressure gradient combined with the low density and heat capacity in the\ngas case compared with the liquid case. We can conclude that the terms Vm¨∇pcannot\nbe neglected in the entropy-based model in the gas case with low specific density.\n(a) Stage 1 (b) Stage 2\n(c) Stage 3\nFigure 12. Discrete solution obtained with the enthalpy-based model on the finest mesh\nm“3: (a) pressure pat the end of stage 1, (b) pressure patt“170 s during stage 2, (c)\ntemperature Tatt“42730 s during stage 3.\n26p T ϕ df JuKτpf Tf\n10´2.610´2.410´2.210´210´510´410´310´210´1100\n1.5\nh (1/total number of fracture faces)L2ErrorConvergence Rates\n10´2.610´2.410´2.210´210´410´310´210´1100\n1.5\nh (1/total number of fracture faces)Convergence Rates\nFigure 13. Relative L2in time errors vs. the mesh step h(1/total number of fracture faces)\nof mean in space p,pf,T,Tfdf,JuKτ,ϕdiscrete solutions for both enthalpy-based (full\nlines) and entropy-based models (dash lines). The errors are computed using the fine mesh\nm“3 reference solutions with the same time stepping and model.\n102103104105012345¨106\nTime (s)mean PressureH-p\nH-pf\nS-p\nS-pf\n(a)102103104105´80´60´40´200\nTime (s)mean TemperatureH-T\nH-Tf\nS-T\nS-Tf\n(b)\nFigure 14. Matrix and fracture (a) mean over-pressures ( p´p0) and (b) mean over-\ntemperatures ( T´T0) as functions of time for the solutions on the finest mesh m“3, for\nthe discretisations of the enthalpy-based (H) and entropy-based (S) models.\n27102103104105012345¨106\nTime (s)mean PressureH-p\nH-pf\nS-p(+Vm¨∇p)\nS-pf(+Vm¨∇p)\n(a)102103104105´15´10´50\nTime (s)mean TemperatureH-T\nH-Tf\nS-T(+Vm¨∇p)\nS-Tf(+Vm¨∇p)\n(b)\nFigure 15. Matrix and fracture (a) mean over-pressures ( p´p0) and (b) mean over-\ntemperatures ( T´T0) as functions of time for the solutions on the finest mesh m“3,\nfor the discretisations of the enthalpy-based (H) model, and the entropy-based model with\nVm¨∇pcorrection (S ( `Vm¨∇p)) discrete models.\n4.2.3 Performances of the nonlinear solver\nFigures 1616and1717exhibit the total numbers of Newton iterations against time for the\nThermo-Hydro and Mechanical models. The total number of time steps is 110 for the\nliquid case in Figure 1616and 87 for the gas case in Figure 1717. We can notice the robustness\nin both cases of the nonlinear solvers w.r.t. the mesh size. Remarkably, in the liquid case, the\nentropy and enthalpy-based discrete models provide similar numbers of iterations through\ntime. In the gas case, the number of Thermo-Hydro Newton iterations gets moderately\nlarger during stage 3 for the entropy-based than for the enthalpy-based discrete models as\na result of a much larger temperature variation for the entropy-based simulation. It has\nbeen checked that adding the Vm¨∇pterms to the entropy-based discrete model gives back\nessentially the same Newton behavior as for the enthalpy-based model.\n28H-Thermo-Hydro H-Mechanics S-Thermo-Hydro S-Mechanics\n10´110010110210310410510601,0002,000\nTime (s)Newton iterationsmesh 3: 182720 cells\n10´110010110210310410510601,0002,000\nTime (s)mesh 1: 11420 cells\nFigure 16. Total numbers of Newton iterations for the Thermo-Hydro and Mechanical\nmodels as a function of time, for the liquid case and both enthalpy-based (H) and entropy-\nbased (S) schemes. (Left) mesh m“3, (right) mesh m“1, with a total number of 110\ntime steps in all cases.\nH-Thermo-Hydro H-Mechanics S-Thermo-Hydro S-Mechanics\n10210310410505001,0001,500\nTime (s)Newton iterationsmesh 3: 182720 cells\n10210310410505001,0001,500\nTime (s)mesh 1: 11420 cells\nFigure 17. Total number of Newton iterations for the Thermo-Hydro and Mechanical models\nas a function of time, for the gas case and both enthalpy-based (H) and entropy-based (S)\nschemes. (Left) mesh m“3, (right) mesh m“1, with a total number of 87 time steps in\nall cases.\n5 Conclusion\nThis work focuses on discretisations of mixed-dimensional THM models preserving energy\nestimates for a general single phase fluid thermodynamical model. Our approach uses a\nFinite Volume scheme for the mass and energy equations with a possible upwinding of the\nconvection terms in order to account for convection dominated regimes. It is combined\nwith a mixed Finite Element discretisation of the contact-mechanics with face-wise con-\n29stant Lagrange multipliers, keeping the dissipative property of the contact terms at matrix\nfracture interfaces. Two formulations of the energy equation are considered and compared\nnumerically. It is built either directly from the energy conservation or obtained from an\napproximate entropy balance equation based on a small Darcy velocity and small tempera-\nture variation assumptions. The Finite Volume discretisation of the entropy-based model,\nand in particular of the non-conservative convection terms, is carefully designed in order\nto preserve the link between both formulations, which in turns guarantees that it satisfies\nan energy estimate. Both discrete models are assessed and compared in terms of conver-\ngence, accuracy and robustness on 2D test cases including a convective dominated regime,\nand either a weakly compressible liquid or highly compressible gas. It is shown that both\napproaches provides similar results in terms of spatial convergence and robustness of the\nnonlinear solver. On the other hand, in the gas case, for low specific density and high\npressure gradient, the terms V¨∇pthat are typically neglected in the entropy balance\napproach must be accounted for in order to provide the physical solution.\nAcknowledgements : the authors are grateful to Andra and BRGM for partially funding\nthis work and to Laurence Beaude, Marc Leconte, Simon Lopez, Antoine Pasteau and Farid\nSmai for fruitful discussions during the elaboration of this work.\nJ. Droniou would like to acknowledge a partial funding by the European Union (ERC\nSynergy, NEMESIS, project number 101115663). Views and opinions expressed are however\nthose of the authors only and do not necessarily reflect those of the European Union or\nthe European Research Council Executive Agency. 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Acta Numerica , 20:569–734, 2011.\n33"    },    {        "title": "2401.12368v1.Bubble_Universe_from_Flat_Spaces.pdf",        "content": "arXiv:2401.12368v1  [gr-qc]  22 Jan 2024Bubble Universe from Flat Spaces\nEduardo Guendelman∗\nDepartment of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel\nFrankfurt Institute for Advanced Studies, Giersch Science Center,\nCampus Riedberg, Frankfurt am Main, Germany\nBahamas Advanced Studies Institute and Conferences,\n4A Ocean Heights, Hill View Circle,\nStella Maris, Long Island, The Bahamas\nJacov Portnoy†\nBahamas Advanced Studies Institute and Conferences,\n4A Ocean Heights, Hill View Circle,\nStella Maris, Long Island, The Bahamas\n(Dated: January 24, 2024)\nAbstract\nWe show by matching two flat spaces one in Minkowski coordinat es ( empty space) and the other\nin Minkowski coordinates after a special conformal transfo rmation (also empty space) through a\nbubble with positive and constant surface tension, that the motion of the bubble is hyperbolic.\nIf the surface tension is very big the initial size of the bubb le is as small as we wish, so that we\ncan indeed obtain an infinite universe out of empty spaces. Th e induced space in the bubble is de\nSitter type.\n∗Electronic address: guendel@bgu.ac.il\n†Electronic address: jacovportnoy@gmail.com\n1I. INTRODUCTION\nThere are many interesting situations where space time is flat every where, except for\nsome lines or surfaces, may be the most famous example being the co smic string [1], where\nspace is flat everywhere except on a line, the string, where a conica l singularity resides.\nNaturally one may ask if instead of a line, we may have a surface match ing two flat\nspaces, expressed nevertheless in different coordinate systems . This seems interesting and\nfundamental because it may shed light into the vacuum structure o f the theory, and finding\nnovel ways to connect different vacuum states. The resulting solu tions also hint to new ways\nto generate universes from nothing when continued to euclidean sp ace.\nWe will study here the matching of Flat Minkowski space and Flat Minko wski space after\na special conformal Transformation and then compare with result s obtained from consider-\nations of string matter and of braneworlds in the context of string theories with dynamical\ntension.\nII. FLAT MINKOWSKI SPACE AND FLAT MINKOWSKI SPACE AFTER A\nSPECIAL CONFORMAL TRANSFORMATION\nThe flat spacetime in Minkowski coordinates is,\nds2\n1=ηαβdxαdxβ(1)\nwhereηαβis the standard Minkowski metric, with η00= 1,η0i= 0 and ηij=−δij. This\nis of course a solution of the vacuum Einstein /acute.ts1s equations.\nWe now consider the conformally transformed metric\nds2\n2= Ω(x)2ηαβdxαdxβ(2)\nwhere the conformal factor coincides with that obtained from the special conformal trans-\nformation\nx′µ=(xµ+aµx2)\n(1+2aνxν+a2x2)(3)\nforacertainDvector aν. which givesΩ2=1\n(1+2aµxµ+a2x2)2Insummary, wehavetwosolutions\nfor the Einstein /acute.ts1s equations, g1\nαβ=ηαβand\n2g2\nαβ= Ω2ηαβ=1\n(1+2aµxµ+a2x2)2ηαβ (4)\nThese two spaces can be matched at the surfaces where Ω2= 1, where Ω2=\n1\n(1+2aµxµ+a2x2)2. We will consider the cases where a2/negationslash= 0. Let us by consider the case\nwhereaµis time like (for aµspacelike similar results are obtained), then without loosing\ngenerality we can take aµ= (A,0,0,...,0). The two solutions of Ω2= 1 are , considering the\ncases Ω = 1 and Ω = −1 ,\n2aµxµ+a2x2= 0 (5)\nand\n2+2aµxµ+a2x2= 0 (6)\nThen for aµ= (A,0,0,...,0), the above condition implies that,\n(2aµxµ+a2x2)(2+2aµxµ+a2x2) = (2At+A2(t2−x2))(2+2At+A2(t2−x2)) (7)\nThis condition, if A/negationslash= 0 implies then the more specific condition,\nx2\n1+x2\n2+x2\n3.....+x2\nD−1−(t+1\nA)2=−1\nA2(8)\nbya similar analysis, the other surface that satisfies the conforma l transformation factor\nis 1, is given by\nx2\n1+x2\n2+x2\n3.....+x2\nD−1−(t+1\nA)2=1\nA2(9)\nsince (7) represents a surface that does not exists for all times, and it can represent propa-\ngation with speed greater than light, we will restrict to the surface (9) which does exist for\nall times, and represents hyperbolic motion with time like propagation , in the next section\nwe will look at the surface tension as given by the Israel matching co nditions.\nIII. SURFACE TENSION AS GIVEN BY THE ISRAEL MATCHING CONDI-\nTIONS IN FOUR SPACE-TIME DIMENSIONS\nLet us consider now D= 4, defining r=/radicalbig\nx2\n1+x2\n2+x2\n3, then, taking the space (4)\noutside, the standard Israel matching conditions [2] apply,\nK+\nθθ−K−\nθθ= 4πGσr2(10)\n3K+\nθθ=1\n2ξµ∂µ(r2Ω2) (11)\nK−\nθθ=1\n2ξµ∂µ(r2) (12)\nK+\nθθ−K−\nθθ=1\n2[ξµ∂µ(r2Ω2)−ξµ∂µ(r2)]/vextendsingle/vextendsingle/vextendsingle\nΩ2=1=1\n2ξµΩ2∂µ(r2)+1\n2ξµr2∂µ(Ω2)−1\n2ξµ∂µ(r2)/vextendsingle/vextendsingle/vextendsingle\nΩ2=1\n(13)\n=1\n2ξµr2∂µΩ2/vextendsingle/vextendsingle/vextendsingle\nΩ2=1= 4πGσr2(14)\nΩ2=1\n(1+2aµxµ+a2x2)2(15)\nξµ=∂µ(Ω2)/radicalbig\n|(∂µ(Ω2)(∂µ(Ω2)|(16)\nσ=1\n8π/radicalBig\n|(∂µ(Ω2)(∂µ(Ω2)| (17)\nσ=|A|\n4π/radicalBig\n|1+2aµxµ+x2| (18)\nand using (5) or (6) we obtain\nσ=|A|\n4π\nNotice however that the surface (8) moves with velocity bigger tha n light, while (9) moves\nwith velocity less than light, so, we must choose (9) as the physical a lternative. Using\nequation σ=|A|\n4πwe can express the motion of the surface separating the two vacu um\nsolutions Ω2= 1 in terms of the surface tension,\nx2\n1+x2\n2+x2\n3.−(t+1\nA)2=1\n16π2σ2(19)\nso in the limit of very large surface tensions we obtain that the minimum radius of the\nbubble becomes very small. This shows similar features with results in [3 ], but here there\nare stronger, since we do not just talk about Universes out of almo st empty space, but\nUniverses out of exactly empty space, already at the classical leve l.\nIV. EUCLIDEAN EXTENSION AND THE QUANTUM CREATION OF A BABY\nUNIVERSE FROM NOTHING\nThe surface Ω2= 1, solved according to (9) allows an Euclidean extension, by defining\nan Euclidean time ( tE+1/A) =−i(t+1/A), which replaced into Ω2= 1 gives the spherical\n4euclidean region\nx2\n1+x2\n2+x2\n3.....+x2\nD−1+(tE+1\nA)2=1\nA2(20)\nThe euclidean and real time regions defined above of the surface Ω2= 1 can be smoothly\nmatched at ( tE+ 1/A) = (t+ 1/A) = 0. The quantum mechanical interpretation of the\neuclidean region is the tunneling region that eventually leads by match ing to the creation of\nthe baby universe. The first surface (8) cannot be extended to E uclidean space and it starts\natt= 0 from zero size in real space.\nV. DISCUSSION AND CONCLUSIONS\nWe have seen that the matching between two patches of flat space , one described by flat\nspace in standard Minkowski coordinates and the other by flat spa ce obtained after a special\nconformal transformation from the standard Minkowski coordin ates. The two spaces can\nbe made to match at the locations where the conformal transform ation equals the identity,\nwhich gives two different possible motions for the surface separatin g the two spaces, we\nchoose the solution which is manifestly time like, we can calculate the su rface tension of the\nwall separating the two spaces and it turns out to be a constant.\nThese type of situation, where we consider two flat spaces, one in M inkowski pace and\nthe other in flat space obtained after a special conformal transf ormation from the standard\nMinkowski coordinates and the surfaces where the conformal tr ansformation is equal to\none plays a special role in the case of string theories with dynamical t ension, and these\nhypersurfaces are in higher dimensions in this case, these surface s appear as surfaces where\nthe string tensions go to infinity causing an effective braneworld sce nario [4]. The approach\nhere is a bit different, we do not rely on any /dieresis.ts1microscopic /dieresis.ts1theory, like string theory with\ndynamical tension, and just simply match the two given spacetimes, obtaining nevertheless\nvery similar situations in this way as well. Notice that although we have s tarted just from\nflat spaces, the induced space in the bubble is a de Sitter type space time.\n5VI. ACKNOWLEDGEMENTS\n, We want to thank The Bahamas Advanced Study Institute and Con ferences (BASIC)\nand Ben Gurion University for support of this project.\n[1] Cosmic Strings A. Vilenkin, Published in: Phys.Rev.D 24 (1981) 2082-2089\n[2] Singular hypersurfaces and thin shells in general relat ivity, W. Israel, Published in: Nuovo\nCim.B 44S10 (1966) 1, Nuovo Cim.B 48 (1967) 463 (erratum), Nu ovo Cim.B 44 (1966) 1\n[3] Universes out of almost empty space Stefano Ansoldi, Edu ardo I. Guendelman, Published in:\nProg.Theor.Phys. 120 (2008) 985-993 •e-Print: 0706.1233 [gr-qc], Almost classical creation of\na universe, E.I. Guendelman, J. Portnoy, Published in: Mod. Phys.Lett.A 16 (2001) 1079-1087\n[4] Light like segment compactification and braneworlds wit h dynamical string tension Eduardo\nGuendelman, Published in: Eur.Phys.J.C 81 (2021) 10, 886 •e-Print: 2107.08005 [hep-th],\nFrom Homogeneous and Isotropic Universes to Braneworlds wi th Dynamical Tension Strings,\nE.I. Guendelman, Published in: Phys.Part.Nucl. 54 (2023) 5 , 913-918, Life of the homogeneous\nand isotropic universe in dynamical string tension theorie s, E.I. Guendelman, Published in:\nEur.Phys.J.C 82 (2022) 10, 857, Relieving string tension by making baby universes in a dy-\nnamical string tension braneworld model, Eduardo I. Guende lman, Zeeya Merali, Published in:\nInt.J.Mod.Phys.D 31 (2022) 14, 2242014 •e-Print: 2205.05261 [hep-th], Brane world creation\nfrom flat or almost flat space in dynamical tension stringtheo ries, E.I. Guendelman, J. Portnoy,\nPublished in: Eur.Phys.J.C 82 (2022) 4, 336 •e-Print: 2202.10457 [hep-th]\n6"    },    {        "title": "2401.12378v1.Shallower_radius_valley_around_low_mass_hosts_provides_evidence_for_icy_planets_or_collisions.pdf",        "content": "MNRAS 000, 1–18 (2023) Preprint 24 January 2024 Compiled using MNRAS L ATEX style file v3.0\nShallower radius valley around low-mass hosts provides evidence for icy\nplanets or collisions\nCynthia S. K. Ho1,2★, James G. Rogers3, Vincent Van Eylen1,2, James E. Owen4and Hilke E. Schlichting3\n1Mullard Space Science Laboratory, University College London, Dorking, RH5 6NT, UK\n2The Centre for Planetary Sciences at UCL/Birkbeck, London, UK\n3Department of Earth, Planetary, and Space Sciences, The University of California, Los Angeles, 595 Charles E. Young Drive East, Los Angeles, CA 90095, USA\n4Imperial Astrophysics, Department of Physics, Imperial College London, Prince Consort Rd, London, SW7 2AZ, UK\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nThe radius valley, i.e., a dearth of planets with radii between 1.5 and 2 Earth radii, provides insights into planetary formation\nandevolution.UsinghomogenouslyrevisedplanetaryparametersfromKepler1-minuteshortcadencelightcurves,weremodel\ntransits of 72 small planets orbiting low-mass stars, improving the precision and accuracy of planet parameters. By combining\nthissample witha similarsample ofplanetsaround higher-massstars, wedetermine thedepth oftheradius valleyas afunction\nof stellar mass. We find that the radius valley is shallower for low-mass stars compared to their higher mass counterparts. Upon\ncomparison,wefindthattheoreticalmodelsofphotoevaporationunder-predictthenumberofplanetsobservedinsidetheradius\nvalleyforlow-massstars:withdecreasingstellarmass,thepredictedfractionofplanetsinsidethevalleyremainsapproximately\nconstant whereas the observed fraction increases. We argue that this provides evidence for the presence of icy planets around\nlow-mass stars. Alternatively, planets orbiting low-mass stars undergo more frequent collisions. We predict that more precise\nmass measurements for planets orbiting low mass stars would be able to distinguish between these scenarios.\nKey words: planets and satellites: composition – planets and satellites: formation – planets and satellites: physical evolution –\nplanets and satellites: terrestrial planets\n1 INTRODUCTION\nThe ‘radius valley’ is an observed paucity of planets between about\n1.5 and 2 Earth radii. Its properties provide insights into planetary\nformationandevolutionmechanisms.Onegroupoftheoriessuggests\nthat small, close-in planets undergo thermally-driven mass loss sce-\nnarios,includingphotoevaporation(e.g.Owen&Wu2013;Lopez&\nFortney2013;Owen&Wu2017),andcore-poweredmassloss(e.g.\nGinzburg et al. 2018; Gupta & Schlichting 2019, 2020), resulting in\na separation between planets without substantial atmospheres, and\nplanets that have retained them. Alternatively, it is suggested that\nunder the late gas-poor formation scenario, planets below the valley\nhave formed after most of the gas has dissipated and hence do not\nhave an atmosphere (e.g. Lopez & Rice 2018).\nTheradiusvalleyaroundFGKstarshasbeenextensivelyobserved\n(e.g. Fulton et al. 2017; Van Eylen et al. 2018; Fulton & Petigura\n2018). These observations all show a separation between planets\nabovethevalley,whicharethoughttopossessasubstantialhydrogen-\nhelium (H-He) atmosphere, and planets below the valley, which are\nexpected to be bare rocky cores. These studies have also found that\nthe radius valley shifts to a larger radius for shorter orbital periods,\nwhichisconsistentwiththermallydrivenmasslossmodels,aslarger\nplanetscouldlosetheiratmospheresclosertotheirhosts.Theradius\nvalley is also found to shift to a larger radius for more massive\n★E-mail: sze.ho.20@ucl.ac.ukhost stars, which again agreeswith both photoevaporation and core-\npowered mass loss models (e.g. Berger et al. 2020b; Petigura et al.\n2022; Ho & Van Eylen 2023).\nHowever, the picture is different when it comes to M-dwarf host\nstars.Cloutier&Menou(2020)studiedtheradiusvalleyaroundlow-\nmass stars, and found that the valley location tends to larger planet\nradii for larger planet-star separation, opposite of the dependence\nfor FGK stars. They suggested that this is evidence for such planets\nto undergo gas-poor formation (e.g. Lopez & Rice 2018; Cloutier\n& Menou 2020). Yet, Van Eylen et al. (2021) analysed small plan-\nets orbiting M-dwarf stars and found the valley location tending to\nsmallerplanetradiiforlargerplanet-starseparation,similartothatof\nFGK stars, contradicting the findings of Cloutier & Menou (2020).\nBonfanti et al. (2023) also similarly analysed the orbital period de-\npendencefortheM-dwarfradiusvalley,andfoundasimilar,butmuch\nweaker dependence, than Van Eylen et al. (2021). Recently, Luque\n& Pallé (2022) investigated the distribution of 43 small transiting\nplanets around M dwarfs with mass measurements and argued that\nthere is no orbital period dependence of the radius valley, but rather\nplanets can be clearly separated into three distinct groups in bulk\ndensity space: rocky planets, ‘water worlds’ with 50% water, and\n‘puffy’ sub-Neptunes with sizeable atmospheres. However, Rogers\net al. (2023) counter-argued that thermally-driven mass loss mecha-\nnisms can still explain the distribution of small planets around low-\nmass stars. If a ‘boil-off’ scenario existed, i.e. a significant amount\nof H-He atmosphere is removed during disk dispersal, the resulting\nplanetdistributionmatcheswellwiththatofthe‘waterworlds’iden-\n©2023 The AuthorsarXiv:2401.12378v1  [astro-ph.EP]  22 Jan 20242 C. S. K. Ho et al.\ntified in Luque & Pallé (2022). Even if the ‘boil-off’ process does\nnothappen,andtheinitialatmosphericlossprocessesarenotknown,\nthe resulting distribution is consistent with the overall arrangement\nnotedinLuque&Pallé(2022).Thisisbecausetheatmosphericmass\nfraction scales with the planet mass, when atmospheric mass-loss is\ntakenintoaccount(Rogersetal.2023).Hence,itisuncertainwhether\nthe difference in the valley’s dependence with orbital period can be\nexplained by the existence of such ‘water worlds’. These discrepan-\nciesinthecharacteristicsoftheradiusvalleyforhoststarsofdifferent\nmasses,aswellasthevariousviewsandinterpretationsoftheradius\nvalley for low-mass stars, suggest the possibility that stellar mass\nplays a role in determining the evolutionary pathways of planets.\nHo&VanEylen(2023)foundthatcarefulfittingofshort-cadence\ntransit light curves to obtain precise planetary radii caused some in-\nferredplanetpropertiestochange(seealsoHernandezCameroetal.\n2023),revealingadeeper,moresharply-determinedradiusvalley.Ho\n&VanEylen(2023)showedthattheradiusvalleypropertiesdepend\non stellar mass, but focused on planets orbiting FGK stars. In this\nwork, we extend the sample by conducting a similar, careful and\nhomogeneous analysis of Keplershort cadence transit observations\ntostudytheradiusvalleyforplanetsorbitinglow-massM-typestars.\nWe compare the observed radius valley for stars of different masses\nand attempt to reproduce the observed radius valley characteristics\nusing photoevaporation models.\nInSection2,wedescribetheplanetsampleandthemethodsusedto\nfitthe Keplertransitlightcurvestoobtainnewplanetaryparameters.\nIn Section 3, we analyse the observed radius valley’s change with\nstellarmassandcomparethemwiththeoreticalmodels.Thephysical\nimplications of the results are discussed in Section 4. Finally, we\nprovide conclusions and outlook in Section 5.\n2 PROPERTIES OF PLANETS ORBITING AROUND\nLOW-MASS STARS\n2.1 Sample of new planets orbiting low-mass stars\nIn this work, we study planets around low-mass stars, using planets\nin the sample from the California- KeplerSurvey (CKS) ‘extended\nmass’(CXM)sample(Petiguraetal.2022),whichhasastellarmass\nrange between 0.4and1.51𝑀⊙. The CXM is a filtered sample of\nthe union of the Kepler Data Release 25 stellar properties catalogue\n(Mathur et al. 2017), the Gaia DR2 catalogue (Gaia Collaboration\netal.2018),andtheGaia-KeplerStellarPropertiesCatalogue(Berger\net al. 2020a).\nWeselectplanetsthatmeetthecriteriadefinedinHo&VanEylen\n(2023), i.e.\n(i)1≤𝑅𝑝/𝑅⊕≤4, where𝑅𝑝is the planetary radius,\n(ii)1≤𝑃/d≤100, where𝑃is the orbital period,\n(iii) at least 6 months of Kepler1-minute short cadence data\navailable, and\n(iv) without transit timing variations (TTV) measurements, as\nreported in Holczer et al. (2016).\nThis gives us 72 planets not previously refitted by Ho & Van Eylen\n(2023) for which we homogeneously refit the planetary transits to\nupdate the planetary parameters.\n2.2 Transit Fitting\nSince Kepler1-minute short cadence light curves produce superior\nplanetary radius precision due to their higher sensitivity towardschanges in the transit shape (e.g. Petigura 2020; Ho & Van Eylen\n2023; Hernandez Camero et al. 2023), we refit the 72 planets that\nhave not been studied in Ho & Van Eylen (2023) using Kepler1-\nminutecadencelightcurves,toobtainupdatedplanetaryparameters.\nWe follow the transit fitting method used in Ho & Van Eylen\n(2023)andsummariseitsmaincharacteristicshere.Westartbyper-\nforming data reduction and light curve detrending, then correcting\nthe light curves with the ‘radius correction factor’ (Furlan et al.\n2017) to account for stellar multiplicities. We then use exoplanet\n(Foreman-Mackeyetal.2021)togenerateatransitlightcurvemodel\nwith quadratic stellar limb darkening, and run a Hamiltonian Monte\nCarlo(HMC)algorithmimplementedin PyMC3(Salvatieretal.2016)\nwithaGaussianProcess(GP)model(Rasmussen&Williams2006)\nincluded to account for undetected noise in the light curves. We ini-\ntialise the HMC chains in the same way and apply the same priors\nas in Ho & Van Eylen (2023). Similarly, we fit for the following\nparameters: orbital period ( 𝑃), transit mid-time ( 𝑡0), planet-to-star\nradius ratio ( 𝑅𝑝/𝑅★), impact parameter ( 𝑏), eccentricity ( 𝑒), argu-\nment of periapsis ( 𝜔), stellar density ( 𝜌★), and two quadratic stellar\nlimb darkening parameters ( 𝑢0and𝑢1).\n2.3 Revised planetary parameters\nWe report the revised planetary parameters of all 72 planets in Ta-\nble 1. For subsequent analysis of the radius valley, we only consider\nthose planets whose properties are well-known. Following Ho &\nVanEylen(2023),we,therefore,applythefollowingparametercon-\nstraints:\n(i) measurement uncertainty on the planet radius ( 𝜎𝑅𝑝)≤10%,\n(ii) radius correction factor (RCF) ≤5%, as reported in Furlan\net al. (2017); this is to account for stellar flux from nearby stars\ncontaminating the light curves (Furlan et al. 2017), and\n(iii) at least 3 transits in the Keplershort cadence data.\nThis results in a sample of 69 planets whose properties are ho-\nmogeneously determined and precisely and accurately known. We\nthencombinetheseplanetswithplanetswithsimilarcharacteristics,\norbitinghigher-massstars,andinclude375suchplanetswithparam-\neters taken from Ho & Van Eylen (2023). This total sample of 444\nplanets is shown in Fig. 1.\nThe sample of planets with properties newly determined in this\nworkprovidesanimportantcomplementtothesamplestudiedbyHo\n&VanEylen(2023).InFig.2,weshowbothsamplesasafunctionof\nstellarmass,highlightinghowthese69newplanetssignificantlyex-\ntendthehomogeneoussamplearoundlow-massMdwarfs.InFig.3,\nwe compare the noise properties of both samples, showing that the\nuncertaintiesinplanet-to-starradiusratio,planetradius,stellarmass\nand radius, are all broadly similar.\n3 RADIUS VALLEY OF LOW-MASS STARS\n3.1 Shallower observed radius valley for lower mass stars\nWenowinvestigatethepropertiesoftheradiusvalleyasafunctionof\nstellarmass.Asastartingpoint,wedividethesampleintofourmass\nbins:𝑀★/𝑀⊙<0.7,0.7≤𝑀★/𝑀⊙<0.9,0.9≤𝑀★/𝑀⊙<1.1,\nand𝑀★/𝑀⊙≥1.1. The resulting sample in those four stellar mass\nbins is shown as a function of planet radius and orbital period in\nFig.4.Onthesamefigure,wealsoshowtheboundariesoftheradius\nvalley, according to the 3-dimensional equation determined in Ho\nMNRAS 000, 1–18 (2023)Shallower radius valley around low-mass hosts 3\nTable 1.TableshowingMCMCparameterestimatesoforbitalperiod( 𝑃),planetary-to-stellar-radiusratio( 𝑅𝑝/𝑅★),planetaryradius( 𝑅𝑝),stellarradius( 𝑅★),\nstellar mass (𝑀★), number of transits in the fitted light curve ( 𝑁tr), of the 72 planets refitted in this work. We convert 𝑅𝑝/𝑅★into𝑅𝑝using𝑅★values from\nPetigura et al. (2022). 𝑀★values are also taken from Petigura et al. (2022). ‘Flag’ refers to whether the planet passes the filter checks and is included in the\nsmaller subset for further analyses (1 for true and 0 for false). Only the first 5 planets are shown here; the full table is available online in a machine-readable\nformat.\nKOI Kepler name 𝑃(d) 𝑅𝑝/𝑅★𝑅𝑝(𝑅⊕)𝑅★(𝑅⊙)𝑀★(𝑀⊙)𝑁trFlag\nK00156.01 Kepler-114 c 8.041329±0.000005 0.0229±0.0004 1.80+0.06\n−0.060.72+0.02\n−0.020.73+0.03\n−0.03152 1\nK00156.02 Kepler-114 b 5.188560±0.000004 0.0172±0.0003 1.35+0.05\n−0.050.72+0.02\n−0.020.73+0.03\n−0.03234 1\nK00156.03 Kepler-114 d 11.776137±0.000005 0.0364±0.0005 2.85+0.09\n−0.090.72+0.02\n−0.020.73+0.03\n−0.03108 1\nK00222.01 Kepler-120 b 6.312501±0.000005 0.0328±0.0007 2.54+0.09\n−0.090.71+0.02\n−0.020.70+0.03\n−0.0357 1\nK00222.02 Kepler-120 c 12.794560±0.000012 0.0267±0.0007 2.07+0.08\n−0.080.71+0.02\n−0.020.70+0.03\n−0.0327 1\n... ... ... ... ... ... ... ... ...\nP (d)131030100\nM (M)\n0.50.711.4Rp (R\n)\n124\nH23\nNew planets\nFigure 1. Planetary radius-orbital period-stellar mass plot of the 444 small\nplanetsanalysedinthiswork,whichincludes375withparametersfromHo&\nVan Eylen (2023) (red) and 69 newly refitted (blue) in this work. The radius\nvalley boundary determined in Ho & Van Eylen (2023) is indicated by the\ngrey planes.\n& Van Eylen (2023), which is defined by a support-vector machine\nseparating the planet population into two groups:\nlog10\u0000𝑅𝑝/𝑅⊕\u0001=𝐴log10(𝑃/d)+𝐵log10\u0000𝑀★/𝑀⊙\u0001+𝐶(1)\nwith𝐴=−0.09,𝐵=0.21,𝐶=0.35,andsetting𝑀★tothemedianof\nthehoststarsmasseswithineachmassrange.Wetake 𝐶upper=0.42\nand𝐶lower=0.29from Ho & Van Eylen (2023), which are deter-\nmined by the parallel lines passing through the supporting vectors.\nHere, we have plotted the valley as determined by Ho & Van Eylen\n(2023)—withoutthecontributionfromthelower-massstars.When\nincludingthelatterandrefittingequation1,wefind 𝐴=−0.10+0.02\n−0.03,\n𝐵=0.17+0.14\n−0.09,𝐶=0.36+0.02\n−0.02, which are in agreement with Ho\n& Van Eylen (2023) well within 1𝜎. As one of the main goals of\nthis work is precisely to test whether the radius valley has the same\nphysical origin around low-mass stars as around their higher-mass\ncounterparts, we choose here not to use these updated radius valley\nparameters.\nVisuallyinspectingthedistributionofplanetradiiandorbitalperi-\nodsinFig.4,weseethatforhighermassstars,therearefewerplanets\nlyingwithintheboundariesoftheradiusvalley.Sincetheradiusval-\n0.4 0.6 0.8 1.0 1.2 1.4\nM (M)\n020406080100Number of planetsH23\nNew planetsFigure 2. Stacked histogram comparing the distribution of the stellar mass\nof the sample of the new planets used in this work, and planets in Ho & Van\nEylen (2023).\nleyisorbital-perioddependent(e.g.VanEylenetal.2018;Ho&Van\nEylen 2023), we divide the plot into multiple tilted bins, as shown\nin Fig. 4, and following the procedure in Ho & Van Eylen (2023),\nshift the planets along the slope of the valley to an equivalent radii\nat𝑃=10d (𝑅𝑝,10). We then plot a histogram of these equivalent\nradii as shown in Fig. 5, separated by different stellar mass ranges.\nAgain, following the procedure in Ho & Van Eylen (2023), we fit\nthe histogram with a curve determined using a Gaussian mixture\nmodel with two components, which is independent of the width and\nboundaries of the histogram bins, and fixes the bimodal distribution\nof small planets. This equation is given by\n𝜆(𝑅𝑝,10)=𝑤1×1√\n2𝜋𝜎1exp \n−\u0000𝑅𝑝,10−𝜇1\u00012\n2𝜎2\n1!\n+𝑤2×1√\n2𝜋𝜎2exp \n−\u0000𝑅𝑝,10−𝜇2\u00012\n2𝜎2\n2!\n,(2)\nwith𝑤1+𝑤2=1. To account for measurement uncertainties, we\nfollowthemethodusedinRogers&Owen(2021)andfittheGaussian\nmixture model as an inhomogeneous Poisson point process. The\nMNRAS 000, 1–18 (2023)4 C. S. K. Ho et al.\n0 2 4 6 8 10\nM (%)\n0255075100125150Number of planetsH23\nNew planets\n0 2 4 6 8 10\nRp/R (%)\n020406080100Number of planetsH23\nNew planets\n0 2 4 6 8 10\nRp (%)\n020406080100Number of planetsH23\nNew planets\n0 2 4 6 8\nR (%)\n050100150200Number of planetsH23\nNew planets\nFigure 3. SameasFig.2,butforpercentageuncertaintyin 𝑀★(𝜎𝑀★),planet-to-starradiusratio( 𝜎𝑅𝑝/𝑅★),planetaryradius( 𝜎𝑅𝑝),andstellarradius( 𝜎𝑅★).\nlikelihood function is given by\n𝐿=e−ΛΛ𝑁pl\n𝑁pl!𝑁plÖ\n𝑖=1𝜆\u0000𝑅𝑝,10,𝑖\u0001\nΛ(3)\nwhereΛ=𝐴∫\n𝜆\u0000𝑅𝑝,10\u0001d𝑅𝑝,10is the underlying occurrence rate,\nwith𝐴being an arbitrary constant which we also fit for, 𝑁plis the\nnumber of observed planets, and 𝜆\u0000𝑅𝑝,10,𝑖\u0001/Λis the probability\ndensity of observing planet 𝑖at𝑅𝑝,10,𝑖. We infer the posteriors for\n𝐴,𝜇1,𝜇2,𝜎1,𝜎2,and𝑤1byinputtingthelog-likelihood, ln(𝐿),into\naMarkovChainMonteCarlo(MCMC)algorithm,usingthePython\npackage emcee(Foreman-Mackeyetal.2013)with100walkersand\n5000 steps. The median and ±1𝜎uncertainties are shown on the\nfitted curves.\nWeobservetwofeaturesinthishistogram.Thefirstoneisthatthe\nradius valley gets deeper for higher stellar mass. We use the metric\ndefined in Ho & Van Eylen (2023):\n𝐸avg=(𝐸SE+𝐸SN)/2 (4)\nto compare the depth of the radius valley, with\n𝐸SE=𝑁super-Earth, peak/𝑁valley (5)\nand\n𝐸SN=𝑁sub-Neptune, peak/𝑁valley (6)where𝑁super-Earth, peak and𝑁sub-Neptune, peak refer to the number\nof planets at the super-Earth and sub-Neptune Gaussian peaks, re-\nspectively, and 𝑁valleyis the lowest number of planets between the\ntwoGaussianpeaks.Theresulting 𝐸valuesfordifferentstellarmass\nrangesarereportedinTable2andplottedinFig.6.FromFig.6,wesee\nthat both𝐸avgand𝐸SNstrictly increases with stellar mass, whereas\n𝐸SEgenerally increases with stellar mass except at the transition at\naround 0.9𝑀⊙;thismayhoweverbeduetothedecreasingdetection\nefficiencyofsmallplanetsforhighermassstars.Weperformalinear\nfit to the𝐸values as a function of stellar mass. The resulting slopes\nshow a clear increasing trend with a slope of 5.36+3.19\n−3.23for𝐸avg,\n4.79+3.44\n−2.19for𝐸SE, and 4.91+3.05\n−3.37for𝐸SN. The p-values from the\nWald Test with t-distribution of the test statistic, as implemented in\nscipy.stats.linregress (Virtanenetal.2020),forthelinearfit\nare0.02,0.09and0.01,respectively,indicatingastrongrejectionof\nthe null hypothesis of no linear correlation between the two param-\neters. We do not expect that observational biases could reproduce\nthese trends. The planet detection efficiency should decrease with\nincreasing stellar radius (and therefore mass). Hence, the deeper ra-\ndiusvalleyforhigher-massstarsshouldnotbecausedbyundetected\nplanets. The relative uncertainty on stellar radius is also similar for\nlow-mass stars as for their higher-mass counterparts (see Figure 3).\nThesecondfeatureweobserveisthatthesub-Neptunepeaksshift\ntohigher𝑅𝑝forhighermassstars,whereasthesuper-Earthpeaksdo\nMNRAS 000, 1–18 (2023)Shallower radius valley around low-mass hosts 5\n1 3 10 30 100\nP (d)1234Rp (R)\nM/M<0.7\n1 3 10 30 100\nP (d)1234Rp (R)\n0.7M/M<0.9\n1 3 10 30 100\nP (d)1234Rp (R)\n0.9M/M<1.1\n1 3 10 30 100\nP (d)1234Rp (R)\nM/M 1.1\nFigure 4. Radius-orbitalperiodplotofplanetsanalysedinthiswork,separatedbyhoststarmass.Theblackdashedlinesindicatetheradiusvalleyboundaryas\ndefined by the parameters in Ho & Van Eylen (2023). The grey dashed lines divide the plot into multiple orbital-period dependent bins, which are then used to\nplot the adjusted histogram in Fig. 5.\nTable 2.𝐸values of the radius valley for planets with different host star\nmasses. The𝐸metrics are calculated from equations 4-6.\n𝑀★(𝑀⊙)𝑁planets𝐸avg𝐸SE𝐸SN\n𝑀★<0.7 63 1.25+0.55\n−0.171.01+0.40\n−0.011.50+0.71\n−0.32\n0.7≤𝑀★<0.9141 2.72+1.10\n−0.742.22+0.95\n−0.633.23+1.25\n−0.84\n0.9≤𝑀★<1.1148 2.87+1.13\n−0.761.91+0.81\n−0.533.83+1.45\n−0.98\n𝑀★≥1.1 92 4.62+2.80\n−1.624.43+2.77\n−1.584.80+2.84\n−1.66\nnotincreaseasmuch.Fig.7showstheradiusofthesuper-Earthand\nsub-Neptune peaks as a function of stellar mass. Again, we perform\nalinearfittotheplanetradiiatthetwopeaksasafunctionofstellar\nmass,andfind 𝑅𝑝,peak=0.13+0.16\n−0.19𝑀★+1.25+0.19\n−0.17forsuper-Earths,\nand𝑅𝑝,peak=1.09+1.30\n−0.29𝑀★+1.58+0.65\n−0.31for sub-Neptunes. The p-\nvalues of 0.45 and 0.005, respectively indicate a strong rejection of\nthenolinearcorrelationhypothesisforsub-Neptunes,butnotforthe\ncase of super-Earths. This effect is also confirmed by other obser-vational studies (e.g. Petigura et al. 2022), which analysed a similar\nsampleofplanets,albeitusing Keplerlong-cadenceobservationsand\ndifferent transit modelling than the ones used here.\n3.2 Photoevaporation models predict a narrower radius valley\nfor lower mass stars\nOwen & Schlichting (2023) suggested that although the observed\nsuper-Earth population likely contains significant fractions of plan-\nets that were stripped by both photoevaporation and core-powered\nmass-loss, photoevaporation was likely responsible for the final at-\nmosphericstrippingofplanetslocatedattheedgeoftheradiusvalley\ntoday, determining the final features observed today. Therefore, we\nonly perform our subsequent comparison between observations and\nphotoevaporation models.\nWe model planets undergoing thermal cooling and photoevapo-\nrative mass loss of their hydrogen-dominated atmospheres with the\nsemi-analyticframeworkofRogers&Owen(2021)andRogersetal.\n(2023).Indoingso,wealsoadopttheunderlyingplanetdistributions\ninferred in Rogers & Owen (2021) for planetary core masses, initial\nMNRAS 000, 1–18 (2023)6 C. S. K. Ho et al.\n1 2 3 45678\nRp at P=10 days (R)\n0.00.20.40.60.81.01.2Relative densityM/M<0.7\n0.7M/M<0.9\n0.9M/M<1.1\nM/M 1.1\nFigure 5. Histogramof𝑅𝑝,adjustedtoanequivalentradiiatP=10daccord-\ning to the valley slope determined in Ho & Van Eylen (2023) ( 𝑚=−0.09),\nforplanetsofdifferenthoststarmasses.Thehistogramsarenormalisedsuch\nthat the peaks of the sub-Neptune curves have a relative density of 1. The\nlinesandshadedregionsshowthemedianand 1𝜎confidenceintervalofthe\nfitted curve respectively.\n0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3\nM (M)\n02468E\nEavg\nESE\nESN\nFigure 6. Plotofthe𝐸metrictomeasurethedepthoftheradiusvalleyfrom\nequations 4-6, as a function of host star mass.\natmospheric mass fractions, orbital periods, as well as an Earth-like\ncore density for all planets of 5.5g cm−3. Changing these distribu-\ntions,whichcomefromstatisticalinferencebetweenthephotoevap-\norationmodeland Keplerdata,willinevitablychangethepopulation\nof planets that might be observed under such conditions. This high-\nlights an important approach in this work, in that, we wish to com-\npare Keplerdata with a realisation of the photoevaporation model\nthat fits the observations as well as reasonably possible. Inevitably,\nthere are multiple parameters within our evolution model that can\nbe tuned within reasonable limits to fit the observations more accu-\nrately. However, if inconsistencies still exist between the model and\nobservations,evenaftertuning,therobustnessoftheresultincreases.\nOnesuchexampleisthat,underthephotoevaporationmodel,thepre-\ndictedslopeoftheradiusvalleyasafunctionoftheorbitalperiodis\n0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3\nM (M)\n1.251.501.752.002.252.502.753.00Rp at peak (R)\nsuper-Earths\nsub-NeptunesFigure 7. Plot of the planetary radius at the super-Earth and sub-Neptune\nhistogram peaks from Fig. 5 as a function of host star mass.\naltered with different assumptions in the escape efficiency 𝜂(Owen\n& Wu 2017), which allows one to calculate mass loss rates:\n¤𝑀=𝜂𝜋𝑅3p𝐹XUV\n𝐺𝑀p, (7)\nwhere𝐹XUVistheincidentstellarXUVflux(followingRogersetal.\n2021),𝐺isthegravitationalconstant,and 𝑅pand𝑀paretheplanet’s\nradius and mass. Typically, the efficiencies are parameterised as a\nfunction of the planet’s escape velocity:\n𝜂=𝜂0\u0012𝑣esc\n23km s−1\u0013𝛼𝜂\n. (8)\nTo begin, we find a pair of 𝜂0and𝛼𝜂that approximately match the\nslope of the observed valley. In practice, we achieve this by finding\nthepairofparametersthatminimisethenumberof observed planets\ninside the model’svalley. We do this only in the highest stellar mass\nbin (1.1≤𝑀★/𝑀⊙<1.3) since the photoevaporation models have\nonly ever been fit to FGK stellar hosts, for which an accurate fit has\nbeen shown to exist (Rogers & Owen 2021). Therefore, we fix the\nmodels to attain this consistent fit at high stellar mass, which then\nyields an extrapolation of the model to lower stellar masses. The\nspecific values were found to be 𝜂0=0.24and𝛼𝜂=−1.11, consis-\ntentwiththeinferenceofRogers&Owen(2021)andhydrodynamic\nsimulations of atmospheric escape.\nWe plot 106modelled planets together with the observed sample\nin Fig. 8. Building on methods discussed in Rogers et al. (2021)\nand Petigura et al. (2022), we locate the boundaries of the radius\nvalley region as the area enclosing 1% of the modelled planets,\nwhile maximising the valley area. Table 3 lists the upper and lower\nboundaries of the radius valley in the photoevaporation model for\neach stellar mass range. We compute the area of the radius valley\nof the photoevaporation model in the 𝑅𝑝−𝑃diagram for different\nstellar mass bins, and report this in Table 3. We see that the area of\ntheradiusvalleyincreaseswithincreasingstellarmass,meaningthat\nhigher-mass stars produce a wider radius valley. To first order, the\nwidth of the valley is controlled by the difference in size between a\nstripped core of a given mass and the same core if it were to host\na small hydrogen-dominated atmospheric mass fraction ∼0.1%.\nThe cause of the increase in valley width is that planets orbiting\nhigher-mass stars experience a higher equilibrium temperature at\nMNRAS 000, 1–18 (2023)Shallower radius valley around low-mass hosts 7\nTable 3.Slopes and intercepts of lower and upper radius valley boundaries\nofthephotoevaporationmodelusedinthiswork,givenby log10\u0000𝑅𝑝/𝑅⊕\u0001=\n𝑚log10(𝑃/d)+𝑐. The area of the radius valley, incorporating 1% of the\nmodelled planets, is also reported.\n𝑀★𝑚lower𝑐lower𝑚upper𝑐upperValley area\n0.5−0.7−0.1846 0.3528−0.1846 0.4021 0.0988\n0.7−0.9−0.1775 0.3898−0.1694 0.4333 0.1031\n0.9−1.1−0.1663 0.4099−0.1559 0.4548 0.1106\n1.1−1.3−0.1704 0.4415−0.1510 0.4787 0.1133\na given orbital separation. Under these circumstances and all else\nheldfixed,thetransitradiusofagivensub-Neptuneismoreinflated\nwhencomparedtothosearoundsmaller-massstars.Onapopulation\nlevel,thisincreasesthedifferenceinsizebetweenastrippedcoreand\nsub-Neptune, thus widening the valley.\nThe model’s widening valley with increasing stellar mass may\nseem to qualitatively match the observations, where a narrower val-\nley could appear shallower due to the uncertainty of the observed\nplanet radius. However, further steps are required to robustly com-\npareourobservationstomodels,asobservationalbiasesexistthatare\nabsent in the models, meaning the observed planet population has a\ndifferentdistributioninradius-periodspacecomparedtothemodels.\nTherefore, we perform a more rigorous comparison in Section 3.3.\n3.3 Comparison with theoretical models of atmospheric mass\nloss\nAs discussed in Section 3.2, we wish to compare Keplerdata with a\nrealisationofthephotoevaporationmodelthatfitstheobservationsas\nwell as reasonably possible. In Section 3.2, we constructed models\nthat minimised the number of observed planets inside the valley,\nthereby matching the approximate valley slope. However, this did\nnotpreciselymatchtherestofthedistributionacrosstheentirerange\nin stellar mass, such as the specific occurrence of super-Earths and\nsub-Neptunesinvariousregionsofparameterspace.InthisSection,\nwe introduce a methodology to fit the planet distribution, which\navoids the extreme computational costs of Rogers & Owen (2021).\nOur goal is to produce a model that is, in fact, perfectly matched to\nthe observations everywhere except inside the valley. This perfectly\nfit model is then used to predict the number of planets inside the\nradius valley for each stellar mass bin, accounting for measurement\nuncertainty, which inevitably scatters some planets into the valley.\nInthisway,weinvestigatewhetherthephotoevaporationframework\nis able to match the observed radius valley trend with stellar mass,\nas a consequence of the model producing a narrower radius valley\nfor lower-mass stars and observational uncertainty. If the model still\ncannot reproduce the observed increase of planets inside the valley\nat lower stellar masses, then the inconsistency is robust. It would\nimplythat photoevaporationmodels –even afterexhausting thefree\nparameters available within these models – are insufficient in their\ncurrent form to explain the trend of the observed radius valley with\n𝑀★.\nFor each stellar mass bin, we divide the radius-period plot into\nmultiple cells, parallel with the valley, as shown in Fig. 8. We con-\nstruct the cells such that the valley is contained within its own cell\nfor each period bin, with boundaries defined from Table 3. The goal\nis to find a numerical weight for each cell outside of the valley\nsuch that the number of planets in the model is weighted to exactly\nmatchthatoftheobservations.Asdiscussed,thisperfectlyweighted\nmodelisthenusedtopredictthenumberofplanetsinsidetheradiusvalley for each stellar mass bin, due to measurement uncertainties\nin planetary radii. To begin, we account for measurement uncer-\ntainty in planetary radius by redrawing each modelled planet from\nN\u0010\n𝑅p, model,𝜎𝑅p, model\u0011\nwhere𝜎𝑅p, modelis a random selection in the\nset of𝜎𝑅p, obsof planets within the same period range. Afterwards,\nfor each cell, we calculate the number of planets that have shifted to\nanother cell as a result of the new draw. We represent this in matrix\nform:\nF=𝑓0,0𝑓0,1···𝑓0,𝑣···𝑓0,𝑛\n𝑓1,0𝑓1,1···𝑓1,𝑣···𝑓1,𝑛\n..................\n𝑓𝑣,0𝑓𝑣,1···𝑓𝑣,𝑣···𝑓𝑣,𝑛\n..................\n𝑓𝑛,0𝑓𝑛,1···𝑓𝑛,𝑣···𝑓𝑛,𝑛(9)\nwhere𝑓𝑖,𝑗represents the number of planets that were in cell 𝑖\noriginally but are now in cell 𝑗after redrawing. Here, 𝑖,𝑗∈\n{0,1,..,𝑣,..,𝑛}and𝑣represents the valley cell index. At this stage,\nweapproximatetozerothorderthatthemodel’sradiusvalleyiscom-\npletely empty, and remove all elements from Fthat associate with\nthe valley cell index 𝑣e.g.𝑓𝑣,𝑖and𝑓𝑣,𝑗, being the cells for which\nplanetshaveshiftedoutof,orintothevalley.Sincethemeasuredun-\ncertainties in planetary orbital periods are very small, we calculate\nFindependentlyforeachperiodbinsinceplanetsdonotshiftacross\nhorizontal cells as a result of the new draws. We can then calculate\nthe weights for each cell\nw=\u0002𝑤0𝑤1... 𝑤𝑣... 𝑤𝑛\u0003\n(10)\nby solving the matrix equation\nnobs=F·w (11)\nwhere\nnobs=\u0002𝑛obs,0𝑛obs,1𝑛obs,2···𝑛obs,𝑛\u0003\n(12)\nis the array of the number of observed planets in each cell. This is\nachieved by finding F−1(the inverse of F), i.e.,\nF−1=1\ndet(F)·adj(F) (13)\nwhichisnumericallytrivialsince Fisasquarematrixwithdominant\ndiagonalelements.Topreventasingularmatrix,weassignavalueof\n1 for the diagonal terms 𝑓𝑖,𝑖if cell𝑖has no planets inside. Our only\nrequirement for the weight inside the valley, 𝑤𝑣, is that the weights\nvary smoothly across the valley for increasing planet size. As such,\nwe determine 𝑤𝑣by taking the average of the two adjacent weights,\n𝑤𝑣−1and𝑤𝑣+1, i.e.\n𝑤𝑣=𝑤𝑣+1+𝑤𝑣−1\n2. (14)\nOur perfectly weighted model is now used to calculate the expected\nnumber of planets in the model’s radius valley with\n𝑛model, valley=fvalley·w (15)\nwhere\nfvalley=\u0002𝑓0,v𝑓1,v𝑓2,v···𝑓𝑛,v\u0003\n(16)\nisthevectorcontainingthenumberofplanetsthathaveshiftedfrom\ndifferent cells into the radius valley from the new draw. In essence,\n𝑛model, valley represents the number of planets the photoevaporation\nmodelwouldpredicttoexistinsidethevalley,giventhatitreproduces\nthe observations everywhere else.\nMNRAS 000, 1–18 (2023)8 C. S. K. Ho et al.\nFigure 8. Radius-orbital period plot of planets for different host star masses. Grey dots indicate planets simulated with the XUV photoevaporation models of\nRogersetal.(2021),andcolouredcirclesindicatetheobservedplanets.Theblacksolidlinesshowtheboundariesofthemodel’sradiusvalley,asdefinedbythe\nparameters in Table 3. The black dotted lines show the cells defined in Section 3.3.\nWe perform a bootstrap of 1000 samples for each stellar\nmass bin, each time randomly redrawing a new set of 𝑅p, obs\nfromN\u0010\n𝑅p,obs,𝜎𝑅p,obs\u0011\n, and solving equation 11 to get a new\n𝑛model, valley , to obtain the median and ±1𝜎uncertainties of\n𝑛model, valley and𝑛obs, valley. To investigate the effect of bin sizes in\nthisanalysis,wetestedthismethodwithmultiplebinsizesinperiod\nandradius.Werepeattheabove-mentionedstepsfordifferentequally-\nspacedlogarithmicbinsinperiodsbetween 1and100d,andvarying\nbinsizesinradius.Wefindthattheresultingvaluesfor 𝑛model, valley\ndo not change substantially, and all values are within 1𝜎of each\nother. We therefore use 5 equally-spaced bins in log10𝑃between 1\nand100d,andabinwidthof0.2in log10𝑅𝑝,forreportingthefinal\nresult. Fig. 9 shows the histogram comparing the number of planets\nin the lowest stellar mass bin ( 0.5≤𝑀★/𝑀⊙<0.7) between the\nobservations and the photoevaporation model after weighting; here\nwe see that the two match everywhere outside the valley, however\nthere is an excess of planets inside the valley for the observations.\nFig. 10 shows the fraction of planets inside the radius valley as\na function of stellar mass, comparing the photoevaporation modelwith observations. The results show that for the observations, this\nfraction largely decreases with increasing stellar mass, with a best-\nfitting slope of 𝑚=−0.038+0.044\n−0.045, whereas in the theoretical photo-\nevaporationscenario,thefractionstaysapproximatelyconstant,with\n𝑚=0.001+0.002\n−0.002. Since the area of the radius valley also decreases\nwith decreasing stellar mass (see Table 3), the stellar mass depen-\ndence of the fraction is less significant compared to the 𝐸metrics\ncomputedinSection3.1(seeFig.6),whichisindependentofthera-\ndius valley area. However, even after taking this effect into account,\nwe continue to see the decrease in fraction with increasing stellar\nmass, meaning that for lower-mass stars, there are more planets in-\nside a narrower valley for the observations. The measured fractions\nbetween the model and observations are more than 1𝜎apart for\nthe 3 lowest stellar mass data points, with the fraction of planets\ninside the radius valley much higher for observations compared to\nthe photoevaporation case. We therefore conclude that current pho-\ntoevaporation models under-predict the number of planets observed\ninside the radius valley for low-mass stars. This suggests that these\nmodels need to be modified or that further physical mechanisms\nMNRAS 000, 1–18 (2023)Shallower radius valley around low-mass hosts 9\n1 2 3 4\nRp at P=10 days (R)\n051015202530Number of planetsObservations\nPhotoevaporation \nmodel\nFigure 9. Histogram of 𝑅𝑝, adjutsed to an equivalent radii at 𝑃=10d,\naccording to the slope of the model’s radius valley as listed in Table 3, after\nweightingthemodelaccordingtothemethoddiscussedinSection3.3.Here,\nthe number of planets match everywhere except inside the valley.\n0.5 0.7 0.9 1.1 1.3\nM (M)\n0.020.040.060.080.10Fraction of planets in valley\nPhotoevaporation model\nObservations\nFigure 10. Fraction of planets inside the radius valley as a function of stel-\nlar mass, after eliminating effects of initial conditions in the model. The\nlines show the linear best fit to the data points. We find a slope and inter-\ncept of−0.038+0.044\n−0.045and0.087+0.042\n−0.045respectively for the observations, and\n0.001+0.002\n−0.002and0.023+0.002\n−0.002for the photoevaporation model.\nneed to be considered to explain the characteristics of small planets\norbiting low-mass stars.\n4 CAUSES FOR DISCREPANCY BETWEEN\nOBSERVATION AND MODEL\nInSection3,wenotedtheincreasedshallownessoftheradiusvalley\nforlow-massstarscomparedtotheirhigh-masscounterparts,aswell\nasthediscrepancyintheemptinessofthevalleybetweentheoretical\nmodels and observations. Given the overall success of photoevapo-\nrationmodelsinexplainingtheobservedpropertiesofclose-insmall\nplanets,weinvestigatehowthesemodelscanbeextendedtoreconcilethemwithobservations.Inparticular,wesuggestthefollowingpossi-\nbleexplanationsfortheobserveddiscrepancy:X-rayandUV(XUV)\nscattering (Section 4.1), dispersion in planetary core composition\n(Section 4.2), and planetary collisions (Section 4.3). Hypothetical\nplots of how each scenario may present in the radius-period plot are\nshown in Fig. 11.\n4.1 Scattering in the stars’ XUV output\nAstar’shigh-energyoutputisknowntobetightlycorrelatedwithits\nrotation period (e.g. Wright et al. 2011; Johnstone et al. 2021). It is\nalso known that at a given age and stellar mass, there is a spread in\nrotation periods, where this spread in rotation periods increases at\nyounger stellar ages (e.g. Reinhold & Hekker 2020). Therefore, em-\npirically, it is anticipated that stars with identical masses could have\ndifferent histories of their XUV output. Studies (e.g. Tu et al. 2015;\nKubyshkinaetal.2019)haveshownthatplausibledifferenthistories\nofastar’shighenergyoutputscouldleadtodifferentplanetaryevolu-\ntionary scenarios, where planets with identical initial properties are\nabletoretainorloseanatmosphere.Thepositionoftheradiusvalley\ndirectly depends on the XUV exposure ( Xxuv, i.e. the total lifetime\nintegrated XUV flux incident upon the planet, Owen & Wu 2017).\nThus,astheXUVexposurewillvaryfromstar-to-starwiththesame\nstellarmass,theradiusvalleywillbecomeblurred.FollowingOwen\n& Wu (2017) with a mass-loss efficiency scaling of 𝜂∝𝑣−2esc, with\n𝑣escthe planet’s escape velocity, if a given star’s XUV exposure is\na fraction𝑓Xof a typical star with the same stellar mass, then the\nposition of the radius valley should scale as:\n𝑅𝑝,valley∝𝑓0.12\nX. (17)\nThis implies a factor of 5 spread in the XUV histories is enough to\nmove the position of the radius valley up or down for an individual\nstar by±10%. This is possible as Tu et al. (2015) has shown that for\nplanets with the same initial planetary and atmospheric masses, the\nplanet retains 45% of its initial atmosphere at 5 Gyr when orbiting\na slowly rotating star with minimal XUV output, whereas the planet\nwouldloseallofitsatmosphereat ∼100Myrifitorbitsafast-rotating\nstar.Thus,variationinastar’sXUVhistorywillfilltheradiusvalley;\nplanets around stars with a higher-than-average XUV output will\nhavearadiusvalleyatlargerplanetaryradii.Therefore,moremassive\ncores can be stripped at a given orbital period, polluting the radius\nvalley with rocky planets. However, stars with a lower-than-average\nXUVoutputwillallowlowermassplanetstoretaintheiratmospheres\natagivenorbitalperiod,pollutingtheradiusvalleywithplanetswith\na H-He dominated atmosphere and thus appear lower density than\na planet with an Earth-like bulk density. Therefore, if XUV scatter\nis the cause of the excess of planets in the radius valley, planet’s in\ntheradiusvalleyshouldhavearangeofdensitiescoveringEarth-like\nbulk densities to more volatile-rich densities.\nThere are two reasons why this should lead to an increasing pol-\nlution factor with decreasing stellar mass. Firstly, as discussed in\nSection3.2,thewidthoftheradiusvalleydecreaseswithdecreasing\nstellarmass.Thus,forafixedspreadinXUVhistories,thiswillnatu-\nrallypolluteanarrowervalleymore,asanyplanetismorelikelytobe\nscatteredintothevalley.Secondly,thereissomeempiricalevidence\nthatthescatterinpossibleXUVhistoriesislargeraroundlowermass\nstars, as evidenced by the larger spread in rotation rates measured at\nagivenage(e.g.Newtonetal.2016;Reinhold&Hekker2020).Pho-\ntoevaporation models could be constructed to take XUV scattering\nintoaccountinordertoinvestigatethiseffectfurther;however,since\nthe rotation evolution of M-dwarfs remains poorly constrained, we\nleave such work for future studies.\nMNRAS 000, 1–18 (2023)10 C. S. K. Ho et al.\nPRpXUV Scatter\nRocky + H-He\nRocky\nPRpCore Composition Dispersion\nRocky + H-He\nRocky\nRocky + ice + H-He\nRocky + ice\nPRpCollisions\nRocky + H-He\nRocky\nFigure 11. Hypothetical𝑅-𝑃plots of XUV scatter (left panel), dispersion in planetary core compositions (middle panel), and collisions (right panel). Planets\nof different types are represented by different colours and shapes. The radius valley is bounded by the black dotted lines.\n4.2 Dispersion in planetary core composition\nAs in the case of scatter in a planet’s XUV history, the bulk density\noftheplanet’scorealsocausesvariationinthepositionoftheradius\nvalley. A planet with a lower-density core, but the same mass, has a\nweaker gravitational potential. Thus, this planet has its atmosphere\nmore easily removed than a planet with a denser core. As such, the\nradius valley, at a fixed period, appears at a larger radius for lower-\ndensitycores.AsdiscussedbyOwen&Wu(2017);Jin&Mordasini\n(2018);Mordasini(2020),anyspreadincore-compositionwillresult\nin a shallower radius valley (Fig. 8 in Owen & Wu 2017). Since\nthe position of the radius valley appears to be in agreement with\npredictions for an Earth-like bulk density, but with an excess of\nplanetsinthevalley,ifaspreadincorecompositioncausedthis,this\nwouldimplythatthedistributionofcoredensitieshasalargertailof\nlower bulk density cores around lower mass stars.\nThe snow line of stars is the distance from the star beyond which\nthe temperature is low enough for volatiles to form. Hence, the ice\nline is closer to the star for lower-mass stars,; planets with a larger\nice fraction in their cores are more likely to form closer to the star,\nwithin the observed orbital period range. Therefore, if there was an\nincreasingpopulationofplanetswithlowerbulkdensitycoresaround\nlower mass stars, this could also explain the excess pollution of the\nradius valley that we observe (e.g. Venturini et al. 2020).\n4.3 Collisions\nCollisionsamongplanetarybodieshavepreviouslybeenproposedto\nbe an integral part of Super-Earth and Sub-Neptune formation (e.g.\nInamdar & Schlichting 2015; Izidoro et al. 2017). Such collisions\namong planets in the super-Earth and Sub-Neptune populations can\nalsoleadtoamorefilledinradiusvalley.Forexample,whentwosub-\nNeptunes collide, this giant impact imparts so much kinetic energy\nthat is subsequently converted into heat that the hydrogen envelope\nis entirely lost while their cores merge (Liu et al. 2015; Biersteker\n& Schlichting 2019; Izidoro et al. 2022). This, therefore, leads to a\nmore massive but smaller planet (due to the lack of the hydrogen\nenvelope) and hence would populate the radius valley. Similarly, a\nmerger of sufficiently large super-Earths could also start to populate\nthe radius valley from the lower edge. Regardless of whether the\ncollisions appear between super-Earths and sub-Neptunes or within\neach of these populations, the net result is generally the loss of any\nhydrogen-envelope,leavingbehindmostlyrocky-cores.Therefore,if\ncollisions are responsible for populating the radius valley, then thepopulationofexoplanetsresidinginitisexpectedtobepredominantly\nrocky in their composition (right panel of Fig. 11).\nForthesameplanetmass( 𝑀𝑝)andsemi-majoraxis(a),planetary\nscale collisions are indeed expected to be more common around\nlower-mass stars because the Hill radius,\n𝑅Hill=𝑎\u0012𝑀𝑝\n3𝑀★\u00131/3\n(18)\nwhich estimates the scale over which neighbouring planets can sig-\nnificantlyperturbeachothergravitationallywhileorbitingtheirhost\nstar(𝑀★),islargerforlowermassstars.Thistherefore,leadstomore\neccentricityexcitation,eventualorbitcrossingsandcollisionsaround\nlower mass stars than higher mass ones, assuming the planet popu-\nlationisidenticalotherwise.Infact,planetswithhighbulkdensities\naroundlow-massstars,whicharethoughttobeproductsofcollision\nevents (e.g. Naponiello et al. 2023) have been discovered.\n4.4 Mass measurements of planets point to a mixture of\nscenarios\nAs discussed in Sections 4.1-4.3, stellar XUV scatter, dispersion in\ncore compositions, and collisions all result in similar views of the\nradiusvalleyinradius-periodspace.Todisentanglethesedegenerate\nviews, one could measure the masses of planets inside the radius\nvalley to infer their compositions.\nInoursampleof444planetsanalysedinthiswork,203orbitaround\nhoststarswith 0.5≤𝑀★/𝑀⊙<0.9,ofwhichonly33havearchival\nmassmeasurements,andonly11ofwhichhavemassprecisionbetter\nthan 30%. This is because the sample studied here originates from\nKeplerobservations, which are suited for highly precise and homo-\ngeneous transit observations, but are less suited for follow-up mass\nobservations due to the relative faintness of Keplerhost stars. We\nincrease the sample size by turning to the NASA Exoplanet Archive\n(Akesonetal.2013)forthesampleofallconfirmedexoplanetswith\nmass measurements, within the period range of 1≤𝑃/d≤100and\nwith planetary radii in the range 1≤𝑅𝑝/𝑅⊕≤4. For planets with\nmultiple reported mass measurements, we select the value with the\nbestmassprecision.Wethenonlyretainplanetswithmassprecision\nbetter than 30%. For radius measurements, we acknowledge the im-\nportance of homogeneity and precise radius measurements (e.g. Ho\n&VanEylen2023),henceweusetheplanetaryradiiandstellarmass\nfrom Berger et al. (2023) if available, otherwise we use the corre-\nspondingplanetaryradiusandstellarmassmeasurementreportedin\nthe NASA Exoplanet Archive for the same mass entry.\nWe calculate an expected envelope fraction for the planets by\nMNRAS 000, 1–18 (2023)Shallower radius valley around low-mass hosts 11\nfollowing the method in Chen & Rogers (2016), which applies the\nModules for Experiments in Stellar Astrophysics (MESA) models\n(Paxton et al. 2011, 2013, 2015) to sub-Neptune atmospheres. This\nmethodtakestheplanetaryradius( 𝑅𝑝),planetarymass( 𝑀𝑝),plan-\netary incident bolometric flux ( 𝐹𝑝), and age as inputs. We use 𝐹𝑝\nfrom Berger et al. (2023) where available, otherwise, we calculate\n𝐹𝑝from the stellar radius and effective temperature, and the orbital\nsemi-major axis. We use the stellar ages from Berger et al. (2023)\nifavailable,elsewefindthevaluewiththelowestpercentageuncer-\ntainty from the NASA Exoplanet Archive. We compute 𝑅coreusing\nthe mass-radius relationship from Fortney et al. (2007) for a given\n𝑀𝑝.Wethensolveforthe 𝑓envrequiredtoproduceaplanetwiththe\ngiven mass and radius using Brent’s method of root finding. Some\nplanets have mean densities consistent with Earth-like, atmosphere-\nfree compositions, as is expected from photoevaporation models for\nplanetsbelowtheradiusvalley.Wherewecannotfindanatmospheric\nsolution, we set a default envelope fraction to 10−4.\nWe plot the resulting 𝑓envfor planets with mass measurements in\na𝑅𝑝−𝑃plot in Fig. 12. A table listing all the planetary masses\nand envelope fractions is provided in the Appendix (Table A4). We\nobservethat,ingeneral,planetswithbarerockycoreslietowardsthe\nlower part of the radius-period plot, while planets with significant\natmosphericfractions( ≳10−2)populatetheupperpart.Wenoticea\nfewplanetswithabarerockycoremoveacrosstheboundaryintothe\nregion with an abundance of planets with large envelope fractions,\nhowever,themassesoftheseplanetsmostlycomefromtransittiming\nvariations (TTV) rather than radial velocity (RV) measurements.\nBy visual inspection, our results do not wholly match the case\nfor XUV scatter, as we do not observe a significant mixing of bare\nrocky planets and planets with predicted substantial atmospheres\ninside the valley as in the left panel of Fig. 11. On the other hand,\nthepresenceofplanetswithpredictedsmall,butnon-zero,envelope\nfractions ( ≲10−2) inside the valley may point to planets with a low\nice fraction in their cores. Finally, if we consider the TTV masses\nto be true, the existence of some high-mass bare rocky planets with\nradii just above the radius valley suggests these are extremely dense\nplanetswhichmaybeby-productsofcollisionevents.Therefore,we\nsuggest that a mixture of these scenarios is the probable scenario\naround low-mass stars.\nHowever, one cannot draw a firm conclusion for the following\nreasons. Most importantly, we suffer from small number statistics\nsince most planets do not have precise mass measurements avail-\nable.Anotherpossibleconcernisthatthecalculationoftheenvelope\nfractionrequiresinformationofthestellarage,whicharehighlyun-\ncertain (with a mean uncertainty of ±90%for0.5≤𝑀★/𝑀⊙<0.9\nin our sample). However, for a sample of main sequence stars, this\nis unlikely to significantly affect our conclusions. Nevertheless, a\nmorecomprehensivestudyisrequiredtoenhanceourunderstanding\naroundlow-massstars,suchasobtainingmoreprecisemassmeasure-\nments of small planets around low-mass stars, with a focus towards\nthose inside the predicted radius valley, as well as more precise age\nestimatesofsuchstars,orthoroughmodellingofXUVscatteringor\ncollision events.\n5 CONCLUSIONS\nInthiswork,werefittedthelightcurvesof72 Keplerplanetsaround\nlow-massstars,using Kepler1-minuteshortcadencedata.Wepresent\ntheir revised planetary parameters, and combining with the parame-\ntersinHo&VanEylen(2023),westatisticallyanalysethechangein\nthe depth of the radius valley as a function of host star mass.We find that the radius valley becomes shallower around lower\nmass host stars. Although the photoevaporation model also predicts\na narrower radius valley for lower mass hosts, this filling-in of the\nobserved valley could not be explained by theoretical models of\nphotoevaporation, even when observation biases and dispersion in\ninitialconditionsaretakenintoaccount.Thissuggeststhatadditional\nmechanisms are involved in shaping the radius valley for planets\naround low-mass stars.\nWesuggestthreepossibletheoreticalscenariosthatcouldexplain\ntheshallowerradiusvalleyforlow-massstars:scatteringinthestars’\nXUVoutput,dispersioninplanetarycorecomposition,andcollision\nevents. Combining our sample with other planets around low-mass\nstars on the NASA Exoplanet Archive, we calculate the expected\nenvelope fraction of planets under the assumption of a rocky core.\nWe note that the resulting population distribution with reference to\ntheradiusvalleylocationdoesnotfavoureitherscenarioexclusively,\nhowever, it provides evidence for the presence of planets with a low\nice fraction, or planetary collisions.\nWith few planets having precise mass and age measurements, we\nareunabletodrawstrongconclusionsaboutthetruenatureofplanets\naround low-mass stars. We suggest potential future work to under-\nstand this picture better, such as obtaining more precise planetary\nmass or stellar age measurements or a comprehensive modelling of\nXUV scatter or planetary collision events. The Transit Exoplanet\nSurvey Satellite (TESS, e.g. Ricker et al. 2015) would yield more\nplanets around bright, nearby stars, which are optimal candidates\nfor radial velocity follow-up observations to obtain precise masses.\nCombining with the future launch of the PLAnetary Transits and\nOscillationsofstars(PLATO)mission(e.g.Raueretal.2014)which\nenables asteroseismic observations of stars to obtain precise stellar\nages, we can deepen our insight into the formation and evolutionary\ntrajectories of small, close-in planets orbiting low-mass stars.\nACKNOWLEDGEMENTS\nThis paper includes data collected by the Kepler mission and ob-\ntained from the MAST data archive at the Space Telescope Science\nInstitute (STScI). Funding for the Kepler mission is provided by the\nNASA Science Mission Directorate. STScI is operated by the Asso-\nciationofUniversitiesforResearchinAstronomy,Inc.,underNASA\ncontract NAS 5–26555. CSKH would like to thank the Science and\nTechnologyFacilitiesCouncil(STFC)forfundingsupportthrougha\nPhDstudentship.JGRissponsoredbytheNationalAeronauticsand\nSpace Administration (NASA) through a contract with Oak Ridge\nAssociated Universities (ORAU). VVE is supported by UK’s Sci-\nence & Technology Facilities Council (STFC) through the STFC\ngrants ST/W001136/1 and ST/S000216/1. HES gratefully acknowl-\nedges support from NASA under grant number 80NSSC21K0392\nissued through the Exoplanet Research Program. JEO is supported\nbyaRoyalSocietyUniversityResearchFellowship.Thisprojecthas\nreceived funding from the European Research Council (ERC) under\nthe European Union’s Horizon 2020 Framework Programme (grant\nagreement no. 853022, PEVAP).\nDATA AVAILABILITY\nThe Kepler1-minute short cadence light curves are available for\ndownload on the NASA Mikulski Archive for Space Telescopes\nMNRAS 000, 1–18 (2023)12 C. S. K. Ho et al.\n1 3 10 30 100\nP (d)1234Rp (R)\n0.5M/M<0.7\n104\n103\n102\n101\nEnvelope fraction\n1 3 10 30 100\nP (d)1234Rp (R)\n0.7M/M<0.9\n104\n103\n102\n101\nEnvelope fraction\nFigure 12. Radius-orbital period plot of all small, close-in planets from the NASA Exoplanet Archive. Coloured points indicate the expected envelope fraction\nfor planets with known mass measurements and with mass precision better than 30%, and grey points show the remaining planets. The radius valley of the\nphotoevaporation model and the observed valley are bounded by the black dashed lines and grey dotted lines, respectively. Planetary masses are determined\nfrom radial velocity (circles) and transit timing variation (triangles) measurements.\n(MAST) database1. The parameter estimates from HMC posteri-\nors for the new planets analysed in this work are provided in the\nAppendix tables.\nREFERENCES\nAkana Murphy J. M., et al., 2021, AJ, 162, 294\nAkeson R. L., et al., 2013, PASP, 125, 989\nAlmenara J. 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V., et al., 2022, AJ, 163, 293\nVanEylenV.,AgentoftC.,LundkvistM.S.,KjeldsenH.,OwenJ.E.,Fulton\nB. J., Petigura E., Snellen I., 2018, MNRAS, 479, 4786\nVan Eylen V., et al., 2021, MNRAS, 507, 2154\nVanderburg A., et al., 2015, ApJ, 800, 59\nVenturiniJ.,GuileraO.M.,HaldemannJ.,RoncoM.P.,MordasiniC.,2020,\nA&A, 643, L1\nVines J. I., et al., 2023, MNRAS, 518, 2627\nVirtanen P., et al., 2020, Nature Methods, 17, 261\nVissapragada S., et al., 2020, AJ, 159, 108\nWest R. G., et al., 2019, MNRAS, 486, 5094\nWilson T. G., et al., 2022, MNRAS, 511, 1043\nWright N. J., Drake J. J., Mamajek E. E., Henry G. W., 2011, ApJ, 743, 48\nXie J.-W., 2014, ApJS, 210, 25\nAPPENDIX A: EXTRA MATERIAL\nMNRAS 000, 1–18 (2023)14 C. S. K. Ho et al.\nTable A1. Host star parameters of the additional planets fitted in this work, that are not in the sample of Ho & Van Eylen (2023). Stellar radius ( 𝑅★), mass\n(𝑀★),effectivetemperature( 𝑇eff),metallicity([Fe/H]),Keplermagnitude(Ksmag),andagevalues,aretakenfromPetiguraetal.(2022).The‘radiuscorrection\nfactors’ (RCFs) are taken from Furlan et al. (2017). Stellar densities ( 𝜌★), and the two quadratic limb darkening parameters ( 𝑢0and𝑢1) are obtained from the\ntransit fits in this work. Only the first 10 rows are shown here; the full table is available online in a machine-readable format.\nKOI𝑅★(𝑅⊙)𝑀★(𝑀⊙)𝑇eff(K) [Fe/H] (dex) Ksmag Age (Gyr) RCF 𝜌★(g cm−3)𝑢0 𝑢1\n156 0.72+0.02\n−0.020.73+0.03\n−0.034511+110\n−1100.11±0.09 11.37±0.02 10.1+6.7\n−6.51.0000 2.039±0.164 0.63±0.07 0.15±0.10\n222 0.71+0.02\n−0.020.70+0.03\n−0.034416+70\n−700.09±0.09 12.31±0.02 10.7+7.3\n−6.61.0000 2.145±0.180 0.56±0.09 0.21±0.13\n247 0.58+0.02\n−0.020.57+0.01\n−0.013840+70\n−700.05±0.09 11.12±0.02 11.6+8.1\n−6.51.0000 2.933±0.157 0.34±0.12 0.20±0.13\n248 0.62+0.02\n−0.020.60+0.01\n−0.014011+70\n−70−0.16±0.09 12.38±0.02 12.8+9.0\n−6.21.0002 2.724±0.184 0.54±0.13 0.31±0.17\n249 0.41+0.01\n−0.010.40+0.01\n−0.013506+70\n−70−0.32±0.09 11.15±0.03 11.8+8.1\n−6.41.0000 5.748±0.253 0.28±0.11 0.44±0.16\n250 0.60+0.02\n−0.020.59+0.01\n−0.024055+70\n−70−0.17±0.09 12.63±0.03 11.6+8.2\n−6.61.0031 2.946±0.245 0.40±0.14 0.20±0.13\n251 0.54+0.02\n−0.020.53+0.01\n−0.013788+70\n−70−0.03±0.09 11.68±0.02 11.6+8.1\n−6.61.0158 3.403±0.161 0.42±0.09 0.16±0.11\n252 0.63+0.02\n−0.020.60+0.01\n−0.013889+70\n−700.10±0.09 12.55±0.03 12.3+8.5\n−6.31.0000 2.552±0.143 0.35±0.10 0.30±0.15\n253 0.66+0.02\n−0.020.63+0.01\n−0.014033+70\n−700.38±0.09 12.29±0.04 9.4+5.9\n−6.41.0000 2.207±0.118 0.49±0.11 0.12±0.09\n312 1.45+0.05\n−0.051.26+0.03\n−0.046246+100\n−1000.07±0.06 10.52±0.01 2.5+0.1\n−0.21.0006 0.419±0.031 0.34±0.15 0.25±0.14\n... ... ... ... ... ... ... ... ... ... ...\nMNRAS 000, 1–18 (2023)Shallower radius valley around low-mass hosts 15Table A2. Planetaryparametersoftheadditionalplanetsfittedinthiswork,thatarenotinthesampleofHo&VanEylen(2023).Orbitalperiod( 𝑃),transitephermeristime( 𝑡0),planet-to-starradiusratio( 𝑅𝑝/𝑅★),\nimpact parameter ( 𝑏), eccentricity ( 𝑒) and argument of periapsis ( 𝜔) are direct results from the transit fitting, while the planetary radius ( 𝑅𝑝), ratio between the semi-major-axis and the stellar radius ( 𝑎/𝑅★) and\nincidentbolometricflux( 𝐹𝑝)areindirectlycalculated.The 𝑅𝑝/𝑅★and𝑅𝑝valuesfromPetiguraetal.(2022)(P22)arealsolistedhereforcomparison.Onlythefirst10rowsareshownhere;thefulltableisavailable\nonline in a machine-readable format.\nKOI Kepler name 𝑃(d) 𝑡0(BJD-2454833) 𝑅𝑝/𝑅★𝑅𝑝/𝑅★(P22)𝑅𝑝(𝑅⊕)𝑅𝑝(𝑅⊕) (P22) 𝑏 𝑒 𝜔 (◦)𝑎/𝑅★𝐹𝑝(𝐹⊕)\nK00156.01 Kepler-114 c 8.041329±0.000005 143 .0401±0.0005 0.0229±0.0004 N/A 1.80+0.06\n−0.06N/A 0.58±0.06 0.05±0.04 18±104 19.42±1.78 45.74±4.33\nK00156.02 Kepler-114 b 5.188560±0.000004 145 .3632±0.0005 0.0172±0.0003 0.0172+0.0005\n−0.00051.35+0.05\n−0.051.37+0.06\n−0.060.51±0.09 0.05±0.04 19±102 14.52±1.31 81.83±7.66\nK00156.03 Kepler-114 d 11.776137±0.000005 142 .7052±0.0004 0.0364±0.0005 0.0343+0.0008\n−0.00082.85+0.09\n−0.092.73+0.11\n−0.110.65±0.04 0.04±0.03−8±102 24.52±1.81 28.68±2.23\nK00222.01 Kepler-120 b 6.312501±0.000005 132 .6551±0.0006 0.0328±0.0007 0.0323+0.0008\n−0.00082.54+0.09\n−0.092.53+0.10\n−0.100.39±0.13 0.05±0.03 13±105 16.73±1.50 56.62±5.15\nK00222.02 Kepler-120 c 12.794560±0.000012 130 .7675±0.0009 0.0267±0.0007 0.0258+0.0009\n−0.00092.07+0.08\n−0.082.02+0.09\n−0.090.42±0.13 0.05±0.04 19±101 26.97±2.49 21.77±2.04\nK00247.01 KOI-247.01 13.815046±0.000014 181 .1253±0.0007 0.0304±0.0010 0.0294+0.0020\n−0.00201.92+0.09\n−0.091.88+0.14\n−0.140.49±0.23 0.37±0.14 87±52 45.74±7.46 4.33±0.71\nK00248.03 Kepler-49 d 2.576569±0.000002 172 .1278±0.0006 0.0267±0.0006 0.0268+0.0004\n−0.00041.81+0.07\n−0.071.81+0.06\n−0.060.59±0.06 0.05±0.04 13±108 9.97±0.92 108.51±10.15\nK00248.04 Kepler-49 e 18.596055±0.000018 141 .2661±0.0008 0.0288±0.0011 0.0263+0.0008\n−0.00081.95+0.10\n−0.101.77+0.08\n−0.080.81±0.03 0.05±0.04 7±105 37.04±3.35 7.86±0.72\nK00249.01 Kepler-504 A b 9.549277±0.000007 175 .7568±0.0005 0.0396±0.0011 0.0392+0.0008\n−0.00081.77+0.07\n−0.071.77+0.07\n−0.070.39±0.21 0.38±0.11 94±38 45.08±6.07 3.10±0.42\nK00250.03 Kepler-26 d 3.543905±0.000004 136 .2606±0.0008 0.0184±0.0006 0.0208+0.0004\n−0.00041.20+0.05\n−0.051.37+0.05\n−0.050.43±0.13 0.05±0.04 13±102 12.67±1.18 70.16±6.67\n... ... ... ... ... ... ... ... ... ... ... ... ...\nMNRAS 000, 1–18 (2023)16 C. S. K. Ho et al.\nTable A3. TransitjitterandGaussianProcess(GP)parametersforthetransit\nfittingofplanetarysystemsperformedinthiswork.Onlythefirst10rowsare\nshown here; the full table is available online in a machine-readable format.\nKOI log𝜎lc log𝜎gp log𝜌gp\n156−7.0732±0.0011−8.8706±0.0167−3.7099±0.0404\n222−6.4394±0.0029−9.4256±1.5430−4.4393±1.7274\n247−6.6126±0.0043−8.4369±0.0699−3.9891±0.1803\n248−6.1755±0.0014−7.9431±0.0193−4.0146±0.0470\n249−6.5927±0.0032−8.6823±0.0644−3.4745±0.1402\n250−6.1652±0.0016−7.9600±0.0210−3.6427±0.0523\n251−6.3174±0.0020−8.3393±0.0431−3.8431±0.1113\n252−5.9738±0.0033−7.6507±0.0399−3.8177±0.0984\n253−5.7038±0.0021−7.9143±0.0578−3.6664±0.1233\n312−8.0250±0.0061−9.6177±0.0736−3.7928±0.2148\n... ... ... ...\nMNRAS 000, 1–18 (2023)Shallower radius valley around low-mass hosts 17Table A4. Planetaryparametersof100planetswith 0.5≤𝑀★/𝑀⊙<0.9,witharchivalplanetarymassmeasurementsavailable.TheenvelopefractioniscalculatedusingthemethoddescribedinSection4.4.Only\nthe first 10 rows are shown here; the full table is available online in a machine-readable format. Sources: 1. This work, 2. Berger et al. (2023), 3. Dai et al. (2019), 4. Rice et al. (2019), 5. Dumusque et al. (2019), 6.\nVines et al. (2023), 7. Orell-Miquel et al. (2023), 8. Barros et al. (2023), 9. Heidari et al. (2022), 10. Gillon et al. (2017), 11. Barros et al. (2022), 12. Bonomo et al. (2023), 13. Bourrier et al. (2022), 14. Barragán\net al. (2022), 15. Díaz et al. (2020), 16. Vanderburg et al. (2015), 17. MacDougall et al. (2021), 18. Mayo et al. (2023), 19. Lopez et al. (2019), 20. Korth et al. (2019), 21. Akana Murphy et al. (2021), 22. Persson\net al. (2018), 23. Mortier et al. (2018), 24. Lam et al. (2018), 25. Palle et al. (2019), 26. Diamond-Lowe et al. (2022), 27. Lillo-Box et al. (2020), 28. Damasso et al. (2019), 29. Ho & Van Eylen (2023), 30. Marcy\net al. (2014), 31. Fulton & Petigura (2018), 32. Van Eylen et al. (2018), 33. Xie (2014), 34. Petigura et al. (2022), 35. Almenara et al. (2018), 36. Hadden & Lithwick (2014), 37. Jontof-Hutter et al. (2016), 38.\nVissapragadaetal.(2020),39.Hadden&Lithwick(2017),40.Sunetal.(2019),41.MacDonaldetal.(2016),42.Astudillo-Defruetal.(2020),43.Westetal.(2019),44.Otegietal.(2021),45.Wilsonetal.(2022),\n46. Luque & Pallé (2022), 47. Turtelboom et al. (2022), 48. Nielsen et al. (2020), 49. Georgieva et al. (2021), 50. Deeg et al. (2023), 51. Espinoza et al. (2022), 52. Leleu et al. (2021), 53. Mallorquín et al. (2023),\n54. Naponiello et al. (2023), 55. Dai et al. (2023), 56. Hoyer et al. (2021), 57. Kunimoto et al. (2023), 58. Carleo et al. (2020), 59. Osborn et al. (2021), 60. Osborne et al. (2023), 61. Brinkman et al. (2023), 62.\nLacedelli et al. (2022), 63. Luque et al. (2021), 64. Burt et al. (2020), 65. Hawthorn et al. (2023), 66. Lillo-Box et al. (2023), 67. Maciejewski et al. (2023), 68. Polanski et al. (2021), 69. Peterson et al. (2018).\nPlanet 𝑃(days) 𝑅𝑝𝑅𝑝source 𝑀𝑝𝑀𝑝source RV/TTV mass 𝑀★𝑀★source𝐹𝑝(𝐹⊕)𝑆source Age (Gyr) Age source Envelope fraction\nGJ 9827 b 1.208912+0.000131\n−0.0001311.58+1.37\n−1.372 4.89+0.43\n−0.423 RV 0.60+0.02\n−0.022 269.58+13.16\n−12.852 13.25+9.74\n−9.492 0.0001\nGJ 9827 d 6.201470+0.000063\n−0.0000612.02+0.05\n−0.044 4.04+0.82\n−0.844 RV 0.60+0.02\n−0.022 30.12+1.65\n−1.652,4 13.25+9.74\n−9.492 0.0137\nHD 15337 b 4.755974+0.000018\n−0.0000181.69+0.69\n−0.692 7.20+0.81\n−0.815 RV 0.83+0.04\n−0.062 161.22+7.06\n−8.862 7.90+9.24\n−5.932 0.0001\nHD 15337 c 17.180721+0.000046\n−0.0000462.15+1.60\n−1.602 8.79+1.68\n−1.685 RV 0.83+0.04\n−0.062 29.09+1.27\n−1.602 7.90+9.24\n−5.932 0.0095\nHD 18599 b 4.137438+0.000004\n−0.0000042.61+0.71\n−0.712 25.50+4.60\n−4.606 RV 0.80+0.04\n−0.052 168.93+7.35\n−7.962 7.93+9.59\n−5.982 0.0095\nHD 191939 b 8.880324+0.000011\n−0.0000113.40+0.31\n−0.292 10.00+0.70\n−0.707 RV 0.89+0.05\n−0.082 98.82+4.68\n−6.332 6.22+7.55\n−4.652 0.0930\nHD 191939 c 28.579673+0.000079\n−0.0000792.91+0.32\n−0.302 8.00+1.00\n−1.007 RV 0.89+0.05\n−0.082 20.80+0.99\n−1.332 6.22+7.55\n−4.652 0.0709\nHD 191939 d 38.353532+0.000149\n−0.0001493.04+0.28\n−0.262 2.80+0.60\n−0.607 RV 0.89+0.05\n−0.082 14.05+0.67\n−0.902 6.22+7.55\n−4.652 0.0847\nHD 207496 b 6.440982+0.000015\n−0.0000152.33+0.40\n−0.402 6.10+1.60\n−1.608 RV 0.77+0.04\n−0.062 77.16+3.36\n−4.472 10.38+10.52\n−7.682 0.0199\nHD 207897 b 16.201703+0.000051\n−0.0000512.68+0.29\n−0.282 14.40+1.60\n−1.609 RV 0.78+0.04\n−0.042 25.94+1.17\n−1.152 7.82+9.91\n−5.972 0.0363\n... ... ... ... ... ... ... ... ... ... ... ... ... ...\nMNRAS 000, 1–18 (2023)18 C. S. K. Ho et al.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–18 (2023)"    },    {        "title": "2401.12390v1.The_Temperature_and_Density_of_a_Solar_Flare_Kernel_Measured_from_Extreme_Ultraviolet_Lines_of_O_IV.pdf",        "content": "Draft version January 24, 2024\nTypeset using L ATEXmodern style in AASTeX631\nThe Temperature and Density of a Solar Flare Kernel Measured from\nExtreme Ultraviolet Lines of O IV\nPeter R. Young\n1, 2\n1NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA\n2Northumbria University, Newcastle upon Tyne, NE1 8ST, UK\nSubmitted to ApJ\nABSTRACT\nPreviously-unexplored diagnostics of O ivin the extreme ultraviolet region 260–\n280˚A are used to derive a temperature and density for a solar flare kernel observed\non 2012 March 9 with the Extreme ultraviolet Imaging Spectrometer on the Hinode\nsatellite. Seven lines from the 2 s2p2–2s2p3stransition array between 271.99 and\n272.31 ˚A are both temperature and density sensitive relative to the line at 279.93 ˚A.\nThe temperature, T, is constrained with the λ268.02/ λ279.93 ratio, giving a value of\nlog (T/K) = 5 .10±0.03. The ratio λ272.13/ λ279.93 then yields an electron number\ndensity, Ne, of log ( Ne/cm−3) = 12 .55 with a lower limit of 11.91, and an upper limit of\n14.40. The O ivemitting volume is estimated to be 0.4′′(300 km) across. Additional\nOivlines at 196, 207 and 260 ˚A are consistent with the derived temperature and\ndensity but have larger uncertainties from the radiometric calibration and blending.\nDensity diagnostics of O vand Mg viifrom the same spectrum are consistent with\na constant pressure of 1017.0K cm−3through the transition region. The temperature\nderived from O ivsupports recent results that O ivis formed around 0.15 dex lower at\nhigh densities compared to standard “zero-density” ionization balance calculations.\n1.INTRODUCTION\nThe standard model for solar flares (Benz 2017) invokes energy release in the corona\nby magnetic reconnection to produce energetic particles and thermal conduction\nfronts that propagate to the solar surface along magnetic field lines. Energy is dis-\nsipated in the chromosphere giving rise to bright flare ribbons in H α, for example.\nEmission along the ribbons is not uniform and there appear bright “kernels” likely\ndue to localized enhanced heating. The chromospheric plasma can be heated to tem-\nperatures >10 MK, and so the ribbons and kernels can be seen in a wide range of\nultraviolet emission lines.arXiv:2401.12390v1  [astro-ph.SR]  22 Jan 20242\nModels of energy transfer and heating along 1D flare loops such as RADYN (Allred\net al. 2015) are able to predict the evolution of temperature and density in a flare\nloop at high cadence. From observations we seek to provide constraints on these\nparameters using line ratio diagnostics. To apply line ratio diagnostics, it is necessary\nto consider optically-thin lines, which limits diagnostics to ions formed above around\n50,000 K.\nGenerally we expect flare loops to evolve under constant pressure, and so the highest\ndensities will be at the lowest temperatures. Previous work has found transition\nregion densities of 1011cm−3to 1013cm−3(Widing & Cook 1987; Polito et al. 2016;\nYoung et al. 2018). The number of line ratio diagnostics sensitive in this range is very\nlimited, but the intercombination lines of O ivand S ivnear 1400 A have been applied\nto data from the currently-operating Interface Region Imaging Spectrograph (IRIS:\nDe Pontieu et al. 2014) mission and the earlier Skylab (Widing & Cook 1987) and\nSolar Maximum missions (Hayes & Shine 1987). The O ivdiagnostics are sensitive\nup to about 1012.5cm−3, and the S ivdiagnostics are sensitive to 1013cm−3. Polito\net al. (2016) provided an example of a flare observed by IRIS that exhibited densities\nat the maxima of these ranges. It is possible to infer higher densities than allowed\nby these ions by comparing an intercombination line with a resonance line, with O iv\nλ1401.16 / Si ivλ1402.77 a particularly attractive ratio as the lines are nearby and\nclose in formation temperature. It has been applied to flare data by Feldman et al.\n(1977), Polito et al. (2016) and Young et al. (2018) but it is subject to many more\nuncertainties than the single-ion ratios, as discussed in these articles.\nModern UV and EUV solar spectrometers are generally confined to relatively narrow\nwavelength ranges due to optical design limitations, the size of electronic detectors,\nand the need for high spectral resolution. There is an approximate dividing line\nat 400 ˚A such that coronal lines formed at temperatures of 1 MK and above lie at\nshorter wavelengths and chromospheric and transition region lines at longer wave-\nlengths (Young 2021). This limits the ability of the spectrometers to make density\nmeasurements over a wide temperature range. The Extreme ultraviolet Imaging Spec-\ntrometer (Culhane et al. 2007) on Hinode falls on the coronal side of this dividing line\nand has a number of density diagnostics in the 0.7–6 MK region that have previously\nbeen applied to flares (e.g., Warren et al. 2008; Milligan 2011). However, the EIS\nwavelength ranges of 170–212 ˚A and 246–292 ˚A contain a number of emission lines\nof Oivthrough O vithat extend the EIS temperature coverage to the mid-TR. The\nlines are much weaker than these ions’ main resonance lines in the 500–1100 ˚A region,\nbut they are measurable in many circumstances and have been used for diagnostics\n(e.g., Muglach et al. 2010).\nIn this article a surprisingly rich spectrum of O ivlines in an EIS spectrum of a\nflare kernel is presented and used to provide constraints on temperature and density.\nThis extends the EIS diagnostic coverage to a temperature region that is normally\nthe preserve of longer wavelength spectrometers such as IRIS.3\nSection 2 describes the availability of data and software used for this article. Sec-\ntion 3 discusses the atomic model for O ivand the available emission lines and diag-\nnostics. The EIS flare dataset is described in Section 4 and the data processing steps\nare given. Section 5 explains how the O ivlines are measured, and Section 6 presents\nthe results from the diagnostic ratios. A comparison with diagnostics from other ions\nis given in Section 7, and Section 8 discusses the results in relation to a flare model.\nFinally, results are summarized in Section 9.\n2.DATA AND SOFTWARE\nSpectra from EIS are analyzed in the present work. EIS was launched on board\ntheHinode spacecraft in 2006, and it is an imaging slit spectrometer operating in\nthe wavelength bands 170–212 ˚A and 246–292 ˚A—referred to as the short-wavelength\n(SW) and long-wavelength (LW) channels, respectively. Four slit options are avail-\nable, including the narrow 1′′and 2′′slits for emission line spectroscopy. Spectral\nresolution is 3000–4000 and spatial resolution is 3′′-4′′(Young & Ugarte-Urra 2022).\nEIS level-0 data are available from the Virtual Solar Observatory1, and are calibrated\nto level-1 using routines in the SolarSoft (Freeland & Handy 1998, 2012) distribution.\nImages from the Atmospheric Imaging Assembly (AIA: Lemen et al. 2012) are used\nto provide context images and identify the position of the EIS slit in relation to the\nrapidly developing flare ribbon. AIA obtains full-disk solar images in seven EUV\nchannels at 12 s cadence. Cutout images were downloaded from the Joint Science\nOperations Center and respiked as flare ribbon pixels are often incorrectly flagged\nas energetic particle hits by the AIA processing pipeline (Young et al. 2013). After\nrespiking, the images were then cleaned of energetic particle hits with the routine\naiaclean cutout sequence (Young et al. 2021b).\nMost software used for data preparation and analysis for this article are available\ninSolarsoft . Additional software of the author for AIA analysis are available in a\nGitHub repository2. Atomic data and software for computing emissivities are taken\nfrom version 10.1 of the CHIANTI atomic database (Young et al. 2016; Dere et al.\n2023), which is distributed through Solarsoft . The spectra and line fit data from this\narticle are available on Zenodo at doi:10.5281/zenodo.6726426.\n3.THE O IV ATOMIC MODEL AND SPECTRUM\nOivhas a rich spectrum of emission lines through the far and extreme ultraviolet.\nTransitions between n= 2 configurations (2 s22p, 2s2p2) give rise to the strongest\nemission lines with three key groups of resonance lines at 553–556 ˚A, 608–610 ˚A and\n787–791 ˚A. The lines observed by EIS and discussed in the present work arise from\nn= 3 and n= 4 configurations. Figure 1 and Table 1 summarize these lines, and\nfurther details are given later in this Section.\n1https://virtualsolar.org.\n2https://github.com/pryoung/sdo-idl.4\nVersion 10.1 of CHIANTI was used for modeling the O ivemission lines in this\narticle. Experimental energies come from the NIST database, Sandlin et al. (1986)\nand Feuchtgruber et al. (1997). Radiative decay rates are from Corr´ eg´ e & Hibbert\n(2004) and Liang et al. (2012), and electron collision strengths are from Liang et al.\n(2012). For the lines in the 260–280 ˚A range, the NIST energies were derived from\nwavelength measurements of Edl´ en (1934) for which the uncertainties range from 6\nto 11 m ˚A.\nOivis formed at middle transition region temperatures. The equilibrium ionization\nbalance data provided by CHIANTI gives a peak ionization at log ( Tiz/K) = 5 .15.\nNon-Maxwellian particle distributions, which are likely during flares, can significantly\nmodify the ionization fraction distribution of O iv(Dud´ ık et al. 2014), but for the\nhigh density found here (Section 6) such distributions would be expected to be quickly\nthermalized by collisions. For a specific O ivtransition, the temperature of formation\nis better represented by the temperature at which the contribution function peaks\n(Tcf). The transitions considered in the present work have high excitation potentials,\ngiving larger emissivities at higher temperatures. They therefore have formation\ntemperatures of log ( Tcf/K) = 5 .25 to 5.30. For a specific solar feature for which a\ndifferential emission measure (DEM) curve is defined, a third formation temperature\ncomes from multiplying the contribution function by the DEM and temperature (see,\nfor example, Young 2023a). This temperature is referred to as Tdemhere. Since the\nstandard quiet Sun, active region and flare DEMs distributed with CHIANTI show\nDEM curves that decrease with increasing temperature through the O ivformation\nregion, then Tdemwill be lower than Tcf. The issue of the formation temperature of\nOivlines is returned to in Section 6.\nEUV density diagnostics arise from population shifting from the ground level to\nhigher excitation levels as the increasing electron density causes greater electron ex-\ncitation rates. The population of an excited level increases until it reaches a quasi-\nBoltzmann equilibrium (QBE) with the ground level. O ivhas two2P1/2,3/2levels in\nthe ground configuration and these reach QBE at log ( Ne/cm−3) = 4 .0, which is too\nlow for solar atmosphere density diagnostics. The next excited levels are the three\n4P1/2,3/2,5/2levels in the 2 s2p3configuration. These come into QBE with the ground\nlevel over log ( Ne/cm−3) = 9 to 12, which is ideal for the solar atmosphere. The\nintercombination transitions coming directly from the quartet levels are found in the\n1397–1408 ˚A wavelength range and show density sensitivity relative to each other.\nThey have been extensively applied in both solar and stellar spectra (e.g., Hayes &\nShine 1987; Keenan et al. 2002; Polito et al. 2016).\nAny quartet term at higher excitation energies will become principally excited from\nthe 2 s2p3 4Plevels as they gain population, and hence transitions from these levels\nwill form useful density diagnostics in relation to lines from doublet levels that are\nprincipally excited from the ground2Pterm. An example is the set of three 2 s2p2\n4PJ– 2p3 4S3/2transitions at 625 ˚A, that form excellent diagnostics with the set of5\nFigure 1. A CHIANTI O ivspectrum computed at log ( T/K) = 5 .15 and log ( Ne/cm−3) =\n9.36 with line widths set to 3 ˚A. Transition arrays (TAs) for the spectral features are in-\ndicated, and the wavelength ranges of the EIS SW and LW channels are highlighted with\nblack horizontal lines.\nfour 2 s22p2PJ– 2s2p2 2PJ′transitions near 554 ˚A. These lines were observed with the\nCoronal Diagnostic Spectrometer (Harrison et al. 1995) and Young & Mason (1997)\npresented an observation of an intense transition region brightening that yielded a\ndensity of log ( Ne/cm−3) = 11 .3.\nFigure 1 shows a synthetic O ivspectrum in the region 160–300 ˚A derived with\nCHIANTI and computed at log ( T/K) = 5 .15 and log ( Ne/cm−3) = 9 .36. Seven\ntransition arrays (TA) are identified corresponding to transitions between different\nsets of configurations. A low resolution is used for the spectrum to better differentiate\nthe TAs. Terms, wavelengths, and theoretical intensities for the lines in the EIS\nwavelength bands are given in Table 1. The theoretical intensities are the same as\nthose used to derive the spectrum in Figure 1, but normalized so that the intensity\nof the line at 279.93 ˚A is 100.\nThe strongest line in the displayed spectrum is from TA2, but is outside the EIS\nwavebands. The next strongest is from TA1 at 279.93 ˚A that is observed by EIS,\nand has been used for Doppler shift measurements by Landi et al. (2010) and Young\net al. (2013), and for temperature and density diagnostics by Muglach et al. (2010)\nand Landi et al. (2010).\nAs discussed above, density diagnostics can be formed from ratios of quartet-quartet\ntransitions to doublet-doublet transitions. The only quartet-quartet transitions be-\nlong to the TA5 multiplet with lines around 272 ˚A, and these lines are conveniently\nclose in wavelength to the 279.93 ��A line. Figure 2(a) shows the predicted variation of\nλ272.13/ λ279.93 with density obtained with CHIANTI. However, it can be seen that\nthe ratio also has some sensitivity to temperature. To accurately derive the density it6\nTable 1. Oivtransitions in the EIS wave-\nlength bands. Transition arrays (TAs) corre-\nspond to those shown on Figure 1.\nTA Terms J–J′λa(˚A) Ib\nTA32P–2D1/2–3/2 195.860 10.7\n3/2–5.2 196.006 19.7\n3/2–3/2 196.008 2.2\nTA42P–2D1/2–3/2 207.183 5.8\n3/2–5/2 207.239 10.6\n3/2–3/2 207.348 1.2\nTA6a2D–2F5/2–7/2 260.389 6.8\n3/2–5/2 260.556 4.7\nTA6b2D–2D5/2–5/2 266.931 3.6\n3/2–3/2 266.981 2.4\nTA72D–2P5/2–3/2 268.024 4.2\n3/2–1/2 268.064 2.3\nTA54P–4P3/2–5/2 271.990 9.6\n1/2–3/2 272.076 8.4\n5/2–5/2 272.127 22.4\n3/2–3/2 272.173 2.7\n1/2–1/2 272.176 1.6\n3/2–1/2 272.273 7.8\n5/2–3/2 272.310 9.1\nTA12P–2S1/2–1/2 279.631 49.9\n3/2–1/2 279.933 100.0\naThe lines used for density and temperature\ndiagnostics (Figure 2) are underlined.\nbIntensity computed with CHIANTI and nor-\nmalized such that λ279.93 has an intensity of\n100.\nis therefore necessary to also constrain the plasma temperature, and this is possible\nwith some of the other lines observed by EIS.\nThe TA6 and TA7 transitions arise from atomic levels with excitation energies\naround 40 % higher than that of the TA1 transitions. The Boltzmann factor in the\nexpression for the electron excitation rate therefore means that the ratios of the TA6\nand TA7 lines to the TA1 lines are strongly sensitive to temperature. Figure 2(b)\nshows the λ268.02/ λ279.93 ratio, which uses a TA7 transition. As both lines are\ndoublet-doublet transitions, then there is negligible density sensitivity.7\nFigure 2. Theoretical O ivline ratios from CHIANTI. (a) The λ272.13/ λ279.93 ratio as a\nfunction of density, calculated for three temperatures: log ( T/K) = 5 .10 (cyan), 5.15 (teal)\nand 5.20 (orange). (b) The λ268.02/ λ279.93 ratio as a function of temperature, calculated\nfor log ( Ne/cm−3) = 12 .0. The crosses with error bars show the values derived from the\nflare kernel (Section 6).\nThe combination of λ268.02, λ279.93 and the lines at 272 ˚A together allow the\ntemperature and density to be constrained. The fact that the lines are relatively\nclose together in the spectrum helps mitigate uncertainties in the EIS radiometric\ncalibration (Appendix A).\n4.OBSERVATION AND DATA PREPARATION\nThe active region with NOAA designation 11429 crossed the solar disk during 2012\nMarch 3–14 and was one of the most intensively-studied of Solar Cycle 24. A search of\nabstracts in the SAO/NASA Astrophysics Data System for “11429” yields 33 journal\narticles, many of which focus on the coronal mass ejections from the region (e.g., Liu\net al. 2014; Chintzoglou et al. 2015; Dhakal et al. 2020). One X-class flare occurred\non March 5 and two more on March 7. Additional flares with classifications of M5\nand higher occurred on March 9, 10 and 13. In the present article EIS data of the\nM6.3 class flare on 9 March are analyzed. The evolution of the X-ray 1–8 ˚A light\ncurve obtained by the GOES-15 spacecraft was complex (Doschek et al. 2013), with\nan initial rise at 03:24 UT, followed by three maxima at 03:27 UT, 03:34 UT and\n03:53 UT.\nImages from AIA are available for the flare, and Figure 3 shows the rapid evolution\nof a section of the flare ribbon during a 1-min period as seen in the 94 ˚A channel. This\nchannel is the only one of the seven EUV channels that does not show saturation and\nso it best shows the rapid evolution of the flare ribbon. Normally in flare conditions\nthis channel is dominated by the Fe xviii 93.93 ˚A (Tiz= 7 MK) line, but the EIS\nspectrum shows that lines formed at 4 MK and higher are weak (see below). In this\ncase the 94 ˚A channel emission mostly comes from Fe xand Fe xiv(Del Zanna 2012).8\nFigure 3. Five consecutive AIA 94 ˚A images showing the development of the flare ribbon.\nA linear intensity scaling is used. Vertical lines on Panel (c) show the location of the EIS\nslit at the time the EIS spectrum was obtained. A blue arrow highlights the time of peak\nbrightness of the flare kernel.\nFigure 3 shows the rapid brightening of one kernel that reaches its peak brightness in\nPanel (e). It is this feature that is observed by EIS and Panel (c) shows the location\nof the EIS slit at the time of this exposure.\nEIS obtained a single raster of the flaring region beginning at 03:09 UT and ending\nat 03:41 UT. Sixty exposures of 30 s were obtained with the 2′′slit and a step size\nof 2′′was used. The raster area was 120′′×160′′. Figure 4 shows EIS raster images\nformed in three emission lines for a sub-region corresponding to exposures 22 to\n60 (EIS rasters from right to left). The compact brightening at the left of Panel (a)\ncorresponds to the flare kernel in Figure 3(e). To determine the spatial location of the\nEIS slit, synthetic raster images were generated from the AIA 94 ˚A images through the\nprocedure described in Young (2023b). These images were compared with the image\nfrom the EIS Fe xiv274.20 ˚A line since Fe xivis one of the two major contributors\nto the AIA channel.3By adjusting the EIS slit in relation to the AIA frames it was\npossible to determine that the EIS slit was positioned as shown in Figure 3(c) at the\ntime EIS exposure 57 was taken. This exposure began at 03:39:07 UT and completed\nat 03:39:37 UT. It thus captured the initial brightening of the kernel. The kernel\ncontinued to brighten after this exposure, when the EIS slit moved to the left by 2′′.\nAlthough the slit was not directly overlying the kernel for this exposure, the lower\nspatial resolution of EIS meant that the kernel still gave a strong signal, hence it is\nthis exposure that is brightest in the EIS images (Figure 4).\nFigure 4 demonstrates that the flare kernel is observed at temperatures between\n0.1 and 3 MK, although it is significantly brighter in O ivin relation to the region\nat (−20,−400) compared to the hotter lines. The kernel is very weak or not visible\nin lines formed at 4 MK or higher, as found by considering Fe xvii λ204.67, Ca xv\nλ200.98 and Ca xviλ208.59.\n3The other is Fe xand the EIS Fe x184.54 ˚A image showed very similar morphology to the Fe xiv\nimage.9\nFigure 4. EIS raster images of AR 11429 in three emission lines. The temperature of\nformation of each ion is indicated, and the image column corresponding to exposure 57 is\nhighlighted on Panel (a). A square-root intensity scaling is applied to each image.\nExposure 57 was chosen rather than the brighter exposure 58 because the latter\nexhibited lines that are asymmetric, with an extended long-wavelength wing making\nspectral modeling more difficult. A spectrum averaged across the brightening in ex-\nposure 57 was obtained by first calibrating the level-0 EIS FITS file using the IDL\nroutine eisprep with the standard options described in Young (2022a) plus the addi-\ntional option /nocr . This prevented the routine from applying the cosmic ray removal\nprocedure as it was found that the intense, compact nature of the brightening led to\nspurious cosmic ray detections. A pixel mask was then defined to select seven y-pixels\nin exposure 57 centered on the brightest pixel. The IDL routine eismask spectrum\nwas then run to create 1D spectra for the kernel for the SW and LW channels based\non this mask. Intensities quoted in this article are derived by summing the intensities\nof the seven individual pixels rather than averaging, as it is assumed the flare kernel\nis not spatially resolved by EIS. This is confirmed in Section 6 where the plasma\nemitting volume is estimated. There are two radiometric calibration options for EIS\nand the EIS data analysis guide does not recommend one over the other. One is due\nto Del Zanna (2013) and the other due to Warren et al. (2014), which are referred to\nas GDZ13 and WUL14 in the remainder of this article. There are significant differ-\nences between the two that impact the O ivdiagnostics. The WUL14 calibration is\npreferred here and further details are given in Appendix A and the following text.\nThe/shift option was used in the call to eismask spectrum , which applies corrections\ndue to the tilt of the EIS slit and the thermally-induced orbit variation of the spectrum\non the detector. This leads to wavelengths that should be accurate to around 5 km s−1\n(Kamio et al. 2010) for lines in the SW channel. There is an additional time-varying\nwavelength variation between the SW and LW channels (Young et al. 2012) that\nwas corrected by forcing Fe viiiλ185.21 and Si viiλ275.37 to have the same Doppler\nshift, as recommended by Young et al. (2012). Doppler velocities for the O ivlines\nare given in Table 2, where they can be compared with other lines formed between\n0.2 and 0.6 MK that are measured in the spectrum.10\nDoschek et al. (2013) used the same EIS dataset to investigate chromospheric evap-\noration during the flare, concentrating on the bright emission around ( −20,−400).\nThey did note, however, that the feature at the left side of the raster was brightest\nin transition region ions.\n5.EMISSION LINE MEASUREMENTS\nEmission line intensities for the seven groups of O ivlines listed in Table 1 are\nderived through fitting Gaussian functions. Due to self-blending within the groups\ncertain constraints are applied to the fits. For example, forcing the O ivlines to\nhave the same width and/or fixing the wavelength separations of the lines using the\nwavelengths from CHIANTI. The Gaussian fit parameters centroid, λmeas, full-width\nat half-maximum, w, and integrated intensity, I, are given in Table 2. The reference\nwavelengths, λref, are used to derive line-of-sight (LOS) Doppler velocities, vLOS. The\nuncertainties combine fitting uncertainties with radiometric calibration uncertainties\n(Appendix A). The two EIS wavelength channels are discussed separately below.\nIn addition to the O ivlines, a number of other emission lines are fit and presented\nin Table 2. These are either discussed in the following text or the lines are strong\ntransition region lines for which the Doppler velocities can be compared with O iv. It\ncan be seen that the transition region line velocities range from +20 to +45 km s−1,\nwith a trend of increasing redshift with ionization level. The coronal lines have smaller\nvelocities, however.\n5.1. Lines in the LW channel\nThe two lines from TA6a at 260.39 ˚A and 260.56 ˚A are weak and partly blended\nwith other species. In addition, the presence of multiple lines close together through\nthis part of the spectrum makes estimating the spectrum background level difficult.\nRather than attempt the fitting of multiple Gaussian functions, the line intensities are\nestimated by simply over-plotting the Gaussian fit function of the stronger 279.93 ˚A\nline on the spectra at the locations of the lines, as shown in Figure 5(a). The TA6a\nlines are assumed to have the same velocity as λ279.93. The relative strengths of the\nTA3a lines are set to the ratios given in CHIANTI, and their absolute magnitude has\nbeen set to 1/16 of the 279.93 ˚A line. The continuum level was estimated from a\nline-free region outside of the displayed wavelength region. It is clear that the two\nprincipal TA6a lines are a good match for two features in the spectra, although both\nare partly blended with nearby stronger lines. The feature at 260.3 ˚A is unknown,\nbut that at 260.7 ˚A is Fe vii260.67 ˚A (Young & Landi 2009). Based on a by-eye\ncomparison of estimated profiles and the observed spectrum, the λ279.93/ λ260.39\nratio is estimated to be 16 ±2. This value is discussed further in Section 6.\nThe TA6b lines at 266.93 ˚A and 266.98 ˚A (Table 1) are blended with each other at\nthe EIS resolution, and a line is clearly seen at this location (Figure 5(b)). However,\nit is too strong to be explained by the O ivlines, as illustrated by the over-plotted11\nTable 2. Gaussian fit parameters and identifications for selected\nemission lines.\nλmeas w I Ion λrefavLOS\n(˚A) (m ˚A) (mW m−2sr−1) ( ˚A) (km s−1)\nTransition region lines\n184.143 89 14987 ±210 O vi 184.117 41.7\n185.240 86 61278 ±329 Fe viii 185.213 44.2\n192.927 92 19908 ±126 O v 192.904 35.3\n195.882 78 896 ±70 O iv 195.860 33.7\n207.251 65 1526 ±126 O iv 207.239 18.1\n248.480b102 10990 ±238 O v 248.460 24.0\n249.154 119 18319 ±294 Si vi 249.124 35.7\n250.157 97 1729 ±91 Al viii 250.134 27.3\n268.046 102 357 ±35 O iv 268.024 24.2\n269.019 105 15813 ±798 Mg vi 268.991 31.7\n272.150 106 7483 ±385 O iv 272.127 25.6\n275.409 118 20811 ±1050 Si vii 275.368c44.2d\n276.178 103 5761 ±301 Mg vii 276.142c39.1\n276.613 97 24171 ±1218 Mg v 276.579 36.3\n279.656 89 4977 ±322 O iv 279.631 27.1\n279.959 96 10941 ±700 O iv 279.933 28.2\n280.769 106 24766 ±210 Mg vii 280.729c43.0\nCoronal lines\n251.956 124 15190 ±224 Fe xiii 251.952 5.1\n261.067 119 7294 ±98 Si x 261.056 12.6\n272.009 138 8694 ±462 Si x 271.992 18.3\naFrom CHIANTI unless otherwise indicated.\nbIncludes contribution of 147 mW m−2sr−1from Al viiiλ284.45 (Sec-\ntion 7).\ncYoung (2023).\ndVelocity forced to be the same as Fe viiiλ185.21 (Section 4).\nblue profile on Figure 5(b). As above, the profiles are Gaussians with the same\nwidth as λ279.93 but shifted to the expected locations of the TA6b lines using the\nCHIANTI wavelengths. Relative amplitudes between the TA6b lines are set by the\nCHIANTI theoretical ratios, and the absolute magnitude is set by using the CHIANTI\nλ279.93/ λ266.93 ratio of 20.8 obtained at log ( Ne/cm−3) = 12 .0 and log ( T/K) = 5 .10\n(Section 6). As the TA6b lines can not be reliably separated from the stronger feature,\nthey are not used in the present analysis.12\nFigure 5. Portions of the flare spectrum (black) showing the O ivlines near 260.5 ˚A (a),\n267.0 ˚A (b), and 268.0 ˚A (c), corresponding to the TA6a,b and TA7 transitions, respectively.\nBlue curves for Panels (a) and (b) show the estimated total line profiles and red curves show\nindividual O ivcomponents. The blue curve on Panel (c) shows the total multi-Gaussian\nfit function, and the red curves show the individual O ivcomponents of the TA7 line group.\nFigure 6. (a) The spectral region containing the O ivmultiplet around 272 ˚A. The multi-\nGaussian fit to the entire spectral feature is plotted in blue, the seven individual O iv\ncomponents are shown in red, and Si xλ271.99 is shown in cyan. (b) The spectral region\ncontaining O ivλλ279.63,279.93, with the two-Gaussian fit to the lines over-plotted in blue.\nA line at 268 ˚A corresponds to a TA7 multiplet that is also blended at the EIS\nresolution. Measurement of this feature is complicated by a broad spectral feature\nto the short-wavelength side, and a weak line on the long-wavelength side. A multi-\nGaussian fit was performed, and the result is shown in Figure 5(c). The three O iv\nlines (shown in red) were fit by forcing the two weaker lines to have fixed wavelength\nseparations relative to the stronger line. Widths were forced to be equal, and the\namplitudes were fixed to the ratios predicted by CHIANTI. The two weaker, unknown\nlines at longer wavelengths were free to vary and the background was forced to be\nflat.13\nThe seven O ivlines around 272 ˚A are part of a complex spectral feature (Fig-\nure 6(a)) that includes Si xλ271.99, which is dominant in most solar conditions. The\nintensity of the complete feature is around 4100 mW m−2sr−1. There is an Fe xiii\nline expected at 272.19 ˚A, but this can be neglected: the Fe xiiiλ272.19/ λ251.96\nratio is predicted from CHIANTI to have a fixed ratio of 0.019, and the measured\nλ251.96 intensity (Table 2) implies a λ272.19 intensity of only 41 mW m−2sr−1, so\nonly 1 % of the total feature. A highly-constrained multi-Gaussian fit was performed\nto estimate the O ivintensities. Since the seven O ivlines show little temperature\nor density sensitivity relative to each other, then the parameters of six of the lines\nwere tied to those of the strongest line at 272.13 ˚A. CHIANTI yields the predicted\nratios of all the lines relative to λ272.13, and their wavelength offsets. Thus the peaks\nof the six lines were forced to be fixed ratios relative to λ272.13, the centroids were\nforced to have fixed offsets relative to λ272.13 and the widths were forced to be the\nsame as λ272.13. Three additional Gaussians were added to the fit function, repre-\nsenting Si xλ271.99, Fe viiλ271.69 and the unknown line at about 271.85 ˚A. The\nspectrum background was fit with a linear function. The auto fitandmask spectrum\nsuites of routines (Young 2022b) were used to prepare the data, set up the template\nand perform the fit. The results are shown in Figure 6(a) where the seven O ivlines\nare displayed as red curves, and the line fit parameters of λ272.13 line are given in\nTable 2. The combined intensity for all of the O ivlines is 2842 ±216 mW m−2sr−1,\nalthough we only consider the λ272.13 intensity for the remainder of this article. The\nwidth and Doppler velocity for λ272.13 are in good agreement with the other LW\nOivlines in Table 2, giving confidence in the fit.\nSixλ271.99 is shown as a cyan curve in Figure 6(a), and the fit parameters are given\nin Table 2. Also given in this table are the fit parameters for Si xλ261.06, which is\nisolated in the spectrum and fit with a single Gaussian. The λ261.06/ λ271.99 ratio is\ninsensitive to density with an empirical value from off-limb quiet Sun spectra of 0.97\nusing the WUL14 calibration. The ratio from the flare spectrum is 0.84, lower than\nexpected, and the λ271.99 width is larger and the redshift larger. This suggests the\nmulti-Gaussian fit for the complex feature at 272 ˚A is over-estimating the Si xline.\nThis could be due to the O ivintensities being under-estimated by around 5%, but\nalso could be due to an unknown blend close to the Si xline. As a 5 % uncertainty\nwas added to the O ivintensity measurements (Appendix A), the multi-Gaussian fit\nto the O ivlines is retained.\nThe two lines at 279 ˚A are isolated in the spectrum, however the background level\nin the spectrum is difficult to estimate due to weak features neighboring the lines.\nFits were performed for two extreme choices of the spectrum background, and for a\ncompromise estimate of the background. The latter fit is shown in Figure 6(b), and\nthe fit parameters are given in Table 2. The uncertainties on the integrated intensities\nhave been increased by 4% of the measured intensities (added in quadrature), based14\non comparisons with the two extreme fits. The λ279.63/ λ279.93 intensity ratio is\n0.455±0.026 which is slightly lower than the expected ratio of 0.499 from CHIANTI.\n5.2. Lines in the SW channel\nThe large uncertainties for the SW–LW cross-calibration as inferred from the GDZ13\nand WUL14 calibrations (Appendix A) mean that the SW O ivlines can not be\nreliably used in diagnostics with the LW lines. However, in this section the SW lines\nare measured and in Section 6 they are assessed in terms of the plasma parameters\ninferred from the LW channel lines.\nTwo lines from TA3 are expected at 195.86 ˚A and 196.01 ˚A, with an expected\nratio of 1:1.8. Figure 7(a) shows this region of the spectrum, which is dominated by\nlines of Fe viiand Fe viii(Young et al. 2021a). The stronger of the two O ivlines is\nblended with Fe viii195.97 ˚A, which is stronger in all solar conditions including the\nflare kernel considered here. A weak feature is seen at the expected location of the\n195.86 ˚A line, and a two-Gaussian fit was performed to fit this feature and an Fe ix\nline of comparable intensity at shorter wavelengths. The fit is shown in Figure 7(a)\nand the O ivfit parameters are given in Table 2. The Doppler velocity is in reasonable\nagreement with the LW channel lines. The main uncertainty (not contained in the\nformal fitting uncertainty in Table 2) is the spectrum background level, which is not\nwell constrained in this region.\nTA4 has two lines at 207.18 and 207.24 ˚A that have a predicted ratio of 1:1.8. Landi\net al. (2010) gave an intensity measurement for λ207.24 for an off-limb coronal mass\nejection observation with EIS, and used the λ207.24/ λ279.93 ratio to obtain lower\nlimits on the density at two times during the event. In the present spectrum, and also\nthe loop footpoint spectra of Landi & Young (2009) and Del Zanna (2009) there is a\nline at 207.15 ˚A due to Fe viii, with O ivresponsible for the weaker emission on the\nlong-wavelength side of this line (Figure 7(b)). A multi-Gaussian fit was performed\nto lines in this region, and is shown as the blue curve on Figure 7(b). The three O iv\nlines are shown in red, and each of the three Gaussian parameters of λ207.18 and\nλ207.35 were tied to those of λ207.24. In particular, the peaks were set to factors\n0.55 and 0.114 of the peak of λ207.24 based on the CHIANTI emissivity model, and\nthe centroid offsets were fixed based on the CHIANTI wavelengths. The fit parameters\nforλ207.24 are given in Table 2, and the Doppler velocity is in reasonable agreement\nwith the other O ivlines, but the width is much narrower. The reduced χ2value\nfor the fit is rather high at 6.2, and there is no evidence for the weak O ivline at\n207.35 ˚A in the spectrum (Figure 7(b)).\n6.DIAGNOSTICS\nFigure 2 shows the temperature and density derived from the λ268.02/ λ279.93 and\nλ272.13/ λ279.93 ratios. The temperature is log ( T/K) = 5 .10±0.03, and the density\nis log ( Ne/cm−3) = 12 .55 with a lower limit of 11.91. The upper limit of the measured\nratio is seen to be above the maximum value of theoretical ratio. However, the15\nFigure 7. (a) The spectral region containing the weak O iv195.86 ˚A line. A two-Gaussian\nfit is shown in blue with the O ivline represented by the red curve. (b) The spectral\nregion containing the O ivTA4 lines, with a five-Gaussian fit shown in blue. The three O iv\ncomponents are shown in red.\ntheoretical ratio reaches a maximum at log ( Ne/cm−3) = 13 and then decreases, thus\nthe lower limit of the measured ratio also yields an upper limit to the density of\nlog (Ne/cm−3)≤14.40.\nThe derived temperature is surprisingly low. The contribution function of λ279.93\npeaks at log ( T/K) = 5 .26. When convolved with the flare differential emission mea-\nsure (DEM) curve distributed with CHIANTI, the function peaks at log ( T/K) = 5 .20\nas the DEM increases with decreasing temperature. CHIANTI has the option to com-\npute a revised ionization equilibrium with density-dependent suppression factors for\ndielectronic recombination (Del Zanna et al. 2021). This has been demonstrated to\nshift the formation temperatures of oxygen ions to lower temperatures (Young et al.\n2018). Computing the O ivfractions using DR suppression at log ( Ne/cm−3) = 12 .0\ngives a λ279.93 contribution function that peaks at log ( T/K) = 5 .18, and convolving\nwith the CHIANTI flare DEM gives a peak at log ( T/K) = 5 .13. Dufresne et al.\n(2020) found that including level-resolved ionization and recombination in the ion\nbalance calculations serves to further lower the formation temperature of O ivby\n0.05 dex. The results from the EIS temperature diagnostic discussed in this article\ntherefore gives support to the addition of density-dependent factors to the standard\nCHIANTI ionization balance calculation.\nFrom the density and temperature an estimate of the size of the flare kernel at O iv\ntemperatures can be made. Assuming an isothermal plasma, the column depth, h, is\ngiven by\nh=4πI\n0.83ϵ(O)G(T, N e)N2\ne(1)\nwhere ϵ(O) is the abundance of oxygen relative to hydrogen, and Gis the contribu-\ntion function. The contribution function is calculated from the CHIANTI ionization\nbalance calculated at an electron pressure of 1017.1K cm−3, and a temperature of16\nlog (T/K) = 5 .10. The density is set to log ( Ne/cm−3) = 12 to be consistent with\nthe assumed pressure. The photospheric abundance file distributed with CHIANTI\nthen gives h= 0.036′′. The EIS pixel size is 2′′×1′′, so if a cubic volume is assumed\nthen the side of this cube is 0.4′′(300 km). This compares with directly measured\nflare ribbon sizes of 0.6′′to 2.0′′obtained from the Goode Solar Telescope at chromo-\nspheric temperatures (Xu et al. 2012). Due to the uncertainties in the density and\nthe element abundance, the EIS size estimate is perhaps accurate to no better than\na factor of two.\nSection 5.1 discussed the O ivlines near 260 ˚A and an estimate for the\nλ279.93/ λ206.39 ratio was given as 16 ±2. The ratio is strongly temperature sensi-\ntive with little density sensitivity, and the observed ratio corresponds to log ( T/K) =\n5.07±0.03, consistent with the value from λ268.02/ λ279.93.\nThe EIS SW lines at 195.86 ˚A and 207.24 ˚A form a ratio that is sensitive to\nboth temperature and density, although much less so than the λ268.02/ λ279.93 and\nλ272.13/ λ279.93 ratios. For a temperature of log ( T/K) = 5 .10, the measured ratio\nof 1.70±0.19 (Table 2) implies a lower limit to the density of log ( Ne/cm−3)≥11.85,\nwhich is consistent with the LW diagnostics discussed above. This gives confidence in\nthe accuracy of the intensity measurements of these lines, despite the issues discussed\nin Section 5.2.\nTheλ207.24/ λ279.93 ratio is strongly temperature sensitive and also has weak den-\nsity sensitive. Assuming log ( Ne/cm−3) = 12 .55, the observed ratio of 0 .139±0.015\nimplies a temperature of log ( T/K) = 5 .12±0.04. Decreasing the density to\nlog (Ne/cm−3) = 11 .91 increases the temperature by 0.01 dex. The ratio is therefore\nconsistent with that derived from the λ268.02/ λ279.93 ratio, and gives confidence in\nthe WUL14 calibration.\n7.COMPARISON WITH OTHER DENSITY DIAGNOSTICS\nThe closest ion to O iv(Tiz= 0.14 MK) in temperature that also has a density diag-\nnostic is O v(Tiz= 0.22 MK). A group of lines between 192.7 and 193.0 ˚A is density\nsensitive relative to λ248.46 (Young et al. 2007), with the former becoming relatively\nstronger at higher densities. The ratio is sensitive to densities up to 1013cm−3and\nthus potentially provides confirmation of the high O ivdensity.\nThere are two complicating factors for the O vratio: (i) the lines are found in the\ntwo different EIS wavelength channels; and (ii) the lines are affected by blending.\nAppendix A shows that the GDZ13 calibration gives a ratio 36 % lower than for\nthe WUL14 calibration, which is preferred in this article. The effect on the derived\ndensity is discussed below.\nBlending issues associated with the group of lines between 192.7 and 193.0 ˚A have\nbeen discussed by Ko et al. (2009). In particular, Fe xiλ192.81 is usually dominant\nin quiet Sun and active region conditions, while Ca xvii 192.85 usually dominates in\nflare conditions. Rather than attempting the complex multi-Gaussian fit performed17\nFigure 8. Portions of the EIS flare kernel spectra showing the O vlines. (a) the fit to O v\nλ192.90 is indicated in blue. (b) a three-Gaussian fit (blue) was performed to the region\ncontaining O vλ248.46 (red), and Al viiiλ250.13 is fit with a single Gaussian (magenta).]\nby Ko et al. (2009) the procedure here is simply to fit the rightmost feature in the\nspectrum (Figure 8) and assume this corresponds to the O vself-blend of λ192.904\nandλ192.911 (the latter contributes less than 15% to the observed feature). This is\njustified because of the strength of O vin the flare kernel spectrum, enabling O vto\nbe clearly separated from the other lines around 192.8 ˚A. The line parameters given\nin Table 2 are identified only with the 192.904 ˚A line, but for computing the density\nboth components are included in the CHIANTI model.\nThe O vline at 248.46 ˚A is blended with the Al viii248.45 ˚A line, but the latter\ncan be estimated by measuring Al viii250.13 ˚A. The branching ratio λ248.45/ λ250.13\nhas a fixed value of 0.594 in all conditions using atomic data in CHIANTI. Fits to\nthe lines are shown in Figure 8(b), and the λ250.13 intensity is given in Table 2. It\nimplies a Al viiiλ248.45 intensity of 147 mW m−2sr−1, and the O vλ248.46 intensity\nis then 1423 ±35 mW m−2sr−1.\nThe measured λ192.90/ λ248.46 ratio is 2.00 with the WUL14 calibration, which\ntranslates to a density of log ( Ne/cm−3) = 11 .43. With the GDZ13 calibration,\nthe ratio is only 1.28, corresponding to log ( Ne/cm−3) = 10 .80. The WUL14 cal-\nibration was preferred for the O ivanalysis (Appendix A), and if an uncertainty\nof 20% is assumed for the SW-LW cross-calibration, then this implies a density of\nlog (Ne/cm−3) = 11 .4±0.3. This density was derived assuming a temperature of\nlog (T/K) = 5 .35, but the result is insensitive to temperature within ±0.2 dex.\nMgvii(Tiz= 0.63 MK) is significantly hotter than O ivbut has density diagnostics\nthat have been widely used (Tripathi et al. 2009; Young et al. 2012; Brosius et al.\n2016). The λ280.73/ λ276.15 observed ratio (Table 2) gives a density close to the high\ndensity limit: log ( Ne/cm−3) = 11 .04+0.37\n−0.20, computed at log ( T/K) = 5 .80.\nTable 3 compares the density results from the three ions and gives the electron pres-\nsure ( Pe=NeT). The Tizvalues have been calculated using the CHIANTI ionization18\nTable 3. Electron pressures for three ions.\nIon log ( Tiz/K) log ( Ne/cm−3) log ( Pe/K cm−3)\nOiv 5.10 ≥11.9 ≥17.0\nOv 5.30 11 .4±0.3 16 .7±0.3\nMgvii 5.75 11 .0+0.4\n−0.2 16.8+0.4\n−0.2\nFigure 9. Plots showing the variation of electron density as a function of time for plasma\nat log ( T/K) = 5 .1 obtained from the F-CHROMA flare models. The colors green, blue,\nmagenta and red correspond to Etot= 3×1010, 1011, 3×1011, and 1012erg cm−2. For each\nmodel δ= 4 and Ec= 20 keV.\nbalance computed at a pressure of log ( Pe/K cm−3) = 17 .1 although the values do\nnot change for pressures up to two orders of magnitude lower than this. The results\nare therefore consistent, within the uncertainties, with a pressure of 1017K cm−3. For\ncomparison, Widing & Cook (1987) found that a number of density diagnostics, in-\ncluding O ivand O v, observed with the Skylab S082B spectrometer were consistent\nwith a pressure of 3 .9×1016K cm−3for a flare kernel observed at the solar limb.\n8.COMPARISON WITH MODEL RESULTS\nThe Flare Chromospheres: Observations, Models and Archives (F-CHROMA) col-\nlaboration provided a number of flare model runs produced with the RADYN code.4\nThe models can be used to investigate how the density and temperature derived from\nthe O ivdiagnostics can help constrain the flare heating process. The models give\nthe electron density and temperature as a function of time, t, and position, z, along\nthe flare loop. Three parameters are varied amongst the models: the spectral index,\nδ, the low-energy cutoff, Ec, and the total energy input, Etot. Full details are given\nin Carlsson et al. (2023).\n4https://star.pst.qub.ac.uk/wiki/public/solarmodels/start.html.19\nThe values δ= 4 and Ec= 20 keV were chosen to represent a relatively strong flare,\nand there are four models with Etot= 3×1010, 1011, 3×1011, and 1012erg cm−2.\nThe data were degraded to 0.5 s time resolution and for each time step the spatial\npixels corresponding to the temperature region log T= 5.0 to 5.2 (spanning the\ntemperature found from the O ivdiagnostic) were identified. An average density for\neach time step is computed asP\niNe,izi/P\nizi, where ziis the size of spatial bin i.\nFigure 9 shows the variation of the average density as a function of time for each\nmodel. As can be seen, the maximum density reached at times t= 10–15 s increases\nstrongly as Etotincreases. The rise in density at 37 s for the Etot= 1012erg cm−2\nmodel is due to downward-propagating wave that has reflected from the loop top.\nThese simulations are not directly tuned to the present observations, and so a\ndirect comparison between the results and the O ivmeasurements can not be made.\nHowever, it demonstrates that the O ivplasma measurements may be a valuable tool\nin constraining flare loop models. In particular, high densities of log ( Ne/cm−3)≥12,\nsuggest a high energy flux into the flare loop.\n9.SUMMARY\nNew O ivdiagnostics for temperature and density have been presented that utilize\nemission lines observed by Hinode/EIS. They have been applied to the spectrum of\na solar flare kernel observed on 2012 March 9. The temperature is strongly con-\nstrained to log ( T/K) = 5 .10±0.03 by the λ268.02/ λ279.93 ratio. This temperature\nis significantly below the temperature given by the default CHIANTI ionization equi-\nlibrium calculation. However, recent work on density dependent ionization balance\ncalculations (Dufresne et al. 2020) places O ivat lower temperatures in high density\nconditions, giving consistency with the flare temperature found here.\nThe measured λ272.13/ λ279.93 ratio is close to the high density limit, and a lower\nlimit of log ( Ne/cm−3) = 11 .9 is found here. Comparisons with O vand Mg vii\ndensity diagnostics suggest a constant pressure of 1017K cm−3could apply through\nthe transition region if the O ivdensity is near the lower density limit. Assuming\na density of log ( Ne/cm−3) = 12 .0 and a cubic emitting volume, the width of the\nvolume is estimated at 0.4′′(300 km).\nInspection of F-CHROMA flare loop models (Section 8) shows that the highest\ndensities reached during the flare heating process are strongly dependent on the total\nenergy input to the loop. Although the flare models are not directly applicable to the\npresent flare, they suggest that the high O ivdensity found here is consistent with a\nhigh energy input to the flare loop footpoint.\nThe upcoming NASA Multi-slit Solar Explorer (MUSE: De Pontieu et al. 2020)\nmission will perform spectroscopy in three EUV channels, one of which is centered at\n284˚A. This channel will have sensitivity to the O ivlines between 268 and 280 ˚A—\nsee Figure 1 of De Pontieu et al. (2020). Due to the weakness of the O ivlines and\noverlapping nature of multi-slit spectra, the lines will likely not be useful in most20\ncircumstances. However, an isolated flare kernel such as discussed in this article may\nyield useful spectra. In addition, the higher spatial resolution of MUSE (about a\nfactor 10 improvement over EIS) may lead to a relative enhancement of transition\nregion lines compared to coronal lines in the spectrum. Evidence has been found for\nmagnetic field lines converging at low heights in flare loops (Kontar et al. 2008). If\nthe flare kernel corresponds to the cross-section of such a loop, and the MUSE spatial\nresolution is sufficient to resolve this cross-section, as would be expected from the\nresults of Xu et al. (2012) who found sizes of 0.6′′to 2.0′′, then transition region lines\nformed deeper in the atmosphere compared to coronal lines would then be expected to\nbe relatively stronger compared to the EIS data where the kernel is likely unresolved.\nIn the current EIS spectrum Fe xvλ284.16 (the dominant line in the MUSE 284 ˚A\nchannel) is a factor 15 stronger than O ivλ279.93.\nThe author thanks Dr. Helen Mason for valuable comments on the manuscript, and\nDr. Joel Allred for help with the F-CHROMA models. The author acknowledges\nfunding from the GSFC Internal Scientist Funding Model competitive work package\nprogram, the Hinode project, and the NASA Heliophysics Data Resource Library.\nHinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ\nas domestic partner and NASA and STFC (UK) as international partners. It is\noperated by these agencies in co-operation with ESA and NSC (Norway).\nFacilities: Hinode(EIS), SDO(AIA)\nREFERENCES\nAllred, J. C., Kowalski, A. F., & Carlsson,\nM. 2015, ApJ, 809, 104,\ndoi: 10.1088/0004-637X/809/1/104\nBenz, A. O. 2017, Living Reviews in Solar\nPhysics, 14, 2,\ndoi: 10.1007/s41116-016-0004-3\nBrosius, J. W., Daw, A. N., & Inglis,\nA. 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P. 2021b,\nSoPh, 296, 181,\ndoi: 10.1007/s11207-021-01929-8\nYoung, P. R., Del Zanna, G., Mason,\nH. E., et al. 2007, PASJ, 59, S857,\ndoi: 10.1093/pasj/59.sp3.S85723\nTable 4. Ratio comparison for different radiometric calibrations.\nIon Ratio Theory Pre-launch GDZ13 WUL14\nOvλ192.90/ λ248.46 ··· 1.0 0.57 0.37\nOivλ207.16/ λ279.93 ··· 1.0 0.50 0.40\nOivλ268.02/ λ279.93 ··· 1.0 0.87 1.00\nλ272.13/ λ279.93 ··· 1.0 0.84 1.04\nSixλ277.26/ λ271.99 0.77 0.83 0.95 0.80\nAPPENDIX\nA.RADIOMETRIC CALIBRATION UNCERTAINTIES\nAn important source of uncertainties for line ratio measurements lies in the EIS\nradiometric calibration. Mariska (2013) found that the instrument sensitivity had\ndeclined after launch, and modeled an exponential decay that was the same at all\nwavelengths. Further work by GDZ13 and WUL14 identified a wavelength depen-\ndence to the sensitivity decay, with the LW channel showing a stronger decay than\nthe SW channel. The authors also derived modifications to the original calibration\nwithin the two channels. Although the results of GDZ13 and WUL14 are broadly\nsimilar, there are important differences and the EIS data analysis guide does not\nrecommend one over the other.\nThe key O ivlines used in the present work are found between 268 and 280 ˚A and\nTable 4 shows the effect of the GDZ13 and WUL14 calibrations on ratio values of 1.0\nin the pre-launch calibration. The WUL14 values are very similar, but the GDZ14\nvalues are around 15% lower. Also shown is the O vλ192.90/ λ248.46 ratio, which\nis used in Section 7. Since the lines are in different channels, the updated ratios\nare significantly different from the pre-launch calibration values. The GDZ13 and\nWUL14 values are also very different, with the GDZ13 ratio 36 % lower.\nAlso shown in Table 4 is a Si xbranching ratio, λ277.26/ λ271.99, which takes a con-\nstant value of 0.77 in all plasma conditions (data from CHIANTI). Since the two lines\nare close in wavelength to the O ivlines then they can be used to estimate the accu-\nracy of the radiometric calibration options in this region. The two lines were measured\nin a quiet Sun off-limb dataset from 2011 August 14, beginning at 05:33 UT, which\nwas the nearest such dataset to the flare observation time. The spectrum and line fit\nparameters are available on Zenodo at DOI:10.5281/zenodo.6726426. Table 4 gives\nthe measured ratios using the three calibration options. The WUL14 calibration gives\nthe best agreement, with the GDZ13 value more than 20% higher than the expected\nvalue. For this reason, the WUL14 calibration is used in the present work. However,24\nan additional 5% uncertainty is added to the measured line intensities to represent\nthe uncertainties due to the radiometric calibration in this wavelength region."    },    {        "title": "2401.12399v1.Renormalized_Solutions_for_Quasilinear_Elliptic_Equations_with_Robin_Boundary_Conditions__Lower_Order_Terms__and__L_1__Data.pdf",        "content": "arXiv:2401.12399v1  [math.AP]  22 Jan 2024RENORMALIZED SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATION S WITH\nROBIN BOUNDARY CONDITIONS, LOWER-ORDER TERMS, AND L1DATA\nJUAN A. APAZA AND MANASS ´ES DE SOUZA\nABSTRACT . In this paper, we establish the existence of a solution for a class of quasilinear equations\ncharacterized by the prototype:\n/braceleftBigg\n−div(ϑα|∇u|p−2∇u)+ϑγb|∇u|p−1+ϑγc|u|r−1u=fϑα inΩ,\nϑα|∇u|p−2∇u·ν+ϑβ|u|p−2u=gϑβ on∂Ω.\nHere,Ωis an open subset of RNwith a Lipschitz boundary, where N≥2and1< p < N . We define\nϑa(x) = (1+ |x|)afora∈(−N,(p−1)N), and the constants α,β,γ,r satisfy suitable conditions.\nAdditionally, fandgare measurable functions, while bandcbelong to a Lorentz space. Our approach\nalso allows us to establish stability results for renormali zed solutions.\n1. I NTRODUCTION\nIn this paper we consider a class of problems with the form\n/braceleftBigg−div((1+ |x|)αA(∇u))+(1+ |x|)γH(x,∇u)+(1+|x|)γG(x,u) =f(x)(1+|x|)αinΩ,\n(1+|x|)αA(∇u)·ν+(1+|x|)βK(u) =g(x)(1+|x|)βon∂Ω,(1.1)\nwhereΩis an open subset of RNwith Lipschitz boundary, and N≥2. Furthermore, f∈L1(Ω;(1+\n|x|)α),g∈L1(∂Ω;(1+|x|)β),A,H,G,K , and the constants α,β,γ satisfy suitable conditions.\nWhenfbelongs to the dual space of the Sobolev space W1,p(Σ), andΣ⊂RNis a bounded\ndomain, the existence and uniqueness (up to additive consta nts) of weak solutions to the problem\n\n\n−div(|∇u|p−2∇u) =finΣ,\n|∇u|p−2∂u\n∂ν= 0 on∂Σ,(1.2)\nare consequences of the classical theory of pseudo-monoton e operators (cf. [32, 33]). However, if\nfis just an L1-function and not an element of the dual space of W1,p(Σ), one has to give meaning\nto the notion of solution. When Dirichlet boundary conditio ns are prescribed, various definitions\nof solutions to nonlinear elliptic equations with the right -hand side in L1or measure have been\nintroduced. In [6, 16, 34, 38], different notions of solutio ns are defined, though they prove to be\nequivalent, at least when the datum is an L1-function. The study of existence or uniqueness for\nDirichlet boundary value problems has been the subject of se veral papers. We recall that the linear\ncase has been studied in [42], while the nonlinear case was in itially addressed in [11, 12] and has\nbeen further explored in various contributions, including [2, 7, 6, 8, 9, 17, 16, 27, 28].\nThe existence for Neumann boundary value problems (1.2) wit hL1-data has been addressed in\nvarious contexts. In [3, 15, 20, 21, 41], the existence of a di stributional solution belonging to a\nsuitable Sobolev space, with a null mean value, is proved. Ho wever, when p≤2−1/N, the\ndistributional solution to problem (1.2) does not belong to a Sobolev space and, in general, is not a\nsummable function. For example, consider the Dirac mass at t he center of a ball as the right-hand\nside; see [17, Example 2.16.].\n2020 Mathematics Subject Classification . 35J62; 35A35; 35J25\nKeywords and phrases . existence; quasilinear elliptic equations; Robin proble ms;L1-data\n12 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nThe concept of a renormalized solution in the context of vari able exponents was studied in [43],\nwhere homogeneous Dirichlet boundary conditions were cons idered. In [5], a concept of renormal-\nized solution was proposed for a Neumann problem in bounded d omains with nonnegative measures\ninL1:\n\n−div(|∇u|p(x)−2∇u)+|u|p(x)−2u+b1(u)|∇u|p(x)=finΣ,\n|∇u|p(x)−2∂u\n∂ν+b2(u) =gon∂Σ,(1.3)\nwherep∈C(¯Σ),1<infΣp,N≥3,f∈L1(Ω),g∈L1(∂Ω), andb1andb2are increasing\ncontinuous functions with b1(0) =b2(0) = 0 , see also [31].\nIn [39], a type of Fourier boundary problem is studied, and th e existence and uniqueness of a\nrenormalized solution to the problem are given when the data f∈L1(Ω)andg∈L1(∂Ω):\n\n\n−div(|∇u|p(x)−2∇u)+b(u) =finΣ,\n|∇u|p(x)−2∂u\n∂ν+cu=gon∂Σ,\nwherep∈C(¯Σ),1<infΣp,N≥3,c >0, andbis a surjective and nondecreasing function such\nthatb(0) = 0 . To obtain their results, they define a new space that helps to account for the boundary\nconditions.\nFor nonlinear equations with a Radon measure in the right-ha nd side,\n/braceleftBigg\n−div(|∇u|p−2∇u)+b(x)|∇u|λ=µinΣ,\nu= 0 on∂Σ,(1.4)\nwhereN≥2,1< p < N ,0≤λ≤p−1, andbbelongs to the Lorentz space LN,1(Σ). In [9], an\nexistence result for the problem (1.4) is provided.\nThe problem (1.4) was studied in [11, 12] (and in [19] where a t ermb(x)|∇u|p−1is considered). In\nthese papers, the existence of a solution that satisfies the e quation in the distributional sense is proven\nwhenp >2−1/N. This assumption on pensures that the solution belongs to the Sobolev space\nW1,q\n0(Σ)withq < N(p−1)/(N−1)(compare this with Remark 2.9 below). To address the problem\n(1.4), two equivalent notions of solutions have been introd uced: the notion of entropy solution in\n[6, 13], and the notion of renormalized solution in [35, 37, 3 8]. In the case where the measure µ\nbelongs to L1(Σ)or toL1(Σ)+W−1,p′(Σ), these papers prove the existence and uniqueness of such\nsolutions.\nBy employing the arguments in [9], we obtain estimates of sol utions for problems in the Lorentz\nspace when the data are in Lp′, wherepis constant. Additionally, we use the techniques from [10] t o\nestablish the existence of a renormalized solution to (1.1) . We deal with the following problems: on\nthe one hand, the right-hand side involves functions in L1; on the other hand, since Ωis unbounded,\nwe cannot use compact embeddings of the form W1,p(Ω;ϑα)֒→Lp(Ω;ϑα).\nWe consider the following conditions:\n(H1) To simplify the representation, we will denote ϑα(x) = (1 + |x|)α, dϑα=ϑαdx, and\nϑα(F) =´\nFϑαdx, whereFis a measurable set. Throughout the paper,\nΩ ={x∈RN||(xd+1,...,x N)|< ρ(|(x1,...,x d)|)},\nwhereρ: [0,∞)→R+isC∞with|ρ′| ≤Con[0,∞)for some positive constant, and\nd∈ {1,...,N−1}.\nAdditionally, we assume that 1< p < N ,α,γ,β∈(−N,(p−1)N),p−1< α−β,\n0≤α−γ < p ,\nϑα(Ω)<∞, ϑβ(∂Ω)<∞,\nN+α−p >0, and(α−γ)N+γp≥0.EXISTENCE OF RENORMALIZED SOLUTIONS 3\n(H2)A:RN→RNandK:R→Rare continuous function satisfying:\nA(ξ)ξ≥σ|ξ|p,|A(ξ)| ≤σ−1|ξ|p−1,[A(ξ)−A(η)][ξ−η]>0,\n|s|p−1≤K(s)sign(s)and|K(s)| ≤σ−1|s|p−1,\nfor every ξ∈RN,η∈RN, ands∈R, where0< σ <1andξ/\\e}atio\\slash=η.\n(H3) Moreover, H: Ω×RN→RandG: Ω×R→Rare Carath´ eodory functions satisfying:\n|H(x,ξ)| ≤b(x)|ξ|p−1, b∈LNα,γ,1(Ω;ϑγ),\nG(x,s)s≥0,|G(x,s)| ≤c(x)|s|r, c∈Lz′,1(Ω;ϑγ),\nfor almost every x∈Ωand for every s∈Randξ∈RN, whereNα,γ= (N+γ)p/(p−α+\nγ),\n0≤r <(N+γ)(p−1)\nN+α−p, z=(N+γ)(p−1)\nN+α−p1\nrand1\nz+1\nz′= 1. (1.5)\nThe main result of the paper is the following theorem:\nTheorem 1.1. Assume that conditions ( H1) - (H3) hold. There exists at least one renormalized\nsolution (in the sense of definition 2.7) to problem (1.1) .\nAdditionally, using the arguments in the proof of Theorem 1. 1, for a sequence (un)of renormal-\nized solutions of (4.40) satisfying suitable conditions, w e have the following stability result.\nTheorem 1.2. Under the assumptions ( H1), (H2), and conditions (4.40) -(4.44) in Section 4.1, up\nto a subsequence (still indexed by n),unconverges to u, whereuis a renormalized solution to (1.1) ,\nand\nun→ua.e inΩ,andun→ua.e on∂Ω, (1.6)\nA(∇Tk(un))∇Tk(un)⇀ A(∇Tk(u))∇Tk(u)inL1(Ω;ϑα), (1.7)\nfor allk >0, whereTkis the truncation at level kdefined in (2.6) .\nThis paper is organized as follows. In Section 2, we gather pr eliminary definitions and results,\nwhich are used several times in the paper: renormalized solu tions, embedding theorems, and equiva-\nlent norms in Sobolev spaces. In Section 3, we provide basic r esults for weak solutions and estimate\nthe norms of these solutions in the Lorentz spaces. In Sectio n 4, we prove Theorem 1.1, which asserts\nthe existence of a renormalized solution to (1.1). Its proof , contained herein, is based on the usual\nprocedure of approximation, involving problems of the type (1.1) with smooth data that strongly con-\nverges to finL1. For such a sequence of problems, weak solutions are obtaine d using fixed-point\narguments (see Appendix A). A priori estimates allow us to pr ove that these weak solutions converge,\nin some sense, to a function u, and a delicate procedure of passage to the limit allows us to establish\nthatuis a renormalized solution to (1.1).\n2. P RELIMINARY DEFINITIONS AND RESULTS\nBefore defining the type of solutions with which we will work, let’s first establish some basic\nproperties of weighted Sobolev and Lorentz spaces.\n2.1. A few properties of weighted Sobolev spaces. LetU⊂RNbe an open set and p >1, define\nLp(U;ϑα) =/braceleftbigg\nu|u: Ω→Ris measurable function andˆ\nU|u|pdϑα<∞/bracerightbigg\n,\nand\nW1,p(U;ϑα) ={u∈Lp(U;ϑα)||∇u| ∈Lp(U;ϑα)},4 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nequipped with the norms\n/ba∇dblu/ba∇dblp,U,α=/parenleftbiggˆ\nU|u|pdϑα/parenrightbigg1\np\nand/ba∇dblu/ba∇dbl1,p,U,α=/ba∇dblu/ba∇dblp,U,α+/ba∇dbl∇u/ba∇dblp,U,α,\nrespectively. For details on these spaces see [30, 29, 40, 36 ].\nProposition 2.1. (see [40, 36] ) Suppose that 1≤p≤q <∞,α,β∈(−N,(p−1)N), and\n0≤(N−1)/q−N/p+ 1≤α/p−β/q. Then, the trace operator W1,p(Ω;ϑα)→Lq(∂Ω;ϑβ)\nis continuous. Moreover, if 0≤(N−1)/q−N/p+ 1< α/p−β/q, then the trace operator is\ncompact.\nRemark 2.2. In condition ( H1) (see [40, 36] ), we can use a general condition related to the boundary\n∂Ω, assuming the existence of a locally finite covering of ∂Ωwith open subsets Ui⊂RNhaving the\nfollowing properties:\n(a) There is a global constant θsuch that/summationtext\niχUi(x)≤θfor every x∈RN;\n(b) There exist cubes Bi⊂RNand Lipschitz diffeomorphisms ϕi:Ui→Bisuch that 0∈Bi\nandϕ−1\ni(RN−1×{0}) =Ui∩∂Ω;\n(c) The partial derivatives of the coordinate functions ϕiandϕ−1\niare uniformly bounded by a\nconstantC(not depending on i).\nDenote\nAn= sup\n/bardblu/bardbl1,p,Ω,α≤1/ba∇dblu/ba∇dblp,Ωn,α,\nwhereΩn={x∈RN||x|> n}∩Ω, andn∈N. Since\n0≤ An+1≤ An≤1,\nthe limit\nA= lim\nn→∞An\nexists, and A ∈[0,1].\nWe have the following proposition:\nProposition 2.3. Assume that p∈(1,N),α,β∈(−N,(p−1)N), andp−1< α−β. For any\nu∈W1,p(Ω;ϑα), let\n/ba∇dblu/ba∇dbl∂=/ba∇dblu/ba∇dblp,∂Ω,β+/ba∇dbl∇u/ba∇dblp,Ω,α.\nThen/ba∇dblu/ba∇dbl∂is a norm on W1,p(Ω;ϑα)that is equivalent to\n/ba∇dblu/ba∇dbl1,p,Ω,α=/ba∇dblu/ba∇dblp,Ω,α+/ba∇dbl∇u/ba∇dblp,Ω,α.\nConsidering that ϑα(Ω)<∞andϑβ(∂Ω)<∞, Proposition 2.3 is a consequence of Proposition\n2.1 and the following lemma.\nLemma 2.4. (see [22, Lemma 3.2] ) Suppose 1/lessorequalslantp <∞. LetFbe a functional on W1,p(Ω;ϑα)\nwith the following properties:\n(i) There is a constant C0such that for all u,v∈W1,p(Ω;ϑα),\n|F(u)−F(v)| ≤C0/ba∇dblu−v/ba∇dbl1,p,Ω,α.\n(ii)F(λu) =λF(u)for allλ >0andu∈W1,p(Ω;ϑα).\n(iii) Ifu∈W1,p(Ω;ϑα)is constant in ΩandF(u) = 0 , thenu= 0.\nLetA<1. Then there is a constant Csuch thatˆ\nΩ|u|pdϑα≤C/parenleftbigg\n|F(u)|p+ˆ\nΩ|∇u|pdϑα/parenrightbigg\n,\nfor allu∈W1,p(Ω;ϑα).EXISTENCE OF RENORMALIZED SOLUTIONS 5\nProposition 2.5. (see [30, Theorem 3] ) Suppose that p∈(1,N),q≥p,α,γ∈(−N,(p−1)N),\nN/p−N/q≤1−α/p+γ/q < N/p and−N/q < γ/q ≤α/p. Then there is a continuous\nembedding W1,p(Ω;ϑα)→Lq(Ω;ϑγ).\n2.2. A few properties of Lorentz spaces. For1< q <∞, the Lorentz space Lq,∞(Ω;ϑγ)is the\nspace of Lebesgue measurable functions such that\nsup\nt>0t1/qf∗\nγ(t)<∞. (2.1)\nHere,f∗denotes the decreasing rearrangement of f, i.e., the decreasing function defined by\nf∗\nγ(t) = inf{s≥0|ϑγ({x∈Ω||f(x)|> s})≤t}. (2.2)\nInLq,∞(Ω;ϑγ), we work with the semi-norm\n/ba∇dblf/ba∇dbl(γ,q,∞,Ω)= sup\nt>0t1/qf∗\nγ(t).\nFurthermore, the Lorentz space Lq,1(Ω;ϑγ)is the space of Lebesgue measurable functions such\nthat\n/ba∇dblf/ba∇dbl(γ,q,1,Ω)=ˆ∞\n0t1/qf∗\nγ(t)dt\nt<∞, (2.3)\nendowed with the norm defined by (2.3). For references about r earrangements, see [14].\nWe will use the following properties of the Lorentz spaces, w hich are intermediate spaces between\nthe Lebesgue spaces, in the sense that, for every 1< q1< q2<∞, the following relationships hold:\nLq2,1(Ω;ϑγ)⊂Lq2,q2(Ω;ϑγ) =Lq2(Ω;ϑγ)⊂Lq2,∞(Ω;ϑγ)⊂Lq1,1(Ω;ϑγ). (2.4)\nOne has the generalized H¨ older inequality:\nProposition 2.6. For anyf∈Lq,∞(Ω;ϑγ)andg∈Lq′,1(Ω;ϑγ),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩfgdϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dblf/ba∇dbl(γ,q,∞,Ω)/ba∇dblg/ba∇dbl(γ,q′,1,Ω). (2.5)\nIn what follows, for simplicity, we write\n/ba∇dblf/ba∇dbl(γ,q,∞,Ω)=/ba∇dblf/ba∇dbl(γ,q,∞)and/ba∇dblf/ba∇dbl(γ,q′,1,Ω)=/ba∇dblf/ba∇dbl(γ,q′,1).\n2.3. The definition of a renormalized solution. Motivated by [9, 10], we present a definition of\nrenormalized solutions for nonlinear elliptic problems.\nFork >0, letTk:R→Rdenote the standard truncation at level k, defined as\nTk(s) =/braceleftBigg\ns if|s| ≤k,\nksign(s)if|s|> k.(2.6)\nForu∈W1,p(Ω;ϑα), we define (cf. [3, 6, 4]):\nT1,p(Ω;ϑα) ={u: Ω→¯Rmeasurable such that Tk(u)∈W1,p(Ω;ϑα)∀k >0}.\nGivenu∈ T1,p(Ω;ϑα), there exists a unique measurable function v: Ω→¯RNsuch that\n∇Tk(u) =vχ{|v|<k},∀k >0. (2.7)\nThis function vis denoted as ∇u. It is evident that if u∈W1,p(Ω;ϑα), thenv∈Lp(Ω;ϑα), and\n∇u=vin the usual sense.\nAs defined in [3], we denote T1,p\ntr(Ω;ϑα)as the set of functions u∈ T1,p(Ω;ϑα)for which there\nexists a sequence (un)⊂W1,p(Ω;ϑα)satisfying the following conditions:\n(a)unconverges to ua.e. inΩ.\n(b)∇Tk(un)→ ∇Tk(u)inL1(V;ϑα)for anyk >0and for any open bounded set V⊂Ω.\n(c) There exists a measurable function w:∂Ω→¯Rsuch that unconverges to wa.e. on∂Ω.6 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nThe function wis the trace of uin a generalized sense. In subsequent discussions, the trac e of\nu∈ T1,p\ntr(Ω;ϑα)on∂Ωwill be denoted by tr (u)or simply u. It is important to note that when\nu∈W1,p(Ω;ϑα), tr(u)coincides with the usual trace of u, denoted as τ(u). Additionally, for every\nu∈ T1,p\ntr(Ω;ϑα)and every k >0,τ(Tk(u)) =Tk(tr(u)). Ifφ∈W1,p(Ω;ϑα)∩L∞(Ω), then\nu−φ∈ T1,p\ntr(Ω;ϑα), and tr(u−φ) =tr(u)−τ(φ).\nDefinition 2.7. Under (H1) - (H3) andp < q≤(N−1)p/(N−p), a function u∈ T1,p\ntr(Ω;ϑα)is\na renormalized solution to (1.1) ifuis finite a.e. in Ω, satisfying the following conditions:\nTk(u)∈W1,p(Ω;ϑα),∀k >0, (2.8)\n|u|p−1∈Lp∗\nα,γ/p,∞(Ω;ϑγ), (2.9)\n|u|p−1∈Lq/p,∞(∂Ω;ϑβ). (2.10)\nThe gradient ∇uintroduced in (2.7) satisfies:\n|∇u|p−1∈LN′\nα,γ,∞(Ω;ϑγ), (2.11)\nlim\nk→∞1\nkˆ\n{|u|<k}A(∇u)∇udϑα= 0, (2.12)\nwherep∗\nα,γ= (N+γ)p/(N+α−p)and1/Nα,γ+1/N′\nα,γ= 1.\nMoreover, for every function h∈W1,∞(R)with compact support and for every ϕ∈L∞(Ω)∩\nW1,p(Ω;ϑα), we haveˆ\nΩh(u)A(∇u)∇ϕdϑα+ˆ\nΩh′(u)A(∇u)∇uϕdϑα+ˆ\nΩH(x,∇u)ϕh(u)dϑγ\n+ˆ\nΩG(x,u)ϕh(u)dϑγ+ˆ\n∂ΩK(u)ϕh(u)dϑβ\n=ˆ\nΩϕh(u)fdϑα+ˆ\n∂Ωϕh(u)gdϑβ.(2.13)\nRemark 2.8. Conditions (2.9) -(2.11) and (H3) onH,G, andKensure that for every renormalized\nsolution,\nG(x,u),H(x,∇u)∈L1(Ω;ϑγ),andK(u)∈L1(∂Ω;ϑβ).\nRemark 2.9. Ifuis a renormalized solution of (1.1) , thenuis also a distributional solution satisfyingˆ\nΩA(∇u)∇ϕdϑα+ˆ\nΩH(x,∇u)ϕdϑγ+ˆ\nΩG(x,u)ϕdϑγ+ˆ\n∂ΩK(u)ϕdϑβ\n=ˆ\nΩϕfdϑα+ˆ\n∂Ωϕgdϑβ,∀ϕ∈C∞\nc(¯Ω).(2.14)\nIndeed, if uis a renormalized solution of (1.1) , we know that uis measurable and a.e. finite in Ω,\nand thatTk(u)∈W1,p(Ω;ϑα)for every k >0. This allows one to define ∇uin the sense of (2.7) .\nWe also know that |∇u|p−1∈LN′\nα,γ,∞(Ω;ϑγ), so that|A(∇u)| ∈LN′\nα,γ,∞(Ω;ϑγ)by the condition\n(H2). Taking φ∈C∞\nc(¯Ω)andhℓdefined by\nhℓ(s) =\n\n0 if|s|>2ℓ,\n2ℓ−|s|\nℓifℓ <|s| ≤2ℓ,\n1 if|s| ≤ℓ,\nand letting ℓ→ ∞ in(2.13) , we obtain (2.14) .\nMoreover, every renormalized solution uof(1.1) belongs to W1,q(Ω;ϑγ)for every q < N′\nα,γ(p−\n1)whenp >2−1/Nα,γ. Indeed,p >2−1/Nα,γimpliesN′\nα,γ(p−1)>1, and therefore, the gradient\n∇udefined by (2.7) , which satisfies (2.11) , belongs to (Lq1(Ω;ϑγ))Nfor every q1< N′\nα,γ(p−1)\nand is the distributional gradient of u(see[17, Remark 2.10] ).EXISTENCE OF RENORMALIZED SOLUTIONS 7\n3. B ASIC RESULTS FOR WEAK SOLUTIONS\nIn this section, we assume more restrictive conditions on f,g,H, andGto prove the existence of\na renormalized solution uto problem (1.1).\nLetfn∈Lp′(Ω;ϑα)andgn∈Lp′(∂Ω;ϑβ)such that\nfn= 0onΩ\\Ωn, gn= 0on∂Ω\\∂Ωn, (3.1)\nfn→finL1(Ω;ϑα), gn→ginL1(∂Ω;ϑβ), (3.2)\n/ba∇dblfn/ba∇dbl1,Ω,α≤ /ba∇dblf/ba∇dbl1,Ω,α,∀nand/ba∇dblgn/ba∇dbl1,∂Ω,β≤ /ba∇dblg/ba∇dbl1,∂Ω,β,∀n, (3.3)\nwhereΩn= Ω∩{|x|< n}andn∈N.\nWe set\nHn(x,ξ) =Tn(H(x,ξ)) andGn(x,s) =Tn(G(x,s)).\nObserve that\n|Hn(x,ξ)| ≤ |H(x,ξ)| ≤b(x)|ξ|p−1, Hn(x,ξ)≤n, (3.4)\nGn(x,s)s≥0,|Gn(x,s)| ≤ |G(x,s)| ≤c(x)|s|r,|Gn(x,s)| ≤n. (3.5)\nLetun∈W1,p(Ω;ϑα)be a weak solution of the following problem (see the Appendix ):/braceleftBigg\n−div(ϑαA(∇un))+ϑγHn(x,∇un)+ϑαGn(x,un) =fninΩ,\nϑαA(∇un)·ν+ϑβK(un) =gnon∂Ω,(3.6)\ni.e.,un∈W1,p(Ω;ϑα), andˆ\nΩA(∇un)∇vdϑα+ˆ\nΩHn(x,∇un)vdϑγ+ˆ\nΩGn(x,un)vdϑγ\n+ˆ\n∂ΩK(un)vdϑβ=ˆ\nΩvfndϑα+ˆ\n∂Ωvgndϑβ,(3.7)\nfor allv∈W1,p(Ω;ϑα).\nLemma 3.1. Letube a measurable function such that Tk(u)∈W1,p(Ω;ϑα)for every k >0, andˆ\nΩ|∇Tk(u)|pdϑα+ˆ\n∂Ω|Tk(u)|pdϑβ≤C0k,∀k >0, (3.8)\nwhereC0>0is a given constant. Then, |u|p−1∈Lp∗\nα,γ/p,∞(Ω;ϑγ),|u|p−1∈Lq/p,∞(∂Ω;ϑβ),\n|∇u|p−1∈LN′\nα,γ,∞(Ω;ϑγ), and\n/ba∇dbl|u|p−1/ba∇dbl(γ,p∗α,γ/p,∞)≤CC0, (3.9)\n/ba∇dbl|u|p−1/ba∇dbl(β,q/p,∞)≤CC0, (3.10)\n/ba∇dbl|∇u|p−1/ba∇dbl(γ,N′α,γ,∞)≤CC0, (3.11)\nwhereC=C(N,p,q,α,γ,β, Ω)>0,p < q≤(N−1)p/(N−p), andp∗\nα,γ= (N+γ)p/(N+\nα−p).\nProof. First, we prove (3.9). Using Propositions 2.3 and 2.5, for ev eryk >0, we have\nkp∗\nα,γϑγ({|u|> k})≤ˆ\nΩ|Tk(u)|p∗\nα,γdϑγ≤C/ba∇dblTk(u)/ba∇dblp∗\nα,γ\n1,p,Ω,α≤C(C0k)p∗\nα,γ/p, (3.12)\nor equivalently, for every s >0,\nsp∗\nα,γ/(p−1)ϑγ({|u|p−1> s})≤C(C0s1/(p−1))p∗\nα,γ/p.\nWe deduce that\nϑγ({|u|p−1> s})≤C(C0s−1)p∗\nα,γ/p,\nand we get\n(|u|p−1)∗\nγ(t)≤C0(t/C)−p/p∗\nα,γ,∀t >0.8 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nHence,\n/ba∇dbl|u|p−1/ba∇dbl(γ,p∗α,γ/p,∞)≤CC0,\nwhich proves (3.9). Applying Proposition 2.1, we can simila rly deduce (3.10).\nNow we prove (3.11). From (3.8), we deduce that for every s >0andk >0\nspϑγ({|∇u|> s,|u|< k})≤ˆ\n{|u|<k}|∇u|pdϑα=ˆ\nΩ|∇Tk(u)|pdϑα≤C0k,\ni.e., for every s >0andk >0,\nsp/(p−1)ϑγ({|∇u|p−1> s,|u|< k} ≤C0k. (3.13)\nFrom (3.12) and (3.13), we obtain that for every s >0andk >0,\nϑγ({|∇u|p−1> s})\n≤ϑγ({|∇u|p−1> s,|u|< k})+ϑγ({|∇u|p−1> s,|u|> k})\n≤C0k\nsp′+C(C0k)p∗\nα,γ/p\nkp∗α,γ=C0k\nsp′+CCp∗\nα,γ/p\n0k−p∗\nα,γ/p′.\nChoosing k >0such that\nC0k\nsp′=Cp∗\nα,γ/p\n0k−p∗\nα,γ/p′,\nyields\nϑγ({x∈Ω||∇u|p−1> s})≤CCN′\nα,γ\n0\nsN′α,γ.\nThen\n(|∇u|p−1)∗\nγ(t)≤CC0\nt1/N′α,γ,∀t >0.\nTherefore,\n/ba∇dbl|∇u|p−1/ba∇dbl(γ,N′α,γ,∞)≤CC0,\nwhich proves (3.11). /square\nLemma 3.2. Every solution unof(3.6) satisfies\n/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞)≤C, (3.14)\n/ba∇dbl|un|p−1/ba∇dbl(γ,p∗α,γ/p,∞)≤C, (3.15)\n/ba∇dbl|un|p−1/ba∇dbl(β,q/p,∞)≤C, (3.16)\nwherep < q≤(N−1)p/(N−p)andCis a positive constant that depends only on p,q,ϑα(Ω),\nϑβ(Ω),N,α,γ,β,/ba∇dblb/ba∇dbl(γ,Nα,γ,1),/ba∇dblc/ba∇dbl(γ,z′,1),/ba∇dblf/ba∇dbl1,Ω,α, and/ba∇dblg/ba∇dbl1,∂Ω,β.\nProof. We will proceed in two cases:\nFirst case: If /ba∇dblb/ba∇dbl(γ,Nα,γ,1)is small enough.\nUsingTk(un),k >0, as a test function in (3.7), we obtain\nˆ\nΩA(∇un)∇Tk(un)dϑα+ˆ\nΩHn(x,∇un)Tk(un)dϑγ+ˆ\nΩGn(x,un)Tk(un)dϑγ\n+ˆ\n∂ΩK(un)Tk(un)dϑβ≤ˆ\nΩTk(un)fndϑα+ˆ\n∂ΩTk(un)gndϑβ.(3.17)EXISTENCE OF RENORMALIZED SOLUTIONS 9\nFrom (H2), we have\nˆ\nΩA(∇un)∇Tk(un)dϑα=ˆ\n{un≤k}A(∇un)∇undϑα≥σˆ\n{un≤k}|∇un|pdϑα\n≥σˆ\nΩ|∇Tk(un)|pdϑα.(3.18)\nOn the other hand, by (3.4), and using the generalized H¨ olde r inequality (2.5) in the Lorentz\nspaces, we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇un)Tk(un)dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤kˆ\nΩ|H(x,∇un)|dϑγ≤kˆ\nΩb|∇un|p−1dϑγ\n≤k/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞).(3.19)\nUsing (3.5), it resultsˆ\nΩGn(x,un)Tk(un)dϑγ≥0. (3.20)\nFinally, we have\nˆ\nΩTk(un)fndϑα≤k/ba∇dblfn/ba∇dbl1,Ω,α andˆ\n∂ΩTk(un)gndϑβ≤k/ba∇dblgn/ba∇dbl1,∂Ω,β. (3.21)\nTherefore, using (3.3) and (3.18) - (3.21), we get\nσˆ\nΩ|∇Tk(un)|pdϑα+σˆ\n∂Ω|Tk(un)|pdϑβ\n≤k/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n.(3.22)\nLet us define\nC0=1\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n. (3.23)\nInequality (3.22) becomes\nˆ\nΩ|∇Tk(un)|pdϑα+ˆ\n∂Ω|Tk(un)|pdϑβ≤C0k,∀k >0. (3.24)\nBy Lemma 3.1, we get\n/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞)≤CC0.\nIf/ba∇dblb/ba∇dbl(γ,Nα,γ,1)is small enough,\nC1\nσ/ba∇dblb/ba∇dbl(γ,N,1)<1\n2,\nwe obtain\n/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞)≤C,\ni.e., (3.14).\nSecond case: If /ba∇dblb/ba∇dbl(γ,Nα,γ,1)can take any value.\nStep 1. As in [9], we define the following set Zn. Sinceϑγ(Ω)is finite, the set Iof the constants\ncsuch that ϑγ({|un|=c})>0is at most countable. Let ˆZn=∪c∈I{|un|=c}. Its complement\nZn= Ω\\ˆZnis the union of sets such that ϑγ({|un|=c}) = 0 . Since, for every c,\n∇un= 0 a.e. on{|un|=c}, (3.25)\nand since ˆZnis at most a countable union, we obtain\n∇un= 0 a.e. onˆZn. (3.26)10 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nSince the constants csuch that ϑγ({|un|=c})>0have been eliminated by considering Zn, it\nresults that for ℓifixed and 0< ℓ < ℓ i, the function\nℓ∈(0,ℓi)/ma√sto→ϑγ(Zn∩{ℓ <|un|< ℓi})is continuous . (3.27)\nDefine for ℓ >0,\nSℓ(s) =/braceleftBigg\n0 if|s| ≤ℓ,\n(|s|−ℓ)sign(s)if|s|> ℓ.\nUsing in (3.7) the test function Tk(Sℓ(un))withℓto be specified later, we obtain\nˆ\nΩA(∇un)∇Tk(Sℓ(un))dϑα+ˆ\nΩHn(x,∇un)Tk(Sℓ(un))dϑγ\n+ˆ\nΩGn(x,un)Tk(Sℓ(un))dϑγ+ˆ\n∂ΩK(un)Tk(Sℓ(un))dϑβ\n≤ˆ\nΩTk(Sℓ(un))fndϑα+ˆ\n∂ΩTk(Sℓ(un))gndϑβ.(3.28)\nWe haveˆ\nΩA(∇un)∇Tk(Sℓ(un))dϑα≥ˆ\n{ℓ≤un≤ℓ+k}A(∇un)∇undϑα\n≥σˆ\nΩ|∇Tk(Sℓ(un))|pdϑα,(3.29)\nˆ\nΩGn(x,un)Tk(Sℓ(un))dϑγ≥0, (3.30)\nˆ\nΩTk(Sℓ(un))fndϑα≤k/ba∇dblfn/ba∇dbl1,Ω,αandˆ\n∂ΩTk(Sℓ(un))gndϑβ≤k/ba∇dblgn/ba∇dbl1,∂Ω,β. (3.31)\nLet us now estimate /vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇un)Tk(Sℓ(un))dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nUsingSℓ(s) = 0 for|s| ≤ℓ, (3.4), and (3.26), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇un)Tk(Sℓ(un))dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤kˆ\n{|un|>ℓ}b|∇un|p−1dϑγ\n=kˆ\nZn∩{|un|>ℓ}b|∇Sℓ(un)|p−1dϑγ\n≤k/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{un>ℓ})/ba∇dbl|∇Sℓ(un)|p−1/ba∇dbl(γ,N′α,γ,∞) (3.32)\nCombining (3.28) - (3.32) we have, for all k >0,\nˆ\nΩ|∇Tk(Sℓ(un))|pdϑα+ˆ\n∂Ω|Tk(Sℓ(un))|pdϑβ≤ℓ1k,\nwhereℓ1is defined by\nℓ1=1\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{un>ℓ})/ba∇dbl|∇Sℓ(un)|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n.\nBy Lemma 3.1, we get\n/ba∇dbl|∇Sℓ(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n≤C\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{un>ℓ})/ba∇dbl|∇Sℓ(un)|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig (3.33)\nSince the decreasing rearrangement of the restrictions b|Eandbsatisfies\n(b|E)∗\nγ(t)≤b∗\nγ(t), t∈[0,ϑγ(E)], (3.34)EXISTENCE OF RENORMALIZED SOLUTIONS 11\nfor any measurable set E, we have\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{un>ℓ})=ˆϑγ(Zn∩{|un|>ℓ})\n0(b|Zn∩{|un|>ℓ})∗\nγ(t)t1/Ndt\nt\n≤ˆϑγ(Zn∩{|un|>ℓ})\n0b∗\nγ(t)t1/Ndt\nt.(3.35)\nIn the case where\nC\nσˆϑγ(Zn∩{|un|>0})\n0b∗\nγ(t)t1/Ndt\nt≤1\n2, (3.36)\nwe choose ℓ=ℓ1= 0. If (3.36) does not hold, we can choose ℓ=ℓ1>0such that\nC\nσˆϑγ(Zn∩{|un|>ℓ1})\n0b∗\nγ(t)t1/Ndt\nt=1\n2;\nindeed, the function ℓ→ϑγ(Zn∩{|un|> ℓ})is continuous (see (3.27)), decreasing, and tends to 0\nwhenℓtends to∞. Note that ℓ1actually depends on nandγ.\nMoreover, we define δby\nC\nσˆδ\n0b∗\nγ(t)t1/Ndt\nt=1\n2, (3.37)\nobserve that δdoes not depend on n. We have\nϑγ(Zn∩{|un|> ℓ1}) =δ. (3.38)\nWith this choice of ℓ=ℓ1, we obtain from (3.33)\n/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)≤2C\nσ(/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β). (3.39)\nStep 2. Define, for 0≤ℓ < ℓ1, the function Sℓ,ℓ1as\nSℓ,ℓ1(s) =\n\nℓ1−ℓifs > ℓ1,\ns−ℓ ifℓ≤s≤ℓ1,\n0 if−ℓ≤s≤ℓ,\ns+ℓ if−ℓ1≤s≤ −ℓ,\nℓ−ℓ1ifs <−ℓ1.(3.40)\nUsing in (3.7) the test function Tk(Sℓ,ℓ1(un))withℓto be specified later, we obtain\nˆ\nΩA(∇un)∇Tk(Sℓ,ℓ1(un))dϑα+ˆ\nΩHn(x,∇un)Tk(Sℓ,ℓ1(un))dϑγ\n+ˆ\nΩGn(x,un)Tk(Sℓ,ℓ1(un))dϑγ+ˆ\n∂ΩK(un)Tk(Sℓ,ℓ1(un))dϑβ\n=ˆ\nΩTk(Sℓ,ℓ1(un))fndϑα+ˆ\n∂ΩTk(Sℓ,ℓ1(un))gndϑβ,(3.41)\nAs in the previous step, we have\nˆ\nΩA(∇un)∇Tk(Sℓ,ℓ1(un))dϑα≥σˆ\nΩ|∇Tk(Sℓ,ℓ1(un))|pdϑα, (3.42)\nˆ\nΩGn(x,un)Tk(Sℓ,ℓ1(un))dϑγ≥0, (3.43)\nˆ\nΩTk(Sℓ,ℓ1(un))fndϑα≤k/ba∇dblf/ba∇dbl1,Ω,αandˆ\n∂ΩTk(Sℓ,ℓ1(un))gndϑβ≤k/ba∇dblg/ba∇dbl1,∂Ω,β. (3.44)12 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nMoreover, using Sℓ,ℓ1(s) = 0 for|s| ≤ℓand (3.4), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇un)Tk(Sℓ,ℓ1(un))dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤kˆ\n{|un|>ℓ}b|∇un|p−1dϑγ\n≤k/parenleftBiggˆ\n{ℓ<|un|<ℓ1}b|∇un|p−1dϑγ+ˆ\n{|un|≥ℓ1}b|∇un|p−1dϑγ/parenrightBigg\n.(3.45)\nLet us estimate each term of the right-hand side of (3.45). Us ing property (3.25) of Znand the\ngeneralized H¨ older inequality in the Lorentz spaces, we ha ve\nˆ\n{ℓ<|un|<ℓ1}b|∇un|p−1dϑγ=ˆ\nZn∩{ℓ<|un|<ℓ1}b|∇Sℓ,ℓ1(un)|p−1dϑγ\n≤ /ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{ℓ<|un|<ℓ1})/ba∇dbl|∇Sℓ,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞).(3.46)\nSimilarly, for the second term of the right-hand side of (3.4 5) we have:\nˆ\n{|un|≥ℓ1}b|∇un|p−1dϑγ=ˆ\nΩb|∇Sℓ1(un)|p−1dϑγ\n≤ /ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞).\nTherefore, we have:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇un)Tk(Sℓ,ℓ1(un))dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤k/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{ℓ<|un|<ℓ1})/ba∇dbl|∇Sℓ,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)/parenrightBig\n.(3.47)\nCombining (3.41) - (3.47) we have, for all k >0\nˆ\nΩ|∇Tk(Sℓ,ℓ1(un))|pdϑα+ˆ\n∂Ω|Tk(Sℓ,ℓ1(un))|pdϑβ≤ℓ2k,\nwhereℓ2is defined by\nℓ2=1\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{ℓ<|un|<ℓ1})/ba∇dbl|∇Sℓ,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n.\nBy Lemma 3.1, we get:\n/ba∇dbl|∇Sℓ,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n≤C\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{ℓ<|un|<ℓ1})/ba∇dbl|∇Sℓ,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n.(3.48)\nSimilarly to (3.35), using (3.34), we have\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{ℓ<|un|<ℓ1})≤ˆϑγ(Zn∩{ℓ<|un|<ℓ1})\n0b∗\nγ(t)t1/Ndt\nt. (3.49)\nIn the case where\nC\nσˆϑγ(Zn∩{0<|un|<ℓ1})\n0b∗\nγ(t)t1/Ndt\nt≤1\n2, (3.50)EXISTENCE OF RENORMALIZED SOLUTIONS 13\nwe choose ℓ=ℓ2= 0. If (3.50) does not hold, since the function ℓ→ϑγ(Zn∩{ℓ <|un|< ℓ1})is\ncontinuous, we can choose ℓ=ℓ2>0such that\nC\nσˆϑγ(Zn∩{ℓ2<|un|<ℓ1})\n0b∗\nγ(t)t1/Ndt\nt=1\n2.\nNote that ℓ2actually depends on nandγ. We have\n|Zn∩{ℓ2<|un|< ℓ1}|=δ, (3.51)\nwhereδis defined by (3.37). With this choice of ℓ=ℓ2, we obtain from (3.48)\n/ba∇dbl|∇Sℓ2,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n≤2C\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n.(3.52)\nStep 3. Define for 0≤ℓ < ℓ2the function Sℓ,ℓ2:\nSℓ,ℓ2(s) =\n\nℓ2−ℓifs > ℓ2,\ns−ℓ ifℓ≤s≤ℓ2,\n0 if−ℓ≤s≤ℓ,\ns+ℓ if−ℓ2≤s≤ −ℓ,\nℓ−ℓ2ifs <−ℓ2,\nfor every s∈R. Using in (3.7) the test function Tk(Sℓ,ℓ2(un))withℓto be specified later, we obtain\nˆ\nΩA(∇un)∇Tk(Sℓ,ℓ2(un))dϑα+ˆ\nΩHn(x,∇un)Tk(Sℓ,ℓ2(un))dϑγ\n+ˆ\nΩGn(x,un)Tk(Sℓ,ℓ2(un))dϑγ+ˆ\n∂ΩK(un)Tk(Sℓ,ℓ2(un))dϑβ\n=ˆ\nΩTk(Sℓ,ℓ2(un))fndϑα+ˆ\n∂ΩTk(Sℓ,ℓ2(un))gndϑβ,(3.53)\nAs in (3.45) - (3.47), we have/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇un)Tk(Sℓ,ℓ2(un))dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤k/parenleftBiggˆ\n{ℓ<|un|<ℓ2}b|∇un|p−1dϑγ+ˆ\n{ℓ2<|un|<ℓ1}b|∇un|p−1dϑγ\n+ˆ\n{|un|≥ℓ1}b|∇un|p−1dϑγ/parenrightBigg\n≤k/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{ℓ<|un|<ℓ2})/ba∇dbl|∇Sℓ,ℓ2(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ2,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)/parenrightBig\n.\nAs in (3.48), we deduce\n/ba∇dbl|∇Sℓ,ℓ2(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n≤C\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,Zn∩{ℓ<|un|<ℓ2})/ba∇dbl|∇Sℓ,ℓ2(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ2,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n.(3.54)14 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nArguing as in (3.49),\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1,(Zn∩{ℓ<|un|<ℓ2})≤ˆϑγ(Zn∩{ℓ<|un|<ℓ2})\n0b∗\nγ(t)t1/Ndt\nt.\nIn the case where\nC\nσˆϑγ(Zn∩{0<|un|<ℓ2})\n0b∗\nγ(t)t1/Ndt\nt≤1\n2, (3.55)\nwe choose ℓ=ℓ3= 0. If (3.55) does not hold, we can choose ℓ=ℓ3>0such that\nC\nσˆϑγ(Zn∩{ℓ3<|un|<ℓ2})\n0b∗\nγ(t)t1/Ndt\nt=1\n2.\nNote that ℓ3actually depends on nandγ. We have\nϑγ(Zn∩{ℓ3<|un|< ℓ2}) =δ, (3.56)\nwhereδis defined by (3.37). With this choice of ℓ=ℓ3, we obtain from (3.54)\n/ba∇dbl|∇Sℓ3,ℓ2(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n≤2C\nσ/parenleftBig\n/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ2,ℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n+/ba∇dblb/ba∇dbl(γ,Nα,γ,1)/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)+/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β/parenrightBig\n.(3.57)\nSteep 4. We repeat this procedure until the time it stops, i.e., when w e arrive to some i=I(which\ndepends on nandγ) for which we have\nC\nσˆϑγ(Zn∩{0<|un|<ℓI−1})\n0b∗\nγ(t)t1/Ndt\nt≤1\n2,\nthen we choose\nℓI= 0.\nLet us now estimate I. We have\nϑγ(Ω)≥ϑγ(Zn)≥ϑγ(Zn∩{|un|> ℓ1})+ϑγ(Zn∩{ℓ2<|un|< ℓ1})\n+ϑγ(Zn∩{ℓ3<|un|< ℓ2})+···+ϑγ(Zn∩{ℓI−1<|un|< ℓI−2})\nand, in view of (3.38), (3.51), and (3.56) we know that\nϑγ(Zn∩{|un|> ℓ1}) =ϑγ(Zn∩{ℓ2<|un|< ℓ1}) =···\n=ϑγ(Zn∩{ℓI−1<|un|< ℓI−2}) =δ,\nwhereδis defined by (3.37), and does not depend on n. Therefore, (I−1)δ≤ϑγ(Ω), and\nI≤I∗= 1+/bracketleftbiggϑγ(Ω)\nδ/bracketrightbigg\n,\nwhere[s]denotes the integer part of s.\nObserve that Iis estimated by the number I∗which does not depend on n, and which depends on\nb∗\nγandδ. We define\nℓ0=∞, Sℓ1,ℓ0=Sℓ1, (3.58)\nand defining\nXi=/ba∇dbl|∇Sℓi,ℓi−1(un)|p−1/ba∇dbl(γ,N′α,γ,∞),for1≤i≤I,\na1=2C\nσ/ba∇dblb/ba∇dbl(γ,Nα,γ,1),\na2=2C\nσ(/ba∇dblf/ba∇dbl1,Ω,α+/ba∇dblg/ba∇dbl1,∂Ω,β).EXISTENCE OF RENORMALIZED SOLUTIONS 15\nObserve that\nX1=/ba∇dbl|∇Sℓ1,ℓ0(un)|p−1/ba∇dbl(γ,N′α,γ,∞)=/ba∇dbl|∇Sℓ1(un)|p−1/ba∇dbl(γ,N′α,γ,∞).\nFrom (3.39), (3.52), (3.57), and (3.58), we see\nX1≤a2, X2≤a1X1+a2, X3≤a1X2+a1X1+a2, ...,\nXI≤a1XI−1+···+a1X1+a2, I≤I∗.\nIt can be proved by induction that\nXi≤(a1+1)i−1a2for1≤i≤I.\nSinceℓI= 0, we have\n|∇un|p−1=I/summationdisplay\ni=1|∇un|p−1χ{ℓi<|un|<ℓi−1}=I/summationdisplay\ni=1|∇Sℓi,ℓi−1(un)|p−1.\nTherefore,\n/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞)≤I/summationdisplay\ni=1/ba∇dbl|∇Sℓi,ℓi−1(un)|p−1/ba∇dbl(γ,N′α,γ,∞)\n≤I/summationdisplay\ni=1Xi≤a2I/summationdisplay\ni=1(a1+1)i−1\n=a2(a1+1)I−1\na1≤a2\na1[(a1+1)I∗−1],\ni.e., the desired result, (3.14).\nLet us finally prove the result (3.15). From (3.24), we have\nˆ\nΩ|∇Tk(un)|pdϑα+ˆ\n∂Ω|Tk(un)|pdϑβ≤C0k,∀k >0,\nwhere the constant C0defined by (3.23) is now bounded independently on nin view of (3.14). The\nresults (3.15) and (3.16) then follow from (3.9) and (3.10), respectively. /square\n4. E XISTENCE RESULTS FOR RENORMALIZED SOLUTIONS\nIn this section, we prove our main results, which give the exi stence of a renormalized solution to\nproblem (1.1).\nProof of Theorem 1.1 .\nStep 1. A priori estimates .\nUsingTk(un)fork >0, as test function in (3.7) we have\nˆ\nΩA(∇un)∇Tk(un)dϑα+ˆ\nΩHn(x,∇un)Tk(un)dϑγ\n+ˆ\nΩGn(x,un)Tk(un)dϑγ+ˆ\n∂ΩK(un)Tk(un)dϑβ\n=ˆ\nΩTk(un)fndϑα+ˆ\n∂ΩTk(un)gndϑβ,\nwhich implies, by ( H2) and (H3),16 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nσˆ\nΩ|∇Tk(un)|pdϑα+ˆ\n∂Ω|Tk(un)|pdϑβ\n≤kˆ\nΩb|∇un|p−1dϑγ+k/ba∇dblfn/ba∇dbl1,Ω,α+k/ba∇dblgn/ba∇dbl1,∂Ω,β.\nBy (3.14) and (3.3) we getˆ\nΩ|∇Tk(un)|pdϑα+ˆ\n∂Ω|Tk(un)|pdϑβ≤Ck (4.1)\nfor a suitable positive constant Cdoes not depend on kandn. We deduce that, for every k >0,\nTk(un)is bounded in W1,p(Ω;ϑα). (4.2)\nMoreover taking into account ( H2) and (4.1), we obtain that for any k >0\nA(∇Tk(un))is bounded in (Lp′(Ω;ϑα))N, (4.3)\nuniformly with respect to n.\nStep 2. We prove\nϑα({x∈Ω||un(x)|> L})≤C\nln(1+L),\nϑβ({x∈∂Ω||un(x)|> L})≤C\nln(1+L),(4.4)\nfor alln, whereC >0is a constant independent of n.\nForq≥1, we consider the function\nΨq(r) =ˆr\n01\n(1+|t|)qdt,∀r∈R.\nForq= 1, we have\nΨ1(r) =ˆr\n01\n1+|t|dt= sign(r)ln(1+|r|), (4.5)\nand by H¨ older inequality,\n|Ψ1(r)|p=/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆr\n01\n1+|t|dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n≤ |r|p/p′/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆr\n01\n(1+|t|)pdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle=|r|p−1|Ψp(r)|. (4.6)\nUsingΨp(un)as a test function in (3.7), we get\nˆ\nΩA(∇un)∇Ψp(un)dϑα+ˆ\nΩHn(x,∇un)Ψp(un)dϑγ\n+ˆ\nΩGn(x,un)Ψp(un)dϑγ+ˆ\n∂ΩK(un)Ψp(un)dϑβ\n=ˆ\nΩΨp(un)fndϑα+ˆ\n∂ΩΨp(un)gndϑβ.(4.7)\nBy (H2), (3.4), (3.5), and since /ba∇dblΨp(un)/ba∇dblL∞(Ω)≤1\np−1, we get\nσˆ\nΩ|∇un|p\n(1+|un|)pdϑα+ˆ\n∂ΩK(un)Ψp(un)dϑβ\n≤1\np−1ˆ\nΩb|∇un|p−1dϑγ+1\np−1/ba∇dblfn/ba∇dbl1,Ω,α+1\np−1/ba∇dblgn/ba∇dbl1,∂Ω,β.\nBy (H2), (3.3), (3.14), (4.5), and (4.6), we haveˆ\nΩ|∇Ψ1(un)|pdϑα+ˆ\n∂Ω|Ψ1(un)|pdϑβ≤C,∀n,EXISTENCE OF RENORMALIZED SOLUTIONS 17\nwhereC >0is a constant independent of n.\nBy Propositions 2.3 and 2.5, we have\n/ba∇dblΨ1(un)/ba∇dblp,Ω,α≤C.\nHence, according to the definition of Ψ1, we obtain (4.4).\nFurthermore, as a consequence of (4.4), we make the followin g assertions:\nBy (4.2), compact embedding theorem, and Proposition 2.1, w e can assume that\nTk(un)|Ωmis a Cauchy sequence in measure , (4.8)\nTk(τ(un)) =τ(Tk(un))is a Cauchy sequence in measure , (4.9)\nwhereΩm= Ω∩{|x|< m}.\nLetε >0be fixed. For every ℓ >0and every n1andn2, we have\n{|un1−un2|> ε} ⊂ {|un1|> ℓ}∪{|un2|> ℓ}∪{|Tℓ(un1)−Tℓ(un2)|> ε}. (4.10)\nCombining (4.10), (4.4), (4.8), and (4.9), we conclude that\nun|Ωmandτ(un)are Cauchy sequences in measure .\nTherefore, there exist measurable functions u: Ω→¯Randw:∂Ω→¯Rfinites a.e. in Ω.\nAdditionally, according to (4.2) and (4.3), for any k >0there exists a function Vk∈(Lp′(Ω;ϑα))N\nsuch that, up to a subsequence still indexed by n,\nun→ua.e. inΩ, (4.11)\nTk(un)⇀ Tk(u)inW1,p(Ω;ϑα), (4.12)\nτ(Tk(u)) =Tk(w), (4.13)\nA(∇Tk(un))⇀ Vkin(Lp′(Ω;ϑα))N, (4.14)\nfor allk >0. Furthermore, since W1,p(Ω;ϑα)֒→Lp(∂Ω;ϑβ), we can assume that\nTk(un)→Tk(w),inLp(∂Ω;ϑβ). (4.15)\nStep 3. The following limits holds.\nlim\nk→∞limsup\nn→∞1\nkˆ\nΩA(∇un)∇Tk(un)dϑα= 0. (4.16)\nUsing the test function1\nkTk(un)in (3.7), we have\n1\nkˆ\nΩA(∇un)∇Tk(un)dϑα+1\nkˆ\nΩHn(x,∇un)Tk(un)dϑγ\n+1\nkˆ\nΩGn(x,un)Tk(un)dϑγ+1\nkˆ\n∂ΩK(un)Tk(un)dϑβ\n=1\nkˆ\nΩTk(un)fndϑα+1\nkˆ\n∂ΩTk(un)gndϑβ,\nwhich yields that\n1\nkˆ\nΩA(∇un)∇Tk(un)dϑα+1\nkˆ\n∂ΩK(un)Tk(un)dϑβ\n≤1\nkˆ\nΩ|Hn(x,∇un)Tk(un)|dϑγ+1\nkˆ\nΩ|fn||Tk(un)|dϑα\n+1\nkˆ\n∂Ω|gn||Tk(un)|dϑβ(4.17)18 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nDue to (4.11), the sequence Tk(un)converges to Tk(u)a.e. inΩ. Sincefnstrongly converges to\nfinL1(Ω,ϑα)it follows that\nlim\nn→∞1\nkˆ\nΩ|fn||Tk(un)|dϑα=1\nkˆ\nΩ|f||Tk(u)|dϑα.\nRecalling that uis finite a.e. in Ω, we deduce that\nlim\nk→∞lim\nn→∞1\nkˆ\nΩ|fn||Tk(un)|dϑα= 0.\nSimilarly, we obtain\nlim\nk→∞lim\nn→∞1\nkˆ\n∂Ω|gn||Tk(un)|dϑβ= 0.\nNow, we procedure to prove\nlim\nk→∞limsup\nn→∞1\nkˆ\nΩ|Hn(x,∇un)Tk(un)|dϑγ= 0. (4.18)\nBy the generalized H¨ older inequality, (3.14) and (4.4), we have\n1\nkˆ\nΩ|Hn(x,∇un)Tk(un)|dϑγ\n≤ /ba∇dblb/ba∇dbl(γ,Nα,γ,1,{|un|>√\nk})/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞,{|un|>√\nk})\n+√\nk\nk/ba∇dblb/ba∇dbl(γ,Nα,γ,1,{|un|≤√\nk})/ba∇dbl|∇un|p−1/ba∇dbl(γ,N′α,γ,∞,{|un|≤√\nk})\n≤Cˆϑγ({|un|>√\nk})\n0t1/Nα,γb∗\nγ(t)dt\nt+C√\nk/ba∇dblb/ba∇dbl(γ,Nα,γ,1)\n≤CˆC/ln(1+√\nk)\n0t1/Nα,γb∗\nγ(t)dt\nt+C√\nk/ba∇dblb/ba∇dbl(γ,Nα,γ,1).\nThis proves (4.18). It follows that (4.16) holds.\nStep 4. We prove that for any k >0\nlim\nn→∞ˆ\nΩ[A(∇Tk(un))−A(∇Tk(u))][∇Tk(un)−∇Tk(u)]dϑα= 0. (4.19)\nLethℓbe defined by\nhℓ(s) =\n\n0 if|s|>2ℓ,\n2ℓ−|s|\nℓifℓ <|s| ≤2ℓ,\n1 if|s| ≤ℓ.\nUsinghℓ(un)[Tk(un)−Tk(u)]in (3.7), we have\nˆ\nΩhℓ(un)A(∇un)(∇Tk(un)−∇Tk(u))dϑα\n=ak,ℓ,n+bk,ℓ,n+ck,ℓ,n+dk,ℓ,n+ek,ℓ,n+fk,ℓ,n,(4.20)EXISTENCE OF RENORMALIZED SOLUTIONS 19\nwith\nak,ℓ,n=ˆ\nΩhℓ(un)fn[Tk(un)−Tk(u)]dϑα,\nbk,ℓ,n=ˆ\n∂Ωhℓ(un)gn[Tk(un)−Tk(u)]dϑβ,\nck,ℓ,n=−ˆ\nΩh′\nℓ(un)A(∇un)∇un[Tk(un)−Tk(u)]dϑα,\ndk,ℓ,n=−ˆ\nΩHn(x,∇un)hℓ(un)[Tk(un)−Tk(u)]dϑγ,\nek,ℓ,n=−ˆ\nΩGn(x,un)hℓ(un)[Tk(un)−Tk(u)]dϑγ,\nfk,ℓ,n=−ˆ\n∂Ωhℓ(un)K(un)[Tk(un)−Tk(u)]dϑβ.\nWe now pass to the limit in (4.20) first as n→ ∞ and then as ℓ→ ∞ . We prove:\nlim\nℓ→∞limsup\nn→∞ˆ\nΩhℓ(un)A(∇un)[∇Tk(un)−∇Tk(u)]dϑα= 0. (4.21)\nDue to the point-wise convergence of unthe sequence Tk(un)−Tk(u)converges to zero a.e. in\nΩ. Sincefn→finL1(Ω;ϑα),gn→ginL1(∂Ω;ϑβ), andTk(un)→Tk(u)inLp(∂Ω;ϑβ), we\nobtain that\nlim\nn→∞ak,ℓ,n= lim\nn→∞bk,ℓ,n= 0, (4.22)\nalso\nlim\nn→∞fk,ℓ,n=ˆ\n∂Ωhℓ(un)K(T2ℓ(un))[Tk(un)−Tk(u)]dϑβ= 0. (4.23)\nSince\n|ck,ℓ,n| ≤2k\nℓˆ\n{|un|≤2ℓ}A(∇un)∇undϑα\nand due to (4.16), we obtain\nlim\nℓ→∞limsup\nn→∞ck,ℓ,n= 0. (4.24)\nNow, we prove that\nlim\nn→∞dk,ℓ,n= 0. (4.25)\nLetq1>1so that1/Nα,γ+(p−1)/p+1/q1= 1. From (3.4), (2.4), and (4.1),\n|dk,ℓ,n|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇un)hℓ(un)[Tk(un)−Tk(u)]dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nΩHn(x,∇T2ℓ(un))hℓ(un)[Tk(un)−Tk(u)]dϑγ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/parenleftbiggˆ\nΩbNα,γdϑγ/parenrightbigg1/Nα,γ/parenleftbiggˆ\nΩ|∇T2ℓ(un)|pdϑγ/parenrightbigg(p−1)/p\n·/bracketleftbiggˆ\nΩ(Tk(un)−Tk(u))q1dϑγ/bracketrightbigg1/q1\n≤(2Cℓ)(p−1)/p/parenleftbiggˆ\nΩbNα,γdϑγ/parenrightbigg1/Nα,γ/bracketleftbiggˆ\nΩ(Tk(un)−Tk(u))q1dϑγ/bracketrightbigg1/q1\n.\nThen, we obtain (4.25).\nOn the other hand, we deduce from (3.15) and from the fact that 0≤r <(N+γ)(p−1)/(N+\nα−p), one has\n/ba∇dbl|un|r/ba∇dbl(γ,z,∞)≤C. (4.26)20 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nUsing (3.5), (4.26), and generalized H¨ older inequality, w e get\n/ba∇dblGn(x,un)/ba∇dbl1,E,γ=ˆ\nE|Gn(x,un)|\n≤ˆ\nE|c||un|rdϑγ≤ /ba∇dblc/ba∇dbl(γ,z′,1,E)/ba∇dbl|un|r/ba∇dbl(γ,z,∞,E)\n≤Cˆϑγ(E)\n0c∗\nγ(t)t1/z′dt\nt,\nwhich is small when ϑγ(E)is small. Hence Gn(x,un)is equi-integrable. Therefore, Vitali’s Theo-\nrem implies that\nGn(x,un)→G(x,u)inL1(Ω,ϑγ). (4.27)\nThen we conclude\nlim\nℓ→∞lim\nn→∞ek,ℓ,n= 0. (4.28)\nFrom (4.22), (4.24), (4.25), and (4.28), we obtain that for a nyk >0\nlim\nℓ→∞limsup\nn→∞ˆ\nΩhℓ(un)A(∇un)[∇Tk(un)−∇Tk(u)]dϑα= 0,\nwith which we prove (4.21).\nRecalling that for any ℓ > k , we have\nhℓ(un)A(∇un)∇Tk(un) =A(∇un)∇Tk(un)a.e. inΩ.\nIt follows that\nlimsup\nn→∞ˆ\nΩA(∇un)∇Tk(un)dϑα\n≤lim\nℓ→∞limsup\nn→∞ˆ\nΩhℓ(un)A(∇un)∇Tk(u)dϑα.(4.29)\nAccording to the definition of hℓ, we have\nhℓ(un)A(∇un) =hℓ(un)A(∇T2ℓ(un)) a.e. inΩ,\nso that (4.11) and (4.14) give\nlim\nn→∞ˆ\nΩhℓ(un)A(∇un)∇Tk(u)dϑα=ˆ\nΩhℓ(u)V2ℓ∇Tk(u)dϑα. (4.30)\nIfℓ > k , we have\nA(∇Tℓ(un))χ{|un|<k}=A(∇Tk(un))χ{|un|<k}\na.e. inΩ. From (4.11) and (4.14), it follows that\nVℓχ{|u|<k}=Vkχ{|u|<k}a.e. inΩ\\{|u|=k},\nand then we obtain for any ℓ > k\nVℓ∇Tk(u) =Vk∇Tk(u)a.e. inΩ.\nTherefore, (4.29) and (4.30) allow us to conclude\nlimsup\nn→∞ˆ\nΩA(∇Tk(un))∇Tk(un)dϑα≤ˆ\nΩVk∇Tk(u)dϑα. (4.31)\nNow, we are in a position to prove (4.19). Indeed, the monoton e character of Aimplies that for\nanyn,\n0≤ˆ\nΩ[A(∇Tk(un))−A(∇Tk(u))][∇Tk(un)−∇Tk(u)]dϑα. (4.32)EXISTENCE OF RENORMALIZED SOLUTIONS 21\nWriting\nˆ\nΩ[A(∇Tk(un))−A(∇Tk(u))][∇Tk(un)−∇Tk(u)]dϑα\n=ˆ\nΩA(∇Tk(un))[∇Tk(un)−∇Tk(u)]dϑα\n−ˆ\nΩA(∇Tk(u))[∇Tk(un)−∇Tk(u)]dϑα.\nUsing (4.31) and (4.32) allows one to conclude that (4.19) ho lds for any k >0.\nStep 5. We prove that for any k >0\nA(∇Tk(u)) =Vk, (4.33)\nA(∇Tk(un))∇Tk(un)⇀ A(∇Tk(u))∇Tk(u)inL1(Ω;ϑα). (4.34)\nFrom (4.19), we have for any k >0\nlim\nn→∞ˆ\nΩA(∇Tk(un))∇Tk(un)dϑα=ˆ\nΩVk∇Tk(u)dϑα. (4.35)\nThe classical arguments, known as Minty arguments (see [32, 33]), allow us to identify Vkwith\nA(∇Tk(u)). Letφ∈(L∞(Ω))N. By (4.14) and (4.35), it follows that for any t∈R\nlim\nn→∞ˆ\nΩ[A(∇Tk(un))−A(∇Tk(u)+tφ)][∇Tk(un)−∇Tk(u)−tφ]dϑα\n=−ˆ\nΩ[Vk−A(∇Tk(u)+tφ)]tφdϑα.\nUsing the monotone character ( H2) ofA, we obtain that for any t/\\e}atio\\slash= 0\n−sign(t)ˆ\nΩ[Vk−A(∇Tk(u)+tφ)]φdϑα≥0.\nSinceA(∇Tk(u) +tφ)converges strongly to A(∇Tk(u))in(Lp′(Ω;ϑα))Nastgoes to zero,\nlettingt→0in the above inequality leads to\nˆ\nΩ[Vk−Tk(∇u)]φdϑα= 0.\nfor anyφ∈(L∞(Ω))N. We conclude that (4.33).\nFrom (4.19), we get\n[A(∇Tk(un))−A(∇Tk(u))][∇Tk(un)−∇Tk(u)]→0,inL1(Ω,ϑα).\nUsing (4.12) leads to (4.34).\nStep 6. We claim that\n∇un|Ωmis a Cauchy sequence in measure . (4.36)22 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nHere, we follow an argument used in the proof of [12, Lemma 1]. To prove (4.36), given ε >0\nandε1>0fixed, we set, for some L >1andδ∈(0,1):\nE1={|∇Tk(un1)|> L}∪{|∇Tk(un2)|> L}∪{|un1|> L}∪{|un2|> L},\nE2={|Tk(un1)−Tk(un2)|> δ2},\nE3={|Tk(un1)−Tk(un2)| ≤δ2,|∇Tk(un1)| ≤L,|∇Tk(un2)| ≤L,\n|un1| ≤L,|un2| ≤L,|∇Tk(un1)−∇Tk(un2)|> ε}.\nObserve that\n{|∇Tk(un1)−∇Tk(un2)|> ε} ⊂E1∪E2∪E3.\nFrom (4.2), (Tk(un))and(|∇Tk(un)|)are bounded in L1(Ωm;ϑα); we have\nϑα(E1∩Ωm)< ε1,\nforL > k large enough, independently of n1andn2.\nNow, let’s consider ϑα(E3∩Ωm). Define\nK0={(ξ,η)∈R2N||ξ| ≤L,|η| ≤L,|ξ−η| ≥ε},\nthen\ninf{[A(ξ)−A(η)][ξ−η]|(ξ,η)∈K0}=C >0,\nsinceK0is compact.\nHence,\nCϑα(E3∩Ωm)\n≤ˆ\nE3∩Ωm[A(∇Tk(un1))−A(∇Tk(un2))][∇Tk(un1)−∇Tk(un2)]dϑα\n≤ˆ\nΩhL(un1)hL(un2)hδ(Tk(un1)−Tk(un2))\n·[A(∇Tk(un1))−A(∇Tk(un2))][∇Tk(un1)−∇Tk(un2)]dϑα.\nSubstituting hL(un1)hL(un2)hδ(Tk(un1)−Tk(un2))[Tk(un1)−Tk(un2)]into (3.7) and considering\nalsoˆ\nΩ{h′\nL(un1)hL(un2)hδ(Tk(un1)−Tk(un2))A(∇un1)∇un1\n+hL(un1)h′\nL(un2)hδ(Tk(un1)−Tk(un2))A(∇un1)∇un2\n+hL(un1)hL(un2)h′\nδ(Tk(un1)−Tk(un2))\n·A(∇un1)[∇Tk(un1)−∇Tk(un2)]}(Tk(un1)−Tk(un2))dϑα\n≤CLδ,\nbecause (H2) and (3.6). Proceeding as in Step 4, for a suitable constant δ∈(0,1), we have that there\nisN0>0such that\nϑα(E3∩Ωm)< ε1,ifn1,n2> N0.\nBy (4.8), there exists N1>0such that\nϑα(E2∩Ωm)< ε1,ifn1,n2> N1.\nHence,\n∇Tk(un)|Ωmis a Cauchy sequence in measure . (4.37)\nFinally, if ε >0is fixed, we have\n{|∇un1−∇un2|> ε}\n⊂ {|un1|> L0}∪{|un2|> L0}∪{|∇TL0(un1)−∇TL0(un2)|> ε}.\nTherefore, (4.4) and (4.37) imply (4.36).EXISTENCE OF RENORMALIZED SOLUTIONS 23\nOn the other hand, from a computation similar to (3.45), usin g (4.36), we see\nHn(x,∇un)→H(x,∇u)inL1(Ω;ϑα). (4.38)\nStep 7. We now proceed to take the limit in the approximated pr oblem.\nLeth∈W1,∞(R)with compact support contained in the interval [−k,k], wherek >0, and let\nϕ∈W1,p(Ω;ϑα)∩L∞(Ω). Usingh(un)ϕas a test function in the approximated problem, we haveˆ\nΩh(un)A(∇un)∇ϕdϑα+ˆ\nΩh′(un)ϕA(∇un)∇undϑα\n+ˆ\nΩHn(x,∇un)h(un)ϕdϑγ+ˆ\nΩGn(x,un)h(un)ϕdϑγ\n+ˆ\n∂ΩK(un)h(un)ϕdϑβ=ˆ\nΩh(un)ϕfndϑα+ˆ\n∂Ωh(un)ϕgndϑβ.(4.39)\nWe want to pass to the limit in this equality. Since supp(h)is contained in the interval [−k,k],\nand considering that fn→finL1(Ω;ϑα),gn→ginL1(∂Ω;ϑβ),Tk(un)→Tk(u)inLp(∂Ω;ϑβ),\nand by (4.11), we obtain\nlim\nn→∞ˆ\nΩfnh(un)ϕdϑα=ˆ\nΩfh(u)ϕdϑα,\nlim\nn→∞ˆ\n∂Ωgnh(un)ϕdϑβ=ˆ\n∂Ωgh(u)ϕdϑβ,\nlim\nn→∞ˆ\n∂ΩK(un)h(un)ϕdϑβ=ˆ\n∂ΩK(u)h(u)ϕdϑβ.\nIn view of (4.14) and (4.33),\nlim\nn→∞ˆ\nΩh(un)A(∇Tk(un))∇ϕdϑα=ˆ\nΩh(u)A(∇Tk(u))∇ϕdϑα.\nFrom (4.34), we get\nlim\nn→∞ˆ\nΩh′(un)ϕA(∇Tk(un))∇Tk(un)dϑα=ˆ\nΩh′(u)ϕA(∇Tk(u))∇Tk(u)dϑα.\nBy (4.27) and (4.38), up to a subsequence still indexed by n,\nlim\nn→∞ˆ\nΩGn(x,un)h(un)ϕdϑγ=ˆ\nΩG(x,u)h(u)ϕdϑγ.\nlim\nn→∞ˆ\nΩHn(x,∇un)h(un)ϕdϑγ=ˆ\nΩHn(x,∇u)h(u)ϕdϑγ.\nTherefore, by passing to the limit in (4.39), we obtain condi tion (2.13) in the definition of a\nrenormalized solution. The decay of the truncated energy (2 .12) is a consequence of (4.16) and\n(4.34).\nAdditionally, from (4.1), (∇un)is equi-integrable, and by ∇Tk(un)→ ∇Tk(u)a.e. inΩ, we\nhave\n∇Tk(un)→ ∇Tk(u)inL1(Ω;ϑα).\nMoreover, by Fatou’s lemma, we deduce from (4.1) thatˆ\nΩ|∇Tk(u)|pdϑα+ˆ\nΩ|∇Tk(u)|pdϑα≤Ck.\nThen, Lemma 3.1 implies |u|p−1∈Lp∗\nα,γ/p,∞(Ω;ϑγ),|u|p−1∈Lq/p,∞(∂Ω;ϑβ), and|∇u|p−1∈\nLN′\nα,γ,∞(Ω;ϑγ). Recall that, from Step 2, uis finite a.e. in Ω, andTk(u)∈W1,p(Ω;ϑα)for any\nk >0. We can conclude that uis a renormalized solution to (1.1). /square24 JUAN A. APAZA AND MANASS ´ES DE SOUZA\n4.1. Stability result. For anyn∈N, letunbe a renormalized solution to the problem\n/braceleftBigg\n−div(ϑαA(∇u))+ϑγHn(x,∇u)+ϑγGn(x,u) =fnϑαinΩ,\nϑαA(∇u)·ν+ϑβK(u) =gnϑβon∂Ω,(4.40)\nwherefn∈L1(Ω,ϑα),gn∈L1(∂Ω;ϑβ), and\nfn→finL1(Ω,ϑα)andgn→ginL1(∂Ω;ϑβ). (4.41)\nAlso,Hn: Ω×RN→R,Gn: Ω×R→Rare Carath´ eodory functions satisfying:\n|Hn(x,ξ)| ≤b(x)|ξ|p−1, b∈LNα,γ,1(Ω;ϑγ), (4.42)\nGn(x,s)s≥0,|Gn(x,s)| ≤c(x)|s|r, c∈Lz′,1(Ω;ϑγ), (4.43)\nfor almost every x∈Ωand for every s∈Randξ∈RN, whereNα,γ,r,zverify (1.5). Furthermore,\nfor almost every x∈Ω\nHn(x,ξn)→H(x,ξ)andGn(x,sn)→G(x,s), (4.44)\nfor every sequences (ξn)⊂RNand(sn)⊂R, whereHandGare Carath´ eodory functions.\nProof of Theorem 1.2 .We mainly follow the arguments developed in the proof of Theo rem 1.1.\nUsingh=hℓ, defined by\nhℓ(s) =\n\n0 if|s|>2ℓ,\n2ℓ−|s|\nℓifℓ <|s| ≤2ℓ,\n1 if|s| ≤ℓ,\nandϕ=Tk(un)in the renormalized formulation (2.13) we have, for any ℓ >0and anyk >0\nˆ\nΩhℓ(un)A(∇un)∇Tk(un)dϑα+ˆ\nΩh′\nℓ(un)A(∇un)∇unTk(un)dϑα\n+ˆ\nΩHn(x,∇un)Tk(un)hℓ(un)dϑγ+ˆ\nΩGn(x,un)Tk(un)hℓ(un)dϑγ\n+ˆ\n∂ΩK(un)Tk(un)hℓ(un)dϑβ\n=ˆ\nΩTk(un)hℓ(un)fndϑα+ˆ\n∂ΩTk(un)hℓ(un)gndϑβ.(4.45)\nWe now pass to the limit as ℓ→ ∞ :\nIn view of the definition of hℓfor anyℓ > k we have\nˆ\nΩhℓ(un)A(∇un)∇Tk(un)dϑα=ˆ\nΩA(∇un)∇Tk(un)dϑα.\nDue to (2.12), we get\nlim\nℓ→∞ˆ\n���h′\nℓ(un)A(∇un)∇unTk(un)dϑα= 0.\nIt follows that passing to the limit as ℓ→ ∞ in (4.45) leads to\nˆ\nΩA(∇un)∇Tk(un)dϑα+ˆ\nΩHn(x,∇un)Tk(un)dϑγ+ˆ\nΩGn(x,un)Tk(un)dϑγ\n+ˆ\n∂ΩK(un)Tk(un)dϑβ=ˆ\nΩTk(un)fndϑα+ˆ\n∂ΩTk(un)gndϑβ.(4.46)EXISTENCE OF RENORMALIZED SOLUTIONS 25\nHence, proceeding as in step 1 of the proof Theorem 1.1, we hav e\nTk(un)is bounded in W1,p(Ω;ϑα),\nA(∇Tk(un))is bounded in (Lp′(Ω;ϑα))Nfor anyk >0.\nUsing a similar process to one used to obtain (4.46) we get\nˆ\nΩA(∇un)∇Ψp(un)dϑα+ˆ\nΩHn(x,∇un)Ψp(un)dϑγ\n+ˆ\nΩGn(x,un)Ψp(un)dϑγ+ˆ\n∂ΩK(un)Ψp(un)dϑβ\n=ˆ\nΩΨp(un)fndϑα+ˆ\n∂ΩΨp(un)gndϑβ,\nwhere\nΨp(r) =ˆr\n01\n(1+|t|)pdt,∀r∈R.\nBy the argument in step 2 of the proof of Theorem 1.1 we obtain\nϑα({x∈Ω||un(x)|> L})≤C\nln(1+L),\nϑβ({x∈∂Ω||un(x)|> L})≤C\nln(1+L),\nfor alln, whereC >0is a constant independent of n.\nFurthermore, there exist a measurable functions u: Ω→¯R,w:∂Ω→¯R, and a field Vk∈\n(Lp′(Ω;ϑα))Nsuch that, up to a subsequence still indexed by n,\nTk(un)|Ωm, un|Ωmand tr(un)are Cauchy sequences in measure ,\nun→ua.e. inΩ,andun→wa.e. on∂Ω,\nTk(un)⇀ Tk(u)inW1,p(Ω;ϑα),\nτ(Tk(u)) =Tk(w),\nA(∇Tk(un))⇀ Vkin(Lp′(Ω;ϑα))N,\nfor allk >0. Also, since W1,p(Ω;ϑα)֒→Lp(∂Ω;ϑβ), we can assume that\nTk(un)→Tk(u)inLp(∂Ω;ϑβ).\nFollowing the arguments developed in steps 3-5 of the proof o f Theorem 1.1, we can conclude\n(1.7). Furthermore, by repeating the same arguments, we can demonstrate that uis a renormalized\nsolution to (1.1). /square\nAPPENDIX A.\nIn this appendix, we prove the existence of weak solutions to the problem (3.6). We study the\nexistence of solutions for the problem\n/braceleftBigg\n−div(ϑαA(∇u))+ϑγHn(x,∇u)+ϑαGn(x,u) =fninΩn,\nϑαA(∇u)·ν+ϑβK(u) =gnon∂Ωn∩∂Ω,(A.1)\nwhereΩn= Ω∩{|x|< n}andn∈N. Recall that fn= 0onΩ\\Ωn, andgn= 0on∂Ω\\∂Ωn.\nDefine\nW|D={u∈W1,p(Ωn;ϑα)|u|∂Ωn\\∂Ω= 0}.26 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nLetw∈W|D. According to the Minty-Browder theorem, there exists a uni queu∈W|Dsuch\nthatˆ\nΩnA(∇u)∇vdϑα+ˆ\nΩnHn(x,∇w)vdϑγ+ˆ\nΩnGn(x,w)vdϑγ\n+ˆ\n∂Ωn∩∂ΩK(w)vdϑβ=ˆ\nΩnvfndϑα+ˆ\n∂Ωn∩∂Ωvgndϑβ,(A.2)\nfor allv∈W|D.\nIt follows that we can consider the functional T:W|D→W|Ddefined by\nT(w) =u,∀w∈W|D,\nwhereuis the unique element of W|Dverifying (A.2). We now prove that Tis a continuous and\ncompact operator.\nStep 1.Tis continuous.\nLetwm∈W|Dsuch that wm→winW|D. Up to a subsequence (still denoted by wn), we have\nwm→w inLp(∂Ωn∩∂Ω;ϑβ), (A.3)\nwm→w a.e. inΩn, (A.4)\nwm→w a.e. in∂Ωn∩∂Ω, (A.5)\n∇wm→ ∇w a.e. in(Ωn)N. (A.6)\nWriteum=T(wm). Withwmin place of wand choosing umas a test function in (A.2), from\n(H2), (3.4), and (3.5), we obtain that\nσˆ\nΩn|∇um|pdϑα≤ˆ\nΩnn|um|dϑγ+ˆ\nΩn|umfn|dϑα+ˆ\n∂Ωn∩∂Ω|umgn|dϑβ.\nFrom the Poincar´ e inequality and Young’s inequality, we ge t\nˆ\nΩn|∇um|pdϑα+ˆ\nΩn|um|pdϑα≤C (A.7)\nwhereC >0is a constant independent of m.\nConsequently, there exists a subsequence (still denoted by um), a measurable function u, and a\nfieldVsuch that\num⇀ u inW1,p(Ωn;ϑα), (A.8)\num→uinLp(Ωn;ϑα), (A.9)\num→uinLp(∂Ωn∩∂Ω;ϑβ), (A.10)\num→ua.e. inΩn, (A.11)\num→ua.e. in∂Ωn∩∂Ω, (A.12)\nA(∇um)⇀ V in(Lp′(Ωn;ϑα))N. (A.13)\nTo prove the continuity of T, it remains to show that u=T(w), i.e.,usatisfies (A.2). Using (A.2)\nwithwmin place of wand the test function um−u, we have\nˆ\nΩnA(∇um)[∇um−∇u]dϑα+ˆ\nΩnHn(x,∇wm)(um−u)dϑγ\n+ˆ\nΩnGn(x,wm)(um−u)dϑγ+ˆ\n∂Ωn∩∂ΩK(wm)(um−u)dϑβ\n=ˆ\nΩn(um−u)fndϑα+ˆ\n∂Ωn∩∂Ω(um−u)gndϑβ.(A.14)EXISTENCE OF RENORMALIZED SOLUTIONS 27\nThen, from (3.4), (3.5), (A.5), and (A.9) - (A.12),\nlim\nm→∞ˆ\nΩnA(∇um)[∇um−∇u]dϑα= 0. (A.15)\nNow, we use the Minty arguments to identify VwithA(∇u). Letφ∈(L∞(Ωn))N. By (A.13)\nand (A.15), it follows that for any t∈R\nlim\nm→∞ˆ\nΩn[A(∇um)−A(∇u+tφ)][∇um−∇u−tφ]dϑα\n=−ˆ\nΩn[V−A(∇u+tφ)]tφdϑα.\nUsing the monotone character A, we obtain that for any t/\\e}atio\\slash= 0,\n−sign(t)ˆ\nΩn[V−A(∇u+tφ)]φdϑα≥0.\nSinceA(∇u+tφ)converges strongly to A(∇u)in(Lp′(Ωn;ϑα))Nastgoes to zero, letting t→0\nin the above inequality leads toˆ\nΩn[V−A(∇u)]φdϑα= 0,\nfor anyφ∈(L∞(Ωn))N. We conclude that\nV=A(∇u). (A.16)\nBy using (A.4) - (A.6), (A.13), and (A.16), we can pass to the l imit asn→ ∞ in (A.2) with wn\nin place of w, and we getˆ\nΩnA(∇u)∇vdϑα+ˆ\nΩnHn(x,∇w)vdϑγ+ˆ\nΩnGn(x,w)vdϑγ\n+ˆ\n∂Ωn∩∂ΩK(w)vdϑβ=ˆ\nΩnvfndϑα+ˆ\n∂Ωn∩∂Ωvgndϑβ,\nfor allv∈W|D. It follows that Tis continuous.\nStep 2.Tis compact.\nLet(wm)⊂W|Dbe a bounded sequence. Up to a subsequence (still denoted by wn), we have\nwm→w inLp(Ωn;ϑα), (A.17)\nwm→w inLp(∂Ωn∩∂Ω;ϑβ), (A.18)\nwm→w a.e. inΩn, (A.19)\nwm→w a.e. in∂Ωn∩∂Ω, (A.20)\n∇wm⇀∇w a.e. in(Ωn)N. (A.21)\nWriteum=T(wm). We haveˆ\nΩnA(∇um)∇vdϑα+ˆ\nΩnHn(x,∇wm)vdϑγ+ˆ\nΩnGn(x,wm)vdϑγ\n+ˆ\n∂Ωn∩∂ΩK(wm)vdϑβ=ˆ\nΩnvfndϑα+ˆ\n∂Ωn∩∂Ωvgndϑβ,(A.22)\nfor allv∈W|D.\nSubstituting v=uminto (A.22), similarly as in the firs step, we obtainˆ\nΩn|∇um|pdϑα+ˆ\nΩn|um|pdϑα≤C, (A.23)28 JUAN A. APAZA AND MANASS ´ES DE SOUZA\nwhereC >0is a constant independent of m.\nHence, there exists a subsequence (still denoted by um), a measurable function u, and a field V\nsuch that\num⇀ u inW1,p(Ωn;ϑα), (A.24)\num→uinLp(Ωn;ϑα), (A.25)\num→uinLp(∂Ωn∩∂Ω;ϑβ), (A.26)\num→ua.e. inΩn, (A.27)\num→ua.e. in∂Ωn∩∂Ω. (A.28)\nNext, we prove\n∇um→ ∇uin(Lp(Ωn;ϑα))N. (A.29)\nTo establish (A.29), first, we prove\n∇umis a Cauchy sequence in measure . (A.30)\nLetε >0andε1>0be fixed. We set, for some L >0andδ >0:\nE1={|∇um1|> L}∪{|∇um2|> L},\nE2={ |∇um1| ≤L,|∇um2| ≤L,|∇um1−∇um2|> ε}.\nObserve that\n{|∇um1−∇um2|> ε} ⊂E1∪E2.\nSince(um)and(|∇um|)are bounded in Lp(Ωn;ϑα), one has\nϑα(E1)< ε1,\nforLlarge enough, independently of m1andm2.\nNow, consider ϑα(E2). Define\nK0={(ξ,η)∈R2N||ξ| ≤L,|η| ≤L,|ξ−η| ≥ε},\nthen\ninf{[A(ξ)−A(η)][ξ−η]|(ξ,η)∈K0}=C >0,\nsinceK0is compact.\nHence,\nCϑα(E2)≤ˆ\nE2[A(∇um1)−A(∇um2)][∇um1−∇um2]dϑα. (A.31)\nSubstituting um1−um2into (A.22), for a suitable constant, we have there is N0>0such that\nϑα(E2)< ε1,ifm1,m2> N0.\nTherefore, we conclude (A.30).\nFinally, substituting v=uminto (A.22), we obtain that (|∇um|p)is uniformly integrable. Then,\nby Vitali’s Convergence Theorem, which proves (A.29). It fo llows that Tis compact.\nTo finalize this appendix, proceeding similarly as in (A.7), we have\nˆ\nΩn|T(w)|pdϑα≤C,∀w∈W|D,\nwhereCis a constant depending on Ωn,n,α,f, andg. Then, using Schauder’s fixed-point theorem\nensures the existence of at least one fixed point T(w) =w, proving the existence of a weak solution\nto the problem (A.1).\nAuthor contributions: All authors have contributed equally to this work for writin g, review and\nediting. All authors have read and agreed to the published ve rsion of the manuscript.EXISTENCE OF RENORMALIZED SOLUTIONS 29\nFunding: This work were supported by National Institute of Science an d Technology of Mathemat-\nics INCT-Mat, CNPq, Grants 170245/2023-3, 160951/2022-4, 308395/2023-9, and by Para´ ıba State\nResearch Foundation (FAPESQ), Grant 3034/2021.\nData Availibility: No data was used for the research described in the article.\nDeclarations\nConflict of interest: The authors declare no conflict of interest.\nDeclaration of generative AI and AI-assisted technologies in the writing process: During the\npreparation of this work, the authors used the AI ChatGPT and the AI Bard to correct grammar and\northography in the text. After using this tool/service, the authors reviewed and edited the content as\nneeded and takes full responsibility for the content of the p ublication.\nREFERENCES\n[1] A. Alvino, A. Cianchi, V .G. Maz’ya, and A. 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A PAZA , DEPARTAMENTO DE MATEM ´ATICA , UNIVERSIDADE FEDERAL DE S˜AOCARLOS , 13565-905,\nS˜AOCARLOS –SP, B RAZIL\nEmail address :juanpabloalconapaza@gmail.com\nMANASS ´ES DE SOUZA , DEPARTAMENTO DE MATEM ´ATICA , UNIVERSIDADE FEDERAL DA PARA´IBA, 58051-900,\nJO˜AOPESSOA –PB, B RAZIL\nEmail address :manasses.xavier@academico.ufpb.br"    },    {        "title": "2401.12434v3.Efficient_near_optimal_decoding_of_the_surface_code_through_ensembling.pdf",        "content": "Efficient near-optimal decoding of the surface code through ensembling\nNoah Shutty,1Michael Newman,2and Benjamin Villalonga1\n1Google Quantum AI, Venice, California 90291, USA\n2Google Quantum AI, Mountain View, California 94043, USA\nWe introduce harmonization , an ensembling method that combines several “noisy” decoders to\ngenerate highly accurate decoding predictions. Harmonized ensembles of MWPM-based decoders\nachieve lower logical error rates than their individual counterparts on repetition and surface code\nbenchmarks, approaching maximum-likelihood accuracy at large ensemble sizes. We can use the\ndegree of consensus among the ensemble as a confidence measure for a layered decoding scheme,\nin which a small ensemble flags high-risk cases to be checked by a larger, more accurate ensemble.\nThis layered scheme can realize the accuracy improvements of large ensembles with a relatively small\nconstant factor of computational overhead. We conclude that harmonization provides a viable path\ntowards highly accurate real-time decoding.\nINTRODUCTION\nA fast and accurate decoder is integral to the con-\nstruction of a fault-tolerant quantum computer. It is\nresponsible for continuously processing syndrome infor-\nmation from the quantum device to give corrected logical\noutcomes. There are several efficient (i.e. polynomial-\ntime) decoding algorithms [diFO+23]. Among the most\npromising are MWPM-based [HG23, HBK+23, Fow13]\nand union-find methods [DLB22, HNB20] due to their\nspeed. On the other hand, there are more accurate but\nless-efficient decoders, such as maximum-likelihood (opti-\nmal) decoding with tensor networks [BSV14, CF21, AI23,\nPCR23] or most-likely-error decoding with integer pro-\ngrams [FWK05, LV18, FGL21, LAR11], whose runtime\ncan scale exponentially with the decoding problem size.\nIdeally, we would like a decoder that is both highly\nefficient and nearly optimal in accuracy. Towards this\nend we introduce Harmony , an assembly of correlated\nminimum-weight perfect matching decoders, each in-\nstantiated with a perturbed prior. Individually, these\ndecoders perform worse than a single matching de-\ncoder instantiated with a perfect prior. Together, they\nform ensembled decisions that can approach maximum-\nlikelihood inference. We note that ensembling has also\nbeen used to boost the performance of decoding with\nmachine learning [SJG20, BSH+23].\nThis technique is embarrassingly parallelizable, and in\nthe regimes we test, converges with reasonable ensemble\nsizes. Querying the full set of decoders can lead to a\nlarge constant overhead, and so we introduce a layered\ndecoding scheme to mitigate this cost. In the layered\nscheme, a small group of watchful sentinels flag high-risk\ncases to be considered by the wider assembly. The result\nis a highly accurate decoder with moments of expensive\ncompute, but overall relatively low amortized cost.\nFirst, we describe the correlated matching decoding\nalgorithm that we use to build our ensembles. Next,\nwe describe Harmony, our ensembled correlated matching\ndecoder. We present our numerical results for the sur-face and repetition codes, comparing logical error rates of\nHarmony against a converged tensor network maximum-\nlikelihood (TNML) decoder [Boh22]. For the repetition\ncode, we optimize for 2D tensor networks using the sweep\nline method [Chu21]. For the surface code, we use a\nTNML decoder developed for the experiments of [AI23],\nand described in the Appendix. Finally, we analyze the\naccuracy and cost of a layered decoding scheme, achiev-\ning the accuracy of a large ensemble with the amortized\ncompute cost of a small ensemble.\nCORRELATED MATCHING\nIn principle, any decoder could be used to build an\nensemble, as long as there is a way of perturbing the error\nmodel prior. In this work, we build our ensembles out of\na correlated matching decoder that accepts as its prior\nan error hypergraph (or equivalently, a Tanner graph).\nThis decoder is similar to those described in [HBK+23,\nFow13].\nRecall that an error hypergraph is a hypergraph for\nwhich each hyperedge is an error mechanism, each detec-\ntor [Gid21] is a vertex, and a hyperedge connects those\ndetectors that the corresponding error activates. Each\nhyperedge is labeled with the probability of the corre-\nsponding error mechanism and the set of observables\nit flips. A minimum-weight perfect matching decoder\ncan provide the most likely error hypothesis for an error\ngraph , on which each error mechanism activates at most\ntwo detectors.\nIn the surface code, the most common example of er-\nror mechanisms producing hyperedges with more than\ntwo detectors are Y-type errors. We can decode the sur-\nface code by regarding Y-type errors as independent X-\ntype and Z-type errors, where each error type forms a\ndistinct, independent error graph. However, this approx-\nimation neglects useful information that can assist de-\ncoding [DKLP02]. A correlated matching decoder tries\nto reincorporate this information by reweighting the X-\nandZ-type error graphs as a function of the candidatearXiv:2401.12434v3  [quant-ph]  15 Mar 20242\nFIG. 1. A toy correlated matching example. The error\nhypergraph is on the left, with the erring hyperedges and de-\ntection events shaded red, and with all edges having equal\nweight. First, we perform a minimum-weight perfect match-\ning (blue edges) that misidentifies the error, in this case guess-\ning the wrong of two equally valid choices. Second, we use\nthe matching from the first step to infer that the hyperedge\nerror mechanism has likely activated, and shorten the com-\nplementary edge. Third, we perform minimum-weight perfect\nmatching on this reweighted graph, correctly identifying the\nerror.\nmatched edges in the opposite graph.\nThe correlated decoder proceeds in three steps: an\ninitial minimum-weight perfect matching, a reweight-\ning of edges based on that initial matching, and a fi-\nnal minimum-weight perfect matching. In the initial\nminimum-weight perfect matching, we distribute the\nprobabilities of edge-like error mechanisms to the cor-\nresponding X- and Z-type error graphs, ignoring the hy-\nperedges. We perform minimum-weight perfect match-\ning on these disjoint error graphs (with weights set\nto−ln(p)) to obtain a set of matched edges. In the\nreweighting step, we assert that the matched edges have\nerred with certainty, and reweight other edges condi-\ntioned on this assertion. This requires that we up-\ndate the probabilities of those edges that together with\nthe matched edge form hyperedges. We renormalize\nthe probabilities of those hyperedges conditioned on the\nmatched edge having erred and reassign these updated\nprobabilities to the complementary edge in each hyper-\nedge. Over all such assignments (including the initial\nprobability of the complementary edge) we select the one\nwith the highest probability. Finally, we run minimum-\nweight perfect matching on this reweighted graph to\ndetermine the correction. In practice, we use Stim’s\nDetectorErrorModel to configure the decoder [Gid21].\nHARMONIZATION\nConventionally, a correlated matching decoder is a de-\nterministic algorithm that is configured with a fixed er-\nror model. To harmonize a correlated matching ensem-\nble, we have to solve two problems: ensemble generation\nand pooling. Ensemble generation consists of perturbing\nthe error model prior given to each decoder so that thepredictions of individual decoders begin to occasionally\ndisagree. Pooling consists of combining together all the\ndecoder outputs into one overall decoding prediction.\nEnsemble Generation\nAs described, correlated matching accepts an error hy-\npergraph prior and transforms it to a graph where each\nedge eis labeled with a probability peand a set of implied\nprobabilities for some other edges S(e) ={(e′, qe′|e) :\ne′has probability qe′|econditioned on the use of e}. In\nthe ensembles described here, we directly perturb\nboth the probabilities and implied probabilities of\neach edge. We produce a randomized graph\nwhere the edges are instead labeled with two per-\nturbed probabilities p(1)\ne, p(2)\ne, where superscripts re-\nfer to the first and second passes of matching,\nand a perturbed reweight set ˜S(e) = {(e′,˜qe′|e) :\ne′has probability ˜ qe′|econditioned on the use of e}.\nWe modify correlated matching as follows. In the first\npass, we use the edge probabilities p(1)\ne. Conditioned on\nthe edges chosen in the first pass matching, we reweight\nsome of the edges in the graph to have new probabilities\nas prescribed by the ˜S(e). As before, we choose the maxi-\nmum probability value for an edge that is reweighted mul-\ntiple times. We set the weights of any un-reweighted edge\netop(2)\ne. Finally, we compute a second pass matching,\nwhich is the final decoding of the randomized ensemble\nmember.\nNow we must describe the probability distribution\nfrom which we sample p(1)\ne, p(2)\ne,and ˜S(e). We fix three\nconstants α1, α2, α3∈[0,1]. We sample the probabilities\nfrom the following intervals uniformly at random:\np(1)\ne∼[(1−α1)·pe,(1 +α1)·pe] (1)\np(2)\ne∼[(1−α2)·pe,(1 +α2)·pe] (2)\n˜qe′|e∼\u0002\n(1−α3)·qe′|e,(1 +α3)·qe′|e\u0003\n. (3)\nNote that the set of edges reweighted by any given edge\ndoes not change, only the reweighted probability values.\nThe values of the constants αiwe use are:\nα1= 1, α 2= 4/5, α 3= 1/2.\nNote that the above values were tuned for correlated\nmatching on the surface code. Other codes and decoders\nmay perform better with different perturbations.\nPooling Methods\nWe consider three different pooling methods. Vote\npooling is perhaps the simplest and most directly gen-\neralizable pooling method. In this method we choose the3\n0 20 40 60 80 100\nEnsemble size1. 75 × 10−21.8 × 10−21.85 × 10−21.9 × 10−21.95 × 10−22 × 10−22.05 × 10−22.1 × 10−2Logical error per rounduncorrela ed ma ching\nvo e pooled\nsum -likelihood pooled\nmost-likely-error pooled\n0 20 40 60 80 100\nEnsemble size5 × 10−36 × 10−3Logical error per roundcorrelated matching\n ote pooled\nsum -lik elihood pooled\nmost-lik ely-error pooled\nFIG. 2. Different pooling methods at distance-7 for circuit-\nlevel noise in the repetition code at p= 0.05 (left) and the\nsurface code at p= 0.008 (right). In the repetition code,\nminimum-weight perfect matching is a most-likely error de-\ncoder, which most-likely-error pooling quickly converges to.\nWe observe that vote pooling, which takes into account de-\ngeneracy, is the most accurate pooling method. For the sur-\nface code, most-likely-error pooling appears to be at least as\neffective as the other pooling methods once the ensemble is\nsufficiently large. This suggests that degeneracy plays a com-\nparatively small role in increasing accuracy within the surface\ncode.\ndecoding prediction given by the largest number of de-\ncoders.\nRather than giving each decoder an equal say in the\npooling process, we may increase the influence of de-\ncoders that found a configuration of errors of relatively\nhigher likelihood. To make use of likelihood in pooling,\neach decoder must return the set of errors it used, rather\nthan a set of edges.\nHow can we recover the errors predicted during de-\ncoding (including those that induce hyperedges), given\nthe edges used by a matching decoder? We use an edge\ndecomposition graph as described in Figure 3. Vertices\nin this graph correspond to edge-like error mechanisms.\nEach edge connects two edge-like error mechanisms that\ncombine to a hyperedge in the error hypergraph, while\nan edge-to-boundary at each vertex provides a way to\nmatch to nothing (representing an edge-like error). In\nthe conventional circuit noise model studied here, all in-\ndependent error mechanisms cause at most one edge-like\nerror in each of the X- and Z-type error graphs. We can\ntherefore apply Blossom to recover the most likely set of\nerrors consistent with the chosen edges.\nWe consider two ways of pooling based on likelihood:\nsum-likelihood pooling andmost-likely-error pooling . In\nsum-likelihood pooling, we sum the likelihoods of all de-\ncoders within each of the two possible logical flip predic-\ntions, and compare these sums to select the prediction\nwith the largest value. In most-likely-error pooling, we\nsimply choose the decoder from the entire ensemble that\nreports the highest likelihood outcome and accept its pre-\ndiction.\nGenerally, we observe that vote pooling gives the high-\nFIG. 3. On the left, an error hypergraph with a set of detec-\ntion events shaded red and the edges chosen by a minimum-\nweight perfect matching decoder overlaid in blue. On the\nright, the corresponding edge decomposition graph, with\nnodes corresponding to chosen edges colored blue. MWPM-\nbased decoding on the edge decomposition graph gives the\nmost likely set of errors, overlaid in green, consistent with\nthe edge set chosen by the matching decoder. In this case, it\nselects the far left edge-like error mechanism, as well as the\nonly hyperedge-like error mechanism.\nest accuracy for the repetition code, while sum-likelihood\nand most-likely-error pooling give the highest accuracy\nfor the surface code - see Figure 2.\nRESULTS\nTo assess the accuracy of ensembled matching, we\nmake three comparisons. We use Stim’s standard circuit\ngeneration tools to create decoding instances [Gid21] -\nsee the Appendix.\nThe first comparison we make is against maximum-\nlikelihood decoding of repetition codes with circuit-level\nnoise. Famously, two-dimensional error graphs can be\ndecoded optimally and efficiently using standard matrix\nproduct state techniques [BSV14]. For circuit-level noise\nin the repetition code, we use the sweep line contractor\ndeveloped in [Chu21].\nThe repetition code is an interesting comparison be-\ncause its decoding problem does not involve any hy-\nperedges. For this reason, minimum-weight perfect\nmatching (when using ln(1−p\np) edge weights) is an ex-\nact minimum-weight error decoder - only degeneracy\nwithin the error graph separates it from behaving as a\nmaximum-likelihood decoder. This allows us to isolate\nthe ability of ensembling to account for degeneracy in de-\ncoding. In principle, this effect also appears in the surface\ncode, where the error graph of the repetition code em-\nbeds as a subgraph. Note that most-likely-error pooling\nperforms similar to minimum-weight perfect matching,\nas they both aim to return an error class with the lowest\nweight error. Consequently, we elect to use vote pool-\ning, which can account for degeneracy and in practice\nperforms best - see Figure 2. The results of this compar-\nison are shown in Figure 4, where we see that ensembling\nachieves near maximum-likelihood performance.\nThe second comparison we make is against a phe-\nnomenological noise model in the surface code. We ap-\nply depolarizing noise to the data qubits (rather than the\nconventional bit-flip noise) to include hyperedges - see the4\n0 20 40 60 80 100\nEnsemble size1. 2 × 10−21.4 × 10−21.6 × 10−21.8 × 10−22 × 10−22. 2 × 10−2Logical error per rounddistance -7\ndistance-9\ndistance-11\nFIG. 4. Decoding circuit-level noise in the repetition code at\np= 0.05. At each distance there are three lines: minimum-\nweight matching (dotted), vote pooled ensembled matching\n(circles), and TNML decoding (dashed). We observe there\nis a relatively small difference between the most-likely error\ndecoder and the TNML decoder, and this difference is mostly\ntraversed at larger ensemble sizes. This is explained by the\nensembled decoder accounting for degeneracy.\nAppendix for more details. Because the error hypergraph\nof this problem is three-dimensional, TNML decoding is\nquite expensive [PCR23, AI23]. To decode, we embed\nthe decoding problem as a planar matrix product state\ncontraction - also see the Appendix [AI23]. We chose\nphenomenological noise to reduce the size of these ten-\nsor networks so that contraction is not too computation-\nally intensive at higher distances. Interestingly, we note\nthat most-likely-error and sum-likelihood pooling meth-\nods achieve similar performance - see Figure 2. As most-\nlikely-error pooling is aimed at finding the most likely\nerror hypothesis, this suggests that in the surface code,\nmost of the benefit of harmonization stems from better\nutilizing correlations rather than accounting for degen-\neracy. We observe in Figure 5 that ensembled match-\ning nearly bridges the accuracy gap between correlated\nmatching and TNML decoding in this simplified error\nmodel.\nFinally, having established that ensembled matching\ncan achieve near maximum-likelihood performance, we\nstudy Harmony using full circuit-level noise in the surface\ncode - see Figure 6 (and also the Appendix for additional\ndata scanning over different error rates). We find that a\nmanageable ensemble size of 100 or so is more than suffi-\ncient to saturate most of the benefit achieved through en-\n0 20 40 60 80 100\nEnsemble si e10−2\n3 × 10−34 × 10−36 × 10−3Logical error per rounddistance -5\ndistance-7\ndistance-9FIG. 5. Decoding phenomenological noise in the surface code\natp= 0.04. At each distance there are three lines: corre-\nlated matching (dotted), most-likely-error pooled ensembled\nmatching (circles), and TNML (dashed). We observe that\nmaximum-likelihood pooled MWPM is nearly as accurate as\nTNML once the ensemble size is large. As the distance in-\ncreases, the gap between maximum-likelihood pooled MWPM\nand TNML increases slightly.\nsembling, while an ensemble size of only 3 is sufficient to\nsurpass correlated matching. Of course, while ensembling\nis embarrassingly parallelizable, running 100 decoders in\nparallel would significantly increase the hardware cost of\ndecoding. However, a layered decoding scheme can reap\nmost of the benefit while minimizing the overall cost.\nCONFIDENCE RATES AND LAYERED\nDECODING\nEmpirically, we observe that a low degree of consensus\namong decoders in the ensemble can suggest a particular\ndecoding task is “high-risk”. The fraction of decoders\nthat vote with the majority can therefore be used as a\nconfidence score for the decoding. In a try-until-success\nprotocol, e.g. in certain magic state preparation rou-\ntines, we could simply restart whenever the confidence\nof decoding is unacceptably low. For general computa-\ntion, we cannot afford to start from scratch every time\nthe ensemble has low confidence. We therefore consider\nan alternative use for the confidence scores: layered de-\ncoding. In a layered decoding scheme, a lighter-weight\nfirst-pass decoder layer handles all decoding for a major-\nity of the shots, only passing “difficult” shots where it is\nnot sufficiently confident in its prediction to a slower but5\n0 20 40 60 80 100\nEnsemble size10−310−2Logical error per rounddistance -7\ndistance-9\ndistance-11\nFIG. 6. Decoding circuit-level noise in the surface code at\np= 0.004. At each distance there are three lines: uncorrelated\nmatching (dotted), correlated matching (dashed), and most-\nlikely-error pooled ensembled matching (circles). The relative\nimprovement grows with code distance.\nmore accurate second-pass decoding layer.\nWe consider a simple two-pass scheme in which an en-\nsemble of N1decoders is used in the first pass, and it\nactivates the second pass only when the confidence is\nless than 100% (i.e., if there is any dissent among the\ndecoders). The improvement factor in logical error rate\nand the overhead are shown in Fig. 7. The results show\nthat we saturate almost all of the improvement in logical\nerror rate with N1≈4. For lower error rates, the amor-\ntized overhead cost per shot is very close to N1, since\nthe first pass ensemble almost always reaches a unani-\nmous decision. We note that one could also use the com-\nplementary gap [GNBJ23] to flag high-risk cases in the\nfirst-pass. Plausibly, this could bring the amortized cost\nclose to N1≈1, but we leave comparing these methods\nto future work.\nCONCLUSION\nWe introduced harmonization as a simple and effec-\ntive form of ensembling for decoders. We showed that\nharmonization gives near-optimal logical error rates for\nthe repetition code and the surface code by benchmark-\ning against TNML decoding. The main improvement of\nensembling comes from degeneracy for the case of the\nrepetition code, and from better use of correlations for\nthe case of the surface code. Moreover, the degree of con-\n0 20 40 60 80 100\nAmortized overhead cost factor1.01.11.21.31.41.51.6Logical error per round improvement factordistance-7 unlayered\ndistance-9 unlayered\ndistance-11 unlayereddistance-7 layered\ndistance-9 layered\ndistance-11 layeredFIG. 7. The improvement in logical error per round versus the\noverhead cost factor for a one-pass unlayered decoding scheme\n(dotted line) and a two-pass layered decoding scheme (solid\nline) for circuit-level noise in the surface code at p= 0.004.\nThe first pass is a Harmony ensemble of size N1(varying)\nand the second pass decoding is given by a most-likely-error\npooled Harmony ensemble of size N2= 100. The second pass\ndecoding is executed if and only if the first pass ensemble\nfails to yield a consensus prediction, which happens with some\nprobability called the “trigger rate”. The horizontal axis is\nthe expected number of decoding instances per shot, which\nincreases with N1because both the first pass ensemble size is\nlarger and because the trigger rate increases due to a higher\nprobability of a dissenting member in the larger first pass\nensemble. Layered decoding achieves the accuracy of a much\nlarger ensemble without significant amortized overhead.\nsensus among the ensemble gives a confidence score that\ncan be exploited to significantly reduce the amortized\noverhead of ensembling in a layered decoding scheme.\nThis suggests that harmonization is a promising path\nto efficiently reduce the logical error rate of real-time\ndecoders. While we have focused on matching in sur-\nface codes, many exciting avenues remain, and we expect\nthese techniques could greatly improve other decoders\nand codes.\nACKNOWLEDGEMENTS\nWe thank Sergio Boixo, Dripto Debroy, Austin Fowler,\nCraig Gidney, Cody Jones, and Adam Zalcman for help-\nful conversations and technical insights.\n[AI23] Google Quantum AI. Suppressing quantum er-\nrors by scaling a surface code logical qubit. Nature ,\n614(7949):676–681, 2023.\n[Boh22] Thomas C Bohdanowicz. Quantum Constructions on\nHamiltonians, Codes, and Circuits . PhD thesis, Califor-6\nnia Institute of Technology, 2022.\n[BSH+23] Johannes Bausch, Andrew W Senior, Francisco JH\nHeras, Thomas Edlich, Alex Davies, Michael Newman,\nCody Jones, Kevin Satzinger, Murphy Yuezhen Niu, Sam\nBlackwell, et al. Learning to decode the surface code with\na recurrent, transformer-based neural network. arXiv\npreprint arXiv:2310.05900 , 2023.\n[BSV14] Sergey Bravyi, Martin Suchara, and Alexander\nVargo. Efficient algorithms for maximum likelihood\ndecoding in the surface code. Physical Review A ,\n90(3):032326, 2014.\n[CF21] Christopher T Chubb and Steven T Flammia. Sta-\ntistical mechanical models for quantum codes with cor-\nrelated noise. Annales de l’Institut Henri Poincar´ e D ,\n8(2):269–321, 2021.\n[Chu21] Christopher T Chubb. General tensor net-\nwork decoding of 2d pauli codes. arXiv preprint\narXiv:2101.04125 , 2021.\n[diFO+23] Antonio deMarti iOlius, Patricio Fuentes, Rom´ an\nOr´ us, Pedro M Crespo, and Josu Etxezarreta Martinez.\nDecoding algorithms for surface codes. arXiv e-prints ,\npages arXiv–2307, 2023.\n[DKLP02] Eric Dennis, Alexei Kitaev, Andrew Landahl, and\nJohn Preskill. Topological quantum memory. Journal of\nMathematical Physics , 43(9):4452–4505, 2002.\n[DLB22] Nicolas Delfosse, Vivien Londe, and Michael E\nBeverland. Toward a union-find decoder for quantum\nldpc codes. IEEE Transactions on Information Theory ,\n68(5):3187–3199, 2022.\n[FGL21] Omar Fawzi, Lucien Grou` es, and Anthony Lever-\nrier. Linear programming decoder for hypergraph prod-\nuct quantum codes. In IEEE ITW 2020 - IEEE Infor-\nmation theory workshop 2020 , Riva del Garda / Virtual,\nItaly, April 2021.\n[Fow13] Austin G Fowler. Optimal complexity correction of\ncorrelated errors in the surface code. arXiv preprint\narXiv:1310.0863 , 2013.\n[FWK05] Jon Feldman, Martin J Wainwright, and David R\nKarger. Using linear programming to decode binary lin-\near codes. IEEE Transactions on Information Theory ,\n51(3):954–972, 2005.\n[Gid21] Craig Gidney. Stim: a fast stabilizer circuit simula-\ntor.Quantum , 5:497, 2021.\n[GNBJ23] Craig Gidney, Michael Newman, Peter Brooks,\nand Cody Jones. Yoked surface codes. arXiv preprint\narXiv:2312.04522 , 2023.\n[HBK+23] Oscar Higgott, Thomas C Bohdanowicz, Alek-\nsander Kubica, Steven T Flammia, and Earl T Camp-\nbell. Improved decoding of circuit noise and fragile\nboundaries of tailored surface codes. Physical Review X ,\n13(3):031007, 2023.\n[HG23] Oscar Higgott and Craig Gidney. Sparse blos-\nsom: correcting a million errors per core sec-\nond with minimum-weight matching. arXiv preprint\narXiv:2303.15933 , 2023.\n[HNB20] Shilin Huang, Michael Newman, and Kenneth R\nBrown. Fault-tolerant weighted union-find decoding on\nthe toric code. Physical Review A , 102(1):012419, 2020.\n[LAR11] Andrew J Landahl, Jonas T Anderson, and\nPatrick R Rice. Fault-tolerant quantum computing with\ncolor codes. arXiv preprint arXiv:1108.5738 , 2011.\n[LV18] July X. Li and Pascal O. Vontobel. Lp decoding\nof quantum stabilizer codes. In 2018 IEEE Interna-\ntional Symposium on Information Theory (ISIT) , page1306–1310. IEEE Press, 2018.\n[PCR23] Christophe Piveteau, Christopher T Chubb, and\nJoseph M Renes. Tensor network decoding beyond 2d.\narXiv preprint arXiv:2310.10722 , 2023.\n[SJG20] Milap Sheth, Sara Zafar Jafarzadeh, and Vlad Gheo-\nrghiu. Neural ensemble decoding for topological quantum\nerror-correcting codes. Physical Review A , 101(3):032338,\n2020.\n[Vid03] Guifr´ e Vidal. Efficient classical simulation of slightly\nentangled quantum computations. Physical review let-\nters, 91(14):147902, 2003.\nGenerating Circuits\nFor reproducibility, we use Stim’s [Gid21] standard cir-\ncuit generation features to create noisy circuits. With\ncircuit-level noise, these are similar but slightly different\nto ‘standard’ circuit-level depolarizing models. They ag-\ngregate measure and reset error into a single error, do\nnot include idling error, and add a depolarizing channel\nbefore each cycle of error-correction. These circuits can\nbe reproduced using the following Stim commands. For\ncircuit-level noise in the repetition code:\nstim gen \\\n− −code r e p e t i t i o n c o d e \\\n− −task memory \\\n− −distance $d i s t a n c e v a r\\\n− −rounds $rounds var\\\n− −b e f o r e r o u n d d a t a d e p o l a r i z a t i o n $pvar\\\n− −b e f o r e m e a s u r e f l i p p r o b a b i l i t y $pvar\\\n− −a f t e r r e s e t f l i p p r o b a b i l i t y $pvar\\\n− −a f t e r c l i f f o r d d e p o l a r i z a t i o n $pvar\nFor phenomenological noise in the surface code:\nstim gen \\\n− −code s u r f a c e c o d e \\\n− −task rotated memory z\\\n− −distance $d i s t a n c e v a r\\\n− −rounds $rounds var\\\n− −b e f o r e r o u n d d a t a d e p o l a r i z a t i o n $pvar\\\n− −b e f o r e m e a s u r e f l i p p r o b a b i l i t y \\\n$( echo ”2 ∗ $pvar /3” |bc−l )\nNote the 2 p/3 probability for the measurement error.\nThis is chosen so that each individual X- orZ-type error\ngraph has uniform edge-weights when treating Y-type er-\nrors as independent X- orZ-type errors. This matches\nthe usual (uncorrelated) phenomenological decoding of a\ncubic error graph. Finally, for circuit-level noise in the\nsurface code:\nstim gen \\\n− −code s u r f a c e c o d e \\\n− −task rotated memory z\\\n− −distance $d i s t a n c e v a r\\\n− −rounds $rounds var\\\n− −b e f o r e r o u n d d a t a d e p o l a r i z a t i o n $pvar\\7\n− −b e f o r e m e a s u r e f l i p p r o b a b i l i t y $pvar\\\n− −a f t e r r e s e t f l i p p r o b a b i l i t y $pvar\\\n− −a f t e r c l i f f o r d d e p o l a r i z a t i o n $pvar\nNote that simulations for the repetition code and\ncircuit-level noise in the surface code use 2 drounds.\nFor phenomenological noise in the surface code, we use\ndrounds to keep the simulations more manageable for\nTNML decoding.\nMaximum-likelihood decoding with tensor networks\nWe benchmark the results shown in the main text\nagainst an (approximate) maximum-likelihood decoder.\nThere are several proposed implementations for a circuit-\nlevel tensor network decoder, including maximum-\nlikelihood decoding of the circuit history code [Boh22].\nHowever, we instead use the tensor network maximum-\nlikelihood (TNML) decoder developed for the experi-\nments of [AI23]. This decoder accepts as input any error\nmodel that can be expressed as an error hypergraph, as\nwell as a list of detection events, and outputs the likeli-\nhood that the local observable either changed parity or\nnot conditioned on the observed detection events. In this\nAppendix we describe our TNML decoder.\nThe error hypergraph\nAn error hypergraph can be represented as a bipartite\n(Tanner) graph where one subset of nodes ( A) represents\nthe error mechanisms contained in the model and the\nother subset of nodes ( B) represents the union of all de-\ntectors and logical operators. For each error mechanism,\nof independent probability pi, we have a binary variable\neithat denotes whether the error is present (1) or not\n(0). Similarly, for each detector we have a binary vari-\nabledjdenoting whether the detection event is present\nor not. Finally, during a memory experiment each logi-\ncal operator might change value (flip), a process that is\nalso represented by a binary variable lk. As an example,\nconsider the following error hypergraph, arising from a\ndistance-3 surface code over a single round and with asingle logical observable:\n0.9%0.5%5.2%0.4%0.8%1.5%0.5%2.7%0.5%0.7%0.6%2.9%0.3%2.0%0.2%2.9%1.6%1.4%4.7%1.9%1.4%0.6%4.1%piei\ndj\nlk\n(4)\nThe connectivity of the error hypergraph represents the\nrelation between an error configuration (bit string over\nA) and the detection event and logical operator config-\nuration (bit string over B): given an error configuration\noverA, the value of a variable on Bis given by the parity\nof the variables on Aincident on it. In other words, both\nthe detector configuration ⃗dand the logical observable\nconfiguration ⃗lare functions of the error configuration:\n⃗d=⃗f(⃗ e) and ⃗l=⃗ g(⃗ e). The following is an example error\nconfiguration over the error hypergraph of (4):\n0\n0\n1\n0\n0\n1\n0\n0\n0\n0\n0\n0\n0\n0\n0\n1\n0\n0\n0\n1\n0\n0\n00\n0\n0\n1\n1\n1\n0\n0\n1\n(5)\nThe probability of an error configuration ⃗ e≡ {ei}iis\nequal to:\nPr(⃗ e) =Y\nipei\ni·(1−pi)1−ei, (6)\ni.e. each error that is present contributes a factor piand\neach error that is not present contributes a factor (1 −pi).\nA tensor network for decoding\nThe goal of a decoder is to infer which value of the\nlogical observable(s) is most likely conditioned on a par-\nticular configuration of detection events (detector con-\nfiguration) observed in an experiment. We denote this\nlikelihood by L\u0010\n⃗l|⃗d\u0011\n. The most likely value of ⃗lis then8\nchosen as the most plausible logical observable configu-\nration. Formally, the likelihood is computed as:\nL\u0010\n⃗l|⃗d\u0011\n∝X\n⃗ e:[⃗f(⃗ e)=⃗d]∧[⃗ g(⃗ e)=⃗l]Pr(⃗ e) (7)\ni.e. as the sum of the probabilities of all error configura-\ntions compatible with the observed detection event con-\nfiguration and a specific logical observable configuration.\nIn the case of having a single logical observable l0, the\ninferred value is equal to 1 if L\u0010\nl0= 0|⃗d\u0011\n≥L\u0010\nl0= 1|⃗d\u0011\nand 0 otherwise.\nIt is possible to evaluate the sum in Eq. (7) by contract-\ning a tensor network. This tensor network has the same\ntopology as the error hypergraph. Each node correspond-\ning to an error mechanism eiwith associated probability\npican be regarded as a tensor of the form\n0\n1\n=\n\npi ifα0=α1=. . .= 1\n1−piifα0=α1=. . .= 0\n0 otherwise(8)\ni.e. a Kronecker delta (also known as a copy tensor)\nweighted with the values piand 1 −pi. Each node cor-\nresponding to a detector dican be regarded as a tensor\nof the form\n0\n1\n2\n=(\n1 if α0+α1+. . .even\n0 if α0+α1+. . .odd(9)\ni.e. a tensor that enforces even parity over the union\nof the errors incident on the detector and the detection\nevent variable itself, di. Each logical operator is also\nsubstituted by a tensor of the from of Eq. (9), enforcing\neven parity over the errors incident to it and the logical\noperator variable lk. For example, the error hypergraphin (4) results in the tensor network\nL\u0010\nl0|⃗d\u0011\n=\nd0\nd1\nd2\nl0 (10)\nGiven a detection event configuration ⃗d, one can project\nall indices djto their corresponding value. For each value\nofl0= 0,1, the contraction of the tensor network yields\na scalar equal to L\u0010\nl0|⃗d\u0011\n. Alternatively, after projecting\nalldj, one can leave the l0index open (not projected), in\nwhich case the contraction of the tensor network yields\na vector with two entries, L\u0010\nl0= 0|⃗d\u0011\nandL\u0010\nl0= 1|⃗d\u0011\n.\nWe can then compare both likelihoods to make a decod-\ning decision.\nUnderstanding why the contraction of the tensor of\nEq. (10) yields the claimed value is relatively straightfor-\nward. To start with, the contraction of a tensor network\nis equal to the sum over all configurations of its indices\n(all binary in our case) of the product of the tensor entries\ncorresponding to each index configuration. Our tensor\nnetwork has two types of tensors. The first one corre-\nsponds to each error mechanism and its tensor entries\nyield 0 unless all indices incident on it are equal, result-\ning in only two contributing configurations. If they are\nall equal to 1, then the error is considered present and\nthe tensor contributes a factor pi. If they are all equal to\n0, then the error is not present and the tensor contributes\na factor (1 −pi). The other type of tensors is parity ten-\nsors. For each of these, if the incident detection event is\npresent (1), then the remaining indices have to add up to\nodd parity, or else the tensor will contribute with a factor\nof 0. If the incident detection event is not present (0),\nthen the remaining indices have to have even parity, or\nelse the tensor will contribute with a 0. This imposes the\nconstraint that the errors incident to each detector must\nbe consistent with the presence or absence of a detection\nevent. This logic applies identically to the logical ob-\nservables. In turn, the contraction of the tensor network\nyields the sum of all products of factors pi(error present)\nand (1 −pi) (error not present) for all configurations of9\nerrors that are compatible with the observed detection\nevent configuration and logical observables, exactly as in\nEq. (7).\nIn practice, contracting the tensor network of Eq. (10)\ncan be challenging. The computational cost of a tensor\nnetwork contraction is dominated by a factor that is ex-\nponential in the tree width of its line graph, a property\nthat is related to the topology of the tensor network, or\nthe error hypergraph in our case. For a surface code,\nthe cost of the tensor network contraction scales expo-\nnentially in O(min( d2, c·dr)), where dis the distance\nof the surface code, ris the number of rounds, and cis\nsome positive scalar (which roughly converts “time” or\n“round” units into spatial units). This makes exact ten-\nsor network contraction decoding of distance ≥5 surface\ncodes impractical. We have to resort to an approximate\ncontraction of the tensor network which, when converged,\nis just as accurate.\nContracting the tensor network in practice: matrix\nproduct states\nGiven the intractability of contracting the tensor net-\nwork L\u0010\n⃗d|⃗l\u0011\n, we rearrange it to a form in which it can be\neasily contracted approximately. Our goal is to rewrite\nthe tensor network of Eq. (10) as a planar graph on a\nsquare grid, which can be contracted approximately as\na matrix product state (MPS) evolution with maximum\nbond dimension χ[Vid03].\nWe first note that a parity tensor like the one in Eq. (9)\ncan be expanded as an MPS of dimension 2:\n0\n1\n2\n=\n1\n2\n0\n (11)\nwhere each tensor on the right is itself a parity tensor as\ndefined in Eq. (9). With the equivalence of Eq. (11) in\nmind, we can rewrite the tensor network of Eq. (10) with\na trivial substitution of parity tensors. We can further\nmanipulate the resulting tensor network so that each ten-\nsor it is easily placed on a 2D grid. In our example, thatis:\nL\u0010\nl0|⃗d\u0011\n=\nd3d0d1d2\nl0 (12)\nwhere we use the modified parity tensors\n0\n1\nx\n=\n\n1 if ( α0+α1+. . .even)\n∧(αx=β)\n0 otherwise(13)\ni.e., a parity tensor over ⃗ αthat also acts as a copy ten-\nsor between an index αxandβ. This is useful so that\neach error mechanism index can “propagate” to the de-\ntector MPS applied further to its right. This way, each\nerror tensor “propagates” its only index indefinitely to\nthe right and a single row is needed per error mecha-\nnism. Note that the resulting 2D grid has as many rows\nas error mechanisms and as many colums as detectors. In\naddition, parity tensors are placed according to the ad-\njacency matrix of the bipartite Tanner graph, i.e. each\ncoordinate of this grid has a parity tensor if and only if\nthe error mechanism corresponding to that row and the\ndetector corresponding to that column have an edge in\ncommon in the error hypergraph.\nFinally, note the crossing indices in the tensor network\nof Eq. (12). In order to obtain a tensor network over a 2D\ngrid that is also a planar graph, we add trivial crossing\ntensors on each intersection of indices:\n0\n 1\n0\n1\n=(\n1 if ( α0=α1)∧(β0=β1)\n0 otherwise\n(14)\nwhich “propagate” indices horizontally andvertically. In10\nour example, this results in the final tensor network\nL\u0010\nl0|⃗d\u0011\n=\nd3d0d1d2\nl0 (15)\nThe planar tensor network of Eq. (15) can now be ap-\nproximately contracted as the evolution of an MPS with\nmaximum bond dimension χfrom left to right. The ini-\ntial MPS is a product state where each site iis in the\n(pi,1−pi) superposition. Each detector MPS (techni-\ncally a matrix product operator or MPO on the indices\nthat are propagated through the copied index) is applied\nto the evolving MPS. When a site ireaches the MPO or\ncolumn where error eireaches its last detector, then that\nsite does not propagate and disappears from the evolu-\ntion.\nApproximate contraction of the tensor network with\nmaximum bond dimension χχ\nAs explained above, the contraction of the tensor net-\nwork of Eq. (15) can be approximated by an MPS evo-\nlution of maximum bond dimension χ. Such contraction\nyields an approximate likelihood\nLχ\u0010\n⃗d|⃗l\u0011\n≈L\u0010\n⃗d|⃗l\u0011\n. (16)\nWe now analyze the convergence of the approximate con-\ntraction Lχdescribed in this section as a function of χ.\nIn particular, we look at the final logical error probabil-\nity (LEP) when decoding using approximate maximum-\nlikelihood decoding with bond dimension χ. Fig. 8 shows\nthe LEP as a function of χ.\nComputational cost of the TNML decoder\nIn this section we restrict our analysis to surface codes.\nThe resulting planar tensor network resembles a circuit\napplied to a 1D register of qubits, albeit describing a\n1\n101\n201\n301\n500\n1/\n103\n102\n101\nLEP\n5x5\np=0.02\np=0.04\np=0.06\n1\n101\n201\n301\n500\n1/\n7x7\n1\n101\n201\n301\n500\n1/\n9x9FIG. 8. Convergence of the logical error probability (LEP)\nusing the TNML decoder as a function of the bond dimension\nχ. We show the results for surface codes of distance d= 5,7,9\noverr=drounds. All results are converged up to statistical\nuncertainty over the 10,000 shots decoded.\nnon-unitary evolution. The width of the circuit, equal to\nthe number of error mechanisms in the error model, is\nproportional to the volume of the computation d2r. For\nan error model that is local in time and both spatial di-\nmensions, such as the ones considered in this paper and\ncommonly used in practice, and for a “good” ordering\nof error mechanisms and detectors, the effective circuit\ndepth for each site (error mechanism) is proportional to\nd2. This is because error mechanisms affect only a small\nnumber of rounds — typically at most two in conven-\ntional circuit-level noise models. See for example the\nfollowing distance-5 surface code over 5 rounds:\ndetectorserror mechanismsd2\nd2r\n(17)\nwhere for simplicity we have just plotted the adjacency\nmatrix between error mechanisms (rows) and detectors\n(columns). For a bond dimension χ, the cost of applying\neach MPO per site grows as χ3. Taking all of this into\naccount, the cost of decoding each shot is O(χ3d4r). Note\nthat we have not studied in depth the scaling of the bond\ndimension χnecessary to achieve a given approximation11\nfor particular values of d,r, and error probabilities ⃗ p. We\nleave this analysis for future work.\nWhy should we expect convergence at small χ?\nThe TNML decoding procedure explained in this sec-\ntion leaves an open question: why should we expect the\napproximate contraction of the tensor network to con-\nverge at low bond dimension χ? In the p→0 limit,\nthe MPS being evolved is at all points a product state:\nthe initial state has all error mechanisms inactive (0) and\nentanglement is never built up, which makes the compu-\ntation efficient. An MPS of bond dimension χ= 1 rep-\nresents this evolution exactly. At low enough values of p,\nthe evolution can be easily approximated as a low-order\nperturbation of the product state evolution. If we do not\ntransition to large values of p, we expect a small bond di-\nmension MPS to represent well the perturbed evolution.12\nAdditional data\n10−1\np10−510−410−310−210−1Logical error rate for r = 4 d  roundsd = 7\nd = 11\nd = 17\n10−1\n2 × 10−2\n3 × 10−2\n4 × 10−2\n6 × 10−2\np10−410−310−210−1Logical error rate for r = 4 d  roundsd = 7\nd = 11\nd = 17\n10−3\n10−2\np10−810−610−410−2100Logical error rate for r = 4 d  roundsd = 7\nd = 11\nd = 17\nFIG. 9. Thresholds of the circuit families considered when decoded using correlated matching. Each shot decoding shot is\ncomprised of r= 4drounds. On the left, the repetition code under circuit-level noise has a threshold of approximately p≈0.05.\nIn the center, the surface code under phenomenological noise has a threshold of approximately p≈.058. On the right, the\nsurface code under circuit-level noise has a threshold of approximately p≈.0088.\n0 20 40 60 80 100\nEnsemble size10−510−4Logical error per roundp=0.0125\ndis ance -7\ndistance-9\ndistance-11\n0 20 40 60 80 100\nEnsemble  size10−3\n3 × 10−44 × 10−46 × 10−42 × 10−3Logical error per roundp=0.025\n0 20 40 60 80 100\nEnsemble size1. 2 × 10−21.4 × 10−21.6 × 10−21.8 × 10−22 × 10−22. 2 × 10−2Logical error per roundp=0.05\nFIG. 10. Decoding circuit-level noise in the repetition code at various error rates. As in the main text, at each distance there\nare three lines: minimum-weight matching (dotted), vote pooled ensembled matching (circles), and tensor network decoding\n(dashed). Note that at distance 11 for p= 0.0125, the minimum-weight matching and tensor network decoder yield the same\nerror rate. This is likely due to the small number of failures, and the minimum-weight matching decoder approaching the\nmaximum-likelihood decoder performance at low error rates (as the most likely error dominates).13\n0 20 40 60 80 100\nEnsemble size10−410−3Logical error per roundp=0.02\ndis ance -5\ndistance-7\ndistance-9\n0 20 40 60 80 100\nEnsemble  size10−2\n3 × 10−34 × 10−36 × 10−3Logical error per roundp=0.04\n0 20 40 60 80 100\nEnsemble size2 × 10−23 × 10−24 × 10−2Logical error per roundp=0.06\nFIG. 11. Decoding phenomenological noise in the surface code at various error rates. As in the main text, at each distance\nthere are three lines: correlated matching (dotted), most-likely-error pooled ensembled matching (circles), and TNML decoding\n(dashed). At higher error rates, the gap between ensembled matching and TNML decoding grows larger. This illustrates an\nimportant point: because the TNML decoder likely boasts a higher threshold, there are error regimes wherein the gap between\nthe TNML decoder and the ensembled matcher will grow. However, these very-near-threshold error rates are not the regime in\nwhich one would build a quantum memory without overwhelming overhead. At more realistic target error rates, the relative\nperformance between the two decoders is similar – e.g., in the leftmost plot where p= 0.02, the error rates of ensembled\nmatching and TNML decoding are indiscernable.\n0 20 40 60 80 100\nEnsemble size10−510−4Logical error per roundp=0.002\ndis ance -7\ndistance-9\ndistance-11\n0 20 40 60 80 100\nEnsemble  size10−310−2Logical error per roundp=0.004\n0 20 40 60 80 100\nEnsemble size10−1\n6 × 10−2Logical error per roundp=0.008\nFIG. 12. Decoding circuit-level noise in the surface code at various error rates. As in the main text, at each distance there\nare three lines: minimum-weight matching (dotted), vote pooled ensembled matching (circles), and TNML decoding (dashed).\nNote the inversion of the lines at p= 0.008, indicating that this error rate is above the threshold of the uncorrelated decoder\nbut below the threshold of the correlated decoder."    },    {        "title": "2401.12448v1.Field_induced_phase_transitions_and_quantum_criticality_in_a_honeycomb_antiferromagnet_Na3Co2SbO6.pdf",        "content": "Field-induced phase transitions and quantum criticality in a honeycomb antiferromagnet\nNa3Co2SbO 6\nZe Hu,1,∗Yue Chen,2,∗Yi Cui,1, 3Shuo Li,1Cong Li,1Xiaoyu Xu,1Ying Chen,1\nXintong Li,2Yuchen Gu,2Rong Yu,1, 3Rui Zhou,4Yuan Li,2, 5,†and Weiqiang Yu1, 3,‡\n1Department of Physics and Beijing Key Laboratory of Opto-electronic Functional\nMaterials &Micro-nano Devices, Renmin University of China, Beijing, 100872, China\n2International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China\n3Key Laboratory of Quantum State Construction and Manipulation (Ministry\nof Education), Renmin University of China, Beijing, 100872, China\n4Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,\nChinese Academy of Sciences, Beijing, 100190, China\n5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China\nWe performed23Na NMR measurements on a single-domain crystal of the Kitaev material Na 3Co2SbO 6,\nwith magnetic field applied along the crystalline aaxis. A positive Curie-Weiss constant is obtained from the\nNMR Knight shift, which suggests the existence of ferromagnetic exchange couplings. The antiferromagnetic\nordering is found to be suppressed at a field of 1.9 T. Inside the ordered phase, our data reveal two additional\nphase transitions. At 1.9 T, the spin-lattice relaxation rate 1/23T1establishes a quantum critical behavior at\nhigh temperatures. However, at low temperatures, a gapped behavior is observed at the “critical” field, which\nsuggests a weakly first-order transition instead and a possible field-induced quantum spin liquid. Our results\nreveal complex microscopic interactions in the system, which may help to search for possible quantum spin\nliquids.\nI. INTRODUCTION\nExperimental search for quantum spin liquids (QSLs), rep-\nresenting a disordered phase beyond Landau’s paradigm, con-\ntaining novel properties such as fractional excitations and\nlong-range entanglement, and promising for unconventional\nsuperconductivity and quantum computation, has been a\nheated frontier in condensed matter physics1–5. In antiferro-\nmagnetic (AFM) systems with triangular, kagome, and py-\nrochlore lattice structures, strong geometric frustration may\nbe sufficient to suppress magnetic ordering and lead to QSLs.\nHowever, it is highly debated if QSLs are established in real\nmaterials, because site or anti-site disorder, which strongly\naffects the nature of the ground states, has been frequently re-\nported6–8.\nIn parallel, the Kitaev model in the honeycomb lat-\ntice, which contains exchange frustration among neighbor-\ning bonds with orthogonal Ising-type couplings, is a rare 2D\ncase where the QSL is an exact ground state with Majo-\nrana and photonic gauge excitations9–11. Theoretical stud-\nies also predict that AFM Kitaev model carries either Z 2or\nU(1) gauge field12–14, whereas ferromagnetic (FM) Kitaev\nmodel only hosts Z 2type. Until recently, it is proposed that\nsome 4d or 5d transition metal ions with hexagonal lattice\nstructures, such as A2IrO3(A=Li, Na)15–18andα-RuCl 319–29,\nmay contain Kitaev interactions arising from strong spin-orbit\ncoupling and bond symmetries30,31. To be more interesting,\nhexagonal cobaltates, such as Na 2Co2TeO6, Na 3Co2SbO 6,\nand BaCo 2(AsO 4)2, are proposed to be a 3d transition metal\nclass of QSL candidate materials, due to interplay among trig-\nonal crystal field, charge transfer, spin-orbit coupling, and\nCoulomb interactions32–36.\nHowever, in all of these Kitaev materails, AFM ground\nstates are usually observed37–44. It is then proposed that these\n(a)( b)a\nabc\ncb\nFIG. 1. Lattice structure of Na 3Co2SbO 6.(a) Side view of one\nunit cell of the compound, where magnetic Co2+ions form the lay-\nered regular honeycomb lattice, with Sb5+ions located at the center\nof each hexagon. Oxygen atoms in the edge-shared CoO 6octahedra\nare omitted for simplicity. Na+ions separate the magnetic layers\nwith two types of inequivalent sites, labeled as Na(1) and Na(2). (b)\nTop view of one half of unit cell along the caxis. The red and brown\ndotted lines depict the in-plane boundary which contains Na+and\nCo2+ions respectively. Na(1) and Na(2) differ slightly by their dis-\ntances to neighboring Co-Co bonds.\nsystems should be described by a combined H-K-Γmodel,\nwhere Hstands for isotropic Heisenberg, Kfor Kitaev, and\nΓfor off-diagonal spin couplings, and QSL may only survive\nin narrow, barely reachable parameter spaces11,45–47.\nNa3Co2SbO 6, as shown in Fig. 1(a), contains edge-shared\nCoO 6octahedra. Co2+ions, with effective spin-1/2, form a\nlayered honeycomb lattice, with SbO 6octahedra located at\nthe center in the same layer. The exchange couplings among\nCo2+are bridged by both edge-shared oxygen (through por-\nbitals) and Sb (through dorbitals) atoms. Dominate FM Ki-\ntaev interactions have been proposed theoretically34, but yet\nto be proved experimentally. At zero field, the compound is\nordered below TN≈6.6 K48, with an AFM pattern and a prop-\nagation vector K= (1/2, 1/2, 0)38,39. The magnetic structure\nof Na 3Co2SbO 6, as well as Na 2Co2TeO6, may not follow thearXiv:2401.12448v1  [cond-mat.str-el]  23 Jan 20242\nsimple zigzag pattern49–52and varies with magnetic field43,48.\nMagnetic excitations in Na 3Co2SbO 6, measured by inelastic\nneutron scattering (INS), µSR and NMR measurements, re-\nveal strong quantum fluctuations and possible coexistence of\nH,KandΓexchange couplings53–58. However, an easy-\nplane XXZ model is also suggested by a recent study52. To\nfurther address these issues and search for QSL, more spec-\ntroscopic studies through tuning are highly desired25,59,60.\nIn this work, we report23Na NMR studies on a high quality,\ntwin-free single crystal of Na 3Co2SbO 6. We identify a posi-\ntive Curie-Weiss constant from the NMR Knight shift in the\nparamagnetic (PM) phase. The magnetic orderings are con-\nfirmed by the NMR spectra and the transition temperatures\nare resolved by the NMR spin-lattice relaxation rates 1/23T1.\nWith field applied along the crystalline aaxis, two additional\nmagnetic phase transitions are found in the ordered phase, as\nshown by the detailed phase diagram (see Fig. 8).\nLow-energy spin dynamics, revealed by the spin-lattice re-\nlaxation rates, 1/23T1, demonstrates a field-suppression of\nmagnetic order. In particular, a funnel shape in the color\nmap of 1/23T1Tis established at a field of 1.9 T, which is\na clear evidence for quantum criticality in the paramagnetic\nphase. However, the low-temperature 1/23T1reveals a possi-\nble weakly first-order quantum phase transition at 1.9 T, and a\npossible QSL at higher fields. Our study reveals the complex-\nity of phases and magnetic exchange couplings in the system,\nand promotes further investigations on field-induced phases,\nsuch as QSLs.\nII. MATERIALS AND TECHNIQUES\nHigh quality single crystals were grown by the chemical\nvapor transport method44. A single domain (twin-free) sam-\nple was selected for the current study. The usage of single\ndomain crystal allows to induce a single magnetic phase with\none field orientation, which also simplifies the NMR spectra.\nA recent low-temperature magnetization study reveals com-\nplicated phase transitions with different in-plane field orienta-\ntion48. In this study, the field is applied along the crystalline\naaxis, which results in a larger critical field48and benefits\nNMR measurements. The magnetic field is calibrated by the\n63Cu resonance frequency from the NMR tank coil.\nA top tuning circuit was used to cover a wide frequency\nrange with fields from 1 T to 10 T. The23Na NMR spectra\nwere collected by the standard spin-echo technique. For ver-\nification, we also performed dcmagnetization measurements\nat selected temperatures in a Magnetic Property Measurement\nSystem (MPMS), with the same field orientation.\n23Na is an isotope with spin I= 3/2 and Zeeman factor\n23γ= 11.262 MHz/T. The NMR Knight shift23Knis calcu-\nlated by Kn= (f−γH)/γH , where fis the peak frequency\nof the center NMR lines. The NMR spin-lattice relaxation rate\n1/23T1is obtained by the inversion-recovery method, with\nspin recovery curve fitted to the exponential function for spin-\n3/2 nuclei, I(t) =a−b[e−(t/T1)β+ 9e−(6t/T1)β], where βis\nthe stretching factor.III. NMR SPECTRA\nIn order to investigate local magnetic properties, the NMR\nspectra were collected at various magnetic fields and tem-\nperatures, with frequencies relative to23γH, as depicted in\nFig. 2(a)-(c). At T=20 K, six23Na NMR lines are identi-\nfied, which correspond to two types of interlayer Na atoms,\nNa(1) and Na(2) as shown in Fig. 1(b), and each has one\ncenter peak and two satellite peaks. With an occupancy ra-\ntio of N1:N2=1:2in the lattice, where N1andN2are the\nnumber of atoms of Na(1) and Na(2), respectively, their\nspectra are resolved accordingly with the spectral weight of\nI1:I2=N1:N2=1:2. The NMR satellites are located at about\n0.83 MHz (0.52 MHz) for Na(1) [Na(2)] away from the cen-\nter transition.\nUpon cooling from 20 K to 8 K, all peaks move toward\nhigh frequencies as shown in Fig. 2(a), which indicates an in-\ncrease of the Knight shift Knin the PM phase, with a positive\nhyperfine coupling constant among23Na and Co2+. At even\nlower temperatures, the NMR spectra broaden significantly,\nfor example, below 6 K at 1 T (Fig. 2(a)) and below 5 K at\n1.8 T(Fig. 2(b)) , which clearly indicates the onset of mag-\nnetic ordering. The satellites become indistinguishable due to\nbroadening of the spectra in the ordered phase. At low fields,\nwith the change of local moment orientations in the lattices,\nthe relative orientation among the ordered moments and the\nprinciple axis of the electric field gradient (EFG) also varies\nand results in additional spectral broadening in the satellites.\nAt high fields, more NMR lines appear at low temperatures\ndue to complicated magnetic structures, which prevents us\nfrom distinguishing the center and the satellite transitions. At\na field of 2.3 T, as shown in Fig. 2(c), all peaks are resolv-\nable again at low temperatures, consistent with the spin polar-\nized phase when magnetic ordering is suppressed by field48.\nNote that the integrated spectral weight, multiplied by tem-\nperature, remains as a constant at 1.9 T and above. At low\nfields, the values drop by 30 %below TNwhen cooled from\nthe PM phase to the ordered phase (data not shown). We be-\nlieve that quenched disorder broadens the spectra and leads to\npartial loss of the signal.\nTo resolve the magnetic structures in the ordered phase, the\nspectra are displayed at a low temperature of 2 K with increas-\ning fields, as shown in Fig. 2(d). Two broad NMR peaks are\nobserved at 1 T, which is a direct evidence of AFM ordering.\nWith field increased from 1.4 T to 1.6 T, more peaks are re-\nsolved, where a change of the magnetic structure is suggested.\nFor fields above 1.9 T, six NMR peaks are seen with overlaps\nof center and satellite lines of23Na from Na(1) and Na(2),\nwhere the fully polarized phase is achieved as the ground\nstate. Therefore, two magnetic phase transitions, at 1.4 T and\n1.9 T respectively, are revealed.\nWe simulated the low-field NMR spectra with two possi-\nble magnetic structures, the zigzag order (Fig. 3(a))39and the\ndouble- qorder (Fig. 3(c))52, respectively. Since23Na nuclei\nare distant from Co2+ions, only dipolar hyperfine couplings\namong them is taken into account. For simplicity, the NMR\nsatellites are not considered in the simulation. The uniform\nmagnetization of Co2+by the external field would induce a3\n101 11 21 31 42 02 12 22 32 42 62 72 82 93 0-\n10 1 2 3 4 5 0 5 01 001 502 0002468101\n1 01 00024681012140\n.000.050 .100.150.000.010.020.030.0420 K1\n4 K1\n0 K8\n K6\n K5\n K4\n Kµ0H = 1 TI0 (arb. unit)f\n (MHz)Na(1)N\na(2)(a)2\n K20 K2\n K3 K5 K10 K20 K(b)µ 0H = 1.8 TI0 (arb. unit)f\n (MHz)2 K5 K10 K20 K(c)µ 0H = 2.3 TI0 (arb. unit)f\n (MHz) \n63Cu1\n.0 T1.1 T1.2 T1.3 T1.4 T1.5 T1.6 T1.8 T1.85 T1.9 T2.0 T2.3 T2.5 T3.0 T4.0 T(d)I0 (arb. unit)f\n - 23γH (MHz)T = 2 K \n3 T, Na(1) \n3 T, Na(2)(e)2\n3K s (%)T\n (K)0.00.51.01.52.02.53.03.51\n/23K s (100) 1.0 T \n1.6 \n1.8 \n1.85 \n1.9 \n2.0 \n2.3 \n2.5 \n3.0 \n4.0 \n5.0 \n7.0 \n10.0TN(f) 23K n (%)T\n (K) Na(2) Na(1) \nNa(2)23K nχ\n (emu/mol/Oe)\nFIG. 2. NMR spectra. (a)-(c)23Na spectra measured at different temperatures, with representative fields of well below (1.0 T), close to\n(1.8 T), and above (2.3 T) the field-induced quantum phase transition. Red and blue arrows label the Na(1) and Na(2) spectra respectively.\nThe orange (magenta) dash lines denote the position of63Cu (65Cu) lines from the NMR coil. (d) Full23Na NMR spectra taken at a constant\ntemperature of 2 K with increasing fields. The red rectangle denotes emerged peaks in the 1/3-AFM phase at fields of 1.4 T and above. The\nup-arrows mark all resolvable peaks across the phase transition. (e) Knight shift23Knas functions of temperatures measured at different\nfields. (f) Knight shift23Ks(left scale) and 1/23Ks(right scale) plotted as functions of temperatures, measured on both Na(1) and Na(2) at\n3 T. Solid lines are linear fits to 1/23Kswith temperature from 25 to 200 K. Inset:23Kn-χplot with temperature from 25 to 200 K. Solid\nlines are linear fits to obtain chemical shift KCandAhf.\n-2- 10 1 2 3 zigzagNa(1)Na(2)Intensity (arb. unit)N\na(2)d\nouble-qNa(1)N\na(2)N\na(2)N a(2)ab\nI\nntensity (arb. unit)\na\nc\n(a)( b)(\nc)(d)N\na(1)N\na(2)6\n5Cu(e)I0 (arb. unit)2\n3H ai\nn (kOe)H = 1.0 TT\n = 2.0 K63Cu\nFIG. 3. Magnetic structures and simulated NMR spectra. (a)-(b)\nSpin pattern and center23Na lines with the zigzag order (see text).\n(c)-(d) Spin pattern and center23Na lines with the double- qorder\n(see text). (e) The measured NMR spectrum at 2 K at 1 T field.\nrigid shift of the whole spectra, but does not affect the line-\nshape, therefore not included in the calculations as well.\nThe total hyperfine field on each23Na nucleus is then ob-\ntained by summing over contributions from ordered momentsof Co2+with their relative coordinates. The calculated cen-\nter transition lines at low field are plotted in Fig. 3(b) and\n(d), with a local moment of 0.9 µB/Co2+assumed for the\nzigzag phase39, and two local moments of√\n3and 1 (in unit\nof 0.9 µB/Co2+) in the double- qphase52, respectively.\nFor the zigzag pattern, Na(1) produces one center line\nat about 0 kOe, and Na(2) produces two center lines at -\n0.326 kOe (-0.37 MHz) and 0.384 kOe (0.43 MHz), with a\nrelative spectral weight of 1:1:1 (Fig. 3(b)). By contrast, the\ndouble- qpattern features one center line for Na(1) at 0 kOe,\nand three center lines for Na(2) at -0.322 kOe (-0.36 MHz),\n0 kOe, and 0.324 kOe (0.36 MHz), which produces three lines\nwith a relative weight of 1:4:1 (Fig. 3(d)).\nThe simulated three-peak spectrum with equal weights for\nthe zigzag pattern is remarkably different from the actual\nspectrum with two peaks at 1 T (Fig. 3(e)) which are assigned\nto be Na(1) and Na(2) respectively (see later). On the other\nhand, for the double- qorder, external field may have different\neffects on local moments in the non-collinear structure, and\nthe assumption of a rigid shift of spectra in the simulation may\nbe invalid. Therefore, our data do not support the zigzag or-\nder, but have no obvious contradiction with the double- qtype.\nWe note that more distinguishable peaks emerge with fields\nabove 1.1 T, which may suggest a very weak incommensurate\ncomponent superimposed on the double- qpattern and smear\nout the detailed spectra at fields of 1 T and below.4\nIV . NMR KNIGHT SHIFT\nThe Knight shifts23Knfor both Na(1) and Na(2) at dif-\nferent fields are obtained from the resonance frequency of\nthe center peaks. As shown in Fig. 2(e),23Knfor Na(2)\nare shown as function of temperatures. Using the relation-\nship between Knight shift and the reported susceptibility\ndata48, the hyperfine interaction Ahfcan be obtained by the\nslope of23Kn−χplot, which is about 0.872 kOe/ µBand\n1.166 kOe/ µBfor Na(1) and Na(2) respectively, as shown in\nthe inset of Fig. 2(f). Note that the orbital contribution to\nχis small and has been subtracted. With this, the chemical\nshifts KC, estimated by the y−intercept, are 0.017 ±0.005%\nand 0.032 ±0.006%for Na(1) and Na(2), respectively.\nThe spin contribution of Knight shift Ksis then obtained\nasKs=Kn−KC. In Fig. 2(f),23Ksof both Na(1) and\nNa(2) are shown at 3 T, where the rapid increase of23Ks\nupon cooling demonstrates PM behaviors. Indeed, the high-\ntemperature data of23Ks(T)follow the Curie-Weiss behav-\nior,23Ks=C/(T−θ).1/23Ks, demonstrated by functions\nof temperatures, are well fitted by straight lines with temper-\natures from 200 K down to 25 K. With this, the Curie-Weiss\ntemperature is obtained as θ= 1.1±0.3 K for both Na(1) and\nNa(2).\nTABLE I. Curie-Weiss temperature θobtained from different mea-\nsurements with field applied along the aaxis.\nMeasurements23Kn[Na(1)]23Kn[Na(2)] M[Low-T]48\nθ(K) 1.1 ±0.3 1.1 ±0.3 1.0\nBy comparison, the magnetic susceptibility data also re-\nveal a positive θ= 1.0 K, shown in Table I, by data fitting\nfrom 20 K to 120 K with the same field orientation48. These\nsmall but positive Curie-Weiss temperatures obtained at high\ntemperatures indicate that the material contains FM intralayer\ncouplings. Given that the ground state has an AFM order38,39,\nhighly competing magnetic FM and AFM interactions are ex-\npected with such a small θ. Then our data support the exis-\ntence of FM interactions as proposed for cobaltates with hon-\neycomb lattices32, although the absolute values of exchange\ncouplings remain to be determined.\nV . SPIN-LATTICE RELAXATION RATES AND\nMAGNETIC TRANSITIONS\n1/23T1was measured to determine the low-energy spin\nfluctuations. 1/T1measures the low-energy spin dynamics\nwith1/T1=TΣqA2\nhf(q)Imχ(q,ω)\nω, where Ahf(q)is the hy-\nperfine coupling constant, χis the dynamic susceptibility of\nelectrons, qis the wave vector, and ωis the NMR frequency.\n1/23T1was measured at various frequencies across the spec-\ntra line at 4 K and 1 T, as shown in Fig. 4(c). 1/23T1on the\nright peak is about 50% larger than that on the left peak, which\nsupports assignment of the right peak to Na(2), because Na(2)\nhas a stronger hyperfine coupling as revealed before.\n11 01 000.0010.010.11101001\n1 01 000.00.20.40.60.81.01\n0.010.511.011.512.012.513.013.514.0024680\n2468( c)(b) \n1.0 T     2.3 \n1.3        2.5 \n1.6        3.0 \n1.8        4.0 \n1.85      5.0 \n1.9        7.0 \n2.0        10.023T1-1(s-1)T\n (K)(a) \n2.3 \n2.5 \n3.0 \n4.0 \n5.0 \n7.0 \n10.0 1.0 T \n1.3 \n1.6 \n1.8 \n1.85 \n1.9 \n2.0/s98T\n (K)N\na(2)6\n5CuH = 1 TT\n = 4 K \n23T1-1 \nspectrumI\nntensity23T1-1 (s-1)f\n (MHz)63CuNa(1)FIG. 4. Spin-lattice relaxation rates. (a)1/23T1of Na(2) as func-\ntions of temperatures measured under typical fields. (b) The stretch-\ning factor βas functions of temperatures. (c) 1/23T1(left scale)\nmeasured on different frequencies of the spectrum (right scale).\nHere we primarily report 1/23T1measured on the center\npeak of Na(2). As shown in Fig. 4(a), 1/23T1is displayed as\nfunctions of temperatures, with fields from 1 T up to 10 T. The\nstretching factor β≈1 (Fig. 4(b)) in the PM phase, indicates\nhigh quality of the sample. Upon cooling, a rapid decrease\nofβis seen in the ordered phase and also in the fully polar-\nized phases, which coincides with the spectral broadening at\nlow temperatures (Fig. 2(a)-(c)). We think that the spectral\nbroadening and decrease of βare caused by quenched dis-\norder which is very effective in the ordered phase and fully\npolarized phase when the local moments are large.\nAt low fields, 1/23T1increases slightly upon cooling\nthrough 100 K, before entering the ordered phase, which ev-\nidences the onset of low-energy spin fluctuations. In the fol-\nlowing, three types of magnetic transitions are revealed.\nFirst, the spin-lattice relaxation rate divided by tempera-\nture, as shown in Fig. 5, reveals all the N ´eel transitions at\nlow fields, characterized by a peaked feature in 1/23T1Tat\nTN.TNis about 6.3 K at 1 T, and barely resolvable at about\n3.5 K with field at 1.8 T, below which 1/23T1Tdrops sharply\nwith a gapped behavior. With fields from 1.85 T to 1.9 T,\nthe peaked behavior in the 1/23T1Tis not resolvable, and the\nphase boundary will be determined later by a field-sweep mea-\nsurement on 1/23T1T. At 2 T and above, 1/23T1Texhibits a\nsmooth decrease with temperature, which indicates the sup-\npression of the AFM ordering. The detailed TNat different\nfields are then added in the phase diagram (Fig. 8). The sup-\npression of TNis consistent with earlier susceptibility mea-\nsurements38,39; however, our values of TNare slightly higher,\nwhich is probably caused either by a shorter time scale of T1\nmeasurements or a small difference in the field alignment.\nSecond, with fields from 1 T to 1.8 T, a “knee”-like feature\nis seen in 1/23T1Tat temperatures far below TN( Fig. 5(a)),\nwith the onset temperatures marked as T∗. This “knee”-like\nbehavior, rather than a continuous gapped behavior below TN\nas expected with an in-plane field, reveals emergence of addi-\ntional enhanced low-energy spin fluctuations. T∗at different\nfields are then determined and shown in Fig. 8, which is about5\n11 00.11101001\n.01 .52 .02 .50.1110100 \n1.0 T \n1.3  \n1.6  \n1.8  \n1.85  \n1.9 \n2.0 (a)23(T1T)-1(s-1K-1)T\n (K)TNT\n * 6.0 K \n5.0 \n4.0 \n3.0 \n2.0 \n1.6  \n1.6 (b)µ\n0H (T)2\n3T1-1(s-1)H\nNH\ns\nFIG. 5. Determination of magnetic phase transitions. (a)\n1/23T1Tas functions of temperatures at low fields. Red and blue\narrows mark the peak and “knee”-like feature in the data, denoting\nthe double transition temperatures of TNandT∗, respectively. (b)\n1/23T1as functions of fields under constant temperatures from 1.6 K\nto 6 K. Up arrows denote the fields of two magnetic transitions oc-\ncurring at HNandHsrespectively. Left and right arrows at 1.6 K\nillustrate the field ramping directions, where magnetic hysteresis is\nseen at about Hswith opposite ramping directions.\n3.5 K at 1 T, decreases to 2 K at 1.8 T, and diminishes at about\n1.85 T. Therefore, T∗follows the same trend as TNwith field.\nBy this, we think that a weak spin reorientation may occur in\nthe ordered phase51. However, such transition is not seen in\nthe spectra, which may be masked by the broad spectra if the\nchange of the magnetic structure is not large.\nThird, 1/23T1, measured as functions of fields at 6 K\nand below, demonstrates a double-peak feature with field, as\nshown in Fig. 5(b). The field values of the peak position in\n1/23T1are labeled as HNandHs, respectively. Each HNwith\nthe corresponding temperature is consistent with TNmeasured\nat the same field, which therefore represents the N ´eel ordering\nas discussed before. At 1.6 K, HN≈1.87 T; by extrapolation,\nthe “critical” field HC≈1.9 T at zero temperature.\nOn the other hand, Hsremains at a constant field of ∼1.4 T,\nbut always lower than HN. At 1.6 K, a hysteresis behavior\nclose to Hsis also revealed, by the data difference with oppo-\nsite field ramping directions as marked in Fig. 5(b), where Hs\ntends to be lower with the field ramped down. Far above or\nbelow Hs, the hysteresis behavior diminishes. This low-field\npeak and its hysteresis should indicate an additional first-order\nmagnetic transition.\nSuch a transition at a constant field is further verified by our\ndcmagnetization measurements. As shown in Fig. 6(a), the\nlow-temperature magnetization M(H)data are shown with\nfield up to 2 T at selected temperatures from 8 K down to\n2 K. At 2 K, double magnetic transitions are clearly seen by\nthe change of the slop in M(H). For accuracy, the deriva-\ntivedM/dH is then calculated and plotted in Fig. 6(b). At\nthe temperatures of 2, 3, and 4 K, dM/dH clearly shows the\ndouble-peak feature, with fields also labeled as HNandHs,\nrespectively. Again, Hsremains at about 1.4 T and barely\nchanges with temperature.\n0.00 .51 .01 .52 .00.00.51.01.52.00\n.00 .51 .01 .52 .005101520 \n8 K \n7 \n6 \n5 \n4 \n3 \n2M (µB/Co)µ\n0H (T)(a) \n7 K \n6 K \n5 K \n4 K \n3 K \n2 K(b) \n8 Kd\nM/dHµ\n0H (T)HsHNFIG. 6. Magnetization data of the sample. (a)M(H)as func-\ntions of fields measured at selected temperatures. (b) Calculated\ndM(H)/dH , at different temperatures. Red and blue arrows mark\nthe peak positions of the data, labeled as HNandHs, respectively.\nData are shifted vertically for clarity.\nFor comparison, the HNandHsdetermined by 1/23T1and\nbydM/dH at each temperature are added in the phase di-\nagram in Fig. 8. It is clearly seen that both measurements\nare consistent. The small difference in the field values ex-\ntracted from the measurements may be due to field-calibration\nerrors and/or a small tilting of the field towards the “harder”\na-axis in the 1/23T1measurement. In fact, the transition at\nHsis consistent with the reported first-order phase transition\nfrom an AFM ordering to a “1/3-AFM” ordering, resolved by\nthe neutron diffraction48. The AFM wave vectors switches\nfrom (±a∗/2,±b∗/2,0)to(±a∗/3,±b∗/3,±c∗/3)through\nthe transition, where the magnetic cell is enlarged.\nVI. SPIN GAPS CLOSE TO THE CRITICAL FIELD\nIn principle, a quantum disordered phase is expected for\nfields above HC, where the spin gap increases monotonically.\nHowever, as we show below, the low-temperature spin gap\nmay exhibit a non-monotonic field dependence in this region,\nwhich suggests an additional phase.\n1/23T1exhibits a downturn behavior at low temperatures,\nwhich suggests the onset of spin gap both in the low-field\nordered phase and the high-field spin polarized phase. By\nwhich, we fit 1/23T1Tto an empirical function, 1/T1T∝\ne−∆/T. In the ordered AFM phase, 1/23T1Tas functions\nof1/T, as shown by the semi-log plots in Fig. 7(a), follow\nstraight lines in the low temperature regime. Then the gaps,\ndefined as ∆1, are obtained by the linear fit and shown in\nFig. 7(c) at different fields. The onset of the gapped behav-\nior in the ordered phase, rather than a power-law temperature\ndependence of 1/T1(T), suggests an ordering with local mo-\nments at least partially along the aaxis, where a gap opens\ndue to longitudinal fields. Similarly, above HC, a gapped be-\nhavior is also followed as shown in Fig. 7(b), and the obtained\ngap values, ∆2, are also added in Fig. 7(c).\nInterestingly, the gap does not drop to zero at the boundary6\n012345678910051015202530350\n.00 .20 .40 .60 .80.0010.010.11100.1110(c)( a)(\nb) Δ1 \nΔ2Δ\n (K)µ\n0H (T)1.85 TQ\nSL? 1.85 T 3.0 \n1.9      4.0 \n2.0      5.0 \n2.3      7.0 \n2.5      10.023(T1T)-1(s-1K-1)T\n -1 (K-1)Δ2Δ1 1.0 T \n1.3 \n1.6 \n1.823(T1T)-1(s-1K-1)\nFIG. 7. Low-temperature spin gaps. (a)-(b) The semi-log plots of\n1/23T1Tas functions of 1/T, below and above the “critical” field\n(see text), respectively. The straight lines are linear fits to data in\nthe low-temperature regime, which determine the gaps ∆1and∆2\nas labeled. (c) ∆1(in the ordered phase) and ∆2(in the disordered\nphase), as functions of fields. Solid lines are guides to the eyes. The\ndown arrow marks the transition field at about 1.85 T.\nof the ordered phase and the disordered phase. ∆1decreases\nwith field and ∆2increases with field. They cross at fields be-\ntween 1.8 and 1.9 T; however, none of them reach zero from\nfield between 1.8 T and 1.9 T. The presence of gap is also di-\nrectly demonstrated by the sharp downturns in the log-log plot\nof1/23T1Tin all of the fields as shown in Fig. 5(a). The ab-\nsence of gapless excitations rules out a quantum critical point\nin the current system; instead, a weakly first-order quantum\nphase transition is suggested with the onset of a small gap at\nthe transition field.\nFurthermore, ∆2demonstrates a non-monotonic field de-\npendence as shown in Fig. 7(c): it increases first with field and\nthen decreases; above 3 T, a large increase emerges again as\nexpected for a fully polarized phase. With fields from 1.85 T\nto 3 T, a dome-like behavior is observed. Such a dome-like\nshape of ∆2strongly supports the existence of an additional\nphase between the AFM phase and the fully polarized phase.\nVII. QUANTUM CRITICALITY AND PHASE DIAGRAM\nA colored contour plot of 1/23T1Tis shown in Fig. 8 as\na function of ( H,T).1/23T1Tis maximized when the tem-\nperature is close to TN, which is a typical feature for a renor-\nmalized classical regime. Below 2 K, 1/23T1Tis maximized\natHC≈1.9 T, which is a signature of quantum phase transi-\ntion with strong quantum fluctuations. At high temperatures, a\nquantum critical region (QCR), as noted in Fig. 8, is also seen\nas a funnel shape in the color map, when it is getting close to\n1.9 T.\nWith detailed data analysis performed in Sec. V, a complete\nmagnetic phase diagram is established as shown in Fig. 8,\nwhere the PM, AFM, 1/3-AFM, disordered phases and their\nphase boundaries are determined. The gap in the disordered\nphase, ∆2, is also added in the phase diagram. A QSL region\n0.00.51.01.52.02.53.03.54.00510150\n51015\nQ\nSL?P olarized TN [1/T1T ] \nHN [1/T1 ] \nHN [dM/dH ] \nT * \n[1/T1T ] \nHs [1/T1 ] \nHs [dM/dH ] \nΔ2 [1/T1T ]A\nFM1\n/3 - AFM  PMΔ\n (K)µ\n0H (T)T (K)lg(1/T1T)Q\nCRFIG. 8. Phase diagram. Colored map represents the contour plot\nof1/23T1Tdata, with a QCR at high temperatures. TN,HN,T∗,\nHs, andHCare phase boundaries (see text) determined by different\nmeasurements as labeled. ∆2is the high-field spin gap. Lines are\nguides to the eyes.\nis speculated just between the ordered phase and the fully po-\nlarized phase where ∆2exhibits a dome-like shape with field.\nWe also compared our results with a previous NMR work\non polycrystals, where the AFM phase, the potential QSL\nphase, and the fully polarized phase are also proposed58. The\nusage of single-domain single crystal allows us to resolve all\nthe phase boundaries precisely. In addition, we identified a\npossible spin reorientation transition below TNand a first-\norder phase transition at 1.4 T. The reported 1/23T1(T)of\npolycrystals exhibits a two-gap feature in a large field range58,\nwhich is absent in our work and therefore should be attributed\nto the magnetic anisotropy of the material.\nVIII. DISCUSSIONS\nThe complicated phase diagram indicates more competing\nexchange interactions in this system, which needs to be fur-\nther studied by different probes. In particular, the nature of the\ntransition at T∗, remains unknown and needs to be verified by\nother measurements. One candidate mechanism for the tran-\nsition is the DM interaction, which could be effective far be-\nlowTN. Given that the inversion symmetry is broken among\nnext-nearest neighboring Co2+ions in the honeycomb lattice,\na weak DM interaction is possible61. Because NMR is very\nsensitive to the low-energy spin fluctuations, such weak DM\ninteraction may lead to the observations below T∗. However,\nwe are cautious that DM interaction has not been reported by\nother studies38,39.\nTheoretically, a first-order phase transition may occur if a\nQSL exists between the ordered phase and the spin-polarized7\n11 00.11103\n456782\n.02 .53 .03 .52.02.53.03.5( c)(b) \n2.3 T \n2.5 T \n3.0 T \n ~ e-Δ2/T \n ~ T α23(T1T)-1(s-1K-1)T\n (K)(a)Δ\n2 (K)α µ\n0H (T)\nFIG. 9. Comparison of gap and power-law fits to 1/23T1T.(a)\n1/23T1Tas functions of temperatures at 2.3, 2.5 and 3.0 T. The solid\nlines and dash lines represent the fit to the gapped function and the\npower-law function below 3 K, respectively. (b) ∆2as a function of\nfield. (c) Power-law exponent αas a function of field.\nphase62–64. We speculate that Kitaev couplings ( K-term) and\noff-diagonal ( Γterm) may exist and strongly affect the spin\ndynamics in the system45,65. For comparison, in two other Ki-\ntaev materials, α-RuCl 3and Na 2Co2TeO6, a new phase seems\nto be established between the ordered phase and the polarized\nphase under in-plane magnetic field, where a QSL has been\nsuggested25,54,66. A gapless behavior is observed in the low-\ntemperature 1/T1data of α-RuCl 3, which supports a proxi-\nmate Kitaev QSL25. For the current compounds, we think that\na QSL may also exist, given the existence of a dome-shape of\n∆2.\nHowever, we found that a power-law fitting is also applica-\nble to the low temperature data 1/23T1Twith field just above\nHC. As shown in Fig. 9(a), function fit to either a gapped\nbehavior and a power-law behavior in the same temperature\nrange is performed at temperatures below 3 K, with fieldsfrom 2.3 to 3 T. The obtained gap ∆2and the power-law\nexponent αare depicted in Fig. 9(b) and (c), respectively,\nas function of field. Notably, both ∆2andαdecrease with\nincreasing field, contradicting to the expected increase of\nboth quantities with field in the fully polarized phase. Such\nanomalous behavior may support a QSL intervals between\nthe ordered phase and the fully polarized phase, although\nwe cannot differentiate a gapped or a gapless behavior with\ncurrent data.\nIX. SUMMARY AND ACKNOWLEDGEMENT\nIn summary, we investigated the static and low-energy dy-\nnamical behavior through NMR experiments on a high-quality\nNa3Co2SbO 6single crystal. With field applied along the a\naxis, our data reveal a positive Curie-Weiss constant at high\ntemperatures, which supports the existence of FM exchange\ncouplings. Given the absence of FM ordering, such a FM\ncoupling may not be the Heisenberg type. The observation of\nthree separate transition lines with field, including TN,T∗and\nHs, further suggests complex magnetic exchange couplings in\nthe system, which may help to establish QSLs. 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We are concerned with the semi-classical limit for ground st ates of the rela-\ntivistic Hartree-Fock energies (HF) under a mass constrain t, which are considered as the\nquantum mean-field model of white dwarfs [16]. In Jang and Seo k [14], fermionic ground\nstates of the relativistic Vlasov-Poisson energy (VP) are c onstructed as a classical mean-\nfield model of white dwarfs, and are shown to be equivalent to t he classical Chandrasekhar\nmodel. In this paper, we prove that as the reduced Planck cons tant/planckover2pi1goes to the zero, the\n/planckover2pi1-parameter family of the ground energies and states of (HF) c onverges to the fermionic\nground energy and state of (VP) with the same mass constraint .\n1.Introduction\n1.1.Backgrounds and setup. White dwarfs are compact and dense celestial objects sup-\nporting themselves by the electron degeneracy pressure aga inst the gravitational collapse.\nThey are considered as final evolutionary states of relative ly light stars that have exhausted\nthe nuclear fuel. In 1931, in his seminal paper [1], Chandras ekhar discovered the existence\nof the maximal mass Mc, the so-called Chandrasekhar limit mass, of non-rotating s table\nwhite dwarfs so that stars with larger mass collapse to other states such as neutron stars\nor black holes.\nTo be precise, the densify function ��of a non-rotating radially symmetric white dwarf\ncan be described by a solution to the equation for a gravitati onal hydrodynamic equilibrium\n1\nr2d\ndrˆr2\nρdP\ndr˙\n“ ´4πρ, (1.1)\nwhererdenotes the radial variable in R3,Pis the degeneracy pressure of the electron gas\nand the gravitational constant is normalized. In [1], the lo cal equation of state\nPpρq “4π\n3żp3ρ{4πq1{3\n0u4\n?\n1`u2du, (1.2)\nis derived for white dwarfs. The equation (1.1) equipped wit h (1.2) is referred to the\nChandrasekhar equation.\nDate: January 24, 2024.\n12 Y. HONG, S. JIN, AND J. SEOK\nFrom a variational formulation, the Chandrasekhar equatio n is derived as the Euler-\nLagrange equation for an energy minimizer subject to the mas s constraint,\nEC;min pMq:“inf\"\nECpρqˇˇˇρě0, ρPL4{3pR3q,ż\nR3ρpxqdx“M*\n, (1.3)\nwhereMą0 and the Chandrasekhar energy is given by\nECpρq:“ż\nR3Apρqdx\nlooooomooooon\nkinetic energy´1\n2ij\nR3ˆR3ρpxqρpx1q\n|x´x1|dxdx1\nloooooooooooooooomoooooooooooooooon\npotential energy\nwith\nApρq “żρ\n0`b\n1` p3\n4πuq2{3´1˘\ndu.\nFor thishydrodynamicmodel, theexistenceofthecritical m assMcą0isestablished inLieb\nand Yau [18]; ´8 ăEC;min pMq ă0 if 0 ăMăMc, whileEC;min pMq “ ´8ifMąMc. It\nis also shown that in the former case, EC;min pMqhas a unique radially symmetric compactly\nsupported minimizer ρC, and thatρCsolves the Chandrasekhar equation.\n1.1.1.Relativistic Hartree-Fock formulation. For the semi-classical description of relativis-\ntic white dwarfs, we follow the mathematical formulation in [8, 12, 13] with the small\nparameter /planckover2pi1ą0, which represents the reduced Planck constant. In the Heis enberg pic-\nture, a relativistic fermionic gas is described by a compact self-adjoint operator γacting on\nL2pR3q. By the Pauli exclusion principle, we assume that 0 ďγď1 as a quadratic form.\nFor such an operator γ, we denote its integral kernel by γpx,x1q, i.e.,\npγφqpxq “ż\nR3γpx,x1qφpx1qdx1,\nand the semi-classical density function by\nρ/planckover2pi1\nγ:“ p2π/planckover2pi1q3ργwithργpxq “γpx,xq.\nWe define the quantum self-generated potential by\nΦ/planckover2pi1\nγ:“ ´1\n|x|˚ρ/planckover2pi1\nγ\nand the exchange term X/planckover2pi1\nγas the operator with kernel\nX/planckover2pi1\nγpx,x1q:“p2π/planckover2pi1q3γpx,x1q\n|x´x1|.\nUsing the above definitions, we introduce the semi-classica lly scaled relativistic Hartree-\nFock energy functional\nE/planckover2pi1\nHFpγq:“Tr/planckover2pi1´\npa\n1´/planckover2pi12∆´1qγ¯\nloooooooooooooomoooooooooooooon\nkinetic energy`1\n2Tr/planckover2pi1`\nΦ/planckover2pi1\nγγ˘\nlooooomooooon\npotential energy`1\n2Tr/planckover2pi1`\nX/planckover2pi1\nγγ˘\nlooooomooooon\nexchange energy(1.4)SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 3\nand the mass\nM/planckover2pi1pγq:“Tr/planckover2pi1pγq,\nwhere\nTr/planckover2pi1p¨q:“ p2π/planckover2pi1q3Trp¨q.\nNote that in the kinetic energy, with an abuse of notation, we write Tr/planckover2pi1p?\n1´/planckover2pi12∆γq “\nTr/planckover2pi1pp1´/planckover2pi12∆q1\n4γp1´/planckover2pi12∆q1\n4q, where p1´/planckover2pi12∆q1\n4is the Fourier multiplier operator with\nsymbol p1`/planckover2pi12|ξ|2q1\n4, because it holds by cyclicity of the trace when γis regular enough.\nWe definethe quantum mechanical admissible class Aqmas the collection of compact self-\nadjointoperators γ:L2pR3q ÑL2pR3qsuchthat0 ďγď1andTr`\np1´∆q1\n4γp1´∆q1\n4˘\nă 8.\nThen, the fermionic gas in a white dwarf with mass Mis formulated a minimizer of the\nvariational problem\nE/planckover2pi1\nHF;min pMq “inf\nγPAqmpMqE/planckover2pi1\nHFpγq, (1.5)\nwhereMą0andAqmpMq:“ tγPAqm|M/planckover2pi1pγq “Mu. In[16], it is shownthat E/planckover2pi1\nHF;min pMq\nhas a minimizer, provided that Mis strictly less than the quantum Chandrasekhar limit\nmass\nMqm:“ p2Kqmq3\n2,\nwhereKqmis the sharp constant for the Lieb-Thirring type inequality , i.e.,\nKqm:“inf\nγPAqm}γ}1\n3`\nTr/planckover2pi1pγq˘2\n3Tr/planckover2pi1`\n|/planckover2pi1∇|1\n2γ|/planckover2pi1∇|1\n2˘\n1\n4π}∇Φ/planckover2pi1γ}2\nL2pR3q. (1.6)\nForµă0, we denote the characteristic function on p´8,µqby/BDpEăµq:“#1 ifEăµ,\n0 ifEěµ.\nTheorem 1.1 ([16, Lenzmann-Lewin]) .Let/planckover2pi1ą0. For0ăMăMqm, the following hold.\npiqThe variational problem E/planckover2pi1\nHF;min pMqhas a minimizer Q/planckover2pi1.\npiiqThe operator?\n1´/planckover2pi12∆´1`Φ/planckover2pi1\nQ/planckover2pi1`X/planckover2pi1\nQ/planckover2pi1has a negative eigenvalue µ/planckover2pi1such that\nQ/planckover2pi1“ /BDp?\n1´/planckover2pi12∆´1`Φ/planckover2pi1\nQ/planckover2pi1`X/planckover2pi1\nQ/planckover2pi1ăµ/planckover2pi1q`R/planckover2pi1, (1.7)\nwhereR/planckover2pi1is a finite rank operator on the eigenspace corresponding to µ/planckover2pi1.\nRemark 1.2.(1) InLenzmann-Lewin[16], theexistenceandpropertiesof minimizersare\nproved for the more delicate Hartree-Fock-Bogoliubov ener gy, but their approach\ncan be adapted to the Hartree-Fock case.\n(2) By homogeneity, the right hand side of (1.6) is invariant under the choice of /planckover2pi1ą0,\nand so is the critical mass Mqm.\n(3) By functional calculus, /BDp?\n1´/planckover2pi12∆´1`Φ/planckover2pi1\nQ/planckover2pi1`X/planckover2pi1\nQ/planckover2pi1ăµ/planckover2pi1qis the spectral projection onto the\neigenspace corresponding to the eigenvalues of γstrictly less than µ/planckover2pi1.4 Y. HONG, S. JIN, AND J. SEOK\n1.1.2.Relativistic Vlasov-Poisson formulation. Next, we review an analogous kinetic de-\nscription for white dwarfs from [14]. In kinetic theory, rel ativistic fermionic gases are de-\nscribed as distributions on the phasespace. For adistribut ionf“fpq,pq:R3ˆR3Ñ r0,1s,\nwe define the mass by\nMpfq:“ij\nR3ˆR3fpq,pqdqdp,\nand the relativistic Vlasov-Poisson energy functional (se e [9, 11, 14, 15]) by\nEVPpfq:“ij\nR3ˆR3`a\n1` |p|2´1˘\nfpq,pqdqdp\nlooooooooooooooooooooomooooooooooooooooooooon\nkinetic energy`1\n2ij\nR3ˆR3Φfpqqfpq,pqdqdp\nlooooooooooooooomooooooooooooooon\npotential energy,\nwhere\nΦf:“ ´1\n|x|˚ρf\nis the classical self-generated potential given by the kine tic density function\nρf“ż\nR3fp¨,pqdp.\nWe define the classical mechanical admissible class by\nAcm:“!\nfPL1pR3ˆR3q: 0ďfpq,pq ď1 a.e., and |p|fPL1pR3ˆR3q)\n,\nwhere the condition 0 ďfď1 is from the Pauli exclusion principle for fermions. Then,\nthe particle distribution in a white dwarf with mass Mis modeled as a minimizer for the\nclassical variational problem\nEVP;min pMq:“inf\nfPAcmpMqEVPpfq, (1.8)\nwhereMą0 andAcmpMq:“ tfPAcm|Mpfq “Mu. Recently, in Jang-Seok [14], it\nis constructed when its mass is strictly less than the classical Chandrasekhar limit mass .\nIndeed, such a critical mass is chosen analogously. Precise ly, it is given by\nMcm:“ p2Kcmq3\n2,\nwhereKcmis the sharp constant for the corresponding kinetic interpo lation inequality\nKcm:“inf\nfPAcm}f}1\n3\nL8pR3ˆR3q}f}2\n3\nL1pR3ˆR3q}|p|f}L1pR3ˆR3q\n1\n4π}∇Φf}2\nL2pR3q. (1.9)\nTheorem 1.3 ([14, Jang-Seok] ) .If0ăMăMcm, the following hold.\npiqThe variational problem EVP;min pMqhas a minimizer Q, which is unique up to a\ntranslation. Moreover, for any minimizing sequence tfnuforEVP;min pMq, one has\n}∇pΦfn´ΦQq}L2pR3qÑ0asnÑ 8, up to a translation.SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 5\npiiqFor someµă0,Qobeys the self-consistent equation\nQ“ /BDp?\n1`|p|2´1`ΦQpqqăµq. (1.10)\npiiiqThe density function ρQattains the minimum Chandrasekhar energy, i.e., ECpρQq “\nEC;min pMq(1.3), and consequently it satisfies the Chandrasekhar equations (1.1)and\n(1.2).\nRemark 1.4.In Theorem 1.3 piq, by the uniqueness, the sequential convergence is obtained\n(see [14, Remark 2.7] and [20]).\n1.2.Statement of the main result. In this paper, we are interested in the connection\nbetween the quantum and the classical variational problems (1.5) and (1.8) via the semi-\nclassical limit. For the statement, we remark that the quant um critical mass is less than or\nequal to the classical one; MqmďMcm(see Lemma 2.4). Our main theorem asserts that if\nMis strictly less than the quantum critical mass, then the gro und energies and minimizers\ncorrespond each other as stated below.\nTheorem 1.5. LetMP p0,Mqmqbe given. Let Q/planckover2pi1andQbe a family of minimizers and\na minimizer of the variational problems (1.5)and(1.8)and respectively. Then, as /planckover2pi1Ñ0,\nafter taking suitable spatial translations, one has\n$\n’’&\n’’%E/planckover2pi1\nHF;min pMq ÑEVP;min pMq\nρQ/planckover2pi1áρQweakly inLqpR3q @qą1\n}ΦQ/planckover2pi1´ΦQ}9H1pR3qXC1pR3qÑ0.\nRemark 1.6.piqIn Lieb and Yau [18, 19], Chandrasekhar’s hydrodynamic whit e dwarf\nmodel is derived directly from the mean-field limit pN“/planckover2pi1´1\n3Ñ 8qof the ground\nenergy of the fermionic N-body relativistic Schr¨ odinger Hamiltonian\nHN“Nÿ\ni“1a\n´∆xi`1´1´1\nN2{3ÿ\n1ďiăjďN1\n|xi´xj|.\nOur main result, together with Theorem 1.3 piiiq, justifies the consistency among\nthree well-known mean-field models; the Hartree-Fock, the V lasov-Poisson and the\nEuler-Poisson energies.\npiiqContrary to the mean-field limit [18, 19], in our setting, one can take advantages\nfrom that minimizers are solutions to the self-consistent e quations (1.7) and (1.10). It\nallows us to obtain a wider range of weak convergence for dens ity functions and the\nstrong convergence for potential functions.\npiiiqIn Choi, Hong and Seok [3], a similar analysis is presented fo r free energy minimizers\nbetween the non-relativistic Hartree and the Vlasov-Poiss on models, where the free\nenergies aredefinedasthesumoftheenergy andaCasimirfunc tional(or ageneralized\nentropy). In this work, the Casimir functional plays a role o f preventing the ground6 Y. HONG, S. JIN, AND J. SEOK\nenergy from diverging to ´8. On the other hand, in the present paper, Pauli’s\nexclusion principle (0 ďγď1 and 0 ďfď1) has the same role.\nTheproofofthemainresultisbasedontheapproachof Choi-H ong-Seok [3]. Inourproof,\na key step is to show the convergence from the quantum to the cl assical minimum energy,\ni.e.,E/planckover2pi1\nHF;min pMq ÑEVP;min pMqas/planckover2pi1Ñ0; then the convergence of the density functions of\nminimizers Q/planckover2pi1andQfollows from uniform regularity (see Lemma 3.1) and the conv ergence\nof ground energy levels between E/planckover2pi1\nHF;min pMqandEVP;min pMq. We notice that this key step\ncannot be directly achieved because the quantum and the clas sical minimizers are different\nmathematical objects – one is an operator and the other is a di stribution function on the\nphase space. So, it is not possible to insert the classical mi nimizerQto the Hartree-Fock\nenergyE/planckover2pi1\nHFpγqor the quantum minimizer Q/planckover2pi1to the Vlasov-Poisson energy EVPfor energy\ncomparison.\nThey nevertheless have common structures, that is, both are characteristic functions\nof local Hamiltonians (see (1.7) and (1.10)). Thus, we intro duce an auxiliary quantum\nstateγ/planckover2pi1“ /BDp?\n1´/planckover2pi12∆´1`ΦQă˜µ/planckover2pi1q`˜R/planckover2pi1mimicking the form of the classical energy minimizer\nQ“ /BDp?\n1`|p|2´1`ΦQăµq, where Φ Qis the potential function of the classical minimizer Q\n(see Lemma 5.1 for the details about the choices of ˜ µ/planckover2pi1and˜R/planckover2pi1). Using this, we prove\nthatEVP;min pMqis asymptotically bounded from below by E/planckover2pi1\nHF;min pMq. For the proof, the\nrelativistic Weyl’s law is crucially used to compare the mas s and the energy of the auxil-\niary state and the classical minimizer. Conversely, constr ucting an auxiliary classical state\nf/planckover2pi1“ /BDp?\n1`|p|2´1`ΦQ/planckover2pi1pqqă˜µ1\n/planckover2pi1q, which is made of the potential function Φ Q/planckover2pi1of the quan-\ntum minimizer Q/planckover2pi1, and using the relativistic Weyl’s law, we show the opposite asymptotic\ninequality.\nWe remark that compared to the previous work [3], several tec hnical arguments are\nsimplified in this paper, and these simplifications also work in the setting of [3]. First of all,\nwe note that for the proof of the minimum energy convergence, we need a version of Weyl’s\nlaw with a /planckover2pi1-parameter family of potentials. It is obtained in [3] modif ying the classical\nproof in Reed-Simon [21] by the profile decomposition. Howev er, in this article, we instead\nadopt the different approach [5, 17] involving Gaussian coher ent states for the relativistic\nWeyl’s law. It turns out that this approach is shorter and mor e direct. Secondly, in [3],\nrather complicated lemmas are introduced for the desired po tential energy convergence.\nHowever, we found that in the gravitational case, the energy functionals have a coercive\nstructure so that the technical proof in the earlier work can be avoided.\n1.3.Notations. Throughout this article, the Fourier transform is defined by\nˆupξq “ż\nR3upxqe´ix¨ξdx.\nFor an operator γ:L2pR3q ÑL2pR3q, we denote the operator norm by }γ}and the Hilbert-\nSchmidt norm by }γ}HS:“ pTrpγ˚γqq1\n2. GivenφPL2pR3q,|φyxφ|denotes the one-particleSEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 7\nprojector\n|φyxφ|:uPL2pR3q ÞÑż\nR3φpxqφpx1qupx1qdx1.\nWe define\nx`“maxtx,0u, x ´“mintx,0u.\n1.4.Outline of the paper. The rest of the paper is organized as follows. In Section 2, we\nrearrange and reformulate two key inequalities, the Lieb-T hirring inequality and the kinetic\ninterpolation inequality, which fit in our setting. The vani shing exchange term estimates\nand the comparison for the quantum and classical critical ma ssesMqmandMcmare also\ngiven in Section 2. The regularity estimates for the quantum white dwarfs are dealt with\nin Section 3. Section 4 is devoted to provide a version of rela tivistic Weyl’s law with a rate\nof convergence, which is suitable for our analysis. We finall y prove the main theorem in\nSection 5 by integrating the aforementioned inequalities, estimates and information.\n1.5.Acknowledgement. Thisworkwas supportedbytheNewFaculty StartupFundfrom\nSeoul National University. This research of the first author was supported by the Basic\nScience Research Program through the National Research Fou ndation of Korea (NRF)\nfunded by the Ministry of Science and ICT (RS-2023-00208824 and RS-2023-00219980).\nThis research of the second author was supported by the Basic Science Research Program\nthrough the National Research Foundation of Korea (NRF) fun ded by the Ministry of\nScience and ICT (RS-2023-00213407). This research of the th ird author is supported by\nBasicScienceResearch Program throughtheNational Resear ch Foundation ofKorea(NRF)\nfunded by the Ministry of Science and ICT (NRF-2020R1C1C1A0 1006415).\n2.Key inequalities and Critical masses\nIn this section, we recall preliminary inequalities for the quantum and the kinetic varia-\ntional problems, and briefly explain why critical masses app ear in our variational problems.\n2.1.Potential energy estimates. The kinetic interpolation and the Lieb-Thirring in-\nequalities are employed to control potential energies.\nLemma 2.1 (Kinetic interpolation inequality; endpoint case) .If0ďfď1, then\n}ρf}4{3\nL4{3pR3qÀ››|p|f››\nL1pR3ˆR3q.\nProof.The density function ρfsatisfies the following trivial inequality\nρf“ż\n|p|ďR`ż\n|p|ěRfp¨,pqdpÀR3`1\nR››|p|fp¨,pq››\nL1ppR3q.\nOptimizing the right hand side, we obtain pρfq4{3À }|p|fp¨,pq}L1ppR3q. Then, integrating,\nwe obtain the desired inequality. /square8 Y. HONG, S. JIN, AND J. SEOK\nAs a quantum analogue of the kinetic interpolation inequali ty, we have the Lieb-Thirring\ninequality. Here, for later analysis (see Lemma 3.1), we giv e a slightly extended one.\nLemma 2.2 (Lieb-Thirring inequality) .Let/planckover2pi1P p0,1s,sě0andαP r0,3\n2q. Ifγis a\ncompact self-adjoint operator on L2pR3qand0ď |/planckover2pi1∇|αγ|/planckover2pi1∇|αď1, then\n}ρ/planckover2pi1\nγ}3`2s´2α\n3´2α\nL3`2s´2α\n3´2αpR3qÀTr/planckover2pi1p|/planckover2pi1∇|sγ|/planckover2pi1∇|sq,\nwhere the implicit constant is independent of /planckover2pi1.\nProof.By scalingγpx,x1q “1\n/planckover2pi13˜γpx\n/planckover2pi1,x1\n/planckover2pi1q, we may assume /planckover2pi1“1. Following the proof in [24],\nwe apply the Littlewood-Paley inequality in [24] to obtain\n}ργ}3`2s´2α\n3´2α\nL3`2s´2α\n3´2αpR3qÀ››››ÿ\nNP2ZρPNγPN››››3`2s´2α\n3´2α\nL3`2s´2α\n3´2αpR3q,\nwherePNis the Fourier multiplier with symbol ψpξ\nNqsuch thatř\nNP2Zψpξ\nNq ”1 onR3and\nψpξqis supported in1\n2ď |ξ| ď4. We note that as quadratic forms,\n0ďPNγPN“PN|∇|´αp|∇|αγ|∇|αq|∇|´αPNÀPN|∇|´2αPN“ |∇|´2αP2\nN,\nwhich implies that\n0ďρPNγPNpxq ďρ|∇|´2αP2\nNpxq “N3´2αp| ¨ |´2αψ2q_pNpx´x1qq|x“x1„N3´2α.\nOn the other hand, we have 0 ďρPNγPNÀ1\nN2sρ|∇|sγ|∇|s. Thus, it follows that\nÿ\nNP2ZρPNγPNÀÿ\nNďRN3´2α`ÿ\nNěR1\nN2sρ|∇|sγ|∇|s„R3´2α`1\nR2sρ|∇|sγ|∇|s.\nThus, optimizing bound as in the proof of Lemma 2.1, we prove t hat\n}ργ}3`2s´2α\n3´2α\nL3`2s´2α\n3´2αpR3qÀż\nR3ρ|∇|sγ|∇|sdx“Trp|∇|sγ|∇|sq.\n/square\n2.2.Exchange term estimates. For a compact self-adjoint operator γhaving a spectral\nrepresentationř8\nj“1λj|φjyxφj|with an orthonormal set tφju8\nj“1, the exchange term X/planckover2pi1\nγis\ndefined as the integral operator with kernelp2π/planckover2pi1q3γpx,x1q\n|x´x1|“ p2π/planckover2pi1q3ř8\nj“1λjφjpxqφjpx1q\n|x´x1|. By the\nFourier transform and the Cauchy-Schwarz inequality, one c an see that if γis non-negative,\nthen 0 ďX/planckover2pi1\nγď1\n|x|˚ρ/planckover2pi1\nγas a quadratic form.\nThe following lemma shows that the exchange terms vanishes i n the semi-classical limit.SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 9\nLemma 2.3 (Exchange term estimates) .Ifγis a compact self-adjoint operator acting on\nL2pR3qand0ďγď1, then\n}X/planckover2pi1\nγ} ď }X/planckover2pi1\nγ}HSÀ?\n/planckover2pi1!\nTr/planckover2pi1`\n|/planckover2pi1∇|γ|/planckover2pi1∇|˘)1\n2,\nij\nR3ˆR3|p2π/planckover2pi1q3γpx,x1q|2\n|x´x1|dxdx1À/planckover2pi12Tr/planckover2pi1`\n|/planckover2pi1∇|1\n2γ|/planckover2pi1∇|1\n2˘\n.\nProof.Fors“1,1{2, by the H¨ older and the Sobolev inequalities in the Lorentz norm (see\n[10] for example), we have\nij|γpx,x1q|2\n|x´x1|2sdxdx1“››››γpx,x1q\n|x´x1|s››››2\nL2xpR3;L2\nx1pR3qqÀ }γpx,x1q}2\nL2xpR3;L6\n3´2s,2\nx1 pR3qq\nÀ››|∇x1|sγpx,x1q››2\nL2xpR3;L2\nx1pR3qq“››γ|∇|s››2\nHS\nď }?γ}2››?γ|∇|s››2\nHSďTr`\n|∇|sγ|∇|s˘\n.\nThen, putting /planckover2pi1, we obtain the lemma. /square\n2.3.Critical masses. The inequalities in the previous subsection are employed to con-\nstruct energy minimizers, but this is possible only under th e assumption that a given mass\nis strictly less than the critical mass.\nTo see this in the kinetic description, we assume that fPAcmpMq. Then, interpolation\nwith the trivial bound }ρf}L1pR3qďMyields\n}ρf}L6{5pR3qÀM1{3››|p|f››1{2\nL1pR3ˆR3q,\nand by the Sobolev inequality,\n››|p|f››\nL1pR3ˆR3qěKcm\nM2{3ij\nR3ˆR3ρfpxqρfpx1q\n|x´x1|dxdx1, (2.1)\nwhereKcmą0 is the sharp constant for the above inequality. An importan t remark is that\nifMăMcm“ p2Kcmq3{2, the variational problem EVP;min pMqis well-formulated, since the\nenergy of an admissible distribution fPAcmpMqis bounded from below;\nEVPpfq ě››|p|f››\nL1pR3ˆR3q´ }f}L1pR3ˆR3q´M2{3\n2Kcm››|p|f››\nL1pR3ˆR3q\něˆ\n1´M2{3\n2Kcm˙››|p|f››\nL1pR3ˆR3q´M.\nOn the other hand, if γPAqmpMq, then by the Sobolev inequality and Lemma 2.2 with\ns“1\n2andα“0, there exists the sharp constant Kqmą0 such that\nTr/planckover2pi1`\n|/planckover2pi1∇|1\n2γ|/planckover2pi1∇|1\n2|˘\něKqm\nM2{3ij\nR3ˆR3ρ/planckover2pi1\nγpxqρ/planckover2pi1\nγpx1q\n|x´x1|dxdx1. (2.2)10 Y. HONG, S. JIN, AND J. SEOK\nThus, it follows that if MăMqm“ p2Kqmq3{2, then\nE/planckover2pi1\nHFpγq ěˆ\n1´M2{3\n2Kqm˙\nTr/planckover2pi1`\n|/planckover2pi1∇|1\n2γ|/planckover2pi1∇|1\n2|˘\n´M,\nwhere the exchange energy is dropped since it is non-negativ e. Note that Mqmis indepen-\ndent of/planckover2pi1P p0,1s.\nLemma 2.4 (Comparison between the quantum and the classical critical masses).\nMqm“ p2Kqmq3{2ďMcm“ p2Kcmq3{2.\nProof.For a Schwartz function fPSpR3ˆR3q, we introduce the operator\nγ/planckover2pi1“1\np2π/planckover2pi1q3ij\n|ϕ/planckover2pi1\npq,pqyxϕ/planckover2pi1\npq,pq|fpq,pqdqdp,\nwhereϕ/planckover2pi1\npq,pqpxq “1\npπ/planckover2pi1q3{4e´|x´q|2\n2/planckover2pi1eip¨px´qq\n/planckover2pi1is a coherent state with pq,pq PR3ˆR3, and insert\nit into the quantum inequality (2.2). Then, by direct calcul ations, taking the limit /planckover2pi1Ñ0,\none can derive the inequality\n››|p|f››\nL1pR3ˆR3qěKqm\nM2{3ij\nR3ˆR3ρfpxqρfpx1q\n|x´x1|dxdx1.\nTherefore, by density, it follows that KcměKqm. /square\n3.Quantum and kinetic white dwarfs and their basic properties\nThe lowest energy states, describing white dwarfs, have bee n constructed both in the\nkinetic and the quantum settings (see Theorem 1.1 and 1.3), e mploying the inequalities in\nthe previous section. From now on, we assume that MăMqm, and let Q/planckover2pi1be the minimum\nenergy state for the quantum variational problem E/planckover2pi1\nHF;min pMq, andQbe that for the kinetic\nvariational problem EVP;min pMq. For notational convenience, we denote\nΦ/planckover2pi1:“Φ/planckover2pi1\nQ/planckover2pi1, X/planckover2pi1:“X/planckover2pi1\nQ/planckover2pi1and Φ : “ΦQ.\nThen, integrating out the momentum variable for the equatio n\nQ“ /BDp?\n1`|p|2´1`Φpqqďµq, (3.1)\nwe obtain the potential equation\n´∆Φ“ ´1\n3!\n2pµ´Φq`` pµ´Φq2\n`)3\n2. (3.2)\nThen, by the elliptic regularity with the kinetic interpola tion inequality (Lemma 2.1), one\ncan show that Φ PL3,8pR3q XC2pR3q. In addition, a minimizing sequence tfpnqu8\nn“1has\na compactness property in the sense that passing to a subsequ ence and up to translation,\n}∇p1\n|x|˚ρfpnq´1\n|x|˚ρ˜Qq}L2pR3qÑ0 for some minimizer ˜Q. For the quantum counterpart,SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 11\nwe note that Q/planckover2pi1obeys the self-consistent equation\nQ/planckover2pi1“ /BDp?\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1ăµ/planckover2pi1q`R/planckover2pi1. (3.3)\nMoreover, if tγpnqu8\nn“1is a minimizing sequence and up to translation, then passing to a\nsubsequence, }∇p1\n|x|˚ρ/planckover2pi1\nγ/planckover2pi1´1\n|x|˚ρ/planckover2pi1\n˜Q/planckover2pi1q}L2pR3qÑ0 for some minimizer ˜Q/planckover2pi1.\nBy construction, a quantum minimizer is a smooth finite-rank operator. To see this, let\nµ/planckover2pi1\n1ďµ/planckover2pi1\n2ďµ/planckover2pi1\n3ď ¨ ¨ ¨ ă0\ndenote the negative eigenvalues (counting multiplicities ) for?\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1. Then,\nit is deduced from (3.3) that Q/planckover2pi1is of the form\nQ/planckover2pi1“ÿ\nµ/planckover2pi1\njăµ/planckover2pi1|φ/planckover2pi1\njyxφ/planckover2pi1\nj| `ÿ\nµ/planckover2pi1\nj“µ/planckover2pi1λ/planckover2pi1\nj|φ/planckover2pi1\njyxφ/planckover2pi1\nj|for someλ/planckover2pi1\njP r0,1s, (3.4)\nwhere each φ/planckover2pi1\njsolves the eigenvalue equation\n`a\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1˘\nφ/planckover2pi1\nj“µ/planckover2pi1\njφ/planckover2pi1\nj. (3.5)\nBy the structure (3.4), Q/planckover2pi1must have finite rank. Note that the equation (3.5), together\nwith (3.4), is simply a finitely many coupled system of energy sub-critical elliptic PDEs.\nStandard elliptic regularity theory ensures that all φ/planckover2pi1\nj’s are smooth, and so are the total\ndensityρ/planckover2pi1\nQ/planckover2pi1and the potential Φ /planckover2pi1. We give a simple proof for this in Appendix. However,\nobtaining uniform bounds is not so obvious.\nIn the following lemma, we prove uniform regularity.\nLemma 3.1 (Regularity of quantum white dwarfs) .ForMP p0,Mqmq, letQ/planckover2pi1be a mini-\nmizer for the variational problem E/planckover2pi1\nminpMqand denote the corresponding potential function\nbyΦ/planckover2pi1. Then, there exists some αP p0,1qsuch that\nsup\n0ă/planckover2pi1ď1}Φ/planckover2pi1}L3,8pR3qXC1,αpR3qă 8.\nProof.By the Sobolev and the Lieb-Thirring (Lemma 2.2) inequaliti es,}Φ/planckover2pi1}L3,8pR3qXL12pR3q\nis uniformly bounded. To upgrade regularity, we write from ( 3.4) that\np2π/planckover2pi1q3ÿ\nµ/planckover2pi1\njăµ/planckover2pi1pµ/planckover2pi1\nj´µ/planckover2pi1q2“Tr/planckover2pi1`\npa\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1´µ/planckover2pi1q2Q/planckover2pi1˘\n.\nNote that all quantities in the above identity are finite, bec auseQ/planckover2pi1is smooth and of finite\nrank. Moreover, we have\np2π/planckover2pi1q3ÿ\nµ/planckover2pi1\njăµ/planckover2pi1pµ/planckover2pi1\nj´µ/planckover2pi1q2“µ2\n/planckover2pi1p2π/planckover2pi1q3ÿ\nµ/planckover2pi1\njăµ/planckover2pi1´µ/planckover2pi1\nj\nµ/planckover2pi1´1¯2\nď }��/planckover2pi1}2\nL8pR3qTr/planckover2pi1pQ/planckover2pi1q12 Y. HONG, S. JIN, AND J. SEOK\nand 0 ďX/planckover2pi1ď ´Φ/planckover2pi1as quadratic forms, and thus }X/planckover2pi1} ď }Φ/planckover2pi1} “ }Φ/planckover2pi1}L8pR3q.Hence, by the\nCauchy-Schwarz inequality, we obtain\n}Φ/planckover2pi1}2\nL8pR3qMěTr/planckover2pi1`\np1´/planckover2pi12∆qQ/planckover2pi1˘\n´ } ´1´µ/planckover2pi1`Φ/planckover2pi1`X/planckover2pi1}2Tr/planckover2pi1pQ/planckover2pi1q\n´2p2π/planckover2pi1q3››a\n1´/planckover2pi12∆a\nQ/planckover2pi1››\nHS}a\nQ/planckover2pi1}HS} ´1´µ/planckover2pi1`Φ/planckover2pi1`X/planckover2pi1}\ně1\n2Tr/planckover2pi1`\np1´/planckover2pi12∆qQ/planckover2pi1˘\n´9p1`3}Φ/planckover2pi1}L8pR3qq2M.\nHowever, by interpolation and the Lieb-Thirring inequalit y (Lemma 2.2), we have\n}Φ/planckover2pi1}L8pR3q“ }1\n|x|˚ρ/planckover2pi1\nQ/planckover2pi1}L8pR3qÀ }ρ/planckover2pi1\nQ/planckover2pi1}4\n9\nL4\n3pR3q}ρ/planckover2pi1\nQ/planckover2pi1}5\n9\nL5\n3pR3qÀ!\nTr/planckover2pi1`\np´/planckover2pi12∆qQ/planckover2pi1˘)1\n3.\nThus, by Young’s inequality, one can show that Tr/planckover2pi1pp1´/planckover2pi12∆qQ/planckover2pi1qand }Φ/planckover2pi1}L8pR3qare\nuniformly bounded. Similarly, by the Cauchy-Schwarz inequ ality and the fact that\n}a\n1´/planckover2pi12∆Q/planckover2pi1p´1`Φ/planckover2pi1`X/planckover2pi1q}2ď }a\n1´/planckover2pi12∆a\nQ/planckover2pi1a\nQ/planckover2pi1}2p1`2}Φ/planckover2pi1}L8pR3qq2\nď }a\n1´/planckover2pi12∆Q/planckover2pi1a\n1´/planckover2pi12∆}p1`2}Φ/planckover2pi1}L8pR3qq2,\none can show that\n1Á pµ/planckover2pi1\n1q2“››pa\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1qQ/planckover2pi1pa\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1q››\ně1\n2}a\n1´/planckover2pi12∆Q/planckover2pi1a\n1´/planckover2pi12∆} ´9p1`2}Φ/planckover2pi1}L8pR3qq2M,\nand so }?\n1´/planckover2pi12∆Q/planckover2pi1?\n1´/planckover2pi12∆}is also uniformly bounded. Then, the Lieb-Thirring in-\nequality (Lemma 2.2) with α“s“1 yields that }ρ/planckover2pi1\nQ/planckover2pi1}L3pR3qÀ1.\nRepeating the same argument to\np2π/planckover2pi1q3ÿ\nµ/planckover2pi1\njăµ/planckover2pi1pµ/planckover2pi1\nj´µ/planckover2pi1q4“Tr/planckover2pi1`\npa\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1´µ/planckover2pi1q4Q/planckover2pi1˘\nand\npµ/planckover2pi1\nj´µ/planckover2pi1q4“Tr/planckover2pi1››pa\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1´µ/planckover2pi1q2Q/planckover2pi1pa\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1´µ/planckover2pi1q2››,\none can show that Tr/planckover2pi1pp1´/planckover2pi12∆q2Q/planckover2pi1qand }p1´/planckover2pi12∆qQ/planckover2pi1p1´/planckover2pi12∆q}are uniformly bounded.\nTherefore, we obtain that as quadratic forms,\n0ďQ/planckover2pi1À p1´/planckover2pi12∆q´2\nso that\n0ďρ/planckover2pi1\nQ/planckover2pi1À p2π/planckover2pi1q3ρp1´/planckover2pi12∆q´2„1. (3.6)\nThen the elliptic estimate shows that there exists some αP p0,1qsuch that Φ /planckover2pi1is uniformly\nbounded in C1,αpR3q. /squareSEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 13\n4.Relativistic Weyl’s law\nWeyl’s law is a fundamental tool for semi-classical analysi s as it provides a precise as-\nymptotic formula for eigenvalue counting for a Schr¨ odinge r operator in terms of the volume\nof level sets with respect to the corresponding classical lo cal Hamiltonian. Weyl’s law has\nbeen established in various settings [5, 7, 17], but we prese nt a version of it is presented in\ndetail (Proposition 4.3), because we could not find the exact formulation we need.\n4.1.Setup.We begin this section with some remarks on our setting. Our go al is to develop\na Weyl’s law, which applies to the quantum minimizer\nQ/planckover2pi1“ /BDp?\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1ăµ/planckover2pi1q`R/planckover2pi1. (4.1)\nHence, we need to handle a family of potentials as well as exch ange terms with the pa-\nrameter which we will take a limit. On the other hand, we may ta ke advantages from that\npotentials are regular (see Lemma 3.1) and that only eigenva lues away from the bottom of\nthe continuous spectrum, not all negative eigenvalues, are taken in account (see the formula\n(3.3)). We also note that the exchange term is not so essentia l in the semi-classical limit,\nbecause it vanishes eventually (see Lemma 2.3).\nFor this reason, the following assumptions are imposed thro ughout this section.\nAssumption 4.1 (Weyl’s law assumptions) .For a family tV/planckover2pi1u/planckover2pi1Pp0,1sof potentials and a\nfamily tA/planckover2pi1u/planckover2pi1Pp0,1sof self-adjoint operators on L2pR3q, the following hold.\n(1)V/planckover2pi1ı0is non-positive;\n(2)}V/planckover2pi1}L3,8pR3qXC1pR3qis bounded uniformly in /planckover2pi1P p0,1s;\n(3) There exists E0ă0such that }pV/planckover2pi1´E0\n2q´}L3{2pR3qis bounded uniformly in /planckover2pi1P p0,1s;\n(4)}A/planckover2pi1}HS“Op?\n/planckover2pi1q.\n4.2.Eigenvalue counting. For each /planckover2pi1ą0, we denote the negative eigenvalues of the\noperator?\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1in non-decreasing order (counting multiplicities) by\nE/planckover2pi1\n1ďE/planckover2pi1\n2ďE/planckover2pi1\n3ď ¨ ¨ ¨ ă0\n(see [21, Theorem XIII.14] and [25, Theorem 4.1]). Given the energy level Eă0, we denote\nthe number of eigenvalues ăEby\nN/planckover2pi1pEq “N/planckover2pi1pE;V/planckover2pi1,A/planckover2pi1q “Tr`/BDp?\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1ăEq˘\n,\nand define the associated sum\nS/planckover2pi1pEq “S/planckover2pi1pE;V/planckover2pi1,A/planckover2pi1q “ p2π/planckover2pi1q3ÿ\nE/planckover2pi1\njăEpE/planckover2pi1\nj´Eq\n“Tr/planckover2pi1`\npa\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1´Eq /BDp?\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1ăEq˘\n.\nBy the Cwikel-Lieb-Rozenblum (CLR) bound below, p2π/planckover2pi1q3N/planckover2pi1pEqis uniformly bounded\nunder Assumption 4.1.14 Y. HONG, S. JIN, AND J. SEOK\nLemma 4.2 (CLR bound) .IfE0ă0, then\np2π/planckover2pi1q3N/planckover2pi1pEq À1\np1´ p1`E0\n2q2\n`q3{2}pE0\n2´V/planckover2pi1q`}3\nL3pR3qfor allEďE0.\nProof.We employ the relativistic Rumin’s inequality [22, 23]; any self-joint operator γwith\n0ďγď p?\n1´/planckover2pi12∆´1´E\n2q´1satisfies\nTr/planckover2pi1`\nγ1{2pa\n1´/planckover2pi12∆´1´E\n2qγ1{2˘\nÁ`\n1´ p1`E\n2q2\n`˘3{4}ρ/planckover2pi1\nγ}3{2\nL3{2pR3q.(4.2)\nIndeed, the above inequality immediately follows from the g eneralized Rumin’s inequality in\nFrank[6, Lemma2.5], whichstates inourcontext that if0 ďγďap´i∇qwithaPL3,8pR3q,\nthen Tr pγ1{2ap´i∇qγ1{2q ě4π3{2\n3?\n3}a}´3{2\nL3,8pR3q}ργ}3{2\nL3{2pR3q. The inequality (4.2) is obtained\ntakingapξq “1 ?\n1`/planckover2pi12|ξ|2´1´E\n2, because }a}3\nL3,8pR3q“4π\n3/planckover2pi13p1´p1`E\n2q2\n`q3{2.\nFortheCLRbound,wefollowtheargumentofFrank[6]. Wetake N/planckover2pi1“N/planckover2pi1pEqeigenfunc-\ntionscorrespondingtotheeigenvalues ďEfortheoperator?\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1. Then,by\ntheGram-Schmidtprocessonthespanofsucheigenfunctions , wecanconstruct tψjuN/planckover2pi1\nj“1such\nthat xp?\n1´/planckover2pi12∆´1´E\n2qψj,ψkyL2pR3q“δjk, and we obtain the operator γ“řN/planckover2pi1\nj“1|ψjyxψj|\nsatisfying 0 ďγď p?\n1´/planckover2pi12∆´1´E\n2q´1and Tr pγ1{2p?\n1´/planckover2pi12∆´1´E\n2qγ1{2q “N/planckover2pi1. Thus,\nit follows that for EďE0,\n0ěTr/planckover2pi1`\nγ1{2pa\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1´Eqγ1{2˘\ně p2π/planckover2pi1q3N/planckover2pi1´ż\nR3pE0\n2´V/planckover2pi1q`ρ/planckover2pi1\nγdx´ }A/planckover2pi1}Tr/planckover2pi1pγq\ně p2π/planckover2pi1q3N/planckover2pi1´ }pE0\n2´V/planckover2pi1q`}L3pR3q}ρ/planckover2pi1\nγ}L3\n2pR3q´?\n/planckover2pi1p2π/planckover2pi1q3N/planckover2pi1,\nwhereLemma2.3isusedfortheexchange term. Then, by (4.2), theCLRboundfollows. /square\n4.3.Relativistic Weyl’s law. We prove the main result of this section.\nProposition 4.3 (Relativistic Weyl’s law) .Suppose that tV/planckover2pi1u/planckover2pi1Pp0,1sand tA/planckover2pi1u/planckover2pi1Pp0,1ssatisfy\nAssumption 4.1 with some E0ă0. Then, for E/planckover2pi1ďE0, we have\nS/planckover2pi1pE/planckover2pi1q “ij\n?\n1`|p|2´1`V/planckover2pi1pqqăE/planckover2pi1`a\n1` |p|2´1`V/planckover2pi1pqq ´E/planckover2pi1˘\ndqdp `Op?\n/planckover2pi1q,(4.3)\np2π/planckover2pi1q3N/planckover2pi1pE/planckover2pi1q “ˇˇ/visualspace\npq,pq PR3ˆR3:a\n1` |p|2´1`V/planckover2pi1pqq ďE/planckover2pi1(ˇˇ`Op/planckover2pi11{4q.(4.4)\nFor the proof, among several approaches, we follow the argum ent in [5, 17] involving\ncoherent states\nϕ/planckover2pi1\npq,pqpxq “1\npπ/planckover2pi1q3{4e´|x´q|2\n2/planckover2pi1eip¨px´qq\n/planckover2pi1,pq,pq PR3ˆR3.SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 15\nLemma 4.4 (Basic properties of coherent states) .\nż\nR3|xϕ/planckover2pi1\npq,pq|uy|2dp“ p2π/planckover2pi1q3ż1\npπ/planckover2pi1q3{2e´|q´x|2\n/planckover2pi1|upxq|2dx,\nż\nR3|xϕ/planckover2pi1\npq,pq|uy|2dq“/planckover2pi13ż\nR31\npπ/planckover2pi1q3{2e´|p´/planckover2pi1ξ|2\n/planckover2pi1|ˆupξq|2dξ.\nProof.By direct calculations, we have\nż\n|xϕ/planckover2pi1\npq,pq|uy|2dp“ż ij1\npπ/planckover2pi1q3{2e´|x´q|2\n2/planckover2pi1e´|x1´q|2\n2/planckover2pi1eip¨px´x1q\n/planckover2pi1upxqupx1qdxdx1dp\n“ij\np4π/planckover2pi1q3\n2e´|x´q|2\n2/planckover2pi1e´|x1´q|2\n2/planckover2pi1upxqupx1qδpx´x1qdxdx1\n“ p2π/planckover2pi1q3ż1\npπ/planckover2pi1q3{2e´|q´x|2\n/planckover2pi1|upxq|2dx.\nOn the other hand, by the Plancherel theorem,\nż\n|xϕ/planckover2pi1\npq,pq|uy|2dq“ż\n|pϕ/planckover2pi1\np0,pq˚uqpqq|2dq“/planckover2pi13ż1\npπ/planckover2pi1q3{2e´|p´/planckover2pi1ξ|2\n/planckover2pi1|ˆupξq|2dξ,\nwhere we usedzϕ/planckover2pi1\npq,pqpξq ��� p4π/planckover2pi1q3{4e|p´/planckover2pi1ξ|2\n2/planckover2pi1e´iq¨ξ. /square\nProof of Proposition 4.3. Step 1(Upper bound for S/planckover2pi1pE/planckover2pi1qin(4.3))We introduce the\noperator\nγ“1\np2π/planckover2pi1q3ij\n?\n1`|p|2´1`V/planckover2pi1pqqďE/planckover2pi1|ϕ/planckover2pi1\npq,pqyxϕ/planckover2pi1\npq,pq|dqdp.\nNote that 0 ďγď1 andγis of trace class, because\nTr/planckover2pi1pγq “ˇˇ/visualspace\npq,pq:a\n1` |p|2´1`V/planckover2pi1pqq ďE/planckover2pi1(ˇˇ\n“4π\n3ż/visualspace\n2pE/planckover2pi1´V/planckover2pi1q`` pE/planckover2pi1´V/planckover2pi1q2\n`(3\n2dxă 8\n(see Section 3). Thus, it follows that\nS/planckover2pi1pE/planckover2pi1q ď p2π/planckover2pi1q3Tr`\npa\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1´E/planckover2pi1qγ˘\n“ij\n?\n1`|p|2´1`V/planckover2pi1pqqďE/planckover2pi1xϕ/planckover2pi1\npq,pq|pa\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1´E/planckover2pi1qϕ/planckover2pi1\npq,pqydqdp.\nOn the right hand side, direct calculations via the Fourier t ransform yield\nxϕ/planckover2pi1\npq,pq|a\n1´/planckover2pi12∆ϕ/planckover2pi1\npq,pqy “żb\n1` |p`?\n/planckover2pi1ξ|21\nπ3{2e´|ξ|2dξ“a\n1` |p|2`Op?\n/planckover2pi1q,\nwhile by Assumption 4.1 (2) and (4),\nxϕ/planckover2pi1\npq,pq|V/planckover2pi1ϕ/planckover2pi1\npq,pqy “ż\nV/planckover2pi1pxq1\npπ/planckover2pi1q3{2e´|x´q|2\n/planckover2pi1dx“V/planckover2pi1pqq `Op?\n/planckover2pi1q16 Y. HONG, S. JIN, AND J. SEOK\nand\n|xϕ/planckover2pi1\npq,pq|A/planckover2pi1ϕ/planckover2pi1\npq,pqy| ď }A/planckover2pi1}}ϕ/planckover2pi1\npq,pq}2\nL2pR3q“Op?\n/planckover2pi1q.\nHere, an important remark is that all Op?\n/planckover2pi1qterms are independent of pq,pq. Therefore, it\nfollows that\nS/planckover2pi1pE/planckover2pi1q ´ij\n?\n1`|p|2´1`V/planckover2pi1pqqăE/planckover2pi1pa\n1` |p|2´1`V/planckover2pi1pqq ´E/planckover2pi1qdqdp\nÀ?\n/planckover2pi1ˇˇ/visualspace\npq,pq PR3ˆR3:a\n1` |p|2´1`V/planckover2pi1pqq ăE/planckover2pi1(ˇˇ\n“?\n/planckover2pi1¨4π\n3ż/visualspace\n2pE/planckover2pi1´V/planckover2pi1q`` pE/planckover2pi1´V/planckover2pi1q2\n`(3\n2dxÀ?\n/planckover2pi1.\nStep 2(Lower bound for S/planckover2pi1pE/planckover2pi1qin(4.3))By theCLRbound(Lemma4.2), theoperator?\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1has only finitely many (say N/planckover2pi1) eigenvalues ăE/planckover2pi1. Let tujuN/planckover2pi1\nj“1be\nthe corresponding orthonormal eigenfunctions. Then, we ha ve\nS/planckover2pi1pE/planckover2pi1q “ p2π/planckover2pi1q3N/planckover2pi1ÿ\nj“1xuj|pa\n1´/planckover2pi12∆´1`V/planckover2pi1`A/planckover2pi1´E/planckover2pi1qujy.\nThen, by Lemma 4.2 and Assumption 4.1 (2) and (4),\nS/planckover2pi1pE/planckover2pi1q “ p2π/planckover2pi1q3N/planckover2pi1ÿ\nj“1@\nuj|pa\n1´/planckover2pi12∆´1`R/planckover2pi1rV/planckover2pi1s ´E/planckover2pi1qujD\n`Op?\n/planckover2pi1q ¨ p2π/planckover2pi1q3N/planckover2pi1pE/planckover2pi1q\n“ p2π/planckover2pi1q3N/planckover2pi1ÿ\nj“1@\nuj|pa\n1´/planckover2pi12∆´1`R/planckover2pi1rV/planckover2pi1s ´E/planckover2pi1qujD\n`Op?\n/planckover2pi1q,\nwhereR/planckover2pi1rV/planckover2pi1s “1\npπ/planckover2pi1q3{2e´|¨|2\n/planckover2pi1˚V/planckover2pi1. Note that by Lemma 4.4,\n1\np2π/planckover2pi1q3ija\n1` |p|2|xϕ/planckover2pi1\npq,pq|ujy|2dqdp\n“1\np2πq3ż\"ż\nR3pa\n1` |p`/planckover2pi1ξ|21\npπ/planckover2pi1q3{2e´|p|2\n/planckover2pi1dp*\n|ˆujpξq|2dξ\n“1\np2πq3ża\n1`/planckover2pi12|ξ|2|ˆujpξq|2dξ`Op?\n/planckover2pi1q “ xuj,a\n1´/planckover2pi12∆ujy `Op?\n/planckover2pi1q\nand1\np2π/planckover2pi1q3ij\n|xϕ/planckover2pi1\npq,pq|ujy|2dqdp “ }uj}2\nL2pR3q,\n1\np2π/planckover2pi1q3ij\nV/planckover2pi1pqq|xϕ/planckover2pi1\npq,pq|ujy|2dqdp “ż\nR/planckover2pi1rV/planckover2pi1s|uj|2dx.\nThus, it follows that\nS/planckover2pi1pE/planckover2pi1q “N/planckover2pi1ÿ\nj“1ij`a\n1` |p|2´1`V/planckover2pi1pqq ´E/planckover2pi1˘\n|xϕ/planckover2pi1\npq,pq|ujy|2dqdp `Op?\n/planckover2pi1q.SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 17\nFinally, applying Bessel’s inequalityřN/planckover2pi1\nj“1|xϕ/planckover2pi1\npq,pq|ujy|2ď }ϕ/planckover2pi1\npq,pq}2\nL2pR3q“1 and the bathtub\nprinciple, we conclude that\nS/planckover2pi1pE/planckover2pi1q ěij\n?\n1`|p|2´1`V/planckover2pi1pqqăE/planckover2pi1`a\n1` |p|2´1`V/planckover2pi1pqq ´E/planckover2pi1˘\ndqdp ´Op?\n/planckover2pi1q.\nStep 3(Proof of (4.4))By the inequality pE´E/planckover2pi1q´´ pE´ pE/planckover2pi1`/planckover2pi11{4qq´ě/planckover2pi11{41tEăE/planckover2pi1u,\nwherea´“minta,0u, we have\np2π/planckover2pi1q3N/planckover2pi1pE/planckover2pi1q ď p2π/planckover2pi1q38ÿ\nj“1pEj´E/planckover2pi1q´´ pEj´ pE/planckover2pi1`/planckover2pi11{4qq´\n/planckover2pi11{4“S/planckover2pi1pE/planckover2pi1q ´S/planckover2pi1pE/planckover2pi1`/planckover2pi11{4q\n/planckover2pi11{4.\nThus, (4.3) deduces that\np2π/planckover2pi1q3N/planckover2pi1pE/planckover2pi1q ď1\n/planckover2pi11{4ij`a\n1` |p|2´1`V/planckover2pi1pqq ´E/planckover2pi1˘\n´\n´`a\n1` |p|2´1`V/planckover2pi1pqq ´E/planckover2pi1´/planckover2pi11{4˘\n´dqdp `Op/planckover2pi11{4q\n“ˇˇ/visualspace\npq,pq PR3ˆR3:a\n1` |p|2´1`V/planckover2pi1pqq ăE/planckover2pi1(ˇˇ`Op/planckover2pi11{4q.\nThe reverse inequality can be proved repeating the above arg ument but with the inequality\npE´E/planckover2pi1`h1{4q´´ pE´E/planckover2pi1q´ď/planckover2pi11{41tEăE/planckover2pi1u. /square\n5.Proof of the main theorem\nWe break the proof into two parts. The first is to show that the q uantum minimum\nenergy is asymptotically bounded from above by the classica l minimum energy (Lemma\n5.1), and the second is to prove the reverse inequality (Lemm a 5.4).\nLemma 5.1 (Asymptotic upper bound on the quantum minimum energy) .\nlimsup\n/planckover2pi1Ñ0E/planckover2pi1\nHF;min pMq ďEVP;min pMq ă0 for allMP p0,Mqmq.\nProof.For the proof, we make use of a minimizer Q“ /BDp?\n1`|p|2´1`Φďµqfor the classical\nvariational problem EVP;min pMq(see Section 3) to construct a quantum state\nγ/planckover2pi1“ /BDp?\n1´/planckover2pi12∆´1`Φă˜µ/planckover2pi1q`˜R/planckover2pi1,\nwhere 0 ď˜R/planckover2pi1ď1 is a self-adjoint operator on the eigenspace of?\n1´/planckover2pi12∆´1`Φ\nassociated to a negative eigenvalue ˜ µ/planckover2pi1. Here, ˜µ/planckover2pi1and˜R/planckover2pi1are chosen so that M/planckover2pi1pγ/planckover2pi1q “\np2π/planckover2pi1q3Trpγ/planckover2pi1q “Mandγ/planckover2pi1is admissible for the variational problem E/planckover2pi1\nHF;min pMq. Indeed,?\n1´/planckover2pi12∆´1`Φ has infinitely many negative eigenvalues, because Φ is a lon g-range\npotential such that if |x|is large, then Φ pxq ď ´c\n|x|for somecą0. Hence, such γ/planckover2pi1,\nhaving mass M, always exists. We also observe that ˜ µ/planckover2pi1ѵ. Indeed, if ˜ µ/planckover2pi1jconverges\nto a different strictly negative limit point ˜ µă0, Weyl’s law (Proposition 4.3) deduces a\ncontradiction to M/planckover2pi1pγ/planckover2pi1q “M. On the other hand, if ˜ µ/planckover2pi1Ñ0, then Weyl’s law implies\nthatM“M/planckover2pi1pγ/planckover2pi1q ěTr/planckover2pi1p /BDp?\n1´/planckover2pi12∆´1´c\n|x|ăǫqq “ } /BDp?\n1`|p|2´1´c\n|q|ăǫq}L1pR3ˆR3qfor anyǫą0.18 Y. HONG, S. JIN, AND J. SEOK\nHowever, since } /BDp?\n1`|p|2´1´c\n|q|ăǫq}L1pR3ˆR3qÑ 8asǫÑ0, taking sufficiently small ǫą0,\nwe can deduce a contradiction.\nWewillshowthat E/planckover2pi1\nHFpγ/planckover2pi1q ďEVPpQq`o/planckover2pi1p1q, whichimmediatelyimpliesthat E/planckover2pi1\nHF;min pMq ď\nEVP;min pMq `o/planckover2pi1p1q. First, we claim that the kinetic energy of γ/planckover2pi1is uniformly bounded so\nthat by Lemma 2.3, its exchange energy vanishes;\nlim\n/planckover2pi1Ñ0ij|p2π/planckover2pi1q3γ/planckover2pi1px,x1q|2\n|x´x1|dxdx1“0.\nIndeed, by the H¨ older inequality and Weyl’s law (Propositi on 4.3), the kinetic energy\nTr/planckover2pi1`\npa\n1´/planckover2pi12∆´1qγ/planckover2pi1˘\n“S/planckover2pi1p˜µ/planckover2pi1;Φ,0q ´ż\nΦρ/planckover2pi1\nγ/planckover2pi1dx`˜µ/planckover2pi1Tr/planckover2pi1pγ/planckover2pi1q\nis bounded from above by\nS/planckover2pi1p˜µ/planckover2pi1;Φ,0q ` }Φ}L8pR3qż\nρ/planckover2pi1\nγ/planckover2pi1dx`˜µ/planckover2pi1Tr/planckover2pi1pγ/planckover2pi1q\n“ij\npa\n1` |p|2´1`Φ´µqQdqdp ``\n}Φ}L8pR3q`µ˘\nM`o/planckover2pi1p1q\n“ij\npa\n1` |p|2´1qQdqdp ´ijρQpxqρQpx1q\n|x´x1|dxdx1` }Φ}L8pR3qM`o/planckover2pi1p1q\n(see Section 3).\nNext, we reorganize the terms in the energy of γ/planckover2pi1in a similar way as\nE/planckover2pi1\nHFpγ/planckover2pi1q “S/planckover2pi1p˜µ/planckover2pi1;Φ,0q `˜µ/planckover2pi1M`1\n2ijρQpxqρQpx1q\n|x´x1|dxdx1\n´1\n2ijpρ/planckover2pi1\nγ/planckover2pi1´ρQqpxqpρ/planckover2pi1\nγ/planckover2pi1´ρQqpx1q\n|x´x1|dxdx1\n`1\n2ij|p2π/planckover2pi1q3γ/planckover2pi1px,x1q|2\n|x´x1|dxdx1.\nFor the upper bound on E/planckover2pi1\nHFpγ/planckover2pi1q, on the right hand side, the fourth negative term can\nbe dropped and the last exchange energy vanishes by the claim . Hence, by Weyl’s law\n(Proposition 4.3) again, it follows that\nE/planckover2pi1\nHFpγ/planckover2pi1q ďij\npa\n1` |p|2´1`Φ´µqQdqdp `µM\n`1\n2ijρQpxqρQpx1q\n|x´x1|dxdx1`o/planckover2pi1p1q “E/planckover2pi1\nVPpQq `o/planckover2pi1p1q.\nFinally, to complete the proof, we recall that EVP;min pMq “EVPpQq ă0 [14, Lemma\n2.2]. /square\nItisshownthatthequantumminimumenergyhasauniformnega tive upperbound. From\nthis, some additional useful information for quantum energ y minimizers can be obtained.SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 19\nLemma 5.2. ForMP p0,Mqmq, letQ/planckover2pi1be a minimizer for the quantum variational problem\nE/planckover2pi1\nHF;min pMqsatisfying the self-consistent equation Q/planckover2pi1“ /BDp?\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1ăµ/planckover2pi1q`R/planckover2pi1. Then,\nthe followings hold.\n(1) The quantum potential Φ/planckover2pi1is uniformly of long-range in the sense that for sufficiently\nsmallδą0, there exists Rδą0, independent of /planckover2pi1, such that Φ/planckover2pi1pxq ď ´δ\n2|x|for all\n|x| ě2Rδ, after a suitable translation.\n(2) The chemical potentials are uniformly away from zero, i.e .,limsup\n/planckover2pi1Ñ0µ/planckover2pi1ă0.\nProof.For (1), it suffices to show that for any small δą0, there exist Rδě1, independent\nof/planckover2pi1ą0, such that up to translation,\nliminf\n/planckover2pi1Ñ0}ρ/planckover2pi1\nQ/planckover2pi1}L1p|x|ďRδqěδ,\nsince it implies that\nΦ/planckover2pi1pxq “ ´żρ/planckover2pi1\nQ/planckover2pi1px1q\n|x´x1|dx1ď ´δ\n2|x|when |x| ě2Rδ.\nIndeed, for contradiction, we assume that for any Rą0, there exists a sequence t/planckover2pi1ju8\nj“1\nsuch that /planckover2pi1jÑ0 and supx1PR3}ρj}L1p|¨´x1|ďRqÑ0 whereρj“ρ/planckover2pi1j\nQ/planckover2pi1j. Then, by (3.6),\nijρjpxqρjpx1q\n|x´x1|dxdx1“ij\n|x´x1|ď1\nR`ij\n1\nRď|x´x1|ďR`ij\n|x´x1|ěRρjpxqρjpx1q\n|x´x1|dxdx1\nÀ1\nR2}ρj}L8pR3q}ρj}L1pR3q\n`R#\nsup\nx1PR3ż\n1\nRď|x´x1|ďRρjpxqdx+\n}ρj}L1pR3q`1\nR}ρj}2\nL1pR3q\nÀ1\nR2`Rsup\nx1PR3ż\n1\nRď|x´x1|ďRρjpxqdx`1\nR.\nThus, since Rą0 can be arbitrarily large, it follows that the potential ene rgy vanishes as\njÑ 8andE/planckover2pi1j\nHF;minpMq “E/planckover2pi1jpQ/planckover2pi1jq ěojp1q. However, it contradicts to Lemma 5.1.\nFor (2), we assume that µ/planckover2pi1jÑ0, and take arbitrarily small ǫą0. Then, for sufficiently\nlargej, we have\nM“Tr/planckover2pi1jpQ/planckover2pi1jq ěTr/planckover2pi1jp /BDp?\n1´/planckover2pi1j2∆´1`Φ/planckover2pi1j`X/planckover2pi1jď´ǫqq.\nThen, Weyl’s law (Proposition 4.3) and Lemma 5.2 (1) yield\nM졡/visualspace\npq,pq:a\n1` |p|2´1`Φ/planckover2pi1jpqq ă ´ǫ(ˇˇ`ojp1q\n“4π\n3ż/visualspace\n2p´ǫ´Φ/planckover2pi1jq`` p´ǫ´Φ/planckover2pi1jq2\n`(3\n2dx`ojp1q\ně4π\n3ż/visualspace\n2pδ\n2|x|´ǫq`` pδ\n2|x|´ǫq2\n`(3\n2dx`ojp1q.20 Y. HONG, S. JIN, AND J. SEOK\nHowever, sinceδ\n2|x|RL3\n2pR3q, taking smaller ǫą0, we can make the lower bound of the\nabove inequality arbitrarily large. It deduces a contradic tion. /square\nNow, we aim to obtain a lower bound for the quantum minimum ene rgy. For this, we\nswitch the role of the quantum and kinetic energy minimizati on problems in the proof of\nLemma 5.1. Using a quantum minimizer Q/planckover2pi1solving the self-consistent equation Q/planckover2pi1“/BDp?\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1ăµ/planckover2pi1q`R/planckover2pi1(see Section 3), we make an auxiliary kinetic distribution f unc-\ntion\nf/planckover2pi1“ /BDp?\n1`|p|2´1`Φ/planckover2pi1pqqă˜µ/planckover2pi1q, (5.1)\nwhere ˜µ/planckover2pi1is chosen so that f/planckover2pi1is admissible for the kinetic variational problem EVP;min pMq,\nthat is, Mpf/planckover2pi1q “M. Indeed, such ˜ µ/planckover2pi1ă0 always exists, because by Lemma 5.2 (1),\nMp /BDp?\n1`|p|2´1`Φ/planckover2pi1pqqăµqqincreases to infinity as µÒ0.\nIn addition, ˜ µ/planckover2pi1is sufficiently close to the quantum chemical potential µ/planckover2pi1.\nLemma 5.3. Forµ/planckover2pi1in(4.1)and˜µ/planckover2pi1in(5.1), we have ˜µ/planckover2pi1´µ/planckover2pi1Ñ0as/planckover2pi1Ñ0.\nProof.Applying Weyl’s law (Proposition 4.3) to Tr/planckover2pi1pQ/planckover2pi1q “M, we obtain\nMp /BDp?\n1`|p|2´1`Φ/planckover2pi1pqqăµ/planckover2pi1qq “Mp /BDp?\n1`|p|2´1`Φ/planckover2pi1pqqă˜µ/planckover2pi1qq `o/planckover2pi1p1q.\nConsequently, since\nMp /BDp?\n1`|p|2´1`Φ/planckover2pi1pqqăµ`ǫqq ´Mp /BDp?\n1`|p|2´1`Φ/planckover2pi1pqqăµqq\něż\ntx:Φ/planckover2pi1ďµu/visualspace\n2pµ`ǫ´Φ/planckover2pi1q ` pµ´Φ/planckover2pi1q2(3\n2´/visualspace\n2pµ´Φ/planckover2pi1q ` pµ´Φ/planckover2pi1q2(3\n2dx\ně2?\n2ǫ3\n2|tx: Φ/planckover2pi1ďµu|\nfor 0 ăǫă ´µ, we deduce ˜ µ/planckover2pi1“µ/planckover2pi1`o/planckover2pi1p1q. /square\nThe following is the reverse inequality for the quantum mini mum energy. We also prove\nthat the potential function of f/planckover2pi1approximates the quantum potential Φ /planckover2pi1. Indeed, this\nadditional information will be useful to complete the proof of the main theorem.\nLemma 5.4 (Asymptoticlowerboundonthequantumminimumenergy) .LetMP p0,Mqmq.\nThen,\nEVP;min pMq ďEVPpf/planckover2pi1q ďliminf\n/planckover2pi1Ñ0ˆ\nE/planckover2pi1\nHF;min pMq ´1\n8π››∇`\nΦ/planckover2pi1´ p´1\n|x|˚ρf/planckover2pi1q˘››2\nL2pR3q˙\n,\nwhereΦ/planckover2pi1“Φ/planckover2pi1\nQ/planckover2pi1andQ/planckover2pi1is a minimizer for the quantum variational problem E/planckover2pi1\nHF;min pMq\nandf/planckover2pi1is the distribution function given by (5.1).SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 21\nProof.The kinetic distribution f/planckover2pi1has the energy\nEVPpf/planckover2pi1q “ij`a\n1` |p|2´1`Φ/planckover2pi1pqq ´µ/planckover2pi1˘\nf/planckover2pi1dqdp `µ/planckover2pi1M\n`1\n2ijρ/planckover2pi1\nQ/planckover2pi1pxqρ/planckover2pi1\nQ/planckover2pi1px1q\n|x´x1|dxdx1´1\n2ijpρf/planckover2pi1´ρ/planckover2pi1\nQ/planckover2pi1qpxqpρf/planckover2pi1´ρ/planckover2pi1\nQ/planckover2pi1qpx1q\n|x´x1|dxdx1.\nWe observe that by Lemma 5.3,\nij`a\n1` |p|2´1`Φ/planckover2pi1pqq ´µ/planckover2pi1˘\nf/planckover2pi1dqdp\n“ij`a\n1` |p|2´1`Φ/planckover2pi1pqq ´µ/planckover2pi1˘/BDp?\n1`|p|2´1`Φ/planckover2pi1pqqăµ/planckover2pi1qdqdp `o/planckover2pi1p1q.\nThen, it follows from Weyl’s law (Proposition 4.3) that\nEVPpf/planckover2pi1q “Tr/planckover2pi1`\npa\n1´/planckover2pi12∆´1`Φ/planckover2pi1`X/planckover2pi1´µ/planckover2pi1qQ/planckover2pi1˘\n`µ/planckover2pi1M\n`1\n2ijρ/planckover2pi1\nQ/planckover2pi1pxqρ/planckover2pi1\nQ/planckover2pi1px1q\n|x´x1|dxdx1´1\n8π››∇`\nΦ/planckover2pi1´ p´1\n|x|˚ρf/planckover2pi1q˘››2\nL2pR3q`o/planckover2pi1p1q.\n“E/planckover2pi1\nHFpQ/planckover2pi1q ´1\n8π››∇`\nΦ/planckover2pi1´ p´1\n|x|˚ρf/planckover2pi1q˘››2\nL2pR3q`o/planckover2pi1p1q.\nBy definition, EVP;min pMq ďEVPpf/planckover2pi1qandE/planckover2pi1\nHFpQ/planckover2pi1q “E/planckover2pi1\nHF;min pMq. Thus, we complete the\nproof. /square\nNow, we are ready to prove the main theorem.\nProof of Theorem 1.5. Combining Lemma 5.1 and Lemma 5.4, we obtain that\nE/planckover2pi1\nHF;min pMq ďEVP;min pMq `o/planckover2pi1p1q ďEVPpf/planckover2pi1q `o/planckover2pi1p1q\n“E/planckover2pi1\nHF;min pMq ´1\n8π}∇pΦ/planckover2pi1´ p´1\n|x|˚ρf/planckover2pi1qq}2\nL2pR3q`o/planckover2pi1p1q.\nTherefore, it follows that\n}∇pΦ/planckover2pi1´ p´1\n|x|˚ρf/planckover2pi1qq}L2pR3qÑ0 (5.2)\nand for any sequence t/planckover2pi1ju8\nj“1such that /planckover2pi1jÑ0,tf/planckover2pi1ju8\nj“1is a minimizing sequence for\nthe variational problem EVP;min pMq. Hence, by Theorem 1.3. piq, we have the potential\nconvergence }∇pp´1\n|x|˚ρf/planckover2pi1jq ´Φq}L2pR3qÑ0 up to a translation. Therefore, together with\n(5.2), we conclude that }∇pΦ/planckover2pi1j´Φq}L2pR3qÑ0. Combining this with Lemma 3.1, the\nSobolev embedding theorem and the uniform estimate (3.6), w e have }ΦQ/planckover2pi1´ΦQ}CpR3qÑ0\nas/planckover2pi1Ñ0. Then, by Lemma 3.1 and the interpolation inequality,\n}ΦQ/planckover2pi1´ΦQ}C1pR3qďǫ}ΦQ/planckover2pi1´ΦQ}C1,αpR3q`Cǫ}ΦQ/planckover2pi1´ΦQ}CpR3qďOpǫq`Cǫ}ΦQ/planckover2pi1´ΦQ}CpR3q,\nwhereαP p0,1q. Thus, sending /planckover2pi1Ñ0 and then ǫÑ0, we prove Therem 1.5. /square22 Y. HONG, S. JIN, AND J. SEOK\nAppendix A.Elliptic regularity for the equations 3.5\nIn this appendix, we prove that for fixed /planckover2pi1ą0, the solutions φ/planckover2pi1\njto the system of elliptic\nequations (3.5) are smooth, even though their high Sobolev n orms may depends on /planckover2pi1.\nBy simple scaling, we may assume /planckover2pi1“1. Then, it is sufficient to prove the following\nproposition.\nProposition A.1. LetKbe a positive integer and tλjuK\nj“1,tνjuK\nj“1be sequences of positive\nreal numbers. Let tφjuK\nj“1PH1{2pR3qbe a solution of the system of equations\n`?\n1´∆´1`νj˘\nφj“Njpφ1,...,φ Kq, j “1,...,K,\nwhere\nNjpφ1,...,φ Kq:“Kÿ\nk“1λkˆ1\n|x|˚ |φk|2˙\nφj´Kÿ\nk“1λkˆ1\n|x|˚ pφjφkq˙\nφk.\nThen, for each j“1,...,K,φjbelongs toHspR3qfor everysě1{2.\nWe need the following lemma for estimating the nonlinear ter mNjpφ1,...,φ Kq. We refer\nto [2] for the proof.\nLemma A.2 (Trilinear estimate for theHartree nonlinearity) .For anysě1{2, there exists\na positive constant Cssatisfying\n›››´1\n|x|˚ pv1v2q¯\nv3›››\nHspR3qďCs3ź\nj“1}vj}HspR3q,@v1,v2,v3PHspR3q.\nProof of Proposition A.1. We claim that if φjPHsfor everyj“1,...,K, thenφjPHs`1\nfor everyj“1,...,K. Since tφjuK\nj“1PH1{2, we are done if the claim is shown to be true.\nSuppose that φjPHsfor everyj“1,...,Kand somesě1{2. We define ν:“\nmintν1,...,νKu. It is easy to see that there is a small constant cą0 such that\ncp|ξ| `1q ďa\n1` |ξ|2´1`ν.\nThen, by combining this and Lemma A.2, we see that for each j“1,...,K\n}φj}Hs`1ď1\nc}p?\n1´∆´1`νjqφj}Hs“1\nc}Njpφ1,...,φ Kq}Hsď2Cs\ncKÿ\nk“1λk}φk}2\nHs}φj}Hs,\nwhich proves the claim.\n/square\nReferences\n[1] S. Chandrasekhar, The maximum mass of ideal white dwarfs , Astrophys. J., 74(1931), 81–82.\n[2] W. Choi, Y. Hong and J. Seok, Optimal convergence rate and regularity of nonrelativisti c limit for the\nnonlinear pseudo-relativistic equations , J. Funct. Anal. 274(2018), no.3, 695–722.\n[3] W. Choi, Y. Hong and J. Seok, Semi-classical limit of quantum free energy minimizers for the gravita-\ntional Hartree equation , Arch. Ration. Mech. Anal. 239(2021), no. 2, 783–829.SEMICLASSICAL EQUIVALENCE OF WHITE DWARF MODELS 23\n[4] I. Daubechies, An uncertainty principle for fermions with generalized kin etic energy , Comm. Math.\nPhys.90(1983), no. 4, 511–520.\n[5] W.D. Evans, R.T. Lewis, H. Siedentop, J.Ph. Solovej, Counting eigenvalues using coherent states with\nan application to Dirac and Schr¨ odinger operators in the se mi-classical limit, Ark. Mat. 34(2), 265–283\n(1996).\n[6] R. Frank, Cwikel’s theorem and the CLR inequality , J. Spectr. Theory 4(2014), no. 1, 1–21.\n[7] R. Frank, Eigenvalue bounds for the fractional Laplacian: a review. R ecent developments in nonlocal\ntheory,210–235, DeGruyter, Berlin, 2018.\n[8] J. Fr¨ ohlich and E. Lenzmann, Dynamical collapse of white dwarfs in Hartree- and Hartree- Fock theory ,\nComm. Math. Phys. 274(2007), no. 3, 737–750.\n[9] R.T. Glassey andJ. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Comm.\nMath. Phys. 101, 459–473 (1985).\n[10] L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, vol. 249, 2nd edn. Springer,\nNew York (2008)\n[11] H.Hadzic andG. Rein, Global existence and nonlinear stability for the relativis tic Vlasov-Poisson system\nin the gravitational case , Indiana Univ. Math. J. 56, 2453–2488 (2007).\n[12] C. Hainzl and B. Schlein, Stellar collapse in the time dependent Hartree-Fock approx imation, Commun.\nMath. Phys. 287(2009), pp. 705–717.\n[13] C. Hainzl, E. Lenzmann, M. Lewin andB. Schlein, On Blowup for Time-Dependent Generalized Hartree-\nFock Equations , Ann. Henri Poincar´ e 111023–1052(2010).\n[14] J. Jang and J. Seok, Kinetic description of stable white dwarfs , Kinet. Relat. Models 15(2022), no. 4,\n605–620.\n[15] M. Lemou, F. M´ ehats and P. Rapha¨ el, Stable ground states for the relativistic gravitational Vl asov-\nPoisson system , Comm. Partial Diff. Eq. 34(7), 703–721 (2009)\n[16] E. Lenzmann and M. Lewin, Minimizers for the Hartree-Fock-Bogoliubov theory of neut ron stars and\nwhite dwarfs , Duke Math. J. 152(2010), no. 2, 257–315.\n[17] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Springer (2001).\n[18] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum me-\nchanics, Comm. Math. Phys. 53(1987),147–174.\n[19] E.H. Lieb and H.-T. Yau, A rigorous examination of the Chandrasekhar theory of stell ar collapse ,\nAstrophys. Jour. 323(1987), 140–144.\n[20] T. Luo, J. Smoller, Nonlinear dynamical stability of Newtonian rotating and no n-rotating white dwarfs\nand rotating supermassive stars, Comm. Math. Phys. 284(2008), no. 2, 425–457\n[21] M. Reed and B. Simon, Methods of Modern Mathematical Physics: Analysis of Operat ors. Vol. 4 ,\nAcademic Press, New York (1978).\n[22] M. Rumin, Spectral density and Sobolev inequalities for pure and mixe d states, Geom. Funct. Anal. 20\n(2010), no. 3, 817–844.\n[23] M. Rumin, Balanced distribution-energy inequalities and related en tropy bounds , Duke Math. J. 160\n(2011), no. 3, 567–597.\n[24] J. Sabin, Littlewood-Paley decomposition of operator densities and application to a new proof of the\nLieb-Thirring inequality , Math. Phys. Anal. Geom. 19(2016), no. 2, Art. 11, 11 pp.\n[25] B. Simon, Trace ideals and their applications , London Math. Soc. Lecture Note Ser., 35Cambridge\nUniversity Press, Cambridge-New York, 1979, viii+134 pp.24 Y. HONG, S. JIN, AND J. SEOK\nDepartment of Mathematics, Chung-Ang University, Seoul 06 974, Korea\nEmail address :yhhong@cau.ac.kr\nDepartment of Mathematics Education, Chungbuk National Un iversity, Cheongju 28644,\nKorea\nEmail address :sangdonjin@cbnu.ac.kr\nDepartment of Mathematics Education, Seoul National Unive rsity, Seoul 08826, Korea\nEmail address :jmseok@snu.ac.kr"    },    {        "title": "2401.12498v1.Understanding_Cellular_Noise_with_Optical_Perturbation_and_Deep_Learning.pdf",        "content": "Understanding Cellular Noise with Optical Perturbation and Deep\nLearning\nChuanbo Liu1, Yu Fu1, 2, Lu Lin1, 2, Elliot L. Elson3, †, and Jin Wang4, ‡\n1State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry,\nChinese Academy of Sciences, Changchun, Jilin, 130022, P.R. China.\n2University of Chinese Academy of Sciences, Beijing, 100000, P.R. China.\n3Department of Biochemistry and Molecular Biophysics, Washington University in St. Louis, Saint\nLouis, MO 63130, United States.\n4Department of Chemistry and of Physics and Astronomy, State University of New York, Stony\nBrook, New York, 11794-3400, United States.\n†elson@wustl.edu\n‡jin.wang.1@stonybrook.edu\n1arXiv:2401.12498v1  [q-bio.MN]  23 Jan 2024Abstract\nNoise plays a crucial role in the regulation of cellular and organismal function and behavior. Exploring\nnoise’s impact is key to understanding fundamental biological processes, such as gene expression, signal\ntransduction, and the mechanisms of development and evolution. Currently, a comprehensive method to\nquantify dynamical behavior of cellular noise within these biochemical systems is lacking. In this study, we\nintroduce an optically-controlled perturbation system utilizing the light-sensitive Phytochrome B (PhyB)\nfrom Arabidopsis thaliana , which enables precise noise modulation with high spatial-temporal resolution.\nOur system exhibits exceptional sensitivity to light, reacting consistently to pulsed light signals, distin-\nguishing it from other photoreceptor-based promoter systems that respond to a single light wavelength. To\ncharacterize our system, we developed a stochastic model for phytochromes that accounts for photoacti-\nvation/deactivation, thermal reversion, and the dynamics of the light-activated gene promoter system. To\nprecisely control our system, we determined the rate constants for this model using an omniscient deep neu-\nral network that can directly map rate constant combinations to time-dependent state joint distributions. By\nadjusting the activation rates through light intensity and degradation rates via N-terminal mutagenesis, we\nillustrate that out optical-controlled perturbation can effectively modulate molecular expression level as well\nas noise. Our results highlight the potential of employing an optically-controlled gene perturbation system\nas a noise-controlled stimulus source. This approach, when combined with the analytical capabilities of a\nsophisticated deep neural network, enables the accurate estimation of rate constants from observational data\nin a broad range of biochemical reaction networks.\n2Introduction\nIn fully differentiated cells under constant environmental conditions, the morphology, function, and number\nof cell-specific organelles, as well as the capacity for cell-to-cell communication, the potential for interaction\nwith extracellular molecules, and the function-specific metabolic adaptations, among other characteristics, ap-\npear unchanged from a macroscopic perspective, suggesting a stable cell state. On the other hand, cell state\ndiversity is frequently observed through single-cell and single-molecule techniques1. Given that the features\nof a cell are defined by its molecular composition, it is reasonable to postulate that the number of molecules\nof proteins, including enzymes, structural proteins, and signaling molecules, fluctuate around a constant mean\nin a stable cell state. These fluctuations are attributed to both intrinsic and extrinsic sources of noise2. It is\ngenerally assumed that the magnitude of noise remains consistent over time in a constant environment, leading\nto a steady-state distribution. Furthermore, the distinct difference of living cell to a thermal equilibrium system\nis the exchange of both energy and materials with the environment at a constant rate apart from stochastic fluc-\ntuations to maintain the steady-state. From a thermodynamic perspective, a living cell in this circumstance is\ncharacterized as a non-equilibrium steady-state (NESS)3.\nThe ability to reach and maintain NESS hinges critically on the chemical and thermal equilibrium of the\nbiochemical reactions within the cell, which defines the phenotype of the cell. Conversely, a cell’s response\nto external stimuli is contingent upon the dynamics of signaling components, with encodes vital biological\ninformation for the cell4,5. Consequently, the cellular response to external stimuli is regulated not solely by\nthe steady-state concentrations of various species but also by the dynamics, which are characterized by both\nthe instantaneous numbers of molecules and their spatial distributions. Thus, to thoroughly understand the\nbehavior of biochemical systems, it is imperative to quantify not just the mean populations of molecules but,\nmore critically, the temporal variations and spatial heterogeneity in the distributions of cellular states.\nAlthough significant progress has been made in understanding the noise of cells in slowly-changing envi-\nronments6–9, the dynamics induced by rapidly-evolving or spatially-specific perturbations remain poorly under-\nstood. One of the primary challenges is the development of efficient labeling techniques for molecular species\nthat enable high-throughput tracking and precise quantification. A further, and perhaps more critical, challenge\nis the absence of high-resolution spatio-temporal modulation methods capable of generating or simulating rapid\nspatio-temporal stimuli for cellular reaction networks. The first challenge has been largely addressed through\nthe use of efficient single-molecule hybridization/tagging techniques or deep sequencing1,10,11. While the sec-\nond challenge may be surmountable by harnessing optogenetic technologies. Over recent decades, optogenetics\nhas emerged as a pivotal technique for exerting precise spatio-temporal control in cells, with applications in neu-\n3roscience, cell physiology, and cardiology12–16. Employing a synergistic approach of genetic engineering and\nbioorthogonal light, which is independent of the native cellular chemical reactions, both natural and synthetic\nphotoreceptors can be utilized to modulate the activity of targeted cells within a highly heterogeneous tissue\nmatrix. The specificity of expression and subcellular localization within the host organism is facilitated through\nthe incorporation of regulatory elements such as promoters, enhancers, and precise targeting sequences into the\nphotoreceptor-encoding DNA constructs. Consequently, cells that are intrinsically non-responsive to light can\nbe rendered photosensitive, thereby becoming amenable to spatially and temporally precise control via targeted\nillumination.\nHowever, mapping the dynamic properties of biological systems remains challenging, despite the avail-\nability of precise spatial-temporal perturbation control. From a modeling perspective, constructing a detailed\nmodel for gene expression regulated by light is arduous. The challenge arises because the rate constants are\nintertwined with cell-state variables, such as the populations of RNA polymerase (RNAP) and ribosomes17,\nthe levels of transcription factors and micro-RNAs18, the binding affinities of genes with RNAP and tran-\nscription factors19, gene copy numbers20, the phase of the cell cycle21, nutrient density22, and chromosomal\nconformations23, among others. These latent variables, typically unmeasured in biochemical experiments, are\nconsidered intrinsic noise, in contrast to extrinsic noise, which is induced by the influence of changing environ-\nments. The interaction between the control protocol and cell states, modulated by these latent variables, adds\nfurther complexity to the modeling of the system.\nOn the other hand, gene expression is an intricate complex chemical reaction system involving the binding\nof transcription factors to DNA regulatory sequences, recruitment of RNA polymerase, synthesis of nascent\nmRNAs, modification of these mRNAs, transport of mRNAs from the nucleus to the cytoplasm, and ulti-\nmately, the translation of mRNAs in ribosomes24. Each of these processes comprises numerous elementary\nsteps, potentially ranging into the tens or hundreds25–28. Many of these elementary steps remain poorly un-\nderstood, either due to the limitations of current technology rendering them invisible or because they occur at\nextremely rapid rates29,30. Furthermore, acquiring accurate reaction rates for these elementary steps is particu-\nlarly challenging. Owing to these complexities, coarse-grained models have been proposed to simulate the gene\nexpression process. Such models employ thermodynamic principles, Boolean logic, and differential equations\nto deterministically or stochastically represent gene expression, each with varying degrees of computational ef-\nficiency and precision31,32. In these models, gene expression is commonly conceptualized as either a one-state\nor two-state chemical reaction system. Transitions between states, as well as transcription and translation, are\noften abstracted into effective complex reactions, each encompassing multiple elementary steps33. While ordi-\nnary differential equations can provide reasonable estimates of mean population levels, a stochastic framework\n4offers a more comprehensive approach. This methodology not only yields insights into the statistical properties\nof the system but also generates the population distribution, which is crucial for understanding the variability\ninherent in gene expression systems34,35.\nFurthermore, since stochastic modeling is indispensable, another emerging challenge is the inferencing of\nthe rate constants, especially for systems with many species where the state space expands exponentially. This\nvast state space precludes the possibility of ergodic integration of the system. To address this challenge, Monte\nCarlo methods are often utilized due to their computational complexity, which is not contingent on the dimen-\nsionality of the state space36–38. Despite this, the computational burden remains substantial. The timescale\nfor an exact one-step simulation of biochemical reaction networks is typically on the order of milliseconds,\nwhereas meaningful phenotypic changes occur over hours or days. Consequently, simulating a single trajectory\nrequires the accumulation of millions of Monte Carlo steps. In addition, constructing the posterior distribution\nnecessitates sampling millions to trillions of trajectories, each corresponding to different sets of rate constants.\nEven with advanced algorithms and hardware that facilitate efficient parallel Monte Carlo sampling, inferring\nrate constants remains a formidable task, particularly in biological systems with a large number of species.\nIn this study, we developed an optically-controlled gene expression system utilizing the light-sensitive Phy-\ntochrome B (PhyB) from Arabidopsis thaliana , engineered for enhanced spatial-temporal precision to probe the\nintrinsic noise within cellular environments. Our system exhibits a continuous response to pulsed light signals,\ndistinguishing it from previously described blue light-regulated gene expression systems39,40. To broaden the\napplicability of our system across various biochemical reaction networks, we constructed a stochastic model\nfor phytochromes and an integrated model for the entire gene expression system. These models incorporate the\nprocesses of photoactivation, deactivation, thermal reversion, and the dynamic behavior of the light-activated\ngene promoter system. For meticulous control of the system, we determined the rate constants of the integrated\nstochastic model using an omniscient deep neural network. This network is capable of mapping the combi-\nnations of rate constants directly to time-dependent joint state distributions. By modulating activation rates\nthrough alterations in light intensity and adjusting degradation rates via N-terminal mutagenesis, we demon-\nstrate that the optically-controlled perturbations can precisely modulate molecular noise. Our findings high-\nlight the potential of employing an optically-controlled gene perturbation system as a noise-managed stimulus\nsource. Furthermore, they suggest the feasibility of inferring rate constants by monitoring system behavior,\nfacilitated by the predictive capabilities of the deep neural network.\n5Results\nPerturbing gene expression dynamics with light\nIn order to achieve high spatial-temporal precision in tuning gene expression, coupled with quantitative\ncontrol, we have designed an optically-controlled perturbation system. This system is based on the light-\nsensitive Phytochrome B (PhyB) from Arabidopsis thaliana , drawing inspiration from the work of Shimizu-\nSato et al.41. The full-length PhyB photosensory module (PSM) is comprised of four domains: the N-terminal\nextension (NTE), the Period/ARNT/Single-minded (PAS) domain (nPAS), the cyclic GMP-regulated cyclic\nnucleotide phosphodiesterase/Adenylate cyclase/FhlA (GAF) domain that binds the tetrapyrrole chromophore\n(PΦB), and the phytochrome-specific domain (PHY). At the interface between nPAS and GAF, there exists\nan intricate figure-eight knot, referred to as the “light-sensing knot,” which is intimately associated with the\nchromophore and plays a critical role in signal transduction. The PSM’s structure is reinforced by a light-\nsensing knot loop (KL) and a hairpin (HP), which interconnect the NTE, GAF, and PHY domains. In our study,\nwe utilized only the N-terminal fragment (residues 1-621) of PhyB, encompassing the essential photosensory\nmodule (PSM) while lacking the dimerization/nuclear import module (DNM; also known as the output module\nin its prokaryotic counterparts), which facilitates dimerization and nuclear translocation of the Pfr form42. In\ncontrast to the full-length PhyB in planta , the PhyB N-terminal (PhyB(NT)) is monomeric in vitro, devoid of\nthe DNM43,44. For the construction of a light-sensitive, switchable molecular system, we selected PIF3 as the\nbinding partner for PhyB(NT). PIF3 comprises three domains: the Active Phytochrome B-binding (APB) motif,\nthe Active phytochrome A-binding (APA) motif, and the basic helix-loop-helix (bHLH) transcription factors.\nThe APB motif preferentially interacts with the light-sensing knot within the N-terminal photosensory module\nof photoactivated PhyB. While PIF3 is capable of forming homodimers via its bHLH domain in its full-length\nstate45, its interaction with monomeric PhyB(NT) promotes a monomeric state in PIF3.\nFor the additional components of the optically-controlled perturbation system, instead of employing the\nGal4 DNA-binding domain (Gal4-DBD) and the Gal4 activation domain (Gal4-AD) fusions for the functional\nmodules, we utilized the LexA DNA-binding domain (LexADBD) and the VP16 activation domain (VP16AD).\nThese combinations are commonly employed in yeast two-hybrid assays. We engineered two fusion proteins,\nLexADBD-PhyB(NT) and PIF3-VP16AD, to regulate gene expression via light induction. Both components\nwere integrated into the yeast genome and expressed under the control of the constitutive promoter PADH1 .\nAfter translation, LexADBD-PhyB(NT) is synthesized in the inactive Pr state and preferentially accumulates\nin the cytosol in the absence of light. Consequently, in dark conditions, the photoreceptor accumulates in this\ninactive state, ready to be activated upon light exposure. For the reporter gene construction, LexA binding sites\n6were fused to a segment of the GAL1 promoter encompassing the TATA box. Thus, the PhyB(NT)/PIF3 in-\nteraction modulates expression from a synthetic promoter comprising four LexA binding sequences, with each\nsequence capable of binding two LexA molecules. Upon light activation, the PSM domain of PhyB(NT) under-\ngoes a conformational change and binds to PIF3-VP16AD. This interaction facilitates the binding of VP16AD\nto the TATA box, triggering the expression of a fluorescent reporter for observation. The degree of photocon-\nversion from Pr to Pfr correlates directly with the number of red photons administered, which is expected to\nprovide precise dosage control in a highly predictable and reproducible manner46. Light also enables high\nspatial-temporal precision in control. Such precise regulation of gene expression is crucial for perturbing cell\nstate dynamics. To modulate the stability of the fluorescent reporter protein for rapid degradation and enhanced\nsignal sensitivity, we employed a ubiquitin fusion technique. This method allows for the alteration of protein\ndegradation rates by revealing a destabilizing residue following the co-translational removal of ubiquitin47. A\nschematic representation of the optically-controlled gene expression system is depicted in Fig. 1(A).\nThe optically-controlled system is expected to respond rapidly in the presence of red light and to be de-\nactivated by far-red light48. Owing to the low basal transcription rate, minimal activation is anticipated in the\nabsence of light41. We monitored the dynamics of reporter expression using a microfluidic chip designed for\nthe long-term incubation of thousands of cells (see Methods for a detailed description of the microfluidic chip\nfabrication). A representative experimental result is depicted in Fig. 1 (B). We observed that pulsed light acti-\nvation induces sustained protein expression, as illustrated in Fig. 1 (C). This behavior contrasts with the blue\nlight-regulated gene expression system, which demonstrated consistent protein expression under continuous\nlight activation and rapid deactivation in the absence of light40,49. Moreover, this system exhibits significant\nfluctuations in gene expression compared to chemically induced gene expression systems. These observa-\ntions suggest that protein expression in our system can be finely tuned by modulating the reaction rates of the\nlight-responsive components and that the underlying mechanisms of light response in our optical perturbation\nsystem are fundamentally different from those in the blue light-regulated gene expression system. Prompted\nby these findings, we proceeded to investigate the photoconversion and thermal reversion processes of PhyB in\nthe presence of PIFs.\nModeling the photoconversion and thermal reversion process of phytochromes\nMost land plants, particularly Arabidopsis , possess several phytochromes, which are prevalent in angiosperms\nand can be categorized into three primary types: Phytochrome A (PhyA), PhyB, and PhyC50. These phy-\ntochromes are bilin-binding photosensory receptors that undergo light-induced conformational changes. These\nchanges facilitate interactions with signaling partners, especially transcription factors or the proteins that reg-\n7ulate them, leading to extensive transcriptional reprogramming. Consequently, phytochromes orchestrate a va-\nriety of physiological processes, including seed germination, chloroplast development, seedling de-etiolation,\nshade avoidance, photoperiodic regulation of flowering time, and ultimately, senescence51. Among these types,\nPhyA and PhyB have similar absorption spectra, yet they exhibit significant differences in their action spectra\nfor physiological responses. PhyB is primarily active under red light, while PhyA functions effectively under\nfar-red light, where PhyB’s activity is suppressed. This dichotomy has spurred extensive research into their\ndistinct roles.\nIn the case of PhyB, bilin phytochromobilin (P ΦB) serves as the chromophore, absorbing energy from\nred light and transferring it to the protein’s structure. During the photoconversion process, light-induced bilin\nisomerization prompts a conformational rearrangement, converting the dark-adapted, biologically inactive Pr\nstate into a metastable, far-red light-absorbing Pfr state, which is biologically active. The partial overlap of\ntheir absorption spectra leads to a dynamic photo-equilibrium between the Pfr and Pr forms, which varies\nwith the light wavelength. Pfr is produced exclusively through photoconversion and can revert to the Pr form\neither through the absorption of far-red light or via spontaneous relaxation—a temperature-dependent, non-\nphotochemical process known as thermal reversion. Thus, the interplay between photoconversion and dark\nreversion dictates the steady-state levels of the active Pfr conformation, enabling the dynamic sensing of light\nquality and quantity. Consequently, signal transduction is modulated by a combination of light intensity, wave-\nlength, and temperature, in accordance with plant requirements52. Upon exposure to inductive wavelengths,\nPhyB adopts a functional dimeric conformation, enabling photosensitive responses and resulting in three po-\ntential phytochrome species: Pr-Pr, Pfr-Pr, and Pfr-Pfr42,44,53. The intracellular protein dynamics among these\nPhyB dimer species dictate their activity potential, with only the nuclear Pfr-Pfr homodimer initiating PhyB\nresponses44.\nAnalysis of in vitro photoconversion induced by red light in Arabidopsis revealed a monomeric process\nfollowing single-exponential kinetics54. In contrast, the Pfr to Pr photoconversion under far-red light illumi-\nnation is more accurately described by a bi-exponential kinetic fit. Additionally, at room temperature (22◦C),\ntotal Pfr reverted entirely to the inactive form (Pr) following bi-exponential decay kinetics in darkness43,54.\nThe bi-exponential kinetics observed in both photoconversion and thermal reversion may result from sequential\nreactions, the presence of multiple conformers, or a combination of these scenarios. Previous studies have iden-\ntified intermediates of phytochromes on the millisecond timescale55,56, yet no intermediates have been detected\non the timescale of hours during the thermal reversion of phytochromes. Consequently, we propose that the\nbi-exponential decay arises from two distinct forms of Pfr within the molecular population42,43,52,57–60. On the\nother hand, thermal reversion is also modulated by protein interactions. The Pfr-Pfr homodimers exhibit greater\n8stability than the Pfr-Pr heterodimers; the additional subunit interactions in Pfr-Pfr homodimers confer a stabi-\nlization of approximately 4 kcal/mol61. Photoconversion of only one subunit in the phytochrome dimer from Pr\nto the metastable Pfr conformation could lead to a reduced stability of the Pfr-Pr heterodimer, thereby favoring\na light-independent back reaction to the Pr-Pr homodimer. This phenomenon results in a more pronounced\ndecline of PhyB responses at longer wavelengths. The rate of thermal reversion for the Pfr-Pr heterodimer is\napproximately 100-fold faster than that of the active Pfr-Pfr homodimer, enabling PhyB to perceive temperature\nfluctuations both diurnally and nocturnally44. Studies of Arabidopsis PhyB expressed in yeast have revealed\nthat the dark reversion rates of Pfr-Pfr homodimers are an order of magnitude lower than those of Pfr-Pr het-\nerodimers61. The interaction of PhyB with ARR4 ( Arabidopsis Response Regulator 4) diminishes the dark\nreversion of the Pfr conformer, thus modulating light sensitivity in Arabidopsis62. Furthermore, phytochromes\ndemonstrate prolonged persistence of the Pfr state when associated with nuclear bodies (NBs), attributable to\nthe suppression of thermal reversion in darkness44. The binding of PhyB(NT) to PIF can also inhibit the ther-\nmal reversion process43,52. Collectively, these findings suggest that the photoconversion rate of phytochrome\nfrom Pfr to Pr is contingent upon the wavelength and fluence rate of light, while the thermal reversion rate is\ninfluenced by temperature and the presence of binding partners.\nTo consolidate the findings of prior studies, we introduce a streamlined model to encapsulate the photocon-\nversion and thermal reversion mechanisms of phytochromes, as depicted in Fig. 2 (A). This model is akin to\nthe dual-pathway framework suggested by63. It posits the existence of two photo-activated conformers, Pfr 1\nand Pfr 2, inferred from the bi-exponential kinetics characteristic of the thermal reversion process54. The model\nallows for the PIF binding partner to interact with both the Pfr and Pr states, with the latter also capable of\nbasal affinity binding to Pa64. Photoconversions are postulated to occur in the dimeric state, with rate constants\nmirroring those of the monomeric state. To determine these rate constants, we devised a score-based bootstrap\ninference algorithm that leverages in vitro equilibrium data from photoconversion and thermal reversion assays,\nas illustrated in Fig. 2 (B, C). The score is defined by the equation\nS=X\ntZ\n∥[xxx(t)]−yyy(t)∥2d[xxx(t)], (1)\nrepresenting the mean L2-norm discrepancy between the model’s theoretical predictions and the empirical data\nacross all observable time points. For each set of experimental conditions, we compute the score using 10\nsimulated trajectories, each generated with the Gillespie exact kinetic sampling algorithm37. The inference\nalgorithm iteratively adjusts the rate constants to minimize the score via the steepest descent approach. The\nphotoconversion rates are influenced by the light’s wavelength and fluence rate; red light catalyzes both the Pr\n9→Pfr and Pfr →Pr transitions in phytochromes. For the sake of simplicity, we characterize photoconversion\nusing effective rates corresponding to specific light wavelengths. That is, when modeling the Pr →Pfr (Pfr\n→Pr) photoconversion, the rate for the Pfr →Pr (Pr →Pfr) transition is set to zero. Additionally, while\nArabidopsis PhyB(NT) exhibits single-exponential kinetics under red light, we assign different photoconversion\nrates to Pr →Pfr1and Pr →Pfr2to achieve a more accurate model, as evidenced by a reduced score.\nThe chemical reactions and rate constants inferred are detailed in the SI Appendix, Table S2. The photo-\nconversion of Pfr 1→Pr occurs approximately 3-fold faster than that of Pfr 2→Pr, while the thermal reversion\nof Pfr 2→Pr is roughly two orders of magnitude slower than that of Pfr 1→Pr. Consequently, the Pfr 2state\nis a photo-activated state that forms slowly but exhibits greater stability compared to Pfr 1. Upon binding to\nPIFs, the thermal reversion rate is reduced by an order of magnitude for both Pfr 1→Pr and Pfr 2→Pr, with\nthe reversion of Pfr 2→Pr remaining two orders of magnitude slower than that of Pfr 1→Pr. The dissociation\nconstants KDfor Pfr 1Pa and Pfr 2Pa are 0.56 and 23.81, respectively, signifying that Pfr 1has a stronger affinity\nfor PIFs than Pfr 2. For PrPa, Kd=6.23×104, which is approximately four orders of magnitude greater than\nthat for Pfr 1and two orders of magnitude greater than that for Pfr 2, aligning with the findings of previous stud-\nies64. Our analysis suggests a model with two photo-activated conformers that can account for the observed\nphotoconversion and thermal reversion behaviors of phytochromes both in isolation and when associated with\nPIFs. The photoconversion rates of Pr →Pfr1and Pr →Pfr2are comparable. However, Pfr 2demonstrates\nenhanced stability in terms of both photoconversion and thermal reversion. The binding of PIFs further stabi-\nlizes the photo-activated state, leading to a reduced rate of deactivation that mirrors the behavior observed in\nplanta42,44,61,62,65,66. As depicted in Fig. 2 (C), our findings surpass those of the prior study by Smith et al.,\nwhich did not accurately predict the thermal reversion process at elevated PIF concentrations43. Moreover, our\nmodel presupposes the existence of only two activated conformers, obviating the need to postulate a secondary\nactivated state for phytochromes. Collectively, these results affirm that our model effectively captures the com-\nprehensive behavior of photoconversion and thermal reversion processes in phytochromes, both independently\nand in the presence of PIFs. We have therefore employed the framework of this streamlined model to analyze\nthe optically controlled gene expression system.\nStochastic modeling and parameter screening with omniscient neural network\nWe monitor the real-time kinetic signal to conclude that the system has a distinct dynamical behavior\ncompared to those systems activated by blue light40,49. In the blue light-activated systems, the light-switchable\npromoter gradually ceases to produce protein upon light removal. However, our system behaves differently: it\nis activated and remains active until exposed to deactivating light. The light filtering pattern in the blue light-\n10activated systems can originate from its light-out dissociation (LOD) behavior. In contrast to these systems,\nwhere the memory of the expression state is gradually lost in darkness, the dimeric dynamics of PhyB lock the\nsystem in the activated state, leading to a robust memory effect. Considering both strong and weak memory,\nand the confidence in these two mechanisms, the biophysical properties of our system endow it with unique\nsignaling capabilities: a strong memory signaling with an enhanced transfer function.\nGiven the highly stochastic fluorescent signals from the gene expression trajectories controlled by PhyB-\nbased optical regulators (Fig. 1 (C)), a deterministic description of the system is precluded34,35. Therefore,\nfrom the perspective of probabilistic modeling, we represent the optically-controlled gene expression regulated\nby phytochromes as a non-Markovian stochastic dynamical system that is responsive to external controls. A\nstochastic reaction network has been constructed for this system67. In light of the evidence presented in the\nprevious section, which reveals two activated conformers through fitting to photoconversion and thermal rever-\nsion processes, we have modeled the system comprising 23 species and 74 reactions, as depicted in Figure 3(A)\nand listed in Table S3.\nThe control of gene expression by light, as described in the previous section, involves altering light exposure\nto red or far-red light, which modulates the rate constants of photoconversions and can initiate or terminate\nmRNA transcription. The activation function of the promoter is modeled as a step function, aligning with\nthe timescale separation where PhyB-PIF3 binding occurs within milliseconds, and gene transcription takes\nplace over seconds. To address the fluorescent background issue associated with the basal expression of the\nfluorescent reporter, we initiate the modeling 12 hours prior to the experimental start time. The complete control\nprotocol is as follows: 12+2 hours of dark incubation, a 2-minute red-light pulse for activation, a subsequent\n5-hour dark incubation, and finally an 8-hour deactivation period under far-red light. The system then reaches a\nNESS prior to the experiment, reflecting the actual system setup. The background cellular fluorescence signal\nintensity is then used to calibrate the steady-state distribution of the fluorescent reporter generated by basal\nexpression. Consequently, the distribution of the background signal also provides insights into the rates of\nbasal transcription and degradation. Furthermore, calibrating the background signal determines the ratio factor\nRfor converting fluorescent intensities to the number of molecules, defined by the equation:\nR=Fluorescent intensity (a.u.)\nNumber of reporter molecules\nwhere Ris expressed in arbitrary units (AU). During the experimental time frame, the ratio factor is assumed\nto remain constant, independent of the number of reporter molecules present. Hence, for each experiment, a\nconsistent ratio factor is applied to accurately fit the stochastic model to the experimental data.\n11For extended two-state stochastic gene expression systems, distributions can be analytically obtained33.\nHowever, solving the PhyB-based optically-controlled gene expression system analytically is infeasible due to\nits large state space. Numerical methods, such as kinetic Monte Carlo methods37, can be employed to approxi-\nmate the joint state probability distribution. Yet, sampling from the posterior distribution given a general prior\nand experimental data remains impractical with these methods because of the prohibitive computational time.\nFor example, estimating the likelihood requires simulating approximately 104trajectories for each set of rate\nconstants. Typically, for a single set of rate constants, each estimation takes about 10 minutes using a parallel\nalgorithm. To accurately estimate the posterior distribution, it is necessary to sample 104combinations of rate\nconstants, not including the potentially more numerous mixing steps. Consequently, even with a reasonably\ninformative prior, one inference can take an estimated 60 days to complete, rendering the process highly in-\nefficient. This limitation is analogous to other numerical methods, such as finite state projection approaches,\nwhich track the joint state probability distribution within a truncated state space68.\nTo surmount this obstacle, we employ a neural network strategy to estimate the time-dependent state dis-\ntribution using variational approximations. The neural network’s architecture is depicted in the SI Appendix,\nFig. S2. The inputs to this network consist of prompts, which include the rate constants σσσ, the initial con-\ndition xxx0, and the time value t−t0, as well as the state vector at the current time xxxt. The network’s outputs\nare the conditional marginal probabilities of the state coordinates given the prompts’ information, denoted as\nˆpθθθ(x(i)\nt|x(1)\nt,..., x(i−1)\nt ,σσσ,xxx0,t−t0). The conditional joint probability of the state is thus expressed as\nˆpθθθ(xxxt|σσσ,xxx0,t−t0) =NY\ni=1ˆpθθθ(x(i)\nt|x(1)\nt,..., x(i−1)\nt ,σσσ,xxx0,t−t0)\nTo delineate the mapping from the amalgamation of prompts and states to the conditional marginal probabilities,\nwe implement a masked multi-head self-attention block that computes the scaled dot-product attentions. The\noutputs from the multi-layer decoder, composed of masked multi-head attention blocks, are subsequently fed\ninto a language model head. This head is designed to project the decoder outputs onto probability weights,\nwhich are then normalized using the softmax function (see Methods for more details).\nTo demonstrate the accuracy and efficiency of the neural network approach, we trained a neural network on\na dataset comprising 256 trajectory ensembles, each consisting of 1,000 trajectories corresponding to a unique\ncombination of rate constants. The trained neural network subsequently provided the conditional likelihood\nˆpθθθ(xxxt|σσσ,xxx0,t−t0), facilitating parameter screening. As illustrated in Fig. 3 (B), despite the neural network being\ntrained on only 256 trajectory ensembles, the means and standard deviations of states generated from the neural\nnetwork’s predictions for 2,400 prompts closely mirror those obtained from stochastic simulations. Owing\n12to the highly parallel computation capabilities of the neural network on General-Purpose Graphics Processing\nUnits (GPGPUs), the computation of the joint state probabilities for 2,400 prompts can be completed within one\nhour. This represents a significant improvement in efficiency compared to traditional methods, which typically\nrequire approximately 40 hours for similar computations.\nInferencing the rate constants with neural network assisted Monte Carlo sampling\nEstimating the likelihood using a trained neural network provides a unified framework for inferring rate\nconstants from a dataset that includes both time-slice and kinetic data (see Methods for more details). This\napproach allows for the integration of information derived from multiple measurement techniques, encompass-\ning both time-slice and real-time kinetics of any species, to enhance the accuracy of inference. In contrast, the\nprevious non-parametric Bayesian method utilized only time-slice data for a limited number of species38. For\nsystems with many species, solving the chemical master equation is computationally formidable, and repre-\nsenting the state distribution is also memory prohibitive. Nevertheless, our neural network-based method is not\nconstrained by these limitations (unpublished data).\nGiven the constraints of computational resources, we have approximated certain rate constants using values\nsourced from prior researches. The basal transcription rate of the reporter gene, absence from any regulatory\nbinding, was established at 0.001 min−1. This rate negligibly impacts the basal reporter distribution. Tran-\nscription rates for PhyB and PIF were inferred from the nascent transcription rates of yeast genes, as reported\nin69,70, which vary between 2 and 30 hour−1. Adopting the upper limit of this range, 0.5 min−1, we approx-\nimated the transcription rates of PhyB and PIF to be 0.31 min−1and 0.33 min−1, respectively. We postulate\nidentical activated transcription rates for the reporter gene when bound to Pfr 1Pa, Pfr 2Pa, and PrPa. Translation\nrates for PhyB, PIF, and the reporter were estimated from translation initiation and elongation rates derived\nfrom ribosome profiling data71, with the translational rate constant ranging between 5 and 8 residues sec−1. By\napplying the highest observed rate, we estimated the translation rates for PhyB(NT)-lexDBD, PIF3-VP16, and\nthe reporter inclusive of a ubiquitin fragment to be 9.573 min−1, 0.775 min−1, and 1.89 min−1, respectively.\nPost-translation, proteins must properly fold to become functional. The median folding time, t50=0.005 sec,\nas sourced from a protein database and adjusted for temperature, equates to a rate constant of 8317.8 min−1is\nused for both PhyB(NT)-lexDBD and PIF3-VP1672. The maturation time for the reporter (sfGFP), t50=6.9\nmin, was derived from in vivo studies of fluorescent protein maturation in yeast cells, corresponding to a rate\nconstant of 0.1 min−1 73. We have assumed uniform mRNA degradation rates for PhyB, PIF, and the reporter.\nAdditionally, we posit that the degradation rates for both immature and mature proteins are equivalent.\nFour rate constants were identified as having the most significant effects on the behavior of the system. The\n13first is the activated transcription rate of the reporter ( σPR), which largely determines the steady-state distribu-\ntion of the number of molecules. Another is the association rate of Pr with Pa ( αPrPa), which greatly influences\nthe basal expression level. Finally, the degradation rates of mRNAs ( βmRNA ) and the reporter ( βPR) define the\nsteady-state distribution after deactivation. Using the trained neural network model, we drew samples from\nthe posterior distribution of the rate constants using a Markov Chain Monte Carlo method (see Methods for\ndetails). As shown in Fig. 3 (C), the four rate constants, σPR,αPrPa,βmRNA , and βPR, are tightly constrained by\nthe experimental data. Utilizing the most probable values of these rate constants for inference, we performed\na stochastic simulation of the system and compared the predicted state distribution with the experimental his-\ntogram, as depicted in Fig. 3 (D). The complete plots across 48 time points are presented in the SI Appendix,\nFig. S5. It is evident that the inferred rate constants accurately capture the state distribution of the fluorescent\nreporter. The complete list of rate constants is provided in the SI Appendix, Table S3.\nModulation of gene expression noise by light\nFor the purpose of applying our optical perturbation system to the study of cellular noise in biological\nsystems, it is essential to be predictive for the output noise of our system. Thus, we investigated the possibility\nof precise noise control by employing variable light intensities and modulation of the degradation rate of the\ntarget molecule. To tune the exposure light intensities, we calibrated the light source of the microscope and\napplied a linear increase in light intensities to activate the system. Additionally, we modulated the degradation\nrate of the reporter protein by expressing a chimeric ubiquitin-X-reporter construct. Upon expression, ubiquitin\nis co-translationally removed from the protein, leaving behind the X-reporter, where X represents the chosen\namino acid residue. The N-terminal residue of a protein significantly influences its half-life, which can vary\nfrom several hours (as in the case of methionine or alanine) to a few minutes (for residues like leucine or\narginine)74,75. Subsequently, we evaluated the behavior of our system under conditions of linearly increasing\nlight intensities and following N-terminal mutagenesis.\nAs shown in Fig. 4 (A, B), the steady-state distributions of the reporter indicate changes in the rate con-\nstants due to varying light intensities and N-terminal mutagenesis. We further sought the optimal rate constants\nfor photoconversions as well as degradation using a score-based bootstrap inference algorithm (Eq. 1). The\ncomparisons between the molecular number distributions predicted by the inferred rate constants and the ex-\nperimental histograms is presented in the SI Appendix, Figs. S3–S7. The Hellinger distances between the\npredicted distributions and the experimental results are around 0.1, which attests to the efficacy of the inference\nalgorithm. From Fig. 4 (C), it is observed that within the range of 20%–60% light intensities, there is an almost\nlinear increase in the rate constant for the reaction Pr →Pfr2. In contrast, the rate constant for the reaction\n14Pr→Pfr1increases exponentially. Consequently, the extent of photoconversion from Pr state to Pfr states is\ndirectly correlated to the quantity of red photons provided, exhibiting high predictability and reproducibility.\nHowever, both rate constants diminish to minimal values when the light intensity reaches 80%, likely due to\nphototoxicity at high light fluences41.\nTo quantify the change in cellular noise during the modulation of rate constants, we calculated the logarithm\nof the coefficient of variation (CV), defined as log10(σ/µ), where σrepresents the standard deviation and µ\ndenotes the mean number of molecules. As depicted in Fig. 4 (D), the logarithm of the CV decreases with\nan increase in activation rates and increases with the degradation rate. The dependence of the logarithm of\nthe CV on the rate constant of Pr →Pfr2is linear, whereas its dependence on the input light intensities is\npredominantly exponential. The slight deviations from linear and exponential dependencies suggest that the\nPr→Pfr2transition contributes significantly to the noise behavior of the system. Furthermore, an exponential\ndependence on light intensities implies that the noise intensity has an intrinsic lower bound in the PhyB-based\noptical perturbation system. Conversely, the noise’s dependence on the degradation rate constant is exponential,\nindicating that the degradation rate constant’s influence on noise is pronounced when its value is small.\nThe complex behavior of the system can be comprehensively understood through the lens of the so-called\ninverse correlation between noise and mean expression level, as illustrated in the SI Appendix Fig. S8. Both\nactivation and deactivation in the optically-controlled perturbation system modulate the expression level and the\nexpression noise simultaneously, with the noise being linearly dependent on the mean population of molecules.\nThis relationship between noise and mean population level has also been observed for both protein and mRNA\nin numerous other systems76–79. However, the PhyB-PIF3 system exhibits a high level of noise relative to its\nmean expression level compared to other light-modulation systems76. Considering the dual-pathway mecha-\nnism of the PhyB system, it is conceivable that an increase in transition pathways maintains the linear depen-\ndency relationship while elevating the ratio of this linear dependency between noise and the mean population\nlevel of molecules.\nDiscussion and conclusion\nUnderstanding the behavior of noise in biological systems is crucial for comprehensively characterizing the\nphenotypic changes in many key processes, including gene expression, signal transduction, development, and\nevolution. While numerous single-molecule labeling methods are available and high spatial-temporal resolution\ncan be achieved through optogenetic techniques, quantitatively describing noise remains a significant challenge.\nTwo major obstacles are the modeling of complex biochemical reactions and the inference of rate constants for\n15these models, both of which have garnered considerable attention.\nIn this work, we present a systematic approach to elucidate the dynamic behavior of cellular noise using\nour optical perturbation system combined with deep learning. We demonstrate that the optically-controllable\nsystem we have developed constitutes a highly sensitive light modulation system. It exhibits distinct behaviors\ncompared to the blue light-regulated gene expression system40,49. The dynamic mechanism of the blue light-\nregulated system can be characterized by Hill functions, whereas our PhyB-based system is more accurately\ndescribed by a multiple-path reaction system involving two activated conformers. Moreover, the blue light-\nregulated system has a faster thermal reversion rate compared to the PhyB-based system. Activation by blue\nlight of these systems leads to a gradual deactivation in these systems, resulting in a decreased population of the\nactivated state. However, the stability of the activated Pfr states of phytochromes is significantly enhanced in\nthe presence of PIFs. As the result, the PhyB-based system not only retains memory for long time duration but\nalso allows for the accumulation of light fluence over prolonged periods. This characteristic renders it a highly\nstable two-state light-regulated gene regulator when compared with the blue light system. Both pulsatile and\ncontinuous activation modes can be achieved through the interplay of red and far-red light, without the need\nfor continuous light illumination. This endows the PhyB-based system with advantages in in vivo applications,\nsuch as the semi-automatic control of glucose homeostasis or immunotherapy by epigenetically engineered\nphotosensitive cells14,80.\nTo further quantitatively characterize the dynamical properties of our system, we developed a stochastic\nmodel comprising 74 reactions. To infer the rate constants of this model, we introduced an omniscient neural\nnetwork capable of mapping the rate constants directly onto joint state distributions. The trained network\nfacilitates efficient computation of likelihoods for various rate constant configurations. We demonstrate that rate\nconstants can be determined with high efficiency and precision using the omniscient neural network, surpassing\nconventional methodologies. Comparing the stochastic model’s predictions with experimental data, we show\nthat our system can not only produce a controllable level of target molecules but also modulate expression noise\nthrough adjustments in light intensity and the degradation rate of the target protein.\nIn conclusion, we showcase a novel integration of optical perturbation with a neural network-based likeli-\nhood estimator to dissect noise in biological systems. Our methodology is readily extendable to diverse optoge-\nnetic systems and organisms, enhancing the elucidation of complex dynamical behaviors in state distributions\nfor expansive systems.\n16Material and methods\nStrains and plasmids construction\nAll experiments were carried out in W303a ( MATa leu2-3,112 trp1-1 can1-100 ura3-1 ade2-1 his3-11,15),\na haploid derivative of W303 obtained from ATCC (#208352). Plasmids were constructed and propagated in E.\ncoliDH5αand verified via Sanger sequencing prior to strain construction. Standard protocols for the growth,\nmaintenance, and transformation of yeast and bacteria, as well as DNA manipulation, were employed. Initial\nplasmids were assembled as described in a previous study41, with functional fragments cloned from cDNA\nby conventional PCR. DNA digestion and ligation utilized restriction endonucleases and T4 DNA ligase from\nNew England Biolabs (USA). Light-responsive strains were generated by integrating pAM206 into the trp1\nlocus of the W303a yeast genome. Given that pAM206 harbors two XbaI sites, partial digestions were exe-\ncuted using the XbaI endonuclease to optimize the yield of linearized plasmids. These linearized plasmids were\nsubsequently transformed into yeast cells following the standard lithium acetate-based transformation protocol.\nTransformed cells were cultured on selective auxotrophic minimal media using the appropriate dropout plates\n(Takara). Integration of plasmids was confirmed by genomic PCR at both the N- and C-terminal sides of the\nrecombination loci. In light of the fact that multiple copies can lead to high expression levels and consider-\nable expression noise, potentially complicating expression kinetics76, single-copy integration was verified us-\ning real-time quantitative PCR. The resultant strains expressed PhyB(NT)-lexADBD. Subsequently, pAM205\nwas linearized with ClaI and integrated into the leu2 locus, resulting in strains that additionally expressed\nPIF3-VP16AD. The strain produced through sequential integration served as the progenitor for all subsequent\nexperiments. To modify the stability of the reporter, pAM236M was engineered by appending an ubiquitin\nsequence upstream of the fluorescent protein in pAM236. The N-terminal methionine (M) of pAM236M was\naltered to arginine (R) through site-directed mutagenesis, thereby modulating the stability of the reporter via the\nN-end rule pathway74. To replace the fluorescent reporter Venus with sfGFP, which offers enhanced photosta-\nbility and accelerated maturation81, the sfGFP sequence was initially PCR-amplified from the GTTg plasmid\n(Addgene #81109) and subsequently fused with the backbone sequences of pAM234 and pAM236 via inverse\nPCR. Comprehensive plasmid information is available in Table S1. The functionality of the final constructs\nwas confirmed by fluorescence microscopy following co-incubation with PCB. Control assays were conducted\nusing PhyB-only and PIF3-only cell strains under identical conditions for validation.\n17Fabrication of microfluidic chip for real-time kinetic measurements\nThe design and fabrication of the microfluidic device for live-cell imaging adhered to protocols from pre-\nviously published work82. Briefly, the process utilized SU-8 2000 series photoresists (MicroChem), chrome\nglass masks (HTA Photomask), and an EVG620 contact mask aligner (EV Group) to construct and pattern the\ndesired features on silicon wafers supplied by University Wafer Inc. The feature heights were confirmed with\na Dektak 150 surface profiler (Veeco). Each chip comprised four identical microfluidic devices, each inspected\nmeticulously to ensure the absence of dust or defects. Prior to cell introduction, the microfluidic chip was\ncleansed with acetone and isopropanol, followed by at least two rinses with sterilized deionized water to re-\nmove any residual solvents. The chip was then positioned on a custom gas chamber situated on the stage of an\ninverted microscope. This setup was enclosed within a thermally stable microscope cage incubator (Okolab),\nmaintaining a constant temperature of 30◦C. The media ports were connected to plastic tubing, which in turn\nwas linked to a high-precision pump supplying fresh media. The waste port of the chip was attached to plastic\ntubing leading to a 50 mL collection tube.\nMedia and growth conditions\nCells were initially cultured in synthetic complete (SC) medium (pH 6.0) overnight until the optical density\nat 600 nm (OD 600) reached approximately 1.0, prior to initiating experiments. For cell loading, the yeast\nculture was transferred using a syringe, driven by a pump at a flow rate of 0.1 mL/min. The majority of the\ntraps were populated with cells within 1-2 minutes. Subsequently, the medium in the syringe was replaced with\nSC medium supplemented with 25 µM of PCB, and the cells were incubated for a minimum of 2 hours in the\nabsence of light. For the light-activated kinetic assays, the flow rate of the medium was set to 0.01 mL/min to\nminimize cell movement and maintain a normal growth rate.\nLight activation procedure and measurements\nThe structural analogs of phytochromobilin (P ΦB), phycocyanobilin (PCB) (CAS No. : 20298-86-6), are\nutilized to induce the photoconversion of PhyB. We employed a Semrock 641/75 nm bandpass filter and a\nSemrock 715 nm long-pass filter for light activation and repression, respectively. For light activation, red light\nat 20%, 40%, 60%, and 80% constant intensities was applied for 2-minute intervals to initiate gene expression.\nConversely, for light repression, far-red light at a constant 50% intensity was used to effectively suppress\ntranscription in the light-regulated system. Fluorescent reporter activity was monitored using 488 nm laser\nillumination with an exposure time of 100 ms.\n18Image analysis with point-prompt cell segmentation\nTo analyze the microscopic images, we have developed a Python-based software that facilitates efficient cell\nsegmentation and cell tracking. The software employs the Segment Anything Model (SAM) as its core model,\nwhich is capable of segmenting yeast cells using point-prompt information83. Following the SAM-based seg-\nmentation, the masks of the segmented cells are manually refined through interactive labeling. Subsequent to\nsegmentation, cell labels are reassigned using automatic tracking via the Hungarian algorithm. The automati-\ncally segmented and tracked cells are subsequently manually reviewed to correct any mislabeling and to exclude\ndead cells. Finally, each cell’s fluorescence intensity, as well as morphological information, are collated and\nexported to a text file for additional analysis.\nSimulation of stochastic reaction network\nTo account for the basal expression of reporter proteins, we simulated the entire process from -12 hours\nuntil the initiation of the experiment. Red light activation was applied at 840 minutes for 2 minutes, and the\nsystem was deactivated using far-red light at 1142 minutes until the conclusion of the experiment. The total\nduration of the simulation was 1624 minutes. We utilized the biocircuits package for parallel simulations\nemploying the Gillespie algorithm. For each set of rate constants, 103– 104trajectories were generated, and\nsystem states were recorded at time points corresponding to the experimental sampling times. To initially\ntrain the autoregressive model in a supervised manner, we simulated trajectory ensembles for 256 sets of rate\nconstants, where the rate constants were sampled using the quasi-Monte Carlo method with the Sobol sequence.\nArchitecture of the omniscient neural network model\nFor modeling the light-regulated gene expression system, we have used the variational approach to approx-\nimate the joint distribution of states given prompts that encode information of rate constants and time. We\ndesigned the omniscient neural network model with masked multi-head self-attention blocks as its core func-\ntional module which describes the connection between prompts and states. For calculating the self-attention\nscore, we use learnable embeddings to convert the state vector xxx∈RNto a tensor in the embedding space\nX∈RN×demb. On the other hand, we collect the logarithm of rate parameters and time value in sequence to\nconstruct the prompt. To handle the continuous variables of rate parameters and times in the prompt, we first\ntransform rate parameters into logarithmic values to reduce value divergences. Then, we employ a one-layer\nperceptron to project the prompt to a vector pppwith fixed length dp. We then map pppto the same embedding\nspace as the discrete-valued states with linear projector, to obtain the tensor Y∈Rdp×demb. The concatenation\n19ofXandYserves as the input tensor Z=Concat (X,Y)∈Rds×dembto the network, where ds=N+dp. The\nmasked multi-head self-attention block calculates the scaled dot-product attention of the input tensor Zwith h\nattention heads\nAttention (Z) =Concat (head 1,..., head h)WO\nwhere head i=Softmax\u0012Mask (QiKT\ni)√dk\u0013\nVi\nwhere the query Qi=ZWi,Q∈Rds×dk, the key Ki=ZWi,K∈Rds×dk, and the value V=ZWi,V∈Rds×dvare all\nlinear projections of the input tensor Z. These projections are obtained using different learnable parameter\nmatrices: Wi,Q∈Rdemb×dk,Wi,K∈Rdemb×dk,Wi,V∈Rdemb×dv, and WO∈Rhdv×demb. Here, dkanddvrepresent the\ndimensions of the number of query (key) elements and value elements for each state value, respectively. When\nhis specified, dkanddvare defined according to the relation dk=dv=demb/h. The attention mask Mask (·)is\nused to conceal the dependency relations between the previous position and subsequent positions of both the\nprompt and the generated sample states. This is achieved by assigning (QiKT\ni)ξ<µ=−∞, where ξandµare\nthe row and column indexes of the QiKT\nimatrix, respectively. We stack multiple attention blocks to create a\nmulti-layer decoder, which enhances network’s representational capability. The output of the masked multi-\nhead attention block A∈Rds×dembis used as the input for another block. Following the multi-layer decoder, a\nlanguage model head is utilized to map the decoder outputs to probability weights, which are then normalized\nusing the softmax function. The predictions of the MET provide joint distributions of states given the input\nprompts.\nCoordinate embedding and training of the omniscient neural network model\nThe synthesis of mRNA and the expression of proteins are significantly influenced by the biochemical rates\nassociated with each reaction. The number of protein molecules ranges from hundreds to millions, necessitating\na large state space for neural network representation. To reduce the number of states, we employed coordinate\nembedding to transform state coordinate values into vectorized representations. This embedding process in-\nvolves base conversion of the coordinate value into a binary or decimal number, followed by the creation of\na vector with each element corresponding to a digit in the number. In the case of a light-controlled gene ex-\npression system, to achieve efficient computations, we constrained the vector sizes for certain coordinates to\nautomatically account for state constraints, such as the number of genes. This reduction in vector sizes also\ndecreased the number of species. Training was conducted using a trajectory ensemble dataset comprising 0.25\nmillion trajectories. To generate this ensemble, 256 different parameter sets were initially sampled using a\nquasi-Monte Carlo method with the Sobol sequence. Subsequently, 1,000 trajectories were sampled for each\n20parameter set using Gillespie’s algorithm. The model underwent training for 40 epochs, totaling 80 hours on a\ncloud server equipped with 4 NVIDIA GeForce RTX 4090 GPUs.\nObtaining posterior probability with the omniscient neural network model\nTo estimate the posterior probability of each set of rate constants, we posited a fundamental assumption:\nthe cells are entrapped randomly within the microfluidic chip’s cavities. The fluorescence emissions emanating\nfrom these cells follow a distribution dictated by the stochastic dynamics inherent to the optical-controlled gene\nexpression system. The trained neural network model provides an accurate approximation for the state joint\ndistribution given the rate constants and the time value ˆ pθθθ(xxxt|σσσ,xxx0,t0),p(xxx|σσσ)for short. For parameter infer-\nencing, we used the Metropolis-Hastings algorithm with the state likelihood provided by the neural network\nmodel. At Monte Carlo step s, we used a Gaussian kernel for generating the next step proposal rate parameters\nσσσs+1=σσσs+lll·N(0,Σ), where covariance matrix Σis diagonal for controlling the Monte Carlo step size, and\nlllis the scale factor. Since the symmetry of Gaussian kernels, it is evident that p(σσσs+1|σσσs) =p(σσσs|σσσs+1). We\nalso use a Gamma probability density function to account for the prior knowledge of σσσ. The accepting criterion\nfor Metropolis-Hastings algorithm is then\nγ=p(σσσs+1|xxx)\np(σσσs|xxx)·p(σσσs|σσσs+1)\np(σσσs+1|σσσs)=p(xxx|σσσs+1)Γ(σσσs+1)\np(xxx|σσσs)Γ(σσσs)\nwhich is the product of the ratio of the joint likelihoods and the ratio of the prior probability. The movement\nfrom the original rate parameters σσσsto the new rate parameters σσσs+1is only allowed with the probability\nmin(1,γ). During the Monte Carlo sampling, nstates are randomly chosen from the observed trajectories that\ncorrespond to different times. The accepting criterion can then be written as:\nγ=nY\nip(xi|σσσs+1)\np(xi|σσσs)Γ(σσσs+1)\nΓ(σσσs)\nThe trained neural network model was then used to approximate p(xi|σσσs,ti)for estimating γ:\nγ≈exp\"nX\ni(ln ˆpθθθ(xi|σσσs+1,ti)−ln ˆpθθθ(xi|σσσs,ti))#\n·Γ(σσσs+1)\nΓ(σσσs)\n21Data & code availability\nThe met package used in this work is available in the git repositories:\nhttps://github.com/cbliu111/cell_seg_track_software\nand\nhttps://github.com/cbliu111/met2\nAcknowledgments\nC.-B.L. thanks the supports from the National Natural Science Foundation of China Grant 32000888, the\nScientific Instrument Developing Project of the Chinese Academy of Sciences Grant YJKYYQ20180038, Jilin\nProvince Science and Technology Development Plan Grant 20230101152JC.\nAuthor contributions\nE.L.E and C.B.L. designed the research and contributed to the experimental materials. Y .F., L.L. and C.B.L.\nconstructed the yeast strains. 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Segment anything (2023). 2304.02643 .\n29nPASGAF\nc\nPΦB 1NTEKL HP115 219 252 439 621PhyB(PSM)\nPHY APB AD APA bHLH27 3991 114 193 210 340 397\n524 1PIF3\nPhotoconversion 8�lexAopPhyB(PSM)PIF3\nVP16\n0\n204\n304\n404\n485\n585\n685\n785\n905A\nB\nCFig. 1 Perturbing gene expression with light. (A) Schematic of the PhyB-based optical-controlled gene expres-\nsion system. (B) Representative time-lapse images of light-activated yeast cell incubated in traps of microfluidic\nchip channels. Signal of sfGFP expression is shown with green colors. The capturing times of the images is labeled\non top of these images with unit minutes. (C) Mean fluorescence intensity of cells as function of observation time.\nIntensity of each cell is plotted as light-gray dotted lines, mean value of these trajectories is denoted by the solid blue\nline and light blue region indicates the standard deviation. Red light activation is represented by red pulse starting at\n120 min and lasts 2 mins. Far-red light deactivation is represented by light-red box starting at 422 min and lasts 490\nmins.\n30Thermal reversionPhotoconversion\nA B CFig. 2 Minimized model for photoconversion and thermal reversion dynamics of phytochromes. (A)\nSchematic of the state conversion model. Pr and Pfr state are labeled with light and dark red, respectively. Fragment\nof PIF was denoted as Pa. In the minimized model, dimer photoconversion and transformation between Pfr1 and Pfr2\nstate are omitted. (B) Fitting of stochastic reaction networks to the photoconversion processes. Experimental data\nrepresenting the photoconversion of PhyB(NT) at 25◦C (empty circle) were obtained from Figure 2 of54. (C) Fitting\nof stochastic reaction networks to the thermal reversion processes. Experimental data representing the thermal rever-\nsion of PhyB(NT) under different concentration of PIFs at 22◦C (empty circle) were obtained from Figure 3 of43.\nThe system was photo-activated under red light for 10 min before darkness for observation. The photoconversion\nrates obtained from (B) was used in the fitting of thermal reversion rates since the photoconversion rates are not\nsensitive to temperature alterations52,54.\n31Thermal reversionPhotoconversion\ne\nA\nB\nCDFig. 3 Stochastic modeling and parameter inference of optical-controlled gene expression system. (A)\nStochastic modeling of the PhyB-based optical-controlled gene expression system. Gene, mRNA, immature pro-\ntein are labeled with capital letters “G”, “R” and “U” for each species, respectively. PhyB, PIF and fluorescence\nreporter are represented by letter “r”, “a” and “R”, respectively. e.g. G Rrepresents the gene of fluorescence reporter,\nand G RPfr1Pa is the trimmer comprised by gene of fluorescence report, the activated Pfr 1and PIF3. Pr and Pfr state\nare labeled with light and dark red, respectively, as in Fig. 2. Fluorescence reporter and of PIF was denoted as Pa.\nThe reporter gene (G R) In the minimized model, dimer photoconversion and transformation between Pfr1 and Pfr2\nstate are omitted. (B) Validation of the trained neural network for prediction of population mean and standard devi-\nations of the reporter protein for 2,400 parameter combinations. Means and standard deviations are calculated using\n1,000 samples from either neural network predictions or stochastic simulation results. (C) Posterior probability dis-\ntributions obtained through Markov Chain Monte Carlo sampling for σPR,αPrPa,βmRNA andβPR. Prior distributions\nare shown by dashed dark red lines. (D) Histograms of predicted reporter protein number compared to experimental\nresults.\n32A B\nD CFig. 4 Modulation of fluctuations with optical-controlled gene expression system. (A) Steady-state probabil-\nity distributions corresponding to different activation light intensities. (B) Steady-state probability distributions of\nwild-type and N-terminal mutagenesis fluorescent reporter. The inset figures in (A) and (B) showed the change of\nmean fluorescence intensities along with observation times. (C) Dependency of rate constants on activation light\nintensities. The inferencing rate constants are represented as solid dots, and dash-dot lines show the smoothed spline\ninterpolation of the data. (D) Dependency of logarithm of coefficient of variations (CV) on activation and degrada-\ntion rate constants (ticks are shown in lower and upper x-axes). The solid dots represent the results obtained from\nstochastic simulations, and colored stars denote the experimental results. The red solid line indicates a linear fitting\nof the rate constant (Pr →Pfr2) to the logarithm of CV and blue solid line indicates an exponentially fitting of the\ndegradation rate constant to logarithm of CV . The red dash-dot line is an exponentially fitting of the rate constant\nto the logarithm of CV . Each experimental result represents a collection of mean fluorescent intensities from about\n20,000 cells.\n33"    },    {        "title": "2401.12499v1.On_the_Fundamental_Tradeoff_of_Joint_Communication_and_Quickest_Change_Detection.pdf",        "content": "arXiv:2401.12499v1  [cs.IT]  23 Jan 20241\nOn the Fundamental Tradeoff of Joint\nCommunication and Quickest Change Detection\nDaewon Seo, Member, IEEE, and Sung Hoon Lim, Senior Member, IEEE\nAbstract —In this work, we take the initiative in studying\nthe fundamental tradeoff between communication and quicke st\nchange detection (QCD) under an integrated sensing and com-\nmunication setting. We formally establish a joint communic ation\nand sensing problem for quickest change detection. Then, by\nutilizing constant subblock-composition codes and a modifi ed\nQuSum detection rule, which we call subblock QuSum (SQS),\nwe provide an inner bound on the fundamental tradeoff betwee n\ncommunication rate and change point detection delay in the\nasymptotic regime of vanishing false alarm rate. We further\nprovide a partial converse that matches our inner bound for a\ncertain class of codes. This implies that the SQS detection s trategy\nis asymptotically optimal for our codes as the false alarm ra te\nconstraint vanishes. We also present some canonical exampl es\nof the tradeoff region for a binary channel, a scalar Gaussia n\nchannel, and a MIMO Gaussian channel.\nIndex Terms —Integrated sensing and communication, Quick-\nest change detection, CuSum test, Constant subblock-compo sition\ncodes\nI. I NTRODUCTION\nThe capability to simultaneously perform communication\nand sensing tasks efficiently within limited resources is on e\nof the key features envisioned for next-generation wireles s\nnetworks [1], [2]. In contrast to conventional systems typi cally\ndesigned with separately dedicated resources and hardware ,\nintegrated sensing and communication (ISAC) systems aim\nto cohesively design both sensing and communication func-\ntionalities. This design approach enables the system to sha re\ncommon resources, such as frequency bands and hardware,\nin order to enhance efficiency and reduce costs [3], [4].\nHenceforth, it is expected that ISAC will become a key tech-\nnology in future wireless systems, supporting many importa nt\napplication scenarios such as device localization, autono mous\nvehicles, surveillance, and healthcare [3], [5], [6].\nIn ISAC, the primary focus of research is on simultaneous\ncommunication of information and target sensing. Conse-\nquently, it is natural to explore the tradeoff between two ke y\nmetrics; one related to data transmission and the other to\nsensing accuracy. For instance, the tradeoff between infor ma-\ntion rate for transmitting data reliably and mean-squared e rror\n(MSE) for estimating an unknown real-valued channel state\nis characterized in [7], [8]. In another instance, the trade off\nbetween information rate and error probability of detectin g a\nD. Seo is with the Department of Electrical Engineering and C omputer\nScience, Daegu Gyeongbuk Institute of Science and Technolo gy (DGIST),\nDaegu 42988, South Korea (e-mail: dwseo@dgist.ac.kr). Sun g Hoon Lim\nis with the School of Information Sciences, Hallym Universi ty, Chuncheon,\nKorea (e-mail: shlim@hallym.ac.kr).discrete target, e.g., obstacle detection, is investigate d in [9]–\n[12]. In addition to these intrinsic ISAC metrics, the metri c\nof secrecy is also considered in [13], [14]. Some surveys and\noverviews on the topic of ISAC are given in [6], [15]–[17].\nBuilding upon such previous studies, we study a new\nmetric of ISAC in which the goal is to detect changes or\nanomalies in the channel as soon as possible subject to some\nfalse alarm constraint. We note that the sole problem of\ndetecting the change point in distribution from a sequence\nof observations (i.e., without a communication requiremen t)\nis alone an active field of research in the context of quickest\nchange detection (QCD) [18]–[22]. In this work, our goal is\nto perform communication and QCD with a common signal\nand provide fundamental tradeoffs and design perspectives for\njoint communication and QCD.\nOne possible motivating scenario for this requirement coul d\narise from a situation in which a mobile device equipped\nwith wireless communication capabilities, such as an un-\nmanned aerial vehicle (UA V), an autonomous vehicle, or a\nrobot arm, must identify sudden obstacles in its trajectory . In\nanother scenario, base stations scattered in a wireless net work\ncan be utilized as a surveillance sensor network to detect\nabnormalities with minimum delay. In these instances, the\ntransmitted signal for communication simultaneously serv es\nas an active sensing signal for detecting anomalies. Thus,\nthe policy for jointly designing communication and quickes t\nchange detection must be jointly designed to provide the bes t\ntradeoff between communication rate and detection delay.\nHowever, there are some inherent fundamental challenges\nto achieving such goals. On the one hand, results on capacity -\nachieving codes for communication over a noisy channel sug-\ngest that the signals should be designed in units of long bloc ks\nthat are distinguishable from other competing codewords wh en\ndecoded over observation sequences. On the other hand, the\ncodewords should also act as potent active sensing signals t hat\nproduce the most distinctive observations that allow detec tion\nof the change point with minimum delay. To put the problem\ninto perspective, we consider a broadcast channel as shown\nin Fig. 1 that consists of one encoder and two receivers. The\nencoder transmits an nlength codeword xnthrough a state-\ndependent memoryless broadcast channel p(yn,˜yn|xn,sn) =/producttextn\ni=1p(yi|xi,si)p(˜yi|xi), wheresnis a state sequence that\nremains constant at the beginning, but at some unknown\nchange point, transitions to another state. Further, ˜yn,ynare\nthe sequences of observations for the information decoder a nd\nQCD detector, respectively. The decoder recovers a message\nsent from the encoder, ˆM, and the QCD detector detects the\nchange point with minimum delay, N, in an online manner.2\nEncoderp˜Y|X\npY|X,SDecoder\nDetectorXn˜Yn\nYnMˆM\nN\nM\nFig. 1: Problem model of joint communication and QCD.\nx1···xν−1xνxν+1···xn\np(0)···p(0)p(1)p(1)···p(1)\ny1···yν−1yνyν+1···yn\np(0)(y|x)p(1)(y|x)\nFig. 2: The distribution of the QCD channel changes at some\nturning point νfrom the baseline distribution p(0)to the\nabnormal distribution p(1).\nThis is known as a bistatic ISAC model in the literature [8],\n[10].\nWe also assume that the message is known at the QCD de-\ntector as side information. This setup can serve as a canonic al\nmodel, such as in a scenario where two base stations with a\nwireline link perform QCD in a wireless network, in which the\nmessage is sent through the wirelink link as side informatio n.\nThe state sequence represents when the change occurs in\nthe distribution of the QCD channel. For example, consider\na binary state channel shown in Fig. 2 where p(0)(y|x) =\np(y|x,s= 0) andp(1)(y|x) =p(y|x,s= 1) . We assume that\nthe QCD channel stays in its base state s= 0until (just before)\na change point ν, and from point νand after, the channel stays\nin states= 1. The goal of the QCD detector is to provide a\nchange point estimate Nwith a minimum delay compared to\nthe true change point νunder a false alarm constraint.\nThis joint communication and QCD setting has several dis-\ntinct features that should be noted. In the context of QCD, th is\nproblem is related to the active sensing QCD problem studied\nin [23], [24], where the main objective is to design a control\nsignal so that observations include as much information as p os-\nsible for quickly detecting the change. The major distincti on\nin our case is that the control signal must be simultaneously\nuseful for communication. In the other perspective, relati ng\nto the field of communications and information theory, our\nwork is also related to state amplification and state estimat ion\nin a state-dependent communication channel [25]–[27]. The\nmajor distinction in our case is that, first of all, we special ize\nour attention to a particular distortion criteria, namely, change\npoint detection delay . Furthermore, we follow the minimax\napproach commonly used in the QCD literature and analyze\nthe worst-case delay over all codewords.\nThis is in contrast to most ISAC scenarios which consideraverage distortion measures over all codewords. As the name\nsuggests, QCD applications are more likely to exhibit some\ndegree of urgency in anomaly detection, which makes worst-\ncase delay the proper criteria. However, to match the worst-\ncase delay criteria, allcodewords must be good for sensing\nsince the codewords are randomly chosen by the message. This\ndisqualifies most standard random i.i.d. generated codeboo ks\nsince even with vanishing probability, the existence of “ba d”\ncodewords for sensing will completely dominate the worst-\ncase performance metric. Finally, another distinction com es\nfrom the criteria of minimizing delay, which suggests that\none should design an online detector since performing an\nestimation after receiving the whole block would simply giv e\nthe worst change point estimation delay.\nIn this work, we contribute to the problem of joint commu-\nnication and QCD as follows. Firstly, we take the initializi ng\nsteps towards addressing the problem and provide a canonica l\nformulation that integrates two heterogeneous criteria wh ich\ncan be readily extended to various scenarios. Most impor-\ntantly, we provide an inner bound on the fundamental tradeof f\nbetween communication and QCD.\nThe precise result is summarized in Thm. 1, where the\nachievable communication rate and the asymptotic delay slo pe\n(with respect to the false alarm rate) are expressed in the\nform of mutual information and Kullback–Leibler (KL) diver -\ngence, respectively. The achievable region can be efficient ly\ncomputed using the Blahut–Arimoto algorithm [28], [29].\nFor the achievability proof, we utilize constant subblock-\ncomposition codes (CSCCs) originally developed in [30],\nwhere the composition of transmitted symbols in every sub-\nblock is identical, and our proposed subblock QuSum (SQS)\ntest, a modified version of the standard QuSum test. As a resul t\nof CSCCs, almost every subblock of observations can be made\nindependent and identically distributed by proper process ing,\nwhich enables us to extend standard QCD analysis techniques .\nMoreover, all the codewords in the CSCC codebook have\nexactly the same empirical distribution on a subblock level ,\nenabling each codeword to be equally good for detection\neven in the worst-case scenario. We also provide a partial\nconverse that matches our inner bound, demonstrating that\nour achievable codes are optimal within a certain class of\ntypical codes. While our converse is limited to a certain cla ss\nof codes, the main implication of the converse result is that\nour SQS detection strategy, detailed in Sec. IV, is indeed th e\nasymptotically optimal detection rule for our codes. Final ly,\nwe present some example tradeoff regions for binary, scalar\nGaussian, and multiple-input multiple-output (MIMO) Gaus -\nsian channels and discuss their implications. Particularl y in the\nMIMO case, we demonstrate an intriguing tradeoff between\ndiversity gains and multiplexing gains that add another deg ree\nof freedom in the tradeoff between information rate and QCD.\nThe remainder of the paper is organized as follows. In the\nnext section, we provide some preliminaries for QCD. Then,\nin Sec. III, we formally give the problem statement. In Sec. I V,\nwe state our main results regarding the fundamental tradeof f\nfor communication and QCD. In Sec. V, we address some\nselective examples. Finally, in Sec. VI, we conclude the pap er\nwith discussions.3\nNotation: We closely follow the notation in [31]. Calli-\ngraphic letters denote alphabet spaces, and uppercase and l ow-\nercase letters respectively denote random variables and th eir\nrealizations. For a sequence xnonXn, the type or composition\nofxnis defined to be π(x|xn) :=/vextendsingle/vextendsingle{i:xi=x}/vextendsingle/vextendsingle/nforx∈ X.\nThen,Pn(X)denotes the set of all types of xn, andTn\npX\nwithpX∈ Pn(X)denotes the set of xnof typepX. We\nalso define [1:n] ={1,...,n}and(x)+= max(0 ,x). The\nnotation for mutual information I(X;Y)can also be written\nasI(pX,pY|X)to clarify its dependency. All logarithms are\nnatural logarithms; hence, the unit of all information-the oretic\nquantities is nats.\nII. P RELIMINARY ON QCD\nIn this section, we briefly summarize the standalone QCD\nproblem. We refer to [21] for a more detailed introduction to\nthe topic.\nConsider a sequence of observations Y1,Y2,... where the\nobservations are initially i.i.d. with some distribution p(0)(y) =\np(y|s= 0) , but at some unknown change point ν, the\ndistribution changes to p(1)(y) =p(y|s= 1)/ne}ationslash=p(0)(y). In\nother words,\nYii.i.d.∼p(0)ifi < ν,\nYii.i.d.∼p(1)ifi≥ν.\nThe goal is to detect the change time νas quickly as possible\nin an online manner but also to minimize the event of declarin g\nthe change before ν, i.e., false alarm.\nBefore formally discussing the QCD, let us define the\nstopping rule as follows.\nDefinition 1 (stopping rule) .A random variable N∈Nis said\nto be a stopping rule with respect to a sequence of random\nvariables Y1,Y2,... if{N=i} ∈σ(Y1,...,Y i), where\nσ(Y1,...,Y i)is theσ-algebra generated by (Y1,...,Y i).\nIn words, the stopping rule Nis a random variable such\nthat the event {N=i}is measurable with respect to the\ngiven information up to that time, i.e., the event {N=i}is\nonly a (possibly stochastic) function of Y1,...,Y i.\nLetNbe a stopping rule with respect to a sequence of\nobservations Y1,Y2,.... If we only consider detection delay,\nthe problem becomes degenerate since to minimize detection\ndelay(N−ν)+, it is best to declare the change point as early\nas possible, i.e., N= 1 always, which will result in a false\nalarm event unless ν= 1. To penalize this, we measure the\nfalse alarm performance in terms of the mean time to a false\nalarm, or equivalently, its reciprocal called the false ala rm rate\n(FAR ) given by\nFAR(N) =1\nE∞(N), (1)\nwhereE∞assumes the probability distribution when the\nchange occurs at ν=∞. For the objective function on\ndetection delay, Lorden [18] considered the worst case (ove r\nall possible change point ν≥1) average detection delay\nconditioned on the worst realization yν−1(WADD ) as\nWADD(N) = sup\nν≥1esssup\nYν−1Eν((N−ν)+|Yν−1), (2)whereEνassumes the probability distribution when the change\noccurs at ν≥1. With both measures at hand, Lorden’s\nformulation characterizes the smallest WADD(N)subject to\naFAR constraint α >0, i.e.,\ninf\nN:FAR(N)≤αWADD(N). (3)\nFor QCD problems, the widely used stopping rules are\nbased on the QuSum statistic, originated by Page [32], and\nits variations. The QuSum stopping rule with threshold bis\ndefined by\nNQS= inf{i≥1 :Wi≥b}, (4)\nwhereWiis the QuSum statistic defined by\nWi= max\n1≤k≤i+1i/summationdisplay\nj=klogp(1)(Yj)\np(0)(Yj),\nwhere the right side is 0by convention if k=i+ 1. It can\nalso be written in a computationally efficient recursive for m\nW0= 0, Wi=/parenleftbigg\nWi−1+logp(1)(Yi)\np(0)(Yi)/parenrightbigg+\n. (5)\nPage’s QuSum stopping rule NQSwas later shown by Lor-\nden [18] that it is indeed asymptotically optimal for Lorden ’s\nformulation (3) as α→0. Specifically, when α→0,\ninf\nN:FAR(N)≤αWADD(N)∼WADD(NQS)∼|logα|\nD(p(1)/⌊ard⌊lp(0)),\nwheref∼gdenotes that f/g→1asα→0. Later,\nMoustakides [33] further showed that the QuSum algorithm\nis exactly optimal under Lorden’s formulation for all α >0.\nIII. P ROBLEM STATEMENT\nIn this section, we give a formal definition of our problem\nstatement.\nConsider a state-dependent memoryless broadcast channel\npY,˜Y|X,S(y,˜y|x,s), shown in Fig. 1, which can be decom-\nposed into two conditionally independent components given\nby\np(yn,˜yn|xn,sn) =n/productdisplay\ni=1p(yi|xi,si)p(˜yi|xi) (6)\nwhereSis the state, Xis the channel input, and Y,˜Yare the\noutputs of the quickest change detection (QCD) channel and\nthe communication channel, respectively.\nLets∈ S={0,1}where we note a base symbol by 0. The\nsequence snis a deterministic sequence\nsn=s1,...,s ν−1,sν,...,s n (7)\nwhereνis a predetermined unknown change point from s= 0\ntos= 1, i.e.,\nsi=/braceleftBigg\n0fori < ν,\n1fori≥ν.\nIn our problem setup, a code serves two distinct roles.\nFirst, it functions as a conventional channel code, used to4\ncommunicate messages reliably over the noisy channel p˜Y|X.\nAdditionally, the transmitted code also acts as a sensing si gnal,\nthrough which we wish to detect the unknown change point ν\noverpY|X,S. Overall, a code for the problem setup is required\nto simultaneously transmit information and produce distin ctive\nobservations for QCD over the broadcast channel.\nFormally, a code for joint communication and QCD has the\nfollowing components. For a given block length n, an(2nR,n)\ncode consists of\n•a message set m∈[1 : 2nR],\n•an encoder that assigns a codeword xn(m)to each\nmessagem∈[1 : 2nR],\n•a decoder that assigns a message estimate ˆm(˜yn)to each\nobservation sequence ˜yn, and\n•a stopping rule Nsuch that the event {N=i}is\nmeasurable with respect to Yiandxn(m). If the stopping\nrule is not activated until the end of the codeword, we\ndefineN=∞.\nA codebook is defined as C(n)={xn(m) :m∈[1 : 2nR]}.\nWe emphasize that Nis dependent on xn(m), however, we\nomit this dependency in our notation for visual clarity.\nThe performance metrics for the problem are given as\nfollows. For the communication component, the performance\nmetric is the maximum probability of error denoted by\nP(n)\ne= max\nm∈[1:2nR]P(ˆM/ne}ationslash=m|M=m).\nWe say that a rate R∈R+is achievable if there exists a\nsequence of codes such that P(n)\ne→0asn→ ∞ .\nNext, for the QCD metrics, we extend Lorden’s formu-\nlation [18] to our setting. To this end, fix a sequence of\ncodebooks C(n). We first define the false alarm rate ( FAR )\nfor the codebooks as\nFAR(N) = limsup\nn→∞max\nxn∈C(n)1\nE∞(N|xn). (8)\nSimilar to (2), we define the WADD objective for our setting,\ni.e., worst-case average detection delay over the worst pos sible\nrealizations as\nWADD(N)\n= sup\nν≥1limsup\nn→∞max\nxn∈C(n)esssup\nYν−1Eν((N−ν+1)+|xn,Yν−1).\n(9)\nThen, we say that WADD ‘D’ at FAR constraint ‘ α’ is\nachievable if there exists a sequence of codebooks C(n)and\nstopping rule Nsuch that\ninf\nN,{C(n)}:FAR(N)≤αWADD(N)≤D.\nThe ultimate goal of the joint communication and QCD\nproblem is to understand the fundamental tradeoff between R\nandDfor some fixed FAR constraint α, i.e., we aim to answer\n“What is the maximum information rate Rand minimum delay\nDthat a code can simultaneously achieve?”. In particular to\nthis work, we are interested in an asymptotic regime of QCD\nwhereα→0. To better characterize the tradeoff in this regime,\nwe define\n∆ := lim\nα→0|logα|\nD(10)and say that ∆is achievable if there exists a pair (α,D)\nthat asymptotically attains ∆asα→0. The reason for an\nadditional definition ∆will be apparent when we introduce\nour achievable region in Thm. 1, in which ∆is simply\ncharacterized in the form of a KL divergence formulation.\nMoreover, ∆represents the asymptotic “slope” of Das a\nfunction of α. Then, the goal of this work can be condensed\nto characterizing the set of achievable (R,∆)pairs, namely,\ntheR-∆region\nR=closure{(R,∆) :Rand∆are achievable }. (11)\nEquivalently, we can express Rusing the R-∆function\nR(∆) := sup {R: (R,∆′)is achievable for some ∆′≤∆}\n(12)\nand the∆-Rfunction\n∆(R) := sup{∆ : (R′,∆)is achievable for some R′≤R}\n(13)\nas\nR={(R,∆) :R≤R(∆),∆∈R+},\n={(R,∆) :R∈R+,∆≤∆(R)}.\nTo give some perspective to the problem, we can imme-\ndiately identify two extreme operating points. On one ex-\ntreme, we can design codes that maximize the communication\nrate while neglecting QCD. By Shannon’s capacity-achievin g\ncodes, the operating point (C,0)whereC= max pXI(X;˜Y)\nis an achievable pair. On the other extreme, we can design\ncodes for pure QCD detection which results in the operation\npoint(0,∆⋆)[21], where\n∆⋆:= max\nx∈XD(p(1)/⌊ard⌊lp(0)|x), (14)\nthat is, the optimal ∆when only QCD is considered. Extend-\ning on the two extreme points, we can alternate transmission\nbetween capacity-achieving codes and the optimal QCD input\nsymbol via timesharing. This results in half of the time send ing\ninformation and half of the time sending a QCD signal in an\nalternating fashion. Since we are communicating informati on\nonly half of the time, C/2is achieved for communication, and\nsince it takes twice as long for QCD, ∆⋆/2is achieved for\ndetection. Generalizing the time sharing strategy for a tim e\nsharing fraction λ∈[0,1],(λC,(1−λ)∆⋆)is achievable. In\naddition, the point (C,∆⋆)will serve as a trivial outer bound\nofR.\nIV. M AINRESULTS\nIn this section, we state our main results and discuss some\nimplications.\nTheorem 1. LetRinbe the set of rate-delay pairs (R,∆)∈\nR2\n+such that\nR < I(X;Y), (15)\n∆< D(p(1)/⌊ard⌊lp(0)|pX), (16)\nfor some pX, wherep(s):=pY|X,s. Then, Rin⊂R. In other\nwords, any point in Rinis achievable.5\nIn the following, after some remarks, we provide an\noverview of the achievability strategy. The detailed proof of\nThm. 1 is deferred to App. A.\nRemark 1. The region Rinis convex since the mutual\ninformation and the KL divergence are concave and linear in\npX, respectively. Thus, including timesharing does not expan d\ntheRinregion.\nRemark 2. The region Rinincludes two extreme points,\n(C,0)and(0,∆⋆), where\nC= max\npXI(X;˜Y),∆⋆= max\nx∈XD(p(1)/⌊ard⌊lp(0)|x).\nRemark 3. By the concavity of the mutual information and\nviewing (16) as a constraint on the distribution, i.e.,\nmax\npX:E(c(X))>∆I(X;Y) (17)\nwherec(x) =D(p(1)/⌊ard⌊lp(0)|X=x), we can easily apply the\nBlahut–Arimoto algorithm [28], [29] with input constraint s\ngiven in [28, Sec. IV] to evaluate the tradeoff region. The\nalgorithm for our case is summarized in App. D.\nFor the achievability proof, we use the constant subblock-\ncomposition codes (CSCCs) originally developed in [30]. Fi x\nan input type pXof length L, i.e.,pX∈ PL(X), and letRbe\nthe rate of the CSCC. Further, let nbe the codeword length\nthat is composed of ksubblocks of length L, i.e.,n=kL.\nThen,\nxn=xk= (x1,...,xk), (18)\nwherexj=xjL\n(j−1)L+1,j∈[1 :k]. Thus, we can view xnas a\nk-length sequence with L-length vector symbols. Let PL(X)\ndenote the set of all compositions of xLand letTL\npXbe the set\nof sequences in XLwith composition pX. Then, the codebook\nis generated as follows.\nCodebook generation: FixpX, and randomly and indepen-\ndently generate 2nRsequences xk(m)such that each vector of\nlengthLis i.i.d. uniformly distributed over TL\npX. The codebook\nis defined as C(n)={xk(m) :m∈[1 : 2nR]}.\nAs shown in [30], [34], by letting k→ ∞ , the following\nrate is achievable using CSCC:\nI(pX,p˜Y|X)−r(L,pX)≤R≤I(pX,p˜Y|X), (19)\nwhere the term r(L,pX)vanishes with L, which will be spec-\nified in App. A. Then, by letting L→ ∞ , rateR < I(X;˜Y)\nis achievable. Incorporating a deliberate detour using CSC C to\nshow that R < I(pX,p˜Y|X) =I(X;˜Y)is achievable serves\nas a crucial component in our QCD analysis, which follows\nbelow.\nFor the QCD analysis, aligning with the vector-symbol\ninterpretation of xn, let us view the QCD channel as a\ndiscrete memoryless channel over x, i.e.,p(s)(yk|xk) =/producttextk\nj=1p(s)(yj|xj), whereyj=yjL\n(j−1)L+1andp(s)(yj|xj) =\n/producttextL\ni=1p(s)\nY|X(y(j−1)L+i)|x(j−1)L+i),s= 0,1. Note that the\nobservations y1,y2,...,ykare produced from subblocks of\nthe same composition and can be made i.i.d. within the pre-\nchange and post-change regimes by proper permutation.Next, we propose to use an adapted QuSum test, which we\ncall a subblock-QuSum (SQS) test. Instead of the standard\nQuSum statistic that updates the statistic whenever it sees a\nnew observation, the subblock-QuSum test updates it only at\nthe end of subblocks. Formally, letting W0= 0, we construct\nthe SQS statistic as follows:\nWi=\n\n/parenleftbigg\nWi−1+logp(1)\nY|X(Yj|Xj)\np(0)\nY|X(Yj|Xj)/parenrightbigg+\nifi=jL,j∈N\nWi−1 otherwise .\n(20)\nWhenWi≥b, we stop updating and declare that a change\noccurred, i.e.,\nNSQS:= inf{i≥1 :Wi≥b}. (21)\nWith the combination of CSCCs and SQS, for some α >0,\nwe formally show in App. A that as α→0, it asymptotically\nholds that\nE∞(NSQS)≥1\nα\nand\nWADD(NSQS)≤|logα|\nD(p(1)/⌊ard⌊lp(0)|pX)(1+o(1)) =:D.\nThen, our CSCC code achieves ∆such that\n∆ = lim\nα→0|logα|\nD=D(p(1)/⌊ard⌊lp(0)|pX).\nThere are some extensions of Thm. 1 that are due. First,\nconsider the case where the state alphabet is |S|>2. The\ndefinition of WADD for this case is given by,\nWADD(N) :=¯Eν,s((N−ν+1)+|xn,Yν−1). (22)\nwhere\n¯Eν,s(·) := max\ns∈S\\{0}sup\nν≥1limsup\nn→∞max\nxn∈C(n)esssup\nYν−1Eν,s(·).\nThen, we can perform |S|−1SQS tests in parallel and pick the\nquickest change point estimate among the tests. This extens ion\nleads to the following corollary.\nCorollary 1. For the case |S|>2, the set of rate-delay pairs\n(R,∆)∈R2\n+such that\nR < I(X;Y), (23)\n∆<min\ns∈S\\{0}D(pY|X,s/⌊ard⌊lp(0)|pX), (24)\nfor some pXis achievable.\nRemark 4. By the standard quantization argument, e.g., [31,\nChap. 3.4.1], constant subblock-composition codes can be\nextended for Gaussian channels, which then allows the rate\nregion Rinto be applied to the Gaussian case. We provide\nsome discussions on some Gaussian settings in Sec. V.\nNext, we discuss some applications to the monostatic model\ngiven in Fig. 3 where p(y,˜y|x,s) =p(y|x,s)p(˜y|x). The\nonly difference from the problem statement of the bistatic\ncase is that the encoder is defined by a mapping that assigns\na codeword xi(m,yi−1),i∈[1 :n]to each message6\nEncoderDetector\npY,˜Y|X,S Decoder\nTransmitterXiYi−1\nMN\n˜YiˆM\nFig. 3: Monostatic model for ISAC. The encoder and QCD\ndetector are the same entity which has access to some feed-\nback.\nm∈[1 : 2nR]as well as a causal observation sequence yi−1.\nFor this case, Thm. 1 remains an inner bound since we can\ndisregard the observation yi−1in the codebook design and use\nthe same codebook and QCD detector defined for the bistatic\ncase. In general, the encoder structure for the bistatic cas e\ncould be improved by incorporating the feedback informatio n\nyi−1at the encoder. Some discussions on extensions to the\nbistatic case are given in Sec. VI.\nIn the following, we show that Thm. 1 is indeed a tight\nevaluation of CSCCs for our joint communication and QCD\nsetting, i.e., no other QCD detectors can be asymptotically\nbetter than the SQS detector. The converse result is shown on\na slightly wider class of codes defined below.\nDefinition 2 (sliding-window typical codes) .A codeC(n)is\nsliding-window typical withpXif for any ǫ >0, there exists\na window size L0(ǫ)such that for every xn∈ C(n)andi∈N\nwithi≤n−L0+1, it holds that\n/vextendsingle/vextendsingle/vextendsingleπ(a|xi+L0−1\ni)−pX(a)/vextendsingle/vextendsingle/vextendsingle≤ǫpX(a)for alla∈ X,(25)\nwhereπ(a|xi+L0−1\ni)is the type of the length L0sequence\nxi+L0−1\ni . Also, a sequence of codes C(n)is sliding-window\ntypical with pXif everyC(n)is sliding-window typical with\npX.\nThen, we have the following lower bound on WADD for\nthe class of sliding-window typical codebooks.\nTheorem 2. For a sequence of codebooks C(n)that is sliding-\nwindow typical with pXand any stopping time N, the QCD\ndelay WADD withFAR(N)≤αconstraint is bounded by\nWADD(N)≥ |logα|/parenleftbigg1\nD(p(1)/⌊ard⌊lp(0)|pX)+o(1)/parenrightbigg\nasα→0.\nProof: The proof is deferred to App. B.\nThe above theorem states that no matter what stopping\nrule is used, WADD cannot be asymptotically smaller than\n|logα|\nD(p(1)/bardblp(0)|pX). As the lower bound coincides with the delay\ncharacterization in Thm. 1, our SQS stopping rule is asymp-\ntotically optimal for CSCCs, or more generally, the class of\nsliding-window typical codes.\nTheorem 3. For the class of sliding-window typical codes,\nRinin Thm. 1 is tight, i.e., on the space of sliding-window\ntypical codes Rin=Rπ, where Rπis the rate-delay region\nfor sliding-window typical codes.0\n10\n11−ǫsǫs\n(a)p(s)\nY|X(y|x)0\n10\n11−ǫ\n1−ǫǫ\nǫ\n(b)p˜Y|X(˜y|x)\nFig. 4: Channels considered in the binary example.\n0 0.1 0.2 0.3 0.4 0.5 0.602468·10−2\n∆Rate [nats/c.u.]\nFig. 5: Plot for the binary channel example where ǫ= 0.3,\nǫ0= 0.1,ǫ1= 0.5.\nProof: The proof is deferred to App. C.\nUnlike standard converse proofs that consider any codes,\nthe above converse is limited to the class of sliding-window\ntypical codes. However, the above converse has an important\nimplication; in particular, it shows that our SQS detection\nalgorithm is in fact asymptotically optimal for the CSCC\ncodes. This is in a similar spirit to the results in [35] where it\nshows a converse only for the class of i.i.d. randomly genera ted\ncodes and proves that the proposed decoder is optimal, i.e.,\nequivalent to maximum likelihood (ML) decoders. In our\nsetting, showing the optimality of the detector is even more\nprominent since we do not have an evident optimal detector\n(c.f. ML for decoding), and defining a sequential detector is\nan essential part of the problem for QCD.\nV. E VALUATION EXAMPLES\nIn this section, we demonstrate how our results can be\napplied through some selection of example cases.\nA. Binary channels\nConsider the case in which the communication channel\np˜Y|Xis the standard binary symmetric channel with crossover\nprobability ǫand the QCD channel p(s)\nY|X=pY|X,S(y|x,s)is\na binaryZ-channel given by\np(0)(0|0) = 1, p(0)(0|1) =ǫ0 (26)\np(1)(0|0) = 1, p(1)(0|1) =ǫ1. (27)\nThe example is contrived so that sending the symbol x= 0\nis useless for the QCD as it always outputs Y= 0 symbol\nin pre- and post-change regimes. However, sending x= 0\nandx= 1 equally likely maximizes the capacity of the7\ncommunication channel, which implies there will be a tradeo ff\nbetween communication and QCD.\nFigure 5 depicts the tradeoff for our proposed strategy at\nǫ= 0.3,ǫ0= 0.1,ǫ1= 0.5. The linear line is a baseline\ntradeoff given by timesharing between pure communication\nand QCD. The capacity of the communication channel is given\nbyH2(pX)−H2(ǫ)withH2(·)being the binary entropy\nfunction, which is attained by a uniform distribution pX.\nOn the other hand, the optimal QCD-only point is ∆⋆=\nD(p(1)/⌊ard⌊lp(0)|X= 1) , i.e., QCD with a constant input symbol\nx= 1. The overall achievable region is given by the tradeoff\n(H2(pX)−H2(ǫ),D(p(1)/⌊ard⌊lp(0)|pX))where\nD(p(1)/⌊ard⌊lp(0)|pX)\n=pX(0)D(p(1)/⌊ard⌊lp(0)|X= 0)+pX(1)D(p(1)/⌊ard⌊lp(0)|X= 1)\n=pX(1)/parenleftbigg\nǫ1logǫ1\nǫ0+(1−ǫ1)log(1−ǫ1)\n(1−ǫ0)/parenrightbigg\n.\nThe circle dot point is the joint communication and QCD\nachievable point attained while maintaining capacity-ach ieving\ncommunication (C,∆(C)), i.e., when pXis uniform.\nB. Scalar Gaussian channels\nConsider the following Gaussian channels for QCD\nYi=Xi+Zi fori < ν,\nYi=hXi+Zifori≥ν,\nand the communication channel given by ˜Yi=Xi+˜Zi, where\nZi,˜Ziare independent standard Gaussian noise. The channel\ninput is subject to an average power constraint1\nn/summationtextn\ni=1x2\ni≤\nP. This setting represents the case, for example, when a user\nis jointly communicating and sensing for some disruption in\nits path, which will alter the channel gain.\nThe capacity of the communication channel is C=\n1\n2log(1 +P), attained by Gaussian input with N(0,P). For\nthe delay of QCD, note that when X=x,D(p(1)/⌊ard⌊lp(0)|X=\nx) =(h−1)2\n2x2. It in turn implies that ∆⋆in (14) is\n∆⋆= max\nx:x2≤PD(p(1)/⌊ard⌊lp(0)|x) =(h−1)2\n2P.\nOn the other hand, due to the power constraint,\n∆ =D(p(1)/⌊ard⌊lp(0)|pX) =/integraldisplay(h−1)2\n2x2p(x)dx≤(h−1)2\n2P,\nwhere the equality holds if the input is Gaussian distribute d\nwithN(0,P). Thus, by choosing X∼ N(0,P), the optimal\ncommunication rate and QCD delay are attained simultane-\nously, i.e., (C,∆⋆)is achievable where\n(C,∆⋆) =/parenleftbigg1\n2log(1+P),(h−1)2\n2P/parenrightbigg\n.\nSince the optimal QCD and the channel capacity are simulta-\nneously achieved, Rin=Rfor this case.\nNext, consider the following Gaussian channels, where the\nsensing channel is given by\nYi=Xi+Z(0)\ni fori≤ν,\nYi=Xi+Z(1)\ni fori > ν,whereZ(s)\ni∼ N(0,σ2\ns),s= 0,1,and the communication\nchannel is the same as the previous example. This setting\nrepresents the case, for example, when a user is jointly com-\nmunicating and sensing the existence of interference signa ls.\nIn this case, we can show that ∆⋆in (14) is given by\nD(p1/⌊ard⌊lp0|pX) =1\n2logσ2\n0\nσ2\n1+σ2\n1\n2σ2\n0−1\n2= ∆⋆,\nwhich is independent of pX. Thus,\n(C,∆⋆) =/parenleftbigg1\n2log(1+P),1\n2logσ2\n0\nσ2\n1+σ2\n1\n2σ2\n0−1\n2/parenrightbigg\nis attainable, which means that the operating point by the\nGaussian input with power Pis optimal for both commu-\nnication and QCD, i.e., Rin=R.\nFor both cases, no tradeoff between rate and delay is\nobserved. Interestingly, it coincides with a prior conclus ion\nfor binary state detection under Gaussian noise [10].\nC. MIMO Gaussian channels\nConsider the MIMO Gaussian sensing channels\nYi=G0xi+Zifori < ν,\nYi=G1xi+Zifori≥ν,\nand the communication channel ˜Yi=˜Gxi+˜Zi, where\nGs,s= 0,1and˜Gare the channel gain matrices for the\nsensing channels with state and the communication channel,\nrespectively. Here, Zi,˜Ziare independent vector Gaussian\nnoise with distribution N(0,I), where Iis the identity\nmatrix. Channel inputs are subject to a power constraint\n1\nn/summationtextn\ni=1/⌊ard⌊lxi/⌊ard⌊l2≤P.\nWe will first explore the pure QCD and pure communication\ncases. We begin by identifying ∆⋆for this case. When X=x,\nD(p(1)/⌊ard⌊lp(0)|X=x) =1\n2xT¯GT¯Gx=1\n2xTΓx.\nwhereΓ :=¯GT¯Gand¯G=G1−G0. Letting Γ =UΛUTbe\nthe singular value decomposition (SVD) of Γand¯x=UTx,\nD(p(1)/⌊ard⌊lp(0)|X=x) =1\n2xTΓx=1\n2¯xTΛ¯x.\nHence, the optimal QCD symbol for ∆⋆in (14) is ¯x=\n[√\nP,0,...,0]which leads to\n∆⋆= max\nx:/bardblx/bardbl2≤PD(p(1)/⌊ard⌊lp(0)|X=x) =1\n2λ1P, (28)\nwhereλ1is the largest singular value of Γ. In the following,\nwe show that a Gaussian distribution can also achieve ∆⋆,\nwhich will allow a positive communication rate. Note that\nsince/⌊ard⌊lUx/⌊ard⌊l=/⌊ard⌊lx/⌊ard⌊lfor any unitary matrix U,\nD(p(1)/⌊ard⌊lp(0)|pX) =E(D(p(1)/⌊ard⌊lp(0)|X))\n=1\n2E/parenleftBig\nXTUΛUTX/parenrightBig\n=1\n2E/parenleftBig\n¯XTΛ¯X/parenrightBig\n≤1\n2λ1P,\nwhere the equality is attained when ¯X∼ N(0,¯ΣX)with\n¯ΣX=diag([P,0,...,0]). It means that for QCD, an alterna-\ntive optimal strategy is to send a vector Gaussian random cod e\nby allocating all the power to the sub-channel with maximum8\n0 2 4 6 8 10 12 14 16 18 20 2200.511.52\n∆Rate [nats/s/Hz]\nFig. 6: Plot for the MIMO Gaussian example with P= 10 .\nsingular value. In a sense, the strategy translates to achie ving\nmaximum diversity gain for detection.\nOn the other hand, the communication rate for a given\ncovariance matrix ΣXis given by\nI(X;˜Y) =1\n2log/vextendsingle/vextendsingle/vextendsingleI+˜GΣX˜GT/vextendsingle/vextendsingle/vextendsingle.\nThe optimal input distribution for the communication-only\ncase is attained by beamforming with respect to SVD and\nwaterfilling over the singular values of ˜G[36].\nThis example entails several interesting observations in t he\ntradeoff between communication and QCD. On the one hand,\nto achieve the best QCD detection delay, the transmitter sho uld\nallocate all its power to the best subchannel with respect to\nthe channel gain difference matrix ¯G. On the other hand, the\nbest communication strategy is to multiplex over subchanne ls\nof˜G, i.e., distribute power across the subchannels of ˜Gwith\na waterfilling policy. Compared to the single antenna case,\nthe MIMO scenario has a more complicated tradeoff in that\nthe mismatch in beamforming and power allocation due to\nthe difference in ¯Gand˜Gresults in a more delicate tradeoff\ninvolving diversity vs. multiplexing gains.\nWe plot an example case for the MIMO setup in Fig. 6\nwhere we assume power constraint P= 10 and\n¯G=/bracketleftbigg\n2 0\n0 1/bracketrightbigg\n,˜G=1√\n2/bracketleftbigg\n1 1\n1−1/bracketrightbigg\n. (29)\nNote that ¯Ghas a strictly maximal singular value while ˜G\nhas two equal singular values. For optimal QCD, the optimal\nstrategy is to allocate all power to the maximum singular\nvalue of ¯Gwhich achieves ∆⋆, while the best strategy for\ncommunication is to allocate equal power to both antennas\nwhich achieves the capacity C. The plot in Fig. 6 shows Rin\nfor the MIMO example. The red dots are the points (C,∆(C))\nand(R(∆⋆),∆⋆).\nVI. D ISCUSSIONS\nThis paper studies the fundamental tradeoff between com-\nmunication rate and quickest change detection delay in the\ncontext of ISAC. Our goal in this work was to establish\nsome firm foundations by taking essential initial steps towa rds\ndetermining the fundamental tradeoff. Nonetheless, sever al\nimportant open problems are still due, for instance, a tight erinner bound and a generic outer bound for all possible codes\non the region R. Since the QCD formulation requires good\ncontrol over the empirical distribution of codewords, deve l-\noping a generic outer bound seems to involve analysis of the\noptimal code statistics with constraints, which is itself a widely\nactive field of study, e.g., [37]–[39]. A step in this directi on\nwould help us better understand the fundamental limits of\njoint communication and QCD. Another interesting directio n\nis to develop a strategy specifically tailored for the monost atic\nmodel in Fig. 3. Here, the main challenge is that the encoder\ncan be interactive, which opens the possibility to encode wi th\nthe knowledge of feedback, i.e., xi(m,yi−1). While Thm. 1\nprovides a valid inner bound for the monostatic case, whethe r\nan interactive encoder can improve the tradeoff performanc e\nis yet another interesting direction for future work.\nAPPENDIX A\nPROOF OF THEOREM 1\nIn this section, we begin with the analysis on the\nachievable rate of constant subblock-composition codes\n(CSCCs) and then provide the analysis of the delay per-\nformance. Notice that it is sufficient to show that a pair\n(I(pX,p˜Y|X),D(p(1)/⌊ard⌊lp(0)|pX))is achievable for an arbitrary\ninput distribution pX. To this end, we fix an arbitrary ǫ >0\nand show that (I(pX,p˜Y|X)−2ǫ,D(p(1)/⌊ard⌊lp(0)|pX)(1+ǫ))is\nachievable. Let Lbe a fixed integer that is large enough so\nthatp(L)\nX∈ PL(X)is “close” to pX. Additional conditions\nwill be stated throughout the proof.\nA. Coding Rate Analysis\nLet CSCC (L,p(L)\nX)be the constant subblock-composition\ncode of subblock length Land composition type p(L)\nX∈\nPL(X). Note that the entire blocklength n=kL, wherek\nis the number of subblocks, and the blocklength will first be\nscaled with kto obtain infinitely long codewords.\nLetr(L,p(L)\nX)be a positive number defined by\nr(L,p(L)\nX) =(s(p(L)\nX)−1)log(2πL)\n2L\n+1\n2L/summationdisplay\na:p(L)\nX(a)>0logp(L)\nX(a)+u(L,p(L)\nX)s(p(L)\nX)\n12Lln2,\nwheres(p(L)\nX)is the number of elements x∈ X such that\np(L)\nX(x)>0and0≤u(L,p(L)\nX)≤1is some bounded function\n[34, p. 26]. Then, the coding rate of CSCC (L,p(L)\nX)is given\nas follows.\nLemma 1 (Thm. 7 in [30]) .The code rate Rof the\nCSCC(L,p(L)\nX)with vanishing maximal block error probability\nasn→ ∞ is bounded by\nI(p(L)\nX,p˜Y|X)−r(L,p(L)\nX)≤R≤I(p(L)\nX,p˜Y|X).\nAsLis large and the mutual information is continuous\nin its input distribution, p(L)\nXcan be chosen close to pXso\nthatI(pX,p˜Y|X)−ǫ < I(p(L)\nX,p˜Y|X). Also, the rate penalty\ntermr(L,p(L)\nX)diminishes with Lsincer(L,p(L)\nX) =O(logL\nL)9\nandp(L)\nXstays around pX. Therefore, r(L,p(L)\nX)< ǫ, which\nimplies that R > I(pX,p˜Y|X)−2ǫis achievable.\nB. QCD Delay Analysis\nFor the delay analysis, we require the following lemma.\nNotice that when Z1,Z2,... are all i.i.d., this lemma reduces\nto the original Wald’s identity [40, Thm. 4.8.6].\nLemma 2 (Modified Wald’s identity) .LetZ1,Z2,... be\nindependent and Z2,Z3,... be identically distributed with\nfinite mean. Then, for any stopping time Nthat is finite with\nprobability 1,\nE/parenleftBiggN/summationdisplay\ni=1Zi/parenrightBigg\n=E(Z2)E(N)+(E(Z1)−E(Z2))P(1≤N).\nFurther, suppose that |Zi|< c for alliwith probability 1.\nThen,\nE/parenleftBiggN/summationdisplay\ni=1Zi/parenrightBigg\n≥E(Z2)E(N)−2c.\nProof of Lemma 2: Note that 1{N≥i}= 1−1{N≤\ni−1}. Then, the right side only depends on Z1,Z2,...,Z i−1\nby the definition of the stopping time, which also implies tha t\nthe left side is independent of Zi. Therefore,\nE/parenleftBiggN/summationdisplay\ni=1Zi/parenrightBigg\n=E/parenleftBigg∞/summationdisplay\ni=1Zi1{i≤N}/parenrightBigg\n=∞/summationdisplay\ni=1E(Zi1{i≤N})\n(a)=∞/summationdisplay\ni=1E(Zi)E(1{i≤N}) =∞/summationdisplay\ni=1E(Zi)P(i≤N)\n=/parenleftBigg∞/summationdisplay\ni=1E(Z2)P(i≤N)/parenrightBigg\n+(E(Z1)−E(Z2))P(1≤N)\n(b)=E(Z2)E(N)+(E(Z1)−E(Z2))P(1≤N),\nwhere(a)follows from the independence, and (b)follows\nfromE(N) =/summationtext∞\ni=1P(i≤N). The first claim is proved.\nThe second claim is proved as follows:\nE/parenleftBiggN/summationdisplay\ni=1Zi/parenrightBigg\n=E(Z2)E(N)+(E(Z1)−E(Z2))P(1≤N)\n≥E(Z2)E(N)−E(|Z1|)−E(|Z2|)\n≥E(Z2)E(N)−2c.\nWe are now ready to analyze the delay asymptotics. Con-\nsider a sequence of CSCCs C(n),n=kL, withk→ ∞ andL\nbeing fixed. For QCD, we use the subblock-QuSum (SQS) test\ndefined in (20) and (21). By its construction, Wionly changes\nati=jL, and therefore, NSQS∈ {jL}j∈N.\nA key idea that connects our CSCC-based QCD and the\nstandard QCD for i.i.d. observations is from the fact that\nby properly permuting transmission symbols of any code-\nword, each subblock of codewords can be thought of as a\nrepetition of x1(1), the first subblock of xn(1). Hence, for\nanyxn(m), the permuted observation vectors are i.i.d. ob-\nservations from x1(1). To be precise, for the j-th subblock\nof them-th codeword, let σj,m be the permutation suchthatσj,m(xj(m)) =x1(1), which always exists since all\nsubblocks in codewords are of the same composition. For\nobservations, let yj′(m) :=σj,m(yj(m))be a vector obtained\nfrom observations in the j-th subblock under permutation\nσj,m. Then, the distributions of subblocks of observations can\nbe categorized into three classes: If the j-th subblock is in the\nprechange period, the j-th random vector after permutation,\nY′\nj(m), is identically distributed to the unpermuted observa-\ntion vector Y1(1)obtained from x1(1), i.e.,\np(0)\nY1|x1(y′\nj) =L/productdisplay\ni=1p(0)\nY|X(y′\n(j−1)L+i|xi(1)).\nSimilarly, if the subblock is strictly after the change poin t,\nthen the permuted observation vectors are distributed by\np(1)\nY1|x1(y′\nj) =L/productdisplay\ni=1p(1)\nY|X(y′\n(j−1)L+i|xi(1)).\nHowever, if the change is in the middle of the subblock, the\npermuted observations’ distribution is not p(1)\nY1|x1norp(0)\nY1|x1.\nLetN′\nSQS:=NSQS/L= inf{j≥1 :WjL≥b}be\nthe equivalent stopping time, which sees the following QCD\nproblem after proper permutations:\nY′\nj∼p(0)\nY1|x1ifjL < ν\nY′\nj∼˜pY1|x1if(j−1)L+1< ν≤jL\nY′\nj∼p(1)\nY1|x1ifν≤(j−1)L+1\nwherex1=x1(1)is the reference transmission vector and\n˜pY1|x1is some distribution that depends on the location of\nνandx1. The first and last intervals are for the vectors\nthat are completely within the pre-change and post-change\nperiods, respectively. The second interval is for the vecto r that\ncontains the change point ν. Note that vectors Y′\n1,Y′\n2,... are\nindependent conditioned on a transmitted codeword since th e\nchannel is memoryless.\nThe QCD setup described above differs from standard cases\nif the change occurs in between a subblock, i.e., νis in the\nmiddle of a subblock. The precise distribution of ˜pY1|x1will\nbe irrelevant in the proof steps, however, what is relevant i s\nthe fact the it is not necessarily distributed by the alterna tive\nhypothesis p(1)\nY1|x1. Fortunately, even in this scenario, one can\nextend Lorden’s analysis to obtain bounds on the performanc e\nin terms of sequential probability ratio tests (SPRTs) [18,\nThm. 2].\nTo be precise, consider the following one-sided SPRT\nwith stopping time N′\nSPRT. LetY′\n1,Y′\n2,... be independent\nobservations conditioned on the transmitted codeword such\nthat\nifH0:Y′\nj∼p(0)\nY1|x1for allj,\nifH1:Y′\n1∼˜pY1|x1,Y′\nj∼p(1)\nY1|x1forj≥2,\nN′\nSPRT:= inf\n\nj≥1 : log˜pY1|x1\np(0)\nY1|x1(Y′\n1)\n+j/summationdisplay\nk=2logp(1)\nY1|x1\np(0)\nY1|x1(Y′\nk)≥b\n\n,10\nwhereH0andH1are the null and alternative hypotheses,\nrespectively. As above, it is slightly different from the st andard\nSPRT in that the first observation under the alternative hypo th-\nesis does not follow p(1)\nY1|x1. Further, note that p(0)\nY1|x1,p(1)\nY1|x1remain unchanged regardless of the choice of codewords if\nproper permutations are available.\nNote that our N′\nSQScan be interpreted as a set of\nSPRTs where N′\nSPRT,jdenotes the SPRT applied to subblocks\nY′\nj,Y′\nj+1,..., by\nN′\nSQS= min{N′\nSPRT,j+j−1,j= 1,2,...}. (30)\nLetν′∈[1 :k]be the subblock index that contains ν. Then,\nthe analysis for the standard i.i.d. QCD/SPRT problem where\n˜p=p(1)[18, Thm. 2] can be extended to our vector-symbol\nQCD/SPRT problem that has ˜p/ne}ationslash=p(1), by which we have the\nfollowing lemma.\nLemma 3. Consider a sequence of CSCC codebooks C(n). If\nlimsup\nn→∞max\nxn∈C(n)P(N′\nSPRT<∞|xn,H0)≤α′,\nthen for our QCD problem,\nlimsup\nn→∞max\nxn∈C(n)E∞(N′\nSQS|xn)≥1\nα′(31)\nand\nsup\nν′≥1limsup\nn→∞max\nxn∈Cnesssup\nYν′−1Eν((N′\nSQS−ν′+1)+|xn,Yν′−1)\n≤max{limsup\nn→∞max\nxn∈C(n)E1(N′\nSPRT,1|xn),E2(N′\nSPRT,2)}.\n(32)\nProof of Lemma 3: The proof of (31) follows exactly the\nsame steps as in the proof of [18, Thm. 2] since the vectors of\nobservations with proper permutation are i.i.d. under H= 0.\nFor the proof of (32), by (30), if ν′= 1, for every xn,\nE1(N′\nSQS|xn)≤E1(N′\nSPRT,1|xn),\nand ifν′≥2,\nEν′((N′\nSQS−ν′+1)+|xn,Yν′−1)≤Eν′(N′\nSPRT,ν′|xn,Yν′−1)\n=Eν′(N′\nSPRT,ν′)\n=E2(N′\nSPRT,2),\nwhere the first equality holds since permuted vectors are\ni.i.d. ifν′≥2and are independent of the past ones. Hence,\nthe claim is proved.\nNow, the problem reduces to bounding E1(N′\nSPRT,1|xn)\nandE2(N′\nSPRT,2). Following Wald’s SPRT analysis [41], [42,\nChap. 9], the false alarm probability is P(false alarm ) :=\nlimsupn→∞maxxn∈C(n)P(N′\nSPRT<∞|xn,H0)≤e−b,\nwherebis the threshold in N′\nSPRT. Also, note that since\n|X|,|Y|are finite,\nmax\nx,y/vextendsingle/vextendsingle/vextendsingle/vextendsinglelogp(1)(y|x)\np(0)(y|x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=:γ <∞. (33)\nLet\nSj:= log˜pY1|x1\np(0)\nY1|x1(Y′\n1)+j/summationdisplay\nk=2logp(1)\nY1|x1\np(0)\nY1|x1(Y′\nk).Then,E1(N′\nSPRT,1|xn)can be bounded by the modified Wald’s\nidentity in Lem. 2 as follows: for a large enough nandxn∈\nC(n),\nb(1+o(1))\n(a)=E1(SN′\nSPRT,1|xn)\n(b)=E1(N′\nSPRT,1|xn)E1\nlogp(1)\nY1|x1\np(0)\nY1|x1(Y′\n2)\n\n+E1\nlog˜pY1|x1\np(0)\nY1|x1(Y′\n1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexn\nP(1≤N′\nSPRT,1|xn)\n−E1\nlogp(1)\nY1|x1\np(0)\nY1|x1(Y′\n2)\nP(1≤N′\nSPRT,1|xn)\n(c)\n≥E1(N′\nSPRT,1|xn)E1\nlogp(1)\nY1|x1\np(0)\nY1|x1(Y′\n2)\n−2γL\n=E1(N′\nSPRT,1|xn)D/parenleftBig\np(1)\nY1x1/vextenddouble/vextenddoublep(0)\nY1|x1/parenrightBig\n−2γL, (34)\nwhere(a)follows since the overshoot is negligible if bis\nlarge, and (b),(c)follow from the modified Wald’s identity\nwith (33). Then, (34) implies that for large enough n,\nE1(N′\nSPRT,1|xn)≤b(1+o(1))+2γL\nD/parenleftBig\np(1)\nY1|x1/vextenddouble/vextenddoublep(0)\nY1|x1/parenrightBig, (35)\nwhich is independent of what xnwas transmitted. Also, Wald’s\noriginal identity [40, Thm. 4.8.6] computes E2(N′\nSPRT,2)as\nfollows:\nE2(N′\nSPRT,2)≤b(1+o(1))\nD/parenleftBig\np(1)\nY1|x1/vextenddouble/vextenddoublep(0)\nY1|x1/parenrightBig.\nLettingα′=e−b, we have b=|logα′|; in other words,\nE∞(N′\nSQS|xn)≥1\nα′for large enough nand\nsup\nν′≥1limsup\nn→∞max\nxn∈Cnesssup\nYν′−1Eν′((N′\nSQS−ν′+1)+|xn,Yν′−1)\n≤|logα′|(1+o(1))+2γL\nD/parenleftBig\np(1)\nY1|x1/vextenddouble/vextenddoublep(0)\nY1|x1/parenrightBig.\nFinally, using two facts that NSQS=L·N′\nSQSand\nD/parenleftBig\np(1)\nY1|x1/vextenddouble/vextenddoublep(0)\nY1|x1/parenrightBig\n=L·D(p(1)/⌊ard⌊lp(0)|p(L)\nX)by the property\nof the KL divergence for product distributions, one can conv ert\nit back to the original QCD bounds given by E∞(NSQS|xn)≥\nL\nα′and\nsup\nν≥1limsup\nn→∞max\nxn∈Cnesssup\nYν−1Eν((NSQS−ν+1)+|xn,Yν−1)\n≤L·|logα′|(1+o(1))+2γL2\nL·D(p(1)/⌊ard⌊lp(0)|p(L)\nX)\n=|logα′|(1+o(1))+2γL\nD(p(1)/⌊ard⌊lp(0)|p(L)\nX).\nBy taking α:=α′\nL⇔logα′= logα+logL,\nE∞(NSQS|xn)≥1\nα11\nand\nWADD(NSQS)≤|logα|(1+o(1))+log L+2γL\nD(p(1)/⌊ard⌊lp(0)|p(L)\nX).\nAsLis fixed but large, one can express the KL divergence\nasD(p(1)/⌊ard⌊lp(0)|p(L)\nX)< D(p(1)/⌊ard⌊lp(0)|pX)/(1 +ǫ). Then, for\nsmall enough α, we have E∞(NSQS|xn)≥1\nαand\nWADD(NSQS)≤|logα|(1+o(1))(1+ ǫ)\nD(p(1)/⌊ard⌊lp(0)|pX)=:D.\nIt proves the claim that\n∆ = lim\nα→0|logα|\nD=D(p(1)/⌊ard⌊lp(0)|pX)(1+ǫ).\nAsǫis arbitrary, we proved the claim.\nAPPENDIX B\nPROOF OF THEOREM 2\nBefore discussing the proof of Thm. 2, we state the fol-\nlowing result by Lai [20], where observations are generic\nsequences that are not necessarily independent nor identic ally\ndistributed. Here, P(ν)denotes the probability measure when\nthe change occurs at ν.\nLemma 4 ( [20, Thm. 1]) .Suppose that a QCD receiver has\nobservations Y1,Y2,... that satisfy\n˜P(ν)/parenleftBigg\nmax\nt≤T1\nTν+t−1/summationdisplay\ni=νlogp(1)(Yi)\np(0)(Yi)≥(1+δ)c/vextendsingle/vextendsingle/vextendsingle/vextendsingleYν−1/parenrightBigg\n= 0\n(36)\nfor anyδ >0with some positive constant c >0, where\n˜P(ν)(·) = lim T→∞supν≥1esssupYν−1P(ν)(·). Then, for any\nstopping rule Nsuch that1\nE∞(N)≤α,\nsup\nν≥1esssup\nYν−1Eν((N−ν+1)+|Yν−1)\n≥ |logα|/parenleftbigg1\nc+o(1)/parenrightbigg\nasα→0.\nSince Lai’s result states a lower bound on the QCD delay for\nmore general observations beyond the i.i.d. case, it is suffi cient\nto show that a sliding-window typical sequence of codes yiel ds\nQCD observations that satisfy the condition (36). Let c=\nD(p(1)/⌊ard⌊lp(0)|pX)andL0be the interval length satisfying (25)\nfor some ǫ >0. To give some preparation for what follows,\nwe note that L0serves as an interval length such that xL0is\ntypical, i.e., it plays the same role as Lin CSCCs. Also, K0L0\nwill serve as an interval length such that YK0L0is typical. To\nbe clear, intervals of length K0L0andL0are denoted by\nsubintervals and microintervals, respectively.\nSuppose that Tin (36) grows as a multiple of K0L0, i.e.,\nT=k(K0L0),k∈N, whereK0is an integer that will be\nspecified later. Then, for a sequence of C(n), the condition to\nshow for our setting is that for any δ >0,\n¯P(ν)/parenleftBigg\nmax\nt≤kK0L01\nkK0L0ν+t−1/summationdisplay\ni=νZi≥(1+δ)c/vextendsingle/vextendsingle/vextendsingle/vextendsinglexν/parenrightBigg\n= 0,\n(37)whereZi:= logp(1)(Yi|xi)\np(0)(Yi|xi),xν=xν+kK0L0−1\nν ,\n¯P(ν)(·) = lim\nk→∞sup\nν≥1limsup\nn→∞max\nxn∈CnP(ν)(·).\nHere,Yν−1is dropped since observations from a memoryless\nchannel are independent of the past observations condition ed\nonxν, and thus, Zi,i≥νis independent of the past.\nNote that [ν:ν+t−1]consists of ⌊t/K0L0⌋subintervals\nof length K0L0and residuals of length less than K0L0, the\neffect of which becomes negligible as ⌊t/K0L0⌋grows, i.e.,\n1\nkK0L0ν+t−1/summationdisplay\ni=νZi\n=1\nk·1\nK0L0\n⌊t/K0L0⌋/summationdisplay\nj=1ν+jK0L0−1/summationdisplay\ni=ν+(j−1)K0L0Zi+ν+t−1/summationdisplay\ni=ν+⌊t/K0L0⌋L0Zi\n\n=1\nk·1\nK0L0⌊t/K0L0⌋/summationdisplay\nj=1ν+jK0L0−1/summationdisplay\ni=ν+(j−1)K0L0Zi+o(1) ask→ ∞.\nFix an arbitrary δ >0and consider the first subinterval of\nlengthK0L0, i.e.,i∈[ν:ν+K0L0−1], that consists of\nK0micro-intervals. Then, the sum over this subinterval can\nbe rewritten by dividing it into |X|partial sums according to\nxand then further dividing it into |Y|sums,\n1\nK0L0ν+K0L0−1/summationdisplay\ni=νlogp(1)(yi|xi)\np(0)(yi|xi)\n=/summationdisplay\nx∈Xnx\nK0L0/parenleftBigg\n1\nnx/summationdisplay\ni:xi=xlogp(1)(yi|x)\np(0)(yi|x)/parenrightBigg\n=/summationdisplay\nx∈Xnx\nK0L0\n/summationdisplay\ny∈Yny|x\nnxlogp(1)(y|x)\np(0)(y|x)\n,\nwherenx:=|{i∈[ν:ν+K0L0−1] :xi=x}|and\nny|x:=|{i∈[ν:ν+K0L0−1] :xi=x,yi=y}|. Since\nxnis sliding-window typical, |na\nK0L0−p(a)| ≤ǫp(a)for all\na∈ X by the definition of L0. Furthermore, by the conditional\ntypicality lemma [31], if K0is sufficiently large, yν+K0L0−1\nν is\nconditionally ǫ′-typical with high probability. In other words,\nfor anyx∈ X,y∈ Y , it holds that |ny|x\nnx−p(1)(y|x)| ≤\nǫ′p(1)(y|x)with probability greater than 1−δ′(ǫ′).\nCombining the above for jointly typical sequences, i.e., wi th\nprobability greater than 1−δ′(ǫ′),\n1\nK0L0ν+K0L0−1/summationdisplay\ni=νlogp(1)(yi|xi)\np(0)(yi|xi)\n≤/summationdisplay\nx∈X(1+ǫ)pX(x)\n/summationdisplay\ny∈Y(1+ǫ′)p(1)(y|x)logp(1)(y|x)\np(0)(y|x)\n\n= (1+ǫ)(1+ǫ′)D(p(1)/⌊ard⌊lp(0)|pX).\nOn the other hand, for atypical observation sequences, i.e. ,\nwith probability less than δ′(ǫ′),\n1\nK0L0ν+K0L0−1/summationdisplay\ni=νlogp(1)(yi|xi)\np(0)(yi|xi)≤γ,12\nwhereγis defined in (33). Then, one can verify that by taking\nK0appropriately so that\nµ1:=E/parenleftBigg\n1\nK0L0ν+K0L0−1/summationdisplay\ni=νlogp(1)(Yi|xi)\np(0)(Yi|xi)/parenrightBigg\n≤(1+ǫ′′)D(p(1)/⌊ard⌊lp(0)|pX)\n<(1+δ)D(p(1)/⌊ard⌊lp(0)|pX). (38)\nNext, noting that other subintervals are independent condi -\ntioned on xn, we apply the same argument to other subinter-\nvals. To simplify notation, define Aj,j∈[1 :k], such that\nAj:=1\nK0L0ν+jK0L0−1/summationdisplay\ni=ν+(j−1)K0L0logp(1)(Yi|xi)\np(0)(Yi|xi)−µj,\nwhereµjare positive values defined in a similar way to µ1.\nNote that Ajhas zero mean and finite variance. Then, the\nprobability in (37) can be rewritten as\nP(ν)\nmax\nt′≤kt′/summationdisplay\nj=1(Aj+µj)≥k(1+δ)D+o(k)/vextendsingle/vextendsingle/vextendsingle/vextendsinglexν\n,\nwhereD=D(p(1)/⌊ard⌊lp(0)|pX). Then,\nP(ν)\nmax\nt′≤kt′/summationdisplay\nj=1(Ai+µj)≥k(1+δ)D+o(k)/vextendsingle/vextendsingle/vextendsingle/vextendsinglexν\n\n≤P(ν)\nmax\nt′≤kt′/summationdisplay\nj=1Ai≥k(1+δ)D−t′/summationdisplay\nj=1µj+o(k)/vextendsingle/vextendsingle/vextendsingle/vextendsinglexν\n\n(a)\n≤P(ν)\nmax\nt′≤kt′/summationdisplay\nj=1Ai≥k(δ−ǫ′′)D+o(k)/vextendsingle/vextendsingle/vextendsingle/vextendsinglexν\n\n≤P(ν)\nmax\nt′≤k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet′/summationdisplay\nj=1Ai/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥k(δ−ǫ′′)D+o(k)/vextendsingle/vextendsingle/vextendsingle/vextendsinglexν\n\n(b)\n≤kmaxjVar(Aj)\nk2(δ−ǫ′′)2D2(p(1)/⌊ard⌊lp(0)|pX)+o(k2)=O(1/k)→0,\nwhere(a)follows from (38) and (b)follows from Kol-\nmogorov’s maximal inequality [40, Thm. 2.5.5]. Since the\nbounds do not depend on νnor a specific choice of xn∈\nC(n), we have proved (37) for T=kK0L0. The proof for\nT/ne}ationslash=kK0L0is immediate since the effect of residuals is\nnegligible.\nAPPENDIX C\nPROOF OF THEOREM 3\nSuppose that (R,∆)such that R >0and∆>0is\nachievable by a sequence of sliding-window typical codes.\nThen, by the relation of ∆andWADD , for any ǫ >0, there\nexists a stopping rule N⋆such that for sufficiently small α\nand sufficiently large n,\n∆≤|logα|\nWADD(N⋆)(1+ǫ)⇒WADD(N⋆)≤|logα|\n∆(1+ǫ).\n(39)Also, from Thm. 2, if the sliding-window typical codes is wit h\npX, then for any stopping rule N,\nWADD(N)≥ |logα|/parenleftbigg1\nD(p(1)/⌊ard⌊lp(0)|pX)+o(1)/parenrightbigg\n.(40)\nCombining (39) and (40), we can conclude that pXsliding-\nwindow typical codes can achieve ∆only if\n∆≤D(p(1)/⌊ard⌊lp(0)|pX). (41)\nViewing the KL divergence for each input symbol as\nan input cost, i.e., c(x) :=D(p(1)/⌊ard⌊lp(0)|x),x∈ X ,\n(41) can be treated as input cost constraint on the code:\nD(p(1)/⌊ard⌊lp(0)|pX) =EpX[c(X)]≥∆. Hence, finding an upper\nbound on the coding rate with the cost constraint gives an\nupper bound on Rπ(∆), where\nRπ(∆) := sup {R: (R,∆′)is achievable using a\nsliding-window typical code for some ∆′≤∆}.\nTo this end, define C(∆) := max pX:E(c(X))≥∆I(X;˜Y).Note\nthatC(∆) is nonincreasing, concave, and continuous in ∆.\nSince our sequence of codes is sliding-window typical, for\nanyδ >0, there exists a large enough n0(δ)such that\n1\nn0n0/summationdisplay\ni=1D(p(1)/⌊ard⌊lp(0)|xi)≥D(p(1)/⌊ard⌊lp(0)|pX)−δ\n=EpX[c(X)]−δ\n≥∆−δfor allxn0∈ C(n0).\nBy standard converse proof steps using Fano’s inequality\nand the data processing inequality, if n≥n0,\nnRπ≤I(Xn;˜Yn)+nǫn\n≤n/summationdisplay\ni=1I(Xi;˜Yi)+nǫn\n(a)\n≤n/summationdisplay\ni=1C(E(c(Xi)))+nǫn\n(b)\n≤nC/parenleftBigg\n1\nnn/summationdisplay\ni=1E(c(Xi))/parenrightBigg\n+nǫn\n(c)\n≤nC(∆−δ)+nǫn\n(d)\n≤nC(∆)+nǫn+ǫ′(δ)\n=n max\npX:D(p(1)/bardblp(0)|pX)≥∆I(X;˜Y)+nǫn+ǫ′(δ),\nwhere step (a)follows from the definition of C(·)and steps\n(b)–(d)follow since C(·)is concave, nonincreasing, and\ncontinuous. Here, ǫ′is a quantity that depends on δand\nvanishes if δ→0. We have derived the upper bound that\ncoincides with Rin, which completes the proof.\nAPPENDIX D\nBLAHUT –ARIMOTO ALGORITHM\nWe present the Blahut–Arimoto algorithm for the achievable\nrate region in Thm. 1, i.e., numerical evaluation of Rin\nin (17). The key idea of the original Blahut–Arimoto algorit hm13\n[28], [29] is that the maximum mutual information can be\nfound by an iterative optimization between input and poster ior\ndistributions, r(x),q(x|˜y), respectively.\nFor our setting, the objective function to maximize is writt en\nas\nJ(r,q) =/summationdisplay\nx/summationdisplay\n˜yr(x)p(˜y|x)logq(x|˜y)\nr(x)+λ/summationdisplay\nxr(x)c(x),\nwhereλ∈[0,∞)andc(x) =D(p(1)/⌊ard⌊lp(0)|X=x). 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Veeravalli, Statistical Inference for Engineers and Data\nScientists . Cambridge, U.K.: Cambridge University Press, 2019."    },    {        "title": "2401.12528v1.Landau_Level_Mixing_and_SU_4__Symmetry_Breaking_in_Graphene.pdf",        "content": "Landau-Level Mixing and SU(4) Symmetry Breaking in Graphene\nNemin Wei,1Guopeng Xu,2Inti Sodemann Villadiego,3and Chunli Huang2\n1Department of Physics, Yale University, New Haven, CT 06520, USA\n2Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA\n3Institut fur Theoretische Physik, Universitat Leipzig, Bruderstrabe 16, 04103, Leipzig, Germany\n(Dated: January 24, 2024)\nRecent scanning tunneling microscopy experiments on graphene at charge neutrality under strong magnetic\nfields have uncovered a ground state characterized by Kekul ´e distortion (KD). In contrast, non-local spin and\ncharge transport experiments in double-encapsulated graphene, which has a higher dielectric constant, have\nidentified an antiferromagnetic (AF) ground state. We propose a mechanism to reconcile these conflicting\nobservations, by showing that Landau-level mixing can drive a transition from AF to KD with the reduction\nof the dielectric screening. Our conclusion is drawn from studying the effect of Landau-level mixing on the\nlattice-scale, valley-dependent interactions to leading order in graphene’s fine structure constant 𝜅=𝑒2/(ℏ𝑣𝐹𝜖).\nThis analysis provides three key insights: 1) Valley-dependent interactions remain predominantly short-range\nwith the𝑚=0 Haldane pseudopotential being at least an order of magnitude greater than the others, affirming\nthe validity of delta-function approximation for these interactions. 2) The phase transition between the AF and\nKD states is driven by the microscopic process in the double-exchange Feynman diagram. 3) The magnitudes of\nthe coupling constants are significantly boosted by remote Landau levels. Our model also provides a theoretical\nbasis for numerical studies of fractional quantum Hall states in graphene.\nIntroduction: The rich interplay between interactions,\ntopology, and the approximate spin-valley SU(4) symmetry\nin graphene quantum Hall physics continues to spark puzzles\neven after two decades of research [1–14] and could provide\ninsights into more complex correlated states such as those\nemerging in moir ´e superlattice flatbands [15–17]. A recent\nnotable discovery is the Kekul ´e distorted (KD) state identi-\nfied by scanning tunneling microscopy (STM) experiments in\ncharge neutral monolayer graphene [18–20], which is in ten-\nsion with the antiferromagnetic (AF) state supported by earlier\nnon-local spin and charge transport experiments [21–26]. We\npresent a model to reconcile these diverging observations.\nKharitonov [27] proposed that the phase diagram of neutral\ngraphene under a strong magnetic field can be conveniently\nrepresented in a two-dimensional parameter space defined by\n𝑢⊥and𝑢𝑧. These phenomenological parameters, describing\nthe strength of valley-exchange ( 𝑢⊥) and valley-antisymmetric\n(𝑢𝑧) interactions that breaks the SU(4) symmetry in the zeroth\nLandau level (LL), are influenced by electron-electron and\nelectron-phonon potentials. Given their complex origins, they\nare often regarded as experimental fitting parameters [11, 19,\n21].\nIn this study, we present a comprehensive analysis of the\neffect of Landau level mixing on the (𝑢⊥,𝑢𝑧)to leading or-\nder of graphene’s fine-structure constant, 𝜅=𝑒2/(ℏ𝑣𝐹𝜖). The\nconstant𝜅, which depends on the electric charge 𝑒, Planck’s\nconstantℏ, the Fermi velocity 𝑣𝐹, and the adjustable environ-\nmental dielectric constant 𝜖, controls the strength of the SU(4)\nsymmetric long-range Coulomb interactions and the extent of\nLandau-Level mixing, as noted in previous studies [9, 28, 29].\nOur study reveals that as 𝜅increases ( i.e., Landau level mixing\nbecomes stronger), 𝑢⊥(𝜅)decreases more rapidly than 𝑢𝑧(𝜅)\nincreases. This triggers a transition from AF to a KD, as il-\nlustrated in Fig. 1. Thus, the variation in Landau-level mixing\nacross different samples leads to the emergence of either the\n1500\n 1250\n 1000\n 750\n 500\n 250\n 0\nu×2l2\nB[meVnm2]\n02004006008001000120014001600uz×2l2\nB[meVnm2]\nKekulé-Distorted (KD)Antiferromagnet (AF)=0.00, 98.06\n=0.50, 131.46\n=1.50, 139.29\n=2.00, 140.49\nAF-KD boundary  135\nFigure 1. The flow of 𝑢⊥(𝜅)and𝑢𝑧(𝜅)with graphene’s fine structure\nconstant,𝜅, indicates a ground state transition from the AF phase to the\nKD phase as 𝜅increases from 0 to 2. The dashed lines serve as visual\nguides and angles in the inset are arctan (𝑢𝑧/𝑢⊥). In the absence of\nZeeman splitting, the AF-KD phase boundary is marked by the black\n135◦line. For comparison, we note Δ𝑍2𝜋𝑙2\n𝐵=492meVnm2where\nΔ𝑍=2𝜇𝐵𝐵is the Zeeman energy.\nKD or AF state. This trend is consistent with experimental\nfindings: double-encapsulated samples used in transport ex-\nperiments, with their larger 𝜖(and consequently smaller 𝜅),\nexhibit AF order, whereas un-encapsulated samples in STM\nexperiments display KD order.\nThe Standard Model of Monolayer Graphene: At the atomic\nscale, interactions between electrons in graphene exhibit weak\nsublattice dependence consistent with its 𝐶6𝑣symmetry. In the\ndilute doping regime, a specific subset of Bloch states –those\nwith energies below an energy cutoff Λand concentrated at\nthe Brillouin zone corners – form the basis for constructing thearXiv:2401.12528v1  [cond-mat.mes-hall]  23 Jan 20242\nlong-wavelength field theory. A practical method to incorpo-\nrate atomic-scale interactions into this long-wavelength theory\nis to model them as Dirac delta functions with corresponding\nbare coupling constants computed from the three dimensional\natomic wavefunctions. This methodology leads to the formu-\nlation of what we refer to as ”graphene’s standard model”,\nencompassing the band Hamiltonian 𝐻0, the SU(4) symmet-\nric long-range Coulomb interaction 𝐻𝑠, and the lattice-scale\ninteractions𝐻𝑎: [30, 31]\n𝐻0=∑︁\n𝑖=𝑥,𝑦𝑣𝐹∫\n𝜓†(r)ˆΠ𝑖𝜎𝑖𝜓(r)𝑑2𝑟 (1a)\n𝐻𝑠=𝑒2\n2𝜖∬𝜓†(r)𝜓†(r′)𝜓(r′)𝜓(r)\n|r−r′|𝑑2𝑟𝑑2𝑟′(1b)\n𝐻𝑎=1\n2∑︁\n𝛼,𝜇=\n0,𝑥,𝑦,𝑧𝑔0\n𝛼𝜇∫\n:\u0002\n𝜓†(r)𝜏𝛼𝜎𝜇𝜓(r)\u00032:𝑑2𝑟(1c)\nHere𝑣𝐹is the Fermi velocity, ˆΠ𝑖=−𝑖ℏ𝜕𝑖+𝑒𝐴𝑖(r)is the\nkinetic momentum operator with the vector potential 𝐴𝑖,𝜖\nis dielectric constant, and the basis 𝜓=(𝜓↑,𝜓↓)𝑇with\n𝜓𝑠=(𝜓𝐾𝐴𝑠,𝜓𝐾𝐵𝑠,𝜓𝐾′𝐵𝑠,−𝜓𝐾′𝐴𝑠)𝑇where𝑠can be↑or↓.\n𝜏𝛼,𝜎𝜇are Pauli matrices operating in the valley and sublattice\nsubspace, respectively.\nEq. (1c) contains 16 valley-and sublattice-dependent inter-\nactions. However, under 𝐶6𝑣and lattice translation symme-\ntry,𝑔0\n𝛼𝑥=𝑔0\n𝛼𝑦≡𝑔0\n𝛼⊥,𝑔0\n𝑥𝜇=𝑔0\n𝑦𝜇≡𝑔0\n⊥𝜇and we neglect\n𝑔0\n00since it is dwarfed by long-range Coulomb interaction.\nThus, Eq. (1c) is characterized by 8 independent parameters\nor coupling-constants[31]:\n𝑔0\n0𝑧, 𝑔0\n0⊥, 𝑔0\n𝑧0, 𝑔0\n𝑧⊥, 𝑔0\n𝑧𝑧, 𝑔0\n⊥0, 𝑔0\n⊥⊥, 𝑔0\n⊥𝑧. (2)\nThe coupling constants 𝑔0\n𝛼𝜇are influenced by electron-electron\nrepulsion and electron-phonon interactions, with the super-\nscript 0 denoting their role as ‘bare’ inputs in the theory with\ncutoffΛ. A key finding of our work is that Landau-level mix-\ning significantly renormalizes these atomic-scale interactions\nwhen reducing the energy from Λto the cyclotron energy.\nIn the Supplementary Material, we provide detailed calcu-\nlations of estimates for the bare coupling constants, based on\nthe assumption that the Bloch functions are linear combina-\ntions of (3D) atomic 𝑝𝑧orbitals. It is important to note that\nconvergence in real space, when summing atomic orbitals,\ncan be notably slow and may lead to inaccuracies [32, 33].\nTo circumvent these issues, performing these calculations in\nmomentum space has proven more effective. Our calculation\nsuggests that the intra-sublattice inter-valley-exchange interac-\ntion𝑔0\n⊥⊥is the dominant lattice-scale interaction, in particular\n𝑔0\n⊥⊥>𝑔0\n𝑧𝑧[33, 34]. Upon incorporating electron-phonon con-\ntributions, we obtain the following estimates (constants not\nlisted below are found to be negligible):\n(𝑔0\n⊥⊥,𝑔0\n𝑧𝑧,𝑔0\n⊥𝑧, 𝑔0\n𝑧⊥)=(269,184,−26,−27)meV·nm2.(3)\n(a) (b) (c)\n(d) (e) (f)Figure 2. (a)-(c) Feynman diagrams representing the O(𝜅)correc-\ntions to two-body valley-dependent Haldane pseudopotentials. Solid\nlines are fermionic propagators and the external lines are in the 𝑛=0\nLandau level. The curly and dashed lines stand for the long-range\nSU(4) Coulomb and the contact sublattice-and valley-dependent in-\nteractions, respectively. Swapping the curly and dashed lines in (a)\nand (b) does not change the corrections to Haldane pseudopotentials\nand for (c) either vertex of the dashed line can be dressed. (d)-(f)\nExamples of diagrams that do not contribute to the valley anisotropic\ninteractions. (d) and (e) vanish. (f) yields SU(4) symmetric correc-\ntions and thus can be neglected.\nLandau-level mixing and the modifications of valley-\ndependent interactions: In a constant magnetic field (i.e. ∇×\nA=−𝐵ˆ𝑧in Eq.(1a)), the eigenstates of 𝐻0for spin-valley\nflavor are denoted as\f\f\f𝜉𝑛,𝑚E\n, where𝜉𝑛represents the Landau\nlevels𝑛=0,1,2...,𝜉=±1, and𝑚enumerates the guiding\ncenter degeneracy of each Landau level. The wavefunction is\na sublattice spinor,\n\f\f\f𝜉𝑛E\n=(√\n2)𝛿𝑛,0−1\u0012|𝑛⟩\n(1−𝛿𝑛,0)𝜉|𝑛−1⟩\u0013\n, (4)\nwhere each state has a kinetic energy of 𝜖𝜉𝑛=𝜉ℏ𝑣𝐹√\n2𝑛/𝑙𝐵.\nThe quantum state|𝑛⟩≡𝑎†𝑛|0⟩/√\n𝑛! and|𝑚⟩≡𝑏†𝑚|0⟩/√\n𝑚!\nare constructed from two sets of ladder operators constructed\nfrom kinematic momentum and guiding centers: 𝑎†=(ˆΠ𝑥−\n𝑖ˆΠ𝑦)𝑙𝐵/√\n2 and𝑏†=(ˆ𝜂𝑥+𝑖ˆ𝜂𝑦)/√\n2𝑙𝐵where ˆ η=r−𝑙2\n𝐵z׈𝚷.\nIn the symmetric gauge, we can loosely call 𝑚the angular\nmomentum although the physical angular momentum is 𝑚−𝑛.\nSince any translationally and rotationally invariant two-body\ninteraction Hamiltonian ˆ𝑉can be decomposed into the Hal-\ndane pseudopotentials using the center of mass and relative\ncoordinates|𝑁𝑀⟩𝑐|𝑛𝑚⟩𝑟[35],\n𝑐⟨𝑁′𝑀′|𝑟⟨𝑛′𝑚′|ˆ𝑉|𝑁𝑀⟩𝑐|𝑛𝑚⟩𝑟\n=𝑟⟨𝑛′𝑚′|ˆ𝑉|𝑛𝑚⟩𝑟𝛿𝑚′−𝑛′,𝑚−𝑛𝛿𝑁′,𝑁𝛿𝑀′,𝑀. (5)\nFor two interacting electrons in the zeroth LL of our inter-\nest,𝑛=𝑛′=0. In the spirit of Larmor’s theorem, we can\neliminate the Zeeman field from the equation of motion in a3\nrotating frame and recover the spin SU(2) symmetry. Thus, the\neffective interactions can be represented in a spin-independent\nform. Moreover, the 𝐶6𝑣, lattice momentum conservation,\nand the valley-sublattice locking of the zeroth LL, enforce\neffectively an U(1) valley conservation, leading to the Hal-\ndane pseudopotentials𝑟⟨0𝑚|𝑉|0𝑚⟩𝑟=𝑉0(𝑚)+𝑉𝑎(𝑚), where\n𝑉0(𝑚)is spin-valley SU(4) Haldane pseudopotential that has\nbeen studied previously [36] and\nˆ𝑉𝑎(𝑚)=𝑢𝑧(𝑚)\n2𝜏𝑧\n𝑖𝜏𝑧\n𝑗+𝑢⊥(𝑚)\n2(𝜏𝑥\n𝑖𝜏𝑥\n𝑗+𝜏𝑦\n𝑖𝜏𝑦\n𝑗), (6)\nare the valley dependent pseudopotentials arising from 𝐻𝑠𝑣\ninteractions in Eq. (1c). At the zeroth order in interactions, the\neffective Hamiltonian is ˆ𝑉(1)=P(ˆ𝐻𝑠+ˆ𝐻𝑎)PwithPbeing\nthe projector into the degenerate non-interacting ground-state\nmanifold where 𝑛 <0 LLs are fully occupied and 𝑛 >0 LLs\nempty. Because a pair of electrons of a nonzero relative an-\ngular momentum do not experience contact interactions [33],\n𝑢(1)\n𝛼(𝑚)=𝑢𝛼𝛿𝑚,0, where𝑢𝛼=(𝑔0\n𝛼0+𝑔0\n𝛼𝑧)/2𝜋𝑙2\n𝐵is the nota-\ntion used in Ref. [27]. However, Landau level mixing generates\nperturbative corrections to these Haldane pseudotentials and\none of our main goals is to compute such corrections.\nThe leading order correction to ˆ𝑉(1)can be derived from\nsecond-order perturbation theory [29]. Our focus here is on the\neffect of dielectric screening on valley-dependent interactions,\nwhich is captured by the following Hamiltonian, comprising\none ˆ𝐻𝑠and one ˆ𝐻𝑎:\nˆ𝑉(2)\n𝑠𝑎=−Pˆ𝐻𝑠P⊥1\nˆ𝐻0−𝐸0P⊥ˆ𝐻𝑎P+ℎ.𝑐. (7)\nHere𝐸0is the energy of degenerate noninteracting ground-\nstates and P⊥=I−P. Because three-body interactions\nare forbidden in the zeroth LL of monolayer graphene by the\nparticle-hole symmetry [36] and the one-body self-energy gen-\nerated by ˆ𝑉(2)\n𝑠𝑎is SU(4) symmetric, the only nontrivial terms in\nEq. (7) are the two-body interactions which can be delineated\ninto Feynman diagrams in Fig. 2 [33]. We call the three non-\nvanishing Feynman diagrams (a) the particle-particle channel,\n(b) the double-exchange channel, and (c) the vertex correction.\nTheir contributions are parametrized as follows:\n𝑢(2)\n𝛼(𝑚)=𝜅\n2𝜋𝑙2\n𝐵∑︁\n𝜇=0,𝑥,𝑦,𝑧(𝑐𝜇\n𝑚,𝑎+𝑐𝜇\n𝑚,𝑏+𝑐𝜇\n𝑚,𝑐)𝑔0\n𝛼𝜇 (8)\nBecause of our choice of pseudospin components (see text\nbelow Eqs. (1)), both valleys sharing identical orbital wave\nfunctions as per Eq. (4), the coefficients 𝑐𝜇\n𝑚,𝑎/𝑏/𝑐are indepen-\ndent of the valley index 𝛼.\nEvaluating the Feynman diagram in Fig. 2a, we find that the\nvalley-sublattice locking unique to the zeroth LL of graphene,\n𝜎𝑧\f\f\f0𝑚E\n=\f\f\f0𝑚E\n, leads to the vanishing interacting matrix ele-\nments\n¯𝑛𝑖,¯𝑛𝑗\f\f𝜎+\n𝑖𝜎−\n𝑗\f\f¯0𝑖,¯0𝑗\u000b\n=0, and𝑐⊥\n𝑚,𝑎≡𝑐𝑥\n𝑚,𝑎+𝑐𝑦\n𝑚,𝑎=0.\nAdditionally, due to the contact valley-dependent interaction,\nthe particle-particle ladder diagram only induces correctionsto𝑢(2)\n𝛼(𝑚=0), with [33]\n𝑐0,𝑧\n𝑚=0,𝑎\n4𝜋=−22𝑁𝑐∑︁\n𝑛=1𝑟⟨00|𝛿(r𝑖𝑗)|𝑛𝑛⟩𝑟𝑟⟨𝑛𝑛|𝑟−1\n𝑖𝑗|00⟩𝑟\n×∞∑︁\n𝑛𝑖,𝑗=−∞Θ(𝑛𝑖𝑛𝑗)\n|𝜖𝑛𝑖+𝜖𝑛𝑗||𝑐⟨0|𝑟⟨𝑛||𝑛𝑖|,|𝑛𝑗|⟩|2,\n=2𝑁𝑐∑︁\n𝑛=1Γ(𝑛+1\n2)\n2𝑛+1𝑛!𝑛∑︁\n𝑛1=0\u0000𝑛\n𝑛1\u0001\n√2𝑛1+√︁\n2(𝑛−𝑛1). (9)\nHere,Θ(𝑥)is the Heaviside function with Θ(0)=1 and we\nmeasure the length in the unit of 𝑙𝐵an the kinetic energy in\nthe unit ofℏ𝑣𝐹/𝑙𝐵to shorten the expressions. Asymptotically,\nΓ(𝑛+1\n2)/𝑛!∼𝑛−1\n2and therefore 𝑐0,𝑧\n0,𝑎is logarithmically diver-\ngent with respect to the LL cutoff 𝑁𝑐, which is expected from\nthe renormalization group analysis [27].\nThe Feynman diagrams in Fig. 2b and c involve particle-\nhole excitations in the intermediate states and need to be cal-\nculated by a different method. We follow Ref. [29, 37] and\nexpress the bare two-body interactions as Fourier integrals,\n𝑟−1\n𝑖𝑗=∫\n𝑑q𝑒𝑖q·r𝑖𝑗/2𝜋𝑞and𝛿(r𝑖𝑗)=∫\n𝑑q′𝑒𝑖q′·r𝑖𝑗/(2𝜋)2. For\neach(q,q′), the scattering amplitude is diagonalized in the\nrelative angular momentum basis {|𝑚⟩𝑟}[33]. The Haldane\npseudopotentials can therefore be expressed as an integral,\nwith\n𝑐𝜇\n𝑚,𝑏\n4𝜋=∬\nq,q′∞∑︁\n𝑛,𝑛′=−∞Θ(𝑛)−Θ(𝑛′)\n𝜖𝑛−𝜖𝑛′V𝜇\n𝑛0,0𝑛′(q′)V0𝑛′,𝑛0(q)\n×𝑟⟨𝑚|𝑒𝑖(q−q′)·η𝑖𝑗|𝑚⟩𝑟+𝑐.𝑐., (10)\n𝑐𝜇\n𝑚,𝑐\n4𝜋=∬\nq,q′∞∑︁\n𝑛,𝑛′=−∞Θ(𝑛′)−Θ(𝑛)\n𝜖𝑛−𝜖𝑛′V𝜇\n0𝑛,0𝑛′(q′)V0𝑛′,0𝑛(q)\n×𝑒𝑖q′∧q\n𝑟⟨𝑚|𝑒−𝑖q′·η𝑖𝑗|𝑚⟩𝑟+𝑐.𝑐.. (11)\nHere∫\nq=∫\n𝑑2𝑞/(2𝜋)2and the matrix elements\nfor the long-range Coulomb interaction V𝑛𝑛′,𝑛1𝑛′\n1(q)=\n2𝜋\n𝑞𝐹0\n𝑛,𝑛 1(q)𝐹0\n𝑛′,𝑛′\n1(−q)and the bare sublattice-and valley-\ndependent interactions V𝜇\n𝑛𝑛′,𝑛1𝑛′\n1(q)=𝐹𝜇\n𝑛,𝑛 1(q)𝐹𝜇\n𝑛′,𝑛′\n1(−q)\nwith the form factors 𝐹𝜇\n𝑛′,𝑛(q)=\n𝑛′\f\f𝜎𝜇𝑒𝑖q·(ˆ𝑧×𝚷)|𝑛⟩for\n𝜇=0,𝑥,𝑦,𝑧 . Due to the valley-sublattice locking in the zeroth\nLL,𝑐⊥\n𝑚,𝑐=0 as in the particle-particle channel.\nDiscussion of main results:– The main results of this work,\nencapsulated in Eq. (S37), (10), and (11), required numerical\ncomputations and they lead to the following three important\nconclusions:\n1) Table I(a) shows the contributions of three Feynman dia-\ngrams to the 𝑚=0 component of 𝑐𝜇\n𝑚(𝑐⊥\n𝑚≡𝑐𝑥\n𝑚+𝑐𝑦\n𝑚). Notably,\nthe inter-valley channel 𝑐⊥\n0, which is entirely derived from the\ndouble-exchange diagram, is found to be attractive ( 𝑐⊥\n0<0).\nThis observation is significant because it implies that a re-\npulsive valley-and sublattice-exchange scattering 𝑔0\n⊥⊥>0 can\ninduce a negative valley-exchange Haldane pseudopotential\n𝑉⊥in the zeroth LL. This contrasts sharply with conventional4\nTable I. Coefficients of the first-order-in- 𝜅corrections to the\nvalley-dependent interactions receive contributions from three non-\nvanishing Feynman diagrams in Fig. 2a (p-p), b (DE), and c(vertex),\nsee Eq. (8). (a) 𝑐𝜇\n𝑚=0,𝑎/𝑏/𝑐are listed in this table. (b) 𝑐𝜇\n𝑚=\n𝑐𝜇\n𝑚,𝑎+𝑐𝜇\n𝑚,𝑏+𝑐𝜇\n𝑚,𝑐for𝑚≤5. The numbers in red depend log-\narithmically on the cutoff 𝑁𝑐of the Landau levels. Here, 𝑁𝑐=160.\n(a)𝑐𝜇\n𝑚=0,𝑎, 𝑐𝜇\n𝑚=0,𝑏, 𝑐𝜇\n𝑚=0,𝑐\n𝜇 p-p DE vertex\n0−1.6773 1.5609 0.3915\n𝑧−1.6773 1.5609 2.7386\n⊥ 0−2.5900 0\n(b)\n𝜇m0 1 2 3 4 5\n0 0.2799−0.1094−0.1283−0.0144−0.0173−0.0051\n𝑧 2.6223 0.1128−0.0627 0.0149−0.0011 0.0051\n⊥−2.5900 0.1064−0.0141 0.0032−0.0001 0.0006\nfermions with parabolic dispersion, where repeated exchange\nscattering enhances opposite-spin repulsion and suppresses\nspin-singlet superconductivity near a ferromagnetic critical\npoint [38]. In graphene, we found that opposite-valley scatter-\ning becomes attractive due to the opposite chirality of electron\nand hole wavefunctions [39]. In the context of Landau levels,\nthe second sublattice components of wavefunctions (Eq.(4))\nfor intermediate electron ( 𝜉 >0) and hole ( 𝜉 <0) have oppo-\nsite signs, leading to an attractive interaction matrix elements\nV⊥\n𝑛0,0𝑛′<0.\n2) The red coefficients in Table I(a) highlight that these\nvalues logarithmically increase with the increasing number\nof Landau levels, a consequence of the linearly increasing\ndensity-of-states in our low-energy theory. This is an impor-\ntant effect as they amplify the magnitude of lattice-scale inter-\nactions, making them 3 to 5 times stronger than the Zeeman\nenergy, consistent with experimental observations [11, 33].\n3) Table I(b) details the values of 𝑢(2)\n𝛼(𝑚)for𝑚=0,...,5,\nincorporating contributions from the aformentioned Feynman\ndiagrams. Notably, the 𝑚=0 component is significantly\nlarger, by at least an order of magnitude, than the 𝑚≠0 com-\nponent. This indicates that even when contact interactions are\ndressed by long-range Coulomb interactions in second-order\nperturbation theory, the resulting interaction remains primar-\nily short-ranged. While this is evident in the particle-particle\nchannel (𝑐𝜇\n𝑚≠0,𝑎=0) the particle-hole channels require further\nexplanation. In the particle-hole channel, the bubble diagram\ncontribution is exactly zero leaving the exchange scattering\ndiagrams as the primary contributors. The bubble diagram\nvanishes because the susceptibility of valley or sublattice po-\nlarization induced by charge perturbation vanishes under 𝐶6𝑣\nsymmetry. The exchange diagrams fail to generate long-range\ninteractions, as they require substantial spatial overlap of the\nwavefunctions. This result suggests that the coexistence phase\nof the KD and AF states, as proposed in Ref. 40 and 41, does\nexist, but it occupies a very small sliver of the phase diagram\nnear the boundary of KD and AF [42, 43].Focusing on the 𝑚=0 component, we obtain the final\nexpression for the corrections to the lattice-scale interactions\nto leading order in 𝜅are given by:\n𝑢⊥(𝜅)≡𝑢⊥+𝑢(2)\n⊥(0)=𝑔0\n⊥𝑧\u0010\n1+𝜅𝑐𝑧\n0\u0011\n−𝜅|𝑐⊥\n0|𝑔0\n⊥⊥\n2𝜋𝑙2\n𝐵,(12)\n𝑢𝑧(𝜅)≡𝑢𝑧+𝑢(2)\n𝑧(0)=𝑔0\n𝑧𝑧\u0010\n1+𝜅𝑐𝑧\n0\u0011\n−𝜅|𝑐⊥\n0|𝑔0\n𝑧⊥\n2𝜋𝑙2\n𝐵. (13)\nBy incorporating the estimates Eq. (3), we derive the main re-\nsults shown in Fig. 1. At 𝜅=0 (i.e.,without Landau-level mix-\ning),𝑢⊥and𝑢𝑧are determined by attractive 𝐾-optical-phonon\npotential (𝑔0\n⊥𝑧) and repulsive sublattice asymmetry electron-\nelectron repulsion ( 𝑔0\n𝑧𝑧) respectively, with the latter being more\ndominant, leading to 𝑢𝑧>−𝑢⊥and thus an AF ground state.\nAt finite𝜅, the respective contributions of 𝑔0\n⊥𝑧and𝑔0\n𝑧𝑧to𝑢⊥\nan𝑢𝑧are enhanced by the same factor 1 +𝜅𝑐𝑧\n0. However,\nthe double-exchange Feynman diagram introduces into 𝑢⊥the\non-site valley-exchange scattering ( 𝑔0\n⊥⊥), the strongest type of\nbare lattice-scale interactions in monolayer graphene [33, 34],\nmaking𝑢⊥decrease faster than the increases in 𝑢𝑧and ulti-\nmately tipping the balance to 𝑢𝑧<−𝑢⊥and the ground-state\ntransition to the KD state, as shown in Fig. 1.\nConclusions and outlook: In summary, our study reveals a\ntransition from antiferromagnetic to Kekul ´e distorted states in\ngraphene as a function of the fine-structure constant 𝜅occur-\nring around𝜅∼0.5. This transition is driven by the enhance-\nment of Landau-level mixing in the double-exchange Feynman\ndiagram. This diagram uniquely produces an attractive 𝑢⊥\nfrom repulsive valley-exchange scattering 𝑔⊥⊥due to matrix-\nelement effects. The logarithmic divergence of 𝑢𝑧and𝑢⊥\nagrees qualitatively with prior renormalization group analyses\nconducted without a magnetic field [27, 31, 33]. This cor-\nrespondence suggests that the notable enhancement of lattice-\nscale interactions is largely attributed to the influence of remote\nLandau levels [33]. Our research advances the understand-\ning of the phase diagram of charge neutral graphene under\nthe strong magnetic field. The calculated valley-dependent\nHaldane pseudopotentials provide a foundation for accurate\nnumerical studies of SU(4) symmetry breaking in graphene’s\nfractional quantum Hall states [36]. 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Martin, Physical review letters\n121, 257001 (2018).\n[49] E. Clementi and D.-L. Raimondi, The Journal of Chemical\nPhysics 38, 2686 (1963).\n[50] We take the convention 𝜎±=(𝜎𝑥±𝑖𝜎𝑦)/2.\n[51] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and\nproducts (Academic press, 2014).1\nSupplementary Material: Landau-Level Mixing and SU(4) Symmetry Breaking in Graphene\nI. ESTIMATION OF THE BARE G-VALUES FROM TIGHT-BINDING CALCULATION\nTo compute the contribution of electron-electron repulsion to 𝑔0\n𝛼𝜇, we adopt a basic tight-binding model, assuming that the\nBloch function of graphene is a linear combination of the 𝑝𝑧atomic orbitals. For this calculation, we only require two Bloch\nfunctions situated at the non-equivalent corners, τ=±K, of the Brillouin zone and their two sublattice projections. They can\nbe grouped into a vector,\n𝑣=[𝑢𝐾𝐴,𝑢𝐾𝐵,𝑢𝐾′𝐵,−𝑢𝐾′𝐴]𝑇, (S1)\n⟨r|𝑢𝜏𝜎⟩=√︂\n𝐴0\n𝑁∑︁\nR𝑒𝑖τ·(R𝜎+R)𝜙(r−R𝜎−R). (S2)\nHereR𝜎represents the sublattice vector so that R𝐴=0 andR𝐵=R𝐴𝐵. In our coordinate system, the origin coincides with\nsublattice𝐴and and R𝐴𝐵is used to denote the relative displacement between the two carbon sublattices inside the unit cell.\nConsequently, the summation over R𝐴covers the set:{R𝐴|𝑚a1+𝑛a2∀𝑚,𝑛∈ℤ}and the summation over R𝐵covers the set:\n{R𝐵|R𝐴𝐵+𝑚a1+𝑛a2∀𝑚,𝑛∈ℤ}where a1anda2are the primitive lattice vectors.\nThe normalized wavefunction for 𝑝𝑧orbitals is given by\n𝜙(𝑟,𝜃,𝜙)=2˜𝑍2√˜𝑍√︃\n96𝑎5\n𝐵𝑟𝑒−˜𝑍𝑟\n2𝑎𝐵𝑌0\n1(𝜃,𝜙), (S3)\nwhere∫\n|𝜙(r)|2𝑑3𝑟=1. Here,𝑎𝐵is the Bohr radius and ˜𝑍is the effective nuclear charge of carbon atom. With this basis in\nplace,𝑔0\n𝛼𝜇can be directly computed as,\n𝑔0\n𝛼𝜇=1\n4𝐴0∑︁\n𝑖𝑗𝑘𝑙𝑉𝑖𝑘,𝑗𝑙(𝜏𝛼𝜎𝜇)𝑖𝑗(𝜏𝛼𝜎𝜇)𝑘𝑙, (S4)\nwhere the Coulomb matrix elements are described by the six-dimensional integrals,\n𝑉𝑖𝑘,𝑗𝑙=𝑒2∫ ∫\n𝑢.𝑐.𝑣∗\n𝑖(r1)𝑣𝑗(r1)𝑣∗\n𝑘(r2)𝑣𝑙(r2)\n|r1−r2|𝑑3𝑟1𝑑3𝑟2. (S5)\nHere𝐴0is the area of the graphene unit cell. Next, we express Eq. (S4) as the Coulomb potential energy between overlap charges:\n𝑔𝑧𝑧=𝑒2\n4𝐴0∑︁\n𝜏,𝜎,𝜏′,𝜎′𝑠𝑔𝑛(𝜎)𝑠𝑔𝑛(𝜎′)∫ ∫\n𝑢.𝑐.𝑑3r1∫\n𝑑3r2𝜌𝜏𝜎,𝜏𝜎(r1)𝜌𝜏′𝜎′,𝜏′𝜎′(r2)\n|r1−r2|(S6)\n𝑔⊥⊥=𝑒2\n4𝐴0∑︁\n𝜏𝜎∫ ∫\n𝑢.𝑐.𝑑3r1∫\n𝑑3r2𝜌𝜏𝜎, ¯𝜏𝜎(r1)𝜌¯𝜏𝜎,𝜏𝜎(r2)\n|r1−r2|(S7)\n𝑔𝑧⊥=𝑒2\n4𝐴0∑︁\n𝜏𝜎2∫ ∫\n𝑢.𝑐.𝑑3r1∫\n𝑑3r2𝜌𝜏¯𝜎,𝜏𝜎(r1)𝜌𝜏𝜎,𝜏 ¯𝜎(r2)\n|r1−r2|(S8)\n𝑔⊥𝑧=𝑒2\n4𝐴0∑︁\n𝜏𝜎2∫ ∫\n𝑢.𝑐.𝑑3r1∫\n𝑑3r2𝜌𝜏𝜎, ¯𝜏¯𝜎(r1)𝜌¯𝜏¯𝜎,𝜏𝜎(r2)\n|r1−r2|. (S9)\nWe define the overlap charge density 𝜌𝑖𝑗(r)as𝑣∗\n𝑖(r)𝑣𝑗(r), where𝑖,𝑗are indices representing 𝐾𝐴,𝐾𝐵,𝐾′𝐴,𝐾′𝐵. Note that\nthe overlap charge within the same sublattice, termed as the intra-sublattice overlap charge, is significantly larger than the\noverlap charge between different sublattices, termed as the inter-sublattice overlap charge. Consequently, we anticipate that the\nmagnitudes of 𝑔0\n𝑧𝑧and𝑔0\n⊥⊥will be greater compared to those of 𝑔0\n𝑧⊥and𝑔0\n⊥𝑧. When considering the intra-sublattice overlap\ncharge, this term can be approximated to leading order by the separation distance between two 𝑝𝑧atomic orbitals:\n𝜌𝜏𝑖𝜎,𝜏𝑗𝜎(r)≡𝑣∗\n𝜏𝑖𝜎(r)𝑣𝜏𝑗𝜎(r)=𝐴0\n𝑁∑︁\nR1,R2𝑒−𝑖τ𝑖·(R1+R𝜎)+𝑖τ𝑗·(R2+R𝜎)𝜙(r−R1−R𝜎)𝜙(r−R2−R𝜎) (S10)\n≈𝐴0∑︁\nR𝑒−𝑖(τ𝑖−τ𝑗)·(R+R𝜎)𝜙2(r−R−R𝜎)2\nThe second line above is equivalent to selecting the 𝑁combinations within the summations where R1is equal to R2. We have\nnumerically confirmed that contributions from subsequent 𝑅shells are notably smaller, showing a reduction of at least an order\nof magnitude. This significant decrease is mainly due to the compact nature of the 𝑝𝑧orbitals in comparison to the distance\nbetween carbon atoms.\nFor inter-sublattice overlap charge, the two 𝑝𝑧orbitals are separated by the carbon-carbon distance R𝐴𝐵. Consequently, the\nleading approximation is given by the following:\n𝜌𝜏𝑖¯𝜎,𝜏𝑗𝜎(r)≡𝑣∗\n𝜏𝑖¯𝜎(r)𝑣𝜏𝑗𝜎(r)=𝐴0\n𝑁∑︁\nR1,R2𝑒−𝑖τ𝑖·(R1+R¯𝜎)+𝑖τ𝑗·(R2+R𝜎)𝜙(r−R1−R¯𝜎)𝜙(r−R2−R𝜎) (S11)\n≈𝐴0∑︁\nR𝑒−𝑖(τ𝑖−τ𝑗)·R𝑒−𝑖𝜏𝑖·R¯𝜎+𝑖𝜏𝑗·R𝜎(𝜙(r−R−R¯𝜎)𝜙(r−R−R𝜎)\n+𝜙(r−R−a1−R¯𝜎)𝜙(r−R−R𝜎)𝑒−𝑖τ𝑖·a1+𝜙(r−R−a2−R¯𝜎)𝜙(r−R−R𝜎)𝑒−𝑖τ𝑖·a2\u0013\nWe have also numerically verified that the contribution from the next order in the 𝑅-shell is smaller by an order of magnitude.\nAfter substituting the expressions for intra- and intersublattice overlap charges into Eq.(S6), we have two independent summations\nover lattice vectors, resulting from the squaring of the overlap charge. We then apply a change of variables to the center-of-mass\ncoordinates, denoted as R𝑐𝑚, and relative coordinates, R. The summation over R𝑐𝑚efficiently replaced the integration over the\nunit cell with an integration across the entire spatial domain. Consequently, we are primarily left with the task of dealing with\nthe summation over R:\n𝑔𝑧𝑧=8∑︁\nR\u0014𝑒2𝐴0\n4∫𝑑3q\n(2𝜋)34𝜋|𝜌𝑎(q)|2\nq2(1−𝑒𝑖q·R𝐴𝐵)𝑒𝑖q·R\u0015\n, (S12)\n𝑔⊥⊥=4∑︁\nR\u0014𝑒2𝐴0\n4∫𝑑3q\n(2𝜋)34𝜋|𝜌𝑎(q)|2\nq2𝑒𝑖q·R𝑒𝑖R·ΔK\u0015\n, (S13)\n𝑔𝑧⊥=8∑︁\nR\u0014𝑒2𝐴0\n4∫𝑑3q\n(2𝜋)34𝜋|𝜌𝑒(q)|2\nq2𝑒𝑖q·R\u0015\n, (S14)\n𝑔⊥𝑧=8∑︁\nR\u0014𝑒2𝐴0\n4∫𝑑3q\n(2𝜋)34𝜋|𝜌𝑒(q)|2\nq2𝑒𝑖q·R\u0015\n, (S15)\n𝜌𝑎(q)=∫\n𝑑3r𝜙(r)𝜙(r)𝑒−𝑖q·r(S16)\n𝜌𝑒(q)=∫\n𝑑3r\u0012\n𝜙(r)𝜙(r−R𝑎𝑏)+𝑒𝑖K·a2𝜙(r)𝜙(r−a2−R𝑎𝑏)+𝑒𝑖K·a1𝜙(r)𝜙(r−a1−R𝑎𝑏)\u0013\n𝑒−𝑖q·r. (S17)\nNote the terms inside the square brackets [...]decay very slowly as Rincreases. For example,\n∫𝑑3q\n(2𝜋)3|𝜌𝑎(q)|2\nq2𝑒𝑖q·R=∫\n𝑑3r1𝑑3r2𝜙2(r1)𝜙2(r2)\n|r1−r2+R|R→∞−−−−−→1\n|R|(S18)\nTherefore, given this poor convergence, it becomes more pragmatic to evaluate them in the Fourier space. UsingÍ\nR𝑒𝑖q·R=\n(2𝜋)2\n𝐴0Í\nG𝛿(2)(q−G)where Gis the 2D reciprocal lattice vectors, we arrive at the following expressions:\n𝑔0\n⊥⊥=4∑︁\nG𝑒2\n4∫𝑑𝑞𝑧\n2𝜋4𝜋|𝜌𝑎(G−ΔK,𝑞𝑧)|2\n(G−ΔK)2+𝑞2𝑧(S19)\n𝑔0\n𝑧𝑧=8∑︁\nG𝑒2\n4∫𝑑𝑞𝑧\n2𝜋4𝜋|𝜌𝑎(G,𝑞𝑧)|2\nG2+𝑞2𝑧(1−cos(G·R𝑎𝑏)) (S20)\n𝑔0\n𝑧⊥=8∑︁\nG𝑒2\n4∫𝑑𝑞𝑧\n2𝜋4𝜋|𝜌𝑒(G,𝑞𝑧)|2\nG2+𝑞2𝑧(S21)\n𝑔0\n⊥𝑧=8∑︁\nG𝑒2\n4∫𝑑𝑞𝑧\n2𝜋4𝜋|𝜌𝑒(G−ΔK,𝑞𝑧)|2\n(G−ΔK)2+𝑞2𝑧(S22)3\n<latexit sha1_base64=\"Qq56qtRsCSVxvty838V29rYlg0g=\">AAACEHicbZDLSsNAFIYn9VbrLerSTbCIFaQkUtSNUHShywr2Ak0Mk+mkHTqZxJmJUEMewY2v4saFIm5duvNtnLQBtfrDwMd/zmHO+b2IEiFN81MrzMzOzS8UF0tLyyura/r6RkuEMUe4iUIa8o4HBaaE4aYkkuJOxDEMPIrb3vAsq7dvMRckZFdyFGEngH1GfIKgVJar79p8EF7Dih1AOfD85Dw9+UY3sUS6f+Pe7TmuXjar5ljGX7ByKINcDVf/sHshigPMJKJQiK5lRtJJIJcEUZyW7FjgCKIh7OOuQgYDLJxkfFBq7CinZ/ghV49JY+z+nEhgIMQo8FRntqyYrmXmf7VuLP1jJyEsiiVmaPKRH1NDhkaWjtEjHCNJRwog4kTtaqAB5BBJlWFJhWBNn/wXWgdV67Bau6yV66d5HEWwBbZBBVjgCNTBBWiAJkDgHjyCZ/CiPWhP2qv2NmktaPnMJvgl7f0LznGdFA==</latexit>⇢a(G=G1s,qz)]<latexit sha1_base64=\"Lfp9Uh/yfobJihaJEeAmyMTX9h8=\">AAACAnicbVDLSsNAFJ3UV62vqCtxEyxCBSmJFHUjFF3osoJ9QBPLZDpph05m4sxEqKG48VfcuFDErV/hzr9x0mahrQcuHM65l3vv8SNKpLLtbyM3N7+wuJRfLqysrq1vmJtbDcljgXAdccpFy4cSU8JwXRFFcSsSGIY+xU1/cJH6zXssJOHsRg0j7IWwx0hAEFRa6pg7rujzW1hyQ6j6fpBcjs7sw7vOw4HXMYt22R7DmiVORoogQ61jfrldjuIQM4UolLLt2JHyEigUQRSPCm4scQTRAPZwW1MGQyy9ZPzCyNrXStcKuNDFlDVWf08kMJRyGPq6M71UTnup+J/XjlVw6iWERbHCDE0WBTG1FLfSPKwuERgpOtQEIkH0rRbqQwGR0qkVdAjO9MuzpHFUdo7LletKsXqexZEHu2APlIADTkAVXIEaqAMEHsEzeAVvxpPxYrwbH5PWnJHNbIM/MD5/ADwjlrA=</latexit>⇢a(G=0,qz)]\nFigure S1. The intrasublattice overlap charge 𝜌𝑎(G,𝑞𝑧)versus𝑞𝑧. As the effective nuclear charge ˜𝑍increases, leading to a more localized\nwave-function, the Fourier components decay slower.\n<latexit sha1_base64=\"z7Mx4EE0AoX54Z5wAa7jEfE31d4=\">AAACGnicbZDLSsNAFIYn9V5vVZdugkWoICURUTeC6EKXVawKTQyT6Uk7OLk4cyLWkOdw46u4caGIO3Hj2zi9gFr9YeDjP+cw5/x+IrhCy/o0CiOjY+MTk1PF6ZnZufnSwuKZilPJoM5iEcsLnyoQPII6chRwkUigoS/g3L866NbPb0AqHken2EnADWkr4gFnFLXllWwH4RazE8gbjmzHl1BxQoptP8gO891v9DJb5evX3t2a65XKVtXqyfwL9gDKZKCaV3p3mjFLQ4iQCapUw7YSdDMqkTMBedFJFSSUXdEWNDRGNATlZr3TcnNVO00ziKV+EZo99+dERkOlOqGvO7vLquFa1/yv1kgx2HEzHiUpQsT6HwWpMDE2uzmZTS6BoehooExyvavJ2lRShjrNog7BHj75L5xtVO2t6ubxZnlvfxDHJFkmK6RCbLJN9sgRqZE6YeSePJJn8mI8GE/Gq/HWby0Yg5kl8kvGxxdRkaGn</latexit>Re[⇢e(G=G1s,qz)]\n<latexit sha1_base64=\"792HH2oN1Qsdhef5mxfIWFYsx2s=\">AAACGnicbZDLSsNAFIYn9V5vVZdugkWoICURUTeC6ELdVbAqNDFMpift4OTizIlYQ57Dja/ixoUi7sSNb+P0Amr1h4GP/5zDnPP7ieAKLevTKIyMjo1PTE4Vp2dm5+ZLC4tnKk4lgzqLRSwvfKpA8AjqyFHARSKBhr6Ac//qoFs/vwGpeBydYicBN6StiAecUdSWV7IdhFvMjsO84ch2fAkVJ6TY9oPsMN/9Ri+zVb5+7d2tuV6pbFWtnsy/YA+gTAaqeaV3pxmzNIQImaBKNWwrQTejEjkTkBedVEFC2RVtQUNjRENQbtY7LTdXtdM0g1jqF6HZc39OZDRUqhP6urO7rBqudc3/ao0Ugx0341GSIkSs/1GQChNjs5uT2eQSGIqOBsok17uarE0lZajTLOoQ7OGT/8LZRtXeqm6ebJb39gdxTJJlskIqxCbbZI8ckRqpE0buySN5Ji/Gg/FkvBpv/daCMZhZIr9kfHwBT9uhpg==</latexit>Im[⇢e(G=G1s,qz)]\nFigure S2. The intersublattice overlap charge 𝜌𝑒(G=G1𝒔,𝑞𝑧)versus𝑞𝑧forG. Note the imaginary part is an order of magnitude stronger\nthan the real part.\nHereΔK=K−K′is the momentum transfer. From the above equations, it is evident that they are all positive definite.\nSpecifically, the positivity of 𝑔0\n⊥⊥and𝑔0\n𝑧⊥originates from the properties of the exchange integral, as discussed Ref. [47]. We\nnote that the estimates presented in Ref. [32] seem to be inaccurate, especially regarding their result indicating 𝑔0\n⊥⊥<0. This\ndiscrepancy might stem from their approach of performing the summation in real space instead of in reciprocal space. Fig. S2\nand Fig. S3 show the Fourier components of the overlap charge vs 𝑞𝑧for different effective nuclear charge ˜𝑍. Note that𝜌𝑎is real\nwhile𝜌𝑒is complex.\nNext, we introduce truncation in G. For𝑔0\n⊥⊥and𝑔0\n⊥𝑧, which involve inter-valley scattering with a momentum transfer of\nΔK, the primary contributions to the reciprocal lattice sum arise mainly from the G=0 and the two corners in the first shell\nclosest toΔK, as indicated by the dashed lines in Fig. (S3). In contrast, for 𝑔0\n𝑧𝑧and𝑔0\n𝑧⊥, corresponding to zero-momentum\nforward scattering, the major contributions arise from the six corners in the first shell, defined as G1𝑠in Fig. (S3). For 𝑔0\n𝑧𝑧,\nthe absence of contribution from G=0 can be attribute to the 1 −cos(G·R𝑎𝑏)factor, a result stemming from AA-scattering\nminus AB-scattering. For 𝑔0\n𝑧⊥the absence of contribution from G=0 is consequence of 𝐶3rotation symmetry, which implies\n𝜌𝑒(0)=0. Numerical verification has confirmed that contributions from the next shell are significantly smaller, by an order of\nmagnitude, compared to those from the first shell. Consequently, evaluating the overlap charge using the leading contributions\ninR′and computing 𝑔0\n𝛼𝜇to leading contributions in Gyield the results summarized in the table below:4\nFigure S3. The Brillouin zone of monolayer graphene. Here b1=2𝜋\n3𝑎(1,√\n3)andb2=2𝜋\n3𝑎(1,−√\n3),𝑎≈1.42˚A is carbon-carbon distance. The\nΓpoint corresponds to G=0 and the first shell corresponds to G1𝑠={𝐺|forG∈(±b1,±b2,±(b1+b2))}. The two corners in the first shell,\nwhich are connected by a dashed line, are equidistant from the zero shell at ΔK.\n𝑚𝑒𝑉𝑛𝑚2˜𝑍=1˜𝑍=1.5˜𝑍=2˜𝑍=2.5˜𝑍=3˜𝑍=4\n𝑔0\n⊥⊥ 8.88 61.5 156.4 268.9 384.3 600.6\n𝑔0\n𝑧𝑧 0.55 12.55 67.46 184.03 353.7 781.15\n𝑔0\n𝑧⊥ 0.234 4.41 15.84 25.11 25.38 12.42\n𝑔0\n⊥𝑧 5.94 29.7 46.08 42.93 30.42 9.72\nNote that the intersublattice ( 𝑔0\n⊥𝑧and𝑔0\n𝑧⊥) contribution is almost an order of magnitude smaller than the intra-sublattice ( 𝑔0\n⊥⊥\nand𝑔0\n𝑧𝑧) contribution. However, the values for 𝑔0\n⊥𝑧and𝑔0\n𝑧⊥actually has a strong contribution coming from electron-optical-phonon\nscattering. The estimates for these interactions [48] are:\n(𝑔0\n⊥𝑧, 𝑔0\n𝑧⊥)𝑝ℎ𝑜𝑛𝑜𝑛=(−69,−52)meV·nm2(S23)\nBy setting the effective nucleus charge ˜𝑍=2.5, a value slightly reduced from the ˜𝑍=3 typically associated with an isolated\ncarbon atom [49], and factoring in the electron-phonon contributions, we derived the following estimates:\n(𝑔0\n⊥⊥,𝑔0\n𝑧𝑧,𝑔0\n⊥𝑧, 𝑔0\n𝑧⊥)=(269,184,−26,−27)meV·nm2. (S24)\nIt is important to point out that our central result, namely that an increase in 𝜅leads to a more pronounced decrease in 𝑢⊥relative\nto the increase of 𝑢𝑧, remains consistent regardless of the choice of ˜𝑍. To leading order in electron-electron and electron-phonon\ncontribution, 𝑔0\n⊥0=𝑔0\n0⊥=𝑔0\n𝑧0=𝑔0\n0𝑧=0 [27].\nII. CALCULATION DETAILS OF LANDAU LEVEL MIXING\nIn the weakly interacting systems, the perturbative expansion of low-energy effective Hamiltonian reads that ˆ𝐻=ˆ𝐻0+ˆ𝑉(1)+\nˆ𝑉(2)+..., where ˆ𝐻0is the non-interacting Hamiltonian, ˆ𝑉(1)is the projection of the bare interactions onto the degenerate\nnon-interactng ground state manifold of energy 𝐸0. In this study, the non-interacting ground states have two electrons in the\nempty𝑛=0 Landau level (LL) along with fully empty(occupied) 𝑛 > 0(𝑛 < 0)LLs [29].The next order correction to the5\neffective Hamiltonian is obtained from the second order perturbation theory,\nˆ𝑉(2)=−P(ˆ𝐻𝑠+ˆ𝐻𝑎)P⊥1\nˆ𝐻0−𝐸0P⊥(ˆ𝐻𝑠+ˆ𝐻𝑎)P\n=−Pˆ𝐻𝑠P⊥1\nˆ𝐻0−𝐸0P⊥ˆ𝐻𝑠P−\u0012\nPˆ𝐻𝑠P⊥1\nˆ𝐻0−𝐸0P⊥ˆ𝐻𝑎P+ℎ.𝑐.\u0013\n−Pˆ𝐻𝑎P⊥1\nˆ𝐻0−𝐸0P⊥ˆ𝐻𝑎P\n≡ˆ𝑉(2)\n𝑠𝑠+ˆ𝑉(2)\n𝑠𝑎+ˆ𝑉(2)\n𝑎𝑎. (S25)\nThe first term ˆ𝑉(2)\n𝑠𝑠has been studied in previous works and could potentially play a crucial role in certain fractional quantum Hall\nstates [28, 36]. However, due to its spin-valley SU(4) symmetry, it cannot differentiate competing spin-valley orders and will\nnot be further discuss here. ˆ𝑉𝑠𝑎gives rise to first-order-in- 𝜅corrections to the valley-dependent effective interactions, while the\nquadratic-in- 𝑔0\n𝛼𝜇corrections ˆ𝑉𝑎𝑎is independent of 𝜅(i.e., dielectric screening environment).\nAs we briefly explained in the main text, ˆ𝑉(2)admits the decomposition ˆ𝑉(2)=ˆ𝑉3𝑏+ˆ𝑉2𝑏+ˆ𝑉1𝑏+𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 based on the\nnumber𝑛𝑣of the virtually excited single-particle states. Note that 𝑛𝑣≠0 since the intermediate many-body states must have\nenergy different from 𝐸0. The𝑛𝑣=1 and 2 terms give rise to three-body interactions ˆ𝑉3𝑏and two-body interactions ˆ𝑉2𝑏[29],\nrespectively. 𝑛𝑣=3 corresponds to a SU(4) symmetric single-particle self-energy ˆ𝑉1𝑏and can be omitted together with the\nconstant energy shift generated by the 𝑛𝑣=4 term. ˆ𝑉3𝑏=0 in the zeroth LL of monolayer graphene thanks to the particle-hole\nsymmetric bare Hamiltonian and the particle-hole symmetry between the 𝑁 > 0 and𝑁 < 0 remote LLs [36]. The corrections to\nvalley-dependent effective interactions are entirely contributed by ˆ𝑉2𝑏.\nTo study the two-body interactions preserving the translational and rotational symmetries, it is convenient to use center of\nmass and relative coordinates,\n𝐴=𝑎𝑖+𝑎𝑗√\n2, 𝐵=𝑏𝑖+𝑏𝑗√\n2, 𝑎=𝑎𝑖−𝑎𝑗√\n2, 𝑏=𝑏𝑖−𝑏𝑗√\n2, (S26)\nwhere𝑎𝑖,𝑏𝑖are the lowering operators in the LL and guiding center space, respectively. We construct states with well defined\ncenter of mass and relative coordinate quantum numbers,\n|𝑁𝑀⟩𝑐|𝑛𝑚⟩𝑟=𝐴†𝑁𝐵†𝑀𝑎†𝑛𝑏†𝑚\n√\n𝑁!𝑀!𝑛!𝑚!|0⟩. (S27)\nFor a two-body interaction ˆ𝑉preserving the translational and rotational symmetries,\n𝑐⟨𝑁′𝑀′|𝑟⟨𝑛′𝑚′|ˆ𝑉|𝑁𝑀⟩𝑐|𝑛𝑚⟩𝑟=𝑟⟨𝑛′𝑚′|ˆ𝑉|𝑛𝑚⟩𝑟𝛿𝑚′−𝑛′,𝑚−𝑛𝛿𝑁′,𝑁𝛿𝑀′,𝑀, (S28)\nwhere the subscript ‘c’ and ‘r’ stands for the center of mass and relative coordinates, respectively. 𝑟⟨0𝑚|ˆ𝑉|0𝑚⟩𝑟are the Haldane\npseudopotentials in the zeroth LL. We follow Ref. [29] to evaluate in the next two sections the corrections to the valley-dependent\nHaldane pseudopotentials, 𝑢(2)\n𝛼(𝑚)/2 induced by 𝑉(2)\n𝑠𝑎and𝑢(2)′\n𝛼(𝑚)/2 induced by 𝑉(2)\n𝑎𝑎.\nA.𝑉𝑠𝑎−induced Haldane pseudopotential corrections 𝑢(2)\n𝛼(𝑚)\n𝑢(2)\n𝛼(𝑚)receive contributions from three types of non-vanishing Feynman diagrams depicted in Fig.2: (a) the particle-particle\nladder diagram, (b) the double-exchange diagram, and (c) the vertex correction to the valley-dependent interactions. As discussed\nin the main text,\n𝑢(2)\n𝛼(𝑚)=𝜅\n2𝜋𝑙2\n𝐵∑︁\n𝜇=0,𝑥,𝑦,𝑧(𝑐𝜇\n𝑚,𝑎+𝑐𝜇\n𝑚,𝑏+𝑐𝜇\n𝑚,𝑐)𝑔0\n𝛼𝜇. (S29)\nBecause𝑔0\n𝛼𝑥=𝑔0\n𝛼𝑦=𝑔0\n𝛼⊥, it will be convenient to define 𝑐⊥\n𝑚,𝑎/𝑏/𝑐=𝑐𝑥\n𝑚,𝑎/𝑏/𝑐+𝑐𝑦\n𝑚,𝑎/𝑏/𝑐.\nIt is straightforward to derive 𝑐0,𝑧\n𝑚,𝑎under two-particle bases ( 𝑐⊥\n𝑚,𝑎=0 due to the valley-sublattice locking in the zeroth LL,\nsee the main text.):\n𝑐0,𝑧\n𝑚,𝑎\n4𝜋=−22𝑁𝑐∑︁\n𝑛=1𝑟⟨0𝑚|𝛿(r𝑖𝑗)|𝑛𝑛+𝑚⟩𝑟𝑟⟨𝑛𝑛+𝑚|1\n𝑟𝑖𝑗|0𝑚⟩𝑟∞∑︁\n𝑛𝑖,𝑗=−∞Θ(𝑛𝑖𝑛𝑗)\n|𝜖𝑛𝑖+𝜖𝑛𝑗||𝑐⟨0|𝑟⟨𝑛||𝑛𝑖|,|𝑛𝑗|⟩|2. (S30)6\nWe measure the length in the unit of 𝑙𝐵and the kinetic energy in the unit of ℏ𝑣𝐹/𝑙𝐵in this section. The interaction matrix\nelements are derived as follows:\n𝑟⟨𝑛(𝑚+𝑛)|1\n𝑟𝑖𝑗|0𝑚⟩𝑟=∫𝑑2𝑞\n(2𝜋)22𝜋\n|q|F𝑛,0(√\n2q)F∗\n𝑚+𝑛,𝑚(−√\n2q) (S31)\n=(−1)𝑛\n2√︂\n(𝑛+𝑚)!\n𝑛!𝑚!Γ(𝑛+1\n2)\n𝑛!2𝐹1(−𝑚,𝑛+1\n2;𝑛+1; 1), (S32)\n𝑟⟨𝑛(𝑚+𝑛)|𝛿(r𝑖𝑗)|0𝑚⟩𝑟=∫𝑑2𝑞\n(2𝜋)2F𝑛,0(√\n2q)F∗\n𝑚+𝑛,𝑚(−√\n2q)\n=(−1)𝑛\n4𝜋√︂\n(𝑛+𝑚)!\n𝑛!𝑚!2𝐹1(−𝑚,𝑛+1;𝑛+1; 1)=(−1)𝑛\n4𝜋𝛿𝑚,0. (S33)\nIn Eq. (S31), we used the density form factor F𝑛′,𝑛(q)=⟨𝑛′|𝑒𝑖q·(ˆ𝑧×𝚷)|𝑛⟩of the LLs of the two dimensional electron gas and\nan analog relation for the guiding center coordinates ⟨𝑚′|𝑒𝑖q·R|𝑚⟩=F∗\n𝑚′,𝑚(−q).F𝑛′,𝑛(q)=F∗\n𝑛,𝑛′(−q)and for𝑛′≥𝑛,\nF𝑛′,𝑛(q)=√︂\n𝑛!\n𝑛′!\u0012𝑞𝑥+𝑖𝑞𝑦√\n2\u0013𝑛′−𝑛\n𝐿𝑛′−𝑛\n𝑛\u0012|𝑞|2\n2\u0013\n𝑒−|𝑞|2/4, (S34)\nwith𝐿𝑚\n𝑛(𝑥)the generalized Laguerre polynomial. The√\n2 factor in Eq. (S31) is because we are using the relative coordinates\nEq. (S26). 2𝐹1is the hypergeometric function. Eq. (S33) indicates that a pair of electrons with nonzero relative angular\nmomentum do not interact via contact interaction because their wave functions vanish at r𝑖𝑗=0.\nThe last term in Eq. (S30) is obtained from the relation\n|𝑛1,𝑛2⟩=𝑛1+𝑛2∑︁\n𝑛=0𝑅𝑛1+𝑛2𝑛2,𝑛×|𝑛1+𝑛2−𝑛⟩𝑐|𝑛⟩𝑟,(𝑛1,2≥0) (S35)\nwhere𝑅𝐿is a𝐿×𝐿real, symmetric and orthogonal matrix whose explicit form reads that\n𝑅𝐿\n𝑛′,𝑛=vt\u0000𝐿\n𝑛′\u0001\n2𝐿\u0000𝐿\n𝑛\u0001min(𝑛,𝑛′)∑︁\n𝜈=max(0,𝑛+𝑛′−𝐿)\u0012𝐿−𝑛′\n𝑛−𝜈\u0013\u0012𝑛′\n𝜈\u0013\n(−1)𝜈. (S36)\nOnly𝑅𝑛\n𝑛′,𝑛=(−1)𝑛√︃\u0000𝑛\n𝑛′\u0001/2𝑛is needed in Eq. (S30) because the center of mass kinetic quantum number 𝑛1+𝑛2−𝑛=0.\nCombining Eq. (S30)-(S36), we obtain 𝑐0,𝑧\n𝑚,𝑎=0 for𝑚≠0 and as shown in Eq.(9) in the main text,\n𝑐0,𝑧\n𝑚=0,𝑎\n4𝜋=2𝑁𝑐∑︁\n𝑛=1Γ(𝑛+1\n2)\n2𝑛+1𝑛!𝑛∑︁\n𝑛1=0\u0000𝑛\n𝑛1\u0001\n√2𝑛1+√︁\n2(𝑛−𝑛1). (S37)\nHere the cut off of the LL numbers is 2 𝑁𝑐because𝑛=|𝑛𝑖|+|𝑛𝑗|and|𝑛𝑖,𝑗|≤𝑁𝑐.\nThe expressions for 𝑐𝜇\n𝑚,𝑏/𝑐are given by Eqs.(10)-(11) in the main text. In the interaction matrix elements VandV𝜇, we\nintroduced the graphene form factors 𝐹𝜇\n𝜉′𝑛′,𝜉𝑛(q)=D\n𝜉′𝑛′\f\f\f𝜎𝜇𝑒𝑖q·(ˆ𝑧×𝚷)\f\f\f𝜉𝑛E\nfor𝜇=0,±,𝑧[50],\n𝐹0/𝑧\n𝜉′𝑛′,𝜉𝑛(q)=1\n21−(𝛿𝜉,0+𝛿𝜉′,0)/2\u0000F𝑛′,𝑛(q)±𝜉𝜉′F𝑛′−1,𝑛−1(q)\u0001, (S38)\n𝐹+\n𝜉′𝑛′,𝜉𝑛(q)=1\n21−(𝛿𝜉,0+𝛿𝜉′,0)/2𝜉F𝑛′,𝑛−1(q), (S39)\n𝐹−\n𝜉′𝑛′,𝜉𝑛(q)=1\n21−(𝛿𝜉,0+𝛿𝜉′,0)/2𝜉′F𝑛′−1,𝑛(q), (S40)\nwhere𝜉,𝜉′=±1 and𝑛,𝑛′≥0. Here we require that the density form factor F𝑛′,𝑛(q)=0 if𝑛′<0 or𝑛<0. Using the relation7\n𝑟⟨𝑚|𝑒𝑖q·η𝑖𝑗|𝑚⟩���=𝑒−q2\n2𝐿𝑚(q2), we find that\n𝑐𝑧\n𝑚,𝑏\n4𝜋=∬𝑑2q1𝑑2q2\n(2𝜋)42𝜋\n|q2|𝑒−q2\n1−q2\n2+q1·q2𝐿𝑚(|q1−q2|2)𝑁𝑐∑︁′\n𝑛,𝑛′=01√\n2𝑛+√\n2𝑛′1\n𝑛!𝑛′!\u0012¯𝑞1¯𝑞∗\n2\n2\u0013𝑛\u0012¯𝑞∗\n1¯𝑞2\n2\u0013𝑛′\n=∫∞\n0∫∞\n0𝑑𝑞1𝑑𝑞2\n(2𝜋)2∫2𝜋\n0𝑑𝜃𝑞 1𝑒−𝑞2\n1−𝑞2\n2−𝑞1𝑞2cos𝜃𝐿𝑚(𝑞2\n1+𝑞2\n2−2𝑞1𝑞2cos𝜃)𝑁𝑐∑︁′\n𝑛,𝑛′=01√\n2𝑛+√\n2𝑛′1\n𝑛!𝑛′!\u0010𝑞1𝑞2\n2\u0011𝑛+𝑛′\n𝑒𝑖(𝑛′−𝑛)𝜃,\n(S41)\n𝑐⊥\n𝑚,𝑏\n4𝜋=−2∬𝑑2q1𝑑2q2\n(2𝜋)42𝜋\n|q2|𝑒−q2\n1−q2\n2+q1·q2𝐿𝑚(|q1−q2|2)𝑁𝑐∑︁\n𝑛,𝑛′=11√\n2𝑛+√\n2𝑛′√\n𝑛𝑛′\n𝑛!𝑛′!\u0012¯𝑞1¯𝑞∗\n2\n2\u0013𝑛\u0012¯𝑞∗\n1¯𝑞2\n2\u0013𝑛′ \nq2\n1\n2!−1\n=−4∫∞\n0∫∞\n0𝑑𝑞1𝑑𝑞2\n(2𝜋)2∫2𝜋\n0𝑑𝜃1\n𝑞1𝑒−𝑞2\n1−𝑞2\n2−𝑞1𝑞2cos𝜃𝐿𝑚(𝑞2\n1+𝑞2\n2−2𝑞1𝑞2cos𝜃)𝑁𝑐∑︁\n𝑛,𝑛′=11√\n2𝑛+√\n2𝑛′√\n𝑛𝑛′\n𝑛!𝑛′!\u0010𝑞1𝑞2\n2\u0011𝑛+𝑛′\n𝑒𝑖(𝑛′−𝑛)𝜃,\n(S42)\nwith ¯𝑞𝑖≡𝑞𝑖,𝑥+𝑖𝑞𝑖,𝑦.\n𝑐0/𝑧\n𝑚,𝑐=−Re∫∞\n0∫∞\n0𝑑𝑞1𝑑𝑞2\n(2𝜋)2∫2𝜋\n0𝑑𝜃𝑞 1𝑒−𝑞2\n1−𝑞2\n2\n2−𝑖𝑞1𝑞2sin𝜃𝐿𝑚(𝑞2\n1)\"∑︁′\n𝑛′≥𝑛≥01√\n2𝑛+√\n2𝑛′1\n2𝑛′𝑛′!𝑞𝑛′−𝑛\n1𝑞𝑛+𝑛′\n2𝑒𝑖(𝑛′−𝑛)𝜃\n× \n𝐿𝑛′−𝑛\n𝑛(𝑞2\n1\n2)∓𝐿𝑛′−𝑛\n𝑛−1(𝑞2\n1\n2)!\n+∑︁\n𝑛>𝑛′≥01√\n2𝑛+√\n2𝑛′1\n2𝑛𝑛!(−𝑞1)𝑛−𝑛′𝑞𝑛+𝑛′\n2𝑒𝑖(𝑛′−𝑛)𝜃 \n𝐿𝑛−𝑛′\n𝑛′(𝑞2\n1\n2)∓𝐿𝑛−𝑛′\n𝑛′−1(𝑞2\n1\n2)!#\n=−Re∫∞\n0𝑑𝑞1\n2𝜋𝑞1𝑒−𝑞2\n1𝐿𝑚(𝑞2\n1)𝑁𝑐∑︁′\n𝑛,𝑛′=01√\n2𝑛+√\n2𝑛′1\n2𝑙𝑙!𝑞|𝑛−𝑛′|\n1𝐼𝑛,𝑛′(𝑞1) \n𝐿|𝑛−𝑛′|\n𝑗(𝑞2\n1\n2)∓𝐿|𝑛−𝑛′|\n𝑗−1(𝑞2\n1\n2)!\n. (S43)\nIn the last line, we defined 𝑗=min(𝑛,𝑛′),𝑙=max(𝑛,𝑛′), and the integral [51]\n𝐼𝑛,𝑛′(𝑞1)=∫∞\n0𝑑𝑞2𝑞𝑛+𝑛′\n2𝐽|𝑛−𝑛′|(𝑞1𝑞2)𝑒−𝑞2\n2\n2=2𝑙+1\n2Γ\u0010\n𝑙+1\n2\u0011\n2|𝑛−𝑛′|+1Γ(|𝑛−𝑛′|+1)1𝐹1 \n𝑙+1\n2;|𝑛−𝑛′|+1;−𝑞2\n1\n2!\n𝑞|𝑛−𝑛′|\n1. (S44)\nHere, 1𝐹1is a confluent hypergeometric function. The summations in Eq. (S41) and (S43) exclude the 𝑛=𝑛′=0 case. Our\nnumerical results of Eqs. (S41)-(S43), summarized in Tables I(a) and (b) in the main text, indicate that 𝑐𝜇\n0,𝑏and𝑐𝑧\n0,𝑐logarithmically\ndiverge with respect to the LL cutoff 𝑁𝑐as𝑐0/𝑧\n0,𝑎do, whereas 𝑐0\n0,𝑐and𝑐𝜇\n𝑚≠0,𝑏/𝑐converge.\nThe logarithmic divergence of the contact valley-dependent interactions in graphene have been studied previously in the\nzero-field with the renormalization group (RG) analysis [27]. It was shown that the sublattice-and valley-dependent short-range\ninteractions obey the RG flow equations 𝑑¯𝑔𝛼0/𝑑𝜉=0 and𝑑¯𝑔𝛼𝑧/𝑑𝜉=4𝐹(𝑤)(¯𝑔𝛼𝑧−¯𝑔𝛼⊥), where𝐹(𝑤)≈𝜅/4 for small𝜅,\n𝜉=ln𝑙/𝑎0and𝑙is the running length scale. Ref. [27] then estimates the effective valley-dependent interactions in the zeroth LL\nin the magnetic field as 2 𝜋𝑙2\n𝐵𝑢𝛼=¯𝑔𝛼0(𝑙=𝑙𝐵)+¯𝑔𝛼𝑧(𝑙=𝑙𝐵). In the limit 𝜅ln𝑙𝐵/𝑎0≪1, the RG equation yields\n2𝜋𝑙2\n𝐵𝑢𝛼≈𝑔𝛼0+𝑔𝛼𝑧(1+𝜅ln𝑙𝐵\n𝑎0)−𝜅𝑔𝛼⊥ln𝑙𝐵\n𝑎0(S45)\n=𝑔𝛼0+𝑔𝛼𝑧(1+0.5𝜅ln𝑁𝑐)−0.5𝜅𝑔𝛼⊥ln𝑁𝑐+𝑐𝑜𝑛𝑠𝑡., (S46)\nTo arrive at the second line, we note that 𝑁𝑐depends on the out-of-plane magnetic field such that√𝑁𝑐/𝑙𝐵∝a fixed energy cutoff\nΛ. The coefficient of the first-order-in- 𝜅ln𝑁𝑐term agrees with our results, see Fig. S4. We note that the ratio 𝑢𝑧/𝑢⊥is affected\nby both𝜅and𝑙𝐵. Thus, there can be a quantum phase transition between the KD and AF states induced by either by out-of-plane\nmagnetic field (via 𝑙𝐵) or the dielectric screening environment (via 𝜅). In the main text, we primarily explore the transition\ninduced by changes in the dielectric environment. However, it’s important to mention that a field-induced phase transition also\noccurs for a moderate fine structure constant ( 𝜅=0.5∼1). Specifically, in a strong field scenario ( 𝑁𝑐=100), the system resides\nin the AF state, whereas in a weak field ( 𝑁𝑐=7000), it transitions to the KD state. Due to the logarithmic dependence of 𝑢𝑧and\n𝑢⊥on the magnetic field, adjusting the dielectric environment is a more effective means of modulating the phase boundary than\nvarying the magnetic field strength.8\n101102103104\nNc1.52.02.53.03.54.04.5|cz,\n0|\nuz\nu\nFigure S4. The logarithmic dependence of |𝑐𝑧,⊥\n0|on the LL cutoff 𝑁𝑐. The blue dots and red triangles represent numerical results for 𝑐𝑧\n0and\n−𝑐⊥\n0, respectively. The dashed lines are linear fit in the semi-log plot, which indicate |𝑐𝑧,⊥\n0|≈0.11+0.5 ln𝑁𝑐.\n(a) (b) (c) (d)\nFigure S5. Feynman diagrams representing the quadratic-in- 𝑔0𝛼𝜇corrections to two-body valley-dependent Haldane pseudopotentials in the\nzeroth LL. The dashed lines stand for the contact sublattice-and valley-dependent interactions.\nHere we have taken a common UV cutoff for all the short-range interactions comparable to the lattice constant. This\nis appropriate for those that originate from the Coulomb forces. For those arising from optical phonons it would be more\nappropriate to take the UV cutoff to be comparable to the phonon gap. However, since the phonon gap is smaller than the band-\nwidth of the 𝑝𝑧orbitals, accounting for this would mean that the contribution to 𝑢𝛼coefficients from the interactions mediated\nby phonons (i.e. 𝑔0\n⊥𝑧,𝑔0\n𝑧⊥as discussed in Eq.(S23)), would have a smaller enhancement factor due to renormalizations than the\none we have used here, because the phonon interactions would effectively renormalize over a shorter RG time. However, since\nthe phonon contribution to interaction coefficients is already smaller than the Coulomb contribution, accounting for this would\nnot significantly alter our estimates. For example, this would mean that the coefficient 𝑐𝑧\n0that accompanies the 𝐾-optical-phonon\npotential𝑔⊥𝑧contribution to 𝑢⊥would be smaller than the one listed before (e.g. in Eq.(S45)), because it would have smaller UV\ncutoff. However, this contribution to the linear in 𝜅enhancement of 𝑢⊥is smaller compared to that coming from the coefficient\n𝑔0\n⊥⊥generated by the Coulomb interactions. Thus we expect accounting for this would not significantly alter our estimates.\nB. ˆ𝑉𝑎𝑎−induced Haldane pseudopotential corrections to 𝑢𝛼(𝑚)\nThe non-vanishing valley-dependent interactions generated ˆ𝑉𝑎𝑎can be represented by Feynman diagrams depicted in Fig. S5a\n(particle-particle), b (double-exchange), c (vertex correction), and d (bubble). The first three types of Feynman diagrams can be\nevaluated according to the same procedure outlined in the previous section and we will explain later the details to compute the\nbubble diagrams, which cannot be generated by ˆ𝑉𝑠𝑎. The results show that the main contributions from ˆ𝑉𝑎𝑎are the corrections9\nto𝑚=0 Haldane pseudopotentials that diverge as√𝑁𝑐,\n𝑢(2)′\n⊥(𝑚=0)=1\n2𝜋𝑙2\n𝐵¯𝑔\u0002\n𝑔0\n𝑧𝑧𝑔0\n⊥𝑧𝑧𝑎−𝑔0\n𝑧𝑧𝑔0\n⊥𝑧𝑧𝑏+2(𝑔0\n𝑧𝑧𝑔0\n⊥⊥+𝑔0\n𝑧⊥𝑔0\n⊥𝑧)𝑦𝑏−4𝑔0\n𝑧⊥𝑔0\n⊥⊥𝑥𝑏−𝑔0\n𝑧𝑧𝑔0\n⊥𝑧𝑧𝑐−2𝑔0\n𝑧⊥𝑔0\n⊥𝑧𝑦𝑐−(𝑔0\n⊥𝑧)2𝑧𝑑\u0003\n,\n(S47)\n𝑢(2)′\n𝑧(𝑚=0)=1\n2𝜋𝑙2\n𝐵¯𝑔\u0002\n(𝑔0\n⊥𝑧)2𝑧𝑎−(𝑔0\n⊥𝑧)2𝑧𝑏+4𝑔0\n⊥𝑧𝑔0\n⊥⊥𝑦𝑏−4(𝑔0\n⊥⊥)2𝑥𝑏+𝑔0\n𝑧𝑧(𝑔0\n𝑧𝑧−2𝑔0\n⊥𝑧)𝑧𝑐+2𝑔0\n𝑧𝑧(𝑔0\n𝑧⊥−2𝑔0\n⊥⊥)𝑦𝑐−(𝑔0\n𝑧𝑧)2𝑧𝑑\u0003\n,\n(S48)\nwhere we have kept 𝑔0𝜇=𝑔𝛼0=0 for simplicity, ¯ 𝑔= ℏ𝑣𝐹𝑙𝐵/√𝑁𝑐, and the coefficients 𝑧𝑎≈0.11,𝑧𝑏≈𝑦𝑏≈𝑥𝑏≈0.1,𝑧𝑐=\n0.2,𝑦𝑐=0.017,and𝑧𝑑=0.45. The subscripts of these coefficients represent the associated Feynman diagrams. For the bubble\ndiagram,\n𝑢(2)′\n𝛼,𝑑(𝑚)\n2=−4×\u0000𝑔0\n𝛼𝑧\u00012\n2𝜋𝑙2\n𝐵∫𝑑2𝑞\n(2𝜋)2𝑟⟨𝑚|𝑒𝑖q·η𝑖𝑗|𝑚⟩𝑟∞∑︁′\n𝑛,𝑛′=−∞Θ(𝜖𝑛)−Θ(𝜖𝑛′)\n𝜖𝑛−𝜖𝑛′|𝐹𝑧\n𝑛′,𝑛(q)𝐹𝑧\n0,0(q)|2\n=−1\n2𝜋𝑙2\n𝐵2\u0000𝑔0\n𝛼𝑧\u00012\nℏ𝑣𝐹𝑙𝐵∫𝑑2q\n(2𝜋)2𝑒−3q2\n2𝐿𝑚(q2)𝑁𝑐∑︁′\n𝑛,𝑛′=01√︁\n2|𝑛|+√︁\n2|𝑛′|𝑗!\n𝑙!\u0012q2\n2\u0013𝑙−𝑗\u0012\n𝐿𝑙−𝑗\n𝑗(q2\n2)+𝐿𝑙−𝑗\n𝑗−1(q2\n2)\u00132\n. (S49)\nwhere𝑗=min(𝑛,𝑛′),𝑙=max(𝑛,𝑛′), and the summation excludes the 𝑛=𝑛′=0 case. Our numerical calculations suggest\nthat𝑢(2)′\n𝛼,𝑑(𝑚)≈−𝑧𝑑(𝑔0\n𝛼𝑧)2\n2𝜋𝑙2\n𝐵¯𝑔for large𝑁𝑐with𝑧𝑑converging to∼0.45.𝑢(2)′\n𝛼,𝑑(𝑚 > 0)is at least one order of magnitude smaller\nthan𝑢(2)′\n𝛼,𝑑(𝑚=0)and thus the valley-dependent interactions remains short-ranged, although we would like to mention that\n|𝑢(2)′\n𝛼,𝑑|≫|𝑢(2)′\n𝛼,𝑎/𝑏/𝑐|for𝑚 > 0, indicating that the effective interactions generated by the bubble diagrams can have a longer\ninteraction range than the other three channels.\nWe comment that the square root divergence of 𝛿𝑢′\n⊥,𝑧(0)causes significant uncertainties on the estimation of their values. For\ninstance, if we fix 𝑔𝑣𝑁𝑐/2𝜋𝑙2\n𝐵=number of graphene unit cell per area ( 𝑔𝑣=2 is the valley degeneracy), ¯ 𝑔=ℏ𝑣𝐹𝑎0√︃√\n3/2𝜋≈\n85meV·nm2and, according to the estimated strength of bare valley-dependent interactions, Eq. (S24), 2 𝜋𝑙2\n𝐵𝑢(2)′\n⊥(𝑚=0)∼\n159meV·nm2and 2𝜋𝑙2\n𝐵𝑢(2)′\n𝑧(𝑚=0)∼− 492meV·nm2. On the other hand, if ℏ𝑣𝐹√2𝑁𝑐/𝑙𝐵= Λ∼2eV, the energy cut off\nof the continuum model of graphene, 2 𝜋𝑙2\n𝐵𝑢(2)′\n⊥(𝑚=0)∼44meV·nm2and 2𝜋𝑙2\n𝐵𝑢(2)′\n𝑧(𝑚=0)∼− 137meV·nm2. Despite\nthese uncertainties on the estimates, we can see that 𝑢(2)′\n𝛼(𝑚=0)are overall smaller than 𝑢(2)\n𝛼(𝑚=0)depicted in Fig.1 for\nlarge𝜅 > 0.5, although their ratio is much larger than the ratio of the bare lattice-scale interaction and long-range Coulomb\ninteraction, O(𝑔0\n𝛼𝜇𝜖/𝑒2𝑙𝐵), due to the√𝑁𝑐divergence in Eqs. (S47) and (S48). More importantly, we emphasize that unlike\n𝑢(2)\n𝛼(𝑚=0)generated by ˆ𝑉(2)\n𝑠𝑎,𝑢(2)′\n𝛼(𝑚=0)are insensitive to the long-wave dielectric screening ( i.e.,𝜅), which justifies\nneglecting𝑢(2)′\n𝛼(𝑚=0)in the study of the dielectric screening effects on the valley-dependent Haldane pseudopotentials in the\nmain text."    },    {        "title": "2401.12530v1.Revisit_on_global_existence_of_solutions_for_semilinear_damped_wave_equations_in___mathbb_R__N__with_noncompactly_supported_initial_data.pdf",        "content": "arXiv:2401.12530v1  [math.AP]  23 Jan 2024Revisit on global existence of solutions for\nsemilinear damped wave equations in R/u1D475with\nnoncompactly supported initial data\nYuta Wakasugi\nAbstract In this note, we study the Cauchy problem of the semilinear da mped wave\nequation and our aim is the small data global existence for no ncompactly supported\ninitial data. For this problem, Ikehata and Tanizawa [5] int roduced the energy method\nwith the exponential-type weight function /u1D452|/u1D465|2/(1+/u1D461), which is the so-called Ikehata–\nTodorova–Yordanov type weight. In this paper, we suggest an other weight function\nof the form (1+|/u1D465|2/(1+/u1D461))/u1D706, which allows us to treat polynomially decaying initial\ndata and give a simpler proof than the previous studies treat ing such initial data.\n1 Introduction\nIn this note, we study the Cauchy problem of the semilinear da mped wave equation\n/braceleftBigg\n/u1D462/u1D461/u1D461−Δ/u1D462+/u1D462/u1D461=|/u1D462|/u1D45D, (/u1D461,/u1D465) ∈ (0,∞) ×R/u1D441,\n/u1D462(0,/u1D465)=/u1D4620(/u1D465), /u1D462/u1D461(0,/u1D465)=/u1D4621(/u1D465), /u1D465∈R/u1D441,(1)\nwhere/u1D462=/u1D462(/u1D461,/u1D465)is a real-valued unknown, /u1D465=(/u1D4651,...,/u1D465/u1D441) ∈R/u1D441,/u1D441≥1,\n/u1D462/u1D461=/u1D715/u1D462//u1D715/u1D461,/u1D462/u1D461/u1D461=/u1D7152/u1D462//u1D715/u1D4612,Δ/u1D462=/u1D7152/u1D462//u1D715/u1D4652\n1+ ···+/u1D7152/u1D462//u1D715/u1D4652\n/u1D441,/u1D45D >1, and/u1D4620,/u1D4621are\ngiven initial data.\nThe purpose is to show the small data global existence of the s olution with non-\ncompactly supported initial data. For this problem, Ikehat a–Tanizawa [5] introduced\nthe energy method with the exponential-type weight functio n/u1D452|/u1D465|2/(1+/u1D461), which is\nthe so-called Ikehata–Todorova–Yordanov type weight. The y proved the small data\nglobal existence for /u1D45D > /u1D45D/u1D439(/u1D441):=1+2//u1D441and exponentially decaying initial data.\nThis result removed the compactness assumption on the suppo rt of initial data in the\nearlier result by Todorova–Yordanov [9].\nLaboratory of Mathematics, Graduate School of Advanced Sci ence and Engineering, Hiroshima\nUniversity, Higashi-Hiroshima, 739-8527, Japan. e-mail: wakasugi@hiroshima-u.ac.jp\n12 Yuta Wakasugi\nIn this note, we suggest another weight function of the form (1+|/u1D465|2\n1+/u1D461)/u1D706with\nsufficiently large /u1D706. This allows us to weaken the assumption of [5] to polynomial ly\ndecaying initial data. Actually, this type of weight was alr eady used in Ikehata–\nNishihara–Zhao [6] to study the case of absorbing nonlinear ity. However, it seems\nto have been overlooked that this technique is also effective in the case of sourcing\nnonlinearity.\nTo state our results, we first define the mild solution. Let D(/u1D461)be the fundamental\nsolution of the linear damped wave equation, that is, D(/u1D461)is defined by the Fourier\nmultiplier\nD(/u1D461)=F−1/u1D452−/u1D461\n2sinh(/u1D461/radicalbig\n1/4− |/u1D709|2)/radicalbig\n1/4− |/u1D709|2F,\nwhereF,F−1denote the Fourier and inverse Fourier transforms, respect ively.\nDefinition 1 Let/u1D447∈ (0,∞]and/u1D43C=[0,/u1D447]if/u1D447 <∞and/u1D43C=[0,∞)if/u1D447=∞. Let\n(/u1D4620,/u1D4621) ∈/u1D43B1(R/u1D441) ×/u1D43F2(R/u1D441). We say that /u1D462∈/u1D436(/u1D43C;/u1D43B1(R/u1D441)) ∩/u1D4361(/u1D43C;/u1D43F2(R/u1D441))is\na mild solution of the Cauchy problem (1) on /u1D43Cif/u1D462satisfies\n/u1D462(/u1D461)=D(/u1D461)(/u1D4620+/u1D4621) +/u1D715\n/u1D715/u1D461D(/u1D461)/u1D4620+/uni222B.dsp/u1D461\n0D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460\nin/u1D436(/u1D43C;/u1D43B1(R/u1D441)) ∩/u1D4361(/u1D43C;/u1D43F2(R/u1D441)). In particular, if /u1D447 <∞, then/u1D462is called a local\n(in time) mild solution; if /u1D447=∞, then/u1D462is called a global (in time) mild solution.\nThe main result of this note is the following.\nTheorem 1 Let/u1D441≥1and let/u1D45D >1satisfy\n/u1D45D/u1D439(/u1D441)< /u1D45D≤/u1D441\n[/u1D441−2]+,\nwhere/u1D45D/u1D439(/u1D441)=1+2//u1D441. Then, there exist constants /u1D706=/u1D706(/u1D441, /u1D45D)>0and/u1D700=\n/u1D700(/u1D441, /u1D45D,/u1D706)>0such that if the initial data (/u1D4620,/u1D4621) ∈/u1D43B1(R/u1D441) ×/u1D43F2(R/u1D441)satisfy\n/ba∇dbl(1+ |/u1D465|)/u1D706/u1D4621/ba∇dbl/u1D43F2+ /ba∇dbl(1+ |/u1D465|)/u1D706∇/u1D465/u1D4620/ba∇dbl/u1D43F2+ /ba∇dbl(1+ |/u1D465|)/u1D706/u1D4620/ba∇dbl/u1D43F2< /u1D700, (2)\nthen the Cauchy problem (1)admits the unique global mild solution.\nRemark 1 The notation /u1D45D≤/u1D441\n[/u1D441−2]+means that /u1D45D <∞for/u1D441≤2 and/u1D45D≤/u1D441\n/u1D441−2for\n/u1D441≥3. Also, we may take /u1D706in the theorem so that /u1D706 >max{1,/u1D441\n2}and\n/u1D706 >max/braceleftbigg2/u1D441− (/u1D441−2)/u1D45D\n2/u1D45D,2/u1D441−8//u1D441− (/u1D441−2)/u1D45D\n4(/u1D45D−/u1D45D/u1D439(/u1D441)),/u1D45E−1\n/u1D45E/u1D441+2− (/u1D441−2)/u1D45D\n2(/u1D45D−/u1D45D/u1D439(/u1D441))/bracerightbigg\nwith/u1D45E=max{2,/u1D441\n2(/u1D45D−1)}hold. Note that /u1D706 >/u1D441\n2ensures/u1D4620,∇/u1D465/u1D4620,/u1D4621∈/u1D43F1(R/u1D441).\nRemark 2 The conclusion of Theorem 1 itself is weaker than those of Hay ashi–\nKaikina–Naumkin [3] and Sobajima [8] in the sense that our or der of the weight /u1D706Semilinear damped wave equation for noncompactly data 3\n(see Remark 1 above) has to be larger than those of [3, 8]. Howe ver, the method of\nthe proof of Theorem 1 is more elementary and it will also be ap plicable to the case\nof time-dependent damping such as\n/u1D462/u1D461/u1D461−Δ/u1D462+/u1D707\n1+/u1D461/u1D462/u1D461=|/u1D462|/u1D45D.\nIt will slightly improve the assumption on the initial data i n the result of D’Abbicco\n[2, Theorem 3]. Moreover, combining with the linear estimat es for damped wave\nequations in general measure spaces proved in [4], we will be able to obtain global\nexistence results for semilinear damped wave equations in m ore general domains or\nwith some potentials, etc.\nFinally, we introduce some notations used in this note. The s et of nonnegative\nintegers is expressed by Z≥0. The symbol /u1D453/lessorsimilar/u1D454stands for /u1D453≤/u1D436/u1D454with some\nconstant/u1D436 >0. The gradient with respect to the spatial variables is writ ten as∇/u1D465,\nthat is,∇/u1D465/u1D462=(/u1D462/u1D4651,...,/u1D462 /u1D465/u1D441)./u1D43F/u1D45E(R/u1D441)and/u1D43B/u1D460(R/u1D441)denote the usual Lebesgue and\nSobolev spaces, respectively.\n2 Energy estimate with a polynomial-type weight\nThe main idea of this note is to use the following weighted ene rgy functional.\nDefinition 2 For/u1D706 >0 and/u1D434≥/u1D706/2, we define the weight functions by\n/u1D713(/u1D461,/u1D465)=/u1D434+|/u1D465|2\n1+/u1D461,Ψ(/u1D461,/u1D465)=/u1D713(/u1D461,/u1D465)/u1D706(3)\nand the weighted energy of a function /u1D462by\n/u1D438Ψ(/u1D461)=/uni222B.dsp\nR/u1D441(|/u1D462/u1D461(/u1D461,/u1D465)|2+ |∇/u1D465/u1D462(/u1D461,/u1D465)|2)Ψ(/u1D461,/u1D465)/u1D451/u1D465.\nLemma 1 The weight function ΨsatisfiesΨ/u1D461≤0and\n|∇/u1D465Ψ|2\n−Ψ/u1D461≤2Ψ.\nProof We calculate\nΨ/u1D461=/u1D706/u1D713/u1D706−1/u1D713/u1D461=−/u1D706|/u1D465|2\n4(1+/u1D461)2/u1D713/u1D706−1<0.\nMoreover, since /u1D434≥/u1D706/2, then we have4 Yuta Wakasugi\n|∇/u1D465Ψ|2\n−Ψ/u1D461=/u1D7062|/u1D465|2\n4(1+/u1D461)2/u1D7132(/u1D706−1)\n/u1D706|/u1D465|2\n4(1+/u1D461)2/u1D713/u1D706−1=/u1D706/u1D713/u1D706−1=/u1D706\n/u1D713Ψ≤/u1D706\n/u1D434Ψ≤2Ψ.\nThis completes the proof. /square\nLemma 2 Let/u1D462be a mild solution to (1)on[0,/u1D447]. Then, we have\n/u1D438Ψ(/u1D461) ≤/u1D438Ψ(0) −2\n/u1D45D+1/uni222B.dsp\nR/u1D441|/u1D4620(/u1D465)|/u1D45D/u1D4620(/u1D465)Ψ(0,/u1D465)/u1D451/u1D465\n+2\n/u1D45D+1/uni222B.dsp\nR/u1D441|/u1D462(/u1D461,/u1D465)|/u1D45D/u1D462(/u1D461,/u1D465)Ψ(/u1D461,/u1D465)/u1D451/u1D465\n−2\n/u1D45D+1/uni222B.dsp/u1D461\n0/uni222B.dsp\nR/u1D441|/u1D462(/u1D460,/u1D465)|/u1D45D/u1D462(/u1D460,/u1D465)Ψ/u1D461(/u1D460,/u1D465)/u1D451/u1D465/u1D451/u1D460,\nprovided that the right-hand side is finite.\nProof Differentiating /u1D438Ψ(/u1D461), applying integration by parts and using the equation\n(1), we have /one.sup\n/u1D451\n/u1D451/u1D461/u1D438Ψ(/u1D461)=2/uni222B.dsp\nR/u1D441(/u1D462/u1D461/u1D462/u1D461/u1D461+∇/u1D465/u1D462·∇/u1D465/u1D462/u1D461)Ψ/u1D451/u1D465+/uni222B.dsp\nR/u1D441/parenleftBig\n|/u1D462/u1D461|2+ |∇/u1D465/u1D462|2/parenrightBig\nΨ/u1D461/u1D451/u1D465\n=−2/uni222B.dsp\nR/u1D441/u1D4622\n/u1D461Ψ/u1D451/u1D465+2/uni222B.dsp\nR/u1D441|/u1D462|/u1D45D/u1D462/u1D461Ψ/u1D451/u1D465\n−2/uni222B.dsp\nR/u1D441/u1D462/u1D461(∇/u1D465/u1D462·∇/u1D465Ψ)/u1D451/u1D465+/uni222B.dsp\nR/u1D441/parenleftBig\n|/u1D462/u1D461|2+ |∇/u1D465/u1D462|2/parenrightBig\nΨ/u1D461/u1D451/u1D465.\nHere, by Lemma 1, we note that Ψ/u1D461≤0 and\n−2/u1D462/u1D461(∇/u1D465/u1D462·∇/u1D465Ψ) + |∇/u1D465/u1D462|2Ψ/u1D461=1\nΨ/u1D461|Ψ/u1D461∇/u1D465/u1D462−/u1D462/u1D461∇/u1D465Ψ|2−|∇/u1D465Ψ|2\nΨ/u1D461/u1D4622\n/u1D461≤2Ψ/u1D4622\n/u1D461\nhold. Using them to the previous identity, we deduce\n/u1D451\n/u1D451/u1D461/u1D438Ψ(/u1D461) ≤2/uni222B.dsp\nR/u1D441|/u1D462|/u1D45D/u1D462/u1D461Ψ/u1D451/u1D465\n=2\n/u1D45D+1/u1D451\n/u1D451/u1D461/uni222B.dsp\nR/u1D441|/u1D462|/u1D45D/u1D462Ψ/u1D451/u1D465−2\n/u1D45D+1/uni222B.dsp\nR/u1D441|/u1D462|/u1D45D/u1D462Ψ/u1D461/u1D451/u1D465.\nIntegrating it over [0,/u1D461]gives the conclusion. /square\n/one.supBy an approximation argument, we can justify the fornal calc lulation here, see e.g., [10]Semilinear damped wave equation for noncompactly data 5\n3 Proof of Theorem 1\n3.1 Preliminaries\nThe following local existence result can be proved in comple tely the same way as\nIkehata–Tanizawa [5, Proposition 2.1], and we omit the proo f.\nLemma 3 Let/u1D441≥1,1< /u1D45D≤/u1D441\n[/u1D441−2]+. Then for each (/u1D4620,/u1D4621) ∈/u1D43B1(R/u1D441)∩/u1D43F2(R/u1D441)\nsatisfying\n/ba∇dbl(1+ |/u1D465|)/u1D706/u1D4621/ba∇dbl/u1D43F2+ /ba∇dbl(1+ |/u1D465|)/u1D706∇/u1D465/u1D4620/ba∇dbl/u1D43F2+ /ba∇dbl(1+ |/u1D465|)/u1D706/u1D4620/ba∇dbl/u1D43F2<∞,\nthere exists /u1D447 >0such that the Cauchy problem (1)admits a unique mild solution\non[0,/u1D447]satisfying\n/ba∇dblΨ1/2/u1D462/u1D461(/u1D461)/ba∇dbl/u1D43F2+ /ba∇dblΨ1/2∇/u1D465/u1D462(/u1D461)/ba∇dbl/u1D43F2+ /ba∇dblΨ1/2/u1D462(/u1D461)/ba∇dbl/u1D43F2<∞\nfor/u1D461∈ [0,/u1D447]. Moreover, if the maximal existence time /u1D447/u1D45A, the supremum of /u1D447such\nthat there is a unique mild solution on [0,/u1D447], is finite, then the solution must satisfy\nlim sup\n/u1D461↑/u1D447/u1D45A/parenleftBig\n/ba∇dblΨ1/2/u1D462/u1D461(/u1D461)/ba∇dbl/u1D43F2+ /ba∇dblΨ1/2∇/u1D465/u1D462(/u1D461)/ba∇dbl/u1D43F2+ /ba∇dblΨ1/2/u1D462(/u1D461)/ba∇dbl/u1D43F2/parenrightBig\n=+∞.\nWe define the following norm to construct the global solution .\nDefinition 3 We define the norm /ba∇dbl · /ba∇dbl/u1D44B(/u1D447)by\n/ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447):=sup\n0</u1D461</u1D447/bracketleftBig\n/u1D438Ψ(/u1D461)1\n2+ (1+/u1D461)/u1D441\n4+1/ba∇dbl/u1D462/u1D461(/u1D461)/ba∇dbl/u1D43F2\n+(1+/u1D461)/u1D441\n4+1\n2/ba∇dbl∇/u1D465/u1D462(/u1D461)/ba∇dbl/u1D43F2+ (1+/u1D461)/u1D441\n4/ba∇dbl/u1D462(/u1D461)/ba∇dbl/u1D43F2/bracketrightBig\n.\nBy the local existence result in Lemma 3 and noting\n/ba∇dblΨ1/2/u1D462(/u1D461)/ba∇dbl/u1D43F2=/bardblex/bardblex/bardblex/bardblexΨ1/2/parenleftbigg\n/u1D4620+/uni222B.dsp/u1D461\n0/u1D462/u1D461(/u1D460)/u1D451/u1D460/parenrightbigg/bardblex/bardblex/bardblex/bardblex\n/u1D43F2/lessorsimilar/ba∇dbl(1+ |/u1D465|)/u1D706/u1D4620/ba∇dbl/u1D43F2+/u1D447/ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447),\nit suffices to show the a priori estimate /ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447)≤/u1D436with some constant /u1D436indepen-\ndent of/u1D447under the smallness assumption (2).\nIn order to estimate the /u1D43F2-norms in /ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447), we apply the following Matsumura\nestimate.\nLemma 4 (Matsumura [7]) For/u1D458∈Z≥0,/u1D6FC∈Z/u1D441\n≥0,/u1D45E∈ [1,2]and/u1D454∈/u1D43F/u1D45E(R/u1D441) ∩\n/u1D43B/u1D458+|/u1D6FC|−1(R/u1D441), we have\n/ba∇dbl/u1D715/u1D458\n/u1D461/u1D715/u1D6FC\n/u1D465D(/u1D461)/u1D454/ba∇dbl/u1D43F2/lessorsimilar(1+/u1D461)−/u1D441\n2/parenleftBig\n1\n/u1D45E−1\n2/parenrightBig\n−/u1D458−|/u1D6FC|\n2/parenleftBig\n/ba∇dbl/u1D454/ba∇dbl/u1D43F/u1D45E+/u1D452−/u1D461\n4/ba∇dbl/u1D454/ba∇dbl/u1D43B/u1D458+|/u1D6FC|−1/parenrightBig6 Yuta Wakasugi\nFinally, to estimate the nonlinear term with polynomial wei ghts, we employ the\nCaffarelli–Kohn–Nirenberg inequality.\nLemma 5 (Caffarelli–Kohn–Nirenberg [1]) Let/u1D45D,/u1D45E,/u1D45F,/u1D6FC,/u1D6FD,/u1D70E,/u1D44E ∈Rsatisfy\n/u1D45D,/u1D45E≥1,/u1D45F >0,0≤/u1D44E≤1,\n1\n/u1D45D+/u1D6FC\n/u1D441,1\n/u1D45E+/u1D6FD\n/u1D441,1\n/u1D45F+/u1D6FE\n/u1D441>0,\nwhere/u1D6FE=/u1D44E/u1D70E+ (1−/u1D44E)/u1D6FD. Then, there exists a positive constant /u1D436such that the\nfollowing inequality holds for all /u1D462∈/u1D436∞\n0(R/u1D441)\n/ba∇dbl|/u1D465|/u1D6FE/u1D462/ba∇dbl/u1D43F/u1D45F≤/u1D436/ba∇dbl|/u1D465|/u1D6FC∇/u1D465/u1D462/ba∇dbl/u1D44E\n/u1D43F/u1D45D/ba∇dbl|/u1D465|/u1D6FD/u1D462/ba∇dbl1−/u1D44E\n/u1D43F/u1D45E\nif and only if the following relations hold:\n1\n/u1D45F+/u1D6FE\n/u1D441=/u1D44E/parenleftbigg1\n/u1D45D+/u1D6FC−1\n/u1D441/parenrightbigg\n+ (1−/u1D44E)/parenleftbigg1\n/u1D45E+/u1D6FD\n/u1D441/parenrightbigg\n(this is the dimensional balance),\n0≤/u1D6FC−/u1D70Eif/u1D44E >0,\nand\n/u1D6FC−/u1D70E≤1if/u1D44E >0and1\n/u1D45D+/u1D6FC−1\n/u1D441=1\n/u1D45F+/u1D6FE\n/u1D441.\nIn particular, the case /u1D6FC=/u1D6FD=/u1D6FE=0 is called the Gagliardo–Nirenberg inequality.\n3.2 Proof of Theorem 1\nWe first prove that the semilinear terms in Lemma 2 is bounded b y/ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447).\nLemma 6 Let/u1D462be a mild solution to (1)on[0,/u1D447]. Then, we have\n/uni222B.dsp\nR/u1D441|/u1D462(/u1D461,/u1D465)|/u1D45D+1Ψ(/u1D461,/u1D465)/u1D451/u1D465+/uni222B.dsp/u1D461\n0/uni222B.dsp\nR/u1D441|/u1D462(/u1D460,/u1D465)|/u1D45D+1|Ψ/u1D461(/u1D460,/u1D465)|/u1D451/u1D465/u1D451/u1D460≤/u1D436/ba∇dbl/u1D462/ba∇dbl/u1D45D+1\n/u1D44B(/u1D447),\nwith some constant /u1D436 >0, provided that the right-hand side is finite.\nProof We consider the first term of the left-hand side. Noting\n/uni222B.dsp\nR/u1D441|/u1D462|/u1D45D+1Ψ/u1D451/u1D465/lessorsimilar/ba∇dbl/u1D462/ba∇dbl/u1D45D+1\n/u1D43F/u1D45D+1+ (1+/u1D461)−/u1D706/ba∇dbl|/u1D465|2/u1D706\n/u1D45D+1/u1D462/ba∇dbl/u1D45D+1\n/u1D43F/u1D45D+1\nand applying Lemma 5, we haveSemilinear damped wave equation for noncompactly data 7\n/ba∇dbl/u1D462/ba∇dbl/u1D43F/u1D45D+1/lessorsimilar/ba∇dbl∇/u1D465/u1D462/ba∇dbl/u1D703\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−/u1D703\n/u1D43F2, /u1D703=/u1D441(/u1D45D−1)\n2(/u1D45D+1),\n/ba∇dbl|/u1D465|2/u1D706\n/u1D45D+1/u1D462/ba∇dbl/u1D43F/u1D45D+1/lessorsimilar/ba∇dbl|/u1D465|/u1D706∇/u1D465/u1D462/ba∇dblΘ\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−Θ\n/u1D43F/u1D45E,Θ =1\n/u1D45D+1+2/u1D706\n/u1D441(/u1D45D+1)−1\n/u1D45E\n1\n2−1\n/u1D45E+/u1D706−1\n/u1D441,\nwhere/u1D45E=max{2,/u1D441\n2(/u1D45D−1)}. Here, we remark that the condition 0 ≤/u1D6FC−/u1D70Eif\n/u1D44E >0 in Lemma 5 requires /u1D45D≤1+2/u1D45E\n/u1D441. Moreover, /u1D43B1(R/u1D441) ⊂/u1D43F/u1D45E(R/u1D441)holds, since\n/u1D45D≤1+4\n[/u1D441−2]+. They are the reasons why we take /u1D45E=max{2,/u1D441\n2(/u1D45D−1)}. The\nGagliardo–Nirenberg inequality further implies\n/ba∇dbl/u1D462/ba∇dbl/u1D43F/u1D45E/lessorsimilar/ba∇dbl∇/u1D465/u1D462/ba∇dbl/u1D707\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−/u1D707\n/u1D43F2, /u1D707=/u1D441/parenleftbigg1\n2−1\n/u1D45E/parenrightbigg\n.\nTherefore, we conclude\n/uni222B.dsp\nR/u1D441|/u1D462|/u1D45D+1Ψ/u1D451/u1D465/lessorsimilar/parenleftBig\n/ba∇dbl∇/u1D465/u1D462/ba∇dbl/u1D703\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−/u1D703\n/u1D43F2/parenrightBig/u1D45D+1\n+ (1+/u1D461)/u1D706/parenleftBig\n/u1D45D+1\n2Θ−1/parenrightBig/parenleftBigg/bardblex/bardblex/bardblex/bardblex|/u1D465|/u1D706\n(1+/u1D461)/u1D706/2∇/u1D465/u1D462/bardblex/bardblex/bardblex/bardblexΘ\n/u1D43F2(/ba∇dbl∇/u1D465/u1D462/ba∇dbl/u1D707\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−/u1D707\n/u1D43F2)1−Θ/parenrightBigg/u1D45D+1\n/lessorsimilar/bracketleftbigg\n(1+/u1D461)−(/u1D441\n4+/u1D703\n2)(/u1D45D+1)+ (1+/u1D461)/u1D706/parenleftBig\n/u1D45D+1\n2Θ−1/parenrightBig\n−(/u1D441\n4+/u1D707\n2)(1−Θ)(/u1D45D+1)/bracketrightbigg\n/ba∇dbl/u1D462/ba∇dbl/u1D45D+1\n/u1D44B(/u1D447)\n/lessorsimilar/ba∇dbl/u1D462/ba∇dbl/u1D45D+1\n/u1D44B(/u1D447),\nsince a straightforward computation shows /u1D706/parenleftBig\n/u1D45D+1\n2Θ−1/parenrightBig\n−/parenleftbig/u1D441\n4+/u1D707\n2/parenrightbig(1−Θ)(/u1D45D+1)<\n0 if/u1D706 >/u1D45E−1\n/u1D45E/u1D441+2− (/u1D441−2)/u1D45D\n2(/u1D45D−/u1D45D/u1D439(/u1D441). Moreover, noticing |Ψ/u1D461|/lessorsimilar(1+/u1D461)−1Ψ, we can\nestimate the second term of the left-hand side of Lemma 6 in th e same way. This\ncompletes the proof. /square\nNext, we estimate the term (1+/u1D461)/u1D441/4/ba∇dbl/u1D462/ba∇dbl/u1D43F2in the norm /ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447).\nLemma 7 Let/u1D462be a mild solution to (1)on[0,/u1D447]. Then, we have\n(1+/u1D461)/u1D441\n4/bardblex/bardblex/bardblex/bardblex/uni222B.dsp/u1D461\n0D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460/bardblex/bardblex/bardblex/bardblex\n/u1D43F2/lessorsimilar/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447).\nProof By Lemma 4, we estimate8 Yuta Wakasugi\n(1+/u1D461)/u1D441\n4/bardblex/bardblex/bardblex/bardblex/uni222B.dsp/u1D461\n0D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460/bardblex/bardblex/bardblex/bardblex\n/u1D43F2\n/lessorsimilar(1+/u1D461)/u1D441\n4/uni222B.dsp/u1D461\n2\n0(1+/u1D461−/u1D460)−/u1D441\n4/parenleftBig\n/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F1+/u1D452−/u1D461−/u1D460\n4/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F2/parenrightBig\n/u1D451/u1D460\n+ (1+/u1D461)/u1D441\n4/uni222B.dsp/u1D461\n/u1D461\n2/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F2/u1D451/u1D460.\nIn order to estimate the term /ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F1=/ba∇dbl/u1D462(/u1D460)/ba∇dbl/u1D45D\n/u1D43F/u1D45D, we divide the case into /u1D45D <2\nand/u1D45D≥2. When/u1D45D <2, applying Lemma 5, we have\n/ba∇dbl/u1D462/ba∇dbl/u1D43F/u1D45D/lessorsimilar/ba∇dbl|/u1D465|/u1D706∇/u1D465/u1D462/ba∇dbl/u1D703\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−/u1D703\n/u1D43F2, /u1D703=/u1D441(2−/u1D45D)\n2/u1D45D(/u1D706−1).\nWe remark that /u1D703 <1 holds, since /u1D706 >2/u1D441− (/u1D441−2)/u1D45D\n2/u1D45D. Thus, we have\n/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D43F/u1D45D/lessorsimilar(1+/u1D461)/u1D706/u1D45D\n2/u1D703−/u1D441\n4/u1D45D(1−/u1D703)/parenlefttpA/parenleftexA\n/parenleftbtA/bardblex/bardblex/bardblex/bardblex/bardblex/parenleftbigg|/u1D465|√\n1+/u1D461/parenrightbigg/u1D706\n∇/u1D465/u1D462/bardblex/bardblex/bardblex/bardblex/bardblex/u1D703\n/u1D43F2/parenleftBig\n(1+/u1D461)/u1D441\n4/ba∇dbl/u1D462/ba∇dbl/u1D43F2/parenrightBig1−/u1D703/parenrighttpA/parenrightexA\n/parenrightbtA/u1D45D\n/lessorsimilar(1+/u1D461)/u1D706/u1D45D\n2/u1D703−/u1D441\n4/u1D45D(1−/u1D703)/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447)\nA direct computation shows that/u1D706/u1D45D\n2/u1D703−/u1D441\n4/u1D45D(1−/u1D703)<−1, since/u1D706 >2/u1D441−8//u1D441−(/u1D441−2)/u1D45D\n4(/u1D45D−/u1D45D/u1D439(/u1D441)).\nWhen/u1D45D≥2, the Gagliardo–Nirenberg inequality implies\n/ba∇dbl/u1D462/ba∇dbl/u1D43F/u1D45D/lessorsimilar/ba∇dbl∇/u1D465/u1D462/ba∇dbl/u1D703\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−/u1D703\n/u1D43F2, /u1D703=/u1D441(/u1D45D−2)\n2/u1D45D.\nThen, we easily obtain\n/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D43F/u1D45D/lessorsimilar(1+/u1D461)−(/u1D441\n4+1\n2)/u1D45D/u1D703−/u1D441\n4/u1D45D(1−/u1D703)/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447)\nwith−/parenleftBig\n/u1D441\n4+1\n2/parenrightBig\n/u1D45D/u1D703−/u1D441\n4/u1D45D(1−/u1D703)<−1, since/u1D45D > /u1D45D/u1D439(/u1D441). Therefore, in both cases,\nwe conclude/uni222B.dsp/u1D461\n2\n0/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F1/u1D451/u1D460/lessorsimilar/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447).\nNext, we estimate the term /ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F2=/ba∇dbl/u1D462(/u1D460)/ba∇dbl/u1D45D\n/u1D43F2/u1D45D. Applying the Gagliardo–\nNirenberg inequality, we have\n/ba∇dbl/u1D462/ba∇dbl/u1D43F2/u1D45D/lessorsimilar/ba∇dbl∇/u1D465/u1D462/ba∇dbl/u1D703\n/u1D43F2/ba∇dbl/u1D462/ba∇dbl1−/u1D703\n/u1D43F2, /u1D703=/u1D441(/u1D45D−1)\n2/u1D45D.\nWe remark that /u1D703≤1 holds, since /u1D45D≤/u1D441\n[/u1D441−2]+. Using the above estimate and repeat\na similar computation as above, we haveSemilinear damped wave equation for noncompactly data 9\n(1+/u1D461)/u1D441\n4/uni222B.dsp/u1D461\n2\n0/u1D452−/u1D461−/u1D460\n4/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F2/u1D451/u1D460+ (1+/u1D461)/u1D441\n4/uni222B.dsp/u1D461\n/u1D461\n2/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F2/u1D451/u1D460/lessorsimilar/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447),\nwhich completes the proof. /square\nIn the same way, we can estimate the other terms in the norm /ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447).\nLemma 8 Let/u1D462be a mild solution to (1)on[0,/u1D447]. For nonnegative integers /u1D458, /u1D457\nsatisfying /u1D458+/u1D457≤1, we have\n(1+/u1D461)/u1D441\n4+/u1D458+/u1D457\n2/bardblex/bardblex/bardblex/bardblex/uni222B.dsp/u1D461\n0/u1D715/u1D458\n/u1D461∇/u1D457\n/u1D465D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460/bardblex/bardblex/bardblex/bardblex\n/u1D43F2/lessorsimilar/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447).\nProof By Lemma 4, we have\n(1+/u1D461)/u1D441\n4+/u1D458+/u1D457\n2/bardblex/bardblex/bardblex/bardblex/uni222B.dsp/u1D461\n0/u1D715/u1D458\n/u1D461∇/u1D457\n/u1D465D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460/bardblex/bardblex/bardblex/bardblex\n/u1D43F2\n/lessorsimilar(1+/u1D461)/u1D441\n4+/u1D458+/u1D457\n2/uni222B.dsp/u1D461\n2\n0(1+/u1D461−/u1D460)−/u1D441\n4−/u1D458−/u1D457\n2/parenleftBig\n/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F1+/u1D452−/u1D461−/u1D460\n4/ba∇dbl|/u1D462(/u1D465)|/u1D45D/ba∇dbl/u1D43F2/parenrightBig\n/u1D451/u1D460\n+ (1+/u1D461)/u1D441\n4+/u1D458+/u1D457\n2/uni222B.dsp/u1D461\n/u1D461\n2(1+/u1D461−/u1D460)−/u1D458−/u1D457\n2/ba∇dbl|/u1D462(/u1D460)|/u1D45D/ba∇dbl/u1D43F2/u1D451/u1D460.\nThe rest part is completely the same as Lemma 7 and we omit the d etail./square\nNow let us prove Theorem 1. By Lemmas 2 and 6, we have\n/u1D438Ψ(/u1D461)1/2/lessorsimilar/u1D438Ψ(0)1/2+ /ba∇dbl(1+ |/u1D465|)/u1D706\n/u1D45D+1/u1D4620/ba∇dbl/u1D45D+1\n2\n/u1D43F/u1D45D+1+ /ba∇dbl/u1D462/ba∇dbl/u1D45D+1\n2\n/u1D44B(/u1D447).\nApplying Lemma 5 to the second term, the right-hand side can b e further bounded\nby/u1D436(/u1D700+/u1D700/u1D45D+1\n2) +/u1D436/ba∇dbl/u1D462/ba∇dbl/u1D45D+1\n2\n/u1D44B(/u1D447). Next, by Lemmas 4, we can easily prove that\n(1+/u1D461)/u1D441\n4+1/ba∇dbl/u1D462/u1D43F\n/u1D461(/u1D461)/ba∇dbl/u1D43F2+ (1+/u1D461)/u1D441\n4+1\n2/ba∇dbl∇/u1D465/u1D462/u1D43F(/u1D461)/ba∇dbl/u1D43F2+ (1+/u1D461)/u1D441\n4/ba∇dbl/u1D462/u1D43F(/u1D461)/ba∇dbl/u1D43F2\n/lessorsimilar/ba∇dbl/u1D4620/ba∇dbl/u1D43F1+ /ba∇dbl/u1D4620/ba∇dbl/u1D43B1+ /ba∇dbl/u1D4621/ba∇dbl/u1D43F1+ /ba∇dbl/u1D4621/ba∇dbl/u1D43F2/lessorsimilar/u1D700\nfor the linear part of the solution /u1D462/u1D43F:=D(/u1D461)(/u1D4620+/u1D4621) +/u1D715\n/u1D715/u1D461D(/u1D461)/u1D4620. Moreover, by\nLemmas 7 and 8, we have\n(1+/u1D461)/u1D441\n4+1/bardblex/bardblex/bardblex/bardblex/uni222B.dsp/u1D461\n0/u1D715/u1D461D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460/bardblex/bardblex/bardblex/bardblex\n/u1D43F2+ (1+/u1D461)/u1D441\n4+1\n2/bardblex/bardblex/bardblex/bardblex/uni222B.dsp/u1D461\n0∇/u1D465D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460/bardblex/bardblex/bardblex/bardblex\n/u1D43F2\n+ (1+/u1D461)/u1D441\n4/bardblex/bardblex/bardblex/bardblex/uni222B.dsp/u1D461\n0D(/u1D461−/u1D460)|/u1D462(/u1D460)|/u1D45D/u1D451/u1D460/bardblex/bardblex/bardblex/bardblex\n/u1D43F2\n/lessorsimilar/ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447).\nPutting all the above estimates together, we conclude10 Yuta Wakasugi\n/ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447)/lessorsimilar/u1D700+/u1D700/u1D45D+1\n2+ /ba∇dbl/u1D462/ba∇dbl/u1D45D+1\n2\n/u1D44B(/u1D447)+ /ba∇dbl/u1D462/ba∇dbl/u1D45D\n/u1D44B(/u1D447),\nwhere the implicit constant is independent of /u1D447. From this and a standard argument\ngive the a priori estimate /ba∇dbl/u1D462/ba∇dbl/u1D44B(/u1D447)≤/u1D436with some constant /u1D436 >0 independent of /u1D447,\nprovided that /u1D700is sufficiently small, and the proof is complete.\nAcknowledgements\nThis work was supported by JSPS KAKENHI Grant Number JP20K14 346. The au-\nthor is deeply grateful to Professors Masahiro Ikeda, Koich i Taniguchi and Motohiro\nSobajima for the fruitful discussion.\nReferences\n1. L. C /a.pc/f.pc/f.pc/a.pc/r.pc/e.pc/l.pc/l.pc/i.pc, R. K/o.pc/h.pc/n.pc, L. N/i.pc/r.pc/e.pc/n.pc/b.pc/e.pc/r.pc/g.pc ,First order interpolation inequalities with weights ,\nCompositio Math. 53(1984), 259–275.\n2.M. D’A/b.pc/b.pc/i.pc/c.pc/c.pc/o.pc ,The threshold of effective damping for semilinear wave equat ions, Math. Meth-\nods Appl. Sci. 38(2015), 1032–1045.\n3.N. H/a.pc/y.pc/a.pc/s.pc/h.pc/i.pc, E. I. K/a.pc/i.pc/k.pc/i.pc/n.pc/a.pc, P. I. 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Sci. 46(2023),\n7067–7107."    },    {        "title": "2401.12548v1.Sobolev_Stability_for_the_2D_MHD_Equations_in_the_Non_Resistive_Limit.pdf",        "content": "SOBOLEV STABILITY FOR THE 2D MHD EQUATIONS IN THE\nNON-RESISTIVE LIMIT\nNIKLAS KNOBEL\nAbstract. In this article, we consider the stability of the 2D magnetohydrody-\nnamics (MHD) equations close to a combination of Couette flow and a constant\nmagnetic field. We consider the ideal conductor limit for the case when viscosity\nνis larger than resistivity κ,ν≥κ> 0. For this regime, we establish a bound\non the Sobolev stability threshold. Furthermore, for κ≤ν3this system exhibits\ninstability, which leads to norm inflation of size νκ−1\n3.\nContents\n1. Introduction 1\n2. Linear Stability 7\n3. Sobolev Stability for the Nonlinear System 10\nAppendix A. Construction of the Weights 27\nAppendix B. Local Wellposedness 29\nReferences 31\n1.Introduction\nThe equations of magnetohydrodynamics (MHD)\n∂tV+V·∇V+∇Π =ν∆V+B·∇B,\n∂tB+V·∇B=κ∆B+B·∇V,\n∇·V=∇·B= 0,\n(t,x,y )∈R+×T×R=: Ω,(1)\nmodel the evolution of a magnetic field B: Ω→R2interacting with the velocity\nV: Ω→R2of a conducting fluid. The MHD equations are a common model\nused in astrophysics, planetary magnetism and controlled nuclear fusion [ Dav16].\nThe quantities ν,κ≥0correspond to fluid viscosity and magnetic resistivity.\nThe pressure Π : Ω→Rensures that the velocity remains divergence-free. A\nfundamental problem of fluid dynamics and plasma physics is the stability and\nDate: January 24, 2024.\n2020Mathematics Subject Classification. 76E25, 76E30, 76E05.\nKey words and phrases. Magnetohydrodynamics, stability threshold.\n1arXiv:2401.12548v1  [math.AP]  23 Jan 20242 NIKLAS KNOBEL\nlong-time behavior of solutions to equation (1)and in particular stability of specific\nsolutions. We consider the combination of an affine shear flow, called Couette flow,\nand a constant magnetic field:\nVs=ye1,\nBs=αe1.\nIn particular, the solution combines the effects of mixing due to shear and coupling\nby the magnetic field. The Couette flow mixes any perturbation, which leads to\nincreased dissipation rates, called enhanced dissipation, and stabilizes the equation.\nThe coupling with a constant magnetic field propagates this mixing to magnetic\nperturbations. However, the magnetic field weakens the mixing, especially if\nviscosity is larger than resistivity, inviscid damping gets counteracted by algebraic\ngrowth for specific time regimes.\nIn the related case of the Navier-Stokes equation, that is when no magnetic field\nis present, one observes turbulent solutions as viscosity reaches small values. In\ncontrast, the linearized problem around Couette flow is stable for all values of the\nviscosity. These phenomena are known as the Sommerfeld paradox [ LL11] and\nhighlight instability due to nonlinear effects. In [ BM15,DM18,DZ21,BM14,IJ13]\nvarious authors show sharp stability in Gevrey 2 spaces (spaces between C∞and\nanalytic). Thenonlinearinstabilitycanbesuppressedbytheviscosityforinitialdata\nsufficiently small in Sobolev spaces, ensuring stability [BVW18, MZ22, BGM17].\nWhen considering the MHD equations without Couette flow, the constant mag-\nnetic field stabilizes the equation. The dynamics of small initial perturbations of\nthe ideal MHD equation around a strong enough magnetic field is close to the\nlinearized system [ BSS88]. For stability in several dissipation regimes we refer to\n[WZ17,RWXZ14 ,HXY18,RWXZ14 ,Sch88,CF23,Koz89] and references therein.\nHowever, global in time wellposedness for the non-resistive case is still open (see\nthe discussion in [ CF23]). Furthermore, a shear flow leads to qualitatively different\nbehavior and instabilities [HT01, HHKL18].\nRecently, the MHD equation around Couette flow has gathered significant interest\n[Lis20,KZ23a,ZZ23,Dol23,KZ23b]. Already on a linear level, the behavior of\nthe MHD changes for different values of νandκ. In [Lis20] Liss proved the first\nstability threshold for the MHD equations. He considered the full dissipative\nregime ofκ=ν > 0and proved the stability of the three-dimensional MHD\nequation for initial data which is sufficiently small in Sobolev spaces. For the\nanalogous two-dimensional problem, Dolce [ Dol23] proved stability in the more\ngeneral setting of 0<κ3≲ν≤κ. In [KZ23b] Zillinger and the author considered\nthe case of only horizontal resistivity and full viscosity and established stability\nfor small data in Sobolev spaces. For the regime of vanishing viscosity ν= 0and\nnon-vanishing resistivity κ>0, in [KZ23a] we constructed a linear stability and\ninstability mechanism around nearby traveling waves in Gevrey 2spaces. In a\ncorresponding nonlinear stability result, Zhao and Zi [ ZZ23] proved the almostMHD STABILITY THRESHOLD 3\nmatching nonlinear result of Gevrey σstability for 1≤σ<2and for sufficiently\nsmall perturbations.\nThe results mentioned above on stability around Couette flow focus on the\nsetting when resistivity is larger than viscosity ν≤κ. Indeed in the setting ν >0\nandκ= 0, the magnetic effects dominate leading to a linear instability mechanism\nand thus a growth of the magnetic field by νtfor specific initial data [KZ23b].\nIn this paper, we consider the setting 0<κ≤ν. In particular, this also includes\nthe non-resistive limit κ↓0independent of ν. To the author’s knowledge the\nstability of the regime κ<νhas not previously been studied for the MHD equation\naround Couette flow. To state the main result, we define the perturbative unknowns\nv(x,y,t ) =V(x+yt,y,t )−Vs,\nb(x,y,t ) =B(x+yt,y,t )−Bs,\nwhere the change of variables x∝⇕⊣√∫⊔≀→x+ytfollows the characteristics of the Couette\nflow. For these unknowns, equation (1) becomes\n∂tv+v2e1−2∂x∆−1\nt∇tv2=ν∆tv+α∂xb+b∇tb−v∇tv−∇tπ,\n∂tb−b2e1 =κ∆tb+α∂xv+b∇tv−v∇tb,\n∇t·v=∇t·b= 0.(2)\nDue to the change of variables the spatial derivatives become time-dependent, i.e.\n∂t\ny=∂y−t∂x,∇t= (∂x,∂t\ny)Tand∆t=∂2\nx+ (∂t\ny)2.\nForequation (2)weestablishLipschitzstabilityforinitialdatawhichissufficiently\nsmall in Sobolev spaces, in the sense that there exists a bound on the initial data\nε0=ε0(ν,κ)and a Lipschitz constant L=L(ν,κ)such that for initial data which\nsatisfies\n∥(v,b)in∥HN=ε≤ε0,\nthe corresponding solution is globally bounded in time by\n∥(v,b)(t)∥HN≤Lε.\nFor the non-resistive case, κ= 0, global wellposedness is an open problem and so\nLipschitz stability in Sobolev spaces is unclear. Thus, naturally the question arises,\nwhichε0andLare optimal and how they behave in the limit ν,κ↓0.\nWe denote a Sobolev stability threshold as γ1,γ2∈R, such that for ε0=c0νγ1κγ2\nwith small c0>0we obtain\n∥(v,b)in∥HN≤c0νγ1κγ2→stability,\n∥(v,b)in∥HN≫c0νγ1κγ2→possible instability .\nThis extends the common convention in the field (eg. see [ BVW18]) to allow for\ntwo independent parameters νandκ. In particular, it agrees with the common\nconvention when restricting to the case ν≈κ. It allows us to discuss cases\nwhereκtends to zero much quicker than ν. Establishing a possible instability is4 NIKLAS KNOBEL\nhighly nontrivial since for the nonlinear setting it is difficult to construct solutions\nthat exhibit norm inflation. To the author’s knowledge, there does not exist any\nnonlinear instability result for the MHD equation around Couette flow in Sobolev\nspaces.\nFor accessibility and simplicity of notation, we state our main result as the\nfollowing theorem (see Theorem 3.1 for a detailed description).\nTheorem1.1. Considerα>1\n2,N≥5and a small enough constant c0=c0(α)>0.\nLet0<κ≤ν≤1\n40(1−1\n2α)6\n5, then we obtain Sobolev stability for initial data which\nis sufficiently small in Sobolev spaces, where the estimates qualitatively differ for\nthe regimes κ≳ν3andκ≲ν3. More precisely:\n•In the regime of ν3≲κ, for all initial data which satisfy\n∥(v,b)in∥HN=ε≤c0ν1\n12κ1\n2,\nthe global in time solution (v,b)of(2)satisfies the Lipschitz bound\nsup\nt>0∥(v,b)(t)∥HN≲ε.\n•In the regime of ν3≳κ, for all initial data which satisfy\n∥(v,b)in∥HN=ε≤c0ν−11\n12κ5\n6,\nthe global in time solution (v,b)of(2)satisfies the Lipschitz bound\nsup\nt>0∥(v,b)(t)∥HN≲νκ−1\n3ε.\nIn particular, we obtain Lipschitz stability for the Lipschitz constant L≈max(1,νκ−1\n3)\nfor the smallness parameter ε0≈min(ν1\n12κ1\n2,ν−11\n12κ5\n6).\nIn the proof, we employ an energy method similar to [ BBZD23 ,MZZ23,Zil21,\nDol23,KZ23b]. In the following, we outline the main challenges and novelties of\nthe proof:\n•The imbalance of resistivity κand viscosity νyields two cases ν3≲κand\nν3≳κ(or equivalently 1≲νκ−1\n3or1≳νκ−1\n3). These cases give different\nvalues forL, namely 1andνκ−1\n3.\n•We consider the case ν3≳κ. On certain time scales the viscosity is so\nstrong that fluid effects get suppressed while the effects of the magnetic\nfield dominate. Thus, the term ∂tb=e1b2in(1)generates algebraic growth\nin specific regimes (see Subsection 2). Estimating this linear effect yields\nthe norm inflation by L=νκ−1\n3. The algebraic growth appears on different\ntime scales depending on the frequency, a precise estimate of the nonlinear\nterms is necessary.\n•For the case ν3≲κthe algebraic growth is bounded by a finite constant.\nIn the subcase ν=κthe sum of the threshold parameters is γ1+γ2=7\n12\nwhich is a slight improvement over2\n3in [Dol23].MHD STABILITY THRESHOLD 5\n•In the proof of Theorem 1.1 we perform a low and high frequency decompo-\nsitiona=ahi+alow. For high frequencies, the nonlinear term consist of\nalow∇tahi, called transport term and ahi∇ta, called reaction term (including\nhi−hiinteractions). Compared to the Navier-Stokes equation, in the case\nof the MHD equation, it is vital to bound the transport term precisely. In\nparticular, for κ≲ν3the previously mentioned algebraic growth affects the\nestimate of the transport term strongly.\n•The threshold is determined by the nonlinear term v∇tb= Λ−1\nt∇⊥p1∇b\nacting onbin(2), for the natural unknown p1= Λ−1\nt∇⊥v(which we discuss\nlater in more detail). In our estimates we rely on two stabilizing effects,\nthe strong viscosity of vand the Λ−1\ntin front ofp1. For the nonlinear term\nv∇tbboth effects fall onto v. Due to the weaker integrability of the bthis\nterm determines the threshold after integrating in time.\nWith the main challenges in mind, let us comment on the results:\n•The size of the constant magnetic field α>1\n2results in a strong interaction\nbetweenvandb. Due to this interaction, the decay in vand growth in b\nare in balance (see Lemma 1). Constants may depend on αand degenerate\nasα↓1\n2. For example we obtain limα↓1\n2c0(α) = 0.\n•Figure 1 shows which areas stability has been proven. The graphic shows\nonly qualitative behavior and after rescaling we obtain the same graphic.\nTheresistivity κisontheverticalaxisandtheviscosity νisonthehorizontal\naxis. We prove stability for the regime 0<κ≤ν, which we divide into two\nsegments:ν3≲��in orange and ν3≳κin red. In [ Dol23] Dolce considered\nthe regime of 0<(16\nακ)3≤ν≤κ, which is in blue. The authors of [ ZZ23]\nconsidered the line ν= 0which is in purple. The black line corresponds to\nν=κ>0of [Lis20].\nStability for the regimes 0<ν≤(16\nακ)3,κ= 0<νandκ=ν= 0remain\nopen. For the set 0<ν≪κ3, we expect that an adjusted application of\nthe methods used in this article yield stability. We expect stability for the\ncaseκ= 0and0<νto be very difficult since we obtain linear growth for\nthep2variable. For Λ−1\ntp2we obtain linear stability but then there is no\ntime decay in the magnetic field and so we lack an important stabilizing\neffect. In the inviscid case, κ=ν= 0the linearized system is stable in the\npvariables. However, due to the lack of dissipation, it is very challenging\nto bound the nonlinear terms.\n•Our threshold consists of parameters γ1andγ2. An alternative notation is\nto impose the relation ν≈κδfor some 0≤δ≤1. With that convention\nwe obtain stability if ε≤c0κγ(δ)for\nγ=/braceleftigg1\n2+δ\n12δ≥1\n35\n6−11\n12δotherwise.\nThe remainder of this article is structured as follows:6 NIKLAS KNOBEL\nFigure 1. Sketch of areas with results for stability.\n•In Section 2 we discuss the linearized system. We identify two different\ntime regions where “circular movement” or “strong viscosity” determine the\nlinearized behavior. We estimate both effects separately and then establish\nthe estimates for the linearized system.\n•In Section 3 we prove the main theorem. We employ a bootstrap approach,\nwhere we control errors in Proposition 3.1. The main difficulty is to bound\nthe linear growth and the nonlinear effect of v∇tbacting onb.\nNotations and Conventions. Fora,b∈Rwe denote their minimum and\nmaximum as\nmin(a,b) =a∧b,\nmax(a,b) =a∨b.\nWe writef≲giff≤Cgfor a constant Cindependent of νandκ. Furthermore,\nwe writef≈giff≲gandg≲f. We denote the Lebesgue spaces Lp=Lp(T×R)\nand the Sobolev spaces HN=HN(T×R)for someN∈N. For time-dependent\nfunctions, we denote LpHs=Lp\ntHsas the space with the norm\n∥f∥LpHs=/vextenddouble/vextenddouble/vextenddouble∥f∥Hs(T×R)/vextenddouble/vextenddouble/vextenddouble\nLp(0,T), (3)MHD STABILITY THRESHOLD 7\nwhere omit writing the T. We write the time-dependent spatial derivatives\n∂t\ny=∂y−t∂x,\n∇t= (∂x,∂t\ny)T,\n∆t=∂2\nx+ (∂t\ny)2,\nand the half Laplacians as\nΛ = (−∆)1\n2,\nΛt= (−∆t)1\n2.\nThe function f∈HNis decomposed into its xaverage and the orthogonal comple-\nment\nf=(y) =/integraldisplay\nf(x,y)dx,\nf̸==f−f=.\nThe adapted unknowns. For the following, it is useful to change to the unknowns\np1,̸== Λ−1\nt∇⊥\ntv̸=andp2,̸== Λ−1\nt∇⊥\ntb̸=. However, since Λ−1\nt∇⊥\ntis not a bounded\noperator on the xaverage, we define\np1,̸== Λ−1\nt∇⊥\ntv̸=,\np1,==v1,=,\np2,̸== Λ−1\nt∇⊥\ntb̸=,\np2,==b1,=.\nThus (2) can be equivalently expressed as\n∂tp1−∂x∂t\ny∆−1\ntp1−α∂xp2=ν∆tp1+ Λ−1\nt∇⊥\nt(b∇tb−v∇tv),\n∂tp2+∂x∂t\ny∆−1\ntp2−α∂xp1=κ∆tp2+ Λ−1\nt∇⊥\nt(b∇tv−v∇tb),\np|t=0=pin.(4)\nThese unknowns are particularly useful since\n∥Ap1∥L2=∥Av∥L2,\n∥Ap2∥L2=∥Ab∥L2,\nfor all Fourier multipliers Asuch that one side is finite.\n2.Linear Stability\nIn this section, we consider the behavior of the linearized version of (4):\n∂tp1−∂x∂t\ny∆−1\ntp1−α∂xp2=ν∆tp1,\n∂tp2+∂x∂t\ny∆−1\ntp2−α∂xp1=κ∆tp2.(5)\nFor this equation, we establish the following proposition:8 NIKLAS KNOBEL\nProposition 1. Considerα >1\n2and0< κ≤ν. Letpin∈HNwithpin,==/integraltextpindx= 0, then the solution pof(5)satisfies the bound\n∥p(t)∥HN≲e−cκ1\n3t(1 +κ−1\n3ν)∥pin∥HN. (6)\nFor the proof of Proposition 1, we perform a Fourier transform (x,y)∝⇕⊣√∫⊔≀→(k,ξ)\nand replace p1byip1(5) to infer for modes k̸= 0\n∂tp1=−t−ξ\nk\n1+(t−ξ\nk)2p1−αkp 2−νk2(1 + (t−ξ\nk)2)p1,\n∂tp2=t−ξ\nk\n1+(t−ξ\nk)2p2+αkp 1−κk2(1 + (t−ξ\nk)2)p2.(7)\nWhere with slight abuse of notation we omit writing the Fourier transformation.\nThis equation has several effects that appear on different regimes of t−ξ\nk, which we\ndiscuss in the following. The effect of circular movement appears on |t−ξ\nk|≲ν−1.\nWe first sketch these effects\nCircular movement. To highlight the effect of the constant magnetic field αin\n(5) we consider the toy model\n∂tp1=−αkp 2,\n∂tp2=αkp 1.(8)\nThis is solved by\np(t) =/parenleftigg\ncos(αt)−sin(αkt)\nsin(αt) cos(αkt)/parenrightigg\npin.\nWe call this effect of the constant magnetic field (8)circular movement, which\nleads to a transfer between p1andp2. This circular movement is counteracted by\nviscosity for times away fromξ\nk.\nEffect of strong viscosity. Let us consider the case when 0<κ≪νand for\nsimplicity of notation let k= 1andξ= 0. Due to the viscosity, we obtain p1≈0\nfor large times t≥t0≫1. Then from (7) we deduce the toy model\n∂tp2= (t\n1+t2−κ(1 +t2))p2. (9)\nThe first term in (9)leads to linear growth until the resistivity is strong enough for\nthe second term to take over. This is seen in the explicit solution of (9)\np2(t) =⟨t⟩\n⟨t0⟩exp(−κ/integraldisplayt\nt01 + ((τ−t))2dτ)p2(t0).\nThis is estimated by\np2(t)≲t−1\n0κ−1\n3e−cκ1\n3(t−t0)p2(t0),\nwhich corresponds to the maximal growth which we obtain. In the following, we\nwill see that t0≈ν−1is the time after which viscosity dominates. The reader mayMHD STABILITY THRESHOLD 9\nexpect that the enhanced dissipation timescale ν−1\n3would be the relevant timescale,\nbut the combination of circular movement and the viscosity gives enough decay for\np2such that the linear growth gets suppressed until the time ν−1.\nProof of Proposition 1.\nProof.For simplicity of notation, we introduce the new variable s=t−ξ\nkand\ninitial time sin=−ξ\nk. Then equation (7) reads\n∂sp1=−s\n1+s2p1−αkp 2−νk2(1 +s2)p1,\n∂sp2=s\n1+s2p2+αkp 1−κk2(1 +s2)p2.\nFurther we change the unknown to ˜p=exp(−κ\n2k2(s−sin+1\n3(s3−s3\nin)))p. For\n˜κ=κ\n2and˜ν=ν−κ\n2, this yield the equation\n∂s˜p1=−s\n1+s2˜p1−αk˜p2−˜νk2(1 +s2)˜p1,\n∂s˜p2=s\n1+s2˜p2+αk˜p1−˜κk2(1 +s2)˜p1.\nLet us denote s0:=ν−1and in the following, we distinguish between times |s|≤s0\nand|s|≥s0. We first consider the case sin≤−s0. For|s|≤s0, the circular\nmovement is not suppressed by the viscosity.\nWe define the energy E=|˜p|2+1\nαk2s\n1+s2˜p1˜p2, thenEis a positive quadratic form\ndue to our assumption α>1\n2and satisfies\n(1−1\n2αk)|˜p|2≤E≤(1 +1\n2αk)|˜p|2.\nWe calculate the time derivative\n∂sE+ ˜νk2(1 +s2)˜p2\n1+ ˜κk2(1 +s2)˜p2\n2\n=1\nαk∂s(2s\n1+s2)˜p1˜p2−2s(˜ν−˜κ)k\nα˜p1˜p2\n≤1\nαk∂s(2s\n1+s2)˜p1˜p2+1\n2˜νk2(1 +s2)˜p2\n1+2˜ν\nα2˜p2\n2\nand so with|˜p|2≤2α\n2α−1Ewe infer\n|∂sE|≤α\nα−1\n2(1\n1+s2+ 2˜ν\nα2)E.\nGronwall’s lemma implies\nE(s0)≤exp/parenleftbigg\nα\nα−1\n2(π+ 2˜ν\nα2|s0|)r/parenrightbigg\nE(−s0).\nSinceνs0= 1, we deduce\nE(s0)≲E(−s0)\nand thus\n|˜p(s0)|≲|˜p(−s0)|. (10)10 NIKLAS KNOBEL\nConsider the case |s|≥s0, we calculate\n1\n2∂s|˜p|2≤(−s\n1+s2−˜νk2(1 +s2))˜p2\n1\n+ (s\n1+s2−˜κk2(1 +s2))˜p2\n2,\nand since (−s\n1+s2−˜νk2(1 +s2))≤0for all|s|≥s0we conclude\n∂s|˜p|2≤(2s\n1+s2−˜κk2(1 +s2))+˜p2\n2.\nThus we obtain the estimate\n|˜p(s)|2≤\n\n|˜p(sin)|2s≤−s0,\n1+s2\n1+s2\n0|˜p(s0)|2s0≤s≤2(κk2)−1\n3,\n(1 + 4ν2κ−2\n3k−4\n3)|˜p(s0)|2s0∨2(κk2)−1\n3≤s.\nCombining this with (10) we infer\n|˜p(s)|≲(1 +νκ−1\n3k−2\n3)|p(sin)|.\nThe casesin≥−s0is established similarly since we only bound the growth. With\nexp(−κ\n2k2(s−sin+1\n3(s3−s3\nin)))≲e−cκ1\n3t\nwe deduce\n|p(s)|≲e−cκ1\n3t|˜p|(s)\n≲(1 +νκ−1\n3k−2\n3)e−cκ1\n3t|p|(sin).\nEquation (7) decouples in ξandk, so we infer the proposition with this estimate.\n□\n3.Sobolev Stability for the Nonlinear System\nThe following theorem is a more general statement of Theorem 1.1. We dedicate\nthe remainder of the section to the proof.\nTheorem 3.1. Letα>1\n2andN≥5, then there exist c0,c> 0, such that for all\n0< κ≤ν≤1\n40(1−1\n2α)6\n5there exist L=max(1,νκ−1\n3), such that for all initial\ndata, which satisfy\n∥(v,b)in,̸=∥HN=ε≤c0L−1ν1\n12κ1\n2,\n∥(v,b)in,=∥HN≤˜ε, withε≤˜ε≤ν−1\n12ε(11)\nthe corresponding solution of (2)satisfies the bound\n∥(v,b)̸=(t)∥L∞HN+∥∇t(νv,κb )̸=∥L2HN≲Le−cκ1\n3tε,\n∥(v,b)=(t)∥L∞HN+∥∂y(νv,κb )=∥L2HN≲˜ε.(12)MHD STABILITY THRESHOLD 11\nFurthermore, we obtain the following enhanced dissipation estimates\n∥v̸=∥L2HN≲Lν−1\n6e−cκ1\n3tε,\n∥b̸=∥L2HN≲Lκ−1\n6e−cκ1\n3tε.\nThis theorem implies Theorem 1.1. With slight abuse of notation, we write Las\ntheνandκdependent part of the Lipschitz constant. We prove this theorem by\nusing a bootstrap method. Let Abe the Fourier weight\nA: =M|∇|Necκ1\n3t1̸=,\nwhereM=MLM1MκMνMν3are defined as\n−˙ML\nML=t−ξ\nk\n1+(ξ\nk−t)21{ν−1≤t−ξ\nk≤(c1κk2)−1\n3}k̸= 0,\n−˙M1\nM1=Cα|k|+ν1\n12|k|2\nk2+(ξ−kt)2 k̸= 0,\n−˙Mν\nMν=ν1\n3\n1+ν2\n3(t−ξ\nk)2k̸= 0,\n−˙Mκ\nMκ=κ1\n3\n1+κ2\n3(t−ξ\nk)2k̸= 0,\n−˙Mν3\nMν3=Cαν\n1+ν2(t−ξ\nk)2 k̸= 0,\nM·(t= 0) =M·(k= 0) = 1.\nThe weight MLis an adaption of the weight m1\n2in [Lis20] to our setting. The\nmethod of using time-dependent Fourier weights is common when working with\nsolutions around Couette flow and the other weights are modifications of previously\nused weights (cf. [ BVW18,MZ22,Lis20,ZZ23]). For simplicity, here we only state\ntheir main properties and refer to Appendix A for a detailed description. The\nconstantsCα=2\nmin(1,α−1\n2),c=1\n200(1−1\n2α)2andc1=1\n20(1−1\n2α)are determined\nthrough the linear estimates. For the weights we obtain\nL−1≤min(1,ν−1κ1\n3k2\n3)≲ML≤1,\nM1≈Mκ≈Mν≈Mν3≈1.(13)\nWe note that the weight MLis distinct from the others due to its lower bound\nL−1, which depends on νandκ. The weight MLis necessary to bound the linear\ngrowth in the region ν−1≲t−ξ\nk≲(κk2)−1\n3. Controlling the effects of MLis one\nof the main challenges in the proof of Theorem 3.1. We recall the unknowns pand\nequation (4)\n∂tp1−∂x∂t\ny∆−1\ntp1−α∂xp2=ν∆tp1+ Λ−1\nt∇⊥\nt(b∇tb−v∇tv),\n∂tp2+∂x∂t\ny∆−1\ntp2−α∂xp1=κ∆tp2+ Λ−1\nt∇⊥\nt(b∇tv−v∇tb),\np|t=0=pin.(14)12 NIKLAS KNOBEL\nLetχ∈C∞(R+×Z×R)be a Fourier multiplier defined by\nχ=χ(k,ξ) =/braceleftigg\n1|t−ξ\nk|≤ν−1\n0|t−ξ\nk|≥2ν−1 (15)\n∂tχ≤2ν. (16)\nWe define the main energy\nE: =∥Ap̸=∥2\nL2+2\nα⟨∂t\ny∆−1\ntχAp 1,̸=,Ap 2,̸=⟩.\nAsα>1\n2, this energy is positive definite and satisfies\n(1−1\n2α)∥Ap∥L2≤E≤(1 +1\n2α)∥Ap∥L2. (17)\nIn the following, we assume initial data as in Theorem 3.1, i.e. (11). We use a\nbootstrap approach to prove the following two estimates globally in time:\nTheenergy estimate without x-average\n∥E∥L∞+∥A∇t⊗(νp1,̸=,κp 2,̸=)∥2\nL2L2\n+/summationdisplay\nj=1,ν,κ,ν3∥/radicalbigg\n−˙Mj\nMjAp̸=∥2\nL2L2≤(Cε)2 (18)\nTheenergy estimate with x-average\n∥p=∥2\nL∞HN+∥∂y(νp1,=,κp 2,=)∥2\nL2HN≤(C˜ε)2. (19)\nWe then prove that the equality in the estimates is not attained at time T. By\nlocal wellposedness, the estimates thus remain valid at least for a short additional\ntime. This contradicts the maximality and thus Thas to be infinite. We note\nthat we suppress in our notation the Tin the estimates (see (3)). With 1≤\nκ−1\n3(−˙Mκ\nMκ+κk2(1 + (t−ξ\nk)2)and1≤ν−1\n3(−˙Mν\nMν+νk2(1 + (t−ξ\nk)2)) we infer from\n(17) and (18) the enhanced dissipation estimates\n∥Ap1,̸=∥L2L2≤2(1−1\n2α)−1ν−1\n6Cε, (20)\n∥Ap2,̸=∥L2L2≤2(1−1\n2α)−1κ−1\n6Cε. (21)\nBy the construction of M1we obtain\n∥∂xΛ−1\ntAp∥L2≲ν−1\n12ε.MHD STABILITY THRESHOLD 13\nWe obtain the energy estimate by deriving the energy E\n∂tE+ 2∥A∇t⊗(νp1,̸=,κp 2,̸=)∥2\nL2+ 2∥/radicalbigg\n−˙M\nMAp̸=∥2\nL2\n=2cκ1\n3∥Ap̸=∥2\nL2\n−2⟨A(1−χ)p1,̸=,∂x∂t\nx∆−1\ntAp1,̸=⟩\n+ 2⟨A(1−χ)p2,̸=,∂x∂t\nx∆−1\ntAp2,̸=⟩\n+4\nα⟨χ∂t\ny∆−1\ntAp1,̸=,˙Ap2,̸=⟩\n+2\nα⟨χ∂t(∂t\ny∆−1\nt)Ap1,̸=,Ap 2,̸=⟩\n+2\nα⟨∂t(χ)∂t\ny∆−1\ntAp1,̸=,Ap 2,̸=⟩\n+2|ν+κ|\nα⟨χ∂t\nyAp1,̸=,Ap 2,̸=⟩\n+ 2⟨Av̸=,A(b∇tb−v∇tv)⟩\n+ 2⟨Ab̸=,A(b∇tv−v∇tb)⟩\n+2\nα⟨χA∂t\ny∆−1\ntb̸=,A(b∇tb−v∇tv)⟩\n+2\nα⟨χA∂t\ny∆−1\ntv̸=,A(b∇tv−v∇tb)⟩\n=L1+LNR+LR+NL̸=+ONL.(22)\nWhere we denoted by ONLall the terms which include the operator ∂t\ny∆−1\ntand\nNLthe one which does not. Furthermore, for the energy of x-averages, we obtain\n∂t∥p=∥HN+∥∂y(νp1,=,κp 2,=)∥HN\n≤⟨⟨∂y⟩Nv1,=,⟨∂y⟩N(b∇tb−v∇tv)=⟩\n+⟨⟨∂y⟩Nb1,=,⟨∂y⟩N(b∇tv−b∇tv)=⟩\n=NL =.(23)\nIn the following subsections, we establish the following proposition:\nProposition 3.1 (Control of errors) .Under the assumptions of Theorem 3.1, there\nexists a constant C=C(α)>0such that if (18)and(19)are satisfied for T >0,\nthen the following estimate holds\n/integraldisplayT\n0L1+LR+LNRdt≤17+3\n2α\n10(Cε)2+ 2∥/radicalbigg\n−˙ML\nMLAp2∥2\nL2L2,\n/integraldisplayT\n0NL̸=+ONL dt ≲Lν−1\n12κ−1\n2ε3+ (Lκ−1\n3+κ−1\n2)˜εε2,\n/integraldisplayT\n0NL =dt≲Lν−1\n12κ−1\n2˜εε2.(24)\nWith this proposition we deduce Theorem 3.1:\nProof of Theorem 3.1. ByastandardapplicationoftheBanachfixed-pointtheorem\nwe obtain local well-posedness, see Appendix B. Thus for all initial data, there14 NIKLAS KNOBEL\nexists a time interval [0,T]such that (18)and(19)hold. LetT∗be the maximal\ntime such that (18)and(19)hold. Letc0be a given, small constant and suppose\nfor the sake of contradiction that T∗<∞. With the estimates (22),(23)and(24)\nand sincec0is small we obtain that the estimates (18)and(19)do not attain\nequality. Thus by local existence, T∗is not the maximal time and thus we obtain\na contradiction. Therefore, for small enough c0, (18) and (19) hold global in time\nand so we infer Theorem 3.1. □\nThe remainder of the section is dedicated to the proof of Proposition 3.1. We\nrearrange and use partial integration to infer that\n⟨Av̸=,b∇tAb̸=−v∇tAv̸=⟩+⟨Ab̸=,b∇tAv̸=−v∇tAb̸=⟩\n=⟨b,∇t(Av̸=Ab̸=)⟩−1\n2⟨v,∇t(Av̸=Av̸=) +∇t(Ab̸=Ab̸=)⟩\n= 0.(25)\nTheNLterm consists of trilinear products with the unknowns\na1a2a3∈{vvv,vbb,bbv,bvb}. (26)\nThus, we denote the nonlinear terms\nNL̸=[a1a2a3] =⟨Aa1\n̸=,A(a2\n̸=∇ta3\n̸=)̸=−a2\n̸=∇tAa3\n̸=⟩,\n+⟨Aa1\n̸=,A(a2\n=∇ta3\n̸=)−a2\n=∇ta3\n̸=⟩,\n+⟨Aa1\n̸=,A(a2\n̸=∇ta3\n=)⟩,\nNL =[a1a2a3] =⟨⟨∂y⟩Na1\n=,⟨∂y⟩N(a2\n̸=∇ta3\n̸=)=⟩.\nIf we do not use specific choices for a1a2a3we write just NL. Similarly, we use\na1a2a3∈{bvv,bbb,vbv,vvb}forONL. Furthermore, we always use isuch that\npi= Λ−1\nt∇⊥\nta2in the sense that i= 1ifa2=vandi= 2ifa2=b. We perform\nthe energy estimates in the next subsections:\n•In Subsection 3.1 we estimate the linear error terms. In this subsection, the\nsplit withχinto resonant and non-resonant regions depending on νis vital.\n•In Subsection 3.2 we conclude the energy estimate for the nonlinear term\nwithoutxaverage. Here it is necessary to perform a low and high frequency\ndecomposition. This gives us a reaction and a transport term. In particular,\nforκ↓0bounding the transport term is very challenging due to the linear\ngrowth.\n•InSubsections3.3, 3.4and3.5weestimatenonlineartermswithan x-average\ncomponent.\n•In Subsection 3.6 we estimate nonlinear term which arise due χin the\nresonant regions. For these terms, we obtain an additional Λ−1\nt, which has\na stabilizing effect. This stabilizing effect is necessary due to a nonlinear\nterm consisting of only magnetic components.MHD STABILITY THRESHOLD 15\n3.1.Linear estimates. In this section, we establish estimates of the linear errors\nL1,LRandLNRof (22). In order to estimate L1, we use (20) and (21) to deduce\n/integraldisplay\nL1dτ= 2cκ1\n3∥Ap̸=∥2\nL2L2≤8(1−1\n2α)−1c(Cε)2.\nFor theLNRterms in (22), we infer\n⟨A(1−χ)p1,̸=,A∂x∂t\nx∆−1\ntp1,̸=⟩=/summationdisplay\nk̸=0/integraldisplay\ndξ(1−χ)t−ξ\nk\n1+(t−ξ\nk)2A2p2\n1\n≤ν3∥A∇tp1,̸=∥2\nL2,\nsince (1−χ)t−ξ\nk\n1+(t−ξ\nk)2≤(1−χ)(ν)3(1 + (t−ξ\nk)2)due toχ= 1for|t−ξ\nk|≤ν−1.\nFurthermore, using (31) we estimate\n⟨A(1−χ)p2,̸=,A∂x∂t\nx∆−1\ntp2⟩=/summationdisplay\nk̸=0/integraldisplay\ndξ(1−χ)t−ξ\nk\n1+(t−ξ\nk)2A2p2\n2\n≤/summationdisplay\nk̸=0/integraldisplay\ndξ(1−χ)(−˙ML\nML+κc1(1 + (t−ξ\nk)2))A2p2\n2\n≤∥/radicalbigg\n−˙ML\nMLAp2,̸=∥2\nL2+κc1∥∇tAp2,̸=∥2\nL2.\nThus with (18) we deduce\n/integraldisplay\nLNRdτ≤(2ν2+ 2c1)(Cε)2+ 2∥/radicalbigg\n−˙ML\nMLAp2,̸=∥2\nL2L2.\nForLR, we estimate in frequency space\n|(1̸=∂t\ny∆−1\nt)∧|=|(ξ−kt\nk2+(ξ−kt)2)k̸=0|≤1\n2,\n−˙ML\nML|(∂t\ny∆t)∧|≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n(t−ξ\nk)2\nk(1+(t−ξ\nk)2)2/parenrightbigg\nk̸=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C−1\nα−˙M1\nM1.\nSo it follows that\n4\nα⟨χ∂t\ny∆−1\ntAp1,̸=,˙Ap2,̸=⟩\n=4\nα⟨χAp 1,̸=,(cκ1\n3+˙M1\nM1+˙ML\nML+˙Mκ\nMκ+˙Mν\nMν+˙Mν3\nMν3)∂t\ny∆−1\ntAp2,̸=⟩\n≤2c\nακ1\n3∥Ap̸=∥2\nL2+ (1 +C−1\nα)1\nα∥/radicalbigg\n−˙M1\nM1Ap̸=∥2\nL2\n+1\nα∥/radicalbigg\n−˙Mκ\nMκAp̸=∥2\nL2+1\nα∥/radicalbigg\n−˙Mν\nMνAp̸=∥2\nL2+1\nα∥/radicalbigg\n−˙Mν3\nMν3Ap̸=∥2\nL2.\nWe use the estimate in frequency space\n|(((∂2\nx−(∂t\ny)2)∆−2\nt)̸=)∧|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\n1−(t−ξ\nk)2\nk2(1+(t−ξ\nk)2)2/parenrightbigg\nk̸=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C−1\nα−˙M1\nM1,16 NIKLAS KNOBEL\nto infer that\n1\nα⟨χAp 1,̸=,A∂−1\nx(∂2\nx−(∂t\ny)2)∆−2\ntp2,̸=⟩≤C−1\nα1\nα∥/radicalbigg\n−˙M1\nM1Ap1,̸=∥L2∥/radicalbigg\n−˙M1\nM1Ap2,̸=∥L2\n≤C−1\nα1\n2α∥/radicalbigg\n−˙M1\nM1Ap̸=∥2\nL2.\nWith (16) we deduce\n⟨∂t\ny∆−1\nt∂t(χ)Ap1,̸=,Ap 2,̸=⟩≤ν∥Ap1,̸=∥L2∥AΛ−1\ntp2,̸=∥L2.\nBy the Fourier support of χ(see(15)) and the definition of Mν3we obtainχ≤\n2C−1\nαν−1−˙Mν3\nMν3χ, which yields\n|ν+κ|\nα⟨χA∂t\nyp1,̸=,Ap 2,̸=⟩≤2C−1\nαν1\n2\nα∥A∂t\nyp1,̸=∥L2∥/radicalbigg\n−Mν3\nMν3Ap2,̸=∥L2.\nThus for the linear error LRwe infer\n/integraldisplay\nLRdτ≤(2c\nα+ν5\n6)(Cε)2\n+1+C−1\nα\nα∥/radicalbigg\n−˙M1\nM1p̸=∥2\nL2L2+1\nα∥/radicalbigg\n−˙Mκ\nMκp̸=∥2\nL2L2+1\nα∥/radicalbigg\n−˙Mν\nMνp̸=∥2\nL2L2\n+1+C−1\nα\n2α∥/radicalbigg\n−˙Mν3\nMν3Ap2,̸=∥L2L2+C−1\nα1\n2αν∥A∂t\nyp1,̸=∥2\nL2L2.\nCombining the estimates for all linear terms, we obtain\n/integraldisplay\nL+LR+LNRdτ\n≤((8 +2\nα)(1−1\n2α)−1c+ 2ν2+ 2c1+ν5\n6)(Cε)2\n+ 2∥/radicalbigg\n−˙ML\nMLAp2∥2\nL2L2\n+ (1 +3\n2C−1\nα)1\nα∥χ/radicalbigg\n−˙M1\nM1p̸=∥2\nL2L2+1\nα∥/radicalbigg\n−˙Mκ\nMκp̸=∥2\nL2L2+1\nα∥/radicalbigg\n−˙Mν\nMνp̸=∥2\nL2L2\n+1+C−1\nα\n2α∥/radicalbigg\n−Mν3\nMν3Ap2,̸=∥L2L2+C−1\nα1\n2αν∥A∂t\nyp1,̸=∥2\nL2L2\n≤(12(1−1\n2α)−1c+ 2ν2+ 2c1+ν5\n6+1+2C−1\nα\n2α)(Cε)2\n+ 2∥/radicalbigg\n−˙ML\nMLAp2,̸=∥2\nL2L2.\nSinceα>1\n2we deduce1+2C−1\nα\nα<1 +1\n2α. Choosing the constants such that\nc=1\n200(1−1\n2α)2,\nc1=1\n20(1−1\n2α),\nand recalling that\nν≤1\n40(1−1\n2α)6\n5,MHD STABILITY THRESHOLD 17\nwe conclude that (12(1−1\n2α)−1c+ 2ν2+ 2c1+ν5\n6+1+2C−1\nα\n2α<17+3\n2α\n20. Thus we obtain\nthe estimate\n/integraldisplay\nL1+LR+LNRdτ≤17+3\n2α\n20(Cε)2+ 2∥/radicalbigg\n−˙ML\nMLAp2,̸=∥2\nL2L2.\nThis yields the first estimate of Proposition 3.1.\n3.2.Nonlinear terms without an x-average. We apply the notation of (26)\nand aim to estimate terms of the form\n⟨Aa1\n̸=,A(a2\n̸=∇ta3\n̸=)−a2\n̸=∇tAa3\n̸=⟩\n=/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)A(k,ξ)−A(l,η)\nA(k−l,ξ−η)A(l,η)ξl−kη\n((k−l)2+(ξ−η−(k−l)t)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n=T+R.\nHere, we split the integral into the reactionRand thetransportTterms which\ncorrespond to the sets\nΩR={|k−l,ξ−η|≥1\n8|l,η|},\nΩT={|k−l,ξ−η|<1\n8|l,η|},\nin Fourier space. We split the weights\nA(k,ξ)−A(l,η) =ectκ1\n3(ML(k,ξ)−ML(l,η))M1(k,ξ)Mκ(k,ξ)Mν(k,ξ)Mν3(k,ξ)|k,ξ|N\n+ectκ1\n3(|k,ξ|N−|l,η|N)ML(l,η)M1(k,ξ)Mκ(k,ξ)Mν(k,ξ)Mν3(k,ξ)\n+ectκ1\n3(M1(k,ξ)−M1(l,η))ML(l,η)Mκ(k,ξ)Mν(k,ξ)Mν3(k,ξ)|l,η|N\n+ectκ1\n3(Mκ(k,ξ)−Mκ(l,η))M1(l,η)ML(l,η)Mν(k,ξ)Mν3(k,ξ)|l,η|N\n+ectκ1\n3(Mν(k,ξ)−Mν(l,η))M1(l,η)ML(l,η)Mκ(l,η)Mν3(k,ξ)|l,η|N\n+ectκ1\n3(Mν3(k,ξ)−Mν3(l,η))M1(l,η)ML(l,η)Mκ(l,η)Mν(l,η)|l,η|N\nand thus by (13) we estimate\nA(k,ξ)−A(l,η)\nA(k−l,ξ−η)A(l,η)≲e−ctκ1\n3|ML(k,ξ)−ML(l,η)|\nML(k−l,ξ−η)ML(l,η)|ξ,η|N\n|l,η|N|k−l,ξ−η|N\n+e−ctκ1\n3||k,ξ|N−|l,η|N|\n|l,η|N|k−l,ξ−η|N1\nML(k−l,ξ−η)\n+e−ctκ1\n3/summationdisplay\nj=1,κ,ν,ν3|Mj(k,ξ)−Mj(l,η)|1\n|k−l,ξ−η|N1\nML(k−l,ξ−η).(27)18 NIKLAS KNOBEL\nReactionterm: Ontheset ΩRitholdsthat|k−l,ξ−η|≥1\n8|l,η|, thus|k,ξ|,|l,η|≲\n|k−l,ξ−η|. From (13), (27) and|k,ξ|N\n|l,η|N|k−l,ξ−η|N,||k,ξ|N−|l,η|N|\n|l,η|N|k−l,ξ−η|N≲1\n|l,η|Nwe infer\nA(k,ξ)−A(l,η)\nA(k−l,ξ−η)A(l,η)≲1\nML(k−l,ξ−η)ML(l,η)1\n|l,η|N.\nWithξl−kη= (ξ−η−(k−l)t)l−(k−l)(η−lt)and Hölder’s inequality we deduce\nR=e−ctκ1\n3/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩRA(k,ξ)−A(l,η)\nA(k−l,ξ−η)A(l,η)ξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲e−ctκ1\n3/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩR1\n|l,η|N1\nML(k−l,ξ−η)ML(l,η)|(ξ−η−(k−l)t)l−(k−l)(η−lt)|\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲∥Aa1\n̸=∥L2∥1\nMLApi,̸=∥L2∥1\nMLAa3\n̸=∥L2\n+∥Aa1\n̸=∥L2∥∂x1\nMLΛ−1\ntApi,̸=∥L2∥1\nMLA∂t\nya3\n̸=∥L2.\nWe use (33) and (34) to infer\nR≲∥Aa1\n̸=∥L2(∥Api,̸=∥L2+ν∥(Λt∧κ−1\n3)Api,̸=∥L2)(∥Aa3\n̸=∥L2+ν∥A(Λt∧κ−1\n3)a3\n̸=∥L2)\n+L∥Aa1\n̸=∥L2(∥A∂xΛ−1\ntpi,̸=∥L2+ν∥Api,̸=∥L2)∥∂t\nyAa3\n̸=∥L2.\nIntegrating in time yields\n/integraldisplay\nRdτ≲Lν−1\n12κ−1\n2ε3.\nTransport term: On the set ΩTit holds that|k−l,ξ−η|<1\n8|l,η|and thus it\nfollows that|k,ξ|≈|l,η|. By the mean value theorem, there exists θ∈[0,1]such\nthat\n/vextendsingle/vextendsingle/vextendsingle|k,ξ|N−|l,η|N/vextendsingle/vextendsingle/vextendsingle≤N|k−l,ξ−η||k−θl,ξ−θη|N−1\n≲|k−l,ξ−η||l,η|N−1.\nThus with (27) and Lemma 4 we conclude, that\nA(k,ξ)−A(l,η)\nA(k−l,ξ−η)A(l,η)≲1\nML(l,η)(1\n|l|+ν1\n12)1\n|k−l,ξ−η|N−1 (28)\n+/summationdisplay\nj=κ,ν,ν3|Mj(k,ξ)−Mj(l,η)|1\nML(l,η)1\n|k−l,ξ−η|N (29)\n+ML(k,ξ)−ML(l,η)\nML(k−l,ξ−η)ML(l,η)1\n|k−l,ξ−η|N. (30)MHD STABILITY THRESHOLD 19\nBased on this estimate, in the following we distinguish between different regimes in\nfrequency,\nT=/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩTA(k,ξ)−A(l,η)\nA(k−l,ξ−η)A(l,η)ξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩT1\nML(l,η)(1\n|l|+ν1\n12)1\n|k−l,ξ−η|N−1ξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n+/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩT1|η\nl−t|≥|ξ−η\nk−l−t|/summationtext\nj=κ,ν,ν 3|Mj(k,ξ)−Mj(l,η)|\nML(l,η)1\n|k−l,ξ−η|Nξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n+/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩT1|η\nl−t|≤|ξ−η\nk−l−t|/summationtext\nj=κ,ν,ν 3|Mj(k,ξ)−Mj(l,η)|\nML(l,η)1\n|k−l,ξ−η|Nξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n+/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩTML(k,ξ)−ML(l,η)\nML(k−l,ξ−η)ML(l,η)1\n|k−l,ξ−η|Nξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n=T1,1+T1,2+T1,3+T2.\nHere, theT1,1term is due to estimate (28). For(29)we distinguish between\nthe frequencies|η\nl−t|≥|ξ−η\nk−l−t|inT1,2and|η\nl−t|≤|ξ−η\nk−l−t|inT1,3. The\nMLcommutator (30)isT2, which requires further splitting. For T1,1we use\nξl−kη= (ξ−η−(k−l)t)l−(k−l)(η−lt), (13) and (34) to estimate\nT1,1=/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)1ΩT1\nML(l,η)(1\n|l|+ν1\n12)1\n|k−l,ξ−η|N−1ξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲∥Aa1\n̸=∥L2∥Api,̸=∥L2∥1\nMLAa3\n̸=∥L2+∥Aa1\n̸=∥L2∥AΛ−1\ntpi,̸=∥L2∥1\nMLA∂t\nya3\n̸=∥L2\n+ν1\n12∥Aa1\n̸=∥L2∥Api,̸=∥L2∥1\nMLA∂xa3\n̸=∥L2\n≤L∥Aa1\n̸=∥L2∥Api,̸=∥L2∥Aa3\n̸=∥L2+L∥Aa1\n̸=∥L2∥AΛ−1\ntpi,̸=∥L2∥A∂t\nya3\n̸=∥L2\n+ν1\n12∥Aa1\n̸=∥L2∥Api,̸=∥L2∥Λta3\n̸=∥L2.\nAfter integrating in time we deduce that\n/integraldisplay\nT1,1dτ≲(L+ν−1\n12)κ−1\n2ε3.\nForT1,2we use|η\nl−t|≥|ξ−η\nk−l−t|to infer that|ξl−kη|=|(ξ−η−(k−l)t)l−(k−\nl)(η−lt)|≤2|(k−l)(η−lt)|. Furthermore, with/summationtext\ni=ν,κ,ν3|Mi(k,ξ)−Mi(l,η)|≈120 NIKLAS KNOBEL\nand (32) we conclude that\nT1,2≲/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩT1|η\nl−t|≥|ξ−η\nk−l−t|1\nML(l,η)1\n|k−l,ξ−η|N|(k−l)(η−lt)|\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲∥Aa1\n̸=∥L2∥AΛ−1\ntpi,̸=∥L2∥1\nMLA∂t\nya3\n̸=∥L2\n≲L∥Aa1\n̸=∥L2∥AΛ−1\ntpi,̸=∥L2∥A∂t\nya3\n̸=∥L2.\nSo after integrating in time, we obtain\n/integraldisplay\nT1,2dτ≲Lκ−1\n2ε3.\nForT1,3, we use|η\nl−t|≤|ξ−η\nk−l−t|to inferξl−kη≤2(ξ−η−(k−l)t)l. Furthermore,\nwith (40) we deduce\n/summationdisplay\nj=κ,ν,ν3|Mj(k,ξ)−Mj(l,η)|≲ν1\n3|ξl−kη|\n|kl|.\nCombining these two estimates by Hölder’s inequality and (32) it follows, that\nT1,3≲/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩT1|η\nl−t|≤|ξ−η\nk−l−t|1\nML(l,η)1\n|k−l,ξ−η|Nν1\n3(ξl−kη)2\nkl((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩT1|η\nl−t|≤|ξ−η\nk−l−t|1\nML(l,η)1\n|k−l,ξ−η|Nν1\n3(ξ−η−(k−l)t)2l2\nkl((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲ν1\n3/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩT1|η\nl−t|≤|ξ−η\nk−l−t|1\nML(l,η)|ξ−η−(k−l)t|\n|k−l,ξ−η|N−1\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲ν1\n3∥Aa1\n̸=∥L2∥AΛtpi,̸=∥L2∥1\nMLAa3\n̸=∥L2\n≲Lν1\n3∥Aa1\n̸=∥L2∥AΛtpi,̸=∥L2∥Aa3\n̸=∥L2.\nThus integrating in time yields\n/integraldisplay\nT1,3dτ≤Lν1\n6κ−1\n2ε3.\nTo estimate the T2term, we split the integral into the sets\nΩ1={min(t−η\nl,t−ξ−η\nk−l)≥ν−1},\nΩ2={t−η\nl≥ν−1≥t−ξ−η\nk−l},\nΩ3={t−ξ−η\nk−l≥ν−1≥t−η\nl},\nΩ4={t−ξ\nk≥ν−1≥max(t−η\nl,t−ξ−η\nk−l)}.MHD STABILITY THRESHOLD 21\nFor frequencies such that ν−1≥max(t−η\nl,t−ξ\nk), thenML(k,ξ)−ML(l,η) = 0\nand hence the commutator vanishes. Thus the sets Ωjcovers all regions of the\nsupport. The sets Ω1,Ω2andΩ3are chosen to distinguish between1\nML= 1and\n1\nML>1for different frequencies and on set Ω4we use strong dissipation in the first\ncomponent. We split the set T2into\nT2=/summationdisplay\nk,l,k−l̸=0/integraldisplay\nd(ξ,η)1ΩTML(k,ξ)−ML(l,η)\nML(k−l,ξ−η)ML(l,η)ξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n21\n|k−l,ξ−η|N\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)(1Ω1+1Ω2+1Ω3+1Ω4)\n=T2,1+T2,2+T2,3+T2,4.\nForT2,1we use (32) to deduce\nT2,1≤ν2/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)1Ω1⟨t−η\nl∧κ−1\n3⟩⟨t−ξ−η\nk−l∧κ−1\n3⟩1\n|k−l,ξ−η|N−1\n(ξ−η−(k−l)t)l−(k−l)(η−lt)\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲ν2∥Aa1\n̸=∥L2∥(Λt∧κ−1\n3)Api,̸=∥L2∥AΛta3\n̸=∥L2\n+Lν∥Aa1\n̸=∥L2∥(Λt∧κ−1\n3)Λ−1\ntΛ−1Api,̸=∥L2∥A∂t\nya3\n̸=∥L2\nand so/integraldisplay\nT2,1dτ≲Lν5\n6κ−1\n2ε3.\nNow we consider T2,2. By (32) we infer\nT2,2≤/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)1Ω2ν⟨t−η\nl∧κ−1\n3⟩1\n|k−l,ξ−η|N−1(ξ−η−(k−l)t)l−(k−l)(η−lt)\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≲ν∥Aa1\n̸=∥L2∥Api,̸=∥L2∥AΛta3\n̸=∥L2\n+L∥Aa1\n̸=∥L2∥AΛ−1\ntpi,̸=∥L2∥A∂t\nya3\n̸=∥L2.\nIntegrating in time yields\n/integraldisplay\nT2,2dτ≲Lκ−1\n2ε3.\nTo estimate T2,3, we need to distinguish between different choices of a. Using(32)\nwe estimate\nT2,3=/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)1Ω3ν⟨t−ξ−η\nk−l∧κ−1\n3⟩1\n|k−l,ξ−η|N−1(ξ−η−(k−l)t)l−(k−l)(η−lt)\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n��ν∥Aa1\n̸=∥L2∥(Λt∧κ−1\n3)Api,̸=∥L2∥A∂xa3\n̸=∥L2\n+ν∥Aa1\n̸=∥L2∥A(Λt∧κ−1\n3)Λ−1Λ−1\ntpi,̸=∥L2∥A∂t\nya3\n̸=∥L222 NIKLAS KNOBEL\nand thus after integrating in time\n/integraldisplay\nT2,3[vvv]dτ≲ν−1\n2ε3,\n/integraldisplay\nT2,3[bvb]dτ≲κ−1\n2ε3,\n/integraldisplay\nT2,3[bbv]dτ≲κ−1\n2ε3.\nIn the case of vbb, we use (32) to estimate\nT2,3[vbb] =/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)ν⟨t−ξ−η\nk−l⟩1\n|k−l,ξ−η|N−1(ξ−η−(k−l)t)k−(k−l)(ξ−kt)\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Av)(k,ξ)(Ap2)(k−l,ξ−η)(Ab)(l,η)\n≲ν∥∂xAv̸=∥L2∥(Λt∧κ−1\n3)Ap2,̸=∥L2∥Ab̸=∥L2\n+ν∥∂t\nyAv̸=∥L2∥(Λt∧κ−1\n3)Λ−1\ntAp2,̸=∥L2∥Ab̸=∥L2\n≤ν∥Ab̸=∥L2∥A∇tv̸=∥L2∥AΛtp2,̸=∥L2.\nThus after integrating in time, we obtain\n/integraldisplay\nT2,3[vbb]dτ≲ν1\n2κ−1\n2ε3.\nForT2,4we obtain that M(l,η) =M(k−l,ξ−η) = 1. We uset−ξ\nk≥ν≥\nmax(t−η\nl,t−ξ−η\nk−l)to deduce that\n1 =kt−ξ\nkt−ξ=k\nkt−ξ\nk\nt−ξ\nk≤ν|ξ−kt|\n|k|.\nWithξl−kη= (ξ−η−(k−l)t)l−(k−l)(η−lt)we infer that\nT2,4=/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)1\n|k−l,ξ−η|Nξl−kη\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n=ν/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)|ξ−kt|1\n|k−l,ξ−η|N−1(ξ−η−(k−l)t)l\n|l|((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n+/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)1\n|k−l,ξ−η|N−1(k−l)(η−lt)\n((k−l)2+(ξ−η−(k−l)τ)2)1\n2\n(Aa1)(k,ξ)(Api)(k−l,ξ−η)(Aa3)(l,η)\n≤ν∥A∂t\nya1\n̸=∥L2∥Api,̸=∥L2∥Aa3\n̸=∥L2\n+∥Aa1\n̸=∥L2∥Λ−1\ntApi,̸=∥L2∥A∂t\nya3\n̸=∥L2.MHD STABILITY THRESHOLD 23\nThus integrating in time yields\n/integraldisplay\nT2,4dτ≲κ−1\n2ε3.\n3.3.Nonlinear terms with an x-average in the second component. We\napply the notation of (26)\n⟨Aa1\n̸=,A(a2\n1,=∂xa3\n̸=)−a2\n1,=∂xAa3\n̸=⟩\n=/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)(Aa1)(k,ξ)(A(k,ξ)−A(k,η))ka2\n1(0,ξ−η)a3(k,η)\n=R+T.\nHere we split into reaction and transport terms according to the sets\nΩR={|ξ−η|≥1\n8|k,η|},\nΩT={|ξ−η|<1\n8|k,η|}.\nReaction term On the set ΩRit holds that|ξ−η|≥1\n8|k,η|, then we obtain\n|A(k,ξ)−A(k,η)|≲|ξ−η|Nand thus with (32), it follows that\n/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)(Aa1)(k,ξ)(A(k,ξ)−A(k,η))ka2\n1(0,ξ−η)a3(k,η)\n≲∥Aa1\n̸=∥L2∥a2\n=∥HN∥∂xa3\n̸=∥L∞\n≲∥Aa1\n̸=∥L2∥a2\n=∥HN∥1\nMLAa3\n̸=∥L2\n≲L∥Aa1\n̸=∥L2∥a2\n=∥HN∥Aa3\n̸=∥L2.\nIntegrating in time yields a bound\n/integraldisplay\nR dτ≲Lκ−1\n3ε2˜ε.\nTransport term On the set ΩLit holds that|k,η|≥1\n8|ξ−η|. By the mean value\ntheorem there exists a θ∈[0,1]\n/vextendsingle/vextendsingle/vextendsingle|k,η|N−|k,ξ|N/vextendsingle/vextendsingle/vextendsingle≲|ξ−η||k,η−θξ|N−1≲|ξ−η||k,η|N−1.\nThus, we can estimate the difference in Aby\n|A(k,ξ)−A(k,η)|≲(ML(k,ξ)−ML(k,η))|k,ξ|N\n+ML(k,η)|k,ξ|N/summationdisplay\nj=1,κ,ν,ν3|Mj(k,ξ)−Mj(k,η)|\n+ML(k,η)(|k,η|N−|k,ξ|N)\n≲1\nk|ξ−η||k,ξ|N24 NIKLAS KNOBEL\nwhere we used (37),(39)and(40)to estimate the differences in Mj. So we infer,\nthat\nT≤/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)1ΩT|Aa1|(k,ξ)|a2\n1|(0,ξ−η)1\nML(k,η)|Aa3|(k,η).\nand thus integrating in time yields\n/integraldisplay\nTdτ≲L∥Aa1\n̸=∥L2L2∥a2\n=∥L∞HN∥Aa3\n̸=∥L2L2≲Lκ−1\n3ε2˜ε.\n3.4.Nonlinear terms with an x-average in the third component. We aim\nto estimate\n⟨Aa1\n1,̸=,A(a2\n2,̸=∂ya3\n1,=)⟩\n=/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)(Aa1\n1)(k,ξ)A(k,ξ)kη√\nk2+(ξ−η−kt)2pi(k,ξ−η)a3\n1(0,η)\n=R+T\nwhere we split into the reaction and transport terms according to the sets\nΩR={|k,ξ−η|≥1\n8|η|},\nΩT={|k,ξ−η|<1\n8|η|}.\nReaction term On the set ΩRit holds that|k,ξ−η|≥1\n8|η|. With(32)we infer\nR=/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)1ΩR(Aa1\n1)(k,ξ)A(k,ξ)kη√\nk2+(ξ−η−kt)2pi(k,ξ−η)a3\n1(0,η)\n≲∥Aa1\n1,̸=∥L2∥A1\nML∂xΛ−1\ntpi,̸=∥L2∥∂ya3\n1,=∥L∞\n≲∥Aa1\n1,̸=∥L2∥Api,̸=∥L2∥a3\n1,=∥HN.\nIntegrating in time then yields\n/integraldisplay\nRdτ≲κ−1\n3ε2˜ε.\nTransport term On the set ΩTit holds that|k,ξ−η|≤1\n8|η|, then with (32)we\nestimate\nT=/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)1ΩT(Aa1\n1)(k,ξ)A(k,ξ)kη√\nk2+(ξ−η−kt)2pi(k,ξ−η)a3\n1(0,η)\n≲∥Aa1\n̸=∥L2∥∂xΛ−1\ntpi,̸=∥L∞∥∂ya3\n=∥HN\n≲∥Aa1\n̸=∥L2∥1\nMLAΛ−1\ntpi,̸=∥L2∥∂ya3\n=∥HN\n≲∥Aa1\n̸=∥L2(∥Λ���1\ntApi∥L2+ν∥Api,̸=∥L2)∥∂ya3\n=∥HN.\nIntegrating in time yields\n/integraldisplay\nTdτ≲κ−1\n2ε2˜ε.MHD STABILITY THRESHOLD 25\n3.5.Nonlinear terms with an x-average in first component. Now we turn\nto\n⟨⟨∂y⟩Na1\n=,1,⟨∂y⟩N(a2\n̸=∇ta3\n̸=,1)=⟩\n=−/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)⟨ξ⟩2Na1\n1(0,ξ)kξ√\nk2+(ξ−η+kt)2pi(−k,ξ−η)a3\n1(k,η).\nApplying Hölder’s inequality, the Sobolev embedding and the definition of Ayields\n⟨⟨∂y⟩Na1\n1,=,⟨∂y⟩N(a2\n̸=∇ta3\n̸=,1)=⟩\n≤∥∂y⟨∂y⟩Na1\n=∥L2(∥∂xΛ−1\ntpi,̸=∥L∞∥⟨∂y⟩Na3\n̸=∥L2+∥⟨∂y⟩NΛ−1\ntpi,̸=∥L2∥∂xa3\n̸=∥L∞\n≤∥∂ya1\n=∥HN∥A1\nMLΛ−1\ntpi,̸=∥L2∥A1\nMLa3\n̸=∥L2\nWith (32) we infer\n⟨⟨∂y⟩Na1\n1,=,⟨∂y⟩N(a2\n̸=∇ta3\n̸=,1)=⟩\n≲∥∂ya1\n=∥HN(∥AΛ−1\ntpi,̸=∥L2+ν∥Api,̸=∥L2)(∥Aa3\n̸=∥L2+ν∥A(Λt∧κ−1\n3)a3\n̸=∥L2)\n≲∥∂ya1\n=∥HN∥AΛ−1\ntpi,̸=∥L2(∥Aa3\n̸=∥L2+ν∥(Λt∧κ−1\n3)a3\n̸=∥L2)\n+ν∥∂ya1\n=∥HN∥Api,̸=∥L2(∥Aa3\n̸=∥L2+ν∥(Λt∧κ−1\n3)a3\n̸=∥L2).\nIntegrating in time yields\n/integraldisplay\n⟨⟨∂y⟩Na1\n=,⟨∂y⟩N(a2\n̸=∇ta3\n̸=,1)=⟩dτ≲Lκ−1\n2ε2˜ε.\n3.6.Other nonlinear terms. In this subsection, we aim to estimate\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2∇ta3)⟩=⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n̸=∇ta3\n̸=)⟩\n+⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n̸=∇ta3\n=)⟩\n+⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n=∇ta3\n̸=)⟩.\nwith the choices a1a2a3∈{bvv,bbb,vbv,vvb}. We start with the case of no x-\naverages and use\nξl−kη= (ξ−kt)(l−k) +k(ξ−η−(k−l)t)\nand (32) to infer\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n̸=∇ta3\n̸=)⟩\n=/summationdisplay\nk,l,k−l̸=0/integraldisplay/integraldisplay\nd(ξ,η)ξ−kt\nk2+(ξ−kt)2ξl−kη√\n(k−l)2+(ξ−η−(k−l)t)2A2(k,ξ)a1(k,ξ)pi(k−l,ξ−η)a3(l,η)\n≲∥Aa1\n̸=∥L2∥1\nML∂xΛ−1\ntApi,̸=∥L2∥1\nMLAa3\n̸=∥L2\n+∥∂xΛ−1\ntAa1\n̸=∥L2∥1\nMLApi,̸=∥L2∥1\nMLAa3\n̸=∥L2\n≲L∥Aa1\n̸=∥L2(∥∂xΛ−1\ntApi,̸=∥L2+ν∥Api,̸=∥L2)∥Aa3\n̸=∥L2\n+L(1 +νκ−1\n3)∥∂xΛ−1\ntAa1\n̸=∥L2∥Api,̸=∥L2∥Aa3\n̸=∥L2.26 NIKLAS KNOBEL\nThus integrating in time yields\n/integraldisplay\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n̸=∇ta3\n̸=)⟩dτ≲Lκ−1\n2ε3.\nForthecase, whentheaverageisinthesecondcomponent, weusepartialintegration\nto estimate\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n1,=∂xa3\n̸=)⟩\n=−⟨χA∂x∂t\ny∆−1\nta1\n̸=,A(a2\n1,=a3\n̸=)⟩\n≲∥∂xΛ−1\nta1\n̸=∥L2∥a2\n=∥HN∥1\nMLAa3\n̸=∥L2\n≲L∥∂xΛ−1\nta1\n̸=∥L2∥a2\n=∥HN∥Aa2\n̸=∥L2\nand thus integrating in time and using L= max(1,νκ−1\n3)yields\n/integraldisplay\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n1,=∂xa3\n̸=)⟩dτ≲κ−1\n2ε2˜ε.\nFor the case when the average is in the third component, we obtain\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n2,̸=∂ya3\n=)⟩\n=/summationdisplay\nk̸=0/integraldisplay/integraldisplay\nd(ξ,η)χξ−kt\nk2+(ξ−kt)2kη√\nk2+(ξ−η−t)2A(ξ,k)\nA(ξ−η,k)(Aa1)(k,ξ)(Api)(k,ξ−η)a3(0,η).\nThus byη=ξ−kt−(ξ−η−kt)we estimate\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n2,̸=∂ya3\n=)⟩\n≤∥(∂t\ny)2∆−1\ntAa1\n̸=∥L2∥∂xΛ−1\nt1\nMLApi,̸=∥L2∥a3\n=∥HN\n+∥∂x∂t\ny∆−1\ntAa1\n̸=∥L2∥1\nMLApi,̸=∥L2∥a3\n=∥HN\n≤L∥Aa1\n̸=∥L2∥∂xΛ−1\ntApi,̸=∥L2∥a3\n=∥HN\n+L∥∂xΛ−1\ntAa1\n̸=∥L2∥Api,̸=∥L2∥a3\n=∥HN\nIntegrating in time and using L= max(1,νκ−1\n3)yields\n/integraldisplay\n⟨χA∂t\ny∆−1\nta1\n̸=,A(a2\n2,̸=∂ya3\n=)⟩dτ≲κ−1\n2ε2˜ε.\nWhich concludes the estimate/integraldisplay\nONLdτ ≲Lκ−1\n2˜εε2.\nCombining the estimates of Subsection 3.2 to 3.6 completes the proof of Proposition\n3.1and thus Theorem 3.1. □\nIn this article, we have shown that the MHD equations around Couette flow with\nmagnetic resistivity smaller than fluid viscosity ν≥κ>0are stable for initial data\nwhich is small enough in Sobolev spaces. If the resistivity is much smaller than\nthe viscosity, νκ−1\n3>0, large viscosity destabilizes the equation, leading to normMHD STABILITY THRESHOLD 27\ninflation of size νκ−1\n3. Controlling this norm inflation is a major new challenge\ncompared to other dissipation regimes.\nAppendix A.Construction of the Weights\nLetAbe the Fourier weight\nA: =M⟨∇⟩ecκ1\n3t1̸=,\nwithM=M1MLMκMνMν3defined as\n−˙ML\nML=t−ξ\nk\n1+(ξ\nk−t)21{ν−1≤t−ξ\nk≤(c1κk2)−1\n3}k̸= 0,\n−˙M1\nM1=Cα|k|+ν1\n12|k|2\nk2+(ξ−kt)2 k̸= 0,\n−˙Mν\nMν=ν1\n3\n1+ν2\n3(t−ξ\nk)2k̸= 0,\n−˙Mκ\nMκ=κ1\n3\n1+κ2\n3(t−ξ\nk)2k̸= 0,\n−˙Mν3\nMν3=Cαν\n1+ν2(t−ξ\nk)2 k̸= 0,\nM·(t= 0) =M·(k= 0) = 1.\nThe weight MLis an adaption of the weight m1\n2in [Lis20] to the present setting\nandMν3we use to differentiate between resonant and non-resonant regions. The\nmethod of using time-dependent Fourier weights is common when working at\nsolutions around Couette flow and the other weights are modifications of previously\nused weights (cf. [ BVW18,MZ22,Lis20,ZZ23] for shear related systems such as\nNavier-Stokes). The constants Cα=2\nmin(1,α−1\n2),c=1\n20(1−1\n2α)2andc1=1\n20(1−1\n2α)\nare determined through the linear estimates. For the weights we obtain that for all\ntimest>0, it holds that\nM1≈Mκ≈Mν≈Mν3≈1,\nL−1≤min(1,ν−1κ1\n3k2\n3)≲ML≤1.\nLemma 2 (MLproperties ) .The weight MLsatisfies the following bounds\n1|t−ξ\nk|≥ν−1t−ξ\nk\n1+(t−ξ\nk)2≤−˙ML\nML+κk2c1(1 + (t−ξ\nk)2), (31)\n1\nML(k,ξ)≤1 +ν1\n2⟨t−ξ\nk⟩∧κ−1\n3. (32)\nFurthermore, it follows for a∈H1, that\n∥1\nMLa̸=∥L2≤∥a̸=∥L2+∥(Λ−1\nt∧κ−1\n3)a̸=∥L2, (33)\n∥1\nML∂xa̸=∥L2≤∥Λta̸=∥L2. (34)\nProof.This follows immediately, from the definition of ML. □28 NIKLAS KNOBEL\nLemma 3 (Enhanced dissipation estimates ) .The weights MνandMκsatisfy the\nfollowing bounds\n1\n2ν1\n3≤−˙Mκ\nMκ+ν(k2+ (ξ−kt)2), (35)\n1\n2κ1\n3≤−˙Mν\nMν+κ(k2+ (ξ−kt)2). (36)\nProof.This follows immediately, from the definition of MνandMκ. □\nLemma 4 (Difference estimates) .Letk,l∈Z\\{0}andξ,η∈R, then there hold\nthe following bounds on differences\n1−M1(k,ξ)\nM1(k,η)≲|ξ−η|\n|k|, (37)\n1−M1(k,ξ)\nM1(l,η)≲|k−l|\n|l|+ν1\n12, (38)\nML(k,η)−ML(k,ξ)≤2|ξ−η|\nk, (39)\n1−Mj(k,ξ)\nMj(l,η)≤2j1\n3|ξl−kη|\n|kl|, j∈{κ,ν,ν3} (40)\nProof.We start with the M1estimate (37) and consider M1(k,ξ)≤M1(k,η)\n1−M1(k,ξ)\nM1(k,η)= 1−exp/parenleftbigg\n−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0|k|+|k|2ν1\n12\nk2+(ξ−kτ)2−|k|+|k|2ν1\n12\nk2+(η−kt)2dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n,\n≤1−exp/parenleftigg\n−/integraldisplay\n[ξ\nk,η\nk]∪t−[ξ\nk,η\nk]1dτ/parenrightigg\n≲|ξ−η|\n|k|.\nThe caseM1(k,ξ)≥M1(k,η)follows by the same argument and M1(k,ξ)≈\nM1(k,η)≈1. For (38) we consider the case M1(k,ξ)≤M1(l,η)and infer that\n1−M1(k,ξ)\nM1(l,η)= 1−exp/parenleftbigg\n−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0|k|+|k|2ν1\n12\nk2+(ξ−kt)2−|l|+|l|2ν1\n12\nl2+(η−lt)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n,\n= 1−exp(−2π(1\nl∧k+ν1\n12))≲1\nl∧k+ν1\n12≲|k−l|\n|l|+ν1\n12.\nThe caseM1(k,ξ)≥M1(l,η)follows by the same argument and M1(k,ξ)≈\nM1(l,η)≈1. For (39) we consider the case ML(k,ξ)≤ML(k,η)and thus\nML(k,η)−ML(k,ξ) =ML(k,η)(1−ML(k,ξ)\nML(k,η))≲1−ML(k,ξ)\nML(k,η).\nWe infer\n1−ML(k,ξ)\nML(k,η)= 1−exp/parenleftbigg\n−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0τ−ξ\nk\n1+(τ−ξ\nk)2dτ−/integraldisplayt\n0τ−η\nk\n1+(τ−η\nk)2dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n,\n= 1−exp/parenleftigg\n−/integraldisplay\n−[ξ\nk,η\nk]∪t−[ξ\nk,η\nk]1dτ/parenrightigg\n,\n≤2|ξ−η|\nk.\nThe caseML(k,ξ)≥ML(k,η)follows by the same argument.MHD STABILITY THRESHOLD 29\nFor(40)we estimate the Mκdifference, since the MνandMν3differences are\ndone similar. Let Mκ(k,ξ)≥Mκ(l,η), then it follows\n1−Mκ(k,ξ)\nMκ(l,η)= 1−exp/parenleftigg\n−κ1\n3|/integraldisplayt\n01\n1+κ2\n3(t−ξ\nk)2−1\n1+κ2\n3(t−η\nl)2|/parenrightigg\n,\n≤1−exp/parenleftbigg\n−κ1\n3|/integraldisplayt\n01−[ξ\nk,η\nl]∪t−[ξ\nk,η\nl](τ)dτ|/parenrightbigg\n,\n≤1−exp/parenleftig\n−2κ1\n3|ξ\nk−η\nl|/parenrightig\n,\n≲κ1\n3|ξl−kη|\n|kl|.\nThe caseMκ(k,ξ)≤Mκ(l,η)follows from the same steps and Mκ(k,ξ)≈Mκ(l,η).\n□\nAppendix B.Local Wellposedness\nWe expect the local wellposedness result to be well-known, but were not able to\nfind it stated in the literature. In the following, we prove the local wellposedness\nby a standard application of the Banach fixed-point theorem.\nProposition 5. Consider equation (14)with initial data pin∈HNforN≥5.\nThen there exists a time Tsuch that there exists a unique solution p(t)∈HNto\n(14)for allt∈[0,T].\nProof.We prove existence with the Banach fixed-point theorem. Let T= 1 +\n2∥pin∥HN(1 +8\nκ)and letXbe the space\nX={p∈L∞HN∩CHN−2:p(t= 0) =pin,∥p∥2\nL∞HN+κ\n2∥∇tp∥2\nL∞HN≤2∥pin∥2\nHN}\nwith the norm\n∥p∥2\nX:=∥p∥2\nL∞HN+κ\n2∥∇tp∥2\nL∞HN.\nWe defineF:X∝⇕⊣√∫⊔≀→Xas a mapping q∝⇕⊣√∫⊔≀→p=F(q)such thatpsolves\n∂tp1−∂x∂t\ny∆−1\ntp1−α∂xp2=ν∆tp1+ Λ−1\nt∇⊥\nt(∇⊥\ntΛ−1\ntq2∇tb−∇⊥\ntΛ−1\ntq1∇tv),\n∂tp2+∂x∂t\ny∆−1\ntp2−α∂xp1=κ∆tp2+ Λ−1\nt∇⊥\nt(∇⊥\ntΛ−1\ntq2∇tv−∇⊥\ntΛ−1\ntq1∇tb),\np|t=0=pin.\nThen the mapping Fsatisfies:\n(1) The mapping F:X→Xis well defined on X.\n(2)The mapping Fis a contraction, i.e. ∥F(p)−F(˜p)∥L∞HN≤1\n2∥p−˜p∥L∞HN.\nSinceXis a complete metric space, if we prove (1) and (2), then it follows that F\nhas a unique fixpoint by the Banach fixed-point theorem.30 NIKLAS KNOBEL\n(1) Letq∈X, then we obtain for p=F(q)\n∂t∥p∥2\nHN+κ∥∇tp∥2\nHN≤∥p∥2\nHN+⟨ΛNv,ΛN(∇⊥\ntΛ−1\ntq2∇tb−∇⊥\ntΛ−1\ntq1∇tv)⟩\n+⟨ΛNb,ΛN(∇⊥\ntΛ−1\ntq2∇tv−∇⊥\ntΛ−1\ntq1∇tb)⟩\n≤∥p∥2\nHN+∥p∥HN∥q∥HN∥∇tpn+1∥HN\n≤∥p∥2\nHN+2\nκ∥p∥2\nHN∥q∥2\nHN+κ\n2∥∇tp∥2\nHN.\nThus we obtain\n∥p∥2\nX≤∥pin∥2\nHN+T(1 +1\nκ∥q∥2\nL∞HN)∥p∥2\nL∞HN\n≤∥pin∥2\nHN+T(1 +2\nκ∥pin∥2\nHN)∥p∥2\nL∞HN,\nSince\nT(1 +4\nκ∥pin∥2\nHN)<1\n2\nwe infer the bound\n∥p∥2\nX≤2∥pin∥2\nHN.\nAs∂tp∈HN−2, it follows that p∈CHN−2and thusp∈X.\n(2)We show that Fis a contraction. Let q,˜q∈Xwe denotep=F(q)and\n˜p=F(˜q). We need to show that\n∥p−˜p∥X<1\n2∥q−˜q∥X,\nby time estimate we obtain\n∂t∥p−˜p∥2\nHN+κ∥∇t(p−˜p)∥2\nHN≤∥p−˜p∥2\nHN\n+⟨ΛN(v−˜v),ΛN(∇⊥\ntΛ−1\ntq2∇tb−∇⊥\ntΛ−1\ntq1∇tv)⟩\n+⟨ΛN(b−˜b),ΛN(∇⊥\ntΛ−1\ntq2∇tv−∇⊥\ntΛ−1\ntq1∇tb)⟩\n−⟨ΛN(v−˜v),ΛN(∇⊥\ntΛ−1\nt˜q2∇t˜b−∇⊥\ntΛ−1\nt˜q1∇tv)⟩\n−⟨ΛN(b−˜b),ΛN(∇⊥\ntΛ−1\nt˜q2∇t˜v−∇⊥\ntΛ−1\nt˜q1∇t˜b)⟩\n≤∥p−˜p∥2\nHN+∥p−˜p∥HN(∥q−˜q∥HN∥∇t˜p∥HN+∥q∥HN∥∇t(p−˜p)∥HN)\n≤∥p−˜p∥2\nHN(1 +2\nκ∥q∥2\nHN)\n+∥p−˜p∥HN∥q−˜q∥HN∥∇t˜p∥HN+κ\n2∥∇t(p−˜p)∥2\nHN.\nIntegrating in time yields\n∥p−˜p∥2\nL∞HN+κ\n2∥∇t(p−˜p)∥2\nL2HN\n≤∥p−˜p∥2\nL∞HNT(1 +2\nκ∥q∥2\nL∞HN)\n+√\nT∥p−˜p∥L∞HN∥q−˜q∥L∞HN∥∇t˜p∥L2HN\n≤∥p−˜p∥2\nL∞HNT(1 +2\nκ∥q∥2\nL∞HN+ 4∥∇t˜p∥2\nL2HN)\n+T\n4∥q−˜q∥2\nL∞HN.MHD STABILITY THRESHOLD 31\nChoosingTsuch that\nT(1 +2\nκ∥q∥2\nHN+∥∇t˜p∥2\nL2HN)≤T+ 2T∥pin∥HN(1 +8\nκ)<1\n2,\nit follows, that\n∥p−˜p∥2\nX≤1\n2∥p−˜p∥HN+T\n4∥q−˜q∥2\nL∞HN.\nWe hence conclude, that\n∥p−˜p∥2\nX≤1\n2∥q−˜q∥2\nX.\n□\nData availability. No data was used for the research described in the article.\nAcknowledgements. Funded by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) – Project-ID 258734477 – SFB 1173. The author\ndeclares that they have no conflict of interest. This article is part of the PhD thesis\nof Niklas Knobel.\nReferences\n[BBZD23] Jacob Bedrossian, Roberta Bianchini, Michele Coti Zelati, and Michele Dolce. Nonlin-\near inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq\nequations. Communications on Pure and Applied Mathematics , 76(12):3685–3768,\n2023.\n[BGM17] Jacob Bedrossian, Pierre Germain, and Nader Masmoudi. On the stability threshold\nfor the 3D Couette flow in Sobolev regularity. Annals of Mathematics , pages 541–608,\n2017.\n[BM14] Jacob Bedrossian and Nader Masmoudi. Asymptotic stability for the Couette flow\nin the 2D Euler equations. 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On the Sobolev stability threshold of 3D Couette flow in a uniform magnetic\nfield.Communications in Mathematical Physics , pages 1–50, 2020.\n[LL11] Y. Charles Li and Zhiwu Lin. A resolution of the Sommerfeld paradox. SIAM Journal\non Mathematical Analysis , 43(4):1923–1954, 2011.\n[MZ22] Nader Masmoudi and Weiren Zhao. Stability threshold of two-dimensional Couette\nflow in Sobolev spaces. Ann. Inst. H. Poincaré C Anal. Non Linéaire , 39(2):245–325,\n2022.\n[MZZ23] Nader Masmoudi, Cuili Zhai, and Weiren Zhao. Asymptotic stability for two-\ndimensional Boussinesq systems around the Couette flow in a finite channel. Journal\nof Functional Analysis , 284(1):109736, 2023.\n[RWXZ14] Xiaoxia Ren, Jiahong Wu, Zhaoyin Xiang, and Zhifei Zhang. Global existence and\ndecay of smooth solution for the 2-D MHD equations without magnetic diffusion.\nJournal of Functional Analysis , 267(2):503–541, 2014.\n[Sch88] Paul Günter Schmidt. On a magnetohydrodynamic problem of Euler type. Journal of\ndifferential equations , 74(2):318–335, 1988.\n[WZ17] Dongyi Wei and Zhifei Zhang. Global well-posedness of the MHD equations in a\nhomogeneous magnetic field. Analysis & PDE , 10(6):1361–1406, 2017.\n[Zil21] Christian Zillinger. On the Boussinesq equations with non-monotone temperature\nprofiles.Journal of Nonlinear Science , 31(4):64, 2021.\n[ZZ23] Weiren Zhao and Ruizhao Zi. Asymptotic stability of Couette flow in a strong uniform\nmagnetic field for the Euler-MHD system. arXiv preprint arXiv:2305.04052 , 2023.\nKarlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany\nEmail address :niklas.knobel@kit.edu"    },    {        "title": "2401.12558v1.Unbounded_convex_polyhedra_as_polynomial_images_of_Euclidean_spaces.pdf",        "content": "arXiv:2401.12558v1  [math.AG]  23 Jan 2024UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL\nIMAGES OF EUCLIDEAN SPACES\nJOS´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nAbstract. In a previous work we proved that each n-dimensional convex polyhedron K⊂Rn\nanditsrelative interiorare regularimages of Rn. Astheimage ofanon-constantpolynomial map\nis an unbounded semialgebraic set, it is not possible to subs titute regular maps by polynomial\nmaps in the previous statement. In this work we determine con structively all unbounded n-\ndimensional convex polyhedra K⊂Rnthat are polynomial images of Rn. We also analyze\nfor which of them the interior Int( K) is a polynomial image of Rn. A discriminating object\nis the recession cone /vectorC(K) ofK. Namely, Kis a polynomial image of Rnif and only if /vectorC(K)\nhas dimension n. In addition, Int( K)is a polynomial image of Rnif and only if /vectorC(K)has\ndimension nandKhas no bounded faces of dimension n−1. A key result is an improvement\nof Pecker’s elimination of inequalities to represent semia lgebraic sets as projections of algebraic\nsets. Empirical approaches suggest us that there are ‘few’ p olynomial maps that have a concrete\nconvex polyhedron as a polynomial image and that there are ev en fewer for which it is affordable\nto show that their images actually correspond to our given co nvex polyhedron. This search of a\n‘needle in the haystack’ justifies somehow the technicaliti es involved in our constructive proofs.\n1.Introduction\nA mapf:= (f1,...,fm) :Rn→Rmispolynomial if its components fk∈R[x] :=R[x1,...,xn]\narepolynomials. Analogously, fisregularifits components can berepresented as quotients fk=\ngk\nhkof two polynomials gk,hk∈R[x] such that hknever vanishes on Rn. By Tarski-Seidenberg’s\nprinciple [BCR, 1.4] the image of an either polynomial or reg ular map is a semialgebraic set.\nA subset S⊂Rnissemialgebraic when it has a description by a finite boolean combination of\npolynomial equalities and inequalities.\nIt is quite natural to wonder about for properties that a set i nRmmust satisfy in order to\nbe the image of a polynomial map f:Rn→Rm. To our knowledge, this question was first\nposedby Gamboa in an Oberwolfach week [G]. A related problem concerns the parameterization\nof semialgebraic sets of dimension dusing continuous semialgebraic maps whose domains are\nsemialgebraic subsets of Rdsatisfying certain nice properties [GRS]. The approach pro posed\nby Gamboa in [G] sacrifices injectivity but chooses the simpl est possible domains (Euclidean\nspaces) and the simplest possible maps (polynomial and regu lar) to represent semialgebraic\nsets. The class of semialgebraic sets that can be represente d as polynomial and regular images\nof Euclidean spaces (even sacrificing injectivity) is surel y much smaller than the one consisting of\nthe images under injective continuous semialgebraic maps o f nice semialgebraic sets. Of course,\nmore general domains than the Euclidean spaces can be consid ered and compact semialgebraic\nsets deserve special attention: balls, spheres, compact co nvex polyhedra, ...For instance, in\n[KPS] the authors develop a computational study of images un derpolynomial maps φ:R3→R2\n(andthecorrespondingconvex hulls) of compact (principal )semialgebraic subsets {f≥0} ⊂R3,\nwheref∈R[x1,x2,x3] (this includes for example the case of a 3-dimensional ball ).\nThe effective representation of a subset S⊂Rmas a polynomial or regular image of Rn\nreduces the study of certain classical problems in Real Geom etry to its study in Rn. Examples\nDate: 17/03/2017.\n2010Mathematics Subject Classification. Primary: 14P10, 14P05; Secondary: 52B99.\nKey words and phrases. Semialgebraic sets, polynomial maps and images, convex pol yhedra, projections of\nalgebraic sets.\nFirst and second authors are supported by Spanish MTM2014-5 5565-P and Grupos UCM 910444 whereas the\nthird author is an external collaborator. Third author was s upported by ‘Scuola Galileo Galilei’ Research Grant\nat the Dipartimento di Matematica of the Universit` a di Pisa while writing a part of this article.\n12 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nof such problems appear in Optimization, with the advantage of avoiding contour conditions\nand reducing optimization problems to the case of Euclidean spaces (see for instance [NDS, PS,\nSch, VS] for relevant tools concerning optimization of poly nomial functions on Rn) or in the\nsearch for Positivstellens¨ atze certificates [S]. These re presentations provide Positivstellensatz\ncertificates for general semialgebraic sets, whenever we ar e able to represent them as regular or\npolynomial images of Rn. Recall that classical Positivstellensatz certificates ar e stated only for\nclosed basic semialgebraic sets. Further details are descr ibed carefully in [FGU1, FU2].\nIfSis a non-compact locally compact semialgebraic set in Rn, it admits a (semialgebraic)\nAlexandrov compactification by one point. In addition, ther e is a doubly exponential (in the\nnumbernof variables describing S) algorithm triangulating each compact semialgebraic set ( see\n[BCR, Ch.9, §2] and [HRR]). Thus, locally compact semialgebraic sets can be considered as\nfinite simplicial complexes (up to losing one vertex), but we remark that the known algorithm\ncan produce a doubly exponential number of simplexes. The al gorithms developed to show that\ncertain semialgebraic sets with piecewise linear boundary are polynomial or regular images of\nRnare constructive (including those provided in this article ), but the degrees of the involved\nmaps are very high; however, it would be interesting to estim ate the smallest degree for which\nthere is a suitable polynomial or regular map, and to compare its complexity with the doubly\nexponential one for the triangulations of semialgebraic se ts.\nSo far we have found partial answers to the representation pr oblem of semialgebraic sets as\npolynomial and regular images of Euclidean spaces [FG1, FG2 , Fe, FU1], but a full geometric\ncharacterization of these sets seems difficult to be obtained at present. On the other hand,\nwe have also focused on finding large families of semialgebra ic sets that can be expressed as\neither polynomial or regular images of Rn, giving constructive methods to obtain explicit maps\nproducing them [Fe, FG1, FGU1, FGU2, FU5, U]. In particular, we have focused our interest in\ndetermining whether convex polyhedra, their interiors and the corresponding complements can\nbe expressed as polynomial or regular images. We understand that these types of semialgebraic\nsets are the simplest among those with piecewise linear boun dary, and their full study is the first\nnatural step to understand which semialgebraic sets whose b oundaries have ‘nice properties’ are\neither polynomial or regular images of Rn.\nIn [FGU1] we proved that every n-dimensional convex polyhedron K⊂Rnand its interior are\nregular images of Rn. This result cannot be extended directly to the polynomial c ase because\nthe image of a non-constant polynomial map is an unbounded se mialgebraic set. Our purpose in\nthis work is to determine all n-dimensional convex polyhedra K⊂Rnsuch that Kand/or Int( K)\nare polynomial images of Rn. Here, Int( K) refers to the relative interior of Kwith respect to\nthe affine subspace of Rnspanned by K, which coincides with the interior of Kas a topological\nmanifold with boundary. For these unbounded convex polyhed ra, their representations as poly-\nnomial images of Euclidean spaces provide a priori simpler P ositivstellens¨ atze certificates and\noptimization approaches that if we use regular maps because polynomial representations do not\ninvolve denominators.\nIn [FG1, FG2, FU1] we found obstructions for a semialgebraic set ofRnto be a polynomial\nimage of some Rn. Two distinguished ones that are relevant to us here are the f ollowing:\nCondition 1: The projections of a polynomial image of a Euclidean space are either singletons\nor unbounded semialgebraic sets .\nCondition 2: If a semialgebraic set S⊂Rnis a polynomial image of RnandZis an irreducible\ncomponent of dimension n−1of the Zariski closure of Cl(S)\\S, thenZ∩Cl(S)is unbounded\n[FG2, Cor. 3.4].\nLet us translate the first condition for convex polyhedra in t erms of the recession cone. Given\na pointpin a convex polyhedron K⊂Rn, the set of vectors /vector v∈Rnsuch that the ray with\noriginpand direction /vector vis contained in Kis called the recession cone ofK(see [Z, Ch.1] and\n[R, II. §8]). This set does not depend on the chosen point p. We will see later in Proposition 2.1\nthat if the dimension of the recession cone /vectorC(K) of a convex polyhedron Kis strictly smallerUNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 3\nthan its dimension, then Khas bounded, non-singleton projections and neither Knor Int(K)\nare polynomial images of an Euclidean space.\nOn the other hand, translating the second condition to our po lyhedral setting turns into the\nfact that if K⊂Rnis ann-dimensional convex polyhedron with a bounded face of dimen sion\nn−1, then Int( K) is not a polynomial image of Rn.\nTaking the previous obstructions in mind our main results in this work, which are the best\npossible ones, are the following:\nTheorem 1.1. LetK⊂Rnbe ann-dimensional convex polyhedron whose recession cone is\nn-dimensional. Then Kis a polynomial image of RnandInt(K)is a polynomial image of Rn+1.\nTheorem 1.2. LetK⊂Rnbe ann-dimensional convex polyhedron without bounded facets and\nwhose recession cone is n-dimensional. Then Int(K)is a polynomial image of Rn.\nThis means that for convex polyhedra, their interiors and th e corresponding complements the\nknown obstructions for the representability of general sem ialgebraic sets as polynomial images\nof Euclidean spaces are enough.\nThe proofs of Theorems 1.1 and 1.2 are rather technical and pa rtly rely on ad hocconstructive\narguments. With respect to the constructions we use to prove both results, it is difficult to\ndetermine how far from being ’optimal’ they are. Even in the s implest non-trivial case of the\nopen quadrant Q:={x>0,y>0}ofR2, we have made several trials [FG1, FGU2, FU5] to\nfind the ‘best’ possible representation of Qas a polynomial image of R2. The criteria to measure\nthe ‘goodness’ of a representation are debatable, and we our selves have been oscillating between\nthe simplicity of the involved polynomial maps and the clear ness of the proofs provided (an\nideal situation would be to find examples where these two prop erties come together). A main\ndifficulty, which permeates this work, is that our proofs are o f constructive nature because we\nlack general principles that could provide a simpler and mor e direct existential approach to\ntackle the problems related to the representation of semial gebraic sets as polynomial images of\nEuclidean spaces. We point out here some obstacles that quic kly arise when confronting them:\n•Therigidity ofpolynomial mapshinderstheir manipulation inordertoobtain thedesired\nimage sets.\n•It is difficult to compute the image of an arbitrary polynomial map and, as far as we\nknow, there are not feasible algorithms to achieve this.\n•The family of polynomial images do not behave nicely with res pect to the usual set-\ntheoretic operations or geometric constructions.\nWe enlighten the latter fact with some examples.\nExample 1.3 (Convex hull of a polynomial image of Rn). The convex hull of a polynomial image\nofRnneeds not be a polynomial image of Rn. The semialgebraic set\nS:={y>(x+1)2(x−1)2} ⊂R2\nis a polynomial image of R2. Indeed, the upper half-plane H:={y>0} ⊂R2is a polynomial\nimage of R2by [FG1, Ex. 1.4, (iv)] whereas Sis the image of Hvia the polynomial map\nH→S,(x,y)/ma√s⊔o→(x,y+(x+1)2(x−1)2).\nThe convex hull of Sis the semialgebraic set\nC:={y>(x+1)2(x−1)2}∪{y>0,−1<x<1},\nwhich is not a polynomial image of R2by [FG2, Thm. 3.8].\nExample 1.4 (Minkowski sum of polynomial images of Rn). The Minkowski sum S+Tof two\npolynomial images SandTofRnneeds not to be a polynomial image of Rn. We take the\nsemialgebraic subsets\nS:={x≥0,y≥0,x+y≥1}andT:={x>0,y>0}4 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nofR2, both of which are polynomial images of R2by [FG2, Thm. 5.1] and [FG1, Thm. 1.7].\nTheir Minkowski sum is\nS+T:={x>0,y>0,x+y>1},\nwhich is not a polynomial image of R2by [FG2, Cor. 3.4].\nExample 1.5 (Connected intersection of polynomial images of Rn). If the intersection S∩Tof\ntwo polynomial images SandTofRnis connected, then S∩Tis not in general a polynomial\nimage of Rn. The semialgebraic subsets S:={x≤1}andT:={x≥ −1}ofR2are polynomial\nimages of R2whereas their intersection S∩T, which is connected, is not a polynomial image\nofR2because it does not satisfy Condition 1 above. Observe that S,TandS∩Tare convex\nsemialgebraic sets.\nExample 1.6 (Connected union of polynomial images of Rn). If the union S∪Tof two polynomial\nimagesSandTofRnis connected, then S∪Tis not in general a polynomial image of Rn. The\nsemialgebraic subsets S:={x≥0,y≥x2}andT:={y≥0,x≥y2}ofR2are polynomial\nimages of R2whereas their union S∪T, which is connected, is not a polynomial image of R2by\n[FU1, Thm. 1.1]. In fact, SandTare convex sets, but their union S∪Tis not.\nWe suspect that with thecurrent knowledgeit is difficult(or e ven plausiblyimpossible)to find\ntwo convex semialgebraic sets which are polynomial images o fRnwhose union is convex but not\na polynomial image of Rn. The reason is the following: if two convex semialgebraic sets Sand\nTsatisfy all known obstructions to be polynomial images of Rnand their union S∪Tis convex,\nthen such union also satisfies all those known obstructions . So we have no known ‘a priori’ tools\nto find such an example. In this regard, it would be relevant to determine whether the union of\ntwo convex polynomial images of Rnis also a polynomial image of Rnwhenever such union is\na convex set . A result of this nature will definitely help to determine all convex semialgebraic\nsets that are polynomial images of Rn. However, at present we feel far from achieving this goal.\nIf we restrict our attention to the family Fofn-dimensional closed convex semialgebraic\nsubsets of Rnwith piecewise linear boundary that are polynomial images o fRn, thenSandT\nare by Theorem 1.1 n-dimensional convex polyhedra whose recession cone has dim ensionn. If\nthe union S∪Tis convex, then S∪Tis again an n-dimensional convex polyhedron with recession\ncone of dimension n, so it is a polynomial image of Rnby Theorem 1.1. Analogously, if we\nare interested in the family Gofn-dimensional open convex semialgebraic subsets of Rnwith\npiecewise linear boundary that are polynomial images of Rn, thenSandTare, by Theorem 1.2,\nn-dimensionalconvex polyhedrawithoutboundedfacets andw hoserecession conehas dimension\nn. If the union S∪Tis convex, then S∪Tis again an n-dimensional convex polyhedron without\nbounded facets and whose recession cone has dimension n. By Theorem 1.2 this union is a\npolynomial image of Rn.\nIn both cases above the result arises ‘a posteriori’ because the union, if convex, of sets of\neither the family ForGis again a set of the family ForG. We guess it is really difficult to\ndevelop a general strategy to prove ‘a priori’ (without know ing the characterizations provided by\nTheorems 1.1 and 1.2) that the union, if convex, of two convex semialgebraic sets with piecewise\nlinear boundary that are polynomial images of Rnis again a polynomial image of Rn.\nIn order to circumvent these difficulties we have developed al ternative strategies that rely\non some constructions introduced in Pecker’s work [P]. The T arski–Seidenberg principle on\nelimination of quantifiers can be also restated geometrical ly by saying that the projection of a\nsemialgebraic set is again semialgebraic. An alternative c onverse problem, to find an algebraic\nset inRn+kwhose projection is a given semialgebraic subset of Rn, is known as the problem\nof eliminating inequalities . Motzkin proved in [Mo] that this problem always has a soluti on\nfork= 1. However, his solution is rather complicated and is gener ally a reducible algebraic\nset. In another direction Andradas–Gamboa proved in [AG1, A G2] that if S⊂Rnis a closed\nsemialgebraic set whose Zariski closure is irreducible, th enSis the projection of an irreducible\nalgebraic set in some Rn+k. In [P] Pecker gives some improvements on both results: for t he\nfirst one by finding a construction of an algebraic set in Rn+1that projects onto the given\nsemialgebraic subset of Rn, far simpler than the original construction of Motzkin; for the secondUNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 5\none by characterizing the semialgebraic sets in Rnwhich are projections of a real variety in\nRn+1. In Section 3 we modify Pecker’s polynomials introduced in [ P,§2] to take advantage of\nthem in order to prove both Theorems 1.1 and 1.2.\nTo ease the presentation of the full picture of what is known [ FGU1, FU1, FU2, FU3, FU4, U]\nabout the representation of semialgebraic sets with piecew ise linear boundary as either polyno-\nmial or regular images of some euclidean space Rmwe introduce the following two invariants.\nGiven a semialgebraic set S⊂Rm, we define\np(S) : = inf{n≥1 :∃f:Rn→Rmpolynomial such that f(Rn) =S},\nr(S) : = inf{n≥1 :∃f:Rn→Rmregular such that f(Rn) =S}.\nThe condition p( S) := +∞expresses the non-representability of Sas a polynomial image of\nsomeRnwhereas r( S) := +∞has the analogous meaning for regular maps. The values of the se\ninvariants for the families of convex polyhedra and their co mplements are shown in Table 1.\nHere,K⊂Rnrepresents an n-dimensional convex polyhedron and its complement S:=Rn\\K\nis assumed to be connected. In addition, we write S:=Rn\\Int(K).\nKbounded Kunbounded\nn= 1n≥2n= 1n≥2\nr(K)1n1n\nr(Int(K))2 2\np(K)+∞1n,+∞(∗)\np(Int(K)) 2n,n+1,+∞(⋆)\nr(S)\n+∞n2\nnr(S) 1\np(S) 2\np(S) 1\nTable 1. Full picture\nLet us explain the (marked) cases in Table 1 which follow from this work:\n(∗) (n,+∞): Ann-dimensional convex polyhedron K⊂Rnhas p(K) =nif and only if its\nrecession cone /vectorC(K) has dimension n. Otherwise, p( K) = +∞.\n(⋆) (n,n+1,+∞): If the recession cone /vectorC(K) of ann-dimensional convex polyhedron Khas\ndimension<n, then p(Int( K)) = +∞. Otherwise, if Khas bounded facets, p(Int( K)) =\nn+1 and if Khas no bounded facets, p(Int( K)) =n.\nStructure of the article. The article is organized as follows. In Section 2 we introduc e some\nbasic notions, notations and tools that will be employed alo ng the article. In Section 3 we\nanalyze further properties of Pecker’s polynomials and we i ntroduce some variations that fit\nthe situation we need. In Section 4 we prove Theorem 1.1 where as Theorem 1.2 is proved in\nSection 5. We end this article with an appendix that collects some useful inequalities for positive\nreal numbers.\n2.Preliminaries and basic tools\nWe proceed first to establish some basic concepts, notations and results. This section can be\nconsidered as a sort of toolkit, where diverse techniques an d auxiliary tools that will be needed\nlater are introduced.6 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\n2.a.Basic notation. Points in the Euclidean space Rnare denoted with the letters x,y,z,p,\nq, ...and vectors by /vector v,/vector w, ...Given two points p,q∈Rn,− →pqrepresents the vector from ptoq\nandpqthe segment joining them. Given an affine subspace W⊂Rn, we use an overlying arrow\n/vectorWto refer to the corresponding linear subspace. This notatio n is extended in the following way:\nGiven a finite union of affine subspaces X:=X1∪···∪Xr, we will denote /vectorXthe union of the\nlinear subspaces /vectorXi, so that\n/vectorX:=/vectorX1∪···∪/vectorXr.\nThe vectors of the standard basis of Rnare denoted /vectorei= (0,...,0,(i)\n1,0,...,0) fori= 1,...,n.\nAnaffine hyperplane ofRnwill usually be written as H:={h= 0}using a non-zero linear\nequationh. It determines two closed half-spaces\nH+:={h≥0}andH−:={h≤0}.\nIn fact, these half-spaces depend on the linear equation hchosen to define H. Whenever needed,\nwe will clearly state the orientation that is being consider ed.\nAn affine subspace WofRnis called verticalif/vectorWcontains the vector /vectoren. Otherwise, we say\nthatWisnon-vertical . In general, whenever an affine object or map is denoted with a s ymbol,\nwe will use an overlying arrow on it to refer to its linear coun terpart.\nGiven a set X⊂Rnand a set of vectors /vectorV⊂Rn, we define\nX+/vectorV:={x+/vector v:x∈X,/vector v∈/vectorV} ⊂Rn.\nWheneverXand/vectorVare convex sets, the set X+/vectorVis also convex. Given a set X⊂Rnand a\nvector/vector v∈Rn, thecylinder of base Xin the direction /vector vis defined as\nX/vector v:={x+λ/vector v:x∈X,λ∈R},\nand thepositive cylinder of base Xin the direction /vector vas\nX/vector v+:={x+λ/vector v:x∈X,λ≥0}.\nWe will use analogous notations /vectorX/vector vand/vectorX/vector v+when/vectorXis a set of vectors instead of a subset\nofRn. As special cases, the line through the point pwith direction /vector vis written as p/vector v, whereas\nthe ray with origin at pand direction /vector vis written as p/vector v+. GivenX1,...,X m⊂Rn, we denote\nSpan(X1,...,X m) the affine span of their union/uniontextm\ni=1Xi.\n2.b.Convex polyhedra and recession cone. A subset K⊂Rnis aconvex polyhedron if it\ncan be described as a finite intersection of closed half-spac es. The dimension dim( K) ofKis\nthe dimension of the smallest affine subspace of Rnthat contains Kand Int(K) represents the\nrelative interior of Kwith respect to this subspace. If Khas non-empty interior there exists\na unique minimal family {H1,...,H m}of affine hyperplanes such that K=/intersectiontextm\ni=1H+\ni. The\nfacetsor (n−1)-facesofKare the intersections Fi:=Hi∩Kfor 1≤i≤m. Each facet\nFi:=Hi∩/intersectiontextm\nj=1H+\njis a convex polyhedron contained in Hi. For 0 ≤j≤n−2 we define\ninductively the j-facesofKas the facets of the ( j+ 1)-faces of K, which are again convex\npolyhedra. The 0-faces are the verticesofKand the 1-faces are the edgesofK. A face Eof\nKisverticalif the affine subspace of Rnspanned by Eis vertical. Otherwise, we say that Eis\nnon-vertical . A convex polyhedron is non-degenerate if it has at least one vertex. Otherwise,\nit is called degenerate . For a detailed study of the main properties of convex sets we refer the\nreader to [Be, R, Z].\nWe associate to each convex polyhedron K⊂Rnitsrecession cone , see [Z, Ch.1] and [R,\nII.§8]. Fix a point p∈Kand denote /vectorC(K) :={/vector v∈Rn:p/vector v+⊂K}. Then/vectorC(K)is a convex\ncone and it does not depend on the choice of p. The set/vectorC(K) is called the recession cone ofK.\nIt holds/vectorC(K) ={/vector0}if and only if Kis bounded. The recession cone of K:=/intersectiontextr\ni=1H+\niis\n/vectorC(K) =r/intersectiondisplay\ni=1/vectorC(H+\ni) =r/intersectiondisplay\ni=1/vectorHi+=/braceleftBigs/summationdisplay\ni=1λi/vector vi:λi≥0/bracerightBigUNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 7\nwhere the non-zero vectors /vector v1,...,/vector vsspan the lines containing the unbounded edges of K. If a\nnon-zero vector /vector v∈Int(/vectorC(K)), thenKdoes not have facets parallel to /vector v.\nIfKis non-degenerate we may write K=K0+/vectorC(K) where K0is the convex hull of the\nset of vertices of K. IfP⊂Rnis a non-degenerate convex polyhedron and k≥1, then\n/vectorC(Rk×P) =Rk×/vectorC(P).\nRecall that each degenerate convex polyhedron can be writte n as the product of a non-\ndegenerate convex polyhedron times an Euclidean space. Bes ides, a convex polyhedron is de-\ngenerate if and only if it contains a line or, equivalently, i f its recession cone contains a line.\nConsequently a convex polyhedron Kis non-degenerate if and only if all its faces are non-\ndegenerate polyhedra.\nThe next result justifies the fact that the recession cone of a polyhedron plays an important\nrole when we are trying to express it as a polynomial image of Rn:\nProposition 2.1. If the dimension of the recession cone /vectorC(K)of ann-dimensional convex\npolyhedron K⊂Rnis strictly smaller than n, then both KandInt(K)have bounded non-\nsingleton projections. Consequently, under the previous hy potheses both KandInt(K)are not\npolynomial images of Rmfor eachm≥1.\nProof.We may assume /vectorC(K) is contained in the hyperplane {xn= 0}. Consider the projection\nη:Rn→R, x:= (x1,...,x n)/ma√s⊔o→xn. Suppose first that Kis non-degenerate. As dim( K) =n,\nwe can choose a set of points W:={p1,...,pk} ⊂Kthat contains all the vertices of Kand\nspans the whole space Rn. ThenK=K′\n0+/vectorC(K) where K′\n0is the convex hull of W. AsK′\n0\nis a compact polyhedron and has dimension n, the projection η(K′\n0) is a non-trivial bounded\ninterval. We have η(K) =η(K′\n0) +/vector η(/vectorC(K)) =η(K′\n0) because/vector η(/vectorC(K)) ={/vector0}. Consequently,\nbothη(K) andη(Int(K)) are bounded non-trivial intervals.\nAssume next that Kis degenerate and suppose K=Rk×K′where 1 ≤k < nandK′⊂\nRn−kis a non-degenerate convex polyhedron of Rn−k. Choose the notation ( xk+1,...,xn) for\nthe coordinates of Rn−k. As/vectorC(K) =Rk×/vectorC(K′), we may assume /vectorC(K′)⊂ {xn= 0}. Let\nτ:Rn→Rn−kdenote the projection onto the last n−kcoordinates and let ¯ η:Rn−k→R\ndenote the projection onto the last coordinate, so that η= ¯η◦τ. We have\nη(K) =η(Rk×K′) = (¯η◦τ)(Rk×K′) = ¯η(K′).\nBy the non-degenerate case ¯ η(K′) and ¯η(Int(K′)) are bounded intervals, as required. /square\nOther results that follow from the use of the recession cone a re the following.\nLemma 2.2. LetK⊂Rnbe a convex polyhedron and let H:={h= 0}be a hyperplane of Rn\nsuch that K⊂ {h>0}. Thendist(K,H) = dist(p0,H)for each point p0contained in one of the\nfaces ofKof minimal dimension and in addition K⊂ {h>h(p0)\n2}.\nProof.Assume first that Kis a non-degenerate convex polyhedron and write K=K0+/vectorC(K)\nwhereK0is the convex hull of the set Vof vertices of K. AsK⊂ {h >0}, thenµ:=\nmin{h(p) :p∈V}>0 and/vectorh(/vector v)≥0 for all/vector v∈/vectorC(K). Observe that h(q)≥µfor allq∈K\nand dist( K,H) = dist(p0,H) wherep0∈Vis a vertex such that h(p0) =µ. In addition,\nK⊂ {h≥h(p0)} ⊂ {h>h(p0)\n2}. As the convex polyhedron Kis non-degenerate, {p0}is a face\nofKof minimal dimension.\nIfKis degenerate, we assume K=K′×RkwhereK′⊂Rn−kis a non-degenerate polyhedron.\nAsK∩H=∅, we haveH=H′×RkwhereH′:={h= 0}is a hyperplane of Rn−kand\nn−k≥1. We abuse notation using the fact that the linear form honly depends on the first\nn−kvariables. Applying the non-degenerate case to K′,H′andhwe find a vertex q0ofK′such\nthat dist( K′,H′) = dist(q0,H′). Observe that E:={q0}×Rkis a face of Kof minimal dimension\nandh(p) =h(q0,0) =h(q0) for eachp∈E. The statement now follows straightforwardly. /square8 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nCorollary 2.3. LetK⊂Rnbe a convex polyhedron and let H1:={h1= 0}andH2:={h2= 0}\nbe hyperplanes of Rn. Suppose that K∩H1⊂ {h2>0}. Then there exists ε >0such that\nK∩{−ε≤h1≤ε} ⊂ {h2>0}.\nProof.DefineP:=K∩{h2≤0}. AsK∩H1∩{h2≤0}=∅, we may assume P⊂ {h1>0}. By\nLemma 2.2 there exists ε >0 such that P⊂ {h1>ε}. Thus,K∩{−ε≤h1≤ε} ⊂ {h2>0},\nas required. /square\n2.c.Vertical cones and convex polyhedra. Along the article we will make frequent use\nof one particular direction in Rn, the one given by the vector /vectoren= (0,...,0,1). Setx′:=\n(x1,...,x n−1)∈Rn−1so that a point in Rn≡Rn−1×Ris written as x:= (x′,xn). Thevertical\ncone of radius δ >0 is defined as\n/vectorCv\nδ:={(v′,vn)∈Rn:/bardblv′/bardbl ≤δvn}.\nGiven a set A⊂Rnwe define the vertical cone of radius δ>0overAas\nCv\nδ(A) :=A+/vectorCv\nδ={x+/vector v:x∈A, /vector v∈/vectorCv\nδ}.\nIfAis a convex set, then Cv\nδ(A) is also a convex set.\nWe establish now some results relating vertical cones and un bounded polyhedra.\nLemma 2.4. LetK⊂Rnbe a convex polyhedron such that /vectoren∈Int(/vectorC(K)). Then there exists\nδ>0such that for each p∈Rnthe inclusion Cv\nδ({p})\\{p} ⊂ {p}+Int(/vectorC(K))holds.\nProof.As/vectoren:= (0,...,0,1)∈Int(/vectorC(K)), there exists δ>0 such that the ball B(/vectoren,δ) of center\n/vectorenand radius δ>0 is contained in Int( /vectorC(K)). As/vectorC(K) is a cone with vertex 0,\n/vectorCv\nδ\\{0} ⊂ {λ/vector v:/vector v∈B(/vectoren,δ), λ>0} ⊂Int(/vectorC(K)).\nFrom this inclusion readily follows that Cv\nδ({p})\\{p} ⊂ {p}+Int(/vectorC(K)) for each p∈K./square\nProposition 2.5. LetK⊂Rnbe a non-degenerate unbounded convex polyhedron. Assume\nK⊂ {xn≥0}, the intersection E:={xn= 0}∩Kis a face of Kand the vector /vectoren∈Int(/vectorC(K)).\nThen there exist positive numbers δ<∆such that Cv\nδ(E)⊂K⊂Cv\n∆(E).\nProof.By Lemma 2.4 we can choose δ >0 such that Cv\nδ({p})⊂ {p}+/vectorC(K) for eachp∈E, so\nthat the inclusions Cv\nδ(E)⊂E+/vectorC(K)⊂Khold.\nWe prove next K⊂Cv\n∆(E) for ∆ large enough. We may assume that 0 ∈Int(E). Observe first\nthat for each p∈Ewe have {p}∪{xn>0}=/uniontext\nk∈NCv\nk({p}), so\nE∪{xn>0}=/uniondisplay\nk∈NCv\nk(E).\nWrite/vectorC(K) ={/summationtexts\ni=1λi/vector vi:λi≥0}where the non-zero vectors /vector v1,...,/vector vsspan the lines spanned\nby the unbounded edges of K. We may assume that the last coordinate of /vector viis positive for\ni= 1,...,rand identically zero for /vector viwithi=r+1,...,s. Consequently, /vectorC(E) ={/summationtexts\ni=r+1λi/vector vi:\nλi≥0}. Pickk0≥δsuch that:\n(1) All the vertices of Kare contained in Cv\nk0(E).\n(2) The rays 0 /vector v+\ni⊂Cv\nk0(E) fori= 1,...,s.\nAsCv\nk0(E) is convex (because Eis convex) and {0}+/vectorC(K) is the convex hull of the rays 0 /vector v+\nifor\ni= 1,...,s, we deduce that both sets {0}+/vectorC(K) and the convex hull K0of the vertices of K\nare contained in Cv\nk0(E). Consequently,\nK=K0+/vectorC(K)⊂Cv\nk0(E)\nand taking ∆ := k0we have K⊂Cv\n∆(E), as required. /squareUNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 9\n2.d.Projections of affine subspaces and convex polyhedra. Given a hyperplane H⊂Rn\nand a vector /vector v∈Rn\\/vectorH, we denote by π/vector v:Rn→Hthe projection onto Hwith direction /vector v.\nFor eachX⊂Rn, the setπ−1\n/vector v(π/vector v(X)) coincides with X/vector v, so it does not depend on the chosen\nprojection hyperplane Hbut only on the vector /vector v. Writex′:= (x1,...,x n−1) andx:= (x′,xn).\nWe use often the vertical projection π/vectoren:Rn→Rn,(x′,xn)/ma√s⊔o→(x′,0) onto the coordinate\nhyperplane {xn= 0}and we reserve the notation πnfor this particular projection.\nProposition 2.6. LetK⊂ {xn≥0} ⊂Rnbe an unbounded convex polyhedron whose recession\ncone/vectorC(K)has dimension nand assume /vectoren∈Int(/vectorC(K)). Then the restriction ρ:=πn|∂K:\n∂K→Rn−1×{0}defines a semialgebraic homeomorphism.\nProof.We prove first: ρis surjective .\nPick a point x:= (x′,0)∈Rn−1× {0}and consider the ray x/vectore+\nn. Choose now y∈K. As\n/vectoren∈Int(/vectorC(K)) and/vectorC(K) has dimension n, there exists ε>0 such that /vector w:=/vectoren+ε− →yx∈/vectorC(K).\nThe rayy/vector w+⊂Kand\nz:=y+1\nε/vector w=x+1\nε/vectoren∈y/vector w+∩x/vectore+\nn⊂K∩x/vectore+\nn.\nConsequently, z/vectore+\nn⊂K∩x/vectore+\nn⊂ {xn≥0}, so there exists a point p∈∂K∩x/vectore+\nn, which satisfies\nπn(p) =x. In addition, x/vectore+\nn∩K=p/vectore+\nn.\nWe show next: ρis injective . It is enough to show: for eachx:= (x′,0)∈Rnthe intersection\nx/vectoren∩∂Kis a singleton .\nWe have already proved that x/vectoren∩K=p/vectore+\nnfor somep∈∂K. If the ray p/vectore+\nnmeets∂Kin\na pointy/\\e}a⊔io\\slash=p, then either K∩p/vectore+\nnis a bounded interval or K∩p/vectore+\nn⊂∂K. As both situations\nare impossible because /vectoren∈Int(/vectorC(K)), we conclude Int( p/vectore+\nn)⊂Int(K), soρ−1(ρ(p)) ={p}.\nTo prove that ρis a homeomorphism, it is enough to check that it is a closed ma p andin fact it\nissufficient thattherestriction ρ|Fisaclosed mapforeach facet FofK. LetHbethehyperplane\nspanned by Fand let us check that πn|His a closed map. As /vectoren∈Int(/vectorC(K))⊂Rn\\/vectorH, the\nrestrictionπn|His an affine bijection and in particular a closed map, as requir ed. /square\nLet us consider now a set X⊂Rnand a projection π/vector v:Rn→H. The setπ−1\n/vector v(π/vector v(X)) =X/vector v\ncontainsX. If we consider now ��nitely many vectors /vector v1,...,/vector vs, the setX′:=/intersectiontexts\ni=1X/vector vialso\ncontainsX. It seems natural to wonder under which conditions can we ass ert thatX′=X.\nWhenXis a finite union of affine subspaces of dimension ≤n−2 we have the following result.\nProposition 2.7. LetX:=/uniontextm\ni=1Xi⊂Rnbe a finite union of affine subspaces Xisuch that\nd:= dim(X)≤n−2andXi/notsubseteqlXjifi/\\e}a⊔io\\slash=j. LetΩbe a non-empty open subset of Rn\\ {0}.\nThen there exist finitely many vectors /vector v1,...,/vector vs∈Ωsuch that/intersectiontexts\ni=1X/vector vi=X. Besides, we can\nchoose these vectors so that /vector vi/∈/uniontexti−1\nj=1/vectorX/vector vjfori= 1,...,s.\nProof.AsX=/uniontextm\ni=1Xiand eachXiis an affine subspace of RnwithXi/notsubseteqlXjifi/\\e}a⊔io\\slash=j, the\naffine subspaces X1,...,X mare the irreducible components of Xas an algebraic subset of Rn.\nGiven/vector v∈Rn\\ {/vector0}, the setX/vector vis also a finite union of affine subspaces of Rn. For each\nirreducible component XiofXthe setXi/vector vis an affine subspace that either coincides with Xi\nor has dimension dim( Xi) +1 and contains Xi. Ifp∈Xi/vector v\\Xi, thenXi/vector v= Span(p,Xi). Set\n/vectorX:=/uniontextm\ni=1/vectorXi.\nForp∈Rn\\Xdefine[p,X] :=/uniontextm\ni=1Span(p,Xi). Theset− −− →[p,X] denotes the unionof the linear\nsubspaces− −−−−−−− →Span(p,Xi) associated to the affine subspaces Span( p,Xi). We have dim([ p,X]) =\ndim(− −− →[p,X])≤d+1 andp /∈X/vector vfor each vector /vector v∈Rn\\− −− →[p,X].\nPick/vector v1∈Ω and letY1\n1,...,Ys\n1be the irreducible components of Y1:=X/vector v1. If eachYi\n1⊂X,\nwe are done. Otherwise, assume Y1\n1,...,Yr\n1are the irreducible components of Y1not contained\ninXand pickpi∈Yi\n1\\Xfori= 1,...,r. AsT1:=/uniontextr\ni=1[pi,X] is a finite union of affine\nsubspaces of Rnwhose dimensions are strictly smaller than n, there exists /vector v2∈Ω\\(/vectorT1∪/vectorY1). We10 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nhavepi/∈Y2:=X/vector v2fori= 1,...,r. LetZbe an irreducible component of Y1∩Y2that is not\ncontained in X. AsZ⊂Y1, there exists an irreducible component Yi\n1ofY1not contained in X\nsuch thatZ⊂Yi\n1. In addition, Z/subsetnoteqlYi\n1becausepi∈Yi\n1\\Z. Consequently, dim( Z)<dim(Yi\n1)\nbecauseZandYi\n1are affine subspaces. Thus, the dimension of every irreducibl e component of\nY1∩Y2not contained in Xis strictly smaller than the dimension of some irreducible c omponent\nofY1that is not contained in X. We conclude dim(( Y1∩Y2)\\X)<dim(Y1\\X).\nNext, for each irreducible component Yj\n12ofY1∩Y2not contained in X(and indexed with\nj= 1,...,ℓ) we choose a point qj∈Yj\n12\\Xand consider the set T2:=/uniontextℓ\nj=1[qj,X]. There\nexists/vector v3∈Ω\\(/vectorT2∪/vectorY1∪/vectorY2) and we have qj/∈Y3:=X/vector v3forj= 1,...,ℓ. The dimension\nof each irreducible component of Y1∩Y2∩Y3not contained in Xis strictly smaller than the\ndimension of some irreducible component of Y1∩Y2that is not contained in X. Again, this\nimplies dim(( Y1∩Y2∩Y3)\\X)<dim((Y1∩Y2)\\X).\nWe repeat the process s≤d+3≤n+1 times to find /vector v1,...,/vector vs∈Ω such that\n/vector vi/∈/vectorY1∪···∪/vectorYi−1=i−1/uniondisplay\nj=1/vectorX/vector vj\nfori= 1,...,sand all irreducible components of/intersectiontexts\ni=1Yi:=/intersectiontexts\ni=1X/vector viare contained in X. This\nholds because in each step dim((/intersectiontextk\ni=1Yi)\\X)<dim((/intersectiontextk−1\ni=1Yi)\\X) fork≥2. Consequently,/intersectiontexts\ni=1X/vector vi=X, as required. /square\n2.e.Separating hyperplanes in convex polyhedra. Given two semialgebraic sets S1,S2⊂\nRn, we say that a hyperplane B:={b= 0} ⊂Rnseparates S1andS2if the semialgebraic sets\nSilie in the different half-spaces {b≥0}and{b≤0}determined by BandB∩Si⊂S1∩S2for\ni= 1,2. Consequently, S1∩S2⊂BandB∩Si=S1∩S2fori= 1,2.\nWe are concerned here about hyperplanes that separate two ad jacent facets of a convex\npolyhedron.\nLemma 2.8. LetF1andF2be two non-parallel facets of a convex polyhedron K⊂Rn. Let\nHi:={hi= 0}be the hyperplane spanned by Fiand assume K⊂ {h1≥0,h2≥0}. For each\nλ>0denoteBλ:={bλ:=h1−λh2= 0}. ThenBλseparates F1andF2and meets Int(K).\nProof.Observe that F1⊂ {bλ≤0},F2⊂ {bλ≥0}andBλ∩Fi={h1= 0,h2= 0}∩K=F1∩F2\nfori= 1,2, soBλseparates F1andF2. Let us check: Bλ∩Int(K)/\\e}a⊔io\\slash=∅.\nPickxi∈Int(Fi). As Int(x1x2)⊂Int(K), it is enough to check: Bλ∩Int(x1x2)/\\e}a⊔io\\slash=∅.\nSet/vector v=− − →x1x2and write each point z∈Int(x1x2) as\nz=zµ:=x1+µ/vector v=x2−(1−µ)/vector v∈Int(x1x2).\nfor some 0<µ<1. Observe that h1(x1) = 0,h2(x2) = 0,/vectorh1(/vector v)>0 and/vectorh2(/vector v)<0. All reduces\nto find a value 0 <µ<1 such that zµ∈Bλ. To that end,\n0 =bλ(zµ) =h1(x1+µ/vector v)−λh2(x2−(1−µ)/vector v) =µ/vectorh1(/vector v)+λ(1−µ)/vectorh2(/vector v)\n❀µ:=−λ/vectorh2(/vector v)\n/vectorh1(/vector v)−λ/vectorh2(/vector v).\nAs 0<µ<1, we have zµ∈Bλ∩Int(x1x2), as required. /square\nWe have denoted πn:Rn→Rn, x:= (x1,...,x n)→(x1,...,x n−1,0) the orthogonal pro-\njection onto the hyperplane {xn= 0}. Let us assume that a convex polyhedron Kis placed so\nthat one of its facets Fis vertical and /vectoren∈/vectorC(K). The following result relates the projection of\nInt(K) underπnwith the union of the projections under πnof the intersections of Int( K) with\na family of separating hyperplanes between Fand its adjacent facets.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 11\nπn(B2∩Int(K))\nπn(B1∩Int(K))π3K\nF1F\nF2B1B2\nFigure 1. πn(Int(K)) =πn(B1∩Int(K))∪πn(B2∩Int(K))\nLemma 2.9. LetK⊂Rnbe an unbounded convex polyhedron and Fone of its facets. Assume\nthatFlies in the hyperplane {xn−1= 0}and the vector /vectoren∈/vectorC(K). LetF1,...,Frbe the\nnon-vertical facets of Kand assume that all of them meet F. LetBibe a hyperplane of Rnthat\nseparates FandFiand meets Int(K). Thenπn(Int(K)) =/uniontextr\ni=1πn(Bi∩Int(K)). Consequently,\nInt(K)/vectoren=/uniontextr\ni=1(Bi∩Int(K))/vectoren.\nProof.We prove first:\nπn(Fi)\\∂πn(K)⊂πn(Bi∩Int(K)) (2.1)\nfori= 1,...,r.\nTakex∈πn(Fi)\\∂πn(K). AsFiis non-vertical, x/vectoren∩K=p/vectore+\nnfor somep∈Fi. We claim:\nx/vectoren∩∂K={p}.\nOtherwise, p/vectore+\nn⊂∂Kand{x}=πn(x/vectoren)⊂∂πn(K). The latter inclusion follows because all\nthe facets that contain p/vectore+\nnare vertical, so their projections are contained in ∂πn(K), which is\na contradiction.\nLet us check: Biis non-vertical .\nOtherwise, pick q∈F∩Fi⊂Bi. AsF⊂ {xn−1= 0}, the rayq/vectore+\nn⊂Bi∩F. AsBiseparates\nFandFi, we haveq/vectore+\nn⊂Bi∩F⊂F∩Fi, soFishould be vertical, which is a contradiction.\nThe linex/vectorenmeetsBiin a point z. We claim: z∈Int(p/vectore+\nn)⊂Int(K), sox=πn(z)∈\nπn(Bi∩Int(K)).\nAsπn(F)⊂∂πn(K) because Fis vertical,x/\\e}a⊔io\\slash∈πn(F). Consequently, p/\\e}a⊔io\\slash∈Bibecause otherwise\np∈Bi∩Fi⊂F∩Fiandx=πn(p)∈πn(F), which is a contradiction.\nLetq∈F∩Fi⊂Biand letbi= 0 be a linear equation of Bi. AsBiis non-vertical, we\nmay assume /vectorbi(/vectoren)>0, so Int(q/vectore+\nn)⊂ {bi>0}becausebi(q) = 0. As q/vectore+\nn⊂F, we deduce12 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nF⊂ {bi≥0}, soFi⊂ {bi≤0}. Asp∈Fi\\Bi, we havebi(p)<0. Writez=p+λ/vectoren, so\n0 =bi(z) =bi(p)+λ/vectorbi(/vectoren)❀0<−bi(p) =λ/vectorbi(/vectoren)\nandλ>0. Thus,z∈Int(p/vectore+\nn), as claimed.\nNotice that πn(K) =/uniontextr\ni=1πn(Fi). By (2.1)\nπn(Int(K)) =πn(K)\\∂πn(K) =/parenleftBigr/uniondisplay\ni=1πn(Fi)/parenrightBig\n\\∂πn(K)\n=r/uniondisplay\ni=1(πn(Fi)\\∂πn(K))⊂r/uniondisplay\ni=1πn(Bi∩Int(K))⊂πn(Int(K)),\nsoπn(Int(K)) =/uniontextr\ni=1πn(Bi∩Int(K)), as required. /square\nTo illustrate the meaning of Lemma 2.9, Figure 1 shows how the projectionπ3:R3→R3\nacts on a polyhedron Kwith two non-vertical facets F1,F2. These facets are separated from F\nby the hyperplanes B1,B2.\n2.f.Nonvertical hyperplanes and polynomial functions. In many of our arguments non-\nvertical affine subspaces play a special role because of the wa y we place our polyhedra in space.\nIf we consider a finite collection of non-vertical hyperplan es, it is intuitively clear that we can\nfind a polynomial function G∈R[x′] :=R[x1,...,xn−1] whose graph {xn=G}lies ‘above’ all\nthese hyperplanes. In fact, we can say more.\nProposition 2.10. Let{Hi}k\ni=1be a finite family of (non-vertical) hyperplanes with linear\nequationsHi:={hi= 0}oriented so that /vectorhi(/vectoren)>0. Then there exists G∈R[x′]such that\nG>1onRn−1and its graph Λ :={xn=G} ⊂Rnsatisfies\nΛ/vectore+\nn={xn≥G} ⊂k/intersectiondisplay\ni=1{hi>1}. (2.2)\nIn particular, Hi∩Λ/vectore+\nn=∅fori= 1,...,kandΛ/vectore+\nn⊂ {/producttextk\ni=1hi>1}.\nProof.Writehi(x′,xn) =/a\\}bracke⊔le{⊔a′\ni,x′/a\\}bracke⊔ri}h⊔+ainxn+biwherea′\ni∈Rn−1,ain,bi∈Randain=/vectorhi(/vectoren)>0.\nDenote\nρi(x′) :=−1\nain(/a\\}bracke⊔le{⊔a′\ni,x′/a\\}bracke⊔ri}h⊔+bi)\nand observe that Hi={xn−ρi(x′) = 0}. For eachi= 1,...,kconsider the polynomial\nGi:= 1+1\nain+ρ2\ni+1\n2∈R[x′].\nWe haveGi(x′)≥1 +1\nain+|ρi(x′)|>1 for eachx′∈Rn−1. DefineG:=/producttextk\ni=1Gi∈R[x′]. It\nholdsG(x′)≥Gi(x′)>1 for eachi= 1,...,kandx′∈Rn−1. Let us check (2.2).\nPick (x′,xn)∈Rnsuch thatxn≥G(x). Then\nhi(x′,xn) =/a\\}bracke⊔le{⊔a′\ni,x′/a\\}bracke⊔ri}h⊔+ainxn+bi≥ainxn−|/a\\}bracke⊔le{⊔a′\ni,x′/a\\}bracke⊔ri}h⊔+bi| ≥ainG(x)−|/a\\}bracke⊔le{⊔a′\ni,x′/a\\}bracke⊔ri}h⊔+bi|\n≥ainGi(x)−|/a\\}bracke⊔le{⊔a′\ni,x′/a\\}bracke⊔ri}h⊔+bi| ≥ain/parenleftbig\n1+1\nain+|ρi(x′)|/parenrightbig\n−|/a\\}bracke⊔le{⊔a′\ni,x′/a\\}bracke⊔ri}h⊔+bi|>1.\nConsequently, Λ /vectore+\nn⊂/intersectiontextk\ni=1{hi>1} ⊂ {/producttextk\ni=1hi>1}andHi∩Λ/vectore+\nn=∅fori= 1,...,k, as\nrequired. /square\nRemark 2.11.By including an extra hyperplane H0of equation xn−b= 0 whereb∈Rwe can\nfinda correspondingpolynomial G(x′) satisfying the previous statement and such that {xn≥G}\nlies in{xn>b+1}.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 13\nIn order to construct polynomial maps f:Rn→Rnwith polyhedral images we will resort\nto maps fixing pointwise finite collections of hyperplanes in Rn. These maps will often leave\nvertical lines invariant. Under these hypotheses the follo wing immediate but useful application\nof Bolzano’s Theorem applies. Given a function ψ:R→Rwe writeψ(±∞) := lim t→±∞ψ(t)\nwhenever the previous limit either exists or is equal to ±∞.\nLemma 2.12. Letψ:R→Rbe a continuous function and let −∞<a<b≤+∞be such that\nψ(a) =aandψ(b) =b. Then]a,b[⊂ψ(]a,b[).\nCorollary 2.13. Letf:= (f′,fn) :Rn→Rnbe a continuous map and let x:= (x′,xn)∈Rn\nbe such that f(x/vectoren)⊂x/vectoren. Then\n(i)For each pair of points p1,p2∈x/vectorenwithf(pi) =pi, it holds Int(p1p2)⊂f(Int(p1p2)).\n(ii)Assume that ψx′(t) :=fn(x′,t)satisfiesψx′(+∞) = +∞. For each p∈x/vectorensuch that\nf(p) =pwe have Int(p/vectore+\nn)⊂f(Int(p/vectore+\nn)).\n3.Variations on Pecker’s polynomials\nOne main result in this section and the key to prove the main re sults of this article is\nLemma 3.2. In 3.d we present some of its consequences that wil l help us to establish a link\nbetween Pecker’s results and Theorems 1.1 and 1.2. Denote x′:= (x1,...,x n−1) so that each\npointx:= (x1,...,x n)∈Rn≡Rn−1×Rwill be written in this section as x:= (x′,xn). As\nbefore,R[x′] :=R[x1,...,xn−1],R[x] :=R[x1,...,xn] andπn:Rn→Rn, x:= (x′,xn)/ma√s⊔o→(x′,0).\nDefinitions 3.1. A tuple g:= (g1,...,gm,gm+1)∈R[x′]m+1such thatgm+1>1 onRn−1is\ncalled an admissible tuple of polynomials of length m+1. We associate to gthe semialgebraic\nsetA(g) :={g1>0,...,gm>0,xn= 0} ⊂Rn, which does not depend on gm+1.\nLemma 3.2. Letg:= (g1,...,gm,gm+1)∈R[x′]m+1be an admissible tuple of length m+ 1.\nThen there exists a polynomial Qg∈R[x]such that:\n(i){Qg≤0} ⊂A(g)/vectoren∩{|xn|>max{gm+1,gm+1√g1···gm}}.\n(ii)For each (x′,0)∈A(g)there exist a positive root r≥gm+1(x′)of the univariate polyno-\nmialQg(x′,t)and a value t≥rsuch thatQg(x′,t) =−1.\n(iii)The set S(g) :={Qg≤0,xn>0}/vectorensatisfiesπn(S(g)) =A(g). In addition, for each\n(x′,xn)∈S(g)there existrn>0andtn≥0such thatxn=rn+tnandQg(x′,rn) = 0.\nA(g)xn= 1xn=gm+1xn=gm+1√g1···gmQg≤0\nQg=−1\nQg= 0(x′,r)(x′,t)\n(x′,0)\nFigure 2. {Qg≤0} ⊂A(g)/vectoren∩{xn>max{gm+1,gm+1√g1···gm}}\nFigure2sketchesthegraphicalmeaningofLemma3.2. Itspro ofreliesonPecker’sconstruction\n[P,§2] that we recall next.14 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\n3.a.Pecker’s construction. Define:\nak(y1,...,yk+1) :=yk+1(y1+···+yk)∈Z[y1,...,yk+1]. (3.1)\nIfyi≥0 fori= 1,...,k+1, it holds ak(y1,...,yk+1)≥0. Consider Pecker’s polynomials defined\nas follows:\nP1(y1,t) :=t−y1,\nPm+1(y1,...,ym+1,t) :=Pm(a1(y1,y2),...,a m(y1,...,ym+1),(t−(y1+···+ym+1))2).\n3.a.1.Basic properties of polynomials Pm.The previous polynomials satisfy the following prop-\nerties [P, Thm.1]:\n(i)Pm∈Z[y1,...,ym,t]is a homogeneous polynomial of degree 2m−1.\n(ii)If eachyi≥0andPm(y1,...,ym,t) = 0, then0≤t≤2/summationtextm\ni=1yi.\n(iii)If all theyiare non-negative, the polynomial Pm(y1,...,ym,t2)in the variable thas\nonly real roots.\n(iv)IfPm(y1,...,ym,t2)has a real root, then all the yiare non-negative.\n(v)Pm(y1,...,yj−1,0,yj+1,...,ym,t) = (Pm−1(y1,...,yj−1,yj+1,...,ym,t))2.\n(vi)Pm(y1,...,ym,t2)is irreducible in R[y1,...,ym,t]and monic in each variable.\n3.a.2.Further properties of polynomials Pm.The polynomials Pmsatisfy in addition the follow-\ning properties:\n(i)If eachyi≥0andPm(y1,...,ym,t) = 0, then\n/parenleftBig\n1−/radicalBig\nm−1\nm/parenrightBigm/summationdisplay\ni=1yi≤t≤/parenleftBig\n1+/radicalBig\nm−1\nm/parenrightBigm/summationdisplay\ni=1yi.\n(ii)Ifm≥2and eachyi>0, thenPm(y1,...,ym,0)>0.\n(iii)DefineA1(y1) :=y1and\nAk+1(y1,...,yk+1) :=Ak(a1(y1,y2),...,a k(y1,...,yk+1))∈Z[y1,...,yk+1].\nThenAmis a homogeneous polynomial of degree 2m−1and there exists a homogeneous\npolynomial Bm−1∈Z[y1,...,ym−1]of degree 2m−1−mwith non-negative coefficients\nsuch thatAm=Bm−1/producttextm\ni=1yi.\n(iv)Given values yi≥0fori= 1,...,m, there exists\ntm/braceleftBigg\n= 0 ifm= 1,\n≥/summationtextm\ni=1yiifm≥2,\nsuch thatPm(y1,...,ym,tm) =−Am(y1,...,ym)<0.\nProof.(i) Itm= 1 the result is clearly true, so let us assume m≥2. Denotesm:=/summationtextm\ni=1yi,\nrm:=/summationtext\n1≤j<k≤myjykandqm:=/summationtextm\ni=1y2\ni. Ass2\nm=qm+2rm, notice that\nqm=m/summationdisplay\ni=1y2\ni=2\nm−1/summationdisplay\n1≤j<k≤my2\nj+y2\nk\n2≥2\nm−1/summationdisplay\n1≤j<k≤myjyk=2rm\nm−1.\nConsequently,\ns2\nm=qm+2rm≥2m\nm−1rm❀2rm≤m−1\nms2\nm. (3.2)\nLett0∈Rbe such that Pm(y1,...,ym,t0) = 0. By definition\nPm(y1,...,ym,t) =Pm−1(a1(y1,y2),...,a m−1(y1,···,ym),(t−(y1+···+ym))2),\nsou0:= (t0−sm)2is a root of the polynomial Pm−1(a1(y1,y2),...,a m−1(y1,...,ym),u). As\neachai(y1,...,yi+1)≥0 (see (3.1)), we deduce from 3.a.1(ii)\nu0= (t0−sm)2≤2m−1/summationdisplay\ni=1ai(y1,...,yi+1) = 2m−1/summationdisplay\ni=1yi+1(y1+···+yi) = 2/summationdisplay\n1≤j<k≤myjyk= 2rm,UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 15\nor equivalently,\nt2\n0−2smt0+qm≤0.\nThe previous condition is equivalent to\nsm−√\n2rm=sm−/radicalbig\ns2m−qm≤t0≤sm+/radicalbig\ns2m−qm=sm+√\n2rm. (3.3)\nBy (3.2) we have√2rm≤/radicalBig\nm−1\nmsmand by (3.3)\n/parenleftBig\n1−/radicalbigg\nm−1\nm/parenrightBig\nsm≤sm−√\n2rm≤t0≤sm+√\n2rm≤/parenleftBig\n1+/radicalbigg\nm−1\nm/parenrightBig\nsm,\nso the statement follows.\n(ii) By (i) the polynomial Pm(y1,...,ym,t)∈R[t] has no real roots in the interval ] −∞,0].\nBy 3.a.1(i) & (vi) Pm(y1,...,ym,t) is a monic polynomial of even degree, so\nlim\nt→−∞Pm(y1,...,ym,t) = +∞.\nConsequently, Pm(y1,...,ym,0)>0.\n(iii) We proceed by induction on m. ForA1(y1) =y1andA2(y1,y2) =y1y2the statement is\ntrue by setting B0=B1= 1. Assume the statement true for m. Then\nAm+1(y1,...,ym+1) =Am(a1(y1,y2),...,a m−1(y1,...,ym),am(y1,...,ym+1))\n=/parenleftBigm/productdisplay\nk=1ak(y1,...,yk+1)/parenrightBig\nBm−1(a1(y1,y2),...,a m−1(y1,...,ym))\n=/parenleftBigm+1/productdisplay\nk=1yk/parenrightBig/parenleftBigm/productdisplay\nk=2(y1+···+yk)/parenrightBig\nBm−1(a1(y1,y2),...,a m−1(y1,...,ym))\n=/parenleftBigm+1/productdisplay\nk=1yk/parenrightBig\nBm(y1,...,ym)\nwhere\nBm(y1,...,ym) :=/parenleftBigm/productdisplay\nk=2(y1+···+yk)/parenrightBig\nBm−1(a1(y1,y2),...,a m−1(y1,...,ym)).\nIn addition, Am+1is by induction a homogeneous polynomial of degree 2m(becauseAmis a\nhomogeneous polynomial of degree 2m−1and eachaiis a homogeneous polynomial of degree 2)\nand the equality\nAm+1(y1,...,ym+1) =y1···ym+1Bm(y1,...,ym)\nshows that Bm(y1,...,ym) is a homogeneous polynomial of degree 2m−(m+ 1). Besides\nBm∈Z[y1,...,ym] has non-negative coefficients by induction hypothesis beca use eachai∈\nZ[y1,...,yi+1] has non-negative coefficients.\n(iv) We work by induction on m. Form= 1 the polynomial P1(y1,t) =t−y1achieves the\nvalue−A1(y1) =−y1fort1:= 0. Form= 2\nP2(y1,y2,t) = (t−y1−y2)2−y1y2,\nand this polynomial attains the value −A2(y1,y2) =−A1(a1(y1,y2)) =−y1y2fort2:=y1+y2.\nGivenyi≥0 fori= 1,...,m, consider the non-negative values\na1(y1,y2),...,a m(y1,...,ym+1).\nSuppose by induction that there exists a real number\nt′\nm≥m/summationdisplay\ni=1ai(y1,...,yi+1)\nsuch that\nPm(a1(y1,y2),...,a m(y1,...,ym+1),t′\nm) =−Am(a1(y1,y2),...,a m(y1,...,ym+1)).(3.4)16 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nIn particular, t′\nm≥0 and\ntm+1:=/radicalbig\nt′m+m+1/summationdisplay\ni=1yi≥m+1/summationdisplay\ni=1yi.\nUsing the definition of Pm+1and (3.4) we have\nPm+1(y1,...,ym,ym+1,tm+1) =Pm/parenleftbig\na1(y1,y2),...,a m(y1,...,ym+1),/parenleftbig\ntm+1−/summationtextm+1\ni=1yi/parenrightbig2/parenrightbig\n=Pm/parenleftbig\na1(y1,y2),...,a m(y1,...,ym+1),t′\nm/parenrightbig\n=−Am(a1(y1,y2),...,a m(y1,...,ym+1))\n=−Am+1(y1,...,ym),\nas required. /square\n3.b.Modified Pecker’s polynomials. Fixm≥2 and denote\nCm:=/parenleftBig\n1−/radicalbigg\nm−1\nm/parenrightBig\nandℓ(m) := (m+1)2m−1−m2+m.\nConsider the polynomial\nQm(y1,...,ym,t) :=/parenleftBigt\nCm+ym/parenrightBigℓ(m)\n(y1···ym)2m−1Pm/parenleftBig\ny1,...,ym−1,1\ny1···ym,t/parenrightBig\n.(3.5)\nThen:\n(i)Qm(y1,...,ym,t2)≥0if someyi≤0. In addition,\n{Qm(y,t2)≤0,ym= 1} ⊂ {y1>0,...,ym−1>0,ym= 1}.\n(ii)If eachyi>0, the polynomial Qm(y1,...,ym,t)achieves the value −1at some\nt≥y1+···+ym−1+1\ny1···ym.\nProof.(i) Asℓ(m) is an even positive integer, the first factor of Qmis non-negative. By 3.a.1\n(i) and (vi)\nFm(y1,...,ym,t) = (y1···ym)2m−1Pm/parenleftBig\ny1,...,ym−1,1\ny1···ym,t/parenrightBig\nis a polynomial of degree ( m+1)2m−1. Consider the projection πm+1:Rm+1→Rm,(y,t)→y.\nBy [P, Cor.1, p.308] the hypersurface {Fm(y,t2) = 0} ⊂Rm+1projects under πm+1onto the\nopen orthant Q:={y1>0,...,ym>0}. Thus, for each t∈Rthe polynomial Fm(y,t2) has\nempty zero-set on Rm\\Q. AsRm\\Qis connected and the origin 0 ∈Rm\\Q, we deduce Fm(y,t2)·\nFm(0,t2)>0 for every y∈Rm\\Q. By 3.a.1(i) & (vi) the polynomial Pmis homogeneous and\nmonic in each variable, so Fm(0,t2) = 1>0. Thus,Fm(y,t2)>0 onRm\\Qand the first part\nof the statement follows.\nIfym= 1, the first factor of Qm(y,t2) is strictly positive. We have proved above that the\nfactorFmis strictly positive on ( Rm\\Q)×R. Consequently,\n{Qm(y,t2)≤0,ym= 1} ⊂ {Fm(y,t2)≤0,ym= 1} ⊂ {y1>0,...,ym−1>0,ym= 1}.\n(ii) Fixy:= (y1,...,ym)∈Q:={y1>0,...,ym>0}. By 3.a.2(iv) there exists\ntm≥y1+···+ym−1+1\ny1···ym(3.6)\nsuch that\nPm(y1,...,ym−1,1\ny1···ym,tm) =−Am(y1,...,ym−1,1\ny1···ym). (3.7)\nBy 3.a.2(iii) we can write\nAm(y1,...,ym−1,1\ny1···ym) =Bm−1(y1,...,ym−1)\nym(3.8)UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 17\nwhereBm−1∈Z[y1,...,ym−1] is a homogeneous polynomial of degree 2m−1−mwith non-\nnegative coefficients. Consider the rational functions\nQm,1(y1,...,ym) :=/parenleftBig\ny1+···+ym+1\ny1···ym/parenrightBig2m−1\n(y1···ym)2m−1\nQm,2(y1,...,ym) :=/parenleftBig\ny1+···+ym+1\ny1···ym/parenrightBigm(2m−1−m)\nBm−1(y1,...,ym−1)\nQm,3(y1,...,ym) :=/parenleftBig\ny1+···+ym+1\ny1···ym/parenrightBigm1\nym\nWe claim:Qm,1(y)>1,Qm,2(y)≥1andQm,3(y)>1.\nThe inequalities Qm,1(y)>1 andQm,3(y)>1 are straightforward. We proceed with\nQm,2(y)≥1. AsBm−1is a homogeneous polynomial of degree 2m−1−mwhose coefficients\nare non-negative integers, we write\nBm−1(y1,...,ym−1) =/summationdisplay\n|ν|=2m−1−maνyν1\n1···yνm−1\nm−1\nwhereν:= (ν1,...,νm−1)∈(N∪{0})m−1,|ν|=ν1+···+νm−1andaν∈N∪{0}.\nFixaν/\\e}a⊔io\\slash= 0. By Lemma A.1(iii) and since ν1+···+νm−1= 2m−1−m, we have\n/parenleftBig\ny1+···+ym+1\ny1···ym/parenrightBigm(2m−1−m)\naνyν1\n1···yνm−1\nm−1\n=aνm−1/productdisplay\ni=1/parenleftBig/parenleftBig\ny1+···+ym+1\ny1···ym/parenrightBigm\nyi/parenrightBigνi≥aν≥1.\nConsequently, Qm,2(y)≥1, as claimed.\nBy (3.6) we have\ntm\nCm+ym≥tm+ym≥y1+···+ym+1\ny1···ym.\nTherefore, by (3.7) and (3.8)\nQm(y,tm) =/parenleftBigtm\nCm+ym/parenrightBigℓ(m)\n(y1···ym)2m−1Pm/parenleftbig\ny1,...,ym−1,1\ny1···ym,tm/parenrightbig\n≤ −Qm,1(y)Qm,2(y)Qm,3(y)≤ −1.\nBy 3.a.1(vi) lim t→+∞Qm(y,t)→+∞. Thus, there exists t≥tmfor whichQm(y,t) =−1, as\nrequired. /square\n3.c.Proof of Lemma 3.2. Consider the polynomial\nQg(x′,xn) :=gℓ0\nm+1(x′)Qm+1/parenleftBig\ng1(x′),...,gm(x′),1,x2\nnCm+1\ng2\nm+1(x′)/parenrightBig\n(3.9)\nwhereQm+1is the polynomial constructed in 3.b, Cm+1:= 1−/radicalBig\nm\nm+1andℓ0:= 2ℓ(m+1)+2m+1\nis large enough to guarantee that Qgis a polynomial.\n(i) We have to show\n{Qg≤0} ⊂A(g)/vectoren∩/braceleftBig\n|xn|>max/braceleftBig\ngm+1,gm+1√g1···gm/bracerightBig/bracerightBig\n.\nBy 3.b(i)\n{Qg≤0}=/braceleftBig\nQm+1/parenleftBig\ng1,...,gm,1,x2\nnCm+1\ng2\nm+1/parenrightBig\n≤0/bracerightBig\n⊂ {g1>0,...,gm>0}=A(g)/vectoren.\nWe check now\n{Qg≤0} ⊂/braceleftBig\n|xn|>max/braceleftBig\ngm+1,gm+1√g1···gm/bracerightBig/bracerightBig\n. (3.10)18 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nFix (x′,0)∈A(g). AsQg(x′,xn) =Qg(x′,−xn) and the leading coefficient of Qg(x′,xn) with\nrespect to xnis positive, lim xn→±∞Qg(x′,xn) = +∞.\nBy 3.a.1(iii) the univariate polynomial Qg(x′,xn) has 2mreal roots. As it defines an even\npolynomial function and by 3.a.2(i) none of its roots is zero , 2m−1of them are positive and 2m−1\nare negative. Let r>0 be the smallest of the positive roots of Qg(x′,xn). We have\nPm+1/parenleftBig\ng1(x′),...,gm(x′),1\ng1(x′)···gm(x′),r2Cm+1\ng2\nm+1(x′)/parenrightBig\n= 0\nand eachgi(x′)>0. By 3.a.2(i) and Lemma A.1(i)\nr2Cm+1\ng2\nm+1(x′)≥Cm+1/parenleftBigm/summationdisplay\ni=1gi(x′)+1\ng1(x′)···gm(x′)/parenrightBig\n≥Cm+1>0.\nThus,r2≥g2\nm+1(x′), sor≥gm+1(x′). In addition,\nr2Cm+1\ng2\nm+1(x′)≥Cm+1\ng1(x′)···gm(x′)❀r≥gm+1(x′)/radicalbig\ng1(x′)···gm(x′).\nConsequently,\nr≥max/braceleftBig\ngm+1(x′),gm+1(x′)/radicalbig\ng1(x′)···gm(x′)/bracerightBig\n. (3.11)\n3.c.1. We claim: ifQg(x′,xn)≤0andxn≥0, thenxn≥r. Asris the smallest positive real\nroot of the univariate polynomial Qg(x′,xn), it is enough to show: Qg(x′,0)>0.\nThis follows from 3.a.2(ii) because gi(x′)>0 fori= 1,...,mand\nQg(x′,0) =gℓ0\nm+1(x′)Qm+1(g1(x′),...,gm(x′),1,0)\n=gℓ0\nm+1(x′)(g1(x′)···gm(x′))2mPm+1/parenleftBig\ng1(x′),...,gm(x′),1\ng1(x′)···gm(x′),0/parenrightBig\n>0.\nBy (3.11) and 3.c.1 the inclussion (3.10) holds.\n(ii) If (x′,0)∈A(g), we haveg1(x′)>0,...,gm(x′)>0. By 3.b(ii) there exists\nt0≥g1(x′)+···+gm(x′)+1\ng1(x′)···gm(x′)>0\nsuch thatQm+1(g1(x′),...,gm(x′),1,t0) =−1. Definet1:=/radicalBig\nt0\nCm+1gm+1(x′) and observe\nQg(x′,t1) =gℓ0\nm+1(x′)Qm+1(g1(x′),...,gm(x′),1,t0) =−gℓ0\nm+1(x′)<−1.\nAsQg(x′,t1)<−1, we know by 3.c.1 that t1>r. We haveQg(x′,t1)<−1<Qg(x′,r) = 0, so\nthere exists r≤t≤t1such thatQg(x′,t) =−1.\n(iii) Statements (i) and (ii) provide the first part of (iii) w hereas 3.c.1 issues the second part\nof (iii), as required /square\n3.d.Consequences of Lemma 3.2. Letg:= (g1,...,gm+1)∈R[x′]m+1be an admissible\ntuple of polynomials and let us consider the corresponding p olynomialQg∈R[x] introduced in\nLemma 3.2 and the associated semialgebraic sets A(g) andS(g). The latter semialgebraic set\nwas introduced in Lemma 3.2, where we also proved some key pro perties of S(g).\nTheorem 3.3. Leth∈R[x′]be positive semidefinite on A(g)and letP∈R[x]be strictly greater\nthan1onS(g). Assume in addition gm+1>honA(g). Define\nf:= (f1,...,fn) :Rn→Rn, x:= (x′,xn)/ma√s⊔o→(x′,xn(1+P(x)Qg(x))2+h(x′)(P(x)Qg(x))2).\nWe have:\n(i)A(g)/vectoren∩{xn≥h(x′)} ⊂f(S(g))⊂A(g)/vectoren∩{2xn≥h(x′)}. In particular, if h= 0, we\nhavef(S(g)) =A(g)/vectore+\nn.\n(ii)Wheneverxn≥0,P(x)Qg(x)≥0andh(x′)≥0, the inequality fn(x)≥xnholds. In\nparticular, this happens if x∈(A(g)/vectore+\nn\\S(g))∩{P≥0}.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 19\n(iii)Ifh(x′)≥0andP(x′,xn)≥0forxnlarge enough, then limxn→+∞fn(x′,xn) = +∞.\nS(g)\nf\nA(g)\n(a) (b)A(g) A(g)f(S(g)) f(S(g))\nxn=h\n2xn=h\nFigure 3. Theorem 3.3: (a) Case h/\\e}a⊔io\\slash= 0. (b) Case h= 0.\nProof.(i) The polynomial map fpreserves vertical lines.\n3.d.1. We prove first: A(g)/vectoren∩{xn≥h(x′)} ⊂f(S(g)).\n3.d.2. Pick a point ( x′,0)∈A(g). We claim: the polynomial Qg(x′,xn)has degree ℓ0:=\n2ℓ(m+1)+2m+1and its leading coefficient is strictly positive .\nBy 3.a.1(i) and (vi), (3.5) and (3.9) the degree of Qg(x′,xn) isℓ0and its leading coefficient is\n(Cm+1g1···gm)2m>0.\n3.d.3. By Lemma 3.2(ii) there exist points p:= (x′,t) andq:= (x′,r) such that t > r≥\ngm+1(x′)>1,Qg(x′,t) =−1 andQg(x′,r) = 0. In particular, p/vectore+\nn⊂S(g), soPis strictly\ngreater than 1 on p/vectore+\nn. Consider the polynomial φx′(xn) := 1+P(x′,xn)Qg(x′,xn). We have\nφx′(r) = 1, φx′(t) = 1+P(x′,t)Qg(x′,t)<0 and lim\nxn→+∞φx′(xn) = +∞.\nConsequently, there exists s∈]r,t[ such that φx′(s) = 0, so [0 ,+∞[⊂φx′([s,+∞[). Consider\nalso the polynomial\nϕx′(xn) :=fn(x′,xn) =xnφ2\nx′(xn)+h(x′)(φx′(xn)−1)2\nand observe that\nϕx′(s) =h(x′) and lim\nxn→+∞ϕx′(xn) = +∞.\nThus, [h(x′),+∞[⊂ϕx′([s,+∞[) and\n{x′}×{xn≥h(x′)} ⊂f(p/vectore+\nn)⊂f(S(g)).\nWe conclude A(g)/vectoren∩{xn≥h(x′)} ⊂f(S(g)).20 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\n3.d.4. Let us check next: f(S(g))⊂A(g)/vectoren∩{2xn≥h(x′)}.\nPick a point ( x′,xn)∈S(g). By Lemma 3.2(iii) ( x′,0)∈A(g) andxn> gm+1(x′)> h(x′).\nConsider the polynomial ψx′(xn) :=P(x′,xn)Qg(x′,xn), so the last component of fcan be\nrewritten as fn(x′,xn) =xn(1+ψx′(xn))2+h(x′)ψ2\nx′(xn). Asxn>h(x′)≥0,\nfn(x′,xn) =xn+ψ2\nx′(xn)(xn+h(x′))+2ψx′(xn)xn\n=/parenleftBig/radicalbig\nxn+h(x′)ψx′(xn)+xn/radicalbig\nxn+h(x′)/parenrightBig2\n+xn−/parenleftBigxn/radicalbig\nxn+h(x′)/parenrightBig2\n≥xn−/parenleftBigxn/radicalbig\nxn+h(x′)/parenrightBig2\n=xnh(x′)\nxn+h(x′)=h(x′)\n1+h(x′)\nxn≥h(x′)\n2.\nConsequently, f(S(g))⊂A(g)/vectoren∩{2xn≥h(x′)}.\n(ii) The statement follows from the required inequalities a nd the definition of the coordinate\nfunctionfn(x).\n(iii) Pickx′∈Rn−1such thath(x′)≥0 andP(x′,xn)≥0 forxnlarge enough. If ( x′,0)/∈\nA(g), thenQg(x′,xn) is positive on {xn>0}. By (ii)fn(x′xn)≥xnifxnis large enough, hence\nlim\nxn→+∞fn(x′,xn) = +∞.\nIf (x′,0)∈A(g), the polynomial Qg(x′,xn) has degree ℓ0:= 2ℓ(m+1) +2m+1and its leading\ncoefficient is strictly positive (see 3.d.2). Consequently, limxn→+∞fn(x′,xn) = +∞, as required.\n/square\nFigure 3 illustrates the action of the polynomial map fin Theorem 3.3 on the semialgebraic\nsetS(g).\n3.e.Lower dimensional semialgebraic sets. Fix 1≤d≤n−2 and write y:= (x1,...,xd),\nz:= (xd+1,...,xn−1) andx′:= (y,z), so that x:= (x1,...,xn) = (x′,xn) = (y,z,xn) and we\nidentifyRn≡Rd×Rn−1−d×R. Letg1,...,gr∈R[y] and letε>0. DenoteIε:= ]−ε,ε[ and set\nm:=r+2(n−1−d). Givengm+1∈R[x′] such that gm+1>1 onRn−1, consider the admissible\ntuple\nˆgε:= (g1,...,gr,xd+1+ε,...,xn−1+ε,ε−xd+1,...,ε−xn−1,gm+1)\nand the polynomial Qˆgεconstructed in Lemma 3.2. Consider also the associated semi algebraic\nsetsA(ˆgε),S(ˆgε) and\nAd(ˆgε) :=A(ˆgε)∩{xd+1= 0,...,xn= 0}={g1>0,...,gr>0,xd+1= 0,...,xn= 0} ⊂Rn.\nRecall that /vectorCv\nδ:={(v′,vn)∈Rn:/bardblv′/bardbl ≤δvn}is the vertical cone of radius δ >0 and given a\nsetT⊂Rnthe setCv\nδ(T) :=T+/vectorCv\nδis the vertical cone of radius δ>0 overT.\nTheorem 3.4. Letε,δ >0and assume gm+1≥1+ε√n−d−1\nδ. LetP∈R[x]be>1onS(ˆgε).\nFor eachk≥1consider the polynomial map\nfk:Rn→Rn, x:= (x′,xn) = (y,z,xn)/ma√s⊔o→(y,A(x)z,Bk(x)xn),\nwhereA:= (1+P2Qˆgε)2andBk:=A+Ak\n2. We have:\n(i) lim xn→+∞Bk(x′,xn)xn= +∞for eachx′∈Rn−1.\n(ii)S(ˆgε)⊂Cv\nδ(Ad(ˆgε)).\n(iii)Ad(ˆgε)/vectore+\nn⊂fk(S(ˆgε))⊂Cv\n2δ(Ad(ˆgε)).\n(iv)For each ∆>2δthere exists k0≥1such that if k≥k0andx∈Cv\n∆(Ad(ˆgε))\\S(ˆgε),\nthenfk(x)∈Cv\n2δ({x}).\nProof.(i) Pickx′∈Rn−1. If (x′,0)/∈A(ˆgε), then by Lemma 3.2(i) Qˆgε(x′,xn) is positive on\n{xn>0}and lim xn→+∞Bk(x′,xn)xn= +∞. If (x′,0)∈A(ˆgε), the polynomial Qg(x′,xn)\nhas positive degree and its leading coefficient is strictly po sitive (see 3.d.2). In addition, by\nLemma 3.2(iii) πn(S(ˆgε)) =A(ˆgε), so lim xn→+∞Bk(x′,xn)xn= +∞.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 21\nε ε ε ε\nA(ˆgε)Cv\nδ(A\nd(ˆgε))Cv\n∆(Ad(ˆgε))S(ˆgε)\nQˆgε= 0\nx\nA(ˆgε)fk(S(ˆgε))Ad(ˆgε)/vectore+\nn\nxfk(x)Cv\n2δ({x})Cv\n2δ(Ad(ˆgε))Cv\n∆(Ad(ˆgε))\nfk\nFigure 4. Behavior of the polynomial map fk(Theorem 3.4).\n(ii) Pick a point x:= (x′,xn) := (y,z,xn)∈S(ˆgε). By Lemma 3.2(iii) we may write x=\n(y,z,rn+tn) wherern>0,tn≥0 andQˆgε(y,z,rn) = 0. By Lemma 3.2(i) ( y,z,rn)∈\nA(ˆgε)/vectoren∩ {xn≥gm+1}, so (y,0,0)∈Ad(ˆgε) andz∈In−d−1\nε. Thus, /bardblz/bardbl ≤ε√\nn−d−1. We\nclaim: (0,z,xn)∈/vectorCv\nδ.\nAs (y,z,rn)∈ {xn≥gm+1}, we deduce\nδxn≥δrn≥δgm+1(y,z)≥δε√\nn−d−1\nδ=ε√\nn−d−1≥ /bardblz/bardbl=/bardbl(0,z)/bardbl,\nso (0,z,xn)∈/vectorCv\nδand\n(y,z,xn) = (y,0,0)+(0,z,xn)∈Cv\nδ(Ad(ˆgε)).\n(iii) We show first: Ad(ˆgε)/vectore+\nn⊂fk(S(ˆgε)).\nPick a point p:= (y,0,0)∈Ad(ˆgε). By Lemma 3.2(ii) there exist values 0 <gm+1(x′)≤r<t\nsuch thatQˆgε(y,0,r) = 0 andQˆgε(y,0,t) =−1. As (y,0,r)∈ {Qˆgε= 0,xn>0}, we deduce\n(y,0,r),(y,0,t)∈S(ˆgε). Defineφy(xn) :=A(y,0,xn) = 1+P2(y,0,xn)Qˆgε(y,0,xn) and observe\nφy(r) = 1 and φy(t)<0. (3.12)\nThus, there exists s∈]r,t[ such that φy(s) = 0. If we set q:= (y,0,s)∈S(ˆgε), thenA(q) = 0\nandBk(q) = 0, sofk(q) =p. In addition, q/vectore+\nn⊂S(ˆgε). Asfk(q) =p, the polynomial map fk\npreserve vertical lines and by (i) lim xn→+∞Bk(y,0,xn)xn= +∞, we deduce\np/vectore+\nn⊂fk(q/vectore+\nn)⊂fk(S(ˆgε)),\nhenceAd(ˆgε)/vectore+\nn⊂fk(S(ˆgε)).\n3.e.1. We prove next: fk(S(ˆgε))⊂Cv\n2δ(Ad(ˆgε)). Pick a point x:= (y,z,xn)∈S(ˆgε) and let us\ncheck: (0,z,xn)∈/vectorCv\nδ.\nBy (ii) (y,z,xn)∈S(ˆgε)⊂Cv\nδ(Ad(ˆgε)), so we write ( y,z,xn) = (y0,0,0) + (y1,z,xn) where\n(y0,0,0)∈Ad(ˆgε) and (y1,z,xn)∈/vectorCv\nδ. Consequently,\nδxn≥ /bardbl(y1,z)/bardbl ≥ /bardblz/bardbl=/bardbl(0,z)/bardbl\nand (0,z,xn)∈/vectorCv\nδ.22 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\n3.e.2. We show next: (0 ,A(x)z,Bk(x)xn)∈/vectorCv\n2δ.\nObserve that\nA(x)\n2Bk(x)=A(x)\nA(x)+Ak(x)=1\n1+Ak−1(x)≤1.\nAs (0,z,xn)∈/vectorCv\nδ,\n2δBk(x)xn≥2Bk(x)/bardbl(0,z)/bardbl ≥A(x)/bardbl(0,z)/bardbl=/bardbl(0,A(x)z)/bardbl,\nhence (0,A(x)z,Bk(x)xn)∈/vectorCv\n2δ.\nAs (y,0,0)∈Ad(ˆgε), we conclude\nfk(x) =fk(y,z,xn) = (y,0,0)+(0,A(x)z,Bk(x)xn)∈Cv\n2δ(Ad(ˆgε)).\n(iv) Take a point x:= (x′,xn) := (y,z,xn)∈Cv\n∆(Ad(ˆgε))\\S(ˆgε). We claim: (0 ,z,xn)∈/vectorCv\n∆.\nWritex= (y1+y2,z,xn) where (y1,0,0)∈Ad(ˆgε) and (y2,z,xn)∈/vectorCv\n∆. This implies that\n∆xn≥ /bardbl(y2,z)/bardbl ≥ /bardblz/bardbl=/bardbl(0,z)/bardbl, hence (0,z,xn)∈/vectorCv\n∆.\nAsxn≥0 andx /∈S(ˆgε), we deduce Qˆgε(x)>0. We have\nfk(x)−x= (0,(A(x)−1)z,(Bk(x)−1)xn).\nLet us write\nA−1 = 2P2Qˆgε+P4Q2\nˆgε,\nBk−1 =1\n2/parenleftBig\n2P2Qˆgε+P4Q2\nˆgε+2k/summationdisplay\nℓ=1/parenleftbigg2k\nℓ/parenrightbigg\n(P2Qˆgε)ℓ/parenrightBig\n.\nConsequently, on {Qˆgε>0}\nBk−1\nA−1=2+2k+(1+/parenleftbig2k\n2/parenrightbig\n)P2Qˆgε+/summationtext2k\nℓ=3/parenleftbig2k\nℓ/parenrightbig\n(P2Qˆgε)ℓ−1\n4+2P2Qˆgε\n≥2+2k+(1+k(2k−1))P2Qˆgε\n4+2P2Qˆgε≥k+1\n2.\nLetk0≥1 be such that k0+1≥∆\nδ. Fork≥k0\nBk−1\nA−1≥k+1\n2≥∆\n2δ.\nBy 3.e.1 (0,z,xn)∈/vectorCv\nδ⊂/vectorCv\n∆. Thus, ∆xn≥ /bardbl(0,z)/bardbl, so fork≥k0\n2δ(Bk(x)−1)xn≥(Bk(x)−1)2δ\n∆/bardbl(0,z)/bardbl ≥(A(x)−1)/bardbl(0,z)/bardbl=/bardbl(0,(A(x)−1)z)/bardbl,\nbecauseA(x)−1≥0 (recall that Qˆgε(x)>0). Therefore, (0 ,(A(x)−1)z,(Bk(x)−1)xn)∈/vectorCv\n2δ,\nsofk(x)−x∈/vectorCv\n2δ, as required. /square\nFigure 4 illustrates the behavior of the polynomial map fkforklarge enough.\n4.Convex polyhedra as polynomial images of Rn\nThe purpose of this section is to prove Theorem 1.1. We prove fi rst this result for pointed\ncones, that is, unbounded convex polyhedra K⊂Rnwith only one vertex p. In such case\nK={p}+/vectorC(K).\nProof of Theorem 1.1for pointed cones. AssumeKis a pointed cone with vertex pand denote\nCp:=Kfor the sake of clearness. Let Hbe a hyperplane such that Cp∩H={p}. LetH′be a\nhyperplane parallel to Hsuch that P:=H′∩Cpis a bounded convex polyhedron of dimension\nn−1 (see [FU3, Lem. 3.2]). Assume pis the origin, H:={xn= 0}andH′:={xn= 1}.\nBy [FGU1, Thm. 1.2] there exists a regular map g:= (g1,...,gn−1,1) :Rn−1→Rn−1× {1}UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 23\nsuch thatg(Rn−1) =P. Writegi:=hi\nh0whereh0,hi∈R[x′] andh0is strictly positive on Rn−1.\nConsider the polynomial map\nf:Rn→Rn,(x′,xn)/ma√s⊔o→x2\nnh(x′)\nwhereh:= (h1,...,h n−1,h0). We claim: f(Rn) =Cp.\nPick a point y∈Cpand consider the vector /vector v:=− →0yand the ray 0 /vector v+. Observe that 0 /vector v+⊂Cp\nand the intersetion H′∩0/vector v+=:{z} ⊂Pis a singleton. Thus, there exist x′∈Rn−1such that\ng(x′) =zandλ≥0 such that y=λz. Denotexn:=/radicalBig\nλ\nh0(x′)and observe that\nf(x′,xn) =λ\nh0(x′)h(x′) =λg(x′) =λz=y.\nConsequently, Cp⊂f(Rn). Conversely, if x∈Rn, then\nf(x) =x2\nnh(x′) =x2\nnh0(x′)h(x′)\nh0(x′)=x2\nnh0(x′)g(x′).\nAsx2\nnh0(x′)≥0 andg(x′)∈Cp, we conclude f(x)∈Cpbecause Cpis a cone with vertex the\norigin. Thus, f(Rn) =Cp, as required. /square\nWe divide the proof of Theorem 1.1 for the general case into th ree parts. The rest of the\nsection is devoted to prove them. As a degenerate convex poly hedronK⊂Rncan be written\nin suitable coordinates as K=P×RkwherePis a non-degenerate convex polyhedron, it is\nenough to approach the non-degenerate case. If k≥n−1, thenKis either Rnor a half-space,\nso it is trivially a polynomial image of Rn. Thus, we assume in addition n≥2. LetK⊂Rn\nbe ann-dimensional non-degenerate convex polyhedron whose rece ssion cone has dimension n.\nLetXbe the union of the affine subspaces of Rnspanned by the faces of Kof dimension n−2.\nWe will prove the following statements.\nProposition 4.1. There exists a polynomial map h:Rn→Rnsuch thath(Rn) =K\\X.\nProposition 4.2. There exists a polynomial map g:Rn→Rnsuch thatg(K\\X) =K.\nCorollary 4.3. There exists a polynomial map f:Rn+1→Rnsuch thatf(Rn+1) = Int(K).\n4.a.Proof of Proposition 4.1. Take a point p∈Int(K). Consider the pointed cone Cp:=\n{p}+/vectorC(K)⊂Int(K). We have already proved that Cpis a polynomial image of Rn, so it is\nenough to show that K\\Xis a polynomial image of Cp. The idea here is to use Cpas a seed to\nfill the polyhedron Kby means of a sequence of polynomial maps whose images make Cpgrow\nuntil we obtain K\\X. We start by placing the polyhedron Kin a convenient position (using\naffine changes of coordinates) in order to make our arguments c learer.\nDenote the facets of KwithF1,...,Fr. By Lemma 2.7 there exist /vector v1,...,/vector vs∈Rnsuch that\ns/intersectiondisplay\nk=1X/vector vk=Xand/vector vl∈Int(/vectorC(K))\\l−1/uniondisplay\nk=1/vectorX/vector vk. (4.1)\n4.a.1. Assume Kis placed in Rnso thatK⊂ {xn≥0}and/vector v1=/vectoren∈Int(/vectorC(K)). Thus, Khas\nno vertical facets. Let Z⊂Rndenote a finite union of non-vertical hyperplanes Wℓ:={wℓ= 0},\nwherewℓdenotes a linear equation of Wℓsuch thatwℓ(/vectoren)>0 for eachℓ. This type of sets will\nbe useful for the inductive process. Choose a facet FiofKand lethi(x′,xn) =hi(x′,0)+xn= 0\nbea non-zero linear equation for the (non-vertical) hyperp lanespanned by Fi. Theaffine change\nof coordinates\nφi:Rn→Rn,(x′,x)/ma√s⊔o→(x′,hi(x′,0)+xn)\nmapsFiontoπn(Fi)⊂ {xn= 0}and keeps the vector /vectoreninvariant. To lighten the presenta-\ntion we preserve the notations for all our geometric objects after applying the affine change of\ncoordinates φi. Write\nFi:={gi,1≥0,...,gi,m≥0,xn= 0}24 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nwhere each gi,jis a non-zero linear polynomial. Let Pi:=Pi,Zbe the square of the product\nof non-zero linear equations of the hyperplanes containing the facets of K, the hyperplanes\ncontaining the facets of Cpand the hyperplanes Wℓ. By Proposition 2.10 there exists gi,m+1∈\nR[x′] strictly greater than 1 on Rn−1such that\n{xn≥gi,m+1} ⊂Cp∩{Pi>1}.\nLetgi:= (gi,1,...,gi,m,gi,m+1). By Lemma 3.2 there exist a polynomial Qgisuch that the\nsemialgebraic sets A(gi) = Int(Fi) andS(gi) ={Qg≤0,xn>0}/vectorensatisfy\nS(gi)⊂A(gi)/vectoren∩{xn≥gi,m+1} ⊂A(gi)/vectoren∩Cp∩{Pi>1}. (4.2)\nConsider now the polynomial map\nfi:=fi,Z:= (fi1,...,fin) :Rn→Rn, x:= (x′,xn)/ma√s⊔o→(x′,xn(1+Pi(x)Qgi(x))2).\n4.a.2. We claim:\n(i)fi(Fj/vectore+\nn\\X) =Fj/vectore+\nn\\Xandfi(Int(Fj)/vectore+\nn) = Int(Fj)/vectore+\nnforj= 1,...,r. In addition,\nfi(K\\X) =K\\X.\n(ii)fi(Cp) = Int(Fi)/vectore+\nn∪Cp.\n(iii)fi|Z= idZ.\nLet us prove the previous statements:\n(i) By Proposition 2.6 πn|∂K:∂K→Rn−1× {0}is a semialgebraic homeomorphism, so\nK=/uniontextr\nj=1Fj/vectore+\nnandK\\X=/uniontextr\nj=1Fj/vectore+\nn\\X. Thus, once we prove the first part of the\nstatement we will have in addition the second.\nPick a point x:= (x′,xn)∈Fj⊂∂Kfor somej= 1,...,r. Ifj/\\e}a⊔io\\slash=i, thenπn(x)/∈Int(Fi) =\nA(gi), soQgiis by Lemma 3.2 strictly positive on x/vectore+\nn. Thus, for each ( x′,t)∈x/vectore+\nnwe have\nfin(x′,t) =t(1+Pi(x′,t)Qgi(x′,t))2≥t≥xn.\nAsfi(x) =x, we havefi(x/vectore+\nn) =x/vectore+\nnandfi(Int(x/vectore+\nn)) = Int(x/vectore+\nn). Consequently,\nfi(Fj/vectore+\nn\\X) =Fj/vectore+\nn\\Xandfi(Int(Fj)/vectore+\nn) = Int(Fj)/vectore+\nn.\nAssume now j=i. As∂Fi=Fi∩/uniontext\nj/\\egatio\\slash=iFj, we have (∂Fi)/vectore+\nn\\X=/uniontext\nj/\\egatio\\slash=i(Fi∩Fj)/vectore+\nn\\X. As\nfipreserves vertical lines,\n(∂Fi)/vectore+\nn\\X=/uniondisplay\nj/\\egatio\\slash=ifi((Fi∩Fj)/vectore+\nn\\X) =fi/parenleftBig/uniondisplay\nj/\\egatio\\slash=i(Fi∩Fj)/vectore+\nn\\X/parenrightBig\n=fi((∂Fi)/vectore+\nn\\X).\nTo finish it is enough to check fi(Int(Fi)/vectore+\nn) = Int(Fi)/vectore+\nn. By Theorem 3.3(i) and (4.2)\nInt(Fi)/vectore+\nn=A(gi)/vectore+\nn=fi(S(gi))⊂fi(A(gi)/vectore+\nn) =fi(Int(Fi)/vectore+\nn)⊂Int(Fi)/vectore+\nn.\nThe latter inclusion follows because fipreserves vertical lines and fi({xn≥0})⊂ {xn≥0}.\n(ii) As/vectoren∈Int(/vectorC(K)) = Int(/vectorC(Cp)), the restriction map πn|∂Cp:∂Cp→Rn−1× {0}is\nby Proposition 2.6 a semialgebraic homeomorphism. Consequ ently,Cp=∂Cp/vectore+\nn. Pickx:=\n(x′,xn)∈∂Cp. Ify:=πn(x)/∈A(gi), thenQgi(x′,t)>0 fort≥xnby Lemma 3.2, so\nfin(x′,t) =t(1+Pi(x′,t)Qgi(x′,t))2≥t≥xn\nfor (x′,t)∈x/vectore+\nn. Asfi(x) =x, we deduce fi(x/vectore+\nn) =x/vectore+\nn. Thus,\nfi(Cp\\A(gi)/vectore+\nn) =Cp\\A(gi)/vectore+\nn=Cp\\Int(Fi)/vectore+\nn.\nBy Theorem 3.3(i), (4.2) and (i)\nInt(Fi)/vectore+\nn=A(gi)/vectore+\nn=fi(S(gi))⊂fi(A(gi)/vectore+\nn∩Cp) =fi(Int(Fi)/vectore+\nn∩Cp)⊂Int(Fi)/vectore+\nn.\nConsequently,\nfi(Cp) =fi(Cp\\A(gi)/vectore+\nn)∪fi(A(gi)/vectore+\nn∩Cp)\n= (Cp\\Int(Fi)/vectore+\nn)∪Int(Fi)/vectore+\nn=Cp∪Int(Fi)/vectore+\nn.\n(iii) This is immediate because Pivanishes identically on the hyperplanes contained in Z.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 25\nFigure 5 illustrates how the polynomial map f1acts on the cone Cp.\nCp\nf1K K f1(Cp)\np\nA(g1) A(g1)\nFigure 5. Behavior of the polynomial map f1overCp.\n4.a.3. Fix 1 ≤k≤sand consider Kplaced in Rn(by means of an affine change of coordinates\nψk) so that K⊂ {xn≥0}and/vector vk=/vectoren. We preserve the names for all our geometric objects\nafter applying the change of coordinates ψk. SetXl:=X/vector vlandZk−1:=/uniontextk−1\nl=1Xl, which are\nunions of hyperplanes because each ( n−2)-affine subspace in Xis parallel to none of the vectors\n/vector vj. In addition, each hyperplane in Zk−1is not parallel to /vector vk(see 4.1). For each i= 1,...,r\nconsidertheaffinechangeofcoordinates φidescribedin4.a.1andthepolynomialmap fi,k:=fi,Z\nintroduced in 4.a.2 taking Z:=φi(Zk−1) andZ0=∅.\nDefine the polynomial map\nFk:=ˆfr,k◦···◦ˆf1,kwhereˆfi,k:=φ−1\ni◦fi,k◦φi.\nWe claim:\n(i)Cp∪(K\\Xk) =Fk(Cp) for 1≤k≤s;\n(ii)Cp∪(K\\/intersectiontextk\nj=1Xj)⊂Fk(Cp∪(K\\/intersectiontextk−1\nj=1Xj)) andFk(K\\X) =K\\Xfor 1≤k≤s.\nTo prove (i) we use recursively 4.a.2. Indeed,\nFk(Cp) = (ˆfr,k◦···◦ˆf2,k)(ˆf1,k(Cp)) = (ˆfr,k◦···◦ˆf2,k)(Cp∪Int(F1)/vectore+\nn)\n=···=Cp∪Int(Fr)/vectore+\nn∪···∪Int(F1)/vectore+\nn=Cp∪(K\\X/vectore+\nn)\n=Cp∪(K\\Xk).\nFigure 6 shows the action of polynomial map F1on the cone Cp.\nCp F1(Cp)\nF1K K\np p\nFigure 6. Behavior of the polynomial map F1overCp.\nWe show now (ii). As each polynomial map ˆfiappearing in the definition of Fksatisfies by\n4.a.2 (i) ˆfi(K\\X) =K\\X, we haveFk(K\\X) =K\\X. By 4.a.2(iii) ˆfi,k|Zk−1= idZk−126 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nfori= 1,...,r. AsXi⊂Zk−1fori= 1,...,k−1, we have ˆfi,k|/intersectiontextk−1\nj=1Xj= id/intersectiontextk−1\nj=1Xj, so\nFk(Y)\\/intersectiontextk−1\nj=1Xj⊂Fk(Y\\/intersectiontextk−1\nj=1Xj) for eachY⊂Rn. AsX⊂/intersectiontextk−1\nj=1Xj, we deduce by 4.a.2\nFk/parenleftBig\nCp∪/parenleftBig\nK\\k−1/intersectiondisplay\nj=1Xj/parenrightBig/parenrightBig\n=Fk(Cp)∪Fk/parenleftBig/parenleftBig\n(K\\X)\\k−1/intersectiondisplay\nj=1Xj/parenrightBig/parenrightBig\n⊃(Cp∪(K\\Xk))∪/parenleftBig\nFk(K\\X)\\k−1/intersectiondisplay\nj=1Xj/parenrightBig\n= (Cp∪(K\\Xk))∪/parenleftBig\n(K\\X)\\k−1/intersectiondisplay\nj=1Xj/parenrightBig\n=Cp∪/parenleftBig\nK\\k/intersectiondisplay\nj=1Xj/parenrightBig\n.\n4.a.4. Let us finish the proof of Proposition 4.1. Define for k= 1,...,sthe polynomial map\nˆFk:=ψ−1\nk◦Fk◦ψk:Rn→Rn.\nAsCpis a pointed cone, we have already constructed a polynomial m aph0:Rn→Rnsuch that\nh0(Rn) =Cp. We claim: the polynomial map\nh:=ˆFs◦···◦ˆF1◦h0\nsatisfiesh(Rn) =K\\X. It is enough to show: ( ˆFs◦···◦ˆF1)(Cp) =K\\X.\nUsing recursively 4.a.3 we deduce\nCp∪(K\\X1) =ˆF1(Cp)⊂K\\X\nCp∪(K\\(X1∩X2))⊂(ˆF2◦ˆF1)(Cp)⊂K\\X\n...\nCp∪/parenleftBig\nK\\s/intersectiondisplay\nj=1Xj/parenrightBig\n⊂(ˆFs◦···◦ˆF1)(Cp)⊂K\\X.\nTo illustrate this process Figure 7 shows how ˆF2acts onˆF1(Cp). AsK\\(/intersectiontexts\nj=1Xj) =K\\Xand\nCp⊂K\\X,\nK\\X=Cp∪/parenleftBig\nK\\/parenleftBigs/intersectiondisplay\nj=1Xj/parenrightBig/parenrightBig\n⊂(ˆFs◦···◦ˆF1)(Cp)⊂K\\X,\nsoh(K\\X) =K\\X, as required. /square\n4.b.Proof of Proposition 4.2. LetEbe a face of Kof dimension d≤n−2. We write\nx:= (x′,xn) := (y,z,xn)∈Rd×Rn−d−1×R. Assume K∩ {xn= 0}=E,K⊂ {xn≥0}\nand/vectoren∈Int(/vectorC(K)). Write Int( E) :={g1>0,...,gr>0} × {0} ⊂Rd× {0}where each\ngi∈R[y] :=R[x1,...,xd]. By Proposition 2.5 there exist positive numbers δ,∆ such that\n/vectorCv\n2δ\\{0} ⊂Int(/vectorC(K)) andCv\n2δ(E)⊂K⊂Cv\n∆(E). As/vectoren∈Int(/vectorC(K)), the hyperplanes spanned\nby the facets of Kare non-vertical. Let P∈R[x] be the product of linear equations of these\nhyperplanes, so ∂K⊂ {P= 0}. Fixε>0. By Proposition 2.10 there exists g∈R[x′] such that\ng >1 +ε√n−d−1\nδonRn−1and{xn≥g} ⊂ {P >1}. DenoteIε:= ]−ε,+ε[ and consider the\nadmissible tuple of polynomials\nˆgε= (g1,...,gr,xd+1+ε,...,xn−1+ε,ε−xd+1,...,ε−xn−1,g).\nIf we writem:=r+2(n−1−d), thenˆgεconsists ofm+1 polynomials. Rename gm+1:=g. The\nadmisible tuple ˆgεhas associated a polynomial Qˆgεconstructed in Lemma 3.2 and semialgebraicUNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 27\nCp\nChange of\ncoordinates\nψ2pCp\np\nZ2\nF2 ˆF2\nChange of\ncoordinates\nψ−1\n2Cp\np pCp\nFigure 7. Behavior of the polynomial map ˆF2.\nsetsA(ˆgε),S(ˆgε) andAd(ˆgε) provided in 3.e. Observe that Int( E) =Ad(ˆgε). By Lemma 3.2\nS(ˆgε)⊂ {xn≥gm+1} ⊂ {P >1}. For eachk≥1 consider the polynomial map\nfk:Rn→Rn, x:= (x′,xn) = (y,z,xn)/ma√s⊔o→(y,A(x)z,Bk(x)xn)\nwhereA:= (1+P2Qˆgε)2andBk:=A+Ak\n2. Note that ε,δ,∆>0,gm+1,ˆgεandPsatisfy the\nhypotheses of Theorem 3.4. Let k0≥1 be the positive integer constructed in Theorem 3.4(iv).\n4.b.1.Main claim: Let Tbe a semialgebraic set such that K\\X⊂T⊂K. Fork≥k0\nfk(T) =T∪Int(E)⊂K. (4.3)\nTo show (4.3) we prove the following facts for k≥k0:\nFact 1:fk(T)\\Int(T) = (T\\Int(T))∪Int(E).\nFact 2: Int(T)⊂fk(Int(T)).\nOnce they are proved and since Int( T) = Int(K), we conclude\nfk(T) = (fk(T)\\Int(K))∪(fk(T)∩Int(K)) = (T\\Int(K))∪Int(E)∪Int(K) =T∪Int(E)\nand equality fk(T) =T∪Int(E) follows.\n4.b.2.Proof of Fact 1. We show first: ( T\\Int(T))∪Int(E)⊂fk(T)\\Int(T).\nAs/vectorCv\nδ\\{0} ⊂Int(/vectorC(K)), Theorem 3.4(ii) provides\nS(ˆgε)⊂Cv\nδ(Ad(ˆgε))∩{xn>0} ⊂Int(E)+Int(/vectorC(K))⊂Int(K) = Int(T)⊂T.\nBy Theorem 3.4(iii) the inclusion Int( E)⊂fk(S(ˆgε))⊂fk(T) holds fork≥1, hence\nInt(E) = Int(E)\\Int(T)⊂fk(T)\\Int(T). (4.4)\nAsfk|T\\Int(T)= idT\\Int(T)because∂T⊂ {P= 0},\nT\\Int(T) =fk(T\\Int(T))⊂fk(T)❀T\\Int(T)⊂fk(T)\\Int(T) (4.5)\nand the inclusion ( T\\Int(T))∪Int(E)⊂fk(T)\\Int(T) follows from (4.4) and (4.5).28 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nTo provefk(T)\\Int(T)⊂(T\\Int(T))∪Int(E), pick a point x∈Tsuch thatfk(x)/\\e}a⊔io\\slash∈Int(T).\nIfx∈T\\Int(T), thenfk(x) =x∈T\\Int(T) becausefk|T\\Int(T)= idT\\Int(T). Ifx∈Int(T), then\nx∈S(ˆgε).\nOtherwise, as Int( T)⊂Cv\n∆(Int(E)), we deduce by Theorem 3.4(iv) that fk(x)∈Cv\n2δ({x})⊂\n{x}+Int(/vectorC(K))⊂Int(K) = Int(T), which is a contradiction.\nAsx∈S(ˆgε), we have by Theorem 3.4(iii)\nfk(x)∈fk(S(ˆgε))⊂Cv\n2δ(Int(E))⊂Int(E)+Int(/vectorC(K))⊂Int(E)∪Int(T),\nsofk(x)∈Int(E). Therefore, the inclusion fk(T)\\Int(T)⊂(T\\Int(T))∪Int(E) holds, as\nrequired. /square\n4.b.3.Proof of Fact 2. As Int(T) = Int(K), we have to check: Int( K)⊂fk(Int(K)). Its proof\nis long and requires a topological argument based on a result by Janiszewski [J].\nPick a point x0:= (x′\n0,x0n) := (y0,z0,x0n)∈Int(K), sox0n>0. Ifz0= 0, consider the\nintersection x0/vectoren∩K=x1/vectore+\nn, where the point x1:= (y0,0,r)∈∂Kmust satisfy 0 ≤r < x0n\nandx0/vectoren∩Int(K) = Int(x1/vectore+\nn). AsA|∂K= 1, we have Bk(y0,0,r)r=A(y0,0,r)+Ak(y0,0,r)\n2r=r.\nBy Theorem 3.4(i) lim xn→+∞Bk(y0,0,xn)xn= +∞. Asr < x 0n, there exists s > rsuch\nthatBk(y0,0,s)s=x0n, sofk(y0,0,s) = (y0,0,x0n) =x0. Note that ( y0,0,s)∈Int(K), so\nx0∈fk(Int(K)).\n4.b.3.1. By Fact 1 for T=Kwe havefk(K)\\Int(K) = (K\\Int(K))∪E⊂K, sofk(K)⊂K.\n4.b.3.2. We assume next that z0/\\e}a⊔io\\slash= 0 and let us prove: there exists x1∈Int(K)such that\nfk(x1) =x0for eachk≥k0.\nThe proof of 4.b.3.2 is conducted in several steps.\n4.b.3.3. Consider the 2-dimensional plane Π determined by t he points (y0,0,0), (y0,0,x0n) and\nx0. Let us show: fk(P)⊂PwhereP:=K∩Π.\nAsfk(x) =fk(y,z,xn) = (y,A(x)z,Bk(x)xn) forx:= (y,z,xn), we have fk(Π)⊂Π. Since\nfk(K)⊂K, it holds\nfk(P) =fk(K∩Π)⊂fk(K)∩fk(Π)⊂K∩Π =P.\n4.b.3.4. Set coordinates ( u,v) in Π with respect to the affine reference\nR:={O:= (y0,0,0);/vector w1= (0,z0\n/bardblz0/bardbl,0),/vector w2= (0,0,1)}.\nObserve that O+u/vector w1+v/vector w2= (y0,z0\n/bardblz0/bardblu,v) and\nfk(O+u/vector w1+v/vector w2) =/parenleftbig\ny0,A(y0,z0\n/bardblz0/bardblu,v)z0\n/bardblz0/bardblu,Bk(y0,z0\n/bardblz0/bardblu,v)v/parenrightbig\n≡/parenleftBig\nα(u,v)u,α(u,v) +α(u,v)k\n2v/parenrightBig\nR,\nwhereα(u,v) :=A(y0,z0\n/bardblz0/bardblu,v)∈R[u,v]. Consider the polynomial map\nGk:= (Gk1,Gk2) :R2→R2,(u,v)/ma√s⊔o→/parenleftBig\nα(u,v)u,α(u,v)+α(u,v)k\n2v/parenrightBig\n.\nNote thatx0≡(/bardblz0/bardbl,x0n)R=: (a,b)R, soa,b>0. Consider the algebraic curve\nYa:={α(u,v)u−a= 0}=G−1\nk1(a)⊂ {u>0}.\n4.b.3.5. We claim: r:= max{ε√\nn−d−1,a} ≥u0for each (u0,v0)∈Ya.\nIfu0> a, thenα(u0,v0)<1. AsP2is the square of a polynomial, Qˆgε(y0,z0\n/bardblz0/bardblu0,v0)<0.\nBy Lemma 3.2 we have ( y0,z0\n/bardblz0/bardblu0,0)∈A(ˆgε). In particular,z0\n/bardblz0/bardblu0∈In−d−1\nε, so\nu0≤ /bardbl(ε,(n−d−1)... ,ε)/bardbl=ε√\nn−d−1≤r.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 29\n4.b.3.6. Consider the convex polygon P0:=P∩ {0≤u≤r+ 1}and the singleton {q}:=\n∂P0∩{u=a}. Writeq:= (a,c)R. Let us check: Ya∩∂P0={q}.\nAs Π meets Int( K), we have∂P=∂K∩Π. AsYa∩({u= 0}∪{u=r+1}) =∅, then\nYa∩∂P0⊂Ya∩∂P⊂Ya∩∂K⊂Ya∩{P= 0} ⊂Ya∩{α= 1}=Ya∩{u=a}.\nThus,Ya∩∂P0=Ya∩∂P0∩{u=a} ⊂ {q}. Asq∈∂P0\\({u= 0}∪{u=r+1})⊂∂K, we\nhaveα(a,c) =A(q) = 1. Asq∈ {u=a}∩{α(u,v) = 1}, we conclude q∈Ya, soYa∩∂P0={q}.\n4.b.3.7. Given a connected topological space Tand different points p,q∈T, we say that K⊂T\nseparatespandqif these points belong to different connected components of T\\K. Given\nS⊂R2, we say that Sis ‘upperly unbounded’ to refer that it is unbounded in the di rection of\nthe second coordinate. We claim: There exists an upperly unbounded connected component Zof\nYa∩P0passing through qsuch thatZ\\{q} ⊂Int(K).\nTo prove this claim we will make use of Janiszewski’s Theorem (see [J] or [Bi, Thm. A]): If\nK1andK2are compact subsets of the plane R2whose intersection is connected, a pair of points\nthat is separated by neither K1norK2is neither separated by their union K1∪K2. The proof\nof our claim is conducted in several steps:\nv=M\n4v=Mu=a\nq0M\n3M\n22M\n3\nu= 0 u=r+1Z1ZℓP\nP′\n0\nYa∩P0\nFigure 8. Description of the fake situation.\nStep1.The line {u= 0} ⊂ {α(u,v)u−a<0}and the line {u=r+1} ⊂ {α(u,v)u−a>0}.\nThe first inclusion is clear. To prove the second denote ζ(u,v) :=α(u,v)u−aand observe\nthat ifζ(r+ 1,v)≤0, thenα(r+ 1,v)<1. AsP2is the square of a polynomial, we deduce\nQˆgε(y0,z0\n/bardblz0/bardbl(r+1),v)<0. By Lemma 3.2 we have\n/parenleftbig\ny0,z0\n/bardblz0/bardbl(r+1),0/parenrightbig\n∈A(ˆgε).\nIn particular,z0\n/bardblz0/bardbl(r+1)∈In−d−1\nεand\nr+1≤ /bardbl(ε,(n−d−1)... ,ε)/bardbl=ε√\nn−d−1≤r,\na contradiction. Consequently, {u=r+1} ⊂ {α(u,v)u−a>0}.\nStep2. LetM >0 be such that all the vertices of P0and all the upperly bounded connected\ncomponents of Ya∩P0are contained in {v<M\n4}. Consider the compact convex polygon\nP′\n0:=P0∩{v≤M} ⊂[0,r+1]×[0,M]. LetZ1,...,Z ℓbe the connected components of Ya∩P0.\nSuppose that none of them meets both ∂P0and{v=M}(see Figure 8). Assume Z1,...,Z sare\nthe upperly bounded connected components of Ya∩P0andZs+1,...,Z ℓare the remaining ones,\nso they meet {v=M}but they do not meet ∂P0. DefineK1:= (∂P0∩{v≤M\n4})∪/uniontexts\ni=1(Zi∩P′\n0)30 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nandK2:=∂P′\n0∪/uniontextℓ\ni=s+1(Zi∩P′\n0) (see Figures 9 and 10). Observe that K1∩K2=∂P0∩{v≤M\n4}\nis connected. Consider the positive real number\nǫ:= min/braceleftbig\ndist/parenleftbig\nK1,∂P′\n0∩/braceleftbig\nv≥M\n3/bracerightbig/parenrightbig\n,dist/parenleftbig/uniontextℓ\ni=s+1(Zi∩P′\n0),∂P′\n0∩/braceleftbig\nv≤2M\n3/bracerightbig/parenrightbig/bracerightbig\n.\nLet 0<ρ<ǫ\n2be such that q1:= (ρ,M\n2)∈ {ζ <0}andq2:= (r+1−ρ,M\n2)∈ {ζ >0}.\nWe haveζ(q1)<0 andζ(q2)>0. Consequently, K1∪K2= (Ya∩P′\n0)∪∂P′\n0separates the\npointsq1andq2.\nu=a\nq0M\n3M\n22M\n3\nu= 0 u=r+1Z1P\nP′\n0\nK1W2\nq1 q2\nFigure 9. Positions of K1and\nW2u=a\nq0M\n3M\n22M\n3\nu= 0 u=r+1ZℓP\nP′\n0\nK2\nW1q1 q2\nFigure 10. Positions of K2\nandW1\nStep3. Let us check: neitherK1norK2separates the points q1andq2.\nThe points q1,q2belong to both open connected subsets\nW1:=/braceleftbig\np∈Int(P′\n0) : 0<dist/parenleftbig\np,/parenleftbig\n∂P′\n0∩/braceleftbig\nv≤2M\n3/bracerightbig/parenrightbig/parenrightbig\n<ǫ\n2/bracerightbig\n,\nW2:=/braceleftbig\np∈Int(P′\n0) : 0<dist/parenleftbig\np,/parenleftbig\n∂P′\n0∩/braceleftbig\nv≥M\n3/bracerightbig/parenrightbig/parenrightbig\n<ǫ\n2/bracerightbig\nof Int(P′\n0) whereasK1∩W2=∅andK2∩W1=∅. Thus,q1,q2are separated neither by K1\nnor byK2, which contradicts Janiszewski’s Theorem.\nStep4. Consequently, there exists a connected component ZjofYa∩P0that meets both ∂P0and\nthe line{v=M}, as shown in Figure 11. As all the upperly bounded connected c omponents\nare contained in {v<M\n4}, we deduce Zjis upperly unbounded. In addition, Ya∩∂P0={q}, so\nq∈Zjand\nZj\\{q} ⊂Zj\\∂P0⊂P\\∂P= Int(P)⊂Int(K),\nas claimed in 4.b.3.7.\n4.b.3.8. We are ready to finish the proof of 4.b.3.2. By [BCR, 2 .9.10]Zjis the union of a finite\nsetFand finitely many Nash paths Γ ithat are Nash diffeomorphic to ]0 ,1[. We may assume\nthat Γ 1is upperly unbounded. Let q′∈Cl(Γ1)\\Γ1and letγ0: [0,1]→Zjbe a semialgebraic\npath such that γ0(0) =qandγ0(1) =q′. Letγ1: [1,+��[→Γ1∪ {q′}be a semialgebraic\nparameterization such that γ1(1) =q′and define\nβ:= (β1,β2) : [0,+∞[→Zj, t/ma√s⊔o→/braceleftBigg\nγ0(t) ift∈[0,1],\nγ1(t) ift∈[1,+∞[.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 31\nAs Γ1is upperly unbounded, lim t→∞β2(t) = +∞. We have\nGk◦β(t) =/parenleftBig\na,/parenleftBig/parenleftBiga\nβ1(t)/parenrightBig\n+/parenleftBiga\nβ1(t)/parenrightBigk/parenrightBigβ2(t)\n2/parenrightBig\nAs 0≤β1(t)≤r+1 fort∈[0,+∞[, we have\n0<a\nr+1+ak\n(r+1)k≤/parenleftBiga\nβ1(t)/parenrightBig\n+/parenleftBiga\nβ1(t)/parenrightBigk\nfort∈[0,+∞[. Consequently,\nlim\nt→∞/parenleftBig/parenleftBig/parenleftBiga\nβ1(t)/parenrightBig\n+/parenleftBiga\nβ1(t)/parenrightBigk/parenrightBigβ2(t)\n2/parenrightBig\n= +∞. (4.6)\nAsx0= (a,b)R∈Int(P0)∩q/vectore+\nn,q= (a,c)R∈∂P0and/vectoren∈/vectorC(P0), we have c < b. As\nq=Gk(q) = (Gk◦β)(0) and using (4.6), there exists t0∈]0,+∞[ such that ( Gk◦β)(t0) =\n(a,b)R=x0, so there exists x1:=β(t0)∈Zj\\{q} ⊂Int(K) such that Gk(x1) =x0and 4.b.3.2\nholds.\nv=Mu=a\nq u= 0 u=r+1Z1P\nP′\n0\nFigure 11. Description of the authentic situation.\n4.b.4.Conclusion of proof for Proposition 4.2. Let E1,...,Embe all the faces of Kof dimension\n≤n−2. We use the symbol ⊔to stress unions that involve only pairwise disjoint sets. W e have\nK∩X=/unionsqtextm\ni=1Int(Ei). By 4.b.1 there exist polynomial maps gℓ:Rn→Rnsuch that\ngℓ/parenleftBig\n(K\\X)⊔ℓ−1/unionsqdisplay\ni=1Int(Ei)/parenrightBig\n= (K\\X)⊔ℓ/unionsqdisplay\ni=1Int(Ei)\nforℓ= 1,...,m. Consequently, g:= (gm◦··· ◦g1) :Rn→Rnsatisfiesg(K\\X) = (K\\X)⊔\n(K∩X) =K, as required. /square\n4.c.Proof of Corollary 4.3. By Propositions 4.1 and 4.2 there exists a polynomial map\nf0:Rn→Rnsuch thatf0(Rn) =K. Letx′:= (x1,...,x n−1) and consider the polynomial map\nf1:Rn+1→Rn+1,\nx:= (x′,xn,xn+1)/ma√s⊔o→(x′,xn+1(xnxn+1−1),(xnxn+1−1)2+x2\nn),\nwhose image is {xn+1>0}(see [FG1, Ex. 1.4 (iv)]). Assume that /vectoren∈/vectorC(K) and let\nf2:Rn+1→Rn,(x1,...,x n,xn+1)/ma√s⊔o→f0(x1,...,x n)+xn+1/vectoren.\nWe havef2({xn+1>0}) = Int(K), so Int( K) is the image of the polynomial map f:= (f2◦f1) :\nRn+1→Rn, as required. /square32 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\n5.Interiors of convex polyhedra as polynomial images of Rn\nIn this section we prove Theorem 1.2. Each degenerate convex polyhedron K⊂Rncan be\nwritten in suitable coordinates as K=P×RkwherePis a non-degenerate convex polyhedron\nand Int(K) = Int(P)×Rk. Ifk≥n−1, then Int( K) is either Rnor an open half-space. The\nsecond case is a polynomial image of Rnby [FG1, Ex. 1.4 (iv)]. Thus, we will prove Theorem 1.2\nassuming in addition that the convex polyhedron Kis non-degenerate and has dimension n≥2.\nThe general strategy is the following. By Proposition 4.1 we know that if K⊂Rnis an\nunbounded non-degenerate convex polyhedron with n-dimensional recession cone /vectorC(K) andX\nis the union of the affine subspaces of Rnspanned by the faces of Kof dimension n−2, then\nK\\Xis a polynomial image of Rn. For each unbounded facet FofKwe devise a procedure to\n‘erase’ it from K\\Xin two steps: (1) firstwe ‘push’Int( F) ‘inside’ Int( K) to obtain a polynomial\nimageSofRncontained in K\\(X∪F) and (2) we fill the gap left between SandFto express\nK\\(X∪F) as a polynomial image of Rn. We ‘erase’ all the facets of the initial image K\\X\nto obtain Int( K) as a polynomial image of Rn. To lighten the proof of Theorem 1.2 we develop\nnext some preliminary work. In the following we write x′′:= (x1,...,x n−2),x′:= (x′′,xn−1)\nandx:= (x′,xn).\n5.a.Preliminary construction. We first introduce the type of polynomial maps that will\nallow us to push the interior of a given facet Fof a convex polyhedron Kinside Int( K).\nLemma 5.1. LetK⊂Rnbe a convex polyhedron of dimension nand letFbe a facet of K.\nAssume F⊂ {xn−1= 0},K⊂ {xn−1≥0}and/vectoren−1∈/vectorC(K). LetTbe a semialgebraic\nset obtained by removing the interior of some facets of KfromK\\Xand letF∈R[x]be a\npolynomial such that {F= 0}∩Int(F) =∅andFis identically zero on the facets of Kdifferent\nfromF. Consider the semialgebraic set R:={F= 0} ∩ {xn−1>0}and the polynomial map\nf0:Rn→Rn, x/ma√s⊔o→x+F2(x)/vectoren−1. Then\n(i)f0(Int(x0/vectore+\nn−1)) = Int(x0/vectore+\nn−1)andf0(x0) =x0for eachx0∈ {F= 0}.\n(ii)T∩R/vectore+\nn−1⊂f0(T)⊂T\\F.\nProof.(i) Writex0:= (x01,...,x 0n). Consider the continuous function\nψ:R→R, t/ma√s⊔o→t+F2(x0+t/vectoren−1).\nAsψ(0) = 0 and ψ(t)≥tfor eacht≥0, we have ψ(]0,+∞[) = ]0,+∞[, sof0(Int(x0/vectore+\nn−1)) =\nInt(x0/vectore+\nn−1) andf0(x0) =x0.\n(ii) Observe that f0(x/vectore+\nn−1)⊂x/vectore+\nn−1for eachx∈Rn. As/vectoren−1∈/vectorC(K) and⊂K, we deduce\nf0(T)⊂T. In addition, if x:= (x′′,xn−1,xn)∈Tis such that f0(x) =x+F2(x)/vectoren−1∈F, then\nxn−1= 0 andF(x) = 0, so\nx∈(K\\X)∩{xn−1= 0}∩{F= 0}= Int(F)∩{F= 0}=∅,\nwhich is a contradiction. Thus, f0(T)⊂T\\F.\nLet us prove now T∩R/vectore+\nn−1⊂f0(T). Pick a point x∈T∩R/vectore+\nn−1and writex=y+λ/vectoren−1\nwherey∈Randλ≥0. Observe that x/vectoren−1∩K=z/vectoren−1where either zbelongs to a facet of\nKdifferent from Forz:= (x′′,0,xn)∈Int(F). In the first case F(z) = 0, so by (i) f0(z) =z\nandx∈z/vectore+\nn−1∩T=f0(z/vectore+\nn−1∩T). In the second case, y/vectore+\nn−1⊂z/vectore+\nn−1∩T. AsF(y) = 0, we\nhave by (i) f0(y) =yandx∈y/vectore+\nn−1=f0(y/vectore+\nn−1)⊂f0(T), as required. /square\nIn order to take advantage of Lemma 5.1 we need a polynomial h∈R[x] with some added\nspecific characteristics, that we proceed to describe below .\n5.b.Pushing an open facet inside the interior of a convex polyhedron. Assume now\nthat a convex, unbounded polyhedron Kwithn-dimensional recession cone is placed in Rnso\nthat/vectoren−1,/vectoren∈/vectorC(K),F={xn−1= 0} ∩K⊂ {xn>0}andK⊂ {xn−1≥0}. Observe thatUNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 33\n/vectoren∈/vectorC(F). Denote the facets of KwithF1,...,Feand lethi= 0 be a non-zero linear equation\nof the hyperplane Hispanned by Fi. Suppose K={h1≥0,...,h e≥0},\n•F1,...,Fsare non-vertical and among them F1,...,Frare those non-vertical facets of\nKthat meet F,\n•Fs+1,...,Feare vertical and Fe=F.\nAs/vectoren∈/vectorC(K), we may assume /vectorhi(/vectoren) = 1 fori= 1,...,r, so thathi=hi(x′,0)+xn. Define\nbi:=hi−xn−1=hi(x′,0)−xn−1+xn, (5.1)\nb′\ni:=hi−2xn−1=hi(x′,0)−2xn−1+xn. (5.2)\nThe hyperplanes Bi:={bi= 0}andB′\ni:={b′\ni= 0}separate by Lemma 2.8 the facets FiandF\nand meet Int( K). Consider now the affine change of coordinates\nφi:Rn→Rn, x:= (x′,xn)/ma√s⊔o→(x′,xn+hi(x′,0)), (5.3)\nwhich satisfies B∗\ni:=φi(Bi) ={xn−xn−1= 0}andB′∗\ni:=φi(B′\ni) ={xn−2xn−1= 0}.\n5.b.1. Denote the union of all the facets of Kthat do not meet FwithG. By Lemma 2.2 and\nCorollary 2.3 there exists ε0∈Rsuch that\n0<ε0<min{1,dist({xn−1= 0},G)}andK∩{xn−1≤ε0} ⊂ {xn>0}.(5.4)\nAs eachBi∩K∩ {xn−1≤ε0} ⊂ {xn>0}, there exists by Lemma 2.2 δ >0 such that\nBi∩K∩{xn−1≤ε0} ⊂ {xn>δ}fori= 1,...,r. Setε:= min{ε0,δ\n2}>0.\n5.b.2. Define K0:=K∩{xn−1≤ε}and observe that Int( K0) = Int(K)∩{xn−1<ε}. Consider\nthe family of hyperplanes containing the non-vertical face ts ofKtogether with all hyperplanes\nB′\ni. By Proposition 2.10 there exists a polynomial Gisuch that\n{xn≥Gi} ⊂r/intersectiondisplay\nj=1{b′\nj◦φ−1\ni>1}∩s/intersectiondisplay\nk=1{hk◦φ−1\ni>1} ⊂ {b′\ni◦φ−1\ni>1}={xn>2xn−1+1}.(5.5)\nDefineBi:=Bi∩Int(K0). We claim:\nBi/vectoren∩Int(K)⊂Bi/vectore+\nn∪({b′\ni≤0}∩Int(K0))⊂Int(K0). (5.6)\nAs/vectoren∈/vectorC(K) andBi⊂Int(K0), we have\nBi/vectore+\nn⊂Int(K0). (5.7)\nIn addition, Bi(−/vectoren)+⊂ {bi≤0} ⊂ {b′\ni≤0}. Consequently,\nBi/vectoren∩Int(K) = (Bi/vectore+\nn∪Bi(−/vectoren)+)∩Int(K0)\n=Bi/vectore+\nn∪(Bi(−/vectoren)+∩Int(K0))⊂Bi/vectore+\nn∪({b′\ni≤0}∩Int(K0))⊂Int(K0).\n5.b.3. Write πn(Bi) ={gi,1>0,...,gi,m>0}where each gi,j∈R[x′] is a polynomial of degree\none. We may assume gi,1=xn−1. Consider the admissible tuple gi:= (gi,1,...,gi,m,gi,m+1)\nwheregi,m+1∈R[x′] is a polynomial satisfying\ngi,m+1>max/braceleftBig\nGi,1+|hi(x′,0)|/radicalBig\n|gi,1···gi,m|, i= 1,...,r/bracerightBig\n(5.8)\nand the associated semialgebraic sets A(gi) =πn(Bi) andS(gi)⊂Bi/vectore+\nn. In addition, by (5.5)\nwe havegi,m+1≥Gi≥2xn−1.\n5.b.4. We claim: hi(x′,0)<0onA(gi)⊂πn(Bi∩K0).\nPick a point x:= (x′,xn)∈Bi∩K0. Thenhi(x′,0) =xn−1−xn<ε−δ<0.34 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\n5.b.5. By the choice of ε >0 the non-vertical facets of K0areFi0:=Fi∩ {xn−1≤ε}for\ni= 1,...,rand all of them meet the facet FofK. By Lemma 2.9\nInt(K)/vectoren∩{xn−1<ε}= Int(K0)/vectoren=r/uniondisplay\ni=1(Bi∩Int(K0))/vectoren=r/uniondisplay\ni=1Bi/vectoren=r/uniondisplay\ni=1A(gi)/vectoren.(5.9)\n5.b.6. Denote S∗(gi) :=φi(S(gi)) ={(x′,xn+hi(x′,0)) : (x′,xn)∈S(gi)}fori= 1,...,r.\nThen there exists a polynomial P∈R[x′′,xn]with empty zero-set such that the zero-set Γof the\npolynomial R(x) :=xn−1P(x′′,xn)−1satisfies Γ⊂ {0<xn−1<ε},Γ/vectore+\nn−1⊂ {R≥0}and\nS(gi)⊂S∗(gi)⊂Γ/vectore+\nn−1∩{R>1}. (5.10)\nProof.The inclusion S(gi)⊂S∗(gi) holds because by 5.b.4 hi(x′,0)≤0 onA(gi). Writegi,j:=\n/a\\}bracke⊔le{⊔/vector aij,(x′,1)/a\\}bracke⊔ri}h⊔where/vector aij∈Rn. PickM0>1 such that /bardbl/vector aij/bardbl ≤M0for each pair ( i,j). We have\n|gi,j(x′)|=|/a\\}bracke⊔le{⊔/vector aij,(x′,1)/a\\}bracke⊔ri}h⊔| ≤ /bardbl/vector aij/bardbl/bardbl(x′,1)/bardbl ≤M0/radicalbig\n/bardblx′/bardbl2+1.\nIfxn−1≤εandM:=M0/radicalBig\n1+1\nε2, then\n|gi,j(x′)| ≤M0/radicalbig\n/bardblx′/bardbl2+1≤M0/radicalbig\n/bardblx′′/bardbl2+ε2+1≤M/radicalbig\n/bardblx′′/bardbl2+ε2≤M\nε(/bardblx′′/bardbl2+ε2).(5.11)\nPickx:= (x′,xn)∈S∗(gi), then (x′,xn−hi(x′,0))∈S(gi)⊂ {xn>0}. By Lemma 3.2(iii) we\nhave (x′,0)∈A(gi), sohi(x′,0)<0. By By Lemma 3.2(i) and (5.8)\nxn−hi(x′,0)≥gm+1(x′)/radicalbig\nxn−1gi,2(x′)···gi,m(x′)≥1√xn−1·1/radicalbig\ngi,2(x′)···gi,m(x′)−hi(x′,0).\nAs 0<ε<1, we deduce by (5.11)\nx2\nn+1≥xn≥1√xn−1·1/radicalbig\ngi,2(x′)···gi,m(x′)≥εm\n2\n√xn−1(√\nM/radicalbig\n/bardblx′′/bardbl2+ε2)m−1.\nConsequently,\nxn−1≥εm\nMm−1(x2n+1)2(/bardblx′′/bardbl2+ε2)m−1(5.12)\nfor each point ( x′,xn)∈S∗(gi). Define\nP:= 3Mm−1(x2\nn+1)2(/bardblx′′/bardbl2+ε2)m−1\nεm\nand observe that by (5.12) each S∗(gi)⊂Γ/vectore+\nn−1where\nΓ :=/braceleftBig\nxn−1=1\nP/bracerightBig\n.\nIn addition, Γ ⊂ {0<xn−1<ε}, Γ/vectore+\nn−1⊂ {R≥0}and\nS∗(gi)⊂/braceleftBig\nxn−1≥3\nP/bracerightBig\n={xn−1P−1≥2} ⊂ {R>1},\nas claimed. /square\n5.b.7. Let F:=R/producttextr\nj=1b′\nj/producttexte−1\nk=1hk∈R[x] be the product of the polynomial R, the linear\nequationsb′\njof thehyperplanes B′\njand thelinear equations hkof thehyperplanes Hkspannedby\nthefacets of Kexceptthat of F. Itholds {F= 0}= Γ∪/uniontextr\nj=1B′\nj∪/uniontexte−1\nk=1Hk. AsB′\njis aseparating\nhyperplane for FandFj, we have F∩B′\nj⊂F∩Fj⊂∂F. In addition, Γ ⊂ {0<xn−1<ε}, so\n{F= 0}∩Int(F) =∅.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 35\n5.b.8. Let Int( K)⊂T⊂K\\Xbe a semialgebraic set obtained by removing the interiors of\nsome facets FiofKfromK\\Xsuch that Fi/\\e}a⊔io\\slash=F. Define\nP:=T∩/parenleftBig\nΓ/vectore+\nn−1∪r/uniondisplay\nj=1{b′\nj≤0}∪e−1/uniondisplay\ni=1Fi/parenrightBig\n,\nT0:=T∩{xn−1≤ε}, (5.13)\nP0:=T0∩P.\nWe claim:\n(i){xn−1≥ε} ⊂Γ/vectore+\nn−1.\n(ii)T0\\F=P0∪/uniontextr\nj=1Bj/vectore+\nn.\nProof.(i) This inclusion follows from the fact that Γ ⊂ {0<xn−1< ε}can be understood as\nthe graph over the hyperplane {xn−1= 0}of the regular function1\nP, which depends on the\nvariables ( x′′,xn).\n(ii) Observe that Int( T0) = Int(K0). By the choice of εthe convex polyhedron K0satisfies\nthe hypothesis of Lemma 2.9, hence Int( T0)/vectoren= Int(K0)/vectoren=/uniontextr\nj=1Bj/vectoren. By (5.6) we have\nInt(T0) = Int(T0)∩(Int(T0)/vectoren) =r/uniondisplay\nj=1(Bj/vectoren∩Int(T0))\n⊂r/uniondisplay\nj=1(Bj/vectore+\nn∪({b′\nj≤0}∩Int(T0)))⊂P0∪r/uniondisplay\nj=1Bj/vectore+\nn.\nIn addition, ∂T0\\F⊂P∩T0=P0(use (i) to guarantee that T0∩{xn−1=ε} ⊂P0), so by (5.7)\nT0\\F= Int(T0)∪(∂T0\\F)⊂P0∪r/uniondisplay\nj=1Bj/vectore+\nn⊂T0\\F,\nas required. /square\nThe interest of the semialgebraic set Pcomes from the following result, which is illustrated\nin Figure 12.\nf0T\nFΓ ΓB′\ni B′\nixn−1=ε\nxn−1=εf0(T)⊃P\nFigure 12. Behavior of the polynomial map f0(Lemma 5.2).\nLemma 5.2. The polynomial map f0:Rn→Rn, x/ma√s⊔o→x+F2(x)/vectoren−1satisfiesP⊂f0(T)⊂T\\F.\nProof.The inclusion f0(T)⊂T\\Ffollows from Lemma 5.1 and 5.b.7. We prove next P⊂f0(T).\nPickx:= (x′′,xn−1,xn)∈Pand consider the intersection P∩x/vectoren−1. This intersection\nconsists of finitely many intervals of the line x/vectoren−1whose endpoints belong to {F= 0}=\nΓ∪/uniontextr\nj=1B′\nj∪/uniontexte−1\ni=1Fi, so they are fixed by f0. As lim xn−1→+∞f0(x′′,xn−1,xn) = +∞, we have\nby Corollary 2.13\nx∈P∩x/vectoren−1⊂f0(P∩x/vectoren−1)⊂f0(T).36 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nThus,P⊂f0(T), as required. /square\nThe image of the polynomial map f0is contained in T\\Fand contains P. The semialgebraic\nsetPleaves a ‘gap’ inside Tin a neighborhood of the facet F. Our next goal is to construct\nanother polynomial map to fill the gap that f0(T) leaves inside T\\F.\n5.c.Filling the interior gap of the convex polyhedron. LetK⊂Rnbe an unbounded\nconvex polyhedron with recession cone /vectorC(K) of dimension n. LetXbe the union of the affine\nsubspaces of Rnspanned by the faces of Kof dimension n−2.\nProposition 5.3. LetFbe one of the unbounded facets of Kand letInt(K)⊂T⊂K\\Xbe\na semialgebraic set obtained by removing the interiors of so me facets FiofKfromK\\Xsuch\nthatFi/\\e}a⊔io\\slash=F. Then there exists a polynomial map F:Rn→Rnsuch thatF(T) =T\\F.\nf0\nTF\nΓΓ\nB′\n1\nB1B′\n1\nB1xn−1=ε\nxn−1=ε\nf0(T)\nA(g1)A(g1)S(g1)S(g1)\nφ1Change of\ncoordinates\nB′∗\n1\nB∗\n1xn−1=εf1(S∗(g1))f∗\n1\nφ−1\n1\nChange of\ncoordinatesΓ∗B′∗\n1\nB∗\n1xn−1=εS∗(g1)\nTB′\n1\nB1\nFigure 13. Erasing an unbounded facet of a convex polyhedron.UNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 37\nProof.Assume first that Kis placed as described in 5.b, take into account all consider ations\ndeveloped thereafter and keep the used notations. We have co nstructed a polynomial map\nf0:Rn→Rnsuch that P⊂f0(T)⊂T\\F(see Lemma 5.2). By 5.b.8(i)\nT1:=T∩{xn−1≥ε}=P∩{xn−1≥ε}.\nFix 1≤i≤rand consider the polynomial\nPi0:=r/productdisplay\nj=1(b′\nj◦φ−1\ni)s/productdisplay\nk=1(hk◦φ−1\ni).\nBy Lemma 3.2 and equations (5.5) and (5.8) we have S∗(gi) =φi(S(gi))⊂ {Pi0>1}. For each\nT⊂Rnwe denoteT∗the setφi(T). It holds\n{Pi0= 0}=r/uniondisplay\nj=1B′∗\nj∪s/uniondisplay\nk=1H∗\nk.\nDefinePi1:=R◦φ−1\ni. By (5.10) S(gi)⊂ {R >1}, henceS∗(gi)⊂ {Pi1>1}. DefinePi:=\n(Pi0Pi1)2and note that\nS(gi)⊂S∗(gi)⊂ {Pi0>1}∩{Pi1>1} ⊂ {Pi>1}.\nConsider the polynomial maps\nfi:= (fi1,...,fin) :Rn→Rn,(x′,xn)/ma√s⊔o→(x′,xn(1+Pi(x)Qgi(x))2+2xn−1(Pi(x)Qgi(x))2)\nand\nˆfi:=φ−1\ni◦fi◦φi. (5.14)\nNotethat g:=gi,P:=Pi,gm+1:=gi,m+1andh:= 2xn−1satisfythehypothesesofTheorem3.3.\n5.c.1. We claim: each polynomial map ˆfisatisfies ˆfi(T1) =T1.\nToprovethat ˆfi(T1) =T1it is enoughtoshow: fi(T∗\n1) =T∗\n1. Itholds T∗\n1⊂ {xn−1≥ε,xn≥0}.\nAsS(gi) ={Qgi≤0,xn≥0} ⊂ {xn−1≤ε}, thepolynomial Qgiispositiveon T∗\n1, aswell asPi,\nwhich is a square, and xn−1. Thus, the inclusion fi(T∗\n1)⊂T∗\n1holds by Theorem 3.3(ii) because\n/vectoren∈/vectorC(K). As the non-vertical facets of K∗are contained in {Pi= 0}and by Theorem 3.3(iii)\nlimxn→∞fi(x′,xn) = +∞for eachx′∈ {xn−1≥0}, we deduce by Corollary 2.13 T∗\n1⊂fi(T∗\n1).\n5.c.2. Let us study the behavior of ˆfionP0∪/uniontexti−1\nj=1Bj/vectore+\nn. We claim:\nP0∪i/uniondisplay\nj=1Bj/vectore+\nn⊂ˆfi/parenleftBig\nP0∪i−1/uniondisplay\nj=1Bj/vectore+\nn/parenrightBig\n⊂ˆfi(T0\\F)⊂T0\\F. (5.15)\nBy 5.b.8(ii) to prove the previous chain of inclusions it is e nough to show\nP∗\n0∪i/uniondisplay\nj=1B∗\nj/vectore+\nn⊂fi/parenleftBig\nP∗\n0∪i−1/uniondisplay\nj=1B∗\nj/vectore+\nn/parenrightBig\nandfi(T∗\n0\\F∗)⊂T∗\n0\\F∗. (5.16)\nPick a point x:= (x′,xn)∈P∗\n0∪/uniontexti−1\nj=1B∗\nj/vectore+\nnand consider the ray P∗\n0,x:=x/vectoren∩P∗\n0⊂ {xn≥0},\nwhich is a finite union of intervals inside the ray x/vectoren∩{xn>0}whose endpoints belong to\nΓ∗∪r/uniondisplay\nj=1B′∗\nj∪s/uniondisplay\nk=1H∗\nk={Pi= 0},\nso they are fixed by fi. In addition, by Theorem 3.3(iii)\nlim\nt→∞fin(x′,t) = +∞,\nbecausex′∈ {xn−1≥0}, hence by Corollary 2.13 P∗\n0x⊂fi(P∗\n0x). Thus,\nP∗\n0∪i−1/uniondisplay\nj=1B∗\nj/vectore+\nn⊂fi/parenleftBig\nP∗\n0∪i−1/uniondisplay\nj=1B∗\nj/vectore+\nn/parenrightBig\n. (5.17)38 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\nObserve that B∗\ni/vectoren=Bi/vectorenbecauseπn(B∗\ni) =πn(Bi) =A(gi). In addition, by (5.3) B∗\ni=\n{xn−xn−1= 0}andB′∗\ni={xn−2xn−1= 0}, soB∗\ni/vectore+\nn={xn−xn−1≥0}andB′∗\ni/vectore+\nn=\n{xn−2xn−1≥0}. By Theorem 3.3(i)\nB∗\ni/vectore+\nn∩B′∗\ni/vectore+\nn=A(gi)/vectore+\nn∩{xn≥2xn−1} ⊂fi(S(gi)).\nAsB′\ni(−/vectoren)+∩T={b′\ni≤0}∩T⊂P, we have B∗\ni/vectore+\nn∩B′∗\ni(−/vectoren)+⊂P∗\n0. By (5.10)\nS(gi)⊂Γ∗/vectore+\nn∩A(gi)/vectore+\nn∩{xn>0} ⊂Γ∗/vectore+\nn∩T∗∩{0<xn−1<ε} ⊂P∗\n0.\nConsequently, by (5.13) and (5.17)\nB∗\ni/vectore+\nn= (B∗\ni/vectore+\nn∩B′∗\ni(−/vectoren)+)∪(B∗\ni/vectore+\nn∩B′∗\ni/vectore+\nn)⊂P∗\n0∪fi(S(gi))⊂fi(P∗\n0).\nTherefore, P∗\n0∪/uniontexti\nj=1B∗\nj/vectore+\nn⊂fi/parenleftBig\nP∗\n0∪/uniontexti−1\nj=1B∗\nj/vectore+\nn/parenrightBig\n.\nLet us check next: fi(T∗\n0\\F∗)⊂T∗\n0\\F∗.\nBy Theorem 3.3(i) and 5.b.8(ii)\nfi(S(gi))⊂A(gi)/vectore+\nn∩{xn≥xn−1}=B∗\ni/vectore+\nn⊂T∗\n0\\F∗.\nThe polynomial Qgiis positive on {xn≥0,xn−1≥0}\\S(gi), as well as Pi, which is a square,\nandxn−1. Thus, the inclusion fi(T∗\n0\\(F∗∪S(gi)))⊂T∗\n0\\F∗holds by Theorem 3.3(ii) because\n/vectoren∈/vectorC(K). We conclude fi(T∗\n0\\F∗)⊂T∗\n0\\F∗.\n5.c.3. Define F:=ˆfr◦···◦ˆf1◦f0. By Lemma 5.2\nP0∪T1⊂P⊂f0(T)⊂T\\F. (5.18)\nBy 5.c.1 and (5.15)\nˆfi(T\\F) =ˆfi(T1∪(T0\\F))⊂T1∪(T0\\F) =T\\F\nfori= 1,...,r. Thus, by (5.18)\nF(T)⊂(ˆfr◦···◦ˆf1)(T\\F)⊂T\\F. (5.19)\nBy 5.b.8, (5.15), (5.18) and (5.19) we deduce\nT0\\F=P0∪r/uniondisplay\nj=1Bj/vectore+\nn⊂(ˆfr◦···◦ˆf1)(P0)⊂F(T)⊂T\\F.\nIn addition, by 5.c.1 and (5.18) we have T1⊂F(T)⊂T\\F. Consequently,\nT\\F= (T0\\F)∪T1⊂F(T)⊂T\\F,\nsoF(T) =T\\F, as required. /square\nFigure 13 shows the combined action of the polynomial map f0:T→f0(T) appearing in\nLemma 5.2 and the polynomial map ˆf1:f0(T)→T\\Fconstructed in (5.14).\n5.d.Proof of Theorem 1.2. By Theorem 4.1 there exists a polynomial map f0:Rn→Rn\nsuch thatf0(Rn) =K\\XwhereXis the union of the affine subspaces of Rnspanned by the\nfaces ofKof dimension n−2. LetF1,...,Fmbe the facets of K. By Proposition 5.3 there exists\na polynomial map Fi:Rn→Rnsuch that\nFi/parenleftBig\n(K\\X)\\i−1/uniondisplay\nj=1Fj/parenrightBig\n= (K\\X)\\i/uniondisplay\nj=1Fj.\nfori= 1,...,m. Consider the polynomial map f:= (Fm◦···◦F1◦f0) :Rn→Rn. Thus,\nf(Rn) = (K\\X)\\r/uniondisplay\nj=1Fj= Int(K),\nas required. /squareUNBOUNDED CONVEX POLYHEDRA AS POLYNOMIAL IMAGES OF EUCLIDE AN SPACES 39\nAppendix A.Some basic inequalities\nSome useful inequalities concerning finite collections of p ositive numbers have been used in\nSection 3. We collect them in the following lemma for easy ref erence.\nLemma A.1. Lety1,...,ymbe positive real numbers and fix 1≤i≤m. Then the following\ninequalities hold:\n(i)y1+···+ym+1\ny1···ym≥m+1>1.\n(ii)y1+···+ym+1\ny1···ym≥yi+mm/radicalbigg1\nyi.\n(iii)/parenleftBig\ny1+···+ym+1\ny1···ym/parenrightBigm\nyi>mm≥1.\nProof.(i) Denotez:=/producttextm\ni=1yi. It is enough to show\nz≤1+z/summationtextm\ni=1yi\nm+1. (A.1)\nConsider the positive real numbers zi:=yizfori= 1,...,mandzm+1= 1. By the arithmetic-\ngeometric inequality\nm+1/radicaltp/radicalvertex/radicalvertex/radicalbtm+1/productdisplay\ni=1zi≤/summationtextm+1\ni=1zi\nm+1.\nAs/producttextm+1\ni=1zi=zm+1and/summationtextm+1\ni=1zi= 1+z/summationtextm\ni=1yi, inequality (A.1) holds.\n(ii) By the arithmetic-geometric inequality\nm/radicalbigg1\nyi=m/radicalBigg\n1\ny1···ym/productdisplay\nj/\\egatio\\slash=iyj≤1\ny1···ym+/summationtext\nj/\\egatio\\slash=iyj\nm,\nso the statement holds.\n(iii) Using (ii) we have\n/parenleftBig\ny1+···+ym+1\ny1···ym/parenrightBigm\nyi≥/parenleftBig\nyi+mm/radicalbigg1\nyi/parenrightBigm\nyi≥ym+1\ni+mm>mm,\nas required. /square\nReferences\n[AG1] C. Andradas, JM. Gamboa: A note on projections of real a lgebraic varieties. Pacific J. Math .115(1984),\n1–11.\n[AG2] C. Andradas, JM. Gamboa: On projections of real algebr aic varieties. Pacific J. Math .121(1986),\n281–291.\n[Be] M. Berger: Geometry. I & II. Universitext . Springer-Verlag, Berlin (1987).\n[Bi] R.H. Bing: Generalizations of two theorems of Janiszew ski.Bull. Amer. Math. Soc. 51(1945), 954–960.\n[BCR] J. Bochnak, M. Coste, M.F. Roy: Real algebraic geometr y.Ergeb. Math. 36. Springer-Verlag, Berlin\n(1998).\n[Fe] J.F. Fernando: On the one dimensional polynomial and re gular images of Rn.J. Pure Appl. Algebra 218\n(2014), no. 9, 1745–1753.\n[FG1] J.F. Fernando, J.M. Gamboa: Polynomial images of Rn.J. Pure Appl. Algebra 179(2003), no. 3, 241–254.\n[FG2] J.F. Fernando, J.M. Gamboa: Polynomial and regular im ages ofRn.Israel J. Math. 153(2006), 61–92.\n[FGU1] J.F. Fernando, J.M. Gamboa, C. Ueno: On convex polyhe dra as regular images of Rn.Proc. London\nMath. Soc. (3)103(2011), 847–878.\n[FGU2] J.F. Fernando, J.M. Gamboa, C. Ueno: The openquadran tproblem: Atopological proof. A Mathematical\ntribute to Professor Jos´ e Mar´ ıa Montesinos Amilibia. Departamento de Geometr´ ıa y Topolog´ ıa. Facultad\nde Ciencias Matem´ aticas, UCM (2016) 137–350.\n[FU1] J.F. Fernando, C. Ueno: On the set of points at infinity o f a polynomial image of Rn.Discrete Comput.\nGeom.52(2014), no. 4, 583–611.\n[FU2] J.F. Fernando, C. Ueno: On complements of convex polyh edra as polynomial and regular images of Rn.\nInt. Math. Res. Not. IMRN2014, no. 18, 5084–5123.40 JOS ´E F. FERNANDO, J.M. GAMBOA, AND CARLOS UENO\n[FU3] J.F. Fernando, C. Ueno: On the complements of 3-dimens ional convex polyhedra as polynomial images\nofR3.Internat. J. Math. 25, no. 7, 1450071, 18 pp, (2014).\n[FU4] J.F. Fernando, C. Ueno: On complements of convex polyh edra as polynomial images of Rn.Preprint\nRAAG (2015). arXiv:1412.5107\n[FU5] J.F. Fernando, C. Ueno: A short proof for the open quadr ant problem. J. Symbolic Comput. 79(2017),\nno. 1, 57–64.\n[G] J.M. Gamboa: Reelle Algebraische Geometrie, June, 10th−16th(1990), Oberwolfach.\n[GRS] M. J. Gonz´ alez-L´ opez, T. Recio, F. Santos: Parametr ization of semialgebraic sets. Symbolic computation,\nnew trends and developments (Lille, 1993). Math. Comput. Simulation 42(1996), no. 4–6, 353–362.\n[HRR] J. Heintz, T. Recio, M.-F. Roy: Algorithms in real alge braic geometry and applications to computational\ngeometry. Discrete and computational geometry (New Brunswick, NJ, 1989/1990), 137–163, DIMACS\nSer.Discrete Math. Theoret. Comput. Sci. ,6, Amer. Math. Soc., Providence, RI (1991).\n[KPS] K. Kubjas, P.A. Parrilo, B. Sturmfels: How to flatten a s occer ball. Preprint (2016).arXiv:1606.02253\n[J] Z. Janiszewski: Sur les coupures du plan faites par les co ntinus,Prace Matematyczno-Fizyczne 26(1913),\n11–63.\n[Mo] T. S. Motzkin: The real solution set of asystem of algebr aic inequalities is the projection of ahypersurface\nin one more dimension. 1970 Inequalities, II (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967)\npp. 251–254 Academic Press, New York.\n[NDS] J. Nie, J. Demmel, B. Sturmfels: Minimizing polynomia ls via sum of squares over the gradient ideal.\nMath. Program. 106(2006), no. 3, Ser. A, 587-606.\n[PS] P.A. Parrilo, B. Sturmfels: Minimizing polynomial fun ctions.Algorithmic and quantitative real algebraic\ngeometry (Piscataway, NJ, 2001), 83-99, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. ,60, Amer.\nMath. Soc., Providence, RI, 2003.\n[P] D. Pecker: On the Elimination of Algebraic Inequalities ,Pacific J. Math. ,146(1990), 305–314.\n[R] T.R. Rockafellar: Convex analysis. Princeton Mathematical Series ,28. Princeton University Press,\nPrinceton, N.J (1970).\n[Sch] M. Schweighofer: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM\nJ. Optim. 17(2006), no. 3, 920–942.\n[S] G. Stengle: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann. 207, (1974)\n87–97.\n[U] C. Ueno: On convex polygons and their complements as imag es of regular and polynomial maps of R2.\nJ. Pure Appl. Algebra 216(2012), no. 11, 2436–2448.\n[VS] H.H. Vui, P.T. So’n: Global optimization of polynomial s using the truncated tangency variety and sums\nof squares. SIAM J. Optim. 19(2008), no. 2, 941-951.\n[Z] G.M. Ziegler: Lectures on Polytopes. Graduate Texts in Mathematics 152. Springer-Verlag, New-York\n(1995).\nDepartamento de ´Algebra, Facultad de Ciencias Matem ´aticas, Universidad Complutense de\nMadrid, 28040 MADRID (SPAIN)\nEmail address :josefer@mat.ucm.es\nDepartamento de ´Algebra, Facultad de Ciencias Matem ´aticas, Universidad Complutense de\nMadrid, 28040 MADRID (SPAIN)\nEmail address :jmgamboa@mat.ucm.es\nDipartimento di Matematica, Universit `a degli Studi di Pisa, Largo Bruno Pontecorvo, 5, 56127\nPISA (ITALY)\nEmail address :cuenjac@gmail.com"    },    {        "title": "2401.12563v1.Pitfalls_of_Exchange_Correlation_Functionals_in_Descriptions_of_Magnetism__Cautionary_Tale_of_the_FeRh_Alloy.pdf",        "content": "Pitfalls of Exchange-Correlation Functionals in Descriptions of Magnetism:\nCautionary Tale of the FeRh Alloy\nShishir Kumar Pandeya, Saikat Debnathb, Zhanghao Zhouyina,c, Qiangqiang Gua,d\naAI for Science Institute, Beijing, China\nbM. V . College, Buxar, Bihar, India\ncCollege of Intelligence and Computing, Tianjin University, Tianjin, 300350, Bihar, China\ndSchool of Mathematical Science, Peking University, Beijing, 100871, China\nAbstract\nThe magnetic ground state of FeRh is highly sensitive towards the lattice constant. This, in addition to partially filled d-shells of\nFe and Rh, posed a significant challenge for Density Functional Theory (DFT) calculations in the past. Here, we have investigated\nthe performance of various exchange-correlation (XC) functionals within the DFT formalism for this challenging binary alloy. We\nhave employed Local Spin Density Approximation (LSDA), various Generalized Gradient Approximations (GGAs), and newly\ndeveloped Strongly Constrained and Appropriately Normed (SCAN) meta-GGA functional. Our results show the limitations of\nany single functional in capturing the intricate interplay of structural, electronic, and magnetic properties in FeRh. While SCAN\ncan accurately describe some magnetic features and phonon dispersion, it significantly overestimates the Fe-Fe magnetic interac-\ntions, leading to an unreasonable magnetic ordering temperature. Conversely, the Perdew-Burke-Ernzerhof (PBE) GGA exhibits\nthe opposite behavior. These findings highlight the challenges in simulating materials with partially filled d-shells using DFT,\nunderscoring the crucial need for developing a versatile XC functional that can e ffectively account for the multifaceted nature of\nsuch systems.\nKeywords: Magnetostructural transition, DFT, Phonon, Monte-Carlo simulations, Binary alloy\n1. Introduction\nTransition metal compounds have become a focal point of\nintense research due to their remarkable and exotic properties.\nSome examples include various forms of magnetism Middey\net al. (2012); Mahana et al. (2020); Gu et al. (2023); Pandey\net al. (2023, 2022); Pandey and Feng (2022); Pandey et al.\n(2021); Zhou et al. (2021) and superconductivity Liu et al.\n(2022); Lee et al. (2006) and di fferent kinds of phase transi-\ntions driven by external perturbations such as doping Pandey\net al. (2017); Lee et al. (2006) and temperature. Magnetic tran-\nsitions in these compounds are typically second-order, charac-\nterized by a continuous evolution from one magnetic state to\nanother without any discontinuous changes in structural param-\neters. While first-order transitions can occur, their presence\nremains relatively rare. In these magnetostructural first-order\nphase transitions, a simultaneous change in the magnetic and\nstructural phase of the material with external stimuli like tem-\nperature, magnetic fields, or strain originates from a strong cou-\npling between lattice and magnetic degrees of freedom. The\npurported technological potential of these first-order magnetic\ntransitions in magnetic recording Thiele et al. (2003) magne-\ntocalorics Liu et al. (2016), spintronics Wang et al. (2020), mag-\nnetorestrictions O’Handley et al. (2000), heat-assisted magnetic\nEmail addresses: shishir.kr.pandey@gmail.com (Shishir Kumar\nPandey), guqq@pku.edu.cn (Qiangqiang Gu)recording Challener et al. (2009), antiferromagnetic (AFM)\nspintronics Yan et al. (2020); Feng et al. (2019), and room-\ntemperature AFM memory resistor Marti et al. (2014) makes\nthem even more fascinating.\nOne particularly interesting example of such magnetostruc-\ntural phase transition occurs for a transition metal binary alloy,\nFeRh Fallot and Hocart (1939); Kouvel and Hartelius (2004).\nBCC-based (CsCl-type) ordered phase of FeRh, upon lower-\ning the temperature, first goes through a paramagnetic to ferro-\nmagnetic (FM) second-order phase transition at ∼TC=670 K.\nFurther lowering of temperature then leads to an FM to AFM\ntransition at∼TN=370 K. A sudden increase in resistivity\nand∼1% volume collapse with lowering of temperature in\nAFM phase reveals first-order nature of the AFM-FM transi-\ntion Kouvel and Hartelius (2004). The reversible nature of this\nphase transition has also been investigated with the application\nof an external magnetic field both in bulk and thin films Kou-\nvel (2004); Maat et al. (2005). What makes this material par-\nticularly interesting in the context of device applications is its\nAFM-FM transition being close to the room temperature.\nEfforts to investigate phase transitions in FeRh have yielded\nconflicting perspectives, with some researchers suggesting si-\nmultaneous structural and magnetic changes Radu et al. (2010);\nDerlet (2012) while others contend that the magnetic transition\nprecedes unit cell expansion Gruner et al. (2003a); Ju et al.\n(2004); Mryasov (2005); Gu and Antropov (2005a); Quirin\net al. (2012). To gain a deeper understanding of such scenar-\nPreprint submitted to Elsevier January 24, 2024arXiv:2401.12563v1  [cond-mat.mtrl-sci]  23 Jan 2024ios, Density Functional Theory (DFT)-based calculations are\noften employed for their precision and reliability. However,\nin the case of FeRh, prior DFT-based calculations indicate an\nAFM-FM transition at a significantly larger volume Yamada\net al. (2006); Gu and Antropov (2005b); Moruzzi and Marcus\n(1992); Gruner et al. (2003b); Mryasov (2005); Aschauer et al.\n(2016) compared to the experimentally observed 1 % change.\nDespite the magnetic moments of Fe /Rh ions, obtained in these\ncalculations, aligning well with experimental data Gruner et al.\n(2003a); Gu and Antropov (2005a); Sandratskii and Mavropou-\nlos (2011); Kudrnovsk ´y et al. (2015); Yamada et al. (2006);\nMoruzzi and Marcus (1992); Aschauer et al. (2016); Wolloch\net al. (2016), it is surprising that DFT calculations fail to cap-\nture the AFM-FM transition. In instances where DFT-derived\nparameters are used for modeling, there is a risk of reaching in-\naccurate conclusions. Therefore, a reassessment to identify the\nissues with DFT calculations in this system is warranted.\nCentral to these DFT calculations are exchange-correlation\n(XC) functionals such as local spin density approxima-\ntion (LSDA) or various generalized gradient approximations\n(GGAs). It has become nearly established that these functionals\ninadequately treat the correlated dorbitals of transition metals,\nleading to issues in describing materials like strongly correlated\nsuperconductors Pickett (1989) and nickel oxide Sawatzky and\nAllen (1984). Such erroneous description arises mainly due\nto self-interaction error Svane (1992) often causing inconsis-\ntency between theory and experiments. In this context, alter-\nnative approaches like DFT +UAnisimov et al. (1991, 1997)\nand semi-local meta-GGA density functionals such as strongly\nconstrained and appropriately normed (SCAN) functional Sun\net al. (2015); Zhang et al. (2018) have demonstrated notable\nsuccess in describing the properties of various materials Sun\net al. (2016); Tran et al. (2016a); Isaacs and Wolverton (2018);\nLane et al. (2018). SCAN, in particular, excels in capturing the\nXC energy of atoms and is computationally e fficient compared\nto accurate hybrid functionals Tran et al. (2016b). Therefore, it\nis worthwhile to investigate whether the previous discrepancies\nof DFT calculations for FeRh can be mitigated by employing\ndifferent XC functionals. If so, understanding the extent of the\ndifferences in results across various functionals is crucial. It’s\nnoteworthy that this issue has also been recently highlighted\nin a comprehensive DFT study by Bosoni et al. Bosoni et al.\n(2023).\nIn this study, we conducted a comparative analysis of re-\nsults obtained from the meta-GGA SCAN functional, LDA, and\nvarious GGA functionals. We find that SCAN provides a bet-\nter description of volume expansion accompanying AFM-FM\ntransition and magnitude of magnetic moments of Fe /Rh atoms\nin FeRh than any other functional. While the phonon disper-\nsions from SCAN and revised-PBE (RPBE) exhibit close agree-\nment, there are qualitative di fferences in the spin-polarized\nband structures between the two cases. We attribute this dispar-\nity to the larger exchange splitting observed in the band struc-\nture calculations using SCAN. However, magnetic interactions\nbetween Fe atoms ( JFe-Fe) in the AFM ground state, calculated\nusing the Frozen Magnon approach, are highly overestimated\nresulting in unreasonably large magnetic ordering temperaturein our Monte-Carlo simulations. In this scenario, the PBE func-\ntional performs reasonably well. Our results emphasize that\nthe intricately intertwined structural, electronic, and magnetic\nproperties of FeRh cannot be accurately described by a single\nfunctional. This underscores the need for further development\nof a versatile exchange-correlation functional that can e ffec-\ntively capture the multifaceted nature of such materials.\n2. Methodology\n2.1. Total energy and phonon calculations\nDensity-functional theory calculations are performed using\nprojector-augmented wave method Kresse and Joubert (1999);\nBl¨ochl (1994), implemented within Vienna ab initio simula-\ntion package (V ASP) Kresse and Furthm ¨uller (1996). Total en-\nergy calculations are done with LDA, GGA-based functionals\nnamely PBE Perdew et al. (1996), revised PBE (RPBE) Ham-\nmer et al. (1999), revised PBE for solids (PBEsol) Perdew\net al. (2008), and meta-GGA functional namely SCAN Sun\net al. (2015). In GGA pseudopotentials of Fe and Rh, 3 d74s1\nand 4 p64d85s1are treated as valence states. We start with a\nsixteen-atom conventional magnetic unit cell of cubic crystal\nstructure (space group: P¯m3m, no: 221) of FeRh made from\nthe two-atom primitive cell with lattice constant 2.99 Å. Given\nthe importance of spin-lattice coupling in this material, full\noptimization of crystal structure was performed with each of\nthe pseudopotentials after imposing the G-type AFM ground\nstate (every Fe-Fe first neighbors are antiferromagnetically cou-\npled) observed experimentally at low temperatures. Energy and\nHellmann-Feynman force convergence criteria of 10−7eV and\n10−4eV/Å respectively are considered during the optimization\nwith the aim of phonon calculations in the next step. Plane-\nwave cuto ffenergy 550 eV and 9 ×9×9Γ-centered k-mesh,\nGaussian smearing width of 0.05 eV are considered in our cal-\nculations. The results obtained are well converged with respect\nto these parameters.\nStructural optimizations with di fferent XC-functional lead to\nquite small changes in the lattice constants except for the LDA\nas shown in Table 1. The total energy of the FM state and\nphonon dispersion for both AFM and FM states are calculated\nwith PBE, RPBE, and SCAN functionals with respective opti-\nmized structures. We employed the finite displacement method\nin phonon calculations and the post-processing is done using\nPhonopy Togo et al. (2023); Togo (2023) software.\n2.2. Estimation of magnetic interactions and Monte Carlo sim-\nulations\nTo quantify the role of XC functionals on the magnetic\nground state, exchange interactions between Fe atoms in the\nAFM ground state are estimated with the Frozen-magnon ap-\nproach (FMA) for both PBE and SCAN functionals. In this\nmethod, the spin spiral state is represented with propagation\nvector ( q) defined in the reciprocal space of the lattice. Each\nspin spiral state is equivalent to a magnetic configuration char-\nacterized by,\nei=(sinθcos (q·Ri),sinθsin (q·Ri),cosθ)\n2where Ri=R+τν=Rν+R, and R,τνare the lattice and the\nbasis vectors respectively. θis the polar angle made by the spin\nwith the zaxis. Using noncollinear magnetism formulation im-\nplemented within the V ASP package, energy corresponding to\neach spin-spiral state E(q) is calculated with 16 atom AFM unit\ncell. Assuming a localized spin at the Fe site, the results are\nthen mapped onto an extended classical Heisenberg Hamilto-\nnian,\nH=−X\ni,jJi jei·ej (1)\nwhere Ji jis the exchange interaction between the Fe spins\nat sites iand j,eiandejare the unit vectors at these two\nsites. Within this Heisenberg model, the energy of any frozen-\nmagnon configurations represented by qcan be given by,\nE(θ,q)=E0(θ)−sin2(θ)J(q)\nFor simplicity, in our calculations, we restricted the spins to lie\nin the xyplane. Fourier transform of Ji jin the above equation\ncan be defined as,\nJµν(q)=δµνJµµ−X\nRJµ[ν+R]e−i(τµ−τν−R)·q\nBy taking the inverse Fourier transform, we can obtain, the real\nspace interaction parameters as,\nJi j=1\nNX\nqJµiνj(q)ei(µi−νj−R)·q\nThe magnitude of spins is absorbed in the obtained J’s which\nare then used in classical Monte Carlo (MC) simulations to es-\ntimate the magnetic ordering temperature. In our in-house MC\ncode, unit vectors on a simple cubic lattice of 16 ×16×16\nare considered, and starting from a random spin configuration,\nthe system is brought into a thermal equilibrium within 2 ×105\nMC steps at each temperature. After the thermal equilibrium\nis achieved, we calculated the magnetization of each sublattice\nwhich was further used to determine the magnetic ordering tem-\nperature. Our results are well-converged with respect to both\nthe lattice size and the number of MC steps.\n3. Results\nOur analysis begins with the optimization of the crystal struc-\nture. For materials exhibiting magnetostructural transitions, a\nstrong lattice-spin coupling is expected. In the context of mate-\nrials undergoing magnetostructural transitions, a robust lattice-\nspin coupling is anticipated. In such scenarios, a comprehen-\nsive structural optimization proves invaluable for gaining in-\nsights. Utilizing various exchange-correlation (XC) function-\nals, we systematically optimize the crystal structure while con-\nsidering the antiferromagnetic (AFM) state, as detailed in the\nmethodology section.\nTable 1 presents the optimized lattice constants, their changes\nobserved during optimization, and the magnetic moments of\nFe atoms. Throughout these calculations, the crystal structuremaintains its cubic symmetry. Consistent with findings from\nprior Density Functional Theory (DFT)-based studies, the mag-\nnetic moments of Rh atoms persist at zero in the AFM ground\nstate.\nTable 1: A comparison of the lattice constants, their variations, and the mag-\nnetic moments of Fe after optimizing the crystal structure using di fferent XC\nfunctionals for the AFM state. Additionally, the di fference in total energy (per\nformula unit) with the ferromagnetic (FM) state is provided. Notably, the re-\nsults obtained with the SCAN functional stand out distinctly from those ob-\ntained with other XC-functionals.\nXC-functional a(Å)δa(%)µFe\nBEFM-EAFM\n(meV)\nLDA 5.832 -2.3 2.810 92\nPBE 5.979 +0.2 3.078 66\nRPBE 6.031 +1.0 3.145 46\nPBEsol 5.984 +0.1 3.074 71\nSCAN 5.921 -0.8 3.341 12\nWith the exception of LDA, the variations in lattice constants\nare relatively small across di fferent XC functionals. However,\nthe behavior of SCAN stands out distinctly from other GGA-\nbased XC functionals on three key parameters. Firstly, it is\ncommonly known that GGAs tend to overestimate lattice con-\nstants due to the underestimation of electron-electron correla-\ntions (Hartree and Hartree-Fock electron-electron correlations),\nwhile LDA exhibits the opposite trend. However, the meta-\nGGA SCAN demonstrates a unique behavior, leading to a de-\ncrease in lattice constant by approximately 0.8%, contrasting\nwith other GGA functionals. Secondly, a closer examination of\nthe Fe magnetic moments in Table 1 reveals that µFe\nBis signifi-\ncantly enhanced (approximately 8%) in the case of SCAN com-\npared to PBE. This value is notably closer to the experimen-\ntally observed 3.3 µBfrom the only neutron di ffraction study in\nRef. Shirane et al. (1964). Thirdly, the energy di fference per\nformula unit with the FM state (E FM-EAFM) in Table 1, calcu-\nlated using optimized structures of respective XC functionals,\nis several times smaller for SCAN compared to other XC func-\ntionals. The AFM-FM transition temperature of 370 K ( ∼32\nmeV) is slightly higher than the value E FM-EAFM=12 meV\nestimated by SCAN. For other functionals, this di fference is\nhigher than the transition temperature, suggesting that SCAN\nmight be closer to describing the AFM-FM transition in FeRh.\nAt this stage, we can assert that SCAN exhibits distinctive be-\nhavior for our current system of interest, FeRh. However, it\nremains unclear whether its performance is superior or inferior\nto other functionals and, in either case, how the results di ffer at\na quantitative level. Addressing these questions requires further\nin-depth analysis, marking the next step in our ongoing study.\nAs the next step in our analysis, we calculate the energy dif-\nference between the FM and AFM states (E FM-EAFM) with the\nchange in volume using di fferent XC-functionals. The corre-\nsponding plot is presented in Fig. 1. Notably, the AFM-FM\ntransition occurs at approximately 2 % volume expansion for\nSCAN, while for PBE /PBEsol, it takes place after a ∼15 %\n3Figure 1: Change in the magnetic ground state with the change of volume cal-\nculated for optimized crystal structure with various XC functionals. E FM-EAFM\ninfers AFM-FM transition. Notably, except SCAN, all other functionals antic-\nipate the transition occurring at a substantially higher volume than the experi-\nmentally observed approximately 1 % change.\nexpansion. RPBE exhibits relatively better performance among\nGGAs, although the transition still occurs after a substantial ∼\n10 % volume expansion. Hence, at first glance, SCAN appears\nto better perform compared to other functionals, particularly in\ndescribing the magnetostructural transition in the FeRh system.\nConsistent with previous DFT-based studies, in the FM state,\nRh magnetic moments at the pristine volume are ∼1.061µB\nwhile no net Rh moment is observed in the AFM state. The\nappearance /disappearance of Rh moments is believed to be at-\ntributed to hybridization among Fe and Rh atoms Sandratskii\nand Mavropoulos (2011); Kudrnovsk ´y et al. (2015). The Fe\nmoment remains largely unchanged throughout the transition.\nTo further assess the reliability of the SCAN functional, we\nevaluate the stability of crystal structures by computing phonon\ndispersion. A similar analysis is conducted for comparison pur-\nposes using PBE and RPBE functionals. The resulting plots for\nthe AFM and FM phases are presented in Fig. 2(a) and Fig. 2(b)\nrespectively. Additional details of the calculations are available\nin the Methodology section. The phonon dispersion of the two\nphases vizFM and AFM, exhibits notable di fferences, partic-\nularly with the softening of optical modes in the AFM phase.\nThis observation underscores the pronounced influence of spin-\nlattice coupling in the FeRh system. In the FM phase, all three\nfunctionals-PBE, RPBE, and SCAN—consistently predict the\ndynamical stability of the crystal lattice. Interestingly, soft\nphonon modes are observed with the PBE functional at the M\npoint for the AFM phase in Fig. 2(a), a feature absent in the case\nof RPBE and SCAN functionals. Similar soft modes have been\nreported in prior DFT-based alculations Aschauer et al. (2016);\nWolloch et al. (2016). In support of this, Wolloch et. al. Wol-\nloch et al. (2016) argued that large Fe magnetic moments can\ndynamically stabilize the crystal structure in the AFM phase.\nBoth Aschauer et. al. Aschauer et al. (2016) and Wolloch et.\nal.Wolloch et al. (2016) manipulated Fe magnetic moments to\nreach this conclusion, either through the DFT +U approach\nor by artificially increasing them. In our current study, both\nRPBE and SCAN functionals (RPBE functional in Ref. Wol-\nloch et al. (2016)) demonstrate success in predicting the dy-\nFigure 2: Phonon dispersion is computed for both the (a) AFM and (b) FM\nphases utilizing the finite displacement method with PBE, RPBE, and SCAN\nfunctionals. For the PBE case in (a), we employ two unit cells of sizes 2 ×2×\n2 and 4×4×2 to ensure convergence concerning unit cell size.\nnamically stable AFM phase of FeRh. However, only SCAN\nappears to reasonably predict the AFM-FM transition with vol-\nume change. This discrepancy prompts a natural question about\nthe di fferences at microscopic level from these two functionals\nin describing the properties of the FeRh alloy.\nTo address the aforementioned point, we conduct a spin-\npolarized band structure analysis using PBE, RPBE, and SCAN\nfunctionals for the FM phase. In this investigation, we consider\nthe primitive cell consisting of two atoms to provide a clear ob-\nservation of the impact of various functionals on the electronic\nstructure. The corresponding plots are presented in Fig. 3. Two\nsignificant observations arise from this analysis, shedding light\non the distinctive behavior of SCAN compared to the two GGA\nfunctionals. Firstly, owing to its improved handling of delo-\ncalization errors, SCAN pushes the occupied bands downward\nand the unoccupied bands upward relative to the Fermi level\n(Fig. 3(c)), contrasting with the band structures obtained with\nPBE and RPBE (which exhibit more or less the same behavior)\nas shown in Fig. 3(a) and (b). This behavior of SCAN suggests\na commendable treatment of electronic correlation, in particular\nof the Fe /Rhdorbitals contributing dominantly (not explicitly\nshown) to the band structure plots in Fig. 3. Secondly, and in-\ntricately connected with the first observation, SCAN reveals an\naugmented exchange splitting at the Fermi level. This enhance-\nment is highlighted by arrows with two di fferent colors for the\ntop two bands in the plots. In the case of PBE and RPBE func-\ntionals, the splitting—depicted by dark and light blue arrows\nin Fig. 3(a),(b)-measures 2.064, 2.603 eV and 2.105, 2.699 eV ,\n4Figure 3: Spin-polarized band structure plots for the FM phase are presented with (a) PBE, (b) RPBE, and (c) SCAN functionals. We used two-atom primitive unit\ncell in our calculations. Bands in the up and down spin channels are depicted in red and green, respectively, with the Fermi level set to zero in the plots. The two\ndifferently colored arrows around Fermi level in the plots illustrate the exchange splitting of the top two bands, measuring 2.064 and 2.603 eV from PBE in (a),\n2.105 and 2.699 eV from RPBE in (b), and 2.809 and 3.307 eV from SCAN in (c).\nrespectively. In contrast, the splitting increases to 2.809 and\n3.307 eV with the SCAN functional in Fig. 3(c). The dispar-\nity in exchange splitting between these GGA functionals and\nSCAN aligns closely with the error by which PBE /RPBE exag-\ngerates the AFM phase stability over the FM phase in Fig. 1.\nConsequently, one can attribute the relatively superior descrip-\ntion of the AFM-FM transition in FeRh by the SCAN functional\nto its capacity to bring a substantial exchange splitting in the\nelectronic structure.\nThe seemingly promising findings regarding the performance\nof the SCAN functional for the FeRh system, particularly con-\ncerning phonon dispersion and the AFM-FM transition with\nvolume, require further examination. To validate the e fficacy of\nthe SCAN functional in accurately describing FeRh, magnetic\ninteractions obtained with it should be able to reproduce the ex-\nperimentally observed paramagnetic transition at ∼670 K. We\nundertake a thorough investigation of this aspect in following\ntwo steps.\nIn the first step, the spin-spiral calculations as the starting\npoint of Frozen Magnon Approximation are performed. As ex-\nplained in the Methodology section, each spin spiral state char-\nacterized by qis equivalent to a magnetic configuration with\nenergy E(q). Fig. 4 depicts the stabilization energy ( E(q) -E(q\n=0) dispersions for various noncollinear magnetic configura-\ntions along the Γ−X−M−Γ−R−Xpath, calculated using\nboth PBE and SCAN XC-functionals. In this plot, Γ-point ( q=\n0) represents the AFM state observed at low temperatures. To\nmodel the system within a Heisenberg model, linear regression\nwas employed to extract Fe-Fe magnetic exchange interactions\nup to the fourth nearest neighbor. Coe fficient of determination\n(R2) are found to be 0.99 and 0.98 for the PBE and SCAN XC-\nfunctionals respectively, indicating the model’s ability to cap-\nture the energy dispersion accurately. This can further be qual-\nitatively verified by looking at the goodness of fitting.\nUpon examining Fig. 4, two straightforward observations\nemerge. Firstly, from both functionals, the FeRh system attains\nits lowest energy at the Γpoint, indicating that the AFM state\nis the ground state. This again is consistent with experimen-\ntal observation and highlight the fact that both these functional\ncorrectly obtain the magnetic ground state. Secondly, while the\nnature of the plots from PBE and SCAN functionals remainslargely similar, there is a significant discrepancy in the energy\nscale. The relative energies [ E(q) -E(q=0)] are notably larger\nwith SCAN compared to PBE. One way to look at this point\nis that SCAN predicts the AFM state at low tenperature to be\nrobustly stabilized against any other competing magnetic or-\nders, particular against the FM state represented by Rpoint in\nFig. 4. However, from Table 1, one can see that E FM-EAFM=12\nmeV meaning FM state is lying in close vicinity to that of the\nAFM state. This observation immediately prompts the question\nof whether their is also discrepancies among magnetic interac-\ntions obtained from the two functionals and, if so, which one\nof them aligns with the experimentally observed paramagnetic\ntransition temperature of ∼670 K. To address these questions\nand assess the validity of SCAN and PBE for this challenging\nsystem, extraction of magnetic interactions and subsequent cal-\nculations of the magnetic ordering temperature are performed\nin the second step.\nFigure 4: Spin-spiral calculations are performed for the low-temperature AFM\nphase using PBE (orange-filled circles) and SCAN (green-filled circles) XC\nfunctionals. In this representation, the Γpoint signifies the AFM ground state.\nThe obtained data is subjected to fitting with the Heisenberg model given in Eq.\n(1), involving up to 4thFe-Fe neighbors, using linear regression. The results\nof the fitting are depicted in the plot with light-green and blue empty diamond\nsymbols for PBE and SCAN functionals, respectively.\nHence, to elucidate the impact of XC functionals, we first\nextract Fe-Fe magnetic interactions ( J’s) up to the fourth near-\nest neighbor using the inverse Fourier transform method within\n5the Frozen Magnon approach, discussed in the Methodology\nsection. Subsequently, Monte Carlo simulations are employed\nto estimate the magnetic ordering temperature. Fig. 5(a) de-\npicts the extracted interactions plotted against di fferent near-\nest neighbor distances. Both PBE and SCAN agree on the\ndominant antiferromagnetic first-neighbor interaction ( J1) at∼\n2.96 Å, consistent with the observed antiferromagnetic ground\nstate. However, their magnitudes di ffer significantly. SCAN\npredicts a substantially larger J1of -151 meV compared to -\n30 meV predicted by PBE. Interestingly, while smaller, the\nsecond-neighbor interaction ( J2) at 4.19 Å exhibits a contrast-\ning behavior. SCAN predicts a ferromagnetic J2of 7 meV ,\nwhereas PBE estimates it as antiferromagnetic with a value of\n-2.6 meV . In the low-temperature AFM configuration, first and\nsecond-neighbor couplings are expected to be AFM and FM,\nrespectively. Therefore, SCAN’s FM J2at first glance appears\nmore consistent. The estimated third-neighbor interactions J3\nat 5.13 Å are -8 and -9.5 meV , and the fourth-neighbor inter-\nactions J4at 5.92 Å are 0 and 2.8 meV for SCAN and PBE,\nrespectively. These discrepancies highlight the significant qual-\nitative di fferences in magnetic interactions extracted using the\ntwo XC functionals.\nFigure 5: (a) Magnetic interactions between Fe atoms in the AFM ground state\nare estimated using the Frozen-magnon approach, as explained in the methodol-\nogy section, with both PBE and SCAN functionals. The 1stneighbor interaction\nis antiferromagnetic, consistent with the magnetic ground state. (b) The varia-\ntion of normalized magnetization with temperature is illustrated in our Monte\nCarlo simulations. The interactions obtained with PBE in (a) yield a magnetic\nordering temperature TN∼700 K, closely matching an experimental estimation\nof 670 K. Conversely, the magnetic interactions from the SCAN functional sig-\nnificantly overestimate it.\nTo check upon the reliability of the two set of interactions\nobtained above in reproducing experimentally observed mag-\nnetic ordering temperature, we perform Monte Carlo simula-\ntions considering the Heisenberg model on the simple cubic\nlattice of Fe atoms (see methodology). Obtained variation innormalized as a function of temperature is plotted in Fig. 5(b)\nfor the two cases, vizPBE and SCAN. The estimated magnetic\nordering temperature with J’s estimated using PBE and SCAN\nfunctionals∼710 K and 3240 K respectively. Though this tem-\nperature is overestimated in both cases, the case of SCAN is in\ncomplete disagreement with the experimentally observed value\nof∼670 K. It an order of magnitude overestimated when com-\npared to PBE-obtained value which is in a better agreement\nwith experiment. This discrepancy can directly be attributed\nto large J1obtained from SCAN in Fig. 5(a). As far the PBE is\nconcerned, one can tune J’s through the application of Uon Fe-\ndstates within the PBE +Uapproach to achieve a closer agree-\nment with experimentally observed magnetic ordering tempera-\nture. However, our presented results su ffice to highlight the be-\nhavior of di fferent XC functional in describing structural, elec-\ntronic, and magnetic properties of the FeRh system.\n4. Conclusion\nIn conclusion, we have investigated the performance of var-\nious XC functionals within the DFT formalism for the chal-\nlenging binary alloy FeRh. Employing LSDA, various GGAs,\nand newly developed SCAN meta-GGA functional, our results\nshow the limitations of any single functional in capturing the\nintricate interplay of structural, electronic, and magnetic prop-\nerties in FeRh. 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URL: https://\nlink.aps.org/doi/10.1103/PhysRevB.103.035137 , doi: 10.1103/\nPhysRevB.103.035137 .\n8"    },    {        "title": "2401.12569v1.Magnetic_Dirac_systems__Violation_of_bulk_edge_correspondence_in_the_zigzag_limit.pdf",        "content": "MAGNETIC DIRAC SYSTEMS: VIOLATION OF BULK-EDGE\nCORRESPONDENCE IN THE ZIGZAG LIMIT\nJ.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND,\nAND E. STOCKMEYER\nAbstract. We consider a Dirac operator with constant magnetic field defined\non a half-plane with boundary conditions that interpolate between infinite mass\nand zigzag. By a detailed study of the energy dispersion curves we show that the\ninfinite mass case generically captures the profile of these curves, which undergoes\na continuous pointwise deformation into the topologically different zigzag profile.\nMoreover, these results are applied to the bulk-edge correspondence. In particular,\nby means of a counterexample, we show that this correspondence does not always\nhold true in the zigzag case.\nContents\n1. Introduction and main results 1\n1.1. The energy dispersion curves 4\n1.1.1. Setting 4\n1.1.2. Main results for the energy dispersion curves 6\n2. Edge conductance formula 9\n3. Energy dispersion curves 12\n3.1. Preliminaries 12\n3.2. A characterization of the eigenvalues for the non-zigzag case 13\n4. The auxiliary quadratic forms 15\n4.1. Study of αÞÑνnpα, ξq 15\n4.2. Study of ξÞÑν˘\nnpα, ξq 17\n5. Proofs for the energy dispersion curves 18\n5.1. Proof of Theorems 1.7 and 1.8 19\nAppendix A. Lemma on trace class operators 20\nAcknowledgments 21\nReferences 21\n1.Introduction and main results\nBulk-edge correspondence is a notion that arises in the study of certain condensed\nmatter physics systems possessing non-trivial topology. It establishes a connection\nbetween the bulk properties of a material (its interior or bulk region) and its edge or\nboundary properties. This correspondence may be given through an equation that\nlinks a Chern number, that depends only on the bulk operator, and an expression\ninvolving the edge-states localized close to the boundary; eventually yielding the\n1arXiv:2401.12569v1  [math-ph]  23 Jan 20242 J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nso-called topological quantization of edge currents [13]. Due to the integer (or topo-\nlogical) nature of the Chern number these relations are very stable under smooth\nchanges of the material parameters and therefore its importance for potential appli-\ncations. In this context, Dirac Hamiltonians are prominent examples that exhibit\ninteresting phenomena. They are used to model various materials, among them,\ngraphene and topological insulators [14].\nIn this article we investigate the bulk-edge correspondence for a two-dimensional\nDirac Hamiltonian with a constant magnetic field defined on a half-plane. This\nmodel was recently considered in [6], where bulk-edge correspondence was shown to\nhold, provided infinite-mass boundary conditions along the edge are imposed. Our\nmotivation is to investigate the validity of these results when any fixed admissible\nlocal boundary condition is allowed. To this end we define a family of Dirac Hamilto-\nniansDγ, where γPRYt`8u characterizes the boundary conditions, in particular,\nγ“˘1corresponds to infinite mass.\nOur main result Theorem 1.2 indicates that bulk-edge correspondence for this\nmodelstillholds, providedtheboundaryconditionsarenotzigzag( i.e.γRt0,`8u).\nMoreover, Theorem1.2showsthatthiscorrespondenceisviolatedforcertainenergies\nwhen zigzag boundary conditions are imposed. In order to show Theorem 1.2 certain\nknowledge on the energy dispersion curves associated to Dγis helpful. In this work\nwe present a fairly detailed analysis of them extending the results of [3] to any local\nboundary condition. We complement our analysis with numerical illustrations of\nthe energy dispersion curves for different values of the boundary parameter and the\nmagnetic field.\nLet us now turn to define our model. We consider a magnetic Dirac system on the\nhalf-plane denoted by R2\n`“tpx1, x2qPR2:x2ą0uin the presence of an orthogonal\nmagnetic field whose component in the x3direction is given by bPRzt0u. The\ncorresponding Hamiltonian acts on functions in L2pR2\n`,C2qas\nσ¨p´i∇´Aq“ˆ\n0´iB1´B2`bx2\n´iB1`B2`bx2 0˙\n. (1.1)\nHereAreferstoavectorpotentialassociatedwiththemagneticfield i.e.rotA“be3.\nWe choose\nA“p´ bx2,0q.\nWe recall that\nσ1“ˆ\n0 1\n1 0˙\n, σ 2“ˆ\n0´i\ni0˙\n, σ 3“ˆ\n1 0\n0´1˙\n.\nThestudyofthemagneticDiracoperatoronthehalf-planewithinfinitemassbound-\nary conditions was recently carried forward in [3]. Here we consider general local\nconditions at the edge interpolating between zigzag and infinite mass. More pre-\ncisely: Let γPRYt`8u, then we consider the self-adjoint realization Dγ”Dγpbq\nacting as (1.1) on a subset of functions ψ“pψ1, ψ2qPL2pR2\n`,C2qsatisfying, for all\nx1PR,#\nψ2px1,0q“γψ1px1,0qifγPR,\nψ1px1,0q“0 ifγ“`8 .(1.2)3\nThe two cases γPt0,`8uare called zigzag, while γ“˘1corresponds to infinite\nmass boundary conditions (see Remark 1.3 bellow).\nRemark 1.1. The domains of self-adjointness of the operators Dγare already\nknown: See [3, Section 1C] for the zigzag cases and [3, Theorem 1.15] for the infi-\nnite mass; the latter result can be easily adapted for the non-zigzag cases. For the\nessential self-adjointness of Dγon the class of Schwartz functions with infinite mass\nboundary conditions see [6, Proposition 1.1], and for the general case see [1].\nWe denote by X1the operator of multiplication with x1, and by J1the current\ndensity operator, we have\nJ1“´irDγ, X1s“´ σ1.\nRecall that the Landau Hamiltonian Dbulkpbqacts on the whole plane as in (1.1) and\nits spectrum consists of the Landau levels given by t˘a\n2n|b|, nPN0u.\nWe say that F:RÑRis equal to aPRnear x0PRif there exists an open\ninterval Iaround x0where Fpxq“a, xPI,holds.\nOur main result is the following.\nTheorem 1.2. Letbą0and let χ“1p0,1qbe the indicator function of the interval\np0,1q. Let γPRYt`8u. Let FPC2\n0pRqbe such that it equals 1near ně1Landau\nlevels, and 0near the others. Then, the operator χpX1qJ1F1pDγqis trace class and\nthe edge Hall conductance is given by\n2πTr`\nχpX1qJ1F1pDγq˘\n“$\n’&\n’%n´1ifγ“0andFequals 1near 0,\nn`1ifγ“`8andFequals 1near 0,\nnotherwise .(1.3)\nComments:\n(1) The fact that the left hand side of this formula can be interpreted as an edge\nconductance is explained for instance in [10].\n(2) Let us make the connection with the bulk-edge correspondence. Let Fandn\nbe as in Theorem 1.2 and define the orthogonal projection Pn“FpDbulkpbqq.\nIn our case, Pncontains exactly nbulk Landau levels and one can show that\nits Chern number equals n, which encapsulates the non-trivial topology of\nthe bulk projection (see e.g.[6]). On the other hand, since F1equals 0near\nthe Landau levels, F1pDγqonly selects edge-states.\n(3) If bă0, the Chern number of Pnbecomes´n, and the right hand side of\nformula (1.3) must be multiplied by ´1.\n(4) Our third alternative in (1.3) indicates that the bulk-edge correspondence\nshould hold for all non-zigzag conditions, a result which in our case is con-\nfirmed by brute force, i.e. by direct computation and comparison with the\nbulk Chern number. The general proof of this fact, under more general con-\nditions than purely constant magnetic field, will be considered in [1]. At\nleast for γ“1, this has been shown to be the case [6]; for Schrödinger-like\noperators see [7].\n(5) If we work with zigzag boundary conditions, and if the zero-energy bulk\nLandau level belongs to the projection Pn, then one of the first two alter-\nnatives in (1.3) occurs. Thus the bulk-edge correspondence does not hold4 J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nin this case. Such an anomaly has been previously observed in other con-\ntinuous models such as shallow-water waves [11, 18], and for \"regularized\"\nnon-magnetic Dirac-like operators [18]. The latter are in fact second order\ndifferential operators, with boundary conditions that are incompatible with\nfirst-order self-adjoint differential operators.\nRemark 1.3 (On the boundary condition) .General local boundary conditions for\nDirac operators are usually written as p´iσ3pσ¨nqcosη`σ3sinηqψ“ψ,onBR2\n`,\nwhere ηP“\n´π\n2,3π\n2˘\n(seee.g.[5]and[4,3]). Inthepresentsituation, wehave n“´e2\nand thus\npσ1cosη`σ3sinηqψ“ψ ,onBR2\n`.\nWe obtain (1.2) by setting γ“cosη\n1`sinη“tan`π\n4´η\n2˘\nwith the convention γ“`8in\nthe case η“´π{2i.e.ψ1“0onBR2\n`.\n1.1.The energy dispersion curves. Our proof of Theorem 1.2 requires certain\nknowledge on the energy dispersion curves and their corresponding eigenfunctions.\nIn what follows we present a description of these curves for different values of the\nboundaryparameter γ. Themaintechnicalissuehereisthelackofsemi-boundedness\nofDγ. This can however be treated using appropriate variational methods [12, 9, 16];\nwe follow the approach proposed in [3]. Before presenting our main results in this\ncontext we establish the basic framework.\n1.1.1.Setting. In view of the translation invariance in the x1-direction, we may\nuse the partial Fourier transform to represent Dγpbqas a family of fiber operators\nDγ,ξpbq”Dγ,ξ, with ξPR. Indeed, we have (see e.g.[3])\nDγ“ż‘\nRDγ,ξdξ ,\nwhere the 1dmagnetic Dirac operator Dγ,ξacts as\nˆ0 dξpbq\nd:\nξpbq0˙\n”ˆ0dξ\nd:\nξ0˙\n,\nwith dξ“ξ´B2`bx2andd:\nξ“ξ`B2`bx2. Its domain is given for γRt0,`8uas\nDompDγ,ξq“tψPH1pR`,C2q:x2ψPL2pR`qandψ2p0q“γψ1p0qu.(1.4)\nAs for the zigzag cases, by denoting\nB1pR`q“tψPH1pR`,C2q:x2ψPL2pR`qu\nwe have\nDompD0,ξq“tuPL2pR`q:d:\nξuPL2pR`quˆH1\n0pR`qXB1pR`q,\nDompD8,ξq“H1\n0pR`qXB1pR`qˆtuPL2pR`q:dξuPL2pR`qu.\nWe have that ( cf.[3]), for any γPRYt`8u andξPR, the operator Dγ,ξ\nis self-adjoint and has compact resolvent (for γ“t0,`8uit follows directly from\nthe compact embedding of H1inL2). We write the spectrum of Dγ,ξas the set\nt´ϑ´\njpγ, ξq:jPNuYtϑ`\njpγ, ξq:jPNusuch that\n. . .ď´ϑ´\n2pγ, ξqď´ ϑ´\n1pγ, ξqă0ďϑ`\n1pγ, ξqďϑ`\n2pγ, ξqď. . . (1.5)5\nFor a given boundary condition γPRYt`8u the map RQξÞÑϑ˘\nnpγ, ξqdefines the\nenergy dispersion relation.\nThe following two propositions are shown in Section 3.1.\nProposition 1.4. LetγPRYt`8u ,ξPR,b“0. We have\n(i)For all ně1, the eigenvalues ϑ˘\nnpγ, ξqare simple.\n(ii)For all ně1,ξÞÑϑ˘\nnpγ, ξqandγÞÑϑ˘\nnpγ, ξqare real-analytic.\n(iii)0belongs to the spectrum of Dγ,ξiffγ“0, in case bą0, orγ“`8, in case\nbă0.\nIn order to study the zigzag case we introduce further the 1dfibers of a magnetic\nDirichlet Pauli operator Hξpbqforbą0. It acts as´B2\n2`pξ`bx2q2`b ,with\nDompHξpbqq“t ψPH2pR`;Cq:x2\n2ψPL2pR`q, ψp0q“0u.\nIt is well-known [8] that Hξpbqis self-adjoint with compact resolvent and that its\nspectrum consists on simple eigenvalues pνDir\nnpb, ξqqnPNwith\n2băνDir\n1pb, ξqăνDir\n2pb, ξqă. . . .\nThe following statements are well-known (see [17] and [2, 3]). We specialize in the\ncasebą0since otherwise one can use the charge conjugation symmetry described\nin Remark 1.9 below.\nProposition 1.5. Consider the case γPt0,`8u. Then, for all ξPRandb“0,\nthe spectrum of Dγ,ξis symmetric with respect to 0. Moreover, for bą0, we have\nϑ`\nnp`8, ξq“a\nνDir\nnpb, ξq´2bpně1q,\nϑ`\n1p0, ξq“0,and ϑ`\nnp0, ξq“b\nνDir\nn´1pb, ξq pně2q.\n \nSpec(D,(b))\n4\n2\n024\n4\n 3\n 2\n 1\n 0 1 2 3 4\n(a)Case γ“0: Here 0is an eigenvalue of Dγ\n \nSpec(D,(b))\n4\n2\n024\n4\n 3\n 2\n 1\n 0 1 2 3 4\n (b)Case γ“8: Here 0is not an eigenvalue.\nDashed lines correspond to the Landau levels\nof the bulk operator.\nFigure 1. Dispersion curves for the zigzag boundary conditions\nRemark 1.6. Recall [8] that, for ně1, the function νDir\nnpb,¨qis increasing and\nlim\nξÑ´8νDir\nnpb, ξq“2nb , lim\nξÑ`8νDir\nnpb, ξq“`8 .6 J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nFigures 1a and 1b give the dispersion curves of the zigzag Dirac operators and on\nthe left of each figure, their spectrum. All illustrations presented in this article are\nobtained thanks to standard finite difference schemes, inverse power and Newton-like\nmethods.\n1.1.2.Main results for the energy dispersion curves. In view of the symmetry it is\nenough to consider the cases of positive magnetic field and non-negative boundary\nparameter i.e. the casepb, γqPp0,`8qˆr 0,`8s(see Remark 1.9).\nThe following theorem gives a description of the dispersion curves when γP\np0,`8qand generalizes the result obtained in [3] for γ“1.\nTheorem 1.7. LetγP p0,`8q,ξPR,bą0. The spectrum of Dγ,ξpbqcan be\ndescribed as follows. Let ně1.\n(i)The function ϑ`\nnpγ,¨qis increasing and\nlim\nξÑ´8ϑ`\nnpγ, ξq“a\n2pn´1qb , lim\nξÑ`8ϑ`\nnpγ, ξq“`8 .\n(ii)The function ϑ´\nnpγ,¨qhas a unique critical point, which is a non-degenerate\nminimum, and\nlim\nξÑ´8ϑ´\nnpγ, ξq“?\n2nb , lim\nξÑ`8ϑ´\nnpγ, ξq“`8 . (1.6)\n \nSpec(D,(b))\n4\n2\na0\n024\n4\n2\na0\n024\nFigure 2. The dispersion curves of Dγ,ξforγ“b“1\nFigure 2 gives the dispersion curves of the infinite mass Dirac magnetic operator\nwith a special focus on the global maxima of the negative dispersion curves. Here\na0Pp0,?\n2q, was introduced in [3], it is the minimum of ϑ´\n1p1,¨q,i.e., it is the size of\nthe spectral gap of the Dirac operator with infinite mass boundary condition. The\nspectral gap of Dγas a function of γcan be read off from Figure 3.7\n0 20 40 60 80 100\n0\na0\n2\n1\nFigure 3. The value of the maximal negative energy of the full operator\nas a function of γforb“1.\n1 2 3 4 5\n1.7\n1.6\n1.5\n1.4\n1.3\nc\nFigure 4. The location of the critical point of the first negative dispersion\ncurve for b“1. The red bullet refers to γ“1.\nOur next result describes the dispersion curves as functions of γand their zigzag\nlimits.\nTheorem 1.8. Letbą0. The families of functions r0,`8q Q γÞÑϑ`\nnpγ,¨qand\nr0,`8q Q γÞÑ ´ ϑ´\nnpγ,¨qare increasing with γ. Moreover, for all ně1and all\nξPR, we have\nlim\nγÑ0ϑ˘\nnpγ, ξq“ϑ˘\nnp0, ξq,lim\nγÑ`8ϑ˘\nnpγ, ξq“ϑ˘\nnp`8, ξq.\nIn Figure 5 we present various pictures of the dispersion curves with varying\nγPp0,`8q. Moreover, Figure 6 illustrates the action of the symmetries, described\nin the following.\nRemark 1.9 (Symmetries) .In view of the underlying symmetries it is enough to\nstudy the dispersion curves when pb, γqPp0,`8qˆr 0,`8s. Indeed, in order to also\nconsider γă0, we notice that\nσ3Dγpbq“´D´γpbqσ3.8 J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\n4\n 2\n 0 2 46\n4\n2\n0246b=1, =0.4\n4\n 2\n 0 2 4b=1, =0.6\n4\n 2\n 0 2 4b=1, =0.8\n4\n 2\n 0 2 46\n4\n2\n0246b=1, =1\n4\n 2\n 0 2 4b=1, =2\n4\n 2\n 0 2 4b=1, =4\nFigure 5. The dispersion curves for b“1and various γ\nMoreover, for bă0we used the charge conjugation Cψ“σ1ψwhich turns the\nboundary conditions into ψ2“γ´1ψ1and hence\nCDγpbq“´Dγ´1p´bqC .\nFor the fiber operators this leads to:\nCDγ,ξpbq“´Dγ´1,´ξp´bqC , σ 3Dγ,ξpbq“´D´γ,ξpbqσ3. (1.7)\nIn particular, we obtain the dispersion curves when bă0from the curves when\nbą0by changing γintoγ´1andξinto´ξ.\n 4\n024\n  b=1, =0.5\n  b=1, =2\n \nSpec(D,(b))\n4\n2\n04\n0\n b=1, =2\n0\n b=1, =0.5\nFigure 6. Action of the symmetries on the dispersion curves.9\nOrganization of this article. In Section 2 we prove Theorem 1.2. Sections 3, 4,\nand 5 are devoted to the description of the energy dispersion relations ϑ˘\nnpγ, ξq.\nIn Section 3 we prove Propositions 1.4 and 1.5. In addition, we state in Theo-\nrem 3.5 a fixed-point characterization of ϑ˘\nnpγ, ξqin terms of a family`\nν˘\nnpα, ξq˘\nαą0of the eigenvalues of certain magnetic Schrödinger-like operators with Robin bound-\nary conditions. We give a proof of this characterization in Section 5. In Section 4\nwe investigate the fundamental mapping properties of ν˘\nnpα, ξqforαą0andξPR.\nFinally, in Section 5, we apply Theorem 3.5, together with the analysis of Section 4,\nto prove Theorems 1.7 and 1.8.\n2.Edge conductance formula\nIn this section we prove Theorem 1.2. In doing so we use various results on the\nenergy dispersion curves which are stated in Section 1.1 and proved in the next\nsections.\nWe let, for all jě1,\nλjpξq“ϑ`\njpγ, ξq,\nand, for all jď´1,\nλjpξq“´ ϑ´\njpγ, ξq.\nLetpΨξ,jqξPRbe an analytic family of normalized eigenfunctions of Dγ,ξassociated\nwith λjpξq(see Proposition 1.4).\nIn view of the asymptotic behavior of λjpξqasξÑ˘8– stated in Theorem 1.7\n(for the non-zigzag case) and Proposition 1.5 and Remark 1.6 (for the zigzag case)\n– the proof of Theorem 1.2 reduces to showing the following result.\nProposition 2.1. We let χ“1p0,1qand consider γPRYt`8u. Let us con-\nsider a function fPC1\n0pRqbeing zero near the Landau levels. Then, the operator\nχpX1qσ1fpDγqis trace class and there exists a finite JĂZzt0usuch that\n2πTr`\nχpX1qσ1fpDγq˘\n“ÿ\njPJż\nRfpλjpξqqλ1\njpξqdξ .\nIn particular, if nPNandFPC2\n0pRqequals 1near nLandau levels and 0near the\nothers, we have\n´2iπTrpχpX1qrDγ, X1sF1pDγqq“ÿ\njPJ´\nF`\nλjp´8q˘\n´F`\nλjp`8q˘¯\n.(2.1)\nLet us first state two useful elementary results. The first one is a direct conse-\nquence of Proposition 1.4 and Theorem 1.7.\nLemma 2.2. Let us consider a function fPC0pRqbeing zero near the Landau\nlevelst˘?\n2n , nPNu. Then, there exists a finite JĂZzt0usuch that for all kRJ,\nf˝λk“0and for all jPJ, the functions ξÞÑfpλjpξqqhave compact supports.\nThe following result might be elementary. We give its proof for the reader’s\nconvenience.10J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nLemma 2.3. Consider an operator TonL2pR2\n`qgiven as a Bochner integral (on a\nfinite interval) of a continuous family of rank one operators\nT“żb\na|ψpξqyxϕpξq|dξ . (2.2)\nAssume further that the trace norm }ψpξq}}ϕpξq}of the above integrand is uniformly\nbounded onra, bs. Then, Tis trace class and\nTrpTq“żb\naTr|ψpξqyxϕpξq|dξ“żb\naxϕpξq, ψpξqydξ . (2.3)\nProof.The integral in (2.2) can be seen as a limit of a Riemann sum Tn, which a\npriori only converges in the operator norm topology. The integrand is a rank-one\ntrace class operator, with a trace norm which is uniformly bounded in ξonra, bs.\nHence, the trace norm of the Tn’s is uniformly bounded in n. By Lemma A.1 from\nthe Appendix we see that T, which a priori is only a compact operator, is actually\ntrace class.\nLettfjujě1be any orthonormal system. Then for all Ně1we have\nNÿ\nj“1xfj, T f jy“żb\naNÿ\nj“1xfj, ψpξqyxϕpξq, fjydξ. (2.4)\nUsing Cauchy-Schwarz and Bessel inequalities we get for every N:\nˇˇˇNÿ\nj“1xfj, ψpξqyxϕpξq, fjyˇˇˇďgffeNÿ\nj“1|xfj, ψpξqy|2gffeNÿ\nj“1|xfj, ϕpξqy|2ď}ψpξq}}ϕpξq}.\nBy Lebesgue’s dominated convergence theorem we can take NÑ8in (2.4) to get\n(2.3). □\nProof of Proposition 2.1. Letusfirstinvestigatetheintegralkernelof fpDγqforsome\nfPC1\n0pRq. We can write\n´\nfpDγqψ¯\npxq“ż\nR2\n`Kfpx, x1qψpx1qdx1,\nwith\nKfpx, x1q“1\n2πż\nRdξeipx1´x1\n1qξkfpξ, x2, x1\n2q,\nwhere\nkfpξ, x2, x1\n2q“ÿ\njPJfpλjpξqqˇˇΨξ,jpx2qD@\nΨξ,jpx1\n2qˇˇ,\nandJĂZzt0uis the finite set from Lemma 2.2. The technical issue here is that,\neven if we multiply by χ”χpX1qfrom the left we can not directly apply Lemma 2.3\nsince the function px1, x2qÞÑeix1ξΨξ,jpx2qis not square integrable on R2\n`. However,\nwe observe that\nσ1χfpDγq“σ1χp1`iX1qfpDγqp1`iX1q´1`σ1χp1`iX1q“\np1`iX1q´1, fpDγq‰\n.(2.5)11\nThe first operator above can be written as\nσ1χp1`iX1qfpDγqp1`iX1q´1\n“1\n2πÿ\njPJż\nRfpλjpξqqˇˇp1`iX1qχ eiX1ξΨξ,jD@\np1´iX1q´1eiX1ξΨξ,jˇˇdξ .(2.6)\nThe second operator has a commutator term r¨,¨swhich can be explicitly computed\nusing the following identity: We get, by doing partial integration,\n2π K fpx, x1q`\n1`i x1\n1˘\n“ż\nRξdξ´\np1`ix1´Bξqeipx1´x1\n1qξ¯\nkfpξ, x2, x1\n2q\n“2π`\n1`i x1˘\nKfpx, x1q`ż\nRdξ eipx1´x1\n1qξBξkfpξ, x2, x1\n2q.\nTherefore, we get as operators on L2pR2\n`q\nσ1χp1`iX1q“\np1`iX1q´1, fpDγq‰\n“1\n2πÿ\njPJż\nRf1pλjpξqqλ1\njpξqˇˇσ1χ eiX1ξΨξ,jD@\np1´iX1q´1eiX1ξΨξ,jˇˇdξ\n`1\n2πÿ\njPJż\nRfpλjpξqqˇˇσ1χ eiX1ξ`\nBξΨξ,j˘D@\np1´iX1q´1eiX1ξΨξ,jˇˇdξ\n`1\n2πÿ\njPJż\nRfpλjpξqqˇˇσ1χ eiX1ξΨξ,jD@\np1´iX1q´1eiX1ξ`\nBξΨξ,j˘ˇˇdξ .(2.7)\nNotice that thanks to Lemma 2.2 the integrals above take place on a finite interval.\nTherefore, each of the four terms appearing in (2.6) and (2.7) can be seen as Bochner\nintegrals involving rank one operators in L2pR2\n`qwhose trace is uniformly bounded\non compact sets. Hence, Lemma 2.3 can be applied to each of the terms involved in\n(2.5). In particular, as a finite sum of trace class operators, σ1χfpDγqis trace class.\nA quick computation using Lemma 2.3 for each term in (2.7) gives\nTr´\nσ1χp1`iX1q“\np1`iX1q´1, fpDγq‰¯\n“1\n2πÿ\njPJż\nRBξ´\nfpλjpξqq@\nσ1χΨξ,j,p1´iX1q´1Ψξ,jD¯\ndξ“0,\nwhere in the last step we used that the term Bξp. . .qhas compact support in ξ. Thus,\nwe get\nTr`\nχσ1fpDγq˘\n“Tr`\nσ1χp1`iX1qfpDγqp1`iX1q´1˘\n“1\n2πÿ\njPJż\nRfpλjpξqqxσ1Ψξ,j,Ψξ,jyL2pR`,C2qdξ .\nNow the conclusion follows since xσ1Ψξ,j,Ψξ,jyL2pR`,C2q“xBξpDγ,ξqΨξ,j,Ψξ,jyL2pR`,C2q\nwhich, by the Feynman-Hellmann theorem, equals λ1\njpξq. In particular, we get (2.1)\nby writing F1“fand integrating in ξ. □12J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\n3.Energy dispersion curves\nWe start this section by showing Propositions 1.4 and 1.5. They state the basic\nproperties of the solutions of the eigenvalue problem, for ξPRandpb, γqPp0,`8qˆ\nr0,`8s\nDγ,ξu“λu . (3.1)\nWe show that these solutions are related to a Schrödinger-like problem with Robin\nboundary conditions. For zigzag boundary conditions this property is already clear\nfrom Proposition 1.5. For γPp0,`8qwe establish this relation in Lemma 3.4 below.\nMoreover, we present in Theorem 3.5 a characterization of the eigenvalues ϑ˘\nnpγ, ξq\nintermsofafixed-pointproblemthatrunsalongafamilyofeigenvalues`\nν˘pα, ξq˘\nαą0of certain Schrödinger-like operators.\n3.1.Preliminaries. Let us investigate some preliminary facts. (Throughout this\nparagraph we assume bą0.) The eigenvalue equation (3.1) can be rewritten as\nd:\nξu1“λu2, d ξu2“λu1.\nThen, we have dξd:\nξu1“λ2u1andd:\nξdξu2“λ2u2. Moreover, from the classical\ntheory of ODEs, we see that u1andu2are smooth on r0,`8q. Since u2p0q“γu1p0q\n(oru1p0q“0when γ“`8) we obtain Robin-type boundary conditions for u1and\nu2, separately. Thus, (3.1) implies\ndξd:\nξu1“λ2u1`\nd:\nξu1p0q“γλu 1p0q˘\n, (3.2)\nd:\nξdξu2“λ2u2`\ndξu2p0q“λ\nγu2p0q˘\n. (3.3)\nProof of Proposition 1.4. Let us consider the eigenvalue equations (3.2) and (3.3).\nFrom the standard theory of initial value problems, we see that ujbelongs to a space\nof dimension at most 1. Therefore, dim ker`\nDγ,ξ´λ˘\nď1. This proves the simplicity\nof the non-zero eigenvalues.\nLet us now discuss the existence of zero modes. For λ“0, we have dξu2“0so\nthat u2is proportional to e1\n2bpξ`bx2q2, which is not in L2pR`qimplying that u2“0\nholds. Moreover, we also check that γu1p0q “0. Using d:\nξu1“0we see that u1\nis proportional to e´1\n2bpξ`bx2q2, which belongs to L2pR`qbut it does not vanish at\nx2“0. Therefore, we find that u1“0unless γ“0.\nThe familypDγ,ξqξPRbeing analytic of type pAq(in the Kato sense, see [15]), the\nsimplicity of the eigenvalues implies their analyticity. □\nNext we discuss the zigzag operators i.e.the cases γPt0,`8u.\nProof of Proposition 1.5. The fact that the eigenvalues are symmetric with respect\nto zero follows from (1.7), hence, we may look at the non-negative ones only.\nLet us consider the case γ“0i.e.u2p0q“0. As we have just seen, we have a zero\nmode and, with our convention, we have ϑ`\n1p0, ξq“0. Let us describe the non-zero\neigenvalues. Let λbe a positive eigenvalue and ua corresponding eigenfunction. In\nview of (3.2), we see that u2cannot be 0and it is an eigenfunction of\nd:\nξdξ“´B2\n2`pξ`bx2q2`b ,13\nwith Dirichlet condition. In particular, λ2belongs to the spectrum of d:\nξdξwith\nDirichlet condition. Conversely, if µą0is an eigenvalue of this operator, we write\nd:\nξdξv“µvwith vp0q“0and we let u“µ´1\n2dξvp‰0qand we have\nd:\nξu“?µv , d ξv“?µu , vp0q“0,\nwhich means that?µis an eigenvalue of D0,ξ.\nThe case γ“`8is quite similar although we have no zero modes. Now, u1is an\neigenfunction (with eigenvalue λ2) of\ndξd:\nξ“´B2\n2`pξ`bx2q2´b ,\nwith Dirichlet condition. Conversely, consider an eigenvalue µą0of this operator.\nProceeding as before, we write dξd:\nξu“µuwith up0q“0and let v“µ´1\n2d:\nξup‰0q\nto get that?µis an eigenvalue of D8,ξ. □\n3.2.A characterization of the eigenvalues for the non-zigzag case. We con-\nsider γPp0,`8q. Let λ‰0be an eigenvalue of Dγ,ξpbq. Multiplying (3.2) by u1\nand integrating by parts yields\nxdξd:\nξu1, u1y“}d:\nξu1}2`d:\nξu1p0qu1p0q“}d:\nξu1}2`λu2p0qu1p0q“}d:\nξu1}2`λγu2\n1p0q.\nMoreover, proceeding analogously for the second component in (3.3) we get\nxd:\nξdξu2, u2y“}dξu2}2´dξu2p0qu2p0q“}dξu2}2´λu1p0qu2p0q“}dξu2}2´λ\nγu2\n1p0q.\nThis suggests to introduce the following family of quadratic forms.\nDefinition3.1. Letαą0. Wedefinetheauxiliaryquadraticforms, for uPB1pR`q,\nas\nq`\nb,α,ξpuq“}d:\nξu}2`αu2p0q,\nq´\nb,α,ξpuq“}dξu}2`αu2p0q.(3.4)\nThey are both non-negative and closed. We denote by h˘\nα,ξthe corresponding self-\nadjoint Schrödinger operators.\nRemark 3.2. By Friedrichs’ extension theorem we have that Domph˘\nα,ξqĂB1pR`q\nand for u˘PDomph˘\nα,ξq\nh`\nα,ξu`“dξd:\nξu`, d:\nξu`p0q“αu`p0q, (3.5)\nh´\nα,ξu´“d:\nξdξu´, d ξu´p0q“´ αu´p0q. (3.6)\nRemark 3.3. Integration by parts yields\nq`\nb,α,ξpuq“}u1}2`}pξ`bx2qu}2´b}u}2`pα´ξqu2p0q,\nq´\nb,α,ξpuq“}u1}2`}pξ`bx2qu}2`b}u}2`pα`ξqu2p0q.(3.7)\nWe also observe that\nq´\nb,α,ξ“q`\n´b,α,´ξ, (3.8)\nwhich reflects the first relation in (1.7). In what follows, we drop the reference to b\nin the notation.14J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nIn relation to our problem we see that, for u“pu1, u2qan eigenfunction of Dγ,ξ,\nwe have\nq`\nλγ,ξpu1q“xu1, dξd:\nξu1y“λ2}u1}2,forλą0and,\nq´\n´λγ´1,ξpu2q“xu2, d:\nξdξu2y“λ2}u2}2,forλă0.\nNext, we describe a bijection existing between the kernels of Dγ,ξ´λandh`\nγλ,ξ´λ2\nprovided γλą0. Ifγλă0analogous statements can be obtained for h´\n´λ{γ,ξ´λ2.\nThe following lemma is a straightforward adaptation of [3, Proposition 2.9]. We\nrecall its proof for the convenience of the reader and we emphasize that it does not\nrequire sign assumptions on bandγ.\nLemma 3.4. Letpb, γqPR2,ξPR. Then, for any λPRzt0uwithγλą0, the map\nJ: kerph`\nγλ,ξ´λ2qQuÞÑpu, λ´1d:\nξuqPkerpDγ,ξ´λq\nis well-defined and it is an isomorphism.\nProof.First, let uPkerph`\nγλ,ξ´λ2qandvPDompDγ,ξ´λq. We have\nxJpuq,pDγ,ξ´λqvy“xu, dξv2´λv1y`λ´1xd:\nξu, d:\nξv1´λv2y,\nso that, by integrating by parts,\nxJpuq,pDγ,ξ´λqvy“λ´1pq`\nγλ,ξpu, v1q´λ2xu, v1yq“ 0,\nwhere we used that uPkerph`\nγλ,ξ´λ2q. Thus, Jis well defined. It is also injective\nfrom the very definition. For the surjectivity, we consider pu1, u2qPkerpDγ,ξ´λq.\nWe have\ndξu2“λu1, d:\nξu1“λu2.\nWe only have to check that u1Pkerph`\nγλ,ξ´λ2q. Take vPB1pR`qand notice that\nq`\nγλ,ξpu1, vq´λ2xu1, vy“xd:\nξu1, d:\nξvy`γλu 1p0qvp0q´λ2xu1, vy\n“λxu2, d:\nξvy`γλu 1p0qvp0q´λ2xu1, vy\n“λxdξu2, vy´λ2xu1, vy\n“0.\nThis finishes the proof. □\nLetusnowturntothecharacterization. Sincethefamily pq˘\nα,ξqpα,ξqPR`ˆRisanalytic\non the common domain B1pR`qthe eigenvalues of h˘\nα,ξ(which are all simple) are\nalso real analytic with respect to αand to ξ. We denote them by ν˘\nnpα, ξqso that\n0ďν˘\n1pα, ξqăν˘\n2pα, ξqă. . .\nThe following result completely characterizes positive and negative eigenvalues of\nDγ,ξin terms of αÞÑν˘\nnpα, ξq. Recall the notation in (1.5).\nTheorem 3.5. Letb‰0andγPp0,`8q. The equation\nν`\nnpγλ, ξq“λ2(3.9)\nhas a unique positive solution λ“ϑ`\nnpγ, ξq. Moreover, the equation\nν´\nnpγ´1λ, ξq“λ2(3.10)15\nhas a unique positive solution λ“ϑ´\nnpγ, ξq.\nTheproofofthistheoremusesLemma3.4andrequirestheanalysisoftheauxiliary\nquadratic forms performed in Section 4; we postpone it to Section 5.\n4.The auxiliary quadratic forms\nIn this section we perform a detailed study of the auxiliary quadratic forms from\nDefinition3.4. Werestricttheanalysistothecaseinwhich pb, γqPp0,`8qˆp 0,`8q.\nForαą0andξPRconsider the eigenvalue problems (recall Remark 3.2)\nh˘\nα,ξu˘\nα,ξ“ν˘pα, ξqu˘\nα,ξ. (4.1)\nMost of the following results can be traced back to [3] (notice, however, the different\nconvention for q´\nα,ξ). For the sake of completeness we present a concise argument.\n4.1.Study of αÞÑνnpα, ξq.\nLemma 4.1. Letν˘pα, ξqbe an eigenvalue as in (4.1). Then, the function αÞÑ\nν˘pα, ξqis increasing and\nBαν˘pα, ξq“pu˘\nα,ξp0qq2ą0. (4.2)\nProof.We only argue for the `case. To simplify notation, we denote the corre-\nsponding normalized solution of (4.1) as up”u`\nα,ξqand we drop the reference to ξ\nandαwhen not relevant.\nLet us first observe that for any smooth function gonr0,8qwe have, integrating\nby parts,\nxu,pdd:´νqgy“xd:u, d:gy`up0qpd:gqp0q´νxu, gy\n“up0qpd:gqp0q´pd:uqp0qgp0q.(4.3)\nIn view of the smoothness of uwith respect to αandx, we see that d:Bαu“d:Bαu.\nHence, since d:up0q“αup0q, we get that\npd:Bαuqp0q“up0q`αBαup0q. (4.4)\nTaking derivative with respect to αin the eigenvalue equation we get\npdd:´νqBαu“pB ανqu .\nAfter multiplying by uand integrating we use (4.3) to get\nBαν“xu, dd:Bαuy´νxu,Bαuy“up0qpd:Bαuqp0q´pd:uqp0qpBαuqp0q.\nThe conclusion follows using (3.5) and (4.4). □\nLemma 4.2. For all ně1, we have\nlim\nαÑ0ν´\nnpα, ξq“ν´\nnp0, ξq“νDir\nnp´b,´ξq (4.5)\nlim\nαÑ`8ν´\nnpα, ξq“νDir\nnpb, ξq. (4.6)16J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nProof.The first equality follows by analyticity. Then, we have q´\n0,ξpuq“}dξu}2. The\ncorresponding operator h´\n0,ξhas no zero mode. Now, if νis a positive eigenvalue, we\nhave\nd:\nξdξu“νu , d ξup0q“0.\nLetting v“dξu, we get dξd:\nξv“νvwith vp0q“0. This shows that νbelongs to the\nspectrum of the Dirichlet realization of dξd:\nξ. Conversely, if νą0is an eigenvalue\nofdξd:\nξassociated with the eigenfunction v, we have\nd:\nξdξu“νu , u“d:\nξv , d ξup0q“0.\nWe can check that u‰0(unless v“0). Thus, νbelongs to the spectrum of the\noperator h´\n0,ξ.\nWhen αÑ`8, we are in a singular regime. By using that H1\n0pR`qXB1pR`qĂ\nB1pR`qand the min-max principle, we see that\nν´\nnpα, ξqďνDir\nnpb, ξq.\nConversely, let us consider\nEnpα, ξq:“span\n1ďkďnu´\nα,ξ,k.\nWe notice that, for all uPEnpα, ξq,\npα`ξqu2p0qďq´\nα,ξpuqďν´\nnpα, ξq}u}2ďνDir\nnpb, ξq}u}2,\nso that, for αlarge enough,\nu2p0qďνDir\nnpb, ξq\nα`ξ}u}2“Opα´1q}u}2.\nThen, we also notice that\n}u1}2`}pξ`bx2qu}2`b}u}2ďq´\nα,ξpuqďνDir\nnpb, ξq}u}2.\nLet us consider a smooth cutoff function χwith compact support equal to 1near 0.\nThe function\n˜upx2q“upx2q´up0qχpx2q\nsatisfies the Dirichlet boundary condition. We notice that\np1´Cα´1\n2q}u}ď} ˜u}ďp 1`Cα´1\n2q}u}.\nThis tells us that, when uruns over Enpα, ξq,˜ualso runs over a space of dimension\nn.\nIn the same way, we get\n}u1}2`}pξ`bx2qu}2`b}u}2ěp1´Cα´1\n2q`\n}˜u1}2`}pξ`bx2q˜u}2`b}˜u}2˘\n.\nWe deduce that\n}˜u1}2`}pξ`bx2q˜u}2`b}˜u}2ďp1`Cα´1\n2qν´\nnpα, ξq}˜u}2.\nUsing the min-max principle, we infer that\nνDir\nnpb, ξqďp 1`Cα´1\n2qν´\nnpα, ξq,\nand the result follows. □17\nLemma 4.3. Forně1, we have\nlim\nαÑ0ν`\nnpα, ξq“ν`\nnp0, ξq“#\n0 n“1\nνDir\nn´1pb, ξqně2, (4.7)\nlim\nαÑ`8ν`\nnpα, ξq“νDir\nnp´b,´ξq. (4.8)\nProof.The proof is similar to that of Lemma 4.2. We have q`\n0,ξpuq “ } d:\nξu}2. In\nparticular, 0is an eigenvalue associated with x2ÞÑe´1\nbpξ`bx2q2. So, ν`\n1p0, ξq “0.\nThen, let us consider a positive eigenvalue ν. We have\ndξd:\nξu“νu , d:\nξup0q“0.\nThis implies that\nd:\nξdξv“νv ,with v“d:\nξu‰0,andvp0q“0.\nConversely, if visaneigenfunctionoftheDirichletrealizationof d:\nξdξwitheigenvalue\nν, we have\ndξd:\nξu“νu ,with u“dξv ,and d:\nξup0q“0.\nThe argument to show the limit in (4.8) follows the same lines as the proof of\n(4.6). □\n4.2.Study of ξÞÑν˘\nnpα, ξq.\nLemma 4.4. Letν˘pα, ξqbe an eigenvalue as in (4.1). Then, we have\nBξν˘pα, ξq“1\nb`\nν˘pα, ξq`α2¯2αξ˘\npu˘\nα,ξp0qq2. (4.9)\nIn addition, if ξαis a critical point of ξÞÑνpα, ξq, we have\nB2\nξν˘pα, ξαq“¯ α2\nbpu˘\nα,ξp0qq2. (4.10)\nIn particular, ν˘pα,¨qhas at most one critical point. This critical point can only be\na local maximum for ν`pα,¨qand a local minimum for ν´.\nProof.We give again the proof only for the `case. We use the notation from the\nproof of the previous Lemma 4.1. (We also replace x2bytin the notation.)\nObserve that since Bξd:u“d:Bξu`u, we get\npd:Bξuqp0q“αpBξuqp0q´up0q. (4.11)\nBy differentiating (4.1) with respect to ξwe get\npdd:´νqBξu“rB ξν´2pξ`btqsu .\nHence, (4.3) and (4.11) yield\nBξν“xu,2pξ`btquy´up0q2. (4.12)18J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nIn addition, integrating by parts and using (4.1), we calculate\nxu,2pξ`btquy“1\nbż\nR`uptq2Btpξ`btq2dt\n“´ξ2\nbup0q2´2\nbż\nR`u1ptqpξ`btq2uptqdt\n“´ξ2\nbup0q2´2\nbż\nR`u1ptqpν`b`B2\ntquptqdt\n“´ξ2\nbup0q2´1\nbż\nR`Btrppν`bquptq2`u1ptq2qsdt .\nUsing that u1p0q“pα´ξqup0qwe readily obtain (4.9). Hence, if a critical point ξα\nexists, it satisfies ν`pα, ξαq`α2´2αξα“0. Taking the derivative of (4.9) with\nrespect to ξand evaluating at ξαwe obtain (4.10). □\nWith the help of the perturbation theory, we get the following (see [3, Lemma\n4.14]).\nLemma 4.5. We have\nlim\nξÑ´8ν`\nnpα, ξq“p 2n´2qb , lim\nξÑ`8ν`\nnpα, ξq“`8 ,\nand\nlim\nξÑ´8ν´\nnpα, ξq“2nb , lim\nξÑ`8ν´\nnpα, ξq“`8 .\nThis allows us to show the following.\nLemma 4.6. The function ν`pα,¨qhas no critical points. Moreover, ν´pα,¨qhas a\nunique critical point, which is a global minimum.\nProof.Since ν`pα,¨q ą 0, from (4.9) we see that it is increasing on p´8,0q. If it\nhas a (unique) critical point for some ξαą0, it must be a non-degenerate global\nmaximum. This contradicts the limit at ξ“`8, hence ν`pα,¨qis increasing on R.\nNow assume that ν´pα,¨qdoes not have critical points. From (4.9) we must have\nν´pα, ξq`α2`2αξă0for all ξ(since it is the case for ξÑ´8). But this would\nimply that ν´pα,¨qis decreasing on R, which contradicts its limit at ξÑ`8.□\n5.Proofs for the energy dispersion curves\nIn this section we start by proving the characterization described in Theorem 3.5.\nNext, we apply that result to show Theorems 1.7 and 1.8.\nProof of Theorem 3.5. It is enough to deal with the positive eigenvalues of Dγ,ξpbq.\nIndeed, due to the charge conjugation (1.7), ϑ´\nnpγ, ξqis the n-th positive eigenvalue\nofDγ´1,´ξp´bq. Thus, if the characterization (3.9) is established, ϑ´\nnpγ, ξqis the\nunique positive solution of ν`\nnp´b, γ´1λ,´ξq“λ2or equivalently of (3.10) (here we\nuse (3.8)).\nLet us now prove that (3.9) has exactly one positive solution. Remember that\nγą0. We let\nfpλq“ν`\nnpγλ, ξq´λ2,19\nand notice that limλÑ`8fpλq“´8,fp0qě0, and f1p0qą0(see (4.2)). Thus (3.9)\nhas at least one positive solution. If Eis such a solution, we have fpEq“0and we\nnotice that\nf1pEq“γBαν`\nnpγE, ξq´2E“γruγE,ξp0qs2´2E .\nTo get the sign of f1pEq, we consider the polynomial of degree two given by\nPpλq“q`\nγλ,ξpuγE,ξq´λ2.\nBecause Pp0qě0,Pp´8q“´8 ,and PpEq“fpEq“0withEą0,thepolynomial\nmust have two roots of opposite sign. Thus, f1pEq“P1pEqă0. This shows that\n(3.9) has at most one positive solution and thus exactly one, which is denoted by\nEnpγ, ξq.\nIn fact,pEnqně1is increasing. Indeed,\n0“ν`\nn`1pγEn`1, ξq´E2\nn`1ąν`\nnpγEn`1, ξq´E2\nn`1“fpEn`1q,\nwhich implies that EnăEn`1.\nFor all ně1, due to Lemma 3.4, Enpγ, ξqis a positive eigenvalue of Dγ,ξ. This\ntells us that\nA:N˚QnÞÑEnpγ, ξqPspecpDγ,ξqXR`\nis well-defined (and it is injective).\nWe now show that the map is surjective. For all ně1, Lemma 3.4 implies that\nh`\nγϑ`\nnpγ,ξq,ξ´pϑ`\nnpγ, ξqq2has a non-zero kernel. This means that, for some mě1, we\nhave\nν`\nmpγϑ`\nnpγ, ξq, ξq“pϑ`\nnpγ, ξqq2,\nand thus ϑ`\nnpγ, ξq “Empγ, ξq. This implies that Ais bijective, hence Enpγ, ξq “\nϑ`\nnpγ, ξqfor all ně1. □\n5.1.Proof of Theorems 1.7 and 1.8. In what follows, in order to ease the read-\nability, we drop the reference to the index nin the notation. In view of Theorem 3.5\nwe have\nν`pγϑ`, ξq“pϑ`q2\nand, due to the analyticity and the chain rule, the derivative with respect to ξgives\n(with α”γϑ`q\npγBαν`pγϑ`, ξq´2ϑ`qBξϑ``Bξν`pγϑ`, ξq“0, (5.1)\nand differentiating with respect to γyields:\npγBαν`pγϑ`, ξq´2ϑ`qBγϑ``Bαν`pγϑ`, ξqϑ`“0. (5.2)\nWe saw in the proof of Theorem 3.5 that\nγBαν`pγϑ`, ξq´2ϑ`ă0.\nIn the`case, we see that ϑ`pγ,¨qhas no critical points and is increasing. We also\nsee that γÞÑϑ`pγ, ξqis increasing (by using (4.2)), which proves the monotonicity\ninγofϑ`announced in Theorem 1.8.\nIn the´case, by performing the same derivatives on (3.10), we have\npγ´1Bαν´pγ´1ϑ´, ξq´2ϑ´qBξϑ´`Bξν´pγ´1ϑ´, ξq“0, (5.3)\nand\npγ´1Bαν´pγ´1ϑ´, ξq´2ϑ´qBγϑ´´γ´2Bαν´pγ´1ϑ´, ξqϑ´“0.(5.4)20J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\nWestillhave γ´1Bαν´pγ´1ϑ´, ξq´2ϑ´ă0. Inparticular, γÞÑϑ´pγ, ξqisdecreasing.\nIfξγis a critical point of ϑ´pγ,¨q, then we have\nBξν´pγ´1ϑ´pγ, ξγq, ξγq“0.\nRemark 5.1. We recall Lemma 4.4 and we have\nν´pα, ξγq`α2`2αξγ“0,with α“γ´1ϑ´pγ, ξγq.\nHence, ϑ´pγ, ξγq“´2γ\nγ2`1ξγandξγ“ξαby the uniqueness of the critical point.\nBeing a non-degenerate minimum, it is necessary that B2\nξν´pγ´1ϑ´pγ, ξγq, ξγqą0\nand, by taking one more derivative in ξof (5.3), we see that B2\nξϑ´pγ, ξγq ą 0.\nTherefore, all the critical points of ϑ´pγ,¨qare local non-degenerate minima and\nthus there is at most one such point. If there is no critical point, we have, for all ξ,\nBξϑ´pγ, ξq“C´\np1`γ´2qϑ´pγ, ξq`2γ´1ξ¯\n‰0, Cą0.\nLet us assume for the moment that (1.6) is true; we will prove that later on. If ξis\nsufficiently negative, then the left-hand side of the above expression is negative, so\nit must remain negative for all ξ. This implies that ϑ´pγ, ξqmust be bounded from\nabove, contradicting the limit ξÑ`8in (1.6). This ends the analysis of critical\npoints announced in Theorem 1.7.\nIt remains to explain why (1.6) holds. We only consider the limit ξÑ´8. We\nrecall Lemma 4.5. Let us fix εą0and define λ1“?\n2nb´εandλ2“?\n2nb`ε.\nThen there exists ξpϵqă0such that for all ξăξpϵqwe have\nν´pγ´1λ1, ξq´λ2\n1ą0 and ν´pγ´1λ2, ξq´λ2\n2ă0.\nThis implies that\nλ1ăϑ´pγ, ξqăλ2,@ξăξpϵq.\nThe limit ξÑ`8can be analyzed similarly (as well as the limits for the ϑ`). This\nends the proof of Theorem 1.7.\nIt remains to discuss the limits in Theorem 1.8. Let us consider ϑ`\nnpγ, ξq. Take\nεą0andλ“ϑnp0, ξq´ε(forně2). We have ν`pγλ, ξq´λ2ą0forγsmall\nenough since ν`p0, ξq“pϑ`p0, ξqq2. Thus, ϑ`p0, ξq´εăϑ`pγ, ξq. In the same way,\nwe get ϑ`pγ, ξqăϑ`p0, ξq`ε. This proves the first limit in Theorem 1.8.\nNext, we consider the limit γÑ `8. We take λ“ϑ`p`8, ξq´ε. We have\nν`pγλ, ξq´λ2ą0forγlarge enough since ν`p`8, ξq “ p ϑ`p`8, ξqq2. Thus,\nϑ`p`8, ξq´εăϑ`pγ, ξq. We easily get the upper bound ϑ`pγ, ξqăϑ`p`8, ξq`ε.\nThe case of ϑ´pγ, ξqis similar.\nAppendix A.Lemma on trace class operators\nLemma A.1. LettTnuně1be a sequence of trace class operators on some separable\nHilbert space, having the property that their trace norms are uniformly bounded, i.e.\nsupně1}Tn}1“că8. Assume that Tnconverges to Tin the operator norm topology.\nThen Tis trace class.21\nProof.Since the Tn’s are compact operators, Tis also compact and admits a singular\nvalue decomposition (SVD), i.e. there exist two orthonormal systems tfjujě1and\ntgjujě1, together with a set of non-increasing singular values sjě0such that\nT“ÿ\njě1sj|fjyxgj|.\nTis trace class ifř\njě1sjă 8. We will show that for every Ně1we haveřN\nj“1sjďc. Let us introduce the SVD of each Tnas\nTn“ÿ\njě1spnq\nj|fpnq\njyxgpnq\nj|,}Tn}1“ÿ\njě1spnq\njďc.\nThen\nNÿ\nk“1sk“lim\nnÑ8Nÿ\nk“1xfk, Tngky“lim\nnÑ8Nÿ\nk“1ÿ\njě1spnq\njxfk, fpnq\njyxgpnq\nj, gky.\nUsing Bessel’s inequality for the orthonormal systems fkandgkwe have\nˇˇˇNÿ\nk“1xfk, fpnq\njyxgpnq\nj, gkyˇˇˇďdÿ\nk|xfk, fpnq\njy|2dÿ\nk|xgk, gpnq\njy|2ď}fpnq\nj}}gpnq\nj}“1\nwhich holds for all j, n, Ně1. This implies\nˇˇˇNÿ\nk“1ÿ\njě1spnq\njxfk, fpnq\njyxgpnq\nj, gkyˇˇˇďÿ\njě1spnq\nj“}Tn}1ďc,\nhenceř8\nj“1sjă8andTis trace class.\n□\nAcknowledgments. This work was partially conducted within the France 2030\nframeworkprogramme, theCentreHenriLebesgueANR-11-LABX-0020-01. Theau-\nthors thank T. Ourmières-Bonafos for useful discussions. They are also very grateful\nto the CIRM (and its staff) where this work was initiated. E.S. acknowledge sup-\nport from Fondecyt (ANID, Chile) through the grant # 123–1539. This work has\nbeen partially supported by CNRS International Research Project Spectral Analy-\nsis of Dirac Operators – SPEDO. H.C. acknowledge support from the Independent\nResearch Fund Denmark–Natural Sciences, grant DFF–10.46540/2032-00005B.\nReferences\n[1] J.-M. Barbaroux, H. Cornean, L. Le Treust, N. Raymond, and E. Stockmeyer. Bulk-edge\ncorrespondence. In preparation , 2024.\n[2] J.-M. Barbaroux, L. Le Treust, N. Raymond, and E. Stockmeyer. On the semiclassical\nspectrum of the Dirichlet–Pauli operator. Journal of the European Mathematical Society ,\n23(10):3279–3321, 2021.\n[3] J.-M. Barbaroux, L. Le Treust, N. Raymond, and E. Stockmeyer. On the Dirac bag model in\nstrong magnetic fields. Pure and Applied Analysis , 2023.\n[4] R. D. Benguria, S. , E. Stockmeyer, and H. Van Den Bosch. Self-Adjointness of Two-\nDimensional Dirac Operators on Domains. Annales Henri Poincaré , pages 1–13, 2017.\n[5] M. V. Berry and R. J. Mondragon. Neutrino billiards: time-reversal symmetry-breaking with-\nout magnetic fields. Proc. Roy. Soc. London Ser. A , 412(1842):53–74, 1987.22J.-M. BARBAROUX, H. D. CORNEAN, L. LE TREUST, N. RAYMOND, AND E. STOCKMEYER\n[6] H. D. Cornean, M. Moscolari, and K. S. Sørensen. Bulk–edge correspondence for unbounded\nDirac–Landau operators. Journal of Mathematical Physics , 64(2):021902, feb 2023.\n[7] H. D. Cornean, M. Moscolari, and S. Teufel. General bulk-edge correspondence at positive\ntemperature, 2022.\n[8] S. De Bièvre and J. V. Pulé. Propagating edge states for a magnetic hamiltonian. In Math-\nematical Physics Electronic Journal: (Print Version) Volumes 5 and 6 , pages 39–55. World\nScientific, 2002.\n[9] J. Dolbeault, M. J. Esteban, and E. Séré. On the eigenvalues of operators with gaps. Applica-\ntion to Dirac operators. J. Funct. Anal. , 174(1):208–226, 2000.\n[10] P. Elbau and G. M. Graf. Equality of bulk and edge Hall conductance revisited. Commun.\nMath. Phys. , 229(3):415–432, 2002.\n[11] G. M. Graf, H. Jud, and C. Tauber. Topology in shallow-water waves: a violation of bulk-edge\ncorrespondence. Communications in Mathematical Physics , 383:731–761, 2021.\n[12] M. Griesemer and H. Siedentop. A minimax principle for the eigenvalues in spectral gaps. J.\nLondon Math. Soc. (2) , 60(2):490–500, 1999.\n[13] J. Kellendonk and H. Schulz-Baldes. Quantization of edge currents for continuous magnetic\noperators. Journal of Functional Analysis , 209(2):388–413, 2004.\n[14] R. S. K. Mong and V. Shivamoggi. Edge states and the bulk-boundary correspondence in Dirac\nHamiltonians. Phys. Rev. B , 83:125109, Mar 2011.\n[15] M. Reed and B. Simon. Methods of modern mathematical physics. IV. Analysis of operators .\nAcademic Press, New York, 1978.\n[16] L. Schimmer, J. P. Solovej, and S. Tokus. Friedrichs extension and min–max principle for\noperators with a gap. 21(2):327–357, 2020.\n[17] K. M. Schmidt. A remark on boundary value problems for the Dirac operator. Quart. J. Math.\nOxford Ser. (2) , 46(184):509–516, 1995.\n[18] C. Tauber, P. Delplace, and A. Venaille. Anomalous bulk-edge correspondence in continuous\nmedia. Phys. Rev. Res. , 2:013147, Feb 2020.\n(J.-M. Barbaroux) Aix Marseille Univ, Université de Toulon, CNRS, CPT, Mar-\nseille, France\nEmail address :barbarou@univ-tln.fr\n(H.D.Cornean) Department of Mathematical Sciences, Aalborg University, Skjern-\nvej 4A, 9220 Aalborg, Denmark\nEmail address :cornean@math.aau.dk\n(L. Le Treust) Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille,\nFrance\nEmail address :loic.le-treust@univ-amu.fr\n(N. Raymond) Univ Angers, CNRS, LAREMA, Institut Universitaire de France,\nSFR MATHSTIC, F-49000 Angers, France\nEmail address :nicolas.raymond@univ-angers.fr\n(E.Stockmeyer) Instituto de Física, Pontificia Universidad Católica de Chile, Vicuña\nMackenna 4860, Santiago 7820436, Chile.\nEmail address :stock@fis.puc.cl"    },    {        "title": "2401.12571v1.The__B_c__meson_and_its_scalar_cousin_with_the_QCD_sum_rules.pdf",        "content": "arXiv:2401.12571v1  [hep-ph]  23 Jan 2024TheBcmeson and its scalar cousin with the QCD sum rules\nZhi-Gang Wang1\nDepartment of Physics, North China Electric Power University, Bao ding 071003, P. R. China\nAbstract\nIn this work, we use optical theorem to calculate the next-to -leading order corrections\nto the spectral densities directly in the QCD sum rules for th e pseudoscalar and scalar Bc\nmesons. We take the experimental data as guides to perform up dated analysis, and obtain\nthe masses and decay constants, therefore the leptonic deca y widths, which can be confronted\nto the experimental data in the future.\nPACS number: 12.38.Bx, 12.38.Lg\nKey words: Next-to-leading order contributions, QCD sum rules\n1 Introduction\nIn 1998, the CDF collaboration observed the pseudoscalar Bcmesons through the decay modes\nB±\nc→J/ψℓ±XandB±\nc→J/ψℓ±¯νℓin thep¯pcollisions at the energy√s= 1.8TeV at the\nFermilab Tevatron, the measured mass is 6 .40±0.39±0.13GeV [1, 2].\nIn 2007, the CDF collaboration confirmed the Bcmesons through the decay modes B±\nc→\nJ/ψπ±with the measured mass 6275 .6±2.9±2.5MeV [3]. In 2008, the D0 collaboration re-\nconstructed the B±\nc→J/ψπ±decays and confirmed the Bcmesons with the measured mass\n6300±14±5MeV [4]. Now the average value listed in the Review of Particle Physics is 6274.47±\n0.27±0.17MeV [5].\nIn 2014, the ATLAS collaboration reported the observation of a st ructure in the B±\ncπ+π−\ninvariant mass spectrum, which is consistent with the predicted B′\ncmeson with a mass of 6842 ±\n4±5MeV [6].\nIn 2019, the CMS collaboration observed two excited ¯bcstates in the B+\ncπ+π−invariant\nmass spectrum, which are consistent with the B′+\ncandB∗′+\nc, respectively [7]. The two states\nare separated in mass by 29 .1±1.5±0.7MeV, and the mass of the B′+\ncis measured to be\n6871.0±1.2±0.8±0.8MeV. Also in 2019, the LHCb collaboration observed the excited B′+\nc\nandB∗′+\ncmesons in the B+\ncπ+π−invariant mass spectrum. The B∗′+\ncmeson has a mass of\n6841.2±0.6±0.1±0.8MeV, which is reconstructed without the low-energy photon emitt ed in the\nB∗+\nc→B+\ncγdecay following the process B′∗+\nc→B∗+\ncπ+π−, while the B′+\ncmeson has a mass of\n6872.1±1.3±0.1±0.8MeV [8].\nIt is odd that the B′\ncmeson emerges as heavier than the mass of the B∗′\ncmeson, which is\nin conflict with all the theoretical estimations, this maybe or maybe n ot due to impossibility of\nreconstruction of the low-energy photon in the B∗+\nc→B+\ncγdecay [9]. Only the BcandB′\ncmesons\nare listed in Review of Particle Physics [5], which are in contrary to the copious (well-established)\nspectroscopy of the charmonium and bottomonium states. Despit e the enormous developments on\nthe heavy quark physics in recent years, the bottom-charm spec troscopy remains poorly known,\nwhich calls for further investigations.\nThe beauty-charm mesons provide an optimal laboratory for explo ring both the perturbative\nand nonperturbative dynamics of the heavy quarks, due to absen ce of the light quark’s contam-\nination, and for exploring the strong and electro-weak interaction s, as they are composed of two\ndifferent heavy flavors and cannot annihilate into gluons or photons . For the excited c¯bstates,\nwhich lie below the BDthreshold, would decay into the Bcmeson through the radiative decays\nor hadronic decays [10, 11], while the ground state Bccan only decay weakly through emitting a\nvirtualW-boson.\nThere have been several theoretical works on the mass spectro scopy of the bottom-charm\nmesons, suchasthe relativized(orrelativistic)quarkmodel withan specialpotential [10,11, 12,13,\n1E-mail,zgwang@aliyun.com.\n114], the nonrelativistic quark model with an special potential [15, 16 , 17, 18, 19, 20, 21, 22, 23], the\nsemi-relativistic quark model using the shifted large- Nexpansion [24, 25], the perturbative QCD\n[26], the nonrelativistic renormalization group [27], the lattice QCD [28 , 29, 30, 31], the Bethe-\nSalpeter equation [32, 33, 34], the full QCD sum rules [35, 36, 37, 38 , 39, 40, 41], the potential\nmodel combined with the QCD sum rules [15, 16], etc.\nWith the continuous development in experimental techniques, we ex pect that more c¯bstates\nwould be observed in the future. The decay constant, which param eterizes the coupling between\na current and a meson, plays an important role in exploring the exclus ive processes, because the\ndecay constants are not only a fundamental parameter describin g the pure leptonic decays, but\nalso are an universal input parameter related to the distribution am plitudes, form-factors, partial\ndecay widths and branching fractions in many processes. By precis ely measuring the branching\nfractions, we can resort to the decay constants to extract the CKM matrix element in the standard\nmodel and search for new physics beyond the standard model [42].\nDecay constants of the bottom-charm mesons have been investig ated in several theoretical\napproaches, such as the full QCD sum rules [35, 36, 37, 38, 39, 40, 41, 43, 44], the potential model\ncombined with the QCD sum rules [15, 16], the QCD sum rule combined with the heavy quark\neffective theory [45, 46, 47, 48, 49, 50, 51, 52], the covariant light -front quark model [53, 54], the\nlattice non-relativistic QCD [31], the shifted large- Nexpansion method [25], the field correlator\nmethod [55], etc. However, the values from different theoretical a pproaches vary in a large range,\nit is interesting to extend our previous works on the vector and axia lvectorBcmesons [40] to\ninvestigate the pseudoscalar and scalar Bcmesons with the full QCD sum rules by including\nnext-to-leading order radiative corrections and choose the upda ted input parameters, thus our\ninvestigations are performed in a consistent way. We take the expe rimental data as guides to\nchoose the suitable Borel parameters and continuum threshold pa rameters, examine the masses\nand decay constants of the pseudoscalar and scalar Bcmesons with the QCD sum rules, therefore\nwe calculate the leptonic decay widths to be confronted to the expe rimental data in the future.\nThe article is arranged: we calculate the next-to-leading order con tributions to the spectral\ndensities and obtain the QCD sum rules in Sect.2; in Sect.3, we present the numerical results and\ndiscussions; and Sect.4 is reserved for our conclusions.\n2 Explicit calculations of the spectral densities at the nex t-\nto-leading order\nWe write down the two-point correlation functions firstly,\nΠP/S(p2) =i/integraldisplay\nd4xeip·x/angbracketleft0|T/braceleftbig\nJ(x)J†(0)/bracerightbig\n|0/angbracketright, (1)\nwhereJ(x) =JP(x) andJS(x),\nJP(x) = ¯c(x)iγ5b(x),\nJS(x) = ¯c(x)b(x). (2)\nThe correlation functions can be written in the form,\nΠP/S(p2) =1\nπ/integraldisplay∞\n(mb+mc)2dsImΠP/S(s)\ns−p2, (3)\nthrough dispersion relation, where\nImΠP/S(s)\nπ=ρ0\nP/S(s)+ρ1\nP/S(s)+ρ2\nP/S(s)+···, (4)\n2Figure 1: The next-to-leading order contributions to the correlat ion functions.\nFigure 2: Six possible cuts correspond to virtual gluon emissions.\ntheρ0\nP/S(s),ρ1\nP/S(s),ρ2\nP/S(s),···are the spectral densities of the leading order, next-to-leading\norder, and next-to-next-to-leading order, ···. At the leading order,\nρ0\nP/S(s) =3\n8π2/radicalbig\nλ(s,m2\nb,m2c)\ns/bracketleftbig\ns−(mb∓mc)2/bracketrightbig\n, (5)\nwhere\nλ(s,m2\nb,m2\nc) =s2+m4\nb+m4\nc−2sm2\nb−2sm2\nc−2m2\nbm2\nc. (6)\nAt the next-to-leading order, there are three Feynman diagrams contribute to the correlation\nfunctions, seeFig.1. We calculatetheimaginarypartsusingthe Cutk osky’sruleoropticaltheorem,\nwhich lead to the same results, then use dispersion relation to obtain the correlation functions\n[39, 56]. There are ten possible cuts, six cuts attribute to virtual g luon emissions and four cuts\nattribute to real gluon emissions.\nThe six cuts shown in Fig.2 make attributions to virtual gluon emissions , and correspond to\nthe self-energy and vertex corrections, respectively. We calcula te the Feynman diagrams directly\nusing the dimensional regularization to regularize both the ultraviole t and infrared divergences,\nand resort to the on-shell renormalization scheme to subtract th e divergences so as to accomplish\nthe wave-function and mass renormalizations. Then we take accou nt of all contributions shown in\nFigure 3: The quark self-energy correction.\n3Figure 4: The vertex correction.\nFig.2 by the simple replacements of the vertexes in the currents,\n¯u(p1)iγ5u(p2)→¯u(p1)iγ5u(p2)+ ¯u(p1)i/tildewideΓ5u(p2)\n=/radicalbig\nZ1/radicalbig\nZ2¯u(p1)iγ5u(p2)+ ¯u(p1)iΓ5u(p2)\n= ¯u(p1)iγ5u(p2)/parenleftbigg\n1+1\n2δZ1+1\n2δZ2/parenrightbigg\n+ ¯u(p1)iΓ5u(p2), (7)\n¯u(p1)u(p2)→¯u(p1)u(p2)+ ¯u(p1)/tildewideΓ0u(p2)\n=/radicalbig\nZ1/radicalbig\nZ2¯u(p1)u(p2)+ ¯u(p1)Γ0u(p2)\n= ¯u(p1)u(p2)/parenleftbigg\n1+1\n2δZ1+1\n2δZ2/parenrightbigg\n+ ¯u(p1)Γ0u(p2), (8)\nwhere\nZi= 1+δZi= 1+4\n3αs\nπ/parenleftbigg\n−1\n4εUV+1\n2εIR+3\n4logm2\ni\n4πµ2+3\n4γ−1/parenrightbigg\n, (9)\nis theiquark’s wave-function renormalization constant from the self-en ergy correction, see Fig.3,\nand\nΓ5/0=γ54\n3g2\ns/integraldisplay1\n0dx/integraldisplay1−x\n0dy/integraldisplaydDkE\n(2π)D\nΓ(3)\n[k2\nE+(xp1+yp2)2]3/braceleftBig\n4k2\nE/parenleftBig\n1−1\n2εUV/parenrightBig\n+2(1−x−y+2xy)(s−m2\nb−m2\nc)\n±2(x+y)mbmc+2x(1−2x)m2\nb+2y(1−2y)m2\nc/bracerightBig\n, (10)\nfor the vertex corrections after the Wick’s rotation, see Fig.4, w hereγis the Euler constant,\nµis the renormalization scale, and the Euclidean four-momentum kE= (k1,k2,k3,k4). We set\nthe dimension D= 4−2εUV= 4 + 2εIRto regularize the ultraviolet and infrared divergences\nrespectively, and would add the scale factors µ2εUVorµ−2εIRif necessary.\nWe accomplish all the integrals and observe that the ultraviolet diver gences1\nεUVin the Γ 5/0,\nδZ1andδZ2are canceled out with each other, which is an outcome of the Ward ide ntity. The\ntotal contributions have no ultraviolet divergence,\n/tildewideΓ5=4\n3αs\n4πγ5fP(s),\n/tildewideΓ0=4\n3αs\n4πfS(s), (11)\n4Figure 5: Four possible cuts correspond to real gluon emissions.\nwhere\nfP/S(s) =fP/S(s)+2\nεIR+3logmbmc\n4πµ2+4log4πµ2\ns−γ+4−2(s−m2\nb−m2\nc)/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω\n1−ω/parenrightbigg\n/parenleftbigg1\nεIR+logs\n4πµ2+γ/parenrightbigg\n,\nfP/S(s) = 4V(s)+2(s−m2\nb−m2\nc)/bracketleftbig\nV00(s)−V10(s)−V01(s)+2V11(s)/bracketrightbig\n±2mbmc\n[V10(s)+V01(s)]+2m2\nb[V10(s)−2V20(s)]+2m2\nc[V01(s)−2V02(s)],\nω=/radicalBigg\ns−(mb+mc)2\ns−(mb−mc)2, (12)\nands=p2, the definitions and explicit expressions of the notations V(s),V00(s) andVij(s) with\ni,j= 0,1,2 are given in the appendix.\nThe total contributions of the virtual gluon emissions to imaginary p arts of the correlation\nfunctions are,\nImΠV\nP/S(s)\nπ=4\n3αs\n4π6\nπ/integraldisplaydD−1/vector p1\n(2π)D−12Ep1dD−1/vector p2\n(2π)D−12Ep2(2π)DδD(p−p1−p2)f(s)/bracketleftbig\ns−(mb∓mc)2/bracketrightbig\n,\n(13)\nthe superscript Vdenotes the virtual gluon emissions. We accomplish the integrals str aightfor-\nwardly inD= 4+2εIRdimension as there is no ultraviolet divergence, and obtain the analyt ical\nexpressions,\nImΠV\nP/S(s)\nπ=4\n3αs\nπρ0\nP/S(s)/braceleftbigg1\nεIR−2log4π+1\n2γ+1\n2logλ2(s,m2\nb,m2\nc)m3\nbm3\nc\nµ8s3+1\n2fP/S(s)\n−s−m2\nb−m2\nc/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω\n1−ω/parenrightbigg/bracketleftbigg1\nεIR−2log4π+2γ−2+logλ(s,m2\nb,m2\nc)\nµ4/bracketrightbigg/bracerightBigg\n.\n(14)\nThe four cuts shown in Fig.5 make contributions to the real gluon emis sions, the corresponding\nscattering amplitudes are shown explicitly in Fig.6. From Fig.6, we write d own the scattering\namplitudes Ta\n5,α(p) andTa\n0,α(p) ,\nTa\n5,α(p) = ¯u(p1)/braceleftbigg\nigsλa\n2γαi\n/negationslashp1+/negationslashk−mbiγ5+iγ5i\n−/negationslashp2−/negationslashk−mcigsλa\n2γα/bracerightbigg\nv(p2),\nTa\n0,α(p) = ¯u(p1)/braceleftbigg\nigsλa\n2γαi\n/negationslashp1+/negationslashk−mb+i\n−/negationslashp2−/negationslashk−mcigsλa\n2γα/bracerightbigg\nv(p2), (15)\n5Figure 6: The amplitudes for the real gluon emissions.\nwhereλais the Gell-Mann matrix. Then we obtain the contributions to the imagin ary parts of\nthe correlation functions with optical theorem,\nImΠR\nP/S(s)\nπ=−1\n2π/integraldisplaydD−1/vectork\n(2π)D−12EkdD−1/vector p1\n(2π)D−12Ep1dD−1/vector p2\n(2π)D−12Ep2(2π)DδD(p−k−p1−p2)\nTr/braceleftBig\nTa\n5/0,α(p)Ta†\n5/0,β(p)/bracerightBig\ngαβ\n=−2g2\ns\nπ/integraldisplaydD−1/vectork\n(2π)D−12EkdD−1/vector p1\n(2π)D−12Ep1dD−1/vector p2\n(2π)D−12Ep2(2π)DδD(p−k−p1−p2)\n/braceleftbigg\n2/bracketleftbig\ns−(mb∓mc)2/bracketrightbig/bracketleftbiggm2\nb\n(k·p1)2+m2\nc\n(k·p2)2−s−m2\nb−m2\nc\nk·p1k·p2\n+s−K2\nk·p1k·p2/bracketrightbigg\n−(s−K2)2\nk·p1k·p2/bracerightbigg\n, (16)\nwhere we have used the formulas/summationtextu(p1)¯u(p1) =/negationslashp1+mband/summationtextv(p2)¯v(p2) =/negationslashp2−mcfor the quark\nand antiquark respectively, K2= (p1+p2)2, and introduce the superscript Rto denote the real\ngluon emissions. We accomplish the integrals in D= 4+2εIRdimension as there is only infrared\ndivergence, and obtain the contributions,\nImΠR\nP/S(s)\nπ=4\n3αs\nπρ0\nP/S(s)/braceleftbigg\n−1\nεIR+2log4π−2γ+2−logλ3(s,m2\nb,m2\nc)\nm2\nbm2cs2µ4+(s−m2\nb−m2\nc)R12(s)\n−R11(s)−R22(s)−R1\n12(s)+R2\n12\n21\ns−(mb∓mc)2+s−m2\nb−m2\nc/radicalbig\nλ(s,m2\nb,m2c)\nlog/parenleftbigg1+ω\n1−ω/parenrightbigg/bracketleftbigg1\nεIR−2log4π+2γ−2+logλ3(s,m2\nb,m2\nc)\nm2\nbm2cs2µ4/bracketrightbigg/bracerightbigg\n, (17)\nthe definitions and explicit expressions of the R11(s),R22(s),R12(s),R1\n12(s) andR2\n12(s) are given\nin the appendix.\nNow we obtain the total spectral densities at the next-to-leading order,\nρ1\nP/S(s) =4\n3αs\nπρ0\nP/S(s)/braceleftbigg1\n2fP/S(s)−R11(s)−R22(s)−R1\n12(s)+(s−m2\nb−m2\nc)R12(s)\n+R2\n12\n21\ns−(mb∓mc)2−3\n2γ+2+1\n2logm7\nbm7\ncs\nλ4(s,m2\nb,m2c)\n+s−m2\nb−m2\nc/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω\n1−ω/parenrightbigg\nlogλ2(s,m2\nb,m2\nc)\nm2\nbm2cs2/bracerightBigg\n. (18)\nThe infrared divergences1\nεIR, log/parenleftBig\n1+ω\n1−ω/parenrightBig\n1\nεIRfrom the virtual and real gluon emissions are canceled\nout with each other, which is guaranteed by the Lee-Nauenberg th eorem [57]. The analytical\nexpressions are applicable in many phenomenological analysis besides the QCD sum rules.\n6Then we calculate the contributions of the gluon condensate direct ly, the calculations are easy\nand no much to say. Finally, we obtain the analytical expressions of t he spectral densities, take\nthe quark-hadron duality below the continuum thresholds s0\nP/Sand perform the Borel transforms\nin regard to the variable P2=−p2to obtain the QCD sum rules,\nf2\nP/SM4\nP/S\n(mb±mc)2exp/parenleftBigg\n−M2\nP/S\nT2/parenrightBigg\n=/integraldisplays0\nP/S\n(mb+mc)2ds/bracketleftBig\nρ0\nP/S(s)+ρ1\nP/S(s)+ρcon\nP/S(s)/bracketrightBig\nexp/parenleftBig\n−s\nT2/parenrightBig\n,\n(19)\nwhere\nρcon\nP/S(s) =∓mbmc\n24T4/angbracketleftαsGG\nπ/angbracketright/integraldisplay1\n0dx/bracketleftbiggm2\nc\nx3+m2\nb\n(1−x)3/bracketrightbigg\nδ(s−/tildewidem2\nQ)\n±mbmc\n8T2/angbracketleftαsGG\nπ/angbracketright/integraldisplay1\n0dx/bracketleftbigg1\nx2+1\n(1−x)2/bracketrightbigg\nδ(s−/tildewidem2\nQ)\n−s\n24T4/angbracketleftαsGG\nπ/angbracketright/integraldisplay1\n0dx/bracketleftbigg(1−x)m2\nc\nx2+xm2\nb\n(1−x)2/bracketrightbigg\nδ(s−/tildewidem2\nQ), (20)\n/tildewidem2\nQ=m2\nb\n1−x+m2\nc\nx, theT2is the Borel parameter, and the decay constants are defined by,\n/angbracketleft0|JP(0)|P(p)/angbracketright=fPM2\nP\nmb+mc,\n/angbracketleft0|JS(0)|S(p)/angbracketright=fSM2\nS\nmb−mc, (21)\nin other words,\n/angbracketleft0|Jα\nA(0)|P(p)/angbracketright=ifPpα,\n/angbracketleft0|Jα\nV(0)|S(p)/angbracketright=ifSpα, (22)\nthe subscripts AandVdenote the axial-vector and vector currents, respectively.\nWe eliminate the decay constants fP/Sand obtain the QCD sum rules for the masses of the\npseudoscalar and scalar Bcmesons,\nM2\nP/S=/integraltexts0\nP/S\n(mb+mc)2dsd\nd(−1/T2)/bracketleftBig\nρ0\nP/S(s)+ρ1\nP/S(s)+ρcon\nP/S(s)/bracketrightBig\nexp/parenleftbig\n−s\nT2/parenrightbig\n/integraltexts0\nP/S\n(mb+mc)2ds/bracketleftBig\nρ0\nP/S(s)+ρ1\nP/S(s)+ρcon\nP/S(s)/bracketrightBig\nexp/parenleftbig\n−s\nT2/parenrightbig.(23)\n3 Numerical results and discussions\nThe value of the gluon condensate /angbracketleftαsGG\nπ/angbracketrighthas been updated from time to time, and changes\ngreatly, we use the updated value /angbracketleftαsGG\nπ/angbracketright= 0.022±0.004GeV4[58]. We take the MSmasses of\nthe heavy quarks mc(mc) = 1.275±0.025GeV and mb(mb) = 4.18±0.03GeV from the Particle\nData Group [5]. In addition, we take account of the energy-scale de pendence of the MSmasses,\nmQ(µ) =mQ(mQ)/bracketleftbiggαs(µ)\nαs(mQ)/bracketrightbigg12\n33−2nf,\nαs(µ) =1\nb0t/bracketleftbigg\n1−b1\nb2\n0logt\nt+b2\n1(log2t−logt−1)+b0b2\nb4\n0t2/bracketrightbigg\n, (24)\n7T2(GeV2)s0(GeV2)pole M(GeV) f(GeV)\nBc(0−)3.0−4.044±1(68−89)%6.274±0.0540.371±0.037\nBc(0+)5.4−6.454±1(69−83)%6.702±0.0600.236±0.017\nTable 1: The Borel parameters, continuum threshold parameters , pole contributions, masses and\ndecay constants of the pseudoscalar and scalar Bcmesons.\n/s51/s46/s48 /s51/s46/s49 /s51/s46/s50 /s51/s46/s51 /s51/s46/s52 /s51/s46/s53 /s51/s46/s54 /s51/s46/s55 /s51/s46/s56 /s51/s46/s57 /s52/s46/s48/s53/s46/s52/s53/s46/s54/s53/s46/s56/s54/s46/s48/s54/s46/s50/s54/s46/s52/s54/s46/s54/s54/s46/s56/s55/s46/s48/s55/s46/s50/s55/s46/s52\n/s80\n/s32/s32/s77/s40/s71/s101/s86/s41\n/s84/s50\n/s40/s71/s101/s86/s50\n/s41/s32/s67/s101/s110/s116/s114/s97/s108/s32/s118/s97/s108/s117/s101\n/s32/s69/s114/s114/s111/s114/s32/s98/s111/s117/s110/s100/s115\n/s53/s46/s52 /s53/s46/s53 /s53/s46/s54 /s53/s46/s55 /s53/s46/s56 /s53/s46/s57 /s54/s46/s48 /s54/s46/s49 /s54/s46/s50 /s54/s46/s51 /s54/s46/s52/s53/s46/s56/s54/s46/s48/s54/s46/s50/s54/s46/s52/s54/s46/s54/s54/s46/s56/s55/s46/s48/s55/s46/s50/s55/s46/s52/s55/s46/s54/s55/s46/s56\n/s83\n/s32/s32/s77/s40/s71/s101/s86/s41\n/s84/s50\n/s40/s71/s101/s86/s50\n/s41/s32/s67/s101/s110/s116/s114/s97/s108/s32/s118/s97/s108/s117/s101\n/s32/s69/s114/s114/s111/s114/s32/s98/s111/s117/s110/s100/s115\nFigure 7: The masses of the pseudoscalar ( P) and scalar ( S)Bcmesons with variations of the\nBorel parameters T2.\nwheret= logµ2\nΛ2,b0=33−2nf\n12π,b1=153−19nf\n24π2,b2=2857−5033\n9nf+325\n27n2\nf\n128π3, Λ = 213MeV, 296MeV\nand 339MeV for the quark flavor numbers nf= 5, 4 and 3, respectively [5]. We choose nf= 4\nand 5 for the candbquarks, respectively, and then evolve the heavy quark masses to the typical\nenergy scale µ= 2GeV.\nThe threshold ( mb+mc)2decreases quickly with increase of the energy scale, the energy\nscale should be larger than 1 .7GeV, which corresponds to the squared mass of the Bcmeson,\n39.4GeV2. If we take the typical energy scale µ= 2GeV, which corresponds to the threshold\n(mb+mc)2≈36.0GeV2<M2\nP, it is reasonable.\nThe experimental masses of the BcandB′\ncmesons are 6274 .47±0.27±0.17MeVand 6871 .2±\n1.0MeV respectively from the Particle Data Group [5]. The scalar Bcmeson still escapes the\nexperimental detection, roughly speaking, the theoretical mass is 6712±18±7MeV from the\nlattice QCD [30] or 6714MeV from the nonrelativistic quark model [23]. We can tentatively take\nthe continuum threshold parameters as s0\nP= (39−47)GeV2ands0\nS= (45−55)GeV2, and search\nfor the ideal values by assuming the energy gap between the groun d state and first radial excited\nstates is about 0 .6GeV if lacking experimental data.\nAfter trial and error, we obtain the ideal Borel windows and contin uum threshold parameters,\nandthe correspondingpolecontributionsabout(70 −85)%,the poledominanceiswellsatisfied. On\nthe other hand, the gluon condensate plays a tiny important role, t he operator product expansion\nis well convergent. It is reliable to extract the masses and pole resid ues, which are shown in Table\n1 and Figs.7-8.\nThe predicted mass MP= 6.274±0.054GeV is in very good agreement with the experimental\ndata 6274.47±0.27±0.17MeV from the Particle Data Group [5], while the predicted mass MS=\n6.702±0.060GeV is consistent with other theoretical calculations [10, 11, 1 2, 13, 14, 15, 16, 18,\n19, 20, 21, 22, 23, 28, 29, 30, 32, 33, 34].\nWhile the decay constants, even for the pseudoscalar Bcmeson, the theoretical values vary in a\nlargerange,forexample, fP= 460±60MeV[16], 300 ±65MeV[36], 360 ±60MeV[37], 270 ±30MeV\n[41], 371±17MeV[43], 528 ±19MeV[44] from the full QCD sum rules, 410 ±40MeV[11], 433MeV\n8/s51/s46/s48 /s51/s46/s49 /s51/s46/s50 /s51/s46/s51 /s51/s46/s52 /s51/s46/s53 /s51/s46/s54 /s51/s46/s55 /s51/s46/s56 /s51/s46/s57 /s52/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56\n/s80\n/s32/s32/s102/s40/s71/s101/s86/s41\n/s84/s50\n/s40/s71/s101/s86/s50\n/s41/s32/s67/s101/s110/s116/s114/s97/s108/s32/s118/s97/s108/s117/s101\n/s32/s69/s114/s114/s111/s114/s32/s98/s111/s117/s110/s100/s115\n/s53/s46/s52 /s53/s46/s53 /s53/s46/s54 /s53/s46/s55 /s53/s46/s56 /s53/s46/s57 /s54/s46/s48 /s54/s46/s49 /s54/s46/s50 /s54/s46/s51 /s54/s46/s52/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48/s48/s46/s51/s53/s48/s46/s52/s48/s48/s46/s52/s53/s48/s46/s53/s48\n/s83\n/s32/s32/s102/s40/s71/s101/s86/s41\n/s84/s50\n/s40/s71/s101/s86/s50\n/s41/s32/s67/s101/s110/s116/s114/s97/s108/s32/s118/s97/s108/s117/s101\n/s32/s69/s114/s114/s111/s114/s32/s98/s111/s117/s110/s100/s115\nFigure 8: The pole residues of the pseudoscalar ( P) and scalar ( S)Bcmesons with variations of\nthe Borel parameters T2.\n[12] from the relativistic quark model, 498MeV [18], 440MeV [19] from t he non-relativistic quark\nmodel, 523 ±62MeV [54] from the light-front quark model, 420 ±13MeV [31] from the lattice\nnon-relativistic QCD, 315+26\n−50MeV [25] from the shifted N-expansion method, 438 ±10MeV [55]\nfrom the field correlator method, 322 ±42MeV [33] from the Bethe-Salpeter equation, etc. At the\npresent time, it is difficult to say which value is superior to others.\nThe present calculation fP= 371±37MeV is in very good agreement with the value 371 ±\n17MeV from the full QCD sum rules [43]. In our previous work, we obta in the values fV=\n384±32MeV and fA= 373±25MeV for the vector and axial-vector Bcmesons, respectively.\nOur calculations indicate that fP≈fV≈fA> fS. While in the QCD sum rule combined\nwith the heavy quark effective theory up to the order α3\ns, the decay constants have the relations\n˜fP=fP>fV>fS>˜fS>fA[47], where the decay constants ˜fPand˜fSare defined by\n/angbracketleft0|JP(0)|P(p)/angbracketright=˜fPMP,\n/angbracketleft0|JS(0)|S(p)/angbracketright=˜fSMS. (25)\nFrom Eq.(21) and Eq.(25), we can obtain the relations,\n˜fP=fPMP\nmb+mc,\n˜fS=fSMS\nmb−mc, (26)\nit is obvious that ˜fP>fPand˜fS<fS, which are in contrary to the relations obtained in Ref.[47],\nso no definite conclusion can be obtained. Naively, we expect that th e vector mesons have larger\ndecay constants than the corresponding pseudoscalar mesons [5 9].\nThe leptonic decay widths Γ ℓ¯νℓof the pseudoscalar and scalar Bcmesons can be written as,\nΓℓ¯νℓ=G2\nF\n8π|Vbc|2f2\nP/SMP/SM2\nℓ/parenleftBigg\n1−M2\nℓ\nM2\nP/S/parenrightBigg2\n, (27)\nwhereℓ=e,µ,τ, the Fermi constant GF= 1.16637×10−5GeV−2, the CKM matrix element\nVcb= 40.8×10−3, the masses of the leptons me= 0.511×10−3GeV,mµ= 105.658×10−3GeV,\nmτ= 1776.86×10−3GeV, the life time of the BcmesonτBc= 0.510×10−12s[5]. We take the\nmasses and decay constants of the pseudoscalar and scalar Bcmesons from the QCD sum rules to\n9obtain the partial decay widths,\nΓP→e¯νe= 2.03×10−12eV,\nΓP→µ¯νµ= 8.68×10−8eV,\nΓP→τ¯ντ= 2.08×10−5eV,\nΓS→e¯νe= 8.78×10−13eV,\nΓS→µ¯νµ= 3.75×10−8eV,\nΓS→τ¯ντ= 9.18×10−6eV, (28)\nand the branching fractions,\nBrP→e¯νe= 1.57×10−9,\nBrP→µ¯νµ= 6.73×10−5,\nBrP→τ¯ντ= 1.61×10−2. (29)\nThe largest branching fractions of the Bc(0−)→ℓ¯νℓare of the order 10−2, the tiny branching\nfractions maybe escape experimental detections.\n4 Conclusion\nIn this work, we extend our previous works on the vector and axialv ectorBcmesons to investigate\nthe pseudoscalar and scalar Bcmesons with the full QCD sum rules by including next-to-leading\norder corrections and choose the updated input parameters. In calculating the next-to-leading\norder corrections, we use optical theorem to obtain the spectra l densities directly, and resort\nto the scheme of the dimensional regularization to regularize both t he ultraviolet and infrared\ndivergences, which are canceled out with each other separately, t he net spectral densities are free\nof divergences. We take the experimental data as guides to choos e the suitable Borel parameters\nand continuum threshold parameters, and make reasonable predic tions for the masses and decay\nconstants, therefore the pure leptonic decay widths, which can b e confronted to the experimental\ndata in the future.\nAcknowledgements\nThis work is supported by National Natural Science Foundation, Gr ant Number 12175068.\nAppendix\nFirstly, we write down the fundamental integrals in calculating the ve rtex corrections,\nVab(s) = 16π2/integraldisplay1\n0dx/integraldisplay1−x\n0dy/integraldisplaydDkE\n(2π)DxaybΓ(3)\n[k2\nE+(xp1+yp2)2]3,\nV(s) = 16π2/parenleftbigg\n1−1\n2εUV/parenrightbigg/integraldisplay1\n0dx/integraldisplay1−x\n0dy/integraldisplaydDkE\n(2π)Dk2\nEΓ(3)\n[k2\nE+(xp1+yp2)2]3,(30)\n10and carry out the integrals to obtain the analytical expressions,\nV00(s) =1/radicalbig\nλ(s,m2\nb,m2c)/braceleftbigg\n−log/parenleftbigg1+ω\n1−ω/parenrightbigg/parenleftbigg1\nεIR+logs\n4πµ2+γ/parenrightbigg\n+log2(1−ω2\n1)\n4−log2(1+ω1)\n+log2(1−ω2\n2)\n4−log2(1+ω2)+2log(ω1+ω2)log/parenleftbigg1+ω\n1−ω/parenrightbigg\n−logω1log/parenleftbigg1+ω2\n1−ω2/parenrightbigg\n−logω2log/parenleftbigg1+ω1\n1−ω1/parenrightbigg\n−Li2/parenleftbigg2ω1\n1+ω1/parenrightbigg\n−Li2/parenleftbigg2ω2\n1+ω2/parenrightbigg\n+π2/bracerightbigg\n,\n=V00(s)−1/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω\n1−ω/parenrightbigg/parenleftbigg1\nεIR+logs\n4πµ2+γ/parenrightbigg\n,\nV10(s) =1\ns/braceleftbigg1\n2log/parenleftbigg1−ω2\n1\n1−ω2\n2/parenrightbigg\n−1\nω2log/parenleftbigg1+ω\n1−ω/parenrightbigg\n+logω2\nω1/bracerightbigg\n,\nV01(s) =V10(s)|ω1↔ω2,\nV20(s) =1\n2s/braceleftbigg\n−ω1ω2\nω1+ω2log/parenleftbigg1+ω\n1−ω/parenrightbigg\n−ω1\nω2(ω1+ω2)log/parenleftbigg1+ω\n1−ω/parenrightbigg\n+ω1\nω1+ω2log/parenleftbigg1−ω2\n1\n1−ω2\n2/parenrightbigg\n+2ω1\nω1+ω2logω2\nω1+1/bracerightbigg\n,\nV02(s) =V20(s)|ω1↔ω2,\nV11(s) =1\n2s/braceleftbiggω1ω2\nω1+ω2log/parenleftbigg1+ω\n1−ω/parenrightbigg\n−ω1−ω2\n2(ω1+ω2)log/parenleftbigg1−ω2\n1\n1−ω2\n2/parenrightbigg\n−1\nω1+ω2log/parenleftbigg1+ω\n1−ω/parenrightbigg\n+ω1\nω1+ω2logω1\nω2+ω2\nω1+ω2logω2\nω1−1/bracerightbigg\n,\nV(s) =1\nεUV+log4πµ2\ns−γ+2−2ω1ω2\nω1+ω2log/parenleftbigg1+ω\n1−ω/parenrightbigg\n−ω2\nω1+ω2log(1−ω2\n1)\n−ω1\nω1+ω2log(1−ω2\n2)−2ω1logω1+ω2logω2\nω1+ω2+2log(ω1+ω2),\n=V(s)+1\nεUV+log4πµ2\ns−γ+2, (31)\nwhere\nω1=/radicalbig\nλ(s,m2\nb,m2c)\ns+m2\nb−m2c,\nω2=/radicalbig\nλ(s,m2\nb,m2c)\ns+m2c−m2\nb,\nM=mb+mc\nmb−mc,\nLi2(x) =−/integraldisplayx\n0dtlog(1−t)\nt. (32)\nSecondly, we introduce the notation\n/integraldisplay\ndps=/integraldisplaydD−1/vectork\n2EkdD−1/vector p1\n2Ep1dD−1/vector p2\n2Ep2δD(p−k−p1−p2),\n11for simplicity, and write down the analytical expressions of the thre e-body phase-space integrals,\nR11(s) =sm2\nb\nπ2/radicalbig\nλ(s,m2\nb,m2c)(2π)−4εIRµ−2εIR/integraldisplay\ndps1\n(k·p1)2\n=1\n2εIR−log4π+γ−1+log/radicalbig\nλ(s,m2\nb,m2c)3\nmbmcsµ2−s+m2\nb−m2\nc\n2/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω1\n1−ω1/parenrightbigg\n−m2\nb−m2\nc/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω1\n1−ω1/parenrightbigg\n−s−m2\nb+m2\nc/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω\n1−ω/parenrightbigg\n=R11(s)+1\n2εIR−log4π+γ−1+log/radicalbig\nλ(s,m2\nb,m2c)3\nmbmcsµ2,\nR22(s) =R11(s)|mb↔mc,\nR12(s) =s\nπ2/radicalbig\nλ(s,m2\nb,m2c)(2π)−4εIRµ−2εIR/integraldisplay\ndps1\nk·p1k·p2\n=1/radicalbig\nλ(s,m2\nb,m2c)/braceleftBigg\nlog/parenleftbigg1+ω\n1−ω/parenrightbigg/bracketleftBigg\n1\nεIR−2log4π+2γ−2+2log/radicalbig\nλ(s,m2\nb,m2c)3\nmbmcsµ2/bracketrightBigg\n−2logmb\nmclog/parenleftbiggM+ω\nM−ω/parenrightbigg\n−log2/parenleftbigg1+ω\n1−ω/parenrightbigg\n+2logs\n¯slog/parenleftbigg1+ω\n1−ω/parenrightbigg\n−4Li2/parenleftbigg2ω\n1+ω/parenrightbigg\n+2Li2/parenleftbiggω−1\nω−M/parenrightbigg\n+2Li2/parenleftbiggω−1\nω+M/parenrightbigg\n−2Li2/parenleftbiggω+1\nω−M/parenrightbigg\n−2Li2/parenleftbiggω+1\nω+M/parenrightbigg\n−1\n2Li2/parenleftbigg1+ω1\n2/parenrightbigg\n−1\n2Li2/parenleftbigg1+ω2\n2/parenrightbigg\n−Li2(ω1)−Li2(ω2)+log2log[(1+ ω1)(1+ω2)]\n2−log22\n2+π2\n12/bracerightbigg\n,\n=R12(s)+1/radicalbig\nλ(s,m2\nb,m2c)log/parenleftbigg1+ω\n1−ω/parenrightbigg/bracketleftBigg\n1\nεIR−2log4π+2γ−2+2log/radicalbig\nλ(s,m2\nb,m2c)3\nmbmcsµ2/bracketrightBigg\n,\nR1\n12(s) =s\nπ2/radicalbig\nλ(s,m2\nb,m2c)/integraldisplay\ndpss−K2\nk·p1k·p2\n=s/radicalbig\nλ(s,m2\nb,m2c)/braceleftbigg\nlog2(1−ω)−log2(1+ω)+2log2s\n¯slog/parenleftbigg1+ω\n1−ω/parenrightbigg\n+2Li2/parenleftbigg1−ω\n2/parenrightbigg\n−2Li2/parenleftbigg1+ω\n2/parenrightbigg\n+2Li2/parenleftbigg1+ω\n1+M/parenrightbigg\n+2Li2/parenleftbigg1+ω\n1−M/parenrightbigg\n−2Li2/parenleftbigg1−ω\n1−M/parenrightbigg\n−2Li2/parenleftbigg1−ω\n1+M/parenrightbigg/bracerightbigg\n,\nR2\n12(s) =s\nπ2/radicalbig\nλ(s,m2\nb,m2c)/integraldisplay\ndps/parenleftbig\ns−K2/parenrightbig2\nk·p1k·p2\n=s2\n/radicalbig\nλ(s,m2\nb,m2c)/braceleftbigg\nlog2(1−ω)−log2(1+ω)+2log4s\n¯slog/parenleftbigg1+ω\n1−ω/parenrightbigg\n+2Li2/parenleftbigg1−ω\n2/parenrightbigg\n−2Li2/parenleftbigg1+ω\n2/parenrightbigg\n+2Li2/parenleftbigg1+ω\n1+M/parenrightbigg\n+2Li2/parenleftbigg1+ω\n1−M/parenrightbigg\n−2Li2/parenleftbigg1−ω\n1−M/parenrightbigg\n−2Li2/parenleftbigg1−ω\n1+M/parenrightbigg\n+2ω¯s\ns−¯s\ns(1+ω2)log/parenleftbigg1+ω\n1−ω/parenrightbigg/bracerightbigg\n, (33)\nwhere ¯s=s−(mb−mc)2.\n12References\n[1] F. 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C75(2015) 427.\n14"    },    {        "title": "2401.12575v2.Linear_stability_and_spectral_modal_decomposition_of_three_dimensional_turbulent_wake_flow_of_a_generic_high_speed_train.pdf",        "content": "arXiv:2401.12575v2  [physics.flu-dyn]  24 Jan 2024This draft was prepared using the LaTeX style file belonging t o the Journal of Fluid Mechanics 1\nLinear stability and spectral modal\ndecomposition of three-dimensional\nturbulent wake flow of a generic high-speed\ntrain\nXiao-Bai Li1,2, Simon Demange2, Guang Chen1, Jia-Bin Wang1,\nXi-Feng Liang1, Oliver T. Schmidt3and Kilian Oberleithner2†\n1Key Laboratory of Traffic Safety on Track of Ministry of Educat ion, School of Traffic &\nTransportation Engineering, Central South University, 41 0075 Changsha, PR China\n2Laboratory for Flow Instability and Dynamics, Technische U niversit¨ at Berlin, 10623 Berlin,\nGermany\n3Department of Mechanical and Aerospace Engineering, Unive rsity of California San Diego, La\nJolla, CA, USA\n(Received xx; revised xx; accepted xx)\nThis work investigates the spatio-temporal evolution of coherent structures in the wake\nof a generic high-speed train, based on a three-dimensional datab ase from large eddy\nsimulation. Spectral proper orthogonal decomposition is used to e xtract energy spectra\nand energy ranked empirical modes for both symmetric and antisym metric components\nof the fluctuating flow field. The spectrum of the symmetric compon ent shows overall\nhigher energy and more pronounced low-rank behavior compared t o the antisymmetric\none, with the most dominant symmetric mode at ω= 3.437 characterized by vortex\nshedding in the near wake and constant streamwise wavenumber st ructures in the far\nwake. The mode bispectrum further reveals the dominant role of se lf-interaction of the\nsymmetriccomponent,leadingtofirstharmonicandsubharmonictr iadsofthefundamen-\ntal frequency, with remarkable deformation of the mean field. The n, the stability of the\nthree dimensional wake flow is analyzed based on two-dimensional loc al linear stability\nanalysis. Temporal stability analysis is first performed for both the near wake and the far\nwake regions, showing a more unstable condition in the near wake reg ion. The absolute\nfrequency of the near-wake eigenmode is determined based on spa tio-temporal analysis,\nthen tracked along the streamwise direction to find out the global m ode growth rate and\nfrequency, which indicate a marginally stable global mode oscillating at a frequency very\nclose to the most dominant SPOD mode. The global mode wavemaker is then located,\nand the structural sensitivity is calculated based on the direct and adjoint modes derived\nfrom a local spatial analysis, with the maximum value localized within the recirculation\nregion close to the train tail. Finally, the global mode shape is compute d by tracking the\nmost spatially unstable eigenmode in the far wake, and the alignment w ith the SPOD\nmode is computed as a function of streamwise location. By combining d ata-driven and\ntheoretical approaches, the mechanisms of coherent structur es in complex wake flows are\nwell identified and isolated.\nKey words: Keywords\n†Email address for correspondence: oberleithner@tu-berli n.de2 X.-B. Li and others\n1. Introduction\nIn the face of climate change, high-speed rail has gradually develop ed to become the\nkey to decarbonizing transportation. As a bluff body operating at h igh Reynolds number\n(Re), the flow around a high-speed train exhibits complex characterist ics of a fully\ndeveloped three-dimensionalturbulent flow( Schetz 2001 ). The unsteadyaerodynamicsof\nhigh-speed trainsisthen directly characterizedby the fluctuating aerodynamicforcesand\ninduced slipstreams. Therefore, understanding the dominant dyn amics in the complex\nturbulent flow around the train is crucial to improving and optimizing a erodynamic\nperformance. For this purpose, the search for and identification of physically significant\ncoherent structures,or modes, is a suitable method ( Tairaet al.2017). In fact, extracting\nand understanding the physical mechanisms of instability in complex t hree-dimensional\nturbulent flow has been attracting research interest for severa l decades but remains\nchallenging. In particular, the complexity of the base flow, as well as the high demand for\ncomputational resources, causes huge difficulties in solving this pro blem (Theofilis 2011 ).\nHowever, due to the increasing demand for transportation efficien cy, passenger comfort,\nand operational safety, extracting and understanding the mech anisms of instability in\nthe flow around the train is still of great research interest and sign ificant engineering\nimportance.\nDespite its aerodynamic design, the high-speed train exhibits bluff-b ody flow char-\nacteristics reminiscent of those of the well-studied Ahmed body ( Ahmedet al.1984;\nLienhart et al.2002). Three important structures can be identified in the wake of the\nbody: a recirculation bubble over the slanted surface, a pair of long itudinal C-pillar\nvortices generated from the two side edges of the slanted surfac e, and a recirculation zone\nbehind the rear vertical base. Several studies have focused on t he interaction and control\nstrategies of these structures. In Zhanget al.(2015);Liuet al.(2021), the unsteady\ncharacteristics of Ahmed bodies in the high- and low-drag regimes we re investigated\nusingmultipleexperimentaltechniques.Onthebasisofthesefinding s,severalsteadyblow\ndrag reduction strategies have been successfully developed Zhanget al.(2018);Liet al.\n(2022). Meanwhile, random switching between two reflectional-symmetry breaking states\nof the wake has been investigated in Grandemange et al.(2013);Heet al.(2021a),\nwith appropriate strategies altering the natural bi-stability of the wake proposed in\nGrandemange et al.(2014);Evstafyeva et al.(2017);Haffneret al.(2020).\nHowever, the full picture of the three-dimensional coherent str uctures in the wake,\nas well as the associated instability information, remains limited. Ther efore, further\ninterpretation and understanding of the physical mechanisms tha t generate disturbances\nin the flow are limited. In particular, for more aerodynamically shaped high-speed\ntrains, the identification of the flow structures mentioned above is less obvious. To\nextend understanding and control of wake dynamics in a more comp lex situation, modal\ndecomposition ( Tairaet al.2017) of the three-dimensional flow must be taken into\nconsideration, which is the objective of this work.\nData-driven analysis has proven to be an effective method to extra ct coherent struc-\ntures from flow snapshots as empirical modes. The most classic and widely used data-\ndriven approach, proper orthogonal decomposition (POD), was fi rst introduced to the\nfield of fluid dynamics by Lumley(1967,1970). In this approach, the flow is represented\nas a mean and a superposition of space-time-dependent modes. Th ese resulting modes\ncan then be used for a variety of purposes, from classification to r educed order modeling\nto control ( Rowley & Dawson 2017 ;Tairaet al.2017). Meanwhile, many other empirical\napproaches have been proposed. For example, the Balanced POD ( Rowley 2005 ) which\nserves as an expansion for linear input-output relationships, and D ynamic Mode De-Linear stability and SPOD of train wake flow 3\ncomposition (DMD) ( Schmid 2010 ) which approximates the dynamics of a higher-order\nsystem through a combination of linearly growing or decaying modes.\nIn this paper, we focus on the application of the spectral form of P OD, which is\ncalled spectral proper orthogonal decomposition (SPOD). This me thod is derived from\nthe space-time formulation of POD for statistically stationary flow. The resulting modes,\nwhich oscillate at a single frequency, are orthogonal under a space -time inner product. In\ngeneral, SPOD combines the advantages of the classical POD and DM D for statistically\nstationary flows, meanwhile, it provides an improved robustness ov er these two methods\nTowneet al.(2018). Furthermore, this method has been further extended to achie ve\nmore features, such as frequency-time ( Nekkanti & Schmidt 2021 ) and triadic interaction\nSchmidt (2020) analysis, or better convergence ( Blancoet al.2022;Schmidt 2022 ) and\nlower computational cost ( Schmidt & Towne 2019 ), and therefore has received increasing\ninterest in identifying coherent turbulent structures that are ph ysically meaningful in\nvarious flow problems. These applications include jet ( Schmidt et al.2018), pipe flow\n(Abreuet al.2020), flow around airfoils ( Abreuet al.2021), disk wake ( Nidhanet al.\n2022), and various industrial flows ( Haffneret al.2020;Heet al.2021b;Liet al.2021a;\nWanget al.2022b).\nConsidering vehicle wake flow, coherent structures represented by empirical modes\nare within the research scope of several studies listed above ( Heet al.2021a;\nGrandemange et al.2013;Haffneret al.2020;Liet al.2021a). However, these\napplications are limited to two-dimensional planes in the wake, which do not fully\ncapture the complex three-dimensional space-time coherent str uctures in the flow.\nTherefore, this study further extends the previous results sho wn inLiet al.(2021a)\nwhere a parameter study using a two-dimensional database from t rain wake flow is\nperformed, to find coherent structures from a global perspect ive. However, although\nSPOD extracts modes related to the dominant fluctuation of the flo w, this method is\npurely data-driven and model-free. As such, it does not reveal th e mechanisms driving\nthe coherent structures. However, it includes all non-linear dyna mics and may reveal\ninteractions between the structures in a quantitative manner. To further search for\nthe instability mechanisms, one may use either stochastic low order d ynamic models\n(Rigaset al.2015;Sieberet al.2021) or a mean field stability analysis.\nMean field linear stability analysis is considered to provide further insig ht into the\nmechanisms driving the flow dynamics. It is known that a self-excited oscillation can\nbe described by an unstable global mode ( Chomaz 2005 ) derived from a global stability\nanalysis. Recently, improved feasibility of large-scale linear algebra c omputations en-\nables tri-global stability analysis of flows with three inhomogeneous d irections ( Theofilis\n2011). Some applications include boundary layer flows with isolated roughn ess elements\n(Loiseauet al.2014;Kurz & Kloker 2016 ;Ma & Mahesh 2022 ), lid-driven cavity flows\n(Paredes 2014 ;G´ omezet al.2014), jets in crossflow ( Regan & Mahesh 2017 ), and wakes\nof rectangular prisms ( Zampogna & Boujo 2023 ). When mean flows display homogeneity\nin the direction normal to convective motion, the Bi-Global stability a pproach can be\nutilized. This approach considers two-dimensional modes with a spat ial wavenumber in\nthe third dimension, and has been applied to a broad range of canonic al flowTheofilis\n(2003,2011), as well as complex technical flows including swirling flows ( Kaiseret al.\n2018;M¨ ulleret al.2020), reacting flows ( Caselet al.2022;Wanget al.2022a), turbo-\nmachineryflows( M¨ ulleret al.2022)andtwo-phaseflows( Schmidt & Oberleithner2023 ).\nFor mean flows weakly evolving in the convection direction, either the tri-global or bi-\nglobal method can be applied locally, i.e. local one-dimensional analysis in the lines\nnormal to the streamwise direction to construct bi-global instabilit y. As reviewed by\nHuerre & Monkewitz (1990), the method is based on a spatio-temporal analysis of the4 X.-B. Li and others\nlocal velocity profile invoking the WKBJ approximation. The relationsh ip between local\nabsolute instability and global modes can be found in Monkewitz et al.(1993);Chomaz\n(2005), which concluded that a regionoflocalabsolute instability is aneces sarycondition\nfor the existence of global instability. Comparisons between result s of local and global\nstability analysis can be found in Giannetti & Luchini (2007);Juniperet al.(2011);\nJuniper & Pier (2015);Kaiseret al.(2018);Demange et al.(2022). In general, local\nstability analyses require less computational memory than global st ability analyzes, since\nthey convert a large matrix eigenvalue problem into several small ind ependent matrix\neigenvalue problems ( Juniper & Pier 2015 ). Meanwhile, as the local linearized Navier-\nStokes equation is solved at each discrete streamwise position, the eigenvalues can be\ncontinuously tracked to provide an accurate spatial description o f the mode. Therefore,\nlocal stability analysis is still widely used for flows beyond the range of global analysis\n(Pier 2008 ;Oberleithner et al.2011;Rukeset al.2016).\nTo the best of the author’s knowledge, there are limited studies reg arding the lin-\near instability mechanisms of fully developed turbulent wake flows beh ind vehicles.\nZampogna & Boujo (2023) investigated the global stability of rectangular prisms with\nrounded front edges, which approximate the geometry of the Ahm ed body. However, this\nstudy considered a laminar flow without ground effect, such that th e problem could be\ntreated with two symmetries. From a more practical perspective, flows behind high-speed\ntrains may consist of a series of vortex structures that are subj ect only to a spanwise\nsymmetry condition. In addition, the spatial and temporal evolutio n of these vortex\nstructures differs greatly from free-evolving structures due to the presence of the ground\n(Schetz 2001 ). Up to now, the instability mechanisms in typical vehicle wakes of high\nindustrial relevance remain an unanswered question. Is there a line ar global mode that\ndrives the instability in the turbulent wake? How does the linear global mode compare\nto the leading SPOD mode? Which part of the wake serves as the origin of the global\ninstability and is most sensitive to external forcing? These question s need to be answered\nto serve as a basis for further optimizations, while extending the ap plications of these\ntwo approaches to more complex flow problems and higher Re.\nSince the flow problem considered is only subject to spanwise symmet ry, the tri-\nglobal stability with three inhomogeneous directions should be consid ered. The WKBJ\napproximation then converts the three-dimensional linearized pro blem into several two-\ndimensional local problems to account for global instability. This is ge nerally a more\ncomplex situation than the researchshown in Juniper & Pier (2015);Rukeset al.(2016);\nKaiseret al.(2018), where local one-dimensional analyzes are used to construct th e bi-\nglobal mode. Meanwhile, the parallelism or weak non-parallelism in local a nalysis could\nbe a strong assumption ( Chomaz 2005 ;Pier 2008 ;Rukeset al.2016;Puckert & Rist\n2018) in the flow problem considered. How nonparallelism would affect the re sults of\nglobal instability and how to deal with nonparallelism in the complex base flow are also\nimportant issues to be solved.\nThe outline of the paper is as follows. The large eddy simulation setup u sed to obtain\nthethree-dimensionalflowfieldaroundthetrainisshownin §2,alongwiththedescription\nof the time-averaged wake-flow structures. In §3, we use SPOD to extract the dominant\nempirical modes and provide a first insight into the spatio-temporal characteristic of the\ncoherent structures. Meanwhile, bispectral mode decomposition is considered to compute\nfrequency triads, which explains the features in the SPOD spectru m. The tri-global\nstability mode obtained from two-dimensional local stability analysis is presented in\n§4. In§5, we compare the SPOD mode with the linear global mode at each strea mwise\nlocation, and the mechanism of the fundamental instability is interpr eted based on the\ncomparison results. The main findings and conclusions are summarize d in§6.Linear stability and SPOD of train wake flow 5\n%RXQGDU\\\u0003/D\\HU\u0003'HYHORSPHQW\u00035HJLPH\nS:DNH\u0003)RUPDWLRQ\n5HJLPH:DNH\u00033URSDJDWLRQ\u00035HJLPH\n\u0010\u0013\u0011\u0015\u0013\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0010\u0013\u0011\u0014\u0017\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 \u0003\u0003\u0003\u0003\u0003\u0010\u0013\u0011\u0013\u001b\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0010\u0013\u0011\u0013\u0015\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 \u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0013\u0011\u0013\u0017\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0013\u0011\u0014\u0013[\n+\n:/\n\u0019\u0011\u0019\u001a +ǻ[ ǻ\\ ǻ] \u0013\u0011\u0013\u0014\u0017 +ǻ[ ǻ\\ ǻ] \u0013\u0011\u0013\u0015\u001b +ǻ[ ǻ\\ ǻ] \u0013\u0011\u0013\u0017\u0015 +\n\u0014\u0011\u0016\u0016 +\u0015\u0011\u0015\u0015 +\u000bD\f \u000bE\f\n\u000bF\f\nFigure 1: Large Eddy Simulation setup. ( a) Geometric model; ( b) Distribution of\ncomputational grid; ( c) Instantaneous snapshot of vortex structures.\n2. Flow problem description\n2.1.Large Eddy Simulation\nThe database for both the data-driven and theoretical mode calc ulations is obtained\nfrom a large eddy simulation performed with the commercial code STA R-CCM+ 14.02.\nA complete description of the numerical setup and validation can be f ound in Liet al.\n(2021b), here only the essential information is presented.\nA simplified version of the Intercity-Express 2 (ICE2), also known a s the aerodynamic\ntrain model (ATM), is considered. The simulation is carried out on the scale of 1:10,\nresulting in the height of the model H= 0.36m, the width of the model W= 0.30m\nand the length of the model L= 2.5m, as shown in figure 1(a). To simulate the relative\nmotion between the train and the surrounding environment, the up stream boundary,\nlocated 10 Hfrom the train head, is assigned the inflow velocity U∞. Meanwhile, the\nground boundary, with a distance to the train bottom surface of 0 .15H, is defined with\nthe same moving velocity. The resulting Rebased on the height of the train and U∞\nis 9.5×104. The downstream boundary is located 30 Hfrom the train tail, with a zero\nstatic pressure outlet condition. On the side and roof of the compu tational domain, the\nsymmetry plane boundary condition is assigned, with a distance of 10 Hfrom the train\nmodel. The computational domain is then discretized using unstruct ured hexahedral\nvolume meshes, with the wall normal and wall parallel distances expr essed in viscous\nunits, respectively ∆y+= 0.16 and∆x+=∆z+= 28 for cells attached to the train\nsurface, as shown in figure 1(b). The total number of volume meshes used in the study\nis 46.8 million.\nInthecurrentresearch,thelargeeddysimulationbasedonthewa ll-adaptinglocal-eddy\nviscosity (WALE) subgrid-scale model is chosen. The use of a novel form of the velocity\ngradient tensor in the WALE subgrid scale model allows for much more universal model\ncoefficients compared to other widely used subgrid scale models. Mea nwhile, the WALE\nsubgrid scale model does not require any form of near-wall damping but automatically6 X.-B. Li and others\nFigure 2: Time-averaged mean flow structures visualized by iso-sur face of vortex\nidentification criterion Ω= 0.6 and streamlines.\nprovides accurate scaling at the walls. More details about the WALE s ubgrid scale model\ncanbe found in Nicoud & Ducros (1999). An implicit unsteadysegregatedincompressible\nfinite-volume solver is used, with the convective terms discretized b ased on a bounded\ncentral-differencing scheme, and the diffusion and turbulence term s are discretized with\nthe second-order upstream scheme. The time marching procedur e is performed using the\nimplicit second-order accurate three-time level scheme, with the d iscretized convective\ntime step set to 0 .0067t∗(t∗=W/U∞), which leads to the Courant-Friedrichs-Lewy\nnumber below unity in most of the computational grids. The simulation was first run for\n66.7t∗in order to reach a statistically stationary state, and then was run for a duration\nof 800t∗to collect field data. An instantaneous scene of the flow structure s around the\ngeneric high-speed train, which briefly illustrates the formation of t he turbulent wake, is\nshown in 1(c).\nThe three-dimensional snapshot database is continuously collecte d from a square box\nextending from x/W=−1.33 upstream of the tail nose tip to x/W= 6.67 downstream\nof the tail nose tip, y/W= 1.33 from the center plane of the train in both spanwise\ndirections, and z/W= 1.33 from the ground in the vertical direction. The time step\nbetweentwoconsecutivesnapshotsis0 .1t∗,resultinginatotalof8000snapshotscollected\nduring the simulation. These values have been shown to be sufficient t o produce well-\nconverged SPOD results according to the parametric study shown inLiet al.(2021a).\n2.2.Mean flow properties\nWeconsiderthetime-averagedfieldasthebasestateforthelinear stabilityanalysisand\nintroduce it in this section. Furthermore, prior knowledge of the me an field will facilitate\nunderstanding of the physics associated with the extracted cohe rent structures. In figure\n2, the time-averaged sectional streamlines are shown at several r epresentative locations.\nMeanwhile, vortex regions are identified by the isosurface of Ω(Liuet al.2016), colored\nbythemagnitudeofthevorticity.The Ω-methodworksasavortexidentificationcriterion\nsimilar to the Q-criterion and λ2-criterion, however, it is less sensitive to threshold valueLinear stability and SPOD of train wake flow 7\nto capture both strong and weak vortices in different cases. Note that the wake flow is\nsymmetric about the central plane after long-term time averaging , so only the y >0 half\nis shown here for better visualization.\nIt can be observed that, the attached flow on the slanted and bot tom surfaces of\nthe tail propagates downstream and approaches the rear end. T hen, the strong adverse\npressure near the tail nose point forces the attached flow partic les to separate from\nthe train surfaces, forming the spanwise vortex pair located just behind the tail nose\npoint. Meanwhile, in addition to the flow structures related to the se paration near the\nrear end, we can also observe the longitudinal vortex located abov e the side edge of\nthe slanted surface of the tail, which is similar in nature to the C-pillar v ortex in the\nAhmed body wake flow Zhanget al.(2015);Liuet al.(2021);Heet al.(2021a);Liet al.\n(2022). Thispairoflongitudinalvortexis formedbyflowseparationfromt he sidesurface,\nwhich exerts a strong pressure-suction effect in this area and con tinuously rolls up flow\nparticles from the slanted surface. As the longitudinal vortex pro pagates downstream, it\ngradually increases in diameter and lifts away from the tail surface, moving toward the\nground along the tail side edge. Due to the strong downwash from t he slanted surface,\nthe trailing vortex can be observed to move away from the central symmetry plane as\nit travels downstream. At x/W≈1.5, the longitudinal vortex structure attaches to\nthe ground and then propagates nearly parallel to the ground in th e downstream wake.\nIn general, the mean field is fully three-dimensional in the near wake r egion, and its\ncomplexity gradually decreases downstream of the solid body, deve loping to be nearly\nparallel downstream of x/W≈2.0. The two main features, the spanwise recirculation\nbubble and the streamwise vortex pair, should be related to the mea n field instability\ndue to the strongvelocity gradient and will therefore be discussed further in the following\ncontent.\n3. Data-driven coherent structure identification\n3.1.Symmetric & antisymmetric decomposition\nBased on the spanwise symmetry of the described flow problem, fluc tuations of the\nsampled flow field can be divided into symmetric and antisymmetric cont ributions\n(Hack & Schmidt 2021 ). To properly isolate coherent structures defined by the two\ndifferent types of fluctuations, an additive decomposition was applie d to the collected\nsnapshot data before extracting empirical modes. Defining the flu ctuating part of the\nflow field as\nq′(t) =q(t)−¯q=\nu(x,y,z,t)−¯u(x,y,z)\nv(x,y,z,t)−¯v(x,y,z)\nw(x,y,z,t)−¯w(x,y,z)\np(x,y,z,t)−¯p(x,y,z)\n=\nu′(x,y,z,t)\nv′(x,y,z,t)\nw′(x,y,z,t)\np′(x,y,z,t)\n(3.1)\nwhere¯(.) denotes the time-averagedflow quantities and ( .)′is the fluctuating part. Then,\nthe spanwise symmetric and antisymmetric contributions can be obt ained using the\nfollowing expression.\nq′\nS(t) =1\n2\nu′(x,y,z,t)+u′(x,−y,z,t)\nv′(x,y,z,t)−v′(x,−y,z,t)\nw′(x,y,z,t)+w′(x,−y,z,t)\np′(x,y,z,t)+p′(x,−y,z,t)\n(3.2)8 X.-B. Li and others\n(a)\n (b)\n (c)\n-0.2 -0.1 0 0.1 0.2u'\nFigure 3: Symmetric & antisymmetric decomposition on instantaneou s snapshot. ( a)\nOriginal field; ( b) Symmetric component; ( c) antisymmetric component.\nq′\nA(t) =1\n2\nu′(x,y,z,t)−u′(x,−y,z,t)\nv′(x,y,z,t)+v′(x,−y,z,t)\nw′(x,y,z,t)−w′(x,−y,z,t)\np′(x,y,z,t)−p′(x,−y,z,t)\n(3.3)\nA visualization of one snapshot of the fluctuating streamwise velocit y field, together\nwith the contributions from the symmetric and antisymmetric compo nents, is shown in\nfigure3.\nThe spanwise symmetrical and anti-symmetrical contributions of a ll collected samples\nare arranged into two separate data matrices as\nQS=/bracketleftbigq′\nS,1q′\nS,2...q′\nS,nt/bracketrightbig\n(3.4a)\nQA=/bracketleftbigq′\nA,1q′\nA,2...q′\nA,nt/bracketrightbig\n(3.4b)\nThe two independent data matrices then serve as the basis for ext racting and analyzing,\nrespectively, the symmetric and antisymmetric empirical modes. No te that, due to the\nzero-integral property of an even and an odd function in the samp ling domain, the\nsymmetric andantisymmetricmodesyield mutualorthogonality( Hack & Schmidt 2021 ).\n3.2.Spectral proper orthogonal decomposition\nThe SPOD approach seeks modes that are optimal in terms of the sp ace-time inner\nproduct defined as/integraldisplay∞\n−∞/integraldisplay\nΩqH(x,t)q(x,t)dxdt (3.5)Linear stability and SPOD of train wake flow 9\nwhere the superscript ( .)Hdenotes the Hermitian transpose. This problem is solved with\nSPOD by extending the database to the frequency domain, and sea rching for spatially\northogonal basis at each discrete frequency, following the stand ard POD routine. The\nderived modes can then optimally represent space-time flow statist ics (Schmidt et al.\n2018;Towneet al.2018).\nTo do this, the snapshot database is first segmented into nblkoverlapping blocks,\nwithnDFTsnapshots in each individual block. Note that to avoid zero padding in the\ndiscrete Fourier transform, nDFT= 256 is used in our study, and the resulting angular\nfrequency resolution is 0.245. An overlap of 50% is used, as a larger o verlap number does\nnot improve the spectral estimation results ( Schmidt & Colonius 2020 ). With the above\nparameters, nblkcan be determined with the value of 61, which is sufficient for well-\nconverged SPOD energy spectrum and modes ( Schmidt & Colonius 2020 ). Along with\nan appropriate window function, a discrete Fourier transform is ap plied to each block.\nˆQ(l)=F/parenleftBig/bracketleftBig\nq′(l)\n1q′(l)\n2...q′(l)\nnDFT/bracketrightBig/parenrightBig\n=/bracketleftBig\nˆ q(l)\n1ˆ q(l)\n2...ˆ q(l)\nnfreq/bracketrightBig (3.6)\nNote that the procedures in this section apply to both symmetric an d antisymmetric\ncomponents, so the subscripts ( .)Sand (.)Aare not shown in the equations for simplicity.\nThen all Fourier realizations at the k-th frequency are collected into a new data matrix.\nˆQk=/bracketleftBig\nˆ q(1)\nkˆ q(2)\nk...ˆ q(nblk)\nk/bracketrightBig\n(3.7)\nAt this point, the workflow would be to find a set of orthogonal basis that gives the\napproximation of\nˆ q(i)\nk=m/summationdisplay\nj=1a(i)\nkjφkj (3.8)\nThis can be easily done by computing the eigenvectors of the cross s pectral density\n(CSD) matrix as shown in Schmidt & Towne (2019). However, this would lead to a\nhigh computational cost, since the number of spatial points is much larger than nblk. In\npractice, a more economical and efficient method consists in calculat ing is\nSk=/parenleftBig\nˆQH\nkWˆQk/parenrightBig\n/nblk (3.9)\nand solving the eigenvalue problem defined by\nSk=UkΛkUH\nk (3.10)\nwhereWis the diagonal matrix containing weight information of each flow quan tity\nat each sampling node, and the resulting eigenvalue matrix Ukserves as coefficients\nthat expand the SPOD modes in Fourier realizations of each block ( Nekkanti & Schmidt\n2021). The SPOD modes can be recovered by\nΦk=1√nblkˆQkUkΛ−1/2\nk(3.11)\nThe matrices Λk= diag/parenleftBig\nλk1,λk2,...,λknblk,/parenrightBig\ncontains the SPOD energies.\nThe weight matrix in 3.9defines the energy norm used in SPOD, thus determining\nphysical process to be highlighted ( Colonius et al.2002). Here, the weight matrix is given10 X.-B. Li and others\n(a)\n10-110010110-710-610-510-4\nj(b)\n50%90%\n0.5 1 2 5 10 201204060\n00.10.20.30.4\n(c)\n10-110010110-710-610-510-4\nj(d)\n50%90%\n0.5 1 2 5 10 201204060\n00.10.20.30.4\nFigure 4: SPOD mode energy spectrum. The left column shows the sp ectral curves with\nthe shading of the line color representing the increase in the mode nu mber. The right\ncolumn shows the percentage of energy that each mode accounts for as a function of\nfrequency, with solid lines indicating that the cumulative energy repr esents 50% and 90%\nof the total energy at each frequency. ( a,b) Symmetric component; ( c,d) antisymmetric\ncomponent.\nas\nW=/integraldisplay\nΩ\n1\n1\n1\n0\ndV (3.12)\nsothat theturbulent kineticenergy(TKE)normcanbe defined.Th e pressurecomponent\nin the data matrix does not contribute to the matrix Sk, therefore, the energy spectrum\nis based only on the turbulent kinetic energy. By applying the snapsh ot method, the\npressure modes associated with coherent structures based on T KE can be recovered.\n3.3.SPOD energy spectra & modes\nSPOD solves the eigenvalue problems at each discrete frequency ind ependently, and\nproduces energy-ranked eigenvalues. Therefore, the energy d istributions of different\nmodes at each frequency can be best visualized in the form of a spec trum (Schmidt et al.\n2018). The left column in figure 4shows SPOD spectra for both symmetric and\nantisymmetric components. Meanwhile, the cumulative energy cont ent and the\npercentage of energy accounted for by each mode as a function o f frequency are\nshown in the right column. The symmetric component displays a higher overall energy\nlevel compared to the antisymmetric component, indicating its domin ant role in the\ndynamics of the turbulent wake. The low-rank behavior, characte rized by a large\nseparation between the first and second modes, also appears to b e more pronounced inLinear stability and SPOD of train wake flow 11\nFigure 5: Spatial distribution of the most dominated SPOD mode ( ω= 3.437) visualized\nbased on iso-surface of the streamwise velocity component.\nthe symmetric component. In particular, in the angular frequency ranges of 2 < ω <4 of\nthe symmetric component, the first modes contribute more than 5 0% of the fluctuating\nenergy according to figure 4(b). The frequency of the dominant symmetric coherent\nstructure is ω= 3.437 as shown in figure 4(a). Additionally, the spectral energy\nconcentration at ω≈6.9, and a less pronounced peak around ω≈1.7 can be identified\nrespectively. These two angular frequencies could be associated w ith the first harmonic\nand subharmonic of the fundamental instability. This will be further investigated by\nchecking the mode bi-spectrum, as the SPOD spectral curves can not give direct insight\ninto the frequency triads.\nIn figure 5, the spatial distribution of the leading SPOD mode is visualized based\non the iso-surface of the streamwise velocity component. Overall, the three-dimensional\nmode is characterized by structures of different scales and shape s at different locations,\nwhich should be attributed to the complexity of the mean flow. In the near wake region,\nspanwise coherent vortex shedding can be observed from the rec irculation region just\nbehind the train tail. The coherent structures generated in span m ove downstream and\ngradually approach the ground due to effect of the moving ground ( Wanget al.2023).\nMeanwhile, we can also observe that the upper spanwise coherent s tructure rolls up at\nboth lateral ends, and gradually stretches to be inclined in downstr eam direction due\nto its interaction with the longitudinal vortex pair. Once fully attach ed to the ground,\nthe spanwise structures vanish and separate at the symmetry pla ne to evolve into far\nwake coherent structures at x/W > 1.5. The far wake coherent structure displays a\nnearly constant streamwise wavelength, indicating that the far wa ke can be considered\nas nearly parallel in the streamwise direction.\nTo better quantify the frequency-wavenumber characteristics , a streamwise Fourier\ntransformisappliedtothestreamwisevelocitycomponentoftheSP ODmodes,toconvert\nthe signal into the domain of streamwise wavenumber α. Due to the dominant role of\nthe symmetric perturbation in the wake, only symmetric modes are c onsidered. First,\nthe power spectral density (PSD) of the leading mode is averaged a long the vertical\ndirection, and plotted as a function of spanwise location and stream wise wavenumber in\nfigure6(a,b). Meanwhile, we isolate the mechanism in near wake and far wake regio ns\nby using two different window functions that constrain the signal to 0< x/W < 1 and\n1< x/W < 6.67. The results are shown in figure 6(a) and figure 6(b), respectively.\nIn the near-wake region, the maximum PSD is located in the symmetric plane, with\nα≈8. We can also observe that the high PSD value occurs over a wide ran ge ofα,\nwhich can be attributed to the growth of the wavelength of the spa nwise vortex shedding12 X.-B. Li and others\n(a)\n-20 -10 0 10 2000.40.81.2y/W(b)\n-20 -10 0 10 2000.40.81.2y/W(c)\nFigure6:StreamwisewavenumberbasedonSPODmodes.( a) 0< x/H < 1atω= 3.437;\n(b) 1< x/H < 6.67 atω= 3.437; (c) Angular frequency - streamwise wavenumber\ndiagram.\nmode. In the far wake region, the coherent structure has a near ly constant streamwise\nwavenumber, with the maximum PSD located away from the symmetry plane. This is\nconsistent with what is shown in figure 5. Then for the first SPOD mode at all discrete\nfrequency points, a window function including both near-wake and f ar-wake is used to\ncompute the PSD distribution, followed by the averaging over all cro ss-flow locations, to\nconstruct the frequency-wavenumberdiagramshown in figure 6(c). In general,a constant\nphase velocity of 0.83 can be observed across the angular frequen cy range. This value is\ntypical of the Kelvin-Helmholtz instability wave and is related to the sh ear layer of the\nfar wake counter-rotating vortex pair. Since the frequency-wa venumber diagram includes\nboth near-wake and far-wake instability waves, the phase velocity tends to decrease at\nfrequencies with a pronounced low rank due to the near-wake mode .\n3.4.Triadic interactions\nAfrequencytriadisdescribedbytheinteractionoftwostructure satfrequencies ωkand\nωl,resultinginathird structureatfrequency ωk+l,obeyingthe condition ωk±ωl±ωk+l=\n0. As introduced previously, the SPOD energy spectrum shows at w hich frequency the\nflow is dominated by coherent structures, but does not contain inf ormation on frequency\ntriads. In our case, the harmonic triads and the broadband low-ra nkbehaviorare of great\nimportance for a more detailed interpretation of the wake dynamics . Here, bispectral\nmode decomposition ( Schmidt 2020 ) is used to estimate the mode bispectrum\nFor a given one-dimensional signal q(t), the bispectrum can be expressed as\nSqqq(k,l) =/integraldisplay∞\n−∞/integraldisplay∞\n−∞E[q(t)q(t−τk)q(t−τl)]exp(−i(ωkτk+ωlτl))dτkdτl\n= lim\nT→∞1\nTE[ˆq(ωk)∗ˆq(ωl)∗ˆq(ωk+ωl)](3.13)\nwhere the superscript ( .)∗is the scalar complex conjugate, E[.] is the expectation value,\nτis the time lag, and ˆ q(ωk) and ˆq(ωl) are the k-th andl-th components of ˆ q.\nThe BMD constitutes an analogy to the classical bispectrum for mult idimensional\nsignals (Schmidt & Oberleithner 2023 ). The principle of BMD is to construct\nb(ωk,ωl)≡E/bracketleftbigg/integraldisplay\nΩˆq∗\nk◦ˆq∗\nl◦ˆqk+ldx/bracketrightbigg\n=E/bracketleftbigˆqH\nk◦lWˆqk+l/bracketrightbig\n(3.14)Linear stability and SPOD of train wake flow 13\nwhere◦denotes the spatial entry-wise product, and ˆqk◦lthe shorthand for ˆqk◦ˆql. To\ngive the approximation of the expectation value, again the DFT realiz ations of ˆqk◦land\nˆqk+lare rearranged into\nˆQk◦l≡/bracketleftBig\nˆ q(1)\nk◦lˆ q(2)\nk◦l...ˆ q(nblk)\nk◦l/bracketrightBig\n(3.15a)\nˆQk+l≡/bracketleftBig\nˆ q(1)\nk+lˆ q(2)\nk+l...ˆ q(nblk)\nk+l/bracketrightBig\n(3.15b)\nThen the bispectral modes φk+l, which are linear combinations of Fourier modes, and\ncross-frequency fields φk◦l, which are maps of phase alignment between two frequency\ncomponents, are introduced. These two fields share a common set of expansion co-\nefficients, ai, which expand the vectors into the spaces spanned by the ensemb les of\nrealizations of ˆ qk+landˆ qk◦l. A compact form can be given as\nφ(i)\nk◦l=ˆQk◦lai (3.16a)\nφ(i)\nk+l=ˆQk+lai (3.16b)\nBy now, the goal of BMD is to seek the set of expansion coefficients a1that maximizes\nthe absolute value of b(ωk,ωl). To find the optimal a1, the weighted bispectral density\nmatrix is introduced.\nB=1\nnblkˆQH\nk◦lWˆQk+l (3.17)\nIt can be the found that the maximize problem is equivalent to finding t he vector a1\nassociated with the numerical radius of B\nr(B) = max/vextendsingle/vextendsingle/vextendsingle/vextendsingleaH\n1Ba1\naH\n1a1/vextendsingle/vextendsingle/vextendsingle/vextendsingle(3.18)\nThen this value is referred as the complex mode bispectrum. For a de tailed mathematical\nderivation and algorithmic implementation, the reader is referred to Schmidt (2020).\nHere, the mode bispectrum is computed using the same spectral es timation parameters\nasinthe SPODof §3.2.Due tothe high-energycontributionofthe symmetriccomponent ,\nboth its self-interaction and the interaction with the antisymmetric component are of\ninterest to study. Note that the third frequency component res ulting from the mutual\ninteraction of symmetric and antisymmetric components is always an tisymmetric.\nIn figure 7, the mode bispectrum are shown. Here, the self-interaction of th e sym-\nmetric component shown in figure 7(a) is identified with several distinct peaks, while\nthe interaction between symmetric and antisymmetric components presented in figure\n7(b,c) contains less information and therefore is not interpreted. We re fer toωfas the\nfundamental frequency detected in SPOD. Conceptually, the trip let (ωf,0,ωf) on the\nmode bispectrum can be regarded as the coherent perturbation d riven by the mean flow.\nHowever, in BMD, only quadratic terms are considered, so the value on theω1-axis\nshould not be interpreted. Then a local maximum of the distribution c an be found that\ncorresponds to the sum interaction of the fundamental instability with itself, generating\nthe first harmonic at twice the fundamental frequency via the tria d (ωf,ωf,2ωf). At the\nsame time, the doublets (2 ωf,−ωf) and (1\n2ωf,1\n2ωf), which include the first harmonic\nand subharmonic, respectively, feed back into ωf. Then the triad ( ωf,−1\n2ωf,1\n2ωf) can\nalso be found that contributes to the subharmonic frequency. Th e mean field distortion,\nindicated by the peak values along line ω1=−ω2can be identified over a wide frequency\nrange.\nBased on the presented results, the important dynamics that cha racterize the\nfrequency-domain energy distribution of the flow can be explained a s follows: As the flow14 X.-B. Li and others\n(a)\n0 5 10-10-50510\n(b)\n0 5 10-10-50510\n(c)\n0 5 10-10-50510\n-20-18-16\nFigure 7: Mode bispectrum in both the sum and difference regions. ( a) Self interaction\nof symmetric component; ( b) Interaction of symmetric and antisymmetric components\nwithω1the symmetric component and ω2the antisymmetric component; ( c) Interaction\nof symmetric and antisymmetric components with ω1the antisymmetric component and\nω2the symmetric component.\noscillates at ωf, the dominant coherent structure meanwhile drives its first harmo nic\nand subharmonic through the energy cascade process. These co herent structures interact\nwith each other through multiple triads and feed back into the relate d frequencies. At\nthe same time, they continuously modify the mean field and lead to dist ortion of the\nbase flow. As the dominant coherent structure is driven by the mea n flow, its oscillating\nfrequency constantly deviates from ωf. With the continuous evolution of the described\nprocess, the flow finally appears to be low-rank in the wide frequenc y range as shown in\nfigure4.\n4. Physics-based coherent structure modelling\nLinear global stability analysis is conducted to identify the mechanism s that drive the\ndominant coherent structure identified in the SPOD. In this work, in stead of directly\nsolving the three-dimensional stability equation, we conduct a two- dimensional local\nanalysis to obtain more local information while reducing computationa l resource. In this\nmanner, the flow is assumed to be weakly nonparallel in the streamwis e direction. Then\nthe full three-dimensionalmatrix eigenvalue problem is replaced by s everallocal indepen-\ndent matrix problems, each based on the two-dimensional cross-fl ow planes at different\nstreamwise locations. To determine the global mode, the concept o f local convective /\nabsolute instability is applied within the WKBJ approximation ( Huerre & Monkewitz\n1990). To be clear, the approach used in the research does not concep tually corre-\nspond to the bi-global analysis ( Theofilis 2011 ), but is similar to the approach used\ninHuerre & Monkewitz (1990);Monkewitz et al.(1993);Juniperet al.(2014), in which\nthe bi-global stability is approximated using one-dimensional local an alysis.\n4.1.Linearized operator and treatment of the nonparallel flow\nCoherentstructurescanbedescribedbythetripledecomposition (Reynolds & Hussain\n1972), which leads to a further decomposition of the fluctuating compon ent into coherentLinear stability and SPOD of train wake flow 15\nand stochastic parts as\nq′(x,t) =/tildewideq(x,t)+q′′(x,t) (4.1)\nwhere/tildewideq(x,t) is the coherent fluctuation part and q′′(x,t) is the stochastic fluctua-\ntion part. This decomposition is substituted into both the momentum and continuity\nequations, and both are time-averaged and phase-averaged. Th en by subtracting the\ntime-averaged set of equations from the phase-averaged set of equations, the equations\ngoverningthe evolution ofcoherent structures can be written ( Reynolds & Hussain 1972 )\n∂/tildewideu\n∂t+(/tildewideu·∇)¯ u+(¯ u·∇)/tildewideu=−∇/tildewidep+∇·/parenleftbigg1\nRe(∇+∇T)/tildewideu/parenrightbigg\n−∇·(τR+τN) (4.2)\n∇·/tildewideu= 0 (4.3)\nHereτNdescribes the quadratic interactions of the coherent perturbat ion. Considering\nthat the energy contribution from this process is higher-order, t his term is neglected in\nthe following. The term τRis the fluctuation of the stochastic Reynolds stresses related\nto the stochastic-coherent interaction, which contributes at lea ding order, according to\nthe energy considerations of Reynolds & Hussain (1972), and is therefore retained in the\nequation. However, this term is not known a priori and needs to be p roperly modeled to\nclose the governing equation. In this paper, we use Boussinesq’s ed dy viscosity model as\nthe closure. The Reynolds stresses are then expressed as\nτR=−νt/parenleftbig\n∇+∇T/parenrightbig/tildewideu. (4.4)\nHereνtis the normalized eddy viscosity, which can be calculated using quantit ies of\nthe LES mean flow. Since this approach yields an eddy viscosity for ea ch independent\nReynolds Stress component, a reasonable compromise would be to t ake the least mean\nsquare of the six values.\nνt=/angbracketleftbig\n−u′u′+2/3kI,¯S/angbracketrightbig\nF\n2/angbracketleftbig¯S,¯S/angbracketrightbig\nF(4.5)\nwith/angb∇acketleft·,·/angb∇acket∇ightFdenoting the Frobenius inner product, kthe kinetic energy, Ithe identity\nmatrix and ¯Sthe mean shear strain rate tensor. This approach has been widely u sed\nin the linear stability analysis of turbulent flows, as presented in Tammisola & Juniper\n(2016);Rukeset al.(2016);Kaiseret al.(2018);M¨ ulleret al.(2020);Kuhnet al.(2021).\nThe linearized momentum and continuity equations for the coherent perturbation can\nbe obtained as\n∂/tildewideu\n∂t+(/tildewideu·∇)¯ u+(¯ u·∇)/tildewideu=−∇/tildewidep+∇·/parenleftbigg/parenleftbigg1\nRe+νt/parenrightbigg\n(∇+∇T)/tildewideu/parenrightbigg\n(4.6)\n∇·/tildewideu= 0. (4.7)\nThese equations can be then rewritten as\nL/tildewideq= 0 (4.8)\nwhereLis the operator of the linearized equations superimposing the spatia l discretiza-\ntion and base state ( Paredeset al.2013).\nBy assuming that the mean field has much smaller derivatives in the str eamwise\ndirection than in the transverse and vertical directions, the syst em can be Fourier-\ntransformed in the streamwise direction, following the streamwise w eakly nonparallel\nflow assumption. The coherent perturbation can be then written a s\n/tildewideq(x,y,z,t) = ˆq(y,z)exp[i(αx−ωt)]+c.c. (4.9)16 X.-B. Li and others\nwhereˆ qisthecomplexeigenfunction, α=αr+iαiisthecomplexstreamwisewavenumber,\nω=ωr+ iωiis the complex angular frequency, and c.c. is the complex conjugate .\nSubstituting equation 4.9into equation 4.8enables stability analysis of the mean field in\nthe cross-flow plane at different streamwise locations. The global s tability characteristics\ncan then be recovered based on the concept of absolute/convec tive instability and the\nglobal mode wavemaker ( Huerre & Monkewitz 1990 ).\nHowever, using the local approach to predict the global mode has b een reported\nto be less accurate than the direct global approach, when the bas e flow is strongly\nnonparallel ( Juniperet al.2011;Juniper & Pier 2015 ). In the current case, where a fully\ndeveloped three-dimensional base flow is considered, the parallel fl ow assumption is likely\nto introduce uncertainty into the results. Therefore, in this work , we also make further\ntreatment to approximate the non-parallelism of the three-dimens ional base flow.\nThe parallel flow assumption is checked by visualizing the streamwise c omponent of\nthe leading SPOD mode on a horizontal plane as shown in figure 8. Here the coherent\nperturbations can be observed with clear wavepacket structure s; however, they do not\ntravel strictly in the streamwise direction but follow oblique paths in b oth the near\nand far wake. This is caused by the downwash flow from the slanted t ail surface which\ngradually separates the wake structures as it propagates downs tream, as can be seen in\nthe mean flow structures shown in figure 2.\nWe intend to account for the obliqueness of the coherent structu re in the linear\nmodeling. To do this, an expression related to the state vector can be written\nˆq(x0,y0,z0) = ˆq(x0+∆x,y0+tanθ∆x,z0+tanγ∆x) (4.10)\nwheretan θandtanγrespectivelyrepresenttheconvectiondirectionrelativetosymme tric\nplane and horizontal plane. By applying a first-order Taylor expans ion to the right-hand\nside of equation 4.10, the left-hand side can be canceled and the expression can be\narranged into\n∂ˆ q\n∂x+tanθ∂ˆ q\n∂y+tanγ∂ˆ q\n∂z= 0 (4.11)\nThis, an artificial x-derivative applied to the state vector ˆ qis used to account for the\noblique traveling direction.\nTo determine tan θand tanγ, we replace ˆ qwith mean field quantities ¯ qin equation\n4.11, by assuming that the perturbation waves follow the mean flow conv ection. Note\nthat, for each node in the computational domain, the four flow var iables are used to\nconstruct the linear equation system for tan θand tanγand the final results are obtained\nusing the least-squares solution of the overdetermined linear equa tion system.\nFinally, the convection direction calculated from the mean field is valida ted by com-\nparison with the oblique path of the SPOD mode. In figure 8, streamlines based on the\ncalculated local angle θare visualized, by decomposing tan θat each computational node\ninto streamwise (cos θ) and transverse (sin θ) components using trigonometric functions.\nIt can be observed that the vector field agrees well with the trave ling direction of the\nSPOD wavepackets. Therefore, the artificial x-derivative is assumed to be reasonable to\naccount for the obliqueness of the coherent structure in the linea r modelling.\n4.2.Spatio-temporal stability approach\nFor the linear stability analysis, the linear operator Lis rearranged to construct the\ngeneralized eigenvalue problem. In this work, both the temporal an d spatial stability\nformulation is needed. Therefore, either the temporal stability fo rm\nA(x,α)ˆ q=ωB(x)ˆ q (4.12)Linear stability and SPOD of train wake flow 17\nFigure8: Two-dimensionalslice ofthe streamwisecomponent ofthr ee-dimensionalSPOD\nmode showing the oblique downstream traveling wavepackets.The s treamlines are drawn\nbased on the vector field (cos θ, sinθ) at each computational node.\nor the spatial stability form\nA(x,ω)ˆ q=αB(x)ˆ q (4.13)\nare constructed.\nIn thetemporalstabilityform(equation 4.12),the streamwisewavenumber αisfixedto\na real value, and the eigenvalue problem is solved for a complex ω, the real part of which\nis the angular frequency and the imaginary part is the temporal gro wth/decay rate. On\nthe contrary, in the spatial stability form (equation 4.13), a real angular frequency ωis\nimposed to search for the complex α, where the real part corresponds to the streamwise\nwavenumber, while the imaginary part is the spatial amplification/dam ping rate. Note\nthat, in the spatial stability equation, quadratic terms appear with respect to α. This\nproblem is solved using the companion matrix method ( Bridges & Morris 1984 ), which\nincreases the size of the eigenvalue problem compared to the tempo ral analysis. The\ncomplete expression of the operators AandBfor both temporal and spatial analysis can\nbe found in Appendix A.\nTo determine the global stability of the mean flow from a local analysis , a so-called\nspatio-temporal analysis using equation 4.12must be performed to distinguish between\nconvective and absolute instability ( Huerre & Monkewitz 1990 ). In this case, both αand\nωarecomplex-valued.Conceptually,the flowis saidto bestableifallpe rturbationsdecay\nin time throughout the entire domain after a localized impulse. Convec tively unstable\nflow gives rise to perturbations that grow in time but convect away f rom the impulse\nlocation, so that the perturbations eventually decayto zeroat ea chspatial location in the\nlong time limit. For an absolutely unstable flow, the perturbations gro w both upstream\nand downstream of the impulse location, contaminating the whole spa tial domain in the\nlong time limit.\nBased on the previous definitions, the convective/absolute natur e of a local velocity\nprofile can be determined from the time-asymptotic behavior of the perturbations that\nremain at the impulse location, that is, for perturbations with zero g roup velocity:\n∂ω/∂α= 0, which is the definition of a saddle point in the complex α−plane. Therefore,\nvalid saddle points on the complex α-plane that satisfy the Briggs-Bers pinch-point\ncriterion ( Briggs 1964 ) must be identified. Therefore, the flow is locally absolutely\nunstable if the absolute growth rate, given by the imaginary part of ωat the most18 X.-B. Li and others\nunstable valid saddle point, is positive. If the absolute growth rate is negative, the\nflow is convectively unstable or stable ( Huerre & Monkewitz 1990 ;Rees & Juniper 2010 ;\nJuniperet al.2014;Rukeset al.2016;Kaiseret al.2018;Demange et al.2020).\nIn a spatially developing flow, a region of absolute instability is a necess ary (but\nnot sufficient) condition for global instability ( Monkewitz et al.1993;Chomaz 2005 ).\nTo further link the local absolute instability to global instability, the a bsolute growth\nrate needs to be tracked along the streamwise direction to determ ine the wavemaker,\nthe location where the global mode is selected. This method has been extensively used\nfor one-dimensional local stability analysis in comparisonwith bi-globa l stability analysis\n(Giannetti & Luchini 2007 ;Juniperet al.2011;Juniper & Pier 2015 ;Kaiseret al.2018).\n4.3.Solving the eigenvalue problem\nTo solve the eigenvalue problem numerically, cross-flow planes with th e dimensions\nof 0/lessorequalslanty/H/lessorequalslant1.3 and 0/lessorequalslantz/H/lessorequalslant1.3 are discretized using Chebyshev spectral collocation\nmethods. This approach has been successively applied to linear stab ility analysis\nbyKhorrami (1991);Parras & Fernandez-Feria (2007);Oberleithner et al.(2011);\nDemange et al.(2020). Detailed descriptions or practical guides to spectral collocation\nmethods can be found in Khorrami et al.(1989);Trefethen (2000).\nTo reduce the numerical cost, we further exploit the symmetry of the mean field with\nrespect to the vertical x-zplane. This allows us to use only half of the wake plane instead\nof the full plane, to compute only transversely symmetric or antisy mmetric eigenmodes\nwhen appropriate boundary conditions are applied ( Zampogna & Boujo 2023 ). As shown\nin§3, the symmetric perturbations are dominant and potentially related to a global\ninstability, so only symmetric eigenmodes are considered. The corre sponding boundary\nconditions are given as\n∂ˆ u\n∂y=∂ˆ w\n∂y=∂ˆ p\n∂y= 0,ˆ v= 0 ony/W= 0 (4.14 a)\n∂ˆ u\n∂y=∂ˆ v\n∂y=∂ˆ w\n∂y=∂ˆ p\n∂y= 0 ony/W= 1.3 (4.14 b)\nˆ u=ˆ v=ˆ w= 0,∂ˆ p\n∂z= 0 onz/W= 0 (4.14 c)\nˆ u=ˆ v=ˆ w=ˆ p= 0 onz/W= 1.3 (4.14 d)\nHere, equation 4.14adetermines eigenmodes to be transversely symmetric. At the wall,\nhomogeneousDirichletboundaryconditionsareimposedforthevelo citycomponents.For\npressure, by substituting the homogeneous Dirichlet conditions fo r velocity into equation\n4.6, a compatibility condition can be obtained ( Theofilis et al.2004). Here, assuming\n∂2ˆ w/∂z2= 0 gives a homogeneous Neumann condition for the pressure. For f ar-field\nboundaries, the upper boundary is set to a homogeneous Dirichlet c ondition, while the\nside boundary is set to a homogeneous Neumann condition which is nec essary to predict\nfar wake eigenmodes.\nThe Krylov-Schur algorithm ( Stewart 2002 ), which serves as an improvement on\ntraditional Krylov subspace methods such as the Arnoldi and Lanc zos algorithms, is\nused to obtain a subset of solutions to the eigenvalue problem. This r equires an initial\nguess of the physical eigenvalue, which can be derived from the disp ersion relation shown\nin§3.3. To discard spurious eigenmodes caused by the discretization, two criteria are\napplied:First,alleigenmodesthatdonotdiminishwhenapproachingt heupperboundary\nare discarded. Second, since spurious eigenvalues are very sensit ive to discretization, aLinear stability and SPOD of train wake flow 19\n(a)\n2 3 4-2-10(b)\n3.6 3.8 4 4.2-1-0.50\nFigure 9: The spectrum of eigenvalues obtained with discretization p oints of 8,100 ( /square)\nand 10,000 ( ◦), the physical eigenvalues are marked with ∗. (a) results based on cross-\nflow plane at X/W= 0.1 and imposing α= 10. (a) results based on cross-flow plane at\nX/W= 3.5 and imposing α= 5.\nconvergence study of eigenvalues computed using different grid re solutions is used as a\ncriterion for filtering spurious eigenvalues.\nThe results of the convergence study based on temporal stability analysis are shown\nin figure 9. Two representative cross-flow planes, located in the near-wake and far-wake,\nrespectively, are considered. Real streamwise wavenumbers of 1 0 and 5 are, respectively,\nimposed in the two eigenvalue problems, which are then discretized us ing two different\ngrid resolutions and solved for the subset of the eigenvalue spectr um. Note that the\nreal streamwise wavenumbers are chosen according to the wavele ngth distribution of the\ndominant SPOD mode shown in figure 6, so that the SPOD peak frequency can be set\nas the initial value of the Krylov-Schur iteration procedures. As sh own in figure 9, both\nspectra feature continuous branches and a set of discrete mode s. In general, continuous\nbranches are made up of spurious eigenmodes caused by discretiza tion and physical\neigenmodes that are highly stable.\nIt can be observed that the locations of these eigenmodes in contin uous branches vary\nsignificantly when the resolution of the grid is changed. On the contr ary, the locations of\nthe discrete modes remain almost stationary with different discretiz ation settings; hence,\nthe discrete modes are considered as physical eigenmodes that ca n potentially contribute\nto global instability. In particular, the near-wake cross-flow plane features one unstable\nmode and two stable modes, while the far-wake plane features two s table modes. Note\nthat these physical eigenvalues do not necessarily correspond be tween the near-wake and\nfar-wake cross-flow planes. When tracking along the streamwise d irection, the physical\neigenvalues may become highly stable and fall into the spurious region , and new physical\neigenvalues may emerge due to the complexity of the base flow.\n4.4.Absolute/convective stability analysis\n4.4.1.Streamwise evolution of temporal growth rate\nBefore considering the absolute / convective nature of the instab ility, it is necessary\nto know which part of the flow is temporally unstable. To this end, a Ne wton-Raphson\niterative method Ypma(1995) is used to vary the real wavenumber αand to determine\nthe maximum temporal growth rate of each previously identified phy sical mode. This\nprocedureisrepeatedateachstreamwisepositiontotrackthema ximumtemporalgrowth20 X.-B. Li and others\n(a)\n(b)\n-1 0 1 2 3\nx/W00.30.6\nFigure 10: Maximum temporal growth rate as functions of streamw ise location. ( a) Mean\nflow distribution in the central plane. ( b) maximum temporal growth rate of near wake\n(), middle wake ( ) and far wake ( ) eigenmodes with real αimposed.\nrates of all physical eigenmode. As mentioned above, due to the co mplexity of the base\nflow, one physical eigenmode may occur only in a certain range of str eamwise location.\nTherefore, several cross-flow planes are used to compute the p hysical eigenvalues, and\nthen the tracking process is repeated to account for the maximum temporal growth rates\nof all physical eigenmodes in the entire wake.\nIn figure 10, the maximum temporal growth rates of these physical eigenmode s are\ndisplayedasafunction ofstreamwiselocation.Onlytheunstablereg ime(ωi>0)isshown\nhere. Three different unstable eigenvalue branches are identified in the entire wake, each\ndominating in different streamwise regions. Accordingly, the flow bec omes temporally\nunstable very close to the tail of the train and remains unstable in th e wake. Based on\nthe spatial locations where these branches become unstable, we c onceptually divide the\nwake into the near, middle, and far wakeregimes and name these bra nchesaccordingly.It\ncan be found in figure 10(b) that the near wake eigenmode is temporally more unstable\nthan all the downstream eigenmodes. According to the mean flow dis tribution shown\nin figure 10(a), this eigenmode branch is attributed to the transverse recircula tion zone\nlocatedrightbehindthetailofthetrain.Thefarwakeeigenmodeca nbeobservedacrossa\nwide streamwise distance and is located closely to the extension of th e streamwise vortex\npair. This branch has the largest temporal growth rate at x/W∼0.4, and gradually\nstabilizes as it extends into the far wake; however, it remains still un stable. The middle\nwake mode becomes unstable at 0 .5< x/W < 1.4. At this location, with respect to the\nmean field, the flow from the slanted tail surface can be observed a ttached to the ground\naccording to the figure 10(a).Linear stability and SPOD of train wake flow 21\n(a)\n (b)\nFigure 11: Contour of the complex angular frequency in the complex α-plane at x/W=\n0.08. Saddle points on the complex α-plane are marked by a red circle. ( a) The real\ncomponent; ( b) The imaginary component, and the ωi= 0 isocontour is highlighted in\nred.\n4.4.2.Spatio-temporal stability analysis in the near wake\nSpatio-temporal analysis is performed to compute the contour of ωin the complex\nα-plane, so as to find valid saddle points following the Briggs-Bers crite rion. Since the\nbranch in the near-wake region is temporally more unstable than the other branches\n(figure10), we start with the near-wake branch.\nFigure11shows the contour of the complex frequency in the complex α-plane, at\nthe streamwise location of x/W= 0.08. At this location, the near-wake eigenmode\nreaches its maximum temporal growth rate. In this figure two sadd le points are found,\nS0andS1. However, valid saddles must be pinched between an α+and anα−branch\n(Huerre & Monkewitz1990 ). Asimple rulebasedonthe Briggs-Berspinch point criterion\n(Briggs 1964 ) can be applied by checking the isocontours of the spatio-tempora l growth\nrate (figure 11(b)): Starting from the saddle point, the contours of the growing ωimust\nreach, respectively, the αi>0 andαi<0 half-planes. Here, the validity of both S0and\nS1can be confirmed, with S0representing the shorter traveling waveat higher frequency,\nwhileS1represents the longer traveling wave at lower frequency. In the lo ng time limit,\nthe nature ofthe instability is determined by S0, which has asignificantly higherabsolute\ngrowth rate than S1. Then the absolute frequency at this location can be determined as\nω0= 3.2904+0.0685i, which shows that the flow is absolutely unstable at this position .\nFurthermore, to ensure that all regions with absolute instability ha ve been taken into\naccount, the absolute growth rate has been determined for all un stable branches at\nseveral streamwise locations. The near wake branch was the only o ne to reveal absolute\ninstability.\n4.5.Determining the global mode wavemaker in the near wake\nTo determine the global mode, the saddle points are tracked in the s tream direction\nto find the global wavemaker. This is done in the local analysis framew ork based on\nthe frequency selection criterion. Several criteria have been eva luated in Pier(2002),\nshowing that the criterion for a linear global mode introduced by Chomaz et al.(1991)\nagrees best with the nonlinear direct numerical simulations. This crit erion is obtained\nfrom an analytical continuationofthe absolute frequency curvein to the complex x-plane.22 X.-B. Li and others\nThe wavemaker region is then represented by the saddle point on th e complex x-plane\n(Huerre & Monkewitz 1990 ) defined as\n∂ω0\n∂x(xs) = 0 (4.15)\nThe global complex angular angular frequency is then given by the va lue of the absolute\nfrequency at this saddle point\nωg=ω0(xs). (4.16)\nFor this purpose, the saddle point S0found in figure 10is tracked along the streamwise\ndirection. The three-point Taylor series expansion algorithm ( Rees 2010 ) is used to\napproach ∂ω/∂α= 0 during the tracking process. Then the absolute frequency as a\nfunction of streamwise location is analytically continued in the complex x-plane using\nthe Pad´ e polynomial, which has been shown to be well behaved in the c omplex plane\n(Cooper & Crighton 2000 ). The Pad´ e polynomial takes the form of\nf(x) =P(x)\nQ(x)=a0+a1x+...+anxn\n1+b1x+...+bmxm(4.17)\nTo determine the order of the polynomials used in the current work, the procedure\ndescribed in Juniperet al.(2011) is followed. Polynomials of order 10 have been proven\nto be sufficient to give a converged approximation of the saddle point .\nFigure12showsthemainresultsofthespatio-temporalstabilityanalysis.Fo rreference,\nthe mean field in the central plane is shown in figure 12(a). The following rows include\nthe results of the procedures described above to determine the g lobal instability using\nlocal analysis. In figure 12(b,c), the imaginary and real parts of the absolute complex\nangular frequency are shown as a function of streamwise location, respectively. A small\nregion of absolute instability is detected, which is located in the recirc ulation region. The\n10th order Pad´ e polynomials show acceptable agreement with the a bsolute frequency\ncurve of the linear stability analysis. The extrapolated complex x-plane is then visualized\nin figure 12(d). The saddle point appears to be very close to the real axis, with xs=\n0.0733+0.0010i,indicating a wavemakerlocated within the region of absolute ins tability.\nThe complex global angular frequency associated with this saddle po int isωg= 3.2702+\n0.0797i.\nThe theoretically predicted global mode frequency can be compare d to the frequency\nof the most energetic SPOD mode. With an empirical value of ωSPOD= 3.437, as shown\nin figure 4, we find that the theoretical prediction deviates by an error of les s than 5%.\nThis is in very good agreement, given the uncertainties introduced b y the nonparallelism\nand eddy viscosity modeling. Moreover, we expect the global mode t o be marginally\nunstable, representing a limit-cycle oscillation when stability analysis is performed on\na temporally averaged mean flow ( Noacket al.2003;Barkley 2006 ;Oberleithner et al.\n2011;Rukeset al.2016;Tammisola & Juniper 2016 ;Kaiseret al.2018). However, as\nreported in Giannetti & Luchini (2007);Juniperet al.(2011);Juniper & Pier (2015), it\nis a common feature of local analysis to overpredict the growth rat e of the linear global\nmode. In our case, with a growth rate value of 0.0797, the relative e rror with respect to\nthe realfrequency(3.2702)islessthan 3%,therefore,it isreaso nabletosaythat themode\nis marginally unstable. Overall, considering the fact that the flow is no t truly parallel\nand the intrinsic assumption of linearized mean field analysis, it can be c oncluded that\nthe theoretical mode identifies the dominant SPOD mode as a global m ode at limit cycle\nwith remarkable accuracy.\nThe global mode is formed by the switching between the α+andα−branches at the\nglobalfrequency.Therefore,tofurthertrulylocalizetheglobal modewavemaker,aspatialLinear stability and SPOD of train wake flow 23\n(a)\n(b)\n   -1.0   -0.5   0.0\nPadé\npolynomials\n(c)\n    3.0    3.5    4.0\nPadé\npolynomials\n(d)\n-0.030.000.03\n-2-10\n(e)\n    -30    -15      0\n(f)\n0.050 0.075 0.1 0.125 0.150       5      10      15\nFigure 12:( a) Streamlines and through-planevorticityofthe mean fiwld in the nea rwake\nregion. (b) Streamwise evolution of the absolute growth rate. ( c) Streamwise evolution\nof the absolute angular frequency. ( d) Contours of ω0,iextruded in the complex x-plane,\nformedbythe fitted Pad´ epolynomial.Thesaddlepoint onthecomple xx-planeis marked\nby red circle. ( e) The imaginary component of local complex streamwise wavenumber of\nα+andα−branches calculated at the global frequency ωg. (f) The same as ( e) with\nthe real component shown. The switching locations between α+andα−branches are\nmarked by a red vertical line in each sub-figure.24 X.-B. Li and others\nstability analysis (equation 4.13) imposing ω=ωgis conducted to compute the α+and\nα−branches ( Juniper & Pier 2015 ). Since the saddle point xson the complex x-plane\nis very close to the real axis, the location of maximum structural se nsitivity should also\nbe close to xs,r. Therefore, in practice, it can be quite straightforward to find th e branch\npairs by computing the eigenvalue problem at x/W=xs,r, and following α+\ni−α−\ni≈0\nin the streamwise direction.\nIn figure 12(e,f), the imaginary and real components of the α+andα−branches are\npresented, respectively. The switch between the two branches c an be identified to take\nplace at x/W= 0.0731. The location of the global mode wavemaker is marked by a red\nvertical line in all sub-figures of figure 12. This location is also within the recirculation\nregionandisnearlyidenticaltothelocationofthesaddlepoint xs.Thedirectglobalmode\nthen follows the α−branch upstream of the wavemaker, and the α+branch downstream.\nIn contrary, the adjoint global mode follows the α+branch upstream of the wavemaker,\nandα−branch downstream ( Juniper & Pier 2015 ). This result also indicates a spatially\ngrowing global mode within the recirculation region, with the growth r ate gradually\ndecaying to zero as it approaches the downstream boundary of th is region.\nIn summary, we identify a global mode with a frequency very close to the dominant\nSPOD mode and a growth rate close to zero. This suggests that the dominant SPOD\nmode is a manifestation of a global mode at limit cycle. The wavemaker o f this mode\nis located in the recirculation zone very close to the tail tip. It acts a s the origin for\nthe entire coherent wavepacket that propagates far downstre am, in a region where the\nflow is convectively unstable. The spatial shape and mechanisms of t he global mode and\ncomparison with the SPOD modes will be shown and discussed in §5.\n4.6.Structural sensitivity based on adjoint method\nTo further analyze where and how intrinsic feedback causes global instability ( Chomaz\n2005;Giannetti & Luchini 2007 ), we calculate the structural sensitivity using adjoint\nlinear stability analysis. The structural sensitivity describes the se nsitivity of the direct\nglobal mode to changes in the linearized Navier-Stokes operator, e .g., base flow modifi-\ncation (Marquet et al.2009). Therefore, this concept is critical for the development of\nefficient flow control strategies ( Giannetti & Luchini 2007 ;M¨ ulleret al.2020).\nIn the global framework, structural sensitivity is proportional t o the overlap between\nthe direct and adjoint global modes ( Chomaz 2005 ). Following Giannetti & Luchini\n(2007), theL2norm of the sensitivity tensor is equivalent to the expression\nλ(x) =/ba∇dblˆ m(x)/ba∇dbl/ba∇dblˆ m†(x)/ba∇dbl, (4.18)\nwhereˆ m(x) andˆ m†(x) represent the direct and adjoint momentum vectors, respectiv ely.\nIn the previous section, the location of the wavemaker was identifie d by computing the\nintersection between the α+andα−branches (at x/W= 0.0731). However, the exact\nlocation of the maximum structural sensitivity at this cross-flow pla ne is still unknown.\nThis requires computation of the adjoint mode at this streamwise loc ation. Therefore,\nthe adjoint linear stability analysis is further pursued in this section.\nDue to the inhomogeneous directions and hence differential depend encies in the lin-\nearized operator matrix, taking the Hermitian transpose of the dir ect operator as the\nadjointoperator,ashasbeendonein Oberleithner et al.(2014);M¨ ulleret al.(2020),does\nnot necessarily apply here. To find the adjoint of the system, the c ontinuous approach is\nused. First, we define the inner product as\n/angb∇acketleftˆ q1,ˆ q2/angb∇acket∇ight ≡/integraldisplay\nSˆ qH\n1ˆ q2dS≡ˆ qH\n1Wˆ q2 (4.19)Linear stability and SPOD of train wake flow 25\nThe adjoint modes depend on this choice of norm, but when recombin ed with the direct\nmodes to give the sensitivity measurement of the eigenvalue to chan ges inL, the effect\nof the norm cancels out ( Chandler et al.2012).\nBy definition, the adjoint operator L†should fulfill\n/angbracketleftbig\nˆ q†,Lˆ q/angbracketrightbig\n≡/angbracketleftbig\nL†ˆ q†,ˆ q/angbracketrightbig\n(4.20)\nThis definition is valid for any pair of vectors, but for convenience, t hey are expressed in\ntermsofthedirectstatevector ˆ qandtheadjointstatevector ˆ q†,asdonein Marquet et al.\n(2009);Qadriet al.(2013). More specifically, we consider the spatial stability form, then\nthe generalized eigenvalue problem is pre-multiplied by the adjoint eige nvector\n/angbracketleftbig\nˆ q†,Aˆ q/angbracketrightbig\n−/angbracketleftbig\nˆ q†,αBˆ q/angbracketrightbig\n= 0. (4.21)\nBy successively integrating the equation by parts using Green’s the orem, equation 4.21\nis rearranged to\n/angbracketleftbig\nA†ˆ q†,ˆ q/angbracketrightbig\n−/angbracketleftbig\nα†B†ˆ q†,ˆ q/angbracketrightbig\n= 0. (4.22)\nNote that the integration-by-partsprocess would leave boundar y terms in the expression.\nTherefore,appropriateadjointboundaryconditionsareappliedt ocanceloutallboundary\nterms. Then the adjoint of the direct eigenvalue problem is obtained from equation 4.22\nas\nA†ˆ q†=α†B†ˆ q†(4.23)\nThe detailed derivations as well as the validation of the adjoint opera tors and boundary\nconditions used in the current study is presented in appendix B.\nIt can be shown that, for well-converged adjoint solutions, the ad joint eigenvalues,\nordered by the same rule as the direct ones, are the complex conju gates of the direct\nones (Schmid & Henningson 2000 ), and the bi-orthogonality condition writes\n/parenleftBig/parenleftbig\nα†\nm/parenrightbig∗−αn/parenrightBig/bracketleftBig/parenleftbig\nˆ q†\nm/parenrightbigHWBˆ qn/bracketrightBig\n= 0. (4.24)\nThe bi-orthogonality enables the characterization of the receptiv ity of the open-loop\nforcing, and the sensitivity to modification of the mean flow ( Marquet et al.2009).\nIn figure 13, the distribution of the structural sensitivity at the streamwise lo cation\nwhereα+andα−branches intersect ( x/W= 0.0731) is shown. This field is obtained\nby first computing the adjoint mode on the considered cross-flow p lane based on the\nmethodology presented above and then calculating the L2norm of the sensitivity tensor\nfollowing equation 4.18. In addition, mean flow streamlines are also included so that the\nstructural sensitivity can be related to specific flow structures. Based on the results, both\nthe upper and lower recirculation zones are highlighted. In particula r, it can be observed\nthat the position of the highest structural sensitivity, enclosed b y the blue solid line in\nfigure13, is located slightly below the stagnation point of the recirculation reg ion. The\nstreamwise vortex pair generated from both sides of the train, on the other hand, seems\nto have no relevance to the sensitivity of the linear global mode. The highest structural\nsensitivity at this location, shown in figure 13, suggests that the interaction between the\nlower and upper shear layers, which drives the self-excitation of th e instability, is the\nmost sensitive to a steady external forcing and is therefore of sig nificant importance in\nterms of controlling of the global mode. Note that in this part, a dee per interpretation of\ndifferent components ofthe structuralsensitivity tensor,asha s been done in ( Qadriet al.\n2013), is beyond the scope of this work.26 X.-B. Li and others\nFigure 13: Structural sensitivity indicating sensitivity to mean flow m odifications, the\nvalues are normalized by the maximum value; Streamlines are drawn ba sed on mean\nflow; The blue line indicates normalized structural sensitivity = 0.90 iso contour.\n0 1 2 3\nx/W-1012\nFigure 14: Spatial amplification rates of the most unstable near wak e (), middle wake\n() and far wake ( ) branches as functions of streamwise location, computed at the\nglobal complex frequency ωg.\n5. Comparison between empirical and theoretical modes\nIn this section, the near wake eigenmode which serves as the origin o f the global mode,\nis tracked further downstream into the far wake to obtain the full picture of the linear\nglobal mode. The linear global mode is then compared with the leading S POD mode to\nshowtheconnectionsbetweenthemandtorevealadditionalphys icaldrivingmechanisms.\nTo reconstruct the shape of the global mode, the α+branch at ωgmust be tracked\nin downstream direction reaching into the far wake. However, as illus trated in §4.3,\nthe complexity of the base flow gives rise to the problem that the phy sical eigenvalue\nmay become highly stable at one location and fall into a spurious region , and then be\nuntraceable at another location. Therefore, we do not only focus on theα+branch, but\nalso include spatial branches that arise and become unstable during the tracking process.\nThe results of the brnach tracking process are shown in figure 14. Accordingly, the\nnear wake branch becomes spatially stable when approaching x/W= 0.3, and cannot be\ntracked further downstream of x/W= 0.5. However, at x/W= 0.4, a new eigenmode,\nwhich corresponds to the middle wake branch, becomes the most sp atially unstable. The\nmiddle wake branch develops to be slightly spatially unstable only within 0 .75< x/W <\n1.05, and remains spatially stable for most of the streamwise range of its occurrence.Linear stability and SPOD of train wake flow 27\nHowever, it is still the most unstable within the streamwise region 0 .4< x/W <\n1.6. Downstream of x/W= 1.6, the far wake branch becomes dominant. The spatial\namplification rate of the far wake eigenmode is observed to grow sligh tly upstream\nofx/W= 2.0, then remains nearly constant with a marginally stable state after this\nlocation, which extends into the farthest streamwise location cons idered in this study.\nTo analyze whether these branches actually represent the empiric al mode observed in\nthe SPOD, we compute the alignment between the leading SPOD mode a nd the linear\nglobal mode based on the L2inner product. Since the wake flow is highly nonparallel,\nwith several different spatial branches contributing to the linear g lobal mode, we do not\nreconstruct the full three-dimensional global mode prior to the a lignment measurement.\nInstead, the alignment between the leading SPOD mode and the thre e eigenmode\nbranches are computed at all streamwise locations.\nIn figure 15(a), the alignment between the leading SPOD mode and the three eigen-\nmode branches are plotted as a function of x/W. It can be observed that the alignment\ncurves generally follow similar trends to the spatial growth rates of the three branches\nshown in figure 14, with the alignment value always being the highest for the most\nunstable eigenmode branch. In particular, the alignment is, in gener al, quite high with\na value above 0.7, except for locations where the spatial branches switch. Therefore,\nthe three eigenmode branches can be regarded as manifestations of the SPOD mode at\ndifferent regions of the wake. This result further supports the mo delling approach in this\nresearch for tracking the linear global mode in a highly complex three -dimensional base\nflow.\nFor better visualization, in figure 15(b,d,f), the three-dimensional SPOD mode is\ndisplayed based on the isosurface of the streamwise component, c olored by the alignment\nof the three eigenmode branches. In addtion, figure 15(c,e,g) shows a direct comparisons\nbetween the cross-flow shapes of the SPOD mode and the linear glob al mode at different\nstreamwise locations. In general, similar structures can be identifie d throughout the\nentire wake region. In the near wake region, the vortex shedding r elated to the transverse\nrecirculationzoneisdominantinboth theSPODandthelinearglobalmo de,withslightly\ndifferent ranges between the two modes. Further downstream in t he middle wake and\nfar wake region, the coherent structures related to the stream wise vortex pair become\ndominant. The SPOD mode generally predicts a higher fluctuation amp litude near the\nground and the central line, compared to the linear global mode.\nIn conclusion,the fundamental mechanism ofinstability in the consid eredflowproblem\ncan be interpreted as follows: Within the recirculationzone, the flow has a small region of\nabsolute instability, which contributesto the globalwavemakerloca ted inside. The global\nfrequency is well matched to the SPOD peak frequency and the linea r global mode in\nthis region has very high alignment with the SPOD mode. Further down stream, the\nflow becomes convectively unstable, amplifying the perturbations r eceived by the global\nwavemaker, with the oscillation frequency synchronized to the glob al frequency. In this\nsituation, the most spatially unstable branch becomes dominant and aligns the best with\nthe SPOD mode.\n6. Summary and Conclusions\nUnderstanding important dynamics in complex technical flow is crucia l in engineering\npractice. In this paper, three-dimensional coherent structure s in the turbulent wake flow\nbehind a generic high-speed train are investigated. A large eddy simu lation has been\nperformed to simulate and collect a database of the flow problem con sidered. For the28 X.-B. Li and others\n(a)\n0 1 2 3\nx/W0.20.61\n(b)\n(c)\n(d)\n(e)\n(f)\n(g)\nFigure 15: Comparison between the most dominated SPOD mode and t he global linear\nstability mode from local analysis. ( a) Alignment between the SPOD mode and the\neigenmodes of the three branches (colors of lines correspond to fi gure14) as functions\nof streamwise location; ( b,d,f) The iso-surface of the streamwise component of SPOD\nmode colored respectively by the alignment with near wake, middle wak e and far wake\nbranches; ( c,e,g) The streamwise component of the near wake, middle wake and far w ake\nSPOD modes and eigenmodes.Linear stability and SPOD of train wake flow 29\npurpose of this research, both the empirical identification approa ch based on SPOD and\nthe theoretical approach using linear stability analysis are used.\nThe turbulent wake is found to be dominated by spanwise symmetric c oherent struc-\ntures, based on the comparison between the SPOD energy spectr um of symmetric and\nantisymmetric fluctuations. The most dominant SPOD mode is found t o oscillate at the\nangular frequency of ω= 3.437. This dominant mode has an increasing wavelength in\nthe near wake and a nearly constant wavelength in the far wake. Th e quadratic nonlinear\ninteraction of the velocity perturbation is further checked by com puting the mode\nbispectrum to explain the spectral distribution. The leading SPOD mo de is identified\nwith strong self-interaction, which generates the first-harmonic and subharmonic triads,\nmeanwhile leading to significant deformation of the mean field. With the continuous\nevolution of this process, the flow finally appears to be low-rank in a w ide frequency\nrange.\nTheglobalinstabilityisanalyzedbyemployingatwo-dimensionallocals patio-temporal\nstability approach, following the WKBJ approximation and the weakly n on-parallel flow\nassumption. Three spatial branches with positive temporal growt h rate are found, with\nthe near wake branch being the most unstable. The absolute frequ ency of the near wake\nbranch is then found on the basis of a valid saddle point on the complex α+plane\nand tracked along the streamwise direction. A confined region of ab solute instability\nis identified in the near wake region. The global frequency is then det ermined to be\nωg= 3.2702+0.0797i based on the frequency selection criterion, indicating a marg inally\nstable global mode. This result is in excellent agreement with the empir ical prediction in\nterms of angular frequency, and the theoretical expectation in t erms of growth rate. The\nnear wake recirculation zone is found to be related to the global mod e wavemaker, which\ndrivesthe self-excitationofthe instability. Theadjoint method is fu rtherused tocompute\nthe structural sensitivity at the location where the α+andα−branches intersect. In the\ncorresponding cross-flow plane, the highest sensitivity is found ne ar the upper and lower\nshear layers of the recirculation bubble, near the tip of the train no se. Accordingly, the\nglobal mode is expected to be most sensitive to mean flow changes in t his region.\nThe linear global mode shape is further computed at each streamwis e location by\nimposing the global frequency. Three spatial branches are found to be dominant re-\nspectively in the near, middle and far wake regions. The alignment bet ween the linear\nglobal and leading SPOD modes is then performed to provide a quantit ative comparison\nbetween mode shapes. Within the recirculation zone where the globa l wavemaker is\nlocated, the linear global mode has very high alignment with the SPOD m ode. Further\ndownstream, the flow becomes convectively unstable and the oscilla tions within these\nregions are synchronized with the global frequency by excitation f rom the wavemaker.\nSpatial branches with the highest spatial amplification rates becom e dominant and show\nhighest alignment with the SPOD mode.\nThis work has two main conclusions, one methodological and one phys ical. Method-\nologically, a framework to track linear global mode in highly complex 3D b ase flows is\nintroduced, and this method is justified a posteriori by the very good agreement with the\nempirical modes. To the authors’ knowledge, this is the first resea rch dealing with the\nlinear global stability in such a complex flow problem. Physically, the dom inant SPOD\nmode is identified as being caused by a linear global mode in the wake of t he train. This\nmode exhibits transverse symmetry and may cause significant fluct uations on the train\nsurface, which can cause operational safety problems. The sens itivity analysis further\nshows that this mode could be quite effectively suppressed by small c hanges of the base\nflow near the tail of the train, which is of significant importance in ter ms of developing\nefficient flow control strategies.30 X.-B. Li and others\nAcknowledgements This work is supported by the National Science Foundation of\nChina (grantno. 52202429),the Hunan ProvincialNaturalScienc e Foundation (grant no.\n2023JJ40747)and the China Scholarship Council (grant no. 20200 6370204).The authors\nacknowledge the computational resources provided by the High Pe rformance Computing\nCentre of Central South University, China.\nDeclaration of Interests The authors report no conflict of interest.\nAppendix A. Direct LSA operator\nThe operator matrices AandBused in the temporal stability analysis are shown as\nfollows\nA=\nC+¯ ux−νtx(Dx+iα)¯ uy−νty(Dx+iα)¯ uz−νtz(Dx+iα)Dx+iα\n¯ vx−νtxDy C+¯ vy−νtyDy¯ vz−νtzDy Dy\n¯ wx−νtxDz ¯ wy−νtyDzC+¯ wz−νtzDzDz\nDx+iα Dy Dz 0\n(A1)\nB=\ni 0 0 0\n0 i 0 0\n0 0 i 0\n0 0 0 0\n(A2)\nwhere subscripts ( .)x, (.)y, and (.)zrelated to flow quantities denoting their first deriva-\ntives respect to the three directions, Dx,DyandDzare the first derivative matrices\nrespect to the three directions. Operator Ccan be written as\nC=¯ u(Dx+iα)+¯ vDy+¯ wDz−/parenleftbigg1\nRe+νt/parenrightbigg\n(Dyy+Dzz+Dxx+2iαDx−α2)\n−νtx(Dx+iα)−νtyDy−νtzDz(A3)\nwithDxx,DyyandDzzbeing the second derivative matrices respect to the three\ndirections.\nIn spatial stability analysis, operator matrices AandBtake the form\nA=\nC+¯ ux−νtxDx¯ uy−νtyDx¯ uz−νtzDxDx0 0 0\n¯ vx−νtxDyC+¯ vy−νtyDy¯ vz−νtzDyDy0 0 0\n¯ wx−νtxDz¯ wy−νtyDzC+¯ wz−νtzDzDz0 0 0\nDx Dy Dz 0 0 0 0\n0 0 0 0 1 0 0\n0 0 0 0 0 1 0\n0 0 0 0 0 0 1\n(A4)\nB=\nE+iνtxiνtyiνtz−i−1/Re−νt0 0\n0E0 0 0 −1/Re−νt0\n0 0 E0 0 0 −1/Re−νt\n−i 0 0 0 0 0 0\n1 0 0 0 0 0 0\n0 1 0 0 0 0 0\n0 0 1 0 0 0 0\n(A5)Linear stability and SPOD of train wake flow 31\nwithCandEbeing\nC=−iω+¯ uDx+¯ vDy+¯ wDz−/parenleftbigg1\nRe+νt/parenrightbigg\n(Dyy+Dzz+Dxx)\n−νtxDx−νtyDy−νtzDz(A6)\nE=−i¯ u+2i/parenleftbigg1\nRe+νt/parenrightbigg\nDx+iνtx (A7)\nAppendix B. Adjoint LSA operator\nThex−momentum equation is shown as an example of the derivation procedu re. By\npremultiplying the equation by the adjoint eigenvector(equation 4.21), the left-hand side\ncan be written as\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[−iω+¯ u(Dx+iα)+¯ vDy+¯ wDz−(1/Re+νt)(Dyy+Dzz+Dxx+2iαDx−α2)\n−νtx(Dx+iα)−νtyDy−νtzDz+¯ ux−νtxDx−iανtx](ˆ u)dydz+/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH(¯ uy−νtyDx−iανty)(ˆ v)dydz+/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH(¯ uz−νtzDx−iανtz)(ˆ w)dydz+\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH(Dx+iα)(ˆ p)dydz\n(B1)\nWe then present the integration-by-parts procedures for differ ent types of terms in the\nequation for simplification. From a mathematical point of view, we can categorize terms\nin the equation based on whether they would have a derivative matrix acting on the state\nvector. For terms that do not have a derivative matrix acting on th e state vector, we\nhave/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[iα¯ u](ˆ u)dydz=/integraldisplay/integraldisplay/parenleftbig\n−iα∗¯ uˆ u†/parenrightbigH(ˆ u)dydz (B2)\nIf the first derivative matrix Dyact on the state vector\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[¯ vDy](ˆ u)dydz=−/integraldisplay/integraldisplay/parenleftbig\nDy¯ vˆ u†/parenrightbigH(ˆ u)dydz+/contintegraldisplay/parenleftbig\n¯ vˆ u†/parenrightbigH(ˆ u)dz(B3)\nThis procedure will leave boundary term which will have to be canceled then. Also for\nDz\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[¯ wDz](ˆ u)dydz=−/integraldisplay/integraldisplay/parenleftbig\nDz¯ wˆ u†/parenrightbigH(ˆ u)dydz−/contintegraldisplay/parenleftbig\n¯ wˆ u†/parenrightbigH(ˆ u)dy(B4)\nForDx, as this operator assume an artificial x−derivative, the form Dx=−tanθDy−\ntanγDzhas to be taken back for the integration-by-parts procedure. H ere for simplifica-\ntion, we refer a=−tanθandb=−tanγ. Then we have\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[¯ uDx](ˆ u)dydz=/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[¯ uaDy+¯ ubDz](ˆ u)dydz\n=−/integraldisplay/integraldisplay/bracketleftbig\n(Dya¯ u+Dzb¯ u)ˆ u†/bracketrightbigH(ˆ u)dydz+/contintegraldisplay/parenleftbig\na¯ uˆ u†/parenrightbigH(ˆ u)dz−/contintegraldisplay/parenleftbig\nb¯ uˆ u†/parenrightbigH(ˆ u)dy\n(B5)32 X.-B. Li and others\nSimilarly, for all second derivative matrices\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[νtDyy](ˆ u)dydz=/integraldisplay/integraldisplay/parenleftbig\nDyyνtˆ u†/parenrightbigH(ˆ u)dydz+/contintegraldisplay/bracketleftBig/parenleftbig\nνtˆ u†/parenrightbigH(Dyˆ u)−/parenleftbig\nDyνtˆ u†/parenrightbigH(ˆ u)/bracketrightBig\ndz\n(B6)\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[νtDzz](ˆ u)dydz=/integraldisplay/integraldisplay/parenleftbig\nDzzνtˆ u†/parenrightbigH(ˆ u)dydz−/contintegraldisplay/bracketleftBig/parenleftbig\nνtˆ u†/parenrightbigH(Dzˆ u)−/parenleftbig\nDzνtˆ u†/parenrightbigH(ˆ u)/bracketrightBig\ndy\n(B7)\n/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH[νtDxx](ˆ u)dydz=/integraldisplay/integraldisplay/parenleftbig\nˆ u†/parenrightbigH/bracketleftbig\nνta2Dyy+2νtabDyDz+νtb2Dzz/bracketrightbig\n(ˆ u)dydz\n=/integraldisplay/integraldisplay/bracketleftbig/parenleftbig\nDyya2νt+2DyDzabνt+Dzzb2νt/parenrightbig\nˆ u†/bracketrightbigH(ˆ u)dydz+\n/contintegraldisplay/bracketleftBig/parenleftbig\na2νtˆ u†/parenrightbigH(Dyˆ u)−/parenleftbig\nDya2νtˆ u†/parenrightbigH(ˆ u)+2/parenleftbig\nabνtˆ u†/parenrightbigH(Dzˆ u)/bracketrightBig\ndz−\n/contintegraldisplay/bracketleftBig/parenleftbig\nb2νtˆ u†/parenrightbigH(Dzˆ u)−/parenleftbig\nDzb2νtˆ u†/parenrightbigH(ˆ u)−2/parenleftbig\nDyabνtˆ u†/parenrightbigH(ˆ u)/bracketrightBig\ndy\n(B8)\nBy applying these procedures to all terms in equation B1and rearranging, the adjoint\noperators A†andB†can be written as\nA†=\nC†+¯ ux+(Dya+Dzb)νtx¯ vx+Dyνtx ¯ wx+Dzνtx−Dya−Dzb0 0 0\n¯ uy+(Dya+Dzb)νtyC†+¯ vy+Dyνty¯ wy+Dzνty −Dy0 0 0\n¯ uz+(Dya+Dzb)νtz ¯ vz+DyνtzC†+¯ wz+Dzνtz−Dz0 0 0\n−Dya−Dzb −Dy −Dz 0 0 0 0\n0 0 0 0 1 0 0\n0 0 0 0 0 1 0\n0 0 0 0 0 0 1\n\n(B9)\nB†=\nE†−iνtx0 0 i −1/Re−νt0 0\n−iνtyE†0 0 0 −1/Re−νt0\n−iνtz0E†0 0 0 −1/Re−νt\ni 0 0 0 0 0 0\n1 0 0 0 0 0 0\n0 1 0 0 0 0 0\n0 0 1 0 0 0 0\n(B10)\nwithC†andE†given the form\nC†=iω∗−Dya¯ u−Dzb¯ u−Dy¯ v−Dz¯ w−Dyy/parenleftbigg1\nRe+νt/parenrightbigg\n−Dzz/parenleftbigg1\nRe+νt/parenrightbigg\n−/parenleftbig\nDyya2+2DyDzab+Dzzb2/parenrightbig/parenleftbigg1\nRe+νt/parenrightbigg\n+Dyaνtx+Dzbνtx\n+Dyνty+Dzνtz(B11)\nE†= i¯ u+2i(Dya+Dzb)/parenleftbigg1\nRe+νt/parenrightbigg\n−iνtx (B12)Linear stability and SPOD of train wake flow 33\nThe boundary terms should then be canceled by applying appropriat e adjoint boundary\nconditions. The expression for boundary terms on side boundaries is\n/contintegraldisplay/parenleftbig\na¯ uˆ u†/parenrightbigHˆ udz+/contintegraldisplay/parenleftbig\n¯ vˆ u†/parenrightbigHˆ udz−/contintegraldisplay/bracketleftBigg/parenleftbigg/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nDyˆ u−/parenleftbigg\nDy/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nˆ u/bracketrightBigg\ndz−\n/contintegraldisplay/bracketleftBigg/parenleftbigg\na2/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nDyˆ u−/parenleftbigg\nDya2/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nˆ u+2/parenleftbigg\nab/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nDzˆ u/bracketrightBigg\ndz−\n2iα∗/contintegraldisplay/parenleftbig\naˆ u†/parenrightbigHˆ udz−/contintegraldisplay/parenleftbig\naνtxˆ u†/parenrightbigHˆ udz−/contintegraldisplay/parenleftbig\nνtyˆ u†/parenrightbigHˆ udz−/contintegraldisplay/parenleftbig\naνtxˆ u†/parenrightbigHˆ udz−\n/contintegraldisplay/parenleftbig\naνtyˆ u†/parenrightbigHˆ vdz−/contintegraldisplay/parenleftbig\naνtzˆ u†/parenrightbigHˆ wdz+/contintegraldisplay/parenleftbig\naˆ u†/parenrightbigHˆ pdz\n(B13)\nand on vertical boundaries is\n−/contintegraldisplay/parenleftbig\nb¯ uˆ u†/parenrightbigHˆ udy−/contintegraldisplay/parenleftbig\n¯ wˆ u†/parenrightbigHˆ udy+/contintegraldisplay/bracketleftBigg/parenleftbigg/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nDzˆ u−/parenleftbigg\nDz/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nˆ u/bracketrightBigg\ndy+\n/contintegraldisplay/bracketleftBigg/parenleftbigg\nb2/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nDzˆ u−/parenleftbigg\nDzb2/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nˆ u−2/parenleftbigg\nDyab/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/parenrightbiggH\nˆ u/bracketrightBigg\ndy+\n2iα∗/contintegraldisplay/parenleftbig\nbˆ u†/parenrightbigHˆ udy+/contintegraldisplay/parenleftbig\nbνtxˆ u†/parenrightbigHˆ udy+/contintegraldisplay/parenleftbig\nνtzˆ u†/parenrightbigHˆ udy+/contintegraldisplay/parenleftbig\nbνtxˆ u†/parenrightbigHˆ udy+\n/contintegraldisplay/parenleftbig\nbνtyˆ u†/parenrightbigHˆ vdy+/contintegraldisplay/parenleftbig\nbνtzˆ u†/parenrightbigHˆ wdy−/contintegraldisplay/parenleftbig\nbˆ u†/parenrightbigHˆ pdy\n(B14)\nAlthough these expressions are given in a huge form, they can be co nverted into much\nsimpler forms, since the adjoint mode only needs to be computed in th e near wake region.\nTherefore we can consider all direct and adjoint perturbations on the far-field boundary\nto be zero, then only the boundary terms on y/W= 0 and z/W= 0 should be further\nconsidered. On y/W= 0 we have the following conditions and equation B13can be\nsimplified into\na=¯ v=νty=ˆ uy= 0 ony/W= 0 (B15 a)\n/contintegraldisplay/bracketleftbigg/parenleftbigg1\nRe+νt/parenrightbigg\nDyˆ u†/bracketrightbiggH\nˆ udz (B15b)\nand onz/W= 0 these conditions can be applied with equation B14can be simplified\ninto\nb=¯ w=ˆ u= 0 onz/W= 0 (B16 a)\n/contintegraldisplay/bracketleftbigg/parenleftbigg1\nRe+νt/parenrightbigg\nˆ u†/bracketrightbiggH\nDzˆ udz (B16b)34 X.-B. Li and others\n(a)\n  -30  -15      0\n(b)\n0.050 0.075 0.100 0.125 0.150\nx/W      4      6    16\nFigure 16: Local complex streamwise wavenumber of the adjoint mo de based on direct\nLNS equation (black solid line), and its complex conjugation based on a djoint LNS\nequation (red dashed line). ( a) The real component; ( b) The imaginary component.\nTherefore all adjoint boundary conditions appropriate to cancel the boundary terms are\nsummarized as follows\n∂ˆ u†\n∂y=∂ˆ w†\n∂y=∂ˆ p†\n∂y= 0,ˆ v†= 0 ony/W= 0 (B17 a)\nˆ u†=ˆ v†=ˆ w†=ˆ p†= 0 ony/W= 1.3 (B17 b)\nˆ u†=ˆ v†=ˆ w†= 0,∂ˆ p†\n∂z= 0 onz/W= 0 (B17 c)\nˆ u†=ˆ v†=ˆ w†=ˆ p†= 0 onz/W= 1.3 (B17 d)\nTo validate the adjoint operators and boundary conditions, we tak e the conjugation\nof the complex streamwise wavenumber of the adjoint mode comput ed based on the\nadjoint method and compare it with the results from the direct appr oach, as shown in\nfigure12(e,f). The comparisons are shown in figure 16. Good agreement can be observed\nbetween the two approaches, with discrepancies being observed o nly downstream of\nx/W= 0.135. This is due to the fact that, downstream of this streamwise loc ation,\nthe adjoint eigenvalue is quite far from the real axis, which would lead to a convergence\nissue ofthe eigenvalueproblem. 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We investigate the work fluctuations in an overdamped non-equilibrium\nprocess that is stopped at a stochastic time. The latter is characterized by a first\npassage event that marks the completion of the non-equilibrium process. In particular,\nwe consider a particle diffusing in one dimension in the presence of a time-dependent\npotential U(x, t) =k|x−vt|n/n, where k >0 is the stiffness and n >0 is the order of the\npotential. Moreover, the particle is confined between two absorbing walls, located at\nL±(t), that move with a constant velocity vand are initially located at L±(0) =±L. As\nsoon as the particle reaches any of the boundaries, the process is said to be completed\nand here, we compute the work done Wby the particle in the modulated trap upto\nthis random time. Employing the Feynman-Kac path integral approach, we find that\nthe typical values of the work scale with Lwith a crucial dependence on the order n.\nWhile for n >1, we show that ⟨W⟩ ∼L1−nexp [( kLn/n−vL)/D] for large L, we get\nan algebraic scaling of the form ⟨W⟩ ∼Lnfor the n <1 case. The marginal case of\nn= 1 is exactly solvable and our analysis unravels three distinct scaling behaviours:\n(i)⟨W⟩ ∼Lforv > k , (ii)⟨W⟩ ∼L2forv=kand (iii) ⟨W⟩ ∼exp [−(v−k)L] for\nv < k . For all cases, we also obtain the probability distribution associated with the\ntypical values of W. Finally, we observe an interesting set of relations between the\nrelative fluctuations of the work done and the first-passage time for different n– which\nwe argue physically. Our results are well supported by the numerical simulations.\n1. Introduction\nStochastic thermodynamics generally provides a thermodynamic framework to nano-\nscaled systems subjected to thermal fluctuations, even when they are driven arbitrarily\nfar from the equilibrium [1–3]. Contrary to classical equilibrium thermodynamics, one\nnow has well-defined probability distributions of the thermodynamic quantities such\nas heat, work and entropy production [4, 5]. Furthermore, stochastic thermodynamicsarXiv:2401.12583v2  [cond-mat.stat-mech]  19 Mar 20242\ncan be used to derive several fundamental relations such as fluctuation theorems [6–12],\nuncertainty relations [13–23] and thermodynamic speed limits [24–36].\nRecently, there has been a growing interest in the study of the thermodynamic\nquantities until a particular event of interest has occurred for the first time [23,37–48].\nExamples include the escape of a colloidal particle from a metastable state or stretching\nof a polymer till a certain length is attained [41]. Since the underlying dynamics is\nstochastic, the time at which these events take place also varies from realisation to\nrealisation. This drastically changes the properties of the thermodynamic quantities\ncompared to situations where the observation time is fixed for all realisations. For\ninstance, the bound on the average work picks up a non-trivial correction term due to\nthe fact that the system is generally out of equilibrium at the end of the first-passage\ntime [41]. Similar observation has also been made for the efficiency of heat engines [40].\nIn fact, based on the martingale theory, many new results on the integral fluctuation\ntheorem and stopping times for entropy production have been analytically obtained [49].\nDespite these general results, the number of thermodynamic first-passage problems that\nhave been solved exactly, seems to be very limited. In this paper, we aim to partially fill\nthis gap by studying an analytically tractable class of models, where one can get exact\nresults for the moment generating function associated with the mechanical work.\nIn order to study this, we reformulate the work done as a first-passage functional\nof the stochastic process [50]. Statistical properties of such functionals have been\nquite extensively studied in the literature by deploying the celebrated Feynman-Kac\nformalism suitably adapted for first-passage problems [51]. Based on this formalism,\nthese functionals have been shown to have many applications in fields ranging from\nqueue theory [52], sandpile and percolation models [53, 54] to disordered systems [55],\namong others [56–58]. Moreover, they have been studied for diverse stochastic processes\nsuch as diffusion and drift-diffusion [59–62], random acceleration [63], L´ evy process [64],\nOrnstein-Uhlenbeck particle [65, 66] and resetting processes [67, 68]. In this paper, we\nare interested in using these tools and techniques to calculate the moments and the\ndistribution of the work done by a diffusing particle subjected to a time-dependent\npotential U(x, t) =k\nn|x−vt|nwith order n > 0. In particular, we show that the\nwork can be reformulated as a first-passage functional and one can then employ the\nFeynman-Kac formalism to investigate its statistical properties.\nThe rest of the paper is structured as follows: In Section 2, we introduce our\nmodel and present a general formalism to study the work done. We first focus on\nthe analytically tractable cases of the linear potential ( n= 1) in Section 3 and the\nharmonic potential ( n= 2) in Section 4. The insights gained from these two cases\nbecome instrumental in dealing with the general nin Section 5. We then discuss an\ninteresting relation between the work done and the first-passage time in Section 6.\nFinally, in Section 7, we present the conclusion and make some future remarks.3\nn=0.5n=1.0n=2.0\n-1 0 1 2 30123\nxU(x,t)\nn=0.5n=1.0n=2.0\n-4 -2 0 2 4 6-4-2024\nxF(x,t)\nFigure 1. Left and right panels: The depiction of the potential U(x, t) (via\nEq. (2)) and the corresponding force F(x, t) for different values of n. Parameters:\nk=v=t= 1.\n2. Model and general formulation of the problem\nWe consider a one-dimensional diffusing particle whose position, in presence of a time-\ndependent potential U(x, t), evolves according to an overdamped Langevin equation\ndx\ndt=−1\n¯γ∂U(x, t)\n∂x+√\n2D η(t), (1)\nwhere η(t) is the Gaussian white noise satisfying the properties ⟨η(t)⟩= 0 and\n⟨η(t)η(t′)⟩=δ(t−t′). Also Dand ¯γdenote the diffusion and damping coefficients,\nrespectively, and are related by the Einstein relation D=kBT/¯γ. Throughout this\npaper, we will fix ¯ γ= 1. The initial position of the particle will be denoted as x0.\nMoreover, we choose the potential U(x, t) to be a confining potential with stiffness\nk(>0) and we let it move with a constant velocity v(>0):\nU(x, t) =k\nn\f\fx−vt\f\fn,with n >0. (2)\nFinally, we will assume the existence of two absorbing walls located at positions L±(t)\nthat themselves move with the same constant velocity v, i.e. L±(t) =±L+vt, with\n−L < x 0< L. Plots of the potential U(x, t) and the associated force F(x, t) are\nprovided in Figure 1. We observe that for n < 1, the magnitude of the force decays\nwith increasing x, while for n >1, it increases with x. Intriguingly, we observe different\nstriking behaviour depending on the order of the potential strength as will be illustrated\nin this paper.\nAs mentioned in the introduction, we are interested in the work distribution of\nfirst-passage time problems. Here, we allow the particle to move until it touches one of\nthe absorbing boundaries at L±(t). Let us denote this time by tfand the corresponding\ntrajectory as {x(τ)}with 0 ≤τ≤tf. We then calculate the work done up to time tf4\nas [4,69–71]\nW(x0) =ˆtf\n0dτ∂U(x(τ), τ)\n∂τ=−kvˆtf\n0dτsgn\u0010\nx(τ)−vτ\u0011\f\fx(τ)−vτ\f\fn−1,(3)\nwith tf= min {τ:x(τ) =L+(τ),orx(τ) =L−(τ)}. (4)\nSince the motion is random, we get different values of tffor different realisations. This\nindicates that in contrast to the usual case of fixed observation time, the definition of\nwork in Eq. (3) possesses two sources of stochasticity - one arises due to the random\ntrajectory {x(τ)}and the other stems from random first-passage time tf.\nNotice that we have considered the absorbing boundaries to be dynamical in our\nmodel. Such scenarios might arise in experiments in which a Brownian particle is\ndragged using a moving potential [69, 70]. Typically, they have been carried out for\na fixed observation time. However, one can also consider scenarios where one waits\ntill the particle comes out of the field of view of the camera or escapes from the trap.\nNow if the camera/trap is moving, the field of view also changes with time and these,\nin turn, mimic a moving boundary scenario. Another example might be investigating\nthe statistics of work for stretching a polymer to a certain threshold length that itself\ndepends on time [41]. This can also be formulated as a moving boundary problem.\nIn order to derive the statistics of W(x0), we reformulate the work done as a first-\npassage functional which has been extensively studied in the literature [50]. To see this,\nlet us first perform a change of variable as\nξ(t) =x(t)−vt, (5)\nand recast the Langevin equation and the work done in terms of this new co-moving\nvariable:\n∂ξ\n∂t=−ksgn (ξ(t))\f\fξ(t)\f\fn−1−v+√\n2D η(t), (6)\nW(x0) =−kvˆtf\n0dτsgn\u0010\nξ(τ)\u0011\n|ξ(τ)|n−1. (7)\nMeanwhile, the first-passage time tf, in the co-moving frame, simply becomes the time\nat which the particle moves out of the fixed interval [ −L, L] for the first time. Therefore,\nby suitably defining the variable ξ(t), we have been able to recast the entire problem\nwith fixed absorbing walls. Following [50,51], one can now derive a backward differential\nequation for the moment generating function\nQ(p, ξ0) =\ne−pW(ξ0)\u000b\n=\u001c\nexp\u0014\nkpvˆtf\n0dτsgn\u0010\nξ(τ)\u0011\f\fξ(τ)\f\fn−1\u0015\u001d\n, (8)\nwith ξ0=ξ(t= 0) = x0and the averaging ⟨...⟩involves averaging both with respect\nto the trajectories as well as the first-passage time tf. To do this, we now look at a\ntrajectory of ξ(τ) from τ= 0 to τ=tfand break it into two parts: (i) a left interval\n[0,∆t] and (ii) a right interval [∆ t, tf] with ∆ t→0. At the end of the left interval, the5\nvariable ξ(t) takes the value ξ(∆t) =ξ0−[ksgn(ξ0)|ξ0|n−1+v] ∆t+√\n2D∆t η(0). In\nthe remaining time interval ( tf−∆t), the particle starting from ξ(∆t) moves out of the\ninterval [ −L, L]. Therefore, decomposing the integral in Eq. (8) as´tf\n0=´∆t\n0+´tf\n∆t, we\nobtain\nQ(p, ξ0)≃D\nexph\nkpvsgn(ξ0)\f\fξ0\f\fn−1∆ti\nQ(p, ξ(∆t))E\n, (9)\n≃h\n1 +kpvsgn(ξ0)\f\fξ0\f\fn−1∆ti\n⟨Q(p, ξ(∆t))⟩. (10)\nInserting the expression of ξ(∆t) and performing the expansion in ∆ t, we obtain\nthe following differential equation for Q(p, ξ0) as ∆ t→0:\nD∂2Q(p, ξ0)\n∂ξ2\n0−\u0010\nkfn(ξ0) +v\u0011∂Q(p, ξ0)\n∂ξ0+kpvf n(ξ0)Q(p, ξ0) = 0 , (11)\nwhere fn(ξ0) = sgn( ξ0)\f\fξ0\f\fn−1. This equation has to be solved in the domain −L≤ξ0≤\nLand it a differential equation in the initial position ξ0. This approach is referred to\nas a“backward” approach as it involves varying the initial position instead of the final\none. Moreover it is a second order differential equation. Thus, we need two boundary\nconditions to solve it. These conditions can be obtained by analysing the behaviour\nofQ(p, ξ0) atξ0=±L. For these two cases, the particle is already at the edge of the\ninterval and gets instantly absorbed. Thus, for both cases, we have tf= 0 which implies\nW(±L) = 0 from Eq. (7). Plugging this in Eq. (8), we obtain\nQ(p, ξ0=±L) = 1 . (12)\nThe goal now is to solve the backward Eq. (11) with these boundary conditions and\nthen utilize it to obtain the moments of W(x0) as\n⟨Wm(ξ0)⟩= (−1)m∂mQ(p, ξ0)\n∂pm\f\f\f\f\np=0. (13)\nOne can also obtain the full probability distribution by performing the inverse\ntransformation with respect to the p-variable in Eq. (8). In what follows, we first\nillustrate this rigorously for the analytically tractable cases n= 1 and n= 2. Then, we\nwill combine the intuition gained in these solvable cases with Eq. (11) to derive some\ngeneral results for arbitrary n.\n3. Linear potential (n= 1)\nLet us first look at the n= 1 case for which we can solve the backward Eq. (11)\nanalytically. For this case, the Langevin equation (6) takes the form\n∂ξ\n∂t=−ksgn (ξ(t))−v+√\n2D η(t). (14)6\nIn essence, this is a drift-diffusion process in which the particle experiences two distinct\ndrifts. The first one involves a drift with a magnitude k, that tries to pull the particle\nback to the origin. The second one involves a drift of magnitude vthat tends to take\nthe particle away from the origin along the negative xdirection. The behaviour of the\nparticle varies depending on which of the two terms dominate. To see this in the context\nof work done, let us proceed to solve the backward Eq. (11). Depending on the sign of\nthe initial position ξ0, we can split the backward equation as\n(\u0002\n∂2\nξ0−µ∂ξ0+kvp\nD\u0003\nQ(p, ξ0) = 0 , for ξ0>0\u0002\n∂2\nξ0−γ∂ξ0−kvp\nD\u0003\nQ(p, ξ0) = 0 , for ξ0<0,(15)\nwhere µ= (v+k)/D,γ= (v−k)/D. Solving this set of equations yields\nQ(p, ξ0) =(\nK1(p)eλ+(p)ξ0+K2(p)eλ−(p)ξ0, forξ0>0\nK3(p)eσ+(p)ξ0+K4(p)eσ−(p)ξ0, forξ0<0,(16)\nwith λ±(p) =1\n2 \nµ±r\nµ2−4kvp\nD!\nandσ±(p) =1\n2 \nγ±r\nγ2+4kvp\nD!\n.(17)\nTo compute the K(p)-functions, we will need four conditions. Two of these are the\nboundary conditions Q(p, ξ0=±L) which are given in Eq. (12). The other two can\nbe obtained by looking at the behaviour of Q(p, ξ0) and its derivative in the vicinity of\nξ0= 0. Integrating the backward Eq. (11) from −ϵto +ϵand then taking ϵ→0+, we\nobtain\n(\nQ(p, ξ0→0+) =Q(p, ξ0→0−),\n∂ξ0Q(p, ξ0)\f\f\nξ0→0+=∂ξ0Q(p, ξ0)\f\f\nξ0→0−.(18)\nUsing these conditions, it is possible to compute all K(p)-functions as\nK1(p) =(σ−−σ+)eλ−L+ (σ+−λ−)e−σ−L+ (λ−−σ−)e−σ+L\neλ+L[(σ+−λ−)e−σ−L+ (λ−−σ−)e−σ+L] +eλ−L[(λ+−σ+)e−σ−L+ (σ−−λ+)e−σ+L],\nK2(p) =(σ+−σ−)eλ+L+ (λ+−σ+)e−σ−L+ (σ−−λ+)e−σ+L\neλ+L[(σ+−λ−)e−σ−L+ (λ−−σ−)e−σ+L] +eλ−L[(λ+−σ+)e−σ−L+ (σ−−λ+)e−σ+L],\nK3(p) =(λ+−σ−)eλ−L+ (σ−−λ−)eλ+L+ (λ−−λ+)e−σ−L\neλ+L[(σ+−λ−)e−σ−L+ (λ−−σ−)e−σ+L] +eλ−L[(λ+−σ+)e−σ−L+ (σ−−λ+)e−σ+L],\nK4(p) =(λ+−σ+)eλ−L+ (σ+−λ−)eλ+L+ (λ−−λ+)e−σ+L\neλ+L[(σ+−λ−)e−σ−L+ (λ−−σ−)e−σ+L] +eλ−L[(λ+−σ+)e−σ−L+ (σ−−λ+)e−σ+L],\nand inserting them in Eq. (16), we get for ξ0= 0\nQ(p,0) = K1(p) +K2(p). (19)\nThis gives us the exact form of the moment-generating function for the case of linear\npotential. By taking derivative of Eq. (19) suitably with respect to p, we can now obtain\nall moments of the work W. For instance, we find the mean as7\nTheoryAsymptoticsSimulation\n0.1 0.4 1.5 5 25 1000.015500\nL〈W〉(a)\nTheoryAsymptoticsSimulation\n0.05 0.4 1.5 5 500.0151000\nL〈W〉(b)\nTheoryAsymptoticsSimulation\n0 2 4 6 8 100.010.10110100\nL〈W〉(c)\nFigure 2. Plots of the mean work ⟨W⟩for the linear n= 1 potential for (a)\nγ > 0 (v= 1, k= 0.5), (b) γ= 0 ( v=k= 1) and γ < 0 (v= 0.5, k= 1).\nFor all panels, the red curve corresponds to the exact result in Eq. (20) and the green\ncurve is the asymptotic expression in Eq. (21). We have chosen D= 1.\n⟨W⟩=\u0012kv\nγDµ\u0013(γ+µ)\u0002\neµL+e−γL−e(µ−γ)L−1\u0003\n−γµL\u0002\neµL−e−γL\u0003\nγ[1−eµL]−µ[1−e−γL]. (20)\nHere, we have introduced the simplified notation ⟨Wm⟩instead of ⟨Wm(0)⟩. To gain\nsome physical insights, let us look at the asymptotic behaviour of this mean as Lbecomes\nlarge. This behaviour turns out to crucially depend on the signature of the parameter\nγand we find\n⟨W⟩ ≃\n\nkvL\nγD, forγ >0,\nv2L2\n2D, forγ= 0,\n2kv2\nγ2D2µe−γL,forγ <0.(21)\nPhysically, for γ <0 (or equivalently v < k ), the drift vin Eq. (6) is not strong enough\nfor the particle to overcome the force k. Thus, the particle typically takes exponentially\nlarge time to reach the absorbing walls which, in turn, gives rise to exponentially large\nvalue of the work. On the other hand, for γ >0 (or equivalently v > k ), the particle can\neasily overcome the constant force and reach the absorbing walls with relatively high\nprobability. Therefore, we get smaller values of the work which scales linearly with L.\nFor the marginal case γ= 0, we anticipate the scaling to lie somewhere in between γ >0\nandγ <0 cases. In Figure 2, we have compared these asymptotic results and their exact\nexpressions in Eq. (20) with the numerical simulation of the Langevin equation (1).\nAfter analysing the mean, we now proceed to compute the full probability\ndistribution of the work. For this, we need to perform the inverse Laplace transformation\nofQ(p,0) in Eq. (19). For large L, we saw that the work typically takes a large positive\nvalue for all values of γ. In terms of p-variable, this translates to taking small plimit of\nthe moment generating function. Taking p→0 limit in Eq. (19), we find that Q(p,0)\nalso takes different forms depending on the signature of γ. In Appendix A, we have8\nTheorySimulation\n0 50 100 150 2000.010.02\nWP(W)(a)\nTheorySimulation\n0 250 500 750 10000.0010.0020.0030.004\nWP(W)(b)\nTheorySimulation\n0250 500 750 10000.0010.0020.0030.004\nWP(W)(c)\nFigure 3. Distribution of the work done by the particle for the linear potential\n(n= 1) for (a) γ >0 (v= 1, k= 0.5, L= 120), (b) γ= 0 ( v=k= 1, L= 25) and\nγ <0 (v= 0.3, k= 1, L= 10). The theoretical expressions are given in Eqs. (23)\nand (25). We have chosen D= 1.\nrigorously derived these forms for large Land found\nQ(p,0)≃\n\nexp\u0012\nγL\n2−L\n2q\nγ2+4vkp\nD\u0013\n, forγ >0,\nµ{exp(2 αL√p)−1}+2β√pexp(αL√p)\nµ{exp(2 αL√p)−1}+β√p{exp(2 αL√p)+1}, forγ= 0,\n\u0014\n1 +p⟨W⟩\u0012\n1 +kvL\nD\f\fγ\f\fp\u0013\u0015−1\n, forγ <0,(22)\nwhere α=v/√\nDandβ=α\u0000\neµL−1\u0001\n. For γ̸= 0, we can perform the inverse Laplace\ntransformation explicitly and obtain the distribution of work to be\nP(W)≃\n\nLq\nkv\n4πDW3exph\n−Dγ2\n4kvW\u0000\nW−kvL\nD\u00012i\n,forγ >0,\n1\n⟨W⟩\u0014\nexp\u0010\n−W\n⟨W⟩\u0011\n−exp\u0012\n−D\f\fγ\f\fW\nkvL\u0013\u0015\n,forγ <0.(23)\nOn the other hand, for γ= 0, we perform the expansion\nQ(p,0)≃1\n1 +L2α2\n2p+L4α4\n24p2, (24)\nand then carry out the inversion with respect to pto obtain\nP(W)≃4√\n3\nL2α2exp\u0012\n−6W\nL2α2\u0013\nsinh \n2√\n3W\nL2α2!\n,forγ= 0. (25)\nIn Figure 3, we have compared our analytic results with the same obtained from\nnumerical simulations. We observe excellent match between the derived results and\nnumerics for γ̸= 0. Contrarily for γ= 0, we see a departure at smaller values of W[see\nmiddle panel in Figure 3]. This stems from the fact that we have truncated the series in9\nEq. (24) at p2and ignored the higher order terms. While this is valid for large values of\nW(which in the Laplace domain corresponds to small p), it becomes less accurate for\nsmaller values of W(which corresponds to large p). Thus, for a more accurate match\nat small W, one needs to consider higher order terms in Eq. (24).\nFurthermore, the distribution of Wfor these different cases turn out to be\ncompletely different. For instance, as seen in panel (a) in Figure 3 for γ > 0, the\ndistribution close to the mean can be effectively approximated by a Gaussian form.\nThis can also be seen in Eq. (23) where we can replace W≃ ⟨W⟩in the vicinity of\nthe mean and the distribution then simply becomes a Gaussian distribution. However,\nas we move towards the tail, this approximation ceases to remain valid and one needs\nto consider the full non-Gaussian form of the distribution. Contrarily, for γ≤0, the\ndistribution is strictly (always) non-Gaussian with exponentially decaying tails. In fact,\nas illustrated in Figure 3, the distributions for γ≤0 become highly skewed compared\nto the γ >0 case.\nOne can also use the simplified expressions of Q(p,0) in Eq. (24) to calculate the\nhigher moments of work. For example, using Eq. (13), we obtain the second moment of\nWto be\n\nW2\u000b\n≃\n\n⟨W⟩2+2k2v2L\nγ3D2, forγ >0,\n5v4L4\n12D2, forγ= 0,\n2⟨W⟩2−2kvL\nD\f\fγ\f\f⟨W⟩, forγ <0.(26)\nOnce again, we see the emergence of different large- Lbehaviours for different\nsignatures of γ, stemming essentially from the same underlying physical reasoning as\ndiscussed in the context of the mean.\n4. Harmonic potential (n= 2)\nWe now consider the other solvable case of harmonic potential. For this case, the\nbackward equation (11) takes the form\n\u0002\nD∂2\nξ0−(kξ0+v)∂ξ0+kpvξ 0)\u0003\nQ(p, ξ0) = 0 . (27)\nSolving this, we obtain\nQ(p, ξ0) =\u0014\nK1(p)HΛ(p)\u0010\nR(p, ξ0)\u0011\n+K2(p)1F1\u0012\n−Λ(p)\n2;1\n2;R(p, ξ0)2\u0013\u0015\nepvξ0,\nwith Λ( p) =pv2(Dp−1)\nk,andR(p, x) =kx−2Dpv+v√\n2Dk, (28)\nwhere Hm(x) stands for the generalised Hermite polynomial with degree m(where mcan\nbe a real number) and 1F1(m; 1/2;x) is the Kummer confluent hypergeometric function.\nThe other two functions K1(p) andK2(p) can be evaluated using the absorbing boundary10\nSimulationTheory(Exact)Large-L\n0. 1. 2. 3. 4. 5. 6.1.×10-411.×1031.×105\nL〈W〉(a)\nSimulationTheory(Exact)\nLarge-L\n0. 1. 2. 3. 4. 5. 6.1.×10-411.×1031.×1071.×1010\nL〈W2〉(b)\nFigure 4. First two moments of the work done by the particle in presence of the\nharmonic potential ( n= 2). In both panels, green curves are obtained by numerically\ndifferentiating the moment-generating function in Eq. (28), while red curves are the\nasymptotic results in Eqs. (31) and (32). We have chosen v=k=D= 1.\nconditions in Eq. (12) and we find\nK1(p) =epvL1F1\u0010\n−Λ(p)\n2;1\n2;R(p, L)2\u0011\n−e−pvL1F1\u0010\n−Λ(p)\n2;1\n2;R(p,−L)2\u0011\nHΛ(p)\u0010\nR(p,−L)\u0011\n1F1\u0010\n−Λ(p)\n2;1\n2;R(p, L)2\u0011\n−HΛ(p)\u0010\nR(p, L)\u0011\n1F1\u0010\n−Λ(p)\n2;1\n2;R(p,−L)2\u0011,\n(29)\nK2(p) =−epvLHΛ(p)\u0010\nR(p, L)\u0011\n−e−pvLHΛ(p)\u0010\nR(p,−L)\u0011\nHΛ(p)\u0010\nR(p,−L)\u0011\n1F1\u0010\n−Λ(p)\n2;1\n2;R(p, L)2\u0011\n−HΛ(p)\u0010\nR(p, L)\u0011\n1F1\u0010\n−Λ(p)\n2;1\n2;R(p,−L)2\u0011.\n(30)\nNow by taking the derivative of Q(p, ξ0), we can find all moments of work done.\nIn Figure 4, we have illustrated this for the mean and the second moment of Wby\nnumerically carrying out the derivative of Eq. (28). As seen from this plot, our results\nare consistent with the numerical simulations. Later, we will derive these moments\nexactly by using a slightly different (but related) method and show that for large L\n⟨W⟩ ∼L−1exp\u00141\nD\u0012kL2\n2−vL\u0013\u0015\n, (31)\nwhile the second moment of Wbehaves as\n\nW2\u000b\n≃2⟨W⟩2. (32)\nWe have verified these scaling behaviours in Figure 4 along with their exact\ncounterparts.\nHaving looked at the first two moments, we next proceed to calculate the\ndistribution of W. Obtaining the distribution using Q(p, ξ0) in Eq. (28) analytically\nseems daunting. Nevertheless, in Appendix B we have provided a heuristic calculation\nthat correctly gives the distribution for typical values of Wat large L. In particular,\nwe find\nP(W)≃2 exp\u0010\n−⟨W⟩\n2⟨W⟩2−⟨W2⟩W\u0011\nq\n2⟨W2⟩ −3⟨W⟩2sinh\nq\n2⟨W2⟩ −3⟨W⟩2\n2⟨W⟩2− ⟨W2⟩W\n, (33)11\n0 10 20 30400.0010.0060.010.014\nWP(W)\n0 100 200 300 4000.0010.0060.01\nWP(W)\nFigure 5. Probability distribution of the work done Wfor the time-dependent\nharmonic potential ( n= 2). Analytic expression in Eq. (33) is plotted in black and\nthe simulation data are shown in red. Inset shows the comparison for small values of\nW. Parameters chosen are v=k=D= 1,L= 4.\nasLbecomes large. We have compared this expression with the numerical simulations\nin Figure 5 and we observe an excellent match between them. For large W, the\ndistribution decays exponentially as ∼exp (−W/ζ W) with the decay constant ζW=\n2⟨W⟩2−⟨W2⟩\n⟨W⟩−√\n2⟨W2⟩−3⟨W⟩2. This is in stark contrast with the case of the fixed observation time,\nwhere distribution of Wforn= 2 case just turns out to be a Gaussian function [69,72].\nIn fact, the exponential decay turns out to be the hallmark property for the work\ndistributions of all potentials with n >1 under a first-passage time as we discuss later.\n5. General n\nHaving examined the analytically tractable cases, we now shift our focus to deriving\ngeneral results applicable to arbitrary n(>0). While solving the backward equation (11)\ndirectly for a general nis a challenging task, we can utilize it to derive a differential\nequation governing the moments of W, which we can subsequently solve. These moments\ncan then be employed to deduce the behaviour of the distribution of Wfor large Las\ndone for the linear and harmonic cases previously. Writing the m-th moment of Was\n⟨Wm(ξ0)⟩=Wm(ξ0), we take the derivative with respect to pin Eq. (11) to obtain\nDd2Wm(ξ0)\ndξ2\n0−\u0010\nkfn(ξ0) +v\u0011dWm(ξ0)\ndξ0=mvkf n(ξ0)Wm−1(ξ0), (34)12\nSimulationTheory(exact)Large-L\n0.1 0.5 2 10 500.00010.0040.11.5\nL〈W〉n=0.75\nSimulationTheory(exact)Large-L\n0 5 10 14 180.0001150050 000\nL〈W〉n=1.5\nFigure 6. Illustration of the average work ⟨W⟩performed by the particle in potentials\nwith nvalues 0 .75 and 1 .5. The theoretical plot is based on the expression in Eq. (36),\npresenting the exact calculation. In addition, the asymptotic large- Lresults are given\nin Eq. (37). Parameters chosen are k= 0.5, v=D= 1.\nwhere W0(ξ0) = 1 and fn(ξ0) = sgn( ξ0)\f\fξ0\f\fn−1. For m= 1, the solution of this equation\nis given by\nW1(ξ0) =B+Asgn (ξ0)ˆ\f\fξ0\f\f\n0dyGsgn(ξ0)(y) + sgn ( ξ0)kv\nDˆ\f\fξ0\f\f\n0dyGsgn(ξ0)(y)ˆy\n0dzzn−1\nGsgn(ξ0)(z),\nwithG±(x) = exp\u00141\nD\u0012kxn\nn±vx\u0013\u0015\n, (35)\nand the constants AandBcan be computed using the condition W1(ξ0=±L) = 0.\nForξ0= 0, the solution in Eq. (35) reduces to\n⟨W⟩=kv\nD\u0010´L\n0dyG+(y)\u0011\u0010´L\n0dyG−(y)´y\n0dzzn−1\nG−(z)\u0011\n−\u0010´L\n0dyG−(y)\u0011\u0010´L\n0dyG+(y)´y\n0dzzn−1\nG+(z)\u0011\n´L\n0dy[G+(y) +G−(y)].\n(36)\nThis is an exact expression of the mean valid for all n. Although analytically integrating\nin this expression turns out to be challenging in general, one can simplify the expression\nand obtain the leading order behaviour of ⟨W⟩in the large- Llimit. As shown in\nAppendix C, this turns out to depend on the exponent nand we get qualitatively\ndifferent results depending on whether nis greater or smaller than 1. In particular, we\nhave shown in Appendix C that\n⟨W⟩ ∼(\nL1−nexp\u00021\nD\u0000kLn\nn−vL\u0001\u0003\n, forn >1,\nLn, forn <1.(37)\nFor the marginal case n= 1, these results are shown in Eq. (21). This can also be\nobtained from Eq. (37) where n→1−gives the γ >0 case in Eq. (21) while n→1+\nleads to the γ <0 case. Next let us try to understand the difference between n >1 and\nn < 1 cases physically. When the potential is sufficiently confining ( n > 1 case), the\nmotion of the particle is heavily restricted due to the confining potential [see Figure 1].\nThus, typically the particle will be located away from the absorbing walls at ±Land13\nit will hit them very rarely. Therefore, for n > 1, the first-passage time to reach the\nwalls is typically high which in turn gives large value for the work. On the other hand,\nforn < 1, while the potential is still confining, the corresponding force decays with\nincreasing distance. Hence, in Eq. (6), the drift vterm is always dominating beyond a\ncertain distance. As a result, we get relatively smaller values of the first-passage time\nand hence the work.\nIn Figure 6, we have compared the exact result in Eq. (36) with the same obtained\nfrom the numerical simulations for n < 1 (left panel) and n > 1 (right panel). We\nobserve an excellent agreement between the theory and the numerics for both cases.\nIn addition to this, we have also presented a comparison of the large- Lexpressions in\nEq. (37) for the two cases. We find that Eq. (37) converges to the exact result as we\ntake larger values of L.\nSo far in this section, we have solved Eq. (34) only for the mean ( m= 1). But one\ncan also utilize it to obtain the higher moments of W. A similar analysis for m= 2\ngives\n\nW2\u000b\n=\u0010´L\n0dyG+(y)\u0011\u0010´L\n0dyG−(y)´y\n0dzzn−1W1(−z)\nG−(z)\u0011\n−\u0010´L\n0dyG−(y)\u0011\u0010´L\n0dyG+(y)´y\n0dzzn−1W1(z)\nG+(z)\u0011\n(2kv)−1D´L\n0dy[G+(y) +G−(y)].\n(38)\nwith the asymptotic behaviour derived in Appendix D to be\n\nW2\u000b\n≃(\n2⟨W⟩2, forn >1,\n⟨W⟩2, forn <1.(39)\nNote that these expressions only provide the leading order behaviour in Land there are\nstill sub-leading corrections that depend on model parameters vandk. Interestingly, it\nturns out that one can use the first two moments to heuristically calculate the probability\ndistribution describing the typical fluctuations of the work [as done before for n= 2].\nWe refer to Appendix B for this calculation where we show\nP(W)≃2 exp\u0010\n−⟨W⟩\n2⟨W⟩2−⟨W2⟩W\u0011\nq\n2⟨W2⟩ −3⟨W⟩2sinh\nq\n2⟨W2⟩ −3⟨W⟩2\n2⟨W⟩2− ⟨W2⟩W\n,forn >1.(40)\nWe emphasize that this expression only describes the typical fluctuations of Waround\nthe mean for n > 1 and will not capture the regimes far away from the mean. In\nFigure 7 (left panel), we have compared this with the numerics for n= 1.5. We notice\nan excellent agreement between them. Next, we address the case where n < 1. As\nobserved in the instance of n= 1 with γ >0, one needs to solve the complete backward\nequation (11) in order to obtain Q(p,0). While achieving this for n= 1 is feasible, doing\nthe same analytically for arbitrary n <1 turns out to be difficult. However, based on\nextensive numerical simulation, we find that for n < 1 also, the distribution has the\nsame form (around the mean) as for n= 1 with γ >0 in Eq. (23). This leads us to14\nassume\nP(W)≃1√A1W3exp\u0014\n−A2\nW(W−A3)2\u0015\n,forn <1, (41)\nfor large L. Here A1andA2andA3are constants that depend on the parameters of\nthe model. To compute them, we use the normalisation condition and the fact that the\nfirst two moments are given. We then obtain\nA1=2π\u0000\n⟨W2⟩ − ⟨W⟩2\u0001\n⟨W⟩3,A2=⟨W⟩\n2\u0002\n⟨W2⟩ − ⟨W⟩2\u0003,A3=⟨W⟩. (42)\nPlugging these constants in Eq. (41) gives\nP(W)≃s\n⟨W⟩3\n2πW3\u0000\n⟨W2⟩ − ⟨W⟩2\u0001exp\"\n−⟨W⟩(W− ⟨W⟩)2\n2W\u0002\n⟨W2⟩ − ⟨W⟩2\u0003#\n,forn <1.(43)\nThe right panel in Figure 7 shows a comparison of Eq. (43) with the same obtained from\nthe numerical simulations for n= 0.75. An excellent match between them validates our\nresults. In summary, we have derived the exact expressions for the first two moments\nand their large- Lforms in this section, applicable for all values of n. We then combined\nthese expressions with a heuristic analysis to obtain the probability distribution of W\nfor large values of Lacross all nvalues.\nBefore ending this section, it is worth noting that although we have focused on\nthe large- Lbehaviour of the work done, one can also use the exact expressions given\nin Eqs. (36) and (38) for computing the first two moments for small values of L.\nTo calculate this, we first observe that for small L, one can perform the expansion\nG±(y)≃1 +1\nD\u0000kyn\nn±vy\u0001\ninside integrals in Eqs. (36) and (38). Now the integrations\ncan be carried out analytically and we obtain\n⟨W⟩ ≃kv2\n2D2Ln+2\n(n+1)(n+2),\n⟨W2⟩ ≃2k2v2\nD2L2n+2\nn(n+1)2(n+2),)\n(for small L). (44)\nCompared to the large- Lexpressions, we find that both ⟨W⟩and⟨W2⟩exhibit algebraic\nscaling with Lfor smaller values of L.\n6. Relation with the first-passage time\nSo far, we have studied the statistics of the work done till the particle reaches one\nof the walls at L±(t) for the first time. Deploying a backward approach in terms of\nthe co-moving variable ξ(t), we have been able to compute the first two moments and\nthe probability distribution of Wfor large L. In the remaining part of our paper, we\nshow that the fluctuations of the work done Wbear an interesting relation with the\nfluctuations of the first-passage time tfdefined in Eq. (4). In order to see this, we first15\nTheorySimulation\n0 100 200 3000.0050.010.015\nWP(W)n=1.5\nTheory\nSimulation\n0 10 20 30 40 500.010.050.0850.12\nWP(W)n=0.75\nFigure 7. Comparison of the probability distribution of Woutlined in Eqs. (40) and\n(43) is presented in the left panel for n >1 and L= 10 and the right panel for n <1\nandL= 100. We have set the parameters as k= 0.5,v=D= 1 for this comparison\nusing numerical simulations.\nwrite a backward equation for the moments Tm(ξ0) =\ntm\nf(ξ0)\u000b\nof the first-passage time\nas done before in Eq. (34) for the work done [50,51]\nDd2Tm(ξ0)\ndξ2\n0−\u0010\nkfn(ξ0) +v\u0011dTm(ξ0)\ndξ0=−mTm−1(ξ0), (45)\nwhere T0(ξ0) = 1 and fn(ξ0) = sgn( ξ0)\f\fξ0\f\fn−1. Meanwhile the moments satisfy the\nboundary conditions Tm(ξ0=±L) = 0, since the particle gets instantly absorbed if its\ninitial position coincides with one of the absorbing walls. Solving Eq. (45) in the same\nway as done in Section 5, we obtain for m= 1\nT1(ξ0) =B1+B2sgn (ξ0)ˆ\f\fξ0\f\f\n0dyGsgn(ξ0)(y)−1\nDˆ\f\fξ0\f\f\n0dyGsgn(ξ0)(y)ˆy\n0dz\nGsgn(ξ0)(z), (46)\nwith functions G±(y) given in Eq. (35). Constants B1andB2can be evaluated using\nthe boundary conditions T1(ξ0=±L) = 0. For ξ0= 0, the mean ⟨tf⟩=T1(0) is given\nby\n⟨tf⟩=1\nD\u0010´L\n0dyG+(y)\u0011\u0010´L\n0dyG−(y)´y\n0dz\nG−(z)\u0011\n+\u0010´L\n0dyG−(y)\u0011\u0010´L\n0dyG+(y)´y\n0dz\nG+(z)\u0011\n´L\n0dy[G+(y) +G−(y)],(47)\nFor large L, the integrations in this expression can be approximately carried out as\ndone for the work done. Proceeding similarly in Appendix C, we find\n⟨tf⟩ ∼(\nL1−nexp\u00021\nD\u0000kLn\nn−vL\u0001\u0003\n, forn >1,\nL, forn <1.(48)\nwhile for the marginal case of n= 1, we obtain\n⟨tf⟩ ≃\n\nL\nγD, forγ >0,\nL2\n2D, forγ= 0,\n2k\nγ2D2µe−γL,forγ <0,(n= 1). (49)16\nk=3, v=2k=3, v=3k=2, v=3\n0 1 2 3 4 5 612345\nLR(L)n=1.5\nk=3, v=2k=3, v=3k=2, v=3\n0 20 40 60 80 1000.91.01.11.21.31.4\nLR(L)n=1.0\nk=3, v=2k=3, v=3k=2,v=3\n0 200 400 600 8000.00.51.01.52.0\nLR(L)\nn=0.8\nFigure 8. Simulation results for the ratio R(L) defined in Eq. (54) for three cases of\nn >1,n= 1 and n <1. For every n, we have considered scenarios with v < k (pink),\nv=k(blue) and v > k (green).\nBoth Eqs. (48) and (49) give the leading order behaviours in L. Interestingly, for n >1,\nwe see that the large- Lscaling of ⟨tf⟩is similar to ⟨W⟩in Eq. (37). We believe this\noccurs because for large tf, the rate of work done W/t fapproaches a constant value (by\nthe law of large numbers), and hence W∼tf. Therefore, the stochasticity of the total\nwork is dominated by the stochasticity of the first-passage time.\nNext, one can solve Eq. (45) for for m= 2 to show that the second moment\nt2\nf\u000b\n=⟨Tm(ξ0= 0)⟩reads\n\nt2\nf\u000b\n=\u0010´L\n0dyG+(y)\u0011\u0010´L\n0dyG−(y)´y\n0dzT1(−z)\nG−(z)\u0011\n+\u0010´L\n0dyG−(y)\u0011\u0010´L\n0dyG+(y)´y\n0dzT1(z)\nG+(z)\u0011\n2−1D´L\n0dy[G+(y) +G−(y)].\n(50)\nFor large L, one can again simplify this expression [see Appendix D] and obtain\n\nt2\nf\u000b\n≃(\n2⟨tf⟩2, forn >1,\n⟨tf⟩2, forn <1,(51)\nand for n= 1, one gets\n\nt2\nf\u000b\n≃\n\n⟨tf⟩2+2L\nD2γ3, forγ >0,\n5L4\n12D2, forγ= 0,\n2⟨tf⟩2, forγ <0.(n= 1). (52)\nOnce again in Eq. (51), we only obtain the leading order asymptotic expression in L.\nHowever, there will be sub-leading corrections that depend on parameters vandkand\nwe do not delve into these corrections in Eq. (51).\nNext, we set out to examine an interesting connection between the first-passage\ntime and the work done. To this end, we define their respective coefficents of variation\nor the variabilities as\nCV\f\f\nW=s\n⟨W2⟩ − ⟨W⟩2\n⟨W⟩2,and CV\f\f\ntf=s\nt2\nf\u000b\n− ⟨tf⟩2\n⟨tf⟩2, (53)17\nand then study the following ratio\nR(L) = \nCV\f\f\nW\nCV\f\f\ntf!2\n. (54)\nAs we show below, this “variability ratio” has intricate behaviors as a function of system\nsizeLand the potential order n.\nForn= 1, one can plug the analytic expressions of first and second moments of W\nandtfderived in the previous sections and show that the ratio R(L) converges to the\nvalue 1 as the length Lbecomes large for all values of vandk. Similarly, for n >1,\none can use the asymptotic results ⟨W2⟩ ≃2⟨W⟩2and\nt2\nf\u000b\n≃2⟨tf⟩2to show that both\nCV\f\f\nWand CV\f\f\ntfindividually go to 1. This indicates that the ratio R(L) also converges\nto 1 for larger Lforn >1. However for n <1, an analytical calculation of R(L) remains\nunlikely. As indicated in Eqs. (39) and (51), for the case of n <1, one needs to have\nthe knowledge of the sub-leading corrections in Lwithin the expressions of ⟨W2⟩and\n⟨t2\nf⟩in order to compute the CV-s. Moreover, extracting these corrections analytically\nfrom the exact expression turns out to be difficult and our analysis in Eqs. (39) and (51)\ncannot capture these details.\nIn Figure 8, we have plotted the ratio R(L) as a function of the initial wall separation\nLfor different values of the exponent n. For n≥1, we find that R(L) indeed saturates\nto the universal value equal to one at large Lindependent of the model parameters k,v\nandD. On the other hand, for n <1, the large- Lbehaviour of R(L) depends on these\nparameters. For instance, in Figure 8 [right panel], we see that the saturation values\nforv= 3, k= 3 (shown in blue) and v= 3, k= 2 (shown in green) are completely\ndifferent for n= 0.8. On the other hand, for v= 2, k= 3 (shown in pink), R(L)\nincreases monotonically at large Land does not saturate to a constant value. This is\npossibly due to the fact that the saturation of R(L) only takes place at large L,i.e.\nforL≫Lth, where Lthis some threshold length. For a given n, the threshold Lthalso\ndepends on the parameters vandk. In Figure 8 [right panel], we find that Lth∼50 for\nthe green curve ( k/v < 1) whereas Lth∼500 for the blue one ( k/v= 1). Thus, we see\na consistent rise in the value of Lthas the ratio k/vstarts to increase. Following this\ntrend, we anticipate Lthto be very large for the pink curve ( k/v > 1). Probing such\nlarge Lvalues in simulations is computationally expensive. Thus, we observe that R(L)\nincreases for the pink curve since we are still in the regime L≪Lth. However, if we go\nto very large Lvalues (beyond what is considered in the figure), we anticipate R(L) to\nconverge to a constant value even for the pink curve ( v= 2, k= 3).\n7. Conclusion\nIn summary, we have analyzed the work statistics in a driven overdamped system in\none dimension. In contrast to the conventional cases where the observables are usually\nmeasured upto a fixed time, here we measure the work upto a random first passage time\nthat is conditioned on a certain criterion. We employ the Feynman-Kac path integral18\napproach to compute the work functional in various set-ups consisting of potentials of\ndifferent configurations. We provide a comprehensive analysis of the work fluctuations\nwhich shows a rich behaviour as a function of the potential strength, the external drive\nand the interval over which the particle moves. Furthermore, we showed an interesting\nsymmetry relation between the signal-to-noise relation or the variability of the work and\nthat of the first-passage time. To understand this phenomena better, we also delved\ndeeper into the attributing physical conditions. Notably, our results illustrate a marked\ndifference in the work statistics between systems driven up to a fixed time and a first-\npassage time. Although we considered a fixed initial condition x0= 0 to calculate\nthese results, our method can, in principle, be extended to a general distribution P(x0).\nIn particular, we anticipate the large- Lexpressions in Eqs. (37) and (39) for work\nfunctionals and in Eqs. (48) and (51) for first-passage time to be valid even for a general\nP(x0) as long as the variance of x0does not scale with the initial separation L.\nMoving forward, it would be interesting to study the “variability ratio” in other\nsystems and to identify any similar pattern as unveiled here. There has been a myriad\nof studies in recent times in the field of stochastic thermodynamics involving various\nother thermodynamic observables such as injected power, dissipated heat or entropy\nproduction. It is only natural to study the same also within our set-up and will be\npursued elsewhere. Furthermore, our model could also be useful to study existing\nthermodynamic bounds for first-passage-time problems [24, 37, 41, 73, 74]. Another\ninteresting direction is to investigate how our method can be extended to incorporate\nother potentials. For instance, we assumed that both the potential and the boundaries\nmove at the same velocity v. It would be interesting to explore what happens if we relax\nthis condition and allow them to move with different velocities. Concluding, we believe\nthat our results can be tested using controlled optical trap experiments [72,75,76] which,\nhopefully, will also unfold new research directions to this problem.\nAcknowledgement\nPS and KP acknowledge support from the European Union’s Horizon 2020 research and\ninnovation program under the Marie Sklodowska-Curie grant agreement No. 847523\n‘INTERACTIONS’ and grant agreement No. 101064626 ‘TSBC’ and from the Novo\nNordisk Foundation (grant No. NNF18SA0035142 and NNF21OC0071284). IM and\nCF acknowledge the financial support from Brazilian agencies CNPq and FAPESP\nunder grants 2021/03372-2, 2021/12551-8 and 2023/00096-0. AP acknowledges research\nsupport from the Department of Science and Technology, India, SERB Start-up Research\nGrant Number SRG/2022/000080 and Department of Atomic Energy, Government of\nIndia.19\nAppendix A. Large- Lbehaviour of Q(p,0)for the linear potential\nIn this appendix, we provide the derivation of the simplified expression of the moment\ngenerating function Q(p,0) for large Las written in Eq. (22). For this, we first rewrite\nthe exact expression of Q(p,0) in Eq. (19) as,\nQ(p,0) =N(p)\nD(p), (A.1)\nwhere the functions N(p) andD(p) are given by\nN(p) = [λ+(p)−λ−(p)]h\ne−σ+(p)L−e−σ−(p)Li\n+ [σ+(p)−σ−(p)]h\neλ−(p)L−eλ+(p)Li\n,(A.2)\nD(p) = [λ−(p)−σ+(p)]e[λ+(p)−σ−(p)]L+ [σ+(p)−λ+(p)]e[λ−(p)−σ−(p)]L\n+ [λ+(p)−σ−(p)]e[λ−(p)−σ+(p)]L+ [σ−(p)−λ−(p)]e[λ+(p)−σ+(p)]L, (A.3)\nwith λ±(p) and σ±(p) given in Eq (17). In order to simplify this expression further, we first\nnote that the work done by the particle typically attains a large positive value which scales\neither algebraically or exponentially with the length L, as seen from the expression of ⟨W⟩in\nEq. (21). In terms of p-variable, this translates to taking the p→0 limit in Q(p,0). However,\nfor small p, the leading order form of Q(p,0) crucially depends on the signature of γ. Below\nwe consider these different cases separately.\nAppendix A.1. γ >0case\nForp→0 limit, we find that σ+(p)≃γ,λ+(p)≃µ, whereas both σ−(p) and λ−(p) are of\norder ∼p. Therefore, we find that N(p)≃ −γeµLandD(p)≃ −γe[µ−σ−(p)]L. Plugging this\nin Eq. (A.1), we find\nQ(p,0)≃exp [σ−(p)L],forγ >0. (A.4)\nAppendix A.2. γ <0case\nOn the other hand, for γ <0, we find σ−(p)≃γ,λ+(p)≃µ, while both σ+(p) and λ−(p) are\nlinear order in p. This implies that to the leading order, we have\nN(p)≃[σ−(p)−σ+(p)]eλ+(p)L, (A.5)\nD(p)≃eλ+(p)Lh\ne−σ+(p)L\u0010\nσ−(p)−λ−(p)\u0011\n+e−σ−(p)L\u0010\nλ−(p)−σ+(p)\u0011i\n. (A.6)\nInserting these expressions in Eq. (A.1) and performing the small pexpansion further, we\nobtain\nQ(p,0)≃\"\n1 +p⟨W⟩ \n1 +kvL\nD\f\fγ\f\fp!#−1\n,forγ <0. (A.7)\nAppendix A.3. γ= 0case\nFor this case, we have λ+(p)≃µ,λ−(p)≃v2p/µD andσ±(p) =±p\nv2p/D. Plugging this in\nEq. (A.1) and performing the small λ−expansion we find\nQ(p,0)≃µ\b\nexp(2 αL√p)−1\t\n+ 2β√pexp(αL√p)\nµ\b\nexp(2 αL√p)−1\t\n+β√p\b\nexp(2 αL√p) + 1\t, forγ= 0. (A.8)20\nAppendix B. Heuristic derivation of the distribution P(W)forn >1\nIn this appendix, we will give a hand-waving derivation of the distribution of the work done\nin Eq. (40) for n >1. To begin with, we first recall that for large L, the work done typically\nattains large positive values as indicated by Eq. (37). In terms of the moment generating\nfunction Q(p,0), this corresponds to performing a small papproximation. Indeed, for the case\nof linear potential, this led us to obtain a simplified expression of Q(p,0) in Eq. (22). In fact,\nfor all n >1, we find that the moment-generating function can be expressed as\nQ(p,0)≃1−p⟨W⟩+\nW2\u000b\n2p2, (B.1)\n≃\"\n1 +p⟨W⟩+p2 \n⟨W⟩2−\nW2\u000b\n2!#−1\n, (B.2)\nfor large Lwith moments ⟨W⟩and\nW2\u000b\ngiven in Eqs. (36) and (38) respectively. Performing\nthe inverse Laplace transformation, we then obtain\nP(W)≃2 exp\u0010\n−⟨W⟩\n2⟨W⟩2−⟨W2⟩W\u0011\nq\n2⟨W2⟩ −3⟨W⟩2sinh\nq\n2⟨W2⟩ −3⟨W⟩2\n2⟨W⟩2− ⟨W2⟩W\n,forn >1. (B.3)\nThis expression has been presented in Eq. (40) in the main text.\nAppendix C. Asymptotic behaviour of the mean work for large L\nHere, we derive the simplified expression of the mean work ⟨W⟩for large L. The exact\nexpression was derived in Eq. (36) which we rewrite here as\n⟨W⟩=kv\nDJ+(L)I−(L)− J−(L)I+(L)\nJ+(L) +J−(L), (C.1)\nwithJ±(L) =ˆL\n0dyG±(y),andI±(L) =ˆL\n0dyG±(y)ˆy\n0dzzn−1\nG±(z), (C.2)\nwhere G±(y) are given in Eq. (35). In order to find the large- Lbehaviour of ⟨W⟩, we have to\ncalculate the corresponding forms of the functions J±(L) andI±(L). These however turn out\nto depend on the exponent nand we get different forms depending on whether nis greater or\nsmaller than 1. Below we consider these two cases separately.\nAppendix C.1. n >1case\nForn >1, both G±(y) functions have faster than exponential growth for large argument. This\nindicates that both J±(L) also grow with L. To see this, we write them as\nJ±(L) =ˆL\n0dyexp\u00141\nD\u0012kyn\nn±vy\u0013\u0015\n, (C.3)\n=Lˆ1\n0dωexp\u00141\nD\u0012kωnLn\nn±vωL\u0013\u0015\n,(ω=y/L). (C.4)\nFor large L, the integrand has a maximum value at ω= 1. Moreover, this value increases\nwith L. Therefore, for large L, the overall integration in Eq. (C.4) will get major contribution21\nfrom the vicinity of ω= 1. Performing an expansion of the integrand in Eq. (C.4) around\ny= 1−uand keeping the leading order in u, we get the large- Lbehaviour of J±(L) as\nJ±(L)∼Lexp\u00141\nD\u0012kLn\nn±vL\u0013\u0015\n׈1\n0duexph\n−u\nD(kLn±vL)i\n, (C.5)\n∼L1−nexp\u00141\nD\u0012kLn\nn±vL\u0013\u0015\n. (C.6)\nWe next turn to I±(L). Rewriting their forms from Eq. (C.2)\nI±(L) =ˆL\n0dyexp\u00141\nD\u0012kyn\nn±vy\u0013\u0015ˆy\n0dz zn−1exp\u0014\n−1\nD\u0012kzn\nn±vz\u0013\u0015\n, (C.7)\n=Lˆ1\n0dωexp\u00141\nD\u0012kωnLn\nn±vωL\u0013\u0015 ˆωL\n0dz zn−1exp\u0014\n−1\nD\u0012kzn\nn±vz\u0013\u0015\n.(C.8)\nLet us first analyse the integration over zfor large values of L. By changing the variable\nu=kzn/nD, this integral can be rewritten as\nˆωL\n0dz zn−1exp\u0014\n−1\nD\u0012kzn\nn±vz\u0013\u0015\n(C.9)\n∼ˆkwnLn\nnD\n0du e−u∞X\ni=0\u0010\n∓v\nD\u0011i\u0012Dn\nk\u0013i\nnui\nn\ni!, (C.10)\n∼∞X\ni=0\u0010\n∓v\nD\u0011i\u0012Dn\nk\u0013i\nnΓ (1 + i/n)\ni!, asL→ ∞ . (C.11)\nThe sum converges for n >1. This implies that the integration over zin Eq. (C.8) becomes\nindependent of ωLasLbecomes very large (for a given ω) and I±(L) accordingly takes the\nform\nI±(L)∼Lˆ1\n0dωexp\u00141\nD\u0012kωnLn\nn±vωL\u0013\u0015\n, (C.12)\n∼L1−nexp\u00141\nD\u0012kLn\nn±vL\u0013\u0015\n, (C.13)\nwhere the second line follows from Eq. (C.6). Combining this result with the expressions of\nJ±(L) in Eq. (C.6) and plugging all of them in Eq. (C.1), we find\n⟨W⟩ ∼L1−nexp\u00141\nD\u0012kLn\nn−vL\u0013\u0015\n,forn >1 and large L. (C.14)\nAppendix C.2. n <1case\nLet us now derive the large- Lexpression of the mean work ⟨W⟩forn <1. Looking at Eq. (C.1),\nthis again translates to calculating the functions J±(L) and I±(L). We first notice that even\nforn <1,G+(y) has a faster than exponential growth for large y. Therefore, one can follow a\nsimilar approach as described in Section Appendix C.1 for J+(L) and I+(L) and obtain\n(\nJ+(L)∼exp\u00021\nD\u0000kLn\nn+vL\u0001\u0003\n,\nI+(L)∼exp\u00021\nD\u0000kLn\nn+vL\u0001\u0003\n,(for large L). (C.15)22\nOn the other hand, G−(y) for n < 1 is a decaying function for large yand requires a\ndifferent approach. To this end, we recast J−(L) as\nJ−(L) =ˆL\n0dy e−vy/D∞X\ni=0\u0012k\nnD\u0013iyni\ni!, (C.16)\nand carry out the integration over yforL→ ∞ to yield\nJ−(L)∼∞X\ni=0\u0012k\nD1−nn vn\u0013iΓ (1 + ni)\ni!. (C.17)\nForn <1, this is a convergent sum. Hence J−(L) becomes independent of Lfor this case.\nNext we turn to I−(L) whose explicit expression follows from Eq. (C.2) as\nI−(L) =ˆL\n0dyexp\u0014\n−1\nD\u0012\nvy−kyn\nn\u0013\u0015ˆy\n0dz zn−1exp\u00141\nD\u0012\nvz−kzn\nn\u0013\u0015\n. (C.18)\nBy changing the variables y=ωLandz=uωL, this expression becomes\nI−(L) =Ln+1ˆ1\n0dω ωnexp\u0014\n−ωL\nD\u0012\nv−k\nn(ωL)1−n\u0013\u0015ˆ1\n0du un−1exp\u0014uωL\nD\u0012\nv−k\nn(uωL)1−n\u0013\u0015\n.\nFor large Land fixed ωandu, one can simplify this expression as\nI−(L)≃Ln+1ˆ1\n0dω ωnexp\u0014\n−vωL\nD\u0015ˆ1\n0du un−1exp\u0014vuωL\nD\u0015\n, (C.19)\n≃Ln+1ˆ1\n0dω ωnexp\u0012\n−vωL\nD\u0013 \u0012\n−D\nvωL\u0013n\u0014\nΓ (n)−Γ\u0012\nn,−vωL\nD\u0013\u0015\n. (C.20)\nUsing the approximate form of the gamma function as Γ ( n,−x)x→∞≃(−x)n−1exand then\nperforming the integration over ωgives\nI−(L)≃D\nnvLn,for large L. (C.21)\nFinally, using this result along with the results derived in Eqs. (C.15) and (C.17) and\ninserting all of them in Eq. (C.2), we obtain the large- Lbehaviour of the mean work as\n⟨W⟩ ≃kv\nDI−(L)∼Ln,forn <1. (C.22)\nEqs. (C.14) and (C.22) have been quoted in Eq. (37) in the main text.\nAppendix D. Variance of the work done for large L\nIn this appendix, we will derive the simplified leading-order expression for the variance\nW2\u000b\nof the work done for large L. The starting point is the exact expression in Eq. (38) which we\nrewrite as\n\nW2\u000b\n=2kv\nDJ+(L)Y−(L)− J−(L)Y+(L)\nJ+(L) +J−(L), (D.1)23\nwhere J±(L) are given in Eq. (C.2) and functions Y±(L) are\nY±(L) =ˆL\n0dyG±(y)ˆy\n0dz W 1(±z)zn−1\nG±(z), (D.2)\nwithG±(y) defined in Eq. (35). In order to find the large- Lexpression of\nW2\u000b\n, we have\nto calculate the corresponding form of these functions. Meanwhile, we computed J±(L) in\nAppendix C and showed that they take different forms depending on whether the exponent\nnis greater than or smaller than one. In the following, we carry out the same analysis for\nY±(L).\nAppendix D.1. n >1case\nLet us write the explicit expression of Y±(L)\nY±(L) =ˆL\n0dyexp\u00141\nD\u0012kyn\nn±vy\u0013\u0015ˆy\n0dz zn−1W1(±z) exp\u0014\n−1\nD\u0012kzn\nn±vz\u0013\u0015\n.\n(D.3)\nLooking at this, we first notice that for yintegration, the integrand has a faster than\nexponential growth. This implies that at large L, this integration will be dominated\npredominantly by the larger values of y. On the other hand, the integration over zis damped\ndue to the presence of the decaying exponential term. Therefore, even though ytakes larger\nvalues, the integration over zwill still be dominated by small values of z(more precisely\nz≪L). This allows us to replace W1(±z)≃W1(0) in Eq. (D.3) and recast it as\nY±(L)≃W1(0)I±(L), (D.4)\nwhere I±(L) are given in Eq. (C.2). Inserting this in Eq. (D.1) then yields\n\nW2\u000b\n≃2kv⟨W⟩\nDJ+(L)I−(L)− J−(L)I+(L)\nJ+(L) +J−(L), (D.5)\nafter which we use Eq. (C.1) to obtain\n\nW2\u000b\n≃2⟨W⟩2,for large L. (D.6)\nAppendix D.2. n <1case\nWe now turn to the other case of n <1. First recall that in Appendix C.2, we showed that\nthe functions J±(L) for this case scale as\n(\nJ+(L)∼exp\u00021\nD\u0000kLn\nn+vL\u0001\u0003\n,\nJ−(L)∼independent of L,(for large L). (D.7)\nSimilarly, for Y+(L), we notice that G+(y) is a growing function in yeven for n < 1. This\nindicates that the approximations used in deriving Eq. (D.4) for Y+(L) will remain valid also\nforn <1 and we have\nY+(L)∼Lnexp\u00141\nD\u0012kLn\nn+vL\u0013\u0015\n. (D.8)24\nHowever Y−(L) has to be treated in a different way since G−(y) is now a decaying function\nat large y. To this end, we first change the variables y=ωLandz=uωL in Eq. (D.2) and\nrewrite it as\nY−(L) =Ln+1ˆ1\n0dω ωnexp\u0014\n−ωL\nD\u0012\nv−k\nn(ωL)1−n\u0013\u0015\n׈1\n0du un−1W1(−uωL) exp\u0014uωL\nD\u0012\nv−k\nn(uωL)1−n\u0013\u0015\n. (D.9)\nTaking large Llimit for fixed ωandu, this simplifies to\nY−(L)≃Ln+1ˆ1\n0dω ωnexp\u0014\n−vωL\nD\u0015ˆ1\n0du un−1W1(−uωL) exp\u0014vuωL\nD\u0015\n. (D.10)\nThis expression is similar to Eq. (C.19) except for the presence of W1(−uωL) term. In\nAppendix D.3, we show that\nW1(−uωL)≃ ⟨W⟩(1−ωnun),for large L. (D.11)\nPlugging this expression in Eq. (D.10) and performing the approximations as done in Eq. (C.19)\nyields\nY−(L)≃⟨W⟩\n2I−(L). (D.12)\nRecall that both ⟨W⟩andI−(L) scale as ∼Lnfor large L[see Eq. (C.22)] which implies that\nY−(L)∼L2n.\nWe now have all functions needed to compute\nW2\u000b\nin Eq. (D.1). Inserting the functions\nJ±(L) andY±(L) from Eqs. (D.7), (D.8) and (D.12) in Eq. (D.1), we finally obtain for n <1\n\nW2\u000b\n≃kv⟨W⟩\nDI−(L), (D.13)\n≃ ⟨W⟩2,[using Eq. (C.22)] . (D.14)\nAppendix D.3. Proof of Eq. (D.11)\nIn the remaining part of this section, we provide a derivation of Eq. (D.11) which was\ninstrumental in deriving\nW2\u000b\nforn < 1 case. Following Eq. (35), we can write the mean\nwork as\nW1(−uωL) =⟨W⟩ − A J −(uωL)−kv\nDI−(uωL). (D.15)\nForn <1,J−(uωL) is independent of Las indicated in Eq. (D.7) whereas I−(uωL) scales as\n∼unωnLn[shown in Eq. (C.21)]. Using this in Eq. (D.15), we obtain\nW1(−uωL)≃ ⟨W⟩ −k\nnunωnLn, (D.16)\n≃ ⟨W⟩(1−unωn). (D.17)\nThis proved the result in Eq. 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KLABUNDE\nGesellschaft f¨ ur Schwerionenforschung (GSI), Postfach 110541,\n6100 Darmstadt, W. Germany\nand\nM. REISER\nElectrical Engineering Department and Department of Physics and Astronomy,\nUniversity of Maryland, College Park, Maryland 20742, U.S.A.\n(Received October 17, 1983)\nThe behavior of K-V, waterbag, parabolic, conical and Gaussian distributions in periodic quadru-\npole channels is studied by particle simulations. It is found that all these different distributions\nexhibit the known K-V instabilities. But the action of the K-V type modes becomes more and\nmore damped in the order of the types of distributions quoted above. This damping is so strong\nfor the Gaussian distribution that the emittance growth factor after a large number of periods is\nconsiderably lower than in the case of an equivalent K-V distribution.\nIn addition, the non K-V distributions experience in only one period of the channel a rapid\ninitial emittance growth, which becomes very significant at high beam intensities. This growth is\nattributed to the homogenization of the space-charge density, resulting in a conversion of electric-\nfield energy into transverse kinetic and potential energy. Two simple analytical formulae are\nderived to estimate the upper and lower boundary values for this effect and are compared with\nthe results obtained from particle simulations.\nPublished in: Particle Accelerators 15, 47–65 (1984)\n1 INTRODUCTION\nRecent analytical and simulation studies by Hofmann, Laslett, Smith and Haber1\nshow that the well-known K-V (Kapchinskij-Vladimirskij) distribution2is subject\nto instabilities in a periodic focusing channel. The region of these instabilities can\nbe defined in terms of the phase advance of the particle oscillation per channel\nperiod without space charge, σ0, and with space charge, σ. From these studies,\nthe above authors concluded that one should impose the restriction σ0≤60◦to\navoid the dangerous second-order (envelope) instabilities (occurring for σ0>90◦)\n1arXiv:2401.12595v1  [physics.plasm-ph]  23 Jan 20242 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\nas well as the third-order mode ( σ0>60◦). In addition, they recommend to limit\nthe beam intensity such that σ≥24◦to avoid fourth- or higher-order instabilities.\nThey point out, however, that this latter restriction may be too conservative in view\nof the fact that simulation studies for σ0≤60◦indicated that these higher-order\nmodes saturate at low levels and that the rms emittance of a K-V distribution is\nnot affected for σvalues as low as 6◦.\nIn connection with the beam transport experiments at the University of Maryland3\nand at GSI,4we studied the behavior of K-V, waterbag, parabolic, conical and\nGaussian distributions by particle simulation with the modified PARMILA code\nat GSI. Our goal was to determine whether more “realistic” non-stationary initial\ndistributions are also affected by the instabilities found in a K-V beam (which\nrepresents the only stationary distribution in a periodic channel). Although each of\nthese distributions is still artificially produced according to the appropriate phase-\nspace density function, one expects that, with an increasing degree non-linearity,\none comes closer and closer to a “real” beam. Of the different types considered, the\ntruncated Gaussian distribution represents undoubtedly the closest approximation\nto a laboratory beam.\nIn a previous paper,5we reported the results of analytical and simulation work\nconcerning envelope oscillations and (second-order) instabilities of mismatched\nbeams. The particle simulation studies were restricted to K-V and Gaussian distri-\nbutions and showed that, for the cases considered ( σ0>90◦), the effect of instabil-\nities on an initial Gaussian distribution is even worse than for a K-V beam. These\nfindings support the results of Ref. 1 that the region σ0>90◦must be avoided in\nthe design and operation of transport channels for intense beams.\nIn the present paper we report the results of systematic studies with different\ninitial particle distributions at values of σ0= 90◦andσ0= 60◦. Of particular\ninterest in the σ0= 90◦case was the third-order instability. In contrast to our\nresults on the envelope modes, we found that the Gaussian distribution remains\nalmost unaffected by this mode. At the same time we discovered that the non-\nK-V distributions considered experience in the first period of the transport channel\na rapid emittance growth which occurs even in regions where no instabilities are\npredicted and which increases approximately proportional to σ−1. It will be shown\nthat the emittance growth can be attributed to an internal redistribution of the\nparticles toward a more uniform density profile, whereby field energy is converted\ninto transverse kinetic energy.EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 3\n2 PROPERTIES OF DIFFERENT PARTICLE DISTRIBUTIONS\nFOR CONTINUOUS BEAMS\n2.1 General Formulae\nTwo particle distributions are identical, if all their moments are the same. To\nallow a comparison of the results obtained with different distribution functions\ncarrying the same current, we use the concept of equivalent beams introduced by\nLapostolle6and Sacherer,7i.e., beams having the same first and second moments.\nSince we are dealing with continuous beams rather than bunched beams, the particle\ndistributions occupy only a 4-dimensional transverse phase space volume, usually\napproximated by a hyperellipsoid. For the sake of simplicity,this hyperellipsoid is\ntransformed into a hypersphere, i.e., the particle density f(r) only depends on the\nradial distance r; hence one has constant particle density on any hypersphere in\nthe phase space\nf(r= const .) = const ., r2=x2+y2+x′2+y′2.\nFor five different types of particle distributions, we have evaluated\n•the ratio of the marginal emittance to the rms emittance,\n•the particle (charge) density g(x, y) in real space,\n•the electric fields E(r) inside the beam,\n•the field energy Uassociated with the charge distribution.\nA given distribution function f(r) is called normalized if\nZa\n0f(r)dV= 1,\nwhere dV=dΩnrn−1drdenotes the volume element of the n-dimensional phase\nspace in spherical coordinates, and athe maximum value of r. In our case of a\ncontinuous beam, we have n= 4. If we assume the beam to be centered, the first\nmoments are zero and the second moments are given by8\n⟨r2⟩=1\nnZa\n0r2f(r)dV\u001eZa\n0f(r)dV.\nSince the particle density only depends on r, the integrals can be simplified:\n⟨r2⟩=1\nnZa\n0f(r)rn+1dr\u001eZa\n0f(r)rn−1dr.\nThe rms emittance of a distribution in the x, x′-plane is defined as6\nεrms= 4·h\n⟨x2⟩⟨x′2⟩ − ⟨xx′⟩2i1/2\n.4 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\nIf the emittance ellipse is upright, one has ⟨xx′⟩= 0 and therefore\nεrms=⟨ε⟩= 4·h\n⟨x2⟩⟨x′2⟩i1/2\n.\nThe factor 4 is chosen so that the rms emittance of a K-V distribution is equal to\nthe area Adivided by πof the minimum circumscribing phase-space ellipse that\nencloses all particles. This number A/π is usually called the maximum emittance\nεmaxof a distribution. The ratio εmax/εrmsis then given by\nεmax/εrms=a2/4⟨r2⟩=1\n4na2·Za\n0f(r)rn−1dr\u001eZa\n0f(r)rn+1dr.\nThe density in real space g(x, y) is obtained by integration over the coordinates\nx′andy′. In our case of a constant particle density on any hypersphere in the 4-\ndimensional transverse phase space, one obtains with the substitution r′2=x′2+y′2\ng(x, y) = 2 π·Z√\na2−x2−y2\n0f\u0000\nx2+y2+r′2\u0001\nr′dr′.\nThe electric field Ei(r) inside the beam is obtained from Poisson’s equation\n1\nr·d\ndr\u0000\nr·Φ′(r)\u0001\n=−q·g(r), q =I/ϵ0cβ\nEi(r) =q\nrZr\n0g(ρ)ρ dρ,\nwhereas the field in the region between beam and wall follows from Laplace’s equa-\ntion. With Ei(a) =Eo(a) one obtains for r≥a\nEo(r) =q\nrZa\n0g(ρ)ρ dρ=q/2πr.\nThe electric field energy Uof such a special charge distribution is finally given by\nU=πϵ0· Za\n0Ei2r dr+ZR\naEo2r dr!\n,\nwhere Rdenotes the wall radius.\nTable 2.1 gives the definitions of the five particle distribution functions that we\nused in our work and their appropriate quantities quoted here.EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 5\nTable 1: Definition and Properties of Different Phase-Space Distribution Functions for Continuous Beams\nDistribution\nFunctionDefinition f(r)\n(normalized)\nr2=x2+y2+x′2+y′2Ratio Marginal-\nto rms-Emittance\nϵmax/ϵrmsParticle Density\nin Real Space\nr2=x2+y2Electric Fields\ninside the Beam\nv0=I\n2πε0cβField Energy U\nu0=I2\n4πε0c2β2\nKapchinskij-\nVladimirskij (KV)1\n2π2a3δ(r−a) 11\nπa2v0r\na2\n=v0\nr/bracketleftbigg\n1−/parenleftbigg\n1−r2\na2/parenrightbigg/bracketrightbiggu0/parenleftbig\nlnR\na+1\n4/parenrightbig\n=u0/parenleftig\nlnR\n⟨x2⟩+1\n4−ln 2/parenrightig\nWaterbag (WB)2\nπ2a41.5 2\nπa2/parenleftbigg\n1−r2\na2/parenrightbigg v0r\na2/parenleftbigg\n2−r2\na2/parenrightbigg\n=\nv0\nr/bracketleftigg\n1−/parenleftbigg\n1−r2\na2/parenrightbigg2/bracketrightiggu0/parenleftbig\nlnR\na+11\n24/parenrightbig\n=u0/parenleftig\nlnR\n⟨x2⟩+11\n24−1\n2ln 6/parenrightig\nParabolic (PA) 6\nπ2a4/parenleftbigg\n1−r2\na2/parenrightbigg\n2 3\nπa2/parenleftbigg\n1−r2\na2/parenrightbigg2v0r\na2/parenleftbigg\n3−3r2\na2+r4\na4/parenrightbigg\n=\nv0\nr/bracketleftigg\n1−/parenleftbigg\n1−r2\na2/parenrightbigg3/bracketrightiggu0/parenleftbig\nlnR\na+73\n120/parenrightbig\n=u0/parenleftig\nlnR\n⟨x2⟩+73\n120−1\n2ln 8/parenrightig\nConical (CO)10\nπ2a4/parenleftig\n1−r\na/parenrightig\n2.1 10\n3πa2/parenleftbigg\n1−3r2\na2+2r3\na3/parenrightbigg\nv0r\na2/parenleftbigg10\n3−5r2\na2+8\n3r3\na3/parenrightbiggu0/parenleftbig\nlnR\na+4801\n7560/parenrightbig\n=u0/parenleftig\nlnR\n⟨x2⟩+4801\n7560−1\n2ln 8.4/parenrightig\nGaussian (GA) 1\n4π2⟨x2⟩2exp/parenleftbigg\n−r2\n2⟨x2⟩/parenrightbigg≃n2/4\nif truncated at\nn/radicalbig\n⟨x2⟩forn≥41\n2π⟨x2⟩exp/parenleftbigg\n−r2\n2⟨x2⟩/parenrightbiggv0\nr/bracketleftbigg\n1−exp/parenleftbigg\n−r2\n2⟨x2⟩/parenrightbigg/bracketrightbigg\nu0/parenleftig\nlnR\n⟨x2⟩+C\n2−ln 2/parenrightig\nC: Euler’s Constant6 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\n2.2 Differences of The Field Energy Associated with Different Distribution Func-\ntions\nAs can be seen in Table 2.1, each charge distribution function is associated with\na specific electric field energy. The minimum field energy of a charge distribution\napplies to the homogeneous charge density in real space associated with a K-V\ndistribution in phase space.\nAssuming that each distribution has the same second moments (or rms emit-\ntance), as stated earlier, one finds that the field energy Uincreases from the K-V\nto the Gauss distribution. One can write the difference between a particular dis-\ntribution and the K-V distribution in the form\n∆U=U−UKV=f·u0,\nwhere\nu0=I2/4πϵ0c2β2= 10−7·(I/β)2[ joules /meter ] .\nThe factors ffor the field-energy difference between each distribution and the\nequivalent K-V beam are listed in Table 2.2. As an example, one finds for the\nadditional field energy stored in a Gaussian distribution compared to an equivalent\nK-V beam\n∆U=UGA−UKV=u0·(1\n2C−1\n4) =u0f,\nwith\nf=1\n2C−1\n4≃0.0386,\nwherein C≃0.5772 denotes Euler’s constant.\nTABLE 1: f-Factors for the transition of different Charge-Distribution Functions\nto KV WB PA CO\nfrom\nWB 0.00560 0 — —\nPA 0.01176 0.00616 0 —\nCO 0.01408 0.00848 0.00232 0\nGA 0.03861 0.03301 0.02685 0.02452\nAs will be explained in the next section, we surmise that this additional field\nenergy is converted into particle kinetic and potential energy (and hence emittance\ngrowth) as the distribution tends to become more homogeneous. The average energy\nper particle of charge state ζavailable for this transformation is\n∆E= 30fIζ/β =fu0/N [ eV ] ,\nwhere N=I/ζe 0βcis the number of particles per unit length.EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 7\n2.3 Emittance Growth Due to a Change of the Particle Distribution\nSince the above defined non-K-V distributions are not stationary, they undergo a\ncomplete change during the transformation through a beam-transport line, leading\nto a more homogeneous charge distribution in real space as shown in Fig. 1 for\ndifferent initial distributions transformed through one period of the GSI quadrupole\nchannel, whose parameters are defined in Ref. 5. This process of a reorganization of\nthe particle phase-space distribution toward a more self-consistent, i.e., stationary,\none yields an increase of the quantities ⟨r2⟩and⟨r′2⟩, hence an increase of the rms\nemittance of the beam, resulting from the transformation of internal field energy\nassociated with the initial charge distribution to additional transverse kinetic and\npotential energy of the beam in the focusing channel.\n(a)σ→σ0\nIn order to obtain an analytical estimate of the emittance growth for low intensi-\nties, an rms-matched non-K-V distribution that changes its charge density in real\nspace towards a homogeneous one is treated as an equivalent K-V distribution that\nbecomes “heated up”. An increase of the total energy of a particle will the result in\nan increase of both the mean kinetic energy (proportional to ⟨x′2⟩), and the mean\npotential energy (proportional to ⟨x2⟩). The total energy Etotof a single particle\nin that harmonic-oscillator potential may be expressed in terms of the tune shift σ,\nthe length of one focusing period S, and transverse rms emittance ⟨ε⟩as\nEtot=1\n4γm0c2β2σ⟨ε⟩/S.\nHere we assume that the motion in the longitudinal direction can be relativistic,\n(hence γm0for the mass) while the transverse motion is non-relativistic. The mean\nenergy gain per unit charge due to a homogenization of the charge distribution as\nevaluated in the previous section is for an arbitrary transverse spatial plane\n∆E=1\n2fIe0/4πϵ0cβγ2,\nwhere the factor1\n2results from the reduction to one degree of freedom. If the\nsuperscript+denotes the appropriate quantities after the homogenization of the\ncharge distribution, we obtain for the total energies\n1\n4γm0c2β2σ⟨ε⟩/S+1\n2fIe0/4πϵ0cβγ2=1\n4γm0c2β2σ⟨ε⟩+/S.\nThus\nσ⟨ε⟩+ 2fIS/I 0β3γ3=σ+⟨ε⟩+,\nwhere I0= 4πϵ0m0c3/e0is the limiting current as defined in Ref. 5. Since σdoes not\nremain constant during reorganization of the particle distribution but increases to\nσ+, we have to express σ+in terms of ⟨ε⟩+. According to the smooth-approximation\ntheory of Ref. 9, this means\nσ+=σ0\u0010p\n1 +u+2−u+\u0011\n, u+=IS/I 0β3γ3σ0⟨ε⟩+.8 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\nFIGURE 1: Real-space ( x, y) projections of K-V, waterbag, parabolic, conical and Gaussian dis-\ntributions before and after the first cell of the GSI FODO channel at σ0= 60◦,σ= 15◦.\nSolving for the ratio ⟨ε⟩+/⟨ε⟩, one obtains after some algebraic transformations\n⟨ε⟩+\n⟨ε⟩=s\u0012σ\nσ0+ 2uf\u0013\u0012σ\nσ0+ 2uf+ 2u\u0013\nwith uas defined above but now pertaining to the initial rms emittance ⟨ε⟩.ucanEMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 9\neasily be expressed in terms of σ0andσas\n2u=σ0\nσ−σ\nσ0.\nThe formula for the relative growth of the transverse rms emittance then reduces\nto\n⟨ε⟩+\n⟨ε⟩=s\u0014\n1 +f·\u0012σ02\nσ2−1\u0013\u0015\u0014\n1 +f·\u0012\n1−σ2\nσ02\u0013\u0015\n.\n(b)σ→0\nFor non-K-V distributions, the effective potential well is no longer harmonic. Es-\npecially for high intensities, the effective potential becomes more and more “flat”\nin the central region of the beam, and has a sharp increase at the beam boundary.\nThis type of potential function may be called a “reflecting-wall potential” since\nthe particles are moving nearly force-free inside the beam and are reflected at its\nboundary. The additional internal electric field energy is then only transformed into\ntransverse kinetic energy, hence resulting in an increase of ⟨r′2⟩. In order to obtain\na formula to estimate these cases, we use again the model of a harmonic oscillator.\nBut now only the mean kinetic energy of a particle is increased, whereas its mean\npotential energy is assumed to remain constant. Since the mean transverse kinetic\nenergy of one particle is given by\nEk=1\n2γm0c2β2⟨x′2⟩,\nthe energy equation may then be written as\n1\n2γm0c2β2⟨x′2⟩+1\n2fIe0/4πϵ0cβγ2=1\n2γm0c2β2⟨x′2⟩+\n⟨x′2⟩+fI/I 0β3γ3=⟨x′2⟩+.\nAt a position where the emittance ellipse is upright ( ⟨xx′⟩= 0), we obtain\n\u0000\n⟨ε⟩+\u00012=⟨ε⟩2+ 16fI⟨x2⟩/I0β3γ3.\nWith the relation\n⟨x2⟩=1\n4⟨ε⟩S/σ,\nwhich is valid a particle moving in a harmonic oscillator potential, this finally leads\nto\n⟨ε⟩+\n⟨ε⟩=s\n1 + 2 f·\u0012σ02\nσ2−1\u0013\n.10 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\n3 SIMULATION STUDIES\n3.1 Third-Order Instability with Equivalent Beams of Different Particle Distribu-\ntions\nThe space-charge potential of a K-V distribution is a quadratic function of the\nspatial coordinates leading to linear forces. As has been shown previously, the\nK-V distribution is unstable against perturbations of this potential distributions\nin some specific regions depending on the external forces (defined by the phase\nadvance σ0) and the beam parameters (defined by σ). The type of perturbation\ncan be classified by the order of additional potential terms. For example, a potential\nproportional to xy2called a “third-order” perturbation potential. For σ0= 90◦,\nthe K-V distribution is unstable against this type of perturbation in the region\n38◦≤σ≤60◦. To evaluate whether and how this instability affects the non-K-\nV distributions defined above, we performed simulation studies for σ0= 90◦and\nσ= 41◦, hence near the maximum growth rate for third-order mode in a K-V beam.\nAll computer runs discussed in this paper were made with the parameters of the\nGSI quadrupole channel (see Ref. 5).\nFigure 2 shows the evolution of an initial K-V distribution. The specific instabil-\nity can easily be recognized by the three “arms” growing out of the initial elliptic\nparticle distribution. After about 50 periods the transverse rms emittance has\ngrown to a factor of 2 .2, remaining nearly constant during the following sections.\nFigure 3 shows the transformation of an equivalent initial waterbag distribution\nunder the same conditions ( σ0= 90◦,σ= 41◦). The resulting type of distortion is\nobviously the same, yet the growing of “arms” is less pronounced in that case. The\nrms emittance growth factor of 2 .2 after 100 periods shows no essential difference\nto that of the K-V distribution.\nFigure 4 shows the evolution of an initial parabolic transverse particle density\ndistribution , where the patterns of the emittance plots are similar to that of the\nwaterbag distribution, yielding a saturated transverse rms emittance growth of 2 .2\nafter 100 periods.\nThe evolution of the conical distribution, plotted in Fig. 5, shows a less pro-\nnounced third-order instability mode compared to the previous types of distribu-\ntions. The growth factor of the rms emittance is lower than in these cases reaching\na value of ≃1.8 after 100 periods.\nThe last density distribution investigated here is the Gaussian distribution trun-\ncated at a= 4p\n⟨x2⟩, whose evolution is shown in Fig. 6. A nearly constant slight\nincrease in rms emittance is obtained and no saturation of that growth can be rec-\nognized after 100 periods, where the growth factor of the rms emittance amounts\nto only ≃1.2.EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 11\n 1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \nPeriod 0 20 40 60 100\nFIGURE 2: Transformation of an initial K-V distribution through the GSI FODO channel at\nσ0= 90◦, σ= 41◦. (The x, x′andy, y′phase-space projections are represented in normal form.)12 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\n 1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \nPeriod 0 20 40 60 100\nFIGURE 3: Transformation of an initial waterbag distribution through the GSI FODO channel at\nσ0= 90◦, σ= 41◦. (The x, x′andy, y′phase-space projections are represented in normal form.)EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 13\n 1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \nPeriod 0 20 40 60 100\nFIGURE 4: Transformation of an initial parabolic distribution through the GSI FODO channel\natσ0= 90◦, σ= 41◦. (The x, x′andy, y′phase-space projections are represented in normal\nform.)14 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\n 1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \nPeriod 0 20 40 60 100\nFIGURE 5: Transformation of an initial conical distribution through the GSI FODO channel at\nσ0= 90◦, σ= 41◦. (The x, x′andy, y′phase-space projections are represented in normal form.)EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 15\n 1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’  1.0 0.0 -1.0\nx   1.0  0.0 -1.0\ny \n 1.0 0.0 -1.0\nx  *e -2 2.0 1.0 0.0-1.0-2.0\nx’ \n 1.0 0.0 -1.0\ny  *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \n2.0 1.0 0.0 -1.0 -2.0 *e -2 2.0 1.0 0.0-1.0-2.0\ny’ \nPeriod 0 20 40 60 100\nFIGURE 6: Transformation of an initial Gaussian distribution through the GSI FODO channel at\nσ0= 90◦, σ= 41◦. (The x, x′andy, y′phase-space projections are represented in normal form.)16 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\nCell Number\nFIGURE 7: Emittance growth factors versus the number of cells obtained from particle simu-\nlations for initial K-V, waterbag, parabolic, conical, and Gaussian distributions at σ0= 90◦,\nσ= 41◦.\nFigure 7 shows the increase of the transverse rms emittance versus the number\nof cells for our five types of distribution functions. The initial offset of the non-K-V\ndistributions is due to the homogenization effect of the particle density in real space\nmentioned in the previous chapter. According to the emittance-growth formulae\nderived in Section 2.3, with which the lower and the upper limits of this effect can\nbe estimated, this leads to an initial growth of 1 .3% to 2 .1% for the waterbag, 2 .7%\n– 4.4% for the parabolic, 3 .2% – 5 .2% for the conical, and 8 .7% – 13 .8% for the\nGaussian distribution. This agrees with the simulation results plotted in Fig. 7,\nsince the obtained initial emittance growth factors lie between these values.\nThe final saturation of the emittance growth leads to the value of ≃2.3, except for\nthe conical and Gaussian distributions, which continue to increase at lower levels.\nThe slope of the emittance growth decreases from the K-V towards the Gaussian\ndistribution, and therefore the number of cells, where a saturation can be observed,\nif there is any, increases. This is due to the increasing spread in the individual\nparticle tunes σ, which minimizes the effect of the resonance. This effect remains\nto be investigated.\nFor comparison, Fig. 8 shows the increase of the rms emittance versus the cell\nnumber of σ0= 60◦,σ= 25◦. No structure resonance is present under these\nconditions in agreement with Ref. 1. So the rms emittance growth factors obtained\nfrom numerical simulations are only due to the initial homogenization of the charge\ndensity, amounting to ≃1.7% for the waterbag, ≃4.0% for the parabolic, ≃5.0%\nfor the conical, and 12 .5% for the Gaussian distribution. These factors lie between\nthe appropriate emittance growth factors evaluated with our formulae (indicated\nby the horizontal lines in Fig. 8).EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 17\nCell Number\nFIGURE 8: Emittance growth factors versus the number of cells obtained from particle simu-\nlations for initial K-V, waterbag, parabolic, conical, and Gaussian distributions at σ0= 60◦,\nσ= 25◦.\n3.2 K-V and Gaussian Distributions at Increasing Intensities\nIn view of the fact that a Gaussian distribution comes perhaps closest to a real\nbeam, we investigated its behavior over a wide parameter range of decreasing σ\nvalues (i.e., increasing intensity) and compared it with a K-V distribution. We\nmade a large number of simulation runs for the two distributions at fixed values of\nσ0= 90◦andσ0= 60◦using as before the geometry of the GSI channel.\n(a)σ0= 90◦\nFigure 9 shows the rms emittance growth after 100 cells for a K-V and a Gaussian\ndistortion at σ0= 90◦and various values of σ. As can be seen, the two types\ndistribution functions behave quite differently:\n•The K-V distribution shows the expected emittance growth due to a third-order\ninstability in the region 60◦≥σ≥38◦. The sharp decrease of the emittance\ngrowth at σ≃38◦indicates that the third-order instability is no longer present.\nAsσdecreases, i.e., at higher intensities, the fourth- and higher-order modes take\neffect in the region σ <37◦.\n•The Gaussian distribution shows a monotonic increase in the emittance growth\nfactors as σdecreases. One part of the emittance growth can be attributed to the\nhomogenization effect discussed in Section 2 .3, since a sharp rise in rms emittance\noccurs within the first cell. The emittance growth as calculated from our formulae\nare indicated by the dashed line for the lower boundary value (Formula 1) and\nby the dotted line for upper one (Formula 2). It is seen that the actual emittance18 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\ngrowth is considerably larger than the formulae predict. This difference suggests\nthat the action of the resonances in the region σ <60◦is still present.\n(b)σ0= 60◦\nFigure 10 shows the rms emittance growth after 50 cells for a K-V and a Gaussian\ndistribution at σ0= 60◦. For the K-V distribution, no emittance growth occurs\nabove σ= 10◦. Below that value, a small growth is obtained, which is due to a\nfourth-order instability mode.\nFIGURE 9: Emittance growth factors versus σafter 100 cells for initial K-V and Gaussian dis-\ntributions at σ0= 90◦.\nFIGURE 10: Emittance growth factors versus σafter 50 cells for initial K-V and Gaussian dis-\ntributions at σ0= 60◦.EMITTANCE GROWTH OF DIFFERENT DISTRIBUTIONS 19\nThe emittance growth factors for Gaussian distribution obtained from particle\nsimulations lie exactly between those calculated from the emittance growth formu-\nlae derived above, as the comparison of dotted analytical curves with the simulation\nresults indicate. This suggests that the Gaussian distribution, like the K-V distri-\nbution is also stable with respect to fourth- and higher-order modes under these\nconditions, and that the emittance growth can be explained in terms of the homog-\nenization effect. This explanation is confirmed by the fact, that in all simulation\nruns at values of σgreater than 10◦no further emittance growth is observed after\nthe initial change of the charge distribution in the first period of the channel.\n4 CONCLUSIONS\nThe K-V distribution has been investigated by many authors, It has been found1,\nthat it shows a series of instabilities due to structure resonances in the case of a\nperiodic transport system. A structure resonance of the nth order can be avoided,\nifσ0≤180◦/n. These results have been confirmed by our simulation studies. The\nK-V distribution shows the typical third- and higher-order modes for σ0= 90◦.\nand the fourth- and higher-order instabilities for σ0= 60◦, leading in both cases to\na growth of the rms emittance. The simulation studies have been extended to four\nnon-K-V distributions: waterbag, parabolic, conical and Gaussian. Two types of\nadditional effects have been found:\n1. A damping of the structure resonances occurs, leading to a smaller slope in emit-\ntance growth. The action of the damping increases from the waterbag, parabolic,\nconical toward the Gaussian distribution.\n2. A homogenization of the particle density in real space takes place within the first\ncell. This leads to an initial growth of the rms emittance due to transformation\nof field energy into kinetic and potential beam energy.\nThe effects of third- and higher-order instabilities are seen in the simulations\nof the non-K-V distribution, yet the typical patterns (a specific number of arms\ngrowing out of two dimensional x, x′- and y, y′-phase space projections) become\nmore and more smoothed out. This effect increases from the waterbag towards the\nGaussian distribution, where these patterns are no longer recognized.\nThe amount of the initial growth depends on the type of the initial distribution\nand also increases from the waterbag over the parabolic and conical towards the\nGaussian distribution. For each distribution we calculated a factor f, which defines\nthe difference between the field energy of the distribution and that of a K-V beam.\nAssuming first that this difference is transformed into kinetic and potential energy\nand secondly into kinetic energy only via homogenization of the charge density in\nreal space, one obtains the estimate for the initial emittance growth\ns\u0014\n1 +f·\u0012σ02\nσ2−1\u0013\u0015\u0014\n1 +f·\u0012\n1−σ2\nσ02\u0013\u0015\n<⟨ε⟩+\n⟨ε⟩<s\n1 + 2 f·\u0012σ02\nσ2−1\u0013\n.20 J. STRUCKMEIER, J. KLABUNDE AND M. REISER\nThe results obtained with these formulae have been found in good agreement with\nthose obtained by the simulation studies for σ0= 60◦where no K-V instabilities are\nobserved and where the “homogenization” is the only source of emittance growth. In\nthe case of σ0= 90◦, the emittance growth factors are greater due to the additional\naction of structure resonances. Nevertheless it has been shown that the types of not\nself-consistent particle distributions treated in this paper show a specific increase of\nthe rms emittance due to the internal matching of the charges to the linear external\nforces, even if they are perfectly rms-matched.\nACKNOWLEDGEMENTS\nThis work was performed while one of us (M. R.) was on sabbatical leave from the\nUniversity of Maryland at GSI, W. Germany. The support from the Alexander von\nHumboldt Foundation and the hospitality of the GSI laboratory during this leave\nare gratefully acknowledged. We would also like to thank W. Lysenko for several\nhelpful comments.\nREFERENCES\n1. I. Hofmann, L. J. Laslett, L. Smith, and I. Haber, Particle Accelerators ,13, 145 (1983).\n2. I. M. Kapchinskij and V. V. Vladimirskij, Proc. Int. Conf. on High Energy Accelerators,\nCERN, Geneva, 1959, p. 274.\n3. J. D. Lawson, E. Chojnacki, P. Loschialpo, W. Namkung, C. R. Prior, T. C. Randle, D.\nReading, and M. Reiser, IEEE Trans. Nucl. Sci., 30, 2537 (1983).\n4. J. Klabunde, M. Reiser, A. Sch¨ onlein, P. Sp¨ adtke, and J. Struckmeier, IEEE Trans. Nucl.\nSci., 30, 2543 (1983).\n5. J. Struckmeier and M. Reiser, “Theoretical Studies of Envelope Oscillations and Instabilities\nof Mismatched Intense Charged Particle Beams in Periodic Focusing Channels”, Particle\nAccelerators ,14, 227 (1984).\n6. P. M. Lapostolle, IEEE Trans. Nucl. Sci., 18, 1101 (1971).\n7. F. J. Sacherer, IEEE Trans. Nucl. Sci., 18, 1105 (1971).\n8. B. Bru and M. Weiss, “Some Useful Formulae Concerning RMS-Values of Various Density\nDistributions”, CERN/MPS/LIN 74-1, Appendix I (1974).\n9. M. Reiser, Particle Accelerators ,8, 167 (1978)."    },    {        "title": "2401.12649v1.Space_time_unfitted_finite_elements_on_moving_explicit_geometry_representations.pdf",        "content": "SPACE-TIME UNFITTED FINITE ELEMENTS ON MOVING EXPLICIT\nGEOMETRY REPRESENTATIONS\nSANTIAGO BADIA1,2, PERE A. MARTORELL3,∗, AND FRANCESC VERDUGO4\nAbstract. This work proposes a novel variational approximation of partial differential equations\non moving geometries determined by explicit boundary representations. The benefits of the\nproposedformulationaretheabilitytohandlelargedisplacementsofexplicitlyrepresenteddomain\nboundaries without generating body-fitted meshes and remeshing techniques. For the space\ndiscretization, we use a background mesh and an unfitted method that relies on integration on\ncut cellsonly. We perform this intersectionby using clipping algorithms. To dealwith the mesh\nmovement,wepullbacktheequationstoareferenceconfiguration(thespatialmeshattheinitial\ntimeslabtimesthetimeinterval)thatisconstantintime. Thisway,thegeometricalintersection\nalgorithm is only required in 3D, another key property of the proposed scheme. At the end of the\ntime slab, we compute the deformed mesh, intersect the deformed boundary with the background\nmesh, and consider an exact transfer operator between meshes to compute jump terms in the time\ndiscontinuous Galerkin integration. The transfer is also computed using geometrical intersection\nalgorithms. Wedemonstratetheapplicabilityofthemethodtofluidproblemsaroundrotating(2D\nand 3D) geometries described by oriented boundary meshes. We also provide a set of numerical\nexperiments that show the optimal convergence of the method.\nKeywords : Unfitted finite elements, embedded finite elements, computer-aided design, compu-\ntationalgeometry,immersedboundaries,boundaryrepresentations,space-timediscretizations,\nlarge displacements.\n1.Introduction\nSpace-timeformulationsforfiniteelementmethods( FEMs)arevaluabletechniquesforsolving\ntransientnumericalproblems. Unlikestandardtimesteppingschemes,whichemploydifferent\ndiscretizationsforspaceandtime,space-timeformulationsdiscretizetheproblemsimultaneously\nin both space and time. However, generating 4D space-time body-fitted meshes becomes\nextremely challenging on moving complex geometries. There are no general-purpose tools\nto generate 4D meshes. Besides, the re-meshing process due to moving domains introduces\nprojection errors when transferring between meshes.\nThebottleneckofmeshgenerationcanbeaddressedbyemployingunfitted FEMs. Unfitted\nFEMs,alsoknownasembeddedorimmersed FEMs,offersasolutionthateliminatestheneed\nfor body-fitted mesh generation, relying instead on a simple background grid, such as a uniform\noradaptiveCartesiangrid. Forlarge-scalesimulationsondistributed-memoryplatforms,one\ncan replace expensive (both in terms of computational time and memory) unstructured mesh\npartitioning[ 1]byefficienttree-meshpartitionerswithloadbalancing[ 2,3]. Unfitted FEMshave\n1School of Mathematics, Monash University, Clayton, Victoria, 3800, Australia.\n2Centre Internacional de Mètodes Numèrics a l’Enginyeria, Campus Nord, 08034, Barcelona, Spain.\n3DepartmentofCivilandEnvironmentalEngineering,UniversitatPolitècnicadeCatalunya,EdificiC1,Campus\nNord UPC, Jordi Girona 1-3, 08034 Barcelona, Spain.\n4Department of Computer Science, Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081 HV Amsterdam, The\nNetherlands.\n∗Corresponding author.\nE-mails: santiago.badia@monash.edu (SB), pmartorell@cimne.upc.edu (PM),\nf.verdugo.rojano@vu.nl (FV) .\n1arXiv:2401.12649v1  [cs.CE]  23 Jan 2024SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 2\ngainedincreasingpopularityinvariousapplications,includingfluid-structureinteraction( FSI)\n[4,5,6], fracture mechanics [ 7,8], additive manufacturing [ 9,10], and stochastic geometry\nproblems[ 11]. Traditionally,unfittedproblemsutilizelevelsetstodescribegeometries. However,\nrecentwork[ 12,13]hasextendedunfitteddiscretizationstosimulategeometriesdescribedby\ncomputer-aided design (CAD) models, i.e., explicit geometry representations.\nThesmallcutcellproblem,commonlydiscussedintheliterature[ 14],isasignificantlimitation\nofunfitted FEMs. Theintersectionbetweenthephysicaldomainandthebackgroundcellscan\nbecome arbitrarily small, thereby giving rise to ill-conditioning issues. While several authors\nhave attempted to address this problem, only a few techniques have demonstrated robustness and\noptimal convergence. The ghostpenalty methods [ 15], employed withinthe so-called CutFEM\nframework [ 16], represent one such approach. Alternatively, cell agglomeration techniques\nhave emerged as viable options to ensure robustness in the presence of cut cells, naturally\nappliedondiscontinuousGalerkin( DG)methods[ 17]. Extensionstothe C0Lagrangianfinite\nelement ( FE) have been introduced in [ 18], while mixed methods have been explored in [ 19],\nwhere the aggregated finite element method ( AgFEM) term was coined. AgFEMexhibits good\nnumerical qualities, including stability, bounds on condition numbers, optimal convergence, and\ncontinuity concerning data. Distributed implementations have been exploited in [ 20,21], while\nAgFEMhasalsobeenextendedto ℎ-adaptivemeshes[ 22]andhigher-order FE[23]. In[24],a\nnovel technique combining ghost penalty methods with AgFEMwas proposed, offering reduced\nsensitivity to stabilization parameters compared to standard ghost methods.\nIn the body-fitted case, one can consider variational space-time formulations [ 25,26,27]\nto avoid the generation of 4D meshes. In these formulations, a body-fitted mesh is extruded\nusing a geometric mapping technique to represent the temporal evolution of the boundary\ndisplacements. However,whendealingwithlargedisplacements,thegeometricmappingprocess\nbecomes ill-posed, often necessitating re-meshing. Similar challenges arise in body-fitted\narbitrary Lagrangian Eulerian ( ALE) [28,29] schemes, where frequent re-meshing becomes\nnecessary due to large changes in topology. Consequently, none of these formulations are robust\nwhen confronted with large displacements. Furthermore, the re-meshing process introduces\nbottlenecks withinthesimulationprocedureandincursinprojectionerrorswhentransferring\nbetween meshes.\nDespite the notable advantages of unfitted FEMs, its application in the space-time domain\n[30,31] is currently limited to implicit level set geometrical representations (in which 4D\ngeometrical treatment is attainable), 2D explicit representations (in which 3D geometrical tools,\ne.g., in [12], can readily be used for space-time) or piecewise constant approximations of the\ngeometry in time [ 32]. This limitation stems from the complexity involved in developing 4D\ngeometrical tools for unfitted methods.\nThe presentwork introducesa novel variationalspace-time formulationfor unfitted FEMs that\neliminatestheneedforcomplex4Dgeometricalalgorithms. Instead,ourformulationrelieson\ntime-extruded FEspaces. Furthermore,weincorporatean exactinter-slabtransfermechanism\nthrough an intersection algorithm, which removes projection errors between time slabs. The\nmain contributions of our work are as follows:\n•We develop space-time formulations for unfitted FEMs on moving 3D geometries\ndescribed explicitly.\n•We propose to pull back the problem to a reference configuration that is constant in each\ntime slab to avoid 4D geometrical algorithms.\n•We propose and implement an exact time-slab mechanism transfer that relies on 3D\nintersection algorithms.\n•We compute the solution of transient problems on complex unfitted domains with large\ndisplacements.SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 3\nTheoutlineofthispaperisasfollows. InSection2,weintroducethegeometrydescription\nand the space-time FEspaces employed in this work. In Section 3, we present the variational\nformulationthroughamodelproblem. Wealsodefinetheintegrationmeasuresandtheinter-slab\ntransfermechanism. InSection4,wedescribetheintersectionalgorithmfortimeslabtransfer.\nThen, in Section 5, we present numerical results for space and time convergence, condition\nnumbertests,andanumericalexampleofamovingdomain. Finally,inSection6,wepresentour\nconclusions and future work.\n2.Space time unfitted finite element method\nIn this section, we introduce the geometry description by utilizing a geometrical map defined\nin a space-time FEspace. We also define the unfitted space-time FEspaces that are necessary to\nsupport our proposed formulation.\n2.1.Geometry description for movingdomains. Let us consideran initial Lipschitz domain\nΩ0in a description suitable for unfitted FEM. Following the methodology described in [ 12], we\nassume thatΩ0is represented by a parameterized surface, e.g., a stereolithography ( STL) mesh\nor aCADmodel. For simplicity, we consider an oriented surface mesh B0\nℎwhere the interior\ncorresponds to the domain Ω0. Let[0,𝑇]be the time interval of interest.\nWe consider the coordinates of the mesh nodes of Bℎto be time-depedendent. Consequently,\nBℎ(𝑡)and its corresponding interior Ω(𝑡). This variation of coordinates can be represented by a\nmap𝐷𝐷𝐷:Bℎ(0)×[ 0,𝑇]→Bℎ(𝑡)such that𝐷𝐷𝐷(·,0)is the identity.\nNext, we define the space-time domain 𝑄={𝑥∈Ω(𝑡):𝑡∈[0,𝑇]}, in accordance with the\nnomenclaturein[ 30],seeFig.1. Additionally,wepartitiontheboundaryofthedomain 𝜕Ω(𝑡)\nintoDirichletandNeumannboundaries, 𝜕Ω𝐷(𝑡)and𝜕Ω𝑁(𝑡),resp. Thispartitionissuchthat\n𝜕Ω(𝑡)=𝜕Ω𝐷(𝑡)∪𝜕Ω𝑁(𝑡)and𝜕Ω𝐷(𝑡)∩𝜕Ω𝑁(𝑡)=∅. Thespace-time boundaries are defined\nas𝜕𝑄∗\u0011Ð\n𝑡∈[0,𝑇]𝜕Ω∗(𝑡)×{𝑡}for∗∈{𝑁,𝐷}. Consequently, the boundary of 𝑄is given by\n𝜕𝑄\u0011Ω(0)∪Ω(𝑇)∪𝜕𝑄𝐷∪𝜕𝑄𝑁.\nΩ(𝑡)Ωart\nΩ(0)𝑡𝑛+1𝑄𝑛\nart𝑄𝑄art\n𝑄𝑛𝜕𝑄𝑛\n𝑄𝑛\nart𝑄𝑛\nart\n𝑡𝑛\n¯T𝑛\n𝑥𝑦𝑡\n𝑦\n𝑥\nFigure 1. Representation of the space-time domain 𝑄embedded within an\nartificialspace-timedomain 𝑄art. Thespacialdomains Ω(𝑡)andΩartin2Dare\nextruded in the time dimension. The solution is computed in each time-slab\n𝐽𝑛=(𝑡𝑛,𝑡𝑛+1)on the space-time domain 𝑄𝑛, which is embedded in 𝑄art.SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 4\nIn addition, we introduce an artificial spatial domain Ωart, such thatΩ(𝑡) ⊂Ωartfor all\n𝑡∈[0,𝑇]. This artificial domain can be a simple geometric shape, such as a bounding box, that\ncan conveniently be meshed with a regular grid, such as a Cartesian mesh. Subsequently, we\ndefine the artificial space-time domain 𝑄art\u0011Ωart×[0,𝑇], such that𝑄⊂𝑄art.\nWedefineapartitionintime, {𝐽𝑛}𝑁\n𝑛=1for[0,𝑇],inwhichthetimedomainisdividedintotime\nslabs. The time slabs are defined as 𝐽𝑛=(𝑡𝑛,𝑡𝑛+1),1< 𝑛 < 𝑁 where𝑡𝑛< 𝑡𝑛+1,∀𝑛∈1,...,𝑁.\nWithineachtimeslab,thetime-stepsizeisdefinedas 𝜏𝑛=𝑡𝑛+1−𝑡𝑛andthedomaintime-step\nsize is defined as 𝜏=max𝑛=1,...,𝑁𝜏𝑛. In each time-slab, we define an artificial space-time\ndomain𝑄𝑛\nart\u0011Ωart×𝐽𝑛. In a time slab, the space-time domain is determined by 𝑄𝑛\u0011𝑄𝑛\nart∩𝑄.\nSimilarly, the space-time boundary is denoted as 𝜕𝑄𝑛\n{𝐷,𝑁}. We also introduce the notation\nΩ𝑛\u0011Ω(𝑡𝑛), 𝑛=1,...,𝑁+1.\nFinally, wedefinean undeformedspace-timedomain ateachtime slabas ˆ𝑄𝑛\u0011Ω𝑛×𝐽𝑛,see\nFig.2. Here,wenotethat Ω𝑛istheinitialspatialdomainofthetimeslab 𝐽𝑛=(𝑡𝑛,𝑡𝑛+1). Inorder\nto define the reference configuration, we require a map 𝜑𝜑𝜑𝑛\nℎ:ˆ𝑄𝑛→𝑄𝑛that must satisfy\n𝜑𝜑𝜑𝑛\nℎ(𝜕Ω(𝑡))=𝐷𝐷𝐷(𝑡)(B0\nℎ),\ni.e., respect the boundary position determined by 𝐷𝐷𝐷.\nˆ𝑄𝑛𝑄𝑛\n𝑄𝑛\nart\n𝝋𝑛\nℎ\n𝑄𝑛\nart\nFigure 2. Representationofthedeformationmap 𝜑𝜑𝜑𝑛\nℎresultingfromextending\nthevariationofthesurfacemap 𝐷𝐷𝐷inthetimeslabdomain 𝑄𝑛(orinthetimeslab\nartifical domain 𝑄𝑛\nart). The undeformed configuration ˆ𝑄𝑛will be used for the FE\nanalysis.\nLet¯T𝑛\nartrepresentasimplepartitionof Ωartattime𝑡𝑛. Eventhough Ωartdoesnotevolveintime,\nthe partition ¯T𝑛\nartcan differ across time slabs, e.g., when using adaptive mesh refinement ( AMR)\ntechniques. We define the partition of 𝑄𝑛\nartas a Cartesian product of ¯T𝑛\nartand𝐽𝑛. Specifically,\nT𝑛\nart\u0011{¯𝐾×𝐽𝑛:¯𝐾∈¯T𝑛\nart}. Wecanclassifythespatialartificialcells 𝐾∈¯T𝑛\nartasinterior,cut,and\nexterior depending on the relative position concerning the domain boundary 𝜕Ω𝑛. Since there is\na one-to-one map between cells in ¯T𝑛\nartandT𝑛\nart, it already provides a classification of the cells in\nthe space-time artificial mesh T𝑛\nart, which corresponds to the in-out classification for the extruded\nspace-time domain boundary 𝜕ˆ𝑄𝑛.\nInthecontextofunfitted FEM,weareinterestedinsolvingpartialdifferentialequations( PDEs)\nusingtheactiveportionof T𝑛\nart,i.e.,thecellstouching Ω𝑛at𝑡𝑛. Thus,weremovetheexteriorcells\n¯T𝑛\nout\u0011{¯𝐾∈¯T𝑛\nart:¯𝐾∩Ω𝑛=∅}fromtheartificialspatialpartition ¯T𝑛\nart,leadingtoanactivespatial\npartition ¯T𝑛\u0011¯T𝑛\nart\\¯T𝑛\nout. From the spatial active partition, we compute the active space-time\npartitionT𝑛\u0011{¯𝐾×𝐽𝑛:¯𝐾∈¯T𝑛}by extrusion.\n2.2.Space-time finite element spaces. In this section, we define space-time unfitted FEspaces\nforthediscretizationofthemodelproblemonmovingdomains. However,amaindifferencewith\nrespectto [ 30], theoriginalproblem on 𝑄𝑛willbe re-statein theundeformeddomain ˆ𝑄𝑛. Asa\nresult, inthis section, we onlyneed a constructionthat works ontime-extruded domains. Thus,SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 5\nthe active meshT𝑛is the one that intersects Ω𝑛at𝑡𝑛, as defined above. We note that T𝑛may\nchange across time slabs in evolving domains.\nFirstly,wedefinethe FEspaceX𝑛\nℎinthespatialdomain. Then,foreachdegreeoffreedom( DOF)\ninXℎ,weintroducea1D FEbasisY𝑛\n𝜏. Thus,weconstructthespace-time FEspacecomposedas\nthe tensor product of these two spaces, V𝑛\nℎ\u0011X𝑛\nℎ⊗Y𝑛\n𝜏.\nIneachspace-timecell 𝑇𝑛=¯𝑇𝑛×𝐽𝑛∈T𝑛,wecandefinethe FElocalinterpolation,which\nconsists of tensor product DOFs Σ\u0011Σ𝑋⊗Σ𝑌and shape functions Φ\u0011Φ𝑋⊗Φ𝑌. Specifically,\nany shape function 𝜙∈Φcan be expressed as 𝜙(𝛼𝑋,𝛼𝑌)(𝑥𝑥𝑥,𝑡)=𝜙𝛼𝑋\n𝑋(𝑥𝑥𝑥)⊗𝜙𝛼𝑌\n𝑌(𝑡).\nTo address the ill-conditioning issues in unfitted FEM, we will utilize as a model example the\nAgFEM,eventhoughtheproposednumericalframeworkcanreadilybeusedforothertechniques,\ne.g., ghost penalty stabilization (see [ 30] for space-time unfitted formulations). The AgFEM\neliminatesproblematicDOFsbyconstrainingthemtowell-posedDOFsusingadiscreteextension\noperatorE. We define the spatial extension operator ¯E:¯Vℎ,in↦→¯Vℎbetween the the spatial\ninterior ¯Vℎ,inandactive ¯VℎFEspaces. Thespatialaggregatedfiniteelement( AgFE)spaceis\ndefined as ¯Vℎ,ag\u0011¯E(¯Vℎ,in)\nWe define a slab-wise space-time discrete operator between V𝑛\nℎ,inandV𝑛\nℎ. In each time slab,\nweaggregatethecutcellsinthespatialactivemesh ¯T𝑛usingtheaggregationtechniquesfrom\ntheAgFEM. For a given time slab 𝐽𝑛, the space-time extension operator E𝑛:V𝑛\nℎ,in↦→V𝑛\nℎis\ndefined as\nE𝑛(V𝑛\nℎ,in)=E𝑛(¯V𝑛\nℎ,in⊗Y𝑛\n𝜏)=¯E𝑛(¯V𝑛\nℎ,in)⊗Y𝑛\n𝜏,\nwhere ¯E𝑛:¯V𝑛\nℎ,in↦→¯V𝑛\nℎrepresentsthespatialextensionoperator,and Y𝑛\n𝜏theFEspaceintime\nat the time slab 𝐽𝑛. We define the space-time AgFEspace on the time-slab. Now, we can define\nthe global AgFEspace asVℎ,ag\u0011V1\nℎ,ag×···×V𝑁\nℎ,ag. Note that no continuity is imposed across\ntime slabs, i.e., the discrete time space is discontinuous.\nIn ghost penalty formulations, one can readily use the non-aggregated spaces on the active\nmesh and add stabilization terms (see, e.g., [ 33]) to make the problem robust with respect to cut\nlocations. Thus, the space-time FE spaces to be used in this case is simply V𝑛\nℎ.\n2.3.Extensionofthedeformationmap. Inthecaseinwhichthedomainisrepresentedintime\nby its explicit boundary representation and its boundary displacement 𝐷𝐷𝐷(𝑡), we must extend\n𝐷𝐷𝐷(𝑡)definedonB(𝑡)tothespace-timedomain ˆ𝑄𝑛ofeachtimeslab 𝐽𝑛. Forthispurpose, we\nsolve the following linear elasticity problem in ˆ𝑄𝑛, even though other extension operators could\nbe considered. Find ˆ𝑢𝑢𝑢∈R𝑑such that,\n \nˆ∇𝑥·𝜎𝜎𝜎(ˆ𝑢𝑢𝑢)=0 in ˆ𝑄𝑛,\nˆ𝑢𝑢𝑢=ˆ𝑢𝑢𝑢𝑛\n𝐷on𝜕ˆ𝑄𝑛\n𝐷,\nˆ𝑢𝑢𝑢=0 on 𝜕ˆ𝑄𝑛\n𝐷0,\nn𝑥·𝜎𝜎𝜎(ˆ𝑢𝑢𝑢)=0 on𝜕ˆ𝑄𝑛\n𝑁.\nwhere𝜎𝜎𝜎isthestresstensor, ˆ∇𝑥isthespatialgradient, n𝑥isthespatialcomponentofoutward\nnormal vector to 𝜕ˆ𝑄𝑛, and\nˆ𝑢𝑢𝑢𝑛\n𝐷(𝑡)=(𝐷𝐷𝐷(𝑡)−𝐷𝐷𝐷(𝑡𝑛))◦(𝐷𝐷𝐷(𝑡𝑛))−1\nis the Dirichlet boundary condition on 𝜕ˆ𝑄𝑛\n𝐷=Bℎ(𝑡𝑛)×𝐽𝑛, which imposes the deformation near\nthe unfitted boundary (see Fig. 3). On the boundary of the artificial domain and at the initial\nspatial domain, 𝜕ˆ𝑄𝑛\n𝐷0=𝜕Ωart×𝐽𝑛∪Ω𝑛×{𝑡𝑛}, the deformation is fixed to zero. The Neumann\nboundary is defined on 𝜕ˆ𝑄𝑛\n𝑁=Ω𝑛×{𝑡𝑛+1}.\nThis problem is approximated with a continuous Galerkin ( CG) method and weak imposition\nof the Dirichlet boundary conditions on the unfitted boundary 𝜕ˆ𝑄𝐷using Nitsche’s method [ 34].SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 6\n𝑫(𝑡𝑛)\n𝑫(𝑡𝑛+1)\n𝑡𝑛+1𝑡1=0 𝑡𝑛\n𝝋𝑛\nℎ(𝑡𝑛+1)Bℎ(0)𝜕Ω𝑛+1\n𝐼 𝜕Ω𝑛\n𝐼\nΩ(0)\n Ω𝑛\n ˜Ω𝑛+1\nFigure 3. Representation of the deformation map 𝐷𝐷𝐷and the extended map 𝜑𝜑𝜑𝑛\nℎin\nthetimeslab 𝐽𝑛=(𝑡𝑛−1,𝑡𝑛). Thisexamplerepresentstheexternaldomain. The\nspatial deformation map 𝐷𝐷𝐷(𝑡):Bℎ(0)↦→Bℎ(𝑡), 𝑡∈[0,𝑇], is defined on the\nsurfacemesh,andtheextendedmap 𝜑𝜑𝜑𝑛\nℎ(𝑡):Ω𝑛↦→˜Ω(𝑡), 𝑡∈𝐽𝑛,isdefinedonthe\nspatialdomain. At 𝑡𝑛+1,thedescriptionoftheinnerboundary 𝜕Ω𝑛+1\n𝐼maydiffer\nacross maps: 𝜑𝜑𝜑𝑛\nℎ(𝑡𝑛+1)(𝜕Ω𝑛\n𝐼)≈𝐷𝐷𝐷(𝑡𝑛+1)(Bℎ(0)), where𝜕Ω𝑛\n𝐼=𝐷𝐷𝐷(𝑡𝑛)(Bℎ(0)).\nThe approximation error decreases with the spatial discretization size ℎ.\nIn the weak formulation, we find 𝑢∈V𝑛\nℎ,agsuch that,\n𝑎(ˆ𝑢𝑢𝑢,ˆ𝑣𝑣𝑣)=𝑙(ˆ𝑣𝑣𝑣),∀ˆ𝑣𝑣𝑣∈V𝑛\nℎ,ag\nwhere the bilinear form and the linear form are defined as,\n𝑎(ˆ𝑢𝑢𝑢,ˆ𝑣𝑣𝑣)=∫\nˆ𝑄𝑛𝜀𝜀𝜀(ˆ𝑣𝑣𝑣):𝜎𝜎𝜎(𝜀𝜀𝜀(ˆ𝑢𝑢𝑢))𝑑ˆ𝑥𝑥𝑥𝑑𝑡+∫\n𝜕ˆ𝑄𝐷(𝜏𝐷ˆ𝑣𝑣𝑣·ˆ𝑢𝑢𝑢−𝑣·(n𝑥·𝜎𝜎𝜎(𝜀𝜀𝜀(ˆ𝑢𝑢𝑢))))𝑑ˆ𝑥𝑥𝑥𝑑𝑡,\nand\n𝑙(ˆ𝑣𝑣𝑣)=∫\n𝜕ˆ𝑄𝐷(ˆ𝑣𝑣𝑣·ˆ𝑢𝑢𝑢𝐷−n𝑥·𝜎𝜎𝜎(𝜀𝜀𝜀(ˆ𝑣𝑣𝑣))·ˆ𝑢𝑢𝑢𝐷)𝑑ˆ𝑥𝑥𝑥𝑑𝑡,\nresp., where 𝜀𝜀𝜀is the symmetric (spatial) gradient operator.\nFrom the approximated solution ˆ𝑢𝑢𝑢𝑛\nℎ∈V𝑛\nℎ,agwe extract the map 𝜑𝜑𝜑𝑛\nℎ(ˆ𝑥𝑥𝑥,𝑡)=ˆ𝑥𝑥𝑥+ˆ𝑢𝑢𝑢𝑛\nℎ(ˆ𝑥𝑥𝑥,𝑡). One\ncan alternatively use a ghost penalty stabilization to solve the elasticity problem. Since the\ndeformation is imposed weakly, the deformation ˆ𝑢𝑢𝑢𝑛\nℎis not equal to ˆ𝑢𝑢𝑢𝑛\n𝐷onBℎ(𝑡). Thus, the\ndeformed space-time domain 𝑄𝑛is approximated by ˜𝑄𝑛=𝜑𝜑𝜑𝑛\nℎ(ˆ𝑄𝑛). In any case, the geometrical\nerror is expected to decrease with the mesh size ℎ. For given time 𝑡∈𝐽𝑛, we can extract the\nspatial map 𝜑𝜑𝜑𝑛\nℎ(𝑡):Ω𝑛↦→˜Ω(𝑡)(see Fig. 3). Here, the spatial domain is also approximated,\nΩ(𝑡)≈˜Ω(𝑡).\nWeassumethatthecomputedmapisone-to-one,i.e., det(∇𝜑𝜑𝜑𝑛\nℎ)>0atalltimes. Assuming\nthat𝐷𝐷𝐷is such that it admits an extension that is diffeomorphic, this requirement can be attained\nforsmallenoughtimestepsinthetime-discreteproblem. OnecanalsomovebacktotheCartesian\nmesh after several time steps, as soon as the map remains bijective.\n2.4.Extended active mesh. As discussed above, in the case in which the deformation map has\nto be computed, 𝜑𝜑𝜑𝑛\nℎis not equal to 𝐷𝐷𝐷onBℎ(𝑡), since unfitted methods usually make use of weak\nimposition of boundary conditions.\nAt the end of the time slab 𝐽𝑛, we can define the solution in the approximated domain ˜𝑄𝑛as\n𝑢𝑛+1,−\nℎ(𝑥𝑥𝑥)=ˆ𝑢𝑛\nℎ(𝜑𝜑𝜑𝑛\nℎ(𝑡𝑛+1)−1(𝑥𝑥𝑥),𝑡𝑛+1),∀𝑥𝑥𝑥∈˜Ω𝑛+1,SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 7\nwhich isa functiondefined on theundeformed domain Ω𝑛; since weintegrate theformson ˆ𝑄𝑛,\nthe inverse map is never computed in practice. On the other side, with the method proposed\nbelow, we have to compute an inter-slab integral on Ω𝑛+1that involves 𝑢𝑛+1,−\nℎ.\nInorderfor𝑢𝑛+1,−\nℎtobedefinedon Ω𝑛+1,weproceedasfollows. First,weextendtheactive\nmesh by ¯T𝑛\next=¯T𝑛∪{𝐾∈¯T𝑛\nart:𝜑𝜑𝜑𝑛(𝑡𝑛+1)(𝐾)∩Ω𝑛+1≠∅}. Assuming that the geometrical map\nis one-to-one, the inverse 𝜑𝜑𝜑𝑛\nℎ(𝑡𝑛+1)−1is well-defined on Ω𝑛+1.\nTo accommodate a FEspace ¯V𝑛\nℎ,extin¯T𝑛\next, one can consider the modification of the extension\noperator. Inthe AgFEMframework,wecansimplyredefinetheextensionoperator ¯E𝑛\next:¯V𝑛\nℎ,in↦→\n¯V𝑛\nℎ,ext. In non-aggregated methods like CutFEM, we can utilize the same extension on the\nsolution ¯E𝑛\next(¯V𝑛\nℎ).\nWe note that the implementation of the extended triangulation and FEspace can be computed\naposteriori,ondemand,wheneveritisneeded. Onecanmarktheadditionalcellsthatareneeded\nfor the extension, and then extend the aggregates to these cells.\n3.Variational formulation on a model problem\nIn this section, we establish the space-time variational formulation by employing a model\nproblem, specificallytheheatequation. Althoughweusetheheatequationfordemonstration\npurposes, it is essential to note that a similar approach can be applied to other PDEs.\n3.1.Weakformulation. Inordertodefinethespace-timevariationalformulation,wefirstdefine\nthe convection-diffusion equation in a space-time domain 𝑄, as find𝑢such that,\n \n𝜕𝑡𝑢+(𝑤𝑤𝑤·∇𝑥)𝑢−𝜇Δ𝑥𝑢=𝑓in𝑄,\n𝑢=𝑢𝐷 on𝜕𝑄𝐷,\n𝜇n𝑥·∇𝑥𝑢=𝑔𝑁 on𝜕𝑄𝑁,\n𝑢=𝑢0 on𝑄(0).(1)\nwhere𝜇is the diffusion coefficient, 𝑤𝑤𝑤the advection velocity, 𝑓the source term, 𝑢𝐷the Dirichlet\nboundarycondition,and 𝑔𝑁boundaryfluxon 𝜕𝑄𝑁.∇𝑥andΔ𝑥denotethespatialgradientand\nspatial Laplacian, resp. Well-posedness requires that 𝑤𝑤𝑤·n𝑥+𝑛𝑡≥0on the Neumann boundary\n𝜕𝑄𝑁. Here, n=(n𝑥,𝑛𝑡)is the outward normal to 𝜕𝑄.\nWediscretizethisproblemwiththe AgFEMinspace(oraghostpenaltystabilization)and\naDGmethod in time. We weakly impose the Dirichlet boundary conditions using Nistche’s\nmethod[34]. Sincethecouplingbetweentimeslabsrespectscausality,weanalyzetheproblemon\nasingletimeslab,assumingweknowthesolutionofthepreviousone(seealso[ 30]). Toanalyze\neach time slab 𝐽𝑛=(𝑡𝑛,𝑡𝑛−1), we define the problem in the reference domain ˆ𝑄𝑛=(𝜑𝜑𝜑𝑛\nℎ)−1(˜𝑄𝑛)\nas: find ˆ𝑢∈V𝑛\nℎ,agsuch that\n𝐵𝑛\nℎ(ˆ𝑢,ˆ𝑣)=𝐿𝑛\nℎ(ˆ𝑣),∀ˆ𝑣∈V𝑛\nℎ,ag,\nwith𝑢=ˆ𝑢◦(𝜑𝜑𝜑𝑛\nℎ)−1. The bilinear form reads as:\n𝐵𝑛\nℎ(ˆ𝑢,ˆ𝑣)=∫\nˆ𝑄𝑛ˆ𝑣𝜕𝑛\n𝑡ˆ𝑢|𝐽˜𝑄𝑛|𝑑ˆ𝑥𝑥𝑥𝑑𝑡+∫\nΩ𝑛ˆ𝑣(𝑡𝑛)ˆ𝑢(𝑡𝑛)|𝐽Ω𝑛|𝑑ˆ𝑥𝑥𝑥+𝑎ℎ(ˆ𝑢,ˆ𝑣),\n𝑎ℎ(ˆ𝑢,ˆ𝑣)=∫\nˆ𝑄𝑛\u0000𝜇∇𝑛\n𝑥ˆ𝑢·∇𝑛\n𝑥ˆ𝑣+ˆ𝑣(𝑤𝑤𝑤◦𝜑𝜑𝜑𝑛\nℎ·∇𝑛\n𝑥)ˆ𝑢\u0001|𝐽˜𝑄𝑛|𝑑ˆ𝑥𝑥𝑥𝑑𝑡\n+∫\n𝜕ˆ𝑄𝑛\n𝐷\u0000𝛽ℎˆ𝑣ˆ𝑢−ˆ𝑣\u0000n𝑛\n𝑥·𝜇∇𝑛\n𝑥ˆ𝑢\u0001−\u0000n𝑛\n𝑥·𝜇∇𝑛\n𝑥ˆ𝑣\u0001ˆ𝑢\u0001|𝐽𝜕˜𝑄𝑛\n𝐷|𝑑ˆ𝑥𝑥𝑥𝑑𝑡,(2)SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 8\nand the linear form reads as,\n𝐿𝑛\nℎ(ˆ𝑣)=∫\nΩ𝑛ˆ𝑣(𝑡𝑛)ˆ𝑢𝑛−1(𝑡𝑛)|𝐽Ω𝑛|𝑑ˆ𝑥𝑥𝑥+𝑙ℎ(ˆ𝑣),\n𝑙ℎ(ˆ𝑣)=∫\nˆ𝑄𝑛ˆ𝑣\u0000𝑓◦𝜑𝜑𝜑𝑛\nℎ\u0001|𝐽˜𝑄𝑛|𝑑ˆ𝑥𝑥𝑥𝑑𝑡+∫\n𝜕ˆ𝑄𝑛\n𝑁ˆ𝑣\u0000𝑔𝑁◦𝜑𝜑𝜑𝑛\nℎ\u0001|𝐽𝜕˜𝑄𝑛\n𝑁|𝑑ˆ𝑥𝑥𝑥𝑑𝑡\n+∫\n𝜕ˆ𝑄𝑛\n𝐷\u0000𝛽ℎˆ𝑣\u0000𝑢𝐷◦𝜑𝜑𝜑𝑛\nℎ\u0001−\u0000n𝑛\n𝑥·𝜇∇𝑛\n𝑥ˆ𝑣\u0001 \u0000𝑢𝐷◦𝜑𝜑𝜑𝑛\nℎ\u0001\u0001|𝐽𝜕˜𝑄𝑛\n𝐷|𝑑ˆ𝑥𝑥𝑥𝑑𝑡,(3)\nLetusdefinethenormsusedin[ 30]toprovestabilityandconvergenceresults,whichwewill\ncomputed in thenumericalexperiments. In [ 30] the space-timeaccumulated DGnorm ofVℎ,ag\nis defined as follows,\n|||𝑣|||2\n𝑛,∗\u0011∥𝑣𝑛(𝑡𝑛+1)∥2\n𝐿2(Ω𝑛)+𝑛∑︁\n𝑖=1∥𝑣𝑖(𝑡𝑖)−𝑣𝑖−1(𝑡𝑖)∥2\n𝐿2(Ω𝑖)+𝑐𝜇∫𝑡𝑛+1\n0∥𝑣∥2\n¯V𝑛(ℎ)𝑑𝑡,\nwhere𝑐𝜇is the coercivity constant and ¯V𝑛(ℎ)\u0011¯V𝑛\nℎ,ag+𝐻2(Ω(𝑡))is the norm of the FEspace\nat a time step 𝑡given by\n|||𝑣|||2\n¯V𝑛(ℎ)\u0011𝜇∥∇𝑣∥2\n𝐿2(Ω(𝑡))+∑︁\n¯𝑇∈¯Tℎ,act𝛽¯𝑇∥𝑣∥2\n𝐿2(¯𝑇∩𝜕Ω𝐷(𝑡))+∑︁\n¯𝑇∈¯T𝑛\nℎ,act𝜇ℎ2\n¯𝑇∥𝑣∥2\n𝐻2(¯𝑇∩Ω(𝑡)).\nThe proof of this stability result follows the ideas in [ 30] in the case in which the deformation\nmap𝜑𝜑𝜑𝑛\nℎ(𝑡)is equal to𝐷𝐷𝐷(𝑡)onΓ(𝑡). Otherwise, the analysis is more technical and would require\nto use ideas similar to the ones in [ 32,35]. The analysis therein assumes a constant deformation\nmap and a level-set description of the domain. In our case, the deformation map can be of\nhigher-order (time-dependent) and the domain is represented by its boundary. We will leave this\nanalysis for future work.\nInthevariationalform,theDirichletboundaryconditionsareimposedweaklywiththeNistche\nmethod,withapenaltyterm 𝛽ℎthatdependsonthespatialcellsize ℎ. Theinitialvalueisalso\nimposedweaklywith DGintime,wherethejumpisgivenby [ˆ𝑢(𝑡𝑛)−ˆ𝑢𝑛−1(𝑡𝑛)]. Theevaluation\nof the solution of the previous time slab ˆ𝑢𝑛−1(𝑡𝑛)inΩ𝑛requires special attention since it is\ncomputed using a different discretization. Further discussion is provided in Sect. 3.2 and Sect. 4.\nThederivativesareprojectionsofthespace-timegradientintospaceandtime, 𝜕𝑛\n𝑡=(∇𝑛)𝑡,and\n∇𝑛\n𝑥=(∇𝑛)𝑥,resp. They are obtained by transporting the space-time gradient to the deformed\ndomain ∇𝑛=F−𝑇ˆ∇,where F=∇𝜑𝜑𝜑𝑛\nℎ. The operators()𝑥and()𝑡represent the projection to\nthespaceandtimedirections, resp. Theboundarynormalisalsotransportedtothedeformed\ndomain. It is computed as follows,\nn𝑛\n𝑥= \nF−𝑇n𝜕ˆ𝑄𝑛\n∥F−𝑇n𝜕ˆ𝑄𝑛∥!\n𝑥,\nwhere n𝜕ˆ𝑄𝑛is the normal vector in the undeformed space-time domain.\nThe integration measures to define the domain change are defined by the jacobians of the\ndeformed domain. The integration measure change of the space-time domain ˜𝑄𝑛is given by\n𝐽𝑄𝑛=𝐽, where𝐽=𝑑𝑒𝑡(F). The initial boundary Jacobian is defined as 𝐽Ω𝑛=𝑑𝑒𝑡(F𝑥(𝑡𝑛)).\nIn the given case 𝐽Ω𝑛=1. Note that for small deformations, where minΩ(𝐽)≈1, a simpler\napproach can be considered by assuming a deformed initial state, e.g., Ω𝑛=𝜑𝜑𝜑𝑛−1\nℎ(𝑡𝑛)(Ω𝑛−1). In\nthissituation,theevaluation ˆ𝑢𝑛−1\nℎ(𝑡𝑛)doesnotrequirethechangeofreferencespacedescribed\nin Sect. 3.2. This approach is equivalent to a time slab with more than one cell in time and\ncontinuous FE spaces in time could be considered within this macro-cell.SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 9\nThe pullback of the area differential form to the reference domain is expressed as\n𝑑𝑎𝜕˜𝑄𝑛=𝐽√︃\nn𝑇\n𝜕ˆ𝑄𝑛·C−1·n𝜕ˆ𝑄𝑛𝑑𝑎𝜕ˆ𝑄𝑛\nwhere C=F𝑇F(see, e.g., [36]).\nThe space-time gradients are transported to the deformed domain, ∇𝑛=F−𝑇ˆ∇. The map\ngradient, F=∇𝜑𝜑𝜑𝑛\nℎ, contains the terms as follows:\nF=\u0014F𝑥 0\n𝜕𝑡𝜑𝜑𝜑𝑛\nℎ,𝑥1\u0015\n=\u0014F𝑥0\nw𝑇1\u0015\n,\nwhere F𝑥is the space gradient and w𝑇=𝜕𝑡𝜑𝜑𝜑𝑛\nℎ,𝑥the deformation velocity. Then, the inverse\ngradient is given by\nF−𝑇=\u0014\nF−1\n𝑥 0\n−w𝑇F−1\n𝑥1\u0015\n.\nThen, by decomposing the space-time gradient into space and time derivatives, we obtain\n\u0014\n∇𝑛\n𝑥\n𝜕𝑛\n𝑡\u0015\n=F−𝑇\u0014ˆ∇𝑥\nˆ𝜕𝑡\u0015\n=\u0014F−1\n𝑥ˆ∇𝑥\nˆ𝜕𝑡−w𝑇F−1\n𝑥ˆ∇𝑥\u0015\n,\nwhich already recovers the derivative terms used in ALE formulations.\n3.2.Inter-slab integration. In the formulation (2)-(3), aDGmethod is used in time, where the\ninitialvalueatthetimeslabisimposedweaklythroughaninter-timeslabjump [ˆ𝑢𝑛(𝑡𝑛)−ˆ𝑢𝑛−1(𝑡𝑛)].\nHowever,integratingthisjumpisnotstraightforward,since ˆ𝑢𝑛(𝑡𝑛)andˆ𝑢𝑛−1(𝑡𝑛)in(3)areexpressed\nin different discrete spaces (and meshes).\nTo address this evaluation, an intermediate unfitted discretization ¯T𝑛\nintis introduced. ¯T𝑛\nintis a partition of Ω𝑛that results of intersecting the embedded discretization T𝑛\ncutofΩ𝑛and\n¯T𝑛\n−=𝜑𝜑𝜑𝑛−1\nℎ(𝑡𝑛)(¯T𝑛−1), i.e., ¯T𝑛\nintresults from intersecting ¯T𝑛,¯T𝑛\n−andΩ𝑛−1. Each cell𝐾int∈¯T𝑛\nint\nhas an injective map to 𝐾𝑛−1∈¯T𝑛−1and𝐾𝑛∈¯T𝑛. A representation of ¯T𝑛\nintis depicted in Fig. 4,\nand its construction is detailed in Sect. 4.\n(a)𝜑𝜑𝜑𝑛−1\nℎ(𝑡𝑛)(¯T𝑛−1)\n (b)¯T𝑛\nint\n(c)¯T𝑛\nFigure 4. Mesh sequence for solution transfer between time slabs. The solution\nobtained in ¯T𝑛−1in (a) is then evaluated at ¯T𝑛(c). Acting as a bridge, the\nintersectedmesh ¯T𝑛\nint=𝜑𝜑𝜑𝑛−1\nℎ(𝑡𝑛)(¯T𝑛−1)∩¯T𝑛∩Ω𝑛in(b)facilitatestheevaluation\nby providing injective cell maps to both active meshes.\nTheintegrationofthejumpinthetimeslabinterfaceisperformedin ¯T𝑛\nint. Thus,weneedto\ndefine the cell maps from ¯T𝑛\nintto¯T𝑛and ¯T𝑛+1. For ˆ𝑢𝑛−1(𝑡𝑛)we need a cell map from ˆ𝐾int∈¯T𝑛\nintSPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 10\ntoˆ𝐾1∈¯T𝑛such that\n𝜓−=\u0000𝜙𝐾1\u0001−1◦\u0010\n𝜑𝜑𝜑𝑛−1\u0011−1\n◦\u0000𝜙𝐾int\u0001,\nwhere𝜙𝐾1mapsfromthereferencespacetotheundeformedphysicalspaceof 𝐾1∈¯T𝑛and𝜙𝐾int\nmaps from the reference space to the physical space of 𝐾int, see Fig. 5. In the case of ˆ𝑢𝑛(𝑡𝑛)we\nneed a cell map from ˆ𝐾int∈¯T𝑛\ninttoˆ𝐾𝑛∈¯T𝑛such that\n𝜓+=\u0000𝜙𝐾2\u0001−1◦\u0000𝜙𝐾int\u0001.\nwhere𝜙𝐾2maps from the reference space to the physical space of 𝐾2∈¯T𝑛. Now, we can\nnumerically integrate the jump by evaluating the following integral,\n∑︁\n𝐾int∈¯Tint∑︁\nˆ𝑞\u0010\nˆ𝑢𝑛(𝑡𝑛)◦𝜓+−ˆ𝑢𝑛−1(𝑡𝑛)◦𝜓−\u0011\n(𝑣𝑛(𝑡𝑛)◦𝜓+(ˆ𝑞))|𝐽𝐾int|𝑤𝐾int,\nwhere𝐽𝐾int,ˆ𝑞and𝑤𝐾intare the Jacobian, quadrature points and quadrature weights of 𝐾int, resp.\nˆ𝐾ˆ𝐾int\n𝐾1 𝐾2𝝋𝑛−1(𝐾1)\n𝐾int\n𝜙𝐾1𝜙𝐾2𝝋𝑛−1\nˆ𝑥ˆ𝑦\n𝑥𝑦𝜙𝐾int\nFigure 5. Representation of the cell maps used in the time slab interface\nintegration. The maps 𝜙𝐾1and𝜙𝐾2send the reference element ˆ𝐾to the physical\nspace of𝐾1∈¯T𝑛−1and𝐾2∈¯T𝑛, resp. The map 𝜙𝐾intsends the reference\nelement ˆ𝐾intto the physical space of 𝐾int∈¯T𝑛−1\nint.\n4.Intersection algorithm for time slab transfer\nIn this section, we will present one of the key novelties of this work. We will develop the\ngeometrical algorithms required in Sect. 3.2 to evaluate the inter-time-slab solution. These\nalgorithms are based on the methods presented in [ 12]. Specifically, we compute a linear\nintersection ¯T𝑛\nintofT𝑛,¯T𝑛\n−=𝜑𝜑𝜑𝑛−1\nℎ(𝑡𝑛)(¯T𝑛−1)andΩ𝑛=int(Bℎ(𝑡𝑛)). It is important to note that,\nto ensure the linearity of the intersections, we need to avoid bilinear terms in 𝜑𝜑𝜑𝑛−1\nℎusing a𝑃𝑘\nspace, e.g., decomposing 𝐾∈¯T𝑛into simplices.\nTo simplify the exposition of the intersection algorithm Alg. 1, we redefine T=¯T𝑛,T−=¯T𝑛\n−\nandB=Bℎ(𝑡𝑛). Within the cell loop of Alg. 1, we first consider the cells close to 𝐾∈Tby\nrestrictingBandT−accordingly (line 3 and line 4). These restriction queries are computed\nduringapreprocessingstagebeforeenteringtheloop. Next,inline5,wecomputetheintersection\nof𝐾with the interior of Busing the algorithms in the loop-body of Algorithm 10 in [12].\nThesealgorithmsassumethat Bisalinearpolytope,whichisnon-convexingeneral. Thus,its\ndomaininterior int(B)isboundedbythesetofplanarfacesof B. Theintersection 𝐾∩int(B)\nrequiresaconvexdecompositionof Band𝐾beforeintersectingthehalf-spacesdefinedbythe\nplanar faces (see [12]).SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 11\nFinally, we intersect each polytope in T𝐾\ncutby the subset ofT−around𝐾(line 8). These\nintersections are performed employing convex linear clipping algorithms [ 37] described in\nAlgorithm2 in [ 12]. Alternatively, in line 10, we intersectthe cells 𝐾∈Twithin thedomain\nint(B)bounded byBthat are not intersected by the domain boundary B. The information about\nthecellsinside int(B)isobtainedfromthepropagationthroughtheuntouchedcells(see[ 12]).\nThe returned triangulation Tintnot only covers the cells cut cells intersected by the boundary\nBbut the entire domain enclosed by B. This process guarantees that each cell 𝐾int∈Tinthas\nan injective map to 𝐾∈Tand𝐾−∈T−. The cells inTintare general polytopes that cannot\nusestandardquadraturerules. Inordertonumericallyintegrateinthesecells,onecanperform\na decomposition of the cells into simplices. Alternatively, one can reduce the dimension of\nthe integrals with Stokes theorem and moment-fitting methods; see more details in [ 13] and\nreferences therein.\nAlgorithm 1T∩T−∩int(B)\n1:Tcut←∅,Tin←∅\n2:for𝐾∈Tdo\n3:B𝐾←restrict(B,𝐾)\n4:T𝐾\n−←restrict(T−,𝐾)\n5:T𝐾\ncut←𝐾∩int(B𝐾)\n6:for𝐾−∈T𝐾\n−do\n7:if𝐾∩B≠∅then\n8:T𝐾,𝐾−\ncut←{𝐾cut∩𝐾−:𝐾cut∈T𝐾\ncut};Tcut←T cut∪T𝐾,𝐾−\ncut\n9:else if𝐾⊂int(B)then\n10:Tin←T in∪(𝐾∩𝐾−)\n11:end if\n12:end for\n13:end for\n14:returnTint←T cut∩Tin\nWeemphasizethattheintersectionalgorithmisasrobustasthecorecutalgorithminline5.\nFurthermore,itisimportanttonotethat,forevaluationpurposes,accuratetolerancemanagement\nis not required in the intersection process of line 8. While the algorithm is described for the\ncore cut algorithm with an exact embedded discretization of explicit domain representation [ 30],\nAlg. 1 is general enough to be used with other unfitted discretizations.\n5.Numerical experiments\n5.1.Objectives. In the experiments of this section, we aim to demonstrate the effectiveness of\nthe presented methods. In particular, we examine the following aspects within our formulation:\n•Weevaluatethe ℎ𝑝-convergenceonboth2Dand3Dspatialdomains,comparedtothe\nresults and analysis in [30].\n•Weexplorenumericalstabilityconcerningthecutlocationandtheapproximationdegrees.\n•Weassesstheapplicabilityofourformulationtocomplex2Dand3Dmovingdomains\nderived from STL models.\nIt is important to note that these experiments focus on comparing our formulation with the\none presented in [ 30]. Recall that [ 30] uses space-time embedded discretizations on implicit\ngeometries determined by level sets, and the results are presented exclusively in 2D+1D domains\n(as the geometrical algorithms for 3D+1D domains are significantly more complex). In contrast,\nwe design a space-time formulation that works on explicit boundary representations and only\nrequire geometrical intersection algorithms in space only, e.g., 2D and 3D vs. 3D and 4D inSPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 12\n[30]. This is possible by pulling back the problem into an extruded space-time domain using the\nformulation in Sect. 3.1.\n5.2.Environment setup. The numerical experiments have been performed on Gadi and Titani\nsupercomputers. Gadiisahigh-endsupercomputerattheNCI(Australia)with4962nodes,3074\nof them powered by a 2 x 24 core Intel Xeon Platinum 8274 (Cascade Lake) at 3.2 GHz and 192\nGBRAM.TitaniislocatedatUPC(Spain)with6nodes,5ofthempoweredbya2x12coreIntel\nXenon E5-2650L v3 at 1.8 GHz and 256 GB RAM. The algorithms presented in this work have\nbeenimplemented intheJulia programminglanguage[ 38]. TheunfittedFE computationshave\nbeen performed using the Julia FE library Gridap.jl [39] version 0.17.17 and the extension\npackageforunfittedmethods GridapEmbedded.jl version0.8.1[ 40].STLCutters.jl version\n0.1.6 [41] has been used to compute intersection computations on STLgeometries. To mitigate\nexcessive computational costs, the condition numbers are computed in the 1-norm using cond()\nJulia function.\n5.3.Space-timeconvergencetests. Todemonstrateoptimalconvergencerateswesolveasimple\nPDEwithamanufacturedsolutionoutofthe FEspace. Inspiredby[ 30],wesolvethePoisson\nequation (1) with the following manufactured solution\n𝑢(𝑥,𝑡)=sin\u0010𝜋𝛼𝑡\n𝑇\u0011𝐷Ö\n𝑖=1sin\u0012𝜋𝑥𝑖\n𝐿𝑖\u0013\n,\nwhere{𝐿1,...,𝐿𝐷,𝑇}are the cartesian dimensions of the space-time domain and the time\nparameterissetto 𝛼=0.5fortheexperimentsshowninFig.6andFig.7. Furthermore,inthe\nequation(1)wesetthediffusiontermas 𝜇=1. Thespacedomainisa 𝑛-cubewitha 𝑛-cubichole\ninthecenter. Theholeisdescribedbythe STLofacube. Thelengthsatthedomainsidesare\n𝐿=3while the lengths of the hole sides are 𝑙=1. The time domain has size 𝑇=1. The hole is\nlinearlytranslatedinthe 𝑥direction. Thedisplacementmapisdescribedas 𝐷𝐷𝐷(𝑥,𝑡)=(0.2𝑡,0,0).\nInallthecases,thedomaindiscretizationisaregularCartesiangrid,withthesamenumberof\nelements𝑛in each direction, both space and time.\nFor implementation reasons, the two-dimensional examples are computed with a three-\ndimensional STL. Thus,webuildapseudo-two-dimensionaldomainthathasonlyonecellinthe\n𝑧direction. Thespace-timedomaindimensionsare {𝐿,𝐿,𝐿\n𝑛,𝑇}whilethenumberofelements\nin each direction is {𝑛,𝑛, 1,𝑛}. The number of elements per direction in convergence tests is\n𝑛=2𝑖,𝑖=3,...,6in two-dimensional runs, while 𝑛=2𝑖,𝑖=3,...,5for three-dimensional runs.\nWe analyze the condition number of the system to be inverted in each time-slab. Since the\nsystemmatrixisnonsymmetric,assuggestedin[ 30],apreconditionerfor DGintime[42]can\nbeconsidered. Theeffectivenessofthispreconditionerdependsontheconditionnumbersthe\nmass and stiffness matrices, which are defined as follows,\nM𝑎𝑏=∫\nˆ𝑄𝑛E𝑛(𝚽𝑎)E𝑛(𝚽𝑏)|𝐽Ω𝑛|𝑑ˆ𝑥𝑥𝑥𝑑𝑡, A𝑎𝑏=𝑎ℎ\u0010\nE𝑛(𝚽𝑎),E𝑛(𝚽𝑏)\u0011\n.\nTheconditionnumberspresentedinFig.6aandFig.6barecomputedintheinitialtimeslabof\nthe two-dimensional convergence experiments. We observe that the condition number of the\nmassmatrixremainsnearlyconstant,whiletheconditionnumberofthestiffnessmatrixscales\nwithO(ℎ−2). These observations align with the behavior expected for AgFEMin space-time\ndomains analyzed in [30].\nNow, let us analyze the convergence of the error norms. In Fig. 6, we can observe the\naccumulated DGerror norm convergence with coercivity constant 𝑐𝜇=1. We observe that\nusing the AgFEMand constant ℎ/𝜏the error converges with O(ℎ𝑟), where the convergence\nrate𝑟=min(𝑝,𝑞). Fig. 7 shows the 𝐿2(Ω𝑛)and𝐻1(Ω𝑛)norms at the final time 𝑡=𝑇. The\nconvergence rate of 𝐿2(Ω𝑛)norm is𝑟=min(𝑝,𝑞)+1while the convergence rate of 𝐻1(Ω𝑛)SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 13\nnorm is𝑟=𝑝. These results are in agreement with the theoretical and analytical space-time\nAgFEM results in [23].\n(a)Mass matrix\n (b)Stiffness matrix\n (c)2D\n (d)3D\nFigure 6. Scaling of the condition numbers of the mass and stiffness matrices in\ntheinitialtimeslab,(a)and(b). Convergenceofthespace-time DGnormerror\n(𝑒=𝑢−𝑢ℎ) in two and three-dimensional space domains (c) and (d). Here, 𝑝\nand𝑞represent the space and time approximation order, resp.\n(a)2D\n (b)3D\n (c)2D\n (d)3D\nFigure 7. Convergenceofthe 𝐿2(Ω𝑛)and𝐻1(Ω𝑛)normsoftheerror( 𝑒=𝑢−𝑢ℎ)\natthefinal time 𝑡=𝑇intwoand three-dimensionalspacedomains. Here, 𝑝and\n𝑞represent the space and time approximation order, resp.\n5.4.Moving domains examples. To demonstrate the applicability of the presented algorithms\nfor the simulation of fluids around moving boundaries, we solve the flow around two rotating\ngeometries in 2D and 3D. We solve the Navier-Stokes and Stokes equations resp. with kinematic\nviscosity𝜈=10−2. Regardless of the dimension, we run similar setups. Both share the same\ndiscretization order for pressure 𝑝𝑝=1, velocity𝑝𝑢=2and time𝑞. We utilize the AgFEM\naggregating all cut cells.\nWe improve the accuracy of both meshes by clustering the cells around the geometry. See\nFig.8andFig.9for2Dand3D,resp. Wemapeachdirection 𝑖∈1,...,𝑑oftheCartesianmeshesSPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 14\nas follows,\n𝜙𝑖\n𝑀(ˆ𝑥)= \nˆ𝑥𝑥𝑥𝑖\n0\u0010\nˆ𝑥𝑖\nˆ𝑥𝑖\n0\u0011𝛼\nifˆ𝑥𝑖<ˆ𝑥𝑖\n0,\n1−(1−ˆ𝑥𝑖\n0)\u0010\n1−ˆ𝑥𝑖\n1−ˆ𝑥𝑖\n0\u0011𝛼\notherwise.(4)\nHere, ˆ𝑥𝑖=𝑥𝑖−𝑥𝑖\n0/𝐿𝑖isthereferenceaxisofthedirection 𝑥𝑖where𝑥𝑖\n𝑜isthelowervalueinthe\n𝑖direction, and 𝐿𝑖is the𝑖-length. In (4), we use ˆ𝑥𝑖\n0=0.5to determine the region of element\ncondensation and the exponential factor 𝛼<1for the smoothness of the map.\nFigure 8. Backgroundspatialmesh ¯Tbgaroundaprismaticgear B0(geometry\nid71711from Thingi10k [43]). Thismeshisthesimplexpartitionofamapped\nCartesian mesh. The uniform elements are mapped in each direction with 𝜙𝑀\n(4)and an exponential factor 𝛼=0.5. The Cartesian mesh has 80×40elements\nbefore the simplex decomposition. The artificial domain Ωartis a box of size\n4.8𝐿𝑥×2.4𝐿𝑦. Here,𝐿𝑥and𝐿𝑦represent the bounding box size of B0.\nFigure 9. Backgroundspatialmesh ¯TbgaroundawingB0(id65604in Thingi10k ).\nThissimplex meshcomesfromamapped Cartesianmesh. Thecoordinatesofthe\nmesh are mapped in the 𝑥and𝑦directions with 𝜙𝑀(4)and an exponential factor\n𝛼=0.3. TheCartesianmeshhas 20×20×8elements. Theartificialdomain Ωart\nisa boxof size 4𝐿𝑥×4𝐿𝑦×0.8𝐿𝑧. Here,𝐿𝑥,𝐿𝑧and𝐿𝑦representthe bounding\nbox size ofB0.\nIn the 2D example of Fig. 8 and Fig. 10, we solve the Navier-Stokes equations with 𝑅𝑒=102.\nWe define a parabolic inlet flow in the 𝑥-direction,\n𝑢(𝑥𝑥𝑥)=𝑈max(4𝑥2−4𝑥2\n2),SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 15\nwhere𝑈max=(1,0,0). Wesetzerovelocityonthe 𝑦-facesandzero 𝑧-velocityinthe 𝑧-faces,i.e.,\nslippingconditions. Weweaklyimposethedisplacementvelocity ∇𝜑𝜑𝜑𝑛withNistche’smethodon\nthe geometryB(𝑡). The initial geometry B0is mapped by 𝐷𝐷𝐷=𝐷𝐷𝐷𝑠◦𝜙𝜙𝜙𝑡,\n𝐷𝐷𝐷𝑠(𝑥𝑥𝑥,𝑡)=𝑥𝑥𝑥0+R(𝑡)·(𝑥𝑥𝑥−𝑥𝑥𝑥0)+A𝑥sin(𝜔𝑥𝑡),\nwhere R(𝑡)is the rotation matrix with angular velocity 𝜔=𝜋/2 rad/s,𝑥𝑥𝑥0is the center of the\nmesh, A𝑥=(0,0.2)and𝜔𝑥=𝜋/2. The time map is defined as, 𝜙𝜙𝜙𝑡(𝑥𝑥𝑥,𝑡)=(𝑥𝑥𝑥,𝜙𝑡(𝑡)),\n𝜙𝑡=(𝑡𝑎\n𝛾\u0010\n𝑡\n𝑡𝑎\u0011𝛾\nif𝑡 <𝑡𝑎,\n(𝑡−𝑡𝑎)+𝑡𝑎\n𝛾otherwise.\nWe set𝛾=2and𝑡𝑎=1/8. With this time map, the initial velocity of the geometry is zero.\n0.0 2 .2\n𝑢ℎ\n(a)𝑡=0.0\n (b)𝑡=0.5\n (c)𝑡=1.0\nFigure 10. Representationofthelineintegralconvolution( LIC)filterofthefluid\nsimulationaroundanevolvinggeometryinFig.8at 𝑅𝑒=102. Thetimestepsize\nis𝜏=1/60. Thegeometrycombinesrotationanddisplacements. Theredtooth\nserves as a reference for visualizing the rotation.\nInthe3DexperimentofFig.9andFig.11,wesimulateaviscousflow( 𝑅𝑒∼0)withthesame\nboundary conditions as in the 2D experiment. However, we define a different inlet profile,\n𝑢(𝑥𝑥𝑥)=𝑈max(4𝑥2−4𝑥2\n2)(4𝑥3−4𝑥2\n3),\nthat utilizes the 3D domain. The differences in the 𝑧-direction are clearly shown in Fig. 11.\nEqually, the 𝑧-faces havea slippingcondition to holdthe rotationof the wingin the 𝑧-direction.\nThe initial geometry B0is mapped by\n𝐷𝐷𝐷(𝑥𝑥𝑥,𝑡)=𝑥𝑥𝑥0+R𝜃(𝑡)(𝑥𝑥𝑥−𝑥𝑥𝑥0),\nwhere𝑥𝑥𝑥0isthecenterofthemeshand R𝜃istherotationmatrixoftheangle 𝜃(𝑡)=𝜃maxsin(𝜔𝜃𝑡)\noverthe𝑧-axis. Here,𝜃max=𝜋/10and𝜔𝜃=𝜋. The3Dexperimentisinitializedwithanon-zero\nvelocity of the geometry.\nWe slightly optimize the number of time slab transfer operations in these experiments. Above\nadeformation threshold, e.g., max(∇𝜑𝜑𝜑𝑛)<0.8, wemaintain thespatial active mesh ¯T𝑛between\ntime slabs. In these cases, the initial value evaluation does not require mesh intersections,\nreducing computational cost.SPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 16\n0.0 1 .0\n𝑢ℎ\n(a)𝑡=0.0,ˆ𝑧=0.5\n (b)𝑡=0.1,ˆ𝑧=0.5\n (c)𝑡=0.2,ˆ𝑧=0.5\n(d)𝑡=0.0,ˆ𝑧=0.75\n (e)𝑡=0.1,ˆ𝑧=0.75\n (f)𝑡=0.2,ˆ𝑧=0.75\n(g)𝑡=0.0,ˆ𝑧=1.0\n (h)𝑡=0.1,ˆ𝑧=1.0\n (i)𝑡=0.2,ˆ𝑧=1.0\nFigure 11. Representation of the LICfilter of the viscous flow around Fig. 9.\nThe matrix representation shows several time slabs on different slices. Here,\nˆ𝑧=(𝑧−𝑧0)/𝐿𝑧represents the relative position in 𝑧where𝑧0is the lower the\n𝑧-coordinate and 𝐿𝑧the𝑧-length. The step size of the simulation is 𝜏=1/120.\nNote thatthe wingis notan extrusionof anairfoil. Thecentral section( ˆ𝑧=0.5)\nhas a flat trailing edge (see Fig. 9).\n6.Conclusions and future work\nIn this work, we have introduced a novel space-time formulation for problems on moving\ndomains with large displacements. The continuous problem at each time slab is pulled back to a\nconstant-in-timedomain(obtainedbyextrusionintimeoftheinitialgeometry). Theresulting\nproblem is discretized using unfitted FEM. Intersection algorithms between the boundary of\nthe initial geometry and background mesh are used to compute the required integrals in the\nformulation. This way, we eliminate the requirement for four-dimensional geometric algorithms.\nThe time discretization relies on DGschemes; the jump terms across time slabs are exactly\nintegrated by computing the intersection of the reference and push-forward background meshes.\nWehavevalidatedthismethodthroughan ℎ𝑝-convergenceanalysisofamanufacturedsolution\nusingAgFEM. We have observed optimal convergence rates for two-dimensional and three-\ndimensional space domains in these experiments. The convergence has been validated by\ncomparing it with another space-time analysis for unfitted FEM[30]. Additionally, we have\nobserved the expected scaling of the condition number of the mass and stiffness matrices.\nFurthermore, we have demonstrated the practical applications of this methodology to theSPACE-TIME UNFITTED FEM ON EXPLICIT GEOMETRIES 17\nsimulation of incompressible flows around rotating geometries in two examples, one with a\ntwo-dimensional geometry and another with a three-dimensional geometry.\nFutureworkinvolvesthenumericalanalysisofthemethodinthecaseinwhichthedeformation\nmaps do not much the boundary displacement by designing a high-order extension of the\nwork in [ 32] and the extension of this method to distributed memory machines [ 44]. The\ngeometrical component of this extension will be highly scalable, given that the presented\nalgorithms are defined cell-wise and thus embarrasingly parallel, as they are in [ 12]. Distributed\ncomputations combined with background mesh refinement, e.g., using octree meshes, will allow\nus to solve larger real-world applications. Additional developments include applying this method\ntoFSIsimulations. This extension will develop the full potential of the method in dynamic\ninterface-coupling multiphysics simulations.\nAcknowledgments\nThis research was partially funded by the Australian Government through the Australian\nResearch Council (project numbers DP210103092 and DP220103160). We acknowledge Grant\nPID2021-123611OB-I00 funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A\nway of making Europe”. P. A. Martorell acknowledges the support received from Universitat\nPolitècnica de Catalunya and Santander Bank through an FPI fellowship (FPI-UPC 2019). This\nwork was also supported by computational resources provided by the Australian Government\nthrough NCI under the National Computational Merit Allocation Scheme.\nReferences\n[1]G. Karypis, K. 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Sen\n ,3and Sunny Vagnozzi\n4, 5\n1INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy\n2Department of Physics, Jamia Millia Islamia, Jamia Nagar, New Delhi-110025, India\n3Centre for Theoretical Physics, Jamia Millia Islamia, Jamia Nagar, New Delhi-110025, India\n4Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy\n5Trento Institute for Fundamental Physics and Applications (TIFPA)-INFN, Via Sommarive 14, 38123 Povo (TN), Italy\nABSTRACT\nEarly observations with the James Webb Space Telescope (JWST) have revealed the existence of an\nunexpectedly large abundance of extremely massive galaxies at redshifts z≳5: these observations\nare in tension with the predictions not only of the standard ΛCDM cosmology, but also with those\nof a wide class of dynamical dark energy (DE) models, and are generally in better agreement with\nmodels characterized by a phantom behaviour. Here we consider a model, inspired by string theory\nand the ubiquity of anti-de Sitter vacua therein, featuring an evolving DE component with positive\nenergy density on top of a negative cosmological constant, argued in an earlier exploratory analysis\nto potentially be able to explain the JWST observations. We perform a robust comparison of this\nmodel against JWST data, considering both photometric observations from the CEERS program, and\nspectroscopic observations from the FRESCO survey. We show that the model is able to accommodate\nthe JWST observations, with a consistency probability of up to 98%, even in the presence of an\nevolving component with a quintessence-like behaviour (easier to accommodate theoretically compared\nto phantom DE), while remaining consistent with standard low-redshift probes. Our results showcase\nthe tremendous potential of measurements of high-redshift galaxy abundances in tests of fundamental\nphysics, and their valuable complementarity with standard cosmological probes.\nKeywords: cosmology: cosmological models — cosmology: cosmological parameters — cosmology:\ncosmological parameters: dark energy — galaxies: abundances — galaxies: high-redshift\ngalaxies\n1.INTRODUCTION\nThe concordance ΛCDM cosmological model consti-\ntutes at present one of the simplest frameworks to de-\nscribe the evolution of our Universe, and owes its suc-\ncess to its ability to describe a wide range of cosmolog-\nical and astrophysical observations. One of the corner-\nstones of current cosmology is dark energy (DE, Pee-\nbles & Ratra 2003; Frieman et al. 2008; Nojiri et al.\n2017), a component with negative pressure and posi-\ntive energy density driving the late-time acceleration of\nthe Universe, whose discovery in 1998 led to one of the\nbiggest paradigm shifts in physics (Riess et al. 1998;\nCorresponding author: Nicola Menci\nnicola.menci@inaf.itPerlmutter et al. 1999): within ΛCDM, DE takes the\nform of a positive cosmological constant (CC) Λ (Car-\nroll 2001). Over the past two and a half decades, the\nnature of DE has to a large extent been investigated by\nmeans of three classes of cosmological observations: low-\nredshift ( z≲2) probes based on Type Ia Supernovae\n(SNeIa) and Baryon Acoustic Oscillations (BAO), the\nlatter observed in the clustering of various tracers of the\nlarge-scale structure, such as galaxies; and high-redshift\n(z∼1100) probes based on the Cosmic Microwave Back-\nground (CMB): see Huterer & Shafer (2018) for a recentarXiv:2401.12659v1  [astro-ph.CO]  23 Jan 20242 Menci et al.\nreview on observational probes of DE.1By virtue of the\nsignificant efforts devoted to the study of observations,\nunderlying theory, and systematics associated to these\nprobes, CMB, BAO, and SNeIa are generally referred to\nas standard cosmological probes and enjoy a (rightly de-\nserved) privileged status for what concerns cosmological\ntests of the expansion of the Universe and of fundamen-\ntal physics, with particular regard to the nature of DE.\nHowever, a series of growing cosmological tensions which\nappear to challenge the ΛCDM model (Perivolaropoulos\n& Skara 2022), and the fact that some of these probes\nare close to being systematics-limited, are making it im-\nperative to start looking for new independent probes,\nwhose complementarity and synergy with the standard\nCMB+BAO+SNeIa can greatly enrich the landscape of\nmethods to study the Universe: in this sense, a number\nof so-called “emerging” cosmological probes are rapidly\ngaining momentum, as beautifully summarized in the\nrecent review of Moresco et al. (2022).2\nIn this context, the long-awaited recent observations\nof massive, distant galaxies delivered by the James Webb\nSpace Telescope (JWST, Gardner et al. 2006) have\nopened a remarkable window not only onto astrophysics\nand the process of galaxy formation, but also onto novel\ncosmological tests of the contents of our Universe and\nmore generally fundamental physics (Labb´ e et al. 2023;\nXiao et al. 2023; Arrabal Haro et al. 2023). This is pos-\nsible because the abundance of galactic haloes of dark\nmatter (DM) mass Mat redshifts z≈5−10 is pre-\ndicted to be exponentially sensitive to the growth fac-\ntor of perturbations D(z), while also being a strongly\ndecreasing function of redshift, and one which depends\ncrucially on the (model-dependent) time-redshift rela-\n1Nevertheless, it is worth emphasizing that the sensitivity of the\nCMB to (late-time) DE is indirectly mostly a low-redshift (ge-\nometrical) one, through the effect of DE on the distance to the\nCMB itself (see e.g. the recent discussion in Escamilla et al. 2023),\nwhereas the sensitivity of the CMB to DE perturbations and the\ndynamics thereof is quite limited, except possibly in the case\nof models which deviate substantially from the positive CC Λ,\nwhich are however ruled out by observations.\n2Of course tests of DE are not limited to cosmological scales. Var-\nious tests of DE on astrophysical, astronomical, and even local\nscales, have been considered over the past decades, see Hamilton\net al. (2015); Burrage & Sakstein (2016, 2018); Vagnozzi et al.\n(2021); Zhang (2022); Ferlito et al. (2022); Brax et al. (2023);\nBenisty & Capozziello (2023); Zhang & Zhang (2023); Benisty\net al. (2023c,a,b); Paraskevas & Perivolaropoulos (2023); Benisty\net al. (2023); Kaneta et al. (2023) for examples in this sense.tion (Del Popolo & Yesilurt 2007).3On the other hand,\nif we denote by Ω band Ω mthe cosmic density param-\neters for baryons and the total matter component re-\nspectively (and with fb≡Ωb/Ωm), the stellar mass of a\ngalaxy M⋆cannot exceed the maximal allowed baryonic\ncontent of the host halo fbM. This implies that compar-\ning the theoretically predicted cumulative comoving stel-\nlar number density of massive ( M⋆≈1010−1011M⊙)\ngalaxies, n(> M ⋆, z), with the observed abundance of\ngalaxies of given stellar mass at redshifts z≈5−10,\ncan potentially provide strong constraints on the expan-\nsion history and growth of cosmological perturbations,\nand therefore also on the nature of DE, in a way which is\ncompletely independent of the complex baryonic physics\ninvolved in the formation of galaxies. Needless to say,\nsuch constraints would be extremely valuable in light of\ntheir high complementarity to state-of-the-art determi-\nnations from CMB, BAO, and SNeIa, both in terms of\nrange of redshifts/cosmic times probed, as well as po-\ntential systematics involved.\nAs hinted to earlier, the possibility of testing the ex-\npansion history of the Universe at such intermediate red-\nshifts is particularly valuable in view of the tension be-\ntween local and high-redshift determinations of the Hub-\nble constant H0(see e.g. Verde et al. 2019; Di Valentino\net al. 2021b,c; Abdalla et al. 2022; Sch¨ oneberg et al.\n2022; Shah et al. 2021; Kamionkowski & Riess 2023;\nHu & Wang 2023; Vagnozzi 2023; Verde et al. 2023, for\nrecent reviews). At a significance of ≳5σ, the Hub-\nble tension poses what is probably one of the most se-\nrious observational challenges to ΛCDM, and can po-\ntentially lead us to completely rethink the nature of\nDE.4Thus, probing the expansion history of the Uni-\n3Distant galaxies are also useful as a test of cosmological models\nthrough their ages, which of course cannot exceed the age of the\nUniverse: see for instance Jimenez et al. (2019); Valcin et al.\n(2021); Bernal et al. (2021); Vagnozzi et al. (2022); Wei & Melia\n(2022); Cimatti & Moresco (2023); Costa et al. (2023) for recent\nworks exploring the important of stellar and galactic ages as a\ntest of cosmological models.\n4With no claims as to completeness, see Yang et al. (2018a);\nGuo et al. (2019); Di Valentino et al. (2020b,c); Zumalacarregui\n(2020); G´ omez-Valent et al. (2020); Di Valentino et al. (2020a);\nGao et al. (2021); Petronikolou et al. (2022); Alestas et al. (2022);\nRoy et al. (2022); Sharma et al. (2022); Moshafi et al. (2022);\nSchiavone et al. (2023); Gao et al. (2022); Bernui et al. (2023);\nBen-Dayan & Kumar (2023); Zhai et al. (2023); Basilakos et al.\n(2023); Petronikolou & Saridakis (2023) for examples of works\nexploring the implications of the Hubble tension for the nature\nof DE. The possibility of a more speculative “early dark energy”\ncomponent, operative at very high redshift, is also a particularly\ninteresting possibility in this context (see e.g. Poulin et al. 2019;\nNiedermann & Sloth 2021; Braglia et al. 2020; Oikonomou 2021a;\nVagnozzi 2021; Karwal et al. 2022; Benevento et al. 2022; Reeves\net al. 2023; Poulin et al. 2023; Odintsov et al. 2023).3\nverse at intermediate redshifts z≈6−10 constitutes a\nkey opportunity to study the validity of ΛCDM model\nin an independent manner, and in an epoch which is\notherwise extremely difficult to access (see also G´ omez-\nValent et al. 2023; Tutusaus et al. 2023, for recents work\nwhich highlighted the relevance of intermediate-redshift\nDE dynamics in the context of the Hubble tension).\nAmong the various aspects of fundamental physics\nwhich can be put to test through measurements of the\nabundance of massive galaxies at very high redshift, of\nparticular interest to the present work are the properties\nof DE. In particular, as explicitly demonstrated by one\nof us in Menci et al. (2020) using data from the CAN-\nDELS survey, the abundances of massive, high-redshift\ngalaxies can provide key constraints on the DE equa-\ntion of state (EoS) w. The reason is that wcontrols\nthe large-scale behaviour of DE at both background\nand perturbation level, thereby dramatically affecting\nthe growth of density perturbations and the formation\nof cosmic structures. In this context, a number of re-\ncent works (see e.g. Lovell et al. 2022; Boylan-Kolchin\n2023; Forconi et al. 2023; Wang & Liu 2022; Forconi\net al. 2023) have demonstrated how initial JWST imag-\ning data from NIRCam observations of the Cosmic Evo-\nlution Early Release Science (CEERS) program, which\nuncovered evidence of a surprisingly abundant popula-\ntion of very massive galaxies at extremely high redshifts\n7≲z≲10, could severely challenge the ΛCDM sce-\nnario. Specializing to the implications for DE, it was\nargued by one of us in Menci et al. (2022) that these\nobservations in fact exclude a significant portion of dy-\nnamical DE parameter space (including the point corre-\nsponding to ΛCDM), and favor models whose equation\nof state takes values w <−1 at some point after recom-\nbination, i.e. phantom models (Caldwell et al. 2003),\nsee also Wang et al. (2023b) for further discussions.5\nAlthough such results are still affected by uncertainties\nrelated to the assumed initial mass function (IMF) of\n5For examples of other works showcasing the enormous potential\nof the early JWST observations in constraining various aspects of\nfundamental physics, see Biagetti et al. (2023); Haslbauer et al.\n(2022); H¨ utsi et al. (2023); Gandolfi et al. (2022); Maio & Viel\n(2023); Yuan et al. (2023); Dayal & Giri (2023); Ilie et al. (2023);\nJiao et al. (2023); Parashari & Laha (2023); Hassan et al. (2023);\nLei et al. (2023); Yoshiura et al. (2023); Padmanabhan & Loeb\n(2023); Lin et al. (2024); Su et al. (2023); Forconi et al. (2023);\nGouttenoire et al. (2023); Guo et al. (2023); Huang et al. (2023);\nBird et al. (2023); Wang et al. (2023b); Libanore et al. (2023);\nWang et al. (2023a); van Putten (2024). Taking a step aside from\nnew fundamental physics, a very important possibility of course\nis that the JWST results may call for a better understanding of\ngalaxy formation, as discussed in a number of recent works (Fer-\nrara et al. 2023; Qin et al. 2023; Pallottini & Ferrara 2023; Wang\net al. 2023; Pacucci et al. 2023).the stellar populations (see e.g. Steinhardt et al. 2023a)\nand to the broadband selection criteria (especially for\ndouble-break sources, see for example Desprez et al.\n2023), it is worth highlighting that recent spectroscopic\nobservations in the redshift range 5 ≲z≲6 (Xiao et al.\n2023) and 8 ≲z≲10 (Arrabal Haro et al. 2023) from\nthe JWST First Reionization Epoch Spectroscopically\nComplete Observations (FRESCO) and CEERS surveys\ncorroborate the above conclusions, while also placing\nthem on a much stronger footing.\nThe implications of the early JWST results for phan-\ntom DE are particularly interesting in view of the Hub-\nble tension. It is now well understood that consis-\ntency with measurements of the acoustic angular scale\nat low redshifts from BAO requires that the sound hori-\nzon be lowered in the presence of a higher Hubble con-\nstant (Bernal et al. 2016; Addison et al. 2018; Lemos\net al. 2019; Aylor et al. 2019; Sch¨ oneberg et al. 2019;\nKnox & Millea 2020; Arendse et al. 2020; Efstathiou\n2021; Cai et al. 2022; Keeley & Shafieloo 2023). This\nimplies that phantom DE, an inherently late-time com-\nponent which does not affect the sound horizon, cannot\non its own completely solve the Hubble tension. Never-\ntheless, within the limits imposed by low-redshift data,\nphantom DE can at least partially alleviate the Hub-\nble tension, in light of its ability to accommodate a\nhigher Hubble constant while keeping the distance to the\nCMB, and thereby the acoustic angular scale observed\nin the CMB θs, fixed (given that the sound horizon is\nunaffected, see Alestas et al. 2020, for a recent explicit\ndiscussion of parameter degeneracies in the presence of\nphantom DE). For this and other reasons, the possibil-\nity of phantom DE playing some role in the context of\nthe Hubble tension is one which has been give serious\nconsideration in the literature (see e.g. Zhao et al. 2017;\nM¨ ortsell & Dhawan 2018; Li & Shafieloo 2019; Vagnozzi\n2020; Alestas et al. 2021; Yang et al. 2021; Kumar 2021;\nTeng et al. 2021; Heisenberg et al. 2023; Chudaykin et al.\n2022; Sharma et al. 2023; Ballardini et al. 2023; Dah-\nmani et al. 2023; Montani et al. 2024; da Costa et al.\n2023, for recent discussions on the subject).\nOur discussions so far have been data-driven. How-\never, theory considerations have a lot to offer to the\ndiscussion on viable DE models. Firstly, the tiny value\nof the positive CC Λ required to explain observations\nis at severe odds with the value expected from theory\nconsiderations, when interpreted in terms of zero-point\nvacuum energy density of quantum fields: this is the\nwell-known CC problem (Weinberg 1989; Padmanab-\nhan 2003). Moving on to the simplest “quintessence”\nmodels for DE, based on a single, minimally coupled\nscalar field in the absence of higher derivative opera-4 Menci et al.\ntors, and with canonical kinetic term (Wetterich 1988;\nPeebles & Ratra 1988; Ratra & Peebles 1988; Wetterich\n1995; Caldwell et al. 1998), it is well known that these\npredict w >−1, and cannot therefore give rise to phan-\ntom DE, which instead requires a violation of the null\nenergy condition (Vikman 2005; Carroll et al. 2005; No-\njiri et al. 2005; Oikonomou & Giannakoudi 2022; Trivedi\n2023). It is also worth noting that, in the absence of ad-\nditional ingredients, these “vanilla” quintessence models\nalso worsen the Hubble tension, and one could there-\nfore argue that they are observationally disfavored if the\nHubble tension is to be taken seriously (Vagnozzi et al.\n2018; ´O Colg´ ain et al. 2019; ´O Colg´ ain & Yavartanoo\n2019; Banerjee et al. 2021; Lee et al. 2022).\nAlthough most quintessence scalar field models fea-\nture a ground state with positive energy density, corre-\nsponding to a de Sitter (dS) vacuum, such a scenario\nhas proven extremely difficult to construct in a con-\ntrolled setting within string theory. In fact, it has been\nconjectured that string theory may be unable to ac-\ncommodate dS vacua (Danielsson & Van Riet 2018),\nas advocated by the swampland program (Vafa 2005;\nPalti 2019; Gra˜ na & Herr´ aez 2021), whose cosmologi-\ncal implications, particularly for inflation and DE, are\nfar-reaching to say the least (see e.g. Obied et al. 2018;\nAgrawal et al. 2018; Ach´ ucarro & Palma 2019; Garg &\nKrishnan 2019; Kehagias & Riotto 2018; Kinney et al.\n2019; Ooguri et al. 2019; Odintsov & Oikonomou 2020;\nOikonomou 2021b; Cicoli et al. 2022).6On the other\nhand, anti-dS (AdS) vacua, which correspond to a neg-\native CC (nCC), appear ubiquitously within string the-\nory, and are among the best understood quantum grav-\nity backgrounds by virtue of the AdS/CFT correspon-\ndence (Maldacena 1998). Of course, a nCC Λ <0 with\nenergy density ρΛ<0 is unable, on its own, to give\nrise to cosmic acceleration. However, an evolving DE\ncomponent with positive energy density ρx>0 on top\nof a nCC can be consistent with the observed late-time\nacceleration, provided ρx+ρΛis positive around the\npresent time, and amounts to about 70% of the total\nenergy budget. Such a scenario, which could be inter-\npreted in terms of a quintessence field whose potential\nfeatures a negative minimum (AdS vacuum), is of great\ntheoretical interest in light of the previous string-driven\nconsiderations. In fact, a number of recent works have\n6Two possible counterexamples to these conjectures are the\nKKLT (Kachru et al. 2003) and Large Volume Compactifica-\ntion (Balasubramanian et al. 2005) scenarios, although there is\nstill no complete consensus on whether the resulting uplifted\n(meta)stable dS vacua are sufficiently long-lived to be able to\nagree with observations.explored similar scenarios in light of standard CMB,\nBAO, and SNeIa observations (see e.g. Cardenas et al.\n2003; Poulin et al. 2018; Dutta et al. 2020; Visinelli et al.\n2019; Ruchika et al. 2023; Di Valentino et al. 2021a;\nCalder´ on et al. 2021; Sen et al. 2022; Malekjani et al.\n2023; Adil et al. 2023), finding that these models per-\nform equally well as ΛCDM, or are even potentially\nstatistically preferred.7The strong theoretical moti-\nvation behind such models, as well as their consistency\nwith standard cosmological probes, strongly motivates\nfurther studies thereof, and in particular a comparison\nagainst other available observations, potentially in dif-\nferent redshift ranges, where the deviations from ΛCDM\ncan be substantially more pronounced.\nIn Adil et al. (2023), it was shown by some of us that\na DE sector featuring an evolving component on top\nof a nCC can, within regions of parameter space con-\nsistent with standard cosmological observations, dras-\ntically alter the growth of structure at high redshifts\nwith respect to ΛCDM to the point of potentially restor-\ning concordance with the (photometric) JWST CEERS\nobservations. We stress the word “potentially” as this\nearlier analysis was merely an exploratory one: in fact,\nbesides only having explored a few benchmark points\nin parameter space, a careful treatment of corrections\nrequired for an accurate comparison to the JWST ob-\nservations was lacking, as well as a more complete as-\nsessment of the complementarity with standard cosmo-\nlogical probes (the last two points were, in fact, both\nexplicitly mentioned as motivation for a more detailed\nfollow-up work in the closing paragraph of Adil et al.\n2023). Given the enormous promise shown by a model\nfeaturing an evolving DE component on top of a nCC,\nit is our goal in the present work to revisit the model\nand go beyond the exploratory analysis of Adil et al.\n(2023), by providing a full comparison of such a model\nagainst the JWST CEERS observations, while also in-\ncluding information from standard cosmological probes.\nTo place our results on a much more solid footing from\nthe observational point of view, we also consider spectro-\nscopic data from the JWST FRESCO survey (Xiao et al.\n2023). Overall, our work confirms and reinforces the\nextremely promising conclusions reached earlier in Adil\net al. (2023), and provides further motivation for ex-\nploring dark sectors featuring components with negative\n7Another related very interesting possibility which has been ex-\nplored in the literature involves an AdS phase around recombina-\ntion, as in the case of so-called AdS-early dark energy models (Ye\n& Piao 2020a,b; Ye et al. 2021; Jiang & Piao 2021; Ye et al. 2023b;\nJiang et al. 2024; Jiang & Piao 2022; Jiang et al. 2023; Ye et al.\n2023a; Peng & Piao 2024).5\nenergy densities, while also highlighting once more the\nenormous potential held by observations of the abun-\ndance of massive galaxies at very high redshift in testing\nfundamental physics.\nThe rest of this paper is then organized as follows.\nIn Sec. 2 we briefly review the DE models considered\nin the rest of the work. Our analysis methods, includ-\ning the adopted datasets, are discussed in Sec. 3. The\nresults of our analysis, and in particular the resulting\nlimits in dynamical DE parameter space, are presented\nin Sec. 4. Finally, in Sec. 5 we draw concluding remarks\nand outline a number of potentially interesting avenues\nfor follow-up work.\n2.DARK ENERGY MODELS\nFollowing the earlier work of Adil et al. (2023), we\nconsider a DE sector consisting of a cosmological con-\nstant Λ ≷0 which in principle can take either sign (with\npositive or negative signs corresponding to a dS or AdS\nvacuum respectively), and with associated energy den-\nsityρΛ= Λ/8πGof the same sign. On top of this dS\nor AdS vacuum we place an evolving DE component\nwith strictly positive energy density ρx(z)>0. Rather\nthan committing to a specific microphysical model for\nthe evolving DE component, we assume that this is de-\nscribed by a time-evolving EoS wx(z) of the Chevallier-\nPolarski-Linder (CPL) form (Chevallier & Polarski 2001;\nLinder 2003):\nwx(z) =w0+waz\n1 +z. (1)\nThere are various reasons why we adopt the widely\nused CPL parametrization, ranging from its highly\nmanageable 2-dimensional nature, to its direct connec-\ntion to several physical DE models (including several\nquintessence DE models, see e.g. Linder 2003, 2006,\n2008a,b; Scherrer 2015) and, last but not least, for ease\nof comparison to the earlier work of Adil et al. (2023).8\nWe refer to the evolving component being quintessence-\nlike (phantom) at a given epoch if, at the redshift in\nquestion, w(z)>−1 (<−1). While w0controls the\ncurrent (quintessence-like or phantom) nature of DE,\nwhether or not this can change in the past is deter-\nmined by the value of wa, given that at asymptotically\n8See Colg´ ain et al. (2021) for a recent discussion on potential\nshortcomings of the CPL parametrization. For completeness, we\nnote that a number of other parametrizations for the EoS of\ndynamical DE components have been proposed in the literature,\nsee e.g. Efstathiou (1999); Jassal et al. (2005); Gong & Zhang\n(2005); Barboza & Alcaniz (2008); Ma & Zhang (2011); Pantazis\net al. (2016); Yang et al. (2018b); Pan et al. (2018); Yang et al.\n(2019); Singh et al. (2023) (see also Perkovic & Stefancic 2020, for\ndiscussions on the theoretical viability of these parametrizations).early times ( z→ ∞ ) the DE EoS tends to the value\nw0+wa. Finally, we note that models crossing which\ncan cross between the two regimes are typically referred\nto as “quintom” models, and have been widely studied\nin the literature (Feng et al. 2006; Guo et al. 2005; Cai\net al. 2007a,b; Zhang 2009; Saridakis 2010; Cai et al.\n2012; Bahamonde et al. 2018; Leon et al. 2018; Pan-\npanich et al. 2021), see Cai et al. (2010) for a review.\nIf we denote by ρ(0)\ncritthe current critical energy density,\nbyρ(0)\nxthe current energy density of the evolving DE\ncomponent, and by Ω x≡ρ(0)\nx/ρ(0)\ncritits density parame-\nter, the energy density of the evolving DE component\nas a function of redshift is given by the following:\nρx(z) = Ω xρ(0)\ncrit(1 +z)3(1+w0+wa)exp\u0012\n−3waz\n1 +z\u0013\n.\n(2)\nWe work under the assumption of a spatially flat\nFriedmann-Lemaˆ ıtre-Robertson-Walker Universe filled,\nbesides the evolving DE component described previ-\nously, by the cosmological constant Λ ≷0, alongside\nthe usual matter and radiation fluids (with density pa-\nrameters Ω mand Ω rrespectively).9Under these as-\nsumptions, and defining the density parameter of the\ncosmological constant Ω Λ≡Λ/3, the evolution of the\nHubble rate is governed by the following equation:\nH2(z)\nH2\n0= Ωr(1 +z)4+ Ωm(1 +z)3+ Ω Λ\n+ Ωx(1 +z)3(1+w0+wa)exp\u0012\n−3waz\n1 +z\u0013\n.(3)\nwhere the density parameters satisfy Ω r+ Ωm+ Ω Λ+\nΩx= 1. Finally, as noted in Adil et al. (2023), it is\nnatural to identify the combination of the evolving CPL\ncomponent and the cosmological constant Λ as compris-\ning the combined DE sector, whose total density param-\neter is Ω DE≡Ωx+ Ω Λ. The important thing to note\nis that, although Λ itself and therefore Ω Λcan be nega-\ntive, the total DE density and therefore Ω DEhave to be\npositive in order to be able to drive cosmic acceleration\nand maintain agreement with cosmological observations.\nRoughly speaking, such a DE sector can in principle\nbe compatible with cosmological observations provided\nΩDE≈0.7, which of course is possible even if Ω Λ<0, as\nnoted in several recent works (Dutta et al. 2020; Visinelli\net al. 2019; Sen et al. 2022). We note that, in order to\nagree with observations, a more negative Ω Λneeds to\n9We neglect neutrinos for simplicity, given their very limited im-\npact at the cosmological epochs of interest.6 Menci et al.\nbe compensated by more negative values of wx, moving\ntowards the phantom regime. These considerations lead\nto comparatively weak, order unity upper limits on |ΩΛ|,\nas discussed for instance in Visinelli et al. (2019) and Sen\net al. (2022), whereas different combinations of wx, Ωx,\nand Ω Λ(the latter two summing to the same value of\nΩDE) can lead to a very rich phenomenology, discussed\nin detail in Calder´ on et al. (2021) (see also Dash et al.\n(2024) for discussions on the expected sensitivity to a\nnCC from future 21-cm observations).\nFrom a theoretical standpoint, such a phenomeno-\nlogical model has been argued to loosely carry string\nmotivation in the case where Λ <0 (Visinelli et al.\n2019; Sen et al. 2022; Adil et al. 2023). While AdS\nvacua appear ubiquitously in string theory, the evolv-\ning DE component on top can, broadly speaking, be\njustified on the grounds that string compactifications\ntypically predict the existence of a plethora of ultra-\nlight (pseudo)scalar particles. This usually goes under\nthe name of “string axiverse” (see for example Svrcek\n& Witten 2006; Arvanitaki et al. 2010; Cicoli et al.\n2012; Visinelli & Vagnozzi 2019; Cicoli et al. 2023), with\nthe axion-like particles arising from Kaluza-Klein reduc-\ntion of higher-dimensional form fields on the topolog-\nical cycles of the compactification space. The topol-\nogy of the compactification manifold fixes the number\nof particles, typically of order hundreds or more, and\nwith masses spread over a huge number of decades,\nwith the rough expectation that the distribution of the\nlogarithms of the masses should be approximately uni-\nform (Kamionkowski et al. 2014; Emami et al. 2016;\nKarwal & Kamionkowski 2016), see also Marsh (2011);\nRuchika et al. (2023) for related studies. The impor-\ntant thing to note is that the effective EoS of multiple\ninteracting scalar fields can be phantom (Carroll et al.\n2003; Vikman 2005; Carroll et al. 2005; Deffayet et al.\n2010; Sawicki & Vikman 2013): this motivates the pos-\nsibility of a component with positive energy density, but\nwith EoS wx(z) potentially crossing −1 [as in Eq. (1) for\nsuitable choices of w0,wa], sitting on top of a nCC.\nConcerning the AdS vacuum itself, the exploratory re-\nsults of Adil et al. (2023) which our work seeks to con-\nfirm show that the relevant region of parameter space\nis one where |ΩΛ| ∼ O (1), i.e. where the nCC is of the\nsame order of the dS vacuum energy in ΛCDM (see also\nDutta et al. 2020; Visinelli et al. 2019; Sen et al. 2022).\nTherefore, the magnitude of the nCC is expected to be\n≲10−123in Planck units, and one could legitimately\nworry that this would introduce a (negative) cosmologi-\ncal constant problem. Intriguingly, recent work in string\ntheory by Demirtas et al. (2021, 2022) has led to the\nexplicit construction of supersymmetric AdS 4vacua ofthe right magnitude. This has been achieved within the\ncontext of type IIB string theory in compactifications\non orientifolds of Calabi-Yau threefold hypersurfaces,\nwith the resulting solution preserving N= 1 supersym-\nmetry, and with the key point of the construction be-\ning the perfect cancellation of all perturbative terms in\nthe superpotential. Such a construction explicitly shows\nthat it is possible to obtain an exponentially small nCC\nwithin string compactifications, thereby providing fur-\nther string motivation for the region of parameter space\nwe shall explore in this work.\nBefore moving on we note that, once the combina-\ntion of evolving CPL component and Λ is identified as\nmaking up the combined DE sector, a natural quantity\ncharacterizing the latter is the effective equation of state\nweff(z), which can be determined through the twice con-\ntracted Bianchi identity, and is given by (see Adil et al.\n2023, for the full calculation):\nweff(z) =Ωx(1+z)2+3w0+3wa[w0+(w0+wa)z] exp \n−3waz\n1 +z!\n−ΩΛ\nΩx(1+z)3(1+ w0+wa)exp\u0010\n−3waz\n1+z\u0011\n+ΩΛ,\nfor which in general weff(z= 0)̸=w0unless Ω Λ= 0.\nIt is also worth noting that if the values of Ω x, ΩΛ,w0,\nandwaare such that the total DE energy density goes\nthrough zero and therefore switches sign at a certain\nredshift, the associated effective EoS given by Eq. (4)\nnecessarily goes through a pole at the same redshift.\nThis has been discussed in detail in Ozulker (2022); Adil\net al. (2023), and does not in itself signal a pathology\ngiven that weff(z) is not associated to the dynamics of\na single microscopical degree of freedom – however, this\ndoes highlight the importance of focusing on the total\nDE energy density rather than the effective EoS.\n3.METHODS\nFor what concerns the properties of high-redshift\ngalaxies, and more generally the formation of structures,\nthe predictions of our model are controlled by seven pa-\nrameters: the matter density parameter Ω m, the baryon\ndensity parameter Ω b(as it controls the maximal bary-\nonic, and therefore stellar, content of a given host DM\nhalo), the density parameter of the (positive or negative)\ncosmological constant Ω Λ, the Hubble constant H0, the\npresent-day linear theory amplitude of matter fluctua-\ntions averaged in spheres of radius 8 h−1Mpc σ8, and\nfinally the parameters characterizing the evolving DE\ncomponent with positive energy density, w0andwa. We\nnote that the radiation density parameter Ω ris essen-\ntially fixed by extremely high-precision measurements\nof the CMB temperature monopole (and is in any case\nnegligible), from which it follows that the density pa-\nrameter of the evolving DE component Ω x>0 is fixed7\nby the closure relation Ω r+ Ωm+ Ω Λ+ Ωx= 1 (which\nbasically reduces to Ω DE= Ω x+ Ω Λ≈1−Ωm), and\ncannot therefore be treated as a free parameter.\nFor each set of cosmological parameters discussed\nabove, following Linder (2005) and as reported in Menci\net al. (2020, 2022) and Adil et al. (2023), we compute the\nevolution of the matter density contrast δ, from which\nwe obtain the linear growth factor of density perturba-\ntions D(z). We note that the evolution of D(z) can\ndiffer significantly from that within ΛCDM. The reason\nis that the equation for the evolution of δ(see e.g. Adil\net al. 2023) depends on both the normalized expansion\nrate E(z)≡H(z)/H0and its first derivative: a dif-\nferent background expansion therefore directly impacts\nthe growth of structure, which within the model at hand\ncan be either suppressed or enhanced depending on the\nchoice of cosmological parameters, with the value of Ω Λ\nplaying an important role (see Adil et al. 2023, for more\ndetailed discussions on this point).\nThe next step is to compute the predicted maximal\nabundance of galaxies with stellar mass M⋆. More pre-\ncisely, the relevant quantities are the comoving cumu-\nlative number or mass (stellar mass or DM halo mass)\ndensities, which quantify the number or mass density of\nhalos above a given threshold, or the number or stellar\nmass density of stars contained in galaxies more massive\nthan a given threshold. Here we proceed as in Menci\net al. (2020, 2022) and just recall the basic steps to\ncompute these quantities, while encouraging the reader\nto consult the above papers for further details (see also\nEqs. (3.7–3.10) of Adil et al. 2023, for explicit expres-\nsions of the four quantities discussed above). For a given\nset of the seven cosmological parameters we compute the\nDM halo mass function dn(M, z)/dM, which quantifies\nthe number of DM haloes of mass Mper unit mass per\nunit comoving volume in the mass range [ M;M+dM]\nat a given redshift. We do so following the prescrip-\ntion of Sheth & Tormen (1999), itself an extension of\nthe Press-Schechter formalism (Press & Schechter 1974)\naccounting for ellipsoidal collapse.\nFrom the DM halo mass function we can obtain the\nmaximal comoving number density of galaxies with stel-\nlar mass in the range [ M⋆;M⋆+dM⋆],dn⋆(M⋆, z)/dM ⋆.\nThis is given by dn⋆(M⋆, z)/dM ⋆=fbdn(M, z)/dM,\nwhere fb≡Ωb/Ωmis the cosmic baryon fraction. We\nstress that this is the maximal density because it is com-\nputed under the extremely optimistic assumption that\nthe entire available baryonic reservoir ends up being con-\nverted into stars. This assumption is of course unreal-\nistic, as in reality the efficiency of converting gas into\nstars, typically denoted by ϵ, is of order ϵ≲0.2 or less,\nwith a moderate redshift dependence (Leroy et al. 2008;Combes et al. 2011; Tacchella et al. 2018). However,\nthis assumption is extremely conservative for the pur-\nposes of our study: given that galaxies cannot outnum-\nber their DM haloes, one can exclude those cosmolog-\nical models for which even when ϵ= 1 the predicted\nnumber density of galaxies of given stellar mass at a\ncertain redshift falls short of the observed abundance of\ngalaxies within the same range of redshift and stellar\nmass. To put it differently, a galaxy of stellar mass M⋆\ncan only form if a DM halo of mass M⋆/ϵfbhas formed\nfirst. Finally, we can compute the corresponding cu-\nmulative comoving maximal number and stellar mass\ndensities of galaxies with stellar masses larger than a\ngiven observational threshold M⋆. These are obtained\nby integrating dn(M, z)/dM andfbMdn (M, z)/dM re-\nspectively, where the lower limit of the integration range\nis given by the threshold DM halo mass M⋆/fb,10and\naveraging over the cosmic volume V(z1, z2) enclosed be-\ntween the redshift range zmin−zmaxcovered by the ob-\nservations (the average is performed within the integral,\nagain see Eqs. (3.7–3.10) of Adil et al. 2023, for explicit\nexpressions of these quantities). Two further corrections\nare required before the computed cumulative comoving\nstellar mass density of galaxies, ρ⋆(>M⋆, z), can be\ncompared against the JWST observations, as we will\ndiscuss shortly.\nOur goal is now to compute ρ⋆(>M⋆, z) as a function\nof various choices of cosmological parameters, while en-\nsuring that the latter are in agreement with standard\ncosmological probes. This will allow us to discuss the\ncomplementarity between the latter, and the observed\nabundance of high-redshift galaxies from JWST. Oper-\nationally, we therefore proceed as follows. The cosmo-\nlogical datasets we consider are given below:11\n•CMB data : measurements of CMB temperature\nanisotropy and polarization power spectra, their\ncross-spectra, and CMB lensing power spectrum\nreconstructed from the temperature 4-point cor-\nrelation function, from the Planck 2018 legacy\n10The upper limit of the integration range is formally ∞, although\nthe integrand drops exponentially quickly for sufficiently large\nDM halo mass.\n11While more recent datasets are available for some of the measure-\nments below (especially for what concerns BAO and SNeIa data),\nthe reason why we chose these datasets was for consistency with\nthe work of Sen et al. (2022), in order to make use of the chains\nwhich were produced therein. Nevertheless, we expect very min-\nimal quantitative changes were we to use the most up-to-date\neBOSS BAO data (Alam et al. 2021) and PantheonPlus SNeIa\ndata (Scolnic et al. 2022), and no qualitative changes, given that\nour conclusions are almost entirely driven by the JWST observa-\ntions. We therefore expect them to be robust against the use of\nslightly more update BAO and SNeIa data.8 Menci et al.\nrelease. This combination is usually referred to\nasTTTEEE +lowl+lowE+lensing , and we anal-\nyse it making use of the official Planck likeli-\nhood (Aghanim et al. 2020a,b,c).\n•BAO data : isotropic and anisotropic distance\nand expansion rate measurements from the\nSDSS-MGS, 6dFGS, and BOSS DR12 collabora-\ntions. (Ross et al. 2015; Beutler et al. 2011; Alam\net al. 2017)\n•SNeIa data : distance moduli measurements from\nthePantheon SNeIa sample within the redshift\nrange 0 .01< z < 2.3 (Scolnic et al. 2018).\n•SH0ES prior : a prior on the Hubble constant\nH0= (73 .30±1.04) km /s/Mpc, as determined by\nthe SH0ES team (Riess et al. 2022).\nConsidering the combination of the above datasets, we\nderive constraints on our set of cosmological parameters\nby making use of Markov Chain Monte Carlo (MCMC)\nmethods, with predictions for the cosmological observ-\nables in question derived using the publicly available\nBoltzmann solver CLASS (Lesgourgues 2011; Blas et al.\n2011). We make use of the publicly available cosmolog-\nical MCMC sampler MontePython 3.3 (Audren et al.\n2013; Brinckmann & Lesgourgues 2019). We monitor\nthe convergence of the generated MCMC chains through\nthe Gelman-Rubin parameter R−1 (Gelman & Ru-\nbin 1992), requiring R−1<0.02 for our chains to\nbe considered converged, and analyze them using the\nGetDist package (Lewis 2019). For a full discussion of\nthe methodology and the resulting constraints, we refer\nthe reader to Sen et al. (2022).\nIn the next stage of the analysis, we introduce the\nJWST observations, in particular the inferred/observed\ncomoving cumulative stellar mass density ρobs(>M⋆).\nMore specifically, we consider the value of ρobs(>M⋆)\ninferred from two different classes of observations:\n•the six most massive, intrinsically red galaxies in\nthe redshift range 9 ≲z≲11, identified in the\nfirst NIRCam observations of the JWST CEERS\nprogram, as reported in Labb´ e et al. (2023), and\nwhich we treat as a measurement of ρobs(>M⋆)\natzeff= 10;\n•the three most massive, optically dark (dust-\nobscured) galaxies with robust spectroscopic red-\nshifts, in the redshift range 5 ≲z≲6, identi-\nfied within the JWST FRESCO NIRCam/grism\nsurvey, as reported in Xiao et al. (2023), and\nwhich we treat as a measurement of ρobs(>M⋆)\natzeff= 5.5.From the two above observations the inferred stellar\nmass density is of the order of ≳106M⊙Mpc−3and\n≳105M⊙Mpc−3respectively.\nFrom the MCMC chains described above, we then se-\nlect only those points which are consistent within 2 σ\nwith the standard cosmological probes discussed ear-\nlier. We stress that in our MCMC analysis we have\nallowed for Ω Λ≷0, i.e. either a dS or AdS vacuum en-\nergy component. Each of the models consistent within\n2σwith the standard cosmological observations is then\ncompared against the inferred values of ρobs(>M⋆)\nfrom the JWST CEERS and FRESCO observations de-\nscribed above. However, in order to properly carry out\nthis comparison, we need to account (and correct) for\nthe fact that both the values of ρobs(>M⋆) reported\nby Labb´ e et al. (2023) and Xiao et al. (2023) have been\nobtained assuming a certain fiducial cosmology, in this\ncase ΛCDM with a given choice of parameters. Let us\nrefer to the vector of fiducial cosmology parameters as\nf, whereas we denote by θthe vector of cosmological\nparameters for each point in our MCMC chain. As dis-\ncussed in detail in Menci et al. (2022), one then needs\nto apply two different corrections:\n•the inferred ρobs(>M⋆) has to be corrected by\na “volume factor” fvol≡Vf/Vθto appropriately\nrescale the volume density, where Vis the cos-\nmic volume computed for the specific model and\nchoice of cosmological parameters in question, at\nthe effective redshift zeff;\n•similarly, the measured masses have to be cor-\nrected by a “luminosity factor” flum≡dL,f/dL,θ\nto account for the fact that the stellar masses\nhave been inferred from the observed luminosities\nwithin the assumption of the given fiducial cosmol-\nogy, where dLis the luminosity distance computed\nfor the specific model and choice of cosmological\nparameters in question, once more at the effective\nredshift zeff.\nThe cosmology-corrected maximal comoving cumulative\nstellar mass densities can then be directly compared to\nthe JWST observations described above.\nWe consider a given cosmological model (which, we re-\ncall, is already ensured to be consistent within 2 σwith\nthe standard cosmological probes) to be excluded if the\nmaximal predicted stellar mass density of galaxies at\nthe given effective redshift is lower than the value in-\nferred from JWST CEERS or FRESCO observations.\nFrom this we can then compute the exclusion probabil-\nityPfor a given model, or conversely the probability of\nconsistency with the JWST observations Q(obviously\nP+Q= 1). This is performed using the procedure de-9\nscribed in Menci et al. (2022), through a Monte Carlo\nprocedure which also accounts for uncertainties in the\nobservational estimates of masses (in particular poten-\ntial systematics related to the spectral energy distri-\nbution fitting procedure), and statistical errors in the\nobservational number densities. We refer the reader\nto Menci et al. (2022) for further details on the calcula-\ntion of the exclusion probability.\n4.RESULTS\nWe begin by comparing our theoretical predictions\nagainst the stellar mass density inferred from the most\nmassive objects identified by NIRCam observations of\nthe JWST CEERS program, as reported in Labb´ e et al.\n(2023). In particular, these observations refer to the red-\nshift range zmin= 9≲z≲zmax= 11, which we treat at\nan effective redshift zeff= 10, focusing on the most mas-\nsive bin considered in Labb´ e et al. (2023), corresponding\ntoM⋆≥M⋆= 1010.5M⊙. As discussed in Sec. 3, from\nthe MCMC chains we extract cosmological parameter\nvectors consistent within 2 σwith the standard CMB,\nBAO, and SNeIa cosmological probes, predict the cor-\nresponding maximal comoving cumulative stellar mass\ndensity, which we then correct for the assumed fiducial\ncosmology, and then compare this against the JWST\nCEERS observations. For each model, we derive the ex-\nclusion probability P, or conversely the probability of\nconsistency with the JWST observations, Q= 1−P.\nThe results of our analysis are reported in terms of\nconsistency probability contour plots as a function of\nthe evolving DE parameters w0andwain Fig. 1, where\neach of the four quadrants refers to different regimes\nfor the vacuum energy density parameter Ω Λ: dS vac-\nuum Ω Λ>0 (lower right quadrant), AdS vacuum with\n−1<ΩΛ<0 (lower left quadrant), AdS vacuum with\neven more negative Ω Λ<−1 (upper right panel), and\nfinally any value of Ω Λ(upper left quadrant). Superim-\nposed on the same plots are 2 σand 3 σjoint confidence\nintervals in the w0-waplane obtained from a combina-\ntion of CMB, BAO, and SNeIa measurements (black\ncontours, as reported in Brout et al. 2022), as well\nas from the combined Hubble diagram of SNeIa and\nquasars (QSOs, blue contours, as reported in Risaliti\n& Lusso 2019), reaching up to redshift z∼5.5.\nThe first noteworthy result is that the existence of a\nregion of w0-waparameter space featuring a DE compo-\nnent with an AdS vacuum which is consistent with all\nthe standard cosmological probes andthe abundance of\nhigh-redshift galaxies inferred from JWST. This can be\ncontrasted to the earlier results of Menci et al. (2022)\nwhich only considered a dS vacuum, finding that a ma-\njor portion of the w0-waparameter space favored by thestandard cosmological probes is excluded at significance\n>2σby the JWST observations. This result on its own,\ntherefore, represents a solid quantitative confirmation of\nthe earlier exploratory results of Adil et al. (2023), and\nconfirms that the presence of a nCC in the DE sector\ncan help accelerate structure formation, thereby aiding\nthe formation of very massive objects at very high red-\nshift beyond what is possible within ΛCDM (for more\ndetailed explanations of why this occurs, we refer the\nreader to Adil et al. 2023).\nIt is also interesting to note that the favored region in\nw0-waparameter space is not far from ( w0,wa)=(−1, 0),\nwhile lying slightly within the phantom regime, in qual-\nitative agreement with Adil et al. (2023). Within this\nregion, we are able to achieve a consistency probability\nwith the JWST measurements of up to 47%. More-\nover, this same region is also consistent within ≈2σ\nwith the region favored by SNeIa+QSOs. As Ω Λis in-\ncreased, and we therefore move from the upper right\nquadrant to the two lower quadrants, we see a gradual\nshift in the required properties of the evolving DE com-\nponent: this progressively shifts from being character-\nized by mostly quintessence-like behaviour (potentially\ncrossing the phantom divide in the past, in agreement\nwith the results of Adil et al. 2023), to a quintessence-\nlike behaviour at present but a phantom behaviour in\nthe past, i.e. moving towards the lower right part of the\nw0-waplane. The latter point is in excellent agreement\nwith the earlier findings of Menci et al. (2022), which in-\ndeed focused precisely on this regime, and showed that\ntherein most of the wa>0 region is excluded, unless\nw0is deep in the phantom regime. Either way, it is\nclear that as Ω Λis increased, the point ( w0,wa)=(−1,\n0) is progressively disfavored, with a probability of con-\nsistency of <10% in the Ω Λ>0 case (lower right quad-\nrant), in agreement with Menci et al. (2022). In the\nlatter case, one should pay attention to the fact that\nFig. 2 in Menci et al. (2022) plots the exclusion proba-\nbility, whereas here we are plotting the probability for\nconsistency : when one accounts for this, it is easy to\nsee that the results are consistent with those of Menci\net al. (2022). On the other hand, an AdS vacuum plus\nevolving DE component scenario displays a good consis-\ntency with the JWST measurements (see especially the\nupper right quadrant): the reason is that the presence\nof the nCC helps accelerate structure formation, thereby\nreducing the need for phantom DE, which also happens\nto be theoretically much harder to accommodate with\nrespect to a quintessence-like component.\nWhile the above conclusions are robust on the theo-\nretical side, critical issues may affect the measurements\nof Labb´ e et al. (2023) we are comparing against. In first10 Menci et al.\nFigure 1. Contours in the w0-waplane colored by the\nprobability of consistency with the stellar mass density in-\nferred from the six most massive galaxies identified in the\nfirst NIRCam observations of the JWST CEERS program,\nin the redshift range 9 ≲z≲11, as reported in Labb´ e et al.\n(2023). The contours are reported considering the dynami-\ncal DE model discussed in Sec. 2, featuring an evolving DE\ncomponent with positive energy density, sitting on top of a\nvacuum energy component with density parameter Ω Λ≷0.\nThe four different quadrants correspond to different regimes\nfor the value of Ω Λ: all values Ω Λ≷0 (upper left quad-\nrant), Ω Λ<−1 (upper right quadrant), −1<ΩΛ<0 (lower\nleft quadrant), and Ω Λ>0 (lower right quadrant). As dis-\ncussed in Sec. 3, we only consider combinations of cosmo-\nlogical parameters consistent within 2 σwith the standard\nCMB+BAO+SNeIa cosmological probes. Our contours are\ncompared to the 2 σand 3 σcontours allowed by the latest\nCMB+BAO+SNeIa measurements (black contours) as re-\nported in Brout et al. (2022), and the combined Hubble dia-\ngram of SNeIa and quasars (QSOs, blue contours) reported\nin Risaliti & Lusso (2019). The green dot corresponds to the\nΛCDM case ( w0=−1,wa= 0).\nplace, potential uncertainties may affect the calibration\nof the JWST photometric data used therein. A second\nissue may concern the Chabrier IMF adopted by Labb´ e\net al. (2023) to derive stellar masses. While we do not\nexpect that assuming other universal forms for the IMF\nbased on low-redshift conditions would change (or even\nstrengthen) the constraints we have derived, the star\nformation process can be significantly different at red-\nshifts as high as the ones we are concerned with (see e.g.\nSteinhardt et al. 2023b).\nFor such reasons, it is important to perform an in-\ndependent test against spectroscopic data in a differ-\nent redshift range. With this in mind, we then pro-\nceed to carry out a comparison against the stellar massdensity inferred from the three most massive, optically\ndark (dust-obscured) galaxies with robust spectroscopic\nredshifts from the JWST FRESCO survey (Xiao et al.\n2023). Note that here as well the stellar masses have\nbeen conservatively inferred using a Chabrier IMF. We\nexpect that assuming other universal forms for the IMF\n(e.g. Salpeter, Kennicut) would yield similar or even\nlarger values of M⋆, and therefore our results can be\ninterpreted as being conservative in this sense. As in\nour previous comparison, we conservatively consider for\nthe measured value M⋆the one corresponding to the\nlower tip of the uncertainty provided by Xiao et al.\n(2023). As for the uncertainties in the number density,\nthe confidence levels thereon are derived using the sta-\ntistical method presented in Ebeling (2004), as discussed\nin Menci et al. (2022).\nThe results of such a comparison are shown in Fig. 2,\nwith the same quadrant structure as in Fig. 1. We see\nthat the results are completely consistent, and very sim-\nilar, to those obtained comparing against the photomet-\nric sample at higher redshift of Labb´ e et al. (2023). The\nonly quantitative difference is that overall we are able\nto achieve higher levels of consistency with the JWST\nFRESCO data, as high as 98%. Moreover, just as ob-\nserved earlier, we see that the fraction of evolving DE\nmodels with phantom behaviour increases with increas-\ning values of Ω Λ, confirming the trend seen with the\nphotometric sample: in other words, introducing a nCC\nhelps accelerate structure formation and reduce the need\nfor a phantom DE component, a welcome reduction from\nthe theory point of view.\nFor completeness, in Fig. 3 we display the associated\ncontours in the h-S8≡σ8p\nΩm/0.3 parameter space,\nwhere h≡H0/(100 km /s/Mpc) is the reduced Hub-\nble constant, and S8controls the overall strength of\nmatter clustering in the late Universe. While the con-\ntours we obtain are, by construction, consistent with the\nstandard cosmological observables we considered, they\ndisplay some tension with weak lensing (and, to some\nextent, galaxy clustering) measurements, which favour\nlower values of S8(see e.g. Di Valentino & Bridle 2018;\nDi Valentino et al. 2021d; Nunes & Vagnozzi 2021). This\nis of course not unexpected.\n5.CONCLUSIONS\nThe puzzling abundance of extremely massive galax-\nies at very high redshift unveiled by the early JWST\nobservations has the potential to upturn the current con-\ncordance ΛCDM model, itself already plagued by other\nobservational tensions. At the same time, the JWST ob-\nservations can be used to test alternative cosmological\nmodels, and can potentially rule out those models which11\nFigure 2. Same as in Fig. 1, but considering the three most\nmassive, optically dark (dust-obscured) galaxies with robust\nspectroscopic redshifts identified within the JWST FRESCO\nsurvey, in the redshift range 5 ≲z≲6, as reported in Xiao\net al. (2023).\nFigure 3. Contours of probability of consistency with the\nsame observations considered in Fig. 2, and with the same\ncolor coding, in the plane of the reduced Hubble constant\nh≡H0/(100 km /s/Mpc) and the clustering parameter S8≡\nσ8p\nΩm/0.3.\ndo not allow for a sufficiently fast growth of structure re-\nquired to explain the formation of the galaxies observed\nby JWST. As argued by various earlier works (Menci\net al. 2020, 2022; Adil et al. 2023), observations of theabundance of high-redshift galaxies can place strong\nconstraints on dynamical DE models, in a way which\nis highly complementary to standard CMB, BAO, and\nSNeIa cosmological probes at low redshifts. In the ear-\nlier exploratory work of Adil et al. (2023) some of us ar-\ngued that a DE sector featuring a negative cosmological\nconstant (AdS vacuum) with an evolving DE component\non top, which carries strong motivation from string the-\nory given the ubiquity of AdS vacua therein, can lead to\na more efficient growth of structure at early times, po-\ntentially explaining the JWST observations. Our goal in\nthis work has been that of going beyond this exploratory\nanalysis, by a)performing a more careful comparison of\nsuch a model against JWST data, while also b)better\nassessing the complementarity with standard cosmolog-\nical probes, and c)considering also spectroscopic data\nfrom the JWST FRESCO survey (Xiao et al. 2023) in\norder to place the results on a more solid footing.\nOur results, which qualitatively confirm the ex-\nploratory findings of Adil et al. (2023), can be summa-\nrized as follows:\n•a DE sector featuring an evolving component with\npositive energy density on top of a negative cos-\nmological constant with density parameter Ω Λ<0\ncan indeed improve consistency with photometric\nJWST observations at redshifts 9 ≲z≲11 (in\nagreement with the findings of Adil et al. 2023),\nwith consistency of up to 47%;\n•as Ω Λis increased, the behaviour evolving DE\ncomponent progressively shifts towards a more\nphantom behaviour – to put it differently, in-\ntroducing a negative cosmological constant al-\nlows for better agreement with the JWST obser-\nvations with a decreased need for phantom be-\nhaviour, which is more problematic compared to\na quintessence-like behaviour from a theoretical\npoint of view;\n•for Ω Λ>0, the JWST observations favor dynami-\ncal DE models which cross the phantom divide, in\nagreement with the findings of Menci et al. (2022);\n•these findings remain intact when consider-\ning spectroscopic observations from the JWST\nFRESCO survey at 5 ≲z≲6, placing all the\nearlier conclusions on a much more robust footing\nfrom the observational point of view;\n•the JWST observations, and more generally obser-\nvations of the abundance of high-redshift galaxies,\nare highly complementary to standard cosmolog-\nical probes from CMB, BAO, and SNeIa, as well\nas observations of high-redshift QSOs.12 Menci et al.\nIn short, we have shown that DE sectors featuring a\nnegative cosmological constant are extremely interest-\ning from the perspective of the high-redshift galaxies\nobserved by JWST. Moreover, this class of models, and\nmore generally models featuring negative energy densi-\nties in the DE sector, also have the potential to partially\n(albeit not completely) alleviate the Hubble tension (see\ne.g. Visinelli et al. 2019; Akarsu et al. 2020; Di Valentino\net al. 2021a; Acquaviva et al. 2021; Akarsu et al. 2021,\n2023a; Sen et al. 2022; Akarsu et al. 2023b; Adil et al.\n2023; Akarsu et al. 2023). This therefore confirms that\nsuch models are interesting from all three the observa-\ntional, phenomenological, and also theoretical perspec-\ntive, in the latter case in light of their strong motivation\nfrom string theory.\nOverall, our results further strengthen the case for illu-\nminating the nature of DE, and more generally testing\nnew fundamental physics, using observations of high-\nredshift galaxies, in a redshift range which cannot be\nreached by standard cosmological probes. We believe\nsuch observations have the potential to become an im-\nportant “emergent probe” (see e.g. Moresco et al. 2022)\nin coming years, especially with the achievable higher\nspatial resolution and improved sensitivity from future\nALMA/NOEMA and deep JWST spectroscopic obser-\nvations, whose complementarity with upcoming CMB\nmeasurements (e.g. Ade et al. 2019) it would be inter-\nesting to explore. From the theory side, it could be in-\nteresting to extend our analysis to one implementing a\nfundamental string scenario from first principles, along\nthe lines of Cicoli et al. (2019) and Ruchika et al. (2023)\n(see also Oikonomou et al. (2023)). The early JWST ob-\nservations have the potential to shake modern cosmology\nand shed new light onto the nature of the dark sector\ncomponents. Our work represents a small step in thelatter direction, and we cannot wait to see what lies in\nstore for cosmology as our long-awaited space telescope\nkeeps gathering data.\nACKNOWLEDGEMENTS\nN.M. acknowledges support from Ministero\ndell’Universit` a e della Ricerca (MUR, Italian Ministry\nfor Universities and Research) through the Progetti di\nRilevante Interesse Nazionale (PRIN) project “Black\nHole winds and the Baryon Life Cycle of Galaxies:\nthe stone-guest at the galaxy evolution supper” (grant\nagreement no. 2017-PH3WAT), and INAF Theory\nGrant ”AGN-driven outflows in cosmological models of\ngalaxy formation”. A.A.S. and U.M. acknowledge sup-\nport from the Science and Engineering Research Board\n(SERB) of the Government of India through research\ngrant no. CRG/2020/004347. S.A.A., U.M., and A.A.S.\nacknowledge the use of the High Performance Com-\nputing facility Pegasus at IUCAA, Pune, India. S.V.\nacknowledges support from the University of Trento and\nthe Provincia Autonoma di Trento (PAT, Autonomous\nProvince of Trento) through the UniTrento Internal Call\nfor Research 2023 grant “Searching for Dark Energy off\nthe beaten track” (DARKTRACK, grant agreement no.\nE63C22000500003), and from the Istituto Nazionale di\nFisica Nucleare (INFN) through the Commissione Sci-\nentifica Nazionale 4 (CSN4) Iniziativa Specifica “Quan-\ntum Fields in Gravity, Cosmology and Black Holes”\n(FLAG). This publication is based upon work from the\nCOST Action CA21136 “Addressing observational ten-\nsions in cosmology with systematics and fundamental\nphysics” (CosmoVerse), supported by COST (European\nCooperation in Science and Technology).\nREFERENCES\nAbdalla, E., et al. 2022, JHEAp, 34, 49,\ndoi: 10.1016/j.jheap.2022.04.002\nAch´ ucarro, A., & Palma, G. A. 2019, JCAP, 02, 041,\ndoi: 10.1088/1475-7516/2019/02/041\nAcquaviva, G., Akarsu, O., Katirci, N., & Vazquez, J. A.\n2021, Phys. Rev. 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D, 102, 023523,\ndoi: 10.1103/PhysRevD.102.023523"    },    {        "title": "2401.12664v1.Polynomial_and_rational_interpolation__potential__barycentric_weights__and_Lebesgue_constants.pdf",        "content": "POLYNOMIAL AND RATIONAL INTERPOLATION: POTENTIAL,\nBARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS∗\nKELONG ZHAO†AND SHUHUANG XIANG‡\nAbstract. In this paper, we focus on barycentric weights and Lebesgue constants for Lagrange\ninterpolation of arbitrary node distributions on [ −1,1]. The following three main works are included:\nestimates of upper and lower bounds on the barycentric weights are given in terms of the logarithmic\npotential function; for interpolation of non-equilibrium potentials, lower bounds with exponentially\ngrowing parts of Lebesgue constants are given; and for interpolation consistent with equilibrium\npotentials, non-exponentially growing upper bounds on their Lebesgue constants are given. Based\non the work in this paper, we can discuss the behavior of the Lebesgue constant and the existence\nof exponential convergence in a unified manner in the framework of potential theory.\nKey words. Barycentric interpolation, Lebesgue constant, potential theory\nAMS subject classifications. 65D05, 41A10, 41A20\n1. Introduction. For a linear interpolant of f(x)∈[−1,1]\nI(n)\nf(x) =nX\nk=0f(n)\nkl(n)\nk(x), f(n)\nk=f(x(n)\nk),−1≤x(n)\n0< x(n)\n1<···< x(n)\nn≤1,\nits Lebesgue constant is defined as\nΛn= max\nx∈[−1,1]Λn(x),Λn(x) =nX\nk=0|l(n)\nk(x)|.\nThis constant can be used to measure the amplification of approximation errors\ndue to rounding. If the Lebesgue constant grows too rapidly, the method fails to\nachieve high-order and high-precision approximations. Therefore, alongside the de-\nvelopment of interpolation theory, the study of Lebesgue constant has a long history\nand remains an important topic, especially for the investigation of Lebesgue constant\nin specific polynomial interpolation [15, 8, 6, 7] and rational interpolation [4, 5].\nIn these studies, the estimation of Lebesgue constant for interpolation methods\non equidistant nodes has long attracted interest [16, 3, 4]. One important reason for\nthis is that, in reality, the data and samples will often not be on Chebyshev points.\nInstead, there is a high probability that they will be on equidistant nodes. However,\nreality is diverse, and it is possible that the data and samples were obtained on other\nnon-isometric nodes. Thus, we can ask a more general question: How would the\nLebesgue constants behave for polynomials and rational interpolations on arbitrary\nnodes?\nSince Lebesgue constant varies with respect to n, these ‘arbitrary’ interpolation\nnodes cannot become irregular as ntends to infinity; otherwise, there is no point in\ndiscussing their Lebesgue constants. Therefore, it is reasonable to impose a restriction\nthat the unit measure µn= 1/(n+ 1)Pn\ni=0δx(n)\niconverges weakly to a unit measure\n∗Submitted to the editors DATE.\nFunding: This work was funded by National Science Foundation of China (No. 12271528).\n†School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, Peo-\nple’s Republic of China (clonezhao.1994@csu.edu.cn).\n‡School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, Peo-\nple’s Republic of China (xiangsh@csu.edu.cn) (corresponding author).\n1arXiv:2401.12664v1  [math.NA]  23 Jan 20242 KELONG ZHAO, AND SHUHUANG XIANG\nµ, i.e., as given in\n(1.1) lim\nn→∞Z1\n−1g(t) dµn(t) =Z1\n−1g(t)w(t) dt\nfor arbitrary continuous function gon [−1,1], where wis the density function of µ.\nThen, the ‘arbitrary’ nodes are transformed into nodes with an ‘arbitrary’ density\nfunction w.\nWe can define a logarithmic potential function\n(1.2) U(x) =Z1\n−1log1\n|x−t|w(t) dt+ϕ(x)\nwhere ϕis an external field. Similarly one can define a discrete potential function for\na rational interpolant rn,mat{x(n)\ni}n\ni=0\n(1.3) Un(x) =1\nn+ 1nX\ni=0log1\n|x−x(n)\ni|+ϕn(x), ϕ n(x) =1\nn+ 1mX\nj=1log|x−p(n)\nj|.\nwhere {p(n)\nj}m\nj=1are all poles of rn,m. When the external field is 0, it corresponds to\npolynomial interpolation.\nIf the discrete potential (1.3) tends to the steady-state potential function (1.2) as\nngrows, then a clear pattern emerges in the rate of growth of the Lebesgue constant\nconcerning the (1.2). We illustrate this with a simple test.\n•The first test involves polynomial interpolation on equidistant nodes, corre-\nsponding to a density function of w(t) = 1 /2 and a potential function given\nby\nUa(x) = 1−1\n2((x+ 1) log( x+ 1)−(x−1) log( x−1)).\n•The second test pertains to rational interpolation rn,nusing the Cheby-\nshev points. In this case, the corresponding density function is w(t) =\n1/(π√\n1−x2), the npoles are located at ±0.5i, and the potential function is\ngiven by\nUb(x) = log 2 + ϕ(x), ϕ(x) =1\n2log|x−0.5i|+1\n2log|x+ 0.5i|.\n•The third test deals with rational interpolation rn,non equidistant nodes.\nThe corresponding density function is w(t) = 1 /2, the npoles are located at\n±0.5i, and the potential function is given by\nUc(x) = 1−1\n2((x+ 1) log( x+ 1)−(x−1) log( x−1)) + ϕ(x).\nSince the nodes {x(n)\ni}n\ni=0and poles {p(n)\nj}m\nj=1of these interpolations are known,\nwe use barycentric interpolation [2]\n(1.4) rn,m(x) =nX\nk=0wkfk\nx−xk.nX\nk=0wk\nx−xk, wk=CQm\nj=1(xk−pj)Qn\ni=0,i̸=k(xk−xi), C̸= 0.POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 3\nfor this purpose. Similar to Lebesgue constant, the quotient of the largest barycentric\nweight by the smallest in absolute values, as given in max |wk|/min|wk|, gives an\nintuitive estimation on the quality of the interpolation method. Based on the upper\nplot of Figure 1b, it can be observed that the maximum ratio of all three interpolations\ngrows exponentially as [exp( d)]n, where dhappens to be the difference in the steady-\nstate potentials, as we will demonstrate in section 3.\n-1 -0.5 0 0.5 1-0.500.51Potential\n(a)\n0 50 100 150 200 250 3001001040108010120\n0 15 30 4510010510101015 (b)\nFig. 1 .Three potential functions, denoted as Ua, Ub, Uc, along with the differences between\ntheir respective maxima and minima: d1,d2,d3. (b): The maximum value of the ratio of absolute\nvalues of barycentric weights (top) and the Lebesgue constant (bottom) for the three interpolants.\nThe reference growth rate is O([exp dk]n)(k= 1,2,3).\nUsing section 3 as a foundation, we will show in section 4 that there exists a lower\nbound for the Lebesgue constants with a similar exponential growth factor [exp( d)]n.\nThus, these Lebesgue constants also exhibit exponential growth (as shown in the lower\nplot of Figure 1b) when the potential function is non-constant on the interval [ −1,1].\nA new question arises: If the potential function (1.2) is constant on the interval\n[−1,1], how rapidly does the Lebesgue constant grow? In section 5, we will provide\nan upper bound on Lebesgue constant in this scenario. This upper bound includes a\nlogarithmic growth term as well as an unknown growth term exp(2( n+ 1)δ±\nn), where\nδ±\nncharacterizes the rate at which the discrete potential function (1.3) converges to\nthe potential function (1.2). When δ−\nn=O(1/n), the Lebesgue constant has an upper\nbound with logarithmic growth.\nOne advantage of polynomial and rational interpolation is the exponential rate\nof convergence for analytic functions. The potential-theoretic explanation of this\nphenomenon is beautiful and classical. In section 6, we will discuss both exponential\nconvergence and Lebesgue constant in terms of potential theory. A sufficient condition\nfor fast and stable rational interpolation is given without proof, based on some rational\ninterpolations [9, 10, 17].\n2. Preliminaries. In this paper, our only requirement for interpolation nodes is\nthat their discrete measures µnconverge weakly to some positive measure µ. However,\nobtaining an intuition for these discrete points from Eq. (1.1) is challenging. A\ndefinition of the discrete points and the density function was provided in our previous\npaper , and here it is restricted to the interval [ −1,1]: A definition of a discrete point\n”obeying” the density function wwas given in our previous paper [17]. Definition 2.1\nis sufficient for the weak star convergence of the measure µ. In fact, it is easy to show4 KELONG ZHAO, AND SHUHUANG XIANG\nthat Definition 2.1 is also necessary for the weak star convergence of the measure µ\non [−1,1]. The definition restricted to [ −1,1] is given here:\nDefinition 2.1.A family of point sets {{x(n)\nk}n\nk=0:n= 1,2,···} ⊂ [−1,1]obeys\nthe density function w(t)>0for all t∈[−1,1]if, for any segment [a, b]⊆[−1,1], the\nfamily of point sets satisfies:\n(2.1) lim\nn→∞n[a,b]\nn+ 1=Zb\naw(t) dt,\nwhere n[a,b]denotes the number of points on [a, b].\nIn addition to the constraints on the nodes, we also have a requirement on the\npoles that the external field ϕngenerated by the poles converges consistently to ϕon\n[−1,1]. It may seem difficult to find poles to satisfy this condition, but in fact, we do\nnot need poles to construct rational interpolants with external fields ϕn. Recalling\nEq. (1.4), we have\n|wk|=CQm\nj=1|xk−pj|Qn\ni=0,i̸=k|xk−xi|=Cexp[( n+ 1)ϕn(xk)]Qn\ni=0,i̸=k|xk−xi|, C̸= 0.\nSince the barycentric weights on the real axis alternate between positive and negative\nvalues, the weights can be expressed as\n(2.2) wk=C(−1)kexp[( n+ 1)ϕn(xk)]. nY\ni=0,i̸=k|xk−xi|, C̸= 0.\n3. Barycentric weights and potential. For any rational function rn,m(where\nm≤n), it can be expressed in the form of barycentric, as described in Eq. (1.4) [2].\nIn this representation, the coefficient Cof the barycentric weights can be any nonzero\nconstant. The quotient of the largest barycentric weight by the smallest in absolute\nvalue is then independent of Cand the rate of growth of this ratio reflects, to some\nextent, the quality of an interpolant [12].\nFor polynomial interpolation, there exists a lower bound for the Lebesgue con-\nstant, as given by\nΛn≥1\n2n2max\n0≤k≤n|wk|.\nmin\n0≤k≤n|wk|.\nThis lower bound is one of the reasons for studying the barycentric weights. In the\ncase of more general rational interpolation, we can establish a connection between the\nLebesgue constant and the barycentric weights as well, but this connection relies on\nthe potential function as a bridge, which we will describe in the next section.\nWe assume that the constant Cof the barycentric weights in Eq. (1.4) is set to\n1, and we express the absolute value of barycentric weight in terms of the discrete\npotential function (1.3)\n(3.1) |w(n)\nk|=Qm\nj=1|x(n)\nk−p(n)\nj|\nQn\ni=0,i̸=k|x(n)\nk−x(n)\ni|= exp[( n+ 1)u(n)\nk(x(n)\nk)],\nwhere\nu(n)\nk(x) =Un(x) +log|x−x(n)\nk|\nn+ 1.\nTherefore, this part will be developed based on the properties of the discrete potential.POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 5\nLemma 3.1.Assume a class of rational interpolants rn,mwith nodes {x(n)\ni}n\ni=0in\nthe interval [−1,1]obeying a positive density function w, and whose poles {p(n)\nj}m\nj=1\nlie outside [−1,1], generating an external field ϕnthat converges to ϕ. Ifϕnis twice\ndifferentiable on [−1,1]and its second derivative has a lower bound M, then there\nexists N > 0such that for n > N , the discrete potential Unof the rational interpolants\nis convex on [−1,1]− {x(n)\ni}n\ni=0.\nProof. Recall the definition of discrete potential\nUn(x) =1\nn+ 1nX\ni=0log1\n|x−x(n)\ni|+ϕn(x),\nwhere\nϕn(x) =1\nn+ 1mX\nj=1log|x−p(n)\nj|,\nwe have\nU′′\nn(x) =1\nn+ 1nX\ni=01\n|x−x(n)\ni|2+ϕ′′\nn(x), (3.2)\n=Z1\n−11\n|x−t|2dµn(t) +ϕ′′\nn(x), (3.3)\nwhere µn=Pn\ni=0δx(n)\ni/(n+ 1) and x /∈ {x(n)\ni}n\ni=0.\nLet a function\n(3.4) FM1(x, t) = min {M1,1/(x−t)2}\nwhere M1such thatR1\n−1FM1(x, t) dt >−Mfor all x∈[−1,1]. Then there exists N1\nsuch thatR1\n−1FM1(x, t) dµn(t)>−Mfor all n > N 1. Therefore, we have\nU′′\nn(x)>Z1\n−1FM1(x, t) dµn(t) +M > 0.\nfor all n > N 1andx∈[−1,1]− {x(n)\ni}n\ni=0.\nRemark 3.2. Clearly, the conclusion of Lemma 3.1 still holds if we remove one\npoint from the n+ 1 interpolation nodes {x(n)\ni}n\ni=0.\nSince for sufficiently large n, the discrete potential Unis a convex function on\neach ( x(n)\ni−1, x(n)\ni). Since the value of the discrete potential on the nodes is + ∞, then\nthere exists a unique minima of Unin each ( x(n)\ni−1, x(n)\ni). Therefore we give a new\ndefinition:\nDefinition 3.3.For sufficiently large n, the discrete potential Unis convex on\neach (x(n)\ni−1, x(n)\ni), so there exists a unique {ζ(n)\ni}n\ni=1∈[−1,1]such that U′\nn(ζ(n)\ni) = 0\nandζ(n)\ni∈(x(n)\ni−1, x(n)\ni). We call {ζ(n)\ni}n\ni=1the inter-potential points of {x(n)\ni}n\ni=0.\nThese inter-potential points have the following property:6 KELONG ZHAO, AND SHUHUANG XIANG\nLemma 3.4.If the rational interpolation’s nodes {x(n)\ni}n\ni=0∈[−1,1]satisfies\na1n−b1≤(x(n)\ni−x(n)\ni−1)≤a2n−b2,\nthe external field ϕnsatisfies\n|ϕ′\nn(x)|< M 2logn,∀x∈(−1 +a1n−b1−1,1−a1n−b1−1).\nThen for sufficiently large n, there is\nζ(n)\ni∈(x(n)\ni−1+a1n−b1−1, x(n)\ni−a1n−b1−1).\nProof. By Definition 3.3, we have U′\nn(ζ(n)\ni) = 0, where\n(3.5) U′\nn(x) =1\nn+ 1nX\ni=0−1\nx−x(n)\ni+ϕ′\nn(x).\nNotice that U′′\nn(x)>0 for x∈[−1,1]− {x(n)\ni}n\ni=0, so we only need to prove\n(3.6) U′\nn(x(n)\ni−1+a1n−b1−1)<0, U′\nn(x(n)\ni−a1n−b1−1)>0.\n•When x=x(n)\ni−1+a1n−b1−1:\nU′\nn(x) =1\nn+ 1nX\nk=0−1\nxi−1−xk+a1n−b1−1+ϕ′(x)\n≤−nb1+1\na1(n+ 1)+1\nn+ 1nX\nk=inb1\n(k−i+ 1)a1−a1n−1+M2logn (3.7)\n=−nb1\na1(n+ 1)(n−n−i+1X\nk=11\nk−n−1) +M2logn. (3.8)\nIt is easy to prove b1≥1, then U′\nn(x)<0 for sufficiently large n.\n•When x=x(n)\ni−a1n−b1−1:\nU′\nn(x) =1\nn+ 1nX\nk=0−1\nxi−1−xk−a1n−b1−1+ϕ′(x)\n≥nb1+1\na1(n+ 1)−1\nn+ 1i−1X\nk=0nb1\n(i−k)a1−a1n−1−M2logn (3.9)\n=nb1\na1(n+ 1)(n−iX\nk=11\nk−n−1)−M2logn. (3.10)\nSoU′\nn(x)>0 for sufficiently large n.\nBased on the above lemmas, we can give upper and lower bound estimates for\nthe barycentric weights as follows:POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 7\nTheorem 3.5.Suppose a family of rational interpolants rn,mon[−1,1]with\nnodes {x(n)\ni}n\ni=0and its poles {p(n)\nj}m\nj=1/∈[−1,1], the node obeys the density func-\ntionw, the external field ϕnconverges to ϕand its inter-potential point is {ζ(n)\ni}n\ni=1.\nIf\n(i)a1n−b1≤(x(n)\ni−x(n)\ni−1)≤a2n−b2;\n(ii)|ϕ′\nn(x)|< M 2logn,∀x∈(−1 +a1n−b1−1,1−a1n−b1−1);\n(iii)−δ−\nn≤Un(ζ(n)\ni)−U(ζ(n)\ni)≤δ+\nn,∀1≤i≤n,\nthena1\nenb1+1[eU(ζ(n)\nki)]n+1e−(n+1)δ−\nn<|wi|<a2\nnb2[eU(ζ(n)\nki)]n+1e(n+1)δ+\nn\nwhere u(n)\ni(x) =Un(x)+1\nn+1log|x−xi|andζki= arg max {u(n)\ni(ζ(n)\ni), u(n)\ni(ζ(n)\ni+1)}for\n0< i < n (k0= 1, kn=n−1).\nProof. From Remark 3.2, we have u(n)\niis convex on ( x(n)\ni−1, x(n)\ni+1) if 0 < i < n .\nThen\nu(n)\ni(ζki) + (u(n)\ni)′(ζ(n)\nki)(−ζ(n)\nki+xi)≤u(n)\ni(xi)≤u(n)\ni(ζ(n)\nki) (3.11)\nUn(ζ(n)\nki) +log| −ζ(n)\nki−xi| −1\nn+ 1≤u(n)\ni(xi)≤Un(ζ(n)\nki) +log| −ζ(n)\nki−xi|\nn+ 1.\nConsidering the barycentric weights\n|w(n)\nk|=Qm\nj=1|xk−pj|Qn\ni=0,i̸=k|xk−xi|= exp[( n+ 1)u(n)\nk(xk)],\nwe have\nUn(ζ(n)\nki) +log| −ζ(n)\nki−xi| −1\nn+ 1≤log|w(n)\ni|\nn+ 1≤Un(ζ(n)\nki) +log| −ζ(n)\nki−xi|\nn+ 1.\nFrom Lemma 3.4, it is easy to prove a1n−b1−1<|ζ(n)\nki−xi|< a2n−b2. Then we have\nUn(ζ(n)\nki) +loga1−lognb1+1−1\nn+ 1<log|w(n)\ni|\nn+ 1< Un(ζ(n)\nki) +loga2−lognb2\nn+ 1.\nTherefore the barycentric weights has\n(3.12)a1e−(n+1)δ−\nn\nenb1+1[eU(ζ(n)\nki)]n+1<|w(n)\ni|<a2e(n+1)δ+\nn\nnb2[eU(ζ(n)\nki)]n+1.\nfor a sufficiently large nand 0 < i < n .\nWhen i= 0,u(n)\niis convex on [ −1, x(n)\n1). It is easy to show that u(n)\n0is monoton-\nically increasing on [ −1, x(n)\n1) when nis large enough. Thus, u(n)\n0satisfies Eq. (3.11).\nSimilarly, it follows that u(n)\nnalso satisfies Eq. (3.11). Then, Eq. (3.12) also holds\nwhen i= 0 or n.\nIfUis constant on the interval [ −1,1], then for all absolute values of the barycen-\ntric weights |wi|, the upper and lower bounds share the same exponential term. This8 KELONG ZHAO, AND SHUHUANG XIANG\nsuggests that we can adjust the value of Cin Eq. (1.4) to eliminate this common\nexponential factor. Consequently, we can derive simplified weights that do not exhibit\nexponential growth with n.\nIfUis continuous on the interval [ −1,1] but not constant, we can let\nx∈arg min\nx∈[−1,1](U(x)),x∈arg max\nx∈[−1,1](U(x)).\nGiven the existence of a consistent upper bound on the distance between neighboring\nnodes, denoted by a2n−b2, we can always find two sequences of nodes {x(n)\ni(n)}∞\nn=1and\n{x(n)\nj(n)}∞\nn=1such that\nlim\nn→∞x(n)\ni(n)=x,lim\nn→∞x(n)\nj(n)=x.\nholds. This leads to the following corollary:\nCorollary 3.6.For sufficiently large n,\na1nb2−b1−1\nea2e−(n+1)(δ−\nn+δ+\nn)ρn+1≤max\n0≤k≤n|w(n)\nk|\nmin\n0≤k≤n|w(n)\nk|≤ea2\na1nb2−b1−1e(n+1)(δ−\nn+δ+\nn)ρn+1\nholds, where ρ=eU(x)−U(x).\n4. Lebesgue constants for exponential growth. The Lebesgue function can\nalso be expressed in terms of the barycentric weights, as shown in\nΛn(x) =nX\nk=0|wk|nY\ni=0,i̸=k|x−xi|.mY\nj=1|x−pj|,\nwhere the barycentric weights are provided in Eq. (3.1). Consequently, the Lebesgue\nfunction can be inscribed by the potential function. Considering the definition of\nLebesgue constant, it is easy to obtain the following rough lower bound on the poten-\ntial function.\nTheorem 4.1.With the same premises as Theorem 3.5, ifUis a continuous\nfunction on [−1,1], then for a sufficiently large n, the Lebesgue constants Λnhave\nΛn≥a1e−(n+1)(δ+\nn+δ−\nn)\n2enb1+1ρ(n+1), ρ =eU(x)−U(x).\nProof. First we represent the Lebesgue function using discrete potentials Unand\nbarycentric weights wk:\nΛn(x) =nX\nk=0|R(n)\nk(x)|=nX\nk=0|wk|nY\ni=0,i̸=k|x−xi|.mY\nj=1|x−pj|\n=nX\nk=0[e−Un(x)]n+1|wk|\n|x−xk|,\nwhere R(n)\nk(xk) = 1 and R(n)\nk(xi) = 0 for 0 ≤i≤n(i̸=k). Considering Theorem 3.5,POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 9\nwe have\nΛn(ζ(n)\nj)>|R(n)\ni(ζ(n)\nj)|= [e−Un(ζ(n)\nj)]n+1|wk|\n|ζ(n)\nj−xk|\n>[e−U(ζ(n)\nj)]n+1e−(n+1)δ+\nna1\nenb1+1[eU(ζ(n)\nki)]n+1e−(n+1)δ−\nn/2\n=a1e−(n+1)(δ+\nn+δ−\nn)\n2enb1+1[eU(ζ(n)\nki)−U(ζ(n)\nj)]n+1\nfor all 0 ≤i, j≤n.\nSince Definition 3.3, we can find {ζ(n)\nαn}∞\nn=1(1≤αn≤n) and {ζ(n)\nβn}∞\nn=1(1≤\nβn≤n) such that\nlim\nn→∞ζ(n)\nαn=x, lim\nn→∞ζ(n)\nβn=x.\nTherefore, the Lebesgue constants Λ nhave\nΛn≥a1e−(n+1)(δ+\nn+δ−\nn)\n2enb1+1ρ(n+1), ρ =eU(x)−U(x).\nfor a sufficiently large n.\nThe result of Theorem 4.1 is rather crude. This limitation arises from our use\nof only one basis function whose absolute value exhibits exponential growth. The\nLebesgue function, however, is the sum of the absolute values of n+1 basis functions.\nDespite this, our choice of a basis function with the fastest exponential growth rate\nallows us to inscribe the exponential growth rate of the Lebesgue constant by the\ndifference of the potential functions, precisely what we aim to demonstrate. The\ncoefficients and algebraic growth part of Theorem 4.1 are not our primary focus.\nReferring to the proof of Theorem 4.1, we can determine the exponential growth\nrate of the Lebesgue function at any point ˆ x∈[−1,1]. Assuming that ˆ xis not always\nan interpolation node, the exponential part of the growth rate of R(n)\ni(ˆx) with respect\ntondoes not exceed [exp( U(ˆx)−U(x))]n, where x(n)\niconverges to x. Since U(x) is a\nminimum, the exponential part of the growth rate of Λ n(ˆx) likewise does not exceed\n[exp( U(ˆx)−U(x))]n.\nTo verify the above conclusions, we first consider an example of polynomial in-\nterpolation (EXAMPLE 1). The density function of its nodes is given by\nw(t) =exp(−t2)√πErf(1),\nwhere Erf is Gauss error function [1]. This density function is the result of a Gaussian\ndistribution restricted to [ −1,1] and normalized. Figure 2a illustrates the correspond-\ning Lebesgue function for this polynomial interpolation at different values of n. It can\nbe observed that the Lebesgue function grows at varying rates at different points.\nFigure 2b then presents the values of the Lebesgue function at 0, 0 .25, 0.5, and\n0.75, along with the reference growth rate we have provided. In the figure, ρ1=\nexp(U(0)−U(0.25)), ρ2= exp( U(0)−U(0.5)), and ρ3= exp( U(0)−U(0.75)), since\nU(0) is the maximum value of Uon [−1,1]. As ngrows, the four points we fixed will\nnot always be in the middle between nodes, causing the Lebesgue functions at these\npoints to fluctuate. However, overall, it coincides with our reference line.10 KELONG ZHAO, AND SHUHUANG XIANG\n-1 -0.5 0 0.5 110010510101015\n(a)\n0 50 100 150 20010010510101015 (b)\nFig. 2 .(a): Lebesgue functions of EXAMPLE 1 at different n. (b): Values of Lebesgue\nfunctions of EXAMPLE 1 at different points.\nThis also indicates that the Lebesgue constant of an interpolation method, even\nif it grows rapidly, does not imply that it will be universally affected by rounding\nerrors. For points where the value of the potential function is close to the maximum,\nthe impact from rounding errors will be small. Conversely, for points other than the\ninterpolation nodes, the lower the value of the potential function, the greater the\namplification of the rounding error.\nReducing the Lebesgue constant of polynomial interpolation can be achieved ef-\nfectively by adding poles. The density function in EXAMPLE 1 has a larger potential\ndifference compared to that of the density function of equidistant nodes, resulting in\na faster exponential growth rate for its Lebesgue constant. However, if we add poles\n(external field), such as the external field given by\nϕ(x) =1\n2log|x−0.5i|+1\n2log|x+ 0.5i|\nin the section 1, the difference in the potential function is reduced (EXAMPLE 2), as\nshown in Figure 3a.\n-1 -0.5 0 0.5 10.20.40.60.811.21.4\nExample 1\nExample 2\n(a)\n0 50 100 150 20010010510101015\nExample 1\nExample 2\nExample 3 (b)\nFig. 3 .(a): Potential functions of EXAMPLE 1 and EXAMPLE 2 along with the difference\nbetween their respective maximum and minimum values: d4,d5. (b): Lebesgue constants of EXAM-\nPLE 1, EXAMPLE 2, and EXAMPLE 3 with their respective growth reference lines.\nTheorem 4.1 shows that the exponentially growing part of the Lebesgue constantPOTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 11\nfor EXAMPLE 1 and EXAMPLE 2 is O([exp d4]n) and O([exp d5]n), respectively.\nIt is straightforward to demonstrate that this density function corresponds to the\nparameter b1= 1. Assuming δn=O(1/n), the lower bounds on the growth rate of\nthe Lebesgue constants for both are given by O([exp d4]n/n2) and O([exp d5]n/n2).\nHowever, the reference lines provided in the figure are given by O([exp d4]n/n) and\nO([exp d5]n/n), which are ntimes larger than the estimates in Theorem 4.1.\nThe reason for this is that the lower bound in Theorem 4.1 arises from one basis\nfunction |R(n)\ni(ζ(n)\nj)|with a growth rate of O(ρn/nb1+1) where ρ= exp( U(α)−U(β)).\nIn reality, there is not just one basis function with a similar exponential growth, but\nan infinite number of them of the same order as n. Let a very small quantity c >0,\nXc={x∈[−1,1] :U(x)> U(α)−c}ifUis continuous on [ −1,1]. Then, for any node\nx(n)\nk∈Xc,|R(n)\nk(ζ(n)\nj)|grows faster than O(ρn\nc/nb1+1) where ρc= exp( U(α)−U(β)−\nc). The number of these nodes ncsatisfies lim n→∞nc/n+ 1 =R\nXcw(t) dt. Thus the\nactual growth rate in Examples 1 and 2 is ntimes the lower bound of Theorem 4.1.\n-1 -0.5 0 0.5 110010510101015\n(a)\n-1 -0.5 0 0.5 110-2100102104 (b)\nFig. 4 .Absolute values of several Lagrangian basis functions for EXAMPLE 1 (a) and EX-\nAMPLE 2 (b) when n= 39.R(n)\nkdenotes the basis functions on x(n)\nk.\nA question arises: Can the exponential growth of the Lebesgue constant be\navoided by adding poles? The answer is yes, as long as the potential function can be\nmade constant on [ −1,1]. We want ϕnto satisfy\nlim\nn→∞Z1\n−1log1\n|x−t|w(t) dt+ϕn(x) =¯U,∀x∈[−1,1]\nwhere ¯Uis a constant. We can set\nϕn(x) =¯U−Z1\n−1log1\n|x−t|w(t) dt.\nThen according Eq. (2.2), we have\nwk=C(−1)kexp[−(n+ 1)Z1\n−1log1\n|xk−t|w(t) dt]. nY\ni=0,i̸=k|xk−xi|, C̸= 0.\nWe used such a rational interpolant for the density function in EXAMPLE 1 to\nobtain EXAMPLE 3. The new rational interpolant avoids the exponential growth of\nLebesgue constant, as illustrated in Figure 3b.12 KELONG ZHAO, AND SHUHUANG XIANG\n5. The Lebesgue constant in an equilibrium potential. In the case of\nthe equilibrium potential, EXAMPLE 3 demonstrates no exponential growth in its\nLebesgue constant. However, Theorem 4.1 only establishes that the exponential\ngrowth part of the lower bound estimate for the Lebesgue constant vanishes for the\nequilibrium potential. To ascertain whether the Lebesgue constant avoids exponential\ngrowth, we need to estimate its upper bound.\nThis proof is divided into two steps. First, it is necessary to demonstrate that the\namplitude of the oscillations of the basis functions does not increase exponentially with\nn, i.e., the phenomenon depicted in Figure 4 does not occur (Lemma 5.1, Lemma 5.2).\nThe second step is then to estimate the accumulation of the absolute values of the\nbasis functions to obtain a consistent upper bound for the Lebesgue function - in other\nwords, to obtain an upper bound estimate for the Lebesgue constant (Theorem 5.3).\nThe results show the existence of a non-exponentially increasing upper bound for the\nLebesgue constant at the equilibrium potential.\nLemma 5.1.Two right triangles ∆ABC and∆DEF are shown in Figure 1. Ex-\ntendABto intersect EFatH. Make HGperpendicular to BEatG. If|BC|=|DE|,\nthen\n|BG|\n|EG|=|DF|\n|AC|.\nA C\nD\nF\nHB\nG\nE\n(a)\nA\nBC\nD\nEF\nG H (b)\nFig. 5 .(a): Schematic representation of Lemma 5.1. (b): Schematic representation of the\nproof of Lemma 5.2.\nLemma 5.2.Under the assumptions of Theorem 3.5, ifUis constant and equal\nto the constant Con [-1,1], then the Lagrangian basis function R(n)\niwith rational\ninterpolation rn,mhas the following upper bound for a sufficiently large n:\n•For all x∈(x(n)\ni−1, x(n)\ni+1)\n|R(n)\ni(x)| ≤e(n+1)(δ−\nn+δ+\nn)+1;\n•For all x∈[−1,1]−(x(n)\ni−1, x(n)\ni+1)\n|R(n)\ni(x)| ≤min(e(n+1)(δ−\nn+δ+\nn)+1|ζ(n)\ni−x(n)\ni|\n|x−x(n)\ni|, e(n+1)(δ−\nn+δ+\nn)+1|ζ(n)\ni+1−x(n)\ni|\n|x−x(n)\ni|).POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 13\nProof. According the assumptions, we have Un(x)> U(x)−δ−\nn=C−δ−\nn. Let\nu(n)\ni(x) =Un(x) +1\nn+ 1log|x−x(n)\ni|.\n•For all x∈(x(n)\ni−1, x(n)\ni+1).\nSince Lemma 3.1 and Remark 3.2, u(n)\niis convex on ( x(n)\ni−1, x(n)\ni+1) for a suf-\nficiently large n. As show in Figure 5b, let A= (ζ(n)\ni, u(n)\ni(ζ(n)\ni)),F=\n(ζ(n)\ni+1, u(n)\ni(ζ(n)\ni+1)) and make tangents AHandFHtou(n)\nithrough AandB.\nThen the convex function u(n)\niwill be above the point H, i.e.,\nu(n)\ni(x)> yH,∀x∈(x(n)\ni−1, x(n)\ni+1).\nNotably\n|BC|=|ζ(n)\ni−x(n)\ni||(u(n)\ni)′(ζ(n)\ni)|=1\nn+ 1\nand\n|DE|=|ζ(n)\ni+1−x(n)\ni||(u(n)\ni)′(ζ(n)\ni+1)|=1\nn+ 1.\nThen from Lemma 5.1, we have\n(5.1) u(n)\ni(x)>(1−λ)yF+λyA−1\nn+ 1,∀x∈(x(n)\ni−1, x(n)\ni+1)\nwhere\nλ=|AC|\n|DF|+|AC|,1−λ=|DF|\n|DF|+|AC|.\nIn addition,\n(5.2) u(n)\ni(x(n)\ni)≤(1−λ)yA+λyF.\nConsidering\nC−δ−\nn+1\nn+ 1log|AC|< yA=u(n)\ni(ζ(n)\ni)< C+δ+\nn+1\nn+ 1log|AC|\nand\nC−δ−\nn+1\nn+ 1log|DF|< yF=u(n)\ni(ζ(n)\ni+1)< C+δ+\nn+1\nn+ 1log|DF|,\nit is easy to prove that\n(5.3)\nyF−yA≤δ−\nn+δ+\nn−1\nn+ 1log|AC|\n|DF|;yA−yF≤δ−\nn+δ+\nn+1\nn+ 1log|AC|\n|DF|.14 KELONG ZHAO, AND SHUHUANG XIANG\nThe basis function R(n)\ni(x) has\nR(n)\ni(x) =Qn\nk=0,k̸=i|x−x(n)\nk|Qm\nj=1|x−pj||wi|\n= exp[ −(n+ 1)u(n)\ni(x)] exp[( n+ 1)u(n)\ni(x(n)\ni)]\n= exp[( n+ 1)( u(n)\ni(x(n)\ni)−u(n)\ni(x))].\nFrom Eq. (5.1), (5.2) and Eq. (5.3), we have\nR(n)\nk(x)<exph\n(n+ 1)[(1 −λ)(yA−yF) +λ(yF−yA)] + 1i\n≤exph|DF| − |AC|\n|DF|+|AC|log|AC|\n|DF|+ (n+ 1)( δ−\nn+δ+\nn) + 1i\n≤exp[( n+ 1)( δ−\nn+δ+\nn) + 1] .\n•For all x∈[−1,1]/[x(n)\ni−1, x(n)\ni+1]. Since Un(x)> C−δ−\nn,\n(5.4) u(n)\nk(x)> C−δ−\nn+1\nn+ 1log|x−x(n)\nk|.\nIt is easy to prove that\nyA=u(n)\ni(ζ(n)\ni)≤C+δ+\nn+1\nn+ 1[log(λ) + log( |AC|+|DF|)];\nyF=u(n)\ni(ζ(n)\ni+1)≤C+δ+\nn+1\nn+ 1[log(1 −λ) + log( |AC|+|DF|)].\nThen we have\nu(n)\ni(x(n)\ni)≤(1−λ)yA+λyF\n≤C+δ+\nn+(1−λ) log( λ) +λlog(1−λ) + log( |AC|+|DF|)\nn+ 1.\nIt is easy to verify that\n(1−λ) log( λ) +λlog(1−λ)<log(λ) + 1;\n(1−λ) log( λ) +λlog(1−λ)<log(1−λ) + 1,\nthen\n(5.5) u(n)\ni(x(n)\ni)< C+δ+\nn+log|AC|+ 1\nn+ 1;\n(5.6) u(n)\ni(x(n)\ni)< C+δ+\nn+log|DF|+ 1\nn+ 1.\nFinally, the lemma is proved due to Eq. (5.4), (5.5), (5.6), and\n|R(n)\ni(x)|= exp[( n+ 1)( u(n)\ni(xi)−u(n)\ni(x))].POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 15\nTheorem 5.3.Under the assumptions of Theorem 3.5, ifUis constant and equal\nto the constant Con [-1,1], then the Lebesgue constant Λnof rational interpolation\nrn,mhas the following upper bound for a sufficiently large n:\nΛn≤e(n+1)(δ−\nn+δ+\nn)+1(5 + 2 b1loga1+ 2b1logn).\nProof. Consider the Lebesgue function Λ n(x), where x∈[−1,1]. If x∈ {x(n)\nk}n\nk=0,\nΛn(x) = 1. Setting x /∈ {x(n)\nk}n\nk=0, then we can find i∈ {1,2, . . . , (n−1)}such that\nx∈(x(n)\ni−1, x(n)\ni+1). From Lemma 5.2, we have\nΛn(x) =nX\nk=0|R(n)\nk(x)|\n≤e(n+1)(δ−\nn+δ+\nn)+1[i−1X\nk=0|ζ(n)\nk+1−x(n)\nk|\n|x−x(n)\nk|+ 1 +nX\nk=i+1|ζ(n)\nk−x(n)\nk|\n|x−x(n)\nk|]\n< e(n+1)(δ−\nn+δ+\nn)+1[i−3X\nk=0|ζ(n)\nk+1−x(n)\nk|\n|x−x(n)\nk|+ 5 +nX\nk=i+3|ζ(n)\nk−x(n)\nk|\n|x−x(n)\nk|]\n< e(n+1)(δ−\nn+δ+\nn)+1[i−3X\nk=0|x(n)\nk+1−x(n)\nk|\n|x−x(n)\nk|+ 5 +nX\nk=i+3|x(n)\nk−1−x(n)\nk|\n|x−x(n)\nk|]\n< e(n+1)(δ−\nn+δ+\nn)+1[Zx(n)\ni−2\n−11\nx−tdt+ 5 +Z1\nx(n)\ni+21\nt−xdt]\n=e(n+1)(δ−\nn+δ+\nn)+1[log(1 −x2) + 5−log(x−x(n)\ni−2)−log(x(n)\ni+2−x)]\n< e(n+1)(δ−\nn+δ+\nn)+1[5−log(x(n)\ni−1−x(n)\ni−2)−log(x(n)\ni+2−x(n)\ni+1)]\n< e(n+1)(δ−\nn+δ+\nn)+1[5−2 log( a1n−b1)]\n< e(n+1)(δ−\nn+δ+\nn)+1[5 + 2 b1loga1+ 2b1logn]\nTherefore Λ n≤e2(n+1)(δ−\nn+δ+\nn)+1(5 + 2 b1loga1+ 2b1logn).\n6. Interpolation Error and Lebesgue Constant. According to the conclu-\nsion of the previous section, if a rational interpolation, whose nodes obey some nonzero\ndensity function w, has an external field ϕnthat satisfies\nlim\nn→∞max\nx∈[−1,1]|ˆUn(x)−¯U|= 0,ˆUn(x) :=Z1\n−1log1\n|x−t|w(t) dt+ϕn(x)\nwhere ¯Uis a constant. Then the Lebesgue constant of the rational interpolation does\nnot grow exponentially.\nHowever, the condition to avoid the exponential growth of the Lebesgue con-\nstant is not sufficient to achieve exponential convergence of the error. An additional\ncondition needed to make rational nterpolation exponentially convergent for analytic\nfunctions is that the external field ϕnconverges consistently on a neighborhood of\n[−1,1]. As shown in Figure 6, if the external field converges consistently on a neigh-\nborhood of [ −1,1], then there exists an enclosing path Γ such that the interior of the\nenclosing path does not contain poles as well as singularities of the interpolated func-\ntion. Then the exponential decay of the error is obtained according to the Hermite\nintegral formula for rational interpolation [17].16 KELONG ZHAO, AND SHUHUANG XIANG\n-1 0 1-1.5-1-0.500.511.5\n-1 0 1 -1 0 1\n00.10.20.30.40.50.6\nFig. 6 .Contours of the potential function Ungenerated by (n+ 1) nodes (blue) and mpoles\n(red). As ngrows, the potential function stabilizes in some neighborhood of [−1,1]. The computation\nof these nodes and poles is detailed in [17].\n6.1. Take the Floater-Hormann interpolation as an example. Barycen-\ntric rational interpolation is often characterized by the barycentric weights and nodes\n[14, 9, 11]. In these methods, the poles of the rational interpolation are implicit but\ncan be uniquely determined from the barycentric weights and nodes. One way to\ncompute the poles using weights and nodes is to translate them into the computation\nof matrix eigenvalues, as detailed in [12, Chapter 2.3.3]. We still denote these poles\nas{p(n)\nj}m\nj=1, which gives us the external field ϕn.\nF-H rational interpolation is one of the very effective methods of rational interpo-\nlation [9, 10, 13]. There are an F-H rational interpolation that achieves exponential\nconvergence to analytic functions at equidistant nodes [10]. The barycentric weights\nof the equidistant nodes are denoted as\n(6.1) w(n,d)\ni= (−1)inX\nk=d\u0012d\nk−i\u0013\n,d(n)\nn→CFH(n→ ∞ )\nfor a fixed CFH∈(0,1]. Figure 7 illustrates the contours of the discrete potential for\nthe equidistant nodes at CFH= 0.25. As nincreases, the external field generated by\nthe poles remains stable in a neighborhood on [ −1,1]. Thus, by directly observing the\npotential function, the method may achieve exponential convergence to the analytic\nfunction.\nHowever, Figure 7 shows that the potential function is not equalized on [ −1,1],\nand the potential is lower at the edges of the interval. The reason for this is that the\npotential generated by the uniform density is weakly singular (with an unbounded\nderivative) at both endpoints. In contrast, the potential generated by the poles con-\nverges on a neighborhood of [ −1,1], so its derivative on [ −1,1] is bounded. As a\nresult, the potential function will not be a constant on [ −1,1].\nAccording to the conclusion in section 4, the Lebesgue constant has an exponential\ngrowth rate of (exp ρ)n, where ρis the difference between the maximum and minimum\nvalues of the potential function on [ −1,1]. The section 3 shows that the growth rate\nof the barycentric weight ratio is likewise (exp ρ)n. Thus, from the barycentric weight\nEq. (6.1), the maximum ratio is 2d≈(2CFH)n, and then the exponential growth rate\nof the Lebesgue constant can be obtained to be (2CFH)n. This is in agreement with\nthe conclusions of [13].POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 17\n-1 0 1-1.5-1-0.500.511.5\n-1 0 1 -1 0 1\n-0.200.20.40.6\nFig. 7 .Contours of the potential function Ungenerated by the nodes (blue) and poles (red) of\nthe F-H rational interpolation [] when d/n= 0.25. As ngrows, the potential function stabilizes in\nsome neighborhood of [−1,1].\nWhile for F-H rational interpolation with a given parameter dat equidistant\nnodes, its Lebesgue constant only grows logarithmically [4]. Numerical tests reveal\nthat the difference of its potential function on [ −1,1] tends to 0. Figure 8a shows\nthe potential function ˆUn(x) for F-H rational interpolation on equidistant nodes for\nd= 4. As ngrows, the potential function gets closer to a constant function on [ −1,1].\nFor different parameters d, Figure 8b shows the difference between the maximum and\nminimum values of the potential function on [ −1,1].\n-1 -0.5 0 0.5 1-0.200.20.40.60.811.2\n(a)\n10210310-210-1 (b)\nFig. 8 .(a): Potential function ˆUncorresponding to different nof F-H rational interpolation\nwhen d= 4. (b): Difference between the maximum and minimum values of potential function ˆUn\non[−1,1]for F-H rational interpolation when d= 0,2,4respectively.\nSince the difference of the potential functions converges to 0, the external field\nϕn(x) converges uniformly to Cn−Uaon [−1,1], where Cnis a constant. Due to\nthe weak singularity of Ua, it is easy to show that the derivative of ϕnon [−1,1]\nhas no upper bound as ngrows. Considering the definition of the external field ϕn,\nit is clear that the poles will grow closer to the real axis as ngrows. As shown in\nFigure 9, unlike the case in Figure 6, there is no integral path Γ to provide a sufficient\ndifference in potential. Thus, the F-H rational interpolation on equidistant nodes has\nno exponential convergence to analytic functions when parameter dis fixed.18 KELONG ZHAO, AND SHUHUANG XIANG\n-1 0 1-1.5-1-0.500.511.5\n-1 0 1 -1 0 1\n00.20.40.6\nFig. 9 .Contours of the potential function Ungenerated by the nodes (blue) and poles (red) of\nthe F-H rational interpolation [9] when d= 4. Asngrows, the poles get closer to the interval [−1,1]\nand the potential function takes on smaller values on the neighborhood of [−1,1].\n6.2. Fast and stable rational interpolation. subsection 6.1 supports the\nwell-established view that rational interpolation on equidistant nodes does not exhibit\nstable exponential convergence to the analytic function. From a potential perspective,\nthe key reason for this is that the density function of equidistant nodes produces a\nlogarithmic potential function Uathat is weakly singular on [ −1,1]. This makes it\ndifficult to match the external fields generated by the poles.\nConversely, if a positive density function wproduces a potential that is analytic\non [−1,1] and can be analytically extended to some neighborhood Ω of [ −1,1]. If there\nexists a harmonic function ϕon some neighborhood Ω′⊆Ω such that the potential\nfunction is constant on [ −1,1] and there exists a family of poles {p(n)\nj}m\nj=1(where\nm≤n), the discrete external field ϕngenerated by {p(n)\nj}m\nj=1converges consistently\ntoϕon Ω′. Then the rational interpolation formed by this density function wwith the\npoles{p(n)\nj}m\nj=1both avoids the exponential growth of Lebesgue constant and achieves\nexponential convergence for the analytic function.\n7. Conclusions. In this work, we focus on the connection between the continu-\nous potential function U, the barycentric weights, and Lebesgue constant for polyno-\nmial and rational interpolation on the interval [ −1,1]. The difference dbetween the\nmaximum and minimum of the potential function induces an exponential growth rate\nlower bound [exp d]nfor the maximum ratio of the absolute values of the barycentric\nweights, which is also reflected in the Lebesgue constant. Moreover, we find that the\nexponential growth rate of the Lebesgue function at ˆ x∈[−1,1] can be approximately\nestimated using the potential function. Finally, when the potential function equi-\nlibrates on [ −1,1], we give a non-exponential growth upper bound on its Leberger\nconstant.\nIt is important to note that the Lebesgue constant is not entirely determined by\nthe continuous potential U. For example, Chebyshev polynomial interpolation and\nLegendre polynomial interpolation, which have the same continuous potential func-\ntion, have different growth rates of Lebesgue constant. In this paper, the potential\nfunction Uis derived from the density function wand the external field ϕ, while the\nLebesgue constant is influenced by the interpolation nodes and poles. The bridge con-\nnecting these two aspects is the discrete potential Un, defined by the nodes and poles.POTENTIAL, BARYCENTRIC WEIGHTS, AND LEBESGUE CONSTANTS 19\nThe manner in which the discrete potential converges to the continuous potential also\nimpacts the growth of the Lebesgue constant. To characterize the convergence of the\ndiscrete potential to the continuous potential, we introduce the parameters δ±\nn.\nBarycentric weights\nLebesgue constantDiscrete potential \u0001\u0002\u0003\u0002±\nContinuous potential \u0001Object of study\nFactors\nFig. 10 .The main research path of this paper.\nSince the focus of this paper is on the connection between the continuous potential\nand the Lebesgue constant, we deliberately ignore the discussion on δ−\nnandδ+\nn. In the\nnumerical examples, all node distributions satisfyRx(n)\ni\n−1w(t) dt=i/n, and all employ\na fixed ϕn. The purpose of these special arrangements is to minimize the effect on\nLebesgue constant of the process by which the discrete potential converges to the\ncontinuous potential. In our tests, these special arrangements make δ−\nn=O(n−1)\nandδ+\nn=O(n−1logn). Thus, the numerical correlation between the continuous\npotential and Lebesgue constant becomes clearer.\nAlthough we ignore δ±\nnin this paper, it may be the key to explaining some phe-\nnomena. Examples include different Lebesgue constants for various Jacobi polynomial\ninterpolations. We believe that discussing the connection between specific node dis-\ntributions and δ±\nnis also an interesting issue that may provide new approaches to the\nstudy of Lebesgue constants for a specific polynomial or rational interpolation.\nREFERENCES\n[1]M. Abramowitz and I. A. Stegun ,Handbook of mathematical functions with formulas,\ngraphs, and mathematical tables , vol. No. 55 of National Bureau of Standards Applied\nMathematics Series, U. S. Government Printing Office, Washington, DC, 1964. For sale by\nthe Superintendent of Documents.\n[2]J.-P. Berrut ,The barycentric weights of rational interpolation with prescribed poles , J. Com-\nput. Appl. 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Xiang ,Barycentric interpolation based on equilibrium potential , arXiv e-\nprints, (2023), arXiv:2303.15222, https://doi.org/10.48550/arXiv.2303.15222."    },    {        "title": "2401.12676v1.Random_Riemannian_Geometry_in_4_Dimensions.pdf",        "content": "arXiv:2401.12676v1  [math.PR]  23 Jan 2024Random Riemannian Geometry in 4 Dimensions\nKarl-Theodor Sturm\nHausdorff Center for Mathematics, University of Bonn, Germa ny\nsturm@uni-bonn.de\nAbstract. Weconstructandanalyzeconformally invariantrandomfield s\non 4-dimensional Riemannian manifolds ( M,g).These centeredGaussian\nfieldsh, calledco-biharmonic Gaussian fields , are characterized by their\ncovariance kernels kdefined as the integral kernel for the inverse of the\nPaneitz operator\np=1\n8π2/bracketleftbigg\n∆2+div/parenleftbigg\n2Ric−2\n3scal/parenrightbigg\n∇/bracketrightbigg\n.\nThe kernel kis invariant (modulo additive corrections) under conforma l\ntransformations, and it exhibits a precise logarithmic div ergence\n/vextendsingle/vextendsingle/vextendsinglek(x,y)−log1\nd(x,y)/vextendsingle/vextendsingle/vextendsingle≤C.\nIn terms of the co-biharmonic Gaussian field h, we define the quantum\nLiouville measure , a random measure on M, heuristically given as\ndµ(x) :=eγh(x)−γ2\n2k(x,x)dvolg(x),\nand rigorously obtained a.s. for |γ|<√\n8 as weak limit of the RHS with\nhreplaced by suitable regular approximations ( hℓ)ℓ∈N.\nFor the flat torus M=T4, we provide discrete approximations of the\nGaussian field and of the Liouville measures in terms of semi- discrete\nrandom objects, based on Gaussian random variables on the di screte\ntorus and piecewise constant functions in the isotropic Haa r system.\nKeywords: random Riemannian geometry, Gaussian field, conformally\ninvariant, Paneitz operator, bi-Laplacian, biharmonic, m embrane model,\nquantum Liouville measure\n2020Mathematics Subject Classification: 60G15, 58J65, 31C25\nAcknowledgement. The author gratefully acknowledges finan cial sup-\nport from the European Research Council through the ERC AdG ‘ Ric-\nciBounds’ (grant agreement 694405) as well as funding by the Deutsche\nForschungsgemeinschaft through the project ‘Random Riema nnian Ge-\nometry’ within the SPP 2265 ‘Random Geometric Systems’. The author\nalso likes to thank the reviewer for his/her careful reading and valuable\ncomments.2 Sturm\n1 Random Riemannian Geometries and Conformal\nInvariance\nThe basic ingredients of any Random Riemannian Geometry are a family Mof\nRiemannian manifolds ( M,g) and a probability measure PMonM. Typically,\nM={(M,g′) :g′=e2hg, h∈ C∞(M)}for some given ( M,g), andPM\nis the push forward of a probability measure PgonC∞(M) under the map\nh/ma√sto→(M,e2hg).\nOfmajorinterestareRandomRiemannianGeometrieswhichare conformally\ninvariant . In the previous setting this means that\n– Pg′=Pgifg′=e2ϕgfor someϕ∈ C∞(M)\n–h(d)=h′◦ΦifΦ: (M,g)→(M′,g′) is an isometry and handh′are\ndistributed according to PgandPg′, resp.\nIn this respect, ofcourse, the 2-dimensionalcaseplays a particu larrole thanks to\nthe powerful Riemannian Mapping Theorem — but the concept of con formally\ninvariant random geometries is by no means restricted to this case.\nMostly, such probability measures Pgare Gaussian fields, informally given\nas\ndPg(h) =1\nZgexp/parenleftBig\n−eg(h,h)/parenrightBig\ndh (1)\nwith some (non-existing) uniform distribution dhonC∞(M), a normalizing con-\nstantZg, and some bilinear form eg. The rigorous definition of such probability\nmeasures Pgoften requirestopassto spacesofdistributions(ratherthansm ooth\nfunctions). It is based on the Bochner–Minlos Theorem and the uniq ue charac-\nterization of Pgas\n/integraldisplay\nei/angbracketleftu,h/angbracketrightdPg(h) = exp/parenleftBig\n−1\n2kg(u,u)/parenrightBig\n∀u∈ C∞(M) (2)\nwherekg(u,u)1/2:= suph/angbracketleftu,h/angbracketright\neg(h,h)1/2denotes the norm dual to eg.\nTheconformalinvariancerequirementfortherandomgeometryt henamounts\nto the requirement\neg(u,u) =ee2ϕg(u,u)∀ϕ,∀u. (3)\nInthetwo-dimensionalcase,thisisawell-knownpropertyofthe Dirichlet energy ,\ncf. [6],\nEg(u,u) :=/integraldisplay\nM/vextendsingle/vextendsingle∇gu/vextendsingle/vextendsingle2dvolg.\nThe conformally invariant random field defined and constructed in th is way\nis the celebrated Gaussian Free Field [23]. It is a particular (and the most\nprominent) case of a log-correlated random field [5] and of a fractio nal GaussianRandom Riemannian Geometry in 4 Dimensions 3\nfield [14]. It naturally arises as the scaling limit of various discrete mode ls of ran-\ndomsurfaces,forinstancediscreteGaussianFreeFieldsorharmo niccrystals[23].\nIt is also deeply related to another planar conformally invariant rand om object\nof fundamental importance, the Schramm–Loewner evolution [10,11,19]. For in-\nstance, level curves of the Discrete Gaussian Free Field converge toSLE4[20],\nand zero contour lines of the Gaussian Free Field are well-defined ran dom curves\ndistributed according to SLE4[21]. The work [15], and subsequent works in its\nseries, thoroughly study the relation between the Schramm-Loew ner evolution\nand Gaussian free field on the plane. The Liouville Quantum Gravity is a ran-\ndom measure, informally obtained as the Riemannian volume measure w hen the\nmetric tensor is conformally transformed with the Gaussian Free Fie ld as con-\nformal weight. Since the Gaussian Free Field is only a distribution, the rigorous\nconstruction of the random measure requires a renormalizationpr ocedure due to\nKahane[9]. Thisrenormalizationdepends onaroughnessparameter γand works\nonly for |γ|<2. In [16] and subsequent work in its series, Miller and Sheffield\nprove that for the value γ=/radicalbig\n8/3 the Liouville Quantum Gravity coincides\nwith the Brownian map, that is a random metric measure space arising as a\nuniversal scaling limit of random trees and random planar graphs (se e [12,13]\nand the references therein). More recently, [4,8] establish the e xistence of the\nLiouville Quantum Gravity metric for γ∈(0,2).\nAll these approaches to conformally invariant random objects so f ar, with\nexception of the recent contribution [1], are limited to the two-dimen sional case.\nThe main reason is not the lack of a Riemannian Mapping Theorem but th e fact\nthat the Dirichlet energy is no longer conformally invariant in dimension n/\\e}atio\\slash= 2.\nOne rather obtains\nEe2ϕg(u,u) =/integraldisplay\nM/vextendsingle/vextendsingle∇gu/vextendsingle/vextendsingle2e(n−2)ϕdvolg.\nIn the four-dimensional case, a more promising candidate appears to be the\nbi-Laplacian energy\n˜eg(u,u) :=/integraldisplay\nM/parenleftbig\n∆gu/parenrightbig2dvolg.\nThis energy functional is still not conformally invariant but it is close t o:\n˜ee2ϕg(u,u) =/integraldisplay\nM/parenleftbig\n∆gu+2∇gϕ∇gu/parenrightbig2dvolg=˜eg(u,u)+ low order terms .\nOur search for a conformally invariant energy functional in dimensio nsn= 4\nfinally will lead us to considering\neg(u,u) =c/integraldisplay\nM/parenleftbig\n∆gu/parenrightbig2dvolg+ low order terms .\nPaneitz [18] found the precise formula for the conformally invariant energy func-\ntional in dimension 4. Subsequently, Graham, Jenne, Mason, and Sp arling [7]\nshowed the existence of a conformally invariant energy functional of the form\neg(u,u) =c/integraldisplay\nM/parenleftbig\n−∆gu/parenrightbign/2dvolg+ low order terms4 Sturm\non Riemannian manifolds of even dimension n.\nBased on these results, jointly with L. Dello Schiavo, R. Herry and E. Kopfer\n[2],weconstructed andanalyzedconformallyinvariantrandomfields onRieman-\nnian manifolds ( M,g) of arbitrary even dimensions. In the subsequent Sections\n2 and 3, we will summarize these results in the particular case n= 4. In Section\n4, we will present a detailed study of approximations of the random fi eld and\nthe random measure on the 4-dimensional flat torus in terms of cor responding\nrandom objects on the discrete tori T4\nℓ,ℓ∈N.\n2 Paneitz Energy on 4-Dimensional Manifolds\nFrom now, we will be more specific. ( M,g) will always be a 4-dimensional\nsmooth, compact,connected Riemannian manifold without boundar y.Integrable\nfunctions (or distributions) uonMwill be called grounded if/integraltext\nMudvolg= 0\n(or/a\\}b∇acketle{tu,1/a\\}b∇acket∇i}ht=0, resp.). Let ( ϕj)j∈N0denote the complete ON-basis of L2(M,volg)\nconsisting of eigenfunctions of −∆gwith corresponding eigenvalues ( λj)j∈N0.\nThen the grounded Sobolev spaces ˚Hs(M,g) = (−∆g)−s/2˚L2(M,volg) fors∈R\nare given by\n˚Hs(M,g) =/braceleftbigg\nu=/summationdisplay\nj∈Nαjϕj:/summationdisplay\nj∈Nλs\nj|αj|2<∞/bracerightbigg\n,\nwhereas the usual Sobolev spaces are Hs(M,g) = (1−∆g)−s/2L2(M,volg) =\n˚Hs(M,g)⊕R·1. Extending the scalar product in L2(M,volg), the pairing be-\ntweenu=/summationtext\nj∈N0αjϕj∈Hsandv=/summationtext\nj∈N0βjϕj∈H−sis given by\n/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}ht:=/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}htHs,H−s:=/summationdisplay\nj∈N0αjβj.\nTheLaplacianactsonthesegroundedspacesby −∆g:˚Hs→˚Hs−2,/summationtext\nj∈Nαjϕj/ma√sto→/summationtext\nj∈Nλjαjϕj. The operator inverse to it is the grounded Green operator\n˚Gg:˚Hs→˚Hs+2,/summationdisplay\nj∈Nαjϕj/ma√sto→/summationdisplay\nj∈Nαj\nλjϕj.\nOn˚H0=˚L2,itisgivenasanintegraloperator ˚Ggu(x) =/integraltext\nM˚Gg(x,y)u(y)dvolg(y)\nin terms of the grounded Green kernel ˚Gg(x,y) onM. The latter is symmetric\ninxandy, it is grounded (i.e./integraltext\nM˚Gg(x,y)dvolg(y) = 0 for all x) and\n/vextendsingle/vextendsingle/vextendsingle˚Gg(x,y)−1\n4π2·dg(x,y)2/vextendsingle/vextendsingle/vextendsingle≤C.\nDefinition 1. ThePaneitzenergy is defined as the bilinear form on L2(M,volg)\nwith domain H2(M)by\neg(u,u) =1\n8π2/integraldisplay\nM/bracketleftbigg\n(∆gu)2−2Ricg(∇gu,∇gu)+2\n3scalg·|∇gu|2/bracketrightbigg\ndvolg.(4)Random Riemannian Geometry in 4 Dimensions 5\nIn particular, for every 4-dimensional Einstein manifold with Ric g=kgfor\nk∈R(which implies scalg= 4k),\neg(u,u) =1\n8π2/integraldisplay\nM/bracketleftbigg\n(∆gu)2+2\n3k|∇gu|2/bracketrightbigg\ndvolg. (5)\nExample 1. a) For the 4-sphere M=S4,\neg(u,u) =1\n8π2/integraldisplay\nM/bracketleftbigg\n(∆gu)2+2|∇gu|2/bracketrightbigg\ndvolg. (6)\nb) For the 4-torus M=T4,\neg(u,u) =1\n8π2/integraldisplay\nM(∆gu)2dvolg. (7)\nTheorem 1 ([18]). The Paneitz energy is conformally invariant:\neg(u,u) =ee2ϕg(u,u)∀ϕ∈ C∞(M),∀u∈H2(M).\nDefinition 2. The 4-manifold (M,g)is called admissible ifeg>0on˚H2(M).\nAs an immediate consequence of Theorem 1, we observe that admiss ibility is a\nconformal invariance. Large classes of 4-manifolds are admissible.\nProposition 1 ([2, Prop. 2.4, 2.5]). a) All compact Einstein 4-manifolds with\nnonnegative Ricci curvature are admissible.\nb) All compact hyperbolic 4-manifolds with spectral gap λ1>2are admissible.\nHowever, not every compact four-dimensional Riemannian manifold is admissi-\nble.\nExample 2 ([2, Prop. 2.7]). LetM1,M2be compact hyperbolic Riemannian sur-\nfaces such that λ1(M1)≤2\n3. Then the Einstein 4-manifold M=M1×M2is not\nadmissible.\nIf (M,g) is admissible, then the Paneitz operator (orco-bilaplacian )\npg=1\n8π2/bracketleftbigg\n∆2\ng+div/parenleftbigg\n2Ricg−2\n3scalg/parenrightbigg\n∇/bracketrightbigg\n(8)\nis a self-adjoint positive operator on L2(M,volg) with domain H4(M). Here the\ncurvature term 2 Ricg−2\n3scalgshould be viewed as an endomorphism of the\ntangent bundle, acting on the gradient of a function. In coordinat es:\npgu=1\n8π2/summationdisplay\ni,j∇i/bracketleftbigg\n∇i∇j+2Ricij\ng−2\n3scalg·gij/bracketrightbigg\n∇ju,∀u∈ C∞(M).6 Sturm\nLet (ψj)j∈N0denote a complete orthonormal basis of L2(M,volg) consisting\nof eigenfunctions for pg, and let (νj)j∈N0denote the corresponding sequence of\neigenvalues. Then the operator kg, inverse to pgon˚L2, is given on H−4(M) by\nkg:u/ma√sto→kgu:=/summationdisplay\nj∈N1\nνj/a\\}b∇acketle{tu,ψj/a\\}b∇acket∇i}htψj,\nand the associated bilinear form with domain H−2(M) is given by\nkg(u,v) :=/a\\}b∇acketle{tu,kgv/a\\}b∇acket∇i}htL2=/summationdisplay\nj∈N1\nνj/a\\}b∇acketle{tu,ψj/a\\}b∇acket∇i}ht/a\\}b∇acketle{tv,ψj/a\\}b∇acket∇i}ht.\nThe crucial properties of the kernel for the co-biharmonic Green operator kgare\nits logarithmic divergence and its conformal invariance.\nTheorem 2 ([2, Thm. 2.18]). If(M,g)is admissible, then kgis an integral\noperator with an integral kernel kgwhich satisfies\n/vextendsingle/vextendsingle/vextendsinglekg(x,y)+logdg(x,y)/vextendsingle/vextendsingle/vextendsingle≤C. (9)\nFurthermore, the kernel kg(x,y)is symmetric in x,yand grounded.\nTheorem 3 ([2, Prop. 2.19]). Assume that (M,g)is admissible and that\ng′:=e2ϕgfor someϕ∈ C∞(M). Then the co-biharmonic Green kernel kg′for\nthe metricg′is given by\nkg′(x,y) =kg(x,y)−1\n2¯φ(x)−1\n2¯φ(y) (10)\nwith¯φ∈ C∞(M)defined by\n¯φ:=2\nvolg′(M)/integraldisplay\nkg(.,z)dvolg′(z)−1\nvolg′(M)2/integraldisplay/integraldisplay\nkg(z,w)dvolg′(z)dvolg′(w).\nExample 3. Assume that ( M,g) is Ricci flat. Then\nkg(x,y) = 8π2˚G(2)\ng(x,y) := 8π2/integraldisplay\nM˚Gg(x,z)˚Gg(z,y)dvolg(z)\nwhere˚Ggdenotes the grounded Green kernel on ( M,g).\n3 Co-biharmonic Gaussian Field and Quantum Liouville\nMeasure\nThroughoutthesequel,assumethat( M,g)isanadmissible4-manifold(compact,\nsmooth, without boundary – as always).Random Riemannian Geometry in 4 Dimensions 7\n3.1 Conformally Invariant Gaussian Field\nDefinition 3. A co-biharmonic Gaussian field hon(M,g)is a linear family\n/parenleftbig\n/a\\}b∇acketle{th,u/a\\}b∇acket∇i}ht/parenrightbig\nu∈H−2\nof centered Gaussian random variables (defined on some proba bility space) with\nE/bracketleftbig\n/a\\}b∇acketle{th,u/a\\}b∇acket∇i}ht2/bracketrightbig\n=kg(u,u)∀u∈H−2(M).\nTheorem 4 ([2, Prop. 3.9, Rem. 3.3]). Let a probability space (Ω,F,P)be\ngiven and an i.i.d. sequence (ξj)j∈NofN(0,1)random variables. Furthermore,\nlet(ψj)j∈N0and(νj)j∈N0denote the sequences of eigenfunctions and eigenvalues\nforpg(counted with multiplicities). Then a co-biharmonic field i s given by\nh:=/summationdisplay\nj∈Nν−1/2\njξjψj. (11)\nMore precisely,\na)For eachℓ∈N, a centered Gaussian random variable hℓwith values in\nC∞(M)is given by\nhℓ:=ℓ/summationdisplay\nj=1ν−1/2\njξjψj. (12)\nb)The convergence hℓ→hholds inL2(P)×H−ǫ(M)for everyǫ >0. In\nparticular, for a.e. ωand everyǫ>0,\nhω∈H−ǫ(M),\nc)For everyu∈H−2(M), the family (/a\\}b∇acketle{tu,hℓ/a\\}b∇acket∇i}ht)ℓ∈Nis a centered L2(P)-bounded\nmartingale and\n/a\\}b∇acketle{tu,hℓ/a\\}b∇acket∇i}ht → /a\\}b∇acketle{tu,h/a\\}b∇acket∇i}htinL2(P)asℓ→ ∞.\nRemark 1. a) A co-biharmonic Gaussian field on ( M,g) can be regarded as a\nrandom variable with values in ˚H−ǫ(M) for anyǫ>0.\nb) Given any ‘grounded’ white noise Ξon (M,g), thenh:=/radicalbig\nkgΞis a\nco-biharmonic Gaussian field on ( M,g).\nTheorem 5 ([2, Thm. 3.11]). Leth:Ω→H−ǫ(M)denote a co-biharmonic\nGaussian field for (M,g)and letg′=e2ϕgwithϕ∈ C∞(M). Then\nh′:=h−1\nvolg′(M)/angbracketleftbig\nh,1/angbracketrightbig\nH−ǫ(M,g′),Hǫ(M,g′)\nis a co-biharmonic Gaussian field for (M,g′).\nBesides the previous eigenfunction approximation, there are nume rous other\nways to approximate a given co-biharmonic Gaussian field hby ‘smooth’ Gaus-\nsian fieldshℓ,ℓ∈N.8 Sturm\nProposition 2. Letρℓforℓ∈Nbe a family of bounded functions on M×M\nsuch thatρℓ(x,.)volgfor eachx∈Mis a family of probability measures on M\nwhich forℓ→ ∞weakly converges to δx. Define centered Gaussian fields hℓfor\nℓ∈Nby\nhℓ(y) :=/a\\}b∇acketle{th|ρℓ(.,y)/a\\}b∇acket∇i}ht. (13)\nThen, for every u∈ C(M), asℓ→ ∞\n/a\\}b∇acketle{thℓ|u/a\\}b∇acket∇i}ht /ma√sto−→ /a\\}b∇acketle{th|u/a\\}b∇acket∇i}htP-a.s. and in L2(P).\nThe associated covariance kernels are given by\nkℓ(x,y) :=/integraldisplay/integraldisplay\nk(x′,y′)ρℓ(x′,x)ρℓ(y′,y)dvolg(x′)dvolg(y′)\nfor allℓ∈N, andkℓ→kasℓ→ ∞on locally uniformly on M×Moff the\ndiagonal.\nProof.Obviously, /a\\}b∇acketle{thℓ|u/a\\}b∇acket∇i}ht=/a\\}b∇acketle{th|ρℓ∗u/a\\}b∇acket∇i}htwith (ρℓ∗u)(x) =/integraltext\nρℓ(x,y)u(y)dvolg(y),\nandρℓ∗u→uinL2asℓ→ ∞. Moreover,\nE/bracketleftBig/vextendsingle/vextendsingle/a\\}b∇acketle{th|u/a\\}b∇acket∇i}ht−/a\\}b∇acketle{thℓu/a\\}b∇acket∇i}ht/vextendsingle/vextendsingle2/bracketrightBig\n=E/bracketleftBig/vextendsingle/vextendsingle/a\\}b∇acketle{th|u−ρℓ∗u/a\\}b∇acket∇i}ht/vextendsingle/vextendsingle2/bracketrightBig\n=∞/summationdisplay\nj=11\nνj/vextendsingle/vextendsingle/a\\}b∇acketle{tψj|u−ρℓ∗u/a\\}b∇acket∇i}ht/vextendsingle/vextendsingle2≤C/vextenddouble/vextenddoubleu−ρℓ∗u/vextenddouble/vextenddouble2\nH−2.\nA particular case of such approximations through convolution kern els will be\nconsidered now.\nProposition 3. Let(Qℓ)ℓ∈Nbe a family of partitions of Mwith∀ℓ,∀Q∈Qℓ:\n∃m∈N,∃Q1,...,Q m∈Qℓ+1:Q=/uniontextm\ni=1Qiand with sup{diam(Q) :Q∈\nQℓ} →0asℓ→ ∞. Forℓ∈Nput\nρℓ(x,y) :=/summationdisplay\nQ∈Qℓ1\nvolg(Q)1Q(x)1Q(y). (14)\nIn other words, for given x∈Mwe haveρℓ(x,.) =1\nvolg(Q)1Qwith the unique\nQ∈Qℓwhich contains x. Defininghℓas before then yields\nhℓ(x) =1\nvolg(Q)/a\\}b∇acketle{th|1Q/a\\}b∇acket∇i}ht ∀x∈Q,∀Q∈Qℓ. (15)\nForℓ∈N, letFℓdenote the σ-field in (Ω,F,Q)generated by the random func-\ntions onMthat are piecewise constant on each of the sets Q∈Qℓ. Then(hℓ)ℓ∈N\nis a(Ω,F,(Fℓ)ℓ∈N,P)-martingale and\nhℓ=E/bracketleftbig\nh/vextendsingle/vextendsingleFℓ/bracketrightbig\n∀ℓ∈N. (16)Random Riemannian Geometry in 4 Dimensions 9\n3.2 Quantum Liouville Measure\nLetanadmissible4-manifold( M,g)begivenaswellasaco-biharmonicGaussian\nfieldhon it. Furthermore, let smooth approximations ( hℓ)ℓ∈Nof it be given —\ninformally defined as hℓ:=ρℓ∗hand formally by (13) — in terms of a sequence\n(ρℓ)ℓ∈N0of bounded convolution densities on M. Fixγ∈R.\nForℓ∈Ndefine a random measure µℓ=ρℓvolgonMwith density\nρℓ(x) := exp/parenleftBig\nγhℓ(x)−γ2\n2kℓ(x,x)/parenrightBig\nwithkℓ(x,y) :=/integraltext/integraltext\nk(x′,y′)ρℓ(x′,x)ρℓ(y′,y)dvolg(x′)dvolg(y′) as before.\nTheorem 6 ([2, Thm. 4.1]). If|γ|<√\n8, then there exists a random measure\nµonMwithµℓ→µ. More precisely, for every u∈ C(M),\n/integraldisplay\nMudµℓ−→/integraldisplay\nMudµinL1(P)andP-a.s. asℓ→ ∞.\nThe random measure µis independent of the choice of the convolution densities\n(ρℓ)∈N.\nIf the(ρℓ)∈Nare chosen according to (14)then for each u∈ C(M)the family\nYℓ:=/integraltext\nMudvolg,ℓ∈N, is a uniformly integrable martingale. If in addition\n|γ|<2, then this martingale is even L2-bounded.\nThe latter claim, indeed, can be seen directly:\nsup\nℓE/bracketleftBig\nYℓ2/bracketrightBig\n= sup\nℓE/integraldisplay/integraldisplay\neγ(hℓ(x)+hℓ(y)−γ2\n2(E[h2\nℓ(x)+h2\nℓ(y)]u(x)u(y)dvolg(x)dvolg(y)\n= sup\nℓ/integraldisplay/integraldisplay\neγ2kℓ(x,y)u(x)u(y)dvolg(x)dvolg(y)\n≤ /ba∇dblu/ba∇dbl2\n∞·sup\nℓ/integraldisplay/integraldisplay/bracketleftbigg/integraldisplay/integraldisplay\nρℓ(x′,x)ρℓ(y′,y)eγ2k(x′,y′)dvolg(x′)dvolg(y′)/bracketrightbigg\ndvolg(x)dvolg(y)\n=/ba∇dblu/ba∇dbl2\n∞·/integraldisplay/integraldisplay\neγ2k(x′,y′)dvolg(x′)dvolg(y′)\n≤ /ba∇dblu/ba∇dbl2\n∞·/integraldisplay/integraldisplay1\nd(x,y)γ2dvolg(x)dvolg(y)+C′\nby means of Jensen’s inequality and the kernel estimate (9). Obviou sly, the final\nintegral is finite if and only if γ2<4.\nDefinition 4. The random measure µ:= lim\nℓ→∞µℓis called quantum Liouville\nmeasure.\nRemark 2 ([2, Cor 4.10, Prop. 4.14]). Assume|γ|<√\n8 and letω/ma√sto→µωdenote\nthe random measure constructed above. Then for P-a.e.ω, the measure µωon\nM10 Sturm\n–does not charge sets of vanishing H2-capacity;\n–does not charge sets of vanishing H1-capacity provided |γ|<2;\n–is singular w.r.t. the volume measure on Mwheneverγ/\\e}atio\\slash= 0.\nMoreover, the random measure µhas finite moments of any negative order,\ni.e. for any p>0,\nE/bracketleftbig\nµ(M)−p/bracketrightbig\n<∞.\nAkeypropertyofthequantumLiouvillemeasureisitsquasi-invarianc eunder\nconformal transformations.\nTheorem 7 ([2, Thm. 4.4]). Letµbe the quantum Liouville measure for\n(M,g), andµ′be the quantum Liouville measure for (M,g′)whereg′=e2ϕg\nfor someϕ∈ C∞(M). Then\nµ′(d)=e−γξ+γ2\n2¯ϕ+4ϕµ (17)\nwhereξ:=1\nv′/a\\}b∇acketle{th,e4ϕ/a\\}b∇acket∇i}htand¯ϕ:=2\nv′kg(e4ϕ)−1\nv′2kg(e4ϕ,e4ϕ)withv′:=volg′(M).\n4 Approximation by Random Fields and Liouville\nMeasures on the Discrete 4-Torus\nFor the remaining part, we now focus on the 4-dimensional torus T4:=R4/Z4,\nequipped with the flat metric. With this choice of ( M,g), we will drop the g\nfrom the notations: k=kg,G=Ggetc.\nFor the 4-torus, we will study approximations of the co-biharmonic field —\nnow briefly called biharmonic field (since the underlying Paneitz operat or or\nco-bilaplacian is now simply the bilaplacian) — and of the quantum Liouville\nmeasure by (semi-) discrete versions of such fields and measures, defined on the\ndiscrete tori T4\nℓasℓ→ ∞.\n4.1 The Isotropic Haar System\nTo begin with, for ℓ∈N0define the parameter sets\nAℓ:=/braceleftbig\n0,1,...,2ℓ−1/bracerightbig4, Bℓ:={0,1}4\\{(0,0,0,0)}, Iℓ:=Aℓ×Bℓ\nand the discrete 4-torus\nT4\nℓ:= 2−ℓ·Aℓ= (2−ℓZ4)/Z4.\nMoreover, let Qℓ:=/braceleftbig\nQℓ,α:α∈Aℓ/bracerightbig\ndenote the set of all dyadic cubes\nQℓ,α:= 2−ℓ·/parenleftBig\n[α1,α1+1)×[α2,α2+1)×[α3,α3+1)×[α4,α4+1)/parenrightBig\n⊂T4\nofedgelength 2−ℓ, andlet Sℓdenote the set ofallgroundedfunctions u:T4→R\nwhich are constant on each of the cubes Q∈Qℓ. With each Qℓ,α∈T4\nℓweRandom Riemannian Geometry in 4 Dimensions 11\nassociate a set {ηℓ,α,β:β∈Bℓ} ⊂ Sℓ+1of 15 multivariate Haar functions with\nsupportQℓ,αgiven by all possible tensor products\nηℓ,α,β(x) := ˜ηℓ,α1,β1(x1)·˜ηℓ,α2,β2(x2)·˜ηℓ,α3,β3(x3)·˜ηℓ,α4,β4(x4)\nwhere\n˜ηℓ,αk,βk(xk) :=/braceleftBigg\n2ℓ/2·1[αk,αk+1)(2ℓxk), ifβk= 0\n2ℓ/2·/parenleftBig\n1[αk,αk+1\n2)−1[αk+1\n2,αk+1)/parenrightBig\n(2ℓxk),ifβk= 1\nfork= 1,2,3,4.\nForℓ∈N0, the block\nHℓ:={ηℓ,α,β:α∈Aℓ, β∈Bℓ}\nconsists of 15 ·24ℓHaar functions which we call Haar functions of level ℓ. The\nunion of all of them,\nH=∞/uniondisplay\nℓ=0Hℓ,\nis a complete orthonormalsystem in ˚L2(T4), calledisotropic 4-dimensional Haar\nsystem, cf. [17]. Moreover,\nSℓ= span/parenleftbiggℓ−1/uniondisplay\nκ=0Hκ/parenrightbigg\n. (18)\nForx∈T4andℓ∈N, the unique cube Q∈Qℓwithx∈Qwill be denoted\nbyQℓ(x). Given a function u∈˚L1(T4), we define the function uℓ∈ Sℓby\nuℓ(x) := 24ℓ/integraldisplay\nQℓ(x)udL4. (19)\nRestricted to ˚L2(T4), the mapπQℓ:u/ma√sto→uℓis theL2-projection onto the linear\nsubspace Sℓ. Moreover,\nuℓ=ℓ−1/summationdisplay\nκ=0/summationdisplay\nι∈Iκ/a\\}b∇acketle{tu,ηκ,ι/a\\}b∇acket∇i}htηκ,ι. (20)\n4.2 The Semi-discrete Gaussian Field\nLet an i.i.d. family of N(0,1) random variables ( ξℓ,ι)ℓ∈N0,ι∈Iℓbe given with\nIℓ=Aℓ×Bℓas before. For ℓ∈Nput\nˆhω\nℓ(x) :=√\n8πℓ−1/summationdisplay\nκ=0/summationdisplay\nι∈Iκξω\nκ,ι·˚Gηκ,ι(x).12 Sturm\nHere˚Gdenotes the grounded Green operator on the 4-torus, given as a n integral\noperator ˚Gu(x) =/integraltext\nT4˚G(x,y)u(y)dL4(y) in terms of the grounded Green kernel\n˚G(x,y) onT4. (For related results with the Green kernel of the torus replaced by\nthe Green kernel of the discrete torus, see Section 4.3 below.) Mor eover, define\nthe non-symmetrickernel ˚Gℓ(x,z) :=/parenleftbig\nπQℓ˚G(x,.)/parenrightbig\n(z) = 24ℓ/integraltext\nQℓ(z)˚G(x,v)dL4(v)\nand put\nˆkℓ(x,y) := 8π2/integraldisplay\nT4˚Gℓ(x,z)˚Gℓ(y,z)dL4(z).\nAsℓ→ ∞, this converges pointwise to k(x,y) := 8π2/integraltext\nT4˚G(x,z)˚G(z,y)dL4(z),\nwhich — up to the pre-factor — is the Green kernel for the bi-Laplac ian∆2.\nProposition 4. For everyℓ∈N,\na)for everyω, the function ˆhω\nℓ\n–is inC1(T4)and grounded (i.e./integraltext\nT4ˆhω\nℓdL4= 0);\n–is smooth off the boundaries of dyadic cubes Q∈Qℓ;\n–has constant Laplacian on the interior of each dyadic cube Q∈Qℓ;\n–is the sum/summationtextℓ−1\nκ=0/summationtext\nι∈Iκˆhω\nκ,ιof functions ˆhω\nκ,ι=√\n8πξω\nκ,ι·˚Gηκ,ιeach of\nwhich is harmonic on T4\\¯Qκ,ιfor the dyadic cube Qκ,ι∈Qℓ;\nb)for everyx∈T4, the random variable ˆhℓ(x)is centered and Gaussian with\nvariance ˆkℓ(x,x), the latter being independent of x;\nc)ˆhℓis a centered Gaussian field with covariance function ˆkℓ(x,y).\nProof.We show c), the rest is straightforward. By the very definition of ˆhℓ, the\ni.i.d. property of the ξκ,ιand the projection properties (19) & (20),\nE/bracketleftBig\nˆhℓ(x)·ˆhℓ(y)/bracketrightBig\n= 8π2·ℓ−1/summationdisplay\nκ=0/summationdisplay\nι∈Iκ/angbracketleftbig˚G(x,.),ηκ,ι/angbracketrightbig\n·/angbracketleftbig˚G(y,.),ηκ,ι/angbracketrightbig\n= 8π2·/angbracketleftBig\nπQℓ˚G(x,.),πQℓ˚G(y,.)/angbracketrightBig\n= 8π2·28ℓ/integraldisplay\nT4/parenleftbigg/integraldisplay\nQℓ(z)˚G(x,v)dL4(v)·/integraldisplay\nQℓ(z)˚G(y,w)dL4(w)/parenrightbigg\ndL4(z)\n=ˆkℓ(x,y).\nTheorem 8. a)The centered Gaussian random field hwith covariance func-\ntionk(as introduced and studied in section 3.1) is given in the cas e of the\n4-torus by\nh:=√\n8π∞/summationdisplay\nκ=0/summationdisplay\nι∈Iκξκ,ι·˚Gηκ,ι\nand called biharmonic Gaussian field.\nb)The convergence ˆhℓ→hholds inL2(P)×H−ǫ(T4)for everyǫ >0. In\nparticular, for a.e. ωand everyǫ>0,\nhω∈H−ǫ(T4).Random Riemannian Geometry in 4 Dimensions 13\nc)For everyu∈H−2(T4), the family (/a\\}b∇acketle{tu,ˆhℓ/a\\}b∇acket∇i}ht)ℓ∈Nis a centered L2(P)-bounded\nmartingale and\n/a\\}b∇acketle{tu,ˆhℓ/a\\}b∇acket∇i}ht → /a\\}b∇acketle{tu,h/a\\}b∇acket∇i}htinL2(P)asℓ→ ∞.\nProof.a) For convergence (and well-definedness) of the infinite sum, see b)\nand/or c) below. To identify the covariance, observe that\nE/bracketleftBig\n/a\\}b∇acketle{tu,h/a\\}b∇acket∇i}ht2/bracketrightBig\n= 8π2·∞/summationdisplay\nκ=0/summationdisplay\nι∈Iκ/angbracketleftbig˚Gu,ηκ,ι/angbracketrightbig2= 8π2·/vextenddouble/vextenddouble˚Gu/vextenddouble/vextenddouble2=/angbracketleftbig\nu,ku/a\\}b∇acket∇i}ht.\nb) Fors >0, let˚Gsdenote the s-th power of the operator ˚Gand let˚G(s)\ndenote its kernel which is given by the formula\n˚G(s)(x,y) =1\nΓ(s)/integraldisplay∞\n0ts−1˚pt(x,y)dt\nin terms of the grounded heat kernel ˚ pt(x,y) =pt(x,y)−1. Then for ǫ>0,\n1\n8π2E/bracketleftBig\n/ba∇dblh/ba∇dbl2\nH−ǫ/bracketrightBig\n=1\n8π2E/bracketleftBig/vextenddouble/vextenddouble˚Gǫh/vextenddouble/vextenddouble2\nL2/bracketrightBig\n=∞/summationdisplay\nκ=0/summationdisplay\nι∈Iκ/vextenddouble/vextenddouble˚G1+ǫηκ,ι/vextenddouble/vextenddouble2\nL2\n=/integraldisplay\nT4/vextenddouble/vextenddouble/vextenddouble˚G(1+ǫ)(.,z)/vextenddouble/vextenddouble/vextenddouble2\ndL4(z) =/integraldisplay\nT4˚G(2+2ǫ)(z,z)dL4(z)\n=˚G(2+2ǫ)(0,0)<∞\nsince˚G(s)fors > n/2 = 2 is a bounded function, see [2]. This proves that\n/ba∇dblh/ba∇dblH−ǫ<∞for a.e.ω. The convergence ˆhℓ→hinH−ǫ(T4) follows similarly.\nc) By construction, for every x∈T4, the family ( ˆhℓ(x))ℓ∈Nis a centered\nmartingale. The martingale property immediately carries over to the family\n(/a\\}b∇acketle{tu,ˆhℓ/a\\}b∇acket∇i}ht)ℓ∈Nfor any function or distribution u. TheL2-boundedness follows from\n1\n8π2·sup\nℓE/bracketleftBig\n/a\\}b∇acketle{tu,ˆhℓ/a\\}b∇acket∇i}ht2/bracketrightBig\n= sup\nℓ/vextenddouble/vextenddoubleπQℓ˚Gu/vextenddouble/vextenddouble2\nL2=/vextenddouble/vextenddouble˚Gu/vextenddouble/vextenddouble2\nL2<∞.\nRemark 3. a) Withηℓ,ιforι= (α,β) inIℓas above and ψℓ,ι:=˚Gηℓ,ι, the family\n(ψℓ,ι)ℓ∈N0,ι∈Iℓis a complete orthonormal system in the Hilbert space H2(T4),\nequipped with the scalar product\n/a\\}b∇acketle{tu,v/a\\}b∇acket∇i}htH2:=/a\\}b∇acketle{t∆u,∆v/a\\}b∇acket∇i}htL2.\nb) Given any complete orthonormal system ( ψk)k∈Nin the Hilbert space\n˚H2(T4) and any i.i.d. sequence of N(0,1) random variables ( ξk)k∈N, with the\nsame arguments as for Theorem 8 one can prove that the Gaussian random field\nhℓ:=√\n8πℓ/summationdisplay\nk=1ξk·ψk\nconverges in L2(P)×H−ǫ(T4) for every ǫ >0 asℓ→ ∞to the biharmonic\nGaussian field hon the 4-torus. Moreover, /a\\}b∇acketle{tu,hℓ/a\\}b∇acket∇i}ht → /a\\}b∇acketle{tu,h/a\\}b∇acket∇i}htinL2(P) for every\nu∈H−2(T4).14 Sturm\n4.3 The Semi-discrete Liouville Measure\nGiven a Gaussian field has considered in Theorem 8, then following Proposition\n3 a semi-discrete approximation of it is defined by\nhℓ:=πQℓh(∀ℓ∈N).\nFor eachℓ, this is a centered Gaussian field with covariance given by\nkℓ(x,y) := 8π228ℓ/integraldisplay\nQℓ(x)/integraldisplay\nQℓ(y)/parenleftbigg/integraldisplay\nT4˚G(z,v)˚G(z,w)dL4(z)/parenrightbigg\ndL4(w)dL4(v)\n= 8π228ℓ/integraldisplay\nQℓ(x)/integraldisplay\nQℓ(y)k(v,w)dL4(w)dL4(v).\nFor anyγ∈Randℓ∈N, we define the semi-discrete quantum Liouville\nmeasureµℓ=ρℓL4as the random measure on T4with density w.r.t. L4given\nby\nρω\nℓ(x) :=eγhω\nℓ(x)−γ2\n2kℓ(x,x).\nCorollary 1. For|γ|<√\n8and a.e.ω, the measures µω\nℓasℓ→ ∞weakly\nconverge to the Borel measure µωintroduced and studied in section 3.2,\nµℓ→µP-a.s.\nFor|γ|<2, the convergence also holds in L2(P).\n4.4 Discrete Random Objects\nTo end up with fully discrete random objects, we have to replace the Green\nfunction on the continuous torus by the Green function on the disc rete torus.\nTo do so, for ℓ∈Nwe define the discrete Laplacian ∆ℓacting on functions\nu∈L2(T4\nℓ) by\n−∆ℓu= 22ℓ+3·/parenleftbig\nu−pℓu/parenrightBig\n, p ℓu(i) :=1\n8/summationdisplay\nj∈Jℓu(i+j)\nwithJℓ:=/braceleftBig\n(k,0,0,0),(0,k,0,0),(0,0,k,0),(0,0,0,k):k∈ {−2−ℓ,2−ℓ}/bracerightBig\n.Note\nthat the associated discrete Dirichlet form Eℓ(u,u) :=−/a\\}b∇acketle{tu,∆ℓu/a\\}b∇acket∇i}htL2onL2(T4\nℓ)\nhas a positive spectral gap λ1\nℓ:= inf/braceleftBig\nEℓ(u,u)\n/bardblu/bardbl2\nL2:u∈˚L2(T4\nℓ)/bracerightBig\n.Furthermore, we\ndefine the grounded transition kernel by ˙ pℓ(i,j) :=1\n81Jℓ(i−j)−2−4ℓ.\nThediscrete Green operator acting on grounded functions u∈˚L2(T4\nℓ) is\ndefined by\n˙Gℓu(i) := 2−2ℓ−3∞/summationdisplay\nk=0pk\nℓu(i) = 2−2ℓ−3∞/summationdisplay\nk=0˙pk\nℓu(i) =/summationdisplay\nj∈T4\nℓ˙Gℓ(i,j)u(j) (21)Random Riemannian Geometry in 4 Dimensions 15\nwhere˙Gℓ(i,j) := 2−2ℓ−3/summationtext∞\nk=0˙pk\nℓ(i,j) denotes the grounded Green kernel. The\nconvergence of the operator sum is granted by the positivity of λ1\nℓ:\n/vextenddouble/vextenddoublepℓ/vextenddouble/vextenddouble˚L2,˚L2≤/vextenddouble/vextenddoublep1/2\nℓ/vextenddouble/vextenddouble2\n˚L2,˚L2= sup\nu∈˚L2/a\\}b∇acketle{tu,pℓu/a\\}b∇acket∇i}ht\n/ba∇dblu/ba∇dbl2\n˚L2= 1−inf\nu∈˚L2Eℓ(u,u)\n/ba∇dblu/ba∇dbl2\n˚L2= 1−λ1\nℓ<1.\nThis operator is then extended to an operator acting on functions u∈Sℓby\n¯Gℓu(x) :=˙Gℓ/parenleftBig\nu|T4\nℓ/parenrightBig/parenleftbig\n2−ℓα/parenrightbig\n∀x∈Qℓ,α,α∈Aℓ.\nIn terms of the (extended) discrete Green operator we define th ediscrete Gaus-\nsian field\n˙hω\nℓ(i) :=√\n8πℓ−1/summationdisplay\nκ=0/summationdisplay\nι∈Iκξω\nκ,ι·˙Gℓ/parenleftBig\nηκ,ι|T4\nℓ/parenrightBig\n(i)\nonT4\nℓand its piecewise constant extension\n¯hω\nℓ(x) :=√\n8πℓ−1/summationdisplay\nκ=0/summationdisplay\nι∈Iκξω\nκ,ι·¯Gℓηκ,ι(x)\nonT4. Forγ∈R, thediscrete quantum Liouville measure is given by\n˙µω\nℓ:= 2−4ℓexp/parenleftBig\n−γ2\n2˙kℓ/parenrightBig/summationdisplay\ni∈T4\nℓexp/parenleftBig\nγ˙hω\nℓ(i)/parenrightBig\nδi\nwith\n˙kℓ:=E/bracketleftbig˙hℓ(i)2/bracketrightbig\n= 8π2·2−4ℓ/summationdisplay\nj∈T4\nℓ˙Gℓ(i,j)2=π2·2−8ℓ−3∞/summationdisplay\nk=0k˙pk\nℓ(i,i),\nindependent of i∈T4\nℓ.\nAlternatively — and equivalent in distribution according to (18) — we ca n\ndefine the discrete Gaussian field by\n˙hω\nℓ(i) :=√\n8π/parenleftbig˙Gℓ˙ξω/parenrightbig\n(i) = 2−2ℓ−3/2π∞/summationdisplay\nk=0/summationdisplay\nj∈T4\nℓ˙pk\nℓ(i,j)˙ξω\nj (22)\nonT4\nℓwith a sequence of N(0,1)-i.i.d. random variables ( ξj)j∈T4\nℓand˙ξi:=\nξi−2−4ℓ/summationtext\nj∈T4ξj. In other words,\n−∆ℓ˙hω\nℓ(i) =√\n8π˙ξω\ni.\nMoreover,\nE/bracketleftBig\n/a\\}b∇acketle{tu,˙hℓ/a\\}b∇acket∇i}ht2\nL2/bracketrightBig\n= 8π2/vextenddouble/vextenddouble˙Gℓu/vextenddouble/vextenddouble2\nL2∀u∈L2(T4\nℓ)16 Sturm\nand thus the distribution of the Gaussian field ˙hℓis explicitly given by the\nprobability measure\ndPℓ(ζ) :=1\nZℓexp/parenleftbigg\n−1\n16π2/vextenddouble/vextenddouble/vextenddouble∆ℓζ/vextenddouble/vextenddouble/vextenddouble2\nL2(T4\nℓ)/parenrightbigg/productdisplay\nj∈T4\nℓL1(dζj), (23)\nconditioned to the hyperplane/braceleftbig/summationtext\niζi= 0/bracerightbig\ninRT4\nℓ. Hereζ= (ζi)i∈T4\nℓ,\n/vextenddouble/vextenddouble/vextenddouble∆ℓζ/vextenddouble/vextenddouble/vextenddouble2\nL2(T4\nℓ)= 2−4ℓ/summationdisplay\ni∈T4\nℓ/vextendsingle/vextendsingle/vextendsingleζi−1\n8/summationdisplay\nj∈Jℓζi+j/vextendsingle/vextendsingle/vextendsingle2\n,\nandZℓ∈(0,∞) denotes a suitable normalization constant.\nThe convergence ¯hℓ→hand ˙µℓ→µasℓ→ ∞will be analyzed in detail\nin the forthcoming paper [3]. For related convergence questions co ncerning bi-\nharmonic Gaussian random fields on the cube [0 ,1]4with Dirichlet boundary\nconditions, see [22].\nReferences\n1. Cercl´ e, B. Liouville Conformal Field Theory on the highe r-dimensional sphere.\narXiv:1912.09219 , 2019.\n2. Dello Schiavo, L., Herry, R., Kopfer, E., and Sturm, K.-T. Conformally invari-\nant random fields, quantum Liouville measures, and random Pa neitz operators on\nRiemannian manifolds of even dimension. Arxiv 2105.13925 , 2021.\n3. Dello Schiavo, L., Herry, R., Kopfer, E., and Sturm, K.-T. 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Sur le Chaos Multiplicatif. Ann. sc. math. Qu´ ebec , 9(2):105–150,\n1985.\n10. Lawler, G. F. Conformally invariant processes in the plane , volume 114 of Mathe-\nmatical Surveys and Monographs . Amer. Math. Soc., Providence, RI, 2005.\n11. Lawler, G. F. Conformally invariant loop measures. In Proceedings of the Interna-\ntional Congress of Mathematicians—Rio de Janeiro 2018. Vol . I. Plenary lectures ,\npages 669–703. World Sci. Publ., Hackensack, NJ, 2018.Random Riemannian Geometry in 4 Dimensions 17\n12. Le Gall, J.-F. Brownian geometry. Jpn. J. Math. , 14(2):135–174, 2019.\n13. Le Gall, J.-F. and Miermont, G. Scaling limits of random t rees and planar maps.\nInProbability and statistical physics in two and more dimensi ons, volume 15 of\nClay Math. Proc. , pages 155–211. Amer. Math. Soc., Providence, RI, 2012.\n14. Lodhia, A., Sheffield, S., Sun, X., and Watson, S.S. Fracti onal Gaussian fields: A\nsurvey.Probab. Surveys , 13:1–56, 2016.\n15. Miller, J. and Sheffield, S. Imaginary geometry I: interac ting SLEs. Probab. Theory\nRelat. Fields , 164(3-4):553–705, 2016.\n16. Miller, J. and Sheffield, S. Liouville quantum gravity and the Brownian map I: the\nQLE(8/3,0) metric. Invent. Math. , 219(1):75–152, 2020.\n17. Oswald:. P. Haar system as Schauder basis in Besov spaces : The limiting cases for\n0< p≤1.INS-Preprint No. 1810, Bonn University .\n18. Paneitz, S. M. A quartic conformally covariant different ial operator for arbitrary\npseudo-Riemannian manifolds (summary). SIGMA Symmetry Integrability Geom.\nMethods Appl. , 4:Paper 036, 3, 1983. Published in 2008.\n19. Schramm, O. Conformally invariant scaling limits: an ov erview and a collection\nof problems. In International Congress of Mathematicians. Vol. I , pages 513–543.\nEur. Math. Soc., Z¨ urich, 2007.\n20. Schramm, O. and Sheffield, S. Contour lines of the two-dime nsional discrete Gaus-\nsian free field. Acta Math. , 202(1):21–137, 2009.\n21. Schramm, O. and Sheffield, S. A contour line of the continuu m Gaussian free field.\nProbab. Theory Relat. Fields , 157(1-2):47–80, 2013.\n22. Schweiger, F. On the membrane model and the discrete Bila placian. PhD Thesis\nBonn University , 2021.\n23. Sheffield, S. Gaussian free fields for mathematicians. Probab. Theory Relat. Fields ,\n139(3-4):521–541, 2007."    },    {        "title": "2401.12685v2.Dynamics_of_inertial_particles_under_velocity_resetting.pdf",        "content": "Dynamics of inertial particles under velocity resetting\nKristian Stølevik Olsen∗and Hartmut L¨ owen\nInstitut f¨ ur Theoretische Physik II - Weiche Materie, Heinrich-Heine-Universit¨ at\nD¨ usseldorf, D-40225 D¨ usseldorf, Germany\n∗e-mail: olsen@thphy.uni-duesseldorf.de\nAbstract: We investigate stochastic resetting in coupled systems involving two degrees\nof freedom, where only one variable is reset. The resetting variable, which we think of\nas hidden, indirectly affects the remaining observable variable through correlations. We\nderive the Fourier-Laplace transform of the observable variable’s propagator and provide\na recursive relation for all the moments, facilitating a comprehensive examination of the\nprocess. We apply this framework to inertial transport processes where we observe the\nparticle position while the velocity is hidden and is being reset at a constant rate. We\nshow that velocity resetting results in a linearly growing spatial mean squared displacement\nat late times, independently of reset-free dynamics, due to resetting-induced tempering of\nvelocity correlations. General expressions for the effective diffusion and drift coefficients\nare derived as function of resetting rate. Non-trivial dependence on the rate may appear\ndue to multiple timescales and crossovers in the reset-free dynamics. An extension that\nincorporates refractory periods after each reset is considered, where the post-resetting pauses\ncan lead to anomalous diffusive behavior. Our results are of relevance to a wide range of\nsystems, including inertial transport where mechanical momentum is lost in collisions with\nthe environment, or the behavior of living organisms where stop-and-go locomotion with\ninertia is ubiquitous. Numerical simulations for underdamped Brownian motion and the\nrandom acceleration process confirm our findings.\n1 Introduction\nMany processes in nature involve degrees of freedom that evolve in seemingly stochas-\ntic ways [1, 2, 3]. In many cases, large jumps in the values of an observed state\nvariable can occur, which may drastically change the overall dynamics. Stochastic\nresetting is one example of large and sudden jumps where a degree of freedom is\nat random times reset to its initial value [4, 5, 6]. Over the past decade, stochastic\nresetting has gained much attention in the non-equilibrium statistical physics com-\nmunity for multiple reasons. First of all, it has been shown to optimize target search\nprocesses, with potential applications ranging from computer science to the under-\nstanding of animal foraging strategies [7, 8, 9]. Furthermore, resetting generates\nnon-equilibrium steady states by trapping the system in a never-ending loop of the\ntransient dynamical regime. Only recently has these non-equilibrium states been\nstudied under the lens of stochastic thermodynamics, giving insights into exactly\nhow far from thermal equilibrium such systems are [10, 11, 12, 13, 14, 15]. The\nmajority of past work is based on the dynamics and resetting of a single degree of\nfreedom. When multiple state variables are present, with resetting only acting on a\nsubset of these, a much richer phenomenology can occur. This paper studies such\npartial resetting1in a coupled two-dimensional system (see Fig. (1)), with particular\nfocus on the case of position and velocity as is pertinent to physics.\n1One should note that the terminology partial resetting is sometimes also used for reset processes\nwhere x→ax, with a∈(0,1) the strength of the resetting. Here we use the phrasing partial in\nstead to refer to the resetting of parts of the set of degrees of freedom.\n1arXiv:2401.12685v2  [cond-mat.stat-mech]  2 Apr 2024Figure 1: Sketch of the system under consideration. A set of coupled stochastic\nvariables ( x, y) where only yis being reset (partial resetting) at random times {ti}.\nWe consider a case where the reset variable is hidden from observations, and infer\nindirect consequences of the resetting on the dynamics of the observable variable x.\nPartial resetting has only been studied for a handful of cases in the past, to the\nbest of our knowledge. In dimensions higher than one, one can considered resetting\nof only one spatial component [16]. For underdamped Brownian particles, reset-\nting of either position alone or simultaneous reset of position and velocity has been\nstudied [17]. Similar types of partial resets have been considered for the random\nacceleration processes, where both the propagator and survival probabilities have\nbeen studied [18, 19]. Since the position of a particle can be seen as the area under\nthe velocity curve, connections between velocity resetting and large deviation theory\ncan be made. In large deviation theory one often studies time-additive observables\nsuch as the area under the curve, which recently have been studied under resetting\n[20, 21, 22, 23]. Velocity resetting is also natural within active matter, where reset-\nting schemes have recently been considered where one can reset both position and\ndirection of motion, or only one of these variables [24, 25, 26, 27, 28]. Furthermore,\nrun and tumble motion is in itself a velocity resetting process [29, 30, 31]. Recently,\na similar problem where an overdamped particle in a potential is driven by a reset-\nting noise was studied using Kesten variables, where both propagators and moments\nwere studied [32].\nWhile much is known regarding the direct effect of resetting on a state variable,\nmuch less is known in general about the indirect effects of partial resetting in coupled\nsystems. An example of this could be that one (or several) degrees of freedom are\neither not experimentally available or simply not of interest. If these un-observed, or\nhidden , degrees of freedom undergo resetting, they may indirectly affect the observed\ndegrees of freedom through cross correlations. A situation where this can occur is in\nthe underdamped dynamics of a Brownian particle. Collisions with the environment\nor with a substrate may induce loss of mechanical momentum, effectively acting as\nresetting on the velocity variable. The position however, remains unaffected by the\nresetting and only changes its dynamics indirectly through its coupling to velocity.\nThis is different from past studies of resets in underdamped dynamics where only\n2position was reset, or both variables reset simultaneously [18, 17].\nStochastic resetting of velocities can also be of relevance to various foraging\nstrategies of animals and insects. An example is the intermittent stopping of flying\nforagers such as bees at flowers, whereby the velocity is reset to zero but position\nremains unchanged. Similar behavior, referred to as stop-and-go locomotion, is\nubiquitous in the behavior of macroscopic living organisms. Here individuals inter-\nmittently stop their motion completely [33, 34], for example in order to save energy\n[35] or to scout for predators. This has for example been observed in fish motion\n[36], in chipmunk foraging strategies [37], and in lizards climbing trees [38]. Such\nmacroscopic systems are typically also heavily prone to inertial effects, which the\nframework presented in this paper incorporates by default.\nIn this paper we study the dynamics of a 2-variable process where only one vari-\nable undergoes resetting. We consider one variable to be the observed variable,\nwhich does not undergo resets, while the other variable undergoes Poissonian reset-\nting at constant rate and is for brevity referred to as the hidden variable. We derive\nexplicit expressions for the Fourier-Laplace transform of the observed variables prop-\nagator, and derive from it a hierarchy of moments. We use this framework to study\nthe dynamics of inertial particles when only velocity is being reset. We show that\nwhen velocity is reset to zero, at late times the mean squared displacement is always\nlinear with an effective diffusivity that depends on the resetting rate. Extensions\nto the case where refractory periods are included after each reset is presented, in\nwhich case the late-time process may become anomalous if the refractory times are\npower-law distributed.\nThis paper is organized as follows. Section 2 discusses the propagator for two\ngeneric coupled variables when only one of them is being reset. Section 3 applies\nthese results to the case of inertial transport processes, and discusses effective trans-\nport coefficients under velocity resetting. Section 4 extends these results to the\ncase where refractory periods are included after each reset. Section 5 provides a\nconcluding discussion as well as potential outlooks.\n2 Propagator in coupled systems with resetting\nFor the sake of simplicity we consider two coupled degrees of freedom ( x, y), where we\nobserve xwhile the variable yexperiences resets. Extension to multiple variables is\nstraight-forward as long as all variables that undergo resetting do so simultaneously,\nin which case ymay be seen as a collective variable. We denote the full propagator\nof the system in the presence of resetting by pr(x, y, t|x0, y0), where ( x0, y0) are the\ninitial conditions at t= 0. The starting point of our analysis is the famous renewal\nframework for resetting processes [39], which we augment to the present case of\npartial resetting. The last renewal equation can be used to express the propagator\nin terms of the propagator of the underlying, or reset-free, system p0(x, y, t|x0, y0)\n3as\npr(x, y, t|x0, y0) =e−rtp0(x, y, t|x0, y0)\n+rZt\n0dτZ\ndx′dy′pr(x′, y′, t−τ|x0, y0)p0(x, y, τ|x′, y0)e−rτ.(1)\nHere the intuition is the same as in most renewal processes; the first term corresponds\nto realizations of the system where no resetting occurred. These paths occur with\nprobability e−rt, and the system evolves with the r= 0 propagator. The second\nterm takes into account general trajectories (with resetting) up to the time of the\nlast resetting t−τ. The resetting takes place with probability rdτ. At this instant,\nthe system is in some arbitrary state ( x′, y′). As only yis reset, the system proceeds\nto evolve towards ( x, y) from the new initial condition ( x′, y0). In this last time\ninterval τthere is no resetting, again taking place with probability e−rτ.\nAnother item of note regarding Eq. (1) is the fact that all memory is deleted at\nthe instance of reset, except for the variable x. Indeed, even if there were additional\ntime dependencies coming from external effects, such as dynamical disorder from\na changing environment or diffusion with a time-dependent drift for example, even\nthe environmental evolution must be reset for the full renewal structure assumed\nin Eq. (1) to be valid. This is simply due to the fact that the free propagator\nwhich links ( x′, y0)→(x, y) over the duration ( t−τ, t) in Eq. (1) only depends on\nthe duration τ. Generally, time-heterogeneity can break the renewal structure one\noften desires when working with resetting processes, and often it is assumed that\nany time-dependent parameters or annealed disorder is simultaneously reset with\nthe system variables to make the system fully renewing [40, 41, 25]. In the case of\noverdamped scaled Brownian motion, where temperature either grows or decreases\nin time, non-renewal resetting has been studied recently, where the time dependence\nof the temperature is allowed to persist through a resetting event [42].\nWe will be interested in the the marginalized propagator of the observable vari-\nablext, which we denote\n℘r(x, t|x0, y0)≡Z\ndypr(x, y, t|x0, y0). (2)\nand similarly for the process without resets. The indirect effect of resetting on the\nvariable xcan then be obtained by integrating over yin the above renewal equation:\n℘r(x, t|x0, y0) =e−rt℘0(x, t|x0, y0) (3)\n+rZt\n0dτe−rτZ\ndx′℘r(x′, t−τ|x0, y0)℘0(x, τ|x′, y0).\nTo proceed we restrict our attention to a class of spatially homogeneous systems,\nwhereby the underlying ( r= 0) propagator satisfies\n℘0(x, τ|x′, y0) =℘0(x−x′, τ|y0). (4)\nThis casts the renewal equation of a convolution form, and we may readily apply a\nFourier transform in space and a Laplace transform in time to solve the equation.\n4We use the conventions\nLs[f(t)]≡˜f(s) =Z∞\n0dte−stf(t), (5)\nFs[g(x)]≡ˆg(k) =Z∞\n0dte−ikxg(x.) (6)\nApplying a Fourier-Laplace transform to the above, we find\nˆ˜℘r(k, s|x0, y0) =ˆ˜℘0(k, s+r|x0, y0)\n1−rˆ˜℘0(k, s+r|y0). (7)\nThis expresses the marginalized propagator of the observable variable xunder re-\nsetting of the non-observable yto the corresponding propagator without resetting.\nNote that only the numerator is conditioned on both initial conditions, while the\nterm in the denominator should be taken with x0= 0 as a consequence of the ho-\nmogeneity assumption. Propagators of this form were recently studied in detail for\nfractional Brownian motion in Ref. [23]. Here we use Eq. (7) to derive a hierarchy\nof moments from which exact expressions for the coefficients governing late-time\nscaling is provided. We later extend these results to also allow refractory periods of\narbitrary durations after each reset.\n2.1 Hierarchy of moments\nInverting the above solution in Eq. (7) exactly can be arduous in many cases, as\nthe marginalized propagator even in simple coupled systems have complex Fourier-\nLaplace transforms. To make further progress, we derive from it a general expression\nfor the moments of the process xt. First, note that moments can be computed from\nFourier transforms of the probability density as\n⟨xn|x0, y0⟩t=∂n\n∂(ik)nˆ℘r(k, t|x0, y0)\f\f\f\f\nk=0. (8)\nAs we have the Fourier-Laplace transform, we will in our case find\ng⟨xn|x0, y0⟩s=∂n\n∂(ik)nˆ˜℘r(k, s|x0, y0)\f\f\f\f\nk=0, (9)\nwhereg⟨xn|x0, y0⟩s=Lt→s[⟨xn|x0, y0⟩t] denotes Laplace transforms. In order to\nobtain expressions for the moments from Eq. (7), we first re-write it as\nˆ˜℘r(k, s|x0, y0)[1−rˆ˜℘0(k, s+r|y0)] = ˆ˜℘0(k, s+r|x0, y0) (10)\nsimply to avoid having to deal with quotients. Using the generalized product rule\nfor higher-order derivatives, we have\n∂n\n∂(ik)nˆ˜℘0(k, s+r|x0, y0) =nX\nℓ=0\u0012n\nℓ\u0013∂(n−ℓ)\n∂(ik)(n−ℓ)ˆ˜℘r(k, s|x0, y0)\n×∂ℓ\n∂(ik)ℓ[1−rˆ˜℘0(k, s+r|y0)]. (11)\n5Setting k= 0 and using the expression for the moments, we have\ng⟨xn|x0, y0⟩(0)\ns+r=nX\nℓ=0\u0012n\nℓ\u0013\ng⟨xn−ℓ|x0, y0⟩s[δℓ,0−rg⟨xℓ|y0⟩(0)\ns+r], (12)\nwhere the superscript (0) denotes moments for the underlying process with r= 0.\nRearranging gives\ng⟨xn|x0, y0⟩s=s+r\nsg⟨xn|x0, y0⟩(0)\ns+r+s+r\ns\"\nrnX\nℓ=1\u0012n\nℓ\u0013\ng⟨xn−ℓ|x0, y0⟩sg⟨xℓ|y0⟩(0)\ns+r#\n.\n(13)\nThis recursive relation can be used to iteratively construct any moment of the\nprocess xtgiven lower-order moments and the moments of the r= 0 case, which are\nassumed to be known. We emphasize that this result can be useful when the Fourier-\nLaplace transform of the propagator itself is hard to obtain, while if ˆ˜p0(k, s|x0, y0) is\nknown one can simply expand Eq. (7) in powers of kto identify the moments. We\nalso note that when r= 0, the above equation reduces to ⟨xn|x0, y0⟩s=⟨xn|x0, y0⟩(0)\ns\nas it should.\n3 Transport processes: crossover from anomalous to normal diffusion\nFor transport processes the relevant physical variables are often position xtand\nvelocity vtof a particle, satisfying ˙ xt=vt. Here we consider the effect of velocity\nresetting on spatial transport by using Eq. (13). This is of relevance to a wide range\nof systems, for example for particles with inelastic collisions with an environment\nor substrate, or in foraging processes where animals intermittently stop (eg. to\ncollect nutrients, or scout for predators) before re-starting their motion from zero\nvelocity but unchanged position. From the hierarchy in Eq. (13) we derive general\nexpressions for the effective drift and diffusivity, and consider in more detail the case\nwhere the underlying process shows anomalous diffusion.\n3.1 Effective transport coefficients\nTo characterize transport, we are interested in the first two spatial moments\ng⟨x|x0, v0⟩s=s+r\ns\u0014\ng⟨x|x0, v0⟩(0)\ns+r+r\nsg⟨x|v0⟩(0)\ns+r\u0015\n, (14)\ng⟨x2|x0, v0⟩s=s+r\nsg⟨x2|x0, v0⟩(0)\ns+r+s+r\nsr\nsg⟨x2|v0⟩(0)\ns+r\n+s+r\ns\u0014\n2rg⟨x|x0, v0⟩sg⟨x|v0⟩(0)\ns+r\u0015\n, (15)\n6which contain information regarding effective drift and dispersion of the spatial\nvariable. Without loss of generality, let us consider x0= 0. Then the first moment\ncan be written compactly as\ng⟨x|0, v0⟩s=\u0012s+r\ns\u00132\ng⟨x|0, v0⟩(0)\ns+r. (16)\nAt late times, corresponding to small values of s, the s−2pole dominates, giving\nrise to a linear growth in time. The inverse Laplace transform can be calculated in\nterms of residues as\n⟨x|0, v0⟩t=L−1\ns→t(\u0012s+r\ns\u00132\ng⟨x|0, v0⟩(0)\ns+r)\n(17)\n=X\npoles{si}Resi\"\u0012s+r\ns\u00132\ng⟨x|0, v0⟩(0)\ns+rest#\n,\nwhere the poles {si}are those of Eq. (16). The poles at non-zero values of sgives\nrise to exponentially decaying terms. Hence the late-time behavior is extracted from\nthe pole at zero, in which case the residue reads\n⟨x|0, v0⟩t= lim\ns→0∂s\u0014\n(s+r)2g⟨x|0, v0⟩(0)\ns+rest\u0015\n=r2g⟨x|0, v0⟩(0)\nrt+... (18)\nwhere the terms + ...represent terms that are either approaching a constant or\nvanishing at late times. The effective drift at late times is then calculated as\nVeff≡lim\nt→∞⟨x|0, v0⟩t\nt=r2g⟨x|0, v0⟩(0)\nr. (19)\nHence velocity resetting to a non-zero value of velocity v0, which typically results\nin non-zero ⟨x|0, v0⟩(0)\nt, will result in a drift at late times, as is expected2. This can\nbe seen as a rectification effect due to velocity resetting. A physical intuition of Eq.\n(19) is obtained by noting that the late-time behavior of the mean position can be\nwritten\n⟨x|0, v0⟩t≃Vefft= (rt)Z\ndtre−rt⟨x|0, v0⟩(0)\nt (20)\nwhich is nothing but the mean number of resets n=rtin time t, times the mean\nstep length during a inter-reset epoch.\nIn the remainder of this section, we consider for simplicity symmetric processes\nwith vanishing odd moments so that no drift is present Veff= 0, and set v0= 0.\nProceeding similarly as for the effective drift, we can derive an expression for the\neffective diffusivity. The second moment in Laplace space reads\ng⟨x2|0,0⟩s=\u0012s+r\ns\u00132\ng⟨x2|0,0⟩(0)\ns+r. (21)\n2It should be noted that one may in principle reset velocity to a non-zero value v0and yet\nobtain a vanishing mean ⟨x|0, v0⟩(0)\nt. For example, in the presence on a constant external drift one\ncould reset to a velocity that exactly opposes the drift.\n7By the same logic, the s−2pole gives rise to a linearly growing mean-squared-\ndisplacement. Again, the real-time mean-squared-displacement can be expressed\nas a sum over poles siof Eq. (21)\n⟨x2|0,0⟩t=L−1\ns→t(\u0012s+r\ns\u00132\ng���x2|0,0⟩(0)\ns+r)\n(22)\n=X\npoles{si}Resi\"\u0012s+r\ns\u00132\ng⟨x2|0,0⟩(0)\ns+rest#\n.\nThe dominant late-time behavior once again comes from the second-order pole at\ns= 0, resulting in\n⟨x2|0,0⟩t= lim\ns→0∂s\u0014\n(s+r)2g⟨x2|0,0⟩(0)\ns+rest\u0015\n=r2g⟨x2|0,0⟩(0)\nrt+... (23)\nHence the effective diffusivity can be calculated as\nDeff≡lim\nt→∞⟨x2|0,0⟩t\n2t=r2\n2g⟨x2|0,0⟩(0)\nr. (24)\nAs for the effective drift, a natural physical interpretation of this in terms of the\nasymptotic behavior of the mean squared displacement can be obtained by noticing\nthat\n⟨x2|0,0⟩t≃2Defft=rtZ∞\n0dtre−rt⟨x2|0,0⟩(0)\nt (25)\nwhich is the mean number of resets ntimes the mean growth of the second moment\nduring a inter-reset epoch. We stress here the generality of this result; as long as\nthe system considered is spatially homogeneous, and the full renewal structure of\nEq. (1) is satisfied, the process under velocity resets will display normal diffusion\nwith the above effective diffusivity, even if the reset-free system shows anomalous\ndiffusion. We discuss this further in later sections.\nUnderdamped Brownian motion: As a simple application of the above for-\nmula, we consider velocity resetting for an underdamped Brownian particle. This\ncan be seen as a minimal model for a particle moving through a complex environ-\nment, where intermittently the particle collides with the environment and looses its\nmomentum [43]. The particle obeys the coupled equations\ndxτ=vτdt, (26)\ndvτ=−γvτdt+p\n2kBTγdW, (27)\nwhere dWis an increment in the Wiener process and γthe friction coefficient. Here\nwe have set mass to unity. The marginalized position distribution reads [17]\n℘0(x, t|x0, v0) =1p\n2πΣ2\ntexp\u0012\n−(x− ⟨xt⟩)2\n2Σ2\nt\u0013\n, (28)\n8Figure 2: Effective diffusivity for an underdamped Brownian motion undergoing\nvelocity resetting, as a function of re-scaled resetting rate ζ=r/γ. Dashed black\nlines show exact theory (Eq. (32)) while points show simulated data, using D0=\nγ= 1.\nwhere the variance and mean of the density reads\nΣ2\nt=D0τγ\u00122t\nτγ+ 4e−t/τγ−e−2t/τγ−3\u0013\n, (29)\n⟨xt⟩=x0+v0τγ\u0000\n1−e−t/τγ\u0001\n, (30)\nwith τγ= 1/γbeing the inertial timescale and D0=kBT/γ. From this one can\neasily calculate the second moment in Laplace space, resulting in\ng⟨x2|0,0⟩(0)\ns=−3D0\nγs+4D0\nγ(γ+s)−D0\nγ(2γ+s)+2D0\ns2, (31)\nwhere we set x0=v0= 0, so that Veff= 0. Using Eq. (24) one immediately finds\nDeff\nD0=\u0012\n1 +ζ3 +ζ\n2\u0013−1\n. (32)\nHere we introduced the dimensionless variable ζ=r/γ. As expected, Deff=D0is\nobtained when ζ= 0, which can be achieved either by r= 0 or in the high-friction\nlimit γ→ ∞ . The latter case highlights the fact that for Brownian particles a non-\ntrivial effective diffusivity only occurs in the presence of inertial effects, whereby\nthe particle must accelerate after each reset. We also see that velocity resetting\nsuppresses spatial transport, and Deff= 0 as r→ ∞ . This is verified using numerical\nsimulations in Fig. (2).\n3.2 When the underlying process is anomalous\nThe above results have interesting implications for transport processes in ( x, v) that\nwould be anomalous in the absence of resetting. Anomalous diffusion processes\nwith resetting have been studied recently for overdamped systems in the context of\n9ergodicity breaking and restoration [44, 45, 46, 47]. Here we investigate the effects\nof velocity resetting in underdamped anomalous systems.\nFor anomalous diffusion processes, the underlying process satisfies in the simplest\ncase\n⟨x2|0,0⟩(0)\nt=D0tα, (33)\nwith diffusion exponent αtaking any positive real value. Such processes are present\nin a wide range of systems, with classical examples being transport of Brownian\nparticles in complex environments such as fractals or media with power-law friction\n[48, 49, 50, 51, 52, 53, 54]. Anomalous diffusion has also been observed in granular\nsystems, such as in the velocity profile of fluid-drive silo discharge [55], in granular\ngasses near a shear instability [56], and in the height fluctuations in graphene [57].\nIn such cases, one often expects subdiffusive processes α <1, while there are also\ncases where superdiffusion with α > 1 occurs, such as in L´ evy flights, random\nacceleration processes, tracers in turbulent flow, active particles with time-dependent\nself-propulsion forces, and in diffusion with density-dependent diffusivity [58, 59, 60,\n61, 62, 63]. For a review of theoretical models of anomalous diffusion, see for example\n[64]. While most of these studies consider particles in the overdamped limit, with\nno coupled variables, many models can be extended to the underdamped case and\nstudied under the framework presented here.\nThe Laplace transform in time of Eq. (33) reads ⟨x2|0,0⟩(0)\ns=D0s−(1+α)Γ(1+ α).\nUsing Eq. (21) this leads directly to\ng⟨x2|0,0⟩s=D0Γ(α+ 1)( r+s)1−α\ns2. (34)\nInverting gives the full time-dependence of the mean-squared-displacement\n⟨x2|x0, v0⟩t= (α−1)D0r−α(αΓ(α, rt)−Γ(α+ 1))\n+ (α−1)αD0r1−α(Γ(α−1)−Γ(α−1, rt))t, (35)\nwhere Γ( x, z) is the (upper) incomplete Gamma function. At early times, as re-\nsetting has not yet had time to occur, the mean-squared-displacement behaves as\nin the underlying system. Generally, the short-time behavior of the mean squared\ndisplacement can be obtained by looking at large values of its Laplace variable. For\ns≫rin Eq. (21) we see that ⟨x2|0,0⟩s=⟨x2|0,0⟩(0)\ns. Hence in this case we have\n⟨x2|0,0⟩t=D0tα. At late times, we see that a linear growth takes over, in agree-\nment with what we predicted more generally (see Eqs. (23) and (24)). The effective\ndiffusivity can be calculated either from Eq. (24) or directly from Eq. (35), yielding\nDeff≡lim\nt→∞⟨x2|0,0⟩t\n2t=D0Γ(α+ 1)\n2rα−1. (36)\nWe see that anomalous diffusion processes with diffusion exponent αbecome normal\nunder velocity resetting, with an effective diffusion coefficient scaling with resetting\nrate as Deff∼r1−α. For superdiffusive processes, the effective diffusivity decreases as\na function of resetting rate, while for subdiffusive processes it grows with increasing\n10Figure 3: Comparisons of theory (dashed black lines) and simulated data for a\nrandom acceleration process. a) Mean squared displacement shows a clear crossover\nfrom anomalous α= 3 diffusion to normal diffusion with α= 1 at times larger than\nthe crossover time t∗(grey vertical line). b) Effective diffusivity at late times scales as\nDeff∼r−2, as predicted by Eq. (36). c) Velocity auto-correlation functions showing\nexponential tempering. Solid colored lines are simulations, while black dashed line\ncorresponds to Eq. (44). Resetting rate is set to unity unless stated otherwise.\nresetting rate. The fact that subdiffusive processes can enhance their diffusivity by\nresetting originates in the full renewal structure of Eq. (1). Since subdiffusion often\noccurs for processes that slow down substantially over time, such as for single-file\ndiffusion [65, 66], restarting to a state of higher motility can be beneficial. The\ncrossover time t∗where the linear growth starts to dominate over the anomalous\ngrowth can be identified by matching the late-time regime with effective diffusivity\ngiven by Eq.(36) with the early-time growth D0tα, resulting in\nt∗=\u0014Γ(α+ 1)\n2\u00151\nα−1\nr−1. (37)\nSince the resetting timescale r−1determines when the mean squared displacement\nshould cross over to linear growth, this proportionality is sensible.\nRandom acceleration process: A concrete example of the above could be the\nrandom acceleration process [60]\n˙x=v, (38)\n˙v=√\n2η(t), (39)\nwhere η(t) is a Gaussian white noise with correlator ⟨η(t1)η(t2)⟩=δ(t1−t2), and\nwhere we set the noise strength to unity for simplicity. The marginalized process\n℘0(x, t|x0, v0) is known to have the Gaussian propagator [60, 18]\n℘0(x, t|0,0) =r\n3\n4πt3exp\u0012\n−3\n4t3x2\u0013\n, (40)\nwith a mean-squared-displacement that grows as a cubic in time ⟨x2|0,0⟩(0)\nt=2\n3t3,\ni.e.α= 3 and D0= 2/3 in the above calculation. The effective diffusivity after\n11resetting reads Deff= 2/r2, with a crossover time t∗=√\n3/r. This result was\nalso reported in Ref. [23] obtained by other methods. Here the full mean-squared-\ndisplacement can easily be obtained from Eq. (35), which we show in Fig. (3 a),\nwhere a clear crossover from anomalous α= 3 to normal α= 1 is seen. The\npredicted Deff∼r−2scaling also matches perfectly with simulated data as seen in\nFig. (3 b).\n3.3 Tempered correlation functions\nThe emergence of normal diffusion independently of the anomalous nature of the un-\nderlying process can be understood as a consequence of resetting-induced tempering\nof the velocity auto-correlation functions. It is known that systems with exponen-\ntial (or sufficiently strong power-law) cutoffs, displays a crossover from anomalous\nto normal diffusion at late times [67].\nFor transport processes, the position is coupled to velocity simply through ˙ x=v,\nimplying that the mean-squared displacement in general comes from the temporal\nbehavior of the velocity correlation functions\n⟨(x−x0)2⟩t= 2Zt\n0dτ2Zτ2\n0dτ1Cr(τ1, τ2), (41)\nwhere we introduced Cr(τ1, τ2) =⟨vτ1vτ2⟩. We will by C0(τ1, τ2) denote the velocity\nauto-correlations in the absence of resetting. For correlations that decay sufficiently\nfast, the system exhibits normal diffusion and the above equation can be turned\ninto a Green-Kubo relation for the diffusion coefficient. If the correlation functions\ndecay too slowly, or even grow in time, one expects anomalous diffusion.\nSimilarly to the probability density, correlation functions are known to satisfy a\nrenewal equation of the type [68, 39]\nCr(t1, t2) =e−rt2C0(t1, t2) (42)\n+re−r(t2−t1)Zt1\n0dτe−rτC0(τ, t2−t1+τ).\nHere we have relied on the assumption that position is coupled to velocity, but not\nvice-versa, allowing a simple renewal equation for the velocity correlations. This is\nnormal for most models of Brownian motion, and has been utilized in the past where\nconsidering underdamped Brownian motion under partial resetting of position alone\n[17]. Already we can see that the inclusion of resetting tempers the correlations by\nincluding an exponential cutoff at a characteristic time r−1already in the first term.\nReturning again to a concrete example where the underlying process is anomalous\nof the type Eq. (33), we consider correlation functions of the form\nC0(t1, t2) =α(α−1)D0\n2min{t1, t2}α−2. (43)\nThis correlation function is chosen so that by applying Eq. (41) gives rise to a\nmean-squared-displacement growing anomalously in time with exponent α. Under\n12resetting, the new correlator Eq. (42) reads\nCr(t1, t2) =α(α−1)D0\n2e−r(t2−t1)\u0000\nr2−α(Γ(α−1)−Γ (α−1, rt1)) +tα−2\n1e−rt1\u0001\n,\nwhere we assumed t2≥t1. To clearly see the tempering, consider Cr(t, t+ ∆) with\n∆≥0 a lag variable. For t≫r−1we then find\nCr(t, t+ ∆)≃Γ(α+ 1)D0r2−α\n2e−r∆, (44)\nwhich shows a clear cutoff scale r−1. This exponential behavior is shown for the\nrandom acceleration process in Fig. (3 c), where normally the velocity correlations\nobey Eq. (43) with α= 3.\n3.4 The effect of real-time crossovers\nAs a final example, we consider a case where the underlying system has a crossover\nfrom one dynamical regime to another at a crossover time tc. We assume\n⟨x2⟩(0)\nt= 2Dαtαθ(tc−t) + 2Dβtβθ(t−tc). (45)\nThe effective diffusivity can again be calculated from Eq. (24), resulting in\nDeff(r) =Dαr1−αγ(α+ 1, rtc) +Dβr1−βΓ(β+ 1, rtc), (46)\nwhere Γ( ·) and γ(·) are the upper and lower incomplete Gamma functions respec-\ntively. We note that while γ(a, z) vanishes as z→0, Γ(a, z) simply approaches Γ( a)\nin this limit. A converse behavior holds for z→ ∞ . Hence the incomplete gamma\nfunctions give more or less weight to the two scaling behaviors depending on the\nvalue of the resetting rate. In particular, the effective diffusivity changes scaling\nbehavior as a function of resetting rate rat a crossover value rc=t−1\nc. In the case\nr≪t−1\nc, the system resets so rarely that the late-time dynamical regime always can\nbe explored, and Deff(r)∼r1−β. Similarly, when r≫t−1\nctypical trajectories only\nexplore the first dynamical regime, and Deff(r)∼r1−α.\nThe underdamped Brownian particle described by Eqs. (26) & (27) yet again\nserves as a good example for a case with a crossover. In this case, it is known\nthat for a particle initially at rest, the short-time mean squared displacement has a\nleading-order cubic behavior [69]\n⟨x2⟩(0)\nt=4γ2D0\n3t3+O(t4), (47)\nwhile at late times it crosses over to normal diffusion ⟨x2⟩(0)\nt∼t. Hence we expect\nan effective diffusivity that first is constant before decaying as r−2. This is indeed\nwhat is observed in Fig. (2), or equivalent in Eq. (32).\n13Figure 4: Sketch of the system we consider. a) Typical dynamics of a forager consists\nin exploration phases interrupted by refractory periods of no motion. b) Sketch of\none dimension time-series for position and velocity, showing a refractory period of\nduration τfollowing a resetting at time t1. Here we assume that velocity is reset to\nzero, so that no motion occurs during the refractory period.\n4 Effects of refractory periods\nIn the case of inertial foragers that perform stop-and-go locomotion, velocity resets\ntov0= 0 are often followed by inactive periods (see Fig. (4)). In the context of\nresetting, such idle times after resetting events are referred to as refractory times\n[70, 71]. In overdamped systems with a single degree of freedom, anomalous diffu-\nsion behavior has been observed for particular choices of refractory and inter-reset\nstatistics [72]. In this section we include the effect of such refractory times in under-\ndamped systems, and show how normal Poissonian resets of velocity can also lead\nto anomalous diffusion.\n4.1 Exact propagator\nAs for the case of velocity resetting without refractory times, we start by investi-\ngating the renewal equation. The first renewal equation for the propagator can be\nwritten\npr(x, v, t|x0, v0) =e−rtp0(x, v, t|x0, v0)\n+rZt\n0dt1e−rt1Zt−t1\n0dτW(τ)Z\ndydup0(y, u, t 1|x0, v0)pr(x, v, t−t1−τ|y, v0)\n+rZt\n0dt1e−rt1Z∞\nt−t1dτW(τ)Z\ndydup0(y, u, t 1|x0, v0)δ(v−v0)δ(x−y). (48)\nHere the first two terms have the same interpretation as before, corresponding re-\nspectively to paths without resets and paths with resets. The only difference is the\ninclusion of a refractory time τafter the first reset event. The third term corresponds\nto paths that at time tend in the refractory phase.\n14As before, we will assume spatial homogeneity, which means we can without loss\nof generality set x0= 0. To ease notation, as before, we suppress this from the\npropagator so that pi(x, v, t|x0= 0, v0) =pi(x, v, t|v0), with i=r,0. Integrating the\nabove first renewal equation over velocity vwe find the position distribution\n℘r(x, t|v0) =e−rt℘0(x, t|v0)\n+rZt\n0dt1e−rt1Zt−t1\n0dτW(τ)Z\ndy℘0(y, t1|v0)℘r(x, t−t1−τ|y, v0)\n+rZt\n0dt1e−rt1Z∞\nt−t1dτW(τ)℘0(x, t1|v0). (49)\nInvoking spatial homogeneity we write ℘r(x, t−t1−τ|y, v0) =℘r(x−y, t−t1−τ|v0).\nTaking a Fourier transform in space, we obtain\nˆ℘r(k, t|v0) =e−rtˆ℘0(k, t|v0)\n+rZt\n0dt1e−rt1ˆ℘0(k, t1|v0)Zt−t1\n0dτW(τ) ˆ℘r(k, t−t1−τ|v0)\n+rZt\n0dt1e−rt1ˆ℘0(k, t1|v0)Z∞\nt−t1dτW(τ). (50)\nLaplace transforming Eq. (50) results in\nˆ˜℘r(k, s|v0) =ˆ˜℘0(k, s+r|v0) +r˜W(s)ˆ˜℘0(k, s+r|v0)ˆ˜℘r(k, s|v0)\n+rˆ˜℘0(k, s+r|v0)\nsh\n1−˜W(s)i\n. (51)\nSolving for the propagator in the presence of resetting, we get the closed form\nexpression\nˆ˜℘r(k, s|v0) =ˆ˜℘0(k, s+r|v0)\n1−r˜W(s)ˆ˜℘0(k, s+r|v0)\u0010\n1−r\ns[1−˜W(s)]\u0011\n. (52)\nThis gives an exact expression for the spatial propagator for any distribution of re-\nfractory times. One can indeed verify that the propagator is normalized ˆ˜℘r(0, s|v0) =\n1/s. Furthermore, in the case without any refractory period W(τ) =δ(τ), we recover\nprevious expressions for the propagator under velocity resets.\nIf one cares only about late-time dynamical behavior, one may naively expect\nthat this reduces to a simple continuous time random walk (CTRW). In this case, the\npropagator in Fourier-Laplace space is given by the famous Montroll-Weiss formula,\nwhich relates the propagator to the distribution of jump lengths and the distribu-\ntion of the waiting times in between each jump [73]. The classical Montroll-Weiss\ntheory assumes independent jump lengths and waiting times, although extension to\ncorrelated cases have been considered [74]. In the present case, the waiting time is\ngiven by the combination tE+τ, with tEthe exploration time and τthe duration of\na refractory phase. The jump length however, is determined both by the underly-\ning propagator in the exploration phase and the duration of the exploration phase.\n15Hence the jump lengths depend only on the time tEof the exploration phase, not the\nduration of the refractory times. Therefore velocity resetting with refractory times\ncan be seen as a form of CTRW with a particular correlation between the jump\nlengths and the waiting times. Of course, another strong contrast to the CTRW\ncase is that the dynamics studied here is fully time-resolved and does not perform\nsudden discrete jumps.\n4.2 Mean-squared displacement\nThe moments of position can be obtained from Eq. (52) by differentiation as before,\nthrough Eq. (8). Assuming we are dealing with a process where the first moment\nalways vanishes, which is typically the case if v0= 0, we can proceed as in earlier\nsections to find\ng⟨x2|0,0⟩s=g⟨x2|0,0⟩(0)\ns+r1 +r\ns[1−˜W(s)]\nh\n1−r˜W(s)\ns+ri2. (53)\nWe consider two scenarios next.\n4.2.1 Mean refractory period is finite (⟨τ⟩<∞):\nAs before, the late-time dynamics can be obtained by considering the small- slimit\nof the Laplace transformed mean squared displacement in Eq. (53). For small s, we\ncan approximate ˜W(s) = 1−s⟨τ⟩+..., in which case the dominant singularity of\nthe MSD is s−2:\ng⟨x2|0,0⟩s=r2g⟨x2|0,0⟩(0)\nr\ns2(1 +r⟨τ⟩). (54)\nHence the diffusion is normal, with an effective diffusion coefficient\nDeff= lim\nt→∞⟨x2(t)|0,0⟩\n2t=r2g⟨x2|0,0⟩(0)\nr\n2 + 2 r⟨τ⟩. (55)\nWe see that refractory times suppress the effective diffusion, and when ⟨τ⟩= 0 we\nrecover previous results.\n4.2.2 Mean refractory period is infinite (⟨τ⟩=∞):\nIf the mean refractory time diverges due to power-law distributed refractory periods\nW(τ)∼1/(τ1+α) with ˜W(s)∼1−asα, for some constant aandα∈(0,1), we in\nstead find the small- sbehavior\ng⟨x2|0,0⟩s=rg⟨x2|0,0⟩(0)\nr\nas1+α. (56)\nIn this case the MSD in real-time reads\n⟨x2(t)⟩ ≃rg⟨x2|0,0⟩(0)\nr\natα, (57)\n16with≃denoting equality at late times. Hence the particle will exhibit anomalous\ndiffusion of the sub-diffusive type independently of its underlying dynamics, simply\nas a consequence of the long refractory periods.\n5 Conclusion and outlook\nIn this paper we have studied stochastic resetting in coupled systems, with particular\nfocus on position and velocity. A general result was derived for the propagator for an\nobservable variable under the indirect effect of resetting of a hidden variable. From\nthis we derived a general recursive equation for the moments, from which one can\nin principle fully characterize the marginalized process. We applied the proposed\nframework to transport processes of inertial particles where velocity undergoes re-\nsetting, and show that generically the late-time dynamics shows normal diffusion\neven if the reset-free system is anomalous. This we attribute to the tempering effect\nstochastic resetting has on velocity auto-correlation functions. We derived a com-\npact expression for the effective drift and diffusivity coefficients. When a temporal\ncrossover between two dynamical regimes exists in the underlying dynamics, this\ntranslates into a crossover as a function of resetting rate in the effective diffusivity,\nwhere different scaling behaviors are observed. We tested the validity of our predic-\ntions in the cases of underdamped Brownian motion and for the random acceleration\nprocess, both showing excellent agreement with simulated data. Extension to the\ncase where refractory periods are included after each reset was also considered, in\nwhich case resetting-induced anomalous diffusion can be observed.\nThe main results of this paper are based on two crucial assumptions. First is the\nhomogeneity in the variable xthat is not reset, and second is the full renewal struc-\nture represented by Eq. (1). Extensions of the results presented here to account for\neither spatial heterogeneity or for example non-renewal reset structure such as in [42]\nwould be very interesting, shining light on anomalous diffusion processes with veloc-\nity resets in systems with quenched or annealed disorder. Generalizations to more\ncomplex types of resetting, such as proportional resetting [43] or non-Poissonian\nwaiting times could also be considered.\nFinally, it would also be interesting to investigate resetting in coupled systems\nfrom a thermodynamic perspective. Recently much effort has been put into under-\nstanding the stochastic thermodynamics of resetting, with both entropy production\nand work having been considered [12, 13, 14]. However, in the present model the\nobservable variable is not the one undergoing resets. Hence it would be interest-\ning to investigate bounds on the thermodynamic cost based on partial accessible\ninformation, a topic which has gained considerable attention in the past decade\n[75, 76, 77, 78, 79, 80, 81, 82, 83, 84].\nAcknowledgments\nInsightful discussions and interactions with Kevin Pierce and Deepak Gupta are\ngratefully acknowledged. 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Received 2024 January 10; in origi nal form 2023 June 29\nABSTRACT\nWe study the surface magnetic field fluctuations due to radial oscillations as a viable cause for the micro structures of th e radio\npulsar pulse patterns. The electrical conductivity of matt er in the outer layer of the crust of a neutron star (NS) plays a crucial\nrole in the resulting surface magnetic field if we assume that the magnetic field is confined to this layer. This outer layer h as\na rapidly varying matter density - that changes the micro-ph ysics of the material affecting the electrical conductivity at every\nstage of the density change. In this study, the varying elect rical conductivity in this rapidly varying density regime o f the outer\nlayer of the NS crust - from ∼1011/u1D454 /u1D450/u1D45A−3to about 104/u1D454 /u1D450/u1D45A−3- has been used to calculate the surface magnetic field using\nthe induction equation. A finite effect of the strong gravitat ional field at the NS surface has also been taken into account. The\nequations have been solved in MATLAB using the method of lines . Any minor radial fluctuation due to stellar oscillation, in\nparticular the radial oscillations, leads to a fluctuation o f the electrical conductivity in the outer layer of the crust . This leads to\nfluctuations in the surface magnetic field with a frequency eq ual to the frequency of the stellar oscillation. We find that n ot only\nthe variation of the surface magnetic field is substantial, b ut also it does not remain constant throughout the lifetime o f the NS.\nKey words: stars: neutron - pulsars: general - stars: magnetic field - st ars: interiors\n1 INTRODUCTION\nIn over five decades since their discovery, the radio pulsars still\npresent us with challenges from theoretical and mathematic al un-\nderstanding of the physical processes going on in that syste m. The\nmagnetic fields of neutron stars play a pivotal role in all of t he ob-\nserved activities of the stars. The magnetospheric structu re of a NS\nis believed to be made of closed and open field lines, where the\nclosed field lines form a torus around the star and co-rotate w ith it.\nThis region is charge saturated - with the charges mainly ori ginat-\ning from pair creation of electron and positrons due to the st rong\nsurface magnetic field. The closed field lines \"open\" up at the light\ncylinder and the observed emission emerge from this conical zone.\nThough mostly the emission from the light cylinder is in the f orm of\na wind of plasma, a small fraction of this plasma is converted to a\ncoherent beam of radio waves arising due to the growth of plas ma\ninstabilities in the inner magnetosphere, from regions bel ow 10% of\nthe light cylinder radius ( Mitra (2017)). This beam is ejected relent-\nlessly, which we finally observe as the pulse profile - also sug gests\nthat the pair creation process generating the plasma has to b e con-\ntinuous. The magnetic field of the magnetosphere is the stron gest at\nthe NS surface, where the pair creation (and hence the genera tion\nof the plasma) is maximum. Since the closed field lines are cha rge\nsaturated, the surface magnetic field at the polar caps plays a seed\nrole in the subsequent processes for the creation of highly p olarised\nradio beams (and the winds).\n★e-mail:kas.sellick@gmail.com\n†e-mail:rays@ukzn.ac.zaThere is no conclusive theory to explain the actual origin of the\nNS magnetic field. It is generally accepted that the strong ma gnetic\nfield of a NS is due to flux conservation of the original star fro m\nwhich it was born. Said so, there isn’t any a priori knowledge about\nthe distribution of the magnetic field inside the NS. There ar e nu-\nmerous mechanisms which may result in field generation in the crust\nof NSs, for instance, Blandford, Applegate & Hernquist (1983) and\nUrpin, Levshakov & Yakovlev (1986) explored the idea that ther-\nmomagnetic instability may have this result. The primary pr oblem in\nhosting a super strong magnetic field in the NS core is due to th e fact\nthe nuclear matter in the core is highly super-conducting, r esulting\nin repulsion of flux to a lower density regime. Another challe nge\nmay come from general relativity (GR), where the vectorial n ature\nof the dipolar magnetic field does not allow interior solutio ns for a\nspherically symmetric configuration for realistic equatio ns of state\n(Rezzolla, Ahmedov & Miller (2001)). Detailed discussions about\nthese problems can be found elsewhere - for our present work, we\nshall focus on the crustal magnetic field evolution only and a ccept\nthat the magnetic field vanishes in the deep layers.\nConsidering the magnetic field being located in the outer lay er\nof the crust, the behaviour of the surface magnetic field will then\ndepend largely on the transport properties, in particular t he electric\nand the thermal conductivities of the matter in the outer cru st and\nhow the field evolves through it. One of the earlier works in co n-\nnection to the transport properties of dense matter was carr ied out\nbyFlowers & Itoh (1976). Thereafter notable studies on the elec-\ntrical and thermal conductivities were done by Yakovlev & Urpin\n(1980),Yakovlev & Urpin (1980a ),Yakovlev & Urpin (1980b ),\nFlowers & Itoh (1981),Itoh et al. (1984), etc., where we find ways\n©2024 The Authors2Sellick & Ray\nof calculating the electrical conductivity due to phonon sc attering\nand impurity scattering.\nThe magnetic field of a NS decays over it’s entire lifetime. Oh mic\ndecay of dipolar magnetic fields, that are confined to the oute r NS\ncrust, was first studied by Sang & Chanmugam (1987) who con-\ncluded that the field does not decay exponentially but by less than\na factor of order 102in the Hubble time. Chanmugam & Sang\n(1989) calculated the braking index of neutron stars from the ohmi c\ndecay of the magnetic fields, and showed that the surface mag-\nnetic fields may fluctuate and even change polarities at short time\nscales. Urpin, Chanmugam & Sang (1994) derived an analytic so-\nlution to the asymptotic behaviour of the dipolar magnetic fi elds.\nUrpin & Muslimov (1992) studied the ohmic decay of the magnetic\nfield confined to the surface layers of the neutron star crust w ith con-\nsideration for the neutron star cooling. They showed that th e initial\ndepth penetrated by the magnetic field is important and has a n otable\neffect on the results. The conductivity of the crust also depe nds largely\non the thermal state of the matter and the cooling process of t he star.\nUrpin & van Riper (1993) compared the magnetic field decay for\ntwo neutron stars consisting of two different core matters an d thus two\ndifferent cooling models, finding that the accelerated cooli ng model\nhas much less field decay than the standard model. Other notab le stud-\nies on neutron star cooling processes include the works of van Riper\n(1991),Yakovlev et al. (2004),Aguilera, Pons & Miralles (2008)\nandPotekhin, Pons & Page (2015).\nBhattacharya & Datta (1996) studied the ohmic diffusion\nof the magnetic flux expelled from the core of a neu-\ntron star to its crust, and inferred it to be caused from\nthe spindown-induced field evolution. This study was done\nfor multiple NS equations of state for which they took the\ncrustal density profiles from Datta, Thampan & Battacharya (1995).\nDatta, Thampan & Battacharya (1995) noted that there are many pa-\npers published which considered cooling models for a specifi c equa-\ntion of state model and then used a crustal model that corresp ond\nto a different equation of state model, which they attribute t o a non-\navailability of detailed numerical crustal density profile s.\nSengupta (1997) similarly considered the ohmic decay of the mag-\nnetic field confined to the crust of the NS and studied for both t he flat\nand curved spacetimes - neglecting the NS cooling. He showed that\nGR has a significant effect on the results, particularly the le ngthening\nof the decay time of the magnetic field. Later Sengupta (1998) in-\ncluded the effects of the neutron star cooling in both flat and c urved\nspacetimes and obtained results comparable to Urpin & Muslimov\n(1992), again showing that the GR effects reduce the decay rate sub-\nstantially. Geppert, Page & Zannias (2000) also performed compar-\nisons of the flat and curved spacetimes and similarly found th at the GR\neffects increased the decay time of the magnetic field. Furthe r studies\non ohmic decay and Hall drift in the NS crust have been done by\nPons & Geppert (2007),Bransgrove, Levin & Beloborodov (2018),\nPons, Miralles & Geppert (2009),Igoshev, Popov & Hollerbach\n(2021) and Pons & Viganò (2019).\nIn order to explain radio emission state changes in pulsars,\nGeppert et al. (2021) proposed the rapid modification of the neu-\ntron star surface magnetic field as the primary contributor t o the\nprocess. According to them, the combined effect of Hall and th ermal\ndrift causes oscillations of the local magnetic field struct ure at the\nsurface of the polar cap of radio pulsars, leading to the rapi d modifi-\ncations of the surface magnetic field which can even be of the o rder\nof a single pulse period.\nBorrowing the idea from Geppert et al. (2021), in this work we\nhave investigated the possibility of generating a variabil ity in the\nsurface magnetic field that can come from the radial fluctuati on ofthe outer layer of the NS crust, if we consider the NS magnetic field\nis confined to this outer later. The incompressibility of nuc lear mat-\nter is extremely high, and hence in any oscillatory mode of a N S,\nthe fluctuations at the outer crust will be more pronounced th an the\nnuclear matter core. This fluctuation in the outer crust will lead to\na density fluctuation and subsequently a fluctuation in the co nduc-\ntivity of matter, thus affecting the resultant surface magne tic field.\nObservationally, pulsars exhibit dynamic pulse patterns c ontaining\ncomplex microstructures, such as sub-pulses, which have du rations\nof a few microseconds to nanoseconds within one single pulse pe-\nriod. This observed behaviour should then be directly relat ed to the\nbehaviour of the magnetic field at the NS surface.\nIn the literature, radial oscillations in neutron stars (in -\ncluding strange stars) for zero temperature EOS have been\nstudied by Chanmugam (1977),Väth & Chanmugam (1992),\nGlass & Lindbolm (1983), etc, and for finite temperature\nEOS by Gondek et al. (1997),Gondek & Zdunik (1999),\nBenvenuto & Horvath (1991), etc. A comprehensive study of\nradial oscillation using multiple NS EOSs has been done by\nKokkotas & Ruoff (2001), where they found that the funda-\nmental mode of such oscillations are of the order of KHz.\nMcDermott, van Horn & Hansen (1988) studied various oscillation\nmodes of a NS and have also shown that the crustal fluctuation d uring\nsuch oscillations are much more than the fluctuations of the c ore.\nIn this paper, in section 2we calculate the magnetic field evolution\nthrough the outer layer of the NS - using the induction equati on, and\nin a general relativistic background. We also show how the el ectrical\nconductivity of matter behaves in the outer layer of the crus t. A short\ndiscussion on the physical nature of the outer crust is made i n section\n3. In section 4we show our numerical findings, where we estimate\nthe surface magnetic fields for two values of surface fluctuat ions∼\n0.2% and 0.4% of the crust width. We notice a sizeable variati on\nof the surface field observed within a short time scale. We fina lly\nconclude in section 5with some plausible inferences.\n2 BASIC FORMALISM\nIn our study we have used the induction equation in a general r ela-\ntivistic background for the evolution of the magnetic field Bthrough\nthe outer layer of the NS crust. We have also shown how the elec trical\nconductivity will behave in the rapidly varying density reg ime of the\nouter crust.\n2.1 Evolution of the magnetic field in a general relativistic\nbackground\nThe induction equation in a flat spacetime is given by:\n/u1D715B\n/u1D715/u1D461=−∇×E (1)\nwhere\nE=/u1D4502\n4/u1D70B/u1D70E∇×B.\nAssuming the magnetic field to be a pure dipole field, the induc tion\nequation in a curved spacetime is derived using the covarian t form\nof Maxwell’s equations,\n1√−/u1D454/u1D715\n/u1D715/u1D465/u1D708/parenleftbig√−/u1D454/u1D439/u1D707/u1D708/parenrightbig=−4/u1D70B\n/u1D450/u1D43D/u1D707(2)\nMNRAS 000, 000–000 (2024)Radio pulsar surface magnetic field 3\nand\n/u1D715/u1D439/u1D707/u1D708\n/u1D715/u1D465/u1D706+/u1D715/u1D439/u1D708/u1D706\n/u1D715/u1D465/u1D707+/u1D715/u1D439/u1D706/u1D707\n/u1D715/u1D465/u1D708=0, (3)\nwith the generalised Ohm’s law\n/u1D43D/u1D707=/u1D70E/u1D454/u1D707/u1D708/u1D439/u1D708/u1D706/u1D462/u1D706(4)\nwhere/u1D439/u1D707/u1D708are the components of the electromagnetic field tensor,\n/u1D43D/u1D707are the components of the four-current density, /u1D462/u1D707are the com-\nponents of the four velocity of the fluid, /u1D454/u1D707/u1D708are the components\nof the spacetime metric describing the background geometry , and\n/u1D454=/u1D451/u1D452/u1D461|/u1D454/u1D707/u1D708|. In our notation Latin indices represent the 3-dim spa-\ntial coordinates and Greek indices represent the 4-dim spac e-time\ncoordinates.\nIf we take into account a stationary gravitational field, the n using\nequations ( 2), (3), and ( 4) and taking /u1D462/u1D456=/u1D451/u1D465/u1D456//u1D451/u1D460=0 we can derive\nthe corresponding induction equation in a curved spacetime as:\n/u1D715/u1D439/u1D458/u1D457\n/u1D450/u1D715/u1D461=/u1D450\n4/u1D70B/braceleftbigg/u1D715\n/u1D715/u1D465/u1D458/bracketleftbigg1√−/u1D4541\n/u1D70E/u1D4620/u1D454/u1D456/u1D457/u1D715\n/u1D715/u1D465/u1D459/parenleftBig√−/u1D454/u1D439/u1D456/u1D459/parenrightBig/bracketrightbigg\n−/u1D715\n/u1D715/u1D465/u1D457/bracketleftbigg1√−/u1D4541\n/u1D70E/u1D4620/u1D454/u1D456/u1D458/u1D715\n/u1D715/u1D465/u1D459/parenleftBig√−/u1D454/u1D439/u1D456/u1D459/parenrightBig/bracketrightbigg/bracerightbigg\n. (5)\nIn the outer crust of a NS, the magnetic field strength will be r el-\natively weaker compared to the gravitational field strength present\nthere, so that the spacetime curvature is not affected by the m agnetic\nfield. Hence it is reasonable to assume the magnetic field confi gu-\nrations of the NS crust in a background Schwarzschild geomet ry -\ndescribed by the metric:\n/u1D451/u1D4602=/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg\n/u1D4502/u1D451/u1D4612−/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg−1\n/u1D451/u1D45F2−/u1D45F2/parenleftBig\n/u1D451/u1D7032+sin2/u1D703/u1D451/u1D7192/parenrightBig\n(6)\nwhere/u1D45A=/u1D440/u1D43A//u1D4502with/u1D440being the total gravitational mass of the\ncore. The crust consists of less than a few percent ( <3%) of the total\ngravitational mass and hence /u1D440can be approximated to the total\nmass of the star. Hence the self-gravitation of the crust is n egligible\nwhen compared to the gravitational field due to the core.\nThe non-zero components of the orthonormal tetrad /u1D452/u1D6FE\n(/u1D6FC)of the\nlocal Lorentz frame for the Schwarzschild geometry is:\n/u1D452/u1D461\n(/u1D461)=/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg−1\n2\n;/u1D452/u1D45F\n(/u1D45F)=/parenleftBig\n1−2/u1D45A\n/u1D45F/parenrightBig1\n2;/u1D452/u1D703\n(/u1D703)=1\n/u1D45F;\n/u1D452/u1D719\n(/u1D719)=1\n/u1D45Fsin/u1D703.\nThe components of the electromagnetic field tensors for the l ocal\nLorentz frame ( /u1D439(/u1D6FC/u1D6FD)) and that of the curved spacetimes ( /u1D439/u1D6FE/u1D6FF) are\nthen connected to each other with the above tetrads as:\n/u1D439(/u1D6FC/u1D6FD)=/u1D452/u1D6FE\n(/u1D6FC)/u1D452/u1D6FF\n(/u1D6FD)/u1D439/u1D6FE/u1D6FF. (7)\nMaking use of the metric in equation ( 6), the induction equation\nin curved space (equation ( 5)) reduces to:\n/u1D715/u1D439/u1D458/u1D457\n/u1D450/u1D715/u1D461=/u1D450\n4/u1D70B/braceleftbigg/u1D715\n/u1D715/u1D465/u1D458/bracketleftbigg1\n/u1D45F2sin/u1D7031\n/u1D70E/u1D4620/u1D454/u1D456/u1D457/u1D715\n/u1D715/u1D465/u1D459/parenleftBig\n/u1D45F2sin/u1D703/u1D439/u1D456/u1D459/parenrightBig/bracketrightbigg\n−/u1D715\n/u1D715/u1D465/u1D457/bracketleftbigg1\n/u1D45F2sin/u1D7031\n/u1D70E/u1D4620/u1D454/u1D456/u1D458/u1D715\n/u1D715/u1D465/u1D459/parenleftBig\n/u1D45F2sin/u1D703/u1D439/u1D456/u1D459/parenrightBig/bracketrightbigg/bracerightbigg\n. (8)Since we are considering the decay of a dipolar magnetic field with\naxial symmetry, the vector potential Acan be written, in spherical\npolar coordinates, as (0,0, /u1D434/u1D719)where/u1D434/u1D719=/u1D434(/u1D45F,/u1D703,/u1D461). The metric\nthen gives /u1D4620as:\n/u1D4620=/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg−1\n2\n. (9)\nUsing this in equation ( 8) we obtain:\n/u1D715/u1D439/u1D45F/u1D719\n/u1D715/u1D461=/u1D4502\n4/u1D70B/u1D715\n/u1D715/u1D45F/bracketleftBigg\n1\n/u1D70E/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg1\n2\nsin/u1D703/braceleftbigg/u1D715\n/u1D715/u1D45F/bracketleftbigg/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg/u1D439/u1D45F/u1D719\nsin/u1D703/bracketrightbigg\n+/u1D715\n/u1D715/u1D703/parenleftbigg/u1D439/u1D703/u1D719\n/u1D45F2sin/u1D703/parenrightbigg/bracerightbigg/bracketrightbigg\n(10)\nand\n/u1D715/u1D439/u1D703/u1D719\n/u1D715/u1D461=/u1D4502\n4/u1D70B/u1D715\n/u1D715/u1D703/bracketleftBigg\n1\n/u1D70E/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg1\n2\nsin/u1D703/braceleftbigg/u1D715\n/u1D715/u1D45F/bracketleftbigg/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg/u1D439/u1D45F/u1D719\nsin/u1D703/bracketrightbigg\n+/u1D715\n/u1D715/u1D703/parenleftbigg/u1D439/u1D703/u1D719\n/u1D45F2sin/u1D703/parenrightbigg/bracerightbigg/bracketrightbigg\n.(11)\nWe then use the definition\n/u1D439/u1D6FC/u1D6FD=/u1D434/u1D6FD,/u1D6FC−/u1D434/u1D6FC,/u1D6FD (12)\nto find the induction equation in Schwarzschild geometry in t erms of\nthe vector potential as:\n/u1D715/u1D434/u1D719\n/u1D715/u1D461=/u1D4502\n4/u1D70B/u1D70E/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg1\n2\nsin/u1D703/braceleftbigg/u1D715\n/u1D715/u1D45F/bracketleftbigg/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg1\nsin/u1D703/u1D715/u1D434/u1D719\n/u1D715/u1D45F/bracketrightbigg\n+/u1D715\n/u1D715/u1D703/parenleftbigg1\n/u1D45F2sin/u1D703/u1D715/u1D434/u1D719\n/u1D715/u1D703/parenrightbigg/bracerightbigg\n.(13)\nWe choose the vector potential for a flat spacetime as:\n/u1D434/u1D719=/u1D453(/u1D45F,/u1D461)\n/u1D45Fsin/u1D703 (14)\nand that for a curved spacetime as:\n/u1D434/u1D719=−/u1D454(/u1D45F,/u1D461)sin2/u1D703. (15)\nUsing this with equations ( 1) and ( 13) we obtain equations for the\nflat and curved spacetimes respectively as:\n/u1D4502\n4/u1D70B/u1D4452/u1D70E(/u1D465)/bracketleftbigg/u1D7152/u1D453(/u1D465,/u1D461)\n/u1D715/u1D4652−2\n/u1D4652/u1D453(/u1D465,/u1D461)/bracketrightbigg\n=/u1D715/u1D453(/u1D465,/u1D461)\n/u1D715/u1D461(16)\nand\n/u1D4502\n4/u1D70B/u1D4452/u1D70E(/u1D465)/parenleftBig\n1−/u1D466\n/u1D465/parenrightBig1\n2/bracketleftbigg/parenleftBig\n1−/u1D466\n/u1D465/parenrightBig/u1D7152/u1D454(/u1D465,/u1D461)\n/u1D715/u1D4652+/u1D466\n/u1D4652/u1D715/u1D454(/u1D465,/u1D461)\n/u1D715/u1D465\n−2\n/u1D4652/u1D454(/u1D465,/u1D461)/bracketrightbigg\n=/u1D715/u1D454(/u1D465,/u1D461)\n/u1D715/u1D461(17)\nwhere/u1D445is the radius of the star, /u1D70E(/u1D465)is the variable conductivity of\nthe outer crust, /u1D465=/u1D45F//u1D445and/u1D466=2/u1D45A//u1D445. In order to solve the above\ntwo differential equations, we need to impose initial condit ions and\nboundary conditions. If the initial potential (at /u1D461=0) for a flat\nspacetime is given by:\nMNRAS 000, 000–000 (2024)4Sellick & Ray\n/u1D434/u1D719(/u1D45F,/u1D703,0)=/u1D434/u1D719(/u1D45F,/u1D703)=/u1D453(/u1D45F,0)\n/u1D45Fsin/u1D703=/u1D453(/u1D45F)\n/u1D45Fsin/u1D703 (18)\nthen for a curved spacetime we have:\n/u1D434/u1D719(/u1D45F,/u1D703,0)=/u1D434/u1D719(/u1D45F,/u1D703)=−3/u1D45F/u1D453(/u1D45F)\n8/u1D45A3/u1D460(/u1D45F)sin2/u1D703 (19)\nwhere/u1D460(/u1D45F)is the general relativistic correction factor. Thus, if we\nassume the initial value for a flat spacetime to be /u1D453(/u1D45F,/u1D461)=/u1D453(/u1D45F), then\nfor a curved spacetime ( Wasserman & Shapiro (1983))\n/u1D454(/u1D45F,0)=/u1D454(/u1D45F)=3/u1D45F /u1D453(/u1D45F)\n8/u1D45A3/bracketleftbigg\n/u1D45F2ln/parenleftbigg\n1−2/u1D45A\n/u1D45F/parenrightbigg\n+2/u1D45A/u1D45F+2/u1D45A2/bracketrightbigg\n. (20)\nThe boundary conditions, which are valid for all times /u1D461, for both\nthe flat and curved spacetimes are given by ( Bhattacharya & Datta\n(1996))\n/u1D445/u1D715/u1D454\n/u1D715/u1D45F+/u1D454=0 at/u1D45F=/u1D445 (21a)\n/u1D454=0 at crust bottom . (21b)\n2.2 Electrical conductivity of the outer crust\nBefore we can determine the nature of the electrical conduct iv-\nity, we need to establish the dependence of the density /u1D70Cwith\nthe depth from the surface /u1D451=/u1D445−/u1D45F. In the outer layers of the\ncrust, there is a drastic variation in the density. Thus one n eeds\nto have a consistent model matching the density with the dept h.\nDatta, Thampan & Battacharya (1995) computed this variation for\nvarious equations of state of nuclear matter. A simple but el egant\nmodel was given by Urpin & Yakovlev (1979) which closely resem-\nbles the results of Datta, Thampan & Battacharya (1995). We have\nadopted this model here in order to find our solutions. The ana lytical\nformula describing this model are as follows:\n/u1D712=/radicalbig\n/u1D467(/u1D467+2)=/parenleftbigg/u1D70C\n/u1D707/u1D452106g cm−3/parenrightbigg1\n3\n, (22)\n/u1D467=/u1D451\n/u1D43B/u1D445, (23)\n/u1D43B/u1D445=/u1D45A/u1D452/u1D4452\n/u1D45A/u1D45A/u1D45D/u1D707/u1D452/parenleftbigg\n1−2/u1D45A\n/u1D445/parenrightbigg1/2\n, (24)\nwhere/u1D45A/u1D452and/u1D45A/u1D45Dare the electron and proton masses respectively,\n/u1D707/u1D452=/u1D434//u1D44Dand/u1D434and/u1D44Dare the atomic number and charge number\nrespectively. For simplicity we assume that only one specie s of ions\nis present.\nFrom ( 22) one can obtain the density in terms of the radius as\n/u1D70C(/u1D45F)=[(/u1D445−/u1D45F) (/u1D445−/u1D45F+2/u1D43B/u1D445)]3/21\n/u1D43B3\n/u1D445/u1D707/u1D452106g cm−3. (25)\nThe electrical conductivity has two components. The first is due\nto the phonon scattering or oscillatory scattering of the el ectrons -\nthat happens more in the denser region, where electrons are m ore\nfree. The second is due to the impurity scattering. The condu ctivity\ndue to phonon scattering /u1D70E/u1D45Dℎis given by Yakovlev & Urpin (1980)\nfrom which we can write /u1D70E/u1D45Dℎas/u1D70E/u1D45Dℎ=1.57×1023/u1D7124\n/u1D4476(2+/u1D7122)√\n0.017+/u1D6FF2\n13/u1D6FFs−1. (26)\nHere/u1D4476=/u1D447/106K and/u1D6FF=0.45(/u1D447//u1D447/u1D437)./u1D447/u1D437is the Debye tempera-\nture given by\n/u1D447/u1D437=0.45/planckover2pi1\n/u1D458/u1D435/parenleftbigg4/u1D70B/u1D44D2/u1D4522/u1D45B/u1D456\n/u1D434/u1D45A/u1D45D/parenrightbigg1\n2\n=2.4×106/parenleftbigg2\n/u1D707/u1D452/parenrightbigg1\n2\n/u1D7123\n2K (27)\nwhere/u1D458/u1D435is the Boltzmann constant and /u1D45B/u1D456is the ion number density.\nThe conductivity for impurity scattering /u1D70E/u1D456/u1D45A/u1D45D, which dominates\nat lower temperatures is given by:\n/u1D70E/u1D456/u1D45A/u1D45D=8.53×1021/u1D7123\nΛ/u1D456/u1D45A/u1D45D/parenleftbig1+/u1D7122/parenrightbig/u1D44D\n/u1D444s−1. (28)\nHereΛ/u1D456/u1D45A/u1D45Dis the Coulomb logarithm which is Λ/u1D456/u1D45A/u1D45D/simequal2 for/u1D70C≥\n105g cm−3and/u1D444is the impurity parameter:\n/u1D444=1\n/u1D45B/summationdisplay.1\n/u1D456/u1D45B/u1D456(/u1D44D−/u1D44D/u1D456)2(29)\nwhere/u1D45Band/u1D44Dare the number density and electric charge respec-\ntively of background ions in the crust lattice without impur ity and\n/u1D45B/u1D456and/u1D44D/u1D456are the density and charge of the /u1D456th impurity species. The\nsummation is extended over all species of impurities. This i mpu-\nrity parameter is independent of the depth and time. Due to la ck of\nan exact understanding of the extent of the impurity content in the\ncrust and the types of impurity species present there, an app roximate\nrange of/u1D444varying from 0.1 to 0.001 has been assumed in most of\nthe earlier studies found in literature.\nThe net electrical conductivity within the crust is then giv en by:\n/u1D70E=/parenleftbigg1\n/u1D70E/u1D45Dℎ+1\n/u1D70E/u1D456/u1D45A/u1D45D/parenrightbigg−1\n. (30)\nMaking substitutions for /u1D70E/u1D45Dℎand/u1D70E/u1D456/u1D45A/u1D45D, we can write the electrical\nconductivity in terms of the density /u1D70Cas:\n/u1D70E(/u1D70C)=/bracketleftBigg\n8.28×10−29/u1D447/parenleftbigg/u1D707/u1D452106\n/u1D70C/parenrightbigg4/3/bracketleftBigg\n2+/parenleftbigg/u1D70C\n/u1D707/u1D452106/parenrightbigg2/3/bracketrightBigg\n×/bracketleftbigg\n9.67×10111\n/u1D4472/u1D7072/u1D452/u1D70C\n106+1/bracketrightbigg−1/2\n+1.17×10−22Λ/u1D456/u1D45A/u1D45D/u1D444\n/u1D44D/u1D707/u1D452106\n/u1D70C/bracketleftBigg\n1+/parenleftbigg/u1D70C\n/u1D707/u1D452106/parenrightbigg2/3/bracketrightBigg/bracketrightBigg−1\ns−1.(31)\nIn Fig. 1, we showed the variation of the conductivity of matter\n(/u1D70E) against the matter density ( /u1D70C) in the outer layer of the crust for\nthree values of the impurity parameter /u1D444= 0.1, 0.01 and 0.001. We\nsee that at higher density ( ∼1011g cm−3) the conductivity changes\nby about one order of magnitude for a similar change in the imp urity\nparameter - which gradually converges and unifies from a dens ity of\n107g cm−3and lower. The temperature has been kept constant at\n/u1D447=105.2K.\nMNRAS 000, 000–000 (2024)Radio pulsar surface magnetic field 5\n 20 21 22 23 24 25 26 27 28\n 4 5 6 7 8 9 10 11 12Log10[σ(ρ)]\nLog10[ρ]T=105.2 K\nQ=0.1\nQ=0.01\nQ=0.001\nFigure 1: Variation of the conductivity ( /u1D70E) with the density of matter\n(/u1D70C) near the stellar surface for different values of the /u1D444parameter,\nand for a temperature /u1D447=105.2K.\nFig.2shows the variation of the conductivity ( /u1D70E) against the matter\ndensity (/u1D70C) in the outer layer of the crust for three different values\nof temperature /u1D447=105K, 105.2K & 105.5K, for a fixed value of the\nimpurity parameter /u1D444=0.001. We note that in the lower density\nregime,/u1D70Evaries substantially with slight variation of temperature .\n 20 21 22 23 24 25 26 27 28\n 4 5 6 7 8 9 10 11 12Log10[σ(ρ)]\nLog10[ρ]Q=0.001\nT=105 K\nT=105.2 K\nT=105.5 K\nFigure 2: Variation of the conductivity ( /u1D70E) with the density of matter\n(/u1D70C) near the stellar surface for three different values of the te mpera-\nture, and for /u1D444=0.001.\nWe also note that the electrical conductivity changes by nea rly 7 to\n8 orders of magnitude in the outer layer of the crust that is sp anning\nonly about 500 metres.\n3 THE OUTER LAYER OF THE NS CRUST\nTypically neutron stars have a nuclear matter core in the den sity range\nof/u1D70C∼1014/u1D454/u1D450/u1D45A−3. The crusts are believed to be about the outer 10%\nof the stellar radius, where the density changes from ∼1014/u1D454/u1D450/u1D45A−3to\nzero at the surface. In such a wide density variation the natu re of the\nparticles inside the crust also change drastically ( Baym & Pethick\n(1975)). Near the surface, above /u1D70C∼104/u1D454 /u1D450/u1D45A−3the electron kinetic\nenergy rises with the density. With increase in the matter de nsity,\nthere is an increase in the electrical conductivity of the cr ust till it\nreaches the neutron drip (/u1D70C∼1011/u1D454 /u1D450/u1D45A−3). Up to this point the\nmatter consists of nuclei of the normal metal in a sea of elect rons -providing the charge neutrality. Beyond the neutron drip , the matter\nbecomes energetically favourable for neutron continuum st ates and\nthe nuclei becomes immersed in a mixture of neutron and elect ron\ngas. The bottom end of the neutron star crust is the normal nuc lear\nmatter density, where there are mostly neutrons and a smalle r fraction\nof protons and electrons.\nThe last few hundred metres, which is defined as the outer crus t,\nhas a very rapid variation in the density of the matter. It is t his region\nwhich is important for our present study. The rapidly varyin g elec-\ntrical conductivity in this rapidly varying density region of the outer\ncrust is also affected by the temperature of the matter. A neut ron\nstar during its birth, attains the temperature of the supern ova pro-\ncess, which is of the order of MeV ( ∼109−10K). However this initial\ntemperature quickly cools down by a few orders of magnitude b y the\nneutrino emission in a direct URCA process ( Yakovlev et al. 2001 )\nfrom the nuclear matter core and by the neutrino bremsstrahl ung from\nelectron-nucleus collision in the crust ( Chamel & Haensel 2008 ). In\na non-linear relativistic mean field model of NS EOS with prot ons\n(/u1D45D), neutrons ( /u1D45B) and hyperons ( Ξ,Σ,Λ), with particle interac-\ntion via the exchange of /u1D70E,/u1D70Cand/u1D714mesons ( Glendenning 1989 ),\nit has also been shown that a relatively high proton fraction trig-\ngers a direct URCA process ( Lattimer et al. 1991), thus inducing\nan enhanced cooling. Modified URCA process involving presen ce of\nadditional nucleon in the direct URCA reactions, have also b een mod-\neled ( Yakovlev & Levenfish 1995 ) as an alternative mechanism for\na NS cooling. Negreiros et al. (2011) and Negreiros et al. (2012)\nhave incorporated rotational deformation to estimate the t hermal\nevolution of a NS to show that from about 102years the polar and\nequatorial temperatures fall to about 105.2K after which the temper-\nature remains relatively constant throughout the lifetime of the NS\n(∼107−108years). Page et al. (2006) and later Viganó et al. (2013)\nhave also shown that around ∼106years, the surface temperature\nremains around 105K. In binary X-ray pulsars, there are constant\nthermo-nuclear reactions taking place at the surface of the star due\nto in-falling accreted matter on the NS. Isolated radio puls ars do not\nhave such external sources to hugely affect their thermal sta tes. Said\nso, there are proposed models about the in-falling plasma of charged\nparticles (the anti-particles of the ones that take part in f orming the\npulsar emission) on the polar caps and heating up the polar ca ps\nconsiderably ( Tsygan (2017);Sznajder & Geppert (2020)). Even if\nwe take into account these effects, since the process is a cont inuous\none, the entire system sets into a thermal equilibrium.\nHence for a short time evolution and variability study of the sur-\nface magnetic field, isothermal state of the NS crust is a reas onable\napproximation. For our present analysis, we have taken a con stant\ntemperature of 105.2K.\n4 RESULTS AND DISCUSSION\nIn our calculations, we initially considered two equation o f states of\nmatter - the Walecka model (tagged as ‘WAL’ in this paper), wh ich\nis a softer equation of state and the Bethe-Johnson model V (t agged\nas ‘BJV’), which is a relatively stiffer equation of state.1The outer\nlayer of the NS crust starts from the nuclear drip region - whi ch is\neffectively from a depth of only a few hundred metres - corresp ond-\ning to only 4% of the stellar radius. Datta, Thampan & Battacharya\n1Walecka (1974) formulated the equation of state for pure nuclear matter us -\ning scalar and vector meson interactions. On the other hand Bethe & Johnson\n(1974) devised phenomenological potentials for neucleon-nucle on interac-\ntions to devise their equation of state.\nMNRAS 000, 000–000 (2024)6Sellick & Ray\n 0 2 4 6 8 10 12 14\n 0.965  0.97 0.975  0.98 0.985  0.99 0.995  1Log10(ρ)\nx(=r/R)WAL\nBJV\nFigure 3. Density profile of the outer crust for Walecka model\n(WAL) and Bethe-Johnson model V (BJV) (data taken from\nDatta, Thampan & Battacharya (1995)).\n 4 5 6 7 8 9 10 11 12\n 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995  1Log10[ρ(x)]\nx (=r/R0)\nFigure 4. Crustal density profile given by equation ( 25).\n(1995) carried out a numerical survey of the crustal density profil es\nfor various EOSs. We used their tabulated data of the crustal density\nprofile for the 1.4 /u1D440⊙stars, and plotted them in Fig. 3.\nThe model of the density profile given in equation ( 25) derived\nfrom Urpin & Yakovlev (1979) has been plotted in Fig. 4. The mass\nand radius parameters chosen in equation ( 25) are 1.4 /u1D440⊙and\n12.28 km respectively, representing values of the Walecka m odel\n(WAL) as tabulated in Datta, Thampan & Battacharya (1995). The\ntwo figures show a close resemblance - this justifies the choic e of the\ndensity profile obtained in equation ( 25). In the subsequent results\nand plots, we will primarily use the NS parameters following WAL.\nWe solved the differential equations ( 16) and ( 17) for the flat and\ncurved spacetime background, in MatLab using the Method of Lines\nbased on finite differencing approximations of spacial deriv atives.\nThe resulting solutions for /u1D453(/u1D465,/u1D461)and/u1D454(/u1D465,/u1D461)gave us the correspond-\ning vector potentials from which we calculated the correspo nding\nsurface magnetic fields. The surface magnetic field decay for both\nthe flat and the curved spacetimes are found to be spanning ove r\na period 108to 109years. In order to verify the consistency of our\nnumerical results, we have compared them with the results of the sur-\nface magnetic field decay study as obtained by Bhattacharya & Datta\n(1996) (for non-GR cases) and Sengupta (1997) (including the GR\neffects). Here we also notice that the general relativistic e ffects de-\nlay the magnetic field decay time. Essentially, the addition al effectsfrom general relativistic terms come mostly from the gravit ational\npotential (1−2/u1D45A//u1D45F)that add up to the stellar compactness. When\ncompared to the flat spacetime (Eq ( 16)), the diffusion equation for\nthe curved spacetime (Eq ( 17)) is affected by the softening of the l.h.s\nof the equation - thus allowing a larger value of time in the so lution\nmatrix. Hence for a given period of time, the surface magneti c field\ndecay calculated in a GR background is slower, thus extendin g the\ndecay time.\nWe next consider the effects radial oscillations may have on t he\nsurface magnetic field. A simple form for the radial oscillat ory motion\ncan be described by\n/u1D6FF/u1D465(/u1D45F,/u1D461)=/u1D44B(/u1D45F)exp(/u1D456/u1D714/u1D461).\nThe frequency ( /u1D714) of radial oscillations of a NS is primarily driven\nby the collective excitation of nucleons leading to the gian t iso-\nscalar monopole oscillations, better known as the breathin g mode\nof oscillations, in the inner core of the star. Hence the freq uency\nof the radial oscillation is dependent on the EoS of matter in the\ncore. This frequency is also affected by a small contribution from\nthe gravitational forces. The amplitude of this oscillatio n/u1D44B(/u1D45F)is\nhowever not so straight forward across the entire radius of t he NS\nas it is expected to have a variable amplitude that is more pro minent\nacross the low density regime below the nuclear drip region o f the\nouter crust. Glass & Lindbolm (1983);Väth & Chanmugam (1992)\n&Kokkotas & Ruoff (2001) calculated the frequencies of radial os-\ncillation for different known nuclear matter EoSs where it ha s been\nshown that these frequencies vary larger than a kilohertz an d more\n(periods of microsecond to nanosecond). Absence of any prev ious es-\ntimation of the amplitude variation across the entire radiu s of the NS\n(from literature) prompted us to consider a two fluid setup wh ere the\ninner fluid is of the dense nuclear matter with high incompres sibility,\nand the outer fluid is of the crustal matter with lower incompr essibil-\nity. Then during radial oscillation of the NS, the outer crus t should\noscillate with a larger amplitude than the rigid inner core. A full\nconsideration of a two fluid model would be of interest in orde r to\nestimate the amplitude of oscillations where the outer crus t oscillates\nwith a larger amplitude than the rigid inner core. However, f or this\nwork we consider only fluctuations in the outer crust (as this is where\nthey will be most prominent) as a first step to see if this consi deration\nis viable.\nIn the next step we study the variation of the surface magneti c\nfield due to variations in the outer crustal radius of 0 .2% and 0 .4%\n(which translates to radial variations of only 1 metre and 2 m etre\nrespectively). In order to fluctuate only the width of the cru st, the\ninitial depth, /u1D4650, is taken to be the same physical distance from the\ncentre of the star for the perturbed fields /u1D435+and/u1D435−as it is for the\nunperturbed field /u1D4350. So for the unperturbed field, the initial depth\ncorresponds to a radius of 0 .96×/u1D445. The initial depth values for the\nperturbed fields /u1D435+and/u1D435−with total radii of /u1D445+and/u1D445−respectively\nare then given by /u1D4650=0.96× (/u1D445//u1D445+)and/u1D4650=0.96× (/u1D445//u1D445−)\nrespectively. The total radii are given by the addition (or s ubtraction)\nof 0.2% and 0 .4% of the crust width such that for a 0 .2% variation\nin crust width the radii are given by /u1D445+=/u1D445+/u1D445×0.04×0.002\nand/u1D445−=/u1D445−/u1D445×0.04×0.002, and for a 0 .4% variation in crust\nwidth the radii are given by /u1D445+=/u1D445+/u1D445×0.04×0.004 and/u1D445−=\n/u1D445−/u1D445×0.04×0.004.\nAs the star oscillates radially the magnetic field strength w ill os-\ncillate between the minimum and maximum values with a freque ncy\nequivalent to the radial oscillation frequency. In Figure 5we have\nshown this resultant variation on the surface magnetic field for a 0.2%\nfluctuation of the crustal width, in a time span of ∼70 years for a NS\naged at 105years in a curved spacetime background. The lines cor-\nMNRAS 000, 000–000 (2024)Radio pulsar surface magnetic field 7\nTable 1. Values of Δ/u1D435in Gauss for a 0 .2% fluctuation in the crust width for WAL given for different po ints in time with two values each of /u1D444and/u1D447for the flat\nand curved spacetimes.\nTime [years] 103105107\nPPPPPPQT [K]105.2105.5105.2105.5105.2105.5\nFlat0.001 1.276×1081.685×1083.178×1083.981×1085.431×1085.094×108\n0.1 1.511×1081.831×1084.652×1084.863×1084.6225×1084.4595×108\nCurved0.001 1.14×1083.27×1083.46×1089.02×1087.009×1081.413×109\n0.1 2.73×1083.55×1081.05×1091.127×1091.1616×1091.1546×109\n-0.0688-0.0686-0.0684-0.0682-0.068-0.0678\n 5 5.00005  5.0001  5.00015  5.0002  5.00025  5.0003B+\nB0\nB-Log10(B/Binitial)\nTime (Log10(t))\nFigure 5. Variation in the surface magnetic field due to the radial fluct uation\nof 0.2% and 0.4% of the outer crust. /u1D4350is the non fluctuating field and /u1D435+and\n/u1D435−are the values for maximum and minimum radial variations res pectively.\nresponding to /u1D4350are the surface magnetic field for the unperturbed\nradius,/u1D435+indicates the surface field for an expanded radius, and /u1D435−\nfor a contracted one. All magnetic field values have been norm alised\nby the initial value of the magnetic field /u1D435/u1D456.\nAs expected, the expanded radius (and hence lowering of the d en-\nsity) reduces the conductivity, resulting in an increased s urface mag-\nnetic field ( /u1D435+), and a contracted radius results in a decreased growth\nof the surface magnetic field ( /u1D435−).\nWe have then studied how the change in the magnetic field ( Δ/u1D435)\nevolves with time over the lifetime of a NS and how this may be\ninterpreted observationally. In Figure 6we have plotted the variation\nof the surface magnetic field Δ/u1D435relative to the unperturbed value of\nthe field/u1D4350due to 0.2% and 0.4% fluctuation of the crustal width, for\nthe entire time period of 102to 108years of a thermally equilibrated\nNS’s life.\nTable 1shows the values of Δ/u1D435in Gauss for a 0 .2% fluctuation\nin the crust width for WAL. The values are shown for three diffe rent\nages in the NS’s lifetime ( /u1D461=103years,/u1D461=105years and /u1D461=107\nyears) to demonstrate the magnitude of the change in the magn etic\nfield at different stages in the evolution. This is done for the flat and\ncurved spacetimes with two values each of /u1D444and/u1D447. It can be seen\nthat this magnitude of Δ/u1D435is of the of the order of ∼108G−109G\nwith values varying by an order of magnitude over the NS lifet ime.\nThe consequences of this change in Δ/u1D435will be further explored and\nmade clearer in the following discussions.\nStudies were done for both the flat and curved spacetime cases\ncomparing the EOSs WAL and BJV and for 0 .2% and 0.4% variation\nin the crust width. The numerous plots of which will be neglec ted herfor the sake of brevity, however, the general findings can be n oted.\nThe fluctuations in the magnetic field are naturally related t o the size\nof the radial fluctuations, but the compactness of the star al so has\nan effect. The softer EoS of WAL has a sightly larger variation in\nmagnitude of the magnetic field than BJV. For the flat spacetim e\nthis can simply be attributed to the fact that for a given mass , WAL\nhas a larger radius than BJV and hence the fluctuations in the c rust\nwidth are also large. However, the effect of curvature enhanc es the\ndifferences in the EoSs as the gravitational potential adds t o the stellar\ncompactness. The more compact the star, the smaller the fluct uations\nin the magnetic field. This is consistent with our assumption that the\ncore of the star will fluctuate less than the crust due to its hi gh density\nand compactness increasing the stiffness of the matter, and h ence the\nmajority of the fluctuations in the radius will occur in the ou ter crust.\nAs well as the compactness of the star having an effect on the ma g-\nnitude of the oscillations, the magnitude of the unperturbe d magnetic\nfield also plays a significant role. The magnetic field of a NS de cays\nas the star ages. As the magnetic field decreases, we find (as se en in\nFigure 6) that the change in the surface magnetic field Δ/u1D435increases.\nThus any microstructure caused by the radial oscillations w ill be\nmore noticeable in the later stages of the star’s evolution. This is con-\nsistent with observational data which shows that drifting s ub-pulses\nare more prominent closer to the deathline ( Song et al. 2023).\nSong et al. (2023) conducted a study on 1198 pulsars using the\nThousand-Pulsar-Array programme on MeerKAT. They found th at\n418 pulsars ( ∼35%) of these pulsars exhibit drifting sub-pulses\nand that these sub-pulses are more pronounced towards the de ath-\nline, as is consistent with other previous studies ( Rankin 1986 ;\nWeltevrede, Edwards & Stappers 2006 ). The detectability of drift-\ning sub-pulses is dependant on the signal to noise ratio (S/N ) which\nis higher for long period pulsars associated with larger cha racteristic\nage/u1D70F/u1D450. It is found that the younger pulsars produce more erratic or\ndiffuse behaviour, whereas the older pulsars exhibit more co herent\nsub-pulse modulations. They also note that drifting sub-pu lses are a\nvery common occurrence and if high quality data were availab le for\nall pulsars then this phenomena should be present in ∼60% them.\nThis suggests that the phenomena should be related to some co mmon\nfeature in NSs.\nThe preference for the detection of drifting sub-pulses in o lder\nand less energetic pulsars suggests that drifting sub-puls es become\nstronger as the pulsars age. Since these drifting sub-pulse s are so\ncommon it is interpreted that the physical processes behind the pro-\nduction of said sub-pulses should be related in some way to th e\nradio emission mechanism and may well be a fundamental prope rty\nof the emission mechanism. In many works, the drifting sub-p ulses\nare explained by the geometric interpretations of the carou sel model,\nhowever, they could be explained by any model that is able to p roduce\nsuch periodic sub-pulse modulations, such as the radial osc illations\nMNRAS 000, 000–000 (2024)8Sellick & Ray\n-6-5-4-3-2-1\n 2  3  4  5  6  7  8Log10(∆ B/B0)\nLog10(t)∆rcr = 0.2%\n∆rcr = 0.4%\nFigure 6. Variation in the surface magnetic field Δ/u1D435due to the radial fluctu-\nation of 0.2% and 0.4% of the outer crust, including the gener al relativistic\neffects. Notable is the increase in the magnetic field fluctuat ion in the later\nstage of the NS’s lifetime\nof the star. We suggest that the physical process behind thes e sub-\npulses could be explained by the radial oscillations and the fact that\nthe sub-pulses are pronounced toward the deathline can be ex plained\nby the fact that, from our study, it is seen that the older puls ars have a\nmore pronounced change in the magnetic field strength due to t he ra-\ndial oscillations, thus any sub-pulses produced from this p henomena\nwould be more pronounced as the pulsar ages.\nIn our analysis, we have considered that the magnetic field of\nthe NS is located in the outer crust. Without imposing such st iff\nconstraint, even if we consider the magnetic field is located in the\ndeeper layers, due to the high electrical conductivity, the field strength\nthere will be relatively weaker than that in the low density r egion of\nthe outer crust. Hence, majority of the contribution to the s urface\nmagnetic field will come from the outer crust, and other than fi ne\ntuning the values for fields present in the deeper layers, qua litatively\nour results will remain the same.\n5 CONCLUSION\nWe considered the effect of radial oscillations on the surfac e magnetic\nfield of a NS and how this may be linked to observational phenom ena.\nWe found the fluctuations of the magnetic field strength due to these\noscillations on a short timescale to be of a significant order such\nthat we may be able to observe these fluctuations in the form of\nmicrostructure and sub-pulses in a pulsars observed emissi on. In the\nlong term, we found the change in the magnitude of these fluctu ations\nin the magnetic field to increase with time and thus any sub-pu lses\ncaused by these fluctuations should be more noticeable as a st ar\nages. In other words older stars should be more effected by the radial\noscillations and would exhibit more pronounced microstruc ture and\nsub-pulse modulations than younger stars. It is exciting to find that\nour results agrees with observational data as is shown in the very\nrecent work of Song et al. (2023) who found that drifting sub-pulses\nare more pronounced in older pulsars towards the deathline. We may\nalso note that the intensity of the varying surface field is di rectly\nrelated to size of the fluctuation of the radial length of the c rust\nas can be seen from the 0 .2% and 0 .4% variations in crust width.\nAdditionally to this, the GR effects add to the compactness of the star\nwhich leads to smaller fluctuations in the magnetic field stre ngth.ACKNOWLEDGEMENTS\nKS acknowledges S Shindin of UKZN for helping with the compu-\ntational codes. KS also acknowledges the NRF for partial res earch\nsupport. SR thanks D Mitra of NCRA, Pune, for many insightful\ndiscussions.\nDATA AVAILABILITY\nNo new data were generated or analysed in support of this rese arch.\nREFERENCES\nAguilera, D. N., Pons, J. A. and Miralles, J. A., 2008, A& A., 486, 255.\nBaym, G and Pethick, C., 1975, Annu. Rev. Nucl. Sci, 25, 27.\nBenvenuto, O. G., and Horvath, J. E., 1991, MNRAS, 250, 679.\nBethe, H. A. and Johnson, M. B., 1974, Nucl. Phys., A230 , 1.\nBattacharya, D. and Datta, B., 1996, MNRAS, 282, 1059.\nBlandford, R., Applegate, J. and Hernquist, L., 1983, MNRAS ,204, 1025.\nBransgrove, A., Levin, Y. and Beloborodov, A., 2018, MNRAS, 473, 2771.\nChamel, N and Haensel, P., 2008, Living Rev. Relativity, 11, 10.\nChanmugam, G. and Sang, Y., 1989, MNRAS, 241, 295.\nChanmugam, G., 1977, ApJ, 217, 799.\nDatta, B., Thampan, A. 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A., 1980, Soviet Astr., 24, 126.\nYakovlev, D. G. and Urpin, V. A., 1980, Soviet Astr., 24, 303.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 000–000 (2024)"    },    {        "title": "2401.12753v1.Optimal_Confidence_Bands_for_Shape_restricted_Regression_in_Multidimensions.pdf",        "content": "Optimal Confidence Bands for Shape-restricted\nRegression in Multidimensions\nAshley (Pratyay) Datta1, Somabha Mukherjee2and Bodhisattva Sen3∗\n1Department of Statistics, Columbia University, e-mail: pd2511@columbia.edu\n2Department of Statistics and Data Science, National University of Singapore, e-mail: somabha@nus.edu.sg\n3Department of Statistics, Columbia University, e-mail: bodhi@stat.columbia.edu\nAbstract: In this paper, we propose and study construction of confidence bands for shape-\nconstrained regression functions when the predictor is multivariate. In particular, we consider\nthe continuous multidimensional white noise model given by dY(t) =n1/2f(t)dt+dW(t),\nwhere Yis the observed stochastic process on [0 ,1]d(d≥1),Wis the standard Brownian sheet\non [0,1]d, and fis the unknown function of interest assumed to belong to a (shape-constrained)\nfunction class, e.g., coordinate-wise monotone functions or convex functions. The constructed\nconfidence bands are based on local kernel averaging with bandwidth chosen automatically\nvia a multivariate multiscale statistic. The confidence bands have guaranteed coverage for\nevery nand for every member of the underlying function class. Under monotonicity/convexity\nconstraints on f, the proposed confidence bands automatically adapt (in terms of width) to the\nglobal and local (H¨ older) smoothness and intrinsic dimensionality of the unknown f; the bands\nare also shown to be optimal in a certain sense. These bands have (almost) parametric ( n−1/2)\nwidths when the underlying function has “low-complexity” (e.g., piecewise constant/affine).\nMSC2020 subject classifications: Primary 62G15, 62G05; secondary 62G20.\nKeywords and phrases: Automatic adaptation, continuous multidimensional white-noise\nmodel, coordinate-wise nondecreasing function, H¨ older smoothness, minimax optimal, multi-\nvariate convex function.\n1. Introduction\nThe area of shape-restricted regression is concerned with nonparametric estimation of a regression\nfunction under shape constraints such as monotonicity, convexity, unimodality/quasiconvexity, etc.\nThis field of statistics has a long history dating back to influential papers such as Hildreth (1954),\nBrunk (1955), Prakasa Rao (1969), Brunk (1970), Groeneboom et al. (2001); also see Barlow et al.\n(1972), Robertson et al. (1988), Groeneboom and Wellner (1992), Groeneboom and Jongbloed\n(2014) for book length treatments on this topic. Indeed, such shape constraints arise naturally in\nvarious contexts: isotonic regression methods are widely employed in many real-life applications\nranging from predicting ad click–through rates (McMahan et al., 2013) to gene–gene interaction\nsearch (Luss et al., 2012); convex regression arises in productivity analysis (Allon et al., 2007),\nefficient frontier methods (Kuosmanen and Johnson, 2010), in stochastic control (Keshavarz et al.,\n2011), etc. Further, in many applications (such as estimation of production and utility functions\nin economics), justifying smoothness assumptions on the regression function is often impractical,\nwhereas qualitative assumptions like monotonicity and/or concavity are available; see e.g., Cham-\nbers (1982), Varian (1992), Allon et al. (2007), Varian (2010), Chen et al. (2018).\n∗Supported by NSF DMS-2311062.\n1arXiv:2401.12753v1  [math.ST]  23 Jan 2024/Confidence Bands for Shape-restricted Regression 2\nIn recent years there has been much activity on estimation of such shape-constrained regres-\nsion functions with multivariate predictors; see e.g., Seijo and Sen (2011), Han et al. (2019), Chen\net al. (2020), Deng and Zhang (2020), Mukherjee et al. (2023). Further, in many recent papers\nthe accuracy of the least squares estimator in these problems has been studied via finite sample\n(adaptive) risk bounds; see e.g., Meyer and Woodroofe (2000), Zhang (2002), Guntuboyina and Sen\n(2015), Chatterjee et al. (2015), Chatterjee et al. (2018), Guntuboyina and Sen (2018), Chatterjee\nand Lafferty (2019), Han et al. (2019), Kur et al. (2020). However, the problem of inference with\nthese multivariate shape-constrained functions, e.g., construction of confidence sets, is largely un-\nexplored. Note that, in contrast, inference for univariate shape-constrained problems is well-studied\nwith a large body of work in the last two decades, see e.g., Banerjee and Wellner (2001), Sen et al.\n(2010), Durot et al. (2012), Banerjee et al. (2019), Deng et al. (2022) and the references therein.\nIn this paper we consider construction of honest confidence bands for shape-constrained regres-\nsion functions with multiple covariates with special emphasis to: (i) coordinate-wise nondecreasing\n(isotonic) functions and (ii) convex functions. Our proposed methodology, which extends the ideas\nof D¨ umbgen (2003) to multiple dimensions, yields asymptotically optimal confidence bands that\npossess various adaptivity properties. In particular, for scenarios (i) and (ii) above, we prove spa-\ntial and local adaptivity of our confidence bands with respect to the (H¨ older) smoothness of the\nunderlying function and its intrinsic dimensionality. Our confidence bands are constructed using\na multidimensional multiscale statistic. Note that in the recent literature many multidimensional\nmultiscale statistics have been developed and studied (see e.g., Chan and Walther (2013), K¨ onig\net al. (2020), Arias-Castro et al. (2005), Walther and Perry (2022), Walther (2010)); however in this\npaper we consider the proposal of Datta and Sen (2021)—which is inspired by the one-dimensional\nmultiscale statistic proposed and studied in D¨ umbgen and Spokoiny (2001)—as it is most convenient\nfor our setup.\nWe consider the following continuous multidimensional white noise (regression) model:\nY(t) =√nˆt1\n0. . .ˆtd\n0f(s1, . . . , s d)dsd. . . ds 1+W(t), (1.1)\nwhere t:= (t1, . . . , t d)∈[0,1]d,d≥1,{Y(t1, . . . , t d) : (t1, . . . , t d)∈[0,1]d}is the observed\ndata, f∈L1([0,1]d) is the unknown (regression) function of interest, W(·) is the unobserved\nd-dimensional Brownian sheet (the multivariate generalization of the usual Brownian motion; see\nDefinition 1 below), and nis a known scale parameter. Estimation and inference in this model is\nclosely related to that of (multivariate) nonparametric regression based on sample size n; see e.g.,\nBrown and Low (1996), Reiß (2008). We work with this white noise model as this formulation is\nmore amiable to rescaling arguments; see e.g., Donoho and Low (1992), D¨ umbgen and Spokoiny\n(2001), Carter (2006).\nGiven a function class F ⊂L1([0,1]d) (e.g., Fcan be the class of coordinate-wise nondecreasing\nfunctions or the class of convex functions, defined on [0 ,1]d), our goal is to construct an honest\nconfidence band [ ˆℓ,ˆu] for the true function f∈ F, i.e., find functions ˆℓand ˆudepending on the\nobserved data Y(·) (see (1.1)) that satisfy:\nPf\u0010\nˆℓ≤f≤ˆu\u0011\n≥1−α, for all f∈ F,for all n≥1, (1.2)\nfor any given confidence level α∈(0,1). Here, by Pf(·) we mean probability computed when the\ntrue function is fin (1.1). Although nonparametric estimation of an unknown regression/density/Confidence Bands for Shape-restricted Regression 3\nfunction based on smoothness assumptions using techniques such as kernels, splines and wavelets\nare abundant in the literature (see e.g., Wahba (1990), Donoho and Johnstone (1994), Wand and\nJones (1995), Donoho and Johnstone (1995), Fan (1996), Hart (1997), Johnstone (1999), Gu (2002),\nBrown et al. (2008), Lian et al. (2019)), it is known that fully adaptive inference for certain smooth-\nness function classes is not possible without making qualitative assumptions of some kind on the\nparameter space; see e.g., Gin´ e and Nickl (2016, Theorem 8.3.11) (also see Donoho (1988), Cai and\nLow (2004)).\nIndeed, shape-constrained functions satisfy a two-sided bias inequality (see (2.5) below)—a crucial\nassumption made in this paper for our proposed method—which enables the construction of uniform\nconfidence bands with inferential guarantees like (1.2). Non-asymptotic confidence bands under such\nshape constraints on the true function are available in the literature but only for one-dimensional\nfunction estimation problems (see e.g., Davies (1995), Hengartner and Stark (1995), D¨ umbgen\n(1998), Freyberger and Reeves (2018)). D¨ umbgen (2003) derived asymptotically optimal confidence\nbands for the true regression function under shape constraints such as monotonicity and convexity\nin the continuous univariate white noise model, based on multiscale tests introduced in D¨ umbgen\nand Spokoiny (2001).\nGeneralizing the approach in D¨ umbgen (2003), we construct a multiscale statistic in the multi-\ndimensional setting (also see Datta and Sen (2021)) which can be written as a supremum of local\nweighted averages of the response Y(·) with weights determined by a kernel function, parametrized\nby a vector of smoothing bandwidths and the centers of the kernel function (see (2.3) below). These\nmultiscale local averages are appropriately penalized to have them in the same footing. This ensures\nthat the random fluctuations of the kernel estimator can be bounded uniformly in the bandwidth\nparameters, i.e., the supremum statistic remains finite almost surely (see Datta and Sen (2021)).\nFurther, working with the supremum avoids the delicate choice of tuning parameters (smoothing\nbandwidths). Using the definition of the supremum statistic T(see (2.3)) and the two-sided bias\ncondition (see (2.4)), one can obtain pointwise lower and upper confidence bounds for the unknown\nfunction fwith guaranteed coverage (i.e., (1.2) holds); see Theorem 1.\nTo intuitively understand the main ideas in our approach let us first look at how kernel averaging\nworks in a neighborhood around a point. If we take a larger neighborhood, the bias of the kernel\nestimator increases whereas the variance of the estimator goes down. The optimal size of the\nneighborhood is the one that balances these two terms. Our method can be interpreted as kernel\naveraging over all possible scales (bandwidths) and then choosing the optimal neighborhood (see\n(2.7) and (2.8)). The multiscale statistic bounds the stochastic fluctuations (i.e., the variance term)\nuniformly over all scales and locations, whereas the bias condition (2.4) uniformly bounds the bias\nterm. This results in confidence bands having guaranteed coverage.\nOur proposed confidence band automatically adapts to the underlying smoothness of the true\nfunction f. To understand why, let us contrast the behavior of our band between the cases where\nthe function is smooth versus when the function is rough. The bias term (obtained from kernel\naveraging) over a neighborhood is much larger when the function is rougher compared to when it is\nsmooth, whereas the variance term does not change (with smoothness). If the function is smooth,\nour multiscale confidence band automatically chooses a much larger neighborhood to average over\n(which balances bias and variance) resulting in a shorter confidence band (with a faster rate of\nconvergence).\nIn particular, for coordinate-wise isotonic and multivariate convex functions, we show that our\nconstructed confidence band is adaptive with respect to the H¨ older smoothness of the underlying/Confidence Bands for Shape-restricted Regression 4\nfunction (see Theorem 4) and the intrinsic dimensionality of the function, i.e., the number of\nvariables/coordinates it truly depends on (see Theorem 6). The confidence band also exhibits local\nadaptivity (as shown in Theorems 2 and 5). To elaborate on the spatial adaptivity property, we\nshow that, as a consequence of Theorem 2, if the true function is monotone and constant (or convex\nand affine) in an open neighborhood of a point then our constructed confidence band achieves the\nparametric ( n−1/2) rate of convergence, uniformly on that neighborhood; see Remark 2.1. This\nin particular, complements the near-parametric risk bounds developed for these “low-complexity”\nfunctions in recent years by various authors; see e.g., Chatterjee et al. (2018), Han and Zhang\n(2020), Kur et al. (2020), Deng et al. (2021). In Theorem 5, we show that the width of our proposed\nconfidence band at a point also adapts to the local H¨ older smoothness of the underlying function.\nMoreover, locally (at t0∈(0,1)d) for a coordinate-wise strictly increasing function, Theorem 7\nshows that the width of our proposed confidence band (constructed using a special kernel function;\nsee Section 2.3 and Theorem 3) attains the minimax rate; moreover, it also essentially attains the\nminimax constant, which is given by the geometric mean of the gradient of the true function f\natt0. Analogously, Theorem 8 shows that we have a similar minimaxity property for multivariate\nconvex functions, modulo slightly different constant factors.\nThe rest of the paper is organized as follows. In Section 2.1, we introduce some notation that\nwill be necessary for stating the main results and analyses presented in this paper. In Section 2.2,\nwe give the construction of our confidence band using our multiscale statistic; in Section 2.3 we\ndiscuss the choice of kernels necessary for this construction, under the natural shape constraints\nof monotonicity and convexity. Section 3 is devoted to proving several adaptivity properties of our\nconstructed confidence band. In Section 4, we show that our proposed confidence band is optimal in\na certain sense. In Section 5 we illustrate our proposed confidence band via simulations for different\nchoices of f; these bands (see e.g., Figures 1-4) illustrate the adaptive properties of our procedure.\nThe proofs of the main results are given in Section 6. Finally, proofs of some technical lemmas are\nprovided in Appendix A.\n2. Confidence bands for multivariate shape-restricted functions\n2.1. Preliminaries\nWe now present some notation that we will use throughout the paper.\nNotation 1 (Function classes F1andF2).We will denote by F1the class of all coordinate-wise\nnondecreasing functions f: [0,1]d→R, i.e., f∈ F 1satisfy:\nf(x1, . . . , x d)⩽f(y1, . . . , y d) if and only if xi⩽yifor all 1 ⩽i⩽d,\nand by F2the class of all convex functions f: [0,1]d→R, i.e., f∈ F 2satisfy:\nf(λx+ (1−λ)y)⩽λf(x) + (1 −λ)f(y) for all x,y∈Rdandλ∈[0,1].\nFor measurable functions g, h:Rd→R, we define:\n⟨g, h⟩|B:=ˆ\nBg(x)h(x)dxand∥g∥|B:=p\n⟨g, g⟩B./Confidence Bands for Shape-restricted Regression 5\nWhen B:=Rd, we just drop the subscripts in the above notation. For vectors a,b∈Rd, we define:\na⋆b:= (a1b1, . . . , a dbd).\nFort= (t1, . . . , t d)∈(0,1)dandh= (h1, . . . , h d)∈(0,1/2)dlet\nB∞(t,h) :={x= (x1, . . . , x d)∈Rd:|ti−xi|⩽hi∀i= 1, . . . , d }.\nNext, we describe the Brownian sheet process (see (1.1)), which will be important in our analysis.\nDefinition 1 (Brownian sheet) .By a d-dimensional Brownian sheet we mean a mean-zero Gaussian\nprocess {W(t) :t∈[0,1]d}with covariance function given by:\nCov(W(t), W(s)) =dY\ni=1min{ti, si}, fors= (s1, . . . , s d),t= (t1, . . . , t d).\nNote that when d= 1,W(·) reduces to the standard Brownian motion on [0,1]. We now mention\nsome useful properties of the Brownian sheet process:\n•Ifg∈L2([0,1]d), then´\ng dW :=´\n[0,1]dg(t)dW(t)∼N(0,∥g∥2).\n•Ifg1, g2∈L2([0,1]d), then Cov\u0000´\ng1dW,´\ng2dW\u0001\n=´\n[0,1]dg1(t)g2(t)dt.\n•Cameron-Martin-Girsanov theorem (Protter (2005)): Let µdenote the measure induced by\nthe Brownian sheet W(·) on the space Ω of all real-valued continuous functions on [0 ,1]d, and\nletνdenote the measure induced by the process Ydefined in (1.1) on Ω. Then νis absolutely\ncontinuous with respect to µand the Radon-Nikodym derivative is given by:\ndν\ndµ= exp\u0012√nˆ\nf dW−n\n2∥f∥2\u0013\n.\nFor a more detailed study of the Brownian sheet see Khoshnevisan (1996).\n2.2. Proposed confidence band\nIn this subsection we construct adaptive and optimal confidence bands ( ˆℓ,ˆu) for f: [0,1]d→R\nwhen fis known to be shape-constrained, e.g., when fis isotonic/convex, and our data is generated\naccording to (1.1). Suppose that ψ:Rd→Ris a measurable function of bounded Hardy-Krause\nvariation (see Datta and Sen (2021, Definition A.1)) such that:\n(i)ψis 0 outside [ −1,1]d;\n(ii)ψ∈L2(Rd), i.e.,´\nRdψ2(x)dx<∞;\n(iii)´\nRdψ(x)dx>0.\nWe call such a function a kernel . For any h:= (h1, . . . , h d)∈I:= (0 ,1/2]dwe define\nAh:=dY\ni=1[hi,1−hi] ={t= (t1, . . . , t d)∈Rd:hi⩽ti⩽1−hifori= 1, . . . , d }.\nFor any t∈Ahwe define the centered (at t) and scaled kernel function ψt,h: [0,1]d→Ras\nψt,h(x) :=ψ\u0012x1−t1\nh1, . . . ,xd−td\nhd\u0013\n, forx= (x1, . . . , x d)∈[0,1]d. (2.1)/Confidence Bands for Shape-restricted Regression 6\nFor a fixed t∈Ahwe construct a kernel estimator ˆfh(t) off(t) as:\nˆfh(t) :=1\n√n(Qd\ni=1hi)⟨1, ψ⟩ˆ\n[0,1]dψt,h(x)dY(x). (2.2)\nElementary calculations show that\nE(ˆfh(t)) =´\n[0,1]dψt,h(x)f(x)dx\n(Qd\ni=1hi)⟨1, ψ⟩=⟨f(t+h⋆·), ψ⟩\n⟨1, ψ⟩\nVar( ˆfh(t)) =∥ψt,h∥2\nn(Qd\ni=1hi)2⟨1, ψ⟩2=∥ψ∥2\nn(Qd\ni=1hi)⟨1, ψ⟩2.\nThe main idea of our approach is to notice that the random fluctuations for these kernel estimators\ncan be bounded uniformly in h. To accomplish this, we look at the following multiscale statistic\n(with kernel ψ):\nT(±ψ) = sup\nh∈Isup\nt∈Ah \n±´\n[0,1]dψt,h(x)dW(x)\n(Qd\ni=1hi)1/2∥ψ∥−Γ(2ddY\ni=1hi)!\n= sup\nh∈Isup\nt∈Ah \n±ˆfh(t)−E(ˆfh(t))\nVar1/2(ˆfh(t))−Γ(2ddY\ni=1hi)!\n(2.3)\nwhere Γ( r) := (2 log( e/r))1/2. Datta and Sen (2021, Theorem 2.1) guarantees that T(±ψ) is finite\nalmost surely.\nWe assume that the unknown fbelongs to the function class F(which could be F1orF2). In\nfact, the results in this section are valid for any function class that satisfies the following two-sided\nbias condition : we assume that we can find kernels ψℓandψusuch that the corresponding kernel\nestimatorsˆfℓ\nhand ˆfu\nh(see (2.2)) satisfy:\nE(ˆfℓ\nh(t))⩽f(t)⩽E(ˆfu\nh(t)) for all h∈I,t∈Ahandf∈ F. (2.4)\nWe will show later that the above condition holds for the function classes F1andF2(see Section 2.3);\nin fact, it holds for most shape-constrained function classes.\nIn view of (2.4) and the definition of T(in (2.3)), we have the following for all h∈Iandt∈Ah:\nf(t) =n\nf(t)−E(ˆfℓ\nh(t))o\n+n\nE(ˆfℓ\nh(t))−ˆfℓ\nh(t)o\n+ˆfℓ\nh(t)\n⩾ˆfℓ\nh(t)−∥ψℓ∥\u0010\nT(ψℓ) + Γ(2dQd\ni=1hi)\u0011\n⟨1, ψℓ⟩(nQd\ni=1hi)1/2(2.5)\nand similarly,\nf(t)⩽ˆfu\nh(t) +∥ψu∥\u0010\nT(−ψu) + Γ(2dQd\ni=1hi)\u0011\n⟨1, ψu⟩(nQd\ni=1hi)1/2. (2.6)\nNow, if καdenotes the (1 −α)thquantile of the statistic:\nT∗:= max {T(ψℓ), T(−ψu)},/Confidence Bands for Shape-restricted Regression 7\nthen in view of (2.5) and (2.6), we can define a 1 −αconfidence band for fas [ˆℓ,ˆu], where:\nˆℓ(t) := sup\nh∈I:t∈Ah\n\nˆfℓ\nh(t)−∥ψℓ∥\u0010\nκα+ Γ(2dQd\ni=1hi)\u0011\n⟨1, ψℓ⟩(nQd\ni=1hi)1/2\n\n, (2.7)\nˆu(t) := inf\nh∈I:t∈Ah\n\nˆfu\nh(t) +∥ψu∥\u0010\nκα+ Γ(2dQd\ni=1hi)\u0011\n⟨1, ψu⟩(nQd\ni=1hi)1/2\n\n. (2.8)\nIn view of (2.5) and (2.6), we have:\nPf\u0010\nˆℓ(t)⩽f(t)⩽ˆu(t) for all t∈[0,1]d\u0011\n≥P\u0010\nT(ψℓ)⩽κα, T(−ψu)⩽κα\u0011\n=P(T∗⩽κα) = 1−α. (2.9)\nThis shows that [ ˆℓ,ˆu] is indeed a confidence band with guaranteed coverage probability 1 −αfor\nalln≥1; we state this formally below.\nTheorem 1. For kernels ψℓandψusatisfying (2.4), we have:\nPf\u0010\nˆℓ(t)⩽f(t)⩽ˆu(t) for all t∈[0,1]d\u0011\n≥1−α (2.10)\nfor all f∈ Fand for all n≥1.\nThe above theorem shows that for any function class Ffor which the two-sided bias bounds\n(2.4) hold, our approach yields an honest finite sample confidence band for any f∈ F. It is natural\nto ask if the above constructed band is conservative in nature. In the following result (proved in\nSection 6.1), we show that if for some function f∈ F, the function −falso belongs to F, then our\nconfidence band has exact coverage at f.\nProposition 1. Suppose f,−f∈ F and (2.4) holds. Then, our 1 −αconfidence band [ ˆℓ,ˆu] has\nexact coverage probability 1 −α, i.e.,\nPf\u0010\nˆℓ(t)⩽f(t)⩽ˆu(t) for all t∈[0,1]d\u0011\n= 1−α.\nProposition 1 shows that if f∈ F 1is a constant or f∈ F 2is an affine function, then the coverage\nprobability of our confidence band is exact. We will now see that for certain functions that exhibit\n“low-complexity” structure locally (for example, f∈ F 1is locally constant or f∈ F 2is locally\naffine), our confidence band exhibits adaptive rates, in particular it can shrink at the parametric\nn−1/2rate locally. The following result is proved in Section 6.2.\nTheorem 2. Assume that (2.4) holds with kernels ψℓandψufor some function class F. Suppose\nfurther that the true f∈ Fsatisfies\nE(ˆfℓ\nε1d(t)) =f(t) =E(ˆfu\nε1d(t)) for all t∈D⊂Aε1d, (2.11)\nfor some ε >0.1Then,\nsup\nt∈D(ˆu(t)−ˆℓ(t))≤Kεn−1/2\u0010\n|κα|+|T(ψu)|+|T(−ψℓ)|\u0011\n(2.12)\n1Here, 1drefers to the d-dimensional vector with all entries 1./Confidence Bands for Shape-restricted Regression 8\nfor some constant Kε>0 (depending on ε, ψℓ, ψu). If (2.11) holds with ε≡εn= (log( en))−1\ndand\nfor all t∈Dn⊂Aεn1dthen\nsup\nt∈Dn(ˆu(t)−ˆℓ(t))≤Kρn \n1 +|κα|+|T(ψu)|+|T(−ψℓ)|p\nlog log( en)!\nfor some constant K > 0 (depending on ψℓ, ψu), where ρn:= (log( en) log log( en)/n)1/2.\nTo understand the implications of Theorem 2, let us consider the special case where both fand\n−f, restricted to a fixed subset Sof [0,1]d, belongs to the class F. In such a case, (2.11) holds\nfor any small ε >0 and D:={x∈(0,1)d:B∞(x, ε1d)⊂S}, the ‘ ε-interior’ of S. In particular,\nthe above result shows that if the true function f∈ F 1is locally constant or if f∈ F 2is locally\naffine in a fixed neighborhood S, then our confidence band automatically adapts to this structure,\nand shrinks at the parametric rate n−1/2onD. The second part of the result shows that a similar\nparametric rate n−1/2(up to multiplicative logarithmic factors) also holds if Dis allowed to grow\ntoSat a certain rate (with n).\nRemark 2.1 (Confidence bands with (locally) parametric widths for “low-complexity” functions) .\nSuppose that f∈ F 1is piecewise constant, or f∈ F 2is piecewise affine. There is a large re-\ncent literature on how the least squares estimator in these shape-constrained problems adapt to\nthis “low-complexity” structure of fand exhibit (almost) parametric rates in terms of risk (see\ne.g,. Guntuboyina and Sen (2018), Han et al. (2019), Kur et al. (2020), Deng et al. (2021)). As an\nimmediate consequence of Theorem 2 we see that our constructed confidence bands locally shrink\nat the parametric rate (see (2.12)) for such “low-complexity” functions.\n2.3. Choice of kernels for function classes F1andF2\nAs we have mentioned in the Introduction, the two prime examples of shape-constrained function\nclasses are: (i) the class of all d-dimensional coordinate-wise nondecreasing functions F1, and (ii)\nthe class of all d-dimensional convex functions F2. In this subsection, we construct kernels ψℓand\nψufor each of these function classes F1andF2, that satisfy the two-sided bias condition (2.4).\nThis would immediately imply that we can construct honest confidence bands for these function\nclasses satisfying (2.10). Moreover, the confidence bands constructed using these kernels will exhibit\ncertain optimality properties (as will be shown in Theorem 7 and 8 below). For the class F1we\ndefine:\nψu\n1(x) := \n1−dX\ni=1xi!\n1x∈[0,∞)d,Pd\ni=1xi⩽1and ψℓ\n1(x) := \n1 +dX\ni=1xi!\n1x∈(−∞,0]d,Pd\ni=1xi≥−1\n(2.13)\nand for the class F2, we define:\nψu\n2(x) := (1 − ∥x∥2) 1∥x∥⩽1 and ψℓ\n2(x) :=\u0012\n1−2d+ 4\nd+ 1∥x∥+d+ 3\nd+ 1∥x∥2\u0013\n1∥x∥⩽1.(2.14)\nNote that ψℓ\n2can take negative values. Theorem 3 below (proved in Section 6.3) shows that (2.4)\nholds for these specific kernel choices.\nTheorem 3. Letˆfℓ\nh,jand ˆfu\nh,jdenote the kernel estimators corresponding to the kernels ψℓ\njand\nψu\njrespectively, for j∈ {1,2}. Then, (2.4) holds for the function classes F1andF2./Confidence Bands for Shape-restricted Regression 9\n3. Adaptivity of the confidence band\nIn this section we show that the width of our confidence band [ ˆℓ,ˆu] (see (2.7) and (2.8)) adapts to\nthe (global and local) smoothness and the intrinsic dimension of the true function f. Let us first\ndefine the rate of convergence for a confidence band as follows: We say that the confidence band\n{(ℓ(t), u(t)) :t∈[0,1]d}, with coverage probability 1 −α(forα∈(0,1)), has rate of convergence\nγnon a set An⊆[0,1]dfor a function class Gif\ninf\nf∈GPf\u0012\nsup\nt∈An(u(t)−ℓ(t))⩽∆γn\u0013\n⩾1−α,for all n,\nwhere ∆ >0 is a constant not depending on n(but may depend on αandG). Clearly we want the\nrate of convergence to be as small as possible.\n3.1. Adaptivity with respect to global smoothness\nTo state our main result in this subsection we need to introduce the notion of H¨ older smoothness\nof a function f: [0,1]d→R, which we define below.\nDefinition 2 (H¨ older smoothness) .For every fixed β > 0 and L > 0, the H¨ older class Hβ,Lon\n[0,1]dis defined as the set of all functions f: [0,1]d→Rthat have all partial derivatives of order\n⌊β⌋(defined as the largest integer strictly less than β) on [0 ,1]d, and satisfy:\nX\nk∈Nd:∥k∥1⩽⌊β⌋sup\nx∈[0,1]d\f\f\f\f\f∂∥k∥1f(x)\n∂xk1\n1. . . ∂xkd\nd\f\f\f\f\f⩽L\nand\nX\nk∈Nd:∥k∥1=⌊β⌋\f\f\f\f\f∂∥k∥1f(y)\n∂xk1\n1. . . ∂xkd\nd−∂∥k∥1f(z)\n∂xk1\n1. . . ∂xkd\nd\f\f\f\f\f⩽L∥y−z∥β−⌊β⌋for all y,z∈[0,1]d. (3.1)\nThe following theorem shows that the rate of convergence of our confidence band [ ˆℓ,ˆu] for the\nclassHβ,L∩ Fj(forj= 1,2) is (log n/n)β/(2β+d). Note that the construction of the confidence\nband does not use knowledge of the smoothness parameter βand it still achieves the minimax rate\nof convergence for the class Hβ,L(see Tsybakov (2009)) if f∈Hβ,L. This highlights the adaptive\nnature of our confidence band.\nTheorem 4. Fixj∈ {1,2},f∈ F j∩Hβ,Lwith j−1< β⩽jandL > 0. Suppose that [ ˆℓ,ˆu]\nis the level αconfidence band constructed as in (2.7) and (2.8) based on kernel functions ψℓ, ψu\nsatisfying the two-sided bias condition (2.4). Then there exists a constant ∆ >0 (depending only\nonL, β, ψℓ, ψu) such that\nsup\nt∈Aεn1d\u0010\nˆu(t)−ˆℓ(t)\u0011\n⩽∆εβ\nn\u0012\n1 +|κα|+|T(−ψℓ)|+|T(ψu)|\n(log(en))1/2\u0013/Confidence Bands for Shape-restricted Regression 10\nwhere εn:= (log( en)/n)1/(2β+d), and 1dis the d-dimensional vector of all ones. This, in particular,\nimplies that\ninf\nf∈Fj∩Hβ,LP \nsup\nt∈Aεn1d(ˆu(t)−ˆℓ(t))⩽∆hlog(en)\nniβ\n2β+d!\n⩾1−α,for all n,\nfor some constant ∆ >0 depending only on L, β, ψℓ, ψu, α.\nTheorem 4 is proved in Section 6.4. Its proof starts by bounding the pointwise deviation of\nthe upper (and lower) band of our constructed confidence set from the true function f, in terms\nof the inner product of the variation of fin a small neighborhood. The variation of fover this\nneighborhood can then be bounded in terms of appropriate powers of the smoothing bandwidth,\nusing the H¨ older smoothness of f.\n3.2. Adaptivity with respect to local smoothness\nOur main result in Section 3.1 showed that our confidence band achieves the optimal rate of\nconvergence when the true function is globally H¨ older smooth. In this subsection we show that our\nadaptivity results also hold when the true function is locally H¨ older smooth. Specifically, we will\nlook at the behavior of ˆ u(t0)−ˆℓ(t0) for a fixed t0∈(0,1)d.\nTheorem 5. Fixj∈ {1,2}andt0∈(0,1)d. Suppose that f∈ Fj, and that there exists ε∈(0,1/2)\nsuch that fis H¨ older smooth on B∞(t0, ε) with smoothness parameter β∈(j−1, j] and L >0.\nWe construct the confidence band [ ˆℓ,ˆu] based on kernel functions ψℓ, ψuthat satisfy the two-sided\nbias condition (2.4). Then there exists a constant Kdepending on β, L, ψℓ, ψusuch that:\nˆu(t0)−ˆℓ(t0)≤Kρn\u0012\n1 +|κα|+|T(−ψℓ)|+|T(ψu)|\n(log(en))1/2\u0013\nwhere ρn= (log( en)/n)β/(2β+d). Note that this implies that\nPf\u0012\nˆu(t0)−ˆℓ(t0)⩽∆hlog(en)\nniβ\n2β+d\u0013\n⩾1−α,for all n,\nfor some constant ∆ >0 depending only on L, β, ψl, ψu, α.\nThe proof of Theorem 5 essentially follows from that of Theorem 4 by noticing that one needs to\ncontrol the bias of the kernel estimators locally which in turn only depends on the local smoothness\noff; see Section 6.5 for a sketch of the proof.\n3.3. Adaptivity with respect to intrinsic dimension\nThe intrinsic dimension of a function refers to the number of coordinates it actually depends on.\nDefinition 3 (Intrinsic dimensionality) .The intrinsic dimension dim( f) of a function f:Rd→R\nisk≥1 if and only if:\n1. there exist 1 ⩽i1< i2< . . . < i k⩽dand a function g:Rk→Rsuch that f(x1, . . . , x d) =\ng(xi1, . . . , x ik) for all ( x1, . . . , x d)∈Rd, and/Confidence Bands for Shape-restricted Regression 11\n2.fis not a function of ( xs)s∈Sfor any strict subset Sof{i1, . . . , i k}.\nIt can be verified easily from Definition 3 that the intrinsic dimensionality of a function fis\nunique. We will now show that our confidence band [ ˆℓ,ˆu] adapts to the intrinsic dimensionality of\nthe true function f.\nTheorem 6. Fixj∈ {1,2},f∈ F j∩Hβ,Lwith j−1< β⩽jandL > 0. Let dim( f) =kand\nsuppose that f(x) =g(xi1, . . . , x ik) for all x= (x1, . . . , x d)∈[0,1]dand some 1 ≤i1< . . . < i k≤d\nand some function g: [0,1]k→R. Suppose that [ ˆℓ,ˆu] is the constructed confidence band based on\nkernel functions ψℓ, ψuthat satisfies the two-sided bias condition (2.4). Then, for every ε∈(0,1/2),\nthere exists a constant ∆ >0 (depending only on L, β, ψℓ, ψu, ε) such that:\nsup\nt∈Aεn,i1,...,ik\u0010\nˆu(t)−ˆℓ(t)\u0011\n≤∆ρn,k\u0012\n1 +|κα|+|T(−ψℓ)|+|T(ψu)|\n(log(en))1/2\u0013\nwhere ρn,k:= (log( en)/n)β/(2β+k),εn:=ερ1/β\nn,kandεn,i1,...,ikis the d-dimensional vector with the\nith\n1, . . . , ith\nkentries all equal to εnand all other entries equal to ε.\nTheorem 6, proved in Section 6.6, states that the rate of convergence of our confidence band when\nthe function flies in Fj∩Hβ,L(forj= 1,2) with j−1< β⩽jandL >0, is (log( en)/n)β/(2β+dim( f)).\nThus, the rate of convergence of the confidence band depends only on the variables which actually\naffect the true function, and not on the redundant variables that the function does not vary with.\nThis is a desirable property, as this shows that our proposed confidence band can avoid the curse of\ndimensionality when dis large, if dim( f) is small. Note that this is another example of automatic\nadaptation of our procedure: the proposed confidence band has no knowledge of dim( f) and it still\nautomatically adjusts to yield a band that shrinks at the correct rate.\n4. Optimality of our confidence band\nIn this section we prove that our proposed confidence band (see (2.7) and (2.8)) is optimal in a\ncertain sense. Our results extend Theorem 4.2 in D¨ umbgen (2003). In order to state our result, we\nneed some notation. For a function g:Rd→RandU⊆Rd, define:\n∥g∥U:= sup\nx∈U|g(x)|.\nWe first state our optimality result for the class of coordinate-wise nondecreasing functions F1.\nTheorem 7. Letf∈ F 1be a continuously differentiable function in an open neighborhood Uof\nt0∈(0,1)dsuch that\nL1≡L1[t0]≡L1[f,t0] :=\"dY\ni=1∂\n∂xif(x)\f\f\nx=t0#1/d\n>0.\nDefine\nρn:=\u0012log(en)\nn\u0013 1\n2+d\nand ∆(z):=\u0012\u0012d+ 2\n2d\u0013\n∥ψz\n1∥2\u0013−1\n2+d\n(4.1)\nwhere zstands for uandℓcorresponding to kernels ψu\n1andψℓ\n1(respectively) as defined in (2.13).\nThen we have the following:/Confidence Bands for Shape-restricted Regression 12\n(a) Let [ ℓ, u] be any confidence band such that, for some α∈(0,1),\nPg\u0000\nℓ(t)⩽g(t)⩽u(t) for all t∈[0,1]d\u0001\n⩾1−αfor all g∈ F 1.\nThen, for any ϵ >0,\nlim inf\nn→∞Pf\u0012\n∥f−ℓ∥U⩾(1−ϵ)∆(ℓ)Ld\n2+d\n1[f,t0]ρn\u0013\n⩾1−α,\nlim inf\nn→∞Pf\u0012\n∥u−f∥U⩾(1−ϵ)∆(u)Ld\n2+d\n1[f,t0]ρn\u0013\n⩾1−α.\n(b) Moreover, let [ ˆℓ,ˆu] be the confidence band with coverage probability 1 −αas defined in (2.7)\nand (2.8), with kernels as in (2.13). Then, for any ϵ >0, we have\nlim\nn→∞Pf\u0012\n(f−ˆℓ)(t0)⩽(1 +ϵ)∆(ℓ)Ld\n2+d\n1[f,t0]ρn\u0013\n= 1,\nlim\nn→∞Pf\u0012\n(ˆu−f)(t0)⩽(1 +ϵ)∆(u)Ld\n2+d\n1[f,t0]ρn\u0013\n= 1.(4.2)\nTheorem 7 (proved in Section 6.7) states that the length of any confidence band for f∈ F 1with\nguaranteed coverage probability 1 −α, is at least (log n/n)1/(2+d)up to a constant factor. Further,\nthis optimal length is essentially achieved locally at t0by our constructed confidence band. We can,\nin particular, exactly compute the constants ∆(ℓ)and ∆(u)appearing in the above result (see (4.1)).\nFor example, for d= 2, ∆(ℓ)= ∆(u)≈1.86121.\nNote that the asymptotic probabilities in the upper bound results in part (b) of Theorem 7 do not\ndepend on α, unlike the corresponding lower bound probabilities in part (a). This can be understood\nfrom the fact that the random variable in part (a) is a supremum of pointwise deviations of the lower\nand upper confidence bands from the true function over a neighborhood, unlike the corresponding\nrandom variable in part (b), which just captures the deviation at t0. To draw a simple analogy,\nnote that the αthquantile q(n)\nαof the maximum of a sequence of i.i.d. Gaussians Z1, . . . , Z nsatisfies\nP(max 1≤i≤nZi≥q(n)\nα) = 1−α, but since q(n)\nαis of the order√logn,P(Z1≤q(n)\nα) = 1−o(1).2\nNext, we state our optimality result for the class of multivariate convex functions F2. Theorem 8\nbelow is proved in Section 6.8.\nTheorem 8. Letf∈ F 2be a twice continuously differentiable function in an open neighborhood\nUoft0∈(0,1)dsuch that\nL2≡L2[t0]≡L2[f,t0] := det( H(t0))1/d>0\nwhere H(t0)∈Rd×ddenotes the Hessian of fatt0, and det( ·) denotes the determinant operator.\nDefine\nρn:=\u0012log(en)\nn\u0013 2\n4+d\n,\n2Another analogy where the quantiles of the supremum have the same order as each individual random variable is\ngiven below: Consider the supremum of the Brownian motion on [0 ,1] which has the same distribution as the absolute\nvalue of a standard Gaussian. In this case, the quantiles of the supremum is of the same order as the corresponding\nquantiles of the process at any single point, but the former is always larger than the latter./Confidence Bands for Shape-restricted Regression 13\n∆(ℓ):=\nd+ 4\n2d r\n2(d+ 3)\nd+ 1!d\n∥ψℓ\n2∥2\n−2\n4+d\n, ∆(ℓ⋆):= ∆(ℓ)dd\nd+4\n∆(u):=\u0012d+ 4\n2d\u0010√\n2\u0011d\n∥ψu\n2∥2\u0013−2\n4+d\nand ∆(u⋆):= ∆(u)dd\nd+4,\nwhere ψu\n2andψℓ\n2(respectively) as defined in (2.14). Also, define\nL2,⋆≡L2,⋆[t0]≡L2,⋆[f,t0] :=\"dY\ni=1H(t0)i,i#1/d\n.\nThen we have the following:\n(a) Let [ ℓ, u] be any confidence band such that, for some α∈(0,1),\nPg\u0000\nℓ(t)⩽g(t)⩽u(t) for all t∈[0,1]d\u0001\n⩾1−αfor all g∈ F 2.\nThen, for any ϵ >0,\nlim inf\nn→∞Pf\u0012\n∥f−ℓ∥U⩾(1−ϵ)∆(ℓ)Ld\n4+d\n2[f,t0]ρn\u0013\n⩾1−α,\nlim inf\nn→∞Pf\u0012\n∥u−f∥U⩾(1−ϵ)∆(u)Ld\n4+d\n2[f,t0]ρn\u0013\n⩾1−α.(4.3)\n(b) Moreover, let [ ˆℓ,ˆu] be the confidence band with coverage probability 1 −αas defined in (2.7)\nand (2.8), with kernels as in (2.14). Then, for any ϵ >0, we have\nlim\nn→∞Pf\u0012\n(f−ˆℓ)(t0)⩽(1 +ϵ)∆(ℓ⋆)Ld\n4+d\n2,⋆[f,t0]ρn\u0013\n= 1,\nlim\nn→∞Pf\u0012\n(ˆu−f)(t0)⩽(1 +ϵ)∆(u⋆)Ld\n4+d\n2,⋆[f,t0]ρn\u0013\n= 1.(4.4)\n(c) In the special case when the Hessian matrix of fatt0is diagonal, one has the following for\nevery ϵ >0,\nlim\nn→∞Pf\u0012\n(f−ˆℓ)(t0)⩽(1 +ϵ)∆(ℓ)Ld\n4+d\n2[f,t0]ρn\u0013\n= 1,\nlim\nn→∞Pf\u0012\n(ˆu−f)(t0)⩽(1 +ϵ)∆(u)Ld\n4+d\n2[f,t0]ρn\u0013\n= 1.(4.5)\nPart (a) of the above result gives a lower bound on the maximal (local) deviation of any honest\nconfidence band (for the class of convex functions) around the true function. We can, in particular,\nexactly compute the constants ∆(ℓ)and ∆(u)appearing in the above result. For example, for d= 2,\n∆(ℓ)≈1.19788 and ∆(u)≈0.68278. Part (c) shows that our confidence band attains this minimal\nlength at t0when the true function fhas a diagonal Hessian matrix at t0. In contrast, if the Hessian\nis not diagonal, our confidence band may have slightly larger width at t0, as indicated in part (b).\nThe following remark discusses this discrepancy./Confidence Bands for Shape-restricted Regression 14\nRemark 1 (On the lower and upper bounds in Theorem 8) .We would like to highlight that the\nconstant terms in the upper and lower bounds appearing in (4.3) and (4.4) are different. This\ndiscrepancy is reflected by the replacement of the geometric mean of the spectrum of the Hessian\noffatt0(i.e., L2[f,t0]) by the geometric mean of the diagonal entries of the Hessian matrix (i.e.,\nL2,⋆[f,t0]). Further, there is an inflation by a multiplicative factor depending on the dimension\nd(see Theorem 8 part (b)). The main two terms L2andL2,⋆in the lower and upper bounds in\n(4.3) and (4.4) satisfy L2≤L2,⋆. Intuitively, this discrepancy is caused because we are limited\nto choosing a diagonal bandwidth matrix, whereas the orientation of the underlying multivariate\nconvex function fis not necessarily restricted along the standard coordinate axes. Note that when\nd= 1, L2=L2,⋆and ∆(z)= ∆(z⋆)forz∈ {u, ℓ}, and hence both the upper and lower bounds\nmatch (as in D¨ umbgen (2003)). Observe that such a discrepancy does not happen for coordinate-\nwise monotone functions (cf. Theorem 7), as these functions are inherently tied to the standard\ncoordinate axes.\nRemark 2 (On the proofs of part (a) of Theorems 7 and 8) .The proofs of the lower bounds (part\n(a) of Theorems 7 and 8) involve the following main ideas. As a first step, one constructs a grid\nin [0,1]dwith spacings given by a bandwidth parameter and centered at t0. For each such grid\npoint t, one defines a function ftby perturbing the true function fsuitably. The second step is to\nshow that all these perturbed functions satisfy the corresponding shape constraint (see Lemmas 6.1\nand 6.4). As a next step, one shows that the probability that the deviation between the upper and\nlower limits (of any honest confidence band with coverage probability 1 −α) exceeds a suitable\nconstant (depending on the bandwidth parameters) is lower bounded by 1 −αminus a remainder\nterm (depending on the perturbation functions). One then argues that this remainder term is\nasymptotically negligible by expressing it in terms of an average of the likelihood ratio between\nthe measures at the perturbed and the true function, and applying the Cameron-Martin-Girsanov\ntheorem in stochastic calculus to evaluate this likelihood ratio. As a final step, several parameters\nare tuned appropriately to obtain the optimal constant (in the statement of the theorems).\nRemark 3 (On the proofs of part (b) of Theorems 7 and 8) .Observe that the random fluctuation\noff(t0)−ˆℓ(t0) is stochastically dominated by the difference between f(t0) and any particular term\n(corresponding to a bandwidth parameter) in the supremum in (2.7). The latter in turn can be\ndecomposed into three terms—a random term having Gaussian fluctuations, a bias-like term arising\nfrom the kernel estimator and the multiscale penalization term. The main crux of the proof involves\na delicate choice of the bandwidth parameter which balances the bias-like term and the multiscale\npenalization term, thereby enabling us to lower bound the probability in (4.2) (and (4.4)).\n5. Simulation studies\nIn this section we construct confidence bands for different shape-restricted regression functions.\nFor simulation purposes, instead of the continuous white noise model (1.1) we consider its discrete\nanalogue as detailed below.\nLet us start with the connection to nonparametric regression on gridded design. Let x1, . . . ,xn∈\nRdbe an enumeration of the m×···× muniform grid Gd\nm:={1/m,2/m, . . . , (m−1)/m,1}dwhere\nmd=n. Let us look at the following nonparametric regression model:\nYi=f(xi) +ϵi, fori= 1, . . . , n, (5.1)/Confidence Bands for Shape-restricted Regression 15\nFig 1 . Confidence band for the function f(x1, x2) =x1+x2assuming fis coordinate-wise isotonic and n= 502.\nwhere f: [0,1]d→Ris the unknown regression function and ϵi’s are i.i.d. standard normal random\nvariables. For a kernel function ψ:Rd→Randh,t∈Gd\nm, such that t−h,t+h∈Gd\nmwe can\ndefine a kernel estimator ˆfhoffas\nˆfh(t) :=P\ni:xi∈B∞(t,h)Yiψ\u0000\n(xi−t)/h\u0001\nP\ni:xi∈B∞(t,h)ψ\u0000\n(xi−t)/h\u0001\nwhere by ( u1, . . . , u d)/(h1, . . . , h d) we mean the vector ( u1/h1, . . . , u d/hd). We can also define the\nstandardized kernel estimator as\nˆΨn(t,h) :=P\ni:xi∈B∞(t,h)Yiψ\u0000\n(xi−t)/h\u0001\nqP\ni:xi∈B∞(t,h)ψ2\u0000\n(xi−t)/h\u0001.\nThen the multiscale statistic for this regression problem reduces to\nTn(Y, ψ) := sup\nh∈Gdm:t−h,t+h∈Gdmsup\nt∈Gdm|ˆΨn(t,h)| −Γ\u0010\n|B∞(t,h)∩Gd\nm|\u0011\n(5.2)\nwhere |B∞(t,h)∩Gd\nm|denotes the number of elements in B∞(t,h)∩Gd\nmandY≡(Y1, . . . , Y n).\nFor our simulation studies we consider d= 2. At first we consider the regression function\nf(x1, x2) =x1+x2. In our simulation studies we consider data on a 50 ×50 grid on [0 ,1]2and\nassume that the underlying regression function is coordinate-wise nondecreasing. Figure 1 shows\nour constructed confidence band with nominal coverage probability 0.95. Here we would like to\npoint out that the confidence band has smaller width around the center of the rectangle and the\nwidth gets larger as we move towards the sides; this is expected as there are smaller number of\ndata points close by to average over near the corners./Confidence Bands for Shape-restricted Regression 16\nFig 2 . Confidence band for the function f(x1, x2) =I(x1⩾0.5)assuming fis coordinate-wise isotonic and n= 502.\nIn our second simulation study, we construct a confidence band for the function f(x1, x2) =\nI(x1⩾0.5); see Figure 2. We can clearly see the local adaptivity of our band in action here. On\nthe regions where the function is constant (regions where x1is away from 0.5) we see that the\nconfidence band has significantly smaller width.\nWe also construct confidence bands under the assumption that the regression function is convex.\nFigure 3 shows the constructed confidence band for the regression function f(x1, x2) =|x1−0.5|,\nwhereas Figure 4 corresponds to the regression function f(x1, x2) = 40 {(x1−0.5)2+ (x2−0.5)2}.\nBoth of these plots show that the upper confidence band is much closer to the true function than\nthe lower band; this behaviour is expected as we have shown in Theorem 8 that the optimal\nseparation constant for the upper bound ∆(u)≈0.68278 is smaller than that for the lower bound\n∆(ℓ)≈1.19788. We also see that the lower confidence band is closer to the actual function around\nthe center of the rectangle than on the sides.\nTable 1 gives the estimated coverage probabilities of our constructed confidence bands for various\ncoordinate-wise nondecreasing and convex functions. It shows that in at least 95% of the cases our\nconfidence bands do contain the actual regression function (which is expected as we have guaranteed\nconverge even in finite samples; cf. Theorem 1). Moreover, when the underlying function is a\nconstant (if f∈ F 1) or affine (if f∈ F 2) we do observe that the nominal coverage coincides with\nthe expected coverage (cf. Proposition 1)./Confidence Bands for Shape-restricted Regression 17\nFig 3 . Confidence band for the function f(x1, x2) =|x1−0.5|assuming fis convex and n= 402.\nFig 4 . Confidence band for the function f(x1, x2) = 40 {(x1−0.5)2+ (x2−0.5)2}assuming fis convex and n= 402./Confidence Bands for Shape-restricted Regression 18\nf(x1, x2) Class Coverage probability\n0 isotonic 0.95\nx1+x2 isotonic 0.97\n20(x1+x2) isotonic 1.00\nI(x1⩾0.5) isotonic 0.97\n0 convex 0.95\nx1+x2 convex 0.96\n10(x1+x2) convex 0.95\n(x1−0.5)2+ (x2−0.5)2convex 0.98\nTable 1\nEstimated coverage probabilities of our constructed confidence bands for different regression functions.\n6. Proofs of the main results\nIn this section, we prove the main results of this paper.\n6.1. Proof of Proposition 1\nWe begin by showing that under the hypothesis of Proposition 1, we have:\nE(ˆfℓ\nh(t)) =f(t) =E(ˆfu\nh(t)) for all t∈Ah,for all h∈I:= (0 ,1/2]d. (6.1)\nTowards this, note that since f,−f∈ F, and since −ˆfhis the kernel estimator of −fas defined in\n(2.2) (since for a standard d-dimensional Brownian sheet W, we have WD=−W), we have:\nE(ˆfℓ\nh(t))⩽f(t)⩽E(ˆfu\nh(t)) and E(−ˆfℓ\nh(t))⩽−f(t)⩽E(−ˆfu\nh(t))\nfor all h∈I,t∈Ah. This proves (6.1).\nNext, observe that in view of (2.9), all it requires to complete the proof of Proposition 1 is to\nshow that:\nn\nˆℓ(t)⩽f(t)⩽ˆu(t) for all t∈[0,1]do\n⊆n\nT(ψℓ)⩽κα, T(−ψu)⩽καo\n. (6.2)\nTowards this, suppose that ˆℓ(t)⩽f(t)⩽ˆu(t) for all t∈[0,1]d. In view of (6.1), we have:\nˆℓ(t)⩽f(t) for all t∈[0,1]d\n=⇒ ˆℓ(t)⩽E(ˆfℓ\nh(t)) for all t∈[0,1]d\n=⇒ ˆfℓ\nh(t)−∥ψℓ∥\u0010\nκα+ Γ(2dQd\ni=1hi)\u0011\n⟨1, ψℓ⟩(nQd\ni=1hi)1/2⩽Eˆfℓ\nh(t) (for all h∈I,t∈Ah)\n=⇒ˆfℓ\nh(t)−Eˆfℓ\nh(t)q\nVar(ˆfℓ\nh(t))−Γ(2ddY\ni=1hi)≤κα (for all h∈I,t∈Ah)\n=⇒ T(ψℓ)≤κα./Confidence Bands for Shape-restricted Regression 19\nSimilarly, one has:\nˆu(t)≥f(t) for all t∈[0,1]d\n=⇒ ˆu(t)≥E(ˆfu\nh(t)) for all t∈[0,1]d\n=⇒ ˆfu\nh(t) +∥ψu∥\u0010\nκα+ Γ(2dQd\ni=1hi)\u0011\n⟨1, ψu⟩(nQd\ni=1hi)1/2≥Eˆfu\nh(t) (for all h∈I,t∈Ah)\n=⇒ˆfu\nh(t)−Eˆfu\nh(t)q\nVar( ˆfu\nh(t))+ Γ(2ddY\ni=1hi)≥ −κα (for all h∈I,t∈Ah)\n=⇒ T(−ψu)≤κα.\nThis proves (6.2) and completes the proof of Proposition 1.\n6.2. Proof of Theorem 2\nTo prove the first result, we take h:=ε1d. Note that the hypothesis (2.11) of Proposition 2 implies\nthat for all t∈D,\n⟨f(t+h⋆·)−f(t), ψu(·)⟩\n⟨1, ψu⟩=E(ˆfu\nh(t))−f(t) = 0 . (6.3)\nAlso note that, for any t∈Aε1d,\nˆu(t)−f(t)\n⩽ˆfu\nh(t)−f(t) +∥ψu∥\u0010\nκα+ Γ(2dQd\ni=1hi)\u0011\n⟨1, ψu⟩(nQd\ni=1hi)1/2\n⩽⟨f(t+h⋆·)−f(t), ψu(·)⟩\n⟨1, ψu⟩+∥ψu∥(κα+ 2Γ(2d(Qd\ni=1hi)) +T(ψu))\n⟨1, ψu⟩(nQd\ni=1hi)1/2. (6.4)\nFor any t∈D⊂Aε1d, we thus have, using (6.3),\nˆu(t)−f(t)≤n−1/2∥ψu∥(κα+ 2Γ(2dεd) +T(ψu))\n⟨1, ψu⟩εd/2≤Kεn−1/2(|κα|/2 +|T(ψu)|)\nfor some constant Kε>0. A similar analysis can be done for f(t)−ˆℓ(t), which concludes the proof\nof the first result by adding the two bounds.\nTo prove the second result, we take ε≡εn:= (log( en))−1\nd, we can conclude from (6.4) (and\nusing (6.3)) that for t∈Dn,\nˆu(t)−f(t)≤K1(log(en))1/2n−1/2∥ψu∥\u0010\n|κα|+|T(ψu)|+p\nlog log( en)\u0011\n/⟨1, ψu⟩\n=K2\u0012log(en) log log( en)\nn\u00131/2 \n1 +|κα|+|T(ψu)|p\nlog log( en)!\nfor some constants K1, K2>0. The bound for f(t)−ˆℓ(t) follows similarly, thereby completing the\nproof of Theorem 2./Confidence Bands for Shape-restricted Regression 20\n6.3. Proof of Theorem 3\nForj= 1, we will show that just the facts that ψu\n1is supported on a subset of [0 ,∞)d,ψℓ\n1is\nsupported on a subset of ( −∞,0]d, and they are non negative, are enough to conclude Theorem 3.\nThis follows easily from the fact that the coordinate-wise nondecreasing nature of fensures that:\n⟨f(t+h⋆·), ψu\n1⟩ ≥f(t)⟨1, ψu\n1⟩ and ⟨f(t+h⋆·), ψℓ\n1⟩⩽f(t)⟨1, ψℓ\n1⟩.\nNext, we consider the case k= 2. If we could show that for every convex function g:Rd→R, we\nhave:\n⟨g, ψu\n2⟩ ≥g(0)⟨1, ψu\n2⟩ and ⟨g, ψℓ\n2⟩⩽g(0)⟨1, ψℓ\n2⟩, (6.5)\nthen we would be done, because then substituting g(x) :=f(t+h⋆x) (which is a convex function)\nin (6.5) will complete the proof. We can also assume, without loss of generality, that g(0) = 0,\nbecause otherwise we can apply (6.5) on the function g(·)−g(0). In view of all these reductions,\nwe just need to show that ⟨g, ψu\n2⟩ ≥0 and ⟨g, ψℓ\n2⟩⩽0. The first inequality in (6.5) is a direct\nconsequence of Jensen’s inequality, because if Udenotes a random vector distributed on the d-\ndimensional sphere Bd:={x∈Rd:∥x∥⩽1}, with density at xbeing proportional to 1 − ∥x∥2,\nthen there exists a constant C >0 such that:\n⟨g, ψu\n2⟩=CE[g(U)]≥Cg(E[U]) =Cg(0) = 0\nwhere we used the fact that E[U] = 0 by symmetry of the distribution of Uaround 0.\nFinally, to prove that ⟨g, ψℓ\n2⟩⩽0, first note that by convexity of gand the fact that g(0) = 0,\nwe have:\ng(αy)⩽αg(y) for all y∈Rdandα∈[0,1].\nWe can now substitute α:= (d+ 3)∥x∥/(d+ 1) and y:= (d+ 1)x/((d+ 3)∥x∥), and have:\ng\u0012(d+ 1)x\n(d+ 3)∥x∥\u0013\n≥(d+ 1)\n(d+ 3)∥x∥g(x) when ∥x∥⩽d+ 1\nd+ 3.\nSimilarly, we can substitute α:= (d+ 1)/((d+ 3)∥x∥) and y:=x, and have:\ng\u0012(d+ 1)x\n(d+ 3)∥x∥\u0013\n⩽(d+ 1)\n(d+ 3)∥x∥g(x) whend+ 1\nd+ 3⩽∥x∥⩽1.\nMoreover, note that ψℓ\n2(x)⩽0 when ( d+ 1)/(d+ 3)⩽∥x∥⩽1 and ψℓ\n2(x)≥0 when ∥x∥⩽\n(d+ 1)/(d+ 3). We have:\n⟨g, ψℓ\n2⟩=ˆ\nBd\u0012\n1−2d+ 4\nd+ 1∥x∥+d+ 3\nd+ 1∥x∥2\u0013\ng(x)dx\n⩽d+ 3\nd+ 1ˆ\nBd∥x∥\u0012\n1−2d+ 4\nd+ 1∥x∥+d+ 3\nd+ 1∥x∥2\u0013\ng\u0012(d+ 1)x\n(d+ 3)∥x∥\u0013\ndx.\nAt this point, for every e∈ {− 1,1}d, define:\nHe:={x∈Bd:eixi≥0 for all 1 ⩽i⩽d}./Confidence Bands for Shape-restricted Regression 21\nNote that {He}e∈{−1,1}dform the 2dorthants of Rd, intersected with Sd−1. We will show that for\nalle∈ {− 1,1}d,\nˆ\nHe∥x∥\u0012\n1−2d+ 4\nd+ 1∥x∥+d+ 3\nd+ 1∥x∥2\u0013\ng\u0012(d+ 1)x\n(d+ 3)∥x∥\u0013\ndx= 0 (6.6)\nwhich is enough to complete the proof. Towards this, fix e∈ {− 1,1}d, and make the following\nchange of variables x7→y:= (y0, y1, . . . , y d−1) onHe:\ny0=∥x∥ and yi=xi\n∥x∥for all 1 ⩽i⩽d−1.\nThis transformation is invertible, and we have:\nxi=y0yifor all 1 ⩽i⩽d−1 and xd:=y0edq\n1−y2\n1−. . .−y2\nd−1.\nThe Jacobian of this transformation is given by:\nJ(y) =\ny1 y0 0 . . . 0\ny2 0 y0 . . . 0\n...............\nedq\n1−Pd−1\ni=1y2\ni−y0y1ed q\n1−Pd−1\ni=1y2\ni−y0y2sd q\n1−Pd−1\ni=1y2\ni. . .−y0yd−1edq\n1−Pd−1\ni=1y2\ni\n\nand hence, we have:\n|det(J(y))|=yd−1\n0q\n1−Pd−1\ni=1y2\ni.\nTherefore, defining ˜y:=\u0010\ny1, . . . , y d−1, edq\n1−y2\n1−. . .−y2\nd−1\u0011\n, we have:\nˆ\nHe∥x∥\u0012\n1−2d+ 4\nd+ 1∥x∥+d+ 3\nd+ 1∥x∥2\u0013\ng\u0012(d+ 1)x\n(d+ 3)∥x∥\u0013\ndx\n=ˆ\nBd−1TQd−1\ni=1ei[0,1]g\u0010\nd+1\nd+3˜y\u0011\nq\n1−Pd−1\ni=1y2\niˆ1\n0yd\n0\u0012\n1−2d+ 4\nd+ 1y0+d+ 3\nd+ 1y2\n0\u0013\ndy0dy1. . . dy d−1\n= 0 (since the inner integral is 0) .\nThis proves (6.6) and completes the proof of Theorem 3.\n6.4. Proof of Theorem 4\nTo begin with, note that for t∈Ah, we have\nˆu(t)−f(t)\n⩽ˆfu\nh(t)−f(t) +∥ψu∥\u0010\nκα+ Γ(2dQd\ni=1hi)\u0011\n⟨1, ψu⟩(nQd\ni=1hi)1/2\n⩽⟨f(t+h⋆·)−f(t), ψu(·)⟩\n⟨1, ψu⟩+∥ψu∥(κα+ 2Γ(2d(Qd\ni=1hi)) +T(ψu))\n⟨1, ψu⟩(nQd\ni=1hi)1/2. (6.7)/Confidence Bands for Shape-restricted Regression 22\nHere the last line follows from the inequality\nT(ψu)⩾ˆfu\nh(t)− ⟨f(t+h⋆·), ψu(·)⟩/⟨1, ψu⟩\n∥ψ∥⟨1, ψu⟩−1(nQd\ni=1hi)−1/2−Γ(2ddY\ni=1hi).\nNow, if f∈Hβ,L∩ F 1(where 0 < β⩽1) we have\n|f(t+h⋆x)−f(t)|⩽L∥h⋆x∥β.\nHence, denoting h:= max {h1, . . . , h d}, we have:\n⟨f(t+h⋆·)−f(t), ψu(·)⟩\n⟨1, ψu⟩⩽L´\n[−1,1]d∥h⋆x∥βψu(x)dx\n⟨1, ψu⟩⩽Lhβ´\n[−1,1]d∥x∥βψu(x)dx\n⟨1, ψu⟩:=K1hβ\n(6.8)\nwhere K1:=L´\n[−1,1]d∥x∥βψu(x)dx/⟨1, ψu⟩. On the other hand, if f∈Hβ,L∩F2(where 1 < β⩽2)\nthen defining g(x) :=f(t+h⋆x), we have the following for some ξxlying in the segment joining\n0andx:\n⟨f(t+h⋆·)−f(t), ψu(·)⟩=ˆ\n[−1,1]dx⊤∇g(ξx)ψu(x)dx\n=ˆ\n[−1,1]dx⊤(∇g(ξx)− ∇g(0))ψu(x)dx\n⩽ˆ\n[−1,1]d∥x∥∥∇g(ξx)− ∇g(0)∥ψu(x)dx\n=ˆ\n[−1,1]d∥x∥∥h⋆(∇f(t+h⋆ ξx)− ∇f(t))∥ψu(x)dx (6.9)\n⩽hˆ\n[−1,1]d∥x∥∥∇f(t+h⋆ ξx)− ∇f(t)∥ψu(x)dx\n⩽Lhβˆ\n[−1,1]d∥x∥βψu(x)dx.\nNote that in the second equality we have used the fact that both the integrals´\n[−1,1]dxiψu(x) and´\n[−1,1]d−xiψu(x) are nonnegative for 1 ≤i≤d, which follows from the bias condition (2.4) (as\nthe functions f(x) =±xiare convex, for i= 1, . . . , d ) and hence´\n[−1,1]dxiψu(x) = 0. The last\ninequality follows from the fact that f∈Hβ,L, and hence,\n∥∇f(t+h⋆ ξx)− ∇f(t)∥ ≤L∥h⋆ ξx∥β−1≤Lhβ−1.\nHence, in this case also, we have:\n⟨f(t+h⋆·)−f(t), ψu(·)⟩\n⟨1, ψu⟩⩽Lhβ´\n[−1,1]d∥x∥βψu(x)dx\n⟨1, ψu⟩=K1hβ.\nTherefore, (6.7) tells us that as long as t∈Ahwe have\nˆu(t)−f(t)⩽K1hβ+∥ψu∥(κα+ 2Γ(2d(Qd\ni=1hi)) +T(ψu))\n⟨1, ψu⟩(nQd\ni=1hi)1/2. (6.10)/Confidence Bands for Shape-restricted Regression 23\nPutting h1=h2=. . .=hd=εn:= (log( en)/n)1/(2β+d)in (6.10), we get K1hβ=K1εβ\nnand\n∥ψu∥(κα+ 2Γ(2d(Qd\ni=1hi)) +T(ψu))\n⟨1, ψu⟩(nQd\ni=1hi)1/2⩽K2|κα|+|T(ψu)|+ 2√2 + 2 log n\nnβ/(2β+d)log(en)d/(4β+2d)\n⩽K3εβ\nn \n|κα|+|T(ψu)|\nlog1/2(en)+ 1!\nfor some constants K2andK3not depending on n. The above two equations tell us that as long\nast∈Aεn1d, we have\nˆu(t)−f(t)⩽Kεβ\nn \n1 +|κα|+|T(ψu)|\nlog1/2(en)!\n(6.11)\nfor some constant Knot depending on n.\nThe steps for bounding f(t)−ˆℓ(t) are similar, but we point out some differences. First, we have:\nf(t)−ˆℓ(t)≤⟨f(t)−f(t+h⋆·), ψℓ(·)⟩\n⟨1, ψℓ⟩+∥ψℓ∥(κα+ 2Γ(2d(Qd\ni=1hi)) +T(−ψℓ))\n⟨1, ψℓ⟩(nQd\ni=1hi)1/2,\nwhich gives:\nf(t)−ˆℓ(t)≤K2hβ+∥ψℓ∥(κα+ 2Γ(2d(Qd\ni=1hi)) +T(−ψℓ))\n⟨1, ψℓ⟩(nQd\ni=1hi)1/2\nwhere K2:=L´\n[−1,1]d∥x∥β|ψℓ(x)|dx/⟨1, ψℓ⟩. Hence, for some constant K′not depending on n, we\nhave:\nf(t)−ˆℓ(t)≤K′εβ\nn \n1 +|κα|+|T(−ψℓ)|\nlog1/2(en)!\n. (6.12)\nTheorem 4 now follows by adding (6.11) and (6.12).\n6.5. Sketch of a proof of Theorem 5\nFor proving Theorem 5, first one needs to observe that for h=ε1d, we have ∥h⋆x∥∞=ε∥x∥∞≤ε,\nand hence, t0+h⋆x∈B∞(t0, ε). The rest of the proof follows exactly as the proof of Theorem 4,\non noting that one only needs the H¨ older smoothness assumption for bounding the terms |f(t0+\nh⋆x)−f(t0)|and∥∇f(t0+h⋆ ξx)− ∇f(t0)∥for some ξxlying in the segment joining 0andx\n(for the classes F1andF2respectively). Consequently, it is enough to have H¨ older smoothness on\nB∞(t0, ε) only.\n6.6. Proof of Theorem 6\nIt follows from (6.7) and (6.10) that for t∈Aεn,i1,...,ik, we have:\nˆu(t)−f(t)⩽K1εβ\nn+∥ψu∥(κα+ 2Γ(2dεk\nnεd−k) +T(ψu))\n⟨1, ψu⟩(nεknεd−k)1/2\n=K1εβρn,k+∥ψu∥(κα+ 2Γ(2dεd(log(en)/n)k/(2β+k)) +T(ψu))\n⟨1, ψu⟩(nεd(log(en)/n)k/(2β+k))1/2\n≤∆1ρn,k\u0012\n1 +|κα|+|T(ψu)|\n(log(en))1/2\u0013/Confidence Bands for Shape-restricted Regression 24\nfor some constants K1and ∆ 1>0. The rest of the proof follows the idea of the proof of Theorem\n4. The only modifications are in (6.8) and (6.9), where one now uses the fact that the function f\nonly depends on kcoordinates, and hence, the vector hcan now be replaced by its restriction on\ntheith\n1, . . . , ith\nkcoordinates.\n6.7. Proof of Theorem 7\n(a) We prove only the bound for ∥f−ℓ∥Uas the other case can be handled similarly. Thus, we will\nshow that for any level 1 −αconfidence band ( ℓ, u) with guaranteed coverage probability for the\nclassF1, and any 0 < γ < 1, we have\nlim inf\nn→∞Pf\u0012\n∥f−ℓ∥U⩾γ∆(ℓ)Ld\n2+d\n1[t0]ρn\u0013\n⩾1−α.\nFor notational simplicity, we will abbreviate ψℓbyψ. By our assumption, fis continuously differ-\nentiable on an open neighborhood Uoft0∈(0,1)dsuch that\nL1[t0]≡L1[f,t0] :=\"dY\ni=1∂\n∂xif(x)\f\f\nx=t0#1/d\n>0.\nLet us define, for i= 1, . . . , d ,\nM⋆\ni:=∂\n∂xif(x)|x=t0.\nWithout loss of generality, let us assume that M⋆\n1⩽M⋆\n2⩽. . .⩽M⋆\nd.Since γ <1, we can find\nϵ >0 and γ∗<1 (depending on γ) such that:\nγLd/(2+d)\n1 [t0] =γ⋆(L1[t0]−ϵ)d/(2+d). (6.13)\nAlso since fis continuously differentiable on U, we can find h0∈[0,1]dandϵ⋆>0 small enough\nsuch that:\n(i)B∞(t0,h0)⊂U,\n(ii)hQd\ni=1(M⋆\ni−ϵ⋆)i1/d\n⩾L1[t0]−ϵ,\n(iii) for all x∈B∞(t0,h0) we have\n∂\n∂xif(x)⩾M⋆\ni−ϵ⋆>0, for all i= 1,2, . . . , d.\nSuppose that h= (h1, h2, . . . , h d) is such that h1∈(0,1/2] and hi:=h1×\u0010M⋆\n1−ϵ⋆\nM⋆\ni−ϵ⋆\u0011\n, fori= 2, . . . , d .\nLet us define a set of grid points Gfor bandwidth has\nG:={t= (t1, . . . , t d) :ti=t0i+hi(2ki) for some integer ki, B∞(t,h)⊂B∞(t0,h0)}\nwhere t0= (t01, . . . , t 0d). For t∈G, let us define the following “perturbation” functions:\nft:=f−h1(M⋆\n1−ϵ⋆)ψt,h,\nwhere ψt,his defined in (2.1). We will now show that for every t∈G,ft∈ F 1./Confidence Bands for Shape-restricted Regression 25\nLemma 6.1. ft∈ F 1for all t∈G.\nProof. Fixt= (t1, . . . , t d)∈G. Suppose that x≤y(coordinate-wise) and x,y∈Tt,hwhere\nTt,h:=(\nu= (u1, . . . , u d)∈[0,1]d:ui⩽tifor all ianddX\ni=1\u0012ui−ti\nhi\u0013\n⩾−1)\n.\nThen, for some ξ∈[x,y]⊂B∞(t0,h0) (here [ x,y] denotes the line segment joining xandy), by\nthe mean value theorem (and (iii) above),\nf(y)−f(x) =▽f(ξ)⊤(y−x)⩾dX\ni=1(M⋆\ni−ϵ⋆)(yi−xi). (6.14)\nThen, using the formula for ψt,h,\n−ψt,h(y) +ψt,h(x) =−dX\ni=1\u0010yi−xi\nhi\u0011\n=−1\nh1(M⋆\n1−ϵ∗)dX\ni=1(M⋆\ni−ϵ⋆)(yi−xi). (6.15)\nIt now follows from (6.14) and (6.15) that:\nft(y)−ft(x)≥dX\ni=1(M⋆\ni−ϵ⋆)(yi−xi)−h1(M⋆\n1−ϵ⋆)\"\n1\nh1(M⋆\n1−ϵ∗)dX\ni=1(M⋆\ni−ϵ⋆)(yi−xi)#\n= 0\nthereby yielding ft(y)⩾ft(x).\nLet us now look at the case x/∈Tt,h,y∈Tt,handx⩽y.Define\na:= 1 +dX\ni=1\u0012xi−ti\nhi\u0013\nand b:= 1 +dX\ni=1\u0012yi−ti\nhi\u0013\n.\nSince y∈Tt,h, we have x⩽y⩽t(coordinate-wise). Hence as x/∈Tt,h,a <0 and as y∈Tt,h,\nb≥0. Define z:=αx+ (1−α)ywhere α:=b/(b−a)∈(0,1).Note thatPd\ni=1(zi−ti)/hi=−1\nandx⩽z⩽y⩽twhich implies that z∈Tt,h. Hence,\nft(y)⩾ft(z) =f(z)⩾f(x) =ft(x),\nthereby yielding ft(y)⩾ft(x). Here, the first inequality follows from the fact that z,y∈Tt,h, the\nthird inequality follows from monotonicity of fand the second and fourth equality follows from the\nfact that ψt,h(z) =ψt,h(x) = 0 (cf. (2.13)).\nNow let us look into the case where x∈Tt,h,y/∈Tt,handx⩽y.In this case\nft(y) =f(y)⩾f(x)⩾f(x)−h1(M⋆\n1−ϵ⋆)ψt,h(x) =ft(x).\nThe only case left is when x/∈Tt,h,y/∈Tt,handx⩽y.In this case ψt,h(x) =ψt,h(y) = 0, hence\nthe monotonicity of ftdirectly follows from the monotonicity of f. This completes the proof of\nLemma 6.1./Confidence Bands for Shape-restricted Regression 26\nNow we continue with the proof of Theorem 7. Let [ ℓ, u] be any honest confidence band for the\nclassF1. Let Abe the event that {ℓ(x)⩽ft(x) for all x∈[0,1]d,for some t∈G}.Now, since\n[ℓ, u] is a confidence band for all f∈ F 1, and since ft∈ F 1, we have\nPft(A)⩾1−αfor all t∈G.\nHence we have\nPf(∥f−ℓ∥U⩾h1(M⋆\n1−ϵ⋆))⩾Pf(A)⩾1−α−min\nt∈G(Pft(A)−Pf(A)). (6.16)\nHere the first inequality follows from the fact that if Ahappens then there exists t∈G⊂Usuch\nthatℓ⩽ft, thereby yielding (as ψt,h(t) = 1):\nℓ(t)⩽ft(t) =f(t)−h1(M⋆\n1−ϵ⋆).\nHence it is enough to bound min t∈G(Pft(A)−Pf(A)). Note that\nmin\nt∈GPft(A)−Pf(A)⩽|G|−1X\nt∈G(Pft(A)−Pf(A))\n=|G|−1X\nt∈GEf\u0012\u0012dPft\ndPf(Y)−1\u0013\nIA(Y)\u0013\n⩽Ef\f\f\f|G|−1X\nt∈G\u0012dPft\ndPf(Y)−1\u0013\f\f\f. (6.17)\nNow by Cameron-Martin-Girsanov’s theorem, we have\nlogdPft\ndPf(Y) =n1/2h1(M⋆\n1−ϵ⋆)q\nΠd\ni=1hi∥ψ∥Xt−n(M⋆\n1−ϵ⋆)2h2\n1(Πd\ni=1hi)∥ψ∥2/2\nwhere\nXt= (Πn\ni=1hi)−1/2∥ψ∥−1ˆ\nψt,hdW\nwith Wbeing the standard Brownian sheet on [0 ,1]d. Note that here Xtfollows a standard normal\ndistribution and for t̸=t′∈G,XtandX′\ntare independent. Now let\nwn:=n1/2h1(M⋆\n1−ϵ⋆)q\nΠd\ni=1hi∥ψ∥.\nAt this point, we need the following lemma (stated and proved in Lemma 6.2 in D¨ umbgen and\nSpokoiny (2001)).\nLemma 6.2. Let{Zn}n≥1be a sequence of independent standard normal variables. If vm:= (1−\nϵm)√2 logmwith limm→∞ϵm= 0andlimm→∞ϵm√logm=∞, then we have\nlim\nm→∞E\f\f\f\f\f1\nmmX\ni=1exp\u0012\nvmZi−v2\nm\n2\u0013\n−1\f\f\f\f\f= 0./Confidence Bands for Shape-restricted Regression 27\nDefine εn:= 1−(wn/p\n2 log|G|). Ifεn→0 and εnp\nlog|G| → ∞ are satisfied, then by Lemma\n6.2 and (6.17), we have the following as |G| → ∞ :\nmin\nt∈GPft(A)−Pf(A)→0. (6.18)\nNow let us choose\nh1:= (1−ϵn)cρnwhere ρn:= (log( en)/n)1/(2+d), (6.19)\nwith ϵn→0 and ϵn√logn→ ∞ andcis a constant to be chosen later. This implies that\np\n2 log|G|= (1−o(1))r\n2d\nd+ 2logn\nand for large n,p\n2 log|G|<q\n2d\nd+2logn. Hence,\nwn= (1−ϵn)(2+d)/2c(2+d)/2∥ψℓ∥(M⋆\n1−ϵ⋆)(2+d)/2\nQd\ni=1(M⋆\ni−ϵ⋆)1/2p\nlog(en).\nLet us now put:\nc:=Qd\ni=1(M⋆\ni−ϵ⋆)1/(2+d)\n(M⋆\n1−ϵ⋆)\u0014(d+ 2)∥ψℓ∥2\n2d\u0015−1/(2+d)\n(6.20)\nwhence we have\nwnp\n2 log|G|∼(1−ϵn)(2+d)/2c(2+d)/2∥ψℓ∥(M⋆\n1−ϵ⋆)(2+d)/2\nΠd\ni=1(M⋆\ni−ϵ⋆)1/2r\nd+ 2\n2d\n= (1 −ϵn)(2+d)/2→1 asn→ ∞ .\nAlso note that for large n, (1−wn/p\n2 log|G|)>0 asp\n2 log|G|<q\n2d\nd+2logn. Further,\np\nlog|G| \n1−wnp\n2 log|G|!\n∼r\nd\n2 +dp\nlogn(1−(1−ϵn)1+d/2)\n=r\nd\n2 +d\u00122 +d\n2+o(1)\u0013\nϵnp\nlogn→ ∞ (by assumption) .\nRecall the definition ∆(ℓ):=\u0000\n(d+ 2)∥ψℓ∥2/2d\u0001−1/(2+d)from the statement of the theorem. Hence,\nusing (6.16), (6.18), (6.19), and (6.20) we have:\n1−α⩽lim inf\nn→∞Pf(∥f−ℓ∥U⩾h1(M⋆\n1−ϵ⋆))\n= lim inf\nn→∞Pf\u0010\n∥f−ℓ∥U⩾(1−ϵn)ρn∆(ℓ)Πd\ni=1(M⋆\ni−ϵ⋆)1/(2+d)\u0011\n⩽lim inf\nn→∞Pf\u0010\n∥f−ℓ∥U⩾γ⋆Πd\ni=1(M⋆\ni−ϵ⋆)1/(2+d)ρn∆(ℓ)\u0011\n(as 1−ϵn→1> γ∗)\n⩽lim inf\nn→∞Pf\u0010\n∥f−ℓ∥U⩾γ⋆(L1[t0]−ϵ)d/(2+d)ρn∆(ℓ)\u0011\n(by (ii) above)\n= lim inf\nn→∞Pf\u0010\n∥f−ℓ∥U⩾γLd/(2+d)\n1 [t0]ρn∆(ℓ)\u0011\n(by (6.13)) ./Confidence Bands for Shape-restricted Regression 28\nThis completes the proof of part (a).\n(b) We again restrict our attention to ( f−ˆℓ)(t0). The other case can be done by a similar argument.\nFori= 1,2, . . . , d , let us recall\nM⋆\ni=∂\n∂xif(x)|x=t0.\nFixϵ >0. As fhas continuous derivative on an open neighborhood Uoft0we can find ϵ⋆>0 and\na hyperrectangle B∞(t0,h0) small enough such that\nsup\nx∈B∞(t0,h0)∂\n∂xif(x)⩽M⋆\ni+ϵ⋆(6.21)\nand\n[Πd\ni=1(M⋆\ni+ϵ⋆)]1/d⩽(1 +ϵ)L1[t0]. (6.22)\nRecall that we have assumed without loss of generality 0 < M⋆\n1⩽M⋆\n2⩽. . .⩽M⋆\nd. Now let\nh= (h1, . . . , h d) be such that hi:=˜h×M⋆\n1+ϵ⋆\nM⋆\ni+ϵ⋆, where ˜hwill be chosen later. Let\nM:=\u0014\nΠd\ni=1M⋆\n1+ϵ⋆\nM⋆\ni+ϵ⋆\u00151/d\n(6.23)\nwhich implies that Πd\ni=1hi=˜hdMd. Recall that\nˆℓ(t0) = sup\nh∈I:t0∈Ah\u001a\nˆfh(t0)−∥ψ∥\n⟨1, ψ⟩(nΠd\ni=1hi)1/2\u0010\nκα+ Γ(2dΠd\ni=1hi)\u0011\u001b\nand\nˆfh(t0) =1\nn1/2(Πd\ni=1hi)⟨1, ψ⟩ˆ\n[0,1]dψt0,h(x)dY(x)\n=1\n⟨1, ψ⟩⟨f(t0+h⋆·), ψ(·)⟩+1\nn1/2(Πd\ni=1hi)⟨1, ψ⟩ˆ\nB∞(t0,h)ψt0,h(x)dW(x)\nwhere for h,x∈Rdwe define h⋆x:= (h1x1, . . . , h dxd).\nNow, it follows from the definition of ˆℓ(t0) that if f(t0)−ˆℓ(t0)⩾(M⋆\n1+ϵ⋆)˜h, then\nˆfh(t0)−∥ψ∥\u0010\nκα+ Γ(2dMd˜hd)\u0011\nn1/2˜hd/2Md/2⟨1, ψ⟩⩽f(t0)−(M⋆\n1+ϵ⋆)˜h,\nwhich can be rewritten as:\n´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2⩽−(n˜hdMd)1/2\n∥ψ∥⟨f(t0+h⋆·)−f(t0) + (M⋆\n1+ϵ⋆)˜h, ψ(·)⟩\n+ Γ(2dMd˜hd) +κα. (6.24)\nLemma 6.3. Suppose h∈(0,1/2]dis such that B∞(t,h)⊆B∞(t0,h0). Then\n⟨f(t+h⋆·)−f(t) + (M⋆\n1+ϵ⋆)˜h, ψ(·)⟩⩾(M⋆\n1+ϵ⋆)˜h∥ψ∥2./Confidence Bands for Shape-restricted Regression 29\nProof of Lemma 6.3: Suppose B∞(t0,h)⊆B∞(t0,h0) holds. Recall that\nψ(x) = \n1 +dX\ni=1xi!\nI \nx⩽0,dX\ni=1xi⩾−1!\n.\nLetx∈[−1,0]dbe such thatPd\ni=1xi⩾−1. Then, by the mean value theorem and (6.21),\nf(t+h⋆x)−f(t) = ( h⋆x)⊤▽f(ξ) (for some ξ∈[t,t+h⋆x])\n⩾dX\ni=1hixi(M⋆\ni+ϵ⋆) (note that h⋆x⩽0)\n=˜h(M⋆\n1+ϵ⋆)dX\ni=1xi.\nHence on the set D:={x⩽0,Pd\ni=1xi⩾−1}, we have\nf(t+h⋆x)−f(t) + (M⋆\n1+ϵ⋆)˜h⩾˜h(M⋆\n1+ϵ⋆) \n1 +dX\ni=1xi!\n⩾0.\nHence, as ψ≥0,\n⟨f(t+h⋆·)−f(t)+(M⋆\n1+ϵ⋆)˜h, ψ(·)⟩⩾ˆ\nD˜h(M⋆\n1+ϵ⋆) \n1 +dX\ni=1xi!2\ndx=˜h(M⋆\n1+ϵ⋆)∥ψ∥2.(6.25)\nThis completes the proof of Lemma 6.3.\nLet us get back to the proof of part (b) of Theorem 7. By (6.24), using (6.25), we get that as long\nasB∞(t0,h)⊆B∞(t0,h0), (f−ˆℓ)(t0)⩾(M⋆\n1+ϵ⋆)˜himplies that\n´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2⩽−√n˜h1+d/2Md/2(M⋆\n1+ϵ⋆)∥ψ∥+ Γ(2dMd˜hd) +κα.\nAlso note that ´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2∼N(0,1).\nNow let us choose\n˜h:=c(M⋆\n1+ϵ⋆)−2\n2+dρn\nwhere ρn= (log( en)/n)1/(2+d)and the constant cis to be chosen later.\nNote that ρn→0 asn→ ∞ . Hence for large enough n,B∞(t0,h)⊆B∞(t0,h0) (here hdepends\nonn). Also we have\nΓ(2dMd˜hd)⩽s\u00122d\n2 +d\u0013\nlog(en) for large n,\n√n˜h1+d/2Md/2(M⋆\n1+ϵ⋆)∥ψ∥=Md/2∥ψ∥c(d+2)/2p\nlog(en)./Confidence Bands for Shape-restricted Regression 30\nNow let us pick\nc:= (1 + ϵ)\u0012(d+ 2)∥ψ∥2\n2d\u0013−1/(d+2)\nM−d\nd+2.\nNote that ∆(ℓ)=\u0010\n(d+2)∥ψ∥2\n2d\u0011−1/(d+2)\nas defined in the statement of the theorem. Hence, for large\nn, we have (with Φ denoting the distribution function of standard normal distribution)\nP\u0010\nf(t0)−ˆℓ(t0)⩾(M⋆\n1+ϵ⋆)˜h\u0011\n⩽P ´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2⩽−√n˜h1+d/2Md/2(M⋆\n1+ϵ⋆)∥ψ∥+ Γ(2dMd˜hd) +κα!\n= Φ\u0010\n−√n˜h1+d/2Md/2(M⋆\n1+ϵ⋆)∥ψ∥+ Γ(2dMd˜hd) +κα\u0011\n⩽Φ \nκα−p\nlog(en)\"\nMd/2∥ψ∥c(d+2)/2−r\n2d\n2 +d#!\n= Φ \nκα−p\nlog(en)r\n2d\n2 +dh\n(1 +ϵ)d+2\n2−1i!\n→0 as n→ ∞ .\nHence,\nlim\nn→∞P\u0010\n(f−ˆℓ)(t0)⩽(M⋆\n1+ϵ⋆)˜h\u0011\n= 1. (6.26)\nNotice that,\n(M⋆\n1+ϵ⋆)˜h= (M⋆\n1+ϵ⋆)c(M⋆\n1+ϵ⋆)−2\n2+dρn\n=ρn(M⋆\n1+ϵ⋆)d/(2+d)(1 +ϵ)∆(ℓ)M−d/(d+2)\n= (1 + ϵ)∆(ℓ)ρn\u0012(M⋆\n1+ϵ⋆)d\nMd\u00131/(d+2)\n= (1 + ϵ)∆(ℓ)ρn\u0010\nΠd\ni=1(M⋆\ni+ϵ⋆)\u00111/(d+2)\n(using (6.23))\n⩽(1 +ϵ)∆(ℓ)ρn(1 +ϵ)d/(d+2)Ld/(d+2)\n1 [t0] (using (6.22))\n= (1 + ϵ)(2d+2)/(d+2)∆(ℓ)ρnLd/(d+2)\n1 [t0].\nAsϵ >0 is arbitrary, the above display along with (6.26) yields the desired result.\n6.8. Proof of Theorem 8\n(a) Once again, we prove only the bound for ∥f−ℓ∥Uand the other case can be handled similarly.\nWe will show that for any level 1 −αconfidence band [ ℓ, u] with guaranteed coverage probability\nfor the class F2, and any 0 < γ < 1, we have\nlim inf\nn→∞Pf\u0012\n∥f−ℓ∥U⩾γ∆(ℓ)Ld\n4+d\n2[t0]ρn\u0013\n⩾1−α.\nWe first introduce a rotation of the coordinate system, so that the Hessian of fatt0with respect\nto this new coordinate system, is diagonal. If ∇2f(t0) =PDP⊤is the spectral decomposition of/Confidence Bands for Shape-restricted Regression 31\nthe Hessian of fatt0(herePdenotes an orthogonal matrix and Dis a diagonal matrix), then for\nany point y∈Rd, we will define y′:=P⊤y, and for any set S⊆Rd, we will define:\nS′:={P⊤s:s∈S}. (6.27)\nFurther, defining\ng(t) :=f(Pt),\nwe note that g(t′) =f(PP⊤t) =f(t) for all t(recall our notation that t′=P⊤t), and ∇2g(t′\n0) =\nD.\nRecall that by assumption, fis twice continuously differentiable on an open neighborhood U\noft0∈(0,1)dsuch that L2[t0]≡L2[f,t0] := det( ∇2f(t0))1/d>0. Hence, gis twice continuously\ndifferentiable on the open neighborhood U′oft′\n0. Denote the ithsmallest eigenvalue of ∇2f(t0) by\nλi(t0), for i= 1, . . . d . Since γ <1, we can find ϵ >0 and γ∗<1 such that:\nγLd/(4+d)\n2 [t0] =γ⋆(L2[t0]−ϵ)d/(4+d). (6.28)\nUsing the twice continuous differentiability of gonU′, we can find h0∈[0,1]dandϵ⋆>0 small\nenough such that\n(i)B′\n∞(t′\n0,h0)⊂U′,\n(ii)hQd\ni=1(λi(t0)−ϵ⋆)i1/d\n⩾L2[t0]−ϵ,\n(iii) for all x′∈B′\n∞(t′\n0,h0), we have\nsup\nv∈B2(0,1)\f\f\fv⊤\u0000\n∇2g(x′)− ∇2g(t′\n0)\u0001\nv\f\f\f< ϵ∗\nwhere B2(0,1) denotes the closed ball around 0∈Rdwith radius 1, and B′\n∞(y,h) :=\n(B∞(Py,h))′; recall the notation in (6.27).\nNext, let h≡(h1, h2, . . . , h d) be such that h1∈(0,1/2] and\nhi=h1×s\nλ1(t0)−ϵ⋆\nλi(t0)−ϵ⋆, fori= 2, . . . , d. (6.29)\nLet us define a set of grid points G⊂[0,1]dfor bandwidth has\nG:=n\nt= (t1, . . . , t d)∈[0,1]d:ti=t0i+αdhi(2ki) for some integer ki, B∞(t, αdh)⊂B∞(t0,h0)o\nwhere t0= (t01, . . . , t 0d) and αd:=p\n2(d+ 3)/(d+ 1). For t′∈G′, let\ngt′(x) :=g(x)−h2\n1(λ1(t0)−ϵ⋆)ψ∗\u0012x1−t′\n1\nh1, . . . ,xd−t′\nd\nhd\u0013\n(6.30)\nwhere\nψ∗(·) := ( GA−G0)(P·) (6.31)/Confidence Bands for Shape-restricted Regression 32\nwith GA,G0defined as follows:\nGA(y) :=∥y∥2\n21∥y∥≤√\n2(d+3)/(d+1)and G0(y) := \n−1 +√\n2(d+ 2)p\n(d+ 1)( d+ 3)∥y∥!\n1∥y∥≤√\n2(d+3)/(d+1).\nNote that ψℓ\n2(x) = (GA−G0)\u0010p\n2(d+ 3)/(d+ 1)x\u0011\n.\nWe will now show that for every t′∈G′, the function gt′∈ F 2.\nLemma 6.4. gt′∈ F 2for all t′∈G′.\nProof. Fixt′∈G′and a vector v∈Bd(0,1). In order to prove Lemma 6.4, it suffices to show that\nthe univariate function h:R→Rdefined as h(s) :=gt′(sv) is convex. To prove this, take scalars\nα > β , such that αvandβv∈B′\n∞(t′\n0,h0), and define\nϕ(x)≡ϕt′,h(x) :=ψ∗\u0012x1−t′\n1\nh1, . . . ,xd−t′\nd\nhd\u0013\n, forx∈[0,1]d.\nThen, we have (using (6.30))\nh′(α)−h′(β)\n= (∇g(αv)− ∇g(βv))⊤v−h2\n1(λ1(t0)−ϵ⋆) [∇ϕ(αv)− ∇ϕ(βv)]⊤v\n= (α−β)h\nv⊤∇2g(ξv)v−h2\n1(λ1(t0)−ϵ⋆)v⊤∇2ϕ(ηv)vi\n(6.32)\nfor some ξ, ηlying between αandβ. First, note that since ξv∈B′\n∞(t′\n0,h0), we have:\nv⊤∇2g(ξv)v≥h\nv⊤∇2g(t′\n0)v−ϵ⋆i\n=dX\ni=1(λi(t0)−ϵ⋆)v2\ni. (6.33)\nHere the inequality above follows from (iii) above and the equality follows from the fact that ∇2g(t′\n0)\nis diagonal. Denote Dh−1to be the diagonal matrix with diagonal entries (1 /hi)d\ni=1. Next, let us\ntry to study the term v⊤∇2ϕ(ηv)v. Note that\nv⊤∇2ϕ(ηv)v=v⊤D⊤\nh−1\u0002\n∇2ψ∗((ηv−t′)/h)\u0003\nDh−1v\n=v⊤D⊤\nh−1P⊤\u0002\n∇2(GA−G0)(P((ηv−t′)/h))\u0003\nPDh−1v\n≤v⊤D⊤\nh−1Dh−1v=dX\ni=1v2\ni\nh2\ni. (6.34)\nHere, the inequality above follows from the fact as long as ∥y∥<p\n2(d+ 3)/(d+ 1),∇2GA(y) is\nthe identity matrix which yields the upper bound. Note that ∇2G0(y) is a positive semidefinite\nmatrix (as G0is a convex function) and hence we can ignore the corresponding term.\nHence, by using (6.32), (6.33) and (6.34) we have:\nh′(α)−h′(β)≥(α−β)\"dX\ni=1(λi(t0)−ϵ⋆)v2\ni−(λ1(t0)−ϵ⋆)dX\ni=1(h1/hi)2v2\ni#\n= 0,\nwhere the equality follows from (6.29). This completes the proof of Lemma 6.4./Confidence Bands for Shape-restricted Regression 33\nLet us get back to the proof of Theorem 8. Recall that [ ℓ, u] is any given confidence band with\nguaranteed coverage. Let us define Ato be the event that\n{ℓ(Px)⩽gt′(x) for all x∈([0,1]d)′,for some t′∈G′}.\nNow, since [ ℓ, u] is a confidence band for all functions in F2, and since the function gt′(P⊤·)∈ F 2,\nwe have\nPgt′(A)⩾1−αfor all t′∈G′.\nHence, defining ℓ∗(x) :=ℓ(Px), we have:\nPg\u0000\n∥g−ℓ∗∥U′⩾h2\n1(λ1(t0)−ϵ⋆)\u0001\n⩾Pg(A)⩾1−α−min\nt′∈G′\u0000\nPgt′(A)−Pg(A)\u0001\n. (6.35)\nNote that the first inequality follows from the fact that if Ahappens then there exists t′∈G′⊂U′\nsuch that ℓ∗(x)⩽gt′(x) on ([0 ,1]d)′, thereby yielding (as ψ∗(0) = 1; see (6.31)):\nℓ∗(t′)⩽gt′(t′) =g(t′)−h2\n1(λ1(t0)−ϵ⋆).\nHence it is enough to bound min t′∈G′\u0000\nPgt′(A)−Pg(A)\u0001\n. Observe that\nmin\nt′∈G′Pgt′(A)−Pg(A)⩽|G′|−1X\nt′∈G′\u0000\nPgt′(A)−Pg(A)\u0001\n=|G′|−1X\nt′∈G′Eg\u0012\u0012dPgt′\ndPg(Y)−1\u0013\nIA(Y)\u0013\n⩽Eg\f\f\f|G′|−1X\nt′∈G′\u0012dPgt′\ndPg(Y)−1\u0013\f\f\f. (6.36)\nNow by Cameron-Martin-Girsanov’s theorem, we have\nlogdPgt′\ndPg(Y) =n1/2h2\n1(λ1(t0)−ϵ⋆)q\nΠd\ni=1hi∥ψ∗∥Xt′−n\n2(λ1(t0)−ϵ⋆)2h4\n1(Πd\ni=1hi)∥ψ∗∥2\nwhere\nXt′:= (Πd\ni=1hi)−1/2∥ψ∗∥−1ˆ\nϕt′,hdW\nwith Wbeing the standard Brownian sheet on [0 ,1]d. Note that Xt′follows a standard normal\ndistribution and for s′̸=t′∈G′,Xs′andXt′are independent. Now let\nwn:=n1/2h2\n1(λ1(t0)−ϵ⋆)q\nΠd\ni=1hi∥ψ∗∥ and εn:= 1−(wn/p\n2 log|G′|).\nIfεn→0 and εnp\nlog|G′| → ∞ are satisfied, then by Lemma 6.2 and (6.36), we have the following\nas|G′| → ∞ :\nmin\nt′∈G′Pgt′(A)−Pg(A)→0. (6.37)\nNow let us choose\nh1:=p\n(1−ϵn)cρn where ρn:= (log( en)/n)2/(4+d), (6.38)/Confidence Bands for Shape-restricted Regression 34\nwith ϵn→0 and ϵn√logn→ ∞ andcis a constant to be chosen later. This implies that\np\n2 log|G′|= (1−o(1))r\n2d\nd+ 4logn\nand for large n,p\n2 log|G′|<q\n2d\nd+4logn. Hence,\nwn= [c(1−ϵn)](4+d)/4∥ψ∗∥(λ1(t0)−ϵ⋆)(4+d)/4\nQd\ni=1(λi(t0)−ϵ⋆)1/4p\nlog(en).\nLet us now put:\nc:=Qd\ni=1(λi(t0)−ϵ⋆)1/(4+d)\n(λ1(t0)−ϵ⋆)\u0014(d+ 4)∥ψ⋆∥2\n2d\u0015−2/(4+d)\n(6.39)\nwhence we have:\nwnp\n2 log|G′|= (1−ϵn)(4+d)/4→1 asn→ ∞ .\nAlso note that for large n, (1−wn/p\n2 log|G|)>0 asp\n2 log|G|<q\n2d\nd+2logn. Also, note that:\np\nlog|G′| \n1−wnp\n2 log|G′|!\n∼r\nd\n4 +dp\nlogn(1−(1−ϵn)1+d/4)\n=r\nd\n4 +d\u00124 +d\n4+o(1)\u0013\nϵnp\nlogn→ ∞ (by assumption) .\nLet Λ(ℓ)\n∗:=\u0000\n(d+ 4)∥ψ∗∥2/2d\u0001−2/(4+d). Hence, by (6.35), (6.37), (6.38) and (6.39), we have:\n1−α⩽lim inf\nn→∞Pg\u0000\n∥g−ℓ∗∥U′⩾h2\n1(λ1(t0)−ϵ⋆)\u0001\n= lim inf\nn→∞Pg\u0010\n∥g−ℓ��∥U′⩾(1−ϵn)ρnΛ(ℓ)\n⋆Πd\ni=1(λi(t0)−ϵ⋆)1/(4+d)\u0011\n⩽lim inf\nn→∞Pg\u0010\n∥g−ℓ∗∥U′⩾γ⋆ρnΛ(ℓ)\n⋆Πd\ni=1(λi(t0)−ϵ⋆)1/(4+d)\u0011\n(as 1−ϵn→1> γ⋆)\n⩽lim inf\nn→∞Pg\u0010\n∥g−ℓ∗∥U′⩾γ⋆(L2[t0]−ϵ)d/(4+d)ρnΛ(ℓ)\n⋆\u0011\n(by (ii) above)\n= lim inf\nn→∞Pg\u0010\n∥g−ℓ∗∥U′⩾γLd/(4+d)\n2 [t0]ρnΛ(ℓ)\n⋆\u0011\n(by (6.28)) .\nNow, note that ∥g−ℓ∗∥U′=∥f−ℓ∥Uand∥ψ∗∥2= (p\n2(d+ 3)/(d+ 1))d∥ψℓ∥2. For bounding\n∥u−f∥U, one works with the transformed kernel (instead of ψ∗in (6.30))\nξu\n2(x) :=\u0012\n1−∥x∥2\n2\u0013\n1∥x∥≤√\n2\nwhich is related to the kernel function ψu\n2by the relation ψu\n2(x) :=ξu\n2(√\n2x). This introduces the\n(√\n2)dterm in ∆(u)and completes the proof of part (a) of Theorem 8./Confidence Bands for Shape-restricted Regression 35\n(b) As in the proof of Theorem 7 (b), we restrict our attention to ( f−ˆℓ)(t0), since the other case\ncan be handled by a similar argument. Fix ϵ >0. For notational convenience, we will abbreviate\nψℓbyψ. Since fis twice continuously differentiable on an open neighborhood Uoft0, we can find\nϵ⋆>0 and a hyperrectangle B∞(t0,h0) small enough such that, for all 1 ≤i≤d,\nsup\nx∈B∞(t0,h0)Hii(x)⩽Hii(t0) +ϵ⋆\nand\n[Πd\ni=1(Hii(t0) +ϵ⋆)]1/d⩽(1 +ϵ)L2,⋆[f,t0]. (6.40)\nNext, let h= (h1, . . . , h d) be such that\nhi:=˜h×s\nH11(t0) +ϵ⋆\nHii(t0) +ϵ⋆, (6.41)\nwhere ˜hwill be chosen later. Let\nM:=\u0014\nΠd\ni=1H11(t0) +ϵ⋆\nHii(t0) +ϵ⋆\u00151/2d\n(6.42)\nwhich implies that Πd\ni=1hi=˜hdMd. Next, as before, recall that\nˆℓ(t0) = sup\nh∈I:t0∈Ah\u001a\nˆfh(t0)−∥ψ∥\n⟨1, ψ⟩(nΠd\ni=1hi)1/2\u0010\nκα+ Γ(2dΠd\ni=1hi)\u0011\u001b\nand\nˆfh(t0) =1\nn1/2(Πd\ni=1hi)⟨1, ψ⟩ˆ\n[0,1]dψt0,h(x)dY(x)\n=1\n⟨1, ψ⟩⟨f(t0+h⋆·), ψ(·)⟩+1\nn1/2(Πd\ni=1hi)⟨1, ψ⟩ˆ\nB∞(t0,h)ψt0,h(x)dW(x)\nwhere for h,x∈Rdwe define h⋆x:= (h1x1, . . . , h dxd).\nRecall that αd:=p\n2(d+ 3)/(d+ 1). Now, it follows from the definition of ˆℓ(t0) that if f(t0)−\nˆℓ(t0)⩾(H11(t0) +ϵ⋆)d˜h2/α2\nd, then\nˆfh(t0)−∥ψ∥\u0010\nκα+ Γ(2dMd˜hd)\u0011\nn1/2˜hd/2Md/2⟨1, ψ⟩⩽f(t0)−(H11(t0) +ϵ⋆)d˜h2/α2\nd,\nwhich can be rewritten as:\n´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2⩽−(n˜hdMd)1/2\n∥ψ∥D\nf(t0+h⋆·)−f(t0) + (H11(t0) +ϵ⋆)d˜h2/α2\nd, ψ(·)E\n+Γ(2dMd˜hd) +κα. (6.43)/Confidence Bands for Shape-restricted Regression 36\nLemma 6.5. Suppose h∈[0,1]dis such that B∞(t,h)⊆B∞(t0,h0). Then\n⟨f(t+h⋆·)−f(t) + (H11(t0) +ϵ⋆)d˜h2/α2\nd, ψ(·)⟩⩾(H11(t0) +ϵ⋆)d˜h2∥ψ∥2/α2\nd.\nProof. Observe that, using (6.41),\n⟨f(t+h⋆·)−f(t) + (H11(t0) +ϵ⋆)d˜h2/α2\nd, ψ(·)⟩\n=(H11(t0)) +ϵ⋆)d˜h2\nα2\nd\u001cf(t+h⋆·)−f(t)\n(Hii(t0) +ϵ⋆)dh2\ni/α2\nd+ 1, ψ(·)\u001d\n(6.44)\nfor every 1 ≤i≤d. Note that the Hessian of the function f(t+h⋆·)−f(t) is given by H⋆∇2f(t+h⋆·),\nwhere H= ((Hi,j)) with Hi,j:=hihjand⋆denotes elementwise product of matrices. As the largest\nentry of a nonnegative definite matrix is always on its diagonal (by Lemma A.1), we next claim\nthat with\nv:= arg max\n1≤i≤d(Hii(t0) +ϵ⋆)h2\ni\nthe function g(·) := ( f(t+h⋆·)−f(t))/(Hvv(t0) +ϵ∗)h2\nvis inH′\n2,1, where we define the superclass\nH′\nβ,L⊇Hβ,Las the set of all functions g: [0,1]d→Rsatisfying (3.1) only. To see this, note that,\nfor any y= (y1, . . . , y d) and z= (z1, . . . , z d)∈Rd, we have the following by a telescopic argument:\ndX\nj=1\f\f\f\f∂g(y)\n∂xj−∂g(z)\n∂xj\f\f\f\f≤dX\nj=1dX\ni=1\f\f\f\f∂\n∂xjg(y1, . . . , y i−1, zi, . . . , z d)−∂\n∂xjg(y1, . . . , y i, zi+1, . . . , z d)\f\f\f\f\n≤ ∥y−z∥1dX\nj=1sup\nu∈L(y,z)max\n1≤i≤d\f\f\f\f∂2\n∂xi∂xjg(u)\f\f\f\f\n≤d∥y−z∥1sup\nu∈L(y,z)max\n1≤i,j≤d\f\f\f\f∂2\n∂xi∂xjg(u)\f\f\f\f(6.45)\nwhere L(y,z) denotes the hyperrectangle defined by the two extreme points yandz. Next we will\nuse the following lemma (proved in Appendix A).\nLemma 6.6. The function ψℓ\n2defined in (2.14) satisfies:\n⟨s, ψℓ\n2⟩ ≥ ∥ ψℓ\n2∥2− ⟨1, ψℓ\n2⟩\nfor all s:Bd→Rwhenever s(0)≥0and the function x→s(x/αd)with support {y:∥y∥ ≤αd}\nis in H′\n2,1.\nWe will take s(·) :=f(t+h⋆·)−f(t)\n(Hvv(t0)+ϵ⋆)dh2v/α2\ndin the above lemma. By (6.45) we see that s(·/αd)∈H′\n2,1\nands(0) = 0. Thus, Lemma 6.5 now follows from Lemma 6.6.\nWe now continue with the proof of Theorem 8 (b). By (6.43) and Lemma 6.5 we get that as long\nasB∞(t0,h)⊆B∞(t0,h0), (f−ˆℓ)(t0)⩾(H11(t0) +ϵ⋆)d˜h2/α2\ndimplies that\n´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2⩽−√n˜h2+d/2Md/2(H11(t0) +ϵ⋆)d∥ψ∥/α2\nd+ Γ(2dMd˜hd) +κα./Confidence Bands for Shape-restricted Regression 37\nAlso note that ´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2∼N(0,1).\nNow let us choose\n˜h:=q\nc(H11(t0) +ϵ⋆)−4\n4+dρn\nwhere ρn:= (log( en)/n)2/(4+d)and the constant cwill be chosen later.\nNote that ρn→0 asn→ ∞ . Hence for large enough n,B∞(t0,h)⊆B∞(t0,h0) (here hdepends\nonn). Also we have\nΓ(2dMd˜hd)⩽s\u00122d\n4 +d\u0013\nlogn for large n\n√n˜h2+d/2Md/2(H11(t0) +ϵ⋆)∥ψ∥=Md/2∥ψ∥c(d+4)/4p\nlog(en).\nNow let us pick\nc:= (1 + ϵ)\u0012(d+ 4)∥ψ∥2\n2d\u0013−2/(d+4)\nM−2d\nd+4α8\n4+d\ndd−4\nd+4. (6.46)\nDefine Λ(ℓ):=\u0010\n(d+4)∥ψ∥2\n2d\u0011−2/(d+4)\n. Hence, for large n, we have\nP\u0010\nf(t0)−ˆℓ(t0)⩾(H11(t0) +ϵ⋆)d˜h2/α2\nd\u0011\n⩽P ´\nB∞(t0,h)ψt0,h(x)dW(x)\n∥ψ∥˜hd/2Md/2⩽−√n˜h2+d/2Md/2(H11(t0) +ϵ⋆)d∥ψ∥/α2\nd+ Γ(2dMd˜hd) +κα!\n= Φ\u0010\n−√n˜h2+d/2Md/2(H11(t0) +ϵ⋆)d∥ψ∥/α2\nd+ Γ(2dMd˜hd) +κα\u0011\n⩽Φ \nκα−p\nlog(en)\"\nMd/2d∥ψ∥c(d+4)/4/α2\nd−r\n2d\n4 +d#!\n= Φ \nκα−p\nlog(en)r\n2d\n4 +dh\n(1 +ϵ)d+4\n4−1i!\n→0 as n→ ∞ .\nThus,\nlim\nn→∞P\u0010\n(f−ˆℓ)(t0)⩽(H11(t0) +ϵ⋆)d˜h2/α2\nd\u0011\n= 1. (6.47)\nNow,\n(H11(t0) +ϵ⋆)d˜h2/α2\nd\n=d(H11(t0) +ϵ⋆)c(H11(t0) +ϵ⋆)−4\n4+dρn/α2\nd\n=ρn(H11(t0) +ϵ⋆)d/(4+d)(1 +ϵ)Λ(ℓ)M−2d/(d+4)dd/(d+4)α−2d\nd+4\nd(by (6.46))\n= (1 + ϵ)Λ(ℓ)ρn\u0010\nΠd\ni=1(Hii(t0) +ϵ⋆)\u00111/(d+4)\ndd/(d+4)α−2d\nd+4\nd(by (6.42))\n⩽(1 +ϵ)Λ(ℓ)ρn(1 +ϵ)d/(d+4)L2,⋆[f,t0]d/(d+4)dd/(d+4)α−2d\nd+4\nd(by (6.40))\n= (1 + ϵ)(2d+4)/(d+4)∆(ℓ⋆)ρnL2,⋆[f,t0]d/(d+4)(6.48)/Confidence Bands for Shape-restricted Regression 38\nAsϵis arbitrary, our assertion is proved by using (6.47) and (6.48). For bounding (ˆ u−f)(t0), one\nagain works with the transformed kernel ξu\n2(x) :=ψu\n2(x/√\n2) to get a result analogous to Lemma\n6.6. This gives rise to the (√\n2)dterm in ∆(u)and completes the proof of part (b) of Theorem 8.\n(c) The proof of part (c) is very similar to that of part (b), so we highlight the differences. To begin\nwith, fix ε >0. Now, as H(t0) is a diagonal matrix and fis twice-continuously differentiable, we\ncan find a hyperrectangle B∞(t0,h0) small enough, such that\nsup\nx∈B∞(t0,h0)|Hij(x)| ≤ε(min 1≤i≤dHii(t0) +ϵ⋆)\ndfor all i̸=j. (6.49)\nAs before, for the function g(x) := ( f(t+h⋆x)−f(t))/(Hvv(t0) +ϵ⋆)h2\nv(cf. (6.44)) note that:\ndX\nj=1\f\f\f\f∂g(y)\n∂xj−∂g(z)\n∂xj\f\f\f\f\n≤dX\nj=1dX\ni=1\f\f\f\f∂\n∂xjg(y1, . . . , y i−1, zi, . . . , z d)−∂\n∂xjg(y1, . . . , y i, zi+1, . . . , z d)\f\f\f\f\n≤dX\nj=1|yj−zj|sup\nu∈L(y,z)\f\f\f\f\f∂2\n∂x2\njg(u)\f\f\f\f\f+dX\nj=1X\ni̸=j|yi−zi|sup\nu∈L(y,z)max\n1≤i≤d\f\f\f\f∂2\n∂xi∂xjg(u)\f\f\f\f\n≤ ∥y−z∥1sup\nu∈L(y,z)max\n1≤i≤d\f\f\f\f∂2\n∂x2\nig(u)\f\f\f\f+dX\ni=1X\nj̸=i|yi−zi|sup\nu∈L(y,z)max\n1≤p̸=q≤d\f\f\f\f∂2\n∂xp∂xqg(u)\f\f\f\f\n≤ ∥y−z∥1 \nsup\nu∈L(y,z)max\n1≤i≤d\f\f\f\f∂2\n∂x2\nig(u)\f\f\f\f+ε!\nwhere L(y,z) denotes the hyperrectangle defined by the two extreme points yandz. The last\ninequality in the above display follows from noticing that\n\f\f\f\f∂2\n∂xp∂xqg(u)\f\f\f\f=\f\f\f∂2\n∂xp∂xqf(t+h⋆u)\f\f\f|hphq|\n(Hvv(t0) +ϵ⋆)h2v⩽ε(min 1≤i≤dHii(t0) +ϵ⋆)\ndp\n(Hpp(t0) +ϵ⋆)(Hqq(t0) +ϵ⋆)⩽ε\nd,\nwhere the second to last inequality follows from (6.41) and (6.49). 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Calibrating the scan statistic: finite sample performance versus\nasymptotics. J. R. Stat. Soc. Ser. B. Stat. Methodol. , 84(5):1608–1639.\nWand, M. P. and Jones, M. C. (1995). Kernel smoothing , volume 60 of Monographs on Statistics\nand Applied Probability . Chapman and Hall, Ltd., London.\nZhang, C.-H. (2002). Risk bounds in isotonic regression. Ann. Statist. , 30(2):528–555.\nAppendix A: Some auxiliary results\nProof of Lemma 6.6 : To begin with, note that ψℓ\n2(x) = (GA−G0)\u0010p\n2(d+ 3)/(d+ 1)x\u0011\n, where\nGA(y) :=∥y∥2\n21∥y∥≤√\n2(d+3)/(d+1)and G0(y) := \n−1 +√\n2(d+ 2)p\n(d+ 1)( d+ 3)∥y∥!\n1∥y∥≤√\n2(d+3)/(d+1).\nWe will now prove the following claim:\nClaim 1. For all g∈H′\n2,1with support Bd:={y:∥y∥ ≤p\n2(d+ 3)/(d+ 1)},GA−gis convex\non the set Bd.\nFor proving Claim 1, it suffices to show that for every v∈Rdsuch thatPd\ni=1vi≥0, the function\nfv:R7→Rdefined as fv(α) :=∥αv∥2\n2−g(αv) is convex. Towards this, note that:\nf′\nv(α) =α∥v∥2−v⊤∇g(αv).\nNow, take any pair ( α, β) such that α < β , and note that:\n\f\f\fv⊤∇g(αv)−v⊤∇g(βv)\f\f\f≤ ∥v∥∥∇g(αv)− ∇g(βv)∥\n≤ ∥v∥dX\ni=1|∇ig(αv)− ∇ ig(βv)|\n≤ ∥v∥∥(α−β)v∥= (β−α)∥v∥2.\nThe last inequality followed from the fact that g∈H′\n2,1. Hence, we have:\nv⊤∇g(βv)−v⊤∇g(αv)≤β∥v∥2−α∥v∥2=⇒f′\nv(α)≤f′\nv(β),\nthereby showing that fvis convex, and completing the proof of Claim 1.\nWith Claim 1 in hand, we are now ready to prove Lemma 6.6. Defining ψ:=GA−G0, we have\nin view of Claim 1 and (6.5), that for any g∈H′\n2,1with support Bd,\n⟨GA−g, ψ⟩ ≤(GA−g)(0)⟨1, ψ⟩\nand hence, we have:\n⟨g, ψ⟩=⟨GA, ψ⟩ − ⟨GA−g, ψ⟩\n≥ ⟨GA, ψ⟩ −(GA−g)(0)⟨1, ψ⟩\n=⟨GA, ψ⟩+g(0)⟨1, ψ⟩\n=∥ψ∥2+⟨G0, ψ⟩+g(0)⟨1, ψ⟩\n=∥ψ∥2+ (g(0)−1)⟨1, ψ⟩/Confidence Bands for Shape-restricted Regression 43\nwhere the last equality followed from the fact that ⟨G0+ 1, ψ⟩= 0, which follows by an argument\nsimilar to the proof of (6.6). Since g(0)≥0, we conclude that:\n⟨g, ψ⟩ ≥ ∥ ψ∥2− ⟨1, ψ⟩\nwhich proves the following claim:\nClaim 2. For all g∈H′\n2,1with support Bd:={y:∥y∥ ≤p\n2(d+ 3)/(d+ 1)}, and ψ:Bd→R\ndefined as: ψ(y) :=ψℓ\n2\u0010q\nd+1\n2(d+3)y\u0011\n, we have:\n⟨g, ψ⟩ ≥ ∥ ψ∥2− ⟨1, ψ⟩\nLemma 6.6 now follows from Claim 2 on observing that ψℓ\n2(y) =ψ\u0010p\n2(d+ 3)/(d+ 1)y\u0011\n.\nLemma A.1. IfA:= (( Ai,j))1≤i,j≤nis an n×nnonnegative definite matrix, then there exists\n1≤i≤nsuch that Ai,i= max 1≤j,k≤nAj,k.\nProof. Suppose that there exists 1 ≤i < j ≤nsuch that Ai,j>max 1≤i≤nAi,i. Note that if ei\ndenotes the vector with the ithentry 1 and all other entries 0, then:\n0≤(ei−ej)⊤A(ei−ej) =Ai,i+Aj,j−2Ai,j=⇒Ai,i+Aj,j≥2Ai,j,\na contradiction! This proves Lemma A.1."    },    {        "title": "2401.12777v2.First_spectral_emissivity_study_of_a_solar_selective_coating_in_the_150_to_600_C_temperature_range.pdf",        "content": "This document is the Accepted Manuscript version of a Published Work that appeared \nin final form in Solar Energy Materials & Solar Cells 117 (2013) 390 –395 after peer \nreview and technical editing by the publisher. To access the final edited and published \nwork see https://doi.org/10.1016/j.solmat.2013.07.002  \n \nFirst spectral emissivity study of a solar selective coating in the 150 –600 \nºC temperature range  \nI. Setién -Fernández a, T. Echániz a, L. Gonzá lez-Fernández a, R.B. Pérez -Sáez a,b,, E. Céspedes \nc, J.A. Sánchez -García c, L. Álvarez -Fraga c, R. Escobar Galindo c, J.M. Albella c, C. Prieto c, M.J. Tello a,b \na Departamento de F ísica de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del \nPaís Vasco, Barrio Sarriena s/n, 48940 Leioa, Bizkaia, Spain  \nb Instituto de Síntesis y Estudio de Materiales, Universidad del País Vasco, Apdo. 644, 48080 Bilbao, \nSpain  \nc Ins tituto de Ciencia de Materiales de Madrid, (ICMM -CSIC), Campus Cantoblanco, 28049 Madrid, Spain  \n \nAbstract  \nA complete experimental study of temperature dependence of the total spectral \nemissivity has been performed, for the first time, for absorber –reflector selective \ncoatings used in concentrated solar power (CSP) systems for energy harvesting. The \ncoating consist of double cermet layers of silicon oxide with different amounts of \nmolybdenum over a silver infrared mirror layer.  The experimental measurements were \ncarried out by a high accurate radiometer (HAIRL) with controlled atmosphere in the \nmid-infrared and for temperatures between 150 and 600 ºC. The spectral emissivity is \nnearly constant in this temperature range. Therefor e, the temperature dependence of \nthe total emissivity is given by Planck function. These results were compared with \nthose obtained with the usual calculus using room temperature re flectance spectrum. \nFinally, the performance of the coating was analyzed by comparison of coated respect \nto non -coated stainless steel.  \n1. Introduction  \nSolar thermal devices are an alternative to produce heat from the sun for heating \nsystems (T <150 ºC) and al so to produce solar thermal electricity (150 ºC < T < 800 \nºC). In these devices both, the thermal energy storage and the solar thermal collectors \n(STC) with different configurations (i.e. flat -plate collector and parabolic concentrated \ncollector), have speci al relevance. In the case of STCs, the solar absorber surface \n(SAS) is the most important part. A surface that facilitates the conversion of solar \nradiation into useful heat should possess two important properties: to absorb the \nincoming solar radiation as  much as possible (i.e. high solar absorptivity, α, at the vis –\nNIR wavelengths) and, at the same time, to retain the collected heat (high thermal \nreflectivity, R, or low emissivity, ε, at NIR –MIR region [1]). The most common type of absorber is based on materials which are b lack in the solar \nradiation range but transparent for the heat, like metal –ceramic nano -composites \n(“cermets”). Among all the existing mechanisms, an absorber –reflector tandem \nconsisting  of small transition metal particles embedded in a dielectric matrix de posited \non a highly infrared reflecting substrate is the most suited method. These thin film \ncoatings offers a high degree of flexibility in order to obtain the desired optical \nproperties to achieve the expected solar selectivity values by changing the thickn ess, \nmetal volume fraction, and the shape of metal nano particles [2]. \nCurrently, most of the commercial SAS are prepared by magne tron sputtering \ntechnology that is a dry, clean and eco -friendly process allowing large area de position \nas compared to the electro chemical methods [3–6]. These SASs are composed of two \nor four homogeneous cermet layers with different metal contents or one cermet layer \nwith a graded refrac tive index [7,8]. The selectivity can be increased adding more \nlayers [9] but, in this case, the price increases and durability decreases [10], being the \ndouble layer cermets the base for the most successful solar selective coatings for \nmedium -high temperature applications [11–14]. \nGenerally the tandem absorbers are degraded  at high operating temperatures due to \ntheir unstable microstructure, which cause a decrease in the solar selectivity (defined \nas α/ε). One of the essential requirements of solar selective absorbers is their stability \nwhen they operate at high temperatures,  from approximately 400 to 600 ºC. Optical \nproperties of th ese coatings should not deterio rate with the rise of the temperature \nduring the period of use.  \nTo accomplish this, new more efficient selective coatings are needed to get both high \nsolar absorptivi ty (α > 0.96) and low thermal emittance (ε < 0.05) at the working \ntemperature range (400 –600 ºC). In fact, for high temperature applications, low ε is the \nkey parameter, because the thermal radiative losses of the absorbers increase \nproportionally to T4 [15]. \nIn order to analyze heat losses, a complete knowledge of the radiative properties of the \ncoating structure is essential for their use in high temperature solar collectors. \nHowever, a systematic study of direct total spectr al emissivity as a function of the \ntemperature for homogeneous cermet of two layers has not been performed yet. For \ninstance, all the measurements reported in the literature were carried out at room \ntemperature or, at most, at 100 ºC. Therefore, the values  of emissivity at working \ntempera tures (400 –600 ºC) are obtained by extrapolation, which can introduce \nsignificant errors in the final result.  \nThis study is focused on the relevance of the high temperature radiometric emissivity \ntechniques in the optical c haracterization of the SAS. In this paper this experimental \ntechnique is applied to study the spectral emissivity behavio r of a coating with double \nlayer cermets of silicon oxide with different amounts of metal. The measurements were \ncarried out using a h igh accurate radiometer with controlled atmosphere in the medium \ninfrared range and for temperatures between 150 and 600 ºC. The results obtained in \nthis study are compared to those obtained with indirect methods.  \n2. Experimental  2 x 2 cm2 plates of stainl ess steel AISI -321 were used as substrates for the coatings. \nThe substrate roughness was measured using a commercial rugosimeter and the \nobtained values are showed in Table 1 , where R a is the roughness average, R z the \naverage maximum height and Rt the maximum height of the profile.  \nTable 1  \nSample s urface roughness.  \nRa (µm) Rz (µm) Rt (µm) \n0.13 0.87 2.14 \nThe selective solar coating was prepared by sequential sputter ing deposition on steel \nsubstrate, air -annealed during 2 h to develop a therma lly grown oxide barrier layer. \nHigh purity silver, and molybdenum targets were sputtered with Ar gas at  5 x 10-3 mbar \nwith RF and DC powers of than 25 W and 1 W to obtain deposition rates of 10 a nd 1.7 \nnm min-1 respectively. Moreover, pure silicon target was also sputtered with a 10% \nO2/Ar gas mixture at 5 x 10-3 mbar by applying a RF power of 100 W to form silicon \noxide layers with a deposition rate of 7 nm min-1. The selective solar coating, pre pared \nusing these growing conditions, is schematically represented in Fig. 1 . From the \nbottom, the deposited stack is formed by four layers: (i) 250 nm thick silver  layer acting \nas IR -mirror, (ii) 85 nm thick layer of high metal volume fraction (HMVF) cermet \ncomposed by Mo and SiO2 with 50% filling factor, (iii) 85 nm thick layer of low metal \nvolume fraction (LMVF) cermet composed by Mo and SiO2 with 20% filling factor and, \non top,  (iv) 53 nm thick antireflective layer of SiO2.  \n \n \nFig. 1. Schematic representation of the selective solar coating used for the emissivity \ncharacterization.  \nFig. 2  shows the measured reflectivity at the UV –vis–IR wavelength range to illustrat e \nthe selective character of such a multilayer coating. UV –vis–IR reflectivity \nmeasurements were performed using both a Shimadzu SolidSpec -3700 \nspectrophot ometer, in the range of 0.19 –3.30 µm, and a Varian 660 -IR FTIR \nspectrometer in the 2.5 –25 µm wavelen gth range. It can be easily observed the abrupt \nchange in the reflectivity spectrum (R(λ)) from very low values at the UV –vis region to \nvery high ones at the IR range, which makes possible to obtain a  total solar absorptivity \nof α = 0.9, and a total thermal  emissi vity at room temperature of ε  = 0.02. These values \nhave been calculated by the well known expressions (1) and (2), using the measured \nnear normal reflec tivity R(λ) in good approximation o f the angle dependent R(λ,θ)  \n \nwhere A(λ) is the Solar emission ASTM G173 -03 Reference Spec trum (AM1.5) and \nL(T, λ) the Planck function.  \n \nFig. 2. UV –vis–IR re flectivity of the selective solar coating of Fig. 1 . \nThe spectral emissiv ity measurements were carried out using a high accuracy infrared \nradiometer (HAIRL) described in Ref. [16], which allows accurate signal detection and \nfast processing. A diaphragm adjusts the sample area viewed by the detecto r and \nensures good temperature homogeneity of the sample measured area. The sample \nholder permits directional measurements, while the sample chamber ensures a \ncontrolled atmosphere (vacuum, inert gas or open atmosphere). The set -up calibration \nwas carried out by using a modified two -temperature method [17] and the emissivity \nwas obtained applying the blacksur method [18]. The combined standard uncertainty of \nthis direct emissivity device was previ ously obtained from the analysis of all uncertainty \nsources [19]. For the measurements presented in this paper, the maximum combined \nstandard uncertainty varies between 1% and 9% depending on wavelength and \ntemperature, its a verage value being around 3.5%. The sample temperature is \nmeasured by means of two K -type thermocouple spot -welded on the sample surface \nout of the area viewed by the detector. Before placing the sample in the sample holder \nits surface is cleaned in an ult rasonic bath of acetone. Once the sample is introduced in \nthe sample chamber, the measurements were carried out in a moderate vacuum or \nwith a slightly reducing atmosphere in order to prevent the oxidation of the sample \nsurface. The measurements are perfor med during five heating cycles between room \ntemperature and nearly 700 ºC. For each heating cycle the emissivity is measured at  \nsix or seven  temperatures. It is interesting to note that in order to ensure the thermal \nequilibrium for each temperature, the time require d for the measurements of a \ncomplete thermal cycle is one day.  \nIn addition, direct emissivity measurements of the samples were obtained at 82 ºC \nusing an emissometer model AE1 from Devices & Service Company.  \n3. Results and discussion  \nThe following sections  show the results and discussion of the total and spectral \nemissivity measurements obtained for the steel substrate ( Section 3.1 ) and for the \ndeposited selective coating  (Section 3.2 ). \n3.1. Total and spectral emissivity measurements of the steel substrate  \nThe spectral emissivity ε(λ, T) measurements were carried out on the steel substrate \nfor five consecutive hea ting cycles in a slightly reduc ing atmosphere. As shown in Fig. \n3 there are no significant differences in the val ues of the normal spectral emis sivity \nbetween the first and fifth heating cycle for this steel.  Therefore, these experimental \nresults confirm that the substrate is free of possible surface tensions [20]. In addition, \nthe substrate  surface was analyzed after the five heating cycle by means of X -ray \ndiffraction and no signs of oxidation were found.  \n \nFig. 3. Normal spectral emissivity ε(λ, T) for substrate at 700 ºC as a function of \nwavelength for the first and fifth heating cycles.  \nFig. 4  shows the normal spectra l emissivity for seven tempera tures,  between 173 and \n688 ºC, during the fifth heating cycle. According the electromagnetic theory, the \nemissivity decreases as wavelength increases [21]. However, it c an be observed that \nthe behavio r of the emissivity with temper ature undergoes a change around λ = 3 µm. \nThis is the so -called X point for the AISI -321 SS  sample. Above the X point the \nemissivity shows a sligh t tempera ture increase with an almost linear dependence, \nwhereas for λ≤3 µm the emissivity decreases with temperature until T = 350 ºC. \nExperimental results in Fig. 4  show that for λ = 10 µm emissivity increases a 20% \nbetwe en 173 and 688 ºC. This result  should be taken into account in the final design of \nthe coated steel system.  \n \nFig. 4. Substra te normal spectral emissivity ε(λ, T) for the fifth heating cycle at  different \ntemperatures.  \nThe temperature dependence of the total normal emissivity ε T (T), given by Eq. (3), is \nneeded in order to calculate the  radiation energy losses of a solar collector. However, \ncalculations of this parameter have not been made at the typical working tempera tures \nof solar collectors (T≥500 ºC). \n \nThe difference with Eq. (1) is the measured temperature  dependence for the spectral \nemissivity. The integration of Eq. (3) requires to know the emissivity for wavelengths \nbetween 0 and 1, but we ha ve experimental data between λ =  1.42 mm and λ  = 22 mm. \nIn this case the integral ca n be calculated from the following equation:  \n \nThus, the total normal emissivity depends on the values of the spectral emissivity used \nas ε 1(λ,T) and ε 2(λ,T) in Eq. (4). To estimate the maximum range of variation of the total \nnormal emissivity two extreme cases have been taken into account. For the highest \nvalue, ε 1(λ,T) = 1 is taken and for ε 2(λ,T) the value of the normal spectral emissivity at \nλ2 is extr apolated. The lowest value is obtained with ε 2(λ, T) = 0 and taking as ε 1(λ, T) \nthe value of the normal spectral e missivity at λ 1. The values for the two limits of the \ntotal normal emissivity for T = 655 ºC are 0.2186 and 0.2163 respectively,  and the \naverage value is 0.217 ±0.001. In Fig. 5  the total normal emissivity is plotted as a \nfunction of temperature. As it can be not ed, it shows the linear behavio r predicted by \nthe electro - magnetic theory for metals.  \nSince the first layer of the selective coating is a steel thermally oxidized, an in situ \nthermal oxidation of the substrate was carr ied out in order to study the variation of the \nsteel emissivity with the oxide thickness. Fig. 6  shows the normal spectral emissivity \nduring the oxidation process of the substrate at 600 ºC for oxidation times up to 22 h. \nAs expected, the emissivity increase with the oxidation time. In addition, it can be \nobserved the first interferential maximum and minimum associated to the growth of the \noxide layer for t > 10 h [22]. The emissivity behavio r of this steel is similar to other \nmetals [22]. \n \nFig. 5. Total normal emissivity ε T(T) of substrate as a function of temperature.  \n \nFig. 6. Spectral normal emissivity ε(λ) of the substrate for various times dur ing the \noxidation process in air at 600 ºC. The first interferential maxima and minima can be \nobserved.  \n3.2. Spectral emissivity measurements of the selective solar coating  \nThe emissivity measurements of the coating were carried out between 150 and 600 ºC \nand under moderate vacuum ( ∼10-3 mbar), according to the working conditions in the \nsolar collectors. The first step has been the study of the emissivity during heating with a \nmaximum heating rate of 2 ºC min-1. In the measurement method used in this study the \ntemperature is stabilized during 20 min after every 30 ºC step approximately, in order to \nmeasure the emissivity. Fig. 7  shows the emissivity values  as a function of the heating \ncycle for two temperatures and four wavelengths . It can be observed that the emissivity \nvariations between the first and fourth heating cycle are lower than the experimental \nuncertainty. It was then checked that, at this heating rate, the coating remains \nunchanged. The results for all wave - length and temperature ranges suggest that the \ncoating is stable over the li fe of the solar collector. In order to verify this statement, \nfurther tests to simulate one year operation are underway, using a programmed heating \nsystem.  \n \nFig. 7. Coating emissivity ε(λ, T) as a function of the heating cycle for 4, 8, 14 and 20 \nµm wavelengths for (a) T = 320 ºC and (b) T = 600 ºC. \nSince the operative heating rate in a solar collector is around 10 ºC min-1, the emissivity \nwas meas ured as a function of the heating rate in a range between 2 and 10 ºC min-1 in \norder to analyze the stability of the optical properties of the SASs The results showed \nemissivity variati ons smaller than the experimen tal uncertainty. Therefore, it can be \nstated that the coating emissivity is independent of the heating rate, which is a critical \nfeature of the coating in order to its application in a real solar plant.  \nFig. 8  shows the normal spectral emissivity for eight tempera tures between 236 and \n600 ºC during the fourth heating cycle. It can be noticed that these emissivit y spectra \nshow the same behavio r of the bare steel substrate, with lower emissivity values ( Fig. \n4). In this case the point X is also observed around 3 µm. From the comparison of the \nnormal spectral emissivity of the substrate and coating ( Figs. 4  and 8) it is concluded \nthat the emissivity of the coating is significan tly lower than that of the substrate, in the \nmost important range for thermal radiation (λ  > 2 µm), between 10% and 30% for low \nand high wavelength respectively. In addition, one can state that this coating presents \nthe optimal emissivity values required in high temperature solar collectors, ε  < 0.05 for \nλ > 10 µm. \n \nFig. 8. Normal spectral emissivity ε(λ, T) of selective coating for eight different \ntemperatures measured in the fourth heating cycle.  \nFig. 9  represents the emissivity as a function of temperature for four different \nwavelengths. The emissivity shows an almost negligible decrease with temperature \nexcept f or small wave lengths. For this spectral range, the emissivity decreases slightly \nuntil 450 ºC and remains constant up to 600 ºC, showing a good performance for high \ntemperature solar harvesting.  \n \nFig. 9. Normal spectral emissivity ε(T) of the selective coating measured at the fourth \nheating cycle as a function of temperature for four different wavelengths.  \nIn order to calculate the radiation energy losses of a sola r collector, it is necessary to \nknow the total normal emissivity.  Fig. 10  shows the total normal emissivity obtained for \nthe c oated SAS sample using three different experimental methods. In first place, the \nemissivity value ob tained with a commercial emiss ometer operating at 82 ºC is \nincluded (black star in Fig. 10 ). Secondly, the emissivity values  obtained from the \nreflectance spectrum measured at room temperature (full triangles in  Fig. 10 ) are \nplotted. In this case, the typical procedure to estimate emissivity at high temperatures \nusing Eq. (2) has been applied, where the temperature depende nce comes from the \nPlanck func tion and the re flectance spectra, R(λ), is considered temperature - \nindependent. In agreement with the discussion of Eq. (4) in Section 3.1 , the integration \nlimits can be taken from 1 to 30 µm. Furthermore, it m ust be mentio ned that the \nreflec tometer refer ence produces small errors in the absolute reflectance values. To \navoid this uncertainty, the reflectance spectrum in the 1.5 –30 µm range has been \nshifted to coincide with the emissometer value at 82 ºC. Finally,  a rigorous calculation \nof the total emissivity was carried out for the first time using Eqs. (3) and (4) (close \ncircles in Fig. 10 ). In this case, both Planck funct ion and experimental normal spectral \nemissivity are temperature dependent. In the same figure, the total normal emissivity of \nthe substrate is plotted for comparison (full squares).  \n \nFig. 10. Total normal emissivity ε T(T) of SAS obtained from radiometric measure ments \n(full circles), from reflectivity measurements (full triangles) and with an emi ssometer at \n82 ºC (open circle). The substrate results are also plotted (full squares).  \nThese results shown in Fig. 10  allow an inte resting analysis of this type of selective \ncoatings as well as a comparison between the experimental measurements methods \nused in this paper. First, it is important to note that the temperature dependence of the \ntotal normal emissivity obtained from the re flectivity at room tempera ture sho ws the \nsame qualitative behavio r of the obtained using the normal spectral emissivity, which is \ntemperature dependent ( Fig. 10 ). Only small quantitative differences are appreciated in \nFig. 10 . The close agreement between radiometric and reflectance measurements in \nFig. 10  is caused by the very weak temperature dependence of the spectral emissivity \nof the SAS (see Fig. 9 ). Just this point justifies the interest of measuring the \ntemperature dependence of the normal and, in some cases, also the directional \nspectral emissivity. In addition, it is essential, for solar selective coatings applications to \nasses s the emissivity changes with heating cycles and heating rate, which can only be \nproven with spectral measurements at different temperatures. Another important \naspect for the applications that can be studied by means of radiometric methods is the \ndetection  of anomalous behavio r of the SAS with the temperature and wavelength. \nFinally, if a coating has a spectral emissivity with a significant decrease with \ntemperature, the total emissivity calculated from Eq. (3) will be lower than those \ncalculated from Eq. (2) on the basis of the same value at room temperature. This is \nbecau se the major temperature depen dence of the emissivity obtained from Eq. (2) \narises from the Pl anck function. Thus, all the curves obtained from this equation will \nhave similar temperature behavio r modulated by the shape of the reflectance \nspectrum.  \nThe results of Fig. 10  also suggest some comments about the coating it self. Firstly, the \ntotal normal emissivity of coated sample shows a slight curvature with the temperature \nwhile the substrate has a quasi -linear behavio r according to electromagnetic theory  \npredictions. Secondly, the difference between the experimental v alues of the total \nnormal emissivity between the substrate and the coating shows the effectiveness of the \nlatter and indicates that it has an appropriate thickness. Finally, it is important to note \nthat the results in Fig. 9  together with the design flexibility and thermal stability of the \nSAS open up the possibility to compensate the Planck function shift with temperature. \nExperimental work in this direction is underway in our laboratory.  \n4. Conclusions  \nIn this paper, it is pr esented, for the first time, a complete radiometric characterization \nof a selective absorber surface by using absolute measurements of spectral emissivity \nat the whole working temperature range (150 –600 ºC). The total emissivity values \nobtained from spectra l emissivity measurements are com - pared with those obtained \nfrom reflectivity data and with direct characterization by a commercial emissometer at \n82 ºC. This spectral emissivity method al lows to know the act ual behavio r of coated \nstainless steel system a t the operation temperatures for CSP applications as it is the \nunique test to detect any anomalous behavio r of the coating with temperature and \nwavelength. Only in the case where the spectral emissivity does not vary with \ntemperature within the wavelength  range in which the radiative transfer is made, a \nqualitative agreement between reflectivity measurements and the radiometer could be \nacceptable.  \nAs expected, for a selective coating, the results show that coated stainless steel \npresent a considerable lower  emissivity when compared with bare stainless steel. It is \nworth noting, the assessment of the emissivity stability with temperature and the \nthermal cycling under different heating rates, which ensures the performance for the \nrequired application.  \nAcknow ledgements  \nThis work was financially su pported by the European Commis sion (project HITECO \nFP7-ENERGY -2010 -ºCollaborative N. 256830) the Spanish Ministry of Science and \nInnovation (projects FUNCOAT CSD2008 -00023, RyC2007 -0026) and program \nETORTEK of the C onsejería de Industria of the Gobierno Vasco in collaboration with \nthe CIC -Energigune Research Center. L. González -Fernández acknowledges the \nBasque Government the support through a Ph.D. fellowship.  \nReferences  \n[1] C.E. Kennedy. Review of Mid -to-High-Temper ature Solar Selective Absorber \nMaterials. NREL/TP -520-31267. Available from: 〈http://www.nrel.gov/docs/  \nfy02osti/31267.pdf 〉. \n[2] G.A. Niklasson, C.G. Granqvist, Selective solar -absorbing surface coatings:  optical \nproperties and degradation, in: C.G. Granqvist (Ed.), Materials Science  for Solar \nEnergy Conversion SystemsPergamon Press, Oxford, 1991, p. 70.  \n[3] M. Nejati. Cermet based solar selective absorbers; further selec tivity improvement \nand developing new fabrication technique, Ph.D. -Dissertation, Saarbrücken, 2008.  \n[4] O.T. Inal, A. Sherer, Optimization and microestructural analysis of electro - \nchemically deposited selective solar absorber coatings, Journal of Materials  Science 21 \n(1986) 729 –736. [5] L. Katulza, A. Surca -Vuc, B. Orel, Structural and IR spectroscopy analysis of sol –\ngel processed CuFeMnO4  spinel and CuFeMnO4 /silica  films for sol ar absorbers, \nJournal  of Sol –Gel Science and Technology 20 (2001) 61 –83. \n[6] Z.C. Orel, M.K.Gunde, Spectral se lective paint coatings preparation and  \ncharacterization, Solar Energy Materials and Solar Cells 68 (2001) 337 –353. \n[7] M. Farooq, M.G. Hutchins, A novel design in composites of various materials in  \nsolar selective coatings, Solar Energy Materials and Solar Cells 71 (2002)  523–535. \n[8] I.T. Ritchie, B. Window, Application of thin graded index  films to solar  absorbers, \nApplied Optics 16 (1977) 1438 –1443.  \n[9] M.R. Nejati, V. Fathollahi , M.K. Asadi, Computer simulation of the optical  \nproperties of high temperature cermet solar selective coatings, Solar Energy  78 (2005) \n235–241. \n[10] B.O. Seraphin, Thin films in phototermal solar energy conversion, Thin Solid  \nFilms 90 (1982) 395 –403. \n[11] N. Selvakumar, H.C Barshilia, Review of physical vapor deposited (PVD)  \nspectrally selective coatings for mid - and high -temperature solar thermal  applications, \nSolar Energy Materials and Solar Cells 98 (2012) 1 –23. \n[12] Q.-C. Zhang, Recent progress in high -temperature solar selective coatings,  Solar \nEnergy Materials and Solar Cells 62 (2000) 63 –74. \n[13] S. Esposito, A. Antonaia, M.L. Addonizio, S. Aprea, Fabrication and optimisation  \nof highly efficient cermet -based spe ctrally selective coatings for high operat - ing \ntemperature, Thin Solid Films 517 (2009) 6000 –6006.  \n[14] C.E. Kennedy, H. Price, Progress in development of high -temperature solar - \nselective coatings, NREL/CP -520-36997. in: Proceedings of the 2005 Interna - tional \nSolar Energy Conference, ISEC'05, Orlando, Florida, USA, ISEC2005 - 76039, August \n6–12, 2005.  \n[15] C.G. Gran qvist, Transparent conductors as solar energy materials: a panoramic  \nreview, Solar Energy Materials and Solar Cells 91 (2007) 1529 –1598.  \n[16] L. del Campo, R.B. Pérez -Sáez, X. Esquisabel, I. Fernández, M.J. Tello, New  \nexperimental device for infrared spectral direction emissivity measurem ents in a \ncontrolled  environment,  Review  of Scientific  Instruments  77 (2006)  113111 . \n[17] L. González -Fernández,  R.B. Pérez -Sáez,  L. del Campo,  M.J. Tello,  Analysis  of \ncalibration  methods  for direct  emissivity  measurements,  Applied  Optics  49 (2010) \n2728 –2735 . \n[18] R.B. Pérez  Sáez,  L. del Campo,  M.J. Tello,  Analysis  of the accuracy  of methods  for the direct  measurement  of emissivity,  International  Journal  of Thermo - physics  29 \n(2008) 1141 –1155 . \n[19] L. del Campo,  R.B. Pérez -Sáez,  L. González -Fernández,  M.J. Tello,  Combined  \nstandar d uncertainty  in direct  emissivity  measurements,  Journal  of Applied  Physics 107 \n(2010) 113510 . \n[20] L. Gonzal ez-Fernández,  E. Risueño,  R.B. Pérez  Sáez,  M.J. Tello,  Infrared  normal  \nspectral  emissivity  of Ti–6Al–4V alloy in the 500–1150  K temperature  range,  Journal of \nAlloys and Compounds 541 (2012)  144–149. \n[21] M.F. Modest,  Radiative  Heat  Transfer,  2nd ed., Academia  Press,  California,  \n2003,  pp. 76-84. \n[22] L. del Campo,  R.B. Pérez -Sáez,  M.J. Tello,  X. Esquisabel,  I. Fernández,  Armco  \niron normal spectral emissivity measurements, International Journal of  Thermophysics \n27 (2006) 1160 –1172 . "    },    {        "title": "2401.12797v1.A_New_Precise_Determination_of_the_Primordial_Abundance_of_Deuterium__Measurement_in_the_metal_poor_sub_DLA_system_at_z_3_42_towards_quasar_J1332_0052.pdf",        "content": "MNRAS 000, 1–12 (2023) Preprint 24 January 2024 Compiled using MNRAS L ATEX style file v3.0\nA New Precise Determination of the Primordial Abundance of Deuterium:\nMeasurement in the metal-poor sub-DLA system at 𝑧=3.42towards\nquasar J1332+0052★\nP.A. Kislitsyn1†, S.A. Balashev1, M.T. Murphy2, C. Ledoux3, P. Noterdaeme4,5and A.V. Ivanchik1\n1Ioffe Institute, Polyteknicheskaya 26, 194021 Saint-Petersburg, Russia\n2Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia\n3European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001, Santiago, Chile\n4Franco-Chilean Laboratory for Astronomy, IRL 3386, CNRS and U. de Chile, Casilla 36-D, Santiago, Chile\n5Institut d’Astrophysique de Paris, CNRS-SU, UMR7095, 98bis bd Arago, 75014 Paris, France\nAccepted 2024 January 18. Received 2024 January 18; in original form 2023 September 30\nABSTRACT\nThe theory of Big Bang nucleosynthesis, coupled with an estimate of the primordial deuterium abundance (D/H) pr, offers\ninsights into the baryon density of the Universe. Independently, the baryon density can be constrained during a different\ncosmological era through the analysis of cosmic microwave background (CMB) anisotropy. The comparison of these estimates\nservesasarigoroustestfortheself-consistencyoftheStandardCosmologicalModelandstandsasapotenttoolinthequestfor\nnew physics beyond theStandard Model of Particle Physics. Fora meaningful comparison, a clear understandingof the various\nsystematic errors affecting deuterium measurements is crucial. Given the limited number of D/H measurements, each new\nestimatecarriessignificantweight.ThisstudypresentsthedetectionofD iabsorptionlinesinametal-poorsub-DampedLyman-\n𝛼system([O/H]=−1.71±0.02,log𝑁(Hi)=19.304±0.004)at𝑧abs=3.42towardsthequasarSDSSJ133254.51+005250.6.\nThrough simultaneous fitting of H iand D iLyman-series lines, as well as low-ionization metal lines, observed at high spectral\nresolution and high signal-to-noise using VLT/UVES and Keck/HIRES, we derive log(Di/Hi)=−4.622±0.014, accounting\nfor statistical and systematic uncertainties of 0.008dex and 0.012dex, respectively. Thanks to negligible ionization corrections\nand minimal deuterium astration at low metallicity, this D/H ratio provides a robust measurement of the primordial deuterium\nabundance, consistent and competitive with previous works. Incorporating all prior measurements, the best estimate of the\nprimordial deuterium abundance is constrained as: (D/H) pr=(2.533±0.024)×10−5. This represents a 5% improvement in\nprecision over previous studies and reveals a moderate tension with the expectation from the Standard Model ( ≈2.2𝜎). This\ndiscrepancy underscores the importance of further measurements in the pursuit of new physics.\nKey words: primordial nucleosynthesis - cosmological parameters - quasars: absorption lines – ISM: clouds.\n1 INTRODUCTION\nPrimordial nucleosynthesis is one of the pillars of the Big Bang\ncosmology that provides the earliest robust observational constraint\non the Standard Cosmological Model. The theory of primordial nu-\ncleosynthesis predicts the values of the relative abundances of light\nnuclei such as H, D,3He,4He, and7Li. According to the Standard\nModel,theseabundancesdependonlyonthebaryon-to-photonratio,\n𝜂b≡𝑛b\n𝑛𝛾(where𝑛b,𝑛𝛾– the number densities of baryons and pho-\ntons, respectively), that is linearly dependent on the baryon density,\nΩb.However,someofthesenucleiaresensitivetoeffectsbeyondthe\nStandard Model. As an example, the primordial helium abundance\n★Based on archival data collected at the European Southern Observatory\nunder programme 0101.A-0061(A) and W.M. Keck Observatory under pro-\ngrammes U152Hr, U088Hb, and U088Hr.\n†E-mail: pavel.kislitsyn@gmail.comcould provide better constraints on the effective number of neutrino\nspecies(Kurichinetal.2021b).Thisdrawsalotofattentiontothede-\nterminationoftheprimordialheliumabundancesanditsestimations\naresteadilyimproving(e.g.,Izotovetal.2014;Fernándezetal.2018;\nHsyuetal.2020;Averetal.2021;Kurichinetal.2021a;Valerdietal.\n2021; Matsumoto et al. 2022). Despite growing observational con-\nstraints on helium, deuterium plays a key role in the modern era of\nprecisecosmologysinceithasthestrongestsensitivityon Ωbamong\nthe other elements and its abundance can be accurately constrained\nusing absorption-line measurements along quasar sightlines.\nWith the advent of high-resolution spectrographs on the largest\noptical telescopes, for more than two decades D/H measurements\nwere derived from the analysis of hydrogen and deuterium absorp-\ntion lines in quasar spectra (Tytler et al. 1996; Noterdaeme et al.\n2012; Cooke et al. 2014, 2018). The first results on D/H possessed\na high dispersion, the reasons of which could be explained by sys-\n©2023 The AuthorsarXiv:2401.12797v1  [astro-ph.CO]  23 Jan 20242Kislitsyn P. et al.\ntematic effects in the analysis or could have physical origins. This\nhas motivated the search for criteria that minimize the systematics\nandprovidethemostpreciseD/Hmeasurements(Cookeetal.2014).\nThemaincriteriaareasimplevelocitystructurefortheabsorber,low\ndeuterium astration and depletion on the dust (Cooke et al. 2018),\nwhich are all satisfied in case of very low metallicities. Therefore,\nonly a few D/H estimations are considered to date to provide robust\nresults (Cooke et al. 2018). Each new estimate is of utmost impor-\ntance, for various combinations of parameters (e.g., metallicity, HI\ncolumndensity,simplicityofthevelocitystructure,redshift),thatal-\nlow to disentangling the influence of the aforementioned systematic\neffects on the measurement of D/H. Such measurements are espe-\ncially important at the high-metallicity end, which can potentially\nincrease the number of suitable targets, especially in the advent of\nnext-generation spectrographs such as CUBES (Evans et al. 2022).\nIn this paper, we present the detection and analysis of\nDilines in the sub-DLA system at 𝑧abs=3.42towards\nSDSS J133254.51+005250.6. This provides a new robust determi-\nnation of the D/H ratio at high redshift.\n2 OBSERVATIONS AND DATA REDUCTION\n2.1 VLT/UVES observations\nSDSSJ133254.51+005250.6(hereafterJ1332 +0052),a𝑧em=3.51,\n𝑉=18.7quasar (a.k.a. Q1330 +0108; Véron-Cetty & Véron 2010)\nwas observed in service mode between April and July 2018 using\nthe Very Large Telescope (VLT) Unit-2, Kueyen, equipped with the\nUltraviolet and Visual Echelle Spectrograph (UVES; Dekker et al.\n2000).Twelveexposuresof3000seachweretakeninDichroicmode\noversevendifferentnights.Thisresultedin24individualspectracov-\nering most of the optical range. Information about instrumental se-\ntupsandobservingconditionsisgiveninTable1.Entrance 1′′-wide\nslits and 2×2pixel binning were used throughout. The observa-\ntions were carried out in dark time (no moon) under clear skies and\nexcellentseeingconditions(seetable1).Foreachexposure,wepro-\ncessedscienceandcalibrationdatausingtheUVESpipelinev6.1on\nthe EsoReflex platform v2.11 (Freudling et al. 2013). Intermediate\ndata products were inspected carefully and data-reduction parame-\nterswereoptimisedstepbystep.Thenominalresolvingpowerofthe\nindividual spectra is 50,000 in the Blue (48,800 in the Red) but can\nbeslightlylargerthanthatwhentheseeingconditionsarebetterthan\n1′′FWHM. The actual resolving power of the combined spectrum\nused in the analysis is determined in Sect. 3.2. Individual exposures\nin each blue/red setup were co-added optimally and the overlapping\nregions of the resulting spectra were stitched together to create the\nfinal data product. The final UVES spectrum has a fairly high S/N\nratio per pixel, peaking at ≈30in the quasar continuum around the\nmost important D ilines.\n2.2 Keck/HIRES observations\nIn addition to the UVES spectra, we used archival spectra from the\nW.M. Keck Observatory using the High-Resolution Echelle Spec-\ntrometer (HIRES; Vogt et al. 1994). The journal of observations is\ngiven in Table 2. The nominal resolving power of the individual\nexposures is 47,700.\nTo reduce the data (eight separate exposures) and combine the\nextracted spectra, we used the approach described in Robert et al.\n(2018).Tosummariseitbriefly,thedatawerereducedandthequasarTable 1.VLT/UVES observations of J1332 +0052\nUT date exp. time setupaairmass IQb\n[s] [nm] [′′]\n09-04-2018 2×3000437+760 1.27 0.83\n13-04-2018 4×3000437+760 1.12/1.16 0.55/0.6\n18-05-2018 2×3000390+564 2.02 1.24\n12-06-2018 4×3000390+564 1.23/1.44 0.98/1.05\n13-06-2018 6×3000437+760 1.19/1.36/1.70 0.78/1.08/1.22\n15-06-2018 4×3000437+760 1.11/1.14 1.09/1.17\n09-07-2018 2×3000390+564 1.70 1.23\nacentral wavelengths of the blue-arm (left) and red-arm (right) spectra.\nbimage quality at 𝜆=650nm measured by the telescope Shack-Hartmann\nwavefront sensor.\nTable 2.Keck/HIRES observations of J1332 +0052\nUT date exp. time wavelength range airmass decker\n[s] [nm]\n07-04-2006 2×3600 424-869 1.08/1.06 C1\n12-04-2015 4×3600 392-686 1.06 - 1.20 C1\n14-05-2015 3288+2678 424-869 1.15/1.07 C1\nspectra extracted using the makee1package, including corrections\nfor the blaze function using flat-field exposures and wavelength cal-\nibration from ThAr lamp exposures. The extracted spectra from all\nechelle orders, and all exposures, were combined into a nearly con-\ntinuous 1-dimensional spectrum using the uves_popler software\n(Murphy 2016; Murphy et al. 2018). All spectra were redispersed\nontothesamewavelengthgridwith2.2kms−1pixelsandcombined\nusinganinverse-variance-weightedmean.Aninitialcontinuumwas\nautomatically fitted to the regions redwards of the Lyman- 𝛼emis-\nsionline,whilstamanually-fittedpolynomialwasusedforthebluer\nregionsaffectedbytheLyman- 𝛼forest.ThecombinedHIRESspec-\ntrumhasaS/Nratioperpixelof ≈20inthequasarcontinuumaround\nthe most important D ilines.\n3 ANALYSIS\n3.1 A method\nTo determine the properties of the sub-DLA system, we used joint\nmulti-component Voigt-profile fitting2of hydrogen, deuterium, and\nlow-ionisation metal absorption lines. Within this approach, each\nconsidered absorption line is described by a collection of compo-\nnents,thataredefinedbytheredshift,Dopplerparameter,andcolumn\ndensity of each species. We used the same number of components\nfor each line (except for Fe iiwhere only the two main components\naredetected),withredshiftstiedbetweenlines.Wealsoassumedthat\nthe Doppler parameters of each species within one component are\ntiedbasedontheassumptionofmicro-turbulence,wherethevelocity\ndistributionofeachspeciesisdescribedbyaGaussianfunctionwith\nDoppler parameter\n1Seehttp://www.astro.caltech.edu/~tb/makee .\n2Weusedthepythonpackagespectro( https://github.com/balashev/\nspectro).\nMNRAS 000, 1–12 (2023)A New Precise Determination of the Primordial Abundance of Deuterium. 3\n𝑏sp=√︂\n𝑏2\nturb+2𝑘𝐵𝑇\n𝑀s, (1)\nwhere𝑘𝐵is the Boltzmann constant, 𝑀sis the atomic mass of\nspecies𝑠,𝑇is the kinetic temperature, and 𝑏turbis the turbulent\nbroadeningparameter.Consequently,thelattertwoparameterswere\nfitted independently in each velocity component.\nTomakethecolumn-densitydeterminationstrustworthy,wecare-\nfullyinspectedeachportionofthespectratowhichthefittingproce-\ndurewasapplied.Wesimultaneouslyfittedanyunrelatedabsorption\nwhichmighthaveanimpactonthefit.Wealsopaidspecialattention\nto including all available D iand H ilines to avoid unconscious bias\nleading to include/exclude lines which may seem to be more/less\nsuitable.\nFor the metal lines and Lyman-series lines, we first reconstructed\na local continuum by spline interpolation from regions without ev-\nident absorption. While for the metal lines, outside Ly 𝛼forest, we\nfoundthatcontinuumcanbewellconstrained,forsomeofLy-series\nlines it is not the case, and assumed continuum may impact on the\nderivedH iandD icolumndensities(see,e.g.,Balashevetal.2016).\nThereforewefittedthecontinuumforseveralselectedLyman-series\nlines simultaneously with line profiles to exclude a bias related to a\nmanualcontinuumplacement.TodothisweusedChebyshevpolyno-\nmialsoforderfive,threeandoneintheregionsnearLy 𝛼,Ly𝛽,other\nLyman-series lines, respectively. In addition, we fitted the effective\nresolutions of the final UVES and HIRES spectra as independent\nparameters.\nWe constrained the probability density distribution of the fit pa-\nrameters within the Bayesian framework, using the Monte Carlo\nMarkov Chain technique with Affine invariant sampler (Goodman\n& Weare 2010). We assumed flat priors on 𝑏turb,𝑇kin,𝑧,log𝑁, the\ncontinuum Chebyshev coefficients, 𝐶𝑖, and resolutions of the spec-\ntra𝑅UVES,𝑅HIRES, with enough width to guarantee independence\nof the posteriors on it. We also assumed a same D/H value for all\ncomponents, i.e. we used a single parameter for the whole subDLA.\nToreporttheresultingfitparametersandtheiruncertainties,weused\nthe maximum posterior probability estimates and highest posterior\ndensity 68.3% credible intervals, respectively.\n3.2 Fit description and results\nWe used a four-component model to fit H i, Di, and low-ionisation\nmetal lines. While the absorption profile is dominated by two main\ncomponents, with a velocity separation smaller than 10 kms−1, we\naddedtwoweakcomponentswithina ≈40kms−1span,barelyseen\ninthestrongC ii1334andSi ii1260transitionlines.Thelineprofiles\nofH iLy𝛼,low-ionisationmetalspecies,andthehigher-orderLyman\nseries of H iand D iare presented in Figs. 1, 2, 3, and 4. The values\nof the fitting parameters are given in Table 3. In the figures, we\ndemonstratetheportionsofUVESandHIRESspectranearthesub-\nDLA absorption system with the best-fit model and the confidence\nintervals for all components inside the sub-DLA system. The first\nthing that could catch the eye is a high signal-to-noise ratio which\nweconsiderthemainsourceofthesmallstatisticalerrorofourD/H\ndetermination. Another point is that despite individual components\nhavingnoticeableuncertaintiesshownbythewidthofcolourfullines,\nthetotalfitshowninredisverywellconstrained.Tomaketheanalysis\nmorerobustweexaminedtheresidualsofthefitwhichwealsoshow\nintheseFigures.Ingeneral,theresidualscomfortablylieintherange\ninside 2𝜎with rare outliers in good agreement with the statistical\npredictions.Thereare,however,someregionswhereonecouldseeaquite significant structure and this is also discussed in Sect. 3.3. We\nalso note that the estimated resolving powers of UVES and HIRES\nspectraare𝑅UVES=50800±1400,𝑅HIRES=49200±1500.These\nvalues are in excellent agreement with the nominal values for these\nspectrographs, and the posterior distribution function indicates no\ncorrelation between the D/H value and resolution.\nThe total H icolumn density is log𝑁=19.304±0.004, indicat-\ning that this system belongs to the class of sub-DLAs, where one\nmay expect substantial ionisation corrections. However, thanks to\nalmost the same ionisation potentials of Hydrogen and Deuterium\nthese ionisation corrections are not suitable for the problem of the\nD/H determination. It has been also shown that the D/H ionisation\ncorrectionistypicallyanorderofmagnitudesmallerthancurrentun-\ncertaintiesofD/Hdeterminationsevenforsub-DLAsystems(Cooke\n& Pettini 2015).\nSince Oxygen has an ionisation potential close to the ionisation\npotential of Hydrogen too, our constraint on the O icolumn density\ncould be used to derive the metallicity of the system: [O/H]=\n−1.71±0.02.Othermetalcolumndensities,suchasC ii,Siii,Aliiand\nFeiihave values which are consistent with the Oxygen metallicity\nwithsmalldepletionfactors(dustcontent),whichisexpectedatlow\nmetallicities.\nFrom the simultaneous fit of the UVES and HIRES spectra, we\nfindlog(𝑁(Di)/𝑁(Hi))=−4.623±0.008. The quoted error is the\nformal statistical uncertainty from the Bayesian analysis.\n3.3 Systematic uncertainty\nThe results derived in the previous section only reported a statisti-\ncal uncertainty since they were obtained using the specific model\nassumptions described in Sect. 3.1. To understand the impact of the\nchosenmodelassumptionsandestimatesystematicuncertainties,we\nsubsequently changed each model assumption and performed inde-\npendent fits. The results are summarized in Fig. 6 where we plot\ntheD/Hratiosderivedthrougheachmodificationandcomparethem\nwith our base model.\nUsingbluesymbolsinFig.6,weshowmodelswhereweexcluded\naregionneareachH iline.Thesedeterminationsarequiteconsistent\nand only the model without a region near H i973Å line is a little bit\noutstanding. Using olive symbols in Fig. 6, we present the model,\nwhere we changed the assumption on the metallicity through the\ncomponentsofthesub-DLAsystem.Weuseddifferentelementsasa\nmetallicitytracer,aswellasfittedthespectrumwithamodelwithout\nthisassumption,andwefoundnosignificantchangesintheD/Hratio\nin comparison with the base model. By purple colour in Fig. 6, we\nplot the models where we use only a wing of H iLy𝛼line. These\nmodels indicate typical systematic uncertainty concerned with the\nmeasurement of N(H i).\nWealsofittedVLT/UVESandKeck/HIRESspectraindependently\nand the results are shown in Fig. 6 by orange error bars. Posterior\ndistributions of the main parameters are presented in Fig. 7. These\n𝑁(Di)/𝑁(Hi)determinations differs by ≈1.7𝜎, which is quite un-\nexpected. As could be seen in Fig. 7, determinations of the total\nlog𝑁(Hi)and [O/H] (as well as other nuisance parameters not\nshown in the figure) are more consistent between two fits, so the\nmainsourceofthediscrepancyislikelythedifferencein log𝑁(Di).\nWe additionally test how the order of the polynomial used for\ncontinuum fitting influences the estimated log𝑁(Hi). To do this,\nwe fit only the region near Ly 𝛼with different polynomial order for\nthe continuum and only a single component for the Ly 𝛼absorption\nfeature.TheresultsarepresentedinFig.8.Ordersof 3and4aretoo\nlow to fit the spectrum in the region while the higher orders fit the\nMNRAS 000, 1–12 (2023)4Kislitsyn P. et al.\nComp. #1 #2 #3 #4 log𝑁tot[X/H]total\nΔv, km/s−17.4+1.1\n−0.80 8.7+0.2\n−0.221.3+0.4\n−0.4\n𝑏turb, km/s 11.7+1.9\n−1.22.8+0.1\n−0.11.0+0.5\n−0.54.2+0.8\n−1.1\n𝑇, K 12600+1900\n−230010100+200\n−20015100+900\n−70022200+1200\n−900\nlog𝑁(Cii) 12.83+0.05\n−0.0413.86+0.02\n−0.0213.18+0.03\n−0.0312.75+0.04\n−0.0314.003+0.01\n−0.011−1.728+0.01\n−0.013\nlog𝑁(Siii) 11.59+0.06\n−0.0513.05+0.01\n−0.0112.33+0.03\n−0.0311.74+0.03\n−0.0313.157+0.009\n−0.008−1.656+0.009\n−0.01\nlog𝑁(Alii) 11.1+0.09\n−0.1111.66+0.03\n−0.0211.29+0.04\n−0.0511.06+0.08\n−0.0811.949+0.017\n−0.017−1.806+0.019\n−0.016\nlog𝑁(Feii) 12.65+0.05\n−0.0410.65+0.5\n−0.5912.664+0.044\n−0.047−2.140+0.043\n−0.048\nlog𝑁(Oi) 11.77+0.49\n−0.5714.25+0.02\n−0.0213.2+0.05\n−0.0612.13+0.13\n−0.1414.29+0.02\n−0.02−1.705+0.021\n−0.022\nlog𝑁(Hi) 16.68+0.06\n−0.0519.25+0.01\n−0.0118.39+0.05\n−0.0916.8+0.09\n−0.0719.304+0.004\n−0.004\nlog(𝑁(Di)/𝑁(Hi)) − 4.623+0.008\n−0.008\n[O/H]individual−1.61+0.49\n−0.58−1.69+0.03\n−0.03−1.86+0.06\n−0.07−1.39+0.16\n−0.15\nTable3.Overallfitresultsfortheabsorption-linesystemat 𝑧=3.4210863(15)towardsJ1332+0052.Theseresultscorrespondtothebasemodel(seesec.3.1).\nWeusedvaluesfromAsplundetal.(2009)forsolarabundances: 12+log(C/H)=8.43,12+log(Si/H)=7.51,12+log(Al/H)=6.45,12+log(Fe/H)=7.50,\n12+log(O/H)=8.69\nFigure 1. Fit to the H iLy-𝛼line at𝑧abs=3.42towards J1332+0052. The black error bars show the pixels of the observed spectra used in the fit. Other pixels\naredisplayedingrey.Thetotalmodelprofileisshownbyredlines,whiletheothercolouredlinesindicatethecontributionofindividualcomponents.Thefilled\nregions between lines correspond to 0.68 credible intervals drawn from the posterior parameter distribution estimated during MCMC fitting. Fit residuals are\ndisplayed above the profiles.\ndataquitewellanddonotshowthesignificantscatterintheestimated\nlog𝑁(Hi). Therefore, we choose five as a compromise between the\nunacceptably simple model and the overfitting.\nWe then combined all posterior distribution functions (estimated\nbyMCMCchains)ofD/Hratiosforthesemodelsintooneanddefined\nthe uncertainty of the 𝑁(Di)/𝑁(Hi)determination of 0.014dex\ndirectly from the final posterior distribution (see Fig. 6). It consists\nof both statistical and systematic uncertainty.\nAnother potential source of systematic uncertainty is the fact that\nthere are some regions in the UVES spectrum where noise does\nnotseemtobeGaussian.Onecouldrecogniseadditionalabsorptionfeatures in such a portion of the spectrum like near the D i𝜆973\nline. For all these cases, comparison with the same regions in the\nHIRESspectrummainlyshowsthatthereisnodemonstrationofany\nadditional absorption in the HIRES spectrum and thus we suppose\nthatsuchanoiseisanartefactoftheprocedureoftheUVESspectrum\nreduction and hence we did not fit all these tiny features as real\nabsorption systems. However, given a noticeable deviation of the\nD/H value obtained with the model where we exclude D i973 line,\nfrom the the majority of other models (see Fig. 6), we also fitted all\nDilines but 973Å for UVES and HIRES spectra independently to\ntest how fluctuations near H i973line in the UVES spectrum may\nMNRAS 000, 1–12 (2023)A New Precise Determination of the Primordial Abundance of Deuterium. 5\nFigure2. Fitstothemetalabsorptionlinesat 𝑧=3.42towardsJ1332+0052in\ntheUVESspectrum.ThegraphicalconventionsofFig.1areused.Transition\nlines are labelled in each panel.\naffecttotheestimatedD i/Hivalue.Usingbothspectraweestimated\nlog(Di/Hi)=−4.647±0.012, while for UVES and HIRES spectra\nonlywegot log(Di/Hi)=−4.647±0.017andlog(Di/Hi)=−4.665±\n0.022,respectively.Itallowsustosupposethatthefluctuationsnear\nHi973line is not related to the relatively low D i/Hivalue for fit\nwithout D i973, since the D i/Hivalue from the fit of the HIRES\nspectrum alone is even lower than from the fit of the UVES one.\nAs a concluding remark on Fig. 6, we would like to highlight two\nkey points. Firstly, all the presented D/H values are demonstrated as\nalternatives to the base model, and we maintain neutrality among\nthese alternatives; there is no preference for any specific option.\nFigure3. Fitstothemetalabsorptionlinesat 𝑧=3.42towardsJ1332+0052in\ntheHIRESspectum.ThegraphicalconventionsofFig.1areused.Transition\nlines are labelled in each panel.\nUpon analysing the results in the figure, it becomes apparent that a\nmajorityoftheD/Hvaluestendtowardsthe\"UVES\"valueorsuggest\nthat the D i973 line plays a decisive role, contributing to a higher\nD/H value. It is crucial to note that, in all cases except \"UVES\" and\n\"HIRES\", we fitted bothspectra. Our base model involves fitting\nallDilines inbothspectra, without favouring any particular metal\nelement and considering bothwings of the H iLy𝛼line. Secondly,\nanyfurtheranalysisisagainstthe\"blind\"methodology,asexplained\nin Sect. 4.2. There is no evident reason to favour one spectrum over\nanother,deemoneD ilinesuperiortoothers,orassertthesuperiority\nof a specific wing of the H iLy𝛼line. Therefore, we amalgamate all\nMNRAS 000, 1–12 (2023)6Kislitsyn P. et al.\nFigure 4. Fits to the H iand D iLyman-series lines at 𝑧abs=3.42towards J1332+0052. The left panels show the UVES spectrum while the right panels show\nthe HIRES spectrum. The graphical elements are the same as in Fig. 1. Transition lines are labelled in each panel.\ntheresultstopresentthefinalposteriordistributionoftheD/Hvalue\nand its associated error.\nWe also checked another fit model where an additional H icom-\nponent was added with the velocity offset of −81.5kms−1relative\nto the major component so H ilines of this additional componentwere overlapped with D ilines. Obtained result is 𝑁(Di)/𝑁(Hi)=\n−4.685±0.014(only statistical uncertainty). In principle, there is\nalwayssuchopportunityforanyconsideredD/Hmeasurement,there-\nfore,whilewedidnotincludethisuncertaintyinourfinalresults,we\nneedtobearinmindthattheresultmaybebiased.Thusthevalueof\nMNRAS 000, 1–12 (2023)A New Precise Determination of the Primordial Abundance of Deuterium. 7\nFigure 5. Fits to the H iand D iLyman-series lines at 𝑧abs=3.42towards J1332+0052 (continuation of Fig. 4).\nΔ=−0.06could be perceived as a bias that might take place in the\ncaseofD ilinesthatareblendedwithanunmodelledH icomponent.\nLastly, we discussed the uncertainty, related to the profiles of\ninterloping absorption systems, which may partially blend the total\nfit near sub-DLA D ilines. In Appendix A, we show a set of figures\nwhich represent the fit to spectral regions around D ilines in theUVES spectrum. In each figure, we present all the lines for each\nadditional H icomponent that are available for fitting and were used\nduring the fit. Looking at, e.g., Figs. A4 and A7, one can notice\nthat all features near the D i𝜆931 line are well constrained by clear\nabsorption by other lines of those absorption systems. The same is\nrelevant for the D i𝜆950 line. With such a reliable constraint on the\nMNRAS 000, 1–12 (2023)8Kislitsyn P. et al.\nFigure6. Di/Hiestimatesusingdifferentmodelassumptionsorspectra(leftpanel)andjointestimate(rightpanel).Differentkeywordsalongthex-axisdenote\na modification of the base model (labelled as \"base\" and described in Sect. 3.2) as follows: \"UVES\" and \"HIRES\": only UVES or HIRES is fitted; \"blue-Ly 𝛼\"\nand\"red-Ly𝛼\":onlythebluerortheredderwingoftheH iLy𝛼lineisfitted;\"met-AlII\",\"met-CII\",\"met-SiII\"and\"met-OI\": 𝑁(Hi)istiedtoeither 𝑁(Alii),\n𝑁(Cii),𝑁(Siii), or𝑁(Oi)along the components; \"wo- 𝜆\": the region near the D i\"𝜆\" (in Å) line is excluded from the fit.\nblending absorption features, we suppose that there is little bias of\nthe usage of the partially blended D ilines, as we did in this work.\n4 DISCUSSION\nThesystempresentedhereallowedustorobustlydeterminetheD/H\nratio thanks to clear damping wings in Ly- 𝛼, especially in the re-\ngions where the flux immediately recovers from zero (DLA core),\nwhich is necessary to accurately constrain the total H icolumn den-\nsity in gas-rich (sub-DLA and DLA) systems. Several D ilines are\nalsodetected,providingstrongconstraintsontheD icolumndensity.\nFinally, this particular sub-DLA system exhibits a quite simple ve-\nlocity structure. With such a precise measurement, one can discuss\nthepossibleastrationofdeuterium,whichiseasilydestroyedinstars.\nChemical evolution models predict little deviation of D/H from the\nprimordialabundanceinpredominantlyunprocessedgas,i.e.,atlow\nmetallicity(e.g.,Dvorkinetal.2016).Weinberg(2017)predictedan\nastrationsmallerthan2%at[O/H ]≈− 1.0whilesimilarresultswere\npresented by van de Voort et al. (2018). The latter authors found a\nmeandeviationofD/Hfromtheprimordialvalueinsimulationsthat\nis smaller than 2% up to [O/H ]≈− 0.6. The system studied here of-\nfersanewopportunitytoobservationallytestthisataslightlyhigher\nmetallicity ([O/H]≈− 1.7) than previously possible, confirming the\nlackofdetectabledependenceofD/Honmetallicity,asalsonoticed\nby Cooke et al. (2016). In short, we showed that astration remains\nnegligible compared to measurement uncertainties for metallicities\nofuptoatleast[O/H ]≈− 1.7.Naturally,largerstatisticsareneeded\ntoobservetinydeviationsfromtheprimordialabundanceatthelevel\npredicted by the models.\nAnotherpointworthmentioningisthatthisnewD/Hmeasurement\nwas performed in a sub-DLA system with outstanding properties\ncompared to previously studied systems. Indeed, this sub-DLA has\nthehighestredshiftamongstsystemsinthe precisionsample ,hasthe\nlowest H icolumn density, and has one of the highest metallicities.\nThese peculiarities make this measurement rather unique and relax\nthe conditions allowing for a robust D/H determination. This is es-pecially important in view of the next generation of ground-based\ntelescopes(e.g.,ELT,TMT,andGMT)whichwillbeabletoobserve\nfainter targets (see also the appendix in Cooke et al. 2016).\nOne may also be interested in how metallicity varies from one\ncomponenttoanotherinthesub-DLAsystemsinceitcanbeusedto\nexplorevariationsinmetallicityandstudythedustcontent(see,e.g.,\nRamburuth-Hurt et al. 2023). In the last row of the table 3 we show\nthe [O/H] for individual components derived using the base model.\nOxygenisthebest-suitedelementformetallicitydetermination,since\nits ionisation potential is very close to the hydrogen one (and so\nthe ionisation correction is negligible) and there are several strong\ntransitions with appropriate wavelengths. [O/H] determinations for\nindividual components differ from the overall oxygen metallicity by\nno more than 0.3 dex. This is slightly smaller than the metallicity\nvariations proposed by De Cia et al. (2021) and is consistent with\nother estimates of the metallicity variations (Esteban et al. 2022).\n4.1 Cosmological implementation\nThe main advantage of the D/H measurements is their direct con-\nnection to the primordial deuterium abundance which can be used\nto infer the cosmological baryon density. Using all available D/H\ndeterminationsfromtheliterature,whicharesummarisedintable4,\none can derive the average value (weighted mean):\n(D/H)all=(2.533±0.024)×10−5. (2)\nThis is our conservative estimate in the sense that it incorporates\nallpreviousdeterminations.Wealsoshowthecomparisonofthenew\nD/H measurement with the previous ones in Fig. 9. In this figure,\nwe show D/H versus [O/H] for all known determinations and we\nusedbluecolourformeasurementswhichmeettheselectioncriteria\nproposedbyCookeetal.(2014),redcolourforournewdetermination\nand grey colour for all other determinations. The new measurement\nis consistent with other precision measurements and has a similar\nuncertainty. Fig. 9 also shows that there is no obvious correlation\nbetween metallicity and D/H ratio.\nMNRAS 000, 1–12 (2023)A New Precise Determination of the Primordial Abundance of Deuterium. 9\nFigure 7. Corner plots showing the posterior distributions of the main parameters in the fitting chains applied to the UVES and HIRES spectra.\nUp to now, there is no robust verification that the spread of the\nD/Hdeterminationsisnotphysical.However,morehigh-qualityob-\nservations can clarify estimates as shown by Cooke et al. (2014).\nMoreover, these authors offered a set of criteria to select the most\nsuitablecasesforD/Hdeterminationswheresomesystematiceffects\nare negligible. Based on the nine systems which agree with these\ncriteria(seetable4),onecanobtainaweightedmeanofpresumably\nbest-suited measurements:\n(D/H)best=(2.501±0.028)×10−5. (3)\nEven if measurements that meet the selection criteria of the pre-\ncision sample are presumed to be the most suitable for D/H deter-\nmination, we suggest that it is important to use all available data\nandmeasurementsmadebydifferentresearchers,andusingdifferent\ntechniques, until this presumption is demonstrated conclusively.\nBoth values (eq. 2, 3) are in moderate tension ( ≈2.2𝜎,≈1.4𝜎respectively)withthetheoretically-predictedvalueoftheprimordial\ndeuterium abundance computed using the PRIMAT code and based\non the cosmological parameters derived from CMB+BAO (Planck\nCollaboration et al. 2020) presented by Pitrou et al. (2021):\n(D/H)th=(2.439±0.037)×10−5(4)\nassuming𝜏𝑛=879,4(6)𝑠and100ℎ2Ω𝑏=2.242(±0.014). An im-\nportant and significant achievement in the D(p,𝛾)3Herate determi-\nnation was reached in the last years by the Laboratory for Under-\ngroundNuclearAstrophysics(LUNA)staff(Mossaetal.2020).This\nreaction was the main source of uncertainty in predicting the theo-\nretical primordial D/H ratio. Pitrou et al. (2021) used the new rate\nof this reaction for the computations and the new rate enhanced the\ntension.\nThis growing discrepancy which was also mentioned by Cooke\nMNRAS 000, 1–12 (2023)10Kislitsyn P. et al.\nQSO [O/H] log(D/H) reference meet the selection criteria?\nHS 0105+1619 -1.771 ±0.021 -4.589±0.026 Cooke et al. (2014) ✓\nQ0913+072 -2.416 ±0.011 -4.597±0.018 Cooke et al. (2014) ✓\nSDSS J1358+6522 -2.335 ±0.022 -4.588±0.012 Cooke et al. (2014) ✓\nSDSS J1419+0829 -1.922 ±0.010 -4.601±0.009 Cooke et al. (2014) ✓\nSDSS J1558-0031 -1.650 ±0.040 -4.619±0.026 Cooke et al. (2014) ✓\nQ1243+307 -2.769 ±0.028 -4.622±0.015 Cooke et al. (2018) ✓\nSDSS J1358+0349 -2.804 ±0.015 -4.582±0.012 Cooke et al. (2016) ✓\nJ1444+2919 -2.042 ±0.005 -4.706±0.067 Balashev et al. (2016) ✓\nCTQ 247 -1.990 ±0.100 -4.560±0.100 Noterdaeme et al. (2012) ×\nPKS 1937-1009 (1) -1.870 ±0.200 -4.610±0.050 Riemer-Sørensen et al. (2015) ×\nJ1337+3152 -2.640 ±0.170 -4.930±0.150 Srianand et al. (2010) ×\nJ1134+5742 <-1.9 -4.690 ±0.130 Fumagalli et al. (2011) ×\nQ2206-199 -2.070 ±0.050a-4.786±0.100 Pettini & Bowen (2001) ×\nQ0347-3819 -0.820 ±0.060 -4.426±0.028 Levshakov et al. (2002) ×\nPKS 1937-1009 (2) -2.250 ±0.250 -4.581±0.008 Riemer-Sørensen et al. (2017) ×\nQ1009+2956 -2.500 ±0.200 -4.606±0.066 Zavarygin et al. (2018) ×\nJ1332+0052 -1.725 ±0.019 -4.622±0.014 this paper ✓\napresented in Pettini et al. (2008)\nTable 4.Current sample of D/H determinations. In the last column we show either a measurement meets or not the selection criteria of the precision sample\nproposed by Cooke et al. (2014).\nFigure 8. Dependence of the total measured log𝑁(Hi) on the order of the\npolynomial used for continuum fitting. The orange dots and error bars show\nthe estimated log𝑁(Hi) using a one-component model for Ly 𝛼absorption\n(left ordinate axis). The blue dots show the reduced 𝜒2of the fits (right\nordinate axis).\netal.(2018)andPitrouetal.(2021)couldbeamanifestationofnew\nphysics beyond the Standard Cosmological Model.\n4.2 Avoiding confirmation bias\nGiven the number of previous works on the primordial D/H ratio\nmeasurements as well as BBN+CMB value, one could expect that a\nnew determination should be consistent with others. This could in\nprinciple influence the decisions made through the analysis of new\nD/H systems.\nFirstofall,itisworthremindingthatD/Hmeasurementsareonlya few so any new measurement is extremely valuable. The detection\nof a clear H iLy𝛼line and several optically thin D ilines lead us to\nmake the analysis to the end, regardless of the obtained D/H value.\nWhilenotperformingacompleteblindanalysis(asdonebyCooke\net al. 2014), where the value of D/H is kept unknown until all deci-\nsionsaboutthemodellingareacceptedbasedonthevisualmatching\nof the data by the fit and 𝜒2values, we did not favour any D/H\nvalue during the fitting process. In fact, Fig. 6 presents only a small\nscatterintheD/Hdeterminationsbetweenourdifferentmodels.Any\npriorbasedonvaluesby,e.g.,Cookeetal.(2018)couldbeanywhere\nwithina∼0.1dexrangearound log(D/H)=−4.61,andallourmod-\nels are comfortably within this range. We would have no reason to\nsubconsciouslyprefersomevaluewithinthatrange.Inotherwords,a\nconfirmationbiaswouldrathertakeplaceforachoicebetween,e.g.,\navalueof−4.6and−4.4,i.e.withadeviationmuchmoresubstantial\nthan observed here. The best way to avoid subjectivity remains to\nperform a D/H determination in each system where it is possible to\ndo and confront analysis obtained by different groups and methods.\n5 CONCLUSIONS\nTo date, there are only 17 determinations of the D/H ratio in metal-\npoor DLA/sub-DLA systems at high redshift. Nine of them agree\nwith the criteria defined by Cooke et al. (2014) to qualify as robust\nmeasurements.Inthisway,eachnewdeterminationiscrucial,aswell\nas any reanalysis of known systems based on new high-quality data.\nOur main conclusions are:\n(i) Basedonhigh-qualityspectraweobtainedanewmeasurement\nof the deuterium abundance in the metal-poor sub-DLA system at\n𝑧=3.42towardsquasarJ1332+0052: D/H=(2.388±0.078)×10−5.\n(ii) This new measurement is consistent with previous determi-\nMNRAS 000, 1–12 (2023)A New Precise Determination of the Primordial Abundance of Deuterium. 11\nFigure 9. Current D/H measurements. The systems matching the criteria defined by Cooke et al. (2014) are shown in blue while other determinations are\ndisplayed in grey. Our new measurement is shown in red. The weighted mean D/H value based on all measurements is represented by the horizontal line. The\nweighted mean values based on either all measurements or only the systems matching the above criteria are displayed at the bottom right.\nnations of the same precision level and meets the selection criteria\nproposed by Cooke et al. (2014).\n(iii) BasedonallavailableD/Hmeasurementsweestimatedanew\nprimordial deuterium abundance D/H pr=(2.533±0.024)×10−5.\nBasedonninemeasurementswhichmeetthecriteriaofapreciseone\nwe find D/H pr=(2.501±0.028)×10−5.\n(iv) We used a new method for accounting the systematic effects\nbased on multiple times fitting, when we do an independent fitting\nfor each model we consider it appropriate for describing the data.\nThe main advantage of this method is that we do not choose which\nmodel is better or worse which might be subconsciously biased by\nour human nature and combine all the results from any possible\nmodel instead. Despite it being very time-consuming, it is possibly\nthe single ability to realise how different aspects could affect the\nresults of the analysis.\n(v) AnewdeterminationoftheD/H prbasedonallmeasurements\nis still marginally inconsistent ( ≈2.2𝜎) with the theoretical value\npredicted by Pitrou et al. (2021) based on the cosmological param-\neters derived from CMB+BAO (Planck Collaboration et al. 2020).\nThe reasons for this remain unsettled and could potentially be an\nindication of new physics.\n(vi) There is still no obvious correlation between D/H and [O/H]\nor N(H i).\nACKNOWLEDGEMENTS\nWe thank the anonymous referee for the careful review and useful\nsuggestions. The authors are grateful to the VLT and Keck observa-\ntorystaffforacquiringtheanalyseddataandtoRyanCookeforhisin-\nvaluable input on this particular system. All performance-expensive\ncalculations were performed on the cluster at Ioffe Institute. This\nwork was supported by RSF grant 23-12-00166.\nSomeofthedatapresentedhereinwereobtainedatKeckObserva-\ntory,whichisaprivate501(c)3non-profitorganizationoperatedasa\nscientific partnership among the California Institute of Technology,\ntheUniversityofCalifornia,andtheNationalAeronauticsandSpace\nAdministration.TheObservatorywasmadepossiblebythegenerousfinancialsupportoftheW.M.KeckFoundation.Theauthorswishto\nrecognizeandacknowledgetheverysignificantculturalroleandrev-\nerencethatthesummitofMaunakeahasalwayshadwithintheNative\nHawaiiancommunity.Wearemostfortunatetohavetheopportunity\ntoconductobservationsfromthismountain.Thisresearchhasmade\nuse of the Keck Observatory Archive (KOA), which is operated by\nthe W. M. Keck Observatory and the NASA Exoplanet Science In-\nstitute (NExScI), under contract with the National Aeronautics and\nSpace Administration.\nDATA AVAILABILITY\nThepipeline-reducedspectraunderlyingthisarticlearereadilyavail-\nable from the ESO science portal ( http://archive.eso.org/\nscienceportal/home ) and partially from the Keck observatory\narchive ( https://koa.ipac.caltech.edu ). Re-reduced spectra\ncan be shared upon request to the corresponding author.\nREFERENCES\nAsplund M., Grevesse N., Sauval A. J., Scott P., 2009, Annual Review of\nAstronomy and Astrophysics, 47, 481\nAver E., Berg D. A., Hirschauer A. S., Olive K. A., Pogge R. W., Rogers N.\nS. J., Salzer J. J., Skillman E. D., 2021, Monthly Notices of the Royal\nAstronomical Society, 510, 373\nBalashevS.A.,ZavaryginE.O.,IvanchikA.V.,TelikovaK.N.,Varshalovich\nD. A., 2016, Monthly Notices of the Royal Astronomical Society, 458,\n2188\nCooke R., Pettini M., 2015, Monthly Notices of the Royal Astronomical\nSociety, 455, 1512\nCooke R.J., PettiniM., JorgensonR. A., MurphyM. T.,Steidel C.C., 2014,\nThe Astrophysical Journal, 781, 31\nCookeR.J.,PettiniM.,NollettK.M.,JorgensonR.,2016,TheAstrophysical\nJournal, 830, 148\nCooke R. J., Pettini M., Steidel C. C., 2018, The Astrophysical Journal, 855,\n102\nDeCiaA.,JenkinsE.B.,FoxA.J.,LedouxC.,Ramburuth-HurtT.,Konstan-\ntopoulou C., Petitjean P., Krogager J.-K., 2021, Nature, 597, 206\nMNRAS 000, 1–12 (2023)12Kislitsyn P. et al.\nDekkerH.,D’OdoricoS.,KauferA.,DelabreB.,KotzlowskiH.,2000,inIye\nM., Moorwood A. F. M., eds, Optical and IR Telescope Instrumentation\nand Detectors. SPIE, pp 534 – 545, doi:10.1117/12.395512, https:\n//doi.org/10.1117/12.395512\nDvorkin I., Vangioni E., Silk J., Petitjean P., Olive K. A., 2016, Monthly\nNotices of the Royal Astronomical Society: Letters, 458, L104–L108\nEstebanC.,Méndez-DelgadoJ.E.,García-RojasJ.,Arellano-CórdovaK.Z.,\n2022, The Astrophysical Journal, 931, 92\nEvans C., et al., 2022, Experimental Astronomy,\nFernández V., Terlevich E., Díaz A. I., Terlevich R., Rosales-Ortega F. F.,\n2018, Monthly Notices of the Royal Astronomical Society, 478, 5301\nFreudlingW.,RomanielloM.,BramichD.M.,BallesterP.,ForchiV.,García-\nDabló C. E., Moehler S., Neeser M. J., 2013, A&A, 559, A96\nFumagalli M., O’Meara J. M., Prochaska J. X., 2011, Science, 334, 1245\nGoodman J., Weare J., 2010, Communications in Applied Mathematics and\nComputational Science, 5, 65\nHsyu T., Cooke R. J., Prochaska J. X., Bolte M., 2020, The Astrophysical\nJournal, 896, 77\nIzotov Y. I., Thuan T. X., Guseva N. G., 2014, Monthly Notices of the Royal\nAstronomical Society, 445, 778\nKurichin O. A., Kislitsyn P. A., Ivanchik A. V., 2021a, Astronomy Letters,\n47, 674\nKurichin O. A., Kislitsyn P. A., Klimenko V. V., Balashev S. A., Ivanchik\nA. V., 2021b, Monthly Notices of the Royal Astronomical Society, 502,\n3045\nLevshakov S. A., Dessauges-Zavadsky M., D’Odorico S., Molaro P., 2002,\nThe Astrophysical Journal, 565, 696\nMatsumoto A., et al., 2022, The Astrophysical Journal, 941, 167\nMossa V., et al., 2020, Nature, 587, 210\nMurphy M. T., 2016, UVES_popler: POst-PipeLine Echelle Reduction soft-\nware.\nMurphy M. T., Kacprzak G. G., Savorgnan G. A. D., Carswell R. F., 2018,\nMonthly Notices of the Royal Astronomical Society, 482, 3458\nNoterdaeme P., López S., Dumont V., Ledoux C., Molaro P., Petitjean P.,\n2012, A&A, 542, L33\nPettini M., Bowen D. V., 2001, The Astrophysical Journal, 560, 41\nPettiniM.,ZychB.J.,SteidelC.C.,ChaffeeF.H.,2008,MonthlyNoticesof\nthe Royal Astronomical Society, 385, 2011\nPitrou C., Coc A., Uzan J.-P., Vangioni E., 2021, Monthly Notices of the\nRoyal Astronomical Society, 502, 2474\nPlanck Collaboration et al., 2020, A&A, 641, A6\nRamburuth-Hurt T., et al., 2023, A&A, 672, A68\nRiemer-SørensenS.,etal.,2015,MonthlyNoticesoftheRoyalAstronomical\nSociety, 447, 2925\nRiemer-SørensenS.,KotušS.,WebbJ.K.,AliK.,DumontV.,MurphyM.T.,\nCarswellR.F.,2017,MonthlyNoticesoftheRoyalAstronomicalSociety,\n468, 3239\nRobertP.F.,MurphyM.T.,O’MearaJ.M.,CrightonN.H.M.,FumagalliM.,\n2018, Monthly Notices of the Royal Astronomical Society, 483, 2736\nSrianandR.,GuptaN.,PetitjeanP.,NoterdaemeP.,LedouxC.,2010,Monthly\nNotices of the Royal Astronomical Society, 405, 1888\nTytler D., Fan X.-M., Burles S., 1996, Nature, 381, 207\nValerdi M., Peimbert A., Peimbert M., 2021, Monthly Notices of the Royal\nAstronomical Society, 505, 3624\nVéron-Cetty M. P., Véron P., 2010, A&A, 518, A10\nVogtS.S.,etal.,1994,inCrawfordD.L.,CraineE.R.,eds,SocietyofPhoto-\nOptical Instrumentation Engineers (SPIE) Conference Series Vol. 2198,\nInstrumentation in Astronomy VIII. p. 362, doi:10.1117/12.176725\nWeinberg D. H., 2017, The Astrophysical Journal, 851, 25\nZavaryginE.O.,WebbJ.K.,DumontV.,Riemer-SørensenS.,2018,Monthly\nNotices of the Royal Astronomical Society, 477, 5536\nvandeVoortF.,QuataertE.,Faucher-GiguèreC.-A.,KerešD.,HopkinsP.F.,\nChanT.K.,FeldmannR.,HafenZ.,2018,MonthlyNoticesoftheRoyal\nAstronomical Society, 477, 80Table A1. Unrelated absorption systems\nline in sub-DLA system Blending system Figure\n𝜆1026 A2\n𝜆973 A3\n𝜆950 A4\n𝜆938 A5, A6\n𝜆931 A7\n𝜆926 A8\n𝜆923 A9\nAPPENDIX A: INTERLOPING SYSTEMS\nInthissection,wepresentthemodellingofunrelatedabsorption-line\nsystems. In figure A1 we present strongest D iabsorption lines of\nthe sub-DLA system and interloping absorption features near them\nin the UVES spectrum. Each following figure shows a portion of\nthe UVES spectrum with the fit in more details. In particular, these\nfigures show additional absorption lines which allow one to better\nconstrainparametersofasystem.Theinterlopingsystemsareshown\nin grey. The other colours are those used in the main figures to\neither depict the total model profile (red) or individual sub-DLA\ncomponents(i.e.,orange,green,blue,andvioletforcomponents#1,\n#2, #3, and #4, respectively). D ilines usually appear in green.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–12 (2023)A New Precise Determination of the Primordial Abundance of Deuterium. 13\nFigure A1. Portion of UVES spectrum with a focus on unrelated absorption features.\nFigure A2. Absorption system interloping with D i𝜆1026. The system is located at redshift 𝑧=2.729383. Other systems which affect total fit near the H iline\nare also shown.\nFigure A3. Absorption system interloping with D i𝜆973. The system is lo-\ncated at redshift 𝑧=2.534616. Another system nearby is also shown.\nMNRAS 000, 1–12 (2023)14Kislitsyn P. et al.\nFigure A4. Absorption systems interloping with D i𝜆950. The velocity offset is counted relative to the system located at redshift 𝑧=3.475422.\nFigureA5. ThefirstabsorptionsysteminterlopingwithD i𝜆938.Thevelocity\noffset is counted relative to the system located at redshift 𝑧=2.409310.\nFigureA6. ThesecondabsorptionsysteminterlopingwithD i𝜆938.Theve-\nlocityoffsetiscountedrelativetothesystemlocatedatredshift 𝑧=3.261862.\nMNRAS 000, 1–12 (2023)A New Precise Determination of the Primordial Abundance of Deuterium. 15\nFigureA7. AbsorptionsysteminterlopingwithD i𝜆931.Thevelocityoffset\nis counted relative to the system located at redshift 𝑧=3.331398.\nFigureA8. AbsorptionsysteminterlopingwithD i𝜆926.Thevelocityoffset\nis counted relative to the system located at redshift 𝑧=2.991426.\nFigureA9. AbsorptionsysteminterlopingwithD i𝜆923.Thevelocityoffset\nis counted relative to the system located at redshift 𝑧=3.351185.\nMNRAS 000, 1–12 (2023)"    },    {        "title": "2401.12802v1.New_lower_bounds_for_three_term_progression_free_sets_in___mathbb_F__p_n_.pdf",        "content": "arXiv:2401.12802v1  [math.CO]  23 Jan 2024New lower bounds for three-term progression free sets in Fn\np\nChristian Elsholtz∗Laura Proske†Lisa Sauermann‡\nJanuary 24, 2024\nAbstract\nWe prove new lower bounds on the maximum size of sets A⊆Fn\nporA⊆Zn\nmnot containing three-term\narithmetic progressions (consisting of three distinct poi nts). More specifically, we prove that for any fixed\nintegerm≥2and sufficiently large n(in terms of m), there exists a three-term progression free subset\nA⊆Zn\nmof size|A| ≥(cm)nfor some absolute constant c >1/2. Such a bound for c= 1/2can be\nobtained with a classical construction of Salem and Spencer from 1942, and improving upon this value of\n1/2has been a well-known open problem (our proof gives c= 0.54).\nOur construction relies on finding a subset S⊆Z2\nmof size at least (7/24)m2with a certain type of\nreducibility property. This property allows us to “lift” Sto a three-term progression free subset of Zn\nmfor\nlargen(even though the original set S⊆Z2\nmdoes contain three-term arithmetic progressions).\n1 Introduction\nStudying the maximum possible size of subsets of {1,...,N}or ofFn\npwithout three-term arithmetic pro-\ngressions is one of the most central problems in additive com binatorics. In a breakthrough result in 2017,\nEllenberg and Gijswijt [10] proved that for any prime p≥3, every subset A⊆Fn\npnot containing a three-term\narithmetic progression (i.e. not containing three distinc t elements x,y,z withx−2y+z= 0) has size at\nmost(cpp)nfor some constant cp<1(wherecpconverges to 0.841...forp→ ∞, see [3, Eq. (4.11)]). In\nthe integer setting, in a more recent breakthrough Kelley an d Meka [20] proved that for N≥3, every subset\nA⊆ {1,...,N}not containing a three-term arithmetic progression has siz e at most Nexp(−c(logN)1/11)\nfor some absolute constant c >0(afterwards, this bound was improved to Nexp(−c(logN)1/9)using a\nmodification of their method by Bloom and Sisask [4]). This ma tches the shape of a classical lower bound\nfor this problem due to Behrend [2], which is of the form Nexp(−C(logN)1/2)for some absolute constant\nC. See also [25] for a recent survey on this problem (both in {1,...,N}and inFn\np), and its analogues for\nlonger arithmetic progressions and polynomial progressio ns.\nTo obtain lower bonds for the maximum size of three-term prog ression free subsets of Fn\npfor large n, one can\nadapt a classical construction of Salem and Spencer [27] fro m 1942, see [1, Theorem 2.13]. Alternatively, one\ncan also adapt Behrend’s construction [2] from 1946 to this s etting, as noted by Tao and Vu in their book on\nadditive combinatorics [28, Exercise 10.1.3] and also obse rved by Alon (see [17, Lemma 17]). Both of these\nconstructions show that for every fixed prime p, there exists a subset A⊆Fn\npof size|A| ≥((p+1)/2)n−o(n)\nwithout a three-term arithmetic progression (here, the asy mptotic notation o(n)is forn→ ∞ withpfixed).\nThe best quantitative bound for the o(n)-term in this statement is due to the first author and Pach [15,\nTheorem 3.10]. For small primes, better bounds were obtaine d with specific constructions depending on\nthe particular prime. For example, in Fn\n3a lower bound of 2.2202n−o(n)was recently obtained by Romera–\nParedes et al. [26] using artifical intelligence building up on traditional methods from previous bound [6, 8, 29].\nNaslund [24] informed us about forthcoming work with an appr oach related to Shannon capacity, proving a\n∗Institute for Analysis and Number Theory, TU Graz, Austria. Email:elsholtz@math.tugraz.at . Supported by the joint\nFWF-ANR project Arithrand: FWF: I 4945-N and ANR-20-CE91-0 006, and FWF grant DOC 183.\n†TU Graz, Austria. Email: proske@student.tugraz.at .\n‡Institute for Applied Mathematics, University of Bonn, Ger many. Email: sauermann@iam.uni-bonn.de . Supported by the\nDFG Heisenberg Program.\n1lower bound of 2.2208n−o(n)for the maximum size of three-term progression free subsets ofFn\n3. InFn\n5, the\nbest known lower bounds is (351/3)n−o(n)due to the first two authors, Pollak, Lipnik and Siebenhofer ( see\n[14]), note that 351/3≈3.271.\nComparing the upper and lower bounds for the maximum possibl e size of a three-term progression free subset\nofFn\npfor a fixed (reasonably large) prime pand large n, there is still a large gap. Both of these bounds are\nroughly of the form (cp)nwith0< c <1, but with different values of c. For the upper bound, the best\nknown constant due to Ellenberg–Gijswijt [10] is c≈0.85(when the fixed prime is large enough), whereas for\nthe lower bound the constant c= 1/2from the Salem–Spencer construction [27] (or alternativel y Behrend’s\nconstruction [2]) has not been improved in more than 80 years . In this paper, we finally improve this constant\nin the lower bound to be strictly larger than 1/2.\nTheorem 1.1. There is a constant c >1/2such that for every prime pand every sufficiently large positive\nintegern(sufficiently large in terms of p), there exists a subset A⊆Fn\npof size|A| ≥(cp)nnot containing a\nthree-term arithmetic progression.\nOur proof shows that one can take any c </radicalbig\n7/24, for example c= 0.54. While quantitatively, this may\nnot seem like a large improvement over 1/2, it is the first qualitative improvement over the constant 1/2\nfrom the constructions of Salem–Spencer and Behrend from th e 1940’s. Both of these constructions lead to\nthree-term progression free subsets of Fn\nponly consisting of vectors with all entries in {0,1,...,(p−1)/2},\ni.e. they only use roughly half of the available elements in Fpin each coordinate. The restriction of all entries\nto{0,1,...,(p−1)/2}is crucial in these constructions, as it ensures that there i s no “wrap-around” over\nFp. However, such an approach cannot be used to obtain three-te rm progression free subset A⊆Fn\npof size\n|A|>((p+1)/2)n, soc= 1/2is a significant barrier for this problem (and it may be consid ered a surprise\nthat it is actually possible to obtain a constant c >1/2).\nIn fact, our construction proving Theorem 1.1 works more gen erally in Zn\nmfor any integer m≥2(not\nnecessarily prime) and sufficiently large n, again showing that there exists a three-term progression f ree\nsubset of Zn\nmof size at least (cm)n. We recall that a three-term arithmetic progression in Zn\nmconsists of\nthree distinct elements x,y,z∈Zn\nmwithx−2y+z= 0(see e.g. the breakthrough work of Croot–Lev–\nPach [7] proving upper bounds for three-term progression fr ee subsets of Zn\n4). For odd m, the best previous\nlower bound for this problem has also been of the form ((m+ 1)/2)n−o(n)(based on the Salem–Spencer\nconstruction [27] or the Behrend construction [2]), and for evenmthe best previous lower bound has been\n((m+2)/2)n−o(n)due to the first author and Pach [15, Theorem 3.11].\nTheorem 1.2. There is a constant c >1/2such that for every integer m≥2and every sufficiently large\npositive integer n(sufficiently large in terms of m), there exists a subset A⊆Zn\nmof size|A| ≥(cm)nnot\ncontaining a three-term arithmetic progression.\nNote that Theorem 1.2 immediately implies Theorem 1.1. So it suffices to prove Theorem 1.2.\nOur construction of the set A⊆Zn\nmin Theorem 1.2 is, in some sense, a higher-dimensional versi on of\nthe Salem–Spencer construction [27]. Generalizing the arg ument of Salem and Spencer requires a certain\nreducibility notion for subsets of Zd\nm. In the one-dimensional case (i.e. for d= 1), a similar type of reducibility\nargument was used by the first author and Lipnik [13] to constr uct subsets of Fn\npwithout three points on\na common line for p∈ {11,17,23,29,41}and large n, and also by the first author, Klahn and Lipnik [12]\nto construct subsets of Zn\nmavoiding arithmetic progressions of a given length k >3(other constructions\nfor this problem with different methods were given in [11, 19, 22]). A higher-dimensional version of this\ntype of argument first appeared in the above-mentioned work o f the first two authors, Pollak, Lipnik and\nSiebenhofer (see [14]) constructing three-term progressi on free subsets of Fn\n5. However, the crucial difficulty\nfor implementing this strategy for larger mis to find suitable higher-dimensional building blocks for t his\nconstruction, i.e. suitable large subsets of Zd\nmwith the required reducibility condition. In that work, a\nsuitable subset of Z3\n5was found via a computer search. While for any given manddone can try to find\nsuitable subsets of Zd\nmvia computer searching, the difficulty in proving Theorems 1. 1 and 1.2 is establishing\nthe existence of such subsets for all m.\nThe problem of finding the maximum possible size of a three-te rm progression free subset has also been\nstudied in the context of general finite abelian groups. Brow n and Buhler [5] proved in 1982 that every\nthree-term progression free subset of an abelian group Ghas sizeo(|G|). Frankl, Graham and Rödl [18] gave\n2a different proof of this statement in 1987, and further upper bounds were obtained by Meshulam [23] and\nLev [21]. Every finite abelian group Gcan be written as a product of factors of the form Zn\nm, and by a simple\nproduct-construction the maximum possible size of a three- term progression free subset of Gis at least the\nproduct of the corresponding maximum subset sizes for these factors. Thus, as long as there are factors Zn\nm\nwithnsufficiently large in terms of m, Theorem 1.2 can also be used to obtain lower bounds for more g eneral\nfinite abelian groups.\nWe will give the proof of Theorem 1.2 in Section 2, apart from p ostponing the crucial construction of the\nabove-mentioned building blocks to Section 3 (recall that T heorem 1.1 follows from Theorem 1.2). We end\nwith some concluding remarks in Section 4.\nAcknowledgements. We would like to thank Eric Naslund for interesting discussi ons.\n2 Proof Overview\nAs mentioned in the introduction, our construction can be vi ewed as a higher-dimensional generalization of\nthe classical Salem–Spencer construction. However, the hi gher-dimensionality makes the construction more\nintricate, and one of the main difficulties is finding higher-d imensional building blocks of that lead to a better\nbound than the classical Salem–Spencer construction with o ne-dimensional building blocks. The following\ndefinitions are used to describe the relevant properties of t hese building blocks of our construction.\nDefinition 2.1. For a subset S⊆Zd\nm, a point y∈Sis called a non-mid-point ofSif there is no solution\nto the equation 2y=x+zwith distinct x,z∈S.\nNote that for odd mand a subset S⊆Zd\nm, a point y∈Sis a non-mid-point of Sif and only if the only\nsolution to 2y=x+zwithx,z∈Sisx=z=y. For even m, on the other hand, if y∈Sis a non-mid-point\nofS, there can be additional solutions to 2y=x+zwithx,z∈Sbesidesx=z=y. To describe these, we\nintroduce the following definition.\nDefinition 2.2. For even m, we say that two points x,y∈Zd\nmarecousins if2x= 2y.\nNote that every point in Zd\nm(for even m) is a cousin of itself. Furthermore, note that the relation o f being\ncousins is an equivalence relation on Zd\nm(in particular, it is transitive), so it divides Zd\nminto equivalence\nclasses. Also note that for a subset S⊆Zd\nmand two cousins x,y∈S, the point xis a non-mid-point of Sif\nand only if the point yis a non-mid-point of S.\nFinally, note that for even mand a subset S⊆Zd\nm, a point y∈Sis a non-mid-point of Sif and only if all\nthe solutions to 2y=x+zwithx,z∈Sare such that x=zis a cousin of y.\nDefinition 2.3. A subset S⊆Zd\nmis called non-mid-point reducible if for every non-empty subset S′⊆S,\nsome point y∈S′is a non-mid-point of S′.\nA subset S⊆Zd\nmis non-mid-point reducible, if and only if it can be reduced t o the empty set by repeatedly\nremoving non-mid-points. More precisely, Sis non-mid-point reducible, if and only if some point y1∈S\nis a non-mid-point of S, and some point y2∈S\\ {y1}is a non-mid-point of S\\ {y1}, and some point\ny3∈S\\{y1,y2}is a non-mid-point of S\\{y1,y2}, and so on, until the final set S\\{y1,y2,...,y |S|}is empty.\nOne can use non-mid-point reducible subsets of Zd\nm(for some fixed mandd) to construct three-term pro-\ngression free subsets of Zn\nm(for large n). Indeed, by appropriately generalizing the arguments of S alem and\nSpencer, one can show the following proposition.\nProposition 2.4. Letmanddbe fixed positive integers, and let S⊆Zd\nmbe a non-mid-point reducible subset\nofZd\nm. Then, there exists a subset A⊆Zn\nmof size|A| ≥ |S|n/d−O(logn)not containing a three-term arithmetic\nprogression.\nHere, the asymptotic notation O(logn)is forn→ ∞ withmanddfixed. The Salem–Spencer construction\nfor three-term progression free subsets of Fn\npcan be viewed as applying this proposition to m=pandd= 1\nwithS={0,1,...,(p−1)/2} ⊆Zp. The above-mentioned work [14] on three-term progression f ree subsets\nofFn\n5used the argument underlying this proposition for d= 3.\n3We will prove Proposition 2.4 at the end of this section. In or der to prove Theorem 1.1 using this proposition,\nwe of course also need to show that there exist reasonably lar ge non-mid-point reducible subsets S⊆Zd\nmfor\nsomed. This is the content of the following proposition, and is one of the main difficulties of this paper.\nProposition 2.5. For any positive integer m, there exists a non-mid-point reducible subset S⊆Z2\nmof size\n|S| ≥(7/24)·m2.\nWe prove Proposition 2.5 in Section 3. We remark that it is fai rly easy to find a non-mid-point reducible\nsubsetS⊆Z2\nmof size|S| ≥ ⌊(m+2)/2⌋2(for example, one can take S={0,1,...,⌊m/2⌋}2). However, this\nwould only lead to a bound of |A| ≥ ⌊(m+ 2)/2⌋n−O(logn)in Theorem 1.2 (compare [15, Theorem 3.11]),\ni.e. it would not give a constant c >1/2. The construction of the set S⊆Z2\nmof size|S| ≥(7/24)·m2\nin Proposition 2.5 is much more involved, and it also takes mo re work to show that this set Sis indeed\nnon-mid-point reducible.\nCombining Propositions 2.4 and 2.5, we can now show that Theo rem 1.2 holds for any constant c </radicalbig\n7/24\n(note that/radicalbig\n7/24≈0.54006...). In fact, for any fixed m≥2, we obtain a three-term progression free subsets\nA⊆Zn\nmof size|A| ≥(/radicalbig\n7/24·m)n−O(logn).\nProof of Theorem 1.2, assuming Propositions 2.4 and 2.5. Noting that/radicalbig\n7/24>/radicalbig\n1/4 = 1/2, choose a con-\nstantcwith1/2< c </radicalbig\n7/24. By Proposition 2.5, for every positive integer mthere exists a non-mid-point\nreducible subset S⊆Z2\nmof size|S| ≥(7/24)·m2. Now, by Proposition 2.4, for large nthere is a progression-\nfree subset A⊆Zn\nmof size\n|A| ≥ |S|n/2−O(logn)=/parenleftBig\n|S|1/2/parenrightBign−O(logn)\n≥/parenleftBig/radicalbig\n7/24·m/parenrightBign−O(logn)\n≥/parenleftBig/radicalbig\n7/24·m/parenrightBig(1−o(1))n\n,\nwhere the o(1)-term converges to zero as n→ ∞ (whilemis fixed). Given that/radicalbig\n7/24·m > cm , for all\nsufficiently large n(in terms of mand the fixed constant c), we can therefore find a progression-free subset\nA⊆Zn\nmof size|A| ≥(cm)n.\nWe end this section by proving Proposition 2.4. First, we pro ve the proposition in the case that mis odd\n(since this case involves fewer technicalities).\nProof of Proposition 2.4 for odd m.First, note that the statement is trivially true if |S| ≤1, so we may\nassume that |S| ≥2.\nFurthermore, noting that |S| ≤md, we may assume that nis divisible by d·|S|. Indeed, assuming that we have\nproved the desired statement under this assumption, for any large positive integer n(large in terms of mandd)\nwe can take some positive integer n′withn−d·|S| ≤n′≤nsuch that n′is divisible by d·|S|, and find a three-\nterm progression free subset A⊆Zn′\nmof size|A| ≥ |S|n′/d−O(logn′)≥ |S|n/d−|S|−O(logn)=|S|n/d−O(logn).\nEmbedding Zn′\nmintoZn\nmby appending n−n′zeroes to each point, this also gives a three-term progressi on\nfree subset of Zn\nmof the same size.\nSo let us from now on assume that n=d·|S|·ℓfor some large positive integer ℓ. We may now view Zn\nmas\na product Zn\nm=Zd\nm×···×Zd\nmofℓ|S|factorsZd\nm. We define\nA⊆S×···×S⊆Zd\nm×···×Zd\nm=Zn\nm\nto be the set of all (ℓ|S|)-tuples(w1,...,w ℓ|S|)∈S× ··· ×S=Sℓ|S|such that each point w∈Sappears\nexactlyℓtimes among w1,...,w ℓ|S|. Then we have\n|A|=/parenleftbiggℓ|S|\nℓ,ℓ,...,ℓ/parenrightbigg\n≥|S|ℓ|S|\n(ℓ|S|+1)|S|≥|S|n/d\n(n+1)|S|≥ |S|n/d−|S|log2(n+1))≥ |S|n/d−O(logn),\nusing that the multinomial coefficient in the second term is th e largest of the at most (ℓ|S|+1)|S|multinomial\ncoefficients with the same value ℓ|S|at the top and |S|parts at the bottom1, and using that |S| ≤mdwith\nmanddbeing fixed for the asymptotic term O(logn).\n1A more careful analysis of the multinomial coefficient with St irling’s formula√\n2π·kk+1/2e−k≤k!≤e·kk+1/2e−k(for all\nk≥1) yields |A| ≥ |S|n/d/(e2ℓ)(|S|−1)/2.\n4It remains to prove that the set Adoes not contain a three-term arithmetic progression. Supp ose for contra-\ndiction that x= (x1,...,x ℓ|S|)andy= (y1,...,y ℓ|S|)andz= (z1,...,z ℓ|S|)are three distinct elements of A\nwithx−2y+z= 0. Then we have 2yi=xi+zifor alli= 1,...,ℓ|S|.\nFor an index i∈ {1,...,ℓ|S|}, let us say that iisboring ifxi=yi=zi, and that iisinteresting otherwise.\nAsx,y,z are distinct, not all indices i∈ {1,...,ℓ|S|}can be boring. Let\nS′={yi|i∈ {1,...,ℓ|S|}interesting } ⊆S,\nthenS′/\\e}atio\\slash=∅, since there must be at least one interesting index i∈ {1,...,ℓ|S|}.\nBy definition of S′, for every interesting index i∈ {1,...,ℓ|S|}we have yi∈S′. We claim that furthermore,\nfor every interesting index i∈ {1,...,ℓ|S|}, we also have xi∈S′. Indeed, if xi=yi, then we trivially have\nxi=yi∈S′. Otherwise, if xi/\\e}atio\\slash=yi, then there are ℓindicesj∈ {1,...,ℓ|S|}\\{i}withyj=xi(asxi∈S\nappears exactly ℓtimes among y1,...,y ℓ|S|). At most ℓ−1of these indices jcan have the property that\nxj=xi, so there must be an index j∈ {1,...,ℓ|S|}\\{i}such that yj=xiandxj/\\e}atio\\slash=xi. Then we must have\nxj/\\e}atio\\slash=yj, so the index jis also interesting and hence xi=yj∈S′as claimed.\nThis shows that for every interesting index i∈ {1,...,ℓ|S|}, we have xi∈S′. Analogously, one can also show\nzi∈S′for every interesting index i∈ {1,...,ℓ|S|}.\nAs the set Sis mid-point-reducible, the non-empty subset S′⊆Smust contain some point that is a non-\nmid-point of S′. So there must be some interesting index i∈ {1,...,ℓ|S|}, such that yiis a non-mid-point of\nS′. On the other hand, we have 2yi=xi+ziandxi,zi∈S′, so by Definition 2.1 this means that xi=zi.\nHence2yi=xi+zi= 2xi, and as mis odd, this implies yi=xi. So we have xi=yi=zi, which is a\ncontradiction to ibeing an interesting index. This contradiction shows that t he setAis indeed three-term\nprogression free.\nNow, we prove Proposition 2.4 for even m. The proof is essentially the same as above, but one needs to b e\nmore careful by handling cousins in a suitable way (since the se can lead to additional solutions to 2y=x+z\nfor a non-mid-point ybesidesx=z=y).\nProof of Proposition 2.4 for even m.For the same reasons as in the previous proof for odd m, we may assume\nthat|S| ≥2and that nis divisible by d·|S|. So letn=d·|S|·ℓfor some large positive integer ℓ. We again\nviewZn\nmas a product Zn\nm=Zd\nm×···×Zd\nmofℓ|S|factorsZd\nm, and define\nA⊆S×···×S⊆Zd\nm×···×Zd\nm=Zd\nm\nto be the set of all (ℓ|S|)-tuples(w1,...,w ℓ|S|)∈S× ··· ×S=Sℓ|S|such that each point w∈Sappears\nexactlyℓtimes among w1,...,w ℓ|S|. Then we again have |A| ≥ |S|n/d−O(logn)by the same calculation as in\nthe previous proof for odd m.\nAgain, it remains to prove that the set Adoes not contain a three-term arithmetic progression. Supp ose\nfor contradiction that x= (x1,...,x ℓ|S|)andy= (y1,...,y ℓ|S|)andz= (z1,...,z ℓ|S|)are three distinct\nelements of Awithx−2y+z= 0, then2yi=xi+zifori= 1,...,ℓ|S|.\nFor an index i∈ {1,...,ℓ|S|}, let us say that iisboring , ifxi=ziandyiis a cousin of xi=zi, and let us say\nthatiisinteresting otherwise. Again, not all indices i∈ {1,...,ℓ|S|}can be boring, since then we would in\nparticular have xi=zifor alli∈ {1,...,ℓ|S|}, meaning that x=z, which contradicts x,y,z being distinct.\nSo there must be at least one interesting index i∈ {1,...,ℓ|S|}. Let\nS′={w∈S|wis a cousin of yifor some interesting i∈ {1,...,ℓ|S|}},\nthenS′⊆SandS′/\\e}atio\\slash=∅.\nClearly, for every interesting index i∈ {1,...,ℓ|S|}, we have yi∈S′(recalling that yiis a cousin of itself).\nWe again claim that we also have xi∈S′for every interesting index i∈ {1,...,ℓ|S|}. Indeed, if xiis a\ncousin of yi, then it follows directly from the definition of S′thatxi∈S′. Ifxiis not a cousin of yi, then\nleth(xi)be the number of points w∈Sthat are cousins of xi. Since each of these points appears exactly ℓ\ntimes among y1,...,y ℓ|S|, in this case there are ℓ·h(xi)indicesj∈ {1,...,ℓ|S|}\\{i}such that yjis a cousin\nofxi. At most ℓ·h(xi)−1of these indices have the property that xjis a cousin of xi, so there must be an\n5indexj∈ {1,...,ℓ|S|}\\{i}such that yjis a cousin of xibutxjis not a cousin of xi. By transitivity of the\ncousin relation, this means that xjis not a cousin of yjand therefore the index jcannot be boring. So jis\ninteresting and xiis a cousin of yj, hencexi∈S′.\nSo we have shown that xi∈S′for every interesting index i∈ {1,...,ℓ|S|}. Analogously, we can also show\nzi∈S′for every interesting index i∈ {1,...,ℓ|S|}.\nAs the set Sis mid-point-reducible, the non-empty subset S′⊆Smust contain some non-mid-point y∈S′\nofS′. By the definition of S′, for this point y∈Sthere must be some interesting index i∈ {1,...,ℓ|S|},\nsuch that yis a cousin of yi. Asy∈S′is a non-mid-point of S′, andyi∈S′is a cousin of y, the point yiis\nalso a non-mid-point of S′. On the other hand, we have 2yi=xi+ziandxi,zi∈S′, so by Definition 2.1\nthis means that xi=zi. Hence2yi=xi+zi= 2xi, and soxi=ziis a cousin of yi. So the index iis boring,\nwhich is a contradiction. Thus, the set Ais indeed three-term progression free.\n3 Construction of a non-mid-point reducible subset of Z2\nm\nIn this section we prove Proposition 2.5, which is one of the m ain difficulties for proving our main results.\nLet us fix a positive integer mthroughout this section.\nIn order to construct the desired non-mid-point reducible s ubsetS⊆Z2\nm, we map Z2\nmto a subset of the\nsquare[0,1)2in a certain way. We start by making some definitions and obser vations.\nDefinition 3.1. For real numbers a,a′∈R, we write a≡a′mod 1 ifa−a′∈Z. For two points\n(a,b),(a′,b′)∈R2, we write (a,b)≡(a′,b′) mod 1 if we have a≡a′mod 1 andb≡b′mod 1 (i.e. if\nwe have a−a′∈Zandb−b′∈Z).\nNote that for each point (a′,b′)∈R2, there exists exactly one point (a,b)∈[0,1)2with(a,b)≡(a′,b′)\nmod 1 .\nDefinition 3.2. For a finite subset S⊆[0,1)2, a point y∈Sis called a non-mid-point ofSif there is no\nsolution to the equation 2y≡x+zmod 1 with distinct x,z∈S.\nGiven real numbers α,β∈R, let us define a map ϕα,β:Z2\nm→[0,1)2as follows. Every point in Z2\nmcan be\nrepresented (uniquely) by a pair (q,r)∈ {0,1,...,m−1}2of residue classes modulo m. Let us now define\nϕα,β(q,r) = (a,b)for the unique point (a,b)∈[0,1)2with(a,b)≡(α+q/m,β+r/m) mod 1 . Note that\nthe map ϕα,β:Z2\nm→[0,1)2is injective for any α,β∈R.\nFact 3.3. Letα,β∈RandS⊆Z2\nm, and consider the image ϕα,β(S)⊆[0,1)2ofSunder the map\nϕα,β:Z2\nm→[0,1)2. Then a point y∈Sis a non-mid-point of Sif and only if its image ϕα,β(y)is a\nnon-mid-point of ϕα,β(S).\nProof. Lety∈S⊆Z2\nmbe represented by the pair (qy,ry)∈ {0,1,...,m−1}2of residue classes modulo m.\nNow, for any points x,z∈S, represented by pairs (qx,rx),(qz,rz)∈ {0,1,...,m−1}2of residue classes, we\nhave the chain of equivalences\n2y=x+z⇔2qy≡qx+qzmodmand2ry≡rx+rzmodm\n⇔2qy/m≡qx/m+qz/mmod 1 and2ry/m≡rx/m+rz/mmod 1\n⇔2(α+qy/m,β+ry/m)≡(α+qx/m,β+rx/m)+(α+qz/m,β+rz/m) mod 1\n⇔2ϕα,β(y)≡ϕα,β(x)+ϕα,β(z) mod 1 .\nThus, there is a solution to 2y=x+zwith distinct x,z∈Sif and only if there is a solution to 2ϕα,β(y)≡\nϕα,β(x)+ϕα,β(z) mod 1 with distinct x,z∈S. Sinceϕα,βis injective, this shows that yis a non-mid-point\nofSif and only if ϕα,β(y)is a non-mid-point of ϕα,β(S).\nThe subset T⊆[0,1)2defined in the following definition plays a crucial role in our proof. Using Fact 3.3, we\nwill construct non-mid-point reducible subsets of Z2\nmfrom this subset T⊆[0,1)2.\n60 7\n121\n2107\n12\n3\n41\n21\n(1\n2,1\n3)(1\n12,1\n2) (1\n3,1\n2)\n(1\n2,1\n12)(2\n3,1\n6)(1\n2,5\n6)(1\n3,1)\nT1\nT2\nFigure 3.1: The set Tdefined in Definition 3.4\nDefinition 3.4. Let us define\nT1=/braceleftBigg\n(a,b)∈/bracketleftBig\n0,1\n2/parenrightBig\n×/bracketleftBig1\n2,1/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle7\n12≤a+b≤4\n3/bracerightBigg\n∪/braceleftBigg\n(a,b)∈/bracketleftBig\n0,1\n2/parenrightBig2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglea+b >5\n6/bracerightBigg\nand\nT2=/braceleftBigg\n(a,b)∈/bracketleftBig1\n2,1/parenrightBig\n×/bracketleftBig\n0,1\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle7\n12≤a+b <5\n6and2a+b <3\n2/bracerightBigg\n,\nas well as T=T1∪T2⊆[0,1)2.\nFor an illustration of these sets, see Figure 3.1. Note that t he area of the set Tcan be computed as follows:\nFirst, the area of T1is given by\n/parenleftBig1\n2/parenrightBig2\n−1\n2·/parenleftBig1\n12/parenrightBig2\n−1\n2·/parenleftBig1\n6/parenrightBig2\n+1\n2·/parenleftBig1\n6/parenrightBig2\n=/parenleftBig1\n2/parenrightBig2\n−1\n2·/parenleftBig1\n12/parenrightBig2\n=72−1\n288=71\n288.\nNext, the area of T2is given by\n1\n2·/parenleftBig1\n3/parenrightBig2\n−1\n2·/parenleftBig1\n12/parenrightBig2\n−1\n2·/parenleftBig1\n12/parenrightBig\n·/parenleftBig1\n6/parenrightBig\n=16−1−2\n288=13\n288.\nThus, the area of Tequals\n71\n288+13\n288=84\n288=7\n24.\nThe key step for proving Proposition 2.5 is to prove the follo wing lemma concerning this subset T⊆[0,1)2.\nLemma 3.5. LetT⊆[0,1)2be defined as in Definition 3.4. Then for any non-empty finite su bsetS⊆T,\nthere is a non-mid-point y∈SofS.\nBefore proving this lemma, let us fist show how it implies Prop osition 2.5. First, we obtain the following\ncorollary from the lemma.\nCorollary 3.6. LetT⊆[0,1)2be defined as in Definition 3.4. Then for any real numbers α,β∈R, the\npreimage ϕ−1\nα,β(T)⊆Z2\nmofTunder the map ϕα,β:Z2\nm→[0,1)2is a non-mid-point reducible subset of Z2\nm.\n7Proof. For every non-empty subset S′⊆ϕ−1\nα,β(T)the image ϕα,β(S′)⊆Tis a non-empty finite subset of\nT. By Lemma 3.5, there exists a non-mid-point of ϕα,β(S′), so there is a point y∈S′such that ϕα,β(y)\nis a non-mid-point of ϕα,β(S′). By Fact 3.3, the point y∈S′is then a non-mid-point of S′. Soϕ−1\nα,β(T)is\nnon-mid-point reducible by Definition 2.3.\nThe last missing piece to deduce Proposition 2.5 is the follo wing simple consequence of the fact that Thas\narea7/24.\nFact 3.7. LetT⊆[0,1)2be defined as in Definition 3.4. There exist real numbers α,β∈R, such that the\npreimage ϕ−1\nα,β(T)ofTunder the map ϕα,β:Z2\nm→[0,1)2has size|ϕ−1\nα,β(T)| ≥(7/24)·m2.\nProof. Letα,β∈[0,1]be independent uniform random variables in the interval [0,1]. Then for every point\nx∈Z2\nm, the probability of having ϕα,β(x)∈Tis precisely the area of T⊆[0,1)2, which is 7/24. Hence\nE[|ϕ−1\nα,β(T)|] =/summationdisplay\nx∈Z2mP[x∈ϕ−1\nα,β(T)] =/summationdisplay\nx∈Z2mP[ϕα,β(x)∈T] =/summationdisplay\nx∈Z2m7/24 = (7/24)·m2,\nwhich implies that for some choice of α,β∈[0,1]we must have |ϕ−1\nα,β(T)| ≥(7/24)·m2.\nNow, combining Corollary 3.6 and Fact 3.7 immediately impli es Proposition 2.5:\nProof of Proposition 2.5. LetT⊆[0,1)2be defined as in Definition 3.4, and choose α,β∈Ras in Fact 3.7\nsuch that |ϕ−1\nα,β(T)| ≥(7/24)·m2. By Corollary 3.6, the set ϕ−1\nα,β(T)⊆Z2\nmis non-mid-point reducible, so\nthere exists a non-mid-point reducible subset of Z2\nmof size at least (7/24)·m2.\nIt remains to prove Lemma 3.5. We first record the following si mple facts that will be used in our proof.\nFact 3.8. For the sets T1,T2andT=T1∪T2defined in Definition 3.4, the following statements hold:\n(i) For all (a,b)∈T, we have 7/12≤a+b≤4/3.\n(ii) For all (a,b)∈T1and(a′,b′)∈T2witha+b=a′+b′, we have a+a′<1.\nProof. Note that (i) follows immediately from the definition of T(noting in particular that for all (a,b)∈\n[0,1/2)2witha+b >5/6we have 7/12<5/6< a+b <1/2+1/2<4/3).\nFor (ii), consider (a,b)∈T1and(a′,b′)∈T2witha+b=a′+b′. By the definition of T2we havea′+b′<5/6,\nand hence a+b <5/6. By the definition of T1this implies that (a,b)/\\e}atio\\slash∈[0,1/2)2and consequently b≥1/2.\nThus, we obtain\na=a′+b′−b≤a′+b′−1\n2\nand therefore\na+a′≤2a′+b′−1\n2<3\n2−1\n2= 1,\nwhere the last inequality follows from the definition of T2.\nFact 3.9. LetT⊆[0,1)2be defined as in Definition 3.4. Consider points (a,b),(a′,b′)∈T, and let (a∗,b∗)\nbe their mid-point (given by a∗= (a+a′)/2andb∗= (b+b′)/2). Then\na∗+b∗≤4\n3.\nFurthermore, if a∗≥1/2, then we have\na∗+b∗<13\n12.\n8Proof. For the first part, we simply observe that by Fact 3.8(i) we hav e\na∗+b∗=a+a′\n2+b+b′\n2=a+b\n2+a′+b′\n2≤4/3\n2+4/3\n2=4\n3.\nFor the second part, let us assume that a∗≥1/2. Then we must have a≥1/2ora′≥1/2, so let us assume\na′≥1/2without loss of generality. As T1⊆[0,1/2)×[0,1), this implies (a′,b′)∈T2and hence a′+b′<5/6\nby the definition of T2. Combining this with Fact 3.8(i), we obtain\na∗+b∗=a+a′\n2+b+b′\n2=a+b\n2+a′+b′\n2<4/3\n2+5/6\n2=2\n3+5\n12=13\n12.\nFact 3.10. LetT⊆[0,1)2be defined as in Definition 3.4. Consider points x,y,z∈Tsuch that 2y≡x+z\nmod 1 . Then, writing x= (ax,bx),y= (ay,by)andz= (az,bz), we have\nay+by≥ax+bx\n2+az+bz\n2. (3.1)\nProof. Let(a∗,b∗)be the mid-point of x= (ax,bx)andz= (az,bz)(given by a∗= (ax+az)/2and\nb∗= (bx+bz)/2). Note that\n(2ay,2by) = 2y≡x+z= (ax,bx)+(az,bz) = (ax+az,bx+bz) = (2a∗,2b∗) mod 1 ,\nmeaning that 2ay≡2a∗mod 1 and2by≡2b∗mod 1 . Hence a∗−ay∈ {−1/2,0,1/2}andb∗−by∈\n{−1/2,0,1/2}, and consequently (a∗+b∗)−(ay+by)∈ {−1,−1/2,0,1/2,1}.\nFurthermore, note that a∗+b∗= (ax+az)/2 + (bx+bz)/2equals the right-hand side of (3.1), and so it\nsuffices to show that ay+by≥a∗+b∗. So let us assume for contradiction that a∗+b∗> ay+by. Then we\nautomatically have a∗+b∗≥ay+by+1/2(recalling that (a∗+b∗)−(ay+by)∈ {−1,−1/2,0,1/2,1}).\nSinceb∗−by≤1/2, we must therefore have a∗≥ay. Similarly, we must have b∗≥by, sincea∗−ay≤1/2.\nIfa∗≥1/2, by the second part of Fact 3.9 we have\na∗+b∗<13\n12=7\n12+1\n2≤ay+by+1\n2,\nwhere the last inequality follows from Fact 3.8(i). This con tradictsa∗+b∗≥ay+by+1/2, so we must have\na∗<1/2.\nNow, we obtain 0≤ay≤a∗<1/2. Together with a∗−ay∈ {−1/2,0,1/2}, this implies a∗=ay. So from\na∗+b∗≥ay+by+1/2, we obtain by≤b∗−1/2<1−1/2 = 1/2.\nThus, we have ay<1/2andby<1/2, meaning that (ay,by)∈[0,1/2)2. By the definition of T, this implies\nay+by>5/6and hence\na∗+b∗≤4\n3=5\n6+1\n2< ay+by+1\n2,\nwhere the first inequality follows from the first part of Fact 3 .9. This is again a contradiction to a∗+b∗≥\nay+by+1/2.\nFor our last fact, we need some more notation. Let us define the function g: [0,1)→[0,1/2)by setting\ng(t) =/braceleftBigg\nt ift∈[0,1/2)\nt−1/2ift∈[1/2,1)\nfor allt∈[0,1). Note that for t,t′∈[0,1)with2t≡2t′mod 1 , we always have g(t) =g(t′).\nFact 3.11. LetT⊆[0,1)2be defined as in Definition 3.4. Consider points x,y,z∈Tsuch that 2y≡x+z\nmod 1 . Let us write x= (ax,bx),y= (ay,by)andz= (az,bz), and assume that ax+bx=az+bz. Then, we\nhave\ng(ay)≥g(ax)+g(az)\n2.\nFurthermore, if g(ax) =g(ay) =g(az), thenx=z.\n9Proof. As in the previous proof, let (a∗,b∗)be the mid-point of x= (ax,bx)andz= (az,bz)(given by\na∗= (ax+az)/2andb∗= (bx+bz)/2) and note that 2ay≡2a∗mod 1 and2by≡2b∗mod 1 . Hence\ng(ay) =g(a∗).\nRecall that x,z∈T=T1∪T2. We now distinguish three cases, first the case that x,z∈T1, second the case\nthatx,z∈T2, and third the case that x∈T1andz∈T2. Note that the fourth possible case that x∈T2and\nz∈T1is analogous to the third case (since we can interchange the r oles ofxandzin the statement).\nCase 1:x,z∈T1.In this case, we have ax,az∈[0,1/2)and hence also a∗= (ax+az)/2∈[0,1/2). Therefore\ng(ax) =ax,g(az) =azandg(a∗) =a∗, and consequently\ng(ay) =g(a∗) =a∗=ax+az\n2=g(ax)+g(az)\n2.\nIf furthermore g(ax) =g(ay) =g(az), thenax=g(ax) =g(az) =az. Together with ax+bx=az+bz, this\nalso implies bx=bz, sox= (ax,bx) = (az,bz) =z.\nCase 2:x,z∈T2.In this case, we have ax,az∈[1/2,1)and hence also a∗= (ax+az)/2∈[1/2,1). Therefore\ng(ax) =ax−1/2,g(az) =az−1/2andg(a∗) =a∗−1/2, and consequently\ng(ay) =g(a∗) =a∗−1/2 =ax+az−1\n2=g(ax)+g(az)\n2.\nIf furthermore g(ax) =g(ay) =g(az), thenax=g(ax)+1/2 =g(az)+1/2 =az. Together with ax+bx=\naz+bz, this also implies bx=bz, sox= (ax,bx) = (az,bz) =z.\nCase 3: x∈T1andz∈T2.In this case, we have ax∈[0,1/2)andaz∈[1/2,1). Furthermore, by Fact\n3.8(ii) we have ax+az<1and hence a∗= (ax+az)/2<1/2. Therefore g(ax) =axandg(a∗) =a∗, while\ng(az) =az−1/2. Thus we obtain\ng(ay) =g(a∗) =a∗=ax+az\n2=g(ax)+g(az)+1/2\n2> .g(ax)+g(az)\n2,\nimplying the desired inequality and showing that g(ax) =g(ay) =g(az)is not possible in this case (so the\nsecond part of the fact is tautologically true in this case).\nFinally, let us prove Lemma 3.5.\nProof of Lemma 3.5. LetS⊆Tbe a non-empty finite subset of T. Letµ= min (a,b)∈Sa+bbe the minimal\nvalue of a+bfor all points (a,b)∈S. Furthermore, define ν= min{g(a)|(a,b)∈S,a+b=µ}to be the\nminimal value of g(a)among all points (a,b)∈Switha+b=µ. Finally, let us choose a point y= (ay,by)∈S\nwithay+by=µandg(ay) =ν. Our goal is to show that yis a non-mid-point of S.\nSuppose for contradiction that yis not a non-mid-point of S, then there exist distinct points x= (ax,bx)∈S\nandz= (az,bz)∈Swith2y≡x+zmod 1 . Now, by Fact 3.10 we have\nµ=ay+by≥ax+bx\n2+az+bz\n2,\nbut on the other hand by the definition of µwe must have ax+bx≥µandaz+bz≥µ. Thus, we obtain\nax+bx=ay+by=az+bz=µ. By Fact 3.11, this implies that\nν=g(ay)≥g(ax)+g(az)\n2,\nbut on the other hand by the definition of νwe must have g(ax)≥νandg(az)≥ν(recalling that ax+bx=µ\nandaz+bz=µ). Consequently, we obtain g(ax) =g(ay) =g(az) =ν. Now, the second part of Fact 3.11\nshows that x=z, which is a contradiction. This contradiction shows that y∈Sis indeed a non-mid-point\nofS.\n104 Concluding remarks\nUnderstanding the size of the largest three-term progressi on free subset of Fn\npfor a fixed prime p≥3and\nlargenis still an open problem. Ellenberg and Gijswijt [10] proved an upper bound of (cpp)nfor some\nconstant cp<1depending on p(withcp→0.841...forp→ ∞). Our results show a lower bound of the form\n(/radicalbig\n7/24·p)n−O(logn)for any fixed prime p≥3. It is plausible that with a similar approach one can get a\nbetter constant than/radicalbig\n7/24≈0.54006..., potentially by applying Proposition 2.4 for some d >2. Of course,\nthe crucial difficulty with this approach is finding a non-mid- point reducible subset of Fd\npof size larger than\n(/radicalbig\n7/24·p)dfor some d.\nIn order to find such a set, it could conceivably be helpful to r elax the reducibility condition for the desired\nsetS⊆Fd\np(see Definition 2.3). Indeed, instead of demanding that ever y non-empty subset S′⊆Scontains\na non-mid-point (i.e. a point that is not the middle term of an arithmetic progression in S′), it is sufficient to\ndemand that every non-empty subset S′⊆Scontains a point that is not the middle term of an arithmetic\nprogression in S′or a point that is not the first or third term of an arithmetic pr ogression in S′. See [15,\nProof of Theorem 3.13] and [12, Proof of Theorem 2.3], where s uch arguments were used (and see also [13],\nwhere in a related situation a further extension based on int eger programming was used).\nFor a fixed choice of a (small) prime p≥3, or more generally for a fixed choice of a (small) integer m≥3,\nit may be possible to obtain a non-mid-point reducible subse tS⊆Z2\nmof size more than (7/24)·m2(i.e.\nlarger than the size guaranteed in Proposition 2.5), by appl ying Corollary 3.6 to the set S=ϕ−1\nα,β(T)⊆Z2\nm\nfor some smartly chosen values of α,β∈R(rather than taking αandβrandomly as in the proof of Fact\n3.7). However, for larger m, the size of the set ϕ−1\nα,β(T)is close to (7/24)·m2for any choice of α,β∈R.\nIn particular, for sufficiently large m, one can make the construction of the set Ain Theorems 1.1 and 1.2\ncompletely explicit by taking S=ϕ−1\n0,0(T)in the proof of Proposition 2.5.\nIt is well-known that for every fixed prime p≥3, the maximum size of a three-term progression free subset of\nFn\npis of the form (γpp)n−o(n)for alln, whereγpis a constant only depending on p(see e.g. [15, Proposition\n3.8]). Ellenberg and Gijswijt [10] showed that γp<1for allpand that limsupp→∞γp<0.842(see [3, Eq.\n(4.11)]). Our work shows that γp>0.54for allpand hence in particular that liminf p→∞γp>1/2. It is an\nopen question whether the limit limp→∞γpexists, see [15, Conjecture 3.9].\nFinally, we remark that the known constructions in the integ er case, i.e. the constructions of large subsets\nof{0,...,N}without three-term arithmetic progressions, are also of a h igh-dimensional nature. In these\nconstructions, one chooses integers tandnsuch that (2t−1)n≤N. One can then obtain a three-term\nprogression free subset of {0,...,N}from a three-term progression free subset of {0,...,t−1}n⊆Znby\nconsidering n-tuples in {0,...,t−1}n⊆Znas numbers written in base 2t−1(for three such numbers x,y,z\nthere cannot be any carries in the equation 2y=x+zwritten in base 2t−1, and hence any three-term\narithmetic progression of such numbers would correspond to a three-term progression in the original subset\nof{0,...,t−1}n). The earliest such construction, described by Erdős and Tu rán [16] and attributed by\nthem to Szekeres, used this idea for t= 2, taking the three-term progression free set {0,1}n(the resulting\nsubset of {0,...,N}then consists of all integers whose base-3 representation d oes not contain the digit 2).\nSzekeres also conjectured that this construction is best po ssible, which turned out to be incorrect. Salem\nand Spencer [27] generalized Szekeres’ idea, by considerin g subsets of {0,...,t−1}nwith those points where\neach of the numbers 0,...,t−1occurs in a prescribed number of coordinates, i.e. when writ ten as integers\nin basem= 2t−1, each of the digits 0,...,t−1occurs a prescribed number of times. Finally, Behrend’s\nconstruction [2] is based on taking the intersection of {0,...,t−1}nwith a suitably chosen sphere (by\nconvexity of the sphere, such an intersection cannot contai n any three-term arithmetic progressions). Elkin\n[9] slightly improved Behrend’s lower bound for the size of t he largest three-term progression free subset of\n{0,...,N}to a bound on the order of N(logN)1/42−2√\n2log2Nby taking a subset of {0,...,t−1}nconsisting\nof points very close to a given sphere.\nGiven the high-dimensional nature of all these constructio ns for progression-free sets in the integers, relying\non three-term progression free subsets of {0,...,t−1}n⊆Zn, it is natural to wonder whether the ideas in\nthis paper might also be useful for this integer version of th e problem. However, one crucial difficulty is that\nin order to avoid carries when performing the additions in th e equation 2y=x+z, one seems to need to\nrestrict oneself to only half of the available digits in base 2t−1.\n11References\n[1] N. Alon, A. Shpilka, and C. Umans, On sunflowers and matrix multiplication , Comput. Complexity 22\n(2013), 219–243.\n[2] F. A. Behrend, On sets of integers which contain no three terms in arithmetical progression , Proc. Nat.\nAcad. Sci. U.S.A. 32(1946), 331–332.\n[3] J. Blasiak, T. Church, H. Cohn, J. A. Grochow, E. Naslund, W. F. Sawin, and C. Umans, On cap sets\nand the group-theoretic approach to matrix multiplication , Discrete Anal. 2017, Paper No. 3, 27pp.\n[4] T. F. Bloom and O. Sisask, An improvement to the Kelley-Meka bounds on three-term arithm etic pro-\ngressions , preprint, 2023, arXiv:2309.02353.\n[5] T. C. Brown and J. P. Buhler, A density version of a geometric Ramsey theorem , J. Combin. Theory\nSer. A 32(1982), 20–34.\n[6] A. R. Calderbank and P. C. Fishburn, Maximal three-independent subsets of {0,1,2}n, Des. Codes\nCryptogr. 4(1994), 203–211.\n[7] E. Croot, V. F. Lev, and P. P. Pach, Progression-free sets in Zn\n4are exponentially small , Ann. of Math.\n185(2017), 331–337.\n[8] Y. Edel, Extensions of generalized product caps , Des. Codes Cryptogr. 31(2004), 5–14.\n[9] M. Elkin, An improved construction of progression-free sets , Israel J. Math. 184(2011), 93–128.\n[10] J. S. Ellenberg and D. Gijswijt, On large subsets of Fn\nqwith no three-term arithmetic progression , Ann.\nof Math. 185(2017), 339–343.\n[11] C. Elsholtz, J. Führer, E. Füredi, B. Kovács, P. P. Pach, D. G. Simon, and N. Velich, Maximal line-free\nsets inFn\np, preprint, 2023, arXiv:2310.03382.\n[12] C. Elsholtz, B. Klahn, and G. F. Lipnik, Large subsets of Zn\nmwithout arithmetic progressions , Des. Codes\nCryptogr. 91(2023), 1443–1452.\n[13] C. Elsholtz and G. F. Lipnik, Exponentially larger affine and projective caps , Mathematika 69(2023),\n232–249.\n[14] C. Elsholtz, G. F. Lipnik, and M. Siebenhofer, New constructions of large caps , chapter 5 in Ph.D.\nThesis of G. F. Lipnik, On Integer Partitions, Caps, Progres sion-Free Sets and q-Regular Sequences,\nGraz University of Technology, March 2023.\n[15] C. Elsholtz and P. P. Pach, Caps and progression-free sets in Zn\nm, Des. Codes Cryptogr. 88(2020),\n2133–2170.\n[16] P. Erdős and P. Turán, On Some Sequences of Integers , J. London Math. Soc. 11(1936), 261–264.\n[17] J. Fox and H. T. Pham, Popular progression differences in vector spaces II , Discrete Anal. 2019, Paper\nNo. 16, 39 pp.\n[18] P. Frankl, R. L. Graham, and V. Rödl, On subsets of abelian groups with no 3-term arithmetic progres -\nsion, J. Combin. Theory Ser. A 45(1987), 157–161.\n[19] J. Führer, More on maximal line-free sets in Fn\np, submitted, 2023.\n[20] Z. Kelley and R. Meka, Strong Bounds for 3-Progressions , preprint, 2023, arXiv:2302.05537.\n[21] V. F. Lev, Progression-free sets in finite abelian groups , J. Number Theory 104(2004), 162–169.\n[22] Y. Lin and J. Wolf, On subsets of Fn\nqcontaining no k-term progressions , European J. Combin. 31(2010),\n1398–1403.\n12[23] R. Meshulam, On subsets of finite abelian groups with no 3-term arithmetic pr ogressions , J. Combin.\nTheory Ser. A 71(1995), 168–172.\n[24] E. Naslund, Lower bounds for the Shannon Capacity of Hypergraphs , manuscript, 2024.\n[25] S. Peluse, Finite field models in arithmetic combinatorics – twenty years o n, preprint, 2023,\narXiv:2312.08100.\n[26] B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog , M. P. Kumar, E. Dupont, F. J. R. Ruiz,\nJ. S. Ellenberg, P. Wang, O. Fawzi, P. Kohli, and A. Fawzi, Mathematical discoveries from program\nsearch with large language models , Nature, to appear.\n[27] R. Salem and D. C. Spencer, On sets of integers which contain no three terms in arithmetical progression ,\nProc. Nat. Acad. Sci. U.S.A. 28(1942), 561–563.\n[28] T. Tao and V. H . Vu, Additive Combinatorics , Cambridge University Press, 2006.\n[29] F. Tyrrell, New lower bounds for cap sets , Discrete Anal. 2023, Paper No. 20, 18pp.\nMSC classification:\n05D05 Extremal set theory\n11B25 Arithmetic progressions\n13"    },    {        "title": "2401.12882v1.Model_Free__δ__Policy_Iteration_Based_on_Damped_Newton_Method_for_Nonlinear_Continuous_Time_H__infty__Tracking_Control.pdf",        "content": "1 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \nModel -Free δ-Policy Iteration Based on Damped \nNewton Method for Nonlinear Continuous -Time H∞ \nTracking Control  \n \nQi Wang  \n \n Abstract—This paper presents a δ-PI algorithm which is based \non damped Newton method for the H∞ tracking control problem \nof unknown continuous -time nonlinear system. A discounted \nperformance function and an augmented system are used to get \nthe tracking Hamilton -Jacobi-Isaac (HJI) equation. Tracking \nHJI equation is a nonlinear partial differential equation, \ntraditional reinforcement learning methods for solving the \ntracking HJI equation are mostly based on the Newton method, \nwhich usually only satisfies local convergence and needs a good \ninitial guess. Based upon the damped Newton iteration operator \nequation, a generalized tracking Bellman equation is derived \nfirstly. The δ-PI algorithm can seek the optimal solution of the \ntracking HJI equation by iteratively solving the generalized \ntracking Bellman equation. On -policy learning and off -policy \nlearning δ-PI reinforcement learning methods are provided, \nrespectively. Off -policy version δ-PI algorithm is a model -free \nalgorithm which can be performed without making use of a \npriori knowledge of the system dynamics. NN -based \nimplementation scheme for the off -policy δ-PI algorithms is \nshown. The suitability of the model -free δ-PI algorithm is \nillustrated with a nonlinear system simulation.  \n \nIndex Terms —Damped Newton method , δ-policy iteration , \ngeneralized tracking Bellman equation , off-policy and on-policy \nlearning , tracking Hamilton -Jacobi -Isaacs equation .  \n \nI. INTRODUCTION  \nHE H∞ optimal control theory is a well -known robust \ncontrol method which has been widely applied in \ncontroller design for systems with disturbance, and the \nadverse impact of disturbance signals on controller \nperformance can be attenuated. The H∞ control method aims \nto find a control policy that the closed -loop system satisfies \ndisturbance attenuation condition, and the system dynamics is \nlocally asymptotically stable while there is no disturbance \nsignal [1]. The sufficient condition for the solution of the H∞ \ncontrol problem is to solve the Hamilton -Jacobi -Isaacs (HJI) \nequation , which is a nonlinear partial differential equation  [2]. \nFor tracking problems, traditional methods to H∞ controller \ndesign involve two steps, first, design a feedforward controller \nto perform tracking, then, design an H∞ feedback controller to \nstabilize the error dynamics of the tracking system [3] -[6]. \n \nManuscript received December  10, 2023.  \nQ. Wang is with the Department of General System Technology, China \nAirborne Missile Academy, Luoyang, HN 471009, China (e -mail: \nwangqibuaa@126.com).   \nColor versions of one or more of the figures in this article are available \nonline at http://ieeexplore.ieee.org  These approaches were suboptimal because the control cost \ngenerated by feedforward control inputs is ignored in the \ndesign of  H∞ controller.  \nIn order to solve the problem of neglecting the control cost \ncaused by feedforward control component in the performance \nindex functional, a new discounted performance function al \nwas proposed  for the tracking  problem H∞ control, and a more \nuniversal definition of the L2-gain was developed, where the \nenergies of the tracking error and the whole control input were  \nweighted by a discount factor of exponential decay type in the \nperformance function [6]. This method corrected the \ntraditional method of only including the control cost of the \nfeedback control component in the performance function, and \nthen a new tracking HJI equation concerned with the \nexponential discounted performance function was presented, \nwhich can give both the feedback component and feedforward \ncomponent of the control input synchronously.  \nHJI equation is a nonlinear partial differential equation, \nwhich is difficult to obtain the closed -form solution for \nnonlinear systems, and the complete knowledge of the \nnonlinear system dynamics is required for solving the \nanalytical solutions. Therefore, i t is even worse  difficult to \nsolve the analytical solution of H∞ control  in practical \napplication scenarios, when the accurate system models \ncannot be obtained or acquisition cost is high. To overcome \nthese difficulties, reinforcement learning (RL) and adaptive \ndynamic programming  (ADP) techniques have been widely \nused in solving H2 optimal control and H∞ optimal control \nproblems in the past few decades,  and have been successfully \napplied in many  practical scenarios . As an interdisciplinary \nsubject in the communities of artificial intelligence and \ncontrol, RL and ADP which can be used for approximately \nsolving the optimal control problems, were widely introduced \nand developed in the computational intelligence and machine \nlearning fields [7], [8]. The solution of the HJB equation or \nHJI equation is not solved directly in the ADP algorithms, \ninstead, random or directional search is made in the strategy \nspace of admissible control. The solution includes a \ntechnology called policy iteration (PI), which was first \npresented in the solution framework of Markov decision \nproblems [9]. In fact, the PI algorithm requires two iterations: \npolicy evaluation is the first step, in this step, the value \nfunction related to the stable control strategy is evaluated and \ncalculated; policy update is the second step, in which control \nstrategy is improved through optimizing the value function. T 2 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \nRepeat these two steps until the policy update step no longer \nachieves optimization results, indicating that the control \nstrategy converges to the optimal one [10]. PI algorithm is in \nessence the Newton’s method [11]. For continuous time linear \ndynamic system and nonlinear dynamic system, the PI \nalgorithm has been widely used to solve the optimal state -\nfeedback control and differential games in [12] -[15]. \nSynchronous policy iteration method was introduced in [16], \nwhich implemented simultaneous continuous -time adjustment \nof both actor and critic neural networks, and a persistence of \nexcitation condition was involve d to guarantee convergence. \nHowever, all of the above -mentioned algorithms are model -\nbased approaches, the complete knowledge of the system \ndynamics is required. It is well known that modeling and \nidentification for an unknown nonlinear system are most often \ntime consuming, which require iterative execution of model \nassumption, parameter identification and model validation. To \nobviate the dependence on complete knowledge of the system \ndynamics, a direct adaptive optimal control algorithm was \npresented, in which a discretized version Bellman equation \n(BE) was introduced to approximate the solution to the HJB \nequation without using the information of the drift dynamics \nof system [17]. This method was extended to optimal tracking \ncontrol of nonlinear constrained -input systems [18], and \nnamed integral reinforcement learning (IRL). A simultaneous \npolicy update algorithm (SPUA) was introduced for solving \nH∞ control problem [19], it was a generalization of the IRL \nalgorithm in H∞ control. SUPA is an on -policy learning \nmethod, which have several drawbacks for solving the H∞ \ncontrol problem, such as disturbance signal should be \nadjustable and exploration space is limited. To overcome the \ndrawbacks, an off -policy policy iteration method was \npresented by using arbitrarily behavior policy in data sampling \non the state and input domain for optimal control design [20] -\n[22]. Subsequently, off -policy learning methods for H∞ control \nproblems were proposed for partially unknown nonlinear \nsystems and completely unknown nonlinear systems, \nrespectively [23] -[25]. The off -policy learning method was \nemployed to solve H∞ tracking control [6]. It is of interest to \nnote that the on -policy and off -policy PI algorithms mentioned \nabove are in essence the Newton method, Newton’s iterative \napproach is locally optimized, and finding suitable initial \nparameters or developing global methods is a difficult \nproblem. In recent years, some new methods, such as \nintegration scheme of policy iteration and value iteration, \nmodel -free λ -PI algorithm, and costate -supplement ADP \nalgorithm, have been proposed to improve convergence [26] -\n[28].  \nTill present, the development of policy iteration  algorithms \nand theories for H∞ tracking control design of continuous -time \nnonlinear system is still an open issue, which promotes this \nresearch. It is worth noting that damped Newton method, \nwhich leverages a non -unit step -size in the Newton direction \nby multiplying a damped parameter δ is an effective algorithm \nfor solving equations. Although damped Newton’s method \nmakes the convergence slower than Newton’s method, damped Newton method could be globally convergent. \nTherefore, the damped Newton method has the advantage of \nenhancing the robustness of the convergence of solving \nprocess with regarding to the initial guess [29].  \nIn this paper, the H∞ tracking control design of nonlinear \ncontinuous -time systems  is consider ed, and a novel policy \niteration algorithm based on damped Newton method  is \ndevelop ed for solving tracking HJI equation, name d δ-policy \niteration ( δ-PI) algorithm . A generalized tracking Bellman \nequation is derived  by constructing a damped Newton iteration \noperator equatio n. The generalized tracking Bellman equation \nis an extension of tracking Bellman equation introduced in [6]. \nAnd then, by iterating on the generalized tracking Bellman \nequation, one can obtain the approximate optimal solution of \nthe tracking HJI equation. On -policy and off -policy δ-PI \nreinforcement learning methods are given respectively.  \nThe organization of this paper is as follows. The \ncontinuous -time nonlinear H∞ tracking control problem \nformulation and preliminaries are presented in section II. The \nNewton method based on -policy PI algorithm  is presented  in \nsection  III. The damped Newton method based on -policy and \noff-policy δ-PI RL methods , and neural -network based \nimplementation schemes  are provided in section  IV. The \neffectiveness of the δ-PI RL algorithm is illustrated with a \nsimulation example in section V. Finally, section VI concludes \nthe paper.  \nII. PROBLEM FORMULATION AND PRELIMINARIES  \nIn this section, the background knowledge reviews and \npreliminar ies of H∞ tracking control problem are provided. \nConsider a class of affine in control input nonlinear \ncontinuous -time system defined as follows  \n𝑥̇=𝑓(𝑥)+𝑔(𝑥)𝑢+𝑘(𝑥)𝑑                       (1) \nwhere 𝑥∈Ω⊂ℜ𝑛 is state vector of the system, and 𝑑∈ℜ𝑞 \nand 𝑢∈ℜ𝑚 represent disturbance input vector and control \ninput vector respectively. 𝑓(𝑥)∈ℜ𝑛 is the internal dynamics, \n𝑔(𝑥)∈ℜ𝑛×𝑚 and 𝑘(𝑥)∈ℜ𝑛×𝑞 are the input -to-state matrix \nand the disturbance coefficient matrix, respectively. On a \ncompact set Ω, 𝑓(𝑥), 𝑔(𝑥) and 𝑘(𝑥) meet locally Lipschitz \ncontinuity and 𝑓(0)=0, and that the system (1) is \nstabilizable.  \nH∞ tracking control is a robust tracking control method, \nwhose goal is to reduce the impact of external disturbance \nwhen system (1) tracks a reference trajectory 𝑟(𝑡)⊂ℜ𝑛.  \nDefine the tracking error 𝑒𝑑 as follows.  \n𝑒𝑑≜𝑥(𝑡)−𝑟(𝑡)                                (2) \nAssumption 1 : Reference trajectory 𝑟(𝑡) is bounded and \nthere exists a Lipschitz continuous command generator \nfunction ℎ𝑑(𝑟(𝑡))∈ℜ𝑛 and ℎ𝑑(0)=0, such that  \n𝑟̇(𝑡)=ℎ𝑑(𝑟(𝑡))                                 (3) \nBy using equations (1) -(3), the error dynamics of the \ntracking system (1) can be obtained as  \n𝑒̇𝑑=𝑓(𝑥)+𝑔(𝑥)𝑢+𝑘(𝑥)𝑑−ℎ𝑑(𝑟)         (4) \nFor any 𝑑∈𝐿2[0,∞), a more general version of \ndisturbance attenuation condition  of H∞ tracking problem is \ngiven as follows [6].   3 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \n∫𝑒−𝛼(𝜏−𝑡)(𝑒𝑑𝑇𝑄𝑒𝑑+𝑢𝑇𝑅𝑢)𝑑𝜏∞\n𝑡\n∫𝑒−𝛼(𝜏−𝑡)(𝑑𝑇𝑑)𝑑τ∞\n𝑡≤𝛾2                  (5) \nwhere 𝛼>0 is a discount factor, 𝑄>0 and 𝑅>0 are \npositive definite symmetric matrix and positive definite \ndiagonal matrix respectively , and 𝛾>0 is a predetermined \ndisturbance attenuation level.  \nThe H∞ optimal control is interested in finding the smallest \n𝛾∗, such that there exists a solution that satisfies the \ndisturbance attenuation condition (5). However, it is \nimpossible to get the smallest value of 𝛾∗ for general nonlinear \nsystems , thus it is assumed that 𝛾>𝛾∗ in this paper [30].  \nDefinition  1 (Bounded L2-Gain): If system (1) satisfies the \ndisturbance attenuation condition (5), then the 𝐿2-gain of \nsystem (1) is less than or equal to 𝛾. \nRemark 1 : The use of discount factor of exponential decay \ntype is crucial in (5). This is because the control input in \ntracking problems usually does not converge to zero, therefore \npunishing the energies of the tracking error and the whole \ncontrol input without a discount factor of exponential decay \ntype will make the performance index function unbounded.  \nThe purpose of the tracking problem H∞ control is to find a \ncontrol strategy 𝑢=𝑢(𝑒𝑑,𝑟) that depends on the tracking \nerror 𝑒𝑑 and the reference trajectory 𝑟(𝑡), such that the closed -\nloop error dynamics  (4) satisfies the disturbance attenuation \ncondition (5), and the tracking error meets local asymptotic \nstability when 𝑑=0. \nDefine the extended system state variable 𝑋(𝑡)=\n[𝑒𝑑(𝑡)𝑇 𝑟(𝑡)𝑇]𝑇∈𝛺⊂ℝ2𝑛, then the augmented system \nwhich is consist ed of tracking error dynamics and reference \nsignal command  generator function  is given as follows.  \n𝑋̇(𝑡)=𝐹(𝑋(𝑡))+𝐺(𝑋(𝑡))𝑢(𝑡)+𝐾(𝑋(𝑡))𝑑(𝑡)          (6) \nwhere 𝐹(𝑋(𝑡))=[𝑓(𝑒𝑑+𝑟)−ℎ𝑑(𝑟)\nℎ𝑑(𝑟)], 𝐺(𝑋)=[𝑔(𝑒𝑑+𝑟)\n0], \nand 𝐾(𝑋(𝑡))=[𝑘(𝑒𝑑+𝑟)\n0]. Based upon the extended system \n(6), the attenuation condition of disturbance (5) becomes  \n∫𝑒−𝛼(𝜏−𝑡)(𝑋𝑇𝑄𝑇𝑋+𝑢𝑇𝑅𝑢)𝑑𝜏∞\n𝑡≤𝛾2∫𝑒−𝛼(𝜏−𝑡)(𝑑𝑇𝑑)𝑑τ∞\n𝑡 \n(7) \nwhere 𝑄𝑇=[𝑄0\n00]. \nDefine the discount performance index function of the \ntracking problem as  \n𝐽(𝑢,𝑑)=∫𝑒−𝛼(𝜏−𝑡)(𝑋𝑇𝑄𝑇𝑋+𝑢𝑇𝑅𝑢−𝛾2𝑑𝑇𝑑)𝑑𝜏∞\n𝑡      (8) \nNote that for the H∞ tracking problem, solving the control \nstrategy which satisfies the bounded L2-gain condition is \nequivalent to solving a differential game problem of the \naugmented system, i.e., a min -max optimization problem. As \nis well known, the two -person  zero-sum differential games \nproblem is closely related to H∞ control [ 31], that is, the \nsolvability of the H∞ control problem is equivalent to the \nsolvability of the following two -person zero-sum differential \ngame s problem . \n𝑉∗(𝑋)=𝐽(𝑢∗,𝑑∗)=min \n𝑢max\n𝑑 𝐽(𝑢,𝑑)                      (9) where 𝑉∗(𝑋) is the optimal value function. If the Nash \ncondition given below holds, then the two -person zero -sum \ndifferential games problem has a unique saddle point solution.  \n𝑉∗(𝑋)=min \n𝑢max\n𝑑 𝐽(𝑢,𝑑)=max \n𝑑min \n𝑢𝐽(𝑢,𝑑)             (10)  \nCalculate the differential on both sides of (8), and note that \n𝑉(𝑋)=𝐽(𝑢,𝑑), then the tracking Bellman equation is \nobtained as follows.  \n𝐻(𝑉,𝑢,𝑑)≜ \n𝑋𝑇𝑄𝑇𝑋+𝑢𝑇𝑅𝑢−𝛾2𝑑𝑇𝑑−𝛼𝑉+(∂𝑉\n∂𝑋)𝑇\n(𝐹+𝐺𝑢+𝐾𝑑)=0  \n(11) \nEmploying the first -order necessary conditions \n∂𝐻(𝑉∗,𝑢,𝑑)/∂𝑑=0and ∂𝐻(𝑉∗,𝑢,𝑑)/∂𝑢=0 for the min -\nmax optimization problem, one can obtain the optimal control \ninput and disturbance input as  \n𝑢∗=−1\n2𝑅−1𝐺𝑇𝑉𝑋∗                                 (12) \n𝑑∗=1\n2𝛾2𝐾𝑇𝑉𝑋∗                                       (13) \nwhere 𝑉𝑋∗=∇𝑉∗=∂𝑉∗/∂𝑋, 𝑉∗ is the optimal value function \ndescribed in (9). By substituting the optimal control policy (12) \nand the optimal disturbance signal (13) into (11), one can \nobtain the tracking problem HJI equation, as follows.  \n𝐻(𝑉∗,𝑢∗,𝑑∗)≜𝑋𝑇𝑄𝑇𝑋−𝛼𝑉∗+(𝑉𝑋∗)𝑇𝐹 \n−1\n4(𝑉𝑋∗)𝑇𝐺𝑅−1𝐺𝑇𝑉𝑋∗+1\n4𝛾2(𝑉𝑋∗)𝑇𝐾𝐾𝑇𝑉𝑋∗=0    (14) \nLemma 1  ([6, Theorem 1]) : Take  the H∞ tracking control \nproblem as a two -person zero-sum differential game s problem  \nof the augmented system  with performance index function (8), \nthen the control strateg y pairs  defined in (12) and (13) offer as  \nNash equilibrium saddle -point solutions.  \nLemma 2 ([6, Theorem 2]) : Assume that the tracking HJI \nequation (14) has a continuous positive -semidefinite solution \n𝑉∗, then the control input given in (12) makes the L2-gain of \nthe closed -loop augmented system (6) less than or equal to 𝛾. \nReference [6] proved the existence of the upper bound 𝛼∗ \non the discount factor 𝛼 to ensure the stability of the solution \nof the tracking HJI equation (14). In practice, a large \nweighting design matrix 𝑄 and/or a very small discount factor \n𝛼 can always be chose to ensure that 𝛼≤𝛼∗. \nIII. NEWTON ’S ITERATIVE METHOD  \nNote that the tracking Bellman equation (11) is linear with \nrespect to the value function 𝑉, whereas the tracking HJI \nequation (14) is nonlinear with respect to the optimal value \nfunction 𝑉∗. Therefore, solving the value function 𝑉 in the \ntracking Bellman equation is easier than solving 𝑉∗ in the \ntracking HJI equation. The PI algorithm for tracking problems \nproposed in [6] decomposes the solution of the tracking HJI \nequation into a series of solutions of tracking Bellman \nequation, instead of directly solving from the tracking HJI \nequation. The PI algorithm is as follows.  \n \n \nAlgorithm 1  PI for H∞ Tracking Control.  \nStep 1: Set 𝑖=0, give an initial admissible control 𝑢𝑖 and 4 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \na disturbance signal 𝑑𝑖. \nStep 2: Solve 𝑉𝑖+1(𝑋) with 𝑉𝑖+1(0)=0 using the tracking \nBellman equation below  \n𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖−𝛼𝑉𝑖+1 \n+(∂𝑉𝑖+1\n∂𝑋)𝑇\n(𝐹+𝐺𝑢����+𝐾𝑑𝑖)=0           (15)  \nStep 3: Update the disturbance policy and control policy \nusing  \n𝑢𝑖+1=−1\n2𝑅−1𝐺T∂𝑉𝑖+1\n∂𝑋                      (16)  \n𝑑𝑖+1=1\n2𝛾2𝐾T∂𝑉𝑖+1\n∂𝑋                              (17)  \nStep 4:  Set 𝑖=𝑖+1, move to Step 2 and go on iteration, \nuntil ‖𝑉𝑖+1 − 𝑉𝑖‖Ω≤𝜀, 𝜀 is a small positive number.  \n \nIn fact, the convergence of Algorithm 1 can be \ndemonstrated by proving that the iterations of (15) -(17) are \nessentially Newton iteration method that converge to the \nunique solution of the tracking HJI equation (14).  \nFirstly, consider a Banach space 𝕍={𝑉(𝑋)|𝑉(𝑋): Ω→ℜ}, \nand define the following mapping 𝒢: 𝕍→𝕍 on 𝕍: \n𝒢(𝑉)≜𝑋𝑇𝑄𝑇𝑋−𝛼𝑉+𝑉𝑋𝑇𝐹−1\n4𝑉𝑋𝑇𝐺𝑅−1𝐺𝑇𝑉𝑋  \n+1\n4𝛾2𝑉𝑋𝑇𝐾𝐾𝑇𝑉𝑋                                        (18) \nThe Fré chet derivative of mapping 𝒢(𝑉) is denoted as \n𝒢′(𝑉), and the Fré chet differential is written as 𝒢′(𝑉)𝑊, \nwhere 𝑊∈𝕍̃⊂𝕍 and 𝕍̃ is a neighborhood of 𝑉. Since the \nFré chet derivative is usually difficult to calculate directly, the \nGâ teaux derivative is an alternative . The Gâ teaux differential \nℒ(𝑊) of  𝒢 at the point 𝑉 is [32]  \nℒ(𝑊)=lim\n𝑠→0𝒢(𝑉+𝑠𝑊)−𝒢(𝑉)\n𝑠                    (19) \nwhere ℒ is the Gâ teaux derivative of 𝒢 at the point 𝑉, 𝑊∈𝕍̃, \n‖𝑊‖=1, 𝑠 is a real number. Equation (19) gives a method to \ncalculate Gâ teaux derivative, rather than Fré chet derivative. \nThe Lemma  3 gives the relationship between Gâ teaux \nderivative and Fré chet derivative, and Lemma  4 provides the \nmethod of calculation for Fré chet differential by using \nGâ teaux differential . \nLemma 3 [33]: If Gâ teaux derivative ℒ exists in some \nneighborhood of 𝑉, and if ℒ is continuous at the point 𝑉, then \nℒ is also a Fré chet derivative at the point 𝑉. \nLemma 4:  On the Banach space 𝕍, ∀𝑉∈𝕍, consider the \nmapping 𝒢 defined in (18), the Fré chet differential of 𝒢 can be \ncomputed as  \n𝒢′(𝑉)𝑊=∇(𝑊)𝑇𝐹−𝛼(𝑊)−1\n4∇(𝑉)𝑇𝐺𝑅−1𝐺𝑇∇(𝑊)  \n                     −1\n4∇(𝑊)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉)+1\n4𝛾2∇(𝑉)𝑇𝐾𝐾𝑇∇(𝑊) \n+1\n4𝛾2∇(𝑊)𝑇𝐾𝐾𝑇∇(𝑉)                                          (20)  \nwhere ∇(∙)= ∂(∙)/∂𝑋. \nProof:  The molecule of (19) can be calculated as  \n𝐺(𝑉+𝑠𝑊)−𝐺(𝑉)=𝑋𝑇𝑄𝑋+∇(𝑉+𝑠𝑊)𝑇𝐹 \n−𝛼(𝑉+𝑠𝑊)−1\n4∇(𝑉+𝑠𝑊)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉+𝑠𝑊)  +1\n4𝛾2∇(𝑉+𝑠𝑊)𝑇𝐾𝐾𝑇∇(𝑉+𝑠𝑊)−[𝑋𝑇𝑄𝑋+∇(𝑉)𝑇𝐹−\n𝛼(𝑉)−1\n4∇(𝑉)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉)+1\n4𝛾2∇(𝑉)𝑇𝐾𝐾𝑇∇(𝑉)]=\n𝑠∇(𝑊)𝑇𝐹−𝛼𝑠(𝑊)−1\n4𝑠∇(𝑉)𝑇𝐺𝑅−1𝐺𝑇∇(𝑊)  \n−1\n4𝑠∇(𝑊)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉)−1\n4𝑠2∇(𝑊)𝑇𝐺𝑅−1𝐺𝑇∇(𝑊)  \n+1\n4𝛾2𝑠∇(𝑉)𝑇𝐾𝐾𝑇∇(𝑊)+1\n4𝛾2𝑠∇(𝑊)𝑇𝐾𝐾𝑇∇(𝑉)  \n+1\n4𝛾2𝑠2∇(𝑊)𝑇𝐾𝐾𝑇∇(𝑊)     \n(21) \nThen  \nℒ(𝑊)=lim\n𝑠→0𝒢(𝑉+𝑠𝑊)−𝒢(𝑉)\n𝑠 \n        =∇(𝑊)𝑇𝐹−𝛼(𝑊)−1\n4∇(𝑉)𝑇𝐺𝑅−1𝐺𝑇∇(𝑊)     \n           −1\n4∇(𝑊)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉)+1\n4𝛾2∇(𝑉)𝑇𝐾𝐾𝑇∇(𝑊)  \n+1\n4𝛾2∇(𝑊)𝑇𝐾𝐾𝑇∇(𝑉)                                               (22)  \nOn the other hand, use the same method as in [15], it can be \nproven that ℒ is a continuous mapping. According to Lemma  3, \nℒ(𝑊) is the Fré chet differential, ℒ is the Fré chet derivative at \n𝑉. This completes the proof.                                                □ \nThe following theorem will prove that the Algorithm 1 \ngiven above is mathematically equivalent to Newton’s \niteration in the Banach space 𝕍. \nTheorem 1 : Define mapping 𝒯:𝕍→𝕍 on Banach space  \n𝒯(𝑉)=𝑉−(𝒢′(𝑉))−1𝒢(𝑉)                        (23) \nThen, the following Newton’s iteration equation with (16) \nand (17) is equivalent to the PI algorithm from (15) to (17).  \n𝑉𝑖+1=𝒯(𝑉𝑖)=𝑉𝑖−(𝒢′(𝑉𝑖))−1𝒢(𝑉𝑖)                  (24) \nProof: Equation (24) can be rewritten as  \n𝒢′(𝑉𝑖)𝑉𝑖+1=𝒢′(𝑉𝑖)𝑉𝑖−𝒢(𝑉𝑖)                      (25) \nBy using Lemma  4, one can get  \n𝒢′(𝑉𝑖)𝑉𝑖+1=∇(𝑉𝑖+1)𝑇𝐹−𝛼(𝑉𝑖+1)  \n−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖+1)−1\n4∇(𝑉𝑖+1)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)  \n+1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖+1)+1\n4𝛾2∇(𝑉𝑖+1)𝑇𝐾𝐾𝑇∇(𝑉𝑖)     (26)  \n𝒢′(𝑉𝑖)𝑉𝑖=∇(���𝑖)𝑇𝐹−𝛼(𝑉𝑖)−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)  \n−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)+1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)  \n+1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)                                                        (27)  \nFrom (18), (16), and (17), one has  \n𝒢(𝑉𝑖)=𝑋𝑇𝑄𝑇𝑋+∇(𝑉𝑖)𝑇𝐹−𝛼(𝑉𝑖)−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖) \n+1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)                                               (28)  \n𝑢𝑖=−1\n2𝑅−1𝐺𝑇∇𝑉𝑖                                     (29)  \n𝑑𝑖=1\n2𝛾2𝐾𝑇∇𝑉𝑖                                             (30)  \nSubstituting (26) -(30) into (25), yields the tracking Bellman \n(15) \n−𝛼(𝑉𝑖+1)+∇(𝑉𝑖+1)𝑇[𝐹+𝐺𝑢𝑖+𝐾𝑑𝑖]= \n−𝑋𝑇𝑄𝑇𝑋−𝑢𝑖𝑇𝑅𝑢𝑖+𝛾2𝑑𝑖𝑇𝑑𝑖                      (31)  \nThe Newton iterative equation (24) with (16) and (17) is \nequivalent to the Algorithm 1 from (15) to (17). This \ncompletes the proof.                                                               □ 5 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \nThe convergence of sequence {𝑉𝑖} given in (24) to the \nunique solution of the fixed -point equation 𝑉∗=𝒯(𝑉∗) can be \nguaranteed by Kantorovich’s theorem [34], then the solution \nof 𝒢(𝑉∗)=0 is obtained, i.e., the solution of the tracking HJI \nequation (14).  \nIV. 𝛿-PI ALGORITHM FOR SOLVING THE TRACKING HJI \nEQUATION  \nThe damped Newton method is an effective method for \nsolving nonlinear equations, which can adjust the calculation \nstep in the Newton direction through damped parameter δ. The \nmain advantage of the damped Newton method is that it \nimproves the robustness of the solution procedure with respect \nto the initial conditions [29]. In this section, a damp ed Newton \niterati on based RL  method is proposed for H∞ tracking \nproblems , named  δ-PI algorithm, which can be  used to pursue \nthe solutions of tracking HJI equation and obtain the  H∞ \ntracking controller.  On-policy learning method and off -policy \nlearning method are presented, respectively. Then, a NN-\nbased implementation scheme  is employed to implement the \noff-policy δ-PI algorithm without making use of any \ninformation of the system dynamics.  \nA. On -policy 𝛿-PI Algorithm  \nBased on the damped Newton method, the on -policy δ-PI is \ngiven in Algorithm 2.  \nAlgorithm 2 On-policy δ-PI for H∞ Tracking Control.  \nStep 1: Set 𝑖=0, give an initial cost function 𝑉0, an initial \nadmissible control 𝑢𝑖 and a disturbance signal 𝑑𝑖. \nStep 2: Solve 𝑉𝑖+1(𝑋) with 𝑉𝑖+1(0)=0 from the \ngeneralized tracking Bellman equation (32).  \n−𝛼(𝑉𝑖+1)+∇(𝑉𝑖+1)𝑇[𝐹+𝐺𝑢𝑖+𝐾𝑑𝑖]= \n(1−𝛿){∇(𝑉𝑖)𝑇[𝐹+𝐺𝑢𝑖+𝐾𝑑𝑖]−𝛼𝑉𝑖} \n−𝛿[𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖]           (32)  \nStep 3: Update the disturbance policy and control policy \nusing  \n𝑢𝑖+1=−1\n2𝑅−1𝐺T∂𝑉𝑖+1\n∂𝑋                      (33)  \n𝑑𝑖+1=1\n2𝛾2𝐾T∂𝑉𝑖+1\n∂𝑋                               (34)  \nStep 4:  Set 𝑖=𝑖+1, move to Step 2 and go on iteration, \nuntil a certain stopping criterion is met.  \n \nDefine a new mapping 𝒯𝛿:𝕍→𝕍 on Banach space 𝕍 \n𝒯𝛿(𝑉)=𝑉−𝛿(𝒢′(𝑉))−1𝒢(𝑉)                 (35)  \nwhere 0<𝛿≤1. \nTheorem 2 : Given the damped Newton mapping 𝒯𝛿 shown \nin (35), then the following damped Newton iteration with (33) \nand (34) is equivalent to the on -policy 𝛿-PI algorithm from \n(32) to (34).  \n𝑉𝑖+1=𝒯𝛿(𝑉𝑖)                                    (36)  \nProof : Substituting (35) into (36) and making some \nchanges, one gets  𝒢′(𝑉𝑖)𝑉𝑖+1=𝒢′(𝑉𝑖)𝑉𝑖−𝛿𝒢(𝑉𝑖).              (37)  \nSubstituting (26), (27), and (18) into (37) yields  \n(𝑉𝑖+1)𝑇𝐹−𝛼(𝑉𝑖+1)−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖+1)  \n−1\n4∇(𝑉𝑖+1)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)+1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖+1)  \n+1\n4𝛾2∇(𝑉𝑖+1)𝑇𝐾𝐾𝑇∇(𝑉𝑖)=∇(𝑉𝑖)𝑇𝐹−𝛼(𝑉𝑖)  \n−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)  \n+1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)+1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)  \n−𝛿[𝑋𝑇𝑄𝑇𝑋+∇(𝑉𝑖)𝑇𝐹−𝛼(𝑉𝑖)−1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)+\n1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)]  \n(38)                                                                         \nMake some mathematical manipulate, one can obtain  \n−𝛼(𝑉𝑖+1)+∇(𝑉𝑖+1)𝑇[��+𝐺𝑢𝑖+𝐾𝑑𝑖]=−(1−𝛿)𝛼𝑉𝑖 \n+(1−𝛿)∇(𝑉𝑖)𝑇𝐹+(1−𝛿)[−1\n2∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)]  \n+(1−𝛿)[1\n2𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)]−𝛿[𝑋𝑇𝑄𝑇𝑋+\n1\n4∇(𝑉𝑖)𝑇𝐺𝑅−1𝐺𝑇∇(𝑉𝑖)−1\n4𝛾2∇(𝑉𝑖)𝑇𝐾𝐾𝑇∇(𝑉𝑖)]  \n(39) \nSubstitute (33) and (34) into (39), one can obtain the \ngeneralized tracking Bellman equation (32). This means that \nthe damped Newton iteration given by (36) with (33) and (34) \nis equivalent to the on -policy 𝛿-PI algorithm from (32) to (34). \nThis completes the proof.                                                     □ \nRemark 2 : Compared with tracking Bellman equation (11), \nthe reinforcement term of the generalized tracking Bellman \nequation (32) is multiplied by the damped coefficient 𝛿. When \n𝛿=1, (32) degenerates to tracking Bellman equation (15) , \nand on-Policy 𝛿-PI algorithm becomes PI algorithm as shown \nin Algorithm 1.  \nRemark 3: The generalized tracking Bellman equation (32) \nis linear  partial differential equation just like the tracking \nbellman equation (11).  \nThe complete knowledge of the augmented system \ndynamics is required for solving 𝑉𝑖+1(𝑋) from (32) . \nMultiplying the two sides of (32) by 𝑒−𝛼(𝜏−𝑡) and integrating \nin the time interval (𝑡,𝑡+𝑇], one can obtain  a discrete version \nof the generalized tracking Bellman equation, as follow s. \n𝑒−𝛼𝑇𝑉𝑖+1(𝑋(𝑡+𝑇))−𝑉𝑖+1(𝑋(𝑡))= \n−𝛿∫𝑒−𝛼(𝜏−𝑡)([𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖])𝑑𝜏𝑡+𝑇\n𝑡  \n+(1−𝛿)[𝑒−𝛼𝑇𝑉𝑖(𝑋(𝑡+𝑇))−𝑉𝑖(𝑋(𝑡))]            (40) \nEquations (40) and (32) have the same solution while \n𝑉𝑖+1(0)=0, which can be pro ved in line with [15]. Using (40) \ninstead of (32) in Algorithm 2 , one can obtain a partially \nmodel -free δ-PI algorithm. The NN implementation scheme of \nAlgorithm 2 can be referred to [15].  \nB. Off -policy δ -PI Algorithm  \nAlgorithm 2 is an on -policy learning method, in which the \ndisturbance signals need to be adjustable  when implemented \nonline, and it is necessary to know the dynamic information of \nthe system. This is not in line with the reality. To overcome 6 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \nthis drawback, motivated by [20] -[25], an off -policy 𝛿-PI \nalgorithm based on the generalized tracking Bellman equation  \nis proposed to solve the tracking HJI equation through \nlearning arbitrary behavior policies , and none of the prior \ninformation of the dynamics of the system is required. Then, \nthree NNs are used to achieve the off-policy  𝛿-PI tracking \nalgorithm proposed.  \nFirstly, the augmented system dynamics (6) can be rewritten \nas \n𝑋̇=𝐹+𝐺𝑢𝑖+𝐾𝑑𝑖+𝐺(𝑢−𝑢𝑖)+𝐾(𝑑−𝑑𝑖)          (41)  \nwhere 𝑢∈ℜ𝑚 and 𝑑∈ℜ𝑞 are behavior policy and actual \ndisturbance, respectively. 𝑢𝑖∈ℜ𝑚 and 𝑑𝑖∈ℜ𝑞 are policies to \nbe evaluated and updated. Let 𝑉𝑖+1(𝒙) be the solution of \ngeneralized tracking Bellman equation (32), and \ndifferentiating 𝑉𝑖+1(𝒙) along with the augmented system \ndynamics (4 1) yields  \n𝑉̇𝑖+1=(∇𝑉𝑖+1)T(𝐹+𝐺𝑢𝑖+𝐾𝑑𝑖)+(∇𝑉𝑖+1)T𝐺(𝑢−𝑢𝑖) \n+(∇𝑉𝑖+1)T𝐾(𝑑−𝑑𝑖)                                               (42)  \nSubstituting (32) into the above equation yields  \n𝑉̇𝑖+1=−𝛿[𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖] \n+(1−𝛿){∇(𝑉𝑖)𝑇[𝐹+𝐺𝑢𝑖+𝐾𝑑𝑖]−𝛼𝑉𝑖}+𝛼(𝑉𝑖+1) \n+(∇𝑉𝑖+1)T𝐺(𝑢−𝑢𝑖)+(∇𝑉𝑖+1)T𝐾(𝑑−𝑑𝑖)                (43)                                 \nSubstituting the augmented system dynamics  (41) into (4 3) \nyields  \n𝑉̇𝑖+1=−𝛿[𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖]  \n+(1−𝛿)∇(𝑉𝑖)𝑇[𝑋̇−𝐺(𝑢−𝑢𝑖)−𝐾(𝑑−𝑑𝑖)]  \n−(1−𝛿)𝛼𝑉𝑖+𝛼(𝑉𝑖+1)+(∇𝑉𝑖+1)T𝐺(𝑢−𝑢𝑖)  \n+(∇𝑉𝑖+1)T𝐾(𝑑−𝑑𝑖)  \n=−𝛿[𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖]  \n+(1−𝛿)∇(𝑉𝑖)𝑇𝑋̇−(1−𝛿)∇(𝑉𝑖)𝑇𝐺(𝑢−𝑢𝑖)  \n−(1−𝛿)∇(𝑉𝑖)𝑇𝐾(𝑑−𝑑𝑖)−(1−𝛿)𝛼𝑉𝑖+𝛼(𝑉𝑖+1)  \n+(∇𝑉𝑖+1)T𝐺(𝑢−𝑢𝑖)+(∇𝑉𝑖+1)T𝐾(𝑑−𝑑𝑖)                  (44) \nSubstituting 𝑑𝑖=1\n2𝛾2𝐾𝑇∇𝑉𝑖, 𝑢𝑖=−1\n2𝑅−1𝐺𝑇∇𝑉𝑖, 𝑑𝑖+1=\n1\n2𝛾2𝐾T∇𝑉𝑖+1, and 𝑢𝑖+1=−1\n2𝑅−1𝐺T∇𝑉𝑖+1 into (4 4) yields an \noff-policy generalized tracking Bellman equation.  \n𝑉̇𝑖+1−𝛼(𝑉𝑖+1)=−𝛿[𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖]  \n+(1−𝛿)[∇(𝑉𝑖)𝑇𝑋̇−𝛼𝑉𝑖]+(1−𝛿)2𝑢𝑖𝑇𝑅(𝑢−𝑢𝑖)             \n−(1−𝛿)2𝛾2𝑑𝑖𝑇(𝑑−𝑑𝑖)−2𝑢𝑖+1𝑇𝑅(𝑢−𝑢𝑖)  \n+2𝛾2𝑑𝑖+1𝑇(𝑑−𝑑𝑖)                                                           (45) \nFrom the derivation process of (4 5), it can be seen that (4 5) \nand the generalized tracking Bellman equation (32) have the \nsame solution. For space reasons the proof is omitted in this \npaper. Multiplying the two sides of (4 5) by 𝑒−𝛼(𝜏−𝑡) and \nintegrating in the time interval (𝑡,𝑡+𝑇], one can obtain  a \ndiscrete version of the off -policy generalized tracking Bellman \nequation, as follows.  \n𝑒−𝛼𝑇𝑉𝑖+1(𝑋(𝑡+𝑇))−𝑉𝑖+1(𝑋(𝑡))=\n−𝛿∫𝑒−𝛼(𝜏−𝑡)([𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖])𝑑𝜏𝑡+𝑇\n𝑡  \n+(1−𝛿)[𝑒−𝛼𝑇𝑉𝑖(𝑋(𝑡+𝑇))−𝑉𝑖(𝑋(𝑡))]  \n+(1−𝛿)∫𝑒−𝛼(𝜏−𝑡)2𝑢𝑖𝑇𝑅(𝑢−𝑢𝑖)𝑑𝜏𝑡+𝑇\n𝑡  −(1−𝛿)∫𝑒−𝛼(𝜏−𝑡)2𝛾2𝑑𝑖𝑇(𝑑−𝑑𝑖)𝑑𝜏𝑡+𝑇\n𝑡  \n−∫𝑒−𝛼(𝜏−𝑡)2𝑢𝑖+1𝑇𝑅(𝑢−𝑢𝑖)𝑑𝜏𝑡+𝑇\n𝑡  \n+∫𝑒−𝛼(𝜏−𝑡)2𝛾2𝑑𝑖+1𝑇(𝑑−𝑑𝑖)𝑑𝜏𝑡+𝑇\n𝑡                                   (46)  \nIt is observed from (4 6) that arbitrary input signals  𝑢 and 𝑑 \ncan be used for learning the value function 𝑉𝑖+1, rather than \nthe policies 𝑢𝑖 and 𝑑𝒊 to be evaluated. Then, replacing (32) in \nAlgorithm 2 with (4 6), one can obtains the off -policy 𝛿-PI \nAlgorithm, as shown in Algorithm 3 . \nAlgorithm  3 Off-Policy 𝛿-PI for H∞ Tracking Control \nProblem . \nStep 1: Use the behavior policy 𝑢 and the actual \ndisturbance 𝑑 to collect N system data which contain \nsystem state, disturbance input and control input at \ndifferent sampling time interval.  \nStep 2: Set 𝑖=0, give an initial cost function 𝑉0 with \n𝑉0(0)=0, and initial control and disturbance policies 𝑢0 \nand 𝑑0. \nStep 3: Reuse the collected data to solve (4 6) for 𝑉𝑖+1(𝑥), \n𝑢𝑖+1 and 𝑑𝑖+1, with 𝑉𝑖+1(0)=0.  \nStep 4: If  a stop iteration condition is met , stop iteration \nand output 𝑉𝑖+1(𝑥) as the approximate optimal solution of \ntracking HJI equation (14), output 𝑢𝑖+1 as the approximate \noptimal control input, i.e., 𝑉∗(𝑥)=𝑉𝑖+1(𝑥) and 𝑢∗=\n𝑢𝑖+1, else set 𝑖=𝑖+1, move to Step 3 and go on \niteration.  \n \nAlgorithm 3 consists of two independent stages. In the first \nstage, behavior strategies are used to collect system data. In \nthe second stage, the data collected in the first stage are reused \nfor policies and value function update. The implementation of \nthe second stage update can be carried out without using any \nknowledge of the system dynamics, that is, Algorithm 3 is a \nmodel -free algorithm. When 𝛿=1, (46) degenerates to the \noff-policy RL Bellman equation as shown in [6].  \nC. NN-Based Implement of Algorithm 3  \nEquation (46) is a scalar equation that can be solved in the \nsense of least -squares after sampling a sufficient number of \ndata from the augmented system. Next, a NN -based actor -\ncritic structure is introduced to implement Algorithm 3 , in \nwhich two actor NNs and one critic NN are adopted to learn \nthe value function, disturbance policy  and control policy, \nrespectively. The three NNs are given as follows  \n𝑉̂𝑖+1(𝑋)=𝑊𝑐T𝜌(𝑋)                                (47) \n𝑢̂𝑖+1(𝑋)=𝑊𝑎T𝜙(𝑋)                               (48) \n𝑑̂𝑖+1(𝑋)=𝑊𝑑T𝜑(𝑋)                               (49) \nwhere 𝜌(𝑋)=[𝜌1(𝑋),…,𝜌𝐿1(𝑋)]T∈ℜ𝐿1 , 𝜙(𝑋)=\n[𝜙1(𝑋),…,𝜙𝐿2(𝑋)]T∈ℜ𝐿2 , and 𝜑(𝑋)=\n[𝜑1(𝑋),…,𝜑𝐿3(𝑋)]T∈ℜ𝐿3 are the linearly independent basis \nfunctions of the three NNs, respectively, which are all defined \non Ω⊂ℜ𝑛. 𝑊𝑐∈ℜ𝐿1, 𝑊𝑎∈ℜ𝐿2×𝑚, and 𝑊𝑑∈ℜ𝐿3×𝑞. \nBased on high -order Weierstrass approximation theories [35], 7 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \nthere exists a complete basis set, that the approximation error \nof NNs tends to zero uniformly when 𝐿1,𝐿2,𝑎𝑛𝑑 𝐿3→∞.  \nSubstituting (4 7)-(49) into (4 6) yields  \n𝑒−𝛼𝑇𝑊𝑐T𝜌(𝑋(𝑡+𝑇))−𝑊𝑐T𝜌(𝑋(𝑡))  \n+2∑𝑟𝑗∫𝑒−𝛼(𝜏−𝑡)𝑊𝑎,𝑗T𝜙(𝑋)𝜈𝑗1 𝑡+𝑇\n𝑡d𝜏𝑚\n𝑗=1   \n−2𝛾2∑∫𝑒−𝛼(𝜏−𝑡)𝑊𝑑,𝑘T𝜑(𝑋)𝜈𝑘2𝑡+𝑇\n𝑡d𝜏𝑞\n𝑘=1 =\n−𝛿∫𝑒−𝛼(𝜏−𝑡)([𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖])𝑑𝜏𝑡+𝑇\n𝑡  \n+(1−𝛿)[𝑒−𝛼𝑇𝑉𝑖(𝑋(𝑡+𝑇))−𝑉𝑖(����(𝑡))]  \n+(1−𝛿)∫𝑒−𝛼(𝜏−𝑡)2𝑢𝑖𝑇𝑅(𝑢−𝑢𝑖)𝑑𝜏𝑡+𝑇\n𝑡  \n−(1−𝛿)∫𝑒−𝛼(𝜏−𝑡)2𝛾2𝑑𝑖𝑇(𝑑−𝑑𝑖)𝑑𝜏𝑡+𝑇\n𝑡                        (50)  \nwhere 𝜈1=[𝜈11,…,𝜈𝑚1]𝑇=𝑢−𝑢𝑖 and 𝜈2=[𝜈12,…,𝜈𝑞2]𝑇=\n𝑑−𝑑𝑖. 𝑊𝑎,𝑗 is the jth column of matrix 𝑊𝑎, 𝑊𝑑,𝑘 is the kth \ncolumn of matrix 𝑊𝑑. \nNote that ( 50) is linear in the NNs weight vectors  𝑊𝑐, \n𝑊𝑎and 𝑊𝑑. Define  \n𝑊=[𝑊𝑐T,𝑊𝑎,1T,…,𝑊𝑎,𝑚T,𝑊𝑑,1T,…,𝑊𝑑,𝑞T]T            (51) \n𝜔(𝑡)=\n[            𝑒−𝛼𝑇𝜌(𝑋(𝑡+∆𝑡))−𝜌(𝑋(𝑡))\n2𝑟1∫𝑒−𝛼(𝜏−𝑡)𝜙(𝑋)𝜈11𝑡+𝑇\n𝑡d𝜏\n⋮\n2𝑟𝑚∫𝑒−𝛼(𝜏−𝑡)𝜙(𝑋)𝜈𝑚1𝑡+𝑇\n𝑡d𝜏\n−2𝛾2∫𝑒−𝛼(𝜏−𝑡)𝜑(𝑋)𝜈12𝑡+𝑇\n𝑡d𝜏\n⋮\n−2𝛾2∫𝑒−𝛼(𝜏−𝑡)𝜑(𝑋)𝜈𝑞2𝑡+𝑇\n𝑡d𝜏]            \n                  (52) \n𝜆(𝑡)=−𝛿∫𝑒−𝛼(𝜏−𝑡)([𝑋𝑇𝑄𝑇𝑋+𝑢𝑖𝑇𝑅𝑢𝑖−𝛾2𝑑𝑖𝑇𝑑𝑖])𝑑𝜏𝑡+𝑇\n𝑡  \n+(1−𝛿)[𝑒−𝛼𝑇𝑉𝑖(𝑋(𝑡+𝑇))−𝑉𝑖(𝑋(𝑡))]  \n+(1−𝛿)∫𝑒−𝛼(𝜏−𝑡)2𝑢𝑖𝑇𝑅(𝑢−𝑢𝑖)𝑑𝜏𝑡+𝑇\n𝑡  \n−(1−𝛿)∫𝑒−𝛼(𝜏−𝑡)2𝛾2𝑑𝑖𝑇(𝑑−𝑑𝑖)𝑑𝜏𝑡+𝑇\n𝑡                        (53) \nThen ( 50) can be rewritten as  \n𝜆(𝑡)=𝑊T𝜔(𝑡)                                (54)  \nNote that (5 4) is linear with respect to the weight 𝑊 of the \nthree NNs, thus 𝑊 can be solved in the sense of least -squares.  \nBecause of 𝑊∈ℝ𝐿1+𝑚×𝐿2+𝑞×𝐿3, therefore one needs to \ncollect 𝑁>𝐿1+𝑚×𝐿2+𝑞×𝐿3 augmented system data \nabout system state, disturbance and control input from 𝑡1 to 𝑡𝑁 \nin the state space. Then, for the given value function 𝑉𝑖, \nevaluated policies 𝑢𝑖and 𝑑𝑖, using this information to \ncalculate (5 2) and (5 3) at the 𝑁 different sampling data point, \none can get  \n𝛺=[𝜔(𝑡1),…,𝜔(𝑡𝑁)]                           (55)  \n𝛬=[𝜆(𝑡1),…,𝜆(𝑡𝑁)]T                           (56)  \nThe least -squares solution of the NNs weight 𝑊 is \n𝑊=(𝛺𝛺T)−1𝛺𝛬                                    (57)  \nThen, 𝑉̂𝑖+1, 𝑢̂𝑖+1, and 𝑑̂𝑖+1 can be obtained by using (4 7)-\n(49). Repeat the above steps , the approximate optimal solution \nof the H∞ tracking control can be obtain ed for the tracking \nproblem.  V. SIMULATION VALIDATION OF THE OFF-POLICY 𝛿-PI \nALGORITHM  \nIn this section, a computer simulation is carried out  to \ndemonstrate the effectiveness of the off -policy 𝛿-PI tracking  \nAlgorithm  3 proposed above.   This is a revised version of \nexample 2 in [18]. The nonlinear system dynamics is as \nfollows  \n[𝑥̇1\n𝑥̇2]=[𝑥2\n−𝑥13−0.5𝑥2]+[0\n1]𝑢+[0\n1]𝑑                  (58)  \nThe reference trajectories to be tracked for system states are \nconsidered as  \n 𝑟1=sin(1.5𝑡)                                        (59) \n𝑟2=1.5cos(1.5𝑡)                                 (60)  \nwhich can be described by the following command generator \nfunction  \n [𝑟̇1\n𝑟̇2]=[01\n−2.250][𝑟1\n𝑟2]                            (61) \nwith initial condition 𝑟=[1,0]. Then, one can get the \naugmented system  \n𝑋̇=[𝑒𝑑2\n−(𝑒𝑑1+𝑟1)3−0.5(𝑒𝑑2+𝑟2)+2.25𝑟1\n𝑟2\n−2.25𝑟1]+[0\n1\n0\n0]𝑢  \n+[0\n1\n0\n0]𝑑                                                                                   (62)  \nwhere 𝑋=[𝑒𝑑1,𝑒𝑑2,𝑟1,𝑟2]𝑇, 𝑒𝑑𝑖=𝑥𝑖−𝑟𝑖,𝑖=1,2. The \nperformance index is chosen as (8) with  𝑄=10 𝐈, I is an \nidentity matrix with appropriate dimension , 𝑅=1, 𝛾=5, and \n𝛼=0.1. \nThe activation function  𝜌(𝑋) of critical NN is selected as \npolynomial function, which contains the even powers of state \nvariable of the augmented system up to order four, and the \nactivation functions  𝜙(𝑋) and 𝜑(𝑋) of actor NNs contain the \nodd powers of state variable of the augmented system up to \norder three, that is  \n𝜌(𝑋)=[𝑋12,𝑋1𝑋2,𝑋1𝑋3,𝑋1𝑋4,𝑋22,𝑋2𝑋3,𝑋2𝑋4, \n𝑋32,𝑋3𝑋4,𝑋42,𝑋13𝑋2,𝑋13𝑋3,𝑋13𝑋4,𝑋12𝑋22, \n𝑋12𝑋2𝑋3,𝑋12𝑋2𝑋4,𝑋12𝑋32,𝑋12𝑋3𝑋4,𝑋12𝑋42,𝑋1𝑋23, \n𝑋1𝑋22𝑋3,𝑋1𝑋22𝑋4,𝑋1𝑋2𝑋32,𝑋1𝑋2𝑋3𝑋4,𝑋1𝑋2𝑋42, \n𝑋1𝑋33,𝑋1𝑋32𝑋4,𝑋1𝑋3𝑋42,𝑋1𝑋43,𝑋24,𝑋23𝑋3,𝑋23𝑋4, \n𝑋22𝑋32,𝑋22𝑋3𝑋4,𝑋22𝑋42,𝑋2𝑋33,𝑋2𝑋32𝑋4,𝑋2𝑋3𝑋42, \n𝑋2𝑋43,𝑋34,𝑋33𝑋4,𝑋32𝑋42,𝑋3𝑋43,𝑋44]𝑇               (63) \n 𝜙(𝑋)=𝜑(𝑋)= \n[𝑋1,𝑋2,𝑋3,𝑋4,𝑋13,𝑋12𝑋2,𝑋12𝑋3,𝑋12𝑋4,𝑋1𝑋22, \n𝑋1𝑋2𝑋3,𝑋1𝑋2𝑋4,𝑋1𝑋32,𝑋1𝑋3𝑋4,𝑋1𝑋42, \n𝑋23,𝑋22𝑋3,𝑋22𝑋4,𝑋2𝑋3𝑋4,𝑋2𝑋42,𝑋33,𝑋32𝑋4,𝑋3𝑋42,𝑋43]𝑇  (64) \nand 𝑊𝑐∈ℜ45,𝑊𝑎 and 𝑊𝑑∈ℜ24. \nDuring the data collection phase, the initial amount of the \nextended system state variable is set as 𝑋=[−1,1,1,0]𝑇, \nrandom signals are leveraged as behavior policies , and \nsampling time step is 𝑇=0.1 sec, then collect N=1000 data. \nThe data collection stage is shown in Fig. 1 and Fig. 2.  8 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \n \nFig. 1.  State variable of the augmented system during the \ndata collection phase . \n \nFig. 2.  Behavior policies u and d for data sampling . \n \nFig. 3.  The weight learning process of critic NN.  \nAfter collecting the required system data, iterative learning \nis performed to  learn  the approximate optimal solution of H∞ \ntracking control. Set 𝑊𝑐0=0, 𝑊𝑎0=0 and 𝑊𝑑0=0, and the \nNewton step -size is taken as 𝛿=0.3. Reuse the sampled data, \nsolve the weights of the neural network iteratively until \n‖𝑊𝑖+1−𝑊𝑖‖<10−7. The learning process is shown in Fig. \n3 and Fig. 4.  Through  96 iteration s, the neural network weights converge to meet the threshold requirements, and the  \nweights  of actor neural network are learned . \n \nFig. 4.  The weight learning process of actor NN.  \nLeveraging t he actor NN weights learn ed, the H∞ tracking \ncontroller can be obtained through (48). Apply this H∞ \ntracking  controller to the system to test performance. The \ninitial value of the reference signal and the system state are set \nas 𝑟0=[0,1.5]𝑇 and 𝑥0=[0,1.5]𝑇. Set the disturbance signal \nas \n𝑑=1.55exp(− 0 . 08 𝑡)cos (0.3t)                  (65)  \nThe dynamic response process of the system is shown in \nFig. 5 -Fig. 7, the tracking error is asymptotically stable when \n𝑑→0. Fig. 8 shows the disturbance attenuation level of the \nH∞ tracking  controller  learned by Algorithm 3. As can be seen \nthat the L2-gain condition (5) is met.  \n \nFig. 5.  The system state x1 versus the reference signal r1. \n9 \n> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE -CLICK HERE TO EDIT) <  \n \n \nFig. 6.  The system state x2 versus the reference signal r2. \n \nFig. 7.  The Control input and disturbance signal.  \n \nFig. 8. Disturbance attenuation for the H∞ tracking \ncontroller obtained.  \nVI. CONCLUSION  \nBecause damped Newton method enhances the robustness \nwith respect to initial condition and has larger convergence \nregion,  a 𝛿-PI algorithm  based on damped Newton iteration \nwas developed for H∞ tracking control of nonlinear \ncontinuous -time systems. Firstly, the damped Newton iteration operator equatio n for solving HJI equation was construct ed, \nthen the generalized tracking Bellman equation which is an \nextension of tracking Bellman equation was derived. Then, by \niteratively solving the generalized tracking Bellman equation, \non-policy 𝛿-PI algorithm and off -policy 𝛿-PI algorithm were \npresented, respectively. The off -policy 𝛿-PI algorithm is a \nmodel -free algorithm for solving H∞ tracking control, which \ncan be implemented without making use of any knowledge of \nthe dynamics of the systems. T he NN -based implementation \nscheme for the off -policy 𝛿-PI algorithms was proposed. 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Bernhard, H∞ optimal control and related minimax design \nproblems , Boston, MA, USA: Springer Science , 1995, pp. 263 -271. \n[32] L. V. Kantorovich and G. P. Akilov, Functional analysis , 2nd ed., Oxford, \nUK: Pergamon Press, 1982, pp. 250 -280. \n[33] E. Zeidler, Nonlinear Functional Analysis: Fixed Point Theorems , New \nYork, NY, USA: Springer -Verlag, 1985, pp. 68 -73. \n[34] R. A. Tapia, “The Kantorovich theorem for Newton’s method,” The \nAmerican Mathematical Monthly,  vol. 78, n o. 4, pp. 389 -392, Aug. 1971.  \n[35] K. Hornik, M. Stinchcombe, and H. White, “Universal approximation of \nan unknown mapping and its derivatives using multilayer feedforward \nnetworks,” Neural Networks , vol. 3, no. 5, pp. 551 -560, May, 1990, DOI: \n10.1016/0893 -6080(90)90005 -6. \n  Qi Wang  was born in Baotou, Inner \nMongolia on November 11, 1982. He \nreceived the B.E. degree in spacecraft \ndesign from Beihang University (BUAA), \nBeijing, in 2005 and the Ph.D. degree in \naircraft design from Chinese \nAeronautical Establishment, Beijing, in \n2023. He is currently a senior engineer in \nChina Airborne Missile Academy. His \nresearch interests are navigation , guidance and control of \ntactical missile, machine learning, adaptive dynamic \nprogramming, and reinforcement learning.  \nDr. Wang was a recipient of the Excellent Doctoral \nDissertation Award of Chinese Aeronautical Establishment in \n2023 . \n \n \n \n \n"    },    {        "title": "2401.12885v2.Damped_kink_motions_in_a_system_of_two_solar_coronal_tubes_with_elliptic_cross_sections.pdf",        "content": "Astronomy &Astrophysics manuscript no. manuV1 ©ESO 2024\nMarch 5, 2024\nDamped kink motions in a system of two solar coronal tubes with\nelliptic cross-sections\nMijie Shi1, Bo Li1, Shaoxia Chen1, Hui Yu1, and Mingzhe Guo1,2\n1Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shan-\ndong University, Weihai 264209, China\ne-mail: bbl@sdu.edu.cn\n2Centre for Mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven,\nBelgium\nReceived ......... /Accepted .........\nABSTRACT\nAims. This study is motivated by observations of coordinated transverse displacements in neighboring solar active region loops,\naddressing specifically how the behavior of kink motions in straight two-tube equilibria is impacted by tube interactions and tube\ncross-sectional shapes.\nMethods. We work with linear, ideal, pressureless magnetohydrodynamics. Axially standing kink motions are examined as an initial\nvalue problem for transversely structured equilibria involving two identical, field-aligned, density-enhanced tubes with elliptic cross-\nsections (elliptic tubes). Continuously nonuniform layers are implemented around both tube boundaries. We numerically follow the\nsystem response to external velocity drivers, largely focusing on the quasi-mode stage of internal flows to derive the pertinent periods\nand damping times.\nResults. The periods and damping times we derive for two-circular-tube setups justify available modal results found with the T-matrix\napproach. Regardless of cross-sectional shapes, our nonuniform layers feature the development of small-scale shears and energy\naccumulation around Alfv ´en resonances, indicative of resonant absorption and phase-mixing. As with two-circular-tube systems, our\nconfigurational symmetries make it still possible to classify lower-order kink motions by the polarization and symmetric properties of\nthe internal flows; hence such mode labels as SxandAx. However, the periods and damping times for two-elliptic-tube setups further\ndepend on cross-sectional aspect ratios, with Axmotions occasionally damped less rapidly than Sxmotions. We find uncertainties\nup to∼20% (∼50%) for the axial Alfv ´en time (the inhomogeneity lengthscale) if the periods (damping times) computed for two-\nelliptic-tube setups are seismologically inverted with canonical theories for isolated circular tubes.\nConclusions. The e ffects of loop interactions and cross-sectional shapes need to be considered when the periods and particularly the\ndamping times are seismologically exploited for coordinated transverse displacements in adjacent coronal loops.\nKey words. magnetohydrodynamics (MHD) — Sun: corona — Sun: magnetic fields — waves\n1. Introduction\nCyclic transverse displacements of solar coronal loops are arguably the most extensively observed collective motion in modern solar\ncoronal seismology (see e.g., Nakariakov & Kolotkov 2020; Nakariakov et al. 2021, for recent reviews). Two regimes have been\nestablished. The decayless regime is such that the displacements show little damping and their magnitudes are usually substantially\nsmaller than visible loop widths, with both features already clear when this regime was first identified in measurements with Hinode\n(Tian et al. 2012) and the Solar Dynamics Observatory /Atmospheric Imaging Assembly (SDO /AIA; Wang et al. 2012; Nisticò et al.\n2013; Anfinogentov et al. 2013). This regime is known not to be connected with eruptive events but ubiquitous in active region (AR)\nloops, as evidenced by a statistical survey of the SDO /AIA data (Anfinogentov et al. 2015) and by the recent series of analyses of the\nmeasurements with the Extreme Ultraviolet Imager (EUI) on board the Solar Orbiter (e.g., Zhong et al. 2022; Petrova et al. 2023;\nZhong et al. 2023). Decaying loop displacements, on the other hand, typically damp over several cycles and are of larger amplitudes\nas revealed by their first imaging observations by the Transition Region and Coronal Explorer (TRACE; Schrijver et al. 1999;\nAschwanden et al. 1999; Nakariakov et al. 1999). These decaying motions are usually associated with lower coronal eruptions\n(Zimovets & Nakariakov 2015). Regardless, there exist ample detections of decaying displacements, an inexhaustive list being\nthose by Hinode (Ofman & Wang 2008; Van Doorsselaere et al. 2008a; Erdélyi & Taroyan 2008), the Solar TErrestrial RElations\nObservatories (STEREO; Verwichte et al. 2009), and by SDO /AIA (Aschwanden & Schrijver 2011; White & Verwichte 2012).\nStatistical studies therefore prove possible either through compiling published results (e.g., Verwichte et al. 2013) or via directly\ncataloging the SDO /AIA events (Goddard et al. 2016; Nechaeva et al. 2019). Decaying cyclic displacements were specifically\nestablished to be nearly exclusively axial fundamentals (e.g., Fig.9 in Goddard et al. 2016 and Fig.5 in Nechaeva et al. 2019).\nSeismological applications of cyclic transverse displacements typically start with their identification as trapped fast kink mo-\ntions. Practically, this identification largely relies on the scheme for classifying collective motions in straight, static, field-aligned\nArticle number, page 1 of 31arXiv:2401.12885v2  [astro-ph.SR]  3 Mar 2024A&A proofs: manuscript no. manuV1\nconfigurations where isolated density-enhanced tubes with circular cross-sections (circular tubes hereafter) are embedded in an\notherwise uniform corona (Edwin & Roberts 1983, ER83; also Zajtsev & Stepanov 1975; Cally 1986). That the configuration is\nstructured only transversely and in a one-dimensional (1D) manner means that the azimuthal wavenumber mmakes physical sense,\nwith kink motions corresponding to m=1 (see the textbooks by Roberts 2019 and Goedbloed et al. 2019). Let “ER83 equilibria”\nlabel specifically those where the transverse structuring is piecewise constant. The relevant theories then enable the measured pe-\nriods of axially standing kink motions to be employed to infer the axial Alfv ´en time and hence the coronal magnetic field strength\nfor decayless (e.g., Anfinogentov & Nakariakov 2019; Li & Long 2023) and decaying regimes alike (Nakariakov & Ofman 2001;\nalso the reviews by e.g., Nakariakov & Verwichte 2005; De Moortel & Nakariakov 2012; Nakariakov et al. 2021). While undamped\nfor ER83 equilibria, trapped fast kink motions are in general resonantly absorbed in the Alfv ´en continuum when the 1D struc-\nturing is allowed to be continuous (Ruderman & Roberts 2002; Goossens et al. 2002, and references therein). Theoretically, the\nconcept of kink quasi-modes then arises and the internal kink motions are damped in conjunction with the accumulation and phase-\nmixing of localized Alfv ´enic motions (e.g., Poedts & Kerner 1991; Tirry & Goossens 1996; Soler et al. 2013; also the review by\nGoossens et al. 2011). Seismologically, resonant absorption has been customarily invoked to interpret the decay of large-amplitude\nloop displacements, thereby allowing the measured damping times to be employed to deduce such key parameters as the transverse\ninhomogeneity lengthscales (e.g., Aschwanden et al. 2003; Goossens et al. 2008; Arregui & Asensio Ramos 2011; Arregui et al.\n2015; Arregui 2022).\nDeviations from the canonical ER83 equilibria are known to impact the behavior of collective motions, and we choose to focus\non two geometrical properties that render the meaning of the azimuthal wavenumber not as clear (e.g., the review by Li et al. 2020).\nOne concerns loop cross-sections, which may actually be tied to coronal heating via their key role in determining the morphology\nof coronal loops in, say, soft X-ray (Klimchuk et al. 1992) and EUV (Watko & Klimchuk 2000) in the first place. Recent imaging\nobservations with Hi-C (Klimchuk & DeForest 2020) and Hi-C 2.1 (Williams et al. 2021) suggest that AR loops may maintain a\ncircular cross-section throughout their visible segments. However, there also exist suggestions that favor elliptic cross-sections to\nbetter account for the morphology of AR loops, as deduced with the aid of coronal magnetic field modeling (e.g., Wang & Sakurai\n1998; Malanushenko & Schrijver 2013) and /or multi-vantage-point measurements with STEREO (e.g., McCarthy et al. 2021). In\nparticular, the derived aspect ratios may readily attain ∼1.5−5 (Malanushenko & Schrijver 2013), a range also compatible with\nthe spectroscopic measurements with Hinode /EUV Imaging Spectrometer (EIS; Kucera et al. 2019). Restrict ourselves to straight\nconfigurations where coronal loops preserve a constant elliptic cross-section along their axes (elliptic tubes hereafter). Substantial\nattention has been paid to collective perturbations in such equilibria from both the modal (e.g., Ruderman 2003; Erdélyi & Morton\n2009; Morton & Ruderman 2011; Aldhafeeri et al. 2021) and initial-value-problem (IVP) perspectives (Guo et al. 2020). Despite the\nlack of axisymmetry, kink motions remain identifiable as those that transversely displace the tube axes, and in general they remain\nsubjected to resonant absorption for continuous transverse structuring. However, as was first shown by Ruderman (2003), one now\nneeds to discriminate two distinct polarizations, where the internal flows are primarily directed along the major (“major-polarized”\nfor brevity) and minor axes (“minor-polarized”), respectively. In addition, the periods of major-(minor-) polarized kink motions\nincrease (decrease) with the major-to-minor-axis ratio, while the damping times for both polarizations tend to increase with this\nratio. It was further demonstrated that the density contrast between elliptic tubes and their surrounding fluids plays a subtle role in\nmediating the di fferences between the dispersive properties of the di fferently polarized motions (Guo et al. 2020).\nAnother geometrical property pertains to systems involving multiple tubular structures. Observationally, it has long been known\nthat perturbations in such systems may evolve collectively, an incomplete list of examples being the coordinated transverse displace-\nments detected either in groups of prominence threads (e.g., Yi et al. 1991; Lin et al. 2003; Okamoto et al. 2007; also the review\nby Arregui et al. 2018) or in neighboring AR loops (e.g., Schrijver & Brown 2000; Schrijver et al. 2002; Verwichte et al. 2004;\nWang et al. 2012; White et al. 2013). Kink motions in straight multi-circular-tube systems have also been extensively examined\ntheoretically. We choose to concentrate on the modal analyses where a harmonic time-dependence is assumed a priori, noting that\nmulti-dimensional IVP studies prove equally informative (e.g., Terradas et al. 2008b; Ofman 2009; Magyar & Van Doorsselaere\n2016; Guo et al. 2019a). The first modal examination was due to Luna et al. (2008, hereafter L08), who numerically solved the\npertinent eigenvalue problem (EVP) for undamped kink modes in a two-tube system. Broadly speaking, two di fferent approaches\nhave been employed in further modal studies. The ensuing EVPs turn out to be analytically tractable in the thin-tube (TT) limit for\ntwo-tube equilibria when formulated in bi-cylindrical coordinates, for both undamped (Van Doorsselaere et al. 2008b; Robertson\net al. 2010; Ruderman & Petrukhin 2023) and damped kink motions (Robertson & Ruderman 2011; Gijsen & Van Doorsselaere\n2014). Similar EVPs have also been formulated with the T-matrix formalism of scattering theory, originally introduced to solar con-\ntexts by Bogdan & Zweibel (1985, 1987). This formalism is su fficiently general for examining kink motions in composite equilibria\ncomprising an arbitrary number of circular tubes, with specific investigations available for undamped motions in systems with two\nor three (Luna et al. 2009) up to tens of tubes (Luna et al. 2010, 2019). Addressing continuous transverse structuring, the T-matrix\napproach proves capable of handling the resonant absorption of collective motions in general (Keppens et al. 1994) and particu-\nlarly that of kink motions in two-tube equilibria (Soler & Luna 2015, SL15 hereafter). The damping is nonetheless assumed to be\nweak by construction, given that it was incorporated in the T-matrix framework exclusively via the thin-boundary (TB) connection\nformulae (see SL15 for more details; also see Sakurai et al. 1991 for the first derivation of the TB formulae). Regardless, kink\nmotions are known to be much more complicated in multi-tube equilibria than for isolated circular tubes, and we restrict ourselves\nto two-identical-tube systems. It was shown by L08 (see Fig.2 therein) that kink motions, namely those that displace both tube axes,\nneed to be classified according to both the polarization and symmetric properties of the two internal flows. The notions of Sx,Ax,\nSy, and Aythen ensue, one subtlety being that the Sx−AyandAx−Sypairs become indistinguishable in terms of frequencies (Van\nDoorsselaere et al. 2008b) and damping rates (SL15) when the tubes are su fficiently thin. This subtlety notwithstanding, Axmotions\nalways turn out to oscillate and damp more rapidly than Sxmotions: the di fferences between the frequencies and particularly the\ndamping rates may be substantial enough to impact seismological deductions (see Fig.4 in SL15).\nArticle number, page 2 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nThis study is intended to examine damped lower-order kink motions in a two-identical-elliptic-tube system, thereby addressing\nthe situation where neither the equilibrium configuration as a whole nor an individual tube allows the azimuthal wavenumber to\nmake exact sense. We will adopt linear, pressureless, ideal magnetohydrodynamics (MHD) throughout, given that quiescent AR\nloops are of primary interest. We will adopt an IVP perspective, paying particular attention to the frequencies and damping rates\nof axially standing motions by largely focusing on the duration where the concept of quasi-modes applies. Our study is new in the\nfollowing two aspects. Firstly, two-elliptic-tube systems have yet to be explored in the context of collective waves, meaning that\nour study will shed new light on how the dispersive properties of kink motions are impacted by the joint e ffects of tube interactions\nand tube cross-sectional shapes. When put to seismological contexts, our results can therefore help assess the uncertainties in the\nkey physical parameters that one deduces with the customary practice where the joints e ffects are absent. Secondly, there exist no\nIVP studies dedicated to the dispersive properties of damped kink motions in two-tube systems to our knowledge. Our numerical\nanalysis, conducted with a self-developed code, will therefore help verify the T-matrix results obtained by SL15.\nThe outline of this manuscript is as follows. Section 2 o ffers the mathematical formulation of our IVP together with a description\nof our numerical code. We then focus on circular tubes in Sect.3, testing our code outputs against available results obtained with\nindependent approaches. Section 4 proceeds to present our numerical results for two-elliptic-tube systems. Section 5 discusses some\nseismological implications of this study. Our findings are summarized in Sect. 6, where some concluding remarks are also given.\n2. Problem Formulation and Solution Method\n2.1. Governing Equations\nWe adopt pressureless, ideal MHD as our theoretical framework, in which the primitive dependents are the mass density ρ, velocity\nv, and magnetic field B. Let ( x,y,z) be a Cartesian system, and let the equilibrium quantities be denoted with a subscript 0. We\ntake the equilibrium magnetic field to be uniform and z-directed ( B0=B0ez). Only straight, static, field-aligned configurations are\nof interest, meaning that v0=0 and the structuring is encapsulated in ρ0(x,y). The Alfv ´en speed vAis defined by v2\nA=B2\n0/µ0ρ0,\nwhereµ0is the magnetic permeability of free space. With magnetically closed structures in mind, we place two dense photospheres\natz=0 and z=L.\nWe formulate an IVP to examine how our equilibrium responds to small-amplitude perturbations (denoted by subscript 1). It\nfollows from linearized, ideal, pressureless MHD equations that\nρ0∂v1x\n∂t=B0\nµ0 ∂B1x\n∂z−∂B1z\n∂x!\n, (1)\nρ0∂v1y\n∂t=B0\nµ0 ∂B1y\n∂z−∂B1z\n∂y!\n, (2)\n∂B1x\n∂t=B0∂v1x\n∂z, (3)\n∂B1y\n∂t=B0∂v1y\n∂z, (4)\n∂B1z\n∂t=−B0 ∂v1x\n∂x+∂v1y\n∂y!\n. (5)\nWe focus on axially standing motions by adopting the ansatz\nv1x(x,y,z;t)=ˆvx(x,y;t) sin( kz),\nv1y(x,y,z;t)=ˆvy(x,y;t) sin( kz),\nB1x(x,y,z;t)=ˆBx(x,y;t) cos( kz),\nB1y(x,y,z;t)=ˆBy(x,y;t) cos( kz),\nB1z(x,y,z;t)=ˆBz(x,y;t) sin( kz), (6)\nwhere k=nπ/Lis the quantized axial wavenumber ( n=1,2,···). Equations (1) to (5) then become\n∂ˆvx\n∂t=−B0\nµ0ρ0 \nkˆBx+∂ˆBz\n∂x!\n, (7)\n∂ˆvy\n∂t=−B0\nµ0ρ0 \nkˆBy+∂ˆBz\n∂y!\n, (8)\n∂ˆBx\n∂t=kB0ˆvx, (9)\n∂ˆBy\n∂t=kB0ˆvy, (10)\n∂ˆBz\n∂t=−B0 ∂ˆvx\n∂x+∂ˆvy\n∂y!\n. (11)\nArticle number, page 3 of 31A&A proofs: manuscript no. manuV1\nAxial fundamentals ( k=π/L) will be examined throughout, even though our analysis can be readily adapted to any axial harmonic\nnumber n.\n2.2. Energy Conservation Law\nEnergetics considerations turn out to be necessary. Consider a volume Vthat laterally occupies an arbitrary area Qand is axially\nbounded by the planes z=0 and z=L. Let the curve enclosing Qbe denoted by ∂Q. Let [ en,et,ez] further define a right-handed set\nof orthonormal system at any point along ∂Q, with the normal direction enpointing away from Q. An energy conservation law can\nbe readily derived from Eqs. (1) to (5), reading\ndE(t)\ndt=−F(t), (12)\nwhere\nE(t)=Z\nQϵdxdy, (13)\nF(t)=I\n∂Qdlten·ˆpT\u0010\nˆvxex+ˆvyey\u0011\n, (14)\nϵ(x,y;t)=1\n2ρ0\u0010\nˆv2\nx+ˆv2\ny\u0011\n+1\n2µ0\u0010ˆB2\nx+ˆB2\ny+ˆB2\nz\u0011\n. (15)\nNote that the ansatz (6) has been employed, and ˆ pT=B0ˆBz/µ0is connected with the Eulerian perturbation of total pressure. Note\nalso that dltdenotes the elementary arclength in the tangential direction defined by et. When multiplied by L/2, the symbols Eand\nFrepresent the instantaneous perturbation energy in Vand the net energy flux out of V, respectively. We nonetheless refer to E(F)\nas the total energy (the net energy flux) for brevity. Likewise, the symbol ϵin Eqs. (13) and (15) will be referred to as the energy\ndensity. We also drop the hat from here onward.\n2.3. Equilibrium Configuration and Initial Perturbation\nOur equilibrium comprises a composite structure embedded in an otherwise uniform external corona (denoted by subscript e). By\n“composite” we mean two identical tubes with elliptic cross-sections and separated by d. Let jlabel a tube, and let a 2D position\nvector Xj=(Xj,Yj) denote the center of tube j. The tube centers are placed on the x-axis without loss of generality, enabling a\ntube to be referred to as either the left ( j=L) or the right tube ( j=R). The tube centers are further taken to be symmetric about\nx=0, resulting in XL,R=∓d/2 and YL,R=0. We consistently use subscript i to denote the equilibrium quantities at either tube axis,\nmeaning in particular that the internal density and Alfv ´en speed are denoted by ρiandvAi, respectively. Likewise, by vAewe denote\nthe Alfv ´en speed evaluated with the external density ρe.\nOur two-tube structure is characterized as follows. The tubes are taken to be identical not only in cross-sectional shapes but\nin their orientations relative to the x-axis. For simplicity, we consider only two orientations by discriminating whether the major\n(hereafter “ x-major”) or minor axis (“ x-minor”) of a tube aligns with the x-direction. Let axandaydenote the spatial extent that\na tube spans in the x- and y-directions, respectively. The x-major ( x-minor) orientation then means that ax>ay(ax<ay). Both\norientations are realized by the density distribution ( j=L or R),\nρj(x,y)=ρi, 0≤¯rj≤1−¯l;\nρi+ρe\n2−ρi−ρe\n2cosπ(¯rj−1)\n¯l, 1−¯l≤¯rj≤1,(16)\nwith the intermediate dimensionless variable ¯ rjdefined by\n¯rj(x,y)Bs x−Xj\nax!2\n+ y\nay!2\n. (17)\nWe take d≥2axsuch that tube jcan be unambiguously identified as where ¯ rj≤1. We also use a(b) to denote the semi-major\n(semi-minor) axis, meaning that [ ax=a,ay=b] ([ax=b,ay=a]) for the x-major ( x-minor) orientation. The symbol bwill be\nfavored when the limiting case of circular tubes is examined ( a=b). Equation (16) represents a tube profile that continuously\nconnects the internal density ρito the external one ρevia some elliptic layer of width ¯la(¯lb) along the direction of the major (minor)\naxis. When a=b, this profile is equivalent to the sinusoidal distribution introduced by Ruderman & Roberts (2002, hereafter\nRR02) for modeling circular inhomogeneities (see e.g., Van Doorsselaere et al. 2004; Soler et al. 2013; Chen et al. 2021, for more\napplications). Somehow subtle is that the RR02 implementation for a tube centered at ( Xj,0) writes\nρj(x,y)=ρi, rj≤R−ℓ/2;\nρi+ρe\n2−ρi−ρe\n2sinπ(rj−R)\nℓ,R−ℓ/2≤rj≤R+ℓ/2,\nρe, rj≥R+ℓ/2.(18)\nArticle number, page 4 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nHere rjBp\n(x−Xj)2+y2. Evidently, the geometrical parameters [ R,ℓ] characterizing the RR02 implementation are connected to\nours via\nb=R+ℓ/2,\n¯l=ℓ\nb=ℓ/R\n1+ℓ/2R. (19)\nOur equilibrium configuration, illustrated in Fig. 1b, is a straightened version of a composite system where two tubes are\nseparated in the x-direction and anchored in the photosphere (Fig. 1a). We see the x- (y-) direction in Fig. 1b as horizontal (vertical)\ngiven that the y−zplane can be identified as the tube plane. For two-tube systems, however, it turns out to be also necessary to\nclassify lower-order kink motions into the Sx,Ax,Sy, and Aytypes by combining the symmetric and polarization properties of the\ninternal flow fields (see Fig. 2 in L08). Proposed for two-identical-circular-tube systems, the subscript x(y) arises when the internal\nvelocities are dominated by their x- (y-)components, while the symbol S(A) is such that the dominant internal velocity component\nis symmetric (anti-symmetric) about x=0. This classification is expected to carry over when elliptic tubes are examined. Somehow\ncomplicated is that one needs to distinguish between the x-major and x-minor arrangements (Figs. 1c and 1d), meaning for instance\nthat the frequencies of the Sxmotions are likely to be di fferent for di fferent tube orientations.\nOur equilibrium is perturbed via velocities. We consider only SxandAxmotions, given that kink motions are commonly\nobserved to be horizontally polarized (see Zhong et al. 2023 and references therein; see e.g., Wang et al. 2008 and Aschwanden &\nSchrijver 2011 for observational instances of vertical kink motions). Regardless of tube orientations, these are excited by imposing\nvini(x,y)\nvAe=e4(\nexp\"\n−\u0012x+xini\nb\u00132#\n±exp\"\n−\u0012x−xini\nb\u00132#)\nex, (20)\nwhere xini=XR+ax+2b, and we recall that [ XR=d/2,b=min(ax,ay)]. The coe fficient e4is introduced only for plotting purposes,\nthe magnitude being immaterial for linear studies. Equation (20) represents a pair of planar compressible perturbations concentrated\nin the ambient corona, exciting Sx(Ax) motions when the plus (minus) sign applies.\n2.4. Parameter Overview and Solution Method\nThe behavior of small-amplitude perturbations is determined by two sets of parameters. We take the dimensional set to be {ρe,b,vAe},\nwhich serves merely as normalizing constants in our context. Accept that we are primarily interested in such timescales as the\noscillation period and some damping time. The importance of the dimensionless set then follows from a straightforward dimensional\nanalysis, which dictates that a timescale tscaleis formally expressible as\nvAetscale\nb=F\"\norientation,ρi\nρe,a\nb,¯l,d\nb;kb; perturbation pattern#\n. (21)\nThe subgrouped parameters characterize the transverse structuring, axial wavenumber, and initial perturbation, respectively. By\n“orientation” we refer to either “ x-major” or “ x-minor”. By “perturbation pattern” we mean either SxorAx. We fix the axial\nwavenumber at kb=π/30 throughout, meaning a tube length L=30bfor axial fundamentals. This L/bsuffices for our purposes,\nlying toward the lower end of but within the accepted range for AR loops imaged in the EUV (e.g., Aschwanden et al. 2004;\nSchrijver 2007). Table 1 briefly overviews our computations, which are grouped into four sets to be detailed later. The density\ncontrastρi/ρeis fixed at either 3 or 5, both values being representative of AR loops (e.g., Aschwanden et al. 2004, and references\ntherein).\nWe solve Eqs. (7) to (11) with the following procedure. A computational domain [ −xM,xM]×[−yM,yM] is discretized into a\nuniform mesh with identical spacing in the x- and y-directions ( ∆x= ∆ y= ∆). The zero-gradient condition is implemented for\nall unknowns at all boundaries. We evolve Eqs. (7) to (11) with the classic MacCormack scheme (MacCormack 1969), a popular\nfinite-di fference algorithm second-order accurate in both space and time (see the textbooks by, e.g., Anderson 1995; Jardin 2010,\nfor more). The time step ∆tis set via some e ffective Courant number cB(vAe∆t/∆x)2/3+(vAe∆t/∆y)2/3as inspired by Hong (1996).\nA value of c=0.99 is employed for all the presented results. We have verified that varying cbetween∼0.7 and∼1.3 introduces\nno discernible di fference, and numerical stability is consistently maintained.\nSome remarks on the grid setup are necessary. We start by noting that the spatial spacing ∆restricts the timeframe in which\nnumerical solutions make physical sense, an aspect raised by Terradas et al. (2008b, hereafter T08) who numerically solved an\nequivalent set of governing equations. Defining the Alfv ´en frequency ωA(x,y)=kvA(x,y), one expects that some time-dependent\nphase-mixing length will emerge as (Mann et al. 1995)\nLph(x,y;t)=2π\n|∇ωA(x,y)|t, (22)\nwhich characterizes the transverse lengthscales of resonantly generated Alfv ´enic motions in the nonuniform portions in the system.\nIt suffices to consider only the nonuniform layer of one tube. Evidently, the shortest phase-mixing length Lmin\nph(t) at a given instant\noccurs at the strongest |∇ωA(x,y)|. Equally evident is that this strongest |∇ωA(x,y)|depends only on ρi/ρeand¯leven when elliptic\ntubes are examined. Let Lmin\nph(500) denote Lmin\nphatt=500b/vAe, before which our computations are consistently terminated. It then\nfollows from the arguments by T08 that our signals are physically relevant provided Lmin\nph(500)≥2∆. As shown by Table 1, this\ncriterion is satisfied by the reference grid setup for any set of our computations.\nArticle number, page 5 of 31A&A proofs: manuscript no. manuV1\nTable 1. Summary of time-dependent solutions presented in the text\nSetρi/ρe¯l Lmin\nph(500) Reference Grid Section Remarks\n1 3≥0.2 ≳0.032b [xM,yM]=[12b,12b],∆ =0.01b §3.1 1-circular-tube\n2 5 ≳0.182 ≳0.019b [xM,yM]=[12b,10b],∆ =0.005b §3.2 2-circular-tube\n3 3 0.4 0.063b [xM,yM]=[30b,25b],∆ =0.01b §4.12-elliptic-tube\nx-major\n4 3 0.4 0.063b [xM,yM]=[30b,25b],∆ =0.01b §4.22-elliptic-tube\nx-minor\nNotes. All computations pertain to kb=π/30 or equivalently L/b=30 for axial fundamentals.\n3. Test Computations for Circular Tubes\nThe MacCormack scheme, while a textbook one, has not been applied to Eqs. (7) to (11). Its applicability is therefore examined\nin this section via some test computations for which the time-dependent behavior can be established or expected with independent\nmethods. Only circular tubes are of interest, and we refer to b(=a) as our tube radius. We start by examining the response of a\none-circular-tube system to axisymmetric (sausage) perturbations, following our previous study (Li et al. 2022) to formulate both\nthe equilibrium and the initial perturbation. The Fourier-integral-based solutions, presented in Fig. 4 therein, agree closely with our\nMacCormack solutions. In particular, no discernible numerical anisotropy shows up even though finite di fferences are performed on\na Cartesian grid to examine a non-planar equilibrium (see e.g., the review by Sescu 2015, for more on numerical anisotropy). The\nrest of this section focuses on kink perturbations.\n3.1. Kink Motions in a One-Circular-Tube Setup\nThis subsection examines kink perturbations in a one-circular-tube configuration, placing the tube center at the origin without loss\nof generality. The equilibrium density is realized through Eqs. (16) and (17) by taking [ ax=ay=b,Xj=0] and then dropping\nthe subscript j. Kink motions in such a configuration have been extensively studied (see the review by Nakariakov et al. 2021, and\nreferences therein), readily allowing our MacCormack computations to be compared with known results. By “known” we refer to\ntwo sets of studies. Set one, presented in Sect. 4 of T08, adopts the RR02 implementation to examine the system response to an\ninitial perturbation of the form\nvini(x,y)=vAeexp\"\n−(y−yini)2\nσ2#\ney. (23)\nFigure 2 in T08 then presents the time sequence of vysampled at the tube center for a combination of parameters [ ρi/ρe=3,kR=\nπ/20,ℓ/R=0.6] and [ yini=3R,σ=R]. We repeat the same experiment, using Eq. (19) to address notational di fferences and\nadopting an identical computational grid. Our finite-di fference results are found to be consistent with T08, despite that the numerical\ncode therein adopts the finite-volume methodology.\nWe proceed to perform an additional series of computations to examine whether our code outputs agree with expectations for\nideal kink quasi-modes. Let ω= Ω−iγdenote the complex-valued quasi-mode frequency, and see Ωandγas positive. With the\nkink speed ckinkdefined as\nc2\nkinkBρiv2\nAi+ρev2\nAe\nρi+ρe, (24)\nit is well established that (RR02; see also Goossens et al. 1992 and Soler et al. 2013)\nΩ≈kckink,\nγ\nΩ≈1\n4¯l\n1−¯l/2ρi/ρe−1\nρi/ρe+1(25)\nin the so-named thin-tube-thin-boundary limit (TTTB, kb≪1 and ¯l≪1). Note that the TTTB expressions are usually formulated in\nterms of the RR02 notations [ R,ℓ]. Note also that the TTTB results may not hold well beyond their range of applicability (e.g., Soler\net al. 2014; Yu et al. 2021). We therefore also evaluate the mode frequencies numerically with the general-purpose finite-element\ncode PDE2D (Sewell 1988), computing ideal kink quasi-modes as resistive eigenmodes (see Goossens et al. 2011 for conceptual\nclarifications; see also Terradas et al. 2005 for the first introduction of PDE2D to solar contexts)1. No restriction is necessary for kb\nor¯l.\n1A uniform resistivity ηis employed for simplicity, the associated magnetic Reynolds number RmBµ0RvAi/η=105. We have verified that\nfurther increasing Rmdoes not introduce any discernible di fference to the oscillation frequencies and damping rates. These η-independent values\nare well known to be physically connected to ideal quasi-modes (e.g., Poedts & Kerner 1991; Terradas et al. 2006; Soler et al. 2013).\nArticle number, page 6 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nThe rest of this subsection is devoted to a fixed combination [ ρi/ρe=3,kb=π/30], allowing only ¯lto vary. Furthermore, the\nreference grid (see Table 1) is consistently employed in our time-dependent computations, where kink motions are excited by a\nperturbation of the fixed form\nvini(x,y)\nvAe=e4exp\"\n−(x+3b)2\nb2#\nex. (26)\nEquation (26) is essentially identical to Eq. (23) except that vxrather than vyis perturbed. Figure 2a examines a representative case\nwith ¯l=0.4, showing the temporal evolution of the x-speed sampled at the tube center ( vx(0,0,t), the solid curve). This vxsignal\nis seen to feature some rapid variations for t≲15b/vAe. By “rapid” we mean timescales on the order of the transverse Alfv ´en time\n(b/vAi), which derives from the multiple reflections o ffthe tube boundaries of the perturbations that are imparted by the external\ndriver to the tube. The vxsignal transitions toward a regular slower-varying pattern afterwards, becoming monochromatic when\nt≳70b/vAe. We choose to leave out the first extremum in this stage for safety, and fit the segment encompassing the next six (the\nred asterisks) with an exponentially damped cosine\nffit(t)∝exp\u0012\n−t\nτ\u0013\ncos 2πt\nP+ϕ!\n. (27)\nThe damping envelope from the best-fit is then plotted for the entire duration in Fig. 2a by the dashed curves. One see that this best-fit\nenvelope, while deduced for some segment, o ffers a good description for the larger-time behavior as well (say, t≳300b/vAe).\nFigure 2b surveys a range of ¯lby plotting the oscillation frequencies ( Ωfit, the black open circles) and damping rates ( γfit, blue),\nwhich are translated from the best-fit periods and damping times ( Ωfit=2π/Pfitandγfit=1/τfit). Note that Ωis measured in units\nof the external Alfv ´en frequency ωAe=kvAe, andγ/Ωis presented rather than γitself. The curves in Fig. 2b further provide the\nquasi-mode expectations with either the analytical TTTB expression (Eq. 25, the solid lines) or the PDE2D computations (labeled\n“Resistive”, dashed). One sees that the TTTB results provide a rather good approximation to the numerical “Resistive” ones, the\nfrequencies being practically the same and the damping rates di ffering by ≳10% only when ¯l≳0.4. One further sees that the\nopen circles agree well with the “Resistive” computations, demonstrating that the internal flow fields practically evolve as an ideal\nquasi-mode at least in the interval where the fitting is performed.\n3.2. Kink Motions in a Two-Circular-Tube Setup\nThis subsection examines kink motions in a two-circular-tube configuration, for which the distinction between SxandAxpatterns\nbecomes necessary. The initial perturbation is therefore chosen to follow Eq. (20), where xinievaluates to d/2+3b. Our code\nconsistently adopts the [ b,¯l] notations. However, the mixed usage of the RR02 convention [ R,ℓ] turns out to be necessary. The\nreason is that this RR02 convention was adopted by SL15 to examine ideal quasi-modes from an eigenvalue-problem perspective\nin a configuration physically identical to ours. Briefly put, the study by SL15 is based on the TB-embedded T-matrix formalism of\nscattering theory, the pertinent results being most straightforward for our time-dependent computations to be compared with. By\n“pertinent” we specifically refer to Fig. 4 therein, which presents the ℓ/R-dependencies of the quasi-mode frequencies and damping\nrates for a fixed combination [ ρi/ρe=5,d/R=2.5,kR=π/100]. Note that varying ℓ/Ractually impacts our ¯l,d/b, and kb\nsimultaneously (see Eq. 19). We adopt a fixed kb=π/30 to save computational time. This does not matter because we will measure\nour timescales or frequencies in appropriate units (say, ΩinωAe) and the resulting readings do not depend on kRwhen kR≲π/20\n(see Fig. 3 in SL15, and note that kR<kb). Equation (19) readily converts the pair [ ℓ/R,d/R] into our [ ¯l,d/b].\nWe now test our time-dependent computations against the SL15 results, to be labeled “T-matrix” for clarity. It su ffices to examine\nthe time sequences of the x-speed sampled at the left tube center (namely, vx(−d/2,0;t)); the right counterpart is either identical\n(forSx) or di fferent only by sign (for Ax). Figure 3 focuses on the choice ℓ/R=0.4, and plots the sampled vxfor (a) the Sxand (b)\ntheAxpatterns by the solid curves. The same analysis as in Fig. 2a is then repeated for both curves, whereby we single out the six\nextrema (the red asterisks in Fig. 3) after the transitory phase to perform a fitting with Eq. (27). The best-fit exponential envelope,\nplotted by the dashed curves for the entire duration, is seen to well reflect the damping of kink motions for much longer time.\nFigure 4 proceeds to examine the ℓ/R-dependencies of (a) the oscillation frequencies ( ΩinωAe) and (b) damping-rate-to-\nfrequency ratios ( γ/Ω). The SxandAxpatterns are discriminated by the di fferent colors. We present the best-fit values from our\ntime-dependent computations by the open circles, and overplot the T-matrix results by the solid curves for comparison. Note that\nthese T-matrix curves are read from Fig. 4 of SL15. Our best-fit results are seen to compare favorably with the T-matrix results,\nwhich is particularly true when ℓ/R≲0.3. This statement holds despite the somewhat visible di fference in Ωat, say,ℓ/R=0.2,\nwhere the best-fit value actually deviates from its T-matrix counterpart by only ∼2.7%. One further sees that the most significant\ndeparture occurs for the Axdamping rate when ℓ/R=0.4. However, this di fference is rather modest and reads ∼21.7% in relative\nterms. We therefore conclude that the dispersive properties of kink quasi-modes can be reasonably computed by incorporating the\nshortcut TB formulae in the T-matrix framework for all ℓ/Rexamined here2. Our time-dependent results, on the other hand, further\ncorroborate the SL15 conclusion that Axmotions posses higher frequencies and damp more rapidly than Sxones.\n2Some discrepancy exists in the literature regarding the damping of lower-order kink motions when the tubes are in contact. The T-matrix results\nby SL15 indicate that these motions remain damped (see Fig. 2 therein), whereas damping was found to disappear in the modal studies formulated\nin bi-cylindrical coordinates (Robertson & Ruderman 2011; Gijsen & Van Doorsselaere 2014). Our IVP study supports the SL15 conclusion. The\nuse of a bi-cylindrical coordinate system was recognized by the referenced studies to be physically problematic for small tube separations.\nArticle number, page 7 of 31A&A proofs: manuscript no. manuV1\n3.3. Interim Summary on Numerical Aspects\nThis subsection further justifies our numerical treatment. An additional series of computations have been performed to address the\neffects of the grid spacing ∆and the domain size [ xM,yM]. These results are collected in Appendix A to streamline the main text.\nOverall, we have verified that the MacCormack scheme is appropriate for our purposes, with no issue arising from the application\nto nonplanar structures of the finite-di fference methodology on a Cartesian grid. The zero-gradient boundary condition is somehow\nnot fully transparent to perturbations excited by our planar drivers, and the domain size somehow impacts how the internal flows\ntransition to a quasi-mode behavior. However, this domain size e ffect is practically negligible on the best-fit periods ( Pfit) and\ndamping times ( τfit) that we derive for the quasi-mode stage. Likewise, the computed internal flows are not a ffected by the grid\nspacing ∆provided that ∆is sufficiently small.\n4. Kink Motions in a Two-Elliptic-Tube Setup\nThis section is devoted to kink motions in a two-elliptic-tube configuration, for which the x-major and x-minor orientations need\nto be discriminated. The reference grid ([ xM=30b,yM=25b,∆ = 0.01b], see Table 1) will be consistently adopted, enabling a\nmeaningful comparison between the tube orientations. The notations [ a,b,¯l] are adopted throughout this section, where all compu-\ntations pertain to a fixed combination of physical parameters [ ρi/ρe=3,¯l=0.4,kb=π/30]. We adjust only the dimensionless tube\nseparation ( d/b) and the ratio of the semi-major to semi-minor axis ( a/b) for a given orientation and a given perturbation pattern\n(see Eq. 21). Tube overlapping is always avoided. The equilibria are consistently perturbed with Eq. (20).\n4.1.SxandAxMotions for the x-major Orientation\nThis subsection addresses the x-major orientation. We start with Fig. 5 to present the temporal evolution of the x-speed sampled at\nthe left tube center ( vx(−d/2,0;t), the solid lines) for both (a) the Sxand (b) the Axmotions. The tube separation is fixed at d=5b,\nwhereas a number of values are examined for a/bas discriminated by the di fferent colors. Only the variations after the rapid-varying\nphase are of interest for any vxsequence. We somehow emphasize the segment exactly encompassing the six extrema starting with\nthe one labeled by the relevant asterisk, performing a fitting procedure with Eq. (27). A best-fit exponential envelope results, with\nan example plotted over the entire duration for the case where a/b=2.5. With this example we demonstrate that an exponentially\ndamped cosine is in general adequate for describing the vxsequences after the rapid-varying phase. The best-fit periods and damping\ntimes thus derived will be be understood as pertaining to an ideal quasi-mode.\nThe temporal attenuation of the internal flow fields means energy redistribution in ideal MHD. Let us take the case with a/b=2.5\nin Fig. 5, and note that the two tubes are actually in contact ( d=5b=2a). Figure 6 presents the spatial distribution of the velocity\nfields ( v=vxex+vyey, the blue arrows) and the instantaneous energy density ( ϵ, filled contours) for (a) the Sxand (b) Axmotions\nat an arbitrarily chosen instant t=260b/vAe. A subarea of the second quadrant is singled out by each inset to emphasize the ϵ\ndistributions in the nonuniform layers of both tubes, with the red curves delineating the layer boundaries. The white curve further\nindicates where the local Alfv ´en frequency equals the quasi-mode frequency. Figure 6 is actually taken from the attached animation.\nThe arrows and filled contours are plotted in a way that it makes sense to compare the vorϵstrengths not only in one snapshot but\nbetween di fferent instants for a given orientation. In particular, the darker a portion is, the larger the value of ϵtherein.\nConsider first the ambient fields for both perturbation patterns, where by “ambient” we refer to the flows some distance away\nfrom the tubes. Likewise, by “tube flow” we mean the field excluding the ambient flow, and by “internal flow” we refer specifically to\nthe uniform portion inside either tube. For both patterns, one sees from the animation that the ambient flows tend to vary more rapidly\nthan the internal ones. This feature is particularly clear for the Sxpattern because the ambient flows tend to be substantially stronger\nthan the internal flows as well. Regardless, a periodogram analysis yields that the ambient flows follow primarily the external\nAlfv ´en frequency ωAe, whereas the variations of the internal flows are by far dominated by the lower quasi-mode frequency. That\nthe ambient motions are not coordinated with the internal ones is a result of the form of the initial exciter given by Eq. (20). The\nrelevant physics is actually very similar to what happens in the T08 study despite the configurational di fferences, the key being the\ny-invariance of Eq. (20). It proves easier to explain this by considering a uniform equilibrium with density ρe, for which Eqs. (7) to\n(11) can be combined to yield a Klein-Gordon equation if the y-dependence is dropped (see Eq. 1 in Terradas et al. 2005). Suppose\nthat an individual component in Eq. (20) is applied. A dispersive fast wave then results, whereby any fluid parcel in the system\neventually ends up in some oscillatory wake at the Alfv ´en frequency ωAe(see Fig. 1 in Terradas et al. 2005). Oscillatory wakes\nturn out to still exist in the ambient flow when the two elliptic tubes are introduced, and when an additional planar perturbation is\nimplemented. Note that the initial perturbations are external to our tubes. Note further that overall the large-time behavior for the\nambient flow actually comprises two component wakes, each being the response to the corresponding component driver. The two\ncomponent wakes tend to interfere constructively (largely destructively) for the Sx(Ax) pattern, thereby explaining the strength of\nthe ambient flow relative to the internal one.\nNow examine the tube flows. Focusing on the blue arrows, one sees from Fig. 6 that the gross patterns for both SxandAx\nare similar to Fig. 2 in L08 despite that a two-circular-tube equilibrium with piece-wise constant profile was addressed therein.\nSpecifically, the internal velocity field is more or less uniform, and a pair of vortical motions develop at the edge of either tube.\nThese two features, combined with the overall symmetric properties of the internal field about x=0, demonstrate the robustness\nof the classification scheme for lower-order kink motions. The primary di fference from L08, on the other hand, is that the vortical\nmotions gradually evolve from a simple dipolar behavior (see the first several instants in the animation) into multiple shearing\nlayers (Fig. 6). This evolution is well known for kink motions in isolated tubes with either circular (e.g., Terradas et al. 2008a;\nPascoe et al. 2010; Antolin et al. 2014) or elliptic cross-sections (e.g., Ruderman 2003; Guo et al. 2020). What Fig. 6 demonstrates\nis then that tube interactions do not compromise the phase-mixing physics behind this evolution. That said, tube interactions do\nArticle number, page 8 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nimpact the detailed development of the small-scale shears, as evidenced by the di fferences in the velocity fields for the two di fferent\nperturbation patterns. Suppose that tube interactions are negligible. The tube flows are then expected to be identical for the SxandAx\npatterns, meaning in particular a symmetric distribution of the energy density ( ϵ) with respect to the minor axis ( x=−d/2=−2.5b\nhere). However, this expectation roughly holds only for the Sxpattern (Fig. 6a inset), whereas the left portion in the left tube is\nfavored in the ϵdistribution for the Axpattern (Fig. 6b inset). Besides phase-mixing, a closely related aspect that is not fundamentally\ncompromised by tube interactions is the resonant interplay between the quasi-mode and the Alfv ´en continuum (see Soler & Terradas\n2015 and references therein for the subtle distinction between resonant absorption and phase-mixing). By this we refer to two\nfeatures common to the SxandAxpatterns, and we concentrate on the insets in the animation. Firstly, the attenuation of the internal\nfield is accompanied by the accumulation of perturbation energies in the nonuniform layer. Secondly, this energy accumulation leads\nto a localized ϵdistribution, with the strongest perturbations tending to the white resonance contour as time proceeds for roughly\none quasi-mode period after the rapid-varying phase. Overall with Fig. 6 we conclude that the notions of phase-mixing and resonant\nabsorption remain applicable to kink motions in our two-elliptic-tube configuration.\nFigure 7 further examines the phase-mixing process by showing the y-cuts of the x-speeds through the left tube center for both\n(a) the Sxand (b) the Axpatterns. A number of instants are rather arbitrarily chosen and discriminated by the di fferent colors.\nThe vertical dotted lines mark the borders of the nonuniform layer, where shearing motions ( ∂vx/∂yhere) with increasingly small\nscales are seen to develop as time proceeds. We choose to quantify this development by relating the number of the vxextrema\n(Nextrm(t)) to the instantaneous phase variation accumulated over the Alfv ´en continuum ( ϕac(t), see the discussion on Eq. A.1). Note\nthatϕac(t)=(ωAe−ωAi)tis identical for both perturbation patterns, corresponding specifically to [2 .82,4.23,5.64] half-cycles for\nthe examined instants. One then expects [2+1\n+0,4+1\n+0,5+1\n+0] extrema in the vxprofiles in Figs. 7a and 7b, which is indeed the case3. As\ncan be readily verified, that Nextrm(t)=⌊ϕac(t)/π⌋at these instants actually constrains the phase φ0in Eq. (A.1) to a very narrow\nrange [ qπ,(q+0.19)π] with q=0,1.\nFigure 8 proceeds to examine (a) the periods Pand (b) damping-time-to-period ratios τ/Pof kink quasi-modes pertaining to both\ntheSx(the solid curves) and the Ax(dashed) patterns, showing how Pandτdepend on the ratio of the semi-major to semi-minor\nradius ( a/b) for a given tube separation ( d). We also examine a number of das discriminated by the di fferent colors. Note that this\nsurvey adopts a fixed combination of physical parameters [ ρi/ρe=3,¯l=0.4,kb=π/30]. The same set of parameters is additionally\nadopted to evaluate Pandτ/Pof the kink quasi-mode for an isolated circular tube with the resistive eigenmode approach (see\nSect. 3.1), the results being plotted by the horizontal lines for comparison. Note that the values for P(in units of the axial Alfv ´en\ntime Taxial=2π/ω Ae=2L/vAe) andτ/Pvary little if kb=πb/Lis further reduced, meaning that little will change if one examines\naxial fundamentals in AR loops with much larger L/bthan adopted here. Note further that a/bis required not to exceed d/2bfor\na given d/bsuch that tube overlapping is avoided. For any given pair [ a/b,d/b], Fig. 8 indicates that the Axmotion possesses a\nshorter period and damps more e fficiently than the Sxmotion. This behavior is identical to what SL15 found for two-circular-tube\nconfigurations. The argument therein is that Axmotions are more “forced”; the two tubes move largely in a synchronous fashion\nfor the Sxpattern, whereas the flows in between the two tubes periodically “collide” when the Axpattern is examined. The same\nargument applies here although elliptic tubes are addressed, with the insets of Fig. 6b already hinting at a more forced behavior for\nAxmotions. Also similar to SL15 is that, for a given a/b, the di fference between the values of P(orτ/P) for the SxandAxmotions\ntends to decrease monotonically with d. This agrees with the intuitive expectation for a weaker tube interaction.\nLet us pay more attention to the overall behavior for Pto increase monotonically with a/bfor a given d/b. Somehow subtle\nis that tube interactions also have some e ffect, because increasing a/bactually brings the two tubes e ffectively closer even if the\ndistance between the tube centers is fixed. Regardless, this tube interaction only plays a minor role, given that the monotonical\na/b-dependence takes place for di fferent dand for the SxandAxmotions alike. One may therefore expect the same a/b-dependence\nfor isolated elliptic tubes (or equivalently d/b→∞ ), in which case it is no longer necessary to distinguish between the SxandAx\npatterns. This was indeed seen in our previous IVP study (Guo et al. 2020), where we o ffered some heuristic argument similar to the\nearlier one by Ruderman (2003). For simplicity, suppose that the equilibrium configuration is transversely structured in a piece-wise\nconstant manner. The heuristic argument then relies on the observation that the velocity normal to the tube edge plays a central\nrole for an isolated density-enhanced tube to communicate with its surroundings. Evidently, a larger a/bmakes the tube edge more\nelongated in the x-direction. Their velocities primarily x-directed, the fluid parcels in the tube interior therefore become less aware\nof the surroundings, meaning some enhanced e ffective inertia and hence a longer period. The same argument can be invoked in this\nstudy, despite the subtlety that the fluids surrounding one tube actually embed another tube.\n4.2.SxandAxMotions for the x-minor Orientation\nThis subsection addresses the x-minor orientation. We start with Fig. 9 where the same set of physical quantities as in Fig. 5 is\nemployed, and the time sequences for the x-speed sampled at the left tube center are presented in an identical format. A fitting\nprocedure is once again performed for any vxsequence over the duration exactly enclosing the six extrema starting with the one\nindicated by the pertinent asterisk. We take the view that the best-fit periods ( P) and damping times ( τ) pertain to the ideal quasi-\nmodes supported by our two-elliptic-tube configuration.\nFigure 10 specializes to the case with [ a/b=2.5,d/b=5], following the same format as in Fig. 6 to present the velocity fields\n(the blue arrows) and the spatial distributions of the perturbation energy density ( ϵ, filled contours) at a representative instant. Note\n3The velocity shears are prone to the Kelvin-Helmholtz instability (KHi), as was first theoretically recognized in the wave context by Heyvaerts\n& Priest (1983) and Browning & Priest (1984). Further nonlinear simulations have demonstrated that the KHi may arise in the early stage of the\nevolution of kink motions for isolated tubes (e.g., Terradas et al. 2008a; Antolin et al. 2015; Shi et al. 2021) and multi-tube setups alike (e.g.,\nMagyar & Van Doorsselaere 2016; Guo et al. 2019a). The applicability of such linear results as presented here may therefore need to be assessed\non a case-by-case basis when real data are analyzed (see Sect. 10 of the review by Nakariakov et al. 2021, for more on the nonlinear aspects).\nArticle number, page 9 of 31A&A proofs: manuscript no. manuV1\nthat the tubes are quite some distance apart in this case. Note also that Fig. 10 is a snapshot extracted from the attached animation.\nFocusing on the velocity fields, one can safely conclude from the animation that the notations of SxandAxmake physical sense as\nproposed by L08 for two-circular-tube systems. For both perturbation patterns, the insets in the animation further indicate that the\nattenuation of the internal flows is accompanied by the enhancement of perturbation energy density ϵin the nonuniform layers, with\nthe localization of ϵstrongly mediated by the Alfv ´en resonance. This latter point can be readily drawn from the close association of\nthe darkest portion in either ϵdistribution with the white curve marking where the quasi-mode frequency matches the local Alfv ´en\nfrequency.\nFigure 11 presents, in a format identical to Fig. 7, the y-cuts of the x-speed through the left tube center for a number of arbitrarily\nchosen instants. The aim is also to show the key role that phase-mixing plays for the gradual development of shearing motions with\nincreasingly small scales in the nonuniform layers. We follow Fig. 7 to quantify this with the aid of Eq. (A.1), once again counting\nthe instantaneous number Nextrm(t) of the vxextrema. The pertinent instantaneous phase variations ( ϕac(t)), on the other hand, are the\nsame as in Fig. 7 and read [2 .82,4.23,5.64]πfor the examined instants. However, somehow di fferent is that the relevant values for\nNextrm now may deviate from ⌊ϕac/π⌋, attaining specifically [3 ,4,5]. This deviation is readily attributable to the phase φ0in Eq. (A.1).\nConversely, this set of deviations can be readily verified to constrain φ0to a rather narrow range of [( q+0.19)π,(q+0.37)π] with\nq=0,1.\nFigure 12 moves on to survey (a) the quasi-mode periods Pand (b) damping-time-to-period ratios τ/Pfor a substantial range\nof combinations [ a/b,d/b]. This figure is identical in format to Fig. 8. The values examined for the tube separation dconsistently\navoid tube overlapping, and hence there is no need to constrain the ratio of the semi-major to semi-minor axis ( a/b) for a given\nd/b. Examine Fig. 12a first, from which one sees the expected monotonic behavior for the di fferences between the SxandAx\nperiods to decrease with d/bfor a given a/b. For a given separation d/b, one further sees that the values of Ptend to decrease\nmonotonically with a/bfor the SxandAxmotions alike. However, tube interactions remain possible to be partly responsible for\nthisa/b-dependence, because the area between the two tubes actually broadens with a/beven if d/bis fixed. This tube-interaction\neffect can hardly be dismissed given that the SxandAxvalues deviate more strongly when a/bincreases for a given d/b. Having\nsaid that, it remains possible to partly explain the monotonic a/b-dependence of Pby invoking the inertia arguments initially o ffered\nfor isolated density-enhanced elliptic tubes. Note that the tube edges are now more elongated in the y-direction, whereas the fluids\nparcels in either tube interior remain to oscillate primarily in the x-direction. It therefore holds that an increase in a/bmakes the\ninternal fluid parcels better aware of the external medium, meaning some reduced e ffective inertia and hence a shorter period.\nConsider now Fig. 12b where the damping-time-to-period ratios ( τ/P) are examined. Two features arise when a/b≲1.5, one\nbeing that Axmotions attenuate more e fficiently than Sxones, the other being that the di fferences between the SxandAxvalues tend\nto decrease with tube separation. These features resemble what happens for two-circular-tube systems, and are expected because the\ncross-sectional shapes are not far from circular. However, the behavior of τ/Pat larger a/bis considerably more complicated than\nin Fig. 8b where the x-major orientation is addressed. By “complicated” we specifically refer to the nonmonotoic d/b-dependence\nofτ/Pfor either perturbation pattern. Take the case with a/b=2.5. When d/bincreases, one sees an increase followed by some\ndecrease in τ/Pfor the Sxmotion, whereas τ/Ptends to decrease for the Axmotion. This latter simplicity is only apparent, because\nτ/Pis bound to increase with d/bagain such that the Axvalues eventually join their Sxcounterparts when the tubes are su fficiently\nseparated. We have verified that the somehow involved behavior of τ/Pis not of numerical origin (see Sect. 3). Furthermore, it\nmatters little as to how to choose the segment of a vxsequence for fitting, provided that this segment starts su fficiently later than\nthe rapid-varying phase (see Fig. 9). The most likely reason is then that the perturbations in between the two tubes depend on a/b\nandd/bin an intricate fashion, and hence an intricate dependence on a/bandd/bof the e fficiency of tube interactions. Regardless,\nthe reason behind the d/b-dependence of τ/Pis also behind, say, the nonmonotonic a/b-dependence of τ/PforAxmotions for a\ngiven d/b(see the dashed curves), and the occasional behavior for Sxmotions to damp more e fficiently (compare the black solid\nand dashed curves).\n5. Implications for Coronal Seismology\nThis section discusses some seismological implications of our study. Only axial fundamentals are considered. We restrict ourselves\nto the inference of the axial Alfv ´en time ( Taxial=2L/vAe) and the dimensionless nonuniform layer width ( ¯l). We further assume that\nneighboring elliptic tubes are imaged in, say, the EUV when the line of sight lies in the tube plane (see Fig. 1a), meaning that the\ntubes may be readily mistaken as circular ones.\nLet us start with Figure 8 where our x-major computations are summarized, and consider only the Sxmotion for [ a/b=\n2.5,d/b=5]. Now assume that the relevant oscillating tubes possess the “true” dimensionless values [ ρi/ρe=3,¯l=0.4] (note that\nL/bis immaterial when su fficiently large). Let Pobsand (τ/P)obsbe the measured values of the period and damping-time-to-period\nratio, and we take ( τ/P)obs=6.39 to comply with Fig. 8. One therefore deduces a “true” value of Pobs/1.6 for Taxial, which is recalled\nto be dimensional. Now suppose that the density contrast is measured to be ρi/ρe=3, and let us follow the customary practice to\ninfer Taxialand¯lby seeing the Sxmotion as a kink quasi-mode in an isolated circular tube. It matters little regarding whether one\nassigns aorbto the tube radius. The dimensionless quasi-mode period P/Taxialdepends essentially only on ρi/ρe, always attaining\n∼1.42 when ¯l≲0.6 (see the green horizontal line in Fig. 8a; see also Fig. 2b). One therefore deduces a value of Pobs/1.42 for\nTaxial, which di ffers from the true value by ∼13%. However, it is considerably more problematic when ( τ/P)obsis invoked to infer\n¯l. Employing the resistive eigenmode approach, we find that a ( τ/P)obs=6.39 results in a ¯l≈0.18, which underestimates the true\nvalue by∼55%. One may argue that this di fference is not that significant; after all the customary seismological practice yields the\ncorrect qualitative picture that the oscillating tube possesses a thin boundary. Our point, however, is that care needs to be exercised\nwhen (τ/P)obsis put to quantitative use, with the uncertainties in the deduced dimensionless layer width ¯lreadily exceeding ∼50%\nif one neglects the combined e ffect of tube cross-sectional shapes and tube interactions.\nArticle number, page 10 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nNow move on to Fig. 12 where our x-minor computations are summarized. We assess what uncertainties the customary practice\nmay introduce to the axial Alfv ´en time Taxial=2L/vAeand dimensionless layer width ¯lby neglecting the joint e ffects of cross-\nsectional shapes and tube interactions. The procedure is identical to what we performed for the x-major orientation; we accordingly\nchoose the Axvalues for [ a/b=2.5,d/b=3] given that they deviate the most from the horizontal lines. Consider the period first.\nOne infers a true value of Taxial=Pobs/1.13 with Fig. 12a, whereas the customary practice remains to yield a Taxial≈Pobs/1.42\nor equivalently some underestimation by ∼20%. This uncertainty in Taxialis larger than for the x-major orientation. Now taking\n(τ/P)obs=3.44 from Fig. 12b as measured, one deduces with the customary practice a dimensionless layer width of ¯l=0.25 or\nsome underestimation of the true value ( ¯l=0.4) by∼38%. Somehow smaller than for the x-major case, this uncertainty remains\nsubstantial enough to corroborate our claim that the inhomogeneity scales deduced with the customary practice need to be treated\nwith caution.\n6. Summary\nWith coordinated transverse displacements in neighboring active region (AR) loops in mind, we have addressed damped kink\nmotions in straight equilibria where two identical parallel density-enhanced tubes with elliptic cross-sections (elliptic tubes) are\nembedded in an ambient corona. Linear, ideal, pressureless MHD was adopted throughout. We concentrated on axially standing\nmotions, formulating the ensuing two-dimensional (2D) initial value problem (IVP) in the plane transverse to the equilibrium\nmagnetic field. We identified the direction connecting the tube centers as horizontal, discriminating two tube orientations where\neither the major (dubbed “ x-major”) or the minor (“ x-minor”) axis is horizontally placed. The system evolution was initiated with\nexternal velocity drivers, implemented in such a way that the internal flow fields are primarily horizontal and are either symmetric\n(Sx) or anti-symmetric ( Ax) with respect to the vertical axis about which our equilibrium configuration is symmetric. The temporal\nevolution of the velocity perturbations at tube centers was of particular interest, allowing us to identify the quasi-mode stage as\nwhere the monochromatic time sequence follows an exponentially damped envelope. We paid special attention to how the quasi-\nmode periods and damping times depend on the tube separation and the cross-sectional aspect ratio, thereby addressing the impact\non the dispersive properties of damped kink motions from the joint e ffects of tube interactions and cross-sectional shapes.\nOur numerical findings are summarized as follows. When two-circular-tube equilibria are examined as a special case, the quasi-\nmode periods and damping times found with our IVP approach are consistent with the modal analysis by Soler & Luna (2015),\nthereby independently justifying both the analytical T-matrix approach and the numerical results therein. We further find that the\nnotions of resonant absorption and phase-mixing are not undermined when elliptic cross-sections are allowed for. Specifically, the\nnonuniform layers around tube boundaries feature both the development of velocity shears with increasingly small scales and the\nenergy accumulation around where the local Alfv ´en frequency matches the quasi-mode frequency. The Axmotions are found to pos-\nsess shorter periods and damp more rapidly than the Sxmotions for the x-major orientation, in which case the periods and damping\ntimes for both perturbation patterns tend to increase with the major-to-minor-axis ratio. When the x-minor orientation is addressed,\ntheAxmotions remain to oscillate more rapidly than the Sxones. However, the periods for both patterns tend to decrease monoton-\nically with the major-to-minor-axis ratio. Somehow subtle are the joint e ffects of tube interactions and cross-sectional shapes on the\ndamping times, which may depend non-monotonically on the tube separation or the major-to-minor-axis ratio. Furthermore, the Ax\nmotions may occasionally damp less e fficiently. Neglecting these joint e ffects may introduce some uncertainty of ∼20% (∼50%)\nto the axial Alfv ´en time (the inhomogeneity lengthscale) deduced with the period (damping time). Consequently, care may need to\nbe exercised when the damping times of kink motions are put to standard seismological practice built on wave theories for isolated\ncircular tubes.\nSome further remarks seem necessary before closing. Firstly, we note that our IVP approach is not restricted to two-tube systems\nbut capable of handling rather generic 2D inhomogeneities. In particular, axially standing motions may be examined for straight\nconfigurations comprising multiple tubular structures with arbitrary cross-sections. Collective motions in multiple neighboring\nresolved loops will then be better understood, together with the motions in those apparently isolated loops that actually involve a\nmultitude of unresolved strands (see e.g., Sect.4.2 of the review by Reale 2014 for more on this multi-stranded nature of AR loops).\nSecondly, that kink motions are resonantly absorbed for configurations with continuous transverse structuring is actually much-\nexpected (e.g., the review by Goossens et al. 2011). However, somehow largely unnoticed is that the decayless regime was also\nobserved for coordinated kink motions in adjacent AR loops (Wang et al. 2012). Accept that the transverse structuring is likely to\nbe continuous in the relevant configurations. It then becomes necessary to address what counteracts the ensuing resonant damping,\nthereby bringing the understanding of decayless kink motions in multi-tube systems closer to the level of sophistication reached for\nthe decayless regime in isolated tubes (e.g., Nakariakov et al. 2016; Guo et al. 2019b; Ruderman et al. 2021). Thirdly, coordinated\nmotions in multiple tubular structures were identified not only in AR loops but also in, say, groups of prominence threads (see e.g.,\nArregui et al. 2018, for a review). To our knowledge, so far the only theoretical investigation in the prominence context was the\none by Soler et al. (2009) with the T-matrix formalism. An IVP perspective is expected to be as fruitful, provided that one further\naccount for, say, the e ffects associated with partial ionization (e.g., the review by Ballester et al. 2018).\nAcknowledgements. 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It su ffices to address only\ntheSxpattern for a fixed combination of physical parameters [ ρi/ρe=5,ℓ/R=0.4,d/R=2.5,kb=π/30].\nWe start with Fig. 13 by displaying the temporal evolution of the x-speed at the left tube center ( vx(−d/2,0;t), the solid curves)\nfor a number of domain sizes [ xM,yM] as discriminated by the di fferent colors. The best-fit damping envelopes are further plotted by\nthe dashed curves for reference. All computations pertain to a fixed grid spacing ∆ =0.01b, which is recalled to be di fferent from the\nreference value in Table 1. Some di fferences are seen among the solid curves, the most prominent one being that the monochromatic\ncycles in the vxsignal are slightly delayed when the domain size increases. We find that this domain size e ffect arises because the\nzero-gradient boundary condition is not fully transparent to impinging motions initiated with the pair of planar perturbations given\nby Eq. (20). Some weakly reflected disturbances reach and interact with the composite structure, resulting in some longer time for\nthe internal flow fields to settle toward a regular pattern for a larger domain. That the domain size e ffect is primarily attributed to the\nform of the initial perturbation is further corroborated by an additional series of computations where we multiply each exponential\nterm in Eq. (20) by a factor exp( −y2/b2). The dependence on the domain size is then found to disappear in the time-dependent\nsolutions responding to this localized perturbation. A monochromatic pattern remains clearly visible after some transitory phase,\nthe period being nearly identical to what we find when Eq. (20) is employed. However, the damping envelope deviates considerably\nfrom an exponential one, making it di fficult to examine ideal quasi-modes from an IVP perspective. The reasons for us to stick to\nEq. (20) are twofold. Firstly, what drives the internal field toward a quasi-mode behavior is beyond our scope. Rather, we content\nourselves with deriving the relevant periods and damping times. Secondly, varying the domain size does not influence the best-fit\nperiod ( Pfit) and only marginally impacts the best-fit damping time ( τfit). In fact, Pfitalways evaluates to ∼112.5b/vAefor the\nexamined domains. The value of τfitvAe/b, on the other hand, changes from 320 .1 for [ xM=12b,yM=10b] only modestly to 313 .3\nfor [xM=18b,yM=15b]. An even weaker variation is found when the domain is further extended to [ xM=24b,yM=20b], in\nwhich case τfitvAe/battains 314.3.\nOne may ask why we chose a spacing ∆ = 0.01bdifferent from the reference value ( ∆ = 0.005b) when examining the domain\nsize e ffect. The answer is that the specific values of ∆do not impact the vx(−d/2,0;t) sequences for a given domain size if ∆is\nsufficiently small. By “su fficiently” we mean some criterion related to the pair of equilibrium quantities [ ρi/ρe,¯l] (see Sect. 2.4). We\nreach this statement by varying only ∆for a substantial fraction of the computations in Fig. 4. Consider only those with the reference\ndomain size. Figure 14 o ffers a representative case with ℓ/R=0.4 by discriminating the vx(−d/2,0;t) sequences for di fferent ∆with\nthe di fferent linestyles. Relative to the computation with the reference ∆(the red dashed curve), no di fference can be discerned when\nthe spacing is halved (blue dash-dotted) or doubled (black dotted). The reference ∆is therefore overly demanding of computational\nresources for ℓ/R=0.4, and the same can be said even for the most stringent case where ℓ/R=0.2 provided that only vx(−d/2,0;t)\nis of interest. That said, varying ∆necessarily entails di fferences in other aspects of the flow fields. We illustrate this point by fixing\nℓ/Ragain at 0.4 and implementing the reference domain size.\nFigure 15 presents the spatial profiles of the x-speed along the y-cut through the left tube center at t=340b/vAe. This instant\nis chosen rather arbitrarily to be marginally inside the interval where our fitting procedure is performed. A cut through a tube\ncenter, on the other hand, is chosen to ease the description of some key dynamics. A number of grid spacings are implemented and\ndiscriminated by the di fferent colors. Only the lower half ( y≤0) of a y-profile is displayed because the other half is symmetric with\nrespect to y=0. The vertical dash-dotted lines represent the borders of the nonuniform layer. Focus for now on this layer, where\none sees a series of ripples in all computations. The most prominent e ffect associated with ∆is that the ripples are better resolved\nwhen the numerical grid gets finer, with the computed magnitudes of the local extrema being larger. Physically speaking, however,\nall computations prove capable of capturing the phase-mixing physics pertaining to the Alfv ´en continuum. To demonstrate this, we\nnote that the Alfv ´en frequency ωA=kvAis locally symmetric about x=−d/2, meaning ∂ωA/∂x=0 for the chosen cut. It then\nfollows from Mann et al. (1995) that the vxprofile in the layer can be written as\nvx∝A(y) cos[ωA(y)t+φ0], (A.1)\nwhereA(y) is some slow-varying envelope. Let ωAi=kvAibe the internal Alfv ´en frequency. Some instantaneous phase variation\nϕac(t)=(ωAe−ωAi)tis therefore accumulated over the continuum, leading to Nhc=⌊ϕac(t)/π⌋half-cycles with⌊·⌋being the floor\nfunction (see Shi et al. 2024, for similar applications). Consequently, one expects NhcorNhc+1 extrema, where the uncertainty\nderives from the phase φ0in Eq. (A.1). Plugging in the numbers yields a ϕac/πof 6.27, and therefore 6 or 7 extrema are expected.\nThis expectation is faithfully reproduced in Fig. 15. The same can actually be said for all instants when the internal field behaves like\na quasi-mode. Now move on to the portions outside the nonuniform layer. The only di fference that one can barely tell for di fferent\n∆appears in the external flow fields immediately adjacent to the layer. On the other hand, no di fference can be discerned in the\ninternal motions.\nFigure 16 examines the grid spacing e ffect from the energetics perspective, di fferentiating ∆by the di fferent colors. Equa-\ntions (12) to (15) are employed to evaluate all energetics-related quantities, and we take the area Qtherein to be a box [ −(d/2+\n1.2b),(d/2+1.2b)]×[−1.2b,1.2b]. Figure 16a performs a gross energy balance analysis over Qby presenting the temporal profiles\nof the time-derivative of the instantaneous total energy (d Etot/dt, the solid curves) and those of the instantaneous energy flux into\nQ(−F, asterisks). The asterisks of di fferent colors cannot be told apart, the reason being that all computations yield an identical set\nof dependents at the boundaries of Q. Some di fference nonetheless shows up among the solid curves, despite that d Etot/dtshould\nbalance F(see Eq. 12). As intuitively expected, gross energy conservation is maintained for a longer duration when a finer grid is\nemployed, with some ∼30% di fference between d Etot/dtand−Fstaring to appear at t∼90b/vAe(∼270b/vAe) when ∆ = 0.01b\n(0.0025 b). Note that the examined Qis the union of three mutually exclusive portions, namely the tube interiors (to be denoted\nas “int”), nonuniform layers (“layer”), and exterior (“ext”). Figure 16b then displays, against time, the energies in these individual\nArticle number, page 14 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nportions. By far the most significant e ffect that the grid spacing has on the energetics is that the energy in the nonuniform layer is\nincreasingly under-estimated when ∆increases. However, there exists only some weak di fference among the Eextcurves because\nonly the flows in the immediate neighborhood of either tube are somehow impacted by choices of ∆. Furthermore, the internal\nenergies ( Eint) consistently remain the same when ∆is varied, reinforcing the notion that ∆has no influence on the internal flow\nfield.\nArticle number, page 15 of 31A&A proofs: manuscript no. manuV1\nFig. 1. Schematics showing both the two-elliptic-tube configuration and lower-order kink motions therein. Illustrated in (b) are two identical tubes\nwith elliptic cross-sections and separated in the x-direction. The y−zplane is identified as the tube plane, the meaning of which is clearer in the\ncurved representation in panel a. This study examines only two tube orientations (panels c and d), with the one labeled “ x-major” (“ x-minor”)\ncorresponding to the situation where the major- (minor-)axis of either tube is aligned with the x-axis. The tube centers are placed symmetrically\nwith respect to the y-axis. Only axial fundamentals are considered, making it possible to examine the three-dimensional (3D) wave fields with a\n2D approach. Lower-order kink motions are classified according to their internal flow fields in the apex plane (panels c or d); hence such symbols\nasSxandAy. See text for more details.\nArticle number, page 16 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 2. Kink motions in isolated tubes with circular cross-sections for a fixed pair of density contrast and dimensionless axial wavenumber\n[ρi/ρe=3,kb=π/30]. A specific dimensionless layer width ¯l=0.4 is examined in (a), the solid curve showing the x-speed sampled at the tube\ncenter ( vx(0,0,t)). An exponentially damped cosine is employed to fit the segment encompassing the extrema shown by the red asterisks, with the\nresulting damping envelope plotted over the entire interval (the dashed curves). Presented in (b) are the ¯l-dependencies of the mode frequencies\n(Ω, the black curves and symbols) and the ratios of the decay rate to the frequency ( γ/Ω, blue) obtained with various methods. The solid curves\nrepresent the analytical thin-tube-thin-boundary (TTTB) expectations with Eq. (25). No assumption on tube length or layer width is imposed in the\n“Resistive” computations (the dashed curves) that evaluate ideal quasi-modes as resistive eigenmodes. Shown by the open circles are the best-fit\nresults from our time-dependent computations, where kink motions are consistently excited with a perturbation given by Eq. (26).\nArticle number, page 17 of 31A&A proofs: manuscript no. manuV1\nFig. 3. Kink motions with (a) Sxand (b) Axpatterns in a two-circular-tube configuration. Plotted are the time sequences of the x-speed sampled\nat the left tube center ( vx(−d/2,0;t), the solid curves) together with the best-fit damping envelopes (dashed). The time-dependent computations\npertain to a fixed combination [ ρi/ρe=5,ℓ/R=0.4,d/R=2.5,kb=π/30], withℓandRbeing the nonuniform layer width and mean tube radius\nimplemented by Ruderman & Roberts (2002). A fitting procedure with an exponentially damped cosine is performed over the duration exactly\nencompassing the extrema indicated by the red asterisks, even though the best-fit envelopes are plotted for the entire time interval. Kink motions\nare excited with the perturbation described by Eq. (20) with appropriate signs.\nArticle number, page 18 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 4. Comparison of (a) the oscillation frequencies Ωand (b) damping rates γof ideal quasi-modes deduced from time-dependent computations\n(the open circles) with those expected with the T-matrix formalism (solid lines). With di fferent colors we discriminate the results for the SxandAx\npatterns. The T-matrix curves are read from Fig. 4 of Soler & Luna (2015, SL15). Shown here is how Ωandγ/Ωdepend onℓ/R, while the rest of\nthe parameters is fixed at [ ρi/ρe=5,d/R=2.5,kb=π/30]. The [ℓ/R,d/R] notations follow from SL15 for the ease of comparison.\nArticle number, page 19 of 31A&A proofs: manuscript no. manuV1\nFig. 5. Kink motions with (a) Sxand (b) Axpatterns in a two-elliptic-tube configuration with x-major orientation. Plotted are the time sequences\nof the x-speed sampled at the left tube center ( vx(−d/2,0;t), the solid curves) for a number of ratios of the semi-major to semi-minor axis ( a/b) as\ndiscriminated by the di fferent colors. A fitting procedure is performed, for each sequence, over the duration encompassing the six extrema starting\nwith the one denoted by an asterisk. The best-fit damping envelope is plotted for the entire sequence, but only for a/b=2.5 to avoid over-crowding\nthe plots. The computations pertain to a fixed combination [ ρi/ρe=3,¯l=0.4,d/b=5,kb=π/30].\nArticle number, page 20 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 6. Representative snapshots of the velocity field (the blue arrows) and energy density distribution ( ϵ, filled contours) for (a) the Sxmotion\nand (b) the Axone in a two-elliptic-tube configuration with the x-major orientation. The inset in each panel emphasizes the distribution of ϵin a\nrepresentative nonuniform layer, whose boundaries are shown by the red solid curves. The white contour indicates where the Alfv ´en frequency\nωA=kvAequals the quasi-mode frequency deduced with the fitting procedure. Both computations pertain to a fixed combination [ ρi/ρe=3,a/b=\n2.5,¯l=0.4,d/b=5,kb=π/30]. This snapshot is extracted from the animation attached to the current figure. The ϵcontours in all snapshots\nare filled in such a way that darker portions correspond to larger ϵ. The blue arrows are also consistently scaled in all snapshots such that longer\narrows correspond to stronger velocities.\nArticle number, page 21 of 31A&A proofs: manuscript no. manuV1\nFig. 7. Distributions of the x-speed along the y-cut through the left tube center at a number of instants as labeled. Panels (a) and (b) correspond\nto the SxandAxmotions, respectively. The vertical dotted lines represent the borders of the nonuniform layer. Only the lower half ( y<0) of a\ny-profile is displayed, the other half being symmetric with respect to y=0. All computations are conducted for the x-major orientation with a\nfixed combination of physical parameters [ ρi/ρe=3,¯l=0.4,d/b=5,kb=π/30].\nArticle number, page 22 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 8. Dependencies of (a) the quasi-mode periods ( P) and (b) damping-time-to-period ratios ( τ/P) on the ratio of semi-major to semi-minor\naxis ( a/b) for kink motions in a two-elliptic-tube configuration with the x-major orientation. The SxandAxpatterns are discriminated by the\nlinestyles. A number of dimensionless tube separations ( d/b) are examined as di fferentiated by the di fferent colors, with d/bconstrained to avoid\ntube overlapping. All results pertain to a fixed combination of physical parameters [ ρi/ρe=3,¯l=0.4,kb=π/30]. The horizontal lines represent\nthe results for an isolated circular tube with the same set of parameters.\nArticle number, page 23 of 31A&A proofs: manuscript no. manuV1\nFig. 9. Similar to Fig. 5 but for the x-minor orientation.\nArticle number, page 24 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 10. Similar to Fig. 6 but for the x-minor orientation. The associated animation is also attached.\nArticle number, page 25 of 31A&A proofs: manuscript no. manuV1\nFig. 11. Similar to Fig. 7 but for the x-minor orientation.\nArticle number, page 26 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 12. Similar to Fig. 8 but for the x-minor orientation.\nArticle number, page 27 of 31A&A proofs: manuscript no. manuV1\nFig. 13. Temporal profiles of the x-speed sampled at the left tube center ( vx(−d/2,0;t), the solid curves) for a number of domain sizes as labeled.\nOverplotted by the dashed curves are the best-fit damping envelopes. All computations pertain to the Sxmotion for a fixed combination of physical\nparameters [ ρi/ρe=5,ℓ/R=0.4,d/R=2.5,kb=π/30]. The grid spacing is fixed at ∆ =0.01b.\nArticle number, page 28 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 14. Temporal profiles of the x-speed sampled at the left tube center for a number of grid spacings as labeled. All computations pertain to the\nSxmotion for a fixed combination of physical parameters [ ρi/ρe=5,ℓ/R=0.4,d/R=2.5,kb=π/30]. The computational domain size is fixed at\n[xM=12b,yM=10b].\nArticle number, page 29 of 31A&A proofs: manuscript no. manuV1\nFig. 15. Distributions of the x-speed at t=340b/vAealong the y-cut through the left tube center for a number of grid spacings as labeled.\nAll computations pertain to the Sxmotion for a fixed combination of physical parameters [ ρi/ρe=5,ℓ/R=0.4,d/R=2.5,kb=π/30]. The\ncomputational domain size is fixed at [ xM=12b,yM=10b]. Only the lower half ( y≤0) is shown for any y-profile, the other half being symmetric\nabout y=0. The vertical dash-dotted lines delineate the borders of the nonuniform layer.\nArticle number, page 30 of 31Shi et al.: Damped kink motions in two-elliptic-tube systems\nFig. 16. Temporal profiles of some energetics-related quantities for several grid spacings as di fferentiated by di fferent colors. Energetics are\nconsistently examined for a box [ −(d/2+1.2b),(d/2+1.2b)]×[−1.2b,1.2b], in which the three mutually exclusive constituents are the tube\ninteriors, nonuniform layers, and exterior. Shown in (a) are the time derivative of the total perturbation energy in the box (d Etot/dt, the sold\ncurves), together with the instantaneous energy flux into this box ( −F, asterisks). Further plotted in (b) are the instantaneous total energy in the\ninteriors ( Eint), nonuniform layers ( Elayer), and the exterior ( Eext). All computations pertain to the Sxpattern for a fixed combination of physical\nparameters [ ρi/ρe=5,ℓ/R=0.4,d/R=2.5,kb=π/30]. The computational domain size is fixed at [ xM=12b,yM=10b].\nArticle number, page 31 of 31"    },    {        "title": "2401.12894v1.Ab_initio_thermodynamic_properties_of_Iridium__A_high_pressure_and_high_temperature_study.pdf",        "content": "1 \n Ab initio t hermodynamic properties of Iridium : A high -pressure and high -temperature  study  \nBalaram Thakur1*, Xuejun Gong1,2, and  Andrea Dal Corso1,2 \n1International School for Advanced Studies (SISSA), Via Bonomea 265,34136 Trieste, Italy . \n2CNR -IOM, Via Bonomea 265, 34136 Trieste, Italy . \n \n \n*Corresponding author: bthakur@sissa.it  \n \nAbstract:    \nThe high-pressure and high -temperature t hermodynamic properties of iridium are studied using \ndensity functional theory  in combination with the quasi -harmonic approximation , where both the \ncontribution s to the free energy of phonons  and of electronic excitation s are considered . The  reliability \nof different exchange and correlational functional s [Perdew -Burke -Ernzerhof generalized gradient \napproximation  (PBE)  (Perdew et al.  Phys. Rev. Lett. 77, 3865 (1996) ), PBE modified for dense solids \n(PBEsol)  (Perdew et al.  Phys. Rev. B 100, 136406  (2008 )) and local density approximation  (LDA)  \n(Perdew et al.  Phys. Rev. B 23, 5048 (1981) )], for studying the equation of state (EOS) , the phonon \ndispersions, the mode -Grüneisen parameter, and different thermodynamic properties  like thermal \npressure , volume thermal expansivity , isobaric heat capacity, bulk  modulus , and the average Grüneisen \nparameter are tested.  Elastic constants are studied at T=0 K as a function of pressure.  The predicted \nresults are compared with the available experiment s and previous  theoretical data. We find generally a \ngood agreement with  experiment s with at least one functional, but none of the three outperform s the \nothers in all the investigated thermodynamic properties. The electronic excitation s contribution is \nminimal in bulk modulus , but it is significant for other thermodynamic properties .  \n \nKeywords :   \nIridium, Density Functional Theory, Quasi -harmonic approximation, thermodynamic properties, high -\npressure and high -temperature (HPHT).  \n \n \nEmail:   Balaram Thakur  (bthakur@sissa.it ), Xuejun Gong  (xgong@sissa.it ), \n             Andrea Dal Corso  (dalcorso@sissa.it ) 2 \n 1. Introduction : \nOnly next to the hexagonal close -packed osmium  (Os), face-centered cubic (fcc) iridium (Ir) is  \na second -dense  material on  the periodic  table with the third -highest bulk modulus after  diamond and  \nOs [1]. As a refractory material, it is ideal for high -pressure and high -temperature (HPHT) \nenvironments.  Experimentally, the thermodynamic  properties of iridium , like heat capacity, thermal \nexpansion, and bulk modulus , are documented in the Refs.  [2–10]. In addition, the shock -wave \nmeasurement provides  the equation of state (EOS) in the extreme environment , described in the recent \ndatabase  [11] and in Refs. [12–15]. However, only a handful of theoretical reports address  the \nexperimental thermophysical properties . These reports , dependent  on the methodology and \napproximation s of the calculation,  do not always agree  among themselves and have varying degree s of \naccuracy with respect to  the experiment .  \nRecently, Luo et al . [16] used first principle molecular dynamics  (FPMD)  and studied the \nthermal equation of state using  the scal ar relativistic  Garrity -Bennett -Rabe -Vanderbilt  (GBRV)  [17] \nultrasoft (US) pseudopotential  and the Perdew -Burke -Ernzerhof (PBE) [18] functional. Though their \nEOS agrees satisfactorily with experiments at 300  K, a significant difference is observed  at high \ntemperature s. Furthermore, at high temperatures,  the authors  [16] found a small  enhancement in the \nvolumetric thermal expansion coefficient  with temperature , which is unusual.   \nThe u se of  density functional theory (DFT) within the quasi -harmonic approximation (QHA) \nto accurately study the thermodynamic  propert ies complements the molecular dynamics up to 2/3rd of \nthe melting temperature  [19]. In th is same line, Fang et al.  [20] used the semi -core Hartwigsen -\nGoedecker -Hutter ( HGH ) norm -conserving pseudopotential  [21] and the local density approximation \n(LDA)  in their DFT with QHA  study of the thermodynamic properties of iridium . However , some of \ntheir results, like the temperature -dependent Grüneisen parameter and the variation of thermal pressure \n(Pth) with the temperature (dPth\ndT), which  increases with decreasing the volume , contradict Luo et al. [16], \nwhere dPth\ndT decreases with decreasing volume. Fang et al.  [20] didn't include  the electron ic excitation  \ncontribution s (EEC)  in the free energy and only consider ed its role in the isobaric heat capacity . These \ndiscrepancies in the literature and the role of electronic excitation s on the thermodynamic properties  of \niridium  need to  be further investigated.  \nMoreover , despite  having many  reports  studying some particular propert y with varying  models \nand approximations , a complete description  of the  thermodynamic  properties of iridium under extreme \nconditions using the same theoretical method  is unavailable . In the present work, we study and compare \nthe anharmonic thermodynamic  properties like thermal expansion, isobaric heat capacity, bulk modulus , \nand thermodynamic average Grüneisen parameter within  the QHA using three  functionals and consider 3 \n the role of electronic excitation . Finally, the performance of different functionals under HPHT \nconditions  is discussed.  \n2. Method : \nThe calculation s shown in the present  work  use Thermo_pw [22], a driver of Quantum \nESPRESSO  [23,24] . The exchange and correlation functional is approximated using the g enerali zed \ngradient approxim ation (GGA) proposed  by Perdew -Burke -Ernzerhof (PBE)  [18], the PBE functional \nmodified for dense solids (PBEsol)  [25], and the local-density approximations  (LDA ) with the Perdew -\nZunger ( PZ) [26] parameterization.  The pseudo -wave functions and the charge densities were  expanded \nin plane wave s with kinetic energy cut-offs of 75 Ry and 650 Ry , respectively. Nuclei and core-electrons  \nwere treated by t he projector augmented wave (PAW)  [27] method with pseudopotentials (PPs) from  \npslibrary  [28,29] . The iridium atom is described with seventeen valence electrons having a \nconfiguration [ Xe] 4f14 6s2 6p0 5d7 and including the 5s2 and 5p6 semi -core states.  For PBE, PBEsol , \nand LDA , we used  the scalar relativistic PPs Ir.pbe-spn-kjpaw_psl.1.0.0.UPF , \nIr.pbesol -spn-kjpaw_psl.1.0.0.UPF , and Ir.pz-spn-kjpaw_psl.1.0.0.UPF .  \nFig.1:  P-V equation of state (EOS) of iridium obtained at 301 K and 2998 K using LDA, PBEsol, and \nPBE functionals. The experimental data from high -pressure compressive measurements by Yusenko et \nal. [30] (at RT), Montesuguro et al.  [31], and Anzellini et al.  [32] (at 300 K and 3000 K) are compared. \nThe experimental 3rd-order BM -EOS at 300 K and 3000 K shown by Anzellini et al.  [32] is displayed. \nFor the DFT study comparison, the EOS of  Luo et al.  [16] (PBE) , Han et al.  [33] (PBE), and Anzellini \net al.  [32] (LDA) are considered.  The inset shows a magnified image at low compression.  \n0 50 100 150 200 250 30010111213141516\nP (GPa)V (Å3)\nDFT Study\n Luo et al.  (300 K) [PBE]\n Han et al.  (300 K) [PBE]\n Luo et al.  (3000 K) [PBE]\n Anzellini et al. (0 K)[LDA]\n Anzellini et al. (3000 K)[LDA]Expt. Data\n Yusenko et al. (RT)\n Monteseguro et al.\n Anzellini et al.  (300 K)\n Anzellini et al.  (3000 K)\nExpt. EOS\n Anzellini et al.  (300 K)\n Anzellini et al.  (3000 K)This work\n301 K       2998 K\n              LDA\n              PBEsol\n              PBEVolume (V) ( Å3)\nPressure (P) (GPa)20 40 60 8012.513.013.514.014.515.04 \n The iridium 's Brillouin zone (BZ)  has been sampled on a k-point mesh using the Monkhorst -\nPack method  [34]. The occurrence  of the Fermi surface  is dealt with by the Methfessel and Paxton  (MP)  \nsmearing approach  [35]. A k-point mesh of 32 × 32 × 32 and a MP smearing parameter of σ = 0.02 Ry \nare used to ensure the calculation's accuracy and efficiency . For phonon calculation s, we use density \nfunctional perturbation theory (DFPT)   [36] extended to PAW   [37] on an 8 × 8 × 8 q-point mesh . The \nanharmonic thermodynamic properties are studied u nder QHA , for which the phonon calculations were \nperformed on a  set of 15 volumes with unit cell  edge  varying  from a = 6.1 a.u. to a = 7.5 a.u. (or primitive \ncell volume from  56.74 a.u.3 (8.408  Å3) to 105.46  a.u.3 (15.628  Å3)) in step s of Δa = 0.1 a.u . The effect \nof choosing steps of Δa = 0.2 a.u. on the thermodynamic properties of iridium is discussed in \nsupplementary data  (S1). The electronic excitation s contribution (EEC) is included within the rigid \nbands ' approximation . The procedure to calculate different thermodynamic properties is described in \nthe previous works of our group  [38–40]. \n3. Results and discussion : \nFig.1 shows the equation of state calculated for LDA, PBEsol, and PBE at 301 K and 2998 K. \nThe experimental data from high -pressure compressive experiments and the theoretical predictions of \ndifferent techniques are compared with the present work's EOS. We  observe that the DAC experiments \ncarried out at room temperature  by Yusenko et al.  [30] and Montesuguro et al.  [31] at low pressure (< \n20 GPa) agree with the PBEsol EOS. Beyond 20 GPa, the agreement with the LDA EOS is better. \nSimilarly, while comparing our EOS at 301 K and 2998 K, the experimental data and the fitted 3rd order \nBirch -Murnaghan EOS (3rd-BM-EOS) of Anzellini et al.  [32] determined at 300 K and 3000 K remain \nclose to our PBEsol EOS.  \nThe theoretical prediction made by Luo et al.  [16] (using PBE) at 300 K and Anzellini et al.  [32] \n(using LDA) at 0 K follow well with our EOS computed using PBE and LDA at 301 K, respectively. \nHowever, EOS calculated by Han et al.  [33] at 300 K using a direct integration approach  (DIA)  to the \npartition function and PBE functional agree with our PBEsol at 0 GPa and later it decrease s with \nincreasing the pressure. To convert Han et al.  [33] 𝑉\n𝑉0(𝑃) results  to V(P) , we used  the experimental  V 0 \nof Montesuguro et al.  [31], for which the predicted results  [33] are very close (relative error is  - 0.55 \n%) to the experiment  [31] (see Fig. 1). Further, at 3000 K, the predicted EOS by Luo et al.  [16] tends \nto follow with our PBE EOS, and interestingly, the EOS computed by Anzellini et al.  [32] (at 3000 K) \nand our LDA EOS (at 2998 K) overlap each other.  \nWe fitted the free energy with the 4th-order BM equation to calculate the equilibrium lattice \nconstant (a 0) from the equilibrium volume (V 0), the bulk modulus (𝐵𝑇), and its pressure derivative ( 𝐵𝑇′). \nTable 1 compares the obtained parameters with experimental and available theoretical results. At 301 \nK, relative to the experimental lattice constant  [32], our lattice constant is higher by 1.2 % and 0.1 % 5 \n for PBE and PBEsol, whereas for LDA, it is lower by 0.4 %. Moreover, the present work's lattice \nconstant values at 0 K and 301 K agree with different theoretical studies  [32,41,42]  (underlined in Table \n1, where the values at 300 K are written in italics)  having an error below by 0.1% (0.2 %), 0.2 % (0.2 \n%) and 0.1 % (0.2 %) at 0 K (300 K) for PBE, PBEsol and LDA, respectively . \nTable 1:  Comparison between the equilibrium parameters like lattice constant (a o), bulk moduli ( BT), \nand its pressure derivative ( 𝐵𝑇′) obtained for different functional s (this work) and other experimental \nand theoretical studies . The values of the parameters at 0 K and 300 K are shown, where the value at \n300 K is in italics. The underlined values are used for the comparison mentioned in the text. The \ndifferent methods mentioned are PAW -Projected Augmented Wave, ae -all-electron, US -Ultrasoft pp, \nLAPW -full-potential Linearized Augmented Plane-Wave, and LMTO -Linear  Muffin -Tin Orbital .  \n \nThe equilibrium bulk modulus at 300 K is compared with the experimental studies of Anzellini \net al.  [32] (Pmax = 35 GPa), and Montesuguro et al.  [31] (Pmax = 140 GPa). The bulk modulus comparison \n(see Table 1) shows that for Anzellini et al.  [32], the PBE underestimates by 7.1 % (23.9 GPa) , whereas \nPBEsol  and LDA overestimate by 3.8 % (14.4 GPa) and 8.2 % (32 GPa), respectively . Instead, for \nMontesuguro et al.  [31], the 𝐵𝑇 is underestimate d by 0.9 % (2.9 GPa)  for PBE, and overestimate d by \n9.5 % (35.4 GPa)  and 13.5 % (53 GPa) for PBEsol and LDA, respectively.  Exchange – \nCorrelation \nFunctional  Lattice constant  \n(Å) Bulk modulus ( 𝐵𝑇)  \n(GPa) Pressure derivative of \n𝐵𝑇 (𝐵𝑇′) \n0 K 301 K  0 K 301 K  0 K 301 K  \nPBE 3.874  3.883 348.7  336.1  5.1 5.2 \nPBEsol  3.834 3.842 387.6  374.4  5.1 5.1 \nLDA  3.816 3.823  404.3  392.0  5.2 5.1 \nOther  \nstudies  Expt (300  K): \n3.837 [31,32]  \nPBE:   \n3.877(PAW)  [41], \n3.891 (PAW)  [42], 3.879(ae)  [41], \n3.887(LAPW)  [43], \n3.876(US)  [16]. \nPBEsol:  \n3.843(PAW)  [42], \n3.851 (PAW)  [42], \n3.847(LAPW)  [43]. \nLDA:  \n3.819(PAW)  [41], \n3.829 (PAW)  [32], 3.819(ae)  [41], \n3.828( FLAPW)  [43], \n3.815(LMTO)  [44]. Expt (300  K):  \n360 [32], 339 [31] \nPBE:  \n342(PAW)  [41], \n348.8(PAW)  [42], 349(ae)  [41], \n361 (US)  [16]. \nPBEsol:  \n387.9(PAW)  [42]. \nLDA:  \n401(PAW)  [41], \n377(PAW)  [32], 406(ae)  [41], \n406.1(PAW)  [42], \n402.7(LMTO)  [44]. Expt (300  K): \n6.0 [32], 5.3 [31] \nPBE:  \n5.2(PAW)  [41], \n5.2(ae)  [41],  \n5.3(US) [16]. \nLDA:  \n5.2(ae)  [41],  \n5.1(PAW)  [41], \n5.3(PAW)  [32], \n4.8(LMTO)  [44]. 6 \n Fig.2: Dependence of thermal pressure (Pth) with the temperature at two volumes 8.408 Å3 and 13.827 \nÅ3 for LDA, PBEsol, and PBE. Filled and empty symbols indicate that the calculation is done while \nconsidering and neglecting the electronic excitation contributions (EEC). The variation of Pth with no \nelectron excitation contribution (no EEC) is shown  only for PBE.  \nThe variation of thermal pressure ( Pth) with temperature is displayed in Fig.2. Here, we studied \nthe Pth on the two same -volume used in the present QHA calculations, 8.408  Å3 (the smallest  volume) \nand 13.827 Å3 (nearest to the equilibrium volume).  Unlike Fang et al.  [20], we observe that at a given \ntemperature, the thermal pressure increases with increasing volume , consistent with Luo et al.  [16]. \nComparing the three functionals, we notice  that for the system near the equilibrium, PBEsol lies in \nbetween PBE and LDA, whereas for the compressed system, all three functionals collapse on a single \ncurve. Furthermore, the EEC on the thermal pressure is studied for each functional, and the dependence \nof Pth with temperature  without EEC  is shown for PBE in Fig.2. At 2000 K , we found that without EEC , \nthe Pth decreases uniformly by ~ 4 % for V = 8.408  Å3 and ~ 7 % for V = 13.827 Å3 for all three \nfunctionals . \nThe variation of volume thermal expansion coefficient ( ) with the temperature at 0 GPa and \n300 GPa is presented in Fig.3. With respect to LDA, the differences of PBEsol and PBE increase with \ntemperature.  The EEC to the  at 0 GPa (see Fig.3) are similar for PBEsol and LDA, whereas it is higher \nfor PBE. At 301 K,  the difference between LDA and PBEsol (PBE) for   (in 10-6 K-1) is 0.62 (2.77), \nand at 1999 K, it is 1.29 (6.87) . Instead, at 301 K and at 0 GPa , the difference of the EEC between LDA \nand PBEsol (PBE) (in 10-6 K-1) is 0.02 (0.08 ), whereas at 1999 K, it is 0.32 (1.54) . Also , for PBEsol, at \n301 K, with increasing the pressure from 0 GPa to 300 GPa, the EEC decreases by ~ 85 % (from 0.3671 \nto 0.05511 in 10-6 K-1), whereas at 1999 K, EEC is reduced by ~ 90 % (from 4.57921 to 0.43535 in      \n10-6 K-1). Thus , the EEC  effect alone does not  explain the difference among the values of  obtained for \n0 500 1000 1500 20000255075100125150\nV (Å3)      8.408     13.827\n                           LDA\n                           PBEsol\n                           PBE\n                           PBE (No EEC)Thermal Pressure (Pth) (kbar)\nTemperature (K)7 \n different functionals. These differences vanish with increasing pressure where the  for different \nfunctionals overlap, as shown in Fig.3.  \nFig.3: Temperature -dependent volumetric thermal expansion ( ) obtained for LDA, PBEsol, and PBE \nat 0 GPa and 300 GPa.  The electronic excitation contribution (EEC) to the  at 0 GPa, and 300 GPa  is \nshown.  The experimental data (symbols) from Arblaster  [6], White et al.  [7], Singh  [8], and \nHalvorson  [9] are included. The simulated results (dash and symbols) from Grabowski et al.  [41] (PBE \nand LDA), Luo et al.  [16] (PBE), and Fang et al.  [20] (LDA) are considered for comparison . \nFig.4:  Pressure -dependent  at 1999 K and 301 K (inset) calculated with including (solid line) and \nexcluding (dash -dot-dot) EEC for three functionals is displayed. The simulated results (dash and \nsymbols) of Luo et al.  [16] (PBE) at 2000 K and 300 K are compared.  \nOn comparing with the experimental data  from  Arblaster  [6], White et al.  [7], Singh  [8], and \nHalvorson et al.  [9], it is observe d that at low temperature s (< 1000 K) , the experimental data follows \nwell with PBEsol . In contrast,  the PBE functional seems to be a better choice at higher temperatures . \n0 50 100 150 200 25061218\n0 50 100 150 200 250 30010203040\nP (GPa)\nPressure (P) (GPa) (10-6  K-1)\nT = 301 K\nT = 1999 K 2000 K (Luo et al. )[PBE]Volume thermal expansion () (10-6  K-1) with EEC   no EEC\n               LDA\n               PBEsol\n               PBE\n 300 K (Luo et al. ) [PBE]\n0 500 1000 1500 2000010203040\nEEC at 0 GPa Grabowski et al.  [PBE]\n Luo et al.  [PBE]\n Grabowski et al.  [LDA]\n Fang et al.  [LDA]Volume thermal expansion () (10-6  K-1)\nP = 300 GPa LDA\n PBEsol\n PBE\n Arblaster\n White et al.\n Singh\n Halvorson et al.\nTemperature (K)EEC at 300 GPaP = 0 GPa8 \n The LDA result of Grabowski et al.  [41] is in good agreement with our work, while their  [41] PBE is \noverestimated.  Fang et al.  [20] (LDA ) and Luo et al.  [16] (PBE)  agree with our curves  only at low \ntemperatures (<  300 K) , whereas , with increasing the temperature, their predicted value s are \nsignificantly lower than the experiment  and of our results . However, at 300 GPa,  our result agrees with \nLuo et al.  [16]. It is noted that Fang et al.  [20] and Luo et al.  [16] neglected the EEC  in the free energy \nand hence in . In part for th is reason , at 0 GPa, Fang et al.  [20] and Luo et al.  [16] disagree with our \nresults and other experimental data .  \nFig.4 presents the pressure -dependent  obtained for the different functionals at 301 K and 1999 \nK, considering and neglecting the EEC. At 301 K (inset of Fig.4), the difference between the  obtained \nwith and without EEC is minimum and becomes negligible with increasing pressure, whereas at 1999 \nK, the difference in  first decreases but does not vanish at higher pressure. The pressure -dependent  \ncalculated in the present work  using PBE agrees well with the result of Luo et al. [16] (PBE) at 300 K,  \neven neglecting the EEC.  In contrast, at 2000 K, there is a significant difference between ours and Luo \net al.  [16] results  at low pressures, which decreases when the pressure is increased  \nFig.5: (a) Temperature -dependent isobaric heat capacity (C P) and the EEC to the heat capacity (C el) at \n0 GPa and 300 GPa  are presented.  The experimental data (symbols) compared are from Touloukian et \nal. [2] TPRC data series, Trukhanova et al.  [3], Furukawa et al.  [4], Arblaster  [5], and Moseley et \nal. [45]. Simulated results from Luo et al.  [16] (PBE) and Fang et al.  [20] (LDA) are included.  \nFig.5 shows the temperature -dependent isobaric heat capacity (C P) and the electronic excitation \ncontributions (EEC) to the heat capacity C el for different functionals at 0 GPa and 300 GPa.  We noticed \nthat at 0 GPa, the variation of C P with T for three functionals is identical up to 1000 K and beyond this, \nthe difference between the PBE and other functionals increases with temperature. A similar trend of C el \nat 0 GPa for different functionals is also observed. At 2000 K relative to LDA, the CP (in J/K.mol) o f \n0 500 1000 1500 20000510152025303540\n Fang et al.  [LDA]\n Luo et al.  [PBE](0 GPa)\n Luo et al.  [PBE](300 GPa)\nCel at 300 GPaCel at 0 GPa Touloukian et al.\n Trukhanova et al.\n Furukawa et al.\n Arblaster\n Moseley et al.0 GPa     300 GPa\n          LDA\n          PBEsol\n          PBE\nTemperature (K)Heat Capacity CP (J/K.mol)Cp at 0 GPa\nCp at  300 GPa9 \n PBEsol (PBE) is 0.35 (1.87) larger, whereas the C el (in J/K.mol) is 0.22 (0.97) more. Therefore, the \ndifference in CP cannot be ascertained totally to  the EEC and is in part due to the presence of higher  \nfor PBE with respect to the PBEsol and LDA, as seen in Fig.3.   \nFig.6: Pressure -dependent C P at 1999 K and 301 K (inset) calculated with including (solid line) and \nexcluding (dash -dot-dot) EEC for three functionals is shown. The simulated result from Luo et al.  [16] \n(PBE) is compared.  \nCompar ed with older literature, Touloukian et al.  [2] TPRC data series, Trukhanova et al.  [3], \nFurukawa et al.  [4], and Arblaster  [5], it is observed that our theoretical prediction  of C P agrees well \nwith PBEsol and LDA functionals in the entire temperature range. The C P and EEC (C el) calculated \nusing LDA by Moseley et al.  [45] and Fang et al.  [20] agree with  the present work (see Fig. 5). Luo et \nal. [16] (PBE) at 0 GPa agrees with our simulated  results  and with experiment only at low temperatures \n(< 300 K) , whereas at 300 GPa, Luo et al.  [16] are consistent with our result only for high temperatures.  \nMoreover, in Fig.5, it is seen that  at 300 GPa, the C P and C el for the different functional overlaps and \ntheir values (for PBE in J/K.mol) at 2000 K  decrease significantly  from their 0 GPa values  by 14.67 \nand 4.61, respectively . \nThe rate of decrease in CP (dC P/dP) with an increase in pressure depends on the temperature, as \nillustrated in Fig.6, where from 0 GPa to 300 GPa (say for PBE) , the dC P/dP (in J/K.mol . GPa) is  about \n0.01 and 0.03  at 301 K  and 1999 K, respectively. The difference in the value of C P at 301 K with and \nwithout considering the EEC remains the same. In contrast, at 1999 K, the C P without EEC decreases \ninitially and remains constant beyond 150 GPa.  Similar to Fig.5, at low  temperature , Luo et al.  [16] \n(PBE) is consistent with our result at  low pressure , while at high temperature , it is consistent only at \nhigh pressure  (see Fig. 6) . \n0 50 100 150 200 250 300253035with EEC     no EEC\n                  LDA\n                  PBEsol\n                  PBE\n Luo et al.  (300 K)\n Luo et al.  (2000 K)Heat Capacity CP (J/K.mol)\nPressure (GPa)T = 301 KT = 1999 K10 \n Fig.7: Variation of isoentropic (B S) and isothermal (B T) bulk modulus with temperature at 0 and 300 \nGPa for LDA, PBEsol , and PBE (shown in solid and dashed lines). Comparison with the experimental \ndata at 300 K of Cynn et al.  [44], Anzellini et al.  [32], and Montesuguro  et al.  [31]  are shown as \nsymbols and theoretical predictions by Ferah et al.  [46] (electron embedded method (EAM) -molecular \ndynamics calculations), Liang et al.  [47] (GGA) and Fang et al.  [20] (LDA) are represented as a line \nwith symbols.  \nThe isoentropic (B S) and the isothermal (B T) bulk moduli for the three functionals at 0 and 300 \nGPa are reported in Fig.7, which shows that, with temperature the bulk moduli B i (BS and B T) decreases.  \nThis decrease in B i depends on the pressure. For all the functionals, at 0 GPa, the decrease in B S and B T \n(dBi/dT) from 0 K to 2000 K is ~ 0.3 GPa/K and ~ 0.5 GPa/K, respectively, whereas at 300 GPa, the \ndBi/dT becomes ~ 0.1 GPa/K and ~ 0.2 GPa/K, respectively. It is noted that on neglecting the EEC (not \nshown in Fig. 7), an insignificant change in the B S and B T is observed.  \nAt 300 K, the experimental bulk modulus studied by Cynn et al.  [44] is in agreement with the \nPBEsol, the one of Montesuguro et al.  [31] is with the PBE, and the one of Anzellini et al.  [32] lies in \nthe mid -way of PBEsol and PBE. Our predictions and others experimental results overestimate the bulk \nmodulus obtained from the MD simulation by Ferah et al.  [46]. Moreover, the B(T) calculated by Liang \net al.  [47] (GGA) is in good agreement with our predicted PBE B T. Also, the theoretical value of B 0 by \nFang et al.  [20] (LDA) matches well with our LDA curves  at up to 500 K; however, their values remain \nconstant with temperature  instead of decreasing.  \n0 500 1000 1500 2000200250300350400450150015501600\n Ferah et al. [EAM]\n Liang  et al. [GGA]\n Fang et al. [LDA]BS         BT\n       LDA\n       PBEsol\n       PBE Cynn et al.\n Anzellini et al.\n Monteseguro et al.300 GPaBulk Modulus (GPa)\nTemperature (K)0 GPa11 \n Fig.8: Phonon dispersions curve obtained at 301 K for LDA, PBEsol, and PBE (lines) is compared with \nthe room temperature inelastic neutron scattering  [48] data (symbols).  \nIn Fig. 8, we compare the phonon dispersions measured using the inelastic neutron \nscattering  [48] and calculated using PBE, PBEsol , and LDA . A comparison between phonon dispersion \nobtained using different k-point and q-point mesh for PBEsol is discussed in supplementary data (S 2). \nThe phonon frequencies  shown here  are interpolated at 301 K , starting from those calculated on  the 15 \ngeometries . The dispersion agrees with Ref.  [42]. Also, f rom Fig. 8, we not e that the phonon frequenc ies \nobtained using the PBEsol and LDA are satisfactory , but for PBE, significant  softening is observed.  The \nerror in the phonon frequency is well correlated with the error in the lattice parameter rather than the \nbulk modulus , as discussed earlier  [42].  \nFig.9: (a) Variation of the mode -Grüneisen parameter ( γqη) for LDA, PBEsol and PBE functionals. The \ntheoretically predicted data of Baria  [49] are shown, and (b) the variation of a thermodynamic average -\nGrüneisen parameter as a function of temperature with considering (solid line) and ignoring (dashed) \nthe electron excitation contribution (EEC) at 0 GPa, 100 GPa, 200 GPa, and 300 GPa.  The compared \nvalues (symbols) are from White et al.  [7], Moseley et al.  [45], and Singh  [8].  \n050100150200\nP\nLT2\nT1L\nT\n(z,z,z) (z,z,0) (1,z,0)Frequency (cm-1)\nG X W X K G L Neutron Scattering \n          Expt. Data\n  LDA\n  PBEsol\n  PBEPhonon dispersions curve  at 301 K\n(z,0,0)LL\nT\n1.21.62.02.42.83.23.6\nAvg. G rüneisen parameter (g) (b) (a)\n LDA\n PBEsol\n PBE\n Bariamode-G rüneisen parameter (gqh)\nG X W XK G L 500 1000 1500 20001.61.82.02.22.42.62.83.0\n0 GPa\n White et al.\n Moseley et al.\n Singhwith EEC     no EEC\n                  LDA\n                  PBEsol\n                  PBE\nTemperature (K)300 GPa200 GPa100 GPa12 \n In Fig. 9(a), we compare  the mode-Grüneisen parameter ( γqη) obtained  for LDA,  PBEsol , and \nPBE. The γqη shows how the phonon mode s are affected by the anharmonicity of the system . The non-\nanalyticity at Γ is an inherent property of γqη, which arises due to the direction dependence of the phonon \nmodes. In Fig.9(a), γqη are similar for LDA a nd PBEsol, while they are  higher for PBE.  For comparison, \nthe γqη calculated by Baria  [49] are shown. A significant difference between the present study and  \nBaria  [49] is observed. Baria  [49] studied the iridium's static and vibrational propert ies using a model \npseudopotential , which incorporates the s-d hybridization effects and depends only on the effective core \nradius.  \nFig.9(b) compares the temperature variation of the thermodynamic average  Grüneisen \nparameter ( γ =  𝐵𝑇 𝑉\n𝐶𝑉) at 0 GPa, 100  GPa, 200  GPa and 300 GPa for the three functionals . The pressure \nderivatives  of γ decrease with increasing pressure. Additionally,  γ remains constant with temperature \nwhen the EEC is considered.  In contrast, ignoring the EECs  leads to the temperature dependence on γ, \nespecially at low pressures . Also, from Fig. 9(b), it  is noticed that  at 0 GPa, the γ from  PBEsol and LDA \nare similar , whereas , for PBE , the value is higher. Th e separation between the values of γ vanishes  with \npressure , where all three functional s collapse  on each  other. A similar trend was discussed previously \nfor the thermal expansion coefficient.   \nFig.10: (a) Variation of elastic constant coefficient (C ij) C 11, C12, and C 44 with pressure at 0 K, obtained \nusing LDA, PBEsol, and PBE. Inset shows the comparison between C ij obtained using the PBE and \nwith Liang et al.  [47] (GGA) predicted data (b) Variation of Pugh ratio (G/B), a ratio of shear moduli \n(G) and bulk modulus (B), with pressure at 0 K for LDA, PBEsol, and PBE. The values of G and B are \nobtained using Voigt -Reuss -Hill approximation. The horizontal dotted line is  at G/B = 0.57, above (or \nbelow); the system is considered brittle (or ductile).  \nOn comparing with the experimental ly obtained  γ, the result of White et al.  [7] is found to \nfollow well with the PBE results of the present work. The γ value calculated by Moseley et al.  [45], \nusing DFT with ultrasoft pseudopotentials and LDA, remains reasonably constant. When compared, we \nfound that our values (for LDA) are overestimated by 4.2 % at 300 K and 6 % at 823 K. The \noverestimation is expected to be due to different technical  ingredients.  Further, the result of Singh  [8] \n0 100 200 3005001000150020002500\n0 100 200 3000.450.500.550.600.65\nPressure (GPa)\nCij (GPa)(a)Elastic constant coefficient Cij (GPa)\nPressure (GPa) LDA\n PBEsol\n PBE\nC44C12\nPressure (GPa)C11\n30 60 903006009001200\n Liang et al.  [GGA]C44C12C11(b)\n LDA\n PBEsol\n PBEPugh ratio (G/B)\nBrittle\nDuctile0.5713 \n deviates significantly from our result s. Singh  [8] calculated the value of γ with a fixed value of the \nvolume V and of the compressibility at  room temperature , while the thermal expansion coefficient  () \nand isochoric heat capacity  (CV) vary with temperature.  For this , we believe that the γ value of Singh  [8] \nincreases linearly with temperature.   \nTable 2: Comparison of equilibrium elastic constant coefficient  Cij (in GPa) at 0 K.  \n C11 C12 C44 \nThis \nStudy  LDA  673.8  270.9  294.6  \nPBEsol  648.1  256.8  286.0  \nPBE 586.7  228.6  258.3  \nExpt. Study  [50] 596.0  252.0  270.0  \n \nIn Fig.10, we report the pressure dependence on the elastic constants. Three independent elastic \ncoefficients, C 11, C 12, and C 44 (collectively assigned as C ij), are present for fcc iridium with cubic \nsymmetry. For each functional, the C ij's are computed for the 15 different geometries used in QHA. The \nvariation of C ij with the pressure is shown in Fig.10, confirming that for the entire range of geometries \nstudied in the present work, the system satisfies the Born stability criteria  [51,52]  (C11 > 0, C 44 > 0, C 11 \n− C 12 > 0, and C 11 + 2C 12 > 0) and therefore all the studied geometries are mechanically stable. The inset \nof Fig.10 shows that the C ij's calculated using the PBE functional is in good agreement with the DFT \nstudy by Liang et al.  [47] using the GGA.  Further, c ompar ing with the experimen t [50] at 0 K , the \nequilibrium C ij obtained using PBE shows  better results  than PBEsol while LDA is worse among the \nthree functionals (see Table 2).  \nMoreover, the nature of the material , whether brittle or ductile , can be studied by studying the \nCauchy pressure  [53], defined as (C12–C44), and the Pugh ratio (G/B)  [54], where G and B are the shear \nand bulk modulus, respectively. For cubic crystal structure s, these two criteria are identical  [55]. The \nmaterial behaves ductile  (brittle)  if the Pugh ratio is less  (more)  than 0.57 . The Pugh ratio at ~ 0 GPa \nfor LDA, PBEsol , and PBE are 0.62, 0.63 , and 0.64, which agrees with  the experimental value of 0.63 \nat 0 K  [50] and with theoretical prediction by Liang et al.  [47] (GGA) . The variation of the Pugh ratio \nwith pressures shown in Fig. 10(b) indicates that iridium is brittle at equilibrium conditions , and at 0 K, \nthe transition from brittleness to ductility occurs at an average pressure of ~ 343 kbar (317 kbar, 357 \nkbar and 355 kbar for LDA, PBEsol , and PBE, respectively).  The transition pressure for PBEsol and \nPBE is considered the same within the tolerance limit . \n4. Conclusion : \nWe studied  the thermodynamic propert ies of iridium using the DFT within the QHA , where \nboth the phonon and electronic excitation  contributions  in the free energy  are considered . Three popular 14 \n functionals , LDA, PBEsol , and PBE , were tested on the  thermodynamic  properties. Comparison of \ndifferent properties with  experiment and other  theoretical models  shows good agreement . The lattice \nconstant obtained using PBEsol gives the minimum error of 0. 1 %, whereas PBE overestimates the \nexperimental value by 1.2 % , and LDA underestimates of 0.4 % . PBEsol and LDA functionals agree \nwell with the experimental results for phonon dispersions , isobaric heat capacity and low -temperature \nthermal expansion calculation, whereas the PBE functional is suitable for high -temperature thermal \nexpansion and bulk modulus .  \n The e lectronic excitation's role is  minimal for bulk  modulus , and the contributions are \nindependent of the functional choice for thermal pressure calculations. This contribution in the \nthermodynamic properties like thermal expansion  coefficient, heat capacit y, and the thermodynamic \naverage Grüneisen parameter  is crucial, particularly at low-pressure  and high -temperature  conditions. \nThe electronic contributions from PBEsol and LDA are similar, whereas PBE is slightly  higher. \nHowever, this difference  diminishes significantly under high pressure.  The elastic constants study at 0 \nK indicates iridium is brittle at equilibrium conditions, with a transition pressure of ~ 343 kbar from \nbrittle to ductile, in agreement with Liang et al.  [47]. Finally, we have shown the mode -Grüneisen \nparameters mapped on the Brillouin zone and the thermodynamic average Grüneisen parameter as a \nfunction of temperature and pressure, which are crucial to understand the anharmonicity in iridium.  \nIt is to be noted that the functionals chosen here are simple and efficient, but our results c ould \nbe further improved  if a single functional could  explain accurately all the thermodynamic properties. \nThere are several possibilities to choose, from a self -interaction corrected (SIC) functional  or hybrid \nfunctionals or functionals within the DFT such as a strongly constrained and appropriately normed \n(SCAN) functional, or also a Hubbard corrected (DFT+U) functional. Using these more sophisticated \nschemes would be very interesting; however, these co rrections require heavy calculations and are \ncomputationally costly. For many  of them, phonon calculations have not yet been implemented. They \nmust, therefore, be reserved for future research.  \nAcknowledg ement : \nComputational facilities were provided by SISSA through its  Linux Cluster and ITCS and \nSISSA -CINECA 2021 -2024 agreement.  This work has been supported by the Italian MUR through the \nNational Centre for HPC, Big Data, and Quantum Computing (grant No. CN00000013).  \nData availability statement:  \nAll data that support the findings of th e present  study are included within the article . \n \n 15 \n References  \n[1] T. Kenichi, Bulk Modulus of Osmium: High -Pressure Powder x -Ray Diffraction Experiments \nunder Quasihydrostatic Conditions , Phys. Rev. B 70, 012101 (2004).  \n[2] Y. S. Touloukian and E. H. Buyco, Thermophysical Properties of Matter - The TPRC Data \nSeries. Volume 4. Specific Heat - Metallic Elements and Alloys , V ol. 4 (New York, 1971).  \n[3] L. N. Trukhanova and L. P. Filippov, Thermal Properties of Iridium at High Temperatures , \nTeplofiz. Vys. Temp 8, 919 (1970).  \n[4] G. T. 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Rep. 11, 4531 (2021).  \n 1 \n Supplementary Data  \nAb initio t hermodynamic properties of Iridium : A high -pressure and high -temperature  study  \nBalaram Thakur1*, Xuejun Gong1,2, and  Andrea Dal Corso1,2 \n1International School for Advanced Studies (SISSA), Via Bonomea 265,34136 Trieste, Italy . \n2CNR -IOM, Via Bonomea 265, 34136 Trieste, Italy . \n \n \n*Corresponding author: bthakur@sissa.it  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nEmail:   Balaram Thakur  (bthakur@sissa.it ), Xuejun Gong  (xgong@sissa.it ), \n             Andrea Dal Corso  (dalcorso@sissa.it ) \n 2 \n S1: Comparison of thermodynamic properties of iridium obtained using Δa = 0. 1 a.u. (15  \ngeometries ) and Δa = 0.2 a.u . (7 geometries ) in quasi -harmonic approximation:  \nThe main article evaluates the thermodynamic properties like the thermal expansion coefficient \nby differentiating the free energy (F) considered on 15 geometries (i.e.  Δa = 0.1 a.u.). Here, we assessed \nthe free energy starting from the second geometry and doubling the step ( Δa = 0.2 a.u.), resulting in 7 \ngeometries. The comparison between the free energy (F), volume thermal expansion ( β), isobaric heat \ncapacity (C P), bulk modulus (B), and thermodynamic average Grüneisen parameter  (γ) obtained from \n15 and 7 geometries for PBEsol  are shown in Fig.S1(A -E). \n•  Free Energy:  \nFig.S1A: Temperature -dependent free energy (F) for PBEsol obtained using 15 and 7 geometries. The \ninset shows the difference (F 15 – F7) in the free energy for the two cases.  \n \n• Volume thermal expansion ( β): \nFig.S1B: Temperature -dependent volumetric thermal expansion ( β) obtained for PBEsol  at 0 GPa  and \n300 GPa  using the derivative of free energies evaluated on 15 and 7 geometries . The electronic \nexcitation contribution (EEC) to the β at 0 GPa is shown . \n0 500 1000 1500 2000-863.86-863.84-863.82-863.80-863.78-863.76-863.74\nPBEsol\nDF (10-5 Ry)T (K)\nTemperature (T) (K)Free energy (F) (Ry) using 15 geometry\n using   7 geometry\n0 1000 2000\n-2-10\n F15 - F7\n0 500 1000 1500 2000010203040\n using 15 geometries\n using 7 geometries 0 GPa\n 300 GPa\nTemperature (K) Volume thermal expansion (b) (10-6 K-1)PBEsol\nEEC at 0 GPa3 \n • Isobaric heat capacity (C P): \nFig.S1C: Temperature -dependent isobaric heat capacity (C P) obtained for PBEsol  at 0 GPa  and 300 \nGPa evaluated on 15 and 7 geometries . The electronic excitation contribution (EEC) to the CP at 0 GPa \nis shown . \n \n• Bulk modulus (B) : \n \nFig.S1 D: Temperature -dependent isoentropic (B S) and isothermal (B T) bulk modulus obtained for \nPBEsol at 0 GPa and 300 GPa evaluated on 15 and 7 geometries.  \n \n \n0 500 1000 1500 2000010203040\nEEC at 0 GPa300 GPa using 15 geometries\n using 7 geometriesHeat Capacity CP (J/K.mol)\nTemperature (K)PBEsol\n0 GPa\n0 500 1000 1500 2000200250300350400450150015501600\nPBEsolBT300 GPaBulk modulus (GPa)\nTemperature (K) using 15 geometries\n using  7 geometries\n0 GPaBSBS\nBT4 \n • Thermodynamic average Grüneisen parameter  (γ): \nFig.S1 E: Temperature -dependent thermodynamic average -Grüneisen parameter obtained for PBEsol at \n0 GPa and 300 GPa evaluated on 15 and 7 geometries.  \n \nS2: Comparison between phonon dispersion obtained using different k -point and q -point mesh:  \nFig.S2: Phonon dispersions curve obtained at 0 K (dashed)  using k-points 16 × 16 × 16 , q-points mesh  \n4 × 4 × 4, and  q-points mesh, and at 0 K (dash -dot-dot) and 301 K (solid lines) using k-points 32 × 32 \n× 32, q-points mesh  8 × 8 × 8 and  q-points mesh. For comparison, the experimental neutron in -elastic \nscattering data is included.  \nThe phonon dispersion interpolated at T=0 K using 16 × 16 × 16 k-points mesh  and 4 × 4 × 4 \nq-points mesh  is presented in Fig.S3. For comparison, we included the phonon dispersion interpolated \nat T = 0 K and 301 K (shown in the main articl e) using 32 × 32 × 32 k-points mesh  and 8 × 8 × 8 q-\npoints mesh , and experimental inelastic scattering data.  From Fig.S3, it is evident that the choice of 32 \n× 32 × 32 k-points mesh  and 8 × 8 × 8 q-points mesh  successfully explains the dispersions between X \nto K to Γ, which is not described well using 16 × 16 × 16 k-points mesh  and 4 × 4 × 4 q-points mesh . \n050100150200\nNeutron scattering expt. data\n k (16 x 16 x 16)_ q (4 x 4 x 4)_0 K\n k (32 x 32 x 32)_ q (8 x 8 x 8)_0 K\n k (32 x 32 x 32)_ q (8 x 8 x 8)_301 K\n(z,z,z)Frequency (cm-1)\nG X W X K G L(z,0,0) (1,z,0) (z,z,0)L\nTP\nL\nT1T2L\nL\nT\n500 1000 1500 20001.501.551.602.402.452.502.552.602.65\nPBEsol\n using 15 geometries\n using  7 geometries\n300 GPa\nTemperature (K)0 GPaAvg. Grüneisen parameter (g)"    },    {        "title": "2401.12907v1.Viability_and_control_of_a_delayed_SIR_epidemic_with_an_ICU_state_constraint.pdf",        "content": "VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC\nWITH AN ICU STATE CONSTRAINT\nDIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nAbstract. This paper studies viability and control synthesis for a delayed SIR epidemic. The model integrates\na constant delay representing an incubation/latency time. The control inputs model non-pharmaceutical\ninterventions, while an intensive care unit (ICU) state-constraint is introduced to reflect the healthcare system’s\ncapacity. The arising delayed control system is analyzed via functional viability tools, providing insights into\nfulfilling the ICU constraint through feedback control maps. In particular, we consider two scenarios: first,\nwe consider the case of general continuous initial conditions. Then, as a further refinement of our analysis,\nwe assume that the initial conditions satisfy a Lipschitz continuity property, consistent with the considered\nmodel. The study compares the (in general, sub-optimal) obtained control policies with the optimal ones for\nthe delay-free case, emphasizing the impact of the delay parameter. The obtained results are supported and\nillustrated, in a concluding section, by numerical examples.\nKeywords: SIR epidemic; state constraints; viability; optimal control; feedback control; delays; functional\ndifferential equations.\n2020 Mathematics Subject Classification: 34K35, 49J21, 49J53, 92D30, 93C23.\n1.Introduction\nThe optimal control of epidemics (see, for instance [2, 6, 21, 31]) starting from the classical SIR model of\nKermack and McKendrick ([25]), has been widely studied in recent years, also because of the COVID-19\npandemic, see [1, 5, 17, 26, 27]. In this paper, we focus on the viability analysis and the synthesis of control\nstrategies for the following two-dimensional SIR model with delay:\n\n\nds\ndt(t) =−b(t)s(t)i(t−h)\ndi\ndt(t) =b(t)s(t)i(t−h)−γi(t).(1.1)\nHere h > 0 is a given positive number representing a constant delay, modeling the presence of an incuba-\ntion/latency time, i.e., assuming a temporal gap between the disease contraction and the development of\ninfectivity of average individuals. The parameter γ∈(0,1) represents the natural (and constant) recovery\nrate, while the control parameters are measurable functions b(t)∈U:= [β⋆, β] for some 0 < β⋆≤β < 1.\nThe value βmodels the natural transmission rate of the disease, while β⋆represents the minimal attainable\ntransmission rate obtained via non-pharmaceutical policies (social distancing, lockdowns, etc.) The initial\nconditions are imposed by prescribing a pair of continuous functions ϕ= (iϕ, sϕ) : [−h,0]→R2, and by\nrequiring that\ni(t) =iϕ(t) and s(t) =sϕ(t) for all t∈[−h,0]. (1.2)\nThe derivatives in (1.1) are meant in a distributional sense and the trajectories are constructed from admissible\ncontrols b∈ U:=L∞((0,+∞);U).\nThe delayed version (1.1) of the classical Kermack and McKendrick SIR model [25], with a constant trasmission\nrate coefficient b≡β, was introduced in [11, 7]. This model has been widely studied, from the stability point\nof view, in several papers, as for instance [43, 32]. Optimization and optimal control problems have been more\n1arXiv:2401.12907v1  [math.OC]  23 Jan 20242 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nrecently considered in [44] but always in the case of a constant transmission rate. In [7], the nonnegative\nconstant hrepresents a time during which the infectious agents develop in a vector and it is only after time h\nthat the infected vector can infect a susceptible individual. On the other hand, hcan also be regarded as a\nlatency time only after which the infected individual becomes able to transmit the infection. In this slightly\ndifferent interpretation, system (1.1) has been considered in the recent Covid-19 studies [29] and [41] but, in\nthe latter, with a constant transmission rate b≡β. We mention [42, 34] among the recent studies about the\nimpact of a latency period on Covid-19 transmission.\nWe consider system (1.1) under an intensive care unit (ICU) constraint on the number of infected (cft.\n[24, 33, 4]), i.e., we require that\ni(t)≤iM∀t≥0, (1.3)\nfor some fixed 0 < iM≤1. In this setting, the prescribed level iMrepresents an upper bound to the capacity\nof the health-care system to treat infected patients.\nFrom a theoretical point of view, the delay system (1.1) together with the constraint (1.3) can be seen as a\ndifferential delayed inclusion , as defined in [18, 22, 14]. Within this approach, the analysis of (1.1)-(1.3) can\nbe tackled by using the functional viability tools introduced in [19] and well summarized in [3, Chapter 12].\nFor recent developments in this area see, for example, [13, 12]. Functional viability analysis provides geometric\ncharacterizations of the notions of uncontrolled forward invariance and viability (i.e., the property of being\nforward invariant under a suitable control ). Under some technical restrictions, these ideas allow us to provide\nan explicit state-space description of subsets of initial conditions for which the constraint (1.3) is satisfied by\nsolutions to (1.1), under arbitrary or suitable controls actions.\nViability analysis, as said, implicitly highlights feasible control actions which allow to keep the solutions of (1.1)\nin the feasible region (defined by the constraint (1.3)). Consequently, it also provides selection mechanisms\nfor implementing such control actions, in the form of feedback control maps . This selection procedure can also\nbe tuned according to the minimization of a given cost functional of the form\nZ+∞\n0G(β−b(t))dt, (1.4)\nwhere Gis a continuous and strictly increasing scalar function. Thus, summarizing, the performed viability\nanalysis allows us to provide a (in general, sub-optimal) feedback strategy, which depends only on the current\nand past position of the state, for the optimal control problem (1.1)-(1.3)-(1.4). We formally compare our\nviability results with those obtained in the delay-free case (see [4, 16]), which can be recovered by taking the\nlimit as h→0.\nWe point out that different approaches to the study of delayed optimal control problems have been proposed\nin the literature. Notably, we underline that recent researches studied various formulations of Pontryagin\nprinciples for delayed optimal control problems with state constraints, see [38, 39, 8] and references therein.\nThe arising conditions are necessary for optimality of feasible control polices and rely on an advance-differential\nequation governing the adjoint states. On the other hand, differently from the delay-free case studied in [15,\n4, 16], these Pontryagin-principle-based conditions have, up to now and for the problem under consideration,\nbeen unable to provide exact closed-form expression for the optimal control policy. On the other hand, viability\nanalysis provides itself a satisfactory alternative and a flexible tool in building feedback control polices that\nonly depends on the present and past values of the solutions.\nThe paper is organized as follows. In Section 2 we introduce the main notation and definitions. We also provide\na self-contained summary of viability analysis tools for delayed differential inclusions, basically borrowed\nfrom [19, 3], together with a novel characterization result for some particular class of (functional) subsets.\nIn Section 3 we provide a preliminary qualitative analysis of system (1.1), comparing and underlying the\npeculiarities of this model with respect to the delay-free case. In Section 4 we provide the applications of\nviability analysis for system (1.1) under the state-constraint (1.3); considering two cases.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 3\n(1) First, we consider initial conditions as general continuous functions ϕ: [−h,0]→R2, satisfying the ICU\nconstraint (1.3), and provide a complete viability analysis and the corresponding feedback control policy.\n(2) We then strengthen our hypothesis, supposing that initial conditions also satisfy a Lipschitz property\ncompatible with the considered model. In this setting, we prove that when the delay converges to 0 we\nare able to recover the viable regions of the delay-free case studied in [4, 16].\nIn Section 5 we illustrate our results with the aid of numerical examples, and some concluding remarks are\nprovided in Section 6. Some technical proofs are postponed in a final Appendix, to avoid breaking the flow of\nthe presentation.\nNotation: We denote by R+= [0,+∞) the set of non-negative reals. Given A⊂RnandB⊂Rm, we\ndenote by C(A;B) and C1(A;B) the set of continuous and continuously differentiable functions from AtoB,\nrespectively. Given x∈Rn, the scalar |x|is its Euclidean norm. Given two sets A, B, the notation g:A⇝B\nstands for the set-valued map g:A→ P(B) where P(B) is the power set of B. Given a norm ∥·∥onRn, and\nR >0, denote by B∥·∥(0, R) the closed ball (w.r.t. ∥ · ∥) of radius R. Given v, w∈Rn,⟨v, w⟩ ∈Rdenotes the\nEuclidean scalar product ofvandw. Given x∈Rn, we denote by |x|max:= max {|x1|,|x2|, . . . ,|xn|}itsmax\n(or infinity) norm .\n2.Notation and Preliminaries\n2.1.The delayed SIR model: notation and existence of solutions. Let us consider the delay system\nintroduced in (1.1) with state constraint (1.3). Besides those already given in the introduction, we use the\nfollowing notation: x:= (s, i)∈R2,C:=C([−h,0];R2). Moreover, the components of a given ϕ∈ Cwill be\ndenoted by sϕandiϕ, that is, ϕ(t) = (sϕ(t), iϕ(t)) for all t∈[−h,0]. Let us define f:C ×U→R2by\nf(ϕ, b) := (−bsϕ(0)iϕ(−h), bsϕ(0)iϕ(−h)−γiϕ(0)), (2.1)\nand introduce the set-valued map FU:C⇝R2defined by\nFU(ϕ) :={f(ϕ, b)|b∈[β⋆, β]}. (2.2)\nIt can be seen that FUhas convex and compact values, it is locally bounded and upper semicontinuous (we\nrefer to [35, 3] for these general concepts).\nFor any t∈R, consider the map\nS(t) :C(R;R2)→ C\nx7→S(t)x\n[S(t)x](s) :=x(t+s)∀s∈[−h,0].(2.3)\nIn the delay-systems community, the simplified notation S(t)x:=xtis often used. As the control changes in\nthe space U, the class of Cauchy problems (1.1)-(1.2) can thus be rewritten as a delayed differential inclusion\nof the form (\nx′(t)∈FU(S(t)x),\nS(0)x=ϕ∈ C.(2.4)\nAccording to [18], a solution to (2.4) is a continuous function xϕ: [−h, τ)→R2, with τ >0, satisfying (2.4) for\nalmost all t∈[0, τ) an such that xϕis absolutely continuous on every compact subinterval of [0 , τ). As done\nfor the initial condition, the two components of xϕwill be denoted by sϕandiϕ, that is, xϕ(t) = (sϕ(t), iϕ(t))\nfor all t∈[−h, τ). This is consistent with the fact that xϕ(t) =ϕ(t) whenever t∈[−h,0].\nAs a preliminary result, we show that, for a remarkable class of initial conditions, solutions exist and are\nglobally defined.\nLet us define the triangle\nT:={x∈R2|s≥0, i≥0, s+i≤1}\nand introduce the set T:=C([−h,0];T).4 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nLemma 2.1. For any initial condition ϕ∈ T and for any control input b∈ U, there exists a unique solution\nxϕ: [−h,+∞)→R2to the Cauchy problem (1.1)-(1.2). Moreover, xϕ(t)∈Tfor every t∈[−h,+∞). We\nwill denote this solution also with xϕ,b(orxϕ,b), when we want to stress the fact that it corresponds to the\nprescribed input b.\nProof. Given any ϕ∈ T and any b∈ U, the local existence and uniqueness of the solution follows by [20,\nChapter 2, Subsection 2.6]. Let us denote by xϕ: [−h, τ)→R2,τ > 0, such solution and prove that\nxϕ(t) = (sϕ,b(t), iϕ(t))∈Tfor all t∈[0, τ). First we note that, by considering iϕ: [−h, τ)→Ras a coefficient\nand integrating the first equation in (1.1), we have\nsϕ(t) =sϕ(0)eRt\n0b(r)iϕ(r−h)dr.\nSince sϕ(0)≥0 by hypothesis, this implies that sϕ(t)≥0 for all t∈[0, τ). Using also the non-negativity of\nthe initial condition, we get\ni′\nϕ(t) =b(t)sϕ(t)iϕ(t−h)−γiϕ(t)≥ −γiϕ(t)∀t∈[0, h)∩[0, τ).\nBy a comparison argument (see for example [37, Lemma 1.2]), the previous inequality implies\niϕ(t)≥iϕ(0)e−γt≥0∀t∈[0, h)∩[0, τ).\nBy iterating the argument on any interval of the form [ kh,(k+ 1)h] fork∈N, we conclude that iϕ(t)≥0 for\nallt∈[0, τ). Now, summing the equations in (1.1) we have\n(sϕ(t) +iϕ(t))′=−γiϕ(t)∀t∈[0, τ).\nSince, by assumption, iϕ(0) + sϕ(0)≤1, this implies that sϕ(t) +iϕ(t)≤1 for all t∈[0, τ). We have thus\nproved that if ϕ∈ T, then xϕ(t)∈T, for all t∈[0, τ). Since Tis compact, the results in [20, Theorem 3.1\nand Subsection 2.6, Chapter 2,] imply that the solution exists in [0 ,+∞), and the proof is concluded. □\nWell-posedness results, like that of Lemma 2.1, can be found also in [11] for the constant control case and\nin [43, 32] for slightly different delayed SIR models. We note that the proof substantially shows that any\ntriangle of the form {x∈R2\n+|s+i≤a}fora≥0 is invariant. It is physically reasonable to consider initial\nconditions in T, assϕ(t) and iϕ(t) are the fractions of susceptible (resp. infectious) population at time t. Thus,\nthe preliminary Lemma 2.1 can be seen as a permanence result: if the initial condition is a “physically feasible”\ncurve, the solution, forward in time, remains in the region of physical feasibility, no matter the external input\nb∈ U.\n2.2.Viability for delay systems: general theory and first results. We recall here the definitions of\nviability/forward invariance for delay systems and related results ([3, Chapter 12], [19]). Given any dimension\nn∈N, consider a set valued map F:C([−h,0];Rn)⇝Rn(in this subsection, we write C=C([−h,0];Rn))\nand a delayed differential inclusion of the form\n(\nx′(t)∈F(S(t)x),\nS(0)x=ϕ∈ C.(2.5)\nFor the SIR model (1.1) we “a priori” know that solutions exist and are globally defined, for suitable initial\nconditions and controls, as proved in Lemma 2.1. For this reason and to simplify the notation, in this subsection\nwe suppose that for any ϕ∈ Cthe solution set of (2.5) is non-empty and maximal solutions are defined on R+.\nDefinition 2.2. Given K ⊆ C , we say that\n(1)Kisforward invariant for(2.5) if for any ϕ∈ Kandanysolution xϕto(2.5), starting at ϕ, we have\nS(t)xϕ∈ Kfor all t∈R+;\n(2)Kisviable for(2.5) if for any ϕ∈ K there exists a solution xϕto(2.5), starting at ϕ, for which it\nholds that S(t)xϕ∈ Kfor all t∈R+.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 5\nThe following concept of feasible directions (see [3, Definition 12.2.1]) is useful in providing a geometric\ncharacterization of forward invariance and viability.\nDefinition 2.3. Given a subset K ⊆ C andϕ∈ K, the set of feasible directions toKatϕis defined by\nDK(ϕ) :=\b\nv∈Rn:∀ε >0,∃τ∈(0, ε)andϕτ∈ C(R,Rn)such that\nS(0)ϕτ=ϕ, S(τ)ϕτ∈ K,ϕτ(τ)−ϕτ(0)\nτ∈B(v, ε)\t\n, (2.6)\nwhere B(v, ε)denotes the closed ball of Rnwith radius εand centered in v.\nIn our setting, a more concise representation of the set of feasible directions can be given in terms of the\nclassical notion of Bouligand contingent cone, recalled below.\nDefinition 2.4. Consider K⊂Rnandx∈K, the Bouligand contingent cone to Katxis defined by\nTK(x) :={v∈Rn| ∃tk→0, tk>0,∃vk→vsuch that x+tkvk∈K∀k∈N}.\nIt is well-known that, if the set Kis convex, the Bouligand contigent cone coincides with the classical tangent\ncone of convex analysis, i.e.,\nTK(x) = cl ( {v∈Rn| ∃λ≥0 and ∃y∈Ksuch that v=λ(y−x)}),\nsee [35, Theorem 6.9]. If x∈int(K), then TK(x) =Rn. Moreover, if the convex set K⊂Rnis defined by\nK={x∈Rn|gi(x)≤0, i∈ {1, . . . , M }}, (2.7)\nfor some g1, . . . , g M∈ C1(Rn;R), then, for any x∈∂Ksuch that gi(x) = 0 for all i∈I⊆ {1, . . . , M }(and\ngj(x)̸= 0 for j /∈I), we have\nTK(x) ={v∈Rn| ⟨∇gi(x), v⟩ ≤0∀i∈I}, (2.8)\nsee [35, Theorem 6.31].\nIn the sequel, the subset K ⊂ C will represent the set of initial conditions of the epidemic model (1.1). As\nanticipated in the Introduction, we are going to consider two different kinds of initial conditions, sharing the\nsame set K⊆S⊆Rnof traces in 0. First, we take the subset KS⊂ Cdefined by\nKS={ϕ∈ C | ϕ(0)∈K, ϕ(t)∈S∀t∈[−h,0]}. (2.9)\nBesides this large set, we consider the following smaller sets of Lipschitz continuous initial conditions, i.e.,\ngiven a constant L >0, we take\nKS,L={ϕ∈ CL,∥·∥|ϕ(0)∈K, ϕ(t)∈S∀t∈[−h,0]}, (2.10)\nwhere CL,∥·∥denotes the set of functions ϕ∈ Cthat are L-Lipschitz in [ −h,0] w.r.t. the norm ∥ · ∥ inRn,\ni.e.,∥ϕ(t1)−ϕ(t2)∥ ≤L|t1−t2|, for all t1, t2∈[−h,0]. In the following statement we characterize the set of\nfeasible directions of Definition 2.3 for these different choices of initial conditions.\nLemma 2.5. Consider a set K⊂Rnand a convex set S⊇K. The following characterization of the set of\nfeasible directions to KSin(2.9) holds:\nDKS(ϕ) =TK(ϕ(0))∀ϕ∈ KS. (2.11)\nMoreover, given L >0, we have\nDKS,L(ϕ) =TK(ϕ(0))∩B∥·∥(0, L)∀ϕ∈ KS,L. (2.12)\nProof. First of all, we note that given x∈Kwe have\nTK(x) ={v∈Rn| ∃tk→0, tk>0, vk→vsuch that ∥vk∥=∥v∥andx+tkvk∈K∀k∈N}, (2.13)\ni.e., we can add, without loss of generality, the constraint ∥vk∥=∥v∥to the sequences of vectors in Defini-\ntion 2.4. Indeed, this is clear if v= 0∈TK(x). When, instead, 0 ̸=v∈TK(x), starting from some sequences6 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\ntk>0 and vk∈Rn\\ {0}such that tk→0,vk→vandx+tkvk∈K,∀k∈N, the claim is proved by taking\nthe modified sequences etk=∥vk∥\n∥v∥tkandevk=∥v∥\n∥vk∥vk.\nLet us note, moreover, that it is enough to prove (2.12), since the proof of (2.11) can be easily obtained from\nthe latter, by sending Lto +∞.\nGiven ϕ∈ KS,L, let us start by proving the inclusion TK(ϕ(0))∩B∥·∥(0, L)⊆DKS,L(ϕ). For v∈TK(ϕ(0))∩\nB∥·∥(0, L), let tkandvkbe two sequences as in (2.13). Given any ε >0 consider a k∈Nlarge enough such\nthattk< εand∥vk−v∥< ε. Define τ=tkand consider ψ∈ C([0, τ],Rn) defined by ψ(s) =ϕ(0)+ svkfor any\ns∈[0, τ]. By convexity of S,ψ(s)∈Sfor all s∈[0, τ]. Moreover, ψisL-Lipschitz, since ∥ψ′(s)∥=∥vk∥=\n∥v∥ ≤L, for any s∈[0, τ]. We then define ϕτ=ϕ∗ψ∈ C([−h, τ],Rn), where ∗denotes the concatenation\nin time. To prove that v∈ DKL,S(ϕ) we have to verify that ϕτsatisfies the conditions (2.6) of Definition 2.3,\nwithK=KS,L. We trivially have S(0)ϕτ=ϕand, moreover, S(τ)ϕτ(0) = ϕτ(τ) =ϕ(0) + tkvk∈K. We\nalso note that S(τ)ϕτ(s)∈Sfor all s∈[−h,0] and that S(τ)ϕτisL-Lipschitz. We have thus proved that\nS(τ)ϕτ∈ KS,L. Also the remaining condition in (2.6) holds since\nϕτ(τ)−ϕτ(0)\nτ=ϕ(0) + tkvk−ψ(0)\ntk=vk∈B∥·∥(v, ε),\nthus concluding the proof of the claimed inclusion.\nWe now prove the opposite inclusion, DKS,L(ϕ)⊆TK(ϕ(0))∩B∥·∥(0, L). Given any v∈ DKL,S(ϕ) and any\nk∈N, consider ε=1\nk,τ∈(0, ε) and ψτas in Definition 2.3. Define tk=τandvk=ψτ(τ)−ψτ(0)\nτ. By\ndefinition of KS,L, we have\n∥vk∥=∥ψτ(τ)−ψτ(0)∥\nτ≤Lτ\nτ=L.\nTo conclude, we recall that tk≤1\nk→0 and vk→vand thus ∥v∥ ≤L. Hence v∈TK(ϕ(0))∩B∥·∥(0, L), and\nthe proof is concluded. □\nRemark 2.6.We note that in Lemma 2.5 we have proven, in particular, that the sets DKS(ϕ) and DKS,L(ϕ)\nare independent of the set S, which constrains the past values of the initial conditions. Indeed, DKS(ϕ) and\nDKS,L(ϕ) only depends on the “arrival” set Kand on the final position ϕ(0) (and on the Lipschitz constant\nL >0, in the case of DKS,L(ϕ)). This property, which holds for sets of the form (2.9) and (2.10), is crucial in\nthe subsequent analysis.\nIn the next statement we specialize the main result concerning viability theory, namely Theorem 12.2.2 [3], to\nthe case in which the set Kis of the form (2.9) or (2.10).\nTheorem 2.7. Consider a set-valued map F:C⇝Rnwhich is upper semicontinuous with nonempty, convex\nand compact values. Given compact and convex sets K⊆S⊂Rn, and L >0, a set K ⊂ C of the form (2.9)\nor(2.10) is forward invariant for (2.5) (resp. viable) if and only if\nF(ϕ)⊆DK(ϕ)∀ϕ∈ K,\n(resp.) F(ϕ)∩DK(ϕ)̸=∅∀ϕ∈ K.\nProof. For the case Kof the form (2.9), we apply [18, Corollary 1.1], recalling by Lemma 2.5 that DK(ϕ) =\nTK(ϕ(0)) for all ϕ∈ K.\nIn the case Kdefined as in (2.10) it suffices to apply Theorem 12.2.2 in [3], after having observed that the\nclosedness of KinRnimplies that Kis closed with respect to the norm of uniform convergence. □\nIn spite of the fact that the sets of feasible directions are independent of S, the characterization of invariant\n(resp. viable) sets, in fact, depend on the past, since the conditions of Theorem 2.7 must by verified on the\n(S, L)-depending set Kof initial conditions. As a consequence, when the set S⊆Rnis fixed in (2.9) and (2.10),\na notion of maximality of the set K(of the traces ϕ(0) of the initial conditions ϕ∈ K) cannot neglect the past\nbehavior of the functions ϕ. We thus give now an appropriate definition which distinguishes between initial\nconditions in KSandKS,L.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 7\nDefinition 2.8. Consider a set H ⊆ C , a set S⊆Rnand an upper semicontinuous set-valued map F:C⇝Rn\nwith nonempty, convex and compact values. A set K⊆Sis said to be\n•themaximal forward invariant set with H-past in Sfor (2.5) ifKis the largest set such that\nK:={ϕ∈ H | ϕ(0)∈K, ϕ (s)∈S∀s∈[−h,0]}\nis forward invariant (in the sense of Definition 2.2).\n•themaximal viable set with H-past in Sfor (2.5) ifKis the largest set such that\nK:={ϕ∈ H | ϕ(0)∈K, ϕ (s)∈S∀s∈[−h,0]}\nis viable (in the sense of Definition 2.2).\nWhen H=CtheH-prefix is dropped.\n2.3.Viability for the delayed SIR model. For the set-valued map FU:C([−h,0],R2)⇝R2of the delayed\nSIR model introduced in (2.2), Theorem 2.7 specializes in the following statement.\nCorollary 2.9. Given compact and convex sets K⊆S⊂R2andL > 0, a set K ⊂ C ([−h,0],R2)of the\nform (2.9) or(2.10) is forward invariant (resp. viable) for (2.4) if and only if\nf(ϕ, b)∈DK(ϕ)∀ϕ∈ K ∀ b∈[β⋆, β],\n(resp.) ∀ϕ∈ K,∃b∈[β⋆, β] :f(ϕ, b)∈DK(ϕ).\n3.Qualitative Analysis of Solutions\nIn this section we collect some properties of (solutions to) system (2.4) under arbitrary inputs b∈ U.\nBy Lemma 2.1, we have that for every initial condition ϕ∈ T a solution to the Cauchy problem for (2.4)\nexists and its values belong to T. Our aim is now to study the asymptotic behavior of such solutions.\nFirst, we note that the case in which ϕ= (sϕ, iϕ)∈ T is such that iϕ(t) = 0 for every t∈[−h,0] is not\nphysically relevant. Nevertheless it is trivial, since in this case ( sϕ(t), iϕ(t))≡(sϕ(0),0) for all t∈R+. For\nthis reason, in the rest of this section we assume that\niϕ̸≡0 in [−h,0]. (3.1)\nLet us call T0the set of all ϕ∈ Tsatisfying (3.1).\nOur first result is inspired by [11], in which the case of a constant b(t) is treated; here we adapt the argument\nto the time-varying case.\nLemma 3.1. Consider any control input b∈ U, and any ϕ∈ T0. Then, sϕis non-increasing in [0,+∞), and\nit holds that\nlim\nt→+∞iϕ(t) = 0 ,lim\nt→+∞sϕ(t) =sϕ,∞∈[0,γ\nβ⋆),\nwhere sϕ,∞depends only on b,ϕ(0)andR0\n−hiϕ(τ)dτ.\nMoreover,\n(1) if sϕ(0)>0, then\n(a)sϕ,∞>0,\n(b)iϕ(t)>0for all t≥h,\n(c)sϕis strictly decreasing in [2h,+∞);\n(2) if the input bis eventually essentially constant, i.e., if there exist T > 0andb∞∈[β⋆, β]such that\nb(t) =b∞for almost all t∈[T,+∞), then sϕ,∞<γ\nb∞.8 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nProof. For every ϕ∈ T0,sϕ(·) :R+→R+is non-increasing since, by the invariance of Tstated in Lemma 2.1,\nwe have s′\nϕ(t)≤0 for all t∈R+. Moreover, sϕ(·) is bounded from below by 0 and thus the limit\nlim\nt→+∞sϕ(t) =:sϕ,∞\nexists and belongs to [0 ,1]. Define Nϕ(t) :=sϕ(t) +iϕ(t). We have that\nN′\nϕ(t) =−γiϕ(t), t∈R+, (3.2)\nand thus also Nϕ(t) is a non-increasing function and admits a non-negative limit, by Lemma 2.1. Since\niϕ(t) =Nϕ(t)−sϕ(t), also iϕadmits a limit iϕ,∞, which is non-negative again by Lemma 2.1.\nOn the other hand, by integrating (3.2) on a time interval [ T,+∞) with T≥0, and using that iϕ,∞≥0, we\ngetZ∞\nTiϕ(τ)d τ=Nϕ(T)−sϕ,∞−iϕ,∞\nγ≤Nϕ(T)−sϕ,∞\nγ. (3.3)\nFrom the last inequality we immediately get that iϕ,∞= lim t→∞iϕ(t) = 0, and the first equality in (3.3)\nbecomes Z∞\nTiϕ(τ)d τ=Nϕ(T)−sϕ,∞\nγfor every T≥0. (3.4)\nTo conclude the proof of the first part of the statement, it remains to show that sϕ,∞<γ\nβ⋆.\nWe use here the notation b(t) =β⋆+v(t) with v∈L∞(R+; [0, β−β⋆]). By integrating the first equation\nin (1.1) for t≥h, and performing a simple change of variable, we obtain\nsϕ(t) =sϕ(0) exp\u0012\n−Zt\n0b(τ)iϕ(τ−h)dτ\u0013\n=sϕ(0) exp \n−Z0\n−hb(τ+h)iϕ(τ)dτ−β⋆Zt−h\n0iϕ(τ)dτ−Zt−h\n0v(τ+h)iϕ(τ)dτ!\n.\nAst→+∞, we have\nsϕ,∞=sϕ(0) exp\u0012\n−Z0\n−hb(τ+h)iϕ(τ)dτ−β⋆Z∞\n0iϕ(τ)dτ−Z∞\n0v(τ+h)iϕ(τ)dτ\u0013\n. (3.5)\nBy using the equation (3.4) with T= 0 to substitute the integralR∞\n0iϕ(τ)dτin (3.5), we obtain\nsϕ,∞=sϕ(0) exp\u0012\n−Z0\n−hb(τ+h)iϕ(τ)dτ−Z∞\n0v(τ+h)iϕ(τ)dτ−β⋆\nγ(Nϕ(0)−sϕ,∞)\u0013\n. (3.6)\nEquation (3.6) can be written in the equivalent form\nsϕ,∞e−β⋆\nγsϕ,∞=M(ϕ, b)sϕ(0)e−β⋆\nγsϕ(0)(3.7)\nwith\nM(ϕ, b) := exp\u0012\n−Z0\n−hb(τ+h)iϕ(τ)dτ−Z∞\n0v(τ+h)iϕ(τ)dτ−β⋆\nγ(iϕ(0))\u0013\n.\nSince iϕ(t) is not identically zero in [ −h,0] then M(ϕ, b)<1. By this inequality and (3.7), then we get\nsϕ,∞e−β⋆\nγsϕ,∞< sϕ(0)e−β⋆\nγsϕ(0)(3.8)\nLet us now introduce the function g(z) :=ze−β⋆\nγz, which is increasing in [0 ,γ\nβ⋆] and decreasing in [γ\nβ⋆,+∞).\nIfsϕ(0)<γ\nβ⋆then the claim ( sϕ,∞<γ\nβ⋆) follows by the monotonicity of sϕ. Let us consider then the case\nsϕ(0)≥γ\nβ⋆and assume by contradiction that sϕ,∞≥γ\nβ⋆. Since sϕ,∞≤sϕ(0), by the monotonicity of gin the\ninterval [γ\nβ⋆,+∞) we have g(sϕ,∞)≥g(sϕ(0)), so contradicting (3.8). This completes the proof of the first\npart of the statement.\nLet us now start proving the “moreover” part (1)of the statement.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 9\nIfsϕ(0)>0, from (3.7), we have sϕ,∞>0, that is (1)(a) .\nTo prove (1)(b) , we first show that there exists a ��∈[0, h] such that iϕ(τ)>0. Since iϕis continuous and not\nidentically zero in [ −h,0], there exist δ∈(0, h) and ε >0 such that iϕ(s)>0 for all s∈(−h+δ−ε,−h+δ).\nSince sϕ(0)>0, we have already proven that sϕ(t)≥sϕ∞>0 for all t∈R+. Suppose by contradiction that\niϕ(s) = 0, for all s∈[0, δ]. We have\niϕ(δ) =Zδ\n0b(s)sϕ(s)iϕ(s−h)ds≥β⋆sϕ,∞Zδ\n0iϕ(s−h)ds≥β⋆sϕ,∞Z−h+δ\n−h+δ−εiϕ(s)ds > 0,\nleading to a contradiction. We can thus fix τ∈[0, h], such that iϕ(τ)>0. Since i′\nϕ(t)≥ −γiϕ(t), by the\ncomparison principle (see [37, Lemma 1.1]) it holds that\niϕ(t)≥iϕ(τ)e−γ(t−τ), (3.9)\nthus proving that iϕ(t)>0 for all t≥h, that is (1)(b) .\nAssertion 1(c)is a direct consequence of (1)(a) and(1)(b) , because sϕ(t) =−b(t)sϕ(t)iϕ(t−h)<0 ift≥2h.\nLet us now prove (2). Suppose that there exists T > h andb∞∈[β⋆, β] such that b(t) =b∞, for almost all\nt∈[T,+∞). By integrating the first equation of (1.1) on the interval [ T, t] for a t≥T+h, we obtain\nsϕ(t) =sϕ(T) exp\u0012\n−Zt\nTb(τ)iϕ(τ−h)dτ\u0013\n=sϕ(T) exp \n−b∞ZT\nT−hiϕ(τ)dτ−b∞Zt−h\nTiϕ(τ)dτ!\n.\nBy taking the limit as t→+∞, we have\nsϕ,∞=sϕ(T) exp \n−b∞ZT\nT−hiϕ(τ)dτ−b∞Z∞\nTiϕ(τ)dτ!\n. (3.10)\nBy using the equation (3.4) to substitute the integralR∞\nTiϕ(τ)dτin (3.10), and manipulating, we see that\nsϕ,∞satisfies the equation\nsϕ,∞e−b∞\nγsϕ,∞=MT(ϕ, b∞)sϕ(T)e−b∞\nγsϕ(T)(3.11)\nwith\nMT(ϕ, b∞) := exp \n−b∞ZT\nT−hiϕ(τ)dτ−b∞\nγ(iϕ(T))!\n<1,\nsince by (1)(b) ifiϕ(t) is not identically zero in [ −h,0], then iϕ(T)>0. Recalling that sϕ,∞≤sϕ(T), and\narguing as in the proof of the first part of the statement we see that (3.11) implies sϕ,∞<γ\nb∞. □\nIn the delay-free case, i.e., when h= 0 the value s=γ\nβis the herd immunity threshold , see for example [4, 16]\nand references therein. In this case, if sϕ(0)≤γ\nβandiϕ(0)>0, (1.1) immediately implies that iϕ:R+→Ris\nstrictly decreasing. In the subsequent statement we show thatγ\nβis an important threshold also in the delay\ncase (when h >0): if at the current instant the susceptible population sis smaller than this value, the number\nof infected people iwill not exceed again the maximum value attained in the last h-period of time.\nLemma 3.2. Consider any control input b∈ U. Given any ϕ∈ T0and any τ∈R+, define bı(ϕ, τ) :=\nmax\nt∈[τ−h,τ]iϕ(t). Then, for any ϕ∈ T0such that sϕ(0)≤γ\nβ, we have\niϕ(t+τ)≤bı(ϕ, τ),∀τ∈R+∀t∈[0, h]. (3.12)\nProof. Consider any τ∈R+, and suppose by contradiction that there exists a ¯ τ∈[τ, τ+h] such that\niϕ(¯τ)>bı(ϕ, τ). Consider\nτ1= max {t∈[τ,¯τ] :iϕ(t) =bı(ϕ, τ)},10 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nwhich exists by continuity and because iϕ(τ)≤bı(ϕ, τ) and iϕ(¯τ)>bı(ϕ, τ). Moreover, τ1<¯τ. Then, by\ndefinition of τ1, it holds that iϕ(t)≥bı(ϕ, τ) for all t∈[τ1,¯τ]. We have\niϕ(¯τ) =iϕ(τ1) +Z¯τ\nτ1b(t)sϕ(t)iϕ(t−h)−γiϕ(t)dt≤bı(ϕ, τ) +γZ¯τ\nτ1(iϕ(t−h)−bı(ϕ, τ))dt≤bı(ϕ, τ),\nwhere, in the last inequality, we used that iϕ(t−h)≤bı(ϕ, τ) for all t∈[τ1,¯τ] because t−h∈[τ−h, τ], and\nthatsϕ(t)≤sϕ(0)≤γ\nβ, for all t≥0. This leads to a contradiction which concludes the proof. □\nIn Lemma 3.2, rephrasing, we have proved that, for any ϕ∈ T0such that sϕ(0)≤γ\nβ, the function t7→\n∥S(t)iϕ∥∞= max s∈[0,h]iϕ(t−s) is non-increasing. This should be compared with the delay-free case (see [4,\n16]), in which, if the initial condition ( sϕ(0), iϕ(0)) is such that sϕ(0)≤γ\nβandiϕ(0)>0, we have that the\nfunction t7→iϕ(t) is strictly decreasing. In this regard, Lemma 3.2 provides a “weak” decreasing property of\nthe infectious component iϕof the state in the delay case. We want to stress the fact that, when h >0, (3.12)\ndoes notimply that iϕbe monotonic (which, in fact, might be not).\n4.Viability analysis and control\nGiven a parameter 0 < iM≤1 (representing the maximal acceptable proportion of infected), we are going to\nanalyze the cases in which the trajectories of (1.1)/(2.4) stay (can be forced to be in) the feasible set\nC:={x∈T|i≤iM}. (4.1)\n4.1.Arbitrary feasible initial conditions. In this subsection we consider the case in which the controller\nhas a limited knowledge on the past evolution of the epidemics: we suppose that it is only known that in the\npast htime-units, the value of iϕdid not exceed the safety threshold iM. More formally, we will assume the\nfollowing.\nAssumption 4.1. The initial condition ϕ∈ Csatisfies ϕ(t)∈Cfor all t∈[−h,0].\nWe now need to introduce some additional state-space curves, in order to characterize (maximal) forward\ninvariant and viable sets with past in Cof (1.1) (see Definition 2.8). In particular, we characterize the\nboundaries of these regions by providing the “ worst case behavior ” of system (1.1), for the two cases b≡β\nandb≡β⋆, respectively. We will thus consider (backward) solutions of (1.1) with an artificial value of i(t−h),\nfixed equal to iM, as formalized in the sequel.\nLet us define Ψ β= (¯sβ,¯ıβ) :R+→R2as the solution to the linear system\n(\ns′(t) = βiMs(t),\ni′(t) =−βiMs(t) +γi(t),(4.2)\nwith initial condition s(0) =γ\nβandi(0) = iM. Since the system is linear, the unique solution can be explicitly\ncomputed to be\n¯sβ(t) =γ\nβeβiMt,¯ıβ(t) =iMeγt(\n1−t, ifβiM=γ,\nβiM−γe(βiM−γ)t\nβiM−γ,ifβiM̸=γ,t∈R+. (4.3)\nLet us denote by Tβ>0 the unique solution of the equation ¯ ıβ(t) = 0 that is the unique time at which Ψ β\ncrosses the s-axis. It turns out to be\nTβ=(\n1, ifβiM=γ,\n1\nβiM−γlnβiM\nγ,ifβiM̸=γ.(4.4)VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 11\nLet us denote\nbsβ:= ¯sβ(Tβ) =γ\nβ\n\neβiM, ifβiM=γ,\n\u0010\nβiM\nγ\u0011βiM\nβiM−γ,ifβiM̸=γ.(4.5)\nSince ¯ sβis strictly increasing, we can define Γ β: [γ\nβ,bsβ]→R+as the function representing the curve Ψ βin\n[0, Tβ] as a graph i(s) = Γ β(s). By dividing the equations in (4.2), we note that Γ βis the solution of the\nCauchy problem(\ni′(s) =−1 +γi(s)\nβiMs,\ni(γ\nβ) =iM.(4.6)\nBy integrating, we have\nΓβ(s) =(\ncsω+s\nω−1, ifω̸= 1,\nc1s−slog(s),ifω= 1,\nwith ω=γ\nβiM,c=\u0000γ\nβ\u0001−ω(iM−γ\nβ(ω−1)) and c1= 1 + log(γ\nβ). By basic calculus arguments one can show that\nΓβis strictly decreasing and concave in [γ\nβ,bsβ], and thus the hypograph {(s, i)∈R2|s∈[γ\nβ,bsβ], i≤Γβ(s)}\nis convex, see Figure 1.\nSimilarly, we denote by Ψ β⋆= (¯sβ⋆,¯ıβ⋆) :R+→R2the unique solution to the system (given by replacing β\nwith β⋆in (4.2))(\ns′(t) = β⋆iMs(t),\ni′(t) =−β⋆iMs(t) +γi(t),\nwith initial condition s(0) =γ\nβ⋆andi(0) = iM. The solution, the interception time Tβ⋆with the s-axis, and\nbsβ⋆:= ¯sβ⋆(Tβ⋆), can be explicitly written by replacing βwith β⋆in (4.3), (4.4) and (4.5), respectively.\nAs before, in the interval [0 , Tβ⋆] we represent the curve Ψ β⋆by a graph i(s) = Γ β⋆(s) with Γ β⋆: [γ\nβ⋆,bsβ⋆]→R+\ndefined by\nΓβ⋆(s) =(\nc⋆sω⋆+s\nω⋆−1, ifω⋆̸= 1,\nc⋆,1s−slog(s),ifω⋆= 1,(4.7)\nwith ω⋆=γ\niMβ⋆,c⋆= (γ\nβ⋆)−ω⋆(iM−γ\nβ⋆(ω⋆−1)) and c⋆,1= 1 + log(γ\nβ⋆). Also Γ β⋆is strictly decreasing and\nconcave in its domain. In Figure 1 we have depicted the curve Γ β⋆for arbitrarily chosen values of γ,β⋆andiM.\nThe next theorem will characterize the maximal forward invariant and viable sets with past in C={x∈\nT|i≤iM}, in terms of the following subsets of C:\nR:= ([0 ,γ\nβ]×[0, iM])∩T, A :=R∪\b\n(s, i)∈T|s∈[γ\nβ,bsβ], i≤Γβ(s)\t\n,\nR⋆:= ([0 ,γ\nβ⋆]×[0, iM])∩T, B :=R⋆∪\b\n(s, i)∈T|s∈[γ\nβ⋆,bsβ⋆], i≤Γβ⋆(s)\t\n.(4.8)\nTheorem 4.2. (1) The set A ⊂ C defined by\nA:={ϕ∈ C | ϕ(0)∈A, ϕ(t)∈C∀t∈[−h,0]}\nis forward invariant, and Ais the maximal forward invariant set with past in C, in the sense of Defini-\ntion 2.8.\n(2) The set B ⊂ C defined by\nB:={ϕ∈ C | ϕ(0)∈B, ϕ (t)∈C∀t∈[−h,0]}\nis viable, and Bis the maximal viable set with past in C, in the sense of Definition 2.8.\nProof. Let us prove the forward invariance of Aby applying Corollary 2.9 with K=A. There, the invariance\ncondition is written in terms of the set DAof feasible directions which, in turn, is characterized in Lemma 2.5\n(with KS=A,S=CandK=A) as\nDA(ϕ) =TA(ϕ(0))∀ϕ∈ A.12 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nS1\nS2\nγ\nβ /hatwidesβ /hatwidesβ⋆Γβ⋆ ΓβiM\nγ\nβ⋆A BC\nFigure 1. Plot of the regions in Theorem 4.2 for the values iM= 0.4,γ/β= 0.2,γ/β⋆= 0.3.\nThe orange region represents the maximal forward invariant set Awith past in C, while the\nwhole colored area represents the set B, that is the maximal viable set with past in C. The red\nlineS1and the orange curve S2represent the sets in which the control policy in Theorem 4.5\nis possibly discontinuous.\nSummarizing, we have to prove that\nf(ϕ, b)∈TA(ϕ(0)) ∀ϕ∈ A ∀ b∈[β⋆, β], (4.9)\nwhere f(ϕ, b) is given in (2.2).\nIt is enough to consider the case ϕ(0)∈∂A, because if ϕ(0)∈Int(A) then TA(ϕ(0)) = Rnand condition (4.9)\nis straightforwardly satisfied.\nThe set Ahas a piecewise C1boundary and can be written in the form (2.7). Thus, we can use the character-\nization of TA(ϕ(0)) given by (2.8). In checking (4.9), by continuity of f, we can restrict ourselves to consider\nonly the ϕ(0) that belong to the (relative) interior of each piece of ∂Aand prove that the scalar product of\nf(ϕ, b) with a (outer) normal vector v̸= 0 to Ainϕ(0) turns out to be non-positive.\nLet us then distinguish the following cases corresponding to the relative interior of different pieces of ∂A.\n(1)iϕ(0) = 0. Since f(ϕ, b) = (−bsϕ(0)iϕ(−h), bs ϕ(0)iϕ(−h)) and v= (0,−1) is a normal vector to the\nhalfplane {i≥0}, then we have ⟨v, f⟩=−bsϕ(0)iϕ(−h)≤0,∀b∈[β⋆, β].\n(2)sϕ(0) = 0. We have that f(ϕ, b) ={(0,−γiϕ(0))}. Since v= (−1,0) is a normal vector to Aat\n(sϕ(0), iϕ(0)) then we have ⟨v, f⟩= 0.\n(3)sϕ(0)<γ\nβandiϕ(0) = iM. Since v= (0,1), we have\n⟨v, f⟩=bsϕ(0)iϕ(−h)−γiϕ(0) = bsϕ(0)iϕ(−h)−γiM<(b\nβ−1)γiM≤0\nfor every b∈[β⋆, β].\n(4)sϕ(0)∈(γ\nβ,bsβ) and iϕ(0) = Γ β(sϕ(0)). Let us denote, for simplicity, ϕ(0) = ( s0, i0) and observe that any\nnormal vector v= (v1, v2)̸= 0 to Aat (s0, i0) satisfies v1≥0 and v2≥0. This vector vis, by definition,\nperpendicular to the tangent vector (1 ,Γ′\nβ(s0)). Since, by (4.6), we have Γ′\nβ(s0) =−1 +γi0\nβs0iM, then\n0 =⟨(v1, v2),(1,Γ′\nβ(s0))⟩=v1+ Γ′\nβ(s0)v2=v1−v2+v2γi0\nβs0iM\nif and only if\n0 =−(v2−v1)βs0iM+v2γi0,\nfrom which we also deduce that v2≥v1. Then, we have\n⟨v, f(ϕ, b)⟩= (v2−v1)bs0iϕ(−h)−v2γi0≤(v2−v1)βs0iM−v2γi0= 0.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 13\nThe proof of forward invariance of Ais thus completed. We now prove the maximality. Consider any set eA̸=A\nsuch that A⊂eA⊆C, and suppose by contradiction that eA:=n\nϕ∈ C | ϕ(0)∈eA, ϕ(s)∈C∀s∈[−h,0]o\nis\nforward invariant. We can now consider a particular ϕ0∈eAsatisfying ϕ0(0)∈eA\\A, i.e.,\nsϕ0(0)>γ\nβ, i ϕ0(0)>Γβ(sϕ0(0)), (4.10)\nandiϕ0(t) =iMfor all t∈[−h,−ε] for a given ε∈(0, h). Since eAis supposed to be forward invariant, the\nsolution ( sϕ0(t), iϕ0(t)) of (1.1) corresponding to the control b≡βbelongs to eAfor every t∈R+. Moreover,\nin the interval [0 , h−ε], it coincides with the solution of the linear system (where iMreplaces i(t−h))\n(\ns′(t) =−βiMs(t),\ni′(t) = + βiMs(t)−γi(t),(4.11)\nwith initial condition ( s(0), i(0)) = ϕ0(0). We note that the system (4.11) is the time-inversion of system (4.2),\ni.e., it is defined by the same vector field with opposite sign. This implies that i(s) = Γ β(s) represents\nthe graph of a solution of (4.2), as well as (4.11). By uniqueness of solutions of (4.11), inequality (4.10)\nimplies iϕ0(t)>Γβ(sϕ0(t)) for all t∈[0, h−ε]. Thus ( sϕ0(t), iϕ0(t))/∈Afor all t∈[0, h−ε], while\n(sϕ0(t), iϕ0(t))∈eA, by forward invariance of eA. We can now iterate the argument, considering a ϕ1∈eA\nsuch that ϕ1(0) = ( sϕ0(h−ε), iϕ0(h−ε)), and iϕ1(t) =iMfort∈[−h,−ε]. The solution ( sϕ1(t), iϕ1(t)) to\nthe system (1.1) corresponding to the control b≡βagain coincides in [0 , h−ε], with the solution of (4.11)\nwith initial condition ( s(0), i(0)) = ϕ1(0). Proceeding similarly to define ϕk∈eA, we have that the solutions\n(sϕk(t), iϕk(t)) have the property that iϕk(t)>Γβ(sϕk(t)) for all t∈[0, h−ε] and all k∈N. We now join all\ncurves ϕkby considering ¯ϕ:R+→R2defined by\n¯ϕ(t) = (s¯ϕ(t), i¯ϕ(t)) = ( sϕk(t−k(h−ε)), iϕk(t−k(h−ε))) if t∈[k(h−ε),(k+ 1)( h−ε)], k∈N.\nBy construction we have that ¯ϕ(t)∈eA\\Afor all t∈R+, and ¯ϕsatisfies the Cauchy problem defined by the\nlinear system (4.11) with initial condition ¯ϕ(0) = ϕ0(0). The Cauchy problem for (4.11) with initial condition\n(s(0), i(0)) = ϕ0(0) can be explicitly solved. In particular, we have s¯ϕ(t) =s(0)e−βiMt. Since s(0)>γ\nβ\n(see (4.10)), in T=1\nβiMlog(s0β\nγ)>0 we have s¯ϕ(T) =γ\nβ. By the uniqueness of solutions of (4.11), we have\ntherefore\ni¯ϕ(T)>Γβ(s¯ϕ(T)) = Γ β(γ\nβ) =iM.\nThus, ¯ϕ(T)/∈C⊇eA, leading to a contradiction.\nThe proof of part (2)of the statement proceeds in a similar way, that is, we prove the viability of Bby applying\nCorollary 2.9 with K=B(see (2.9)) and Lemma 2.5 with KS=B,S=CandK=B. In the current case,\nwe have to prove that\n∀ϕ∈ B,∃b∈[β⋆, β] :f(ϕ, b)∈TB(ϕ(0)), (4.12)\nwhere f(ϕ, b) is given in (2.2), and it is enough to consider the case ϕ(0)∈∂B(since otherwise TB(ϕ(0)) = Rn).\nAs before, since the set Bhas a piecewise C1boundary we can restrict ourselves to consider only the ϕ(0)\nthat belong to the (relative) interior of each piece of ∂Band prove that the scalar product of f(ϕ, b) with a\nnormal vector v̸= 0 to Binϕ(0) turns out to be non-positive.\nWe explicitly develop the non-trivial cases only. In the case 0 < sϕ(0)<γ\nβ⋆andiϕ(0) = iM, by computing\nthe scalar product of f(ϕ, b) with the normal vector v= (0,1), we obtain\n⟨v, f⟩=bsϕ(0)iϕ(−h)−γiϕ(0) = bsϕ(0)iϕ(−h)−γiM≤(bγ\nβ⋆−γ)iM,\nwhich is non-positive if b≤β⋆, and thus f(ϕ, β⋆)∈TB(ϕ(0)). Now suppose that ϕ(0) = ( s0, i0) is such that\ns0∈(γ\nβ⋆,bsβ⋆) and Γ β⋆(s0) =i0. Again, a normal vector to Bat (s0, i0) is of the form w= (w1, w2)̸= 0, with\nw1, w2≥0, and, by definition of Γ β⋆it satisfies\n(w2−w1)β⋆s0iM−w2γi0= 0, (4.13)14 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nwhich also implies w2≥w1. We have\n⟨w, f(ϕ, β⋆)⟩= (w2−w1)β⋆s0iϕ(−h)−w2γi0≤(w2−w1)β⋆s0iM−w2γi0= 0,\nand (4.12) is satisfied. This proves that Bis viable. The maximality follows by an argument analogous to the\none used in proving (1). □\nBesides the state equation (1.1) and the state constraint (1.3), we consider now a cost functional of the form\nJ(u) :=Z∞\n0G(u(t))dt, (4.14)\nwhere u(t) :=β−b(t) and G:R+→R+is a convex and strictly increasing function, satisfying G(0) = 0.\nProblem 4.3 (Optimal Control Problem Pϕ.).Given the initial data ϕ∈ C, minimize over all admissible\ncontrols u∈ U\n•the cost functional (4.14) ,\n•under the ICU constraint (1.3) on the trajectory of (1.1).\nIn the sequel we refer to this formulation as problem Pϕ.\nWe first prove that a solution to the optimal control problem Pϕexists, for initial conditions in C.\nTheorem 4.4. For any initial condition ϕ∈ C, the optimal control problem Pϕadmits a solution.\nProof. Let us denote for simplicity I= (0,+∞). To prove the existence of an optimal solution we observe that\nit is equivalent to prove the existence of a minimizer of the functional J∞:L∞(I; [0, β−β⋆])×W1,∞(I;R2)→\n[0,+∞] defined by\nJ∞(u, s, i ) :=J(u) +χΛ(u, s, i ) +χi≤iM(i), (4.15)\nwhere Λ is the set of admissible pairs, that is all control-state vectors ( u, s, i ) that satisfy the initial value\nproblem for the state equation (1.1) with initial condition ϕ∈ C, while χΛdenotes the indicator function of\nΛ that takes the value 0 on Λ and + ∞otherwise; similarly, the function χi≤iM(i) is 0 if i(t)≤iMfor every\nt∈I, and + ∞otherwise.\nOn the domain of J∞we consider the topology given by the product of the weak* topologies of the spaces L∞\nandW1,∞, and aim to prove sequential lower semicontinuity and coercivity of the functional J∞with respect\nto this topology. By the Direct Method of the Calculus of Variations (see, for instance, Buttazzo [10, Sec. 1.2]),\nthese properties imply the existence of a solution to the minimum problem. They are direct consequences of\nthe fact that the space of controls is weakly* compact, that the assumptions on the integrand imply that the\ncost functional Jis weakly* lower semicontinuous (see, for instance, [16, Theorem 5.1]) and the fact that the\nsets Λ and {i≤iM}are closed with respect to the weak* convergence. The claimed closedness of such sets\nfollows by the application of Rellich compactness theorem, which ensures that weakly* converging sequences\ninW1,∞(I) are, up to subsequences, uniformly converging on every bounded subinterval of I(see for instance\n[9, Theorem 8.8 and Remark 10]). To prove it in details, let us consider a sequence ( un, sn, in)∈Λ weakly*\nconverging in L∞(I)×W1,∞(I,R2) to ( u, s, i ) and satisfying in≤iMfor every n∈N. Then, we easily get\nthatin(·−h) =iϕ1[0,h]+in1(h,+∞)weakly* converges to i(·−h) =iϕ1[0,h]+i1(h,+∞)inL∞(I). Then we can\npass to the limit in the state equations and, by uniqueness of the limit, we obtain that ( u, s, i )∈Λ. Moreover,\ni≤iMby the local uniform convergence. □\nThe performed viability analysis provides a route for designing a state-dependent control policy in order to\nminimize/bound the cost (4.14) and to fulfill the state constraint (1.3). Indeed, in Theorem 4.2 we have proven\nthat, for any initial condition ϕ∈ B, there exists at least a control action for which the corresponding solution\nsatisfies S(t)xϕ∈ B for all t∈R+. Rephrasing, we have proven that the so-called regulation map (see [3,\nDefinition 6.1.2])\nUB(ϕ) :={b∈[β⋆, β]|f(ϕ, b)∈DB(ϕ)},VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 15\nin non-empty for every ϕ∈ B.\nIn what follows, we explicitly select a (in general, sub-optimal) control b(ϕ)∈UB(ϕ) by minimizing the\nfunction Gin (4.14), i.e., we consider\nb(ϕ) = arg min\nb∈UB(ϕ)G(β−b) = max {b∈UB(ϕ)}.\nThis idea is formalized in the subsequent statement.\nTheorem 4.5 (greedy control policy) .LetΓβ⋆: [γ\nβ⋆,bsβ⋆]→[0, iM]be the curve defined in (4.7) andB⊂C\nbe the set defined in (4.8). Let us introduce the sets\nS1:= co\b\n(γ\nβ, iM),(γ\nβ⋆, iM)\t\n⊂∂B,\nS2:=\b\n(s, i)∈T|s∈[γ\nβ⋆,bsβ⋆]andi= Γβ⋆(s)\t\n⊂∂B,\nand consider the state-feedback control policy eb:B → [β⋆, β]given by\neb(ϕ) =\n\nβ, ifϕ(0)∈B\\(S1∪S2),\nmin\b\nβ,γiM\nsϕ(0)iϕ(−h)\t\n, ifϕ(0)∈S1,\nmin\b\nβ, β ⋆iM\niϕ(−h)\t\n, ifϕ(0)∈S2,(4.16)\nwith the convention 1/0 = +∞. Given the function eF:B →R2defined by eF(ϕ) =f(ϕ,eb(ϕ))(with fdefined\nin(2.1)), there exists a solution xϕ: [−h,+∞)→R2to\n(\nx′(t) =eF(S(t)x),\nS(0)x=ϕ∈ B,(4.17)\nwhere the operator S(t)has been introduced in (2.3).\nGiven any ϕ∈ B, we have that\n(1)xϕ(t)∈Bfor all t∈R+;\n(2) defining ebϕ:R+→[β⋆, β]by\nebϕ(t) :=eb(S(t)xϕ), (4.18)\nwithebdefined as in (4.16) , it holds that\n(a)xϕis the unique solution to the Cauchy problem (1.1)-(1.2) with b=ebϕ;\n(b)ebϕis eventually constant equal to β;\n(c) the corresponding cost (4.14) is finite.\nMoreover, if ϕ∈ B ∩ T 0(i.e., avoiding the trivial case of iϕidentically zero in [−h,0]), we also have\n(3) there exists a T=T(ϕ)≥0such that xϕ(t)∈R(see(4.8)) for all t≥T.\nRemark 4.6.\n(1) The control eb(ϕ) is well-defined for ϕsuch that ϕ(0)∈S1∩S2. Indeed, if ϕ(0)∈S1∩S2={(γ\nβ⋆, iM)}we\nhaveγiM\nsϕ(0)iϕ(−h)=β⋆iM\niϕ(−h).\n(2) The case in which eb(ϕ) =β⋆only occurs when ϕ(0)∈S2andiϕ(−h) =iM. Hence, the case in which\nebϕ(t) =β⋆for a.e. tin an interval Jonly occurs when ( sϕ(t), iϕ(t))∈S2andiϕ(t−h) =iMfor all\nt∈J. This scenario may happen only in the time interval [0 , h] and for a suitable initial condition. Then,\nin general, one cannot expect that a constant control regime ebϕ=β⋆occurs, except than in some very\nparticular cases.16 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nProof. First of all we observe that the feedback control policy eb, defined in (4.16), takes values in [ β⋆, β]. This\nis not trivial only in the case in which ϕ(0)∈S1. In this case we have sϕ(0)≤γ\nβ⋆and, recalling also that\nϕ∈ Bimplies iϕ(−h)≤iM, we have\nγiM\nsϕ(0)iϕ(−h)≥β⋆,\nthus proving that eb(ϕ)∈[β⋆, β], as claimed.\nAs a preliminary step to the proof of existence of a solution to the Cauchy problem (4.17), we prove that\neF(ϕ)∈ DB(ϕ)∀ϕ∈ B. (4.19)\nTo this aim we note that, by Lemma 2.5, we have DB(ϕ) =TB(ϕ(0)) for every ϕ∈ B. Since ϕ(0)∈Int(B)\nimplies DB(ϕ) =TB(ϕ(0)) = Rn, it suffices to consider the case ϕ(0)∈∂B. Given ϕ∈ B, it is convenient to\ndistinguish the following four cases.\n(a)ϕ(0)∈∂B\\(S1∪S2). Ifϕ(0) lies on the horizontal axis, the statement is trivial. In the case ϕ(0)∈A∩∂B,\nsinceAis forward invariant (see Theorem 4.2), Corollary 2.9 implies that eF(ϕ) =f(ϕ, β)∈DA(ϕ) which,\nin turn, is contained in DB(ϕ). Therefore, (4.19) is proved in this case.\n(b)ϕ(0)∈S1\\S2. In this case we have sϕ(0)∈[γ\nβ,γ\nβ⋆) and iϕ(0) = iM. At the point ϕ(0), we compute the\nscalar product of the vector field with the (unique, modulo positive scalar multiplication) normal vector\nv= (0,1) to Batϕ(0), so obtaining\n⟨v, f(ϕ,eb(ϕ))⟩=eb(ϕ)sϕ(0)iϕ(−h)−γiM=eb(ϕ)sϕ(0)iϕ(−h)−γiM. (4.20)\nLet us suppose, first, that eb(ϕ) = minn\nβ,γiM\nsϕ(0)iϕ(−h)o\n=γiM\nsϕ(0)iϕ(−h). Then we have\neb(ϕ)sϕ(0)iϕ(−h)−γiM=γiM\nsϕ(0)iϕ(−h)sϕ(0)iϕ(−h)−γiM= 0,\nproving that the vector field is tangent to the line {(s, i)∈R2|i=iM}. If, on the contrary, we have\neb(ϕ) =β, this implies that β≤γiM\nsϕ(0)iϕ(−h)and thus\neb(ϕ)sϕ(0)iϕ(−h)−γiM=βsϕ(0)iϕ(−h)−γiM≤0,\nand the equality holds only if β=γiM\nsϕ(0)iϕ(−h). Then, we have proven that f(ϕ,eb(ϕ))∈TB(ϕ(0)) = DB(ϕ),\nthat is (4.19).\n(c)ϕ(0)∈S2\\S1. In this case sϕ(0)∈(γ\nβ⋆,bsβ⋆] and iϕ(0) = Γ β⋆(sϕ(0)). At the point ( sϕ(0), iϕ(0)) the unique\n(modulo positive scalar multiplication) normal to the set Bis a vector v= (v1, v2) for some v1, v2≥0\nsuch that\n(v2−v1)β⋆sϕ(0)iM−v2γiϕ(0) = 0 ,\nby definition of Γ β⋆. Suppose first that eb(ϕ) =β⋆iM\niϕ(−h), i.e., β≥β⋆iM\niϕ(−h). By the previous equality, we\nhave\n⟨v, f(ϕ,eb(ϕ))⟩= (v2−v1)β⋆iM\niϕ(−h)sϕ(0)iϕ(−h)−v2γiϕ(0) = 0 .\nSuppose now that eb(ϕ) =β, i.e., β≤β⋆iM\niϕ(−h). Then\n⟨v, f(ϕ,eb(ϕ))⟩= (v2−v1)βsϕ(0)iϕ(−h)−v2γiϕ(0)≤(v2−v1)β⋆sϕ(0)iM−v2γiϕ(0) = 0 ,\nand, again, the equality holds only if and only if β=β⋆iM\niϕ(−h). This proves that f(ϕ,eb(ϕ))∈TB(ϕ(0) =\nDB(ϕ), that is (4.19) also in the current case.\n(d)ϕ(0)∈S1∩S2. In this case, ϕ(0) = (γ\nβ⋆, iM) and the claim follows by items (b) and (c), through a\ncontinuity argument.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 17\nWe have thus proven (4.19). Under this condition, the existence of solutions xϕto (4.17) is proved in\nLemma A.1 in Appendix, together with the property S(t)xϕ∈ B, for all t∈R+, which implies S(t)xϕ(0) =\nxϕ(t)∈B, for all t∈R+. Then, part (1)of the statement is proved.\nTo prove (2)(a) , it is enough to observe that\neF(S(t)xϕ) =f(S(t)xϕ,ebϕ(t)) = (−ebϕ(t)sϕ(t)iϕ(t−h),ebϕ(t)sϕ(t)iϕ(t−h)−γiϕ(t)),\nwhich means that the solution xϕto (4.17) is also a solution to (1.1)-(1.2) with b=ebϕ, and such Cauchy\nproblem has a unique solution by Lemma 2.1. Of course, this implies (a posteriori) that also (4.17) has a\nunique solution.\nLet us now prove (2)(b) . In Lemma 3.1 we have proven that lim t→+∞iϕ(t) = 0 and sϕ,∞= lim t→+∞sϕ(t)∈\n[0,γ\nβ⋆). Then, there exists a T1(ϕ)≥0 such that xϕ(t)/∈S1∪S2for all t≥T1(ϕ). By (4.16) and (4.18), this\nimplies that bϕ(t) =b(S(t)xϕ) =βfor all t≥T1(ϕ), since S(t)xϕ(0) = xϕ(t)/∈S1∪S2. This proves (2)(b) .\n(2)(c) holds since\nZ+∞\n0G(β−ebϕ(t))dt=ZT1(ϕ)\n0G(β−ebϕ(t))dt≤ZT1(ϕ)\n0G(β−β⋆)dt=T1(ϕ)G(β−β⋆).\nTo prove part (3), suppose ϕ∈ B ∩ T 0. Since we have proven that bϕis eventually equal to β, by part (2)\nof Lemma 3.1 we have that sϕ,∞∈[0,γ\nβ), which in turns implies that there exists a T=T(ϕ)≥0 such that\nsϕ(t)≤γ\nβ(and thus xϕ(t)∈R) for all t≥T, and the proposition is completely proved. □\nA first direct consequence of the existence of the “greedy” feedback control policy defined in (4.16) is stated\nbelow.\nCorollary 4.7. For any prescribed initial condition ϕ∈ B, the optimal control problem Pϕadmits a solution\nwith a finite cost.\nProof. The proof directly follows by Theorem 4.4 and assertion (3)of Theorem 4.5. □\nRemark 4.8.We note that the invariance/viability regions in Theorem 4.2 are independent of h. In particular,\nthey do not converge, as h→0, to the regions obtained in the delay-free case (see [15, 4, 16]), recalled also in\nthe subsequent Theorem 4.13. In the next subsection we strenghten Assumption 4.1 by bounding the velocity\nof the past evolution of the epidemic. This allows us to obtain a viability analysis and a control policy which do\ndepend on the parameter h >0, and that converge, in a sense that will be clarified, to the solutions provided\nfor the delay-free case.\n4.2.Lipschitz continuous initial conditions. In the previous subsection, we supposed to have limited\ninformation on the past evolution of the epidemic, only considering Assumption 4.1. We hereafter assume\nthat, in the past htime units, the epidemic dynamic not only was under the warning level (i.e. iϕ(t)≤iM\nfor all t∈[−h,0]), but also was evolving with a limited “speed”, compatible with the epidemic model. More\nformally we assume the following.\nAssumption 4.9. Given a threshold 0< iM≤1consider any L∈Rsuch that\nL≥iMmax{β, γ}>0. (4.21)\nWe assume that the initial condition ϕ∈ Cof(1.1) satisfies\nϕ(t)∈C∀t∈[−h,0],\nwith C={(s, i)∈T|i≤iM}, and\nϕ∈ CL,|·|max={φ∈ C | | φ(t1)−φ(t2)|max≤L|t1−t2| ∀t1, t2∈[−h,0]},\nwhere |x|max:= max {|x1|,|x2|}, for any x= (x1, x2)∈R2.18 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nRemark 4.10.The choice of the infinity/max norm is motivated by the fact that the dynamics in (1.1) are\nonly affected by the delayed value of the i-component ( i(−h)), while the delayed s-component ( s(−h)) does\nnot play any role. The max norm, which, intuitively, is only affected by the “worst case” of any component\n(considered separately), is thus a natural choice. Nevertheless, the subsequent analysis can be adapted to the\nchoice of any other norm in R2.\nRemark 4.11.The lower bound (4.21) on the Lipschitz constant L >0 is motivated by noting that, for any\nx= (s, i)∈Csuch that i≤iMwe have\n| −βsi| ≤βiMand|βsi−γi| ≤iMmax{β, γ}.\nThus,\n|f(ϕ, b)|max≤L∀ϕ∈ C,∀b∈U, (4.22)\nand, hence, L≥iMmax{β, γ}represents a uniform upper bound to the speed modulus of the solution to (1.1)\nwith initial condition ϕ∈ Csuch that ϕ(s)∈C, for all s∈[−h,0]. Intuitively, in Assumption 4.9, we are\nsupposing that the epidemic, in the past uncontrolled interval of time [ −h,0], has evolved at a bounded speed,\nand this bound is assumed to be not smaller than the one holding forward in time, according with the model.\nIn order to retrace the analysis performed in Subsection 4.1, we need to introduce auxiliary (non-delayed)\nsystems that mimic the “worst-case” behaviour of the delay system (1.1). Since the considered initial conditions\nare supposed to satisfy Assumption 4.9, we first define a scalar function that models the maximal gap between\ni(0) and i(−h). Namely, for any L, h > 0, we consider the Lipschitz continuous function ψL,h:R→R(see\nFigure 2) defined by\nψL,h(i) :=\n\n−iM, ifi≤ −iM−Lh,\ni+Lh, if−iM−Lh < i ≤iM−Lh,\niM, ifiM−Lh < i.(4.23)\niM\niMiM−LhiψL,h(i)\nFigure 2. The truncating function ψL,hdefined in (4.23).\nLet us now consider the non-linear differential equation(\ns′(t) = βs(t)ψL,h(i(t)),\ni′(t) =−βs(t)ψL,h(i(t)) +γi(t).(4.24)\nThe right-hand side in (4.24) is locally Lipschitz, and since |ψL,h(i)| ≤iMfor all i∈R, we also have that\n|(βsψL,h(i),−βsψL,h(i) +γi)|max≤A|(s, i)|max,∀(s, i)∈R2, (4.25)\nfor a suitable A > 0. This implies that, for any prescribed initial condition, the solution to (4.24) is unique\nand globally defined, see [37, Theorem 2.17]. We thus denote by Ψ β,L,h = (¯sβ,L,h,¯ıβ,L,h) :R+→R2the\nsolution to (4.24) corresponding to the initial condition ( s(0), i(0)) = (γ\nβ, iM). We define\nTβ,L,h := sup {¯t≥0|¯ıβ,L,h(t)>0∀t∈[0,¯t)}.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 19\nLet us denote bsβ,L,h := limt→T−\nβ,L,h¯sβ,L,h(t). Since the solutions to (4.24) satisfy s(t) =s0eβRt\n0ψL,h(i(τ))dτ\nandψL,h(i)>0 for any i∈R+, we have that ¯ sβ,L,h is strictly positive and strictly increasing in [0 , Tβ,L,h).\nThus, we can define Γ β,L,h : [γ\nβ,bsβ,L,h)→R+as the function representing the curve Ψ β,L,h in [0, Tβ,L,h) as a\ngraph i(s) = Γ β,L,h(s). In Lemma A.2 in Appendix we show that Γ β,L,h is strictly decreasing and concave in\n[γ\nβ,bsβ,L,h). Since lims→bs−\nβ,L,hΓβ,L,h(s)≥0, we have that bsβ,L,h is finite. In turn, this implies that Tβ,L,h <+∞.\nIndeed, suppose by contradiction that Tβ,L,h = +∞; then ¯ s′\nβ,L,h(t) =β¯sβ,L,h(t)ψL,h(¯ıβ,L,h(t))≥γLh for all\nt∈(0, Tβ,L,h) = (0 ,+∞), in contradiction with bsβ,L,h = limt→T−\nβ,L,h¯sβ,L,h(t)<+∞. Moreover, recalling that\n¯sβ,L,h(t) is continuous and strictly increasing, Tβ,L,h <+∞implies that\nlim\ns→bs−\nβ,L,hΓβ,L,h(s) = lim\nt→T−\nβ,L,hΓβ,L,h(¯sβ,L,h(t)) = lim\nt→T−\nβ,L,h¯ıβ,L,h(t) = 0 ,\nwhere the last equality follows by definition of Tβ,L,h. We can thus extend Γ β,L,h on the closed interval\n[γ\nβ,bsβ,L,h] by setting Γ β,L,h(bsβ,L,h) = 0. See Figure 3 for a graphical representation of such curve.\nSimilarly, denote by Ψ β⋆,L,h= (¯sβ⋆,L,h,¯ıβ⋆,L,h) :R+→R2the solution to the differential equation\n(\ns′(t) = β⋆s(t)ψL,h(i(t))\ni′(t) =−β⋆s(t)ψL,h(i(t)) +γi(t)(4.26)\nwith initial condition ( s(0), i(0)) = (γ\nβ⋆, iM). We consider Tβ⋆,L,h:= sup {¯t≥0|¯ıβ⋆,L,h(t)>0∀t∈[0,¯t)},\ndefinebsβ⋆,L,h:= limt→T−\nβ⋆,L,h¯sβ⋆,L,h(t), and represent the curve Ψ β⋆,L,h: [0, Tβ⋆,L,h)→R2as a graph i(s) =\nΓβ⋆,L,h(s), with Γ β⋆,L,h: [γ\nβ⋆,bsβ⋆,L,h)→R+, see Figure 3. In Lemma A.2, it is proved that Γ β⋆,L,his concave\nand strictly decreasing, also implying that bsβ⋆,L,his finite. Arguing as before, we can thus extend Γ β⋆,L,hby\ncontinuity on [γ\nβ⋆,bsβ⋆,L,h] by setting Γ β⋆,L,h(bsβ⋆,L,h) = 0.\nWe can now provide the viability result.\nTheorem 4.12. Consider the sets R, R ⋆⊂Cdefined in (4.8), and define the sets AL,h, BL,h⊂Cby\nAL,h:=R∪n\n(s, i)∈T|s∈[γ\nβ,bsβ,L,h], i≤Γβ,L,h(s)o\n,\nBL,h:=R⋆∪n\n(s, i)∈T|s∈[γ\nβ⋆,bsβ⋆,L,h], i≤Γβ⋆,L,h(s)o\n.(4.27)\nThe following propositions hold.\n(1)AL,his the maximal forward invariant set with CL,|·|max-past in C.\n(2)BL,his the maximal viable set with CL,|·|max-past in C.\nProof. The idea behind the proof is essentially the same of the proof of Theorem 4.2, and it relies on Lemma 2.5\nand Corollary 2.9. We start by proving that AL,h=\b\nϕ∈ CL,|·|max|ϕ(0)∈AL,h, ϕ(s)∈C∀s∈[−h,0]\t\nis\nforward invariant. As always, it is enough to consider ϕ∈ A L,hsuch that ϕ(0)∈∂AL,hand prove that\nFU(ϕ)⊆TAL,h(ϕ(0))∩B|·|max(0, L). By (4.22), it sufficies to prove that\nFU(ϕ)⊆TAL,h(ϕ(0)). (4.28)\nLet us consider two cases.\n(1) Given the set D={(s, i)∈T|s≤γ\nβ}, we note that A∩D=AL,h∩D. Hence, in Dthe analysis\nperformed for Ain Theorem 4.2 applies also to AL,h. Therefore, the inclusion (4.28) holds true in the\ncaseϕ(0)∈∂AL,h∩D.\n(2) In the remaining case, in which ϕ(0) = ( s0, i0) with s0∈(γ\nβ,bsβ,L,h] and i0= Γ β,L,h(s0), we denote the\nunique normal vector (up to positive scalar multiplication) to AL,hat (s0, i0) byv= (v1, v2)∈R2\\{0}. It20 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nis, by definition, perpendicular to the tangent vector (1 ,Γ′\nβ,L,h(s0)). Since Γ′\nβ,L,h(s0)<0 (by Lemma A.2),\nwe have that v1≥0, v2≥0, and (by normality)\n0 =v1βs0ψL,h(i0)−v2βs0ψL,h(i0) +v2γi0=−(v2−v1)βs0ψL,h(i0) +v2γi0, (4.29)\nalso proving that v2≥v1. We note that, by Assumption 4.9, we have iϕ(−h)≤min{iM, i0+Lh}=\nψL,h(i0). Considering any b∈[β⋆, β] and using (4.29), we obtain\n⟨v, f(ϕ, b)⟩= (v2−v1)bs0iϕ(−h)−v2γi0≤(v2−v1)βs0ψL,h(i0)−v2γi0= 0,\nwhich proves (4.28) also in this case.\nThis concludes the proof of forward invariance of AL,h. The proofs of maximality of AL,hand assertion (2)\nfollow by arguments similar to the ones provided in the proof of Theorem 4.2, and are left to the reader. □\nDifferently from the viability analysis performed in Theorem 4.2, the regions defined in Theorem 4.12 do\ndepend on the delay parameter h >0. We will show that they converge, in a sense we are going to clarify,\nto the invariance/viability regions for the delay-free case, as h→0. To this aim, let us summarize here the\nmain viability and invariance results obtained in [4, Section 2] for the delay-free system of ordinary differential\nequations(\ns′(t) =−b(t)s(t)i(t),\ni′(t) =b(t)s(t)i(t)−γi(t).(4.30)\nTheorem 4.13 (Theorem 2.3, [4]) .The following propositions hold true for the delay-free system (4.30) .\n(1) The maximal forward invariant set contained in C(the all-control zone Ain[4]) , is given by\nA0:={x= (s, i)∈C|s≤Γβ,0(s)},\nwhere\nΓβ,0(s) =(\niM, if0≤s≤γ\nβ,\nγ\nβ+iM−s+γ\nβlog(β\nγs), ifs≥γ\nβ.(4.31)\n(2) The maximal viable set contained in C(the feasible set Bin[4]), is given by\nB0:={x= (s, i)∈C|s≤Γβ⋆,0(s)},\nwhere\nΓβ⋆,0(s) =(\niM, if0≤s≤γ\nβ⋆,\nγ\nβ⋆+iM−s+γ\nβ⋆log(β⋆\nγs), ifs≥γ\nβ⋆.\nRemark 4.14.We have chosen to denote the maximal sets by A0andB0instead than AandBas in [4] to\nstress the fact that these sets corresponds to the delay h= 0.\nWe now study and characterize the dependence of the sets AL,handBL,hon the parameter h >0, and their\nrelations with the sets A, B defined in Subsection 4.1 for an arbitary continuous initial condition, and the sets\nA0, B0corresponding to the delay-free case. To these aims, we introduce the Cauchy problems\n(S0) :(\nx′(t) =g(x(t)),\nx(0) = (γ\nβ, iM),(4.32a)\n(SL,h) :(\nx′(t) =gL,h(x(t)),\nx(0) = (γ\nβ, iM),(4.32b)\nwith x= (s, i),g(x) := ( βsi,−βsi+γi) and gL,h(x) := ( βsψL,h(i),−βsψL,h(i)+γi). We note that the system\nin (4.32b) has been already considered in (4.24), and the function Γ β,L,h, introduced immediately after, is the\ngraph, in the ( s, i)-plane, of the solution to SL,h. Similarly, for s≥γ\nβ, the function Γ β,0appearing in (4.31)\nis the graph of the solution to the delay-free problem (4.30) with b(t) =β, as well as to its time-reversed\nversion S0.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 21\nS1\nS2S2,L,hΓβ⋆,0 Γβ,0\nγ\nβ /hatwidesβ/hatwidesβ,L,h /hatwidesβ,0/hatwidesβ⋆/hatwidesβ⋆,L,h /hatwidesβ⋆,0iM\niM−Lh\nγ\nβ⋆\nFigure 3. The figure shows qualitatite graphs curves introduced in Theo-\nrem 4.12, Lemma 4.17 and Proposition 4.18. In orange we plot the graphs of the\nfunctions Γ β,L,h and Γ β⋆,L,h. In black the curves Γ βand Γ β⋆corresponding to the case\nstudied in Subsection 4.1, while in blue the curves Γ β,0and Γ β⋆,0corresponding to the\n(invariant and viable) sets in the delay-free case.\nAs a preliminary step, we study the asymptotic behaviour of the solution to S0.\nLemma 4.15. The Cauchy problem S0admits a unique solution x(t) = ( s(t), i(t)),t∈R+. Moreover, it\nsatisfies the following properties:\n(1)sandiare strictly positive;\n(2)sis strictly increasing, and iis non increasing;\n(3)limt→+∞s(t) =:bsβ,0<+∞andlimt→+∞i(t) = 0 ;\n(4)bsβ,0is equal to the unique element of [γ\nβ,+∞)such that Γβ,0(bsβ,0) = 0 .\nProof. Since gis a smooth function, then it is locally Lipschitz and there exists a unique solution defined on\na maximal interval [0 , τ),τ >0.\n(1)By integration we have s(t) =γ\nβeRt\n0βi(ξ)dξ>0 and i(t) =iMeRt\n0(−βs(ξ)+γ)dξ>0, for all t∈[0, τ).\n(2)Since s′(t) =βs(t)i(t)>0 for all t∈[0, τ), we have that sis strictly increasing. Similarly, i′(t) =\n−βs(t)i(t) +γi(t)≤ −βγ\nβi(t) +γi(t)≤0, proving that iis non-increasing.\nAs a consequence, we have\f\fg\u0000\ns(t), i(t)\u0001\f\f≤βiM|s(t)|+γiM\nfor every t∈[0, τ), which implies that τ=∞(otherwise, the solution could be extended on a right neighbor-\nhood of τ) and the solution exists on [0 ,+∞) as claimed in the first part of the statement.\n(3)By summing the equations, integrating in [0 , t] and substituting the function i(t) under the integral with\nits expression obtained by the first equation, we get\ni(t) =γ\nβ+iM−s(t) +γ\nβlog(β\nγs(t))∀t∈R+. (4.33)\nThe limits in (3)exist by monotonicity. If, by contradiction, bsβ,0= +∞then the previous equation\nwould imply lim t→+∞i(t) =−∞, leading to a contradiction with (1). Assuming, by contradiction, that22 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nlimt→+∞i(t) =:bıβ,0>0, we have that there exists ¯t >0 such that\ns′(t)≥βγ\nβbıβ,0\n2=γbıβ,0\n2>0,∀t≥¯t,\ncontradicting bsβ,0<+∞.\n(4)By taking the limit as t→+∞in (4.33), we get 0 =γ\nβ+iM−bsβ,0+γ\nβlog(β\nγbsβ,0), that is, Γ β,0(bsβ,0) = 0\n(see (4.31)). The uniqueness simply follows by the strict monotonicity of Γ β,0in the interval [γ\nβ,∞). □\nRemark 4.16.Also the Cauchy problem SL,hadmits a unique solution xL,h(t) = ( sL,h(t), iL,h(t)),t∈R+.\nMoreover, sL,his strictly positive but (it is easy to prove that) iL,hmust take also negative values.\nWe are now able to prove the aforementioned convergence result.\nTheorem 4.17. Let us consider h >0,h′>0,L > 0and the sets defined in Theorem 4.2, Theorem 4.12\nand Lemma 4.13. The following propositions hold.\n(1) If Lh≥iM, then A=AL,handB=BL,h.\n(2)A⊆AL,handB⊆BL,h.\n(3) If h≥h′, then AL,h⊆AL,h′andBL,h⊆BL,h′.\n(4)S\nh>0AL,h=A0, andS\nh>0BL,h=B0.\nProof. Proposition (1)follows by the fact that, if Lh≥iM, then ψL,h(i) =iMfor all i≥0 and thus the\ncurves defining A, A L,handB, B L,hcoincide, i.e., Γ β≡Γβ,L,h and Γ β⋆≡Γβ⋆,L,hin their respective domains.\nAbout (2), we recall from Subsection 4.1 that Γ βis the restriction to [γ\nβ,bsβ] of the solution to the scalar\ndifferential equation i′(s) =−1 +γi(s)\nβsiM, with initial condition i(γ\nβ) =iM, see (4.6). Similarly, by dividing the\nequations in (4.24), Γ β,L,h turns out to be the solution to\ni′(s) =−1 +γi(s)\nβsψL,h(i(s))(4.34)\nwith the same initial condition. Since ψL,h(i)≤iMfor all i∈R, by the Comparison Lemma (see for\nexample [37, Lemma 1.2]) we have\nΓβ(s)≤Γβ,L,h(s)∀s∈[γ\nβ,bsβ],\nproving that A⊆AL,h. The proof of the inclusion B⊆BL,hfollows the same argument.\nProposition (3)follows again by a similar comparison argument, noting that, if h′≤h, then ψL,h′(i)≤ψL,h(i)\nfor all i∈R.\nTo prove (4), we first show thatS\nh>0AL,h⊆A0. Let us start by recalling that Γ β,0(s), for s∈[γ\nβ,bsβ,0], is\nthe graph of the solution to the delay-free problem (4.30) with b≡β. Hence, it is the solution in [γ\nβ,bsβ,0] to\nthe scalar differential equation\ni′(s) =−1 +γ\nβs\nwith initial condition i(γ\nβ) =iM. On the other hand, we have already seen that Γ β,L,h, in [γ\nβ,bsβ,L,h], is the\nsolution to (4.34) with the same initial condition i(γ\nβ) =iM. Sincei\nψL,h(i)≤1 for all i∈[0, iM] and all h≥0,\nwe have, again by a comparison argument, that Γ β,L,h(s)≤Γβ,0(s) for all h >0 and for all s∈[γ\nβ,bsβ,L,h].\nThis in particular yields AL,h⊂A0for all h >0, implyingS\nh>0AL,h⊆A0.\nSince A0=int(A0), to prove the converse inclusion it is enough to show that\nint(A0)⊆[\nh>0AL,h. (4.35)VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 23\nTo this aim, it is useful to recall that, as s≥γ\nβ, the curves Γ β,0and Γ β,L,h, defining the (boundaries of)\nthe sets A0andAL,h, are the graphs, in the ( s, i)-plane, of the solutions to the problems ( S0) and ( SL,h),\nrespectively. Thus, as a preliminary step, we prove that the solutions to ( SL,h) defined in (4.32b) pointwise\nconverge, as h→0, to the solution to ( S0) defined in (4.32a), by verifying the hypotheses of Lemma A.3. Let\nus denote by ¯ x= (¯s,¯ı) :R+→R2the solution to ( S0) and by xL,h= (sL,h, iL,h) :R+→R2the solution to\n(SL,h). It can be proved that the vector fields gL,h:R2→R2are uniformly locally Lipschitz, i.e., for any\ncompact set K⊂Rnthere exists a MK≥0 such that\n|gL,h(x1)−gL,h(x2)|max≤MK|x1−x2|max,∀h >0,∀x1, x2∈K, (4.36)\nby using the inequality |ψL,h(i2)−ψL,h(i1)| ≤min{2iM,|i2−i1|}for all i1, i2∈Rand all h > 0. By\nLemma 4.15, there exists a compact set Q⊂[γ\nβ,+∞)×[0, iM] such that that ¯ x(t)∈Q, for all t∈R+. We also\nnote that ψL,hconverges uniformly to the identity Id( i) =iin [0, iM] ashgoes to 0. Thus, gL,his converging\nuniformly to ginQ. Recalling the sublinear bound in (4.25), all the hypotheses of Lemma A.3 are verified.\nWe can thus conclude that lim\nh→0xL,h(t) = ¯x(t), for all t∈R+.\nLet us now consider any point z= (sz, iz)∈int(A0). Ifz∈A, then (4.35) trivially follows by point (2). Let\nus thus suppose that z∈int(A0)\\A, i.e.,γ\nβ< sz<bsβ,0and max {0,Γβ(sz)}< iz<Γβ,0(sz). By Lemma 4.15,\nsince sz<bsβ,0there exists Tz>0 such that ¯ s(Tz)> sz. By Lemma 4.15 we also have ¯ ı(Tz)>0. By pointwise\nconvergence of xL,hto ¯x, there exists h >0 small enough such that sL,h(Tz)> szfor all h≤h. Thus, by\ncontinuity of sL,h, for any h≤hwe can choose τh∈(0, Tz) such that sL,h(τh) =sz. In (0 ,h), the function\nh7→τhis bounded from below by 0 and bounded from above by Tz, and thus there exists a sequence hn→0+\nsuch that lim\nn→+∞τhn= ¯τfor some ¯ τ∈[0, Tz]. By Lemma A.3 we have that, up to a subsequence, xL,hn→x\nuniformly in [0 , Tz], asn→+∞. We thus have:\nlim\nn→∞Γβ,L,h n(sz) = lim\nn→∞Γβ,L,h n(sL,hn(τhn)) = lim\nn→+∞iL,hn(τhn) = ¯ı(¯τ) = Γ β,0(sz).\nSince iz<Γβ,0(sz) this implies that there exists hz>0 such that\niz≤Γβ,L,h z(sz)<Γβ,0(sz),\nimplying that z∈AL,hz⊂S\nh>0AL,h. We have thus proved that int( A0)⊆S\nh>0AL,h, as claimed.\nThe argument for BL,handB0is similar. □\nIn Figure 3 the results of Lemma 4.17 can be visualized in a particular case.\nAs in the case studied in Subsection 4.1, Theorem 4.12 provides a tool in designing a feedback control policy,\nas detailed in what follows.\nTheorem 4.18 (greedy control policy for Lipschitz continuous initial conditions) .Consider the graph Γβ⋆,L,h:\n[γ\nβ⋆,bsβ⋆,L,h]→[0, iM]representing the solution of (4.26) in the plane (s, i), the set BL,h⊂Cdefined in\nTheorem 4.12, and the set of functions\nBL,h:=\b\nϕ∈ CL,|·|max|ϕ(0)∈BL,h, ϕ(s)∈C∀s∈[−h,0]\t\n.\nConsider the set S1⊂R2defined in Theorem 4.5 and the set S2,L,hdefined by\nS2,L,h:=\b\n(s, i)∈T|s∈[γ\nβ⋆,bsβ⋆,L,h]andi= Γβ⋆,L,h(s)\t\n.\nDefine the state-feedback control policy bL,h:BL,h→[β⋆, β]by\nbL,h(ϕ) =\n\nβ, ifϕ(0)∈BL,h\\(S1∪S2,L,h),\nmin\b\nβ,γiM\nsϕ(0)iϕ(−h)\t\n, ifϕ(0)∈S1,\nmin\b\nβ, β⋆ψL,h(iϕ(0))\niϕ(−h)\t\n, ifϕ(0)∈S2,L,h,(4.37)24 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nwith the convention 1/0 = + ∞. Given the function FL,h:BL,h→R2defined by FL,h(ϕ) =f(ϕ, bL,h(ϕ))\n(see(2.1)), there exists a solution xϕ: [−h,+∞)→R2to the Cauchy problem\n(\nx′(t) =FL,h(S(t)x),\nS(0)x=ϕ∈ BL,h.(4.38)\nGiven any ϕ∈ BL,h, we have that\n(1)xϕ(t)∈BL,hfor all t∈R+.\n(2) Defining bL,h,ϕ :R+→[β⋆, β]by\nbL,h,ϕ(t) :=bL,h(S(t)xϕ), (4.39)\nit holds that\n(a)xϕis the unique solution to the Cauchy problem (1.1)-(1.2) with b=bL,h,ϕ;\n(b)bL,h,ϕ is eventually constant equal to β;\n(c) the corresponding cost (4.14) is finite.\nMoreover, if ϕ∈ BL,h∩ T0(i.e. avoiding the trivial case of iϕidentically zero in [−h,0]), then\n(3) there exists a T=T(ϕ)≥0such that xϕ(t)∈Rfor all t≥T.\nProof. First of all we observe that the feedback control policy bL,h, defined in (4.37), takes values in [ β⋆, β].\nThis is trivial if ϕ(0)∈BL,h\\(S1∪S2,L,h). If ϕ(0)∈S1we have sϕ(0)≤γ\nβ⋆and, since ϕ∈ B implies\niϕ(−h)≤iM, we obtain\nγiM\nsϕ(0)iϕ(−h)≥β⋆,\nthus proving that bL,h(ϕ)∈[β⋆, β], as claimed. In the remaining case ϕ(0)∈S2,L,h, the claim follows by\nobserving that ϕ∈ BL,himplies ψL,h(iϕ(0))≥iϕ(−h)≥0.\nThe rest of the proof relies on arguments similar to the ones of Theorem 4.5, using the viability analysis\nperformed in Theorem 4.12.\nA preliminary step to the proof of existence of a solution to the Cauchy problem (4.38), consists in proving\nthat\nFL,h(ϕ)∈ DBL,h(ϕ) =TBL,h(ϕ(0))∩B|·|max(0, L)∀ϕ∈ BL,h. (4.40)\nSince, by (4.22), we have FL,h(ϕ) =f(ϕ, bL,h(ϕ))∈B|·|max(0, L), then we have just to check that\nf(ϕ, bL,h(ϕ))∈TBL,h(ϕ(0)) ∀ϕ∈ BL,h. (4.41)\nLet us then consider an initial condition ϕ∈ BL,h. Given the control policy eb:B → [β⋆, β] defined in (4.16), we\nnote that bL,h(ϕ) =eb(ϕ) whenever sϕ(0)≤γ\nβ⋆. Since BL,h∩ {x|s≤γ\nβ⋆}=B∩ {x|s≤γ\nβ⋆}, by Theorem 4.5,\nin this case we directly have that\nf(ϕ, bL,h(ϕ)) =f(ϕ, b(ϕ))∈TB(ϕ(0)) = TBL,h(ϕ(0)).\nSince the case ϕ(0)∈Int(BL,h) is trivial, it only remains to check condition (4.41) when ϕ(0)∈S2,L,h.\nIn this case we have sϕ(0)∈[γ\nβ⋆,bsβ⋆,L,h] and iϕ(0) = Γ β⋆,L,h(sϕ(0)). The unique (modulo positive scalar\nmultiplication) normal vector to the set BL,hatϕ(0) is a vector v= (v1, v2) for some v1, v2≥0 such that\n(v2−v1)β⋆sϕ(0)ψL,h(iϕ(0))−v2γiϕ(0) = 0 , (4.42)\nalso proving that v2≥v1. When bL,h(ϕ) =β⋆ψL,h(iϕ(0))\niϕ(−h), we have\n⟨v, f(ϕ, bL,h(ϕ))⟩= (v2−v1)β⋆ψL,h(iϕ(0))\niϕ(−h)sϕ(0)iϕ(−h)−v2γiϕ(0) = 0 ,VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 25\nby (4.42). The remaining case bL,h(ϕ) =βholds when β≤β⋆ψL,h(iϕ(0))\niϕ(−h), and we have\n⟨v, f(ϕ, bL,h(ϕ))⟩= (v2−v1)βsϕ(0)iϕ(−h)−v2γiϕ(0)≤(v2−v1)β⋆sϕ(0)ψL,h(iϕ(0))−v2γiϕ(0) = 0 .\nThis concludes the proof of (4.41) and, hence, of (4.40).\nUnder this condition, the existence of solutions xϕto (4.38) is proved in Lemma A.1 in Appendix, together\nwith the property S(t)xϕ∈ BL,h, for all t∈R+, which implies S(t)xϕ(0) = xϕ(t)∈BL,h, for all t∈R+.\nThen, part (1)of the statement is proved.\nThe proof of the remaining part of the statement proceeds exactly as done in the proof of Theorem 4.5, just\nunder the necessary notational adaptation. □\nA direct consequence of Proposition 4.18 in terms of the optimal control problem Pϕdefined in Problem 4.3\nis stated below.\nCorollary 4.19. Given any h >0and any L >0, the optimal control problem Pϕadmits a solution with a\nfinite cost, for any initial condition ϕ∈ BL,h.\n5.Examples and numerical simulations\nIn the sequel, with the aid of numerical examples, we illustrate the greedy control schemes introduced in the\nprevious section (Theorem 4.5 and Theorem 4.18).\nExample 5.1. The epidemic parameters we are choosing in this example are compatible with the state\nof knowledge of the COVID-19 epidemic in Italy in autumn 2021 (see [16], Example 4.3). Actually, we\nspecialize (1.1) by choosing γ= 0.0714, β= 0.5,iM= 0.021, β⋆= 0.186. An appropriate latency period\nwould be between 4 and 6 ([28, 30, 40]). We consider here a delay h= 6, which is large enough to appreciate\nthe effect of the delay in numerical computations.\nWe start by considering a constant initial condition ϕ0(t)≡x0:= (s0, i0) := (0 .45,0.001) for all t∈[−h,0].\nThis choice represents the case in which there is an isolated-in-time spreading event at time −h; then, nothing\nhappens ( sandistay constant) till time 0 after which the infection can be transmitted.\nSince hmax{β, γ} ≥1, for every L >0 satisfying (4.21) we have Lh≥iM. By Theorem 4.17 (1), this implies\nthat the control strategy described in Subsection 4.2 is equivalent to the one introduced in Subsection 4.1.\nRoughly speaking, the delay parameter h= 6 is large enough to prevent any gain in imposing the Lipschitz\ncondition in Assumption 4.9. We thus consider the “greedy” control policy ebϕ0described in Theorem 4.5.\nThis is possible, since it can be verified that ϕ0belongs to viable set B(see Theorem 4.2).\nThe numerical computation of the greedy control uϕ0(t) :=β−ebϕ0(t) and the corresponding solution xϕ0=\n(sϕ0, iϕ0) are performed in Matlab and plotted in red in Figure 4. In the same figure, we compare the\nperformance of this control strategy with the “optimal” one numerically obtained by the algorithmic optimal\ncontrol toolbox Bocop ([36]), and denoted by ubcp. The latter is obtained by choosing the Gauss’s II method\nand a discretization detail of 10 time steps per day. It can be observed that the two control actions appear to\nbe close one to the other. On the other hand, it can be noted that uϕ0starts its action earlier with respect to\nubcp, as illustrated in zoomed detail inside the small bottom-right box of Figure 4. Moreover, differently from\nthe optimal one ubcp, the control strategy uϕ0does not provide any time interval at the maximum regime β⋆\n(corresponding to a lockdown policy), according to what already expected in Remark 4.6 (2).\nThe suboptimality of uϕ0is also highlighted by the numerical computation of the costs. Considering for\nsimplicity G≡Id in (4.14), it reads\nJ(uϕ0) =Z+∞\n0uϕ0(t)dt=Z+∞\n0β−bϕ0(t)dt, (5.1)\nandMatlab returns the value J(uϕ0) = 38 .766, while Bocop gives J(ubcp) = 38 .5263.26 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\n0 200 40000.20.314\n0 200 40000.010.021\n0 0.15 0.3 0.4500.0050.010.0150.020.021\n0 0.15 0.3 0.4500.0050.010.0150.020.021\nFigure 4. Solutions for the constant initial condition ϕ0(t)≡x0= (0.45,0.001). The dotted\nblack lines in the pictures above represent the optimal control ubcp(t) and the value of ibcp(t),\nrespectively, as computed by Bocop . The red lines represent the greedy control uϕ0(t) =β−\nbϕ0(t) described in Theorem 4.5, and the i-component of the solution, respectively, computed\nwithMatlab . In the pictures below, on the left, we plotted the trajectory of the solution\nxϕ0, corresponding to the greedy control uϕ0, with the curve S2in Theorem 4.5 plotted in\ndotted black line. On the right, the same plot for the solution xbcpcorresponding to the\ncontrol ubcp.\nIt is worth noting also that, while Bocop provides an approximated optimal solution in some minutes, the\ncomputation time of the suboptimal greedy strategy with Matlab is of the order of a couple of seconds.\nIn order to further analyze the effectiveness and performance of the greedy control strategy, in Figure 5 we plot\nthe function x7→¯J(x) :=J(uϕx) by considering constant initial conditions ϕx(t)≡xin [−h,0], for x∈B,\nand where uϕx=β−bϕxwith bϕxdefined in (4.18). As expected, the value of ¯J(x) is 0 when x∈A, while J\nincreases in value as the initial condition approaches the set S1∪S2⊂∂B(using the notation in Theorem 4.2,\nsee also Figure 1).\nWe then apply the greedy control policy described in Theorem 4.5 to different initial conditions (reaching the\nsame point x0= (0.45,0.001)∈Tat time 0). This allows us to evaluate the dependence of the control, the\ncorresponding solution and the corresponding cost, on the past/delayed behaviour of the initial condition.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 27\nFigure 5. The value of the cost Jfor different constant initial conditions in B. On the\nright, the viable zone Bof the ( s, i)-plane is represented, with different colors for different\nvalues of J. The color scale goes from dark green, corresponding to a null cost, to light yellow\ncorresponding to the highest values. The curve in black, crossing the set B, represents the\nrelative boundary (Γ β) of the set A. On the left, the 3D graph of the map x7→J(x).\nWe first consider the situation in which at time t=−hthere is an initial proportion of exposed population\nthat is not infectious, but it is recovering with rate equal to γ. We thus consider ϕ1: [−h,0]→R2defined by\nϕ1(t) = (s0, i0e−γt), t∈[−h,0]. (5.2)\nThe constancy of the s-component of the initial condition can be assumed without loss of generality, because\nthe dynamics in system (1.1) does not depend on the delayed value of s. It can be numerically verified that\nϕ1∈ B, and thus the control introduced in Theorem 4.5 is feasible also for this initial condition. The control\nuϕ1and the corresponding i-component of the solution, iϕ1, are depicted in Figure 6. Compared to the\n0 200 40000.20.314\n0 200 40000.010.021\nFigure 6. The control uϕ1obtained, via Theorem 4.5, for the initial condition ϕ1defined\nin (5.2), and the corresponding infected population iϕ1.\nconstant initial condition case depicted in Figure 4, we note that we have a slightly “perturbed behaviour” of28 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nthei-component in the first time steps, due to the greater value of the initial condition in [ −h,0]. Besides this\nslight discrepance in the very short term, the proposed feedback control uϕ1and the corresponding behaviour\nof the solution are qualitatively equivalent to the ones obtained with the constant initial condition ϕ0. Also\nthe cost J(ϕ1) = 38 .799 is again close to the cost J(ϕ0) corresponding to the constant initial condition.\nWe finally consider a third initial condition motivated by mathematical interest. We take ϕ2: [−h,0]→R2\ndefined by\nϕ2(t) = (s0, i0+iM−i0\n1−e−5h(1−e5t)), t∈[−h,0], (5.3)\ni.e., we consider an exponentially decreasing i-component, starting at i=iMatt=−hand reaching i=i0at\nt= 0. The function iϕ2: [−h,0]→Ris illustrated in Figure 7, and it models an initial condition close to the\n“worst case” used in the proof of Theorem 4.5: it remains close to the maximal admissible iMbefore rapidly\ndecreasing to the initial value i0.\niM\ni0\n−h tiϕ2(t)\nFigure 7. Thei-component of ϕ2defined in (5.3), in the interval [ −h,0].\nThe resulting solution and control are depicted in Figure 8. With respect to the previous cases of Figure 4\n0 200 40000.20.314\n0 200 40000.010.021\nFigure 8. The control uϕ2obtained, via Theorem 4.5, for the initial condition ϕ2defined\nin (5.3), and the corresponding infected population iϕ2. In the upper-right squares, we high-\nlight the non-monotonic behaviour of uϕ2andiϕ2in the initial part of the epidemic horizon.\nand Figure 6, we note that we have an initial non-monotonic evolution of the i-component and of the control\nuϕ2. We also note that the control action starts earlier, since the solution reaches earlier the curve S2defined\nin Theorem 4.5 (as can be numerically verified). This discrepancy is also reflected by the value J(ϕ2) = 41 .004,\nwhich is larger than the costs corresponding to the initial conditions ϕ0andϕ1previously computed.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 29\n6.Conclusions\nThis paper is concerned with a viability analysis and control synthesis of a delayed SIR model under an\nICU state-constraint, where a constant delay represents an incubation/latency time. As a by-product of the\nconsidered functional viability tools, we provided feasible feedback control maps driving the solutions to a\nsafe set, according to a cost functional to be minimized. Two scenarios were examined: in the first the initial\nconditions are simply continuous, while in the other they satisfy a suitable Lipschitz continuity assumption.\nIn the latter case, we studied the dependence of the obtained results on the delay parameter. The theoretical\ndevelopments have been illustrated via a numerical example inspired by the recent COVID-19 epidemic.\nAppendix A.Technical Proofs\nIn this Appendix we collect some technical results.\nLemma A.1. Assume that eF(ϕ)∈ DB(ϕ)for any ϕ∈ B. Then, there exists a solution xϕto the Cauchy\nproblem (4.17) . Moreover, any maximal solution is global, i.e., dom (xϕ) = [−h,+∞), for all ϕ∈ B, and it\nholds that S(t)xϕ∈ B,∀t∈R+. A similar result holds when Bis replaced by BL,h, andeFis replaced by FL,h.\nProof. To prove the existence of solutions, we consider first a convex regularization of the (possibly discontin-\nuous) map eF:B →R2, by considering the set-valued map eG:B⇝R2, defined by\neG(ϕ) =(\n{eF(ϕ)}={f(ϕ, β)},ifϕ(0)∈B\\(S1∪S2),\ncon\nf(ϕ, β),eF(ϕ)o\n,ifϕ(0)∈S1∪S2.(A.1)\nNote that if ϕ(0)∈S1, we can also write eG(ϕ) = co {f(ϕ, β), f(ϕ,min{β,γiM\nsϕ(0)iϕ(−h)})}, while, if ϕ(0)∈S2\nwe have eG(ϕ) = co {f(ϕ, β), f(ϕ,min{β, β⋆iM\niϕ(−h)})}. The map eG:B⇝R2is obtained by considering\nthe Krasovskii regularization (see [23, Definition 2.2] for the definition for finite dimensional maps) of the\neF:B →R2inB, defined by\neG(ϕ) =\\\nε>0co{eF(φ)|φ∈B∞(ϕ, ε)∩ B},\nwhere B∞(ϕ, ε) :={φ∈ C | supt∈[−h,0]|φ(t)−ϕ(t)|max≤ε}. By definition, eG:B →R2has non-empty,\ncompact and convex values, and eF(ϕ)∈eG(ϕ) for all ϕ∈ B. We now prove that eGis also upper semicontinuous,\ni.e.,\n∀ϕ∈ B,∀ε >0∃δ >0 such that φ∈B∞(ϕ, δ)∩ B ⇒ eG(φ)⊆eG(ϕ) +B|·|max(0, ε).\nLet us take ϕ∈ Band proceed by cases.\n(1) If ϕ(0)∈B\\(S1∪S2), the conclusion easily follows by observing that eG(ϕ) ={f(ϕ, β)}, the map\nf(·, β) :B →R2is continuous with respect to ∥·∥∞inBand the set B\\(S1∪S2) is relatively open in B.\n(2) If ϕ(0)∈S1\\S2, we distinguish the following two sub-cases.\n(a) If iϕ(−h) = 0, there exists δ >0 such that, if ∥φ−ϕ∥∞≤δthen\nβ≤γiM\n(γ\nβ⋆+δ)δ≤γiM\nsφ(0)iφ(−h),\nand thus b(φ) =β, implying eG(φ) ={f(φ, β)}, and concluding the proof in this case.\n(b) In the case iϕ(−h)>0, we note that for any ε′>0 there exists δ′>0 such that ∥ϕ−φ∥∞≤δ′\nimplies\f\f\fγiM\nsϕ(0)iϕ(−h)−γiM\nsφ(0)iφ(−h)\f\f\f≤ε′.30 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nGiven ε >0, by continuity of f:C ×U→R2, of the min operator, and by the previous inequality,\nthere exists a δ >0 such that ∥ϕ−φ∥∞≤δimplies\neG(φ) = co {f(φ, β), f(φ,min{β,γiM\nsφ(0)iφ(−h)})}\n⊆co\u001a\nf(ϕ, β) +B|·|max(0, ε), f(ϕ,min{β,γiM\nsϕ(0)iϕ(−h)}+B|·|max(0, ε)\u001b\n= co{f(ϕ, β), f(ϕ,min{β,γiM\nsϕ(0)iϕ(−h)})}+B|·|max(0, ε) =eG(ϕ) +B|·|max(0, ε),\nwhere we used the fact that, for any z1, z2∈Rnand any convex set K⊂Rnit holds that\nco{z1+K, z 2+K}= co{z1, z2}+K.\nWe have thus proved the upper semicontinuity in this case.\n(3) Let ϕ(0)∈S2\\S1. The case iϕ(−h) = 0 can be treated as in the previous case. If iϕ(−h)>0, the claim\nfollows by continuity of the function i7→β⋆iM\niwhen i̸= 0, by adapting the argument of the previous\ncase.\n(4) For ϕ(0)∈S1∩S2={(γ\nβ⋆, iM)}, it suffices to recall that in this case we have\nγiM\nsϕ(0)iϕ(−h)=β⋆iM\niϕ(−h),\nand the same continuity argument can thus be applied.\nSummarizing, the set-valued map eG:C →Rnis upper semicontinuous with non-empty, compact and convex\nvalues. By assumption, eF(ϕ)∈ DB(ϕ) for all ϕ∈ B. Since eF(ϕ)∈eG(ϕ) for all ϕ∈ B, this implies\neG(ϕ)∩DB(ϕ)̸=∅∀ϕ∈ B.\nWe can thus apply [18, Theorem 1.1] (which is the local version of Theorem 2.7) proving that, for any ϕ∈ B\nthere exist a τ >0 and a function xϕ: [−h, τ)→R2satisfying\nx′\nϕ(t)∈eG(S(t)xϕ) for a.e. t∈[0, τ),\nS(0)xϕ=ϕ,\nS(t)xϕ∈ B ∀ t∈[0, τ).(A.2)\nThe fact that τ= +∞, i.e. that maximal solutions are defined on [ −h,+∞), follows again by viability analysis,\nsince Bis compact and thus solutions cannot explode in finite time, see [18, Page 12, Proof of Theorem 1.1].\nTo conclude the proof we show that, for any ϕ∈ B, the only viable direction in eG(ϕ) is given by the vector\neF(ϕ), that is\n{eF(ϕ)}=eG(ϕ)∩DB(ϕ)∀ϕ∈ B, (A.3)\nthus proving that any solution to (A.2) is a solution to (4.17).\nBy definition of eG(see (A.1) and the two lines after) the claim is trivially true if, either,\n•ϕ(0)∈B\\(S1∪S2), or\n•ϕ(0)∈S1andβ≤γiM\nsϕ(0)iϕ(−h), or\n•ϕ(0)∈S2andβ≤β⋆iM\niϕ(−h).\nWe then proceed by analyzing the following two remaining cases, only. To this aim it is useful to note that,\nby Lemma 2.5, the claimed proposition (A.3) is equivalent to\n{eF(ϕ)}=eG(ϕ)∩TB(ϕ(0))∀ϕ∈ B. (A.4)VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 31\n(1) Let ϕ(0)∈S1andβ >γiM\nsϕ(0)iϕ(−h). Considering the vector v= (0,1), normal to Batϕ(0), and recalling\n(2.1), we have\n⟨v, f(ϕ,γiM\nsϕ(0)iϕ(−h))⟩= 0,\nand\n⟨v, f(ϕ, β)⟩=βsϕ(0)iϕ(−h)−γiϕ(0)>0.\nHence, by convexity, ⟨v, g⟩>0 for all g∈eG(ϕ),g̸=eF(ϕ). This proves (A.4), and hence (A.3), in the\ncurrent case.\n(2) If ϕ(0)∈S2andβ > β ⋆iM\niϕ(−h), we can argue as in the first part of the proof of Theorem 4.5, case (c). There,\nwe proved that, denoted by w∈R2the unique normal vector (modulo positive scalar multiplication) to\nBatϕ(0), it holds\n⟨w, f(ϕ, b(ϕ))⟩=⟨w, f(ϕ, β⋆iM\niϕ(−h))⟩= 0,\nand⟨w, f(ϕ, β)⟩>0. Now, by convexity, we have ⟨w, f⟩>0 for all f∈eG(ϕ),f̸=eF(ϕ), concluding the\nproof.\n□\nLemma A.2. For any value of γ > 0,β > β ⋆>0,0< iM≤1and for any h > 0, the functions\nΓβ,L,h : [γ\nβ,bsβ,L,h)→R+andΓβ⋆,L,h: [γ\nβ⋆,bsβ⋆,L,h)→R+defined in Subsection 4.2 are strictly decreasing and\nconcave. Moreover, the sets AL,handBL,hdefined in Theorem 4.12 are convex.\nProof. We recall that Γ β,L,h : [γ\nβ,bsβ,L,h)→Ris defined as the graph in the plane ( s, i) of the solution of (4.24)\nwith initial condition ( s(0), i(0)) = (γ\nβ, iM). Since sψL,h(i)>0 for all s∈[γ\nβ,bsβ,L,h) and i∈[0,+∞), we can\ndivide the equations in (4.24) by this quantity, obtaining that Γ β,L,h is solution to the differential equation\nd\ndsi(s) =−1 +γi(s)\nβsψL,h(i(s)),\nwith initial condition i(γ\nβ) =iM, in the interval [γ\nβ,bsβ,L,h). For simplicity, in what follows we write Γ β,L,h(s) :=\ni(s), for all s∈[γ\nβ,bsβ,L,h).\nFirst we note thatdi\nds(γ\nβ) = 0 and\nd2i\nds2(γ\nβ) =−1\nγ<0.\nThis implies that iis strictly decreasing in a right-neighborhood Nofs(0) =γ\nβand, hence, i(s) = Γ β,L,h(s)<\niMfor all s∈N. On the other hand, as long as i≤iMands >γ\nβ, we have\ndi\nds(s)≤ −1 +γ\nβs<0,\nsincei\nψL,h(i)≤1 for all i≤iM. This implies that iis strictly decreasing in [γ\nβ,bsβ,L,h), and by definition of\nbsβ,L,h we also have that i(s)>0 for all s∈[γ\nβ,bsβ,L,h). We now show that its derivative is also decreasing,\nproving concavity. By definition of ψL,h(i(s)), we have\ni(s)\nψL,h(i(s))=\n\ni(s)\niM, ifiM−Lh≤i(s)≤iM,\ni(s)\ni(s)+Lh,if 0≤i(s)≤iM−Lh.32 DIMITRI BREDA, MATTEO DELLA ROSSA, AND LORENZO FREDDI\nWe note that both the functions z7→z\niMandz7→z\nz+Lhare strictly increasing in [0 ,+∞); since we have\nproved that i(s) is strictly decreasing in [γ\nβ,bsβ,L,h), so is s7→i(s)\nψL,h(i(s)). Then the function\ns7→di\nds(s) =−1 +γi(s)\nβsψL,h(i(s)),\nis also decreasing, concluding the proof of concavity of Γ β,L,h in [γ\nβ,bsβ,L,h).\nThe set AL,hdefined in (4.27) is then convex, since defined as the intersection of the convex set Twith the\nhypo-graph of a concave function.\nThe proof for Γ β⋆,L,h: [γ\nβ⋆,bsβ⋆,L,h)→R+is similar. □\nLemma A.3. Given an interval I= [0, τ)(with, possibly, τ= +∞), consider a continuous function g:Rn→\nRnandx0∈Rn. Suppose there exists x:I→Rn, solution to\nx′(t) =g(x(t)), x(0) = x0,\nand that there exists a compact set Q⊂Rnsuch that x(t)∈Qfor all t∈I. For a given a >0, for any\nh∈(0, a)consider gh:Rn→Rnsuch that:\n•ghis uniformly converging to ginQ, ash→0,\n•ghare uniformly locally Lipschitz, i.e., for any compact set K⊂Rnthere exists MK≥0such that\n|gh(x1)−gh(x2)| ≤MK|x1−x2|,∀h∈(0, a),∀x1, x2∈K. (A.5)\n•ghare uniformly sublinear in norm, i.e., there exist A >0, B > 0such that\n|gh(x)| ≤A|x|+B,∀h∈(0, a),∀x∈Rn. (A.6)\nLet us denote by yh:I→Rnthe solution to\ny′(t) =gh(y(t)), y(0) = x0.\nThen we have\nlim\nh→0+|x(t)−yh(t)|= 0∀t∈I.\nMoreover, for every compact interval J⊂I, there exists a sequence hn→0such that yhnconverges uniformly\ntoxonJ.\nProof. Let us fix t∈I. By (A.6), using Gr¨ onwall’s inequality (see [37, Lemma 2.7]) we have that\n|yh(t)| ≤¯R:=|x0|eAt+B\nA(eAt−1),∀h∈(0, a). (A.7)\nGiven the compact set K=Q∪B∥·∥(0,¯R), consider MK>0 as in (A.5). Now, for all h∈(0, a), we have\n|yh(t)−x(t)| ≤Zt\n0|gh(yh(θ))−g(x(θ))|dθ\n≤Zt\n0|gh(yh(θ))−gh(x(θ))|dθ+Zt\n0|gh(x(θ))−g(x(θ))|dθ\n≤MKZt\n0|yh(θ)−x(θ)|dθ+Zt\n0|gh(x(θ))−g(x(θ))|dθ\nBy uniform convergence, for every ε >0 there exists h∈(0, a) such that, for any h≤h, it holds that (recall\nthattis fixed)\n|yh(t)−x(t)| ≤MKZt\n0|yh(θ)−x(θ)|dθ+ε.\nApplying again the Gr¨ onwall’s Inequality ([37, Lemma 2.7]) this implies that\n|yh(t)−x(t)| ≤εeMKt.VIABILITY AND CONTROL OF A DELAYED SIR EPIDEMIC 33\nWe have, thus, proved that\nlim\nh→0|yh(t)−x(t)|= 0,\nas required. Let us now take a compact subinterval J⊂I. By (A.7), the functions yhare uniformly bounded\nonJ, i.e., there exists C > 0 such that |yh(t)| ≤Cfor all h∈(0, a) and for all t∈J. Moreover, by (A.6) we\nhave|gh(t)| ≤AC+Bfor all for all h∈(0, a) for all t∈J. In other words, the derivatives of the functions yh\nare uniformly bounded in Jand this implies that the functions yhare equicontinuous on J. 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Biosystems , 98(1):43–50,\n2009.\n(Dimitri Breda, Matteo Della Rossa, Lorenzo Freddi) Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Uni-\nversit `a di Udine, via delle Scienze 206, 33100 Udine, Italy\nEmail address :dimitri.breda@uniud.it, matteo.dellarossa@uniud.it, lorenzo.freddi@uniud.it"    },    {        "title": "2401.12922v1.Flow_structure_beneath_periodic_waves_with_constant_vorticity_under_strong_horizontal_electric_Fields.pdf",        "content": "Flow structure beneath periodic waves with constant vorticity\nunder strong horizontal electric fields\nM. V. Flamarion1, E. Kochurin2,3, R. Ribeiro-Jr4, N. Zubarev2,5\n1Department of Mathematics, Aeronautics Institute of Technology (ITA), São José dos Campos, SP 12228-900, Brazil\nmarcelovflamarion@gmail.com\n2UInstitute of Electrophysics, Ural Branch of Russian Academy of Sciences, Yekaterinburg, 620016, Russia\n3Skolkovo Institute of Science and Technology, Moscow, 121205, Russia\nkochurin@iep.uran.ru\n4UFPR/Federal University of Paraná, Departamento de Matemática, Centro Politécnico, Jardim das Américas, Caixa Postal\n19081, Curitiba, PR, 81531-980, Brazil\nrobertoribeiro@ufpr.br\n5P. N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia\nnick@iep.uran.ru\nAbstract\nWhile several articles have been written on Electrohydrodynamics (EHD) flows or flows with constant\nvorticity separately, little is known about the extent to which the combined effects of EHD and constant\nvorticity affect the flow. This study aims to fill this gap by investigating how a horizontal electric field and\nconstant vorticity jointly influence the free surface and the emergence of stagnation points. Using the Euler\nequations framework, we employ conformal mapping and pseudo-spectral numerical methods. Our findings\nreveal that increasing the electric field intensity eliminates stagnation points and smoothen the wave profile.\nThis implies that a horizontal electric field acts as a mechanism for the elimination of stagnation points\nwithin the fluid body.\n1 Introduction\nElectrohydrodynamics (EHD) flows and flows with constant vorticity are two topics of interest to researchers\nfrom different areas. EHD flows consist of the coupling between fluid and electromagnetism. Its motivation\ncomesfromtheindustrialinterestincontrollingfluidmotiontroughelectricfieldmanipulation[15,23,3,7,19,2].\nThis application extends to cooling systems in conducting pumps [13], coating processes [14], among others.\nConstant vorticity flows serve as a model for waves within linearly sheared currents, representing realistic flow\nscenarios in certain conditions as for instances in flows when waves are long compared with the depth [25].\nMost studies on EHD and flows with constant vorticity have typically examined these topics separately. The\nliterature in those topics is broad and therefore it is difficult to give a comprehensive overview of contributions.\nFor the interested reader, we mention a few references from which the bibliography may be useful.\nThe paper by Papageorgiou [22] offers a good review on EHD flows, perfectly describing the effects of electric\nfields on immiscible film flows. Specifically, it highlights that normal electric fields have the capability to desta-\nbilize the interface between two fluids[21], whereas tangential electric fields, conversely, can stabilize interfacial\nwaves due to these effects in a way that Rayleigh-Taylor[3] and Kelvin-Helmholtz[33, 34, 35] instabilities can\nbe suppressed.\nRegarding flows with constant vorticity, one of their primary characteristics is the presence of overhanging\nwaves and the emergence of stagnation points. Overhanging waves are free surface waves that are not a graph of\na function. They have been demonstrated numerically [27, 28, 5, 6], and their existence has also been rigorously\nproven for periodic waves [4, 17]. Stagnation points represent fluid particles that remain stationary in the wave\nmoving frame. In irrotational flows, these points are charactersed by occuring at wave crests, forming a sharp\ncrest [29]. By adjusting the vorticity in constant vorticity flows, stagnation points may emerge within the fluid\nbulk, creating a recirculation zone whose profile resembles Kelvin’s cat eye flow [24, 30, 25, 12].\nAll previously mentioned studies have primarily focused on either EHD flows or flows with constant vorticity.\nOnly recently research efforts have shifted towards investigating the behavior of electrohydrodynamic flows\nwithin the context of constant vorticity. This approach commenced with the work of Hunt and Dutykh [16]\nwhich concerned in the derivation of linear and weakly nonlinear models for rotational waves in the presence of\n1arXiv:2401.12922v1  [physics.flu-dyn]  23 Jan 2024vertical electric fields. The underlying flow structure beneath waves with constant vorticity under the influence\nof vertical electric fields has been investigated more recently by Flamarion and collaborators [9, 10]. Their\nfindings highlighted that by maintaining a fixed vorticity, the increase in the intensity of the electric field can\nlead to the appearance of stagnation points. This phenomenon subsequently induces the formation of cat-eye\nstructure. This result underscored the potential for controlling stagnation points through the manipulation of\nelectric field strength rather than adjusting vorticity. The first investigation into the combination of a horizontal\nelectric field and linear shear flow was conducted by Flamarion et al. [11] . Their work demonstrates that in the\nhigh vorticity regime in the absence of an external electric field, the evolution of the free surface exhibits notably\nintricate nonlinear behavior. Specifically, observations include the occurrence of wave breaking resulting in the\nformation of an air bubble within the fluid. In contrast, the introduction of a horizontal electric field ensures\nthe preservation of the free surface wave profile.\nAs far as we are aware, the effect of horizontal electric fields on the flow structure beneath a surface wave\nremains unexplored. The current study aims to contribute as an additional effort towards addressing this gap\nin the literature. In particular, we focus on numerically investigating the combined influence of a horizontal\nelectric field and constant vorticity on the behavior of the free surface and the emergence of stagnation points.\nOur investigation is carried out within the framework of Euler equations, utilizing computation made through a\nconformal mapping technique and pseudo-spectral numerical methods. We find that the increase of the intensity\nof the electric field leads to the elimination of stagnation points and tends to smoothen the wave profile. This\nimplies that a horizontal electric field acts as a mechanism for the elimination of stagnation points within the\nfluid body. This behaviour is the opposite of the effect of normal electric field where an increase in electric field\nstimulates the emergence of stagnation points [9].\nThis paper is organized as follows. Section 2 presents the mathematical formulation, while Section 3 provides\na succinct overview of the conformal mapping technique and the numerical approach employed. Section 4\noutlines the linear theory, followed by the presentation of results obtained from the full Euler equations in\nSection5. Finally,Section6encompassesacomprehensivediscussionoftheresultswithasummaryofconcluding\nremarks.\n2 Mathematical Formulation\nLet us consider an inviscid and incompressible dielectric fluid (there are no electric charges inside the fluid)\nwith density ρand electric permittivity ϵ1on channel of mean depth hin a two-dimensional space. The\nfluid is bounded laterally by two wall electrode and at the top by a non-conducting gas. The electric field\nacts horizontally in the fluid, with a magnitude represented by E0=V0/L, where Lsignifies the system\nhorizontal size and V0represents the potential difference along the horizontal axis. In the electrostatic limit of\nMaxwell’s equations, the induced magnetic fields are negligible, then we obtain an irrotational electric field due\nto Faraday’s law. This is expressed via a potential function V(x, y)governing the electric field ⃗E, such that\n⃗E=−∇V. Consequently, Vsatisfies the Laplace Equation within the dielectric fluid layer. The schematic\nrepresentation of this problem is depicted in Figure 1. Establishing a Cartesian coordinate system {x, y}with\ngravity oriented in the negative y-direction and y= 0denoting the undisturbed free surface, the unperturbed\nstate reveals the electric field potential to adopt a linear form: V(x, y) =−(V0/L)x. 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sha1_base64=\"UeM476q28ANHjVLhZ7FbVPzTSrw=\">AAAB73icbVBNSwMxEJ2tX7V+VT16CRbBi2W3CnoRil48VrDbQruUbJptQ7PJmmSFsvRPePGgiFf/jjf/jWm7B219MPB4b4aZeWHCmTau++0UVlbX1jeKm6Wt7Z3dvfL+ga9lqghtEsmlaodYU84EbRpmOG0niuI45LQVjm6nfuuJKs2keDDjhAYxHggWMYKNldo+ukZnfs/tlStu1Z0BLRMvJxXI0eiVv7p9SdKYCkM41rrjuYkJMqwMI5xOSt1U0wSTER7QjqUCx1QH2ezeCTqxSh9FUtkSBs3U3xMZjrUex6HtjLEZ6kVvKv7ndVITXQUZE0lqqCDzRVHKkZFo+jzqM0WJ4WNLMFHM3orIECtMjI2oZEPwFl9eJn6t6p1Xa/cXlfpNHkcRjuAYTsGDS6jDHTSgCQQ4PMMrvDmPzovz7nzMWwtOPnMIf+B8/gAZuo6z</latexit>left electrod<latexit 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sha1_base64=\"TLwfXTXr/JYQfpMujh4XaQGouZg=\">AAAB/nicbVBNSwMxEJ2tX7V+rYonL8EieCq7VdBj0YvHCvYD2qVks2kbmmyWJCuWpeBf8eJBEa/+Dm/+G9N2D9r6YODx3gwz88KEM20879sprKyurW8UN0tb2zu7e+7+QVPLVBHaIJJL1Q6xppzFtGGY4bSdKIpFyGkrHN1M/dYDVZrJ+N6MExoIPIhZnxFsrNRzj7oilI+ZYoOhQZRTYpSMJj237FW8GdAy8XNShhz1nvvVjSRJBY0N4Vjrju8lJsiwMoxwOil1U00TTEZ4QDuWxlhQHWSz8yfo1CoR6ktlKzZopv6eyLDQeixC2ymwGepFbyr+53VS078KMhYnqaExmS/qpxwZiaZZoIgp+zAfW4KJYvZWRIZYYWJsYiUbgr/48jJpViv+eaV6d1GuXedxFOEYTuAMfLiEGtxCHRpAIINneIU358l5cd6dj3lrwclnDuEPnM8f7U+WGw==</latexit>h<latexit sha1_base64=\"Avqj6DgOR2NBV6dY7Rsio1T0XiY=\">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOl5qhfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8fz0eM8A==</latexit>\nFigure 1: Schematic illustration of the geometry of the problem.\nWe consider that the flow in the dieletric fluid is rotational with constant vorticity γ. In the unperturbed\nstate, the velocity field in the fluid body is given by\n⃗U0= (−γ(y+h),0).\nThe background flow ⃗U0is a linear shear flow with horizontal velocity equal to zero at the bottom. Let us\nconsider ϕas an potential velocity of an irrotational perturbation of the shear flow, i.e, the velocity field in the\nfluid body is given as\n⃗U=⃗U0+∇ϕ. (1)\n2The goal of this work is to identify steady waves with wavelength Lin a frame of reference that moves with\nthe wave speed c. This is achieved by introducing the change of variables X=x−ct,Y=y, which yields\nη(x, t) =η(X). In this context, the dimensionless governing equation can be written in terms of the potential\nϕand the voltage potential Vas follows\n∇2ϕ= 0 in−1< Y < η (X), (2)\n∇2V= 0 in−1< Y < η (X), (3)\n−cηX+ (ϕX−γ(η+ 1)) ηX=ϕY for Y=η(X), (4)\nϕY= 0 for Y=−1, (5)\n∂V/∂n = 0 for Y=η(X), (6)\nVy= 0 for Y=−1, (7)\nwhere ∂/∂ndenotes the normal derivative at the free surface η(x, t). Additionally, the dynamic boundary\nconditionisobtainedbyensuringthecontinuityofpressureonthefreesurface,leadingtothefollowingexpression\n−cϕX+1\n2(ϕ2\nX+ϕ2\nY)−γ(η+ 1)ϕX+γψ−Eb\n2|∇V|2=B, (8)\nwhere ψis the harmonic conjugate of ϕ,Eb=ϵ1V2\n0/ρgh3is the nondimensional parameter characterizing the\nintensity of the electric field and Bis the Bernoulli constant. Here, we consider the case of a strong electric\nfield, i.e., the effects of gravity and capillarity are not taken into account. Besides, we highlight that this set\nof equations is written in the dimensionless form where we have chosen h,h/gandV0as the reference length,\ntime and voltage potential, respectively.\nThe stream function can be written as\nψs(X, Y) =ψ(X, Y)−γY\u0012Y\n2+ 1\u0013\n−cY. (9)\nIn the scenario of a fluid with a high dielectric constant, the electric field lines are align tangentially to\nthe free surface, indicating a tangential field orientation [31, 32]. Consequently, the electrostatic matter can\nbe entirely resolved using the method of conformal transformation, a topic that will be elaborated upon in the\nsubsequent section.\n3 Conformal mapping and numerical method\nThe present study employs a pseudo-spectral numerical method to solve the equations (2)-(8). The chosen\napproach is based on conformal mapping, which facilitates the transformation of the physical domain into a\ncanonical one with simpler geometry. This technique has been previously applied in the study of rotational\nperiodic waves [24], as well as rotational solitary waves [12]. Addionally, it has been employed to investigate the\neffect of a normal electric field on the flow structure beneath periodic rotational waves [9] and, more recently,\nin examining the effects of external horizontal electric fields on the propagation of free surface waves in shear\nflow[11]. Essentially, the physical domain is mapped onto a horizontal strip of uniform thickness denoted as D,\nand all computations are carried out within this canonical domain. A concise overview of the numerical scheme\nis provided here.\nLet us denote by {ξ, ζ}the coordinate system in the canonical domain and by X(ξ, ζ)andY(ξ, ζ)the\nhorizontal and vertical component of the conformal map under which a strip of width Dis mapped onto the\nphysical domain. By elementary computation it is easy to find that\nY(ξ, ζ) =F−1\u0014sinh( kj(ζ+D))\nsinh( kjD)Fkj[Y]\u0015\n+ζ\nD, (10)\nand\nX(ξ, ζ) =ξ−F−1\u0014\nicosh( kj(ζ+D))\nsinh( kjD)Fkj[Y]\u0015\n, j̸= 0 (11)\nwhere Y(ξ,0) =Y(ξ)≡η(X(ξ,0),0),kj= 2πj/L, forj∈Z, and\nFkj[g(ξ)] = ˆg(kj) =1\nLZL/2\n−L/2g(ξ)e−ikjξdξ,\nF−1[{ˆg(kj)}j∈Z] =+∞X\nj=−∞ˆg(kj)eikjξ.\n3Note that the profile of the free surface wave in the physical domain is determined by the trace of the curve\n(X(ξ),Y(ξ)), where X(ξ) :=X(ξ,0)andY(ξ) :=Y(ξ,0). Besides, we obtain from equation (10) and (11) that\nX(ξ) =ξ− C[Y(ξ)], (12)\nwithC[·] :=F−1\u0002\nicoth( kjD)Fkj[·]\u0003\n.\nWe choose the the wavelength in both physical and canonical domain to be equal to L. This yields the\nequation\nD=⟨Y⟩+ 1, (13)\nwhere\n⟨Y⟩=1\nLZL/2\n−L/2Y(ξ)dξ. (14)\nUpon establishing the conformal mapping formulas, the governing equations (2)-(8) are reformulated in\ncanonical coordinates. Subsequent manipulations are applied, resulting in the derivation of a single governing\nequation for the free surface wave presented below\n−c2\n2−c2\n2J+γ2(C[(Y+ 1)Yξ])2−((Y+ 1)Yξ)2\n2J+γ2Y\u0012Y\n2+b\u0013\n+(c+γC[(Y+ 1)Yξ]) (c+γ(Y+ 1)(1 − C[Yξ]))\nJ−γc−Eb\n2J=B,(15)\nwhere J=X2\nξ+Y2\nξis the Jacobian evaluated at ζ= 0, andXξis the ξ-derivative of X(ξ), which is given\nbyXξ= 1− C[Yξ)]. It should be noted that in the absence of the shear flow, γ= 0, the equation (15) has\nexact solution for any shape of the surface profile: c2=Eb,B=−c2/2. In such fully electrohydrodynamic\nregime of motion, the fluid moves along the electric field lines. We refer to the work of Ribeiro-Jr et al. [24]\nand Flamarion et al. [11] respectively for further details regarding the derivation of the rotational and electric\ncomponents associated with equation (15).\nThe free surface wave is assumed to be an even function with crest at X= 0. Therefore, we consider that\nthe crest is located at ξ= 0. We fix the Hheight of the free-surface elevation Y(ξ) =Y(ξ,0)through the\nequation\nY(0)−Y(−L/2) = H. (16)\nTo ensure that the mean level of the free surface wave is zero in the physical domain, we set\nZ0\n−L/2YXξ, dξ= 0. (17)\nHere, [−L/2, L/2)is the computational domain, and recall that Lis chosen such that the wavelength in the\nphysical and canonical domain are the same.\nThe system of equations comprising (13), (15), (16) and (17) constitutes four equations with four unknowns:\nY,c,D, and B. The system is discretized in ξby introducing collocation points uniformly distributed along\nξ. By evaluating all the ξ-derivatives via a Fourier spectral method using the FFT, the partial differential\nequations are transformed into algebraic equations, which can be solved using Newton’s method, as detailed in\n[24, 20].\n4 Results from the Linear Theory\nIn this section, we will develop a linear theory based on the governing equations. We begin by considering a\ntrivial solution characterized by\nϕ0(X, Y) = 0 , V 0(X, Y) =X, η 0(X) = 0 . (18)\nThis solution is then perturbed by a small disturbance given by\nn\nη(X) =ϵˆη, ϕ(X, Y) =ϵˆϕ, V (X, Y) =X+ϵˆV , (19)\nwhere ϵis a small parameter that measures the wave amplitude. By solving the Laplace equations (2) and (3),\nalong with the boundary conditions (5) and (7), we obtain the following expressions\n\n\nˆη(X) =ℜ\b\nAeikX\t\n,\nˆϕ(X, Y) =ℜ\b\nMeikXcoshk(Y+ 1)\t\n,\nˆV(X, Y) =ℜ\b\nNeikXcoshk(Y+ 1)\t\n,(20)\n4where A,M, and Nare unknown constants, and k= 2π/Lrepresents the wavenumber. By neglecting the\nnonlinear terms in the kinematic condition (4) and the dynamic boundary condition (8), the equations can be\nlinearized as follows\n−(c+γb) ˆηX=ˆϕY, (21)\nˆVY=−ˆηX, (22)\n−(c+γb)ˆϕX+γˆψ−EbˆVX= 0. (23)\nBy substituting equations (20) into equations (21)–(23), it is revealed that the unknown constants Mand\nNare given by\nM=−iA(c+γb)\nsinhk, (24)\nN=iA\nsinhk, (25)\nrespectively. Furthermore, the linear dispersion relation is obtained as\nc±=−γ+1\n2kh\nγtanh k±√\n∆i\n, (26)\nwhere the discriminant ∆is defined as\n∆ = γ2tanh2k+ 4k2Eb. (27)\nIn the reference frame moving with the speed of the wave, the trajectory of a fluid particle is described by\n(X(t), Y(t)), satisfying the equation\n\n\ndX\ndt=ϕX(X, Y)−γ(Y+ 1)−c\ndY\ndt=ϕY(X, Y).(28)\nUnder the linear theory, this system can be approximated as\n\n\ndX\ndt≈ −γ(Y+ 1)−c\ndY\ndt≈ −0.(29)\nHence, if Y∗denotes the location of a stagnation point we have that\n−γ(Y∗+ 1)−c= 0, (30)\nwhere cis given by the formula (26).\nTo ensure that the stagnation point lies within the physical domain, we impose the condition Y∗∈[−1,0].\nThis condition is guaranteed if and only if\nY∗=−tanh( k)\n2k−s\u0012tanh( k)\n2k\u00132\n+Eb\nγ2(31)\nand\nkEb−γ2(k−tanh( k))<0. (32)\nTherefore, based on equations (31) and (32), we can draw the following conclusions:\n•When kandγare fixed, increasing the value of Ebfrom zero to higher values leads to the destruction of\nstagnation points.\n•When γandEbare fixed, in the shallow water limit ( k≈0), the depth of the stagnation point is given by\nY∗=−1\n2−s\u00121\n2\u00132\n+Eb\nγ2<−1.\nThis implies that in the linear long wave regime, there are no stagnation points within the bulk of the\nfluid.\n5•When γandEbare fixed, in the deep water regime ( k→ ∞), the depth of the stagnation point is given\nby\nY∗=−s\nEb\nγ2and Eb< γ2.\nThe aforementioned observations, although valid only for linear theory, provide valuable insights into the\nbehaviour of stagnation points in nonlinear waves, as will be elaborated upon in the next section.\n5 Results from the Full Euler Equations\nInthissection, weexploresolutionsofthefullEulerequations(2)-(8)acrossarangeofwaveamplitude, vorticity,\nand electric Bond number parameters.\nThe amplitude parameter, denoted by ε, is defined as the wave steepness, given by the ratio of wave height\nHto wavelength L. In our experiments, the depth is normalized to 1and we analyse setups of depth regime\nthat corresponds to shallow, intermediate and deep water. Our main purpose is to investigate the impact of\nthe intensity of the electric Bond number Ebon the free surface, in the emergence of stagnation points and on\nthe shape of the level curves of the voltage potential.\n-20 0 20-0.100.20.4\n- 0-0.4-0.200.20.4\n-0.5 0 0.5-0.2-0.100.1\n-20 0 20-0.6-0.4-0.200.2\n- 0-0.6-0.4-0.200.2\n-0.5 0 0.5-0.2-0.100.1\nFigure 2: The effect of the intensity of the electric field ( Eb) on the shape of the free surface wave in different\ndepth regimes: shallow water (left), intermediate water (middle) and deep water (right). The parameter εis\nthe steepness of the wave.\n5.1 The effect of the electric field on the free surface\nThe effect of horizontal electric field on the free surface wave is a complex and active area of research. Among\nthe facts that can influence the interaction between the electric field and the wave, we mention the strength\nof the electric field and the wavenumber. We refer to the work of Papageorgiu [22] for a enlighting review\non experimental, numerical and mathematical studies on electrohydrodynamic flows. In general, a horizontal\nelectric field tend to stabilise the wave.\nWe focus on investigating how the intensity of the electric Bond number ( Eb) affects wave profiles and wave\nspeeds for a range of amplitude, wavelength and vorticity parameters. Figure 2 illustrates the influence of Eb\non the shape of waves in different water depth regimes: shallow water regime (left column), intermediate regime\n(center column), and deep water regime (right column). The simulations are performed for flows with vorticity\nvalues γ= 2,and−2. Here we do not present surface profiles for the irrotational regime γ= 0, since in this case\n6waves with arbitrary shape propagate along the liquid boundary without distortions. Upon fixing the depth\nregime, we observe that waves in the shallow water regime are particularly sensitive to variations in the electric\nfield, with this impact being more pronounced in flows with negative vorticity. As the intensity of the electric\nfield increases, it tends to gradually smooth the wave profile. Specifically, the sharp troughs noticeable at lower\nvalues of Ebbecome more smooth with the increase of the intensity of the electric field. This phenomenon can\nbe read as a kind of stabilization of the flow caused by the horizontal electric field.\n01234500.511.522.5\n012345-0.500.511.5\n0123451.522.533.54\n01234500.511.522.5\n01234500.511.5\n01234522.533.5\n01234500.511.522.5\n012345-2-1.5-1-0.500.5\n01234522.533.544.5\nFigure 3: The wave speed cas a function of Ebfor waves in a intermediate depth regime ( λ= 2π). The\nparameter εis the steepness of the wave.\nFigure 3 shows the relationship between wave speed candEbfor flows with vorticity γ= 0,2,−2for\nintermediate regime ( L= 2π), shallow water ( L= 40) and deep water ( L= 1). Notably, the speed for ε= 0.01\naligns well with the linear speed which is given by the formula (26). Additionally, we notice that the velocity\nincreases as Ebincreases. We note that when vorticity is zero, the velocity remains constant irrespective of the\nεvalue. This observation ties with the discussion in equation (15), where we noticed that for irrotational flows\nthe wave speed remains the same regardless of the wave profile.\n-0.5 0 0.5-0.1-0.0500.02\nFigure 4: Effect of increasing electric field intensity on the destruction of self-intersecting wave profiles.\nIn addition to the wave profiles shown in Figure 2, we observe self-intersecting waves, as depicted in Figure 4.\n7Our simulations suggest that higher intensities of the electric field tend to disrupt the self-intersecting wave\nprofiles. This observation is in agreement with the findings of Flamarion et al. [11], who previously observed\nthe spontaneous formation of self-intersecting wave profiles in flows with high vorticity and in the absence of an\nelectric field. Their study highlighted that the inclusion of a horizontal electric field avoid the formation of such\nwave profile. It is important to point out that streamlines and stagnation points are only computed beneath\nfree surface waves that do not self-intersect.\n5.2 Streamlines and stagnation points\nIn this section, we investigate the influence of a horizontal electric field on the velocity field beneath periodic free\nsurface waves. Specifically, we focus on analyzing the emergence and disappearance of stagnation points within\nthe fluid bulk for different wave regimes. To achieve this, we start by analyzing waves within the intermediate\ndepth regime, where the wavelength is fixed at L= 2π. Subsequently, we observe the transition between\ndifferent regimes by varying the wavelength. It is important to say that the phase portraits presented in this\nsection are computed as level curves of the stream function (9) and the precise identification of stagnation points\nis established by employing the velocity field formula in the canonical domain (see [24] for more details on that).\nIn our first experiment, we keep the vorticity fixed at γ= 2and the wave steepness at ε= 0.1, while\nvarying the value of Eb. The resulting phase portrait, obtained by solving the full Euler equation, is presented\nin Figure 5. Initially, three stagnation points are observed for Eb= 0. However, as the value of Ebincreases, the\nstagnation points gradually move closer, coalesce at the bottom, and finally disappear. These findings suggest\nthat a horizontal electric field acts as a mechanism for the elimination of stagnation points within the flow. This\nbehaviour is the opposite of the effect of normal electric field where the increase of Ebstimulates the emergence\nof stagnation points. We refer to [9] for further details on normal electric fields.\n- 0-1-0.500.5\n- 0-1-0.500.5\n- 0-1-0.500.5\nFigure 5: Phase portrait for a flow with vorticity γ= 2at varying electric Bond numbers, E∗\nb≈2.751.\nA more comprehensive understanding of the bifurcations occurring in the parameter space (Eb, ε)forγ= 2\nis presented in Figure 6 (top). This parameter space is divided into three distinct regions:\nRegion 1: In this region, the flow exhibits two stagnation points within the fluid bulk: one centre beneath the\ncrest and one saddle beneath the trough. A typical phase portrait for choices of (Eb, ε)in this region is\ndisplayed in Figure 6 -1.\nRegion 2: Flows in this region possess three stagnation points: one centre within the bulk of the fluid and two\nsaddle points located at the bottom (see Figure 6 2iand2ii)\nRegion 3: In this region, there are no stagnation points beneath the free surface wave.\nWhen ε= 0, according to the linear theory formula (31), we find that the flow admits stagnation points at\nthe bottom boundary beneath both the trough and crest of the wave when Eb≈0.953. Notably, the point\n(0,0.953)lies on the boundary of regions 1, 2, and 3. A dashed line, marked as line A, separates regions 1 and\n2. Along line A, there exist two stagnation points per wavelength: one centre beneath the crest and one saddle\nbeneath the trough at the bottom – see Figure 6- A. Moving from region 1 to 2 along line A, the saddle points,\nwhich are initially one wavelenght apart, reach the bottom and subsequently split into two other ones. On the\ndashed line B, a bifurcation occurs when crossing from region 2, with three stagnation points, to region 3, with\nno stagnation in the fluid. On this bifurcation line, there exists a single stagnation point located at the bottom\nright under the crest – see Figure 6- B. As the nonlinearity of the wave is increased by elevating the values\nofε, the free surface wave tends to assume a profile with almost flat troughs that nearly reach the bottom,\nalong with a dominant recirculation zone within the flow (see Figure 6- 2ii). This feature indicates that, in this\nscenario, vorticity controls the flow. This observation leads to the conclusion that increasing nonlinearity results\nin vorticity dominating over the electric field.\nIt is interesting to discuss the nature of the flow shown in the Figure 6- 2ii. It can be assumed that in the\nabsence of an electric field ( Eb= 0), the solution with the maximum possible nonlinearity is a circular cylinder\nwith a radius of√\n2≈1.41, rolling along the bottom: see Figure 7. The regime of fluid motion shown in Figure 7\n80 1 2 3 4 500.10.20.3\n- 0-1-0.500.3\n- 0-1-0.500.3\n- 0-1-0.500.3\n- 0-1-0.500.3\n- 0-1-0.500.3\nFigure 6: Variation in the flow structure as the wave steepness ( ε) and the intensity of the electric field ( Eb)\nchange, while maintaining fixed values for vorticity ( γ= 2), depth ( 1), and wavelength ( L= 2π).\nis not a result of numerical simulation but the exact analytical solution of the problem. For such a limiting\nsolution, the XandYvelocity components are {−γY/2,+γX/2}, which corresponds to the rigid rotation of the\nfluid (in a coordinate system moving with the wave) around the stagnation point. Note that the flows shown\nin Figs. 5 and 6 can be interpreted as a nonlinear superposition of exact solutions of the problem, realized in\nthree different limiting cases: (i) trivial solution ε= 0(γandEbare arbitrary), (ii) solution with arbitrary ε\nforγ= 0, and (iii) solution shown in Figure 7 with the maximum possible steepness for Eb= 0. These three\nregimes of motion represent three limiting cases in which the problem has analytical solutions.\nFigure 7: The limiting wave configuration corresponding to the maximum possible nonlinearity. The solution\nis a rigid rotation of the fluid around the stagnation point.\nIn Figure 8, we illustrate the parameter space (Eb, ε)and provide typical phase portraits for flows with\nvorticity γ=−2. The dashed line Acorresponds to bifurcation points at which the number of stagnation points\nchanges. This line demarcates region 2, which there is no stagnation points in the fluid, from region 1, which\ncontains three stagnation points: two saddles at the free surface and a centre beneath the crest. Along line A,\na single stagnation point is located at the wave crest. It is well-established that in the absence of the electric\nfield, flows with negative vorticity do not have stagnation points (see [24]). Therefore, the type of recirculation\n9zone observed in Figure 8- 1can be read as consequence of the electric field. To the best of our knowledge, this\nconfiguration has not been reported in the literature before.\n0 1 2 3 4 500.050.10.15\n- 0-1-0.500.1\n- 0-1-0.500.1\n- 0-1-0.500.1\nFigure 8: Variation in the flow structure as the wave steepness ( ε) and the intensity of the electric field ( Eb)\nchange, while maintaining fixed values for vorticity ( γ=−2), depth ( 1), and wavelength ( L= 2π).\nWhen we vary the depth regime from intermediate to deep or shallow water, we observe that the electric\nfield causes the same effect in the flow: the destruction of stagnation points. The primary difference between\nthese regimes and the intermediate regime is mainly characterized by the following: in deep water, waves with\npositive vorticity may feature saddle points closer to the trough of the wave, and when these saddle points\nreach the bottom, the resulting recirculation zone is notably narrower. In the shallow water regime, flows with\npositive vorticity only have saddle points at the bottom. Figure 9 depicts some typical phase portraits. It\nis noteworthy that these characteristics arise from the interplay between the wave regime and the intensity of\nvorticity, rather than being attributed to any discernible effects of the electric field. This observation aligns with\nanalogous characteristics highlighted by Ribeiro-Jr et al. [24] in their investigation of periodic waves within\nflows characterized solely by constant vorticity.\n5.3 Electric potential\nThe aim of this section is to clarify the physical reason for the destruction of stagnation points in a strong\nhorizontal electric field. As was mentioned earlier, in the irrotational case, the governing equation system has\nan exact analytical solution in the form of a stationary wave with arbitrary shape, for more details, see [31, 32].\nIn this regime of fluid motion, the electric field lines coincide with the stream lines inside the fluid (the velocity\npotential equals the electric field potential). Thus, the question arises: how are the velocity and electric field\npotentials related in the constant vorticity regime? To answer it, Figure 10 shows the equipotential line (dashed\nred line) along which the electric potential Vis constant and the level curves (solid black lines) of the quantity\nϕT(X, Y)≡ϕ(X, Y)−cX.\nWhen the flow is irrotatinal this quantity ϕT(X, Y)corresponds to the velocity potential in the wave moving\nframe associated with the flow. Remarkably, for γ= 0, the curves appear nearly identical indicating that\nϕT∝V, which corresponds to Zubarev’s solution [31]. In the regime of a strong field but finite vorticity, the\nequipotentials for V(X, Y)andϕT(X, Y)deviate slightly from each other, see Figure 10.Thus, as the electric\nfield increases, the velocity potential tends to take the form of the electric potential. Since, the electric field in\na liquid is potential (irrotational) it leads to the destruction of stagnation points.\n6 Discussion of the results and conclusions\nIn this study, the examination focused on traveling periodic waves occurring on the surface of a finite depth\nlayer composed of non-conducting fluid characterized by a high dielectric constant. Our investigation concen-\ntrated on scenarios where the sole driving forces behind the fluid motion are the tangential electric field and a\nconstant vorticity. Comprehending the behaviour of periodic waves influenced solely by these specific driving\nforces, without the interference of other mechanisms contributes to fundamental knowledge in fluid dynamics.\n10-0.5 0 0.5-1-0.5-0.200.1\n-0.5 0 0.5-1-0.5-0.200.1\n    -1-0.99\n-20 0 20-1-0.500.5Figure 9: Typical phase portrait beneath waves in deep and shallow water for flows with vorticity γ= 2. Here\nE∗\nb1≈2.999\nFurthermore, this understanding offers insights for configuring laboratory experiments and comprehending the\ndynamics of real-world systems. Such insights are particularly relevant in scenarios where these forces operate\nin conjunction with other complex forces, providing a clearer understanding of their individual impacts and\ncollective interactions.\nThe results obtained in this study pave the way for a comprehensive discussion on the role of the vorticity\nand a tangential electric field on the characteristics of the flow beneath a periodic free surface wave.\nOur investigation was executed through a pseudo-spectral numerical method founded on a conformal map-\nping technique. Upon observing simulations where the electric field was set to zero, thus the only driving force\nis the vorticity, we notice that the emergence of saddle points within the bulk of the fluid was comparatively\nmore feasible in this scenario in contrast to rotational-gravity flows. This contrast is evident when examining\nboth linear theory and nonlinear solution. In the context of linear theory, as described in equations (31) and\n(32), it is easy to see that setting Eb= 0leads to the existence of stagnation points for any given value of\nvorticity. Additionally, the nonlinear simulation, as depicted in Figure 6 (top), illustrates well-defined saddle\npoints within the fluid bulk for a vorticity value of γ= 2. This observation stands in contrast to the findings\npresented by Ribeiro-Jr et al. [24] (Figure 3 and 5), where configurations featuring similar characteristics in\nrotational-gravity flows were reported exclusively for considerably higher vorticity values.\nFigure 9 and the linear theory investigation reveal a shared characteristic between rotational-gravity flows\nand pure rotational flows: in the context of linear waves in shallow water regime, stagnation points beneath\nthem are absent. Conversely, nonlinear waves in this depth regime demonstrate a phase portrait characterized\nby two saddle points at the bottom. This observation underscores the pivotal role of the depth regime in\ndetermining the presence or absence of stagnation points, as demonstrated in others studies [9, 8].\nIn our analysis of the combined influence of vorticity and a horizontal electric field, a notable observation\nemerges: the increase of the intensity of the electric field leads to the elimination of stagnation points and\ntends to smoothen the wave profile, as visually depicted in figures 2, 5- 8. This characteristic and the fact that\nself-intersecting are destroyed in the presence of high horizontal electric field (see Figure 4) aligns with existing\nunderstanding that horizontal electric fields have a stabilizing effect on flow dynamics [22].\nThe influence of the electric field on smoothening the free surface is particularly noteworthy. In contrast to\nrotational-gravity flows, where negative vorticity typically results in waves with sharp crests (as depicted in [24],\nfigure 13), the introduction of a horizontal electric field serves to attenuate the crest sharpness, concurrently\ngenerating stagnation points at the free surface (see Figure 8). The physical reason for the destruction of\nstagnation points by a strong electric field is that the velocity potential tends to take a form of the electric\npotential, as shown in Figure 10. Thus, the irrotational (potential) electric field leads to disappearing the\n11- 0-1-0.500.5\n- 0-1-0.500.5\n- 0-1-0.500.5\n- 0-1-0.500.5\n- 0-1-0.500.5Figure 10: Level curves of the voltage potential V(solid lines) and of the quantity ϕT(dashed lines). Both\npotentials are normalized to one for comparative analysis. Parameter: ε= 0.1.\nstagnation points.\nThe identification of a stagnation point at the crest of a rounded wave presents an unexpected phenomenon.\nTypically, a wave featuring a stagnation point at its crest represents the highest irrotational wave characterised\nby a sharp crest with an angle of 120 degrees [1, 26, 29]. Hence, the phase portraits depicted in Figure 8 is an\nunexpected occurrence in this context.\nAcknowledgments\nThe work of M.V.F. and R.R.Jr was supported in part by National Council Scientific and Technological De-\nvelopment (CNPq) under Chamada CNPq/MCTI/N◦10/2023-Universal. The work of E.K. on Section 2 is\nsupported by Russian Science Foundation project No. 23-71-10012.\nData Availability Statement\nReferences\n[1] Amick C J, Fraenkel L E & Toland, J. F. 1982. On the Stokes conjecture for the wave of extreme form.\nActa Mathematica 148, 193-214.\n[2] Chen X, Cheng J, Yin X. 2003. Advances and applications of electrohydrodynamics, Chin. Sci. 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The S H0ESH0≃73 km/s/Mpc has\nremained relatively unaltered across 18 years (2005-2023), despite marked shifts in maser and Cepheid distances to the keystone\ngalaxy NGC4258 (M106), and changes in the slope, zeropoint, metallicity, and extinction terms tied to the Leavitt Law, and\nnotwithstanding uncertain photometry for remote Cepheids spanning galaxies with highly inhomogeneous crowding and surface\nbrightness profiles. Concerns raised regarding the S H0ES findings by fellow researchers are likewise highlighted. An independent\nblind assessment of the entire suite of raw HST Cepheid images is warranted, while being mindful of a priori constraints and\nconfirmation bias that unwittingly impact conclusions.\nPart of a talk given at the VVVX Survey conference, hosted at the Vatican Observatory, Castel Gandolfo (Oct. 2023).\n1. Introduction\nThe reputed ‘ Hubble tension ’ stems in part from an offset\nbetween the Planck CMB H0relative to that reported by\nthe SH0ES team. The latter argue the difference separating\nthe estimates is significant ( ≃67 vs. 73 km/s/Mpc), and\nthe Planck team’s determination is erroneous. The S H0ES\nteam purports their conclusion is robust, and uncertainties\nassociated with the Cepheid distance scale were mitigated\n(e.g., Riess et al. 2016).\nPresent claims favoring a bona fide offset between Cepheid\nand CMB H0determinations are premature given: a lack\nof consensus on the Leavitt Law (coefficients, zeropoint,\nand form), errors an anomalies endemic to S H0ES find-\nings, ongoing challenges to secure uncontaminated Cepheid\nphotometry across remote spiral galaxies (e.g., NGC4258,\n§2, Fig. 3), a history of underestimated H0uncertainties\n(Fig. 1), and owing to a spread in CMB results (Fig. 4).\n2. SH0ES results & NGC4258 Cepheids\nFor example, Fig. 12 in Riess et al. (2009, S H0ES) im-\nplies that the slope ( α) of the classical Cepheid WV Ifunc-\ntion for near-solar Cepheids is α=−2.98±0.07, as in-\nferred mainly from SN-host galaxies (e.g., NGC3370). Ma-\njaess (2010) and Majaess et al. (2011a) countered that the\nWV Islope is comparatively constant across a sizable metal-\nlicity baseline ( α≃ −3.3, ∆[Fe /H]≃1), as established from\nLocal Group Cepheids in the Magellanic Clouds, Milky Way,\nNGC6822, and IC1613 (Fig. 5). The S H0ES team ostensibly\nrecanted their earlier position by citing a WV Islope tied to\nnear-solar Cepheids of α≃ −3.3 (Riess et al. 2022, S H0ES),\nwhereas for example Riess et al. (2009, S H0ES) deduced\nα=−2.60±0.24 for NGC3021 Cepheids.\nThe history of Cepheid and maser distances to NGC4258\nis disconcerting. Maoz et al. (1999) assessed HST images\nto determine a Cepheid distance for NGC4258 of 8 .1±0.4\nMpc, which was discrepant relative to the Herrnstein et al.\n(1999) maser distance of 7 .2±0.3 Mpc. The Maoz et al.\n(1999) team revisited their Cepheid distance in Newman\net al. (2001), and revised it downward (7 .8±0.3±0.5 Mpc)\nand marginally closer to the Herrnstein et al. (1999) maser\nresult. At the time various scenarios were proposed aiming\nFigure 1. Historical H0estimates from the Huchra database.\nLow uncertainties have been reported for decades. Fig. 2 in Steer\n(2020) features H0data beyond 2010.\nto reconcile the two estimates (e.g., Newman et al. 2001; Ca-\nputo et al. 2002). Macri et al. (2006) analyzed HST images\nof Cepheids in two separate NGC4258 fields, and discovered\na substantial distance offset between them (0m.16), which\nis readily discernible in Fig. 3, and that was established\nin this instance using a Galactic WV Icalibration (Majaess\net al. 2011a, and references therein). Importantly, an in-\nsidious degeneracy exists whereby more crowded fields at\nsmaller galactocentric radii are comparatively metal-rich,\nwhereas increasingly metal-poor Cepheids extend outward\nto the galaxy’s lower surface brightness periphery. That de-\ngeneracy can compromise determinations of the impact of\nchemical composition on Cepheid distances (e.g., Macri et al.\n1arXiv:2401.12964v1  [astro-ph.CO]  23 Jan 20242\nFigure 2. A subsample of S H0ESH0estimates across ∼18 years (e.g., Riess et al. 2005). Their H0remained comparatively\nunchanged ( ≃73 km/s/Mpc) despite ambiguities in the Leavitt Law, maser and WV ICepheid distances for NGC4258, etc. ( §2).\n2001, their §5). Indeed, the lack of consensus on the Leavitt\nLaw partly stems from the aforementioned degeneracy. Ma-\njaess et al. (2012a,b, 2016) noted that photometric contam-\nination was likewise problematic for certain globular clus-\nters and the Galactic Center sightline. †Yuan et al. (2020)\nidentified that Cepheids in high surface brightness regions\nof the Seyfert 1 galaxy NGC4151 exhibited a systematic\nshift (their Fig. 9, right panel). Rather than favoring the\ncontamination scenario, Macri et al. (2006) attributed their\nNGC4258 variations to a Leavitt Law zeropoint that’s metal-\nlicity dependent ( γ=−0.29±0.09±0.05 mag/dex). How-\never, Majaess (2010) and Bresolin (2011) argued that the\nabundance gradient across NGC4258 may be shallower ‡than\nthat finally selected by Macri et al. (2006, §4.3), which would\nmarkedly expand their cited Cepheid metallicity effect to an\nunrealistic value. Yet there are those favoring an extreme\nCepheid metallicity effect, such as Shappee & Stanek (2011,\n−0.80±0.21 mag/dex) and Fausnaugh et al. (2015, −0.61±\n0.21 mag/dex). Critically, Majaess et al. (2011a) disagreed\nwith those conclusions by relaying that applying such an\nimmense metallicity dependence yielded spurious Cepheid\ndistances for the Magellanic Clouds (e.g., µ0,LMC ̸= 18.1).\nUdalski et al. (2001), Majaess et al. (2011a), Wielg´ orski\net al. (2017), and Madore & Freedman (2023) concluded\nthat WV Ifunctions are relatively insensitive to metallic-\nity,¶whereas Riess & Breuval (2023, S H0ES) advocate for\na larger zeropoint dependence by comparison (e.g., −0.22±\n0.04 mag/dex). Madore & Freedman (2023) relayed TRGB-\n†Paradoxically, Majaess (2020) confirmed that blending with\nsay red clump giants may advantageously thrust otherwise faint\nextragalactic targets (e.g., RR Lyrae variables) into the range of\ndetectability (see also §2.4 of Majaess et al. 2018).\n‡Table 12 in Riess et al. (2009) and Fig. 4 in Majaess (2010).\n¶Bono et al. (2008) and Anderson et al. (2016) disagree re-\ngarding what models produced relative to the Leavitt Law’s de-\npendence on metallicity. §5.5 of Breuval et al. (2022) presents\ntheir viewpoint concerning the Wielg´ orski et al. (2017) result.Cepheid distances (their Fig. 1) which overturn the existing\nSakai et al. (2004) analysis, and likewise contest Breuval\net al. (2022, −0.201±0.071 mag/dex). Moreover, the reader\nis encouraged to examine Fig. 6 in Yuan et al. (2022, S H0ES)\nwhere a constant (indicating a null-dependence) can rep-\nresent their latest NGC4258 analysis rather than the fits\nthey overlaid. Yuan et al. (2022, S H0ES) constrained the\ndependence to −0.07±0.21 mag/dex, which together with\ntheir overall data are in stark contrast to the Macri et al.\n(2006) and Hoffmann et al. (2016, S H0ES) interpretations\nof NGC4258, and the reader can arrive at their own conclu-\nsion by inspecting Fig. 3 (see also §4 in Yuan et al. 2022).\nAlarmingly, there’s a significant I-band (F814W) and WV I\ndiscrepancy between Hoffmann et al. (2016, S H0ES) and\nYuan et al. (2022, S H0ES), which is characterized by a con-\nsiderable mean difference ( ≳0m.15). Regardless of applying\na Galactic WV ICepheid calibration (Benedict et al. 2007;\nMajaess et al. 2011a) or the Breuval et al. (2022) metallicity-\ndependent WV Ifunction to the NGC4258 datasets of Maoz\net al. (1999), Newman et al. (2001), Macri et al. (2006),\nFausnaugh et al. (2015), Hoffmann et al. (2016, S H0ES),\nand Yuan et al. (2022, S H0ES): discrepant results arise.\nOffsets in Cepheid distances across remote galaxies pos-\nsessing non-uniform crowding and surface brightness are\ncomparatively unassociated with a WV Imetallicity effect,\nbut rather point to serious shortcomings in establishing ho-\nmogeneous uncontaminated photometry, as echoed previ-\nously (e.g., Majaess et al. 2011a). The offset highlighted by\nMacri et al. (2001, their §5) resides elsewhere in extragalac-\ntic Cepheid data.\nThe Herrnstein et al. (1999, 7 .2±0.3 Mpc) maser distance\nfor NGC4258 was superseded by Humphreys et al. (2013,\n7.60±0.17±0.15 Mpc, see also §3 in Riess et al. 2016). Reid\net al. (2019) subsequently determined d= 7.576±0.082±\n0.076 Mpc.The Concerning S H0ES Hubble Constant 3\nFigure 3. NGC4258 Cepheid datasets exhibit striking differences with regards to: their distances as a function of apparent galac-\ntocentric radius (x-axis), a (null-)dependence on abundance, and the mean distance. Discrepancies emerge irrespective of whether a\nWV IGalactic Cepheid calibration (Majaess et al. 2011a, and references therein) is used (shown), or the metallicity-dependent WV I\nrelationship of Breuval et al. (2022).\nFigure 4. H0estimates from diverse CMB surveys. There exist\nsolutions beyond the Planck CMB, and which are comparable\nwithin the uncertainties to the Freedman & Madore (2023a, and\nreferences therein) TRGB analysis.\n3. Independent Cepheid-TRGB Analyses\nSH0ES-independent Cepheid and TRGB based analyses\ninclude the HST key project to measure H0, namely 68 ±\n5 and 72 ±8 km/s/Mpc (Gibson et al. 2000; Freedman\net al. 2001). Moreover, Tammann & Reindl (2013) cite afinal determination of H0= 64.1±2.0 km/s/Mpc (see also\nTammann & Sandage 2010). The ‘ Hubble tension’ debate\nemerged because of the latter’s passing. Admittedly, Ma-\njaess (2010) and others argued that the Sandage et al. (2004)\ndistance scale was too remote (e.g., Benedict et al. 2007;\nRiess et al. 2009), and hence their H0was underestimated.\nHowever, the impact of photometric contamination was not\ncomprehensively addressed by that team and would shift H0\nin the opposite direction.\nConcerns emerged near the conclusion of the HST key\nproject to secure H0that extraneous flux from stars along\nthe Cepheid’s sightline could lead to an overestimated ex-\npansion rate (e.g., Stanek & Udalski 1999), and the reader\nis encouraged to review the debate and rebuttals in §8.5 of\nFreedman et al. (2001), §7 in Mochejska et al. (2000), and §8\nin Mochejska et al. (2001). Freedman et al. (2001) attributed\na sizable uncertainty to the phenomenon (0m.1), and the\nFreedman et al. (2012, CHP) Cepheid effort shouldn’t be\nleveraged to support the S H0ES Cepheid H0since it re-\nlies on the earlier potentially contaminated HST key project\ndata (c2000). †Importantly, the current Freedman & Madore\n(2023a) TRGB efforts are pertinent in part because the\nteam’s modus operandi is to avoid crowded regions. Their\nH0is comparable to diverse CMB measurements (Fig. 4).\nIndeed, a consensus framework on the topic of photometric\ncontamination remains outstanding, in tandem with the ter-\nminology employed (Majaess 2020, §4 in Yuan et al. 2022,\nand see also the disparate discussions in Riess & Breuval\n†Freedman et al. (2012) advance a separate position in their\n§3.3 and appendix.4\nFigure 5. Majaess (2010) indicated the WV ILeavitt Law’s slope\n(α≃ −3.3) is insensitive to metallicity. Riess et al. (2009, S H0ES)\nargued that near-solar Cepheids adhered to α=−2.98±0.07, a\nresult they subsequently overturned ( α≃ −3.3, Riess et al. 2022,\nSH0ES). For example, Riess et al. (2009, S H0ES) determined\nthat NGC3021 Cepheids follow α=−2.60±0.24.\n2023 and Freedman & Madore 2023b). The reader can promptly\ngrasp the difficulties faced by inspecting Fig. 1 in Mochejska\net al. (2001), Fig. 2 in Majaess et al. (2012b), and critically\nFig. 8 in Freedman & Madore (2023b), which conveys HST\ncrowding relative to JWST.\n4. Conclusion\nHistorical H0uncertainties (Fig. 1), errors and anomalies\nexisting within S H0ES and NGC4258 data ( §2), a relatively\nunchanged S H0ESH0across ∼18 years (Fig. 2), diver-\nsity among CMB measurements (Fig. 4), and independent\nresults from Madore & Freedman (2023) and Freedman &\nMadore (2023a,b): indicate in concert that claims of a bona\nfideoffset between Cepheid and CMB H0findings should\nbe viewed cautiously.\nThe aforementioned ambiguities continue a long estab-\nlished trend tied to H0research (Fig. 1), from say Sandage\nand de Vaucouleurs (Overbye 1991), to Freedman et al.\n(2001) and Sandage et al. (2004), and now between Riess\net al. (2016, S H0ES), Freedman & Madore (2023a), and oth-\ners. TRGB and Cepheid research by groups and (co)authors\nbeyond S H0ES and (C)CHP are desirable, and that includes\na reassessment of the entire raw HST Cepheid imagery with-\nout biased constraints that inadvertently sway conclusions.\nThe case of NGC4258 broadly illustrates that a chief con-\ncern regarding H0determinations is the challenging task\nof obtaining precise, commonly standardized, multiepoch,\nmultiband, uncontaminated remote extragalactic Cepheid\nphotometry (e.g., Majaess 2010).\nContinued research is needed to assess Gaia’s data. The\nspurious Hipparcos distances to the Pleiades and Blanco\n1 are key lessons pointing toward enhanced vigilance (Ma-\njaess et al. 2011b). Moreover, oft-cited NGC4258 and LMC\ndistances could likewise be incorrect, and certain results\nmay require adjustment that conspire to sway H0unidi-\nrectionally. 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G., Casertano,\nS., Filippenko, A. V., Hoffmann, S. L., Huang, C. D., &\nScolnic, D. 2022, ApJ, 940, 64"    },    {        "title": "2401.13002v1.Theorem_Discovery_Amongst_Cyclic_Polygons.pdf",        "content": "P. Quaresma and Z. Kovács (Ed.): Automated Deduction\nin Geometry 2023 (ADG 2023).\nEPTCS 398, 2024, pp. 153–164, doi:10.4204/EPTCS.398.18© P.H. Todd\nThis work is licensed under the\nCreative Commons Attribution License.Theorem Discovery Amongst Cyclic Polygons\nPhilip Todd\nSaltire Software\nPortland OR USA\nphilt@saltire.com\nWe examine a class of geometric theorems on cyclic 2 n-gons. We prove that if we take n disjoint pairs\nof sides, each pair separated by an even number of polygon sides, then there is a linear combination\nof the angles between those sides which is constant. We present a formula for the linear combination,\nwhich provides a theorem statement in terms of those angles. We describe a program which uses this\nresult to generate new geometry proof problems and their solutions.\n1 Introduction\nIn [8], a characterization is made of linear systems involving angle bisection conditions which are not\nfull rank. In such a system, one of the conditions is implied by the remainder, and, if the angle bisections\nare interpreted geometrically, this dependence may be stated in a number of different ways as a geometry\ntheorem. The characterization leads to a catalog of such linear systems. An approach to theorem dis-\ncovery is proposed wherein a linear system is initially selected, and then interpreted geometrically as a\ntheorem. In [7], a program is described which applies this approach, constructing a particular geometry\ntheorem corresponding to a randomly selected linear system from the catalog. In order to reduce diagram\ncomplexity, the program is biased in favor of constructing cyclic polygons wherever possible. In the case\nwhere it is able to construct a cyclic polygon using all the rows of the linear system, the theorem which is\nproduced has the following form. Given a cyclic 2 n-gon, where n−1 specified pairs of sides are parallel,\nthen a final specified pair of sides is also parallel. For example, in a cyclic hexagon, with two pairs of\nopposite sides parallel, the third pair of sides is also parallel. (In passing, we note thet this theorem is not\ntrue for a cyclic octagon, but is for a cyclic decagon.)\nIn this presentation, we consider a generalization of the above class of theorems, where instead of\nmaking line pairs parallel, we allow line pairs to be given non zero, but determined angles. While there\nis no geometric meaning to a parallel relationship between consecutive sides, replacing the parallelism\nby a defined non-zero angle permits adjacent sides of the polygon to be related. We examine the case of\na 2n-gon, with the angles between npairs of sides named. We show that if the pairs share no side, and\nif the sides in each pair are either adjacent or seperated by an even number of polygon sides, then the\nnamed angles satisfy a particular linear relation.\nAn approach to theorem discovery in this context mirrors and illustrates that described in [8] and [7].\nAs a first step, a value of nis chosen, and a set of line pairs conforming to our criterion selected from\na catalog containing all such sets. A diagram is produced directly which allows the coefficients of the\nconstant linear combination to be computed.\n2 Cyclic Polygons and Angle Bisectors\nLetPbe a polygon whose vertices lie on the unit circle centered at the origin whose vertices p1,..., pn\nhave position vectors u1,..., un.154 Theorem Discovery\nFigure 1: θ0...θ6are angles of position vectors. θ6=θ1+2π.φi=1\n2(θi−1+θi)\nWe define u0=un.\nLetα(u,v)be the directed angle between vector uand vector v. We define θ0=0 and for ifrom 1 to\nn:\nθi=θi−1+α(ui−1,ui)\nAsu0=un,θn=θ0+2πWwhere Wis the winding number of the polygon about the origin. For ifrom\n1 tonwe define\nφi=1\n2(θi+θi−1)\nForiandjfrom 1 to nwe define\nδi j=φj−φi\nDefine Lito be the line passing through points pi−1andpi. We will define qi jto be the intersection of\nLiandLj. We define the angle\nψi j≠pi−1qi jpj\n2.1 Cyclic Quadrilateral\nWe first examine the cyclic quadrilateral (Figure 2).\nIn the figure, the quadrilateral has winding number 1 about the circle center, hence θ4=θ0+2π.\nThe two indicated opposite angles of the quadrilateral have values π−δ12andπ−δ34.P.H. Todd 155\nFigure 2: A cyclic quadrilateral with winding number 1.\nThe figure may be expressed by the following matrix equation:\n\n1 1 0 0 0 −2 0 0 0\n0 1 1 0 0 0 −2 0 0\n0 0 1 1 0 0 0 −2 0\n0 0 0 1 1 0 0 0 −2\n−1 0 0 0 1 0 0 0 0\n0 0 0 0 0 −1 1 0 0\n0 0 0 0 0 0 0 −1 1\n\nθ0\nθ1\nθ2\nθ3\nθ4\nφ1\nφ2\nφ3\nφ4\n=\n0\n0\n0\n0\n2π\nδ12\nδ34\n(1)\nWe transform the matrix equation by performing the following row operations:\nR1←R1+R5,R2←R2+2×R6, and R3←R3+2×R7, which, after eliminating the zero columns,\ngives this matrix equation:\n\n1 0 0 1 −2 0\n1 1 0 0 −2 0\n0 1 1 0 0 −2\n0 0 1 1 0 −2\n\nθ1\nθ2\nθ3\nθ4\nφ1\nφ3\n=\n2π\n2δ12\n0\n2δ34\n(2)\nThe matrix can be triangularized using the algorithm of [2]. Let Ribe the i’th row of the original matrix,\nandTithe i’th row of the triangularized matrix, then T1=R1, and for ifrom 2 to n:\nTi=Ri−Ti−1 (3)156 Theorem Discovery\nUsing this algorithm, our triangularized matrix equation is:\n\n1 0 0 1 −2 0\n0 1 0 −1 0 0\n0 0 1 1 0 −2\n0 0 0 0 0 0\n\nθ1\nθ2\nθ3\nθ4\nφ1\nφ3\n=\n2π\n2δ12−2π\n−2δ12+2π\n2δ34+2δ12−2π\n(4)\nConsistency of this system requires\n2δ34+2δ12−2π=0 (5)\nor\nδ34+δ12=π (6)\nIn terms of ψ12andψ34\nπ−ψ12+π−ψ34=π (7)\nHence ψ12+ψ34=π, which is the familiar result that the opposite angles of a cyclic quadrilateral are\nsupplementary.\n2.2 General Theorem\nThe following general theorem may be proved by an analogous approach to that employed in the above\nsection.\nTheorem 2.1. Given a cyclic 2n-gon with winding number W about the circumcircle’s center, and n\nordered pairs (a1,b1)...(an,bn)such that\n{ai}∪{ bi}={1...2n}, bi>aiand b i−aiis odd for each i\nn\n∑\ni=1(−1)biδaibi=Wπ (8)\nProof. LetPi,j=−2 where j=aiorj=biand 0 otherwise, let Q1=2πW. For i>1, let Qj=2δai,biif\nj=biand 0 otherwise.\nAnalogous to equation (2) we have:\n\n1 0 ··· 0 1 P1,1··· Pn,1\n1 1 ··· 0 0 P1,2··· Pn,2\n........................\n0 0 ··· 1 1 P1,2n··· Pn,2n\n\nθ1\nθ2\n...\nθ2n−1\nθ2n\nφ1\n...\nφn\n=\nQ1\nQ2\n...\nQ2n\n(9)P.H. Todd 157\nTriangulation yields a matrix equation analogous to that of (4(\n ..................\n0··· 0R1··· Rn!\nθ1\n...\nθ2n\nφ1\n...\nφn\n= ...\nS!\n(10)\nwhere\nRi=2n\n∑\nj=1(−1)jPi,j (11)\nand\nS=2n\n∑\nj=1(−1)jQj (12)\nHence\nRi= (−1)bi+ (−1)ai(13)\nand\nS=n\n∑\ni=1(−1)bi2δaibi−2Wπ (14)\nAsbi−aiis odd, Ri=0 for all i. Hence for consistency S=0. Hence the result.\n3 Automated Discovery of Cyclic Polygon Theorems\nWe describe a mechanized process for automatic theorem generation for angles in cyclic polygons. Each\ntheorem will establish a relationship between angles in a cyclic polygon. The process breaks down into\nthree steps. First a set of side pairs is chosen which satisfy the criteria of Theorem 2.1. Secondly, a\nspecific location is decided for the vertices of the polygon.and a geometry diagram created. Angles\nbetween the side pairs are given names and drawn on the diagram. Thirdly, (8) is expressed in terms of\nthese angle names, yielding the conclusion of our theorem statement.\n3.1 Choosing Side Pairs\nIn order to use theorem 2.1 to create cyclic polygon theorems, we need to first find a set of ordered pairs\n(a1,b1)...(an,bn)such that {ai}∪{ bi}={1,..., 2n}and|bi−ai|is odd for each i\nWe then sort the elements so that ai<bi. Each pair (ai,bi)contains an odd and an even integer. Hence\nany set of pairs corresponds to a 1-1 mapping between A={1,3,..., 2n−1}andB={2,4,..., 2n}. A\nrandom such mapping is selected. Applying theorem 2.1 yields an expression for a linear combination\nof the differences in direction δaibi.\nThe number of such mappings is n!. However, when considered as pairings between sides of a cyclic\npolygons, there are many which are simply rotated or reflected images. The number of non-isomorphic\npatterns for n=2 to 7 are 1 ,3,5,17,53,260 [3]. Figure 3 depicts these patterns for nequal to 3 and 4.158 Theorem Discovery\nFigure 3: Graphs representing possible conforming sets of edge pairs for n=3,4. Vertices on these\ngraphs correspond to polygon sides. Edges indicate the side pairs whose angles will be specified\n3.2 Diagram Creation\nA diagram should be constructed containing the circle and the cyclic polygon. Points qaibishould be\ncomputed and angles ψaibishould be marked on the diagram and named. We need to relate the named\nangles: ψaibito the angles treated in our theorem: δaibi.\nOne straightforward approach is to create a polygon whose vertices are arranged clockwise around\nthe circle, making a single circuit, but at non-regular intervals. Constraints may be set on the location\nof the polygon vertices to ensure that the angle between chosen side-pairs is not too small for their\nintersection to appear on the diagram. This intersection would be displayed, along with a named angle\n(Figure 5).\n3.3 Geometric Angle Conclusion\nThe relation between δi jandψi jdepends on whether the circle center is on the same side of wiandwj\nand whether a rotation from witowjis clockwise or counter-clockwise.\nWhen the polygon is convex and has a winding number of 1 around the circumcenter, the formula is\nas follows\nLets=sgn(wi∧wj)and\nδi j=π−s·(ψi j) (15)\n4 Example\nAs an example, we start with one of the sets of pairings for the decagon: {(1,2),(3,10),(4,7),(5,8),(6,9)}\n(Figure 4).P.H. Todd 159\nFigure 4: Graph for the set of side pairings {(1,2),(3,10),(4,7),(5,8),(6,9)}\nApplying Theorem 2.1 to this set of side pairings, and assuming a winding number of 1 gives the\nfollowing:\nδ1,2+δ3,10−δ4,7+δ5,8−δ6,9=π (16)\nA cyclic decagon is drawn, where the vertices are placed at random subject to constraints that the\npolygon does not self-intersect, the sides are not too small, and the winding number is 1. Angles between\nthe five side pairs are marked and named. In the case of non-contiguous sides, the sides are extended in\nthe diagram to their intersection (Figure 5). As their angles are counter-clockwise, we have:\nFigure 5: Cyclic decagon with angles marked for side pairings {(1,2),(3,10),(4,7),(5,8),(6,9)}\nδ1,2=π−ψ1,2\nδ4,7=π−ψ4,7\nδ5,8=π−ψ5,8\nδ6,9=π−ψ6,9(17)160 Theorem Discovery\nHowever, the angle for side pair (3,10) is clockwise, so\nδ3,10=π+ψ3,10 (18)\nSubstituting these values into (16) gives the following\nπ−ψ1,2+π+ψ3,10−(π−ψ4,7) +π−ψ5,8−(π−ψ6,9) =π (19)\nwhich can be presented as this theorem conclusion\nψ3,10+ψ4,7+ψ6,9=ψ1,2+ψ5,8 (20)\nWe note that it would be possible to draw a polygon such that the angle between lines 3 and 10 is\nclockwise. In this case (21) becomes\nδ3,10=π−ψ3,10 (21)\nand the theorem conclusion (20) becomes\nψ4,7+ψ6,9=ψ1,2+ψ5,8+ψ3,10 (22)\n5 An Automated Problem Generator\nFigure 6: (a) Given convex cyclic hexagon ABCDEF ,̸ABC +̸CDE +̸EFA =2π. (b) Given convex\ncyclic pentagon ABCDE ,̸ABC +̸CDE−̸ECA =π. (c) Given convex cyclic hexagon ABCDE with\ncenter O,̸ABC +̸CDE +̸OEA =3π\n2.\nAn automated problem generator based on the above approach was created [5], and introduces further\nelaborations. First, rather than the vertices of the polygon lying in order around the circle, we allow any\npermutation of the vertices. With any but the identity permutation, the polygon will be self-intersecting\nand will, in fact, be made up of some subset of the sides and diagonals of a convex polygon.\nA theorem for a polygon with an odd number of sides may be constructed from a polygon with one\nmore side by merging a pair of points in the larger polygon. For example, figure 6 (a) shows a theoremP.H. Todd 161\nlinking three alternating angles of cyclic polygon ABCDEF . Figure 6(b) shows the result of merging\npoints FandCof the hexagon. Edges EFandFAnow become the diagonals ECandCAof a cyclic\npentagon ABCDE , and the theorem relates two angles of the pentagon and an angle between two of its\ndiagonals. If the points which are merged share an edge of the larger polygon, that edge will be replaced\nby a line joining the center of the circle to the merged point, and the theorem statement will be altered\nbyπ\n2. Figure 6(c) shows the result of merging points FandEof the hexagon ABCDEF . Angle AFE\nbecomes, in the limit, the angle between AEand the tangent to the circle at E, which differs from OEby\nπ\n2(where Ois the circle center).\nIn addition to presenting the theorems, our problem generator [5] can create a human readable proof\nby propagating a handful of simple angle generation rules. A single step of the proof generator proceeds\nby finding all the angles determined by the application of the following set of rules to the known angles:\n1. If ADis between ABandACthen ̸BAC ≠DAB +̸DAC .\n2. The angles of a triangle add to π.\n3. Two angles which form a line add to π.\n4. Two angles on the same side of the same chord of a circle are the same.\n5. Opposite angles of a cyclic quadrilateral add to π.\n6. The angle at the center of a circle is twice the angle on the same chord at the circumference.\n7. If ABis a chord of circle centered Othen ̸OAB ≠OBA =1\n2(π−̸AOB ).\nFigure 7: Automatically generated diagram with problem statement: Let ABCDEF be a cyclic hexagon\nwith center O. Prove that BDE =ACE +AFB162 Theorem Discovery\nOur theorems can all be expressed as a linear combination of the angles and π. The result of each\nangle generation rule is a linear combination of its inputs and π. Hence a convenient way of storing the\ncomputed value of any angle is as a vector of coefficients of the given angles and π. The proof generator\nis started with the known angles and proceeds to compute values for all angles which can be computed\nfrom those by application of the above rules (breadth first). The angle value is stored, as a vector of\ncoefficients of the known angles and π. Along with the value, a reference is stored to the rule which was\nfired, and the arguments it was applied to. If the process finds a new value for a given angle, this new\nvalue can be equated to the given angle yielding a linear expression. The sequence of rule applications\nused to reach the final expression may be turned into the words of a proof.\nFor example, in figure 7, we set ̸BDE = (1,0,0,0),̸ACE = (0,1,0,0),̸AFB = (0,0,1,0).\nApplying (4) to ̸AFB gives ̸ACB = (0,0,1,0).\nApplying (4) to ̸BDE gives ̸BCE = (1,0,0,0).\nApplying (1) to ̸BCE and̸ACE gives ̸ACB = (1,−1,0,0).\nBut we already have ̸ACB = (0,0,1,0)so our linear relation can be expressed as (1,−1,−1,0) =0.\nThis would result in the following proof.\nLetBDE =x. Let ACE =y.LetAFB =z.\nAsAFB andACB stand on the same chord, ACB =AFB , soACB =z.\nAsBDE andBCE stand on the same chord, BCE =BDE , soBCE =x.\nAsACE =y,ACB =x−y.\nButACB =z, sox−y=z, orx=y+z, orBDE =ACE +AFB .\nIf the proof generator does not run to completion, additional geometry is added to the diagram in\ntwo phases. First, missing line segments are added, which allow additional angles to be derived (angles\nare only computed when the line segments joining their vertices are present). A second step is for an\nadditional angle to be specified as known, and given a name. As long as the new angle is not in the span\nof the known angles, it cannot show up in the eventual linear relation, and hence is guaranteed to simplify\nout in the course of the proof.\nFor example the following is the proof generated for the theorem stated in figure 8.\nTheorem 5.1. Let ABCDE be a cyclic pentagon with center O. Let F be the intersection of AE and\nDC. Let G be the intersection of OE and CB. Let H be the intersection of ED and BA. BGE +AHE =\nDFE +90.\nProof. Draw line BE.\nLet̸DFE =x. Let ̸BGE =y. Let ̸AHE =z.\nLet̸AEH =w.\nAs̸AHE =z,̸EAH =180−z−w.\nAs̸EAH =180−z−w,̸EAB =z+w.\nAs̸AEH =w,̸HEF =180−w.\nAs̸FEH =180−w,̸FED =w.\nAs̸DEF =w,̸EDF =180−x−w.\nAs̸EDF =180−x−w,̸EDC =x+w.\nAsCDEB is a cyclic quadrilateral, ̸CBE =180−̸CDE , so̸CBE =180−x−w.\nAs̸EBG =180−x−w,̸BEG =x+w−y.\nAs triangle BEO is isosceles, ̸BOE =2y−2x−2w+180.\nAs̸BOE is at the center of a circle on the same chord, but in the opposite direction to ̸BAE ,\n̸BOE =360−2BAE , so̸BAE =x+w−y+90.\nBut̸BAE =z+w, sox+w−y+90=z+w, orx+90=y+z, or̸DFE +90≠BGE +̸AHE .P.H. Todd 163\nFigure 8: Let ABCDE be a cyclic pentagon with center O. Let Fbe the intersection of AEandDC. Let\nGbe the intersection of OEandCB. Let Hbe the intersection of EDandBA. Prove that BGE +AHE =\nDFE +90.\nAutomated proofs can be simply adapted to provide step-by-step solutions to problems framed in\nterms of determining an unknown angle, either numerically or algebraically.\nOur generator can be set to create problem collections. It is important for such collections to eliminate\nproblems which are simply rotated or reflected duplicates. For a hexagon, for example, we see from\nfigure 3 that there are 3 different pair patterns. When we combine this with the different permutations of\n6 points, and remove rotated and reflected duplicates, we end up with 49 distinct diagrams.\nFor the pentagon, we can choose any pair of points to merge, but again need to avoid duplication by\nkeeping track not only of the diagrams which have been produced, but also of their rotated and reflected\nimages. The pentagon yields 54 distinct diagrams. [4] shows the complete collection of pentagon and\nhexagon theorems generated in this way.\nA heptagon theorem may be generated from an octagon theorem by merging a pair of points. A\nhexagon theorem may be formed from the heptagon theorem by merging a further pair of points. Ad-\nmitting such “four angle” hexagon theorems expands the number of possibilities to hundreds. Admitting\noctagons and heptagons expands the possible theorem count to thousands. [6] is a collection of 200\nrandom theorems and their proofs generated by [5] .\n6 Conclusion\nThe angles considered in this paper are not the full angles which have their own place in automated\ndeduction in geometry [1], and which are largely impervious to changes in the diagram caused by simply164 Theorem Discovery\npositioning the points in different locations. The angles we use are well defined: an angle ABC means\nthe non-reflex undirected angle defined by the line segments ABandBC. While the angle is well defined,\nits relationship to the directed angle, which is the basis of, for example, (16) is highly dependent on the\nrelative location of the points on the diagram. When an angle passes through π, the relation between\nthe ordered angle and the unordered non-reflex angle changes and, in our setting, the theorem statement.\nThis is both a blessing and a curse. A curse because it is difficult to give a succinct general theorem\nstatement without a complicated set of side conditions. In the context of automated problem generation,\nhowever, the curse becomes a blessing, as it proliferates the number of distinct examples which may be\ngenerated.\nFrom a pedagogical standpoint, the problems generated have several advantages. They can all be\nsolved by applying a small set of geometric facts, however some ingenuity in applying them is called\nfor. Algebraically, the problems require only the manipulation of linear equations. Problems can be\nphrased as the determination of unknown quantities, either numerically or symbolically, or as proof\nproblems. Finally the availability of machine generated human-readable proofs gives a convenient means\nof scaffolding.\nReferences\n[1] Shang-Ching Chou, Xiao-Shan Gao & Jing-Zhong Zhang (1996): Automated generation of readable proofs\nwith geometric invariants. II. Theorem proving with full-angles .Journal of Automated Reasoning 17(3), pp.\n349–370, doi:10.1007/BF00283134.\n[2] B. D. Saunders (2015): Matrices with two nonzero entries per row . In: Proceedings of the 2015 ACM on Inter-\nnational Symposium on Symbolic and Algebraic Computation , pp. 323–330, doi:10.1109/TIT.1986.1057137.\n[3] N. J. A. Sloane: Sequences A357442 in \"The On-Line Encyclopedia of Integer Sequences.\" . Available at\nhttps://oeis.org/A357442 .\n[4] P. H. Todd (2023): The Complete Set of Automatically Generated Angle Problems for Cyclic Hexagon and\nPentagon. Saltire Technical Report 23-1 . Available at https://saltire.com/download/TR2023-1.pdf .\n[5] P. H. Todd (2023): Cyclic Polygon Angle Problem Generator . Available at https://www.saltire.com/\nProblemGenerator .\n[6] P. H. Todd (2023): A Set of 200 Cyclic Polygon Angle Problems. Saltire Technical Report 23-2 . Available at\nhttps://saltire.com/download/TR2023-2.pdf .\n[7] P. H. Todd & D. Aley (2023): A program to create new geometry proof problems .Annals of Mathematics and\nArtificial Intelligence , pp. 779–795, doi:10.1007/s10472-023-09854-1.\n[8] Philip Todd (2023): Automated Discovery of Angle Theorems .Annals of Mathematics and Artificial Intelli-\ngence , doi:10.1007/s11786-020-00490-0."    },    {        "title": "2401.13005v1.Viscous_universe_with_cosmological_constant.pdf",        "content": "Viscous universe with cosmological constant\nJinwen Hua, Huan Hua\naDepartment of physics and technology, Wuhan university, Wuhan, 430060, China\nAbstract\nWe investigated a bulk viscous fluid universe with cosmological constant Λ by assuming that the\nbulk viscosity to be proportional to the Hubble parameter. We found that for an expanding uni-\nverse, the (relative) matter density will be always greater than a non-zero constant, and tends to\nthis non-zero constant in the future. We show that the bulk viscosity model has a significantly\nbetter fitting to the combined SNe Ia+ CMB + BAO + H(z) data than the ΛCDM model. Gen-\nerally, the evolution or values of some cosmological parameters predicted by the bulk viscosity\nmodel do not deviate significantly from which are obtained from the ΛCDM model since the bulk\nviscosity coefficient obtained from the astronomical observational data is so small. We also made\na statefinder analysis of the bulk viscosity model and found that the evolution of the {r,s}pa-\nrameters behaves in such a way that 0 < s < 1,0.945< r < 1, indicating the bulk viscosity model\nis different from the ΛCDM model and the other “dark energy” model.\nKeywords: Cosmological constant, Bulk viscosity, Dark energy, ΛCDM model, Statefinder\nanalysis\nPACS: 95 .36.+ x, 98.80.-k, 98.80.Es\n1. Introduction\nObservations data, for instance the data from type Ia supernovae [1, 2], the large-scale structure\nof universe [3] and cosmic microwave background (CMB) [4], indicated that our universe is spatially\nflat and accelerating, which is considered that there exists an exotic cosmic fluid (called the dark\nenergy) that accounts for about 2 /3 of the total energy of the universe. Despite the observational\nevidence on this existence of “dark energy”, its fundamental and nature origin is still unknown,\nresulting in many models have been proposed. Generally there are two different types of model.\nThe first one is to modify the right side of the Einstein equation by introducing a special energy-\nmomentum tensor Tµνwith a negative pressure. The simplest model for this type of model is\nthe ΛCDM model, which introduce a cosmological constant Λ as the dark energy, and the ΛCDM\nmodel is characterized by the equation of state, i.e. , ωΛ=−1. However, there exists some\nproblems in the ΛCDM model, such as the coincidence problem [5], which refers to the coincidence\nthat in today’s universe the matter density and the dark energy density happen to be the same\norder of magnitude although they evolve at a different way. A possible answer to this question is\nthat the dark energy is dynamical.\nOn the other hand, the process of dissipation arose in the viscous fluid, including both shear\nviscosity and bulk viscosity, as many studies shown, may be involved in the evolution of universe\nas an important role [6–8]. Eckart [9], Landau and Lifshitz [10] first studied the relativistic viscous\nEmail address: 200731890025@whu.edu.cn/2007.hujinwen@163.com (Jinwen Hu)\nPreprint submitted to Elsevier January 25, 2024arXiv:2401.13005v1  [gr-qc]  23 Jan 2024fluid and derived the parabolic differential equations. But they just considered the first-order\ndeviation from equilibrium, and from the obtained equations one can obtain that the heat flow\nand viscosity have an infinite speed of propagation, which is in contradiction with the principle\nof causality. Subsequently, Israel and Stewart [11–14] studied the second-order deviation from\nequilibrium, and applied it to study the evolution of the early universe [15]. However, in the\ncontext of the full causal theory the character of the evolution equation is so complicated that\nthe simplified phenomenon, i.e., the quasi-stationary phenomenon, are studied by the conventional\ntheory [12–14]. In the isotropic and homogeneous universe, the dissipative process is usually\ncharacterized by a bulk viscosity parameter ζ. And as the usual practice [16], the effect of the\nshear viscosity ηwill be neglected. The dissipative effect caused by the bulk viscosity can be\nredefined by the effective pressure, Pvisc=−3ζH, where ζis the bulk viscosity coefficient and\nHis the Hubble parameter. And with the requirement of the second law of thermodynamics, it\nrequires that ζ >0, which assures that the entropy production is positive [17, 18].\nIn the late accelerating universe, the effect of bulk viscous fluid was studied in Refs. [19–23].\nBut a drawback faced by the bulk viscous fluid is to find a valid mechanism for the origin of\nbulk viscosity in the expanding universe. According to some theoretical point of view, when the\nlocal thermodynamics equilibrium is broken, the bulk viscosity arises [24]. And the bulk viscosity\ncan be regarded as an effective pressure to keep the system back to its thermal equilibrium state.\nThe bulk viscosity pressure arises when the cosmological fluid expands or contract too fast (i.e.,\nthe state deviated from the local thermodynamics equilibrium) [25–27], and ceases when the fluid\nreaches the thermal equilibrium again.\nIn an expanding universe, the expansion process is likely to be a collection of states that lose\ntheir thermal equilibrium in a small fraction of time [28]. It is therefore natural to assume that there\nexists a bulk viscosity coefficient in the description of the universe. In this paper, we investigated\na bulk viscous matter universe with cosmological constant Λ, presenting a model with equivalent\ndynamical “dark energy”. This paper is organized as follows. In Sect. 2 we present the basic\nformalism of the flat universe with bulk viscous fluid (called the bulk viscosity model) and show\nthe interesting conclusions from this model. And we derived the evolution equation of the Hubble\nparameter. In Sect. 3, we investigated the ΛCDM model and the bulk viscosity model with the\nobservational data from SNe Ia, CMB, BAO and Hubble parameter. In Sect. 4 we present the\nstatefinder analysis of the bulk viscosity model. In Sect. 5 we summarized this paper.\n2. FLRW universe with bulk viscous matter\nWe consider a spatially flat universe described by the Friedmann-Lemaitre-RobertsonWalker\n(FLRW) metric [29, 30]\nds2=−dt2+a(t)2\u0000\ndr2+r2dθ2+r2sin2θdϕ2\u0001\n(1)\nwhere ( r, θ, ϕ ) are defined as the co-moving coordinates, tdenotes the proper time of cosmic, and\na(t) denotes the scale factor of the universe with bulk viscous matter, which produce an equivalent\npressure [10, 31]\nPeff=P−3ζH (2)\nwhere His the Hubble parameter, Pdenotes the normal pressure, which is equal to 0 for the\nnonrelativistic matter, and ζis generated in the viscous fluid that deviates from the local thermal\n2equilibrium, and ζcan be a function of Hubble parameter and its derivatives. Here we didn’t\nconsider the radiation component for it is a reasonable simplification to the late universe.\nBased on Eq. (2), for an imperfect fluid, and with the first-order deviation from the thermo-\ndynamic equilibrium, we can obtain the form of the energy-momentum tensor [32]\nTµv=ρuµuv+ (gµv+uµuv)Peff (3)\nwhere ρdenotes the density of the fluid.\nIt can be seen that the above energy-momentum tensor is similar to that of a perfect but with\nan equivalent pressure Peff, which is composed by the pressure Pof the fluid plus the viscous\npressure Pvisc=−3ζH. And as stated above that the viscous pressure, Pvisc, can be regarded as\na “measurement” of the pressure to restore the local thermodynamic equilibrium.\nTaking the cosmological constant Λ into consideration, we conclude that for a pressureless\nmatter universe ( P= 0) governed by General Relativity in the presence of a viscous fluid, and\nwith the FLRW geometry the two Friedman equations can be written as\nH2=8πG\n3ρm+Λ\n3(4)\n¨a\na=−4πG\n3(ρm−9Hζ) +Λ\n3(5)\nWhere ρmis the total matter density, Gis the gravitational constant, Λ denotes the cosmological\nconstant, and the dot means time derivative. The conservation equation corresponding to Eq. (3)\nis\n˙ρm+ 3H(ρm−3Hζ) = 0 (6)\nFor an expanding universe (i.e., H > 0 ), we have\n˙ρm<0 (7)\nTherefor, based on Eq. (6), we obtain\nρm>3Hζ (8)\nSubstituting Eq. (4) into Eq. (8), we obtain\nH2−8πGζH −Λ\n3>0 (9)\nNote that Eq. (9) always holds in an expanding universe. Here we defined the dimension-\nless Hubble parameter E, the present (relative) vacuum density parameter Ω Λ0and the present\n(relative) matter density parameter Ω m0as\nE=H\nH0,ΩΛ0=Λ\n3H2\n0Ωm0=8πG\n3H2\n0ρm0 (10)\nWhere H0is the present value of the Hubble parameter, ρm0is the present matter density.\nOn the other hand, based on Eqs. (4) and (5), we have\n˙H=−4πG(ρm−3Hζ) (11)\n3Obviously, it can be seen that the evolution of His related to the parameter ζ. But so far we\nstill don’t know the explicit form of the bulk viscosity, which has to be usually assumed a priori\n(see for instance Refs. [33–38], the bulk viscosity coefficient is assumed a prior without being\nderived from known physics). In Refs. [39–48], it is assumed that the parameter ζis relate to the\nHubble parameter, i.e., ζ=ζ(H), and one of the assumed forms is\nζ=ζ0+ζ1H (12)\nWhere ζ0, ζ1are constants. The motivation for considering this form of bulk viscosity comes\nfrom the fluid mechanics, which holds that the viscosity phenomenon is related to the “velocity”.\nAnd both ζ=ζ0andζ=ζ1Hare separately studied by many authors.\nHere we take the simple case of ζ=ζ1Has the bulk viscosity model, which follows the\nassumption on the parameter ζin Refs. [39, 43, 45, 48, 49].\nThe other reason why we take the simple case of ζ=ζ1His that, substituting ζ=ζ1Hinto\nEq. (9), we can obtain\nΩm>8πGζ 1 (13)\nEq. (13) indicates that for an expanding universe, the (relative) matter density Ω mwill be\nalways greater than a non-zero constant, i.e., 8 πGζ1, and tends to this non-zero constant in the\nfuture, which is an interesting model, and based on Eq. (11), it indicates that the universe in the\nbulk viscosity model tend to the de Sitter time-space in the future, which is similar to the ΛCDM\nmodel, but the expansion rate of the future’s universe predicted by the bulk viscosity model is\nlower than which predicted by the ΛCDM model.\nBased on Eqs. (4), (5) and (10), we can obtain\ndE\ndz=(1.5−Ωτ)E\n1 +z−1.5ΩΛ0\nE(1 +z)(14)\nWhere Ω τ= 12πGζ 1is a dimensionless parameter, zdenotes the cosmological red shift, and\na= 1/(1 +z).\n3. Parameters estimation\n3.1. Model constraints from SNe Ia alone\nIt is well known that the ΛCDM model has been constrained by the observational data, such\nas the SNe Ia data, Power spectrum of the Cosmic Microwave Background fluctuations (CMB),\nBaryon Acoustic Oscillations (BAO), etc. Now we first consider the SNe Ia data alone. In Ref.\n[50], with the JLA SNe Ia sample, the authors found that the best fit value for Ω m0is 0.295±0.034\nwith a fixed fiducial value of H0= 70 km /s/Mpc. So here we first take Ω m0= 0.295,ΩΛ0= 1\nΩm0= 0.705 in Eq. (14) to study the evolution of the parameter E, which is shown in figure 1.\nIn a spatially flat universe, the luminosity distance dLis defined as [50]\ndL= (1 + z)Zz\n0dz′\nH(15)\nAnd the theoretical distance module µtfor the kth Supernova at red shift zkis defined as\nµt=m−M= 5 log10\u0012dL\nMpc\u0013\n+ 25 (16)\n4Figure 1: Behavior of the parameter Ewhen taking Ω m0=0.295,ΩΛ0= 0.705 for the two models. The solid line\ncorresponds to the Λ CDM model, the dotted line corresponds to the bulk viscosity model with Ω τ= 0.01, the\ndashed line corresponds to the bulk viscosity model with Ω τ=0.02.\nFigure 2: The comparison of the z ∼µtcurve when taking Ω m0=0.295,ΩΛ0= 0.705, H0= 70 km /s/Mpc for\nthe two models. The solid line corresponds to the ΛCDM model, the dashed line corresponds to the bulk viscosity\nmodel with Ω τ= 0.02.\nWhere mandMare the apparent and absolute magnitudes of the SNe, respectively.\nFigure 2 shows the comparison of the z ∼µtcurves derived from the ΛCDM model and bulk\nviscosity model with the same fiducial value of Ω m0= 0.295,ΩΛ0= 1−Ωm0= 0.705, H0=\n70 km /s/Mpc.\nNote that the JLA SNe Ia sample adopted in this paper cover the red shift range 0 .01< z < 1.2.\nAnd from Figure 1 and Figure 2 it can be seen that the two curves are basically coincident or have\nlittle deviation ( <1%) from each other within the range 0 .01< z < 1.2. With the negligible\ndeviation ( <1%) of the parameter Eandµt, we can conclude that within the red shift range\n0.01< z < 1.2, the best estimated parameters corresponding to the two models are almost the\nsame. That is, for the bulk viscosity model proposed in this paper, with a fixed fiducial value\nofH0= 70 km /s/Mpc, the best estimated parameters obtained from the JLA SNe Ia sample are\nΩm0= 0.295±0.034, Ω Λ0= 0.705±0.034,Ωτ= 0.01±0.01.\n3.2. Data from Power spectrum of the Cosmic Microwave Background fluctuations\nThe temperature power spectrum of CMB is sensitive to the physics of decoupling epoch and\nthe physics between today and the decoupling epoch. The former mainly affects the ratio of the\n5peak heights and the Silk damping, i.e., the amplitude of acoustic peaks, while the latter affects the\nlocations of peaks via the angular diameter distance out to the decoupling epoch. The “acoustic\nscale” lAand the “shift parameter” Ris defined as [51]\nlA=πr(z∗)\nrs(z∗), R =p\nΩm0Zz∗\n0dz′\nE(z′)(17)\nWhere z∗is the red shift at decoupling, r(z) is the co-moving distance, which is defined as\nr(z) =1\nH0Zz\n0dz′\nE(z′)(18)\nandrs(z∗) is the co-moving sound horizon at recombination, which is defined by\nrs(z∗) =Za(z∗)\n0cs(a)\na2H(a)da (19)\nWhere the sound speed cs(a) is defined by\ncs(a) =\u0014\n3\u0012\n1 +3Ωb0\n4Ωγ0\u0013\u0015−1/2\n(20)\nTo determine the optimal value of the parameters (with 1 σerror at least) involved in the dark\nenergy model, we will adopt the maximum likelihood method. That is, when the observation data\nsatisfy the Gaussian distribution, then the corresponding likelihood function will satisfy L∝e−x2/2,\nwhere the statistical function χ2represents the deviation between the theoretical value derived\nfrom a certain model and the observed value. The maximum likelihood method refer to finding\nthe optimal value of the parameter involved in the theoretical model to minimize the value of χ2,\nand when the observation data we use to restrict the theoretical model are not independent of each\nother, such as the above three observation data from CMB, i.e., lA, R, z∗, then the χ2function\ncorresponding to the CMB data can be given as [52]\nχ2\nCMB =XTC−1\nCMBX (21)\nWhere\nX=\nlA−302.40\nR−1.7246\nz∗−1090.88\n (22)\nand the inverse covariance matrix is given by [52]\nC−1\nCMB =\n3.182 18 .253 −1.429\n18.253 11887 .879−193.808\n−1.429−193.808 4 .556\n (23)\nBased on Eqs. (17) ∼(23), and with the best estimated parameters obtained from the JLA\nSNe Ia sample above, we can calculate the χ2, which is shown in Table 1.\nTable 1 shows the value of χ2obtained from the bulk viscosity model is significantly less than\nthe value of χ2andχ2min ( χ2min corresponds to the ΛCDM model with the best estimated\nparameters obtained from the CMB data alone) obtained from the ΛCDM model, which imply the\nbulk viscosity has a significantly better fitting to the combined JLA SNe Ia and CMB data than\nthe ΛCDM model.\n6Table 1: Computation of the parameters in CMBa)\nParametersb)Theoretical value\nfrom modelsObserved\nmean value\nin Ref. [42]χ2χ2min\nΛCDM\nmodelBulk viscosity\nmodelΛCDM\nmodelBulk viscosity\nmodelΛCDM\nmodel\nz∗1090.88 1090.88 1090.88\nlA 337.22 302.56 302.40 3887.78 13.75 1280.06\nR 1.7468 1.8293 1.7246\na) Here we adopt the best estimated parameters obtained from the JLA SNe Ia sample in Sect. 3.1, i.e., in the Λ\nCDM model Ω m0= 0.295,ΩΛ0= 0.705 and in the bulk viscosity model Ω m0= 0.295, Ω Λ0= 0.705,Ωτ= 0.018\n(within the range of Ω τ= 0.01±0.01 ), instead of computing the “best estimated” for parameters from the\nCMB sample alone, for the purpose to directly compare the two models.\nb)The computation method of z∗can be seen in Ref. [53].\n3.3. Data from Baryon Acoustic Oscillations\nIn Ref. [52], it shows that the acoustic peak in the galaxy correlation function has been detected\nin the red shift range z= 0.1∼0.7. This linear feature in the galaxy data provide a standard\nruler with which to measure the distance ratio, rs/Dv. And the position of the BAO can be used\nto constrain dz≡rs(zd)/Dv(z), where rs(zd) is the co-moving sound horizon at the baryon drag\nepoch, and the Dv(z) is defined by [54]\nDv(z)≡\"\u0012Zz\n0dz′\nH(z′)\u00132z\nH(z)#1/3\n(24)\nSince the release of the seven-year WMAP data, the acoustic scale has been precisely measured\nby the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) galaxy surveys and Sloan Digital\nSky Survey (SDSS), and the WiggleZ and 6dFGRS survey. The measurement results summarized\nfrom the previous work are shown in Table 2.\nTable 2: BAO data in the Nine-Year analysis\nRed shift z Data Set rs(zd)/Dv(z) Reference\n0.1 6dFGRS 0 .336±0.015 Beutler et al. (2011) [55]\n0.35 SDSS-DR7-rec 0 .113±0.002 Padmanabhan et al. (2012) [56]\n0.57 SDSS-DR9-rec 0 .073±0.001 Anderson et al. (2012) [57]\n0.44 WiggleZ 0 .0916±0.0071 Blake et al. (2012) [58]\n0.60 WiggleZ 0 .0726±0.0034 Blake et al. (2012) [58]\n0.73 WiggleZ 0 .0592±0.0032 Blake et al. (2012) [58]\nSimilarly, here we used the parameters obtained from the SNe Ia data to study the BAO data.\nFigure 3 shows the comparison of the z ∼rs(zd)/Dv(z) curves derived from the two models.\nIn addition, the obtained value of χ2corresponding to the ΛCDM model (the solid line in\nFigure 3) is 131.87 , while for the dotted line, the obtained value of χ2is 65.10 , and for the bulk\n7Figure 3: The comparison of the z ∼rs(zd)/Dv(z) curves. The solid line corresponds to the ΛCDM model with\nΩm0= 0.295,ΩΛ0= 0.705, the dotted line corresponds to the Λ CDM model with the best estimated parameters\n(i.e., Ω m0=0.266,ΩΛ0=0.734) obtained from the BAO data alone, the dashed line corresponds to the bulk\nviscosity model with Ωm0=0.295,ΩΛ0=0.705, Ωτ=0.018. The observational data rs(zd)/Dv(z) are also\nplotted with their error bars.\nviscosity model (the dashed line), the obtained value of χ2is 11.33 , which imply that the bulk\nviscosity model has a significantly better fitting to the combined SNe Ia and BAO data than the\nΛCDM model.\n3.4. Data from Hubble parameter\nBased on different ages of galaxies, we can obtain the observational Hubble parameter (see for\ninstance Ref. [59]). The function of H(z) plays an important role in understanding the evolution of\nuniverse and the property of dark energy, since its value is directly obtained from the cosmological\nobservations. So the Hubble parameter can be applied to constrain the cosmological parameters\nindependently. Here we take the data coming from Ref. [60–74], which are shown in Table 3, to\nstudy the two models.\nTheχ2function of the Hubble parameter is defined by\nχ2\nH=41X\ni=1[H(zi)obs−H(zi)th]2\nσ2(zi)(25)\nAnd the corresponding goodness-of-fit χ2dof is defined as\nχ2\ndof=χ2/(N−n) (26)\nWhere Ndenotes the number of the observed data and nis the number of parameters involved\nin the applied model. And χ2\ndof≤1 represents the fit is good and the data are consistent with the\napplied model.\nNow, with the above best estimated parameters obtained from the SNe Ia data alone, i.e., in\nthe ΛCDM model Ω m0= 0.295,ΩΛ0= 0.705, H0= 70 km /s/Mpc and in the bulk viscosity model\nΩm0= 0.295,ΩΛ0= 0.705,Ωτ= 0.018 (within the range of Ω τ= 0.01±0.01 ), H0= 70 km /s/Mpc,\nwe would like to compare the two models. The comparison of the z ∼H(z) curves derived from\nthe two models are shown in Figure 4.\nCorrespondingly, for the Λ CDM model, χ2= 30.81, χ2\ndof= 30.81/(41−3) = 0 .81<1. While\nfor the bulk viscosity model, χ2= 20.80, χ2\ndof= 20.80/(41−4) = 0 .56<1. Note that in Ref. [65],\n8it present that with the above 41 H(z) data alone the obtained best estimated parameters involved\nin the ΛCDM model is Ω m0= 0.25,ΩΛ0= 0.75, H0= 70 .6 km /s/Mpc, and correspondingly,\nχ2\nmin= 18.55, which is very close to the value of χ2obtained from the bulk viscosity model with\nthe best estimated parameters obtained from the SNe Ia data. It imply that the bulk viscosity\nmodel has a better fitting to the combined SNe Ia and 41 H(z) data than the ΛCDM model.\nFigure 4: The comparison of the z ∼H(z) curves. The solid line corresponds to the ACDM model with Ω m0=\n0.295,ΩΛ0= 0.705, H0= 70 km /s/Mpc, the dashed line corresponds to the bulk viscosity model with Ω m0=\n0.295,ΩΛ0= 0.705,Ωτ= 0.018, H0= 70 km /s/Mpc. The Hobs(z) data are also plotted with their error bars.\nTable 3: The observational Hubble parameter data (in units [ km /s/Mpc] )\nz H (z)σH Ref. z H (z)σH Ref.\n0.070 69 19.6 [60] 0.480 97 62 [69]\n0.090 69 12 [61] 0.570 92.9 7.855 [70]\n0.120 68.6 26.2 [60] 0.593 104 13 [62]\n0.170 83 8 [61] 0.6 87.9 6.1 [68]\n0.179 75 4 [62] 0.68 92 8 [62]\n0.199 75 5 [62] 0.73 97.3 7.0 [68]\n0.200 72.9 29.6 [60] 0.781 105 12 [62]\n0.240 79.69 6.65 [63] 0.875 125 17 [62]\n0.270 77 14 [60] 0.88 90 40 [69]\n0.280 88.8 36.6 [61] 0.9 117 23 [61]\n0.300 81.7 6.22 [64] 1.037 154 20 [62]\n0.350 82.7 8.4 [65] 1.300 168 17 [61]\n0.352 83 14 [62] 1.363 160 33.6 [71]\n0.3802 83 13.5 [66] 1.43 177 18 [61]\n0.400 95 17 [60] 1.53 140 14 [61]\n0.4004 77 10.02 [66] 1.75 202 40 [61]\n0.4247 87.1 11.2 [66] 1.965 186.5 50.4 [71]\n0.43 86.45 3.68 [67] 2.300 224 8 [72]\n0.440 82.6 7.8 [68] 2.34 222 7 [73]\n0.44497 92.8 12.9 [66] 2.36 226 8 [74]\n0.4783 80.9 9 [66]\n94. Statefinder analysis\nIn this section, the statefinder diagnostic pair {r, s}parameters, which is introduced by Sahni\net al. [75], will be appiled to study the viscous universe with the cosmological constant. The\nstatefinder is a geometrical diagnostic tool that can characterize the properties of dark energy in\na model-independent manner. The statefinder parameters {r, s}are usually defined by\nr=...a\naH3, s =r−1\n3\u0000\nq−1\n2\u0001 (27)\nWhere qdenotes the deceleration parameter, which is defined as\nq(z)≡ −¨a\na1\nH2(28)\nAnd the z∼qcurves derived from the two models with the best estimated parameters obtained\nfrom the SNe Ia are shown in Figure 5.\nFigure 5: The comparison of the z ∼qcurves. The solid line corresponds to the Λ CDM model with Ω m0=\n0.295,ΩΛ0= 0.705, the dashed line corresponds to the bulk viscosity model with Ω m0=0.295,ΩΛ0= 0.705,���τ=\n0.018.\nFigure 6: The evolution of the bulk viscosity model in the r-s plane for the best estimates of the parameters from\nthe SNe Ia data, i.e., Ω m0= 0.295,ΩΛ0 = 0 .705,Ωτ= 0.018. In the r-s plane the Λ CDM model corresponds to\nthe point {0,1}. The arrow denotes the time direction of the evolution. The dashed line is the asymptote line of\nthe solid line.\n10From Figure 5 we can obtain that q0=−0.557, zT= 0.68 for the ΛCDM model, and q0=\n−0.575,zT= 0.74 for the bulk viscosity model, where q0is the present deceleration parameter and\nzTis the transition red shift, at which qis equal to 0 . And the latest observations indicates that\nq0=−0.63±0.12 [76, 77], which imply that the bulk viscosity model cannot be discriminated\nfrom the ΛCDM model by the deceleration parameter due to the insufficient observation accuracy.\nBased on Eq. (27), we can obtain the {r, s}plane trajectories of the bulk viscosity model,\nwhich is shown in Figure 6.\nIt can be seen from Fig. 6 that the r-s evolution starts from a region r∼0.945, s∼1 and\ntends to the ΛCDM point in the future, and the present position of the universe corresponds to\n{r0,s0}={0.984,0.0047}, which is close to the Λ CDM point. Fig. 6 shows that the bulk viscosity\nmodel is obviously different from the ΛCDM.\n5. Conclusions\nIn this paper, we performed a detailed study of a bulk viscous matter universe with cosmological\nconstant Λ. The explicit form of the bulk viscosity is still unknown, but we found that if we follow\nthe assumption that ζ=ζ1H, where His the Hubble parameter, ζ1is a constant, then in an\nexpanding universe, the (relative) matter density will be always greater than 8 πGζ 1, i.e., 0.012\n(the value is obtained from Eq. (13) and the obtained Ω τ= 0.018 ), and in the future the\n(relative) matter density tends to 0.012 . The universe in the bulk viscosity model also tend to\nthe de Sitter time-space in the future, but the expansion rate of the future’s universe predicted by\nthe bulk viscosity model is lower than which predicted by the ΛCDM model.\nSecondly, with the JLA SNe Ia sample we estimated the values of parameters involved in the\nbulk viscosity model. And subsequently, we applied the best estimated parameters obtained from\nthe SNe Ia data to study the CMB data, BAO data and Hubble data. Note that in this paper we\nused a fixed values of Ω m0,ΩΛ0,Ωτ(i.e., Ω m0= 0.295,ΩΛ0= 0.705,Ωτ= 0.018 ), to study the SNe\nIa, CMB ,BAO and Hubble data with the purpose to directly compare the two models, and we\nfound that the bulk viscosity model has a significantly better fitting to these data than the ΛCDM\nmodel, which indicates that the bulk viscosity model is a competitive model to fit the combined\nSNe Ia + CMB + BAO + H(z) data.\nIn addition, it should be noted that the obtained best estimated value of ζis positive, which\nis in consistent with the requirement of the second law of thermodynamics [17, 18].\nOn the other hand, it shows that there is no significant difference between the curves of z∼q\npredicted by the two models with the same parameters. And the bulk viscosity model with the best\nestimated parameters obtained from the SNe Ia data predict the present deceleration parameter\nq0= -0.575 , which is within the range of the observational results, i.e., −0.64±0.12.\nSince the universe predicted by the bulk viscosity model is similar to the standard forms of\ndark energy, we have analyzed the model with the statefinder parameters to distinguish it from\nthe ΛCDM model. It shows that the evolution of the {r, s}parameters behaves in such a way\nthat 0 < s < 1,0.945< r < 1. And the present position of the bulk viscosity model in the r-s\nplane corresponds to {s0, r0}={0.0047,0.984}, which is close to the ΛCDM point {0,1}. Hence\nthe model is obviously different from the ΛCDM model and the other dark energy model.\nØ\nHere it should be noted that, in this paper we have not listed the calculation results of other\ncosmological parameters based on the bulk viscosity model, such as the age of the universe (the\n11age predicted by the bulk viscosity model is slightly higher than the age predicted by the ΛCDM\nmodel), the curvature scalar, et al., for that the bulk viscosity coefficient ζobtained from the SNe\nIa data is so small that the evolution or values of these cosmological parameters do not deviate\nsignificantly from which are obtained from the ΛCDM model (when ζ= 0 it returns to the ΛCDM\nmodel), which is exactly what we expected, as that the ΛCDM model has already achieved great\nsuccess in explaining some astronomical phenomenon. And in this paper we just made a little\nimprovement to the ΛCDM model.\nReferences\n[1] A. G. Riess, A. V. Filippenko, P. 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From the dynamic \nprogramming principle, the solution of optimal control problems or H∞ optimal control problems can be pursued by \nsolving Hamilton –Jacobi –Bellman (HJB) equation or Hamilton –Jacobi –Isaacs (HJI) equation respectively [1], [2]. \nThe HJB and HJI equations are formulated using performance value function, once the value function is  solved, a \nfeedback controller which results in closed -loop systems asymptotic stability can be synthesized. For linear systems \nand quadratic performance functionals, the HJB and HJI equations become the well -known algebraic Riccati equation \nand game algeb raic Riccati equation [3] -[5], respectively. But for nonlinear systems or nonquadratic performance \nfunctionals, due to the inherent nonlinear properties, the solution s of HJB and HJI equation are challenging. Therefore, \nit is a formidable task to compute the value function of nonlinear systems by solving the HJB or HJI equation. \nMoreover, another disadvantage of solving HJB or HJI equation is that the full knowledge of the system dynamics \nneeds to  be available.  \nA great deal of effort has been made to solve these equations. Because it is intractable to get the accurate analytical \nsolutions of HJB and HJI equations, approximate solutions of HJB and HJI have been found using many techniques, \nsuch as Galerkin’s approximation of solving generalized HJB equation [6], adaptive dynamic programming [7], \nintelligent optimal control [8] and Taylor series solution [9]. For nonlinear continuous -time sys tems with control input \nconstraint, an offline neural -networks -based HJB te chnique was proposed in [10], this method requires the full \nknowledge of systems model. To relax the requirement of internal dynamics knowledge of the nonlinear system, an \nonline direct adaptive optimal control algorithm was presented in [11]. The experien ce replay technique was \nsubsequently used to learn the solutions of HJB equation online for constrained -input systems with unknown internal \ndynamics in [12]. To overcome the difficulties of full knowledge requirements of the system model, a data -driven \n \n* Senior Engineer, Department of General System Technology , wangqibuaa@126.com.  T model -free approximate policy iteration (PI) algorithm was introduced in [13], [14], in which the off -policy learning \nmethod of reinforcement learning (RL) was used.  \nThe optimal H∞ control theory of nonlinear system was developed in [2], in which the notion of dissipativity \nintroduced in [15] was utilized, and Schaft proofed that the smooth solution of the HJI equation can be pursued by a \nsequence of iterative procedur es. Several offline algorithms of policy iteration for solving HJI equation were proposed \nin [16] -[19], in which solving the HJI equation was converted into solving a series of linear partial differential equation. \nIn the offline policy iteration approach,  the control i nput and disturbance input were computed iteratively in the inner -\nloop and outer -loop, respectively. Vamvoudakis et al . [20] introduced an online synchronous policy iteration \nalgorithm to solve HJI equation, and Modares et al. [21] extended this method to constrained input systems. These \nmethods need to tune synchronously three NN approximators, i.e., actor, disturbance, and critic, for calculating the \napproximate solution of the HJI equation. In [22] -[24], Luo et al. presented a simultaneous policy update algorithm. \nThere need only one critic neural network to approximate the value function and only one iterative loop, thus, the \nsimultaneous policy update algorithm  is easy to offline or online implement. But the simultaneous policy update \nalgorithm is an on -policy integral reinforcement learning algorithm, sampling data used in the policy evaluation phase \nare generated from evaluating policies. It requires that the disturbance input is adjustable, which is generally unrealistic \nfor most practical sys tems. An d then, an off -policy integral RL approac h was introduced for H∞ control of nonlinear \ncontinuous -time system without requiring knowledge of the system internal dynamics [25]. The sampling data which \nare used for learning value function are generated with behavior policies rather than the target policies to be evaluated  \nin the off -policy integral RL approach. A model -free version of off -policy integral RL approach for H∞ optimal control \nwas proposed in [26]. Modares and Kiumarsi et al. extended the results o f the simultaneous policy update algorithm  \nand off -policy integral RL algorithm to the H∞ tracking control for nonlinear affine and nonaffine systems without \nusing any knowledge of the dynamics of the systems [27], [28]. The algorithms for solving optimal control and H∞ \ncontrol problems  mentioned above are essentially Newton ’s iterative algorithms when a fixed -point equation is \nconstructed in a Banach space. Hence, the simultaneous policy update algorithm and off -policy integral RL have the \nsame disadvantage as the Newton iterative approach, that is, the algorithm is locally optimized, and finding suitable \ninitial parameters or developing global methods is a difficult problem.  \nIn recent years, Luo et al. presented an integration scheme of the policy iteration and value iteration methods to \nrelax the requirement of the initial admissible control input and speed up the training convergence process for nonlinear discrete -time systems [29]. Besides, a model -free λ -PI algorithm was presented in [30], which can balance \nthe control performance and convergence rate between PI algorithm and value iteration algorithm for linear discrete -\ntime systems.  Till present, the development of policy iteration  algorithms and theories for H∞ control design of \ncontinuous -time nonlinear system is still an open issue, which promotes this research. It is worth noting that damped \nNewton ’s method, which leverages a non -unit step -size in the Newton direction by  multiplying a damping parameter  \nα, is an effective algorithm for solving equations. Although damped Newton method makes the convergence slower \nthan Newton method, damped Newton’s method could be globally convergent. Therefore, the damped Newton’s \nmethod has the advantage of enhancing the robustness of the convergence of solving process with regarding to the \ninitial guess.  \nIn this paper, the H∞ control design of nonlinear continuous -time systems  is consider ed, and a novel policy iteration \nalgorithm based on damped Newton ’s method  is develop ed for solving HJI equation , name d α-policy iteration ( α-PI) \nalgorithm . By constructing a damped Newton iteration operator equatio n, a generalized Bellman equation is given, \nthe generalized Bellman equation is an extension of Bellman equation. And then, by iterating on the generalized \nBellman equation , the optimal solution of the HJI equation can be obtained. Under this framework, first ly, an on -\npolicy α -PI integral reinforcement learning method is given, and then the off -policy α -PI reinforcement learning \nmethod is provided , namely a model -free method . \nThe organization of this paper is as follows. The continuous -time nonlinear H∞ control problem statement and \npreliminaries are presented in section II. The damped Newton method based on -policy and off -policy α-PI integral \nRL methods are provided in section  III and IV  respectively , and neural -network based implementation schemes of \nthese α -PI algorithms are derived. Two c omputer simulation example s are presented in section V, one is for the general \nnonlinear system H∞ control problem, and the other is missile interception simulation . Finally, section VI concludes \nthe paper.  \nII. H∞ Control  Problem  and Simultaneous Policy Update Algorithm  \nIn this section, the background knowledge reviews and preliminary results of H∞ control problem are provided. \nConsider a class affine in control input nonlinear continuous -time system defined as follows:  \n𝒙̇=𝒇(𝒙)+𝒈(𝒙)𝒖+𝒌(𝒙)𝒘                                                                        (1) \n𝒛=𝒉(𝒙)                                                                                                            (2) where 𝒙∈Ω⊂ℜ𝑛 is state vector of the system, and 𝒘∈ℜ𝑞 and 𝒖∈ℜ𝑚 represent disturbance input vector and \ncontrol input vector respectively, and 𝒛∈ℜ𝑝 is the output of system (1). 𝒇(𝒙)∈ℜ𝑛 is internal dynamics, 𝒈(𝒙)∈\nℜ𝑛×𝑚 and 𝒌(𝒙)∈ℜ𝑛×𝑞 are input -to-state matrix and disturbance coefficient matrix, respectively. On a compact set \nΩ, 𝒇(𝒙), 𝒈(𝒙) and 𝒌(𝒙) meet locally Lipschitz continuity and 𝒇(0)=0, that is 𝒙=0 is an equilibrium point of the \nsystem. Assume that 𝒘(𝑡)∈𝐿2[0 ,∞) and 𝒖(𝑡)∈𝐿2[0 ,∞). \nThe objective of H∞ control problem is to reduce the influence of disturbance signal 𝒘 on the system , i.e., find a \nfeedback control law 𝒖 such that system (1) meets a disturbance attenuation condition with starting from 𝒙(0)=0, \nas follows:  \n∫(𝑄 (𝒙) + 𝒖T 𝑹 𝒖)∞\n0d𝑡≤∫𝛾2𝒘T𝒘∞\n0d𝑡                                                                    (3) \nwhere 𝑄(𝒙)=‖𝒛‖2, 𝑹∈ℜ𝑚×𝑚 a positive definite symmetric matrix , and 𝛾>0 is a predetermined disturbance \nattenuation level.  \nIf the condition (3) is satisfied, then the 𝐿2-gain of system (1) is less than or equal to 𝛾>0, that is,  \n∫(𝑄 (𝒙) + 𝒖T 𝑹 𝒖)∞\n0d𝑡\n∫𝒘T𝒘∞\n0d𝑡⩽𝛾2.                                                                            (4) \nThe H∞ optimal control problem is concerned with finding the minimum 𝛾∗ in the presence of solutions for \nbounded 𝐿2-gain control problem. However, for general nonlinear system, there is no way to find the minimum value \nof the disturbance attenuation. Therefore, this paper focuses on suboptimal H∞ control problem, it is assumed that 𝛾 \nis prescribed and 𝛾>𝛾∗. It is worth noting that the two -person zero -sum differential games problem is equivalent to \nthe suboptimal H∞ control problem [31]. Game theory deals  with the strategic interaction between the two participants, \neach of which has an objective contained in a performance value function that the two participants try to maximize \nand minimize respectively.  \nDefine the performance cost function for the policies of the two players as follows:  \n𝑉𝒖,𝒘(𝒙 (𝑡))=∫(𝑄 (𝒙) + 𝒖T 𝑹 𝒖 − 𝛾2 𝒘T 𝒘)∞\n𝑡d𝑡.                                                 (5) \nIn two -person zero -sum differential games, the objective of participant 𝒖 is to minimize the performance cost \nfunction, whereas the objective of participant 𝒘 is to maximize it, hence,  \n𝑉∗(𝒙0)=min \n𝒖max\n𝒘∫(𝑄 (𝒙) + 𝒖T 𝑹 𝒖 − 𝛾2 𝒘T 𝒘)∞\n0d𝑡.                                              (6) \nDefine the Hamiltonian function related to the minimax design problem as  𝐻(𝒙 , 𝒖 , 𝒘 ,∇ 𝑉)=𝑄(𝒙)+𝒖T𝑹𝒖−𝛾2𝒘T𝒘+∇𝑉T(𝒇 (𝒙) + 𝒈 (𝒙) 𝒖 + 𝒌 (𝒙) 𝒘)                         (7) \nwhere ∇V=∂V\n∂𝒙. \nIn the sight of the Bellman optimality principle, if there exists a nonnegative solution 𝑉∗(𝑥)≥0:ℜ𝑛→ℜ of HJI \nequation shown as follows:  \n0=𝑄(𝒙)+(∂𝑉∗\n∂𝒙)T\n𝒇(𝒙)−1\n4(∂𝑉∗\n∂𝒙)T\n𝒈𝑹−1𝒈T∂𝑉∗\n∂𝒙+1\n4𝛾2(∂𝑉∗\n∂𝒙)T\n𝒌𝒌T∂𝑉∗\n∂𝒙.                                   (8) \nThen, the control strategy pairs given in (9) and (10) are Nash equilibrium saddle -point solution [21].  \n𝒖∗=−1\n2𝑹−1𝒈T∂𝑉∗\n∂𝒙                                                                              (9) \n𝒘∗=1\n2𝛾2𝒌T∂𝑉∗\n∂𝒙                                                                                   (10) \nThe Nash equilibrium strategy is inherently robust, once the equilibrium is reached, no participant can unilaterally \ndeviate from its Nash strategy to improve its payoff. And, the 𝐿2-gain of closed -loop system (1) with feedback control \nlaw (9) is less than or equal to 𝛾. \nBecause the HJI equation is nonlinear with respect to 𝑉∗, finding the closed -form solution is usually intractable. \nThe Bellman equation (11) which is linear in the cost function is used in simultaneous policy update algorithm  for \npursuing the solution of the HJI equation. Algorithm 1 shows the  simultaneous policy update algorithm  which is used \nto solve iteratively  the H∞ control problem [24].  \nAlgorithm 1 Simultaneous Policy Update Algorithm  for H∞ Control Problem  \nStep 1: Set 𝑖=0, give an initial value function 𝑉0, and initial disturbance and \ncontrol policies associated.  \n𝒖0=−1\n2𝑹−1𝒈T∂𝑉𝑖\n∂𝒙, 𝒘0=1\n2𝛾2𝒌T∂𝑉𝑖\n∂𝒙 . \nStep 2: Solve for 𝑉𝑖+1(𝒙) with 𝑉𝑖+1(0)=0 by using  \n0=𝑄(𝒙)+𝒖𝑖T𝑹𝒖𝑖−𝛾2𝒘𝑖T𝒘𝑖+∇𝑉𝑖+1T(𝒇 (𝒙) + 𝒈 (𝒙) 𝒖𝑖 + 𝒌 (𝒙) 𝒘𝑖).         (11) \nStep 3: Update the disturbance policy and control policy using  \n𝒖𝑖+1=−1\n2𝑹−1𝒈T∂𝑉𝑖+1\n∂𝒙                                              (12) \n𝒘𝑖+1=1\n2𝛾2𝒌T∂𝑉𝑖+1\n∂𝒙.                                                    (13) \nStep 4: If   ‖𝑉𝑖+1 − 𝑉𝑖‖Ω≤𝜀, 𝜀>0 (positive real number and small), stop \ncalculation, else set 𝑖=𝑖+1, move to Step 2 and go on iteration.  \n Luo et al . proofed that simultaneous policy update algorithm  is essentially a Newton iteration algorithm for \npursuing a solution of the fixed -point equation in a Banach space, The convergence can be established by the \nKantorovich’s theorem [23]. The proof method is briefly described as follows.  \nFirstly, considering a Banach space 𝕍, 𝕍={𝑉 (𝒙) | 𝑉 (𝒙) :Ω → ℜ ,  𝑉 (0) = 0}, and ‖𝑉‖ is the norm of 𝑉. Define the \nfollowing mapping 𝒢: 𝕍→𝕍 on 𝕍: \n𝒢(𝑉)=𝑄(𝒙)+(∂𝑉\n∂𝒙)T\n𝒇(𝒙)−1\n4(∂𝑉\n∂𝒙)T\n𝒈𝑹−1𝒈T∂𝑉\n∂𝒙 +1\n4𝛾2(∂𝑉\n∂𝒙)T\n𝒌𝒌T∂𝑉\n∂𝒙                       (14) \nThe Fré chet derivative of mapping 𝒢(𝑉) is denoted as 𝒢′(𝑉), and the Fré chet differential is written as 𝒢′(𝑉)𝑍, \nwhere 𝑍∈𝕍̃⊂𝕍 and 𝕍̃ is a neighborhood of 𝑉. \nReference [23] showed that how to compute the Fré chet differential, as shown in Lemma 1.  \nLemma 1 ([23, Lemma 3]) : On the Banach space 𝕍, ∀𝑉∈𝕍, consider the mapping 𝒢 defined in (14), the Fré chet \ndifferential of 𝒢 is computed as:  \n𝒢′(𝑉)𝑍=(∇ 𝑍)T𝒇(𝒙)−1\n4(∇ 𝑍)T𝒈𝑹−1𝒈T∇𝑉−1\n4(∇ 𝑉)T𝒈𝑹−1𝒈T∇𝑍+1\n4𝛾2(∇ 𝑉)T𝒌𝒌T∇𝑍+1\n4𝛾2(∇ 𝑍)T𝒌𝒌T∇𝑉  \n   (15) \nThen, in Theorem 1 of [23], Luo et al . proofed that the simultaneous policy update algorithm  is equivalent \nmathematically to Newton method.  \nLemma 2 ([23, Theorem  1]): The iteration procedure from (11) to (13) is equivalent to Newton method with (12) \nand (13), as follows:  \n𝑉𝑖+1=𝑉𝑖−(𝒢′ (𝑉𝑖))−1𝒢(𝑉𝑖),    𝑖=0,1,2,⋯.                                                      (16) \nIn a Banach space, under some proper assumptions, the convergence of Newton method can be guaranteed by the \nKantorovich’s theorem [32]. And the solution of 𝒢(𝑉∗)=0 can be obtained, thus, HJI equation (8) is solved.  \nIII. On-Policy 𝜶-PI for Learning the Solution of HJI Equation  \nIn this section, an  on-policy 𝛼-PI algorithm which is based on damped Newton iteration method is presented to \npursue the solution of the HJI equation, and the  H∞ controller is obtained. Then, NN -based critic –actor –disturbance \nscheme is leveraged to implement the on -policy 𝛼-PI algorithm proposed without making use of the information about \nthe internal dynamics of the system.  A. On-Policy 𝜶-PI Algorithm  \nDamped Newton’s method is an effective approach for solving nonlinear equation, in which a non -unit step size \nin the Newton direction is leveraged by introducing a damping parameter. The major advantage of the damped \nNewton’s method is that the convergence  of solving process with regarding to the initial condition is enhanced.  \nBased on damped Newton method, the on -policy 𝛼-PI is given in Algorithm 2.  \nAlgorithm 2 On-Policy 𝜶-PI for H∞ Control Design  \nStep 1: Set 𝑖=0, give an initial value function 𝑉0 with 𝑉0(0)=0, associated \ncontrol and disturbance policies  \n𝒖0=−1\n2𝑹−1𝒈T∂𝑉0\n∂𝒙,𝒘0=1\n2𝛾2𝒌T∂𝑉0\n∂𝒙. \nStep 2: Solve for 𝑉𝑖+1(𝒙) with 𝑉𝑖+1(0)=0 by using  \n                  [𝑉𝑖+1 (𝒙 (𝑡 + ∆ 𝑡)) − 𝑉𝑖+1 (𝒙 (𝑡))]+𝛼∫(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)𝑡+∆𝑡\n𝑡d𝑡 \n=(1 − 𝛼)[𝑉𝑖 (𝒙 (𝑡 + ∆ 𝑡)) − 𝑉𝑖 (𝒙 (𝑡))].                     (17) \nStep 3: Update the disturbance and control policies using  \n𝒖𝑖+1=−1\n2𝑹−1𝒈T∂𝑉𝑖+1\n∂𝒙                                             (18) \n𝒘𝑖+1=1\n2𝛾2𝒌T∂𝑉𝑖+1\n∂𝒙.                                                   (19) \nStep 4: If  ‖𝑉𝑖+1 − 𝑉𝑖‖Ω≤𝜀, 𝜀>0 (small positive number), stop calculation, else set \n𝑖=𝑖+1, move to Step 2 and go on iteration.  \n \nIn a Banach space 𝕍, the above on -policy 𝛼-PI algorithm is equivalent to damped Newton iteration method \nmathematically.  \nConsidering the Banach space mentioned above, 𝕍={𝑉 (𝒙) | 𝑉 (𝒙) :Ω → ℜ ,  𝑉 (0) = 0}, a new mapping  𝒯:𝕍→𝕍 is \ndefined as follows:  \n𝒯(𝑉)=𝑉−𝛼(𝒢′ (𝑉))−1𝒢(𝑉)                                                            (20) \nwhere 𝛼 is Newton step-size, 0<𝛼≤1. \nTheorem 1 : Given a mapping 𝒯:𝕍→𝕍 shown in (20), then the following damped Newton iteration with (18) and \n(19) is equivalent to the on -policy 𝛼-PI algorithm from (17) to (19).  \n𝑉𝑖+1=𝒯(𝑉𝑖)                                                                            (21) \nProof : Substituting (20) into (21), one can obtain the following equation  𝑉𝑖+1=𝒯(𝑉𝑖)=𝑉𝑖−𝛼(𝒢′ (𝑉𝑖))−1𝒢(𝑉𝑖).                                                  (22) \nEquation  (22) can be rewritten as  \n𝒢′(𝑉𝑖)𝑉𝑖+1=𝒢′(𝑉𝑖)𝑉𝑖−𝛼𝒢(𝑉𝑖).                                                        (23) \nFrom (18), (19), and (15), one has  \n𝒢′(𝑉𝑖)𝑉𝑖+1=(∇ 𝑉𝑖+1)T𝒇(𝒙)−1\n4(∇ 𝑉𝑖+1)T𝒈𝑹−𝟏𝒈T∇𝑉𝑖−1\n4(∇ 𝑉𝑖)T𝒈𝑹−𝟏𝒈T∇𝑉𝑖+1+1\n4𝛾2(∇ 𝑉𝑖)T𝒌𝒌T∇𝑉𝑖+1     \n+1\n4𝛾2(∇ 𝑉𝑖+1)T𝒌𝒌T∇𝑉𝑖 \n=(∇ 𝑉𝑖+1)T[𝒇 (𝒙) − 1\n2 𝒈 𝑹−1 𝒈T ∇ 𝑉𝑖 + 1\n2𝛾2 𝒌 𝒌T ∇ 𝑉𝑖]                                                                \n=(∇ 𝑉𝑖+1)T[𝒇 (𝒙) + 𝒈 𝒖𝑖 + 𝒌 𝒘𝑖]                                                                                                                     (24) \nand \n𝒢′(𝑉𝑖)𝑉𝑖=(∇ 𝑉𝑖)T𝒇(𝒙)−1\n4(∇ 𝑉𝑖)T𝒈𝑹−𝟏𝒈T∇𝑉𝑖−1\n4(∇ 𝑉𝑖)T𝒈𝑹−𝟏𝒈T∇𝑉𝑖+1\n4𝛾2(∇ 𝑉𝑖)T𝒌𝒌T∇𝑉𝑖\n+1\n4𝛾2(∇ 𝑉𝑖)T𝒌𝒌T∇𝑉𝑖                                      \n=(∇ 𝑉𝑖)T𝒇(𝒙)−2𝒖𝑖T𝑹𝒖𝑖+2𝛾2𝒘𝑖T𝒘𝑖 .                                                                                             (25) \nSubstitute (24), (25) and (14) into (23), and make some mathematical manipulate, one can obtain the generalized \nBellman equation, as follows:  \n(∇ 𝑉𝑖+1)T[𝒇 (𝒙) + 𝒈 𝒖𝑖 + 𝒌 𝒘𝑖]=(1 − 𝛼)∇𝑉𝑖T[𝒇 (𝒙) + 𝒈 𝒖𝑖 + 𝒌 𝒘𝑖]−𝛼(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖).              (26) \nAlong the trajectory of system 𝒙̇=𝒇(𝒙)+𝒈𝒖𝑖+𝒌𝒘𝑖, integrating both sides of (26) from 𝑡 to 𝑡+∆𝑡 yields the \ndiscretized version of generalized Bellman equation (17). In line with [3] and [25], with 𝑉𝑖+1(0)=0, the differential \nversion of generalized Bellman equation (26) and the discretized version of generalized Bellman equation (17) have \nthe same solution.  \nThis means that the damped Newton iteration given by (2 1) with (18) and (19) is equivalent to the on -policy 𝛼-PI \nalgorithm from (17) to (19). This completes the proof.                                                                               □ \nRemark 1: The generalized Bellman equation (26) is linear  partial differential equation just like the Bellman \nequation (11).  \nRemark 2 : Compared with Bellman equation (11), the integral reinforcement term of the generalized Bellman \nequation (26) is multiplied by the damped coefficient 𝛼, and reinforcement term needs to consider the influence of the \ncurrent value function 𝑉𝑖. Remark 3 : When 𝛼=1, (26) degenerates to Bellman equation (11) , and on-Policy 𝛼-PI algorithm becomes the \nsimultaneous policy update algorithm  shown in Algorithm 1.  \nNewton method has a faster convergence rate near the solution, but smaller convergence region, it usually only \nsatisfies local convergence and needs a good initial guess. On the other hand, damped Newton method converges \nslowly, but convergence condition can be relaxed. By choosing α smaller one can obtain larger convergence domain \n[33].   \nIn order to obtain the optimal control policy , the value function 𝑉𝑖+1 needs to be solved in algorithm 2 . Due to the \nunknown form of the value function , a NN approximator is used to approximate the value function.  A NN -based \nimplementation scheme of the iteration algorithm 2 is provided in the next section.  \nB. NN-Based Online Implement of Algorithm 2  \nBecause of the universal approximation property, NNs are natural candidates for approximating smooth functions \non compact sets. In this section, we use a single critic NN structure to implement on -policy 𝛼-PI algorithm online.  \nLet 𝝆(𝒙)=[𝜌1(𝒙),⋯,𝜌𝑁(𝒙)]T be the critic  NN activation functions, it means that there are N neurons  on the \nhidden layer. The cost function is approximated by  \n𝑉(𝒙)=𝑉̂(𝒙)+𝜀(𝒙)=𝑾T𝝆(𝒙)+𝜀(𝒙)                                                         (27) \nwhere 𝑾∈ℜ𝑁 is the weight vector of NN, 𝜀(𝒙) is the error of NN approximator. Based on high -order Weierstrass \napproximation theories [34], if 𝑉(𝒙) is smooth enough, there exists a complete basis set, that 𝜀→0 uniformly with \n𝑁→∞. \nBy using NN approximator 𝑉̂𝑖+1(𝒙)=𝑾𝑖+1T𝝆(𝒙) and 𝑉̂𝑖(𝒙)=𝑾𝑖T𝝆(𝒙), given control 𝒖𝑖 and disturbance 𝒘𝑖, \ngeneralized Bellman equation (17) becomes  \n[𝑾𝑖+1T 𝝆 (𝒙 (𝑡 + ∆ 𝑡)) − 𝑾𝑖+1T 𝝆 (𝒙 (𝑡))]+𝛼∫(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)𝑡+∆𝑡\n𝑡d𝑡 \n=(1 − 𝛼)[𝑾𝑖T 𝝆 (𝒙 (𝑡 + ∆ 𝑡)) − 𝑾𝑖T 𝝆 (𝒙 (𝑡))].                                                                                              (28) \nThat is,  \n[𝝆T (𝒙 (𝑡)) − 𝝆T (𝒙 (𝑡 + ∆ 𝑡))]𝑾𝑖+1=(1 − 𝛼)[𝝆T (𝒙 (𝑡)) − 𝝆T (𝒙 (𝑡 + ∆ 𝑡))]𝑾𝑖 \n+𝛼∫(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)𝑡+∆𝑡\n𝑡d𝑡.                          (29) It is worth noting that there are N unknown parameters in the NN approximator. Therefore, at least N sampling \ndata points are needed to solve the weight vector of neural network while  the batch least -squares (BLS) method is \nused. When 𝑁̅ (𝑁̅>𝑁) data are collected along the system state trajectories in the domain of Ω, the BLS solution of \nthe NN weights is  \n𝑾𝑖+1=(𝑿 𝑿T)−1𝑿𝒀                                                                        (30) \nwhere  \n𝑿=[𝝆(𝒙 (𝑡))−𝝆(𝒙 (𝑡 + ∆ 𝑡)),⋯,𝝆(𝒙 (𝑡 + (𝑁̅ − 1) ∆ 𝑡))−𝝆(𝒙 (𝑡 + 𝑁̅ ∆ 𝑡))]                  (31) \n𝒀=[𝑦𝒖𝑖,𝒘𝑖 (𝑥 (𝑡)) , ⋯ , 𝑦𝒖𝑖,𝒘𝑖 (𝑥 (𝑡 + (𝑁̅ − 1) ∆ 𝑡))]T                                           (32) \nand \n𝑦𝒖𝑖,𝒘𝑖(𝑥 (𝑡 + 𝑘 ∆ 𝑡))=(1 − 𝛼)[𝝆T (𝒙 (𝑡 + 𝑘 ∆ 𝑡)) − 𝝆T (𝒙 (𝑡 + (𝑘 + 1) ∆ 𝑡))]𝑾𝑖 \n         +𝛼∫ (𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)𝑡+(𝑘 + 1)∆𝑡\n𝑡+𝑘∆𝑡d𝑡                                         (33) \nwith 𝑘=0,1,⋯,𝑁̅−1. \nHaving solved for the approximate value function 𝑉̂𝑖+1(𝒙) associated with the disturbance 𝒘𝑖 and control 𝒖𝑖, the \ndisturbance and control policies update step can be executed as follows:  \n𝒘̂𝑖+1=1\n2𝛾2𝒌T(∇ 𝝆)T𝑾𝑖+1                                                                    (34) \n𝒖̂𝑖+1=−1\n2𝑹−1𝒈T(∇ 𝝆)T𝑾𝑖+1                                                             (35) \nwhere ∇𝝆=∂𝝆\n∂𝒙, is the Jacobian matrix of 𝝆(𝒙). \nSince Bellman equation (11) and generalized Bellman equation (26) are first order linear partial differential \nequation , utilizing some important theorems and properties that have been formed in the field of computational \nintelligence, the convergence proof of the presented BLS NN algorithm for implementing Algorithm 2 is almost the \nsame as that in [10] and [24].  \nIV. Off-Policy 𝜶-PI for Learning the Solution of HJI Equation  \nFor on -policy 𝛼-PI algorithm, the disturbance policy needs to be adjustable, which is usually unrealistic for \npractical applications. To overcome this drawback, motivated by [25] -[28], an off -policy 𝛼-PI algorithm is proposed \nto pursue the solution of HJI equation without using any prior information about the dynamics of the system. Then, a \nNN-based online implementation scheme is given.  A. Off-Policy 𝜶-PI Algorithm  \nFirstly, the system dynamics (1) can be rewritten as  \n𝒙̇=𝒇(𝒙)+𝒈(𝒙)𝒖𝑖+𝒌(𝒙)𝒘𝒊+𝒈(𝒙)(𝒖 − 𝒖𝑖)+𝒌(𝒙)(𝒘 − 𝒘𝒊)                              (36) \nwhere 𝒖∈ℜ𝑚 and 𝒘∈ℜ𝑞 are behavior policy and actual disturbance, respectively. 𝒖𝑖∈ℜ𝑚 and 𝒘𝑖∈ℜ𝑞 are \nstrategies to be evaluated and updated. Let 𝑉𝑖+1(𝒙) be the solution of generalized Bellman equation (26), and \ndifferentiating 𝑉𝑖+1(𝒙) along with the system dynamics (36) yields  \n𝑉̇𝑖+1=(∇ 𝑉𝑖+1)T(𝒇 + 𝒈 𝒖𝑖 + 𝒌 𝒘𝒊)+(∇ 𝑉𝑖+1)T𝒈(𝒖 − 𝒖𝑖)+(∇ 𝑉𝑖+1)T𝒌(𝒘 − 𝒘𝒊).              (37) \nUsing (26) one can obtain  \n𝑉̇𝑖+1=−𝛼(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)+(1 − 𝛼)∇𝑉𝑖T[𝒇 (𝒙) + 𝒈 𝒖𝑖 + 𝒌 𝒘𝑖]+(∇ 𝑉𝑖+1)T𝒈(𝒖 − 𝒖𝑖)+(∇ 𝑉𝑖+1)T𝒌(𝒘 − 𝒘𝒊).  \n    (38) \nSubstituting (36) into (38), one can obtain  \n𝑉̇𝑖+1=−𝛼(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)+(1 − 𝛼)∇𝑉𝑖T[𝒙̇ − 𝒈 (𝒙) (𝒖 − 𝒖𝑖) − 𝒌 (𝒙) (𝒘 − 𝒘𝒊)] \n+(∇ 𝑉𝑖+1)T𝒈(𝒖 − 𝒖𝑖)+(∇ 𝑉𝑖+1)T𝒌(𝒘 − 𝒘𝒊) \n=−𝛼(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)+(1 − 𝛼)∇𝑉𝑖T𝒙̇−(1 − 𝛼)∇𝑉𝑖T𝒈(𝒙)(𝒖 − 𝒖𝑖)−(1 − 𝛼)∇𝑉𝑖T𝒌(𝒙)(𝒘 − 𝒘𝒊) \n+(∇ 𝑉𝑖+1)T𝒈(𝒖 − 𝒖𝑖)+(∇ 𝑉𝑖+1)T𝒌(𝒘 − 𝒘𝒊).                                                                                                       (39) \nSubstituting 𝒖𝑖+1=−1\n2𝑹−1𝒈T∂𝑉𝑖+1\n∂𝒙, 𝒖𝑖=−1\n2𝑹−1𝒈T∂𝑉𝑖\n∂𝒙, 𝒘𝑖+1=1\n2𝛾2𝒌T∂𝑉𝑖+1\n∂𝒙 and 𝒘𝑖=1\n2𝛾2𝒌T∂𝑉𝑖\n∂𝒙 into (39) \nyields the following off -policy generalized Bellman equation:  \n𝑉̇𝑖+1=−𝛼(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)+(1 − 𝛼)𝑉̇𝑖+(1 − 𝛼)2𝒖𝑖T𝑹(𝒖 − 𝒖𝑖)−(1 − 𝛼)2𝛾2𝒘𝑖T(𝒘 − 𝒘𝒊) \n−2𝒖𝑖+1T𝑹(𝒖 − 𝒖𝑖)+2𝛾2𝒘𝑖+1T(𝒘 − 𝒘𝒊).                                                                                       (40) \nIt can be seen from the derivation process of (40) that it has the same solution as (26). Integrating both sides of \n(40) in interval [𝑡 ,  𝑡 + ∆ 𝑡) forms the discretized version of off -policy generalized Bellman equation (41).  \n     [𝑉𝑖+1 (𝒙 (𝑡 + ∆ 𝑡)) − 𝑉𝑖+1 (𝒙 (𝑡))]+∫2𝒖𝑖+1T𝑹(𝒖 − 𝒖𝑖)𝑡+∆𝑡\n𝑡d𝑡−∫2𝛾2𝒘𝑖+1T(𝒘 − 𝒘𝒊)𝑡+∆𝑡\n𝑡d𝑡                                \n=−𝛼∫(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)𝑡+∆𝑡\n𝑡d𝑡+(1 − 𝛼)[𝑉𝑖 (𝒙 (𝑡 + ∆ 𝑡)) − 𝑉𝑖 (𝒙 (𝑡))]+(1 − 𝛼)∫2𝒖𝑖T𝑹(𝒖 − 𝒖𝑖)𝑡+∆𝑡\n𝑡d𝑡 \n−(1 − 𝛼)∫2𝛾2𝒘𝑖T(𝒘 − 𝒘𝒊)𝑡+∆𝑡\n𝑡d𝑡                                                                                                                                    (41) Equation  (41) and generalized Bellman equation (26) have the same solution for the cost function. For space \nreasons the proof is omitted in this paper. It is observed from (41) that arbitrary input signals 𝒖 and 𝒘 can be used for \nlearning the value function 𝑉𝑖+1, rather than the policies 𝒖𝑖 and 𝒘𝒊 to be evaluated. Then, replacing (17) in Algorithm \n2 with (41), one can obtains the off -policy 𝛼-PI Algorithm, as shown in Algorithm 3 . \nAlgorithm 3 Off-Policy 𝜶-PI for H∞ Control Problem  \nStep 1: Use the behavior policy 𝒖 and the actual disturbance 𝒘 to collect M system data \nwhich contain system state, disturbance input and control input at different sampling time \ninterval.  \nStep 2: Set 𝑖=0, give an initial cost function 𝑉0 with 𝑉0(0)=0, and initial control and \ndisturbance policies  \n𝒖0 and 𝒘0. \nStep 3: Reuse the collected data to solve (41) for 𝑉𝑖+1(𝑥), 𝒖𝑖+1 and 𝒘𝑖+1, with 𝑉𝑖+1(0)=\n0.  \nStep 4: If  a stop iteration condition is met , stop iteration and output 𝑉𝑖+1(𝑥) as the \napproximate optimal solution of HJI equation (8), output 𝒖𝑖+1 as the approximate optimal \ncontrol input, i.e., 𝑉∗(𝑥)=𝑉𝑖+1(𝑥) and 𝒖∗=𝒖𝑖+1, else set 𝑖=𝑖+1, move to Step 3 and \ngo on iteration.  \n \nAlgorithm 3 is a model -free off -policy integral RL method, none of the prior information about system dynamics \nis needed. When 𝛼=1, (41) degenerates to off -policy integral reinforcement learning Bellman equation as shown in \n[26] and [27].  \nB. NN-Based Online Implement of Algorithm 3 \nIn order to solve 𝑉𝑖+1(𝒙), 𝒖𝑖+1 and 𝒘𝑖+1 in (41) by using the system sampling data, the Neural Networks based \nactor -critic structure is introduced. Here three NNs, i.e., two actor NNs and one critic NN, are adopted to learn the \ncost function, disturbance policy  and control policy, respectively. The three NNs are given as follows:  \n𝑉̂𝑖+1(𝒙)=(𝑾𝑐𝑖+1)T𝝆(𝒙)                                                             (42) \n𝒖̂𝑖+1(𝒙)=(𝑾𝑎𝑖+1)T𝝋(𝒙)                                                             (43) \n𝒘̂𝑖+1(𝒙)=(𝑾𝑑𝑖+1)T𝝓(𝒙)                                                             (44) \nwhere 𝝆(𝒙)=[𝜌1 (𝑥) , … , 𝜌𝐿1 (𝑥)]T∈ℜ𝐿1 are the linearly independent basis functions for the critic NN, 𝝋(𝒙)=\n[𝜑1 (𝒙) , … , 𝜑𝐿2 (𝒙)]T∈ℜ𝐿2 and 𝝓(𝒙)=[𝜙1 (𝒙) , … , 𝜙𝐿3 (𝒙)]T∈ℜ𝐿3 are the linearly independent basis functions for \nactor and disturbance NNs, respectively, which are all defined on Ω⊂ℜ𝑛. 𝐿1, 𝐿2 and 𝐿3 are the number of neurons in the hidden layer of the three neural networks, respectively. 𝑾𝑐𝑖+1∈ℜ𝐿1, 𝑾𝑎𝑖+1∈ℜ𝐿2×𝑚 and 𝑾𝑑𝑖+1∈ℜ𝐿3×𝑞 are \nweight vectors. Define 𝑹=diag(𝑟1 , … , 𝑟𝑚), substituting (42) -(44) into (41) yields  \n(𝑾𝑐𝑖+1)T[𝝆 (𝒙 (𝑡 + ∆ 𝑡)) − 𝝆 (𝒙 (𝑡))]+2∑𝑟𝑗∫(𝑾𝑎,𝑗𝑖+1)T𝝋(𝒙)𝜇𝑗𝑡+∆𝑡\n𝑡d𝑡𝑚\n𝑗=1−2𝛾2∑∫(𝑾𝑑,𝑘𝑖+1)T𝝓(𝒙)𝜈𝑘𝑡+∆𝑡\n𝑡d𝑡𝑞\n𝑘=1\n=−𝛼∫(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)𝑡+∆𝑡\n𝑡d𝑡+(1 − 𝛼)(𝑾𝑐𝑖)T[𝝆 (𝒙 (𝑡 + ∆ 𝑡)) − 𝝆 (𝒙 (𝑡))]\n+2(1 − 𝛼)∑𝑟𝑗∫(𝑾𝑎,𝑗𝑖)T𝝋(𝒙)𝜇𝑗𝑡+∆𝑡\n𝑡d𝑡𝑚\n𝑗=1−2(1 − 𝛼)𝛾2∑∫(𝑾𝑑,𝑘𝑖)T𝝓(𝒙)𝜈𝑘𝑡+∆𝑡\n𝑡d𝑡𝑞\n𝑘=1    (45) \nwhere 𝝁=[𝜇1,…,𝜇𝑚]T=𝒖−𝒖𝑖, 𝝂=[𝜈1 , … , 𝜈𝑞]T=𝒘−𝒘𝑖, 𝑾𝑎,𝑗𝑖+1 and 𝑾𝑎,𝑗𝑖 are the jth column of matrix 𝑾𝑎𝑖+1 and \n𝑾𝑎𝑖 respectively, 𝑾𝑑,𝑘𝑖+1 and 𝑾𝑑,𝑘𝑖 are the kth column of matrix 𝑾𝑑𝑖+1 and 𝑾𝑑𝑖 respectively. Note that (45) is linear in \nthe Neural Networks weight vectors 𝑾𝑐𝑖+1, 𝑾𝑎𝑖+1 and 𝑾𝑑𝑖+1. Define  \n𝑾1𝑖+1=[(𝑾𝑐𝑖+1)T,(𝑾𝑎,1𝑖+1)T,…,(𝑾𝑎,m𝑖+1)T,(𝑾𝑑,1𝑖+1)T,…,(𝑾𝑑,𝑞𝑖+1)T]T\n                                 (46) \n𝝕(𝑡)=\n[            𝝆(𝒙 (𝑡 + ∆ 𝑡))−𝝆(𝒙 (𝑡))\n2𝑟1∫𝝋(𝒙)𝜇1𝑡+∆𝑡\n𝑡d𝑡\n⋮\n2𝑟𝑚∫𝝋(𝒙)𝜇𝑚𝑡+∆𝑡\n𝑡d𝑡\n−2𝛾2∫𝝓(𝒙)𝜈1𝑡+∆𝑡\n𝑡d𝑡\n⋮\n−2𝛾2∫𝝓(𝒙)𝜈𝑞𝑡+∆𝑡\n𝑡d𝑡]            \n                                                                    (47) \nand \n𝜆(𝑡)=−𝛼∫(𝑄 (𝒙) + 𝒖𝑖T 𝑹 𝒖𝑖 − 𝛾2 𝒘𝑖T 𝒘𝑖)𝑡+∆𝑡\n𝑡d𝑡+(1 − 𝛼)(𝑾𝑐𝑖)T[𝝆 (𝒙 (𝑡 + ∆ 𝑡)) − 𝝆 (𝒙 (𝑡))] \n+2(1 − 𝛼)∑𝑟𝑗∫(𝑾𝑎,𝑗𝑖)T𝝋(𝒙)𝜇𝑗𝑡+∆𝑡\n𝑡d𝑡𝑚\n𝑗=1 −2(1 − 𝛼)𝛾2∑∫(𝑾𝑑,𝑘𝑖)T𝝓(𝒙)𝜈𝑘𝑡+∆𝑡\n𝑡d𝑡.𝑞\n𝑘=1         (48) \nThen (45) can be rewritten as  \n𝜆(𝑡)=(𝑾1𝑖+1)T𝝕(𝑡).                                                                     (49) \nNote that (49) is linear in the 𝑾1𝑖+1, therefore 𝑾1𝑖+1 can be solved in the sense of  least-squares. Because of 𝑾1𝑖+1∈\nℜ𝐿1+𝑚×𝐿2+𝑞×𝐿3, therefore one needs to collect 𝑀>𝐿1+𝑚×𝐿2+𝑞×𝐿3 system data about system state, disturbance and control input from 𝑡1 to 𝑡𝑀 in the state space. Then, for the given evaluating policies 𝒘̂𝑖 and 𝒖̂𝑖, using \nthis information to calculate (47) and (48) at the 𝑀 different sampling data point, one can get  \n𝜫=[𝝕 (𝑡1) , … , 𝝕 (𝑡𝑀)]                                                                     (50) \n𝜦=[𝜆 (𝑡1) , … , 𝜆 (𝑡𝑀)]T.                                                                     (51) \nThe least -squares solution of (49) is given as  \n𝑾1𝑖+1=(𝜫 𝜫T)−1𝜫𝜦.                                                                            (52) \nThen one can obtain 𝑉̂𝑖+1, 𝒘̂𝑖+1 and 𝒖̂𝑖+1 using (42 -44). Reusing the collect data to solve (49), one can get the \napproximate optimal control input pairs 𝒖∗ and 𝒘∗. \nRemark 4: Although (47) contains 𝒙(𝑡 + ∆ 𝑡), the data can be sampled continuously over the same time interval ∆𝑡, \ni.e., 𝒙(𝑡),𝒙(𝑡 + ∆ 𝑡),…,𝒙(𝑡 + 𝑀 ∆ 𝑡). Thus, none of the information about the dynamics of the system is needed for \ncalculating the system state vector 𝒙(𝑡 + ∆ 𝑡) at instant 𝑡. \nV. Simulation Validation  \nIn this section, the effectiveness of the off -policy α -PI algorithm is verified through two c omputer simulation \nexample s, one is H∞ control problem  for the general nonlinear system, and the other is missile interception simulation . \nA. Nonlinear System H∞ Control Simulation  \nThis is a revised version of example 2 in [23]. The model is as follows:  \n𝒙̇=[−𝑥1+𝑥2\n−0.5𝑥1−0.5𝑥2+0.5𝑥2sin2(𝑥1)]+[0\nsin(𝑥1)]𝑢+[0\ncos(𝑥1)]𝑤                            (53) \n𝒛=𝒙.                                                                                          (54) \nSelect 𝑅=1 and 𝛾=2 for performance value function (5). Polynomial functions are selected as the basis functions \nof neural network ，that is,  𝝆(𝒙)=[𝑥12 , 𝑥22 , 𝑥1 𝑥2 , 𝑥14 , 𝑥24]T with 𝐿1=5, 𝝋(𝒙)=[𝑥1 , 𝑥2 , 𝑥12 , 𝑥1 𝑥2 , 𝑥22 , 𝑥12 𝑥2 , 𝑥1 𝑥22 , 𝑥23 , 𝑥13]T \nwith 𝐿2=9, and disturbance NN and control NN have the same basis functions, i.e., 𝝓(𝒙)=𝝋(𝒙) with 𝐿3=𝐿2. \nAlgorithm 3 is used to learn online the solution of HJI equation. During the data sampling phase, the initial state of \nthe system is set as 𝒙(0)=[0 . 4 , 0 . 5]T, sampling time step is ∆𝑡=0.05 sec, and control input signal and disturbance \ninput signal are set as random signal with uniform distributed in [− 1 , 1]. And then collect 50 (i.e., M=50) sample data \nfrom 𝑡=0 sec to 𝑡=2.5 sec. At the learning stage , the initial weight vectors of the three NNs are set as zero,  𝑾𝑐0=\n0, 𝑾𝑎0=0 and 𝑾𝑑0=0, it means that 𝑉0=0, 𝒖0=0 and 𝒘0=0. The Newton step -size is set as 𝛼=0.3. By reusing the sampled data, the weight vector of NNs can be solved iteratively, when  ‖𝑾1𝑖+1−𝑾1𝑖‖<10−7 stop \niteration.  \nFigs. 1 and 2 show the weight vector 𝑾𝑐 of critic NN and the weight vector 𝑾𝑎 of actor NN at each iteration. It \ncan be seen that the weight vectors of NNs converge at 33rd iteration, 𝑾𝑐33=\n[0.5215,−0.0298,1.1289,−0.0987,0.0245]T, 𝑾𝑎33=[-0.0056,0.0157,0.0719, -1.4079,0.2066,0.6638, -0.4448, -\n0.0721, -0.1117]T. \n \nFig. 1 Weight vector 𝑾𝒄 of critic NN.  \n \nFig. 2 Weight vector 𝑾𝒂 of actor NN.  \nLeveraging t he NN s weights learn ed, the solution of HJI equation and the corresponding H∞ controller can be \nobtained through (42) and (43). Restate the system with the controller obtained and set the disturbance signal as:  \n𝑤=5exp(− 0 . 2 (𝑡 − 𝑡0))cos(𝑡 − 𝑡0)                                                  (55) \nwhere 𝑡0=2.5. The state trajectories of the closed -loop system, control input and disturbance signal are shown in \nFigs. 3 and 4. The first 2.5 seconds in Figs. 3 and 4 correspond to the data collected phase. At the instant 2.5 sec, off -\npolicy α -PI is leveraged to pursue the solution of HJI equation . It also can be seen form Fig. 5 that the disturbance \nattenuation level is less than 0.6, which meets the L2-gain condition.  \n \nFig. 3 State trajectories of the closed -loop system . \n \nFig. 4 Control input and disturbance signals.  \n \nFig. 5 Attenuation level of disturbance with the controller  obtained.  \nB. Missile Interception Simulation  \nThe missile interception problem can be regarded as a two-person zero -sum differential games  problem, which \ndepends on the solution of HJI equation.  The engagement geometry of missile  interception  problem  is shown in  Fig. \n6. \n \nFig. 6 Missile interception target scenario . \nwhere 𝑉M represents missile  velocit y, 𝜂 is the flight -path angles  (FPA) of missile,  𝑎M is missile  acceleration which is \nperpendicular to its velocity . 𝑉T, 𝛽 and 𝑎T are the corresponding variable s of the target.  \n and 𝑟 represent line-of-\nsight angle and missile -target  relative distance , respectively.  The relative motion equations  of missile  and target  are \nas follows . \n𝑟̇=𝑉𝑟=𝑉Tcos(𝛽−𝜃)−𝑉Mcos(𝜂−𝜃)                                                       (55) \n𝜃̇=[𝑉Tsin(𝛽−𝜃)−𝑉Msin(𝜂−𝜃)]/𝑟                                                          (56) \nwhere  𝑉𝑟 is closing velocity, i.e., range rate along the line -of-sight , 𝜃̇ is line-of-sight  rate. Assume that the velocity of \nthe missile and target is constant , select 𝑥1=𝜃 and 𝑥2=𝜃̇ as the system state variables , take the derivative of both \nsides of ( 56), one can get \n{𝑥̇1=𝑥2\n𝑥̇2=−2𝑉𝑟\n𝑟𝑥2+cos(𝜂−𝜃)\n𝑟𝑎M−cos(𝛽−𝜃)\n𝑟𝑎T                                                    (57) \nIn the missile interception target problem,  the expectation of missile is to minimize  𝜃̇, however, the objective of \ntarget which is  an opponent is to maximize  𝜃̇. Therefore, the missile interception target problem  can be regarded as a  \ntwo-person  zero-sum differential games  problem [35].  \nThe performance cost function for the policies of missile and target is defined as follows  \n𝑉𝒖,𝒘(𝒙 (𝑡))=∫(𝑄(𝑥1,𝑥2)+𝑅𝑎M2−𝛾2𝑎T2)∞\n𝑡d𝑡.                                                   (58) \nwhere 𝑄(𝑥1,𝑥2)=𝑞1𝑥12+𝑞2𝑥22, 𝑞1=0, 𝑞2=108, 𝑅=1 and 𝛾=10. \nThe hidden layer neurons of critic NN, actor NN and disturber NN  on \n  are set as follows, respectively.  \n𝝆(𝑥1,𝑥2)=[𝑥14,𝑥13𝑥2,𝑥12𝑥22,𝑥1𝑥23,𝑥24,𝑥12,𝑥1𝑥2,𝑥22]T∈ℝ8                                        (59) 𝝓(𝑥1,𝑥2)=[𝑥2,𝑥12𝑥2,𝑥23]T∈ℝ3                                                             (60) \n𝝋(𝑥1,𝑥2)=[𝑥2,𝑥12𝑥2,𝑥23]T∈ℝ3                                                             (61) \nThus, 𝑊𝑐∈ℝ8, 𝑊𝑎∈ℝ3×1 and 𝑊𝑑∈ℝ3×1, then 𝑊∈ℝ14.  \nNext , the off-policy  α-PI algorithm proposed in this paper is used to solve this missile interception problem.  \nBecause 𝑟, 𝑉𝑟, 𝜙 and 𝛽 change over time, (57) is  a time -varying differential equation . Therefore, the off-policy  α-PI \nalgorithm  is periodically investigated to solve  the optimal control policy 𝑎M∗ of the missile.  The numerical simulation \ncalculation step ∆𝑡 is select to be 0.005 s, and take 𝑁=100, i.e., c ollect 100 data points every 0. 5 s along the system \nstate trajectories, therefore Algorithm 3 needs to be used for solving the solution of HJI equation every 0.5 s. The \napproximate optimal control strategy of the missile obtained  in the previous cycle will be used as behavior policy  of \nthe missile in the next cycle . \nNumerical simulation conditions  of missile and target engagement scenario  are shown in Table 1. \nTable 1 Numerical simulation conditions  \nParameters  Symbol  Value \nInitial position of missile  (𝑥M,𝑧M) (0, 0) m  \nInitial flight -path angle  of missile  𝜂 0 deg  \nVelocity magnitude of missile  𝑉M 600 m/s  \nInitial position of target  (𝑥T,𝑧T) (10000, 0) m  \nInitial flight -path angle  of target  𝛽 170 deg   \nTarget v elocity magnitude  𝑉T 300 m/s  \nTarget acceleration  𝑎T 9 g, S-type maneuver  \n \nThe n umerical simulation results of Algorithm 3  are compared with the adaptive  dynamic surface guidance (ADSG) \nlaw and linear quadratic differential game  (LQDG ) introduced in [36] and [37]. N umerical simulation results of missile \ntarget engagement process are shown in Figs. 7-10. Fig. 7 shows the missile and target trajectories, the miss distance \nis 0.082 m, 0.200  and 1.122 m by using α-PI algorithm , ADSG and LQDG respectively . α-PI algorithm  and ADSG  \ncan achieve a better  guidance accuracy. The acceleration instruction  histories of the three guidance algorithm s are \ndemonstrate d in Fig. 8. In the initial  phase , the target is non -maneuver ing, and after 1.5  s, the target actuates a  S-type \nmaneuver with acceleration  of 9 g, i.e., the maneuvering acceleration is 9 times that of gravity . In the initial, the \nacceleration instruction of missile guided by α-PI algorithm  is driven by  the white  noise . Compared with LQDG , α-\nPI algorithm  and ASDG have a lower demand for missile acceleration . Because the α-PI algorithm  solves the \napproximate solution of the nonlinear differential game guidance problem , which has the advantages of low demand \nof acceleration  and high guidance accuracy.  Furthermore , compared to ASDG, there is no chattering phenomenon  in the acceleration instruction  of α-PI algorithm . The actor NN weights learned during  the online learning guidance \nprocess are shown in Fig. 9.  Fig. 10 shows the n umber of iterations per calculation cycle , as can be seen that the \nmaximum iteration number is 46 at 𝑡=2.0 s. \n \nFig. 7 Missile and target trajectories . \n \nFig. 8 Missile acceleration  instruction  for intercepting  S-type m aneuvering  target . \n \nFig. 9 Actor  NN weights learned  during the online learning guidance process . \n \nFig. 10 Number of iterations per calculation cycle . \nVI. Conclusion  \nTill present, the development of r einforcement learning algorithms and theories for H∞ control design problem of \ncontinuous -time nonlinear system is still an open issue. Most integral reinforcement learning algorithms were based \non Newton iterative method . We proposed a novel 𝛼-PI algorithm which  is based on the damped Newton iterative \nmethod in this paper. Damped Newton iterative method makes the convergence slower than Newton’s method, but \nmight be globally convergent. Thus 𝛼-PI algorithm has better convergence theoretically.  The generalized Bellman \nequation was obtained first, then, on -policy and off -policy 𝛼-PI algorithms were derived, respectively. The on -policy \n𝛼-PI algorithm can be implemented online  for pursuing the solution of H∞ control problem without using information \nabout the system internal dynamics. The off -policy version of 𝛼-PI approach is a model -free algorithm for solving H∞ \ncontrol online. 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In power plant control, one often needs to obtain a precise representation of the system\ndynamics and carefully design the control scheme accordingly. Model-free Reinforcement learning\n(RL) has emerged as a promising solution for control tasks due to its ability to learn from trial-and-\nerror interactions with the environment. It eliminates the need for explicitly modeling the environ-\nment’s dynamics, which is potentially inaccurate. However, the direct imposition of state constraints\nin power plant control raises challenges for standard RL methods. To address this, we propose a\nchance-constrained RL algorithm based on Proximal Policy Optimization for supervisory control.\nOur method employs Lagrangian relaxation to convert the constrained optimization problem into\nan unconstrained objective, where trainable Lagrange multipliers enforce the state constraints. Our\napproach achieves the smallest distance of violation and violation rate in a load-follow maneuver for\nan advanced Nuclear Power Plant design.\nKeywords Data-driven control ·Safe reinforcement\nlearning ·Constrained optimization ·Power plants\n1 Introduction\nTraditionally, control systems have relied on classical\ncontrol theory and feedback control loops, which can be\ncomplex and require significant manual engineering [1].\nIn particular, power plant control requires accurate system\ndynamics models to ensure safety, efficiency, and reliabil-\nity. The construction of these models requires consider-\nable effort. Recently, there has been a growing interest\nin using machine learning and artificial intelligence tech-\nniques for power plant control, including RL [2, 3]. RL\nmethods can potentially reduce the need for manual en-\ngineering and improve the performance, safety, and effi-\n*Sami conducted a majority of the work while he was affil-\niated with the Mathematics and Computer Science Division at\nArgonne National Laboratory.\n†Corresponding Author, ajd@anl.govciency of power plant operations by allowing control sys-\ntems to learn from data and adapt to changing operating\nconditions more efficiently [4, 5, 6]. These previous stud-\nies focused on training an agent to perform tasks, which\ndid not consider the operational constraints of states.\nMeeting constraints allows the unit to operate optimally\nby reducing excessive wear & tear, thus, making the unit\nmore cost-efficient. To learn an optimal control policy that\nincorporates state-level constraints, in this work, we for-\nmulate the “Safe RL” [7] problem as a chance-constrained\noptimization problem and use Lagrangian relaxation to\nconvert the problem to an unconstrained setting. More-\nover, we adopt the state-of-the-art policy gradient method\nto train an intelligent agent to control an advanced Nuclear\nPower Plant (NPP) during routine operational transients.\nThe contributions of this work are four-fold:\n1. We create a physics-based learning environment\nfor training an intelligent agent using reinforce-\nment learning to control a nuclear power plant.arXiv:2401.13020v1  [cs.SY]  23 Jan 2024A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\nThe learning environment is a reduced-order\nSINDYc [8] model, significantly reducing the\ncomputational time to obtain environment feed-\nback.\n2. State-level constraints are included in learn-\ning the optimal policy, where the problem is\nconverted to an unconstrained setting with La-\ngrangian relaxation, and the corresponding La-\ngrange multipliers are trainable parameters.\n3. We propose a new learning scheme where we up-\ndate the policy and value networks on a faster\ntime scale and learn the Lagrange multipliers on\na slower time scale. The faster time scale al-\nlows agents to sufficiently learn a policy to fol-\nlow the load demand under a fixed level of con-\nstraint awareness before a necessary increase in\nthe constraint penalty.\n4. By learning the Lagrangians, the proposed\nmodel shows the best performance in control-\nling an NPP during demanding power transients\n(up to 50 % reduction in total power for a load-\nfollow).\n2 Background and Related Work\n2.1 Reinforcement Learning\nRL is a subfield of machine learning that has gained sig-\nnificant attention in recent years due to its potential to\nenable intelligent agents to learn and adapt to new envi-\nronments and tasks. At its core, reinforcement learning\ninvolves an agent interacting with an environment. The\nagent takes actions based on its current state and receives\nfeedback through rewards or penalties. Optimizing its ac-\ntions to maximize its cumulative reward over time allows\nthe agent to make better decisions and achieve its objec-\ntives more effectively [9]. RL has been the alternative to\nclassic control methods due to its flexibility and no need\nfor much manual engineering and has been successfully\napplied to various domains, including robotics [10, 11],\ngame play [12, 13], and natural language processing [14],\nand has shown promise in addressing complex real-world\nproblems.\nThe RL problem can be mathematically formulated as\na Markov Decision Process (MDP). An MDP consists\nof a state space, S, action space, A, rewards, R, pro-\ncess dynamics determined by the state-transition proba-\nbility, P, and a discount factor, γ. The objective is to\nmaximize an agent’s expected cumulative reward (return)\nwhile navigating through an MDP. Under a policy, π,\nthe state value function describes such expected return,\nvπ(s) =E[P∞\nk=0γkRt+k+1|St=s]. The Bellman equa-\ntion provides a recursive decomposition of the problem.\nv(st) =X\na∈Aπ(a|s)X\ns′∈S,r∈Rp(s′, r|s, a)[rt+v(s′)],\n(1)where sis the current state and s′is the next state. We\naim to find a policy function, π∗, that maximizes the value\nfunction such that\nv∗(s) = max\nπvπ(s) =vπ∗(s). (2)\nIn this work, we adopt policy gradient methods. Unlike\nmethods that deduce a policy from state-action values,\npolicy gradient methods aim to directly learn a policy π\nusing a parametrized model. This approach is particularly\nuseful for problems with continuous action spaces as it\nallows the model to provide a probability distribution for\nactions. Furthermore, policy gradient methods are often\nmore sample-efficient than other methods, making them\nwell-suited for our problem setting.\n2.2 Constrained Deep Reinforcement Learning\nOne of the primary motivations for deep RL with re-\nstricted safety is the increasing deployment of RL agents\nin safety-critical applications such as autonomous driv-\ning, robotics, and healthcare care [15, 16, 17]. These\napplications require that agents operate safely and avoid\nactions that could cause harm to humans or the environ-\nment. However, traditional RL algorithms are often de-\nsigned to optimize a single objective, such as maximiz-\ning a reward signal, without explicitly considering safety.\nSafe RL seeks to overcome this limitation by integrat-\ning safety constraints into the RL framework and design-\ning algorithms that balance safety and performance ob-\njectives. This involves developing methods for modeling\nand predicting safety hazards, enforcing safety constraints\nduring training and testing, and ensuring that the agents’\nbehavior is interpretable and understandable to humans.\nSafe RL has the potential to revolutionize many indus-\ntries by enabling the safe and reliable deployment of au-\ntonomous systems.\nIn the MDP framework, there are two significant cate-\ngories of Safe RL [18], modifying the optimality criterion\nand changing the exploration process based on the guid-\nance of a risk metric. The general idea behind modifying\nthe optimization criterion and the reward function is to\nconsider the risk in the objective function RL models tend\nto optimize. Additional criteria can include worst-case,\nrisk-sensitive, and constrained criteria [19, 20, 21]. On the\nother hand, with safe exploration, the agent seeks to fol-\nlow a safety-aware exploration and exploitation scheme\nby incorporating external knowledge, such as teacher ad-\nvice [22], or follow a risk-directed exploration [23].\nAnother way of dealing with safe RL problems is\nto formulate a constrained Markov Decision Pro-\ncess (cMDP) [24]. In a cMDP, the agent must satisfy\nconstraints while maximizing the cumulative reward. In\naddition to the elements of an MDP, cMDPs contain a\nconstraint function C, specifying the cost or penalty for\ntransitioning from one state to another by taking an ac-\ntion. The constraints are known a priori, and linear pro-\ngramming and Lagrangian methods are commonly used\n2A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\nto solve cMDPs [7]. In our work, we formulate the\nconstraint function as a chance constraint and adopt La-\ngrangian relaxation to convert it into an unconstrained\nlearning problem. Section 4 describes the problem setting\nand formulation details.\n2.3 Power Plant Control (constrained continuous\ncontrol)\nControlling power plants is challenging due to the com-\nplexity of coordinating multiple systems, such as boilers,\nturbines, generators, and auxiliary systems. Power plants\nmust balance the electricity supply with its demand to en-\nsure the electrical grid’s stability. The electricity demand\nvaries depending on weather patterns, the availability of\ncompeting plants, and the energy demand. Nuclear power\nplants (NPPs) have additional layers of complexity to pre-\nvent accidents. Multiple safety systems address the safety\nrisks that have led to NPPs being the safest baseload en-\nergy generators [25]. However, NPPs also have high Op-\neration & Maintenance (O&M) costs. The O&M costs for\nexisting plants are almost an order of magnitude higher\nthan those of a conventional combined-cycle gas turbine\nplant [26].\nA proposal to address the high O&M costs has been\nto shift the operation of NPPs from a baseload opera-\ntion paradigm to a “load-following” one [27]. In load-\nfollowing, the operators would adjust the plant’s electri-\ncal output to meet requested changes in load from the\ngrid. To achieve this, the total thermal energy generated\nby the plant must be reduced or increased. In an NPP,\nthis is primarily achieved by controlling the reactivity of\nthe nuclear core by inserting or removing control blades.\nBecause NPPs are underactuated, i.e., the number of ac-\ntuators is lower than the number of states, some states\nvary between high and low power levels. There are no\nadditional actuators available to constrain the deviation of\nsuch states. Constraining the deviation of such states is\nessential to optimize the plant’s performance. For exam-\nple, if an excessive deviation in temperature occurs during\nroutine load-follows, the cyclical thermal stresses can fail\ncomponents [28]. Therefore, an additional control algo-\nrithm is needed to constrain the changes in power levels\nto ensure that the plant operates optimally. This work pro-\nposes to develop and demonstrate an RL algorithm to ad-\ndress this task.\n3 Supervisory NPP Control\nThis section describes a physics-based environment de-\nsigned for training RL agents that provide supervisory\ncontrol of an advanced NPP.\n3.1 RL Environment Design\nTo design next-generation NPPs, significant efforts focus\non developing advanced simulators. The System Analy-\nsis Module (SAM) code is a state-of-the-art physics-basedtool developed at Argonne National Laboratory to enable\nhigh-fidelity transient analyses of the whole plant for ad-\nvanced NPP designs [29]. In this work, SAM is embedded\ninto an RL environment as a digital twin to simulate the\ndynamics of the NPP. The SAM-RL environment is pre-\nsented in Figure 1. The environment has two prominent\nfeatures. First, SAM will provide a physics-based sim-\nulation of NPPs by modeling heat transfer, fluid dynam-\nics, and nuclear dynamics. The SAM model is formed\nby the thermophysical properties of the system’s fluid, the\nsystem’s physical characteristics (e.g., coefficients defin-\ning dynamics of the nuclear reactor), and the specification\nand arrangement of physical components (e.g., pipes, heat\nexchangers, pumps). Second, lower-level controllers will\ninterface actions provided by the RL agent and the NPP’s\nactuators. This controller arrangement intentionally main-\ntains overrideability and allows physical constraints to be\napplied to the actuators.\nSUPERVISORY NPP\nAGENT\nSAM-RL E NVIRONMENT\nSAM\nNPP Model\n•Thermophysical\nProper�es\n•Plant Components\n•Thermal Hydraulic\nand Neutronic\nCharacteris�cs•Physical actuator\nconstraints\nAc�onVector-valued Reward\nNPP State\nLower-level\nControllers\n(PID, on/off)\nFigure 1: Application of RL for supervisory NPP Control\nusing the proposed SAM-RL environment\nThe RL agent interacts with the SAM-RL environment in\ndiscrete time steps t={0,1,···, T−1}by exchang-\ning system states st, actions at, and vector-valued rewards\nrt. Vector-valued rewards encode the primary ( e.g., load-\nfollowing reward) and secondary objectives ( e.g., avoid\nconstraint violation) of the controller. They can enforce\nphysical constraints on system states or actuators as dis-\ncussed in subsection 4.1.\nIn this work, we adopt a 320 MW pebble-bed fluoride-\ncooled high-temperature reactor design [30]. The layout\nof the power plant is presented in Figure 2. There are four\ncircuits in total. The primary side has a pump that cir-\nculates FLiBe molten salt through the intermediate heat\nexchanger and the nuclear core. The secondary side has a\npump that circulates FLiNaK molten salt through a steam\ngenerator and the intermediate heat exchanger. Fixed heat\ntransfer boundary conditions are imposed on the steam\ngenerator to mimic the presence of an energy conversion\ncycle. Additionally, a passive safety system is present to\naid in cooling the reactor vessel in the case of an accident.\n3A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\nThis reactor design is modeled in SAM and then embed-\nded in the SAM-RL environment.\nPassive\nSafety\nSystem\nPebble-bed\nNuclear\nCorePrimary\nPumpSecondary\nPump\nIntermediate Heat \nExchanger\nSteam\nGeneratorImposed\nBoundary\nConditions\nFigure 2: Layout of the Advanced NPP studied in this\nwork. Measurement locations of various states are anno-\ntated. The major actuators are listed in blue text below the\ncomponent labels.\n3.2 Control Strategy\nThis work adopts a control strategy that reduces ther-\nmal cycling during load-follow operation [31, §2.1]. The\nquantitative goal of this control strategy is to regulate the\ncore inlet and outlet temperatures. The reactor’s power\nwill be manipulated through external reactivity insertion\nto accommodate load increase or decrease requests. Then,\npumps on both the primary and secondary sides will gov-\nern the heat transfer rates across the primary heat ex-\nchanger. The association between controlled and manipu-\nlated variables has been selected to optimize the impact of\ncontrol actions and lessen cross-talk. These associations\nare noted below:\n• The reactor power ( ˙QRX) is controlled by exter-\nnal reactivity insertion ( ρext).\n• The core outlet temperature ( Tc,out) is controlled\nby the primary pump head ( ∆Pp).\n• The core inlet temperature ( Tc,in) is controlled\nby the solar salt pump head ( ∆Ps).\nThree Proportional-Integral-Derivative (PID) controllers\nwere designed, with two regulating the temperatures at\nthe core outlet and inlet and the third tracking the re-\nactor power. The control strategy closely manages the\ndynamics of the primary loop (red circuit in 2). Mean-\nwhile, the secondary side of the plant is not directly con-\ntrolled (orange circuit in 2). In 5, the dashed lines show\nthat the secondary-side inlet and outlet temperatures ( Ts,in\nandTs,out) achieve new equilibrium values once the load\nfollows transient has completed. To prevent temperature\nfluctuations and optimize the dynamics of the NPP, two\nconstraints are set on the secondary side: a minimum tem-\nperature constraint on the heat exchanger inlet and a max-\nimum temperature constraint on the heat exchanger outlet.\nA common approach is to use a model that accurately rep-\nresents the system dynamics to perform model predictive\ncontrol. However, achieving the level of precision of themodel for effective control is not trivial, and a different\nmodel must be established for each system. Model-free\nreinforcement learning algorithms relax this requirement\nby interacting with the environment and learning by trial\nand error.\n3.3 Reduced-order Environment\nGenerating a transient of a power plant using SAM re-\nquires a computation time on the order of days. Training\nRL agents from scratch on SAM is, therefore, infeasible,\nas RL agents typically require hundreds of thousands of\nenvironment interactions. To avoid exhausting compute\nresources and speeding up the training of RL agents, we\nadopt a transfer learning approach whereby RL agents are\nfirst trained on a reduced-order model (ROM) of SAM.\nThe ROM of SAM can be learned in a data-driven man-\nner by generating Ktransients in parallel using SAM and\napplying a sparse identification of the nonlinear dynami-\ncal system (SINDYc) [32] on the data. The learned ROM\nis used next to train the RL agents. To facilitate algo-\nrithm development and efficient training, we adopt a ROM\nsub-sampled every five steps from the original time scale\n(episode length T= 11250 ), resulting in a shortened\nT= 2250 for the experiments.\n4 Chance Constrained RL with\nLagrangian Primal-Dual Optimization\n4.1 Load-following Problem Formulation\nThis work focuses on creating a Safe RL framework,\nwhich allows the agent to execute actions that opti-\nmize the cumulative rewards while adhering to state con-\nstraints. Load-following transient falls under this formu-\nlation, which is an important maneuver for advanced reac-\ntors to adopt due to the increasing proportion of intermit-\ntent energy sources. In the load-following transient, the\ntask is to satisfy the requested changes in power generated\nby the operating unit while adhering to constraints placed\non the system states. Prediction of violation is challeng-\ning, as the entire system has nonlinear behavior and lags\nassociated with transport phenomena.\nMathematically, the problem can be formulated as a\nchance-constrained optimization problem over the space\nof parameterized policies πθ(a|s)with parameters θ,\nwhich follows the safety level 1, constraint satisfaction\nencouraged, in the Safe RL context [7]. The objective is\nto maximize the expected cumulative rewards subject to a\nchance constraint that ensures the system’s state remains\nwithin a safe region over the operating horizon Twith a\nhigh probability,\nmax\nπEat∼π,st+1∼P,s0∼βhT−1X\nt=0γtR(st, at)i\ns.t.ps(π) =PrhT−1\\\nt=0st∈ S safei\n≥1−Kδ,(3)\n4A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\nwhere Ssafeis the set of states that are within the safe oper-\nation range. In its current form, the optimization problem\nin (3) is hard to solve because the gradient of the chance\nconstraint with respect to θis discontinuous, nondiffer-\nentiable, and crucially intractable. Notice that the Monte\nCarlo (MC) sample average to estimate ps(π)for a given\nπisˆps(π) =1\nMPM−1\nm=0 1\u0000TT−1\nt=0sm\nt∈ S safe\u0001\n, where sm\nt\nis the state of the system at time step tin the m-th sampled\ntrajectory, and 1\u0000TT−1\nt=0sm\nt∈ S safe\u0001\nis an indicator func-\ntion that is 1if the system remains within the safe region\nat all time steps Tand is 0otherwise.\nTo solve equation (3) using model-free RL methods, we\nreformulate the chance constraint by applying Boolean al-\ngebra laws and probability bounds to arrive at a tractable\nform. First, notice that,\nPr\"T−1\\\nt=0st∈ S safe#\n≤Kδ⇒PrhT−1\\\nt=0st∈ S safei\n≥1−Kδ,\n(4)\nthat is, if we guarantee that the probability of the comple-\nmentary event is at most Kδ, then the joint chance con-\nstraint will be satisfied with probability at least 1−Kδ.\nBy applying DeMorgan’s law followed by a union bound,\nPr\"T−1\\\nt=0st∈ S safe#\n=Pr\"T−1[\nt=0st̸∈ S safe#\n≤T−1X\nt=0Prh\nst̸∈ S safei(5)\nIn practice, the event st∈ S safeis usually expressed as K\n(non)-linear constraint functions of the state,\nst∈ S safe⇔ T k(st)≤0,∀k∈ {1,···, K},(6)\nwhere Tkis a (non)-linear operator on st. Define an indi-\ncator function,\nCk(st) =\u001a0,Tk(st)≤0\n1,Tk(st)>0,(7)\nthat is, the indicator function Ck(st)tracks whether or not\nthek-th safety constraint is violated at st. Notice that\nPrh\nst̸∈ S safei(i)=PrhK[\nk=1\u0000\nCk(st) = 1\u0001i\n(ii)\n≤KX\nk=1Prh\nCk(st) = 1i\n,(8)\nwhere (i)again follows by applying DeMorgan’s law, and\n(ii)follows by applying a union bound. Combining the\nresults of (5) with that of (8), we obtain the following im-\nplications,\nT−1X\nt=0Prh\nst̸∈ S safei\n≤T−1X\nt=0KX\nk=1Prh\nCk(st) = 1i\n≤Kδ\n⇒ps(π)≥1−Kδ.\n(9)It is also true that ifPT−1\nt=0Prh\nCk(st) = 1i\n≤δ,∀k∈\n{1,···, K}, the original chance constraint will be satis-\nfied with probability at least 1−Kδ. Notice that,\nEhT−1X\nt=0Ck(st)i\n=T−1X\nt=0Eh\nCk(st)i\n=T−1X\nt=0Prh\nCk(st) = 1i\n∗1 +Prh\nCk(st) = 0i\n∗0\n=T−1X\nt=0Prh\nCk(st) = 1i\n(10)\nFurthermore, EhPT−1\nt=0Ck(st)i\n≤\n1\n1−γEhPT−1\nt=0γtCk(st)i\nas long asγt\n1−γ≥1,∀t∈\n{0,···, T−1}(e.g. for T= 2250 , γ⪆0.9974 ). Hence,\na more conservative yet tractable reformulation of (3) is,\nmax\nπEhT−1X\nt=0γtR(st, at)i\ns.t.EhT−1X\nt=0γtCk(st)i\n≤δ(1−γ),∀k∈ {1,···, K}.\n(11)\nBy learning the optimal policy for (11), we ensure that this\npolicy is feasible, albeit being over-conservative, with re-\nspect to (3), i.e. ps(π)≥1−Kδ. The optimization prob-\nlem of (11) can be readily formulated as a discrete-time\ncMDP with continuous state-action spaces as follows,\n1.∀st∈ S, st= [xt,ct, pt]∈Rd, where xtis\na vector denoting the state of the power plant at\ntime step t,ctis a vector denoting NPP state con-\nstraint bounds, and ptis a scalar that denotes the\npower demand,\n2.∀at∈ A, at∈R+is the action that denotes the\npower output from the plant,\n3.rt= [r0\nt, r1\nt,···, rK\nt]∈RK+1is a vector val-\nued reward function which encodes the primary\n(r0\nt)andK-secondary objectives (r1\nt,···, rK\nt)\nof the RL-based controller. r0\ntis the negative\nof the squared L 2norm between the power de-\nmand ptand power plant supply at, i.e. r0\nt=\nR(st, at) =−||pt−at||2\n2. By maximizing a cu-\nmulative sum of the primary reward signals, the\nagent is trained to learn a policy that satisfies the\npower demand as much as possible. On the other\nhand, the k-th secondary objectives rk\nt=Ck(st)\nis the indicator function that is 1ifstviolates the\nk-th constraint on the state of the system and is 0\notherwise, as given by Equation (22).\n5A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\n4.2 Lagrangian Primal-Dual Optimization\nIn this subsection, we design an on-policy RL algorithm to\nsolve (11). The on-policy algorithm is based on proximal\npolicy optimization (PPO) [33]. PPO has shown the ad-\nvantages of its scalability, data efficiency, and robustness\nin various applications [34, 15]. To incorporate chance-\nconstrained state optimization under the framework of\nPPO, we leverage the idea of Lagrangian relaxation in\nwhich (11) is converted to an equivalent unconstrained\nproblem,\nmax\nπmin\n{λk≥0}EhT−1X\nt=0γt\u0010\nR(st, at)−KX\nk=1λkCk(st)\u0011i\n+KX\nk=1λkδ(1−γ),\n(12)\nand invoking the min-max theorem to interchange the max\nand min terms,\nmin\n{λk≥0}max\nπEhT−1X\nt=0γt\u0010\nR(st, at)−KX\nk=1λkCk(st)\u0011i\n+KX\nk=1λkδ(1−γ),\n(13)\nwhich can be solved on two-time scales: on a faster time\nscale, gradient ascent is performed on state values to find\nthe optimal policy for a given set of Lagrangian variables,\nand on a slower time scale, gradient descent is performed\non dual variables [35]. Notice that for a fixed set of La-\ngrange variables, the problem reduces to a standard MDP\nwith a penalized reward function [36],\nˆR(st, at) =R(st, at)−KX\nk=1λkCk(st) +constant .(14)\nwhere constant =PK\nk=1λkδ(1−γ)is a constant for pol-\nicy and value network updates at the faster time scale, thus\ncan be dropped from the optimization objective. Opti-\nmization time scales are controlled by choosing the max-\nimum learning rate of the stochastic gradient optimizer\nused, e.g., adaptive moment estimation [37], and the num-\nber of gradient steps performed at the end of each training\nepoch.\n4.3 Constrained PPO\nWe use PPO to train an RL agent to maximize the pro-\nposed objective (14). PPO is a state-of-the-art on-policy\nRL algorithm in which the parameterized policy πθ(a|s)\nis directly learned by maximizing the PPO-clip objective\nfunction,\nOclip(θ) =Eth\nmin(πθ(at|st)\nπθold(at|st)ˆAt,clip(πθ(at|st)\nπθold(at|st),1 +ϵ,1−ϵ)ˆAt)i\n,\n(15)\nwhere θare the policy parameters, ϵis a clip fraction, and\nˆAtis the generalized advantage estimator (GAE) [38],\nˆAt=∞X\nl=0(γξ)l\u0000ˆRt+l+γVϕ(st+l+1)−Vϕ(st+l)\u0001\n,(16)where ξis a hyper-parameter to strike a balance between\nbias and variance in estimating the advantage, and Vϕ(st)\nis the estimated parameterized value function with param-\netersϕ. The clipped surrogate advantage objective (15)\nensures that gradient-based policy updates yield improved\npolicies by limiting (by clipping) how much the new pol-\nicy can move away from the old policy while improving\nthe training objective. To further constrain policy updates,\nthe gradient optimization subroutine running on a batch\nof trajectories is typically terminated when the expected\nKL-divergence between the new policy and the old policy\nreaches a predefined threshold KL Th.\nIn (16), ˆRnis the Lagrangian penalized reward function\ngiven by (14). The Lagrangian penalty multipliers are up-\ndated according to policy feasibility by gradient descent\non the original constraints. Given that λk,∀kare ini-\ntially set to 0, i.e., the agent is initially indifferent to the\ncost constraints, the Lagrangian penalty multipliers are\nupdated at a slower time scale than that of updating the\npolicy by minimizing the following loss function with re-\nspect to λk,∀k,\nOL(λ1,···, λK) =PK\nk=1λkclip\u0010\nδ(1−γ)−EhPT−1\nt=0γtCk(st)i\n,−∞,0\u0011\n(17)\nAs long as a constraint kis violated, then∂OL\n∂λk<0, and\nsoλkwill be increased to enforce the constraint. Due\nto clipping, λkwill not be updated if the constraint kis\nsatisfied. By restricting λk,∀kto increase monotonously\nduring training, the agent avoids oscillations between fea-\nsible and infeasible policy spaces when optimizing pol-\nicy parameters (15), which improves learning stability and\npromotes convergence to a local saddle point.\nFinally, the state-value function is learned by minimiz-\ning the mean squared error loss against the policy’s dis-\ncounted penalized rewards-to-go.\nOV(ϕ) =Eh\u0010\nVϕ(st)−∞X\nl=0γlˆRt+l(st+l, at+l)\u00112i\n.\n(18)\n4.4 RL Agent Training\n4.4.1 Policy/Value Network Architecture\nThe policy is a parameterized Gaussian policy, shown in\nEquation 19. During training, we sampled the action from\nthe Gaussian policy and completed episodes. An invert-\nible squashing function tanh was applied to the policy to\nenforce action bounds. The distribution of the squashed\npolicy and its log-likelihood can be easily derived as in\n[39].\nπ(at|st,θπ) =1\nσ(st,θπ)√\n2πexp\u0010\n−(at−µ(st,θπ))2\n2σ(st,θπ)2\u0011\n.\n(19)\nWe parallelize the training process with multiple work-\ners through MPI and randomly sample a demand curve\n6A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\nfor each worker per epoch. During training, for each\nepisode, the policy network inputs the state variables\n(primary-side mass flow rate, secondary-side mass flow\nrate, core inlet temperature, core outlet temperature, core\noutlet pressure, secondary-side heat-exchanger inlet tem-\nperature, secondary-side heat-exchanger outlet tempera-\nture, delayed neutron precursor concentrations, total heat-\nexchanger energy transfer, and energy generated by the\nsteam generator), demand, and inlet/outlet temperature\nconstraints, and outputs the Gaussian policy’s mean and\nstandard deviation functions. The SINDYc environment\nthen takes a sampled action from the policy and returns\nthe state variables and demand for the next time step un-\ntil it reaches the end of the episode. The value networks\napproximate the value of current and future states. Fi-\nnally, the model calculates the reward values and updates\nthe model parameter via gradient ascent over the reward\nfunction on the faster time scale and gradient descent to\nupdate the Lagrangians on a slower time scale. We control\nthe randomization of the generated power demand curves\nand designate 36 ×600 unique episodes for training, 10 for\nvalidation and 30 for testing.\n5 Results\nThis section shows the experimental results using the pro-\nposed model for performing load-following in generic flu-\noride high-temperature reactors. The model evaluation is\nbased on reporting three metrics scores. We first show the\ntraining and testing results of the proposed models using\nmulti-layer perceptron (MLP) and Long-short term mem-\nory (LSTM) actors, respectively. Furthermore, we com-\npare the proposed model with other baselines where we\nfix the Lagrangian multipliers in (14) as a regularization\nterm to the load-following reward function, which can be\nregarded as a means of reward shaping [40]. Finally, we\ndeploy the trained λ-PPO agent to the full-fidelity SAM\nmodel with action clipping for stability and show control\nresults of the direct application of the trained agent.\nThe implementation and models were in TensorFlow with\nan MPI setup, and the training and evaluation were on the\nBebop cluster with 36 workers from Argonne’s Labora-\ntory Resource and Computing Center (LCRC).\n5.1 Evaluation Metrics\nWe use three metrics to evaluate model performance, the\nreward-cost score, ¯R(st, at), the mean distance of viola-\ntion,D, and the mean proportion of violation or violation\nrate,ω.\n¯R(st, at), defined in (20), is simply the load-following\nreward minus the cost values for violating the constraints.\nA higher value indicates better performance.\n¯R(st, at) =R(st, at)−2X\nk=1Ck(st, at). (20)Since the violation costs are binary values, that is, viola-\ntions of constraints contribute a value of 1 and 0; other-\nwise, costs do not directly capture the magnitude of the\nviolation a model causes. Therefore, to account for the\nmagnitude of violation, we also introduce the mean dis-\ntance of violation, D, defined in (21).\nDin=NX\ni=1I(Cin−Tin)|Tin− Cin|,\nDout=NX\ni=1I(Tout− Cin)|Tout− Cout|,\nD=Din+Dout\n2N,(21)\nwhere Nis the number of trajectories in the testing set,\nandIis an indicator function (22), which takes value 1\nwhen there is a violation of constraints.\nI(x) =\u001a0ifx⩽0\n1ifx >0. (22)\nThe third metric is the mean violation rate defined as (23),\nω=1\nNNX\ni=1t(i)\nv\nτ(i), (23)\nwhere t(i)\nvis the time duration during which a constraint is\nviolated, and τ(i)is the total duration of the ith trajectory.\nWe use all three metrics to evaluate the models in the fol-\nlowing sections. A lower number indicates better model\nperformance for all these metrics.\n5.2 Load-following in generic fluoride\nhigh-temperature reactors (gFHR)\nWe trained the proposed chance-constrained PPO agent\non36×600 (600 epochs and 36 workers per epoch)\nunique episodes and validated it using 10 other unique\nepisodes. The objective of the model was to learn a policy\nthat maximizes the cumulative reward over each episode,\nensuring that the inlet/outlet temperature does not breach\nthe time-dependent constraints. We used an MLP-based\nand LSTM-based actor to further analyze the impact of\nthe policy network structure on the agent’s performance.\nWe trained MLP- and LSTM-actor PPO agents on the\nsame training trajectories. Figure 3 shows the training\ndynamics of the proposed models. Both models present\na drastic jump in reward at the beginning of training and\na gradual decrease due to the enforcement of safety con-\nstraints as training progresses. As a result, the correspond-\ning costs CinandCout, decrease, suggesting that agents\nare learning policies that follow demand while staying in\nsafe regions. The trainable Lagrangian multipliers, λin\nandλout, in both MLP-actor and LSTM-actor models,\nincrease sharply at the early stage of training when the\nagents simply try to follow the demand only. At a later\n7A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\n0 200 400 600\nEpochs500\n400\n300\n200\n100\n0(st,at)\nMLP\nLSTM\n0 200 400 600\nEpochs025050075010001250\nMLP\nin\nLSTM\nin\nMLP\nout\nLSTM\nout\n0 200 400 600\nEpochs0.00.10.20.30.4\nMLP\nin\nLSTM\nin\nMLP\nout\nLSTM\nout\n0 200 400 600\nEpochs0.20.40.60.81.0HHMLPHLSTM\nFigure 3: The training curves of the proposed models with MLP- and LSTM-actors. From left to right, it shows the\nchanges in rewards, costs associated with two safety constraints, the magnitude of learned Lagrangian multipliers, and\nthe policy entropy.\nab\ncd\nFigure 4: Visualization of agents’ performance on a testing trajectory. a.&c.show the model performance of the\nMLP actor; b.&d.show the model performance of the LSTM actor. Both models resulted in compliance with safety\nconstraints while closely following demand when possible. The MLP actor, in comparison, adhered more closely to\nthe demand and safety constraints, whereas the LSTM actor led to fewer oscillations in actions. In c.&d., both\ntrained RL agents adapt successfully to time-varying constraints. The resulting states adhere closely to the changing\nconstraints and return to following the demand-induced trajectory when potential violations no longer present.\ntime, the rate of increase of λinandλoutslows down as\nthe agents learn to satisfy the constraints simultaneously.\nThe entropy of the Gaussian policy implies the agent’s ex-\nploration rate. Sufficient exploration is necessary for the\nagent to obtain different experiences to find the optimal\npolicy. However, excessive exploration can diverge from\noptimal policy and make recovery difficult. In training\nthe proposed models, the policy entropy started with high\nvalues, ensuring abundant exploration, and gradually de-\ncreased to exploit the best possible policy.\nIn comparison, the MLP actor model showed a higher\nload-following reward, lower constraint costs, and lower\nentropy at the end of training. The learned λinandλout\nof the MLP-actor lie between them of the LSTM-actor,\nwhich suggests a harsher regularization for Tinand less\nforToutin the MLP-actor. Regarding the policy entropy,\nthe MLP-actor ended up with a stabilized value of around0.2 while the LSTM-actor presents a more oscillatory and\nhigher policy entropy with a value of around 0.4. To eval-\nuate the performance of the RL model, we calculated ¯R,\nD, and V R(Table 1) over 30 testing episodes and visual-\nized several trajectories. We compared the state variable\ntrajectories controlled by the RL agent with the trajecto-\nries obtained by following the demands. Figure 4 shows\nthat the actions controlled by the RL agent lead to non-\nviolation of the constraints while closely following the\ndemand where possible. In comparison, the inlet tempera-\nture controlled by the reference governor is slightly lower\nthan the constraint lower bound. Meanwhile, the expected\nstates from simply following the demand evidently violate\nboth upper and lower bounds of the constraints.\nThe performance difference between the MLP and LSTM\nactors was because, on the one hand, the system was\nMarkovian, where the states at the next step only depend\n8A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\n426427428Ts,in [°C]\n470.0472.5Ts,out [°C]\n80010001200mp [kg/s]\n0 2500 5000\nTime [s]300040005000ms [kg/s]\n0 2500 5000\nTime [s]819.5820.0Tc,in [°C]\n0 2500 5000\nTime [s]250300vQRX [MW]\nDemand\nRef. Gov.\n-PPO\n-PPO States\n Ref. Gov. States Reference Constraints\nFigure 5: The visualization of the response of other state variables to load demand and λ-PPO agent. Compared to the\nreference governor, λ-PPO agent adheres to the constraints more closely without violation.\nTable 1: Model performance of the proposed models\nModels ¯R D ω\nMLP-actor −251.84 3 .2657×10−50.0015\nLSTM-actor −450.27 0.0016 0.0796\non the current states. The MLP actor depended only on\nthe current-state input and the output of the corresponding\naction. This led to a faster response and closer adherence\nto the constraints of the MLP actor. On the other hand,\nthe LSTM actor considered the hidden states in the recur-\nrent layer, of which the weights were trained on a certain\nlength of experience history (trace length). Therefore, the\nLSTM actor resulted in a slower response to constraint\nchanges and a smoother action output.\nGiven the superior performance of the MLP-actor PPO\nagent, we call it λ-PPO model and use it for the subse-\nquent results. Fig. 5 shows the performance of the trained\nPPO agent. The agent is given the task of reducing the\npower of the plant through several ramps, shown in the\nv˙QRXsubplot. When reducing power, the agent is re-\nquired to adhere to constraints on the inlet and outlet of\nthe secondary heat exchanger ( Ts,inandTs,out). Due to\nthe control strategy, Section 3.2, the temperatures at the\nprimary circuit are regulated ( i.e., to be kept constant).\nThe secondary side settles to a new value, while the reg-\nulated temperatures revert to the initial values once the\npower change is completed. The agent attempts to meet\nthe requested load change while satisfying the constraints\non the changes in secondary-side temperatures. At the be-\nginning of the transient, the agent correctly follows the\nrequested load because no constraints are anticipated to\nbe violated. However, at around 2100 s, a constraint vi-\nolation would occur if the agent continues to follow thereduction in load. The agent correctly intervenes to avoid\na constraint violation while attempting to meet the reduc-\ntion in load.\nInstead of fixing the Lagrangian multipliers in (14), learn-\ning is essential to maximize the possible cumulative re-\nward while satisfying constraints. The learned Lagrangian\nmultipliers effectively regulate the action of the agent to\nsatisfy the safety constraints. Figure 6 shows the progres-\nsion of the Lagrangian effect during training as they grad-\nually increase. At the start of training, as updating the\nLagrangians was on a slower time scale, the policy was\nmerely to follow the demand without safety concerns. Af-\nter a few epochs, the learned Lagrangians increased, re-\nsulting in better compliance with the constraints.\nTable 2: Model performance comparison\nModels ¯R D ω\nλ-PPO −251.84 3 .2658×10−50.0015\nλ= 0.1−1600.28 0.6652 0.3698\nλ= 0.2−746.00 0.2611 0.0923\nλ= 0.3−261.85 5 .1481×10−50.0039\nλ= 0.4−258.42 3 .5640×10−50.0033\nλ= 0.5−273.03 9 .5473×10−50.0049\n5.3 Learned Lagrangians\nWe compared the performance of our proposed model,\nwhich we call λ-PPO, with baseline models with fixed\nLagrangian multipliers. Baseline models can be viewed\nfollowing a reward-shaping scheme [40], where we ex-\nplicitly impose constraints on the objective function. Fig-\nure 7 and Table 2 show the comparison of these models\nin terms of ¯R,Dandω. The results suggest a distinctly\nsuperior performance of λ-PPO with the lowest violation\n9A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\n0 1000 2000 3000 4000 5000\nTime [s]426.0426.5427.0427.5428.0428.5Tin (C)\n0 1000 2000 3000 4000 5000\nTime [s]468469470471472473Tout (C)\nin\nLoad-following Tin\nEpoch 30\nEpoch 80Epoch 120\nEpoch 300\nout\nLoad-following ToutEpoch 30\nEpoch 80\nEpoch 120\nEpoch 300\nFigure 6: Agent performance controlling the inlet and out-\nlet temperature during training. The magnitude of the La-\ngrangian multipliers increased as the training went on. At\nthe initial stage of training, the model closely follows the\ndemand, violating the constraints. However, as the num-\nber of epochs increased, the model acted within safety\nconstraints while following the demand when it was fea-\nsible.\ndistance Dand violation rate V R. Meanwhile, with a\nsmall fixed λ(e.g. 0.1 and 0.2), the agent resulted in a\nlow reward-cost score, ¯R, and high constraints violation\ncosts,CinandCout. With a larger fixed λ(e.g. 0.4 and\n0.5), which happened to be close to the learned values,\nthe agent gained an evident improvement in satisfying the\nconstraints. However, the degree of violation of the con-\nstraint and the violation rate are higher than the λ-PPO\nmodel. The proposed λ-PPO model was shown to have\nthe ability to learn the most appropriate Lagrangian mul-\ntipliers in (14) from the given environment, eliminating\nthe extra effort of manual fine-tuning.\n5.4 Transfer Learning on the full SAM model\nGiven the fact that the SAM model needs to run on the\noriginal time resolution, we trained a new λ-PPO agent\nin the reduced-order SINDYc environment with fine time\nresolution. Due to the much longer episode length com-\npared to the previous environment (11250 vs. 2250), to\nreduce the variance in value and advantage function cal-\nculation, the model treated a long episode as 5 shorter sub-episodes (same length as the previous episode). The boot-\nstrapped values of the value network output replaced the\nvalues of the last state in the sub-episodes. Figure 8 shows\nthe training dynamics of such λ-PPO agent.\nDirect application of the agent. We investigated the\nagent’s performance on controlling the high-fidelity SAM\nmodel trained using the reduced-order SINDYc environ-\nment. The SINDYc environment is a linear approximation\nof the full-fidelity SAM environment, hence the changes\nin actions need to be further constrained to prevent insta-\nbility in the system. As a result, at rollout time, the RL\nagent followed a designed action clipping scheme. The\naction clipping is as follows.\nat←at+ ∆at,t−1\n∆at,t−1=clip(−η, η, a t−at−1)(24)\nIn this particular experiment η= 5×10−4, determined by\nthe model performance on the validation trajectories from\nSINDYc environment. Clipped actions mitigate oscilla-\ntions that could cause instability while allowing the agent\nto follow the demand closely. Figure 9 shows the λ-PPO\nagent’s performance in full-fidelity SAM environment\nIn the initial phase of the trajectory, the λ-PPO agent ad-\nheres closely to the demand. However, when the state\nTs,inapproaches violation, the agent ceases to follow the\ndemand, leading to a transient phase characterized by os-\ncillations in the agent’s actions. These oscillations sub-\nsequently induce instability in other states. This unsta-\nble behavior can be traced back to the contrasting dynam-\nics between two environments: the SINDYc and SAM. In\nthe SINDYc environment, a surrogate model is trained on\ndata that includes load demand curves but excludes any di-\nrect interventions or abrupt changes from an agent. Con-\nversely, the SAM environment is a high-fidelity, physics-\nbased model where multiple coupled Partial Differential\nEquations (PDEs) are solved to simulate the system’s be-\nhavior. A key equation in SAM is the point-kinetics\nequation (PKE), which governs the propagation of power\nwithin a Nuclear Power Plant. In this context, power in-\ncreases typically follow an exponential pattern. Conse-\nquently, any sharp alterations in reactivity insertion (the\nvariable controlling reactor power) can lead to exponen-\ntial changes in the generated power. We hypothesize that\nthe SINDYc model does not capture this nuanced relation-\nship between input and reactor power, as it is trained only\non data with “smooth” changes. This limitation manifests\nin the agent’s behavior when trained on the SINDYc envi-\nronment, resulting in small oscillations in the power set-\npoint as it attempts to reconcile the conflicting demands\nof load and temperature constraints. To rectify this be-\nhavior, future research will explore transfer learning tech-\nniques, allowing the SINDYc-trained agent to adapt and\nlearn within the more complex SAM environment. This\napproach aims to bridge the gap between the simplified\ndynamics of the SINDYc model and the intricate, real-\nworld physics represented in the SAM environment.\n10A Safe Reinforcement Learning Algorithm for Supervisory Control of Power Plants A P REPRINT\n0 200 400 600\nEpochs2000\n1500\n1000\n500\n(st,at)\n0 200 400 600\nEpochs02505007501000in\n0 200 400 600\nEpochs05001000out\n-PPO\n=0.1\n=0.2\n=0.3\n=0.4\n=0.5\nFigure 7: Learning dynamics of the proposed λ-PPO and other baseline models.\n0 100 200 300\nEpochs6000\n5000\n4000\n3000\n2000\n1000\n0(st,at)\nMLP\n0 100 200 300\nEpochs010002000300040005000\nMLP\nin\nMLP\nout\n0 100 200 300\nEpochs0.00.51.01.52.0\nMLP\nin\nMLP\nout\n0 100 200 300\nEpochs0.40.60.81.0HHMLP\nFigure 8: Learning curve for MLP λ-PPO agent on fine resolution reduced-order SINDYc environment.\n426.5427.0427.5428.0428.5Ts,in [°C]\n468469470471472473474Ts,out [°C]\n8009001000110012001300mp [kg/s]\n0 2000 4000\nTime [s]250030003500400045005000ms [kg/s]\n0 2000 4000\nTime [s]819.5820.0820.5821.0Tc,in [°C]\n0 2000 4000\nTime [s]220240260280300320vQRX [MW]\nFigure 9: λ-PPO agent’s performance as a result of direct\napplication to the full-fidelity SAM environment.\n6 Conclusion\nWe proposed λ-PPO model for applying principles of\nSafe RL to the control of a complex dynamical system.\nWe formulated Safe RL problem as a chance-constrained\nMarkov Decision Process and transformed the intractable\noriginal form into an unconstrained standard MDP with\nthe primal-dual Lagrangian method. As a result, we pro-\nposed λ-PPO model that adopts Proximal Policy Opti-\nmization to train agents in an on-policy manner. The\nmodel training followed two timescales - the policy and\nvalue networks were updated on a faster scale with a fixed\nset of Lagrangian multipliers, while the Lagrangians were\nlearned on a slower timescale given the current policy. We\napplied the proposed model to a load-following transient\nproblem where the agent was trained and evaluated using\nreduced-order SINDYc models. The results show that theλ-PPO agent followed the load demand and adhered to\nthe temperature restrictions when following the demand\ncould result in safety violations. Furthermore, we demon-\nstrated the superiority of learning Lagrangians over fixing\nthem to constant values, where the agent with the learned\nLagrangians had the best performance regarding the dis-\ntance from the violation, D, and the violation rate ω.\nThe application of the trained model to full-fidelity SAM\nsuggests its efficacy in real-world scenarios. However,\ndue to the lost nonlinearity in the reduced-order SINDYc\nenvironments, the agent’s actions resulted in oscillations\nwith different degrees in some states. Therefore, in fu-\nture work, we propose to address this issue by 1) building\na nonlinear ROM for SAM using physics-informed neu-\nral networks and 2) offline training of RL models that can\nbenefit from existing data collected from the SAM model\nthat contains the nonlinear information.\nAcknowledgment\nThis work is supported by Argonne National Laboratory\nunder the Laboratory Directed Research & Development\nproject 2022-0077.\nReferences\n[1] B. Karanayil, M. 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Laud, Theory and application of reward shap-\ning in reinforcement learning, University of Illinois\nat Urbana-Champaign, 2004.\n13"    },    {        "title": "2401.13028v1.HOMERUN_a_new_approach_to_photoionization_modelling__I____reproducing_observed_emission_lines_with_percent_accuracy_and_obtaining_accurate_physical_properties_of_the_ionized_gas.pdf",        "content": "Astronomy &Astrophysics manuscript no. paper_sub ©ESO 2024\nJanuary 25, 2024\nHOMERUN⋆a new approach to photoionization modelling\nI - reproducing observed emission lines with percent accuracy and obtaining\naccurate physical properties of the ionized gas\nA. Marconi1,2,⋆⋆, A. Amiri3,1,2, A. Feltre2, F. Belfiore2, G. Cresci2, M. Curti4, F. Mannucci2, E. Bertola2, M.\nBrazzini1,2, S. Carniani7,2, E. Cataldi1,2, Q. D’Amato2, G. de Rosa8, E. Di Teodoro1, M. Ginolfi1,2, N. Kumari8, C.\nMarconcini1,2, R. Maiolino5,6, L. Magrini2, A. Marasco9, M. Mingozzi8, B. Moreschini1,2, T. Nagao10, E. Oliva2, M.\nScialpi1,2, N. Tomicic11,1,2, G. Tozzi1,2, L. Ulivi1,2,12, and G. Venturi7,2\n1Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I-50019, Sesto F.no (Firenze), Italy\n2INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy\n3Department of Physics, University of Arkansas, 226 Physics Building, 825 West Dickson Street, Fayetteville, AR 72701, USA\n4European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany\n5Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 OHA, UK\n6Cavendish Laboratory - Astrophysics Group, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 OHE, UK\n7Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy\n8Space Telescope Science Institute, 3700 San Martin Drive. Baltimore, MD, 21210\n9INAF-Padova Astronomical Observatory, Vicolo Osservatorio 5, 35122, Padova, Italy\n10Research Center for Space and Cosmic Evolution, Ehime University, Matsuyama, Ehime 790-8577, Japan\n11Department of Physics, Faculty of Science, University of Zagreb, Bijeni ˇcka 32, 10 000 Zagreb, Croatia\n12Department of Physics, University of Trento, Via Sommarive 14, I-38123 Povo (Trento), Italy\nReceived XXXX; accepted XXXXX\nABSTRACT\nWe present HOMERUN (Highly Optimized Multi-cloud Emission-line Ratios Using photo-ionizatioN), a new approach to modelling\nemission lines from photoionized gas that can simultaneously reproduce all observed line intensities from a wide range of ionization\nlevels and with high accuracy. Our approach is based on the weighted combination of multiple single-cloud photoionization models\nand, contrary to previous works, the novelty of our approach consists in using the weights as free parameters of the fit and constraining\nthem with the observed data. One of the main applications of HOMERUN is the accurate determination of gas-phase metallicities\nand we show that a critical point is to allow for a variation of the N /O and S /O abundance ratios which can significantly improve the\nquality of the fit and the accuracy of the results. Moreover, our approach provides a major improvement compared to the single-cloud,\nconstant-pressure models commonly used in the literature. By using high-quality literature spectra of H iiregions where 10 to 20\nemission lines (including several auroral lines) are detected with high signal-to-noise ratio, we show that all lines are reproduced by\nthe model with an accuracy better than 10%. In particular, the model is able to simultaneously reproduce [O i], [O ii], [O iii], [S ii], and\n[S iii] emission lines which, to our knowledge, is an unprecedented result. Finally, we show that the gas metallicities estimated with\nour models for HII regions in the Milky Way are in agreement with the stellar metallicities than the estimates based on the Te-method.\nOverall, our method provides a new accurate tool to estimate the metallicity and the physical conditions of the ionized gas. It can be\napplied to many di fferent science cases from HII regions to AGN and wherever there are emission lines from photoionized gas.\nKey words. Atomic processes – Line: formation – ISM: abundances – HII regions – Galaxies: abundances – Galaxies: ISM\n1. Introduction\nMetallicity is a key observable of galaxies as element abun-\ndances and their evolution across cosmic epochs provide unique\ninformation on the physical processes driving the evolution of\ngalaxies (see, e.g., Maiolino & Mannucci 2019 for a review).\nIn particular, since all elements heavier than Lithium and up to\nIron and Nickel are created in stellar cores, the amount of met-\nals provides a measure of the integrated star formation history of\ngalaxies. Furthermore, the existence of scaling relations between\nmetallicity and galaxy properties such as the mass-metallicity\n⋆Highly Optimized Multi-cloud Emission-line Ratios Using photo-\nionizatioN\n⋆⋆e-mail: alessandro.marconi@unifi.it(e.g. Tremonti et al. 2004) and fundamental metallicity relations\n(e.g. Mannucci et al. 2010) also indicates that metallicity traces\nthe infall and outflow of gas to and from galaxies. Metallicity can\ntherefore be used to constrain feedback /fueling processes and\ngrowth timescales in galaxies (e.g., Mannucci et al. 2009; Cresci\net al. 2010; Kewley et al. 2019b; Kumari et al. 2021; Curti et al.\n2023 and references therein).\nThe metallicity of a galaxy and of the surrounding medium\ncan be measured from a range of observables: from the inte-\ngrated emission of stars, from the warm ( Te∼104K) ionized\ninterstellar medium (ISM), from the hot ( Te∼106K) ionized in-\ntracluster (ICM) and circumgalactic medium (CGM), and from\nthe absorption features caused by cold /warm ICM /CGM on the\nspectra of distant quasars and galaxies. In this paper, we fo-\nArticle number, page 1 of 16arXiv:2401.13028v1  [astro-ph.GA]  23 Jan 2024A&A proofs: manuscript no. paper_sub\ncus on the gas-phase metallicity of the ISM warm ionized gas\n(Te∼104K) mainly located within the HII regions surrounding\nmassive O and B stars but also in the form of di ffused ionized\ngas (DIG).\nThe emission line spectrum of photoionized gas can be used\nto constrain the properties of the ionizing source(s) and provides\ninformation on gas physical properties like density, tempera-\nture, and metallicity. In particular, the measurement of metallic-\nity depends on the density, temperature, and ionization status of\nthe ionized gas (e.g., Osterbrock & Ferland 2006; Netzer 2008;\nStasi ´nska 2009; Kewley et al. 2019b). Three methods are usually\nadopted: two observational ones, based on direct electron tem-\nperature determinations and on metal recombination lines, re-\nspectively; and a theoretical one based on photoionization mod-\nels (see, section 3 of Maiolino & Mannucci (2019) and refer-\nences therein for a detailed description and discussion on these\nmethods).\nThe first observational method, the so-called Te-method ,\nis based on density and temperature estimates from the ra-\ntios of collisionally excited forbidden emission lines. It as-\nsumes that the complex stratification of the physical prop-\nerties in the ionized gas volume can be modeled with a\nfew (2-4) homogeneous regions characterized by constant val-\nues of electron density ( Ne), temperature ( Te), and ioniza-\ntion fractions of any ion X+i;NeandTeof each region can\nthen be determined from one or more line ratios sensitive\nto density, like [S ii]6730 /6716, [O ii]3729 /3728, and tempera-\nture like [O iii]4363 /5007, [O ii]7330 /3728, [N ii]5755 /6548, and\n[S iii]6312 /9532 (e.g., Osterbrock & Ferland 2006; Berg et al.\n2020a; Mingozzi et al. 2022 and references therein). The emis-\nsivity Jk,lof a specific emission line originating from the k,ltran-\nsition, whether due to collisional excitation or recombination,\ncan be determined using these NeandTevalues. Subsequently,\nthe relative abundance of the emitting ion can be obtained from\nthe reddening corrected observed ratio between the emission line\nofX+iand a hydrogen line, typically H β. Both lines are assumed\nto be emitted from the same volume ∆V. In formulae,\nLk,l(X+i)≃N(X+i)NeJk,l(X+i;Ne,Te)∆V(X+i), (1)\nL(Hβ)≃N(H+)NeJHβ(Ne,Te)∆V(Hβ), (2)\nN(X+i)\nN(H)≃Lk,l(X+i)\nL(Hβ)JHβ(Ne,Te)\nJk,l(X+i;Ne,Te), (3)\nwhere Lis a line luminosity and it is assumed that N(H+)≃N(H)\nand∆V(X+i)≃∆V(Hβ). The abundance of an element Xcan\nthen be determined by adding up the abundances of its ions:\nN(X)\nN(H)=X\niN(X+i)\nN(H). (4)\nReliable values of Teare needed because of the strong, ex-\nponential dependence of Jk,lon temperature. Hereafter, we refer\nto these abundances as Te-abundances.\nThe second observational method is based on metal recom-\nbination lines and is similar to the aforementioned approach,\nwith the only fundamental di fference that the line emissivity\nJk,l(X+i;Ne,Te) has only a weak, roughly linear dependence on\ntemperature whose exact value is therefore much less important\nthan for the Te-method (e.g. Peimbert & Peimbert 2013; Esteban\net al. 2014; Toribio San Cipriano et al. 2017).\nAlthough commonly considered observational methods,\nboth approaches necessarily rely on photoionization calcula-\ntions to estimate ionic temperatures and abundances not di-\nrectly accessible from the observations (see, e.g., Gutkin et al.2016; Amayo et al. 2021 and references therein). Moreover,\nabundances derived from the Te-method are known to disagree\nwith those derived from metal recombination lines. A possible\nexplanation is the existence of temperature fluctuations which\nmostly a ffect the determinations based on collisionally excited\nlines (e.g., Peimbert 1967; Peimbert & Peimbert 2013; Toribio\nSan Cipriano et al. 2017; Nicholls et al. 2020; Méndez-Delgado\net al. 2023b). Also, especially for distant galaxies, the tempera-\nture is usually derived for only one of the zones of the HII re-\ngions and locally calibrated relations are used to extrapolate the\ntemperatures of the other zone(s) (e.g. Curti et al. 2017), poten-\ntially introducing large uncertainties.\nThe theoretical method is instead based on a comparison be-\ntween the observed line ratios and those predicted from pho-\ntoionization models. In practice, given a set of observed emis-\nsion lines, a suite of photoionization models ( single-cloud mod-\nels) for di fferent ionizing continua and physical properties of\nthe gas (including metallicity) is created to best match the\nobserved emission lines thus determining the gas metallicity\namong the other physical properties (e.g., López-Sánchez et al.\n2012; Nicholls et al. 2012; Binette et al. 2012; Pilyugin et al.\n2012; Dopita et al. 2013; Blanc et al. 2015; Pilyugin & Grebel\n2016; Vale Asari et al. 2016; Pérez-Montero & Amorín 2017;\nStrom et al. 2018; Mignoli et al. 2019; Papovich et al. 2022).\nHereafter, we refer to this method as the Theoretical method and\nto the metal abundances derived with it as the Model abundances.\nOverall, all these methods require the measurement of sev-\neral emission lines to constrain metallicities. The Te-method,\nin particular, requires the measurement of the so-called auro-\nrallines, notable examples of which are [S ii]4069, [O iii]4363,\n[O ii]7330, [N ii]5755, and [S iii]6312, which are needed for tem-\nperature determination but are usually quite faint (less than a\nfew percent of the brightest lines). Instead, most photoioniza-\ntion models, at least in the extragalactic field, use exclusively\nthe brightest emission lines which are the only ones usually de-\ntected: these are lines like H β, Hα, [O iii]5007, [O ii]3727, 3729,\n[N ii]6584, [S ii]6716, 6730, [S iii]9532 and are usually referred\nto as strong lines .\nFor the sources where it is possible to measure only these\nstrong emission lines, it is customary to use the so-called Strong-\nLine method, where metallicity is estimated from one or more\nstrong line ratios, whose dependence on metallicity has been\ncalibrated using Te-based and /orModel -based metallicities from\nHII-regions and galaxies with high S /N spectra (e.g., Kewley &\nDopita 2002; Pettini & Pagel 2004; Tremonti et al. 2004; Kewley\n& Ellison 2008; López-Sánchez et al. 2012; Dopita et al. 2013;\nCurti et al. 2017; Kewley et al. 2019b; Sanders et al. 2023).\nThese line ratios have clearly defined metallicity dependencies\nbecause of existing correlations between the physical parame-\nters of gas clouds, like the anti-correlation between ionization\nparameter and metallicity (see, e.g., Carton et al. 2017; Ji & Yan\n2022) or star formation rate (e.g., Papovich et al. 2022). These\nmethods, and especially the Strong-Line one, have extensively\nbeen used in studies of galaxy evolution, providing evidence for\nthe well-known metallicity scaling relations mentioned above.\nHowever, there are still important open issues in metallicity\nestimates: contradicting calibrations of the Strong-Line method\nexist in the literature. These disagreements are due to the dif-\nferent reference HII or galaxy samples used by the di fferent au-\nthors, the di fferences in the application of the Temethod (i.e.\ncorrections for temperature fluctuations, ionization corrections),\nand the di fferent assumptions used for the photoionization mod-\nels (abundance ratios, ionizing continua, dust properties etc.).\nMoreover, the Temethod and Theoretical methods based on pho-\nArticle number, page 2 of 16A. Marconi et al.: HOMERUN a new approach to photoionization modelling\ntoionization models usually provide discrepant results, with the\nlatter suggesting metallicities which can be larger by 0.3 dex or\nmore, leading to significant uncertainties on all scaling relations\ninvolving metallicity (e.g., Kewley & Ellison 2008; Stasi ´nska\n2010; López-Sánchez et al. 2012; Curti et al. 2020).\nIn general, it is clear that both approaches aim to reproduce\nthe complexity of ionized clouds under oversimplified assump-\ntions: a single HII region is a complex structure made of conden-\nsations and filaments of gas with non-trivial gradients of physi-\ncal properties, possibly exposed to the ionizing photons emitted\nby di fferent sources, and the emission line spectrum of a star-\nforming galaxy is the superposition of several hundreds of HII\nregions, each with di fferent physical conditions (e.g. Charlot &\nFall 2000).\nOverall, the recent important advances in spectroscopic in-\nstrumentation, in terms of both quality and quantity of the\ndata brought forward by integral field spectrographs, have not\nbeen matched by similar improvements in the methods used\nfor metallicity measurements. On one hand, the Te-method has\nnot changed in decades, apart from (semi)empirical corrections\nor improved theoretical ionization corrections (e.g., Méndez-\nDelgado et al. 2023b. On the other hand, the physics within pho-\ntoionization models has been greatly improved, but the modeling\nof the observed data is still mostly based on single-cloud models\nor on simple combinations of multiple clouds: there have been\na few attempts to model the complexity of real gaseous nebu-\nlae using, e.g., Montecarlo techniques (Ercolano et al. 2003; Jin\net al. 2022) but at the expense of the accuracy of the physical\nprocesses included in codes. Still, the successes and improve-\nments of photoionization models over the last years indicate that\nthey could represent the way forward to improve metallicity es-\ntimates.\nHere we present HOMERUN (Highly Optimized Multi-\ncloud Emission-line Ratios Using photo-ionizatioN) a new ap-\nproach to photoionization modelling which builds upon a grid\nofsingle-cloud photoionization models but, in contrast to other\nmodels presented in the literature, combines them with a weight-\ning function which is determined from the observations, not as-\nsumed a-priori . In section 2 we describe the current approaches\nto photoionization modeling while in 3 we present our new\nHOMERUN model. In in sections 3.6 through 5, we describe\nand discuss its application to the spectra of samples of HII re-\ngions. Finally, in section 6 we show how our model resolves the\napparent discrepancy between the observed metallicities of HII\nregions and stars in the Milky Way. We summarise our findings\nand draw our conclusions in section 7.\n2. Photoionization Models\nThere are broadly two kinds of approaches in photoioniza-\ntion modelling of HII regions, and in both cases the publicly\navailable CLOUDY1(Ferland 1993; Ferland et al. 2017) and\nMAPPINGS2(Sutherland & Dopita 1993; Dopita & Suther-\nland 1996) codes are widely used. One approach is based on\ncomparing single-cloud constant-pressure models with observa-\ntions (e.g., Dopita et al. 2016), also taking into account the star-\nformation history (e.g., Charlot & Longhetti 2001; Chevallard\net al. 2018). The other is based on the combination of di ffer-\nent single-cloud constant-density models through an assumed\nweighting function: this is for instance the case of the so-called\nLocally Optimally-emitting Clouds (LOC; Baldwin et al. 1995;\n1https: //nublado.org\n2https: //mappings.readthedocs.io /en/latest /index.htmlKorista et al. 1998; Nagao et al. 2006). The former approach is\nextensively used in the analysis of galaxy spectra and in the de-\ntermination of metal abundances in their ISM, while the latter\napproaches are mostly used in the analysis of the emission lines\nfrom the Broad Line Regions of Active Galactic Nuclei (AGN)\nalthough it has also been applied to the spectra of star-forming\ngalaxies (Richardson et al. 2016). These models are successful\nin reproducing the general emission-line properties of popula-\ntions of galaxies and AGN but they struggle in reproducing the\nemission-line spectra of single objects: di fferent line ratios in the\nsame galaxy are often reproduced by models with di fferent input\nparameters because the models fail to reproduce the observed\nwide range of excitation levels which is manifested by the ex-\nistence of strong emission lines from di fferent ionization states,\nlike [O iii]λ5007, [O ii]λ3727 and [O i]λ6300 (e.g., Martin 1997;\nIglesias-Páramo & Muñoz-Tuñón 2002; Stasi ´nska et al. 2015;\nRamambason et al. 2020). More in detail, a single-cloud model\nis described by a slab of gas in plane-parallel or spherical geom-\netry, ionized by a central source. A single-cloud model therefore\nis defined by:\n–the spectrum of the ionizing continuum, that can vary widely\ndepending on the model used for the ionizing source (stel-\nlar age, stellar and gas-phase metallicity, presence of binary\nstars, etc.);\n–the hydrogen density NHat the illuminated face of the cloud;\n–the radial distance rinof the illuminated face of the cloud\nfrom the ionizing source;\n–the ionization parameter U=Q(H)/(4πr2\ninc NH), where\nQ(H) is the rate of ionizing photons and cthe speed of light;\nsometimes di fferent authors use a di fferent definition of U\nwhich is computed, e.g., at a di fferent radius or as a volume\naverage (see for example Charlot & Longhetti 2001; Gutkin\net al. 2016; Plat et al. 2019).\n–the abundances of all the elements relative to hydrogen;\n–whether the cloud is at constant pressure or constant density;\n–whether the cloud contains dust, and what are its properties\nand the fraction of chemical elements depleted into dust;\n–the assumption of stationary equilibrium, i.e. no time evolu-\ntion;\n–a criterion to define the outer edge of the cloud, which could\nbe a limit on any parameter among hydrogen column density,\nionization fraction, and electron temperature.\nThe model is integrated starting from the illuminated face of the\ncloud, numerically solving the equations of ionization equilib-\nrium for all ions, the energy balance equation, and the radiative\ntransfer equation; all other relevant physical processes are also\nconsidered (e.g, Osterbrock & Ferland 2006; Stasi ´nska 2009;\nNetzer 2008; see volume 3 of Cloudy manual \"Hazy\" for an ex-\nhausting description3). The model predicts, among other things,\nthe fluxes or luminosities of all the considered transitions which\ncan then be compared with observations. The metallicity of the\nmodel which provides the best match with the observations is\nthen the estimated gas metallicity.\nThe ionization level of the gas at a given point in the cloud is\nset, among other things, by the value of the ionization parameter\nat that point, i.e. computed from Q(H) and NHatr, where ris\nthe distance from the ionizing source. In constant-pressure cloud\nmodels, as rincreases, the temperature decreases and the density\ngrows, leading to a radial decrease in Uwhich is faster than that\npredicted by constant-density models. This wider Urange allows\nfor a better match with the observed emission lines from di fferent\n3Available with the CLOUDY software at nublado.org\nArticle number, page 3 of 16A&A proofs: manuscript no. paper_sub\nionization stages, and is the reason why constant-pressure, single\ncloud models are typically more successful than constant-density\nones (e.g., Dopita et al. 2002; Stern et al. 2014; Kewley et al.\n2019a).\nIn order to provide a better match to the observed emission\nlines, combinations of single-cloud models have been consid-\nered. One notable example is that of the LOC models mentioned\nabove. Here, a grid of single clouds is computed, each char-\nacterized by a single value of NHandΦ(H)=Q(H)/(4πr2\nin)\n(the ionizing photon flux), and the emission lines predicted by\nthe single-cloud models are combined with a weighting func-\ntion which is usually proportional to Φ(H)−αN−β\nH, withαandβ\ntwo slopes of order of unity which can be free parameters of the\nmodel (Nagao et al. 2006). As in the case of single-cloud mod-\nels, the metallicity of the model which provides the best match is\nthen the estimated gas metallicity. A di fferent approach, mostly\napplied to AGN, assumes that the clouds are confined by radia-\ntion or magnetic pressure and that their covering factor varies as\na power law of distance rfrom the ionizing source, resulting in\nNHandUwhich vary similarly as a power law of r(e.g., Netzer\n2020). LOC models are better suited to reproduce the complexity\nof a real HII region and its irregular geometrical and ionization\nstructure, but the use of a fixed weighting scheme is still a severe\nlimitation.\n3. The HOMERUN model\n3.1. The basic assumption: a combination of multiple\nsingle-cloud models\nThe luminosity of an emission line lof ion X+ifrom either re-\ncombination or collisional excitation can be written as\nLl(X+i)obs=Z\nVNeN(X+i)Jl(X+i,Te) dV, (5)\nwhere Vis the volume of the ionized gas, NeandN(X+i) are\nthe electron and ion densities, respectively, Teis the electron\ntemperature and Jl(X+i,Te) is the emissivity of line l. The line\nemissivity Jlis the same as in Eq. 1 but here we simply indicate\nthe transition as land we do not explicitly mention the usually\nweak dependence on the electron density. Ne,N(X+i) and Teare\nphysical quantities which depend on the position within the vol-\nume, yet the luminosity, which is an integrated quantity, does\nnot depend on the detailed density and temperature distribution\nof the gas but on their weighted average values over the entire\ngas volume. Therefore, any model that has the correct volume,\naverage density and temperature of the X+iregion, will repro-\nduce Ll(X+i)obs, even if the spatial distribution of the gas physical\nproperties does not match the real one.\nIn this study we assume that the ion emission line luminosity\ndefined in Eq. 5 can be su fficiently well approximated by a linear\ncombination of msingle cloud models, each characterized by its\nown ionisation parameter and density\nLl(X+i)mod =mX\nj=1wjLl(X+i)mod,j (6)\n=mX\nj=1wjZ\nVjNeN(X+i)Jl(X+i,Te)dV (7)\nwhere Ll(X+i)mod,jis a set of simple , single cloud photoionization\nmodels which can be considered as the basic building blocks of\nthe model. The non-negative weights wjcan either be assumed(like in the LOC models) or can be left as free parameters and\ndetermined from fitting the observed lines. The main underlying\nassumption is that if a model is able to accurately reproduce the\nluminosity of many di fferent emission lines from a few di ffer-\nent ions, then it has the correct combinations of volume, aver-\nage density and temperature for each ion, regardless on their real\nspatial distribution. With respect to single cloud models, we ex-\npect this approach to provide a more realistic description of the\ncomplexity of the warm ionized ISM.\n3.2. The single-cloud models\nThe single-cloud models used in the following were computed\nusing CLOUDY v22.01 (Ferland et al. 2017) with the following\ninput parameters:\n–BPASS simple stellar population models (v2.3, Byrne et al.\n2022; Stanway & Eldridge 2018) including binary stellar\nevolution, characterized by a Kroupa (2001) initial mass\nfunction from 0.1 to 300 M ⊙, [α/Fe]= +0.0, metallicities\nSolar and 0.1 Solar, a single burst of star formation and ages\n(log yrs) 6, 6.3, 6.7 7.1 (for Solar met.) and 6, 6.4, 6.7, 7.3\n(for 0.1 Z⊙)\n–Abundances Zfrom -2 to +0.4 in steps of 0.2 dex relative\nto Solar; the Solar abundance pattern is taken from Grevesse\net al. (2010).\n–Nitrogen is rescaled as a secondary element with respect\nto oxygen as [ N/O]=−1.5+1.4∗([O/H]−[O/H]c) for\n[O/H]≤[O/H]critwith [ O/H]c=8.3 (see, eg., fig. 49\nof Maiolino & Mannucci 2019); we also consider cases in\nwhich [ N/O] varies by±0.6 dex with respect to this scaling\nrelation.\n–Sulphur scales linearly with oxygen following the Solar\n[S/O] abundances ratio, but we also consider the case in\nwhich [ S/O] varies by±0.3 dex.\n–Gas with and without dust; dust is included with the\n\"grains ism Z log\" CLOUDY command line where Z\nis the log of the abundance relative to Solar; we then use\nthe command \"metals deplete\" to take into account the\ndepletion of metals into dust (adopted Depletion factors are\nthe default values of Cloudy - see Table 7.8 and references\ntherein of Hazy1 of Cloudy v22.1).\n–A grid in UandNHwith log Ufrom -5 to -0.5 in steps of\n0.5 dex and log NHfrom 0 to 7 in steps of 1.0 dex. Here, U\nrepresents the ionization parameter at the inner face of the\ncloud.\n–All single cloud models have constant density.\n–All single cloud models have a plane parallel, open geome-\ntry and are radiation bounded with the following integration\nstopping criteria: maximum H column density 1023cm−2,\nminimum temperature 3000 K, minimum fraction of elec-\ntrons 0.05.\nOverall, these choices of parameters require the computation of\n2×4×25×3×3×10×8×2=134,784 CLOUDY models. Then, to\nincrease the sampling, the model line luminosities relative to H β\nare log-linearly interpolated across metallicity in steps of 0.02\ndex.\nThe choice of constant density was made in order to have the\nsimplest possible physical structure of a single cloud model and\nit does not a ffect the final results. Indeed, we have verified that\nconstant pressure models can be reproduced by a combination of\na constant density models.\nThe stellar metallicity, which directly a ffects the shape of the\nstellar continuum, has not been tied to the metallicity of the gas\nArticle number, page 4 of 16A. Marconi et al.: HOMERUN a new approach to photoionization modelling\nto allow more freedom to the modeling. We have additionally\nverified that the fit results are not much sensitive to the stellar\ncontinuum shape which cannot then be accurately constrained;\nthis issue is beyond the scope of this paper and will be further\ninvestigated elsewhere.\nA single-cloud CLOUDY model is characterized by a to-\ntal Hβsurface brightness. Considering the radial symmetry in\na spherical model, it is possible to obtain the volume that the\ncloud must occupy to reproduce the observed luminosity. Mod-\nels with low density and low ionization parameters may there-\nfore become too large to be physical or simply to be included\nin the apertures from which the spectra are obtained. It is easy\nto show that this implies that these low-density, low-ionization-\nparameter models have an H βsurface brightness which is too\nlow compared to the observed one. The CHAOS HII regions\nspectra described below (Sect. 5) are obtained through aper-\ntures (e.g., slits) whose linear size is ∼10′′and the ob-\nserved Hβsurface brightness is, on average, of the order of\n∼10−4erg cm−2s−1arcsec−2. Therefore, in this paper, we have\nconsidered∼10−5erg cm−2s−1arcsec−2as the minimum surface\nbrightness that a single-cloud model must have to be retained\nin the computations. This ensures that single-cloud models with\nweights larger than 10% of the total account for at least 10%\nof the observed surface brightness. The e ffect of this choice is\nto discard some models in the low density, low ionization pa-\nrameter corners of the U,NHgrids computed for a given abun-\ndance Zand ionizing continuum. In principle, one should adopt\nan iterative procedure: fit the data - as described below - with all\nsingle-cloud models in a grid and find the model weights, com-\npute the surface brightness of the single cloud models given their\nweights and discard the models with surface brightness lower\nthan the observed surface brightness multiplied by the model\nweight, redo the fit and iterate until no single cloud model is\ndiscarded.\n3.3. The loss function L\nThe agreement of a multi-cloud model with the observations is\nquantified with the loss function Ldefined below. We consider\na grid of single-cloud constant-density, radiation-bound models\nin ionization parameter Uand H-density NHcharacterized by\n–the spectrum of the ionizing source: Sν\n–the gas metallicity, parameterised with the Oxygen abun-\ndance:AO\n–the relative abundances of metal elements, Z;\nWe identify this grid of models with G=G(Sν,AO,Z). Each\nsingle-cloud model of the grid Gpredicts a set of emission line\nluminosities which we normalise to an arbitrary dimensionless\nHβluminosity of, e.g., 100. All emission lines are then combined\nwith weights which are found by minimising the loss function\n(L)\nL=1\nnnX\ni=1 Ll(X+i)obs−Ll(X+i;w1,w2,..., wm)mod\n∆Ll(X+i)obs!2\n(8)\nwhere nis the number of observed emission lines and mis the\nnumbers of single cloud models in the grid. ∆Ll(X+i)obscan be\neither the error on the observed emission line luminosity or what\nwe arbitrarily consider an acceptable discrepancy between the\nobserved and model luminosities: in the following, we will con-\nsider ∆Ll(X+i)obsas the maximum of the two.The values of the loss function can be approximately in-\nterpreted by assuming that the average relative discrepancy be-\ntween models and observations is εand that for all emission lines\nand all ions\nLl,mod =Ll,obs(1+εrδl), (9)\n∆Ll,obs=εaLl,obs, (10)\nwhereδlare random numbers extracted from a normal distri-\nbution with zero average, and unit standard deviation. We have\ndistinguished between “real” relative discrepancy εrbetween the\nmodels and the observations, and the acceptable discrepancy εa,\ni.e. the one we adopt when computing the loss function. There-\nfore the loss function is given by\nL=1\nnX\nl εrδl\nεa!2\n. (11)\nFor example, if the acceptable discrepancy is equal to the real\naverage discrepancy, the loss function is L≃ 1.0. Conversely,\nassuming an acceptable discrepancy εa=0.1, real average dis-\ncrepancies of εr=0.01,0.05,0.1,0.11,0.15,0.2 will correspond\nto loss function values of L ≃ 0.01,0.25,1.0,1.2,2.3,4.0, re-\nspectively. This numbers should serve as a guideline to estimate\nthe real average discrepancy for any given value of the loss func-\ntion.\n3.4. Varying the N/O and S/O abundance ratios\nUsually, the relative abundances of all elements relative to Oxy-\ngen are constant and fixed to some predetermined value, like\nthe Solar one. However, in our model fitting we will allow for\na scaling of the relative abundances of the elements: valid es-\npecially for N and S, which have the strongest observed lines\nin the optical, this choice stems from the evidence presented in\nthe literature, and is physically expected given the wide range of\nconditions in which stars and galaxies form and evolve (Steidel\net al. 2016; Morisset et al. 2016; Strom et al. 2018; Maiolino &\nMannucci 2019; Berg et al. 2020b).\nIn order to allow for di fferent relative abundances other than\nthe Solar ones used in the computation of the single cloud mod-\nels, we rescale each set of model emission lines of a given ele-\nment by a scaling factor which is a free parameter of the fit. The\nonly element for which we do not apply such rescaling is oxy-\ngen since some of its forbidden emission lines provide the main\ncooling mechanism of the ionized zone and therefore the line\nluminosities are not trivially proportional to abundance. Consid-\nering, for instance, observed lines from oxygen, N, and S the loss\nfunction becomes\nL=nO,HX\ni=1 Ll(X+i)obs−Ll(X+i;w)mod\n∆Ll(X+i)obs!2\n+nNX\ni=1 Ll(X+i)obs−N∗ Ll(X+i;w)mod\n∆Ll(X+i)obs!2\n+nSX\ni=1 Ll(X+i)obs−S∗ Ll(X+i;w)mod\n∆Ll(X+i)obs!2\n(12)\nwhere w=(w1,w2,..., wm) are the weights of the msingle-\ncloud models, nO,Hindicates that the sum is made on all Hand\nO lines while nNandnSindicate summation on N and S lines\nrespectively, andN,Sare the scaling factors which are free pa-\nrameters of the fit.\nArticle number, page 5 of 16A&A proofs: manuscript no. paper_sub\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁\n☁w2w3w1U, NH grid of CLOUDY models \n(single-cloud ) with \nMetallicity  and \nIonizing Continuum   12+log(OH)Sν\n×w3×w1×w2Observed spectrumRepeat for all   12+log(OH)\nFig. 1. A schematic representation of the fitting procedure: for any U,NHgrid of single-cloud models, we compute the weighted combination of\nclouds in the grid that gives the best match with the observed lines. Usually only a small fraction of the clouds in the grid have non-zero weights.\nThe goodness of the fit is then quantified with a value of the loss function L. The procedure is repeated for all U,NHgrids computed with di fferent\ngas metallicities and the behaviour of Las a function of metallicity is obtained. The metallicity of the observed spectrum is identified by the\nminimum value of L. This procedure can be repeated with di fferent ionizing continua thus estimating the e ffect of varying the ionizing continuum\non the metallicity estimate.\nThis procedure assumes that the N and S emission lines scale\nlinearly for small variations of the abundances of these elements.\nThis is not always true, but we have verified that if the abundance\nof N is varied by less than ±0.3 dex with respect to the solar\nvalue, and S by less than ±0.15 dex, the line luminosities scale\nlinearly to better than ∼10%.\nTherefore, to correct for the non-linearity of the N and S\nline intensities with the elements’ abundances, we consider only\nthe models where the scaling factors NandSare smaller than\n±0.3 and±0.15 dex, respectively. With the choices of [ N/O] and\n[S/O] adopted in the computation of the CLOUDY models, this\nensures the accuracy of our computations when varying [ N/O]\nand [ S/O] within±0.9 and±0.45 dex of the Solar values, re-\nspectively.\n3.5. The fitting procedure\nIn the fitting procedure we consider first a U,NHgrid of single-\ncloud models computed with a given ionizing continuum and\nset of element abundances, G(Sν,AO,Z), and find the multi-\ncloud model which provides the minimum value of the loss func-\ntionLmin(Sν,AO,Z). For a given set of scaling factors N,S,\nthe best weights ( w1,w2,..., wm) are found by solving the non-\nnegative least square (NNLS) problem, in a similar way as in the\npPXF python procedure by Cappellari & Emsellem (2004) and\nby Cappellari (2017). Indeed, this model fitting is similar in prin-\nciple to the fitting of the stellar continua of galaxies where many\nhundreds or thousands of stellar continuum templates are com-\nbined with non-negative weights. Here, emission-line templates\nfrom the single-cloud models are combined with non-negative\nweights. Since in the single-cloud models the emission lines are\nnormalized such that L(Hβ)=100, the sum of the weights will\nbe equal to the total observed H βluminosity. Finally, in finding\nthe best weights we do not apply any regularization, a standardmathematical procedure to mitigate the high-frequency varia-\ntions in the solutions (e.g. Cappellari 2017).\nWe repeat for all U,NHgrids of single-cloud models avail-\nable, iterating over abundances and ionizing continua and, fi-\nnally, we find the minimum of all Lminvalues, thus identify-\ning the combination of Sν,AO,Zwhich provides the best fit to\nthe observed line ratios. A schematic representation of the fitting\nprocedure is presented in Fig. 1.\n3.6. An example of a model fit\nWe describe the model by presenting an example of its applica-\ntion to the emission line fluxes of an HII region. The test HII\nregion is randomly extracted from the sample presented by Zu-\nrita et al. (2021) and it has all the emission lines listed in the\nbottom right panel of Figure 2 measured with Signal-to-Noise\nratio (S /N) larger than 5. The results from the application of the\nmodel to the entire Zurita et al. (2021) sample of HII regions will\nbe presented in section 4.\nWe fit the observed lines with the HOMERUN model us-\ning as ionizing continuum a BPASS model with age 106yr and\nSolar stellar metallicity (Section 3). As mentioned in the previ-\nous section, the nitrogen lines ([N ii]λ5755, [N ii]λ6583) and sul-\nphur lines ([S ii]λλ4069,4076, [S ii]λλ6716,6731], [S iii]λ6312,\n[S iii]λ9531) predicted by the model are rescaled by the factors\nNandSwhich are free parameters of the fit and which take into\naccount that the relative N /O and S /O abundances may be di ffer-\nent than those assumed in the photoionization calculations. The\nsame rescaling is not applied to the oxygen lines ([O iii]λ5007,\n[O iii]λ4363, [O ii]λλ3726,3729, [O ii]λλ7320,7330) since they\ndo not depend linearly on abundances. For the grids defined by\na specific ionizing continuum and gas metallicity value, we find\nthe weights wiandN,Sfactors which minimise the loss func-\nArticle number, page 6 of 16A. Marconi et al.: HOMERUN a new approach to photoionization modelling\n7.8 8.0 8.2 8.4 8.6 8.8 9.0101\n100101\n7.8 8.0 8.2 8.4 8.6 8.8 9.0\n12+log(O/H)678912+ log(N/H) log(S/H) log(N/H) input\nlog(S/H) input\nlog(N/H) fit\nlog(S/H) fit\n0 2 4 6\nlog(NH/cm3)\n5\n4\n3\n2\n1\nlog(U)\n020406080100\nweight\n0.8 1.0 1.2\nModel/ObservedH\nH\n[OIII]5007Å\n[OII]3726,3729Å\n[OIII]4363Å\n[OII]7320,7330Å\n[NII]6583Å\n[NII]5755Å\n[SII]6731Å\n[SII]6716Å\n[SII]4069,4076Å\n[SIII]6312Å\n[SIII]9531Å\nbpass_ch300_z_-1.6_age_6.3_dust_N_0.0_S_-0.3    min=0.114\nFig. 2. Top left panel: variation of the loss function Las a function of the Oxygen abundance, 12 +log(O/H), for the emission line spectrum of the\ntest HII region. In this scale the Solar metallicity is 8.69. The blue vertical dashed line represents the metallicity of the model with the minimum\nvalue ofL. Bottom left panel: variation of the N and S abundances as a function of the Oxygen one. The dashed lines represent the values adopted\nfor the computation of the photoionization models, while the filled circles correspond to the abundances rescaled as described in the text. Top\nright panel: grid of single cloud models in log( U) and log( NH). The colors represent the weights of each single cloud model, as indicated by the\ncolorbar, whenLreaches the minimum value. In this case only 4 out of 72 models of the grid have non zero weights. The black dot represents the\nweighted density and ionization parameter of the single cloud models. Bottom right panel: ratio between model and observed line luminosities.\nThe gray areas represent the acceptable discrepancy of ±10% or, when larger, the relative error on the observed value.\ntionL; finally we select the model with the minimum Lacross\nall metallicities considered.\nThe top left panel of Fig. 2 shows the variation of the loss\nfunction as a function of the gas-phase metallicity, i.e. after\ndust depletions have been taken into account; the blue vertical\ndashed line represents the estimated model metallicity, which\ntrivially corresponds to that of the model minimizing L. For\nall lines, we have adopted an acceptable discrepancy of 10%\n(∆Ll(X+i)obs/Ll(X+i)obs=0.1) when computing L.\nThe bottom left panel of Figure 2 represents the nitrogen and\nsulphur metallicities provided as input in the CLOUDY simula-tions (dashed coloured lines) and derived from the fit after cor-\nrecting the input model values with the N,Sfactors (filled cir-\ncles). The values derived from the fit are close to the relative\nmetal abundances assumed in the Cloudy models.\nThe top right panel of Figure 2 shows the weights of the sin-\ngle cloud models with di fferent UandNHwhich were derived\nfor the best fit: four single clouds out of the 72 in the U,NHgrid\nhave non-zero weights, three of them account for more than 90%\nof the total H βflux. Using the weights, it is possible to com-\npute the average density and ionization parameter of the clouds,\nArticle number, page 7 of 16A&A proofs: manuscript no. paper_sub\nwhich are identified by the black filled circle in top right panel\nof Figure 2.\nWhile it is not possible to a ffirm that the single clouds used\nin the model have a direct correspondence to the real NH,Udis-\ntribution, the average density and ionization parameter derived\nfrom the fit are representative of the average density and ioniza-\ntion parameter of the HII region.\nIn the bottom right panel of Figure 2, we compare the model\npredictions with the observations: the x-axis represents the ratio\nbetween model and observed line fluxes for a certain line, which\nis indicated on the y-axis. The length of the gray bar represents a\n±10% discrepancy. In the case of the [O iii]4363 line the gray bar\nreflects the relative error of the observations which is larger than\nthe adopted 10% discrepancy. Finally, the green color indicates a\nratio which is within 0.9 and 1.1 (10% discrepancy max), while\norange bars (when present) indicate values outside that range.\nAs discussed in section 3, a loss function value of 0.11,\nwhich is the minimum value in the top left panel, can be in-\nterpreted as an average discrepancy of ∼3.3% between observa-\ntions and models which is visually confirmed in Figure 2.\nThe model is able to reproduce a large number of emis-\nsion lines from di fferent ionization stages within 5% discrep-\nancy (with only the [O iii] 4363 line reproduced with a 10% dis-\ncrepancy but still consistent with the observational error): this is\na definite improvement with respect to previous models. [S iii]\nlines are in very good agreement with observations, at variance\nwith previous work in which [S iii] lines are usually so di fficult\nto reproduce that problems with their atomic parameters have\nbeen suggested (Mingozzi et al. 2020). The physical parameters\nof the HII region inferred from the model, such as metallicity,\naverage ionization parameter and hydrogen density, are also re-\nalistic and in agreement with our expectations for HII regions.\nFinally, the plot of the loss function in Figure 2 presents a min-\nimum so deep to require a logarithmic scale and indicates that\nthe problem is well defined and not a ffected by degeneracies, as\none might naively expect given the freedom in the choice of the\nweights. In other words, this model has potentially many param-\neters (the weights of each single CLOUDY model) but it is built\nin such a way that if the metallicity is not the right one, the lines\ncan not be well reproduced.\n4. The sample of HII regions by Zurita et al.\nAfter presenting the model and its characteristics with a few se-\nlected examples, we present its application to larger samples of\nemission-line sources. In this paper, we focus on HII regions\nwithin the Milky Way or local spiral galaxies.\nWe consider the sample by Zurita et al. (2021), who\ncollected the emission line fluxes of 2831 published HII\nregions from 51 nearby galaxies to study how the pres-\nence of bars a ffects metallicity gradients. The compiled line\nfluxes, reddening corrected and normalised to H β, include both\nbright emission lines ([O ii]λλ3726,3729, [O iii]λλ4959,5007,\n[N ii]λλ6548,6583, H α, [S ii]λλ6717,6731, [S iii]λλ9069,9532)\nand auroral lines ([S ii]λλ4068,4076, [O iii]λ4363, [N ii]λ5755,\n[S iii]λ6312, and [O ii]λλ7320, 7330) when available.\nFor our analysis we selected only the sources where the\nfluxes of all bright and auroral emission lines are measured with\nS/N>5, resulting in a sub sample of 24 HII regions. We then per-\nformed model fitting with both dusty and non-dusty clouds - i.e.\nusing a weigthed combination of single-cloud photoionization\nmodels either with or without dust mixed to the gas - and with all\nthe ionizing continua presented in section 3. Each model set M\nis then characterized by the ionizing continuum and by whetherthere is dust mixed with the gas, and consists of U,NHgrids of\nsingle-cloud models, each computed for a given metallicity Z.\nFor each model set M, we then find the minimum value of Lby\nvarying Z, which we labelLM,min. We then consider Lminas the\nsmallest of allLM,minvalues, selected by having scaling factors\nfor N and S abundances within ±0.3 and±0.15 dex, respectively.\nFinally, we consider acceptable all the models for which\nL≤L min+0.25 (13)\nFollowing equation 11, the increase on average \"discrepancy\"\nbetween model and observations depends on εaand onεrof\nthe best model with L=Lmin. For instance, given εa=0.1,\nmodels withL=Lmin+0.25 have real \"discrepancies\" εr=\n0.05,0.07,0.11 if the best models have εr=0.01,0.05,0.1, re-\nspectively.\n4.1. Observed vs Model Lines\nWe now compare models and observations by comparing the ob-\nserved and model line fluxes. Figure 3a shows the distribution of\ntheLminvalues for all the HII regions in the selected subsam-\nple. We distinguish the cases where, for each HII region, Lminis\nfound considering all models, only models with dust, and those\nwithout. When considering models with or without dust, we find\nLmin<1.1 except for three cases each where Lminis∼2 or\nlarger. However, when finding Lminover all models we always\nhaveLmin<1.1 and the best fit models are given in 20 out of\n24 sources by dusty models, while in the rest by models without\ndust. Indeed, figure 3a shows that, overall, dusty models pro-\nvide in general lower Lminvalues. The distributions of Lminval-\nues thus imply that our multi-cloud models are able to repro-\nduce allemission lines with an accuracy better than 10%; this\nis an extremely good achievement very rarely matched in previ-\nous works (see, e.g., Chevallard et al. 2018, Blanc et al. 2015,\nMorisset et al. 2016, D’Agostino et al. 2019, Mingozzi et al.\n2020, Olivier et al. 2022, Fernández et al. 2022). It is important\nto mention that only a small fraction of the single-cloud models\nin a 8×9 (U,NH)-grid have non zero weights: in 21 out of 24\nbest fit models, the numbers of clouds with non-zero weights is\nbetween 4 and 9, with only three models being made of 11 or 12\nclouds.\nThe performance of the model with each line is shown in\nFigure 3b where we present the distributions of the ratios be-\ntween observed and model line fluxes for all the selected HII\nregions and all the models with L≤L min+0.25. The HOME-\nRUN model is able to reproduce to within 10% both strong and\nauroral lines, from high ([O iii]), intermediate ([S iii]) and low ex-\ncitation ([O ii], [S ii], [N ii]) lines. In particular, the model can re-\nproduce both [S ii] and [S iii] lines at variance with what is found\nin other studies as already mention in Sec. 4.1. In our case [S iii]\nlines are those for which we observe the largest discrepancies\nwhich are still below 20% for [S iii]λ9531 in all models while\nonly 11% of the models present discrepancies larger than 20%\nfor [S iii]λ6312.\n4.2. Observed vs Model Temetallicities\nWe now compare models and observations by comparing the Te-\nmetallicities obtained by both observed and model line fluxes.\nWe estimated Te-metallicities using the python package pyneb ,\nwhich computes the physical conditions and ionic and elemen-\ntal abundances of photoionized gas by solving the equilibrium\nequations for an n-level atom (Luridiana et al. 2015). We se-\nlected the same sets of atomic parameters used in CLOUDY and\nArticle number, page 8 of 16A. Marconi et al.: HOMERUN a new approach to photoionization modelling\n0.1 0.5 0.9 1.3 1.7 2.1 2.5 2.9\nmin\n0.02.55.07.510.012.515.017.5# of modelsAll\nw/ dust\nw/o dust\n0.5 1.0 1.501002003001.01\n0.04H\n0.5 1.0 1.50100200300 1.00\n0.04H\n0.5 1.0 1.501002000.98\n0.06[OIII]5007Å\n0.5 1.0 1.50100200300400\n1.00\n0.05[OII]3726,3729Å\n0.5 1.0 1.50100200300400\n1.02\n0.04[OIII]4363Å\n0.5 1.0 1.50100200300400\n1.00\n0.02[OII]7320,7330Å\n0.5 1.0 1.501002003001.00\n0.03[NII]6583Å\n0.5 1.0 1.501002003001.01\n0.06[NII]5755Å\n0.5 1.0 1.501002003004001.00\n0.01[SII]6731Å\n0.5 1.0 1.502004001.00\n0.02[SII]6716Å\n0.5 1.0 1.5\nFlux(model)/Flux(observed)0100200300400\n1.00\n0.01[SII]4069,4076Å\n0.5 1.0 1.5050100150\n0.95\n0.10[SIII]6312Å\n0.5 1.0 1.50501001501.03\n0.07[SIII]9531Å\nFig. 3. Left: distribution of Lminvalues for the subsample of HII regions selected from Zurita et al. (2021). Right: distributions of the ratios between\nmodel and observed line fluxes for all the selected HII regions and all the models with L≤L min+0.25. The numbers in the top right corner of\nthe panels represent, from top to bottom, median and standard deviation of distribution of Lvalues. The vertical dashed lines represent the ±10%\ndiscrepancy and show how the large majority of models reproduce all lines within ±10% of the observed values.\nwe considered a two-zone cloud where the density of both zones\nis given by Ne(S+), the temperature of the low-ionization zone is\ngiven by Te(N+) and that of the high-ionization zone by Te(O+3).\nThe choice of Te(N+) for the temperature of the low-ionization\nzone is justified by the fact that Te(O+) can lead to underesti-\nmated Te-metallicities and is the same adopted by Zurita et al.\n(2021) (see also Pérez-Montero & Díaz 2003; Méndez-Delgado\net al. 2023a). In particular, we adopt the following procedure:\n–Te(O+3) and Ne(S+) are computed simultaneously by com-\nbining the [O iii]λ5007/λ4363 and [S ii]λ6731/λ6716 ratios\nwith the getCrossTemDen procedure of pyneb . The [S ii]\nflux ratio is not an accurate tracer of density below the low-\ndensity limit (less than a few ×10 cm−3) which characterises\nmost HII regions (e.g., Nagao et al. 2006); however the tem-\nperature sensitive line ratios that we used are almost inde-\npendent of density in this regime.\n–we used Ne(S+) to compute Te(N+) from [N ii]λ5755/λ6584,\nTe(S+) from [S ii]λ4069/λ6720, Te(S+2) from\n[S iii]λ6312/λ9531 and Te(O+) from [O ii]λ7325/λ3727;\n–we used Te(N+) to compute Ne(O+) from [O ii]λ3726/λ3729;\n–[O+/H] and [O+2/H] abundance ratios are computed\nfrom [O ii]λ3727/Hβ, using Te(N+),Ne(S+), and from\n[O iii]λ5007/Hβline ratios, using Te(O+2),Ne(S+); [O/H] is\nthen the sum of the abundances of the two ions, neglecting\nany contribution from either neutral Oxygen or O+3which\nare typically negligible for HII regions (e.g. Zurita et al.\n2021).–[N/O] and [S/O] abundance ratios are similarly computed\nfrom [N ii]λ6584/[O ii]λ3727 and [S ii]λ6720/[O ii]λ3727\nline ratios. Here we assume that in the regions emitting these\nlines, N, S and O are mostly single ionized ions.\nErrors are estimated with 100 Montecarlo realisations where the\nobserved line sets are replicated by adding to line fluxes random\nnumbers extracted from a Gaussian distribution with 0 mean and\nstandard deviation equal to the observed relative error (i.e., equal\nto the assumed discrepancy of ε=10%). Figure 4 (left) com-\npares the Temetallicities derived from the observed line fluxes\nto the ones predicted by the models: considering all acceptable\nmodels (L<Lmin+0.25) for all objects, the agreement is good\nas model and observed metallicities have a median di fference\nof -0.03 dex with a 0.07 dex standard deviation. Similarly, as\nshown in Figure 4 (right), when considering only the best mod-\nels (L=Lmin) the median di fference is -0.03 dex with a 0.06 dex\nstandard deviation. These values are consistent with the <10%\ndiscrepancy between model and observed line flux that we re-\nquired for the HOMERUN model and provide a di fferent mea-\nsure of the (dis)agreement between models and observations.\nFig. 3 of Chevallard et al. (2018) shows a similar agreement be-\ntween O /H from Te and (single cloud) models.\nFigure 5 presents the comparison between the metallicities\nused as input of the HOMERUN model calculations, labeled as\n\"log(O/H)HR[in models ]\", and the Temetallicities derived with\npyneb from the line fluxes predicted by the models, labels as\n\"log(O/H)HR[Temethod ]\". This is not a comparison between\nArticle number, page 9 of 16A&A proofs: manuscript no. paper_sub\n0.4\n 0.2\n 0.0 0.2 0.4\nlog(O/H)Te,HR log(O/H)Te,Observed\n050100150200 med: -0.03\nstd: 0.07All HR models with <min+0.25\n8.1 8.2 8.3 8.4 8.5 8.6 8.7\nlog(O/H)Te,Observed0.3\n0.2\n0.1\n0.00.10.20.3log(O/H)Te,HR log(O/H)Te,Observed\nAll objects, HR models with =min\nmed: -0.03 std 0.06\nFig. 4. Comparison between the Oxygen abundances derived with the Temethod using observed and HOMERUN model line fluxes from the\nselected HII regions from the sample of Zurita et al. (2021). In the left panel, we consider, for all sources, all the models with L<Lmin+0.25; the\nvertical dashed line represents the median of the distribution. In the right panel we consider only the models with L=Lmin.\n8.2 8.4 8.6 8.8\nlog(O/H)HR[inmodels]8.18.28.38.48.58.68.7log(O/H)HR[Temethod]Models with <min+0.25\nModels with =min\nFig. 5. Comparison between the model metallicities inferred from the\nmulti-cloud modelling of the Zurita et al. 2021 sample (x axis) and the\nTemetallicities derived from the line fluxes predicted by the same mod-\nels (y models). The red dots represent the best models for each HII\nregion of the sample, while the green dots represent all \"acceptable\"\nmodels, i.e. those with L<Lmin+0.25. The solid blue line represents\nthe 1:1 relation, while the dotted blue line represent the case where Te\nmetallicities are 0.1 dex lower than model metallicities.\nmodels and observations, as it only involves model quantities,\nbut provides an indication on how (non-)accurate the Te-method\nis in retrieving the metallicity of a system. The Temetallicities\nare always smaller than the true values and can be discrepant\nup to∼0.5 dex: the median discrepancy and standard deviation\nof the di fference between Teand model metallicities consider-\ning only the best models are -0.09 and 0.07 dex, respectively.\nThese values increase to -0.13 and 0.14 dex when considering all\nmodels withL<Lmin+0.25. We believe that this discrepancy\nis due to the simplified assumptions underlying the Temethod\n(e.g., 2-3 zones with constant density and temperature), confirm-\ning the findings by other authors (see, e.g., Esteban et al. 2014;Cameron et al. 2022), and showing that, with the computational\ncapabilities and the quality and amount of data now available, it\nis important to adopt more accurate methods for abundance de-\nterminations from emission line ratios. A companion paper by\nAmiri et al. (2024, in preparation) will address in more details\nthe accuracy problems of the Temethod.\n4.3. The effect of varying N/O and S/O relative abundances\nFigure 6 outlines the importance of allowing for a scaling fac-\ntor of the N and S abundances. The left panel shows how in the\nexample model the minimum value of the Lfunction increases\nby a factor≃10 when no abundance scaling is taken into ac-\ncount. Overall, when fitting all the HII regions in the Zurita et al.\n(2021) sample (see Section 4), minimum Lvalues increase by\nfactors up to over 100 (center /middle panel), while, at the same\ntime, metallicities are over- or under-estimated by up to 0.4 dex\nin case of no abundance scaling (right panel). Clearly, rescaling\nthe N and S abundances with respect to the ones assumed in the\nphotoionization computations is extremely important to improve\nthe comparison of models and observations and the reliability of\nthe metallicity estimates (e.g., Strom et al. 2018).\n4.4. Comparison with constant-pressure single-cloud models\nFigure 7 shows the large improvement of our multi-cloud ap-\nproach with respect to the case of the single cloud constant pres-\nsure models which are used in the literature by many authors\n(e.g., Blanc et al. 2015; Dopita et al. 2016; Pérez-Montero et al.\n2021 and references therein).\nThe left panel compares the Lcurves obtained with the sin-\ngle (orange) and multi-cloud (blue) approach applied to the test\nHII region from Zurita et al. (2021) used before. The blue curve\nis the same as in Figures 2 and 6.\nIn order to obtain the orange curve, we have obtained grids of\nconstant-pressure, single-cloud models following the same ap-\nproach as before: for each given ionising continuum and gas\nmetallicity, we have computed a grid of models with varying\nArticle number, page 10 of 16A. Marconi et al.: HOMERUN a new approach to photoionization modelling\n8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8\n12+log(O\nH)101\n100101\nHOMERUN standard\nHOMERUN No N, S abundance scaling\n103\n102\n101\n100\nHR\n100101102103HRnoscNS/HR\n8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9\n12+log(O\nH)HR0.4\n0.2\n0.00.20.4log(O\nH)HRnoscNS log(O\nH)HR\nFig. 6. The e ffect of rescaling N /O and S /O relative abundances with respect to the values used in photoionization calculations. Left: Variation\nof the loss function Las a function of metallicity when fitting the emission line spectrum of the test HII region in Figure 1. The blue dots /line\nrepresent the HOMERUN model with N and S rescaling (standard model), while the red dots /line represent the same model but without rescaling\n(No N,S abundance scaling). Note that the minimum value of the loss function when excluding re-scaling is ∼ten times higher than for the standard\nmodel. The red vertical dashed and dotted lines represent the metallicity of the models with the minimum value of the loss function L. Middle:\nratios ofLminimum values from models with no N,S abundance scaling and standard models when applied to the HII regions of the Zurita et al.\n(2021) sample. The e ffect of non rescaling N and S is to increase the minimum Lvalue by up to a factor 100. Right: variation of abundances of\nmodels with no N,S abundance scaling with respect to standard models compared to the abundances of standard models.\n8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8\n12+log(O\nH)101\n100101\nHR Standard\nConstant-P Single Cloud\n103\n102\n101\n100\nHR\n100101102103singleCP/HR\n8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9\n12+log(O\nH)HR0.6\n0.4\n0.2\n0.00.20.40.6log(O\nH)singleCP log(O\nH)HR\nFig. 7. Comparison of HOMERUN with single cloud, constant pressure models. Left: Variation of the loss function Las a function of metallicity\nfor the emission line spectrum of the test HII region. The blue dots /line represent the HOMERUN model presented here (standard model), while\nthe orange dots /line represent the case of constant pressure single cloud models (Constant-P single cloud). The red vertical dashed lines represent\nthe metallicity of the models with the minimum value of the loss function L. Middle: ratios of Lminimum values from Constant-P single cloud\nand HOMERUN standard models. Right: variation of abundances of Constant-P single cloud models with respect to HOMERUN standard models.\nUandNH. For each single-cloud model in the grid we have\nexplored the same ( U,NH) grid used in the multi-cloud model\nto determine the optimal ( U,NH) pair that minimises the loss\nfunction, along with the best scalings for N and S abundances,\nat given gas metallicity and ionising continuum. We have then\nconsidered the single cloud model with the lowest Lvalue. In\npractice, we have selected the single-cloud, constant-pressure\nmodel which provides the minimum value of the loss function\nfor a given continuum, gas abundance, ionization parameter and\ndensity. The di fference with respect to other models in the lit-\nerature is that we allow for a variation of N /O and S /O and /or\nwe consider a much wider grid in density (e.g., van Zee et al.\n1998; Dopita et al. 2013; Blanc et al. 2015; Strom et al. 2018;\nYan 2018).\nIn the case of this particular example, the minimum Lvalue\nof our multi-cloud approach is more than a factor ten lower al-\nthough the estimated gas metallicity is similar. The comparison\nbetween the orange curves in Figures 2 and 6 shows that rescal-\ning N and S abundances is crucial for having a low Lminimum\nvalue. Indeed, the single-cloud constant pressure model has a\nsimilarLminvalue as the multi-cloud model where the N and\nS abundances are those assumed in the photoionization calcula-\ntions.\nThe center and right panels in the figure compare the mini-\nmumLvalues and the oxygen abundances obtained by applying\nthe two methods to the HII regions of the Zurita et al. (2021)\nsample:Lvalues obtained with the multi-cloud approach aremore than a factor 10 lower, but the discrepancy can increase\nby two orders of magnitude. On the other hand, discrepancies in\noxygen abundances are less severe and are up to ±0.3−0.4 dex.\nFinally, we note that the di fference between the ionization\nparameter of the single-cloud, constant-pressure models and the\naverage Uof the HOMERUN models is on average -0.2 dex with\na standard deviation of 0.5 dex. Similarly, the di fferences in den-\nsities are on average 0.2 dex but with a standard deviation of 1.2\ndex.\n5. The sample of HII regions from the CHAOS\nproject\nWe now apply the model to a sample of HII regions with more\nhomogeneous data, higher S /N and a larger number of emission\nlines.\nWe consider the HII regions of six galaxies from the CHem-\nical Abundances of Spirals (CHAOS) project (Berg et al. 2015;\nCroxall et al. 2015, 2016; Berg et al. 2020a; Rogers et al. 2021,\n2022) and we apply the same procedure followed in the previous\nsection. At variance with the compilation by Zurita et al. 2021,\nthe data now include 11 additional lines (see figure 8 for the list)\nincluding [O i]λ6300, a line which has always been di fficult to\naccount for in photoionization models (e.g., Dopita et al. 2013).\nWe select the HII regions with the same criterion used for the\nZurita et al. (2021) sample, i.e. the auroral lines detected with\nS/N>5.\nArticle number, page 11 of 16A&A proofs: manuscript no. paper_sub\n0.1 0.5 0.9 1.3 1.7\nmin\n0.02.55.07.510.012.515.017.5# of modelsAll\nw/ dust\nw/o dust\n0.50 0.75 1.00 1.25 1.50050010001500H\n0.50 0.75 1.00 1.25 1.5005001000H\n0.50 0.75 1.00 1.25 1.5005001000[OIII]5007Å\n0.50 0.75 1.00 1.25 1.5005001000[OI]6300Å\n0.50 0.75 1.00 1.25 1.500250500750[OII]3726,3729Å\n0.50 0.75 1.00 1.25 1.5005001000[OIII]4363Å\n0.50 0.75 1.00 1.25 1.50050010001500[OII]7320,7330Å\n0.50 0.75 1.00 1.25 1.50050010001500[NII]6583Å\n0.50 0.75 1.00 1.25 1.500200400600[NII]5755Å\n0.50 0.75 1.00 1.25 1.50050010001500[SII]6731Å\n0.50 0.75 1.00 1.25 1.50050010001500[SII]6716Å\n0.50 0.75 1.00 1.25 1.5005001000[SII]4069,4076Å\n0.50 0.75 1.00 1.25 1.500200400[SIII]6312Å\n0.50 0.75 1.00 1.25 1.500250500750[SIII]9531Å\n0.50 0.75 1.00 1.25 1.50050010001500H\n0.50 0.75 1.00 1.25 1.50050010001500H\n0.50 0.75 1.00 1.25 1.50050010001500[ArIII]7136Å\n0.50 0.75 1.00 1.25 1.5005001000[ArIII]7751Å\nFig. 8. Left: distribution of Lvalues for the best /acceptable models of the HII regions from CHAOS sample. Right: distributions of the contributions\ntoLfrom each of the lines used for model fitting (the adopted error on line fluxes is the maximum between the observed error and 10%).\n0.4\n 0.2\n 0.0 0.2 0.4\nlog(O/H)Te,HR log(O/H)Te,Observed\n050100150200 med: -0.03\nstd: 0.07All HR models with <min+0.25\n8.1 8.2 8.3 8.4 8.5 8.6 8.7\nlog(O/H)Te,Observed0.3\n0.2\n0.1\n0.00.10.20.3log(O/H)Te,HR log(O/H)Te,Observed\nAll objects, HR models with =min\nmed: -0.02 std 0.06\nFig. 9. Comparison between the Oxygen abundances derived with the Temethod using observed and HOMERUN model line fluxes from the\nselected HII regions from the CHAOS sample. In the left panel, we consider, for all sources, all the models with L<Lmin+0.25; the vertical\ndashed line represents the median of the distribution. In the right panel we consider only the models with L=Lmin.\nFigures 8 and 9 confirm the results of the previous section:\ntheLminvalues of the best models are all lower than 1 (red bars in\nfigure) indicating an average agreement between model and ob-\nserved line fluxes better than 10%. All lines are well reproduced\nand in particular, the models are able to reproduce at the same\ntime [O i], [O ii] and [O iii], as well as [S ii] and [S iii], which\nis a significant improvement compared to previous works (see,\ne.g., Mingozzi et al. 2020 on [S iii] and Dopita et al. 2013 on\n[O i]) Note that the [NeIII] λ3869 line does not provide any con-\nstraint to the model since it is the only Neon line, and it is used\nonly to estimate Ne abundance. It is remarkable that when usingonly the emission lines from the Zurita et al. (2021) sample and\n[O i]λ6300, theLminvalues are all lower than 0.3. Similarly to\nthe case of the Zurita et al. (2021) sample, only a small fraction\nof the single-cloud models in a 8 ×9 (U, NH)-grid have non\nzero weights: in 23 out of 24 best fit models, the numbers of\nclouds with non-zero weights is be- tween 5 and 10, with only\none models being made of 12 clouds.\nWe have estimated Temetallicities as in the previous section\nand found that the agreement between estimates from observed\nand model line fluxes is similar to what found for the Zurita et al.\n(2021) sample (figure 9). Finally, we confirm as well the discrep-\nArticle number, page 12 of 16A. Marconi et al.: HOMERUN a new approach to photoionization modelling\n8.2 8.4 8.6 8.8\nlog(O/H)HR[inmodels]8.18.28.38.48.58.68.7log(O/H)HR[Temethod]Models with <min+0.25\nModels with =min\nFig. 10. Comparison between the model metallicities inferred from the\nmulti-cloud modelling of the CHAOS sample (x axis) and the Temetal-\nlicities derived from the line fluxes predicted by the same models (y\nmodels). The red dots represent the best models for each HII region of\nthe sample, while the green dots represent all \"acceptable\" models, i.e.\nthose withL<Lmin+0.25. The solid blue line represents the 1:1 rela-\ntion, while the dotted blue line represent the case where Temetallicities\nare 0.1 dex lower than model metallicities.\nancy of up to 0.2 dex between the metallicities used in the pho-\ntoionization model computations and the Temetallicities derived\nfrom the line fluxes predicted by the same models (figure 10).\nOutside of H iiregions, the warm ( Te∼104K), ionized com-\nponent of the ISM is made of a more di ffuse medium, the so-\ncalled DIG, particularly evident above and below the galactic\nplane. Such medium can significantly a ffect the integrated spec-\ntra of H iiregions and star forming galaxies, depending on the\nsize of the aperture from which spectra are extracted. Mannucci\net al. (2021) investigated the contribution of the DIG which is of-\nten invoked to explain the di fferences between the spectra of HII\nregions in local galaxies and the total or kpc-scale spectra of star-\nforming galaxies when the flux ratios involving low-ionization\nemission lines are considered (e.g., Belfiore et al. 2016; Ku-\nmari et al. 2019; Vale Asari et al. 2019; Della Bruna et al. 2020;\nSanders et al. 2020). Comparing single slit spectra of HII regions\nfrom CHAOS with those extracted with di fferent aperture sizes\nfrom integral field spectra, they concluded that these di fferences\nare mostly ascribed to the small anglular size of the slits, which\nis not covering the full extent of the HII region, rather than to\nthe DIG contamination. In principle, our model approach, based\non combining multiple clouds to reproduce the correct volume\nof gas occupied by any given ion X+iwith the correct Teand\nNe, can naturally take into account both aperture e ffects and the\ncontributions from regions with di fferent physical properties like\nthe DIG. However, this issue will be addressed in a forthcoming\npaper.6. Gas and stellar metallicities in the Milky Way\nWe present the last application of our method and its potential to\nsolve the observed discrepancy between the metallicities of HII\nregions and those of the stars in our Galaxy.\nThe metallicity of the ionized gas is generally is good agree-\nment with that of the stars, at least in external galaxies (see, e.g.,\nFig. 11 of Bresolin et al. 2016). However, there is a discrepancy\nbetween the metallicities of HII regions in our Galaxy, derived\nfrom the application of the Te-method to gas emission lines,\nand the stellar metallicities derived from B-type stars, Classical\nCepheids and Open clusters, which are ∼0.2 dex larger (e.g.,\nFig. 4 of Esteban et al. 2022). Such di fference is not present\nwhen considering gas-phase metallicities from recombination\nlines and is part of the so-called abundance discrepancy prob-\nlem which, as described in Sec. 1, is quite a common occurrence\nwhen comparing Teand recombination-line-based metallicities.\nAs mentioned by Esteban et al. (2022), such discrepancies might\nbe related to temperature fluctuations which bias auroral lines to-\nwards the hotter parts of the HII regions providing higher than\naverage temperatures and consequently lower abundances (Pe-\nimbert 1967). Indeed, this discrepancy disappears when correc-\ntion for temperature fluctuations and ionization fractions are ap-\nplied (Méndez-Delgado et al. 2022). Still corrections for temper-\nature fluctuations are di fficult to estimate and ionization correc-\ntions must be derived from photoionization models, thus adding\nup additional assumptions and uncertainties to the Temethod.\nWe consider all the Galactic HII regions from table 1 of\n(Méndez-Delgado et al. 2022) which have at least three of the\nfive auroral lines considered so far detected with S/N>5 and\nestimate their metallicities by applying our method to the emis-\nsion lines fluxes from Arellano-Córdova et al. (2020), 2021 and\nZurita et al. (2021). Figure 11 shows the HII region gas-phase\nmetallicities plotted as a function of GAIA-based galactocen-\ntric distances from Méndez-Delgado et al. (2022), and compare\nthem with the metallicity gradient of the same HII regions es-\ntimated with the Te-method without any ionization correction\n(Arellano-Córdova et al. 2021), the Cepheid metallicity gradi-\nent from Mollá et al. (2019) and the intermediate-age (1-3 Gyr)\nopen clusters metallicity gradient from Magrini et al. (2023). The\nlinear fit considers all HII regions and was performed with the\nltsfit4python package which implements a Robust linear re-\ngression with scatter in one or two dimensions (Cappellari et al.\n2013).\nThe plot clearly shows how our model-based metallicities\nare, on average, in agreement with the stellar metallicities espe-\ncially when considering that oxygen is a refractory element and\nis likely depleted into stars: indeed, it is important to recall that\nthese are gas-phase metallicities which do not take into account\nthe depletion of metals into dust. Oxygen depletion can vary be-\ntween 10-20% (Psaradaki et al. 2023; Amayo et al. 2021) and the\n40% assumed by CLOUDY . Therefore, any abundance discrep-\nancy here can be easily explained with a ≲0.1 dex depletion of\noxygen into dust without having to rely on uncertain temperature\nfluctuations or ionization corrections.\n7. Summary and Conclusions\nWe have presented a new approach to modelling emission lines\nfrom photoionized gas which allows to reproduce observed line\nratios from a wide range of ionization and with an accuracy to\nbetter than 10%, and whose main application is the accurate de-\ntermination of gas metallicity.\n4Available at https: //www-astro.physics.ox.ac.uk /∼cappellari /software /\nArticle number, page 13 of 16A&A proofs: manuscript no. paper_sub\n0 2 4 6 8 10 12 14 16 18\nRG(kpc)7.67.88.08.28.48.68.89.012+log(O/H)\nSun\nCepheids (Molla et al. 2019)\nOpen clusters 1-3 Gyr (Magrini et al. 2023)\nHII regions Te (Arellano-Cordova+2021)\nThis work (fit all)\nThis work (5 auroral lines w/ S/N>5)\nThis work (3 auroral lines or more w/ S/N>5)\nFig. 11. Comparison between our metallicity estimates of HII regions in our Galaxy, the metallicity gradient from the Teabundances of HII regions\n(Arellano-Córdova et al. 2021), the Cepheids (Mollá et al. 2019) and intermediate age open clusters (Magrini et al. 2023). The blue line surrounded\nby the blue shaded area represents the linear fit, with uncertainties, to all our metallicity estimates.\n–Our approach is based on the weighted combination of mul-\ntiple single-cloud photoionization models computed with\nCLOUDY , as in previous works. The novelty of our approach\nis that the weights of the single-cloud models are not para-\nmetric nor assumed a priori, but are free parameters of the fit.\nThe resulting best fit model accounts for the varying physi-\ncal conditions (temperatures, densities, ionisation conditions\nand gas metallicities) observed along the line of sight of the\ninterstellar gas of HII regions and galaxies.\n–Since observed line luminosities are integrated quantities\nover the entire volume of the emitting source, the single-\ncloud models contributing to a best fit do not necessarily cor-\nrespond to physical entities but are the basic building blocks\nof the model. Therefore, if a best fit model has a single cloud\nwith specific U,NH, this does not necessarily mean that such\ncloud exists but that the total line emission has a contribution\nfrom gas with those physical parameters, regardless of where\nand how it is spatially distributed.\n–Our model has as many free parameters as the single-cloud\nmodels considered, however the fitting of the observed data\nis not degenerate and the loss function ( L) curves, computed\nas a function of gas metallicity, have very deep and well de-\nfined minima. Indeed our approach is similar to the one used\nfor fitting the stellar continua of galaxies where many hun-\ndreds of templates can be combined with weights which are\nfree parameters of the fit (see, e.g., the ppxf code to analyze\nstellar continuum spectra).\n–We have shown that a critical point for our modelling is to al-\nlow for di fferent abundances for elements like N and S than\nthose used in the photoionisation calculations: N and S emis-\nsion lines are not the major coolants of the photoionised re-gion like O lines and, therefore, their luminosities scale al-\nmost linearly with gas abundances. Not accounting for N and\nS abundance variations can significantly a ffect the quality of\nthe fit (with minimum Lvalues worse by even a couple of\norders of magnitude) and the estimated gas metallicity.\n–We have also shown that our approach provides a signifi-\ncant improvement compared to the single-cloud, constant-\npressure models commonly used in the literature, as well as\nbeing able to model a larger set of emission lines. Indeed,\nseveral works based on single-cloud photoionization mod-\nelling claimed to pin down the metallicity with high accuracy\nby using just a small set of lines, typically the optical strong\nlines and they rarely, if ever, include auroral lines which are\ninstead successfully reproduced in HOMERUN.\n–We have applied our model to the HII regions from the\nsamples by Zurita et al. (2021) and from the CHAOS\nproject. For the sample by Zurita et al. (2021), we\nhave considered HII regions where all the following\nlines were detected with S/N>5: [O ii]λλ3726,3729,\n[O iii]λλ4959,5007, [N ii]λλ6548,6583, H α,\n[S ii]λλ6717,6731, [S iii]λλ9069,9532, [S ii]λλ4068,4076,\n[O iii]λ4363, [N ii]λ5755, [S iii]λ6312, and [O ii]λλ7320,\n7330. All the observed emission lines for the 24 selected\nHII regions were reproduced by the model to better than\n10%. In particular the model was able to reproduce also\nthe [S iii]9069,9532 lines which are well known to be\nproblematic. The Te-metallicities derived from the observed\nemission lines and those derived from the model emission\nlines agree to within 0.05 dex, another indication of the\nextremely good performance of the models.\nArticle number, page 14 of 16A. Marconi et al.: HOMERUN a new approach to photoionization modelling\n–However, we showed that Te-metallicities su ffer from un-\ncertainties which are related to the simplified assumptions\nunderlying the method: the Te-metallicities computed from\nthe emission lines predicted by the models can be more than\n0.1 dex smaller than the real gas metallicities of the models\nthemselves.\n–The CHAOS sample is more homogeneous in terms of data\nquality, is characterized by better S /N and provides the fluxes\nof many more emission lines. Overall, we have modeled the\nsame emission lines used for the Zurita et al. (2021) sam-\nple but we have added also other lines, doubling their total\nnumber.\n–Even with these increased constraints, the models reproduce\nthe observed emission lines to better than 10%. It is worth\nnoting that the HOMERUN models are able to simultane-\nously reproduce the emission lines from very di fferent ion-\nization stages, like [O i], [O ii] and [O iii].\n–Finally, we have shown that the gas metallicities estimated\nwith our models in HII regions of the Milky Way are in bet-\nter agreement with the stellar metallicities than the estimates\nbased on the Te-method.\nOverall, our method provides a new accurate tool to estimate the\nmetallicity and the physical conditions of the ionized gas. It can\nbe applied in many di fferent science cases and wherever there\nare emission lines from photoionized gas.\nThere are many applications of our method which will be\npresented in a series of forthcoming papers. In particular, the\nsecond paper in this series (Amiri et al. 2024, in preparation) will\npresent the application of this method to the spectra of galaxies,\nwill determine their metallicities and will provide a new self-\nconsistent calibration of the strong line method.\nAcknowledgements. We are grateful to Gary Ferland and his collaborators for\nthe development of CLOUDY and for allowing its free use for the community.\nThis work has been partially financed by the European Union with the Next Gen-\neration EU plan, through PRIN-MUR project \"PROMETEUS\" (202223XPZM).\nWe acknowledge the support of the INAF Large Grant 2022 “The metal circle: a\nnew sharp view of the baryon cycle up to Cosmic Dawn with the latest genera-\ntion IFU facilities” and INAF Large Grant 2022 \"Dual and binary SMBH in the\nmulti-messenger era\". 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Antonelli15,16, Matteo Bachetti17, Luca Baldini18,19,\nWayne H. Baumgartner13, Ronaldo Bellazzini18, Stephen D. Bongiorno13, Raffaella Bonino20,21, Alessandro Brez18,\nNiccolò Bucciantini22,23,24, Simone Castellano18, Elisabetta Cavazzuti25, Chien-Ting Chen26, Stefano Ciprini27,16,\nEnrico Costa2, Alessandra De Rosa2, Ettore Del Monte2, Laura Di Gesu25, Niccolò Di Lalla28,\nImmacolata Donnarumma25, Victor Doroshenko29, Michal Dov ˇciak30, Steven R. Ehlert13, Teruaki Enoto31,\nYuri Evangelista2, Riccardo Ferrazzoli2, Javier A. García32, Shuichi Gunji33, Kiyoshi Hayashida34, Wataru Iwakiri35,\nSvetlana G. Jorstad36,37, Vladimir Karas30, Fabian Kislat38, Takao Kitaguchi31, Jeffery J. Kolodziejczak13,\nHenric Krawczynski39, Luca Latronico20, Ioannis Liodakis13, Simone Maldera20, Alberto Manfreda40,\nFrédéric Marin41, Andrea Marinucci25, Alan P. Marscher36, Herman L. Marshall5, Francesco Massaro20,21,\nIkuyuki Mitsuishi42, Tsunefumi Mizuno43, Michela Negro44, Chi-Yung Ng45, Stephen L. O’Dell13, Nicola Omodei28,\nChiara Oppedisano20, Alessandro Papitto15, George G. Pavlov46, Abel L. Peirson28, Matteo Perri16,15,\nMelissa Pesce-Rollins18, Pierre-Olivier Petrucci47, Maura Pilia17, Andrea Possenti17, Simonetta Puccetti16,\nBrian D. Ramsey13, Ajay Ratheesh2, Oliver J. Roberts26, Roger W. Romani28, Carmelo Sgrò18, Patrick Slane48,\nPaolo So ffitta2, Gloria Spandre18, Douglas A. Swartz26, Toru Tamagawa31, Fabrizio Tavecchio49, Roberto Taverna50,\nYuzuru Tawara42, Allyn F. Tennant13, Nicholas E. Thomas13, Francesco Tombesi3,27,51, Alessio Trois17,\nSergey S. Tsygankov1, Roberto Turolla50,52, Jacco Vink9, Kinwah Wu52, Fei Xie53,2, and Silvia Zane52\n(Affiliations can be found after the references)\n2024\nABSTRACT\nWeakly magnetized neutron stars in X-ray binaries show complex phenomenology with several spectral components that can be associated with\nthe accretion disk, boundary and /or spreading layer, a corona, and a wind. Spectroscopic information alone is, however, not enough to disentangle\nthese components. The analysis of the timing data revealed that most of the variability and, in particular, kilohertz quasi-period oscillations are\nassociated with the high-energy component corresponding to the boundary /spreading layer. Additional information about the nature of the spectral\ncomponents and in particular the geometry of the emission region can be provided by X-ray polarimetry. One of the objects of the class, a bright,\npersistent, and rather peculiar galactic Type I X-ray burster GX 13 +1, was observed with the Imaging X-ray Polarimetry Explorer ( IXPE ) and\nthe X-ray Multi-Mirror Mission Newton ( XMM–Newton ). Using the XMM–Newton data we estimated the current state of the source as well as\ndetected strong absorption lines associated with the accretion disk wind. IXPE data showed the source to be significantly polarized in the 2–8 keV\nenergy band with the overall polarization degree (PD) of 1 .4%±0.3% at polarization angle (PA) of −2◦±6◦(errors at 68% confidence level).\nDuring the two-day long observation, we detected rotation of the PA by about 70◦with the corresponding changes in the PD from 2% to non-\ndetectable and then up to 5%. These variations in polarization properties are not accompanied by visible changes in spectroscopic characteristics.\nThe energy-resolved polarimetric analysis showed a significant change in polarization, from being strongly dependent on energy at the beginning\nof the observation to being almost constant with energy in the later parts of the observation. As a possible interpretation, we suggest the presence\nof a constant component of polarization, strong wind scattering, or di fferent polarization of the two main spectral components with individually\npeculiar behavior. The rotation of the PA suggests a 30◦–40◦misalignment of the neutron star spin from the orbital axis.\nKey words. accretion, accretion disks – polarization – stars: neutron – X-rays: binaries\n1. Introduction\nWeakly magnetized neutron stars (WMNSs) reside in low-mass\nX-ray binary (LMXB) systems and accrete material from com-\npanion stars via an accretion disk. With their luminosity rang-\ning in the 1036−1038erg s−1, they are among the brightest X-\nray sources in the sky. WMNSs can be classified as Z- or atoll-\nsources, based on the shape of their tracks in the X-ray hardness-intensity (HID) diagram or color-color diagram (CCD), their\ntiming proprieties in the 1–10 keV band, their radio emission,\nand their luminosity and mass accretion rate. WMNSs are also\nknown for their fast X-ray variability: quasi-periodic oscillations\nin the seconds, milliseconds, and kHz ranges were observed\nfrom many sources of the class and are traditionally interpreted\nArticle number, page 1 of 12arXiv:2401.13058v1  [astro-ph.HE]  23 Jan 2024A&A proofs: manuscript no. ixpe_AA\nas the result of the luminosity modulation at the neutron star sur-\nface (Gilfanov et al. 2003).\nIn the absence of a strong magnetic field surrounding the\nNS in such systems, the matter from the accretion disk falls\non the equator of the star via the boundary layer (BL, see e.g.\nShakura & Sunyaev 1988) and spreads further along the NS sur-\nface forming the so-called spreading layer (SL, see e.g. Lapidus\n& Sunyaev 1985; Inogamov & Sunyaev 1999). Both the geom-\netry and physics of the processes happening within these layers\nhave an impact on the emission of WMNSs and their polarization\n(see e.g. Lapidus & Sunyaev 1985; Gnarini et al. 2022). Emis-\nsion of WMNSs in the X-ray band has two main components: a\nsoft thermal emission, which can be a blackbody-like emission\nfrom the NS surface or a multicolor emission coming from the\ndisk, and a harder component associated with the Comptoniza-\ntion happening in the BL or SL. In addition to these two main\ncomponents, one can sometimes see the signs of reflection of the\nSL emission from the accretion disk (usually a broad emission\nline around 6.4 keV) and a set of absorption lines that indicate\nthe interaction of the emission in the wind above the disk.\nPolarimetry is a useful tool to understand both the geome-\ntry of the source and the mechanisms of emission by studying\nthe two additional variables it provides, the polarization degree\n(PD) and polarization angle (PA). The Imaging X-ray Polarime-\ntry Explorer ( IXPE , Weisskopf et al. 2022), launched in Decem-\nber 2021, is a pathfinder that gives a unique opportunity to mea-\nsure the X-ray polarimetric properties of WMNSs with high pre-\ncision. In the last two years, nine objects of the class were ob-\nserved (see Capitanio et al. 2023; Farinelli et al. 2023; Cocchi\net al. 2023; Ursini et al. 2023; Fabiani et al. 2023; Di Marco\net al. 2023a; Rankin et al. 2024; La Monaca et al. 2024; Saade\net al. 2023). These observations shed new light on the mecha-\nnisms of emission of such sources, as well as on the changes PD\nand PA experience while the source changes the spectral state. In\nthis article, we aim to add another important piece of evidence\nto the global picture, with a special focus on the scattering of the\nradiation in the wind above the accretion disk.\nGX 13 +1 is a peculiar source with a unique position between\natolls and Z-sources and is located at the distance of 7 ±1 kpc\nin a binary system with a late-evolved K5 III giant as a compan-\nion (Bandyopadhyay et al. 1999). The source shows several fea-\ntures traditionally associated with Z-sources: it shines almost as\nbrightly as Z-sources (up to 0 .5LEdd, D’Aí et al. 2014), it has per-\nsistent radio emission (Grindlay & Seaquist 1986), and it shows\n57–69 Hz quasi-periodic oscillations (Homan et al. 1998). Still,\nthe path it draws on the CCD is closer to the one we normally\nsee from the atolls (Schnerr et al. 2003). However, in the latest\npapers, even the CCD of the source is interpreted as if GX 13 +1\nwas a Z-source (Saavedra et al. 2023; Allen et al. 2018). Another\nunique property of GX 13 +1 is the complex spectrum (e.g., Díaz\nTrigo et al. 2012). In addition to the common softer accretion\ndisk and harder comptonized components, the sign of reflection\nof the comptonized emission from the disk is present, as well as\nseven absorption features coming from the interaction of the ra-\ndiation in the wind above the disk. It was shown in Allen et al.\n(2018) that these absorption features could not be produced by\na single absorbing region, which indicates a complicated struc-\nture of the wind. GX 13 +1 also exhibits periodic variations of the\nemission with a period of 24.5 d (Corbet et al. 2010). These mod-\nulations are associated with the orbital movement of the source,\nmaking GX 13 +1 the NS–LMXB with the longest orbital period.\nShort dips of∼1 ks in the X-ray spectrum are associated with\nthis orbital movement (Diaz Trigo et al. 2010; Iaria et al. 2014),\nyet they are occasional events and are not present in every orbit.These dips are traditionally interpreted as an indication of the\nhigh source inclination (60◦−85◦) and as a sign of the presence\nof the mostly neutral bulge located at the outer accretion disk\nand formed as a result of the accretion stream interaction with\nthe disk (White & Holt 1982; D’Aí et al. 2014).\nThe wind above the accretion disk of GX 13 +1 was carefully\nstudied with several observatories. For instance, using Chandra\ndata, Tomaru et al. (2020) gave constraints on the radial and az-\nimuthal velocity of the wind, and Allen et al. (2018) constrained\nthe wind launching radius showing that the radiation pressure is\nlikely responsible for the wind launching. Maiolino et al. (2019)\ndeveloped a non-relativistic framework capable of fitting and in-\nterpreting the asymmetric K αlines in XMM–Newton data and\nreported that this non-relativistic approach performs as well as\nthe known relativistic framework, adding an alternative expla-\nnation for the red-skewed Fe line in the spectrum of GX 13 +1.\nSaavedra et al. (2023) using NuSTAR data put some estimations\nof the distance between the source and the ionized absorbing\nmaterial of (4−40)×105km, constrained the magnetic field,\natB≲1.8×108G, and pointed out the overabundance of Fe\nand Ni in the source. GX 13 +1 is also in the list of targets for\nthe recently launched X-ray Imaging and Spectroscopy Mission\n(XRISM ; XRISM Science Team 2022). The aim of our project is\nto obtain constraints on the source geometry and radiation mech-\nanisms using IXPE polarimetric capabilities.\nThe remainder of the paper is structured as follows. We in-\ntroduce the observations performed with IXPE and the XMM–\nNewton observatories in Sect. 2. In Sect. 3 we present the data\nanalysis. We provide an interpretation of the results in Sect. 4\nand summarize in Sect. 5.\n2. Observations\n2.1. IXPE\nThe Imaging X-ray Polarimetry Explorer ( IXPE ) was launched\non 2021 December 9 as a result of joint e fforts of NASA and\nthe Italian Space Agency. It has three identical grazing inci-\ndence telescopes on board, each of them comprising an X-ray\nmirror assembly and a polarization-sensitive detector unit (DU)\nequipped with a gas-pixel detector (So ffitta et al. 2021; Baldini\net al. 2021). IXPE is capable of performing imaging polarime-\ntry over the 2–8 keV energy band with a time resolution of the\norder of 10 µs. A detailed description of the observatory and its\nperformance is given in Weisskopf et al. (2022).\nIXPE observed GX 13 +1 on 2023 October 17–19 with a to-\ntal effective exposure of ≃100 ks for each of the three telescopes\n(see Table 1). For the data analysis, we used ixpeobssim software\nversion 30.6.3 (Baldini et al. 2022) and the heasoft package ver-\nsion 6.31.1 (Nasa Heasarc 2014). We used the latest version of\ntheIXPE response matrices, released at the H easarc CALDB on\n2023 June 16. We extracted the I,Q, and Uspectra from a region\nwith an 80′′radius. Following Di Marco et al. (2023b), we did\nnot perform background subtraction or rejection, as GX 13 +1\nis sufficiently bright that the background is negligible. The po-\nlarimetric analysis was performed using the formalism of Kislat\net al. (2015) implemented in the ixpeobssim package (Baldini\net al. 2022) under the pcube algorithm in the xpbin tool. We\nused ixpeobssim to investigate the time-dependency of the nor-\nmalized Stokes parameters q=Q/Iandu=U/I, as well as\nthe PD =p\nq2+u2and PA =1\n2arctan( u/q) (see Fig. 1a-d). We\napplied an unweighted analysis implemented in the ixpeobssim .\nFor the spectral and spectro-polarimetric analysis, we used xspec\nversion 12.13.0c (Arnaud 1996) and applied a weighted analysis\nArticle number, page 2 of 12Bobrikova et al.: First detection of polarized emission from GX 13 +1\nTable 1. Observations of GX 13 +1 presented in the paper.\nObservatory Date ObsID Instrument Duration (s)\nIXPE 17–19.10.2023 02006801 DU1 98204\nDU2 98299\nDU3 98299\nXMM–Newton 16.10.2023 0932390601 EPIC /PN 11336\nas introduced in Di Marco et al. (2022). For the spectral analy-\nsis, the data were binned to have at least 30 counts in each energy\nbin. For the spectro-polarimetric analysis, we applied a constant\nenergy binning of 80 eV for the three Stokes parameters.\n2.2. XMM–Newton\nXMM–Newton observed GX 13 +1 on 2023 October 16 (see\nTable 1). This observation could not be scheduled simultane-\nously with IXPE because the visibility window for GX 13 +1\nwith XMM–Newton closed on October 16. As GX 13 +1 is very\nbright, EPIC /PN was operated in timing mode. The thick filter\nwas used to reduce the count rate, which was, on average, about\n690 cnt s−1during the observation. The EPIC /MOS instruments\nwere not used, and their telemetry was allocated to EPIC /PN.\nYet, despite this e ffort, PN never-the-less su ffered from telemetry\ngaps as the science bu ffer was full throughout the observation.\nFor this reason, the data are not suitable for timing analysis but\nare still useful for measuring the spectral shape and the known\nwind lines in the 6–9 keV range (Díaz Trigo et al. 2012).\nThe data were reduced using xmm–sasv.21 with the latest\ncalibration files as of October 2023. The clean event files were\ngenerated from the ODF files with the epchain tool. They were\nfiltered further using the #XMMEA_EP and PATTERN ==0\nflags in the 0.3–12 keV range. No soft proton flares were seen\nin the light curves above 10 keV , giving us a total e ffective expo-\nsure of≃10 ks. For extracting the source spectrum, we selected\nevents from a 9-pixel wide box around the RAWX column with\nthe highest count rate. We also extracted a background spectrum\nfrom the RAWX columns 3–5, but – as expected for such a bright\ntarget – these columns were still dominated by the source itself,\nand thus, we did not subtract the background at all. The ancillary\nresponse file and the redistribution matrix were generated using\nthearfgen andrmfgen tools, respectively. Finally, we grouped\nthe data to have at least 20 counts per channel and added 1% sys-\ntematic errors in quadrature to each channel using the grppha\ntool.\n3. Data analysis\n3.1. Model-independent polarimetric analysis\nThe light curve of GX 13 +1 is shown in Fig. 1f. The source\nshowed some variations, with the count rate dropping almost by\na factor of two in the wide dip, but some more rapid dips are\nnoticeable. The hardness ratio (Fig. 1e) changes with time only\nby a small margin; the biggest increase slightly after MJD 60236\ncoincides with the wide dip in the light curve and is of ≈10%.\nThe first thing to note is the significant variability of the\nStokes parameters. This variability is reflected in the changes in\nPD and PA in Fig. 1c,d. While the PA slowly increases through-\nout the observation, the PD changes significantly and without an\nobvious pattern.\nThe HID presented in Fig. 2 illustrates a lack of state vari-\nability within the more than two days of IXPE observation. From\n10\n010 [%]\na\n10\n010 [%]\nb\n100\n0100PA [deg]\nc\n0510PD [%]\nd\n0.200.250.304-8 keV/2-4 keV\ne\n4.5 5.0 5.5 6.0 6.5 7.0\nTime [MJD60230]\n2040Rate [cnt/s]\nBin 1 Bin 2 Bin 3 Bin 4 Bin 5\nfFig. 1. Normalized Stokes q(a) and u(b) parameters, PA (c), PD (d),\nhardness ratio (e), and the count rate (f) as functions of time as ob-\nserved by IXPE (all three detectors combined in the energy range 2–8\nkeV). The light curve is binned in ∼200 s; the blue dashed vertical lines\nseparate the observation into five equal, 10.5 h long, time bins (Bin 1–\n5). The region highlighted in yellow corresponds to the “dip” part of\nthe observation, according to the data separation into three parts by the\nlight curve (between MJD −60230 =6.05 and 6.30). Uncertainties are re-\nported at 68% CL.\nthe MAXI (Matsuoka et al. 2009) data, we discovered that the\nsource was in the so-called lower left banana (LLB) state, a soft\nstate common for the atoll sources. In the two days of observa-\nArticle number, page 3 of 12A&A proofs: manuscript no. ixpe_AA\n20 30 40\nRate (2-8 keV) [cnt/s]0.200.250.304-8 keV/2-4 keV\n5.05.56.06.5\nMJD60230\nFig. 2. Hardness-intensity diagram derived from the IXPE data in 200 s\ntime bins. Colors correspond to the evolution of the data with time, from\ndark blue at the beginning of the observation to yellow at the end of the\nobservation.\n0246PD [%]\nPD  1.4%\n2 3 4 5 6 7 8\nEnergy [keV]100\n50\n0PA [deg]\nPA 2°\nFig. 3. Energy-resolved PD and PA averaged over all the IXPE obser-\nvation, obtained with the pcube algorithm. Data are divided into six\n1-keV wide energy bins. Uncertainties are reported at the 68% CL.\ntion, GX 13 +1 moved in the lower-left part of the CCD, but the\nuncertainties are too high to derive any conclusions about the\nstate variability. Polarimetric analysis of the averaged data in the\n2–8 keV energy band resulted in low PD values of 1 .4%±0.3%\nand a PA of−2◦±6◦(from here onward the errors are at 68% con-\nfidence level, CL) with a significance of the detection exceeding\n99.99% (see Fig. 3 and Table 2).\nAs the HID shows no change of state, we decided to per-\nform a more detailed study of di fferent parts of the observation\nbased on the light curve. We separated the dip-state (highlighted\nin yellow in Fig. 1f) from the non-dip states and divided the ob-\nservation into three non-equal parts: pre-dip, dip, and post-dip.\nThe results are gathered in Fig. 4. The PD in the dip state is\n2.2%±1.1%, and PA is−49◦±14◦at 68% CL. Due to the lack\nof statistics, we chose not to perform an energy-resolved analysis\nof this part of the observation.\nHowever, the pre-dip and post-dip parts show interesting and\ncompletely di fferent behavior, see Table 2. The energy-resolved\nPA and PD for these parts are presented in Fig. 5, and the corre-Table 2. Results of the pcube analysis performed with various ap-\nproaches to separate the data.\nPart of the observation PD (%) PA (deg)\noverall 1 .4±0.3−2±6\npre-dip 1 .8±0.4−29±6\ndip 2 .2±1.1−49±14\npost-dip 5 .3±0.6 26±3\n“dip core” 7 .5±1.7−51±7\nBin 1 3 .2±0.7−44±6\nBin 2 1 .5±0.7−21±13\nBin 3 1 .6±0.7 4±13\nBin 4 1 .6±0.7 12±13\nBin 5 5 .0±0.7 26±4\nDimmer state, <25 cnt s−11.3±0.9−36±18\nBrighter state, >25 cnt s−11.6±0.3 2±6\nNotes. The errors are at 68% CL. Results are presented for the overall\npolarization during the observation, separation by the light curve into\npre-dip, dip, and post-dip parts with a special mention of the “dip core”,\none bin at≈6.15 MJD–60230 in Fig. 1, separation into five equal time\nbins, and separation by the count rate into dimmer and brighter states.\n80\n60\n40\n20\n020\n40\n60\n80\n012345678Polarization angle (deg)East WestNorth\nPolarization degree (%)pre-dip\ndip\npost-dip\nFig. 4. Polar plot of the PD and PA in the 2–8 keV energy band for the\nthree parts of the observation (pre-dip, dip, and post-dip). Contours are\nat 68% CL (dashed lines) and 99% CL (solid lines).\nsponding Stokes qanduparameters are shown in Fig. 6. We also\nadditionally studied the “dip core”, one specific bin in Fig. 1 at\n6.15 MJD-60230 that falls out of patterns in Fig. 1a and Fig. 1b.\nThis bin lies in the middle of the hardening in Fig. 1e and of the\ndip in Fig. 1f, which makes it particularly interesting. As men-\ntioned in Table 2, PD there exceeds 7%, and the PA is below\n−50◦with a significance of the detection higher than 99%.\nThe first thing to note here is the change in PD behavior. In\nthe pre-dip part, the PD strongly depended on energy and in-\ncreased from an upper limit of 1 .7% at 2σCL in the 2–3 keV\nbin to a strong detection of 5 .5% in the 5.5–8 keV bin. In the\npost-dip part, however, the PD was almost constant with energy\nand averaged at about 5 .3%, exceeding 7% in the 5.5–8 keV bin.\nNo significant PA dependence on energy is observed in either\npart of the observation, yet the value of the angle shifts by al-\nmost 60◦between the pre-dip and post-dip parts. This complete\nchange of behavior between the two parts of the observation is\nalso visible in the Stokes q−uplane (see Fig. 6), the polarization\nvector changes both in amplitude and direction.\nGoing even deeper into the separation of the observation, we\nsplit it into five equal parts of 10.5 h each (“Bin 1, 2, ..., 5” are de-\nArticle number, page 4 of 12Bobrikova et al.: First detection of polarized emission from GX 13 +1\n0246PD [%]\nPD  1.8%\n2 3 4 5 6 7 8\nEnergy [keV]50\n050PA [deg]\nPA 29°\n0.02.55.07.510.0PD [%]\nPD  5.3%\n2 3 4 5 6 7 8\nEnergy [keV]02040PA [deg]\nPA  26°\nFig. 5. Energy dependence of the PD and PA, obtained by the pcube algorithm. Left: the pre-dip part of the observation, right: the post-dip part of\nthe observation. Data are divided into four energy bins: 2–3, 3–4, 4–5.5, and 5.5–8 keV . Uncertainties are reported at 68% CL.\n8\n 6\n 4\n 2\n 0 2 4\n [%]\n8\n6\n4\n2\n024 [%]\n2.0-3.0 keV\n3.0-4.0 keV\n4.0-5.5 keV\n5.5-8.0 keV\n2\n 0 2 4 6 8 10\n [%]\n2\n0246810 [%]\n2.0-3.0 keV\n3.0-4.0 keV\n4.0-5.5 keV5.5-8.0 keV\nFig. 6. Same as Fig. 5, but for the normalized Stokes parameters qandu. Arrows are only pointing the label of the energy band. Uncertainties are\nreported at 68% CL.\nfined with blue vertical dashed lines in the light curve in Fig. 1f).\nThe results are shown in Fig. 7 and Table 2. We performed a fit of\nthe five PA values with a straight line, see Fig. 8, and confirmed\nthat we see the clear rotation of the PA with time with a slope of\n1◦.64 per hour and the total rotation angle exceeding 70◦. We also\nnote that the PD is relatively high at the beginning and in the end\nof the observation. We note here that the points corresponding to\nBins 2, 3, and 4 in Fig. 7 have the origin within their error bars\nat 99% CL. Thus, as three individual measurements, they are not\nsignificant, yet as a part of the trend, they are still meaningful.\nLast but not least, we attempted to divide the observation into\ntwo parts by the count rate: below and above 25 cnt s−1. Data be-\nlow the threshold fall into one of the many dips we see in the\nlight curve. In the brighter state (above 25 cnt s−1), we saw a\npolarization similar to that shown in Fig. 3 and in Table 2 for\nthe overall polarization, with PD of ≈1.6% and PA of≈2◦in\nthe brighter state. The dimmer state (below 15 cnt s−1) showed\nsomething in between the dip state of the previous analysis andthe overall polarization: in the dimmer state, we saw PD ≈1.3%\nand PA≈ −36◦. The nature of the wide dip (between MJD-\n60230 =6.05 and 6.30, as shown in Fig. 1f) is probably di ffer-\nent from the nature of all other dips in the light curve, as adding\nthose to the data set changes the behavior of the polarimetric\nproperties towards the averaged polarization values.\n3.2. Spectroscopic analysis\n3.2.1. XMM–Newton\nIn the XMM–Newton data, below 2 keV , the count rate drops\nsignificantly due to the high interstellar absorption. We observe\na significant excess below 2 keV with any continuum model\ncombination. Also, large residuals were seen near instrumental\nedges. These facts, in combination with the inability to subtract\nthe background, lead us to discard these low-energy data. After\nall, they do not a ffect the modeling at higher energies.\nArticle number, page 5 of 12A&A proofs: manuscript no. ixpe_AA\n4\n 2\n 0 2 4\n (%)\n4\n2\n024 (%)\nBin 1Bin 2Bin 3Bin 4Bin 5\nFig. 7. Time dependence of the normalized Stokes parameters qand\nuobtained by the pcube algorithm using the data separation into five\nequal time bins. Uncertainties are reported at 68% CL.\n0 10 20 30 40 50\nElapsed time (h)60\n40\n20\n0204060PA (deg)\nFig. 8. Linear fit of the PA obtained for time Bins 1–5 in Fig.1f. The plot\nillustrates the rotation of the polarization plane with time. Uncertainties\nare reported at 68% CL.\nThus, we modeled the spectrum of the XMM–Newton\nobservation in the 2–10 keV range. As we tradition-\nally expect two continuum components to be present in\nthe spectrum of a WMNS with some additional reflec-\ntion and absorption features, we performed a fit with a\ntbabs*(diskbb+bbodyrad+gaussian) model modified by\nfive gaussian absorption lines, gabs , and two interstellar dust\nabsorption edges, edge . The results are shown in Table 3 and in\nFig. 9. To resolve the absorption features in the 6–9 keV energy\nrange properly, we chose to fix the parameters of the gaussian\ncorresponding to the broad emission line.\nThe resulting fit illustrates that the spectrum of GX 13 +1 is\ndominated by the disk. The harder component is modeled with\nthe blackbody (as suggested in Schnerr et al. 2003; Díaz Trigo\net al. 2012; Revnivtsev et al. 2013) and dominates at energies\nhigher than 7 keV . We also see the presence of four strong ab-\nsorption lines at 6.90, 7.18, 8.10, and 8.48 keV . These lines are\nthe Fe XXV , Fe XXVI, Ni XXVII, and Ni XXVIII lines, and\nthey are well-known in the source (see, e.g., Tomaru et al. 2020);\nthree more of the known lines are not resolved in our fit. If com-\npared with the previous results, the energies of the lines we ob-\ntain are higher than in Díaz Trigo et al. (2012) by several percent,\nwhile for the two lines present in the Saavedra et al. (2023), the\nenergy values are almost the same. We also note the anomalously\n1234EFE (keV cm2 s1)\ndiskbb\nbbodyrad\ngaussian\nModel\nXMM-Newton\n2 3 4 5 678910\nEnergy (keV)5\n05(data-model)/error\nFig. 9. Spectral energy distribution of GX 13 +1 inEF Erepresentation.\nThe blue crosses show the data from XMM–Newton . The di fferent spec-\ntral model components are reported in black lines for diskbb (short-\ndashed), bbodyrad (long-dashed), and gaussian (dotted). The total\nmodel is shown with a solid black line. The bottom panel shows the\nresiduals between the data and the best fit. Data are rebinned only for\nplotting purposes.\nTable 3. Best-fit parameters of GX 13 +1 spectrum from XMM–Newton\ndata.\nParameter Value\ntbabs NH(1022cm−2) 5.94+0.06\n���0.07\ndiskbb kT(keV) 1 .39+0.12\n−0.06\nnorm 130+20\n−30\nbbodyrad kT(keV) 2 .0+0.3\n−0.1norm 20 .08±0.03\nRbb(km) 3 .13±0.02\ngaussian E(keV) 6 .6 (frozen)\nσ(keV) 0 .8 (frozen)\nnorm 0 .0048+0.0009\n−0.0007\ngabs E(keV) 8 .476+0.005\n−0.009σ(keV) 0 .05 (frozen)\ndepth (keV) 0 .067±0.004\ngabs E(keV) 4 .29 (frozen)\nσ(keV) 0 .05 (frozen)\ndepth (keV) 1e-6 (frozen)\ngabs E(keV) 8 .10±0.01\nσ(keV) 0 .05 (frozen)\ndepth (keV) 0 .040±0.003\ngabs E(keV) 7 .180+0.006\n−0.004σ(keV) 0 .05 (frozen)\ndepth (keV) 0 .040±0.001\ngabs E(keV) 6 .90±0.01\nσ(keV) 0 .050 (frozen)\ndepth (keV) 0 .0150+0.0017\n−0.0013edge E(keV) 8 .83 (frozen)\nτ 0.20±0.01\nedge E(keV) 9 .28 (frozen)\nτ 0.08±0.01\nχ2/d.o.f. 1504 /1483\nNotes. The errors are at 68% CL. Rbbis calculated from the normaliza-\ntion of the bbodyrad component, assuming the distance to the source\nof 7 kpc.\nArticle number, page 6 of 12Bobrikova et al.: First detection of polarized emission from GX 13 +1\n0.51.05.0 [keV cm s]\nBin 1\nBin 2Bin 3\nBin 4Bin 5\n2 3 4 5 6 7 8\nEnergy (keV)4\n2\n024(data-model)/error\nFig. 10. Spectral energy distribution of the source in EF Erepresenta-\ntion during five equal parts of the observation. Here only IXPE DU1 is\nreported as an example, and the data are rebinned for plotting purposes.\nhigh equivalent hydrogen column density parameter NHof the\nabsorption component.\nAdditionally, we tried to model the XMM–Newton spectrum\nwith only one continuum component. That is, we removed the\nbbodyrad component, and fit the whole continuum using a sin-\nglediskbb component with high kTdbb≈1.9 keV resulting in\na badχ2/d.o.f.=2058/1485 confirmed by the F-test between this\none-component model and the one described above (the F-test\nresult is 273).\n3.2.2. IXPE\nAs the HID obtained from the IXPE data in Fig. 2 has rather\nlarge uncertainties and is obtained over a small energy range, it\nis not the most informative representation of the spectral varia-\ntion of the data with time. We used xspec to fit the Ispectrum ob-\ntained by IXPE . For this study, we chose the same five equal time\nbins as for the pcube analysis presented in Fig. 7. All five spec-\ntra are fitted with the same model tbabs*(diskbb+bbodyrad) .\nConsidering the spectral capabilities of IXPE and the small ef-\nfective area above 6 keV , we did not attempt to add the gabs\norgaussian components that were prominent in the XMM–\nNewton spectrum. We multiplied the model by constant com-\nponent to scale the data from the three detectors of IXPE . We\nalso fixed the value of the equivalent hydrogen column density\nNH, asIXPE does not have the capability to estimate it, and the\ntemperature of the bbodyrad component to the best-fit values\nobtained with the XMM–Newton . We applied the weighted anal-\nysis introduced in Di Marco et al. (2023b). The results are pre-\nsented in Table 4 and Fig. 10.\nWe note the lack of change in the spectral parameters with\ntime. The temperature of the disk does not vary with time, and\nonly the normalization of the continuum components decreases\ntowards the end of the observation. In Bin 5 of the observation,\nwe also see a drop in the contribution of the blackbody compo-\nnent by a factor of two. Bin 4 almost coincides with the broad\ndip and has a lower flux, and this is seen in Fig. 10.3.3. Spectro-polarimetric analysis\nFor the spectro-polarimetric analysis, we used the results of both\nthe model-independent polarimetric analysis and the spectro-\nscopic analysis as a driver. As we learned that both spectral and\npolarimetric properties are rather similar in Bins 2–4 (see Ta-\nble 2 and Table 4), we combined these three in one bigger time\nbin. We chose to do so to obtain a stronger detection in this part\nof the observation, as in polarimetric analysis we got the results\nonly meaningful as a part of the trend for Bins 2–4. We stud-\nied Bin 1 and Bin 5 separately. Instead of studying the polar-\nization of each spectral component independently, we applied\nthepolconst model to the entire continuum, as the data quality\ndoes not allow us to determine the polarization of the two spec-\ntral components. For Bin 1 we also attempted to apply polllin\npolarimetric model, as we previously saw the dependency of PD\non energy at the beginning of the observation, see Fig. 5, left.\nWe again fixed the hydrogen column density NHand the black-\nbody temperature. The results are presented in Table 5, and the\nfit performed for the Bin 1 data with the pollin model is shown\nin Fig. 11.\nWe note that, for Bins 1 and 5, both spectral and polari-\nmetric properties obtained in the spectro-polarimetric analysis\nare in agreement with the previous results presented in Tables 2\nand 4. As we do not have strong detection for the polarimetric\nproperties of Bins 2–4, we decided not to present the spectro-\npolarimetric analysis of these bins independently in Table 5,\nas this does not improve the quality of our results. In Bin 1,\ndue to the strong dependency of PD on energy, we applied the\npollin model with a constant PA which gives an improvement\nby∆χ2=9 compared to the fit with the polconst model (that\nhas one less parameter). This improvement corresponds to the\nF-test probability of ≈3×10−3.\n4. Discussion\n4.1. Correlation of the polarimetric behavior with the dip in\nthe light curve\nThe data analysis presented in Sect. 3 provided us with several\nsignificant insights that require interpretation and further reflec-\ntion. To begin with, we observed significant and anomalous vari-\nation in the light curve: the broad dip we see closer to the end of\nthe observation has nothing to do with the regular dips known\nin the source: the folding of the MAXI data and formulae from\nIaria et al. (2014) suggest that our observation was one week\nbefore this particular feature. During the dip, we also observed\na slight increase in the hardness ratio. However, the behavior\nof the polarimetric properties is not in alignment with this vari-\nation: the PA is slowly increasing, and the PD first decreases\nand then increases after the dip. Inconsistency in the behavior\nof spectral and polarimetric properties invites us to search for\nan explanation for the PA and PD variation outside the regular\nhardness /intensity changes.\nLooking back at the results of the previous IXPE observa-\ntions of the X-ray binaries, one can see that the concept of a\nrapid and significant change in the PA is not new to this class of\nobjects. The high luminosity of GX 13 +1 allowed us to separate\nthe observation into smaller parts and see the slow monotonous\nrotation of the polarization plane, but the results presented in\nFig. 5 are similar to several previously obtained. It is tempting\nto assume that similar mechanisms could be responsible for the\npolarimetric behavior of GX 13 +1 and a slow rotation similar to\nthe one presented in Fig. 7 has not been observed previously just\nArticle number, page 7 of 12A&A proofs: manuscript no. ixpe_AA\nTable 4. Best-fit parameters of GX 13 +1 spectrum from IXPE data divided into five equal time bins.\nComponent Parameter (units) Bin 1 Bin 2 Bin 3 Bin 4 Bin 5\ntbabs NH(1022cm−2) 5.9 (frozen) 5.9 (frozen) 5.9 (frozen) 5.9 (frozen) 5.9 (frozen)\ndiskbb kTin(keV) 1.49+0.03\n−0.051.44+0.04\n−0.031.45±0.04 1.57+0.06\n−0.071.55±0.06\nnorm 131+14\n−8142+13\n−12130+10\n−1179+10\n−8102+10\n−7bbodyrad kT(keV) 2.0 (frozen) 2.0 (frozen) 2.0 (frozen) 2.0 (frozen) 2.0 (frozen)\nnorm 14±4 14+2\n−514±3 10±4 8+4\n−5\nRbb(km) 2.6 ±0.4 2.6+0.2\n−0.52.6±0.3 2.2±0.4 2 .0+0.5\n−0.6constant DU1 1 (frozen) 1 (frozen) 1 (frozen) 1 (frozen) 1 (frozen)\nDU2 0 .951±0.004 0.956±0.004 0.948±0.004 0.951±0.004 0.950±0.006\nDU3 0 .908±0.003 0.910±0.004 0.904±0.004 0.912±0.004 0.911±0.005\nχ2/dof 392 /420 415 /416 389 /411 420 /411 403 /409\nFlux 2-8 keV (10−9erg s−1cm−2) 6.42 5.98 5.77 4.96 5.65\nFlux diskbb /Flux 2-8 keV 0.82 0.83 0.81 0.81 0.91\nFlux bbodyrad /Flux 2-8 keV 0.18 0.17 0.19 0.19 0.09\nNotes. The errors are at 68% CL. A gain slope of ∼0.96 and gain o ffsets∼2–80 eV for the three DUs were needed to obtain a good reduced χ2.\nRbbis calculated from the normalization of the bbodyrad component, assuming the distance to the source of 7 kpc.\n0.11.02.0 (keV cm s)\nI\ndiskbb\nbbodyrad\nModelDU1\nDU2\nDU3\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nQ\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nU\n2 3 4 5 6 7 8\nEnergy (keV)4\n2\n024(data-model)/error\n2 3 4 5 6 7 8\nEnergy (keV)4\n2\n024\n2 3 4 5 6 7 8\nEnergy (keV)4\n2\n024\nFig. 11. Spectral energy distribution of GX 13 +1 inEF Erepresentation as observed by IXPE in Bin 1. The left, middle, and right panels are for\nthe Stokes parameters I,Q, and U, respectively. The fit is performed in the 2–8 keV energy band using the three IXPE detectors and applying\nthe model tbabs*pollin*(diskbb+bbodyrad) . The total model is shown with the solid black line, while the diskbb andbbodyrad with the\ndashed and dotted lines, respectively. The lower sub-panels show the residuals between the data and the best fit. The data are rebinned only for\nplotting purposes.\ndue to the lack of statistics or a di fferent approach used in the\ndata analysis. In Sco X-1 (La Monaca et al. 2024), for instance,\nthe rotation of the PA with respect to the previous observations\n(Long et al. 1979) and the jet position angle was around 54◦,\nand the authors suggested explaining it with the precession of\nthe jet or with the change of corona geometry with the transi-\ntion between the states. However, we see no sign of X-ray color\nvariability in GX 13 +1 that could be associated with the state\nchanges. In the case of Cir X-1 (Rankin et al. 2024), the PA\nchanged by 60◦between the two parts of a single observation,\nbut in this case, a clear sign of a corresponding state transition\nwas observed. The di fference of 60◦between the PA of hard and\nsoft components of the observation was explained by a switch\nfrom the BL to the SL in a tilted NS, as Cir X-1 is a young sys-\ntem. GX 13 +1, on the other hand, is an old stable system witha high accretion rate; chances for the rotation axis of the NS to\nbe significantly misaligned relative to the orbital axis are small.\nThe results are also, in many ways, very similar to the polariza-\ntion we usually see in the black hole X-ray binaries (see, e.g.,\nFig. S6 in Krawczynski et al. 2022). As for Cyg X-1, we see the\nPD from the higher flux rising with energy, while the PD from\na (slightly) lower flux is constant. The PA is constant with en-\nergy in both cases, but the change of PA by 60◦between the two\ncases is significantly di fferent from what we see in black holes.\nWe conclude that we need to search for a new mechanism re-\nsponsible for the polarimetric behavior of GX 13 +1.\nArticle number, page 8 of 12Bobrikova et al.: First detection of polarized emission from GX 13 +1\nTable 5. Best-fit parameters of the spectro-polarimetric model for GX 13 +1.\nComponent Parameter (units) Bin 1, polconst Bin 1, pollin Bin 2–4 Bin 5\ntbabs NH(1022cm−2) 5.9 (frozen) 5.9 (frozen) 5.9 (frozen) 5.9 (frozen)\ndiskbb kTin(keV) 1 .51+0.05\n−0.041.51+0.05\n−0.041.50+0.02\n−0.031.58+0.05\n−0.06\nnorm 120 ±11 120±11 106+6\n−592±10\nbbodyrad kT(keV) 2.0 (frozen) 2.0 (frozen) 2.0 (frozen) 2.0 (frozen)\nnorm 13+6\n−313+4\n−312±2 <14\nRbb(km) 2 .5+0.6\n−0.32.5+0.4\n−0.32.4±0.2 <2.6\npolconst A(%) 3 .0±0.5 – 1 .2±0.3 4.5±0.5\nψ(deg) −42±5 – 7 ±7 32±3\npollin A1(%) – −0.75±1.4 – –\nAslope(% keV−1) – 1 .3±0.5 – –\nψ1(deg) – −42±4 – –\nψslope(deg keV−1) – 0 (frozen) – –\nconstant DU1 1 (frozen) 1 (frozen) 1 (frozen) 1 (frozen)\nDU2 0 .953±0.004 0.953±0.004 0.956±0.002 0.950±0.004\nDU3 0 .910±0.004 0.910±0.004 0.910±0.002 0.909±0.004\nχ2/d.o.f. 634 /653 626 /652 600 /653 630 /653\nNotes. IXPE I ,Q, and Uspectra are divided into five equal time bins and regrouped based on their spectral and polarimetric behavior. The errors\nare at 68% CL. A gain slope of ∼0.97 and gain o ffsets∼130–200 eV for the three DUs were needed to obtain a good reduced χ2.Rbbis calculated\nfrom the normalization of the bbodyrad component, assuming the distance to the source of 7 kpc.\n4\n 2\n 0 2 4\n (%)\n4\n2\n024 (%)\nBin 1Bin 2Bin 3Bin 4Bin 5\nFig. 12. Time dependence of the normalized Stokes parameters qand\nu, obtained by the pcube algorithm using the data separation into five\nequal time bins. Grey points illustrate the observed Stokes parameters\nshown previously in Fig. 7. Colored points show the result of the sub-\ntraction of the (absolute) Stokes parameters corresponding to Bin 4. Un-\ncertainties are reported at 68% CL.\n4.2. Additional polarimetric component\nAs we saw in Fig. 7 the evolution of the source shows a straight\nline (within the uncertainties) in the q−uplane shifted from\nthe origin. Considering this, we can try decomposing the polar-\nized flux into di fferent components with di fferent polarization\norientations. Interpreting emission in Bin 1 and Bin 5 as com-\ning from two distinct components, di ffering in PA by 70◦, with\nthe rest representing a transitional phase, would be problematic,\nas such a transition is unlikely to result in a straight-line trajec-\ntory in the q−uplane, particularly given that the flux during\nBin 4 is lower than during Bin 1 and Bin 5. Thus we suggest\nan alternative speculation that the polarization nature in Bin 1\nand Bin 5 is predominantly the same, and it is the misaligned\npolarization component that is observed in between. A similarmodel that involves a constant polarized component has been\npreviously proposed for the interpretations of X-ray polarization\ndata from a bright X-ray pulsar LS V +44 17 /RX J0440.9 +4431\n(Doroshenko et al. 2023).\nWe define a constant component corresponding to Bin 4 (a\nperiod nearly overlapping with the dip, as shown in Fig. 1f).\nThus we subtract absolute Stokes parameters corresponding to\nBin 4 from the absolute Stokes parameters of each time Bin.\nThe resulting normalized Stokes parameters are shown in Fig. 12\nwith colored points, while the originally observed values are\nshown with grey points. We see that their positions are still con-\nsistent with a straight line which is now going near the origin.\nIn this scenario, the polarization vectors during Bin 1 and Bin 5\nare essentially perpendicular to each other (since they lie on the\nsame line in the q−uplane at di fferent sides around the origin),\nand the radiation observed in Bin 2 and Bin 3 appears statisti-\ncally consistent with being unpolarized.\nThe picture we see after the subtraction is physically mean-\ningful. A 90◦difference in PAs of Bin 1 and Bin 5 could be at-\ntributed to changes in the geometry of the scattering or reflecting\nmedium, change in the scattering optical depth, and the relative\nbrightness changes of the accretion disk, the BL, and the SL.\nIn order to explain the rotation of the PA by 70◦, we need to\nassume that the PA of the constant component di ffers from the\nPA of the variable component by 30◦–40◦. If we associate the\nconstant component with the scattering in the wind (see below)\nand the variable component with the BL /SL, we are forced to\nconclude that there needs to be a strong misalignment of the NS\nspin from the orbital axis.\n4.3. Influence of the variations in the wind\nOne other possible explanation of the rotation of the polarization\nplane with time could be the impact of the orbital movement on\nthe polarization arising from the scattering in the accretion bulge.\nWith the high accretion rate known in GX 13 +1, one can expect\nsome asymmetry in the accretion disk surrounding the NS. How-\never, GX 13 +1 is recognized for having the largest disk among\nall observed LMXBs. Therefore, if a bulge exists, it should be\nArticle number, page 9 of 12A&A proofs: manuscript no. ixpe_AA\nquite distant from the source. Another thing to note is that we\nobserved a rotation of the polarization plane by ≈70◦in the two\ndays of observation. The orbital period of GX 13 +1 is 24.5 days.\nThe unknown mechanism responsible for the rotation of the po-\nlarization plane would need to operate on a smaller timescale.\nGX 13 +1 is known to have a strong and rapidly changing\nwind, and we see the evidence of the absorption in the wind in\ntheXMM–Newton observation. Scattering in the equatorial wind,\nin theory, can produce di fferent polarization patterns, and with\nthis mechanism and with the high inclination of the source, rel-\natively high polarization can be easily produced (see Sunyaev &\nTitarchuk 1985; Tomaru et al. 2024; Nitindala et al., in prep.).\nThe nature of the polarization variability could be that the ob-\nserved emission is scattered in the di fferent parts of the wind\nabove the accretion disk. As di fferent absorption features are as-\nsumed to come from several distinct parts of the wind with vari-\nous physical, optical, and geometrical properties, we can expect\nsignificant di fferences in the polarimetric behavior. If the line-of-\nsight passes close to the edge of the wind, then small variations\nin the wind opening angle would lead to large variations in the\ncontribution of the scattered component and the polarization de-\ngree. We hope to study the correlation between the behavior of\nthe wind and polarimetric properties in the future with simulta-\nneous XRISM andIXPE observations.\n4.4. Spectral features\nComparison of the spectrum observed with XMM–Newton with\nthe previous campaigns on GX 13 +1 reported in Díaz Trigo et al.\n(2012) and Saavedra et al. (2023) allowed us to assume that the\nstate of the source was somewhere between the one of Obs. 6\nand Obs. 7 in Díaz Trigo et al. (2012). However, these two ob-\nservations are separated by 11 days, so it is hard to predict the\nspectrum of the source during the IXPE observation.\nAll the studies we performed on the spectra of GX 13 +1 con-\nfirmed the original results coming from the HID: there are no\nsignificant changes in the spectral behavior with time. The study\nof the radius of the blackbody component Rbbshowed no vari-\nation in the geometry throughout the observation, with only a\nslight decrease in the last two bins. In Bin 5, we also observed a\ndrop in the contribution of this component to the total flux. This\nadds a challenge to the data interpretation. If we assume that in\nBin 5, the polarization is produced by the dominating soft disk\ncomponent, we need to find a way for the disk emission to be\npolarized at the detected level. The PD coming from the disk is\nestimated at up to 4% for the inclination of 80◦and up to only\n1.5% for the inclination of 60◦(Loktev et al. 2022) assuming\nsemi-infinite electron-scattering dominated atmosphere (Chan-\ndrasekhar 1960). This means some scattering in the wind above\nthe disk might need to be introduced. As we have strong absorp-\ntion features in the spectrum, and GX 13 +1 is known to have a\nstrong wind with several absorbing areas, we can assume that the\nwind adds to the polarization of this component.\nFor the harder component, we can assume both the SL and\nthe BL here. Optically thick SL can produce polarization up to\n1.5%, and optically thin one can produce PD up to 6% (Bo-\nbrikova et al, in prep.), yet it is challenging to understand the\nnature of an optically thin SL in the source with a high and sta-\nble accretion rate.\nWe also explored the possibility that the two continua\ncomponents ( diskbb and bbodyrad ) each have di fferent\npolarization properties by fitting the data with the xspec model\ntbabs*(polconst*diskbb+polconst*bbodyrad)*const .\nWe only attempted to apply this model to Bin 1, because thisis the only one with the strong dependence of PD on energy,\nso it would be easier to resolve the two components. The fit\nsuggested that the diskbb component is almost unpolarized (not\nsignificant polarization with the PD <2.9% is obtained at 99%\nCL), and the bbodyrad component has the PD of 10% and the\nPA of−42◦. However interesting, this suggestion makes it even\nharder to understand the relatively high polarization in Bin 5,\nwhere we see the disk completely dominating the spectrum.\nThe results of the observation, promising and exciting, require\nfurther study supported by the modeling of the emission coming\nfrom di fferent components as well as scattered in the wind\nabove the disk.\n5. Summary\nIn the present article, we report the highly significant detection\nof polarization from GX 13 +1 obtained by IXPE . We performed\nthe polarimetric, spectroscopic, and spectro-polarimetric analy-\nsis of the IXPE data. The two main components of the spec-\ntrum, soft disk emission and harder comptonized component, we\ndescribed by the diskbb andbbodyrad models, respectively.\nSpectral analysis was supported by a non-simultaneous observa-\ntion performed by XMM–Newton . Due to the high energy reso-\nlution of XMM–Newton , we added the gaussian component to\ndescribe the broad emission line corresponding to the reflection\nof the NS surface emission from the disk, five Gaussian absorp-\ntion lines gabs and two edges edge . For di fferent parts of the\nobservation, we used polconst andpollin models for the po-\nlarimetric properties. Overall PD of the IXPE data was measured\nat 1.4% at >5σCL, and PA at−2◦.\nFrom the in-depth study of the time variability of the po-\nlarimetric properties, we reported the strong rotation of the PA\nby 70◦during the two days of the observation. The PD varied\nfrom 3.2% to the non-detectable and then up to 5 .0%. More-\nover, we report the change of dependency of the PD on energy:\nat the beginning of the observation, we detect polarization lin-\nearly dependent on energy, while in the middle and at the end\nof the observation the PD is constant with energy. Some of the\nprevious studies of the WMNSs with IXPE reported the rapid\nswitch of the PA by up to 60◦, yet the persistent rotation of the\npolarization plane is presented for the first time in this paper.\nWe suggested several ways to interpret the results of the data\nanalysis. We studied the correlation of the dip in the light curve\nwith the polarimetric properties and concluded that there is no\ndirect connection between them. We defined a constant polar-\nized component and subtracted it from all the observations and\nconcluded that the second, time variable component then is ex-\npected to switch sign of the PD (i.e. experience rotation of the\nPA by 90◦) in the middle of the observation. We also discussed\nthe possible impact of the wind and the unusual state in which\nwe found the source. The observed variations in the PA likely\nimply that there is a strong (30◦–40◦) misalignment between the\nNS spin and the orbital axis. Future polarimetric observations\nwill help uncover the nature of this unique phenomenon.\nAcknowledgements. The Imaging X-ray Polarimetry Explorer (IXPE) is a joint\nUS and Italian mission. The US contribution is supported by the National Aero-\nnautics and Space Administration (NASA) and led and managed by its Mar-\nshall Space Flight Center (MSFC), with industry partner Ball Aerospace (con-\ntract NNM15AA18C). The Italian contribution is supported by the Italian Space\nAgency (Agenzia Spaziale Italiana, ASI) through contract ASI-OHBI-2022-\n13-I.0, agreements ASI-INAF-2022-19-HH.0 and ASI-INFN-2017.13-H0, and\nits Space Science Data Center (SSDC) with agreements ASI-INAF-2022-14-\nHH.0 and ASI-INFN 2021-43-HH.0, and by the Istituto Nazionale di Astrofisica\n(INAF) and the Istituto Nazionale di Fisica Nucleare (INFN) in Italy. This re-\nsearch used data products provided by the IXPE Team (MSFC, SSDC, INAF,\nArticle number, page 10 of 12Bobrikova et al.: First detection of polarized emission from GX 13 +1\nand INFN) and distributed with additional software tools by the High-Energy\nAstrophysics Science Archive Research Center (HEASARC), at NASA God-\ndard Space Flight Center (GSFC). We thank the XMM–Newton Project Scientist\nfor approving our DDT request to observe GX 13 +1. The research leading to\nthese results has received funding from the European Union’s Horizon 2020 Pro-\ngramme under the AHEAD2020 project (grant agreement 871158). This research\nhas been supported by the Academy of Finland grants 333112, 349144, and\n355672 (AB, SVF, JP, VL, A V , APN, SST) and the German Academic Exchange\nService (DAAD) travel grant 57525212 (VD). 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S. 1982, ApJ, 257, 318\nXRISM Science Team. 2022, arXiv e-prints, arXiv:2202.053991Department of Physics and Astronomy, FI-20014 University of\nTurku, Finland\ne-mail: anna.a.bobrikova@utu.fi\n2INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso\ndel Cavaliere 100, 00133 Roma, Italy\n3Dipartimento di Fisica, Università degli Studi di Roma “Tor Ver-\ngata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy\n4Dipartimento di Fisica, Università degli Studi di Roma “La\nSapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy\n5MIT Kavli Institute for Astrophysics and Space Research, Mas-\nsachusetts Institute of Technology, 77 Massachusetts Avenue, Cam-\nbridge, MA 02139, USA\n6Serco for the European Space Agency (ESA), European Space As-\ntronomy Centre, Camino Bajo del Castillo s /n, E-28692 Villanueva\nde la Cañada, Madrid, Spain\n7Dipartimento di Matematica e Fisica, Università degli Studi Roma\nTre, via della Vasca Navale 84, 00146 Roma, Italy\n8Nordita, KTH Royal Institute of Technology and Stockholm Univer-\nsity, Hannes Alfvéns väg 12, SE-106 91, Sweden\n9Anton Pannekoek Institute for Astronomy & GRAPPA, Univer-\nsity of Amsterdam, Science Park 904, 1098 XH Amsterdam, The\nNetherlands\n10Centre for Extragalactic Astronomy, Department of Physics,\nDurham University, South Road, Durham DH1 3LE, UK\n11Kavli Institute for Physics and Mathematics of the Universe (WPI),\nUniversity of Tokyo, Kashiwa, Chiba 277-8583, Japan\n12University of British Columbia, Vancouver, BC V6T 1Z4, Canada\n13NASA Marshall Space Flight Center, Huntsville, AL 35812, USA\n14Instituto de Astrofísica de Andalucía – CSIC, Glorieta de la As-\ntronomía s /n, 18008 Granada, Spain\n15INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00040\nMonte Porzio Catone (RM), Italy\n16Space Science Data Center, Agenzia Spaziale Italiana, Via del Po-\nlitecnico snc, 00133 Roma, Italy\n17INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5,\n09047 Selargius (CA), Italy\n18Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo B. Pon-\ntecorvo 3, 56127 Pisa, Italy\n19Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo 3,\n56127 Pisa, Italy\n20Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via Pietro\nGiuria 1, 10125 Torino, Italy\n21Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro\nGiuria 1, 10125 Torino, Italy\n22INAF Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5,\n50125 Firenze, Italy\n23Dipartimento di Fisica e Astronomia, Università degli Studi di\nFirenze, Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy\n24Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Via San-\nsone 1, 50019 Sesto Fiorentino (FI), Italy\n25Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma,\nItaly\n26Science and Technology Institute, Universities Space Research As-\nsociation, Huntsville, AL 35805, USA\n27Istituto Nazionale di Fisica Nucleare, Sezione di Roma “Tor Ver-\ngata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy\n28Department of Physics and Kavli Institute for Particle Astrophysics\nand Cosmology, Stanford University, Stanford, California 94305,\nUSA\n29Institut für Astronomie und Astrophysik, Universität Tübingen,\nSand 1, D-72076 Tübingen, Germany\n30Astronomical Institute of the Czech Academy of Sciences, Bo ˇcní II\n1401/1, 14100 Praha 4, Czech Republic\n31RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako,\nSaitama 351-0198, Japan\n32X-ray Astrophysics Laboratory, NASA Goddard Space Flight Cen-\nter, Greenbelt, MD 20771, USA\n33Yamagata University,1-4-12 Kojirakawa-machi, Yamagata-shi 990-\n8560, Japan\nArticle number, page 11 of 12A&A proofs: manuscript no. ixpe_AA\n34Osaka University, 1-1 Yamadaoka, Suita, Osaka 565-0871, Japan\n35International Center for Hadron Astrophysics, Chiba University,\nChiba 263-8522, Japan\n36Institute for Astrophysical Research, Boston University, 725 Com-\nmonwealth Avenue, Boston, MA 02215, USA\n37Department of Astrophysics, St. Petersburg State University, Uni-\nversitetsky pr. 28, Petrodvoretz, 198504 St. Petersburg, Russia\n38Department of Physics and Astronomy and Space Science Center,\nUniversity of New Hampshire, Durham, NH 03824, USA\n39Physics Department and McDonnell Center for the Space Sciences,\nWashington University in St. Louis, St. Louis, MO 63130, USA\n40Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Strada Co-\nmunale Cinthia, 80126 Napoli, Italy\n41Université de Strasbourg, CNRS, Observatoire Astronomique de\nStrasbourg, UMR 7550, 67000 Strasbourg, France\n42Graduate School of Science, Division of Particle and Astrophysical\nScience, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi\n464-8602, Japan\n43Hiroshima Astrophysical Science Center, Hiroshima University, 1-\n3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan\n44Department of Physics and Astronomy, Louisiana State University,\nBaton Rouge, LA 70803, USA\n45Department of Physics, University of Hong Kong, Pokfulam, Hong\nKong\n46Department of Astronomy and Astrophysics, Pennsylvania State\nUniversity, University Park, PA 16801, USA\n47Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France\n48Center for Astrophysics, Harvard & Smithsonian, 60 Garden St,\nCambridge, MA 02138, USA\n49INAF Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807\nMerate (LC), Italy\n50Dipartimento di Fisica e Astronomia, Università degli Studi di\nPadova, Via Marzolo 8, 35131 Padova, Italy\n51Department of Astronomy, University of Maryland, College Park,\nMaryland 20742, USA\n52Mullard Space Science Laboratory, University College London,\nHolmbury St Mary, Dorking, Surrey RH5 6NT, UK\n53Guangxi Key Laboratory for Relativistic Astrophysics, School of\nPhysical Science and Technology, Guangxi University, Nanning\n530004, China\nArticle number, page 12 of 12"    },    {        "title": "2401.13070v1.Chaos_and_quantization_of_the_three_particle_generic_Fermi_Pasta_Ulam_Tsingou_model_II__phenomenology_of_quantum_eigenstates.pdf",        "content": "Chaos and quantization of the three-particle generic Fermi-Pasta-Ulam-Tsingou\nmodel II: phenomenology of quantum eigenstates\nHua Yan∗and Marko Robnik†\nCAMTP - Center for Applied Mathematics and Theoretical Physics,\nUniversity of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia, European Union\n(Dated: January 25, 2024)\nWe undertake a thorough investigation into the phenomenology of quantum eigenstates, in the\nthree-particle FPUT model. Employing different Husimi functions, our study focuses on both the\nα-type, which is canonically equivalent to the celebrated H´ enon-Heiles Hamiltonian, a nonintegrable\nand mixed-type system, and the general case at the saddle energy where the system is fully chaotic.\nBased on Husimi quantum surface of sections (QSOS), we find that in the mixed-type system, the\nfraction of mixed eigenstates in an energy shell [ E−δE/2, E+δE/2] with δE≪Eshows a power-\nlaw decay with respect to the decreasing Planck constant ℏ. Defining the localization measures in\nterms of the R´ enyi-Wehrl entropy, in both the mixed-type and fully chaotic systems, we find a better\nfit with the beta distribution and a lesser degree of localization, in the distribution of localization\nmeasures of chaotic eigenstates, as the controlling ratio αL=tH/tTbetween the Heisenberg time\ntHand the classical transport time tTincreases. This transition with respect to αLand the power-\nlaw decay of the mixed states, together provide supporting evidence for the principle of uniform\nsemiclassical condensation (PUSC) in the semiclassical limit. Moreover, we find that in the general\ncase which is fully chaotic, the maximally localized state, is influenced by the stable and unstable\nmanifold of the saddles (hyperbolic fixed points), while the maximally extended state notably avoids\nthese points, extending across the remaining space, complementing each other.\nPACS numbers: 01.55.+b, 02.50.Cw, 02.60.Cb, 05.45.Pq, 05.45.Mt\nI. INTRODUCTION\nThis paper is a continuation of our recent paper (pa-\nper I) [1] in which we have studied in detail classical and\nquantum aspects of the well known paradigmatic Fermi-\nPasta-Ulam-Tsingou (FPUT) model [2]. We are special-\nizing on the few-body case, the three-particle FPUT sys-\ntem, which is reduced to one particle in two-dimensional\npotential, described as two identical harmonic oscillators\ncoupled by cubic and quartic potential terms. This is so\ncalled α-FPUT (system) in the case of purely cubic cou-\nplings (it is precisely the celebrated H´ enon-Heiles system\n[3]), or β-FPUT (in the case of purely quartic couplings),\nandαβ-FPUT in the general case.\nOur focus in paper I was the relevance of classical\nchaos, in particular of the mixed-type structure of the\nphase space and the degree of chaos as measured by the\nsmaller alignment index (SALI) method [4–8], for the\nquantal energy spectra and their statistical properties.\nWe analyzed the semiclassical limiting behavior of the\nenergy spectra. First, our explicit analytic results for\nthe quantum density of states (DOS), using the Thomas-\nFermi rule, are derived and compared with the exact (nu-\nmerical) quantum DOS. They perfectly agree, as well as\nthey do agree with the DOS based on the quantum typi-\ncality method [9–11]. Next, the energy spectra were stud-\nied, in the case of mixed-type classical (divided) phase\nspace. In the case of no quantum (or dynamical) localiza-\n∗yanhua@ustc.edu.cn\n†Robnik@uni-mb.sition the Berry-Robnik (BR) picture [12] is confirmed, the\nlevel spacing distribution follows the BR formula: The\nextracted quantum BR parameter (the relative size of\nthe chaotic component) agrees with the classical value\nwithin better than one percent. If the localization effects\nof the chaotic eigenstates are included, the level spacing\ndistribution is generalized to Berry-Robnik-Brody (BRB)\ndistribution, as the localized chaotic eigenstates are (em-\npirically) well captured by the Brody level spacing distri-\nbution [13–15]. The BRB distribution is found to excel-\nlently fit the numerical distribution. In this way we have\nfollowed the classical and quantum transition from reg-\nularity to chaos, by increasing the energy of the system.\nThis transition also has been described by the ratio of\nconsecutive level spacings (so-called spacing ratio, which\ndoes not depend on the unfolding procedure).\nIn the present paper we study in detail the structure\nand statistical properties of the eigenstates, by means\nof the Husimi functions, which are Gaussian smoothed\nWigner functions, or equivalently, absolute squared pro-\njections of the eigenstates on the coherent states. We\nstudy the quantum Husimi surfaces of section (QSOS)\n[16, 17], which are precise analog of the classical Poincar´ e\nSOS, as well as various projections of the Husimi func-\ntions. In doing so, the classical structure of the mixed-\ntype phase space is clearly revealed. We use the overlap\nindex M, introduced by Batisti´ c and Robnik [18], which\nmeasures the degree of the overlap of the QSOS with the\nclasical regular ( M=−1) or chaotic ( M= +1) region,\nor it measures the partial overlap with both regions such\nthat|M|<1. Using this index we look at its distribu-\ntion at given energy (in a very narrow energy shell, for the\nfirst time, in a genuinely continuous Hamiltonian systemarXiv:2401.13070v1  [quant-ph]  23 Jan 20242\nwith a smooth potential) and sufficiently small Planck\nconstant, comprising about 1200 eigenstates. We show\nagain that the fraction of mixed states with |M|<1 de-\ncreases as a power law with decreasing Planck constant,\nlike it was shown before in billiards [19], kicked top [20]\nand Dicke model [21].\nUsing the index Mwe separate the chaotic eigen-\nstates M≥0.8 (those which live inside the classical\nchaotic component, and can be localized there) and study\ntheir quantum phase space localization phenomena. We\nintroduce various entropy localization measures (ELM)\nbased on α-R´ enyi-Wehrl entropy: The information en-\ntropy at α= 1 and inverse participation ratio at α= 2.\nThen we study the statistical properties of ELM, show-\ning that their distribution is well captured by the beta\ndistribution, in perfect agreement with the results of the\nsame analysis in many other systems, e.g. billiard sys-\ntems [19, 22–24], kicked top [25] and Dicke model [26].\nWe also observe that most localized eigenstates in the\nfully chaotic (ergodic) general αβ-FPUT system, with\nλ= 1/16 at the energy E= 1/3, are strongly local-\nized around the hyperbolic fixed point and its stable and\nunstable manifolds, contributing to the lower tail of the\nlocalization measure distribution. This is an analogy of\nthe strong localization in the classical stickiness regions\nobserved in the ergodic lemon billiard [23]. This effect,\nhowever, is expected to disappear in the strict semiclassi-\ncal limit, at smaller Planck constant, according to PUSC.\nThe paper is structured as follows. In Sec. II we de-\nfine the details of the classical FPUT system, specialize\nto the three particle case and perform the quantization\nprocedure by introducing the rotated bosonic operators\nand the corresponding circular two-mode basis and re-\nlating them to the Cartesian-mode basis. In Sec. III\nwe define the Husimi functions, the quantum surface of\nsection, i.e. Husimi QSOS, as well as the various pro-\njections of the Husimi function. In Sec. IV we introduce\nthe overlap index M, look at its distributions and demon-\nstrate the power-law decay of the fraction of the mixed\neigenstates with the decreasing Planck constant. In Sec.\nV we introduce the localization measures based on the\nR´ enyi-Wehrl entropy, and look at their distribution, also\nin relation to the controlling parameter, the ratio of the\nHeisenberg time and the classical transport time. In Sec.\nVI we conclude and discuss the results and the outlook.\nIn the Appendices A-D we present mathematical details,\nas well as the gallery of eigenstates, represented by the\nHusimi functions both for the H´ enon-Heiles system and\nthe general FPUT system.\nII. HAMILTONIAN AND CIRCULAR\nTWO-MODE BASIS\nIn this section, we present a brief overview of the\ncanonically equivalent Hamiltonian that governs the clas-\nsical dynamics of a three-particle FPUT, and its quan-\ntization using rotated bosonic operators. Then, for thediscussion of properties of circular two-mode basis, in\nwhich the matrix form of Hamiltonian is expressed, we\nderive a Wigner d-matrix decomposition of the unitary\ntransformation from this circular basis to the Cartesian\ntwo-mode basis.\nA. Canonically Equivalent Hamiltonian of generic\nthree-particle FPUT\nThe generic three-particle FPUT is a chain of N= 3\nmoving particles with nearest neighbor interaction given\nby a potential V, the Hamiltonian of such a system is\ngiven by\nH=NX\nj=1\u0012y2\ni\n2+V(xj+1−xj)\u0013\n, (1)\nwith the periodic boundary condition (PBC) x1=xN+1,\nwhere xjis the displacement of the particles with respect\nto the equilibrium positions and yjis the corresponding\nmomentum, and the potential is\nV(s) =1\n2s2+α\n3s3+β\n4s4. (2)\nα-FPUT refers to the case β= 0, while in the β-FPUT\ncaseα= 0. In paper I, after transforming to the normal\nmode representation, we have rescaled the Hamiltonians\nin the center of mass frame and obtained the canonically\nequivalent Hamiltonians\nHα=1\n22X\ni=1(p2\ni+q2\ni) +α(q2\n1q2−1\n3q3\n2) (3)\nforα-FPUT, which is also known as the H´ enon-Heiles\nHamiltonian, and\nHβ=1\n22X\ni=1(p2\ni+q2\ni) +3β\n4(q2\n1+q2\n2)2(4)\nforβ-FPUT, while for the general case ( αβ-FPUT)\nH=1\n22X\ni=1(p2\ni+q2\ni) +q2\n1q2−1\n3q3\n2+λ(q2\n1+q2\n2)2,(5)\nwhere we denote the coupling parameter of the quartic\nterm as λ= 3β/(4α2). In Fig. 1 we show several exam-\nples of classical Poincar´ e surface of section (SOS).\nFor the quantization, we define ˆ q±= ˆq1±iˆq2,ˆp±=\nˆp1±iˆp2, and introduce the rotated bosonic operators as\na±=1√\n2(a1∓ia2) =1\n2√\nℏ(ˆq∓+iˆp∓),\na†\n±=1√\n2(a†\n1±ia†\n2) =1\n2√\nℏ(ˆq±−iˆp±),(6)3\nFIG. 1. Classical Poincar´ e surface of section for α-FPUT with\nα= 1 at two energies, E= 0.14 (a) and E= 0.16 (b), and for\nthe general case with λ= 1/16 at E= 1/3 the saddle energy\n(c), generated by one single chaotic orbit.\nwhere the annihilation and creation operator fulfill the\ncanonical commutation relations [ ai, a†\nj] = δij, with\ni, j∈ {1,2,±}. We can then check that for ˆ nthe\nnumber operator and the angular momentum operator\nˆℓ= (ˆq1ˆp2−ˆq2ˆp1)/ℏfulfill\nˆn=a†\n1a1+a†\n2a2=a†\n+a++a†\n−a−,\nˆℓ=i(a†\n2a1−a†\n1a2) =a†\n+a+−a†\n−a−,(7)\nand the quantized Hamiltonian of α-FPUT can be com-\npactly written as\nˆHα=ℏ(ˆn+ 1)−iα(ˆq3\n+−ˆq3\n−)/6, (8)\nwhile for the general case\nˆH=ℏ(ˆn+ 1)−i(ˆq3\n+−ˆq3\n−)/6 +λ(ˆq+ˆq−)2. (9)\nIn circular two-mode basis states |n, l⟩, defined as\nthe simultaneous eigenfunctions of ˆ nandˆℓ, where l=\n−n,−n+ 2,···, n(n∈N0),nis the radial quantum\nnumber and lis the orbital angular momentum (OAM),\nwe have the expressions for matrix elements of the cubic\nand quartic coupling terms (see Appendix A). The cubic\nterm in quantum α-FPUT couples states with ∆ l=±3,\ntherefore the coupling takes place in three decoupled sets\nof basis states: the singlet {a}l∈ {···− 6,−3,0,3,6. . .},\ntwo doublets with the same eigenspectra {b}l∈ {··· −\n5,−2,1,4. . .}and{c}l∈ {··· − 4,−1,2,5. . .}. This\nproperty of eigenspectra can also be verified from the\nC3vsymmetry of the classical Hamiltonian, that they\nmust belong to the irreducible representations of the\npoint symmetry group C3v: the subspaces of two dou-\nblets are of Esymmetry and degenerate, and the singlet\nis a combination of ( A1,A2) symmetry. The quartic term\nin quantum β-FPUT does not introduce any coupling\nbetween different l, due to the conservation of angular\nmomentum.\nB. Properties of the circular two-mode basis\nGiven the definition of rotated-bosonic operators and\nthe corresponding circular two-mode basis |n, l⟩, the Fockspace can be spanned as\n|n, l⟩=|n++n−, n+−n−⟩=(a†\n+)n+(a†\n−)n−\np\nn+!n−!|00⟩,(10)\nwhere |00⟩is two-mode vacuum and n=n++n−, l=\nn+−n−. The interpretation of these relations is direct:\nˆn+=a†\n+a+,ˆn−=a†\n−a−,ˆℓ= ˆn+−ˆn−,(11)\nˆn+stands for counterclockwise rotation with positive an-\ngular momentum, and ˆ n−refers to clockwise rotation\nwith negative angular momentum.\nThe expansion of Eq. (10) defines the unitary trans-\nformation Ω between the circular-mode basis |n, l⟩and\nthe Cartesian-mode basis |n1, n2⟩\n|n, l⟩=X\nn1,n2Ωnl\nn1n2|n1, n2⟩, n=n1+n2, (12)\nwhere Ωnl\nn1n2, the elements of Ω, are all sums of many\nfactorials of large numbers with alternating signs for n≫\n1, because of which a direct numerical evaluation would\nsuffer from serious errors and instability at large n, very\nsimilar to the longstanding problem of the evaluation of\nthe Wigner d-matrix at high spins [27].\nTo avoid these large factorials in the numerical evalua-\ntion, a further analytical expression of Ωnl\nn1n2is essential.\nSetting n= 2j, l= 2m, in Appendix B we have proven\nthat\n|n, l⟩:=|2j,2m⟩=jX\nm′=−jOj\nm,m′|j+m′, j−m′⟩,(13)\nwith the correspondence\nΩnl\nn1n2=Oj\nm,m′, n1=j+m′, n2=j−m′, (14)\nwhere the coefficients Oj\nm,m′are particular instances of\nthe rotation matrix elements, which we write as\nOj\nm,m′=im′−jdj\nm,m′(π/2)\n=im′−j⟨jm|e−iπJy/2|jm′⟩, (15)\nwhere dj\nm,m′(θ) is the Wigner d-matrix, Jyis one gener-\nator of the Lie algebra of SU(2) and SO(3), component\nof the angular momentum operator. Oj\nm,m′then can be\ncalculated quite effectively by the sparse matrix expo-\nnential, using the kernel polynomial method [10].\nIII. HUSIMI FUNCTION\nAs a Gaussian coarse graining of the Wigner func-\ntion, Husimi function is a powerful tool to study the\nquantum-classical correspondence in quantum systems.\nEquivalently, it can be defined as the projection of the4\nwave function onto the coherent state. In quantum three-\nparticle FPUT, for the k-th energy eigenstate |Ek⟩, the\nHusimi function is given by\nHk(α1, α2)≡ H k(q1, p1;q2, p2) =|⟨α1, α2|Ek⟩|2,(16)\nwhere |α1, α2⟩=|α1⟩⊗|α2⟩is the product coherent state,\nand\n1\nπ2Z\n|α1, α2⟩⟨α1, α2|d2α1d2α2=I. (17)\nThe coherent state |αi⟩defined as ai|αi⟩=αi|αi⟩with\nαi=p\n1/2ℏ(qi+ipi), can be expanded in terms of Fock\nstates as follows\n|αi⟩=e−1\n2|αi|2∞X\nni=0αni\ni√ni!|ni⟩, i= 1,2. (18)\nThe Husimi function for |Ek⟩can be then written as\nHk(α1, α2) =|X\nn1,n2Bk\nn1n2⟨α1|n1⟩⟨α2|n2⟩|2,\nBk\nn1n2=X\nn,lCk\nnlΩnl\nn1n2=X\nj,m,m′Ck\njmOj\nm,m′,(19)\nwhere Ck\njm:=Ck\nnl=⟨n, l|Ek⟩are the expanding coeffi-\ncients. From the properties of circular two-mode basis,\nwe have verified in Appendix C that the Husimi function\ncan be equivalently expressed as\nHk(α1, α2) =|X\nn,lCk\nnl⟨α+, α−|n, l⟩|2, (20)\nwhere α±= (α1∓iα2)/√\n2, and\n⟨α+, α−|n, l⟩:=(α∗\n+)n+(α∗\n−)n−\np\nn+!n−!e−(|α+|2+|α−|2)/2,(21)\nprovides coherent state |α±⟩defined to be the eigenstate\nof the (rotated) annihilation operator a±.\nA. Husimi Quantum Poincar´ e surface of section\nTo facilitate a comprehensive comparison with classi-\ncal calculations, especially the classical SOS, the quan-\ntum Poincar´ e surface of section (QSOS) based on Husimi\nfunction as the quantum analog of the classical SOS is\nnecessary to be defined. Similar to the classical SOS on\nq1= 0 plane with p1>0, one can define the (normalized)\nHusimi QSOS namely as\n¯Qk(α2) =1\nAkHk(¯α1, α2),¯α1= (¯q1+i¯p1)/√\n2ℏ,(22)\nwhere Ak=1\nπR\nSHk(¯α1, α2)d2α2is a normalization con-\nstant and Sdenotes the surface of section of the q1= 0\nplane at Ek. ¯q1= 0, ¯ p1=p+\n1(q1= 0, q2, p2, Ek) is thepositive classical momentum obtained from the classical\nHamiltonian at the surface of section,\np+\n1=p\n2Ek−2H(q1= 0, p1= 0, q2, p2). (23)\nThe resulting Husimi QSOS of |Ek⟩then reads\n¯Qk(α2) =1\nAk|X\nn1,n2Bk\nn1n2⟨¯α1|n1⟩⟨α2|n2⟩|2. (24)\nIt can also be obtained directly from the circular basis\naccording to the expression given in Eq. (20)\n¯Qk(α2) =1\nAk|X\nn,lCk\nnl⟨¯α+,¯α−|n, l⟩|2, (25)\nwhere ¯ α±= (¯α1∓iα2)/√\n2. The Husimi QSOS proves\nvaluable for investigating quantum systems with classical\ncorrespondence. It can serve to unveil the phenomenol-\nogy of quantum eigenstates in mixed-type systems, such\nas the study of statistical properties of the localization\nmeasure of chaotic eigenstates, and the clarification of\nthe effects of stickiness in quantum states in billiard sys-\ntems [22–24].\nIn Fig. 2( a1)-(a6) we show the Husimi QSOS of\nsix consecutive eigenstates in the energy interval [ E−\nδE/2, E+δE/2] at E= 0.14 with δE/E ≃0.0014, for\nthe singlet of α-FPUT with α= 1 (in all the follow-\ning we set α= 1 as default), where Nthe cutoff of n\nis 600 and ℏ= 1×10−3. The narrow energy window\nfulfills δE/E ≪1, and this ensures that these consec-\nutive levels are associated with the same classical SOS\nofE= 0.14 in the semiclassical limit. From the semi-\nclassical counting the number of energy levels, there are\napproximately 16 times more levels provided above this\nenergy for the numerical calculation, enough to achieve\na good convergence of the numerical results with respect\nto the cutoff. A clear correspondence is observable when\ncomparing plots of the Husimi QSOS with the classical\nSOS shown in Fig. 1( a). For this mixed-type system, the\ndisplayed eigenstates do not solely concentrate on the\ninvariant tori within the integrable region nor do they\nevenly distribute across the chaotic area:\n(i) There are chaotic eigenstates with different degrees\nof localization, located almost totally in the chaotic\nregion such as the ones shown in Fig. 2( a1)-(a3),\nor flooding into the regular areas such as the cases\nshown in Fig. 2( a4)-(a5).\n(ii) Some eigenstates are strongly localized around reg-\nular regions as shown in Fig. 2( a6). Surrounding\nthe boundaries that separate the chaotic and regular\nregion, visible patterns of Husimi QSOS manifest as\na result of quantum tunneling.\nOne interpretation of these observations is that the\nPlanck constant we have chosen here, ℏ= 1×10−3, is not\nso deep in the semiclassical limit. In the present context,5\nFIG. 2. Examples of Husimi functions of six eigenstates in the energy interval [ E−δE/2, E+δE/2] at E= 0.14 with\nδE/E ≃0.0014, for the singlet of α-FPUT with α= 1, where the irreducible Hilbert space is of size NS= 60301, with the\ncutoff of nset to N= 600 and the Planck constant ℏ= 1×10−3. Panels ( a1)-(a6): Husimi QSOS given by Eq. (25) plotted\nin the logarithmic scale, the darkest blues show the area where ¯Qk(α2)<10−11. Panels ( b1)-(b6) are plots of the completely\nprojected Husimi functions Pk(α2) of Eq. (26) and panels ( c1)-(c6) are the projected Husimi functions from the classical energy\nshell ePk(α2) given by Eq. (27). Panels ( d1)-(d6): projected Husimi functions in configuration space ePk(q1, q2) of Eq. (30)\nplotted in linear scale. The darker colors in each panel for projected Husimi functions indicate larger density, in a linear scale.\nPanels from the same column correspond to the same eigenstate.\nwe have not shown eigenstates entirely confined within\nregular regions. A more comprehensive exploration of\nthis phenomenon is reserved for detailed exposition in\nSection IV. Further numerous illustrative examples can\nbe found in Appendix D, specifically showcased in Fig.\n13. These instances have been derived from a larger en-\nsemble of eigenstates, a procedure undertaken after in-\ntroducing the overlap index as a meaure to separate the\nchaotic and regular eigenstates. According to the prin-\nciple of uniform semiclassical condensation (PUSC) of\nWigner or Husimi functions [28–33], the mixed states\nshould disappear in the semiclassical limit according to a\npower law described below, leaving behind only regular\nand chaotic states.\nFrom Fig. 3, a replot with enhanced resolution of\nFig. 2( a1), we have another important observation: the\nnon-uniformity (localization) of chaotic eigenstates is at-\ntributed to the classical boundaries between regular and\nchaotic regions, where variations in the magnitude of the\nHusimi QSOS are evident across these boundaries. Fur-\nthermore, within the chaotic region, there are embedded\nweak (approximate) zeros of the Husimi QSOS [34–36],\nshown in Fig. 3 inside the light spots some magnitude\nsmaller than the surroundings, which also contribute to\nFIG. 3. Husimi QSOS plotted in the logarithmic scale, of the\nsame eigenstate as the one shown in Fig. 2( a1), but with much\nhigher resolution about ∼1×106grid points. It corresponds\nto SOS of Fig. 1( a).\nthe non-uniformity. These weak zeros are owing to the\nmathematical property of Husimi functions as the square6\nmoduli of complex analytic functions, also referred to as\nthe Bargmann-Husimi representation of quantum states.\nFurther factorization of the Bargmann transform [37, 38]\nproves that the Husimi functions are completely deter-\nmined by its zeros, therefore the geometry of the quan-\ntum state can be essentially described by the distribution\nof the zeros, the so-called stellar representation of quan-\ntum states. The distributions of zeros show contrasting\nbehaviors for regular and chaotic states: they form reg-\nular ordered patterns along the lines of invariant curves\nfor regular states and exhibit more disordered patterns\nfor chaotic states. Qualitatively, this can be seen in Fig.\n3, where regular and chaotic regions coexist: The ze-\nros around the regular island seem to lie on the outmost\ninvariant torus, while in the chaotic region they are scat-\ntered over the entire area.\nMoreover, we show in Fig. 4( a1)-(a6) examples of\nthe Husimi QSOS for the singlet of general FPUT, with\nλ= 1/16 and ℏ= 5×10−3, in an energy interval at\nE= 1/3, the saddle energy (for more details of saddles,\nsee paper I), with δE/E ≃0.002 (in Appendix D there\nis a gallery of more quantum states). The classical SOS\nof the general case is shown in Fig. 1( c), and indicates\nthat the system is classically fully chaotic at this energy,\nwhich has also been verified by our calculation of µc, the\nrelative phase space volume of the chaotic region (vol-\nume in the energy surface of the phase space by Liouville\nmeasure, see paper I). All Husimi QSOSs of the chaotic\neigenstates are non-uniformly extended, and show differ-\nent disordered pattern of weak zeros. The non-uniformity\ndisappears in the semiclassical limit according to PUSC\n(see below).\nB. Projected Husimi function\nTo identify quantum scars (or periodic orbits) embed-\nded in quantum states [39–41] and enhance the visualiza-\ntion of the Husimi function, we project the Husimi func-\ntionHk(α1, α2) completely into the phase space ( q2, p2)\nof the second oscillator, as\nPk(α2) =1\nπZ\nHk(α1, α2)d2α1\n=X\nn1,n2,n′\n2Bk\nn1n2Bk∗\nn1n′\n2⟨α2|n2⟩⟨n′\n2|α2⟩,(26)\nby first integrate out the variable α1.Pk(α2) can be\nreferred to as the completely projected Husimi function,\nbecause it includes all energy shells, bringing about a\nblurred visualization of quantum scars [42].\nYet another definition of the projected Husimi func-\ntion is by first intersecting the Husimi function with the\nclassical energy shell at the respective eigenenergy Ek,and finalize the integration thereafter [43], yielding\nePk(α2) =1\n2πℏZ\nδ(Ek−H(q,p))Hk(α1, α2)dq1dp1\n=1\n2ℏZp+\n1\np−\n1dp1X\nq1=q±\n1Hk(q1, p1;q2, p2)\n|∂H(q,p)/∂q1|\n=1\nℏZp+\n1\np−\n1dp1Hk(q+\n1, p1;q2, p2)\n|∂H(q,p)/∂q1|q1=q+\n1, (27)\nwhere q±\n1are two roots of the equation H(q,p) =Ek,\np±\n1roots of the equation\n∂H(q1, p1, q2, p2)\n∂q1|q=q±\n1= 0, (28)\nand the Husimi function\nHk(q+\n1, p1;q2, p2) =|X\nn,lCk\nnl⟨eα+,eα−|n, l⟩|2, (29)\nwhere eα1= (q+\n1+ip1)/√\n2ℏ,eα±= (eα1∓iα2)/√\n2, ac-\ncording to Eq. (20). It should be noted that the last\nintegral for projected Husimi function can be computed\neffectively by a Chebyshev-Gauss quadrature method. In\nFig. 2( b1)-(b6) we show the plots of Pk(α2) and in Fig.\n2(c1)-(c6) the plots of ePk(α2), for the same six consec-\nutive eigenstates. Comparing these two type of plots, it\nis evident that the completely projected Husimi function\nPk(α2) is just the blurred projected Husimi function. In\nall panels, the scaring of chaotic eigenstates is evident.\nTo further clarify the corresponding family of peri-\nodic orbits contributing to the scaring, it is much bet-\nter to project the Husimi function into the configuration\nspace. Similar to the method used to obtain the pro-\njected Husimi function in phase space of the subsystem,\nwe can have the projected Husimi function in configura-\ntion space as\nePk(q1, q2) =1\n2πℏZ\nδ(Ek−H(q,p))Hk(α1, α2)dp1dp2\n=1\nℏZp+\n2\np−\n2dp2Hk(q1, p+\n1;q2, p2)/p+\n1, (30)\nwhere p±\n2=±p\n2(Ek−V),p+\n1=p\n2(Ek−V)−p2\n2,\nandV(q1, q2) being the potential of the classical Hamil-\ntonian. It is worth noting that the projected Husimi\nfunction ePk(q1, q2) we have defined above, is related to\nthe configuration-space probability density |ψk(q1, q2)|2\nof the eigenstate (see the derivation and flowchart shown\nin Fig. 12, Appendix C), where ψk(q1, q2) can be written\nas a summation of the product of Hermite-Gauss modes\nψk(q1, q2) =⟨q1, q2|Ek⟩=X\nn1,n2Bk\nn1n2⟨q1, q2|n1, n2⟩\n=X\nn1,n2Bk\nn1n22Y\ni=1(ℏ/π)−1/4\n√2nini!Hni(qi√\nℏ) exp(−q2\ni\n2ℏ),(31)7\nFIG. 4. Examples of Husimi functions of six eigenstates in the energy interval [ E−δE/2, E+δE/2] at E= 1/3 with\nδE/E ≃0.002, for the singlet of general FPUT with λ= 1/16, where the cutoff of nis set to N= 600 and the Planck\nconstant ℏ= 5×10−3. Panels ( a1)-(a6): Husimi QSOS given by Eq. (25) plotted in the logarithmic scale. Panels ( b1)-(b6):\nprojected Husimi functions in configuration space ePk(q1, q2) given by Eq. (30) plotted in linear scale. Panels in the same\ncolumn correspond to the same eigenstate.\nwhere Hni(x) stands for the Hermite polynomial. Thus,\nfor large N, the numerical evaluation of ψk(q1, q2) de-\npends on the unavoidable direct calculation of large fac-\ntorials, which make the computation much harder for\nN > 100, while one can easily calculate the projected\nHusimi functions, even for N > 1000.\nIn Fig. 2( d1)-(d6) we present ePk(q1, q2) the projected\nHusimi functions in configuration space for mixed-type\nsystem, of the same eigenstates which have been plot-\nted in other panels using other Husimi functions. As a\ncomparison, we also show in Fig. 4( b1)-(b6)ePk(q1, q2)\nfor the general FPUT, of which the corresponding clas-\nsical dynamics is fully chaotic. The chaotic eigenstates\nin all projections, either for mixed-type system, or for\nfully chaotic system, display scarring that may be asso-\nciated with one single particular periodic orbit from the\ncorresponding classical dynamics, or as a superposition\nof several periodic orbits, similar to the case of Hydrogen\natom in a uniform strong magnetic field [44–46]. In both\nα-FPUT and the general case, because of the same C3v\nsymmetry of the potentials, we see the scars in Fig. 2( d3)\nand Fig. 4( b2), supported by the unstable periodic orbit\nfrom the same B(“base”) family, which has been cal-\nculated by the monodromy method [47]. Other periodic\norbits underlying different quantum scars, which have not\nbeen calculated by the monodromy method, are possible\nto be classified using the frequency analysis [48, 49], is\nsubject of our forthcoming papers.\nWe close section III B with one of the main conclusions\nof this work: For a much more detailed study of the struc-\nture and the statistics of localization measure of chaotic\neigenstates, one needs both Husimi QSOS and projected\nHusimi functions. The former is essential for the classifi-\ncation of different behaviors of eigenstates in mixed-type\nsystems, because eigenstates in these systems can spreadeither over the integrable region, or a chaotic compo-\nnent, or there is the flooding of chaotic eigenstates from\nthe chaotic region into the regular region, the so-called\nmixed states, if the semiclassical limit is not reached. The\nlatter serves as a valuable tool for identifying highly local-\nized states, and especially the periodic orbits underlying\nquantum scarred states. These different types of Husimi\nfunctions work together to investigate the mechanisms\nof localization, enabling a comprehensive examination of\nthis phenomenon.\nIV. FRACTION OF MIXED EIGENSTATES\nDue to the correspondence between the Husimi QSOS\nand classical SOS, as shown in Sec. III A, we employ\nHusimi QSOS for the identification of regular, mixed,\nand chaotic eigenstates, by the criterion of overlap with\nthe classical SOS, where one can use the SALI method to\nidentify whether an initial condition on the classical SOS\nbelongs to the chaotic or regular regions. As it was intro-\nduced and implemented in previous works, for the k-th\neigenstate |Ek⟩, the overlap is quantified by the overlap\nindex Mk, defined as [18, 50, 51]\nMk=Z\nSdq2dp2C(q2, p2)¯Qk(α2), (32)\nwhere C(q2, p2) is +1 if ( q2, p2)∈ Sbelongs to the chaotic\nregion, and −1 if lies in the regular regions. By discretiz-\ning the classical SOS into a rectangular grid of points\nindexed by ( i, j) centered in cells of equal area, Mkcan\nbe calculated numerically as\nMk=X\ni,jCi,j¯Qi,j\nk/X\ni,j¯Qi,j\nk, (33)8\nwhere Ci,j:=C(q2,i, p2,j),¯Qi,j\nk:=¯Qk(q2,i, p2,j).\nAccording to PUSC of Husimi functions, in the suffi-\nciently deep semiclassical limit, Mkshould be either +1\nor−1, for chaotic or regular Husimi QSOS, respectively.\nHowever, as it has been revealed in systems such as quan-\ntum billiards, the actual value of Mkvaries from −1 to\n+1, if the semiclassical limit is not yet reached. This is\nconfirmed by the histogram shown in Fig. 5 ( a), where\nwe show the distribution of the overlap index Mfor an\nensemble of approximately 1200 eigenstates, in the en-\nergy interval [ E−δE/2, E+δE/2] at E= 0.14 with\nδE/E ≃0.01, for the singlet of α-FPUT. It exhibits a\nbimodal distribution, where the two peaks are located\nat each end with M=−1 and M= +1. In the cal-\nculation of M, we use the SALI method to numerically\ncompute Ci.jacross the SOS. The criterion for the classi-\nfication of initial conditions belonging to a chaotic region\nis that SALI ≤10−8at t = 1000 (the unit of dimension-\nless time is one period of the linear oscillator, for more\ndetails about this criterion, see paper I). Fig. 5( b) shows\nthe SALI plot on the classical SOS, where the value of\nSALI of each point is plotted using an assigned color ac-\ncordingly. It shows some similarity compared with clas-\nsical SOS shown in Fig. 1( a), but there are more details\nvisible especially the stickiness on the border between\nthe regular and chaotic region. The relative area of the\nchaotic components on the classical SOS ηccomputed by\nSALI is 0.674 at E= 0.14, where\nηc=R\nSdq2dp2χc(q,p)\nA(S), (34)\nwhere A( S) is the surface area of Sthe classical SOS,\nχc(q,p) denotes the characteristic function of the chaotic\ncomponent, which takes the value of 1 on chaotic region\nand zero otherwise.\nIn Fig. 5( b1)-(b4) we show four typical examples of\nHusimi QSOS from the ensemble, with different values\nofM. Except the M=−1 state living in the regu-\nlar islands and the M= +1 state extending across the\nchaotic region, there are mixed eigenstates with |M|<1:\ntheM≃ −0.5 state shown in Fig. 5( b2) lives predomi-\nnantly in the vicinity of the regular islands with a small\ncontribution in the main part of the chaotic sea, which is\nreferred to as the hierarchical state in the study of quan-\ntum standard map [52]. It also exhibits the tunneling\nprocess between the different islands. Most importantly,\nthere is the M≃0.5 state shown in Fig. 5( b3) that\nis flooding from the chaotic sea to the regular region.\nWhile, it must be pointed out, the state shown in Fig.\n2(a6) is of M≃0.52, although predominantly inside the\nchaotic region, it is much less extended, due to dynamical\nlocalization. This reveals that the overlap index Mis not\ndirectly correlated with the extendedness, which we will\ndiscuss in more detail in Sec. V, studying the statistics\nof the localization measures.\nTo illustrate a general trend in the reduction of mixed\neigenstates as one approaches a deeper semiclassical\nlimit, we have calculated the Mindex of an ensemble of\nFIG. 5. ( a) The histogram of distribution of the overlap index\nMfor an ensemble of approximately 1200 eigenstates, in the\nenergy interval [ E−δE/2, E+δE/2] atE= 0.14 with δE/E ≃\n0.01, for the singlet of α-FPUT, where the irreducible Hilbert\nspace is of size NS∼3.75×105, with the cutoff N= 1500 and\nthe Planck constant ℏ= (4±0.016)×10−4. (b) Regions of\ndifferent (logarithmic) values of the SALI on the classical SOS\nforE= 0.14 the same as Fig. 1(a), at t= 1000: the initial\nconditions colored dark blue correspond to chaotic orbits, the\nyellowish color indicates ordered motion, and the intermediate\ncolor suggests sticky orbits. Panels ( b1)-(b4): Husimi QSOS\nplotted in the logarithmic scale, for four eigenstates selected\nfrom the ensemble, with different values of Mindex, from left\nto right M≃ −1,−0.5,0.5,1 respectively, where the darkest\nblue indicates area where ¯Qk(α2)<10−11.\nFIG. 6. The histogram of distribution of overlap index M\nfor an ensemble of approximately 1200 eigenstates from the\nsinglet of α-FPUT, in the energy interval [ E−δE/2, E+δE/2]\nwith δE/E ≃0.01, for energies E= 0.14 (top panels), and\nE= 0.16 (bottom panels). The Planck constant is ℏ=ℏ0±\njℏ0/1000, where from left to right ℏ0= 1×10−3(with j=\n0,1,2,3,4,5),7×10−4(with j= 0,2,4),4×10−4(with j= 4).\nabout 1200 eigenstates, in narrow energy intervals with\nδE/E ≃0.01, for different values of the Planck constant.\nIt is worth emphasizing here in the numerical calculation\nof Husimi QSOSs, there is an unavoidable computing lim-\nitation of the smallness of ℏ, due to the exponential term\ne−|α|2/2=e−(p2+q2)/2ℏin the definition of coherent state\nbasis. From the analytical expression of the semiclassi-\ncal density of states, consequently there is a limitation of\nthe number of energy levels in the energy intervals. To\nachieve a statistically reasonable size of each ensemble,\nwe set ℏ=ℏ0±jℏ0/1000, where the chosen jensures9\ntheℏ∈[ℏ0−δℏ0/2,ℏ0+δℏ0/2] with δℏ0/ℏ0≤0.01.\nThe narrow energy window and narrow window of Planck\nconstant ensure that eigenstates from the same ensemble\nhave approximately the same classical correspondence.\nFig. 6 shows the distribution of Mindex at two ener-\ngies, for three different ℏ. The relative area of chaotic\ncomponents is ηc= 0.865 on classical SOS at E= 0.16,\ntherefore the relative fraction of M= 1 states shown in\nFig. 6( b) is larger than in the case E= 0.14 shown in\nFig. 6( a), where ηc= 0.674, and for the M=−1 states\nthe opposite.\nWith the decreasing ℏ, the relative fraction of M=±1\nstates increases accordingly, meaning that increasingly\nmore states either live in the purely regular region or\nthe chaotic sea, while the fraction of mixed eigenstates\ndecreases, when approaching a deeper semiclassical limit.\nTo quantify the decay of the fraction of mixed states, we\ndefine\nχM(E,ℏ) =N(M∈ {M0, M1} |E, δE,ℏ)\nN(E, δE,ℏ)(35)\nas the fraction of eigenstates with the overlap index M0≤\nM≤M1, given the total number of states N(E, δE,ℏ)\nin an energy interval, within δE≪E. For the case\n−0.8≤M≤0, the mixed states with more regular region\noccupation, we see in Fig. 7( a) a power-law decay of\nχM(E,ℏ)∼ℏξwith respect to the decreasing value of ℏ,\nwhere ξ= 0.421 for E= 0.14 and ξ= 0.320 for E=\n0.16. For mixed eigenstates with |M| ≤0.8, with states\nneither predominantly in the regular islands nor in the\nchaotic sea, χM(E,ℏ) also follows a power law of ℏ, but\nwith slightly different power exponents, as shown in Fig.\n7(b). It should be noted that the power-law decay of the\nfraction of mixed states we have found here is different\nfrom the result in Ref. [52] for quantum standard map,\nregarding the power-law decay of the hierarchical states,\nwhich generally predominantly live in the vicinity of the\nregular islands, by definition the M≤0 states.\nUpon examining various power-law decays in Fig. 7,\nit was observed that the states with M≤0 exhibit a\nslower decay rate than those with M > 0 atE= 0.14.\nConversely, at E= 0.16, the trend reverses, with the\nM > 0 states demonstrating a slower decay compared to\ntheM≤0 states. While we have shown in Sec. III A and\nFig. 13 in Appendix D, for states with M≤0, the pre-\ndominant mechanism involves (chaos-assisted) tunneling\nbetween regular islands, whereas for states with M > 0,\nthe primary phenomenon is the flooding of chaotic states\ninto regular regions. It delineates the disparities between\nthese two tunneling mechanisms as one approaches the\nsemiclassical limit.\nV. LOCALIZATION MEASURES OF CHAOTIC\nEIGENSTATES\nThe study of Husimi functions in Sec. III, either the\nHusimi QSOS or the projected Husimi functions, clearly\nFIG. 7. Decay of the fraction of mixed eigenstates χM(E,ℏ)\nwith respect to the Planck constant ℏ, at two energies E=\n0.14 (squares) and E= 0.16 (circles), where δE/E ≃0.01 and\nN(E, δE,ℏ)≈1200. The dash-dotted lines and dashed lines\nshow the power law χM(E,ℏ)∼ℏξ, where ( a) is for mixed\neigenstates with −0.8≤M≤0, and in ( b)−0.8≤M≤0.8.\ndemonstrates the localization of chaotic eigenstates to\nvarying degrees within the same narrow energy interval,\nin both α-FPUT the mixed-type system and the general\ncase with λ= 1/16 which is fully chaotic, ergodic. To\nfurther quantify the extent of localization, following the\ndeveloped method that has been used to reveal the scal-\ning behavior of localization [53, 54] in natural basis, and\nrecently for the study of Poincar´ e-Husimi functions in\ndifferent type of quantum billiards (see Ref. [22, 24]), we\ndefine localization measures based on the R´ enyi-Wehrl\nentropy [55] of the Husimi QSOS. While the entropy of\nHusimi QSOS of the chaotic eigenstate |Ek⟩is given as\nH(α)\nk:=1\n1−αlnZ\nScdq2dp2[Qk(α2)]α, (36)\nwhere ( q2, p2)∈ ScandSc(compact phase space) denotes\nthe chaotic region on the classical SOS, Qk(α2) is the\nrescaled Husimi QSOS\nQk(α2) =¯Qk(α2)/Z\nScdq2dp2¯Qk(α2). (37)\nThe R´ enyi-Wehrl entropy localization measure (ELM) is\nthen defined as\nLα\nk=expH(α)\nk\nA(Sc), (38)\nwhere A( Sc) is the surface area of Sc. Obviously, for fully\nchaotic systems Sc≡ SandQk(α2)≡¯Qk(α2). The lat-\nter equivalence is also true for mixed-type systems, if it is10\nin the semiclassical limit. But as we have shown in Sec.\nIV, for a regime that is not deeply semiclassical, there ex-\nist states that live both in the regular and chaotic region,\nwith the overlap index |M|<1, thus Qk(α2)̸=¯Qk(α2),\nand it becomes necessary to establish a criterion for iden-\ntifying chaotic eigenstates: states with M≥Mccan be\ndeclared chaotic. Ref. [18] has demonstrated that there\nexist two natural criteria, one from classical referring to\nthe relative Liouville measure of the classical part of the\nphase space µc, the other from quantum, referring to the\nagreement of the chaotic level statistics with the Brody\ndistribution. In the following, we choose Mc= 0.8, so\nthat the states predominantly live in the chaotic region,\nto just make sure that only chaotic eigenstates are being\nused.\nIn the limit α→1, Eq. (36) gives the Wehrl entropy,\nan information entropy based on Husimi functions. The\nα= 2 R´ enyi-Wehrl ELM is the inverse participation ra-\ntio (IPR). The R´ eyni-Wehrl entropy is a non-increasing\nfunction of the parameter α, and as αtends towards in-\nfinity, it is increasingly determined by events of highest\nprobability, therefore quite straightforwardly, one can use\nhigher order R´ enyi-Wehrl ELM to detect highly local-\nized or scarred states, as demonstrated in recent studies\nof the Dicke model with projected Husimi functions (see\nRef. [43]). There are asymptotic upper bounds Lα\n|R⟩for\nthe R´ enyi-Wehrl ELMs, from the eigenvector statistics\nof random unitary matrices, where |R⟩is a random pure\nstate of size N≫1. First, it leads to an ensemble average\nof the R´ enyi-Wehrl entropy [56, 57]\n⟨H(α)\n|R⟩⟩N=1\n1−αlnΓ(N+ 1)Γ( α+ 1)\nΓ(N+α). (39)\nWe can then define the corresponding R´ enyi-Wehrl ELM\nin large Nlimit as the upper bounds\nLα\n|R⟩= lim\nN→∞exp⟨H(α)\n|R⟩⟩N\nN= Γ(1 + α)1/(1−α).(40)\nThen, one has\nL1\n|R⟩=eγ−1,L2\n|R⟩= 0.5, (41)\nwhere γdenotes the Euler constant so that L1\n|R⟩≈0.66.\nIt is necessary to point out that these upper bounds do\nnot contradict the microcanonical assumption [58] for an\nergodic system and the PUSC for the mixed-type system,\nnamely that in the semiclassical limit, the local averaged\nWigner (or Husimi) function can be written as\n¯Ψk(q,p) =δ(Ek−H(q,p))χc(q,p)R\ndqdpδ(Ek−H(q,p))χc(q,p),(42)\nwhich means that the state is uniformly extended over\nthe entire phase space in the fully chaotic system, and\nover chaotic invariant components in the mixed-type.\nThis uniformity results in αR´ enyi-Wehrl ELM of the\naveraged state to be Lα\nk= 1. The “ local” means that theaverage is taken over many oscillations of the wavefunc-\ntion, while the scale of these oscillations is of order ℏ, as\npointed out by Berry (see Ref. [30]).\nThus, following the statistics given by random ma-\ntrix theory (RMT), the localization measure of the (fully\nextended) chaotic state is Lα\nk=Lα\n|R⟩, calculated from\nHusimi functions without local average. The localiza-\ntion phenomena in the chaotic eigenstates implying Lα\nk<\nLα\n|R⟩, are determined by the ratio of the Heisenberg time\ntHand the classical transport time tT, as the localization\ncontrol parameter, denoted as\nαL=tH/tT, (43)\nwhere tH= 2πℏρ(E) with ρ(E) the quantum density\nof states (DOS). The chaotic eigenstates are maximally\nlocalized if αL≪1, and maximally extended if αL≫\n1, in between there is a distribution of the localization\nmeasures.\nA. Classical transport time and Heisenberg time\nThe classical transport (diffusion) time tTas a classical\ntimescale, is the typical time needed for an ensemble of\nsharply distributed initial momentum with zero variance,\nto spread uniformly over the classical chaotic component.\nThat is, one needs to estimate the timescale tTof the mo-\nmentum spreading characterized by σ2\np(t), the variance\nof momentum, at which it reaches a certain saturation\nvalue. While the saturation value can be obtained from\nthe longtime average σ2p, the estimation of the exact time\nof reaching it is not obvious, as shown in Fig. 8( a1)-(b1),\nbecause in practice there are fluctuations around σ2p. As\nan analog to the definition of quantum relaxation time\n[59, 60], we define the classical transport time as\n|σ2\np(t)−σ2p| ≤ϵ1,for all tT≤t≤trec, (44)\nwhere ϵ1is some small quantity and t≤trecis the classi-\ncal recurrence time. Another equivalent definition is that\nµ2\np(t) the temporal fluctuation of σ2\np(t) across time win-\ndows [ t−∆t/2, t+ ∆t/2], as shown in Fig. 8( a2)-(b2),\nfulfills that µ2\np≤ϵ2fort≥tT,ϵ2as the threshold is also\na small quantity.\nComparing these two definitions and the correspon-\ndent numerical results shown in Fig. 8, we found that\npractically it is more convenient to estimate tTaccord-\ning to the temporal fluctuation. In Fig. 8(c) we show\ntTas a function of energy Einα-FPUT, with two dif-\nferent thresholds: 1% and 2% of the peak value of µ2\np(t).\nThe general behavior of the transport time versus energy\nis similar under two thresholds, as expected, and it ex-\nhibits an almost monotonic decrease as energy increases,\nbecause chaos is more pronounced (larger µc) at higher\nenergies within this range of energy. In the following, we\nopt for the 2% threshold in our estimation for two spe-\ncific reasons: it aligns more consistently with the results\nobtained from the definition in Eq. (44) and exhibits11\nFIG. 8. ( a1) The variance of momentum σ2\np=⟨p2⟩ − ⟨p⟩2\nvs. tforα-FPUT with α= 1 at E= 0.14, where σ2\np\nis calculated with 4000 initial conditions from the classical\nSOS that are uniformly distributed in the chaotic region with\nq2∈[−0.25,−0.15] and p2= 0. The red dashed horizontal\nline indicates σ2p, the longtime averaged value of σ2\np. (a2) The\ntemporal fluctuation of σ2\np, as a function of time, across the\ntime windows [ t−∆t/2, t+ ∆t/2], with ∆ t= 30, and the\nblue dashed horizontal line indicates the threshold 2% of the\npeak value of µ2\np. (b1)-(b2) Analogous data as ( a1)-(a2) re-\nspectively, at E= 0.16. ( c) Classical transport time tTas a\nfunction of energy Einα-FPUT (H´ enon-Heiles), using two\ndifferent thresholds. ( d)tTversus Ein general FPUT (2%\nthreshold).\nfewer anomalies compared to the 1% threshold. For the\ngeneral FPUT with λ= 1/16, numerical results of tTare\nshown in Fig. 8( d).\nRegarding the Heisenberg time tH= 2πℏρ(E), the\nquantum DOS ρ(E) has a semiclassical approximation\ng(E), obtained from the Thomas-Fermi rule\ng(E) =1\n(2πℏ)2Z\ndqdpδ(E−H(q,p)), (45)\nwhich is the volume of the available classical phase space\nfor a given energy Edivided by (2 πℏ)2. For α-FPUT of\nthe H´ enon-Heiles potential, when E≤1/6α2, the result-\ning Thomas-Fermi expression [1] is\ng(E) =3\n4πℏ2\u0002Zπ/3\n0r2\n+dϕ+Z2π/3\nπ/3r2\n−dϕ\u0003\n, (46)\nwhere\n(\nr+(ϕ) =1\nαsin 3ϕ(cosθ\n3−1\n2), if sin 3 ϕ≥0,\nr−(ϕ) =−1\nαsin 3ϕ(cosθ+π\n3+1\n2),if sin 3 ϕ≤0,(47)\nFIG. 9. Scaled semiclassical DOS 2 πℏ2g(E) for α-FPUT ( a)\nand the general case ( b), where the red dashed vertical lines\ndenote the saddle energies (see paper I). ( c) Ratio αLas a\nfunction of 1 /ℏ, where black dashed line is for E= 0.14 and\nblue dash-dotted line for E= 0.16, in α-FPUT, with respect\nto the bottom x-axis (colored in black). On the lines the\nblack and blue dot indicate αLatℏ= 4×10−4. The red solid\nline is for E= 1/3 in general FPUT, with respect to the top\nx-axis in red color, where the red dot indicates the ratio at\nℏ= 2×10−3.\nand cos θ= 12Eα2sin23ϕ−1,θ∈[0, π]. For the general\ncase with λ= 1/16, an analytical expression of g(E) de-\npends on the solving of a quartic equation (for a detailed\nderivation, see paper I). The scaled semiclassical DOS\nf(E) = 2 πℏ2g(E) for two cases, as a function of energy,\nis shown in Fig. 9( a)-(b). Therefore, the controlling ratio\nαL=2πℏρ(E)\ntT≃2πℏg(E)\ntT=f(E)\nℏtT. (48)\nIn Fig. 9( c) we show αLas a function of 1 /ℏ, at two\nenergies for α-FPUT, and at the saddle energy E= 1/3\nfor the general case. The increase in αLin relation to\n1/ℏatE= 0.14 is slower compared to E= 0.16, at-\ntributable to the larger classical transport time at the\nformer energy. While, in the general case at E= 1/3\nwithℏ= 2×10−3, the ratio is comparable to that of the\nα-FPUT system at E= 0.16 with ℏ= 4×10−4. These\ndistinctions and comparisons will manifest in the distri-\nbution of localization measures, as we will illustrate in\nthe subsequent discussion.\nB. The distribution of localization measures\nThe distributions of R´ enyi-Wehrl ELMs have been\nstudied thoroughly in different families of billiards (with\nor without stickiness, mixed or ergodic), demonstrating12\nFIG. 10. Distributions of the α= 1 and α= 2 R´ enyi-Wehrl ELMs, for the M≥0.8 eigenstates, as a function of the Planck\nconstant ℏ(indicated at the top of each column), at two energies for the singlet of α-FPUT. The main figures show the histogram\nand its best-fitting beta distribution, while the insets show the corresponding cumulative distribution. (Colors online: black\nare data, red is the best-fitting). The vertical dash lines indicate L1\n|R⟩≈0.66 for α= 1 cases and L2\n|R⟩= 0.5 for α= 2, from\nrandom pure states. Parameters of the best fit beta distribution for P(L1) from left to right: at E= 0.14 are (6 .17,1.83),\n(10.72,3.98), (7 .48,2.26), (7 .18,2.35) with L1\n0= 0.76 and at E= 0.16 are (8 .87,2.33), (10 .29,3.11), (11 .27,3.62), (13 .17,4.55)\nrespectively with L2\n0= 0.64. Eigenstates for the statistics in each panel, are selected from an ensemble of N(E, δE,ℏ)≈1200\nstates with δE/E ≃0.01.\nthat the ELMs of a sequence of consecutive eigenstates\nconform to a shared empirical distribution, well described\nby the beta distribution [22, 61], namely\nP(Lα) =1\nC(Lα)βa(Lα\n0− Lα)βb, (49)\nwhere Lα\n0is the upper limit of the interval [0 ,Lα\n0] on\nwhich P(Lα) is defined, βaandβbare two positive shape\nparameters, the normalization constant Cis given by\nC= (Lα\n0)βa+βb+1B(βa+ 1, βb+ 1), (50)\nwhere B(x, y) =R1\n0tx−1(1−t)y−1dtis the beta function.\nThis has also been confirmed in the Dicke model and in\nthe kicked-top model [25, 26]. Given the variance of betadistribution\nσ2(Lα) = (Lα\n0)2 (βa+ 2)( βb+ 2)\n(βa+βb+ 4)( βa+βb+ 3)2,(51)\ntherefore the variation in shape parameters is directly\nassociated with the degree of localization of the eigen-\nstates, controlled by the ratio αL: A greater value of the\nparameter βagenerally results in a more compact distri-\nbution with a reduced width, and in the limit βa→ ∞ ,\nP(Lα) = δ(Lα− Lα\n0), while the statistics from RMT\nshows that P(Lα) =δ(Lα− Lα\n|R⟩) for N≫1, in agree-\nment with Shnirelman’s theorem [28].\nIn a genuine continuous Hamiltonian system with a\nsmooth potential, obtaining a statistically meaningful\nnumber of consecutive eigenstates within a narrow en-\nergy window, for the study of distribution of localization13\nmeasures, poses a significant challenge. Owing to the\nsparse matrix representation of quantum three-particle\nFPUT Hamiltonian, for the first time, we can employ the\nKrylov subspace method to obtain N(E, δE,ℏ)∼1×103\nconsecutive states of the singlet, with δE≪Efor the\nstatistics of the R´ enyi-Wehrl ELMs. And for the numer-\nical calculation of ELMs, we need to discretize the Eqs.\n(36)-(38), so that\nH(α)\nk=1\n1−αlnX\ni,j\u0010\nQi,j\nk\u0011α\n,Lα\nk=expH(α)\nk\nNc,(52)\nwhere Qi,j\nk=¯Qi,j\nk/P\ni,j¯Qi,j\nk, and ( i, j) indicates grid\npoints ( q2,i, p2,j)∈ Sc,Ncis the number of grid points of\nthe chaotic parts from SOS.\nIn Fig. 10 we show the distributions of the α= 1\nandα= 2 R´ enyi-Wehrl ELMs, for the M≥0.8 eigen-\nstates, as a function of the Planck constant ℏ, at two en-\nergies for the singlet of α-FPUT. The comparison is made\nwith the best-fitting beta distribution in the main figures,\ncomplemented by an evaluation of cumulative distribu-\ntions, as illustrated in the inset. At each energy, as ℏ\ndecreases, the agreement between P(Lα) and the beta\ndistribution improves. This improvement is character-\nized by reduced fluctuations and a decline in the propor-\ntion of ELMs with Lα>Lα\n|R⟩, mostly ascribed to the\ninfluence of finite-size effects, because our numerical cal-\nculations involve the discretization of SOS with a grid\nsize ∆ q2= ∆p2= 5×10−3. A larger grid size (larger\nthan the Planck cell) is equivalent to local averaging, re-\nsulting Lα= 1 for the fully extended (chaotic) states.\nAt each energy, the value of the parameter βain the\nbest-fitting parameters for P(L1) generally increases as\nℏdecreases, except in the case with ℏ= 8×10−4at\nE= 0.14, where the fitting is less satisfactory. More-\nover, the value of βais higher at E= 0.16 compared\ntoE= 0.14 for the same Planck constant, even when\ncomparing the case of the former with ℏ= 9×10−4to\nthe latter with ℏ= 4×10−4. This observation can be\nexplained by examining the controlling ratio αLas de-\npicted in Fig. 9( c): In all cases, the ratio at E= 0.16\nis greater than that at E= 0.14, resulting in better fit-\ntings and a reduced degree of localization, characterized\nby the shrinking of the distributions. To provide addi-\ntional evidence supporting the exclusive correlation be-\ntween βaandαL, we illustrate the distribution of ELMs\nin Fig. 11 (top panels), for the singlet of the general case\nwith λ= 1/16, at the saddle energy E= 1/3. While\nthe Planck constant ℏis set to 2 ×10−3, a higher value\ncompared to the α-FPUT case where ℏwas 4 ×10−4\natE= 0.16, it’s noteworthy that the parameter βafalls\nwithin the range observed for the latter with ℏ= 5×10−4\nand that of ℏ= 4×10−4. This observation is explained\nby the findings depicted (denoted by colored dots) in Fig.\n9(c), which illustrate that the value of αLis comparable\nin the two cases. In conclusion, the value of βaexhibits\na monotonically increasing behavior with respect to αL.\nAs one expects, when αL≫1, the trend suggests that\nFIG. 11. Top panels: distributions of the α= 1 and α= 2\nR´ enyi-Wehrl ELMs of N(E, δE,ℏ)≃1200 eigenstates, from\nthe singlet of general FPUT with λ= 1/16, at energy E= 1/3\nwith δE/E ≃0.01 and ℏ= 2×10−3. Parameters of the best-\nfitting beta distribution for P(L1) are (11 .60,2.55),L1\n0= 0.72\nandL2\n0= 0.59. The vertical dashed lines indicate Lα\n|R⟩. Mid-\ndle panels: Husimi QSOSs given by Eq. (25) (plotted in the\nlogarithmic scale) of the maximally localized and extended\nstates, which are pointed out by red arrows in the left top\npanel. Bottom panels: Projected Husimi functions in con-\nfiguration space ePk(q1, q2) given by Eq. (30) (plotted in the\nlinear scale), for the same two states plotted previously in the\nmiddle panels.\nβatends towards infinity, but as unveiled in the study\nof quantum billiards [22], this transition might be very\nslow.\nAs discussed earlier, higher order R´ enyi-Wehrl ELMs\namplify differences in probability densities, resulting in\na broader distribution compared to the lower orders.\nMeanwhile, the more localized states in the lower tail of\nthe distribution become more prominent, as illustrated\nin Figs. 10-11, where the difference between P(L1) and\nP(L2) of IPR is demonstrated. Having chosen the max-\nimally localized and extended states from the distribu-\ntion of α= 1 ELMs in the general case, as indicated in\nthe top left panel of Fig. 11, we then depict both the\nHusimi QSOS and projected Husimi functions for these\ntwo states. Remarkably, the results demonstrate that\nthe maximally localized state concentrates around the\nsaddle points, while the maximally extended state no-\ntably avoids these points, extending across the remain-14\ning space, complementing each other. Identified as unsta-\nble (hyperbolic) fixed points, these saddles are elucidated\nin paper I, in polar coordinates ( r, ϕ) as (2 , ϕm), where\nsin 3ϕm=−1,q1=rcosϕ,q2=rsinϕ. Even more\nstrikingly, the Husimi QSOS of the maximally localized\nstate shows that the localization is influenced by the sta-\nble and unstable manifold of these saddles, which exist in\nisolation and embedded in a sea of chaos, similar to the\nfinding in Ref. [62]. An extensive study of the classical\nperiodic orbits and the stable and unstable manifolds of\nhyperbolic points, in relation to quantum scars, will be\npresent in a forthcoming paper.\nIn the gallery of states presented in Fig. 14 in Ap-\npendix D, one can observe more examples of states, which\nare localized around the saddle points. It is this spe-\ncific type of strong localization that contributes to the\nbulge, causing a deviation from the beta distribution in\nthe lower tail. While most states notably avoid the sad-\ndles, displaying a much lesser degree of localization, some\nexhibit clear evidence of quantum scarring, indicating the\npresence of underlying unstable periodic orbits.\nVI. CONCLUSIONS AND DISCUSSION\nIn conclusion, this work presented a systematic study\nof the phenomenology of quantum eigenstates in the\nthree-particle FPUT model, considering both the α-type,\nwhich is canonically equivalent to the celebrated H´ enon-\nHeiles Hamiltonian, a nonintegrable and mixed type sys-\ntem, and the general case at the saddle energy where\nthe system is fully chaotic (ergodic). While employing\nthe circular (two-mode) basis for quantization, we have\nderived a Wigner d-matrix decomposition of the unitary\ntransformation from this circular basis to the Cartesian\ntwo-mode basis. The analytical expression of this unitary\ntransformation plays a vital role for the discussion and\ncomparison of various Husimi functions, with a partic-\nular focus on the completely projected Husimi function\nand the projected Husimi function from the energy shell\nin (q2, p2) phase space. It is worth noting that the lat-\nter can be regarded as a blurred version of the former,\nas pointed out in Ref [42], and we make use of this ef-\nfect to establish a connection (equivalence) between the\nprojected Husimi function in configuration space and the\nprobability density in configuration space from the wave-\nfunction.\nThrough a comprehensive examination of Husimi func-\ntions, representing various eigenstates, we have identified\nthe correspondence between the Husimi QSOS and classi-\ncal SOS. This correspondence establishes a valuable tool\nfor distinguishing regular, mixed, and chaotic eigenstates\nin mixed-type systems. Furthermore, it allows for the\nquantification of localization of the chaotic eigenstates.\nOn the other hand, various projected Husimi functions\nprove to be effective tools for identifying highly localized\nstates, and especially the periodic orbits underlying the\nquantum scarred states.By introducing the overlap index M, a measure to\nquantify the overlap between the Husimi QSOS and the\nclassical SOS, we define in mixed-type system, a quan-\ntityχM(E,ℏ), the fraction of mixed eigenstates within an\nensemble of states in an energy interval [ E−δE/2, E+\nδE/2], where δE≪E. For states with |M| ≤0.8, i.e.\nstates not entirely supported by either the regular islands\nor the chaotic region, there is a power-law decay concern-\ning the decreasing value of ℏ, that χM(E,ℏ)∼ℏξ. For\nstates with −0.8≤M≤0, a power-law decay is also\nobserved, yet with a slightly different power exponent.\nMoreover, the exponent varies with energy, while the rel-\native phase space volume of the chaotic region varies with\nrespect to energy, in the mixed-type system.\nUpon examining various power-law decays, it was ob-\nserved that the mixed states with M≤0 exhibit a dif-\nferent decay rate than those with M > 0. For mixed\nstates with M≤0, the predominant mechanism in-\nvolves (chaos-assisted) tunneling between regular islands,\nwhereas for mixed states with M > 0, the primary phe-\nnomenon is the flooding of chaotic states into regular\nregions. It delineates the disparities between these two\ntunneling mechanisms as one approaches the semiclassi-\ncal limit. It is noteworthy to mention that within the\ngallery of states, we have identified also a recurring pat-\ntern of states with M≃ −1. Integrating this observed\npattern of M≃ −1 states with the PUSC picture poses\nan intriguing open question for our future work.\nWe investigate the localization measure Lαof chaotic\neigenstates, as defined in terms of the R´ enyi-Wehrl en-\ntropy, in both mixed-type system and the fully chaotic\nsystem. Lαis normalized exponential of the α-R´ enyi-\nWehrl entropy. While α= 1 refers to the Wehrl (in-\nformation) entropy localization measure, the α= 2 case\nrefers to the inverse participation ratio. The analysis\nis based on Husimi QSOS. For the first time, we explore\nthe distribution of localization measures P(Lα) for an en-\nsemble of eigenstates from a narrow energy window, in a\ngenuinely continuous Hamiltonian system with a smooth\npotential. The degree of localization in the chaotic eigen-\nstates is determined by the controlling ratio αL=tH/tT,\nbetween the Heisenberg time tHand the classical trans-\nport time tT. We have shown that a larger value of αL\nleads to a more accurate fit with a beta distribution of\nP(Lα), where βaserves as the fitting parameter, and to\nits larger value. The monotonically increasing trend ob-\nserved in the values of βaconcerning αLimplies that\nwhen αL≫1,βatends towards infinity, although this\ntransition may be very slow. The condition βa≫1 leads\ntoP(Lα) =δ(Lα− Lα\n|R⟩), where Lα\n|R⟩is the localization\nmeasure of the random pure state from RMT. This tran-\nsition with respect to αL(∼1/ℏ) and the power-law decay\nof the mixed states with decreasing ℏ, together provide\nsupporting evidence for the PUSC in the semiclassical\nlimit. A theoretical analysis the of localization properties\nof chaotic eigenstates, especially a mathematical proof of\nthe beta distribution describing the distribution of local-\nization measures of chaotic eigenstates, is still an open15\nquestion. We will leave this study for future work.\nMoreover, we have found that the maximally localized\nstate in the general case, which is a fully chaotic system,\nis influenced by the stable and unstable manifold of these\nsaddles, which exist in isolation and embedded in a sea of\nchaos, while the maximally extended state notably avoids\nthese points, extending across the remaining space, com-\nplementing each other. It is this specific type of strong\nlocalization that contributes to the bulge, causing a devi-\nation from the beta distribution in the lower tail, which\nis expected to disappear in the strict semiclassical limit,\naccording to PUSC.\nIn the examination of projected Husimi functions,\nwhether in configuration space or the phase space of a\nsubspace, we have demonstrated the existence of distinct\nquantum scars. An extensive study of the classical peri-\nodic orbits, including the stable and unstable manifolds\nof hyperbolic points, in relation to quantum scars, will\nbe present in our forthcoming paper.\nVII. ACKNOWLEDGEMENT\nWe express our gratitude to Dr. Jiaozi Wang for dis-\ncussions and extend our appreciation to Matic Orel for\nhis technical assistance. This work was supported by the\nSlovenian Research and Innovation Agency (ARIS) under\nthe grant J1-4387.\nAppendix A: Matrix elements in circular Fock basis\nfor different coupling terms\nMatrix elements in the circular basis of the cubic cou-\npling terms are (for detailed derivation, see paper I)\n⟨n′, l′|ˆq3\n±|n, l⟩=ℏ3/2δl′,l±3X\nm∈M 1k±\nm(n, l)δn′,n+m,(A1)where M1={±1,±3},k+\nm(n, l) =k−\nm(n,−l), and\nk+\n−1= 3p\n(n−l)(n−l−2)(n+l+ 2)/8,\nk+\n1= 3p\n(n−l)(n+l+ 2)( n+l+ 4)/8,\nk+\n−3=p\n(n−l)(n−l−2)(n−l−4)/8,\nk+\n3=p\n(n+l+ 2)( n+l+ 4)( n+l+ 6)/8.(A2)\nFor the quartic coupling, there is\n⟨n′, l′|(ˆq+ˆq−)2|n, l⟩=ℏ2δl′lX\nm∈M 2km(n, l)δn′,n+m(A3)\nwhere M2={0,±2,±4}, and the coefficients\nk0=3\n2n2−1\n2l2+ 3n+ 2,\nk−2=np\nn2−l2, k2= (n+ 2)p\n(n+ 2)2−l2,\nk−4=1\n4p\nn2−l2p\n(n−2)2−l2,\nk4=1\n4p\n(n+ 2)2−l2p\n(n+ 4)2−l2.(A4)\nSo in the circular basis |n, l⟩, the cubic terms in quantum\nα-FPUT couple states with ∆ l=±3. Therefore, the cu-\nbic coupling takes place in three decoupled sets of basis\nstates, two doublets and one singlet, agrees with the fact\nthat C3vsymmetry exists in the classical H´ enon-Helies\npotential. The quartic term in the three-particle quan-\ntumβ-FPUT does not introduce any coupling between\ndifferent l, due to the conservation of angular momentum.\nAppendix B: Unitary transformation between the Cartesian- and circular-mode\nTo get the unitary transformation between the circular-mode basis |n, l⟩and the Cartesian-mode basis |n1, n2⟩, first\nwe expand Eq. (10) by using the definition of rotated bosonic operator given in Eq. (6)\n|n, l⟩=X\nn1,n2Ωnl\nn1n2|n1, n2⟩=(a†\n+)n+(a†\n−)n−\np\nn+!n−!|00⟩=1\n2n/2(a†\n1+ia†\n2)n+(a†\n1−ia†\n2)n−\np\nn+!n−!|00⟩\n=n+l\n2X\nk1=0n−l\n2X\nk2=0ik1−k2\n2n/2\u0012n+l\n2\nk1\u0013\u0012n−l\n2\nk2\u0013s\n(n−k1−k2)!(k1+k2)!\n(n+l\n2)!(n−l\n2)!|n−k1−k2, k1+k2⟩. (B1)\nDenoting n= 2j, l= 2m, one can rewrite Eq. (B1) as\n|n, l⟩=|2j,2m⟩=j+mX\nk1=0j−mX\nk2=0ik1−k2\n2jp\n(j+m)!(j−m)!(2j−k1−k2)!(k1+k2)!\nk1!k2!(j+m−k1)!(j−m−k2)!|2j−k1−k2, k1+k2⟩. (B2)16\nReplacing k1+k2=j−m′, where −j≤m′≤jobtained directly from 0 ≤k1+k2≤2j,\n|n, l⟩=j+mX\nk1=0j−mX\nk2=0i2k1−j+m′\n2jp\n(j+m)!(j−m)!(j+m′)!(j−m′)!\n(j+m−k1)!k1!(j−k1−m′)!(k1−m+m′)!|j+m′, j−m′⟩,\n=jX\nm′=−jX\nki2k−j+m′\n2jp\n(j+m)!(j−m)!(j+m′)!(j−m′)!\n(j+m−k)!k!(j−k−m′)!(k−m+m′)!|j+m′, j−m′⟩. (B3)\nThe reordering of the double summation introduces a\nnew range of k,\nkmin= max(0 , m−m′),\nkmax= min( j−m′, j+m).(B4)\nThus, the unitary transformation can be further ex-\npressed as the following:\n|n, l⟩=jX\nm′=−jOj\nm,m′|j+m′, j−m′⟩, (B5)\nwhere the coefficients Oj\nm,m′are expressed as\nOj\nm,m′=k=kmaxX\nk=kmini2k−j+m′[cos(π/4)]2j−2k+m−m′\n×[sin(π/4)]2k−m+m′Wjmm′\nk(B6)\nwith\nWjmm′\nk=p\n(j+m)!(j−m)!(j+m′)!(j−m′)!\n(j+m−k)!k!(j−k−m′)!(k−m+m′)!.\nComparing with the Wigner’s formula for Wigner d-\nmatrix [63]\ndj\nm,m′(θ) =X\nk(−1)k\u0012\ncosθ\n2\u00132j−2k+m′−m\n×\u0012\n−sinθ\n2\u00132k+m−m′\nwjmm′\nk,(B7)\nwhere\nwjmm′\nk=p\n(j+m)!(j−m)!(j+m′)!(j−m′)!\n(j−m−k)!k!(j−k+m′)!(k+m−m′)!,\nandk∈[max(0 , m′−m),min(j−m, j+m′)], we get\nOj\nm,m′=im′−jdj\nm′,m(−π/2) = im′−jdj\nm,m′(π/2),(B8)\nusing one of the symmetries dj\nm′,m(−θ) =dj\nm,m′(θ). So\nfinally we have\n|n, l⟩=jX\nm′=−jim′−jdj\nm,m′(π/2)|j+m′, j−m′⟩.(B9)Recently, it was found that the employing of Jacobi\nmethod can avoid the error and instability introduced\nby the summation over many terms with large numbers\nof alternating signs, based on the expansion of Wigner\nd-function in terms of the Jacobi polynomials [64],\ndj\nm,m′(θ) =ξm,m′\u0014s!(s+µ+ν)!\n(s+µ)!(s+ν)!\u00151/2\n×\u0012\nsinθ\n2\u0013µ\u0012\ncosθ\n2\u0013ν\nP(µ,ν)\ns(cosθ),(B10)\nwhere µ, νandsare related to m, m′andjby\nµ=|m−m′|, ν=|m+m′|, s=j−1\n2(µ+ν),(B11)\nand\nξm,m′=(\n1 if m′≥m,\n(−1)m′−mifm′< m,(B12)\nwhere the Jacobi polynomial can be calculated by the\nrecurrence relations. Here we generate all the elements of\nWigner d-matrix straightly from another definition with\ndj\nm,m′(θ) =⟨jm|e−iθJy|jm′⟩, (B13)\nwhere Jyis one generator of the Lie algebra of SU(2)\nand SO(3), also a component of the angular momentum\noperator. The sparse matrix exponential (by employing\nthe kernel polynomial method) would be more effective\nthan the Jacobi method to generate the unitary matrix\nor the Wigner d-matrix for large j.\nAppendix C: Equivalent Husimi functions\nThe equivalence between two Husimi functions results\ndirectly from the following equation:\n⟨α1, α2|n, l⟩=⟨α+, α−|n, l⟩, (C1)\ngiven the coherent state a±|α±⟩=α±|α±⟩, written as\n|α±⟩=e−|α±|2/2eα±a†\n±|0±⟩, (C2)\nwhere the vacuum two-mode state in two different repre-\nsentations |0+0−⟩ ≡ |00⟩. Eq. (C1) can easily be verified17\nWk(α1, α2) Hk(α1, α2)\n|ψ(q1, q2)|2 /tildewidePk(q1, q2)Coarse\ngraining\nIntegration over\nthe whole spaceIntegration\nfrom\nthe energy shell\nFIG. 12. A flowchart illustrating the relation between the\nprojected Husimi function in configuration space and the\nconfiguration-space probability density |ψk(q1, q2)|2, linked\nthrough the Wigner function.\nfrom the following derivation:\n|α+, α−⟩=e−(|α+|2+|α−|2)/2eα+a†\n+eα−a†\n−|0+0−⟩\n=e−(|α+|2+|α−|2)/2eα+a†\n++α−a†\n−|0+0−⟩\n=e−(|α1|2+|α2|2)/2eα1a†\n1+α2a†\n2|00⟩=|α1, α2⟩,(C3)\nPhysically, it is a quite obvious result, because |α+, α−⟩\nand|α1, α2⟩correspond to the same point in phase space.\nThe relationship between the projected Husimi\nfunction in configuration space ePk(q1, q2), and the\nconfiguration-space probability density |ψk(q1, q2)|2from\nwavefunction can be elucidated by drawing connections\nto the Wigner function. The projected Husimi function\nis defined as in Eq. (30)\nePk(q1, q2) =1\n2πℏZ\nδ(Ek−H(q,p))Hk(α1, α2)dp1dp2.\nSimultaneously, the configuration-space probability den-\nsity|ψk(q1, q2)|2can be expressed in terms of the Wigner\nfunction Wk(α1, α2) as\n|ψk(q1, q2)|2=Z∞\n−∞Wk(α1, α2)dp1dp2. (C4)\nWhile the Husimi function Hk(α1, α2) can be under-\nstood as the Weierstrass transform of the Wigner func-\ntion (smoothing by a Gaussian filter),\nHk(α1, α2) =4\nπ2Z\ne−2|α1−β1|2e−2|α2−β2|2\n×Wk(β1, β2)d2β1d2β2. (C5)\nHence, ePk(q1, q2) represents the integration over mo-\nmentum within the energy shell EkofHk(α1, α2). On\nthe other hand, |ψk(q1, q2)|2corresponds to the integra-\ntion over momentum across the entire momentum space\nof the Wigner function. The connection is visually de-\npicted in the flowchart presented in Fig. 12. In our anal-\nysis of ePk(q2, p2) the projection from energy shell and\nthe completely projected Husimi function Pk(q2, p2) of\nEq. (26), as discussed in Section III B, we observed thatintegrating over the entire space leads to a blurring ef-\nfect when compared to the integration specifically over\nthe energy shell. This blurring effect can be interpreted\nas a form of coarse graining. Following the flowchart, one\nmight reasonably conjecture that ePk(q1, q2) is equivalent\nto|ψk(q1, q2)|2, though lacking a rigorous mathematical\nproof.\nAppendix D: Gallery of states\nIn the main text, we have conducted a comprehensive\nphenomenological study of an ensemble of eigenstates, in\nan energy shell [ E−δE/2, E+δE/2] close to the energy\nsurface, with δE≪E. This investigation applies to both\nthe mixed-type system and the fully chaotic system. One\ntechnical issue needs to be addressed here: the number\nof consecutive energy levels for the singlet is constrained,\napproximately determined by N(E, δE,ℏ)≃g(E)δE/3,\nwhere g(E) is the semiclassical DOS. To ensure the con-\nvergence of the eigenspectra, the cutoff N, represent-\ning the limit of the quantum number n, needs to satisfy\nthe condition that the total dimensions of the truncated\nHilbert space, denoted as N= (N+ 1)( N+ 2)/2, are\nsignificantly larger than NE=R\ng(E)dE. In all our cal-\nculations in this work, Nis ten or more times larger than\nNE, ensuring good convergence.\nIn Fig. 13, we present examples of Husimi QSOSs\ngiven by Eq. (25) for 80 consecutive eigenstates within\nthe energy interval [ E−δE/2, E+δE/2] at E= 0.14,\nwhere δE/E ≃0.01. These examples correspond to the\nsinglet of the α-FPUT system with α= 1, each is la-\nbeled with its respective overlap index M. It displays a\ndistinct pattern illustrating the recurrence of states with\nM≃ −1, as shown in Fig. 13( a11-a12), ( a28-a29), ( a48-\na49), ( a65-a66). The doubling can be explained that\nthe singlet is a combination of ( A1, A2) symmetry, from\nthe point symmetry group C3ν. While the phenomenon\nof repetitiveness has not been elucidated previously in\nquantum mixed-type systems, it may be associated with\nthe same classical periodic orbit and can occur when the\naction difference satisfies |∆I|= 2πℏn(n= 1,2,···),\njust like the scarring pattern in quantum chaotic systems\n[41]. According to the PUSC, in the semiclassical limit,\nthe states with M≃ −1 would occupy a notable percent-\nage within the ensemble of states if 1 −µcis not small.\nHow to combine the PUSC picture with the M≃ −1\npattern, will be an exciting open problem for our future\nworks. In Fig. 14, we show the gallery of states of both\nHusimi QSOSs of Eq. (25), each is labeled with its re-\nspective α= 1 ELM and projected Husimi functions in\nconfiguration space given by Eq. (30), for the general\nFPUT with λ= 1/16, at saddle energy E= 1/3. 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Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Univ. del País Vasco UPV/EHU, Barrio Sarriena s/n, E-48940 Leioa, Bizkaia, Spain\ncDpto. Matemática Aplicada, Escuela de Ingeniería de Bilbao, Univ. del País Vasco UPV/EHU, Alda. Urquijo s/n, E-48013 Bilbao, Spain\ndCEMHTI-UPR3079, University of Orléans, CNRS Site Haute Température CS 90055, 1D Avenue de la Recherche Scientifi que, 45071 Orléans Cedex 2, France\neInstituto de Síntesis y Estudio de Materiales, Univ. del País Vasco, Apdo. 644, E-48080 Bilbao, Spain\nabstract\nThis paper shows a systematic experimental and theoretical study on the temperature dependence of the infrared optical properties of pyrolytic boron \nnitride (pBN), from 390 to 1050 °C for wavelengths between\n4 and 16 μm. The temperature dependence of these properties has never been analyzed before. The measured emissivity spectra were fitted to a dielectric \nfunction model and an effective medium ap-proximation. The phonon frequencies and dielectric constants agreed well with room temperature ex-\nperimental values from the literature, as well as with ab initio and first principles calculations. In ad-dition, the phonon frequencies of the perpendicular \nmode and the dielectric constants of the parallel mode showed an appreciable parabolic temperature dependence, which justifies the interest of more \ntheoretical efforts in order to explain this behavior. Finally, the results of this work demonstrate that thermal emission spectroscopy allows obtaining the \nvalues of the optical and dielectric parameters of impure ceramic materials in a simple manner as a function of temperature.\n1. Introduction\nBoron nitride shows four polymorphic variants whose special\nproperties make this compound suitable for a large number of\ntechnological applications [1–4]. The hexagonal form (h-BN) is the\nmost common phase and the one that shows the most unique\nelectronic and optical properties. It is a very good electrical in-\nsulator, has great chemical stability and oxidation resistance, lowthermal conductivity and can withstand extreme temperature\nconditions. Due to these properties, it is widely used in micro-\nelectronic devices, nuclear energy, X-ray lithography, lubrication,vacuum technology or high-performance oxidation-resistant\ncoatings for metals and graphene up to 1100 °C in an oxidizing\natmosphere [5–10]. Recently, the discovery of several new features\nhas increased its interest. Firstly, the achievement of h-BN nano-tubes with this simple hexagonal structure [11]. Secondly, its\nlasing capacity at a wavelength of 210 nm, which shows great\npotential for optoelectronic applications in the ultraviolet range\n[12]. Thirdly, its very good optical selectivity (infrared emitter andvisible re flector) has made it a good candidate for coating heat\nshields of solar probes [13]. Finally, the presence of two phonon\nbands in the mid-infrared can be exploited to tune its optical\nproperties by coupling them to plasmon polaritons from graphene\nor from a metal grating [14–16].\nHexagonal BN crystallizes in the P6\n3/mmc space group with\nfour atoms in the unit cell and with the following irreducible re-presentation of the phonon modes at the Brillouin zone center:\nΓ= ( )+ ( )+ ( )+ ( ) A IR E IR 2E Raman 2B silentvib u u g g 21 2 1 . This h-BN poly-\nmorphic modi fication consists of sp2-bonded 2D layers [1]. Within\neach layer the atoms are bound by strong covalent bonds, whereas\nthe layers are held together by weak Van der Waals\nforces. Since 1950, a continued effort has been made for thecharacterization of the physical properties of this compound. In\nparticular, the optical properties of h-BN were studied by means of\na large number of experimental techniques, mainly for wave-\nnumbers above 3000 cm\n/C01, using single crystals, polycrystals and\nthin films deposited on different substrates (see for example re-\nferences in [7]). First principles and ab initio calculations were\nmade using different theories and approximations in order to\nanalyze the experimental results. However, infrared and Raman\nspectroscopic experimental data that characterize the optical\nproperties in the mid-infrared range or that allow obtaining theThis document is the Accepted Manuscript version of a Published Work that appeared in \nfinal form in Journal of Quantitative Spectroscopy & Radiative Transfer 194 (2017) 1–6  \nafter peer review and technical editing by the publisher. To access the final edited and \npublished work see http://dx.doi.org/10.1016/j.jqsrt.2017.02.016energies of the phonons at the Γ-point of the Brillouin zone are\nscarce [17–24]. The same scarcity occurs with respect to the\nnumber of publications that give theoretical values of the di-\nelectric constants and the frequencies of the Γ-point phonons\nbased on ab initio calculations [25–30]. Moreover, some dis-\ncrepancies may be found among data from the literature, mainly in\nthe parallel components of the static and high frequency dielectricconstants, but also in some of the IR active phonon frequencies\n[17,19–21,24]. These discrepancies are also observed between ex-\nperimental and theoretical values [25–30].\nIn this paper, we report an experimental study on the emis-\nsivity spectra of pyrolytic boron nitride (pBN), as well as its opticalconstants,\nΓphonon frequencies and dielectric properties (high-\nfrequency and static dielectric constants) obtained by thermal\nemission spectroscopy. The theoretical interpretation of the ex-\nperimental results is carried out by using a four-parameter di-\nelectric function model as well as a Kramers –Kronig analysis. In\naddition to the basic interest of these measurements for the ana-lysis of the discrepancies indicated above, this material shows\npromising properties in the mid-infrared range that make it at-\ntractive for many applications. As noted above, pBN may be anappropriate candidate for the main thermal protection system of\nthe Solar Probe Plus (NASA) [10,31]or, from a broader viewpoint,\nas a high temperature passive thermal controller. Besides, tunableinfrared absorption has also been achieved with a h-BN metama-\nterial, and it is known that numerical simulations for these ap-plications require accurate values of the temperature-dependent\ndielectric function [32]. In order to implement these possible\napplications, it is necessary to expand the scarce number of pub-lished data on the temperature dependence of its optical proper-\nties [31,33,34]. The paper also presents the first study on the\ntemperature dependence of the optical constants,\nΓphonon fre-\nquencies and dielectric properties (high-frequency and static di-electric constants) of pBN samples.\n2. Experimental\nThe pBN sample studied in this paper consisted of a 1 mm thick\nsquare plate of hexagonal BN deposited on a graphite substrate by\npyrolysis of precursor gases at 1800 °C[35]. The route employed\nensures good homogeneity and high purity, since it does not in-\nvolve binding phases. The composition and microstructure of the\nsample were characterized by chemical analysis, X-ray diffraction\n(XRD) and scanning electron microscopy (SEM). The X-ray dif-\nfractogram (shown in Fig. 1 ) and the SEM images are in complete\nagreement with those of a previous sample [24]. The sample\ndensity was measured to be 2.03 g/cm\n3, a value 11% lower than thesingle-crystal value. The surface roughness parameters, measured\nwith a conventional pro filometer, are =μR 0.90 ma , =μR 1.08 mq ,\n=μR 4.35 mz and =μR 5.15 mt .\nThe emissivity measurements were made with a radiometer\nwhich has been described in the literature [36]. The sample\nchamber was purged with dry air in order to avoid CO 2and water\nabsorption. The sample was heated with a CO 2laser ( λ=μ10.6 m ),\nwhich allows reaching temperatures beyond 1000 °C. The sample\nthermal emission was collected with a Fourier transform infraredspectrometer calibrated by a black-body. The emissivity was thendetermined in the whole spectrum by comparison of the sampleand black-body signals through the following formula:\n=(− )\n(−)×−\n− ()EFT I I\nFT I IPP\nPPE\n1SR T\nBB RTBB RT\nSR TBB\nwhere FTmeans Fourier transform; IS,IBBand IRTare the inter-\nferograms recorded for the sample, black-body and room tem-\nperature background of the apparatus; PS,PBBand PRTare the\nPlanck functions calculated at the temperatures of the sample,\nblack-body and apparatus; and EBBis the emissivity of the black-\nbody.\nThe temperature of the sample was determined at the Chris-\ntiansen point ( λChris), a wavelength at which ceramic materials\nbehave like a black-body ( E¼1). This point does not depend on\ntemperature and is thus used as a common non-contact methodfor surface temperature measurement in ceramic materials [37].I n\nthis paper, it was determined by a room-temperature diffuse re-\nflectance measurement using an integrating sphere, with a value\nof\nλ =μ6.0 mChris .\n3. Results and discussion\nThe X-ray diffractogram of Fig. 1 confirms the polycrystalline\nnature of the material, with no preferential orientation of thecrystallites. It also reveals, together with the results from SEM\nmicroscopy, the presence of a small amount of graphite, which\noriginates in the deposition process and has been found in similarsamples in the literature [38]. The size of the graphite inclusions\nwas observed to be smaller than the wavelength range of interest\nin this paper, which justi fies the choice of the theoretical models\nfor the interpretation of the data.\nThe emissivity spectrum of pBN was measured in the 390 –\n1050 °C temperature range and for wavelengths between 4 and\n16\nμm. The wavelength range of study was limited by the onset of\ntransmittance at both extremes, which makes the emissivity of thematerial to fall abruptly to 0. The maximum temperature was\nchosen to avoid any possible oxidation, while the minimum tem-perature chosen was the lowest one that ensured an acceptable\nsignal-to-noise ratio in the selected spectral range. The emissivity\nspectra for three temperatures are shown in Fig. 2 .\nThe spectra show two reststrahlen bands at 6 –8 and 12 –14\nμm,\ncorresponding to the two infrared active phonons of h-BN. Thesespectra show a great similarity to those observed by re flectance\nmeasurements at room temperature [17,24]and with the only\nemissivity spectrum found in the literature [34]. No experimental\ndata in the infrared was found for single crystals. However, thereare slight differences between the measured spectrum on poly-crystalline samples [17] or highly oriented ones (hopBN) [24] with\nthe measured spectrum in this paper. Group theory predicts thatthe observation of each phonon requires a determined direction of\npolarization (\n→∥Ecand→⊥Ec). This occurs for hopBN samples but\nnot for pBN, for which, as shown in Fig. 2 , both phonons are ob-\nserved in a near-normal measurement without polarization. This\nbehavior is associated, as mentioned before, with the presence of\n Fig. 1. X-ray diffractogram of the sample, taken with Cu-αKradiation.randomly oriented crystallites observed from X-ray diffraction and\nSEM microscopy.\nA minor peak occurs at 6.4 μm, which becomes progressively\nweaker with temperature. This peak has been observed in re-flectivity measurements on polycrystalline samples at room tem-\nperature and was attributed to 2-phonon combination processes[17]. However, more realistic recent results attribute this feature to\nthe presence of pores and impurities in similar ceramic materials,such as NiO [39], SiC [40] and GaN [41].\nThe observed Christiansen wavelength in this sample was\nλ =μ6.0 mChris , a value that agrees with that obtained with a\nhopBN sample ( λ =μ5.9 mChris )[24]. The characterization of this\npoint is very useful for pyrometric temperature measurements in\nvacuum or in a dry atmosphere (as in this paper), while the\nmeasured value of ε≃0.93 at the atmospheric window of 4 μmi s\nmore relevant for industrial applications of this material.\nIn order to analyze the pBN spectra, the data was fitted to a\ndielectric function model, which is related to the emissivity and\nreflectance through the Fresnel relations [42]. The model chosen,\ncalled the four-parameter or Kurosawa model [43], has been suc-\ncessfully employed to fit optical spectra of a great number of\nceramic materials [44,45]:\nωΩω γ ω\nΩω γ ωϵ( ) = ϵ−+\n−+()⊥∥ ∞⊥∥ ⊥∥\n⊥∥ ⊥∥ii2LO LO\nTO TO,,22\n,\n,22\n,\nwhere ϵ∞is the high-frequency electronic contribution to the di-\nelectric function, Ωis the frequency of an optical phonon (trans-\nverse or longitudinal), γis the damping frequency of that phonon,\nand⊥and∥are the two possible electric field polarizations for a\nhexagonal crystal (perpendicular or parallel to the optical c-axis).\nAs the sample is a randomly oriented polycrystal, the global\ndielectric function should be an average of the two polarizations,\nin which the perpendicular accounts for 2/3 of the total because of\nits higher multiplicity:\nϵ=ϵ +ϵ()∥⊥1\n32\n3 3av\nFinally, the effect of the microstructure and the presence of\ncarbon impurities observed by the XRD and SEM experimental\ntechniques, in agreement with data in the literature for similar\nsamples, should be taken into account. To this end, and given thesmall size of the voids and defects ( o1\nμm), the Maxwell –Garnett\neffective medium theory was applied [46]. This model takes into\naccount the presence of spherical pores of dielectric function ϵi\nand its volume fraction fi, and returns an effective value of the\ndielectric function of the global medium:ϵ− ϵ\nϵ+ϵ=ϵ− ϵ\nϵ+ ϵ ()⎛\n⎝⎜⎞\n⎠⎟ f22 4ef av\nef aviia v\nia v\nwhere ϵefstands for the effective dielectric function, ϵifor one of\nthe impurities, fifor their volume concentration and ϵavfor the\naverage dielectric function of the polycrystal, given by Eq. (3).\nThe experimental data for an intermediate temperature of\n691 °C have been fitted to this model, with the result shown in\nFig. 3 . There is an excellent agreement, even around the 6.4 μm\nminor peak, which could not be accounted for without the effec-\ntive medium theory.\nThe parameters obtained with this model (phonon frequencies\nand dielectric constants) are shown in Table 1 , together with other\nliterature experimental and theoretical parameters obtained usingfirst principles calculations. The temperature dependence of these\nparameters is plotted in Figs. 4 and 5.Afirst view of Table 1 in-\ndicates that there is a small variation between the values of thinfilms and those of bulk samples. Among the bulk samples, the only\nnoticeable discrepancy appears in the values of the dielectricconstants obtained with a polycrystalline sample [17]. These data\nwere the only experimental values available until 1999 and dis-agree with all theoretical predictions prior to this date [25,26,28].\nDue to the need of an accurate prediction of the band-gap for first\nprinciples calculations, it was argued that the discrepancy could be\nassociated with the difference between the calculated and\nFig. 2. Emissivity spectrum of pBN for three selected temperatures.\n Fig. 3. Theoretical fit of the experimental data at 691 °C to Eqs. (2)–(4). Only some\nexperimental points are shown.\nTable 1\nCompilation of experimental and theoretical values of the phonon frequencies (incm\n/C01) and dielectric constants of pBN. The values shown for this paper correspond\nto the lowest temperature measured (392 °C) to allow comparison with room\ntemperature values. All theoretical calculations referenced rely on the local densityapproximation (LDA) of the density functional theory. The kind of atomic bases orpseudopotentials used are specifi ed in the first column.\nSample\nΩ∥TO,Ω∥LO,Ω⊥TO,Ω⊥LO,ϵ⊥0, ϵ∥0, ϵ∞⊥,ϵ∞∥,Ref.\nExperimentalThis paper 773 829 1373 1601 7.06 2.52 5.19 2.19hopBN 770 826 1365 1622 6.3 2.3 4.4 2.0 [24]\npBN 783 828 1367 1610 7.04 5.09 4.95 4.1 [17]\npBN 770 – 1383 –– – – – [20]\nThin film\n(sputtering)766 811 1400 1586 –––– [21]\nThin film (PECVD) 790 828 1386 1599 –– 4.71 2.72 [22]\nTheoryOLCAO ––– –– – 4.32 2.21 [25]\nUltrasoft 754 823 1382 1614 6.61 3.38 4.85 2.84 [26]\nUltrasoft 757 783 1366 1587 –––– [27]\nNorm-conserving 746 819 1372 1610 6.71 3.57 4.87 2.95 [28]\nNorm-conserving 745 – 1375 1607 –––– [29]\nNorm-conserving 760 824 1376 1610 –––– [30]measured values for the gap. For the rest of the experimental va-\nlues of the parameters of Table 1 , the discrepancies ( <1.5%)a r e\nwithin the experimental error, even for thin films. Besides, the\nvalues of the dielectric constants are insensitive to the method\nused (e.g., ultra soft or norm-conserving pseudopotentials). How-\never, the same is not true for all phonon frequencies. All theore-\ntical predictions give values of Ω∥TO,between 3% and 5 %lower than\nthose found experimentally. For the other modes there is a slight\ndiscrepancy between the theoretical values, which are reasonably\nclose to the experimental ones. The only exception is the Ω⊥LO,in\nReference [27].\nThe other parameters of the Kurosawa model have not been\nthoroughly studied. The errors introduced by the fitting procedure\nin the values of the damping constants are greater than the slight\npositive temperature dependence that could be observed. The\nmean values for the perpendicular modes are γ =±⊥42 2TO,cm/C01\nand γ =±⊥99 8LO,cm/C01. No conclusions could be extracted from\nthe parallel damping constants, since the fitting algorithm in-\ntroduced noticeable errors near the end of the spectral range. Thevalues reported above are in good agreement with the data given\nin the literature for room temperature [22,24]. The effective\nmedium parameters have been found to be equal at all tempera-tures, within the expected uncertainty and with values of\nϵ= 5.0i\nand =f 8%i. These parameters are crucial for an accurate de-\nscription of the global emission spectra, but the approximation\nchosen is not precise enough to allow any discussion of the\nproperties of the impurities themselves.\nThe temperature dependence of the phonon frequencies and\ndielectric constants are plotted in Figs. 4 and 5. All data in thefigures are normalized to the value at 392 °C (given in Table 1 ) and\nshifted for clarity. Two facts should be highlighted. The frequencies\nof the perpendicular mode and the dielectric constants of the\nparallel one show an appreciable temperature dependence. On the\ncontrary, the other components show a very slight dependence.\nThe parallel dielectric constants show a parabolic behavior that\npasses through a minimum around 900 °C, whereas for the per-\npendicular components of the frequencies the behavior is concavedownwards. No explanation was found for the behavior of thedielectric constant, but the perpendicular frequencies show the\ntypical shape due to three- and four-phonon interactions [47]. This\nhighlights the need for theoretical calculations that take tem-perature into account, as well as more experimental studies of the\ntemperature-dependent optical parameters in other ceramic\ncompounds.\nIn order to obtain more precise values of the dielectric function\nof the material, a Kramers –Kronig inversion of the re flectivity of the\nsample (\n=−RE 1 ) was performed. Contrary to the model- fitting\napproach, this method involves purely mathematical operations onthe data without any physical input. The imaginary part of the\nFresnel re flection coef ficient (\n*=θrr ei)i sc a l c u l a t e df r o mt h er e a l\npart ( =rR ) by means of the following equation [48]:\n∫ θωπω\nω() = −+−(( ) )\n()∞\nPssrs\nsds1lndl n\nd 5\n0\nwhere Pstands for principal value of the integral.\nThe result of the Kramers –Kronig is shown in Fig. 6 , together\nwith the dielectric functions generated with the parameters con-\ntained in Table 1 .\nIt can be seen that the two methods lead to curves which\noverlap for most of the spectral range considered. However, thereis a noticeable difference around the main phonon at 7.5\nμm, more\nclearly seen in the imaginary part of the dielectric function. Thismay reveal an inadequacy of the semiclassical models when trying\nto obtain precise values of the optical constants of materials, even\nFig. 4. Temperature dependence of the phonon frequencies, normalized to the\nvalue for the lowest temperature (indicated in Table 1 ). The parallel components\nhave been shifted 0.01 upwards for clarity.\nFig. 5. Temperature dependence of the static and high-frequency dielectric con-\nstants, normalized to the value for the lowest temperature (indicated in Table 1 ).\nThe perpendicular components have been shifted 0.05 upwards for clarity.\nFig. 6. Comparison of the real (a) and imaginary (b) parts of the dielectric functions\nobtained from the experimental emissivity data at 691 °C from the fitting of the\nKurosawa model and the Kramers –Kronig inversion.when they reproduce successfully the experimental emissivity or\nreflectivity data. A similar problem was observed in a pBN sample\nfrom the literature, for which only the effect of porosity ( ϵ= 1i)\nwas considered and not that of general impurities [24]. The\nmodelling present on this paper improves the quality of the fitting,\nand thus the discrepancies between the dielectric functions arelower than in that previous paper, but still signi ficant. However, no\ndefinitive conclusions can be made in the absence of an in-\ndependent determination of the optical constants.\n4. Conclusions\nThe emissivity spectra of pBN were measured as a function of\ntemperature and fitted to a four-parameter dielectric function\nmodel taking into account the random orientation of the crystals\nand the presence of carbon impurities, which were described byan effective medium approximation. This modelling approachshows a better agreement between the theoretical and experi-mental spectra than previous attempts.\nThe temperature dependence of the optical and dielectric\nparameters of this material has been reported for the first time.\nThe values obtained by the fitting procedure agree well with ex-\nperimental data at room temperature and with first-principles\ncalculations. No systematic differences have been found for dif-ferent sample types, except for those of thin films. A parabolic\ntemperature dependence was found for the frequencies of theperpendicular components of the phonons and for the parallel\ncomponents of the dielectric constants. This temperature depen-\ndence could not be explained by the current ab initio calculations,\nwhich rely almost exclusively on the T¼0 approximation.\nThe quality of the experimental results presented in this work\nconfirms that thermal emission spectroscopy is a good alternative\nfor the study of optical properties of ceramics beyond room tem-perature. Knowledge of the emissivity spectra at high temperatures\nis necessary for many interesting applications, such as protective\nand selective coatings for the aerospace and solar energy industries.Moreover, accurate temperature-dependent values of the dielectricfunction are also crucial for numerical simulations of complex ma-terials based on h-BN, such as tunable infrared absorbers.\nAcknowledgments\nThis research was partially supported by the research program\nof UPV/EHU. T. Echániz would like to acknowledge the UPV/EHU\nfor its support through a postdoctoral fellowship. I. 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Appl Spectrosc 2007;61:1390 –1397.\nhttp://dx.doi.org/10.1366/000370207783292163 ."    },    {        "title": "2401.13116v2.Sharp_second_order_regularity_for_widely_degenerate_elliptic_equations.pdf",        "content": "arXiv:2401.13116v2  [math.AP]  29 Jan 2024Sharp second-order regularity for widely\ndegenerate elliptic equations\nPasquale Ambrosio, Antonio Giuseppe Grimaldi, Antonia Pas sarelli di Napoli\nJanuary 29, 2024\nAbstract\nWe consider local weak solutions of widely degenerate or sin gular elliptic PDEs of the type\n−div/parenleftbigg\n(|Du|−λ)p−1\n+Du\n|Du|/parenrightbigg\n=finΩ,\nwhereΩis an open subset of Rnforn≥2,λis a positive constant and (·)+stands for the\npositive part. We establish some higher differentiability r esults, under essentially sharp\nconditions on the datum f. Our results improve the one contained in [ 8] forλ= 0and\np >2, and give back a result similar to that in [ 12] for1< p≤2.\nMathematics Subject Classification: 35J70, 35J75, 35J92, 49K20.\nKeywords: Degenerate elliptic equations; singular elliptic equatio ns; sharp second-order reg-\nularity.\n1 Introduction\nThe aim of this paper is to establish a second-order regulari ty result for weak solutions to\nstrongly singular or degenerate elliptic equations of the t ype\n−div/parenleftbigg\n(|Du|−λ)p−1\n+Du\n|Du|/parenrightbigg\n=fin Ω, (1.1)\nwhere1< p <∞,Ωis an open subset of Rn(n≥2),λ >0is a fixed parameter and (·)+stands\nfor the positive part. The peculiarity of equation ( 1.1) is that it is uniformly elliptic only outside\nthe ball centered at the origin with radius λ, where its principal part behaves asymptotically\nas the classical p-Laplace operator. Therefore, the study of such an equation fits into the wider\nclass of the asymptotically regular problems that have been extensively studied starting from\nthe pioneering paper [ 9] (see also [ 14,15,16,18,19,20,22,24,26,27] for extensions to various\nother settings).\nIt is well known that for the solutions of ( 1.1), even in the case f= 0, no more than C0,1\nregularity can be obtained. Actually, any Lipschitz functi on with Lipschitz constant less than\nor equal to λsolves ( 1.1) withf= 0.\nOn the other hand, higher regularity for the solutions such a s the higher differentiability of\ninteger or fractional order is known to hold outside the dege neracy set of the equation. We\nrefer the reader to [ 2,6,7,13]. More precisely, when the source term possesses a suitable\n12 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\nSobolev regularity, the higher differentiability is establ ished for a nonlinear function of the\ngradient Duof the weak solutions that vanishes in the degeneracy set {|Du| ≤λ}.\nHere, our results tail the ones mentioned above with the purp ose of identifying the sharp\nassumptions on the datum f, as already done for the p-Poisson equation in [ 8]. More precisely,\nforp >2, our result reads as follows. We refer to Sections 2and3for notation and definitions.\nTheorem 1.1. Letn≥2,p >2,α≥p+1\n2(p−1)andf∈Bp−2\np\np′,1,loc(Ω), wherep′=p/(p−1)is the\nconjugate exponent of p. Moreover, let u∈W1,p\nloc(Ω)be a local weak solution of equation (1.1).\nThen, setting\nGα,p(t) :=ˆt\n0ωp−1+2α\n2\n(ω+λ)1+2α\n2dωfort≥0,\nwe have\nGα,p((|Du|−λ)+)∈W1,2\nloc(Ω). (1.2)\nFurthermore, the following estimate\nˆ\nBr/4|DGα,p((|Du|−λ)+)|2dx\n≤C\nr2/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nLp′(BR)/bracketrightBig\n+C/bardblf/bardblp′\nBp−2\np\np′,1(BR)(1.3)\nholds for every pair of concentric balls Br⊂BR⋐Ωand for a positive constant Cdepending\nonly onn,p,αandR.\nThe proof of the previous theorem is achieved combining an a priori estimate for the solution\nof a suitable approximating problem with a comparison estim ate. In establishing the a priori\nestimate, we first need to identify a suitable function of the gradient that vanishes in the\ndegeneracy set, for which the second-order a priori estimate holds true. Next, we need to\nestimate the right-hand side in terms of the derivatives of s uch function, without assuming any\nSobolev regularity for the datum f. This is done by virtue of the following implication\n|g|p−2\n2g∈W1,2\nloc(Ω)⇒g∈B2\np\np,∞,loc(Ω), (1.4)\nwhich allows us to use the duality of Besov spaces, provided o ne imposes a suitable Besov\nregularity on the right-hand side f. Finally, we use a comparison argument to transfer the\nhigher differentiability of the approximating solutions to the solution of equation ( 1.1).\nFor the sake of clarity, for p >2we remark that\n|DGα,p((|Du|−λ)+)| ≈(|Du|−λ)p−1+2α\n2\n+\n|Du|1+2α\n2|D2u|=(|Du|−λ)1+2α\n2\n+\n|Du|1+2α\n2(|Du|−λ)p−2\n2\n+|D2u|.\nTherefore, for λ= 0, Theorem 1.1improves [ 8, Theorem 1.1], since |DGα,p(|Du|)| ≈\n|D(|Du|p−2\n2Du)|belongs to L2\nloc(Ω)under the assumption f∈Bp−2\np\np′,1,loc(Ω), which is weaker\nthanf∈Ws,p′\nloc(Ω)for(p−2)/p < s≤1, due to Lemma 3.8below.\nMoreover, our assumption on the datum fis sharp, in the sense that assertion ( 1.2) is false\nifλ= 0and the order of differentiation of fgoes below (p−2)/p. Indeed, Brasco and San-\ntambrogio [ 8, Section 5] showed with an explicit example that an extensio n of our Theorem\n1.1is not possible when λ= 0andfbelongs to a fractional Sobolev space Wσ,p′\nloc(Rn), withP. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 3\n0< σ <(p−2)/p, which is continuously embedded into Bs\np′,1,loc(Rn)whenever s∈(0,σ)(see\nLemma 3.8below).\nHowever, we would like to mention that in [ 10,11], the authors proved that the assumption\nf∈L2is sufficient to prove the W1,2regularity of |Du|p−2Du, which is of course a different\nfunction of the gradient. At the moment, we do not know whethe r the analogous result can be\nobtained for the solutions of widely degenerate equations.\nNow we turn our attention to the sub-quadratic case, i.e. whe n1< p≤2. It is well known\nthat, already for the less degenerate case of the p-Poisson equation, the higher differentiability\nof the solutions can be achieved without assuming any differe ntiability on the datum f. This\ndifferent behaviour can be easily explained by observing tha t, if1< p≤2,\n|g|p−2\n2g∈W1,2\nloc(Ω)⇒g∈W1,p\nloc(Ω),\nwhich of course does not hold true for p >2(compare with ( 1.4)). Therefore, the right-hand\nside can be estimated without assuming any differentiabilit y forf(neither of integer nor of\nfractional order), but only a suitable degree of integrabil ity. The sharp assumption on fin the\nscale of Lebesgue spaces has been recently found in [ 12].\nHere, we prove that a result analogous to [ 12, Theorem 1.1] holds true also when dealing\nwith solutions of widely singular equations. More precisel y, we have:\nTheorem 1.2. Letn≥2,1< p≤2,α≥p+1\n2(p−1)andf∈Lnp\nn(p−1)+2−p\nloc (Ω). Moreover, let\nu∈W1,p\nloc(Ω)be a local weak solution of equation (1.1). Then\nGα,p((|Du|−λ)+)∈W1,2\nloc(Ω).\nFurthermore, for every pair of concentric balls Br⊂BR⋐Ω, we have\nˆ\nBr/4|DGα,p((|Du|−λ)+)|2dx\n≤C\nr2/bracketleftbigg\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nLnp\nn(p−1)+2−p(BR)/bracketrightbigg\n+C/bardblf/bardblp′\nLnp\nn(p−1)+2−p(BR)(1.5)\nfor a positive constant Cdepending only on n,p,αand R.\nAs an easy consequence of the higher differentiability resul ts in Theorems 1.1and1.2, since\nthe gradient of the solution is bounded in the degeneracy set andGα,p(t)≈tp/2for large values\noft(see Lemma 2.4below), we are able to establish the following higher integr ability result for\nthe gradient of the local weak solutions of ( 1.1):\nCorollary 1.3. Under the assumptions of Theorem 1.1or Theorem 1.2, we have\nDu∈Lq\nloc(Ω),\nwhere\nq=/braceleftBigg\nany value in [1,∞)ifn= 2,\nnp\nn−2ifn≥3.\nThe paper is organized as follows. Section 2is devoted to the preliminaries: after a list\nof some classic notations and some essentials estimates, we recall the basic properties of the\ndifference quotients of Sobolev functions. Section 3is entirely devoted to the definitions and\nproperties of Besov spaces that will be useful to prove our re sults. In Section 4, we establish\nsomea priori estimates that will be needed to demonstrate Theorems 1.1and1.2, whose proofs\nare contained in Sections 5and6, respectively.4 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\n2 Notations and preliminaries\nIn this paper we shall denote by Corca general positive constant that may vary on different\noccasions. Relevant dependencies on parameters and specia l constants will be suitably empha-\nsized using parentheses or subscripts. The norm we use on Rnwill be the standard Euclidean\none and it will be denoted by |·|. In particular, for the vectors ξ,η∈Rn, we write /a\\}bracketle{tξ,η/a\\}bracketri}htfor\nthe usual inner product and |ξ|:=/a\\}bracketle{tξ,ξ/a\\}bracketri}ht1\n2for the corresponding Euclidean norm.\nIn what follows, Br(x0) ={x∈Rn:|x−x0|< r}will denote the n-dimensional open ball\ncentered at x0with radius r. We shall sometimes omit the dependence on the center when al l\nballs occurring in a proof are concentric. Unless otherwise stated, different balls in the same\ncontext will have the same center.\nWe now recall some tools that will be useful to prove our resul ts. Combining [ 1, Lemma 2.2]\nwith [ 22, Formula (2.4)], we obtain the following\nLemma 2.1. Let1< p <∞. There exists a constant c≡c(n,p)>0such that\nc−1(|ξ|2+|η|2)p−2\n2≤/vextendsingle/vextendsingle/vextendsingle|ξ|p−2\n2ξ−|η|p−2\n2η/vextendsingle/vextendsingle/vextendsingle2\n|ξ−η|2≤c(|ξ|2+|η|2)p−2\n2\nfor every ξ,η∈Rnwithξ/\\e}atio\\slash=η.\nFor the auxiliary function Hγ:Rn→Rndefined by\nHγ(ξ) :=/braceleftBigg\n(|ξ|−λ)γ\n+ξ\n|ξ|ifξ∈Rn\\{0},\n0 if ξ= 0,\nwhereγ >0is a parameter, we record the following estimate, whose proo f can be found in [ 6,\nLemma 4.1] for the case p≥2, and in [ 2, Lemma 2.5] for the case 1< p <2.\nLemma 2.2. Let1< p <∞. Then, there exists a constant β≡β(n,p)>0such that\n/a\\}bracketle{tHp−1(ξ)−Hp−1(η),ξ−η/a\\}bracketri}ht ≥β/vextendsingle/vextendsingle/vextendsingleHp\n2(ξ)−Hp\n2(η)/vextendsingle/vextendsingle/vextendsingle2\n,\nfor every ξ,η∈Rn.\nFor further needs, we introduce the function\nGα,p(t) :=ˆt\n0ωp−1+2α\n2\n(ω+λ)1+2α\n2dωfort≥0,\nwhereα≥0. The next lemma relates the function Gα,p((|ξ|−λ)+)withHp\n2(ξ).\nLemma 2.3. If1< p <∞, then for every α≥0we have\n|Gα,p((|ξ|−λ)+)−Gα,p((|η|−λ)+)| ≤2\np/vextendsingle/vextendsingle/vextendsingleHp\n2(ξ)−Hp\n2(η)/vextendsingle/vextendsingle/vextendsingle.\nfor every ξ,η∈Rn.P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 5\nProof. The above inequality is trivially satisfied when |ξ|,|η| ≤λ. Therefore, we shall assume\nthat|ξ|> λand, without loss of generality, we may suppose that |η| ≤ |ξ|. SinceGα,pis an\nincreasing function, we have\n|Gα,p((|ξ|−λ)+)−Gα,p((|η|−λ)+)|\n=Gα,p(|ξ|−λ)−Gα,p((|η|−λ)+) =ˆ|ξ|−λ\n(|η|−λ)+ωp−1+2α\n2\n(ω+λ)1+2α\n2dω\n≤ˆ|ξ|−λ\n(|η|−λ)+ωp−2\n2dω=2\np/bracketleftBig\n|Hp\n2(ξ)|−|Hp\n2(η)|/bracketrightBig\n≤2\np/vextendsingle/vextendsingle/vextendsingleHp\n2(ξ)−Hp\n2(η)/vextendsingle/vextendsingle/vextendsingle.\nThis completes the proof.\nWe conclude this section with the proof of the following lemm a, on which the conclusion of\nCorollary 1.3is based.\nLemma 2.4. Letλ >0,α >1\n2and1< p <∞. Then, there exist two positive constants\nc≡c(p,α)and˜c≡˜c(p,α)such that\nc(λ+t)p/2−˜cλp/2≤Gα,p(t)≤2\nptp/2\nfor allt≥0.\nProof. From the very definition of the function Gα,p, we easily get the upper bound\nGα,p(t)≤ˆt\n0ωp−2\n2dω=2\nptp/2for allt≥0.\nFor the derivation of the lower bound, we calculate the integ ral in the definition of Gα,p(t). By\nthe change of variable r=ω+λ, we have\nˆt\n0ωp−1+2α\n2\n(ω+λ)1+2α\n2dω=ˆλ+t\nλ(r−λ)p−1+2α\n2\nr1+2α\n2dr.\nNow we recall that for every γ >0it holds\n2−γaγ−bγ≤(a−b)γ,∀a≥b≥0. (2.1)\nUsing inequality ( 2.1) withγ=p−1+2α\n2,a=randb=λ, we find that\nˆλ+t\nλ(r−λ)p−1+2α\n2\nr1+2α\n2dr≥c(p,α)ˆλ+t\nλrp−1+2α\n2\nr1+2α\n2dr−ˆλ+t\nλλp−1+2α\n2\nr1+2α\n2dr\n=c(p,α)ˆλ+t\nλrp−2\n2dr−λp−1+2α\n2/bracketleftBigg\n2r1−2α\n2\n1−2α/bracketrightBiggλ+t\nλ\n=c(p,α)/bracketleftbigg2rp\n2\np/bracketrightbiggλ+t\nλ+2λp−1+2α\n2\n(2α−1)(λ+t)2α−1\n2−2λp−1+2α\n2\n(2α−1)λ2α−1\n2\n≥c(p,α)(λ+t)p/2−c(p,α)λp/2−2λp/2\n2α−1, (2.2)\nwhere, in the last line, we have used that α >1\n2. Finally, setting ˜c:=c(p,α)+2\n2α−1, we obtain\nthe asserted lower bound.6 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\n2.1 Difference quotients\nWe recall here the definition and some elementary properties of the difference quotients that\nwill be useful in the following (see, for example, [ 23]).\nDefinition 2.5. For every vector-valued function F:Rn→Rkthefinite difference operator\nin the direction xjis defined by\nτj,hF(x) =F(x+hej)−F(x),\nwhereh∈R,ejis the unit vector in the direction xjandj∈ {1,...,n}.\nThedifference quotient ofFwith respect to xjis defined for h∈R\\{0}by\n∆j,hF(x) =τj,hF(x)\nh.\nWhen no confusion can arise, we shall omit the index jand simply write τhor∆hinstead of\nτj,hor∆j,h, respectively.\nProposition 2.6. LetFbe a function such that F∈W1,q(Ω;Rk), withq≥1, and let us\nconsider the set\nΩ|h|:={x∈Ω : dist(x,∂Ω)>|h|}.\nThen:\n(i) ∆hF∈W1,q/parenleftbig\nΩ|h|/parenrightbig\nand∂i(∆hF) = ∆h(∂iF)for every i∈ {1,...,n}.\n(ii)If at least one of the functions ForGhas support contained in Ω|h|, then\nˆ\nΩF∆hGdx=−ˆ\nΩG∆−hF dx.\n(iii)We have\n∆h(FG)(x) =F(x+hej)∆hG(x) +G(x)∆hF(x).\nThe next result about the finite difference operator is a kind o f integral version of the Lagrange\nTheorem and its proof can be found in [ 23, Lemma 8.1].\nLemma 2.7. If0< ρ < R ,|h|<R−ρ\n2,1< q <+∞andF∈Lq(BR,Rk)is such that\nDF∈Lq(BR,Rk×n), then\nˆ\nBρ|τhF(x)|qdx≤cq(n)|h|qˆ\nBR|DF(x)|qdx.\nMoreover ˆ\nBρ|F(x+hej)|qdx≤ˆ\nBR|F(x)|qdx.\nFinally, we recall the following fundamental result, whose proof can be found in [ 23, Lemma\n8.2]:P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 7\nLemma 2.8. LetF:Rn→Rk,F∈Lq(BR,Rk)with1< q <+∞. Suppose that there exist\nρ∈(0,R)and a constant M >0such that\nn/summationdisplay\nj=1ˆ\nBρ|τj,hF(x)|qdx≤Mq|h|q\nfor every hwith|h|<R−ρ\n2. ThenF∈W1,q(Bρ,Rk). Moreover\n/bardblDF/bardblLq(Bρ)≤M\nand\n∆j,hF→∂jF in Lq\nloc(BR,Rk)as h→0,\nfor each j∈ {1,...,n}.\n3 Besov spaces\nHere we recall some essential facts on the Besov spaces invol ved in this paper.\nWe denote by S(Rn)andS′(Rn)the Schwartz space and the space of tempered distributions o n\nRn, respectively. If v∈S(Rn), then\nˆv(ξ) = (Fv)(ξ) = (2π)−n/2ˆ\nRne−i/an}bracketle{tx,ξ/an}bracketri}htv(x)dx, ξ∈Rn, (3.1)\ndenotes the Fourier transform of v. As usual, F−1vandv∨stand for the inverse Fourier\ntransform, given by the right-hand side of ( 3.1) withiin place of −i. BothFandF−1are\nextended to S′(Rn)in the standard way.\nNow, let Γ(Rn)be the collection of all sequences ϕ={ϕj}∞\nj=0⊂S(Rn)such that\n/braceleftBigg\nsuppϕ0⊂ {x∈Rn:|x| ≤2}\nsuppϕj⊂ {x∈Rn: 2j−1≤ |x| ≤2j+1}ifj∈N,\nfor every multi-index βthere exists a positive number cβsuch that\n2j|β||Dβϕj(x)| ≤cβ,∀j∈N0,∀x∈Rn\nand∞/summationdisplay\nj=0ϕj(x) = 1,∀x∈Rn.\nThen, it is well known that Γ(Rn)is not empty (see [ 28, Section 2.3.1, Remark 1]). Moreover,\nif{ϕj}∞\nj=0∈Γ(Rn), the entire analytic functions (ϕjˆv)∨(x)make sense pointwise in Rnfor any\nv∈S′(Rn). Therefore, the following definition makes sense:\nDefinition 3.1. Lets∈R,0< p,q≤ ∞andϕ={ϕj}∞\nj=0∈Γ(Rn). We define the inhomoge-\nneous Besov space Bs\np,q(Rn)as the set of all v∈S′(Rn)such that\n/bardblv/bardblBsp,q(Rn):=/parenleftBigg∞/summationdisplay\nj=02jsq/bardbl(ϕjˆv)∨/bardblq\nLp(Rn)/parenrightBigg1\nq\n<+∞ (3.2)\n(with the usual modification if q=∞).8 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\nRemark 3.2. Obviously, the quasi-norm /bardblv/bardblBsp,q(Rn)depends on the chosen sequence ϕ∈\nΓ(Rn), but this is not the case for the spaces Bs\np,q(Rn)themselves, in the sense that two\ndifferent choices for the sequence ϕgive rise to equivalent quasi-norms (see [ 28, Section 2.3.2,\nProposition 1]). This justifies our omission of the dependen ce onϕin the left-hand side of ( 3.2)\nand in the sequel.\nWe also know that the norms of the classical Besov spaces Bs\np,q(Rn)withs∈(0,1),1≤p <∞\nand1≤q≤ ∞ can be characterized via differences of the functions involv ed, cf. [ 28, Section\n2.5.12, Theorem 1]. More precisely, for h∈Rnand a measurable function v:Rn→Rk, let us\ndefine\nδhv(x) :=v(x+h)−v(x).\nThen we have the equivalence\n/bardblv/bardblBsp,q(Rn)≈ /bardblv/bardblLp(Rn)+[v]Bsp,q(Rn),\nwhere\n[v]Bsp,q(Rn):=/parenleftbiggˆ\nRn/parenleftbiggˆ\nRn|δhv(x)|p\n|h|spdx/parenrightbiggq\npdh\n|h|n/parenrightbigg1\nq\n,if1≤q <∞, (3.3)\nand\n[v]Bsp,∞(Rn):= sup\nh∈Rn/parenleftbiggˆ\nRn|δhv(x)|p\n|h|spdx/parenrightbigg1\np\n. (3.4)\nAs usual, in ( 3.3) if one simply integrates for |h|< rfor a fixed r >0, then an equivalent norm\nis obtained, since\n/parenleftbiggˆ\n{|h|≥r}/parenleftbiggˆ\nRn|δhv(x)|p\n|h|spdx/parenrightbiggq\npdh\n|h|n/parenrightbigg1\nq\n≤c(n,s,p,q,r )/bardblv/bardblLp(Rn).\nSimilarly, in ( 3.4) one can simply take supremum over |h| ≤rand obtain an equivalent norm.\nBy construction, Bs\np,q(Rn)⊂Lp(Rn).\nIn the case of an arbitrary domain Ω⊂Rn, the spaces Bs\np,q(Ω)are defined by restriction.\nMore precisely, denoting by D′(Ω)the dual space of all distributions in Ω, we set\nBs\np,q(Ω) :=/braceleftbig\nv∈D′(Ω) :v=g|Ωfor some g∈Bs\np,q(Rn)/bracerightbig\n,\nwith the quasi-norm /bardblv/bardblBsp,q(Ω):= inf/bardblg/bardblBsp,q(Rn), where the infimum is taken over all g∈\nBs\np,q(Rn)such that g|Ω=v.\nIfΩis a bounded C∞-domain in Rn, then the restriction operator\nreΩ:S′(Rn)֒→D′(Ω),reΩ(v) =v|Ω\ngenerates a linear and bounded map from Bs\np,q(Rn)ontoBs\np,q(Ω). Furthermore, the spaces\nBs\np,q(Ω)satisfy the so-called extension property , as ensured by the next theorem.\nTheorem 3.3. Lets∈R, let0< p,q≤ ∞ and letΩbe a bounded C∞-domain in Rn.\nThen, there exists a linear and bounded extension operator extΩ:Bs\np,q(Ω)֒→Bs\np,q(Rn)such that\nreΩ◦extΩ= id, whereidis the identity in Bs\np,q(Ω).\nFor the proof we refer to [ 29, Theorem 2.82] . For further needs, we now give the following\nDefinition 3.4. For a bounded domain Ω⊂Rn,s∈(0,1)and1≤p,q≤ ∞, thehomogeneous\nBesov space ˚Bs\np,q(Ω)is defined as the completion of C∞\n0(Ω)inBs\np,q(Ω)with respect to the\nseminorm\nv/mapsto→ /bardblv/bardbl˚Bsp,q(Ω):= [v]Bsp,q(Rn).P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 9\nFors∈(0,1)and1≤p,q≤ ∞, we shall denote by (˚Bs\np,q(Ω))′the topological dual of ˚Bs\np,q(Ω),\nwhich is endowed with the natural dual norm\n/bardblF/bardbl(˚Bsp,q(Ω))′= sup/braceleftBig\n|/a\\}bracketle{tF,ϕ/a\\}bracketri}ht|:ϕ∈C∞\n0(Ω) and /bardblϕ/bardbl˚Bsp,q(Ω)≤1/bracerightBig\n, F∈(˚Bs\np,q(Ω))′.\nNow we recall the following duality formula, which has to be m eant as an isomorphism of\nquasi-normed spaces (see [ 28, Section 2.11.2, Remark 2]).\nTheorem 3.5. Lets∈Rand1≤p,q≤ ∞. Then\n(˚Bs\np,q(Rn))′=B−s\np′,q′(Rn).\nThe next result is a key ingredient for the proof of Theorem 1.1and its proof can be found in\n[28, Section 3.3.5].\nTheorem 3.6. Lets∈Rand0< p,q≤ ∞. Moreover, assume that Ωis a bounded C∞-domain\ninRn. Then, for every v∈Bs\np,q(Ω)and every j∈ {1,...,n}we have\n/bardbl∂jv/bardblBs−1\np,q(Ω)≤c/bardblv/bardblBsp,q(Ω)\nfor a positive constant cwhich is independent of v.\nWe can also define local Besov spaces as follows. Given a domai nΩ⊂Rn, we say that a\nfunction vbelongs to Bs\np,q,loc(Ω)ifφv∈Bs\np,q(Rn)whenever φ∈C∞\n0(Ω).\nDefinition 3.7. LetΩ⊆Rnbe an open set. For any s∈(0,1)and for any q∈[1,+∞), we\ndefine the fractional Sobolev space Ws,q(Ω,Rk)as follows:\nWs,q(Ω,Rk) :=/braceleftBigg\nv∈Lq(Ω,Rk) :|v(x)−v(y)|\n|x−y|n\nq+s∈Lq(Ω×Ω)/bracerightBigg\n,\ni.e. an intermerdiate Banach space between Lq(Ω,Rk)andW1,q(Ω,Rk), endowed with the norm\n/bardblv/bardblWs,q(Ω):=/bardblv/bardblLq(Ω)+[v]Ws,q(Rn),\nwhere the term\n[v]Ws,q(Rn):=/parenleftbiggˆ\nΩˆ\nΩ|v(x)−v(y)|q\n|x−y|n+sqdxdy/parenrightbigg1\nq\nis the so-called Gagliardo seminorm ofv.\nWe conclude this section with the following embedding resul t, whose proof can be obtained\nby combining [ 28, Section 2.2.2, Remark 3] with [ 28, Section 2.3.2, Proposition 2(ii)].\nLemma 3.8. Lets∈(0,1)andq≥1. Then, for every σ∈(0,1−s)we have the continuous\nembedding Ws+σ,q\nloc(Rn)֒→Bs\nq,1,loc(Rn).10 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\n4 Estimates for a regularized problem\nThe aim of this section is to establish some uniform estimate s for the gradient of the weak\nsolutions of a family of suitable approximating problems. M ore precisely, let u∈W1,p\nloc(Ω)be\na local weak solution of ( 1.1), for some p >1. Fix an open ball BR⋐Ωand assume without\nloss of generality that R≤1. Forε∈(0,1], we consider the problem\n/braceleftBigg\n−div(DGε(Duε)) =fεinBR,\nuε=u on∂BR,(4.1)\nwhere:\n•Gε(z) :=1\np(|z|−λ)p\n++ε\np(1+|z|2)p\n2,for every z∈Rn;\n•fε:=f∗φεand{φε}ε>0is a family of standard compactly supported C∞mollifiers.\nObserve that\nDzGε(z) =Hp−1(z) +ε(1+|z|2)p−2\n2z. (4.2)\nNow we set, for s > λ,\nλ(s) :=\n\n(s−λ)p−1\nsifp >2,\n(p−1)(s−λ)p−1\nsif 1< p≤2,\nand\nΛ(s) :=/braceleftBigg\n(p−1)(s−λ)p−2ifp >2,\n(s−λ)p−2if 1< p≤2,\nandλ(s) = 0 =Λ(s)for0≤s≤λ. These definitions prove to be useful in the formulation of\nthe next lemma, whose proof follows from [ 4, Lemma 2.7] (see also [ 5]) together with standard\nestimates for the p-Laplace operator.\nLemma 4.1. Letε∈[0,1]andz∈Rn\\{0}. Then, for every ζ∈Rnwe have\n/bracketleftbig\nεc0(1+|z|2)p−2\n2+λ(|z|)/bracketrightbig\n|ζ|2≤ /a\\}bracketle{tD2Gε(z)ζ,ζ/a\\}bracketri}ht ≤/bracketleftbig\nεc1(1+|z|2)p−2\n2+Λ(|z|)/bracketrightbig\n|ζ|2,\nwherec0= min{1,p−1}andc1= max{1,p−1}.\nProof. Actually, setting for t >0\nh(t) =(t−λ)p−1\n+\ntanda(t) = (1+ t2)p−2\n2,\nwe can write\nDzGε(z) =/bracketleftBig\nh(|z|)+εa(|z|)/bracketrightBig\nz\nand we can easily calculate\nD2Gε(z) =/bracketleftBig\nh(|z|)+εa(|z|)/bracketrightBig\nI+/bracketleftbig\nh′(|z|)+εa′(|z|)/bracketrightBigz⊗z\n|z|.P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 11\nThus we get\n/a\\}bracketle{tD2Gε(z)η,ζ/a\\}bracketri}ht=/bracketleftBig\nh(|z|)+εa(|z|)/bracketrightBig\n/a\\}bracketle{tη,ζ/a\\}bracketri}ht+/bracketleftBig\nh′(|z|)+εa′(|z|)/bracketrightBign/summationdisplay\ni,j=1ziηizjζj\n|z|,\nfor anyη,ζ∈Rn. By the Cauchy-Schwarz inequality, we have\n0≤n/summationdisplay\ni,j=1ziζizjζj\n|z|≤ |z||ζ|2. (4.3)\nAt this point, if h′(|z|)+εa′(|z|)≥0(which occurs when p≥2), from the lower bound in ( 4.3)\nwe immediately obtain\n/a\\}bracketle{tD2Gε(z)ζ,ζ/a\\}bracketri}ht ≥/bracketleftbig\nh(|z|)+εa(|z|)/bracketrightbig\n|ζ|2=(|z|−λ)p−1\n+\n|z||ζ|2+ε(1+|z|2)p−2\n2|ζ|2.\nOn the other hand, using the upper bound in ( 4.3), forp≥2we deduce\n/a\\}bracketle{tD2Gε(z)ζ,ζ/a\\}bracketri}ht ≤(p−1)(|z|−λ)p−2\n+|ζ|2+ε(p−1)(1+|z|2)p−2\n2|ζ|2,\nwhere we have also used that\nh(t)+th′(t) = (p−1)(t−λ)p−2\n+ (4.4)\nand\na(t)+ta′(t)≤(p−1)(1+t2)p−2\n2whenp≥2.\nOtherwise, if h′(|z|)+εa′(|z|)<0(which may happen only when 1< p <2), we easily get\n/a\\}bracketle{tD2Gε(z)ζ,ζ/a\\}bracketri}ht ≤/bracketleftbig\nh(|z|)+εa(|z|)/bracketrightbig\n|ζ|2.\nFor the derivation of the lower bound, we use the right inequa lity in ( 4.3) to deduce that\n/a\\}bracketle{tD2Gε(z)ζ,ζ/a\\}bracketri}ht ≥/bracketleftbig\nh(|z|)+εa(|z|)/bracketrightbig\n|ζ|2+/bracketleftbig\nh′(|z|)+εa′(|z|)/bracketrightbig\n|z||ζ|2\n=/bracketleftbig\nh(|z|)+h′(|z|)|z|/bracketrightbig\n|ζ|2+/bracketleftbig\na(|z|)+εa′(|z|)|z|/bracketrightbig\n|ζ|2\n≥(p−1)(|z|−λ)p−2\n+|ζ|2+ε(p−1)(1+|z|2)p−2\n2|ζ|2,\nwhere, in the last line, we have used ( 4.4) and the fact that\na(t)+ta′(t)≥(p−1)(1+t2)p−2\n2when1< p <2.\nThis proves the claim.\nIn what follows, uε∈u+W1,p\n0(BR)will be the unique weak solution to ( 4.1). By standard\nelliptic regularity [ 23, Chapter 8], we know that (1+|Duε|2)p−2\n4Duε∈W1,2\nloc(Ω)and therefore\nuε∈W2,2\nloc(Ω). Moreover, as usual, we shall denote by p∗the Sobolev conjugate exponent of p,\ndefined as\np∗:=/braceleftBigg\nnp\nn−pifp < n,\nany value in (p,∞)ifp≥n,\nand denote by (p∗)′its Hölder conjugate exponent.\nThe proofs of Theorems 1.1and1.2are crucially based on the following results.12 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\nProposition 4.2 (Uniform energy estimate ).With the notation and under the assumptions\nabove, if f∈L(p∗)′(BR), there exist two positive constants ε0≤1andC≡C(n,p)such that\nˆ\nBR|Duε|pdx≤C/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n(4.5)\nfor allε∈(0,ε0].\nProof .We insert in the weak formulation of ( 4.1)\nˆ\nBR/a\\}bracketle{tDGε(Duε),Dϕ/a\\}bracketri}htdx=ˆ\nBRfεϕdxfor every ϕ∈W1,p\n0(BR),\nthe test function ϕ=uε−u. Recalling ( 4.2), this gives\nˆ\nBR/a\\}bracketle{tHp−1(Duε)+ε(1+|Duε|2)p−2\n2Duε,Duε/a\\}bracketri}htdx (4.6)\n=ˆ\nBR/a\\}bracketle{tHp−1(Duε)+ε(1+|Duε|2)p−2\n2Duε,Du/a\\}bracketri}htdx+ˆ\nBRfε(uε−u)dx.\nSince\n/a\\}bracketle{tHp−1(z)+ε(1+|z|2)p−2\n2z,z/a\\}bracketri}ht ≥(|z|−λ)p\n++ε(1+|z|2)p−2\n2|z|2≥(|z|−λ)p\n+\nfor every z∈Rn, we can estimate the integrals in ( 4.6), thus obtaining\nˆ\nBR(|Duε|−λ)p\n+dx\n≤ˆ\nBR|Duε|p−1|Du|dx+εˆ\nBR(1+|Duε|2)p−1\n2|Du|dx+/bardblfε/bardblL(p∗)′(BR)/bardbluε−u/bardblLp∗(BR)\n≤/parenleftBig\n1+2p−1\n2/parenrightBigˆ\nBR|Duε|p−1|Du|dx+ 2p−1\n2ˆ\nBR|Du|dx\n+c(n,p)/bardblfε/bardblL(p∗)′(BR)/bardblDuε−Du/bardblLp(BR),\nwhere we have used Hölder’s and Sobolev inequalities and the fact that ε,R≤1. Now, applying\nYoung’s inequality with σ >0, we arrive at\nˆ\nBR(|Duε|−λ)p\n+dx\n≤σˆ\nBR|Duε|pdx+c(n,p,σ)ˆ\nBR|Du|pdx+c(n,p,σ)/bracketleftBig\n1+/bardblfε/bardblp′\nL(p∗)′(BR)/bracketrightBig\n,\nwhere we have used again that R≤1. Since\nfε→fstrongly in L(p∗)′(BR) asε→0+, (4.7)\nthere exists a positive number ε0≤1such that\n/bardblfε/bardblL(p∗)′(BR)≤1+/bardblf/bardblL(p∗)′(BR)for allε∈(0,ε0].\nThen, for ε∈(0,ε0], we have\nˆ\nBR(|Duε|−λ)p\n+dxP. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 13\n≤σˆ\nBR|Duε|pdx+c/bardblDu/bardblp\nLp(BR)+c/bracketleftBig\n1+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n≤σˆ\nBR[λ+(|Duε|−λ)+]pdx+c/bardblDu/bardblp\nLp(BR)+c/bracketleftBig\n1+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n≤2p−1σˆ\nBR(|Duε|−λ)p\n+dx+cλp+c/bardblDu/bardblp\nLp(BR)+c/bracketleftBig\n1+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n,(4.8)\nwherec≡c(n,p,σ)>0. Choosing σ=1\n2pand absorbing the first term on the right-hand side\nof (4.8) into the left-hand side, we obtain\nˆ\nBR(|Duε|−λ)p\n+dx≤C/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\nfor some finite positive constant Cdepending on nandp, but not on ε. This estimate is\nsufficient to ensure the validity of the assertion.\nProposition 4.3 (Comparison estimate ).With the notation and under the assumptions\nabove, if f∈L(p∗)′(BR), there exists a positive constant Cdepending only on nandpsuch that\nthe estimate\nˆ\nBR|Gα,p((|Duε|−λ)+)−Gα,p((|Du|−λ)+)|2dx\n≤C/bardblfε−f/bardblp′\nL(p∗)′(BR)+C/bardblfε−f/bardblL(p∗)′(BR)/bracketleftbiggˆ\nBR(λp+|Du|p)dx/bracketrightbigg1\np\n+Cε/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n(4.9)\nholds for every ε∈(0,ε0], whereε0is the constant from Proposition 4.2.\nProof .We proceed by testing equations ( 1.1) and(4.1)1with the map ϕ=uε−u. Thus we\nfind\nˆ\nBR/a\\}bracketle{tHp−1(Duε)−Hp−1(Du),Duε−Du/a\\}bracketri}htdx+εˆ\nBR/a\\}bracketle{t(1+|Duε|2)p−2\n2Duε,Duε−Du/a\\}bracketri}htdx\n=ˆ\nBR(fε−f)(uε−u)dx. (4.10)\nUsing Lemma 2.2, the Cauchy-Schwarz inequality as well as Hölder’s and Youn g’s inequalities,\nfrom ( 4.10) we obtain\nCˆ\nBR/vextendsingle/vextendsingle/vextendsingleHp\n2(Duε)−Hp\n2(Du)/vextendsingle/vextendsingle/vextendsingle2\ndx+εˆ\nBR(1+|Duε|2)p−2\n2|Duε|2dx\n≤ /bardblfε−f/bardblL(p∗)′(BR)/bardbluε−u/bardblLp∗(BR)+εˆ\nBR(1+|Duε|2)p−2\n2|Duε||Du|dx\n≤ /bardblfε−f/bardblL(p∗)′(BR)/bardbluε−u/bardblLp∗(BR)+εˆ\nBR(1+|Duε|2)p−1\n2|Du|dx\n≤ /bardblfε−f/bardblL(p∗)′(BR)/bardbluε−u/bardblLp∗(BR)+ε\np′ˆ\nBR(1+|Duε|2)p\n2dx+ε\npˆ\nBR|Du|pdx,14 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\nwhereCis a positive constant depending at most on nandp. We now apply the Sobolev\ninequalities and Proposition 4.2, thus getting\nˆ\nBR/vextendsingle/vextendsingle/vextendsingleHp\n2(Duε)−Hp\n2(Du)/vextendsingle/vextendsingle/vextendsingle2\ndx≤c(n,p)/bardblfε−f/bardblL(p∗)′(BR)/bardblDuε−Du/bardblLp(BR)\n+εc(n,p)/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n(4.11)\nfor every ε∈(0,ε0]. Now, arguing as in [ 3, Formula (4.20)], we have\nˆ\nBR|Duε−Du|pdx≤2p−1ˆ\nBR|Duε|pdx+ 2p−1ˆ\nBR|Du|pdx\n≤c(p)ˆ\nBR/vextendsingle/vextendsingle/vextendsingleHp\n2(Duε)−Hp\n2(Du)/vextendsingle/vextendsingle/vextendsingle2\ndx+c(p)ˆ\nBR(λp+|Du|p)dx.\nInserting this estimate into ( 4.11) and applying Young’s inequality, for every ε∈(0,ε0]we\nobtain\nˆ\nBR/vextendsingle/vextendsingle/vextendsingleHp\n2(Duε)−Hp\n2(Du)/vextendsingle/vextendsingle/vextendsingle2\ndx≤1\n2ˆ\nBR/vextendsingle/vextendsingle/vextendsingleHp\n2(Duε)−Hp\n2(Du)/vextendsingle/vextendsingle/vextendsingle2\ndx+c/bardblfε−f/bardblp′\nL(p∗)′(BR)\n+c/bardblfε−f/bardblL(p∗)′(BR)/parenleftbiggˆ\nBR(λp+|Du|p)dx/parenrightbigg1\np\n+cε/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n, (4.12)\nwherec≡c(n,p)>0. Absorbing the first term in the right-hand side of ( 4.12) by the left-hand\nside, and then applying Lemma 2.3, the statement is proved.\n5 Proof of Theorem 1.1\nIn this section we prove Theorem 1.1, by dividing the proof into two steps. First, we shall\nderive a suitable uniform a priori estimate for the solution uεto problem ( 4.1). Then, we\nconclude with a standard comparison argument (see e.g. [ 3,17,21]) which, combined with the\nestimates from Propositions 4.2,4.3and5.1, yields the local Sobolev regularity of the function\nGα,p((|Du|−λ)+). We begin with the following result.\nProposition 5.1. Under the assumptions of Theorem 1.1and with the notations above, for\nevery pair of concentric balls Br/2⊂Br⊂BRwe have\nˆ\nBr/2|DGα,p((|Duε|−λ)+)|2dx≤C\nr2ˆ\nBr(1+|Duε|2)p\n2dx+C/bardblfε/bardblp′\nBp−2\np\np′,1(Br)(5.1)\nfor a positive constant Cdepending only on n,pandα.\nProof .Differentiating the equation in ( 4.1) with respect to xjfor some j∈ {1,...,n}and\nthen integrating by parts, we obtain\nˆ\nBR/a\\}bracketle{tD2Gε(Duε)D(∂juε),Dϕ/a\\}bracketri}htdx=ˆ\nBR(∂jfε)ϕdx, ∀ϕ∈W1,p\n0(BR). (5.2)P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 15\nLetη∈C∞\n0(Br)be a standard cut-off function such that\n0≤η≤1, η≡1 onBr/2,/bardblDη/bardbl∞≤˜c\nr(5.3)\nand consider\nϕ=η2(∂juε)Φ/parenleftbig\n(|Duε|−λ)+/parenrightbig\n,\nwhereΦ : [0,∞)→[0,∞)is an increasing, locally Lipschitz continuous function, s uch that Φ\nandΦ′are bounded on [0,∞),Φ(0) = 0 and\nΦ′(t)t≤cΦΦ(t) (5.4)\nfor a suitable constant cΦ>0. Usingϕas a test function in ( 5.2), we get\nˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),D(∂juε)/a\\}bracketri}htη2Φ((|Duε|−λ)+)dx\n+ˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),D[(|Duε|−λ)+]/a\\}bracketri}htη2(∂juε)Φ′((|Duε|−λ)+)dx\n=−2ˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),Dη/a\\}bracketri}htη(∂juε)Φ((|Duε|−λ)+)dx\n+ˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx.(5.5)\nAs for the first term on the right-hand side of ( 5.5), we have\n−2ˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),Dη/a\\}bracketri}htη(∂juε)Φ((|Duε|−λ)+)dx\n≤2ˆ\nBr|/a\\}bracketle{tD2Gε(Duε)D(∂juε),Dη/a\\}bracketri}ht|η|∂juε|Φ((|Duε|−λ)+)dx\n≤2ˆ\nBr/radicalBig\n/a\\}bracketle{tD2Gε(Duε)D(∂juε),D(∂juε)/a\\}bracketri}ht/radicalbig\n/a\\}bracketle{tD2Gε(Duε)Dη,Dη/a\\}bracketri}htη|∂juε|Φ((|Duε|−λ)+)dx\n≤1\n2ˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),D(∂juε)/a\\}bracketri}htη2Φ((|Duε|−λ)+)dx\n+ 2ˆ\nBr/a\\}bracketle{tD2Gε(Duε)Dη,Dη/a\\}bracketri}ht|∂juε|2Φ((|Duε|−λ)+)dx, (5.6)\nwhere we have used Cauchy-Schwarz and Young’s inequalities . Joining ( 5.5) and ( 5.6), we get\nˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),D(∂juε)/a\\}bracketri}htη2Φ((|Duε|−λ)+)dx\n+ˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),D[(|Duε|−λ)+]/a\\}bracketri}htη2(∂juε)Φ′((|Duε|−λ)+)dx\n≤1\n2ˆ\nBr/a\\}bracketle{tD2Gε(Duε)D(∂juε),D(∂juε)/a\\}bracketri}htη2Φ((|Duε|−λ)+)dx\n+ 2ˆ\nBr/a\\}bracketle{tD2Gε(Duε)Dη,Dη/a\\}bracketri}ht|∂juε|2Φ((|Duε|−λ)+)dx\n+ˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx. (5.7)16 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\nReabsorbing the first integral in the right-hand side of ( 5.7) by the left-hand side and summing\nthe resulting inequalities with respect to jfrom1ton, we obtain\nI1+I2≤I3+I4, (5.8)\nwhere\nI1:=ˆ\nBrn/summationdisplay\nj=1/a\\}bracketle{tD2Gε(Duε)D(∂juε),D(∂juε)/a\\}bracketri}htη2Φ((|Duε|−λ)+)dx,\nI2:= 2ˆ\nBrn/summationdisplay\nj=1/a\\}bracketle{tD2Gε(Duε)D(∂juε),D[(|Duε|−λ)+]/a\\}bracketri}htη2(∂juε)Φ′((|Duε|−λ)+)dx,\nI3:= 4ˆ\nBrn/summationdisplay\nj=1/a\\}bracketle{tD2Gε(Duε)Dη,Dη/a\\}bracketri}ht|∂juε|2Φ((|Duε|−λ)+)dx,\nI4:= 2ˆ\nBrn/summationdisplay\nj=1(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx.\nWe now prove that I2is non-negative, so that it can be dropped in the following. F irst of all,\nfor|Duε|> λwe have\nn/summationdisplay\nj=1/a\\}bracketle{tD2Gε(Duε)D(∂juε),D[(|Duε|−λ)+]/a\\}bracketri}ht(∂juε)\n=/bracketleftbigg\n(p−1)(|Duε|−λ)p−2\n+\n|Duε|2−(|Duε|−λ)p−1\n+\n|Duε|3+ε(p−2)(1+|Duε|2)p−4\n2/bracketrightbigg\n·/summationdisplay\ni,j,k(∂juε)(∂iuε)(∂kuε)(∂2\nijuε)∂k[(|Duε|−λ)+]\n+/bracketleftbigg(|Duε|−λ)p−1\n+\n|Duε|+ε(1+|Duε|2)p−2\n2/bracketrightbigg/summationdisplay\ni,j(∂juε)(∂2\nijuε)∂i[(|Duε|−λ)+]\n=/bracketleftbigg\n(p−1)(|Duε|−λ)p−2\n+\n|Duε|−(|Duε|−λ)p−1\n+\n|Duε|2+ε(p−2)(1+|Duε|2)p−4\n2|Duε|/bracketrightbigg\n·/bracketleftBiggn/summationdisplay\nk=1(∂kuε)∂k[(|Duε|−λ)+]/bracketrightBigg2\n+/bracketleftBig\n(|Duε|−λ)p−1\n++ε(1+|Duε|2)p−2\n2|Duε|/bracketrightBig/vextendsingle/vextendsingleD[(|Duε|−λ)+]/vextendsingle/vextendsingle2,(5.9)\nwhere we have used the fact that\n∂k[(|Duε|−λ)+] =∂k(|Duε|) =1\n|Duε|n/summationdisplay\nj=1(∂juε)(∂2\nkjuε) when |Duε|> λ.\nThus, coming back to the estimate of I2, from ( 5.9) we deduce\nI2≥2ˆ\nBrη2Φ′((|Duε|−λ)+)/braceleftBigg/bracketleftbigg\n(p−1)(|Duε|−λ)p−2\n+\n|Duε|−(|Duε|−λ)p−1\n+\n|Duε|2/bracketrightbigg\n·/bracketleftBiggn/summationdisplay\nk=1(∂kuε)∂k[(|Duε|−λ)+]/bracketrightBigg2\n+ (|Duε|−λ)p−1\n+/vextendsingle/vextendsingleD[(|Duε|−λ)+]/vextendsingle/vextendsingle2/bracerightBigg\ndx.P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 17\nNow, arguing as in the proof of [ 25, Lemma 4.1], for |Duε|> λwe have\n/bracketleftBiggn/summationdisplay\nk=1(∂kuε)∂k[(|Duε|−λ)+]/bracketrightBigg2\n≤ |Duε|2/vextendsingle/vextendsingleD(|Duε|)/vextendsingle/vextendsingle2=|Duε|2/vextendsingle/vextendsingleD[(|Duε|−λ)+]/vextendsingle/vextendsingle2.(5.10)\nThis implies\nI2≥2(p−1)ˆ\nBrη2Φ′((|Duε|−λ)+)(|Duε|−λ)p−2\n+\n|Duε|/bracketleftBiggn/summationdisplay\nk=1(∂kuε)∂k[(|Duε|−λ)+]/bracketrightBigg2\ndx≥0,\nwhere we have used the fact that Φ′((|Duε|−λ)+)≥0. From estimate ( 5.8) we thus obtain\nˆ\nBrn/summationdisplay\nj=1/a\\}bracketle{tD2Gε(Duε)D(∂juε),D(∂juε)/a\\}bracketri}htη2Φ((|Duε|−λ)+)dx\n≤4n/summationdisplay\nj=1ˆ\nBr/a\\}bracketle{tD2Gε(Duε)Dη,Dη/a\\}bracketri}ht|∂juε|2Φ((|Duε|−λ)+)dx\n+ 2n/summationdisplay\nj=1ˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx.(5.11)\nNow we choose\nΦ(t) :=t2α\n(t2+λ2)αfort≥0, (5.12)\nand therefore\nΦ′(t) =2αλ2t2α−1\n(t2+λ2)α+1.\nThis function satisfies ( 5.4) withcΦ= 2α. Using ( 5.3), (5.12) and Lemma 4.1, estimate ( 5.11)\nturns into\nˆ\nBr(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α|D2uε|2η2dx\n≤c(p)\nr2ˆ\nBr(1+|Duε|2)p\n2dx+c(p)n/summationdisplay\nj=1ˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx.(5.13)\nBy Theorem 3.3, there exists a linear and bounded extension operator\nextBr:B−2/p\np′,1(Br)֒→B−2/p\np′,1(Rn)\nsuch that reBr◦extBr= id, wherereBris the restriction operator defined in Section 3andidis\nthe identity in B−2/p\np′,1(Br). Since∂jfε= extBr(∂jfε)almost everywhere in Br, we have\nˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx=ˆ\nBrextBr(∂jfε)·(∂juε)η2Φ((|Duε|−λ)+)dx.\nAt this point, we need to estimate the integral containing extBr(∂jfε). To this aim, we argue\nas in [ 8, Proposition 3.2]. By definition of dual norm, we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nBrextBr(∂jfε)·(∂juε)η2Φ((|Duε|−λ)+)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /bardblextBr(∂jfε)/bardbl(˚B2/p\np,∞(Rn))′/bracketleftbig\n(∂juε)η2Φ((|Duε|−λ)+)/bracketrightbig\nB2/p\np,∞(Rn).(5.14)18 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\nUsing Theorem 3.5, we obtain\n/bardblextBr(∂jfε)/bardbl(˚B2/p\np,∞(Rn))′≤c/bardblextBr(∂jfε)/bardblB−2/p\np′,1(Rn),\nfor some positive constant c≡c(n,p). Combining the above inequality and the boundedness\nof the operator extBryields\n/bardblextBr(∂jfε)/bardbl(˚B2/p\np,∞(Rn))′≤c/bardbl∂jfε/bardblB−2/p\np′,1(Br).\nMoreover, applying Theorem 3.6, we find that\n/bardbl∂jfε/bardblB−2/p\np′,1(Br)≤c/bardblfε/bardbl\nBp−2\np\np′,1(Br).\nCombining the preceding inequalities, we infer\n/bardblextBr(∂jfε)/bardbl(˚B2/p\np,∞(Rn))′≤c/bardblfε/bardbl\nBp−2\np\np′,1(Br), (5.15)\nfor a positive constant cdepending only on nandp. Now, recalling that\n/bracketleftbig\n(∂juε)η2Φ((|Duε|−λ)+)/bracketrightbigp\nB2/p\np,∞(Rn)= sup\n|h|>0ˆ\nRn|δh((∂juε)η2Φ((|Duε|−λ)+))|p\n|h|2dx\nand using the well-known inequality\n|a−b|p≤c(p)/vextendsingle/vextendsingle/vextendsingle|a|p−2\n2a−|b|p−2\n2b/vextendsingle/vextendsingle/vextendsingle2\nfor every a,b∈R,\nwe deduce\nˆ\nRn|δh((∂juε)η2Φ((|Duε|−λ)+))|p\n|h|2dx\n≤c\n|h|2ˆ\nRn/vextendsingle/vextendsingle/vextendsingleδh/parenleftBig\n|∂juε|p−2\n2(∂juε)ηp[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx\n≤c(n,p)ˆ\nRn/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|∂juε|p−2\n2(∂juε)ηp[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx,\nwhere, in the last line, we have used the first statement in Lem ma2.7. By the properties of η\nat (5.3) and the boundedness of Φ, one can easily obtain\n/bracketleftbig\n(∂juε)η2Φ((|Duε|−λ)+)/bracketrightbigp\nB2/p\np,∞(Rn)≤cˆ\nBr/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\nη2dx\n+c\nr2ˆ\nBr|Duε|pdx, (5.16)\nwherec≡c(n,p)>0. Now, a straightforward computation reveals that, for ever yk∈\n{1,...,n}, we have\n∂k/bracketleftBig\n|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p\n2/bracketrightBig\n=p\n2|∂juε|p−2\n2(∂2\nkjuε)[Φ((|Duε|−λ)+)]p\n2P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 19\n+p\n2|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p−2\n2Φ′((|Duε|−λ)+)/a\\}bracketle{tDuε,∂kDuε/a\\}bracketri}ht\n|Duε|.\nThis yields/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\n≤c(n,p)(A1+A2), (5.17)\nwhere we set\nA1:=|Duε|p−2|D2uε|2[Φ((|Duε|−λ)+)]p\nand\nA2:=|Duε|p|D2uε|2[Φ((|Duε|−λ)+)]p−2[Φ′((|Duε|−λ)+)]2.\nWe now estimate A1andA2separately in the set where |Duε|> λ, since both A1andA2\nvanish in the set BR∩{|Duε| ≤λ}. Recalling the definition of Φin (5.12), we can write A1as\nthe product of two terms:\nA1=|Duε|p−1(|Duε|−λ)(2α−1)(p−1)\n+\n[λ2+(|Duε|−λ)2\n+]α(p−1)·|D2uε|2(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α.\nThen we have\n|Duε|p−1(|Duε|−λ)(2α−1)(p−1)\n+\n[λ2+(|Duε|−λ)2\n+]α(p−1)≤2α(p−1)|Duε|2α(p−1)\n|Duε|2α(p−1)= 2α(p−1),\nwhere we have used that 2α−1>0, since by assumption α≥p+1\n2(p−1)>1\n2. This implies that\nA1≤2α(p−1)|D2uε|2(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α. (5.18)\nNow we deal with A2. First, we observe that\n[Φ(t)]p−2[Φ′(t)]2=4α2λ4t2(αp−1)\n(t2+λ2)αp+2.\nTherefore, A2can be written as follows:\nA2= 4α2λ4|Duε|p+1(|Duε|−λ)2α(p−1)−p−1\n+\n[λ2+(|Duε|−λ)2\n+]α(p−1)+2·|D2uε|2(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α.\nAgain, the assumption α≥p+1\n2(p−1)implies(|Duε|−λ)2α(p−1)−p−1\n+ ≤ |Duε|2α(p−1)−p−1, and so\nλ4|Duε|p+1(|Duε|−λ)2α(p−1)−p−1\n+\n[λ2+(|Duε|−λ)2\n+]α(p−1)+2≤2α(p−1)+2λ4|Duε|2α(p−1)\n|Duε|4|Duε|2α(p−1)<2α(p−1)+2,\nwhere we have used that |Duε|> λ. Thus we have\nA2≤2α(p−1)+4α2|D2uε|2(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α. (5.19)\nJoining estimates ( 5.16)−(5.19), we find\n/bracketleftbig\n(∂juε)η2Φ((|Duε|−λ)+)/bracketrightbigp\nB2/p\np,∞(Rn)≤cˆ\nBr(|Duε|−λ)p−1+2α\n+|D2uε|2\n|Duε|[λ2+(|Duε|−λ)2\n+]αη2dx20 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\n+c\nr2ˆ\nBr|Duε|pdx,\nwherec≡c(n,p,α)>0. Inserting the previous inequality and ( 5.15) into ( 5.14), we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤c/bardblfε/bardbl\nBp−2\np\np′,1(Br)/bracketleftBigg/parenleftbiggˆ\nBr(|Duε|−λ)p−1+2α\n+|D2uε|2\n|Duε|[λ2+(|Duε|−λ)2\n+]αη2dx/parenrightbigg1\np\n+1\nr2/p/bardblDuε/bardblLp(Br)/bracketrightBigg\n,\nfor a constant c≡c(n,p,α)>0. Going back to ( 5.13) and using the above estimate gives\nˆ\nBr(|Duε|−λ)p−1+2α\n+|D2uε|2\n|Duε|[λ2+(|Duε|−λ)2\n+]αη2dx\n≤c/parenleftbiggˆ\nBr(|Duε|−λ)p−1+2α\n+|D2uε|2\n|Duε|[λ2+(|Duε|−λ)2\n+]αη2dx/parenrightbigg1\np\n/bardblfε/bardbl\nBp−2\np\np′,1(Br)\n+c\nr2/p/bardblDuε/bardblLp(Br)/bardblfε/bardbl\nBp−2\np\np′,1(Br)+c\nr2ˆ\nBr(1+|Duε|2)p\n2dx,\nwherec≡c(n,p,α)>0. At this point, we apply Young’s inequality to the first integ ral on the\nright-hand side of the previous inequality. This yields\nˆ\nBr(|Duε|−λ)p−1+2α\n+|D2uε|2\n|Duε|[λ2+(|Duε|−λ)2\n+]αη2dx≤C\nr2ˆ\nBr(1+|Duε|2)p\n2dx+C/bardblf��/bardblp′\nBp−2\np\np′,1(Br),(5.20)\nfor some constant C≡C(n,p,α)>0. Notice that\n∂jGα,p((|Duε|−λ)+) =G′\nα,p((|Duε|−λ)+)·∂j[(|Duε|−λ)+]\n=(|Duε|−λ)p−1+2α\n2\n+\n|Duε|[λ+(|Duε|−λ)+]2α+1\n2/a\\}bracketle{tDuε,∂jDuε/a\\}bracketri}ht,\nand therefore, using the Cauchy-Schwarz inequality, we ded uce\nˆ\nBr|DGα,p((|Duε|−λ)+)|2η2dx≤ˆ\nBr(|Duε|−λ)p−1+2α\n+\n[λ+(|Duε|−λ)+]2α+1|D2uε|2η2dx\n≤ˆ\nBr(|Duε|−λ)p−1+2α\n+\n[λ+(|Duε|−λ)+] [λ2+(|Duε|−λ)2\n+]α|D2uε|2η2dx\n≤ˆ\nBr(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α|D2uε|2η2dx. (5.21)\nJoining estimates ( 5.20) and ( 5.21), and recalling that η≡1onBr/2, we finally obtain the\nconclusion.\nCombining Lemma 2.7with estimate ( 5.1), we obtain the following\nCorollary 5.2. Under the assumptions of Theorem 1.1and with the notations above, for every\npair of concentric balls Br/4⊂Br⊂BRwe have\nˆ\nBr/4|τj,hGα,p((|Duε|−λ)+)|2dx≤C|h|2\nr2ˆ\nBr(1+|Duε|2)p\n2dx+C|h|2/bardblfε/bardblp′\nBp−2\np\np′,1(Br)(5.22)P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 21\nfor every j∈ {1,...,n}, for every h∈Rsuch that |h|<r\n8and a positive constant C≡\nC(n,p,α).\nWe are now in a position to prove Theorem 1.1.\nProof of Theorem 1.1.Letε∈(0,ε0]withε0∈(0,1]determined in Proposition 4.2. More-\nover, let uεbe the unique energy solution of the Dirichlet problem ( 4.1). Now we fix three\nconcentric balls Br/4,Br/2andBr, withBr⊂BR⋐Ω,R≤1, and use the finite difference\noperator τj,hdefined in Section 2.1, for increments h∈R\\ {0}such that |h|<r\n8. In what\nfollows, we will denote by candCsome positive constants which neither depend on εnor on\nh. In order to obtain an estimate for the finite difference τj,hGα,p((|Du| −λ)+), we use the\nfollowing comparison argument:\nˆ\nBr/4|τj,hGα,p((|Du|−λ)+)|2dx\n≤4ˆ\nBr/4|τj,hGα,p((|Duε|−λ)+)|2dx+ 4ˆ\nBr/4|Gα,p((|Duε|−λ)+)−Gα,p((|Du|−λ)+)|2dx\n+ 4ˆ\nBr/4|Gα,p((|Duε(x+hej)|−λ)+)−Gα,p((|Du(x+hej)|−λ)+)|2dx\n≤4ˆ\nBr/4|τj,hGα,p((|Duε|−λ)+)|2dx+ 8ˆ\nBR|Gα,p((|Duε|−λ)+)−Gα,p((|Du|−λ)+)|2dx,\nwhere we have used the second statement in Lemma 2.7. Since(p∗)′< p′, we can combine the\nprevious estimate with ( 5.22) and ( 4.9), thus obtaining\nˆ\nBr/4|τj,hGα,p((|Du|−λ)+)|2dx\n≤c\nr2|h|2ˆ\nBr(1+|Duε|2)p\n2dx+c|h|2/bardblfε/bardblp′\nBp−2\np\np′,1(Br)\n+c/bardblfε−f/bardblp′\nL(p∗)′(BR)+c/bardblfε−f/bardblL(p∗)′(BR)/parenleftbiggˆ\nBR(λp+|Du|p)dx/parenrightbigg1\np\n+cε/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n,\nwhich holds for every j∈ {1,...,n}, for every sufficiently small h∈R\\ {0}and a positive\nconstant c≡c(n,p,α). Thanks to estimate ( 4.5) we have\nˆ\nBr(1 +|Duε|2)p\n2dx≤c(n,p)/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n.\nFurthermore, there exists a positive number ε1≤ε0such that\n/bardblfε/bardbl\nBp−2\np\np′,1(Br)≤ /bardblf/bardbl\nBp−2\np\np′,1(BR)<+∞for every ε∈(0,ε1].\nJoining the last three estimates for ε∈(0,ε1], we find\nˆ\nBr/4|τj,hGα,p((|Du|−λ)+)|2dx22 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\n≤C/parenleftbigg|h|2\nr2+ε/parenrightbigg/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n+C|h|2/bardblf/bardblp′\nBp−2\np\np′,1(BR)\n+C/bardblfε−f/bardblp′\nL(p∗)′(BR)+C/bardblfε−f/bardblL(p∗)′(BR)/parenleftbiggˆ\nBR(λp+|Du|p)dx/parenrightbigg1/p\n, (5.23)\nwhich holds for every suitably small h/\\e}atio\\slash= 0and a constant C≡C(n,p,α)>0. Therefore,\nrecalling ( 4.7) and letting εց0in (5.23), we obtain\nˆ\nBr/4|∆j,hGα,p((|Du|−λ)+)|2dx\n≤C\nr2/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nL(p∗)′(BR)/bracketrightBig\n+C/bardblf/bardblp′\nBp−2\np\np′,1(BR).\nSince(p∗)′< p′, using Hölder’s inequality, from the previous estimate we g et\nˆ\nBr/4|∆j,hGα,p((|Du|−λ)+)|2dx\n≤C\nr2/bracketleftBig\n1+λp+/bardblDu/bardblp\nLp(BR)+/bardblf/bardblp′\nLp′(BR)/bracketrightBig\n+C/bardblf/bardblp′\nBp−2\np\np′,1(BR).\nSince the above inequality holds for every j∈ {1,...,n}and every sufficiently small h/\\e}atio\\slash= 0, by\nLemma 2.8we may conclude that Gα,p((|Du|−λ)+)∈W1,2\nloc(Ω). Moreover, letting h→0in the\nprevious inequality, we obtain estimate ( 1.3) for every ball BR⋐Ω, withR≤1. The validity\nof (1.3) for arbitrary balls follows from a standard covering argum ent.\n6 Proof of Theorem 1.2\nThis section is devoted to the proof of Theorem 1.2. Actually, here we limit ourselves to\nderiving the a priori estimates, since inequality ( 1.5) can be obtained using the same arguments\npresented in Section 5. In what follows, we shall keep the notations used for the pro of of\nProposition 5.1.\nProof of Theorem 1.2.Arguing as in the proof of Proposition 5.1, we define the integrals\nI1–I4exactly as in ( 5.8). We need to treat differently only the integrals I2andI4, in which the\nnew assumptions 1< p≤2andf∈Lnp\nn(p−1)+2−p\nloc (Ω)are involved.\nIt comes out that I2is non-negative, as in the super-quadratic case. Indeed, es timates ( 5.9)\nand (5.10) lead us to\nI2≥2ˆ\nBrη2Φ′((|Duε|−λ)+)\n·/braceleftBigg/bracketleftbigg\n(p−1)(|Duε|−λ)p−2\n+\n|Duε|−(|Duε|−λ)p−1\n+\n|Duε|2+ε(p−2)(1+|Duε|2)p−4\n2|Duε|/bracketrightbigg\n·/bracketleftBiggn/summationdisplay\nk=1(∂kuε)∂k[(|Duε|−λ)+]/bracketrightBigg2\n+ (|Duε|−λ)p−1\n+/vextendsingle/vextendsingleD[(|Duε|−λ)+]/vextendsingle/vextendsingle2\n+ε(1+|Duε|2)p−4\n2|Duε|3/vextendsingle/vextendsingleD[(|Duε|−λ)+]/vextendsingle/vextendsingle2/bracerightBigg\ndxP. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 23\n≥2ˆ\nBrη2Φ′((|Duε|−λ)+)/bracketleftbigg\n(p−1)(|Duε|−λ)p−2\n+\n|Duε|+ε(p−1)(1+|Duε|2)p−4\n2|Duε|/bracketrightbigg\n·/bracketleftBiggn/summationdisplay\nk=1(∂kuε)∂k[(|Duε|−λ)+]/bracketrightBigg2\ndx≥0.\nTherefore, using ( 5.3), (5.12) and Lemma 4.1, from ( 5.8) we now obtain\nˆ\nBr(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α|D2uε|2η2dx\n≤c(p)\nr2ˆ\nBr(1+|Duε|2)p\n2dx+c(p)\nr2ˆ\nBr(|Duε|−λ)p−2+2α\n+\n[λ2+(|Duε|−λ)2\n+]α|Duε|2dx\n+c(p)n/summationdisplay\nj=1ˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx\n≤c(p,α)\nr2ˆ\nBr(1+|Duε|2)p\n2dx+c(p)n/summationdisplay\nj=1ˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx, (6.1)\nwhere the function Φis defined in ( 5.12). At this point, we integrate by parts and then apply\nHölder’s inequality in the second integral on right-hand si de of ( 6.1). This gives\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /bardblfε/bardbl\nLnp\nn(p−1)+2−p(Br)/bardbl∂j[(∂juε)η2Φ((|Duε|−λ)+)]/bardblLnp\nn−2+p(Rn).(6.2)\nFrom now on, we will only deal with the case n≥3and1< p <2, since the remaining cases\nimply thatnp\nn(p−1)+2−p=np\nn−2+p= 2\nand can be addressed by suitably modifying the arguments use d in Section 5. Note that in the\ncasep=n= 2, we have (p∗)′< p′= 2, and therefore we can continue to argue as in the proof\nof Theorem 1.1.\nFor ease of notation, we now set\nZ(x) :=η2(∂juε)Φ((|Duε|−λ)+).\nSincenp\nn−2+p<2forn≥3and1< p <2, an application of Hölder’s inequality and Lemma 2.1\nyield\nˆ\nRn|τj,hZ(x)|np\nn−2+pdx\n=ˆ\nRn|τj,hZ(x)|np\nn−2+p/parenleftbig\n|Z(x+hej)|2+|Z(x)|2/parenrightbig(p−2)\n4np\nn−2+p/parenleftbig\n|Z(x+hej)|2+|Z(x)|2/parenrightbig(2−p)\n4np\nn−2+pdx\n≤c(n,p)/parenleftbiggˆ\nRn|τj,hZ(x)|2/parenleftbig\n|Z(x+hej)|2+|Z(x)|2/parenrightbigp−2\n2dx/parenrightbiggnp\n2(n−2+p)/parenleftbiggˆ\nRn|Z(x)|np\nn−2dx/parenrightbigg(n−2)(2−p)\n2(n−2+p)\n≤c(n,p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingleτj,h/parenleftBig\n|Z(x)|p−2\n2Z(x)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbiggnp\n2(n−2+p)/parenleftbiggˆ\nRn|Z(x)|np\nn−2dx/parenrightbigg(n−2)(2−p)\n2(n−2+p)24 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\nfor every h∈R\\{0}. Dividing both sides by |h|np\nn−2+pand letting h→0, by virtue of Lemma\n2.8we obtain\nˆ\nRn|∂jZ(x)|np\nn−2+pdx\n≤c(n,p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingle∂j/parenleftBig\n|Z(x)|p−2\n2Z(x)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbiggnp\n2(n−2+p)/parenleftbiggˆ\nRn|Z(x)|np\nn−2dx/parenrightbigg(n−2)(2−p)\n2(n−2+p)\n≤c(n,p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|Z(x)|p−2\n2Z(x)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbiggnp\n2(n−2+p)/parenleftbiggˆ\nRn|Z(x)|np\nn−2dx/parenrightbigg(n−2)(2−p)\n2(n−2+p)\n=c(n,p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|Z(x)|p−2\n2Z(x)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbiggnp\n2(n−2+p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingle|Z(x)|p−2\n2Z(x)/vextendsingle/vextendsingle/vextendsingle2n\nn−2dx/parenrightbigg(n−2)(2−p)\n2(n−2+p)\n≤c(n,p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|Z(x)|p−2\n2Z(x)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbiggnp\n2(n−2+p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|Z(x)|p−2\n2Z(x)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbiggn(2−p)\n2(n−2+p)\n=c(n,p)/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|Z(x)|p−2\n2Z(x)/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbiggn\nn−2+p\n. (6.3)\nRecalling the definition of Z, calculating the gradient in the right-hand side of ( 6.3), using the\nproperties of ηand recalling that Φ≤1, we get\n/bardbl∂j[η2(∂juε)Φ((|Duε|−λ)+)]/bardblLnp\nn−2+p(Rn)\n≤c/parenleftbiggˆ\nRn/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\nηp|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbigg1\np\n≤c/parenleftbiggˆ\nBrη2p/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbigg1\np\n+c/parenleftbiggˆ\nBrη2p−2|Dη|2|Duε|pdx/parenrightbigg1\np\n. (6.4)\nInserting ( 6.4) into ( 6.2) and using the properties of η, we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nBr(∂jfε)(∂juε)η2Φ((|Duε|−λ)+)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤c/bardblfε/bardbl\nLnp\nn(p−1)+2−p(Br)/parenleftbiggˆ\nBrη2/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbigg1\np\n+c\nr2/p/bardblfε/bardbl\nLnp\nn(p−1)+2−p(Br)/parenleftbiggˆ\nBr|Duε|pdx/parenrightbigg1\np\n. (6.5)\nNow, combining ( 6.5) with ( 6.1), we have\nˆ\nBr(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α|D2uε|2η2dx\n≤c\nr2ˆ\nBr(1+|Duε|2)p\n2dx+c\nr2/p/bardblfε/bardbl\nLnp\nn(p−1)+2−p(Br)/bardblDuε/bardblLp(Br)\n+c/bardblfε/bardbl\nLnp\nn(p−1)+2−p(Br)n/summationdisplay\nj=1/parenleftbiggˆ\nBrη2/vextendsingle/vextendsingle/vextendsingleD/parenleftBig\n|∂juε|p−2\n2(∂juε)[Φ((|Duε|−λ)+)]p\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2\ndx/parenrightbigg1\np\n,P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI 25\nwhere now c≡c(n,p,α)>0. The last integral can be estimated using ( 5.17), (5.18) and ( 5.19).\nThus we infer\nˆ\nBr(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α|D2uε|2η2dx\n≤c\nr2ˆ\nBr(1+|Duε|2)p\n2dx+c\nr2/p/bardblfε/bardbl\nLnp\nn(p−1)+2−p(Br)/bardblDuε/bardblLp(Br)\n+c/bardblfε/bardbl\nLnp\nn(p−1)+2−p(Br)/parenleftbiggˆ\nBr(|Duε|−λ)p−1+2α\n+\n|Duε|[λ2+(|Duε|−λ)2\n+]α|D2uε|2η2dx/parenrightbigg1\np\n.\nFinally, applying Young’s inequality in order to reabsorb t he term containing D2uεon the\nright-hand side by the left-hand side, and then using inequa lity (5.21), we derive\nˆ\nBr|DGα,p((|Duε|−λ)+)|2η2dx≤C\nr2ˆ\nBr(1+|Duε|2)p\n2dx+C/bardblfε/bardblp′\nLnp\nn(p−1)+2−p(Br),\nfor a positive constant Cdepending only on n,pandα. The desired conclusion follows by\narguing as in the proof of Theorem 1.1, observing that\n(p∗)′=np\nnp−n+p<np\nn(p−1)+2−pfor every p >1.\nAcknowledgements. We gratefully acknowledge Lorenzo Brasco for pointing out t o us a cru-\ncial point in the proof of Theorem 1.1. The authors are members of the Gruppo Nazionale per\nl’Analisi Matematica, la Probabilità e le loro Applicazion i (GNAMPA) of the Istituto Nazionale\ndi Alta Matematica (INdAM). P. Ambrosio and A.G. Grimaldi ha ve been partially supported\nthrough the INdAM −GNAMPA 2023 Project “Risultati di regolarità per PDEs in spa zi di fun-\nzione non-standard” (CUP: E53C22001930001). P. Ambrosio h as also been supported through\nthe INdAM −GNAMPA 2024 Project “Fenomeno di Lavrentiev, Bounded Slope Condition e re-\ngolarità per minimi di funzionali integrali con crescite no n standard e lagrangiane non uniforme-\nmente convesse” (CUP: E53C23001670001). A.G. Grimaldi and A. Passarelli di Napoli have\nbeen partially supported through the INdAM −GNAMPA 2024 Project “Interazione ottimale\ntra la regolarità dei coefficienti e l’anisotropia del proble ma in funzionali integrali a crescite non\nstandard” (CUP: E53C23001670001). A. Passarelli di Napoli has also been supported through\nthe INdAM −GNAMPA 2023 Project “Su alcuni problemi di regolarità del ca lcolo delle vari-\nazioni con convessità degenere” (CUP: E53C22001930001). I n addition, A. Passarelli di Napoli\nhas been supported by the Università degli Studi di Napoli “F ederico II” through the project\nFRA-000022-ALTRI-CDA-752021-FRA-PASSARELLI and by the C entro Nazionale per la Mo-\nbilità Sostenibile (CN00000023) - Spoke 10 Logistica Merci (CUP: E63C22000930007).\nDeclarations. The authors have no relevant financial or non-financial compe ting interests.\nReferences\n[1] E. Acerbi, N. Fusco, Regularity for minimizers of non-quadratic functionals: t he case\n1< p <2, J. Math. Anal. Appl., 140(1989), 115-135.\n[2] P. Ambrosio, Besov regularity for a class of singular or degenerate ellip tic equations , J.\nMath. Anal. Appl. 505(2022), no. 2, Paper No. 125636.26 P. AMBROSIO, A.G. GRIMALDI, A. PASSARELLI DI NAPOLI\n[3] P. Ambrosio, A. 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Marcellini, An existence result for a nonconvex variational problem\nvia regularity , ESAIM Control Optim. Calc. Var. 7, 69-95 (2002).\n[20] M. Foss, A. Passarelli di Napoli, A. Verde, Global Lipschitz regularity for almost minimizers\nof asymptotically convex variational problems , Ann. Mat. Pura Appl. (4) 189, 127-162\n(2010).\n[21] A. Gentile, A. Passarelli di Napoli, Higher regularity for weak solutions to degenerate\nparabolic problems , Calc. Var. 62, 225 (2023).\n[22] M. Giaquinta, G. Modica, Remarks on the regularity of the minimizers of certain degen erate\nfunctionals , Manuscripta Math. 57, 55-99 (1986).\n[23] E. Giusti, Direct Methods in the Calculus of Variations , World Scientific Publishing Co.,\n2003.\n[24] C. Leone, A. Passarelli di Napoli, A. Verde, Lipschitz regularity for some asymptotically\nsubquadratic problems , Nonlinear Anal. 67, 1532-1539 (2007).\n[25] P. Marcellini, G. Papi, Nonlinear elliptic systems with general growth , J. Diff. Equ. 221,\n412-443 (2006).\n[26] J.P. Raymond, Lipschitz regularity of solutions of some asymptotically c onvex problems ,\nProc. R. Soc. Edinburgh Sect. A 117, 59-73 (1991).\n[27] C. Scheven, T. Schmidt, Asymptotically regular problems I: Higher integrability , J. Differ.\nEqu.248, 745-791 (2010).\n[28] H. Triebel, Theory of Function Spaces , Monogr. Math. 78, Birkhäuser, Basel, 1983.\n[29] H. Triebel, Theory of Function Spaces IV , Monogr. Math. 107, Birkhäuser, Basel, 2020.\nPasquale Ambrosio\nDipartimento di Matematica e Applicazioni “R. Caccioppoli ”\nUniversità degli Studi di Napoli “Federico II”\nVia Cintia, 80126 Napoli, Italy.\nE-mail address : pasquale.ambrosio2@unina.it\nAntonio Giuseppe Grimaldi\nDipartimento di Matematica e Applicazioni “R. Caccioppoli ”\nUniversità degli Studi di Napoli “Federico II”\nVia Cintia, 80126 Napoli, Italy.\nE-mail address : antoniogiuseppe.grimaldi@unina.it\nAntonia Passarelli di Napoli\nDipartimento di Matematica e Applicazioni “R. Caccioppoli ”\nUniversità degli Studi di Napoli “Federico II”\nVia Cintia, 80126 Napoli, Italy.\nE-mail address : antpassa@unina.it"    },    {        "title": "2401.13139v1.Factorizable_convergence_of_random_variables_in_Grand_Lebesgue_Spaces.pdf",        "content": "arXiv:2401.13139v1  [math.PR]  23 Jan 2024Factorizable convergence of random variables\nin Grand Lebesgue Spaces\nMaria Rosaria Formica1, Eugeny Ostrovsky2and Leonid Sirota3\n1Universit` a degli Studi di Napoli “Parthenope”, via Generale Paris i 13,\nPalazzo Pacanowsky, 80132, Napoli, Italy.\ne-mail: mara.formica@uniparthenope.it\n2,3Bar-Ilan University, Department of Mathematics and Statistics,\n52900, Ramat Gan, Israel.\ne-mail: eugostrovsky@list.ru\ne-mail: sirota3@bezeqint.net\nAbstract\nWe obtain results concerning the so-called factorization for theconvergence\nof random variables almost everywhere (almost surely or with probability one),\nbelonging to the classical Lebesgue-Riesz spaces and we ext end these results to\nthe Grand Lebesgue Spaces.\nWe also give exact estimates for the parameters involved and provide several\nexamples.\nWe also show that the obtained estimates are, in the general c ase, essentially\nnon-improvable, of course up to a multiplicative constant.\nKeywords: Probability, random variables, separable random process, conver gence almost surely,\nregulator, Lebesgue-Riesz spaces, Grand Lebesgue Spaces, ex pectation, classical and exponential\nmoments, Young-Fenchel or Legendre transform, slowly varying function, Bonferroni’s inequality,\nTchebychev-Markov’s and Lyapunov’s inequalities, tail of distribut ion, Borel-Cantelli lemma.\n2020 Mathematics Subject Classification :primary 60G17; secondary 60E07; 60G70.\n1 Introduction.\nP.E. Kloeden and A. Neuenkirch in [24] proved a result which establishe s a connection\nbetween the convergence rates in the p-th mean ( p≥1) and the pathwise convergence\nratesforsequencesofrandomvariablesbelongingtotheclassical Lebesgue-Rieszspaces.\nThe importance of this result is also due to applications, given by the a uthors, for Itˆ o\nstochastic differential equations.\nInthispaperweextend theaboveresult torandomvariablesbelong ingtothegener-\nalized Grand Lebesgue Spaces. These spaces have a crucial role in s everal applications,\nas in Probability theory, Functional Analysis, interpolation theory, PDEs.\n1Factorizable convergence of random variables in Grand Lebesgue S paces 2\nLet(Ω={ω},B,P)be a probability space. Let {ξn}, n=1,2,3,..., be a\nsequence of numerical valued random variables (r.v.). We say that {ξn}converges\nto zero as n→∞almost everywhere (a.e.) , or equally almost surely (a.s) , orwith\nprobability one (w.p.o.) if\nP/parenleft.alt2lim\nn→∞ξn=0/parenright.alt2=1. (1)\nThe following theorem can be found, e.g., in [23], chapter 1, section 6, Theorem 5,\npart 2, known as Theorem of regulator of the convergence a.e. or, in the other words,\nfactorizable convergence.\nTheorem 1.1. (Regulator) Let{ξn}, n=1,2,3,..., be a sequence of numerical\nvalued random variables converging to zero almost everywhe re (a.e.). Then there exists\na non-negative finite a.e. random variable ζand a sequence of non-random real valued\nnon-negative variables ǫntending to zero ( ǫn→0) such that\n/divides.alt0ξn/divides.alt0≤ǫnζ. (2)\nOf course, the converse implication is also true.\nThe pair{ǫn, ζ},non unique, may be named by definition as the regulator for\nthe sequence{ξn}.\nRemark 1.1. Note that the famous book [23] does not contain the proof of this\nassertion as well as the references of the works where this propo sition is ground. For\nthe filling of this gap see [34, 35]. For instance, (1) holds true if and on ly if\nlim\nn→∞Esup\nm≥n[/divid⟩s.alt0ξm/divid⟩s.alt0/slash.l⟩ft(1+/divid⟩s.alt0ξm/divid⟩s.alt0)]=0. (3)\nReally, if the relation (1) is satisfied, then (3) is grounded on the bas is of the\ndominated convergence theorem.\nConversely, let the condition (3) be satisfied. There exists a deter ministic positive\nsequence{δn}for which lim\nn→∞δn=0 and\nlim\nn→∞Esup\nm≥n[/divid⟩s.alt0ξm/slash.l⟩ftδm/divid⟩s.alt0/slash.l⟩ft(1+/divid⟩s.alt0ξm/divid⟩s.alt0/slash.l⟩ftδm)]=0.\nTherefore lim\nn→∞[/divid⟩s.alt0ξn/divid⟩s.alt0/slash.l⟩ftδn]=0,with probability one, and denoting υ∶=sup\nn[/divid⟩s.alt0ξn/divid⟩s.alt0/slash.l⟩ftδn]\nwe have\nP(0≤υ<∞)=1\nand finally\n/divid⟩s.alt0ξn/divid⟩s.alt0≤υ⋅δn,\ni.e. the existence of a regulator of the convergence.\nAn useful estimation for this functional, in the case when the r.v. {ξn}are sums\nof dependent variables, is offered in the recent article [8].\nMoreover, inequality (2) remains true if the r.v. {ξn}take values in a separable\nBanach space B=(B,/divid⟩s.alt0/divid⟩s.alt0⋅/divid⟩s.alt0/divid⟩s.alt0B),where the absolute value /divid⟩s.alt0⋅/divid⟩s.alt0in the inequality is replaced\nby the norm/divid⟩s.alt0/divid⟩s.alt0⋅/divid⟩s.alt0/divid⟩s.alt0B(see [32, 34]).Factorizable convergence of random variables in Grand Lebesgue S paces 3\nRemark 1.2. Another criterion for the convergence a.e. (a.s.) states that a se quence\nof numerical valued r.v. {Xn}, n=1,2,..., converges a.e. to zero iff\n∀ǫ>0⇒lim\nn→∞P/par⟩nl⟩ft.alt4/uni22C3.disp\nm≥n{/divid⟩s.alt0Xm/divid⟩s.alt0≤ǫ}/par⟩nright.alt4=1. (4)\nBut our approach is more convenient for us.\nRemark 1.3. Let{Xn}be a sequence of numerical valued r.v. and consider the\nsequence\nηn∶=/divid⟩s.alt0Xn/divid⟩s.alt0\n1+/divid⟩s.alt0Xn/divid⟩s.alt0,\nwhich is positive and bounded, 0 ≤ηn≤1.\nThe almost everywhere convergence to zero of {Xn}, n→∞, is completely equiva-\nlent to the following convergence\nlim\nn→∞E/brac⟩l⟩ft.alt3sup\nm≥nηm/brac⟩right.alt3=0.\nNotethatthemetric(symmetry, positivity, triangleinequality) and otherproperties\nof the distance function\nρ(X,Y)∶=E/divid⟩s.alt0X−Y/divid⟩s.alt0\n1+/divid⟩s.alt0X−Y/divid⟩s.alt0\nin more than used in this report (one-dimensional case dim (X)=dim(Y)=1) are in\ndetail investigated in the recent articles [3, 4].\nRemark 1.4. Continuum case. Let nowη(t), t∈[0,1], bea separable bi -measurable\nrandom process such that η(1)=0.Question : under what conditions imposed on\nthe distribution of the random process η(t), t∈[0,1],\nP(lim\nt→1−η(t)=0)=1 ? (5)\nNote first of all that there exists a simple (but not convenient in pra ctice)necessary\nand sufficient condition for (5): indeed, when there exists a constant δ∈(0,1)such\nthat almost all trajectories of the random process η(⋅)are bounded in the interval\n[1−δ,1], i.e.\n∃δ∈(0,1)∶P(sup\nt∈[1−δ,1]/divid⟩s.alt0η(t)/divid⟩s.alt0<∞)=1. (6)\nFurther, suppose in addition that\n∃δ∈(0,1)∶lim\nt→1−Esup\nt∈[1−δ,1][/divid⟩s.alt0η(t)/divid⟩s.alt0/slash.l⟩ft(1+/divid⟩s.alt0η(t))/divid⟩s.alt0]=0. (7)\nProposition 1.2. Both the conditions (6)and(7)commonare necessaryand sufficient\nfor the convergence a.e. in (5).Factorizable convergence of random variables in Grand Lebesgue S paces 4\nRemark 1.5. Evidently, if the random process η=η(t), t∈[0,1], is such that\nη(1)=0, is continuous w.p.o., then the condition (5) is satisfied.\nNote that the exact exponential tail estimations of the maximum dis tribution\nfor thecontinuous random fields are obtained, e.g., in [10, 30], as well as for the\ndiscontinuous random fields are in detail investigated in [36].\nOur aim in this report is to extend previous results in this direction and to give\nquantitative estimates of the norm and tail for the random variable s involved in the\nright-hand side of (2), based on ones in the left-hand side, and to e xtrapolate these\nestimates into the Grand Lebesgue Spaces of random variables.\nRemark 1.6. Note that the systematic investigation of the convergence a.e.in\nmany different scientific directions, as operator and ergodic theor y, functional anal-\nysis, physics, etc, is treated in [5].\n2 Lebesgue - Riesz norm estimations.\nLet(Ω={ω},B,P)be a probability space. Recall that the classical Lebesgue-Riesz\nLp-norm (p≥1)of a numerical-valued random variable ξ=ξ(ω)is defined by\n/divid⟩s.alt0/divid⟩s.alt0ξ/divid⟩s.alt0/divid⟩s.alt0p=/divid⟩s.alt0/divid⟩s.alt0ξ/divid⟩s.alt0/divid⟩s.alt0p,Ω=(E/divid⟩s.alt0ξ/divid⟩s.alt0p)1/slash.leftp=/par⟩nl⟩ft.alt3/integral.dispΩ/divid⟩s.alt0ξ(ω)/divid⟩s.alt0pP(dω)/par⟩nright.alt31/slash.leftp\n, p≥1,\n/divid⟩s.alt0/divid⟩s.alt0ξ/divid⟩s.alt0/divid⟩s.alt0∞∶=esssup\nω∈Ω/divid⟩s.alt0ξ(ω)/divid⟩s.alt0, p=∞.\nLet us mention a previous result (see, e.g., [34, Theorem 3.1]). Let {κn}, n∈N,\nbe a sequence of random variables such that, for some singlevaluep∈(1,∞),\nlim\nn→∞/divid⟩s.alt0/divid⟩s.alt0κn/divid⟩s.alt0/divid⟩s.alt0p=0.\nThen a factorization /divid⟩s.alt0κn/divid⟩s.alt0≤δn⋅τholds, where δnis a certain positive deterministic\nsequence tending to zero and the random variable τbelongs to the space Lp(Ω),\nsuch that\nlim\nn→∞δn=0,/divid⟩s.alt0/divid⟩s.alt0τ/divid⟩s.alt0/divid⟩s.alt0p=1.\nWe follow further one of the auxiliary results in [24, Lemma 2.1], (see a lso [39,\nLemma 33]).\nLemma 2.1. Letα>0andK(p)∈[0,∞)a non-negative function for p≥1. LetZn,\nn∈N, be a sequence of random variables such that\n(E/divid⟩s.alt0Zn/divid⟩s.alt0p)1/slash.leftp≤K(p)⋅n−α, (8)\nfor allp≥1and alln∈N. Then, for all ǫ∈(0,α), there exists a non-negative random\nvariableηα,ǫsuch that\n/divid⟩s.alt0Zn/divid⟩s.alt0≤ηα,ǫ⋅n−α+ǫalmost surely (9)\nfor alln∈N.Factorizable convergence of random variables in Grand Lebesgue S paces 5\nIn other words,\n∀ǫ∈(0,α)⇒ηα,ǫdef=sup\nn∈N[nα−ǫ/divid⟩s.alt0Zn/divid⟩s.alt0]<∞,with probability one .(10)\nMoreover, one can conclude in addition that\n∃p0=p0(α,ǫ)≥1∶ ∀p≥p0⇒E/divid⟩s.alt0ηα,ǫ/divid⟩s.alt0p<∞. (11)\nOf course, only the case 0 <ǫ<αis not trivial for us.\nInterestingapplicationsoftheseresultsinthetheoryofstatistic ofrandomprocesses\nand fields may be found in [9]. Moreover in [24] applications of Lemma 2.1 are given\nfor Itˆ o stochastic differential equations.\nWe want to clarify slightly the last result, under the same conditions, and extend\none under additional assumptions.\nTheorem 2.2. Letα>0andK(p)∈[0,∞)a non-negative function for p≥1. LetZn,\nn∈N, be a sequence of r.v. satisfying the condition (8). In addition let pandǫbe such\nthat\nǫ∈(0,min{1,α}), p>1/slash.l⟩ftǫ. (12)\nThen the r.v.\nηα,ǫdef=sup\nn≥2{nα−ǫ/divid⟩s.alt0Zn/divid⟩s.alt0}\nsatisfies\n/divid⟩s.alt0/divid⟩s.alt0ηα,ǫ/divid⟩s.alt0/divid⟩s.alt0p≤K(p)(pǫ−1)−1/slash.leftp. (13)\nProof. We follow the same arguments as in [24]. Indeed, using Tchebychev-\nMarkov’s inequality, for all δ>0 ,ǫ∈(0,α)andp>1/slash.l⟩ftǫ, we have\nP(nα−ǫ/divid⟩s.alt0Zn/divid⟩s.alt0>δ)≤E/divid⟩s.alt0Zn/divid⟩s.alt0p\nδp⋅n(α−ǫ)p≤Kp(p)\nδp⋅n−pǫ,\ntherefore∞\n/summation.disp\nn=2P(nα−ǫ/divid⟩s.alt0Zn/divid⟩s.alt0>δ)<∞.\nIt follows from Borel-Cantelli lemma\nlim\nn→∞nα−ǫ/divid⟩s.alt0Zn/divid⟩s.alt0=0\nwithprobabilityone. Consequently, ther.v. ηα,ǫisfinitea.e. Moreover, foranarbitrary\nnumberqsuch thatq>1/slash.l⟩ftǫand using (8) and the integral test for the series, we get\n/divid⟩s.alt0/divid⟩s.alt0ηα,ǫ/divid⟩s.alt0/divid⟩s.alt0q\nq=E[ηα,ǫ]q=E/brac⟩l⟩ft.alt3sup\nn≥2/brack⟩tl⟩ft.alt1n(α−ǫ)q/divid⟩s.alt0Zn/divid⟩s.alt0q/brack⟩tright.alt/brac⟩right.alt3\n≤E/brac⟩l⟩ft.alt4/summation.disp\nn≥2n(α−ǫ)q/divid⟩s.alt0Zn/divid⟩s.alt0q/brac⟩right.alt4=/summation.disp\nn≥2E/brac⟩l⟩ft.alt1n(α−ǫ)q/divid⟩s.alt0Zn/divid⟩s.alt0q/brac⟩right.alt1\n≤∞\n/summation.disp\nn=2n(α−ǫ)qE/divid⟩s.alt0Zn/divid⟩s.alt0q≤Kq(q)∞\n/summation.disp\nn=2n−qǫ\n≤Kq(q)/integral.disp∞\n1x−qǫdx=Kq(q)/slash.l⟩ft(qǫ−1).\nSubstituting q=p,we obtain the absert. ✷Factorizable convergence of random variables in Grand Lebesgue S paces 6\nRemark 2.1. It is sufficient for the conclusion of Theorem 2.2 to suppose only the\nfiniteness of the function K=K(p)at least for a singlevaluepsuch thatp>1/slash.l⟩ftǫ.\n3 Grand Lebesgue Spaces norm estimations.\nWe recall here for the reader convenience some known definitions a nd facts from the\ntheory of Grand Lebesgue Spaces (GLS). Let b∈(1,∞]andψ=ψ(p), p∈[1,b), be a\npositive measurable numerical valued function, not necessarily finit e inb, such that\ninf\np∈[1,b)ψ(p)>0.\nWe denote with GΨ(b)the set of all such functions ψ(p), p∈[1,b), for some 1<b≤∞,\nand put\nGΨ∶=/uni22C3.disp\n1<b≤∞GΨ(b).\nFor instance, if m=const>0,\nψm(p)∶=p1/slash.leftm, p∈[1,∞)\nor, for 1<b<∞, α,β=const≥0,\nψb;α,β(p)∶=(p−1)−α(b−p)−β, p∈(1,b),\nthe above functions belong to GΨ(b).\nThe (Banach) Grand Lebesgue Space Gψ=Gψ(b)consists of all the real (or com-\nplex) numerical valued measurable functions f∶Ω→Rhaving finite norm\n/divid⟩s.alt0/divid⟩s.alt0f/divid⟩s.alt0/divid⟩s.alt0Gψdef=sup\np∈[1,b)/brack⟩tl⟩ft.alt4/divid⟩s.alt0/divid⟩s.alt0f/divid⟩s.alt0/divid⟩s.alt0p\nψ(p)/brack⟩tright.alt4. (14)\nWe writeGψwhenb=∞.The function ψ=ψ(p)is named the generating function\nfor the space Gψ.\nIf for instance\nψ(p)=ψr(p)=1, p=r;ψr(p)=+∞, p≠r,\nwherer=const∈[1,∞), C/slash.l⟩ft∞∶=0, C∈R,(an extremal case), then the correspondent\nGψ(r)(p)space coincides with the classical Lebesgue - Riesz space Lr=Lr(Ω).\nRemark 3.1. Let 1<q<∞andb=q. Defineψ(p)=(q−p)−1/slash.leftp,p∈(1,q), and take\nΩ⊂Rn,n≥1, a measurable set with finite Lebesgue measure; then replacing in ( 14)p\nwithq−ε,ε∈(0,q−1), the space Gψreduces to the classical Grand Lebesgue space\nLq)(Ω)defined by the norm\n/divid⟩s.alt0/divid⟩s.alt0f/divid⟩s.alt0/divid⟩s.alt0Lq)(Ω)=/divid⟩s.alt0/divid⟩s.alt0f/divid⟩s.alt0/divid⟩s.alt0q)=sup\n0<ε<q−1ε1\nq−ε/divid⟩s.alt0/divid⟩s.alt0f/divid⟩s.alt0/divid⟩s.alt0q−ε.Factorizable convergence of random variables in Grand Lebesgue S paces 7\nThese spaces and their particular cases Lq)(Ω)(Remark 3.1) are investigated in\nmany works (e.g. [11, 26, 28, 31, 32, 34, 37, 38]). For example they play an important\nrole in the theory of Partial Differential Equations (PDEs) (see, e.g ., [2, 13, 15, 22]), in\ninterpolationtheory(see, e.g., [1,14,16]), inFunctionalAnalysis(s ee, e.g., [12,18,19]),\nin the theory of Probability ([10, 20, 21, 17, 33, 34]), in Statistics ([3 0, chapter 5]), in\ntheory of random fields ([26, 32]).\nThese spaces are rearrangement invariant (r.i.) Banach function s paces; the funda-\nmental function has been studied in [32]. They not coincide, in the gen eral case, with\nthe classical Banach rearrangement functional spaces: Orlicz, L orentz, Marcinkiewicz,\netc., (see [28, 31]).\nIn the general case these spaces are non - separable.\nThe belonging of a function f∶Ω→Rto someGψspace is closely related to its\ntail function behavior\nTf(t)def=P(/divid⟩s.alt0f/divid⟩s.alt0≥t), t≥0,\nast→0+as well as when t→∞(see [26, 27, 21]).\nIn detail, let υbe a non-zero r.v. belonging to some Grand Lebesgue Space Gψ\nand suppose/divid⟩s.alt0/divid⟩s.alt0υ/divid⟩s.alt0/divid⟩s.alt0Gψ=1. Define the Young-Fenchel (or Legendre) transform of the\nfunctionh(p)=h[ψ](p)=plnψ(p), (see, e.g., [7] and references therein)\nh∗(v)=h∗[ψ](v)∶=sup\np∈Dom[ψ](pv−plnψ(p))\nwhere Dom[ψ]denotesthedomainofdefinition(andfiniteness) forthefunction ψ(⋅).\nWe get\nTυ(t)≤exp(−h∗(lnt)), t≥e. (15)\nNote that the inverse conclusion, under suitable natural condition s, is true ([26, 27]).\nNotice also that these spaces coincide, up to equivalence of the nor ms and under\nappropriate conditions, with the so-called exponential Orlicz spaces, see e.g. [26, 25,\n29, 30, 33].\nRemark 3.2. Suppose that for some r.v. ζis satisfied the following condition\n∃a∈(1,b)∶ν(r)def=/divid⟩s.alt0/divid⟩s.alt0ζ/divid⟩s.alt0/divid⟩s.alt0r<∞,∀r≥a.\nFrom Lyapunov’s inequality follows\n/divid⟩s.alt0/divid⟩s.alt0ζ/divid⟩s.alt0/divid⟩s.alt0p≤ν(a),∀p∈[1,a].\nTherefore this r.v. ζbelongs to the Grand Lebesgue space Gθwith the generating\nfunctionθ(⋅)such that\nθ(p)=ν(a), p∈[1,a];θ(p)=ν(p), p∈(a,b).\nLet now the r.v. ζsuch that/divid⟩s.alt0/divid⟩s.alt0ζ/divid⟩s.alt0/divid⟩s.alt0b<∞, b∈(1,∞].\nThe so-called natural function forζis the function ψ(p)=ψ[ζ](p)defined by\nψ(p)=ψ[ζ](p)def=/divid⟩s.alt0/divid⟩s.alt0ζ/divid⟩s.alt0/divid⟩s.alt0p,Factorizable convergence of random variables in Grand Lebesgue S paces 8\nwith the corresponding domain of definition Dom ψ[ζ](⋅).\nAnalogously, letVbe an arbitrary set and let {ζv}, v∈V,be afamilyof random\nvariables such that\nsup\nv∈V/divid⟩s.alt0/divid⟩s.alt0ζv/divid⟩s.alt0/divid⟩s.alt0p<∞,∀p∈[1,b), b∈(1,∞]. (16)\nThenatural function for this family is defined by\nψ[ζv](p)def=sup\nv∈V/divid⟩s.alt0/divid⟩s.alt0ζv/divid⟩s.alt0/divid⟩s.alt0p, (17)\nwith the corresponding domain of definition Dom ψ[ζv].\nEvidently,\nsup\nv∈V/divid⟩s.alt0/divid⟩s.alt0ζv/divid⟩s.alt0/divid⟩s.alt0Gψ[ζv]=1.\nLet us return to the initial relation (8), which we rewrite as follows: le tα>0\nandǫ∈(0,min{1,α}); assume that there exists a generating function ψ(p),p∈(1/slash.l⟩ftǫ,b),\n1/slash.l⟩ftǫ<b≤∞such that\n(E/divid⟩s.alt0Zn/divid⟩s.alt0p)1/slash.leftp≤ψ(p)⋅n−α,1/slash.l⟩ftǫ<p<b. (18)\nNote that the last relation may be rewritten on the language of Gψspaces as\nfollows\n/divid⟩s.alt0/divid⟩s.alt0Zn/divid⟩s.alt0/divid⟩s.alt0Gψ≤n−α. (19)\nDefine as before the keyr.v.\nηǫdef=sup\nn≥1{nα−ǫ/divid⟩s.alt0Zn/divid⟩s.alt0},\nwhich is finite almost everywhere. Moreover, by virtue of Theorem 2 .2, we have\n/divid⟩s.alt0/divid⟩s.alt0ηǫ/divid⟩s.alt0/divid⟩s.alt0p≤ψ(p)(pǫ−1)−1/slash.leftp, p∈(1/slash.l⟩ftǫ,b). (20)\nIntroduce the following new generating function\nκα,ǫ[ψ](p)def=ψ(p)(pǫ−1)−1/slash.leftp, p∈(1/slash.l⟩ftǫ,b), (21)\nwhich define the corresponding Grand Lebesgue space Gκα,ǫ[ψ].\nTo summarize, we have the following result.\nTheorem 3.1. Letα>0,b∈(1,∞]andǫ∈(0,min{1,α}). LetZn,n∈N, be a\nsequence of r.v. and let ψ(p),p∈(1/slash.l⟩ftǫ,b), be a function satisfying the condition (18).\nLetκα,ǫ[ψ]the function defined in (21). Then the r.v. ηǫdefined above satisfies\n/divid⟩s.alt0/divid⟩s.alt0ηǫ/divid⟩s.alt0/divid⟩s.alt0Gκα,ǫ[ψ]≤1, (22)\nand the corresponding exponential tail estimate (15).Factorizable convergence of random variables in Grand Lebesgue S paces 9\nRemark 3.3. Note that one can also consider the space Gψ(a,b),1<a<b≤∞,\nwherep∈(a,b).In this case the norm is defined alike above\n/divid⟩s.alt0/divid⟩s.alt0f/divid⟩s.alt0/divid⟩s.alt0Gψ(a,b)def=sup\np∈(a,b)/brack⟩tl⟩ft.alt4/divid⟩s.alt0/divid⟩s.alt0f/divid⟩s.alt0/divid⟩s.alt0p\nψ(p)/brack⟩tright.alt4. (23)\n4 A slight generalization.\nConsider the source random sequence, which we denote now as {Vn}, n∈N, taking\nvalues in a certain (complete) Banach space Xequipped with the norm ⟨⋅⟩=⟨⋅⟩X,\nnon necessarily separable, and let us impose the following condition\n/divid⟩s.alt0/divid⟩s.alt0⟨Vn⟩X/divid⟩s.alt0/divid⟩s.alt0p≤ψ(p)⋅ǫn, (24)\nwhereψ=ψ(p),p∈[1,b), b∈(1,∞], is some positive generating function and {ǫn}\nis a numerical positive sequence tending to zero as n→∞∶lim\nn→∞ǫn=0.\nMoreover, we suppose that\n∃p∈[1,b)∶∞\n/summation.disp\nn=1ǫp\nn<∞. (25)\nLet also{βn}be another numerical positive sequence also tending to zero and\nsuch that the following function\nσp(p)def=∞\n/summation.disp\nn=1ǫp\nn\nβp\nn, (26)\nis finite at least for some value p∈[1,b).It is easy to ground the existence of this\nfunction under the condition (25).\nExample 4.1. Let here\nǫn=n−αlnm(n+1), α>0, m∈R;\nthen one can choose\nβn∶=n−θln−ν(n+1), θ∈(0,α), ν∈R,\nand correspondingly one can choose pan arbitrary number such that p>1/slash.l⟩ft(α−θ), for\narbitrary values of the parameters m,ν; but in the case when\nm+ν\nα−θ<−1\none can choose even p∶=1/slash.l⟩ft(α−θ).\nOfcourse, onecanusearbitrary slowlyvarying atinfinityfunctions, strictlypositive,\nL=L(n), M=M(n), instead of lnk(⋅).Factorizable convergence of random variables in Grand Lebesgue S paces 10\nIn detail, let\nǫn∶=n−αL(n), α>0,\nthen one can analogously choose\nβn∶=n−θM(n), θ∈(0,α),\nwithp>1/slash.l⟩ft(α−θ).\nPut also\nζdef=sup\nn/brack⟩tl⟩ft.alt3⟨Vn⟩X\nβn/brack⟩tright.alt3. (27)\nWe deduce quite alike in the proof of Theorem 2.2\nEζp≤ψp(p)∞\n/summation.disp\nn=1/brack⟩tl⟩ft.alt4ǫp\nn\nβp\nn/brack⟩tright.alt4=ψp(p)⋅σp(p).\nTo summarize:\nTheorem 4.1. Under the above restrictions and notations the following es timate holds\n/divid⟩s.alt0/divid⟩s.alt0ζ/divid⟩s.alt0/divid⟩s.alt0p,Ω≤ψ(p)⋅σ(p), (28)\nor equally, if we introduce the generating function γ(p)∶=ψ(p)⋅σ(p),in the interval\nof its finiteness, then the norm of ζin the Grand Lebesgue space Gγsatisfies\n/divid⟩s.alt0/divid⟩s.alt0ζ/divid⟩s.alt0/divid⟩s.alt0Gγ≤1. (29)\nExample 4.2. Let nowǫn=qn, n=0,1,2,...;q=const∈(0,1).One can choose\nQ=const∈(q,1),so thatδ∶=q/slash.l⟩ftQ∈(0,1).We get\nσp(p)=∞\n/summation.disp\nn=0qpn\nQpn=∞\n/summation.disp\nn=0δpn=1\n1−δp<∞\nfor all the positivevaluesp.Butfor the values p≥1 we have\nσ(p)=(1−δp)−1/slash.leftp≤(1−δ)−1,\nand we conclude, on the basis of Theorem 4.1,\n/divid⟩s.alt0/divid⟩s.alt0ζ/divid⟩s.alt0/divid⟩s.alt0p,Ω≤ψ(p)/slash.l⟩ft(1−δ). (30)Factorizable convergence of random variables in Grand Lebesgue S paces 11\n5 Examples. Lower estimates.\n1.Let us ground that the estimate of Theorem 2.2 is essentially non-imp rovable,\nof course, under our restrictions. Namely, let α=const>0 and put\nZn∶=θn\nnα, n=2,3,...,\nwhere{θn}is a sequence of independent positive identical distributed r.v. having\nstandard exponential distribution\nP(θn>t)=e−t, t>0.\nNote first of all that, since the r.v. {Zn}are independent, the convergence\nZn→0 asn→∞is completely equivalent to the following restriction\n∀ǫ>0⇒∞\n/summation.disp\nn=1P(Zn>ǫ)<∞,\nor equally\n∀ǫ>0⇒∞\n/summation.disp\nn=1exp(−ǫ nα)<∞.\nEvidently, both these conditions are satisfied, therefore really\nP(lim\nn→∞Zn→0)=1,\nand consequently supnZn<∞a.e.\nDefine, as before, η∶=sup\nn≥1[nα−ǫZn], ǫ∈(0,α).It is not hard to prove that as\nu→∞,(u≥3),\nP(η>u)≍C(ǫ)u−1/slash.leftǫ, C(ǫ)=1\nǫΓ/par⟩nl⟩ft.alt31\nǫ/par⟩nright.alt3=Γ/par⟩nl⟩ft.alt31+1\nǫ/par⟩nright.alt3, (31)\nwhere Γ(⋅)is the Euler’s Gamma function.\nIn detail, denoting κn=θn/slash.l⟩ftnǫ=nα−ǫZn, by integral test for the series and the\nchange of variable z=uxǫ, asu→∞we have\nΣ1def=/summation.disp\nnP(κn>u)=/summation.disp\nnexp(−u nǫ)∼/integral.disp∞\n0exp(−u xǫ)dx\n=u−1/slash.leftǫ⋅1\nǫ/integral.disp∞\n0z1\nǫ−1e−zdz≍1\nǫΓ/par⟩nl⟩ft.alt31\nǫ/par⟩nright.alt3u−1/slash.leftǫ=C(ǫ)u−1/slash.leftǫ.\nFrom the Bonferroni’s inequality (upper bound), as u→∞, we have\nP(η>u)=P(sup\nnκn>u)=P/par⟩nl⟩ft.alt4/uni22C3.disp\nnκn>u/par⟩nright.alt4≤Σ1.\nOn the other hand, taking into account the independence of the ra ndom variables\n{κn}, we have\nΣ2def=/summation.disp\nn/summation.disp\nm>nP(κn>u, κm>u)=/summation.disp\nn/summation.disp\nm>nP(κn>u)P(κm>u)\n≤/summation.disp\nn/summation.disp\nmP(κn>u)P(κm>u)=Σ2\n1.Factorizable convergence of random variables in Grand Lebesgue S paces 12\nThe relation (31) follows from the bilateral Bonferroni’s estimate and Σ 2≤Σ2\n1\nsince, foru→∞, we have\nΣ1≍Σ1−Σ2\n1≤Σ1−Σ2≤P(sup\nnκn>u)≤Σ1.\nFurther, we have therefore\n/divid⟩s.alt0/divid⟩s.alt0η/divid⟩s.alt0/divid⟩s.alt0p≍(1/slash.l⟩ftǫ−p)−1/slash.leftp, p∈[1,1/slash.l⟩ftǫ). (32)\n2.On the other hand, η≥Z1,therefore also\n/divid⟩s.alt0/divid⟩s.alt0η/divid⟩s.alt0/divid⟩s.alt0p≥/divid⟩s.alt0/divid⟩s.alt0Z1/divid⟩s.alt0/divid⟩s.alt0p≥Cp, p≥1. (33)\nThus, in this case\n/divid⟩s.alt0/divid⟩s.alt0η/divid⟩s.alt0/divid⟩s.alt0p≥Cmax{p,(1/slash.l⟩ftǫ−p)−1/slash.leftp}, p∈[1,1/slash.l⟩ftǫ). (34)\n3.Let the inequality (20) be given under conditions of the first section . Let\nus assume also that the generating function ψ=ψ(p)is the natural function for the\nvariableZ1∶\nψ(p)=/divid⟩s.alt0/divid⟩s.alt0Z1/divid⟩s.alt0/divid⟩s.alt0p, p∈[1,b).\nSuppose nowthat the pbelongs tothe interval (a,b),wherea≥4/slash.l⟩ftǫ, b∈(a,∞].\nThen one has for such values p\n/divid⟩s.alt0/divid⟩s.alt0η/divid⟩s.alt0/divid⟩s.alt0p≤31/slash.leftǫ⋅ψ(p). (35)\nOn the other hand,\n/divid⟩s.alt0/divid⟩s.alt0η/divid⟩s.alt0/divid⟩s.alt0p≥/divid⟩s.alt0/divid⟩s.alt0Z1/divid⟩s.alt0/divid⟩s.alt0p=ψ(p). (36)\n6 Concluding remarks.\nIt is interesting in our opinion to investigate a more general converg ence of random\nvariables almost everywhere in the metric spaces, even in the topolo gical spaces, in the\nspirit of Remark 1.2.\nAcknowledgements .M.R. Formica is member of Gruppo Nazionale per l’Analisi Matematica,\nla Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (IN-\ndAM) and member of the UMI group “Teoria dell’Approssimazione e App licazioni (T.A.A.)”and is\npartially supported by the INdAM-GNAMPA project, Risultati di regolarit` a per PDEs in spazi di fun-\nzione non-standard , codice CUP E53C22001930001 and partially supported by University of Naples\n“Parthenope”, Dept. of Economic and Legal Studies, project Co RNDiS, DM MUR 737/2021, CUP\nI55F21003620001.\nAuthor contributions: All authors contribute equally to the manuscript.\nConflict of interest : The authors declare that they have no conflict of interest.Factorizable convergence of random variables in Grand Lebesgue S paces 13\nData Availability : Data sharing is not applicable to this article as no datasets\nwere generated or analyzed during the current study.\nReferences\n[1] I. Ahmed, A. Fiorenza and M.R. 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Appl. 129(2019), no. 9, 3018–3054."    },    {        "title": "2401.13141v2.Nonlinear_bandgap_transmission_by_discrete_rogue_waves_induced_in_a_pendulum_chain.pdf",        "content": "arXiv:2401.13141v2  [nlin.PS]  26 Jan 2024Nonlinear bandgap transmission by discrete rogue waves ind uced in a pendulum chain\nAlain B. Togueu Motcheyo1,2,∗, Masayuki Kimura3, Yusuke Doi4, Juan F. R. Archilla5\n1Department of Mechanical Engineering, Higher Technical Te acher’s Training College (ENSET) Ebolowa,\nUniversity of Ebolowa, P.O. Box 886, Ebolowa-Cameroon\n2Laboratory of Mechanics, Department of Physics, Faculty of Science,\nUniversity of Yaounde I, P.O. Box 812, Yaounde, Cameroon\n3Department of Electrical and Electronic Engineering,\nFaculty of Science and Engineering, Setsunan University, J apan\n4Department of Mechanical Engineering, Graduate School of E ngineering,\nOsaka University 2-1 Yamadaoka, Suita, Osaka 565-0871, Jap an\n5Group of Nonlinear Physics, Universidad de Sevilla,\nETSI Inform´ atica, Avda Reina Mercedes s/n, 41012-Sevilla , Spain∗\nWestudynumericallyadiscrete, nonlinearlattice, whichi sformedbyachainofpendulasubmitted\nto a harmonic-driving source with constant amplitude and pa rametrical excitation. A supratrans-\nmission phenomenon is obtained after the derivation of the h omoclinic threshold for the case when\nthe lattice is driven at one edge. The lattice traps gap solit ons when the chain is subjected to a\nperiodic horizontal displacement of the pivot. Discrete ro gue waves are generated for the case when\nthe pendulum is simultaneously driven and shaken. This work may pave the way for experimental\ngeneration of discrete rogue waves within simple devices.\nKeywords: Discrete rogue wave, Nonlinear supratransmissi on, Nonlinear pendulum\nI. INTRODUCTION\nSince the pioneering work by Geniet and L´ eon [1] on\nthenonlinearsupratransmissionphenomenon, thebehav-\nior of a plane wave within the forbidden band has fasci-\nnated a number of researchers. Nowadays, there is an-\nother important consequence of this phenomenon, which\nis the capacity to reveal the types of waves that can\npropagate in the dispersive nonlinear lattice with a pe-\nriodically driven boundary edge. Breathers have been\ngenerated in mechanical systems [1–7], in the Fermi-\nPasta-Ulam Tsingou (FPUT) model [8–10], in a discrete\ninductance-capacitance electrical line [11], in Josephson\njunctions [12], and in molecular dynamics models [13].\nAn envelope soliton has been created in optical wave-\nguide arrays [14–16] and in electrical lattices [17, 18]. A\ntrain of dark solitons has been generated in a discrete\nSchr¨ odinger lattice with cubic-quintic nonlinearity [19].\nA travelling asymmetric bright soliton has been gener-\nated for the α,β-FPUT [10]. Up to now, to the best of\nour knowledge, the transmission of gap solitons is done\ncontinuously in time due to the periodic excitation at\nthe edge of the lattice. There exist waves that seem to\nappear out of nowhere and then vanish without a trace\n[20]. They are called rogue waves, a phenomenon that\nhas been observed in water [21, 22], nonlinear optics [23],\nphotonic lattices [24], metamaterials [25], and in beam–\nplasma interactions [26], to mention a few systems.\nGenerally, rogue waves occur within systems modeled\nby an analytical integrable equation. For the continu-\nous integrable equations, we can name the Sine-Gordon\n∗Corresponding author: alain.togueu@univ-yaounde1.cm;\nAlternate electronic addresses:\nabtogueu@yahoo.fr; alainbertrandtogueu@gmail.comequation[27], the one-dimensionalNonlinear Schr¨ odinger\n(NLS) equation [20, 21], the coupled NLS equation [28],\nthe Coupled Higgs Equation [29], and the Sasa-Satsuma\nequation [30]. For the analytically integrable discrete\nequation, rogue waves have been found in the discrete\nAblowitz-Ladik [31], the coupled Ablowitz-Ladik equa-\ntions [32], and the Ablowitz-Musslimani equation [33].\nForthe non-analyticalintegrablediscreteequation, rogue\nwaves have been simulated numerically in the discrete\nNLS equations with cubic [34], and saturable [35] nonlin-\nearities. We have recently pointed out the first idea for\nthe creation of a rogue wave within a nonlinear band gap\nin the conference paper [36]. The purpose of this letter is\nto consider a way of creating discrete rogue waves using\na relatively simple device.\nThe outline of the paper is the following: in Section\nII, we present the model under investigation. The ho-\nmoclinic supratransmission threshold is derived for the\ndiscrete Sine-Gordon equation. In Section III, we nu-\nmerically integrate the dimensionless equation governing\nthe physical model. Firstly, the chain of discrete pendula\nis studied without the parametric excitation in order to\nvalidate the supratransmission threshold; secondly, the\nlattice is shaken without a driven edge and thirdly, the\nbehavior of the lattice is observed with a shaken and a\ndriven edge simultaneously. The spectral analysis for the\nlatter system is also included. Section IV concludes the\nletter.2\nII. MODEL AND SUPRATRANSMISSION\nTHRESHOLD\nA. Model description\nThe model under consideration consists of a pendulum\nchain connected by torsional springs and subjected to a\nhorizontaldriving force with frequency ωdand amplitude\nA, asillustratedinFig.1 reproducedfromRef.[37]. Each\nrigid rod of length land mass msupports the pendulum\nbobwithmass Matitsend. Theexperimentalsystemfor\nthe pendulum chain shown in Fig.1 had been proposed\nin Ref. [38]. The Lagrangian for a chain of pendula in\nabsence of damping can be written as follows [37]:\nL=N/summationtext\nn=1{1\n2I˙θn2+1\n2/parenleftbigg\nMl+ml\n2/parenrightbigg\ngcos(θn)\n+1\n2/parenleftbigg\nMl+ml\n2/parenrightbigg/bracketleftBig\n2Aωd˙θnsin(ωdt1)cos(θn)/bracketrightBig\n−1\n4β/bracketleftBig\n(θn−θn−1)2+(θn−θn+1)2/bracketrightBig\n},\n(1)\nwithI=Ml2+1\n3ml2beingthesystemmomentofinertia.\nThe angle θnmeasures the deviation from the vertical\nfor the nth pendulum, ˙θnis the corresponding angular\nspeed,gdenotes the acceleration due to gravity, and β\nis the linear coupling coefficient between pendula due to\na torsion spring. The equation of motion for the nth\npendulum, derived using the Euler-Lagrange’s equation,\nis given by [38, 39]:\n¨θn−β\nI(θn+1+θn−1−2θn)+ω2\n0sin(θn)+\nfω2\ndcos(ωdt1)cos(θn) = 0,\n(2)\nwithω2\n0=g\nI(Ml+ml\n2), andf=ω2\n0A\ng, being the dimen-\nsionless forcing coefficient. Equation (2) can be further\nsimplified by scaling the time using the transformation.\nt1=t\nω0. In this way, the dimensionless form of the equa-\ntion of motion for the nth pendulum can be written as\nfollows:\n¨θn−c(θn+1+θn−1−2θn)+sin(θn)+\nfω2\n1cos(ω1t)cos(θn) = 0,(3)\nwherec=β\nIis the dimensionless coupling parameter,\nandω1=ωd\nω0is the dimensionless frequency for the pe-\nriodic horizontal displacement of the pivot. The linear\ndispersion relation for Eq.(3) is given by\nω2= 1+2c(1−cos(k)), (4)\nfrom which the linear band 1 ≤ω≤√1+4cis obtained.\nFIG. 1. Schematic representation of a pendulum chain con-\nnected by torsional springs subjected to a horizontal drivi ng\nforce. (a) front view; (b) profile view. Reproduced with per-\nmission from Ref.[37], copyright by APS 2014.\nB. Supratransmission threshold\nHere, we consider the model without the forcing coeffi-\ncient, i.e., f= 0, then Eq.(3) becomes the discrete Sine-\nGordon equation. Expanding sin( θn) as a Taylor series\nup to the third order, the equations of motion become\n¨θn−c(θn+1+θn−1−2θn)+θn−θ3\nn\n6= 0.(5)\nThe time-periodic solution of the equation can be ob-\ntained by proposing a harmonic solution in the form\nθn=xncos(ωt) (see Refs. [40–42]). The map corre-\nsponding to the stationary equation can be written as\n[40–48]\nxn+1=axn−bx3\nn−yn, yn+1=xn.(6)\nwitha= 2+1\nc/parenleftbig\n1−ω2/parenrightbig\nandb=1\n8c. This map possesses\nthree fixed points: x0= 0 and x±=±/radicalbig\n8(1−ω2). The\nlatter two exist only for ω <1. This frequency band\ncorresponds to the lower forbidden band, which is in\nagreementwiththebandforwhichthesupratransmission\nphenomenon has been observed by Geniet and L´ eon [1].\nFixed points are depicted by a cross in Fig.2. The sta-\nbility of a fixed point is obtained by linearizing the map\naround the fixed point using the procedure described in\nRef.[47]. A necessary condition for the existence of man-\nifolds is that the fixed point must be a saddle point. A\nhomoclinic orbit corresponds to an orbit that connects,\nboth in forward and backward time, a saddle fixed point\nwith itself [46]. Within the lower forbidden band ( ω <1)\nonly the fixed point x0= 0 is a saddle point. Figure 2 de-\npicts the progression of the stable (blue line) and unsta-\nble (red line) manifolds emanating from the fixed point\nx0= 0. Stable and unstable manifolds emanating from\nthe fixed point x0= 0 intersect and form a homoclinic\norbit, which can be identified by the loop in Fig. 2. The\nfirst-orderconnectionsoftheloopcorrespondtothemain\nhomoclinicorbit ascanbe clearlyobservedin Fig. 2. The3\nsupratransmission threshold corresponds to the value of\nthe turning point of the main homoclinic orbit [15]. Its\nvalue can be identified in this work by the dashed green\nline in Fig.2.\nxn-1.5 -1 -0.5 0 0.5 1 1.5yn\n-1.5-1-0.500.511.5\nAmplitude threshold (1.255)\nFIG. 2. Homoclinic orbit of the 2D map (6) for c= 1, and\nω= 0.95.. The dashed green line corresponds to the supra-\ntransmission threshold: Athr= 1.255.\nIII. NUMERICAL EXPERIMENTS\nIn this section, numericalstudies arecarriedouton the\ndiscrete equation (3) with the ode45 solver of MatLab.\nThis solver uses variable time step to keep the desired\nrelative and absolute tolerance, which are set at 10−10.\nThe left boundary of the lattice depends on whether or\nnot the lattice is driven. The reflection at the right edge\nof the lattice will be avoided by choosing a large value\nof N=101 and appropriate time for the full integration of\nEq.(3).\nA. Driven pendulum chain without shaking\nThe unshaken lattice ( f= 0) will be consider here and\nthe following harmonic boundary condition is imposed to\nthe chain for the full integration of Eq.(3):\nθ0(t) =Acos(ωt), (7)\nwhereAis the driving amplitude smoothly growing from\nthe value 0 to Aandωis the dimensionless frequency.\nFigure 3 depicts the behavior of the chain with driving\namplitude A= 1.254 (slightly below the threshold 1.255)\nand with the dimensionless frequency ω= 0.95 within\nthe lower forbidden band, while Fig.4 shows a train of\ngap solitons generated by driving the lattice with ampli-\ntude A=1.256 (slightly above the threshold 1.255) and\nthe same frequency. The evanescent wave seen shown in\nFig.3 and the energy flow shown in Fig.4 confirm the\nagreement of the homoclinic threshold shown in Fig2\nwith the numerical simulation.\nFIG. 3. Spatiotemporal evolution for the discrete equation\n(3) with boundary driving condition (7). The parameters are\nf= 0,c= 1,ω= 0.95.andA= 1.254< Athr\nB. Horizontally shaken pendulum chain\nThe lattice is subjected to a periodic horizontal dis-\nplacement of the pivot point with dimensionless fre-\nquencyω1and amplitude f. Figure 5 depicts the spa-\ntiotemporal evolution of the wave that results from the\nfull integration of Eq.(3) with a shaking dimensionless\nfrequency in the lower forbidden band ( ω1= 0.95) and\nthe other parameters given by: f= 0.2 andc= 1. It\nis observed, similarly to Ref.[37], that the lattice traps\nthe gap soliton as and that, therefore, there is no trans-\nmission process. The same phenomenon is observed (not\nshown here) in the upper forbidden band.\nThe trapping of the gap soliton here is an analogywith\nthe light being trapped into several neighboring waveg-\nuides obtained in Ref.[14], but it is done at different\ntimes. The trapping here is different from the localiza-\ntion in space, which is a consequence of the staggeringly\ndriving force obtained in Ref.[49]. Another form of local-\nization can be obtained using random fluctuations [50].\nBy periodically driving the lattice, the gap transmission\nis observed while the soliton is trapped at a different\ntime when the chain is subjected to a periodic horizontal4\nFIG. 4. Spatiotemporal evolution for the discrete equation\n(3) with boundary driving condition (7). The parameters are\nf= 0,c= 1,ω= 0.95.andA= 1.256> Athr.\nshaking. Below, the behavior of the chain will be ex-\nplored when it is subjected to being both simultaneously\ndriven and shaken.\nC. Simultaneously driven and shaken pendulum\nchain\nHere we have two excitations: periodically driven edge\nand parametric excitation. The periodically driven edge\nfrequency is taken in the lower forbidden band ( ω=\n0.95). The shaking frequency ω1will be taken firstly\nwithin the linear phonon band (1 ≤ω≤√1+4c) and\nsecondly in the lower forbidden band ( ω <1).\nFigure 6 shows the spatiotemporal evolution of the\nchain with a driving frequency ( ω= 0.95) within the\nlower forbidden band and with driving amplitude ( A=\n1.22) below the supratransmission threshold ( Athr=\n1.255). The shaking frequency ( ω1= 1.22) is taken\nwithin the linear phonon band. The generation of a train\nof solitons is observed, although the driving amplitude is\nbelow the threshold. Therefore, the presence of the shak-\ning reduces the supratransmission threshold. The same\nphenomenonisobserved(notshownhere)whentheshak-\ning frequency is taken in the upper forbidden band.\nLet us now consider the case where the driving ampli-\ntude isA= 1.256> Athrand the frequency is ω= 0.95.\nFIG. 5. Spatiotemporal evolution of the lattice submitted t o\nperiodic horizontal shaking: f= 0.2;c= 1;ω1= 0.95.\nFIG. 6. Spatiotemporal evolution of the lattice with coupli ng\nconstant c= 1 submitted simultaneously to a periodically\ndriven edge ( ω= 0.95,A= 1.22< Athr), and a periodic\nhorizontal shaking with frequency ω1= 1.22 in the linear\nband. The forcing coefficient is f= 0.02.\nWithout the shaking phenomenon ( f= 0), numerical\nintegration of Eq.(3) with these parameters generates\nband-gap transmission as can be seen in Fig.4. Fig-\nure 7 (a) shows that the presence of shaking with fre-\nquencyω1= 0.8, within the lower forbidden band, de-\nstroys the supratransmission phenomenon. This is con-\ntrary to the case with the shaking frequency within the\nphonon band. When the forcing coefficient fincreases5\na)\nb)\nFIG. 7. Spatiotemporal evolution of the lattice with coupli ng\nconstant c= 1 submitted simultaneously to a periodically\ndriven edge ( ω= 0.95,A= 1.256> Athr), and a periodic\nhorizontal shaking frequency ω1= 0.8 in the lower forbidden\nband. Forcing coefficient: a) f= 0.02, b)f= 0.08\ntof= 0.08, the nonlinear bandgap transmission oc-\ncurs as can be seen in Fig.7 (b). The presence of the\nshaking phenomenon here increases the supratransmis-\nsion threshold.\nIn Fig.8, we depict a progression of the spatiotempo-\nral evolution of the driven lattice in the presence of the\nshaking phenomenon for different forcing coefficients f.\nThe shaking frequency is equal to the driving frequency\nω. For small f(f= 0.02), the band-gap transmission\ndisappears, although the driving frequency is within the\ngap and the amplitude is above the supratransmission\nthreshold, as can be seen in Fig.8 (a). The disappear-\nance of the supratransmission phenomenon means that\nthe threshold increases in the presence of shaking with\nfrequency within the lower forbidden band.\nForf= 0.026 (See Fig.8 (b)), an unpredictable\nand unexpectedly localized wave appears and disappears\nwithout a trace. The phenomenon is similar to that ob-\ntained by Akhmediev et al. [20] in a continuous lattice.\nThediscreteroguewaveisproducedasaresultofasimul-\ntaneously driven and shaken pendulum. For f= 0.027\n(Fig.8 (c)), the number of unpredictable localized waves\nincreases and the spatiotemporal dynamic of the lattice\nis similar to the second-order discrete rogue wave found\nanalytically in Ref.[32]. For a slightly larger value of theforcing coefficient (see Fig.8 (d)), several localized waves\nappear and the configuration is similar to that obtained\nin Ref.[51].\nA discrete rogue wave is generated here with a plane\nwave as the initial condition. To the best of our knowl-\nedge, this is the first time that this phenomenon is ob-\nserved. It is worth pointing out that presently it is not\npossible to discard that a rogue wave can also be gener-\nated in the just driven, not shaken case, although it has\nnot yet been observed.\nD. Spectral analysis\nWehaveperformedthespectralanalysisofroguewaves\nfollowing Refs.[53]. In particular we present here the\nanalysis of the case presented in Fig.8 (d) for f= 0.03,\nwhere we can observe several rogue waves. As rogue\nwaves are limited in time and space the precision of the\ntwo dimensional fast Fourier transform in space and time\n(XTFFT) is small as it depends on the time interval and\nsublattice size, which cannot be enlarged, but it provides\nvaluable information nonetheless. For example, we con-\ncentrate in the roguewave appearingin the first 30 parti-\nclesforthetimebetween250and400,showninFig.9(a).\nThe dispersion relation and the XTFFT are represented\nin Fig.9 (b). The breather band can be seen slightly\nbelow the forcing frequency near k= 0. The correspond-\ning breather line is also represented, its slope being the\nrogue wave velocity Vb≃0.11 which can also be ob-\ntained from the x(t) curve in Fig.9 (a). The breather\nband value for k= 0 provides the frequency in the mov-\ning frame Ω b≃0.85. As this frequency is not zero, the\nrogue wave is breather–like and comparing Ω bwith the\nvelocity frequency ωV= 2πVb(the frequency at which\na rogue wave encounters different sites), we obtain that\nΩb/ωV≃5/4. This means that the rogue wave performs\napproximately 5 oscillations while moving 4 sites. As the\ncentral wave vector of the breather band is near k= 0,\nthedifferentpendulahaveasmallphasedifference, mean-\ning that the breather profile is bell shaped. Therefore, it\nis also confirmed that in this system rogue waves are be-\nlow the phonon band. Similar analysis can be performed\nfor other rogue waves with qualitatively similar results.\nIV. CONCLUSIONS\nIn this work, we have studied the spatiotemporal be-\nhavior of the discrete pendulum chain, firstly excited\nby a harmonic wave, then by a parametric excitation,\nand finally by simultaneous driving and shaking exci-\ntations. The one-edge-driven lattice produces the well-\nknownsupratransmissionphenomenoninagreementwith\nthe homoclinic threshold. Localization in space of the\nwave is obtained after shaking the lattice. The threshold\nof the supratransmissionis reduced when the lattice is si-\nmultaneously driven with a frequency within the forbid-6\nden band and shakenwith afrequency within the phonon\nband. The threshold value increases when it is shaken\nwith a frequency in the lower forbidden band. Discrete\nrogue waves are obtained after simultaneously driving\nand shaking the lattice with the same frequency. Theexperimental device for the shaken pendulum has been\nrealized in order to derive discrete breathers at Dickin-\nson College[52]. We hope that the result presented in\nthis letter will allow the modification of the experimen-\ntal setup in order to obtain discrete rogue waves.\na)\n b)\nc)\n d)\nFIG. 8. Spatiotemporal evolution of the lattice with coupli ng constant c= 1 submitted simultaneously to a periodically driven\nedge (ω= 0.95,A= 1.256) and a periodic horizontal shaking ( ω1=ω= 0.95) with the forcing coefficient fincreasing from\nleft to right and from top to bottom: a) f= 0.02, b)f= 0.026, c)f= 0.027, d)f= 0.03.\nACKNOWLEDGMENTS\nAlain Bertrand Togueu Motcheyo wishes to express its\ndeepest gratitude to Prof. Masayuki Kimura and to all\nthe organizers of the ”International Symposium on Non-\nlinear Theory and Its Applications” (NOLTA 2022), De-\ncember 12-15, 2022, for the opportunity they provided to\nhim for presenting a part of this work. JFRA acknowl-edgestheUniversitiesofOsakaandLatviaforhospitality.\nFUNDING\nMK acknowledges support from grants JSPS Kak-\nenhi (C) No. 21K03935. YD acknowledges the support\nfrom grant JSPS Kakenhi (C) No. 19K03654. 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Different Mpriors lead to relative changes in\nthe mean values of H0from 2% to 7%. Loose priors on Myield H0estimates consistent with both the\nPlanck 2018 result and the SH0ES result at the 68% confidence level. We also examine the potential\nimpact of peculiar velocity subtraction on the value of H0, and show that it is insignificant for the\nSNe Ia observations with redshift z >0.01 used in our analyses. We also repeat the analysis in the\ncosmography model and find very similar results. This suggests that our results are robust and model\nindependent.\nKeywords: Cosmology (343) — Cosmological parameters (339) — Hubble constant (758) — Observa-\ntional cosmology (1146)\n1.INTRODUCTION\nThe Hubble constant, H0, that is the current value\nof the cosmological expansion rate, is inversely propor-\ntional to the age of the universe, t0∝1/H0, and is\na key cosmological parameter, see e.g., Freedman &\nMadore (2010). In addition, when estimating cosmolog-\nical model parameters from observational data, H0can\nbe degenerate with other cosmological parameters such\nas the nonrelativistic matter density parameter or the\nspatial curvature density parameter, and so the value of\nH0can affect observational constraints on other param-\neters (see e.g. Chen & Xu 2016; Chen et al. 2016, 2017;\nPark & Ratra 2020; Cao et al. 2020; Khadka & Ratra\n2020; Qi et al. 2021).\nMeasuring H0is challenging, especially because of\nthe difficulty in determining distances to astronomical\nobjects (Shah et al. 2021). Many methods have been\nused to improve the accuracy and precision in the mea-\nchenyun@bao.ac.cn\nsuresh.math@igu.ac.insurement of H0since Hubble’s first measurement in\n1929. Nevertheless, there have been many differing esti-\nmates of H0over the following more than half a century\nuntil measured values began to converge at the start\nof the 21st century, led by the 2001 measurement of\nH0= 72±8 km s−1Mpc−1(1σerror including sys-\ntematics) from the Hubble Space Telescope (HST) Key\nProject (Freedman et al. 2001).\nIn the two decades since, H0measurements with\nsmaller error bars have become available. For example,\nthe Planck satellite 2013 results included an estimate of\nH0= 67.9±1.5 km s−1Mpc−1(Planck Collaboration\n2014) inferred from measurements of (higher-redshift)\ncosmic microwave background (CMB) temperature and\nlensing-potential power spectra in the framework of the\nstandard spatially-flat ΛCDM cosmological model (Pee-\nbles 1984). Slightly earlier, Riess et al. (2011) estimated\nH0= 73 .8±2.4 km s−1Mpc−1from (lower-redshift)\napparent magnitude versus redshift measurements of\nSNe Ia that were calibrated using HST observations of\nCepheid variables in host galaxies of eight nearby SNe\nIa. Statistically, the difference between these two re-\nsults is 2 .1σ. In the same year, from a median statisticsarXiv:2401.13187v2  [astro-ph.CO]  28 Feb 20242\nanalysis of Huchra’s compilation of 553 measurements\nofH0, Chen & Ratra (2011) found H0= 68±2.8 km\ns−1Mpc−1, at 1 σand including systematic errors (also\nsee Gott et al. 2001; Chen et al. 2003; Calabrese et al.\n2012), 1 .6σlower than the Cepheid-calibrated SNe Ia\nestimate (Riess et al. 2011).\nMore recently, the difference between some local and\nhigh-redshift determinations of H0has increased to ∼\n5σ, for instance, H0= 73.04±1.04 km s−1Mpc−1from\nthe Supernovae and H0 for the Equation of State of\ndark energy (SH0ES) project that use (lower-redshift)\nCepheid-calibrated SNe Ia data (Riess et al. 2022),\nandH0= 67 .36±0.54 km s−1Mpc−1from (higher-\nredshift) Planck 2018 CMB observations under the stan-\ndard ΛCDM model (Planck Collaboration 2020). There\nare however other local H0estimates that are not as\nhigh, for instance, H0= 69.8±1.7 km s−1Mpc−1from\nSNe Ia calibrated using tip of the red-giant branch data\n(Freedman 2021) or the flat ΛCDM model value pro-\nvided in Cao & Ratra (2023), H0= 69 .5±2.4 km\ns−1Mpc−1, from a joint analysis of Hubble parame-\nter, baryon acoustic oscillation (BAO), Pantheon+ SNe\nIa, quasar angular size, reverberation-measured Mg ii\nand C ivquasar, and 118 Amati correlation gamma-ray\nburst data (that is independent of CMB data).\nThe difference between the higher-redshift (Planck\nand other CMB) H0values and some of the lower-\nredshift H0values is known as the Hubble tension. A\nnumber of schemes have been proposed to explain or\nresolve the H0tension. In general, these fall into two\ncategories (Shah et al. 2021; Di Valentino et al. 2021;\nPerivolaropoulos & Skara 2022; Abdalla et al. 2022; Hu\n& Wang 2023; Vagnozzi 2023; Freedman & Madore 2023;\nWang et al. 2024; Dainotti et al. 2021; Bargiacchi et al.\n2023): (i) the difference is due to unresolved or unknown\nsystematics in the analyses of the higher-redshift CMB\ndata and/or some of the lower-redshift local distance-\nladder data; or (ii) the difference indicates that the stan-\ndard flat ΛCDM model is an inadequate cosmological\nmodel. The second category includes a variety of pro-\nposed cosmological models, some with modified, relative\nto flat ΛCDM, low-redshift or late-time evolution, e.g.,\nsome modified gravity theories (D’Agostino & Nunes\n2020); some specific dark energy models (Dai et al. 2020;\nCai et al. 2021; Belgacem & Prokopec 2022; Krishnan\net al. 2021); models with a significant local inhomogene-\nity (Marra et al. 2013; Kenworthy et al. 2019; Huang\net al. 2024); and early dark energy models with different\nhigh-redshift or early-time evolution (Poulin et al. 2019;\nTian & Zhu 2021; Rossi et al. 2019; Braglia et al. 2020;\nAbadi & Kovetz 2021); as well as models that include\ndark matter-neutrino interactions (Di Valentino et al.2018), or neutrino self-interactions (Kreisch et al. 2020);\nmodels that include dark radiation (Lu et al. 2023; Gari-\nazzo & Mena 2023); and models that rapidly transition\nfrom an anti-de Sitter vacuum to a de Sitter vacuum at\nredshift z∼2 (Akarsu et al. 2020, 2021, 2023; Akarsu\net al. 2023a).\nAlthough the proposed alternate cosmologies are valu-\nable in the context of the H0tension (Liao et al. 2019;\nSun et al. 2021; Jia et al. 2023), there also are a class\nof solutions that indicate that prior knowledge of the\npeak absolute magnitude Mof SNe Ia could have a non-\nignorable influence on the value of H0determined from\nSNe Ia observations. For example, Perivolaropoulos &\nSkara (2023) show that introducing a single new de-\ngree of freedom in the cosmological analysis of the Pan-\ntheon+ SNe Ia compilation can change the best-fitting\nvalue of H0; here instead of the usual single peak abso-\nlute magnitude parameter M, they allow for a change\nof the absolute magnitude at a transition distance dcrit,\ni.e.,M=M<when distance d < d crit, and M=M>\nwhen d > d crit. Also, some studies claim that intrinsic\ndispersion or evolution of Mcan sufficiently alter the\nestimate of H0from SNe Ia data (Tutusaus et al. 2017,\n2019; Di Valentino et al. 2020).\nWhile the correlation between H0and Mis well\nknown and widely discussed in the literature, previous\nresearch has not systematically investigated the impact\nof the prior of Mon the measurement of H0from SNe\nIa data. Here we try to systematically assess how the M\nprior affects the determination of H0from SNe Ia data,\nby considering five typical data-motivated priors for M\nthat have previously been discussed in the literature.\nThe rest of the paper is organized as follows. In Sec-\ntion 2, we summarize the methodology of using the\nrecent Pantheon+ SNe Ia compilation in cosmological\nanalyses, and discuss how two potential factors may af-\nfect the constraint on H0in such analyses. We present\nand discuss our results in Section 3. In the last section,\nwe summarize our main findings.\n2.DATA AND METHODOLOGY\n2.1. Cosmological application of SNe Ia\nSNe Ia apparent magnitude versus redshift measure-\nments provided the first convincing evidence for cur-\nrently accelerating cosmological expansion (Riess et al.\n1998; Perlmutter et al. 1999). Over the past quarter\ncentury, many more SNe Ia have been discovered, and\nwith improved data reduction techniques this has re-\nsulted in refinements of this neoclassical cosmological\ntest (Guy et al. 2010; Betoule et al. 2014; Scolnic et al.\n2018; Abbott et al. 2019; Gao et al. 2020; M¨ oller et al.\n2022; Rubin et al. 2023; DES Collaboration 2024). In3\nthis work, we use the recent Pantheon+ sample that in-\ncludes 1701 light curves of 1550 distinct SNe Ia in the\nredshift range 0 .001< z < 2.26 (Brout et al. 2022a).\nThe Pantheon+ sample is compiled from 18 different\nsurveys, but with the SNe Ia light curves now uniformly\nfit to SALT2 model light curves (Guy et al. 2010; Brout\net al. 2022b). The SALT2 light-curve fit returns four\nparameters for each supernova: i) the light-curve ampli-\ntude x0, which is related to the apparent B-band peak\nmagnitude mB≡ −2.5 log10(x0); ii) the stretch param-\neterx1, which corresponds to the light-curve width; iii)\nthe light-curve color parameter c, which has contribu-\ntions from both intrinsic color and dust; and, iv) t0,\nwhich denotes the time of peak brightness.\nAfter fitting the light curves of SNe Ia in the Pan-\ntheon+ sample with the SALT2 model, Brout et al.\n(2022a) determined the Hubble diagram and the cor-\nresponding covariance matrices by using the “BEAMS\nwith Bias Corrections” (BBC) method (Kessler & Scol-\nnic 2017), where BEAMS (Bayesian Estimation Applied\nto Multiple Species) provides a unified classification and\nparameter estimation methodology (Kunz et al. 2007).\nIn practice, the corrected apparent magnitude is used as\nan observable quantity for each Pantheon+ SN Ia (see\nTable 7 of Brout et al. 2022a), i.e.,\nmcorr\nB≡µ+M\n=mB+ (αx1−βc−δbias+δhost),(1)\nwhere µis the distance modulus and Mis the peak\nabsolute B-band magnitude. Here αandβare global\nnuisance parameters relating stretch and color to lumi-\nnosity, δbiasis a correction term related to selection bi-\nases, and δhostis a correction term originating from the\nresidual correlation between the standardized brightness\nof the supernova and the host galaxy mass.\nIn a given cosmological model, the predicted apparent\nmagnitude at redshift zcan be computed from\nmth(z;p) =µth(z;p) +M, (2)\nwhere pis the set of cosmological model parameters and\nthe predicted distance modulus is\nµth(z;p) = 5 log10\u0014dL(z;p)\nMpc\u0015\n+ 25. (3)\nHere dL(z;p) is the predicted model-based luminosity\ndistance which is related to the Hubble parameter H(z).\nIn our analysis here we use the standard spatially-\nflat ΛCDM cosmological model in which the luminosity\ndistance can be expressed as\ndL(z;p) =c0(1 +z)Zz\n0dz′\nH(z′;p), (4)where c0is the speed of light. In the flat ΛCDM model\nthe expansion rate or Hubble parameter is\nH(z;p) =H0p\nΩm0(1 +z)3+ (1−Ωm0),(5)\nwhere the model parameter set p= (Ω m0, H0) consists\nof the current value of the non-relativistic matter density\nparameter Ω m0andH0=H(z= 0).\nIn our analyses, the likelihood of the SNe Ia sample is\ncomputed using\nL(p)∝e−χ2/2, (6)\nwhere χ2is\nχ2= ∆− →mTC−1∆− →m, (7)\nwith the residual vector ∆− →mi=mcorr\nB,i−mth(zi;p). Here\nmcorr\nB,iandmth(zi;p) for the ithsupernova can be com-\nputed from equations (1) and (2), respectively. The\ncovariance matrix Cincludes contributions from both\nstatistical and systematic errors and is available on the\nPantheon+ website.1\nFollowing Akarsu et al. (2019, 2023b), we use the Py-\nMultiNest code (Buchner et al. 2014) to compute poste-\nrior probability distributions for the model parameters,\nwhere the likelihood is defined in equation (6). PyMulti-\nNest is a Python interface for the MultiNest algorithm\n(Feroz et al. 2009), which is a generic Bayesian inference\ntool that uses nested sampling (Skilling 2004) to obtain\nboth posterior samples and Bayesian evidence estimates.\n2.2. Effects of assumed absolute magnitudes and\ntruncation redshifts\nThe degeneracy between the peak absolute magnitude\nMand the Hubble constant H0is a well-known issue\nin cosmological analyses with SNe Ia data, and how to\ntreat this degeneracy — besides the traditional approach\nof using local calibrators to break it — has also been\ndiscussed in the literature (Camarena & Marra 2021,\n2023; Efstathiou 2021; Perivolaropoulos & Skara 2022,\n2023; Mahtessian et al. 2023). Here we use five different\ndata-motivated priors on Mto break this degeneracy.\nIn addition, the relative uncertainties of the cosmolog-\nical redshifts arising from peculiar velocity corrections\nare larger for low- zSNe Ia than for those in the high- z\nrealm. To assess the effect of peculiar velocity subtrac-\ntion, we choose to remove from the analysis low- zSNe\nIa at different redshift thresholds.\n2.2.1. Priors on the peak absolute magnitude of SNe Ia\n1https://github.com/PantheonPlusSH0ES/DataRelease4\nTo explore the effects of the prior on Min the cosmo-\nlogical application of SNe Ia we consider five different\npriors, that have been discussed in the literature, which\nwe label as Prior I–V .\n•Prior I: A Gaussian prior from the Cepheid-\ncalibrated distance ladder method. Riess\net al. (2022) obtain M=−19.253±0.027 by using\nthe Cepheid calibration of SNe Ia based on obser-\nvations of the SH0ES program. As discussed in\nEfstathiou (2021), when combining SH0ES data\nwith other astrophysical data to constrain late-\ntime physics, one should impose a SH0ES prior\nonMand not on H0.\n•Prior II: A Gaussian prior from the BAO-\ncalibrated inverse distance ladder method.\nBy using Gaussian process regression, Dinda &\nBanerjee (2023) obtain M=−19.396±0.016.\nHere Gaussian process cosmological reconstruction\nuse Pantheon SNe Ia and BAO data. Since higher-\nzBAO data are used, this process is sometimes\ncalled the “inverse distance ladder” method, to\ndistinguish it from the traditional distance lad-\nder method which uses nearby calibrators such as\nCepheid variables.\n•Prior III: A Gaussian prior from the Mul-\nticolor Light Curve Shape method. Wang\n(2000) finds M=−19.33±0.25 from observa-\ntions of 18 SNe Ia with distance moduli obtained\nusing the Multicolor Light Curve Shape (MLCS)\nmethod (Riess et al. 1998).\n•Prior IV: A loose top-hat prior. We also con-\nsider a loose top-hat prior with −20.0< M <\n−18.0, which is broad enough to cover the cor-\nresponding values for almost all known SNe Ia\n(Richardson et al. 2002; Li et al. 2011; Ashall et al.\n2016).\n•Prior V: A much looser top-hat prior. To\nexamine how the width of a top-hat prior on M\nimpacts the constraint on H0, we also consider a\nmuch looser top-hat prior compared to Prior IV ,\ni.e.,−21.0< M < −17.0.\nIt is believed that the formation mechanism of SNe\nIa implies that they possess an almost identical peak\nabsolute magnitude (Colgate 1979; Branch & Khokhlov\n1995), however, the diversity of progenitors of SNe Ia\nmay result in a dispersion in Mamong the SNe Ia (How-\nell 2011; Livio & Mazzali 2018; Wang 2018; Fitz Axen\n& Nugent 2023). Additionally, observationally, absolutemagnitudes of SNe Ia depend on properties of their host\ngalaxies (Phillips 1993; Sullivan et al. 2010; Kelly et al.\n2010; Kim 2011; Ashall et al. 2016; Kang et al. 2020;\nRigault et al. 2020; Meldorf et al. 2023). Therefore, an\nintrinsic dispersion of Mamong the SNe Ia should be\ntaken into account. The five priors discussed above rep-\nresent five different levels of dispersion on M. Among\nthem, Priors I–III are Gaussian priors with different\nlevels of uncertainties, with the first two having ∼0.1%\nuncertainties, and the third a ∼1% uncertainty.\n2.2.2. Truncation redshifts\nAs noted above, the relative uncertainties of the cos-\nmological redshifts for the SNe Ia arising from peculiar\nvelocity subtraction are larger in the low-redshift uni-\nverse than in the high-redshift realm (Davis et al. 2011;\nBhattacharya et al. 2011; Huterer 2020). As discussed in\nBrout et al. (2022a), the Pantheon+ collaboration uses\nthez >0.01 part of the sample in cosmological analysis\nto avoid dependence on the peculiar velocity correction\nmodel. To study potential effects of this correction, we\nstudy five different redshift-trucated SNe Ia samples, re-\nmoving from the complete Pantheon+ compilation low-\nredshift SNe Ia at five different redshift thresholds, i.e.,\natz >0.01,z >0.02,z >0.05,z >0.1 and z >0.2.\n3.RESULTS\nThe SNe Ia observational constraints on Ω m0andH0\nare presented in Table 1 and Figure 1, including the\nresults obtained from the truncated samples with five\ndifferent redshift thresholds (Section 2.2.2), for the five\npriors of M(Section 2.2.1). In Table 1 we summarize the\nmean values of Ω m0andH0together with their 68% con-\nfidence limits, where the five Mprior results from each\n(redshift-truncated) sample are grouped in the same ver-\ntical subpanel of the table. There are six panels in Fig-\nure 1. The first panel displays the mean values of Ω m0\nandH0obtained from the five redshift-truncated sam-\nples, for the five priors of M. The other five panels show\nthe mean values with marginalized 68% confidence level\nerrors of Ω m0andH0obtained for each prior of M.\nWe first focus on the effect of prior knowledge of M.\nThe relevant information can be extracted from Table\n1. We study relative changes of the Ω m0andH0val-\nues when changing the prior on M, with the results ob-\ntained with Prior I , i.e., M=−19.253±0.027, taken\nas reference values. We see that relative differences in\nmean values of Ω m0are less than 1% for the five redshift-\ntruncated samples. Relative changes in mean values of\nH0are larger, ranging from ∼2% to ∼7%. The changes\nof the uncertainties of Ω m0are almost negligible. As ex-\npected, the uncertainty of H0is significantly positively\nrelated to the corresponding uncertainty of M.5\nTable 1. Observational constraints (mean values with 68% confidence level errors) on the parameters Ω m0andH0(ΛCDM\nmodel as well as model-independent cosmography approach) obtained from the five redshift-truncated samples, for the five\ndifferent priors of M.H0has units of km s−1Mpc−1. Log-Bayesian evidence (ln Bi) and relative log-Bayesian evidence\n(lnBi0= lnBi−lnB0) are also displayed in each case.\nData Sample Mprior ΛCDM model Cosmography approach\nΩm0 H0 lnBilnBi0 H0 q0 j0 s0 lnBilnBi0\n−19.253±0.027 0 .350±0.017 72 .84±0.81 -974.83 0.00 72 .84±0.85 −0.488±0.043 1 .01±0.30 −0.11±0.72 -977.19 0.00\ntruncated sample −19.396±0.016 0 .352±0.017 68 .09±0.50 -975.47 -0.64 68 .21±0.50 −0.490±0.444 1 .03±0.30 −0.05±0.73 -977.43 -0.24\nwith z >0.01 −19.33±0.25 0 .351±0.017 71 .58±6.24 -975.37 -0.54 70 .90±5.32 −0.490±0.044 1 .03±0.30 −0.65±0.72 -977.73 -0.54\n(N= 1590) [ −20.0,−18.0] 0 .350±0.018 71 .00±11.35 -975.86 -1.03 70 .72±11.07−0.492±0.044 1 .04±0.30 −0.51±0.72 -977.75 -0.56\n[−21.0,−17.0] 0 .351±0.017 70 .56±11.74 -976.57 -1.74 71 .18±11.26−0.488±0.046 1 .04±0.31 −0.06±0.73 -978.70 -1.51\n−19.253±0.027 0 .358±0.017 72 .86±0.83 -855.80 0.00 72 .89±0.79 −0.474±0.048 1 .00±0.33 −0.19±0.75 -858.43 0.00\ntruncated sample −19.396±0.016 0 .359±0.017 68 .24±0.47 -856.33 -0.53 68 .23±0.50 −0.476±0.047 1 .01±0.32 −0.15±0.74 -858.53 -0.10\nwith z >0.02 −19.33±0.25 0 .359±0.017 71 .01±6.23 -856.34 -0.54 70 .37±5.21 −0.475±0.045 1 .01±0.31 −0.18±0.73 -858.94 -0.51\n(N= 1436) [ −20.0,−18.0] 0 .359±0.017 71 .89±10.93 -856.89 -1.09 71 .39±11.09−0.474±0.048 1 .00±0.33 −0.17±0.76 -859.05 -0.62\n[−21.0,−17.0] 0 .359±0.017 70 .39±11.79 -857.16 -1.36 71 .21±11.87−0.477±0.046 1 .03±0.31 −0.12±0.73 -859.63 -1.20\n−19.253±0.027 0 .364±0.021 72 .58±0.89 -617.92 0.00 72 .61±0.93 −0.451±0.605 0 .90±0.36 −0.39±0.79 -620.67 0.00\ntruncated sample −19.396±0.016 0 .364±0.020 67 .96±0.55 -618.36 -0.44 68 .03±0.58 −0.452±0.060 0 .90±0.36 −0.37±0.77 -620.72 -0.05\nwith z >0.05 −19.33±0.25 0 .364±0.020 71 .39±6.07 -618.28 -0.36 71 .33±5.65 −0.451±0.058 0 .90±0.35 −0.38±0.78 -620.84 -0.17\n(N= 1056) [ −20.0,−18.0] 0 .364±0.020 70 .42±11.40 -618.89 -0.97 70 .75±11.37−0.456±0.058 0 .94±0.36 −0.30±0.79 -621.17 -0.50\n[−21.0,−17.0] 0 .364±0.021 70 .69±11.67 -619.43 -1.51 70 .09±12.05−0.451±0.060 0 .89±0.36 −0.40±0.78 -621.84 -1.17\n−19.253±0.027 0 .373±0.024 72 .43±0.87 -555.71 0.00 72 .41±0.94 −0.422±0.072 0 .80±0.39 −0.60±0.79 -558.15 0.00\ntruncated sample −19.396±0.016 0 .374±0.024 67 .77±0.60 -555.85 -0.14 67 .80±0.66 −0.433±0.071 0 .87±0.38 −0.45±0.81 -558.30 -0.15\nwith z >0.1 −19.33±0.25 0 .371±0.024 70 .97±6.61 -555.83 -0.12 70 .30±5.24 −0.429±0.070 0 .86±0.39 −0.48±0.79 -558.33 -0.18\n(N= 960) [ −20.0,−18.0] 0 .373±0.024 70 .29±11.15 -556.43 -0.72 70 .52±11.13−0.427±0.070 0 .84±0.38 −0.52±0.79 -558.73 -0.58\n[−21.0,−17.0] 0 .374±0.025 70 .89±11.32 -557.07 -1.36 70 .72±11.75−0.427±0.073 0 .83±0.40 −0.52±0.82 -558.76 -0.61\n−19.253±0.027 0 .369±0.028 72 .49±1.02 -414.51 0.00 72 .46±1.02 −0.430±0.831 0 .83±0.40 −0.48±0.82 -417.34 0.00\ntruncated sample −19.396±0.016 0 .369±0.029 67 .82±0.68 -414.83 -0.32 67 .83±0.72 −0.437±0.078 0 .87±0.39 −0.39±0.81 -417.42 -0.08\nwith z >0.2 −19.33±0.25 0 .370±0.029 70 .83±6.37 -414.96 -0.45 70 .44±5.74 −0.435±0.084 0 .86±0.39 −0.42±0.82 -417.44 -0.10\n(N= 753) [ −20.0,−18.0] 0 .370±0.028 70 .49±11.17 −415.25 -0.74 71 .11±11.54−0.428±0.082 0 .84±0.39 −0.48±0.82 -417.89 -0.55\n[−21.0,−17.0] 0 .370±0.030 70 .25±11.42 -415.87 -1.36 69 .81±11.42−0.433±0.080 0 .85±0.38 −0.46±0.80 -418.53 -1.19\nIn passing, we note that the mean values of Ω m0ob-\ntained here are somewhat larger than most other esti-\nmates, e.g. Planck Collaboration (2020) find Ω m0=\n0.3153±0.0073 from CMB data, de Cruz P´ erez et al.\n(2023) measure Ω m0= 0.3053±0.0050 from CMB and\nnon-CMB data, and Cao & Ratra (2023) find Ω m0=\n0.313±0.012 from non-CMB data, all in the flat ΛCDM\nmodel. This seems to be the case when SNe Ia data are\nused to constrain flat ΛCDM model parameters, for in-\nstance Rubin et al. (2023) find Ω m0= 0.356+0.028\n−0.026while\nDES Collaboration (2024) get Ω m0= 0.352±0.017.\nWe next investigate the effect of the truncation red-\nshift used to eliminate lower-redshift SNe and so study\nthe potential impact of the peculiar velocity subtrac-\ntion. The relevant information can be extracted from\nFigure 1. From panel (a) of Figure 1 we see two impor-\ntant trends: (i) independent of which redshift-truncated\nsample is used, mean values of H0obtained with Prior\nI,M=−19.253±0.027, are larger than those obtained\nusing the other four priors; conversely, mean values of\nH0obtained with Prior II ,M=−19.396±0.016, are\nobviously smaller than those obtained using the other\nfour priors; (ii) under a given Mprior, different trunca-\ntion redshifts result in only small relative changes in the\nmean values of H0, less than 1%, while relative changes\nin the mean values of Ω m0range from 0% to ∼6%.\nPanel (b) of Figure 1 indicates that the estimates\nofH0obtained with MPrior I are consistent with\nthe SH0ES project (Riess et al. 2022) value, H0=\n73.04±1.04 km s−1Mpc−1, independent of the trun-cation redshift. In turn, panel (c) of Figure 1 indicates\nthat the estimates of H0obtained with MPrior II are\nconsistent with the Planck 2018 (Planck Collaboration\n2020) result, H0= 67.36±0.54 km s−1Mpc−1. Finally,\npanels (d-f) of Figure 1 show that the estimates of H0\nobtained with MPriors III–V are consistent with both\nthe Planck 2018 result and the SH0ES result. Addition-\nally, from panels (b-f), we see that the estimates of Ω m0\nare consistent at 68% confidence level, regardless of the\nchosen truncation redshift.\nWe also use Bayesian evidence (Kass & Raftery 1995;\nTrotta 2008) to judge which prior on Mis preferred by\nthe dataset. We use ln Bito denote the natural loga-\nrithm of the Bayesian evidence for each scenario, and\nlnBi0= lnBi−lnB0for the relative log-Bayesian evi-\ndence, where “0” corresponds to the model with Prior\nI. The model with the smallest |lnBi|is the preferred\nmodel, and therefore we use the model with Prior I\nas the reference model in model comparison. Accord-\ning to the empirical scale for evaluating the strength\nof evidence, |lnBi0| ∈(0,1.0),(1.0,2.5),(2.5,5.0), and\n(5.0,∞) correspond to inconclusive, weak, moderate,\nand strong evidence, respectively (Trotta 2008). The\ncorresponding values of ln Biand ln Bi0are listed in Ta-\nble 1, where we notice |lnBi0|<2.0 for all cases, which\nmeans that there is weak evidence that Prior I is pre-\nferred.\nIt is also necessary to test whether the overall find-\nings are model-independent. Hence, we repeat the above\nanalysis by replacing the flat ΛCDM model with the6\ncosmography model (Turner & Riess 2002; Visser 2004;\nShapiro & Turner 2006; Xu & Wang 2011). In the spa-\ntially flat scenario, the cosmography model parameter\nset is p= (H0, q0, j0, s0), where q0,j0, and s0denote\nthe present-day values of the deceleration, jerk, and snap\nparameters, respectively. The observational constraints\non the cosmography model parameters, along with the\nBayesian evidence in each case, are listed in the right\npart of Table 1. The cosmography model results are\nvery similar to the flat ΛCDM model results, which sug-\ngests that our results are robust and model independent.\n4.CONCLUSION\nWe have investigated the impact of different data-\nmoitvated priors for the peak absolute magnitude Mof\nSNe Ia on the measurement of the Hubble constant H0\nfrom apparent magnitude-redshift data from the Pan-\ntheon+ sample with 1701 light curves of 1550 distinct\nSNe Ia in the redshift range 0 .001< z < 2.26. We\nuse five distinct Mpriors, discussed in the literature,\nand show that the choice of Mprior can significantly\naffect the measured H0value, but has marginal impact\non the estimate of Ω m0. The variations in mean H0val-\nues range between 2% and 7%, with uncertainty in H0\npositively correlated to Mprior uncertainty. Conversely,\nchanges in the mean Ω m0values are less than 1%, and\nchanges in Ω m0uncertainties remain nearly negligible.\nWe have also explored the influence of peculiar veloc-\nity subtraction by using five different low- z-truncated\nSNe Ia samples. The redshift threshold of the trun-\ncated sample can significantly impact the estimate ofΩm0, while it has only a small effect on the derived H0\nvalue.\nThe loose priors for Myield H0values consistent with\nboth Planck 2018 and SH0ES results at a 68% confi-\ndence level. Considering the critical role of a reasonable\nand effective Mprior in obtaining reliable constraints\nonH0, our study advocates the need for accurate esti-\nmates of the intrinsic dispersion of Mamong SNe Ia,\nrequiring a thorough understanding of contributing fac-\ntors, including contributions from the diversity of SNe\nIa progenitors (Howell 2011; Livio & Mazzali 2018; Fitz\nAxen & Nugent 2023), and from correlations between\nSNe Ia absolute magnitudes and host galaxy properties\n(Rigault et al. 2020; Meldorf et al. 2023). Finally, the\nsimilarity between the cosmography model results and\nthe ΛCDM model results indicate that our results are\nrobust and model independent.\nYC would like to thank Sheng Yang for helpful discus-\nsions on the observed SN Ia light curves. This work has\nbeen supported by the National Key Research and De-\nvelopment Program of China (No. 2022YFA1602903),\nthe National Natural Science Foundation of China\n(Nos. 11988101 and 12033008), the China Manned\nSpace Project (No. CMS-CSST-2021-A01), and the K.\nC. Wong Education Foundation. SK gratefully ac-\nknowledges support from the Science and Engineer-\ning Research Board (SERB), Govt. of India (File\nNo. CRG/2021/004658).1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\nREFERENCES\nAbadi, T., & Kovetz, E. D. 2021, PhRvD, 103, 023530,\ndoi: 10.1103/PhysRevD.103.023530\nAbbott, T. M. C., Allam, S., Andersen, P., et al. 2019,\nApJL, 872, L30, doi: 10.3847/2041-8213/ab04fa\nAbdalla, E., et al. 2022, JHEAp, 34, 49,\ndoi: 10.1016/j.jheap.2022.04.002\nAkarsu, O., Barrow, J. D., Escamilla, L. A., & Vazquez,\nJ. A. 2020, Phys. Rev. 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When the laser intensity is\nin the intermediate regime, the interferences of pair produ ction in different formation lengths induce an en-\nhancement of pair production probability for spin-down pos itrons, which significantly a ffects the polarization\nof created positrons. The polarization features are distin ct from that obtained with LCFA, revealing the inva-\nlidity of LCFA in this regime. Meanwhile, the angular distri bution for different spin states varies, resulting in\nan angular-dependent polarization of positrons. The avera ge polarization of positrons at beam center is highly\nsensitive to the laser’s carrier-envelope phase (CEP), whi ch provides a potential alternative way of determining\nthe CEP of strong lasers. The verification of the observed int erference phenomenon is possible for the upcoming\nexperiments.\nI. INTRODUCTION\nIn the presence of a strong electromagnetic field, high-\nenergy photons can be converted into an electron-positron\npair via the nonlinear Breit-Wheeler process [1–11]. This\nan appealing physical process as it allows for the conversio n\nof light into particles of matter, and has practical implica -\ntions for astrophysics, high-energy physics, and laser-pl asma\ninteractions [12, 13]. The E-144 experiment conducted at\nSLAC in the 1960s provided the first experimental evidence\nfor the nonlinear Breit-Wheeler process (NBW) [3, 4]. Pair\nproduction involving multiphoton is observed by colliding a\n46.6GeV electron beam with a terawatt laser with peak inten-\nsity of∼1018W/cm2. The polarized (linear) Breit-Wheeler\nprocess has been measured at the Relativistic Heavy Ion Col-\nlider, which reveals a large fourth-order angular modulati on\nof the observed yields [14]. The advancement of high-power\nlaser technology [15–17] has sparked interest in revisitin g the\nE-144 experiment with lasers of higher intensity. The up-\ncoming experiments at FACET II at SLAC [18] and LUXE\nat DESY [19] aim to explore NBW in the tunneling regime,\nwhere the production rate follows an exponential scaling la w\nof exp(−πEcr/E) [20]. Here Eis the electric field strength\nandEcr=m2/|e|the critical field of quantum electrodynamics\n(QED), with e<0 and mbeing the electron charge and mass,\nrespectively. However, the experiments will be initially c on-\nducted in the intermediate laser intensity regime, where th e\ntransition from multiphoton to tunneling regime is expecte d to\noccur. The separation of regimes is determined by the nonlin -\near parameter a0=|e|E/mω0, withω0being the laser carrier\nfrequency.\nIn the tunneling regime of a0≫1, the formation length of\npair creation is much smaller than the spatial and temporal i n-\nhomogeneities of the laser field [1, 2, 21]. The probability c an\nbe calculated under the locally constant field approximatio n\n∗tangsuo@ouc.edu.cn\n†yue-yue.chen@shnu.edu.cn(LCFA) [1]. Recently, various LCFA probabilities have been\nderived incorporating electron-positron spins and photon po-\nlarization [22–27], and has been applied to QED-simulation s\ncodes for studying polarization e ffects in strong-field QED\nprocesses [24, 26, 28–36]. Ultrarelativistic positrons (e lec-\ntrons) beam with high degree of polarization can be generate d\nvia NBW in asymmetric laser fields [31], as a result of spin\npreferences parallel (antiparallel) to the instantaneous quanti-\nzation axis during creation. However, the current or upcomi ng\nexperiments are performed with intermediate values of lase r\nintensity ( a0∼1), where the formation length of pair pro-\nduction lf∼λ/a0is comparable with the laser field inhomo-\ngeneities. In this case, interferences e ffects arising from the\npair creation process may significantly modify the positron s\npolarization.\nThe interference e ffects has been investigated in several\nworks [11, 37–44]. It has been reported that the interferenc es\noccurring at macroscoptically separated space-times poin ts\ncould induce interference structures in the momentum distr i-\nbution of the the final particles [11], and interference betw een\npair creation channels of di fferent photon number combina-\ntions could enhance the yield of pairs in bichromatic fields\n[45]. Meanwhile, the carrier-envelope phase (CEP) e ffects are\npronounced for a few-cycle pulses with moderate laser inten -\nsity [46], which induces shifts of spectra and angular distr ibu-\ntion due to interferences [11, 38, 39, 45]. Moreover, the pol ar-\nization effects in pair production has been investigated in the\ntransition regime ( a0∼1) [45, 47, 48]. The oscillation pattern\nin angular spectra of positrons [47] and positron yield [49, 50]\ncan be enhanced by tuning photon polarization. The relevanc e\nof spin effects on NBW has been investigated by comparing\nthe production of spinor and scalar particles [42, 51]. How-\never, a comprehensive study of the positron polarization in\nthis regime has yet to be conducted.\nTo improve the accuracy of simulation codes in the interme-\ndiate intensity regime, a method based on locally monochro-\nmatic approximation (LMA) has been developed taking into\naccount interference e ffects at the scale of the laser wave-\nlength [4, 52, 53]. In contrast to LCFA, LMA is more suitable\nfor exploring the transition region for planewave-like fiel ds2\nwith slowly varying envelope. While for a short laser pulse,\nLMA would underestimate the pair production as it loses the\nhigh-frequency components induced by the finite-pulse enve -\nlope. In fact, these overlooked components have the potenti al\nto effectively enhance pair production [10]. While for a short\nlaser pulse, the fully spin-resolved pair-creation probab ility\ncould be obtained using the semiclassical operator method o f\nBaier et al. [2, 54], which is applicable to non-plane-wave\nbackground. The probability associated with particular sp in\nstates and photon polarization can be calculated by coheren tly\nintegrating over the electron trajectory. The numerical re sults\nobtained for plane waves exhibit good agreement with the pre -\ndictions of the V olkov state approach [55], implying that th e\nsemiclassical method is capable of describing the polarize d\nBreit-Wheeler process in the intermediate intensity regim e\n[49, 56, 57].\nIn this paper, we investigate the NBW in a linearly polar-\nized ultrastrong laser pulse using the semiclassical metho d be-\nyond LCFA. It allowed us to identify interference e ffects in\npositrons polarization and angular distribution. We show t hat\nthe positrons polarization exhibits unique features due to in-\nterferences that cannot be captured by LCFA. The spectra and\nangular distribution of final particles are depend sensitiv ely on\nthe spin states, leading to an angular dependent polarizati on of\npositrons. By cutting the positrons distributed at small an gle\nregion, one could obtain polarized positrons with polariza tion\ndegree∼25%. Moreover, the polarization degree is propor-\ntional to CEP, which is attractive for relativistic positro n gen-\neration with controlled polarization and CEP measurement i n\nstrong laser fields.\nII. THEORETICAL FRAMEWORK\nA. Semiclassical approach beyond LCFA\nWe consider the head-on collision of a 93.5GeV photon\nwith a linearly polarized laser pulse propagating along −z\ndirection. XCELS [58] envisions advanced accelerator com-\nplexes capable of achieving particle energies in the 1-10 Te V\nrange, which could be utilized to generate gamma quanta with\nenergies of hundreds of GeV through Compton backscatter-\ning [45, 49]. The background field takes the form of Ex=\nE0sin4(ϕ/2N)cos(ϕ+ϕ0)ifϕ∈{0,2πN}andEx=0 oth-\nerwise, where ϕis the laser phase of its carrier frequency\nω0. The peak intensity is I∼1018W/cm2(a0=1), wave-\nlength of the laser λ0=800nm, N=5 corresponding to a\npulse duration of∼3T0with period T0, and CEP ϕ0=0.\nAs the emitted photon number for a single electron is much\nsmaller than 1 ( Nγ∼αa0τp/T0≪1 [1, 59]), the photon emis-\nsions of produced particles can be disregarded. Otherwise, the\nproduced pairs could undergoes further radiation in the fiel d,\nwhich could change the spin states due to radiative polariza -\ntion and radiative corrections. In this case, the polarizat ion\nfeatures predicted by pair production probability itself m ay\nnot be accurate.\nFor the semiclassical approach without LCFA, the angle-\nresolved spectral probability for electron can be obtained byintegrating the trajectories of electrons with di fferent final mo-\nmentumpf\n−=(pf\n−cosψsinθ,pf\n−sinψsinθ,pf\n−cosθ). One\nshould go through all possible choices of angles and electro n\nmomenta, obtain the trajectories for each final momentum by\nsolving Lorentz equation with initial electron momentum of\npi\n−=pf\n−. Note that, in the cases where the initial and final\nmomenta are different due to radiation reaction or laser con-\nfiguration, one need to resort the obtained probabilities wi th\nthe final momentum pf\n−. The Lorentz equation is solved us-\ning the Runge-Kutta algorithm with a time step dt=10−19s,\nwhich is small enough to ensure the convergence of integrals .\nThe integrals are evaluated with summations over finite dif-\nferenced points N=tf/dt. At each discrete point, the time-\ndependent momenta and coordinates of the electrons are sub-\nstituted into the angle-resolved spectrum for electrons [4 9, 60].\nThe rapidly oscillating function ei(ε−/ε+)kµxµ\n−in the integrals is\nsplit into real and imaginary parts, which are calculated se p-\narately in each time step. Since the energy (momentum) of\nultrarelativistic electron and positron pairs is correlat ed, the\nangle-resolved spectral probability for positrons can be o b-\ntained through the electron trajectory:\nd2P\ndε+dΩ=e2\n(2π)2ε2\n−N2\nωω′2/vextendsingle/vextendsingle/vextendsingleφ†\n−{ε−ω[nJ−I]·(ǫ×n)−\niσ·/bracketleftBig\nǫωmJ−2ε2\n−K−ε−ωn(ǫ·I)/bracketrightBig/bracerightBig\nφ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,(1)\nwhere\nN=1√4ε−ε+(ε−+m) (ε++m),\nI=/integraldisplayn×[(n−v−)×a+]\n(1−n·v−)2eiε−\nε+kµxµ\n−dt,\nJ=/integraldisplayn·a−\n(1−n·v−)2eiε−\nε+kµxµ\n−dt,\nK=/integraldisplay/parenleftBigg[a−(ǫ·v−)+v−(ǫ·a−)]\n(1−n·v−)+\n+[v−(ǫ·v−) (n·a−)]\n(1−n·v−)2/parenrightBigg\neiε−\nε+kµxµ\n−dt.\nHere kµ=ωm{1,n}is the four-momentum of the parent pho-\nton with n=(0,0,1) being the direction of motion of the\nparent photon, ω′=ε−\nε+ωwithε+(ε−) being the energy of the\ncreated positron (electron) and ω=(ε++ε−)/mthe energy of\nthe parent photon. The photon polarization vector is defined\nasǫ={e∝bardbl,e⊥}with\ne∝bardbl=ˆy×n,e⊥=n×e∝bardbl, (2)\nwhere ˆy=(0,1,0) being the unit vector along magnetic\nfield direction. For the considered setup, e∝bardblande⊥are the\nlaser electric and magnetic field direction, respectively. xµ\n+=\n{t,r(t)},v−anda−are the four-coordinate, velocity and accel-\neration of the produced electron, respectively. φ−andφ+are3\n(a) (b)\nδ+ δ+dP/dδ+\nζ+\nfSemi.↑\nSemi.↓\nLCFA↑\nLCFA↓Semi.\nLCFA\nFIG. 1. (a) The pair production probability dP/dδ+versusδ+after averaging over photon polarization: for spin-up posi trons using semiclasscial\napproach (red-dotted) or the LCFA approach (blue-solid); f or spin-down positrons using semiclasscial approach (gree n-dot-dashed) or the\nLCFA approach (black-dashed). (b) The spectra of positron p olarization with semiclasscial approach (blue-solid) or t he LCFA approach (red-\ndashed). The arrow denotes the positron spin with respect to ˆy=(0,1,0).\nthe bispinor of the electrons and positrons, respectively, and\nσare the Pauli matrices. If the y-axis is selected as the spin\nquantization axis, then φ−=/parenleftBig1\n2−i1\n2,1\n2+i1\n2/parenrightBigTcorresponds to\nthe electron’s spin-up state, and φ+=/parenleftBig\n−1\n2+i1\n2,1\n2+i1\n2/parenrightBigTrep-\nresents the positron’s spin-up state.\nB. Spin-resolved LCFA approach\nTo illustrate the interference e ffects, we also investigated\nNBW using the spin-resolved LCFA probability. The LCFA\nassumed the formation length for pair production is much\nsmaller than the wavelength and period of the field, which al-\nlows one to regard the field inside the formation length as con -\nstant, i.e. ignore the interference within a formation leng th,\nand drop out the rapidly oscillating terms related to interf er-\nences between different formation lengths. Therefore, a com-\nparison between semiclassical and LCFA results could revea l\nthe role of interference. The spin-resolved LCFA probabili ty\ncan be written as P±=(P±∆P)/2, where Pis the total unpo-\nlarized probability [61]:\nd2P\ndε+dΩ=α\n2πδ2\n+/integraldisplaydϕ\nω0/bracketleftBigg\nν/parenleftBiggε−\nε++ε+\nε−/parenrightBigg\n+µ/bracketrightBigg\nAi(ν),(3)\nand∆Pis the difference in probability due to creating the\npositron into±ˆyspin-polarization state:\nd2∆P\ndε+dΩ=α\n2πδ+/integraldisplaydϕ\nω0√µAi′(ν)sgn( Ex), (4)\nwhereδ+=ε+/ωis the fraction of the energy taken by the\npositron from the parent photon, and\nµ=[χℓδ+(1−δ+)]−2/3, ν=µ(1+ε2\n+sin2θ/m2),(5)withχℓ=ωE/Ecr.\nIII. NUMERICAL CALCULATIONS\nUsing the semiclassical method and the parameters given\nin Sec. II A, one could obtain the spectrum for di fferent parti-\ncles’ polarization states (see Appendix B). For a more intui tive\nperspective, we average over photon polarizations and sum\nover electron polarizations to specifically examine the spi n-\nresolved probability of positron with respect to ˆy, see Fig. 1\n(a). It is evident that the probability for positron spin-do wn\nis higher than that for positron spin-up as a result of asymme -\ntry of the short laser pulse, while the LCFA underestimates\nthis difference in the probabilities for positron spin-up and\nspin-down. In the semiclassical method, interference peak s\nappear in the spectra [Fig.1 (a)]. The interferences lead to\nan enhancement of pair production probabilities and a more\npronounced asymmetry between di fferent polarization states\nof positrons. Counterintuitively, the probability for pos itron\nspin-down exceeds spin-up. This is because interference ef -\nfects arising from pair production at negative half-cycles with\nBy<0 are more significant than that at positive half-cycles,\nleading to a notable enhancement of pair production probabi l-\nity with negative positron spin, i.e.,ζ+\nf<0.\nThe disparities in probability give rise to distinct polar-\nization characteristics across di fferent approaches. For the\nLCFA approach, the polarization of positron decreases with\nthe increase of positron energy, which coincides with that i n\na constant field with By>0. The average polarization of\nthe positron beam is 3 .8%. However, the polarization under-\ngoes significant changes in the results from the semiclassic al\nmethod, in which interferences are introduced [see Fig. 1\n(b)]. Surprisingly, the average polarization becomes −1.5%.4\n1 2 3 4-101108\n0 0.5 102410-4\n0 0.5 102410-4\n0 0.5 1012310-4\n0 0.5 1012310-40 1 2 3 4 502410-5\n024\n 10-4\nSemi. Semi.\nLCFA LCFA(a) (b)\n(c) (d)\n(e) (f)ϕ/2πByI+ II+ III+\nI− II−\nϕ/2πdP/dϕLCFA\nSemi.P(ϕ)\nδ+dP/dδ+↑I+\n↑II+\n↑III+\n↑I+,II+,III+\nδ+dP/dδ+↓I−\n↓II−\n↓I−,II−dP/dδ+↑I+\n↑II+\n↑III+\n↑I+,II+,III+\nδ+\ndP/dδ+↓I−\n↓II−\n↓I−,II−\nδ+\nFIG. 2. (a) The magnetic field and labels for di fferent half-cycles. (b)\nProbability density d P/dϕat laser phase ϕusing LCFA (blue-solid)\nand probability at each half-cycles using semiclassical ap proach (red-\ncircle). Spectral probabilities for spin-up positrons in t he half-cycle\nI+(blue-solid), II +(red-dashed), III +(black-dotted) and the superpo-\nsition of I+, II+and III+(green-dot-dashed) for: (c) semiclassical ap-\nproach and (e) LCFA approach. Spectral probabilities for sp in-down\npositrons in the half-cycle I −(blue-solid), II −(red-dashed), and the\nsuperposition of I +and II+(black-dotted) for: (d) semiclassical ap-\nproach and (f) LCFA approach. Here the spin-up and spin-down is\ndefined with respect to ˆy=(0,1,0). The parameters are same as Fig.\n1.\nThe positron polarization ζ+\nfdecreases with large oscillations\nat small position energy δ+, exhibits a negative dip around\nδ+=0.5 with|ζ+\nf|reaching to∼10%, and finally diminishes\nto zero when δ+>0.6. The LCFA approach fails to capture\nthe interference structures, resulting in an overestimati on of\npolarization for low energy positrons. Conversely, it unde res-\ntimates the polarization near the density peak. Therefore, for\na0=1, the LCFA performs poorly in accurately describing\nthe polarization of the created particle.\nFor the LCFA approach, the positron polarization arises due\nto the asymmetry of the ultrashort laser pulse and the prefer -\nence for pair production in spin-up positrons, as the LCFA\nprobability is solely depend on the local quantum parameter\nχγ∝a0ωand the positrons are mostly produced at the intense\npositive half-cycle II +. Since the pair production is more prob-\nable for spin-up positron, the positrons produced at the hal f-\ncycle II+are more likely to be polarized with ζ+\nf>0 [Fig. 2\n(b)]. However, due to the cancellation of polarization e ffect\nbetween positive and negative half-cycles, the average pol ar-\nization of positron in an ultrashort laser pulse is rather sm all\n(∼3.8%).\nWhen interferences are taken into account, the density of\npositrons with ζ+\nf<0 gets larger than that of ζ+\nf>0, indi-cating more pairs are produced at the negative half-cycles I −\nand II−than the positive peak II +as shown in Fig. 2. This\nis because interferences play an essential role in determin ing\nthe spin-resolved pair production probabilities. The influ ences\nof interferences within each laser cycle and between macro-\nscopically separated cycles are illustrated in Fig. 2 (c) an d\n(d). It can be seen that the positrons with spin parallel with ˆy\nare mostly comes from pair production at the pulse peak II +.\nThe positive half-cycles I +and III+also contribute but with a\nmuch lower amplitude [Fig. 2 (c)]. Due to the interferences\nwithin each laser cycle, the pair production amplitude at I +,\nII+and III+is all enhanced compared to the results obtained\nfrom the LCFA [Fig. 2 (e)]. Meanwhile, at a0/lessorsimilar1, the co-\nherence length is of the order of the period of the electron\ntrajectory, which means that the process is formed along all\nmulticycle dynamics, i.e., e ffectively the formation length is\nthe whole trajectory length. In this case, the interference s be-\ntween cycles also make an important role. When considering\npair productions during all the positive half-cycles, inte rfer-\nence structures appear in the spectrum.\nIn contrast, the spin-down positrons mainly come from pair\nproduction at negative half-cycles I −and II−, which contribute\nequally to positron spectrum. The interferences within eac h\nhalf-cycle I−and II−increase the corresponding probability\ncompared with the LCFA results in Fig. 2 (e). More im-\nportantly, the constructive interferences arising from pa ir pro-\nduction occurring at I −and II−result in an enhancement of\npositrons density with ζ+\nf>0 andδ+≈0.5 [Fig. 2 (d)]. With\nrespect to ˆy, the pair production is more probable for spin-\ndown positron around δ+∈(0.4,0.6). However, outside of\nthis region, probability for spin-up dominates over spin-d own.\nThis is because the half-cycle II +is more intense than the neg-\native half-cycles [Fig. 2 (a)], there is a higher number of ph o-\n0 0.2 0.4 0.6 0.8 100.511.522.5310-4\nSemi.↓0\nSemi.↓π/2\nSemi.↓π/10\nLCFA↓\nLCFA↓π/2\nLCFA↓π/10\nδ+dP/dδ+\nFIG. 3. Spectral probabilities for spin-down positrons in t he half-\ncycle II−for laser pulse with CEP ϕ0=0 (blue solid line), ϕ0=π/2\n(red danshed line), and ϕ0=7π/10 (green dotted line): for semiclas-\nsical approach (thick line) and LCFA approach (thin line). T he rest\nparameters are same with Fig. 1.5\ntons ninvolved in pair production during II +compared to the\nother cycles. The width of the spectrum increases as the har-\nmonic order nincreases [62]. The maximum of nare deter-\nmined by the field intensity where pairs are produced. The\ninterference between di fferent half-cycles only a ffects the am-\nplitude of harmonics through an interference pattern, but i t\ndoesn’t change the harmonic order. Consequently, the spec-\ntrum for spin-down positrons exhibits a broader width com-\npared to the negative half-cycles, leading to a positive pol ar-\nization outside the range of δ+∈(0.4,0.6). It is important\nto note that the observed interference e ffects, including inter-\nference structures in spectrum and negative polarization, pre-\ndominantly arise from pair production events that occur at\nmacroscopically separated formation regions. The interfe r-\nences within each cycle, on the other hand, only result in an\noverall enhancement of the respective probability.\nThe interferences within each half-cycle always result in\nan increase in pair production probability compared with\nLCFA, see Fig. 3. This is because the laser field in a finite\ntime domain has a wide spectrum. The high frequency\ncomponents make a significant contribution to the pair\nproduction probability, which are ignored by LCFA. In\ncontrast, the interferences between macroscopically sepa rated\ncycles could be destructive or constructive determined by\nSemi. LCFA\n(a)\n(c)\n(e)\nθx θxθy θy θy(b)\n(d)\n(f)\nFIG. 4. Angular distribution of positron density d2P/dθx/dθy(rad−2)\nversusθx(mrad) and θy(mrad): for positron spin-up (a) and spin-\ndown (c) calculated with semiclassical approach without LC FA; for\npositron spin-up (b) and spin-down (d) calculated with LCFA ap-\nproach. Angular distribution of positron polarization ζ+\nfversusθx\n(mrad) andθy(mrad) calculated with semiclassical approach without\nLCFA (e) and LCFA probability (f). The parameters are same as Fig.\n1.0.2 0.4 0.6 0.8024610-4\n-0.0500.050.10.15\n0°50°100°150°200°250°300°350°-0.200.20 2 4-1-0.500.51107\n(a) (b)\n(c)By\nϕ/2π δ+dP/dδ+ζ+\nf↑\n↓\nζ+\nf\nϕ0\nFIG. 5. The variation of the magnetic field with respect to the laser\nphaseϕ(a) and the spectral probabilities for spin-up and spin-dow n\npositrons, with respect to ˆy(b) for a CEP of ϕ0=π/2. (c) The scaling\nlaw of average polarization ζ+\nfwithinθ∈[−0.02mrad,0.02mrad]\nwith CEPϕ0. The rest parameters are same with Fig. 1.\nthe phase differences. See the discussion in Ref. [63]. In\nour case, the interferences patten of two adjacent nega-\ntive (positive) half-cycles can be quantitatively interpr eted\nby the interferences from two trajectory segments that\nhave the same velocity and acceleration. In this case, the\nintegral over the entire trajectory can be estimated with\nP∝ |ei(ε−/ε+)kµxµ\n1−+ei(ε−/ε+)kµxµ\n2−|2, where xµ\n1−and xµ\n2−are\nfour-coordinates for the two trajectory segments with a\n2πdifference. The destructive interferences occur when\nnπ=(ε−/ε+)ω∆t(1−β−cosθ)≈m2π/δ+(1−δ+)ωω 0,\nresulting in the emergence of valleys in the spec-\ntrum atδ+=nωω 0±/radicalBig\n−4m2nωω 0+n2ω2ω2\n0/2nωω 0\n[2, 59, 64]. Similarly, constructive peaks emerge when\n2nπ=(ε−/ε+)ω∆t(1−β−cosθ). Therefore, the probability\nof pair production is enhanced mainly due to the interferenc e\nwithin each half-cycles, and interference between cycles\nresults in destructive valleys and constructive peaks supe rim-\nposed on the enhanced probability.\nThe angular distributions of the positrons in di fferent spin\nstates are illustrated in Fig. 4. For the semiclassical appr oach\nwithout LCFA, the distribution of positrons exhibits two sy m-\nmetric circular regions around θx=0. The separation is in-\nduced by longitudinal acceleration appeared in J∝˙vz∼vxBy.\nAtθx≈0, the positron density decreases as Japproaches zero,\nand increases to a maximum at θx≈±a0/2γ∼±3×10−6rad.\nThe separation between the two regions is larger for the\npositrons with spin-up than spin-down with respect to ˆy[Fig.\n4 (a) and (c)]. This because the final momentum of positrons is\ndetermined by the laser intensity at creation, i.e. px\nf≈eA(ϕ+)\nwithϕ+being the creation phase. The positrons with ζ+\nf>0\nare mostly generated in the high laser intensity region, con se-6\nquently, they have higher transverse momentum pxcompared\nto the positrons with ζ+\nf<0. Therefore, the small angle re-\ngion is dominated by the spin-down positrons created at neg-\native half-cycles, while the large angle region is determin ed\nby the spin-up positrons created at positive half-cycles, l ead-\ning to an angle dependent polarization [see Fig. 4 (e)]. By\ncollecting the positrons at small angle regions, one could o b-\ntain polarized positrons with average polarization reachi ng to\n∼26%. In contrast, the angular distribution obtained with\nLCFA approach exhibits an elliptical shape for di fferent spin\nstates [Fig. 4 (b) and (d)]. It is because positrons are most\ncreated at laser peaks with px(ϕ+)≈0 [Fig.2 (b)], and con-\nsequently have a small θxout of laser. Moreover, the inter-\nference ring structure, which is present in the semiclassic al\napproach, disappears in the LCFA approach. The probability\nfor spin-up always dominates over spin-down, which is more\npronounced for low-energy positrons deflected to large angl e\nregion [Fig. 4 (f)].\nFor the polarized Breit-wheeler process in short laser puls e,\nit is possible to identify the CEP e ffects in the positron polar-\nization. In the case of ϕ0=π/2 [Fig. 5 (a)], the pair produc-\ntion probabilities for di fferent spin states are equivalent, lead-\ning to a vanished polarization throughout the spectrum [Fig .\n5 (b)]. This is because the ultrashort laser pulse becomes\nsymmetric when ϕ0=π/2. Polarization effects in different\nhalf-cycles cancel out due to the symmetry of the laser field.\nBy collecting the small angle positrons, we obtain the scali ng\nlaw of average polarization on the CEP of laser [Fig. 5 (c)].\nThe polarization increases monotonically from -25% to 25%\nwhen the CEP increases from 0 to π, and decreases from 25%\nto -25% when the CEP increases from πto 2π. The sensitive\ndependence of polarization on CEP provides an potential way\nof detecting strong laser’s CEP.\nIV . CONCLUSION\nIn this paper, we have investigated the polarization e ffects\nof the NBW using a short laser pulse in the intermediate inten -\nsity regime. By comparing the semiclassical approach beyon d\nLCFA and the LCFA approach, we scrutinized the validity of\nthe LCFA in predicting the polarization of created pairs. Th e\npositrons produced in a short pulse exhibit a net polarizati on\nas a result of the asymmetry of the background field. However,\nthe polarization features obtained from di fferent approaches\ncan be completely di fferent, even when considering the same\nasymmetric field. The LCFA approach demonstrates a posi-\ntive polarization due to the dominant influence of the peak po s-\nitive half-cycle, and the polarization degree decreases mo no-\ntonically with the increase of positron energy. When taking\ninterference effects into account, it is observed that the den-\nsity of positrons with ζ+<0 surpasses that of positrons with\nζ+>0 aroundδ+=0.5, leading to a negative polarization\nin the spectrum center. The variation of polarization featu re\nis result from the interferences arising from separated for ma-\ntion lengths. The pair production occurring at negative hal f-\ncycles interferences constructively, leading to an enhanc ement\nof pair production for positrons with ζ+<0. Interferencesstructures appear in spectrum for spin-up positrons but wit h-\nout a substantial enhancement of probability. Consequentl y,\na negative polarization emerges around δ+=0.5, which is\ncontrast to the predictions of LCFA. Meanwhile, positrons\npresents an angular dependent polarization. The positrons de-\nflected towards the small angle region exhibit a negative po-\nlarization of−25% forϕ0=0. This negative polarization in-\ncreases as the carrier-envelope phase is increased before r each-\ningϕ0=π. However, after ϕ0=π, the trend is reversed and\nthe polarization starts to decrease. The sensitive depende nce\nof polarization on CEP provides an alternative way of genera t-\ning and manipulating the positron polarization, as well as C EP\nmeasurement in strong laser fields.\nAPPENDIX A: COMPARISON BETWEEN SEMICLASSICAL\nAND QED APPROACHES\nIn this section, we compared the semiclassical results with\nthe full QED calculation derived from the standard S-matrix\napproach, see Eqs (20) and (21) in Ref. [10]. The spin-\nresolved probabilities from the QED calculation display ex -\ncellent agreement with the semiclassical approach, marked ly\ndiffering from the results yielded by the LCFA [Fig. 6 (a)].\nThe polarizations obtained from di fferent approaches also co-\nincide with each other except for the low energy region of\nδ+<0.6 [Fig. 6 (b)]. The relative di fference between the\nQED and Semi. results within the range of δ+∈[0.4,0.6]\nis represented by Ri=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggdPi\nQED\ndδ+−dPi\nSemi.\ndδ+/parenrightbigg\n/dPi\nQED\ndδ+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<1% with\ni∈{↑,↓}. Due to the excellent agreement within this range,\nthe average polarization remains consistent at approximat ely\nthe level of -1.5% over the entire spectrum. The QED calcu-\nlation is derived using the parameter s=kp+/kk′, while semi-\nclassical method is derived using the parameter δ+=ε+/ω.\nIn the case of large positron energy ε+, these two parameters\nare equivalent, which allows for a comparison of results ob-\ntained using different approaches. However, in the low-energy\nregime, the approximation of s=kp+/kk′≈δ+may not hold,\n(a) (b)\nδ+ δ+dP/dδ+Semi.↑\nSemi.↓\nQED↑\nQED↓\nζ+\nfSemi.\nQED\nFIG. 6. (a) The pair production probability dP/dδ+versusδ+:\nfor spin-up positrons using semiclasscial approach (blue- solid) or\nthe QED approach (green-dashed); for spin-down positrons u sing\nsemiclasscial approach (black-dashed) or the QED approach (red-\ndot-dashed). (b) The spectra of positron polarization with semiclass-\ncial approach (blue-solid) or the QED approach (red-dashed ).7\n(a) (b) (c)\nθx θx θxθy\nFIG. 7. Angular distribution of positron density d2N/dθx/dθy(rad−2) versusθx(mrad) and θy(mrad) calculated with QED approach: for\npositron spin-up (a) and spin-down (b). (c) Angular distrib ution of positron polarization ζ+\nfversusθx(mrad) and θy(mrad) calculated with\nQED approach.\nwhich could be responsible for the discrepancy between di ffer-\nent approaches. Moreover, the discrepancy could be induced\nby the numerical error of semiclassical method. To accurate ly\ndescribe the spectrum in the low-energy region, it is neces-\nsary to extend the trajectory integral to infinity. However, this\nrequires an infinite number of discrete points for numerical in-\ntegration, which is not possible in practice. Therefore, a fi nite\nintegration range could result in discrepancies in the resu lts.\nThis deviation can be considered inconsequential when eval -\nuating the average polarization due to the negligible posit ron\ndensity in this region. Furthermore, the angular distribut ion of\npositrons obtained using the semiclassical approach close ly\nmatches that of the QED approach [Figs. 7 (a) and (b)], as\nwell as the interferences rings exhibited in the polarizati on dis-\ntribution [Fig. 7 (c)].APPENDIX B: PHOTON POLARIZATION\nThe semiclassical method allows one to obtain the spec-\ntrum for arbitrary particles’ polarization states, see Fig . 8. All\ncurves are symmetric about δ+=0.5 except↑↑∝bardbland↓↓∝bardbl, and\nthe curves for↑↑∝bardbland↓↓∝bardblare mirror images of each other\naboutδ+=0.5. The probability is dominated by dP↑↓e⊥, which\ncorresponds to the scenario where the incoming photon is po-\nlarized perpendicularly to the laser polarization along th e di-\nrection of magnetic field ˆy=(0,1,0), while the final electron\nhas spin-up and the positron has spin-down with respect to ˆy.\nIn the case that the photon is initially polarized along magn etic\nfield direction, the spin-resolved probabilities and polar ization\nof created positron are roughly the same with the unpolarize d\ncase. On the other hand, when the photon has initial polariza -\ntion along the electric field direction, the positron’s pola riza-\ntion is negligible, making it unsuitable for studying posit ron\npolarization.\nAcknowledgement: We gratefully acknowledge helpful dis-\ncussions with Karen Z. Hatsagortsyan. 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A lot of different approaches have been p ut forward to deal with this, ranging from\nthe conventional cosmological constant to various models o f dark energy and beyond. Recently one very\ninteresting approach to explaining the late time accelerat ion has been put forward, where the the expan-\nsion of the universe is driven by mergers with other ”baby” un iverses and has been shown to be quite\nviable as well from the point of view of recent observational data. So in this work we examine the possi-\nbility of various rip scenarios and other future cosmologic al singularities in such ”multiversal” scenario,\nprobing such singularities for the first time in a multi unive rse scenario. We examine two models of such\na baby universe merging cosmology, and show that remarkably no rip scenario or future cosmological\nsingularity is possible in such models.\n1 Introduction\nThe surprising revelation of the late-time acceleration of the Univer se posed a significant challenge for the\nfield ofcosmology[1]. Since this discovery, extensive investigationsh avebeen undertaken to unravelthe mys-\ntery behind this unexpected expansion phenomenon. Scientists ha ve approached the cosmological expansion\nproblem from diverse perspectives, employing traditional methods such as the Cosmological constant [2–4]\nand venturing into more unconventional theories like Modified Gravit y [5–7]. Additionally, scenarios involv-\ningscalarfieldsdrivinglate-timecosmicaccelerationhavebeen explor ed[8–14]. Quantumgravityapproaches\nhave also played a role in addressing the cosmic acceleration puzzle, s panning from Braneworld cosmology in\nstring theory to theories like loop quantum cosmology and asymptot ically safe cosmology [15–25]. However,\nthese endeavors have uncovered certain inconsistencies, under scoring the limitations of our current under-\nstandingoftheuniverse. Oneofthemostprominentchallengesisth eHubbletension,indicatingdiscrepancies\nin the values of the Hubble constant derived from detailed Cosmic Micr owave Background (CMB) maps,\ncombined with Baryon Acoustic Oscillations data and Supernovae Typ e Ia (SNeIa) data [26–29]. Therefore,\nthe present epoch of the universe has presented us with a broad s pectrum of questions and appears poised\nto become a domain where advanced gravitational physics will pave t he way for a deeper comprehension of\ncosmology.\nVarious approaches to Quantum gravity have come forward in rece nt times and more often than not, they\nhave also been shown to be of some insight into tackling the issues of la te time cosmology and one such\nexciting approach to quantum gravity is that of Causal dynamical T riangulations or CDTs, which represent\ncurved spacetimes with Lorentzian signature and serve as regular ized lattice configurations in contemporary\n∗oem.t@ahduni.edu.in\n†khlopov@apc.in2p3.fr\n1lattice gravity models [30–37] . The fundamental idea behind CDT is to formulate a quantum gravity theory\nby taking a suitable scaling limit of a lattice theory. The dynamics of this lattice theory involve a nonper-\nturbative path integral over geometric lattices, providing a direct approximation of curved spacetimes from\nclassical general relativity. Similar to lattice QCD, the path integral relies on bare coupling constants and a\nUV lattice cutoff (denoted as ’a’) representing the shortest length unit on the lattice.\nAs the lattice spacing ’a’ is removed ( a→0), one seeks continuum limits near a critical point where\na certain correlation length in lattice units diverges. Achieving these continuum limits typically requires\nfine-tuning and renormalization of dimensionless bare couplings in the lattice model.\nFor pure gravity within the CDT framework, the path integral ZCDTtakes the form of a continuum limit\nof a regularized lattice expression. Schematically, it is given by:\nZCDT(GN,Λ) = lim\na→0/summationdisplay\nT∈T1\nC(T)eiSCDT[T],\nwhere the sum is over inequivalent Lorentzian triangulations T, following specific causal gluing rules. Each\nterm in the sum contributes with an amplitude determined by a lattice d iscretization SCDTof the continuum\nEinstein-Hilbert action, given by:\nSEH=1\nGN/integraldisplay\nd4x/radicalbig\n−detg(R[g,∂g,∂2g]−2Λ),\nwhereGNand Λ represent Newton’s constant and the cosmological constan t, respectively, and Rdenotes\nthe Ricci scalar of the metric tensor g. The quantity C(T) in the expression for ZCDTis the number of\nelements in the automorphism group of the triangulation T. This number is equal to 1 when Tlacks any\nsymmetries, which is the typical scenario. The chosen version for t he lattice action, denoted as SCDT, often\nfollows the Regge version of the gravitational action, but this choic e is not fundamental and is akin to a\ndiscrete approximation of general relativity.\nAnother interesting thing is that Monte Carlo simulations have also be en employed to explore the prop-\nerties of the CDT model. Despite being defined for Lorentzian signat ure, numerical methods are applicable\ndue to the presence ofa well-defined Wick rotation—ararityin quant um gravitybeyond perturbation theory.\nOne of the outcomes of these simulations reveals that when the spa tial topology is T3, the effective action\nas a function of the three-volume V(t) at proper time tis given by [38,39]:\nS=1\nΓ/integraldisplay\ndt/parenleftig˙V2\n2V+ΛV/parenrightig\n. (1)\nThis finding is noteworthy as it essentially mirrors the Hartle-Hawking minisuperspace action, including the\nrotation of the conformal factor . The rotation of the conforma l factor was proposed as a solution to the\nproblem of the unboundedness from below of the Euclidean Einstein- Hilbert action. In CDT, the effective\naction arises through the integration (via Monte Carlo simulations) o f all other degrees of freedom than V(t),\nwhile Hartle and Hawking manually constrained the geometry to depen d solely on V(t).\nRecalling the standard minisuperspace approximation, where the me tric is given by:\nds2=−N2(t)dt2+a2(t)dΩ3, dΩ3=3/summationdisplay\ni=1dx2\ni. (2)\nWe introduce:\nv(t) =1\nκa3(t), κ= 8πG, (3)\nwhereGis the gravitational constant. The minisuperspace Einstein-Hilbert action becomes:\nS=/integraldisplay\ndt/parenleftig\n−˙v2\n3Nv−λNv/parenrightig\n, (4)\nwithλbeing the cosmological constant. The Hamiltonian corresponding to (4) is\nH(v,p) =Nv/parenleftbigg\n−3\n4p2+λ/parenrightbigg\n, (5)\n2wherepdenotes the momentum conjugate to v. The analytic continuation of (4) by Hartle and Hawking,\nincluding the rotation of the conformal factor, results in the follow ing action:\nShh=/integraldisplay\ndt/parenleftig˙v2\n3Nv+λNv/parenrightig\n. (6)\nHartle and Hawking primarily aimed to use (6) in the path integral. Rega rdless of the obtained result, one\nwould eventually need to rotate back to Lorentzian signature for c osmological applications. If the quantum\ntheory has a classical limit, this limit should be given by (4) and (5). Thu s, we do not anticipate the\nlate-time aspects of cosmology to be directly impacted by the quant um aspects of gravity. Similarly, since\nCDT is designed as a quantum gravity theory, the result (1) sugges ts that CDT may not offer new insights\ninto late-time cosmology. ”Traditional” quantum gravity might influen ce the early-time universe, addressing\nBig Bang singularities, etc., but not the late-time universe. However , if we allow for more ”untraditional”\nquantum phenomena, such as the absorption and emission of so-ca lled baby-universes1, this situation can\nchange as shown in [52]. So in this work we shall explore this scenario an d shall see whether or not one\ncan obtain rips and other cosmological singularities as we know them, in a universe where such a continual\nmerger of other universes is happening as well. Essentially we are try ing to tackle the issues of rips and\nsingularities in the context of a Multiversal theory, which does seem to us as the first such endeavour in this\ndirection. In the next section we shall describe this baby universe m erging scenario in more detail yet briefly,\nwhile in section III we shall discuss the status of rips and other singu larities in such a cosmology. We shall\nconclude our work in section IV.\n2 Cosmic expansion with universe merger\nHere we shall briefly overview the scenario where late time expansion of the universe is through merging\nwith other baby universes. Note that even though the word baby u niverse has been used here, it does not\nnecessarily mean that the other universes are very small compare d to ours but we use such nomenclature to\nbe in line with the work in [52], which we are following here. Given our consid eration for other universes\nmerging with our Universe, we are essentially discussing a multi-univer se theory. Similar to a many-particle\ntheory, it is natural to introduce creation and annihilation operato rs Ψ†(v) and Ψ(v) for single universes with\nspatial volume v. In a complete theory of four-dimensional quantum gravity, the s patial volume alone would\nnot fully characterize a state at a given time t. Here, we simplify drastically by working in a minisuperspace\napproximation, where the spatial universe is entirely characterize d by the spatial volume v. Thus, we\ndenote the quantum state of a spatial universe with volume vas|v/angbracketright. Considering the multi-universe Fock\nspace constructed from these single universe states, we denote the Fock vacuum state as |0/angbracketright. The following\ncommutation relation and operator actions apply:\n[Ψ(v),Ψ†(v′)] =δ(v−v′), (7)\nΨ†(v)|0/angbracketright=|v/angbracketright, (8)\nΨ(v)|0/angbracketright= 0. (9)\nIn this manner, the (minisuperspace) quantum Hamiltonian, incorpo rating the creation and destruction\nof universes, can be expressed as:\nˆH=ˆH(0)−g/integraldisplay\ndv1/integraldisplay\ndv2Ψ†(v1)Ψ†(v2) (v1+v2)Ψ(v1+v2)− (10)\ng/integraldisplay\ndv1/integraldisplay\ndv2Ψ†(v1+v2)v2Ψ(v2)v1Ψ(v1)−/integraldisplaydv\nvρ(v)Ψ†(v),\nˆH(0)=/integraldisplay∞\n0dv\nvΨ†(v)ˆH(0)vΨ(v),ˆH(0)=v/parenleftbigg\n−3\n4d2\ndv2+λ/parenrightbigg\n. (11)\n1for more works on baby universes [40–51]\n3Here,ˆH(0)describesthe quantumHamiltoniancorrespondingtothe action(1) (withβ= 2/3), portraying\nthe propagation of a single universe and note that we are taking g >0 in our work2. The two cubic terms\ndescribe the splitting of a universe into two and the merging of two un iverses into one, respectively. The last\nterm suggests that a universe can be created from the Fock vacu um|0/angbracketrightprovided the spatial volume is zero.\nWithout this term, ˆH|0/angbracketright= 0, and the Fock vacuum would be stable. In our minisuperspace app roximation,\nwe do not attempt to describe howsuch mergingor splitting realistica llytakes place; our primaryinterest lies\nin how the volume of space can be influenced by such processes, whe re the minisuperspace model provides\nvaluable insights.\nEven the minisuperspace Hamiltonian ˆHis too complex to be solved in general. A universe can suc-\ncessively split and be joined by many others, and a part that splits off can later rejoin, thereby changing\nthe topology of spacetime. The Hamiltonian is essentially dimension-ind ependent (dimension dependence\nis absorbed in the coupling constants κ,λ, andg). Here we are dealing with string theory theory of 2 di-\nmensional CDT and there exists a truncation that can be solved ana lytically of this called generalized CDT\n(GDCT). This theory has our main interest from a cosmological point of view as it follows the evolution of\nour Universe over time, considering the merging with other universe s created at various times. After some\nanalysis, we obtain:\nH=v/parenleftig\n−3\n4(p2+λ−2gF(p)/parenrightig\n=3\n4v/parenleftig\n(p+α)/radicalbigg\n(p−α)2+2g\nα/parenrightig\n, (12)\nwherepis the classical momentum conjugate to v. It was then shown that the Hamiltonian in (12) can\ngive the expansion of the universe without any cosmological consta nt, the expansion instead being driven by\nmergers with other universes. One can then include matter in the Ha miltonian, after which it can take the\nform\nH[v,p] =v(−f(p)+κρm(v)), vρm(v) =vptρm(vpt), (13)\nwherevptandρm(vpt) denote the values at the present time tptThe equations of motion for arbitrary f(p)\nin (13) can be written as\n˙v=∂H\n∂p=−vf′(p),i.e.3˙a\na=˙v\nv=−f′(p) (14)\n˙p=−∂H\n∂v=f(p),i.e. t−t0=/integraldisplayp\np0dp\nf(p)(15)\np0is p at some initial time t0, which is not necessarily the start of the universe here and the prim es denote\ndifferentiation with respect to p. From (14) we see that the Hubble p arameter can be written as\nH=−f′(p)\n3(16)\nWe are now at a point from which we can slowly transition to analyzie rips and singularities, which we shall\nnow do in the next section.\n3 Impossibility of rips and singularities in this ”Mulivers e”\nIn recent times, a substantial body of literature has emerged, fo cusing on the exploration of various types\nof singularities that may arise in the present and distant future of t he Universe. The detection of late-time\nacceleration has significantly propelled such investigations [53–68]. T he term ”singularity” encompasses di-\nverse definitions, and until the late 20th century, the primary inst ances of singularity formation in cosmology\nwere the initial Big Bang singularity and, in spatially closed cosmological models, the eventual Big Crunch\nsingularity. Hawking and Penrose provided the definition of a singular point in cosmology, and many of the\ntheorems they established rely on the null energy condition. Moreo ver, at a singular point in spacetime,\ngeodesic incompleteness and the divergence of curvature scalars occur. Even though modified gravity might\nalter the null energycondition comparedto the Einstein-Hilbert cas e, it is widely acknowledgedthat geodesic\n2Although g=0 also form a class of solutions, which correspon d to the simple dS space-times [52].\n4incompleteness and the divergence of curvature invariants stron gly suggest the existence of a crushing sin-\ngularity.\nThe effects of singularities in cosmology vary, and a comprehensive c lassification was conducted in [56,69].\nWhile one can interpret singularities as points where a cosmological th eory encounters challenges, they can\nalso be seen as gateways to new physics, adding a different kind of int riguing interest. In particular, finite-\ntime singularities (those occurring within a finite time) could be perceiv ed as either imperfections in classical\ntheory or as portals to a quantum description of general relativity . This distinction arises because these\nsingularities cannot be treated similarly to the spacelike singularities o f black holes. Therefore, questions\nabout the accuracy of predictions made by classical gravitational theories arise. Consequently, the study\nof singularities in cosmological contexts and exploring methods to po tentially eliminate them contribute to\na deeper understanding of the connection between quantum desc riptions of cosmology and their classical\ncounterparts. The rip scenarios and singularities we are talking of a re\n•Big rip (Type I singularity) : A well known scenario, where for t→tf, wheretfis finite, we have\nboth the effective energy density and pressure density of the univ erse diverging, peff→ ∞,ρeff→ ∞,\nwhile we also have a diverging Hubble parameter H→ ∞[70]. This results in a scenario of universal\ndeath, where everything within the universe undergoes progress ive disintegration. [70].\n•Sudden/Pressure singularity (Type II singularity): In this case, peffdiverges, as well as the derivatives\nof the scale factor beyond the second derivative [71,72] (Big Brak e singularities are a special case of\nthis [73] ).\n•Big Freeze singularity (Type III singularity): In this case, the deriv ative of the scale factor from the\nfirst derivative onwards diverges. These were detected in genera lized Chaplygin gas models [74].\n•Generalized sudden singularities (Type IV singularity): These are fin ite time singularities with finite\ndensity and pressure instead of diverging pressure. In this case, the derivative of the scale factor\ndiverges from a derivative higher than the second [56,75] .\n•Little rip : It is similar to the big rip, but here where for t→ ∞, we have the effective energy density,\npressure density and Hubble parameter diverging, peff→ ∞,ρeff→ ∞,H→ ∞. It is effectively a\nbig rip which happens at infinite time [76].\n•Pseudo rip : The Pseudo rip also takes place at finite time, but in this ca se we have a finite Hubble\nparameter, so H→Hffor a finite Hfast→ ∞[77]\n•Little sibling of the Big rip : In this case at infinite times, we have a diverg ing Hubble parameter again\njust like the Little rip but the derivatives of the Hubble parameter ar e finite [78].\nWe are interested to see which of these scenario can be allowed for in the models prepared in [52]. The\ninterest here stems from the very exciting notion of this universe, where smaller baby universes are merging\ninto it and so seeing what possibilities it could have in the future realistic ally seems to be something very\nworth knowing.\nWith the singularities discussed in detail, we need to know inculcate the pressure and energy terms in\nour formulation and this can be done in a similar way to usual cosmology as achieved in [52]. One can\nwrite various important cosmological parameters in terms of p using the formulation as discussed in section\nII, for example Redshift, angular diameter etc. but what we shall b e interested in here are the so called\n”formal pressure” and ”formal energy”, which are the equivalen ts of the pressure and energy densities in this\ncosmology. Any solution to (14)-(15) is inherently subject to the c ondition H= const by construction. Our\nfocus lies on the ”on-shell” solutions where H= 0, implying that\nf(p) =κρm(v) =κρm(vtp)vtp\nv=f(ttp)ttp\nv, (17)\nwherep0represents the value of pat the present time ttp, with (17) being recognized as the ”Generalized\nFriedmann equation” [52]. The ”formal density” ρf(t) orρf(p) associated with the function f(p) is obtained\n5by expressing the generalized Friedmann equation as\n/parenleftbigg˙a(t)\na(t)/parenrightbigg2\n=κρm(v)\n3+κρf(v)\n3, (18)\nfrom which, utilizing the equations of motion, we derive\nκρf(p) =1\n3(f′(p))2−f(p). (19)\nκdρf\ndt=f(p)f′(p)/parenleftbigg2\n3f′′(p)−1/parenrightbigg\n. (20)\nWe introduce the ”formal pressure” Pfassociated with ρfthrough the energy conservation equation\nd\ndt(vρf)+Pfd\ndtv= 0. (21)\nThis leads to\nPf=f(p)/parenleftbigg2\n3f′′(p)−1/parenrightbigg\n−ρf(v), (22)\nand the ”formal equation of state parameter” wfis defined (for ρf/negationslash= 0) as\nwf=Pf\nρf=f(p)/parenleftbig2\n3f′′(p)−1/parenrightbig\n1\n3(f′(p))2−f(p)−1. (23)\nThe definitions of ρfandPfensure that our equations of motion can be expressed in the stand ard GR form,\nusinga,ρm,ρf, andP. Till now we have maintained the notation used in [52] for formal pre ssure and\nformal density, but from here onwards formal pressure would re fer to ˜pand formal density to ˜ ρin this work.\nFurthermore, differentiating (16) with respect to time, and using ( 14) in doing so, we then arrive at\n/integraldisplayHf\nHodH\nH=/integraldisplaypf\npof′′(p)\nf(p)f′(p)dp (24)\nDividing (20) by (19) and again using (14), we arrive at the following fo r the formal density\n/integraldisplay˜ρf\n˜ρod˜ρ\n˜ρ=/integraldisplaypf\npo3f′(p)/parenleftig\n2f′′(p)\n3−1/parenrightig\nf′2(p)−3f(p)dp (25)\nDifferentiating(22)ofthepaper[52]withrespecttotimeandusing( 14), wethenarriveatasimilarexpression\nfor formal pressure\n/integraldisplay˜pf\n˜pod˜p\n˜p=/integraldisplaypf\npof′(p)/parenleftig\n2f′′(p)\n3−1/parenrightig\n+2f′′′(p)f(p)\n3−f′(p)\nκ/parenleftig\n2f′′(p)\n3−1/parenrightig\nf(p)/parenleftig\n2f′′(p)\n3−1/parenrightig\n−1\nκ/parenleftig\nf′(p)2\n3−f(p)/parenrightigdp (26)\nSimilarly we can write for the derivatives of H as\n/integraldisplay˙Hf\n˙Hod˙H\n˙H=/integraldisplaypf\npof′(p)f′′(p)+f(p)f′′′(p)\nf′′(p)f(p)dp (27)\n/integraldisplay¨Hf\n¨Hod¨H\n¨H=f′(p)2f′′(p)+f(p)f′′(p)2+f(p)2f′′′′(p)+3f′′(p)f(p)f′′′(p)\nf(p)(f′(p)f′′(p)+f(p)f′′′(p))dp (28)\nNow we can identify the realism of the various rip scenarios and futur e singularities we have outlined above\nas follows\n6•Big rip (type I) : We firstly check if there is a pffor which the RHS in (24) diverges, and if we do find\nsuch, we then apply that p in (14) to check if the LHS in it is non diverge nt. If both conditions are\nsatisfied, alongside having diverging RHS in (25) and (26), then we ca n say that for that particular\nf(p), a big rip is a realistic scenario.\n•Pressure / Big Brake/Sudden singularities (Type II) : In this case w e would just have to check if there\nis apffor which (26) diverges, while others remain finite.\n•Big Freeze (Type III) : In this case we would have to check if there is apffor which (26) and (25)\ndiverge, but (24) and (14) do not.\n•Generalized sudden singularities (Type IV) : In this case, we would ha ve to check if the RHS in (28)\ndiverges for certain pfwhile other integrals do not.\n•Little rip and Pseudo rip: We firstly check if there is a pffor which the LHS in (14) diverges, and if\nwe do find such, we then apply that p in (24) to check if the RHS in it dive rges or not. We would also\ncheck whether the RHS in (25) and (26) diverges as well. If the RHS in (24) diverges, we have a Little\nrip in this scenario while if it is non divergent, we then have a Pseudo rip f or that particular f(p)\n•Little sibling of the big rip : In this case we would have to check whether the RHS in (27) diverges or\nnot for the p for which t→ ∞, besides the checks as above.\nWe now need to consider particular models of baby universe merger d riven cosmic expansion to fully under-\nstand what we want to and for that we need to finally specify f(p) and we will focus on the baby universe\nmerging models provided in [52], those being\nfgcdt(p) =3\n4(p+α)/radicalbig\n(p−α)2+2α2 (29)\nand\nfmod(p) =3\n4/parenleftbigg\np2+2g\np/parenrightbigg\n(30)\nwhereα=g1/3andg≥0 We will firstly talk of the model in (30). For this model it was already s hown\nin [52] that t→ ∞,p−>−21/3α, so that is the highest value any p can take, which gives us a constra int\npf≤ −21/3α\nThis is a negative value as g≥0, in fact it was shown in [52] how g= 0 just corresponds to the simple dS\nsolution, which they also discussed a toy model in their paper. But fo r actually relevant purposes from the\npoint of a view of a cosmos with merging universes, it is imperative to ta keg >0, which was again showed\nto still give an expanding universe even without a cosmological const ant.\nNow, we calculate the relevant integrals (24),(25), (26), (27), (2 8) in this case, starting with (24)\n/integraldisplayHf\nHodH\nH= ln/parenleftigg\np3\nf−g\np2\nf/parenrightigg\n+q1(p0) (31)\n/integraldisplay˜pf\n˜pod˜p\n˜p= ln/parenleftigg\n4gκ−g+2κp3\nf+4p3\nf\np4\nf/parenrightigg\n+q2(p0) (32)\n/integraldisplay˜ρf\n˜ρod˜ρ\n˜ρ= ln/parenleftigg\n4p3\nf−g\np4\nf/parenrightigg\n+q3(p0) (33)\n/integraldisplay˙Hf\n˙Hod˙H\n˙H= ln/parenleftigg\n(p3\nf+2g)2\np4\nf/parenrightigg\n+q4(p0) (34)\n7/integraldisplay¨Hf\n¨Hod¨H\n¨H=1\n8\n33√\n2/parenleftigg\nln/parenleftbigg\n4g2/3+22/3p2\nf\n(23√g−3√\n2pf)2+23√\n23√gpf/parenrightbigg\n−2√\n3tan−1/parenleftigg3√\n2pf\n3√g+1\n√\n3/parenrightigg/parenrightigg\n3√g\n\n+1\n8/parenleftigg\nln/parenleftig/parenleftbig\n2g+p3\nf/parenrightbig2/parenleftbig\np3\nf−4g/parenrightbig14/parenrightig\n+ln/parenleftigg\n1\np30\nf/parenrightigg\n−36\npf/parenrightigg\n+q5(p0) (35)\nwhere the qi(p0),i= 1,2..5 are some constants based on what we choose p0. In [52] it was considered that\np0→ ∞but we here are primarily only concerned with the properties of pfso we do not need to specify any\np0as well.\nWe firstly note the impossibility of the big rip in this scenario, as the inte gral (31) would not diverge\nfor any allowed pfas it can only diverge for pf→0 . It is important to realize that pfcannot attain positive\nvalues or zero, as we have taken g >0 and time reaches infinity as p→ −21/3α. So going to values like\n0 or positive for p would mean going ”beyond infinity” in some way, which does not make sense here. So\nthat would mean H would not diverge in (31). Similarly, we see that ˜ p, ˜ρand˙Hwould also not diverge\nfor any value of p, as all the expressions in (32),(33) and (34) would only diverge for p→0, which is not a\npossibility as we have seen here. This would mean that we do not have t he possibilities of the Little rip and\nPseudo rip either, the energy density and pressure density does n ot diverge in any case. The Little sibling\nof the big rip is not a possibility as well as H does not diverge at infinite tim es. The Sudden singularity (\ntype II ) and Big Freeze singularity (type III) are not possible as we ll hence. For type IV singularity, we\nsee that ¨Hcan diverge as in (28) for p→0 and also for p→22/3g= 22/3α. But we again note that p is\nconstrained as p≤ −21/3α, which means that the value of p for which the EOS parameter diverg es is not\nattainable and hence a type IV singularity is also not possible. We now c ome to the very intriguing conclu-\nsion that in such a cosmic scenario where baby universes keep contin ually merging with our universe, we do\nnot observe any future cosmological singularities or at least, not t he ones we are aware of. This is incredibly\nexciting, as this is the first time the question of cosmological singular ities has been addressed from the point\nof view of a multiverse and seeing that all the prominent cosmological singularities do not exist in a multiver-\nsal setting is certainly bewildering. Furthermore seeing in particular how all the main rip scenarios are not\npossible in such auniverse, it alsomakesonewonderhow sucha babym erginguniversewould eventuallyend.\nNow for the model (29), we note that in this case, as shown in [52] it is not entirely clear for what value of\np t→ ∞, so in this case the Little rip, Pseudo rip and Little sibling of the big rip lie b eyond our scope to\nbegin with. The integral (24) reads as\n/integraldisplayHf\nHodH\nH= ln\nα2+p2\nf−αpf/radicalig\n3α2+p2\nf−2αpf\n+ ˜q1(p0) (36)\nwhere the ˜ qi(p0),i= 1,2..4 are again some constants based on what we choose p0. We note here that the\nintegral above can only diverge for a complex value of p, to be precis e forpf=α±i√\n2α, which is of course\nunrealistic. Similarly, the integrals (26), (25), (27) read\n/integraldisplay˜pf\n˜pod˜p\n˜p= ln/bracketleftigg\n−ακ/radicalig\n3α2+p2\nf−2αpf−κpf/radicalig\n3α2+p2\nf−2αpf−α3(α+2ακ−2(2κ+1)pf)\n3α2+p2\nf−2αpf+\n(κ−1)p2\nf+α/radicalig\n3α2+p2\nf−2αpf+pf/radicalig\n3α2+p2\nf−2αpf/bracketrightigg\n+ ˜q2(p0) (37)\n/integraldisplay˜ρf\n˜ρod˜ρ\n˜ρ= ln/parenleftigg\n−α/radicalig\n3α2+p2\nf−2αpf−pf/radicalig\n3α2+p2\nf−2αpf+α3(α−2pf)\n3α2+p2\nf−2αpf+p2\nf/parenrightigg\n+ ˜q3(p0) (38)\n8/integraldisplay˙Hf\n˙Hod˙H\n˙H= ln\n(α+pf)/parenleftig\n−2α3+p3\nf−3αp2\nf+6α2pf/parenrightig\n3α2+p2\nf−2αpf\n+ ˜q4(p0) (39)\nWe again note that all the above integrals will only diverge for a comple x values of p, which is the same\nas that for H which is pf=α±i√\n2α. The expression for (28) is awfully long and so we dont write that\nhere, but similarly it will also blow up only for this complex value. This is of c ourse unrealistic again and so\nexpecting these quantities to blow up is also outside our realm of poss ibilities. This tells us that there would\nbe no finite time blow ups of the energy density, pressure density, H ubble parameter and its derivatives even\nup till the second order. This tells us that the big rip , Pressure/ Big B rake/ Sudden singularity, Big Freeze\nand generalized sudden singularities are all beyond the realm of poss ibilities in this model as well.\n4 Conclusions\nIn this work, we have pondered about the realism of rips and other c osmological singularities in a universe\nwhich is continually merging with other universes and driving its expans ion via these mergers. We firstly\ndiscussed how such universes can come into being with the exciting ap proach of Causal dynamical trian-\ngulations to quantum gravity, discussing the universal wavefunct ion approach in this scenario and other\nsubtleties. We then formulated conditions on how we can attest to w hether or not the rip scenarios (big\nrip, little rip, quasi rip, pseudo rip ) and other cosmological singularit ies could occur in such a universe,\nafter which we analyzed this occurrence for two such viable multiver sal models. The conclusion we see\nfrom analyzing the two models with baby universe mergers into our ow n universe in (30) and (29) is that\nsuch a multiversal scenario somehow gives us rise to a cosmos where we do not see the prominent future\ncosmological singularities taking place. It is especially interesting, be cause one might imagine that if one has\nnew universes merging with another universe (in this case our own) t hen such blow ups may become more\nfrequent but what we end up with is a way more refined cosmology. It is even more remarkable given the\nfact that this scenario develops out after considering just the Ein stein-Hilbert Action in the framework of\nCDT, as the conventional EH action has been frequently shown to b e riddled with these singularities and\nattempts have always been made to smoothen them out. Even thou gh many such attempts, from quantum\ngravity perspectives like conformal anomalies or modified gravity eff ects, have been pursued one sees that\nnot all the singularities can be delayed or avoided even in the most opt imistic of cases. So this springs a huge\nsurprise, as by considering a Multiversal scenario one ends up with a universe which is apparently free from\nthe prominent cosmological singularities and rips. 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D , 24(10):1550078, 2015.\n13"    },    {        "title": "2401.13288v1.Black_hole_no_hair_theorem_for_self_gravitating_time_dependent_spherically_symmetric_multiple_scalar_fields.pdf",        "content": "arXiv:2401.13288v1  [gr-qc]  24 Jan 2024Black hole no-hair theorem for self-gravitating time-depe ndent spherically\nsymmetric multiple scalar fields\nStoytcho S. Yazadjiev1, 2,∗and Daniela D. Doneva3, 4, †\n1Department of Theoretical Physics, Faculty of Physics, Sofi a University, Sofia 1164, Bulgaria\n2Institute of Mathematics and Informatics, Bulgarian Acade my of Sciences,\nAcad. G. Bonchev St. 8, Sofia 1113, Bulgaria\n3Theoretical Astrophysics, Eberhard Karls University of T¨ ubingen, T¨ ubingen 72076, Germany\n4INRNE - Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria\nWe prove under certain weak assumptions a black hole no-hair theorem in spherically\nsymmetric spacetimes for self-gravitating time-dependen t multiple scalar fields with an\narbitrary target space admitting a Killing field with a non-e mpty axis and arbitrary non-\nnegative potential invariant under the flow of the Killing fie ld. It is shown that for such\nconfigurations the only spherically symmetric and asymptot ically flat black hole solutions\nconsist of the Schwarzschild metric and a constant multi-sc alar map. In due course of the\nproof we also unveil the intrinsic connection of the time-de pendence of the scalar fields\nwith the symmetries of the target space.\nI. INTRODUCTION\nScalar fields play an important role in modern theoretical phy sics, especially in the context of\ngravitational theories, cosmology, and astrophysics. Of s pecial interest for black hole physics is\nthe question of whether black holes can support scalar hair. In the case of a single scalar field it\nis well known that black holes can not support scalar hair in p hysically realistic scenarios [1],[2].\nThe same applies to the case with multiple static scalar fields [3],[4]. The picture changes drasti-\ncally in the case of time-dependent multiple scalar fields. Ev en for Einstein’s equations minimally\ncoupled to two (real) scalar fields or equivalently one comple x scalar field, stationary black hole\nsolutions with scalar hair exist when the scalar fields are tim e-dependent but with a stationary\nenergy-momentum tensor [5],[6]. There are serious numeric al indications that the picture of the\npossible stationary black hole solutions in the presence of two real time-dependent scalar fields\n(one time-dependent complex scalar field) could be very compl icated [7]. This fact naturally raises\nthe problem for the classification of the possible stationary black hole solutions of the Einstein\nequations coupled to time-dependent multiple scalar fields. In its full generality, this mathemat-\nical problem is extremely difficult and that is why it is natura l as a first step to focus on static\nblack hole solutions with spherical symmetry. A partial res ult, in this case, has been achieved in\n[8] where a no-hair theorem for spherically symmetric scala r fields with special time dependence\nin the particular case of a flat target space was proven.\nThe purpose of the present paper is to prove a black hole non-h air theorem in spherically\nsymmetric spacetimes for self-gravitating time-dependen t multiple scalar fields with an arbitrary\ntarget space and an arbitrary non-negative potential. The o nly non-avoidable restriction we im-\npose on the target space metric is to admit a Killing field with a n on-empty axis whose action\n∗yazad@phys.uni-sofia.bg\n†daniela.doneva@uni-tuebingen.de2\nleaves the potential invariant. This restriction, as expla ined below, ensures the stationarity of the\nenergy-momentum tensor of the scalar fields. In due course of t he proof we also unveil the in-\ntrinsic connection of the time-dependence of the scalar field s with the symmetries of the target\nspace.\nII. BLACK HOLE NO-HAIR THEOREM FOR SELF-GRA VITATING TIME-D EPENDENT\nSPHERICALLY SYMMETRIC MULTIPLE SCALAR FIELDS\nWe consider the 4-dimensional spacetime (M,gµν), an N-dimensional Riemannian manifold\n(EN,γab), the so-called target space, and a map ϕ:(M,gµν)→(EN,γab). The differential dϕ\ninduces a map between the tangent spaces of MandEN,dϕ:TM→TEN. The norm of the\ndifferential <dϕ,dϕ>in local coordinate patches on MandENis given by\n<dϕ,dϕ>=gµν(x)γab(ϕ(x))∂µϕa(x)∂νϕb(x). (1)\nWe shall also consider multi-scalar self-gravitating theo ries defined by the following action\nS=1\n16πG/integraldisplay\nMd4x/radicalbig−g[R−2<dϕ,dϕ>−4V(ϕ)], (2)\nwhere Ris the Ricci scalar curvature and V(ϕ)is the scalar fields potential. The field equations\nassociated with the action (2) and written in local coordina te patches of MandENare\nRµν−1\n2Rgµν=2γab(ϕ)∇µϕa∇νϕb−γab(ϕ)∇σϕa∇σϕbgµν−2V(ϕ)gµν, (3)\n∇µ∇µϕa=−γa\ncd(ϕ)∇µϕc∇µϕd+1\n4γab(ϕ)∂V(ϕ)\n∂ϕb, (4)\nwhere∇µis the covariant derivative with respect to the spacetime me tricgµνand γa\ncd(ϕ)are\nthe Christoffel symbols with respect to the target space met ricγab(ϕ). We shall assume that the\npotential V(ϕ)is non-negative and that the system∂V(ϕ)\n∂ϕa=0 possesses at least one solution ϕ0at\nwhich V(ϕ0) =0.\nWe focus here on static and spherically symmetric spacetime s with a metric\nds2=−e2Φ(r)dt2+e2Λ(r)dr2+r2sijdxidxj, (5)\nwhere sijis the metric on the unit 2D sphere, namely sijdxidxj=dθ2+sin2θdφ2.\nWe require the scalar fields to be spherically symmetric, i.e. invariant under the flow of the\nKilling fields generating the spherical symmetry. In simple wo rds this means that the scalar fields\nare independent from the angular coordinate, i.e. ∂iϕa=0. Contrary to that, the scalar fields\nare not required to depend only on the radial coordinate r. They can depend on the time in such\na way that the energy-momentum tensor to be time-independen t. In order to guarantee that the\nenergy-momentum tensor of the scalar fields is stationary (ti me-independent), we assume that the\ntarget space metric γab(ϕ)admits a Killing field kawhose flow leaves the potential V(ϕ)invariant,\nnamelyLkV(ϕ) =ka∂aV(ϕ) =0 and the dynamics of scalar fields is confined on the flow of ka. In\nmore formal language we require the Lie derivative Lξϕaalong the timelike Killing field ξ=∂\n∂t3\nsatisfiesLξϕa=∂tϕa=−ωkawhere ω/negationslash=0 is a real number. Indeed, using that Lξϕa=−ωkawe\nhave\nLξTµν=−ω(Lkγab(ϕ))/bracketleftBig\n2∇µϕa∇νϕb−∇ σϕa∇σϕbgµν/bracketrightBig\n+2ωLkV(ϕ)gµν=0. (6)\nIn order to further guarantee the staticity of the metric we m ust impose the condition ξσRσ[µξν]=\n0, which, taking into account that ∂iϕa=0, reduces to ka∂rϕa=0. The norm of kawill be denoted\nby|k|, i.e.|k|2=γab(ϕ)kakb.\nFrom now on we assume the existence of such a Killing field kaand in addition we assume\nthat the axis of ka, i.e. the points where ka=0, is non-empty. The additional requirement for the\naxis of kais important because of the following. First, as we discuss b elow, the asymptotic flatness\nrequires lim r→∞|k|=0. Second, in order for the black hole to be static on the black hole horizon\nwe must have Rµνξµξν|h=Tµνξµξν|h=0 which, using the Einstein equations and Lξϕa=−ωka,\nreduces to Rµνξµξν=ω2|k|2|h=0. All this clearly shows that the axis of kamust be non-empty.\nAnother way to derive the conditions ka∂rϕa=0 and|k||h=0 is as follows. When the target\nspace metric admits a Killing field leaving the scalar potentia l invariant, one can show by using\nthe field equations that there exists a conserved current jµsatisfying ∇µjµ=0 and explicitly given\nby\njµ=ka∇µϕa. (7)\nThen the condition ka∂rϕa=0 means that the scalar flux in radial direction vanishes. The\nscalar flux through the horizon is ξµjµ=ξµka∇µϕa=−ω|k|2|h. Therefore the condition |k||h=0\nguarantees that there is no scalar flux through the horizon an d the geometry can be static.\nThe assumptions we made above are very natural from mathemat ical and physical points of\nview. In particular, if the orbits of the Killing field kaare periodic, then the time-dependence\nwill also be periodic. This is the case for example with boson stars which constitute of two time-\ndependent scalar fields periodic in time, or equivalently one periodic in time complex scalar field.\nLet us demonstrate this on a simple but instructive example. Consider the flat target space (R,δab).\nIt admits three Killing vectors k(1)=∂\n∂ϕ1,k(2)=∂\n∂ϕ2and k(3)=ϕ2∂\n∂ϕ1−ϕ1∂\n∂ϕ2. Only the third\nKilling vector has periodic orbits and non-empty axis corres ponding to ϕ1=ϕ2=0. In this\ncase, the conserved current associated with the Killing vect ork(3)isjµ=k(3)a∇µϕa=ϕ2∇µϕ1−\nϕ2∇µϕ1=i\n2(Ψ∗∇µΨ−Ψ∇µΨ∗)which is the standard conserved current for the complex scal ar\nfield Ψ=ϕ1+iϕ2. Solving then the equations∂ϕa\n∂t=−ωka\n(3)and taking into account that ϕado\nnot depend on the angular coordinates, we find\nϕ1(r,t) =ψ(r)cos(ωt+f(r)),ϕ2(r,t) =ψ(r)sin(ωt+f(r)), (8)\nwhere ψ(r)and f(r)are functions of the radial coordinate only. Imposing furth er the staticity\ncondition ka∂rϕa=ϕ2∂rϕ1−ϕ1∂rϕ2=0 we obtain ψ2(r)∂rf(r) = 0 which means that f(r) =\nC=constant . Without loss of generality we can put C=0. In terms of the complex scalar field\nΨ=ϕ1+iϕ2we have Ψ(r,t) = ψ(r)eiωt. In this way we derived the standard ansatz for the\ncomplex scalar field in studying the boson stars and the black h oles.\nLet us go back to the general case. We shall make one more techn ical assumption. In order to\nsatisfy the asymptotic flatness we require the axis of kato contain the solution ϕ0of the algebraic\nsystem∂V(ϕ)\n∂ϕa=0. This can be easily achieved by assuming that the potential depends properly4\non the scalar field through the norm of the Killing filed ka,V(ϕ) =V(|k|2), which is the common\ncase with two (real) scalar fields with flat target space or equi valently one complex scalar field,\nconsidered in the literature1.\nUnder the assumptions we made, the dimensionally reduced fiel d equations are the following\n2\nre−2ΛΛ′+1\nr2/parenleftBig\n1−e−2Λ/parenrightBig\n=ω2e−2Φ|k|2+e−2Λγab(ϕ)∂rϕa∂rϕb+2V(ϕ),\n(9)\n2\nre−2ΛΦ′−1\nr2/parenleftBig\n1−e−2Λ/parenrightBig\n=ω2e−2Φ|k|2+e−2Λγab(ϕ)∂rϕa∂rϕb−2V(ϕ),\n(10)\ne−2Λ/bracketleftbigg\nΦ′′+(Φ′+1\nr)(Φ′−Λ′)/bracketrightbigg\n=−1\n2K−2V(ϕ),\n(11)\n∂r/parenleftBig\neΦ−Λr2γab(ϕ)∂rϕb/parenrightBig\n=1\n4eΦ+Λr2/parenleftbigg∂V(ϕ)\n∂ϕa+∂K\n∂ϕa/parenrightbigg\n+ω2r2eΛ−Φkb∂ka\n∂ϕb,\n(12)\nwhere Kis defined by\nK=2γab(ϕ)∇µϕa∇µϕb=−2ω2e−2Φ|k|2+2e−2Λγab(ϕ)∂rϕa∂rϕb.\n(13)\nTaking into account that the metric functions Φand Λdepend on ronly, from the dimension-\nally reduced field equations it follows that γab(ϕ)∂rϕa∂rϕband|k|are functions of ronly. Then\nwe can define\nP2(r) =γab(ϕ)∂rϕa∂rϕb. (14)\nIn the present paper we are interested in asymptotically flat spacetimes. In this case, by using\nthe dimensionally reduced field equations, one can easily see that\nlim\nr→∞P2(r) =0, lim\nr→∞V(ϕ) =0, lim\nr→∞|k|=0. (15)\nOn the regular horizons the function (Φ+Λ)and its derivative are regular. Using this fact, and\nby adding the equation for Φto the equation for Λ, one can see that\nlim\nr→rhe−2Φ|k|2=0. (16)\nThe strategy for proving the no-hair theorem is based on a pro per divergence identity. In order\ntoderive it we used the following methodology. We multiply the equation (12) for ϕaby∂rϕaand\nafter some algebra we obtain\n∂r/parenleftBig\neΦ−Λr2γad(ϕ)∂rϕa∂rϕd/parenrightBig\n−/parenleftBig\neΦ−Λr2γad(ϕ)∂rϕd/parenrightBig\n∂2\nrϕa\n=1\n4eΦ+Λr2/parenleftbigg∂V(ϕ)\n∂ϕa+∂K\n∂ϕa/parenrightbigg\n∂rϕa+ω2r2eΛ−Φkb∂ka\n∂ϕb∂rϕa.\n1In the literature the potential V(ϕ), expressed in terms of the norm of the complex scalar field Ψ, is usually in the\nform V(ϕ) =2m2|Ψ|2+λ|ψ|4with|Ψ|2= (ϕ1)2+(ϕ2)2=|k(3)|2.5\nWe then express eΦ−Λr2∂2\nrϕaagain from (12) and substitute into the above equation (17). Doing\nso we get\n∂r/parenleftBig\neΦ−Λr2γad(ϕ)∂rϕa∂rϕd/parenrightBig\n+∂r/parenleftBig\nr2eΦ−Λγad(ϕ)/parenrightBig\n∂rϕa∂rϕd\n=1\n2eΦ+Λr2/parenleftbigg∂V(ϕ)\n∂ϕa+∂K\n∂ϕa/parenrightbigg\n∂rϕa+2ω2r2eΛ−Φkb∂ka\n∂ϕb∂rϕa.\nThe next step is to make use of the explicit form (13) of Kand after long algebra we find\nd\ndr/bracketleftBig\n2r2eΦ+ΛV(ϕ)−r2eΦ−ΛP2−ω2r2eΛ−Φ|k|2/bracketrightBig\n(17)\n=reΦ+Λ/bracketleftBig\n4V(ϕ)+( 1+e−2Λ)P2+ω2e−2Φ|k|2(e2Λ−3)/bracketrightBig\n.\nThis divergence identity is however not what we want since th e right hand side is not mani-\nfestly non-negative. Another divergence identity with the desired property can be derived in the\nfollowing way. We write our identity (17) in the form\ndB\ndr=A. (18)\nThen if Ω(r)is an everywhere positive function, we can construct a new id entity of the form\nd(ΩB)\ndr=Ω/parenleftbigg\nA+Ω′\nΩB/parenrightbigg\n. (19)\nApplying this to our identity (17) with Ω(r) =r−2we finally obtain the desired divergence iden-\ntity, namely\nd\ndr/bracketleftBig\n2eΦ+ΛV(ϕ)−eΦ−ΛP2−ω2eΛ−Φ|k|2/bracketrightBig\n(20)\n=1\nreΦ+Λ/bracketleftBig\n(1+3e−2Λ)P2+ω2e2Λ−2Φ|k|2(1−e−2Λ)/bracketrightBig\n.\nThe right side of the above equation is non-negative which fo llows from the fact that (1−e−2Λ)>\n0 for r∈[rH,∞)and this can be seen from the equation for Λwritten in the form\nd\ndr/bracketleftBig\nr(1−e−2Λ)/bracketrightBig\n=r2/bracketleftBig\nω2e−2Φ|k|2+e−2ΛP2(r)+2V(ϕ)/bracketrightBig\n≥0.\nTherefore r(1−e−2Λ)is non-decreasing function and taking into account that it i s positive on the\nhorizon and r>0, we conclude that (1−e−2Λ)>0 for r∈[rH,+∞).\nThe last step is to integrate the identity (20) from the horiz on to infinity and we get\n−2e(Φ+Λ)hV(ϕh) =/integraldisplay+∞\nrHdr1\nreΦ+Λ/bracketleftBig\n(1+3e−2Λ)P2+ω2e2Λ−2Φ|k|2(1−e−2Λ)/bracketrightBig\n,\nwhere we have taken into account (15) and that lim r→rheΦ−ΛP2(r) = lim r→rheΛ−Φ|k|2=0.\nThe left hand side is non-positive since by assumption V(ϕ)≥0, while the right hand side is\nnon-negative. Therefore we can conclude that both sides van ish. Consequently we have that\nP2(r) = γab(ϕ)∂rϕa∂rϕb=0 and ω2|k|2=γab(ϕ)∂tϕa∂tϕb=0 everywhere which means that the\nscalar fields are constant in the domain of outer communicatio ns. Therefore the right hand side6\nof the dimensionally reduced field equations (9)–(12) vanish and the equations are reduced to the\nvacuum, static and spherically symmetric Einstein equatio ns whose unique black hole solution\nwith regular horizon is the Schwarzschild one.\nSummarizing we proved the following theorem\nTheorem Let us consider self-gravitating map ϕ:(M,gµν)→(EN,γab)with the action (2) and\nassume that the scalar field potential is non-negative, V (ϕ)≥0, and the system∂V(ϕ)\n∂ϕa=0possesses at\nleast one solution ϕ0at which V (ϕ0) =0. Assume further that the scalar fields are invariant under the fl ow\nof the Killing fields generating the spherical symmetry, the ta rget space metric admits a Killing field kawith\nnon-empty axis containing roots ϕ0of the algebraic system∂V(ϕ)\n∂ϕa=0and the time dependence of the scalar\nfields satisfies Lξϕa=−ωkawith ω/negationslash=0being a real number. Then every static and spherically symmet ric\nblack hole solution to the field equations (3)-(4) with regular horizon consists of the Schwarzschild solution\nand a constant map ϕ0with∂V\n∂ϕa(ϕ0) =V(ϕ0) =|k|(ϕ0) =0.\nIII. DISCUSSION\nIn the present paper we studied the black hole solutions to th e Einstein gravity coupled to\ntime-dependent multiple scalar fields with an arbitrary targ et space and non-negative potential\nin spherical symmetry. We first showed the intrinsic connecti on of the symmetries of the target\nspace with the time-dependence of the scalar fields which ensu res the stationarity of the energy-\nmomentum tensor of the scalar field. Using that connection we d erived a proper divergence\nidentity with the help of which we proved a no-hair theorem. I n this way, we achieved a full clas-\nsification of the black hole solutions in the considered secto r of the theory. The natural question is\nwhether such a classification is possible for more general sec tors of the theory, namely the static\nsector without spherical symmetry and stationary sector. I t turns out that, under certain addi-\ntional assumptions, the full classification is also possible for the mentioned sectors of the theory.\nThey require different mathematical techniques and will be presented elsewhere.\nACKNOWLEDGMENTS\nThis study is financed by the European Union-NextGenerationE U, through the National Re-\ncovery and Resilience Plan of the Republic of Bulgaria, proj ect No. BG-RRP-2.004-0008-C01. DD\nacknowledges financial support via an Emmy Noether Research G roup funded by the German\nResearch Foundation (DFG) under grant no. DO 1771/1-1.\n[1] J. D. Bekenstein, Phys. Rev. Lett. 28, 452 (1972).\n[2] C. Herdeiro and E. Radu, Int. J. Mod. Phys. D24, 09, 154201 4 (2015).\n[3] M. Heusler, J. Math. Phys. 33, 3497 (1992).\n[4] M. Heusler, Class. Quantum Grav. 12, 2021 (1995).\n[5] C. Herdeiro and E. Radu, Phys. Rev. Lett. 112, 221101 (201 4).\n[6] L. Collodel, D. Doneva and S. Yazadjiev, Phys. Rev. D 102, 084032 (2020).\n[7] C. Herdeiro and E. Radu, Phys. Rev. Lett. 131, 121401 (202 3).7\n[8] I. Pena and D. Sudarsky, Class. Quantum Grav. 14, 3131 (19 97)."    },    {        "title": "2401.13291v1.Investigation_of_the_hyperfine_coupling_constant_of_locally_noncentrosymmetric_heavy_fermion_superconductor_CeRh___2__As___2__.pdf",        "content": "arXiv:2401.13291v1  [cond-mat.supr-con]  24 Jan 2024Investigation of the hyperfine coupling constant of locally noncentrosymmetric\nheavy-fermion superconductor CeRh 2As2\nShiki Ogata∗and Shunsaku Kitagawa, Mayu Kibune, Kenji Ishida\nDepartment of Physics, Kyoto University, Kyoto 606-8502, J apan\nKatsuki Kinjo\nInstitute of Multidisciplinary Research for Advanced Mate rials,\nTohoku University, Sendai, Miyagi 980-8577, Japan\nManuel Brando, Christoph Geibel, Seunghyun Khim\nMax Planck Institute for Chemical Physics of Solids, D-0118 7 Dresden, Germany\nElena Hassinger\nMax Planck Institute for Chemical Physics of Solids, D-0118 7 Dresden, Germany and\nTechnical University Dresden, Institute for Solid State an d Materials Physics, 01062 Dresden, Germany\nWe performed75As-NMR measurements in H/bardblabto investigate the normal-state magnetic\nproperties of CeRh 2As2, a recently-discovered heavy-fermion superconductor. We compared the\nNMR Knight shift Kwith the magnetic susceptibility χab, and estimated the hyperfine coupling\nconstant Ahffrom the slope of the K−χplot. We observed that the magnitude of Ahf,abat the\nAs(1) site changes at around 20 K owing to emerging the heavy- fermion state, which was also\nobserved in Ahfat the As(2) site and in H/bardblc. The sign of Ahf,abat the As(1) site is negative\nin low temperature. These are important for the analysis of t he NMR results of CeRh 2As2in the\nsuperconducting state.\nPACS numbers: 74.70.Tx\nI. INTRODUCTION\nIn conventional superconductors, a superconducting\norder parameter can be classified as the multiplication\nof the states of spin sand momentum k, that is, even-\nparityspin-singlet andodd-parityspin-triplet states. Re-\ncently, superconductors with the additional degrees of\nfreedomshavebeen considered,suchasfrequency, atomic\norbital state and sublattice [1–3]. In such systems, the\nnovel superconducting states of odd-parity spin-singlet\nand even-parity spin-triplet states can be realized, be-\ncause the fermionic antisymmetry should be held even in\nan additional degrees of freedom.\nCeRh2As2is a recently discovered heavy-fermion su-\nperconductor, whose TSCis approximately 0.3 K [4].\nCeRh2As2has the crystal structure of the tetragonal\nCaBe2Ge2-type with space group P4/nmm(No.129,\nD7\n4h). CeRh 2As2has two crystallographically inequiv-\nalent As and Rh sites; As(1) [Rh(1)] is tetrahedrally co-\nordinated by Rh(2) [As(2)], as shown in Fig.1(a). The\nheavy-fermion superconductivity is characterized by a\nbroad maximum in resistivity at Tcoh∼40 K and a large\nspecific heat jump at TSC. In the magnetic field parallel\nto thec-axis, the superconductingcritical field Hc2shows\na clear kink at H∗∼4 T, and magnetic anomalies also\nappear at H∗inside the superconducting phase, which\n∗Electronic address: ogata.shiki.86c@st.kyoto-u.ac.jpsuggests the superconducting multiphase. The high-field\nsuperconducting phase has a very large upper critical\nfieldHc2∼14 T. For this superconducting multiphase,\nthe local inversion-symmetry breaking is considered to\nbe important. Theoretical study predicted that low-\nfield even-parity and high-field odd-parity superconduct-\ning phases would be realized in locally inversion-breaking\nsuperconductors[3].\nAlong with this superconducting multiphase, it was re-\nportedthatCeRh 2As2showsnon-magneticandmagnetic\nordersjustaboveandbelow TSC, respectively[4–10]. The\nspecific heat shows a relatively small kink at T0∼0.4 K\ntogetherwith largepeakat TSC, and the resistivityshows\nan upturn at T0[7, 8] without the magnetic anomaly\natT0.T0is suppressed in H/bardblc, while it increases in\nH/bardblabwith transition to another non-magnetic ordered\nstates [5]. Based on the field responses of T0[6, 7],T0\nis considered to be decoupled from the superconducting\nmultiphase.\nIn addition to T0, we have reported antiferromag-\nnetic order inside the superconducting state from nuclear\nquadrupole resonance (NQR) [9] and nuclear magnetic\nresonance(NMR)[10]measurements. Thisantiferromag-\nnetic order breaks the inversion symmetry. Remarkably,\nthe antiferromagnetic order disappears almost simulta-\nneously with H∗, which suggests the relation with the\nsuperconducting multiphase. Therefore, CeRh 2As2is\na promising compound to study how the breaking of\nlocal inversion symmetry induces or influences uncon-\nventional nonmagnetic, antiferromagnetic and supercon-\nducting states, as well as their interaction.2\nIt is important to investigate the spin state of the\nCooper pair for understanding unconventional supercon-\nductors. The Knight-shift measurements using NMR\nprovide critical information about the spin state. To es-\ntimate the spin susceptibility from the Knight shift mea-\nsurement, it is necessary to estimate the hyperfine cou-\npling constant Ahf. Previously, Ahfhas been estimated\nfor two As sites in H/bardblcand one As site [As(2) site] in\nH/bardblab. However, Ahffor the As(1) site in H/bardblabcould\nnot been estimated due to the limitations of the previous\nexperimental setup [11].\nIn this study, we report the75As-NMR results of the\nAs(1)sitein H/bardblab. WecomparedtheNMRKnightshift\nKwith the magnetic susceptibility χab, and estimated\nthe hyperfine coupling constant Ahffrom the slope of\ntheK−χplot. It was evaluated that Ahf,abat the As(1)\nsite is 0.24 T/ µBabove 20 K, and -0.22 T/ µBbelow 20\nK. The change in Ahfis attributed to the development of\nthe heavy-fermion state due to the c−fhybridization.\nNegative Ahfresults in an increase in the Knight shift\nwhenthespinsusceptibilitydecreasesbyformationofthe\nspin-singlet pairs in superconducting state. This result\nis important to evaluate the spin susceptibility in the\nsuperconducting state of H/bardblab.\na b\nAs(2)\nAs(1)As(2)\nRh(2)Rh(2)Rh(1)\nFIG.1: (a)CrystalstructureofCeRh 2As2. (b)NMRspectrum\nof the As(1) site in µ0Hab∼5.0 T. The dashed line indicate\nthe position of K= 0 when the contribution of the nuclear\nquadrupole interaction is taken into account.\nII. EXPERIMENTS\nSingle crystals of CeRh 2As2were grown using the bis-\nmuth flux method [4]. The bulk magnetic susceptibility\nwas measured using a commercial magnetometer with\na superconducting quantum interference device (Quan-\ntum Design, MPMS). Aconventionalspin-echotechnique\nwas used for the NMR measurements. The75As-NMR\nspectra (nuclear spin I= 3/2, nuclear gyromagnetic ra-\ntioγ/2π= 7.29 MHz/T, and natural abundance 100% )\nwere obtained as a function of frequency at fixed mag-\nnetic fields ( ∼5.0 T). We observed the As(1) site around\nf∼41.9 MHz at µ0H∼5.0 T, as shown in Fig. 1(b).The site assignment of the NMR peaks has been de-\nscribed in a previous paper [11]. To estimate the Knight\nshiftKi(i=candab), we computed the resonance fre-\nquency by diagonalizing the following nuclear Hamilto-\nnian;\nH=−/parenleftBigγ\n2π/parenrightBig\nh(1+Ki)I·H\n+hνQ\n6/bracketleftBig\n3I2\nz−I(I+1)+η\n2(I2\n++I2\n−)/bracketrightBig\n,(1)\nwhereh,νQ=3eQVzz\n2I(2I−1), andη=|Vyy−Vxx\nVzz|arethePlanck\nconstant, NQR frequency, and asymmetry parameter, re-\nspectively. ηis zero at each As site because of the 4-\nfold symmetry of the atomic position. The temperature\ndependence of νQwas determined from NQR measure-\nments. The magnetic field was calibrated using63Cu\n(63γn/2π= 11.289MHz/T) and65Cu (65γn/2π= 12.093\nMHz/T) NMR signals from the NMR coil.\nIII. RESULTS AND DISCUSSION\nFigure 2 shows the temperature dependence of the\nKnight shift Kabat the As(1) site (measured in this\nstudy) and the As(2) site (measured in previous study\n[11]). For comparison, the temperature dependence of\nthemagneticsusceptibility χabisalsoshown. TheKnight\nshift is estimated from the peak frequency of the Gaus-\nsianfittings. TheKnightshift isproportionaltothemag-\nnetic susceptibility as given by,\nKi=Ahf,iχspin,i+Korb,i, (2)\nwhereAhf,i,χspin,i, andKorb,iare the hyperfine cou-\npling constant, spin susceptibility, and orbital part of\nthe Knight shift in each direction, respectively. Korb,iis\ngenerally temperature independent. Above 20 K, Kabof\nboth As sites increases on cooling, while Kabof the As(1)\nsite shows broad maximum at 20 K. Such behaviors are\nattributed to the development of the heavy-fermion state\ndue to the c−fhybridization, which are often observed\nin Ce-based heavy-fermion systems [12, 13]. Indeed, the\ntemperature where the Kabshows the broad maximum is\nroughlymatch Tcoh∼40Kestimated fromthe resistivity\nmeasurement.\nThe hyperfine coupling constant Ahf,ican be deter-\nmined from the slope of the K−χplot, as shown in Fig.\n3. For both sites, the slopes of K−χplot are changed\nbetween the regions with small and large χAC. This be-\nhaviorwas observedalsoin H/bardblc. Curroet al.suggested\nthat the magnitude of Ahfis changed at T∗, below which\nheavy-fermion state starts to developed [14]. We ob-\ntainedAhf,abfrom fitting the slopes in each region. Table\nI summarizes the estimated hyperfine coupling constants\nAhfandKorbby this study and [11]. Ahf,cat the As(1)\nsite becomes larger on cooling, while Ahf,cat the As(2)\nsite and Ahf,abat both As sites become smaller on cool-\ning. Notably, Ahf,abatthe As(1) site becomesnegativein3\nTABLE I: Estimated hyperfine coupling constants AhfandKorb. Temperatures in parentheses indicate the fitted temperatu re\nrange. The data of Ref. [11] is also listed.\nSite Direction of HAhf(T/µB)\nin lowTAhf(T/µB)\nin highTKorb(%)\nin highT\nAs(1) H/bardblc1.55\n(2-100 K)0.46\n(125-200 K)0.57\n(125-200 K)\nH/bardblab\n(New)-0.22\n(4.2-20 K)0.24\n(20-50 K)0.91\n(20-50 K)\nAs(2) H/bardblc0.27\n(2-30 K)0.55\n(40-200 K)0.10\n(40-200 K)\nH/bardblab0.16\n(2-10 K)0.50\n(15-200 K)0.09\n(15-200 K)\n0 20 40 601.01.21.41.6Kab (%)\nT (K)H||ab\n75As-NMR\nμ0H~5.0 TAs(1)\nAs(2)\n0.0100.0120.0140.0160.018\nχab [emu/(Oe・mol)]\nFIG. 2: The temperature dependence of the Knight shift\ninab-plane field Kab. For comparison, the temperature de-\npendence of magnetic susceptibility χabis also depicted (the\ndashed line).\nFIG. 3:K−χplot forH/bardblab. The dashed lines indicate the\nresults of linear fitting.lowtemperatureregion. Assumingtheclassicaldipolein-\nteraction, Ahf,iwas calculated to be 0.02-0.06 T/ µB[11],\nwhich is of one order of magnitude smaller than the ex-\nperimentally estimated Ahf,i. Therefore, the anomalous\ntransferred hyperfine field is dominant in CeRh 2As2. It\nis noteworthy that the similar change in the sign of Ahf\nwas observed in Ahf,cat the Si site in CeCu 2Si2[12] and\ninAhf,bat the In(2) site in CeCoIn 5[13]. Although the\nobvious change in AhfatT∗was observed in most of\nCe-based heavy-fermion compounds, it seems that the\nmechanism and the estimation of the change have not\nbeen well understood.\nIV. CONCLUSION\nIn conclusion, we performed75As-NMR measurements\nto investigate the hyperfine coupling constant Ahf,abat\nthe As(1) site of CeRh 2As2, which was not measured in\npreviousstudy[11]. Ahf,abatthe As(1)sitechangedfrom\npositive (0.24 T/ µB) to negative (-0.22 T/ µB) at around\n20 K due to the formation of the heavy-fermion state;\nchanges in Ahfhave been observed at other sites and\ninH/bardblc. These results are essential for NMR studies of\nCeRh2As2inH/bardblab, which is necessary to determine the\nmagnetic structure of antiferromagnetic state, and the\ndecrease of the spin susceptibility in the superconducting\nstate.\nACKNOWLEDGEMENTS\nThis work was partially supported by the Kyoto Uni-\nversity LTM Center and Grants-in-Aid for Scientific\nResearch (KAKENHI) (Grants No. JP19K14657, No.\nJP19H04696, No. JP20KK0061, No. JP20H00130, No.\nJP21K18600, No. JP22H04933, No. JP22H01168, and\nNo.JP23H01124). C. G. and E. H. acknowledge sup-\nport from the DFG program Fermi-NESt through Grant\nNo. GE 602/4-1. Additionally, E. H. acknowledges\nfunding by the DFG through CRC1143 (Project No.\n247310070) and the W¨ urzburg-Dresden Cluster of Ex-\ncellence on Complexity and Topology in Quantum Mat-\nter—ct.qmat (EXC 2147, Project ID 390858490). S. O.4\nwould like to acknowledgethe support from the Motizuki Fund of Yuk awa Memorial Foundation.\n[1] J. Linder and A. V. Balatsky, Reviews of Modern Physics\n91, 045005 (2019).\n[2] H. G. Suh, H. Menke, P. Brydon, C. Timm, A. Ramires,\nandD.F. Agterberg,PhysicalReviewResearch 2,032023\n(2020).\n[3] T. Yoshida, M. Sigrist, and Y. Yanase, Physical Review\nB86, 134514 (2012).\n[4] S. Khim, J. F. Landaeta, J. Banda, N. Bannor,\nM. Brando, et al., Science 373, 1012 (2021).\n[5] D. Hafner, P. Khanenko, E.-O. Eljaouhari, R. K¨ uchler,\nJ. Banda, et al., Physical Review X 12, 011023 (2022).\n[6] J. Landaeta, P. Khanenko, D. Cavanagh, C. Geibel,\nS. Khim, et al., Physical Review X 12, 031001 (2022).\n[7] K. Semeniuk, D. Hafner, P. Khanenko, T. L¨ uhmann,\nJ. Banda, et al., Physical Review B 107, L220504 (2023).\n[8] S. Mishra, Y. Liu, E. D. Bauer, F. Ronning, and S. M.Thomas, Physical Review B 106, L140502 (2022).\n[9] M. Kibune, S.Kitagawa, K.Kinjo, S.Ogata, M. Manago,\net al., Physical review letters 128, 057002 (2022).\n[10] S. Ogata, S. Kitagawa, K. Kinjo, K. Ishida, M. Brando,\net al., Physical Review Letters 130, 166001 (2023).\n[11] S. Kitagawa, M. Kibune, K. Kinjo, M. Manago,\nT. Taniguchi, et al., journal of the physical society of\njapan91, 043702 (2022).\n[12] T. Ohama, H. Yasuoka, D. Mandrus, Z. Fisk, and\nJ. Smith, Journal of the Physical Society of Japan 64,\n2628 (1995).\n[13] N. Curro, B. Simovic, P. Hammel, P. Pagliuso, J. Sarrao,\net al., Physical Review B 64, 180514 (2001).\n[14] N. Curro, B.-L. Young, J. Schmalian, and D. Pines,\nPhysical Review B 70, 235117 (2004)."    },    {        "title": "2401.13348v1.Eigenmode_analysis_of_the_damped_Jaynes_Cummings_model.pdf",        "content": "Eigenmode analysis of the damped Jaynes-Cummings model\nL.G. Suttorp∗\nInstitute of Physics, University of Amsterdam,\nScience Park 904, NL-1098 XH Amsterdam, The Netherlands\nJanuary 25, 2024\nAbstract\nThe generating functions for density matrix elements of the Jaynes-Cummings model with\ncavity damping are analysed in terms of their eigenmodes, which are characterised by a specific\ntemporal behaviour. These eigenmodes are shown to be proportional to particular generalised\nhypergeometric functions. The relative weights of these eigenmodes in the generating functions are\ndetermined by the initial conditions of the model. These weights are found by deriving orthogonality\nrelations involving adjoint modes. In an example it is shown how the time-dependent density matrix\nelements and the related factorial moments can be extracted from the eigenmode decompositions\nof the generating functions.\n1 Introduction\nThe model introduced by Jaynes and Cummings [1] in 1963 continues to draw attention, as is illustrated\nby the publication of a collection of papers on the occasion of its 50th anniversary [2]. The original\nmodel, which describes the interaction of a two-state atom with photons in a cavity mode, has been\nextended in several ways. In particular, interesting phenomena show up when damping effects by\nthe escape of photons from the cavity are included. To incorporate these cavity damping effects a\nmaster equation approach has frequently been employed. It follows by supplementing the equation\ngoverning the time dependence of the density operator with Lindblad terms [3]-[4]. Other types of\nmaster equations for the damped Jaynes-Cummings model have been studied recently as well [5]-[8].\nVarious techniques have been employed to solve the master equation for the damped Jaynes-\nCummings model in the Lindblad form. Solutions have been obtained by using quasi-probability\ndistributions [9]-[12] or damping bases [13], or by starting from the coupled equations for the density\noperator matrix elements [14]. All of these methods lead to rather complicated expressions for the\ntime-dependent matrix elements of the density operator. Simpler results have been derived by making\nvarious assumptions about the relative magnitude of the parameters in the model and the initial form\nof the density operator [15]-[17].\nIn [16] an attempt has been made to simplify matters by making use of suitable special functions.\nIt is stated in that paper that the results for the density operator of the Jaynes-Cummings model with\ndamping can not be fitted to its initial value in a rigorous way, the reason being that no orthogonality\nrelations are said to be available for the relevant special functions. In the following, however, we\nshall obtain the eigenmodes of the generating functions for the density operator matrix elements of\nthe damped Jaynes-Cummings model in terms of generalised hypergeometric functions and derive\nsuitable orthogonality relations for these functions. In this way it will be demonstrated that the full\ntime dependence of the density operator can be derived, with an exact fitting to the initial conditions.\n∗email: l.g.suttorp2@contact.uva.nl\n1arXiv:2401.13348v1  [quant-ph]  24 Jan 20242 Jaynes-Cummings model with cavity damping\nThe Lindblad master equation that governs the time evolution of the density operator ρfor the\nJaynes-Cummings model at resonance and with cavity damping reads:\n∂ρ\n∂t=−i[H, ρ] +κ(2aρa†−a†aρ−ρa†a) (1)\nwith aanda†the annihilation and creation operators of the field mode with frequency ω0>0, and\nκ >0 the damping rate. The Hamiltonian His:\nH=1\n2ω0(|e⟩⟨e| − |g⟩⟨g|) +ω0a†a+f(a|e⟩⟨g|+a†|g⟩⟨e|) (2)\nwith|g⟩and|e⟩denoting the atomic ground and excited states, respectively, and f >0 the coupling\nconstant. From the master equation one may derive coupled differential equations for the matrix\nelements of ρon the basis of the states |g, n⟩and|e, n⟩of atom and field, with nthe number of\nphotons in the field mode. For arbitrary nand fixed values of m−nthe equations couple the time\nevolution of the matrix elements ⟨g, n|ρ|g, m⟩,⟨g, n+ 1|ρ|e, m⟩,⟨e, n|ρ|g, m+ 1⟩and⟨e, n|ρ|e, m⟩. In\nthe following we will concentrate on the case m=n.\nUpon introducing the abbreviations gn(τ) =⟨g, n|ρ(τ)|g, n⟩,en(τ) =⟨e, n|ρ(τ)|e, n⟩,fn(τ) =\n2√n+ 1 Re[ ⟨e, n|ρ(τ)|g, n+ 1⟩] and hn(τ) = 2√n+ 1 Im[ ⟨e, n|ρ(τ)|g, n+ 1⟩] one arrives at a set of\ndifferential equations for en(τ),fn(τ),gn(τ), and hn(τ). It turns out that these equations simplify by\nintroducing instead of gnthe combinations dn=gn+en−1forn >0, and d0=g0. With the scaled\nvariables τ=κtandα=f/κwe get for n≥0:\nd\ndτdn= 2(n+ 1)dn+1−2ndn−2en+ 2en−1, (3)\nd\ndτen= 2(n+ 1)en+1−2nen−αhn, (4)\nd\ndτfn= 2(n+ 1)fn+1−(2n+ 1)fn, (5)\nd\ndτhn= 2(n+ 1)hn+1−(2n+ 1)hn−2α(n+ 1)dn+1+ 4α(n+ 1)en. (6)\nIn the first equation the last term should be omitted for n= 0. The first and the third equation do\nnot contain the coupling constant α. Furthermore, the equation for fnis decoupled from those for dn,\nenandhn.\nTo solve the coupled differential equations (3)-(6) we introduce the generating functions D(z, τ) =P∞\nn=0zndn(τ) and similarly E(z, τ),F(z, τ) and H(z, τ). The time evolution of these functions is\ndetermined by a set of partial differential equations that follow from (3)–(6) as\n∂D\n∂τ= 2(1 −z)∂D\n∂z−2(1−z)E , (7)\n∂E\n∂τ= 2(1 −z)∂E\n∂z−αH , (8)\n∂F\n∂τ= 2(1 −z)∂F\n∂z−F , (9)\n∂H\n∂τ= 2(1 −z)∂H\n∂z−H−2α∂D\n∂z+ 4α∂(zE)\n∂z. (10)\nThe function D(z, τ) is determined up to an additive constant, since only its derivatives appear in the\nequations.\nIn the following the differential equations (7)-(10) will be solved in terms of eigenmodes. It should\nbe noted that the equation (9) for F(z, τ) is decoupled from those for the other three functions.\nAlthough it can easily be solved directly, it will be analysed in terms of eigenmodes as well, so as to\npreserve the analogy in the treatment of the four equations.\n23 Eigenmode solutions\nThe eigenmode solution of equation (9) wil be discussed first. The form of the equations (7)-(10)\nsuggests a change of variable from ztou= 1−z, so that (9) becomes an equation for ¯F(u, τ) =\nF(1−u, τ). An equation for its eigenmodes is obtained by writing F(u, τ) =Fλ(u)eλτ, where we\nsuppressed the bar above Fagain. The resulting eigenmode equation gets the form:\n−2udFλ(u)\ndu−Fλ(u) =λFλ(u). (11)\nWhen the generating function F(u, τ) is taken to be regular in the interval 0 ≤u≤1, one may expand\nFλ(u) asP∞\nn=0cnunnear u= 0. Substituting this form in (11) one finds ( λ+ 2n+ 1)cn= 0 for all\nn≥0, so that a non-trivial solution Fλ(u) =ukis obtained for λ=−2k−1, with non-negative integer\nk. Upon choosing kas a new label, the set of solutions is found as\nFk(u) =uk(12)\nwith non-negative integer k. The generating function F(u, τ) gets the form\nF(u, τ) =∞X\nk=0AkFk(u)e−(2k+1)τ(13)\nin terms of its eigenmodes. The coefficients Akcan be found from the initial conditions, as will be\nshown in the following section.\nNext, the eigenmode equations that follow from the coupled partial differential equations for\nD(u, τ),E(u, τ) and H(u, τ) will be considered. Upon changing variables from ztouin (7), (8)\nand (10), and assuming an exponential time dependence as before, one gets:\n−2udDλ(u)\ndu−2uEλ(u) =λDλ(u), (14)\n−2udEλ(u)\ndu−αHλ(u) =λEλ(u), (15)\n−2udHλ(u)\ndu−Hλ(u) + 2αdDλ(u)\ndu−4α(1−u)dEλ\ndu+ 4αEλ(u) =λHλ(u). (16)\nAfter eliminating Dλ(u) and Hλ(u), one arrives at a third-order differential equation for Eλ(u):\n8u3d3Eλ(u)\ndu3+ 4u[u(3λ+ 2a+ 9)−2a]d2Eλ(u)\ndu2+ 2\u0002\nu(3λ2+ 2aλ+ 12λ+ 10a+ 12)\n−2aλ−4a]dEλ(u)\ndu+ (λ3+ 3λ2+ 4aλ+ 2λ+ 4a)Eλ(u) = 0 (17)\nwith a=α2. Insertion of a series of the form Eλ(u) =P∞\nn=0cnunleads to a recursion relation for the\ncoefficients:\n(n+ 1)( n+1\n2λ+ 1)a cn+1= (n+1\n2λ+1\n2)(n+1\n2λ+1\n2a+1\n2wλ+1\n2)(n+1\n2λ+1\n2a−1\n2wλ+1\n2)cn(18)\nwith the abbreviation wλ=p\n(a−1)2+ 2aλ. For general values of λthe series representing Eλ(u)\ndiverges near u= 0. A convergent result is found only if the series terminates after a finite number of\nterms. This may happen in several different ways: either one has1\n2λ+1\n2=−kor1\n2λ+1\n2a∓1\n2wλ+1\n2=\n−k, with non-negative integer k.\nIn the first case, for λ=−2k−1, one has wλ=p\na2−4a(k+ 1) + 1 ≡wk. The solution for Eλ(u)\nis proportional to a terminating generalised hypergeometric function 3F1:\n3F1(−k,−k+1\n2a+1\n2wk,−k+1\n2a−1\n2wk;−k+1\n2;u\na). (19)\nBy inverting the order of the terms in the finite series one may write the functions E0,k(u) of this first\nset of eigenmodes in terms of terminating generalised hypergeometric functions 2F2:\nE0,k(u) =\u0010u\na\u0011k\n2F2(−k,1\n2;−1\n2a+ 1 +1\n2wk,−1\n2a+ 1−1\n2wk;−a\nu). (20)\n3The functions D0,k(u) and H0,k(u) of these eigenmodes are obtained from (14)–(16) as\nD0,k(u) =−2a\u0010u\na\u0011k+1\n2F2(−k,−1\n2;−1\n2a+ 1 +1\n2wk,−1\n2a+ 1−1\n2wk;−a\nu), (21)\nH0,k(u) =1√a\u0010u\na\u0011k\n2F2(−k,3\n2;−1\n2a+ 1 +1\n2wk,−1\n2a+ 1−1\n2wk;−a\nu). (22)\nIn the other two cases, with1\n2λ+1\n2a∓1\n2wλ+1\n2=−k,the eigenvalues get the form λ=−2k−1±¯wk\nwith ¯ wk≡p\n1−4a(k+ 1). Again, the solution for Eλ(u) is proportional to a terminating generalised\nhypergeometric function:\n3F1(−k,−k±1\n2¯wk,−k+a±1\n2¯wk;−k+1\n2±1\n2¯wk;u\na). (23)\nAs before, the order of the terms can be inverted, with the result\nE±,k(u) =\u0010u\na\u0011k\n2F2(−k,1\n2∓1\n2¯wk; 1∓1\n2¯wk,1−a∓¯wk;−a\nu). (24)\nThe associated functions D±,k(u) and H±,k(u) are\nD±,k(u) =−2a\n1±¯wk\u0010u\na\u0011k+1\n2F2(−k,−1\n2∓1\n2¯wk; 1∓1\n2¯wk,1−a∓¯wk;−a\nu), (25)\nH±,k(u) =1√a(1∓¯wk)\u0010u\na\u0011k\n2F2(−k,3\n2∓1\n2¯wk; 1∓1\n2¯wk,1−a∓¯wk;−a\nu). (26)\nThe generating function E(u, τ) can be expanded in terms of the eigenmodes (20) and (24):\nE(u, τ) =X\ns=0,±∞X\nk=0As,kEs,k(u)e(−2k−1+s¯wk)τ(27)\nwith coefficients As,kthat follow from the initial conditions. The generating functions D(u, τ) and\nH(u, τ) get analogous forms:\nD(u, τ) =X\ns=0,±∞X\nk=0As,kDs,k(u)e(−2k−1+s¯wk)τ+ 1, (28)\nH(u, τ) =X\ns=0,±∞X\nk=0As,kHs,k(u)e(−2k−1+s¯wk)τ. (29)\nThe generating function G(u, τ) follows from E(u, τ) and D(u, τ) as\nG(u, τ) =D(u, τ)−(1−u)E(u, τ) (30)\nsince gnequals dn−en−1forn >0 and d0=g0. For u= 0 the relation (30) implies G(0, τ)+E(0, τ) =\nD(0, τ). Because the normalisation of the density operator impliesP∞\nn=0[gn(τ) +en(τ)] = 1 for all τ,\nthe function D(u, τ) should equal 1 for u= 0. For that reason a constant term has been added to the\nright-hand side of (28).\nThe function E(u, τ) depending on uis the generating function of the factorial moments ¯ emasso-\nciated to en. In fact, from the relation u= 1−zit follows that the definition E(z, τ) =P∞\nn=0znen(τ)\nleads to the expansion\nE(u, τ) =∞X\nm=0(−1)m\nm!um¯em(τ) (31)\nwith the factorial moments defined as\n¯em(τ) =∞X\nn=mn!\n(n−m)!en(τ). (32)\n4The lowest-order factorial moment is ¯ e0(τ) =P∞\nn=0en(τ). It is obtained from E(u, τ) by putting\nuequal to 0. The functions Es,k(u) in (27) are polynomials in u, which are finite at u= 0. The\nexpansion of G(u, τ) in factorial moments ¯ gm(τ) is similar to (31) with (32). The normalisation of the\ndensity operator can be written in terms of the factorial moments as ¯ g0(τ) + ¯e0(τ) = 1 for all τ.\nThe expressions (20-(22) and (24)-(26) for the eigenmodes can easily be rewritten as polynomials\ninz= 1−u. One finds for instance from (20) with the help of the binomial theorem:\nE0,k(z) =k!\nakkX\nn=0(−1)n\nn!(k−n)!2F2(−k+n,1\n2;−1\n2a+ 1 +1\n2wk,−1\n2a+ 1−1\n2wk;−a)zn. (33)\nAs the dependence on zis now made explicit the contribution of the eigenmode with label 0 , kto the\ndensity matrix element enfollows directly.\nThe results (27)-(29) with (20)-(22) and (24)-(26) give the complete eigenmode expansions of the\ngenerating functions in terms of generalised hypergeometric functions. They contain coefficients As,k,\nwhich still have to be determined.\n4 Adjoint modes\nThe relative weights As,kin the eigenmode expansions may be obtained from the initial conditions at\nτ= 0. Their form can be found by considering the adjoint differential equations and their eigenmodes.\nThe solutions (20)–(22) and (24)–(26) suggest that it is convenient to choose the new independent\nvariable x=a/uinstead of u. For analogy, we shall use the variable xto determine the adjoint\neigenmodes associated to (12) as well, although its simple form does not point in that direction.\nIn terms of xthe eigenmode equation (11) reads:\n2xdFλ(x)\ndx−Fλ(x) =λFλ(x). (34)\nIts adjoint equation is\n−2xdˆFλ(x)\ndx−3ˆFλ(x) =λˆFλ(x) (35)\nwith the solution ˆFλ(x) =x−(λ+3)/2up to a constant factor. Upon choosing the same eigenvalue\nspectrum as in (12) by writing λ=−2m−1 with non-negative integer m, we write the adjoint\neigenmodes as ˆFm(x) =cmxm−1.\nThe eigenmodes Fk(x) and their adjoints ˆFm(x) satisfy an orthogonality relation involving a con-\ntour integral in the complex x-plane around x= 0:\n1\n2πiI\ndxˆFm(x)Fk(x) =δm,k (36)\nifcmis chosen to be equal to a−m. This identity may be proven directly by substituting the explicit\nexpressions for ˆFm(x) and Fk(x). A formal proof for m̸=kstarts by evaluating the integralH\ndxˆFm(x)\n[2xdFk(x)/dx−Fk(x)] in two ways, either by employing (34) or by using (35) after an integration by\nparts. Equating the two results one arrives at (36) for m̸=k. The orthogonality relation (36) can be\nemployed to determine the coefficients Akin (13) as:\nAk=1\n2πiI\ndxˆFk(x)F(x,0). (37)\nAfter this rather simple case we now turn to the coupled set of equations (7), (8) and (10). They\nhave led to the eigenmode equations (14)–(16). Rewriting these equations in terms of xwe get\n2xdDλ(x)\ndx−2a\nxEλ(x) =λDλ(x), (38)\n2xdEλ(x)\ndx−√aHλ(x) =λEλ(x), (39)\n2xdHλ(x)\ndx−Hλ(x)−2√ax2dDλ(x)\ndx+4√ax(x−a)dEλ(x)\ndx+ 4√aEλ(x) =λHλ(x).(40)\n5The adjoint differential equations are\n−2xdˆDλ(x)\ndx−2ˆDλ(x) +2√ax2dˆHλ(x)\ndx+4√axˆHλ(x) =λˆDλ(x), (41)\n−2xdˆEλ(x)\ndx−2ˆEλ(x)−2a\nxˆDλ(x)−4√ax(x−a)dˆHλ(x)\ndx−8√a(x−a)ˆHλ(x) =λˆEλ(x),(42)\n−2xdˆHλ(x)\ndx−3ˆHλ(x)−√aˆEλ(x) =λˆHλ(x). (43)\nElimination of ˆDλand ˆHλyields the third-order differential equation\n8x3d3ˆEλ(x)\ndx3+ 4x2(−2x+ 3λ+ 2a+ 13)d2ˆEλ(x)\ndx2+ 2x\u0002\n−2(λ+ 12) x+ 3λ2+ 2aλ+ 20λ\n+14a+ 32]dˆEλ(x)\ndx+\u0002\n−8(λ+ 6)x+λ3+ 7λ2+ 8aλ+ 14λ+ 8a+ 8\u0003ˆEλ(x) = 0 . (44)\nFor arbitrary values of λthree independent solutions are\nˆE0,λ(x) =x−(λ+1)/22F2(−1\n2λ+3\n2,5\n2;1\n2a+ 2 +1\n2wλ,1\n2a+ 2−1\n2wλ;x), (45)\nˆE±,λ(x) =x−(λ+a+3∓wλ)/22F2(−1\n2a+3\n2±1\n2wλ,−1\n2a−1\n2λ+1\n2±1\n2wλ;−1\n2a±1\n2wλ,1±wλ;x) (46)\nwith wλ=p\n(a−1)2+ 2aλas before.\nFrom the solutions for general λa suitable set of adjoint eigenmodes will be obtained by imposing\nthe condition that the spectrum is the same as that found for the eigenmodes in the previous section.\nHence, one should take either λ=−2m−1 orλ=−2m−1±¯wm, with non-negative integer m. Upon\nchoosing solutions that are either analytic or having a simple pole at x= 0 we get from (45) with\nλ=−2m−1:\nˆE0,m(x) =c0,mxm2F2(m+ 2,5\n2;1\n2a+ 2 +1\n2wm,1\n2a+ 2−1\n2wm;x) (47)\nwith wm=p\na2−4a(m+ 1) + 1 and with an as yet arbitrary constant c0,m. From (41)-(43) the\nassociated functions ˆD0,mand ˆH0,mare obtained as\nˆD0,m(x) =c′\n0,mxm2F2(m+ 2,1\n2;1\n2a+ 1 +1\n2wm,1\n2a+ 1−1\n2wm;x), (48)\nˆH0,m(x) =c′′\n0,mxm−12F2(m+ 1,3\n2;1\n2a+ 1 +1\n2wm,1\n2a+ 1−1\n2wm;x) (49)\nwith the coefficients c′\n0,m=−1\n6(4am+ 8a+ 3)c0,mandc′′\n0,m=1\n2[√a/(m+ 1)]c′\n0,m.\nLikewise, we find from (46) for λ=−2m−1±¯wm, with ¯ wm=p\n1−4a(m+ 1):\nˆE±,m(x) =c±,mxm−12F2(m+ 1,3\n2±1\n2¯wm;±1\n2¯wm,1 +a±¯wm;x). (50)\nThe associated functions ˆD±,mand ˆH±,mare:\nˆD±,m(x) =c′\n±,mxm2F2(m+ 2,1\n2±1\n2¯wm; 1±1\n2¯wm,1 +a±¯wm;x), (51)\nˆH±,m(x) =c′′\n±,mxm−12F2(m+ 1,3\n2±1\n2¯wm; 1±1\n2¯wm,1 +a±¯wm;x) (52)\nwith c′\n±,m=∓[(1∓¯wm)/(2a¯wm)]c±,mandc′′\n±,m= [2a3/2/(1∓¯wm)]c′\n±,m.\nThe eigenmodes (20)–(22), (24)–(26) and their adjoints (47)–(52) satisfy orthogonality relations of\nthe form:\n1\n2πiI\ndxh\nˆEr,m(x)Es,k(x) +ˆDr,m(x)Ds,k(x) +ˆHr,m(x)Hs,k(x)i\n=δr,sδk,m (53)\nforr= 0,±ands= 0,±and for all non-negative integers k, m. The contour integral in the complex\nx-plane encircles the origin x= 0. The normalisation constants of the adjoint modes have to be\nchosen as c′\n0,m= 2(m+ 1)/[1−4a(m+ 1)] and c′\n±,m=−(m+ 1)/[1−4a(m+ 1)]. The proof of the\northogonality relations for r̸=sand/or k̸=mfollows by multiplying the left-hand side of (53) by\n6the factor −2k−1 +s¯wk, using (38)–(40), integrating by parts and employing (41)–(43), with a result\nthat is again proportional to the left-hand side of (53), with a different factor −2m−1 +r¯wm, so\nthat the integral must vanish. For the diagonal case r=sandk=mthe relation may be verified\nby inserting the explicit forms of the eigenmodes and their adjoints. The result of the integration is\nfound from the leading terms of the generalised hypergeometric functions.\nOnce the orthogonality relations have been established we may use them to find the coefficients\nAs,kin the expansions (27)–(29). In fact, upon changing variables from utox=a/u, putting τ= 0,\nmultiplying each of these expansions by the corresponding expression of an adjoint mode (with fixed\nparameters randm), summing the results and integrating over xone gets from (53):\nAr,m=1\n2πiI\ndxh\nˆEr,m(x)E(x,0) + ˆDr,m(x)D(x,0) + ˆHr,m(x)H(x,0)i\n. (54)\nSince the functions ˆDr,m(x) are analytic in x= 0 the last term in (28) does not contribute to Ar,m.\n5 Example\nAs an example of the use of eigenmodes in analysing the behaviour of the damped Jaynes-Cummings\nmodel a special case will be considered. It follows by assuming that at τ= 0 the atom is in its ground\nstate, with n0photons present. Hence, the initial value of dnis given by dn(0) = δn,n0. Furthermore,\nen(0) and hn(0) vanish for all n. The generating function D(x,0) gets the form\nD(x,0) =n0X\np=0(−1)pn0!\np!(n0−p)!\u0010a\nx\u0011p\n(55)\nwhile E(x,0) en H(x,0) both vanish. The coefficient As,kas given by (54) becomes:\nAs,k=n0X\np=0(−1)pn0!\np!(n0−p)!1\n2πiI\ndx\u0010a\nx\u0011pˆDs,k(x). (56)\nFors= 0 one gets after substituting (48) and performing the contour integral around the origin:\nA0,k=c′\n0,k(−a)k+1n0!\n(k+ 1)!( n0−k−1)!2F2(−n0+k+ 1,1\n2;1\n2a+ 1 +1\n2wk,1\n2a+ 1−1\n2wk;a) (57)\nfor 0≤k≤n0−1. Likewise, one obtains for s=±and 0 ≤k≤n0−1:\nA±,k=c′\n±,k(−a)k+1n0!\n(k+ 1)!( n0−k−1)!2F2(−n0+k+ 1,1\n2±1\n2¯wk; 1±1\n2¯wk,1 +a±¯wk;a). (58)\nThe normalisation constants c′\n0,kandc′\n±,khave been defined below (53).\nThe generating function E(u, τ) of the factorial moments ¯ em(τ) follows by insertion of (20), (24)\nand (57)-(58) in (27). The resulting expression is a sum over k(with 0 ≤k≤n0−1) of products of\ntwo terminating generalised hypergeometric functions (one with the argument aand the other with\nthe argument −a/uand a pre-factor uk), and a time-dependent exponential factor.\nAs an illustration, the characteristic wavelike behaviour of the generating function E(u, τ) is shown\nin Figure 1 for n0= 6 and a= 5.\nThe lowest-order factorial moment ¯ e0(τ) is obtained from E(u, τ) by setting uequal to 0. Its value,\nas given in Figure 2, determines the probability of the atom being in its excited state for any number\nof photons in the cavity. Starting from 0 at τ= 0 it returns to that value in the course of time. On\nthe other hand, the lowest-order density matrix element e0(τ) follows from E(u, τ) by taking u= 1,\norz= 0. As shown in Figure 3, it gives the probability of finding the atom at time τin its excited\nstate and no photons present. Clearly e0(τ) is less than (or equal to) ¯ e0(τ) for all τ.\nNext, we turn to the generating function G(u, τ) of the density matrix elements gn(τ). It can be\nfound by considering a suitable combination of E(u, τ) and D(u, τ), as given by (30). The function\nD(u, τ) may be obtained by substitution of (21), (25) and (57)-(58) in (28). Combining E(u, τ) and\n7Figure 1: The generating function E(u, τ) as a function of u(for 0 ≤u≤1) and τ(for 0 ≤τ≤3),\nforn0= 6 and a= 5.\n¯e0\n0.5 1.0 1.5 2.0 2.5 3.00.20.40.60.81.0\nτ\n1\nFigure 2: The factorial moment ¯ e0(τ) as a function of τ(for 0 ≤τ≤3), for n0= 6 and a= 5.\nD(u, τ), as in (30), we get an expression for G(u, τ). Its behaviour is shown in Figure 4 for the same\nvalues of n0andaas above.\nAgain, one may consider the lowest-order factorial moment ¯ g0(τ) (see Figure 5). It gives the prob-\nability of the atom being in its ground state regardless of the number of photons present. Comparison\nof the Figures 2 and 5 shows that the two lowest-order factorial moments ¯ e0(τ) and ¯ g0(τ) add up to\n1 for all τ, as expected. Finally, the time behaviour of the lowest-order density matrix element g0(τ)\nis shown in Figure 6. It is rising from its initial value 0 to (nearly) its final value 1 in the time span\ne0\n0.5 1.0 1.5 2.0 2.5 3.00.10.20.3\nτ\n1\nFigure 3: The density matrix element e0(τ) as a function of τ(for 0 ≤τ≤3), for n0= 6 and a= 5.\n8Figure 4: The generating function G(u, τ) as a function of u(for 0 ≤u≤1) and τ(for 0 ≤τ≤3),\nforn0= 6 and a= 5.\n¯g0\n0.5 1.0 1.5 2.0 2.5 3.00.20.40.60.81.0\nτ\n1\nFigure 5: The factorial moment ¯ g0(τ) as a function of τ(for 0 ≤τ≤3), for n0= 6 and a= 5.\nconsidered here.\nThe expressions for the generating functions that follow by inserting the coefficients (57)-(58) in\n(27)-(29) are valid for arbitrary values of n0anda. In the special case a≫n0the results for en(τ),\ngn(τ) and hn(τ) agree with those given in [7].\ng0\n0.5 1.0 1.5 2.0 2.5 3.00.20.40.60.8\nτ\n1\nFigure 6: The density matrix element g0(τ) as a function of τ(for 0 ≤τ≤3), for n0= 6 and a= 5.\n96 Final remarks\nIn conclusion, it has been demonstrated how the generating functions for density matrix elements of the\nJaynes-Cummings model with cavity damping may be written as sums over eigenmodes with a fixed\ntime dependence. The results (13) and (27)-(29) contain coefficients (37) and (54) that are adjusted\nto the initial conditions by means of the orthogonality relations (36) and (53). The eigenmodes have\nbeen written in terms of generalised hypergeometric functions.\nThe above analysis has been limited to a study of the generating functions for a suitable subset\nof the density matrix elements, namely those with m=n, as discussed below (2). This is allowed as\nthe complete collection of density matrix elements falls apart in decoupled subsets, each with its own\nfixed value of m−n. For values m−n̸= 0 a similar analysis can be performed, although the details\nare somewhat more complicated. In fact, one has to solve a set of four coupled equations instead of\nthe single equation (9) and the three coupled ones given in (7), (8) and (10).\nReferences\n[1] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51, 89 (1963)\n[2] A.D. Greentree, J. Koch, J. Larson (eds.), J. Phys. B 46, 220201 (2013)\n[3] G. Lindblad, Commun. Math. Phys. 48, 119 (1976)\n[4] V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976)\n[5] M. Scala, B. Militello, A. Messina, J. Piilo, S. Maniscalco, Phys. Rev. A 75, 013811 (2007)\n[6] C.A. Gonz´ alez-Guti´ errez, D. Solis-Valles, B.M. Rodriguez-Lara, J. Phys. A 51, 015301 (2018)\n[7] S. Ozaki, H. Nakazato, Phys. Rev. A 103, 053713 (2021)\n[8] A. Watanabe, H. Nakazato, Ann. Phys. 441, 168890 (2022)\n[9] J. Eiselt, H. Risken, Opt. Commun. 72, 351 (1989)\n[10] J. Eiselt, H. Risken, Phys. Rev. A 43, 346 (1991)\n[11] B. Daeubler, H. Risken, L. Schoendorff, Phys. Rev. A 46, 1654 (1992)\n[12] B. Daeubler, H. Risken, L. Schoendorff, Phys. Rev. A 48, 3955 (1993)\n[13] H.-J. Briegel, B.-G. Englert, Phys. Rev. A 47, 3311 (1993)\n[14] A.J. van Wonderen, Phys. Rev. A 56, 3116 (1997)\n[15] J. Gea-Banacloche, Phys. Rev. A 47, 2221 (1993)\n[16] A. Lindner, H. Freese, G. Quehl, D. Reiß, K. Schiller, Eur. Phys. J. D 17, 99 (2001)\n[17] S.M. Barnett, J. Jeffers, J. Mod. Optics 54, 2033 (2007)\n10"    },    {        "title": "2401.13361v1.A_note_on_the_numerical_approximation_of_Greeks_for_American_style_options.pdf",        "content": "arXiv:2401.13361v1  [math.NA]  24 Jan 2024A note on the numerical approximation\nof Greeks for American-style options\nKarel J. in ’t Hout∗\nJanuary 25, 2024\nAbstract\nIn this note we consider the approximation of the Greeks Delt a and Gamma of American-\nstyleoptions throughthe numerical solution of time-depen dentpartial differential complemen-\ntarity problems (PDCPs). This approach is very attractive a s it can yield accurate approxi-\nmations to these Greeks at essentially no additional comput ational cost during the numerical\nsolution of the PDCP for the pertinent option value function . For the temporal discretization,\nthe Crank–Nicolson method is arguably the most popular meth od in computational finance.\nIt is well-known, however, that this method can have an undes irable convergence behaviour\nin the approximation of the Greeks Delta and Gamma for Americ an-style options, even when\nbackward Euler damping (Rannacher smoothing) is employed.\nIn this note we study for the temporal discretization an inte resting family of diagonally\nimplicit Runge–Kutta (DIRK) methods together with the two- stage Lobatto IIIC method.\nThrough ample numerical experiments for one- and two-asset American-style options, it is\nshown that these methods can yield a regular second-order co nvergence behaviour for the\noption value as well as for the Greeks Delta and Gamma. A mutua l comparison reveals that\nthe DIRK method with suitably chosen parameter θis preferable.\nKeywords: American option, Greeks, partial differential complementa rity problem, Runge–\nKutta methods, convergence\n1 Introduction\nAmerican-style options, which can be exercised at any given single tim e up to and including\nmaturity, are widely traded derivatives in the financial markets. It is thus of major importance\nto know their fair values as well as their Greeks, in particular Delta an d Gamma. Expressions in\n(semi-)closed analytical form for these quantities are in general n ot available in the literature and\none therefore relies on methods for their approximation.\nThe present note deals with the approximation of American option va lues and their Greeks\nDelta and Gamma via the numerical solution of the pertinent time-dep endent partial differential\ncomplementarity problem (PDCP). This well-known approach is highly a ttractive. It can generate\naccurate approximations to these Greeks at essentially no addition al computational cost during\nthe numerical solution of the PDCP for the American option value fun ction.\nFor the numerical solution of PDCPs, the method of lines is ubiquitous in the computational\nfinance literature. Here one first discretizes in the spatial variable s by finite differences and subse-\nquently discretizes in the temporal variable by means of a suitable imp licit time stepping method.\nFor the temporal discretization, the Crank–Nicolson scheme is arg uably the most popular method\nin computational finance. When applied with backward Euler damping, also called Rannacher\nsmoothing, it yields a desired, second-order convergence behavio ur for many types of European-\nstyle options and their Greeks under many different asset price mod els. It is well-known, however,\n∗Department of Mathematics, University of Antwerp, Middelh eimlaan 1, B-2020 Antwerp, Belgium. Email:\nkarel.inthout@uantwerpen.be .\n1that in the case of American-style options the Crank–Nicolson sche me can lead to less satisfactory\nresults. For example, for an American put under the Black–Scholes model, Forsyth & Vetzal [4]\nreport spurious oscillations in the Gamma near the early exercise bou ndary that are attributed to\nthe Crank–Nicolson scheme and which are not resolved by employing b ackward Euler damping.\nIn view of the above, for the numerical approximation of American o ption values and their\nGreeksviaPDCPs, variousauthorshaveconsideredalternativese cond-orderimplicittimestepping\nmethods that possess a stronger damping property than the Cra nk–Nicolson scheme. Khaliq et\nal. [13, 17, 18, 19], Ikonen & Toivanen [8, 9] and Le Floc’h [15] propos e several second-order\nRunge–Kutta methods that are L-stable. This means that the pertinent stability function is\nbounded in modulus by one in the closed left half-plane in C, which is referred to as A-stability,\nand in addition vanishes at infinity, see e.g. [5, 7]. The Crank–Nicolson s cheme is A-stable, but\nnotL-stable, because the value of its stability function at infinity equals m inus one.\nIn this paper we considera detailed numericalconvergenceanalysis and comparisonofa variety\nof temporal discretization methods, in particular from the refere nces above, in the approximation\nof the fair values and the Greeks Delta and Gamma via PDCPs for Amer ican-style options on one\nand two assets under the Black–Scholes model. Our interest is in the ir temporal discretization\nerrors in the maximum norm on a region of interest around the strike price of the option that lies\nwell within the continuation region.\nAn outline of the rest of this paper is as follows.\nIn Section 2 the PDCP for the fair value of a two-asset American-st yle option under the Black–\nScholes framework is formulated. We describe a standard semidiscr etization of this PDCP where\nsecond-order central finite differences are applied on a suitable no nuniform Cartesian grid in the\nasset price domain and discuss the approximation of the Greeks Delt a and Gamma.\nIn Section 3 the temporal discretization methods that are studied in this paper are introduced.\nThe first is the familiar θ-method. For θ“1\n2this becomes the Crank–Nicolson (CN) scheme\nand forθ“1 the backward Euler (BE) method. The next is a second-order diag onally implicit\nRunge–Kutta (DIRK) method. This method also contains a paramet erθ. In particular, the choice\nθ“1´1\n2?\n2 yields an L-stable method that has been independently studied in [13] and [8, 9]\nand is further related to the method proposed in [15]. The final is the second-order Lobatto IIIC\nmethod, which is L-stable and has been considered in e.g. [17, 18, 19]. For the adapta tion of\nthese three Runge–Kutta methods to the semidiscrete PDCP syst em from Section 2 we employ\nthe well-known penalty approach and, complementary to this, discu ss the use of variable step sizes\nin time.\nIn Section 4 ample numerical experiments are presented to obtain in sight into the temporal\nconvergence behaviour of the above three methods in the applicat ion to the semidiscrete PDCP.\nSubsection 4.1 deals with the special case of the one-asset America n put option. First we illustrate\nthe importance of variable step sizes to avoid the order reduction p henomenon obtained with\nconstant step sizes, as first noted in [4] for the CN scheme. Next , using suitable variable step\nsizes, we numerically examine in detail the temporal convergence be haviour of the BE method,\nthe CN method, the DIRK method for θ“1´1\n2?\n2 andθ“1\n3and the Lobatto IIIC method.\nHere the convergence behaviour for both the option value and the Greeks Delta and Gamma is\ninvestigated. In Subsection 4.2 the numerical study is then extend ed to the two-asset American\noption case. As a typical example, an American put-on-the-avera ge option is considered.\nIn the final Section 5 we give our conclusions.\n2 American option valuation model\nDenote by ups1,s2,tqthe fair value of a two-asset American-style option under the Black –Scholes\nframework if at ttime units before the given maturity time Tthe two underlying asset prices are\nequal to s1ě0 ands2ě0. Letφps1,s2qdenote the payoff of the option and consider the spatial\ndifferential operator\nA“1\n2σ2\n1s2\n1B2\nBs2\n1`ρσ1σ2s1s2B2\nBs1Bs2`1\n2σ2\n2s2\n2B2\nBs2\n2`rs1B\nBs1`rs2B\nBs2´r. (2.1)\n2Here constant rą0 is the risk-free interest rate, constant σią0 is the volatility of the price of\nassetifori“1,2 and the constant ρP r´1,1sstands for the correlation factor pertinent to the\ntwo underlying asset price processes. It is a well-known result from the literature that the function\nusatisfies the PDCP\nuěφ,Bu\nBtěAu, pu´φqˆBu\nBt´Au˙\n“0, (2.2)\nvalid pointwise for ps1,s2,tqwhenever s1ą0,s2ą0, 0 ătďT. The initial condition is provided\nby the payoff,\nups1,s2,0q “φps1,s2q (2.3)\nfors1ě0,s2ě0 and boundary conditions are given by imposing (2.2) for s1“0 ands2“0,\nrespectively. As already alluded to above, along with the American op tion value function u, we are\nparticularly interested in this paper in the Greeks Delta and Gamma. T hese key risk quantities\nare defined by\n∆1“Bu\nBs1,∆2“Bu\nBs2,Γ11“B2u\nBs2\n1,Γ12“B2u\nBs1Bs2,Γ22“B2u\nBs2\n2. (2.4)\nClearly, the Greeks (2.4) all appear in the operator Aabove.\nThe three conditions in (2.2) naturally give rise to a decomposition of t heps1,s2,tq-domain:\nthe early exercise region is the set of all points ps1,s2,tqwhereu“φholds and the continuation\nregion is the set of all points ps1,s2,tqwhere Bu{Bt“Auholds. The joint boundary of these two\nregions is called the early exercise boundary or free boundary.\nFor most American-style options their value function, Greeks and e arly exercise boundary are\nunknown in (semi-)closed analytical form and one therefore often resorts to numerical methods\nfor their approximation. As a typical example we deal in this paper wit h an American put-on-\nthe-average option, which has the payoff\nφps1,s2q “maxˆ\n0, K´s1`s2\n2˙\n(2.5)\nwith given strike price Ką0.\nFollowing the popular method of lines, one first discretizes the PDCP ( 2.2) in the spatial vari-\nables ps1,s2qand subsequently discretizes in the temporal variable t. For the spatial discretization\nof (2.2), we consider a standard approach, which is described in det ail in [10, Sect. 3.1]. Here\nsecond-ordercentral finite differences for all derivatives in Aare applied on a smooth, nonuniform,\nCartesian grid in a truncated spatial domain r0,Smaxs ˆ r0,Smaxswith fixed value Smax ą2K\nchosen sufficiently large. For the put-on-the-averageoption, ho mogeneous Dirichlet conditions are\ntaken at the far field boundaries s1“Smaxands2“Smax, respectively.\nLetmě3 be any given integer. In each spatial direction ( s1ors2) the mesh under considera-\ntion has mpoints. It is uniform in the interval r0,2Ksand relatively fine. Outside this interval, it\nis nonuniform and relatively coarse. Figure 1 displays a sample grid whe rem“50 andK“100,\nSmax“5K.\nThe given spatial discretization leads to a semidiscrete PDCP system of the form\nUptq ěU0, U1ptq ěAUptq,pUptq ´U0qTpU1ptq ´AUptqq “0 (2.6)\nfor 0 ătďTwithUp0q “U0. HereUptqdenotes the Mˆ1 vector representing the semidiscrete\napproximation to the option value function up¨,¨,tqon the spatial grid, where M“m2. The\nMˆMmatrixAand theMˆ1 vector U0are given. The latter represents the payoff function\nφon the spatial grid. Here cell averaging is used near the line segment s1`s2“2Kwhereφis\nnonsmooth, cf. e.g. [10]. The vector inequalities in (2.6) are to be und erstood componentwise and\nthe symbolTdenotes the transpose.\nSemidiscrete approximations to the Greeks Delta and Gamma are dire ctly acquired by appli-\ncation of the pertinent second-order central finite differences f ormulas to Uptq. Note that as such\nthey form terms in the matrix-vector product AUptq.\n30 100 200 300 400 5000100200300400500\nFigure 1: Sample spatial grid for m“50,K“100,Smax“500.\n3 Temporal discretization\nFor the numerical solution of the semidiscrete PDCP system (2.6), w e consider in this paper\nthe adaptation of three temporal discretization methods for the system of ordinary differential\nequations (ODEs)\nU1ptq “AUptq p0ătďTq. (3.1)\nLet parameter θą0 be given. Let 0 “t0ăt1ăt2ă...ătN“Tbe any given sequence of\ntemporal grid points with (constant or variable) step sizes ∆ tn“tn´tn´1(1ďnďN).\nThe first method under consideration is the familiar θ-method. When applied to the ODE\nsystem (3.1), it generates approximations UntoUptnqsuccessively for n“1,2,3,...,Nby\npI´θ∆tnAqUn“Un´1` p1´θq∆tnAUn´1. (3.2)\nForθ“1 the backward Euler method (BE) is obtained and for θ“1\n2the trapezoidal rule or\nCrank–Nicolson (CN) method. The classical order of consistency o f theθ-method equals two for\nθ“1\n2and it is one otherwise. If θě1\n2, theθ-method is A-stable, because its stability function\nRpzq “1` p1´θqz\n1´θzpzPCq\nsatisfies |Rpzq| ď1 whenever ℜzď0. Moreover, it is L-stable if and only if θ“1.\nThesecondtemporaldiscretizationmethodthatwestudyinthispa peristhe diagonally implicit\nRunge–Kutta (DIRK) method given by the Butcher tableau\n00\n11´θ θ\n11\n21\n2´θ θ\n1\n21\n2´θ θ(3.3)\nThe DIRK method (3.3) was introduced by Cash [3] and has been indep endently considered by\nvarious authors in the computational finance literature. It has be en employed in the numerical\nvaluation of American options by Khaliq, Voss & Kazmi [13] and Ikonen & Toivanen [8, 9]. Also, it\n4is interesting to note that it forms the underlying implicit method ofbo th the Hundsdorfer–Verwer\n(HV) scheme and the modified Craig–Sneyd (MCS) scheme, two popu lar alternating direction\nimplicit (ADI) schemes that have been introduced in the computation al finance literature in [11]\nand [12], respectively. Application of the above DIRK method to the O DE system (3.1) yields\n$\n&\n%pI´θ∆tnAqY“Un´1` p1´θq∆tnAUn´1,\npI´θ∆tnAqZ“Un´1`1\n2∆tnAUn´1``1\n2´θ˘\n∆tnAY,\nUn“Z.(3.4)\nMethod (3.3) has classical order of consistency equal to two for a ny value θ. Its stability function\nis given by\nRpzq “1` p1´2θqz` p1\n2´2θ`θ2qz2\np1´θzq2pzPCq. (3.5)\nThe method is A-stable whenever θě1\n4and it is L-stable if and only if θ“1˘1\n2?\n2, cf. Cash [3]\nand also e.g. Hundsdorfer & Verwer [7].\nKhaliq, Voss & Kazmi [13] and Ikonen & Toivanen [8, 9] both selected t he value θ“1´1\n2?\n2,\nwhich yields L-stabilityand a relativelysmallerrorconstant. Forthis value θthe stability function\n(3.5) is further identical to that of the TR-BDF2 method. This comb ination of the trapezoidal\nrule and the two-step backward differentiation formula was introdu ced by Bank et al. [1] and\nsubsequently studied by e.g. Hosea & Shampine [6]. The TR-BDF2 met hod has been advocated\nfor the numerical approximation of American option values by Le Floc ’h [15]. When applied to\nthe ODE system (3.1), it is equivalent to (3.4) with θ“1´1\n2?\n2.\nIn the case of the MCS scheme, the value θ“1\n3has often been (judiciously) chosen in the\nliterature, cf. [10]. For this choice of θthe DIRK method (3.5) is A-stable, but not L-stable, since\nRp8q “ ´1\n2.\nThe third and final temporal discretization method that we conside r in this paper is the two-\nstageLobatto IIIC method , given by the Butcher tableau\n01\n2´1\n2\n11\n21\n2\n1\n21\n2(3.6)\nMethod (3.6) also possesses a classical order of consistency equa l to two, and it is L-stable. Its\nstability function is\nRpzq “1\n1´z`1\n2z2pzPCq. (3.7)\nThis Runge–Kutta method has been applied for a variety of problems in computational finance,\nnotably American option valuation, by Khaliq et al. [14, 17, 18, 19]. Ap plication of (3.6) to (3.1)\nyields$\n&\n%`\nI´1\n2∆tnA˘\nY`1\n2∆tnAZ“Un´1,`\nI´1\n2∆tnA˘\nZ´1\n2∆tnAY“Un´1,\nUn“Z.(3.8)\nIn order to arrive at the adaptation of the three temporal discre tization methods formulated\nabove to the semidiscrete PDCP (2.6), we employ the popular penalty approach. The main idea\nis to approximate (2.6) by the nonlinear system of ODEs\nU1ptq “AUptq `ρmaxtU0´Uptq,0u p0ătďTq (3.9)\nwith large penalty parameter ρą0, where the maximum of two vectors is to be taken componen-\ntwise. This approach has first been proposed in the literature for A merican option valuation by\nForsyth et al. [4, 20, 21].\nLetLarge ą0 be a given, fixed large number and let tolą0 be a given, fixed small tolerance.\nSetpU0“U0. Then adaptation of (3.2) to (2.6) by means of the penalty approac h gives rise to the\n5θ-P method :\n#`\nI´θ∆tnA`Ppkq˘\nYpk`1q“pUn´1` p1´θq∆tnApUn´1`PpkqU0\nfork“0,1,...,κ ´1 and pUn“Ypκq.(3.10)\nIt generates approximations pUnto the exact solution Uof (2.6) at t“tnsuccessively for n“\n1,2,3,...,N. In each time step an iteration is performed. Here Yp0q“pUn´1is the starting vector\nandPpkq(for 0 ďkăκ) is defined as the diagonal matrix with l-th diagonal entry equal to Large\nifYpkq\nlăU0,land zero otherwise. A natural stopping criterion for the penalty it eration is\nmax\nl|Ypκq\nl´Ypκ´1q\nl|\nmaxt1,|Ypκq\nl|uătolorPpκq“Ppκ´1q. (3.11)\nIn this paper we shall make the common choice Large “107andtol“10´7.\nIn a similar fashion, adaptation of (3.4) to (2.6) by the penalty appro ach leads to the\nDIRK-P method :\n$\n’’’&\n’’’%`\nI´θ∆tnA`Ppkq˘\nYpk`1q“pUn´1` p1´θq∆tnApUn´1`PpkqU0\nfork“0,1,...,κ 1´1 and pY“Ypκ1q,`\nI´θ∆tnA`Qpkq˘\nZpk`1q“pUn´1`1\n2∆tnApUn´1``1\n2´θ˘\n∆tnApY`QpkqU0\nfork“0,1,...,κ 2´1 and pUn“Zpκ2q.(3.12)\nHereYp0qandPpkqare defined analogously as above. Next, Zp0q“pUn´1andQpkqis the diagonal\nmatrix with l-th diagonal entry equal to LargeifZpkq\nlăU0,land zero otherwise.\nFinally, adapting (3.8) to (2.6) by the penalty approach we obtain\nLobatto-P method :\n$\n’&\n’%`\nI´1\n2∆tnA`Ppkq˘\nYpk`1q``1\n2∆tnA´Qpkq˘\nZpk`1q“pUn´1``\nPpkq´Qpkq˘\nU0`\nI´1\n2∆tnA`Qpkq˘\nZpk`1q´`1\n2∆tnA´Ppkq˘\nYpk`1q“pUn´1``\nPpkq`Qpkq˘\nU0\nfork“0,1,...,κ ´1 and pUn“Zpκq,(3.13)\nwhere the starting vectors Yp0q,Zp0qand diagonal matrices Ppkq,Qpkqare defined analogously as\nabove.\nIt is well-known in the literature on American option valuation that the use of constant step\nsizes can lead, for second-order consistent temporal discretiza tion methods, to a reduced con-\nvergence order that is significantly lower than two, see e.g. [4, 9, 16 ]. It has been shown that\nsecond-order convergence can be restored by choosing suitable variable step sizes [4]. In this\npaper, we shall consider the nonuniform temporal grid defined by [9 , 16]\ntn“´n\nN¯2\nTforn“0,1,2,...,N. (3.14)\nHere the variable step size ∆ tnis smallest for n“1 and grows linearly in n. The effectiveness of\nthe above nonuniform temporal grid will be illustrated in the following s ection.\nUnless stated otherwise, in view of the nonsmoothness of the initial (payoff) function, the first\ntwo time steps from t“0 will always be performed using the BE-P method, i.e., (3.10) with θ“1.\nThis well-known procedure is referred to as backward Euler damping or Rannacher smoothing.\n64 Numerical study\nIn this section we numerically study the convergence behaviour of t he temporal discretization\nmethods (3.10), (3.12), (3.13)in theapplicationtothe semidiscrete two-assetBlack–ScholesPDCP\n(2.6). The following key instances will be considered:\nBE-P: (3.10) with θ“1\nCN-P: (3.10) with θ“1\n2\nDIRKa-P: (3.12) with θ“1´1\n2?\n2\nDIRKb-P: (3.12) with θ“1\n3\nLobatto-P: (3.13).\nWe shall investigate the temporal discretization errors of these methods at t“tN“T, on a\ngiven region of interest ROI in the ps1,s2q-domain, defined by\npepN;mq “maxt |UlpTq ´pUN,l|: 0ďi,jďm,ps1,i, s2,jq PROI u. (4.1)\nHereUpTqrepresents the exact solution vector to (2.6) at t“Tandl“lpi,jqdenotes the index\nsuch that the components UlpTqand pUN,lcorrespond to the (same) spatial grid point ps1,i, s2,jq.\nIn addition to (4.1), we shall devote much attention to the tempora l discretization errors in the\ncase of the Greeks Delta and Gamma. These errors are defined com pletely analogously to (4.1).\nClearly, the temporal discretization error is measured in the import ant maximum-norm. We\nremark that the error due to the spatial discretization is not part of our study here; the present\npaper is devoted to an analysis of the temporal discretization erro r by itself.\n4.1 One-dimensional PDCP\nWe start by considering the special case of the one-asset America n put option under the Black–\nScholes framework. Then a one-dimensional PDCP (2.2) holds for th e option value function\nu“ups,tqwith spatial differential operator\nA“1\n2σ2s2B2\nBs2`rsB\nBs´r\nand payoff φpsq “maxpK´s,0q(forsě0). As a representative set of financial parameter values\nwe take\nσ“0.40, r“0.02, T “0.5, K “100, Smax“5K (4.2)\nand choose for the ROI the interval p0.8K,1.2Kq. Figure 2 displays the graphs of the pertinent\noption value function and Delta and Gamma functions ∆ “ Bu{Bsand Γ “ B2u{Bs2on the\ns-domain r0,2Ksfort“T. Observe in particular that the early exercise point for t“Tis\napproximately equal to 58, and hence, the selected ROI lies well with in the continuation region.\nOurfirstnumericalexperimentillustratesthesubstantialimprove mentintemporalconvergence\nbehaviour that is obtained by using the nonuniform temporal grid (3 .14) instead of a uniform\ntemporal grid, cf. the end of Section 3. Figure 3 displays the tempo ral discretization errors1of\nthe BE-P (dark blue bullets), CN-P (green diamonds), DIRKa-P (ligh t blue squares), DIRKb-P\n(orange squares) and Lobatto-P (red bullets) methods versus 1 {NforN“10,11,12,...,100 and\nm“200 in the case of a uniform temporal grid (left) and the nonuniform temporal grid (3.14)\n(right). For the BE-P method, a first-order convergence behav iour is found on both grids, as\nexpected. Furthermore, for any given N, the temporal errors obtained with the two grids are\nof similar size. For each of the other methods, however, a significan t difference in convergence\nbehaviour is observed between the two grids. In the case of a unifo rm grid a reduced convergence\norder of 1.5 is found, whereas it is 2.0 in the case of the nonuniform gr id (3.14). Further, for\nany given N, the temporal error obtained with the nonuniform grid is substant ially smaller than\n1A reference solution UpTqto the semidiscrete one-dimensional PDCP in this subsectio n has been computed by\napplying the DIRKa method combined with the Brennan–Schwar tz algorithm using (3.14) with N“2000.\n70 50 100 150 2000255075100\n0 50 100 150 200-1-0.75-0.5-0.250\n0 50 100 150 20000.0050.010.0150.02\nFigure 2: The value, Delta and Gamma functions of an American put op tion for t“Tand\nparameter set (4.2).\n10-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\nFigure 3: American put and parameter set (4.2). Temporal discret ization errorsof the BE-P (dark\nblue bullets), CN-P (green diamonds), DIRKa-P (light blue squares) , DIRKb-P (orange squares)\nand Lobatto-P (red bullets) methods for m“200 in the case of a uniform temporal grid (left)\nand the nonuniform temporal grid (3.14) (right).\n8that obtained with the uniform grid. For the CN-P method, these co nclusions have been made\nbefore in [4]. We note that the observed lower order of convergenc e for the CN-P, DIRKa-P,\nDIRKb-P and Lobatto-P methods on a uniform grid is not due to the u se of the penalty approach.\nIndeed, for the one-asset American put, the CN, DIRKa and DIRK b methods2can also be (easily)\ncombined with the classical Brennan–Schwartz algorithm [2] that e xactly solves the underlying\nlinear complementarity problem in each time step and this leads to visua lly identical results to\nthose in Figure 3.\nIn view of the above, we shall always employ the nonuniform tempora l grid (3.14) in the\nremainder of this paper. For the methods under consideration, we next study their temporal\nerrors pepN;mqfor different (smaller and larger) values mtogether with those for the Greeks Delta\nand Gamma, denoted here by pe∆pN;mqand peΓpN;mq, respectively. Figure 4 displays pepN;mq\n(top row), pe∆pN;mq(middle row) and peΓpN;mq(bottom row) for N“10,11,12,...,100 if\nm“100 (left column) and m“200 (right column). Figure 5 displays these errors for m“300\n(left column) and m“400 (right column).\nClearly, the BE-P method shows a neat first-order convergence b ehaviour for the option value\nandforboth GreeksDelta andGamma. Moreover,the threepertin ent errorconstantsareindepen-\ndent ofm, which is a favourable property and is often referred to in the litera ture asconvergence\nin the stiff sense orin the PDE sense .\nThe CN-P method, on the other hand, shows an undesirable conver gence behaviour for both\nDelta and Gamma on the ROI, with often large temporal errors, whic h becomes increasingly more\npronounced as mincreases. For these Greeks, a regular, second-order converg ence behaviour on\nthe ROI is obtained only if the number of time steps satisfies Něm{λwithλ«4. Such a\nrestriction on Nfor second-order convergence in the case of Delta and Gamma is fo und in many\nother numerical experiments for the CN-P method and is in line with pr evious observations for\nthis method when applied to American-style options [15, 16]. Here the valueλą0 is problem-\ndependent and a priori unknown. Increasing the (fixed) number o f backward Euler damping steps,\ne.g. from two to four, does not remove this restriction on N.\nAs a highly positive result, the DIRKa-P method reveals a second-or derconvergencebehaviour\nfor the option value as well as the Greeks Delta and Gamma, with erro r constants that are\nessentially independent of m. The DIRKb-P method also shows such a favourable, second-orde r\nconvergence behaviour, where the error constants are slightly la rger than the corresponding ones\nfor the DIRKa-P method. It is interesting to recall here that the D IRKa method is L-stable,\nwhereas the DIRKb method is not, cf. Section 3.\nThe Lobatto-P method is found to yield a second-order convergen ce behaviour for the option\nvalue and the Greeks Delta and Gamma as well. The pertinent error co nstants, which are also\nessentially independent of m, are substantially larger however than those for the DIRKa-P and\nDIRKb-P methods.\nThe aboveconclusionson the convergencebehaviourthe BE-P,CN -P, DIRKa-P,DIRKb-Pand\nLobatto-P methods – for the option value and Delta and Gamma on a R OI around the strike that\nlies well with the continuation region – are obtained in many subsequen t numerical experiments\nfor the one-asset American put option under the Black–Scholes mo del.\n2For the Lobatto method such a combination is not directly cle ar to us and we have thus not considered it here.\n910-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\nFigure 4: American put and parameter set (4.2). Temporal discret ization errors of the BE-P,\nCN-P, DIRKa-P, DIRKb-P and Lobatto-P methods for m“100 (left column) and m“200 (right\ncolumn) for the option value (top row), the Delta (middle row) and th e Gamma (bottom row).\nNonuniform temporal grid (3.14).\n1010-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nLobatto-P\nFigure 5: American put and parameter set (4.2). Temporal discret ization errors of the BE-P,\nCN-P, DIRKa-P, DIRKb-P and Lobatto-P methods for m“300 (left column) and m“400 (right\ncolumn) for the option value (top row), the Delta (middle row) and th e Gamma (bottom row).\nNonuniform temporal grid (3.14).\n114.2 Two-dimensional PDCP\nWe subsequently consider the American put-on-the-average opt ion under the two-asset Black–\nScholes model. As a representative set of financial parameter valu es we take\nσ1“0.30, σ2“0.40, ρ “0.50, r“0.01, T “0.5, K “100, Smax“5K. (4.3)\nFigure 6 displays, for t“Ton thes-domain r0,2Ks2, the graphs of the corresponding option\nvalue function uand the five Delta and Gamma functions (2.4). Figure 7 shows in grey t he early\nexercise region. For the ROI in the temporal discretization error ( 4.1) we consider the square\np0.9K,1.1Kq2, which has been indicated in blue in Figure 7. Again, this ROI lies well within the\ncontinuation region, i.e., at a significant distance from the early exer cise boundary.\nWe discuss numerical experiments for the BE-P, CN-P, DIRKa-P an d DIRKb-P methods,\nwhere the latter two will now be applied without backward Euler dampin g. For the solution of\nthe large linear systems in each time step, the BiCGSTAB iterative met hod3is employed with an\nILU preconditioner. Here a small tolerance4tol“10´15is (heuristically) selected to render the\napproximation error due to the iterative solution of the linear syste ms in each time step negligible.\nWe shall not consider the Lobatto-P method in this subsection in view of both its relatively large\nerror constant and the relatively big computational effort involved in solving the pertinent linear\nsystems (3.13).\nFigures 8, 9, 10, 11 display the obtained results for the temporal d iscretization errors5of the\nBE-P, CN-P, DIRKa-P and DIRKb-P methods for m“100,200,300,400, respectively. The left\ncolumn in each figure shows these errors in the case of the option va lue (top), ∆ 1(middle) and ∆ 2\n(bottom) and the right colums shows these in the case of Γ 11(top), Γ 12(middle) and Γ 22(bottom).\nThe same conclusions concerning the temporal convergence of th e above methods are found to\nhold as in the case of the one-asset American put option from Subse ction 4.1.\nThe BE-P method shows a neat first-order convergence behaviou r for the option value and for\nall Greeks Delta and Gamma, with the desirable property that the pe rtinent error constants are\nindependent of m. The CN-P method, on the other hand, exhibits for each of these fi ve Greeks\na similar undesirable convergence behaviour as observed in Subsect ion 4.1 and only turns out to\nyield regular, second-orderconvergenceunder a restriction on t he number of time steps of the type\nNěm{λwith certain (problem-dependent) constant λą0.\nAs a very favourable result, the DIRKa-P method shows again a sec ond-order convergence\nbehaviour for the option value and all Greeks Delta and Gamma with er ror constants that are\nessentially independent of m. The DIRKb-P method also shows such a favourable, second-orde r\nconvergence behaviour, provided a mild lower bound on Nholds in the case of the Gamma\nGreeks. The lower bound is attributed to the lack of L-stability of the DIRKb method. Additional\nexperimentsshowthat byapplyingbackwardEulerdamping, thiswea kconditionon Ndisappears.\n3As implemented in Matlab version R2020b through the functio nbicgstab .\n4For the relative residual error in the Euclidean norm.\n5A reference solution UpTqto the semidiscrete two-dimensional PDCP in this subsectio n has been computed by\napplying the DIRKa-P method using (3.14) with N“500.\n12Figure 6: The value, Delta and Gamma functions of an American put-o n-the-average option for\nt“Tand parameter set (4.3).\n130 50 100 150 200050100150200\nFigure 7: In grey: the early exercise region of an American put-on- the-average option for t“T\nand parameter set (4.3). In blue: the region of interest.\n1410-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nFigure 8: American put-on-the-average and parameter set (4.3) . Temporal discretization errors\nof the BE-P, CN-P, DIRKa-P and DIRKb-P methods for m“100. Option value (top left), ∆ 1\n(middleleft), ∆ 2(bottomleft), Γ 11(topright),Γ 12(middleright),Γ 22(bottomright). Nonuniform\ntemporal grid (3.14).\n1510-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nFigure 9: American put-on-the-average and parameter set (4.3) . Temporal discretization errors\nof the BE-P, CN-P, DIRKa-P and DIRKb-P methods for m“200. Option value (top left), ∆ 1\n(middleleft), ∆ 2(bottomleft), Γ 11(topright),Γ 12(middleright),Γ 22(bottomright). Nonuniform\ntemporal grid (3.14).\n1610-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nFigure 10: American put-on-the-average and parameter set (4.3 ). Temporal discretization errors\nof the BE-P, CN-P, DIRKa-P and DIRKb-P methods for m“300. Option value (top left), ∆ 1\n(middleleft), ∆ 2(bottomleft), Γ 11(topright),Γ 12(middleright),Γ 22(bottomright). Nonuniform\ntemporal grid (3.14).\n1710-210-110-610-510-410-310-210-1100\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-710-610-510-410-310-210-1\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\n10-210-110-810-710-610-510-410-310-2\nBE-P\nCN-P\nDIRKa-P\nDIRKb-P\nFigure 11: American put-on-the-average and parameter set (4.3 ). Temporal discretization errors\nof the BE-P, CN-P, DIRKa-P and DIRKb-P methods for m“400. Option value (top left), ∆ 1\n(middleleft), ∆ 2(bottomleft), Γ 11(topright),Γ 12(middleright),Γ 22(bottomright). Nonuniform\ntemporal grid (3.14).\n185 Conclusions\nThrough ample numerical experiments we have studied the converg ence of a variety of temporal\ndiscretization methods of the Runge–Kutta kind in the approximatio n of the option values and\nthe Greeks Delta and Gamma for one- and two-asset American-sty le options via the numerical\nsolution of the pertinent one- and two-dimensional PDCPs. The met hods under consideration are\nthe backward Euler method, the Crank–Nicolson method, two memb ers of a family of second-\norder DIRK methods and the second-order Lobatto IIIC method . Their adaptation to PDCPs\nis obtained by employing the popular penalty approach. A specific non uniform temporal grid is\nused to avoid the well-known order reduction phenomenon for seco nd-order methods.\nThebackwardEulermethod showsaneatfirst-orderconvergenc ebehaviourin thestiffsensefor\nthe option value and all Greeks Delta and Gamma. In line with previous r esults in the literature,\nthe Crank–Nicolsonmethod showsan undesirable convergencebeh aviourfor all these Greeks, with\noften large temporal errors, which becomes increasingly more pro nounced as mincreases. Here\nregular, second-order convergence in the case of Delta and Gamm a is only obtained if the number\nof time steps NsatisfiesNěm{λwith certain problem-dependent constant λą0.\nAs a very favourable result, for both choices θ“1´1\n2?\n2 andθ“1\n3, the DIRK method (3.3)\nreveals a second-order convergence behaviour in the stiff sense f or the option value as well as all\nGreeks Delta and Gamma, provided backward Euler damping is applied if θ“1\n3. The Lobatto\nIIIC method (3.6) also yields a second-order convergence behavio ur in the stiff sense. It has a\nsubstantially largererrorconstant, however, and is alsocomputa tionally much more intensive than\n(3.3).\nWe conclude that the DIRK methods with θ“1´1\n2?\n2 andθ“1\n3are preferable among the\ntemporal discretization methods under investigation in this paper.\nReferences\n[1] R. E. Bank, W. M. Coughran, W. Fichtner, E. H. Grosse, D. J. Ro se, and R. K. Smith.\nTransient simulation of silicon devices and circuits. IEEE Trans. Comput.-Aided Design Int.\nCirc. Sys. , 4:436–451, 1985.\n[2] M. J. Brennan and E. S. Schwartz. The valuation of American put options. J. Finance ,\n32:449–462, 1977.\n[3] J. R. Cash. Two new finite difference schemes for parabolic equat ions.SIAM J. Numer.\nAnal., 21:433–446, 1984.\n[4] P. A. Forsyth and K. R. Vetzal. Quadratic convergence for valu ing American options using a\npenalty method. SIAM J. Sci. Comp. , 23:2095–2122, 2002.\n[5] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II . Springer, 1991.\n[6] M. E. Hosea and L. F. Shampine. Analysis and implementation of TR- BDF2.Appl. Numer.\nMath., 20:21–37, 1996.\n[7] W. Hundsdorfer and J. G. Verwer. Numerical Solution of Time-Dependent Advection-\nDiffusion-Reaction Equations . Springer, 2003.\n[8] S. Ikonen and J. Toivanen. Pricing American options using LU deco mposition. Appl. Math.\nSc., 1:2529–2551, 2007.\n[9] S. Ikonen and J. Toivanen. Operator splitting methods for pricin g American options under\nstochastic volatility. Numer. Math. , 113:299–324, 2009.\n[10] K. J. in ’t Hout and P. Lamotte. Efficient numerical valuation of Eu ropean options under the\ntwo-asset Kou jump-diffusion model. J. Comp. Finan. , 26:101–137, 2023.\n19[11] K. J. in ’t Hout and B. D. Welfert. Stability of ADI schemes applied t o convection-diffusion\nequations with mixed derivative terms. Appl. Numer. Math. , 57:19–35, 2007.\n[12] K. J. in ’t Hout and B. D. Welfert. Unconditional stability of secon d-order ADI schemes\napplied to multi-dimensional diffusion equations with mixed derivative te rms.Appl. Numer.\nMath., 59:677–692, 2009.\n[13] A. Q. M. Khaliq, D. A. Voss, and S. H. K. Kazmi. A linearly implicit pred ictor-corrector\nscheme for pricing American options using a penalty method approac h.J. Bank Finan. ,\n30:489–502, 2006.\n[14] A. Q. M. Khaliq, D. A. Voss, and M. Yousuf. Pricing exotic options withL-stable Pad´ e\nschemes. J. Bank Finan. , 31:3438–3461, 2007.\n[15] F. Le Floc’h. TR-BDF2 for fast stable American option pricing. J. Comp. Finan. , 17:31–56,\n2014.\n[16] C. Reisinger and A. Whitley. The impact of a natural time change o n the convergence of the\nCrank–Nicolson scheme. IMA J. Numer. Anal. , 34:1156–1192, 2014.\n[17] M. Yousuf and A. Q. M. Khaliq. Partial differential integral equa tion model for pricing\nAmerican option under multi state regime switching with jumps. Numer. Meth. Part. Diff.\nEq., 39:890–912, 2023.\n[18] M. Yousuf, A. Q. M. Khaliq, and B. Kleefeld. The numerical appro ximation of nonlinear\nBlack–Scholes model for exotic path-dependent American options with transaction cost. Int.\nJ. Comp. Math. , 89:1239–1254, 2012.\n[19] M. Yousuf, A. Q. M. Khaliq, and R. H. Liu. Pricing American options under multi-state\nregime switching with an efficient L-stable method. Int. J. Comp. Math. , 92:2530–2550,\n2015.\n[20] R. Zvan, P. A. Forsyth, and K. R. Vetzal. Penalty methods for American options with\nstochastic volatility. J. Comp. Appl. Math. , 91:199–218, 1998.\n[21] R. Zvan, P. A. Forsyth, and K. R. Vetzal. A finite volume approa ch for contingent claims\nvaluation. IMA J. Numer. Anal. , 21:703–731, 2001.\n20"    },    {        "title": "2401.13415v2.Least_response_method_to_separate_CMB_spectral_distortions_from_foregrounds.pdf",        "content": "Phys. Rev. D 109, 023523 Published 19 January 2024 DOI: 10.1103/PhysRevD.109.023523\nLeast response method to separate CMB spectral distortions from foregrounds\nJ.-P. Maillard,1A. Mihalchenko,2, 3D. Novikov,2A. Osipova,2, 4S. Pilipenko,2and J. Silk1, 5, 6\n1Institut d’Astrophysique de Paris, UMR 7095, CNRS,\nSorbonne Universit´ e, 98bis Boulevard Arago, 75014 Paris, France\n2Astro-Space Center of P.N. Lebedev Physical Institute, Profsoyusnaya 84/32, Moscow 117997, Russia\n3Moscow Institute of Physics and Technology, Institutskiy pereulok, d.9, Dolgoprudny, Moscow 141701, Russia\n4National Research University Higher School of Economics, 11 Pokrovsky Bulvar, Moscow 109028, Russia\n5Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA\n6Beecroft Institute for Particle Astrophysics and Cosmology,\nDepartment of Physics, University of Oxford, Oxford OX1 3RH, United Kingdom\nWe present a signal-foreground separation algorithm for filtering observational data to extract\nspectral distortions of the cosmic microwave background (CMB). Our linear method, called the\nleast response method (LRM), is based on the idea of simultaneously minimizing the response to\nall possible foregrounds with poorly defined spectral shapes and random noise while maintaining\na constant response to the signal of interest. This idea was introduced in detail in our previous\npaper. Here, we have expanded our analysis by taking into consideration all the main foregrounds.\nWe draw a detailed comparison between our approach and the moment internal linear combina-\ntion method, which is a modification of the internal linear combination technique previously used\nfor CMB anisotropy maps. We demonstrate advantages of LRM and evaluate the prospects for\nmeasuring various types of spectral distortions.\nBesides, we show that LRM suggests the possibility of its improvements if we use an iterative\napproach with sequential separation and partial subtraction of foreground components from the\nobserved signal.\nIn addition, we estimate the optimal temperature that the telescope’s optical system should have\nin order to detect the chemical type µdistortions. We present a design of an instrument where,\naccording to our estimates, the optimal contrast between its thermal emission and the CMB allows\nus to measure such distortions.\nPACS numbers:\nKeywords: Cosmic Microwave Background, spectral distortions, data analysis, components separation, ob-\nservations\nI. INTRODUCTION\nThe recent space missions related to the study of relic\nradiation WMAP [1] and Planck [2, 3] has substantially\nbroadened our knowledge of cosmic microwave back-\nground anisotropy: the power spectrum of ∆ T/T fluctua-\ntions and almost Gaussian nature of their distribution on\nthe celestial sphere. Ultimately, these observational data\nhelped to estimate the main cosmological parameters.\nHowever, even more information can be gained by delv-\ning into cosmic microwave background (CMB) spectral\ndistortions (SDs). Being one of the key goals of obser-\nvational cosmology [4–11], measuring deviations of the\nCMB spectrum from a black body shape shall reveal a\nvast amount of information about the early Universe, un-\nobtainable by other observational methods [4, 12–15].\nThe chemical potential µ-type distortions [16] are cre-\nated if either energy is injected into the cosmic plasma\nor photon number density is modified [6, 17–22] when\nthe redshift is less than ∼2×106. Thus, they can be\nused as a tool to trace the energy history of the Uni-\nverse. A later type of distortions ( ydistortions) [12]\ncontain information about the structure of intracluster\nmedium [23–26]. In addition, very specific deviations in\nthe frequency spectrum of radiation coming from galaxyclusters can be used for independent measurements of\nthe amplitudes and orientations of low CMB anisotropy\nmultipoles [27, 28]. The COBE/FIRAS mission [29, 30]\nmeasured the CMB frequency spectrum and established\nits Planckian shape with good accuracy. Future detec-\ntion of the spectral features is expected from FIRAS-type\nmissions [31–34], experiments with a large primary mir-\nror with good angular resolution [35] (which is essential\nfory-type deviations) or even using a Moon-based tele-\nscope [36], which is revised in this paper.\nAn exhaustive review of the theory behind SDs, their\nobservational challenges, computational techniques and\npossible new directions in this field are given in [17]. The\nmain difficulty in the task of measuring small spectral dis-\ntortions is the presence of foregrounds of cosmic and in-\nstrumental origin, which are several orders of magnitude\nlarger in amplitude than the signals of interest. Thus, in\norder to ensure detection of such signals, it is necessary\nnot only to achieve high sensitivity of the experiment,\nbut also to learn the most optimal way to separate SDs\nfrom the rest of the observed signal. The spectra of some\nastronomical foregrounds as well as the spectrum emit-\nted by the optical system of the instrument are poorly\ndefined and cannot be modeled and predicted with the\naccuracy required for reliable SDs measurements. ThisarXiv:2401.13415v2  [astro-ph.CO]  15 Feb 20242\nmeans that sufficiently advanced and efficient data pro-\ncessing methods must be used to solve such a problem.\nIn particular, any methods used must take into account\npossible variations in foreground frequency spectra.\nThe well-known blind internal linear combination\n(ILC) method [37] has been successfully applied to pro-\ncess multifrequency analysis of CMB anisotropy maps\n[38–40]. However, the application of this method is lim-\nited by the presence of large foreground components that\nhave a nonzero projection onto the signal of interest,\nwhich leads to a bias. To avoid this, the constrained\nILC (cILC) [41–43] method was proposed as a modifi-\ncation of the ILC. This approach completely eliminates\nthe contribution of foregrounds with well-known spectral\nshapes, treating the remaining components as unmod-\neled noise. However, to solve the problem of getting rid\nof all foreground components, it is necessary to take into\naccount the spectral variations of dust, cosmic infrared\nbackground (CIB), synchrotron radiation and other sky\nforegrounds along the line of site and from one direction\nto another one, as well as time variations of the fore-\nground contribution from the instrument emission. The\nspectra of these components depend on the parameters\nand, therefore, spectral variations are equivalent to vari-\nations in the parameters.\nTo take this into account, a quite effective method,\ncalled MILC (Moments ILC) [44–46], was introduced,\nwhere the foreground spectra are expanded into Taylor\nseries in terms of parameters in the vicinity of some av-\nerage reference parameter values. Data filtering, which\ninvolves zeroing out the expansion moments, ensures that\nthe contribution from such foregrounds is eliminated.\nHowever, this method also has a rather serious disadvan-\ntage. A large number of constraints ensuring the zeroing\nof the foreground contribution during the filtering pro-\ncess inevitably leads to an unacceptably large response to\nunmodeled noise (including random photon noise). This\nis not surprising, since a large number of imposed strict\nconditions limits the number of degrees of freedom and\nincreases the contribution of noise to the estimation of\nthe signal of interest.\nIn work [47], another modification of the cILC method,\ncalled partially constrained ILC (pcILC), was proposed.\nInstead of completely eliminating the contribution from a\nforeground with a well-defined spectrum (as cILC does),\npcILC reduces this contribution to some empirically de-\nterminable level. As a result the strict cILC constraints\nare somewhat softened. This made it possible to reduce\nthe response to noise when processing ∆ T/T anisotropy\ndata in the presence of the Sunyaev-Zel’dovich effect or\nCMB lensing. However, application of this approach in\nthe presence of many different foregrounds is very com-\nplicated and its effectiveness in this case has not been\nproven. Besides, unlike MILC, this method does not al-\nlow for possible spectral shape variations of foregrounds.\nWe recently proposed a data filtering method called\nleast response method (LRM) [48], which is quite sim-\nple and easy to implement. This approach involves theoptimization of a single functional for all components of\nthe observed signal. During the process of data filtering\nwe minimize the response to all foregrounds and random\nnoise simultaneously, while keeping the response to the\nsignal of interest equal to unity. We assume that we know\nthe following information about foreground signals:\n1. The foreground spectral parameters can change\nwithin a limited range of their possible variations;\n2. The amplitudes of the foregrounds are limited from\nabove by known values.\nNote that such information is available to us from pre-\nvious observational data and characteristics of the in-\nstrument optical system for a particular experiment. In\nthis article, we compare LRM and MILC approaches and\nshow the advantages and prospects of LRM for measur-\ning various spectral distortions of the relic radiation in\nthe presence of all main foregrounds, including the signal\nfrom the instrument optics.\nIn addition, we show that when measuring µdistor-\ntions, the temperature of the instrument optical system\nshould not approach the CMB temperature. Measur-\ning such distortions requires calibration of the instrument\nand in this case the signal component generated by the\noptics becomes a part of the observed signal. This com-\nponent is close in shape to a black body signal, and if its\ntemperature is close to 2.7 K, then it becomes ’poisoned’\nto some extent, since the device itself begins to create\nspectral features simulating CMB distortions. We show\nthat, according to our estimates, the optimal tempera-\nture of the instrument for µspectral distortion measure-\nments should be close to 9 K.\nThe outline of this paper is as follows. In Sec. II,\nwe review linear methods for observational data filtering.\nWe describe MILC and LRM approaches for CMB spec-\ntral distortions detection in the case when the observed\nsignal contains foregrounds with poorly defined spectral\nshapes. In Sec. III, we describe models of signals of\ninterest, foregrounds and photon noise and present a de-\ntailed comparative numerical analysis of MILC and LRM\nmethods. In this Section we also estimate the instrument\noptics temperature that is optimal for detecting µ-type\nCMB spectral distortions. In the same section we give an\nexample of an instrument for measuring the relic radia-\ntion spectral features. Brief conclusions and our sugges-\ntions about possible improvements of the LRM approach\nare given in Sec. IV.\nII. LINEAR DATA FILTERING METHODS\nThe frequency spectrum S(ν) we observe consists of\nthe signal of interest and a set of other components that\nwe would like to get rid of during data processing. There-\nfore, the total observed spectrum can be written as fol-\nlows:\nS(ν) =adId(ν) +MX\nm=1Im(ν), (1)3\nwhere Idis the certain kind of CMB spectral distortions,\nwhich we would like to separate from other components.\nIndex “ d” can denote the µdistortions: Iµ, the Sunyaev-\nZel’dovich effect ( ydistortions): Iy0, the first or second\nrelativistic corrections to this effect: Iy1orIy2. There-\nfore, adis the frequency-independent amplitude of the\nsignal of interest we want to estimate. The rest of the\nsignal S(ν) consists of Mforegrounds of various physi-\ncal origins Im(ν), which may include spectral distortions\nthemselves. For example, if we are interested in signal\nIµ, then signals Iy0, Iy1Iy2will be part of the total fore-\nground.\nUsing a Fourier-transform spectrometer (FTS)\nmounted on a space telescope, one can obtain discrete\nvalues of the total signal (or row vector) S= (S1, .., S J)\ninJequally wide frequency channels in a broad\nfrequency range from νmintoνmax:\nSj=adIj\nd+P\nmIj\nm+Nj, j= 1, .., J,\nIj\nx=νj+∆ν\n2R\nνj−∆ν\n2Ix(ν)dν\n∆ν,(2)\nwhere indices jindicate the frequency channel number,\n∆νis the channel width and Njis the random photon\nnoise with zero mean and covariance matrix [ Cij] =C=\n⟨NTN⟩. The spectral shapes of CMB distortions Iµ,Iy,\nIy1andIy2are well-known, while the foreground spectra\nImcan depend on various parameters and in fact are su-\nperpositions of spectra integrated along the line of sight\nor/and obtained as a result of averaging over spatial pix-\nels of a sky map. Therefore, the foreground components\ncan be written as\nIj\nm=R\nΩam(P)fm(νj,P)dP,\ndP=dp1dp2· ·dpL,(3)\nwhere P=p1, .., pLis the set of Lparameters, fm(νj,P)\nare the functions representing the foreground spectra, Ω\nis the region of possible parameter variations, and amare\nthe amplitudes of the foreground radiation as functions of\nparameters P. Thus, if, for example, am(P) has the form\nof a delta function am(P) =Am·δ(P−Pm), then the\nforeground spectrum with index mwill have a template\nwith well-defined parameters Pmand the amplitude Am:\nIj\nm=Am·fm(νj,Pm).\nThe total observed signal S= (S1, .., SJ) can be di-\nvided into three parts (three vectors):\nS=adId+F+N,\nF= (F1, .., F J), Fj=P\nmIj\nm,\nN= (N1, .., N J)(4)\nwhere Idis some particular kind of spectral distortion\nwe want to separate from the rest of the signal, Fis the\ntotal foreground and Nrepresents the random noise.The task of any linear algorithm is to find the optimal\nvector of weights ω= (ω1, .., ω J) for frequency channels\nthat should have the following property:\nωST=JX\nj=1ωjSj→adfor C ij→0, i, j = 1, .., J. (5)\nThus, in the ideal case, when the noise tends to zero\n(Cij→0), it is desirable that the algorithm accurately\nreproduces the amplitude of the signal of interest adwith-\nout any bias. With nonzero noise, the algorithm should\nminimize the deviation of the estimate of this amplitude\nfrom its true value.\nLet us denote the scalar product ωST=R(S) as the\nresponse to the signal:\nR(S) =adR(Id) +R(F) +R(N). (6)\nThe first condition imposed on the weights is common to\nall linear algorithms and is quite obvious:\nR(Id) =ωIT\nd=X\njωjIj\nd= 1. (7)\nThus, it is necessary to minimize the response to the\nrest of the signal R(F) +R(N) while maintaining the\ncondition in Eq. 7. To denote the expected total average\nvalue of the response to foreground and noise, we use the\nfollowing definition:\nR(F+N) :=q\nR2(F) +⟨R2(N)⟩. (8)\nTherefore, R(F+N) is the value that must be minimized\nunder condition in Eq. (7).\nA. Internal linear combination method\nThe ILC method assumes that we know the spectral\nenergy distribution (SED) of the signal of interest, but\ndoes not assume knowledge of any information about\nforegrounds and noise. With this approach all compo-\nnents of the observed signal, apart from the signal of\ninterest, are considered as unmodeled noise. That is, in\nthis case in Eqs. (4) and (8) no distinction is made be-\ntween the components FandN. Thus, the problem is\nreduced to minimizing the response to such noise while\nmaintaining a constant response to Id. As a result, the\nweights ωjare the solution of the following system of\nequations:\n1.ωIT\nd= 1,\n2. ∂(ωDωT)/∂ω= 0,(9)\nwhere D=⟨STS⟩−⟨ST⟩⟨S⟩is the data covariance matrix\nand⟨⟩means averaging over the spatial pixels of the sky\nmap. The solution of such a system is:\nω=IdD−1·\u0000\nIdD−1IT\nd\u0001−1. (10)4\nThis completely blind approach can be biased due to\nnonzero projections of foregrounds on the considered sig-\nnal,FIT\nd̸= 0. Taking into account that in our case the\nforeground amplitudes can exceed the amplitude of the\nsignal of interest by several orders of magnitude, this\nmethod is not suitable.\nB. Moments approach (MILC)\nIn order to avoid biasing in the process of signal Id\nextraction, the constrained ILC (or cILC) method was\nproposed [41, 42], which nullifies the response to some\nmodeled foregrounds with known spectra. Thus, a num-\nber of constraints are added to Eq. (9) that ensure a zero\nresponse to certain foregrounds. This modification of the\nblind ILC approach was extended with the MILC method\n[44–46], where foregrounds with badly defined spectra in\nEq. (3) were considered. The idea behind the MILC\nmethod is quite simple and effective. Since the spectral\nshapes of some signal components depend on the param-\netersP, they can be expanded in a Taylor series up to\nsome order nin the vicinity of a certain reference point\ncorresponding to the average preestimated value P0:\nfm(νj,P)≈fm(νj,P0)+\nP\nn1,.,nL\u0014\n∂n1+..+nL\n∂Pn1\n1··PnL\nLfm(νj,P)\u0015\nP=P0∆Pn1\n1· ·∆PnL\nL,\n(11)\nwhere the summation is performed over all positive\nn1, .., nLsatisfying the condition 0 ≤n1+..+nL≤n. In\norder to remove the influence of such foregrounds on data\nprocessing, it is sufficient to require a zero response to all\nderivatives of the spectra with respect to parameters up\nto the nth order. As a result, we get the following system\nof equations:\n1.ωIT\nd= 1,\n2.P\njωj∂n1+..+nL\n∂Pn1\n1··PnL\nLfm(νj,P)|P=P0= 0,\nm= 1, .., M, 0≤n1+..+nL≤n,\n3. ∂(ωCωT)/∂ω= 0.(12)\nTherefore, this method ensures that the response to all\nforegrounds is zeroed to within\nR(F) =O\nX\njωj∂n+1fm(νj,P)\n∂Pn+1\n\nP=P0=Rn+1(F).\n(13)\nHowever, the disadvantage of MILC is that with a large\nnumber of constraints in formula 2 of Eq. (12), the re-\nsponse to noise R(N) can become unacceptably large.\nIndeed, this formula is equivalent to the system of equa-\ntionsωφT\nk= 0, k= 1, ..K, where the vectors φkcorre-\nspond to the derivatives of the foregrounds with respectto the parameters Pand the capital Kdenotes the to-\ntal number of constraints (or derivatives). The system of\nvectors φkcan be orthogonalized in such a way that the\nnew vectors ˜φkare a linear combinations of the original\nvectors φkand represent an orthonormal system:\n˜φk=˜φk(φ1, ..,φK),\n˜φk˜φT\nk′=δk′\nk, k, k′= 1, ..., K.(14)\nThus, the normalized signal ˜Id=Id·(IdIT\nd)−1\n2can be\nwritten in the following form:\n˜Id=KP\nk=1γk˜φk+∆d,\nγk=˜φk·˜IT\nd,(15)\nwhere ∆dis part of signal ˜Idorthogonal to all mod-\neled foregrounds: ˜φk∆T\nd= 0, k= 1, .., K . That is, the\nsmaller the length of the vector ∆d, the more difficult\nit is to separate the vector Idfrom the rest of the total\nsignal. We can define the value of Γ =\u0000\n∆d∆T\nd\u00011/2as a\nmeasure of the orthogonality of the normalized signal ˜Id\nto all foregrounds. It is easy to show, that Γ2= 1−KP\nk=1γ2\nk.\nFinally, the solution of the system of Eq. (12) for weights\nωis\nω=∆dC−1·\u0000\n∆dC−1IT\nd\u0001−1, (16)\nand the noise response can be estimated as follows:\n⟨R2(N)⟩=ωCωT∼σ2/Γ2,\nσ2=⟨NNT⟩\nJ.(17)\nHere, σis equivalent to the mean sensitivity per fre-\nquency channel. Thus, a small value of Γ means a large\nresponse to random noise. In case when Γ →0, the signal\nIdbecomes a linear combination of foregrounds, and the\nseparation of such a signal from them with this method\nbecomes impossible.\nC. Least response method\nRecently, a new approach to the component separa-\ntion problem has been proposed [48], which can be called\nthe LRM. Unlike MILC, this method implies the avail-\nability of information not only about the possible fore-\nground spectral shapes variations, but also about their\nmaximum possible amplitudes. It is important to note\nthat such information about main known foregrounds is\navailable to us. In addition, overestimation of the upper\nlimit of the foreground amplitudes is not critical for our\napproach (unlike underestimation). This helps to avoid\nimposing strict conditions on the weight vector ωand\nthereby significantly reduce the noise response. The only5\nfairly mild assumption about the foregrounds described\nin Eq. (3) is that the amplitudes aminside the parame-\nter domain Ω should be less than certain (preestimated)\nvalues Am:\n|am(P)|≤AmforP∈Ω, (18)\nandam(P) = 0 otherwise.\nIt is easy to show, that the mean square of the response\nto foreground has an upper limit:\n⟨R2(F)⟩ ≤ ⟨MX\nm=1M·a2\nm(P)\nJX\nj=1fm(νj,P)·ωj\n2\n⟩,(19)\nand according to Eq. (18) the following inequality is\nalways true:\n⟨R2(F)⟩ ≤ωΦωT,Φ=M\u0014MP\nm=1A2\nmqm\nij\u0015\n,\nqm\nij=1\nVΩR\nΩfm(νi,P)fm(νj,P)dP(20)\nwhere integrals qm\nijcan be precalculated for all types of\nforeground ( m= 1, .., M ) numerically or in some partic-\nular cases analytically depending on the configuration of\nΩ. It is worth noting that the prefactor Mfor the matrix\nΦin Eqs. (19) and (20) is used to ensure that these in-\nequalities are certainly correct. In general, the prefactor\nfor this matrix can vary from 1 (when there are no corre-\nlations between foregrounds) to M (for 100% correlation\nbetween all foregrounds). Thus, using the value M as a\nprefactor provides a complete guarantee of taking into ac-\ncount all possible correlations. In reality, foregrounds of\ndifferent physical origins are weakly correlated with each\nother. In this case, the coefficient M can be replaced by\none. In our estimates we use uncorrelated foregrounds\nmodel and, therefore, Φ=\u0014MP\nm=1A2\nmqm\nij\u0015\n. For a more de-\ntailed analysis, possible correlations between individual\nforegrounds can be taken into account in the calculation\nof the Φmatrix.\nSince ⟨R2(N)⟩=ωCωT=P\ni,jCijωiωjand the fore-\ngrounds are not correlated with noise we have\n⟨(R(F) +R(N))2⟩ ≤ω[Φ+C]ωT. (21)\nTherefore the minimization of the response to the fore-\nground and to the noise is achieved with weights ωj\ncorresponding to the minimum of the quadratic form\nω[Φ+C]ωTunder condition in Eq. (7):\n1.ωIT\nd= 1,\n2. ∂(ω[Φ+C]ωT)/∂ω= 0.(22)\nThe solution of the system Eq. (22) is\nω=Id[Φ+C]−1\n·\u0010\nId[Φ+C]−1\nIdT\u0011−1\n. (23)Thus, LRM, unlike MILC, does not require complete\northogonality of the signal of interest to all foregrounds\n[compare Eqs. (12) and (22)]. In the next section we will\ndemonstrate the advantages of this approach and show\nthat it allows to detect a signal in the observational data\nat a much lower sensitivity.\nIII. METHODS COMPARISON AND\nPROSPECTS FOR MEASURING SPECTRAL\nDISTORTIONS\nA. Modeling spectral distortions and foregrounds\nOur numerical calculations were performed for a sim-\nulated observed frequency spectrum that included the\nsignal of interest, all possible foregrounds with varying\nparameters and photon noise. We did not use any in-\nformation about the spatial distribution of foreground\nsources in the sky. In order to carry out a numerical ex-\nperiment on the application of the two methods (MILC\nand LRM) described in the previous section, we used the\nfollowing models of signal components.\nSignals of CMB origin areµdistortion Iµ,ydistortion\n(SZ effect) Iy0, first and second relativistic corrections to\nthermal SZ effect Iy1,Iy2and CMB anisotropy ICMBA .\n(It is assumed that the CMB monopole can be easily re-\nmoved from the observational data). All these spectra\nhave well-defined shapes that do not depend on any pa-\nrameters and look as follows:\nIµ(ν) =I0x4ex\n(ex−1)2\u00001\nb−1\nx\u0001\nµ,\nIy0(ν) =I0x4ex\n(ex−1)2\u0000\nxcoth(x\n2)−4\u0001\ny,\nIy1(ν) =I0x4ex\n(ex−1)2Y1(x)θey,\nIy2(ν) =I0x4ex\n(ex−1)2Y2(x)θ2\ney,\nICMBA (ν) =2(kT0)3\n(hc)2x4\n(ex−1)2∆T\nT0(24)\nwhere x=hν/kTCMB and the CMB temperature is\nTCMB = 2.72548 K, [29, 30] and ∆ T/TCMB<10−4. The\nsame estimated values for constants b,I0,µandyas in\n[33] are used: I0= 270 MJy/sr, b= 2.1923, µ= 2×10−8,\ny= 1.77×10−6andθe=kTSZ/mec2∼2.44×10−3. The\nfunctions Y1(x) and Y2(x) for the first and second cor-\nrections have a rather cumbersome form and analytical\nformulas for them can be found in [49].\nDust and CIB foregrounds are considered together and\nmodeled as a modified blackbody radiation with two\nfloating parameters: temperature Tand spectral index\nβ,\nIDust,CIB (ν, T, β ) =τDC(ν/νDC)βB(ν, T),\nB(ν, T) =2(kT)3\n(hc)2x3\nex−1,(25)\nwhere νDC= 353 GHz. The boundaries of the parame-\nters ( T, β) domain were determined in [48] using Planck6\ndata [50, 51]. The probability distribution function for\nthese parameters was calculated for 10 degrees circular\nsky part centered at l= 13.731o,b=−73.946o, see Fig.\n1(a). Two isocontour solid lines limit the region Ω( T, β)\nof possible parameters variations for both dust and CIB.\nThe probability to find parameters outside these two\nspots is less than 0.0002. The maximum allowable value\nof emissivity τDCfor the data we used does not exceed\n10−6.\nSynchrotron radiation is modeled according to [52] and\nits spectrum has a power-law form with the single free\nparameter βs:\nIsync(ν, βs) =As(ν/νs)−βs, (26)\nwhere νs= 30 GHz and As<1000 Jy/sr. In accordance\nwith the results of [53], βscan vary from 0.9 to 1.4.\nFree-free emission is given by the following formula in\n[54]:\nIff(ν) =Aff\u0012\n1 + ln\u0014\n1 +\u0010νff\nν\u0011√\n3/π\u0015\u0013\n, (27)\nwhere, Aff<500 Jy/sr, νff= 255 .33\u0000Te\n1000K\u00013/2GHz.\nAccording to [54], the parameter Te= 7000 K±3Kdoes\nnot vary across the sky strongly enough to noticeably\nmake any difference in the shape of the spectrum. There-\nfore, in our calculations we consider free-free spectrum as\nwell-defined one without any parameter variations.\nThe instrument optics emission is considered in this\npaper as a graybody radiation with varying temperature\nTopt:\nIopt=τoptB(ν, Topt) (28)\nTemperature variations can depend on the stability of\nthe cooling system, design features and quality of the\noptical system. In our numerical calculations we use a\npossible range of temperature Toptvariations as Tmin≤\nTopt≤TmaxandTmax−Tmin≤2K. The emissivity τopt\ndepends on the quality of the polishing of the reflecting\nsurfaces of the optical system. We use three different\nupper limits for emissivity: τopt≤0.001,τopt≤0.01 and\nτopt≤0.05.\nThe photon noise comes from the CMB radiation, a\nnumber of main foregrounds and emission from the in-\nstrument. All these sources of fluctuations can be char-\nacterized by the noise equivalent power ( NEP ) which is\nmeasured in units of W/Hz1/2. For the FTS, intensity of\nthe noise (i.e. the 1 σsensitivity) is given by the following\nequation [55]:\nσ= 0.61NEP\n∆ν√\ntG, (29)\nwhere tis the integration time and Gis the throughput of\nthe system. The value Gis the solid angle of the entrance\npupil seen from the detector multiplied by the area of theinstrument feedhorn. For the diffraction-limited instru-\nment it can be estimated as\nGdif= (0.61πc/νmin)2. (30)\nHowever, the instruments we analyze are not necessarily\ndiffraction-limited, so generally it is possible to have G≫\nGdif.\nTheNEP is computed as\nNEP2=NEP2\ndet+NEP2\nF, (31)\nwhere NEPdetis the intrinsic NEP of the detector while\nNEPFis created by Mvarious foregrounds [see Eq. (1)].\nWe assume that NEPdet≤10−19W·Hz−1/2≪NEPF\nin our estimates. Therefore:\nNEP2≈NEP2\nF=MP\nm=1NEP2\nm,\nNEP2\nm=νmaxR\nνmin4Gh2\nc2ν4nm(ν) [1 + nm(ν)]dν,(32)\nwhere nm(ν) =c2\n2hν3Im(ν) is the photon concentration\nin phase space for mth foreground component. Here we\nassume ideal optical and main beam efficiency.\nIf the noise is determined by a weak constant fore-\nground I(ν) =const with nm≪1, from (32) and (29)\none can obtain:\nσ∝r\nI\ntνmax−νmin\n∆ν,\nwhich coincides with the often used estimate of FTS sen-\nsitivity, see, e.g. [56].\nAs one can see from Eqs. (29)-(32), the noise ampli-\ntude depends on the intensity of radiation coming from\nthe sky and from the all optical system, on the FTS fre-\nquency range and on the spectral resolution. The main\nsources of the noise are CMB monopole, dust radiation,\nCIB and the radiation emitted by the instrument optics.\nIn this section we use a white noise model. Therefore the\nnoise covariance matrix has a diagonal form C=σ2E,\nwhere σis the sensitivity. This kind of noise distribu-\ntion corresponds to a single frequency band of the FTS\ninstrument.\nWe consider the single band receiver for 384 channels\n(7.5 GHz each) from 15 GHz to 2895 GHz. The through-\nput is computed as for a diffraction-limited system, i.e.\nfrom Eq. (30).\nB. Numerical results\n1. Dust and CIB foregrounds\nWe start our analysis by comparing the efficiency of the\nMILC and LRM methods for Iµ(ν) distortion detection\nwith a simple example when only dust and infrared radi-\nation are taken into account as foregrounds. Noise level7\n10 20 300.41.22.0\nT, Ka\nDust\nCIB\n0.00.20.40.60.8\n10 20 300.41.22.0\nT, Kb\nMILC, n=2 |R(F)|\n0.00.20.40.60.81.0\n10 20 300.41.22.0\nT, Kc\nMILC, n=2 R(F+N)\n0.00.20.40.60.81.0\n10 20 300.41.22.0\nT, Kd\nMILC, n=3 |R(F)|\n0.00.20.40.60.81.0\n10 20 300.41.22.0\nT, Ke\nMILC, n=3 R(F+N)\n0.00.20.40.60.81.0\n10 20 300.41.22.0\nT, Kf\nMILC, n=4 |R(F)|\n0.00.20.40.60.81.0\n10 20 300.41.22.0\nT, Kg\nMILC, n=4 R(F+N)\n0.00.20.40.60.81.0\n10 20 300.41.22.0\nT, Kh\nLRM |R(F)|\n0.00.20.40.60.81.0\n10 20 300.41.22.0\nT, Ki\nLRM R(F+N)\n0.00.20.40.60.81.0\nFIG. 1: MILC and LRM methods for separating µdistortions when only dust and CIB are taken as foregrounds. Panel (a)\nshows the probability distribution function for Tandβparameters. Isocontour lines limit the Ω region of parameter variations.\nPanel (b) shows MILC response to foreground |R(F)|if n=2. Dark red indicates the region where the response to the\nforeground exceeds the response to the µsignal: |R(F)|≥1.Panel (c) shows the total MILC noise+signal response R(F+N)\nif n=2. Panel (d): |R(F)|for MILC, n=3. Panel (e): R(F+N) for MILC, n=3. Panel (f): |R(F)|for MILC, n=4. Panel\n(g):R(F+N) for MILC, n=4. Panel (h): |R(F)|for LRM. Panel (i): R(F+N) for LRM.\n(or sensitivity) in this example is σ= 1 Jy/sr per single\nchannel. The MILC method was applied for three differ-\nent versions of the Taylor series expansion of the modified\nblack body in terms of parameters Tandβ[see Eq. (25)]\nto the second, third and fourth order respectively in the\nvicinity of the reference value P0of the vector P(T, β):\nP0= (T0, β0),T0= 21.2 Kβ0= 1.11 in Eq. (12). So the\nnumber of constraints zeroing derivatives over two vari-\nables up to the nth order are as follows: 6 constraints for\nn=2, 10 constraints for n=3 and 15 constraints for n=4.\nThe results are shown in Fig. 1.\nForn= 2, the response to foreground |R(F)|and\nthe total response to foreground and noise R(F+N) are\nshown in Fig. 1(b) and in Fig. 1(c) respectively in com-parison with the response to the µsignal R(Iµ) = 1.\nObviously, the constraints imposed in Eq. (12) for n= 2\nare not enough to clean the area Ω from the influ-\nence of the foreground, that is, to ensure the condition\n|R(F)|<< R (Iµ) inside two spots limited by isocontour\nlines.\nForn= 3, the foreground response is quite low com-\npared to the signal response, although the noise response\nbecomes relatively large and as a result R(F+N)≈\n0.8R(Iµ), see Fig. 1(d) and Fig. 1(e).\nIn the case when n= 4 [Figs. 1(f) and 1(g)], the\nregion Ω is excessively clean from the influence of the\nforeground, but at the same time 15 constraints in Eq.\n(12) lead to an extremely high response to random noise8\nfor the given sensitivity, |R(N)|>> R (Iµ).\nThus, the choice of n= 3 is the optimal number of\nmoments in the decomposition of dust and CIB spectra\nwhen using the MILC method. Figure 2 shows the depen-\ndence of responses to foreground and noise on the num-\nber of constraints imposed. As this number increases,\nthe response to the foreground gets smaller and, at the\nsame time, the response to random noise rises due to a\ndecrease in the index Γ which is a measure of orthogo-\nnality between foreground and µsignal [see Eq. (17)].\nSo the total response to the foreground+noise reaches its\nminimum at a certain number of constraints.\nLRM, in contrast to MILC, assumes one single con-\ndition on foregrounds and noise and provides us with\nthe optimal weight vector ωto minimize the ratio (fore-\nground+noise)/signal. The result can be seen in Fig.\n1(h) and Fig. 1(i) for foreground and foreground+noise\nresponses, respectively and in Fig. 2.\nAn interesting approach is to study the responses to\nthe foreground and random noise on the sensitivity of\nthe experiment: R(F, σ) and R(N, σ). This makes it\npossible to estimate the necessary sensitivity for reliable\ndetection of certain spectral distortions in the presence\nof foregrounds. In addition, this dependence allows one\nto compare the effectiveness of different methods of data\ncleaning.\nFigure 2 (right panel) shows the response to foreground\nand noise as a function of σfor the MILC and LRM meth-\nods. It is important to note that for the MILC method,\nthe weight vector ωdoes not depend on the sensitivity.\nThus, the dependence of the response to noise on σis\nlinear ⟨R2(N)⟩1\n2=σ(ω·ωT)1\n2. Since the response to the\nforeground is constant and given by Eq. (13), the over-\nall response to noise and signal as a function of σhas a\nsimple analytical form:\nRMILC (F+N, σ) =\u0002\n(ω·ωT)σ2+R2\nn+1(F)\u00031\n2.(33)\nUnlike the MILC method, when applying the LRM fil-\ntering, the weights ωjdepend on the magnitude of the\nphoton noise: ω=ω(σ). As can be seen from Fig. 2,\nthe total response to noise and foreground for the LRM\napproach is always significantly less than for the MILS\nand provides a better signal/(foreground+noise) ratio:\nRLRM(F+N, σ)< RMILC (F+N, σ). (34)\nThis inequality is true for any sensitivity σ. The rea-\nson for this is that LRM, unlike MILC, does not require\ncomplete orthogonality of the investigated signal to the\nforeground.\n2. All foregrounds\nBelow we perform an analysis of two approaches to the\nproblem of signal separation, taking into account all the\nmain foregrounds, adding them to the already consideredcomponents of dust and infrared radiation. As mentioned\nabove, the signals of relic origin and free-free emission are\nconsidered as well-defined components of the spectrum\nwithout parameter variations. They are only subject to\nrestrictions from above in terms of amplitudes.\nThe model of synchrotron radiation depends on one\nparameter βswhich varies from 0.9 to 1.4. Empiri-\ncally, it was found that three derivatives of the expansion\nin a Taylor series in the vicinity of the reference point\nβs= 1.15 are sufficient to get rid of the response to this\ncomponent for MILC method. Thus, the synchrotron re-\nquires four constraints for the weights ω(the function\nitself and three derivatives).\nThe spectrum emitted by the optical system of the in-\nstrument in our example depends only on its temperature\nin the vicinity of Topt= 10 K and can vary between 9\nK and 11 K. To reliably get rid of the influence of this\nradiation on data analysis, it is necessary to zero the re-\nsponse to the function B(ν, Topt) and seven derivatives\nover temperature: ∂nB(ν, Topt)/∂Tn\nopt|Topt=10K,n≤7\n(eight constraints in total).\nThe results for µsignal extraction for both methods\nare shown in Fig. 3 (left panel). The more components\nare taken into account, the larger the foreground+noise\nresponse becomes for both methods. Nevertheless LRM\nalways shows an acceptable result, while MILC gives al-\nmost infinitely large response to noise when trying to get\nrid of all foregrounds. In the same figure (right panel)\nwe show the orthogonality measures of individual compo-\nnents to the µsignal: Γ m= Γdust+CIB,ΓCMBA , ...,Γoptics\nalong with a orthogonality measure of the aggregate of\ncomponents ΓΣ. It is important to note that there is no\ndirect analytical relation between ΓΣand Γ msince the\nforeground signals are not orthogonal to each other. It is\neasy to see, that RMILC (F+N)∼Γ−1\nΣas it should be.\nFinally, Fig. 4 shows the dependence of the response\nR(F+N, σ) on the sensitivity of the experiment in the\npresence of all foregrounds for both methods for µsignal\ndetection (left panel). The right panel shows the same\nfunctions for the LRM approach in cases when the signals\nof interest are Iy0, Iy1, Iy2. In this case, the foreground\ncreated by the instrument is not taken into account, since\nwhen observing ydistortions, it is possible to use the sky\ndifference, which automatically excludes this component\nfrom the observed signal.\n3. Optimal temperature for an instrument’s optical system\nFor most space experiments, it is preferable to cool the\ntelescope’s primary mirror and other mirrors as much as\npossible to avoid creating additional photon noise and de-\ngrading the sensitivity. Below we will demonstrate that\ncooling the system of mirrors to too much low tempera-\nture, that is, close to the CMB temperature ∼3 K, can\nsignificantly worsen the result if the signal of interest is\nµ-type distortion. Indeed, on the one hand, a decrease\nin the instrument optics temperature leads to a decrease9\n12345678910111213141510-210-1100101102\nDust+CIB only\nn=0 n=1 n=2 n=3 n=4\nMILC\nLRMµsignal\nnumber of constraintsResponse\n10-610-510-410-310-210-110010110-610-510-410-310-210-1100101 Dust+CIB only\nSensitivity, Jy/srResponse\nµ signal\nMILC, n=3\nLRM\nFIG. 2: Left panel : red solid line: total response to foreground+noise for MILC as a function of the number of constraints.\nRed dashed line: Noise response for MILC. Red dash-dotted line: MILC response to foreground. The blue solid, dashed, and\ndash-dotted lines show LRM responses to foreground+noise, noise, and foreground, respectively. Right panel : responses to\nnoise and foreground as functions of sensitivity for MILC (n=3) and LRM approaches. Solid, dashed, and dash-dotted lines\nrepresent the same as in the left panel.\n10-1100101\ndust\n+CIBCMBA y0y1y2synch fr-fr opticsResponseMILC\nLRM\n12345\ndust\n+CIBCMBA y0y1y2synch fr-fr opticsΓ−1\n100101102103104105106107\nΓ−1\nΣ\nFIG. 3: Left panel : the histogram represents the response to foreground+noise with the sequential addition (one by one from\nleft to right) of various components to the studied signal. Thus, the leftmost column shows the response to foreground+noise in\nthe presence of only dust and CIB radiation as foreground, and the rightmost column shows this response if all the components\nlisted along the horizontal axis are taken into account. Calculations were made for MILC and LRM to separate µdistortions\nand correspond to the sensitivity σ= 1 Jy/sr. Right panel : the green histogram shows the measure of orthogonality Γ cof the\nµsignal to each individual component. The red step line is the measure of orthogonality Γ Σofµdistortion to all components\nto the left of the step in question (similar to the left panel).\nin photon noise, on the other hand, the approach of this\ntemperature to the temperature of CMB reduces the de-\ngree of orthogonality of the µsignal to the foreground\ncreated by the optics. Despite the fact that LRM does\nnot assume complete orthogonality to the foregrounds\n(including orthogonality to the signal created by the op-\ntics), the combination of CMB related signals together\nwith a poorly defined signal from the optics can, to a cer-\ntain extent, mimic the desired µsignal. This inevitablyincreases the response to foreground+noise and, thereby,\nreduces the sensitivity of the experiment to µdistortion\nmeasurements. Figure 5 demonstrates this effect. Shown\nhere are the responses R(F+N, Topt) as a functions of\nthe temperature of the optics for three different values\nof optics emissivity τopt. The responses are normalized\nin such a way that for all three considered τopt, a sen-\nsitivity of 1 Jy/sr is achieved at Topt= 10 K. In the\nsame figure we show the orthogonality measure of the10\n10-510-410-310-210-110010110-210-1100101102103\nSensitivity, Jy/srResponseµ signal\nMILC, total\nMILC, no optics\nLRM, total\nLRM, no optics\n10-310-210-110010110210310410510-310-210-1100101\nSensitivity, Jy/srResponsey0 LRM\ny0 MILC\ny1 LRM\ny1 MILC\ny2 LRM\ny2 MILC\nFIG. 4: Left panel : the result of applying the MILC and LRM methods to separate the µsignal. The dependencies of the total\nnoise+foreground response on the sensitivity of the experiment in the presence of all the main foregrounds are shown. The\ndashed lines show results for an ’ideal’ experiment with no influence from the instrument optical system. Right panel shows the\nresults of extracting y0,y1andy2signals for the LRM and for MILC approaches. We show the responses to noise+foreground\nwhen taking into account all the main foregrounds except the instrument optics.\n3 5 7 9 11 13 15 17 195101520\nTopt , KResponse   and   Γ−1\nΣΓ−1\nCMB+opticsτopt=0.05\nτopt=0.01\nτopt=0.001\nFIG. 5: Solid lines show the response to foreground+noise\nas a function of the temperature of the instrument optics for\ndifferent emissivity values. The dashed line corresponds to\nthe value which is inverse to the measure of orthogonality\nbetween Iµand a set of signals associated with CMB and\noptics: ICMBA , Iy0, Iy1, Iy2, Ioptics .\nIµto the combination of CMB related signals and op-\ntics: ICMBA , Iy0, Iy1, Iy2, Iopt. We denote this measure\nas Γ CMB +optics . It is assumed that the temperature can\nvary within Topt±1 K. At low temperatures, this response\nbehaves in a similar way as Γ−1\nCMB +optics and, therefore,\nincreases strongly if Topt→TCMB. At high temperatures\nthe response increases due to an increase in photon noise.\nThis effect obviously depends on the emissivity of the op-\nticsτopt. Nevertheless, even for small emissivity, cooling\nthe mirror to temperatures close to TCMB is undesirable.Finally the minimum response is reached when Topt∼9\nK.\nIt should be noted that the dependence we calculated\nwill change for a different FTS configuration, but the\neffect of increasing the response to foreground+noise at\nTopt∼TCMB will remain relatively strong in any case.\nC. The instrument\nBelow we present the example of a possible space mis-\nsion for measuring CMB spectral distortions of µtype.\nFigure 6 shows the top view of the instrument called\nSIMBAD (Spectroscopic Interferometer for Microwave\nBAckground Distortions). The concept, originally de-\nveloped to be placed at L2, has been then proposed to\nbe deposited in a permanently shadowed region (PSR)\npresent in some polar lunar craters [36] since they could\noffer a very low temperature, passive cooling for the in-\nstrument. The property of a few spots within the PSRs\nwhich reached 18 K has been put forward [57], as provid-\ning a much lower temperature than is obtained at L2 for\nJWST [58] to the price of a huge sunscreen. The paper\n[59] is part of a special issue devoted to the “Astronomy\nfrom the Moon: the next decades”. In this version of\nSIMBAD (Fig. 6) the diameter of the telescope primary\nmirror is pushed to 1.5 m, feeding a dual-input imaging\nFTS [56].\nAll the instrument is placed in a cryostat designed to\nfit into the internal fairing of the largest cargo launcher\nin development to go to the Moon [60]. To bend the\nbeam, the telescope primary mirror and the following\nflat mirror must be actively cooled at 9 K according to\nour results (Fig. 5). But the IFTS behind (right part of11\nthe cryostat) is actively cooled at a lower temperature,\nclose to the CMB temperature, since its contribution to\nthe foregrounds is cancelled by the differential concept of\nthe instrument, reducing the photon noise budget. If the\nSIMBAD cryostat is installed on one of the cold lunar\nspots at 18 K, as described in [36], then such a low tem-\nperature passive cooling could reduce the power needed\nfor the active cooling of the whole instrument.\nAt the two IFTS outputs there are 2 ×2 bolometers\nwith their feeding horn as shown on Fig. 6, each of 1 .5◦\naperture on the sky, providing a total field coverage of\n3◦×3◦for each data acquisition. The spectral coverage\nof the IFTS is going from 15 to 2895 GHz, to cover the\nfull CMB domain. The maximum motion of the moving\nmirror of the IFTS (6 mm) is adjusted to reach a 7.5\nGHz resolution on all the frequency domain, providing\n384 equal frequency channels. To be as simple as possi-\nble, the telescope has no tracking motion. The direction\nof pointing is determined by three adjustable legs which\nsupport the SIMBAD cryostat on the lunar ground. The\nscanning of a full ring of 3◦width is simply obtained by\nthe rotation of the Moon. Thanks to the IFTS [56], the\nfour bolometers are integrating simultaneously, making\na gain of a factor 2 in sensitivity for µcompared to an\nFTS with a single detector. In paper [36] the sensitivity\nof 1 Jy/sr is reached by a SIMBAD instrument with a\nfrequency resolution of 15 GHz and a 72 cm telescope\ndiameter, but already a total field-of-view of 3◦×3◦ob-\ntained by four detectors observing simultaneously for a\ntotal integration time of 4 years. With the new SIMBAD\nFIG. 6: SIMBAD instrument\nwith a frequency resolution divided by 2 and a collect-\ning area multiplied by 4, from Eq. (29), the same total\nintegration time is needed. As proposed in [36] the sen-\nsitivity and the field of view can be improved by placing\nin the same lunar crater several identical SIMBAD units\npointing on adjacent directions scanning a wider ring.\nIt is important to note that the instrument require-\nments for measuring µ,ysignals and recombination lines\nvary. For a µsignal, it is not important for us to havegood angular resolution, but we need reliable calibration\nof the instrument and the temperature of its optical sys-\ntem should be about 9 K. At the same time, y distortions\nand especially relativistic corrections y1,y2have a small\nangular size. For their observations, it is desirable to\nhave good angular resolution, but there is no need to\nhave a contrast between the CMB temperature and the\ntemperature of the instrument. Besides the calibration\nof the instrument for this task is not required. To detect\nextremely weak signals from hydrogen and helium recom-\nbination lines, it is desirable to cool the instrument optics\nas much as possible to reduce photon noise.\nThus, it is necessary to look for a very complex com-\npromise to combine all these requirements in one device\nif the goal of the mission is to simultaneously solve all\nthree problems. SIMBAD is mostly targeted for µsignal\nmeasurements.\nIV. CONCLUSIONS AND DISCUSSION\nDetecting CMB spectral distortions is one of the key\ngoals of modern observational cosmology. Hidden in fore-\ngrounds of cosmic and instrumental origin, these devia-\ntions of the CMB spectrum from a perfect black body\nshape contain a wealth of information unobtainable by\nother observational methods. To extract CMB frequency\nspectral distortions, a signal-foreground separation algo-\nrithm was introduced. The key feature of this algorithm\nis its weak sensitivity to the foregrounds spectral shapes.\nGiven the areas of foreground parameters variation,\nwhich are relatively easy to determine, and the upper\nbounds of the foreground amplitudes, whose overestima-\ntion would not skew our results, we are able to minimize\nthe response both to the foregrounds and to noise. With\nthis in mind, we have compared the performance of our\nLRM algorithm with the MILC method.\nStarting off with a limited number of foregrounds\n(dust and CIB) we define the minimal number of mo-\nments/constraints optimal for the MILC analysis. Since\nLRM allows for the investigated signal to be nonorthog-\nonal to the foreground, the total response to noise and\nforeground for our method is significantly less than for\nthe MILC for any sensitivity value. Subsequently, we\nbroaden the scope of our analysis by including CMB\nanisotropy, y-distortions, synchrotron and free-free emis-\nsion and radiation created by the instrument optics.\nEven though adding more components increases the to-\ntal response for both methods, the response to noise ob-\ntained with MILC exceeds LRM results by a wide margin.\nIn this paper we did not take into account Zodiacal light\nand CO lines as foregrounds in our modelling.\nIt is important to note that our LRM method allows\nfor an improvement by subtracting foreground compo-\nnents one by one from the observed signal. Applying our\nmethod sequentially to all foreground components, one\ncan estimate the real foreground amplitudes am. By sub-\ntracting foregrounds with estimated amplitudes from the12\ntotal signal, new restrictions can be made on the max-\nimum possible amplitudes of foreground residuals Am.\nReducing the upper limits of foreground amplitudes leads\nto a decrease in the response to foreground+noise and,\ntherefore, more accurate estimation of the spectral distor-\ntion amplitudes. Implementation of such improvement is\nout of the scope of this paper.\nFinally, we have established that cooling the instru-\nment down to the temperature close to the temperature\nof relic radiation spoils the results drastically when mea-\nsuring µtype distortions. This occurs mainly due to a\ndecrease in the measure of orthogonality of the µsignal\nto the instrument optics component. If the system of\nmirrors is cooled to the temperature of the relic, it itself\nbegins to create distortions close in shape to the CMB\nspectral distortions. 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Zarnecki, Discussion\nmeeting issue on Astronomy from the Moon: the next\ndecades, Philosophical Transactions of the Royal Soci-\nety A: Mathematical, Physical and Engineering Sciences\n(2023), in progress.\n[60]Nasa space launch system, 0080-sls-fact-sheet-jun2023-\n508-1.pdf ."    },    {        "title": "2401.13417v1.Interaction_of_intense_ultrashort_laser_pulses_with_solid_targets__A_systematic_analysis_using_first_principles_calculations.pdf",        "content": "Interaction of intense ultrashort laser pulses with solid targets: A systematic analysis\nusing first-principles calculations\nAtsushi Yamada\nDepartment of Applied Chemistry, National Defense Academy, Yokosuka 239-8686, Japan\nKazuhiro Yabana\nCenter for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan\n(Dated: January 25, 2024)\nIntense ultrashort laser pulse irradiation of solid targets was systematically investigated at the\nfirst-principles level, both theoretically and computationally. In the method, the propagation of a\npulsed light through a thin film is described by a one-dimensional Maxwell’s equation, and the mi-\ncroscopic electronic motion at different positions in the film is described by employing first-principles\ntime-dependent density functional theory (TDDFT). The method uses a coarse-graining approxi-\nmation to couple light propagation and electronic motion, and is termed the multiscale Maxwell-\nTDDFT method. The reflectance, transmittance, and absorbance of pulsed light incident normally\non thin films of 50–200 nm thickness were calculated for materials with different optical properties,\nsuch as aluminum (simple metal), graphite (semi-metal), silicon (small-gap dielectric), and quartz\n(wide-gap dielectric). Optical response transitions were explored as the light intensity shifted from\nthe linear regime, represented by the dielectric function for weak light, to the extremely nonlinear\nregime, represented by plasma reflection under intense light conditions. Numerous mechanisms that\ndepend on the laser pulse intensity and material type were found to contribute to these changes.\nThese include multiphoton absorption, saturable absorption, sign change of the effective dielectric\nconstant, and transition from quantum occupation to classical Boltzmann distribution. Thus, the\ncalculations provide a unified understanding of the interaction of intense pulsed light with solids,\noccurring on an extremely short time scale.\nI. INTRODUCTION\nThe interaction of light with solids varies significantly\ndepending on various factors, such as frequency, inten-\nsity, polarization, and pulse duration of the light and the\nelectronic structure of solids, dielectrics, or metals. These\nfactors determine the characteristics of the light-matter\ninteraction. To theoretically describe the interaction of\nintense ultrashort pulsed light with various materials, it\nis important to clarify the basic physical pictures that\nchange drastically depending on the temporal duration\nand intensity of the pulsed light.\nWe first considered the variation in the optical response\nas the light intensity increased. For ordinary weak light,\nthe dielectric function given by the perturbation theory\nof quantum mechanics determines the light-matter inter-\nactions. For moderately intense pulses, perturbative non-\nlinear optics phenomena such as the optical Kerr effect\nand second- and third-harmonic generation are exten-\nsively treated in nonlinear optics textbooks [1, 2]. As the\nintensity of pulsed light approaches the damage threshold\nof materials (but is still below the threshold), nonlinear\nresponses, which cannot be described by the perturba-\ntion theory, become significant, and they have been ex-\ntensively explored in the last decade. For example, high\nharmonic generation from solids [3–5] has attracted sig-\nnificant interest, possibly leading to the development of\ncompact XUV sources. Ultrafast changes in the opti-\ncal properties of materials, such as the metallization of\ndielectrics [6], have also attracted interest in the devel-\nopment of new optical device principles, such as peta-hertz optoelectronics [7]. Above a certain critical inten-\nsity, materials suffer permanent and irreversible damage.\nThe formation of periodic microstructures at surfaces has\nbeen observed and attracting interest in the mechanism\nof ultrafast energy deposition [8]. Femtosecond-laser pro-\ncessing using ultrashort laser pulses has attracted consid-\nerable interest as a tool for precise material processing\n[9]. Further intense laser pulses instantaneously created\nplasma on the surface of materials. Strongly excited mat-\nter on a surface is characterized as warm dense matter\n[10], which is an intermediate state between ordinary con-\ndensed matter and hot plasma.\nNext, we considered the variation in the optical re-\nsponse as the temporal duration of the pulse decreased.\nWhen the pulse duration is sufficiently longer than a cer-\ntain relaxation time, the irradiated material can be con-\nsidered to be in a quasi-equilibrium state with respect to\nthe related relaxation dynamics. Thus, there is a balance\nbetween excitation by the laser pulse and de-excitation by\nthe relaxation process. As the pulse duration decreases\nand approaches the relaxation time, the system is con-\nsidered to be in a nonequilibrium state. On further de-\ncreasing the pulse duration, if the pulse duration is suf-\nficiently shorter than any kind of relaxation, a ballistic\npicture becomes appropriate for electronic motion. This\nentails treating them as independent electrons moving at\nan average one-body potential.\nThere have been numerous reports on systematic mea-\nsurements of the reflectance, transmittance, and ab-\nsorbance of intense ultrashort pulsed light from the sur-\nface or thin films of various solids [11–31]. For example,arXiv:2401.13417v1  [physics.optics]  24 Jan 20242\nin [14], measurements were reported using laser pulses\nwith a wavelength of 800 nm, a duration of 120 fs, and\nintensities ranging 1013∼1018W/cm2for materials such\nas wide-gap dielectrics, quartz, and metals (Al, Au, etc.).\nWhile all materials eventually reflected the incident pulse\nby plasma reflection at intensities above 1015W/cm2, a\nstrong absorption was observed below this intensity, with\nthe strongest absorption observed in quartz. Numerous\ntheoretical approaches have been developed to describe\nthe carrier dynamics and light propagation in solids. In\nmetals, the increase in electronic temperature due to laser\nirradiation and resulting change in the dielectric constant\nhave been considered [17, 19, 32, 33]. In dielectrics, gen-\neration of free carriers is the most significant process, and\nit is described by multiple rate equations [16, 34–38]. In\neither case, the carrier density and collision frequency of\nthe electrons are the decisive parameters that determine\nthe propagation of the intense laser pulses within solids.\nIn this paper, we report a systematic theoretical in-\nvestigation of the reflectance, transmittance, and ab-\nsorbance of the intense ultrashort pulsed light from thin\nfilms of various materials, using the first-principles com-\nputational method in materials science. To obtain a\nunified view of the light-matter interaction in extremely\nshort time scales for wide intensities, we performed sys-\ntematic calculations of the propagation of a few-cycle\npulsed light of optical frequency in thin films of vari-\nous materials, including metals (aluminum), semi-metals\n(graphite), small-gap dielectrics (silicon), and wide-gap\ndielectrics ( α-quartz). The calculated results reveal the\nbasic features of the optical response, including a linear\nresponse for a weakly pulsed light, where the dielectric\nfunction determines the response, and an extremely non-\nlinear response for an intense pulsed light, where plasma\nreflection eventually dominates.\nWe employed multiscale formalism to describe linear\nand nonlinear light propagation. At the macroscopic\nscale, we solved a one-dimensional Maxwell’s equation to\ncalculate the reflectance, transmittance, and absorbance\nof the pulsed light that normally irradiate thin films of\na wide variety of materials. At the microscopic scale,\nwe employed the first-principles time-dependent density\nfunctional theory (TDDFT) [39, 40] to calculate the elec-\ntronic motion in a unit cell of crystalline solids at each\ngrid point in macroscopic coordinates. By solving the\ntime-dependent Kohn–Sham (TDKS) equation in real\ntime with the electric field of the pulsed light as an ex-\nternal potential, the Bloch orbitals can be evolved in\ntime to obtain the electric current induced by the pulsed\nlight [41, 42]. By simultaneously solving the Maxwell\nand TDKS equations, we can self-consistently describe\nthe light propagation and electronic motion. We call this\nframework the multiscale Maxwell-TDDFT method [43].\nFor weakly pulsed light, this calculation is equivalent to\nthe description of light propagation by macroscopic elec-\ntromagnetism, where a dielectric function given by the\nTDDFT is utilized. For an intense pulsed light, this\nmethod can describe nonlinear light propagation with-out any perturbation theory expansions.\nThe remainder of this paper is organized as follows.\nIn Section II, we present theoretical and computational\nmethods to describe nonlinear light propagation as well\nas first-principles light-electron interactions. In Section\nIII, the results of systematic calculations are presented\nand discussed. Section IV presents a summary of the\nresults.\nII. THEORETICAL AND COMPUTATIONAL\nMETHOD\nWe investigated nonlinear light propagation through\nthin films of four typical materials, namely aluminum\n(Al), graphite, silicon (Si), and α-quartz (SiO 2), with\ndifferent optical properties. For a thin film of thick-\nness 50–200 nm placed in a vacuum, a linearly polar-\nized, few-cycle pulsed light irradiates normally on the\nfilm. The multiscale Maxwell-TDDFT method developed\nin Ref. [43] is used to describe light propagation and\nelectronic motion inside the thin film. In this section,\nwe first describe the formalism based on the TDDFT\nfor calculating the electronic motion in a unit cell of a\ncrystalline solid, driven by an external, spatially-uniform\nelectric field [41, 42]. We then explain the coupling of the\nTDDFT framework in a unit cell with a one-dimensional\nMaxwell’s equation that describes light propagation.\nA. Electronic motion in a unit cell\nWe describe the electronic motion in a unit cell of\na crystalline solid driven by a spatially uniform pulsed\nelectric field with a given time profile, using the orbitals\n{unk(r, t)}as functions of the spatial vector rand time\nt. Here, nis the band index, and kis the crystalline\nmomentum. The orbitals satisfy the TDKS equation,\n[41, 42]\niℏ∂\n∂tunk(r, t) =hKSh\nk+e\nℏcA(t)i\nunk(r, t),(1)\nwith the Kohn-Sham Hamiltonian\nhKS[k] =1\n2m(−iℏ∇+ℏk)2−eϕ(r, t)\n+ ˆvk\nNL+Vxc(r, t). (2)\nHere, the vector potential A(t) is related to the applied\nelectric field E(t) as follows:\nE(t) =−1\ncd\ndtA(t). (3)\nWe treat the interaction of valence electrons with ions us-\ning a norm-conserving pseudopotential [44]. The scalar\npotential ϕ(r, t) exhibits the same periodicity as the lat-\ntice and satisfies the following equation:\n∇2ϕ(r, t) =−4πe(nion(r)−ne(r, t)), (4)3\nwhere the ionic charge density nion(r) is included to pro-\nduce the local part of the pseudopotential in ϕ. In this\nstudy, the atomic positions are fixed during the time evo-\nlution calculations. The electron density ne(r, t) is given\nby\nne(r, t) =1\nNkoccX\nn,k|unk(r, t)|2, (5)\nwhere Nkdenotes the number of k-points. The sum n\nis taken over the occupied bands. The nonlocal part of\nthe pseudopotential is modified for a k-point system as\nˆvk\nNL≡e−ik·rˆvNLeik·r. Here, ˆ vNLis the nonlocal part of\nthe pseudopotential for which we take a usual separa-\nble form [45]. Vxc(r, t) is the exchange-correlation poten-\ntial, for which the adiabatic approximation is assumed\nbased on either the local density approximation [46] or\nmeta-GGA approximation [47]. The Bloch orbitals were\ninitially set as ground-state solutions.\nBy solving the TDKS equation for a given electric field\nspecified by the vector potential A(t), we obtain various\nphysical quantities, including the electric current density,\nelectronic excited energy, number density of excited elec-\ntrons, and dielectric function after the pulse ends. The\nquantities are defined as follows:\nThe electric current density averaged over the unit cell\nvolume Ω is given by\nJ[A(t)](t) =−e\nmZ\nΩdr\nNkΩoccX\nn,ku∗\nnk(r, t)\n×n\n−iℏ∇+ℏk+e\ncA(t) +m\niℏh\nr,ˆvk+e\nℏcA(t)\nNLio\nunk(r, t).\n(6)\nThis can be regarded as the macroscopic electric current\nfor a given electric field of pulsed light and be utilized as\nthe constitutive relationship in the multiscale Maxwell-\nTDDFT method. For a sufficiently weak pulse, the cal-\nculated macroscopic electric current coincides with that\ngiven by the linear constitutive relationship with the con-\nductivity tensor σ0\nµν(t).\nJµ(t) =X\nνZt\ndt′σ0\nµν(t−t′)Eν(t′), (7)\nwhere µ, νdenote the Cartesian components, x, y, z . To\nindicate that this conductivity is related to the ground\nstate, we used the superscript 0 in the conductivity. This\nrelationship is regarded as the definition of the TDDFT\nconductivity in the ground state of the material. By solv-\ning the TDKS equation in real-time with a strong electric\nfield, we can obtain a macroscopic electric current that\nincludes the effects of nonlinear electronic motion with-\nout perturbative approximations.\nWe adopted the following definition for the number\ndensity of the excited electrons in the unit cell.\nnex(t) =1\nΩ(\nNe−X\nnn′k\f\f\f⟨u0\nnk+(e/c)A(t)|un′k(t)⟩\f\f\f2)\n,(8)where Neis the total number of electrons in the unit cell\nandu0\nnk+(e/c)A(t)is the Houston function, which is the\nBloch wave function in the ground state with a shifted\nvector potential at time t. It is also feasible to calcu-\nlate the electronic excitation energy at time tin the unit\ncell, which is given by the difference between the energy\nat time tand that in the ground state. A systematic\ncalculation is reported in [48].\nFinally, we introduce a dielectric function at the end of\nthe applied pulse. To introduce this concept, we regard\nthe applied pulsed electric field as a pump pulse and de-\nnote it as Epump(t). The current density induced by the\npump pulse is expressed as Jpump(t). In addition to the\npump pulse, a weak impulsive electric field is applied as\nthe probe pulse.\nEprobe(t) =−∆A\ncδ(t−t0), (9)\nwhere t0denotes the time at which the pump pulse ended.\nAn impulsive electric field can be computationally ap-\nplied using a small shift in the vector potential, A(t)→\nA(t) + ∆Aθ(t−t0). We denote the current density due\nto both pump and probe pulses as Jpump+probe(t). The\nconductivity of the electronic state excited by a pump\npulse is introduced in [49]\nJpump+probe\nµ (t)−Jpump\nµ(t) =X\nνZt\ndt′σpump\nµν(t−t′)Eprobe\nν(t′).\n(10)\nUsing the conductivity, we obtained the dielectric func-\ntion after the pump pulse ended.\nϵpump\nµν(ω) =δµν+4πiσpump\nµν(ω)\nω, (11)\nwhere σpump\nµν(ω) is the Fourier transform of σpump\nµν(t).\nThe defined dielectric function depends on t0. However,\nthis dependence is weak [49].\nB. Multiscale Maxwell-TDDFT method\nNext, we explain the multiscale Maxwell-TDDFT\nmethod [43], which describes light propagation by cou-\npling the formalism of the microscopic electronic motion\ndescribed above with the macroscopic one-dimensional\nMaxwell’s equation. We consider the irradiation of a thin\nfilm of thickness din vacuum by ultrashort pulsed light of\na linearly polarized plane wave at normal incidence. We\nassume that the thin film is in the yz-plane with polar-\nization in the z-direction and that the pulsed light prop-\nagates along the x-axis. For one-dimensional light propa-\ngation, Maxwell’s equation yields a one-dimensional wave\nequation for the vector potential.\n\u00121\nc2∂2\n∂t2−∂2\n∂X2\u0013\nAX(t) =4π\ncJX(t), (12)4\nwhere Xis the coordinate describing the light propaga-\ntion. The material is assumed to be placed in the region\n0< X < d .AX(t) and JX(t) are the vector poten-\ntial and current density, respectively, at position X. To\nsolve this equation, we introduce a uniform spatial grid\nfor coordinate X.\nTo solve Eq. (12), we must determine JX(t) at each\ngrid point specified by Xinside the thin film for a given\nvector potential AX(t). For this purpose, we used the\nmicroscopic electronic motion described in the previous\nsubsection. At each point Xinside the film, we consider\nan infinitely extended electronic system under a time-\ndependent electric field EX(t) =−(1/c)(d/dt)AX(t).\nHere, the electric field EX(t) is regarded as spatially uni-\nform, and the dependence on microscopic coordinate ris\nignored. Then, the electronic motion at point Xis de-\nscribed by the Bloch orbitals unk,X(r, t) that satisfy the\nTDKS equation.\niℏ∂\n∂tunk,X(r, t) =hKSh\nk+e\nℏcAX(t)i\nunk,X(r, t),\n(13)\nwhere the Kohn–Sham Hamiltonian is given by Eq. (2).\nThe scalar and exchange correlation potentials also de-\npend on X. The macroscopic current density JX(t) at\neach Xis expressed using Eq. (6) as\nJX(t) =J[AX(t)](t). (14)\nBy simultaneously solving Eqs. (12), (13), and (14), we\nobtain the solution AX(t) for a given initial pulse gen-\nerated in the vacuum region in front of the thin film.\nAt the beginning of the calculation, the Bloch orbitals\nunk,X(r, t), for all grid points of Xinside the thin film,\nwere set to the ground-state solution.\nFor the incident laser pulse, the following time profile\nwas adopted:\nA(t) =−cE0\nω0sin\u0014\nω0\u0012\nt−T\n2\u0013\u0015\nsin2\u0012πt\nT\u0013\nˆz,(0< t < T ).\n(15)\nwhere E0is the maximum amplitude of the electric field,\nω0is the fundamental frequency, and Tis the total\npulse duration. For the calculations, we set T= 20 fs\nand ℏω0= 1.55 eV. The full width at half-maximum\n(FWHM) of the pulse is approximately 7 fs. The initial\nvector potential at t= 0 is expressed as\nAX(t= 0) = A\u0012\n−X\nc\u0013\n. (16)\nThe reflectance Rand transmittance Twere evaluated\nas the ratio of the energies of the reflected and trans-\nmitted waves to the energy of the incident pulse. The\nenergies of the respective waves were evaluated at the fi-\nnal time of the propagation. The absorbance Awas then\ngiven by A= 1−R−T.\nThe multiscale Maxwell-TDDFT method has been suc-\ncessfully utilized for various nonlinear optical responses,\nincluding attosecond transient absorption spectroscopy[50], energy transfer from intense few-cycle pulsed light to\ndielectrics [51–53], propagation effects in high-harmonic\ngeneration [54, 55], and the generation and propagation\nof coherent phonons [56].\nC. Numerical detail\nWe utilized an open-source software package SALMON\n(Scalable Ab initio Light-Matter Simulator for Optics\nand Nanoscience) [58, 59], for which the authors are\namong the leading developers. In this code, the finite-\ndifference method was used to solve for the electronic\norbitals and electromagnetic fields. The temporal evo-\nlution of the electron orbitals was determined using the\nTaylor expansion method [60].\nTo solve the TDKS equation, we use orthogonal coordi-\nnate systems for four materials:, namely Al, Si, graphite,\nand SiO 2. The crystalline structures, crystalline pa-\nrameters, and grid parameters of the unit cell and light\npropagation calculations, number of k-points, time steps,\nnumber of time steps, and exchange-correlation poten-\ntials employed are summarized in Table I. We adopted\nan adiabatic approximation, utilizing the ground-state\nexchange-correlation potential in the time evolution cal-\nculation. For Al and graphite, we employed a simple local\ndensity approximation (LDA) [46]. For Si and SiO 2that\nhave finite band gaps, we employed the meta-GGA po-\ntential of Tran and Blaha [47] (abbreviated TB-mBJ),\nwhich reproduces the bandgap energies of various di-\nelectrics reasonably.\nIn first-principles optical response calculations, it is of-\nten necessary to use a large number of k-points to obtain\na convergent result. This is particularly significant when\nlong pulses with sharply defined frequencies are used.\nBecause multiscale Maxwell-TDDFT calculations require\nthe extensive use of supercomputer resources and we em-\nploy a relatively short incident pulse of 7 fs at FWHM,\nwe use moderately fine grid points in the k-space. We\nconsider the results obtained here using the grid parame-\nters shown in Table I to be sufficiently accurate, at least\nfor the discussion here.\nIII. RESULTS AND DISCUSSION\nA. Typical time-evolution calculation\nA typical calculation using the multiscale Maxwell-\nTDDFT method with a Si thin film as an example is\nshown in Fig. 1. Fig. 1(a) shows the grid system used\nto calculate light propagation. The Si thin film exists\nin the ”material region” of the ”macroscopic space.” Be-\ncause we set the film thickness to 200 nm and grid spacing\nforXto 10 nm, as described in Table I, the number of\ngrid points for Xin the material region is 20. At each\ngrid point, indicated as ”microscopic space,” electronic\nmotion is calculated using the TDDFT.5\nAl Graphite Si α-quartz (SiO 2)\nCrystalline structure fcc hexagonal diamond rhombic\nLattice constants ( ˚A) 4.049 a= 6.696,b= 4.254,c= 2.456 5.429 a= 4.913,b= 8.510,c= 5.405\nNumber of atoms 4 8 8 18\nNumber of grid points 30332×20×16 32328×50×32\nk-points 16 ×16×44 adaptive [57] 12343\nFunctional LDA [46] LDA [46] TB-mBJ [47] TB-mBJ [47]\nFilm thickness (nm) 50 200 200 200\ngrid spacing for X(nm) 2.5 10 10 10\nTime step (as) 0.5 0.5 0.5 0.5 - 0.3\nNumber of time step 60,000 60,000 60,000 60,000\nTABLE I. Summary of the crystal structures, grid information, and exchange-correlation potential used for the four materials.\nFor all systems, a rectangular unit cell is used, with the ( a, b, c ) axes set parallel to the ( x, y, z ) directions. The light propagates\nalong the x-direction and polarizes along the z-direction. The thin film is in the yz-plane.\nFigs. 1(b–d) show the snapshots of the electric field\nEX(t) =−(1/c)dAX(t)/dtat three different times: (b)\natt= 0, when the incident pulse is placed in the vacuum\nregion in front of the film; (c) at t= 12 fs, when the\npulse passes through the film; and (d) at t= 24 fs, when\nthe pulse separates into transmitted and reflected pulses.\nThe three cases of pulse propagation with different maxi-\nmum intensities of the incident pulse are compared in the\npanels shown in (b)—(d). Grey lines indicate the elec-\ntric field for a weak incident pulse, where the maximum\nintensity is set to Imax= 1011W/cm2. At this inten-\nsity, light propagation can be regarded as almost linear\nand be equivalently described by the dielectric function\nin the ground state. Red lines indicate the electric field\nfor a moderately intense incident pulse with Imax= 1013\nW/cm2. Blue lines indicate the electric field for a more\nintense incident pulse with Imax= 1015W/cm2. The\ncentral frequency of the pulse was set to 1.55 eV, which\nis below the direct bandgap of Si. The present calcula-\ntion, which uses the TB-mBJ functional, yields 3.4 eV,\nwhich is close to the measured value. The grey (red) lines\nare multiplied by a factor of 100 (10) such that the inci-\ndent pulses of the three cases coincide with each other.\nThe difference between the gray and other colored lines\nshows the nonlinear effects on propagation. As shown\nin the figure, nonlinear effects are more pronounced in\nthe transmission than in the reflection. For the case of\nImax= 1015W/cm2, the transmitted pulse almost disap-\npeared owing to the strong nonlinear absorption, whereas\nthe amplitude of the reflected pulse was the largest be-\ncause of the plasma reflection that occurred at the front\nsurface of the thin film.\nIn Fig. 1(e), the electric field at the front surface of the\nthin film ( X= 0) is plotted as a function of time. The\nelectric fields for the three cases with different incident\nintensities were compared. The electric field at the front\nsurface was given by the sum of the electric fields of the\nincident and reflected waves. Although the reflected wave\nbecame stronger in the higher-intensity case, as shown\nin Fig. 1(d), the electric field at the front surface was\nweaker than that expected in the linear response, owing\nto the destructive interference between the incident andreflected waves.\nFig. 1(f) shows the number density of excited electrons\nin the unit cell at the front surface of the thin film, calcu-\nlated using Eq. (8). Again, the results for the three cases\nwith different initial intensities were compared. These\nwere multiplied by the ratio of the intensity, 102(104),\nforImax= 1013(1011) W/cm2. If electronic excitations\noccur owing to linear absorption, the three lines should\nbe identical. Because the average frequency of the pulse\n(1.55 eV) is lower than the direct bandgap (3.4 eV), the\nexcitation is dominated by multiphoton absorption, and\nthe three lines show different behavior after multiplica-\ntion. At a weak intensity of Imax= 1011W/cm2, the\noscillations are proportional to E(t)2, which corresponds\nto the virtual electronic excitation. When Imax= 1015\nW/cm2, the number density becomes as high as approx-\nimately 1 .6×1023/cm3, which is close to the typical\ncarrier density of metals.\nFigs. 1(g) and (h) show the real and imaginary parts\nof the dielectric function, respectively, at t= 30 fs. At\nthis time, most of the reflected and transmitted waves\nhave exited the thin film, as shown in Fig. 1(d). The\ndielectric function after pulse propagation was calculated\nusing the procedure explained in Eqs. (9)–(11), where\nthe value of t0is set to t0= 30 fs. Black lines indicate\nthe dielectric function of Si in the ground state. Red\nlines indicate the dielectric function with a pump pulse\nofImax= 1013W/cm2. Green line in Fig. 1(g) shows\nthe dielectric function of the following model adding the\nDrude contribution to that in the ground state.\nϵD(ω) = Re [ ϵgs(ω)]−4πnex\nm∗ω2, (17)\nwhere nexis the number density of the excited electrons\nat the front surface of the thin film at t= 30 fs, and m∗\nis the effective mass of the electron. We use the value\nofnexobtained from Fig. 1(f) ( nex= 4.3×1021/cm3)\nand set m∗= 0.18 as employed in Ref. [16]. The green\nand red lines in Fig. 1(g) agree well indicating that the\nDrude model reasonably describes the change in the real\npart of the dielectric function, caused by the nonlinear\nelectronic excitation. A detailed discussion of this anal-\nysis is provided in Ref. [52].6\nBecause the medium is gradually excited by the elec-\ntric field of the pulsed light, the dielectric function also\nchanges over time from that in the ground state to that\natt= 30 fs. Therefore, the reflectance of a pulse is not\ndetermined by the dielectric function shown in Figs. 1(g)\nand (h) but by the transient dielectric function.\nB. Systematic analysis: overview\nWe performed systematic calculations of light propaga-\ntion through the thin films of four materials: Al (metal),\ngraphite (semi-metal), Si (small-gap dielectric), and SiO 2\n(wide-gap dielectric). As in the previous subsection, lin-\nearly polarized light with a duration of T= 20 fs (7 fs\nFWHM) and an average frequency of ℏω= 1.55 eV was\nused as the incident pulse for all calculations. The film\nthickness was set to 200 nm for all materials except for\nAl, for which a thickness of 50 nm was used. No transmit-\nting wave was observed for the 50 nm thick Al thin film.\nTherefore, the results would not change when calculated\nfor an Al thin film with a thickness of 200 nm.\nWe first present an overview of the reflectance and ab-\nsorbance of the four materials when the maximum inten-\nsity of the incident pulse is changed from Imax= 109\nW/cm2to 1017W/cm2. Figs. 2 (a) and (b) show\nthe results for the reflectance and absorbance, respec-\ntively. First, we examined both sides of the weak and\nintense pulses. For sufficiently weak incident pulses with\nImax<1010W/cm2, each material responds to light,\nas determined by its dielectric function in the ground\nstate, metallic reflection in Al, and transparent propa-\ngation in SiO 2. For sufficiently intense pulsed light with\nImax>1016W/cm2, reflectance dominates in all materi-\nals, indicating that the materials lose their individuality\nand behave as a plasma of electrons. A closer exam-\nination of the results presented in Fig. 2 reveals that\nthe change with the intensity of the light, the process\nby which a material loses its individuality and becomes\na plasma, is different for each material. In particular,\nthe reflectance and absorbance of graphite and Si exhibit\ncomplex and non-monotonic changes in intensity.\nIn the following subsections, we discuss the changes\nin the optical properties of each material in detail. Be-\nfore that, we discuss the adequacy and limitations of our\ntheoretical and computational methods for light-matter\ninteractions. Our calculations are ab initio and do not\ninclude empirical parameters. However, in practice, this\nis accompanied by significant approximations. Since the\nTDKS equation (1) is a one-body Schr¨ odinger-like equa-\ntion, non-relativistic kinematics is assumed for the elec-\ntronic motion. The effects of the magnetic field on the\nelectronic motion, which are significant at extremely high\nintensities, were ignored. These approximations can be\njustified if the speed of electrons is much lower than\nthat of light. The light intensity in the atomic unit was\nI= 3.51×1016W/cm2. At approximately or above this\nintensity, we can expect the significance of the relativistickinematics and Lorentz force due to the magnetic field for\nelectronic motion. However, the light field inside the ma-\nterial is different from that of the incident pulse. For the\nsufficiently short pulse employed here, the electric field at\nthe front surface of the thin film ( Esurface ) is given by the\nsum of the electric fields of the incident ( Eincident ) and re-\nflected ( Ereflected ) pulses ( Esurface =Eincident +Ereflected ).\nFigure 2 shows the plasma reflection dominates for ex-\ntremely intense incident pulses. As seen in Fig. 1(e), a\nstrong cancellation occurs between the incident and re-\nflected waves, such that the electric field inside the ma-\nterial ( Esurface ) is much smaller than the electric field of\nthe incident pulse ( Eincident ). In our calculations of the\nSi thin film, using an incident pulse with Imax= 1017\nW/cm2, the maximum electric field at the front sur-\nface was 12 V/m, corresponding to 0.23 in atomic unit.\nTherefore, the present calculation that uses nonrelativis-\ntic kinematics and ignores the magnetic field can be jus-\ntified even for the highest incident pulse of Imax= 1017\nW/cm2used in our calculations.\nThe TDKS equation (1) is a one-body Schr¨ odinger-like\nequation, where electron-electron interaction is treated\nonly partially. While part of the electron-electron inter-\naction can be incorporated into the one-body electronic\nHamiltonian of Eq. (2), such as the screening effect in the\nHartree potential, electron–electron collision effects ac-\ncompanying high-momentum transfer, such as avalanche\nand Auger recombination effects, are not included. Be-\ncause we fixed the atomic positions during the propaga-\ntion calculation, no energy transfer from electrons to ions\nwas considered. These approximations can be justified in\nthe present calculation using a very short incident pulse\nof 7 fs at FWHM.\nC. Systematic analysis: each material\n1. Aluminum\nFirst, we discuss the results of Al, which is a simple\nmetal. An adiabatic local density approximation was\nadopted for the calculation. Fig. 3(a) shows the re-\nflectance, transmittance, and absorbance of an Al thin\nfilm with a thickness of 50 nm against the maximum in-\ntensity of the incident pulse. Fig. 3(b) shows the electron\noccupation distribution in the unit cell at the front sur-\nface of the thin film at t= 30 fs for the four different in-\ncident intensities. The total density of states is indicated\nby the black line. Figs. 3(c) and (d) show the real and\nimaginary parts of the dielectric function, respectively.\nIn addition to the dielectric function in the ground state\nthat is indicated as ”no pulse,” the dielectric function in\nthe unit cell at the front surface of the thin film at t= 30\nfs is shown for the four different incident pulse intensities.\nThe dashed vertical lines in Fig. 3(a) are drawn at the\nintensities at which the lines are shown in Figs. 3(b)–(d),\nusing the same colors.\nAs seen in Fig. 3(a), reflection dominates at all inten-7\n0 10 20 30Time[fs]4\n2\n0\n-2\n-4Ez(t) [V/Å]E×10(Imax=1013)\nE(Imax=1015)E×102(Imax=1011)\n[W/cm2]\n0 10 20 30Time[fs]nex(t) [1023cm-3]\n012345\nnex×102(Imax=1013)nex×104(Imax=1011)\nnex(Imax=1015)\n[W/cm2]\nX\nxyzmaterialregion\nMicroscopicspace\n(TDDFT)Macrocopicspace\n(Maxwellequation)(a)\n-7-6-5-4-3-2-1 76543210\nX [μm]Ez(t,X) ×10 (Imax=1013W/cm2)\nEz(t,X) (Imax=1015W/cm2)Electricfield [arb. unit](b)t= 0 fs\n(c)t= 12 fs\n(d)t= 24 fsEz(t,X) ×102(Imax=1011W/cm2)\n(f)\n(X= 0 μm)(h) (e) \n(X= 0μm)(X= 0 μm) (g) \n012345678910\nEnergy [eV]Im[ ε(ω) ]60\n40\n20\n0Re[ ε(ω) ]\n-2002040\n-40Ground state\nImax=1013W/cm2\nEq. (17)\nFIG. 1. Overview of the multiscale Maxwell-TDDFT method for a pulsed light irradiation normally on a Si thin film of 200\nnm thickness. (a) Grid system for the macroscopic coordinate Xand microscopic space that is prepared for each macroscopic\ngrid point. Snapshots of the electric field (b) at the initial time t= 0; (c) at t= 12 fs, when the pulse passes through the film;\nand (d) at t= 24 fs when the pulse is separated into the transmitted and reflected waves. Propagations of the three cases with\ndifferent initial intensities are compared. (e) Electric fields as functions of time and (f) number density of the excited electrons\nat the front surface ( X= 0) of the thin film for the three cases with different initial intensities. (g) Real and (h) imaginary\nparts of the dielectric function at the front surface at t= 30 fs. See text for the detail.8\n109Al\nGraphite\nSi\nSiO 21.0\n0.8\n0.6\n0.4\n0.2\n010101011101210131014101510161017\nImax[W/cm2]Reflectance\n1.0\n0.8\n0.6\n0.4\n0.2\n0Absorbance\n10910101011101210131014101510161017\nImax[W/cm2]Al\nGraphite\nSi\nSiO 2(a)\n(b)\nFIG. 2. (a) Reflectance and (b) absorbance for a pulse irra-\ndiation of thin films of Al (red), graphite (blue), Si (green),\nand SiO 2(purple) against the maximum intensity of the in-\ncident pulse.\nsities. There was essentially no transmission in the film\nof 50 nm thickness at all intensity. Slight absorptions\nappear at weak intensities ( Imax<1011W/cm2). As\nthe intensity increased, the absorbance decreased, and\nthe reflectance increased. Furthermore, the film exhib-\nited almost complete reflection at an intensity of ap-\nproximately 1013∼1015W/cm2. Subsequently, the ab-\nsorbance increased to a peak value of 0.2 at around 1016\nW/cm2and then decreased. Next, we consider the phys-\nical mechanisms of these variations in the reflectance and\nabsorbance.\nThe slight absorbance at weak intensities ( <1011\nW/cm2) were caused by the interband transition, as in-\ndicated by the black line in Fig. 3(d), showing the fi-\nnite imaginary part at ℏω= 1.55 eV. We consider that\nan increase in the reflectance and a decrease in the ab-\nsorbance that appear as the intensity increases (up to\napproximately 1013W/cm2) are caused by saturable ab-\nsorption and are related to the change in the occupation\ndistribution. As seen in Fig. 3(b), a slight decrease in the\noccupation just below the Fermi level and slight increase\njust above the Fermi level can be observed at an incident\nintensity of 1013W/cm2. This small change in occupa-\ntion is considered to be responsible for the reduction of\nthe interband transition owing to the saturable absorp-\ntion. This hindrance of the absorption is caused by thedecrease in valence electrons that can be excited by the\nlinear absorption and by the increase in conduction elec-\ntrons, which hinders the linear absorption by Pauli block-\ning. The change in the occupation shown in Fig. 3(b) ap-\npears rather small at 1013W/cm2after integration over\nthek-space. Because linearly polarized light excites elec-\ntrons in a specific direction, the change in occupation at\na specific k-value can be substantial, even if the change\nin the occupation distribution is small.\nWe consider that the increase in absorption at approx-\nimately 1015−1016W/cm2is related to the significant\nchange in the electron occupation distribution from a\nquantum Fermi distribution to a classical Boltzmann dis-\ntribution. As seen in Fig. 3(b), although the occupation\nshows a clear Fermi surface up to an incident intensity of\n1013W/cm2, it becomes an intermediate state between\nthe quantum and classical distributions at an intensity of\n1015W/cm2. A classical Boltzmann distribution is ob-\nserved at 7 ×1015W/cm2. The dielectric function at the\nfront surface at t= 30 fs is shown in Figs. 3(c) and (d).\nThe real part of the dielectric function is dominated by a\nDrude-like negative contribution, irrespective of the inci-\ndent intensity, and the imaginary part shows a complex\nchange as the intensity increases. The imaginary part of\nthe dielectric function at ℏω= 1.55 eV shows a decrease\nof 1013W/cm2from that of the ground state. It contin-\nued to decrease until 1015W/cm2and then increased at\n7×1015W/cm2, which is consistent with the intensity\ndependence of the absorbance. Therefore, the change in\nthe absorbance is related to the change in the imaginary\npart of the dielectric function. Thus, during the occu-\npation change from the quantum Fermi distribution to\nthe classical Boltzmann distribution, the imaginary part\nof the dielectric function increases, and the absorbance\nshows a maximum.\nWe now compare our results with the measurements in\nthe literature. Although the setups of reported experi-\nments differ from the present calculation in many aspects,\nthe comparison of calculations and experiments is con-\nsidered important for clarifying the physical mechanism\nby which the light-matter interaction changes. Price et\nal. [14] reported the absorbance of Al for an incident\npulse with a wavelength and pulse duration of 800 nm\nand 120 fs, respectively, for a wide range of intensities,\n1013∼1018W/cm2. It shows that the absorbance in-\ncreases as the intensity increases, with a maximum value\nof approximately 0.3 at an intensity 3 ×1014W/cm2.\nFig. 3(a) shows that the absorbance in our calculation\nincreases above 1014W/cm2and reaches a maximum\nvalue of approximately 0.2 at approximately 7 ×1015\nW/cm2. Therefore, the intensity of the maximum ab-\nsorption shifted by a factor of 20, based on the measure-\nment of [14]. The pulse duration used in our calculation\nis about 7 fs at the FWHM, which is about 20 times\nshorter than that of the measurement. If we compare the\nposition of the maximum absorbance using the fluence of\nthe incident pulse rather than the maximum intensity,\nour calculations and measurements will reasonably agree9\n0123456\nEnergy[eV]Im[ε(ω)]125\n100\n75\n50\n25\n-250Re[ε(ω)]\n-125-100-75-50\n-150-250\nnopulse\nImax=1×1013\nImax=1×1015\nImax=7×1015\nImax=5×1016\n1.0\n0.8\n0.6\n0.4\n0.2\n010910101011101210131014101510161017\nImax[W/cm2]Absorbance\nTransmittance\nReflectanceAl (a)\n(b) (d)(c)\n0.8\n0.6\n0.4\n0.2\n0-10 -5 0 5 10 15\nEnergy[eV]Occupation[/eV/atom]DOS\nImax=1×1013\nImax=1×1015\nImax=7×1015\nImax=5×1016\nFIG. 3. (a) Reflectance (green), absorbance (red), and transmittance (blue) of a pulsed light through an Al thin film with\na thickness of 50 nm, shown against the maximum intensity of the incident pulse. Vertical dashed lines are drawn at the\nintensities where the lines are shown in (b–d) using the same colors. (b) Occupation distribution of electrons and (c) real and\n(d) imaginary parts of the dielectric function in the unit cell at the front surface of the thin film ( X= 0) at t= 30 fs, for the\ncases of incident pulses with different intensities as indicated in the figure. The black line in (b) shows the density of states.\nwith each other.\nIn Ref. [19], measurements and theoretical analyses\nhave been reported for the absorbance of a pulsed light\nfrom an Al surface with a wavelength and pulse duration\n800 nm and 20 fs, respectively, for the intensity range\nof 1011∼1015W/cm2. They observed a decrease of\nthe absorbance from 0.15 to 0.1 when the intensity in-\ncreased from 1011W/cm2to 1013W/cm2and attributed\nit to the decrease of the interband transition. This trend\nof decrease in absorbance, along with its interpretation,\nis consistent with our results, although the minimum\nabsorbance in the measurement (approximately 0.1) is\nlarger than that in our calculation, which is much smaller\nthan 0.1 and close to zero, as seen in Fig. 3(a).\nRecently, the reflectance from Al surfaces was reported\nusing pulses with wavelengths and durations of 800 nm\nand 15 ∼100 fs, respectively, for a fluence range of\n0.05∼10 J/cm2[30]. The measurement shows a strong\ndecrease in the reflectance as the fluence increases, with\na reflectance as small as 0 .2∼0.3 for a 15 fs pulse with\napproximately 5 J/cm2, that is, approximately 3 ×1014\nW/cm2. These results are inconsistent with our calcu-\nlations or the previous measurements described above.Currently, the cause of this discrepancy is not under-\nstood, and further studies are warranted.\n2.α-quartz (SiO 2)\nNext, we discuss the results for α-quartz (SiO 2), which\nis a typical wide-gap dielectric. The calculation uses the\nmeta-GGA potential of Tran and Blaha [47]. Using this\npotential, the calculated bandgap was 8.6 eV, which is\nclose to the measured value. For SiO 2, calculations for a\nsmaller range of intensities have been reported using the\nmultiscale Maxwell-TDDFT method [52].\nFig. 4(a) shows the reflectance, transmittance, and ab-\nsorbance of SiO 2thin film with a thickness of 200 nm\nagainst the maximum intensity of the incident pulse. As\nshown in the figure, the high transmittance at a low in-\ntensity decreases rapidly when the laser intensity exceeds\n1013W/cm2. The decrease in transmittance is first com-\npensated for by the absorbance, which shows a maximum\nof approximately 0.6 at approximately 1015W/cm2and\nthen decreases monotonically. The reflectance gradually\nincreases above the mid-1014W/cm2and becomes dom-10\n1.0\n0.8\n0.6\n0.4\n0.2\n010910101011101210131014101510161017\nImax[W/cm2]Absorbance\nTransmittance\nReflectance\n0.81.0\n0.6\n0.4\n0.2\n01.2\n-5 0 5 10 15\nEnergy [eV]Occupation [/eV/atom]DOS\nImax=1×1013\nImax=1×1015\nImax=1×1017SiO 2(a)\n(b) (d) (c) \n0Im[ ε(ω) ]81012\n6\n4\n2\n0\n-2Re[ ε(ω) ]5\n-15-100\n-5\n0 2 4 6 810 12 14 16\nEnergy [eV]no pulse\nImax=1×1013\nImax=1×1015\nImax=1×1017\nFIG. 4. Same as Fig. 3 except that this figure is for a thin film of α-quartz with a thickness of 200 nm.\ninant at and above 1016W/cm2. The absorbance above\nthe mid-1013W/cm2originates from a highly nonlinear\nmultiphoton absorption process. Because the bandgap of\nSiO2is 8.6 eV in the present calculation, absorption of\nmore than five photons is required for the valence elec-\ntrons to be excited into the conduction bands.\nFig. 4(b) shows the occupation distribution of an unit\ncell at the front surface of the thin film at t= 30 fs. At an\nincident intensity of 1015W/cm2, where the absorbance\nshows a maximum, the occupation distribution does not\nappear much different from that of the ground state, al-\nthough a slight decrease is observed for the occupied or-\nbitals. Figs. 4(c) and (d) show the real and imaginary\nparts of the dielectric function, respectively. In addition\nto that in the ground state, shown by a black line and\nindicated as ”no pulse,” those at the front surface of the\nthin film at t= 30 fs are shown. As the incident intensity\nincreases, the real part of the dielectric function gradu-\nally receives a Drude-like negative contribution owing to\nelectronic excitations.\nIn Fig. 4(a), a small dip is observed in the reflectance\nat approximately 1014W/cm2. This behavior is related\nto the decrease in the real part of the dielectric function.\nAt an intensity of approximately 1014W/cm2, the real\npart of the dielectric function changes sign around the\nfrequency of the laser pulse ( ℏω= 1.55 eV). This co-\nincidence is known as the resonant condition, and it ex-plains the dip in the reflection. Above an intensity of 1014\nW/cm2, the real part of the dielectric function becomes\nnegative at a frequency of ℏω= 1.55 eV (Fig. 4(c)). This\nimplies that during pulse irradiation, the optical prop-\nerty of the material changes from dielectric to plasma,\nand the plasma-like reflection becomes dominant as the\nintensity of the incident pulse increases.\nTo summarize, multiphoton absorption plays a dom-\ninant role in the intensity dependence in SiO 2. It first\ninduces an increase in absorbance above an intensity of\n1014W/cm2. It also creates free carriers in the con-\nduction band, which add the Drude-like negative contri-\nbution to the real part of the effective dielectric func-\ntion. The change in the real part of the dielectric func-\ntion induces an increase in the reflectance due to plasma\nreflection and a decrease in the absorbance above 1015\nW/cm2. As the occupation distribution did not change\nsignificantly from the ground state at an intensity of 1015\nW/cm2, it did not play an important role in SiO 2, unlike\nthe case of Al.\nWe compared the results of our calculations with the\nmeasurements and other theoretical analyses. Price et\nal. [14] reported the absorbance of SiO 2, as well as Al.\nThis shows that the absorbance was quite high, reach-\ning a maximum value of approximately 0.9 at an inten-\nsity of approximately 2 ×1013W/cm2and then decreas-\ning rapidly. In our calculations, Fig. 4(a) shows that11\nthe absorbance reaches a maximum value of approxi-\nmately 0.6 at approximately 7 ×1014W/cm2and then\ndecreases. Therefore, the intensity of the maximum ab-\nsorption shifted by a factor of 30 ∼40 compared to the\nmeasurement. Regarding the difference in the maximum\nvalue of the absorbance, there is still a transmittance of\napproximately 0.2 at the intensity that gives the maxi-\nmum absorbance. This is due to our calculation being\nperformed for a thin film of 200 nm. This implies that\nthe maximum absorbance can be as high as 0.8, which is\nthe sum of the absorbance and transmittance, if calcu-\nlations were performed for thicker films. Regarding the\nshift in the intensity that gives the maximum absorbance,\na similar shift was also observed for Al, as discussed in\nthe previous subsection. This shift can be attributed to\nthe differences in the pulse duration of approximately 7\nand 120 fs FWHM in the present calculation and mea-\nsurement, respectively.\nZiener et al. [21] reported the reflectance for incident\npulses of wavelength and duration of 800 nm and 90 fs,\nrespectively, from SiO 2surface for incident intensities\nof 1013∼5×1017W/cm2. In the measurement, the\nreflectance started to increase above 1014W/cm2and\nrapidly increased to 0.8. Above 1017W/cm2, a rapid de-\ncrease in the reflectance was reported. Considering the\ndifference in the pulse lengths used in the experiments\nand calculations, we can conclude that our calculations\nagree reasonably well with the measurements. Our cal-\nculations do not indicate any decrease in the reflectance\nat high intensities around 1017W/cm2. The difference\nbetween the measurements and our calculation may be\neither because the pulse duration used in our calculation\nis much shorter than that used in the measurement, or\nthe decrease in the measured reflectance originates from\nphysical effects that were ignored in the present calcula-\ntion.\nKaiser et.al. [34] reported the theoretical analysis of\nabsorption of intense pulses in SiO 2using time-dependent\nBoltzmann equation that includes electron-electron col-\nlision effects. They concluded that multiphoton absorp-\ntion dominates in transparent solids, and the effects of\nthe electron-electron collisions are small for a time scale\nof less than 100 fs. This justifies our computational\nmethod, which ignores hard electron-electron collisions\nfor pulses employed in the present work.\n3. Graphite\nIn Fig. 2, the reflectance and absorbance of graphite\nand Si exhibit complex changes with incident intensity,\ncontrary to those of Al and SiO 2. In this subsection, we\ndiscuss the results for graphite, which is a semi-metal.\nThe direction of light propagation was chosen to be per-\npendicular to the plane of the honeycomb lattice, and the\nelectric field was in this plane. It is noted that, in Table\nI, the axis perpendicular to the honeycomb lattice was\nchosen as the a-axis. An adiabatic local-density approxi-mation was used in the calculations. Fig. 5(a) shows the\nreflectance, transmittance, and absorbance of a graphite\nthin film of 200 nm thickness against the maximum in-\ntensity of the incident pulse. Fig. 5(b) shows the electron\noccupation distribution at the front surface of the thin\nfilm at t= 30 fs for the four cases with different incident\nintensities. Figs. 5(c) and (d) show the real and imag-\ninary parts of the dielectric function, respectively. In\naddition to the dielectric function in the ground state in-\ndicated as ”no pulse,” the dielectric function at the front\nsurface at t= 30 fs is shown for the four different incident\nintensities.\nAs seen in Fig. 5(a), the reflectance increases and ab-\nsorbance decreases as the intensity increases. A small\namount of transmittance appeared in the limited inten-\nsity region of approximately 1012∼1013W/cm2. This\nwas due to saturable absorption, which was also observed\nin Al. It occurs due to a decrease in the valence elec-\ntrons that can be excited by linear absorption and an\nincrease in the conduction electrons, which hinders lin-\near absorption by Pauli blocking. Figs. 5(c) and (d) show\nthat the real part of the dielectric function at ℏω= 1.55\neV becomes negative and the absolute value increases to\n1014W/cm2as intensity increases, while the imaginary\npart decreases. These changes are consistent with the ob-\nserved decrease in absorbance and increase in reflectance\nin this intensity region. In Ref. [57], which is consistent\nwith the present results, saturable absorption in graphite\nhas been reported for a limited range of incident intensi-\nties using the multiscale Maxwell-TDDFT method.\nThe reflectance begins to increase at approximately\n1011W/cm2and reaches a maximum at an incident in-\ntensity of 1014W/cm2. This was caused by the large\nnegative value of the real part of the dielectric function\n(Fig. 5(c)). Above 1014W/cm2, the reflectance decreased\nand reached a minimum at 1015W/cm2. The absorbance\nshows a maximum at the same intensity. Fig. 5(b) shows\na change from almost full occupation below Fermi level\nto a classical Boltzmann distribution taking place across\nthe intensity of 1015W/cm2. The imaginary part of the\ndielectric function, shown in Fig. 5(d), exhibits a large\nvalue of ℏω= 1.55 eV for this intensity. From these\nobservations, the appearance of the minimum reflectance\nand maximum absorbance at an approximate incident in-\ntensity of 1015W/cm2is related to the change in the oc-\ncupation distribution from a quantum Fermi distribution\nto a classical Boltzmann distribution. A similar explana-\ntion has been provided for the absorption peak of Al at\nan incident intensity of 1016W/cm2.\nSummarizing the results for graphite, three physical\nmechanisms play a role in intensity dependence. Sat-\nurable absorption, which first appears at an intensity\nof 1012∼1013W/cm2, reduces the absorbance and in-\ncreases the transmittance. Above this intensity region,\nelectronic excitations induce a negative contribution to\nthe real part of the effective dielectric function, and\nplasma reflection becomes dominant. Upon further in-\ncreasing the intensity, a drastic change in the occupation12\nAbsorbance\nTransmittance\nReflectance1.0\n0.8\n0.6\n0.4\n0.2\n010910101011101210131014101510161017\nImax[W/cm2]\n0.8\n0.6\n0.4\n0.2\n0-10 -5 0 5 10 15\nEnergy [eV]Occupation [/eV/atom]DOS\nImax=3×1012\nImax=1×1014\nImax=1×1015\nImax=1×1017Graphite(a)\n(b) (d) (c) \nRe[ ε(ω) ]\n-20-10010\n-30\n0Im[ ε(ω) ]\n10\n-1030\n20\n0no pulse\nImax=3×1012\nImax=1×1014\nImax=1×1015\nImax=1×1017\n02468101214\nEnergy [eV]\nFIG. 5. Same as Fig. 3 except that this figure is for a graphite thin film of 200 nm thickness.\ndistribution from a quantum Fermi distribution to a clas-\nsical Boltzmann distribution occurred at an intensity of\n1015W/cm2, and the absorbance (reflectance) reached\nthe maximum (minimum). Upon further increasing the\nintensity, the plasma reflection became dominant.\nWe compared our calculated results with the measure-\nments. Breusing et al. [25] reported pump-probe mea-\nsurements of graphite flakes of 20 ∼30 nm thickness\nusing a 7 fs pulse with a central frequency of 1.55 eV\nand showed an increase in the transmittance of 20 % at\na pump intensity of 5 ×1010W/cm2. Winzer et al. [26]\nreported the theoretical and experimental results of sat-\nurable absorption for a graphite thin film with 60 layers\n(approximately 20 nm). For pulses with frequency and\nduration of 1.5 eV and 56 fs, respectively, absorption sat-\nuration was observed at 5 .7×1010W/cm2. Under such\nconditions, the measured values did not coincide with\nthe present calculations. The intensities of the incident\npulses that showed an increase in the transmittance were\nlower than those in our calculations, and the amount of\ntransmittance was small in our calculations. We consider\nthese differences to originate from the difference in the\nfilm thickness between the measurements (20 ∼30 nm)\nand our calculation (200 nm). In Ref. [57], we report\nthe calculated results for a graphite thin film of 50 nm\nthickness using the multiscale Maxwell-TDDFT method.An increase in transmittance was found to occur around\nthe incident intensity of 1011W/cm2, and the amount\nof increase in transmittance was approximately 0.4 at\napproximately 1012W/cm2, which were in better agree-\nment with the measurements.\n4. Silicon\nFinally, we present the results obtained for Si, which\nis a small-gap dielectric. The calculation used the meta-\nGGA potential of Tran and Blaha [47]. The bandgap\ncalculated using this potential was 3.4 eV, which is close\nto the measured value. The first application for the ir-\nradiation of the bulk surface of Si was carried out in\nRef. [43], where the multiscale Maxwell-TDDFT method\nwas developed. Recently, high harmonic generation from\nSi thin films was reported using the same method [54, 55].\nFig. 6(a) shows the reflectance, transmittance, and ab-\nsorbance of Si thin film with a thickness of 200 nm against\nthe maximum intensity of the incident pulse. Silicon\nshould have weak absorption at ℏω= 1.55 eV in the lin-\near response owing to the one-photon absorption accom-\npanying phonons. However, in the present calculation,\nsuch processes are ignored because we fix the atomic po-\nsitions at the equilibrium positions in the primitive cell.13\n0.81.0\n0.6\n0.4\n0.2\n01.2\n-10 -5 0 5 10 15\nEnergy[eV]Occupation[/eV/atom]DOS Imax=1×1013\nImax=3×1014\nImax=1×1015\nImax=5×1016\nAbsorbance\nTransmittance\nReflectance1.0\n0.8\n0.6\n0.4\n0.2\n010910101011101210131014101510161017\nImax[W/cm2]Si(a)\n(b)(d)(c)\n012345678910Energy[eV]Re[ε(ω)]\n-2002040\n-40Im[ε(ω)]60\n40\n20\n0nopulse\nImax=1×1013\nImax=3×1014\nImax=1×1015\nImax=5×1016\nFIG. 6. Same as Fig. 3 except that this figure is for a Si thin film of 200 nm thickness.\nThus, the thin film was transparent at low intensities.\nIn Fig. 6(b), the electron occupation distribution at the\nfront surface of the thin film at t= 30 fs is shown for\nthe four incident intensities. Figs. 6(c) and (d) show the\nreal and imaginary parts of the dielectric function, re-\nspectively. In addition to the dielectric function in the\nground state, denoted as ”no pulse,” dielectric function\nat the front surface at t= 30 fs is shown for the four\ndifferent incident intensities.\nAs the light intensity increased, the transmittance\ngradually decreased, and the absorbance increased ow-\ning to multiphoton absorption. The reflectance decreased\nslightly at an intensity of 1012∼1013W/cm2and then\nincreased. This can be explained by the behavior of the\nreal part of the dielectric function at ℏω= 1.55 eV as\nshown in Fig. 6(c). The real part of the dielectric func-\ntion approaches zero at an intensity of 1013W/cm2and\nbecomes negative above this intensity. This behavior of\nthe real part of the dielectric function causes a slight de-\ncrease in the reflectance at an intensity of 1012∼1013\nW/cm2and then an increase. A similar behavior was\nobserved for SiO 2.\nAbove 1013W/cm2, reflectance increases, while ab-\nsorbance and transmittance decrease. However, changes\nin the incident intensity do not follow a monotonic pat-\ntern but exhibit complex behavior. The reflectance\nshows a maximum and then a minimum at approximately1014∼1015W/cm2, whereas the absorbance shows the\nopposite trend. Occupational distributions are shown in\nFig. 6(b). The occupation of the valence orbitals ex-\nhibits a large change in this intensity region. The imag-\ninary part of the dielectric function, shown in Fig. 6(d),\nexhibits a significant enhancement at ℏω= 1.55 eV, at\nan incident intensity of 1015W/cm2. We consider the\ncomplex change of the reflectance and absorbance as the\nincrease of the intensity to be related to the rapid change\nof the occupation distribution, from mostly occupied va-\nlence orbitals to a classical Boltzmann-like occupation.\nA similar mechanism for producing the maximum ab-\nsorbance was also observed in aluminum and graphite,\nas shown in Figs. 3(a),(b), and(d) and 5(a),(b), and(d).\nThus, we consider that such enhanced absorption linked\nto the drastic change in occupation distribution is a uni-\nversal phenomenon in the interaction between intense\nand ultrashort pulsed light.\nIn summary, multiphoton absorption first induces a\nnonlinear absorption and then decreases the transmit-\ntance above an intensity of 1011W/cm2. Electronic ex-\ncitation contributes negatively to the real part of the\neffective dielectric function. As the intensity increases,\nplasma reflection gradually becomes significant and in-\nduces a decrease (increase) in absorbance (reflectance)\nabove an intensity of 1013W/cm2. Upon further in-\ncreasing the intensity, a drastic change in the occupa-14\ntional distribution from a quantum distribution to a clas-\nsical Boltzmann distribution occurred at an intensity\nof 1014∼1015W/cm2. The absorbance (reflectance)\nshowed the maximum (minimum) value. Plasma reflec-\ntion dominated upon further increase of intensity.\nWe compared our calculated results with the measure-\nments. Hulin et al. [11] reported a change in the re-\nflectance of the Si surface for a pulse of duration and\nwavelength of 100 fs and 620 nm, respectively. As the\nintensity of the pulse increased, a small decrease of ap-\nproximately 5% occurred in the reflectance, followed by\nan increase of approximately 50% when the intensity in-\ncreased by a factor of 10. These behaviors are consistent\nwith our calculations, although the pulse durations and\nfrequencies are different. In their analysis, Drude model\nwas used to describe the changes in optical properties\ncaused by electronic excitations. The Drude model in-\ncludes two parameters: carrier density and collision time.\nA number density of free carriers between 5 ×1021cm−3\nand 2×1022cm−3was adopted. This value is consistent\nwith our calculations. As shown in Fig. 1(f), the num-\nber density of excited electrons at t= 30 fs is 4 .4×1021\ncm−3at an intensity of Imax= 1013W/cm2. This in-\ntensity is close to the intensity at which reflectance is\nminimized. The collision time is set as 3 ×10−16s. The\nauthors state that this very short collision time origi-\nnates from electron-hole interactions. A small electron\nrelaxation time has also been adopted in the literature\n[16, 37]. The mechanism of the collision, specifically the\nelectron-hole interaction effect, is taken into account in\nour multiscale Maxwell-TDDFT method. When solving\nthe TDKS equation in real time, the scattering of elec-\ntrons by ions, which induces interactions between elec-\ntrons and holes, is fully taken into account.\nPump-probe measurements of the reflectance of Si and\ntheoretical analyses have been reported in Ref. [16]. In\ntheir theoretical analysis, a minimum of reflectance was\npredicted at the excited electron density of about 7 ×1021\ncm−3, and a rapid increase of reflectance was predicted\nabove this value of the excited electron density. These\nestimates are consistent with our calculations and those\nof Ref. [11]. A similar behavior of reflectance against in-\ntensity was reported in Ref. [27] using pulses with wave-\nlengths and duration of 800 nm and 100 fs, respectively.\nHowever, Ref. [15] reported that no clear evidence of en-\nhanced reflectance was observed at high intensities when\nusing an incident pulse with a wavelength and duration\n750 nm and 60 fs, respectively.\nIV. SUMMARY\nIn this study, we present a systematic theoretical anal-\nysis of the propagation of intense ultrashort pulsed light\nthrough thin films of various materials. We employed\nthe first-principles multiscale Maxwell-TDDFT method,\nin which microscopic electronic motion and macroscopic\nlight propagation are coupled using a coarse-graining ap-proximation. The reflectance, transmittance, and ab-\nsorbance of a thin film with a thickness of 50 ∼200 nm\nwere calculated for four materials, namely Al (metal),\ngraphite (semi-metal), Si (small-gap dielectric), and SiO 2\n(wide-gap dielectric), with different band structures and\noptical properties. Linearly polarized pulsed light with\nan average frequency of 1.55 eV, a pulse duration of 7\nfs at FWHM, and maximum intensities of 109∼1017\nW/cm2normally irradiated the thin films.\nFrom the systematic calculations of the four materials,\nwe found the following characteristics in nonlinear optical\nresponse: While the optical responses to weak light are\ndescribed by the dielectric function, which varies signif-\nicantly depending on the material, the response to very\nintense light is dominated by plasma reflection, regardless\nof the material. The process of plasma reflection becomes\ndominant as the intensity increases non-monotonically.\nAs the light intensity increases, perturbative nonlineari-\nties first appear. Multiphoton absorption appears in di-\nelectrics, where the bandgap energy is larger than the\nfrequency of light, whereas saturable absorption appears\nin metals and semi-metals where the linear excitation\noccurs at the frequency of the light. In dielectrics, a\nsmall minimum reflectance appears when the real part\nof the effective dielectric function that incorporates the\neffects of excited carriers crosses zero. The maxima in\nabsorbance and minima in reflectance appear for all four\nmaterials at an intensity of 1015∼1016W/cm2. In Al,\ngraphite, and Si, this originates from a significant change\nin the occupation distribution from a quantum-fully oc-\ncupied distribution of valence orbitals to a classical Boltz-\nmann distribution. The imaginary part of the effective\ndielectric function is enhanced during the change in the\ndistribution. In SiO 2, although maximum absorption ap-\npears at a similar intensity, it is not related to the drastic\nchange in the occupation distribution but to the interplay\nbetween multiphoton absorption and plasma reflection.\nAbove an incident intensity of 1016W/cm2, all the ma-\nterials showed plasma reflection, characterized by large\nnegative values in the real part of the effective dielectric\nfunction.\nNote that the calculations are ab initio and employ\nno empirical parameters. For weak light, our formalism\nof the multiscale Maxwell-TDDFT method results in or-\ndinary macroscopic electromagnetism with the dielectric\nfunctions given by TDDFT. Because the TDKS equation\nis solved in real time and no perturbative approximation\nis employed for the electronic motion, the nonlinearities\nin the electronic response are fully considered in the cal-\nculation. Because we employed a three-dimensional grid\nsystem in solving the TDKS equation, there was no trun-\ncation in the spatial degrees of freedom, that is, the un-\noccupied orbitals were not truncated.\nHowever, numerous physical effects are ignored in our\nframework. Because we assume fixed atomic positions\nat the equilibrium positions in the primitive cell, the\nthermal effects related to ionic motion and energy trans-\nfer between electrons and ions are not included. For15\nthe very short pulse with a duration of a few tens of\nfemtoseconds used here, the energy transfer from elec-\ntrons to ions may not be important. Another serious ap-\nproximation in the present TDDFT calculations for elec-\ntronic motion is the adiabatic approximation, in which\nthe exchange-correlation functional in the ground state is\nemployed for time-dependent motion. As we are essen-\ntially solving a one-body TDKS equation, only limited\nmany-body correlation effects are incorporated. Because\nwe calculated Hartree and exchange-correlation poten-\ntials at each time step, dynamic screening effects were\nincluded through the time-dependent Hartree potential.\nHowever, the electron-electron collision effects accompa-\nnying large momentum or energy transfer were not in-\ncluded. In particular, the avalanche mechanism, which\nis important for pulses with durations of a few tens of\nfemtoseconds or longer, was not included. Notably, de-\nspite these many approximations, the present calculation\nplausibly captures the variation in the optical responsein a wide intensity region for various materials, reaching\nthe plasma reflection, all within the framework of first-\nprinciples calculations.\nACKNOWLEDGMENTS\nThis research was supported by JST-CREST under the\ngrant number JP-MJCR16N5, MEXT Quantum Leap\nFlagship Program (MEXT Q-LEAP) (Grant number:\nJPMXS0118068681), and JSPS KAKENHI (Grant num-\nber: 20H2649). 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And what predicts doing so? Over six months,\nwe tracked the frequency and predictors of ∼900K followers un-\nfollowing∼5K health misinformation spreaders on Twitter. We\nfound that misinformation ties are persistent. Monthly unfollowing\nrates are just 0.52%. In other words, 99.5% of misinformation ties\npersist each month. Users are also 31% more likely to unfollow\nnon-misinformation spreaders than they are to unfollow misinfor-\nmation spreaders. Although generally infrequent, the factors most\nassociated with unfollowing misinformation spreaders are (1) re-\ndundancy and (2) ideology. First, users initially following many\nspreaders, or who follow spreaders that tweet often, are most likely\nto unfollow later. Second, liberals are more likely to unfollow than\nconservatives. Overall, we observe a strong persistence of misin-\nformation ties. The fact that users rarely unfollow misinformation\nspreaders suggests a need for external nudges and the importance\nof preventing exposure from arising in the first place.\nCCS CONCEPTS\n•Human-centered computing →Collaborative and social\ncomputing ;Empirical studies in collaborative and social com-\nputing ;Social networks ;Social media ;Social content sharing ;\nSocial network analysis ;•Applied computing ;\nKEYWORDS\nmisinformation; social media; social networks; unfollowing; twitter\nACM Reference Format:\nJoshua Ashkinaze, Eric Gilbert, and Ceren Budak. 2024. The Dynamics of\n(Not) Unfollowing Misinformation Spreaders. In Proceedings of the ACM Web\nConference 2024 (WWW ’24), May 13–17, 2024, Singapore, Singapore. ACM,\nNew York, NY, USA, 11 pages. https://doi.org/10.1145/3589334.3645445\n1 INTRODUCTION\nMisinformation exposure is associated with important decisions like\nvaccine hesitancy [ 21] and compliance with health regulations [ 12].\nDue to its possible harms, many studies explore how users become\nexposed to misinformation. But much less is known about when\nPermission to make digital or hard copies of all or part of this work for personal or\nclassroom use is granted without fee provided that copies are not made or distributed\nfor profit or commercial advantage and that copies bear this notice and the full citation\non the first page. Copyrights for components of this work owned by others than the\nauthor(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or\nrepublish, to post on servers or to redistribute to lists, requires prior specific permission\nand/or a fee. Request permissions from permissions@acm.org.\nWWW ’24, May 13–17, 2024, Singapore, Singapore\n©2024 Copyright held by the owner/author(s). Publication rights licensed to ACM.\nACM ISBN 979-8-4007-0171-9/24/05\nhttps://doi.org/10.1145/3589334.3645445\nStep 1Collect health misinformation URLs and tweets flagged by PolitiFact.Step 2Find users who share this content on Twitter. Denote these users misinformation spreaders. Also pull the followers of spreaders.Study 1RQ1What is the frequency of unfollowing misinformation spreaders?  \nRQ2Are misinformation spreaders unfollowed at a different rate?\nProcedure: We selected a panel of followers from Step 2, pulled users who they followed at T1 (March 2023) and T2 (Oct 2023), then measured their unfollowing rates of both spreaders and non-spreaders.Study 2RQ3What are the predictors of unfollowing a misinformation spreader?Procedure: In two periods (March 2023 and Oct 2023), we pulled the followers of misinformation spreaders identified in Step 2. We then modeled the predictors of unfollowing these misinformation spreaders.Figure 1: Graphical summary of studies.\nusers choose to unfollow misinformation sources. Even some of\nthe most fundamental empirical questions around misinformation\nunfollowing are still largely unanswered. For example, is unfol-\nlowing misinformation sources generally common or rare? And is\nmisinformation exposure self-reinforcing orself-correcting ? That is:\nIs high initial exposure predictive of higher orlower unfollowing\nrates?\nLittle is known about misinformation unfollowing—but under-\nstanding its frequency and predictors benefits researchers and plat-\nforms interested in a more responsible Web. First, regarding fre-\nquency: If unfollowing is rare, then this suggests content modera-\ntion and interventions to stop connections from forming in the first\nplace are crucial for stopping exposure. But if unfollowing is com-\nmon, then perhaps users organically reduce their misinformation\nexposure—reducing the content moderation burden. If unfollowing\nis common, this also complicates the interpretation of studies de-\nscribing a ‘snapshot’ of misinformation exposure at a moment in\ntime. Second, understanding the predictors of unfollowing misinfor-\nmation spreaders can help both researchers and platforms design\ninterventions to further increase unfollowing.\nHere, we provide the first large-scale account of misinforma-\ntion unfollowing. We modeled the frequency and predictors of\nTwitter users (∼900K followers,∼5K spreaders,∼3M edges) unfol-\nlowing health misinformation spreaders. We identified misinforma-\ntion spreaders as Twitter users who shared content that PolitiFactarXiv:2401.13480v2  [cs.SI]  17 Feb 2024WWW ’24, May 13–17, 2024, Singapore, Singapore Joshua Ashkinaze, Eric Gilbert, & Ceren Budak\nflagged as health misinformation. The study period ran from March\n2023 to October 2023. We focus on health misinformation for two\nreasons. First, claims related to health are often more straightfor-\nwardly falsifiable than claims in other domains. Second, the cost of\nfalse health misinformation can be high.\nAcross two studies, we answer three research questions related\nto the frequency and predictors of unfollowing misinformation\nspreaders. See Figure 1 for a graphical overview.\nRQ1 (Study 1): How often are misinformation spreaders un-\nfollowed? We track how often spreaders are unfollowed be-\ntween two data collection points. We find that misinforma-\ntion ties are rarely severed, with unfollowing rates of 0.52%\nper month.\nRQ2 (Study 1): Are misinformation spreaders unfollowed\nat a different rate than non-misinformation spreaders? We\ncompared how often a subset of followers unfollows non-\nspreaders vs spreaders. Users are 31% more likely to unfollow\nnon-misinformation spreaders than they are to unfollow\nmisinformation spreaders.\nRQ3 (Study 2): What predicts unfollowing a misinformation\nspreader? We model unfollowing as a function of initial ex-\nposure, ideology, edge characteristics, and platform activity.\nWe find that reciprocity, initial exposure, and ideology are\nthe most important factors.\n2 RELATED WORK\nWe review related work on (1) factors perpetuating misinformation\nexposure and (2) predictors of unfollowing on social media.\n2.1 Misinformation Exposure\nMany studies explore how misinformation exposure arises in the\nfirst place. There is no uniform pathway. Factors perpetuating mis-\ninformation exposure can be grouped into two buckets: individual\n(e.g., ideology, cognitive reflection, attention) and environmental\n(e.g., algorithms, defaults).\n2.1.1 Individual Factors. Misinformation exposure is driven by\nindividual-level factors, such as selective exposure [ 8], cognitive\nreflection [26], and inattention to accuracy [24].\nSelective Exposure. People consume (mis)information consistent\nwith their ideology. A large study based on browsing data found that\nconservatives consumed more untrustworthy news content than\nliberals in the 2016 election [ 8]. But for both Trump and Clinton\nsupporters, users were more likely to visit untrustworthy web-\nsites consistent with their political ideology—with an especially\nlarge effect for Trump supporters [ 8]. Another large-scale study\nalso supports selective exposure: people are less likely to click on\ncross-ideology links in newsfeeds [ 1]. Moreover, selective exposure\nappears to be a stronger driver of misinformation consumption for\nusers with extreme political ideologies. In 2016, consumption of\nfake news with respect to ideology was ‘v-shaped’: both extreme lib-\nerals and extreme conservatives consumed larger amounts of fake\nnews [ 19]. Although fake news exposure decreased during the 2020\nelection, it was still the case that conservatives—and particularly, ex-\ntreme conservatives—consumed more fake news than liberals [ 19].\nOne study [ 28] compared the amount of unreliable and partisannews users were exposed to in search results versus the amount of\nunreliable and partisan news they consumed. Consumption of the\nlatter was significantly higher than the former. This also suggests\nusers are seeking unreliable news (over and above what is being\nshown to them by platforms). Overall, there is large-scale evidence\nthat selective exposure drives misinformation consumption.\nCognitive Reflection. Cognitive reflection is implicated in misin-\nformation exposure and belief. The Cognitive Reflection Test (CRT)\nmeasures a person’s tendency towards analytical thinking [ 7]. Peo-\nple who are less likely to engage in such thinking are more likely to\nconsume and believe misinformation. Low-CRT participants in lab\nexperiments are more likely to believe fake news [ 25,26]. Also, CRT\ncorrelates with on-platform behavior [ 20]. Low-CRT Twitter users\nare more likely to share low-quality news, and low vs high-CRT\nusers follow different sets of Twitter accounts.\nInattention. Users may share misinformation because they are\nnot paying attention to the accuracy of what they share [ 24,27].\nThis line of research is principally concerned with sharing of misin-\nformation but also has implications for exposure to misinformation.\nUsers likely encounter more misinformation in their feeds because\nother users share content inattentively.\n2.1.2 Environmental Factors. Misinformation exposure also may\narise more incidentally through environmental factors. Here, we\nrefer to both environments built for users (e.g., social media algo-\nrithms) and environments users build for themselves (e.g., defaults).\nAlgorithms. There is mixed evidence regarding the role of rec-\nommendation algorithms in promoting misinformation. Several\nauditing studies found consuming misinformation content leads to\nmore misinformation content being recommended by algorithms\n[10,22,30]. These studies suggest recommendation algorithms may\namplify already-existing misinformation exposure. Yet, the largest\nfield experiment to date on this topic suggests algorithms do not\npromote misinformation on Facebook and Instagram [ 9]. In a dif-\nferent domain (extremist content on YouTube), [ 4] also questions\nthe significance of algorithmic effects and points to demand effects\nas a more important mechanism.\nDefaults. Misinformation exposure can arise through the defaults\nthat users set. Users curate a set of sources they regularly consume,\nwhat [ 13] calls a ‘media repertoire’. This repertoire can serve as a\ndefault filter for news. For example, [ 6] showed that most online\nnews consumption was driven by users visiting their homepage.\nThat is, the ‘defaults’ that users created (i.e., setting a homepage)\nstrongly influenced the content that users saw. Analogously, the\ncontent from the people that a user chooses to follow online can be\nthought of as the ‘default’ content that the user sees. Combined with\nhomophily, this can create ideologically segregated filter bubbles or\necho chambers [ 3,5,29]. This is a different dynamic than selective\nexposure, where one is seeking out the information itself. Of course,\nthe initial choosing of defaults is an individual-level decision. But\nthis decision then creates a content environment that may expose\none to misinformation.The Dynamics of (Not) Unfollowing Misinformation Spreaders WWW ’24, May 13–17, 2024, Singapore, Singapore\nTable 1: How samples map to studies and research questions.\nSample Name Description Used in Study Answers Research Question(s)\nInitial Sample Initial pull of spreaders and followers - -\nModeling Sample Subset of the Initial Sample used for predictors of unfollowing Study 2 RQ3\nRate Sample Subset of the Modeling Sample used for rates of unfollowing Study 1 RQ1, RQ2\n2.2 Unfollowing on Social Media\nSeveral variables have emerged as key predictors of unfollowing.\nA reciprocal tie between Twitter users A and B is associated with\nsignificantly lower odds of A unfollowing B [ 14–16,32]. Reciprocity\nmay be a cause or effect of tie strength. For example, [ 15] argues that\nreciprocal relationships on social media cause emotional closeness\nbetween two users since they see each other’s posts. Alternatively,\nreciprocity may signal two users are already friends or acquain-\ntances [ 18]. Additionally, redundancy —either similar content to\nwhat the user follows or burst-tweeting—is a predictor of unfollow-\ning [ 15,17]. More relevant, [ 11] found participants reported higher\ntheoretical intentions to unfollow or block an imagined cross-party\nfriend for posting misinformation, and [ 33] found liberals were\nmore likely to report political unfriending.\n3 SPECIFIC HYPOTHESES\nConsidering unfollowing of misinformation spreaders , we are specif-\nically interested in two key predictors. The first predictor is T1\nExposure . This is the number of spreaders followed at Time 1, the\ndate of the first data pull. (From here on, ‘T1’ and ‘T2’ denote the\ndates of the first and second data pulls.) The second predictor is\nPartisan Ideology , the ideology of the follower. These are network\nand individual-level variables, respectively.\n3.1 Initial (T1) Exposure Effect on Unfollowing\nIt is not clear, based on the existing literature, if T1 exposure (the\nnumber of misinformation spreaders, hereafter ‘spreaders’, followed\nat the first time point) should be positively or negatively related\nto unfollowing. And yet, this empirical relationship is important\nsince it can speak to the extent to which misinformation exposure\nis self-correcting or self-reinforcing:\n•Reversion Hypothesis (H1) : Higher exposure at T1 is as-\nsociated with higher unfollowing at T2—meaning misinfor-\nmation is partially self-correcting.\n•Inertia Hypothesis (H2) : Higher exposure at T1 is associ-\nated with lower unfollowing at T2—meaning misinformation\nis partially self-reinforcing.\nThe logic for the reversion hypothesis (H1) is that high T1 expo-\nsure would signal high redundancy. And if misinformation ties\nare redundant, the probability of each being unfollowed should\nincrease. Other mechanisms point to the same conclusion. If the\nfollower came to be exposed to misinformation through some inci-\ndental mechanism and not an intentional one, or if their interest\nin misinformation wanes over time, then regression to the mean\nis a likely prediction. These mechanisms would suggest increased\nunfollowing in T2 if T1 exposure is high.The logic for the inertia hypothesis (H2) is that high misinforma-\ntion exposure at T1 may be (1) a consequence of selective exposure\ndue to extreme ideology [ 8] or (2) a cause of believing in the mis-\ninformation (since exposure1to misinformation increases its per-\nceived accuracy [ 23]). Both mechanisms would suggest decreased\nunfollowing in T2 if T1 exposure is high.\n3.2 Partisan Ideology Effect on Unfollowing\nWe hypothesized that ideology (left/right, moderate/extreme) would\naffect if a user unfollowed a misinformation spreader. Some evi-\ndence suggests liberals have a higher political unfriending rate [ 33]\nso we hypothesized that (H3) liberals may be more likely to unfol-\nlow here (though political unfriending is a different phenomenon).\nWe also hypothesized that (H4) politically extreme users would\nbe less likely to unfollow since ideological extremity is correlated\nwith misinformation exposure [ 19]. But in 2020, extreme liberals\ndecreased consumption of fake news [ 19]. And even in 2016, the\ntop decile of conservatives had larger misinformation consumption\nthan the top decile of liberals [ 8]. Consequently, we hypothesized\nthat there would be a negative interaction effect (H5) between\nconservatism and ideological strength; an equivalent increase in\nideological strength would reduce the probability of unfollowing\nmore for conservatives than it would for liberals. We were also\ninterested in ( Q) if there was an interaction effect between T1 expo-\nsure and ideological strength such that the effect of T1 exposure on\nunfollowing might differ for ideologically moderate versus extreme\nusers.\n4 DATA\nThe data for Studies 1 and 2 come from an Initial Sample of mis-\ninformation spreaders and their followers, which we describe in\nSection 4.1. We then created two subsets of this Initial Sample —a\nRate Sample used for Study 1 (rates of unfollowing) and a Model-\ning Sample used for Study 2 (modeling predictors of unfollowing).\nThe creation of these samples is described in sections 4.2 and 4.3.\nSee Table 1 for how samples relate.\nTheModeling Sample is composed of users for whom we could\nobtain covariate information to include in our unfollowing model\nand who did not leave the platform between T1 and T2. The Rate\nSample is an active, smaller sample of followers derived from the\nModeling Sample . We pull the users who this sample follows at\nT1 and T2 so we can estimate unfollowing rates.\n1Though of course, it may be the case that not all misinformation spreaders share the\nsame misinformation.WWW ’24, May 13–17, 2024, Singapore, Singapore Joshua Ashkinaze, Eric Gilbert, & Ceren Budak\n4.1 Initial Sample\nTo construct the Initial Sample of spreaders and followers, we: (1)\nidentified health misinformation rumors from PolitiFact, (2) found\nusers who shared URLs corresponding to these rumors (‘spreaders’),\nand then (3) collected users who followed ‘spreaders’ (‘followers’).\nCollecting Misinformation URLs and Tweets. In March 2023 we\ncollected all health2misinformation3rumors on PolitiFact since\nJune 2021. We refer to the clean set of blog post URLs and tweets\nas Misinformation URLs and Misinformation Tweets, respectively.\nThere were 84 such URLs and tweets.\nIdentifying Eligible Spreaders. We next identified ‘Eligible Spread-\ners’. We conducted a search of Twitter to find all users who either\n(A) posted a Misinformation URL, (B) retweeted a post contain-\ning a Misinformation URL, (C) posted a Misinformation Tweet, or\n(D) retweeted a Misinformation Tweet. We denote the union of\n𝐴,𝐵,𝐶,𝐷 as Eligible Spreaders.\nFiltering Eligible Spreaders. We applied three filters to Eligible\nSpreaders. First, we removed any post or retweet of a post that\ncontained a series of debunking words. We generated debunking\nwords (Appendix Table 2) in a human-AI hybrid approach, where\nwe prompted GPT-3 to expand on a list of common debunking\nwords. We inspected the resultant words for coherence. Using GPT-\n3 allowed us to include words we may have missed due to our\nbiases. This step removed 1.6% of tweets. We inspected a sample\nof resultant tweets and none were debunking. Second, we selected\nspreaders with follower and friend counts of over ten and a follower\ncount of less than 20K. The lower bound was applied to make sure\nspreaders had some activity on the platform. The upper bound on\nfollowers was partially a resource constraint (since we had to pull\nall of the followers), but it also allowed us to understand ‘regular’\nusers and not celebrities. After the first two filters, there were\n58 Misinformation URLs and Misinformation Tweets, since not\nall Misinformation URLs were ever tweeted. Some of these URLs\nhad a disproportionate number of associated tweets. Consequently,\nin the third filter, we used a simple greedy algorithm to retrieve\n5,600 misinformation spreaders4while minimizing the number of\nspreaders that came from any specific story (Appendix Algorithm\n1). The motivation is that we did not want our entire sample and\nresults to be dominated by a few misinformation stories. These steps\nresulted in 5,613 misinformation spreaders (hereafter ‘Spreaders’),\nand the rumor, or misinformation URL, with the largest number of\nassociated spreaders made up just 3.6% of the total.\nCollecting Followers. We then pulled all followers of the spreaders\nat two time points: (T1) March 2023 and (T2) October 2023.\n4.2 Modeling Sample\nThe participant pool for the Modeling Sample (used in Study 2)\nbegan from the followers and spreaders from the Initial Sample ,\nand then two filters were applied. In the first filter, we restricted our\nanalysis to only followers whose ideology could be estimated via\n2PolitiFact categories: [‘abortion’, ‘autism’, ‘coronavirus’, ‘drugs’, ‘disability’, ‘health-\ncare’, ‘health-check’, ‘public-health’]\n3PolitiFact truth values: [‘pants on fire’, ‘false’, ‘mostly false’]\n4This number (5,600) was based on a power analysis conducted for a concurrent project\nthat required detecting misinformation spreaders.\nFigure 2: Distribution of the political ideology of followers\nfrom the Modeling Sample, measured using the method from\n[2]. Most followers are conservative.\nFigure 3: Number of misinformation spreaders from the Mod-\neling Sample followed at T1. Ideology is cut at zero using [ 2].\nMisinformation exposure is right-skewed.\ndata files provided by the first author of the ‘Bayesian Ideal Point\nEstimation’ [ 2] method. This method estimates a user’s ideology by\nwho the user follows. Applying this restriction yielded 944,972 fol-\nlowers and 5,593 spreaders. As a robustness check (Appendix Figure\n11), we show that relationships between non-ideology variables and\nunfollowing hold for users whose ideology we could not estimate,\nsuggesting the structural predictors of unfollowing do not differ\nFigure 4: Characteristics of those Modeling Sample follow-\ners who are in the top 10% for following misinformation\nspreaders. Ideology is a continuous measure where positive\nis conservative. ‘Recip’ is the proportion of a follower’s ties\nto spreaders that are reciprocated.\n.The Dynamics of (Not) Unfollowing Misinformation Spreaders WWW ’24, May 13–17, 2024, Singapore, Singapore\nFigure 5: Comparing unfollowing rates across studies, misin-\nformation spreaders are unfollowed relatively infrequently.\nFigure 6: In Study 1, misinformation spreaders were unfol-\nlowed less than non-misinformation spreaders.\nfor users whose ideology we could not estimate. In the second fil-\nter, we removed any follower or spreader whose basic information\n(e.g., followers, tweet count, friend count) could not be pulled at\neither T1 or T2. This can happen for multiple reasons—voluntarily\nexiting the platform, getting banned, etc. It is important to remove\nedges where either the follower or spreader left the platform since\nincluding them would distort unfollowing rates. For the Modeling\nSample , we also had to omit users whose accounts were protected\nsince their information could not be pulled and hence could not be\nincluded in the model. After these two filters, there were 898,701\nfollowers, 5,334 spreaders, and 3,376,785 edges. See Appendix Table\n3 for variable explanations and descriptive statistics.\nMost followers in the Modeling Sample are conservative (Fig-\nure 2), but there is a bimodal distribution similar to the ‘v-shaped‘\ndistribution of fake news exposure in 2016 [ 19]. Misinformation\nexposure is right-skewed, and conservatives follow more misinfor-\nmation spreaders than liberals (Figure 3). The followers in the top\n10% for the number of misinformation spreaders followed at T1\nare generally (1) ideologically extreme and (2) conservative (Fig-\nure 4), which tracks browsing studies on selective exposure [ 8,19].\nThere is a high reciprocity rate (69%), driven by a few spreaders.\nIn fact, the top 10% of spreaders by reciprocal ties are responsible\nfor 66% of the reciprocal ties; the Gini coefficient of reciprocal ties\n(0.79) is even larger than that of followers (0.74). This could suggest\neither that the most active misinformation spreaders engage in\nfrequent ‘follow-back’ behavior or that the most popular misinfor-\nmation spreaders are embedded in close-knit communities where\nreciprocation rates are high.\n4.3 Rate Sample\nTo estimate unfollowing rates of misinformation spreaders and\nnon-spreaders (Study 1), we selected a panel of 2,500 followers\n(Rate Sample ) from the Modeling Sample and pulled who these\nusers followed at roughly5the same time two points as in Study\n2. The Rate Sample was constructed by applying several filters\nto followers in the Modeling Sample . First, we filtered followers\n5Due to Twitter API instability, Study 2 started 11 days earlier and ended four days\nearlier than Study 1.in the Modeling Sample by activity level and network size (last\ntweet within 14 days, tweet count greater than 20, follower count\nbetween 20-20K, friends count between 20-20K) to ensure these\nusers were active on Twitter. Second, we capped the number of\nfriends at the 85th percentile (4,989) to constrain network size. This\nwas done due to resource constraints. Finally, 2,500 participants\nwere randomly sampled from the remaining filtered pool to serve\nas a panel of egos.\n5 STUDY 1: UNFOLLOWING OF SPREADERS\nAND NON-SPREADERS\n5.1 Objectives\nIn Study 1, we analyze both (RQ1) the overall unfollowing rates\nof misinformation spreaders and (RQ2) if the unfollowing rate of\nmisinformation spreaders differs from that of non-spreaders.\n5.2 Participants\nThe participants started from the 2,500 misinformation followers\nfrom the Rate Sample . See Table 1.\n5.3 Methods\nFor each of these 2,500 follower ‘egos’, we pulled who they followed\n(their ‘alters’) at two time points—March 2023 (T1) and October 2023\n(T2). We were interested in the proportion of (follower⇒alter)\nties that were dissolved from T1 to T2 and if a tie is more likely\nto dissolve if the alter is a misinformation spreader. To avoid exit\nrates confounding unfollowing estimates—we might count a user as\nunfollowed if they instead deleted their account in T2—we removed\nall egos and alters who exited the platform because they were either\nsuspended or their account was deleted.6This process yielded 2,467\negos, 2,245,645 T1 alters, and 5,087,256 T1 (follower⇒alter)edges.\n8,052 of the initial edges were with misinformation spreaders.\n6We got this information by querying Twitter’s compliance endpoint and removing\nany accounts where the compliance status was ‘suspended’ or ‘deleted’.WWW ’24, May 13–17, 2024, Singapore, Singapore Joshua Ashkinaze, Eric Gilbert, & Ceren Budak\nFigure 7: Fewer misinformation spreaders in Study 1 were\nunfollowed than would be expected under a null hypothesis\nof random unfollowing ( 𝑝=1.6×10−6). We modeled the null\nhypothesis with a hypergeometric distribution.\n5.4 Results\n5.4.1 RQ1: How often are misinformation spreaders unfollowed?\nMisinformation spreaders are very rarely unfollowed. The unfol-\nlowing rate of misinformation spreaders was 3.3% (95% CI = [2.93%,\n3.72%]), or a monthly rate of 0.52% (95% CI = [0.46%, 0.58%]). We\ncomputed confidence intervals (CIs) using the Wilson [31] method\nfor binomial proportions. We computed monthly unfollowing rates\nby dividing the total unfollowing rate bystudy duration in days\n30. This\nunfollowing rate is low relative to Twitter unfollowing rates ob-\nserved in prior studies78(Figure 5) or what would be expected\nbased on [11].\nCIs for monthly unfollowing rates were overlapping, and point\nestimates were similarly low, across (A) the entire sample (0.52%,\n95% CI = [0.46%, 0.58%]), (B) those following one misinformation\nspreader (0.54%, 95% CI = [0.47%, 0.62%]), (C) those following more\nthan one misinformation spreader (0.49%, 95% CI = [0.39%, 0.60%]),\nand (D) those following more than two misinformation spreaders\n(0.55%, 95% CI = [0.4%, 0.75%]). The latter estimates are less precise\ndue to the smaller sample.\n5.4.2 RQ2: Are misinformation spreaders unfollowed at a different\nrate than non-spreaders? The unfollowing rate of non-misinformation\nspreaders (4.33%, 95% CI = [4.31%, 4.34%]) was 31% higher than the\nequivalent rate of misinformation spreaders (3.3%, 95% CI = [2.93%,\n3.72%]). See Figure 6. Note that the 95% CIs for unfollowing rates\nbetween the two groups do not overlap. (The wider confidence in-\nterval for the spreader unfollowing rate reflects the fact that there\nwere fewer spreader edges in Study 1.)\nWe ruled out an alternative explanation for our results. We ap-\nplied several filtering criteria to misinformation spreaders, so it\nmay be that these (non-misinformation related) differences in alter\ncharacteristics of the spreaders vs non-spreaders drive our finding.\nWe fit cluster-robust logistic regressions with unfollowing as the\nDV and a dummy variable for IsSpreader (1 if alter is misinforma-\ntion spreader, else 0). The coefficient on IsSpreader is similar with\n(𝛽=−0.329, 𝑆𝐸 =0.110) or without ( 𝛽=−0.331, 𝑆𝐸 =0.111)\n7Prior unfollowing studies often differ in some way (e.g., sample, time period, exact\nmeasure, exclusion criteria) to our study, but they provide a rough baseline.\n8Kwak et al. [15] was based on a shorter duration than the other studies.alter controls (friends, followers, tweets). This suggests these alter\ncharacteristics do not explain the result.\nWe also tested the hypothesis that fewer spreaders were un-\nfollowed than would be expected if unfollowing was random (or\nindependent of whether the alter was a spreader). Under a null hy-\npothesis of random unfollowing, we would expect the proportion of\nsevered spreader ties (0.12%) to be the same as the original propor-\ntion of spreader ties (0.16%). But spreaders were a lower fraction of\nsevered ties than initial ties. The null hypothesis of random unfol-\nlowing can be thought of as randomly ‘sampling’ users to unfollow\nwithout replacement. Then the probability of observing 𝑘spreader\nunfollows out of 𝑛total unfollows, given spreaders are𝐾\n𝑁propor-\ntion of initial ties, is described by the hypergeometric distribution\nwith parameters ( 𝑁=5,087,256,𝐾=8,052,𝑛=219,979,𝑘=266):\n𝑃(𝑋=𝑘)=\u0000𝐾\n𝑘\u0001\u0000𝑁−𝐾\n𝑛−𝑘\u0001\n\u0000𝑁\n𝑛\u0001\nand\n𝑃(𝑋≤266)=266∑︁\n𝑖=0𝑃(𝑋=𝑖)\nprovides an exact p-value on the probability of observing 266 or\nfewer spreaders being unfollowed. That p-value is 1.6×10−6(Figure\n7). We conclude that fewer spreaders were unfollowed than one\nwould expect if unfollowing was random.\n6 STUDY 2: PREDICTORS OF UNFOLLOWING\n6.1 Objectives\nIn Study 2, we answered what predicts a user unfollowing a misin-\nformation spreader (RQ3).\n6.2 Participants\nThe participants were the spreaders and followers from the Model-\ning Sample . See Table 1.\n6.3 Methods\nWe modeled a follower 𝑓unfollowing a misinformation spreader 𝑠\nusing cluster-robust logistic regression. The data was at an edge\nlevel(follower f⇒spreader s). We predicted if an edge that ex-\nisted at T1 (March 2023) would be dissolved at T2 (October 2023).\nThat is, we predicted if a follower would unfollow a spreader. To\naccount for dependencies within spreaders, we report HC1 cluster-\nrobust errors, clustered at the spreader level.9See Appendix Table\n4 for regression results. The covariates were the variables listed in\nAppendix Table 3, plus two-way interactions between our variables\nof interest: initial exposure (measured by number of spreaders fol-\nlowed at T1), ideological strength (measured by absolute value of\n[2]’s ideology measure), and an indicator for liberal ideology (equal\nto 1 if our ideology [ 2] measure is below 0). We used the marginal-\neffects R package to compute average marginal effects (AME) from\nour logistic regression model.\n9We also conducted a Bayesian hierarchical logistic regression with random intercepts\nfor spreaders, but the model yielded a convergence error. Nonetheless, the results\nof the Bayesian model were very similar to the results of our cluster-robust logistic\nregression (Appendix Table 4).The Dynamics of (Not) Unfollowing Misinformation Spreaders WWW ’24, May 13–17, 2024, Singapore, Singapore\nFigure 8: Average marginal effects of predictors on the probability of unfollowing a misinformation spreader. Note that these\neffects are the average effects across the sample, taking into account interaction effects involving predictors.\nFigure 9: The effect of T1 exposure (how many spreaders a\nfollower followed at T1) on the probability of unfollowing\ndiffered for liberals vs conservatives. T1 exposure has a larger\neffect on unfollowing for liberal users.\n6.4 Results\n6.4.1 RQ3: What are the predictors of unfollowing a misinformation\nspreader? See Figure 8 for non-interaction AME estimates. See\nAppendix Table 4 for regression results.\nReciprocity. Reciprocity was the biggest predictor of unfollowing.\nA tie being reciprocal was associated with a significantly lower prob-\nability of unfollowing, AME = -0.0134 (95% CI = [-0.0138, -0.0131]).\nThe unfollowing rate of non-reciprocated misinformation-spreader\nties (2.14%) was 2.4 times the unfollowing rate of reciprocated\nmisinformation-spreader ties (0.89%).\nInitial Exposure. The results were consistent with the reversion\nhypothesis ( H1) and not the inertia hypothesis (H2) : T1 exposure\n(AME = 0.0028, 95% CI = [0.0026, 0.0029]) and the spreader tweeting\noften from T1 to T2 (AME = 0.0022, 95% CI = [0.0021, 0.0023]) were\nFigure 10: The effect of ideological strength on the proba-\nbility of unfollowing differed for liberals vs conservatives.\nIdeological strength is associated with higher unfollowing\nfor liberals and lower unfollowing for conservatives.\nassociated with unfollowing. These can both be considered mea-\nsures of ‘redundancy’. There was a significant interaction between\nT1 exposure and partisan ideology, although the effects were direc-\ntionally the same (see Figure 9). T1 exposure had a larger effect for\nliberals (AME = 0.0040, 95% CI = [0.0035, 0.0046]) than for conserva-\ntives (AME = 0.0024, 95% CI = [0.0023, 0.0025]). Interestingly, there\nwas also a positive interaction between T1 exposure and ideological\nstrength (Appendix Table 4). Overall, the reversion hypothesis (high\nexposure at T1 is associated with high unfollowing at T2) holds on\naverage and separately for both liberals and conservatives—though\nthis effect is roughly 1.7 times as strong for liberals.\nPartisan Ideology. Misinformation unfollowing shows partisan\nasymmetries. Liberals are more likely to unfollow than conserva-\ntives. And at higher levels of ideological strength, this gap widens.\nFirst, (H3) was supported: liberals were more likely to unfollowWWW ’24, May 13–17, 2024, Singapore, Singapore Joshua Ashkinaze, Eric Gilbert, & Ceren Budak\nthan conservatives (AME = 0.0082, 95% CI = [0.0063, 0.0101]). Sec-\nond, (H4) was technically refuted: Averaging across the sample,\nthere was a near-zero, positive AME of ideological extremity on\nunfollowing (AME = 0.0004, 95% CI = [0.0001, 0.0006]). However,\nthis near-zero, average effect masked large partisan asymmetries\n(H5): For liberals, an increase in ideological strength was associated\nwith an increase in unfollowing (AME = 0.0026, 95% CI = [0.0018,\n0.0035]). Yet for conservatives, an increase in ideological strength\nwas associated with a decrease in unfollowing (AME = -0.0004, 95%\nCI = [-0.0005, -0.0002]). That is, extreme liberals are more likely to\nunfollow than moderate liberals, while extreme conservatives are\nlesslikely to unfollow than moderate conservatives. See Figure 10.\n7 DISCUSSION\nWhile much is known about misinformation exposure, little is\nknown about misinformation unfollowing. We provided a large-\nscale analysis of the frequency and predictors of unfollowing mis-\ninformation spreaders.\nRegarding frequency, we found misinformation spreaders are\nrarely unfollowed. Monthly unfollowing rates are just 0.52%. Users\nare also 31% more likely to unfollow non-misinformation spreaders\nthan they are to unfollow misinformation spreaders. The low overall\nunfollowing rate suggests a role for interventions or design changes\nthat either prevent the initial formation or encourage the dissolution\nof these ties. Additionally, misinformation spreaders are unfollowed\nless than would be expected from [ 11], which asked laboratory\nparticipants their hypothetical intentions to unfollow an imagined\nmisinformation spreader. This discrepancy has two implications.\nFirst, hypothetical exercises do not capture the realities of actual\nunfollowing behavior, highlighting the importance of large-scale\nstudies ‘in the field’. Second, there is room for interventions that\nmove individual behaviors more toward individuals’ stated goals\nregarding their information environments. Indeed, interventions\ncan make users (more) aware they follow misinformation producers.\nAlthough rare overall, some factors did (non-trivially) predict\nunfollowing. We found that reciprocity had a large downward\neffect on the probability of unfollowing. And likely because con-\nnections to misinformation spreaders were highly reciprocal, there\nwas a low overall unfollowing rate. This suggests that cost-effective\ninterventions to limit the spread of misinformation might target\nnon-reciprocated connections, which are easier to dissolve.\nConsidering initial exposure, we found more evidence for a rever-\nsionaccount than an inertia account: Overall, higher initial exposure\nwas associated with higher unfollowing and not lower unfollowing.\nThis could be driven by redundancy (indeed, spreaders tweeting\noften10were also predictive of unfollowing) or a regression to the\nmean effect if high initial exposure was incidental rather than in-\ntentional or if the user lost interest in misinformation. Of course,\ndetermining whether the higher rate of unfollowing was due to\nredundancy or regression to the mean has important implications\nfor identifying the right messaging interventions to encourage un-\nfollowing. Highlighting information overload costs might be more\neffective for the former, while messages that simply remind users\nof the information pollution in their networks might be more ef-\nfective for the latter. Future work (e.g., interviews of individuals\n10Though this may also be indicative of information overload.who unfollow misinformation producers) can help determine the\nmechanisms at play and inform messaging strategies.\nWe found partisan asymmetries. Liberals were more likely to\nunfollow than conservatives. Future work could explore possible\nmechanisms for this main effect. At more extreme ends of the\nspectrum, the gap between liberal and conservative unfollowing\nwidened: Extreme liberals are more likely to unfollow than moder-\nate liberals, but extreme conservatives are lesslikely to unfollow\nthan moderate conservatives. Additionally, the reversion effect\nis stronger for liberals. That is, initial exposure has a larger self-\ncorrection effect for liberals than for conservatives. It is worrying\nthat extreme conservatives are both (1) more likely to consume\nmisinformation [ 8,19]and(2) less likely to sever ties with those\nwho spread it. These two dynamics suggest that misinformation\nexposure among conservatives is likely to stay at a high level. Our\nfindings suggest that future work aiming to dissolve connections to\nmisinformation spreaders should consider the user’s ideology. Tar-\ngeting left-leaning users is more likely to result in a tie dissolution,\nbut external interventions are more needed for conservative users.\nTo summarize, we observed a high persistence of misinforma-\ntion ties and asymmetries across partisan lines. The stability of\nmisinformation ties also points to the importance of stopping mis-\ninformation exposure or tie formation in the first place.\n8 LIMITATIONS\nThis work has limitations. First, our method of identifying misin-\nformation was ‘high precision’ but ‘low recall’; our results do not\ngeneralize to allspreaders of misinformation. Our sample is skewed\nby the English-language and North American focus of the PolitiFact\nhealth misinformation fact checks. Second, different dynamics may\nhold for misinformation spreaders with more than 20K followers.\nWe were interested in ‘ordinary’ and ‘non-celebrity’ users. Third,\nwe refer to cases where the edge between (User A⇒User B)disap-\npeared as unfollowing . But API limits prevent us from determining\nif User B blocked User A or manually removed User A as a follower.\nWe suspect these cases are less common, and they are tie dissolu-\ntions, nonetheless. Fourth, our analysis concerns one stretch of time\non one platform. It is important to replicate findings across time\nand platforms. However, API restrictions make this increasingly\ndifficult. Despite these limitations, this work provides a large-scale\nview of misinformation unfollowing.\n9 ETHICS\nThere are ethical consequences of categorizing people as misin-\nformation spreaders of followers. We took steps to address these\nrisks. First, we perform analysis at the aggregated level to minimize\nthe risk of exposing potentially sensitive personal data about in-\ndividuals. Second, we refrain from sharing any information at the\nindividual level. Third, we emphasize that there are ethical concerns\nabout incorrectly generalizing the findings (see Limitations).\n10 ACKNOWLEDGEMENTS\nThis project was supported by the Social Science Research Council’s\nMercury Project and the National Science Foundation [2045432].The Dynamics of (Not) Unfollowing Misinformation Spreaders WWW ’24, May 13–17, 2024, Singapore, Singapore\nREFERENCES\n[1]Eytan Bakshy, Solomon Messing, and Lada A. Adamic. 2015. Exposure to ideo-\nlogically diverse news and opinion on Facebook. 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Structures of\nbroken ties: exploring unfollow behavior on twitter. In Proceedings of the 2013\nconference on Computer supported cooperative work - CSCW ’13 . ACM Press, San\nAntonio, Texas, USA, 871. https://doi.org/10.1145/2441776.2441875\n[33] Joseph Yoo, Margaret Yee Man Ng, and Thomas Johnson. 2018. Social networking\nsite as a Political Filtering Machine: Predicting the Act of Political Unfriending\nand Hiding on Social Networking Sites. The Journal of Social Media in Society 7, 2\n(Dec. 2018), 92–119. https://www.thejsms.org/index.php/JSMS/article/view/359WWW ’24, May 13–17, 2024, Singapore, Singapore Joshua Ashkinaze, Eric Gilbert, & Ceren Budak\nAlgorithm 1 The aim is to find the minimal number of spreaders\ncoming from a single story that allows us to retain at least a target\nnumber of spreaders.\nRequire:𝑇, the target number of unique spreaders\n1:Initialize𝜆, the maximum number of spreaders per story, to 1\n2:Initialize𝑆, the set of spreaders, by randomly sampling 1 user\nper story without replacement\n3:while|𝑆|≤𝑇do\n4: Re-initialize 𝑆as the set of spreaders derived from randomly\nsampling𝜆spreaders per story without replacement\n5: Increment𝜆by 1\n6:end while\nA APPENDIX\nTable 2: These words were generated by prompting GPT-3\nwith ‘Here are a list of words people use when they do not\nagree with a statement: lie, misinformation, debunked, false.\nWhat are other words?’ We used the text-davinci-003 model\nwith a temperature of 0.7.\nDebunking Words\nBaseless Disproved Deceit\nDeceptive Disputed Distorted\nDelusion Erroneous Fabricated\nFalse Falsehood Fictitious\nFlawed Hoax Implausible\nInaccurate Incorrect Lie\nMisinformation Misleading Misrepresent\nMyth No Evidence Not True\nRefuted Unfounded Unreliable\nUnsubstantiated Untrue Unverified\nFake Fake News Dubious\nFigure 11: We estimated a truncated version of our baseline\nspecification (removing any ideology variables) on both the\nModeling Sample and a full sample (i.e: including followers\nexcluded from the Modeling Sample because their ideology\ncould not be matched). Non-binary variables were z-scored.\nCoefficients are the effect of a +1SD change on the log odds\nof unfollowing. Coefficient estimates for the sample of all\nfollowers (red) were similar to those for the sample of fol-\nlowers whose ideology we could match (blue). Error bars are\n95% CIs.\nReceived 12 October 2023; revised 23 January 2024; accepted 23 January\n2024The Dynamics of (Not) Unfollowing Misinformation Spreaders WWW ’24, May 13–17, 2024, Singapore, Singapore\nTable 3: Descriptive Statistics of Followers and Spreaders in the Modeling Sample. There are 898,701 followers, 5,334 spreaders,\n3,376,785 edges. ‘Is Reciprocal Tie’ is a binary variable denoting if a tie is reciprocated. ‘Is Liberal’ is an indicator variable for\nwhen the ideology measure from [ 2] is less than zero. The change in spreader tweet count was computed by subtracting the\nspreader’s total count of tweets as of T2 from the equivalent tweet count in T1. In certain cases, the API returned fewer total\ntweets for a spreader in T2 than in T1. In these cases, we changed the ‘Change Spreader Tweet Count’ to zero.\nMetric Mean SD 25% 50% 75%\nFollower Tweet Count 18116 51193 566 3673 14984\nFollower Follower Count 4311 208369 130 461 1510\nFollower Following Count 2896 13362 628 1442 3107\nN Spreader Following 4 13 1 1 3\nSpreader Tweet Count 52311 97719 7084 20664 56644\nSpreader Follower Count 1413 2389 161 536 1560\nSpreader Following Count 1816 2223 387 962 2484\nChange Spreader Tweet Count 4773 10009 131 1291 5018\nIdeology (Positive Is Conservative) 1.65 1.57 0.06 2.29 3.0\nIs Reciprocal Tie 0.69 0.46 . . .\nIs Liberal 0.24 0.43 . . .\nTable 4: Logistic regression with HC1 cluster-robust errors at the spreader level and Bayesian hierarchical logistic regression with\nrandom intercepts for spreaders. Note: Non-binary variables are z-scored so coefficients can be interpreted as the accompanying\nchange in the log odds of unfollowing with a +1SD increase in the predictor variable.\nDependent variable:\nUnfollowed\nCluster-Robust Logistic Regression Bayesian Multilevel Logistic Regression\n(1) (2)\nrecip −0.948∗∗∗(0.030) −0.983∗∗∗(0.012)\nis_liberal 0.420∗∗∗(0.079) 0.488∗∗∗(0.046)\nfollower_tweet_count 0.027∗∗∗(0.004) 0.030∗∗∗(0.004)\nfollower_following_count −0.050∗∗∗(0.007) −0.048∗∗∗(0.007)\nfollower_follower_count 0.005∗∗∗(0.001) 0.005∗∗(0.002)\nspreader_tweet_count −0.109∗∗∗(0.021) −0.150∗∗∗(0.017)\nspreader_following_count −0.032 (0.049) 0.053∗(0.028)\nspreader_follower_count −0.114∗∗(0.052) −0.206∗∗∗(0.028)\nn_spreader_following 0.182∗∗∗(0.006) 0.172∗∗∗(0.005)\nchange_spreader_tweet_count 0.177∗∗∗(0.014) 0.223∗∗∗(0.013)\nabs_ideo −0.057∗∗∗(0.010) −0.067∗∗∗(0.007)\nis_liberal:abs_ideo 0.264∗∗∗(0.052) 0.313∗∗∗(0.035)\nis_liberal:n_spreader_following 0.174∗∗∗(0.020) 0.166∗∗∗(0.021)\nn_spreader_following:abs_ideo 0.053∗∗∗(0.005) 0.061∗∗∗(0.005)\nConstant −3.872∗∗∗(0.020) −3.916∗∗∗(0.018)\nsd(spreader) 0.492\nN 3376785 3376785\nNote:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01"    },    {        "title": "2401.13481v1.How_AI_Ideas_Affect_the_Creativity__Diversity__and_Evolution_of_Human_Ideas__Evidence_From_a_Large__Dynamic_Experiment.pdf",        "content": "How AI Ideas Affect the Creativity, Diversity, and Evolution of Human Ideas:\nEvidence From a Large, Dynamic Experiment\nJOSHUA ASHKINAZE, University of Michigan, USA\nJULIA MENDELSOHN, University of Michigan, USA\nLI QIWEI, University of Michigan, USA\nCEREN BUDAK, , USA University of Michigan\nERIC GILBERT, University of Michigan, USA\nExposure to large language model output is rapidly increasing. How will seeing AI-generated ideas affect human ideas? We conducted an\nexperiment (800+ participants, 40+ countries) where participants viewed creative ideas that were from ChatGPT or prior experimental\nparticipants and then brainstormed their own idea. We varied the number of AI-generated examples (none, low, or high exposure ) and\nif the examples were labeled as “AI” ( disclosure ). Our dynamic experiment design—ideas from prior participants in an experimental\ncondition are used as stimuli for future participants in the same experimental condition—mimics the interdependent process of cultural\ncreation: creative ideas are built upon prior ideas. Hence, we capture the compounding effects of having LLMs “in the culture loop”.\nWe find that high AI exposure (but not low AI exposure) did not affect the creativity of individual ideas but did increase the average\namount and rate of change of collective idea diversity. AI made ideas different, not better. There were no main effects of disclosure.\nWe also found that self-reported creative people were less influenced by knowing an idea was from AI, and that participants were\nmore likely to knowingly adopt AI ideas when the task was difficult. Our findings suggest that introducing AI ideas into society may\nincrease collective diversity but not individual creativity.\nCCS Concepts: •Human-centered computing →Human computer interaction (HCI) ;Empirical studies in HCI ;Labo-\nratory experiments ;Empirical studies in collaborative and social computing ;Computer supported cooperative work ;\nCollaborative content creation .\nAdditional Key Words and Phrases: creativity, large language models, generative artificial intelligence, chatgpt, co-creation, experiments\nACM Reference Format:\nJoshua Ashkinaze, Julia Mendelsohn, Li Qiwei, Ceren Budak, and Eric Gilbert. 2023. How AI Ideas Affect the Creativity, Diversity,\nand Evolution of Human Ideas: Evidence From a Large, Dynamic Experiment. In tbd.ACM, New York, NY, USA, 42 pages. https:\n//doi.org/XXXXXXX.XXXXXXX\n1 INTRODUCTION\nIf we think of culture as a “loop” where individuals and societies shape each other through exchanges of ideas and\npractices [ 8,55], then a fundamental question emerges: What happens when generative AI joins the “culture loop?” This\nquestion is not hypothetical. Exposure to LLMs (large language models) is increasing rapidly: When released, ChatGPT\nwas the fastest-growing consumer application in history [ 25]. Moreover, we are likely exposed to even more AI content\nthan we realize: Humans overestimate their ability to distinguish AI from human content [ 28]. This exposure likely\nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not\nmade or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components\nof this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to\nredistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.\n©2023 Association for Computing Machinery.\nManuscript submitted to ACM\n1arXiv:2401.13481v1  [cs.CY]  24 Jan 2024Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nmatters: Ideas we see affect the ideas we create [ 44]. How, then, will the rapid rise of exposure to LLM-generated ideas\naffect the creativity, diversity, and evolution of human ideas? And to what extent do AI ideas influence human ideas?\nAs AI exposure has increased, so have concerns over AI disclosure [23] (whether AI systems should disclose the fact\nthat they are AI). California, for instance, considered passing a law requiring disclosure on behalf of anyone using bots\non social media [ 67]. Concerns regarding the disclosure of LLMs are only more likely to grow. In domains ranging from\npoetry [ 31] to online social media profiles [ 28], LLM output is increasingly indistinguishable from that of humans. We\nare interested, then, if disclosing ideas as coming from AI moderates the effect of AI exposure.\n1TaskParticipants submit creative uses for an item after viewing example uses. Examples are from humans or AI.Participants view example creative uses for an item\nThen participants submit their own idea 2Conditions Examples vary by exposure (number of AI ideas) and disclosure (if AI ideas are labeled as such).Exposure = High \n(4 AI ideas,  2 Human ideas) Disclosure = YesExposure = High  (4 AI ideas,  2 Human ideas) Disclosure = NoExposure = Low  (2 AI ideas,  4 Human ideas) Disclosure = YesExposure = Low  (2 AI ideas,  4 Human ideas) Disclosure = NoControl Condition\n(0 AI ideas,  6 Human ideas) Disclosure = No\n…3Dynamic ComponentThe experiment has a dynamic component since the human examples are from prior participants in the same condition. View the most recent human examples (solid) and random AI examples (dashed) \nTrial 1Trial 2Trial 3Submit a new idea, which becomes an example idea for the next personAI[\nFig. 1. Graphical depiction of experiment. The task (Panel 1) is to submit a creative idea after seeing examples, where examples\nare from humans or AI. We vary (Panel 2) the amount of AI ideas in the example set ( exposure ) and if AI ideas are labeled as such\n(disclosure ). The experiment is dynamic (Panel 3). Responses from prior participants serve as examples for future participants.\nMotivated by these dynamics, we conducted a large-scale experiment to systematically test how AI exposure and\ndisclosure affect the creativity, diversity, and evolution of human ideas. We employ a variant of the Alternate Uses\nTask (AUT, [ 22]), a common measure of creativity, and manipulate exposure to LLM ideas. In the AUT, participants are\ntold to think of non-obvious uses of an item. For example: What is a creative use for a tire? In our variant, participants\ncomplete the AUT for an item after viewing example ideas. These examples constitute our manipulation. Examples vary\nin AI exposure (none, few, or many AI examples) and AI disclosure (whether AI-generated ideas are labeled as such).\nThe human-generated ideas in each example set come from prior participants in the same experimental condition. See\nFigure 1 for a graphical depiction of the experiment.\nOur dynamic experiment design—ideas from prior participants are used as stimuli for future participants—mimics the\ninterdependent process of cultural creation: creative ideas are built upon prior ideas. Hence, we capture the compounding\neffects of having LLMs “in the culture loop”. Our design allows us to observe not just average levels but also temporal\n2How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\ndynamics of creativity and diversity in each condition. Taken together, our results provide insights into the role of\nLLMs in shaping collective thought.\nConcretely, our main findings are:\n(1)High AI exposure increases collective diversity but not individual creativity. Passive exposure to AI\nideas is important to understand. Due to the interconnected nature of social media, many more people may be\nexposed to AI content than actively engaged with it. As far as we know, this is the first large-scale experiment\nto measure how passive exposure to (rather than active engagement with) off-the-shelf LLMs affects collective\nbrainstorming. We find that high AI exposure increases collective idea diversity, but does not affect individual\ncreativity. Our high-powered null finding around creativity can inform public debates over the creative impact\nof AI ideas on an individual’s creativity. Maybe, our results suggest, there is little impact on individual creativity.\nHowever, we found conditions with high levels of AI exposure had more collective idea diversity. That is, ideas\nin the high AI exposure conditions were more different from each other. Our findings around creativity and\ndiversity suggest the effect of AI exposure may be nuanced: The introduction of AI ideas into human society\nmay yield more diverse but no better human ideas.\n(2)High AI exposure increases the speed at which idea diversity develops. Culture is constantly evolving\n[8,9], yet many laboratory experiments are not designed to model this evolution. Through our dynamic design,\nwe find that high AI exposure increases not only the average levels of collective idea diversity but also the rate\nof change in idea diversity. This is a consequential finding since even small differences in rates of change can\nlead to large cumulative differences over time.\n(3)People who identify as creative are less influenced by AI disclosure . Prior work argues that attitudes and\nexpectations shape engagement with human-AI co-creation systems [ 19]. Due to our large sample size, we can\nmodel this heterogeneity. We find that for users who self-identify as highly creative, adoption of AI ideas is not\ninfluenced by AI disclosure. But AI disclosure did affect the adoption of AI ideas for users who self-identified as\nlow in creativity. This finding suggests that highly creative people will not be “duped” into adopting AI ideas.\n(4)Participants adopted AI ideas more for harder prompts. We find that when AI ideas are disclosed, partici-\npants are more likely to adopt the ideas of AI for difficult AUT prompts. This suggests that users will rely on\nAI ideas not for trivial creative tasks but for difficult ones. But since this finding is based on a small number of\nprompts, we view this finding as speculative/preliminary and encourage more work on the topic.\n1.1 Defining Concepts and Variables\n1.1.1 Creativity. Creativity is defined in many ways [ 64]. But one common conception is divergent thinking [ 21]. This\nis when “an individual solves a problem or reaches a decision using strategies that deviate from commonly used or\npreviously taught strategies” [46]. One of the most common [1] tests of divergent thinking is the Alternate Uses Task\n(AUT) [ 22]1, where participants are asked to think of an original use for an everyday object. Traditionally, responses\nto the AUT are measured along four dimensions: originality (how original the idea is), elaboration (how much the\nparticipant elaborates on the idea), fluency (how many ideas), and flexibility (different categories of ideas). The latter\ntwo can only be measured if the participant provides multiple responses to the same question. Due to our research\ndesign2, we have participants generate just one creative idea (as in [4]), and we focus on originality.\n1https://www.mindgarden.com/67-alternate-uses\n2Participants see the most recent responses in the condition as stimuli, so if one participant brainstorms many responses that participant would be\nover-represented in future participants’ example sets.\n3Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nWe follow a long tradition of scoring responses to the AUT computationally [ 3,4,15,48,69,72]. Specifically, we\nmeasure the creativity of AUT ideas with an existing fine-tuned GPT-3 classifier [ 48,49], which has an r=0.81 overall\ncorrelation with human judgments of AUT originality. Moreover, we chose AUT items for our experiment where the\nclassifier had the highest accuracy3. Note that our task is highly ‘in-domain’ for the classifier: we ask participants to\ndo the same exact task for the same exact items the model was trained on. We refer to the originality score from this\nclassifier as individual-level creativity , though we note that future work can explore other dimensions of creativity\n(such as fluency). We discuss this classifier in more detail in Section 3.\n1.1.2 Idea Diversity & AI Adoption. In addition to creativity, we measure how our experimental factors (LLM exposure\nand LLM disclosure) shape the diversity of ideas that participants produce. This is a complementary measure to\ncreativity. Creativity is often thought of as an individual-level outcome. Diversity is a collective outcome. Put another\nway, creativity is a property of an idea while diversity is a property of an idea set.4We measure two sides of diversity—\nsemantic divergence (which we refer to as idea diversity ) and semantic convergence towards AI ideas (which we refer\nto as AI adoption ).\nTo measure idea diversity and AI adoption, we first embed all ideas using SBERT [ 52], which are transformer-\nbased embeddings designed for sentences. SBERT excels at capturing semantic similarity [ 52]. Prior work uses neural\nembeddings to compute similarity for AUT responses [2] and other creative tasks [56].\nIdea diversity is the median pairwise cosine distance between idea embeddings in an idea set. As robustness\nchecks, we also measure the mean pairwise distance and average distance to the centroid of a set.\nAI adoption is the maximum cosine similarity between the embedding of the idea a participant submits and the\nembeddings of AI examples that the participants see. Following Roemmele [56], we use the max rather than a\nmeasure of central tendency because if a participant is inspired by an idea, it would likely be a single idea. As\nrobustness checks, we also measure the mean and median pairwise similarity between the submitted idea and an\nAI example, but these are noisier measures of adoption.\n2 RELATED WORK\nOur work bridges two research streams: human-AI co-creation and crowd-sourced creativity. AI ideas are scattered\namongst human ideas, whether or not we can tell [ 28]. This exposure presumably affects the ideas we create (co-creation).\nAnd our ideas presumably affect the ideas others create (crowdsourced creativity). After discussing how our study\nbridges these two streams, we turn to the particular kind of creativity and diversity our experiment captures and what\nis known about how our two factors (LLM exposure; LLM disclosure) would affect these outcome variables. However,\nmuch of the relevant literature gives conflicting predictions, a key motivation for conducting the current study.\n2.1 Situating Our Work Between Co-Creation and Crowd Creativity\n2.1.1 Human-AI Co-Creation. As the creative ability of AI advances [ 35], researchers explored how co-creating with\nAI affects human creativity. Much of this research explores creative writing with language models, in particular [ 14,17–\n19,24,33,36,37,56,68,73]. While most prior work in this domain involves users actively engaging with custom systems,\nour study is concerned with passive exposure to outputs from off-the-shelf models. (By ‘passive exposure’ we mean\nthat (1) users are shown LLM outputs but did not have an active role in the creation of these outputs and that (2) users\n3tire(r=0.91), pants (r=0.91), shoe (r=0.91), table (r=0.9), and bottle (r=0.88)\n4Hence, it is possible in our framework to have a diverse set of non-creative ideas if each individual idea is not creative (by some metric) but different\nfrom one another.\n4How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nwere given no instructions to actively engage with these outputs; they were merely shown the LLM outputs. ‘Passive\nexposure’ approximates how users often encounter LLM outputs in the real world.) Human-AI co-creation shows\nthat the relationship between AI ideas and their effect on human creativity is nuanced, with task-level and attitudinal\nfactors playing a role. Roemmele [56] found seeing AI examples influenced outcomes for hard, but not easy, prompts.\nGero et al . [19] found the quality of LLM outputs did not correlate with perceived usefulness. This is consistent with\nother research showing large variance in the perceived usefulness of outputs from co-creation systems [ 12], suggesting\nhuman attitudes partially determine the utility of AI creativity aids. We extend this predominantly qualitative work\nwith a large-scale quantitative study.\nSeveral studies [ 27,57,65] explored the most relevant co-creation task to our study: brainstorming with chatbots.\nHwang and Won [27] developed a study design where participants generated ideas with a partner who was either a\nhuman confederate or a scripted chatbot. Participants produced more creative ideas when they believed their partner\nwas a chatbot. Hwang and Won [27] explain this using social facilitation theory—the perceived novelty of collaborating\nwith an AI system enhanced performance. This core finding was replicated in Wieland et al . [65] . We build on this work\nby viewing the effect of AI disclosure through social facilitation. But our study has two key differences from Hwang\nand Won [27]. First, we consider exposure to real AI ideas rather than a scripted chatbot. Second, we focus on passive\nexposure and not direct engagement; perhaps the novelty of AI is lower when users are passively exposed to its ideas\nvs. directly engaging with it.\n2.1.2 Crowdsourced Creativity. Crowdsourcing can enhance creative outcomes [ 26,43,60,70,71]. For example, Yu\nand Nickerson [71] devised a method where crowds build on each other’s ideas by combining ideas from previous\ngenerations. Later generations of ideas were rated as more creative compared to earlier generations. Siangliulue et al .\n[60] found that the creativity and diversity of idea sets that participants saw influenced the creativity and diversity of\nwhat these participants produced. This supports a main contention of our paper: AI exposure matters because the ideas\nwe see affect the ideas we create. We incorporate elements of crowdsourced creativity, particularly in measuring how\ncreativity and diversity unfold over subsequent generations.\n2.1.3 Our Contributions. Our study incorporates elements of human-AI co-creation and crowd creativity, complement-\ning both fields. We note that co-creation studies often confound the effect of exposure with the effect of disclosure:\nIf one is creating with an AI system, it is impossible to separate the content of an AI system from the knowledge that\nthe content is from an AI system. Our factorial design lets us estimate the marginal effect of exposure and disclosure\nseparately. Co-creation studies typically employ a small number of specialized participants actively engaged with a\nsystem. From the perspective of validating a system, this is reasonable. But we are interested in the effects of (1) passive\nexposure on (2) a general public. For this reason, we adopt a large-scale experimental design—similar to crowd-sourced\ncreativity studies—that lets us estimate effects on the general public rather than specialized users. A key benefit of\nour large sample size is that we can precisely estimate how participant attitudes affect human-AI outcomes. This is\nimportant because, as Gero et al . [19, pg. 1016 ]write: “[P]articipant attitudes are a major unknown factor when studying\nhuman-AI collaboration.” Our experiment design also lets us understand the effect of AI over time since responses feed\nforward, allowing us to observe differences in rates of change between conditions.\n2.2 Factor 1: LLM Exposure\n2.2.1 Effects on Creativity. Intuitively, the effect of exposure to ChatGPT ideas will depend on how creative ChatGPT\nanswers are relative to human ideas. In preliminary testing, we found that the answers to the AUT generated by our\n5Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nprompt were scored as roughly 12 percentage points more creative than the ideas generated by humans (see Appendix\nE) via the Organisciak et al . [48] classifier. LLM generations may be increasing in creativity: while GPT-3 (an earlier\nmodel than ChatGPT-3.5) scored lower in AUT creativity than humans on the AUT [ 62], GPT-4 (a more recent model\nthan ChatGPT-3.5) scored among the top percentile of humans on a similar verbal creativity task [45].\nEven if language models can generate creative ideas, it is unclear from prior work if exposure to these ideas can\nincrease human creativity. On one hand, the associative model of brainstorming suggests that exposure to others’ ideas\ncan stimulate idea generation by activating a non-accessible concept of a participant’s memory [ 11,44,51]. For example,\nChatGPT may come up with a use for a bottle that you never associated with bottles. This can then inspire you to come\nup with creative uses along this line. In this way, ChatGPT can stimulate creativity. On the other hand, there is also\nevidence that seeing the ideas of others inhibits a participant’s idea generation if “one is exposed to an idea that has few\nconnections to other ideas in an individual’s semantic network” [ 51, pg. 10]. Indeed, this appeared to be the case in\nYang et al . [68] . There is a possibility that AI ideas are creative but so divorced from how humans generate ideas that\nseeing these ideas actually has an inhibiting effect. Separate from prior academic work, there are public debates about\nthe impact of LLMs (such as ChatGPT) on creativity (e.g., [ 16,29,32,42,54,63,66]). Many of these debates assume\nChatGPT will have some impact on an individual’s creativity—either good or bad. Our work contributes empirical\nresults to this broader public conversation.\n2.2.2 Effects on Diversity. Prior work in AI co-creation finds that actively collaborating with AI can lead to more diverse\nor unexpected outputs [ 10,18,33,50,68]. But note that these studies are testing active engagement with intentionally\nconstructed systems. This is different from the passive, incidental exposure to AI ideas that now occurs in everyday life.\nWriters call ChatGPT ‘a blurry JPEG of the internet’ [ 13] and discuss its ‘incredible blandness’ [ 34]; researchers call\nit a ‘stochastic parrot’ [ 6]. It is not clear, then, if ideas from off-the-shelf LLMs—precisely the kind we are inundated\nwith— alsoincrease the diversity of human ideas.\n2.3 Factor 2: LLM Disclosure\n2.3.1 Effects on Creativity. Building on Hwang and Won [27], we employ the theory of social facilitation [ 7] to\nunderstand how LLM disclosure can affect human creativity. Facilitation theory is concerned with how the presence of\nothers affects one’s performance. Hwang and Won [27]asked participants to brainstorm with chatbots (which gave\npre-programmed responses) and experimentally varied whether or not participants were told that their partner was a\nchatbot. Disclosing that the partner was a chatbot led to higher creativity in participant responses, which Hwang and\nWon [27] attributes to the novelty of brainstorming with a chatbot. We build on this notion of facilitation as a theoretical\nlens. However, it is not clear if Hwang and Won [27]’s finding (that telling people they are brainstorming with a chatbot\nincreases creativity) would replicate in our study, especially in a post-ChatGPT era. First, we are measuring exposure\nand not direct engagement with chatbots. The novelty of a chatbot may be higher when you are the one working\nwith it to generate ideas. Second, presumably, the novelty of talking to a chatbot may be lower due to the widespread\npopularity of ChatGPT. Moreover, we may expect heterogeneity in disclosure’s effect on creativity and diversity. It may\nbe that users who have lower self-perceived creative abilities may feel ‘competition’ with AI due to its presence and, in\nturn, submit more creative responses when they know the ideas they are exposed to are from AI.\n6How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\n2.3.2 Effects on Diversity. It is not clear how knowing content is from AI will affect the diversity of ideas participants\nproduce. But prior work suggests heterogeneity along two lines: the difficulty of the prompt5, and the attitude of the\nparticipant. Prior work suggests that disclosing ideas as AI-generated would decrease diversity due to automation bias,\nthe tendency to over-rely on AI systems [ 20,38,58]. Increased reliance on AI ideas (when labeled as such) could lead to\nlower idea diversity and higher AI adoption. Conversely, some evidence suggests people display algorithmic aversion\nto creative products such as haikus [ 24] or art [ 30]. This aversion would yield the opposite prediction. Roemmele [56]\nfound that seeing AI examples only affected the participant’s writing on a key measure for difficult prompts—suggesting\ncreative task difficulty might moderate the effect of disclosure on AI adoption. Task confidence decreases reliance on\nautomated systems and trust in a system increases reliance on automated systems [ 20]. Although this literature is not\nusually applied to creativity, we might then suspect that people self-reporting low creativity (i.e., low task confidence)\nand those who think AI is more creative than humans (i.e., high system trust) are most likely to increase adoption of AI\nideas when the source is disclosed.\n3 PRE-EXPERIMENT\nBefore describing the experiment, we discuss how we chose the five specific AUT (Alternate Uses Test) items and\nconstructed our ChatGPT prompt.\n3.1 Stimuli Construction\n3.1.1 Choosing AUT Items. We had to choose a selection of items that people would brainstorm creative uses for. We\nchose five items for which the creativity classifier that we used had the highest accuracy. Previously, Organisciak et al .\n[48] fine-tuned GPT-3 Davinci to predict the creativity of AUT items. This dataset contains 20,121 responses from 2,025\nparticipants, across 21 distinct AUT items and nine distinct studies [ 48].6Each response was graded for creativity by\nhumans and normalized to a scale of 1-5. Then Organisciak et al . [48] fine-tuned GPT-3 Davinci on this dataset to\npredict creativity scores. Here, fine-tuning involves providing { Input (an AUT response), Output (human rating)} pairs\nto a pre-trained LLM. Then the LLM adjusts its parameters to produce a similar output given an input, proxying human\njudgements. Overall, the fine-tuned GPT-3 classifier had a correlation7of𝑟=0.81with human judgment. Accuracy\nvaried by item (Appendix Table 5). For our experiment, we picked the five items for which the classifier had the highest\naccuracy: tire(r=0.91), pants (r=0.91), shoe (r=0.91), table (r=0.9), and bottle (r=0.88). Our task is ‘in-domain’ for the\nclassifier since we ask participants to do the same task for the same items the classifier was trained on.\n3.1.2 Generating GPT Ideas. We generated AI ideas with ChatGPT-3.5 and a zero-shot prompt based on prior work.\nThese decisions followed two principles: ecological validity and precedent.\nModel & Prompting Strategy. Our model and prompting strategy were driven by a desire to approximate how ordinary\nusers would use large language models to generate ideas. First, we used ChatGPT-3.5, the latest ChatGPT model freely\navailable at the time of the study. Because ChatGPT has a popular and accessible UI, we assume users would be more\nlikely to use ChatGPT rather than a model available only through an API or on a limited basis. Second, we used zero-shot\n5As discussed later, we measure the ‘difficulty’ of a prompt by the inverse rank of the average creativity in the control condition. If participants tended to\nsubmit lower creativity ideas in the control condition for item X, we said item X was difficult .\n6We obtained this dataset by direct correspondence with Dr. Organisciak on February 23, 2023; the code that Dr. Organisciak used to generate this dataset\nis available at https://github.com/massivetexts/llm_aut_study/blob/main/notebooks/Process_AUT_GT.ipynb\n7We obtained scores for this classifier by downloading the zip file from (https://github.com/massivetexts/llm_aut_study/blob/main/results/evaluation.zip),\nthen navigating to gt_main2/gpt-ft-davinci-1.csv\n7Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nprompting rather than few-shot prompting. Because zero-shot prompting requires no labeled data, this would be a\nmore natural use case for most users.\nPrompt Construction. Our specific zero-shot prompt was informed by prior work on LLMs and creativity. Stevenson\net al. [62] administered the AUT to GPT-3 through a zero-shot prompt. However, this prompt generated much wordier\nresponses(𝑀=24.1,𝑆𝐷=8.8)than the human responses in the Organisciak Dataset (𝑀=4.6,𝑆𝐷=5). Such a\ndiscrepancy would alert participants to what was AI vs. human generated, which would nullify the disclosure factor\n(whether the source of an idea is disclosed). Hence, we appended a request (Figure 2) to use roughly the same number\nof words (5) as the average human response.\nWord Length Prompt Experiment. We conducted a Monte Carlo experiment to confirm our modified zero-shot prompt\nresulted in responses with a similar word length to humans. Using parameters from Stevenson et al . [62] ’s experiments:\nFor𝑛=1000 trials, we fixed presence penalty and frequency penalty at 1, randomly chose a temperature (higher values\nlead to more randomness) in [0.65, 0.7, 0.75, 0.8], and randomly chose one of our 5 AUT items. For each trial, ChatGPT\ngenerated five ideas. The modified prompt resulted in responses with an average word length (M=4.44, SD = 1.34) much\ncloser to human responses (M=4.56, SD = 4.97) than the original zero-shot prompt (M = 25.38, SD = 8.55). A permutation\ntest (Appendix B), further shows that this difference was significant at 𝑝<0.001. We used the ideas generated by our\nmodified zero-shot prompt as stimuli for the main experiment.\nWhat are some creative uses for a [OBJECT]? The\ngoal is to come up with creative ideas, which\nare ideas that strike people as clever, unusual,\ninteresting, uncommon, humorous, innovative, or\ndifferent. List creative uses for a [OBJECT]. Make\nsure each response is [MEAN HUMAN WORDS] words.\nFig. 2. To generate AUT ideas, we used the zero-shot prompt from Stevenson et al . [62] with an additional instruction at the end to\nmatch the mean length of human responses from prior work.\n4 EXPERIMENT\n4.1 Summary\nWe recruited participants from a mixture of social media and newsletters. Once participants clicked on the link to\nthe experiment, they were taken to a landing page. In addition to a consent button, that landing page asked several\nquestions. Participants were asked (1) to rate their creativity relative to other humans and to AI, (2) to rate their attitude\ntowards AI [ 40,41], (3) age, (4) country, and (5) gender. After consenting, participants engaged in 5 trials. For each\ntrial, a participant generated a creative use for an item under a specific experimental condition, after seeing example\nideas. These example ideas constituted our experimental manipulation. Ideas fed forward to future trials such that if a\nparticipant was in the {[ Control ],tire} condition the example ideas the participant saw were the most recent ideas from\nprior participants in the {[ Control ],tire} condition. See Table 1 for experimental conditions. The experiment took place\nin the summer of 2023.\n8How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nTable 1. Experiment conditions and associated factors. Participants complete the Alternate Uses Task in each condition after being\nexposed to prior responses generated in each condition. Conditions vary by LLM exposure (none, high or low) and LLM disclosure\n(source is labeled or not).\nCondition Number of AI Examples Number of Human Examples Source of Ideas Disclosed\nControl 0 6 N\nHigh Exposure;\nDisclosed 4 2 Y\nHigh Exposure;\nNot Disclosed 4 2 N\nLow Exposure;\nDisclosed 2 4 Y\nLow Exposure;\nNot Disclosed 2 4 N\n4.2 Ethics\nThe experiment was approved by the [REDACTED FOR REVIEW] IRB. The experiment was deemed exempt from\nongoing oversight under Exemption 3: Benign Behavioral Intervention.\n4.3 Recruitment\nWe recruited volunteer participants through three sources: (1) Facebook ads, (2) Reddit, and (3) the weekly newsletter\nof Creative Mornings89, which is ‘the world’s largest face-to-face creative community’. Creative Mornings is an\norganization geared towards creative professionals that organizes (e.g.) talks and meetups. We ensured all participants\nwere above 18. While we did not offer monetary compensation, we offered to give participants information about\nthemselves, such as their creativity relative to both humans and AI, and their ability to spot creative ideas. Providing\ninformation to participants about themselves is often effective for recruiting volunteer participants since it makes the\ntask intrinsically rewarding [53]. Appendix F describes the information we provided to participants.\nWe recruited volunteer participants instead of crowdsourced workers for several reasons. First, we wanted participants\nto be intrinsically motivated since (1) many theories suggest intrinsic motivation helps creativity [ 39] and (2) we did\nnot want low-quality engagement to confound results (especially since ideas propagate forward). Second, we were\ninterested in an international sample. Because we did not pay participants, we did not need to collect any personally\nidentifiable information. Each user was assigned a random identifier. The experiment being anonymous created a lower\nbarrier to recruiting international participants since GDPR was not operative. Finally, we recruited participants in a\ntargeted manner. In particular, we wanted to generalize this experiment to two key groups: individuals who have a\ndemonstrated interest in technology and those who have a demonstrated interest in creativity. These groups are most\nrelevant to the phenomena in question. To this end, we reached technology-oriented users by posting the experiment\nin the following subreddits: r/InternetIsBeautiful, r/chatgpt, r/singularity, and r/artifical. We reached creativity-oriented\nusers by posting the experiment in r/writing, r/poetry, and the Creative Mornings newsletter. We also used several\n8https://creativemornings.com/\n9The first author contacted Creative Mornings, who agreed to include the experiment in the newsletter.\n9Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\n‘neutral’ sources to test the experiment: r/samplesize and Facebook ads. If a participant completed the experiment, then\nthe participant was given a shareable link to their results so they could spread the study.\n4.4 Experiment Procedure\nOnce participants clicked on our link, they were taken to a landing page that included a consent form, task description,\nand pre-treatment questions.\n4.4.1 Study Description. The description read as follows:\nWhat you will do:\nWe’ll show you 5 common items, and you’ll come up with creative uses for each item. To spark your\nimagination, you’ll see ideas from prior participants and even from AI (i.e., ChatGPT). You’ll be asked\nto rank these ideas in order of creativity. The ideas you write may be anonymously shown to future\nparticipants to spark their imagination. The study takes 3-6 minutes to complete. The goal is to learn\nabout how humans and AI brainstorm.\nWhat you will learn:\n•How creative you are compared to other humans\n•How creative you are compared to AI\n•How well you can rank creative ideas\nWe will give you a shareable link with results at the end.\nSee Appendix F for more details on how each of these three pieces of information was calculated.\n4.4.2 Pre-Treatment Questions. Participants were asked several pre-treatment questions:\n(1) (required) A slider ranging from 0 to 100 that says ‘I am more creative than X% of AI‘\n(2) (required) A slider ranging from 0 to 100 that says ‘I am more creative than X% of Humans‘\n(3)(required) ‘Artificial intelligence computer programs are designed to learn tasks that humans typically do. Would\nyou say the increased use of artificial intelligence computer programs in daily life makes you feel...[‘More\nconcerned than excited’, ‘More excited than concerned’, ‘Equally excited and concerned’]\n(4) (optional) What country are you from?\n(5) (optional) What is your age?\n(6) (optional) What is your gender?\nThe third question was from Pew [ 40,41]. Our gender question was based on guidance from Spiel et al . [61] . We\nchose the Pew question instead of a longer battery of questions about AI to minimize the response burden. See Appendix\nC for more details about these questions.\n4.4.3 Randomization. Participants were assigned a sequence of 5 trials, where each trial was a {[ condition ],item} pair.\nFor example, one trial might be a creative idea for pants in the [ High Exposure ,Disclosed ] condition. We mapped each\nAUT item (pants, tire, shoe, bottle, table) to one of the five conditions such that neither conditions nor items repeated in\na 5-item sequence. See Figure 3 for a visual explanation.\n10How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nFig. 3. Participants are randomized to a sequence of 5 trials. In each trial, participants generate a creative use for an item under a\nspecific experimental condition. Neither items nor conditions repeat in a 5-trial sequence.\n4.4.4 Task Instructions. For each trial, participants were asked to first rank a list of example ideas in order of creativity\nand then submit their own idea:\nTask\nFor this task, you will submit a creative use for a [ITEM]. But before submitting your idea, here are some\nideas for inspiration. Rank them by creativity.\nRank Previous Ideas\n•Rank these ideas in order of creativity, with the most creative use on top. Drag ideas to rank them.\n•We’ll show you how your rankings compare to rankings from a highly accurate model.\n[SORTABLE EXAMPLE IDEAS HERE]\nSubmit Your Idea\nYour turn! What is a creative use for a [ITEM]? The goal is to come up with a creative idea, which is an\nidea that strikes people as clever, unusual, interesting, uncommon, humorous, innovative, or different.\nList a creative use for a [ITEM].\nSee Appendix M for screenshots. We asked participants to rank ideas to ensure that they would engage with the\nexample ideas.10Depending on the condition, (1) either there were or were not AI ideas in this example set ( exposure );\n(2) AI ideas were or were not labeled ( disclosure ). We use the same prompt for humans (the text under Submit Your\nIdea ) as with ChatGPT [ 62], but with a slight modification to request a single idea. This prompt contains language\nconsistent with best practices in divergent thinking assessment Beaty et al. [5]. After submitting an idea, participants\nreceived feedback on their idea’s uniqueness and how accurately they ranked the example ideas (Appendix M).\n4.4.5 Response Chains.\nLogic. The human ideas that participants saw came from prior participants in the same {[ condition ],item} combination.\nSee Figure 4. For instance, if a user was placed in the [ Control ] condition for a tire, that user would see six human\n10We did not use these rankings as a DV since—because a participant ranks the examples they are shown, and all examples are from the same\ncondition—these ranks could not speak to between-condition differences, which is the focus of the paper.\n11Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nResponse Chain LogicResponses feed forward to future participants. This is an example for the control condition but we do this for all conditions.\nT1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…T1T20…20 trials make up one response chain. After 20 trials, the chain starts again.On trial 20, this participant sees ideas generated in trials 14-19 in the response chain. In each trial, participants generate a creative use for an item after seeing prior responses in the chain. Control Condition\nFig. 4. Participants see example ideas from prior participants in the same condition. These ‘response chains’ reset every 20 responses.\nideas—the most recent six ideas for a tireunder the [ Control ] condition. In order to avoid overfitting to a specific idea\nsequence, we reset this ‘response chain’ every 20 trials. So, the first 20 participants in the {[ Control ],tire} combination\nwould see each other’s ideas, but the chain would reset for the 21st respondent. We use the logic described in this\nparagraph and Figure 4 for the human ideas in all conditions.\n(Note that because human ideas are propagated at the {[ condition ],item} level, the human ideas in the [ Control ]\ncondition are ‘clean’ from AI contamination. They were brainstormed after seeing sets of human-only ideas, also from\nthe [ Control ] condition.)\nWe ran seven response chains for each of the 25 (5 items x 5 conditions) combinations, corresponding to 175 response\nchains in all and 3500 targeted responses (175response chains×20trials per chain).\nHuman Seeds. Of course, there is a bootstrapping problem—what human ideas does the first person in the {[ Control ],\ntire} condition see? The seeds for each {[ condition ],item} combination came from prior responses from the Organisciak\nDataset. That is, Participant 1 for a {[ Control ],tire} response chain would see 6 seed items. Then Participant 2 in the\nsame response chain would see 5 seed items plus Participant 1’s idea (the order of ideas is randomized). Participant 3\nwould see 4 seed items plus Participant 1 and Participant 2’s ideas, etc. We chose a random sample of seeds for each\n{[condition ],item} combination from the Organisciak Dataset. The dataset labeled ideas with gold-standard human\n12How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nTable 2. Summary Statistics of Experiment\nValue\nUnique Countries 48.00\nTotal Responses 3414.00\nUnique Participants 844.00\nAvg Responses/Participant 4.05\nAvg Duration/Response 144.31\nTable 3. Sources of participants and trials. For analysis, we categorized each source into a higher-level interest group (technology,\ncreativity, neutral).\nInterest Group source Participants (N, % of total) Trials (N, % of total)\ncreative Creative Mornings newsletter 343 (40.6%) 1470 (43.1%)\ntechnology r/InternetIsBeautiful 298 (35.3%) 1115 (32.7%)\nneutral r/samplesize 94 (11.1%) 389 (11.4%)\nneutral share 61 (7.2%) 250 (7.3%)\ntechnology r/chatgpt 19 (2.3%) 79 (2.3%)\ncreative r/writing 7 (0.8%) 30 (0.9%)\nneutral other 6 (0.7%) 22 (0.6%)\ntechnology r/singularity 6 (0.7%) 13 (0.4%)\ntechnology r/artificial 5 (0.6%) 24 (0.7%)\ncreative r/poetry 3 (0.4%) 15 (0.4%)\nneutral facebook 2 (0.2%) 7 (0.2%)\nratings of originality. We conducted an ANOVA and found no significant condition-level difference in the originality of\nthe seeds we used.\n5 RECRUITED PARTICIPANTS\nWe received over 3000 responses from 48 countries. See Appendix G for sample characteristics. Out of a total of five\ntrials, participants finished four trials on average (Table 2), suggesting the experiment was engaging. Most participants\ncame from the Creative Mornings newsletter or r/InternetIsBeautiful (Table 3 for source counts and categorization).\nThe sample was 50% women, 43% men, 4% non-binary, 3% not disclosed, 1% self-described. The mean age was 34.92\n(SD = 10.86). Regarding AI, the sample was 48% neutral, 28% excited, 24% concerned. Participants said they were more\ncreative than 57.86% (SD = 26.66) of AI and 58.67% (SD = 23.65) of humans. See Appendix Figure 11 for kernel density\nplots. Users from neutral interest groups who were concerned about AI tended to have low self-reported creativity.\n6 OUTCOME MEASURES\nWe have three outcome measures (idea diversity, creativity, and AI adoption) and three levels of analysis (local, evolution,\nand global). See Table 4. The local level measures outcomes at the level of an individual trial (e.g., how a submitted\nresponse relates to example responses). The evolution level measures the rate of change of outcome variables with\nrespect to the trial number in the response chain (i.e., experiment iteration). The global level compares all submitted\nresponses in a condition to each other. For all pairwise comparisons, we use a Holm-Bonferroni adjustment for multiple\n13Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\ncomparisons. For idea diversity and AI adoption, we scale the dependent variable (cosine distance or cosine similarity,\nrespectively) by 100 for easier interpretation.\nTable 4. We measure three outcome measures (idea diversity, creativity, AI adoption) and three levels of analysis (local, global,\nand evolution). If a level of analysis is not appropriate for an outcome measure, we put a ‘Not applicable’ in that cell. All ideas are\nembedded using SBERT.\nLocal Global Evolution\nCreativity How creative is the submitted\nresponse? This is measured by\nthe prediction of the classifier\nfrom Organisciak et al. (2022).Not applicable Does the creativity of submitted\nideas change over time? This is\nmeasured by the slope of the re-\nsponse chain’s trial number (i.e.,\niteration in the response chain)\non creativity (the metric from\nOrganisciak et al. (2022)).\nIdea Diversity How different is a partici-\npant’s response from example\nresponses? This is measured by\nthe median pairwise semantic\ndistance between ideas a partic-\nipant sees and their response.How diverse were all the partici-\npant’s ideas in a condition? This\nis measured by the median pair-\nwise distance between all sub-\nmitted ideas in a condition.Do ideas become more different\nfrom each other as the exper-\niment goes on? We first mea-\nsure the median pairwise dis-\ntance (‘idea diversity’) of ideas\nat each trial number (i.e., itera-\ntion in the response chain). We\nthen measure the slope of the\ntrial number on idea diversity.\nAI Adoption How similar is a participant’s\nresponse to AI example re-\nsponses? This is measured by\nthe maximum semantic dis-\ntance between a participant’s re-\nsponse and AI examples.Not applicable Not applicable\n6.1 Local Level\nOutcomes at the local level—the level of an individual trial—are useful for two reasons. First, this level shows how a\nparticipant’s response relates to the examples they see. Second, this level lets us model whether individual differences\nmoderate the effect of either disclosure or transparency. For each of our local outcomes, we have a baseline model that\nuses crossed random intercepts to account for the multilevel structure of the experiment. The first random intercept\nis for participants, accounting for clustering due to repeated measures. This random intercept is then crossed with\na second random intercept for response chains, which we nest inside of {[ condition ],item} combinations.11Models\n11In R syntax, the random effect structure was ... + (1|ParticipantID) + (1|ItemCondition/ResponseChainID) ; See Figure 4 for a visual\nexplanation of how response chains are nested in items and conditions.\n14How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nwere fit in the lme4 R package. We computed profile likelihood confidence intervals for coefficients using the confint R\npackage. We used estimated marginal means ( emmeans R package) to conduct model-adjusted F-tests, linear contrasts,\npredictions, and pairwise comparisons. We apply Holm-Bonferroni adjustments to pairwise comparison p-values. Our\nbaseline ‘local’ model is:\nvariable 𝑖 𝑗𝑘=𝛽0+𝛽1condition 𝑗+𝛽2CreativityHuman𝑖+𝛽3AiRelCreate 𝑖+\n𝛽4AiFeeling𝑖+𝛽5InterestGroup𝑖+𝛽6ConditionOrder 𝑖 𝑗𝑘\n𝛽7LogDuration𝑖 𝑗𝑘+𝛽8nSeedsPresent 𝑖 𝑗𝑘+𝛽9TrialNo 𝑗𝑘\n𝑢0𝑖+𝑣0𝑗𝑘+𝑒𝑖 𝑗𝑘\nwhere\n•𝑖indexes participants, 𝑗indexes item-condition combinations, 𝑘indexes response chains.\n•CreativityHuman is self-perceived creativity relative to AI.\n•AiRelCreate is constructed as (self-perceived creativity to humans) - (self-perceived creativity to AI). Note\nthat this is an implicit measure of AI’s creativity relative to humans. For example, if you say you are more\ncreative than 40% of humans and 60% of AI, then AiRelCreate = -20, as the implicit belief is AI is less creative (-20\npercentile points) than humans. Conversely, if you say you are more creative than 20% of AI but 50% of humans\nthen the implicit belief is humans are more creative (50% - 30% = +20).\n•AiFeeling refers to the AI sentiment question.\n•InterestGroup maps each source of the experiment to categories: creative, neutral, or technology. These\ncategories are described in Table 3.\n•ConditionOrder denotes the sequence in which the participant was assigned to complete the trial (e.g., the 1st\ntime a participant took part, etc.).\n•LogDuration is the natural logarithm of the time (in seconds) a participant spent before submitting their answer.\n•nSeedsPresent controls for the number of examples the participant saw that were seed ideas from the Organisciak\nDataset.\n•TrialNo indicates the trial number within a specific response chain. For example: the 18th response for {[ Control ],\ntire, response chain 5}\nWe balanced interest in testing experimental hypotheses that conditions differed by subgroups with caution around\nan over-fitted model. We considered interactions between the treatment condition and four potential moderators:\nself-perceived human creativity ( 𝐶𝑟𝑒𝑎𝑡𝑒𝐻𝑢𝑚𝑎𝑛 ), AI - Human creativity ( 𝐴𝑖𝑅𝑒𝑙𝐶𝑟𝑒𝑎𝑡𝑒 ), feeling towards AI ( 𝐴𝐼𝐹𝑒𝑒𝑙𝑖𝑛𝑔),\nand interest group ( 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝐺𝑟𝑜𝑢𝑝 ). We first conducted likelihood ratio tests to test if adding each moderator improved\nour baseline model. Moderators were kept only if they significantly improved the fit (𝑝<0.05). See Appendix Table 11\nfor retained moderators. Then, we used emmeans to probe and interpret moderating effects.\n6.2 Global\nIntuitively, the global diversity of ideas in a condition measures how similar or different submitted ideas in a condition\ntend to be. The relevant level of aggregation here is allof the submitted ideas at a {[ condition ],item} level. For example,\nconsider the total set of ideas participants submitted for a tirein the [ Control ] condition. Is this set of ideas more diverse\nfrom each other than the set of submitted ideas for a tire in the [ High Exposure ,Disclosed ] condition?\n15Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nWe used a Monte Carlo procedure and permutation tests to assess if conditions differed with respect to these metrics.\nFor 50 Monte Carlo runs, for each {[ condition ],item} combination, we randomly sampled 50 ideas and computed idea\ndiversity metrics. We then conducted pairwise paired (at the level of Monte Carlo seeds and items) permutation tests\nwith 10,000 iterations to see if the two conditions differed on these metrics. As a non-parametric measure of effect size,\nwe also calculate Cliff’s Delta (𝛿), which ranges from -1 to 1. A value of 0 indicates no difference between the two\nconditions, +1 indicates values from the first condition are always larger , and -1 indicates the opposite. See Appendix\nI.1 for more details.\n6.3 Evolution\n6.3.1 Creativity. To test if conditions differed in their evolution of creativity, we conducted a likelihood ratio test on\nwhether an interaction between condition andTrialNo significantly improved the fit of the local creativity model.\n6.3.2 Idea Diversity. Intuitively, we are interested in if—as the experiment goes on—ideas that participants submit\ntend to become more or less similar to each other. We use the trial number in a response chain to index time in the\nexperiment. For example, is the set of submitted responses at trial number 4 more or less similar to each other as the\nset of submitted responses at trial number 20? Here, the diversity of interest is not between a submitted response and\nexample responses but between all submitted responses at a given ‘time point’ (i.e., trial number). The question is if the\ndiversity increases or decreases as the experiment goes on and if this rate of change differs by condition. Here is the\nmechanics of our process. See Appendix I.3 for more details.\n(1)We first ‘pooled’ together all ideas at the {[ condition ],item,trial number } level, across response chains. For\nexample, consider all ideas for a tirefor the [ Control ] condition that were the fourth response in a response chain.\nWe refer to this set as a ‘pool’ of ideas.\n(2)We next computed idea diversity measures for each pool of ideas, where idea pools were defined in (1). We\nuse the same metrics that we measure at a local level for idea diversity. Median pairwise distance is our main\nmeasure. We conduct robustness checks using mean pairwise distance and mean distance from the centroid.\nEach metric shows qualitatively similar results.\n(3)We then fit a mixed model (items as random intercepts) to test if the slope of trial number on idea diversity\ndiffered by condition. That is: Are submitted responses in some conditions changing at a faster rate?\n7 RESULTS\n7.1 Creativity\nWe found no effect of conditions on creativity. Average individual creativity did not significantly differ by condition\n(𝐹(4,19.86)=0.12,𝑝=0.97)and no condition coefficient differed from zero in our regression (Appendix Table 19).\nHence, we conclude that neither AI exposure nor AI disclosure affected individual creativity. Additionally, we tested for\nwhether the evolution of creativity differed by condition via a likelihood ratio test on whether interacting trial number\nand experimental condition would improve the model fit. The likelihood ratio test indicated that allowing for these\ninteractions did not significantly improve the model fit (𝜒2(4)=6.52,𝑝=0.16)12. In short, we do not find enough\nevidence to conclude creativity was affected by experimental conditions.\n12However, there was a small, negative interaction effect ( 𝛽=−0.015,𝑡(3248)=−2.21, 95% CI =[−0.03,−0.002]) between trial number and the [High\nExposure, Disclosed] condition when adding this interaction. However, due to the (1) size of the interaction combined with (2) no overall differences and\n(3) a null likelihood ratio test, we do not interpret this interaction.\n16How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\n7.2 Idea Diversity\n7.2.1 Local Level. Intuitively, local idea diversity is how different a response is from the examples a participant sees.\nThere was no main effect of condition (𝐹(4,19.95)=0.09,𝑝=0.98), and the effect of self-perceived creativity did\nnot differ by condition (𝐹(4,2650.44)=1.59,𝑝=0.18). But the effect of belief in AI’s relative creativity did differ by\ncondition,𝐹(4,2635.80)=2.93,𝑝=0.02. As robustness checks, we ran the same specification with two alternative\nmeasures of idea diversity, mean pairwise distance and distance from the centroid. Regression results are broadly similar.\nSee Appendix I.2 for regression results and pairwise comparisons.\nWe found mixed evidence that belief in AI’s relative creativity moderates local idea diversity. Regression results\nshowed a small but significant interaction effect between the [ High Exposure, Undisclosed ] condition and relative AI\ncreativity (𝛽=0.038,𝑡(3244)=2.149, 95% CI =[0.003,0.073],𝑝=0.03). We probed this effect with estimated marginal\nmeans, predicting local idea diversity for the bottom and top decile of participants by perception of AI creativity.\nTop-decile participants had slightly higher local idea diversity than bottom-decile participants in the [ High Exposure,\nUndisclosed ] condition ( Δ=2.62,𝑑=−0.34) but although this difference was significant before multiple comparisons\n(𝑝=0.01), it was not significant after adjusting for multiple comparisons, ( 𝑝=0.06; see Appendix Table 17). Hence, we\nconclude there is mixed evidence for the role of belief in AI’s relative creativity as a moderator of local idea diversity.\nFig. 5. Median pairwise distance of submitted ideas in a condition. There was more global diversity of ideas in the high AI exposure\nconditions than in the control condition.\n7.2.2 Global. By measuring global idea diversity, we capture how different the submitted ideas in a condition are\nfrom one another. This can be thought of as a measure of collective idea diversity. See Appendix I.1 for more details\non the procedure. Across a range of different metrics, high AI exposure conditions had more global idea diversity\nthan the control condition (Figure 5; Appendix Tables 12, 13, 14). The median pairwise distance provides the most\nconservative estimate of the metrics that we measured. But even for median pairwise distance, both the [ High Exposure,\nDisclosed ] (Cliff’s𝛿=0.32on a scale of -1 to 1 )and [ High Exposure, Undisclosed ](𝛿=0.27)condition had more idea\ndiversity than the control condition. But of the low exposure conditions, only the [ Low Exposure, Undisclosed ](𝛿=0.12)\n17Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\ncondition had higher global diversity than the control condition, with a much smaller effect size than the high exposure\nconditions. Hence, high AI exposure (but not necessarily low AI exposure) increases global idea diversity.\nFig. 6. High exposure to AI ideas increased the rate of change in idea diversity.\n7.2.3 Evolution. By measuring the evolution of idea diversity, we capture the rate of change in idea diversity across\ntrials. See Appendix I.3 for more details on the procedure. Relative to the control condition, the conditions with high\nexposure to AI ideas (but not low exposure to AI) had increased rates of change in idea diversity. See Figure 6 for\nestimated marginal means predictions and Appendix Table 18 for regression results. As with global idea diversity,\ndifferent metrics yielded similar regression coefficients. In the control condition, idea diversity decreased over trials\n(𝛽=−0.39,𝑡(349)=−2.23, 95% CI =[−0.73,−0.05],𝑝=0.03). That is, submitted ideas were becoming more similar\nto each other as the experiment went on. Relative to the control condition, however, the slope of idea diversity with\nrespect to trial number was more positive for the [ High Exposure, Undisclosed ] condition ( 𝛽=0.53,𝑡(349)=2.2, 95%\nCI =[0.06,0.99],𝑝=0.03) and the[ High Exposure, Disclosed ] condition ( 𝛽=0.57,𝑡(349)=2.37, 95% CI =[0.1,1.03],\n𝑝=0.02). The rate of change in idea diversity for the low AI exposure conditions did not differ from therate of change\nin the control condition. Thus, we conclude that high exposure to AI ideas increased the rate of idea diversity relative\nto the no-AI, control condition.\n7.3 AI Adoption\n7.3.1 Local Level. At the local level, we measured AI adoption by the maximum cosine similarity between a participant’s\nresponse and AI examples the participant saw. There was a main effect of condition (𝐹(3,16.59)=4.33,𝑝=0.02). But\nwe would expect higher similarity to AI ideas in the high-exposure condition even by chance (since there are more AI\nideas), so we do not interpret main effects and instead focus on subgroup differences and effects of disclosure in the high-\nexposure condition. We found that the effect of conditions did not differ by interest groups (𝐹(6,719.77)=1.98,𝑝=0.07),\n18How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nbut the effect of conditions did differ by self-perceived creativity (𝐹(3,1984.95)=5.18,𝑝=0.001)and relative AI\ncreativity(𝐹(3,1974.58)=2.9,𝑝=0.03). As robustness checks, we ran the same specification with two alternative\nmeasures of AI adoption, mean and median AI adoption. The coefficients of our regression are broadly similar. See\nAppendix K for regression results and post-hoc contrasts.\nFig. 7. In the [ High Exposure, Disclosed ] condition, participants\nhigh in self-reported creativity had higher AI adoption (raw\ndata).\nFig. 8. Adoption of AI ideas by self-perceived creativity In the\n[High Exposure ] conditions (estimated marginal means).\nFig. 9. Higher creativity participants adopted ideas solely based on exposure, not disclosure (estimated\nmarginal means).\nExposure to AI ideas increased adoption for (self-perceived) high-creativity participants regardless of disclosure,\nbut this was not the case for (self-perceived) low-creativity participants. There was a significant interaction between\nself-perceived human creativity and the [ High Exposure, Disclosed ] condition ( 𝛽=0.11,𝑡(2588)=3.93, 95% CI =\n[0.06,0.17],𝑝=0.0001 ; Appendix Table 23). To probe this interaction, we used our model to predict AI adoption by\ncondition for both the top 10% and bottom 10% of participants by self-perceived creativity (Figures 7, 8 and 9). For\nhigh-creativity participants (Figure 9), adoption rates appear to differ only by exposure (color) and not disclosure (shape).\nMore formally, we tested whether the effect of exposure on adoption is larger when AI ideas are disclosed vs undisclosed.\n19Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nWe find that for high-creativity participants, there is no difference in adoption between ([ High Exposure, Undisclosed ] -\n[Low Exposure, Undisclosed ]) and ([ High Exposure, Disclosed ] - [Low Exposure, Disclosed ]),Δ=1.69,𝑑=−0.14,𝑝=0.59.\nThat is, the effect of exposure is notmoderated by disclosure. But for low-creativity participants, the difference in\nadoption for the undisclosed conditions ([ High Exposure, Undisclosed ] - [Low Exposure, Undisclosed ]) was larger than\nthe equivalent difference in adoption for disclosed conditions ([ High Exposure, Disclosed ] - [Low Exposure, Disclosed ]),\nΔ=7.77,𝑑=0.65,𝑝=0.03. That is, disclosing ideas as from AI reduced the effect of exposure on adoption for lower\n(self-reported) creativity participants. In summary, higher (self-reported) creativity people adopt AI ideas solely based\non content, and not disclosure.\nWe also found that one’s attitude about AI’s creativity affected adoption, though this had a smaller effect than\nself-perceived creativity. There was a significant interaction between the [ High Exposure, Undisclosed ] condition and\nrelative AI creativity (positive values imply AI is more creative than humans), 𝛽=−0.07,𝑡(2588)=−2.61, 95% CI =\n[−0.13,−0.02],𝑝=0.01. We used estimated marginal means to probe this interaction by predicting AI adoption for the\ntop and bottom decile of participants by belief in relative AI creativity. We found that in the [ High Exposure, Undisclosed ]\ncondition, people who believed AI was uncreative (bottom decile of AiRelCreate ) were slightly more likely to adopt\nAI ideas than people who believed AI was creative (top decile of AiRelCreate ),Δ=4.66,𝑑=0.39,𝑝=0.005. But no\nsuch difference existed in the [ High Exposure, Disclosed ] condition. This may suggest labeling sources as AI neutralizes\nadoption among users who do not think AI is creative.\nFig. 10. Adoption of AI ideas in the [ High Exposure, Disclosed ] condition versus ‘difficulty’ of prompt. When ideas were disclosed as\nfrom AI, participants adopted AI ideas for difficult prompts.\nIn addition to who adopts AI ideas, we also measured when AI ideas are adopted. We found that people adopt AI\nideas for difficult prompts rather than easier prompts (Figure 10). To measure the ‘difficulty’ of an item prompt, we first\ncalculated the mean creativity of a response to an item in the control condition. Then we reverse-ranked items such\n20How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nthat high mean creativity implies lowdifficulty and vice versa. We measured ‘AI adoption‘ of an item by the average\nof the trial-level maximum similarity to AI examples. We then examined the rank-order correlation between item\ndifficulty and AI adoption in high-exposure conditions. If task difficulty leads people to rely on AI, then we should see a\nlarger correlation between item difficulty and AI adoption in the [ High Exposure, Disclosed ] condition than in the [ High\nExposure, Undisclosed ] condition. That is what we find. The rank-order correlation between difficulty and adoption\nwas𝜌=0.8for the [ High Exposure, Disclosed ] condition but only 𝜌=0.3for the [ High Exposure, Undisclosed ]. That is,\nwhen people were told ideas were from AI, they were more likely to adopt AI ideas if the prompt was difficult. Since we\nemployed only five items, we view this finding as speculative; future work should test this relationship with a larger\nnumber of stimuli.\n8 DISCUSSION\nAgainst the backdrop of a massive increase in LLM exposure, we asked: How does exposure to ideas generated by LLMs\naffect the creativity, diversity, and evolution of human ideas? To answer this, we conducted a large-scale experiment\nwhere participants submitted ideas in response to the Alternate Uses Task (a measure of creativity where people\nbrainstorm novel uses of an item) after seeing a set of example ideas. The examples were from prior participants in the\nsame experimental condition or—in some conditions—ChatGPT. The evolving aspect of our experiment, that ideas in a\ncondition feed forward to subsequent trials in that condition, captures the interdependent nature of idea formation and\nlets us model the evolutionary effects of having AI ‘in the culture loop’. Here are three takeaways from our experiment.\n8.1 AI makes ideas different but not better.\nMost notably, exposure to AI ideas did not, on average, make human ideas any ‘better’ or ‘worse’ (by creativity). Our\nhigh-powered, null finding around average creativity by condition can inform debates about the effect of AI ideas\non individual human creativity. Maybe there is little effect. Of course, our experiment is measuring just a single task.\nBut these results suggest that perhaps both worry and optimism around the effect of AI ideas on individual human\ncreativity should be tempered.\nOur null finding around creativity contrasts with some prior work suggesting human-AI co-creation enhances the\nquality of creative outputs [ 24,37,56,73]. But our study differs from prior studies in its aim and design: We test passive\nexposure to off-the-shelf LLMs—not active engagement with optimized-for-creativity AI aides. The latter is useful for\nunderstanding how AI could affect creativity. But we aim to approximate how ordinary, existing, and pervasive AI tools\ndoaffect the creativity of ideas. At least for this task, we find no evidence of such an effect.\nOn the other hand, the presence of AI ideas increased the diversity of human ideas. This is consistent with prior\nwork suggesting collaborating with AI leads to more diverse or unexpected outputs [ 10,18,33,50,68]. Crucially,\nhigh AI exposure increased both average amounts of diversity andrates of change in idea diversity. The latter result\nis especially important. Small differences in rates of change can yield large aggregate differences over time. Future\nwork—both simulations and dynamic experiments—can explore the implications of this increase in collective idea\ndiversity unaccompanied by an increase in individual creativity. For instance, can this dynamic generate ‘innovation’?\nOur finding around the evolution of diversity (Figure 6) is instructive. Seeing other people’s ideas reduced idea\ndiversity in the control condition over time. This may suggest that successive participants were converging on a\nparticular idea sequence. But then injecting AI ideas into the example set increased the diversity of submitted responses\nby partially ‘resetting’ this convergence. A promising avenue for future work: Can AI input reduce ‘groupthink’?\n21Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\n8.2 High creativity people are less influenced by the source label of ideas.\nParticipants who viewed themselves as highly creative had the same levels of adoption of AI ideas in both disclosed and\nnon-disclosed conditions. But for lower-creativity participants, knowing the source of an idea didaffect the adoption\nof that idea. Perhaps people high in self-reported creativity relied less on source cues when adopting ideas because\nthey were more confident in their ability to judge an idea’s creative merit. Future work can employ think-alouds to\nbetter understand how AI disclosure affects the idea-generation process, itself. Regardless, our results suggest that\n(self-reported) creative people will adopt ideas on the basis of their content. Knowing the source does not matter. In\na world where humans have difficulty distinguishing if the content was human or AI-generated [ 28], these findings\nsuggest people high in (self-reported) creativity will not be ‘duped’ into adopting AI ideas.\n8.3 Participants adopted AI ideas when the prompt was hard.\nWhen AI ideas were labeled, participants were more likely to adopt AI ideas for difficult prompts rather than easy\nprompts. This finding is similar to what Roemmele [56] observed, where seeing AI examples only influenced creative\noutput when the task was difficult. Both our and Roemmele [56]’s results are consistent with a more general finding\nthat task difficulty is associated with increased reliance on automation [ 20]. However, since this finding is based on a\nsmall number of AUT prompts we view this finding as speculative.\nIf users turn to AI for difficult rather than trivial tasks, this would have several implications. On one hand, AI can\naugment human creativity where human imaginations falter. At the same time, do we really want our most difficult\ncreative tasks to be assisted by LLMs? Separate from an ethical question is a practical one. Researchers raised concerns\nover ‘model collapse’ [ 59]—the deteriorating performance of LLMs when trained on their outputs. If reliance on AI\nfor creative tasks becomes routine, this may contribute to model collapse, ironically decreasing the efficacy of such\nreliance. More work is needed to understand LLM reliance and task difficulty.\n8.4 Conclusion: Passive exposure to AI ideas affects collective thought.\nWe conclude that passive exposure to AI ideas—the kind of passive exposure we are inundated with in a post-ChatGPT\nera—does affect collective thought. Even small effects are meaningful since this exposure is both pervasive and growing.\nBut the effects of AI ideas are nuanced. Seeing AI ideas did not increase individual creativity, though it did increase\ncollective diversity. The effects of AI ideas vary across individuals and tasks. There is still much to learn. We hope our\nstudy inspires more research on how passive exposure to AI ideas affects collective thought.\n9 LIMITATIONS & FUTURE WORK\nOur study has several limitations that can inform future work. First, we measured the effect of AI ideas for a single\ntask. We chose this task because it is one of the most common creativity tasks [ 1]. But future work could explore if our\nresults replicate for other kinds of tasks. Second, we had to operationalize ‘ChatGPT’ in some concrete way. The logic\nfor our prompt was driven by ecological validity and prior work: We used a zero-shot prompt because that is what\nusers would likely use, and the specific prompt we used was derived from prior research. We chose not to vary prompts\nin order not to further increase the complexity of an already complex experiment. Future work could explore if different\nprompts elicit different results. Another avenue for future work is only propagating the ‘best’ AI ideas forward. Third,\nfuture work should test if alternative classifiers or ways of conceiving variables yield different results. For idea diversity\nand AI adoption, we addressed this problem by showing that conceptually similar ways of measuring variables yielded\n22How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nqualitatively similar results. For our creativity measure, we used a highly accurate classifier (correlation with human\njudgments greater than 𝑟=0.88for items we used) trained for this exact task , for these exact items . But of course, all\nmodels have some error and future work based on this model propagates these errors. Incidentally, human judges of\ncreativity only correlate with other human judges at 𝑟=0.88[49], suggesting the classifier we used may be approaching\n‘the approximate ceiling at which we could expect a model to correlate with human judgments’ [ 49, pg. 11] of creativity.\nFourth, our finding about AI adoption and task difficulty is based on five AUT items. Future work should explore this\nrelationship with a larger number of stimuli. Fifth, we focus on one facet of creativity: originality. Future work can\nalso explore whether AI ideas have different effects on other facets of creativity. Sixth, we employed a convenience\nsample of technology-interested users and creative professionals. While these two groups are most relevant to the\nphenomena in question, our sample also limits generalizability. Future work can explore these dynamics with different\nsamples. Finally, we conducted this experiment close to the launch of ChatGPT. As AI becomes increasingly embedded\nin everyday life, attitudes towards AI and ways of engaging with AI may also change. Despite these limitations, our\nwork offers the first large-scale, dynamic account of how ideas from LLMs affect collective thought.\n10 ACKNOWLEDGEMENTS\nWe thank Joyce Chai, Misha Teplitskiy, and Harman Kaur for their feedback.\nREFERENCES\n[1]Anna Abraham. 2016. Gender and creativity: an overview of psychological and neuroscientific literature. 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Psychology of Aesthetics, Creativity, and the Arts (2023), No\nPagination Specified–No Pagination Specified. https://doi.org/10.1037/aca0000573\n[73] Ann Yuan, Andy Coenen, Emily Reif, and Daphne Ippolito. 2022. Wordcraft: Story Writing With Large Language Models. In 27th International\nConference on Intelligent User Interfaces (IUI ’22) . Association for Computing Machinery, New York, NY, USA, 841–852. https://doi.org/10.1145/\n3490099.3511105\nA AUT ITEMS\nTable 5. AUT items by frequency of occurrence in dataset and classifier accuracy. Accuracy is defined as the correlation between human\nratings of creativity and model predictions. The overall accuracy was r = 0.81. The accuracy of responses from the best-performing\n5-item subset was r= 0.90. The data and model is from Organisciak et al. [48].\nAUT Item Classifier Accuracy (r) Frequency in Test Set\ntire 0.91 412\npants 0.91 443\nshoe 0.91 382\ntable 0.90 461\nbottle 0.88 839\npencil 0.85 384\nball 0.84 393\nfork 0.83 407\nlightbulb 0.83 383\ntoothbrush 0.81 379\nknife 0.81 2163\nbackpack 0.80 34\nshovel 0.79 339\npaperclip 0.79 1385\nhat 0.76 380\nbox 0.74 2842\nspoon 0.73 386\nbook 0.71 487\nsock 0.69 380\nbrick 0.64 5162\nrope 0.56 2080\nB AUT PROMPTS\nWe conducted a permutation test of whether the limited-length zero shot prompt (‘ZeroShotLimit’) yielded more similar\nword counts to the human responses than the original zero shot prompt (‘ZeroShot’) from [ 62]. Note that permutation\ntests are non-parametric, so they are robust to (e.g.) violations of normality. The observed difference is computed as:\n26How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\n𝑇𝑜𝑏𝑠=\n|𝐴𝑣𝑔(ZeroShot)−𝐴𝑣𝑔(HumanResponses)|−\n|𝐴𝑣𝑔(ZeroShotLimit)−𝐴𝑣𝑔(HumanResponses)|\nwhere𝐴𝑣𝑔is the average word count of a response and the human responses are from Organisciak et al. [48].\nUnder the null hypothesis, we assume prompts do not differ in respective word counts so we can swap the labels of\n‘ZeroShot’ and ‘ZeroShotLimit’. To simulate draws from this null distribution, we permuted the labels of the source\ncolumn 10000 times, each time calculating a new value for the difference of differences defined in 𝑇. The p-value is the\nproportion of permutations where the null 𝑇is greater than or equal to the observed difference, 𝑇𝑜𝑏𝑠. As in Ojala and\nGarriga [47], we apply a conservative adjustment and add 1to the numerator and denominator (this means we never\nget a p-value of 0).\np=[Í𝑛\n𝑖=1𝐼(𝑇𝑛𝑢𝑙𝑙>=𝑇𝑜𝑏𝑠)]+ 1\n𝑛+1\nThis p-value indicates the likelihood of observing a difference as extreme as our observed difference if the null\nhypothesis—that prompts did not differ in word counts—is true. We conclude the observed difference is significant at\nthe𝑝<0.001level.\nTable 6. Summary statistics of AUT prompt experiment. Human ideas are from Organisciak et al . [48] and include only those ideas in\nresponse to the chosen AUT items. Note that in some cases ChatGPT did not return the desired number of ideas, leading to a slight\ndiscrepancy between ideas generated between the two prompts.\nN Average Words SD Words\nCondition\nHuman Ideas 2537 4.56 4.97\nZero Shot Length Limited 7500 4.44 1.34\nZero Shot 8153 25.38 8.55\nC PRE-TREATMENT QUESTIONS\nPew asked about feeling towards AI [ 40,41] and we used the specific phrasing and choice ordering from [ 40]. We\nrandomized the first two options and kept neutral last. Our gender question was based on guidance from Spiel et al .\n[61]. The options were: ’woman’, ’man’, ’non-binary’, ’prefer to self-describe’, ’prefer not to disclose’. We added a text\nbox meant for those who preferred to self-describe. The only deviation from Spiel et al . [61] is that we did not allow for\nparticipants to select multiple options. We note that gender (as well as age and country) were optional.\nD EXCLUSION CRITERIA FOR ANALYSIS\nParticipants could have consented and answered pre-treatment questions but failed to complete any trial. We only\nanalyze data from participants who completed at least one trial. In (𝑛=4)cases, users submitted ages that were\nimplausible. We replaced these age values with missing for the purpose of summarizing participants but kept the\nresponses. In(𝑛=2)cases, responses that should not have been shown were shown. We remove these two responses\nfrom analysis. As discussed in L, we instituted content moderation after receiving several troll responses. After the\n27Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nstudy, we manually inspected each response flagged by our system. There were 46ideas labeled as profane, and we\ndetermined 36were true positives. We remove the true positives (𝑛=36)from analysis, resulting in a final set of 3414\nresponses for analysis from an initial set of 3452 responses. Importantly, we conducted chi-squared tests and found that\ncondition was unrelated to the number of flagged ideas ( 𝜒2(4) = 6.06, p = 0.19), number of flagged ideas minus false\npositives (𝜒2(4) = 2.92, p = 0.57) or total number of excluded ideas ( 𝜒2(4) = 3.87, p = 0.42).\n28How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nE HUMAN VS AI IDEAS\nWe compared a sample of 1500 ideas from our modified Stevenson prompt in the prompt experiment and a random\nsample of 1500 ideas from the Organisciak Dataset for our 5 items. For each set, we used the model’s predicted originality\nscores. Originality ranges from 1-5. Overall, ChatGPT ideas had higher ( 𝛽=0.62,𝑡(2994)=22.49, 95% CI =[0.56,0.67])\noriginality. In the paper, we refer to this as a 12 p.p difference since 0.62/5=12%.\nTable 7. Comparing predicted originality of ChatGPT generated ideas to ideas from a dataset of prior human responses\nDependent variable:\noriginality\nsourcechatgpt 0.618∗∗∗\n(0.027)\npromptpants −0.072∗\n(0.041)\npromptshoe 0.096∗∗\n(0.043)\nprompttable −0.007\n(0.042)\nprompttire −0.196∗∗∗\n(0.041)\nConstant 2.751∗∗∗\n(0.029)\nObservations 3,000\nR20.159\nAdjusted R20.157\nResidual Std. Error 0.746 (df = 2994)\nF Statistic 112.903∗∗∗(df = 5; 2994)\nNote:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01\nF PARTICIPANT FEEDBACK\nWe encouraged participants to start the study in the first place by saying that—if they finished all 5 trials—we would\nshow them how creative they are relative to humans and AI. At the end of the experiment, we first computed a\nparticipant’s average score from the Organisciak et al . [48] classifier as their ‘creativity score’. We then graphically and\nverbally showed participants what percentile this score would be for both humans and AI (where the human and AI\nscores come from applying the Organisciak et al . [48] classifier to a sample of AI ideas we generated and prior human\nideas from the Organisciak Dataset.) We also provided a graph that compared a participant’s scores in the AI condition\nto their scores in the no-AI conditions.\n29Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nAdditionally, we wanted to minimize attrition for participants once they started. We gave participants two pieces of\nfeedback after each trial so they would continue taking the study. See M for screenshots.\n•First, we calculated how unique a participant’s response was relative to the last person’s response. We did this\nby calculating the cosine distance between a word2vec embedding of the participant’s response and a word2vec\nembedding of the last response in a given {[ condition ],item}. Due to resource constraints, we used a truncated\nword2vec model—the top 15k words in English.\n•We also compared the accuracy of participants’ rankings to the rankings of ideas by the classifier [ 48] we used.\nTo do this, we calculated the rank-order correlation between a participant’s rankings of items and the rank order\ngenerated by the Organisciak et al. [48] model.\nIn certain cases, either of these metrics could not be calculated, and we returned an arbitrary, random number.\nG SAMPLE CHARACTERISTICS\nFig. 11. Distributions of self-perceived creativity relative to humans and relative to AI, by both interest group and sentiment towards\nAI.\nAlthough we did not assess English language proficiency, the top five countries by responses (77.85% of responses)\nwere the United States, Canada, Germany, United Kingdom, and Australia— countries with high English proficiency.\nThe median response length was six words, which is relatively short, also suggesting English language proficiency is\nnot a likely confounder.\n30How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nTable 8. Descriptive Stats (Non-Missing Values)\nMean SD 25th Percentile Median 75th Percentile\nage 34.92 10.86 27.0 33.0 40.0\ncreativity_ai 57.86 26.66 40.0 60.0 76.0\ncreativity_human 58.67 23.65 44.0 62.0 75.0\nTable 9. Distribution of Gender\nCounts (% of total)\ngender\nwoman 308 (36%)\nman 268 (32%)\nMissing 222 (26%)\nnon-binary 23 (3%)\nprefer_not_disclose 16 (2%)\nprefer_self_describe 7 (1%)Table 10. Distribution of AI Feeling\nCounts (% of total)\nai_feeling\nneutral 403 (48%)\nexcited 232 (27%)\nconcerned 198 (23%)\nMissing 11 (1%)\nH MODEL SELECTION\nDV Potential Moderator 𝜒2Df𝑝<𝜒2Added Interaction\nIdea Diversity Self-Perceived Human Creativity 10.32 4.00 0.04 YES\nAI - Human Creativity 15.70 4.00 0.00 YES\nAI Feeling 8.20 8.00 0.41 NO\nInterest Group 12.02 8.00 0.15 NO\nCreativity Self-Perceived Human Creativity 3.28 4.00 0.51 NO\nAI - Human Creativity 1.11 4.00 0.89 NO\nAI Feeling 8.19 8.00 0.41 NO\nInterest Group 1.57 8.00 0.99 NO\nAI Adoption Self-Perceived Human Creativity 18.24 3.00 0.00 YES\nAI - Human Creativity 9.94 3.00 0.02 YES\nAI Feeling 4.14 6.00 0.66 NO\nInterest Group 13.05 6.00 0.04 YES\nTable 11. To determine which moderating variables to include, we conducted likelihood ratio tests comparing the baseline specification\nto a model including an interaction between a potential moderator and the treatment condition. If the likelihood ratio test indicated\nthe interaction improved the fit at 𝑝<0.05, we included this interaction in our model.\nSelected models already include ‘Interest Group’ to control for participant source (neutral, creative, technical). As a\nrobustness check, we subsequently created an additional participant source variable, ‘IsSocialMedia’, indicating if the\nrespondent was from social media. Likelihood ratio tests found adding ‘IsSocialMedia’ and its interaction with the\ntreatment condition did not improve the fit of the selected models (𝑝>0.39for all models).\n31Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nTable 12. Global idea diversity measured by mean pairwise distance\nContrast Diff in Means Adj P Value Cliff’s Delta\n0 HighExposureDisclosed-Control 1.29 0.0010 0.35\n1 HighExposureUndisclosed-Control 0.97 0.0010 0.28\n2 LowExposureDisclosed-Control 0.38 0.0066 0.10\n3 LowExposureUndisclosed-Control -0.40 0.0748 -0.03\n4 HighExposureDisclosed-HighExposureUndisclosed 0.32 0.0748 0.03\n5 HighExposureDisclosed-LowExposureDisclosed 0.91 0.0010 0.27\n6 HighExposureDisclosed-LowExposureUndisclosed 1.69 0.0010 0.30\n7 HighExposureUndisclosed-LowExposureDisclosed 0.59 0.0010 0.22\n8 HighExposureUndisclosed-LowExposureUndisclosed 1.37 0.0010 0.27\n9 LowExposureDisclosed-LowExposureUndisclosed 0.78 0.0010 0.09\nI IDEA DIVERSITY\nI.1 Global\nFor 50 Monte Carlo runs with different seed values, we sampled 50 ideas for each {[ condition ],item} combination.\nFor each 50-idea set, we computed various idea diversity measures. First, we calculated all pairwise SBERT distances.\nNext, we measured the mean, median, and max pairwise distance. We also computed the centroid of each 50-idea\nset and calculated the mean distance from the centroid. After calculating these metrics, we conducted two-tailed,\npaired permutation tests (10,000 iterations) to test if two conditions differed on these metrics. To conduct the paired\npermutation test, we randomly swapped the sign of the difference between pairs of values, equivalent to randomly\nswapping the condition labels of rows—simulating the null hypothesis that conditions do not differ. We then counted\nthe proportion of null distribution iterations where one would observe a larger absolute difference in means than the\nobserved difference. We added a 1 to the numerator and denominator, which is a common, conservative adjustment\n[47] and stops p-values from being 0. Because the test is paired (equivalent to swapping the condition label within each\n‘row’), our permutation tests are controlling for both AUT items and Monte Carlo seeds, since each row shares these\nattributes. We controlled for multiple pairwise comparisons by applying a Holm-Bonferroni adjustment to p-values. As\na non-parametric measure of effect size, we used Cliff’s Delta. This metric ranges from -1 to +1 where 0 indicates no\ndifference between conditions, +1 indicates that all Monte Carlo runs for the first condition are larger than those for the\nsecond, and vice versa for -1. As with evolution, to reduce any effect of seeds, we consider ideas after the sixth trial (see\nAppendix I.3 for details.)\n32How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nTable 13. Global idea diversity measured by median pairwise distance\nContrast Diff in Means Adj P Value Cliff’s Delta\n0 HighExposureDisclosed-Control 1.18 0.0010 0.32\n1 HighExposureUndisclosed-Control 0.87 0.0010 0.27\n2 LowExposureDisclosed-Control 0.45 0.0024 0.12\n3 LowExposureUndisclosed-Control -0.54 0.0086 -0.05\n4 HighExposureDisclosed-HighExposureUndisclosed 0.31 0.0747 0.03\n5 HighExposureDisclosed-LowExposureDisclosed 0.73 0.0010 0.24\n6 HighExposureDisclosed-LowExposureUndisclosed 1.72 0.0010 0.30\n7 HighExposureUndisclosed-LowExposureDisclosed 0.42 0.0060 0.19\n8 HighExposureUndisclosed-LowExposureUndisclosed 1.41 0.0010 0.27\n9 LowExposureDisclosed-LowExposureUndisclosed 0.99 0.0010 0.13\nTable 14. Global idea diversity measured by mean centroid distance\nContrast Diff in Means Adj P Value Cliff’s Delta\n0 HighExposureDisclosed-Control 1.42 0.0010 0.35\n1 HighExposureUndisclosed-Control 1.10 0.0010 0.28\n2 LowExposureDisclosed-Control 0.40 0.0072 0.10\n3 LowExposureUndisclosed-Control -0.36 0.1032 -0.03\n4 HighExposureDisclosed-HighExposureUndisclosed 0.32 0.1032 0.03\n5 HighExposureDisclosed-LowExposureDisclosed 1.02 0.0010 0.27\n6 HighExposureDisclosed-LowExposureUndisclosed 1.79 0.0010 0.30\n7 HighExposureUndisclosed-LowExposureDisclosed 0.70 0.0010 0.22\n8 HighExposureUndisclosed-LowExposureUndisclosed 1.46 0.0010 0.27\n9 LowExposureDisclosed-LowExposureUndisclosed 0.76 0.0010 0.09\n33Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nI.2 Local\nTable 15. Predictors of local idea diversity with coefficients and SEs in parentheses. The DV for models (1) and (2) are the median\nand mean pairwise distances between a participant’s response and examples. Model (3) uses the distance between a participant’s\nresponse and the centroid of examples. Ideas are embedded using SBERT. All three models have a random intercept for participants\ncrossed with a random intercept for response chains, nested in (item, condition) combinations.\nDependent variable:\nMedian PW Distance Mean PW Distance Centroid Distance\n(1) (2) (3)\nconditionLoExposure_Disclosed −1.467 (1.864) −2.177 (1.701) −3.791 (2.438)\nt =−0.787 t = −1.280 t = −1.555\nconditionLoExposure_Undisclosed 0.272 (1.861) −0.098 (1.698) −0.414 (2.433)\nt = 0.146 t = −0.057 t = −0.170\nconditionHiExposure_Disclosed 1.051 (1.864) 0.814 (1.701) 3.354 (2.438)\nt = 0.564 t = 0.479 t = 1.376\nconditionHiExposure_Undisclosed −0.744 (1.866) −0.958 (1.703) 0.442 (2.441)\nt =−0.399 t = −0.563 t = 0.181\ncreativity_human −0.006 (0.013) −0.008 (0.013) −0.015 (0.021)\nt =−0.484 t = −0.647 t = −0.723\nai_rel_create −0.006 (0.013) −0.005 (0.012) −0.008 (0.020)\nt =−0.425 t = −0.401 t = −0.388\ntrial_no −0.023 (0.029) −0.020 (0.027) −0.007 (0.045)\nt =−0.814 t = −0.725 t = −0.166\nai_feelingconcerned 0.354 (0.358) 0.317 (0.339) 0.455 (0.565)\nt = 0.990 t = 0.935 t = 0.807\nai_feelingexcited 0.270 (0.349) 0.107 (0.331) 0.207 (0.550)\nt = 0.773 t = 0.324 t = 0.377\ninterest_groupcreative −0.573 (0.453) −0.492 (0.431) −0.694 (0.693)\nt =−1.263 t = −1.142 t = −1.002\ninterest_grouptechnology 0.081 (0.465) 0.161 (0.443) 0.139 (0.711)\nt = 0.173 t = 0.363 t = 0.196\ncondition_order 0.182∗∗(0.092) 0.165∗(0.087) 0.256∗(0.143)\nt = 1.982 t = 1.897 t = 1.798\nlog_duration −0.765∗∗∗(0.200)−0.718∗∗∗(0.189)−1.192∗∗∗(0.313)\nt =−3.820 t = −3.791 t = −3.809\nn_seeds 0.398∗∗∗(0.136) 0.440∗∗∗(0.128) 0.597∗∗∗(0.213)\nt = 2.927 t = 3.429 t = 2.808\nconditionLoExposure_Disclosed:creativity_human 0.034∗(0.018) 0.040∗∗(0.017) 0.070∗∗(0.028)\nt = 1.884 t = 2.313 t = 2.456\nconditionLoExposure_Undisclosed:creativity_human 0.006 (0.018) 0.005 (0.017) 0.015 (0.028)\nt = 0.304 t = 0.285 t = 0.519\nconditionHiExposure_Disclosed:creativity_human −0.003 (0.018) −0.004 (0.017) −0.013 (0.028)\nt =−0.169 t = −0.248 t = −0.455\nconditionHiExposure_Undisclosed:creativity_human 0.024 (0.018) 0.022 (0.017) 0.030 (0.028)\nt = 1.298 t = 1.256 t = 1.065\nconditionLoExposure_Disclosed:ai_rel_create −0.018 (0.018) −0.017 (0.017) −0.031 (0.028)\nt =−1.027 t = −0.991 t = −1.136\nconditionLoExposure_Undisclosed:ai_rel_create 0.007 (0.018) 0.005 (0.017) 0.006 (0.028)\nt = 0.365 t = 0.296 t = 0.207\nconditionHiExposure_Disclosed:ai_rel_create −0.008 (0.018) −0.007 (0.017) −0.005 (0.027)\nt =−0.462 t = −0.414 t = −0.199\nconditionHiExposure_Undisclosed:ai_rel_create 0.038∗∗(0.018) 0.040∗∗(0.017) 0.065∗∗(0.028)\nt = 2.149 t = 2.384 t = 2.331\nConstant 85.104∗∗∗(1.733) 84.644∗∗∗(1.607) 72.776∗∗∗(2.466)\nt = 49.099 t = 52.676 t = 29.508\nObservations 3,271 3,271 3,271\nLog Likelihood −11,201.060 −11,014.890 −12,630.840\nAkaike Inf. Crit. 22,456.120 22,083.790 25,315.680\nBayesian Inf. Crit. 22,620.630 22,248.300 25,480.180\nNote:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01\n34How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nTable 16. Estimated marginal means contrasts of local idea diversity, using a mixed model to compare predictions for top 10 percentile\nand bottom 10 percentile of participants by belief in relative AI creativity. Local idea diversity is computed as the median pairwise\ndistance between a participant’s idea and the example ideas. P-values adjusted for multiple comparisons using the Holm-Bonferroni\nmethod.\ncontrast Relative AI Creativity Percentile estimate SE df t.ratio Adjusted P Value d\nLoExposure_Undisclosed - HiExposure_Undisclosed 10 -0.368 1.540 20.311 -0.239 1.000 -0.048\nLoExposure_Undisclosed - LoExposure_Disclosed 10 0.298 1.539 20.279 0.193 1.000 0.038\nLoExposure_Undisclosed - HiExposure_Disclosed 10 -0.129 1.539 20.299 -0.084 1.000 -0.017\nLoExposure_Undisclosed - Control 10 0.660 1.541 20.381 0.428 1.000 0.085\nHiExposure_Undisclosed - LoExposure_Disclosed 10 0.666 1.540 20.356 0.432 1.000 0.086\nHiExposure_Undisclosed - HiExposure_Disclosed 10 0.239 1.539 20.276 0.155 1.000 0.031\nHiExposure_Undisclosed - Control 10 1.029 1.545 20.585 0.666 1.000 0.133\nLoExposure_Disclosed - HiExposure_Disclosed 10 -0.427 1.540 20.346 -0.277 1.000 -0.055\nLoExposure_Disclosed - Control 10 0.363 1.541 20.390 0.235 1.000 0.047\nHiExposure_Disclosed - Control 10 0.790 1.545 20.591 0.511 1.000 0.102\nLoExposure_Undisclosed - HiExposure_Undisclosed 90 -2.915 2.203 84.275 -1.323 1.000 -0.377\nLoExposure_Undisclosed - LoExposure_Disclosed 90 2.284 2.202 84.112 1.037 1.000 0.295\nLoExposure_Undisclosed - HiExposure_Disclosed 90 1.042 2.187 81.952 0.476 1.000 0.135\nLoExposure_Undisclosed - Control 90 1.181 2.206 84.767 0.535 1.000 0.153\nHiExposure_Undisclosed - LoExposure_Disclosed 90 5.198 2.206 84.630 2.357 0.207 0.672\nHiExposure_Undisclosed - HiExposure_Disclosed 90 3.957 2.188 81.948 1.809 0.608 0.512\nHiExposure_Undisclosed - Control 90 4.096 2.213 85.683 1.851 0.608 0.530\nLoExposure_Disclosed - HiExposure_Disclosed 90 -1.242 2.190 82.342 -0.567 1.000 -0.161\nLoExposure_Disclosed - Control 90 -1.102 2.207 84.791 -0.500 1.000 -0.143\nHiExposure_Disclosed - Control 90 0.139 2.197 83.314 0.063 1.000 0.018\nTable 17. Estimated marginal means contrasts of local idea diversity, using a mixed model to compare predictions for the top\n10 percentile and bottom ten percentile of participants by belief in relative AI creativity. Local idea diversity is computed as the\nmedian pairwise distance between a participant’s idea and the example ideas. P-values adjusted for multiple comparisons using the\nHolm-Bonferroni method\ncontrast condition estimate SE df t.ratio P Value d Adjusted P Value\nai_rel_create10 - ai_rel_create90 LoExposure_Undisclosed -0.077 1.043 3156.269 -0.074 0.941 -0.010 1.000\nai_rel_create10 - ai_rel_create90 HiExposure_Undisclosed -2.624 1.040 3136.309 -2.523 0.012 -0.339 0.058\nai_rel_create10 - ai_rel_create90 LoExposure_Disclosed 1.909 1.041 3159.288 1.833 0.067 0.247 0.268\nai_rel_create10 - ai_rel_create90 HiExposure_Disclosed 1.094 1.013 3181.870 1.080 0.280 0.141 0.840\nai_rel_create10 - ai_rel_create90 Control 0.444 1.046 3177.974 0.424 0.671 0.057 1.000\n35Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nI.3 Evolution\nTo model the evolution of idea diversity, we pooled together submitted ideas at the level of (item, condition, trial\nnumber). We then computed the median pairwise distance, mean pairwise distance, and mean distance from centroid\nfor each pool of ideas. We fit a model to test if idea diversity changed at a different rate for different conditions:\nvariable 𝑐𝑡𝑖=𝛽0+𝛽1Condition 𝑐+𝛽2TrialNo 𝑡+𝛽3TrialNo X Condition 𝑡𝑐+\n𝛽4Nobs 𝑐𝑡𝑖+𝑢0𝑖+𝑒𝑐𝑡𝑖\nWhere:\n•𝑐indexes conditions.\n•𝑡indexes trial number.\n•𝑖indexes items\n•𝛽0is the global intercept.\n•𝑢0𝑖∼𝑁(0,𝜎2𝑢)are random intercepts for items\n•𝑒𝑐𝑡𝑖∼𝑁(0,𝜎2)is the residual\nWe took two additional steps to make sure our results were not driven by confounding factors. First, 𝑁𝑜𝑏𝑠 controls\nfor how many ideas are in the set that is being analyzed. Recall that we designed the experiment so that each (item,\ncondition) combination was replicated exactly seven times in response chains of exactly 20 trials. However, there were\nsome minor deviations (discussed in Appendix L) in response chains, resulting in some (item, condition, trial number)\nsets having fewer items than others. Hence, we control for the number of ideas in a set. Second, we only ran this\nanalysis on data after the sixth trial in a response chain to rule out the effect of seeds on evolution. The logic here is\nthat the condition with the most initial seeds (6) was the control condition. The experiment is designed to ‘shed’ all\nseeds after trial six since by that time there would have been six experiment responses, meaning the most recent six\nideas in the control condition would now all be from the experiment, and hence no seeds present in the example sets.\n(Note that for all local analyses, we directly control for the number of seeds present in the example set as a fixed effect.)\n36How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nTable 18. Evolution of idea diversity by condition. Each model has a random intercept for item. The reference level for experimental\nconditions is the control condition.\nDependent variable:\nMedian PW Distance Mean PW Distance Centroid Distance\n(1) (2) (3)\nnobs 0.827∗∗(0.337) 0.894∗∗∗(0.317) 3.940∗∗∗(0.214)\nt = 2.454 t = 2.818 t = 18.450\nconditionLow ExposureUndisclosed −1.558 (3.349) −1.048 (3.151) −0.534 (2.121)\nt =−0.465 t = −0.333 t = −0.252\nconditionLow ExposureDisclosed −4.087 (3.368) −4.163 (3.168) −1.884 (2.133)\nt =−1.213 t = −1.314 t = −0.883\nconditionHigh ExposureUndisclosed −5.575 (3.417) −4.853 (3.214) −3.517 (2.164)\nt =−1.632 t = −1.510 t = −1.625\nconditionHigh ExposureDisclosed −6.003∗(3.416)−5.573∗(3.213)−3.608∗(2.163)\nt =−1.757 t = −1.734 t = −1.668\ntrial_no −0.391∗∗(0.175)−0.321∗(0.165)−0.141 (0.111)\nt =−2.232 t = −1.948 t = −1.273\nconditionLow ExposureUndisclosed:trial_no 0.140 (0.231) 0.111 (0.218) 0.069 (0.146)\nt = 0.605 t = 0.512 t = 0.472\nconditionLow ExposureDisclosed:trial_no 0.368 (0.233) 0.379∗(0.220) 0.196 (0.148)\nt = 1.578 t = 1.728 t = 1.328\nconditionHigh ExposureUndisclosed:trial_no 0.525∗∗(0.239) 0.461∗∗(0.225) 0.335∗∗(0.151)\nt = 2.200 t = 2.051 t = 2.213\nconditionHigh ExposureDisclosed:trial_no 0.566∗∗(0.239) 0.561∗∗(0.225) 0.378∗∗(0.151)\nt = 2.371 t = 2.498 t = 2.500\nConstant 81.500∗∗∗(3.899) 78.538∗∗∗(3.669) 19.040∗∗∗(2.479)\nt = 20.904 t = 21.404 t = 7.681\nObservations 362 362 362\nLog Likelihood −1,158.987 −1,137.571 −998.930\nAkaike Inf. Crit. 2,343.974 2,301.141 2,023.861\nBayesian Inf. Crit. 2,394.566 2,351.733 2,074.452\nNote:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01\n37Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nJ CREATIVITY\nTable 19. Predictors of creativity with coefficients and SEs in parentheses. This model has a random intercept for participants crossed\nwith a random intercept for response chains, nested in (item, condition) combinations.\nDependent variable:\nCreativity\nconditionLoExposure_Disclosed 0.011 (0.112)\nt = 0.095\nconditionLoExposure_Undisclosed 0.019 (0.112)\nt = 0.166\nconditionHiExposure_Disclosed −0.025 (0.113)\nt =−0.221\nconditionHiExposure_Undisclosed 0.050 (0.113)\nt = 0.443\ncreativity_human 0.001 (0.001)\nt = 1.587\nai_rel_create 0.001 (0.001)\nt = 1.506\ninterest_groupcreative −0.069∗(0.039)\nt =−1.760\ninterest_grouptechnology −0.010 (0.040)\nt =−0.249\ntrial_no −0.002 (0.003)\nt =−0.797\nai_feelingconcerned −0.017 (0.033)\nt =−0.526\nai_feelingexcited −0.046 (0.032)\nt =−1.448\ncondition_order 0.003 (0.008)\nt = 0.448\nlog_duration 0.097∗∗∗(0.018)\nt = 5.525\nn_seeds −0.014 (0.012)\nt =−1.215\nConstant 3.192∗∗∗(0.128)\nt = 24.932\nObservations 3,271\nLog Likelihood −3,196.244\nAkaike Inf. Crit. 6,430.488\nBayesian Inf. Crit. 6,546.252\nNote:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01\n38How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nK AI ADOPTION\nTable 20. Estimated marginal means contrasts of AI adoption, using a mixed model to compare predictions for top 10 percentile and\nbottom 10 percentile of participants by self-perceived human creativity. AI Adoption is the max cosine similarity of a participant’s\nresponse and AI examples. P-values are adjusted for multiple comparisons using Holm-Bonferroni method.\ncontrast Perceived Creativity Percentile estimate SE df t.ratio Adjusted P Value Cohen’s d\nLoExposure_Undisclosed - HiExposure_Undisclosed 10 -6.021 2.369 37.026 -2.542 0.092 -0.502\nLoExposure_Undisclosed - LoExposure_Disclosed 10 -3.669 2.369 37.048 -1.549 0.520 -0.306\nLoExposure_Undisclosed - HiExposure_Disclosed 10 -1.924 2.367 36.890 -0.813 0.983 -0.160\nHiExposure_Undisclosed - LoExposure_Disclosed 10 2.352 2.372 37.212 0.992 0.983 0.196\nHiExposure_Undisclosed - HiExposure_Disclosed 10 4.097 2.365 36.779 1.732 0.458 0.341\nLoExposure_Disclosed - HiExposure_Disclosed 10 1.744 2.370 37.092 0.736 0.983 0.145\nLoExposure_Undisclosed - HiExposure_Undisclosed 90 -5.757 2.172 26.231 -2.650 0.040 -0.480\nLoExposure_Undisclosed - LoExposure_Disclosed 90 0.599 2.170 26.127 0.276 1.000 0.050\nLoExposure_Undisclosed - HiExposure_Disclosed 90 -6.852 2.168 26.058 -3.160 0.020 -0.571\nHiExposure_Undisclosed - LoExposure_Disclosed 90 6.356 2.174 26.323 2.924 0.028 0.529\nHiExposure_Undisclosed - HiExposure_Disclosed 90 -1.095 2.166 25.913 -0.506 1.000 -0.091\nLoExposure_Disclosed - HiExposure_Disclosed 90 -7.451 2.171 26.171 -3.433 0.012 -0.621\nTable 21. Estimated marginal means contrasts of AI adoption, using a mixed model to compare predictions for the top 10 percentile\nand bottom 10 percentile of participants by belief in relative AI creativity. This metric captures how creative participants think AI is\nrelative to humans (higher values means more creative than humans). AI Adoption is the max cosine similarity of a participant’s\nresponse and AI examples. P-values adjusted for multiple comparisons using Holm-Bonferroni method.\ncontrast Relative AI Creativity Percentile estimate SE df t.ratio Adjusted P Value d\nLoExposure_Undisclosed - HiExposure_Undisclosed 10 -5.225 1.952 17.121 -2.677 0.095 -0.435\nLoExposure_Undisclosed - LoExposure_Disclosed 10 -1.115 1.949 17.038 -0.572 1.000 -0.093\nLoExposure_Undisclosed - HiExposure_Disclosed 10 -4.651 1.951 17.100 -2.384 0.145 -0.387\nHiExposure_Undisclosed - LoExposure_Disclosed 10 4.110 1.953 17.186 2.104 0.201 0.342\nHiExposure_Undisclosed - HiExposure_Disclosed 10 0.574 1.949 17.013 0.295 1.000 0.048\nLoExposure_Disclosed - HiExposure_Disclosed 10 -3.536 1.953 17.169 -1.811 0.263 -0.295\nLoExposure_Undisclosed - HiExposure_Undisclosed 90 0.422 3.168 115.486 0.133 1.000 0.035\nLoExposure_Undisclosed - LoExposure_Disclosed 90 -1.388 3.161 114.586 -0.439 1.000 -0.116\nLoExposure_Undisclosed - HiExposure_Disclosed 90 -2.363 3.130 110.316 -0.755 1.000 -0.197\nHiExposure_Undisclosed - LoExposure_Disclosed 90 -1.811 3.172 115.993 -0.571 1.000 -0.151\nHiExposure_Undisclosed - HiExposure_Disclosed 90 -2.785 3.134 110.601 -0.889 1.000 -0.232\nLoExposure_Disclosed - HiExposure_Disclosed 90 -0.974 3.135 110.956 -0.311 1.000 -0.081\nTable 22. Estimated marginal means contrasts of AI adoption, using a mixed model to compare predictions for top 10 percentile\nand bottom 10 percentile of participants by belief in relative AI creativity. This metric captures how creative participants think AI is\nrelative to humans (higher values means more creative than humans). AI Adoption is the max cosine similarity of a participant’s\nresponse and AI examples. P-values adjusted for multiple comparisons using Holm-Bonferroni method.\ncontrast condition estimate SE df t.ratio Adjusted P Value d\nai_rel_create10 - ai_rel_create90 LoExposure_Undisclosed -0.987 1.673 2545.523 -0.590 0.555 -0.082\nai_rel_create10 - ai_rel_create90 HiExposure_Undisclosed 4.660 1.669 2538.494 2.792 0.005 0.388\nai_rel_create10 - ai_rel_create90 LoExposure_Disclosed -1.261 1.668 2542.774 -0.756 0.450 -0.105\nai_rel_create10 - ai_rel_create90 HiExposure_Disclosed 1.301 1.611 2480.927 0.808 0.419 0.108\n39Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nTable 23. Predictors of AI adoption with coefficients and SEs in parentheses. The respective dependent variables are the max, mean,\nand median cosine similarities between the SBERT embedding of a participant’s response and the SBERT embeddings of AI examples\nthe participant saw. All three models have a random intercept for participants crossed with a random intercept for response chains,\nnested in (item, condition) combinations.\nDependent variable:\nMax AI Similarity Mean AI Similarity Median AI Similarity\n(1) (2) (3)\nconditionLoExposure_Undisclosed −5.481∗(2.801)−3.962∗(2.356)−3.960 (2.412)\nt =−1.957 t = −1.682 t = −1.642\nconditionHiExposure_Disclosed −2.243 (2.799) −3.498 (2.355) −3.415 (2.411)\nt =−0.801 t = −1.485 t = −1.416\nconditionHiExposure_Undisclosed 3.507 (2.799) 0.098 (2.355) −0.584 (2.411)\nt = 1.253 t = 0.041 t = −0.242\ncreativity_human −0.059∗∗∗(0.022)−0.041∗∗(0.017)−0.041∗∗(0.017)\nt =−2.726 t = −2.430 t = −2.363\nai_rel_create 0.016 (0.021) 0.015 (0.016) 0.015 (0.017)\nt = 0.757 t = 0.911 t = 0.885\ninterest_groupcreative 2.425∗(1.360) 1.853∗(1.052) 1.828∗(1.074)\nt = 1.784 t = 1.761 t = 1.702\ninterest_grouptechnology 0.708 (1.389) 0.979 (1.074) 0.945 (1.097)\nt = 0.510 t = 0.912 t = 0.862\ntrial_no −0.037 (0.050) 0.005 (0.039) 0.022 (0.040)\nt =−0.734 t = 0.122 t = 0.556\nai_feelingconcerned −0.261 (0.630) −0.211 (0.479) −0.189 (0.489)\nt =−0.413 t = −0.441 t = −0.388\nai_feelingexcited 0.569 (0.615) 0.165 (0.467) 0.149 (0.477)\nt = 0.925 t = 0.354 t = 0.312\ncondition_order −0.204 (0.164) −0.201 (0.128) −0.212 (0.131)\nt =−1.247 t = −1.569 t = −1.619\nlog_duration 0.529 (0.354) 0.539∗∗(0.273) 0.510∗(0.279)\nt = 1.494 t = 1.974 t = 1.828\nn_seeds −0.435 (0.309) −0.265 (0.240) −0.214 (0.245)\nt =−1.407 t = −1.104 t = −0.875\nconditionLoExposure_Undisclosed:creativity_human 0.053∗(0.029) 0.037 (0.023) 0.037 (0.023)\nt = 1.820 t = 1.601 t = 1.557\nconditionHiExposure_Disclosed:creativity_human 0.115∗∗∗(0.029) 0.066∗∗∗(0.023) 0.061∗∗∗(0.023)\nt = 3.931 t = 2.868 t = 2.601\nconditionHiExposure_Undisclosed:creativity_human 0.050∗(0.029) 0.030 (0.023) 0.039∗(0.024)\nt = 1.702 t = 1.293 t = 1.655\nconditionLoExposure_Undisclosed:ai_rel_create −0.003 (0.028) −0.004 (0.022) −0.004 (0.023)\nt =−0.121 t = −0.178 t = −0.174\nconditionHiExposure_Disclosed:ai_rel_create −0.032 (0.028) −0.011 (0.022) −0.007 (0.022)\nt =−1.151 t = −0.523 t = −0.330\nconditionHiExposure_Undisclosed:ai_rel_create −0.074∗∗∗(0.028)−0.065∗∗∗(0.022)−0.071∗∗∗(0.023)\nt =−2.612 t = −2.924 t = −3.146\nconditionLoExposure_Undisclosed:interest_groupcreative 1.036 (1.851) 0.819 (1.446) 0.834 (1.476)\nt = 0.560 t = 0.566 t = 0.565\nconditionHiExposure_Disclosed:interest_groupcreative −0.465 (1.844) −0.421 (1.440) −0.523 (1.470)\nt =−0.252 t = −0.293 t = −0.356\nconditionHiExposure_Undisclosed:interest_groupcreative −3.224∗(1.843)−2.329 (1.440) −2.344 (1.470)\nt =−1.750 t = −1.617 t = −1.595\nconditionLoExposure_Undisclosed:interest_grouptechnology 2.810 (1.888) 1.516 (1.474) 1.535 (1.504)\nt = 1.488 t = 1.028 t = 1.020\nconditionHiExposure_Disclosed:interest_grouptechnology −1.391 (1.871) −1.550 (1.461) −1.731 (1.491)\nt =−0.743 t = −1.061 t = −1.161\nconditionHiExposure_Undisclosed:interest_grouptechnology −1.522 (1.875) −1.870 (1.465) −1.926 (1.495)\nt =−0.811 t = −1.277 t = −1.288\nConstant 23.789∗∗∗(2.704) 17.655∗∗∗(2.185) 17.609∗∗∗(2.235)\nt = 8.796 t = 8.080 t = 7.880\nObservations 2,618 2,618 2,618\nLog Likelihood −10,155.720 −9,508.078 −9,561.834\nAkaike Inf. Crit. 20,371.430 19,076.160 19,183.670\nBayesian Inf. Crit. 20,547.540 19,252.260 19,359.770\nNote:∗p<0.1;∗∗p<0.05;∗∗∗p<0.01\n40How AI Ideas Affect Human Ideas Conference acronym ’XX, June 03–05, 2024, Woodstock, NY\nL IMPLEMENTATION COMPLICATIONS\nRunning a massive online networked experiment open to any user on the Internet will often invite implementation\nchallenges. In the interest of disclosure—and for the benefit of future researchers running similar experiments—we\nshare the challenges we faced, solutions we implemented, and rationales for our decisions.\nL.1 Server Capacity\nOverall, we received far more responses than we expected. At several points throughout the experiment, we experienced\nmore concurrent traffic than the application was designed to handle. Hence, we had to temporarily turn off the experiment\nto wait out high demand, add more resources, or implement and test changes described in Content Moderation. We\nnote that these capacity issues did not affect the responses we collected.\nL.2 Content Moderation\nInitially, we did not implement any content moderation. But on July 8th, we received an influx of responses from\nReddit. Several participants were trolls, providing repetitive profane responses. We then implemented a form of content\nmoderation, flagging any idea that contained a word in a list of words banned by Google as of July 8, 202313and\nsubsequently added two more words to the list. If an idea was flagged, it was written to our database but not shown to\nfuture participants. Some profane ideas were already shown to participants in between the time when the responses\nwere submitted and we saw and implemented our solution. Also, the content moderation strategy was imperfect: 10 of\n46 flagged ideas were false positives. As discussed earlier, condition was unrelated to the number of flagged ideas ( 𝜒2(4)\n= 6.06, p = 0.19) or total number of excluded ideas ( 𝜒2(4) = 3.87, p = 0.42).\nWe acknowledge there are more advanced and nuanced content moderation strategies, but this one was the best\noption given our specific circumstances and constraints. First, this bag-of-words method is very transparent. Second, we\ndeployed this experiment through Heroku, which imposes a CPU limit on the project, precluding the use of pre-trained\nclassifiers such as BERT. Third, we did not want to use APIs like Jigsaw or OpenAI moderation endpoints because these\nAPIs have rate limits, which can slow down the experiment.\nL.3 Small Deviations From 20 Trials Per Chain\nWe intended for each response chain to contain 20 responses. The average number of trials per response chain was 19.73\n(SD = 1.45) and the median number per chain was 20. We concluded the experiment before the last round of response\nchains was completely finished for all condition and item combinations, so the minimum number of trials in a response\nchain (occurring for an item and condition combination in the last round) was 14. The maximum number of trials was\n24. These minor deviations occurred due to server overload, very high traffic leading to race conditions and excluding\nseveral responses based on the criteria described in D. Based on a two-way ANOVA, we concluded that response chain\nlengths did not differ by item, (𝐹(4)=0.38,𝑝=0.82), condition(𝐹(4)=0.01,𝑝=1.00), or the interaction between\nitems and conditions (𝐹(16)=0.01,𝑝=1.00),\nM EXPERIMENT SCREENSHOTS\nReceived September 14 2023; revised 12 March 2009; accepted 5 June 2009\n13https://github.com/coffee-and-fun/google-profanity-words\n41Conference acronym ’XX, June 03–05, 2024, Woodstock, NY Ashkinaze et al.\nItem 2 of 5: Generate a creative use for a tire.TaskFor this task, you will submit a creative use for a tire. But before submitting your idea, here are some ideasfor inspiration. Rank them by creativity.Rank Previous IdeasFor this object, we also asked AI to come up with ideas!Rank these ideas in order of creativity, with the most creative use on top. Drag ideas to rank them.We'll show you how your rankings compare to rankings from a highly accurate model.1) garden planter for plants (Source: A.I)2) Turn it into steps for a steep hill. The rubber would provide traction. (Source: Human)3) cut in half and use as a climbing area for kids (Source: Human)4) exercise sled for workouts (Source: A.I)5) Carry it around so you get the opportunity to make \"I'm tired\" jokes anywhere you go (Source:Human)6) Put some dirt in it and use it as an above-ground planter. (Source: Human)Submit Your IdeaYour turn! What is a creative use for a tire? The goal is to come up with a creative idea, which is an idea thatstrikes people as clever, unusual, interesting, uncommon, humorous, innovative, or different. List a creativeuse for a tire.Write your idea hereSubmit\nFig. 12. Screenshot of a trial participants are told to complete.\nThis is the [ Low Exposure ,Disclosed ] condition since there\nare two AI ideas and the AI ideas are labeled.\nFig. 13. After each trial, participants were given feedback as\nmotivation to continue.\nFig. 14. Screenshots of the experiment and feedback we provided after each response\n42"    },    {        "title": "2401.13514v1._ρ__meson_spectroscopy_and_diffractive_production_using_the_holographic_light_front_Schrödinger_equation_and_the__t_Hooft_equation.pdf",        "content": "ρ-meson spectroscopy and diffractive production using the holographic light-front\nSchr¨ odinger equation and the ’t Hooft equation\nBheemsehan Gurjar\n ,1,∗Chandan Mondal\n ,2, 3,†and Satvir Kaur\n2, 3,‡\n1Department of Physics, Indian Institute of Technology Kanpur, Kanpur-208016, India\n2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China\n3School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China\nWe determine the mass spectroscopy and light-front wave functions (LFWFs) of the ρ-meson by\nsolving the holographic Schr¨ odinger equation of light-front chiral QCD along with the ’t Hooft equa-\ntion of (1+1)-dimensional QCD in the large Nclimit. Subsequently, we utilize the obtained LFWFs\nin conjunction with the color glass condensate dipole cross-section to calculate the cross sections for\nthe diffractive ρ-meson electroproduction. Our spectroscopic results align well with the experimental\ndata. Predictions for the diffractive cross sections demonstrate good consistency with the available\nexperimental data at different energies from H1 and ZEUS collaborations. Additionally, we show\nthat the resulting LFWFs for the ρ-meson can effectively describe various properties, including its\ndecay constant, distribution amplitudes, electromagnetic form factors, charge radius, magnetic and\nquadrupole moments. Comparative analyses are conducted with experimental measurements and\nthe available theoretical predictions.\nI. INTRODUCTION\nExperimental processes such as deep inelastic scattering (DIS), deeply virtual Compton scattering (DVCS), and ex-\nclusive diffractive vector meson production serve as effective tools to investigate Quantum Chromodynamics (QCD) [1].\nParticularly, at small x, these processes are predominantly influenced by gluon saturation. This phenomenon has ex-\ntensively been explored within the framework of the Color Glass Condensate (CGC) effective field theory [2–6]. The\nCGC theory describes the balance of gluons through recombination and multiple scattering limitations within a dipole\npicture [7–10]. In this dipole model, a virtual photon splits into a quark and an anti-quark pair (dipole), interacts\nwith proton through gluon exchange, and reforms into a vector meson or photon as illustrated in Fig. 1.\nThe goal of this paper is to predict the cross section for diffractive ρ-meson electroproduction, observed at the\nHERA collider [11–16], using the QCD color dipole model and the nonperturbative holographic meson light-front\nwave functions (LFWFs) [17] by taking into account the longitudinal dynamics generated by the ’t Hooft Equation\nin (1 + 1)-dim QCD at large Nc[18].\nThe holographic light-front QCD (hLFQCD) is developed within the chiral limit of light-front QCD, establish-\ning an exact correspondence between strongly coupled (1+3)-dimensional light-front QCD and weakly interacting\nstring modes in (1+4)-dimensional anti-de-Sitter (AdS) space. For a review of hLFQCD, see Ref. [19]. The primary\nnontrivial prediction of this approach leads to the lightest bound state, i.e., the pion is massless. Another crucial\nprediction asserts that the meson masses align along the universal Regge trajectories, mirroring experimental obser-\nvations. Note that the predicted slopes are dictated by the strength of the confining potential, κ. The form of the\nconfining potential in physical spacetime is determined by a dilaton field that breaks the conformal symmetry of AdS\nspace. A phenomenologically successful choice involves a quadratic dilaton in the fifth dimension of AdS space, which\ncorresponds to a light-front harmonic oscillator in physical spacetime. The mass scale parameter, κ, is fixed by fitting\nthe experimentally observed slopes of the meson mass spectrum Regge trajectories for different meson groups. It is\nfound that for all the light mesons, κ≃0.5 GeV [17].\nGoing beyond the semiclassical approximation, Brodsky and de T´ eramond proposed an invariant mass ansatz (IMA)\nfor including nonzero quark masses [20]. Using IMA, one can calculate the shift in the meson masses as a first order\nperturbation. The predicted mass shift for the pion (and kaon) appears as same as the physical mass of the meson.\nThese results have been obtained by fixing the scale parameter as κ= 0.54 GeV, and the light quark masses as\nmu/d= 0.046 GeV and ms= 0.357 GeV, which vanish in the chiral limit [17].\nPrevious works [21, 22] reported predictions for vector mesons production by utilizing the holographic wave function\nwith IMA together with the CGC dipole cross section [23]. Reference [21] has investigated the process of ρ-meson\nproduction using a light quark mass of mq= 0.14 GeV, which is consistent with the fitted parameters of CGC dipole\n∗gbheem@iitk.ac.in\n†mondal@impcas.ac.cn\n‡satvir@impcas.ac.cnarXiv:2401.13514v1  [hep-ph]  24 Jan 20242\ncross section [24, 25] from the inclusive DIS data [26, 27]. The most recent analyses of dipole cross sections have\nutilized the 2010 DIS data at HERA [28]. It has been acknowledged in Ref. [23] that the DIS data prefers the lower\nlight quark masses, but also noted that the use of effective quark mass, mq= 0.14 GeV, yields satisfactory fit to the\n2001 DIS structure function data. In a more recent study [29], a novel dipole model has demonstrated that both\ncurrent quark mass and effective quark mass mq= 0.14 GeV accurately fit the 2010 DIS structure function data [28].\nIn Ref. [22], the authors revisited the CGC dipole model and fit the conclusive 2015 HERA data of inclusive DIS\nwith the light quark masses. They studied cross sections for the diffractive ρandϕmeson production using the fitted\ndipole cross section [30], the perturbatively calculated photon LFWFs [31, 32], and the holographic meson LFWFs\nwith IMA, which contains no dynamical information of the meson in the longitudinal direction [33].\nIn this work, the nonzero light quark masses are incorporated through the chiral symmetry breaking and the\nlongitudinal dynamics governed by the ’t Hooft equation of (1+1)-dimensional QCD in the large Nclimit. The\ncombined holographic Schr¨ odinger equation and the ’t Hooft equation provide a reliable picture of the data to the\nρ-meson spectrum with the universal κ. We show that, together, they can simultaneously describe various properties\nof the ρ-meson, including its decay constant, parton distribution amplitude (PDA), electromagnetic form factors,\ncharge radius etc., as well as, in conjunction with the color glass condensate dipole cross-section, the resulting wave\nfunctions can provide good description of the HERA data of the diffractive ρ-meson electroproduction.\nThe rest of the paper is organized as follows: In Sec. II, we review the color dipole model. The holographic meson\nLFWFs followed by the longitudinal dynamics using the ’t Hooft equation are discussed in Sec. III. In Sec. IV, we\ndiscuss the numerical results for the mass spectroscopy, the diffractive cross sections using the dipole cross section,\nPDAs, electromagnetic form factors, decay constant, charge radius, magnetic and quadrupole moments for the ρ-meson\nwith the holographic meson wave function. Finally, we conclude the paper in Sec. V.\nII. THE DIPOLE MODEL OF EXCLUSIVE VECTOR MESON PRODUCTION\nIn the dipole picture, the scattering amplitude for the diffractive process γ∗p→V pis expressed as the convolution\nof the overlap of the LFWFs of the photon and the vector meson, and the proton-dipole scattering amplitude [24],\nℑmAγ∗p→V p\nΛ (s, t;Q2) =X\nh,¯hZ\nd2r⊥dxΨγ∗,Λ\nh,¯h(x,r⊥;Q2)ΨV,Λ\nh,¯h(x,r⊥)∗e−ixr⊥·∆N(xm,r⊥,∆), (1)\nwhere Q2is the virtuality of the photon and t=−∆2is the squared of the transverse momentum transfer at\nthe proton vertex. The substantial value of the center-of-mass energy squared, s, ensures the factorization of the\nscattering amplitude for the diffractive process into a convolution of the LFWFs of the photon, Ψγ,Λ\nh,¯h(x,r⊥;Q2), and\nvector meson, ΨV,Λ\nh,¯h(x,r⊥), and a dipole cross section, N(xm,r⊥,∆). The variable r⊥is the transverse size of the\ndipole and xdefines the longitudinal momentum fraction carried by the quark as shown in Fig. 1. The indices h\nand¯hare the helicities of the quark and the antiquark. The symbol Λ in the superscript of the LFWFs denotes\nthe polarization of the photon and the vector meson. The systems can be longitudinally polarized or transversely\npolarized; symbolically, Λ = 0 or ±1, respectively. The dipole-proton scattering amplitude depends upon the center-of-\nmass energy of the photon-proton system ( W), related to the modified Bjorken variable as [34]: xm=xBj\u0010\n1 +M2\nV\nQ2\u0011\n,\nwhere xBj=Q2\nW2. The dipole-proton scattering amplitude encapsulates the high-energy QCD dynamics associated\nwith the dipole-proton interaction. Being a universal object, it can be obtained via an approximate solution of the\nBalitsky-Kovchegov equation [35–37] within the CGC formalism [38–42].\nThe differential cross section for the exclusive vector meson production is given by [15, 23],\ndσγ∗p→V p\nΛ\ndt=1\n16π[ℑmAγ∗p→V p\nΛ (s, t= 0)]2(1 +β2\nΛ) exp(−BDt) (2)\nwith the parameter βΛbeing the ratio of the real to imaginary components of the scattering amplitude expressed\nas [23, 43],\nβΛ= tan\u0010π\n2αΛ\u0011\nwith αΛ=∂log|ℑmAΛ|\n∂log(1/xm), (3)\nand the diffractive slope parameter BDparameterized as [22],\nBD=N\"\n14.0\u00121 GeV2\nQ2+M2\nV\u00130.2\n+ 1#\n(4)3\n1−\nγ∗γ∗\n1−\nγ∗(1−)\nFIG. 1. The γ∗pelastic scattering amplitude for DIS (left) and for exclusive vector meson production (right) in dipole model.\nHere, xis the fraction of the photon momentum carried by the quark and r⊥is the transverse size of the dipole.\nwith N= 0.55 GeV−2. The parametrization of this slope parameter is consistent with the ZEUS data for ρ-meson\nproduction [15, 43]. However, the most recent H1 data [12] favours somewhat larger value of BDaccompanied by the\nlarge uncertainty.\nA simplified model for the dipole cross-section was proposed a long ago in Ref. [30], known as the CGC dipole\nmodel. The dipole cross section is given by\nˆσ(xm, r⊥) =σ0N(xm, r⊥Qs,0) (5)\nwith\nN(xm, r⊥Qs,0) =\n\nN0\u0010\nr⊥Qs\n2\u00112h\nγs+ln(2/r⊥Qs)\nκ0λln(1/xm)i\nfor r⊥Qs≤2,\n1−exp[−Aln2(Br⊥Qs)] for r⊥Qs>2(6)\nwhere r⊥=|r⊥|and the saturation scale is Qs= (x0/xm)λ/2GeV. The coefficients AandBin Eq. (6) are determined\nuniquely from the condition that N(xm, r⊥Qs,0) and its derivative with respect to r⊥Qsare continuous at r⊥Qs= 2.\nThis leads to\nA=−N2\n0γ2\ns\n(1− N 0)2ln(1− N 0),B=1\n2(1− N 0)−(1−N0)\nN0γs. (7)\nThe free parameters of the CGC dipole model, σ0, λ, x 0, and γsare determined by a fit to the H1 and ZEUS (2015)\nF2structure function data [44] (for xBj≤0.01 and Q2∈[0.045,45] GeV2) with a χ2/d.o.f = 1 .03 [22]. Here, we\nuse the parameters as determined in Ref. [22] as: σ0= 26.3 mb, γs= 0.741, λ= 0.219, and x0= 1.81×10−5for\nmu,d∼0.046 GeV. The parameters N0andκ0are fixed as 0 .7 and 9 .9 (leading order Balitsky-Fadin-Kuraev-Lipatov\nprediction), respectively.\nTo compute the scattering amplitude for exclusive ρ-meson production, Eq. (1), we need to employ the LFWFs\nof the incoming virtual photon and the outgoing vector meson. In practice, the expressions for the photon LFWFs\nare obtained perturbatively in light-front QED. The lowest order perturbative LFWFs for the longitudinally and\ntransversely polarized photons are given by [31, 32, 45],\nΨγ,Λ=0\nh,¯h(x,r⊥;Q2, mq) =r\nNc\n4πδh,−¯he eq2x(1−x)QK0(ϵr⊥)\n2π,\nΨγ,Λ=±1\nh,¯h(x,r⊥;Q2, mq) =±r\nNc\n2πe eq\u0002\nie±iθr⊥(xδh±,¯h∓−(1−x)δh∓,¯h±)∂r⊥+mqδh±,¯h±\u0003K0(ϵr⊥)\n2π, (8)\nwhere ϵ2=x(1−x)Q2+m2\nqande2= 4παemwith αembeing the QED coupling constant, eqandmqrepresent the\neffective charge and mass of the quark, respectively. Nccorresponds to the color factor, K0denotes the second kind of\nBessel function and r⊥eiθr⊥is the complex notation for the transverse distance between the quark and the antiquark.\nOn the other hand, a nonperturbative model for the meson LFWFs is discussed in Sec. III.\nThe total cross section is expressed as the linear combination of the transverse and longitudinal cross sections by\nintegrating them (given in Eq. (2)) over t. Therefore,\nσγ∗p→V p\ntot (x, Q2) =σγ∗p→V p\nΛ=±1(x, Q2) +ωσγ∗p→V p\nΛ=0 (x, Q2), (9)\nwhere ωis the photon polarization parameter, with ⟨ω⟩= 0.98 in the kinematic domain corresponding to the HERA\nmeasurement for the ρ-meson production [12]. We consider the same value of ωfor predicting the total cross section\nand compare it with the HERA data.4\nIII. HOLOGRAPHIC MESON WAVE FUNCTIONS WITH LONGITUDINAL CONFINEMENT\nIn the previous section, we reviewed the photon wave functions, which are obtained from the perturbative QED.\nThe vector meson LFWFs appearing in Eq. (1) can not be obtained in perturbation theory. Nevertheless, they can\nbe considered to have the same spinor and polarization structure as in the photon case, together with an unidentified\nnonperturbative wave function. Various ansatz for the meson nonperturbative wave functions have been reported in\nliterature [8, 21, 32]. However, the most popular is the boosted Gaussian wave function [8, 46], which has recently\nbeen employed in Refs. [43, 47] to simultaneously reproduce the cross-section data for diffractive ρ, ϕandJ/Ψ\nproduction. Explicitly, the spin-improved LFWFs for longitudinally and transversely polarized vector meson can\nbe written as [21, 46, 48, 49]\nΨV,Λ=0\nh,¯h(x,r⊥) =1\n2δh,−¯h\u0014\n1 +m2\nq− ∇2\nr⊥\nx(1−x)M2\nV\u0015\nΨ(x,r⊥), (10)\nand\nΨV,Λ=±1\nh,¯h(x,r⊥) =±\u0014\nie±iθr⊥(xδh±,¯h∓−(1−x)δh∓,¯h±)∂r⊥+mqδh±,¯h±\u0015Ψ(x,r⊥)\n2x(1−x), (11)\nrespectively, where Ψ represents the spin independent part of the vector meson wave functions.\nBrodsky and de T´ eramond proposed a nonperturbative approach to construct the hadronic LFWFs based on\nhLFQCD [33, 50, 51]. To connect with AdS space, a holographic variable, ζ=p\nx(1−x)r⊥is introduced, and the\nwave function is written in a factorized form in terms of x,ζandφvariables :\nΨ(x, ζ, φ ) =ϕ(ζ)√2πζeiLφX(x), (12)\nwhere X(x) =p\nx(1−x)χ(x) and ϕ(ζ) are referred as the longitudinal and transverse modes, respectively and Lis\nthe orbital quantum number. In hLFQCD, only the transverse mode, ϕ(ζ), is dynamical and it is generated by the\nholographic Schr¨ odinger-like equation [19, 52–54],\n\u0012\n−d2\ndζ2+4L2−1\n4ζ2+U⊥(ζ)\u0013\nϕ(ζ) =M2\n⊥ϕ(ζ), (13)\nwhere the confinement potential is given by\nU⊥(ζ) =κ4ζ2+ 2κ2(J−1), (14)\nwith J=L+Sbeing the total angular momentum of the meson. The analytical expression of Eq. (14) is uniquely\ndetermined by a holographic mapping to AdS 5, where light-front variable ζmaps onto the fifth dimension of AdS\nspace and the underlying conformal symmetry [55]. The emerging mass scale, κ, fixes the confinement scale and\nproduces meson masses in the chiral limit. Using Eq. (14) in Eq. (13) yields\nM2\n⊥(n⊥, J, L) = 4 κ2\u0012\nn⊥+J+L\n2\u0013\n, (15)\nand\nϕn⊥L(ζ)∝ζ1/2+Lexp\u0012−κ2ζ2\n2\u0013\nLL\nn⊥(κ2ζ2), (16)\nwith n⊥being the transverse principle quantum number. An important outcome of Eq. (15) is that the lowest-lying\nhadronic bound state, with n⊥=L=S= 0, is massless. This is inherently recognized as the pion, which is\nanticipated to exhibit zero mass in the chiral limit of QCD.\nMeanwhile, the longitudinal mode, χ(x), is not dynamical in hLFQCD. The longitudinal wave function, X(x) =p\nx(1−x) is explicitly obtained by the holographic mapping of the electromagnetic or gravitational form factor in\nphysical spacetime and AdS 5[56, 57]. This results χ(x) = 1. Inserting Eq. (16) in Eq. (12) yields the holographic\nLFWFs in the chiral-limit. For the ground state mesons ( n= 0, L= 0), the holographic LFWFs become\nΨmq=0(x, ζ2)∝p\nx(1−x) exp\u0012−κ2ζ2\n2\u0013\n. (17)5\nA two-dimensional Fourier transform results\nΦmq=0(x, k2\n⊥)∝1p\nx(1−x)exp\u0012\n−k2\n⊥\n2κ2x(1−x)\u0013\n, (18)\nwhere k⊥=|k⊥|, the Fourier conjugate of r⊥, defines the transverse momentum of the quark. Going beyond the\nchiral limit, Brodsky and de T´ eramond proposed a prescription to describe the longitudinal mode as [20]:\nX(x) =p\nx(1−x) exp\"\n−1\n2κ2 \nm2\nq\nx+m2\n¯q\n1−x!#\n, (19)\nbased on the observation that the chiral-limit of invariant mass of quark-antiquark pair,\nM2\nq¯q=k2\n⊥\nx(1−x)+m2\nq\nx+m2\n¯q\n1−x, (20)\nappears in Eq. (18). Consequently, the bound-state mass eigenvalue receives a first-order correction such that\n∆M2=Zdx\nx(1−x)X2(x) \nm2\nq\nx+m2\n¯q\n1−x!\n. (21)\nNote that there are two shortcomings with the above prescription. First, it indicates that [58] M2\nπ= ∆M2∝\n2m2\nq(ln(κ2/m2\nq)−γE) with γE= 0.577216 being the Euler’s constant, in contrast to Gell-Mann-Oakes-Renner (GMOR)\nrelation, M2\nπ∝mq. Second, the longitudinal mode, given by Eq. (19), with no nodes, remains same for all the radially\nexcited states. However, this prescription has successfully been implemented to describe the light as well as heavy\nmesons [19, 20, 59–65].\nMeanwhile, Refs. [66, 67] consider longitudinal dynamics, generated by the ’t Hooft Equation, in order to describe\nthe full meson spectrum, while the pion dynamics has been predicted in Ref. [68]. The concept of employing the\n’t Hooft equation to extend beyond the invariant mass prescription was initially suggested in Ref. [69], aiming to\nforecast meson decay constants and parton distribution functions. Recently, in Refs. [58, 59], the prescription was\nsurpassed using a phenomenological longitudinal confinement potential, which was initially introduced in Ref. [70]\nwithin the framework of basis light-front quantization. While both Refs. [58, 59] concentrate on the chiral limit and the\noccurrence of chiral symmetry breaking, Ref. [59] broadens their investigation to heavy mesons in their ground state\nand explores the connection of their approach to the ’t Hooft equation. It is worth mentioning that there has been a\nnotable surge in interest regarding the incorporation of longitudinal dynamics within hLFQCD [58, 59, 66, 71–73].\nThe ’t Hooft equation can be derived by using the QCD Lagrangian in (1+1)-dim with large Ncapproximations\nas [18]\n \nm2\nq\nx+m2\n¯q\n1−x!\nχ(x) +g2\nπPZ\ndy|χ(x)−χ(y)|\n(x−y)2=M∥χ(x), (22)\nwhere gis the longitudinal confinement scale and Pdenotes the Cauchy principal value. It is important to note\nthat in the conformal limit, the ’t Hooft equation has a gravity dual on AdS 3[74] and has been widely studied\nin the literature [66, 68, 75–81]. Unlike the holographic light-front Schr¨ odinger equation, the ’t Hooft does not\nadmit analytical solutions. We solve it numerically using the matrix method illustrated in Ref. [69]. Using both the\nholographic Schr¨ odinger equation and the ’t Hooft equation, the meson mass is then given by\nM2(n⊥, n∥, J, L) = 4 κ2\u0012\nn⊥+J+L\n2\u0013\n+M2\n∥(n∥, mq, m¯q, g), (23)\nwhere n∥defines the longitudinal quantum number. Since, the holographic Schr¨ odinger equation predicts a massless\npion, it follows that the only contribution to the pion mass is produced by the ’t Hooft equation. Note that together, the\nholographic Schr¨ odinger equation and the ’t Hooft equation correctly predicts the GMOR relation M2\nπ∼mu,d[59, 68].\nWith the universal transverse confinement scale κ= 0.523 GeV and the light quark mass mu/d= 0.046 GeV, which\nis the value considered in hLFQCD together with the IMA [19], the longitudinal confining scale g= 0.109 GeV leads\nto excellent agreement of the mass spectroscopy for the pion family with the experimental data [68]. In this work,\nwe use the same set of parameters in order to predict the spectroscopic data for the ρ-meson family and obtain the\ncorresponding wave functions.6\nTABLE I. Quantum numbers and masses of the ρ-meson family with S= 1.\nJP(C)Name n⊥n∥L M (MeV)\n1−−ρ(770) 0 0 0 752\n2++a2(1320) 0 2 1 1315\n3−−ρ3(1690) 0 4 2 1702\n4++a4(1970) 0 6 3 2016\n1−−ρ(1450) 1 2 0 1315\n2++a2(1700) 1 4 1 1702\n1−−ρ(1700) 2 4 0 1702\nThe complete spin-independent part of the meson LFWFs can then be expressed as,\nΨ(x, ζ) =Np\nx(1−x)χ(x) exp\u0014\n−κ2ζ2\n2\u0015\n, (24)\nwhere χ(x) is the solution of the ’t Hooft equation. Nis a normalization constant, which depends on polarization of\nthe meson and can be fixed by using the normalization condition as,\nX\nh,¯hZ\nd2r⊥dx|ΨV,Λ\nh,¯h(x,r⊥)|2= 1, (25)\nwhere the forms of the spin-improved wave functions ΨV,Λ\nh,¯hare given in Eqs. (10) and (11). The numerical solutions\nfor the longitudinal modes χ(x) of the ground state meson can approximately be fitted to the following polynomial\nform:\nχ(x)≃xβ1(1−x)β2, (26)\nwith βibeing the quark mass dependent variables, which vanish in the chiral limit. For the ground state of ρ-meson,\nwe find that β1,2= 0.51.\nIV. RESULTS AND DISCUSSION\nA. Mass spectroscopy\nThe parity and charge conjugation quantum numbers of meson states are given by [66–68]:\nP= (−1)L+1, C= (−1)L+S+n∥. (27)\nUsing Eq. (23) and the parameters determined for the pion family: mu/d= 0.046 GeV, κ= 0.523 GeV, and g= 0.109\nGeV [68], we are able to describe the spectroscopic data for the ρ-meson family. We present our computed masses of\ntheρ-meson and its excited states in Table I. Our results (last column) are in good agreement with the experimental\ndata (second column, in parentheses). Note that an emerging condition n∥≥n⊥+Lin Table I is observed to remain\ntrue across the full hadron spectrum [67]. The resulting Regge trajectories for the ρ-meson family in our calculation\nare shown in Fig. 2.\nB. ρ-meson diffractive cross-section\nThe exclusive vector meson production cross section depends upon the overlap of the q¯qcomponent of the virtual\nphoton wave functions with the vector meson wave functions as illustrated in Eq. (1). We present the overlap of the\nLFWFs after integrating over xat different photon virtualities Q2= 2,4.8 and 19 .7 GeV2in Fig. 3. We compare two\ndifferent overlap functions, which are different in considering the longitudinal modes in the meson wave functions:\n(i) the IMA that does not contain dynamical mode along the longitudinal direction, and (ii) the one containing\nlongitudinal dynamics incorporated through the ’t Hooft equation. However, we find that they lead to more or less7\nn⟂=0n⟂=2\nn⟂=1\n▲▲ ▲ρ(770)a2(1320 )▲▲\n▲ ρ3(1690 )a4(1970 )\nρ(1450 )▲ ▲ ▲\na2(1700 ) ρ(1700 )\n0 1 2 302468\nLM2[GeV2]\nFIG. 2. Our Regge trajectories for the ρ-meson family.\nsolid: 't Hooftdashed: IMAQ2=2 GeV2Q2=4.8 GeV2Q2=19.7 GeV2\n0.01 0.05 0.10 0.50 10.00000.00050.00100.00150.00200.00250.0030Transverselypolarized ρmesons\nsolid: 't Hooftdashed: IMAQ2=2 GeV2\nQ2=4.8 GeV2\nQ2=19.7 GeV2\n0.01 0.05 0.10 0.50 10.0000.0020.0040.0060.0080.0100.012Longitudinally polarized ρmesons\nFIG. 3. The transverse (left) and the longitudinal (right) overlap functions between the photon and the ρ-meson LFWFs\nintegrated over xas a function of the dipole transverse size r⊥(in fm) at different photon virtualities predicted by the light\nfront holography ’t Hooft (solid) and the light front holography IMA (dashed) approaches, respectively.\nsimilar behavior of the overlap functions. The peaks of the distributions undergo a shift towards lower values of r⊥\nand decrease in magnitude as the virtuality of the photon increases.\nIn Fig. 4, we illustrate the three-dimensional probabilistic distributions, |ΨΛ\nh,¯h(x,r⊥)|2, as a function of xandr⊥\nfor a longitudinally (left panel) and a transversely (right panel) polarized ρ-meson from our resulting holographic\nLFWFs incorporated with longitudinal modes generated by the ’t Hooft equation. We notice that the wave function\npeaks at x= 0.5 and r⊥= 0, and go rapidly to zero as x→0,1 and r⊥increases. Our results behaves similar to the\nwave functions reported in Ref. [46].\nWe have now all the ingredients (photon and vector meson LFWFs, and the dipole cross-section) to compute the\ndiffractive vector meson production cross section. We calculate ρ-meson production utilizing the CGC dipole model\nwith parameters fitted to DIS data from HERA, as detailed in Ref. [22]. In Fig. 5, we show the total cross section\nas a function of Wfor different Q2bins. On the left panel, we compare our predictions with the experimental data\nfor 0≤Q2≤35.6 GeV2from the H1 Collaboration [12, 13], whereas, the right panel compares our results with the\nexperimental data from the ZEUS Collaboration [12] in the range 0 .47≤Q2≤27.0 GeV2. From the comparison, we\nobserve that our predictions are in reasonable agreement with the experiments within the range of allowed uncertainty.8\nFIG. 4. Distributions of the longitudinal |ΨΛ=0\nh,¯h(x,r⊥)|2(left) and transverse |ΨΛ=±1\nh,¯h(x,r⊥)|2(right) LFWFs of the ρ-meson\nas functions of longitudinal momentum fraction carried by the quark xand the dipole separation r⊥(in fm).\nH1(2010)This workZEUS (2005)\n50 100 150 2000.11101001000104\nZEUS (2010)This work\n40 60 80 100 120 140 1601101001000104\nFIG. 5. Our predictions of the total diffractive cross-section for γ∗p→ρpas a function of W(in GeV) in different Q2bins (in\nGeV2) and compared with experimental data from H1 2010 [12, 13], ZEUS 2000 data (at Q2= 0) [16] (left) and ZEUS 2010\ndata [12] (right).\nWe also note that the longitudinal dynamics implemented through the ’t Hooft equation improves the predictions\ncompared to those calculated using IMA in Ref. [22]. The differential cross-section, dσ/dt , as a function of tfor\nthe elastic ρ-meson production are shown in Fig. 6, where we compare our results with the experimental data from\nthe H1 [12] (left panel) and ZEUS [15] (right panel) Collaboration at different values of Q2. Again, we observe that\nour predictions show a good agreement with the measurements. However, at large tand small Q2, our results are\nsomewhat underestimated for the ZEUS data.\nThe Fig. 7(a) and 7(b) present the longitudinal and transverse cross-sections, respectively, for the diffractive ρ-\nmeson production as a function of photon virtually for the fixed value of W= 75 GeV. We find good agreement with\nthe experiment at HERA [12]. The total γ∗pelastic cross-section for ρ-meson production is also found to be in good\nagreement with the experiments [11, 15] as can be seen in Fig. 7(c). Finally, in Fig. 7(d), we illustrate the ratio of the\nlongitudinal to transverse cross section, σL/σT, forρproduction. We compare our prediction with the H1 2010 [12],\nZEUS 2007 [15] and old 2000 H1 [11] data. Here, we notice that for the low values of Q2, i.e., Q2≤10 GeV2, our\nresults show a good agreement with all the three data sets, whereas it slightly deviates at large Q2from the measured\ndata.9\nH1(2010)This work\n0.0 0.2 0.4 0.6 0.8 1.00.11101001000\nZEUS (2007)This work\n0.0 0.2 0.4 0.6 0.8 1.00.11101001000\nFIG. 6. Our predictions of differential cross-section, dσ/dt (in nb/GeV2) forγ∗p→ρ0pas a function of |t|(in GeV2) compared\nwith H1 2010 [12] (left) and ZEUS 2007 [15] (right) data, respectively.\nH1(2010)\nThiswork\n10 20 30 40 500.010.101101001000\n(a)\nH1(2010)\nThiswork\n10 20 30 40 500.010.101101001000\n(b)\nH1(2010)\nZEUS (2007)\nThiswork\n20 40 60 80 1000.010.101101001000\n(c)\nH1(2010)\nZEUS (2007)\nH1(2000)\nThiswork\n0 10 20 30 40 500246810\n(d)\nFIG. 7. Our predictions for the Q2dependence of (a) longitudinal, (b) transverse, (c) Total γ∗pcross sections for elastic ρ\nmeson production with W= 75 GeV and (d) longitudinal to transverse cross-section ratio compared to the H1 at W= 75\nGeV [11, 12] and ZEUS at W= 75 GeV [15] data.10\nTABLE II. Our predictions for the electronic decay widths of the ρ-meson using ’t Hooft equation with mu,d= 0.046 GeV.\nThe LF holography IMA predictions correspond to mu,d= [0.046,0.14] GeV.\nρ-meson fV[MeV] Γe+e−[KeV]\nThis work 208 6.294\nLFH (IMA)[21, 22] [210,211] [6.355,6.66]\nExp. (PDG) [84] 216(5) 7.04±0.06\nC. Decay Constants and Distribution Amplitudes\nIn this section, we compute the decay constants and distribution amplitudes for the ρ-meson and compare them\nwith the experimental data along with the theoretical predictions from other models. The vector and tensor coupling\nconstants, fVandfT\nVrespectively, are defined as the local vacuum-to-hadron matrix elements [82]\n⟨0|¯q(0)γµq(0)|V(P,Λ)⟩=fVMVϵµ\nΛ, (28)\nand\n⟨0|¯q(0)[γµ, γν]q(0)|V(P,Λ)⟩= 2fT\nV(ϵµ\nΛPν−ϵν\nΛPµ), (29)\nwhere q(¯q) are the quark(anti-quark) field operators at same space-time points. The momentum and polarization\nvectors are denoted as Pµandϵµ\nΛ, respectively. In terms of LFWFs, the decay constant can be expressed as [83]\nfV=r\nNc\nπZ1\n0dx\"\n1 +m2\nq− ∇2\nr⊥\nx(1−x)M2\nV#\nΨ(x,r⊥)|r⊥=0, (30)\nand\nfT\nV(µ) =r\nNc\n2πmqZ1\n0dxZ\ndr⊥µJ1(µr⊥)Ψ(x,r⊥)\nx(1−x), (31)\nwhere Ψ is the meson LFWF given in Eq. (24) and µis the ultraviolet cut-off scale. We note that our predictions for\nthe tensor coupling are scale independent for µ2≥1. However, it is sensitive to the quark mass mq, as can be seen\nfrom Eq. (31). In the chiral limit, mq→0, the tensor coupling vanishes, whereas the vector coupling has a nonzero\nvalue. The vector coupling can be used to calculate the electronic decay width\nΓV→e+e−=4πα2\nemC2\nV\n3MVf2\nV, (32)\nwhere, for the ρ-meson, Cρ= 1/√\n2. In Table II, we present our predictions for the vector coupling constant, which\nis associated with the decay width of the ρ-meson. These predictions are compared with the results from LFhQCD\nwith IMA approach [21, 22] and experimental data [84]. Additionally, in Table III, we compare our model predictions\nfor decay constants when ρ-meson is considered to be longitudinally and transversely polarized, as well as their ratio\nf⊥\nρ/fρ. We compare our results with the predictions from other theoretical approaches in Table III. Our prediction\nfor the vector coupling constant demonstrates good agreement with both the theoretical and experimental studies.\nHowever, the tensor coupling constant is significantly smaller as compared to the other predictions in the literature.\nPDAs are obtainable through the vacuum-to-meson transition matrix elements of quark-antiquark nonlocal gauge-\ninvariant operators. The longitudinal and transverse components of the distribution amplitude for the vector meson\nare defined as [94]\nfρϕ∥\nρ(x, µ) =Z\ndx−eixP+x−⟨0|¯q(0)γ+q(x−)|V(P,Λ = 0) ⟩, (33)\nand\nf⊥\nρϕ⊥\nρ(x, µ) =1\n2Z\ndx−eixP+x−⟨0|¯q(0)[ϵ∗\nΛ(±).γ, γ+]q(x−)|V(P,Λ =±1)⟩. (34)11\nTABLE III. Using the value of light quark masses, mu,d= 0.046 GeV, we predict the longitudinal and transverse decay\nconstants for the ρ-meson, as well as the ratio between them and compared with other model estimate. Our predictions are at\na scale of µ= 1 GeV.\nReference Approach fρ[MeV] f⊥\nρ[MeV] f⊥\nρ/fρ\nThis work LFH (‘t Hooft) 208 36 0.17\nRef. [22, 85] LFH (IMA) [211,214] [95,36] [0.45,0.17]\nRef. [86] LFQM 208±7 152±9\nRef. [87] Sum Rules 198±7 152±9\nRef. [88] Sum Rules 206±7 145±9 0.70±0.04\nRef. [89] Lattice (continuum) 0.72±0.02\nRef. [90] Lattice (finite) 0.742±0.014\nRef. [91] Lattice (unquenched) 159±0.008 0.76±0.04\nRef. [92] Dyson-Schwinger 212 156 0.73\nRef. [93] LFQM: Linear [HO] 246 [215] 187 [163] 0.76 [0.80]\n't Hooft\nIMA\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.5\n't Hooft\nIMA\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.0\nFIG. 8. Our results for the PDAs for longitudinally (left) and transversely (right) polarized ρ-mesons (in solid blue). We\ncompared them with IMA PDAs (magenta-dashed).\nAfter calculating the matrix elements, the above Eqs. (33) and (34) lead to\nϕ∥\nρ(x, µ) =Nc\nπfρMρZ\ndr⊥µJ1(µr⊥)\u0002\nM2\nVx(1−x) +m2\nq− ∇2\nr⊥\u0003Ψ(x,r⊥)\nx(1−x), (35)\nand\nϕ⊥\nρ(x, µ) =Ncmq\nπf⊥ρZ\ndr⊥µJ1(µr⊥)Ψ(x,r⊥)\nx(1−x), (36)\nwhere fρandf⊥\nρare the vector and tensor couplings, which are given in Eq. (30) and (31), respectively. The\nlongitudinal ( ϕ∥\nρ) and transverse ( ϕ⊥\nρ) components of the PDAs can be normalized as [95]\nZ1\n0dxϕ∥\nρ(x, µ) = 1 ,andZ1\n0dxϕ⊥\nρ(x, µ) = 1 . (37)\nFig. 8 illustrates the normalized longitudinal and transverse components of ρ-meson PDAs and their comparison\nwith hLFQCD associated with IMA predictions at nonzero light quark mass, mq≃0.046 GeV. We find that our12\nTABLE IV. Computed moments of the ρ-meson PDAs, Eq. (38), compared with selected results obtained elsewhere, using:\nAdS/QCD models for LFWFs fitted to HERA data [96]; QCD sum rules [94, 97]; nonlocal condensates [98, 99]; light-front\nquark model [93]; light-front holographic QCD [22], and lattice-QCD [100, 101].\n⟨(2x−1)m⟩ m= 2 4 6 8 10\nThis work ∥ 0.20 0.087 0.048 0.031 0.022\n⊥ 0.25 0.13 0.079 0.055 0.042\nφ=φasy0.20 0.086 0.048 0.030 0.021\nφ= constant 0.33 0.2 0.14 0.11 0.091\n[22] ∥ 0.25 0.12 0.075 0.052 0.038\n⊥ 0.26 0.13 0.079 0.054 0.039\n[102] ∥ 0.22 0.103 0.066 0.046 0.035\n[96] ∥ 0.23 0.11 0.062 0.041 0.029\n⊥ 0.26 0.13 0.079 0.054 0.039\n[97] ∥ 0.26\n⊥ 0.27\n[98] ∥ 0.23(1) 0.095(5) 0.051(4) 0.030(2) 0.020(5)\n⊥ 0.33(1)\n[99] ∥ 0.22(2) 0.089(9) 0.048(5) 0.030(3) 0.022(2)\n⊥ 0.11(1) 0.022(2)\n[93] ∥ 0.20(1) 0.085(5) 0.045(5)\n⊥ 0.21(1) 0.095(5) 0.05(1)\n[100, 101] 0.25(2)(2)\nϕ∥\nρ(x) exhibits narrower distribution, while ϕ⊥\nρ(x) shows flat distribution compared to those evaluated using hLFQCD\nassociated with IMA.\nWe calculate the moments of the PDAs, also known as ξ-moments, in order to quantitatively compare with other\napproaches. The nthmoment is defined as [95],\n⟨ξn⟩=⟨(2x−1)n⟩=Z1\n0dx(2x−1)nϕ(x). (38)\nIn Table IV, we compare our model results for the computed ξ-moments at µ∼1 GeV with other theoretical\nestimations for n= 2,4,6,8 and 10. For the odd values of n, the moments vanish due to the isospin symmetry. Our\nξ-moments are more or less consistent with other theoretical studies.\nD. ρ-meson form factors\nThe LFWFs also provide direct access to electromagnetic form factors. The Lorentz-invariant electromagnetic\nForm factors Fi(i= 1,2,3) for a vector meson (spin-1) can be obtained by calculating the matrix elements of the\nelectromagnetic current Jµas [103, 104],\n⟨V(P′,Λ′)|Jµ|V(P,Λ)⟩=−ϵ∗\nΛ′·ϵΛ(P+P′)µF1(Q2) +\u0000\nϵµ\nΛq·ϵ∗\nΛ′−ϵ∗µ\nΛ′q·ϵΛ′\u0001\nF2(Q2)\n+(ϵ∗\nΛ′·q)(ϵΛ·q)\n2M2\nV(P+P′)µF3(Q2) (39)\nwhere ϵΛandϵΛ′are the polarization vectors of the initial and final mesons, respectively. We employ the Breit frame,\nwhere the momentum transfer occurs only in one transverse direction, i.e., ( q+= 0, qx=Q, q y= 0), and P⊥=\n−P′\n⊥[105, 106]. The momenta of the initial and final states are defined as: Pµ= (MV√1 +η, M V√1 +η,−Q/2,0)\nandP′µ= (MV√1 +η, M V√1 +η, Q/ 2,0), respectively with η=Q2/4M2\nV. We follow the notation pµ=\n(p+, p−, p1, p2). We compute the form factors by considering the plus component of the electromagnetic current,13\nJ+(0). The matrix elements of J+(0) can be expressed as [107, 108]\nI+\nΛ′,Λ(Q2)≜⟨V(P′,Λ′)|J+(0)\n2P+|V(P,Λ)⟩\n=X\nh,¯hZ1\n0Z∞\n0dxd2k⊥\n16π3ΨΛ′∗\nh¯h(x,k⊥+ (1−x)q⊥)ΨΛ\nh¯h(x,k⊥), (40)\nwhere Λ and Λ′denote the helicities of the incoming and outgoing vector mesons, respectively. There are a total of\nnine matrix elements of the electromagnetic current, I+\nΛΛ′for Λ ,Λ′= 0,±1. Using the light-front parity and time\nreversal invariance, one can reduce it to only four matrix elements: I+\n1,1, I+\n1,−1, I+\n1,0, and I+\n0,0. Note that the physical\ncharge ( GC), magnetic ( GM), and quadrupole ( GQ) form factors are often employed to describe the electromagnetic\nproperties of a hadron, instead of the Lorentz invariant electromagnetic form factors, Fi. However, these two types\nof form factors are related to each other such that\nGC=F1+2\n3ηGQ, G M=−F2, G Q=F1+F2+ (1 + η)F3.\nWe obtain the static charge ( e), magnetic moment ( µ) and quadrupole moment ( Q) of the hadron from the above\nform factors at zero momentum transfer,\neGC(0) = e , eG M(0) = 2 MVµ , −eGQ=M2\nVQ.\nNotably, there are different prescriptions, for example, Grach and Kondratyuk (GK) [109], and Brodsky and Hiller\n(BH) [110], to calculate such type of form factors. Nevertheless, we compute these physical form factors following the\nBH prescription, which includes the zero-mode contributions. In the BH prescription, the form factors are defined as,\nGBH\nC(Q2) =1\n2P+(1 + 2 η)\"\n3−2η\n3I+\n0,0+16\n3ηI+\n1,0√2η+2\n3(2η−1)I+\n1,−1#\n,\nGBH\nM(Q2) =2\n2P+(1 + 2 η)\u0014\nI+\n0,0+(2η−1)√2ηI+\n1,0−I+\n1,−1\u0015\n,\nGBH\nQ(Q2) =−1\n2P+(1 + 2 η)\"\nI+\n0,0−2I+\n1,0√2η+1 +η\nηI+\n1,−1#\n.(41)\nWe show the variation of the the charge, magnetic, and quadrupole elastic form factors with Q2in Fig. 9, where\nwe include the results generated using the IMA and Lattice QCD [111, 112] for comparison. We observe a good\nagreement of our results with the latice QCD simulations. From the charge form factor, we further calculate the\ncharge root-mean-squared (rms) radius of the ρmeson, which is defined as [113],\n⟨r2\nρ⟩=−6\nGC(0)lim\nQ2→0∂GC(Q2)\n∂Q2. (42)\nWe present our results for the static properties of the ρmeson: rms charge radius, magnetic moment, and quadrupole\nmoment in Table V, where we compare them with the predictions from various theoretical approaches. We observe\nthat our result for the charge radius is close to the results from BSE [114], lattice QCD [115], and NJL model [116].\nOn the other hand, our magnetic moment is more or less consistent with all other studies summarized in Table V.\nThe quadrupole moment agrees well with the results from BSE [114], LFQM [104], and lattice QCD [117], and they\ndiffer from other predictions.\nV. CONCLUSION\nThe ’t Hooft equation is complementary to the light-front holographic Schr¨ odinger equation, in governing the\nlongitudinal dynamics of quark-antiquark mesons. We have shown that together, they predict remarkably well the\nmass spectroscopy of ρ-meson family without further adjusting parameters: the universal transverse confinement\nscale κ= 0.523 GeV, the longitudinal confinement scale g= 0.109 GeV, and the light quark mass mq= 0.046\nGeV, which were determined to predict the pion spectroscopy and its structure [68]. In conjunction with the color\nglass condensate dipole cross-section, the ρ-meson holographic LFWFs after incorporating the longitudinal mode14\n't Hooft\nIMA\nLattice2008\nLattice2015\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0\n't Hooft\nIMA\nLattice2008\nLattice2015\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.01.52.02.53.0\n't Hooft\nIMA\nLattice2008\nLattice2015\n0.0 0.5 1.0 1.5 2.0 2.5 3.0- 1.0- 0.8- 0.6- 0.4- 0.20.00.2\nFIG. 9. Left: the charge ( Gρ\nC); middle: the magnetic ( Gρ\nM); and right: the quadrupole ( Gρ\nQ) form factors for the ρ-meson as a\nfunctions of Q2. Our results are compared with Lattice QCD predictions [111, 112].\nTABLE V. Comparison of the ρ-meson charge radiip\n⟨r2ρ⟩(in fm), magnetic moments µρ(in units of Bohr magneton) and\nquadrupole moments Qρ(in fm2) with various theoretical approaches.\nThis work LFH-IMA BLFQ [107] BSE [114] Lattice QCD [115] Lattice QCD [117] LFQM [104] NJL model [116]p\n⟨r2ρ⟩ 0.75 0 .94 0 .44 0 .73 0 .819(42) 0 .55(5) 0 .52 0 .82\nµρ 2.40 2 .90 2.15 2.01 2.067(76) 2.17(10) 1.92 2.48\nQρ−0.027 −0.023 -0.063 -0.026 -0.0452(61) -0.035 -0.028 -0.070\ngenerated by the ’t Hooft equation lead to a good description of the cross-section data for the diffractive ρ-meson\nelectroproduction at different energies. Using the resulting LFWFs, we have calculated the decay constant, distribution\namplitude, electromagnetic form factors, charge radius, magnetic moment, and quadrupole moment of the ρ-meson.\nInterestingly, we have noticed that although, the electromagnetic form factors in our approach agree well with the\nLFH-IMA predictions, they differ from each other in describing the distribution amplitudes. We have found that the\nvector coupling is close to the experimentally measured data and various theoretical predictions; however, the tensor\ncoupling constant is significantly smaller compared to the other predictions in the literature. Meanwhile, the moments\nof distribution amplitudes and the static properties: charge radius, magnetic moment and quadrupole moment have\nbeen found to be consistent with other theoretical as well as lattice QCD results.\nACKNOWLEDGEMENT\nWe thank Ruben Sandapen and Mohammad Ahmady for fruitful discussions. 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D 91, 114501 (2015)."    },    {        "title": "2401.13525v1.Flaring_Stars_in_a_Non_targeted_mm_wave_Survey_with_SPT_3G.pdf",        "content": "Draft version January 25, 2024\nTypeset using L ATEXtwocolumn style in AASTeX63\nFlaring Stars in a Non-targeted mm-wave Survey with SPT-3G\nC. Tandoi,1S. Guns,2A. Foster,3P. A. R. Ade,4A. J. Anderson,5, 6, 7B. Ansarinejad,8M. Archipley,1, 9\nL. Balkenhol,10K. Benabed,10A. N. Bender,11, 6B. A. Benson,5, 6, 7F. Bianchini,12, 13, 14L. E. Bleem,11, 6\nF. R. Bouchet,10L. Bryant,15E. Camphuis,10J. E. Carlstrom,6, 15, 16, 11, 7T. W. Cecil,11C. L. Chang,11, 6, 7\nP. Chaubal,8P. M. Chichura,16, 6T.-L. Chou,16, 6A. Coerver,2T. M. Crawford,6, 7A. Cukierman,12, 14, 13\nC. Daley,1T. de Haan,17K. R. Dibert,7, 6M. A. Dobbs,18, 19A. Doussot,10D. Dutcher,3W. Everett,20C. Feng,21\nK. R. Ferguson,22K. Fichman,16, 6S. Galli,10A. E. Gambrel,6R. W. Gardner,15F. Ge,23N. Goeckner-Wald,13, 12\nR. Gualtieri,11F. Guidi,10N. W. Halverson,24, 25E. Hivon,10G. P. Holder,21W. L. Holzapfel,2J. C. Hood,6\nN. Huang,2F. K´eruzor ´e,11L. Knox,23M. Korman,26K. Kornoelje,7, 6, 11C.-L. Kuo,12, 13, 14A. T. Lee,2, 27K. Levy,8\nA. E. Lowitz,6C. Lu,21A. Maniyar,12, 13, 14F. Menanteau,1, 9M. Millea,2J. Montgomery,18Y. Moon,1\nY. Nakato,13T. Natoli,6G. I. Noble,28, 29V. Novosad,30Y. Omori,7, 6S. Padin,6, 31Z. Pan,11, 6, 16P. Paschos,15\nK. A. Phadke,1, 9K. Prabhu,23Z. Qu,1W. Quan,16, 6M. Rahimi,8A. Rahlin,5, 6C. L. Reichardt,8C. Reuter,1\nM. Rouble,18J. E. Ruhl,26E. Schiappucci,8G. Smecher,32J. A. Sobrin,5, 6A. A. Stark,33J. Stephen,15\nA. Suzuki,27K. L. Thompson,12, 13, 14B. Thorne,23C. Trendafilova,9C. Tucker,4C. Umilta,21J. D. Vieira,1, 21, 9\nY. Wan,1, 9G. Wang,11N. Whitehorn,34W. L. K. Wu,12, 14V. Yefremenko,11M. R. Young,5, 6and\nJ. A. Zebrowski6, 7, 5\n1Department of Astronomy, University of Illinois Urbana-Champaign, 1002 West Green Street, Urbana, IL, 61801, USA\n2Department of Physics, University of California, Berkeley, CA, 94720, USA\n3Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA\n4School of Physics and Astronomy, Cardiff University, Cardiff CF24 3YB, United Kingdom\n5Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL, 60510, USA\n6Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA\n7Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA\n8School of Physics, University of Melbourne, Parkville, VIC 3010, Australia\n9Center for AstroPhysical Surveys, National Center for Supercomputing Applications, Urbana, IL, 61801, USA\n10Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Universit´ e, 98 bis boulevard Arago, 75014 Paris, France\n11High-Energy Physics Division, Argonne National Laboratory, 9700 South Cass Avenue., Lemont, IL, 60439, USA\n12Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA, 94305, USA\n13Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA, 94305, USA\n14SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA, 94025, USA\n15Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA\n16Department of Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA\n17High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan\n18Department of Physics and McGill Space Institute, McGill University, 3600 Rue University, Montreal, Quebec H3A 2T8, Canada\n19Canadian Institute for Advanced Research, CIFAR Program in Gravity and the Extreme Universe, Toronto, ON, M5G 1Z8, Canada\n20Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, 80309, USA\n21Department of Physics, University of Illinois Urbana-Champaign, 1110 West Green Street, Urbana, IL, 61801, USA\n22Department of Physics and Astronomy, University of California, Los Angeles, CA, 90095, USA\n23Department of Physics & Astronomy, University of California, One Shields Avenue, Davis, CA 95616, USA\n24CASA, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, 80309, USA\n25Department of Physics, University of Colorado, Boulder, CO, 80309, USA\n26Department of Physics, Case Western Reserve University, Cleveland, OH, 44106, USA\n27Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA\n28Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada\n29David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON, M5S 3H4,\nCanada\n30Materials Sciences Division, Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL, 60439, USA\n31California Institute of Technology, 1200 East California Boulevard., Pasadena, CA, 91125, USA\n32Three-Speed Logic, Inc., Victoria, B.C., V8S 3Z5, Canada\nCorresponding author: Chris Tandoi\nctandoi2@illinois.eduarXiv:2401.13525v1  [astro-ph.SR]  24 Jan 20242 Tandoi et al.\n33Center for Astrophysics |Harvard &Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA\n34Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA\nABSTRACT\nWe present a flare star catalog from four years of non-targeted millimeter-wave survey data from the\nSouth Pole Telescope (SPT). The data were taken with the SPT-3G camera and cover a 1500-square-\ndegree region of the sky from 20h40m0sto 3h20m0sin right ascension and −42◦to−70◦in declination.\nThis region was observed on a nearly daily cadence from 2019-2022 and chosen to avoid the plane of\nthe galaxy. A short-duration transient search of this survey yields 111 flaring events from 66 stars,\nincreasing the number of both flaring events and detected flare stars by an order of magnitude from\nthe previous SPT-3G data release. We provide cross-matching to Gaia DR3, as well as matches to\nX-ray point sources found in the second ROSAT all-sky survey. We have detected flaring stars across\nthe main sequence, from early-type A stars to M dwarfs, as well as a large population of evolved stars.\nThese stars are mostly nearby, spanning 10 to 1000 parsecs in distance. Most of the flare spectral\nindices are constant or gently rising as a function of frequency at 95/150/220 GHz. The timescale of\nthese events can range from minutes to hours, and the peak νLνluminosities range from 1027to 1031\nerg s−1in the SPT-3G frequency bands.\nKeywords: Stellar flares (1603), Millimeter astronomy (1061), Transient detection (1957), Transient\nsources (1851)\n1.INTRODUCTION\nFlaring activity in stars can be found across a wide\nrange of spectral types (Balona 2015) and is thought to\nresult from a release of magnetic energy. The Sun is\nalso known to flare, and has been extensively studied\nfrom radio wavelengths to gamma-rays (Fletcher et al.\n2011; Benz 2017) including the millimeter wavelengths\n(Silva et al. 1996) studied in this paper. Solar flares\nhave been used to serve as a framework for our under-\nstanding of flaring activities on other stars (Pettersen\n1989). While there are similarities in processes and\nthe physics between many solar and stellar flares (As-\nchwanden et al. 2008), the utility of these comparisons\nreaches a limit as the Sun is atypical when looking at\nlarger stellar populations. For example, even when con-\nsidering other “Sun-like” stars (stars with similarities\nsuch as spectral type, age, rotation period, and position\nalong the main sequence), the Sun has been shown to ex-\nhibit substantially lower photometric variability (Rein-\nhold et al. 2020). Typical solar flares have optical en-\nergies of ∼1029erg with a maximum of ∼1031erg in\nX-ray. On the extreme side, flares as strong as ∼1038\nerg at optical wavelengths have been observed on “Sun-\nlike” stars (Schaefer et al. 2000).\nThe generally accepted model of stellar flares (Brown\n1971; Hudson 1972) involves a sudden release of energy\ntriggered by reconnection of magnetic field lines, result-\ning in acceleration of charged particles that emit radia-\ntion across the electromagnetic spectrum (Allred et al.\n2015; Benz 2017). Different wavelengths track different\nphysics of the flare: charged particles are acceleratedfrom this area of reconnection in an impulsive event,\nseen simultaneously as gyrosynchrotron in radio wave-\nlengths and bremsstrahlung in hard X-rays (HXR, ≳10\nkeV) (Bastian et al. 1998; Massi et al. 2002; Benz &\nG¨ udel 2010). The particles travel along the field lines,\neventually depositing their energy lower in the stellar\natmosphere. This energy heats and evaporates plasma\nback into the upper atmosphere and can be seen in op-\ntical/ultraviolet (UV) (Krucker et al. 2015; Henry &\nNewsom 1996). As this plasma reaches the corona, the\nemission transitions from a sudden and fast rise in en-\nergy released to a gradual rise that can be seen in soft\nX-ray (SXR, ≲10 keV) as cooling starts (G¨ udel 2004).\nExternal interactions between stars can also produce\nflaring events. RS Canum Venaticorum variable stars\n(RS CVn) and other binary systems have shown flares\nacross the electromagnetic spectrum which can be due to\ninteractions in the magnetic field shared between compo-\nnents of the system (Catalano 1986; Massi et al. 2002).\nProtostars and young, pre-main sequence stars (e.g. T\nTauri) that still have an accretion disk can exhibit flares\ndue to magnetic field interactions between the star and\nthe disk itself (Feigelson & Montmerle 1999).\nAt radio wavelengths, solar flares typically show a\npeak at 1-10 GHz which is thought to correspond to\nmildly relativistic synchrotron radiation as the source\ntransitions from from optically thick to optically thin,\nwith the optically thin part of the spectrum extending\ninto millimeter wavelengths (Bastian et al. 1998). Peak\nflux spectra of solar flares show flux at 5 GHz to be\nover an order of magnitude larger than flux at 30 GHzSPT-3G Flaring Stars 3\n(Kundu & Vlahos 1982), with more recent observations\nexhibiting a similar falling spectrum between flux at 3.4-\n17 GHz and 86 GHz (Raulin et al. 1999) and even out to\n345 GHz (L¨ uthi et al. 2004). Rising spectra at submil-\nlimeter wavelengths have been occasionally observed in\nsolar flares (Zhou et al. 2011), with radiative processes\nsuch as bremsstrahlung and gyrosynchrotron requiring\nextreme parameters as possible explanations (Krucker\net al. 2013).\nStellar flare astronomy has historically involved tar-\ngeted observations of individual stars (e.g. Large et al.\n1989; Webber et al. 1973). Recent years have seen mas-\nsive growth in the field of time-domain astronomy with\nexoplanet detecting experiments Kepler (Koch et al.\n2010), and TESS (Ricker et al. 2014) providing an ob-\nserving cadence as high as 2 minutes, or even 20 seconds\nfor a small number of bright candidates. This high ca-\ndence data is invaluable for monitoring the variability\nof stars, and has resulted in enormous catalogs of stellar\nflares at optical wavelengths (Davenport 2016; G¨ unther\net al. 2020). Transient surveys have also been carried\nout at infrared (Davenport et al. 2012), UV (Brasseur\net al. 2019; Loyd et al. 2018a), and X-ray (Pye et al.\n2015; Getman & Feigelson 2021).\nAt longer wavelengths, radio observations of stellar\nflares have been mainly composed of targeted observa-\ntions (Loyd et al. 2018b; Smith et al. 2005) and recent\narchival surveys (Levinson et al. 2002). In the future,\nradio wavelength surveys such as SKA pathfinders, and\neventually SKA, will monitor the southern sky for radio\ntransients (Murphy et al. 2013).\nWhile observations of stellar flares at or near millime-\nter wavelengths have been made in the past (Burton\net al. 2022; MacGregor et al. 2021, 2020, 2018; Massi\net al. 2006; Umemoto et al. 2009; Brown & Brown 2006;\nBeasley & Bastian 1998; Mairs et al. 2019), there have\nnot been any dedicated transient surveys at these wave-\nlengths until those conducted with the South Pole Tele-\nscope (SPT Guns et al. 2021; Whitehorn et al. 2016)\nwhich yielded 13 stellar flares from 8 stars, and the At-\nacama Cosmology Telescope (ACT Li et al. 2023; Naess\net al. 2021) which yielded a total of 17 stellar flares from\n14 stars.\nStellar flares detected by SPT and the ACT have been\nin the range of νLν∼1026−1030erg s−1at 150 GHz\nwith durations on the scale of minutes to hours and spec-\ntral indices typically in the range of -1 to +2. These\nflares are orders of magnitude brighter and longer than\nrecent ALMA observations of millimeter stellar flares\nprovided in Table 1 of Burton et al. (2022): 10 flares\nfrom the stars AU Mic, Proxima Centauri, and Epsilon\nEridani which have luminosities of νLν∼1024−1026erg s−1with durations on the scale of seconds. Burton\net al. (2022) also report a wider range of spectral in-\ndices for these ALMA flares, ranging from roughly -3 to\n7. Solar flares as a comparison are even weaker: one of\nthe brightest solar flares ever recorded – an X28 class\nflare detected on November 4, 2003 – had a 212 GHz\nluminosity of νLν∼1023erg s−1(Zhou et al. 2011).\nIn this paper we perform a non-targeted search us-\ning data from the 2019–2022 observing seasons of the\nSPT-3G survey and present a catalog of 111 flares from\n66 stars, an increase of almost a factor of 10 over the\nprevious transient analysis of the 2020 observing sea-\nson presented in Guns et al. (2021). Providing such a\nlarge sample of these very strong millimeter flares can\nhelp improve our understanding of the physics during\nthe impulsive particle acceleration phase of stellar flares.\nM Dwarfs are of particular interest from the standpoint\nof exoplanet habitability (Shields et al. 2016), and mil-\nlimeter emission as a proxy for FUV emission during\nstellar flares may help constrain the radiation environ-\nment around exoplanets (MacGregor et al. 2021). Ad-\nditionally, characterizing this population of flaring stars\nwill be useful for future millimeter wavelength transient\nsurveys, such as CMB-S4 (Abazajian et al. 2019) and\nthe Simons Observatory (SO) (Ade et al. 2019).\nThe format of the paper is as follows: mapmaking\nand filtering along with transient detection methods are\ndescribed in Section 2. The flare star catalog, full 4-\nyear single-observation lightcurves at 95 and 150 GHz\nfor each associated source, and single-scan lightcurves\nat 95, 150, and 220 GHz of each flaring event are pro-\nvided in Section 3, as well as characterizations of the\nstellar populations and analysis of flaring properties. A\nconclusion including directions for further study is de-\nscribed in Section 4.\n2.METHODS\n2.1. Instrument and Survey\nThe SPT is a 10-meter telescope located at\nAmundsen-Scott South Pole Station, Antarctica (Carl-\nstrom et al. 2011). The SPT-3G receiver is the third\ncamera to be mounted on the SPT and has been op-\nerational since 2017. The SPT-3G focal plane con-\ntains 2690 dual-polarization, tri-chroic pixels sensitive\nin three frequency bands centered at 95 GHz, 150 GHz,\nand 220 GHz. Each pixel contains six bolometric de-\ntectors, for a total of ∼16000 detectors read out using\na frequency-domain multiplexing system. The large 10\nmeter primary on the SPT allows for high-resolution ob-\nservations, with beam sizes on the order of 1′FWHM\n(Sobrin et al. 2022).4 Tandoi et al.\nThe data used in this work are from four years of the\nSPT-3G winter survey, which comprises a 1500 square\ndegree patch of the southern sky and is observed annu-\nally from late March through November with an average\ncadence of 12 hours. The SPT-3G winter field spans a\nrectangle in sky coordinates from 20h40m0sto 3h20m0s\nin right ascension (R.A.) and from −70◦to−42◦in decli-\nnation (decl.). The field is split horizontally into 4 equal\nheight subfields. A single SPT-3G observation consists\nof a series of scans across a given subfield that last ap-\nproximately 2 hours. During an observation, the SPT\nrasters back and forth across the sky at constant ele-\nvation and scan speed (constant on the bearing, i.e. in\nazimuth angle per second) before taking a 12 .5′step in\nelevation and repeating the process, until the entire sub-\nfield has been observed. A full description of the design,\noperation and performance of the SPT-3G camera can\nbe found in Sobrin et al. (2022).\nWe use a set of SPT-3G winter survey maps that\nhave been optimized for source finding. Each single-\nfrequency, single-observation subfield map is made by\nfiltering the detector time-ordered data and using indi-\nvidual detector pointing to bin the data into a pixelated\n2D skymap with 0 .25′pixels in a Lambert azimuthal\nequal-area projection. The map filtering is described in\ndetail in Dutcher et al. (2021), and we follow the same\nprocedure but with different high-pass and low-pass cut-\noff values for the time-domain filter, which are set to\nfilter out modes with spatial scales below angular mul-\ntipoles of ℓ= 500 and above ℓ= 20,000. Bright point\nsources (defined as having average flux ≥50 mJy at\n150 GHz) are linearly interpolated over before the time\ndomain filter in order to suppress filtering artifacts, and\nthe interpolated samples are removed from the search.\nSimilar to Guns et al. (2021), we subtract a three-year\naverage map from each individual observation in order\nto remove the cosmic microwave background (CMB),\ngalaxy clusters, static point sources, and other static\nbackgrounds. The background-subtracted maps thus\ncontain a combination of red atmospheric noise that\ndominates at large angular scales (typically ℓ <2000,\nbut up to ℓ= 3000 for adverse weather conditions),\ndetector white noise, and time-varying point source sig-\nnals. We apply a map-space pixel mask to remove a 5′\nradius disk around all 2465 known point sources with\n3-year average fluxes of 5 mJy or higher at 150 GHz\nfrom the search area. In order to remove the atmo-\nspheric noise, we apply an isotropic 2D high-pass filter\nas a weighted convolution in map space (using inverse-\nvariance weights), and filter the resulting white noise-\ndominated maps with a Gaussian beam template to\nmaximize sensitivity to point sources. The beam tem-plate is constructed as a Gaussian fit to the central lobe\nof the SPT-3G beams, with widths σbeam = 0.67′,\n0.48′, and 0 .45′at 95 GHz, 150 GHz, and 220 GHz. The\nisotropic high-pass filter kernel is an annulus with an in-\nner radius of 3 σbeam and an outer radius of 5′at 95 GHz\nand 3′at 150 GHz and 220 GHz. More aggressive fil-\ntering is required at higher frequencies because of the\nrising atmospheric noise spectrum. Unlike static SPT\nsource analyses, which use multi-year averaged maps\nand carry out all map-space filtering operations in the\nFourier domain, we choose to filter in real space be-\ncause the pixel weights in our single-observation maps\nare non-uniform (varying by as much as 20% across the\nmap) which is properly accounted for with a weighted\nreal-space convolution filter. The time-domain filter im-\nprints additional large-scale structure onto the beam,\nadding degree-scale filtering artifacts along lines of con-\nstant declination. As the computing time for a real-\nspace convolution increases linearly with the kernel area,\nwe choose not to account for this large-scale structure.\nCompared to the full-size filter kernel, this choice sac-\nrifices ∼3.5% in optimality in order to speed up the\nfiltering operations by an order of magnitude.\nWe summarize the pixel noise for each single obser-\nvation map by taking the median noise across all the\npixels in the map. The results are shown in Figure\n1. Across all observations in all subfields, we achieve\nmedian noise values of 7 .8 mJy root-mean-square (rms)\nat 95 GHz and 8 .0 mJy rms at 150 GHz. The median\nnoise at 220 GHz is much higher at ∼27 mJy rms. As\na result we do not include the 220 GHz maps in our\ntransient finding method and only use them to measure\nfluxes of sources detected in the other two observing\nbands. The constant scan speed in azimuth results in\na slower on-sky scan speed and longer observation time\nper map area with increasing elevation (decreasing dec-\nlination). This, together with higher atmospheric load-\ning (and corresponding higher detector noise), results\nin higher map noise at lower elevations. To achieve\nuniform noise across the whole winter survey field on\nlong timescales, the lower elevation fields are observed\nmore frequently. Although the cadence averages out to\none full field observation every 12 hours, the actual ca-\ndence consists of clusters of 2 or 3 observations of the\nsame subfield followed by either switching to a differ-\nent elevation or a period of downtime to cycle the cryo-\ngenic fridge and observe calibration sources. As a con-\nsequence, the actual time between subsequent observa-\ntions of the same source can range from 2 hours to ap-\nproximately 36 hours.\nThe non-uniform observing cadence and uneven sub-\nfield coverage between re-observations of the same sub-SPT-3G Flaring Stars 5\n0100200300count-44.75°\n-52.25°\n-59.75°\n-67.25°\n5 6 7 8 9 10 11 12\nPixel noise [mJy]0100200300count\nFigure 1. Histograms of noise values for every observa-\ntion processed in the archival transient pipeline, grouped by\nsubfield. Histograms are colored by subfield and referred to\nby the center declination of that subfield. The noise value\nfor each observation is the median pixel noise across all the\npixels in the corresponding skymap. The top and bottom\npanes show the noise values for the 95 GHz and 150 GHz\nmaps, respectively. Dotted colored lines mark the median\nnoise across all observations within a given subfield, while\nthe dashed black line shows the overall median across all\nobservations.\nfield make it necessary to use simulations to calculate\nthe observing efficiency for bright transients of specific\ndurations. By injecting simple tophat flares brighter\nthan the single-observation detection threshold, we find\nthat our instrument and observing strategy detects ap-\nproximately 80% of 24 hour flares, 20% of 4 hour flares,\nand less than 5% of flares in our field lasting 30 minutes\nor less.\n2.2. Transient Finding\nThe transient finding method is described in detail in\nGuns et al. (2021) and we provide a brief overview here\nas well as noting any pertinent changes. For each un-\nmasked location on the pixelated sky, we construct a se-\nries of two-band lightcurves spanning consecutive 12-day\nintervals and using only 95 GHz and 150 GHz data. The\n12-day interval covers short-duration signals of interest –\nmillimeter stellar flares from previous and ongoing anal-\nyses (i.e. Guns et al. 2021) as well as predicted short\nduration extragalactic millimeter transients (Eftekhari\net al. 2022). We apply a preliminary cut on the data,\nselecting all lightcurves which show a >3σexcursion in\nboth bands at the same point in time. To each remaining\nlightcurve we fit a four-parameter Gaussian flare model\nwith two independent flux parameters S95andS150at\n95 and 150 GHz, a peak time t0, and an event full width\nat half maximum (FWHM) w. The Test Statistic (TS)is constructed by maximizing the likelihood ratio of the\nbest-fit flare model to the null model (zero amplitude\nin both bands), with an additional logarithmic penalty\nterm. This penalty term corrects for a likelihood bias to-\nwards short flare widths: since there are many more un-\ncorrelated starting times for short flare widths than for\nlong ones, the effective volume of searchable parameter\nspace grows as the flare width decreases. The logarith-\nmic penalty term effectively imposes a flat prior on the\nflare duration in order to correct for this effect (Braun\net al. 2010).\nThe TS is thus defined as:\nTS = 2 ln L\u0010\nˆS95,ˆS150,ˆt0,ˆw\u0011\n−2 lnL(0)+2 ln\u0012ˆw\n12 days\u0013\n,\n(1)\nwhere hatted quantities denote best-fit parameters as\nfound by a non-linear optimizer using the Nelder-Mead\nsimplex method, as implemented in the GNU Scientific\nLibrary (Galassi et al. 2009).\nFocusing our search on short-duration events, we im-\npose a significance cut of TS >45 and a best-fit FWHM\nduration <10 days. The TS cut is set by demanding\na negligible background of false detections from back-\nground noise ( ≪1 events over four years [S. Guns et al,\nin preparation]). In the short duration regime, the main\nsystematic background for SPT transients are weather\nballoons that are launched daily by the South Pole Me-\nteorological Office (Met). Through a data sharing agree-\nment established in 2022, SPT now receives Met GPS\ndata soon after each balloon launch and cuts any tele-\nscope scans that intersect with a weather balloon trajec-\ntory, vastly reducing the number of detected balloons in\nthe SPT-3G transient detection pipeline at the cost of a\nnegligible reduction in data volume. The balloon avoid-\nance system is not infallible however, and fails when the\ntracking data arrives too late to be integrated into the\nSPT-3G data processing pipeline, or when the balloon\nruptures at high altitude and debris that is not tracked\nby the balloon’s GPS module enters the SPT-3G field\nof view. Remaining balloon detections number approxi-\nmately one per month and are identified by their obser-\nvational signature: due to their fast movement they are\ndetected with a short duration in single telescope scans\n(lasting 2 or fewer seconds), high brightness ( >1 Jy),\nthermal or steeper-than-thermal spectra, and within 2\nhours of a logged balloon launch.\nAfter removing events with best-fit durations >10\ndays and TS <45 we are left with a dataset of can-\ndidate events which we inspect by eye. We manually\nanalyze the lightcurves and thumbnails to reject dupli-\ncate observations of the same event, filtering artifacts\nfrom bright nearby transients, untracked balloons, and6 Tandoi et al.\n11 13 15 17 19 21 23\nApril 201920\n020406080100Flux [mJy]\n0 10 20\nTime since start of scan [minutes]0200400600Flux [mJy]\n150GHz\n95GHz\n30\"\n10\n5\n 0\n 5\n 10\n10\n5\n0\n5\n10\n2019-04-19\n 04:37:36\n10\n5\n 0\n 5\n 10\n2019-04-19\n 07:05:38\n10\n5\n 0\n 5\n 10\n2019-04-20\n 19:55:03\n20\n020406080\nFlux [mJy]\n10\n5\n 0\n 5\n 10\n10\n5\n0\n5\n10\n2019-2021\ncoadded background\n2\n02\nFlux [mJy]\nFigure 2. Example lightcurves and thumbnails for the SPT event on MJD 58592.3. Top left : Lightcurves for the highest\nTS pixel on the triggered event. The main plot shows the peak single-observation flux while the inset shows the single-scan\nflux within the peak single-observation. Top right : unWISE 3.4 µm W1 grayscale image (Lang 2014) with blue SPT 150 GHz\nflux density contours in 5 σsteps from the peak signal. Bottom : 150 GHz thumbnails for the single-observation data points\n(immediately before, during, and immediately after the triggered event), and the three-year average map that has been subtracted\nto create these thumbnails. These thumbnails show that this flaring star is undetectable in single SPT-3G observations at its\nquiescent level.\ndetector glitches (large map streaks in the scan direc-\ntion). Events with any ambiguity are excluded from\nfurther analysis. After these cuts, we keep 111 short du-\nration ( <10 day) events with robust morphologies that\nwe use to cross-match to the Gaia Data Release 3 (DR3)\ncatalog (van Leeuwen et al. 2022).\nFigure 2 shows an example event detected by the tran-\nsient pipeline. The main lightcurve plot shows the data\nthat is input into the lightcurve fitter and the result-\ning Gaussian fit, with a TS of 194.8. Each data point\nrepresents the average flux over a 2-hour SPT subfield\nobservation, in which any given point on the sky is ob-\nserved for about 8 consecutive left/right scans (16 scans\ntotal) over the course of 20 minutes. We construct post-\ndetection lightcurves of the single-scan fluxes (each scan\nrepresenting one pass of the focal plane, or roughly 2\nseconds of integrated data) that make up the peak ob-servation of the flare, shown as an inset to the main\nlightcurve. Figure 3 shows thumbnails of the single-\nscans for this event. Thumbnails of scans 1, 2, 19, and\n20 are discarded because the sky location of interest is\nnot observed by the focal plane. Scans at the beginning\nand end (such as scans 3 and 4) are noisier due to only\npart of the focal plane seeing the sky location of interest.\nIn order to localize events at finer scales than the 15′′\npixel grid, we determine an event centroid location from\nthe TS map associated with each flare. The TS map is\nconstructed by fitting the lightcurve model to each pixel\nin a 5x5 pixel (1 .25′x1.25′) box centered on the event.\nThe TS map combines all the available information from\nevery time point and both frequency bands (under the\nassumption of a Gaussian flare model). For each event,\nwe fit a cubic spline to to the TS map and choose the\nspline maximum as the event location.SPT-3G Flaring Stars 7\nscan 1\n scan 2\n scan 3\n scan 4\nscan 5\n scan 6\n scan 7\n scan 8\nscan 9\n scan 10\n scan 11\n scan 12\nscan 13\n scan 14\n scan 15\n scan 16\nscan 17\n scan 18\n scan 19\n scan 20\n100\n0100200300400500600\nFlux [mJy]\nFigure 3. 15′′x15′′single-scan 150 GHz thumbnails cen-\ntered on the highest TS pixel from the event in Figure 2.\nThe source is only clearly visible in scans 5 and 6, and has\nreturned to near noise flux values in scans 7 and 8.\nThe uncertainties in the SPT R.A. and decl., σRAand\nσDec, come from a combination of systematic pointing\nuncertainty and statistical uncertainty in localizing a\nsource given the SPT beam and noise:\nσ2\nRA= 3.73′′2+\u001250′′\n√\nTS\u00132\n, (2)\nσ2\nDec= 4.35′′2+\u001249′′\n√\nTS\u00132\n. (3)\nThe first term is the systematic contribution to the er-\nror budget and comes from uncertainties in the pointing\nof the physical instrument, whereas the second term is\nthe statistical contribution, and is sourced by the noise\nin the observations. We estimate the first term by mea-\nsuring the positions of one year of daily observations\nof bright point sources and comparing to a reference\ncatalog. The second term is calculated from injecting\nN=10,000 simulated sources of various amplitudes at\nknown locations in the SPT field and recovering the\nmeasured position. In a small fraction ( <1%) of cases,\nheavy winds and other inclement weather leads to sys-\ntematic pointing errors that are much larger than the\nmean estimate in Equations 2 and 3. In those cases, we\nestimate the systematic pointing error from the bright\n105\n104\n103\n102\n101\np100101countFigure 4. The distribution of p-values from matching\nSPT flaring events to stars in the Gaia DR3 catalog. The\n5 events at p >0.02 have been investigated and are discussed\nin Section 3.8.3.\npoint source positions as measured in that observation\nonly.\n3.RESULTS\n3.1. Stellar Association\n3.1.1. Candidate Sources\nTo find candidate stars for our events we only consider\nGaia sources within our observing field that have a max-\nimum G magnitude of 20 and exclude sources that are\nfound in the qso candidates or galaxy candidates tables\nin Gaia DR3 to eliminate known extragalactic sources.\nThis leaves about 7.3 million stars as possible matches\nfor SPT-3G transients. X-ray associations with sources\nfrom the second ROSAT all-sky survey (2RXS) are pro-\nvided by the main table of the ROSAT stellar content\ncatalog (Freund et al. 2022), matching only on Gaia DR3\nsource ID. For Gaia stars without a match in the stel-\nlar content catalog, we check the full 2RXS point source\ncatalog (Boller et al. 2016) for any sources within 1′and\nhave separately distinguished them.\n3.1.2. Cross-Matching Algorithm\nTo associate flaring events with sources, the cross-\nmatching algorithm we use compares the likelihood of\nan event originating from a candidate source to the like-\nlihood of a random event having a chance association,\nusing a 1◦radius centered on each candidate to deter-\nmine a local source density in the Gaia catalog. This\nresults in the figure of merit in association Λ:\nΛ = ln\u0012\nBNtotal\nN(>S)\u0013\n, (4)\nwhere Ntotalis the number of sources within the local\narea, and N( >S) is the number of sources within the8 Tandoi et al.\nColumn Label Units Description\n1 spt id IAU-approved SPT source name, corresponding to the SPT\ndetected location.\n2 mjd [days] The approximate time in MJD corresponding to the beginning\nof the observation the detection took place in.\n3 ts The test statistic of the event, used to flag a possible flaring\nevent occurring during that observation. Minimum values are\n45, with a higher value having a more significant detection.\n4,5 ra [deg] The R.A. and Declination (J2000) of the SPT event.\ndec\n6-11 95 flux [mJy] Flux values and their errors for the three SPT bands.\n95fluxerr\n150flux\n150fluxerr\n220flux\n220fluxerr\n12-15 spectral index 95150 2-band spectral index values and their errors for the 95/150\nspectral index 95150err and 150/220 SPT bands.\nspectral index 150220\nspectral index 150220err\n16 dr3 source id Gaia DR3 unique source identifier.\n17 p The probability of a random event cross-matching to this\nsource.\n18,19 source ra [deg] The R.A. and Declination (J2000) of the Gaia DR3 source.\nsource dec\n20-22 phot gmean mag Magnitudes of the three Gaia passbands.\nphot bpmean mag\nphot rpmean mag\n23,24 parallax [mas] Parallax and error of the Gaia source.\nparallax err\n25 binary Boolean flag for binary candidacy.\n26 2rxs id 2RXS identifier.\n27 2rxs ref Flag for source association: “Freund” for an established asso-\nciationa, or “DR3/2RXS Proximity” for nearby sources.\n28 2rxs flux [mW m−2] 2RXS flux.\naFreund et al. (2022)\nTable 1. Columns and descriptions for the SPT-3G flare star catalog.\nlocal area that are brighter in Gaia magnitude than the\ncandidate source. Bconsiders the uncertainties of the\nSPT beam and the angular separation:\nB= exp\u0014\n−1\n2\u0012\n(∆RA\nσRA)2+ (∆Dec\nσDec)2\u0013\u0015\n. (5)\n∆RAand ∆ Decare the angular separations in R.A.\nand decl. between the transient centroid location and\ncandidate source in the Gaia catalog while σRAandσDec\nare defined in Equations 2 and 3.\nWe create a probability distribution function for ran-\ndom matches by sampling 400,000 random points in the\nSPT-3G field and matching them to Gaia using this al-\ngorithm. We then construct a cumulative distribution\nfunction (CDF) based on the distribution of Λ values for\nthis random sample, and then create a one-to-one map-ping of Λ to p-values in the form of p(Λ) = 1 −CDF(Λ).\nWe interpolate this mapping to find the p-value corre-\nsponding to the observed Λ for each event, telling us\nthe chance of a random event having that same associ-\nation value or higher. The distribution of p-values for\nour events is shown in Figure 4.\n3.2. SPT-3G Flare Star Catalog\nAs described in Section 2.2, this flare star catalog is\na result of searching only for short-duration transient\nevents in SPT-3G archival data, using a cutoff of 10 days\non the best-fit FWHM duration. With our current anal-\nysis we have recovered 12 of the 13 stellar flaring events\nin Guns et al. (2021), and include them in the flare star\ncatalog. The missing flaring event, previously referred\nto as event 6(a) by the star CC Eri, was a weak flareSPT-3G Flaring Stars 9\n0 1 2 3 4 5\nGBP GRP\n0\n2\n4\n6\n8\n10\n12\n14MG\nEvolved\nUpper main sequence\nLower main sequence\nOutlier main sequence\n101102103104\nDistance [pc]\nFigure 5. Left : CMD of SPT flaring stars, with contours showing density of Gaia stars in the SPT-3G winter field. Right :\nMagnitude vs distance of the same populations. The two inverted triangles are outlier main sequence stars and considered to\nbe incorrect matches, which are discussed in Section 3.8.3.\ndetected at 29/21 mJy in the 95/150 GHz bands with re-\nspective luminosities of 4.4 ×1026/5.0×1026erg s−1and\na TS of 29.5, below the current detected threshold of\n45. The other flares from CC Eri were successfully re-\ncovered. The previous analysis in Guns et al. (2021)\nhad a detection threshold of TS = 100. Any location\nin the sky exhibiting a flare with a TS ≥100 was then\nmanually scrutinized over the entire observing season to\nsee if there were smaller flares below the threshold that\nwere not detected. This method was how event 6(a),\nat TS = 29 .5, was found originally. We do not search\nfor below-treshold flares in the current analysis and only\nshow events above the new TS = 45 threshold. The two\nlong duration events from Guns et al. (2021) are longer\nthan the 10 day best-fit FWHM cut and thus were not\nrecovered.\nTable 1 shows the full column names and descriptions\nfor the flare star catalog. A full machine readable table\ncan be found at https://pole.uchicago.edu/public/data/\ntandoi24/.\nAfter matching with the external catalogs described in\nSection 3.1.1, the full 4 year SPT flare star catalog has\nassociated all 111 flaring events with 66 unique stars.\nWe find that 17 stars have flared multiple times and are\nresponsible for 62 events, leaving 49 stars with single\nevents associated with them. Distances for stars are\nderived from Gaia DR3 parallax when available. 5 stars,\nassociated with 6 events, have no reported parallax fromGaia DR3 ID Parallax\n[mas]Distance\n[pc]Source\n4914632365579875200 21a\n4726387007013852672 59 ±1b\n6580179279486143744 44 ±2c\n4850405974392619648 44 ±1d\n4942856382389905408 19 ±4e\naRiedel et al. (2014)\nbvan Leeuwen (2007)\ncBell et al. (2015)\ndStassun et al. (2019)\neFinch et al. (2014)\nTable 2. Alternate sources for parallax or distance for stars\nthat have missing parallax in Gaia, with rounded values.\nGaia. Table 2 shows alternate sources of parallax, or\ndistance if parallax is not available.\nFigure 5 shows the population of SPT detected flare\nstars with Gaia parallax in a color-magnitude diagram\n(CMD), and a magnitude vs distance plot. Differ-\nent marker shapes denote our classifications of spectral\ntypes with “Evolved” containing giants and subgiants,\n“Upper main sequence” containing early-type and sun-\nlike stars, and “Lower main sequence” mainly composed\nof M dwarfs. These classifications are based purely on\nposition in the CMD. We note that these classifications\nmay be inaccurate due to the possibility of misrepresent-10 Tandoi et al.\n1.5\n 1.0\n 0.5\n 0.0 0.5 1.0 1.5 2.0 2.5\n95\n150\n8\n6\n4\n2\n024150\n220\n100101102103\nS150 [mJy]100101102103S220 [mJy]\n100101102103\nS95 [mJy]100101102103S150 [mJy]\nSelf-absorption (=2.5)\nRayleigh-Jeans limit (=2)\nFlat (=0)\nOptically thin\n(=0.75)\nEvolved\nUpper main sequence\nLower main sequence\nconstant \n220GHz>3\n220GHz>3 median\nAll 220GHz\nAll 220GHz median\nFigure 6. Left : Spectral indices for all stellar flare events, with KDEs of the probability densities along the insets. Shaded\nregions mark separate quadrants of positive and negative values of alpha, while the dotted line indicates a spectral index that\nis constant in frequency. Events with 220 GHz SNR <3σhave their y-error bars omitted and marker sizes reduced for visual\nclarity. These spectral indices are within the typical values for synchrotron radiation more commonly seen at radio wavelengths.\nRight : Comparisons of flux densities for all stellar flare events, across the three SPT bands. Typical spectral indices of relevant\nphenomena are shown as lines.\ning a binary system which we discuss in Section 3.8. In\naddition, the two main sequence outlier stars which we\nshow in Section 3.8.3 are likely misidentifications and\nare not used in analysis.\nM dwarf stars, having a Gaia GBP−GRPcolor of\n>∼2.1 are interesting due to both their tendency for\nlarge and frequent flares (France et al. 2016) and abun-\ndance — accounting for ∼75% of stars in our local\nneighborhood (Bochanski et al. 2010). The M dwarf\npopulation that the SPT has seen flare is largely above\nthe main sequence, indicating these could be young stars\nthat have not yet settled onto the main sequence.\nEarly type stars (spectral types B5-F5 with a Gaia\nGBP−GRPcolor of <∼0.5) are also represented in the\nSPT flaring population. While initially believed to not\nexhibit flares, there has been recent evidence of flaring\nactivity in early type stars at other wavelengths (Balona\n2012; Van Doorsselaere et al. 2017). The upper main se-\nquence stars that have been detected flaring by SPT are\ngenerally closer in distance than typical main sequencestars. This could indicate that flaring in these types of\nstar could be very common, and SPT is preferentially\ndetecting the sample that happens to be nearby.\n3.3. Spectral Index\nSpectral indices for the events are shown in the main\nplot in Figure 6. Events are separated by their 220 GHz\nSNR, with a focus on events with strong ( >3σ) detec-\ntions. Events with weak 220 GHz SNR have their y-error\nbars omitted and marker sizes reduced for visual clarity.\nThe spectral indices are defined through the relation:\nSν∝να, (6)\nsuch that\nαν1\nν2=log(S1/S2)\nlog(ν1/ν2). (7)\nTo represent probability densities of the observed\nspectral indices, we have calculated kernel density es-\ntimations (KDEs) for two populations: a KDE for theSPT-3G Flaring Stars 11\n102103\nS150GHz [mJy]103\n102\nN(>S) obs1\nLower main sequence\nUpper main sequence\nEvolved\nS3/2\n1027102810291030\n L,150GHz [erg s1]\n104\n103\n102\nN(>L) obs1\nEvolved median\nUpper ms median\nLower ms median\nFigure 7. Cumulative number counts per SPT observation for flaring events at 150 GHz. Left: Number counts as a function\nof source flux densities. Dashed lines are examples of the Euclidean source count expected power law of S−3/2.Right : Number\ncounts as a function of peak flare luminosities. Vertical lines show the median luminosity value for each population of stars.\nsubset of events with strong 220 GHz detections as well\nas a KDE for the entire population of flaring events re-\ngardless of 220 GHz SNR, in both α95\n150andα150\n220. Gaus-\nsian kernels are used with the width set by the uncer-\ntainty on the spectral index measurement. We show\nthese KDEs along the x- and y-insets, as well as median\nspectral index values for the two populations.\nMost of the events detected by SPT cluster around\nthe constant spectral index line, with median spectral\nindices of α≈0 to 0.5. Measurements of αgenerally do\nnot exceed 5/2 which is the spectral index for optically\nthick synchrotron radiation (Pacholczyk 1970), but most\ndo not seem consistent with a falling spectrum expected\nfor the optically thin side of synchrotron radiation that\npeaked at radio wavelengths around ∼5 GHz.\nIn the flux subplots, spectral indices with typical val-\nues in the radio regime are marked with black lines.\nα=−0.75 represents an optically thin synchrotron\nsource, α= 0 represents a flat spectral index, α= 2\nrepresents the Rayleigh-Jeans tail of thermal emission,\nandα= 5/2 represents an optically thick synchrotron\nsource due to self-absorption. The SPT detected events\nfall within the bounds of typical synchrotron spectra for\nα95\n150as well as α150\n220, suggesting this as a plausible emis-\nsion mechanism.\n3.4. Number Counts\nFigure 7 shows cumulative number counts of both flux\ndensity and 150 GHz peak flare luminosity, split for the\nthree populations of stars we have established.\nThe flux density number count can be approximated\nas a power law of N(> S)≈S−3/2where the index\ncomes from assuming a uniform density of objects within\na sphere: N∝r3andS∝r−2. This approximation is\nshown as a series of dashed lines in the flux density num-ber count plot. At distances that exceed the thickness\nof the galactic disk the power law approximation will\nbreak down. The lower main sequence stars are <100\npc in distance and are not detected outside of the disk.\nHowever, the evolved stars are seen at distances up to 1\nkpc and may be detected at distances where a drop in\ndensity is expected.\nThe luminosity number count shows that the main se-\nquence stars have similar luminosity distributions, while\nthe evolved stars have much brighter flares as well as\nhigher flaring rates in comparison. Incorporating the\ntransient detection efficiency results of flares lasting 30\nminutes or less as discussed in Section 2.1, we find an ex-\npected flaring rate N( >L) deg−2yr−1of 0.2/0.3/0.4 for\nthe upper main sequence/lower main sequence/evolved\npopulations of stars at a νLν150 GHz luminosity of 1027\nerg s−1.\n3.5. X-ray\nStudies of the Sun and other active main sequence\nstars have shown that a linear relationship exists be-\ntween soft X-ray (SXR) emission and magnetic flux\n(Pevtsov et al. 2003), linking thermal coronal emission\nto magnetic fields. Flaring rates and luminosity have\nbeen linked to quiescent SXR emission (Audard et al.\n2000) as well as other proxies for stellar magnetic activ-\nity such as H αemission and rotation period (Davenport\n2016; Yang et al. 2017).\nFigure 8 shows the SPT flaring stars at 150 GHz along\nwith quiescent SXR (0.1-2.4 keV) from 2RXS in lumi-\nnosity, as well as a histogram of the ratios of luminosity.\nStars that are not associated with any 2RXS sources\nare shown as faint markers at an approximate upper\nlimit of 1 .5×10−13erg s−1. The upper limit is only a\nrough estimate: the exposure time strongly varies and12 Tandoi et al.\n102710281029103010311032\nLX [erg s1]\n10271028102910301031L,150GHz [erg s1]\nEvolved\nUpper main sequence\nLower main sequence\n2RXS proximity\n101\n100101102\nLX/L,150GHz\n02468CountEvolved\nUpper main sequence\nLower main sequence\n2RXS non-detection upper limits\nFigure 8. Left : Comparison of mm flaring and X-ray quiescent luminosities. A dotted line shows equal luminosities, and\nSPT-detected flare stars without a 2RXS association are shown as transparent markers at the luminosity corresponding to the\nmean X-ray flux upper limit of 1 .5×10−13erg s−1cm−2calculated in Freund et al. (2022). Right : A histogram of luminosity\nratios for the three populations of stars that have 2RXS associations through Freund et al. (2022) and proximity to objects in\nBoller et al. (2016). Stars without an X-ray association are binned by their upper limits and are marked with arrows pointing\nleft. These plots show a wide spread in the ratio of quiescent SXR to flaring mm-wave luminosities.\nthus no uniform detection limit exists (Freund et al.\n2022). Additionally, four sources not matched in Freund\net al. (2022) but within 1′of sources included in Boller\net al. (2016) are shown with a dashed circle around the\nmarker. Their flux ( FX) is calculated following Schmitt\net al. (1995) using the hardness ratio (HR) and photon\ncount rate (ctr):\nFX= (5.30HR + 8 .31)ctr ×10−12ergs cm−2s−1.(8)\nWe have used this equation to recover the same flux\nvalues for sources given in Freund et al. (2022), using the\nhardness ratios and photon count rates supplied within.\nConsidering only the SPT-detected flare stars with an\nassociation in 2RXS (i.e., excluding the matches within\n1 arcmin proximity), the percentage of stars that are X-\nray bright is 54.5%. For comparison, we select a popula-\ntion of 330 stars that are representative of the SPT flar-\ning star population. Using the parameters of magnitude\nand distance (see the right plot in Figure 5), we selected\n5 nearest neighbors for each SPT flare star. The nearest\nneighbor stars have a percentage of X-ray bright stars of\n4.8%. These rates provide evidence that stars observed\nto flare at mm wavelengths are much more likely to be\nX-ray bright than non-flaring stars.\nThe histograms in Figure 8 show a large scatter in the\nratio of quiescent SXR luminosity to mm-wave flaring\nluminosity for the population of stars with 2RXS asso-ciations. Simultaneous observations of SXR and mm-\nwave emission during flaring states may reveal a tighter\ncorrelation between these luminosities.\n3.6. Single-Scan lightcurves\nFlare morphologies on short timescales can tell us\nabout the physics of the flare. However, binning or\ngaps in data can obscure important features of these\nflares. At optical wavelengths, higher cadence observa-\ntions from TESS have shown a degeneracy in data even\nat the 2 minute scale with subsequent pulsations after\nthe peak and during the decay phase possible in a larger\nportion of the population (Howard & MacGregor 2022).\nLightcurves at radio wavelengths typically describe\nthe impulsive phase of the flare, but a phase of grad-\nual change at these wavelengths has also been de-\ntected (Zhang et al. 2018). The impulsive stellar flare\nlightcurve seen in radio, optical, and HXR wavelengths\nis characterized by a sharp rise followed by an exponen-\ntial decay with some small complexities occurring dur-\ning both phases (Davenport et al. 2014; Kowalski et al.\n2019). Different types of stars can also exhibit unusual\nflare morphologies: flares from RS CVn binary systems\nhave longer rise times and sustained peak emission that\nplateaus before decaying (Osten & Brown 1999).\nFlaring activity can also trigger additional flares in\nneighboring active regions of the star in a process known\nas sympathetic flaring. Sympathetic flaring has beenSPT-3G Flaring Stars 13\n0 10 20 300200400600Flux [mJy]MJD 59110.27,  DR3 4727920241619206912\n0 10 20 30MJD 58592.3,  DR3 4738200056863233536\n150GHz\n95GHz\nTime since start of scan [minutes]\nFigure 9. Possible stellar flare models on well-constrained single-scan lightcurves. As shown in the appendix, most flares\ndetected by SPT are not as clear in their structure. Both of these lightcurves are of flares from lower main sequence stars. Left:\nA simple Gaussian model showing a gradual rise in flux that is comparable to the decay period. Right : A sharp Gaussian rise\nwith a longer exponential decay.\nobserved on the Sun (Pearce & Harrison 1990), with ev-\nidence of this phenomena happening in other stars (An-\nfinogentov et al. 2013). This could explain the appear-\nance of a complex flare as the superposition of separate\nflares on the surface of the star.\nMost descriptions of the duration of stellar flares at\nshorter wavelengths benefit from higher cadence obser-\nvations, longer times spent viewing the star, or some\ncombination of both. Due to the SPT scanning strat-\negy, we are limited to the binned single-scan data points\nroughly 100 seconds apart on average with the actual\ntime spent on a given sky location depending on where\nthat location falls in the focal plane’s approximately oval\nshape. As these are not targeted observations, we rarely\nobserve the duration of a flare in its entirety. Such ob-\nservations can only give us a lower limit on the duration\nof these events.\nFigure 9 shows lightcurves of 2 events with clear mor-\nphologies, including including a long Gaussian rise time\nand an exponential decay. Visual inspection resulted in\naround 15 detected flares being considered to have clear\nstructure, while most flares did not have a well defined\nevolution of the lightcurve during the time observed.\nAdditional single-scan lightcurves for all SPT stars are\nincluded in the appendix; these lightcurves also contain\n220 GHz data which has approximately 5 ×higher noise\nthan the 95 and 150 GHz bands.\nThe peak flux values of flares reported in the cata-\nlog are averaged over full observations and thus have\na lower peak flux value than the corresponding single-\nscan lightcurve. Shorter duration events have larger bi-\nases since typical observations contain 16 single-scan\ndata points. An event that is shorter than 100 sec-\nonds in duration may only be caught in one single-scan\n10271028102910301031\n150 GHz single observation L [erg s1]\n0246810150 GHz single scan / single observation L\nEvolved\nUpper Main Sequence\nLower Main SequenceFigure 10. Comparison of single-scan and single-\nobservation peak 150 GHz luminosities for each event. The\nflux values from observations are a factor of 2 lower than\nthe peak from single-scans for the majority of flares.\nand will be reduced by a factor of 16 when using the\nsingle-observation flux instead. We choose to represent\nthe stellar flares with single-observation flux values due\nto our transient detection pipeline running on single-\nobservation lightcurves as well as the increased noise val-\nues inherent to the single-scan data. Figure 10 shows the\ndifference in 150 GHz luminosities for flares using peak\nsingle-observation and peak single-scan fluxes, showing\nthat the average difference in recovered luminosity is\nroughly a factor of 2.14 Tandoi et al.\n100101102103104105\nDuration [minutes]1027102810291030150 GHz single observation L [erg s1]\nSingle scan range\nFull single observation\nEvolved\nUpper main sequence\nLower main sequence\nFigure 11. Constraints on durations for stellar flares, us-\ning single-scan lightcurves to determine lower limits and\nsingle-observation lightcurves to determine upper limits. 5\nevents are found to be fully contained within the single-scan\nlightcurve and have single markers at a larger size.\nOur single-scan lightcurves occasionally have addi-\ntional peaks in them. When considering the typical du-\nration for the impulsive phase of a stellar flare on the\norder of seconds, we can not confidently classify these\npeaks as a separate event occurring nearby temporally,\nor a pre/post flare pulsation as described in Howard &\nMacGregor (2022) where similarly binned TESS data\nat 120 second cadence is sufficient enough time to hide\nstructure of the flare morphology.\n3.7. Flaring Activity Durations\nWe constrain the duration of flaring activity by us-\ning the single-scan and single-observation lightcurves for\neach event. Lower limits on duration are set by the\nsingle-scan lightcurve, with a flare considered to be at\nquiescence (either before or after the peak) by having a\nsingle-scan SNR below 4 σsimultaneously in both bands.\nFor a flare to be considered fully contained within a\nsingle-scan lightcurve, quiescent points need to exist on\neither side of the flare. For flares with only one quies-\ncent point in their single-scan lightcurve, the minimum\nduration is considered to be from this quiescent point to\neither the beginning or end of the lightcurve depending\non where it is in relation to the peak. Flares with no\nquiescent points in their single-scan lightcurve use thefull time between the first and last scan as a minimum\nduration. Upper limits on duration are constrained in\na similar way: quiescence is considered to be when the\nclosest single-observation returns to 1 σbefore and after\nthe peak.\nFigure 11 shows the constraints on duration for all\nflaring events. We find 5 flares to be fully constrained\nwithin the single-scan lightcurve and have measured\ntheir durations: these are plotted as larger marker sizes\nwith an uncertainty of one full single-scan (100 s). The\nshaded area denotes the range in possible times that a\nlocation will be observed by the SPT, from one scan (100\ns) to 20 scans including the time for the SPT to acceler-\nate and change elevation while it is not collecting data\n(2060 s). The vertical line shows the typical duration of\na full single-observation (7668 s).\nGoing from single-observation to single-scan\nlightcurves increases our time resolution, but still suffers\na low cadence, as there are 100 seconds on average be-\ntween data points. Depending on where in the subfield\nthe flaring event occurs, subsequent single-scan data\npoints may represent just a few seconds of the focal\nplane passing over that part of the sky (see Figure 3),\nwith large gaps in time possible as it takes the telescope\n100 seconds to complete a scan in one direction. This\ncadence is more than sufficient to hide features in a\nflare.\nSimilarly, large gaps between single-observation data\npoints can span anywhere from 2 to 24+ hours depend-\ning on the subfield observation schedule. For most flares,\nthese temporal gaps in SPT data are large enough that\nit is likely that the true duration of the flare is much\ncloser to the lower limit. Some flares exhibit multi-\nple single-observation data points that are likely part\nof the same long duration flare, but could also possibly\nbe separate back-to-back flares. The transient detec-\ntion pipeline only fits for a single flare in each 12-day\nlightcurve, so additional flares in the same lightcurve\nwill go unreported. Lightcurves for all events have been\nmanually analyzed for obvious secondary flares and none\nhave been found. This fitting method will be improved\non in the future to allow for finding multiple flares within\nthe same lightcurve.\n3.8. Additional Classifications\n3.8.1. Binary Systems\nWe have checked for binarity in the sample of SPT-\ndetected flare stars. Binary systems can lead to confu-\nsion in determining the correct source — there is a pos-\nsibility of matching to the wrong star due to undetected\nsecondaries or preferentially selecting the brightest star\nin Gaia. Stars in binary systems can also exhibit flaresSPT-3G Flaring Stars 15\n0 1 2 3 4 5\nGBP GRP\n2\n0\n2\n4\n6\n8\n10\n12\n14MGEvolved\nUpper MS\nLower MS\n2345678\nNumber of flares\nFigure 12. Gaia CMD showing stars that have multiple\nflares detected by the SPT. Colored and marker size scaled\nby number of flares. Most of the multiple flares detected\noriginate from evolved stars.\ndue to interactions between the stars, such as RS CVn\nsystems which have been detected at millimeter wave-\nlengths (Brown & Brown 2006) and are known to flare\nfrequently (Osten & Brown 1999).\nUsing Vizier (Ochsenbein et al. 2000), we have\nsearched all flaring stars for inclusion in catalogs of bi-\nnary stars as well as classification by Gaia (Rimoldini\net al. 2023). We use the same process described in Sec-\ntion 3.5 to compare SPT-detected stars to a similar rep-\nresentative population in Gaia.\nWe find that the SPT population of stars is over-\nrepresented in RS CVn classification by Gaia (10.6%\ncompared to 0.6% for the nearest-neighbor population)\nas well as probable binary candidates (51.5% compared\nto 22.7%). While many of these classifications require\nmore definitive proof such as spectroscopic observations,\nthis propensity for binarity could be important to the\nunderlying mechanisms found in millimeter flares.\n3.8.2. Multiple Flares\nFigure 12 shows a CMD of the stars we have detected\nflaring multiple times. While these stars sample the en-\ntirety of our population, we find that our detections of\nmultiple flares are dominated by evolved stars with 30\nof the 62 events coming from just 5 stars. From prelimi-\nnary simulations of injected flare efficiency (Section 2.1),\nit is likely the true rate of above-threshold flares from\nthese stars ranges from 2 ×to 20×higher than observed.\nFigure 11 shows only 5 flares — from upper and lower\nmain sequence stars — to be fully constrained and to\nhave durations on the scale of 30 minutes or less, sug-\ngesting the main sequence stars in Figure 12 have a trueflaring rate closer to 20 ×higher. Table A1 shows a full\nlisting of these stars and all of their associated events.\n3.8.3. Outlier Events\nFlaring events in Figure 4 having p-values greater than\n0.02 have been individually investigated. In most cases\nthe higher than typical p-value is due to the associated\nstar being in a high local density region and having\nmultiple nearby candidate stars, and we believe these\nmatches to be correct.\nFigure 13 shows the local Gaia density for the two\nevents on MJD 58706.43 (p: 0.038) and MJD 59767.07\n(p: 0.086). For both of these events, our cross-matching\nalgorithm picked a dimmer star closer in angular sepa-\nration rather than brighter stars at slightly larger sep-\naration. In addition, both events are matched to ex-\ntreme outliers: they are main sequence stars at distances\nover 1kpc, significantly further away than other stars we\nmatch to. Figure 5 shows these two anomalous stars in\nthe middle of the main sequence as having much further\ndistances than similar main sequence stars the SPT has\ndetected, resulting in extremely high flaring luminosi-\nties. For both of these events, the other potential can-\ndidates are nearby M Dwarfs: the candidate for MJD\n58706.43 is a potential binary system (Reiners et al.\n2010), and the two candidates for MJD 59767.07 have\nan angular separation of 6 .2′′and distances of 51.13 and\n51.17 pc giving a distance from each other of ∼0.04 pc.\nWe believe that the M Dwarfs are a more likely match for\nboth events, and have changed the associated sources.\nThese cases suggest some improvements in our cross-\nmatching algorithm can be made. The events were pref-\nerentially matched to a distant main sequence star —\nrather than a brighter M Dwarf at slightly higher an-\ngular separation — it is possible that including a prior\non parallax based on our confirmed distributions in the\nCMD can steer the event to a more plausible match.\nLikewise, incorporating a prior that favors stars that\nhave been observed flaring previously might be consid-\nered. Further optimization of this algorithm will be nec-\nessary to inform future analysis of surveys such as CMB-\nS4 and SO, and especially for surveys that include the\ngalactic plane.16 Tandoi et al.\n-30 0 +30-300+30MJD 58706.43, p: 0.038\n-30 0 +30-300+30MJD 59767.07, p: 0.086\n15.0\n15.5\n16.0\n16.5\n17.0\n17.5\n18.0\nmG\nRA [arcsec]\nDec [arcsec]\nFigure 13. Two events with outlier distances and their potential Gaia matches shown at their angular separation. Stars\nrepresent candidate sources, and are colored by brightness and sized scaling inversely with p-value.\n4.CONCLUSION\nWe have presented a new catalog of short duration\n(<10 days) transient events found in a non-targeted\nSPT-3G survey using an updated transient detection\nand matching method. This catalog is produced from\nanalyzing 4 years of daily observations of a 1500 square\ndegree field at 95 and 150 GHz by the SPT-3G sur-\nvey, with the inclusion of 220 GHz data coming af-\nter events have been detected. The catalog is are\nprovided in machine readable format at https://pole.\nuchicago.edu/public/data/tandoi24/, along with 4-year\nsingle-observation lightcurves at 95 and 150 GHz of each\nunique star, and single-scan lightcurves at 95, 150, and\n220 GHz of each flaring event.\nWe have found that two-band spectral index compar-\nisons for 95/150 and 150/220 GHz generally fall between\nan optically thick synchrotron spectrum ( α=5/2) and a\ntypical value of the optically thin tail of synchrotron\n(α=-0.75). While optically thick synchrotron radiation\ntypically peaks at radio frequencies ( ∼5 GHz), evidence\nof rising spectra at sub-mm frequencies have been ob-\nserved in solar flares.\nWe have found these flaring events to be nearby stars\nthat we have matched to Gaia Data Release 3. Many of\nthese stars have counterparts in the 2RXS stellar content\ncatalog. A CMD shows our population of matched stars,\nindicating that we are seeing flaring events from stars\nspanning a wide range of spectral types with most of\nthese flares occurring in M Dwarfs and along the Giant\nbranch.Quiescent SXR emission has been shown to have a\nlinear relationship with magnetic flux in main sequence\nstars, assumed to be the cause for the flaring activity\nwe have observed, along with measures of direct flaring\nactivity. We have shown quiescent SXR emission along\nwith peak flare 150 GHz luminosity, and a comparison\nof this ratio, and found strong evidence that mm-wave\nflaring stars tend to be quiescently bright in SXR. A\nwide spread in this ratio exists that may be tightened\nup with simultaneous SXR and mm emissions during a\nflare.\nWe have shown cumulative number counts of the flar-\ning stars in flux densities and peak flare luminosities.\nThe flux density number count has a slope consistent\nwith observing sources in a uniform density, i.e. the\nMilky Way disk, at higher flux values, with this assumed\npower law of N(> S)≈S−3/2breaking as flux density\napproaches the SPT transient detection threshold. The\nnumber count as a function of luminosity shows lower\nand upper main sequence stars having similar median\nluminosities. Evolved stars have a median peak flare\nluminosity over one magnitude brighter than the SPT-\ndetected main sequence stars.\nWe have flagged all probable binary system candi-\ndates, known through Gaia variability tables and a lit-\nerature search in Vizier. It is not clear if these flares\noriginate from the associated star, a companion, or an\ninteraction between the stars in the system.\nWe have observed 17 out of 66 stars flare multiple\ntimes, comprising over half of our events and a wideSPT-3G Flaring Stars 17\nrange of spectral types, with most of these multi-flare\nevents originating from just a few evolved stars.\nWe believe that our cross-matching algorithm has er-\nroneously matched two events with distant stars result-\ning in anomalously high luminosities. Both matches are\nto main sequence stars at distances of >1 kpc (greater\nthan any other Gaia star that SPT has observed flar-\ning) and are closer in angular separation to the SPT\nevent compared to nearby M Dwarfs at ≈50 pc which\nare brighter in apparent magnitude. We believe that\nthe M Dwarfs are the source of the flaring activity for\nboth of these events. In the future, we are considering\nadding a prior based on location in the Gaia CMD to\nour cross-matching algorithm.\nWe have shown finer time resolution single-scan\nlightcurves of events, using these as a lower limit con-\nstraint on durations. Upper limits on the duration are\nfound using single-observation lightcurves and we have\nfound typical constraints on duration range from ∼101\nto∼104minutes. These constraints are weak, and are\ngenerally dominated by the SPT observation cadence.\nTwo potential models are shown to fit to some of the\nshorter duration single-scan lightcurves: a sharp rise\nwith exponential decay, and a Gaussian fit with a rise\ntime comparable to the decay time.\nSimultaneous observations will be possible for future\nSPT observing seasons with TESS having revisited the\nSPT-3G field in July 2023, and the ASKAP Variables\nand Slow Transients (VAST) survey covering the same\narea of the sky over parts of the next few years. With\nadditional wavelengths, and polarization in the case of\nVAST, these surveys will deliver important information\nas to the nature of these flaring events. Targeted ob-\nservations of known multi-flaring SPT stars from ra-\ndio instruments such as the Atacama Large Millime-\nter/submillimeter Array (ALMA) in conjunction with\nthe SPT can extend the radio SED and provide the finer\ntime resolution that can constrain the durations of these\nevents. We can also use SPT-3G data to search for lin-\nearly polarized emission.\nThe flaring events in this catalog have greatly in-\ncreased the known number of millimeter wavelength\nflare stars. Additionally, the methods used to create\nthis catalog —map filtering, transient detection, flare ef-\nficiency simulations, and cross-matching to Gaia— have\nbuilt upon previous SPT transient analysis and will con-\ntinue to be improved. These methods and the resulting\ncatalog from 4 years of the non-targeted SPT-3G survey\nwill form a basis for what to expect while searching for\nshort duration transient events and are important for\nplanning future millimeter wavelength surveys such as\nCMB-S4 and SO.ACKNOWLEDGMENTS\nThe authors thank Rachel Osten, Alex Gagliano, Kon-\nstantin Malanchev, Gautham Narayan, and Brian Fields\nfor helpful discussions. The South Pole Telescope pro-\ngram is supported by the National Science Foundation\n(NSF) through award OPP-1852617. Partial support\nis also provided by the Kavli Institute of Cosmologi-\ncal Physics at the University of Chicago. Work at Ar-\ngonne National Lab is supported by UChicago Argonne\nLLC, Operator of Argonne National Laboratory (Ar-\ngonne). Argonne, a U.S. Department of Energy Of-\nfice of Science Laboratory, is operated under contract\nno. DE-AC02-06CH11357. Work at Fermi National\nAccelerator Laboratory, a DOE-OS, HEP User Facil-\nity managed by the Fermi Research Alliance, LLC, was\nsupported under Contract No. DE-AC02-07CH11359.\nThe IAP authors acknowledge support from the Centre\nNational d’ ´Etudes Spatiales (CNES). This project has\nreceived funding from the European Research Council\n(ERC) under the European Union’s Horizon 2020 re-\nsearch and innovation programme (grant agreement No\n101001897). The Melbourne authors acknowledge sup-\nport from the Australian Research Council’s Discovery\nProject scheme (No. DP210102386). The McGill au-\nthors acknowledge funding from the Natural Sciences\nand Engineering Research Council of Canada, Cana-\ndian Institute for Advanced Research, and the Fonds\nde recherche du Qu´ ebec Nature et technologies. The\nSLAC authors acknowledge support by the Department\nof Energy, Contract DE-AC02-76SF00515. The UCLA\nand MSU authors acknowledge support from NSF AST-\n1716965 and CSSI-1835865. M.A., K.A.P., J.V., and\nY.W. acknowledge support from the Center for Astro-\nPhysical Surveys at the National Center for Supercom-\nputing Applications in Urbana, IL. K.F. acknowledges\nsupport from the Department of Energy Office of Science\nGraduate Student Research (SCGSR) Program. Z.P.\nwas supported by Laboratory Directed Research and\nDevelopment (LDRD) funding from Argonne National\nLaboratory, provided by the Director, Office of Science,\nof the U.S. Department of Energy under Contract No.\nLDRD-2021-0186. J.V. acknowledges support from the\nSloan Foundation. W.L.K.W is supported in part by the\nDepartment of Energy, Laboratory Directed Research\nand Development program and as part of the Panofsky\nFellowship program at SLAC National Accelerator Lab-\noratory, under contract DE-AC02-76SF00515. This re-\nsearch has made use of the SIMBAD database, operated\nat CDS, Strasbourg, France. This research has made use\nof the VizieR catalogue access tool, CDS, Strasbourg,\nFrance (DOI : 10.26093/cds/vizier). The original de-\nscription of the VizieR service was published in 2000,18 Tandoi et al.\nA&AS 143, 23. This research was done using resources\nprovided by the Open Science Grid (Pordes et al. 2007;\nSfiligoi et al. 2009), which is supported by the NSF\naward 1148698, and the U.S. Department of Energy’s\nOffice of Science. The data analysis pipeline also uses\nthe scientific python stack (Hunter 2007; Jones et al.\n2001; van der Walt et al. 2011). This work has made\nuse of data from the European Space Agency (ESA)\nmission Gaia (https://www.cosmos.esa.int/gaia), pro-\ncessed by the Gaia Data Processing and Analysis\nConsortium (DPAC, https://www.cosmos.esa.int/web/\ngaia/dpac/consortium). Funding for the DPAC hasbeen provided by national institutions, in particular\nthe institutions participating in the Gaia Multilateral\nAgreement. 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Table A1 shows luminosities and spectral indices for all stars that were detected\nflaring multiple times.\n0500\nMJD 58576.02,  DR3 4909009394396454144\n250\n0250\nMJD 58576.15,  DR3 4697144528904715008220 GHz 150 GHz 95 GHz\n0500\nMJD 58579.23,  DR3 4727935836644070656\n500\n0500\nMJD 58590.05,  DR3 4928347433108236416\n0500\nMJD 58591.69,  DR3 6471134835842209792\n0500\nMJD 58592.3,  DR3 4738200056863233536\n0500\nMJD 58594.72,  DR3 4979859106192681344\n0200\nMJD 58595.56,  DR3 6493575352770287488\n0500\nMJD 58645.25,  DR3 6478107251392112640\n0200400\nMJD 58655.8,  DR3 6471134835842209792\n0200\nMJD 58667.89,  DR3 6452221724017193216\n0200400\nMJD 58705.72,  DR3 4742040410461492096\n0100200\nMJD 58706.43,  DR3 4920330653311504256\n0500\nMJD 58716.35,  DR3 4902014095262270336\n0250500\nMJD 58719.74,  DR3 4742040410461492096\n250\n0250\nMJD 58726.21,  DR3 4918091463521598592\n0200\nMJD 58747.57,  DR3 4723685987981062656\n0500\nMJD 58754.31,  DR3 4905813212878537216\n0500\nMJD 58755.38,  DR3 6463546350186454144\n250\n0\nMJD 58761.09,  DR3 4691426694779173376\n100\n0100\nMJD 58767.67,  DR3 6456528236185330816\n0500\nMJD 58778.27,  DR3 4910314961376339584\n0200400\nMJD 58798.28,  DR3 4713771622913507328\n0250500\nMJD 58804.31,  DR3 6527965293348044288\n500\n0\nMJD 58808.36,  DR3 4979859106192681344\n0500\nMJD 58808.69,  DR3 6478107251392112640\n0200\nMJD 58934.1,  DR3 6388245295629220608\n0200\nMJD 58940.76,  DR3 6387778729037646080\n0500\nMJD 58941.62,  DR3 6478107251392112640\n250\n0250\nMJD 58948.07,  DR3 6524079447458513536\n5\n 0 5 10 15 20 25 300250\nMJD 58949.56,  DR3 4910693055937045888\n5\n 0 5 10 15 20 25 300200\nMJD 58958.14,  DR3 6452221724017193216\n5\n 0 5 10 15 20 25 300200400\nMJD 58961.91,  DR3 6471134835842209792\nTime since first scan [minutes]Flux [mJy]22 Tandoi et al.\n200\n0200\nMJD 58966.06,  DR3 6471134835842209792\n500\n0500\nMJD 58982.09,  DR3 4726387007013852672\n0500\nMJD 58997.58,  DR3 6495448989302214784\n010002000\nMJD 59003.11,  DR3 6463546350186454144\n200\n0200\nMJD 59005.5,  DR3 6463546350186454144\n01000\nMJD 59017.39,  DR3 6559052816392332288\n500750\nMJD 59024.26,  DR3 6559052816392332288\n0500\nMJD 59040.35,  DR3 4946938113149426944\n0250500\nMJD 59065.99,  DR3 4745195905754074496\n0500\nMJD 59067.63,  DR3 6388014157668558080\n500\n0\nMJD 59068.28,  DR3 6567445491726233216\n200\n0200\nMJD 59096.39,  DR3 6463546350186454144\n01000\nMJD 59109.2,  DR3 4946938113149426944\n500\n0500\nMJD 59110.27,  DR3 4727920241619206912\n0500\nMJD 59118.44,  DR3 4933323685296675584\n200\n0200\nMJD 59120.33,  DR3 6470742481994348416\n250\n0250\nMJD 59131.78,  DR3 4954453580066220800\n0200400\nMJD 59149.76,  DR3 4914632365579875200\n250\n0250\nMJD 59150.31,  DR3 4901934140151066624\n010002000\nMJD 59158.65,  DR3 4946938113149426944\n250500\nMJD 59167.07,  DR3 4900780511935295872\n0250\nMJD 59175.55,  DR3 6400464275724294784\n200\n0200\nMJD 59308.02,  DR3 6463546350186454144\n250\n0250\nMJD 59338.65,  DR3 4900997562402605952\n0250\nMJD 59338.88,  DR3 6396977135940047616\n0200400\nMJD 59345.82,  DR3 4708842443566593536\n250\n0250\nMJD 59346.44,  DR3 6478107251392112640\n0500\nMJD 59348.69,  DR3 6580179279486143744\n500\n0500\nMJD 59353.05,  DR3 6559052816392332288\n200\n0200\nMJD 59361.0,  DR3 4723685987981062656\n0500\nMJD 59373.1,  DR3 4742040410461492096\n0500\nMJD 59379.72,  DR3 4743636248509810944\n250\n0250\nMJD 59382.16,  DR3 6478107251392112640\n0200\nMJD 59393.69,  DR3 6396977135940047616\n250\n0250\nMJD 59408.63,  DR3 6577998398172195840\n0500\nMJD 59413.11,  DR3 4742040410461492096\n5\n 0 5 10 15 20 25 300200\nMJD 59428.56,  DR3 4910693055937045888\n5\n 0 5 10 15 20 25 30250500750\nMJD 59430.21,  DR3 6403279953204895360\n5\n 0 5 10 15 20 25 30250\n0250\nMJD 59447.56,  DR3 4928347433108236416\nTime since first scan [minutes]Flux [mJy]SPT-3G Flaring Stars 23\n0200\nMJD 59464.06,  DR3 6569006626440192512\n0500\nMJD 59499.04,  DR3 6528936333914427392\n01000\nMJD 59507.78,  DR3 4726387007013852672\n0200400\nMJD 59525.13,  DR3 6559052816392332288\n0250\nMJD 59526.67,  DR3 4743636248509810944\n05001000\nMJD 59536.24,  DR3 4745459616746179200\n0200400\nMJD 59547.09,  DR3 4743636248509810944\n0500\nMJD 59676.48,  DR3 4946938113149426944\n0500\nMJD 59701.38,  DR3 4850405974392619648\n250\n0250\nMJD 59710.62,  DR3 4713771622913507328\n250\n0250\nMJD 59714.85,  DR3 6464757870268513152\n05001000\nMJD 59718.96,  DR3 6559052816392332288\n0500\nMJD 59720.24,  DR3 4991772314678700416\n200\n0200\nMJD 59726.28,  DR3 6404861806838482944\n0500\nMJD 59734.26,  DR3 4942856382389905408\n0200400\nMJD 59736.28,  DR3 6559052816392332288\n0200\nMJD 59739.27,  DR3 4723685987981062656\n250\n0250\nMJD 59741.04,  DR3 4742040410461492096\n0500\nMJD 59742.94,  DR3 6463546350186454144\n50010001500\nMJD 59757.48,  DR3 6463546350186454144\n250\n0250\nMJD 59763.23,  DR3 4920732799689111424\n0200\nMJD 59767.07,  DR3 4851182435759863296\n250\n0250\nMJD 59769.96,  DR3 4741424615230777216\n05001000\nMJD 59776.63,  DR3 6478107251392112640\n250\n0250\nMJD 59778.0,  DR3 6452221724017193216\n0500\nMJD 59781.46,  DR3 4743636248509810944\n200\n0200\nMJD 59788.74,  DR3 4710627809932009344\n0100\nMJD 59793.29,  DR3 6375292709353220992\n0250500\nMJD 59819.21,  DR3 6463546350186454144\n0200400\nMJD 59826.49,  DR3 6490166213890463616\n0200\nMJD 59834.56,  DR3 4910693055937045888\n250\n0250\nMJD 59839.07,  DR3 4743636248509810944\n200\n0200\nMJD 59842.17,  DR3 6560463421092689024\n0200\nMJD 59851.23,  DR3 4946915641880235904\n0200\nMJD 59854.89,  DR3 4708842443566593536\n200\n0200\nMJD 59871.16,  DR3 6481430353489019776\n5\n 0 5 10 15 20 25 300250\nMJD 59872.95,  DR3 6404861806838482944\n5\n 0 5 10 15 20 25 30250\n0250\nMJD 59877.83,  DR3 6561698791124472960\n5\n 0 5 10 15 20 25 30250\n0250\nMJD 59887.43,  DR3 6471134835842209792\nTime since first scan [minutes]Flux [mJy]\nFigure A1. Additional single scan lightcurves.24 Tandoi et al.\nTable A1. Luminosities and spectral indices for all stars with multiple detected flares.\nGaia DR3 ID MJD νLν,90\n[erg s−1]νLν,150\n[erg s−1]νLν,220\n[erg s−1]α90\n150 α150\n220 Flares\n6463546350186454144 59757.48 2.19 ×10305.81×10301.15×10311.14±0.04 0.78 ±0.07 8\n59742.94 8.32 ×10291.40×10301.93×10300.14±0.14 -0.17 ±0.47\n59819.21 6.34 ×10291.30×10301.92×10300.58±0.15 0.0 ±0.43\n59003.11 4.00 ×10298.92×10291.58×10300.76±0.35 0.49 ±1.01\n59096.39 5.42 ×10297.58×10294.73×1029-0.27±0.21 -2.23 ±1.63\n58755.38 4.70 ×10297.42×10291.31×10300.0±0.24 0.48 ±0.61\n59005.50 2.32 ×10293.35×10293.15×1029-0.2±0.5 -1.16 ±2.39\n59308.02 1.86 ×10292.64×10293.06×1029-0.23±0.59 -0.61 ±2.51\n6559052816392332288 59024.26 2.79 ×10304.54×10306.25×10300.06±0.04 -0.16 ±0.14 6\n59017.39 2.37 ×10303.84×10306.58×10300.06±0.04 0.41 ±0.13\n59718.96 5.32 ×10291.62×10302.83×10301.44±0.16 0.45 ±0.28\n59353.05 9.69 ×10291.18×10307.91×1029-0.56±0.16 -2.06 ±1.25\n59525.13 4.01 ×10295.27×10295.20×1029-0.4±0.3 -1.04 ±1.59\n59736.28 2.13 ×10292.29×10294.68×1028-0.84±0.65 -5.14 ±16.47\n6478107251392112640 59776.63 6.40 ×10291.40×10301.81×10300.71±0.19 -0.33 ±0.58 6\n58645.25 6.63 ×10291.14×10301.51×10300.18±0.26 -0.26 ±0.89\n58941.62 5.74 ×10291.06×10301.96×10300.34±0.25 0.61 ±0.72\n59382.16 3.80 ×10295.12×10297.59×1029-0.35±0.42 0.03 ±1.49\n58808.69 3.49 ×10294.57×10293.02×1028-0.41±0.52 -8.1 ±38.87\n59346.44 2.55 ×10292.75×10297.74×1028-0.84±0.69 -4.31 ±13.19\n6471134835842209792 58961.91 2.24 ×10304.92×10306.11×10300.73±0.14 -0.43 ±0.41 5\n58655.80 7.54 ×10291.20×10302.20×10300.02±0.47 0.59 ±1.2\n58966.06 5.51 ×10298.38×10294.47×1029-0.08±0.71 -2.64 ±6.81\n59887.43 7.24 ×10297.66×10291.45×1030-0.88±0.66 0.66 ±2.08\n58591.69 6.20 ×10296.88×10298.49×1028-0.77±0.76 -6.46 ±32.38\n4743636248509810944 59526.67 8.00 ×10291.36×10301.17×10300.15±0.14 -1.38 ±0.7 5\n59547.09 8.03 ×10298.69×10291.06×1030-0.83±0.19 -0.47 ±0.83\n59379.72 2.37 ×10292.78×10293.38×1029-0.65±0.58 -0.5 ±2.97\n59839.07 2.82 ×10292.61×10296.20×1029-1.17±0.57 1.26 ±1.41\n59781.46 2.46 ×10292.50×10294.74×1029-0.97±0.63 0.67 ±1.85\n4742040410461492096 58705.72 2.27 ×10283.87×10288.66×10280.17±0.22 1.1 ±0.46 5\n59373.10 1.64 ×10283.34×10287.71×10280.56±0.26 1.18 ±0.5\n58719.74 1.03 ×10282.00×10287.43×10280.44±0.46 2.43 ±0.6\n59741.04 1.37 ×10281.84×10287.67×1027-0.36±0.44 -3.28 ±5.48\n59413.11 6.80 ×10271.82×10287.61×10271.16±0.57 -3.28 ±4.73\n4946938113149426944 59158.65 4.11 ×10279.07×10271.74×10280.74±0.08 0.7 ±0.19 4\n59109.20 2.44 ×10277.05×10271.53×10281.32±0.13 1.02 ±0.22\n59040.35 1.01 ×10271.96×10271.33×10270.44±0.34 -2.02 ±2.06\n59676.48 1.06 ×10271.43×10271.26×1027-0.36±0.39 -1.34 ±2.27\n6452221724017193216 59778.00 1.55 ×10293.64×10295.42×10280.87±0.45 -5.97 ±10.39 3\n58667.89 1.39 ×10292.92×10294.16×10290.61±0.51 -0.07 ±1.37\n58958.14 1.62 ×10292.69×10294.46×10290.11±0.5 0.32 ±1.36\n4910693055937045888 59834.56 7.41 ×10281.40×10292.70×10290.39±0.44 0.72 ±0.96 3\n58949.56 5.83 ×10281.35×10292.93×10290.83±0.5 1.03 ±0.88\n59428.56 7.81 ×10281.15×10292.90×1028-0.15±0.46 -4.6 ±8.6\n4723685987981062656 58747.57 7.91 ×10292.31×10303.20×10301.34±0.24 -0.14 ±0.53 3\n59739.27 4.52 ×10291.02×10302.96×10300.79±0.47 1.77 ±0.62\n59361.00 3.21 ×10296.80×10295.93×10290.65±0.59 -1.36 ±2.68\n6404861806838482944 59872.95 1.25 ×10302.50×10302.83×10300.52±0.32 -0.67 ±1.06 2\n59726.28 7.35 ×10291.87×10302.12×10301.05±0.52 -0.68 ±1.52\n6396977135940047616 59338.88 1.80 ×10282.64×10283.30×1028-0.16±0.33 -0.42 ±1.37 2\n59393.69 8.83 ×10271.85×10284.51×10280.62±0.55 1.33 ±0.96\n4979859106192681344 58594.72 3.44 ×10287.56×10281.27×10290.72±0.56 0.35 ±1.42 2\n58808.36 2.93 ×10287.16×10285.35×10270.96±0.63 -7.78 ±30.45\n4928347433108236416 59447.56 5.48 ×10271.35×10282.26×10280.97±0.39 0.36 ±0.85 2\n58590.05 5.00 ×10277.40×10271.30×1028-0.14±0.63 0.46 ±1.94\n4726387007013852672 59507.78 5.84 ×10271.67×10284.38×10281.3±0.09 1.52 ±0.13 2\n58982.09 1.35 ×10272.37×10271.47×10270.23±0.49 -2.26 ±3.39\n4713771622913507328 58798.28 1.02 ×10281.57×10281.88×1028-0.06±0.21 -0.53 ±0.79 2\n59710.62 3.84 ×10277.03×10272.15×10280.32±0.49 1.92 ±0.72\n4708842443566593536 59345.82 1.72 ×10305.16×10301.03×10311.4±0.18 0.8 ±0.32 2\n59854.89 1.25 ×10301.71×10301.74×1030-0.32±0.39 -0.96 ±1.7"    },    {        "title": "2401.13526v1.Tuning_of_Charge_Order_by_Uniaxial_Stress_in_a_Cuprate_Superconductor.pdf",        "content": "Tuning of Charge Order by Uniaxial Stress in a Cuprate Superconductor  \n \nAuthors  \nLaure Thomarat,1, 2 Frank Elson,3 Elisabetta Nocerino,4,5 Debarchan Das,6 Oleh Ivashko,7 \nMarek Bartkowiak,1 Martin Månsson,3 Yasmine Sassa,8 Tadashi Adachi,9 Martin v. \nZimmermann,7 Hubertus Luetkens,10 Johan Chang,11 Marc Janoschek,1,11 Zurab Guguchia,10∗ \nGediminas Simutis1,8∗ \n \nAffiliations   \n1Laboratory for Neutron and Muon Instrumentation, Paul Scherrer Institut, CH -5232 Villigen \nPSI, Switzerland  \n2École norma le supérieure Paris -Saclay, 91190 Gif -sur-Yvette, France  \n3Department of Applied Physics, KTH Royal Institute of Technology, SE -106 91 Stockholm, \nSweden  \n4Department of Materials and Environmental Chemistry,  Stockholm University, SE -106 91 \nStockholm, Sweden  \n5Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, CH -5232 Villigen \nPSI, Switzerland  \n6Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH -5232 Villigen PSI, \nSwitzerland  \n7Deutsches Elektronen -Synchrotro n DESY, Notkestraße 85, 22607 Hamburg, Germany  \n8Department of Physics, Chalmers University of Technology, SE -41296 Göteborg, Sweden  \n9Department of Engineering and Applied Sciences, Sophia University, Chiyoda, Tokyo 102 -\n8554, Japan  \n10Laboratory for Muon Spi n Spectroscopy, Paul Scherrer Institute, Villigen PSI, Switzerland  \n11Physik -Institut, Universität Zürich, Winterthurerstrasse 190, CH -8057 Zürrich, Switzerland  \n*Corresponding  author s. Email:  zurab.guguchia@psi.ch , gediminas.simutis@psi.ch   \n \nAbstract  \nStrongly correlated electron materials are often characterized by competition and interplay of \nmultiple  quantum states. For example, in high -temperature cuprate superconductors  \nunconventional superconductivity,  spin- and charge -density wave order s coexist. A key \nquestion  is whether competing states coexist on  the atomic scale or if they segregate in to \ndistinct “islands”. Using X -ray diffraction, we investigate the  competition between charge \norder and  superconductivity in the archetypal cuprate La 2−xBaxCuO 4, around x  = 1/8 -doping, \nwhere uniaxial stress restores optimal 3D superconductivity at σ3D ≈ 0.06 GPa. We find  that \nthe charge order peaks and the correlation length along the stripe are strongly reduced up to \nσ3D, above which they stay constant. Simultaneously , the charge  order onset temperature only \nshows a modest decrease. Our findings suggest th at optimal  3D superconductivity is not linked \nto the absence of charge stripes but instead requires their arrangement  into smaller \"islands”. \nOur results provide insight into the  length scales over which the interplay between  \nsuperconductivity and charge o rder takes place . Introduction  \n \nAn intriguing feature of high temperature cuprate  superconductors is the competition and \ncooperation between stripe order and superconductivity. Stripe order refers to a pattern of \ncharge and spin density modulations that emerge at low temperatures in these materials (1 –5). \nThis ordering phenomenon has substantial impact on the properties of the cuprates . Yet, the \nexact nature of the stripe order and its relationship to the underlying electronic structure is still \nthe subject of intense debate . There is strong evidence that stripe order competes with thre e \ndimensional superconductivity in the cuprates, leading to a suppression of the critical \ntemperature for superconductivity (6, 7). However, there are also indications that the stripe \norder may play a more subtle role in the superconducting state, such as stabilizing certain exotic \ntypes of superconducting pairing (8, 9). Understanding the interplay between stripe order and \nsuperconductivity in the cuprates is thus a crucial step towards a comprehensive understanding \nof these fascinating materials.  \nAs the d ifferent ground states in these systems are subtly balanced, small external perturbation \ncan be used to select between the emergent phases. Recently, uniaxial stress has emerged as an \neffective surgical tool to manipulate these systems. In a striking examp le, a three -dimensional \ncharge density wave was observed in YBa 2Cu3O7−x at zero magnetic field (10, 11). On the other \nhand, in La 2−xSrxCuO 4 the population of different stripe order domains could be achieved to \nunequivocally demonstrate the uniaxial nature of the stripe order and exclude any reasonable \ncheckerboard models (12 –14). \nThe best way to understand the complex interplay between these competing states is to study \na system, where the lattice, charge and spin degrees of freedom as well as superconducti vity \nhas well -defined states that clearly change in the phase diagram. As such, La 2−xBaxCuO 4 is an \nexcellent  test system - a dramatic suppression of 3D superconducting transition temperature \nTSC is observed around 1/8 doping, where the charge and spin order is most pronounced as seen \nin the phase diagram of Fig. 1 a (15-18). In this range, however, superconductivity persist in \nthe two -dimensional layers of copper -oxygen planes (19). At the same dop ing levels, the \nsystem undergoes a structural phase transition towards a low -temperature tetragonal (LTT) \nstructure, where the orientation of copper -oxygen octahedra alternates between adjacent layers \n(20). Such a restructuring of the lattice potentially l ocks in the stripe order, consequently \nfrustrating the Josephson coupling between the individual superconducting layers (3, 21).  Thus, the presence of multiple structural phases across the temperature and doping range raises \nthe question of what are the un derlying driving phases. On one hand it has been suggested that \nthe LTT phase is a host structure for the stripe formation (22). However, charge order has been \nrepeatedly observed outside this structural phase throughout various cuprate systems (23 –31). \nAs the stripe order has been shown to be incompatible with the orthorhombic phase ( 32), some \nother deviations from the pure orthorhombic symmetry must be present in the system. The \nestablishment of the Low -Temperature Less -Orthorhombic (LTLO) phase was there fore \nsuggested to take place in La 2−xBaxCuO 4 (15). Here, the tilt axis of the copper -oxygen \noctahedra points along an intermediate in -plane direction ( 33). \nThe utility of La 2−xBaxCuO 4 as a model system has been further proven by the recent \nobservation that 3D superconducting transition temperature is vastly increased upon moderate \nuniaxial compression of the superconducting planes along the copper -copper direction (17), as \nseen in Fig.  1b. Similarly, u niaxial -stress induced enhancement of T C and effect on charge \norder was also reported in a Nd -doped La 2−x SrxCuO 4 (34-36). Concomitant with the increase \nof superconductivity, a suppression of the magnetically ordered volume fraction in \nLa2−xBaxCuO 4 has been observed by both muon spin rotation (16, 17) as well as neutron \nscattering (3 7). A recent structural study has revealed that the applied stress also suppresses \nthe tetragonal structure and lifts the geometric frustration (16). The remai ning question is: what \nhappens to the charge order upon this compression? In this study, we use hard X -ray diffraction \ncoupled to in -situ uniaxial stress tuning (3 8) to directly determine the effect on the charge order \nas illustrated in Fig. 2 a. This allow s us to complete the phase diagram of La 2−xBaxCuO 4 under \nuniaxial stress and describe the evolution of its structural, charge and magnetic phases.  \nWe find that the intensity of the charge order peaks as well as the correlation length along the \nstripe direc tion is strongly reduced with uniaxial stress up to the critical value σ3D, defined as \nthe pressure at which the LTT phase is completely suppressed,  above which they stay constant. \nOn the other hand, the transition temperature of the charge order shows only a modest decrease \nupon approaching σ3D, above which it also stays unchanged. Our findings reveal that \nestablishment of optimal 3D superconductivit y does not require a full suppression of charge -\nstripes but rather weakening them to an optimal limit. Our findings point toward a remarkable \ncooperation between superconducting pairing/coherence and spin/charge/lattice degrees of \nfreedom.  \n  \nResults   \n \nA ke y characteristic of charge stripe order is the emergence of diffraction peaks at QCDW= τ ± \nqCDW with τ being fundamental Bragg reflections and q CDW = (δ,0,1/2) and (0,δ, 1/2) describing \nthe charge order incommensurability, which depends on doping (15). The structure factor \nimplies that the charge order peak amplitudes vary in strength across different Brillouin zones \n(39). Due to our experimental setup that includes a cryostat and a magnet with only a small \naluminum windows for the X -ray path, the background scattering also varies with momentum. \nGenerally, contaminating powder rings originating from the sample environment are less \npronounced in zones with large momenta (| Q| > 15 r.l.u.). At ambient stress, we have inspected \nmultiple charge order peaks and identified zones with the best intensity to background ratio. \nTherefore, we concentrate on a Brillouin zone defined by τ = (2,2,17) with respect to the high -\ntemperature tetragonal nota tion (a = b = 3.78 Å). Additionally, for cross checks, a second peak \nin the (1, 1, 9) Brillouin  zone was measured for consistency ( 40). \nOur main results are summarized in Fig. 2 b, where the 3 -dimensional reciprocal scattering \nvolume is projected onto the ( 2,k,l) plane with an integration of 0.005 r.l.u. perpendicular to \nthe plane. First, we note that in all of the measurements a charge order peak can be identified \nand is gradually suppressed with strain. For a clearer illustration of the suppression, we als o \npresent the one -dimensional cuts through the peak in Fig. 3 a. Analogous measurements under \nmagnetic field are presented in Fig. 3 b. The data in these plots is obtained by integrating the \nintensity around the peak, perpendicular to the cut by 0.005 r.l.u.  in h direction and 0.05 r.l.u. \nin l direction. We further note that due to the setup of our experiment, we likely do not have \nthe resolution to determine possible subtle shifts of the peak position, but we can identify the \nupper limit of the shift as 0.03  r.l.u. for the h position, 0.01 r.l.u for the k position and 0.1 r.l.u \nfor the l position. For the purpose of presentation, we therefore fix the positions to the ambient \nstress value.  \nWhen we consider both the zero -field and applied -field cases, the peak amplitude is reduced \nas already seen in the 2D -cuts and the width of the peak increases as the stress is applied. In \norder to perform a more detailed study of this trend, we have fit th e data to a three -dimensional \nmodel as well as in several different limited cuts, obtaining a consistent picture, independent \non the analysis methods.  The obtained fit results are presented in Fig. 4. At ambient stress, the peak intensity increases \nwith de creasing temperature, consistent with earlier measurements (15). As the stress is \nincreased, the onset temperature of the charge order decreases, as shown in Fig. 4 a. At ambient \nstress, the charge order is found to arise at temperatures that coincide with the onset of the \nstructural transition towards the tetragonal symmetry (15, 16). Remarkably, even though the \nfull low -temperature tetragonal structure is suppressed with a modest critical stress of σ3D ≈ \n0.06 GPa (16), the charge order persists up to the h ighest applied stress, with the change \nsaturating above σ3D. We note, that the same behavior qualitatively is observed for spin order \n(17). Inspecting the structure evolution in detail, we observe that at elevated stress value, the \ncharge order tracks the onset of the LTLO structure, where the orthorhombicity is reduced (Fig. \n1b). Moreover, as stress is increased, the intensity of the scattering saturates at lower values \nthan in the unperturbed case. This is explicitly shown in Fig. 4 b, where it becomes cle ar that \nthe fast suppression of the intensities saturates beyond σ3D and the scattering with a finite value \npersists up to the highest measurable stress. In addition to the amplitude of the peaks, we also \ninvestigate the widths of the peak in all three dir ections in the reciprocal space as the stress is \nincreased. This is shown in Fig. 4 c and reveals a well -defined trend - the correlation length \n(defined as the inverse of HWHM) along the stripe propagation direction decreases with the \nincrease of the stress , whereas the correlation lengths of the charge order in the other two \ndirections remain unchanged.  \nDiscussion   \nSuperconductivity in La 1.885Ba0.115CuO 4 is enhanced with uniaxial stress (16, 17) as the \nmagnetic and charge order is reduced. This reduction do es not result in a complete suppression \nhowever, the magnetic and charge modulation phases are persistently present in the system \nthroughout the studied stress range.  \nWe first address the differences between the previously studied  spin channel  (16) and th e \npresently measured charge channel  of the stripe order. In contrast with the recovery of the \nvolume fraction of the spin order reported previously (16) towards low temperatures, we find \nthat the intensity of the charge order peaks saturates and stays cons tant to the lowest measured \ntemperatures. This suggests that the superconductivity and charge order manifest effectively in \nthe same stripe, whereas the spin order concerns a larger volume of space.  \nWe now turn towards understanding the mechanism that increases the 3D d -wave \nsuperconductivity upon the uniaxial compression of the system. A plausible mechanism is the reduction of the frustration of the Josephson coupling (8) when we pressurize and suppr ess the \nLTT phase (16). However, a careful examination of the relevant temperatures for structural \ndistortions and the emergence of the charge order points to a different scenario. While the full \nformation of the LTT phase is suppressed with moderate stres s of σ3D ≈ 0.06 GPa, the charge \norder persists up to the highest measured values, at least three times of σ3D. Moreover, we find \nthat the onset of the charge order tracks the onset of the LTLO structure. As such, this \nobservation indicates that the LTT phase is not a requirement for the formation of stripe order. \nHowever, it enhances the macroscopic extent of the  stripe order islands. Such a conclusion is \nin line with the persistence of finite -volume stripe order phases in Sr -doped cuprate systems \n(12, 41, 42), where the LTT phase is not present. Yet, it has been recently pointed out that the \nstripe order symmetry  is incompatible with the low -temperature orthorhombic crystal structure \n(32). It is therefore conceivable that the orthorhombic structure is accompanied by weak \nmonoclinic distortions that enable the formation of the stripe order. As local distortions can  be \naveraged over the whole sample, it is possible that they are “disguised ” as an average LTLO \nstructure.  \nThis leads us towards answering the main question: what is the mechanism behind the \ncompetition between the superconductivity and the charge -spin str ipes? One scenario would \nbe that the stripes align between the adjacent layers. This would shift the peak intensity away \nfrom the half -integer towards integer values, perpendicular to the cuprate planes. We, however, \nobserve that there is no change in the position or the width of the CDW peak along c-direction, \neven though the LTT phase is completely suppressed at this value of stress (16). Similarly, \ncharge order persists also when the four -fold rotational symmetry is recovered upon application \nof high hyd rostatic pressure ( 43). The effect may be much more subtle as the charge stripes \ncan be pinned by the domains walls between different orthorhombic twin domains (4 4) and can \neven be impacted by the structural properties at elevated temperatures (4 5). Theref ore, we \nconclude that the key element that allows the increase of the superconducting temperature and \nformation of coherent d-wave superconductivity is the optimization of the sample volume \nwhere the stripe order is unfavorable.  \nThis is further corroborate d by our measurements investigating the role of magnetic field. In \nthe doping regime where the superconductivity is favorable compared to the pinned stripe \norder, the magnetic field can greatly increase the intensity of stripe order peaks (4 6). In \nLa1.885Ba0.115CuO 4, the magnetic field has very little effect as the stripes are already fully \nformed. In the \"competition -only\" scenario, the application of magnetic field could recover the stripes after they are reduced due to uniaxial stress. We find that the m agnetic field has no \neffect on the stripe order in the uniaxially pressurized phase. This observation implies  that \nlikely the stripe order and superconductivity achieve a n energetically favorable  balanced state \nupon the application of uniaxial stress and t herefore the field effect is absent. Our conclusion  \nis also consistent  with, and a natural co -explanation of the recent measurements, where proton \nirradiation was used to induce disorder in La 1.875Ba0.125CuO 4 crystals (4 7). The irradiation \nsimultaneously i ncreased the superconducting transition temperature and decreased the \ncorrelation length of the stripe order, pointing to the importance of the optimal length scale \nupon which the stripes have to be correlated for the maximization of coherent 3D \nsupercondu ctivity in cuprates.  Similar conclusion has been reached in a recent study where an \ninverse effect has been reported – the enhancement of charge order through quenching of \nsuperconductivity in  YBa 2Cu3O6+x (48). Finally, very recent RIXS measurements have \nsuggested that among the multiple intertwined orders in cuprates, pair density wave  (PDW)  \nexhibits a separate subharmonic order at temperatures above the long -range superconductivity \n(49). In the future, it woul d be revealing to study the evolution of such signatures using uniaxial \npressure methods and compare with the changes in stripe order.  \nIn summary, we have directly shown that the increased 3D -superconductivity temperature in \nLa1.885Ba0.115CuO 4 is accompani ed by the suppression of the charge ordered phase, which \nnevertheless persists in a finite volume of the sample. The state obtained above the critical \npressure is very stable to further compression as well as magnetic fields and suggests a subtle \ncooperati on between the intertwined orders. This points towards the need to include real -space \nvolume inhomogeneity in the theoretical description of emergent phases in cuprates.  \n \nMaterials and Methods  \n \nFollowing the method devised in Ref. 50, polycrystalline sampl es of La 2−xBaxCuO 4 with x = \n0.115 were synthesized using the conventional solid -state reaction method, which involved the \nuse of La 2O3, BaCO 3, and CuO  as starting materials. The powders obtained were then subjected \nto scintering and verified to be single -phase through powder X -ray diffraction analysis. The \nsingle crystals of La 2−xBaxCuO 4 with x = 0.115 were grown using the traveling solvent floating -\nzone method. In this technique, a small molten zone is created between two counter -rotating \nrods, which melt the material and then cool it down to form a single crystal. The crystal was \nthen aligned and cut into a cuboid shape with the tetragonal [0,0,1] and [1,1,0] directions spanning the scattering plane. Stress was applied to the crystal at an angle of 45◦ to the Cu -O \nbond direction.  \nX-ray experiments were performed on P21.1 beamline at Petra -III synchrotron radiation source \nin DESY. Photon energy of 101626  eV (corresponding to 0.122 Å) was used to penetrate the \ncomplex sample environment that allowed us to cool down to 4.5 K and apply horizontal field \nup to 10 T. The sample was pressurized using a recently developed in -situ uniaxial device, \nbased on linear motor actuator (32). The constant force on the sample was maintained using a \nseparate feedback system and the temperature was measured independently on the uniaxial cell \nnext to the sample. As the magnet has limited window (opening of 10◦ in four perpendic ular \ndirections of the scattering plane), the sample was fixed in an orientation that allowed us to \nreach several most intense charge order peaks. This meant that the center of the window was \noriented at 30◦ with respect to the crystal c axis. We have appl ied a magnetic field of 10 T to \nthe sample, which is the value quoted in Fig. 3 b, but it is important to note that the magnet is \ndesigned in a way, where the field can only be applied perpendicular to the windows that allow \nthe X -rays to get through. Taking into account the previously mentioned rotation of the sample \nmeant That t he ef fective field along the c axis was therefore B = 8.7 T. 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Y.S. is funded by the  Swedish Research Council (VR) through a Starting \nGrant  (Dnr. 2017 -05078). Y.S. and G.S. further acknowledge  funding from the Area of \nAdvance -Material Sciences from  Chalmers University of Technology. E.N. is support ed by the \nSwedish Foundation  for Strategic Research (SSF) through the SwedNess  grant SNP21 -0004. \nJ.C. would like to thank the Swiss National  Science Foundation for the financial support.  \n Author contributions:  Conceptualization: JC, MJ, ZG, GS, Experimenta l work: FE, EN, \nDD, OI, MB, TA, MvZ, GS Data Analysis: LT, FE, ZG, GS, Writing —original draft: GS, \nLT, JC, ZG, Writing —discussions & editing: LT, FE, EN, DD, OI, MB, MM, YS, TA, MvZ, \nHL, JC, MJ, ZG, GS.  \n \nCompeting interests:  Authors declare that they have no competing interests . \n \nData and materials availability:  All data used for discussing results are available in the \nmain text or the supplementary information . The materials are available from the authors \nupon reasonable requests.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figures  \n \nFig. 1.  Phase Diagram of La 2-xBaxCuO 4. (a) Temperature versus hole -doping phase diagram \nfrom (15), showing the multiple electronic and structural phases and indicating with a grey \ndashed line the newly -explored part of the phase diagram, accessed by uniaxial tuning. (b) The \ncomplete phase diagram a s a function of uniaxial stress, around the doping of x = 0.115, where \nthe multiple degrees of freedom and corresponding phases are modified. As superconducting \ntemperature is increased with stress, the transition temperatures to Charge Density Wave \n(CDW) and the spin order (SO) are suppressed, together with the reduction of macroscopically \noccupied volume fraction as discussed in the text. Notably, the structural LTT phase is \ncompletely suppressed at moderate stress, yet the electronic stripe order persist s throughout the \nstudied pressure range, albeit with a reduced volume fraction. All changes saturate above the \nmeager stress of  σ3D ≈ 0.06 GPa . The transition temperatures for the superconductivity, spin \norder and structural transformations for panel (b) are taken from  (16), where exactly the same \nsample was used.  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 2.  Charge Order Under Uniaxial Stress . (a) The experimental setup used to measure the \ncharge order peaks in La 1.885Ba0.115CuO 4. The incoming X -rays that hit the uniaxially \npressurized sample and the resulting diffraction pattern is collected by a two -dimensional  \ndetector, which allows a reconstruction to intensity as a function of the position in the reciprocal \nspace. The close -up of the sample region  illustrates that stress is applied along copper -copper \ndirection, diagonally to the stripe propagation direction. (b) Development of the CDW  as a \nfunction  of temperature (horizontal axis) and uniaxial stress (vertical axis). Elastic scattering \narising from the charge order manifests  as a cigar -shaped peak in the reciprocal space. The \ntwo-dimensional reciprocal -space maps show how the intensity of the scatte ring decreases as \na function of temperature at all stress values. Further, the intensity of the charge order peak \ndecreases upon the increase of  uniaxial stress.  The black arcs and a sharp spot at k = 1.7 and l \n= 16.5 are due to background arising from sam ple environment.  \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3.  Tunability of charge order with uniaxial stress . A one dimensional  cut at base T = \n4.6 K through the charge order peak at zero ( a) and  applied magnetic field ( b) at various values \nof applied stress after  subtracting the background. In both cases, the peak amplitude is  \ndrastically reduced and the widths of the peaks increase, corresponding  to the reduction of the \ncorrelation length in the propagation  direction.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4.  The evolution of charge -order parameters as obtained from the (2, 1.77, 16.5) peak.  \n(a) shows the amplitude of the peak as a function of temperature for the various values of \napplied stress. As a comparison, we also show the temperature dependence from Ref. (15), \nwhich is normalized to overlap with our zero -stress data. (b) shows how the base temperature \npeak amplitude varies as a function of applied stress. The stress -dependence of the correlation \nlengths for the different directions is shown in ( c). The lines in all the panels are guides to the \neye. \n \n"    },    {        "title": "2401.13533v1.Moving_crystal_phases_of_a_quantum_Wigner_solid_in_an_ultra_high_quality_2D_electron_system.pdf",        "content": "Moving crystal phases of a quantum Wigner solid in an ultra-high-quality 2D electron\nsystem\nP. T. Madathil,1K. A. Villegas Rosales,1Y. J. Chung,1K. W. West,1\nK. W. Baldwin,1L. N. Pfeiffer,1L. W. Engel,2and M. Shayegan1\n1Department of Electrical and Computer Engineering,\nPrinceton University, Princeton, New Jersey 08544, USA\n2National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA\n(Dated: January 25, 2024)\nIn low-disorder, two-dimensional electron systems (2DESs), the fractional quantum Hall states\nat very small Landau level fillings ( ν) terminate in a Wigner solid (WS) phase, where electrons\narrange themselves in a periodic array. The WS is typically pinned by the residual disorder sites\nand manifests an insulating behavior, with non-linear current-voltage ( I-V) and noise characteristics.\nWe report here, measurements on an ultra-low-disorder, dilute 2DES, confined to a GaAs quantum\nwell. In the ν <1/5 range, superimposed on a highly-insulating longitudinal resistance, the 2DES\nexhibits a developing fractional quantum Hall state at ν= 1/7, attesting to its exceptional high\nquality, and dominance of electron-electron interaction in the low filling regime. In the nearby\ninsulating phases, we observe remarkable non-linear I-Vand noise characteristics as a function of\nincreasing current, with current thresholds delineating three distinct phases of the WS: a pinned\nphase (P1) with very small noise, a second phase (P2) in which dV/dI fluctuates between positive\nand negative values and is accompanied by very high noise, and a third phase (P3) where dV/dI\nis nearly constant and small, and noise is about an order of magnitude lower than in P2. In the\ndepinned (P2 and P3) phases, the noise spectrum also reveals well-defined peaks at frequencies that\nvary linearly with the applied current, suggestive of washboard frequencies. We discuss the data in\nlight of a recent theory that proposes different dynamic phases for a driven WS.\nTwo-dimensional electron systems (2DESs) in a suffi-\nciently strong, perpendicular magnetic field ( B) at low\ntemperatures and very small Landau level filling factors\n(ν <1/5), form a many-body, ordered array of electrons,\nknown as the quantum Wigner solid (WS) [1–4]. The\nelectrons, having their kinetic energy quenched signifi-\ncantly, succumb to the repulsive Coulomb interaction,\nand form a triangular lattice, maximizing the distance\nbetween each pair. In the presence of disorder, the WS\nbreaks into domains and gets pinned to the local dis-\norder sites which significantly alter the bulk transport\nproperties [5–10]. The longitudinal resistance ( Rxx) of\na pinned WS exhibits an insulating, typically activated\nbehavior, Rxx∝eEA/2kBT[11], where EAis commonly\nassociated with the WS defect formation energy [12–15].\nThe pinned WS displays microwave or radio-frequency\nresonances that can be understood as collective excita-\ntions caused by the oscillations of the WS domains within\nthe pinning potential [16, 17]. It can also show non-linear\ncurrent-voltage ( I-V) and noise characteristics, revealing\nthe complex dynamics of a moving WS [9, 11, 18–24].\nWhile the presence of disorder is a crucial ingredient\nin the transport properties of a pinned WS, less dis-\nordered 2DESs exhibit additional remarkable structure\nin transport measurements. Recent magnetoresistance\nmeasurements in ultra-high-quality 2DESs [25] indeed\nshow clear signatures of fragile fractional quantum Hall\nstates (FQHSs) at ν= 1/7 and other fillings embed-\nded deep within the insulating phase [14], clearly sig-\nnaling the presence of interaction-driven phenomena at\nvery small fillings. One would expect the quality to bereflected in the WS phase as well, and yield novel trans-\nport signatures. Our Letter reports striking non-linear\nI-V and noise characteristics of the driven WS in this\nextremely high-quality regime.\nThe ultra-high-quality 2DES in our study is realized\nby confining electrons to an 89-nm-wide GaAs quantum\nwell. The sample was grown using molecular beam epi-\ntaxy, following an extensive campaign to improve the\ncrystal purity via an optimization of the growth envi-\nronment and sample structure [25, 26]. The quantum\nwell has flanking Al xGa1−xAs barriers with stepped val-\nues of x, and Si dopants inside a doping-well structure\n[14, 25–27]. The very dilute 2DES has an areal den-\nsity of ≃2.83×1010cm−2and a record-high mobility of\n≃15×106cm2/Vs at such low density. We performed\nelectrical transport measurements on a 4 ×4 mm2van\nder Pauw geometry, with alloyed In:Sn contacts at its\ncorners and edge midpoints. The sample was cooled in\na dilution refrigerator. For details of our measurement\nprotocols, see Supplemental Material [28].\nFigure 1 presents Rxxvs.Bfor our 2DES. The sam-\nple’s exceptionally high quality is evinced from the well-\ndeveloped FQHSs up to ν= 8/17 and ν= 8/15 on the\nflanks of ν= 1/2 (Fig. 1 inset), as well as an emerging\nFQHS between ν= 1/3 and 2/5, at ν= 4/11, which\ncan be understood as the FQHS of interacting composite\nfermions [29–31]. The ν= 4/11 FQHS is extremely frag-\nile and has been reported before only in the highest qual-\nity 2DESs and at much higher densities [14, 25, 32, 33].\nIts presence in our dilute 2DES is a testament to its un-\nprecedented quality. The data in the high-field rangearXiv:2401.13533v1  [cond-mat.mes-hall]  24 Jan 20242\nFIG. 1. Longitudinal resistance ( Rxx) vs. magnetic field ( B)\nfor a dilute 2DES confined to an 89-nm-wide GaAs quantum\nwell at T≃72 mK in the small filling range. The position of\nan emerging FQHS at ν= 1/7 is marked. Inset: RxxatT≃\n30 mK in the higher filling range.\n(Fig. 1 main) show that the resistance is approximately\nfour orders of magnitude larger than in the low-field\ntrace, indicating a highly insulating behavior when the\nfilling factor is small ( ν <1/5). Nevertheless, we can see\na clear Rxxminimum at ν= 1/7, strongly corroborating\nthe high quality of this dilute 2DES.\nFigure 2(a) displays the non-linear transport traces\nat four different temperatures below the melting tem-\nperature of the WS which we deduce to be Tm≃120\nmK [28, 34, 35]. We measure the differential resistance\n(dV/dI ) as a function of the driving dccurrent ( I). A\nsmall sinusoidal ( ac) excitation current of 0.05 nA is su-\nperimposed on I, and the ratio of the measured differen-\ntial voltage to the small signal excitation gives the value\nofdV/dI . The frequency for the accurrent is kept small\n(0.1 Hz) to maximize the in-phase (resistive) component.\nThe data are obtained at B= 8.40 T ( ν= 0.139), past\ntheν=1/7 FQHS.\nAt the lowest T[80 mK, top trace in Fig. 2(a)], dV/dI\nvs.Iexhibits a strong non-linear behavior which can\nbe broken into three distinct regimes, separated by two\nthreshold currents. At very small I,dV/dI drops fairly\nsharply, by a factor of about 5 within ≃1 nA, and then\nshows a very abrupt drop at a first threshold current\n(Ith1) to negative values. We associate Ith1with the\ndepinning of the pinned WS. For I > I th1,dV/dI be-\ncomes non-monotonic and fluctuates around zero. Be-\nyond a second threshold current ( Ith2), the amplitude\nof the fluctuations significantly diminishes, and dV/dI\nattains a small, relatively constant value, with little de-\nFIG. 2. (a) dV/dI vs.IatB= 8.40 T ( ν= 0.139).\nThe different traces, corresponding to different temperatures\nT≃80,85,90,and 96 mK, are offset vertically by 12, 8, 4, and\n0 MΩ, respectively. (b) Color-scale plot summarizing dV/dI\nvs.Icurves for all temperatures. The dashed white line\nfollows the first minimum corresponding to the negative dif-\nferential resistance seen just beyond Ith1. (c) The integrated\nI-Vcharacteristics at different temperatures.\npendence on current. We associate the three different\nregimes separated by Ith1andIth2with three different\ndynamic phases of the WS; we refer to these as P1, P2,\nand P3, as a function of increasing I, and show later in\nthe manuscript that they each possess a distinct noise\nspectrum.\nThe traces shown in Fig. 2(a) reveal that the main\npeak becomes broader and Ith1shifts towards higher val-\nues as Tincreases. In addition, the local minima past\nIth1, corresponding to negative dV/dI , become less neg-\native and the amplitude of the fluctuations diminishes\nuntil it fully disappears at T= 96 mK. Figure 2(b) is\na color-scale plot that summarizes dV/dI vs.Icurves\nat all temperatures. Figure 2(c) shows the I-Vcharac-\nteristics obtained by integrating the dV/dI vs.Icurves.\nAt the lowest temperature, I-Vis step-like, with a large\nvoltage developed for a small amount of current. This\nsharp jump can be understood as the voltage, or elec-\ntric field, required to overcome the pinning potential and3\nP1P2 P3P1\nP2 P3\nFIG. 3. Noise data at B= 7.50 T ( ν= 0.155). (a,b) dV/dI (blue trace) and noise power obtained by averaging over a\nfrequency window of ±10 Hz centered at 40 Hz, defined as S40(red trace), vs. IatT≃61 and ≃78 mK, respectively. (c)\nS40vs.Iat different temperatures. The scale for S40is logarithmic and spans over four decades. (d,e) Color-scale plots\nsummarizing the noise power as a function of fandIatT≃61 and ≃78 mK, respectively. (f,g) Noise power vs. ffor different\nvalues of Ias indicated.\ncause motion of the WS. Beyond Ith1, the voltage fluctu-\nates about the threshold voltage value, and increases very\ngradually with I, indicating that once depinned, a negli-\ngible amount of extra electric field is required to sustain\na current. Increasing temperature makes the step-like\ntransitions more smooth, and the fluctuations become\nweaker, eventually vanishing at the highest temperatures\n[36].\nTo further investigate the driven phases of the WS, we\nmeasured the noise spectrum in response to a driving dc\ncurrent, concomitantly with dV/dI . Figure 3 summarizes\nthe noise data at B= 7.50 T ( ν= 0.155). In Fig. 3(a),\nwe show the non-linear dV/dI (blue trace) together with\nthe noise power averaged between 30 and 50 Hz, defined\nasS40(red trace) vs. IatT≃61 mK. Qualitatively\nsimilar to Fig. 2(a) data, the dV/dI vs.Idata in Fig.\n3(a) exhibit two current thresholds that delineate three\nphases. (Note that the data in Figs. 2(a) and 3(a) were\ntaken at different fillings and temperatures.) There is a\nremarkable correlation between the behaviors of S40and\ndV/dI with Iin Fig. 3(a) in the three phases. At low\ndrives ( I < I th1≃0.4 nA), the noise is low ( S40<10−7\nV2/Hz). We identify this pinned, low-noise phase as P1.\nFor currents between Ith1andIth2(0.4≲I≲1.2 nA),\nwhere dV/dI fluctuates, S40rises dramatically, by about\nfour orders of magnitude, and reaches a peak value of ≃\n10−4V2/Hz. This is the second phase, P2. Past P2, inphase P3, dV/dI is very low and constant, noise decreases\nand, for large driving currents ( I≳2 nA), it becomes\nroughly constant and settles at a value of ≃3×10−6\nV2/Hz.\nIn Fig. 3(b) we present non-linear dV/dI andS40data\nat a higher temperature of T≃78 mK. The dV/dI trace\nindicates larger threshold currents, with suppressed fluc-\ntuating dV/dI features in the P2 phase. The latter is also\nreflected in S40data, where the peak in S40in phase P2\nis strongly suppressed compared to Fig. 3(a) data. Data\nshown in Fig. 3(c) at intermediate temperatures display\nthe evolution of S40with temperature.\nBesides S40, we also measured the noise power spec-\ntrum between 5 Hz and 1 kHz as a function of frequency\n(f) at different driving currents. Figure 3(f) contains\nsuch data at T≃61 mK, for I= 0, 1, and 3 nA. A\nsummary of the full noise response vs. fandIatT≃\n61 mK is presented as a color-scale plot in Fig. 3(d).\nThe noise spectrum reveals a distinct peak, accompanied\nby additional harmonics, that shift to higher frequen-\ncies with increasing Iin the P2 and P3 phases. The\npeak frequencies are linear in I, consistent with a signal\nthat is of the washboard type [37]. However, the fre-\nquency values we observe are ∼103times smaller than\nthe expected washboard frequencies estimated from the\nexpression f=J/nea 0where Jis the current density,\nnis the electron density, eis the electron charge and a04\nP1P2P3B = 7.50 T (𝜈= 0.155)\nFIG. 4. Phase diagram for the WS as a function of TandI\natB= 7.50 T ( ν= 0.155).\nis the WS lattice constant [22, 37]. Figure 3(g) contains\nthe noise response vs. fat a higher temperature of T≃\n78 mK, for I= 0, 4, and 6 nA. A summarized color-scale\nplot is presented in Fig. 3(e). At this higher temper-\nature, only a broad peak is observed for I= 6 nA. In\nthe Supplemental Material [28], we report the tempera-\nture dependence of the slope of the fvs.Iplots [see\nFigs. 3(d) and (e)], and of the power associated with the\nfundamental frequency of the narrow-band noise spectra.\nOur experimental observations can be summarized in\na phase diagram for the WS as presented in Fig. 4. The\ndata are taken at B= 7.50 T ( ν= 0.155), and Iis\npresented in a log-scale to visualize the width of the P2\nregion clearly. The black data points are obtained from\nthe two thresholds, Ith1andIth2, extracted from dV/dI\nvs.Idata. The extent of the P2 phase in Iis widest at\nthe lowest temperatures. It narrows as Tincreases and\ncompletely disappears above T≃85 mK (see Supple-\nmental Material [28] for dV/dI andS40data at T≃85\nmK).\nOur observations of noise and the dV/dI data follow\nclosely the numerical calculations of Reichhardt and Re-\nichhardt [38–40], which suggest the presence of a pinned\nphase, followed by a moving WS glass phase, and then a\nmoving smectic phase, with increasing drive. At low tem-\nperatures, below the melting temperature of the WS, the\ndisordered WS exhibits channel flow just above the first\nthreshold. In this phase, the conduction is filamentary,\nand can show jumps in dV/dI which can be negative\n[38–40]. Our dV/dI data also show fluctuations, with\nnegative dV/dI just above depinning which disappears\natT≃85 mK. Note that Tm≃120 mK for the WS in\nour sample [28], and thus the P2 phase is lost before the\nWS melts. Our experimental observation of fluctuating\nand negative dV/dI in phase P2 is highly suggestive of\nthe filamentary phase reported in Refs. [39, 40].\nReichhardt and Reichhardt [39, 40] also report on noise\npower (which we associate with S40in our work) as a\nfunction of the drive. Their calculations show suppressednoise in the pinned phase, high noise in the moving glass\nphase, and low noise in the moving smectic phase as the\ndrive increases. Our S40data also reveal a similar quali-\ntative picture, with S40being very low in phase P1, high\nin phase P2, and suppressed again in phase P3, with in-\ncreasing current.\nDespite the above qualitative agreements, there are\nsome notable departures. While theory [39, 40] predicts\nthe emergence of the washboard signal only in the P3\n(moving smectic) phase, we observe the noise peaks in\nphases P2 and P3. Also, although the linear dependence\nof the peak frequency on Ithat we observe [Fig. 3(d)]\nis consistent with the washboard frequency, the three or-\nders of magnitude discrepancy between the experimental\nand estimated frequencies is puzzling and leaves room\nfor speculation. One possibility is that the origin of the\nnoise signal is the motion of WS domains and its inter-\naction with the disorder potential. If we assume that the\ndomains have a reasonably uniform size, and since the\ndomain size is much larger than the WS lattice constant,\nthe resulting peak frequencies can be much smaller than\nthe washboard frequency [41]. Based on our observed\nEth≃1 V/m, and the expression L2\n0≃0.02e/4πϵϵ0Eth\n[42], we obtain a rough estimate of nL2\n0≃600 electrons\nper domain [43–46]. It is worth mentioning that narrow-\nband-noise peaks have been reported in the reentrant in-\nteger quantum Hall states, and there too, the observed\nfrequencies are much smaller than the expected wash-\nboard value [47, 48]. In the Supplemental Material [28],\nwe also report voltage ( V) vs. time ( t) traces of the noise\nmeasured with an oscilloscope. The real-time waveforms\nare saw-tooth like, suggesting a stick-slip motion of the\ndomains as they encounter disorder sites [49, 50].\nThere have been previous experiments reporting the\nnon-linear I-Vand noise characteristics of the high-field\n(ν <1/5) WSs [11, 18–21, 24]. While the data presented\nin these studies have distinct dissimilarities, a common\nconclusion has been the existence of two phases, a pinned\nphase at low drives, followed by a sliding phase at high\ndrives. In our ultra-high-quality samples, we report an\nadditional phase where dV/dI is negative and fluctuat-\ning, and is accompanied by a large, low-frequency noise\n[Fig. 3(c)]. This is different from the data in previous\nwork; so is the presence of a current-dependent peak in\nthe noise spectrum in the depinned phases [Figs. 3(d)\nand (e)] [51].\nWe also note that a rich collection of experiments on\nthe negative dV/dI exists in different platforms, suggest-\ning a variety of possible mechanisms. In Refs. [52–54],\ne.g., transport measurements of classical WSs formed on\nthe surface of superfluid helium at zero magnetic field\nare reported. Reference [55] reports two-threshold I-V\ncharacteristics past the metal-insulator transition in Si-\nMOSFETs (metal-oxide-semiconductor field-effect tran-\nsistors). While the phenomena reported in Refs. [52–55]\nhappen in the insulating phases, a more surprising ob-5\nservation is the similarity of our I-V features to those\nreported in Ref. [56] in a GaAs 2DES, but in vastly\ndifferent parameter ranges: ∼30 times larger density, ∼\n100 times higher temperatures, ∼10,000 times larger cur-\nrents, and ∼10 times lower magnetic fields ( B∼0.8 T,\nν∼50)! The I-Vcharacteristics were attributed to a re-\ndistribution of electrons in energy space, induced by the\ncurrent [56]. These observations highlight the complexi-\nties and yet certain qualitative similarities of the electri-\ncal transport properties in remarkably different platforms\nand circumstances. Finally, it has been theoretically\nproposed that composite fermion correlations present in\nthe FQHSs persist in the WS phase, and that compos-\nite fermion WSs are more stable than electron WSs at\nvery low fillings [13, 57]. Our data presented here should\nprove useful in testing future theories dealing with the\ntransport characteristics of composite fermion WSs.\nWe acknowledge support by the National Science\nFoundation (NSF) Grant No. DMR 2104771 for mea-\nsurements. 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Lett. 122, 116601\n(2019).\n[36] It is worth noting that, while the current threshold\nIth1increases with increasing temperature, the voltage\nthreshold remains approximately fixed at ≃5 mV [see\nFig. 2(c)].\n[37] G. Gr¨ uner, The dynamics of charge-density waves, Rev.\nMod. Phys. 60, 1129 (1988).\n[38] C. Reichhardt, C. J. Olson, N. Grønbech-Jensen, and\nFranco Nori, Moving Wigner Glasses and Smectics: Dy-\nnamics of Disordered Wigner Crystals, Phys. Rev. Lett.\n86, 4354 (2001).\n[39] C. Reichhardt and C. J. O. Reichhardt, Nonlinear dy-\nnamics, avalanches, and noise for driven Wigner crystals,\nPhys. Rev. B 106, 235417 (2022).\n[40] C. Reichhardt and C. J. O. Reichhardt, Noise and ther-\nmal depinning of Wigner crystals, J. Phys.: Condens.\nMatter 35, 325603 (2023).\n[41] P. Monceau, Electronic Crystals: An Experimental\nOverview, Adv. Phys. 61, 325 (2012).\n[42] H. Fukuyama and P. A. Lee, Dynamics of the charge-\ndensity wave. I. Impurity pinning in a single chain, Phys.\nRev. B 17, 535 (1978).\n[43] Our estimate of the WS domain size, L0≃1.5µm,\nis consistent with the recent results of microwave mea-\nsurements [44, 45] where WS pinning mode resonances\nat very small frequencies ( ≃30 MHz) are observed in\nultra-high-quality samples similar to ours. These are the\nsmallest pinning mode resonance frequencies reported for\nmagnetic-field-induced WSs, and attest to the exception-ally high quality of the samples.\n[44] Lili Zhao, Wenlu Lin, Yoon Jang Chung, Adbhut Gupta,\nKirk W. Baldwin, Loren N. Pfeiffer, and Yang Liu, Dy-\nnamic Response of Wigner Crystals, Phys. Rev. Lett.\n130, 246401 (2023).\n[45] Matthew L. Freeman, P. T. Madathil, L. N. Pfeiffer, K.\nW. Baldwin, Y. J. Chung, L. W. Engel, R. Winkler, and\nM. Shayegan, unpublished.\n[46] We also obtain the activation energies ( EA) by measuring\nthe temperature dependence of Rxxin the filling factor\nrange 6 ≲1/ν≲7 using the expression Rxx∝eEA/2kBT.\nWe report EA≃2 K for our sample [14]. This is twice\nlarger than the values reported previously for samples\nwith similar densities but of lower quality. It is close to\nthe theoretically calculated EAfor defect energies in a\nclean WS [13], reflecting the extremely high quality, and\nlarge domains of the WS realized in our system.\n[47] K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W.\nWest, Observation of Narrow-Band Noise Accompanying\nthe Breakdown of Insulating States in High Landau Lev-\nels, Phys. Rev. Lett. 90, 226803 (2003).\n[48] Jian Sun, Jiasen Niu , Yifan Li, Yang Liu, L.N. Pfeif-\nfer, K.W. West, Pengjie Wang, Xi Lin, Dynamic order-\ning transitions in charged solid, Fundamental Research\n2, 178 (2022).\n[49] G. Gr¨ uner, Density Waves in Solids (Perseus, Cam-\nbridge, Massachusetts), 1994.\n[50] Tomoyuki Sekine, Norikazu Satoh, Mitsuhiro Nakazawa,\nand Toshikazu Nakamura, Sliding spin-density wave in\n(TMTSF) 2PF6studied with narrow-band noise, Phys.\nRev. B 70, 214201 (2004).\n[51] While a negative dV/dI and frequency-dependent noise\nin WS was reported in Ref. [24], the negative dV/dI\nregime was devoid of any fluctuations, and the noise peak\nin the spectra was restricted only to the negative dV/dI\nregime. The frequency-dependent noise in our experi-\nments extends well beyond the negative and fluctuating\ndV/dI regime and into the P3 phase, clearly differentiat-\ning our data in this new, ultra-high-quality 2DES.\n[52] P. Glasson, V. Dotsenko, P. Fozooni, M. J. Lea, W.\nBailey, G. Papageorgiou, S. E. Andresen, and A. Kris-\ntensen, Observation of Dynamical Ordering in a Confined\nWigner Crystal, Phys. Rev. Lett. 87, 176802 (2001).\n[53] David G. Rees, Niyaz R. Beysengulov, Juhn-Jong Lin,\nand Kimitoshi Kono, Stick-Slip Motion of the Wigner\nSolid on Liquid Helium, Phys. Rev. Lett. 116, 206801\n(2016).\n[54] Shan Zou, Denis Konstantinov, and David G. Rees, Dy-\nnamical ordering in a two-dimensional electron crystal\nconfined in a narrow channel geometry, Phys. Rev. B\n104, 045427 (2021).\n[55] Pedro Brussarski, S. Li, S. V. Kravchenko, A. A.\nShashkin, and M. P. Sarachik , Transport evidence for\na sliding two-dimensional quantum electron solid, Nat.\nCommun. 9, 3803 (2018).\n[56] A. A. Bykov, Jing-qiao Zhang, Sergey Vitkalov, A. K.\nKalagin, and A. K. Bakarov, Zero-Differential Resistance\nState of Two-Dimensional Electron Systems in Strong\nMagnetic Fields, Phys. Rev. Lett. 99, 116801 (2007).\n[57] A. C. Archer, K. Park and J. K. Jain, Competing Crystal\nPhases in the Lowest Landau Level, Phys. Rev. Lett.\n111, 146804 (2013).Supplemental Material for “Moving crystal phases of a quantum Wigner solid in an\nultra-high-quality 2D electron system”\nP. T. Madathil,1K. A. Villegas Rosales,1Y. J. Chung,1K. W. West,1\nK. W. Baldwin,1L. N. Pfeiffer,1L. W. Engel,2and M. Shayegan1\n1Department of Electrical and Computer Engineering,\nPrinceton University, Princeton, New Jersey 08544, USA\n2National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA\n(Dated: January 25, 2024)\nI. Experimental setup for dV/dI , noise, and voltage\n(V) vs. time ( t) measurements\nV+\nV-\nLock -in    Amplifier\nLock -in Amplifier𝑉𝑎𝑐OUT\n𝐼𝑎𝑐OUT 𝑉𝑎𝑐IN𝑉𝑑𝑐IN\n~  \n~  A; 𝐼𝑑𝑐OUT\n1 GΩ100 MΩSpectrum \nAnalyzer100 MΩ\nOscilloscopePre-\nAmplifier(a) (b)\n(c)SampleSample𝑉𝑑𝑐IN; 𝐼𝑑𝑐OUT\n𝑉𝑑𝑐IN; 𝐼𝑑𝑐OUTSample100 MΩ\nFIG. S1. Measurement setup for: (a) dV/dI vs.I, (b) noise\nspectrum, and (c) Vvs.t.\nFigure S1(a) shows the schematic for the dV/dI vs.\nImeasurements. A voltage supply (Keithley 2410) is\nused to source a dcvoltage and measure the dccurrent.\nThe circuit is driven in the constant-current mode by\nconnecting a 100 MΩ resistor in series to the sample.\nAdditionally, an acvoltage source with a 1 GΩ resistor\nis used to source the acexcitation current for the differ-\nential resistance measurements. The ratio of VOUT\nac and\nIOUT\nac measured using lock-in amplifiers gives the value\nofdV/dI . The noise measurement setup is presented in\nFig. S1(b). The protocol for the dcdrive remains the\nsame. We use a spectrum analyzer (HP 3562A) to mea-\nsure the noise power spectral density. Figure S1(c) shows\nthe scheme for Vvs.tmeasurements. A pre-amplifier\nis used to boost the voltage signal from the sample in\ntheaccoupling mode which is then monitored with an\noscilloscope.\nII. Melting temperature of the Wigner solid\nIn our measurements, we can deduce the melting tem-\nperature ( Tm) of the Wigner solid (WS) from the tem-\nperature dependence of the S40noise in the high-current\nregime (P3 phase). Figure S2(a) contains S40data at\nT≃61 and 85 mK. The average S40power in the P1\nphase (near I= 0), and in the P3 phase, where S40sat-\nurates, are marked by the green lines for T= 61 mK,\nFIG. S2. (a) S40data at T≃61 and 85 mK at B= 7.50\nT (ν= 0.155) as a function of the driving current ( I). The\ndifference between the averaged noise power in the P1 and\nP3 phases is marked as ∆ N61and ∆ N85for the two different\ntemperatures. (b) ∆ Nvs.T. From the x-intercept of the\nlinear fit (red line), we estimate a melting temperature, Tm≃\n120 mK.\nand red lines for T= 85 mK. We define the power differ-\nence between the P3 and P1 phases as ∆N. Figure S2(b)\nshows the ∆N values for various temperatures. The error\nbars are computed from the standard deviation of the S40\ndata points involved in obtaining the average noise pow-\ners in the P1 and P3 phases. We observe that, despite\nthe error bars, ∆N falls off approximately linearly with\nT, and extrapolates to zero at a temperature which we\nassociate with Tm. By fitting a straight line to the data\npoints in Fig. S2(b), we estimate Tm≃120 mK from the\nx-intercept of the fitted line.\nIII.dV/dI andS40vs.IatB= 7.50 T ( ν= 0.155)\nandT≃85 mK\nFigure S3 shows the dV/dI and noise power, defined as\nS40in the main text, vs. Iat a high temperature of T≃\n85 mK. The system makes a transition from the pinned\nphase to the constant, linear phase directly, without ex-\nhibiting any fluctuating or negative dV/dI features. S40\ndata also support this, revealing a single transition from\na low-noise ( S40≃10−8V2/Hz) pinned phase (P1) to a\nconstant, higher-noise phase ( S40≃10−6V2/Hz), char-arXiv:2401.13533v1  [cond-mat.mes-hall]  24 Jan 20242\nP1 P3\nFIG. S3. dV/dI (blue trace) and noise power, defined as S40\n(red trace) vs. IatT≃85 mK.\nacteristic of the P3 phase. The dominant peak in S40,\nattributed to the P2 phase at low temperatures, seen in\nFig. 3(c) of the main text, is absent in Fig. S3. The\nzero-drive resistance in Fig. S3 is of the order of 1 MΩ\nand the I-Vis significantly non-linear, which can be seen\nfrom the dV/dI vs.Itrace. The noise also increases\nby∼2 orders of magnitude across the P1 to P3 transi-\ntion. These observations confirm the persistence of the\nsolid phase at this temperature and the absence of the P2\nphase before the solid melts at T≃120 mK (see section\nII).\nIV. Current dependence of the narrow-band noise\npeaks\nFIG. S4. Noise power vs. ffor different values of Ias\nindicated, at T≃61 mK and B= 7.50 T ( ν= 0.155). The\nyellow bands highlight the width of the first harmonic of the\nnarrow-band noise peaks.\nIn Fig. S4, we show the noise power spectra for four\ndifferent values of IatT≃61 mK, and B= 7.50 T ( ν\n= 0.155). The traces shown are deep in the P3 phase,\nwhich can be concluded from Figs. 3(a) and 3(d) of the\nmain text. We define the width of the first harmonic\nof the narrow-band noise peaks as the frequency range\nof the yellow bands presented in Fig. S4. The widths\nobtained for the four different currents ( I= 2.0, 2.8, 3.4,\nand 4.0 nA) are ≃140, 220, 300, and 370 Hz, respectively.Despite this increase in the peak width, the integrated\nnoise power in the yellow bands stays approximately the\nsame (0 .011±0.001 V2) for all the four current values.\nV. Temperature dependence of the narrow-band\nnoise peaks\nFIG. S5. (a) Slope ( α) of the dependence of the fundamental\nfrequency peak on Iin the fvs.Iplots is shown as a func-\ntion of Ton a log-log scale. (b) Temperature dependence of\nthe power associated with the fundamental frequency of the\nnarrow-band noise spectra in a log-log, and linear scale plot\n(inset).\nIn our measurements, we observe a strong dependence of\nthe slope ( α) of the fundamental frequency in the fvs.\nIplots on temperature (see Figs. 3(d) and 3(e) of the\nmain text). In Fig. S5(a), we show this dependence in\na log-log plot. We observe an increasing trend of αwith\nan approximate power-law dependence of α∝T3.3. We\nalso measure the power contained within the fundamen-\ntal frequency by integrating the noise power spectrum\ncurves in the frequency range corresponding to the yel-\nlow bands shown in Fig. S4, and then subtracting the\npower of the noise floor calculated for the same window.\nWe repeat this process for different Iwhen the system\nis deep in the P3 phase (see, for example, Fig. S4). We\nthen obtain the average of these power values, and the\nstandard deviation. We present the average values in a\nlog-log plot shown in Fig. S5(b); the error bars reflect\nthe standard deviation. The magnitude of the slope in\nthe log-log scale plot increases with T, implying that the\npower decreases drastically at high temperatures. We\nalso plot the power in a linear scale, which is presented\nin Fig. S5(b) inset. An explanation for the temperature\ndependencies of αand the power of the first harmonic,\nalong with the current dependence of the narrow-band\nnoise peaks described in Section IV, awaits future theo-\nretical work that treats the pinned WS at a microscopic\nlevel.3\nVI.Vvs.tatB= 7.50 T ( ν= 0.155) and T≃61\nmK\nFIG. S6. Vvs.tforI= 2.5, 5.0, and 7.5 nA at B= 7.50 T\n(ν= 0.155) and T≃61 mK. For clarity, he traces are shifted\nvertically by 0, 3, and 6 mV.\nFigure S6 shows the Vvs. time ( t) traces taken at three\ndifferent currents, I= 2.5, 5.0, and 7.5 nA, at 7.50 T ( ν=\n0.155) and T≃61 mK. The traces, which were measured\nusing an oscilloscope (Tektronix TDS 1002), are shifted\nvertically by 0, 3, and 6 mV, respectively. The signal\nwasac-coupled at the pre-amplifier stage and so, the dc\nvoltage was filtered out, as shown in Fig. S1(c). For I\nmentioned above, the WS is deep in the P3 phase, which\ncan be seen from Fig. 3(a) of the main text.\nThe waveform of the real-time signals is saw-tooth like,\ncomprised of a slow rise and a fast drop, suggesting a\nstick-slip motion of the WS, similar to what has been\npreviously reported for the sliding dynamics of the charge\ndensity waves (CDWs) [1, 2]. However, because of an\nuncertainty in determining the polarity of the V(t) traces\nin the experimental setup, it is possible that the saw-\ntooth profile is flipped, in which case there would be a\nfast rise followed by a slow decay. This type of waveform\nhas also been previously observed in CDW motion and\nis attributed to the stick-slip motion of the CDW as it\nslides over a disordered background [3].VII. Reproducibility of the non-linear I-Vdata\nIn this section, we discuss the reproducibility of the data\nafter thermal cycling. In each panel of Fig. S7 we show\ntwo traces (blue and red) taken at a given magnetic field.\nEach blue trace is obtained after warming the sample up\ntoT= 6 K and then cooling it back to the same low tem-\nFIG. S7. Reproducibility of the non-linear I-Vdata. In each\npanel, the blue trace was taken after a thermal cycling to 6 K.\nThe red traces are shifted up by 2 MΩ. The temperatures at\nwhich the two traces in each panel were taken are very close\nbut not identical (see text).\nperature as in the red trace, as indicted in each panel.\nThe red traces are shifted up by 2 MΩ for clarity and ease\nof comparison. We note that each pair of traces, although\nnot identical, follow each other closely. They show similar\nthreshold currents, and the subsequent fluctuations are\nalso repeatable. There are however, some quantitative\ndifferences, which can be attributed to the fact that the\nnew state of the sample after a thermal cycle is not iden-\ntical to the old state. Also, the temperatures at which the\ntwo traces were measured in each panel are not exactly\nthe same (e.g. the blue and the red traces at B= 8.2 T\nin panel (d) were taken at nominal temperatures of 79.6\nmK and 80.6 mK, respectively).\n[1] G. Gr¨ uner, Density Waves in Solids (Perseus, Cambridge,\nMassachusetts), 1994.\n[2] P. Monceau, Electronic Crystals: An Experimental\nOverview, Adv. Phys. 61, 325 (2012).\n[3] Tomoyuki Sekine, Norikazu Satoh, Mitsuhiro Nakazawa,\nand Toshikazu Nakamura, Sliding spin-density wave in\n(TMTSF) 2PF6studied with narrow-band noise, Phys.\nRev. B 70, 214201 (2004)."    },    {        "title": "2401.13557v1.Scalarized_Hybrid_Neutron_Stars_in_Scalar_Tensor_Gravity.pdf",        "content": "arXiv:2401.13557v1  [astro-ph.HE]  24 Jan 2024Scalarized Hybrid Neutron Stars in Scalar Tensor Gravity\nFahimeh Rahimi1,2andZeinab Rezaei1,2∗\n1Department of Physics, School of Science,\nShiraz University, Shiraz 71454, Iran.\n2Biruni Observatory, School of Science,\nShiraz University, Shiraz 71454, Iran.\nAbstract\nHybrid neutron stars, the compact objects consisting hadro nic matter and strange quark mat-\nter, can be considered as the probes for the scalar tensor gra vity. In this work, we explore the\nscalarization of hybrid neutron stars in the scalar tensor g ravity. For the hadronic phase, we apply\na piecewise polytropic equation of state constrained by the observational data of GW170817 and\nthe data of six low-mass X-ray binaries with thermonuclear b urst or the symmetry energy of the\nnuclear interaction. In addition, to describe the strange q uark matter inside the hybrid neutron\nstar, different MIT bag models are employed. We study the effects of the value of bag constant, the\nmass of s quark, the perturbative quantum chromodynamics co rrection parameter, and the density\njump at the surface of quark-hadronic phase transition on th e scalarization of hybrid neutron stars.\nOur results confirm that the scalarization is more sensitive to the value of bag constant, the mass of\ns quark, and the density jump compared to the perturbative qu antum chromodynamics correction\nparameter.\n∗Corresponding author. E-mail: zrezaei@shirazu.ac.ir\n1I. INTRODUCTION\nIn hybrid neutron stars (HNSs), the remnants of massive stars, the nucleons get decom-\nposed into quarks andtherefore formdeconfined strangequark matter. Quantum chromody-\nnamics (QCD) explains such phase transition from baryonic to decon fined quark matter at\nultra-high densities [1]. The strange quark matter in these stars is s urrounded by hadronic\nmatter. With slow conversion speed at the interface, a new class of dynamically stable HNSs\ncan be exist [2]. Weakening the deconfinement phase transition leads to axions stabilizes\nmassive HNSs against gravitational collapse [3]. HNSs can experience two phase transitions\nfrom hadronic matter to low- and high-density quark matter phase s [4]. The stars with\nsufficient amount of strange quark matter in the core can convert into the strange quark\nstars [5]. No core singularity in the formations of the HNS can be foun d [6].\nAstrophysical observations are in agreement with the results for the HNSs within the\nMIT bag model [7], QCD motivated chiral approach [8], vector bag mod el [9], self-consistent\nNambu-Jona-Lasinio model [10–15], and Field Correlator Method [16]. In Dyson-Schwinger\nquark model, HNS mass and radius are affected by the quark-gluon v ertex and gluon prop-\nagator [17]. Considering Nambu-Jona-Lasinio model for HNSs, the v ector-isovector terms\nlead to larger quark cores and smaller deconfinement density [18]. As trophysical data verify\nthe high values of the bag parameter in MIT bag model for HNSs [19]. B ag pressure in MIT\nbag model has important influence in the emergence of the special p oints in mass radius\nrelation of HNSs [20]. HNSs can be self-bound rather than gravitatio nally bound [15]. The\nstrength of the vector interaction affects the equation of state (EoS) of HNS matter and\nvector strength results in the stiffness of the EoS [21, 22]. The mat ter in the intermediate\ndensity range is expected to be soft, while it is stiff in higher density ra nge [9]. The quark-\nhadron density jump and the EoS determine the crust-breaking fr equency and maximum\nfiducial ellipticity of HNSs [23].\nContribution of HNS quark core is more related to the radius compar ed to the star mass\nwhich is affected by the EoS of the quark matter [14]. Exothermic pha se transition in cold\nneutron stars can lead to HNSs with masses smaller than a parent ne utron star [24]. The\nmass and tidal deformability constraints from the observational d ata are in agreement with\nlarge strangeness content and large quark cores for HNSs in binar y systems [25]. Radii\nof HNSs can constrain the symmetry energy [26]. Besides, the maxim um masses of HNSs\n2constrain the EoS of symmetric nuclear and quark matter [26]. The d ark matter in HNSs\nalters the energy density discontinuity and results in the decrease of the star minimum mass\n[27]. Besides, it affects the star mass radius relation as well as the co re radius of the star\n[27, 28]. Hybrid equations of state describing the quark matter inne r core and the nuclear\nouter core can lead to star with mass of M= 0.77+0.20\n−0.17M/circledottextand radius of R= 10.4+0.86\n−0.78km\nunlike the nuclear models [29]. Lower transition energy density, smalle r energy density\ndiscontinuity, and higher sound speed of quark matter lead to supe rmassive HNSs [30].\nIn relativistic stars, the tachyonic instability as a result of the scala r field and curvature\nnonminimal coupling results inthespontaneousscalarization. X-Ray andgravitational-wave\nobservations of neutron stars can constrain the massive scalar t ensor theories [31]. Applying\nBayesian analysis, the spontaneous scalarization parameters hav e been constrained using\nthe neutron star mass and radius measurements [32]. Employing the parameters in accor-\ndance with the observational data, the scalarization results in sign ificant deviations from\nthe general relativistic universal relations of neutron stars [33]. T he scalar field in neutron\nstars affects the pulse profile’s overall shape leading to strong dev iations from the General\nRelativity (GR) one [34]. Mutual interplay between magnetic and scala r fields has influence\non the magnetic and the scalarisation properties of neutron stars [35]. Spontaneous scalari-\nsation alters the neutron star magnetic deformation and emitted q uadrupolar gravitational\nwaves [36].\nScalarfieldinneutronstarsissuppressed duetotheself-interact ioninscalartensortheory\n[37]. Topological neutron stars in tensor-multi-scalar theories of g ravity are characterized\nby topological charge [38]. In scalar tensor theories, a quasi-unive rsal relation between the\nscalar charge and stellar binding energy has been presented [39]. Sc alar-gauge coupling in\nEinstein gravity induces spontaneous scalarization in charged star s [40]. The scalar field\ncouplings to Ricci scalar and Gauss-Bonnet invariant affect the dom ain of existence and the\namount of scalar charge in scalarized neutron stars [41].\nThe neutron star maximum compactness in GR is larger than the one in scalar tensor\ngravity [42]. In mixed configurations of tensor-multi-scalar soliton s and relativistic neutron\nstars, the stability region extends over larger central energy de nsities and central values\nof the scalar field [43]. In tensor-multi-scalar theories of gravity, a family of scalarized\nbranches specified by the number of the scalar field nodes appears and separates from the\ngeneral relativistic solution at the points with new unstable modes [4 4]. Nonscalarized stars\n3can convert to scalarized ones through the core collapse [45]. Neu tron stars in degenerate\nhigher-order scalar tensor theories have masses and radii larger than the ones in GR [46].\nIn this work, we investigate the properties of scalarized HNSs. In s ection II, the EoSs which\ndescribe the HNSs arepresented. Section III is related to scalar t ensor theory. The structure\nof scalarized HNSs is explained in section IV. Section V is devoted to co nclusion.\nII. HYBRID NEUTRON STAR EQUATION OF STATE\nIn this paper, we utilize a model for HNS similar the one described in Ref . [47]. The\nHNS includes a quark core and a hadronic layer. In this model, we supp ose that a sharp\nphase transition surface without a mixed phase divides two parts an d the density can be\ndiscontinuous at the phase splitting surface [48]. For the hadronic p hase, we employ the\ndense neutron star matter EoS in the form of a piecewise polytropic expansion constrained\nby the observational data related to GW170817 as well as the data of six low-mass X-ray\nbinaries with thermonuclear burst or the symmetry energy of the n uclear interaction [49].\nIn order to describe the quark core, we apply different MIT bag mod els including massless\nquark approximation [50], massive strange quark matter considerin g the QCD coupling\nconstant [51–53], and interacting strange quark matter specified by effective bag constant\nand perturbative QCD corrections term [51]. These models for the q uark phase will be\ndescribed in the following. We take the density jump at the quark-ha dronic phase transition\nsurface as a free parameter. This jump is described by the parame terηdefined as\nη≡ǫq\nǫh−1, (1)\nwith the density at the top of the quark phase, ǫq, and the density at the bottom of the\nhadronic phase, ǫh. We can calculate ǫqorǫhandthe pressure at the phase transition surface\nusingpq=phat the phase transition surface.\nIn the massless quark approximation, a system containing u,d, andsquarks that are\nnoninteracting and massless is considered. Applying this approximat ion in MIT bag model,\nthe quark pressure Ptakes the following form\nP=1\n3(ǫ−4B), (2)\ninwhichǫdenotestheenergydensityofthequarkmatterdistributionand Bpresentsthebag\nconstant that acts as the inward pressure appropriate to confin e quarks inside the bag. Fig.\n41 shows the EoS of HNS inthe massless quark approximation with differ ent values of the bag\nconstantBand the density jump η. We have assumed ρ0= 1.66×1014g/cm3. The pressure\ntakes higher values as the density jump increases, especially at lowe r densities. The pressure\ndecreases as the bag constant increases from B= 70MeV/fm3toB= 75MeV/fm3.\nB=70MeV/fm3\nη0.0\nη0.45\nη0.85\n24681012140.00.51.01.52.0\nρ˜/ρ0P˜/ρ0B=75MeV/fm3\nη=0.0\nη=0.45\nη=0.85\n24681012140.00.51.01.52.0\nρ/ρ0P/ρ0\nFIG. 1: EoS of HNS in the massless quark approximation consid ering different values of the bag\nconstant, B, and the density jump, η.\nThe second MIT bag model that we employ in this work is the massive st range quark\nmatter considering the QCDcoupling constant. In thismodel, thequ ark matter iscomposed\nof the massless uanddquarks, electrons e, and massive squarks [51]. The EoS of quark\nmatter in this system depends on the bag constant, B, the QCD coupling constant, αc, the\nmass of the strange quark, ms, and the renormalization point, ρN. We select the values\nρN= 313MeVsimilar to Ref. [51] and αc= 0.2 following Ref. [53]. In Fig. 2, we have\npresented the EoS of HNS considering the massive strange quark m atter with the QCD\ncoupling constant. With lower values of the bag constant and the st range quark mass,\nthe effect of the density jump on the EoS is more significant. Besides , the EoS is softer\nconsidering higher values of the bag constant and the strange qua rk mass.\nInthethirdmodel, theinteracting strangequarkmattercharact erized bytheeffective bag\nconstant and the perturbative QCD corrections term is explored. For this model, a mixture\nof quarks,u,d,s, and electrons e, with the transformation due to weak interaction between\nquarks and leptons is considered [51]. The grand canonical potentia l per unit volume is\ngiven by\nΩ =/summationdisplay\ni=u,d,s,eΩ0\ni+3(1−a4)\n4π2µ4+Beff. (3)\n5B=45MeV/fm3\nms=185MeV\nη=0.0\nη=0.45\nη=0.85\n24681012140.00.51.01.52.0\nρ˜/ρ0P˜/ρ0B=5 \u0000MeV/fm3\nms=200MeV\nη=0.0\nη=0.45\nη=0.85\n24681012140.00.51.01.52.0\nρ˜/ρ0P˜/ρ0\nFIG. 2: EoSofHNSconsideringthemassivestrangequarkmatt er withtheQCDcouplingconstant.\nDifferent values of the bag constant, B, the mass of the strange quark, ms, and the density jump,\nη, have been applied.\nIn the above equation, the grand canonical potential for u,d,squarks and electrons as\nthe ideal relativistic Fermi gases is presented by Ω0\ni. Besides, the average quark chemical\npotential is denoted by µ= (µu+µd+µs)/3. We present the contributions from the QCD\nvacuum using Beffand the perturbative QCD contribution from one-gluon exchange f or\ngluon interaction by a4. The number density of particles in strange quark matter is written\nas\nni=−∂Ω\n∂µi, (4)\nin whichµi(i=u,d,s,e) is the chemical potential of particles. The weak interactions\ndetermine the conditions for the quark matter at the equilibrium sta te,\nµd=µu+µe, (5)\nµd=µs. (6)\nThe charge neutrality condition is also given by\n2\n3nu=1\n3[nd+ns]+ne. (7)\nThe pressure of quark matter is then as follows\nP=−Ω, (8)\n6and the energy density is given by\nε= Ω+/summationdisplay\ni=u,d,s,eµini. (9)\nFig. 3 shows the EoS of HNS in the third model with different values of Beffanda4.\nWe have fixed the strange quark mass ms= 100MeVand the density jump η= 0. The\npressure decreases by increasing the effective bag constant. Th e influence of the effective bag\nconstant is more significant at higher densities. In this paper, we inv estigate the scalarized\nHNSs which described by the above EoSs.\na4=0 \u0001 \u0002 \u0003\nBeff=42.1MeV /fm3\nBeff=46.8MeV /fm3\nBeff=52.0MeV /fm3\n4681012140.00.51.01.52.02.53.0\n˜/0P˜/0a4=\u0004 \u0005 \u0006 \u0007\nBeff=42.1MeV /fm3\nBeff=46.8MeV /fm3\nBeff=52.0MeV /fm3\n4681012140.00.51.01.52.02.53.0\n˜/0P˜/0\na4=\b \t \n \u000b\nBeff=42.1MeV /fm3\nBeff=46.8MeV /fm3\nBeff=52.0MeV /fm3\n4681012140.00.51.01.52.02.53.0\n˜/0P˜/0\nFIG. 3: EoS of HNS in the third model considering different valu es ofBeffanda4with fixed values\nof the strange quark mass ms= 100MeVand the density jump η= 0.\nIII. SCALAR TENSOR GRAVITY\nScalar tensor theories in Einstein frame can be defined by the followin g action\nS[gµν,φ,Ψm] =1\n16π/integraldisplay\nd4x√−g(R−2∇µφ∇µφ)+Sm[Ψm,a(φ)2gµν].(10)\n7In the above equation, g=det(gµν),Ris Ricci scalar, φshows the scalar field, Ψ mdenotes\nthe matter field, and a(φ) is the coupling function satisfying the relation ˜ gµν=a(φ)gµν\nbetweengµνin Einstein frame and ˜ gµνin Jordan frame. Here, we assume the form of the\ncoupling function as a(φ) =e1\n2β(φ−φ0)2with the coupling constant βandφ0= 0. To describe\na spherical symmetric static HNS in the scalar tensor theory in Einst ein frame, we apply\nthe following form of the spacetime line element\nds2=−N(r)2dt2+A(r)2dr2+r2(dθ2+sin2θdφ2), (11)\nwith the metric functions N(r) andA(r) = [1−2m(r)/r]−1/2and the mass profile m(r).\nTherefore, the field equations result in the following differential equ ations [28],\ndm\ndr= 4πr2a4˜ǫ+r\n2(r−2m)/parenleftBigdφ\ndr/parenrightBig2\n, (12)\ndlnN\ndr=4πr2a4˜p\nr−2m+r\n2/parenleftBigdφ\ndr/parenrightBig2\n+m\nr(r−2m), (13)\nd2φ\ndr2=4πra4\nr−2m/bracketleftbigg\nα(˜ǫ−3˜p)+r(˜ǫ−˜p)dφ\ndr/bracketrightbigg\n−2(r−m)\nr(r−2m)dφ\ndr, (14)\nd˜p\ndr=−(˜ǫ+ ˜p)/bracketleftbigg4πr2a4˜p\nr−2m+r\n2/parenleftBigdφ\ndr/parenrightBig2\n+m\nr(r−2m)+αdφ\ndr/bracketrightbigg\n, (15)\ndmb\ndr=4πr2a3˜ρ/radicalBig\n1−2m\nr. (16)\nHere, ˜ǫis the energy density, ˜ pshows the pressure, and mbdenotes the baryonic mass. The\nboundary conditions which used for solving these equations are\nm(0) =mb(0) = 0,lim\nr→∞N(r) = 1, φ(0) =φc,lim\nr→∞φ(r) = 0,\ndφ\ndr(0) = 0,˜p(0) =pc,˜p(Rs) = 0. (17)\nHere, the star radius is presented by Rsandcshows the center of star. Considering an\nappropriate boundary conditions at r= 0 as well as a guess φ(0) =φcat the center, the\niteration on φcis done getting the condition [54, 55],\nφs+2ψs/radicalbig\n˙ν2\ns+4ψ2\nsarctanh/bracketleftBigg/radicalbig\n˙ν2s+4ψ2s\n˙νs+2/Rs/bracketrightBigg\n= 0, (18)\nin which s indicates the quantities on the surface of the star as well a sψs= (dφ/dr)sand\n˙νs= 2(dlnN/dr)|s=Rsψ2\ns+2ms/[Rs(Rs−2ms)]. The ADM mass, MADM, and the scalar\n8charge,ω, are also calculated as follows [26,28],\nMADM=R2\ns˙νs\n2/parenleftbigg\n1−2ms\nRs/parenrightbigg1\n2\nexp/bracketleftBigg\n−˙νs/radicalbig\n˙ν2s+4ψ2sarctanh/parenleftBigg/radicalbig\n˙ν2\ns+4ψ2\ns\n˙νs+2/Rs/parenrightBigg/bracketrightBigg\n,(19)\nω=−2MADMψs/˙νs. (20)\nHere, we report the values of the HNS ADM mass. In the following, we study the HNS in\nthe scalar tensor gravity.\nIV. RESULTS AND DISCUSSION\nA. Mass of scalarized hybrid neutron star\nWe have presented the mass of HNS applying different EoS models in Fig s. 4-6. Figs. 4\nand 5 confirm that with the higher values of the density jump, the st ar mass grows. This\neffect becomes more important by increasing η= 0.45 toη= 0.85. Fig. 6 verifies that the\nstar mass reduces as Beffincreases. In all cases, the deviation of scalarized star mass from\nthe GR one is more clear with smaller values of β. The range of star mass at which the HNS\nis scalarized becomes larger by decreasing the coupling constant. A pplying three models for\nthe HNS EoS, the most massive stars are the scalarized ones. With lo wer values of β, the\nmaximum star mass grows. The central densities corresponding to most massive scalarized\nstars decrease as the density jump increases. Our results show t hat considering the higher\nvalues of the bag constant leads to the scalarization of the HNSs wit h larger central densities\n(see Figs. 4 and 5). It is obvious from Fig. 6 that the most massive st ars in the third model\nare the scalarized ones with lower values of Beffandβ. Besides, the central density related\nto the maximum mass of scalarized stars in the third model increases byBeff.\nIn Tables I-III, the maximum mass of HNS considering different mode ls for the HNS EoS\nemploying the scalar tensor gravity and GR is presented. These tab les confirm that the\nmaximum mass of scalarized stars with β=−4.5 is almost equal to the GR one. Besides,\nfor both scalarized stars and the stars in GR, the maximum mass dec reases by increasing the\ncoupling constant. Table II shows that in the second model, the max imum mass decreases\nby increasing the bag constant and the mass of the strange quark . In addition, it is obvious\nfrom Table III that the maximum mass of HNS is not significantly affect ed by the model\nparametera4.\n9B=70MeV/fm3\nη=0.0,β4.5\n=0.45,β4.5\n=0.85,β4.5\nGR(=0.0)\nGR(=0.45)\nGR(=0.85)\n510 15 201.21.41.61.82.0\n˜\nc/0M(M⊙)B=70MeV/fm3\nη=0.0,β5\nη0.45,β5\nη0.85,β5\nGR(η0.0)\nGR(η=0.45)\nGR(η=0.85)\n510 15 201.21.41.61.82.0\n˜\nc/0M(M⊙)\nB=75MeV/fm3\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\nGR(η=0.0)\nGR(η=0.45)\nGR(η=0.85)\n510 15 201.21.41.61.82.0\n˜\nc/0M(M⊙)B=75MeV/fm3\nη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nGR(η=0.0)\nGR(η=0.45)\nGR(η=0.85)\n510 15 201.21.41.61.82.0\n˜\nc/0M(M⊙)\nFIG. 4: HNS mass in the first model for the HNS EoS applying the s calar tensor gravity and GR.\nDifferent values of the bag constant, B, the density jump, η, and the coupling constant, β, have\nbeen considered.\nB= 70MeV/fm3B= 75MeV/fm3\nβη= 0 η= 0.45 η= 0.85η= 0 η= 0.45 η= 0.85\n−4.51.79 1.80 1.84 1.75 1.76 1.80\n−51.90 1.91 1.95 1.86 1.87 1.91\nGR1.79 1.80 1.84 1.75 1.76 1.80\nTABLE I: Maximum mass (in solar mass unit) of HNS in the first mo del for the HNS EoS with\ndifferent values of the bag constant, B, the density jump, η, and the coupling constant, β.\nB. Central scalar field in hybrid neutron star\nFigs. 7-9 demonstrate the central scalar field in HNS employing differ ent models for the\nHNS EoS. The coupling constant affects the spontaneous scalariza tion with higher values\n10B=4 \fMeV/fm3\nms=185MeVη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\nGR(,η=0.0)\nGR(η=0.45)\nGR(η=0.85)\n5101520251.01.21.41.61.82.02.2\n˜\nc/0M(M⊙)\nB=\r \u000eMeV/fm3\nms=185MeVη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nGR(η=0.0)\nGR(η=0.45)\nGR(η=0.85)\n5101520251.01.21.41.61.82.02.2\n˜\nc/0M(M⊙)\nB=\u000f \u0010MeV/fm3\nms=200MeVη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\nGR(η=0.0)\nGR(η=0.45)\nGR(η=0.85)\n5101520251.01.21.41.61.82.02.2\n˜\nc/0M(M⊙)\nB=\u0011 \u0012MeV/fm3\nms=200MeVη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nGR(η=0.0)\nGR(η=0.45)\nGR(η=0.85)\n5101520251.01.21.41.61.82.02.2\n˜\nc/0M(M⊙)\nFIG. 5: Same as Fig. 4 but for the second model of the HNS EoS. We have applied different values\nof the bag constant, B, the mass of the strange quark, ms, and the density jump, η.\nB= 45MeV/fm3B= 56MeV/fm3\nms= 185MeV ms= 200MeV\nβη= 0 η= 0.45η= 0.85η= 0 η= 0.45 η= 0.85\n−4.51.92 1.98 2.05 1.76 1.77 1.82\n−52.04 2.10 2.17 1.88 1.88 1.92\nGR1.92 1.98 2.04 1.76 1.77 1.81\nTABLE II: Same as Table I but for the second model of the HNS EoS .\nof the central scalar field considering the lower values of the couplin g constant. The range\nof the central density at which the central scalar field is nonzero b ecomes larger applying\nthe smaller values of β. Figs. 7 and 8 verify that the central density at which φcbecomes\nnonzero (first critical density) decreases as the density jump in H NS grows. The central\ndensity at which the spontaneous scalarization terminates, i.e. φc= 0, (second critical\n11a4=0.56\nBeff=42.1MeV /fm3,=-4.5\nBeff=46.8MeV /fm3,=-4.5\nBeff=52.0MeV /fm3,=-4.5\nGR(Beff=42.1MeV /fm3)\nGR(Beff=46.8MeV /fm3)\nGR(Beff=52.0MeV /fm3)\n681012141618201.21.41.61.82.02.2\nρ˜\nc/ρ0M(M\n\u0013)a4=0.56\nBeff=42.1MeV /fm3,=-5\nBeff=46.8MeV /fm3,=-5\nBeff=52.0MeV /fm3,=-5\nGR(Beff=42.1MeV /fm3)\nGR(Beff=46.8MeV /fm3)\nGR(Beff=52.0MeV /fm3)\n681012141618201.21.41.61.82.02.2\nρ˜\nc/ρ0M(M\n\u0014)\na4= \u0015 \u0016 \u0017 \u0018\nBeff=42.1MeV /fm3,=-4.5\nBeff=46.8MeV /fm3,=-4.5\nBeff=52.0MeV /fm3,=-4.5\nGR(Beff=42.1MeV /fm3)\nGR(Beff=46.8MeV /fm3)\nGR(Beff=52.0MeV /fm3)\n681012141618201.21.41.61.82.02.2\nρ˜\nc/ρ0M(M\n\u0019)a4= \u001a \u001b \u001c \u001d\nBeff=42.1MeV /fm3,=-5\nBeff=46.8MeV /fm3,=-5\nBeff=52.0MeV /fm3,=-5\nGR(Beff=42.1MeV /fm3)\nGR(Beff=46.8MeV /fm3)\nGR(Beff=52.0MeV /fm3)\n681012141618201.21.41.61.82.02.2\nρ˜\nc/ρ0M(M\n\u001e)\na4=0.91\nBeff=42.1MeV /fm3,=-4.5\nBeff=46.8MeV /fm3,=-4.5\nBeff=52.0MeV /fm3,=-4.5\nGR(Beff=42.1MeV /fm3)\nGR(Beff=46.8MeV /fm3)\nGR(Beff=52.0MeV /fm3)\n681012141618201.21.41.61.82.02.2\nρ˜\nc/ρ0M(M\n\u001f)a4=0.91\nBeff=42.1MeV /fm3,=-5\nBeff=46.8MeV /fm3,=-5\nBeff=52.0MeV /fm3,=-5\nGR(Beff=42.1MeV /fm3)\nGR(Beff=46.8MeV /fm3)\nGR(Beff=52.0MeV /fm3)\n681012141618201.21.41.61.82.02.2\nρ˜\nc/ρ0M(M\n )\nFIG. 6: Same as Fig. 4 but for the third model of the HNS EoS. Diffe rent values of Beffanda4\nwith fixed values of the strange quark mass, ms= 100MeV, and the density jump, η= 0, have\nbeen considered.\ndensity) is also smaller when the density jump is larger. Fig. 9 shows th at the effective\nbag constant alters the spontaneous scalarization. Our calculatio ns indicate that the second\ncritical density grows by increasing the effective bag constant, wh ile the first one is not\nalmost affected by the effective bag constant. The increase of the second critical density by\nBeffis more important when the coupling constant is smaller. In the third m odel of the\n12a4= 0.56 a4= 0.73 a4= 0.91\nBeff(MeV\nfm3)42.1 46.8 52.042.1 46.8 52.042.1 46.8 52.0\nβ\n−4.5 1.89 1.88 1.85 1.90 1.88 1.86 1.90 1.87 1.86\n−5 2.01 1.99 1.97 2.02 1.99 1.98 2.02 1.99 1.98\nGR 1.89 1.88 1.85 1.90 1.88 1.86 1.90 1.87 1.86\nTABLE III: Same as Table I but for the third model of the HNS EoS .\nHNS EoS, with higher values of the effective bag constant, the rang e of the scalarization\nincreases. In all cases, the reduction of the coupling constant lea ds to the decrement in the\nfirst critical density and increment in the second one.\nTables IV-VI give the first critical density of the spontaneous sca larization for different\nmodels of the HNS EoS. Table IV confirms that in the first model of th e HNS EoS, the first\ncritical density grows as the bag constant increases. This is also tr ue for the second critical\ndensity, with higher values of the second critical density by increas ing the bag constant\n(see Fig. 7). It is clear from Table V that in the second model of the H NS EoS, the first\ncritical density grows as the bag constant and the mass of the str ange quark increase. This\nenhancement also takes place for the second critical density (see Fig. 8). Our results show\nthat the effects of the bag constant and the mass of the strange quark on the spontaneous\nscalarizationaremoreremarkablewhenthedensityjumpinHNSislarg er. TableVIapproves\nthat the influence of the effective bag constant as well as the mode l parameter a4on the\nfirst critical density is nearly negligible. Fig. 10 presents the critical value of the coupling\nconstant,βcr, at which the spontaneous scalarization takes place. It is obvious t hat this\ncritical value is nearly the same in different models of the HNS EoS, i.e. t he valueβcr≃\n−4.35.\nC. Hybrid neutron star scalar charge\nIn Figs. 11-13, we have plotted the scalar charge of HNS in different models of the\nHNS EoS. Scalar charge of scalarized HNS increases by decreasing t he coupling constant.\nWith higher values of the density jump, the scalar charge is larger. C onsidering the stars\n13B=70MeV/fm3\nη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\n51015200.000.050.100.150.20\n˜\nc/0ϕcB=75MeV/fm3\nη0.0,β5\nη0.45,β5\nη0.85,β5\nη0.0,β4.5\nη0.45,β4.5\nη0.85,β4.5\n51015200.000.050.100.150.20\nρ˜\nc/ρ0ϕc\nFIG. 7: Central scalar field, φc, versus the central density, ρc, applying first model of the HNS EoS\nwith different values of the bag constant, B, the density jump, η, and the coupling constant, β.\nβηB= 70MeV/fm3B= 75MeV/fm3\n0 6.7 7.1\n−4.50.45 6.6 6.9\n0.85 6.3 6.6\n0 5.4 5.6\n−50.45 5.3 5.5\n0.85 5 5.1\nTABLE IV: First critical density (in unit of ρ0) of the spontaneous scalarization in the first model\nof the HNS EoS.\nwith larger η, the less values of the compactness are needed for appearance o f the scalar\ncharge (Figs. 11 and 12). Fig. 12 confirms that in the second model of the HNS EoS, the\nscalar charge reduces when the bag constant and the mass of the strange quark become\nlarger. Besides, applying the lower values of the bag constant and t he mass of the strange\nquark, the influences of the density jump on the scalar charge are more significant. Fig.\n13 confirms that with the lower values of the coupling constant, the effective bag constant\nalters the scalar charge more significantly.\n14B=45MeV/fm3\nms=185MeV\nη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\n510152025300.000.050.100.150.20\nρ˜\nc/ρ0ϕcB=56MeV/fm3\nms=200MeV\nη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\n510152025300.000.050.100.150.20\nρ˜\nc/ρ0ϕc\nFIG. 8: Same as Fig. 7 but for the second model of the HNS EoS con sidering different values of\nthe bag constant, B, the mass of the strange quark, ms, and the density jump, η.\nβηB= 45MeV/fm3B= 56MeV/fm3\nms= 185MeV ms= 200MeV\n0 6.4 6.8\n−4.50.45 5.4 6.7\n0.85 5 6.4\n0 4.7 5.4\n−50.45 4.3 5.3\n0.85 3.9 5\nTABLE V: Same as Table IV but for the second model of the HNS EoS .\nβBeff(MeV/fm3)a4= 0.56a4= 0.73a4= 0.91\n42.1 6.2 6.2 6.2\n��4.5 46.8 6.3 6.3 6.3\n52.0 6.3 6.3 6.3\n42.1 5.1 5.1 5.1\n−0.5 46.8 5.1 5.2 5.2\n52.0 5.2 5.1 5.1\nTABLE VI: Same as Table IV but for the third model of the HNS EoS .\n15a4=0.56Beff=42.1MeV /fm3,=-5\nBeff=46.8MeV /fm3,=-5\nBeff=52.0MeV /fm3,=-5\nBeff=42.1MeV /fm3,=-4.5\nBeff=46.8MeV /fm3,=-4.5\nBeff=52.0MeV /fm3,=-4.5\n510152025300.00.10.20.30.4\nρ˜\nc/ρ0ϕc\na4=! \" # $Beff=42.1MeV /fm3,5\nBeff=46.8MeV /fm3,5\nBeff=52.0MeV /fm3,5\nBeff=42.1MeV /fm3,4.5\nBeff=46.8MeV /fm3,4.5\nBeff=52.0MeV /fm3,4.5\n510152025300.00.10.20.30.4\nρ˜\nc/ρ0ϕc\na4=0.91Beff=42.1MeV /fm3,=-5\nBeff=46.8MeV /fm3,=-5\nBeff=52.0MeV /fm3,=-5\nBeff=42.1MeV /fm3,=-4.5\nBeff=46.8MeV /fm3,=-4.5\nBeff=52.0MeV /fm3,=-4.5\n510152025300.00.10.20.30.4\nρ˜\nc/ρ0ϕc\nFIG. 9: Same as Fig. 7 but for the third model of the HNS EoS appl ying different values of Beff\nanda4with fixed values of the strange quark mass, ms= 100MeV, and the density jump, η= 0.\nV. SUMMARY AND CONCLUDING REMARKS\nIn the present work, we have studied the hybrid neutron stars (H NSs) in the scalar\ntensor gravity. For this aim, a piecewise polytropic EoS constrained by the observational\ndata and different MIT bag models have been employed to describe th e hadronic phase and\nthe strange quark matter, respectively. Our calculations approv e that the density jump\nin the HNS affects the central density of most massive scalarized st ars. The effective bag\nconstant altersthemass ofscalarized starsaswell asthe centra l density corresponding tothe\nmaximum mass. In addition, the density jump in HNS leads to the reduc tion of the first and\nthe second critical densities of the spontaneous scalarization. Ho wever, the second critical\ndensity increases as the effective bag constant grows. The range of the HNS scalarization\nbecomes larger by increasing the effective bag constant. Besides, the scalar charge of HNS\ngrows as the density jump increases.\n16● ● ■ ■ ◆ ◆▲ ▲▼ ▼ ○ ○ □ □◇ ◇ △ △ ▽ ▽● ● ■ ■ ◆ ◆▲ ▲ ▼ ▼ ○ ○□ □◇ ◇△ △▽ ▽● ●\n●B =70MeV /fm3,η =0.0 ■B =56MeV /fm3,ms\n=200MeV, η =0.85\n■B =70MeV /fm3,η =0.45 ◆Beff\n=42.1MeV /fm3,a4\n=0.56\n◆B =70MeV /fm3,η =0.85 ▲Beff\n=46.8MeV /fm3,a4\n=0.56\n▲B =75MeV /fm3,η =0.0 ▼Beff\n=52.0MeV /fm3,a4\n=0.56\n▼B=75MeV/fm3,η=0.45 ○Beff\n=42.1MeV/fm3,a4\n=0.73\n○B =75MeV /fm3,η =0.85 □Beff\n=46.8MeV /fm3,a4\n=0.73\n□B=45MeV/fm3,ms\n=185MeV, η=0.0◇Beff\n=52.0MeV/fm3,a4\n=0.73\n◇B =45MeV /fm3,ms\n=185MeV, η =0.45△Beff\n=42.1MeV /fm3,a4\n=0.91\n△B=45MeV/fm3,ms\n=185MeV, η=0.85▽Beff\n=46.8MeV/fm3,a4\n=0.91\n▽B =56MeV /fm3,ms\n=200MeV, η =0.0●Beff\n=52.0MeV /fm3,a4\n=0.91\n●B=56MeV/fm3,ms\n=200MeV, η=0.45\n0.1560.1580.1600.1620.1640.1666.05.55.04.54.0\nMRβcr\nFIG. 10: Critical value of the coupling constant, βcr, versus the star compactness, M/R, in\ndifferent models of the HNS EoS considered in this paper.\nB=70MeV/fm3η=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\n0.080.100.120.140.160.180.00.20.40.60.8\nM/Rω\nB=75MeV/fm3η=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\n0.080.100.120.140.160.180.00.20.40.60.8\nM/Rω\nFIG. 11: Scalar charge of HNS, ω, versus the star compactness, M/R, in the firstmodel of the HNS\nEoS with different values of the bag constant, B, the density jump, η, and the coupling constant,\nβ.\nAcknowledgments\nThe authors wish to thank the Shiraz University Research Council.\n[1] M. G. Alford, S. Han, and K. Schwenzer, J. Phys. G, Nucl. Pa rt. Phys. 46, 114001 (2019).\n[2] G. Lugones, M. Mariani, and I. F. Ranea-Sandoval, J. Cosm ol. Astropart. Phys. 03, 028\n(2023).\n17B=45MeV/fm3\nms=185MeVη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\n0.080.100.120.140.160.180.00.20.40.60.8\nM/Rω\nB=56MeV/fm3\nms=200MeVη=0.0,β=-5\nη=0.45,β=-5\nη=0.85,β=-5\nη=0.0,β=-4.5\nη=0.45,β=-4.5\nη=0.85,β=-4.5\n0.080.100.120.140.160.180.00.20.40.60.8\nM/Rω\nFIG. 12: Same as Fig. 11 but in the second model of the HNS EoS wi th different values of the bag\nconstant, B, the mass of the strange quark, ms, and the density jump, η.\na4=0.56Beff=42.1MeV /fm3, %=-5\nBeff=46.8MeV /fm3,%=-5\nBeff=52.0MeV /fm3,%=-5\nBeff=42.1MeV /fm3, %=-4.5\nBeff=46.8MeV /fm3, %=-4.5\nBeff=52.0MeV /fm3,%=-4.5\n0.05 0.10 0.15 0.200.00.20.40.60.8\nM/Ra4=& ' ( )Beff=42.1MeV /fm3, *=-5\nBeff=46.8MeV /fm3,*=-5\nBeff=52.0MeV /fm3,*=-5\nBeff=42.1MeV /fm3, *=-4.5\nBeff=46.8MeV /fm3, *=-4.5\nBeff=52.0MeV /fm3,*=-4.5\n0.05 0.10 0.15 0.200.00.20.40.60.8\nM/R\na4=0.91Beff=42.1MeV /fm3, +=-5\nBeff=46.8MeV /fm3,+=-5\nBeff=52.0MeV /fm3,+=-5\nBeff=42.1MeV /fm3, +=-4.5\nBeff=46.8MeV /fm3, +=-4.5\nBeff=52.0MeV /fm3,+=-4.5\n0.05 0.10 0.15 0.200.00.20.40.60.8\nM/R\nFIG. 13: Same as Fig. 11 but in the third model of the HNS EoS app lying different values of Beff\nanda4with fixed values of the strange quark mass, ms= 100MeV, and the density jump, η= 0.\n[3] B. 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To do so, we propose a multi-Dirac/Weyl Kondo lattice\nmodel which is analyzed through a mean-field approach appropriate to the paramagnetic phase. We\nstudy both the broken time-reversal and the broken inversion-symmetry Weyl cases. The multi-\nDirac and broken-time reversal multi-Weyl cases have similar behavior, which is in contrast to the\nbroken-parity case. For the former, low-energy particle-hole symmetry leads to the emergence of a\ncritical coupling constant below which there is no Kondo quenching, reminiscent of the pseudogap\nKondo impurity problem. Away from particle-hole symmetry, there is always Kondo quenching.\nFor the broken inversion symmetry, there is no critical coupling. Depending on the conduction\nelectron filling, Kondo insulator, heavy fermion metal or semimetal phases can be realized. In\nthe last two cases, quasiparticle renormalizations can differ widely between opposite chirality sec-\ntors, with characteristic dependences on microscopic parameters that could in principle be detected\nexperimentally.\nI. INTRODUCTION\nIn the last several years much attention has been de-\nvoted to systems that show non-trivial topological states\nof matter [1, 2]. The main examples of topological sys-\ntems are topological insulators and topological supercon-\nductors [3, 4]. Besides their fascinating physics, this fo-\ncus is also due to some of their characteristic features\nthat can find applications in a wide range of areas, from\nelectronic transport to quantum computation [5–7].\nOther important examples of topological systems are\ntopological semimetals [8]. When time-reversal and in-\nversion symmetries are present, topological semimetals\nare characterized by degenerate 3D Dirac cones. If,\non the other hand, time-reversal symmetry (TRS) or\ninversion symmetry (IS) is broken, the Dirac nodes\nsplit into Weyl nodes and the system becomes a Weyl\nsemimetal [9, 10]. Weyl nodes act as monopoles of 3D\nBery curvature characterized by the topological charge\nn. In 3D translationally invariant systems only topolog-\nical charges n= 1,2 and 3 are allowed [11]. Most known\nWeyl semimetals are single-Weyl semimetals, character-\nized by unitary topological charge ( n= 1) [12]. How-\never, double-Weyl nodes ( n= 2) are predicted to occur,\nfor example, in HgCr2Se4and SrSi 2[11, 13]. Moreover,\ntriple-Weyl nodes ( n= 3) can be found in transition-\nmetal monochalcogenides A(MoX)3, with A = Rb ,Tl;\nX = Te [14].\nThe presence of single-Weyl nodes in the heavy fermion\ncompound Ce 3Bi4Pd3[15] has been proposed as an ex-\nplanation of some its exotic physical properties, such\nas a giant spontaneous Hall effect in a nonmagnetic\nregime [16–18]. These systems are referred to as Weyl-\nKondo semimetals [16]. Furthermore, the presence of\ndouble-Weyl nodes has been predicted in Weyl semimet-\nals without inversion symmetry [19]. Finally, the interest\nin Dirac(Weyl)-Kondo semimetals goes beyond the con-\ndensed matter community, as they are a platform for the\nstudy of relativistic fermions in the presence of quantum\nimpurities, such as the “QCD Kondo effect”, which maybe realized in quark matter systems [20–23].\nDespite the existence of previous studies of multi-\nDirac(Weyl) systems in the presence of single and double\nmagnetic impurities [24–28], less attention has been de-\nvoted to multi-Dirac(Weyl) semimetals in the presence\nof an ordered lattice of magnetic moments [15–18]. It is\nthe aim of this paper address this question. We propose\nhere the study of a multi-Weyl Kondo lattice model with\narbitrary topological charge n. As a first step in that di-\nrection, we focus on the paramagnetic phase of the model,\nwhere a large- Ninspired mean-field approach has proved\nextremely useful for the understanding of experiments in\nthe usual Kondo lattice case. Prominent among the lat-\nter are the interaction-induced renormalizations that lead\nto the large effective masses in heavy fermion metals and\nthe reduced hybridization gaps in Kondo insulators. In\nthe multi-Dirac(Weyl) Kondo lattice, we find three pos-\nsible phases: the multi-Dirac(Weyl) Kondo insulator, the\nheavy fermion semimetal and the heavy fermion metal.\nWe distinguish, on the one hand, the multi-Dirac and the\nbroken-TRS multi-Weyl Kondo lattices, both of which\nhave similar properties, from the broken-IS multi-Weyl\ncase, on the other hand. Our main results are a thor-\nough characterization of the different renormalizations of\nKondo gaps, quasiparticle masses and velocities. These\nwill depend on whether particle-hole symmetry is present\nor not. In particular, in the broken-IS multi-Weyl Kondo\nlattice, different chirality sectors will suffer vastly dif-\nferent renormalizations, which should lead to discernible\nexperimental signatures that we will discuss.\nThe paper is organized as follows. In Sec. II we present\nthe multi-Dirac(Weyl) Kondo lattice model. In Sec. III\nwe describe the general mean-field approach used. In\nSec. IV we present the results of this approach as applied\nto the multi-Dirac and multi-Weyl Kondo systems. We\nanalyze the mean-field results in Sec. V in light of some\nprevious theoretical results and point to the measurable\nsignatures of our findings. Finally, in Sec. VI, we wrap\nup with some concluding remarks. Some mathematical\ndevelopments are left to the Appendices.arXiv:2401.13607v1  [cond-mat.str-el]  24 Jan 20242\nII. MODEL\nWe focus on a model of a multi-Weyl conduction band\ncoupled to a lattice of quantum spins, aptly described as\na multi-Weyl-Kondo lattice. The Hamiltonian consists\nthen of two terms\nH=H0+HK, (1)\nwhere H0corresponds to the multi-Weyl semimetal de-\nscribed in k-space by the minimal model [27, 29, 30]\nH0=X\nkΨ†\nkHkΨk−µ, (2)\nwhere Ψ k= (ck+↑, ck+↓, ck−↑, ck−↓)T,c†\nksαis the cre-\nation operator for an electron in state kof orbital sand\nspinα,µis chemical potential,\nHk=τz⊗[v⊥k(1−n)\n0 (kn\n−σ++kn\n+σ−) +vzkzσz−Q0σ0]\n−vzQτ0⊗σz, (3)\nk0is a dimensionful reference wave vector, k±=kx±iky,\nσ±= (σx±iσy)/2, and n= 1,2,3 is the topological\ncharge that characterizes the multi-Weyl semimetal in-\ndex. Besides, σ= (σx, σy, σz) are the Pauli matrices\nwhich act on the (pseudo)spin-space, τ= (τx, τy, τz) act\non the orbital space, and σ0andτ0denote the iden-\ntity matrices in spin and orbital spaces, respectively.\nThe term in Qbreaks time reversal symmetry (TRS),\nsince for Q= 0 we have that T Hk(k)T−1=Hk(−k),\nwhere T=τ0⊗(iσy)Kis the time reversal operator\n(with complex conjugation K). On the other hand, the\nterm in Q0breaks inversion symmetry (IS). For Q0= 0,\nPHk(k)P−1=Hk(−k), where P=τx⊗σ0is the in-\nversion operator. When both time reversal and inversion\nsymmetries are present ( Q=Q0= 0) the model re-\nduces to a 3D multi-Dirac semimetal [31, 32]. A generic\nKondo Hamiltonian HKhas two kinds of contributions,\nassociated with intra-orbital processes and inter-orbital\nprocesses [33, 34]. Thus, it is given by\nHK=1\n2X\njs,α,β(JSjs+WSj¯s)·(c†\njsασαβcjsβ)\n+1\n2X\njs,α,βK(Sjs+Sj¯s)·h\n(c†\njsασαβcj¯sβ)\n+(c†\nj¯sασαβcjsβ)i\n(4)\nwhere α, βare spin components, s=±are the orbital\nindices, ¯ s=−s,c†\njsα=1√\nVP\nkc†\nksαe−ik·Rj, andVis the\nnumber of unit cells. The first term of HKis associated\nwith intra-orbital processes, while the second refers to\ninter-orbital ones, both mediated by the interaction with\nthe local magnetic moments. The latter are assumed to\nhave, in general, orbital indices as well.III. MEAN-FIELD APPROACH\nThe local magnetic moment operators that appear in (4)\ncan be written using Abrikosov’s pseudo-fermion opera-\ntorsfjsαwith both spin and orbital indices as\nSjs=1\n2X\nαβf†\njsασαβfjsβ (5)\nA faithful representation of spin-1/2 operators requires\nthe enforcement of the single-occupancy constraint (per\nsite and orbital)\nX\nαf†\njsαfjsα=nfjs= 1. (6)\nUsing this we can write HKup to constant factors as\nHK=−J\n2X\njs,α,β(c†\njsβfjsβ)(f†\njsαcjsα)\n−W\n2X\njs,α,β(c†\nj¯sβfjsβ)(f†\njsαcj¯sα)\n−K\n2X\njs,α,βh\n(c†\nj+βfjsβ)(f†\njsαcj−α) + H .ci\n+˜λX\njs(nfjs−1). (7)\nIn the last line, we introduced the Lagrange multiplier\n˜λanticipating the later enforcement of the constraint\nnfjs= 1 at the mean field level.\nA. Mean-field Hamiltonian and Free energy\nWe now proceed along the lines of the large- N-inspired\nmean-field treatment of Kondo lattices [35–37]. We do\nso in Hamiltonian language through the following decou-\nplings of the quartic terms in the usual Kondo channels\n(c†\njsβfjsβ)(f†\njsαcjsα)→ ⟨c†\njsβfjsβ⟩f†\njsαcjsα\n+⟨f†\njsαcjsα⟩c†\njsβfjsβ\n−⟨c†\njsβfjsβ⟩⟨f†\njsαcjsα⟩,\n(c†\nj¯sβfjsβ)(f†\njsαcj¯sα)→ ⟨c†\nj¯sβfjsβ⟩f†\njsαcj¯sα\n+⟨f†\njsαcj¯sα⟩c†\nj¯sβfjsβ\n−⟨c†\nj¯sβfjsβ⟩⟨f†\njsαcj¯sα⟩,\n(c†\nj+βfjsβ)(f†\njsαcj−α)→ ⟨c†\nj+βfjsβ⟩f†\njsαcj−α\n+⟨f†\njsαcj−α⟩c†\nj+βfjsβ\n−⟨c†\nj+βfjsβ⟩⟨f†\njsαcj−α⟩.\nDefining the spinors Ψ†\nks= (c†\nks↑, c†\nks↓) and Ψ†\nkfs=\n(f†\nks↑, f†\nks↓), where f†\nksσ=1√\nVP\njf†\njsσeik·Rjwe can\nwrite the mean-field Hamiltonian in the terms of Φ†\nk=\n(Ψ†\nk+,Ψ†\nkf+,Ψ†\nk−,Ψ†\nkf−) as3\nHMF=X\nkΦ†\nkHHS\nkΦk+V\"X\ns(J+K)|vs|2+X\ns(W+K)|ws|2−λ#\n, (8)\nwith λ= 2˜λ, enforcing the constraint of Eq. (6) on the average, and\nHHS\nk=\nHk+−µ1 Jv+\n2×2+Kw−\n2×2 02×2 Ww+\n2×2+Kv−\n2×2\nJv∗+\n2×2+Kw∗−\n2×2 λ2×2 Ww∗+\n2×2+Kv∗−\n2×2 02×2\n02×2 Ww−\n2×2+Kv+\n2×2 Hk−−µ1 Jv−\n2×2+Kw+\n2×2\nWw∗−\n2×2+Kv∗+\n2×2 02×2 Jv∗−\n2×2+Kw∗+\n2×2 λ2×2\n. (9)\nAbove\nHks=s[v⊥k(1−n)\n0 (kn\n−σ++kn\n+σ−) +vzkzσz−Q0σ0]\n−vzQσz,\n(10)\nand the matrices are given by\nvs\n2×2=\u0012\nvs0\n0vs\u0013\n,ws\n2×2=\u0012\nws0\n0ws\u0013\n,\nλ2×2=\u0012\nλ0\n0λ\u0013\n,02×2=\u0012\n0 0\n0 0\u0013\n, (11)\nwhere we defined the Kondo order parameters\nvs=⟨c†\njsσfjsσ⟩, w s=⟨c†\njsσfj¯sσ⟩. (12)\nThe equilibrium values of the order parameters are ob-\ntained by minimization of the mean-field free-energy den-sity with respect to their values\nFMF=−1\nβX\niωnZd3k\n(2π)3lnDet[ −iωn1+HHS\nk]\n+\"X\ns(J+K)|vs|2\n+X\ns(W+K)|ws|2−λ#\n, (13)\nwhere β= 1/kBT, and ωn= (2n+1)π/βis the fermionic\nMatsubara frequency.\nThe most generic mean-field equations do not allow\nan analytical treatment, although a numerical analysis\nis straightforward. In order to obtain a deeper physical\nunderstanding, we will focus in this paper on the cases\nin which analytical insight can be gained. This happens\nwhen there is only intra-orbital Kondo interactions ( J̸=\n0 and W=K= 0), which we take up in the remainder\nof the main text. Furthermore, in presence of particle-\nhole and time-reversal symmetry ( λ=µ=Q= 0), cases\nwith W̸= 0 and/or K̸= 0 also allow for an analytical\napproach. These are listed in Appendix E. We will not\ndwell on them because the mean-field equations have a\nform quite similar to the cases discussed in detail below.\nIV. MEAN-FIELD THEORY OF THE\nMULTI-DIRAC/WEYL KONDO LATTICE\nWe will explore in detail the case in which only the\nintra-orbital Kondo interaction is present, that is J̸= 0\nandW=K= 0 in the Hamiltonian of Eq. (4). Using\nEqs. (9) and (13) we obtain a block-diagonal mean-field\nfree energy given by\nFMF=−1\nβX\niωnX\nsZd3k\n(2π)3lnDet\u0014\n−iωn1+\u0012\nHks−µ1 Vs\n2×2\nV∗s\n2×2λ2×2\u0013\u0015\n+ \n1\nJX\ns|Vs|2−λ!\n, (14)\nwhere Vs\n2×2=Jvs\n2×2. This can be written in terms of the\ndispersion relations as\nFMF=−1\nβX\nsZd3k\n(2π)3X\nα,ξln[1 + e−βEαξ\ns(k)]\n+ \n1\nJX\ns|Vs|2−λ!\n, (15)where\nEαξ\ns(k) =1\n2(εα\nks−µ+λ) +ξ\n2q\n(εα\nks−µ−λ)2+ 4V2s,\n(16)4\n−1 0 1\nk⊥−101\n−1 0 1\nk⊥−101Eαξ(k⊥,kz= 0)\n−1 0 101−1 0 1\nk⊥\n−1 0 1\nk⊥\n−1 0 1\nk⊥−1 0 1\n−1 0 1\nk⊥\n−1 0 1n= 1 n= 2 n= 3\n(a) (d) (g)\n(b) (e) (h)\n(c) (f) (i)\nFIG. 1. Dispersion relations for the multi-Dirac Kondo system (in the kz= 0 plane) for n= 1,2,3. Upper panels [(a),(d),(g))]\nshow the non-hybridized bare dispersions. Middle panels [((b),(e),(h))] show the multi-Dirac Kondo insulator. Bottom pan-\nels [((c),(f),(i))] show the multi-Dirac Kondo semimetal regime. The parameter values used are V= 0.15Λ, λ= 0.1Λ. The\nhorizontal dashed grey line indicates the Fermi energy.\nand\nεα\nks=αq\nA2nk2n\n⊥+v2z(kz−sQ)2−sQ0, (17)\nwith An=k(1−n)\n0 v⊥,k2\n⊥=k2\nx+k2\ny,α=±andξ=±.\nThe mean-field parameters Vs=Jvsandλare deter-\nmined by the saddle point equations\n∂FMF\n∂Vs=∂FMF\n∂λ= 0, (18)\nresulting in\nZd3k\n(2π)3\u00141\nE++s(k)−E+−s(k)[f(E++\ns(k))−f(E+−\ns(k))]\n+1\nE−+s(k)−E−−s(k)(k)[f(E−+\ns(k))−f(E−−\ns(k))]\u0015\n=−1\n2J(19)\nand\nX\ns,α,ξZd3k\n(2π)3f\u0000\nEαξ\ns(k)\u0001\n×\"\n1−ξ(εα\nks−µ−λ)p\n(εα\nks−µ−λ)2+ 4V2s#\n= 1.(20)Above, f(x) = (1 + eβx)−1is the Fermi-Dirac distribu-\ntion. We will now solve these self-consistent equations at\nzero temperature for different physical situations.\nA. The multi-Dirac Kondo lattice (Preserved TRS\nand IS)\nWhen both TRS and IS are present ( Q=Q0= 0)\nH0in Eq. (2) describes a 3D multi-Dirac system with\ngapless Kramers-degenerate Dirac nodes at kD= 0, see\nFig.1[(a),(d),(g)]. First, let us analyze the particle-hole\nsymmetric case, µ=λ= 0. In the absence magnetic\nmoments, the system is a multi-Dirac semimetal. In the\npresence of these moments, however, due to the semimet-\nalic nature of the system, a finite mean-field Vsonly oc-\ncurs above a critical value ( J > Jc\nn), which is character-\nistic of the pseudo-gap Kondo problem [38]. Therefore,\nforJ > Jc\nnthef-electrons hybridize with itinerant multi-\nDirac c-electrons giving rise to heavy quasiparticles. The\nDirac nodes are energetically split, opening an energy\ngap in the spectra, Fig.1[(b),(e),(h)]. Thus, the system\nbecomes a multi-Dirac Kondo insulator . If both TRS an\nIS are present, we have Vs=V, and at T= 0 we obtain5\n1.0 1.2 1.4 1.6 1.8 2.0\njn0.00.10.20.30.40.5mn= 1\nn= 2\nn= 3\nFIG. 2. Numerical solution for the multi-Dirac Kondo insu-\nlator gap equation (21) for n= 1,2,3.\nthe gap equation\nZ1\n0dxx2/n\n√\nx2+m2=n\n2jn. (21)\nAbove, jn=J/Jc\nn, where Jc\nn= (2/n)/ρn(Λ) is the crit-\nical Kondo coupling, ρn(Λ) is the multi-Dirac density of\nstates at the high-energy cutoff Λ, and m= 2V/Λ (see\nAppendix B for details). Numerical solutions of the gap\nequation for n= 1,2,3 are show in Fig. 2. A similar gap\nequation is found in the study of the axionic insulator\nphase for general multi-Weyl systems [29].\nNow, to study in detail the order parameter critical\nbehavior near the quantum phase transition (QPT) point\njn≈1 (J≈Jc\nn), we write jn= 1 + δ, with δ=|J−\nJc\nn|/Jc\nn≪1. For n >1, (see the details in Appendix C)\nm(δ) =Cnδn/2, (n >1) (22)\nwith Cn=\u0014\n−n√π\nΓ(1\n2+1\nn)Γ(−1\nn)\u0015n/2\n. Specifically, C2= 1,\nC3≈1.249. In the single-Dirac case ( n= 1) the critical\nbehavior is modified by\nm(δ) =f(δ)δ1/2, (n= 1), (23)\nwhere f(δ) carries logarithmic corrections, calculated in\nAppendix C. The comparison between numerical solu-\ntions and the analytic expressions (22) and (23) is shown\nin Fig. 3. The mean-field quantum critical exponent is\nthus ν=n/2, in agreement with early work [38].\nAway from particle-hole symmetry ( µ̸= 0), the density\nof states of the unhybridized system ( J= 0) is always fi-\nnite and the Kondo effect occurs for any finite Kondo cou-\npling. Therefore, we obtain a multi-Dirac Kondo metal .\nThe self-consistent equations at T= 0 are given by\nZ1\n0dxx(2/n)+1\np\n(x−α)2+m2=1\nρn(Λ)Λ\u0014\n1−α\nJ/Λ\u0015\n,(24a)\nZ1\n0dxx2/n\np\n(x−α)2+m2=1\nρn(Λ)J. (24b)\n10−610−510−410−310−210−1\nδ10−410−310−2m\nn= 1\nn= 2\nn= 3O Numerical\n— AnalyticalFIG. 3. Critical behavior of the multi-Dirac Kondo insulator\norder parameter mforn= 1,2,3. Symbols show numerical\nresults, while solid lines are the analytical results of Eqs. (22)\nand (23).\n0.4 0.6 0.8 1.0\nρn(Λ)J0.00.10.20.3¯λn= 1\nn= 2\nn= 3\n0.4 0.6 0.8 1.0\nρn(Λ)J0.120.240.360.48(a) (b) Metal Semimetal\n¯µ= 0.1Λ\nFIG. 4. Numerical solution of the self-consistent mean-field\nparameter ¯λforn= 1,2,3 for (a) the multi-Dirac Kondo\nmetal and (b) the multi-Dirac Kondo semimetal.\nwhere α= (µ+λ)/Λ (see Appendix B).\nA particular case of special interest occurs when µ=\n−V2/λ(λ > 0). In this case, the lower hybridized\nDirac node is pinned to the Fermi energy: E+−\ns(kD) =\nE−−\ns(kD) = 0. This particular choice of chemical\npotential realizes a multi-Dirac Kondo semimetal , see\nFig.1[(c),(f),(i)]. This is reminiscent of the choice of\nRef. [16] for the single Kondo-Weyl semimetal. The\nself-consistency equations of the multi-Dirac Kondo\nsemimetal are given by (24), but with α=α(m,¯λ) =\n−m2/4¯λ+¯λ, where m= 2V/Λ,¯λ=λ/Λ. The nu-\nmerical solution of the self-consistent equations both for\nthe multi-Dirac Kondo metal and semimetal are shown\nin Figs. 4 and 5. Their behavior is quite similar.6\n1 2 3 4 5\n1/ρn(Λ)J10−810−610−410−2100mn= 1\nn= 2\nn= 3\n1 2 3 4 5\n1/ρn(Λ)J(a) (b) Metal Semimetal\nµ= 0.1Λ\nFIG. 5. Semi-log plot of the numerical solution of the mean-\nfield parameter mfor the multi-Dirac Kondo metal and\nsemimetal for n= 1,2,3.\nB. The multi-Weyl Kondo lattice (Broken TRS or\nIS)\n1. Broken-TRS multi-Weyl Kondo lattice\nWhen TRS is broken, and IS is preserved ( Q̸= 0,\nQ0= 0) the Kramers-degenerate Dirac nodes of the\nunhybridized bands split in momentum space into two\nmulti-Weyl nodes with opposite chiralities at the same\nenergy and distinct k-points k±\nW= (0,0,∓Q) [39, 40],\nsee Fig. 6(a). If the energy of those Weyl nodes coin-\ncides with the Fermi energy we have a nodal semimetal\nbefore coupling to the local moments. For strong enough\ninteractions with the lattice of local moments and at the\nparticle-hole symmetric point ( µ=λ= 0), the sys-\ntem becomes a broken-TRS multi-Weyl Kondo insula-\ntor, because the hybridization opens a gap at the Weyl\nnodes, see Figs.6(b). At first, it would appear that the\nhybridized bands would define two distinct order pa-\nrameters, V±. After integration of the self-consistent\nequations, however, as in the multi-Dirac Kondo case,\nVs=V(see Appendix B). Thus, the mean-field equa-\ntions of this broken-TRS multi-Weyl Kondo lattice have\na form similar to those of the previously analyzed multi-\nDirac Kondo lattice. In particular, if we now, as we did\nin the multi-Dirac Kondo lattice case, set the chemical\npotential at µ=−V2/λ, we obtain a broken-TRS multi-\nWeyl Kondo semimetal , see Fig. 6(c).\n2. Broken-IS multi-Weyl Kondo lattice\nFinally, we consider the case where there is TRS but IS\nis broken ( Q= 0, Q0̸= 0). In this case, the multi-Weylnodes occur at the same k-point kW=0, but with dif-\nferent energies ε±W=∓Q0[39, 40]. Thus, IS breaking\ndestroys the nodal semimetal phase generically produc-\ning a phase with electron and hole Fermi surfaces [39].\nThe dispersion relations for the non-hybridized broken-\nIS single and double-Weyl metals are shown in the upper\npanels of Fig. 7 ( n= 1) and 8 ( n= 2). In Figs. 7(a) and\n8(a), the particle-hole symmetric case ( µ= 0) is shown,\nwhereas in Figs. 7(b) and 8(b), particle-hole symmetry\nis broken by setting µ=−Q0, which pins the Weyl node\nwith positive chirality to the Fermi energy.\nWe first analyze the effect of coupling the lattice of\nlocal moments to the broken-IS multi-Weyl Kondo lat-\ntice at particle-hole symmetry ( µ=λ= 0). In con-\ntrast to the multi-Dirac and broken-TRS multi-Weyl\ncases, now the Fermi surface is always finite and the\nlocal moments Kondo-bind to the itinerant c-electrons\nfor any finite Kondo coupling: thus, the critical value\nJc= 0. Because a gap opens at the chemical poten-\ntial, we will call this a broken-IS multi-Weyl Kondo in-\nsulator , as shown in Figs. 7(c) and 8(c). Moreover,\nV+=V−=V, and, at T= 0, f(E++\nks) =f(E−+\nks) = 0,\nandf(E+−\nks) =f(E−−\nks) = 1, so the mean-field equation\nbecomes\nX\ns=±Z1\n0dxx2/n\np\n(x−s¯Q0)2+m2=2\nρn(Λ)J.(25)\nHere, m= 2V/Λ,¯Q0=Q0/Λ. Numerical solutions for\nthe order parameter mforn= 1,2,3 are shown in Fig. 9,\nwhere the usual Kondo exponential dependence can be\ndiscerned.\nIn the limit ρn(Λ)J≪1 and m≪¯Q0, it is possible to\nextract analytically the exponential dependence on Jfor\nn= 1,2. We find\nm=fKI\n1(Q0/Λ)exp[ −1/(2ρW\n1(0)J)] (n= 1),(26a)\nm=fKI\n2(Q0/Λ)exp[ −1/(2ρW\n2(0)J] (n= 2),(26b)\nwhere\nfKI\n1(x) =r\nx(1−x)\n2exp[(1 −3x2)/2x2],(27a)\nfKI\n2(x) =r\nx(1−2x)\n2exp[(2 −2x)/2x],(27b)\nandρW\nn(0) is the multi-Weyl density of states at the\nFermi energy (see Appendix A for details). As expected,\nsince IS breaking drives the critical coupling Jcto zero,\nthe order parameter is non-analytic in both JandQ0.\nThe explicit exponential dependence of monQ0becomes\nclear when we note that ρW\nn(0)∝Q2/n\n0(see Eq.(A7) and\nEq.(A8) of Appendix A). Moreover, it is reminiscent of\nthe Kondo temperature obtained previously by one of us\nusing the numerical renormalization group in the study\nof multi-Weyl systems in the presence of a single quan-\ntum impurity [28]. We have checked that these analytical\nexpressions perfectly match the numerical results in their\nvalidity region.7\n−0.50.00.5\nkz−0.50.00.5Eαξ\ns(k⊥= 0,kz)\ns= +\ns=−\n−0.50.00.5\nkz−0.50.00.5\nkz(a) (b) (c)\nFIG. 6. Dispersion relations for the broken-TRS multi-Weyl Kondo system (in the k⊥= 0 plane), for n= 1. (a) The non-\nhybridized case ( J= 0). (b) The broken-TRS multi-Weyl Kondo insulator ( λ=µ= 0). (c) The broken-TRS multi-Weyl\nKondo semimetal ( µ=−V2/λ). The parameters used are Q= 0.1Λ,V= 0.15Λ, λ= 0.1Λ. The horizontal dashed grey line\nindicates the Fermi energy. The vertical dashed black lines indicate the Weyl node k-points k±\nW= (0,0±Q).\n−0.5 0.0 0.5\nk⊥−0.50.00.5Eαξ\ns(k⊥,kz= 0)s= +\ns=−\n−0.5 0.0 0.5\nk⊥\n−0.5 0.0 0.5\nk⊥−1.0−0.50.00.51.0Eαξ\ns(k⊥,kz= 0)\n−0.5 0.0 0.5\nk⊥(a) (b)\n(c) (d)n= 1\nPH−Symmetric PH−Asymmetric\nPH−Symmetric PH−Asymmetric\nFIG. 7. Dispersion relations for the broken-IS ( Q0=−0.1Λ)\nsingle-Weyl Kondo system (in the kz= 0 plane). The non-\nhybridized limit ( J= 0) is shown in the upper panels (a) and\n(b), with ( µ= 0) and without ( µ=−Q0) particle-hole sym-\nmetry, respectively. The interacting regimes are shown in the\nbottom panels (c) and (d), with µ= 0 and µ=−V2\n+/λ−Q0,\nrespectively. The parameters used in the interacting particle-\nhole symmetric case are V+=V−= 0.15Λ, λ= 0, whereas in\nthe asymmetric case we have V+= 0.05Λ, V−= 0.25Λ, λ=\n0.1Λ. The horizontal dashed grey line shows the Fermi en-\nergy.\nWe now consider the particle-hole asymmetric case. In\nthis case, V+̸=V−. Inspired by the choice of Ref. [16], we\nset the chemical potential to µ=−(V2\n+/λ)−Q0, which\npins the unhybridized multi-Weyl node with positive chi-\nrality to the Fermi energy, i.e. E+−\n+(kW) =E−−\n+(kW) =\n−0.5 0.0 0.5\nk⊥−0.50.00.5Eαξ\ns(k⊥,kz= 0)s= +\ns=−\n−0.5 0.0 0.5\nk⊥\n−0.5 0.0 0.5\nk⊥−0.50.00.5Eαξ\ns(k⊥,kz= 0)\n−0.5 0.0 0.5\nk⊥(a) (b)\n(c) (d)n= 2\nPH−Symmetric PH−Asymmetric\nPH−Symmetric PH−AsymmetricFIG. 8. Dispersion relations for the broken-IS double-Weyl\nKondo system (in the kz= 0 plane). The description and\nparameters used are the same as those of Fig. 7.\n0, see Figs. 7(d) and 8(d). Unlike the broken-TRS multi-\nWeyl Kondo lattice, in broken-IS Weyl systems with both\nWeyl nodes at the same k-point, as we have here, a nodal\nsemimetallic phase is not possible. This is because, at\nthe Fermi energy, although the density of states associ-\nated with a particular Weyl chirality might go to zero,\nthe density of states associated with the opposite chiral-\nity is always non-vanishing. Moreover, as we will see,\nthis difference will lead to a huge difference between the\nmean-field order parameters associated with each chiral-\nity.\nThe self-consistency equations in the broken-IS8\n2 4 6 8 10 12 14 16\n1/ρn(Λ)J10−1010−810−610−410−2100mn= 1\nn= 2\nn= 3Q0= 0.05Λ\nFIG. 9. Semi-log plot of the order parameter mfor the\nbroken-IS multi-Weyl Kondo insulator ( Q0= 0.05Λ) for\nn= 1,2,3.\nparticle-hole asymmetric Weyl-Kondo lattice setting µ=\n−(V2\n+/λ)−Q0are (see Appendix B for details)\nZ1\n0dxx(2/n)+1\nq\u0002\nx+α(m+,¯λ)\u00032+m2\n+\n+Z1\n0dxx(2/n)+1\nq\u0002\nx+α(m+,¯λ) + 2 ¯Q0\u00032+m2\n−\n=2\nρn(Λ)Λ\u0014\n1−α(m+,¯λ) +¯Q0\nJ/Λ\u0015\n, (28a)\nZ1\n0dxx2/n\nq\u0002\nx+α(m+,¯λ)\u00032+m2\n+=1\nρn(Λ)J,(28b)\nZ1\n0dxx2/n\nq\u0002\nx+α(m+,¯λ) + 2 ¯Q0\u00032+m2\n−=1\nρn(Λ)J.\n(28c)\nAbove, m±= 2V±/Λ,¯Q0=Q0/Λ, and α(m+,¯λ) =\n−(m2\n+/4¯λ) +¯λ. Note that, for Q0= 0, IS is restored\nand Eqs. (28) reduce to Eqs. (24) for the multi-Dirac\nKondo case, as expected. Besides, for Q0̸= 0, like in the\nparticle-hole symmetric case, the total density of states at\nthe Fermi level is finite and the critical coupling constant\nis zero. The numerical solutions of Eqs. (28) are plotted\nin Figs. 10 to 12. Like in the multi-Dirac and broken-\nTRS multi-Weyl lattice, the self-consistent parameter ¯λ\ndoes not present any exponential behavior, nor does it\nchange significantly with the increase of the IS-breaking\nparameter ¯Q0. In contrast, from Fig. 11 we can see that\nthe order parameters m+andm−both behave exponen-\ntially with a marked hierarchy: m+≪m−. In Fig. 12, we\nshow the Kondo parameters for the broken-IS multi-Weyl\nKondo lattice for several values of ¯Q0forn= 2. Clearly,\nalthough they are equal when there IS ( ¯Q0= 0), they\ndiffer significantly as ¯Q0= 0 is increased, even thoughthey both decrease exponentially for decreasing ρn(Λ)J.\nThis is a reflection of the semimetallic character of the\npositive chirality electronic band, contrasting with the\nfully metallic nature of the negative chirality sector.\nV. DISCUSSION\nThe mean-field description of heavy fermion met-\nals [41] and Kondo insulators [42] has played a pivotal\nrole in our understanding of these large families of com-\npounds. Complications due to magnetic ordering [43],\nin particular quantum critical behavior [44], and heavy\nfermion superconductivity [45] are beyond the scope of\nthe theory. However, on the paramagnetic side of the Do-\nniach phase diagram [43], considerable insight has been\ngained, from the origin of the heavy effective masses to\nthe nature of the renormalized Kondo insulating gap.\nThis work offers an analogous phenomenology for multi-\nDirac and multi-Weyl Kondo systems.\nWe have seen that, at particle-hole symmetry, the\nbare multi-Dirac and TRS-broken multi-Weyl densities\nof states have the power law form ρ(ω)∝ωr, with\nr= 2/n(see Eq.(A9) in Appendix A). Then, in the pres-\nence of Kondo spins, these systems become archetypes\nof pseudogap Kondo systems. The pseudogap problem\nfor a single Kondo impurity was originally proposed by\nWithoff and Fradkin (WF) [38]. That and subsequent\nworks focused on a large-N approach similar to the one\nwe used here [46–48]. The problem has also been ex-\ntensively analyzed using the perturbative renormaliza-\ntion group (PRG) [49–52] and the numerical renormal-\nization group (NRG) [53–57]. It has been thoroughly\nreviewed in [58]. Although we are dealing here with a\nlattice of spins instead of a single impurity, the local\nnature of the large- Ninspired approach should lead to\nsimilar results. In particular, the critical behavior close\ntoJcshould be similar. In this respect, subtle compli-\ncations arise due to the absence of exact particle-hole\nsymmetry at all scales in any realistic description [46–\n48, 58]. We have skirted these issues by working within\na simplified particle-hole symmetric description close to\nthe Dirac/Weyl nodes that ignores higher-energy devia-\ntions from that symmetry. This simplification leads to\nthe critical behavior we found for the multi-Dirac Kondo\nin Section IV A and the TRS-broken multi-Weyl case in\nSectionn IV B 1, namely, that the characteristic energy\nscale m∝ |J−Jc|ν, with ν= 1/r. We expect that a\nmore microscopic description of the Dirac/Weyl bands\nwould change the detailed critical behavior but not the\npresence of a critical coupling constant in the cases con-\nsidered.\nIf the coupling constant exceeds the critical value,\nthen the usual opening of a Kondo insulating gap en-\nsues. In contrast to usual Kondo insulators [42], though,\nthe renormalized energy gap is not an exponential func-\ntion of J. This may have detectable experimental con-\nsequences, e.g., in pressure studies, which can tune the9\n0.75 1.00 1.25 1.50\nρn(Λ)J0.20.40.60.8¯λ¯Q0= 0.0\n¯Q0=−0.05\n¯Q0=−0.075\n¯Q0=−0.1\n0.50 0.75 1.00 1.25 1.50\nρn(Λ)J0.00.20.40.60.81.0\n0.50 0.75 1.00 1.25 1.50\nρn(Λ)J0.00.20.40.60.81.0\n(a) (b) (c)n= 1 n= 2 n= 3\nFIG. 10. Self-consistent parameter ¯λfor the broken-IS multi-Weyl Kondo lattice for n= 1,2,3 for several values of ¯Q0.\n1.21.62.02.4\n1/ρn(Λ)J10−710−610−510−410−310−210−1100mm+\nm−\n1.21.62.02.4\n1/ρn(Λ)J2.5 5.0 7.5\n1/ρn(Λ)J(a)(b) (c) n= 1 n= 2 n= 3\n¯Q0=−0.1\nFIG. 11. Semi-log plot of Kondo parameters m+andm−\nfor the broken-IS multi-Weyl Kondo lattice for n= 1,2,3 for\n¯Q0=−0.1.\nmicroscopic value of J. Away from particle-hole symme-\ntry, the generic behavior is that of a heavy fermion metal,\nwith the usual phenomenology.\nA particular fine-tuned case of possible interest, how-\never, occurs when the chemical potential of the multi-\nDirac or TRS-broken multi-Weyl Kondo lattice is at or\nclose to the Dirac or Weyl nodes, see Figs. 1[(c),(f),(i)]\nand 6(c). Then, the hybridized multi-Dirac/Weyl nodes\nwill have renormalized velocities, a consequence of the\ncomposite nature of fermions in the Kondo condensate.\nAs a result, physical quantities such as the specific heat\nwill be deeply affected. In general, it will depend on\ntemperature as a higher power law than in a metal\nCv(T)\nT=γnT2/n, (29)\nwhere γnwill reflect the nodal velocity renormaliza-\ntion (see Appendix F). Thus, specific heat measurements\n1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0\n1/ρn(Λ)J10−610−510−410−310−210−1100mm+(¯Q0= 0.0)\nm−(¯Q0= 0.0)\nm+(¯Q0=−0.05)\nm−(¯Q0=−0.05)\nm+(¯Q0=−0.1)\nm−(¯Q0=−0.1)n= 2FIG. 12. Semi-log plot of Kondo parameters m+andm−for\nthe broken-IS multi-Weyl Kondo lattice for n= 2 for several\nvalues of ¯Q0.\ncould play an important role in the search for signatures\nof strongly correlated topological materials [18, 59].\nIn sharp contrast, for broken-IS multi-Weyl systems,\nthe bare density of states is always non-zero at the\nFermi level, see Fig.7 and Fig.8, and Kondo condensation\nwill occur for any value of J. At particle-hole symme-\ntry, a broken-IS multi-Weyl Kondo insulator ensues, see\nFig.7(c) and Fig.8(c). The renormalized Kondo gap has\nthe usual exponential dependence on J, as can be seen\nin Fig. 9 and from Eq. (26). Note, however, that the\nIS-breaking momentum Q0also appears in the argument\nof the exponential, which might offer an opportunity for\nexperimental detection if Q0can be externally tuned.\nThe heavy fermion metal phenomenology applies away\nfrom particle-hole symmetry, but now there are differ-\nent mean-field order parameters m+̸=m−, each one\nresponsible for renormalizing a different chirality sector.\nEven for modest IS-breaking, these two can be paramet-\nrically widely separated, as seen in Figs. 11 and 12. These\nwidely different energy scales are expected to be reflected10\nin the temperature dependence, the smallest mscross-\ning over to zero at a much lower temperature than the\nother. Evidently, these should give rise to detectable con-\nsequences in thermodynamic and transport properties.\nIf the chemical potential happens to fall at or close\nto one of Weyl nodes there will be both metallic and\nsemimetallic bands, as shown in Figs. 7(d) and 8(d).\nThe specific heat will be given by CV/T=γ0+γnT2/n,\nthe first and second terms coming from the metallic\nand semimetallic bands, respectively (see Appendix F).\nWhether the presence of these two terms can be distin-\nguished from other contributions, such as phonons, re-\nmains to be seen.\nOne final remark is worth making. Whenever any of\nthe order parameters mshas the familiar exponential de-\npendence on 1 /J, we expect its value to be on the order of\nthe ratio between the Kondo temperature and the band\nwidth, TK/D, in conventional heavy-fermion materials,\nnamely, ms∼10−3−10−2. The inverse of this ratio\nwill determine the renormalizations of the corresponding\nquasiparticle masses or Kondo insulating gap, whichever\napplies. In the semimetallic phases we analyzed here,\nhowever, the Dirac/Weyl masses are only weakly renor-\nmalized, as can be easily shown. This is because we chose\na model with Dirac/Weyl points at the zone center . Had\nwe chosen them closer to the zone boundaries, they would\nalso have been strongly renormalized on a 1 /msscale\n(see, e.g., references [15–18, 60]).\nVI. CONCLUDING REMARKS\nIn this manuscript we have studied multi-Dirac/Weyl\nsystems in the presence of a lattice of local moments\nusing the multi-Dirac/Weyl Kondo lattice model. We\nhave done so based on the large- Ninspired mean-field\napproach, adequate to the paramagnetic side of the Do-\nniach phase diagram. At particle-hole symmetry and in\nthe multi-Dirac and broken-TRS Weyl cases, there is a\nnon-zero coupling constant Jc, below which no Kondo\ncompensation occurs. For J > J c, however, there is\nKondo quenching and Kondo insulating behavior, but\nwith a non-exponential characteristic energy scale. Away\nfrom particle-hole symmetry Jc= 0, and either heavy\nfermion metallic or semimetallic behavior can occur, de-\npending on the position of the chemical potential. For\nthe broken-IS multi-Weyl Kondo lattice Jc= 0 always.\nDepending on the conduction electron filling, we can have\na Kondo insulator or a heavy fermion metal. Different\nquasi-particle renormalizations are expected for different\nchirality sectors.\nOur results have experimental consequences that could\nin principle be sought in some compounds. In partic-\nular, we have found a complex phenomenology for the\nmass and velocity renormalizations of the quasi-particles,\nwhich we have described in detail, that could be directly\ninvestigated. More detailed studies of transport prop-\nerties and the effects of disorder are promising futuredirections to pursue. Beyond mean-field theory, there re-\nmain important questions we have not touched, such as\nmagnetic ordering, quantum criticality and superconduc-\ntivity. We hope our results will serve as an initial guide\nin those directions.\nACKNOWLEDGMENTS\nJ. F. S acknowledges the support of the postdoctoral\nfellowship from FAPESP No. 2021/04078-0. E. M. ac-\nknowledges the support of CNPq through Grant No.\n309584/2021-3 and Capes through Grant No. 0899/2018.\nAppendix A: The Multi-Weyl density of states\nThe multi-Weyl density of states can be computed us-\ning the multi-Weyl matrix Green’s function Gk(ω) =\n[(ω+µ)1−Hk]−1, where the Hamiltonian Hkis given by\nEq. (3). The spin-diagonal part of the Green’s function\nis given by\nGW\nkσs(ω) =ω+µ+sQ0\n[ω+µ+sQ0]2−ϵ2\nks\n+sσvz(kz−sQ)\n[ω+µ+sQ0]2−ϵ2\nks. (A1)\nAbove, σ=±,s=±,ϵks=p\nA2nk2n\n⊥+v2z(kz−sQ)2,\nwith k2\n⊥=k2\nx+k2\ny, and An=k(1−n)\n0 v⊥. The spectral\nfunction is\nAW\nkσs(ω) =1\nπIm[GW\nkσs(ω−iδ)]\n=|ω+µ+sQ0|δ[(ω+µ+sQ0)2−ϵ2\nks]\n+sσvz(kz−sQ)\n2εks[δ(ω+µ+sQ0−εks)\n−δ(ω+µ+sQ0+εks)].\n(A2)\nThe density of states is\nρW\nσs(ω) =Zd3k\n(2π)3AW\nkσs(ω)\n=1\n(2π)2ZΛ′\n−Λ′dkzZΛ′\n0dk⊥k⊥AW\nkσs(ω),\n(A3)\nwhere Λ′is a high-momentum cutoff. The density of\nstates is independent of the TRS-breaking parameter Q.\nTo see that, we shift kz−sQ→kz, which leads to\nρW\nσs(ω) =1\n(2π)2ZΛ′−sQ\n−Λ′−sQdkzZΛ′\n0dk⊥k⊥AW\nkσs(ω),11\nfrom which we can drop Qin the limit of moderate TR\nbreaking Λ′≫Q. We now make the change of variables\nAnkn\n⊥=ρcosθ, k zvz=ρsinθ, (A4)\nand use a different regularization at high energies (which\ndoes not affect the low energy physics), with 0 ≤ρ≤\nΛ,−π/2≤θ≤π/2 and\ndkzdk⊥=ρ1/n\nnA1/n\nnvzcos(1\nn−1)(θ)dρdθ.\nThe integrals in Eq. (A3) can now be easily computed as\nZd3k\n(2π)3δ[(ω+µ+sQ0)2−ε2\nks]\n=1\n(2π)2nA2/n\nnvz√πΓ(1/n)\nΓ\u00001\n2+1\nn\u0001|ω+µ+sQ0|(2/n)−1\n2,\n(A5)\nwhere Γ( z) is the Gamma function, and\nZd3k\n(2π)3(kz−sQ)\n2εks[δ(ω+µ+sQ0−εks)\n−δ(ω+µ+sQ0−εks)] = 0 . (A6)\nThe final expression for the density of states per spin and\nchirality is\nρW\nnσs(ω) =1\n2(2π)2nA2/n\nnvz√πΓ(1/n)\nΓ[(n+ 2)/2n]|ω+µ+sQ0|2/n,\n(A7)\nwhere we added a subscript “ n” to highlight the depen-\ndence on the topological charge. Thus, the total multi-\nWeyl density of states with both broken TRS and IS is\ngiven by\nρW\nn(ω) =X\nσsρW\nσs(ω). (A8)\nWe notice from Eqs. (A7) and (A8) that if both TRS\nand IS are preserved (multi-Dirac system) or if only TRS\nsymmetry is broken (broken-TRS multi-Weyl system),\nthen the density of states does not depend on either spin\nor chirality indices, and we have ρD\nn(ω) =ρW\nnBTRS (ω)≡\nρn(ω) with\nρn(ω) =2\n(2π)2nA2/n\nnvz√πΓ(1/n)\nΓ[(n+ 2)/2n]|ω+µ|2/n.(A9)\nIn particular, for n= 1,2,3, the multi-Weyl densities of\nstates (A7) are\nρW\n1σs(ω) =1\n4π2v2\n⊥vz|ω+µ+sQ0|2, (A10)\nρW\n2σs(ω) =1\n16π(k−1\n0v⊥)vz|ω+µ+sQ0|, (A11)\nρW\n3σs(ω) =1\n24π2(k−2\n0v⊥)2/3vzΓ(1/3)\nΓ[5/6]|ω+µ+sQ0|2/3.\n(A12)Appendix B: Mean-Field equations for the\nMulti-Dirac/Weyl Kondo lattice\nIn the multi-Dirac case with particle-hole symmetry\nwe have only one mean-field parameter Vs=V. More-\nover, at T= 0 f(E++(k)) = f(E−+(k)) = 0, and\nf(E+−(k)) =f(E−−(k)) = 1, yielding\nZd3k\n(2π)31p\nϵ2\nk+ 4V2=1\n2J, (B1)\nwhere ϵk=p\nA2nk2n\n⊥+v2zk2z. Introducing the high-\nmomentum cutoff Λ and performing the change of vari-\nables (A4) of Appendix A, we obtain the dimensionless\nequation\nZ1\n0dxx2/n\n√\nx2+m2=1\nρn(Λ)J, (B2)\nwhere m= 2V/Λ and ρn(Λ), given by (A9), is here com-\nputed at Λ with µ= 0. Eq. (B2) has a non-trivial solu-\ntion only above a critical Kondo coupling Jc\nn, determined\nby letting m= 0 in Eq. (B2), resulting in\nJc\nn=(2/n)\nρn(Λ), (B3)\nwhich is consistent with the Withoff-Fradkin result for\nthe pseudogap Kondo problem with r= 2/n[38].\nIn the absence of particle-hole symmetry ( µ̸= 0,\nλ̸= 0), using the same procedure as before, at T= 0\nf(E++(k)) =f(E+−(k)) =f(E−+(k)) = 0, and we ob-\ntain the self-consistency equations\nZ1\n0dxx(2/n)+1\np\n(x−α)2+m2=1\nρn(Λ)Λ\u0014\n1−α\nJ/Λ\u0015\n,(B4)\nZ1\n0dxx2/n\np\n(x−α)2+m2=1\nρn(Λ)J. (B5)\nwhere α= (µ+λ)/Λ.\nWhen the TRS is broken but inversion symmetry is\npreserved, we have Q̸= 0 and Q0= 0. Following\nalong the same lines as in Appendix A, we can shift\nkz−sQ→kzin the mean-field equations and, for\nweak TRS breaking Λ ≫Q, the dependence on the\nTRS breaking parameter Qdisappears, and we conclude\nthatVs=Vfor the broken-TRS multi-Weyl Kondo sys-\ntem. Thus, the broken-TRS multi-Weyl Kondo mean-\nfield equations are identical to those of the multi-Dirac\nsystem [Eqs. (B4)-(B5)].\nFor the broken-IS multi-Weyl Kondo lattice, mini-\nmization of the mean-field free energy leads to the self-12\nconsistency equations\nZ1\n0dxx2/n\nq\n(x−α−¯Q0)2+m2\n+=1\nρn(Λ)J,(B6)\nZ1\n0dxx2/n\nq\n(x−α+¯Q0)2+m2\n−=1\nρn(Λ)J,(B7)\nZ1\n0dxx(2/n)+1\nq\n(x−α−¯Q0)2+m2\n+\n+Z1\n0dxx(2/n)+1\nq\n(x−α+¯Q0)2+m2\n−\n=2\nρn(Λ)Λ\u0014\n1−α\nJ/Λ\u0015\n, (B8)\nwhere α= (µ+λ)/Λ,˜Q0=Q0/Λ and m±= 2V±/Λ. In\nthe limit Q0= 0 we recover Eqs. (B4)-(B5), as expected.\nAppendix C: Detailed calculations of the\nmulti-Dirac Kondo quantum critical behavior\nIn the vicinity of the quantum phase transition point\njn= 1 ( J=Jc\nn) the Kondo coupling can be written in\nterms of a small perturbation δasjn= 1+ δ, with δ≪1.\nUsing this in Eq. (21) and expanding up to linear order\ninδwe get\nfn(m) =−n\n2δ, (C1)\nwhere\nfn(m) =Z1\n0dxx2/n\u00141√\nx2+m2−1\nx\u0015\n.\nPerforming the variable change x=my\nfn(m) =m2/nZ1/m\n0dyy2/n\"\n1p\ny2+ 1−1\ny#\n.(C2)\nForn >1, we can let m→0 since the integral converges.\nIn this case,\nm(δ) =Cnδn/2, (n >1) (C3)\nwhere Cn=\u0014\n−n√π\nΓ(1\n2+1\nn)Γ(−1\nn)\u0015n/2\n. Thus, for n > 1, the\nmean-field critical exponent is ν=n/2.\nForn= 1, we split the relevant integral as\nZ1/m\n0dyy2\"\n1p\ny2+ 1−1\ny#\n=Z1\n0dyy2\"\n1p\ny2+ 1−1\ny#\n+Z1/m\n1dy(\ny2\"\n1p\ny2+ 1−1\ny#\n+1\n2y)\n−1\n2Z1/m\n1dy\ny.We can safely let m→0 in the second term on right-\nhand side since that integral is now convergent. Using\nthis in Eq. (C1) results in\nm2ln\u0010m\nA\u0011\n=−δ, (C4)\nwhere A=√e/2. Eq. (C4) can be solved in the limit\nδ≪1 yielding\nm(δ) =f(δ)δ1/2, (n= 1), (C5)\nwhere f(δ) =q\n−2\nW−1[−A2δ]. Here, W−1(x) is one of the\nbranches of the Lambert function [61], which has the fol-\nlowing asymptotic behavior for 0 < x≪1\nW−1(−x) = ln ( x)−ln [−ln (x)] +O(1).(C6)\nThus, the critical exponent is ν= 1/2, with logarithmic\ncorrections built into the function f(δ).\nAppendix D: Analytical results for the multi-Dirac\nand broken-TRS multi-Weyl Kondo lattice\nWe can extract analytical results from the self-\nconsistency Eqs. (B4)-(B5) in the n= 1,2 cases. To\ndo so, we use the expressions\nZ1\n0dxxp\n(x−α)2+m2=p\nm2+ (1−α)2−p\nm2+α2\n+αh\nln\u0010\n1−α+p\nm2+ (1−α)2\u0011\n−ln\u0010p\nm2+α2−α\u0011i\n,\nZ1\n0dxx2\np\n(x−α)2+m2=1\n2hp\nm2+ (1−α)2\n+3α\u0010p\nm2+ (1−α)2−p\nm2+α2\u0011i\n−1\n2(m2−2α2)h\nln\u0010\n1−α+p\nm2+ (1−α)2\u0011\n−ln\u0010p\nm2+α2−α\u0011i\n,\nZ1\n0dxx3\np\n(x−α)2+m2=1\n6p\n(m2+α2)[m2+ (1−α)2]×\nn\n2p\nm2+α2+ 4m4\u0010p\nm2+ (1−α)2−p\nm2+α2\u0011\n+m2h\n−2p\nm2+α2+ 13αp\nm2+α2\n+7α2\u0010p\nm2+α2−p\nm2+ (1−α)2\u0011i\nαhp\nm2+α2+ 3αp\nm2+α2−17α2p\nm2+α2\n+11α3\u0010p\nm2+α2−p\nm2+ (1−α)2\u0011io\n+α\n2(−3m2+ 2α2)h\nln\u0010\n1−α+p\nm2+ (1−α)2\u0011\n−ln\u0010p\nm2+α2−α\u0011i\n.13\nIn the regime in which the Kondo parameter mis ex-\nponentially small and m≪α, these equations simplify\nconsiderably. Then, if n= 1,\n−α2ln\u0012m2\n2α\u0013\n+f1(α) =1\nρ1(Λ)J, (D1)\n2α\nρ1(Λ)J+1\n3(1−2α3) =1\nρ1(Λ)Λ, (D2)\nwhere f1(α) =1\n2{1 + 2 α−6α2+ 2α2ln[2(1 −α)]}. For\nn= 2 we obtain\n−αln\u0012m2\n2α\u0013\n+f2(α) =1\nρ2(Λ)J, (D3)\n2α\nρ2(Λ)J+1\n2(1−2α2) =1\nρ2(Λ)Λ, (D4)\nwhere f2(α) = 1−2α+αln[2(1−α)]. Closed-form expres-\nsions can be obtained when 0 < α≪1, with exponential\naccuracy for m. For n= 1,\nα≈1\n2\u00121\nρ1(Λ)Λ−1\n3\u0013\nρ1(Λ)J, (D5)\nm≈exp\u0014\n−1\n2ρ1(α)J\u0015\n.\n(D6)\nAnalogously for n= 2,\nα≈\u00121\nρ2(Λ)Λ−1\n2\u0013\nρ2(Λ)J, (D7)\nm≈exp\u0014\n−1\n2ρ2(α)J\u0015\n.\n(D8)\nAppendix E: General Hamiltonian at the\nParticle-Hole Symmetric Point\nAt the particle-hole symmetric point ( λ=µ= 0) and\nin the presence of TRS ( Q= 0), some choices of the\ncoupling parameters J,W, and Kallow for an analytical\ntreatment, which we now describe. In these cases, the\ndispersion relations of Hamiltonian (9) are given by\nEαξ\ns(k) =1\n2α\u0014q\nε2\nks+ 4r2\n1+ξq\nε2\nks+ 4r2\n2\u0015\n,(E1)\nwhere α=±,ξ=±,εks=p\nA2nk2n\n⊥+v2zk2z−sQ0, and\nr1andr2are order parameters which depend on the case\nin question.\nThe first interesting case is K= 0, for which r1=Jv,\nr2=Ww. By minimization of the free energy at T= 0\nifJ > W we have\nX\ns=±Z1\n0dxx2/n\np\n(x−sQ0)2+m2=2\nρn(Λ)J, (E2)where m= 2Jv/Λ,ρn(Λ) is the multi-Weyl density of\nstates (see Appendix B) and w= 0 ( r2= 0). When\nW > J , we should exchange ( J, v) and ( W, w). In both\ncases, the dispersion relations are those of Eq. (16) with\nµ=λ=Q= 0,V=Jv(J > W ) orV=Ww(W > J ).\nThe second case is J=W̸=K, for which r1=\nJv+Kwandr2=Kv+Jw. After the free energy min-\nimization we obtain v/J=w/K and again the problem\nreduces to a single order parameter ruled by Eq. (E2).\nHowever, the dispersion relation structure is slightly dif-\nferent from Eq. (16) since r1andr2are both finite.\nAppendix F: Multi-Weyl Kondo semimetal specific\nheat\nTo compute the multi-Weyl Kondo semimetal spe-\ncific heat we will follow Hsin-Hua Lai et al. in the\nstudy of the single Kondo-Weyl semimetal [16]. We\nwill take the effective multi-Weyl dispersion εks=\nℏp\nA∗2nk2n\n⊥+v∗2z(kz−sQ)2, where we restore the funda-\nmental constant ℏ,A∗\nn=k(1−n)\n0 v∗\n⊥,v∗\n⊥andv∗\nzare the\nrenormalized velocities of the heavy Fermi liquid. The\nfree-energy can be computed as\nF=−N\nβX\ns=±Zd3k\n(2π)3ln[1 + e−βεks]. (F1)\nPerforming the momentum integral above using the same\nvariable transformations (A4), and using\nCv(T) =−T∂2F\n∂T2,\nwe obtain the specific heat for the multi-Weyl Kondo\nsemimetal\nCv(T)\nT=γnT2/n, (F2)\nwhere\nγn=kB\nπ2\u0012kB\nℏ\u0013(2+n)/n(1 +n)(2 + n)\nn4[Γ(1/n)]2\n×ζ\u0012\n2 +2\nn\u0013[2(2+n)/n−1]\nkn(1−n)/n\n0 v∗2/n\n⊥v∗z, (F3)\nandζ(s) is the Riemann zeta function.\nThis result agrees with reference [16] in the particular\ncase of n= 1. In general, we see that the specific heat is\nenhanced relative to the non-interacting value by a factor\nof (v⊥/v∗\n⊥)2/n(vz/v∗\nz).14\n[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[2] J. E. Moore, Nature 464, 194 (2010).\n[3] B. Bernevig and T. Hughes, Topological Insulators and\nTopological Superconductors (Princeton University Press,\n2013).\n[4] M. Sato and Y. Ando, Reports on Progress in Physics\n80, 076501 (2017).\n[5] F. Wilczek, Physics World 19, 22 (2006).\n[6] C. Nayak, S. H. Simon, A. Stern, M. 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Cheon et al., Effects of beam spinning on th e fourth-order particle resonance of\n3D bunched beams in high-intensity linear accelerators, Ph ys. Rev. Accel. & Beams 25, 064002\n(2022)]. Although not further treated therein, spinning ca n be quantified by angular momentum\nor by vorticity. Considering vorticity revealed that the la tter has remarkable similarity w.r.t. its\nmodelling along solenoid channels to modelling the beam env elope. Matrices of vorticity trans-\nport, corresponding phase advances, and Twiss parameters l ook very similar and are partially even\nidentical to their counterparts concerning envelopes. Cor responding to emittance, the quantity of\nvortissance, being a constant of motion, is defined. Unlike e mittance, for vorticity-dominated beams\nit may take imaginary values, causing Twiss parameters, and negative or zero phase advances along\na finite beam line section. This imposes considerable conseq uences on respective periodic solutions.\nI. INTRODUCTION\nPreservationofbeam quality is a majorconcern within\nthe design of almost all linear particle accelerators. The\nquality can be degraded along the accelerator by vari-\nous perturbations. The probably most known and feared\nperturbation are non-linear forces from the beam’s self\nfields. Many techniques have been applied successfully\nin order to minimize the impact of space charge. Among\nthese is provision of very regular, i.e., periodic focusing\nlattices as well as matching of the beam envelope to the\nlatter. Matching is achieved if the beam’s spatial pa-\nrameters reproduce the periodicity of the lattice. Since\ndecades matching has been restricted to the horizontal,\nvertical, and longitudinal bunched beam dimension, dis-\nregarding eventual coupling among the three planes of\nphase space (planes for brevity). Examples for this can\nbe found for instance in [1, 2]. During the last years, ex-\ntension of matching towards consideration of inter-plane\ncoupling has been started [3–5]. A very first conceptual\napproach has been sketched already in the 1980ies by [6].\nApart from matching, the amount of focusing strength\nhas been optimized in order to minimize emittance\ngrowth along the lattice. To a large extend this implies\navoiding single-particle resonances and collective insta-\nbilities, being reviewed in [7] for instance.\nRecently, an additional tool for further reduction of re-\nmainingemittancegrowthhasbeenproposed[8]. Itisthe\ncontrolled spinning of the beam, in analogy to stabiliza-\ntion of flying objects against turbulences. Some evidence\nfor mitigation of emittance growth with increased spin-\nning has been provided, hence paving path to a broad\nfield of further research.\nThereportedactivitieshereinaimforprovisionoftools\nfor better understanding of what type of spinning or ro-\ntation stabilizes particle beams. Apart from angular mo-\nmentum, rotation may be quantified by vorticity. For\nrigidly rotating objects with cylindrical symmetry, like\nballs, bullets or frisbees, the angular momentum is equal\nto the vorticity. However, particle beams are generallyneither rigid nor cylindrically symmetric. Accordingly,\nthe question rises, what type of rotation causes stabiliza-\ntion of beams.\nWithin the pursue of this question, the behaviour of\nvorticity has been investigated along solenoid channels,\nsince the latter preserve angular momentum. It has been\nfound that vorticity has some remarkable features lead-\ning to vorticity dynamics being very similar to envelope\ndynamics. The related findings shall be reported here.\nThe manuscript commences by mentioning some rel-\nevant features of angular momentum and vorticity re-\nlated to particle beams. Afterwards, a beam line is\nsketched that can form beams having either angular mo-\nmentum, or vorticity, or both. The fourth section is\non properties of vorticity along linear lattice elements.\nThe resulting vorticity beam dynamics modelling is de-\nscribed subsequently, followed by a section on properties\nofmatchedbeamsdependingontheiramountofvorticity.\nFinally, the properties w.r.t. vorticity of a special pair of\nquadrupole triplets are briefly reported. The manuscript\ncloses with preliminary conclusions and an outlook. Sev-\neral well-known issues and equations from envelope dy-\nnamics are referred to throughout the manuscript. They\nhave been placed into the appendices, in order to focus\nthe main body of the report on the new topics.\nII. ANGULAR MOMENTUM, VORTICITY,\nAND EIGEN-EMITTANCES\nOne issue not being addressed so far is the proper def-\ninition of spinning. For the time being, spinning or ro-\ntation has been associated and even been set equal to\nthe amount of angular momentum. This seemed quite\njustified, considering the angular momentum’s outstand-\ning role in physics. In beam physics, the rms angular\nmomentum can be properly defined through two second\norder beam moments as\nL:=/angb∇acketleftxy′/angb∇acket∇ight−/angb∇acketleftx′y/angb∇acket∇ight, (1)2\nwhere the brackets indicate the mean value of their con-\ntent for a given ensemble, here the product of two par-\nticle coordinates. The above equation is the translation\nof/vector r×/vector pinto beam physics in transverse coordinates.\nThroughoutthemanuscript,thehorizontalparticlepo-\nsitionxpoints towards the left and yis upwards. The\nbeam propagation sfollows a right-handed coordinate\nsystem as /vector s:=/vector x×/vector y. The notation u′indicates the\nderivative of the coordinate uw.r.t.s.\nSpinning orrotation may alsobe quantified by the vor-\nticity for instance. It is defined by the local rotation of\nthe velocity /vector∇×/vector v, being integrated over the extension of\nthe ensemble.\n˜VA=ˆ\nA/bracketleftBig\n/vector∇×/vector v/bracketrightBig\n·d/vectorA, (2)\nTranslated into beam physics and using [9]\nV=/angb∇acketlefty2/angb∇acket∇ight/angb∇acketleftxy′/angb∇acket∇ight−/angb∇acketleftx2/angb∇acket∇ight/angb∇acketleftyx′/angb∇acket∇ight+/angb∇acketleftxy/angb∇acket∇ight(/angb∇acketleftxx′/angb∇acket∇ight−/angb∇acketleftyy′/angb∇acket∇ight) (3)\nresults into\nVA:=V\nA, (4)\nwithA2:=/angb∇acketleftx2/angb∇acket∇ight/angb∇acketlefty2/angb∇acket∇ight−/angb∇acketleftxy/angb∇acket∇ight2as the beam rms area. The\nquantity VAhas the same dimension as angular momen-\ntum and projected transverse emittance.\nFor instance, a shearing motion has finite angular mo-\nmentum but no vorticity. The simplest example is given\nby twoobjects passingby eachotherwith constantveloc-\nity vectors each. There is no rotation in this movement\nand it has no vorticity. Anyway, it has a well defined\nfinite angular momentum.\nAnother example is sketched in Fig. 1. Particles form-\ning ellipses perform rotations of different kind. The first\nrotation is rigid and all particles spin at angular veloc-\nityω. The second scenario is an intrinsic rotation. Parti-\ncles move around the centre without changing the ellipse\nshape. Determining the angular momentum and VAre-\nveals for the rigid rotation [10]\nLrig=ω\n4a2(1+r2), (5)\nVA,rig=ω\n2a2r, (6)\nwhile for the intrinsic rotation one obtains\nLint=ω\n2a2r, (7)\nVA,int=ω\n4a2(1+r2). (8)\nThe expressions for LandVAflip when flipping from\nrigid to intrinsic rotation. Additionally, for extreme as-\npect ratios of r≪1 orr≫1, the rigid rotation has just\nangular momentum and vanishing VA(relatively), while\nFIG. 1. Ellipse with aspect ratio rperforming a rigid rotation\n(upper) and an intrinsic rotation (lower).\nthe intrinsic rotation has just VAbut vanishing angular\nmomentum (relatively). Another special case is the cir-\ncle (r= 1) with L=VA. As shown in section III, beams\ncan be created having considerable angular momentum\nand zero vorticity and vice versa.\nVorticity received attention in beam physics thanks to\nits tight relation to eigen-emittances. The latter have\nbeen introduced in [11] and are two constants of motion\nalong symplectic beam line elements, which may cou-\nple the horizontal and vertical planes. The two eigen-\nemittances are equal to the projected rms emittances,\nonce inter-planecoupling has been fully removedby sym-\nplectic elements. Their product is equal to the four-\ndimensional (4d) rms emittance.\nThe eigen-emittances of the two ellipses are calculated\nas (see App. Aand [10])\nǫ1,rig=VA,rig≤Lrig, (9)\nǫ2,rig= 0 (10)\nand\nǫ1,int=VA,int≥Lint, (11)\nǫ2,int= 0. (12)\nThese relations are one example illustrating that eigen-\nemittances are related to vorticity rather than to angular\nmomentum. Vorticity also occurs within the extension3\nof Busch’s theorem [12] to particle beams [9]. Finally, as\nshown in [10], the change of eigen-emittances along short\nbut non-symplectic beam line elements is related to the\nchange of vorticity VAthrough\n(∆ε)2−V2\nA=const, (13)\nwhere ∆ εis the difference of the two eigen-emittances.\nAs pointed out in [10], the above equation is the gener-\nalization of Kim’s relation [13]\nǫ1/2=ǫrms±L\n2(14)\nbetweeneigen-emittances,projectedrms-emittances,and\nangularmomentum. Kim’s relation applies to the special\ncaseoffull cylindrical symmetry, i.e., L=VAand with the\nconstant of Eq. ( 13) being equal to zero.\nIII. FORMATION OF L- AND VA-BEAMS\nAs pre-requisites shall be just re-called the statements\nof [10] concerning one property of regular and skewed\nquadrupoles. Both practically do not change the vortic-\nityVA. Figure 2depicts a beam line that can form beams\nwith adjustable amounts of vorticity VAand angular mo-\nmentum L. The creation of ions is accomplished inside\nof a solenoid. At creation, the beam is fully uncoupled,\ni.e., all off-diagonal beam moments are zero. It shall be\na dc-beam of protons with a sharp energy of 95 keV.\nThe 4d-emittance is set to 200 mm2mrad2, accordingly\nboth eigen-emittances are equal to 14.1 mm mrad. The\nbeam rms width is chosen to be 5.3 mm in both trans-\nverse planes. Extraction of the beam is through the exit\nfringe field of the solenoid. The exit fringe field imposes\nrotation to the cylindrical symmetric beam with [10]\nL=VA=−2κA (15)\nandκ:=B/(2(Bρ)), with Bas solenoid magnet field\nstrength and Bρas beam rigidity.\nAfterwards, a regular quadrupole removes the cylin-\ndrical symmetry. The quadrupole does practically\nnot change Lsince the beam features /angb∇acketleftxy/angb∇acket∇ight=0 at the\nquadrupole entrance ( Lis not changed strictly just in\nthe thin lens approximation). As mentioned above,\nit does not change the vorticity either. The regular\nquadrupole is followed by a skewed quadrupole. The\nbeam at its entrance has no cylindrical symmetry, hence\nthis quadrupole changes L(but not VA). For a short\nskewedquadrupole, ∆ Liscalculatedas(usingEq.( B10))\n∆L=k·(/angb∇acketlefty2/angb∇acket∇ight−/angb∇acketleftx2/angb∇acket∇ight), (16)\nwherek:=GLqs/(Bρ)withGasthefieldgradientand Lqs\nas the effective field length.\nAccordingly, the sketched beam line allows for impos-\ning arbitrary amounts of LandVAto the beam. In sum-\nmary, the method is to impose the required VAthroughthe solenoid field strength and to adapt the quadrupole\nstrengths to the desired L. Although the method has\nbeen describedassumingshortquadrupoles, the principle\nworks also in general with quadrupoles of finite lengths.\nTableIlists the settings of the beam line which form\nvarious amounts of angular momentum and vorticity VA.\nThe resulting phase space distribution for a beam with\nangular momentum but without VAis plotted in Fig. 3.\nSuch a beam is referred to as an L-beam. It has equal\neigen-emittances in agreement to Eq. ( 13) with the con-\nstant being equal to zero. Figure 4instead depicts the\ndistribution of a beam without angular momentum but\nwithVA, which will be referred to as a VA-beam. This\nbeam has different eigen-emittances, again in agreement\nto Eq. ( 13). For completeness, the distribution for a\nbeam with angular momentum equal to vorticity VAis\nshowninFig. 5. Thisbeamhascylindricalsymmetryand\nis referred to as an L=VA-beam. It shall be mentioned\nTABLE I. Magnet settings for the beam line to form various\namounts of angular momentum Land vorticity VA.\nL/VAsolenoid field reg. quad. grad. sk. quad. grad.\nµm/µmmT mT/m mT/m\n0.0 / 0.0 0 0 0\n-20 / 0.0 0 417 -280\n0.0 / 20 321 417 -273\n20 / 20 314 0 0\n-40 / 0.0 0 417 -565\n0.0 / 40 664 417 -479\n40 / 40 630 0 0\nFIG. 2. Beam line to form beams with arbitrary amounts of\nangular momentum Land vorticity VA.\nthat the beam parameters listed within the captions of\nFigs.3,4, and5, all meet the general Eq. ( 13), while\njust the special case of cylindrical symmetry (Fig. 5)\nmeets Eq. ( 14).4\nFIG. 3. Two-dimensional projections of a L-beam,\nε1=ε2=14 mm mrad, L=-20 mm mrad, VA=0,\nεx=εy=46 mm mrad.\nIV. PROPERTIES OF BEAM VORTICITY\nThe beam vorticity and its derivatives w.r.t. salong a\ndrift are\nV=/angb∇acketlefty2/angb∇acket∇ight/angb∇acketleftxy′/angb∇acket∇ight−/angb∇acketleftx2/angb∇acket∇ight/angb∇acketleftyx′/angb∇acket∇ight+/angb∇acketleftxy/angb∇acket∇ight(/angb∇acketleftxx′/angb∇acket∇ight−/angb∇acketleftyy′/angb∇acket∇ight),(17)\nV′= [/angb∇acketleftxx′/angb∇acket∇ight+/angb∇acketleftyy′/angb∇acket∇ight][/angb∇acketleftxy′/angb∇acket∇ight−/angb∇acketleftx′y/angb∇acket∇ight]\n+/angb∇acketleftx′y′/angb∇acket∇ight/bracketleftbig\n/angb∇acketlefty2/angb∇acket∇ight−/angb∇acketleftx2/angb∇acket∇ight/bracketrightbig\n+/angb∇acketleftxy/angb∇acket∇ight/bracketleftbig\n/angb∇acketleftx′2/angb∇acket∇ight−/angb∇acketlefty′2/angb∇acket∇ight/bracketrightbig\n,(18)\nV′′=2/bracketleftbig\n/angb∇acketleftx′2/angb∇acket∇ight/angb∇acketleftxy′/angb∇acket∇ight−/angb∇acketlefty′2/angb∇acket∇ight/angb∇acketleftx′y/angb∇acket∇ight/bracketrightbig\n+2/angb∇acketleftx′y′/angb∇acket∇ight[/angb∇acketleftyy′/angb∇acket∇ight−/angb∇acketleftxx′/angb∇acket∇ight],(19)\nand\nV′′′= 0. (20)\nAs for the beam envelope /angb∇acketleftx2/angb∇acket∇ight, the second derivative of V\nisaconstantalongadrift. Accordingly, vorticityistrans-\nformed througha drift oflength dby the matrixequation\n\nV\nV′\nV′′\n→\n1dd2\n2\n0 1d\n0 0 1\n·\nV\nV′\nV′′\n, (21)\nFIG. 4. Two-dimensional projections of a VA-beam.\nε1=27 mm mrad, ε2=7 mm mrad, L=0,VA=20 mm mrad,\nεx=59 mm mrad, εy=40 mm mrad.\nwhich is in full analogy to the transformation of the en-\nvelope by a drift given in Eq. ( C4). Equation ( 21) intro-\nduces the vorticity vector /vectorV.\nAngular momentum is strictly preserved along\nsolenoids, while vorticity is changed. First, the transport\nof vorticity along a short solenoid shall be determined.\nThis starts from the transformation of beam moments\nby a short solenoid by the matrix of Eq. ( B13). Through\nre-groupingofthe initialandfinalbeam moments, the re-\nsulting final vorticity vector Vcan be expressed through\nthe initial vorticity vector. The transformation can be\nstated as\n/vectorV →\n1 0 0\n−2κ2Ls1 0\n2κ4L2\ns−2κ2Ls1\n·/vectorV.(22)\nDoing so reveals, that the transformation of the beam\nenvelopethroughashortquadrupoleby Eq.( C5) and the\ntransformationof Vthroughashortsolenoidareidentical\nifk=κ2Ls.\nThe same method is applied to derive the transforma-\ntion of vorticity by a solenoid of arbitrary length. Equa-\ntion (B11) delivers the transformation of beam moments\nthrough a general solenoid. As for the short solenoid,5\nFIG. 5. Two-dimensional projections of a L=VA-beam.\nε1=27 mm mrad, ε2=7 mm mrad, L=VA=20 mm mrad,\nεx=εy=17 mm mrad.\nre-grouping of the final beam moments results into the\ndesired transformation of V. This re-grouping is very\nlengthyandittakescarefulbookkeepingofthenumerous\nterms. The final result is the transformation of vorticity\nas\n/vectorV →\nC2CS/κ S2/(2κ2)\n−2κCS C2−S2CS/κ\n2κ2S2−2κCS C2\n·/vectorV.(23)\nThe above equation is the counterpart of Eq. ( C6) which\ntransports the beam envelope through a finite regular\nand focusing quadrupole.\nTo summarize, the transformation of vorticity along\ndriftsandsolenoidsismodelledbymatrixequations. The\ninvolved matrices have the determinant of 1 and have\ncounterparts within the transformation of the beam en-\nvelope.\nAdditionally, the transformation of vorticity through\ninstantaneous angular kicks as\nx′→x′−kxx (24)\ny′→y′−kyy (25)shall be reported. If kx=ky=k >0, this corresponds\nto a short solenoid with κ2=(k/Ls). If both are different\nfromeachotherbut positive, it modelsspacechargekicks\nfor an elliptical but non-round beam with homogeneous\nspatial distribution. The vorticity vector is changed as\nV → V (26)\nV′→ V′−[kx+ky]V (27)\nV′′→ V′′+2kx·/bracketleftbig\n/angb∇acketleftx2/angb∇acket∇ight/angb∇acketleftx′y′/angb∇acket∇ight+/angb∇acketleftxy/angb∇acket∇ight/angb∇acketlefty′2/angb∇acket∇ight/bracketrightbig\n−2kx·/angb∇acketleftxy′/angb∇acket∇ight·[/angb∇acketleftxx′/angb∇acket∇ight+/angb∇acketleftyy′/angb∇acket∇ight]\n−2ky·/bracketleftbig\n/angb∇acketlefty2/angb∇acket∇ight/angb∇acketleftx′y′/angb∇acket∇ight+/angb∇acketleftxy/angb∇acket∇ight/angb∇acketleftx′2/angb∇acket∇ight/bracketrightbig\n+2ky·/angb∇acketleftx′y/angb∇acket∇ight·[/angb∇acketleftxx′/angb∇acket∇ight+/angb∇acketleftyy′/angb∇acket∇ight]\n+2kxky·V.(28)\nIn general, this transformation cannot be expressed\nthrough a matrix operation on /vectorV. It can be done just\nifkx=ky=k, i.e., if the k-terms in V′′sum up to -2 kV′.\nAccordingly, if kx/negationslash=ky, the effect of linear angular kicks\nand hence of quadrupoles on /vectorVcannot be modelled by a\nmatrix.\nThis section closes by considering the transformation\nof the vorticity vectorthrough a solenoid exit fringe field.\nFor the time being, no general expressions have been\nfound, which are reasonably compact. For the simple\ncase of an initially fully uncoupled beam, i.e., of zero\noff-diagonal moments, the change is given by\nV → −2κA2, (29)\nV′→0, (30)\nV′′→ −2κ/bracketleftbig\nε2\n1+ε2\n2+2κ2A2/bracketrightbig\n, (31)\nwhich cannot be expressed by a matrix equation.\nV. VORTICITY BEAM DYNAMICS AND\nVORTISSANCE\nThis section introduces the term of beam vortissance,\nwhich is a quantity being preserved along drifts and\nsolenoids. Corresponding transport matrices are defined\nas well as phase advances and Twiss parameters. Spe-\ncial emphasis is put on the fact and consequences that\nvortissance can be purely imaginary.\nA. Beam vortissance\nThe paragraph commences by realizing the similar-\nity of Eq. ( C4) to Eq. ( 21), Eq. (C5) to Eq. ( 22), and\nEq. (C6) to Eq. ( 23). These similarities suggest con-\nstruction of a quantity from /vectorVwhich is preserved along6\ndrifts and by solenoids. To this end, the horizontal rms\nemittance shall be re-expressed as\nǫx=/radicalbigg\n/angb∇acketleftx2/angb∇acket∇ight/angb∇acketleftx2/angb∇acket∇ight′′\n2−/angb∇acketleftx2/angb∇acket∇ight′2\n4. (32)\nIt is preserved along drifts and regular quadrupoles. Ac-\ncordingly, the ”rms vortissance” shall be defined in ana-\nlogue way as\nV:=/radicalbigg\nVV′′\n2−V′2\n4. (33)\nUsing Eq. ( 21), Eq. (22), and Eq. ( 23) confirms preser-\nvation of Vby each of these transformations. It shall be\nexplicitly mentioned that V2can be negative, hence the\nvortissance can take purely imaginary values. The unit\nofVis mm2mrad2. Since it is not positive definite, Vis\nneither equal to the 4d-rms emittance nor to the square\nof the angular momentum. In fact, even a beam without\nangular momentum can feature a considerable amount of\nvortissance Vas shown in section III. The vortissance is\nnot preserved along quadrupoles as indicated by Eq. ( 28)\nwithky=−kxsince then the latter is not a matrix equa-\ntion which involves just derivatives of V.\nForthe time being, areasonablyshortexpressionof V2\nthroughsecondbeammomentshasnotbeenfound. How-\never,V2can be expressed as the determinant of the ma-\ntrix\nWv:=/bracketleftbigg\nV V′/2\nV′/2V′′/2/bracketrightbigg\n, (34)\nin straight analogy to the relation of beam moments ma-\ntrix to emittance for the beam envelope.\nB. Vorticity transport matrices\nExploiting the analogies derived above, transport ma-\ntricesMeare defined such, that for a given beam line\nelement the transport of Wvis mathematically identical\nto the transport of the beam envelope, i.e.,\nWv,f:=Me·Wv,i·MT\ne, det(Me) = 1.(35)\nUsing Eqs. ( 21), (22), and (23) together with claiming\nthe determinant of each Meto be equal to 1, these matri-\nces can be derived in a straight forward way. For a drift\none obtains\nMdrift=/bracketleftbigg\n1d\n0 1/bracketrightbigg\n(36)\nand for a short solenoid\nMsol,short=/bracketleftbigg\n1 0\n−κ2Ls1/bracketrightbigg\n, (37)\nwhile for a general solenoid it is\nMsol=/bracketleftbigg\nC S/κ\n−κS C/bracketrightbigg\n. (38)Finally, the matrix Msdscorresponding to the single\nparticle transportmatrix msds(Eq. (B16)) is simply con-\nstructed through\nMsds(κ,d) :=Msol(−κ)·Mdrift(d)·Msol(κ) (39)\nand is found to read as\nMsds(κ,d) =/bracketleftbigg\nC2−κdCS−S22CS\nκ+dC2\n−2κCS+κ2dS2C2−κdCS−S2/bracketrightbigg\n.\n(40)\nThere is identity of mdrift,xxandMdrift, as well\nas ofmsol,short,xx andMsol,shortas well as of msds,xx\nandMsds. These identities imply very convenient con-\nsequences for lattices comprising drifts and pairs of\nsolenoids with opposite field directions. For instance, the\nperiodic phase advancesof /angb∇acketleftx2/angb∇acket∇ight(s) and ofV(s) along such\nlattices are identical, i.e.,\ncos(∆Φm,xx,latt) =1\n2Tr(mxx,latt)\n=1\n2Tr(Mlatt) =cos(∆ΦM,latt).\n(41)\nModelling the transport of the beam envelope /angb∇acketleftx2/angb∇acket∇ightand\nof the vorticity Vis through the same matrices. As seen\nthroughout the manuscript, the corresponding periodic\nTwiss parameters are equal as well. Accordingly, peri-\nodic solutions of Vwithin a sequence of solenoids and\ndrifts can be constructed or determined using the same\nmatrices as for periodic solutions for the beam envelope\nwithin a sequence of focusing quadrupoles and drifts.\nC. Vortissance Twiss parameters\nThe vortissance Vrelates to the vorticity as the emit-\ntance relates to the beam envelope. However, since V2\nmay be negative, Vcan take purely imaginary values.\nThis issue is discussed in the next subsection. The case\nofV= 0 is not considered, since if it is equal to zero,\nthere is no need to consider vorticity dynamics.\nThanks to the above correspondences, vorticity Twiss\nparameters can be defined as for the beam envelope\nthrough\nβv:=V\nV, (42)\nαv:=−β′\nv\n2=−V′\n2V, (43)\nγv:=1+α2\nv\nβv=V′′\n2V. (44)\nThe units of the vorticity Twiss parameters are the same\nas of the beam envelope Twiss parameters, i.e, m, 1,\nand 1/m.7\nVorticity betatron phase advances between two loca-\ntionsaandbalongsmay be defined as for the envelope\nthrough\n∆Φv,a,b:=ˆb\nads\nβv(s), (45)\nwhich along a drift turns into\n∆Φv,a,b:=±/bracketleftbigg\natan/bracketleftbiggV′′(s)s+V′(s)\n2V/bracketrightbigg/bracketrightbiggb\na.(46)\nUnlike the beam envelope and squared emittance, the\nvorticity and squared vortissance can take negative val-\nues. This has some consequences, which shall be dis-\ncussed in the following.\n1. Real vortissance V\nIfV2is positive, the vortissance is real, resulting into\nreal Twiss parameters and phase advances. The deter-\nminant of the corresponding vorticity Twiss parameter\nmatrix\n/bracketleftbigg\nβv−αv\n−αvγv/bracketrightbigg\n(47)\nis equal to 1. However, the vorticity Vand its second\nderivative V′′may be negative. In consequence, the vor-\nticity Twiss parameters βvandγvwill be negativeas well\nas the phase advances. This does not occur for beam en-\nvelopes, which are positive by construction. Apart from\nthis ambiguity in sign, there is full equivalence of vor-\nticity dynamics and envelope dynamics along channels\nmade from solenoids pairs of opposite field directions.\nFor such channels, there exist periodic solutions of the\nTwiss parameters with corresponding phase advances for\nboth, the beam envelope and for the vorticity. One set\nof solutions is equal to each other,\n∆Φv,per= ∆Φ e,per>0 (48)\nβv,per=βe,per>0 (49)\nαv,per=αe,per. (50)\nSince vortissance beam dynamics permits also for neg-\nativeβvandγv, there is the second periodic vorticity\nsolution with\n∆Φv,per=−∆Φe,per<0 (51)\nβv,per=−βe,per<0 (52)\nαv,per=−αe,per. (53)Physically,thesecondsolutionjustdescribesabeamwith\nvortex of same amount but into the opposite direction\nw.r.t. the first solution. Inverting the vortex direction\nwill change the sign of /vectorVand hence preserve the vortis-\nsanceV.\nThe constance of V2>0 enforces both, VandV′′,\nto be of same sign and being different from zero. Hence,\nV2>0 imposes an intrinsic defocusing of VandV′′away\nfrom zero. The sign of both is preserved, i.e., the zeros\ncannot be crossed. In envelope dynamics, this is the well\nknown emittance defocusing term.\n2. Imaginary vortissance V\nIn case of V2<0, the Twiss parameters are purely\nimaginary and shall be defined as:\nβv:=−iV\n|V|, (54)\nαv:=−β′\nv\n2=iV′\n2|V|, (55)\nγv:=1+α2\nv\nβv=−iV′′\n2|V|. (56)\nThedeterminantofthe correspondingvorticityTwisspa-\nrameter matrix is equal to -1. As for real vortissance, the\nTwiss parameters can assume negative (but imaginary)\nvalues. Phase advances are purely imaginary and can\ntake negative values as well. In contrast to real vortis-\nsance, there is no intrinsic defocusing of vorticity. Ac-\ncordingly, Vand/orV′′as well as the Twiss parameters\nβvand/orγvmay be zero, or different in sign.\nHowever, V=βv=0 does not cause an ill-defined γv,\nsince the constance of V2<0 imposes V′=2iVand hence\nα2\nv=-1, thus preventing the singularity of γv. Addition-\nally, the singularityisphysicallypreventedby γv=V′′/2V\nwithV/negationslash= 0.\nEventual zero crossings of βvdo not harm determi-\nnation of the imaginary phase advance according to\nEq. (45). But they may result into zero phase advance\nbetween two locations. This does not occur in envelope\ndynamics nor in vorticity dynamics at real vortissance.\nThe possibilityofzerocrossingsofvorticityand ofvan-\nishing phase advances has impact on the nature of peri-\nodic solutions. For instance, the phase advance along a\nperiodic cell is given by the trace of the matrix modelling\nthis cell. The traces of the vortissance matrices Meare\npositive and real each. But anyway, at imaginary vor-\ntissance, phase advances are imaginary. Accordingly, the\ndefinition of phase advance through the trace of the pe-\nriodic cell matrix shall be extended, such that for imagi-\nnary vortissances, it must be multiplied by ±ias\ncos(∆Φv,per) =±i1\n2Tr(Me),ifV2<0.(57)8\nTraces of periodic cell matrices are always positive.\nThis applies to periodic envelope cells as well as to pe-\nriodic vorticity cells. Hence, periodic solutions with\nzero phase advances cannot exist neither in envelope\nnor in vorticity dynamics. This applies even for the\nextended definition of periodic vorticity phase advance\nthrough Eq. ( 57). Hence, there are no periodic vortic-\nity solutions that represent symmetric quasi-oscillations\naround zero of the kind\nβv(s+Lc/2) =−βv(s), (58)\nwithLcas the length of one cell.\nD. Sources of vortissance\nTransformation of vorticity by quadrupoles and by\nsolenoid fringe fields cannot be expressed by symplec-\ntic matrices. In fact, these two beam line elements are\nsources (or sinks) of vortissance. For instance, a beam\nwith just diagonalmoments being different from zerowill\nacquire real vortissance according to Eqs. ( 29) to (31) as\nV= 2κA/radicalBig\nε2\n1+ε2\n2+2κ2A2. (59)\nImaginary vortissance can be created through skewed\nquadrupoles. The beam line introduced in sec-\ntionIIIcomprises a solenoid fringe field and (skewed)\nquadrupoles and it can form beams with real or with\nimaginary vortissance.\nVI. PERIODIC SOLUTIONS OF VALONG\nPAIRS OF SOLENOIDS\nThanks to the analogy of transport of Vthrough\nsolenoids and drifts to the transport of beam envelope\nthrough focusing quadrupoles and drifts, periodic solu-\ntions for both can be constructed for a given solenoid\nchannel. Especially, for a channel made from solenoids\nwith alternating field directions, the matrices transport-\ning the envelope and Vare identical. Accordingly, the\ncorresponding Twiss parameters are transported in the\nsame way.\nStarting from a given location s > s0along the beam\nline, for beams with real vortissance V, the Twiss param-\neters can be even equal (modulo the sign of the vorticity\nTwiss parameters), i.e.,\nβv(s) =±βe(s) (60)\nαv(s) =±αe(s). (61)\nHence for real V, permanent identity of envelope and\nvorticity Twiss parameters can be achieved.\nThis is not the case for imaginary V. Although the\nvorticity Twiss parameters are transformed in the sameway as the envelope Twiss parameters for any V, their\nrelationβvγv−α2\nv=-1 forbids the vorticity Twiss param-\neters to be equal to the envelope Twiss parameters at\nany location of the beam line. Accordingly, these two\nsets of Twiss parameters are intrinsically and perma-\nnently different from each other. Periodic solutions for\nthe Twiss parameters can be constructed for the enve-\nlope and for the vorticity. For real vortissance they are\nidentical (modulo sign), while for imaginary vortissance\nthey are intrinsically different.\nIn the following, matched solutions shall be presented\nfor the case of real Vand of imaginary V. The peri-\nodic channel comprises solenoids with effective lengths of\n0.2 m being separated by 0.6 m from each other. Their\nmagnetic field strength is 89 mT and its direction alter-\nnates from one solenoid to the next. Figure 6depicts one\nsolenoid pair comprising the smallest unit of the periodic\nchannel. The matrices of this unit are calculated from\nFIG. 6. Pair of solenoids with opposite polarity.\nthe individual transport matrices from subsection VBto\n(in m, 1, 1/m)\nmu=\n0.695 1.42 0.00 0.00\n−0.365 0.695 0.00 0.00\n0.00 0.00 0.695 1.42\n0.00 0.00−0.365 0.695\n,(62)\nMu=/bracketleftbigg\n0.695 1.42\n−0.365 0.695/bracketrightbigg\n, (63)\nand the unit’s phase advance (horizontal, vertical, and\nvorticity) is accordingly |∆Φ|=46◦. The periodic enve-\nlope Twiss parameters in both planes are βe=1.97 m\nandαe=0.\nInitially, the spinning beam is formed as described in\nsectionIII. Afterwards, four solenoids match the beam\nto the periodic channel. Here, 2d-matching aims just for\nbest envelope matching for each plane separately, i.e, full\n4d-matching of all ten beam moments is not considered.\nSolenoidsareused, sinceunlikequadrupoles,theystrictly\npreserve angular momentum Land vortissance V. The\nchannel comprises eight solenoid pairs.\nFigure7plots the three β-functions along the beam\nline for the L=VA-beam shown in section III. The vor-\ntissance is real, i.e., V=345 mm2mrad2. Matching to9\nthe periodic channel is perfect. The three β-functions\n(horizontal, vertical, vorticity) are equal to each other\nall along the beam line. The periodicity is just the half\nof one solenoid pair. The Figure does not plot the β-\nfunctions inside of solenoids since they perform steep\nvariations at the solenoids’ entrance and exit. The latter\ncause non-symplectic transformations and just the com-\nplete solenoid comprises a symplectic element.\nFIG. 7. Beta functions along the full beam line compris-\ning initial beam spinning, envelope matching to, and the\nperiodic channel itself. Shown is the case of a beam with\nreal vortissance, namely the L=VA-beam from section III.\nGrey: horizontal, red: vertical, blue: vorticity V. The lines\nat the bottom indicate the beam line elements; from left\nto right: solenoid exit fringe field, regular quadrupole, sk ew\nquadrupole, four matching solenoids, eight solenoid pairs .\nInstead, the VA-beam with imaginary vortissance\nofV=902i mm2mrad2behaves considerably different\nw.r.t. the previous one. Figure 8plots the respective β-\nfunctionsalongthebeamline. Thisbeamhasnocylindri-\ncal symmetry and hence the two transverse β-functions\ndifferfromeachother. Additionally, asexpected, thevor-\nticityβ-function differs from the transverse ones. Apart\nfrom being imaginary, also its amount is different from\ntheoneofthespatial β-functions. Theextensionofquasi-\nperiodicity ofthe three matched solutions is foursolenoid\npairs. It is called quasi-periodic, since the remaining vor-\nticity mismatch parameter is still 0.05. Mismatch is de-\nfined analogue to the horizontal envelope mismatch pa-\nrameter [14]. For the vorticity Vthis is due to the fact\nmentioned in subsection VC2: for imaginary V, zero\ncrossings of a periodic βv-function may cause zero vor-\nticity phase advance which is in contradiction to the non-\nzero trace of the respective matrix Mu. Construction of\na periodic vorticity βv-function from less then four units\nwould imply such crossings. Accordingly, such functions\ndo not exist.\nThe vorticity β-function performs a cosine-like oscil-\nFIG. 8. Beta functions along the full beam line compris-\ning initial beam spinning, envelope matching to, and the\nperiodic channel itself. Shown is the case of a beam with\nimaginary vortissance, namely the VA-beam from section III.\nGrey: horizontal, red: vertical, blue: vorticity V. The lines\nat the bottom indicate the beam line elements; from left\nto right: solenoid exit fringe field, regular quadrupole, sk ew\nquadrupole, four matching solenoids, eight solenoid pairs .\nlation. This is in full analogy to an oscillating enve-\nlopeβ-function alongacontinuouslyfocusingquadrupole\nchannel. It confirms the finding that vorticity Vbehaves\nalong solenoid channels as beam envelope behaves along\nquadrupole channels.\nFinally, it shall be mentioned that matching to a pe-\nriodic channel of solenoid pairs can be done either with\nthe periodic envelope Twiss parameters or with the peri-\nodic vorticity Twiss parameters. Figures 7and8showed\nexamples for matching with the envelope Twiss param-\neters. In the case of a L=VA-beam of Fig. 7there is\nno difference. However, there is a difference for the VA-\nbeam. Figure 9plots the vorticity matched counterpart\nof the case shown in Fig. 8. Both cases result into peri-\nodic solutions but these solutions are different from each\nother.\nVII. PAIR OF QUADRUPOLE TRIPLETS\nThis section introduces a cell made from regular\nquadrupoles that is fully equivalent to a pair of solenoids\nw.r.t. vorticity beam dynamics. The cell preserves the\nangular momentum and the vortissance. Its transport\nmatrix reads as\nmtp∗=\na b0 0\nc a0 0\n0 0a b\n0 0c a\n, (64)10\nFIG. 9. Beta functions along the full beam line compris-\ning initial beam spinning, vorticity matching to, and the\nperiodic channel itself. Shown is the case of a beam with\nimaginary vortissance, namely the VA-beam from section III.\nGrey: horizontal, red: vertical, blue: vorticity V. The lines\nat the bottom indicate the beam line elements; from left\nto right: solenoid exit fringe field, regular quadrupole, sk ew\nquadrupole, four matching solenoids, eight solenoid pairs .\nwithc= (a2−1)/b. This matrix is fully equivalent to\nthe matrix of a pair of solenoids msdsof Eq. (B16).\nThe cell comprises two triplets of regular quadrupoles\nas sketched in Fig. 10.\nFIG. 10. Pair of triplets from regular quadrupoles with op-\nposite polarity. Each triplet comprises quadrupoles of sam e\nlengthLT. Seperation of the quadrupoles within a triplet is\nby a drift of length dTand the triplets are separated by a drift\nof length DT. Outer quadrupoles have the gradient ±G1and\ncenter quadrupoles have the gradient ∓G2.\nThe general transport matrix of such a pair of triplets\nis\nmtp=\nd e0 0\ng f0 0\n0 0f e\n0 0g d\n, (65)\nwithg= (df−1)/e. This relation shall be stated here\nwithout presenting a stringent proof, which to the best\nof our knowledge is not at hand for the time being. It\nhas been rather found with a computer code. Addition-\nally, as expected, it has been found that by choosing anappropriateratio G2/G1, the triplet pair is set such, that\nEq. (64) is met. Also this statement is not yet backed by\nan appropriate proof. However, for the time being, no\nexample being in contradiction to one of the two state-\nments has been found. Assuming that these statements\nare correct, all findings presented on vorticity beam dy-\nnamics along channels of solenoid pairs apply also to the\ndynamics along pairs of triplets.\nVIII. CONCLUSION AND OUTLOOK\nAlong channels from solenoids and properly set regu-\nlar quadrupole triplets, the beam vorticity Vis modelled\nwith remarkable similarity to the beam envelope. Trans-\nportmatricesareidenticalandTwissparametersarevery\nsimilar. Thevorticitycounterparttothermsemittanceis\nthe rms vortissance. However, the latter may be purely\nimaginary resulting into imaginary and negative Twiss\nparameters and even into zero current phase advances.\nPeriodic vorticity can be achieved by lattices comprising\npairs of solenoids and of appropriate quadrupole triplets.\nMatching of L=VA-beams with real vortissance is\nstraight forward as for envelopes. Instead, matching of\nbeams with imaginary vortissance ( VA-beams) requires\nmany more super-periods.\nThe presented studies are far from being concluded.\nFurther activities shall aim on including the effects of\nspace charge. The ultimate goal is application of the\nfindings to reduction of emittance growth due to space\ncharge through appropriate spinning and matching.\nIX. ACKNOWLEDGEMENT\nWe thank Moses Chung (UNIST/Korea) and Chen\nXiao (GSI/Germany) for valuable suggestions and ad-\nvices during preparation of this report.\nAppendix A: Definition of eigen-emittances and\nsymplectic transformations\nBeam eigen-emittances are calculated through beam\nrms moments as\nε1/2=1\n2/radicalBig\n−tr[(CJ)2]±/radicalbig\ntr2[(CJ)2]−16det(C),\n(A1)\nwith\nC=\n/angb∇acketleftx2/angb∇acket∇ight /angb∇acketleftxx′/angb∇acket∇ight /angb∇acketleftxy/angb∇acket∇ight /angb∇acketleftxy′/angb∇acket∇ight\n/angb∇acketleftxx′/angb∇acket∇ight /angb∇acketleftx′2/angb∇acket∇ight /angb∇acketleftyx′/angb∇acket∇ight /angb∇acketleftx′y′/angb∇acket∇ight\n/angb∇acketleftxy/angb∇acket∇ight /angb∇acketleftyx′/angb∇acket∇ight /angb∇acketlefty2/angb∇acket∇ight /angb∇acketleftyy′/angb∇acket∇ight\n/angb∇acketleftxy′/angb∇acket∇ight /angb∇acketleftx′y′/angb∇acket∇ight /angb∇acketleftyy′/angb∇acket∇ight /angb∇acketlefty′2/angb∇acket∇ight\n,(A2)\nJ=\n0 1 0 0\n−1 0 0 0\n0 0 0 1\n0 0−1 0\n, (A3)11\nandε4d:=(detC)1\n2=ε1·ε2. Projected transverse beam\nrms emittances are defined as [15]\nεx=/radicalbig\n/angb∇acketleftx2/angb∇acket∇ight/angb∇acketleftx′2/angb∇acket∇ight − /angb∇acketleftxx′/angb∇acket∇ight2, (A4)\nεy=/radicalbig\n/angb∇acketlefty2/angb∇acket∇ight/angb∇acketlefty′2/angb∇acket∇ight − /angb∇acketleftyy′/angb∇acket∇ight2. (A5)\nA matrix Msis called symplectic if it satisfies the equa-\ntion\nMT\nsJMs=J . (A6)\nAppendix B: Relevant particle transport matrices\nand examples for transport of beam rms moments\nThis appendix lists particle transport matrices being\nrelevant for the manuscript. For some of them it gives\nexamples for the corresponding transformation of some\nbeam rms moments. In general, the full 4d-matrices are\nlisted. In case that just the horizontal phase space is re-\nferred to, it is indicated by the subscript xx. The beam\nrigidity is given by Bρand the coordinate system de-\nfinesxtowards the left, yupwards, and the beam di-\nrectionsinto forward direction, i.e., it is a right-handed\nsystem.\nThe matrix of a drift of length dis given by\nmdrift=\n1d0 0\n0 1 0 0\n0 0 1d\n0 0 1 1\n. (B1)\nFor instance, the horizontal beam rms moments are\nchanged by\n/angb∇acketleftx2/angb∇acket∇ight → /angb∇acketleftx2/angb∇acket∇ight+2d/angb∇acketleftxx′/angb∇acket∇ight+d2/angb∇acketleftx′2/angb∇acket∇ight,(B2)\n/angb∇acketleftxx′/angb∇acket∇ight → /angb∇acketleftxx′/angb∇acket∇ight+d/angb∇acketleftx′2/angb∇acket∇ight, (B3)\n/angb∇acketleftx′2/angb∇acket∇ight → /angb∇acketleftx′2/angb∇acket∇ight. (B4)\nA horizontally focusing quadrupole with effective\nlengthland magnetic field gradient Gis modelled by\nmq,xx=/bracketleftbigg\ncosΩu−1sinΩ\n−usinΩcosΩ/bracketrightbigg\n,(B5)\nwithΩ:= ulandu:=|G/(Bρ)|1\n2. Ifthequadrupoleisshort\n(ul≪1), this matrix is well approximated by\nmsq,xx=/bracketleftbigg\n1 0\n−k1/bracketrightbigg\n, (B6)\nwherek:=ul. The short quadrupole transforms the hor-\nizontal moments as\n/angb∇acketleftx2/angb∇acket∇ight → /angb∇acketleftx2/angb∇acket∇ight, (B7)/angb∇acketleftxx′/angb∇acket∇ight → /angb∇acketleftxx′/angb∇acket∇ight−k/angb∇acketleftx2/angb∇acket∇ight, (B8)\n/angb∇acketleftx′2/angb∇acket∇ight → /angb∇acketleftx′2/angb∇acket∇ight−2k/angb∇acketleftxx′/angb∇acket∇ight+k2/angb∇acketleftx2/angb∇acket∇ight.(B9)\nThe skewed quadrupole is a quadrupole being rotated\nby 45◦clockwise around the positive beam direction. For\na short skew quadrupole the matrix reads as\nmq,skew=\n1 0 0 0\n0 1−k0\n0 0 1 1\n−k0 0 1\n. (B10)\nA solenoid of length Lsshall have the longitudinal\nmagnetic field strength Balong the positive beam direc-\ntion. The solenoid’s strength is defined by κ:=B\n2(Bρ).\nAdditionally, C:=cos(κLs), andS:=sin(κLs). The\nsolenoid transport matrix is\nmsol=\nC2CS\nκCSS2\nκ\n−κCS C2−κS2CS\n−CS−S2\nκC2CS\nκ\nκS2−CS−κCS C2\n.(B11)\nThe following example gives the lengthy transformation\nof the beam moment /angb∇acketleftxy′/angb∇acket∇ightby a solenoid\n/angb∇acketleftxy′/angb∇acket∇ight →+/angb∇acketleftx2/angb∇acket∇ight·κS2C2+/angb∇acketleftxx′/angb∇acket∇ight·/bracketleftbig\nCS3−C3S/bracketrightbig\n+/angb∇acketleftxy/angb∇acket∇ight·/bracketleftbig\nκCS3−κC3S/bracketrightbig\n+/angb∇acketleftxy′/angb∇acket∇ight·/bracketleftbig\nC4+S4/bracketrightbig\n−/angb∇acketleftx′2/angb∇acket∇ight·C2S2/κ−/angb∇acketleftx′y/angb∇acket∇ight·2C2S2\n+/angb∇acketleftx′y′/angb∇acket∇ight·/bracketleftbig\nC3S/κ−CS3/κ/bracketrightbig\n−/angb∇acketlefty2/angb∇acket∇ight·κC2S2\n+/angb∇acketleftyy′/angb∇acket∇ight·/bracketleftbig\nC3S−CS3/bracketrightbig\n+/angb∇acketlefty′2/angb∇acket∇ight·/bracketleftbig\nC2S2/κ/bracketrightbig\n.\n(B12)\nFor a short solenoid holds κLs≪1 which results into\nmsol,short=\n1 0 0 0\n−κ2Ls1 0 0\n0 0 1 0\n0 0−κ2Ls1\n.(B13)\nThe solenoid’s entrance and exit field are part of the\nsolenoid matrix. However, they can be modelled stand-\nalone through:\nmsol,fringe =\n1 0 0 0\n0 1±κ0\n0 0 1 0\n∓κ0 0 1\n,(B14)\nwhere the upper (lower) sign refers to the entrance (exit)\nfringe field. This fringe field matrix is the only one be-\ning mentioned within this study which is not symplectic.\nIt changes the two beam eigen-emittances. However, it\npreserves the 4d-rms emittance which is the product of\nboth.\nOfspecialrelevancehereisthesequenceofasolenoid,a\ndrift, and a second solenoid with opposite field w.r.t. the12\ndirection of the first one. The corresponding transport\nmatrix does not couple the two transverse planes and\nreads as\nmsds:=msol(−B)·mdrift·msol(B),(B15)\nmsds=/bracketleftbigg\nmsds,xx 0\n0msds,yy/bracketrightbigg\n, (B16)\nmsds,xx=msds,yy=/bracketleftbigg\nC2−κdCS−S22CS\nκ+dC2\n−2κCS+κ2dS2C2−κdCS−S2/bracketrightbigg\n.(B17)\nAppendix C: Transport of transverse beam envelope\nIn the following, the discussion is restricted to the\nhorizontal plane, however, the vertical one could have\nbeen chosen as well. The beam moments are expressed\nthrough the derivatives of the squared beam size /angb∇acketleftx2/angb∇acket∇ight\nw.r.t.salong a drift:\n/angb∇acketleftx2/angb∇acket∇ight′= 2/angb∇acketleftxx′/angb∇acket∇ight, (C1)\n/angb∇acketleftx2/angb∇acket∇ight′′= 2/angb∇acketleftx′2/angb∇acket∇ight, (C2)\nand explicitly mentioning\n/angb∇acketleftx2/angb∇acket∇ight′′′= 0. (C3)Defining the envelope vector /vectorX:= [/angb∇acketleftx2/angb∇acket∇ight,/angb∇acketleftx2/angb∇acket∇ight′,/angb∇acketleftx2/angb∇acket∇ight′′],\nthe transport of the beam envelope along a drift is ex-\npressed through a matrix equation\n/vectorX→\n1dd2\n20 1d\n0 0 1\n·/vectorX, (C4)\nwhich is just the Taylor series.\nIn an analogue way, the transport of the envelope\nthrough a short quadrupole is stated as\n/vectorX→\n1 0 0\n−2k1 0\n2k2−2k1\n·/vectorX. (C5)\nFinally, thetransportoftheenvelopethroughageneral\nhorizontally focusing quadrupole is\n/vectorX→\nC2\nqCqSq\nuS2\nq\n2u2\n−2uCqSqC2\nq−S2\nqCqSq\nu\n2u2S2\nq−2uCqSqC2\nq\n·/vectorX, (C6)\nwhereCqandSqrefer to the trigonometrical functions\nstated in Eq. ( B5).\n[1]F. Sacherer, RMS Envelope Equations with Space\nCharge, IEEE Trans. Nucl. Sci. 18, 3, 1105 (1971).\n[2]L. Groening, W. Barth, W. Bayer, G. Clemente, L. Dahl,\nP. Forck, P. Gerhard, I. Hofmann, G. Riehl, and\nS. Yaramyshev, Benchmarking of measurement and sim-\nulation of transverse rms-emittance growth, Phys. Rev.\nAccel. & Beams 11, 094201 (2008).\n[3]A. Khan, O. Boine-Frankenheim, F. Hug, C. Stoll, Beam\nmatching with space charge in energy recovery linacs,\nNucl. Instrum. & Meth. in Phys. Res. A 948, 162822\n(2019).\n[4]A. Hoover, N.J. Evans, and J.A. Holmes, Computation of\nthe matched envelope of the Danilov distribution, Phys.\nRev. Accel. & Beams 24, 044201 (2021).\n[5]C. Xiao and L. Groening, Periodic solution for trans-\nport of intense and coupled coasting beams through\nquadrupole channels, arXiv 2309.11277 (2023).\n[6]D. Chernin, Evolution of rms beam envelopes in trans-\nport systems with linear x-ycoupling, Part. Accel. 24,\npp. 24-44, (1988).\n[7]Y. L. Cheon, S. H. Moon, M. Chung, and D. Jeon,\nAnalysis on the stopband of fourth-order resonance in\nhigh-intensity linear accelerators, Phys. Plasmas 27,063105 (2020).\n[8]Yoo-Lim Cheon, Seok-Ho Moon, and Moses Chung, Ef-\nfects of beam spinning on the fourth-order particle reso-\nnance of 3D bunched beams in high-intensity linear ac-\ncelerator, Phys. Rev. Accel. & Beams 25, 064002 (2022).\n[9]L. Groening, C. Xiao, M. Chung, Extension of Busch’s\ntheorem to particle beams, Phys. Rev. Accel. & Beams\n21, 014201 (2018).\n[10]L. Groening, C. Xiao, M. Chung, Particle beam eigene-\nmittances, phase integral, vorticity, and rotations, Phys .\nRev. Accel. & Beams 24, 054201 (2021).\n[11]A.J. Dragt, General moment invariants for linear Hamil-\ntonian systems, Phys. Rev. A 45, 4 (1992).\n[12]H.Busch, BerechnungderBahnvonKathodenstrahlenin\naxial symmetrischen electromagnetischen Felde, Z. Phys.\n81, (5) p. 974, (1926).\n[13]K.-J. Kim, Round-to-flat transformation of angular-\nmomentum-dominated beams, Phys. Rev. ST Accel.\nBeams6, 104002 (2003).\n[14]T.P. Wangler, Rf Linear Accelerators, second edition\n(Wiley-VCH, Mannheim/Germany, 2008), p. 223.\n[15]K. Floettmann, Some basic features of the beam emit-\ntance, Phys. Rev. ST Accel. Beams 6, 034202 (2013)."    },    {        "title": "2401.13655v1.Bi_Hamiltonian_in_Semiflexible_Polymer_as_Strongly_Coupled_System.pdf",        "content": "Bi-Hamiltonian in Semiflexible Polymer as Strongly Coupled System\nHeeyuen Koh∗\nSoft Foundry Institute, Seoul National University,\n1 Gwanak-ro, Gwanak-gu, Seoul, 08826, South Korea\nShigeo Maruyama\nMechanical Engineering Dept., The University of Tokyo, Hongo 3-7-1, Bunkyoku, Tokyo, Japan\n(Dated: January 25, 2024)\nWhen many body system in Classical Mechanics is described using bi-Hamiltonian for its group\nof particles which forms a coarse grained bead, the damping energy from the heat bath perturbs the\nmotion of a particle which is along the trajectory of multiple Hamiltonian system simultaneously.\nAs managing the coarse grained modeling of the suspended semiflexible polymer like single walled\ncarbon nanotubes(SWCNT) in vacuum, the authors discovered that such interconnection between\nHamiltonian systems as sharing a heat bath builds a strongly coupled system. The consequence\nof this interconnection is revealed to form certain form of cross correlated momentum in coarsed\ngrained system as well as in the original atomic scales. In this paper, we adapt the framework of\nStochastic Thermodynamics to describe the cross correlation of bi-Hamiltonian system which repli-\ncates the macroscopic behaviour of SWCNT under the Smoluchowski picture for cross correlation\nbetween Hamiltonian systems. The numerical experiments using collision between tubes confirms\nthe derivation and the justification of the usage of the heat diffusion to compensate the abnormaly\nfrom cross correlated momentum.\nI. INTRODUCTION\nPerturbation of Hamiltonian in Classical Mechanics\nfrom KAM theory for celestial system to its extention\nof thermalization of the molecular system is invented to\nassure the evolution of the dynamics of many body sys-\ntem to be in the ergodic condition. Such perturbation\nfrom the heat bath, especially to engage the systemic be-\nhaviour based on the dynamic characteristics from ther-\nmal fluctuation of microscale system is thoroughly being\ndeduced under the Stochastic Thermodynamics regime\nfrom Seifert[1], which covers the scope of many body sys-\ntem in Classical and Quantum Mechanics[2].\nTo accurately describe thermodynamic properties\nin microscale system at both equilibrium and non-\nequilibrium conditions, harmonic oscillators and molec-\nular system in coarsed grained modeling has interpre-\ntated using Hamiltonian of mean force.[3, 4] Sekimoto\nproves the energetics which links Langevin equation to\nThermodynamics[5]. Forkker-Plank equation and en-\ntropy production[1] and the localization of the effect\nof the mean force of Hamiltonian to a coarse graind\nparticle[6, 7] are lively studied. The projection for slow\nand fast variables and the framework to connect the de-\nscription of strongly coupled system with its heat bath\nare an essential description to characterize the molecule\nin the multiscale scope[8]. The nature of the perturba-\ntion as work rather than a heat energy[9], and the frame-\nwork to distinguish thermal fluctuation of the molecular\nsystem upon Thermodynamics[10] are thoroughly inves-\ntigated.\n∗hy_koh@snu.ac.krRecently, Koh et al.[11], found that the bi-Hamiltonian\nsystemhascrosscorrelatedmomentumwhichinducesab-\nnormal memory effect and the diffusion of related energy\nadjusted in their equation of motion based on the La-\ngrangian dynamics which resolved the discrepancy. Bi-\nHamiltonian system, here, are the simple beads system\nto replicate the SWCNT’s nonlinear bending motion[12],\nwhich is almost perfectly syncronized with atomic scale\nsimulation result. The interconnection between Hamil-\ntonian system and heat bath which brings the memory\neffect has been presented by Zwanzig[13] for the gener-\nalilzed Langevin dynamics. Inspired from the energetics\nof Lagevin equation following Sekimoto’s frameworks, we\ncan presume the stochstic nature of the equation of mo-\ntion modified for bi-Hamiltonian mentioned above is in\nthe boundary of thermodynamic descrptors. Applying\nStochastic Thermodynamics to crarify cross correlation\nmomentum of SWCNT in atomic scale trajectory expli-\ncates the intramoleculat thermal energy mechanism as\nwellasthejustificationoftheusageofheatdiffusionterm\nin the equation of motion in coarse-grained molecular dy-\nnamcis (CGMD), which corrects the memore effect from\ncross correlation.\nIn this paper, we adapt the frame work suggested by\nJarzynski[10] whose arrangement for quantities to de-\nscribe the microscale systems are derived for real molec-\nular system in thermodynamic environment. First, the\ncross correlated momentum is interpretated as ther-\nmodynamic quantities in Jarzynski’s framework[10] and\nSmoluchowski equation[14] for its evolution. And then\nSWCNT caculated using molecular dynamics in nonequi-\nlibriumstatecausedbyexternalcollisionisdirectlyinter-\npreated with Stochastic Thermodynamic frameworks. In\nsection II. Theoretical modeling is derived based on the\nframework by Jarzynski. The data provided by molecu-arXiv:2401.13655v1  [physics.comp-ph]  24 Jan 20242\nlardynamicsandcoarsegraindmoleculardynamicsisex-\nplainedinsectionIII.Simulation. Themeaningofderiva-\ntion which is confirmed from Simulation is described in\nIV Results and Discussion. The summary and further\nworks are included in V. Conclusion.\nII. THEORETICAL MODELING\nA. Cross correlation between Hamiltonians\nThe momentum# »piof the ith atom is defined with\nthe Cartesian coordinates as# »pi=# »piτ+# »pir+# »pil. Each\nterm represents the component of the momentum along\nthe axis defined in the system of the interest. ˆeis the\nunit vector along the tube length. ˆerandˆeτare those\nfor radial direction and twist motion on the circumfer-\nence of the part of the tube. The simplified model of\nSWCNT is supposed to have two degree of freedom along\nˆel, the average of# »piland its skewed symmetry to form\nthe bending, which are noted with subscript of ℓand\nθ. The macroscopic motion along those two direction is\nregarded as bond-length and angle-wise motion of sim-\nplified beads model which has a particle representing the\nsystem of the interest shaded in blue in Fig. 1A-a and\n1A-b. The momentum of an atom# »piis can be composed\nof three different terms as below:\n# »pi=# »pi,θ+# »pi,ℓ+# »pi∆. (1)\n# »pi,θand# »pi,ℓare the component of momentum belong\nto elongation and bending deformation.# »pi∆is remained\nlevel of momentum, which is not included neither of pro-\njections.\nWhen the target system is described using two indepe-\ndent Hamiltonian systems for bond length deformation\nand angle deformation with N coarse grained particles,\nwe are going to regard the system of interest is repre-\nsented by a CG particle. Then during the evolution of\neach Hamiltonian system, the damping for each momen-\ntum# »pi,θand# ��pi,ℓare caused by# »pℓ,dampand# »pθ,dampas\nbelow:\n# »pℓ,damp =# »pi,θ+# »pi,∆, (2)\n# »pθ,damp =# »pi,ℓ+# »pi,∆. (3)\nSince# »pℓ,dampand# »pθ,dampis shares by# »pi,∆, we have\na certain correlation in atomic scale between two inde-\npendent Hamiltonians:\n# »pℓ,damp ·# »pθ̸= 0, (4)\n# »pθ,damp ·# »pℓ̸= 0. (5)\nThe existanceof this kindof correlation canresolvethe\nabnormaldiscrepancyobservedincoarsedgrainedsystem\nFIG. 1. A. a. Schematic figure of suspended SWCNT and\nthe system of the interest in blue shade, b. The cartoon of\nthe coarse grained system from suspended SWCNT with two\nindependent potential energy which builds bi-Hamiltonian\nsystem. B. The system of the interest with deformation of\nelogation( ∆ℓ) and bending( ∆θ). C. Evolution of the defor-\nmation along bond length and angle direction at tandt+∆t.\nThe coordinate system is also developed along the macro-\nscopic motion of the tube.\nwhich is defined as bi-Hamiltonian as shown in Fig. 1A-\nb. The damping momentum for the set of pj=j=θor\nℓis paired with the perturbation of potential energy Uℓ\nandUθas below:\nUℓ=Uℓ0+U′\nθ, (6)\nUθ=Uθ0+U′\nℓ. (7)\nUℓ0andUθ0is the potential function of bond length\nand angle when SWCNT is modeled as simple beads sys-\ntem. Whenthetubeisthermalized, theinflucedofdamp-\ning momentum perturbs the harmonic potential energy\nwell from Eq. (4). The existance of such alternation has\nwell known in Classical Mechanics since the elongation of\nthe tube disturbs the neutral axis of Euler beam and vice\nversa from the imperfect bending symmetry also known\nas Timoshenko beam theory.\nWhen the system of the interest is regarded as one\nlumped mass, the evolution of the Cartesian coordinate3\nFIG. 2. A. Dissipation and random force effect on harmonic\noscillator(left) and the deformed harmonic oscillator by cross\ncorrelation in bi-Hamiltonian system. B. Effect of cross cor-\nrelation on each Hamiltonian system HℓandHtheta. The\npotential energy curvature is deformed affected by another\nHamiltonian system as well as its momentum. The red and\nyellow dot has interconnectivity from cross correoation and\nits reconsiliation.\ncauses the cross correlation of the deformation change\nalong the bond-length and angle, as shown in Fig. 1C.\nThe vectors along the deformation measure is perpendic-\nular at a moment, but the altered deformation along the\nmacroscopic motion changes the coordinate axis, there-\nfore, the momentum or the deformation rate along each\naxis should be correlated as below:\n∆ℓt·∆θt+∆t̸= 0, (8)\n∆θt·∆ℓt+∆t̸= 0. (9)\nHere, ∆ℓtand∆θtis the deformation caused at time\nt. The nature of perturbation from cross correlation is\ndifferent from the Langevin equation discussed in Seki-\nmoto’s energetics whose is the dissipation and random\nforce is evolved on the Harmonic oscillaotor as shown\nin Fig. 2A-a. The cause of the alternation of har-\nmonic potential energy well and the damping from an-\nother Hamiltnonian system is rather to be in Fig. 2A-b.\nIn previous study, we show these cross correlations de-\nvelops the memory effect[11], but the atomic scale recon-\nciliation to maintain the two Hamiltonian system is not\nfully considered. In next subsection, the perturbation\ncaused by bi-Hamiltonian condition is regarded as the\nthermal environment according to the study of Zwanzig,\nwhich has revealed the evolution of Hamiltonian in heat\nbath causing the memory effect, and the governing equa-\ntion of the evolution of atomic scale reconsiliation.B. Jarzynski’s frameworks for cross correlation\nDue to the SWCNT which is to be replicated by the\nsimple beads system which has two Hamiltonian system\nfor the variables of bond length and angle deformation\nbetween simple beads as shown in Fig. 1A-b, the state\nvariables which indicates a point in the phase space,\nxj= (qj,pj)are separately described for each system\nalong j=θorℓ.θrepresents the angle and ℓrepre-\nsent the bond length. The state variable xjrepresent\nthe information from the set of atoms in the target sys-\ntem emerged as one typical type of Hamiltonian in sim-\nple bead system. Then the heat bath environment has\nits state variable yj= (q−qj,p−pj). The concept of\nseparation, as seen in Zwanzig’s heat bath definition, is\npresent in this paper as well, however, the interconnec-\ntivity of the system is more pronounced, causing the heat\nbath variable to no longer have complete orthogonality\nwith the state variable of yj. The total energy of the\nsystem defined by Jarzynski[10] is as below:\nUTOT=us(xj;λ) +UE(yj) +uSE(xj, yj) +PVE.(10)\nus(xj;λ)is the Hamiltonian of the system of the in-\nterest which is represented in the blue shade of the tube.\nλis the external control parameter over the Hamilto-\nnian of the system. Uϵ(yj)is that of the thermal envi-\nronment, therefore, it becomes the rest of the SWCNT\ntube. uSE(xj, yj)is the energy related both of the sys-\ntem and the environment. Pis the pressure of thermal\nenvironment and VEis the volume of the pistone with\nfull of solution in the reference.The isothermal-isobaric\nensemble as as below:\nπeq\nλNPT(ζ) =1\nYAe−βUS+E(ζ;λ)+PVE,\n=1\nYAe−βUS+E,(11)\nYA=Z\ndζe−βUtot=Z\ndxe−βus(x;λ)ZE\nx=ZxZE\n0.\nHere, US+Eisus(xj;λ) +Uϵ(yj) +uSE(xj, yj), and ζ\nis(xj, yj). Then, the ensemble of the system of interest\ncan be derived as below:\npeq\nλNPT=Z\ndyjπeq(xj, yj),\n=1\nZλe−β(us(xj;λ)+ϕ(xj)), (12)\nϕ(xj) =ϕ(xj:N, P, T ),\n=−β−1R\ndyjexp[−β(UE(yj) +uSE+PVE)]R\ndyexp [−β(UE(yj) +PVE)].(13)4\nZλ(N, P, T ) =Z\ndxje−β(us+ϕ(xj)).(14)\nHere, ϕ(xj)is the perturbation of energy defined by\nthe thermal environment, and defined as the difference\nbetween Gibbs free energy −β−2ln(ZE\nxjZE\n0). In the ref-\nerence, ϕis, therefore, the energy of conformation of the\ntarget system in the solution of the pistone, whose pres-\nsure is given from the environment. When vsindicates\nthe deformation of the contour of the target molecular\nsystem caused by thermodynamic environment E,ϕcan\nbe deduced as Pvs.\nIn case of the part of the SWCNT, which is the system\nof interest is defined as a part of suspended SWCNT ly-\ning along the axis yas shown in Fig. 1A-a, the definition\nofPvsis accomodated from the thermal environment de-\nfined by another Hamiltonian as explained in next sub-\nsection. For the simplicity, SWCNT in this study has\nboth ends fixed, and the fixation is supposed to be adia-\nbatic in the vacuum chamber. In our system, PVEis zero\nfor the adiabatic condition.\nNote that the system of the interest is a part of\nSWCNT which is connected to the rest of the tubes, and\nwe are considering the motion either of bending or elon-\ngation. The thermal environment Efor the Hamiltonian\nHθ, is therefore, Hℓand connected tubes on both ends\nof the system. xℓis the state variable for the system, Hθ\nbelongs to E. The influence from the connection of the\ntube structure is belong to the potential energy. Then,\nϕis the quantity that perturbed by another Hamiltonian\nif the tube is suspended in the vacuum chamber. The\nquantity is already shown to be η# »pℓ# »pθin previous study\nfor coarsed grained model of SWCNT. ηis the damp-\ning coefficient for cross correlated momentum.# »pθand# »pℓare the momentum of CG particle for each type of\nHamiltonain, HθandHℓ, respectively.\nIn next subsection, the evolution of the ensemble mod-\nified with ϕis discussed.\nC. Evolution of Cross Correlation under\nSmoluchowski picture\nInatomicscalesystem, wepresumethatthesmallscale\nmotion of the atoms, which is averaged as zero, resolves\nthe cross correlated condition. Since such condition has\nno momentum, the evolution equation in for CG scale\nparticles becomes Smoluchowski equation. The probabil-\nity density function has its governing equation as below:\n∂tρ=LCρ+LDρ+LSρ. (15)\nLCis the Liouville operator with conservative force\nfrom potential energy. LDis the Fokker-Plank opera-\ntor for the dissipative and ramdom force. Lastly, LSis\nSmoluchowski operator which has its evolution as below:LSρ=∇(ρ∇U) +T∇2ρ. (16)\nThe substitution of Eq. (13) to Smoluchowski eq. is\nas follows in equilibrium case:\n∆us+P∆vs=1\nξ∂2\n∂x2(Eke+Pvs) = 0 .(17)\nThe derivation is in Appendix. A. Intuitively, Eq.\n(17) should be zero in equilibrium condition to maintain\nthe harmonic oscillator, and such condition can be from\n∂2\n∂x2Ekewhen equilvalent to −∂2\n∂x2Pvs. This indicates the\nboundary condition of the target system and how the\ntarget system is interconnected with geometrically con-\ntacted heat bath. In the case of SWCNT in vacuum, the\ngeometrical heat bath is the tube segment at both sides\nof the target system.\n∆us+P∆vs=∂2\n∂x2(−Udef+Eke+Pvs).(18)\nWhen the system has external stimuli which provokes de-\nformation, The first term in RHS of Eq. (18) remained\nwith unresolved deformation energy, Udef, which is de-\nviated from the potential energy in equilbrium state as\npart of peq. When the system is close to equlibrium from\nexternal stimuli, RHS of Eq. (18) becomes zero. Accord-\ning to Eq. (18), it is EkeandPvswhich compenstate\nthe−Udef. To confirm this process of thermalization of\nUdef, the molecular dynamics simulation is conducted.\nIII. RESULTS AND DISCUSSION\nTo examine the response of the bi-Hamiltonian from\nEq. (18), the collision between SWCNT is conducted us-\ning molecular dynamic(MD) simultion. Two (5,5) SWC-\nNTs with 20 nm length is arranged with 90 degree to\nbe intercepted at the middle of the tube during collision.\nTo invoke the collision, one of the tube is artificially bent\nwith 5 nm depth and released to make a collision with\nanother tube underneath the bent SWCNT as shwon in\nFig. 3A.\nThe consequence of the deformation caused by colli-\nsion which draws a bending curvature on the tube is far\nlarger than the curvature of bending from equilbrium.\nThe trajectory of the SWCNT which is affected by the\ncollisionisanalyzedafterthepostprocessingofaveraging\nthe displacement from a ring, which has 10 atoms shap-\ning the circunference as shown in Fig. 1B. The angle\nfrom the deformation of collision on each bead is irreg-\nular, ∆θi̸= ∆θj, i̸=j, and it brings the perturbation\nPvs=η∆tθ∆tℓforHℓ.\nThrough the diffusion term, the double derivative∂2\n∂x2\nalong the x=qℓ, pℓvariable, offers the information of\nevolutionalongtimevariable. Amongthemodesthatthe5\nFIG.3. A.ThesnaptionoftheMDsimulation, B.Thenormal\nmode decomposition results of 1st bending mode(upper), x,y\nand z direction of nth mode, respectively.\nenthalpy Udef+us+Pvs, the highest wavevector will be\nemphasized. The expected response from the collision,\ntherefore, maximze the shorted wavelength mode, k=\nπ\nL/Nxwhere L is the length of the tube, N is the total\nnumber of the node and x is the location of the node due\nto the∂2\n∂x2.\nThe level of the amplitude analyzed by projecting\nexp(ikXi), k=π\nL/NXiwhere Xiis the coordinate of each\nsimple bead along yaxis into simple beads system from\nthe trajectory data of the collided tube is as shown in\nFig. 3B. Lis the total length of the tube and Nis the\nnumber of the bead which is connected in a series as\nshown in Fig. 1A-b. In this numerical experiment, we\nhave 200 simple beads which is from 10 atoms in a ring of\n20nm SWCNT which has 2000 atoms. The effect of dou-\nble partial derivative is confirmed through the amplitude\nofk=π\nL/Nis spikes up along the collison from another\ntube. The further analysis with k=π\nL/N−1,π\nL/N−3also\nshows similar spikes, yet the amplitude becomes undis-\ntinguishable when the wave vector k <π\nL/N−3.\nAs another validation, the collision simulation using\nFIG. 4. Normal mode decomposition of the result from\nCGMD with same configuration in Fig. 3A., A. 1st bend-\ning mode amplitude, B. N th mode along x and y direction.\nCGMD with heat diffusion damping is conducted. The\nCGMDthatweadaptedhasitsgoverningequationasthe\nequation of motion derived from the reference[11] whose\nthermalization is considering the cross correlation with\nheat diffusion term, so that the response from the exter-\nnal collision governed by the diffusion term in EOM is\nobserbed. The result is in Fig. 4.\nThe damping speed of the tube is not identical to the\natomic scale numerical experiment, yet the response of\nthe n th mode in simple beads system shows certain\namplification following the collision between two tubes.\nHowever, the governing equation is derived from La-\ngrangian withdiffusion term forthe coarse grainedmodel\nof former study, which shows identical nonlinear motion\nof SWCNT from CGMD with atomic MD. The evolution\nof enthalpy, h=us+Pvshas the double partial, yet it\nis not included as the perturbation. The discrepancy to\nimprove the governing equation need further study.\n∆us+P∆vsis equivalent to heat qand work wgiven\nto the system. According to the definition of the depen-\ndency of variable xandλ, each term in Eq. ??becomes\nas below:\nhs=us+Pvs, (19)\nq=Z\ndt˙x∂h\n∂x≈∂2\n∂x2Eke, (20)\nw=Z\ndt˙λ∂h\n∂λ≈∂2\n∂x2Pvs. (21)\nHere, hsis the enthapy of the system, and qandw\nare heat and work applied to the system, respectively.\nNote that the evolution caused by Pvsfrom double par-\ntial derivative acts as work in the system. Therefore, the\nadditional term of double partial in EOM derived from\nthe Lagrangian in Koh et al. [11] can be validated. In\nprevious study, we assume that the abnormal cross corre-6\nlation is resolved as heat diffusion. According to VI Par-\ntial Molar Representation in Jarzynski’s framework[10],\nthe certain amout of the heat is induced when the ϕis\nthe function of temperature as below:\nϕT=T∂ϕ\n∂T, (22)\n¯q=q−∆ϕT, (23)\n¯s=s−< ϕT>eq\nT. (24)\nThe temperature dependenc of Pvswhich is equivalent\ntoϕand the cross correlation term η# »pθ# »pℓis clear when\nthe macroscopic motion of bending is regarded as a slow\nvariable which absorbes certain amount of kinetic energy\nfrom the system during its oscillating. The fluctuation\nof# »pθ, therefore, depedent on the macroscopic deforma-\ntion during its bending motion so that we can conjecture\nthat∂ϕ\n∂T̸= 0. This implies that the ¯qis dependent on\nη# »pθ# »pℓandq= 0for adiabatic fixation. The irregularity\nofη# »pθ# »pℓalong the tube can cause the diffusion as the\nsource of heat.\nThis derivation is valid for HθandHℓ. The descrip-\ntion provided by the framework of Stochastic Thermody-\nnamics [10] justfies the perturbation of Hamiltonian Pvs\ncan be regarded as the cross correlation term in atomic\nscale system. Applying Smoluchowski equation for mod-\nified PDF from the perturbation shows the evolution of\nenthalpy of the target system is managed from double\npartial derivatives. The existance of the double partial\nderivatives are discussed from two independent studies.\nFirst, the trajectory from the numerical experiment of\ncollision between two SWCNTs is analyzed using nor-\nmal mode analysis. The response of n th mode of sim-\nple beads system is well shown. The similar response is\nobserved in its replication using heat diffusion damping\nCGMD.\nIV. CONCLUSION\nIn this paper, we tried to localize the scope of the ob-\nservation and description of the target molecular system\nfurther from the framework suggested by Jarzynski as\npixelizing the target of system to be a part of the semi-\nflexiblepolymer, SWCNT.Thetargetsystemissupposed\nto be averaged as a group of mass for a particle in coarsed\ngrained model, which is simple beads system. These sim-\nplebeadsconnectedbythetwopotentialenergyfunctions\nusing bond length and angle which are the variable for\neach Hamiltonian system separately as sharing a mass.\nThe disturbances of the trajectory caused by co-\nexisting Hamiltonian system for coarse grained molecular\ndynamicsiswellexplainedatKohetal. [11]. Empirically\nresolved discrepancy using heat diffusion for abnormally\nformed cross correlated momentum bewteen two Hamil-\ntonian systems managed the CGMD simulation, whichstarts from the initial velocity and displacement informa-\ntion given from atomic simulation, dictates the nonlinear\nbending motion almost identically. From this finding, we\ninvestigate the origin of the cross correlation shown in\ndiscretized coarsed grained model in the full atomic scale\nview point.\nFrom these two simulations we can conclude that the\ndiffusion term with double partial derivatives are generic\nform based on the atomic scale dynamics and justifies\nthe usage of diffusion term as damping in the equation of\nmotion. Further study to prove the direct use of the heat\ndiffusion damping from evolution of probability density\nfunction is remained as further study.\nACKNOWLEDGMENTS\nThis research is supported by Basic Science Research\nProgram through the National Research Foundation of\nKorea(NRF) funded by the Ministry of Education (NRF-\n2022R1I1A1A01063582). Itscomputationalresourcesare\nfrom National Supercomputing Center with supercom-\nputing resources including technical support (KSC-2020-\nCRE-0345). There are no conflicts to declare.\nAppendix A: Simulation Condition\nThe AiREBO potential energy function[15] is used for\ncarbon atoms in the SWCNT. Both tubes are arranged\nat 300 K with 1 ns of thermalization using Langevin\nthermostat[16] and 1 ns of relaxation. The artificial\nbending is applied after the relaxation. Fixation at both\nends has zero kelvin for a unit cell of the tube. Lammps\npackaged[17] is adapted. The sampling is conducted for\neach 1 ps during total simulation time which is 25 ns.\nThe same arrangement is prepared using coarse graind\nmolecular dynamics model. The heat diffusion damping\nwhich considers the abnormal cross correlation between\ntwo independent Hamiltonian in simple beads system is\nadapted. More detail is in the reference[11]. The ini-\ntial configuration of the simple beads system is from the\nmolecular dynamic simulation by averaging the location\nof the beads and its velocity for each unit cell which be-\ncomes a bead.\nAppendix B: Probability Density Function in\nSmoluchoswki equation\nFor resolving the discrepancy of cross correlation be-\ntween Hamiltonian systems, it is supposed that the rear-\nangement of atoms in the target system is not inducing\nany motion of the group of atoms. The evolution of state7\nvariable from probability density function is, then gov-\nerned by Smoluchowski equation as below:\nξ∂ρ\n∂t=∂\n∂x(U′ρ) +1\nβ∂2ρ\n∂x2. (B1)\nHere, ξis damping coefficient. Uis the potential func-\ntion. ρ, the probability density function can be derived\nfrom Jarzynski’s frameworks as the ensemble for the tar-\nget system:\nρ=peq=e−β[us+Pvs]\nR\ndxe−β[us+Pvs]=A\nZ.(B2)\nAise−β[us+Pvs], and ZisR\ndxe−β[us+Pvs]. The LHS\ndivided by ξbecomes:\n∂ρ\n∂t=A\nZ\u0012\n−β∂\n∂t(us+Pvs)\u0013\n. (B3)The first term of RHS of Eq. (B1) is:\n∂\n∂x(U′ρ) =∂2U\n∂x2ρ+U′∂ρ\n∂x(B4)\n∂ρ\n∂x=−A2\nZ2−βA\nZ∂\n∂x(ux+Pvs).(B5)\nThe second term of RHS of Eq.(B1) is as below:\n1\nβ∂2f\n∂x2=1\nβ∂\n∂x\u0012∂\n∂x\u0012A\nZ\u0013\u0013\n, (B6)\n= 2A3\nZ3+A2\nZ2∂2Z\n∂x2+1\nZ∂2A\n∂x2. (B7)\nTo make the assumption ofA\nZ<<1, we need the re-\nstriction on |∂A\n∂x|<1/β=kT. The restrain indicates\nthat the possible number of state should be varied in\nrange of kTbetween its neighbors. OtherwiseA\nZ<<1\nis not satisfied so that the high order ofA\nZcan not be\neliminated. The full derivation of Eq.(B1) is as below:\nξA\nZ∂\n∂t(us+Pvs),\n=∂2U\n∂x2A\nZ−1\nβ\u0012\n−βA\nZ∂2\n∂x2(us+Pvs)\u0013\n,\n∂\n∂t(us+Pvs) =1\nξ∂2\n∂x2(pVs+us).(B8)\n[1] U. Seifert, Reports on Progress in Physics 75(2012),\n10.1088/0034-4885/75/12/126001.\n[2] P. Talkner and P. Hänggi, Reviews of Modern Physics 92\n(2020), 10.1103/RevModPhys.92.041002.\n[3] J. G. Kirkwood, J. Chem. Phys. 3, 300 (1935).\n[4] M. Campisi, P. Talkner, and P. Hänggi, Physical Review\nLetters 102(2009), 10.1103/PhysRevLett.102.210401.\n[5] K.Sekimoto,ProgressofTheoreticalPhysicsSupplement\n130(1998), 10.1143/PTPS.130.17/1842313.\n[6] P. Strasberg and M. Esposito, Physical Review E 95\n(2017), 10.1103/PhysRevE.95.062101.\n[7] P. Strasberg and M. Esposito, Physical Review E 101\n(2020), 10.1103/PhysRevE.101.050101.\n[8] M.DingandX.Xing,PhysicalReviewResearch 5(2023),\n10.1103/PhysRevResearch.5.013193.\n[9] P. Talkner and P. Hänggi, Physical Review E 94(2016),\n10.1103/PhysRevE.94.022143.\n[10] C. Jarzynski, Physical Review X 7(2017), 10.1103/Phys-\nRevX.7.011008.[11] H. Koh, S. Chiashi, J. Shiomi, and S. Maruyama, Scien-\ntific Reports 11(2021), 10.1038/s41598-020-79200-6.\n[12] H. Koh, J. J. Cannon, T. Shiga, J. Shiomi, S. Chiashi,\nand S. Maruyama, Physical Review B 92, 024306 (2015).\n[13] R. Zwanzig, Journal of Statistical Physics 9, 215 (1973).\n[14] R. Zwanzig, Nonequilibrium Statistical Mechanics (Ox-\nford University Press, 2001).\n[15] S. J. Stuart, A. B. Tutein, and J. A. Harrison, Journal\nof Chemical Physics 112, 6472 (2000).\n[16] T. Schneider and E. Stoll, Physical Review B 17, 1302\n(1978).\n[17] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolin-\ntineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld,\nA. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan,\nM. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimp-\nton, Computer Physics Communications 271(2022),\n10.1016/j.cpc.2021.108171."    },    {        "title": "2401.13664v1.Hamiltonian__Geometric_Momentum_and_Force_Operators_for_a_Spin_Zero_Particle_on_a_Curve__Physical_Approach.pdf",        "content": "arXiv:2401.13664v1  [quant-ph]  24 Jan 2024Hamiltonian, Geometric Momentum and Force Operators for a S pin Zero Particle on\na Curve: Physical Approach\nM. S. Shikakhwa\nDepartment of Basic Sciences, TED University\nZiya G¨ okalp Caddesi No:48, 06420, Ankara, T¨ urkiye\nN.Chair\nDepartment of Physics,University of Jordan,\nQueen Rania Street,\nAmman, Jordan\nThe Hamiltonian for a spin zero particle that is confined to a c urve embedded in the 3D space\nis constructed by squeezing the coordinates spanning a tube normal to the curve onto the curve\nassuming strong normal forces. We follow the new approach th at we applied to confine a particle to\na surface, in that we start with an expression for the 3D momen tum operators whose components\nalong and normal to the curve directions are separately Herm itian. The kinetic energy operator\nexpressed in terms of the momentum operator in the normal dir ection is then a Hermitian operator\nin this case. When this operator is dropped and the thickness of the tube surrounding the curve\nis set to zero, one automatically gets the Hermitian curve Ha miltonian that contains the geometric\npotential term as expected. It is demonstrated that the orig in of this potential lies in the ordering\nor symmetrization of the original 3D momentum operators in o rder to render them Hermitian. The\nHermitian momentum operator for the particle as it is confine d to the curve is also constructed\nand is seen to be similar to what is known as the geometric mome ntum of a particle confined to a\nsurface in that it has a term proportional to the curvature th at is along the normal to the curve.\nThe force operator of the particle on the curve is also derive d, and is shown to reduce, for a curve\nwith a constant curvature and torsion, to a -apparently- sin gle component normal to the curve that\nis a symmetrization of the classical expression plus a quant um term. All the above quantities are\nthen derived for the specific case of a particle confined to a cy lindrical helix embedded in 3D space.\nI. INTRODUCTION\nThe research on the non-relativistic quantum mechanics on genera l 2D surfaces and 1D curves, despite being\nintrinsically interesting, has received a substantial boost in the last decades due to the fast technological advances\nin the fabrication of nano-scale structures, like nano-spheres, n ano-tubes...etc. A - by now- popular approach to the\nproblem of constructing the Hamiltonian is such systems was first de veloped by Jensen and Koppe [1] for confining\na particle to a spherical surface, then was generalized to confinem ent to an arbitrary 2D surface embedded in 3D\nspace by da Costa [2] and we refer to it now on by the CJK approach ( also known in the literature as Thin Layer\nQuantization (TLQ)). da Costa in [2] applied the approach to confine a spin zero particle to an arbitrary curve, too.\nLater works [3–5] applied the same approach to construct the Ham iltonian of a spin one-half particle in the presence\nof a variety of interactions and couplings confined to a 1D curve emb edded in 3D Euclidean space. The essential idea\nis to start with the Schr¨ odinger equation for a particle in a 3D layer ( a 3D tube for confinement to a 1D curve)\nsurrounding a surface (a curve). A very strong confining potent ial depending only on the coordinate(s) normal to, but\nnot those along the surface (curve) is assumed to pin the particle t o the surface ( curve). Eventually the two (one)\ncoordinate(s) normal to the surface (curve) are (is) set to zer o and the resulting Schr¨ odinger equation is separated\ninto two equations for the normal and surface (curve) degrees o f freedom. The key point in the CJK approach is a\ntransformation of the wavefunction that isolates a finite term tha t would otherwise be mistakenly dropped from the\nHamiltonian once the thickness of the layer ( tube) is set to zero. Th e authors of [1] call this the dangerous term.\nThis finite term gives in the zero thickness limit what is known as the geo metrical potential or the geometrical kinetic\nenergy contribution to the Hamiltonian. This potential appears whe n pinning a particle to both a surface and a curve\nand is essential to render the Hamiltonian Hermitian. Recently, we int roduced [6] a twist to the CJK approach to\nconstraint a particle to a surface, which renders it more intuitive an d transparent, we believe. Our idea is to note that\nwhile, intuitively, dropping the derivatives with respect to the coord inates normal to the surface when pinning the\nparticle to it, by setting the coordinates normal to the surface to zero should work well, we need to drop Hermitian\nquantities to end up with a Hermitian Hamiltonian on the surface. We ac hieve this by working from the onset with\nmomentum operators along and normal to the surface that are se parately Hermitian. This way, we drop, once the\nparticle is confined to the reduced manifold, the Hermitian normal mo mentum operator and are left with a Hermitian\nHamiltonian with the geometric potential appearing naturally. We hav e applied our approach to confine a spin zero\nparticle to an arbitrary surface embedded in 3D space spanned by g eneral curvilinear coordinates in [6] and to confine2\na spin one-half particle with various interactions to a 2D surface emb edded in a 3D space spanned by orthogonal\ncurvilinear coordinates in [7, 8].\nIn the present work, we show that our approach can also be applied to the problem of confining a spin zero particle to\nan arbitrary space curve embedded in 3D Euclidean space spanned b y general curvilinear coordinates. We construct\nthe Hermitian 3D momentum operators along and normal to the curv e that we need to start with (section 1)and\nderive the Hermitian Hamiltonian upon pinning the particle to the curve in section 2. In section 3, we derive the\nexpression of the Hermitian momentum operator of the particle onc e it is pinned to the curve and show that it is\nexactly similar to the so called geometrical momentum on a surface [9, 10]. In section 4, we derive the force operator\nfor the particle on the curve and in section 5 we derive all the above q uantities for the specific example of a particle\nconfined to a cylindrical helix embedded in 3D. We sum up and give our co nclusions in section 6.\nII. THE 3D HERMITIAN MOMENTUM OPERATOR\nTo confine a particle in 3D space to an arbitrary space curve embedd ed in this space, we start with the coordinate\nsystem -adopted also in [3, 4]- where a vector a(q1) centered at the origin of the coordinates traces the curve as th e\ncoordinate q1varies along the curve. At each point on the curve, i.e. for each valu e ofq1, the coordinates q2and\nq3vary over the normal ˆ n(q1) and binormal ˆb(q1) unit vectors, respectively, defining a curve-centered 3D coord inate\nsystem. The position vector R(q1,q2,q3) of any point in space in the neighbourhood of the curve is then given as:\nR(q1,q2,q3) =a(q1)+q2ˆn(q1)+q3ˆb(q1) (1)\nˆt(q1) =∂a/∂q1is a unit vectortangent to the curve forming with ˆ n(q1) andˆb(q1) a set of three orthogonalunit vectors\ndefining the so called Fresnel frame. These three unit vectors are related by the well-known Frenet-Serret equations:\n\n∂1ˆt(q1)\n∂1ˆn(q1)\n∂1ˆb(q1)\n=\n0κ(q1) 0\n−κ(q1) 0τ(q1)\n0−τ(q1) 0\n\nˆt(q1)\nˆn(q1)\nˆb(q1)\n (2)\nwithκ(q1) andτ(q1) denoting the curvature and torsion of the space curve, respec tively. The vectors tangent to\nthe coordinates ui≡∂R\n∂qi=∂iRare defined as usual, and for our coordinate system they read:\nu1≡∂1R=∂1a+q2∂1ˆn(q1)+q3∂1ˆb(q1)\n=ˆt(q1)+q2∂1ˆn(q1)+q3∂1ˆb(q1)\n=ˆt(1−q2κ)−q3τˆn+q2τˆb(3)\nand,\nu2≡∂2R= ˆn(q1)\nu3≡∂3R=ˆb(q1)(4)\nwhere Eq.(2) was used to express u1in terms of κ(q1) andτ(q1) in Eq.(3). The metric tensor Gij=ui·ujfollowing\nfrom the above coordinate system then reads:\nGij=\n(1−κq2)2+τ2/parenleftbig\nq2\n2+q2\n3/parenrightbig\n−τq3τq2\n−τq3 1 0\nτq2 0 1\n (5)\nThe determinant Gof this metric tensor is then G=detGij= (1−κq2)2. The inverse metric Gij=Kij\nGwithKij\nbeing the cofactor of Gijis then calculated to read:\nGij=1\n(1−κq2)2\n(1−κq2)2+τ2/parenleftbig\nq2\n2+q2\n3/parenrightbig\n−τ2q32−τ2q22\n−τ2q32(1−κq2)2+τ2q320\n−τ2q220 (1 −κq2)2+τ2q22\n (6)\nWe now construct the Hermitian momentum operatorsalongand per pendicular to the curve. The Hermicity of the 3D\nmomentum operator p=−i/planckover2pi1∇should evidently be preserved when it is expressed in general curvilin ear coordinates\nwhere it reads p=−i/planckover2pi1∇=−i/planckover2pi1ui∂i(recall that ui≡∂iRand soui=Gijuj). Checking, we find:3\n/an}b∇acketle{tΨ|pΨ/an}b∇acket∇i}ht=/an}b∇acketle{tpΨ|Ψ/an}b∇acket∇i}ht+/angbracketleftbigg\nΨ|−i/planckover2pi1√\nG∂i/parenleftBig√\nGui/parenrightBig\nΨ/angbracketrightbigg\nwhere integration is over all space with the measure√\nGd3uand the wavefunction was assumed to satisfy boundary\nconditions that allows the surface term to be dropped. Hermicity of pdemands the vanishing of the second term on\nthe r.h.s, i.e.\n1√\nG∂i/parenleftBig√\nGui/parenrightBig\n=1√\nG∂1/parenleftBig√\nGu1/parenrightBig\n+1√\nG∂a/parenleftBig√\nGua/parenrightBig\n= 0 (7)\nwhere the tangent vectors ua,a= 1,2 are, respectively, just ˆ n(q1) andˆb(q1) defined in Eq.(4). The relation\n1√\nG∂i/parenleftBig√\nGui/parenrightBig\n= 0 is in fact an identity as was discussed in [6].The 3D momentum operato rpwhen expressed as a\nsum of two momentum operators p1andp⊥along and normal to the curve, respectively, reads:\np=p1+p⊥\n=−i/planckover2pi1u1∂1+−i/planckover2pi1ua∂a(8)\nIt is straightforwardto check that neither the momentum operat oralongnor that normal to the curve areHermitian\nin 3D space as they stand; only their sum is. Now adding (half) of the z ero-valued expression in Eq. (7) multiplied\nby−i/planckover2pi1split among p1andp⊥to the above p, we have:\np=p1+p⊥=p1H+p⊥H (9)\nwith the operators p1Handp⊥Hbeing now:\np1H=p1−i/planckover2pi1\n2√\nG∂1/parenleftBig√\nGu1/parenrightBig\n=−i/planckover2pi1/parenleftbig\nu1∂1+Γ/parenrightbig(10)\nand,\np⊥H=p⊥−i/planckover2pi1\n2√\nG∂a/parenleftBig√\nGua/parenrightBig\n=p⊥+i/planckover2pi1\n2√\nG∂1/parenleftBig√\nGu1/parenrightBig\n=−i/planckover2pi1(ua∂a−Γ)(11)\nand we have defined Γas:\nΓ=1\n2√\nG∂1(√\nGu1) (12)\nThe newly defined momentum operator along the curve p1Hand the momentum operator normal to the curve p⊥H\ncan be readily checked to be Hermitian over the 3D space. Therefor e, by adding a zero-valued quantity to the full 3D\nHermitian momentum operator, we managed to express it as the sum of two Hermitian momentum operators; along\nand normal to the curve. This a key step for the following analysis.\nIII. THE HERMITIAN HAMILTONIAN ON THE CURVE\nIn this section, we will construct the Hamiltonian for a spin zero part icle confined to the space curve and is\notherwise free. The approach [6] is based on the intuitive argument that if one starts from the full Hamiltonian\nin the 3D space spanned by the coordinate system given in Eq. (1) an d then confine the particle to the curve by\nintroducing a strong confining potential (force) along the directio n normal to the curve; then, the excitation along\nthis direction will need an infinite energy and so the dynamics is essent ially along the curve. This amounts to freezing\nthe normal degree of freedom and dropping it from the Hamiltonian w hich is achieved by setting q2,q3to zero and\ndropping the normal momentum operator from the kinetic energy o perator. The critical point of the present approach4\nis to use in the Hamiltonian momentum (and so kinetic energy) operato rs along and normal to the curve that are\nseparately Hermitian, so that as one drops the Hermitian kinetic energy operat or ( Hermitian momentum normal\nto the curve) one is left with a Hermitian Hamiltonian on the curve. While this intuition might seem obvious and\ntrivial, blindly dropping the normal degrees of freedom without obse rving for Hermicity has led to the reporting of\nnon-Hermitian surface Hamiltonians in the literature [11]. Therefore , the essential starting point is the expression of\nthe 3D momentum operator as a sum of the two Hermitian surface an d normal momentum operators as in Eqs. (10)\nand (11). The free particle Hamiltonian in 3D with the momentum opera tor given by these two equations reads:\nH3D=p2\n2m=1\n2m/parenleftbig\np2\n1H+p2\n⊥H+p1H·p⊥H+p⊥H·p1H/parenrightbig\n=p2\n⊥H\n2m−/planckover2pi12\n2m/parenleftbig\nu1·(∂1ua)∂a+ua·/parenleftbig\n∂au1/parenrightbig\n∂1\n+u1·/parenleftbig\n∂1u1/parenrightbig\n∂1+2u1·ua∂1∂a\n+u1·u1∂2\n1+2ua·Γ∂a+ua·∂aΓ\n−Γ·Γ)(13)\nIn the limit q2,q3→0, we note that\nGij,Gij−→δij,δij\nG−→1\nu2=G2juj−→ˆn\nu3=G3juj→ˆb\nu1=G1juj→ˆt(14)\nleading in this limit to:\nu1·(∂1ua)∂a−→ −κ∂2\n2ua·Γ∂a−→κ∂2\nu1·u1∂2\n1−→∂2\n1\nua·∂aΓ−→κ2/2\nΓ·Γ−→κ2/4(15)\nand all of ua·/parenleftbig\n∂au1/parenrightbig\n∂1;u1·/parenleftbig\n∂1u1/parenrightbig\n∂1;2u1·ua∂1∂a−→0. Along with taking the limit q2,q3→0 we also drop\nthe Hermitian kinetic energy operator representing the degrees o f freedom normal to curve, i.e.p2\n⊥H\n2mfrom the\nHamiltonian. This immediately gives the same Hamiltonian as in TLQ with the geometric potential −/planckover2pi12\n2m/parenleftBig\nκ2\n4/parenrightBig\nappearing automatically:\nH=−/planckover2pi12\n2m/parenleftbigg\n∂2\n1+κ2\n4/parenrightbigg\n(16)\nIV. THE GEOMETRIC MOMENTUM ON THE CURVE\nIn Eq.(10) we have given the Hermitian momentum vector along the cu rve in 3D space, i.e. before pinning the\nparticle to the curve as:\np1H=−i/planckover2pi1/parenleftbigg\nu1∂1+1\n2√\nG∂1(√\nGu1)/parenrightbigg\n(17)\nHere, we find the form to which this operator reduces in the limit q2,q3→0 and check for its Hermicity in the reduced\n1D space. In the limit q2,q3→0,1\n2√\nG∂1√\nGu1=Γ→κ\n2ˆnreducing the momentum along the curve to the expression:\np1H−→pc\n1H=−i/planckover2pi1/parenleftbigg\nˆt∂1+ˆnκ\n2/parenrightbigg\n(18)5\nThis curve momentum operator pc\n1His Hermitian on the reduced 1-D space, i.e\n/integraldisplay\ndq1ψ∗pc\n1Hψ=/integraldisplay\ndq1(pc\n1Hψ)∗ψ (19)\nTheˆnκ\n2term is essential to establish Hermicity; −i/planckover2pi1ˆt∂1is not Hermitian by itself. The Hermitian momentum on a\ncurved surface embedded in 3D space has a form similar to this curve momentum, Eq.(18), and is sometimes called\nthe geometric momentum in the literature. It was derived using Dirac quantization in [10] and using an approach\nsimilar to the current one in [6, 9]. This is - to the best of our knowledge - the first time it is reported for a curve. The\nfact that the normal vector ˆ nappears in the momentum operator on the curve, Eq.(18), does no t mean that it has a\nnormal component. In fact one can easily check that ˆ n·pc\n1H+pc\n1H·ˆn= 0 indicating that it is purely tangential. In\nfact, just as is the case with the geometric momentum on a surface [6], this curve geometric momentum operator can\nbe expressed in a symmetric form that makes it manifestly tangentia l:\npc\n1H=−i/planckover2pi1\n2/parenleftbigˆt∂1+∂1ˆt/parenrightbig\n(20)\nwhere in the second term of the above equation, the ∂1is to be understood as acting on everything to its right. It is\ninteresting to note that the curve geometric momentum operator ,pc\n1H, is the kinematical momentum operator that\nwe get when taking the time derivative of a(q1), the position operator of the particle on the curve ( see Eq.(1)).\nIndeed:\nmda\ndt=[a,H]\ni/planckover2pi1=−i/planckover2pi1/parenleftbigg\nˆt∂1+ˆnκ\n2/parenrightbigg\n=pc\n1H\nClosing this section we would like to note it is not only the Hermitian momen tum on the curve that can be written\nin a symmetric form as in Eq.(20) above. In fact the Hermitian moment um operators in 3D curvilinear coordinate,\nEq.(10) and Eq.(11), can also be symmetrized as was discussed in [6] to read :\np1H=p1−i/planckover2pi1\n2√\nG∂1/parenleftBig√\nGu1/parenrightBig\n=−i/planckover2pi1(u1∂1+Γ)\n=−i/planckover2pi1\n2√\nG(√\nGu1∂1+∂1√\nGu1) =−i/planckover2pi1\n2√\nG{√\nGu1,∂1}+(21)\nand,\np⊥H=p⊥−i/planckover2pi1\n2√\nG∂a/parenleftBig√\nG/parenrightBig\n=−i/planckover2pi1(ua∂a−Γ)\n=−i/planckover2pi1\n2√\nG(√\nGua∂a+∂a√\nGua) =−i/planckover2pi1\n2√\nG{√\nGua,∂a}+(22)\nwitharunning over 2 and 3, and the bracket {.,.}+denotes anti-commutator. Now, we have seen in the derivations\nmade in the Hamiltonian, Eq.(13), that it is the Γthat gives rise to the appearance of the geometrical potential in\nthe Hamiltonian. But this Γ, which is added to the two momentum operators in Eqs.(10) and (11) to render each\nHermitian is nothing but a recipe for symmetrizing the derivatives in th e specific manner given in these equations\nin order to construct the 3D Hermitian operator p. This means that the geometric potential has its origin in the\nsymmetrization or ordering of the derivatives in the momentum oper ators. The same argument was made in the case\nof a spin-zero particle confined to a surface [6].6\nV. THE FORCE OPERATOR\nThe force operatoron the particle on the curveis one more quantit y that we can calculate. The Heisenberg equation\nfor the curve geometric momentum gives:\nF=dpc\n1H\ndt=[pc\n1H,H]\ni/planckover2pi1\n=/planckover2pi12\n2m/parenleftbigg\n[ˆt∂1,∂2\n1]+[ˆt∂1,κ2\n4]+1\n2[ˆnκ,∂2\n1]/parenrightbigg\n=/planckover2pi12\n2m/parenleftBigg\nˆt(2κ2∂1+2κκ′)+ ˆn(−2κ∂2\n1−2κ′∂1+κ3\n2+τ2κ\n2−κ′′\n2)\n+ˆb(−2τκ∂1−κτ′\n2−τκ′)/parenrightbigg(23)\nTheκκ′- term along ˆthas two origins: The standard/bracketleftBig\nˆtˆ∂1,κ2\nu/bracketrightBig\nterm which is the gradient of the geometric potential,\nplus the/bracketleftbig\nˆnκ,∂2\n1/bracketrightbig\n-term. The ˆt2κ2∂1originates from this last term, too. This means that, even if the cur vature is\nconstant so that it has no dependence on q1, the force operator for a particle on a curve will always have a tang ential\ncomponent, in contrast to the classical case. The torsion and its d erivatives have their origin in the1\n2[ˆnκ,∂2\n1]-term.\nThe above expression can be brought to a more symmetrical and tr ansparent form:\nF=ˆn\n2κmv2+mv2κˆn\n2−ˆn/planckover2pi12κ\n4m/parenleftbigg\n2κ2+τ2−κ′′\nκ/parenrightbigg\n−/planckover2pi12\n2m/parenleftbigg\nˆt(κκ′)−ˆb(τκ′\n2+κ′τ)/parenrightbigg (24)\nwhere we have defined\nv2=(pc\n1H)2\nm2=−/planckover2pi12\nm2/parenleftbigg\n∂2\n1−κ2\n4/parenrightbigg\nNon -constant curvature and torsion give rise to a force with explic it components along the tangential and binormal\ndirections. WhiletheforcealongthenormaldirectioninEq.(24) seem slikeasymmetrizationoftheclassicalexpression\nplus a quantum term, we need to keep in mind that vcontains derivative with respect to q1and thatκ=κ(q1) and\nˆn= ˆn(q1). In particular, this symmetrization hides a tangential component of the force. For constant κandτthe\nforce reduces to a simpler expression that is -again, apparently- a symmetrization of the classical one plus a quantum\nterm:\nF=ˆn\n2κmv2+mv2κˆn\n2−ˆn/planckover2pi12κ\n4m/parenleftbig\n2κ2+τ2/parenrightbig\n(25)\nWe derived similar symmetrical expressions for the force on a partic le confined to specific two-dimensional surfaces\nembedded in 3D space in [9].\nVI. MOMENTUM, HAMILTONIAN AND FORCE FOR A PARTICLE ON A HELIX\nA cylindrical helix characterized by radius Rand apexCis given in cylindrical coordinates as r=R,θ=θ(s),z=\nCθ(s) wheres=√\nR2+C2θis the arc subtended by θ. The unit vectors ˆt, ˆnandˆbof the Fresnel coordinate system\non the curve are given for the helix in terms of the cylindrical unit vec tors as\nˆt=Rˆθ+Cˆz\n(R2+C2)1\n2= (κˆθ+τˆz)/parenleftbig\nR2+C2/parenrightbig1\n2\nˆn=−ˆr\nˆb=Rˆz−Cˆθ\n(R2+C2)1\n2= (κˆz−τˆθ)/parenleftbig\nR2+C2/parenrightbig1\n2(26)7\nso that\n∂1=/parenleftbig\nR2+C2/parenrightbig−1\n2∂θ=∂s (27)\nIn this coordinate system, we have κ=R\nR2+C2andτ=C\nR2+C2. It is convenient to define the constant angle α\nby tanα=κ\nτ=R\nC. For the helix, the geometric momentum, Eq.(18), and the Hamiltonia n, Eq.(16) are given,\nrespectively, by:\nphlx=−i/planckover2pi1/parenleftbigg\n(κˆθ+τˆz)∂θ−ˆrκ\n2/parenrightbigg\n(28)\nHhlx=−/planckover2pi12\n2m(R2+C2)/parenleftbigg\n∂2\nθ+sin2α\n4/parenrightbigg\n(29)\nwhere sinα=R2\nR2+C2. The force operator, with κandτbeing constants for the helix can be readily written from\nEq,(25), and reads:\nFhlx=sinα\n2/parenleftbigg\n(−ˆr)mv2\nR+mv2\nR(−ˆr)/parenrightbigg\n+/planckover2pi12Rˆr\n4m(R2+c2)3/parenleftbig\n2sin2α+cos2α/parenrightbig\n(30)\nwith\nv2=p2\nhlx\nm2=−/planckover2pi12\nm2(R2+c2)/parenleftbigg\n∂2\nθ−sin2α\n4/parenrightbigg\n(31)\nVII. CONCLUSION\nThe Hermitian Hamiltonian of a spin zero particle on an arbitrary space curve embedded in Euclidean 3D space can\nbe constructed applying the following intuitive and physical procedu re: Start with the 3D Hamiltonian of a particle\nin the vicinity of the curve formulated using the specific coordinate s ystem, Eq.(1), with one coordinate q1along\nthe curve and the other two, q2andq3normal to the curve. Then, re-express the Hamiltonian so that ea ch of the\nmomentum operators along and normal to the curve are separate ly Hermitian, which can be done by adding to the\n3D momentum operator the null term ( Eq.(12)) split among the two c omponents. Now, assume that a very strong\npotentialV(q2,q3) confines the particle to the curve making the energyneeded to ex cite the normal degrees of freedom\nvery high, thus rendering the dynamics essentially 1-dimensional alo ng the curve. Thus, take the limit q2,q3→0 and\nat the same time drop the part of the kinetic energy operator that has the normal momentum operator, which is now\nHermitian by construction, from the Hamiltonian. The resulting Hamilt onian -same as the one derived using the CJK\napproach - will be the Hermitian Hamiltonian on the curve with the geom etric potential term that comes up being\nthe piece that guarantees Hermicity. The origin of this geometric po tential can be seen to stem from the ordering of\nthe derivatives in the momentum operator in curvilinear coordinates .\nThe explicit form of the curve momentum operator that results whe n one pins the particle to the curve by takeing the\nlimitq2,q3→0 of the Hermitian 3D momentum operator parallel to the surface sh ows that it contains in addition\nto the derivative term a term proportional to the curvature of th e curve and is - apparently- along the normal to the\ncurve, Eq.(18), in complete analogy with the so called geometric mome ntum on a 2D surface embedded in 3D space.\nThis additional term is seen to result, again, from ordering of the de rivatives, and this curve geometrical momentum\ncan be expressed as a symmetrical expression which is manifestly alo ng the curve, Eq.(20). This geometrical curve\nmomentum is at the same time the kinematical momentum of the partic le that comes out when one calculates the\ntime derivative of the position operator (times the mass) using the H eisenberg equations of motion. The most general\nforce operator of the particle is also found by taking the derivative of the geometrical curve momentum, and turns out\nto have components along and normal to the curve. For a curve wit h constant curvature and torsion, this operator is\nshown to reduce to a -apparently- normal one, with a term that is a symmetrization of the classical centripetal force\nplus a quantum correction. All of the Hamiltonian, geometric curve m omentum and force operators are calculated\nfor the specific example of a spin zero particle confined to a cylindrica l helix embedded in 3D space.\nThe above results are consistent with our previous results [6] for the construction of the Hamiltonian of a spin zero\nparticle, both free and in an electromagnetic field, confined to an ar bitrary 2D surface embedded in 3D Euclidean\nspace applying the same approach, as well as the results for the ge ometric momentum and force operators derived for8\nfree spin zero particle confined to a 2D surface [6, 9]. The question o f applying the current approach to the problem\nof spin one-half particle confined to a 2D surface embedded in 3D spa ce was carried out in [7, 8] for orthogonal\ncurvilinear coordinates; while that of confining such a particle to 1D c urve embedded in 3D space is currently under\ninvestigation.\n[1] H. Jensen and H. Koppe, Ann. Phys. 63, 586(1971)\n[2] R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).\n[3] Carmine Ortix, Phys. Rev. B 91. 245412 (2015).\n[4] Guo-Hua Liang, Yong-Long Wang, Meng-Yun Lai, hao Zhao, H ong-Shi Zong and Hui Liu, Phys. Rev. A 101, 053632\n(2020)).\n[5] L.I.Magarill and M.V.Entin, JETP 96,766 (2003)\n[6] Shikakhwa, M.S., Chair, N., European Physical Journal P lus137,560 (2022)\n[7] M.S.Shikakhwa and N.Chair, Phys.Lett.A 380,1985 (2016).\n[8] M.S.Shikakhwa and N.Chair, Phys.Lett.A 380,2876 (2016).\n[9] Shikakhwa, M.S., Communications in Theoretical Physic s70,263 (2018).\n[10] Q. H. Liu, C. L. Tong, and M. M. Lai, J. Phys. A 40,4161 (2007)\n[11] A.G. Aronov, Y.B. Lyanda-Geller, Phys. Rev. B 70, 343 (1993)"    },    {        "title": "2401.13718v1.Rainbow_gravity_effects_on_quantum_dynamics_of_scalar_and_oscillator_fields_in_a_topological_defect_cosmological_space_time.pdf",        "content": "Rainbow gravity effects on quantum dynamics of scalar and oscillator fields\nin a topological defect cosmological space-time\nFaizuddin Ahmed1\nDepartment of Physics, University of Science & Technology Meghalaya, Ri-Bhoi, 793101, India\nAbdelmalek Bouzenada2\nLaboratory of Theoretical and Applied Physics, Echahid Cheikh Larbi Tebessi University, Algeria\nAbstract\nIn this paper, we investigate the quantum dynamics of scalar and oscillator fields in a topological\ndefect space-time background under the influence of rainbow gravity’s. The rainbow gravity’s are\nintroduced into the considered cosmological space-time geometry by replacing the temporal part\ndt→dt\nF(χ)and the spatial part dxi→dxi\nH(χ), where F,Hare the rainbow functions and χ=E/E p.\nWe derived the radial equation of the Klein-Gordon equation and its oscillator equation under\nrainbow gravity’s in topological space-time. To obtain eigenvalue of the quantum systems under\ninvestigations, we set the rainbow functions F(χ) = 1 and H(χ) =√1−β χp, where p= 1,2.\nWe solve the radial equations through special functions using these rainbow functions and analyze\nthe results. In fact, it is shown that the presence of cosmological constant, the topological defect\nparameter α, and the rainbow parameter βmodified the energy spectrum of scalar and oscillator\nfields in comparison to the results obtained in flat space.\nKeywords : Modified theories of gravity; Quantum fields in curved space-time; Relativistic wave\nequations, Solutions of wave equations: bound-state, special functions.\nPACS numbers: 04.50.Kd; 04.62.+v; 03.65.Pm; 03.65.Ge; 02.30.Gp\n1 Introduction\nRainbow gravity represents a semi-classical exploration of high-energy phenomena in quantum gravity.\nIt achieves this by introducing higher-order terms in the energy-momentum dispersion relation through\nthe use of rainbow functions. These functions, linked to the ratio between the energy of a test particle\n(such as a boson or fermion) and the Planck energy, lead to a breakdown of Lorentz symmetry at\nthis particular energy scale [1, 2, 3]. The rainbow functions play a crucial role in governing deviations\nfrom the conventional relativistic, Minkowskian expressions. Consequently, to establish this theoretical\nframework, the spacetime metric must also exhibit energy dependence, with this dependence intensifying\nas the energy of the probing particle approaches the Planck scale.\nThe concept known as rainbow gravity, which is essentially an effective theory of quantum gravity,\ncan be viewed as an endeavor to embody the generalized curvature of spacetime in the context of\nDoubly Special Relativity [1, 2, 4, 5]. In the realm of Doubly Special Relativity, modifications in\ndispersion relations are accompanied by alterations in Lorentz symmetries to establish an invariant\nenergy (length) scale, alongside the conventional invariant velocity of light (especially at lower energies).\nAs such, rainbow gravity is grounded on the foundational principle that these two quantities remain\ninvariant, with the fixed energy scale being identified as the Planck energy [1, 2, 3, 4, 5].\n1faizuddinahmed15@gmail.com; faizuddin@ustm.ac.in\n2abdelmalekbouzenada@gmail.com ; abdelmalek.bouzenada@univ-tebessa.dz\n1arXiv:2401.13718v1  [gr-qc]  24 Jan 2024Rainbow gravity holds significant importance as it finds applications across various physical sys-\ntems, bringing modifications to classical theories. An illustrative example is its integration with Fried-\nmann–Robertson–Walker (FRW) cosmologies, explored in [6], where the potential resolution of the Big\nBang singularity is investigated. Furthermore, [7] delves into the exploration of the modified Starobin-\nsky model and the inflationary solution to motion equations using rainbow gravity. This investigation\nextends to the calculation of crucial parameters such as the spectral index of curvature perturbation and\nthe tensor-to-scalar ratio. Expanding its scope, rainbow gravity has also been employed to study phe-\nnomena like the deflection of light, photon time delay, gravitational red-shift, and the weak equivalence\nprinciple, as detailed in [8].\nRecent investigations into the behavior of relativistic Klein-Gordon (KG) and Dirac particles have\ndelved into diverse spacetimes within the framework of rainbow gravity. Noteworthy examples include\nstudies conducted in a cosmic string spacetime background, where Bezerra et al. [9] examined Landau\nlevels through Schr¨ odinger and KG equations. Additionally, Bakke and Mota [10, 11] explored Dirac\noscillators, Sogut et al. [12] investigated the quantum dynamics of photons, and Kangal et al. [13] studied\nKG particles in a topologically trivial G¨ odel-type spacetime under rainbow gravity. Mustafa explored\nKG-Coulomb particles in a cosmic string rainbow gravity spacetime [14], and massless KG-oscillators in\nSom-Raychaudhuri cosmic string spacetime within finely tuned rainbow gravity [15]. These studies aim\nto unravel the dynamics of both relativistic and non-relativistic particles in curved spacetime, providing\ninsights into the intricate interplay between quantum mechanics and general relativity. The pursuit of\nexact or quasi-exact solutions for such systems illuminates the captivating influence of curved spacetime\non the spectroscopic structure of relativistic particles, with implications in cosmology, the geometrical\ntheory of topological defects, and the study of black holes and wormholes, among other areas.\nIn gravity’s rainbow, the metric describing the geometry of spacetime depends on the energy of\nthe test particle used to probe the structure of that space-time. So, the geometry of space-time is\nrepresented by a family of energy dependent metrics forming a rainbow of metrics. In gravity’s rainbow,\nthe energy–momentum dispersion relation is modified by energy dependent rainbow functions, F(χ) and\nH(χ), where χ=E/E p, such that [16]\nE2F2(χ)−p2H2(χ) =m2. (1)\nAs it is required that the usual energy-momentum dispersion relation is recovered in the infrared (IR)\nlimit, these rainbow functions are required to satisfy\nlim\nχ→0F(χ)→1,lim\nχ→0H(χ)→1. (2)\nThe energy dependent metric in gravity’s rainbow can be written as [17]\ngµν(χ) =eµ\na(χ)eν\nb(χ)ηab, (3)\nwhere ηµ\nais the orthonormal tetrad fields, ηabis the Minkowski metric, and Eis the energy of the\nprobing particle. The rainbow functions are defined using this energy Ethat cannot exceed the Planck\nenergy Ep.\nIn the literature of gravity’s rainbow, many proposals exist for the rainbow functions F(χ) andH(χ)\n[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. The choice of the rainbow\nfunctions is supposed to be based on phenomenological motivations. We use one of the most interesting\nand most studied rainbow functions that was proposed in Refs. [16, 38, 39]\nF(χ) = 1 ,H=p\n1−β χp(p= 1,2). (4)\n2The modified dispersion relation with these functions is compatible with some results from non-critical\nstring theory, loop quantum gravity and κ-Minkowski non-commutative space-time. This modified\ndispersion relation has been used to study the dispersion of electromagnetic waves from gamma ray\nbursters [39]. It also solves the ultra high energy gamma rays paradox [40, 41], and the paradox of the\n20 TeV gamma rays from the galaxy Markarian 501 [40, 42]. In addition, this MDR provides stringent\nconstraints on deformations of special relativity and Lorentz violations [43, 44].\nAddressing the Einstein-Maxwell equations, Bonnor formulated an exact static solution, discussed\nin detail for its physical implications [45]. Melvin later revisited this solution, leading to the currently\nrecognized Bonnor-Melvin magnetic universe [46]. An axisymmetric Einstein-Maxwell solution, incorpo-\nrating a varying magnetic field and a cosmological constant, was constructed in [47]. This electrovacuum\nsolution was subsequently expanded upon in [47, 48]. This analysis primary focus on Bonnor-Melvin-\ntype universe featuring a cosmological constant, discussed in detailed in Ref. [49]. The line-element\ndescribing this magnetic universe in the chart ( t, ρ, ϕ, z ) is given by [49]\nds2=−dt2+dz2+1\n2 Λ\u0010\ndρ2+ sin2ρ dϕ2\u0011\n, (5)\nwhere Λ denotes the cosmological constant and the magnetic field strength is given by B=√\n2 sin ρ.\nWe now introduces a cosmic string into the above line-element by performing a coordinate transfor-\nmation ϕ→α ϕ, where αis the cosmic string parameter which produces an angular deficit with the\nranges 0 < α < 1. Therefore, the above line-element (5) under this transformation becomes\nds2=−dt2+dz2+1\n2 Λ\u0010\ndρ2+α2sin2ρ dϕ2\u0011\n, (6)\nwhere now the magnetic field strength becomes B=√\n2αsinρwhich is αtimes the original field\nstrength but decreases by this amount.\nFinally, the rainbow gravity’s effect is introduced into this space-time (6) by replacing the temporal\npart dt→dt\nF(χ)and the spatial part dxi→dxi\nH(χ), where F,Hare the rainbow functions stated earlier.\nTherefore, the space-time (6) under rainbow gravity’s can be described by the following line-element\n[50, 51]\nds2=−dt2\nF2(χ)+1\nH2(χ)\"\ndz2+1\n2 Λ\u0010\ndρ2+α2sin2ρ dϕ2\u0011#\n. (7)\nOne can evaluate the magnetic field strength for the modified topological defect cosmological space-time\n(7) and it is given by B=α√\n2H(χ)sinρwhich depends on the rainbow function. For χ→0, we will get\nback the topological defect cosmological space-time (6).\nThe exploration of rainbow gravity’s implications in quantum mechanical problems has attracted\nmuch attention in current times. Various studies have delved into the effects of rainbow gravity in\ndifferent quantum systems, including the Dirac oscillator in cosmic string space-time [52], scalar fields\nin a wormhole background with cosmic strings [53], quantum motions of scalar particles [54], and the\nbehavior of spin-1/2 particles in a topologically trivial G¨ odel-type space-time [55]. Additionally, inves-\ntigations have extended to the motions of photons in cosmic string space-time [56], and the generalized\nDuffin-Kemmer-Petiau equation with non-minimal coupling in cosmic string space-time [57].\nOur motivation is to study the relativistic quantum motions of scalar and oscillator fields in a\ntopological defect cosmological space-time under the influence of rainbow gravity’s. It is well-known\nthat the presence of rainbow gravity’s in a space-time alter its geometrical characteristics, and hence,\nbehaviours of quantum particles in this geometry background would definitely changes. We study this\nquantum mechanical problems by choosing two pairs of rainbow functions by considering p= 1,2 in Eq.\n(4). In fact, we show that the solutions of the relativistic wave equation via the Klein-Gordon and its\n3oscillator fields are influenced by the cosmological constant Λ, the topological defect parameter αand\nthe rainbow parameter β. Throughout this paper, we use the system of units, where ¯ h=c=G= 1.\n2 Scalar fields in topological defect cosmological space-time\nIn this part, we consider the physical environment of rainbow gravity effects on the relativistic quantum\nmotions of scalar particles described by the Klein-Gordon equation in the background of axisymmetric\nEinstein-Maxwell solution with a cosmological. We begin this part by writing the relativistic wave\nequation describing the quantum motions of scalar particles given by [50, 51, 58]\n\"\n−1√−g∂µ\u0010√−g gµν∂ν\u0011\n+M2#\nΨ = 0 , (8)\nwhere Mis the rest mass of the particles, gis the determinant of the metric tensor gµνwith its inverse\ngµν.\nThe covariant ( gµν) and contravariant form ( gµν) of the metric tensor for the space-time (7) are\ngiven by\ngµν= diag\u0010\n−1\nF2(χ),1\n2 ΛH2(χ),α2sin2ρ\n2 ΛH2(χ),1\nH2(χ)\u0011\n,\ngµν= diag\u0010\n− F2(χ),2 ΛH2(χ),2 ΛH2(χ)\nα2sin2ρ,H2(χ)\u0011\n. (9)\nThe determinant of the metric tensor for the space-time (7) is given by\ndet(gµν) =g=−α2sin2ρ\n4 Λ2H6(χ)F2(χ). (10)\nTherefore, the KG-equation (8) in the space-time (6) can be expressed in the following differential\nequation:\n\"\n− F2d2\ndt2+ 2 ΛH2(\nd2\ndρ2+1\ntanρd\ndρ+1\nα2sin2ρd2\ndϕ2)\n+H2d2\ndz2−M2#\nΨ = 0 . (11)\nIn quantum mechanical system, the total wave function is always expressible in terms of different\nvariables. In our case, we choose the following ansatz of the total wave function Ψ in terms of different\nvariable functions ψ(x) and Θ( θ) as follows:\nΨ = exp( −i E t) exp( i ℓ ϕ) exp( i k z)ψ(ρ), (12)\nwhere Eis the particles energy, ℓ= 0,±1,±2, ....are the eigenvalues of the angular quantum number,\nandkis an arbitrary constant.\nSubstituting the total wave function (12) into the differential equation (11), we obtain the differential\nequations for ψ(ρ) as follows:\nψ′′+1\ntanρψ′+\"\n(F2E2−M2)\n2 ΛH2−k2\n2 Λ−ι2\nsin2ρ#\nψ= 0, ι=|ℓ|\nα. (13)\nWe solve the above equation taking an approximation of the trigonometric function up to the first\norder, that is, sin ρ≈ρ, tan ρ≈ρ. Therefore, the radial wave equation (13) reduces to the following\nform:\nψ′′+1\nρψ′+\u0010\nη2−ι2\nρ2\u0011\nψ= 0, (14)\n4where\nη=r\n(F2E2−M2)\n2 ΛH2−k2\n2 Λ. (15)\nEquation (14) is the second-order Bessel equation whose solutions are well-known given by ψ(ρ) =\nc1Jι(η ρ) +c2Yι(η ρ). It si also known that the Bessel function of first kind Jis finite and the second\nkind Ydiverge at the origin ρ= 0. Therefore, employing the condition that the wave function ψ→0\natρ= 0 implies that c2= 0. Therefore, the regular solution at the origin is given by\nψ(ρ) =c1Jι(η ρ). (16)\nAt large distance, the Bessel function of the first kind in the asymptotic form will be [59, 60]\nJι(η ρ)∝cos\u0010\nη ρ−π ι\n2−π\n4\u0011\n. (17)\nNow, we want to restrict the motion of quantum particles to a region where a hard-wall confining\npotential is present. This type of confinement is important because it is a very good approximation to\nconsider when discussing the quantum properties of a gas molecule system and other particles, which are\nnecessarily confined in a box of certain dimensions. The hard-wall coffining potential has been studied\nby many researchers in the quantum systems Refs. [61, 62, 63, 64, 65, 66, 67, 68, 69, 70]. According to\nthe Dirichlet boundary condition, the radial wave function vanishes at some finite distance ρ=ρ0, that\nis,\nψ(ρ=ρ0) = 0 . (18)\nThus, using Eq. (17) into the Eq. (16) and finally employing condition (18), we obtain\n\u0010\nη ρ−π ι\n2−π\n4\u0011\n= (2n+ 1)π\n2(n= 0,1,2, ...). (19)\nSimplification of the above relation results\nF2(χ)E2−M2=H2(χ)\"\nk2+ 2 Λ\u0010\nn+|m|\n2α+3\n4\u00112π2\nρ2\n0#\n. (20)\nIn this analysis, we choose two pairs of rainbow functions as follows and obtain the energy spectrum\nof the scalar particles.\nCase A: F(χ) = 1 and H(χ) =√1−β χ, where χ=|E|\nEp\nIn this case, we choose a pair of rainbow functions given by\nF(χ) = 1 ,H(χ) =p\n1−β χ, χ =E\nEp. (21)\nSubstituting this pair of rainbow function into the energy eigenvalue relation (20), we obtain\nE2\nn,ℓ−M2=\u0010\n1−β\nEp|En,ℓ|\u0011\"\nk2+ 2 Λ\u0010\nn+|ℓ|\n2α+3\n4\u00112π2\nρ2\n0#\n. (22)\nFor|En,ℓ|=En,ℓ, from the above relation we obtain the compact expression of the energy eigenvalue\ngiven by\nE±\nn,ℓ=−\u0010β\nEp\u0011∆n,ℓ\n2±s\u0010β\nEp\u00112∆2\nn,ℓ\n4+M2+ ∆ n,ℓ, (23)\n5Λ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0.0 0.2 0.4 0.6 0.8 1.024681012\nβ(Ep)-1En,ℓ+(a) here α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0.0 0.2 0.4 0.6 0.8 1.024681012\nβ(Ep)-1En,ℓ+ (b) here Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0.0 0.2 0.4 0.6 0.8 1.02468101214\nβ(Ep)-1En,ℓ+\n(c) here α= Λ = 0 .5\nFigure 1: Positive energy E+\nn,ℓin (23) where n=K=M= 1\nΛ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0.0 0.2 0.4 0.6 0.8 1.0-150-100-500\nβ(Ep)-1En,ℓ-\n(a) here α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0.0 0.2 0.4 0.6 0.8 1.0-140-120-100-80-60-40-200\nβ(Ep)-1En,ℓ- (b) here Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0.0 0.2 0.4 0.6 0.8 1.0-200-150-100-500\nβ(Ep)-1En,ℓ-\n(c) here α= Λ = 0 .5\nFigure 2: Negative energy E−\nn,ℓin (23) where n=K=M= 1\n6where\n∆n,ℓ=k2+ 2 Λ\u0010\nn+|ℓ|\n2α+3\n4\u00112π2\nρ2\n0. (24)\nFor|En,ℓ|=−En,ℓ, from the above relation we obtain the compact expression of the energy eigenvalue\ngiven by\nE±\nn,ℓ=\u0010β\nEp\u0011∆n,ℓ\n2±s\u0010β\nEp\u00112∆2\nn,ℓ\n4+M2+ ∆ n,ℓ. (25)\nEquations (23) and (25) are the relativistic approximate energy eigenvalue of scalar particles associ-\nated with the modes {n, ℓ}in the background of topological defect cosmological space-time (7) for the\nchosen rainbow function F(χ) = 1 and H(χ) =√1−β χ.\nWe have plotted the energy spectrum of scalar particles given in Eq. (23) in Figures 1-2 and Eq.\n(25) in Figures 3-4 for different values of the cosmological constant Λ, the topological parameter α, and\nthe angular quantum number ℓand shows their behaviour either increasing or decreasing trend.\nCase B: F(χ) = 1 and H(χ) =p\n1−β χ2, where χ=|E|\nEp\nIn this case, we choose a pair of rainbow functions given by\nF(χ) = 1 ,H(χ) =p\n1−β χ2, χ =|E|\nEp. (26)\nSubstituting this pair of rainbow function into the energy eigenvalue relation (20), we obtain\nE2\nn,ℓ−M2= (1−β χ2)\"\nk2+ 2 Λ\u0010\nn+|ℓ|\n2α+3\n4\u00112π2\nρ2\n0#\n. (27)\nSimplification of the above relation gives us the compact expression of the energy eigenvalue given\nby\nE±\nn,ℓ=±vuuuutM2+k2+ 2 Λ\u0010\nn+|ℓ|\n2α+3\n4\u00112π2\nρ2\n0\n1 +β\nE2ph\nk2+ 2 Λ\u0010\nn+|ℓ|\n2α+3\n4\u00112π2\nρ2\n0i. (28)\nEquation (28) is the relativistic approximate energy eigenvalue of scalar particles associated with the\nmodes {n, ℓ}in the background of topological defect cosmological space-time (7) for the chosen rainbow\nfunction F(χ) = 1 and H(χ) =p\n1−β χ2.\nWe have plotted the energy spectrum of scalar particles given in Eq. (28) in Figure 5 for different\nvalues of the cosmological constant Λ, the topological parameter α, and the angular quantum number\nℓand shows their behaviour.\n3 Quantum oscillator field in topological defect cosmological\nspace-time\nIn this part, dynamics of quantum oscillator field described by the Klein-Gordon oscillator under the\nenvironment of rainbow gravity’s is investigated in curved space-time produced by a magnetic field given\n7Λ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0 2 4 6 8 10050010001500\nβ(Ep)-1En,ℓ+(a)α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0 2 4 6 8 100200400600800100012001400\nβ(Ep)-1En,ℓ+ (b)Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0 2 4 6 8 100500100015002000\nβ(Ep)-1En,ℓ+\n(c)α= Λ = 0 .5\nFigure 3: The energy spectrum E+\nn,ℓof Eq. (25), where n=ρ0=k=M= 1\nΛ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0.0 0.2 0.4 0.6 0.8 1.0-12-10-8-6-4-2\nβ(Ep)-1En,ℓ-\n(a)α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0.0 0.2 0.4 0.6 0.8 1.0-12-10-8-6-4-2\nβ(Ep)-1En,ℓ- (b)Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0.0 0.2 0.4 0.6 0.8 1.0-14-12-10-8-6-4-2\nβ(Ep)-1En,ℓ-\n(c)α= Λ = 0 .5\nFigure 4: The energy spectrum E−\nn,ℓof Eq. (25), where n=ρ0=k=M= 1\n8Λ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0 20 40 60 80 100-505\n(Ep)2β-1En,ℓ±(a)α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0 20 40 60 80 100-505\n(Ep)2β-1En,ℓ± (b)Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0 20 40 60 80 100-505\n(Ep)2β-1En,ℓ±\n(c)α= Λ = 0 .5\nFigure 5: The energy spectrum E±\nn,ℓof Eq. (28), where n=ρ0=k=M= 1\nby the metric (7). The Klein-Gordon oscillator equation is studied by substituting the momentum four-\nvector ∂µ→(∂µ+M ω X µ), with Xµ= (0, r,0,0) and ωrepresents the oscillator frequency. Extensive\nexploration investigating the dynamics of quantum oscillator field in various space-time backgrounds\nincluding G¨ odel and G¨ odel-type space-times both with and without cosmic strings have been done.\nMoreover, investigations have been conducted in global monopoles, cosmic string space-times (both\nstandard and spinning), as well as in topologically trivial and non-trivial geometries (see, Refs. [71,\n72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82]). The examination of the relativistic quantum oscillator field\nwithin diverse space-time backgrounds provides valuable insights into the interplay between quantum\noscillation fields and the underlying cosmological geometry, offering a comprehensive understanding of\nthe intricate dynamics involved in these scenarios.\nTherefore, the relativistic wave equation describing the quantum oscillator field is given by\nh1√−g(∂µ+M ω X µ) (√−g gµν) (∂ν−M ω X ν)i\nΨ =M2Ψ. (29)\nExpressing this wave equation (29) in the space-time background (7), we obtain the following equa-\ntion\"\n− F2d2\ndt2+ 2 ΛH2(\nd2\ndρ2+1\ntanρd\ndρ−M ω−M ω ρ\ntanρ−M2ω2ρ2+1\nα2sin2ρd2\ndϕ2)\n+H2d2\ndz2−M2#\nΨ = 0 . (30)\nSubstituting the wave function ansatz Eq. (12), we obtain\nψ′′(ρ) +1\ntanρψ′(ρ) +\"\n(F2E2−M2)\n2 ΛH2−M ω−M ω ρ\ntanρ−M2ω2ρ2−k2\n2 Λ−ι2\nsin2ρ#\nψ(ρ) = 0 .(31)\n9We solve the above equation taking an approximation of up to the first order, that is, for small values\nof the radial distance r. Therefore, the radial wave equation (31) reduces to the following form:\nψ′′(ρ) +1\nρψ′(ρ) +\u0010\nξ2−M2ω2ρ2−ι2\nρ2\u0011\nψ(ρ) = 0 , (32)\nwhere we have defined\nξ2=(F2E2−M2)\n2 ΛH2−k2\n2 Λ−2M ω . (33)\nTransforming the above equation (32) to a new variable via\ns=M ω ρ2(34)\nwe obtain\nψ′′(s) +1\nsψ′(s) + \nξ2\n4M ω1\ns−1\n4−ι2/4\ns2!\nψ(s) = 0 . (35)\nThe requirement of the wave function is that the wave function ψ(s) must be finite everywhere\nincluding at the origin s= 0. Suppose a possible solution to this equation (35) is given as follows:\nψ(s) =sι/2e−s\n2F(s), (36)\nwhere F(s) is an unknown function.\nThereby, substituting this possible solution into the equation (35) results the following second-order\ndifferential equation\ns F′′(s) + (1 + ι−s)F′(s) + \nξ2\n4M ω−1\n2−ι\n2!\nF(s) = 0 . (37)\nThis is the confluent hypergeometric equation [59], which is a second-order linear homogeneous differ-\nential equation where two independent solutions can be obtained. The solution of Eq. (37) which is\nregular at the origin s= 0 is given by the confluent hypergeometric function as follows:\nF(s) = 1F1 \n1\n2+ι\n2−ξ2\n4M ω; 1 +ι;s!\n. (38)\nThe solution of F(s) is a polynomial of degree ncan be obtained by the Frobenius method. Note\nthat we require regularity of the wave function in the origin s= 0 and at infinity s→ ∞ in Eq. (36).\nThe exponential term in Eq. (36) implies that M ω > 0 in order to ensure the condition of regularity of\nthe wave function at infinity ( s→ ∞ ). In terms of the coordinate ρ, we see that Eq. (36) is written in\nthe form ψ∝ριe−1\n2M ω ρ2\n1F1(M ω ρ2). Naturally, for all possible values of ρ, it is easy to observe that\nour solution is divergent at the extremal point. However, this blow-up can be eliminated by a truncation\nof the power series. This procedure is equivalent to imposing the following condition:\n1\n2+ι\n2−ξ2\n4M ω=−n(n= 0,1,2,3, ....). (39)\nAfter simplification of the above condition, we obtain the energy relation as follows:\nF2(χ)E2−M2=H2(χ)\"\nk2+ 8M ωΛ\u0010\nn+|ℓ|\n2α+ 1\u0011#\n. (40)\nThe corresponding wave function (36) becomes\nψn,ℓ(s) =s|ℓ|\n2αe−s\n21F1\u0010\n−n; 1 +|ℓ|\nα;s\u0011\n. (41)\n10Λ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0.0 0.2 0.4 0.6 0.8 1.012345\nβ(Ep)-1En,ℓ+(a) here α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0.0 0.2 0.4 0.6 0.8 1.01.01.52.02.53.03.54.0\nβ(Ep)-1En,ℓ+ (b) here Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0.0 0.2 0.4 0.6 0.8 1.01234\nβ(Ep)-1En,ℓ+\n(c) here α= Λ = 0 .5\nFigure 6: Positive energy E+\nn,ℓin (45) where n=ω=K=M= 1\nIn terms of ρ, we obtain the radial wave function\nψn,ℓ(ρ) =C(M ω)|ℓ|\n2αρ|ℓ|\nαe−1\n2M ω ρ2\n1F1\u0010\n−n; 1 +|ℓ|\nα;M ω ρ2\u0011\n, (42)\nCis the normalization constant.\nBelow, we evaluate the energy spectrum of oscillator fields by choosing two pairs of rainbow functions\nconsidered earlier.\nCase A: F(χ) = 1 and H(χ) =√1−β χ, where χ=E\nEp\nIn this case, we choose a pair of rainbow functions given by\nF= 1,H=p\n1−β χ, χ =E\nEp. (43)\nSubstituting this pair of rainbow function into the energy eigenvalue relation (40), we obtain\nE2\nn,ℓ−M2=\u0010\n1−β\nEp|En,ℓ|\u0011\"\nk2+ 8M ωΛ\u0010\nn+|ℓ|\n2α+ 1\u0011#\n. (44)\nFor|En,ℓ|=En,ℓ, simplification of the above relation gives us the compact expression of the energy\neigenvalue given by\nE±\nn,ℓ=−\u0010β\nEp\u0011Θn,ℓ\n2±s\u0010β\nEp\u00112Θ2\nn,ℓ\n4+M2+ Θ n,ℓ, (45)\n11Λ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0.0 0.2 0.4 0.6 0.8 1.0-25-20-15-10-5\nβ(Ep)-1En,ℓ-(a) here α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0.0 0.2 0.4 0.6 0.8 1.0-15-10-5\nβ(Ep)-1En,ℓ- (b) here Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0.0 0.2 0.4 0.6 0.8 1.0-20-15-10-5\nβ(Ep)-1En,ℓ-\n(c) here α= Λ = 0 .5\nFigure 7: Negative energy E−\nn,ℓin (45) where n=ω=K=M= 1\nwhere\nΘn,ℓ=k2+ 8M ωΛ\u0010\nn+|ℓ|\n2α+ 1\u0011\n. (46)\nFor|En,ℓ|=−En,ℓ, simplification of the above relation gives us the compact expression of the energy\neigenvalue given by\nE±\nn,ℓ=\u0010β\nEp\u0011Θn,ℓ\n2±s\u0010β\nEp\u00112Θ2\nn,ℓ\n4+M2+ Θ n,ℓ. (47)\nEquations (45) and (47) are the relativistic approximate energy eigenvalue of quantum oscillator\nfields associated with the modes {n, ℓ}in the background of topological defect cosmological space-time\n(7) for the chosen rainbow function F(χ) = 1 and H(χ) =√1−β χ.\nWe have plotted the energy spectrum of quantum oscillator fields given in Eq. (45) in Figures\n6-7 and Eq. (47) in Figures 8-9 for different values of the cosmological constant Λ, the topological\nparameter α, and the angular quantum number ℓand shows their behaviour with increasing values of\nthese parameters.\nCase B: F(χ) = 1 and H(χ) =p\n1−β χ2, where χ=E\nEp\n12Λ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0 2 4 6 8 10050100150200250\nβ(Ep)-1En,ℓ+(a)α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0 2 4 6 8 10050100150\nβ(Ep)-1En,ℓ+ (b)Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0 2 4 6 8 10050100150200\nβ(Ep)-1En,ℓ+\n(c)α= Λ = 0 .5\nFigure 8: The energy spectrum E+\nn,ℓof Eq. (47), where n=ω=k=M= 1\nΛ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0.0 0.2 0.4 0.6 0.8 1.0-5-4-3-2-1\nβ(Ep)-1En,ℓ-\n(a)α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0.0 0.2 0.4 0.6 0.8 1.0-4.0-3.5-3.0-2.5-2.0-1.5-1.0\nβ(Ep)-1En,ℓ- (b)Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0.0 0.2 0.4 0.6 0.8 1.0-4-3-2-1\nβ(Ep)-1En,ℓ-\n(c)α= Λ = 0 .5\nFigure 9: The energy spectrum E−\nn,ℓof Eq. (47), where n=ω=k=M= 1\n13Λ=0.25\nΛ=0.5\nΛ=0.75\nΛ=1\n0 20 40 60 80 100-4-2024\nβ-1(Ep)2En,ℓ±(a)α= 0.5, ℓ= 1\nα=0.25\nα=0.5\nα=0.75\nα=1\n0 20 40 60 80 100-4-2024\nβ-1(Ep)2En,ℓ± (b)Λ= 0.5, ℓ= 1\nℓ=0\nℓ=1\nℓ=2\nℓ=3\n0 20 40 60 80 100-4-2024\nβ-1(Ep)2En,ℓ±\n(c)α= Λ = 0 .5\nFigure 10: The energy spectrum E±\nn,ℓof Eq. (50), where n=ω=k=M= 1\nIn this case, we choose a pair of rainbow functions given by\nF(χ) = 1 ,H(χ) =p\n1−β χ2, χ =E\nEp. (48)\nSubstituting this pair of rainbow function into the energy eigenvalue relation (40), we obtain\nE2\nn,ℓ−M2=\u0010\n1−β\nE2pE2\nn,ℓ\u0011\"\nk2+ 8M ωΛ\u0010\nn+|ℓ|\n2α+ 1\u0011#\n. (49)\nSimplification of the above relation gives us the compact expression of the energy eigenvalue given by\nE±\nn,ℓ=±vuuutM2+k2+ 8M ωΛ\u0010\nn+|ℓ|\n2α+ 1\u0011\n1 +β\nE2ph\nk2+ 8M ωΛ\u0010\nn+|ℓ|\n2α+ 1\u0011i. (50)\nEquation (50) is the relativistic approximate energy eigenvalue of quantum oscillator fields associated\nwith the modes {n, ℓ}in the background of topological defect cosmological space-time (7) for the chosen\nrainbow function F(χ) = 1 and H(χ) =p\n1−β χ2.\nWe have plotted the energy spectrum of the quantum oscillator fields given in Eq. (50) in Figure\n10 for different values of the cosmological constant Λ, the topological parameter α, and the angular\nquantum number ℓand shows their behaviour with increasing values of these parameters.\n4 Conclusions\nThe exploration of scalar fields in topological defect cosmological spacetime and the investigation of\nquantum oscillator fields within the framework of rainbow gravity have provided valuable insights into\n14the intricate interplay between quantum mechanics and general relativity. The examination of scalar\nfields in the context of topological defect cosmological spacetime has unveiled unique characteristics,\nshedding light on the influence of gravitational effects on scalar particles. Similarly, the study of quantum\noscillator fields in this spacetime, particularly within the paradigm of rainbow gravity, has expanded\nour understanding of how fundamental particles behave in diverse gravitational environments. These\nfindings contribute to the broader exploration of unconventional aspects in the relationship between\ngravity and quantum physics, offering a nuanced perspective on the nature of spacetime and its impact\non scalar and oscillator fields.\nIn future research, exploring the dynamics of relativistic particles in rainbow gravity backgrounds\ncould involve broadening the scope to encompass a wider range of particles and interactions. Addi-\ntionally, there is a potential for developing more accurate computational models to predict quantum\nbehavior under these conditions. Investigating the environmental influences on relativistic particles\nand exploring practical applications in areas like material design and technology could further enhance\nour understanding and application of the intricate interplay between quantum mechanics and general\nrelativity.\nIn conclusion, these recent studies on the dynamics of relativistic Klein-Gordon (KG) particles have\nexplored diverse spacetimes within the framework of rainbow gravity. These investigations represent a\nkey point in the modern understanding of the gravitational impact on particle behavior in the realm\nof physics. Providing precise or quasi-exact solutions for these systems contributes to describing the\nrelativistic and non-relativistic dynamics of particles in curved space, shedding light on the intricate\ninterplay between quantum mechanics and general relativity. 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In this paper we\ncompute the supersymmetric index of such elementary string states using gravitational\npath integral. The dominant contribution to the path integral comes from an Euclidean\nrotating black hole solution of the supergravity theory with a finite area event horizon,\nbut the logarithm of the index, computed from the saddle point, vanishes. Nevertheless\nwe show that the solution is singular on certain subspaces of the horizon where higher\nderivative corrections can be important, and once the higher derivative corrections are\ntaken into account the solution could yield a finite result for the logarithm of the index\nwhose form agrees with the microscopic results up to an overall numerical constant. While\nthe numerical constant is not determined in our analysis, we show that it is independent\nof the details of the compactification and even the number of non-compact dimensions,\nin agreement with the microscopic results.arXiv:2401.13730v1  [hep-th]  24 Jan 2024Contents\n1 Introduction 2\n2 Microscopic results for small black hole entropy 5\n3 Strategy for macroscopic computation of small black hole entropy 6\n4 The black hole solution 9\n4.1 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n4.2 BPS limit and Euclidean continuation . . . . . . . . . . . . . . . . . . . . . . . 11\n4.3 Periodicity of the τcircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n4.4 ( D−1)-dimensional base and the singular subspace . . . . . . . . . . . . . . . 15\n5 The geometry near the singularity 17\n5.1 Coordinates centered at the singular surface . . . . . . . . . . . . . . . . . . . 17\n5.2 Large charge limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n6 Wald entropy and the index 20\n7 Generalization to other compactifications 23\n1 Introduction\nThe excitations of a string lead to an infinite tower of massive states with large degeneracy. It\nhas long been suspected that sufficiently heavy states of the string can be identified as black\nholes and that the logarithm of the degeneracy of such string states may provide an explanation\nof the black hole entropy [1–6]. However since the counting of states of the elementary string\ncan be carried out reliably at weak string coupling, while the description of the system as a\nblack hole holds at large string coupling, quantitative comparison between the microscopic and\nmacroscopic calculations is difficult.\nThis problem can be overcome by considering a special class of elementary string states\nthat preserve some of the supersymmetries of the theory and hence saturate the BPS bound [7].\nFor such states there are general arguments that the log of the degeneracy computed at weak\ncoupling remains the same at strong coupling and hence can be compared with the entropy\nof the black hole carrying the same charges. One finds however that the corresponding black\n2hole has vanishing area of the event horizon and hence the entropy vanishes. It was argued\nin [8–10] that once we take into account the higher derivative corrections to the effective action,\nthe black hole may acquire a finite entropy. Using the symmetries of the theory and a scaling\nargument, one could determine the dependence of the entropy on the charges up to an overall\nnumerical factor whose value depends on the details of the higher derivative corrections [8].\nThe dependence of the entropy on the charges, obtained this way, precisely matches with the\nknown result from the microscopic counting [8–10]. The numerical constant was determined\nlater in [11] under the assumption that only certain four derivative terms contribute to the\nentropy of the black hole.\nThere is however one caveat in this analysis. The analysis described above relies on the\ndegeneracy of BPS states being protected under quantum corrections so that it takes the same\nvalue for weak and strong coupling. However it is not the degeneracy, but a closely related\nquantity called the supersymmetric index, that is protected this way. So we need to make an\nextra assumption, namely that the degeneracy and the index should be equal both at strong\nand weak coupling. In the weak coupling limit, where the states can be described as excitations\nof the fundamental string, this is known to be the case for compactification of heterotic string\ntheory. However for type II string compactification this fails – the index vanishes while the\ndegeneracy continues to be large. At strong coupling where the black hole description is valid,\nthere is no general argument showing the equality of degeneracy and the index either in the\nheterotic string theory or in type II string theory. See e.g.section 5 of [12] for some discussion\non these issues.\nThe goal of this paper is to circumvent this problem by directly analyzing the index of\nfundamental string states in heterotic string theory compactified on (10 −D) dimensional\ntorus and other manifolds. On the microscopic side the computation is straightforward and\nthe result has been known for many years. On the macroscopic side we use a procedure for\ncomputing index developed recently in [13], following earlier work on AdS spaces [14]. This\nprocedure has been successful in reproducing correctly the leading result [13, 15] as well as\nlogarithmic correction to the logarithm of the index of black holes with regular finite area event\nhorizon [16,17]. We use this method to calculate the leading contribution to the logarithm of\nthe index of small black holes.\nIt turns out that the saddle point that contributes to the log of the index of small black\nholes is not a small black hole with zero area event horizon but a rotating Euclidean black hole\nwith finite area event horizon. Nevetheless, when one follows the algorithm for computing the\n3index using this black hole saddle point, one finds that in the two derivative theory the index\nvanishes, just as the entropy of the small black holes vanished in the two derivative theory. One\nfinds however that despite having an event horizon of finite area, this black hole solution is not\ncompletely smooth, – it has singularity on a subspace of the horizon where higher derivative\ncorrections may become important. We carefully analyze the geometry near the singularity,\nand by using the symmetries of the theory and a scaling argument very similar to the ones\nused in [8], we can completely determine the corrected logarithm of the index up to an overall\nnumerical factor. This dependence of the index on the charges found this way turns out to be\nin perfect agreement with the microscopic results for the index. We furthermore show that the\nundetermined numerical factor is independent of the dimension Dof the non-compact part of\nthe space-time even though the full black hole solution depends non-trivially on D. This is\nalso in agreement with the microscopic results.\nThe rest of the paper is organized as follows. In section 2 we review the microscopic results\nfor the index of fundamental string states. In section 3 we describe the general strategy that\nwe shall use for the macroscopic computation of the log of the index. In section 4 we describe\nthe rotating black hole solution that contributes to the index and show that the log of the\nindex computed from this solution vanishes in the two derivative theory. We also show that\nthe solution is singular on a subspace of the horizon. In section 5 we zoom in on the region\nnear the singularity and express the solution in a rescaled coordinate system that is useful for\nstudying the solution near the singularities. In section 6 we present the argument based on\nscaling and the symmetries that determines the higher derivative correction to the log of the\nindex up to an overall numerical constant. We show that the dependence of this correction on\nthe charges carried by the state is precisely what is expected from the microscopic analysis. In\nsection 7 we generalize these results to other compactifications of heterotic string theory.\nSince the paper is technical, for the benefit of the reader we would like to point to the final\nresults of our paper that are contained in section 6. Eqs.(6.1)-(6.6) describe the solution near\nthe singularity. These configurations are universal, i.e. independent of the parameters of the\nsolution like the charges or the asymptotic values of the moduli, except for two factors. First\nthe string metric (6.1) contains a direct sum of a universal four dimensional metric and a nearly\nflat (D−4) dimensional metric b2dΩD−4, whose only role in our analysis is to provide a factor\nof the volume of the ( D−4) dimensional space, proportional to bD−4, in the evaluation of the\naction. The second source of non-universality is an additive constant −ln(g−2\nsm0bD−4) in the\nexpression for the dilaton field Φ as given in (6.6), which also provides an overall multiplicative\n4factor in the action but does not affect our analysis in any other way. Expressing these non-\nuniversal factors in terms of the charges carried by the black hole, we arrive at the result (6.12)\nwhich is our final result for the logarithm of the index. Here Kis an unknown numerical factor\nthat can in principle be determined with a better understanding of the higher derivative terms\nin the action, or, equivalently, a better understanding of the world-sheet σmodel with target\nspace determined by the solutions (6.1)-(6.6).\n2 Microscopic results for small black hole entropy\nWe consider heterotic string theory compactified on a (10 −D) dimensional torus so that\nwe have a theory in Dnon-compact space-time directions. The vacuum of this theory is\ncharacterized by the string coupling gsthat can be identified as the vacuum expectation value\nofeΦ/2, Φ being the dilaton field and the vacuum expectation value of a (36 −2D)×(36−2D)\nmatrix valued scalar field M, satisfying\nMLMT=L, MT=M, L =\u0012\n−I26−D 0\n0 I10−D\u0013\n, (2.1)\nwhere Indenotes n×nidentity matrix. We shall denote by Φ ∞andM∞the asymptotic values\nof the fields Φ and M.\nThe elementary string states in this theory carry 36 −2Ddifferent charges, which we can\ndivide into a (10 −D) dimensional charge vector QRand a (26 −D) dimensional charge vector\nQL. If we denote by Qthe full (36 −D) dimensional charge vector, then we can write\nQR=1\n2(M∞+L)Q, Q L=1\n2(M∞−L)Q . (2.2)\nWe also define,\nQ2\nR≡1\n2QT(M∞+L)Q, Q2\nL≡1\n2QT(M∞−L)Q, Q2≡QTLQ=Q2\nR−Q2\nL.(2.3)\nAmong the elementary string states there are a special class of states, known as half BPS\nstates, that are invariant under eight of the sixteen unbroken supersymmetries of the theory.\nThe mass of such a state, measured in the string metric, is given by\nmBPS=p\nQT(M∞+L)Q=q\n2Q2\nR. (2.4)\nAt this point we need to describe our convention. We shall work in the α′= 1 unit so that if\nwe have a fundamental string carrying nunits of momentum and wunits of winding along an\n5internal circle of radius R, it has mass wR+n/R. In our convention QR= (wR+n/R)/√\n2\nandQL= (wR−n/R)/√\n2 for this state so that (2.4) holds and we have Q2\nR−Q2\nL= 2nw.\nThe microscopic entropy of such BPS states, defined as the log of the degeneracy of the\nstates, can be computed and yields the result (see e.g.[4,8])\nSmicro = 4πr\nQ2\nR−Q2\nL\n2= 2√\n2πp\nQ2, (2.5)\nin the limit of large charges. However this is not protected under quantum corrections. The\nmore relevant quantity for our analysis is an appropriate index, known as the fourth helicity\ntrace index, defined as\n1\n4!TrQ\u0002\ne−β(H−mBPS)(−1)F(2J)4\u0003\n, (2.6)\nwhere TrQdenotes trace over all states carrying charge Qand zero momentum, ( −1)Ftakes\nvalue 1 and −1 for bosonic and fermionic states respectively, His the Hamiltonian and Jis the\ncomponent of the angular momentum carried by the state in some fixed two dimensional plane.\nβis an arbitrary parameter labelling the inverse temperature of the system but the index is\nsupposed to be independent of βsince only those states whose Heigenvalue is equal to mBPS\ncontribute to the trace. For heterotic string theory on T10−Dthis quantity can be shown to\nbe equal to the degeneracy in the weak coupling limit, and so the log of the index is given by\n(2.5). However this index can also be shown to be protected under quantum corrections and\nhence takes the same value at strong coupling.\nOur goal is to try to reproduce (2.5) from macroscopic analysis, i.e. from the analysis of\nthe classical geometry describing a solution carrying these charges.\n3 Strategy for macroscopic computation of small black\nhole entropy\nIn any two derivative theory in Dspace-time dimensions containing the metric gµν, 2-form field\nBµν,U(1) gauge fields {A(i)\nµ}and scalar fields {ϕα}, there is a scaling relation under which\ngµν→λ2gµν, B µν→λ2Bµν, A(i)\nµ→λ A(i)\nµ, ϕ α→ϕα, (3.1)\nand the action scales as λD−2. For a classical black hole solution carrying mass m, charges Q\nand angular momenta J, if we scale the fields as in (3.1), the various quantities parametrizing\nthe solution scale as\nm→λD−3m, Q →λD−3Q, J →λD−2J . (3.2)\n6Under this scaling the Bekenstein-Hawking entropy SBHof a regular black hole scales in the\nsame way as the action:\nSBH→λD−2SBH. (3.3)\nWe can take the macroscopic limit by taking λto be large. Note that under the same scaling\n(2.5) scales by a factor of λD−3, leading to an apparent contradiction. This issue was resolved\nin [8] by noting that the classical BPS black hole in the two derivative supergravity theory has\nzero area event horizon and hence vanishing entropy and we need to take into account higher\nderivative corrections to get the correct scaling of the entropy. We shall carry out a similar\nanalysis in this paper, but focussing from the beginning on the computation of the index on\nthe macroscopic side as well.\nWe now outline the main steps in our analysis.\n1. Since we are interested in computing the index, we follow the strategy described in [13],\nwhere we begin with an electrically charged black hole carrying a charge vector Qand\nangular momentum Jin a two dimensional plane. The angular momentum Jis adjusted\nso that the chemical potential Ω dual to this angular momentum, also called the angular\nvelocity, is given by\nΩ =−2πi/β , (3.4)\nwhere βis the inverse temperature of the black hole. The effect of this chemical potential\nis to insert a factor of e2πiJinto the path integral, which in turn plays the role of ( −1)F\nthat appears in (2.6). Therefore this rotating black hole solution provides the dominant\nsaddle point in the computation of the index. Then the leading semiclassical result Smacro\nfor the log of the index, computed from the gravitational path integral, is given by\nSmacro =Swald+ 2πiJ , (3.5)\nwhere Swaldis the Wald entropy of the black hole, which can also be computed following\nthe approach of Gibbons and Hawking where we first compute the partition function by\ncomputing the path integral over all the fields and then use the appropriate thermody-\nnamic relations to extract the entropy [18]. We shall denote by Swaldthe expression for\nthe black hole entropy including the effect of higher derivative corrections, irrespective of\nwhich way it is calculated. For two derivative theories this agrees with the Bekenstein-\nHawking entropy SBHof the black hole given by A/(4GN) where Ais the horizon area\nandGNis the Newton’s gravitational constant. The 2 πiJterm in (3.5) is the effect of\n7the chemical potential that inserts the explicit factor of e2πiJin the path integral. Note\nthat according to (2.6) we also need to insert a factor of (2 J)4into the path integral, but\nas discussed in [16,17], this factor just soaks up the fermion zero modes associated with\nbroken supersymmetries in the path integral and does not have any other effect.\n2. We then show that for this solution\nSBH+ 2πiJ= 0. (3.6)\nIf this had been non-zero, then according to (3.2), (3.3) this would scale as λD−2. On\nthe other hand (2.5) and (3.2) shows that Smicro scales as λD−3. This would lead to an\nobvious contradiction which is avoided by (3.6).\n3. We can now ask whether higher derivative corrections to (3.5) could resolve the dis-\nagreement. Since Jis measured from the fields at infinity where the higher derivative\ncorrections can be ignored, we focus on Swald. If the horizon had been smooth, then the\nfour derivative corrections to Swaldwould be of order λD−4according to (3.1) since two\nextra derivatives need to be contracted with a gµν. This would be inconsistent with (2.5)\nwhich scales as λD−3. We show that the solution is singular near a subspace, and that\nin the neighbourhood of this subspace we can choose a coordinate system in which the\nstring frame metric becomes independent of the charges and the dilaton takes the form\ne−Φ=q\nQ2\nR−Q2\nLF (3.7)\nwhere Fis a function of the coordinates that does not depend on the charges. Since\neΦ/2expectation value gives the string coupling, (3.7) shows that for large charges, the\nstring coupling at the horizon is small. Hence we can restrict our analysis to tree level\nstring theory. On the other hand since the metric and ∂µΦ are of order unity (having\nno dependence on the charges) we see that the α′corrections are of order unity, and we\nneed to take into account all order corrections in α′to compute the entropy. However\nthese corrections are completely universal, independent of the charges.\n4. This argument shows that if the higher derivative corrections generate a finite entropy,\nthen the dependence of the entropy on the charges comes solely from thep\nQ2\nR−Q2\nL\nfactor in (3.7). This will appear as an overall multiplicative factor in the expression for\nthe on-shell action and hence the entropy. So we can write\nSmacro =Cq\nQ2\nR−Q2\nL, (3.8)\n8in agreement with (2.5). Here Cis an unknown numerical constant that depends on\nthe details of how the α′corrections modify the near horizon geometry. However Cis\nindependent of Φ ∞andM∞. We also show that Cis independent of D, in agreement\nwith (2.5).\nFinally we would like to remark that since the macroscopic computation of the index will\nbe done using the full space-time geometry, it implicitly uses the grand canonical ensemble, i.e.\ninstead of computing the trace over fixed charge states as in (2.6) we sum over all charged states\nweighted by e−βµ.Qand then perform the Legendre transform of the logarithm of the result\nwith respect to the chemical potential µ. Of course we shall not see this being done explicitly,\nbut the gravitational formula (3.5) for the index that we shall use does this automatically. If\nthe index is a smooth function of the charges then the result of this computation differs from\nthe actual index by subleading logarithmic terms that we do not keep track of. However if the\nindex is not a smooth function of the charges, e.g. if it switches sign as we move from one\nelement of the charge lattice to its neighbouring element, then the procedure described above\ncan give results that differ significantly from the actual index, – see e.g.[14] for similar issues\nfor AdS back holes. For heterotic string compactification we know that the microscopic index\nis a smooth function of the charges for large charge. We shall assume that this holds on the\nmacroscopic side as well.\n4 The black hole solution\nWe begin by writing down the two derivative action describing the massless bosonic fields in\nheterotic string theory compactified on T10−D[19]:\nS=CDZ\ndDx√\n−detG e−Φh\nRG+Gµν∂µΦ∂νΦ +1\n8GµνTr(∂µML∂ νML)\n−1\n12Gµµ′Gνν′Gρρ′HµνρHµ′ν′ρ′−Gµµ′Gνν′F(j)\nµν(LML )jkF(k)\nµ′ν′i\n, (4.1)\nwhere,\nF(j)\nµν=∂µA(j)\nν−∂νA(j)\nµ, (4.2)\nHµνρ=∂µBνρ+ 2A(j)\nµLjkF(k)\nνρ+∂νBρµ+ 2A(j)\nνLjkF(k)\nρµ+∂ρBµν+ 2A(j)\nρLjkF(k)\nµν, (4.3)\nandRGis the Ricci scalar associated with the metric Gµν.MandLhave been introduced in\n(2.1). CDis an arbitrary D-dependent constant, which can be absorbed into the dilaton field\n9Φ. This action is invariant under an O(26−D,10−D) symmetry under which\nM→ΩMΩT, A(j)\nµ→ΩjkA(k)\nµ,∀Ω such that ΩTLΩ =L . (4.4)\nFurthermore this symmetry is known to survive α′correction to the effective action, i.e. this\nis an exact symmetry of tree level string theory [20,21].\nGµνappearing in (4.1) is the string frame metric. The canonical Einstein frame metric gµν\nis related to this via the equation:\ng(can)µν=g4/(D−2)\ns e−2Φ/(D−2)Gµν, (4.5)\nwhere g2\nsis the asymptotic value of eΦ. The gsdependent factor is normally not included in the\ndefinition of the Einstein frame metric, but we have included it here so that if Gµνapproaches\nηµνasymptotically, so does gµν. We shall always work with the string frame metric unless\nmentioned otherwise.\n4.1 The solution\nLetαandγbe two boost parameters. Let ⃗ nbe a (26 −D) dimensional unit vector and ⃗ pbe\na (10−D) dimensional unit vector. The general non-extremal solution in D-dimensions with\nsingle rotation takes the form [22]:\nds2≡Gµνdxµdxν\n= (ρ2+a2cos2θ)n\n−∆−1(ρ2+a2cos2θ−2mρ5−D)dt2+ (ρ2+a2−2mρ5−D)−1dρ2+dθ2\n+∆−1sin2θ[∆ + a2sin2θ(ρ2+a2cos2θ+ 2mρ5−Dcoshαcoshγ)]dϕ2\n−2∆−1mρ5−Dasin2θ(cosh α+ cosh γ)dtdϕ +ρ2cos2θ(ρ2+a2cos2θ)−1dΩD−4o\n,(4.6)\nwhere,\n∆ = ( ρ2+a2cos2θ)2+ 2mρ5−D(ρ2+a2cos2θ)(cosh αcoshγ−1) +m2ρ10−2D(cosh α−coshγ)2.\n(4.7)\n{t, ρ, θ, ϕ }are the four spacetime coordinates and dΩD−4is the round metric on a ( D−4)-\ndimensional unit sphere. The dilaton supporting the solution is:\nΦ = 2 ln gs+1\n2ln(ρ2+a2cos2θ)2\n∆. (4.8)\n10The U(1) vector fields A(j)for 1≤j≤(26−D) are determined by the unit vector ⃗ n:\nA(j)\nt=−n(j)\n√\n2∆−1mρ5−Dsinhα{(ρ2+a2cos2θ) cosh γ+mρ5−D(cosh α−coshγ)},\nA(j)\nϕ=n(j)\n√\n2∆−1mρ5−Dasinhαsin2θ{ρ2+a2cos2θ+mρ5−Dcoshγ(cosh α−coshγ)},\n(4.9)\nand the remaining vector fields A(j)forj≥(27−D) are determined by the unit vector ⃗ p:\nA(j)\nt=−p(j−26+D)\n√\n2∆−1mρ5−Dsinhγ{(ρ2+a2cos2θ) cosh α+mρ5−D(cosh γ−coshα)},\nA(j)\nϕ=p(j−26+D)\n√\n2∆−1mρ5−Dasinhγsin2θ{ρ2+a2cos2θ+mρ5−Dcoshα(cosh γ−coshα)}.\n(4.10)\nThe two-form field takes the form,\nBtϕ= ∆−1mρ5−Dasin2θ(cosh α−coshγ){ρ2+a2cos2θ+mρ5−D(cosh αcoshγ−1)},(4.11)\nand the matrix of scalars takes the form,\nM=I36−2D+\u0012\nPnnTUnpT\nUpnTPppT\u0013\n, (4.12)\nwhere,\nP= 2∆−1m2ρ10−2Dsinh2αsinh2γ ,\nU=−2∆−1mρ5−Dsinhαsinhγ{ρ2+a2cos2θ+mρ5−D(cosh αcoshγ−1)}.(4.13)\nSince PandUfall off for large ρ, we see from (4.12) that M∞=I36−2D. However using the\nsymmetry transformation (4.4) we can generate any other M∞from this solution, and hence\nsetting M∞toI36−2Ddoes not restrict the solution in any way.\n4.2 BPS limit and Euclidean continuation\nThe BPS limit is obtained by taking m→0,γ→ ∞ keeping m0≡mcoshγfinite. The\nBPS solution is also written in [22]. Let us in addition consider the analytic continuation to\n11Euclidean space, together with a redefinition of the rotation parameter aand the fields:\nt= iτ,\na= ib, J = iJE, Ω =−i ΩE,\nA(j)= iA(j)\n(E),\nB=−B(E). (4.14)\nThe last analytic continuation is required for consistency with (4.3). This gives the following\nEuclidean solution,\nds2= (ρ2−b2cos2θ)n\n∆−1(ρ2−b2cos2θ)dτ2+ (ρ2−b2)−1dρ2+dθ2\n+∆−1sin2θ[∆−b2sin2θ(ρ2−b2cos2θ+ 2m0ρ5−Dcoshα)]dϕ2\n+2∆−1m0ρ5−Dbsin2θdτdϕ +ρ2cos2θ(ρ2−b2cos2θ)−1dΩD−4o\n, (4.15)\nwhere now\n∆ = ( ρ2−b2cos2θ)2+ 2m0ρ5−Dcoshα(ρ2−b2cos2θ) +m2\n0ρ10−2D, (4.16)\nΦ = 2 ln gs+1\n2ln(ρ2−b2cos2θ)2\n∆. (4.17)\nThe gauge fields are given by\nA(j)\n(E)τ=−n(j)\n√\n2∆−1m0ρ5−Dsinhα(ρ2−b2cos2θ),\nA(j)\n(E)ϕ=−n(j)\n√\n2∆−1(m0)2ρ10−2Dbsinhαsin2θ, for 1≤j≤26−D , (4.18)\nand\nA(j)\n(E)τ=−p(j−26+D)\n√\n2∆−1m0ρ5−D{(ρ2−b2cos2θ) cosh α+m0ρ5−D},\nA(j)\n(E)ϕ=p(j−26+D)\n√\n2∆−1m0ρ5−Dbsin2θ{ρ2−b2cos2θ+m0ρ5−Dcoshα},\nforj≥27−D . (4.19)\nThe two form field takes the form,\nB(E)τϕ=−∆−1m0ρ5−Dbsin2θ{ρ2−b2cos2θ+m0ρ5−Dcoshα}, (4.20)\n12and the scalar fields are given by,\nM=I36−2D+\u0012\nPnnTUnpT\nUpnTPppT\u0013\n, (4.21)\nwhere\nP= 2∆−1m2\n0ρ10−2Dsinh2α,\nU=−2∆−1m0ρ5−Dsinhα{ρ2−b2cos2θ+m0ρ5−Dcoshα}. (4.22)\nNote that with the redefinitions given in (4.14) the solution is real. The horizon is at ρ=b\nand the relevant region of space-time is ρ≥b. Also the solution is singular at ρ=b, cos2θ= 1.\nThis is easiest to see by examining the dilaton given in (4.17), which blows up at these values.\nWe now compute the electric charges carried by the solution. For this we note that for\nlarge ρ, we have\nF(j)\n(E)ρτ≃∂ρA(j)\n(E)τ=(\n(D−3)n(j)√\n2m0ρ2−Dsinhαfor 1≤j≤26−D\n(D−3)p(j−26+D)\n√\n2m0ρ2−Dcoshαfor 27 −D≤j≤36−D.(4.23)\nWe define the electric charges Q(j)as the integral of1\n2δS\nδF(j)\n(E)ρτover the ( D−2) dimensional\nsphere at a large but fixed value of ρ.1Using the fact that e−Φ∞= 1/g2\nsandM∞=I36−2D,\nQ=√\n2 (D−3)CDg−2\nsm0ωD−2\u0012\nnsinhα\npcoshα\u0013\n≡\u0012\nQL\nQR\u0013\n, (4.24)\nwhere ωkrepresents the volume of a unit sphere in kdimensions. Since nandpare unit\nvectors, we get q\nQ2\nR−Q2\nL=p\nQTLQ=√\n2 (D−3)CDg−2\nsm0ωD−2. (4.25)\nWe can also calculate the ADM mass of the black hole. For this we use (4.15), (4.17) and\n(4.5) to write down the expression for the canonical Einstein metric:\nds2\n(can)= (ρ2−b2cos2θ)\u0012ρ2−b2cos2θ\n∆1/2\u0013−2/(D−2)\nn\n∆−1(ρ2−b2cos2θ)dτ2+ (ρ2−b2)−1dρ2+dθ2\n+∆−1sin2θ[∆−b2sin2θ(ρ2−b2cos2θ+ 2m0ρ5−Dcoshα)]dϕ2\n+2∆−1m0ρ5−Dbsin2θdτdϕ +ρ2cos2θ(ρ2−b2cos2θ)−1dΩD−4o\n,(4.26)\n1We have used the factor of 2 in the denominator to compensate for the fact that the gauge field kinetic\nterm in (4.1) does not have the usual 1 /4. We shall see that this normalization is consistent with the convention\nof section 2.\n13Using this and eq.(14) of [23], we get\nmADM =(D−2)ωD−2\n16πGNlim\nρ→∞ρD−3(1−gττ) =ωD−2\n8πGN(D−3)m0coshα , (4.27)\nwhere the Newton’s constant GNfor the action (4.1) is given by\nGN=1\n16πCDg−2\ns. (4.28)\nThis gives\nmADM = 2ωD−2CDg−2\ns(D−3)m0coshα=q\n2Q2\nR, (4.29)\nin agreement with (2.4).\nThe overall normalization of the charges depends on the normalization of the U(1) gauge\nfields appearing in (4.1). We have chosen the normalization so that we get (4.29). We have\nalso carefully kept track of all dependence on the asymptotic moduli and the dimension D. In\nparticular, if we use the transformation (4.4) to generate a solution with some other M∞, the\ndependence of mADM onM∞and the charge Qwill be captured by the dependence of Q2\nRon\nM∞andQvia (2.3), since (2.3) is invariant under O(26−D,10−D) transformation.\nThe area of the ρ=bsurface in the Einstein frame is\nAH=m0b ωD−2. (4.30)\nThe Bekenstein-Hawking entropy of the black hole is given by\nSBH=AH\n4GN=m0b ωD−2\n4GN. (4.31)\nThe Euclidean angular momentum JEof the black hole is,\nJE=ωD−2\n8πGNm0b. (4.32)\nClearly,\nSBH−2πJE= 0. (4.33)\nThis reproduces (3.6) with the identification given in (4.14).\n144.3 Periodicity of the τcircle\nTo understand the periodicity of the τcircle, let us introduce\neρ=√ρ−b,\nϕ=eϕ−bD−4\nm0τ. (4.34)\nWith these coordinate changes, the relevant part of the metric near eρ= 0 is\nds2≃2bsin2θ\u001a\u0012\ndeρ2+eρ2b2D−8\nm2\n0dτ2\u0013\n(4.35)\n+b\n2\u0012\ndθ2+m2\n0\nm2\n0+ 2bD−3m0coshαsin2θ+b2D−6sin4θsin2θdeϕ2+ cot2θdΩD−4\u0013\u001b\n.\nIn writing this metric, we have dropped the terms of order eρ2dτdeϕand various other higher\norder terms in the expansion in eρ. Away from the singular subspace on which sin θvanishes,\nthe (eρ, τ) part of the metric is smooth at eρ= 0 provided τhas periodicity\n(τ,eϕ)≡(τ+β,eϕ) where β= 2πm0\nbD−4. (4.36)\nUsing (4.34), this translates to\n(τ, ϕ)≡\u0012\nτ+β, ϕ−bD−4\nm0β\u0013\n. (4.37)\nOf course ϕitself has a periodic identification under shift by 2 πin order that the metric (4.41)\nis non-singular for ρ > b . From this we can determine the angular velocity of Euclidean black\nhole to be\nΩE=bD−4\nm0. (4.38)\n(4.36) and (4.38) give,\nβΩE= 2π, (4.39)\nin agreement with (3.4).\n4.4 (D−1)-dimensional base and the singular subspace\nRemarkably, the string frame metric (4.15) takes the following form in general dimensions2\nds2=f(dτ+ωϕdϕ)2+ds2\nbase, (4.40)\n2Ref. [22] also noted this with α= 0.\n15where\nds2\nbase=\u0012ρ2−b2cos2θ\nρ2−b2\u0013\ndρ2+(ρ2−b2cos2θ)dθ2+(ρ2−b2) sin2θdϕ2+ρ2cos2θdΩD−4,(4.41)\nf=(ρ2−b2cos2θ)2\n∆=\u0012\n1 +1\nρD−5m0eα\nρ2−b2cos2θ\u0013−1\u0012\n1 +1\nρD−5m0e−α\nρ2−b2cos2θ\u0013−1\n,(4.42)\nand\nωϕ=bm0ρ5−D\nρ2−b2cos2θsin2θ. (4.43)\nThe base metric ds2\nbasedoes not depend on the charges and is in fact flat. To see this, let us\nintroduce,\ny1=p\nρ2−b2sinθcosϕ,\ny2=p\nρ2−b2sinθsinϕ,\ny3=ρcosθcosψ1\ny4=ρcosθsinψ1cosψ2, (4.44)\n·\n·\nyD−2=ρcosθsinψ1···sinψD−5cosψD−4,\nyD−1=ρcosθsinψ1···sinψD−5sinψD−4,\nwhere ψ1, . . . ψ D−4are the angles labelling points on the ( D−4) sphere. In this coordinate\nsystem the base metric takes the form:\nds2\nbase=dy2\n1+dy2\n2+dy2\n3+. . .+dy2\nD−1. (4.45)\nDespite having a flat base, the metric is singular is at ρ=b, cos2θ= 1 since fvanishes\nandωϕblows up there. To determine the geometry of the singular surface, note from (4.44)\nthaty1andy2vanish on the singular subspace and y3,···, yD−1lie on the surface of a ( D−4)\ndimensional sphere of radius b. The range of the coordinate θis (0, π) for D= 4 whereas for\nD > 4 it is (0 ,π\n2). Therefore for D > 4, we have a connected singular surface at ρ=b,θ= 0\nwhich is a SD−4sphere. In D= 4 the singular subspace is a pair of points corresponding to\nρ=b,θ= 0, π. Since S0consists of a pair of points, we can describe the singular surface as\nSD−4for all D. This will be important later.\n16fcomputed from (4.42) can be written as\nf−1=\u0012\n1 +m0eα\n2ρD−4\u00121\nR++1\nR−\u0013\u0013\u0012\n1 +m0e−α\n2ρD−4\u00121\nR++1\nR−\u0013\u0013\n, (4.46)\nwhere\nR±=ρ±bcosθ . (4.47)\nInD= 4, R±have the interpretation of the distance of a point from the locations of the\nsingularities at ρ=b,θ= 0, π. In this case f−1is simply a product of two harmonic functions\nand the solution can be written in a standard IWP form [15,24].\n5 The geometry near the singularity\nEq. (4.33) shows that in the two derivative approximation to the effective action, the macro-\nscopic index vanishes. According to the scaling argument given between (3.6) and (3.7), higher\nderivative corrections to Swaldfrom the non-singular regions cannot produce the answer known\nfrom the microscopic counting. We shall now focus on the singular regions near the horizon\nand explore whether higher derivative correction to Swaldfrom the singular region can produce\nthe desired result.\n5.1 Coordinates centered at the singular surface\nIn order to analyse the solution near the singular surface cos θ= 1, we introduce the following\ncoordinates near the singular subspace. Define,\neR=ρ−bcosθ, coseθ=ρcosθ−b\nρ−bcosθ. (5.1)\nThe inverse coordinate transformations are,\nρ=1\n2\u0012\neR+q\neR2+ 4beRcoseθ+ 4b2\u0013\n, cosθ=1\n2b\u0012\n−eR+q\neR2+ 4beRcoseθ+ 4b2\u0013\n.\n(5.2)\neθhas range (0 , π) for all D. These coordinates have the important property that when eR= 0,\nwe have ρ=bandcosθ= 1, i.e., we are at the singular surface. For D= 4 the second copy\nof the singular subspace is at eR= 2b,eθ=π, but near that region it is more appropriate to\nintroduce new coordinate system by replacing cos θby−cosθin (5.1).\n175.2 Large charge limit\nWe now make a final coordinate transformation,\neθ= 2Θ ,eϕ=ϕ+τ\nm0b4−D,eτ=τ\nm0b4−D, (5.3)\nand take the limit m0→ ∞ andb→ ∞ by taking\nm0∼λD−3, b ∼��, λ → ∞ , (5.4)\nkeeping fixed eR,eθ,eτ,eϕ,ΩD−4. The scaling of bfollows from (3.2) after noting that JE∝m0b.\nThe limit gives\nρ≃b+eR\n2(1 + cos eθ) (5.5)\ncosθ≃1−eR\n2b(1−coseθ) (5.6)\nsin2θ≃eR\nb(1−coseθ) (5.7)\n(ρ2−b2cos2θ)≃2beR (5.8)\n∆≃m2\n0b10−2D−b6−2Dm0eRn\n(D−5)b3m0(1 + cos eθ)−4bDcoshαo\n(5.9)\nf≃4eR2\nm2\n0b8−2D(5.10)\nωϕ≃m0b4−Dsin2eθ\n2. (5.11)\nWe have kept terms in the expansion to the order that will be needed in our computation of\nthe solution up to terms that vanish for large λ. Combining all this together and using (4.40),\nwe arrive at a universal form for the metric:\nds2≃deR2+ 4eR2dΘ2+ 4eR2sin2Θdeϕ2+ 4eR2cos2Θdeτ2+b2dΩD−4. (5.12)\nΘ has range\u0000\n0,π\n2\u0001\nandeϕandeτhave periodicities 2 π. The metric is singular at eR= 0. Since\nbis large, we can replace b2dΩD−4withPD−4\ni=1dx2\nilocally.\nFrom this we see that near the singularity the metric has a universal form independent\nofD. If on the other hand we take a spherically symmetric solution by taking the rotation\nparameter bto zero, the size of the ( D−4) dimensional sphere goes to zero and the nature of\nthe singularity becomes dependent on D[9].\n18We now turn to the matter fields in the large charge limit at fixed eR,eτ,eϕ, Θ. In this limit\n(4.22) gives\nP≃2 sinh2α ,\nU≃ − 2 sinh αcoshα . (5.13)\n(4.21) now implies,\nM≃I36−2D+\u0012\n2 sinh2α(nnT) −2 sinh αcoshα(npT)\n−2 sinh αcoshα(pnT) 2 sinh2α(ppT)\u0013\n. (5.14)\nFor the gauge fields, we get, for 1 ≤j≤26−D,\nA(j)\n(E)eτ≃ −√\n2n(j)cos2Θ sinh αeR,\nA(j)\n(E)eϕ≃ −√\n2n(j)sin2Θ sinh αeR , (5.15)\nand for j≥27−D,\nA(j)\n(E)eτ≃ −p(j−26+D)\n√\n2b4−Dm0+√\n2p(j−26+D)cos2Θ cosh αeR,\nA(j)\n(E)eϕ≃√\n2p(j−26+D)sin2Θ cosh αeR . (5.16)\nThe constant additive term in the expression for A(j)\n(E)eτcontributes to the three form field\nstrength defined in (4.3), but can be dropped after computing this field strength.\nFor the two form field, we shall directly write down the expression for the gauge invariant\nthree form field strength after taking into account the contribution from the Chern-Simons\nterm in (4.3). This takes the form:\nH(E)≡1\n6H(E)µνρdxµ∧dxν∧dxρ≃8eR2sin Θ cos Θ dΘ∧deϕ∧deτ , (5.17)\nwhere\nH(E)µνρ≡∂µB(E)νρ+2A(j)\n(E)µLjkF(k)\n(E)νρ+∂νB(E)ρµ+2A(j)\n(E)νLjkF(k)\n(E)ρµ+∂ρB(E)µν+2A(j)\n(E)ρLjkF(k)\n(E)µν.\n(5.18)\nFinally, the dilaton behaves as follows,\nexp[−Φ]≃g−2\nsb4−Dm0\n2eR. (5.19)\n19If we regard this near horizon geometry as a solution in four dimensional heterotic string\ntheory by regarding the coordinates along ΩD−4as describing an almost flat compact space,\nthen the canonical metric in four dimensions is given by e−ΦGµν. This describes a flat four\ndimensional metric, as can be seen by making a change of coordinates eR=er2. The significance\nof this observation is not clear to us.\nNote that the matrix valued scalar field Mand the gauge fields depend of the parameter\nα. The αdependence on the background Mcan be removed by doing a O(26−D,10−D)\ntransformation described in (4.4) by choosing,\nΩ =I36−2D+\u0012\n(cosh α−1) (nnT) sinh α(npT)\nsinhα(pnT) (cosh α−1) (ppT)\u0013\n. (5.20)\nThis gives\nM≃I36−2D. (5.21)\nThe vector fields also change under this. We get, for 1 ≤j≤26−D,\nA(j)\n(E)eτ≃0,\nA(j)\n(E)eϕ≃0, (5.22)\nand for j≥27−D,\nA(j)\n(E)eτ≃√\n2p(j−26+D)eRcos2Θ,\nA(j)\n(E)eϕ≃√\n2p(j−26+D)eRsin2Θ, (5.23)\nwhere we have dropped the constant term in the expression for A(j)\n(E)eτ. Since we no longer need\nto use the gauge field for computing H(E), this is an allowed operation. The new fields do not\ndepend on the parameter α. Furthermore using an O(10−D)⊂O(26−D,10−D) rotation\nwe could bring the unit vector pto any standard form, e.g.having only the first component\nnon-zero.\n6 Wald entropy and the index\nLet us summarize the solution in the large charge limit after the O(26−D,10−D) rotation\n(5.20). We have,\nds2≃deR2+ 4eR2dΘ2+ 4eR2sin2Θdeϕ2+ 4eR2cos2Θdeτ2+b2dΩD−4,\n0≤Θ≤π\n2,eϕ≡eϕ+ 2π, eτ≡eτ+ 2π , (6.1)\n20M≃I36−2D, (6.2)\nA(j)\n(E)eτ≃0,\nA(j)\n(E)eϕ≃0, for 1≤j≤26−D , (6.3)\nA(j)\n(E)eτ≃√\n2p(j−26+D)eRcos2Θ,\nA(j)\n(E)eϕ≃√\n2p(j−26+D)eRsin2Θ, forj≥27−D , (6.4)\nH(E)≃8eR2sin Θ cos Θ dΘ∧deϕ∧deτ , (6.5)\nand\nexp[−Φ]≃g−2\nsb4−Dm0\n2eR. (6.6)\nWe now note the following features of the solution:\n1. Since the original configuration described in section 4.2 solved the equations of motion of\nthe two derivative theory for all values of the parameters, and the configuration given in\n(6.1)-(6.6) is obtained as a limit of the original solution for large m0,b, (6.1)-(6.6) should\nalso satisfy the equations of motion of the two derivative theory. We have checked this\nby explicit computation, after replacing the b2dΩD−4part of the metric by flat metric.\nHowever since for finite eRthe curvature invariants and other invariants associated with\nthis solution are of order unity, we expect the solution to be modified by higher derivative\ncorrections. Such corrections should fall off for large eRwhere the invariants are small.\n2. In the large charge limit, since b∼λandm0∼λD−3,e−Φgiven in (6.6) is of order λ.\nSince e−Φcan be identified as the inverse square of the string coupling locally, we see\nthat near the singularity the string coupling is small. Hence we can ignore string loop\ncorrections and focus only on the α′correction to the solution.\n3. Since α′correction to the effective action does not destroy the O(26−D,10−D) symmetry,\ntheO(26−D,10−D) transformation that was used to bring the solution to the form\n(6.1)-(6.6) is a valid symmetry and leaves the α′corrected formula for Swaldinvariant. In\nfact we can use the O(10−D) subgroup of O(26−D,10−D) to bring the vector pto some\ncanonical form, e.g. only with the first component non-zero. In this case the solution\nis universal except for the b2dΩD−4term in the metric (6.1) and the overall factor of\ng−2\nsb4−Dm0in the expression for e−Φin (6.6). In the limit of large btheb2dΩD−4part of\n21the metric describes almost locally flat ( D−4) dimensional space and the α′corrections\nare suppressed by inverse power of λ. At the leading order this part of the metric will\ngive an overall multiplicative factor of bD−4ωD−4to the action and hence to Swaldbut\nhas no effect on the α′correction to the rest of the field configuration. On the other\nhand from (4.1) we see that the effect of the g−2\nsb4−Dm0in the expression for e−Φis to\nmultiply the action by a factor of g−2\nsb4−Dm0but it also does not affect the α′corrections\nto the solution since these corrections involve Φ only through its derivatives. Therefore\nα′corrections to the solution take a universal form, even independent of D, and modify\nonly the eRand Θ dependence of various fields in (6.1)-(6.6).\nTaking into account the overall factor of CDin (4.1) we see that the net α′correction to\nSwald, and hence to Smacro according to (3.5), (3.6), takes the form,\nSmacro =K×CD×bD−4ωD−4×g−2\nsb4−Dm0=K C Dg−2\nsm0ωD−4, (6.7)\nwhere Kis a universal numerical constant that can in principle be computed from α′correction\nto the solution (6.1)-(6.6). In particular, Kis independent of D.\nWe now recall from (4.25):\nq\nQ2\nR−Q2\nL=p\nQTLQ=√\n2 (D−3)CDg−2\nsm0ωD−2. (6.8)\nUsing this we can express (6.7) as\nSmacro =K√\n2ωD−4\n(D−3)ωD−2q\nQ2\nR−Q2\nL. (6.9)\nUsing the result\nωk= 2π(k+1)/2/Γ[(k+ 1)/2], (6.10)\nwe get\nωD−2/ωD−4= 2π/(D−3). (6.11)\nSubstituting this into (6.9) we get\nSmacro =K\n2π√\n2q\nQ2\nR−Q2\nL. (6.12)\nThis would agree with (2.5) if K= 8π2. While we do not have a way to calculate Kat present,\ntheD-independence of (6.12) is consistent with the D-independence of (2.5).\n22Note that the dependence on the parameter b, that controls the temperature of the black\nhole via (4.36), has disappeared. This is consistent with the fact that the index should be\nindependent of the temperature. Note also that the result is independent of the combination\nQTM∞Q=Q2\nL+Q2\nR. This can be traced to the α-independence of the result, – the O(26−\nD,10−D) transformation that removes the αdependence of Mnear the singularity also\nremoves the αdependence of the gauge fields. Such a dependence would have been inconsistent\nwith the microscopic results which do not depend on M∞. This was also a feature in the analysis\nin [8].\n7 Generalization to other compactifications\nThe above analysis can be easily generalized to other N= 4 or N= 2 supersymmetric\ncompactifications of heterotic string theory. For this we use the observation of [10] that the\nblack hole solution describing BPS elementary string states can be embedded in a universal\nconsistent truncation of the supergravity theory used above. Furthermore the universality is\nmaintained to all orders in α′. Therefore the analysis given above can be carried out without any\nfurther modification and we shall arrive at the expression (6.12) for the macroscopic entropy\nwith the same K. The microscopic counting also remains unchanged, since the asymptotic\ngrowth of BPS states is controlled by the central charge of the left-moving matter sector of\nthe world-sheet theory and this is always equal to 26. Note however that in order to get BPS\nfundamental string in the first place, we must have at least one internal circle on which the\nstring can wrap, i.e. we cannot have such states in ten dimensional heterotic string theory or\nthe six dimensional theory obtained by compactifying heterotic string theory on K3.\nFinally we shall make a few remarks on the type II string compactification. For definiteness\nlet us consider the case of torus compactification. The microscopic index for the fundamental\nstring states vanishes in this case due to the cancellation between the contributions from the\nbosonic and fermionic states. On the macroscopic side the relevant part of the two derivative\naction is very similar to the one we have analyzed here and identical arguments lead to a\nform (6.12) for the logarithm of the index. Therefore for consistency the coefficient Kmust\nvanish for type II theory. This may be related to the fact that higher derivative corrections\nare relatively rare in type II string theory. For example the four derivative term, that was\nused in [11] to get a finite result for Kin the heterotic string theory, is absent in type II string\ntheory. However to get a complete picture we need to learn how to compute Kby taking into\n23account the effect of all the α′corrections to the effective action.\nAcknowledgement : We would like to thank A H Anupam for discussions and collabora-\ntion during the initial stages of this work. CC thanks David Turton and Shasha Tyukov for\ndiscussions. AS thanks Sameer Murthy for discussions. AV thanks David Chow for email\ncorrespondence. 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Virmani, “Killing Spinors for Finite Temperature Euclidean Solutions at\nthe BPS Bound,” [arXiv:2311.09427 [hep-th]].\n26"    },    {        "title": "2401.13731v1.Long_Plateau_Doth_So__How_Internal_Heating_Sources_Affect_Hydrogen_Rich_Supernova_Light_Curves.pdf",        "content": "Draft version January 26, 2024\nTypeset using L ATEXtwocolumn style in AASTeX63\nLong Plateau Doth So: How Internal Heating Sources Affect Hydrogen-Rich Supernova Light Curves\nTatsuya Matsumoto,1, 2, 3Brian D. Metzger,1, 4and Jared A. Goldberg4\n1Department of Physics and Columbia Astrophysics Laboratory, Columbia University, Pupin Hall, New York, NY 10027, USA\n2Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan\n3Hakubi Center, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan\n4Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA\nABSTRACT\nSome hydrogen-rich core-collapse supernovae (type IIP SNe) exhibit evidence for a sustained energy\nsource powering their light curves, resulting in a brighter and/or longer-lasting hydrogen-recombination\nplateau phase. We present a semi-analytic SNIIP light curve model that accounts for the effects of\nan arbitrary internal heating source, considering as special cases56Ni/56Co decay, a central engine\n(millisecond magnetar or accreting compact object), and shock interaction with a dense circumstellar\ndisk. While a sustained internal power source can boost the plateau luminosity commensurate with\nthe magnitude of the power, the duration of the recombination plateau can typically be increased by\nat most a factor ∼2−3 compared to the zero-heating case. For a given ejecta mass and initial kinetic\nenergy, the longest plateau duration is achieved for a constant heating rate at the highest magnitude\nthat does not appreciably accelerate the ejecta. This finding has implications for the minimum ejecta\nmass required to explain particularly long-lasting supernovae such as iPTF14hls, and for confidently\nidentifying rare explosions of the most-massive hydrogen-rich (e.g. population III) stars. We present\na number of analytic estimates which elucidate the key features of the detailed model.\nKeywords: XXX\n1.INTRODUCTION\nRoughly 75% of all stellar explosions are core-collapse\nsupernovae (SNe; e.g., Mannucci et al. 2007), of which\n≈70% are hydrogen-rich, i.e. type II SNe (e.g., Perley\net al. 2020). Roughly 3/4 of the type II SNe are fur-\nther of the IIP class (hereafter SNIIP), with a plateau-\nshaped light curve phase lasting typically around 100\ndays (Barbon et al. 1979; Anderson et al. 2014; Faran\net al. 2014; Valenti et al. 2016; Martinez et al. 2022a).\nIn most cases, the luminosity during the plateau phase\nis supported mainly by the thermal energy deposited by\nthe supernova shock, its evolution dictated by a cool-\ning and recombination wave receding inwards through\nthe expanding envelope (Grassberg et al. 1971; Gras-\nberg & Nadezhin 1976; Falk & Arnett 1977; Litvinova\n& Nadezhin 1983, 1985; Chugai 1991; Popov 1993; Faran\net al. 2019).\nThe properties of SNIIP light curves, such as the du-\nration and luminosity of the plateau phase, can along\nwith spectral information such as photosphere veloci-\nties, be used to constrain the ejecta properties (e.g.,\nPejcha & Prieto 2015; Martinez et al. 2022b), though\nwith substantial degeneracies (e.g., Dessart & Hillier\n2019,Goldberg et al. 2019). These constraints frequently\nmake use of analytic scaling relations calibrated to nu-\nmerical simulations (e.g., Popov 1993; Zha et al. 2023).\nIt is also known that the sustained heating from the56Ni→56Co→56Fe decay chain acts to flatten and extend\nthe duration of the plateau (e.g., Young 2004; Kasen &\nWoosley 2009; Bersten et al. 2011; Nakar et al. 2016;\nSukhbold et al. 2016; Goldberg et al. 2019; Kozyreva\net al. 2019), which can also be included in numerically-\ncalibrated analytic estimates.\nHowever, there is increasing evidence for the need for\nextra energy sources beyond56Ni in a growing subset\nof supernovae, the most extreme cases being superlu-\nminous supernovae (SLSNe-II; Inserra 2019; Gal-Yam\n2019). This “extra” heating source has variously been\nattributed to circumstellar (CSM) interaction (e.g., Fas-\nsia et al. 2000; Smith & McCray 2007; Miller et al. 2009;\nInserra et al. 2012; Chatzopoulos et al. 2012; Mauer-\nhan et al. 2013; Fransson et al. 2014; Smith et al. 2014;\nBostroem et al. 2019; Nyholm et al. 2020; see Fraser\n2020 for a review) or the injection of rotational energy\nfrom a rapidly spinning magnetized neutron star (Kasen\n& Bildsten 2010; Woosley 2010; Sukhbold & Thompson\n2017; Dessart 2018) or accretion energy from a black\nhole (Dexter & Kasen 2013; Perna et al. 2018) or neu-\ntron star (Metzger et al. 2018). An extreme example\nis iPTF14hls, which produced a plateau-like (optically-\nthick) phase lasting for ∼1000 days (Arcavi et al. 2017;\nSollerman et al. 2019), only finally to reveal the appear-\nance of narrow emission lines and hence the presence of\nCSM interaction (Andrews & Smith 2018).arXiv:2401.13731v1  [astro-ph.HE]  24 Jan 20242 Matsumoto et al.\nExtremely luminous or long-lasting SNIIP can also re-\nsult from rare or exotic explosions of extremely massive\nprogenitors, such as pair-instability supernovae (PISNe)\nfrom stars of ≳100M⊙(Rakavy & Shaviv 1967; Heger\n& Woosley 2002; Woosley et al. 2007; Chen et al. 2023).\nHowever, the standard analytic expressions commonly\nused to estimate the ejecta properties from observables\nlike plateau duration and luminosity (e.g., Popov 1993;\nSukhbold et al. 2016), and hence identify such rare ex-\nplosion classes, in general do not apply in the presence of\nadditional heating sources. This motivates the present\nstudy to explore the full landscape of theoretically per-\nmitted SNIIP properties, allowing for an arbitrary heat-\ning source evolution, in order to ascertain what robust\nconstraints can be placed by supernova observations on\nthe stellar progenitor properties (particularly the ejecta\nmass). Such a framework would help address whether\nexceptionally long-duration light curves like iPTF14hls\nnecessitate an atypically massive stellar progenitor, or\nwhether an otherwise ordinary-mass explosion with a\npowerful internal heating source (e.g., from CSM inter-\naction) is alone sufficient.\nThe Vera C. Rubin Observatory (Ivezi´ c et al. 2019;\nhereafter Rubin ) will monitor roughly half of the extra-\ngalactic sky over ten years, during which it is expected to\nobserve millions of SNIIP (LSST Science Collaboration\net al. 2009). This unprecedented sample will enable de-\ntailed studies of the distribution of explosion properties\n(e.g., Murphy et al. 2019,Goldberg & Bildsten 2020,Mar-\ntinez et al. 2022a), and how they map onto progenitor\nstar (e.g., Strotjohann et al. 2024) and host galaxy prop-\nerties (e.g., Gagliano et al. 2023). In addition to the\nmany relatively nearby supernovae observed by Rubin ,\nrare explosions arising from the first generations of stars\n(e.g., Population III) are targets for high-redshift sur-\nveys such as Euclid (e.g., Moriya et al. 2022) and the\nNancy Grace Roman Space Telescope (e.g., Rose et al.\n2021). The analytic estimates derived in this paper will\nbe usefully applied to constrain the ejecta properties or\nenergy sources for future large samples of SNIIP, or for\nobtaining robust conclusions for individual exceptional\nevents like iPTF14hls.\nThis paper is organized as follows. In Sec. 2 we gen-\neralize existing semi-analytic SNIIP light curve models\nto include the presence of an arbitrary internal heat-\ning source. In Sec. 3 we first consider the case of\na temporally-constant heating source, providing ample\nanalytic estimates which help interpret our numerical\nresults. In Sec. 4 we expand our considerations to\nthe case of a broken-power law heating rate, which we\nshow approximates the behavior of most physical heat-\ning sources, ranging from radioactive56Ni decay to a\ncentral magnetar or accreting engine to CSM interac-\ntion. In Sec. 5 we discuss implications of our results\nand in Sec. 6 we summarize our findings and conclude.\nReaders who are not interested in analytical derivationscan go directly to Sec. 5 and Fig. 12, where our main\nfindings and their implication are discussed.\n2.GENERALIZED POPOV MODEL\nWe generalize the Popov (1993) model to include a\ncentral heating source of arbitrary magnitude, which\nwill in essence act to lengthen the optically-thick pho-\ntospheric phase, i.e. “plateau”, phase (at least for non-\nextremal heating rates). Throughout this paper, we re-\nfer to phases during which the ejecta shell is still opaque\nto electron scattering as the “plateau”, even when the\nlight curve is not flat. Prior to when the ejecta cools suf-\nficiently for hydrogen recombination to set in, its evolu-\ntion follows the well known Arnett model (Arnett 1980,\n1982) with an additional heating source (e.g., Kasen &\nBildsten 2010; Chatzopoulos et al. 2012; Metzger et al.\n2015). The first law of thermodynamics applied to a\none-zone model for the ejecta reads:\ndE\ndt=−PdV\ndt−L+H , (1)\nwhere E,P,V=4π\n3R3,R,L, and Hare the internal\nthermal energy, pressure, volume, radius, radiated lu-\nminosity, and (specified) heating rate, respectively. We\nassume that radiation pressure dominates over gas pres-\nsure, such that E=aT4VandP=E/(3V), respec-\ntively, where ais the radiation constant and Tis the\ninternal temperature. The radiated luminosity follows\nfrom the diffusion approximation:\nL≃4πR2c\n3κρd(aT4)\ndR∼tdynE\nt2\ndiff, (2)\nwhere the dynamical and diffusion timescales are, re-\nspectively, defined by\ntdyn≡R\nv, (3)\ntdiff���\u00123κM\n4πcv\u00131/2\n≃110 day M1/2\n10v−1/2\n6000 .(4)\nHere κ= 0.34 cm2g−1is Thomson opacity for fully ion-\nized solar composition material, cis the speed of light,\nand we have introduced the shorthand notation M=\n10M10M⊙andv= 6000 v6000km s−1for the ejecta\nmass and characteristic velocity, respectively. Eq. (1)\ncan now be written:\ndE\ndt=−E\ntdyn−tdynE\nt2\ndiff+H . (5)\nThe first term in Eq. (1) accounts for the loss of inter-\nnal energy due to PdV work done on the ejecta by itself.\nThis term acts to increase the ejecta kinetic energy Ekin,\naccording to:\ndEkin\ndt=PdV\ndt. (6)Maximal Plateau Duration 3\nFor homologously expanding ejecta of assumed constant\ndensity, ρ=M/V , and velocity v′=v(r/R), we have\nEkin=ZR\n0\u00121\n2ρv′2\u0013\n4πr2dr=3\n10Mv2(7)\n≃2.2×1051ergM10v2\n6000,\nNote that we have neglected any contribution to Ekin\nfrom a high-velocity tail v′≫v, which contributes to\nthe early time shock cooling emission but not to the\nplateau. Eq. (6) thus becomes:\nMdv\ndt=20π\n3R2P . (8)\nEqs. (5) and (8) determine the time evolution of ejecta\nuntil the hydrogen recombination phase begins.\nThe recombination phase starts when the effective\ntemperature of the photosphere decreases to the hydro-\ngen recombination temperature, Ti≃6000 K. The pho-\ntosphere temperature is defined as:\nTeff≡\u0012L\n4πσSBR2\u00131/4\n≃1.1T\nτ1/4, (9)\nwhere σSBis the Stefan-Boltzmann constant, τ=κRρ\nis the Thompson optical depth through the ejecta (i.e.,\nassuming it is fully ionized), and the second equality\nmakes use of Eq. (2).\nAfter recombination begins at t=ti, a sharp recombi-\nnation front forms and begins to recede back through the\nejecta shell. Outside this front, the ejecta is neutral and\ntransparent to radiation. Therefore we regard the pho-\ntosphere radius Rphas coinciding with the recombina-\ntion front, at dimensionless coordinate x≡Rph/R≤1.\nWe retain a one-zone model similar to that described\nearlier to calculate the evolution of fully ionized re-\ngion (within the photosphere) during the recombination\nphase. All quantities retain the same meaning, except\nthe ejecta radius Ris now replaced with Rph=xR.\nFor instance, the volume of the ionized region becomes\nV=4π\n3(xR)3, while the radiated luminosity becomes:\nL≃4π(xR)2c\n3κρaT4\nxR=tdynE\nx2t2\ndiff. (10)\nEquating this to the photosphere luminosity, L=\n4π(xR)2σSBT4\ni, gives an expression analogous to Eq. (9)\nbut with replacements Teff→Tiandτ→xτ. Combined\nwith Eq. (5) this gives the evolution of the recombina-\ntion depth (Dexter & Kasen 2013):\ndx\ndt=−2x\n5tdyn−tdyn\n5t2\ndiffx+H\n5Hcr1\ntdynx3, (11)\nwhere we have defined\nHcr= 4π(vtdiff)2σSBT4\ni (12)\n≃3.0×1043erg s−1M1v6000T4\ni,6000,andTi= 6000 Ti,6000K. This critical heating rate con-\ntrols the recombination time and will be used in the\nanalytic estimates below.\nWe have neglected acceleration of the ejecta during the\nrecombination phase. This is usually justified because\nthe acceleration timescale at recombination ( t≃ti),\nv\ndv/dt\f\f\f\f\nti=v\n5\n4κaT4\ni≃1700 day v6000T−4\ni,6000, (13)\ngreatly exceeds the dynamical timescale,1where we have\nused Eqs. (8), (9) and set Teff=Ti. Acceleration also\nbecomes negligible after radiation begins to diffuse out\nof the ejecta ( t≳tdiff) due to the associated loss of\nthermal pressure.\nIn the following sections, we solve the above equa-\ntions for different central heating sources H(t) and ejecta\nproperties. The latter includes the total mass M, initial\nsize (usually, progenitor star radius) R0, initial internal\nenergy E0, and initial kinetic energy Ekin,0.2Through-\nout, we assume solar abundances and κ= 0.34 cm2g−1\n(though this could readily be generalized to different\nabundances by modifying µandκaccordingly). We\nalso assume that the ejecta internal and kinetic energies\nare initially comparable, E0≃Ekin,0att= 0, with the\ntotal explosion energy equal to their sum. After just a\nfew dynamical times, most of the initial internal energy\nis converted into kinetic energy by PdV work, causing\nthe ejecta to reach a terminal speed v=√\n2v0, where\nv0is the initial velocity calculated from Ekin,0andM\n(Eq. (7)). For Ekin,0= 1051erg and M= 10M⊙, this\ngives v0≃4100 km s−1andv≃5800 km s−1. Except\nwhere noted (e.g., Sec. 3.2), we hereafter express ana-\nlytic estimates in terms of the terminal speed, v.\nBefore describing our results for specific heating\nsources, we review the original Popov estimates (Popov\n1993; Dexter & Kasen 2013; Matsumoto et al. 2016),\nwhich are obtained neglecting any heating or accelera-\ntion of the ejecta after the initial explosion. As shown\nbelow (around Eq. 28), the recombination phase begins\non the timescale\nti≃19 day E1/2\n0,51M−1/2\n10R1/2\n0,500v−1\n6000T−2\ni,6000, (15)\nwhere E0= 1051E0,51erg,R0= 500 R0,500R⊙, and we\nhave implicitly assumed ti≪tdiff. Under these assump-\n1This may break down for low-velocity events such as supernova\nprecursors (e.g., Strotjohann et al. 2021; Matsumoto & Metzger\n2022a). See Tsuna et al. (2024) for an attempt to include the\nacceleration during the recombination phase.\n2Radiation pressure dominates over gas pressure for large explo-\nsion energies obeying:\nE0≳1.3×1048erg (µ/0.60)−1M4/3\n10R−1\n0,500, (14)\nwhere µis the mean molecular weight.4 Matsumoto et al.\ntions, Eq. (11) has the following analytic solution:\nx=\u0012t\nti\u0013−2\n5\"\n1 +1\n7\u0012ti\ntdiff\u00132\n−1\n7\u0012ti\ntdiff\u00132\u0012t\nti\u001314\n5#1\n2\n.\n(16)\nThe plateau duration corresponds to when the ejecta\nfully recombines ( x(tpl,Popov ) = 0), giving:\ntpl,Popov =ti\"\n1 + 7\u0012tdiff\nti\u00132#5/14\n≃75/14t2/7\nit5/7\ndiff(17)\n≃130 day E1/7\n0,51M3/14\n10R1/7\n0,500v−9/14\n6000 T−4/7\ni,6000,\nwhere the second equality on the first line assumes\nti≪tdiff. At the same level of approximation, the to-\ntal radiated energy and average plateau luminosity are\ngiven, respectively, by\nEpl=Ztpl\ntiLdt≃56π\n55σSBT4\niv2t3\ni\u0012ti\n71\n2tdiff\u0013−11\n7\n,(18)\n≃2.8×1049ergE5/7\n0,51M1/14\n10R5/7\n0,500v−3/14\n6000 T8/7\ni,6000,\nLpl≡Epl\ntpl≃73/756\n55πσSBT4\niv2t8/7\nit6/7\ndiff(19)\n≃2.4×1042erg s−1E4/7\n0,51M−1/7\n10R4/7\n0,500v3/7\n6000T12/7\ni,6000,\nwhere we have used tpl,Popov fortpl. These expres-\nsions exhibit slightly different exponents and normal-\nization from those originally derived by Popov (1993),\nwho assumed a self-similar radial temperature profile,\nwhich is not necessarily achieved (e.g., Khatami & Kasen\n2019). These expressions have been compared with the\nresults of radiation hydrodynamic simulations (Kasen\n& Woosley 2009; Sukhbold et al. 2016; Goldberg et al.\n2019), their accuracy confirmed to within a factor of a\nfew.3\n3.CONSTANT HEATING RATE\nWe now solve the equations presented in the previ-\nous section to explore how the supernova plateau phase\nis modified by a temporally-constant heating source of\n3Numerical results in previous works can be summarized as:\nL50=eL50erg s−1E5/6\nSN,51M−1/2\n10R2/3\n0,500, (20)\ntpl=etpldayE−1/6\nSN,51M1/2\n10R1/6\n0,500, (21)\nwhere L50is the bolometric luminosity at 50 days after the ex-\nplosion, and ESN= 1051ESN,51erg the total explosion energy.\nKasen & Woosley (2009) obtained eL50≃1.3×1042andetpl≃122\nwith a different dependence of tpl∝E−1/4\nSN,51; Sukhbold et al.\n(2016) found eL50≃1.8×1042andetpl≃96; while Goldberg\net al. (2019) found eL50≃1.4×1042and a slightly different scal-\ning (see their Eq. 8).arbitrary magnitude. While a constant heating rate\nonly represents a crude approximation to a more phys-\nical time-evolving power source, this simple assumption\nenables us to obtain general physical insights into the\nimpact of heating and to derive useful analytical for-\nmulae. Furthermore, we show later that the constant\nheating case bounds the light curve behavior for more\ncomplex, physically-motivated heating curves, in partic-\nular by defining the maximal plateau duration.\nA given model is specified by the (constant) heat-\ning rate Hand the initial conditions of the explosion,\nas mentioned above. At early times ( t < t i), the\neffective temperature exceeds the recombination value\n(Teff> Ti), and we solve Eq. (5) and (8) until Teff=Tiat\nti. During the recombination phase, we solve Eq. (11) for\nx(t). The evolution is followed until the nebular phase,\nwhich we define as when the optical depth decreases to\nunity: xτ= 1.\nFig. 1 depicts the time evolution of effective temper-\nature, luminosity, and the dimensionless photospheric\nradius for various values of the heating rate H= 1040−\n1046erg s−1, adopting ejecta properties of M= 10 M ⊙,\nR0= 500 R ⊙, and E0=Ekin,0= 1051erg (correspond-\ning to v0≃4100 km s−1andv≃5800 km s−1), typical of\ntype II SNe (e.g., Kasen & Woosley 2009). The results\nare qualitatively summarized as follows: When the heat-\ning rate is sufficiently small ( H≲1041erg s−1), its effect\nis negligible and the result is indistinguishable from the\nzero heating case H= 0. As the heating rate increases,\nthe plateau duration (equivalently in our formulation,\nbeginning of the nebular phase) becomes longer but\nthe initial onset tiof the recombination phase remains\nlargely unchanged up to H≲Hcr≃3×1043erg s−1.\nThe recombination time grows substantially for H≳\nHcr. For yet larger heating rates H≫Hcr, the recombi-\nnation phase terminates earlier and the plateau duration\nshrinks because the ejecta is now significantly acceler-\nated by PdV work. The plateau duration thus achieves\nits maximal value for a given ejecta mass/explosion en-\nergy at the maximum heating rate that does not appre-\nciably impact the ejecta dynamics ( H≃1044erg s−1,\nfor the adopted parameters). In the following, we derive\nanalytical formulae for these timescales and the critical\nheating rates which delineate these regimes.\n3.1. Without Acceleration\nWe first consider cases in which PdV acceleration is\nnegligible following the initial dynamical timescale; this\ngreatly simplifies the analysis and is a good approxima-\ntion for small heating rates. For constant velocity, the\ndynamical timescale is identical to the time since explo-\nsion, tdyn=t(or equivalently R=vt), and Eq. (5) canMaximal Plateau Duration 5\n10000\n3000500070002000030000Teff [K]\nTi=6000 K\n104110421043104410451046L [erg/s]ti (Recombination)\ntpl (End of Plateau)\ntrec\ndiff (LH)\n101\n100x=Rph/R\nH=1040erg/s\n1041\n1042\n1043\n1044\n1045\n1046\n101102103\nTime : t [day]103104vph(=xv) [km/s]\nFigure 1. Time evolution of the effective temperature, bolo-\nmetric luminosity, and location and velocity of the photo-\nsphere (from top to bottom) for different constant heating\nrates as marked. The recombination phase begins when\nTeff=Tiatti(marked as squares). The ejecta enters the\nnebular phase when the optical depth becomes smaller than\nunity at tpl(marked as circles), providing one definition of\nthe plateau duration. After time trec\ndiff(marked as triangles)\nduring the recombination phase, the luminosity starts to\ntrack the internal heating rate, a transition which may also\nbe observationally associated with the end of plateau. The\nadopted ejecta parameters are M= 10 M ⊙,R0= 500 R ⊙,\nE0=Ekin,0= 1051erg (v0≃4100 km s−1).be integrated to obtain:\nE=E0\u0012t0\nt\u0013\ne−t2−t2\n0\n2t2\ndiff\"\n1 +t2\ndiffH\nt0E0 \net2−t2\n0\n2t2\ndiff−1!#\n,\n(22)\nwhere\nt0=R0\nv0≃0.95 day R0,500v−1\n6000, (23)\nis the initial dynamical/expansion timescale and we\nagain relate the initial and terminal ejecta speed, v0=\nv/√\n2. The second term within square brackets in\nEq. (22) represents the effect of the heating source. For\ntypical supernova ejecta properties, the diffusion time\nexceeds the initial expansion timescale, tdiff≫t0; since\ninitially the heating does not impact the ejecta evolu-\ntion, the internal energy declines adiabatically, E∝t−1.\nThis evolution is modified once the deposited energy be-\ncomes comparable to the internal energy, tH≃t0E0/t.\nThis occurs on the timescale:\nth=\u0012t0E0\nH\u00131/2\n≃33 day E1/2\n0,51R1/2\n0,500v−1/2\n6000H−1/2\n43 ,\n(24)\nwhere H= 10xHxerg s−1. When this timescale is\nshorter than the diffusion timescale, which is true for\nlarge heating rates\nH≳t0E0\nt2\ndiff≃9.1×1041erg s−1E0,51M−1\n10R0,500,(25)\nthe time evolution of the internal energy can be summa-\nrized as:\nE≃\n\nt0E0\nt:t < t h,\ntH :th< t < t diff,\nt2\ndiffH\nt:tdiff< t .(26)\nWhen th> tdiff, the middle regime disappears.\nThe effective temperature evolves following Eq. (9),\nTeff≃Ti\u0012H\nHcr\u00131/4\n\n\u0010\nt\nth\u0011−1/2\n:t < t h,\n1 : th< t < t diff,\u0010\nt\ntdiff\u0011−1/2\n:tdiff< t ,\n(27)\nwhere Hcris the critical heating rate introduced earlier\n(Eq. (12)). The recombination time tiis obtained by\nsetting Teff=Tiwith Eq. (27). Noting that the effective6 Matsumoto et al.\ntemperature is constant for th< t < t diff, we obtain\nti=\n\n\u0010\nt0E0\nHcr\u00111/2\n:H≤Hcr,\n\u0010\nt2\ndiffH\nHcr\u00111/2\n:Hcr< H ,(28)\n≃(\n19 day E1/2\n0,51M−1/2\n10R1/2\n0,500v−1\n6000T−2\ni,6000 :H≤Hcr,\n66 day v−1\n6000T−2\ni,6000H1/2\n43 :Hcr< H .\nThis timescale for H < H cris notably identical to that\nin the H= 0 limit (Eq. (15)). Interestingly, the recom-\nbination time is discontinuous at H=Hcrbecause of\nthe static effective temperature for th< t < t diff, thus\naccounting for the sudden jump in the recombination\ntime above H≃Hcr≃3×1043erg s−1.\nFor the recombination phase, while Eq. (11) cannot\nbe solved analytically for H̸= 0, the behavior of the so-\nlution can still be understood quantitatively. The three\nterms on the right hand side of Eq. (11) correspond to\nadiabatic cooling, radiative cooling (photon diffusion),\nand heating, respectively. First consider the limit of\nweak heating H < H cr. In such cases, just after the\nrecombination starts, neither heating or radiative losses\nare important, and the photosphere thus shrinks in time\nasx≃(t/ti)−2/5corresponding to adiabatic evolution\n(Eq. (16)). However, as xdecreases, radiative cooling\nand heating eventually come to compete with, or bal-\nance, respectively, the adiabatic cooling. For parameter\nregimes in which radiative cooling is subdominant, adia-\nbatic cooling balances heating. This balance is achieved\non the timescale\ntrec\nh=\u00122Hcr\nH\u00135/8\nti (29)\n≃59 day E1/2\n0,51M1/8\n10R1/2\n0,500v−3/8\n6000T1/2\ni,6000H−5/8\n43 .\nAt times t > trec\nh, the photosphere stalls at x≃\n(H/2Hcr)1/4until radiative cooling takes over from adi-\nabatic cooling in balancing heating. This occurs after\non the timescale:\ntrec\ndiff=\u00122H\nHcr\u00131/4\ntdiff≃99 day M1/4\n10v−3/4\n6000T−2\ni,6000H1/4\n43.\n(30)\nAfter this transition t > trec\ndiff, the photosphere continues\nto shrink again, this time as x≃(H/H cr)1/2tdiff/t. In\nsummary, for H < H crandti< th< tdiff, the photo-\nsphere location evolves as:\nx≃\n\n\u0010\nt\nti\u0011−2/5\n:t < trec\nh,\n\u0010\nH\n2Hcr\u00111/4\n:trec\nh< t < trec\ndiff,\n\u0010\nH\nHcr\u00111/2tdiff\nt:trec\ndiff< t .(31)The superscripts “rec” in Eqs. (29) and (30) denote\nheating and diffusion timescales during the recombina-\ntion phase, respectively. After the diffusion timescale,\nthe escaping luminosity roughly tracks the heating rate\nL≃H, as illustrated by triangles in Fig. 1. While we\nhave defined the end of plateau as when the ejecta be-\ncomes optically thin, one could instead define this by\nthe change in light curve shape near trec\ndiff(e.g. the loca-\ntion of the triangles for the H= 1041and 1042erg s−1\ncases in Fig. 1). We discuss later in this section how our\nresults changes if we instead identify trec\ndiffas the plateau\nduration.\nFor the opposite regime, radiative cooling dominates\nadiabatic cooling even before either balances heating. In\nthese cases the photosphere evolution follows Eq. (16)\nuntil heating balances cooling at t≃tpl,Popov , after\nwhich xdecreases following the final case in Eq. (31).\nSuch an evolutionary sequence is realized for low heat-\ning rates, below a critical value:\nH≲0.66\u0012ti\ntdiff\u00138/7\nHcr (32)\n≃2.7×1042erg s−1E4/7\n0,51M−1/7\n10R4/7\n0,500v3/7\n6000T12/7\ni,6000,\nobtained by equating the plateau duration neglecting\nheating ( H= 0; Eq. (17)) with the timescale (29). For\nlarge heating rates H > H cr, the first regime in Eq. (31)\nvanishes because heating has already become important\neven prior to the recombination phase.\nIn either case, as long as the heating is present, the\nphotosphere at late times t > trec\ndiffortpl,Popov begins to\nshrink as x∝t−1(following the final regime of Eq. (31))\nand never reaches x= 0 as in the no-heating case H= 0.\nAgain defining the plateau duration at when the ejecta\nbecomes optically thin (or enters the nebular phase):\nxτ= 1, in the presence of a heating source we find:\ntpl≃\u0010v\nc\u00111/6\u0012H\nHcr\u00131/6\ntthin=\u001232κ2M2H\n26π3σSBT4\niv6\u00131/6\n(33)\n≃340 day M1/3\n10v−1\n6000T−2/3\ni,6000H1/6\n43,\nwhere\ntthin=\u00123κM\n4πv2\u00131/2\n≃780 day M1/2\n10v−1\n6000, (34)\nis the time the ejecta becomes optically thin assuming\nfull ionization. Equating Eq. (33) with Eq. (17) gives\nthe heating rate above which the plateau duration is\nsignificantly boosted relative to the zero-heating case:\nHmin=\u0010c\nv\u0011\u0012tpl,Popov\ntthin\u00136\nHcr (35)\n≃3.9×1040erg s−1E6/7\n0,51M−5/7\n10R6/7\n0,500v15/7\n6000T4/7\ni,6000.Maximal Plateau Duration 7\nNote that while heating can extend the plateau dura-\ntion in accordance with Eq. (33), the duration is ulti-\nmately limited by when the ejecta becomes optically-\nthin/nebular (Eq. (34)). The latter limit is achieved\nfor very powerful heating sources, which keep the ejecta\nfully ionized to late times. However, as we shall discuss,\nsufficiently powerful energy sources also act to accelerate\nthe ejecta, hastening the nebular phase (Eq. (34)).\n3.2. With Acceleration\nWe now consider the effects of ejecta acceleration,\nwhich modifies the characteristic timescales ( tdiffand\ntpl) which depend on v. Because the internal energy\nevolution is not modified from Eq. (26) until the pho-\nton diffusion time, Eq. (8) can be rewritten using the\napproximation R≃vt:\ndv\ndt=5E\n3MR≃(\n5t0E0\n3Mvt2:t < t h,\n5H\n3Mv:th< t < tacc\ndiff.(36)\nHere tacc\ndiffis the diffusion timescale, now including the\neffects of acceleration, which in general will differ from\nEq. (4) as described below. Eq. (36) can be integrated\nto obtain:\nv≃\n\nq\nv2\n0+v2\n0\u0000\n1−t0\nt\u0001\n:t < t h,r\nv2\n0+v2\n0\u0010\n1−t0\nth\u0011\n+10H(t−th)\n3M:th< t < tacc\ndiff,\n(37)\nwhere v0is the initial ejecta velocity. As discussed ear-\nlier, for t < t h, the velocity increases and asymptotes\ntov=√\n2v0at the expense of the initial internal en-\nergy, regardless of any heating source. For t > t h, the\nacceleration by PdV work becomes significant on the\ntimescale\ntacc=3Mv2\n0\n5H≃22 day M10v2\n0,4000H−1\n45,(38)\nas determined by the condition that the injected energy\nbe comparable to the ejecta kinetic energy ( taccH≃\nEkin). Here v0= 4000 v0,4000km s−1. Therefore, the\nvelocity evolution until t < tacc\ndiffcan be summarized as:\nv≃√\n2v0\n\n1 : t < t acc,\u0010\nt\ntacc\u00111/2\n:tacc< t < tacc\ndiff.(39)\nThe diffusion timescale is obtained by substituting\nEq. (39) into Eq. (4) and setting t=tacc\ndiff:\ntacc\ndiff=\u001227κ2M3\n160π2c2H\u00131/5\n≃82 day M3/5\n10H−1/5\n45 .(40)\nWe find that after radiative cooling becomes important\nfort > tacc\ndiff, the internal energy starts to decline again,following Eq. (26). The equation of motion then takes\nthe form dv/dt ∝1/(vt)2and gives a similar solution to\nEq. (37) for t < t h. We can thus regard that acceleration\neffectively ceases at tacc\ndiff. The maximal velocity attained\nforth< t < tacc\ndiffis given by Eqs. (39) and (40):\nvmax≃\u001225κH2\n3πcM\u00131/5\n≃11000 km s−1M−1/5\n10H2/5\n45.\n(41)\nThe corresponding effective temperature evolution is ob-\ntained by substituting Eqs. (39) or (41) into Eq. (27):\nTeff=\n\n\u0010\nc2H\n30κ2σ2\nSBMt\u00111/8\n:tacc< t < tacc\ndiff,\n\u0010\n9c2M2H\n21054π3σ5\nSBκ2t10\u00111/20\n:tacc\ndiff< t .\n(42)\nDuring the acceleration phase, the effective temperature\nremains almost constant and recombination starts after\ntacc\ndiff, as in the constant velocity case.4This timescale is\nobtained by equating Eq. (42) with Ti:\ntacc\ni=\u00129c2M2H\n21054π3σ5\nSBκ2T20\ni\u00131/10\n(43)\n≃350 day M1/5\n10T−1/2\ni,6000H1/10\n45.\nOnce recombination begins at tacc\ni, the following evo-\nlution is identical to the constant velocity case. In par-\nticular, the plateau duration is obtained by substituting\nthe maximal velocity, Eq. (41), into Eq. (33):\ntacc\npl=\u0012316κ4c6M16\n230512π9σ5\nSBT20\niH7\u00131/30\n(44)\n≃400 day M8/15\n10T−2/3\ni,6000H−7/30\n45 ,\nContrary to the constant-velocity case, the plateau du-\nration now shrinks with increasing Has a result of the\nhigher ejecta expansion speed.\nThe maximal plateau duration is thus obtained by the\nmaximal heating rate which does not appreciably affect\nthe ejecta dynamics. Since acceleration of the ejecta\nceases on the diffusion time, we can obtain this critical\nheating rate by equating taccwith tacc\ndiff(Eq. (40)):\nHpl,max=\u00123πcMv5\n25κ\u00131/2\n(45)\n≃2.3×1044erg s−1M1/2\n10v5/2\n6000,\n4Recombination occurs during the acceleration phase ( tacc< t <\ntacc\ndiff) for\nH≲ \n3452σ5\nSBκ6M4T20\ni\nπc6!1/3\n≃7.8×1042erg s−1M4/3\n10T20/3\ni,6000.\nSuch small heating rates do not significantly impact the ejecta’s\ndynamics.8 Matsumoto et al.\nwhich when substituted into Eq. (33) gives the corre-\nsponding maximal plateau duration:\ntpl,max=\u001235κ3cM5\n21252π5σ2\nSBT8\niv7\u00131/12\n(46)\n≃570 day M5/12\n10v−7/12\n6000 T−2/3\ni,6000.\nAbove we have assumed that the acceleration termi-\nnates on the diffusion time assuming fully ionized ejecta;\nhowever, this transition can occur earlier if the ejecta\nrecombines first. Because recombination occurs when\nTeff=Tion, the maximal heating rate in this case is Hcr\n(Eq. (12)). When the condition\nEkin< H crtdiff, (47)\nis satisfied, the critical heating rate obeys Hcr>\nHpl,max, resulting in a maximal plateau duration:\ntpl,max=\u001233κ3M3\n26π3cv5\u00131/6\n≃410 day M1/2\n10v−5/6\n6000 .(48)\nFor very large heating rates, the ejecta bypasses the re-\ncombination phase entirely and proceeds to become op-\ntically thin while still fully ionized. By equating Eq. (44)\nwith Eq. (43), we obtain the critical heating rate above\nwhich the recombination phase vanishes:\nHthin=\u0010c\nv\u0011\nHcr= 3κσSBT4\niM (49)\n≃1.5×1045erg s−1M10T4\ni,6000.\nThe nebular phase in this case begins at\ntacc\nthin=\u001237κ3c2M7\n21054π3H4\u00131/10\n≃430 day M7/10\n10H−2/5\n45 ,\n(50)\nobtained by substituting Eq. (41) into Eq. (34).\nFig. 2 illustrates the key timescales as a function of\nthe heating rate. Black dotted and solid curves show\nthe recombination time tiand plateau duration tpl, re-\nspectively, as obtained by solving the equations directly\nas in Fig. 1. Their behavior agrees with our analytical\nformulae, supporting the above arguments. A dotted\ncurve denotes the recombination time tiat which the ef-\nfective temperature equals the recombination tempera-\nture. For small heating rates H < H cr≃3×1043erg s−1\n(Eq. (12)), this timescale is independent of Hand iden-\ntical to the Popov formula (Eqs. (15), (28)). For heat-\ning rates approaching the critical value, H≃Hcr, the\nrecombination time increases rapidly (representing the\ngap in tiin Eq. (28)), because heating begins to af-\nfect the thermal evolution for t≳th(Eq. (24)), which\nsignificantly delays the recombination. Once heating\nstarts to affect the dynamics of the ejecta as well for\ntacc< t < tacc\ndiff(Eqs. 38 and 40), the recombination time\nincreases more gradually, as tacc\ni∝H1/10(Eq. (43)).\n1040104110421043104410451046\nHeating rate : H [erg/s]101102103Time : t [day]Nebular (Optically Thin)\nRecombination\n(Teff=Ti)\nTeff>Titacc(tH>Ekin)taccdiff\nth(tH>E)10 M, 500 R, 1051 erg\nFigure 2. Key timescales as a function of the assumed con-\nstant ejecta heating rate. Dotted and solid curves denote\nthe recombination time tiand plateau duration tpl, respec-\ntively. Light blue, orange, and gray shaded regions represent\nthe photospheric phase with Teff> T i, recombination phase\n(Teff=Ti), and nebular phase (optically thin), respectively.\nThe dash-dotted curve shows trec\ndiff, after which the luminosity\nbegins to track the input heating rate. Dashed lines represent\nthree characteristic timescales, on which the heating appre-\nciably alters the thermal ( th, Eq. (24)) and dynamical ( tacc,\nEq. (38)) evolution of the ejecta, as well as the diffusion time\nfor maximally-accelerated ejecta ( tacc\ndiff, Eq. (40)).\nA solid curve in Fig. 2 depicts the plateau time tpl\nat which the ejecta becomes optically thin and enters\nthe nebular phase. For heating rates H≳Hmin≃\n4×1040erg s−1(Eq. (35)), the plateau duration is pro-\nlonged from the Popov formula (Eq. (17)) as tpl∝H1/6\n(Eq. (28)), where Hminis smaller than the critical rate\nHcrbecause the heating plays a role only during the\nrecombination phase. When the acceleration becomes\nsignificant for higher heating rates, the plateau duration\nshrinks as tacc\npl∝H−7/30(Eq. (44)) upon further heating\ndue to the higher ejecta velocity. The maximal allowed\nplateau duration for a given ejecta mass (Eq. (46)) is\nthus obtained for the maximal heating rate (Eq. (45))\nwhich does not appreciably accelerate the ejecta. For\nthe highest heating rates H > H thin≃2×1045erg s−1\n(Eq. (49)), the ejecta remains ionized throughout its en-\ntire evolution and the recombination phase disappears.\nThe nebular phase also begins earlier for largest hearing\nrate as tacc\nthin∝H−2/5(Eq. (50)) because of the higher\nvelocity.\nAs discussed around Eq. (40), the plateau duration\ncan observationally be identified as when the luminos-\nity drops rapidly and begins to track the ejecta heating\nrate (e.g., Valenti et al. 2016). In our framework, this\ntransition happens at trec\ndiff, as indicated by a dash-dottedMaximal Plateau Duration 9\n100 200 300 400 600 800\nDuration : tpl [day]10421043104410451046Lpl [erg/s]t2/5\nt6t41044H=1045erg/s1046\n10421041104010 M, 500 R, 1051 erg\n10080 200 300 400 600 8001000\nDuration : tpl [day]10421043104410451046Lpl [erg/s]20M\n 5M\n1000R\n100R\n3×1051erg\n3×1050ergt30/7\nFigure 3. (Top) Trajectory of plateau duration and lu-\nminosity for different constant heating rates (increasing in\norder-of-magnitude increments from blue to red) for an\ncanonical explosion of ejecta mass 10 M⊙, energy 1051erg,\nand 500 R⊙progenitor radius. Circle and triangles along\nthe curve shows the results corresponding to calculations in\nFig. 1 with the same color scheme. ( Bottom ) The same as\nthe top panel, but for different ejecta properties (total mass,\ninitial energy, and progenitor radius) as marked.\ncurve in Fig. 2. By definition this transition occurs be-\nforetpland thus provides a conservative lower limit on\nthe plateau duration, while it does not change our quan-\ntitative results significantly.\nFig. 3 shows the “trajectory” of plateau duration and\nluminosity, achieved for different heating rates, in the\nduration-luminosity phase space diagram of optical tran-\nsients (Kasliwal 2011). As with the Popov formulae\n(Eq. (18)), we define the plateau luminosity by divid-\ning the total radiated energy up to the nebular phase by\nthe plateau duration (when the ejecta become opticallythin):\nLpl≡Erad\ntplandErad=Ztpl\n0Ldt . (51)\nForH≲Hthin, the total radiated energy is dominated\nby the recombination phase. Initially, as the heating rate\nincreases, only the plateau duration increases (Fig. 1,\nEq. (33)). This incremental change reflects as only\na modest flattening of the light curve after the main\nplateau for trec\ndiff< t < t pl, which does not contribute\nto the radiated energy, causing the plateau luminosity\nto remain unchanged or to slightly decrease. As Hin-\ncreases further, both the luminosity and duration in-\ncrease up to the critical heating rate which gives the\nmaximal plateau duration. In this regime, the radiated\nenergy is dominated by the recombination phase:\nErad≃Ztpl\ntiLdt= 4πv2σSBT4\niZtpl\nti(xt)2dt≃Htpl,\n(52)\nwhere the last equality makes use of x≃\n(H/H cr)1/2tdiff/t(Eq. (31)) and assumes tpl≫ti.\nPerhaps surprisingly, this expression also provides a\nreasonable estimate of the the plateau luminosity even\nfor larger heating rates, because the supernova luminos-\nity closely tracks Hafter the diffusion timescale (see\nFig. 1). By solving Eq. (33), (44), and (50) for H, we\nobtain the following scaling relations:\nLpl≃\n\n3.9×1042erg s−1E3\nkin,0,51M−5\n10T4\ni,6000\u0010\ntpl\n300 day\u00116\n:H < H pl,max,\n3.5×1045erg s−1M16/7\n10T−20/7\ni,6000\u0010\ntpl\n300 day\u0011−30/7\n:Hpl,max< H < H thin,\n2.5×1045erg s−1M7/4\n10\u0010\ntpl\n300 day\u0011−5/2\n:Hthin< H ,\n(53)\nwhere we have again replaced tacc\nplin the middle and\ntacc\nthinin the bottom regimes with tpl, retaining the same\nnotation. We have also substitute vfrom Eq. (7).\nThe top panel of Fig. 3 depicts the Lpl−tpltrajec-\ntories of our light curve models from Fig. 1, illustrating\nhow our analytic scaling relations (Eq. (53)) can reason-\nably reproduce the results of our full numerical calcula-\ntions. For the specific ejecta properties chosen, the mid-\ndle regime of Eq. (53) does not clearly appear, because\nthe two characteristic heating rates, Hpl,max∼Hthin,\nare not well separated (though this regime appears more\nclearly for other parameter choices; see below). A dash-\ndotted curve shows how the trajectory changes if one\nwere to instead take trec\ndiff(the time at which the light\ncurve starts to track the central heating source) as the10 Matsumoto et al.\ndefinition of plateau duration. This alternative defini-\ntion predicts a slightly different scaling relationship:\nLpl≃2.3×1043erg s−1E3/2\nkin,0,51M−5/2\n10T4\ni,6000\u0012tpl\n300 day\u00134\n.\nThe bottom panel of Fig. 3 shows similar Lpl−tpl\ntrajectories, but for different ejecta and explosion prop-\nerties. While the shape of the trajectory does not change\nsignificantly (except for the middle regime in Eq. (53)\nappearing in some cases), it shifts side to side, depend-\ning on the kinetic energy and ejecta mass. These general\nbehavior can be simply understood by the longer dif-\nfusion timescale/plateau duration that arises for more\nmassive and/or less-energetic/slower ejecta. Consistent\nwith Eq. (53), the trajectory depends only weakly on\nthe initial progenitor radius R0, except in the weakly\nheated regime H < H mincorresponding to the standard\nPopov limit.\n4.PHYSICALLY MOTIVATED HEATING\nSOURCES\nWe now apply our model to several physically-\nmotivated ejecta heating sources: radioactive56Ni/56Co\ndecay; rotational or accretion power from a central com-\npact object (neutron star or black hole); and shock in-\nteraction with circumstellar gas.\nThe evolution of the heating rate can in most of these\ncases be approximately described as a constant heating\nrate up to some break time tbr, followed by a power-law\ndecay, viz.\nH(t) =\n\neH :t < t br,\neH\u0010\nt\ntbr\u0011−α\n:tbr< t .(54)\nThe normalization eH, break-time tbrand power-law ex-\nponent α > 0 depend on the energy source in ques-\ntion (see Fig. 4). In Appendix A we rederive several of\nthe analytic expressions from the constant-heating case\n(Sec. 3), for this more complex heating evolution.\n4.1. Radioactive decay heating\nIt is well known that heating from radioactive decay\nextends the plateau duration in SNIIP (e.g., Young 2004;\nKasen & Woosley 2009; Bersten et al. 2011; Nakar et al.\n2016; Goldberg et al. 2019; Kozyreva et al. 2019). In\nparticular, the heating rate of the56Ni→56Co→56Fe de-\ncay chain can be written (Nadyozhin 1994)\nH(t) =MNih\n(εNi−εCo)e−t/tNi+εCoe−t/tCoi\n,(55)\nwhere MNiis the nickel mass, εNi = 3 .9×\n1010erg g−1s−1,εCo= 6.8×109erg g−1s−1,tNi=\n8.8 day, and tCo= 111 .3 day. The total energy released\n101\n100101102\nTime : t [day]1041104210431044H [erg/s]H\nt\ntbr\nNickel (MNi=1M)\nFigure 4. The physically-motivated ejecta heating sources\npresented in this paper can in most cases be approximated as\na constant heating rate eHfollowed after a break at time tbr\nby a power-law decay ∝t−αof the form (Eq. (54)), shown\nhere schematically with a black solid line. A gray dashed\nline denotes such a form fit to the56Ni→56Co→56Fe decay\nchain, the latter shown as dark(light) solid blue lines in a\ncase with(without) the gamma-ray energy deposition factor\nfdep.\nby radioactive decay,\nENi≡Z∞\n0H(t)dt≃MNi[tNi(εNi−εCo) +tCoεCo],\n≃1.8×1048erg\u0012MNi\n10−2M⊙\u0013\n, (56)\nis typically much smaller than the ejecta kinetic energy\nfor values MNi∼10−3−0.1M⊙characteristic of SNIIP\n(e.g., Hamuy 2003; Anderson et al. 2014; Valenti et al.\n2016; M¨ uller et al. 2017; Anderson 2019; Martinez et al.\n2022b). Because ejecta acceleration by radioactive de-\ncay energy is generally negligible, it is safe to assume\nconstant ejecta speed in analytic estimates.\nMost of the energy released by the56Ni decay chain is\ncarried by gamma-rays, which at sufficiently late times\nleak out of the ejecta without depositing energy. This\nsuppression effect on the heating rate (Eq. (55)) can\nbe included by a multiplying deposition factor (Jeffery\n1999):\nfdep(t) = 1−e−xτγ, (57)\nwhere τγ=κγρRis the gamma-ray optical depth\nandκγ= 0.03 cm2g−1the associated opacity (Swartz\net al. 1995). After the ejecta becomes optically thin to\ngamma-rays at ≃(κγ/κ)1/2tthin≃230 days, the deposi-\ntion efficiency steeply declines fdep(t)∝t−2.\nFig. 5 depicts the time evolution of the same quantities\n(luminosity, photosphere radius, photosphere velocity)Maximal Plateau Duration 11\n1040104110421043L [erg/s]\nti (Recombination)\ntpl (End of Plateau)\n102\n101\n100x=Rph/R\nNickel Free\nMNi=0.01M\n0.03\n0.1\n0.3\n1\n0 50 100 150 200\nTime : t [day]103104vph(=xv) [km/s]\nFigure 5. Time evolution of the bolometric luminosity, lo-\ncation and velocity of the photosphere (similar to Fig. 1) for\nmodels incorporating the exact56Ni decay chain heating, for\ndifferent values of the56Ni mass as marked.\n103\n102\n101\n100101\nNickel Mass : MNi [M]\n101102Time : t [day]Nebular (Optically Thin)\nRecombination\n(Teff=Tion)\nTeff>TionTypical MNi for type IIPKW09\nG19\nS1610 M, 500 R, 1051 erg\nFigure 6. The same as Fig. 2, but for56Ni decay chain heat-\ning. Thin solid lines show analytic estimates for the plateau\nduration from the literature (Kasen & Woosley 2009; Gold-\nberg et al. 2019; Sukhbold et al. 2016, from top to bottom).\nA double-headed arrow denotes the typical range of56Ni\nmasses inferred for SNIIP, MNi≃10−3−0.1 M⊙.shown in Fig. 1 for the time-dependent H(t) and fdep(t)\ndescribed by Eqs. (55) and (57). As expected, the re-\ncombination phase grows longer for larger56Ni masses.\nHowever, different from the case of constant heating,\nthe ejecta becomes optically thin prior to when the lu-\nminosity begins to track the input heating rate (i.e., the\ntimescale trec\ndiffis not reached) because of the sharp drop-\noff in the heating rate at t≳100 days (Fig. 4).\nFig. 6 shows the key timescales from Fig. 2 but now as\na function of56Ni mass instead of generic heating rate.\nAlthough we consider nickel masses which range up to\nthe total ejecta mass ( M= 10 M ⊙), such extreme yields\nof radioactive material are obviously unphysical, except\nperhaps in some PISNe. For nickel masses typical of\nSNIIP, the onset time of the recombination phase tiis\nessentially unaffected by radioactive heating. Though\nstill relatively modest, lengthening of the plateau dura-\ntion with increasing56Ni mass, is more readily apparent.\nThe above results obtained employing the exact heat-\ning rate (Eq. (55)) can also be understood quantitatively\nwithin our analytical framework, following Appendix A.\nFig. 4 shows that the heating rate of the56Ni decay\nchain at t≲100 days is well-approximated as a broken\npower-law function of the form Eq. (54), with parame-\nters:\neH=MNiεNi≃7.8×1041erg s−1\u0012MNi\n10−2M⊙\u0013\n,\ntbr= 3.3 day ,\nα= 0.77.\nNeglecting the energy deposition factor, our analytic es-\ntimate for the plateau duration for these parameters fol-\nlows from Eq. (A8),\ntpl≃200 day\u0012MNi\n0.1 M⊙\u00130.15\nE−0.44\n0,51M0.74\n10, (58)\nwhere we have omitted the dependence on Ti, and elim-\ninated the v−dependence using Eq. (7) and by equating\nthe initial internal and kinetic energies E0=Ekin,0. By\nequating Eq. (58) with Eq. (17), we obtain the minimal\nnickel mass to modify the plateau duration:\nMNi,min≃8.1×10−3M⊙R0.97\n0,500E1.8\n0,51M−1.4\n10.(59)\nWe now compare above results with previous works.\nKasen & Woosley (2009) present a heuristic argument\nfor how to include56Ni heating as a correction to the\nanalytic plateau duration of Popov (1993), described by:\ntpl=tpl,Popov×\u0014\n1 +Cf\u0012MNi\nM⊙\u0013\nE−1/2\nSN,51M−1/2\n10R−1\n0,500\u00151/6\n,\n(60)\nwhere ESN(=E0+Ekin,0) is the total explosion energy.\nThe constant Cfis sensitive to the ejecta density profile12 Matsumoto et al.\nand distribution of56Ni in the ejecta, and can be cali-\nbrated by radiation transport simulations, with Kasen &\nWoosley (2009) finding Cf≃21, Sukhbold et al. (2016)\nfinding ≃50, and Goldberg et al. (2019) finding ≃87.\nEq. (60) is shown as thin solid curves in Fig. 6 for differ-\nent values of Cf. Here, the nickel-free plateau duration\ntpl,Popov is not given by Eq. (17) but rather is taken di-\nrectly from each reference (see the footnote at the end of\nSec. 2). Our result (thick solid curve) is broadly consis-\ntent with these numerically-calibrated analytic expres-\nsions.\nWhile our results and previous studies show reason-\nable agreement, there are differences between them.\nFrom Eq. (60), the minimum nickel mass required to\nimpact the plateau duration can estimated by equating\nthe second term in the square bracket with unity:\nMNi,min≃4.8×10−2M⊙\u0012Cf\n21\u0013−1\nE1/2\nSN,51M1/2\n10R500.\n(61)\nFor nickel masses larger than this value (which corre-\nsponds to a 21/6≃12% increase in tplversus the Ni-free\ncase), the plateau duration is considerably increased,\nnow scaling with the ejecta properties according to:\ntpl∝M1/6\nNiE−1/4\nSNM5/12≃M0.17\nNiE−0.25\nSNM0.42,(62)\nwhere we used Eq. (21) for tpl,Popov . The exponents on\nthe ejecta mass and explosion energy here differ from\nour analytical expressions in Eqs. (58) and (59), with\nin particular the dependence of MNi,minonMchanging\nsign. This disagreement results from the different ways\nthat56Ni heating is treated in our formula versus that\nof Kasen & Woosley (2009). We define tplbyxτ= 1,\nand account for56Ni heating during the plateau phase,\nwhich delays the recession of the recombination front\nx(t). By contrast, Kasen & Woosley (2009) consider the\nheat from radioactive decay to boost the internal en-\nergy such that heating effectively occurs immediately at\ntNiandtCo(see their Eq. (9)). Therefore in our for-\nmulation increasing Mslows the intrinsic evolution of\nx, thus allowing a lower MNito significantly increase\ntpl> tpl,Popov . In the Kasen & Woosley (2009) sce-\nnario, increasing Mboosts the internal energy available\nat a given t(> t NiortCo) due to the smaller adia-\nbatic losses experienced for lower expansion velocity (at\nfixed ESN), which thus requires higher MNito gener-\nate enough heating to boost tpl> tpl,Popov . We remark\nthat the exponent of Min our MNi,mindepends on αas\nMNi,min∝M5(3α−10)\n28.\n4.2. Magnetar\nAnother frequently invoked central energy source is\nthe spin-down energy of a newly-formed, rapidly spin-\nning magnetized neutron star (“magnetar”; e.g., Maeda\net al. 2007; Kasen & Bildsten 2010; Woosley 2010; Chat-\nzopoulos et al. 2012; Metzger et al. 2015; Sukhbold &Thompson 2017). Assuming the Poynting luminosity of\nthe magnetar wind, and the high-energy radiation it gen-\nerates, to be thermalized within the ejecta (likely a good\napproximation at least at early times; Vurm & Metzger\n2021), the heating rate evolution will follow the stan-\ndard magnetic dipole spin-down rate (e.g., Spitkovsky\n2006). The latter can be expressed in the form:\nH(t) =Erot\ntsd\u0012\n1 +t\ntsd\u0013−2\n, (63)\nwhere Erotandtsdare the magnetar’s initial rotational\nenergy and spin-down timescale, respectively. These\nquantities are roughly related to the initial spin period\nPiand surface magnetic field strength Bby\nPi≃\u00122π2I\nErot\u00131/2\n≃1.4 msE−1/2\nrot,52, (64)\nB≃\u0012c3I2\n4R6\nNSErottsd\u00131/2\n≃8.8×1013GE−1/2\nrot,52\u0012tsd\nday\u0013−1/2\n,\n(65)\nwhere IandRNSare the magnetar’s moment of inertia\nand radius, respectively, and we adopt I= 1045g cm2\nandRNS= 10 km. However, we neglect this correspon-\ndence for simplicity, instead referring directly to tsdand\nErot. One hard constraint is that a magnetar cannot\nspin faster than its centrifugal break-up rate, which lim-\nitsErot≲1053erg (Metzger et al. 2015). Additionally,\ndifferent from heating due to radioactive decay or CSM\ninteraction, energy input from the magnetar can appre-\nciably accelerate the supernova ejecta (e.g., Kasen et al.\n2016; Suzuki & Maeda 2021).\nFig. 7 shows the plateau duration and luminosity for\na range of values Erot= 1048−1053erg and tsd=\n10−3−104day, corresponding to physically allowed birth\nspin periods and magnetic field strengths. At a fixed\nspin down time, the plateau luminosity increases mono-\ntonically with Erot, while the plateau duration initially\ngrows but then turns over and decreases for Erot≳\nEkin,0= 1051erg once acceleration of the ejecta becomes\nsignificant.\nContours of fixed tsdaccumulate in the two extreme\nlimits tsd→0 and∞. For spin down times much shorter\nthan the initial dynamical time tsd≲t0∼1 day, heat-\ning of the ejecta is effectively instantaneous. This leads\nto acceleration of the ejecta at the expense of the de-\nposited energy for t≳t0, with the subsequent evolu-\ntion practically described by replacing E0→Erotand\nv→p\n10Erot/3Min the Popov formula. Because for\nearly energy injection the ejecta loses memory of when\nenergy is deposited, the contours becomes independent\noftsdfor small tsd≲10−2day.\nIn the opposite limit tsd→ ∞ , the contours asymptote\nto the constant heating case. For heating times longer\nthan the plateau duration tsd≳tpl,Popov∼102day, the\nmagnetar provides what is effectively a constant heatingMaximal Plateau Duration 13\n100 60 80 200 300 400 500 700\nDuration : tpl [day]1042104310441045Lpl [erg/s]\n103\n102\n0.11tsd=10day102\n103\n104\n1053\nErot=1052erg\n1051\n105010 M, 500 R, 1051 erg\nFigure 7. The same as Fig. 3 but for heating by magne-\ntar spin-down. Black solid (dashed) curves show contours\nfor fixed spin down timescale, tsd(and fixed magnetar’s ro-\ntation energy, Erot). A red thick curve shows the constant\nheating case (see Fig. 3), which bounds the high- tpledge of\nthe parameter space.\nsource during the plateau. A red curve shows the con-\nstant heating rate limit (see Fig. 3), which bounds the\nallowed plateau duration in the magnetar model.\nMost previous applications of magnetar engines are\nto hydrogen-poor superluminous supernovae; however,\na few works have explored the impact of a magnetar\non hydrogen-rich supernova light curves using numeri-\ncal calculations (Sukhbold & Thompson 2017; Dessart\n& Audit 2018; Orellana et al. 2018). While these works\nadopt different definitions for the plateau duration and\nluminosity, precluding a quantitative comparison, their\nfindings are broadly compatible with the simpler ana-\nlytic estimates presented here.\n4.3. Accreting Compact Object\nAnother potential heating source of the supernova\nejecta is a relativistic jet5or accretion disk outflow,\npowered by mass accretion on the central black hole or\nneutron star (e.g., Dexter & Kasen 2013; Moriya et al.\n2018). The rate of fall-back accretion is expected to de-\ncline at late times as ˙Mfb∝t−5/3(Michel 1988; Cheva-\nlier 1989), following an initial phase defined by the radial\nstructure of the progenitor star (Dexter & Kasen 2013;\nPerna et al. 2014; Moriya et al. 2019), particularly its ra-\ndius (e.g., Perna et al. 2018). The manner in which the\n5For red supergiants, their massive thick envelopes prevent a jet\nfrom breaking out of the stellar surface and producing high-\nenergy emission akin to typical long gamma-ray bursts (e.g.,\nMatzner 2003; Matsumoto et al. 2015 but see Quataert & Kasen\n2012 for the case of very-long duration events).mass fall-back is processed by the central accretion disk\nor its outflows, can also modify the accretion rate reach-\ning the central compact object and hence the heating\nrate (e.g., Metzger et al. 2008). We encapsulate these\nuncertainties by adopting a broken power-law heating\nrate, similar to the magnetar case:\nH(t) =(αfb−1)Eacc\ntfb\u0012\n1 +t\ntfb\u0013−αfb\n, (66)\nwhere the total available energy Eaccnow scales with\nthe accreted mass Maccfor some assumed efficiency η=\nEacc/Maccc2, and tfbis the characteristic fallback time.\nThe latter is generally expected to scale with the fall-\nback time of the stellar envelope,\ntfb∼1√Gρ⋆≃130 day\u0012M⋆\n10 M⊙\u0013−1/2\u0012R⋆\n500 R ⊙\u00133/2\n,\n(67)\nwhere ρ⋆≡3M⋆/4πR3\n⋆is the mean density of the star\nof mass M⋆and radius R⋆. We consider the post-\nbreak exponent αfb, as a free parameter. The αfb−1\nprefactor in Eq. (66) follows from the normalization\nEacc=R∞\n0H(t)dt.\nFig. 8 depicts the plateau duration and luminosity\nfor different αfb,Eacc, and tfb, similar to the format of\nFig. 7. The colors of each shaded region correspond to\nthe range of tplandLplfor different ranges of αfb= 4/3\nto 4 as marked. The contours follow a similar shape to\nthat of the magnetar heating (corresponding to αfb= 2).\nBlack solid and dashed curves represent the contours for\nfixed Eaccandtfbfor the special case αfb= 5/3 (corre-\nsponding to a canonical fallback rate). For smaller expo-\nnents, the heating rate declines more slowly, increasing\nthe plateau luminosity. As in the case of magnetar heat-\ning, the plateau duration is maximized for longer fall-\nback timescales, but the attainable region is bounded by\nthe limit of a constant heating rate.\n4.4. CSM Shock Interaction\nAs a final heating source, we consider shock interac-\ntion between the supernova ejecta and a circum-stellar\nmedium (CSM). While most of the modeling literature\nfocuses on CSM which distributed spherically symmet-\nrically around the explosion site (e.g., Khatami & Kasen\n2023), this geometry confines effects of shock heating to\nthe outermost layers of the ejecta. While shock heating\ncan in that case contribute to the early-time supernova\nlight curve by impacting the initial thermal energy im-\nparted to these outer layers, it does not provide a sus-\ntained heating source embedded behind the photosphere\nwhich would affect the duration of the plateau and later-\ntime SN evolution. We are thus motivated to consider\nthe case of aspherical CSM, confined into a disk or other-\nwise equatorially-focused configuration surrounding the\nprogenitor (Blondin et al. 1996; van Marle et al. 2010;14 Matsumoto et al.\n100 40 60 80 200 300 400 600\nDuration : tpl [day]10421043104410451046Lpl [erg/s]\n103\n102\n0.11tfb=10day102\n103\n104\n1053\nEacc=1052erg\n1051\n105010 M, 500 R, 1051 erg\nfb=4/3\n5/3\n2\n3\n4\nFigure 8. The same as Fig. 3 but for heating by accre-\ntion onto a central compact object, following Eq. (66). Each\nshaded region corresponds to the parameter space of plateau\nduration and luminosity for different values of the post-break\nindex αfb. Shaded regions are overlaid such that the curves\nfor larger αfbcover part of the smaller αfbcurves. All values\nofαfbare bounded at the bottom by the constant-heating\ncase. The region for αfb= 2 essentially follows the magnetar\nheating case (see Fig. 7). Black solid and dashed contours\ndenote cases of fixed fallback timescale and total accretion\nenergy for αfb= 5/3. A red thick curve shows the case of a\nconstant heating rate (see Fig. 3), which bounds the high- tpl\nedge of the parameter space (see also Fig. 3).\nFigure 9. A schematic diagram of the envisioned scenario\nfor shock interaction between the supernova ejecta and an\nequatorial CSM disk. Radiation released by the ejecta-CSM\nshocks in the equatorial regions provide a source of internal\n(sub-photospheric) heating for bulk of the faster, effectively\nfreely-expanding ejecta at higher latitudes.\nVlasis et al. 2016; McDowell et al. 2018; Kurf¨ urst &\nKrtiˇ cka 2019; Suzuki et al. 2019; Kurf¨ urst et al. 2020;\nNagao et al. 2020), a geometry which is supported by\nsome observations of type IIn or otherwise peculiar SNe(Chugai & Danziger 1994; Leonard et al. 2000; Andrews\net al. 2017; Andrews & Smith 2018; Andrews et al. 2019;\nBilinski et al. 2020, 2023). Indeed, the most luminous\nSLSNe-II show no early-time evidence for narrow lines\n(e.g., Gezari et al. 2009; Miller et al. 2009), supporting\na configuration in which the CSM shock is initially em-\nbedded behind the photosphere. Additionally, a fraction\nof SLSNe-II show no narrow lines during photospheric\nphase (Inserra et al. 2018; Kangas et al. 2022).\nFig. 9 provides a schematic depiction of the envisioned\nshock interaction. The supernova ejecta collides with a\nCSM disk starting at its inner edge, forming forward and\nreverse shocks, sandwiched between a swept-up shell of\nshocked gas, which expands outwards in time. Since the\nportion of the supernova ejecta directed along the po-\nlar region expands relatively freely without encountering\nsignificant mass, it can envelope the slower-expanding\nshocked region (e.g., Metzger 2010; Andrews & Smith\n2018). Radiation generated at the equatorial shock thus\ndiffuses vertically into the ejecta, serving as an effective\ninternal heating source for a (one-dimensional) super-\nnova light curve model (e.g., Metzger & Pejcha 2017).\nWe assume that the shocked CSM expands only into the\nradial direction, retaining its initial solid angle. We also\nneglect the finite time required for radiation to diffuse\nradially out of the disk and into the surrounding super-\nnova ejecta; our model thus provides a maximal heating\nluminosity and hence a conservative upper limit on the\ncorresponding heating-extended plateau duration.\nWe adopt a radial density profile for the equatorial\ndisk corresponding to that of a steady-wind,\nρCSM=˙M\n4πfΩvCSMr2, (68)\nwhere fΩ<1 is the fraction of total solid angle sub-\ntended by the disk, vCSM is the CSM radial velocity,\nand ˙Mis the wind mass-loss rate. We estimate the ef-\nfective heating rate due to CSM shock interaction as\nfollows (a more detailed description of the shock evo-\nlution, for a generic density profile index, is given in\nAppendix B). The time evolution of the shocked region\ncan be described by mass and momentum conservation\n(e.g., Metzger & Pejcha 2017). We assume sufficiently\ndense CSM that both forward and reverse shocks are\nradiative. Since the radial width of the shocked region\nis small in this case, we characterize the shell of shocked\ngas by a representative radius Rshand velocity vsh.\nInitially, the shock expands outwards at a roughly con-\nstant speed close to that of the supernova ejecta, the\nswept-up mass growing as Msh≃˙Mvt/v CSM.Once Msh\nbecomes comparable to the shocked ejecta mass, fΩM,\nthe shell of shocked gas starts to appreciably decelerate,\non a characteristic timescale\ntdec≃fΩMvCSM\n˙Mv≃610 dayfΩ,−1M10vCSM,100\nv6000˙M−1\n−2,(69)Maximal Plateau Duration 15\nwhere vCSM = 100 vCSM,100km s−1, ˙M =\n10−2˙M−2M⊙yr−1, and fΩ= 0.1fΩ,−1. From momen-\ntum conservation, the radius and velocity of the shell\nevolve according to,\nRsh≃\n\nvt :t < t dec,\nvtdec\u0010\nt\ntdec\u00111/2\n:t > t dec,(70)\nvsh≃\n\nv :t < t dec,\nv\u0010\nt\ntdec\u0011−1/2\n:t > t dec.(71)\nThe kinetic luminosity of the forward shock dominates\nthat of the reverse shock, and thus the forward shock\ndominates the ejecta heating. The forward shock lumi-\nnosity evolves according to:\nH(t)∼4πfΩR2\nshρCSMv3\nsh=˙Mv3\nsh\nvCSM(72)\n≃1.4×1043erg s−1v3\n6000˙M−2\nvCSM,100\n\n1 : t < t dec,\u0010\nt\ntdec\u0011−3\n2:t > t dec.\nInterestingly then, for a wind-like CSM profile ρCSM∝\nr−2, the heating rate evolution again follows the func-\ntional form of Eq. (54).\nOur treatment of the shock as an internal heating\nsource requires the shocked region to remain inside the\nsupernova photosphere, i.e.,\nRsh≤Rph(=xR). (73)\nAfter this condition becomes violated, we truncate the\nheating rate in our model, i.e. H= 0 for Rsh> R ph.\nFig. 10 shows light curves, as well as the evolution\nof photosphere radius and velocity, for various assumed\nCSM wind mass-loss rates, obtained by calculating the\nshock evolution numerically (see Appendix B for the ex-\nact equations). The inner edge of the CSM disk is as-\nsumed to coincide with the progenitor star radius, and\nwe fix the other CSM properties according to fΩ= 0.1\nandvCSM = 100 km s−1(the results depend only on\nthe overall CSM density normalization ∝˙M/v CSM, pro-\nvided that vCSM≪vsh). As a general rule, the ejecta\ncannot be appreciably accelerated as a result of CSM in-\nteraction because the ultimate source of the ejecta heat-\ning is the initial kinetic energy of the ejecta which inter-\ncepts the CSM, ∼fΩEkin,0.\nWhile the lengthening of the plateau phase from shock\nheating is typically only moderate, we identify an “op-\ntimal” pre-explosion progenitor mass-loss rate, ˙M∼\n0.1 M⊙yr−1(forvCSM= 100 km s−1), which maximizes\nthe plateau duration. While the shock power increases\nfor larger ˙M, the duration of this heating phase is cor-\nrespondingly shorter because of the shorter deceleration\ntime, which results in a lower shock luminosity during\nthe recombination phase. On the other hand, for less\n1040104110421043L [erg/s]\nti (Recombination)\ntpl (End of Plateau)\n101\n100x=Rph/RRsh/R\nM=103M/yr\n102\n0.1110100\n0 50 100 150 200\nTime : t [day]103104vph(=xv) [km/s]Figure 10. The same as Fig. 5 but for energy injection\ndue to shock interaction between the supernova ejecta and\na disk-like CSM. For the CSM velocity and solid angle cov-\nering fraction, we adopt vCSM = 100 km s−1andfΩ= 0.1,\nrespectively. In the middle panel, dashed curves show the\nshock radius in units of the ejecta radius, Rsh/R. Once the\nshocks emerge from the photosphere such that Rsh> R ph\n(intersection of solid and dashed curves), the shock can no\nlonger supply a source of internal heating to the ejecta and\ninternal heating is truncated. This truncation can best be\nseen in the yellow line ( ˙M= 10M⊙/yr).\nmassive CSM (smaller ˙M), the shock heating is sus-\ntained for a longer timescale, but at a value too low to\nsustain the ejecta ionization and appreciably lengthen\nthe plateau. The shocked region also emerges faster\nfrom the photosphere in the freely-coasting case, violat-\ning condition (73) and terminating the ejecta heating.\nThe above description of the optimal mass loss rate\nis supported by Fig. 11, which again shows the key\ntimescales, this time as a function of ˙M. For ˙M≳\n0.1 M⊙yr−1deceleration of the ejecta is significant dur-\ning the recombination phase (right of the dashed line),\nand resulting weaker shock power, Lsh∝˙M−1/2, causes\nthe plateau duration to shorten. For ˙M≲0.1 M⊙yr−1,16 Matsumoto et al.\n103\n102\n101\n100101102\nM [M/yr]\n101102Time : t [day]Nebular (Optically Thin)\nRecombination\n(Teff=Tion)\nTeff>Tiontdec>t\nRsh>Rph10 M, 500 R, 1051 erg\nFigure 11. The same as Fig. 2 but for shock interac-\ntion heating for different values of the CSM density profile.\nDashed diagonal lines denote the deceleration timescale be-\nfore which the shock expands freely and the shock luminos-\nity is approximately constant. A dash-dotted line shows the\ntime at which the shock radius becomes larger than the re-\ntreating supernova photosphere.\nthe shock does not experience significant deceleration\nduring the recombination phase, and eventually over-\ntakes the photosphere. The small rise in the plateau\nduration around ˙M∼10−100 M ⊙yr−1is caused by\nthe tail of the light curve tracing the shock luminosity,\nas shown in Fig. 10. In summary, the optimal mass\nloss rate for lengthening the plateau duration is that for\nwhich ejecta deceleration and recombination occur si-\nmultaneously. However, even in this optimal case, the\nplateau duration is extended by only a factor ∼2 com-\npared to the no-heating case. In Appendix B we show\nthis conclusion also holds for generic (i.e., non wind-like)\nCSM density profiles.\nIt is not straightforward to estimate analytically the\noptimal tpl−maximizing mass-loss rate because the ef-\nfects of shock heating on the recombination process are\nsubtle for ˙M∼10−2−10−1M⊙yr−1. As an alternative,\nwe adopt the mass-loss rate above which the resulting\nshock heating appreciably slows the ejecta recombina-\ntion (the analog of Hcrfrom the constant heating rate\ncase). Equating Eqs. (12) with (72, t < t dec), we find:\n˙Mcr≃3.4×10−2M⊙yr−1M10v−2\n6000vCSM,100\u0012ζ\n0.83\u0013−3\n.\n(74)\nHere, we have written vsh=ζvto account for the mod-\nerate deceleration of the ejecta even prior to the formal\ndeceleration time, where ζ≃0.83 is motivated by our\nnumerical results. This estimate slightly underestimatesthe true ˙Mwhich maximizes tplin our full numerical\ncalculations (Fig. 11).\n5.DISCUSSION\nFig. 12 summarizes the allowed space of light curve\nproperties, for the wide range of heating sources consid-\nered in this paper, for fiducial progenitor star/explosion\nproperties. The presence of a sustained heating source\ngenerally acts to boost the plateau luminosity and du-\nration above those of the no-heating case (the latter of\nwhich describes most SNIIP, shown for comparison as\nbrown crosses). While increases in the luminosity of\nup to several orders of magnitude are possible for suffi-\nciently powerful heating rates, the corresponding boost\nto the plateau duration is more tightly bounded, to\nwithin a factor ≲3 of the duration in the zero-heating\ncase. The attainable plateau duration is capped by the\nlimiting constant heating-rate case (depicted as a thick\nred curve). This maximal duration is approximately\ngiven by Eq. (46), which expressed in terms of ejecta\nmass and explosion energy gives:\ntpl,max≃580 day T−2/3\ni,6000M17/24\n10E−7/24\n0,51.(75)\nFig. 12 also shows for comparison a sample of type IIn\nsupernovae, i.e. those showing narrow hydrogen lines in-\ndicative of CSM interaction, including the outlier event\nSN2016aps. We also show two other “extreme” explo-\nsions, iPTF14hls (Arcavi et al. 2017) and AT2021lwx\n(Subrayan et al. 2023; Wiseman et al. 2023), though the\nlatter has been interpreted as a tidal disruption event\nby some authors.\n5.1. Minimum ejecta mass of long-plateau transients\nThe existence of a maximal plateau duration implies\nthat a minimum ejecta mass is required to realize the\nplateau duration of a given observed event. Since the\nduration-luminosity trajectory for the limiting constant\nheating-rate case depends on the ejecta properties (bot-\ntom panel of Fig. 3) we show in the top panel of Fig. 13\nthese curves for an expanded parameter range of ini-\ntial explosion energies 1050erg to 1053erg (thin to thick\ncurves), and ejecta masses from 1 to 103M⊙in steps of\n0.2 dex.6As expected from Sec. 3, the maximal plateau-\nduration increases with ejecta mass and decreases with\nhigher explosion energy.\nThe bottom panel of Fig. 13 depicts the minimum\nejecta mass, Mmin, needed to produce a given plateau\nduration. Based on the Lpl−tpltrajectory (Eq. 53, for\n6We remove models satisfying E0= 1050erg and M≳400 M ⊙\nbecause gas pressure dominates over radiation pressure in the\nejecta for these parameters, violating our assumption (Eq. (14)).Maximal Plateau Duration 17\n100 1000 50 70 200 300 500 700\nDuration : tpl [day]1042104310441045Luminosity : Lpl [erg/s]\nIIn\nLRNiPTF14hlsSN2016apsAT2021lwx\n56Ni decayAccreting compact object\nConstant heating10 M, 500 R, 1051 erg\nCSM shock (CSM rq)\nq=0\n12\n3\nIIP\nFigure 12. Allowed space of plateau duration and luminosity for different heating sources. The colored shaded regions\ncorrespond to an accreting compact object (see Figs. 3, 8), bounded by a solid red curve corresponding to constant heating-rate\ncase. Gray dashed and colored solid curves denote, respectively, radioactive56Ni heating and CSM shock interaction (for a range\nof different radial density profiles). The ranges of nickel masses and CSM density normalizations follow those in Figs. 6 and\n15, respectively. Brown crosses, magenta circles, orange squares show the parameters for type IIP SNe, IIn SNe, and luminous\nred novae (LRNe) (data are taken from Martinez et al. 2022a; Nyholm et al. 2020; Matsumoto & Metzger 2022b). Three stars\nrepresent peculiar events, iPTF14hls (Arcavi et al. 2017; Andrews & Smith 2018), SN 2016aps (Nicholl et al. 2020), and AT\n2021lwx (Subrayan et al. 2023; Wiseman et al. 2023). For type IIn, SN 2016aps, and AT 2021lwx, we use peak luminosities and\nshow durations corresponding to the timescale the light-curve remains within ≲1 magnitude of peak.\ntheH < H pl,maxcase), we estimate:\nMmin≃180 M ⊙E3/5\n0,52\u0012Lpl\n3×1042erg s−1\u0013−1/5\u0012tpl\n103day\u00136/5\n,\n(76)\nin rough agreement with the full calculation. If the\nejecta velocity can be measured observationally, the ex-\nplosion energy and hence Mmincan be more robustly\nconstrained. Along each curve, a small circle denotes\nthe minimum explosion energy in shock-powered scenar-\nios,E0≳tplLpl/fΩ, which is consistent with generating\nthe radiated energy.\nAs expected from the Lpl−tpltrajectories, curves of\nMminterminate at high energies (e.g., at E0≃1052erg\nforLpl= 1043erg s−1andtpl= 300 day), and flatten forlower energy (e.g., E0≲1051erg for Lpl= 1043erg s−1\nandtpl= 1000 day). The former behavior occurs be-\ncause the Lpl−tpltrajectory rises with increasing E0,\nuntil it no longer touches the given Lpl. The flatten-\ning at low E0occurs because at high heating rates\nH≳Hpl,max, the Lpl−tpltrajectory loses sensitivity to\nthe initial explosion energy (Eq. 53 for the H > H pl,max\ncases). This flattening occurs once the ejecta experi-\nences significant acceleration, leading to a breakdown of\nthe relations E0= 3Mv2/20,Ekin,0=E0andv=√\n2v0\n(Eqs. (7)). Contours of constant ejecta velocity are de-\nnoted by gray dashed diagonal lines in the bottom panel\nof Fig. 13).\nAs an example, we show the Mmintrajectory for\niPTF14hls ( tpl∼103days; Lpl≃3×1042erg s−1) as\na red dashed line in the bottom panel of Fig. 13 (along18 Matsumoto et al.\n101102103104\nDuration : tpl [day]10411042104310441045Lpl [erg/s]\nE0=1050erg105110521053\n110102103\nMass : M [M]\n1050105110521053\nInitial Energy : E0 [erg]101102Minimal Mass : Mmin [M]\ntpl=3000day1000\n300\n100\n4000km/sv=8000km/s16000km/s\nLpl=1042erg/s\n3×1042\n1043\n1044\nFigure 13. (Top) Trajectories of plateau duration and lu-\nminosity for the bounding constant heating rate case ( H=\nconst ), similar to Fig. 3 but now for a range of the energies\nE0= 1050erg−1053erg and ejecta masses M= 1−103M⊙\nas labeled. The range of heating rate values Halong each\ntrajectory, is set to cover Hmin(Eq. 35) so that the heating-\nfree solutions are present for each parameter set. Two stars\nrepresent iPTF14hls and SN 2016aps, as in Fig. 12. ( Bot-\ntom) Minimal ejecta mass Mminto produce a given plateau\nduration (different colored lines) and luminosity (different\nlinestyles) as a function of the initial explosion energies E0.\nGray dashed lines show the corresponding mean ejecta ve-\nlocity. Along each curve, a circle denotes the minimum\nenergy in shock interaction scenarios, E0≳tplLpl/fΩ. A\nstar represents the location of iPTF14hls with tpl≃103day,\nLpl≃3×1042erg s−1, and v≃4000 km s−1.\nthis line, acceleration of the ejecta is negligible). If the\nvelocity obtained from the Fe line absorption minimum,\nv≃4000 km s−1(Arcavi et al. 2017), is representative\nof the unshocked ejecta, then we find the initial ex-\nplosion energy and ejecta mass are constrained to be\nE0≃1052erg and Mej≳Mmin≃200 M ⊙, respectively.\nSuch a large ejecta mass would obviously require a verymassive progenitor star, possibly supporting a pulsa-\ntional pair-instability explanation for this event (e.g.,\nWoosley et al. 2007; Woosley 2018; Woosley & Smith\n2022; Wang et al. 2022; Chen et al. 2023).\nThe minimum ejecta mass (Eq. (76)) can be expressed\nin terms of vinstead of E0:\nMmin≃140 M ⊙\u0012Lpl\n3×1042erg s−1\u0013−1/2\n\u0010v\n4000 km s−1\u00113\u0012tpl\n103day\u00133\n, (77)\nwhere again acceleration of the ejecta is neglected. The\nbottom panel of Fig. 14 shows Mminas a function of\nplateau duration for different values of Lplandv. Each\nMmincurve terminates at both short and long plateau\ndurations, the latter corresponding to the regime of ap-\npreciable acceleration of the ejecta ( H≳Hpl,max). This\nis illustrated in the top panel of Fig. 14, which is similar\nto Fig. 13 but shows Lpl−tpltrajectories for different\nassumed mean ejecta velocities.\nThe above demonstrates a methodology for placing\nlower limits on the ejecta mass for given supernovae.\nThis can help identify the explosions of very mas-\nsive stars, including PISNe caused by pair-production-\ninduced collapse leading to explosive oxygen burning\nfor progenitors with ≃140−260 M ⊙(Rakavy & Sha-\nviv 1967; Barkat et al. 1967; Heger & Woosley 2002;\nHeger et al. 2003; Kasen et al. 2011; Dessart et al.\n2013; Chen et al. 2014a; Kozyreva et al. 2014a,b; Gilmer\net al. 2017) and general-relativistic instability super-\nnovae (GRSNe) for supermassive population III stars\n∼104−5M⊙(Chandrasekhar 1964; Fuller et al. 1986;\nMontero et al. 2012; Chen et al. 2014b; Matsumoto et al.\n2016; Uchida et al. 2017; Moriya et al. 2021a; Nagele\net al. 2020, 2022). These explosions should be rare\nand current observations only weakly constrain their\nrates. For instance, Moriya et al. (2021b) constrain\nthe rate of long-lasting H-rich supernovae observation-\nally, finding ∼40% of type IIn exhibit light-curves last-\ning longer than a year, while the rate of PISN is con-\nstrained to ≲0.01−0.1% of the total core-collapse su-\npernova rate. More recently, using JWST observations,\nMoriya et al. (2023) constrain the rate of GRSNe to\n≲800 Gpc−3yr−1. With current and upcoming facil-\nities such as JWST ,Rubin ,Euclid , and Roman in an\nexcellent position to discover additional candidate ex-\nplosions in the near future, our framework may prove\nuseful to help identify these events and constrain the\nejecta properties.\n5.2. Application to Luminous Red Novae\nLuminous red novae (LRNe) are weeks- to months-\nlong transients with luminosities between those of clas-\nsical novae and supernovae (e.g., Bond et al. 2003;\nTylenda et al. 2011; Blagorodnova et al. 2017), and\nwhich are believed to arise from stellar mergers (TylendaMaximal Plateau Duration 19\n101102103\nDuration : tpl [day]1042104310441045Lpl [erg/s]\nv=4000km/s800016000\n110102103\nMass : M [M]\n102103\nDuration : tpl [day]100101102103Minimal Mass : Mmin [M]\nLpl=1042erg/s\n3×1042\n1043\n1044\nv=4000km/s\n8000\n16000iPTF14hls\nFigure 14. (Top) The same as the top panel of Fig. 13 but\nfor fixed v. The trajectories terminates at high luminosity\n(roughly corresponding to Hpl,max, where the ejecta accel-\nerates. ( Bottom ) Minimal ejecta mass Mminto produce a\ngiven plateau duration tplfor different luminosity and ve-\nlocity. As in the bottom panel in Fig. 13, a star represents\niPTF14hls.\n& Soker 2006; Ivanova et al. 2013). While dimmer\nmembers of the LRN class with ≲1040erg s−1can be\npowered exclusively by hydrogen recombination energy,\nthe brightest and longest events may require an addi-\ntional energy source (Matsumoto & Metzger 2022b),\nmost likely shock interaction between the merger ejecta\nand stellar material unbound from the binary prior to\nthe dynamical merger (e.g., Pejcha et al. 2016a,b; Met-\nzger & Pejcha 2017; MacLeod & Loeb 2020a).\nOur results illustrate that shock interaction can in-\ncrease the plateau duration and luminosity by at most a\nfactor ∼2 and ∼10, respectively, boosting the radiated\nenergy by a factor ≲20 compared to the zero-heating\ncase. Scaling to properties typical of stellar mergers, the\nduration, luminosity, and radiated energy in the zero-heating case are estimated by Eqs. (17), (19), and (18):7\ntpl,Popov≃78 day\u0012M\n3 M⊙\u00135/14\n\u0010v\n103km s−1\u0011−5/14\u0012R0\n30 R⊙\u00131/7\n,(78)\nLpl,Popov≃2.6×1040erg s−1\u0012M\n3 M⊙\u00133/7\n\u0010v\n103km s−1\u001111/7\u0012R0\n30 R⊙\u00134/7\n, (79)\nEpl,Popov≃1.7×1047erg\u0012M\n3 M⊙\u001311/14\n\u0010v\n103km s−1\u001117/14\u0012R0\n30 R⊙\u00135/7\n. (80)\nWhile these estimated fall short of explaining the most\nluminous LRNe by an order of magnitude for reason-\nable ejecta masses (Matsumoto & Metzger 2022b), the\npresence of CSM interaction could alleviate these ten-\nsions by boosting the plateau luminosity and duration\nfor optimally-chosen values of the (pre-merger) binary\nmass-loss rate, ˙M∼˙Mcr(Sec. 4.4). In particular, from\nEq. (74) we find:\n˙Mcr≃1.34 M⊙yr−1\u0012M\n3 M⊙\u0013\n\u0010v\n103km s−1\u0011−2\nvCSM,100\u0012ζ\n0.56\u0013−3\n, (81)\nwhere here we take ζ≡vsh/v≃0.56, appropriate for\nLRN parameters. Since mass-loss prior to the merger\n(e.g., from the binary L2 Lagrange point) typically\namounts to at most tens of percent of the donor star’s\nmass (MacLeod et al. 2017; Pejcha et al. 2017; MacLeod\n& Loeb 2020b), this optimal mass-loss rate could be re-\nalized in a massive ∼10 M⊙binary which loses ∼M⊙\nover the year preceding the dynamical merger event.\n6.SUMMARY AND CONCLUSIONS\nThe basic observables of SNeIIP contain information\non the properties of the exploding star, which must be\nunderstood to take advantage of the growing samples of\nsupernovae to be discovered, e.g. by Rubin ,JWST , and\nRoman . One such readily-measured property is the du-\nration of the optically-thick plateau phase, which canon-\nically depends (for a given explosion energy and progen-\nitor radius) on the ejecta mass. All else being equal, a\n7Note that for these parameters, the ejecta energy,\nE0≃9.0×1048erg\u0012M\n3 M⊙\u0013\u0010v\n103km s−1\u00112\n,\nis close to the limit below which gas pressure is comparable to\nradiation pressure (Eq. (14)).20 Matsumoto et al.\nlonger plateau requires a higher ejecta mass, and hence\nmore massive progenitor star. However, the plateau du-\nration can also be prolonged by the presence of a sus-\ntained internal heating source, which keeps the ejecta\nionized −and hence optically-thick −longer than in a pas-\nsively cooling case. Indeed, growing evidence points to\nadditional heating sources being required to power a\nlarge number of peculiar supernovae and related opti-\ncal transients.\nTo understand the interplay between these processes,\nand how to break degeneracies to better constrain the\nejecta properties of individual events, we have studied\nthe effects of an internal heating source on light curves of\nhydrogen-rich explosions by generalizing the analytical\nSNIIP light curve model of Popov (1993). Our findings\nare summarized as follows:\n•While sufficiently large internal heating rates can\nboost the plateau luminosity by several orders of\nmagnitude, the plateau duration can be prolonged\nby at most a factor of ∼2−3 times compared to\nzero-heating case. For a temporally-constant heat-\ning source, the maximal plateau duration is real-\nized for the maximal heating rate which does not\nappreciably alter the ejecta dynamics. Above this\ncritical heating rate, the ejecta experiences signif-\nicant acceleration, reducing the photon diffusion\ntimescale and hence shortening the plateau.\n•Heating from56Ni decay is found to boost the\nplateau duration by at most a few tens of percent\nfor nickel masses typical of SNIIP, consistent with\nprevious findings in the literature. This agreement\nwith more sophisticated radiation transport cal-\nculations supports our model capturing at least\nsemi-quantitatively the effects of an internal heat-\ning source on supernova light-curves.\n•Insofar that sustained energy injection from a cen-\ntral engine, such as millisecond magnetar or ac-\ncreting compact object, can be described as heat-\ning with a characteristic duration and total de-\nposited energy as in Eq. (63) or (66), the result-\ning supernova plateau phase can exhibit a wide\nrange of luminosities and durations depending on\nthese parameters. While a longer engine heat-\ning timescale (e.g., magnetar spin-down time tsd,\nor the mass fall-back timescale tfb) results in a\nlonger plateau duration, the latter is again capped\nat a few times the duration in the zero-heatingcase in the limit of temporally-constant heating\n(tsd, tfb→ ∞ ).\n•Shock interaction between the fast supernova\nejecta and slower equatorially-confined (“disk-\nlike”) CSM can provide an additional source of\nsub-photospheric heating for sustaining a longer\nplateau (Fig. 9). However, for typical parameters,\nshock heating can boost the plateau duration by at\nmost a factor of ∼2 for an optimal mass loss rate\n(Eq. (74)). While a very massive CSM results in a\nhigh shock luminosity, concomitant strong deceler-\nation of the ejecta limits this phase to early times.\nBy contrast, while ejecta deceleration is weaker for\nlow CSM masses, in that case the shocks quickly\nexpand beyond the retreating photosphere radius,\ntherefore no longer serving as a sustained inter-\nnalheating source capable of keeping the ejecta\nionized.\nOur results are conveniently summarized in the space\nof plateau duration and luminosity, as shown in Fig. 12\nand compared to observed hydrogen-rich supernovae.\n•Motivated by our finding that the temporally-\nconstant heating case defines an absolute bound-\nary on the attainable duration and luminosity, we\ndeveloped a framework to constrain the minimal\nmass for observed events, as captured in Eq. (77)\nin Sec. 5.1. As a proof of principle, we used the\nobserved duration, luminosity, and ejecta veloc-\nity, to constrain the ejecta mass in iPTF14hls to\n≳200 M ⊙, supporting an explosion of very mas-\nsive star, such as a PISN.\n•Our results can also be applied to shock interac-\ntion heating in LRNe from stellar mergers. We\nconfirmed that shock interaction between the LRN\nejecta and circumbinary material ejected prior to\nthe merger can indeed explain the otherwise puz-\nzling properties of the most luminous events, for\nan assumed pre-dynamical binary mass-loss rate\nof∼M⊙yr−1.\nACKNOWLEDGMENTS\nT.M. acknowledges supports from JSPS Overseas Re-\nsearch Fellowship and the Hakubi project at Kyoto Uni-\nversity. B.D.M. acknowledges support from the National\nScience Foundation (grant number AST-2009255). The\nFlatiron Institute is supported by the Simons Founda-\ntion.\nAPPENDIXMaximal Plateau Duration 21\nA.ANALYTIC ESTIMATES FOR BROKEN POWER-LAW HEATING PROFILE\nWe now extend our analytic estimates from Sec. 3 for a constant heating rate to the broken power-law heating rate\nof the form (54). We first consider cases in which the heating affects only the thermal evolution of ejecta as in Sec. 3.1.\nA quantitatively different evolution from the H=const case occurs for tbr< t < t i. In this regime, as long as α <2\nthe internal energy grows as E≃tH∝t1−αafter the deposited energy has become comparable to the internal energy\nat time\neth=\u0012th\ntbr\u0013α\n2−α\nth, (A1)\nwhere the definition for thfollows Eq. (24) but now for H→eH. The evolution of the internal energy and effective\ntemperature subsequently obey:\nE≃\u0012t\ntbr\u00131−α\neHtbr, (A2)\nTeff≃ eH\nHcr!1/4\u0012t\ntbr\u0013−α/4\nTi, (A3)\nrespectively. For sufficiently high heating rates, eH > H cr, recombination begins at time\neti= eH\nHcr!1/α\ntbr. (A4)\nWhen α≥2, the evolution until recombination completes is identical to the zero-heating case.\nFrom Eq. (A1) we find that for heating rates smaller than a critical rate, eH < t 0E0/t2\nbr, the thermal evolution of the\nejecta is affected only after the heating rate has begun to decline ( eth> tbr). For higher heating rates, the evolution is\nmodified before the heating breaks ( th< tbr), and for α <2 it is identical to the H=const case until t < t br. The\nevolution at t > t bris identical to that given in Eqs. (A2), (A3). For α≥2, the internal energy is effectively reset\nby the injected heat on the time tbr, such that E≃eHtbr(t/tbr)−1. And the effective temperature and recombination\ntime follow the same expressions as in Eqs. (27) and (28) for the regime of tdiff< tandHcr< H, respectively, but\nmaking the replacements tdiff→tbrandH→eH.\nThe photosphere evolution during the recombination phase can be derived following similar arguments as in the\nconstant heating case. In particular, when the heating rate declines before recombination begins ( tbr< ti), the\nphotosphere location follows\nx≃\n\n\u0010\nt\nti\u0011−2/5\n:t <etrec\nh,\n\u0010eH\n2Hcr\u00111/4\u0010\nt\ntbr\u0011−α/4\n:etrec\nh< t <etrec\ndiff,\n\u0010eH\nHcr\u00111/2\u0010\ntdiff\ntbr\u0011\u0010\nt\ntbr\u0011−α+2\n2:etrec\ndiff< t ,(A5)\nwhere\netrec\nh= eH\n2Hcr! 5\n5α−8\u0012tbr\nti\u00135α\n5α−8\nti, (A6)\netrec\ndiff= \n2eH\nHcr!1\nα+4\u0012tbr\ntdiff\u0013α\nα+4\ntdiff. (A7)\nDepending on the parameters, some regimes disappear; for instance, when the break in the heating rate occurs after\nrecombination completes ( tbr> ti), the evolution follows Eq. (31) until the break. However, for any set of parameters,\nthe final regime of evolution always exists, with the condition xτ= 1 again giving the plateau duration:\netpl≃ eH\nHcr!1\nα+6\u0010v\nc\u00111\nα+6\u0012tbr\ntthin\u0013α\nα+6\ntthin. (A8)22 Matsumoto et al.\nAs expected, all of the expressions given here except for Eq. (A4) reduce to the H=const case in the limit α→0.\nWe now discuss cases in which the heating impacts the ejecta dynamics, as considered in Sec. 3.2. For such large\nheating rates, the ejecta experiences appreciable acceleration at times t > t acc, where taccis again defined by Eq. (38)\nbut for eHinstead of H. While in the H=const case the acceleration was found to terminate when recombination\noccurs or photon diffusion becomes important, we find that acceleration can also terminate after the break in the\nheating-rate. This is because the internal energy falls as E∝t1−α(α < 2) or ∝t−1(α≥2), which results in\nan equation of motion: dv/dt ∝1/(tαv) or∝1/(t2v), respectively. Therefore, as long as α > 1, as obeyed by the\nphysically-motivated heating rates we consider, the ejecta is no longer accelerated significantly after the break. This\nsimplifies the analysis by allowing us to treat the acceleration in the same way as in the constant-heating case. In\nparticular, we need only consider two cases, in which the ejecta acceleration terminates due to the break at tbror\nphoton diffusion at tdiff. In the former case, the terminal velocity is given by\nevmax≃ \n10eHtbr\n3M!1/2\n, (A9)\nand the corresponding photon diffusion timescale is given by\netacc\ndiff=\u001227κ2M3\n160π2c2tbreH\u00131/4\n. (A10)\nTherefore, the condition tbr<etacc\ndiffis satisfied for\neH <27κ2M3\n160π2c2t5\nbr. (A11)\nThe plateau duration is obtained in the same way as the H=const case, namely, following Eq. (A8)/(34) with evmax\nreplacing vmaxwhen recombination occurs after/before tbr. In cases of yet stronger heating, acceleration terminates\nonly once photon diffusion becomes important. The corresponding velocity and plateau duration in this case again\nfollow the Eqs. (41) and (44), respectively.\nB.DETAILS OF THE EJECTA-CSM SHOCK DYNAMICS\nWe describe our treatment of the dynamics of the shock interaction between the supernova ejecta and equatorially-\nconcentrated (disk-like) CSM for the case of a generic power-law density profile (see also Moriya et al. 2013; Metzger\n& Pejcha 2017; Hiramatsu et al. 2023). We assume that both the forward shock (FS) and reverse shock (RS) are\nradiative and hence both share a single representative radius Rshand (lab frame) velocity vsh. The unshocked CSM\nis characterized by its solid angle 4 πfΩ, velocity vCSM≪vsh, and a radial density profile\nρCSM=Ar−q, (B12)\nrespectively. The mass Mshand momentum Mshvshof the shocked gas increase as the FS and RS sweep up the CSM\nand supernova ejecta, respectively, their evolution obeying:\ndMsh\ndt= 4πfΩR2\nshρSN(vSN−vsh) + 4πfΩR2−q\nshA(vsh−vCSM), (B13)\nd(Mshvsh)\ndt= 4πfΩR2\nshρSNvSN(vSN−vsh) + 4πfΩR2−q\nshAvCSM(vsh−vCSM), (B14)\nwhere ρSN=M/(4πR2/3) is the ejecta density (taken to be a homogeneous sphere of radius R) and vSN=v(Rsh/R)\nits velocity (assuming homologous expansion). Momentum conservation (Eq. B14) can be rewritten:\nMshdvsh\ndt= 4πfΩR2\nshρSN(vSN−vsh)2−4πfΩR2−q\nshA(vsh−vCSM)2. (B15)\nThe dynamics of the swept-up shell is thus determined by solving Eqs. (B13) and (B15) with initial conditions Rsh=R0,\nvsh= 0, and Msh≪Mfor assumed values of A,q,vCSM, and fΩ.Maximal Plateau Duration 23\n1014\n1013\n1012\n1011\n1010\n109\nCSM@1015cm [g/cm3]\n101102Time : t [day]10 M, 500 R, 1051 erg\nq=0\n1\n2\n3\nFigure 15. The same as Fig. 11 but for different slopes of CSM radial density profile, ρCSM∝r−q(indicated by different\ncolors). Thick solid lines denote the plateau duration and dotted lines denote the time at which Teff=Ti. Dashed diagonal\nlines denote the shock deceleration timescale. The horizontal axis denotes the CSM density at 1015cm.\nOnce the shock dynamics are determined, we estimate the total shock luminosity (ejecta heating rate) as the sum\nof the kinetic luminosities of the FS and RS shocks, given, respectively, by:\nLkin,FS= 4πfΩR2\nsh(vsh−vCSM)eFS=9π\n2fΩR2−q\nshA(vsh−vCSM)3, (B16)\nLkin,RS= 4πfΩR2\nsh(vSN−vsh)eRS=27fΩMR2\nsh(vSN−vsh)3\n8R3, (B17)\nwhere the internal energy density of the post FS (RS) region is given by\neFS(RS) =2ρCSM(SN) (vsh−vCSM(SN) )2\n(γ+ 1)( γ−1). (B18)\nHere γis the adiabatic index and we adopt γ= 5/3. In a more detailed calculation, the dynamics of the shocks would\nbe coupled to that of the supernova ejecta via heating, such that their evolution should be calculated self-consistently.\nFig. 15 shows the key timescales plot (cf. Fig. 11) as a function of the CSM density normalization at r= 1015cm\nfor different values of the CSM power-law slope qas marked. The qualitative features in all cases follow those in\nthe standard q= 2 wind-like profile. The impact on the recombination time (thick dashed curves) and the plateau\nduration (thick solid curves) are slightly larger for smaller q. This may be because for q <2, the shock luminosity\nincreases with time, peaking on the deceleration time.8Furthermore, the fact that deceleration starts earlier helps to\nsatisfy the condition (73) for a longer time. Nonetheless, the plateau duration is not extended more than a few times\nthe zero-heating case ( ρCSM→0), even for q̸= 2.\nC.GLOSSARY OF SYMBOLS AND NOTATIONS\nTable 1 summarizes the symbols and notations of characteristic quantities.\nREFERENCES\n8An increasing heating rate potentially pushes the photosphere\noutward during the recombination phase, dx/dt > 0. This oc-\ncures when the dimensionless phoptosphere becomes smaller than\nxcr=1\n2\n\n−\u0012t\ntdiff\u00132\n+s\u0012t\ntdiff\u00134\n+ 8\u0012H\nHcr\u0013\n\n1/2\n,(B19)\nwhile we confirm that the photophere always shrinks in our cal-\nculations.Anderson, J. P. 2019, A&A, 628, A7,\ndoi: 10.1051/0004-6361/20193502724 Matsumoto et al.\nTable 1. Definitions of Key Timescales and Heating Rates\nNotation Eq. Meaning\ntdyn (3) Dynamical time\ntdiff (4) Diffusion time\nti (28) Recombination time (beginning of recombination phase)\nt0 (23) Initial dynamical time\nth (24) Heating timescale (heating starts to modify thermal evolution)\ntrec\nh (29) Heating timescale during recombination phase\ntrec\ndiff (30) Diffusion timescale during recombination phase (after which L≃H)\ntpl (33) Plateau duration (when ejecta becomes optically thin) including heating effects\ntthin (34) When fully ionized ejecta becomes optically thin\ntacc (38) Acceleration timescale (heating source starts to significantly accelerate ejecta)\ntacc\ndiff (40) Diffusion timescale for accelerated ejecta\ntacc\ni (43) Recombination timescale for accelerated ejecta\ntacc\npl (44) Plateau duration for accelerated ejecta\ntpl,max (46)/(48) Maximal plateau duration ((48) if Eq. 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Our\nmodel is based on the neutrino-portal Affleck-Dine (AD) mechanism, which generates the asymmetry\nof the AD sector during the radiation-dominated era and subsequently transfers it to the baryon\nnumber before the electroweak phase transition. The observed baryon asymmetry is then a natural\noutcome of this scenario. The model is testable as it predicts the existence of a Majoron with a keV\nmass and an electroweak scale decay constant. The impact of the relic Majoron on ∆ Neffcan be\nmeasured through near-future CMB observations.\nIntroduction.— The observed baryon asymmetry\n[1] is often parameterized with\nYB≡¯nB\ns= (0.82−0.92)×10−10, (1)\nwhere ¯ nBis the net baryon number density and sis the\nentropy density. This value is significantly larger than the\nbaryon asymmetry naturally anticipated in the Standard\nModel (SM), requiring new physics beyond the SM. How-\never, if the baryon asymmetry arises from new physics,\nit is essential to realize the observed baryon asymmetry\nwith the model parameters of the new physics. Previous\nworks have typically relied on small couplings or large\nwash-out effects to yield the observed baryon asymmetry\n[2–4].\nIn this Letter , we propose a scenario of baryogenesis\nwhere the baryon asymmetry results directly from a scale\nhierarchy between the two important mass scales in phe-\nnomenology: the electroweak scale v= 246 GeV and the\nreduced Planck mass MP= 2.4×1018GeV, in the form\nof\nYB=O(0.01)rv\nMP∼10−10. (2)\nWe utilize the Affleck-Dine (AD) mechanism [5, 6], where\nthe asymmetry is generated from the dynamics of a B−L\ncharged complex scalar field ϕ.ϕnaturally has an elec-\ntroweak scale mass, mϕ=O(100 GeV), if the scalar po-\ntential of ϕis governed by the same mechanism that\nensures the stability of the Higgs boson mass against\nquantum corrections. In contrast to previous works on\nAD baryogenesis [7], we explore the generation of B−L\nasymmetry during the radiation-dominated era and its\nsubsequent transfer to the SM sector through neutrino-\nportal interactions. Therefore in our scenario, there is no\ndependence on the reheating temperature.A simple model realizing our scenario is constructed by\nemploying supersymmetry (SUSY) [8–10]. The relevant\ninteractions of the model in terms of superpotential are\ngiven by\nW=yνℓhN+1\n2λNϕN2+κ\n4MPϕ4+···, (3)\nwhere ℓis the SM lepton doublet, his the Higgs field,\nNis a right-handed (RH) neutrino with B−L= 1,\nandϕis a SM singlet scalar field with B−L=−2.\nyνis the neutrino Yukawa coupling that gives neutrino\nmasses mνthrough the seesaw mechanism [11–15], while\nλNand κareO(1) coefficients. A global U(1)B−L\nsymmetry is preserved at the renormalizable level,\nallowing only the seesaw operators, but is explicitly\nbroken by Planck-scale suppressed operators as generally\nexpected due to quantum gravity effects [16–18]. We\nhighlight how supersymmetry breaking can provide\na well-organized scalar potential in the presence of\nexplicit U(1)B−Lbreaking terms and show how the two\nfundamental mass scales are combined to determine the\nbaryon asymmetry. Moreover, we establish an interest-\ning connection between the observable dark radiation\nand the baryon asymmetry with certain implications\nof the electroweak scale, emerging as a generic prediction.\nSummary of cosmological history.— Initially,\nthe SM plasma dominates the energy density of the\nUniverse, while the abundances of the novel particles ϕ\nandNare negligible. When the temperature drops to\nTAD∼p\nmϕMP, the AD mechanism becomes active,\ngenerating the asymmetry of ϕ. The temperature contin-\nues to decrease, leading to the thermalization of the AD\nsector ( ϕandN) with the SM plasma at the temperature\nTNvia the neutrino Yukawa interactions. Once thermal-\nized, the asymmetry is transmitted to the lepton sector\nthrough the neutrino portal, and the baryon asymmetryarXiv:2401.13734v2  [hep-ph]  5 Feb 20242\nis induced via the weak sphaleron process and frozen\nafter the electroweak phase transition. After the B−L\nphase transition happens, all novel particles decay to\nthe Majoron J, the pseudo-Nambu-Goldstone boson\nassociated with the spontaneous U(1)B−Lbreaking.\nThe Majoron, along with its decay products, contributes\nto ∆ Neff, providing a phenomenological signal for the\nmodel.\nAsymmetry generation.— With the field decom-\nposition of ϕ=1√\n2rϕeiθϕ, the net charge density of the\nscalar field ϕbecomes\n¯nϕ=i(˙ϕ∗ϕ−ϕ∗˙ϕ) =r2\nϕ˙θϕ, (4)\nwhich can be interpreted as a rotating scalar field carry-\ning angular momentum in the field space. In the early\nUniverse, a nonzero charge density can arise from the\n“kick” along the θϕdirection induced by the B−Lbreak-\ning potential term for a large value of rϕ. In addition to\nthe supersymmetric contribution given by |∂W/∂ϕ |2, the\nscalar potential incorporates soft supersymmetry break-\ning terms [19–23],\n∆Vsoft=m2\nϕ|ϕ|2−(αmϕW(ϕ) +h.c.), (5)\nwhere the model-dependent constant αnaturally assumes\na value of O(0.1−1). Coupled to the SM fermions ψa\nthrough the higher dimensional operator arising from the\nK¨ ahler potential,\n∆L=κa|ϕ|2\nM2\nP¯ψaiγµDµψa, (6)\nthe field ϕacquires an additional mass. During the\nradiation-dominated era, the above interaction generates\nan effective mass squared of ϕproportional to H2=\nρtot/(3M2\nP) =π2g∗T4/(90M2\nP). Here His the Hubble\nexpansion rate, and g∗=O(100) is relativistic degrees of\nfreedom. The overall coefficient of this Hubble term κH\nis typically the order of unity [24]. Including all these\ncontributions, the relevant potential of ϕis given by\nV= (m2\nϕ−κHH2)|ϕ|2−καm ϕ\n4MP\u0000\nϕ4+ϕ∗4\u0001\n+κ2\nM2\nP|ϕ|6\n=1\n2(m2\nϕ−κHH2)r2\nϕ−καm ϕ\n8MPr4\nϕcos 4θϕ+κ2\n8M2\nPr6\nϕ,\n(7)\nin the field basis where αis real. Here we take m2\nϕ>0\nandκH>0. It is worth noting that, owing to supersym-\nmetry, the quadratic term remains shielded from large\nquantum corrections while the B−Lbreaking quartic\nterm is highly suppressed by a factor of mϕ/MP, which\nare critical aspects in our scenario.\nAt high temperatures where H≫mϕ, the mini-\nmum of the potential is predominantly determined bythe Hubble-induced mass term. Having a mass similar\ntoH, the radial field rϕ(x) resides at\n⟨rϕ⟩=\u00124κH\n3κ2\u00131\n4p\nHM P. (8)\nOn the other hand, the angular field a(x) =⟨rϕ⟩θϕ(x)\nhas an effective mass smaller than H, resulting in its\nposition in the field space being nearly frozen at an arbi-\ntrary value. As the Universe expands and Hdecreases,\nthe sign of the quadratic term inverts, and the scalar\npotential is lifted subsequently. Due to the rapid lifting\ncompared to the Hubble time, there is an increase solely\nin the potential energy, with a negligible change in the ra-\ndial field value. As the quadratic term approaches zero,\nthe quartic potential term imparts a mass of O(H) to\nthe angular field. Then, for a typical value of θin, an ini-\ntial misalignment angle of θϕ, the angular field starts to\nroll toward the minimum. However, with the diminish-\ning radial field value due to rolling, the potential barrier\nheight along the radial direction also decreases, allowing\nthe traversal of the angular field across the barriers. The\nradial and angular motions can be described through ro-\ntation within the two-dimensional complex field space,\ncharacterized by a specific angular momentum.\nThe equation of motion for Yϕ≡¯nϕ/sis given by\ndYϕ\ndt=−1\ns∂V\n∂θϕ, (9)\nwhere the RHS of Eq. (9) serves as the origin of the\nnetϕcharge density. Its impact is maximized just af-\nter the lifting of the scalar potential at temperature\nTAD∼p\nmϕMPand gives ˙θϕ∼αmϕatrϕ∼p\nmϕMP.\nAfter the onset of scalar field rotation, the B−Lbreak-\ning effect gets suppressed, freezing Yϕ. Incorporating the\nparametric dependence on α,κHandκmore accurately,\nwe get\nYϕ≈ −0.1ακH\nκsin 4θin\u0012200\ng∗(TAD)\u00131/4rmϕ\nMP.(10)\nSee Supplemental Material for details. The quantity Yϕ\nremains conserved throughout the rotational evolution\nof the scalar field. Following the decay of ϕ, the net ϕ\nnumber efficiently transfers to the actual baryon number\nin the visible sector. For an electroweak scale mϕ, a\ntypical value of θin, and the naturally expected range\nof model parameters α, κH, κ=O(0.1−1),Yϕattains\nO(10−10), which agrees with the experimentally observed\nYB.\nWe have checked that Eq. (10) is consistent with the\nnumerical calculations for interesting parameter ranges,\nand the summarized results are depicted in Fig. 1.\nAsymmetry transfer.— The thermalization of ϕ\nwith the SM thermal bath and the subsequent transfer of3\nFIG. 1. Analytic and numerical results for |Yϕ|. The fidu-\ncial parameters are shown at the bottom of the plot, and\nthe colored lines and bands show the case where one of the\nparameters is varied. We show analytic results in Eq. (10)\nforθin=π/8 (dashed line) and θin=π/40 (dotted line),\nand numerical results (bands) for θinvalues between them.\nθin=π/8 corresponds to the maximum Yϕ, while θin=π/40\nis the minimum within 10% tuning. Note that the observed\nasymmetry value is shown in the green band and the labels\nof the horizontal axis for different colors are shown below the\nplot. See Supplemental Material for details.\nasymmetry to the SM sector take place via the neutrino\nportal. From Eq. (3), the relevant interactions are given\nby\n∆L=¯Ni¯σµ∂µN−\u0012\nyνℓhN+1\n2λNϕNN +h.c.\u0013\n.(11)\nThrough the Yukawa coupling yν, the RH neutrinos are\nproduced from the SM thermal bath. The production\nofNbecomes cosmologically important when Γ N(TN) =\nH(TN), where\nΓN≈4×10−3y2\nνT (12)\nis the production rate of Nfrom the SM plasma [25–\n27]. The neutrino Yukawa coupling can be written as\nyν=√mNmν/vfor the RH neutrino mass mNin the\nvacuum. Thus, TNis given by [28]\nTN∼5mN\u0012Pmν\n0.05 eV\u0013\n. (13)\nIfTN> T AD, the population of Nleads to the thermal\npotential of ϕas∼λ2\nNT2|ϕ|2that spoils the Affleck-Dine\nmechanism. However, we can easily control the thermal\neffect by taking a relatively small neutrino Yukawa cou-\npling. For mN≪p\nmϕMP∼1010GeV, we can safely\nignore the effect of Nfor the scalar potential of ϕ.\nDuring the thermalization, we can neglect the slight\nchange of the entropy density sfrom new degrees of free-\ndom in the thermal bath. Before the onset of asymmetry\ngeneration, the radial component of ϕis positioned at thepotential minimum given by Eq. (8). During this phase,\nthe energy density of ϕis dominated by the homogeneous\nkinetic energy and is given by ρϕ∼H3MP. Thus, it\nis negligible in comparison to the radiation energy den-\nsity, ρrad∼H2M2\nP. After the asymmetry generation,\nρϕscales as a−4shortly during free-rolling and as a−3\nafter it starts to oscillate near the origin. Although ρϕ\ndecreases slower than ρrad∝a−4,ρϕremains negligible\ncompared to ρraduntil they get thermalized.\nAs the temperature becomes lower than TN,ϕand\nNare all thermalized with the SM bath through the\nλNϕNN term, and asymmetry of ϕis distributed to\nthe SM lepton sector. The baryon asymmetry is also\ngenerated through the weak sphaleron process when the\nneutrino-portal interaction is in equilibrium before the\nelectroweak phase transition ( TEW≈159 GeV), more\nprecisely before the freeze-out of the sphaleron process\n(Tsp≈132 GeV) [29]. This translates to the lower bound\non the RH neutrino mass as mN≳26 GeV. The bound\nonmN=λN⟨ϕ⟩T=0in the simplest model of neutrino\nimplies that the neutrino mass is Majorana type, i.e. the\nB−Lnumber should be spontaneously broken in the vac-\nuum. The spontaneous breaking of U(1)B−Lin the early\nUniverse should not lead to a wash-out of the existing\nasymmetry.\nAfter all AD sector particles decay, the asymmetry\nof the AD sector is evenly distributed to leptons and\nbaryons due to the sphaleron process. Because ϕcarries\nB−Lcharge −2, the final baryon asymmetry will have\nthe same magnitude and opposite sign as the initial ϕ\nasymmetry Eq. (10), i.e. YB=−Yϕ.\nMajoron phenomenology— The model has two\nSM singlet scalar fields: ϕand ˜N, where ˜Nis the super-\npartner of N. The scalar dynamics at high temperatures\nis mostly dominated by ϕ, while ˜Nplays an important\nrole at low temperatures. Our simple assumption is that\n˜Nhas a tachyonic soft mass, which is set to be the elec-\ntroweak scale for the same reason that mϕis the elec-\ntroweak scale. Then, the scalar potential from Eq. (3)\nfor˜Nat low energy is\n∆V= (λ2\nN|ϕ|2−m2\n˜N)|˜N|2+\u0012αλNmϕ\n2ϕ˜N2+h.c.\u0013\n+λ2\nN\n4|˜N|4+O\u0012αmϕ\nMP˜N4\u0013\n, (14)\nwhere the last term is interactions suppressed by MPand\nis not relevant for scalar dynamics. At high temperatures\nT > T AD,˜Nis trapped at the origin because of a large\npositive mass contribution from λ2\nN|ϕ|2∼HM P. After\ntheYϕgeneration, ϕrolls to the origin with the behavior\nof|ϕ|2∝a−3. Because of the negative mass squared of\n˜N,|˜N|gradually develops a nonzero expectation value as\nλ2\nN|ϕ|2becomes smaller than m2\n˜N. Thereafter, the asym-\nmetry of ϕsmoothly transfers to that of ˜N=1√\n2rNeiθN.4\nFIG. 2. Constraints (shaded region) and future sensitivities\n(dashed and dot-dashed lines) in the mJ−λν(fJ) plane. The\nbrown region is excluded by Planck [32], and the orange region\nis excluded from ∆ Neff>0.3 [1]. The constraint TN> T sp\n(purple region) gives the upper bound on fJ, while the per-\nturbativity condition for λN(gray region) gives the lower\nbound. The allowed parameter space (white region) is near\nfJ=O(100 GeV) and mJ=O(0.1−1 keV), which are pre-\ndicted values by the model. The dashed and dot-dashed lines\nare future sensitivities for ∆ Neff>0.1 and ∆ Neff>0.06, re-\nspectively. ∆ Nth\neffindicates the contribution from the thermal\nproduction, while ∆ Neffdenotes the contribution from the\nrelic Majoron. Note all the allowed parameter space can be\nprobed with the sensitivity ∆ Neff<0.06.\nThe dynamics of ˜Nconserves the B−Lnumber, and thenet density of the AD sector\n¯nB−L|AD=−2¯nϕ+ ¯nN=−2r2\nϕ˙θϕ+r2\nN˙θN≡f2\nJ˙θJ(15)\nis decreasing as a−3until the explicit breaking effect be-\ncomes important. The U(1)B−Lspontaneous breaking\nscale fJ=q\n4r2\nϕ+r2\nNis gradually dominated by rN.\nThe associated Nambu-Goldstone boson, Majoron, cor-\nresponds to J(x) =fJθJ(x), and the B−Lasymmetry\nis carried by the kinetic energy of J[30, 31].\nThe explicit breaking term of the scalar potential pro-\nvides a damping effect. First of all, it gives a scalar po-\ntential for Majoron as VJ∼m2\nJf2\nJcos(J/fJ), where\nmJ∼fJrαmϕ\nMP=O(0.1−1 keV) . (16)\nWe note that ˙θJremains much larger than mJwhen\nthe AD sector is thermalized with the SM sector, which\nmakes the damping effect negligible so the total B−L\nnumber is nearly conserved. For Tsp≲T≲TN, the ther-\nmal potential for rϕandrNcould lead to the trap of the\nscalar fields at the origin ⟨fJ⟩= 0 depending on the size\nof their thermal mass of O(λNT). After the sphaleoron\nprocess freezes out, YBis entirely frozen.\nEventually, U(1)B−Lis spontaneously broken with the\nvacuum expectation values\n⟨|ϕ|⟩ ∼αmϕ\nλNand (17)\n⟨|˜N|⟩ ∼fJ∼mϕ\nλN, (18)\nassuming m˜N∼mϕandα≲1. The Majorons are co-\npiously produced around the B−Lphase transition and\ntheir density easily becomes an equilibrium value [33].\nThe Majoron decays after BBN and contributes to\n∆Neff=4\n7\u001211\n4g∗S(T0)\ng∗S(Td)\u00134/3\nMax\"\n1,30ζ(3)\nπ4\u0012g∗S(T0)\ng∗S(Td)\u0013−1/3mJ\nTdecay#\n, (19)\nwhere T0is the temperature today, Td∼0.1mNis the\nthe temperature at Majoron decoupling [34], and Tdecay\nis the temperature at Majoron decay. See Supplemen-\ntal Material for details. If fJis excessively large, the\nMajoron decays much later than when it becomes non-\nrelativistic. Since the energy density of non-relativistic\nparticles redshifts slower than radiation, it contributes\nmore to Neffafter decay, as shown in Eq. (19). The Neff\nbound, ∆ Neff<0.3 (2σ) from [1], constrains fJnot to\nsignificantly exceed the electroweak scale.\nIn Fig. 2, we show the constrained parameter space\nalong with future ∆ Neffsensitivities in the planeofmJand λν(fJ), where λν≈0.05 eV /fJis the\nMajoron-neutrino coupling. We have used O(1) model\nparameters to get mJandfJfrom the model parameters\nas shown in the plot, so note it can be modified by\nO(1) for different parameter choices. The orange region\nand lines come from the ∆ Neffcontribution from the\nrelic Majoron, while the brown region and line are\nfrom the late-time production from J↔νν. We\nrequire TN> T sp, otherwise no baryon asymmetry is\ngenerated since thermalization of Nhappens after the\nweak sphaleron process ceases. mN< f Jis imposed\nbecause the RH neutrino mass mN=λN⟨|ϕ|⟩needs to5\nbe smaller than the Majoron decay constant fJ>⟨|ϕ|⟩\nwith λN<O(1). The allowed parameter space nearly\npoints fJ=O(100 GeV) and mJ=O(0.1−1 keV),\nwhich can be well predicted by our scenario. The future\nCMB observations for ∆ Nefffrom Simons Observatory\n[35] or CMB-S4 [36] can probe all the allowed parameter\nspace.\nDiscussion.— The neutrino-portal Affleck-Dine\nmechanism yields the observed small baryon asymmetry\nin the Universe as a direct consequence of a hierarchy be-\ntween two mass parameters inherent in the scenario, mϕ\nandMP. Both parameters are linked to the symmetry-\nbreaking scales and provide an organizing principle of the\nscalar potential. MPemerges as a cutoff scale in higher-\ndimensional operators that break global symmetries due\nto the quantum gravity effects. On the other hand, mϕ\nrepresents the soft supersymmetry breaking scale, which\nis the energy scale beyond which the potential of a scalar\nfield remains shielded against UV-sensitive quantum cor-\nrections. This naturally prompts the exploration of the\ncorrelation between mϕand the electroweak scale v.\nAn interesting relation between mϕandvarises in phe-\nnomenological observables as well. As shown in Fig. 2,\nthe phenomenologically allowed values for the Majoron\ndecay constant fJ, hence also for mϕ, are close to the\nelectroweak scale. This result is independent of the the-\noretical motivation for linking mϕtov. Our scenario\nestablishes a fundamental correlation between the elec-\ntroweak scale and the origin of asymmetry in the Uni-\nverse.\nIn contrast to conventional high-scale baryogenesis\nmodels, which typically do not predict any observable\nfor new physics, our model has distinct low-energy ob-\nservable implications in cosmology, yet asymmetry is still\ngenerated at a high scale, TAD∼1010GeV. The pres-\nence of a light Majoron with a keV mass is well predicted\nin the model, and the consequential effects on Neffcan\nbe measured in the near future.\nThe reheating temperature needs to be higher than\nTADfor the AD mechanism to work, but it cannot be\nsignificantly higher for two reasons. Firstly, a higher re-\nheating temperature leads to a larger baryon isocurvature\nperturbation induced by a light field, the angular com-\nponent of ϕin this case, during inflation [37]. Secondly,\nthe scalar field ϕundergoes negative damping at temper-\natures above TADif it is displaced from the fixed point\n[6, 38]. The AD mechanism occurring shortly after re-\nheating successfully exhibits all the previously discussed\nproperties without these issues.\nOur minimal scenario with SUSY has another phe-\nnomenological observable, which is that the lightest\nneutrino is almost massless, mlight∼(mN/MP)Pmν.\nThe future observations of the upper bound on neutrino\nmasses from various experiments [39, 40] may provide a\nstrong hint of the model. The more detailed connectionto the spectrum of superpartners of the SM particles\nis also an important question, and it requires a more\ndedicated study of the SUSY-breaking mediation mech-\nanism that will be revealed from the future observation\nof SUSY particles [41, 42]. We leave this aspect to\nfuture works because all other aspects presented in this\nwork yield consistent results as long as the same scalar\npotentials are employed. Here, SUSY only serves as a\ntool for organizing the scalar potentials.\nAcknowledgments.— We are grateful to Arushi\nBodas and Keisuke Harigaya for helpful comments. JHC\nis supported by Fermi Research Alliance, LLC under\nContract No. DE-AC02-07CH11359 with the U.S. De-\npartment of Energy, Office of Science, Office of High\nEnergy Physics. KSJ is supported by the National Re-\nsearch Foundation (NRF) of Korea grants funded by\nthe Korea government: Grants No. 2021R1A4A5031460\nand RS-2023-00249330. CHL and CSS are also sup-\nported by the NRF of Korea (NRF-2022R1C1C1011840,\nNRF-2022R1A4A5030362). This work was performed in\npart at the workshop “Dark Matter as a Portal to New\nPhysics” supported by APCTP.\n∗jhchang@fnal.gov\n†ksjeong@pusan.ac.kr\n‡chlee@o.cnu.ac.kr\n§csshin@cnu.ac.kr\n[1] N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6\n(2020), [Erratum: Astron.Astrophys. 652, C4 (2021)],\narXiv:1807.06209 [astro-ph.CO].\n[2] D. Bodeker and W. Buchmuller, Rev. Mod. 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Niibo, JHEP 07, 132 (2023),\narXiv:2304.04430 [hep-ph].\nSupplemental Material\nSCALAR POTENTIAL WITH SOFTLY BROKEN SUPERSYMMETRY\nSupersymmetry (SUSY) is a generalization of the spacetime symmetries relating fermions and bosons. Because\nSUSY guarantees that quadratic divergences of all orders will cancel out in perturbation theory, the scalar potential\ncan be well organized without worrying about various quantum corrections from UV physics. The supersymmetric\nLagrangian can be systematically derived with the concept of superspace and superfield [S43]. Superspace is the\ncoordinate space ( x, θ,¯θ) that contains anticommuting coordinates θα,¯θ˙α(α= 1,2) in addition to the usual spacetime\ncoordinate x. Superfield is a field defined on superspace that is valued in a supermultiplet, an irreducible representation\nof the SUSY algebra. This contains both fermion and boson. The relevant superfield in our discussion is a chiral\nsuperfield, which can be written as\nΦi(y, θ) =ϕi(y) +√\n2θψi(y) +θ2Fi(y) (S1)\nfory=x+iθσµ¯θ.ϕiis a complex scalar, and ψiis a chiral fermion, where ϕiandψiare superpartners of each other.\nFiis the auxiliary field whose nonzero expectation value contributes to the spontaneous breaking of SUSY.\nThe supersymmetric model of these fields ϕiandψican be constructed as follows. For a given superpotential W(ϕ)\n(a holomorphic function of ϕi),\nW(ϕ) =µijϕiϕj+λijkϕiϕjϕk+κijkl\nMPϕiϕjϕkϕl+··· (S2)\nand the K¨ ahler potential K(ϕ, ϕ†) (a real-valued function of ϕiandϕ∗\ni),\nK(ϕ, ϕ†) =ϕ†\niϕi+κij\nM2\nPϕ†\niϕiϕ†\njϕj+···, (S3)2\nthe supersymmetric Lagrangian is given by\nLsusy=Z\nd2θd2¯θ K(Φ,Φ†) +Z\nd2θ W(Φ) +Z\nd2¯θW(Φ†)\n=Ki¯j\u0012\n∂µϕi∂µϕ∗\nj+i\n2Dµψiσµ¯ψj−i\n2ψiDµ¯ψj\u0013\n(S4)\n−(K−1)¯ijWiWj−1\n2Wijψiψj−1\n2Wij¯ψi¯ψj+O\u0012∂3K\n∂ϕi∂ϕj∂ϕ∗\nk\u0013\n, (S5)\nwhere θ2=ϵαβθβθα,¯θ2=ϵ˙α˙β¯θ˙α¯θ˙βand\nWi=∂W(ϕ)\n∂ϕi= (Wi)∗, W ij=∂2W(ϕ)\n∂ϕi∂ϕj= (Wij)∗,\nKi¯j=∂2K(ϕ, ϕ†)\n∂ϕi∂ϕ∗\nj, Ki¯j(K−1)¯jk= (K−1)¯ijKj¯k=δik. (S6)\nThe metric of the kinetic term is given by Ki¯j≃δij, so the supersymmetric part of the scalar potential becomes\nVsusy≃X\ni\f\f\f\f∂W\n∂ϕi\f\f\f\f2\n. (S7)\nThe higher order term in Ki¯jprovides the interactions between the scalar fields and the kinetic terms as Eq. (6).\nBesides the supersymmetric part, the scalar potential also contains the SUSY-breaking part as the result of the\nspontaneous breaking of SUSY. When SUSY is spontaneously broken, its effect must not lead to quadratic divergences\nin scalar masses. Such a class of supersymmetry breaking is called Soft SUSY breaking [S20]. A phenomenological\napproach that takes these effects into account is to decompose the sectors into the visible sector which includes the\nAD sector (Φ), the SUSY-breaking sector ( X), and the messengers between them [S44]. For the discussion about the\ndynamics of the visible sector in the SUSY breaking background, we do not need to know the detailed mechanism of\nSUSY breaking. Its effect can be simply parameterized by the SUSY breaking spurion superfield\nX=θ2FX, (S8)\nwith a nonzero expectation value of FX. The interaction between the SUSY breaking sector and the visible sector is\nmediated by some interactions suppressed by the messenger scale M(10 TeV ≲M≲MP). For some reason, if the\nvisible sector is completely decoupled from the SUSY breaking sector, it implies M→ ∞ .\nThe interactions between Xand the visible sector fields at scales well below Mcan be written as\n∆Wsoft=1\nMX\u0012\nαW(Φ) + αiΦi∂W\n∂Φi+···\u0013\n, (S9)\n∆Ksoft=1\nM2X†X\u0010\nκiΦ†\niΦi+···\u0011\n. (S10)\nThe overall soft SUSY breaking mass scale in the visible sector is parameterized by\nm=FX\nM. (S11)\nThe messenger scale Mand the size of dimensionless couplings are determined by the detailed mediation mechanism.\nHowever, it is natural that the overall mass scale mis universal if the SUSY breaking terms originate from the same\nmediation mechanism. The corresponding soft SUSY breaking potential is\n∆Vsoft=−X\niκim2|ϕi|2− \nαmW (ϕ) +X\niαim ϕi∂W\n∂ϕi+h.c.!\n(S12)\nthat provides contributions of Eqs. (5) and (14).3\nCALCULATIONS OF Yϕ\nWe provide more detailed explanations for the parameter dependence of Yϕexpressed in Eq. (10). The value of\nYϕ(t) can be obtained by integrating the equation of motion of nϕin Eq. (9) over twith the potential Eq. (7) and the\ninitial condition Yϕ(tin) = 0 in the radiation-dominated era:\nYϕ(t) =−Zt\ntin1\ns∂V\n∂θϕdt′(S13)\n=−1\n4\u001290\nπ2g∗\u00131/4\u0012καm ϕ\n2MP\u0013Zt\ntinr4\nϕ(t′) sin 4 θϕ(t′)\n(H(t′)MP)3/2dt′. (S14)\nFor the generation of asymmetry, the relevant dynamics arises when H(t) =1\n2t∼mϕ, so we define t∗with H(t∗) =mϕ\nand rewrite Eq. (S14) with a dimensionless time variable ˜t=t/t∗,\n˜Yϕ(˜t)≡Yϕ(t=˜t t∗) =−1\n6\u001290\nπ2g∗\u00131/4\u0010\nακH\nκsin 4θin\u0011rmϕ\nMPZ˜t\n˜tin˜t′3/2\n2˜r4\nϕ(˜t′) sin 4 ˜θϕ(˜t′)\nr4∗sin 4θind˜t′. (S15)\nHere, r∗=⟨rϕ⟩in Eq. (8) with H=mϕ,˜tin=tin/t∗, and ˜ rϕ(˜t) and ˜θϕ(˜t) are rϕ(˜t t∗) and θϕ(˜t t∗), respectively.\nOne can perform a numerical calculation for the integration part, but we can start with analytic calculations with\nseveral approximations. The radial and angular fields show different characteristics before and after the rolling of the\nfields. For ˜t≪1, the angular field is frozen, and the radial field follows Eq. (8), which gives rϕ(t)∝t−1/2, so we get\n˜r4\nϕ(˜t) sin 4 ˜θϕ(˜t)/(r4\n∗sin 4θin)≈˜t−2. On the other hand, for ˜t≫1, the real and imaginary parts of the complex scalar\nfieldϕare oscillating as the independent harmonic oscillators with the angular frequency mϕ. At the same time, the\naverage value of rϕ(t) decreases as a−3/2∝t−3/4, therefore rϕ(t)∝t−3/4cosmϕt. The initial eccentricity is close\nto the unity so θϕ(t) remains θinnear the maximum of the oscillations. Thus we get ˜ r4\nϕ(˜t) sin 4 ˜θϕ(˜t)/(r4\n∗sin 4θin)≃\n˜t−3(cos1\n2(˜t−1))4for˜t≫1. Then the final Yϕatt→ ∞ can be separated and analyzed as\nYϕ≈ −1\n6\u001290\nπ2g∗\u00131/4\u0010\nακH\nκsin 4θin\u0011rmϕ\nMP Z1\n01\n2˜t1/2d˜t+Z∞\n1(cos1\n2(˜t−1))4\n2˜t3/2d˜t!\n≈ −1\n6\u001290\nπ2g∗\u00131/4\u0010\nακH\nκsin 4θin\u0011rmϕ\nMP\u0010\n1 + 0 .48\u0011\n(S16)\n≈ − 0.1ακH\nκsin 4θin\u0012200\ng∗(TAD)\u00131/4rmϕ\nMP,\nwhere the integration range of is simplified to (0 ,∞) because the initial and final time values are far from ˜t= 1.\nThe numerical value of Yϕis obtained from the following process. From the Eq. (7), the equation of ϕ(t) is\n¨ϕ+ 3H˙ϕ+ (m2\nϕ−κHH2)ϕ−καm ϕ\nMPϕ∗3+3κ2\nM2\nPϕ3ϕ∗2= 0. (S17)\nThen, we can evaluate Eq. (S14) with\nr4\nϕsin 4θϕ=−2i(ϕ4−ϕ∗4) (S18)\nfor the numerical solution of Yϕ(t). Fig. S1 shows analytic and numerical evolutions of\n˙˜yϕ(˜t)≡˜t5/2\n2˜r4\nϕ(˜t) sin 4 ˜θϕ(˜t)\nr4∗sin 4θin≈d\ndln˜t ˜Yϕ(˜t)\n˜Yϕ(˜t= 1)!\n. (S19)\nThe black dashed line represents the analytic approximation given by Eq. (S16) and colored real lines are numerical\nvalues with the solution of ϕ(t). Here, we note that the quadratic potential of rϕ(t) changes its sign at t=tflip, where\nκHH2(tflip) =m2\nϕ. Depending on the size of κH,tflipis different from t∗, but we confirm that t∗, rather than tflip, is\nthe time when the behavior of Yϕchanges significantly, as shown in Fig. S1. We also display the final analytical and\nnumerical results for Yϕin Fig. 1, which shows how well the parameter-dependence of Yϕin the analytical expression\nagrees with the numerical results.4\n-1.0-0.50.0.51.0\nFIG. S1. The κH-dependence of ˙˜yϕ(˜t) defined by Eq. (S19) in radiation dominated era. The black dashed line is the analytic\napproximation given by Eq. (S16) that shows a good agreement with the actual value from the solution of ϕ(t) to Eq. (S17)\nwith a small deviation around ˜t= 1. This is true for other values of κH.\nLATE-TIME EVOLUTION OF MAJORONS\nThe Majoron emerges after the B−Lsymmetry is spontaneously broken. Its mass in our scenario is O(0.1−1)\nkeV, which is much smaller than all B−Lcharged particles except the active neutrinos. The efficient way to write the\nrelevant Lagrangian for Majoron cosmology is to take a field basis of heavy particles Φ i(except the active neutrinos\nν) as\nΦi→Φiexp(iqΦiθJ), (S20)\nwhere qΦiis the B−Lcharge of Φ i. Then, only the Majoron and active neutrinos are transformed as\nU(1)B−L:θJ→θJ+c, ν i→e−icνi. (S21)\nAfter integrating out these massive fields, the relevant effective Lagrangian for the Majoron becomes\nLeff=1\n2f2\nJ(∂µθJ)2−\u00121\n2(mν)ije2iθJνiνj+h.c.\u0013\n−αm\n16MP\u0010\nκ⟨r4\nϕ⟩e−8iθJ+κ2⟨r3\nϕrN⟩e−5iθJ+κ3⟨r2\nϕr2\nN⟩e−2iθJ+κ4⟨rϕr3\nN⟩eiθJ+κ5⟨r4\nN⟩e4iθJ+h.c.\u0011\n(S22)\n=1\n2(∂µJ)2−1\n2m2\nJJ2−J\u0012i(mν)ij\nfJνiνj+h.c.\u0013\n+···,\nwhere rϕ=√\n2|ϕ|,rN=√\n2|˜N|, the Majoron decay constant fJ=qP\ni2q2\nΦi⟨|Φi|2⟩=q\n4⟨r2\nϕ⟩+⟨r2\nN⟩, and the\ncanonically normalized Majoron J(x) =fJθJ(x). There are several contributions to the Majoron mass from the\nexplicit B−Lbreaking terms in the superpotential suppressed by 1 /MP. Although the correct form of the Majoron\nmass is quite complicated and includes many parameters, we can easily identify its order of magnitude as m2\nJ∼\n(αm/M P)f2\nJ.\nIn our scenario, the Majoron Jis thermally produced after B−Lphase transition and its abundance is frozen at\nTd∼0.1mN, where mNis the right-handed neutrino mass and Tdis the freeze-out temperature of the Majoron. Then\nrelativistic Majoron contribution to ∆ Nefftoday is given by\n∆Neff=4\n7\u001211\n4g∗S(T0)\ng∗S(Td)\u00134/3\n. (S23)\nSince the Majoron mass is much smaller than mN, the initial Majorons are relativistic. If the Majoron becomes\nnon-relativistic before it decays away, an additional factor FNRshould be included as matter redshifts slower than5\nradiation,\nFNR=Zd3p\n(2π)3mJ\nexp (p/TJ,decay)−1\u001eZd3p\n(2π)3p\nexp (p/TJ,decay)−1(S24)\n=30ζ(3)\nπ4\u0012g∗S(T0)\ng∗S(Td)\u0013−1/3mJ\nTdecay, (S25)\nwhere TJ,decay andTdecay are the temperatures of the Majoron and the photon, respectively, at which the Majoron\ndecays. The rate of Majoron decay to neutrinos Γ Jis [S45]\nΓJ=mJ\n16πf2\nJX\nm2\nν (S26)\n≈1\n3.3×108second\u0010mJ\n1 keV\u0011\u0012100 GeV\nfJ\u00132\u0012Pm2\nν\n10−3eV2\u0013\n. (S27)\nBy using Γ J=H1, we get\nTdecay≈141 eV\u0010mJ\n1 keV\u00111/2\u0012100 GeV\nfJ\u0013\u0012Pm2\nν\n(0.05 eV)2\u00131/2\n, (S28)\nassuming the decay happens in the radiation-dominated era. The final contribution to ∆ Neffis given by\n∆Neff=4\n7\u001211\n4g∗S(T0)\ng∗S(Td)\u00134/3\nMax[1 , FNR] (S29)\n≈\n\n0.029\u0010\n100\ng∗S(Td)\u00114/3\n, FNR≤1\n0.23\u0000mJ\n1 keV\u00011/2\u0010\nfJ\n100 GeV\u0011\u0010\n(0.05 eV)2\nPm2\ni\u00111/2\u0010\n100\ng∗S(Td)\u0011\n, FNR>1(S30)\n1For more precise calculation, we need to include the time dilation\neffect for the lifetime of the Majoron as Γ JmJ/EJ=H. Howeverthis effect on ∆ Neffis negligible because ∆ Neffdoes not depend\nonTdecay if Majorons decay while relativistic."    },    {        "title": "2401.13736v1.Structure__kinematics_and_time_evolution_of_the_Galactic_Warp_from_Classical_Cepheids.pdf",        "content": "MNRAS 000, 1–24 (2023) Preprint 26 January 2024 Compiled using MNRAS L ATEX style file v3.0\nStructure, kinematics and time evolution of the Galactic Warp from\nClassical Cepheids\nMauro Cabrera-Gadea1★, Cecilia Mateu1, Pau Ramos2, Mercè Romero-Gómez3,4,5, Teresa Antoja3,4,5,\nand Luis Aguilar6\n1Departamento de Astronomía, Instituto de Física, Universidad de la República, Iguá 4225, CP 11400 Montevideo, Uruguay\n2National Astronomical Observatory of Japan, Mitaka-shi, Tokyo 181-8588, Japan\n3Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), Martí i Franquès 1, E-08028 Barcelona, Spain\n4Departament de Física Quàntica i Astrofísica (FQA), Universitat de Barcelona (UB), Martí i Franquès 1, E-08028 Barcelona, Spain\n5Institut d’Estudis Espacials de Catalunya (IEEC), c. Gran Capità, 2-4, E-08034 Barcelona, Spain.\n6Instituto de Astronomía, Universidad Nacional Autonóma de México, Apdo. Postal 877, Ensenada, 22800 Baja California, Mexico\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nThewarpisawell-knownundulationoftheMilkyWaydisc.Itsstructurehasbeenwidelystudied,butonlysinceGaiaDR2has\nitbeenpossibletorevealitskinematicsignaturebeyondthesolarneighbourhood.Inthisworkwepresentananalysisofthewarp\ntraced by Classical Cepheids by means of a Fourier decomposition of their height ( 𝑍) and, for the first time, of their vertical\nvelocity (𝑉𝑧). We find a clear but complex signal that in both variables reveals an asymmetrical warp. In 𝑍we find the warp to\nbealmostsymmetricinamplitudeatthedisc’soutskirts,withthetwoextremesneverbeingdiametricallyopposedatanyradius\nand the line of nodes presenting a twist in the direction of stellar rotation for 𝑅>11kpc. For𝑉𝑧, in addition to the usual 𝑚=1\nmode, an𝑚=2mode is needed to represent the kinematic signal of the warp, reflecting its azimuthal asymmetry. The line of\nmaximum vertical velocity is similarly twisted as the line of nodes and trails behind by ≈25◦. We develop a new formalism to\nderive the pattern speed and change in amplitude with time ¤𝐴of each Fourier mode at each radius, via a joint analysis of the\nFourier decomposition in 𝑍and𝑉𝑧. By applying it to the Cepheids we find, for the 𝑚=1mode, a constant pattern speed in\nthe direction of stellar rotation of 9.2±3.1km/s/kpc, a negligible ¤𝐴up to𝑅≈14kpc and a slight increase at larger radii, in\nagreement with previous works.\nKeywords: Galaxy:disc–Galaxy:structure–Galaxy:kinematicsanddynamics–Galaxy:evolution–stars:variables:Cepheids\n1 INTRODUCTION\nThe warp is an undulation in a galactic disc that makes its mean\nvertical height deviate from the mid plane in the outskirts of the\ngalaxy. Between 40−50%of edge-on disc galaxies are found to\nbe warped (Sanchez-Saavedra et al. 1990; Reshetnikov & Combes\n1998), which implies that warps should be long lived phenomena\nor the formation mechanism a very recurrent one in the history of\ngalactic discs. The Milky Way is not an exception, having a warp\nwhose structure has been widely studied with different tracers like\nHI (Levine et al. 2006), dust (Marshall et al. 2006) as well as with\ndifferentstellarpopulations(López-Corredoiraetal.2002;Romero-\nGómez et al. 2019; Skowron et al. 2019a; Chen et al. 2019; Cheng\net al. 2020; Chrobáková et al. 2020; Li et al. 2023). Although the\nGalacticwarphasbeenknownforalongtime(Burke1957),itsorigin\nisstillpuzzling.Inordertoelucidatethehistoryandformationofthe\nMilkyWay’swarp,itisimportanttocharacteriseitsmainproperties,\nas its structure and kinematics.\nClassicalCepheidshaveprovenexceptionallyusefulintracingthe\nstructureandkinematicsofthewarpofferingseveralkeyadvantages\n★E-mail: mauro.cabrera@pedeciba.edu.uyto study the Galactic disc (Bobylev 2013; Skowron et al. 2019b,a;\nChen et al. 2019). Being very young stars (with ages up to a few\nhundred million years, e.g. Catelan & Smith 2015) it is expected\nthat they have recently inherited the warped structure of the HI gas\nwhere they have formed, while still having cold kinematics (vertical\nvelocity dispersion <5km s−1, Chen et al. 2019) making it easier\nto observe the warp signal as secular dynamics has not had time to\n’heat’ordisturbit,asitwouldhaveforolderpopulations(Binney&\nTremaine2008,Sec.8.1).Also,belongingtosuchayoungpopulation\nmeansthereisnocontaminationfromanyotherGalacticcomponent,\ne.g. the thick disc or halo, which means they exclusively trace the\nGalactic thin disc. In addition, Classical Cepheids are well-known\nstandard candles (Leavitt 1908; Leavitt & Pickering 1912), offering\nextremelyprecisedistancemeasurements( ∼3%errors);theycanbe\nreliably identified based on their variability, making contamination\nfrom other stars negligible (e.g. Jayasinghe et al. 2019; Rimoldini\net al. 2019, 2023); and being luminous ( 500< 𝐿/𝐿⊙<20,000,\nCatelan & Smith 2015), makes them observable throughout a large\nextent of the disc even with the optical surveys used to identify\nthem at present (Udalski et al. 2018; Ripepi et al. 2023). Their only\ndisadvantage is that they are relatively scarce, with fewer than 2500\nClassical Cepheids in the deepest and most complete catalogues of\n©2023 The AuthorsarXiv:2401.13736v1  [astro-ph.GA]  24 Jan 20242M. Cabrera-Gadea et al.\ntheGalacticdisctodateprovidedbyOGLE-IV(Udalskietal.2018)\nand the Third Gaia Data Release (DR3, Gaia Collaboration et al.\n2020; Ripepi et al. 2023). For these reasons, Cepheids have been\nused to study the 3D structure of the warp in more detail than any\nother stellar population (Skowron et al. 2019b), providing evidence\nforatwistedlineofnodes(Chenetal.2019;Dehnenetal.2023)and\nasymmetry in height between both extremes (Skowron et al. 2019a)\nsimilartotheHIwarp(Levineetal.2006).Theyhavealsobeenused\nto study the kinematics of the warp revealing its characteristic bulk\nvertical motion in the outskirts of the disc (Skowron et al. 2019a).\nTo describe the structure of the warp, several studies have shown\nFourierseriestobeofgreatuseduetotheirversatilitytosummarise\nany warp signal if enough modes are considered (e.g. Levine et al.\n2006; Chen et al. 2019; Skowron et al. 2019a). These studies have\nfocused on describing the structure of the warp, i.e. its mean height\nas a function of radius and azimuth. Using a catalogue of Classical\nCepheidsidentifiedmainlywithOGLE-IVandcombinedwithGaia\nDR2 astrometry, Skowron et al. (2019b) used a Fourier decomposi-\ntion with up to 2 modes ( 𝑚≤2) and a fixed line of nodes (LON) to\npresent the first map of the Galactic warp in the young population\ncoveringoverhalfthedisc.Chenetal.(2019),usingacompilationof\nopticalplusWide-fieldInfraredSurveyExplorer(WISE,Chenetal.\n2018) Cepheid catalogues, studied the azimuthal dependence of the\nLON with radius finding it does not coincide with the Sun-Galactic\nCentre direction and that it presents a leading pattern, following\nBriggs’s rules for HI warps in spiral galaxies (Briggs 1990). For the\nkinematics,Fourierserieshavebeenusedtocharacterisethechanges\nin mean vertical velocity ( 𝑉𝑧) in simulations (Chequers et al. 2018;\nPoggioetal.2021),butinsofartherehavebeennoFourierdecompo-\nsition studies of the warp’s kinematic signal with Cepheids (or any\ntracer) which can reflect and quantify its plausible azimuthal asym-\nmetriesandchangeswithradius.Previousstudieswithother–older–\nstellarpopulationshaveassumedthekinematicsignaltobewellrep-\nresented by an 𝑚=1mode (Poggio et al. 2020; Cheng et al. 2020;\nWang et al. 2020; Chrobáková & López-Corredoira 2021; Dehnen\net al. 2023), as expected from a tilted rings model ( 𝑚=1mode),\nbut Romero-Gómez et al. (2019) argue this model is insufficient to\nexplain the more complex kinematic signature they observed with\nRed Clump stars.\nIn this work we use a Fourier Decomposition method to study\nthe structure and kinematics of the Galactic warp using Classical\nCepheids as tracers. We use the Cepheid catalogue from Skowron\net al. (2019b) combined with kinematic data from Gaia DR3 (Gaia\nCollaboration et al. 2020) to explore the dependence of the ampli-\ntudesandtheazimuthsofthemodesasfreeparametersasafunction\nof radius, which allows us to infer the position of the LON and Line\nof Maximum Vertical Velocity (LMV 𝑧) for a general warp model\nthat accounts for the lopsidedness of the warp. The new method we\npresenthere(Sec.5.2),basedonajointanalysisoftheFourierseries\nfor𝑍and𝑉𝑧,allowsustoinferthetimeevolutionoftheFouriercom-\nponentsofthewarp:i.e.theirpatternspeedandinstantaneouschange\ninamplitude.TheinferenceoftheevolutionarytermsoftheGalactic\nwarphasbeentackledrecentlyusingdifferenttracers,butmostlyun-\nder the tilted rings model which assumes a symmetric warp. Poggio\net al. (2020) and Cheng et al. (2020) have focused on inferring the\npattern speed, while Wang et al. (2020) derived the change in am-\nplitude. These works use general samples of stars with good quality\nGaia DR2 and DR3 astrometry and available radial velocities; by\nnot been focused on a specific tracer or having any age constraints,\nthese parameters are representative of the general population of the\ndiscasweighedbyitsstarformationhistory,i.e.astellarpopulation\nof intermediate age (several Gyrs old). The recent work by Dehnenet al. (2023) derives both evolutionary terms for different radii for\na sample of Cepheids via a tilted rings model, finding differential\nrotation and change in inclination of the rings. Our work uses the\nsame stellar population to derive the time evolution parameters of\nthe warp with a completely independent method.\nThe structure of the present paper is as follows. In Section 2\nwe present the Fourier decomposition method used to describe the\nGalacticwarp’sheightandverticalvelocity(Sec.2.2)andthemethod\nshowinghowthesearecombinedtoderiveeachmode’spatternspeed\nand amplitude change (Section 2.3). In Section 2.4 we present the\ninference model used to estimate the warp model’s parameters, in-\ncluding main conclusions from the inference validation performed\nusingamockcatalogue.InSec.3wedescribethecatalogueofClassi-\ncalCepheidsusedinthiswork.InSection4weapplythemethodsto\nthissampleandsummariseourresultsforthestructureandkinemat-\nicsoftheCepheid’swarp(Sec.4.1)andthoseforthetimeevolution\n(Section 4.2). In Section 5 we discuss our results and compare with\nthepreviousliterature.OurconclusionsaresummarisedinSection6.\n2 FOURIER DECOMPOSITION METHOD\n2.1 Reference Frame\nWebeginbydescribingthecoordinatesystemandreferenceframewe\nusethroughoutthispaper.Theoriginofthereferenceframeisatthe\ngalacticcenter(GC),fixedwithrespecttoanexternalinertialframe.\nPositions can be given in Cartesian, or cylindrical coordinates. The\nX-axis points from the GC away from the Sun, the Y-axis is parallel\ntotherotationvelocityofthediscattheSunpositionandtheZ-axis\nis perpendicular to the Galactic plane forming a right-handed triad.\nIncylindricalcoordinatesweusetheGalactocentricazimuthalangle\n𝜙measuredfromtheX-axistowardtheY-axis(i.e.oppositetoGalac-\ntic rotation). In this coordinate system the Sun is at 𝑅⊙=8.277kpc\n(GRAVITY Collaboration et al. 2022), 𝜙⊙=180◦and𝑍⊙=0.027\nkpc (Chen et al. 2001). For velocities we use a Cartesian system\nwhose origin is at rest with the GC and their axes parallel to the di-\nrections in which the X-Y-Z axes increase. This is an inertial system\nand thus does not rotate with the Galaxy, the Sun being along the\nnegative X-axis only at present. This facilitates the kinematical and\ndynamicaldescriptions.WeassumetheSunhasGalactocentriccarte-\nsian velocity(𝑉𝑥,𝑉𝑦,𝑉𝑧)=(11.10,232.24,7.25)km/s (Schönrich\net al. 2010; Bovy 2015).\n2.2 Fourier Decomposition of the Structure and Kinematics\nWe implement the Fourier decomposition method following Levine\net al. (2006) and Chequers et al. (2018). The disc is divided into\nconcentric Galactocentric rings, in each ring the mean behaviour as\na function of the azimuth for 𝑍and𝑉𝑧described by a Fourier sums\nup to𝑀modes as\n𝑍(𝜙)=𝑀∑︁\n𝑚=0𝐴𝑚sin(𝑚𝜙−𝜑𝑚) (1)\n𝑉𝑧(𝜙)=𝑀∑︁\n𝑚=0𝑉𝑚sin(𝑚𝜙−𝜑𝑉\n𝑚). (2)\nThe amplitudes(𝐴𝑚,𝑉𝑚)and phases(𝜑𝑚,𝜑𝑉𝑚)are free param-\neters, obtained as a function of 𝑅. In what follows we describe the\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 3\nmethod only for 𝑍(𝜙), being analogous for 𝑉𝑧(𝜙). The Fourier rep-\nresentation is flexible enough, even with a small number of modes,\ntodescribemanyknownwarpshapes:forexampleaU-shapedwarp\nwill be mainly described by an 𝑚=0mode with increasing ampli-\ntude as a function of radius; an integral or S-shaped warp, will be\nmainly described by an 𝑚=1mode with increasing amplitude as a\nfunction of radius, and asymmetries will be mainly described by a\ncombination of 𝑚=1and𝑚≥2modes.\nAs we will see in Sec 2.4, it is also convenient to rewrite Eq. 1 in\nlinear form with free parameters 𝑎𝑚,𝑏𝑚as\n𝑍(𝜙)=𝑀∑︁\n𝑚=0𝑎𝑚cos(𝑚𝜙)+𝑏𝑚sin(𝑚𝜙), (3)\nwhere the transformation between 𝑎𝑚,𝑏𝑚and𝐴𝑚,𝜙𝑚is given by\n𝐴𝑚=√︃\n𝑎2𝑚+𝑏2𝑚;𝜑𝑚=arctan2(−𝑎𝑚,𝑏𝑚). (4)\n2.3 Deriving Time Evolution\nInthissectionwepresentanewformalismtoderivetheevolutionary\nterms from the warp, its pattern speed and the change in amplitude\nofeachmodeateachradii,disentangledfromthemotionofthestars.\nFromnowonwedenotethestar’sverticalheightandverticalvelocity\nas𝑧and𝑣𝑧(lowercase) and the Fourier fits to the warp as 𝑍and𝑉𝑧\n(uppercase).\nWe begin by taking a ring at a radius 𝑅and considering a star\nthat has no radial motion and constant angular velocity Ω, that it\nsimply rotates around the Galactic Centre but following the warp of\na razor thin disc. These assumptions are reasonable for dynamically\ncold populations such as the Cepheid stars we use in our analysis.\nGiventhatthestarsfollowthewarp’sshape,theirheight 𝑧(𝑡)attime\n𝑡isgivenbythefunctionalexpressionofthewarp 𝑍(𝜙,𝑡),whichwe\ncanexpressasaFourierseriesevaluatedatthestar’sazimuth 𝜙(𝑡)as\nfollows\n𝑧(𝑡)=𝑍\u0010\n𝜙(𝑡),𝑡\u0011\n=𝑀∑︁\n𝑚=0𝐴𝑚(𝑡)sin(𝑚𝜙(𝑡)−𝜑𝑚(𝑡)). (5)\nWe allow the amplitude and phase of each mode to evolve in\ntime because we are interested in determining their instantaneous\nderivatives¤𝐴𝑚and¤𝜑𝑚. If we take the total derivative of 𝑧(𝑡)with\nrespecttotimeweobtaintheverticalvelocity 𝑣𝑧ofthestar–notthe\nwarp– given by\n𝑣𝑧(𝑡)=𝑀∑︁\n𝑚=0¤𝐴𝑚(𝑡)sin(𝑚𝜙(𝑡)−𝜑𝑚(𝑡))+\n𝐴𝑚(𝑡)cos(𝑚𝜙(𝑡)−𝜑𝑚(𝑡))[𝑚¤𝜙(𝑡)−¤𝜑𝑚(𝑡)].(6)\nAs expected, Eq. 6 involves terms regarding the time evolution of\nthe warp (¤𝐴𝑚and¤𝜑𝑚), and a term regarding the motion of the star\ndue to its own angular velocity ( ¤𝜙(𝑡)= Ω). Now we want to link\nEq. 6, which describes the velocity of just one star at azimuth 𝜙(𝑡),\ntothe𝑍(𝜙)and𝑉𝑧(𝜙)fitsfromtheprevioussection,whichdescribe\nthe mean motion of all stars in the ring at a given time 𝑡0(today).\nIn a razor thin disc the height of the disc at an arbitrary azimuth\nandthepositionofastaratthesameazimuthmustexactlycoincide.\nThus, it follows that the vertical height 𝑧(𝑡0)and vertical velocity\n𝑣𝑧(𝑡0)of a star at𝑡0and azimuth 𝜙(𝑡0)=𝜙0must coincide withthe Fourier fits ( 𝑍(𝜙0),𝑉𝑧(𝜙0)) we obtained at that same 𝑡0time.\nTaking𝑡0as today,𝐴𝑚(𝑡0)=𝐴𝑚,0and𝜑𝑚=𝜑𝑚,0, the amplitudes\nand phases obtained from the Fourier fits from Eqs. 1 and 2.\nSimilarly,theverticalvelocity 𝑣𝑧(𝑡0)ofthestarmustalsocoincide\nwiththemeanverticalvelocityobtainedfromourFourierfit 𝑉𝑧(𝜙0),\nevaluated at the star’s azimuth. Setting 𝑉𝑧(𝜙0)=𝑣𝑧(𝑡0)in the left\nhand side of Eq. 6 and expressing 𝑉𝑧(𝑡0)as the Fourier fit for 𝑣𝑧in\nits linear form (as shown in Eq. 3 for 𝑍) we obtain\n𝑀∑︁\n𝑚=0𝑎𝑉\n𝑚cos(𝑚𝜙0)+𝑏𝑉\n𝑚sin(𝑚𝜙0)=𝑀∑︁\n𝑚=0¤𝐴𝑚(𝑡0)sin(𝑚𝜙0−𝜑𝑚)+\n𝐴𝑚cos(𝑚𝜙0−𝜑𝑚)[𝑚Ω−¤𝜑𝑚(𝑡0)],(7)\nwhere𝑎𝑉𝑚and𝑏𝑉𝑚arethelinearamplitudesresultingfromtheFourier\nfits in velocity, calculated from the 𝑉𝑚and𝜑𝑉𝑚obtained via Eq. 4.\nTheterms sin(𝑚𝜙0−𝜑𝑚)andcos(𝑚𝜙0−𝜑𝑚)intherighthandside\nof Eq. 7 can be rewritten as\nsin(𝑚𝜙0−𝜑𝑚)=sin(𝑚𝜙0)cos(𝜑𝑚)−cos(𝑚𝜙0)sin(𝜑𝑚)(8)\ncos(𝑚𝜙0−𝜑𝑚)=cos(𝑚𝜙0)cos(𝜑𝑚)+sin(𝑚𝜙0)sin(𝜑𝑚).(9)\nRegrouping the terms as a function of 𝜙0and using the orthogo-\nnality of the Fourier modes, we obtain that the amplitudes 𝑎𝑉𝑚,𝑏𝑉𝑚\nfrom the𝑉𝑧fit are related to the amplitudes 𝐴𝑚and𝜑𝑚from the𝑍\nand the warp evolutionary terms as\n𝑎𝑉\n𝑚=𝐴𝑚[𝑚Ω−¤𝜑𝑚(𝑡0)]cos(𝜑𝑚)−¤𝐴𝑚(𝑡0)sin(𝜑𝑚) (10)\n𝑏𝑉\n𝑚=𝐴𝑚[𝑚Ω−¤𝜑𝑚(𝑡0)]sin(𝜑𝑚)+¤𝐴𝑚(𝑡0)cos(𝜑𝑚). (11)\nSolvingthislinearsystemofequationsfor ¤𝐴𝑚(𝑡0)and[𝑚¤𝜙(𝑡0)−\n¤𝜑𝑚(𝑡0)]and writing back 𝑎𝑉𝑚,𝑏𝑉𝑚in terms of the amplitude and\nphase (𝑉𝑚,𝜑𝑉𝑚), the evolutionary terms of the warp are given by\n[𝑚Ω−¤𝜑𝑚(𝑡0)]=𝑉𝑚\n𝐴𝑚sin(𝜑𝑚−𝜑𝑉\n𝑚) (12)\nand\n¤𝐴𝑚(𝑡0)=𝑉𝑚cos(𝜑𝑚−𝜑𝑉\n𝑚). (13)\nAssumingthatthe 𝑚-thmodehasangularvelocity 𝜔𝑚,thensetting\n𝜑𝑚=𝑚𝜔𝑚𝑡+𝜑𝑚,0inEq.12,wegetthepatternspeedforeachmode\nas\n𝜔𝑚=Ω−𝑉𝑚\n𝑚𝐴𝑚sin(𝜑𝑚−𝜑𝑉\n𝑚). (14)\nTherefore,havingconnectedtheFourierfitsin 𝑍and𝑉𝑧atagiven\nradius,Eqs.13and14describehoweachpatternspeedandamplitude\nchangeintime,allowingareconstructthetimeevolutionofthewarp\nas a function of radius.\nWeleaveforafutureworkthepublicationofamoregeneralframe-\nwork that consider an azimuthal dependence not only of the vertical\nmotionofthestars,butalsotheirradialandazimuthalvelocity,which\nwould presumably result in a better inference of the time evolution\nof individual Fourier modes of the warp.1\n1Using this framework with standard values allow us to conclude that the\nradial bulk motions and spiral arms can be ignored in a first-order approxi-\nmation to derive the pattern speed and change in amplitude of the warp.\nMNRAS 000, 1–24 (2023)4M. Cabrera-Gadea et al.\n2.4 The Inference\nWe have used Bayesian Inference to infer the 𝑎𝑚,𝑏𝑚(or𝐴𝑚,𝜙𝑚)\nthatbestdescribethemeanbehaviourofthestarsinagivenringwhen\napplying the methods from Secs. 2.2 and 2.3 to a particular sample.\nBayes’theorem(e.g.Sivia2006)relatesthePosteriordistributionto\nthe Likelihood (L) and the Prior ( 𝑝) probability densities functions\n(PDF) as\n𝑃(X|𝐷,𝐼)∝L(𝐷|X,𝐼)𝑝(X|𝐼), (15)\nwhere𝐷referstothedata, Xtothemodelparametersand 𝐼standsfor\nanyotheravailableinformation.Inourcase, Xisavectorcontaining\nthe linear amplitudes:\nX=[𝑎0,𝑎1,...,𝑎𝑀,𝑏1,...,𝑏𝑀]. (16)\nFor the model parameters we assume uniform priors, with suffi-\nciently and arbitrarily large limits. Assuming that the observations\nareindependent,thelikelihoodisexpressedastheproductofthein-\ndividuallikelihoodofeachsingledatapoint 𝑧����,forwhichweassume\na Gaussian distribution\nL({𝑧}|X,𝐼)=𝑁Ö\n𝑖=11√︃\n2𝜋𝜎2\n𝑖exp\u0014\n−(𝑧𝑖−𝑍(𝜙𝑖,X))2\n2𝜎2\n𝑖\u0015\n(17)\nwhere we take 𝜎2\n𝑖to be the square sum of the uncertainty in the\nmeasurement 𝑧𝑖and theintrinsic dispersion 𝜎𝐼𝐷of the variable, at\nthat ring. This 𝜎𝐼𝐷is introduced to take into account the natural\ndispersion around the mean value that arises from the dynamics of\ntheGalacticdisc;in 𝑣𝑧,itmeasuresthevelocitydispersionand,in 𝑧,\nitmeasureshowthickthediscisatthatring.Theintrinsicdispersion\nisnotafreeparameterinthefit.Weestimateitasthemeandispersion\ninthevariableofinterestinequallyspacedazimuthalbins,weighted\nby the number of stars in each bin because low number statistics\ndominate over observational errors.\nIn our case, because the model is linear in all parameters and we\nhave assumed a uniform prior, the maximum a posteriori (MAP)\nX0coincides with the maximum of Land the posterior is exactly a\nGaussiandistributionwithmean X0andcovariancematrix 𝚺(seee.g.\nSec. 1 in Hogg et al. 2010, for a detailed discussion). The posterior\nPDF can, therefore, be expressed as\n𝑃(X|𝐷,𝐼)=exp\u0002\n−1\n2(X−X0)𝑇𝚺−1(X−X0)\u0003\n(2𝜋)𝑁+1/2\u0002\ndet𝚺\u00031/2(18)\nwhere X0is given by\nAX0=p. (19)\nThe covariance matrix 𝚺is the inverse of A: the matrix that con-\ntains in its entries the \"projection\" of each mode into the other ones\n(see Eq. A2 in Appendix A) weighted by the dispersion in the data,\nand the vector phas the \"projection\" of the data in each mode (see\nEq. A4).\nBecauseweuseaFourierseriestorepresentavariable,onewould\nexpect the modes to be mutually independent and therefore not cor-\nrelated. This is not usually the case. When we have discrete mea-\nsurements, the modes are not mutually orthogonal unless the mea-\nsurements are equally spaced in azimuth and have the same 𝜎𝑖. In\nthis special case Ais diagonal and, in consequence, the covariancematrix is too. This particular distribution allows the modes to be\nmutuallyindependent.Naturally,wewillnevergetthisconfiguration\nfromthedataitself,butthismethodshowsanalyticallythatdatathat\nare more or less uniformly distributed in azimuth are preferred for a\nFourieranalysisofthewholedisc:studieswithasparseandirregular\nazimuthalcoveragewillgetmodesthatarenot\"fundamental\",inthe\nsense that they are not describing the modes of the warp itself . The\neffect gets worse with high-frequency modes 𝑚≥2. This should be\nkept in mind when interpreting results for individual modes, never-\ntheless, it will not affect our conclusions on the description of the\nwarpasawhole(thesumofthemodes)intheregionswellsampled\nby the data.\nFinally,thediscisdividedinringssuchthatwegeta\"continuum\"\nviewofhowthemodesandthewarpchangewiththeradius.Todoso\nwetakeeachringtocontainafixednumber 𝑁ofstarsoutofthetotal\n𝑁totstars,thefirstringstartingwiththestaratthesmallestradius.The\nsecondringwillstartattheradiusofthesecondstarandhaveawidth\nsuchthatitalsocontains 𝑁stars,andsoonforsubsequentrings.This\nscheme implies that the rings will have a varying width, depending\nonthesample’sradialdistribution.Wetaketheradiusassociatedwith\neachringasthemeanradiusofthestarsinit.Thisprocedureallows\nustohaveacontinuousview,withallringshavingthesamenumber\nofstars𝑁and,therefore,constantstochasticnoise.Itmustbekeptin\nmind, however, that only one out of every 𝑁consecutive rings will\nbe independent. Changing the number of stars in each ring changes\nthe smoothened parameters inferred as a function of the radius (the\nbiggerN,thesmootheritgets).Also,thechangeinNmovesthemean\nradius of each ring, the tendency is that a bigger N makes the rings\nto move inwards (smaller radius), as expected for a density profile\nthat decreases with radius.\n2.4.1 Validation with simulations\nHere we present our main conclusions about the performance of the\nmethodsdescribedintheprevioussection,assessedbyapplyingthem\nto mock catalogues constructed from test particle simulations. As\ndiscussedindetailinAppendix.2,weusedatestparticlesimulation\nofawarpeddiscfromRomero-Gómezetal.(2019)tocreateamock\ncatalogue affected by the Gaia DR3 selection function (SF) and\nobservational errors. A fiducial model, unaffected by the SF or by\nerrors, is used as a baseline for comparison of the results of the\nFourier Decomposition. The interested reader may find full details\nand discussion of these results in Appendix. 2.\nOur main results on how the SF affects the recovery of the warp\nas a whole, in different regions of the disc, are summarised here as\nfollows:\n•For𝑍the best sampled region, the quadrants I and III ( 𝑋 <\n0kpc)isrecoveredwell(differencesbetweentherealandtherecov-\nered warp are smaller than 𝜎𝐼𝐷) and the general tendency for all\nradii is recovered for both series summing up to 𝑀=1and𝑀=2.\nFor𝑋 > 0kpc (quadrants I and IV) the SF causes the warp to be\nexaggerated.Thisbiasisreducedforouterradiiasthemainmodeof\nthe warp (𝑚=1) becomes greater than the intrinsic dispersion (see\nFig. B2).\n•For𝑉𝑧the recovery is better than 𝑍, although for the inner\ndisc (𝑅⪅9kpc) the recovery is poor for 𝑋 > 0. The recovery\nin the sampled area is better than in 𝑍for both𝑀=1and𝑀=\n2(differences are smaller than 𝜎𝑉\n𝐼𝐷in most of the disc area, see\nFig. B2).\n•Main conclusion: The recovery of the full model (the Fourier\nsum) in both variables is robust in the well sampled regions for\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 5\n𝑅 > 10kpc, i.e. second and third quadrants. In this region all the\nwarp features are well recovered within the uncertainties given by\nthe Posterior realisations.\nWe also tested how the individual modes are recovered, our main\nconclusion being:\n•The𝑚=0mode is well recovered throughout the disc.\n•The𝑚=1mode tends to be overestimated in amplitude at the\ninnerdisc.Intheouterregions( 𝑅>10kpc)itsamplitudeandphase\nare well recovered in both variables (i.e 𝑍and𝑉𝑧).\n•The𝑚=2modecapturestheasymmetriesandiswellrecovered\nwherethereisdata,i.e.quadrantsIIandIIIand 𝑅>10kpc,buttends\ntobeunderestimatedinamplitudeandthegeneraltrendofthephase\nis poorly recovered in the whole disc.\n•Mainconclusion: Theuncertaintyontherecoveryoftheindivid-\nual modes stems from the correlations between the modes which, in\nturn, appear as a consequence of the imperfect azimuthal coverage.\nThe degeneracies introduced by the correlations mean that different\ncombinationsofamplitudesandphasesfortheindividualmodescan\ngive the same sum model, in a finite azimuth range. A full azimuthal\ncoveragewouldbreakthisdegeneracyandmaketheinferenceonthe\nindividualmodesunique.Themodelessaffectedbythisdegeneracies\nis the𝑚=1mode due to its large amplitude.\nThe intrinsic dispersion in 𝑧is well recovered in the outer disc\nwhere the warp amplitude is larger than the dispersion. In the inner\ndisc𝜎𝐼𝐷tends to be off by 10%. For𝑣𝑧, we find𝜎𝑉\n𝐼𝐷is underesti-\nmated by 3%without dependency on the radius.\nGiven these results, we decide to include up to the 𝑚=2\nmode in the Fourier fits for this work because it offers the least\nbiased recovery for the region of the disc where the warp is\nmost prominent (i.e. outer radii). Reliable results for the inner\nregion of the disc are limited to |𝜙|≳90◦, the region least affected\nbytheSFwithbestcoverage,wherebiasesintherecoveryarelowest.\nWe also tested the inference of the time evolution parameters ¤𝐴𝑚\nand𝜔𝑚. We concluded that the recovery of the ¤𝐴𝑚for𝑚=2is\nunreliable due to the biases and noise. For the 𝑚=1mode we\nconclude:\n•¤𝐴1iswellrecoveredwithinitsuncertaintiesparticularlyforthe\nouter disc\n•The recovered 𝜔1tends to be overestimated due to a slight\noverestimation in 𝐴1, but the mean difference is ≈4km s−1kpc−1.\nIn the outer disc ( 𝑅 > 14kpc) we recover the values of the fiducial\nmodel within the uncertainty.\n3 THE CEPHEIDS SAMPLE\nWe use the catalogue of Milky Way Cepheids from Skowron et al.\n(2019a). The catalogue contains 2385Classical (Type I) Cepheids\nidentified mainly with the OGLE survey (for more details see\nSkowronetal.2019b,a)withphotometricdistancescomputedbased\nonmid-IRphotometryfromtheWide-fieldInfraredSurveyExplorer\n(WISE)andtheSpitzerSpaceTelescope,resultingindistanceuncer-\ntainties of 3%on average. We cross-matched the Cepheid catalogue\n(at1arcsec tolerance) with Gaia DR3 to retrieve proper motions\nfor these stars. Out of the 2381Cepheids with Gaia proper motions,\nonly 860 stars have radial velocities in DR3. In order to curate a ho-\nmogeneous catalogue with full velocity information allowing us to\ncompute𝑣𝑧,weinferthemissinglineofsightvelocityforallstarsin\nthe catalogue by assuming the Cepheid rotation curve from Ablimitet al. (2020) which has a slope of −1.33km s−1kpc−1and takes the\nvalue 232km s−1at the solar radius2.\nWe clean this sample by keeping stars with RUWE <1.4,\n𝜎𝑧≤0.1kpc and𝜎𝑣𝑧≤13km s−1. These upper bounds in 𝑧\nand𝑣𝑧uncertainties guarantee a significant amount of stars whose\nuncertainties are at most of the order of 𝜎𝐼𝐷. To avoid clear out-\nliers due to probable contaminants and the Magellanic Clouds we\nrestrict the analysis to stars with |𝑧|≤2kpc,|𝑣𝑧|≤30km s−1and\n3kpc< 𝑅 < 18kpc. These are very broad cuts that only remove\nveryfew(≈3%)clearoutliers(5sigmas)mostofthemduetothecut\nin𝑉𝑧(only one star is removed for the cut in 𝑍). These constraints\nreduce the sample to a total of 𝑁tot=1997stars.\n4 RESULTS\nHere we present the results obtained by applying the methods de-\nscribed in Sec. 2 to the final sample with 𝑀=2and𝑁=200stars\nin each ring. To calculate 𝜎𝐼𝐷in both variables we use 8azimuth\nbins.Theresultingamplitudesandphasesasafunctionofradiusfor\nthe best fitting (Maximum a Posteriori, MAP) models for 𝑍and𝑉𝑧\nareprovidedinTable1and100posteriorrealisationsareprovidedin\nTable 2. Figures C3 and C4 in the Appendix C show the amplitudes\nand phase (respectively) of each mode in 𝑍and in𝑉𝑧as a function\nof the radius.\n4.1 Structure and kinematics of the warp\nIn the following sections we analyse different features of the warp\nstructureandkinematics.WeanalysethefullFourierseriesobtained.\nSince the validation with simulations indicated results for the indi-\nvidual modes are prone to be biased due to correlations between\nthe modes, we discuss and summarise this in Appendix C for the\ninterested reader.\n4.1.1 General structure of the warp\nWe show in the upper panels of Fig. 1 the results of three fits in\n𝑍for different Galactocentric radii. Each panel shows, for rings of\nincreasingradius,theCepheidspresentinthering,thebestFourierfit\n(blackcurve)and500randomrealisationsfromthePosteriorPDF,the\ngraycurvesarefitsto 200bootstrappingrealisation.Theplotsclearly\nshowagrowthinamplitudetypicalofanS-shapedwarp,reachinga\nmaximum of≈1.1kpc in the outskirts of the disc. The effect of the\nSFisevident,theazimuthrangesampledincreaseswithradius.Other\nfeatures like the change of the warp as a function of 𝜙become clear\nin the second panel ( 𝑅=11.0kpc), where a plateau is noticeable\naround𝜙=180◦. The third panel ( 𝑅=15kpc) shows how from\n𝜙≈60◦to𝜙≈240◦the change in the warp between the extremes\nresemblesastraightlinemorethanasinusoidalcurvecorresponding\nto a pure𝑚=1mode would. This feature is correctly reproduced\nby the model thanks to the 𝑚=2mode; a simple tilted rings model\n(𝑀=1) cannot reflect it. The fits to the bootstrapping realisations\nshows that for 𝑅 < 10kpc the fits are affected by statistical noise\nas shown in the first panel of Fig. 1 at 𝑅=8kpc in the first and\nfourth quadrant of the galactic plane, this became more clear in the\nresiduals plot in the bottom panels. For the outer radii the fits are\nless sensitive to statistical noise as we see in the second and third\n2We have tested different values for the rotation curve around this one and\nour results are not affected.\nMNRAS 000, 1–24 (2023)6M. Cabrera-Gadea et al.\nTable 1.Amplitudes and phases as a function of radius for the best fitting (MAP) models for 𝑍and𝑉𝑍. The first few rows of the table are shown to provide\nguidance regarding its form and content. The full version of the table is available in the electronic version.\n𝑅 𝐴 0sin(−𝜑0)𝐴1𝐴2𝜑1𝜑2𝑉0sin(−𝜑𝑉\n0)𝑉1𝑉2𝜑𝑉\n1𝜑𝑉\n2\n(kpc) (kpc) (kpc) (kpc) (rad) (rad) (km s−1) (km s−1) (km s−1) (rad) (rad)\n5.171 -0.106 0.171 0.059 1.971705 2.337575 -0.422 2.638 1.976 -2.887806 -2.663989\n5.186 -0.107 0.176 0.063 1.979711 2.40411 -0.327 2.455 1.865 -2.875007 -2.720106\n5.202 -0.107 0.175 0.062 1.978782 2.403354 -0.363 2.296 1.941 -2.777056 -2.595807\nTable 2.Amplitudes and phases as a function of radius for 100 posterior realisations for the 𝑍and𝑉𝑍models. The first few rows of the table are shown to\nprovide guidance regarding its form and content. The full version of the table is available in the electronic version.\n𝑅 𝐴 0sin(−𝜑0)𝐴1𝐴2𝜑1𝜑2𝑉0sin(−𝜑𝑉\n0)𝑉1𝑉2𝜑𝑉\n1𝜑𝑉\n2\n(kpc) (kpc) (kpc) (kpc) (rad) (rad) (km s−1) (km s−1) (km s−1) (rad) (rad)\n5.171 -0.109 0.163 0.042 2.000795 2.192443 -0.951 3.761 3.294 2.232128 1.993639\n5.171 -0.097 0.155 0.053 1.848048 2.048916 0.626 5.258 4.114 -2.853366 -3.10008\n5.171 -0.111 0.19 0.076 1.897841 2.256799 -3.56 3.122 1.602 2.53726 -2.602042\npanel where the posterior realisations coincide with the bootstrap\nrealisations. For this reason we focus our analysis on the second a\nthird quadrants.\nFigure2shows 𝑉𝑧asafunctionof 𝜙forthesamethreeringsshown\nin Fig. 1. The first panel ( 𝑅=8.0kpc) of this figure, as well as in\nthepreviousone,showshowthefewobserveddatapointsinregions\nmostaffectedbytheSF(e.g. 𝜙∼300◦)stronglydrivethefitinthose\nregions. As discussed in Sec. C, this makes the inference unreliable\nfor the inner disc at 𝑅 < 10kpc, except around the azimuth of the\nsolar neighbourhood. Therefore, in what follows we will restrict our\nanalysisto𝑅≥10kpc.Asradiusincreases(secondandthirdpanels)\ntheamplitudeofthewarpinvelocitygrowsbutonlymildly,asitisat\nmostoftheorderoftheintrinsicdispersion 𝜎𝑉\n𝐼𝐷≈8kms−1evenat\ntheouterdisc.Thisisincontrastwith 𝑍,wheretheamplitudeofthe\nwarp exceeds the intrinsic dispersion by a factor of ≈3in the outer\ndisc. This low amplitude in comparison with 𝜎𝑉\n𝐼𝐷makes it harder\nto detect the kinematic signature of the warp, but at the outer disc\nit is clear there is a complex and asymmetrical behaviour, as seen\nin the third panel in Fig. 2. The bootstrap realisations for 𝑉𝑧show\nthe same conclusions as in 𝑍, but due to the low amplitude of the\nkinematicsignalinthefirstandfourthquadranttherealisationsshow\na greater dispersion than the posterior, illustrating that due to low\nnumber statistics noise is larger. For this reason we will focus the\nanalysis of the kinematic signal to the second and third quadrants.\n4.1.2 Asymmetries in height\nFirst, we explore the asymmetries of the warp in height above and\nbelow the plane. The left panel of Figure 3 shows the difference be-\ntweenthemaximumandminimumheightreachedbythewarpabove\nand below the plane in the North and South Galactic hemispheres\nrespectively. Positive values in this plot, at any given radius, imply\nthattheNorthernextremeofthewarpdeviatesmorefromtheGalac-\ntic plane than the South. Up to 𝑅≈12kpc the northern extreme\nis larger than the southern, even within the uncertainties, showing\nan asymmetrical warp. This asymmetry decreases towards the outer\ndisc, with the warp being almost symmetrical to within the uncer-\ntainties (≈100pc) at𝑅≳13.5kpc. We should keep in mind that\nbecauseoftheSF,theextremesofthewarptendtobeoverestimated\nin the internal regions. However, a more accurate and reliable mea-surement of this asymmetry is expected at the outskirts of the disc\nfrom our validation tests (Sec. 2.4.1).\nSince we set the phases of each mode free, we can also track\nthe azimuth of each extreme of the warp to explore the azimuthal\nasymmetryasafunctionof 𝑅.TherightpanelofFig.3exploresthe\nazimuthal asymmetry of the extremes of the warp as a function of\nradiusbyshowingthesmallestangulardifferenceintheazimuthsof\nthewarpextremesin 𝑍.InasimpletiltedringsmodelofanS-shaped\nwarp these extremes are always separated 180◦, even if the line of\nnodesistwisted.Theplotclearlyshows theextremesoftheCepheid\nwarp are never diametrically opposed . The difference in azimuth\nstartsatitslowestvalueof ≈120◦at𝑅≈10−11.5kpcandincreases\nup to≈145◦at𝑅≈12.5kpc after which it remains approximately\nconstant. This is a robust feature that cannot be reproduced by an\n𝑚=1warp, reinforcing the need for an 𝑚=2mode to describe the\nfull warp.\n4.1.3 Line of nodes and Line of Maximum 𝑉𝑧\nThe overall behaviour of the best fitting (MAP) warp model for the\nCepheidsisshowninFig.4inaface-onviewofthediscwithacolour\nscaleindicatingthemeanheightabove/belowthemid-plane.Theline\nofnodes(fromnowonLON)andlineofmaximumverticalvelocity\n(LMV𝑧)areindicatedwiththeblackandgreenlinesrespectively.A\nleading twist (i.e. in the direction of Galactic rotation) in both the\nLON and LMV 𝑧is evident, as well as an offset between the two.\nFig.5showstheLONandLMV 𝑧azimuths(for 𝑋 <0)asafunc-\ntion of radius. The figure shows that the azimuth of the LON is well\nrepresented by the straight line (in the plane 𝜙,𝑅) with the parame-\nters presented in Eq. 20, obtained from a fit to data in independent\nrings with𝑅 > 11. The LMV 𝑧also follows a linear tendency, well\ndescribed by an almost constant azimuthal difference of 25.4◦with\nrespect to the LON.\n𝜙LON(𝑅)=(−12.7±0.3)deg\nkpc𝑅+(347.◦5±3.◦5)for𝑅>11kpc.\n(20)\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 7\nFigure 1. The upper panels are the vertical height 𝑍as a function of galactocentric azimuth 𝜙for three different rings with radius 8.0kpc (left), 11.0kpc\n(middle) and 15kpc (right). Gray dots represent the Cepheids in each ring, the black solid line represents the Fourier curve MAP fit to the gray dots, and the\norangescurvesare 500randomrealisationsofthePosteriorPDF.Thegraycurvesarefitsto 200bootstrappingrealisations.Thebottompanelsshowtheresiduals\nbetween the best fit and each bootstrap realisation.\nFigure 2. The upper panels are the vertical velocity 𝑉𝑧as a function of galactocentric azimuth 𝜙for three different rings with radius 8.0kpc (left), 11.0kpc\n(middle) and 15kpc (right). Gray dots represent the Cepheids in each ring, the black solid line represents the Fourier curve MAP fit to the gray dots, and the\norangescurvesare 500randomrealisationsofthePosteriorPDF.Thegraycurvesarefitsto 200bootstrappingrealisation.Thebottompanelsshowtheresiduals\nbetween the best fit and each bootstrap realisation.\nMNRAS 000, 1–24 (2023)8M. Cabrera-Gadea et al.\nFigure 3. Left: Difference between the north and south extreme of the warp as a function of galactocentric radii from our results (black curve), the same is\ncalculatedforthewarpmodelbySkowronetal.(2019a)(dotedorangecurve).Right:Leastangulardifferencebetweenthenorthandsouthextremesasafunction\nof galactocentric radii from our results (black curve), the same is calculated for the warp model of Skowron et al. (2019a) (doted orange curve). The grey dots\nare500random realisation at each ring taken from the Posterior.\nFigure4. Face-onviewofthebestfitting(MAP)warpmodelfortheCepheids\n(graydots).ThecolourscalerepresentsthemeanZofthedisc(blueabovethe\nplaneandredbelowit).Thelineofnodes(LON,i.e. 𝑍=0)isindicatedwith\nthe black curve. The line of maximum vertical velocity (LMV 𝑧) is indicated\nby the dark green curve. The different coloured lines correspond to lines of\nconstant galactocentric azimuth.\n4.2 The time evolution of the warp traced by Cepheid\nHere we present results for the pattern speed (from Eq. 14) and the\nchange in amplitude with time (from Eq. 13) for the 𝑚=1mode\nobtained for the Cepheids. We ignore the 𝑚=0mode, since its\npattern speed is ill-defined and its amplitude change is 𝑉0sin(−𝜑𝑉\n0)\n(this is shown in the right panel of Fig.C3).\nAlthoughtheFourierseriesfor 𝑍and𝑉𝑧havebeenfitwith 𝑀=2,\nwefocusthisanalysisinthe 𝑚=1mode,becauseitisthedominant\nmodeofthewarpandtherecoveryoftheevolutionarytermsfor 𝑚=2\nare biased and noisy due to selection function effects (as shown in\nSec. B1.2). From Sec. 2.4.1 we recall that, for our simulation, 𝜔1\nand¤𝐴1are well-defined for 𝑅≳12kpc where𝑉1is non-zero and\nFigure 5. Azimuth as a function of galactocentric radius for: the LON from\nthis work (black curve), Chen et al. (2019) (cyan dashed curve) and Dehnen\net al. (2023) (red curve); the LMV 𝑧(thick green curve) and 𝜑1for𝑚=1\nmode from our 𝑍fits (red dots). The gray and olive green dots are 500\nrealisations of the LON and the LMV 𝑧taken from the posterior at each\nring,respectively.TheverticaldashedlineindicatestheHolmbergradiusfor\nthe Milky Way from Chen et al. (2019). The crosses indicate a sample of\nindependent (disjoint) rings.\nalso well-defined (as shown in Sec. C). Therefore, we will restrict\nthis part of the analysis to 𝑅≳12kpc.\n4.2.1 Amplitudes\nIn Fig. 6 we present results for ¤𝐴1as a function of 𝑅. In the range\n𝑅 <14.5kpc the change in amplitude is negligible, for 𝑅 >15kpc\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 9\nFigure 6. The change in amplitude ¤𝐴1for the𝑚=1mode as a function\nof the galactocentric radii from our fits (black curve) and the ¤𝐴1by Dehnen\netal.(2023)(cyancurve).Reddotsindicatemeasuresforindependentrings.\nThegraydotsaroundeach ¤𝐴1are500realisationtakenfromtheposteriorat\neach ring.\nit shows a growing tendency3reaching a maximum in the external\ndiscof≈5kms−1≈5kpc Gyr−1,thistendencyisalsopresentinthe\nresultsbyDehnenetal.(2023).Basedonourvalidationsummarised\nin Sec. 2.4.1 we expect these results to be unbiased over this radial\nrange.\n4.2.2 Pattern speed\nAssumingtheangularvelocity ΩfromtherotationcurvebyAblimit\net al. (2020), we obtained the pattern speed for the 𝑚=1mode\nfrom Eq. 14. Because in our reference frame the stars rotate in the\ndirectioninwhich 𝜙decreases,theangularvelocitiesinthedirection\nof stellar rotation are negative. To avoid confusion we present the\nangular velocities with their sign changed . Figure 7 presents, as a\nfunction of Galactocentric radius, the pattern speed of the 𝑚=1\nmode𝜔1(black curve), the angular velocity of the rotation curve Ω\n(redcurve),theupperandlowerlimitgivenbythemeasurementsby\nPoggioetal.(2020)(dottedbluelines)andresultsfromDehnenetal.\n(2023) (solid cyan line and dots).\nWe find that 𝜔1decreases for 𝑅≳11kpc and shows a small\noscillation for 13< 𝑅/kpc<16. This overall behaviour, both the\ndecreaseandtheoscillation,areobservedbyDehnenetal.(2023)but\nataslightlydifferentradius.Thisdifferencemayarisefromtheiruse\nofguidingradiusandalsobecauseweuseameanradiustorepresent\neachring,whichtendstodrivetheresultsfromtheoutertotheinner\nradii. We would need smaller uncertainties to ensure this oscillation\nisaphysicalphenomenoninthediscandnotanartefactfromourfits.\nHowever,thefactthatitisalsoobservedbyDehnenetal.(2023),with\na sample that includes radial velocities, increases our confidence in\nthe result. Our mean value observed for 𝑅 >12kpc is in agreement\nwiththeresultsfrompreviousworksonmeasuringthepatternspeed\nby Poggio et al. (2020) and Cheng et al. (2020), who assumed rigid\nbody rotation for the warp.\n3We consider this result to be extended beyond 𝑅=15kpc and not only\nup to𝑅=15.5kpc because the rings at this radius contain stars beyond\n𝑅=15.5kpc\nFigure 7. Minus the angular frequency as a function of the galactocentric\nradius for the pattern speed for the 𝑚=1mode𝜔1(black curve) from our\nfits, the angular velocity given by the rotation curve Ω(red curve) (Ablimit\net al. 2020), the upper and lower limit found by Poggio et al. (2020), and\nresults from Dehnen et al. (2023). The dots around 𝜔1are500posterior\nrealisations at each ring.\n5 DISCUSSION\n5.1 Structure and kinematics\n5.1.1 Comparison with different warp observations\nIn Fig. 8 we compare different warp models in the literature to our\nresultsfor𝜙=90◦(northernregion)and 𝜙=270◦(southernregion)\nfor𝑅>10kpc.Thevariousworkscitedherehavedifferentazimuthal\nand/or radial coverage, use different tracers, and have used different\nmethods to fit for the warp. Table 3 summarises this information for\nthe works presented in the figure. We have selected these works in\nordertocompareagainstotherdynamicallyyoungtracerslikethegas,\ndust and OB stars. We also include results from a few warp models\nfor dynamically older populations for which the time evolution of\nthe warp has been inferred. Since the warp followed by the older\npopulation may differ from that of the young, in Sec. 5.2 we will\ndiscuss the effect due to the assumed structure on the inference of\nthe time evolution of the warp.\nWe begin by comparing our results against those from Skowron\net al. (2019a), obtained for the same Cepheid sample as used here.\nWithin the uncertainties the two coincide at almost all radii. The\nSkowronetal.(2019a)modelbehaveslikeanaveragesmoothmodel\naroundourresults.Themeandifferencebetweenbothmodelsforthe\nnorthern region (for 𝑅⪆10kpc) is 0.054kpc, and for the southern\nregion is 0.043kpc. This level of agreement is expected because\nwe are using a subset of their sample, the differences being in how\nwe model the warp. Skowron et al. (2019a) model the warp as a\nFourier sum with 𝑀=2(as we do) but assume a constant phase for\neach mode as a function of radius ( 𝜕𝑅𝜑𝑖=0) and a second-degree\npolynomialforeachamplitude( 𝐴𝑚(𝑅)=𝛾𝑚(𝑅−𝑅𝑑)2where𝛾𝑚is\naconstant)asafunctionof 𝑅.Undertheseassumptionstheresulting\nmodel has the form\n𝑍(𝜙,𝑅)=𝐴0+(𝑅−𝑅𝑑)22∑︁\n𝑚=1𝛾𝑚sin(𝑚𝜙−𝜑𝑚). (21)\nIn consequence, there is a single Fourier sum that expresses the\nmeanazimuthalbehaviourofthewarpatall 𝑅> 𝑅𝑑whichisscaled\nMNRAS 000, 1–24 (2023)10M. Cabrera-Gadea et al.\nTable 3.Models from the literature. The asterisk ( ∗) indicates the model shown in Fig. 8.\nWork Tracer/Method Symmetric (yes or no) LON (fixed or free) Disc range\nYusifov (2004) Pulsars Yes Fixed 𝑅⪅15kpc\nLevine et al. (2006) HI No Fixed∗/Free 10<𝑅/kpc<30\nMarshall et al. (2006) Dust No Fixed [𝑑⪅13]kpc𝑙∈[− 90◦,90◦]\nSkowron et al. (2019a) Cepheids No Fixed 𝑅< 20kpc\nChen et al. (2019) Cepheids Yes∗/No Fixed∗/Free 5⪅𝑅/kpc<20\nLi et al. (2023) OB Yes Fixed 8.3<𝑅/kpc<14|𝑧|<1kpc\nAmôres et al. (2017) 2MASS star counts No Fixed 𝑅< 18kpc\nLópez-Corredoira et al. (2002) Red Clump Yes Fixed 𝑅< 13kpc\nWang et al. (2020) Red Clump Yes Fixed 8.3<𝑅/kpc<14,|𝑧|<1kpc\nChrobáková et al. (2020) Gaia DR2 Yes∗/No Fixed∗/Free 𝑅< 20kpc\nCheng et al. (2020) K type stars Yes Fixed 𝑅< 16kpc\nbythefunction(𝑅−𝑅𝑑)2.Inourmodel,withouttheseassumptions,\nwe can represent how the azimuthal geometry of the warp changes\nwith the radius, giving rise to the differences between both models.\nThe Skowron et al. (2019a) model has the ability to reproduce the\nmean asymmetries observed in the warp, but not the LON twist or\nazimuthal changes in the different modes, which affect where the\nmaxima are located.\nWe also compare our results with those from other warp models\nobtained for dynamically cold tracers like HI (Levine et al. 2006),\nDust(Marshalletal.2006),OBstars(Lietal.2023),Cepheid(Chen\net al. 2019) and pulsars (Yusifov 2004). Because Cepheids are a\nyoungpopulation( <500Myr,e.g.Catelan&Smith2015),theyare\nexpected to still retain the warp shape inherited from the gas and\nits star-forming regions, so the agreement among young tracers is\nexpected.WealsoshowtheresultsfromAmôresetal.(2017)selected\nforayoungpopulationwithanageof 400Myrcompatiblewiththat\nof Cepheids. In the northern region, within uncertainties, we found\nexcellent agreement with all previous results for young tracers, and\nacleardisagreementwithresultsfromAmôresetal.(2017)inferred\nfromstarcountsmodellingusingtheBesançonGalacticmodel.The\nwarp model from pulsars departs the most from ours, with a mean\ndifferenceof 0.14kpc(lessthantheintrinsicdispersionofCepheids,\nseeFig.C3).FortheHImodelwefounddifferencesfor 𝑅<12kpc,\nwhich may be due to the underestimation by the amplitude fitted\nto its own results by Levine et al. (2006) between 10< 𝑅 < 12.\nCompared to our results in the southern region, these works tend\nto underestimate the amplitude of the warp for 𝑅⪆13kpc. The\nwarptracedbypulsarsunderestimatestheheightthemost,compared\nto ours, with a maximum difference of 0.42kpc. These differences\nmayariseduetothesymmetryimposedinthemodelsforthisradial\nrange. The models from Levine et al. (2006), Chen et al. (2019)\nand Li et al. (2023) are strictly symmetric in this radial range, in\nconsequence,theasymmetrygivenbythe 𝑚=2modebetweenboth\nregions can’t be represented. The difference with the model from\nMarshall et al. (2006) may be due to its radial coverage which does\nnot extend beyond 𝑅∼13.\nAlthoughthedegreeofagreementinthesouthernregionisnotas\ngood as in the north, its clear that all young tracers follow a similar\nwarp (Chen et al. 2019; Skowron et al. 2019a; Li et al. 2023). The\nclear exception to this agreement is the result from Amôres et al.\n(2017). Although the disagreement with the Amôres et al. results\nin the south is not as strong as in the north, they still found a warp\namplitude that is systematically lower than ours as well as all other\nworks for Cepheids and similarly young tracers like dust and HI.\nWe now focus our attention on the intermediate population: Red\nClump stars (López-Corredoira et al. 2002; Wang et al. 2020), Atype stars (Ardèvol et al. 2023), K type stars (Cheng et al. 2020)\nand the full Gaia DR2 population (Chrobáková et al. 2020). Results\nfrom López-Corredoira et al. (2002) in the radial range 𝑅⪅13kpc\nspannedbyitsobservations(thickpartoftheline)showsagreement\nwith our results and, as with the young populations, the agreement\nis better for the northern region. However, extrapolating this warp\nmodel (thin part of the line) for the outer region of the disc would\nyield increasing differences that would grow up to the order of a\nfew kpc. Also, the models by Cheng et al. (2020) and Wang et al.\n(2020)fora1-3Gyrpopulationareinagreementwithinuncertainties\nfor the northern region in 𝑅⪅12. In the southern region both\nmodelsareinagreementwithourresultsfor 𝑅⪅11.5kpc,afterthis\nradiusthedifferencesincreaseuptoseveralkpcintheouterregions.\nThe warp model presented by Chrobáková et al. (2020) is in clear\ndisagreement in both the northern/southern regions with all other\nwarpmodelsusingsimilarlyoldtracers(likeChengetal.2020)and\nwith ours and all other results for young tracers. As we will discuss\nin Sec. 5.2, these differences in amplitudes between the models will\nbecome important in the determination of the pattern speed of the\nwarp. Results from Ardèvol et al. (2023) for the kinematics of A-\ntype stars population have shown a clear signal of the warp in the\nanticentre direction ( 𝜙=180), the increasing vertical velocity as a\nfunction of the radius from 𝑅≈12kpc, reaching≈6−7km s−1at\n𝑅=14kpc, similar to our results.4\nThe issue of the warp’s dependence on age of the stellar tracer\nremains an open question. Older stellar populations like RGB stars,\nRed Clump stars and other tracers older than Cepheids may trace a\nsimilar warp considering the uncertainty in the parameters of each\nmodel and their validity range. Also, Cantat-Gaudin et al. (2020)\nreported that stellar clusters typically olderthan 1Gyr trace the\nsouthern region of the warp similarly to the Cepheids. Thus, it is\nunclearwhethertherearesignificantdiscrepanciesbetweenthewarps\ntraced by older and younger populations.\nAmongpreviousresultsavailableatpresent,eitherthepredictions\nof models with age dependency deviate significantly from the warp\nobserved for bona fide young tracers like the Cepheids, as seen in\nthe case of Amôres et al. (2017), or there is not enough discrepancy\nin the differences (considering uncertainties) to determine an age\ndependency for the warp, as in the case of results from Wang et al.\n(2020) shown in Fig 8. The uncertainties in the parameters obtained\nby Wang et al. (2020) for all ages are large enough to allow for the\nagreement of all models from 1 to 12 Gyrs with our result with\n4Because Ardèvol et al. (2023) do not present a model of the warp traced\nby the A-type stars we can not include it in Fig. 8.\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 11\nFigure 8. Vertical height of the warp as a function of galactocentric radius for slices at 𝜙=90◦and𝜙=270◦, for this work and warp models in the literature\nsummarised in Table 3. The shaded region represents the uncertainty in the warp model from Wang et al. (2020) (see text for more details).\nCepheids. In particular the model for 9 Gyrs (not shown), an age\ncompletelyincompatiblewiththatofClassicalCepheids,istheonein\nbest agreement with our results. Taking into account the restrictions\npresent in some of the models regarding the asymmetry and radial\ndependence of the warp, each model’s validity range in distance\nand azimuth, and the current precision of the observed warp using\ndifferent tracers, it remains unclear whether or not there is an age\ndependency in the warp.\n5.1.2 Asymmetries and deviations from the tilted rings model\nOur results, as well as several previous ones, showed that a tilted\nrings model ( 𝑍=𝐴(𝑅)sin(𝜙−𝜑(𝑅))or𝑀=1) does not explain\nmany of the features observed at different radii in position and in\nkinematics.For 𝑍,thepresenceofaplateauat 10kpc⪅𝑅⪅11kpc\nand𝜙≈180◦showninthesecondpanelofFig.1,wherethewarpin\n𝑍isalreadypresent,cannotbeexplainedwithoutan 𝑚=2mode.At\nthat distance the tendency of the disc to warp towards the southern\nhemisphere is clear at 𝜙≈240◦, still far enough in azimuth from\nthe strong obscuration towards the bulge ( |𝜙|<90) to be an effect\nof the selection function.The bootstrapping realisations shown in\nFig.1andFig.2thattheplateauiswellrecovered,forthisreasonwe\nconsider unlikely to be an artefact of statistical noise. The northern\nextremeliesinthefirstquadrantandsoitsinferenceismoreaffected\nby selection function effects due mainly to obscuration, hence, it is\nless well constrained than the southern extreme. Nevertheless, the\nextremes of the warp (in 𝑍) are found to be≈120◦apart, while in a\ntilted rings model this difference must be 180◦by construction. The\nobserved shape resembles the \"S-Lopsided\" warp model presented\nby Romero-Gómez et al. (2019). A better azimuthal coverage in\nthe first quadrant and behind the bulge (currently unavailable due\nto extinction) would provide better constraints for this model. Ourresult is robust, however, since better coverage can only make the\ndifference between the warp extremes even smaller if the northern\nextreme lies closer to the bulge.\nIn the kinematics, a staticwarp (i.e.𝜔𝑚=0and¤𝐴𝑚=0) with a\nplateau would create a distinctive shape in 𝑉𝑧. If we consider a star\nrotatingwithangularvelocity Ωfollowingtheshapepresentedinthe\nsecond panel of Fig. 1, then, because the star rotates in the direction\ninwhich𝜙decreases,afterpassingtheminimumin 𝜙≈300◦thestar\nincreases its vertical velocity until it reaches the plateau ( 𝜙≈180◦)\nwhere𝑉𝑧≈0, then, on its way to the maximum 𝑍close to𝜙≈60◦\nthe star gains 𝑉𝑧until a certain point after which its 𝑉𝑧decreases to\nzerowhenitreachesthemaximum 𝑍.Thiscreatestwomaximain 𝑉𝑧,\nonebeforetheplateauandanotheroneafterit.Atoyrepresentationof\naplateauwouldbe 𝑍(𝜙)=𝐴1sin(𝜙)+𝐴1\n2sin(2𝜙);forastarrotating\nwith angular velocity Ωin a static plateau, this will give 𝑉𝑧(𝜙)=\nΩ𝐴1(cos(𝜙)+cos(2𝜙)),whichshowsthegeometrydescribedbefore.\nThisshapeisobservedinthesecondpanelinFig.2.Wealsotakethe\nratiobetweentheamplitudesofthemodes 𝑚=1and𝑚=2in𝑍and\n𝑉𝑧and found consistency with what is expected from the toy model\n(𝐴1\n𝐴2≈2and𝑉1\n𝑉2≈1around𝑅≈10.5).Thispeculiarsignalwasalso\nobserved in proper motions by Romero-Gómez et al. (2019) in the\nRGB population, who interpret it was a signal of the lopsidedness\nof the warp. As we see here it is actually a characteristic signal of\ntheS-Lopsidedmodelduetoitsplateau.Anindirectevidenceofthe\nplateauisalsoillustratedinFig.5bythelargedispersionoftheLON\nfor𝑅≲11.5kpcwheretheLONisill-defined.For 𝑅>11.5kpcthe\nplateau disappears, and the dispersion in Fig. 5 is sharply reduced\nas the disc is significantly inclined and the LON becomes well-\ndefined.For𝑅>11.5kpcotherfeaturesthatdifferfromatiltedrings\nmodel are still present, like the azimuthal asymmetry between the\ntwoextremes.Theangulardifferencebetweenthemgrowsbutnever\nreaches 180◦,meaningthatan 𝑚=2modeisneededtodescribethe\nMNRAS 000, 1–24 (2023)12M. Cabrera-Gadea et al.\ngalactic warp. In consequence the tilted rings (i.e. 𝑀=1) model is\nunabletoaccuratelydescribetheobservedazimuthallocationofthe\nwarp extremes at any radius.\nIn Sec. 4.1.2 we presented our results of the asymmetry between\nthe north and south extremes in the Cepheid’s warp. Asymmetry\nbetween the height of the warp extremes, or lopsidedness, has also\nbeenreportedbyChenetal.(2019)andSkowronetal.(2019a)forthe\nCepheids sample, by Levine et al. (2006) for the HI component and\nalsobyRomero-Gómezetal.(2019)fortheOBandRGBpopulations.\nAll these works seem to agree in the existence of an asymmetrical\ndistribution, with the HI as the best exponent of this feature. In our\nresults, the northern extreme is larger by ≈0.25kpc at 11.5<\n𝑅/kpc<13which declines to a mean difference ≈0.1kpc for\n𝑅 > 13.5kpc as shown in Fig. 3. For comparison the figure also\nshows the North/South asymmetry for the warp model obtained by\nSkowronetal.(2019a).Thisdifferencebehaveslikeameantrendof\nourresultasaconsequenceoftheassumptionofconstantphasesfor\nthe modes and the polynomial radial dependence of the amplitudes.\nTheobservedasymmetryintheouterdiscissimilartothatfoundfor\nthe OB population at 𝑅≈14kpc by Romero-Gómez et al. (2019),\nbut note that Romero-Gómez et al. (2019) report an amplitude for\nthe warp traced by OB of 0.3kpc, much lower than the 0.8kpc\nwe observe for the Cepheids warp. For the RGB stars, which are\nolderthantheCepheids,Romero-Gómezetal.(2019)reportalarger\nasymmetry(redline)butwiththeoppositesign.Thiswouldmeanthe\nRGBpresentawarpwithsimilaramplitudetotheCepheidsbutlarger\natthesouthernextreme.Asweshowinthefollowingdiscussion,this\nmay be due to an azimuth dependency of the asymmetry measured.\nWe have also found an azimuthal dependency in the asymmetry.\nFig. 8 shows how the warp at 𝜙=90◦(north) and𝜙=270◦(south)\npresentsasouthernregionwithalargerdeparturefromthemidplane\nintheoutskirtsofthedisc( 𝑅≈15.5kpc),butinFig.3comparingthe\nnorthandsouthextremeswegetanorthernwarpheightthatislarger\nat all radii (see also Fig. 4), with a decline in asymmetry towards\na minimum almost constant value at the outer disc ( 𝑅 > 13kpc).\nHere, the twisted LON can create misleading interpretations in the\nmeasurementoftheasymmetry,dependingonhowthismeasurement\nis made. Because the LON is leading and closely centred around\n𝜙≈180◦(see Fig. 4), a sample which covers the region 90◦<\n𝜙<270◦will tend to cover mostly regions below the galactic plane\n(the mean𝑍in azimuth between 90◦< 𝜙 < 270◦is below the\nplane for𝑅 >13kpc). Hence, comparing 𝑍at symmetric azimuths\n𝜙=180◦±Δ, rather than symmetric with respect to the LON, will\ntendtoshowawarpwithlargeramplitudebelowtheplane.InFig.9\nweshowtheresultoftakingdifferencesbetweentheabsolutevalues\nin height above/below the plane at lines of sight symmetric with\nrespecttotheSun-Anticentreline,i.e. |𝑍(180◦−Δ)|and|𝑍(180◦+\nΔ)|for different Δ. This clearly shows how measurements of the\nasymmetryatazimuthssymmetricwithrespecttotheSun-Anticentre\nline will yield a different result than when the extremes of the warp\nare compared. This is a consequence of both the twisted LON and\ntheextremesofthewarpneverbeingdiametricallyopposed(Fig.3).\nIftheeffectintroducedbythetwistintheLONfoundwithCepheids\nis also present in other populations, then different values for the\nasymmetrymaynotbeenoughtoensuredifferentwarpsfordifferent\npopulationsiftheazimuthaldependencyoftheLONisnottakeninto\naccount.\nTheresultsfortheRGBsampleobtainedbyRomero-Gómezetal.\n(2019),whoreportedawarplargerinthesouththaninthenorth,are\nalso shown in Fig. 9. The north/south extremes found by Romero-\nGómez et al. (2019) are roughly symmetric with respect to the Sun-\nAnticentre line, so comparison in Fig. 9 is appropriate, and this\nFigure 9. DifferenceΔ𝑍=|𝑍(180◦−Δ𝜙)|−|𝑍(180◦+Δ𝜙)|between\ntwo constant azimuths symmetric with respect to the anticentre direction\n𝜙=180◦, as a function of galactocentric radius. Results for RGB stars\nreported by Romero-Gómez et al. (2019) are also shown.\nshows their results are consistent with ours for various Δ𝑍at their\nmeasured distance of 𝑅≈14kpc. Results for the OB population\nthatwereshowninFig.3tobeinagreementwithoursregardingthe\nasymmetry are not shown here because they do not correspond to\nmeasurements made symmetric with respect to the Sun-Anticentre\nline.\n5.1.3 Line of nodes and line of maximum 𝑉𝑧\nIn Sec. 4.1.3 we presented results for the LON. As was previously\nreported by Chen et al. (2019) and more recently also by Dehnen\net al. (2023), the LON in the warp traced by Cepheids is twisted\nin the direction of the stellar rotation, meaning a leading LON, as\nshown in Figs. 4 and 5. This leading LON is in accordance with\nBriggs’sThirdRuleforwarps(Briggs1990),whichstatesthatwarp’s\nLONs are straight for 𝑅 < 𝑅𝐻0and twist for 𝑅 > 𝑅𝐻0, where𝑅𝐻0\nis the Holmberg radius. Although these rules are derived for the\nwarps traced by HI, they are expected to also apply for warps in the\nyoungpopulation.Chenetal.(2019)estimatetheHolmbergradiusat\n𝑅𝐻0=11.4kpc,itsLONandits 𝑅𝐻0areplottedinFig.5(cyancurve\nand dashed vertical line, respectively). We found better agreement\nbetween the 𝑅𝐻0and our twist’s starting radius, than that of Chen\net al. (2019), which starts further out in the disc, as shown in Fig. 5.\nFor𝑅⪆12.5kpc, the LON obtained by Chen et al. (2019) is in\nquitegoodagreementwithourresults.Webelievethatthedifference\nfor𝑅 < 12.5kpc between both works is because, fitting only with\none mode, the 𝑚=1mode in Chen et al. (2019) has to represent\nthe whole warp despite its asymmetries. For this reason, and its\nrelatively low amplitude in 𝑅 < 12.5kpc, the𝑚=1mode in Chen\net al. (2019) behaves as a mean between our LON (the full fit) and\nthe phase of our 𝑚=1mode (blue dots). The LON twist is also\nsuggested by Romero-Gómez et al. (2019) to be present in the warp\ntraced by RGB stars, but with an opposite direction, i.e. a trailing\nLON. However, Romero-Gómez et al. (2019) warn that this result\nmay be driven by selection effects due to extinction.\nInSec.4.1.3weshowhowtheLONandtheLMV 𝑧haveasimilar\ntwistbuttheydonotoverlap,havinganalmostconstantphaseoffset\nof25.◦4between them. Both lines lie in the region of the disc best\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 13\npopulated by our data (as seen in Fig. 4) and best recovered in our\ntests with simulations from Sec. B1.1. Therefore, we consider both\nlinestoberobustandnotaffectedbybiases.Thedifferenceinphase\nbetween the two lines could be due to the presence of 𝑚≥2modes\nintheoverallwarp.Romero-Gómezetal.(2019)alsofoundanoffset\nbetween the LON and the maximum vertical proper motion for the\nRGBsandattributedittothelopsidednessofthewarp.Accordingto\nRomero-Gómezetal.(2019,seetheirFig.8andSec.5.1)theLMV 𝑧\nfor the RGBs may lie at 𝜙≈160◦−170◦(they observe 𝜇𝑏,𝐿𝑆𝑅\nrather than𝑉𝑧), leading their LON (at 𝜙≈180◦−200◦) and also\nours, but with a twist opposite to our results with Cepheids. Again,\nthis result for the RGBs may be subject to selection effects which\nmay have affected the inference of the LON.\nInawarpdominatedbythe 𝑚=1warpinboth 𝑍and𝑉𝑧,achange\nin amplitude with time could be responsible for the out-of-phase\nLONandLMV 𝑧.Thephaseoffset 𝛿betweentheLONandLMV 𝑧is\ngiven by𝜑1−𝜑𝑉\n1+𝜋/2, so Eq. 13 translates into\n¤𝐴1=𝑉1sin𝛿 (22)\ndirectly relating the phase offset with the amplitude change. There\nare several caveats, however. First, as we have shown, for Galactic\nCepheids the 𝑚=1mode dominates the warp in 𝑍but not in𝑉𝑧,\nin which the 𝑚=2mode has a comparable amplitude at all radii.\nSecond,anLMV 𝑧trailingtheLONimplies 𝛿<0,andEq.22would\nrequire¤𝐴1<0in contradiction with our results and those from\nDehnen et al. (2023) shown in Fig. 6, which show that ¤𝐴1≈0up to\n𝑅∼14kpcand¤𝐴1>0atlargerradii.Thereforetheevolutionofthe\n𝑚=1modealonecannotexplaintheobservedphaseoffsetbetween\nthe LON and LMV 𝑧.\nWetested whethershiftingthe discmid-planecanmove theLON\ntocoincidewiththeLMV 𝑧.Todosowewouldneedtoshiftthestars\nby−240pc, the mean vertical height of the stars along the LMV 𝑧.\nThis would be too big a shift compared to the typical uncertainties\nof the position of the Sun above the Galactic plane (of the order\nof tenths of pc, see e.g. Chen et al. 2001), making this explanation\nunlikely.\nInconclusion,wefindthemostplausibleexplanationforthephase\noffset between the LON and LMV 𝑧to be the presence of 𝑚 > 1\nmodeswhichdeviatetheLMV 𝑧fromtheLON,meaningthatlopsid-\nedness would indeedbe the main driver of this offset as suggested\nalso by Romero-Gómez et al. (2019) for the RGB sample. Samples\nwith larger azimuthal coverage and also with measured line of sight\nvelocities may help to confirm modes with higher frequencies in\n𝑍and𝑉𝑧, and settle the reason behind this out-of-phase LON and\nLMV𝑧.\n5.1.4 The twist and velocity arcs\nIn Sec. 4.1.3 we showed the LON and LMV 𝑧are twisted and found\nthese are well represented by straight lines in the plane 𝜙,𝑅(Eq. 20\nfortheLONandplusanoffsetforLMV 𝑧).Theseparametersarealso\nexploredbyDehnenetal.(2023)whopresenttwoLONsasafunction\nofguidingradiusforthewarptracedbyCepheids,obtainedfromtwo\ndifferent methods (mean orbital plane and mean position plane) and\nfindarateofchangeintheLONof −14.7±0.7deg/kpc(meanorbital\nplane) and−10.6±0.8deg/kpc (mean orbital position). Our result\n−12.7±0.3deg/kpc lies between the two values.\nAsthediscrotatesandtheLONwrapsupandgetsmoreandmore\ntwisted,thedisccouldappeartohaveripplesiftheLONwrapsaround\nmorethanoncearoundthedisc.TherateofchangeoftheLONwith\n𝑅can be associated with the inverse of a radial wavelength of theseripples.Ifwetakeasimpletiltedringsmodel( 𝑍(𝑅)∝sin(𝜙−𝜑(𝑅)))\nandlookathowitchangesradiallyforaconstantazimuth,e.g. 𝜙=0,\nthen the warp will cross the plane at 𝜑(𝑅𝑗)=𝑗𝜋. Therefore, if the\nphase is described by 𝜑=𝛼𝑅+𝛽,𝛼it can be easily associated\nwith a wavelength by 𝜆LON=2𝜋\n𝛼5. This wavelength is the radial\ndistance between two warp peaks at a constant 𝜙, if the LON and\nthe amplitudes do not change its behaviour. For our LON we obtain\n𝜆LON≈28.4kpc. In the left panel of Fig. 10 we show how this\ntwisted LON creates long arcs in 𝑍for different azimuths.\nThe right panel of Figure 10 shows 𝑉𝑧as a function of 𝑅for\ndifferent azimuthal cuts, in which the velocity is seen to create arcs\nwhose peak changes in radius for different azimuths. These arcs are\nexplainedbythetwistedLMV 𝑧togetherwiththegrowthinamplitude\nin the kinematic signal as a function of 𝑅. These arcs are a direct\nconsequence of the twist in the LMV 𝑧because the kinematic signal\ndoesnotdecline,andalsobecausethepeaksofthearcsmoveoutward\nas phi decreases, as expected for the leading LMV 𝑧. Of course the\nchange in amplitude and the asymmetries play a role in the position\nof the maximum, but the main driver of this arcs is the twist in the\nLMV𝑧.\nThese arcs in 𝑉𝑧have been observed by previous works using\nGaia DR2 and DR3 with other stellar tracers (Cheng et al. 2020;\nGaiaCollaborationetal.2021).Chengetal.(2020)pointedoutthat\nthese arcs in 𝑉𝑧are a consequence of the pattern speed in a tilted\nring model, and indeed an arc can be created with just a constant\npatternspeedandagrowingkinematicsignalwithoutatwistedLON.\nThis is because the growing amplitude gets modulated by the factor\n(Ω−𝜔) so𝑉𝑧grows and then starts to decline as the co-rotation\nradius is achieved where 𝑉𝑧is null (if¤𝐴=0), creating an arc. But\nthis explanation can’t take into account the change of the arc shape\nasafunctionoftheazimuth(asshownintherightpanelofFig.10),\nwhichcanonlybeduetothetwistedLMV 𝑧,whichisaconsequence\nofthetwistedLONandthecombinationofthedifferentevolutionary\nterms of the warp modes, not only the pattern speed of the 𝑚=1\nmode.\nHere we have shown these arcs in 𝑉𝑧are a direct consequence of\nthetwistedLMV 𝑧.ItisworthnoticingthattheLMV 𝑧doesnottrack\nthelineofmaximumin 𝑉𝑧ofthearcspresentedinFig.10duetothe\nchange in amplitude as a function of 𝑅.\nBecause the LMV 𝑧has the same twist as the LON, 𝜆LONrepre-\nsents also the radial distance between two 𝑉𝑧peaks at a constant 𝜙.\nUsing the same 𝜆LONfor the LMV 𝑧we may expect from a extrap-\nolation of this oscillatory behaviour the minimum in 𝑉𝑧at the anti\ncentre direction to be around 𝑅≈28kpc and the point of null 𝑉𝑧to\nbearound𝑅≈21kpc.Wangetal.(2023)usedGaiaDR3tomapthe\ndisc population out to 𝑅≈23kpc. In their Fig. 3 the disc’s vertical\nvelocity goes from positive to negative values at 𝑅≳20kpc. These\nresultsseemtosupportourprediction,assuming 𝜆LONholdsforthe\nentireGaiaDR3sampleusedbyWangetal.(2023).Futureextended\nmaps of𝑉𝑧may prove helpful to explore whether this analysis also\nholds for𝑅>20kpc and for other stellar populations.\nFinally, Poggio et al. (2021) have shown with an N-body simu-\nlation of the Milky Way affected by the Sagittarius dwarf galaxy,\nthat the𝑚=1mode has prograde rotation if the Milky Way disc\nandSagittariusareclosetoaninteraction.Afterapproximatelyafew\nMyrtheprogrademotioncoherentwiththe 𝑚=2modedisappears.\nPerhaps the coherent rotation between the 𝑚=1and𝑚=2modes\n5In this approach we have ignored the amplitude ( 𝜕𝑅𝐴/𝐴≈0) which\ncan change the distance between the peaks, but this change is negligible in\ncomparison with our uncertainties\nMNRAS 000, 1–24 (2023)14M. Cabrera-Gadea et al.\nFigure10.𝑍(leftpanel)and 𝑉𝑧(rightpanel)asafunctionofgalactocentricradiusfornineevenlyspacedazimuthalcutsfrom 𝜙=120◦to240◦.Therangeof\nazimuthisselectedtobeintheregionofthediscmorepopulatedbydataandlessaffectedbytheSF.Intherightpaneltheblackcurverepresentthemeanfrom\nthe coloured ones\nTable 4.Pattern speed for the 𝑚=1mode in (km/s)/kpc given by our mean\nvalue for𝑅> 12kpc, Poggio et al. (2020) and Cheng et al. (2020).\nThis work⟨𝜔1⟩𝑅>12kpcPoggio et al. (2020) Cheng et al. (2020)\n−9.18±3.12 −10.86±3.23−13.57±0.2\nclose to an interaction found by Poggio et al. (2021) is the reason\nwhytheazimuthoftheLONandtheLMV 𝑧arewellapproximatedby\na monotonic (linear) dependence as a function of radius; without a\ncoherentmovementintheoutskirtsofthedisctheLONmaybehave\nmore erratically than is observed.\n5.2 Time evolution\nIn Sec. 4.2 we provided a new formalism to derive the time change\nofeachmode’samplitude( ¤𝐴𝑚)andpatternspeed( 𝜔𝑚)ateachring,\nfromtheFourierdecompositionofits 𝑍and𝑉𝑧.Thisnewformalism\nis free from assumptions on how the amplitude and phase of each\nmode depends on the radius. By applying it to the Cepheids we\nderived the pattern speed and change in the amplitude of the 𝑚=1\nmode for𝑅>12kpc.\nThe dominant mode in 𝑍for the warp is the 𝑚=1mode as ex-\npectedforanS-typewarp,soitsevolutionmaydrivemostofthetime\nevolution of the warp. In Fig. 7 we show that, within uncertainties,\n𝜔1shows a mean rotation of 9.2±3.1km/s/kpc6and its error cor-\nresponds to the standard deviation of posterior realisations from the\nindependent rings, in agreement with previous reports from Poggio\net al. (2020) and Cheng et al. (2020), presented in Table. 4. Some\noscillations are present, similar to the results obtained by Dehnen\net al. (2023), the main difference being that for 𝑅 < 12kpc they\nfound differential rotation slightly larger than we do, perhaps as a\nconsequence of our use of the galactocentric radius as opposed to\ntheir use of the guiding centre. We note, however, our results from\nsimulations (Appendix B) suggest 𝜔1may be overestimated in this\nradial range.\nChrobáková&López-Corredoira(2021)presentargumentsabout\n6The mean was obtained with measurements in independent rings for 𝑅>\n12kpchowtheover-estimationintheamplitudeofthewarpleadstoaover-\nestimationinitspatternspeed7,thereforegettingaloweramplitudeof\nthewarpwilltranslateintoaslowerprecession.Thisiswellreflected\nbyourEq.14.However,theverylowamplitudeofthewarppresented\nbyChrobákováetal.(2020)seemsunrealisticwhencomparedtothe\nrestofresultsfromtheliterature,evencomparedtothosewithsimilar\ntracers as Cheng et al. (2020). Chrobáková & López-Corredoira\n(2021) also present a warp model for the younger population in\nitssample,withverysimilarresultsasobtainedforthetotalsample.\nThis particular disagreement in amplitude with the models of the\nyoungpopulationsmayindicatethatthemodelbyChrobákováetal.\n(2020) may be significantly underestimating precession rate of the\nwarp, as a consequence of the underestimated amplitude.\nEquation14alsomakesitclearwhyourresultsfor 𝜔1aresimilar\nto those of Poggio et al. (2020) and Cheng et al. (2020), despite\ndifferent assumptions in the three warp models, like the amplitude\nor the fixed phase 𝜙1. This equation shows that 𝜔1depends on the\ndifference between the phase of the mode in 𝑍and in𝑉𝑧, therefore,\nit doesn’t matter where they are located or if they are twisted, as\nlong as the phase difference is the same. Because in the Milky Way\nthe assumption that 𝜙1−𝜙𝑉\n1≈−𝜋\n2seems to hold at least up to\n𝑅≈14kpc, which is the same as assuming ¤𝐴1≈0, independently\nofwhich𝜙1themodeladoptsorifitistwistedorfixed 𝜔1willnotbe\ninfluencedbythisassumptionaslongthemodeladopts ¤𝐴1≈0.Also,\nthe assumed amplitude affects 𝜔1as was previously mentioned, but\n𝜔1getssaturatedbyover-estimationsin 𝐴1,becauseas𝐴1→∞then\n𝜔1→Ω(because the kinematic signature 𝑉1makes𝜔1<Ω). This\nmaybethereasonwhyChengetal.(2020),withitslargeramplitude,\ngetsalarger𝜔1thanours,andalsowhyPoggioetal.(2020)withthe\nsamekinematicsignaturegetsalarger 𝜔1,asituseslargeramplitudes.\nInthisanalysisof 𝜔1wehaveleft𝑉1constantbecausethekinematic\namplitude of the warp seems similar for different tracers as shown\nbyGaiaCollaborationetal.(2021).Thesecouldbethereasonswhy\nPoggio et al. (2020) and Cheng et al. (2020) get similar results to\nours, even when they do not consider a twisted 𝜙1and when their\namplitudesarelargerthanours.Weshouldaddthatthedifferencein\namplitudeis notthe onlyparameter thatplaysa rolein thisanalysis,\ntherotationcurveandthekinematicsignalarenotthesamebetween\n7This relation holds for a warp with prograde rotation.\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 15\ntheworkscitedandtheycanchangethepatternspeedmeasurements,\nso we expect that these differences to also play a role.\nPrevious works on the time evolution of the warp neglect the\ncontribution by the change in amplitude to the warp’s kinematics\n(Poggio et al. 2020; Cheng et al. 2020). Poggio et al. (2020) argue\nthat the effect of¤𝐴1may be a second order effect in the kinematics.\nOur results shows empirically that the change in amplitude can be\nneglected at least up to 𝑅≈15kpc. Wang et al. (2020), finds the\nchange in amplitude derived from the young population ( ≈1Gy)\nto be null, which within uncertainties is consistent with our mean\nmeasurement up to the radial limit to which Wang et al. (2020)\nrestricted its sample, i.e. 𝑅=14kpc. For𝑅 > 14kpc we found\n¤𝐴1>0, reaching a maximum ¤𝐴1≈5km s−1, this tendency is also\nobservedinthechangeoftheinclinationinthetiltedringsmodelby\nDehnen et al. (2023) with similar values.\nThe prograde rotation of the 𝑚=1mode found with Cepheids is\nexpectedinthecontextofadiscembeddedinaprolatehaloasshown\nby Ideta et al. (2000) and Jeon et al. (2009). However, if this were\nthe case, the prograde motion should be much slower ( 0.1km/s/kpc\nto1.5km/s/kpc) than our result.\nAlthough the 𝑚=1mode rotates almost rigidly, this does not\nguarantee a rigid rotation of the LON, because the 𝑚=2mode\nalso plays a role in the LON evolution, and in 𝑉𝑧its amplitude is\ncomparable to that of the 𝑚=1mode. Due to the poor recovery\nexpected for the 𝑚=2mode (Sec.B1.2), a derivation of 𝜔2and¤𝐴2\nwith our data would be biased, so we cannot ensure the evolution\nof the LON or of the whole warp to be one with rigid rotation. The\n𝑚=1modealsopresentsagrowingamplitudefor 𝑅>15kpc,asis\nalsoreportedbyDehnenetal.(2023).For 𝑅<14.5kpcthechanges\nin amplitude are insignificant within the uncertainties, therefore, we\npresent a warp which, at first order, shows a stable behaviour for\n𝑅<14.5kpc but still evolving in the outskirts of the disc.\nInourderivationof 𝜔𝑚and¤𝐴𝑚wehaveignoredtheradialvelocity\nandazimuthalchangesin Ω.Consideringaradialmotionof 10km/s\neven when radial velocities may seem to be slower (Cheng et al.\n2020), we found that ¤𝐴1may change by about 1km/s and𝜔1by\n2km/s/kpc, which are in the order of the uncertainties. Also, that\nthe radial bulk motion reported by Cheng et al. (2020) is inwards\nfor𝑅⪆14kpc, will translate into a decrease in the measurement\nof¤𝐴1unless it is considered. Therefore, the growth in amplitude\nfor𝑅⪆14kpc cannot be reduced by taking the radial motion into\nconsideration(infact,itshouldincrease).Thesechangesaresmaller\nthan the uncertainties in the results presented in this work, therefore\nwedonottakethemintoaccountinouranalysis.Thesefeaturescould\nbeaddedtotheanalysisbyconsideringafieldofradialvelocityand\nΩdescribed by Fourier sums at different radii. The extension of our\nformalismtoaccountfortheradialcomponentwillbepresentedina\nfuture work.\n6 CONCLUSIONS\nIn this work we have used the Skowron et al. (2019b) catalogue of\nClassicalCepheidstostudythestructureandkinematicsoftheMilky\nWay warp by means of Fourier Decomposition methods. These are\nthe first results presented in the literature for the Fourier Decompo-\nsitionofthewarpin 𝑉𝑧.Ourmainresultsregardingthestructureand\nkinematics of the warp are the following:\n•The warp is clearly lopsided, both in 𝑍and𝑉𝑧. In𝑍, the am-\nplitudes of the 𝑚=1and𝑚=2modes are comparable up to\n𝑅∼13kpc. At larger radii the 𝑚=1mode dominates, as found\npreviously by Chen et al. (2019); Skowron et al. (2019b). In 𝑉𝑧, theamplitudesofthe 𝑚=1and𝑚=2modesarecomparableatallradii.\nThe𝑚=0modedoesnotplayamajorroleintheoverallwarpshape,\nwe detect a bowl-like shape in the radial range 11.5< 𝑅/kpc<13\nwith a maximum amplitude of ≈200pc. In𝑉𝑧the𝑚=0mode is\nalmost null for 𝑅>10kpc.\n•Thewarppresentsaplateauat 10< 𝑅/kpc<11.Theobserved\nshape resembles that of the S-lopsided model from Romero-Gómez\netal.(2019).Thedoublepeakobservedin 𝑉𝑧atthisradiusisakine-\nmatic signal associated with this plateau. It has also been observed\nin the proper motions of Red Clump stars by Romero-Gómez et al.\n(2019).\n•The warp is clearly asymmetric up to 𝑅∼13, with a Northern\nwarplargerthantheSouthernwarp.Intheouterdisc( 𝑅≳13.5kpc)\nthe warp becomes symmetric to within uncertainties.\n•TheextremesoftheCepheidwarpin 𝑍areneverdiametrically\nopposed. The difference in azimuth between the warp extremes is\n∼120◦at𝑅∼10−11.5kpcandincreasesupto 140◦at𝑅≈12.5kpc,\nremaining constant at larger radii.\n•The LON begins to twist at around 𝑅≈11, which is close to\ntheHolmbergradiusfortheMilkyWay( 11.4kpc,Chenetal.2019),\nin agreement with Briggs’ rules (Briggs 1990). The LON’s azimuth\nfollows a linear relationship with radius, presented in Equation 20.\nWe found a twist of −12.7±0.3◦\nkpc.\n•TheLMV𝑧doesnotcoincidewiththeLON,buttrailsbehindit\nwith a constant offset of 25.◦4. We rule out that this offset is due to\nthe change in amplitude with time of the 𝑚=1mode, and explain\nthis offset as a consequence of the lopsidedness also present in the\nkinematics.\n•The arcs in𝑉𝑧as function of 𝑅observed in other stellar popu-\nlations (Cheng et al. 2020; Gaia Collaboration et al. 2021) are also\npresentintheCepheidssample.Weshowtheseareaconsequenceof\nthe twisted LMV 𝑧(see Figure 10).\nWe have also introduced a new formalism (Section 4.2), based\non the joint analysis of the Fourier series in 𝑍and𝑉𝑧, from which\nthe pattern speed and instantaneous change in amplitude for each\nindividualFouriermodecanbederived.Byapplyingthisformalism\ntotheFourierDecompositionobtainedfortheCepheidsin 𝑍and𝑉𝑧,\nwederivethepatternspeedandamplitudechangeofthe 𝑚=1mode\nas a function of radius. Our main results are as follows:\n•The𝑚=1mode shows a prograde differential rotation for\n11< 𝑅(kpc)<13with𝜔1going from∼−20km s−1kpc−1at\n𝑅∼10−11kpcto−9.18kms−1kpc−1at𝑅∼13.Ourresultsfrom\nsimulations, however, suggest 𝜔1may be overestimated in 11<\n𝑅(kpc)<13this radial range.\n•The amplitude of the 𝑚=1mode remains approximately con-\nstant, with¤𝐴1≈0km s−1for𝑅 < 14.5kpc. The amplitude change\nhas a growing tendency for 𝑅 > 15kpc, reaching¤𝐴1≈5km s−1at\n𝑅≈15.5.\nThankstotheprecisemeasurementsfromGaiaDR3anddistances\nfromSkowronetal.(2019a)toitssampleofCepheids,wecanexplore\nthe complex signal of the warp in both its structure and kinematics.\nFuture Cepheid samples with increased coverage in the first and\nfourthquadrantswillcontributetobetterrestricttheparametersofthe\nwarp.Abetterunderstandingofthewarpkinematicsisnecessaryto\nmakemorerobustcomparisonswithsimulationsandwithanalytical\nmodels of its dynamics, which can lead to better constraints on the\npossiblehistoryofthewarpanditsroleintheevolutionoftheMilky\nWay disc’s dynamics. Furthermore, the complexity revealed may\nnot be unique to the Galactic warp, understanding it will help also\nunderstand warps in external galaxies.\nMNRAS 000, 1–24 (2023)16M. Cabrera-Gadea et al.\nACKNOWLEDGEMENTS\nThe authors thank the referee for the clear and succinct review\nwhich has enriched the presentation of our work. MC and CM\nkindly thank Marcin Semczuk and Walter Dehnen for an early re-\nview of this manuscript and for sharing their results prior to pub-\nlication. This research has been supported by funding from project\nFCE_1_2021_1_167524 of the Fondo Clemente Estable, Agencia\nNacional de Innovación e Investigación (ANII) and project C120-\n347 of the Comisión Sectorial de Investigación Científica (CSIC)\nat the Universidad de la República, Uruguay. MRG acknowledge\nfunding by the Spanish MICIN/AEI/10.13039/501100011033 and\nby \"ERDF A way of making Europe\" by the “European Union”\nthroughgrantPID2021-122842OB-C21.TAacknowledgesthegrant\nRYC2018-025968-IfundedbyMCIN/AEI/10.13039/501100011033\nand by “ESF Investing in your future”. This work was (partially)\nsupportedbytheSpanishMICIN/AEI/10.13039/501100011033and\nby \"ERDF A way of making Europe\" by the “European Union”\nand the European Union «Next Generation EU»/PRTR, through\ngrants PID2021-125451NA-I00 and CNS2022-135232. MRG and\nTA acknowledge the Institute of Cosmos Sciences University of\nBarcelona (ICCUB, Unidad de Excelencia ’María de Maeztu’)\nthroughgrantCEX2019-000918-M.Thisworkhasmadeuseofdata\nfrom the European Space Agency (ESA) mission Gaia(https:\n//www.cosmos.esa.int/gaia ), processed by the GaiaData Pro-\ncessing and Analysis Consortium (DPAC, https://www.cosmos.\nesa.int/web/gaia/dpac/consortium ). Funding for the DPAC\nhas been provided by national institutions, in particular the institu-\ntions participating in the GaiaMultilateral Agreement.\nSoftware: Astropy(AstropyCollaborationetal.2018),Matplotlib\n(Hunter 2007), Numpy (Walt et al. 2011), Jupyter (Kluyver et al.\n2016), and TOPCAT (Taylor 2005, 2006)\nDATA AVAILABILITY\nThecatalogueofClassicalCepheidsusedinthisresearchispublicly\navailable in Skowron et al. (2019a). Theresults from our best fitting\nmodels as well as posterior samples are provided in Tables 1 and 2.\nREFERENCES\nAblimit I., Zhao G., Flynn C., Bird S. A., 2020, ApJ, 895, L12\nAllen C., Santillan A., 1991, Rev. Mex. Astron. Astrofis., 22, 255\nAmôres E. B., Robin A. C., Reylé C., 2017, A&A, 602, A67\nArdèvol J., Monguió M., Figueras F., Romero-Gómez M., Carrasco J. M.,\n2023, arXiv e-prints, p. arXiv:2308.01901\nAstropy Collaboration et al., 2018, AJ, 156, 123\nBinney J., Tremaine S., 2008, Galactic Dynamics: Second Edition\nBobylev V. 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F., et al., 2020, ApJ, 897, 119\nWang H.-F., Chrobáková Ž., López-Corredoira M., Sylos Labini F., 2023,\nApJ, 942, 12\nYusifovI.,2004,inUyanikerB.,ReichW.,WielebinskiR.,eds,TheMagne-\ntized Interstellar Medium. pp 165–169 ( arXiv:astro-ph/0405517 )\nAPPENDIX A: MATHEMATICAL DEFINITIONS FOR THE\nINFERENCE\nInthissectionweexpandonthedefinitionsofvariousmathematical\nobjects used in Sec. 2. We begin defining the vector ®CS(𝜙)as\n®CS(𝜙)=[1,cos(𝜙),cos(2𝜙),...,cos(𝑀𝜙),sin(𝜙),...,sin(𝑀𝜙)],\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 17\n(A1)\nwe can write the matrix Aas\nA=𝑁∑︁\n𝑖=1®CS(𝜙𝑖)⊗®CS(𝜙𝑖)\n𝜎2\n𝑖, (A2)\nwhere⊗denotes the outer product, 𝜙𝑖the azimuth of the 𝑖-th star\nand the𝜎𝑖its dispersion defined as\n𝜎2\n𝑖=𝜎2\n𝑧𝑖+𝜎2\n𝐼𝐷, (A3)\nwhere𝜎𝑧𝑖is the uncertainty in 𝑧and𝜎𝐼𝐷the intrinsic dispersion.\nThe vector pis defended as\np=𝑁∑︁\n𝑖=1𝑧𝑖\n𝜎2\n𝑖®CS(𝜙𝑖). (A4)\nAPPENDIX B: VALIDATION WITH SIMULATIONS\nIn this section we use a warped galactic disc simulation to analyse\nthe performance of the method described in Sec. 2 when applied to\nmock data. We analyse how observational errors and the selection\nfunction of the data affect the recovery of each mode’s parameters\nin𝑍and in𝑉𝑧, the full Fourier sum, and the intrinsic dispersion in\ndifferent regions of the disc.\nB1 Structure and Kinematics\nFor our warped galactic disc model (without observational errors or\nselection function) we use the test particle simulation of the Sine\nLopsided warp from Romero-Gómez et al. (2019). This is an S-\nshaped warp modified from a simple tilted rings model to allow for\nan warp with an arbitrary asymmetry (a 3D representation is shown\nin Fig. B1 in Romero-Gómez et al. 2019). The warp is such that the\nheight of the mean plane of the disc is given by\n⟨𝑧(𝑅,𝜙)⟩=𝑅sin(𝜙)sin(𝜓(𝑅,𝜙)) (B1)\nwhere\n𝜓(𝑅,𝜙;𝑅1,𝑅2,𝛼,𝜓𝑢𝑝,𝜓𝑑𝑜𝑤𝑛)=[𝐴+𝐵sin(𝜙)]𝑓(𝑅;𝑅1,𝑅2,𝛼)\n(B2)\nwith𝐴=1\n2(𝜓𝑢𝑝+𝜓𝑑𝑜𝑤𝑛),𝐵=1\n2(𝜓𝑢𝑝−𝜓𝑑𝑜𝑤𝑛)and𝑓having\nthe following expression\n𝑓(𝑅;𝑅1,𝑅2,𝛼)= \n0 𝑅≤𝑅1\u0000𝑅−𝑅1\n𝑅2−𝑅1\u0001𝛼𝑅1< 𝑅< 𝑅 2\n1 𝑅≥𝑅2(B3)\nwhere Romero-Gómez et al. (2019) set 𝑅1=10.1kpc,𝑅2=\n14kpc,𝛼=1.1,𝜓𝑢𝑝=7.5◦and𝜓𝑑𝑜𝑤𝑛=4.25◦.Theseparameters\nwere chosen so that they would represent a plausible model of the\nasymmetry observed in the Galactic warp.\nFor the test particle simulation the strategy followed by Romero-\nGómez et al. (2019) was to initialise test particles in a a flat disc\nrelaxedinanAllen&Santillan(1991)Galacticpotential,thenwarp\nthe potential adiabatically for five periods of the circular orbit at\na radius𝑅=𝑅2and finally let the stars relax for a further two\nperiods (at𝑅=𝑅2). The resulting configuration is such that stars at𝑅 < 𝑅 2kpc are in statistical equilibrium with the imposed potential\nand their mean 𝑧is described by Eq. B1.\nThetestparticleswereinitialisedwithaverticalvelocitydispersion\nof16.6kms−1representativeofRedClumpstarsatthesolarradius.\nIn this work we will apply the method to a sample of Cepheids, a\nkinematicallycolderpopulationinwhichthedetectionofthewarpis\nmorefavourable.TheresultsobtainedfortheRedClumpsimulation\nwill,nonetheless,stillbeusefultounderstandthegeneraladvantages\nand flaws of the method described in Sec.2.\nThe mock catalogue from Romero-Gómez et al. (2019) includes\nthe simulation of the Gaia DR2 observational errors and selection\nfunction (hereafter SF), as described in their Appendix D. In what\nfollows we will use this mock catalogue down-sampled to 𝑁𝑡𝑜𝑡=\n1997tomatchthenumberofCepheidsinourfinalcatalogue(Sec.3),\nkeeping the simulated errors in proper motion, and assuming a 3%\nerror in distance, representative of the photometric distances for the\nCepheidsinoursample(Sec.3).Errorsfor 𝑉𝑧werepropagatedfrom\ndistance and proper motion errors, assuming the radial velocity is\ninferred from the rotation curve, as described in Sec. 3.\nSince the test particles in the Romero-Gómez et al. (2019) simu-\nlation are only relaxed up to 𝑅=𝑅2=14kpc, we need to establish\na ground truth model representative of the whole disc that can be\nused as a fiducial model against which results for mock catalogues\nare compared. We take as our ground truth model (from now on GT\nmodel)theFourierfitsfor 𝑍and𝑉𝑧obtainedbyapplyingthemethod\ndescribed in the previous section to an arbitrarily large sample of\nthe simulation, without errors or SF and fitting up to the second or-\nder mode (𝑀=2)8. Other combinations for 𝑁,𝑁𝑡𝑜𝑡,𝑀for the GT\nmodel were tested and we find that the selected one optimises the\ncomputational time required and gives us the detailed information\nneeded.Finally,wewillcalltheSFmodeltheFourierfitsobtainedby\napplyingthemethodtothemockcatalogueaffectedbyobservational\nerrorsandselectionfunction.Inallcases,tocompute 𝜎𝐼𝐷wedivide\neach ring in 15equally spaced azimuthal bins and find the weighted\naverage of the standard deviation.\nFigureB1comparesthedistributionoftestparticlesintheGT(gray\ndots)andSF(bluedots)simulatedsamplestotheresultingGTandSF\nmodels (black and blue lines) in 𝑍versus𝑅plots for three different\nazimuths. The figure shows in all cases the GT model does indeed\ncapture the behaviour of the test particles in the full radial range\nand coincides with the analytical prediction for 𝑅 < 𝑅 2. Beyond\nthisradiusthestarsceasetobeinequilibriumwiththepotentialand\nthe mean⟨𝑧⟩traced by the stars is not expected to follow Eq. B1.\nThe SF model agrees very well with the GT model over the whole\ndisc,capturingtheoverallbehaviourofthewarp.Themeanabsolute\ndifference between the SF and GT models for each 𝜙is reported\nin the top label of each panel in Fig. B1 ( ⟨|Δ𝑍|⟩𝑅), the maximum\n⟨|Δ𝑍|⟩𝑅≈0.40kpc corresponding to the region most affected by\nthebulgeextinction( 𝜙=0◦),andtheminimum ⟨|Δ𝑍|⟩𝑅≈0.08kpc\nwhich corresponds to the region towards the anticentre ( 𝜙=180◦).\nIn Fig. B1 we notice that the general trend of the warp is recovered\nin all directions for the external region of the disc ( 𝑅⪆10) where\nthe warp begins.\nThe right panel of Figure B1 shows the corresponding results for\n𝑉𝑧. In this case, we compare only the results for the GT and the\nSF models, since there is no simple analytical form for 𝑉𝑧(𝑅)as\ndiscussed in Appendix C in Romero-Gómez et al. (2019). Again, as\nin𝑍, the best-recovered region is around 𝜙=180◦because it is less\n8We tested a larger 𝑀and find that𝑀 > 2does not improve significantly\nour results.\nMNRAS 000, 1–24 (2023)18M. Cabrera-Gadea et al.\naffected by the SF. All differences between the GT and SF models\nare much smaller than the corresponding velocity dispersion, which\nhas a mean of 19.2km/s throughout the disc. For both 𝑍and𝑉𝑧the\nreduced chi square 𝜒2𝑛𝑢shows that the GT model fits for 𝑍and𝑉𝑧\nare good (𝜒2𝜈≈1∀𝑅).\nB1.1 Azimuthal and radial biases\nFigureB1hintedtheexistenceofregionsinwhichthereconstruction\nof the warp given by the SF model lacks accuracy. We argue this\nis due to the correlation between modes introduced by not having\na uniformly distributed sample in azimuth and by stochastic clumps\nin regions with fewer stars in the sample due to the SF. To illustrate\nthis, Figure B2 shows, in each panel, a residual plot between the\nSF and the GT model in 𝑍and𝑉𝑧(respectively top and bottom)\nnormalised by the intrinsic dispersion given by the GT model (with\nfixed𝑀=2), for SF model fits with up to 1 (left) and 2 modes\n(right).GreydotsshowtheSFsample,theblackstarshowstheSun’s\nposition, the inner ring is 𝑅=10kpc where the warp begins and\nthe outer ring is the radius in which the amplitude of 𝑚=1mode\nis bigger than the intrinsic dispersion in the variable. The red and\nblue colours correspond to over/under estimations by the SF model\nrespectively.\nIn𝑍, for both𝑀=1and𝑀=2the differences are greater at\n𝑋 >0kpcintheinnerregionbeforethewarpbegins.Thediscrepancy\nis larger for 𝑀=2because when fitting with a higher number of\nmodes, in the areas most affected by the SF the higher frequency\nmodes tend to drive the fit towards the few data points available\nintroducing spurious oscillations where there is less data. For both\n𝑀=1and𝑀=2the recovery is best for outer radii, where the\nwarp amplitude is larger than the intrinsic dispersion. For 𝑀=1,\nthedifferencesstartgrowingwithradiusduetothesimulatedwarp’s\nasymmetry which is not well represented by the Fourier series with\n𝑚≤1modes, generating an 𝑚=2pattern in the differences. By\ncontrast, the asymmetry is better captured by the series for 𝑀=2\nforwhichthediscrepanciesintheouterregionaresmaller.However,\nahintofthe𝑚=2patterninthedifferencesstillremains;thisisdue\nto a lower amplitude of the 𝑚=2mode recovered by the SF model\n(this is illustrated in left panel of Fig. B3).\nFor𝑉𝑧we analyse the bottom panels in Fig.B2, the left plot for\n𝑀=1and the right for 𝑀=2. The differences between the SF and\nGT models are always smaller for 𝑉𝑧than the intrinsic dispersion\nin the whole disc, both for 𝑀=1and𝑀=2, in contrast with the\nrecoveryin𝑍wheredifferencesexceedtheintrinsicdispersioninthe\ninnerregion.Thebestandworstrecoveryfor 𝑉𝑧arefoundinthesame\nregions as for 𝑍because the azimuthal distribution is the same for\nboth samples; with the best recovery at negative 𝑋, and the worst in\ntheinternaldiscatpositive 𝑋.Finally,thedifferencesbetweentheSF\nandGTmodelsaremuchlowerin 𝑉𝑧thanin𝑍.Asalsodiscussedin\nRomero-Gómezetal.(2019),thisisexpectedbecausetheSFcreates\nexclusion zones in 𝑍due to high extinction near the Galactic plane,\nbut doesn’t in 𝑉𝑧because the correlation between 𝑧and𝑣𝑧is weak\nfor a given star.\nGiven these results, we decide to use 𝑀=2for Fourier fits for\nthisworkbecauseitofferstheleastbiasedrecoveryfortheregionof\nthediscwherethewarpismostprominent(i.e.outerradii).Reliable\nresults for the inner region of the disc are limited to 90≳𝜙≳\n270kpc, the region least affected by the SF with best coverage,\nwhere biases in the recovery are lowest.B1.2 Recovery of individual modes\nSofarwehaveanalysedtherecoveryoftheshapeandkinematicsof\nthe warp as a whole, given by the sum of the 𝑀individual modes\nin the Fourier series. Now we will analyse how well each mode is\nrecovered.\nEach mode𝑚is characterised by its amplitude 𝐴𝑚and its phase\n𝜑𝑚in𝑍, and in𝑉𝑧with𝑉𝑚and𝜑𝑉𝑚. In Fig. B3 we compare the\namplitudesfor 𝑍(left)and𝑉𝑧(right)asafunctionof 𝑅recoveredfor\nthe SF model (dark solid lines) against the values given by the GT\nmodel (pale solid lines) for each mode. The intrinsic dispersion as a\nfunction of radius is also plotted in each panel.\nThe left panel of Fig. B3 shows how for inner radii ( 𝑅 <10kpc)\nthediscisflatbeforetheonsetofthewarp,asshownbythenearzero\namplitudes for all modes in the GT model. Particularly for 𝑚=1,2,\nthe SF model finds non-zero amplitudes of the order of the intrinsic\ndispersion. Amplitudes are overestimated in the inner disc because\nthemodesmakethefullFourierseriesflatintheregionlessaffected\nbytheSF(𝜙≈180◦),butitalsotriestofitstochasticclumpsfarfrom\nthemidplaneat 𝜙≈0◦wheretheSFhasremovedstarspreferentially\nin the disc plane. At the outer parts of the disc, the 𝑚=1mode is\noverestimatedbytheSFmodelbutthebiasisreducedattheexternal\npart of the warp ( 𝑅 > 13kpc) where the 𝐴1amplitudes are larger.\nThe𝑚=2mode is overestimated due to correlations with other\nmodeswhenthewholefitoftheseriesisdrivenbystochasticclumps\nat𝑅≲10kpc, as for the 𝑚=1mode. The𝑚=0mode is well\nrecovered over the whole disc.\nSomefeaturesobservedfortheamplitudesin 𝑍arepresentalsoin\n𝑉𝑧.Forexampletheamplitudesarenot 0kms−1for𝑅< 𝑅 1duetothe\nsparse azimuthal coverage caused by the SF. For 𝑉𝑧, the amplitude\nof𝑚=1is underestimated but the general trend is well recovered\nby the SF model for 𝑅 > 𝑅 1=10kpc as in𝑍. The amplitudes of\n𝑚=0,2have differences between the SF and the GT model, also as\nin𝑍, which is expected because both amplitudes in the GT model\naresmallerthan 𝜎𝐼𝐷,whichmakesthemharderfortheSFmodelto\nrecover.\nSimilarly to Figure B3, in Figure B4 we compare how the phase\nofeachmodein 𝑍(left)and𝑉𝑧(right)isrecoveredbytheSFmodel\nas a function of radius. We do not plot the phase for 𝑚=0because\nit can only take two possible values ( −90◦and90◦).\nFor the inner disc at radii 𝑅≲𝑅1=10kpc before the onset\nof the warp, it is normal that the phase is badly recovered for all\nmodes because the (true) amplitudes are near zero at these radii and\nthe phase becomes meaningless. For the 𝑚=1mode the phase\nfor both𝑍and𝑉𝑧are very well recovered, with no significant bias,\nfor𝑅≳12kpc where𝐴1> 𝜎𝐼𝐷. For𝑚=2, the general trends\nare recovered for 𝑅≳12kpc, e.g. the twist in 𝑍and𝑉𝑧showing\nthe change of phase as a function of radius. However, we must be\ncautious in any particular analysis of 𝑚=2as an individual mode\nduetothelackofrecoverybytheSFmodelwiththismode,itsphase\nrecovers some of its tendency but without accuracy.\nB1.3 Intrinsic dispersion\nFinally, we analyse the bias introduced by the SF to the intrinsic\ndispersion that our method calculates. To do so, we compute the\nfractional difference between the 𝜎𝐼𝐷obtained with the GT and SF\nsamples. These differences for 𝑍(blue curve) and 𝑉𝑧(green curve)\nare plotted in Fig. B5 as a function of radius. The black vertical\ndotted line at 10kpc indicates the beginning of the warp, the blue\none when the mode 𝑚=1for𝑍starts to be greater than 𝜎𝐼𝐷, the\ngreen one is the same as the blue but for 𝑉𝑧.\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 19\nFigure B1. Each panel in the left column shows 𝑍as a function of 𝑅centred at four different Galactocentric azimuths (from top to bottom 𝜙=0.0◦,90.0◦,\n180.0◦and270.0◦)withΔ𝜙=5◦.Thesameisshownintherightcolumnfor 𝑉𝑧.Thedifferentsolidcurvesshow:theanalyticalpredictionforthemeanposition\nof test particles in the warped potential (red curve), the ground truth model (black curve) and the selection function model (blue curve) for the variable 𝑍. The\ngray dots represent the stars used for the GT model, and the blue ones represent the star used for the SF model. The mean absolute difference between the SF\nand GT models for each 𝜙is reported in the title.\nMNRAS 000, 1–24 (2023)20M. Cabrera-Gadea et al.\nFigureB2. Eachplotshowstheresidualsbetweenthefiducialmodel( 𝐺𝑇)andthemodelrecoveredwiththemockcatalogue( 𝑆𝐹),normalisedbytheintrinsic\ndispersion of the variable obtained in the fiducial model. The left and right panels show the residuals for 𝑍and𝑉𝑧, respectively, for 𝑀=1(top) and𝑀=2\n(bottom).Ineachoftheplotstheinnerblackringat 𝑅=10kpcindicateswherethewarpsstarts,theouterringiswherethe 𝑚=1modebeginstobegreaterin\namplitudethantheintrinsicdispersion.Thishappensat 𝑅=12kpcfor𝑍and𝑅=14kpcfor𝑉𝑧.Thepalegraydotsarethestarsinthemockcatalogueusedin\nthe fit, and the black star indicates the position of the Sun.\nFigureB3. Eachpanelshowtheamplitudeforthe 𝑚=0(yellow),𝑚=1(blue),𝑚=2(violet)modesand 𝜎𝐼𝐷(black)fortheSFmodel(darksolidlines)and\nthe GT model (pale solid lines) obtained for 𝑍(left panel) and 𝑉𝑧(right panel). Each amplitude and 𝜎𝐼𝐷is plotted as a function of radii.\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 21\nFigure B4. Each panel shows the phase for the modes 𝑚=1(blue) and𝑚=2(violet) for the SF model (solid dark line) and the GT model (solid pale line)\nobtained for𝑍(left panel) and 𝑉𝑧(right panel), as a function of galactocentric radius 𝑅.\nFigureB5. TheratiobetweenthedifferenceinintrinsicdispersionbytheGT\nmodel and the one obtained with the SF model, over the intrinsic dispersion\nof the GT model as a function of the galactocentric radii. The blue and\ngreen continuous curves correspond to the intrinsic dispersion of 𝑍and𝑉𝑧,\nrespectively. The black, blue, and green vertical dashed lines indicate the\nradius at which the warp begins, when the amplitude of the mode 𝑚=1is\nbigger than the intrinsic dispersion for 𝑍and for𝑉𝑧, respectively.\nFirst, for𝑍the recovered 𝜎𝐼𝐷is increasingly overestimated at\ninner radii until the warp becomes greater than the disc’s thickness;\nfor larger radii, the recovered 𝜎𝐼𝐷decreases and is off just by 10%\nof the GT value. Both effects are due to the combination of the\nincreased warp amplitude and the SF. The SF makes the stars near\nthe plane very unlikely to be observed due to high extinction, while\nthestarsawayfromtheplanearelessaffectedbyit;sincethesestars\nare further away from the disc plane (because of the amplitude of\nthe warp) this tends to inflate 𝜎𝐼𝐷for𝑍. This effect is expected to\nbesmallerforadynamicallycolderstellarpopulationlikeCepheids.\nFor𝑉𝑧, on the other hand, we find a mean underestimation of 3%,\nmuch smaller than for 𝑍. We find the appearance of the warp signal\nin𝑉𝑧hasnoeffectintheabilitytorecover 𝜎𝐼𝐷.Overalltherecovery\noftheintrinsicdispersionaffectstheinferenceontheamplitudesand\nphases in terms only of the dispersion of the posterior PDF, it does\nnot introduce any systematic biases in the parameters themselves.\nB1.4 Assumptions on the rotation curve\nWetestedhowtheassumedrotationcurvemayaffecttheinferencein\nthesimulationsandwiththerealdata.Wedidnotfindanysystematic\nbiasinthe amplitudes,phasesandintrinsic dispersioninferredfromthe mock catalogues when we used the 𝑣𝑧derived from the rotation\ncurve, even when using different rotation curves.\nIn the case of the Cepheids, we tested whether changing the rota-\ntion curve offset by ±10km s−1could change our main results. We\nfound that different offsets change the amplitudes of the 𝑉𝑧arcs by\n∼1kms−1butdonotchangethegeneraltrendofthekinematicsignal\nofthewarp.Thechangesin 𝜔1and¤𝐴1duetochangesintherotation\ncurve are insignificant in comparison with the uncertainties.\nB2 Time Evolution\nIn this Section, we validate the inference of 𝜔𝑚and¤𝐴𝑚by apply-\ning the formalism developed in Sec. 2.3 to the simulated sample\naffected by the SF and comparing it to results for the GT model (as\nin Sec. B1.2). By doing this we’re assuming that the formalism de-\nveloped holds and will yield correct results for the GT model. Since\nthe test particle simulation we are using has a fixed warp, we expect\nfrom this test to recover a constant amplitude and null pattern speed\nintheregionatequilibriumwiththepotential(i.e. 𝑅< 𝑅 2=14kpc).\nIntheouterpartsthewarpwouldbeexpectedtoevolvewithtimeas\nthe stars relax in the potential. Because the warp model used in the\ntest particle simulation is not constructed by definition as a Fourier\nseries it is not straightforward to use this data to test the recovery\nof specific values of the time evolution parameters. More involved\ntests in this direction could be done in a future work to validate the\nmethod.\nInwhatfollowsweanalysethedifferencebetweentheparameters\nfrom both models for 𝑅 > 10kpc where the warp is present. We\napply this formalism only to the 𝑚=1mode due to the bias and\nnoisyrecoveryinthe 𝑚=2modeparametersdiscussedinSec.B1.2.\nFigure B6 shows the results for the GT and SF models for 𝜔1\nas a function of radius. For the GT model we get 𝜔1=0for𝑅 >\n12kpc (black curve in Fig. B6). The variations observed in 𝜔1at\n10< 𝑅/kpc<12are expected in this region were the amplitude of\nthe mode is still very low and it’s pattern speed ill-defined. As the\namplitude of 𝑚=1mode increases the pattern speed recovered for\nthe SF model converges to results for the GT at the outermost radii.\nThe mean overestimation in 𝜔1for𝑅 > 12kpc is of the order of\n4kms−1/kpc,whichiswithintheuncertaintiesgivenbytheposterior\nrealisations (gray dots).\nFigure B7 shows the result for ¤𝐴1for the GT (black curve) and\nthe SF models (blue curve). The difference between the two for\n𝑅 < 12kpc is due to the poor recovery in 𝜑𝑉\n1as shown in Fig. B4.\nMNRAS 000, 1–24 (2023)22M. Cabrera-Gadea et al.\nFigureB6. Angularfrequencyasafunctionofthegalactocentricradiiofthe\npattern speed 𝜔1for the𝑚=1mode, calculated from Eq. 14 for the GT\nmodel(blackcurve)andfortheSFmodel(redcurve),theangularvelocityof\nthe starsΩ(green curve) is derived analytically from the Allen & Santillan\n(1991)potential.Thereddotsaroundeach 𝜔1are500realisationtakenfrom\nthe posterior at each ring for the SF model.\nFigure B7. The change in amplitude ¤𝐴1for the𝑚=1mode as a function\nof the galactocentric radius calculated from Eq. 13 for the GT model (black\ncurve) and for the SF model (red curve). The red dots around each 𝜔1are\n500realisation taken from the posterior at each ring for the SF model.\nThemeandifferencefor 𝑅<12kpcbetweentherecoverywiththeSF\nandGTmodelsislessthan 2kms−1,whichiswithintheuncertainties\ngiven by the posterior realisations (gray dots). For 12< 𝑅/kpc<\n17, the general tendency for ¤𝐴1is recovered within the uncertainty\nwith not appreciable bias. The relatively large uncertainty in the\nrecoveryon¤𝐴1stemsfromsmalldifferencesin 𝜑1−𝜑𝑉\n1,whichnear\n𝜋/2translate in large differences in ¤𝐴1due to it dependence on the\ndifference via a cosine function (Eq. 13). The opposite happens for\n𝜔𝑚because it depends on the difference via a sine function.\nAPPENDIX C: GOODNESS OF FIT AND RESULTS FOR\nINDIVIDUAL MODES\nThis appendix presents the results for the goodness of fits and sum-\nmarises the results of Sec. 4.1 for the individual modes.\nFigure C1. Reduced Chi-square for the fits in 𝑍done with𝑀=1(green\ncurve) and𝑀=2(blue curve) as a function of galactocentric radius.\nFigure C2. Reduced Chi-square for the fits in 𝑉𝑧done with𝑀=1(green\ncurve) and𝑀=2(blue curve) as a function of galactocentric radius.\nC1 Goodness of fit\nWehavetestedwiththereducedChi-squarehowmeanymodewhere\nneeded to do the fits in 𝑍and𝑉𝑧for the Cepheids sample. Fig. C1\nand Fig. C2 shows as a function of the radius the results for 𝑍and\n𝑉𝑧. Clearly the result favoured the fits for 𝑀=2for both variables,\nshowing the need of the 𝑚=2mode to reflect the asymmetries\npresentinthewarp.WehavealsocomputedtheBayesianinformation\ncriteria (BIC, Ivezić et al. (2014)) for different radii and we found\nthat𝑀=2isalwaysclearlythebestmodelfor 𝑍atallradii>10kpc.\nThisisofspecialimportancesince 𝑍ismoresensitivetobiasesdue\nto the selection function problems. For 𝑉𝑧the fits with 𝑀=2is\nalsothebestcasefortheouterdiscwheretheamplitudeofthewarp\nis significant. We have therefore chose the 𝑀=2model for both\nvariables.\nC2 Individual modes\nC2.1 Fits in 𝑧\nIn Fig. C3 (left panel), we present the results of the amplitudes for\neach mode and the intrinsic dispersion in 𝑍as a function of radius.\nClearlythe𝑚=1mode(red)dominatesthefit(ithasamaximumof\n≈1.1kpc), as expected from an almost S-type like the Milky Way\nwarp. The main mode that takes into account the asymmetries is\n𝑚=2(violet),itsamplitudebeginstogrowat 𝑅≈10kpcbutnever\nexceeds 250pc.For𝑚=0(yellow)wehaveamaximumof ≈200pc.\nMNRAS 000, 1–24 (2023)Time Evolution of the Warp 23\nFigureC3. Amplitudesofeachmodefor 𝑍(leftpanel)and𝑉𝑧(rightpanel)asafunctionofgalactocentricradius.Theblackcurveshowstheintrinsicdispersion\nfor each radius in the respective variable. The dotted line for the amplitudes in 𝑍shows the results from Skowron et al. (2019a). The colour dots around each\nmode are 500realisation taken from the posterior at each ring.\nFigure C4. Each panel shows the phases of each mode as a function of galactocentric radius. The first two top panels are the results for 𝑍and the bottom two\nfor𝑉𝑧(left𝑚=1andright𝑚=2).Thedotedlineforthephasesin 𝑍aretheconstantphasesobtainedbySkowronetal.(2019a).Thecolourdotsaroundeach\nmode are 500realisation taken from the posterior at each ring.\nThis mode can give asymmetry between both extremes of the warp,\nbut its main purpose is to set the mean height in each ring, so it has\ntheabilitytorepresentradialrippleswithnoazimuthaldependence.\nForcomparison,weplottheamplitudesforeachmodefromSkowron\netal.(2019a)(dottedcurves)obtainedwithexactlythesameCepheid\nsamplebutundertheassumptionofamonotonicdependencyof 𝐴𝑚\nwith𝑅2. For the𝑚=1mode at𝑅 > 10kpc both amplitudes are\npractically the same; for the other modes the amplitudes obtained\nby Skowron et al. (2019a) are similar to the mean behaviour of our\nresults.\nThe wavy pattern in the amplitudes for 𝑅 < 10kpc should notbe fully taken as real corrugations in the modes. In Sec. B1.2 we\nconcluded that stochastic clumps in the 𝜙−𝑧plane due to the SF\ngenerate correlations between the modes. This wavy pattern in 𝐴1\nis removed if we take 𝑀=1, so the wavy pattern is mainly due to\ncorrelations between 𝑚=1and𝑚=2.\nInFig.C4wepresentthephasesofthe 𝑚=1,2modesfor𝑍(top\nright for𝑚=1and top left for 𝑚=2) as a function of 𝑅. First, lets\nconsider𝜑1, our results and those of Skowron et al. (2019a) (dotted\nline) coincide in their general trends for the external region of the\ndisc (>10kpc). For𝑚=1, a twist in the direction of the galactic\nrotation is well defined, beginning at 𝑅≈13kpc. For the internal\nMNRAS 000, 1–24 (2023)24M. Cabrera-Gadea et al.\nregionbothphasesaredifficulttodetermineduetothelowamplitude\nofthewarpandbecausetheazimuthalcoverageisaffectedbytheSF.\nFor𝜑2thereismoreuncertaintythanfor 𝜑1because𝑚=1isbetter\ndefined and dominates the warp. Within its uncertainty 𝜑2agrees\nwith the phase obtained by Skowron et al. (2019a) (red dotted line).\nFor𝑅 > 10kpc the phases, like the amplitudes, are better behaved\nthan in the internal disc as we expected from Sec. B1.2.\nFinally, given that we calculate the intrinsic dispersion for 𝑍in\neach ring, we can see how the disc traced by Cepheids becomes\nthicker at larger radius, as its shown with the black curve in the left\npanel of Fig. C3. This shows how the flare in this young population\nstartsataround 𝑅≈8kpcwithaheight≈100pctoendupataheight\n≈390pc at𝑅≈15kpc. Previous measurements on how thick the\ndisc traced by Cepheid is (Skowron et al. 2019b; Chen et al. 2019)\nagree with our results for the scale and trend found from 𝜎𝐼𝐷.\nC2.2 Fits in 𝑣𝑧\nTherightpanelofFig.C3presentsamplitudesforthefitsinvertical\nvelocityasafunctionofgalactocentricradius.For 𝑉𝑧theamplitudes\nshowasmoothoscillatingpattern.Theimportantdifferencebetween\nthe𝐴𝑚and𝑉𝑚is that in𝑍the𝑚=1mode dominates the warp at\nallradii;in𝑉𝑧thekinematicsignalofthewarpisdominatedbyboth\n𝑚=1and𝑚=2, a result unexpected for a tilted rings model. The\n𝑚=1modein𝑉𝑧startstoappearat 𝑅≈12kpcandatitsmaximum\nreaches an amplitude similar to the value of 𝜎𝐼𝐷≈7.2km s−1. For\n𝑚=2in𝑉𝑧thereisanoscillation,asinfor 𝑚=1too.Theamplitude\nofnoneofthekinematicmodesneverexceedstheintrinsicdispersion,\nbycontrasttothewarpin 𝑍,inwhichtheydo.However,theamplitude\nof the oscillations in 𝑚=1,2is larger than the uncertainty in each\nmode, making the result more significant.\nForthephases, 𝜑𝑉\n1risesfor𝑅>11kpcand𝜑𝑉\n2isnearlyconstant,\ndeclining for 𝑅 >14kpc. Since the amplitudes in 𝑉𝑧for𝑚=1and\n𝑚=2arecomparable,theconnectionbetweenthesebehavioursand\nthe twisting of the LMV 𝑧is not as straightforward as in 𝑍where\n𝑚=1clearly dominates and the LON twist is evident in the decline\nof𝜑1for outer radii.\nFinally, the intrinsic dispersion for 𝑉𝑧(black curve, Fig. C3 right\npanel) is found to be almost constant with radius at 𝜎𝐼𝐷≈7.2km\ns−1.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–24 (2023)"    },    {        "title": "2401.13742v1.The__M___rm_BH__M____relation_up_to__z_sim2__through_decomposition_of_COSMOS_Web_NIRCam_images.pdf",        "content": "Draft version January 26, 2024\nTypeset using L ATEXtwocolumn style in AASTeX631\nThe MBH−M∗relation up to z∼2through decomposition of COSMOS-Web NIRCam images\nTakumi S. Tanaka\n ,1, 2, 3John D. Silverman\n ,1, 2, 3, 4Xuheng Ding\n ,1, 2, 3Knud Jahnke\n ,5\nBenny Trakhtenbrot\n ,6Erini Lambrides\n ,7,∗Masafusa Onoue\n ,2, 8Irham Taufik Andika\n ,9, 10\nAngela Bongiorno\n ,11Andreas L. Faisst\n ,12Steven Gillman\n ,13, 14Christopher C. Hayward\n ,15\nMichaela Hirschmann\n ,16Anton Koekemoer\n ,17Vasily Kokorev\n ,18Zhaoxuan Liu\n ,1, 2, 3\nGeorgios E. Magdis\n ,13, 19, 20Alvio Renzini\n ,21Caitlin Casey\n ,13, 22Nicole E. Drakos\n ,23\nMaximilien Franco\n ,22Ghassem Gozaliasl\n ,24, 25Jeyhan Kartaltepe\n ,26Daizhong Liu\n ,27\nHenry Joy McCracken\n ,28Jason Rhodes\n ,29Brant Robertson\n ,30and Sune Toft\n13, 20\n1Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan\n2Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The\nUniversity of Tokyo, Kashiwa, Chiba 277-8583, Japan\n3Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan\n4Center for Astrophysical Sciences, Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA\n5Max-Planck-Institut f¨ ur Astronomie, K¨ onigstuhl 17, D-69117 Heidelberg, Germany\n6School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel\n7NASA-Goddard Space Flight Center, Code 662, Greenbelt, MD, 20771, USA\n8Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China\n9Technical University of Munich, TUM School of Natural Sciences, Department of Physics, James-Franck-Str. 1, D-85748 Garching,\nGermany\n10Max-Planck-Institut f¨ ur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany\n11INAF–Osservatorio Astronomico di Roma, Via Frascati 33, I-00078, Monte Porzio Catone, Italy\n12Caltech/IPAC, 1200 E. California Blvd. Pasadena, CA 91125, USA\n13Cosmic Dawn Center (DAWN), Denmark\n14DTU Space, Technical University of Denmark, Elektrovej, Building 328, 2800, Kgs. Lyngby, Denmark\n15Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\n16Institute for Physics, Laboratory for Galaxy Evolution and Spectral Modelling, EPFL, Observatoire de Sauverny, Chemin Pegasi 51,\n1290 Versoix, Switzerland\n17Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\n18Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, NL-9700AV Groningen, the Netherlands\n19DTU-Space, Technical University of Denmark, Elektrovej 327, DK2800 Kgs. Lyngby, Denmark\n20Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen N, Denmark\n21Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, I-35122, Italy\n22The University of Texas at Austin, 2515 Speedway Boulevard Stop C1400, Austin, TX 78712, USA\n23Department of Physics and Astronomy, University of Hawaii, Hilo, 200 W Kawili St, Hilo, HI 96720, USA\n24Department of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00 076, Finland\n25Department of Physics, Faculty of Science, University of Helsinki, 00014-Helsinki, Finland\n26Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial\nDrive, Rochester, NY 14623, USA\n27Max-Planck-Institut f¨ ur extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching b. M¨ unchen, Germany\n28Institut d’Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Universit ´e, 98 bis boulevard Arago, F-75014 Paris, France\n29Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91001, USA\n30Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA\nABSTRACT\nOur knowledge of relations between supermassive black holes and their host galaxies at z≳1\nis still limited, even though being actively sought out to z∼6. Here, we use the high resolution\nand sensitivity of JWST to measure the host galaxy properties for 61 X-ray-selected type-I AGNs\nCorresponding author: Takumi S. Tanaka\ntakumi.tanaka@ipmu.jparXiv:2401.13742v1  [astro-ph.GA]  24 Jan 20242\nat 0.7< z < 2.5 with rest-frame optical/near-infrared imaging from COSMOS-Web and PRIMER.\nBlack hole masses (log ( MBH/M⊙)∼7.5−9.5) are available from previous spectroscopic campaigns.\nWe extract the host galaxy components from four NIRCam broadband images and the HST/ACS\nF814W image by applying a 2D image decomposition technique. We detect the host galaxy for ∼90%\nof the sample after subtracting the unresolved AGN emission. With host photometry free of AGN\nemission, we determine the stellar mass of the host galaxies to be log ( M∗/M⊙)∼10−11.5 through\nSED fitting and measure the evolution of the mass relation between SMBHs and their host galaxies.\nConsidering selection biases and measurement uncertainties, we find that the MBH/M∗ratio evolves as\n(1 +z)0.37+0.35\n−0.60thus remains essentially constant or exhibits mild evolution up to z∼2.5. We also see\nan amount of scatter ( σµ= 0.28±0.13) is similar to the local relation and consistent with low- zstudies;\nthis appears to not rule out non-causal cosmic assembly where mergers contribute to the statistical\naveraging towards the local relation. We highlight improvements to come with larger samples from\nJWST and, particularly, Euclid, which will exceed the statistical power of wide and deep surveys such\nas Subaru Hyper Suprime-Cam.\nKeywords: AGN host galaxies (2017) — Active galactic nuclei (16) — Active galaxies (17) — Galaxy\nevolution (594)\n1.INTRODUCTION\nWith our understanding that galaxies grow by increas-\ning their stellar mass through mergers and in situ star\nformation from gas accretion, there are still many unre-\nsolved questions in galaxy evolution. One of the most\nimportant challenges in galaxy formation is understand-\ning the physical processes that relate the growth of su-\npermassive black holes (SMBHs) alongside the growth\nof the galaxies that harbor them. Observational studies,\nmainly in the local universe, have unveiled tight cor-\nrelations between the mass of SMBHs ( MBH) and the\nphysical properties of their host galaxies, such as stellar\nvelocity dispersion σ∗and stellar mass M∗(Magorrian\net al. 1998; Ferrarese & Merritt 2000; Marconi & Hunt\n2003; H¨ aring & Rix 2004; G¨ ultekin et al. 2009; Gra-\nham et al. 2011; Beifiori et al. 2012; Kormendy & Ho\n2013; Reines & Volonteri 2015). How and through what\nphysical processes such a tight relation is formed is still\nunclear, and the origin of the mass relation can shed\nlight on the evolution history of not only SMBHs but\nalso galaxies.\nA widely considered scenario, in answer to this ques-\ntion, is a co-evolution scheme, where galaxies and black\nholes mutually increase their mass at a correlated pace.\nAs a potential physical cause for a co-evolution scenario,\nsome studies implement active galactic nuclei (AGN)\nfeedback, where the energy released from AGNs heats\nthe gas and controls star formation or gas accretion\nthrough radio jets or an AGN winds (e.g. Springel et al.\n2005; Di Matteo et al. 2008; Hopkins et al. 2008; Fabian\n∗NPP Fellow.2012; DeGraf et al. 2015; Harrison 2017). Addition-\nally, studies support a common gas supply simultane-\nously fueling both SMBHs and their host galaxies by in-\ncreasing the BH accretion rate and star formation rate\n(SFR, Cen 2015; Menci et al. 2016). On the other hand,\nothers have shown that, even in the absence of a close\nphysical connection between SMBHs and host galaxies,\nthe mass relation can be achieved through a non-casual\nconnection; major mergers have averaged the mass rela-\ntion statistically (cosmic averaging scenario; Peng 2007;\nHirschmann et al. 2010; Jahnke & Macci` o 2011).\nTo unravel the cause of the mass relation, an effective\napproach is observing the relation between MBHandM∗\nthroughout cosmic history. With such observations, we\ncan directly determine whether the relation in the lo-\ncal universe, both its ratio and dispersion, evolves with\nredshift. Then, comparisons of the observational results\nwith simulations (e.g., Ding et al. 2020; Habouzit et al.\n2021; Ding et al. 2022b) based on various physical mod-\nels can allow us to discuss the physical processes that\nestablish the galaxy-BH relations and further constrain\nthe physics of black hole formation and galaxy evolution.\nBefore the advent of James Webb Space Telescope\n(JWST), statistical studies using 2D image decomposi-\ntion analyses were conducted using images obtained by\nHubble Space Telescope (HST) (e.g., Peng et al. 2006;\nJahnke et al. 2009; Bennert et al. 2011a; Cisternas et al.\n2011; Simmons et al. 2011, 2012; Schramm & Silverman\n2013; Mechtley et al. 2016; Ding et al. 2020; Bennert\net al. 2021; Li et al. 2023b) and ground-based telescopes\nincluding Subaru’s Hyper Suprime-Cam (HSC) (Ishino\net al. 2020; Li et al. 2021), and Pan-STARRS1 (PS1)\n(Zhuang & Ho 2023). In summary, these studies have3\nconcluded that the relation between MBHandM∗does\nnot evolve with redshift at z≲2. However, studies using\na large statistical and universal sample have yet to be\nachieved at z >1. At these redshifts, we can obtain in-\nformation longer than the 4000 ˚A break to constrain M∗\nfrom observations at near-infrared wavelengths. How-\never, there are a limited number of previous studies us-\ning near-infrared data at z≳1 (e.g., Jahnke et al. 2009;\nSimmons et al. 2012; Mechtley et al. 2016; Ding et al.\n2020, utilizing HST/NICMOS or HST/WFC3). Other\nstudies also studied statistical AGN samples using spec-\ntral energy distribution (SED) fitting based 1D decom-\nposition method (e.g., Merloni et al. 2010; Sun et al.\n2015; Suh et al. 2020).\nJWST is now revolutionizing the field of AGN - host\ngalaxy relations up to z∼6 and beyond based on its\nhigh spatial resolution and unprecedented sensitivity.\nFor instance, Ding et al. (2022a) applied a 2D decom-\nposition analysis on early JWST NIRCam data from\nCEERS that successfully detected the host galaxies of\nfive quasars at z∼1.6−3.5 from the SDSS DR17Q cata-\nlog (Lyke et al. 2020). They also succeeded in detecting\nclear substructure and performed pixel-by-pixel SED fit-\nting for one of the five targets, SDSSJ1420+5300A, at\nz∼1.6. Other studies have also performed 2D decom-\npositions of AGN host galaxies using JWST imaging;\nfor instance, Li et al. (2023a) have analyzed a galaxy\nthat is one of the most promising candidates for hav-\ning a recoiling SMBH ( z∼0.36) while Kocevski et al.\n(2023) present the host properties of five X-ray-luminous\nAGNs (3 < z < 5) in CEERS. Zhang et al. (2023) also\nutilized JWST NIRCam data to assess the validity of\nM∗estimated from 1D decomposition (spectrum-based)\nmethod for the HETDEX type-I AGNs (2 < z < 2.5).\nAtz >6, Ding et al. (2023) conducted decomposition\nanalysis of two low-luminosity quasars, thus represent-\ning the highest-redshift record for detection of host stel-\nlar emission. They suggest that z∼6 low luminosity\nquasars have a mass relation consistent with the local\nrelation after considering selection biases and measure-\nment uncertainties, albeit with a small sample. Equally\nremarkable, JWST studies of high- zAGNs are revealing\na higher abundance of lower mass black holes that are ac-\ntively accreting within very dusty and compact galaxies\n(Onoue et al. 2023; Kocevski et al. 2023; Matthee et al.\n2023; Harikane et al. 2023; Maiolino et al. 2023; Greene\net al. 2023). However, in general, the sample sizes of\nthese high- zAGNs and the accuracy of M∗estimation\nare still limited. Therefore, the redshift evolution of the\nmass relation remains highly uncertain.\nTherefore, as an important next step to investigate\nthe evolution of the mass relation, we perform a 2Ddecomposition analysis with JWST/NIRCam data for\na sample whose redshift range significantly improves\nupon what was statistically analyzed before the JWST\nera, further bridging the gap between low- zstatistical\nand limited high- zstudies. We use a sample larger\nthan previous studies using JWST, and that reaches\nup to z∼2−3 when AGN and star formation ac-\ntivities peaked in cosmic history. With NIRCam im-\nages of N= 61 AGNs in COSMOS-Web (Casey et al.\n2023) and PRIMER-COSMOS, we conduct 2D decom-\nposition analyses and then statistically discuss the evo-\nlution of the MBH−M∗relation with consideration of\nthe selection bias and measurement uncertainty (Lauer\net al. 2007; Shen & Kelly 2010; Schulze & Wisotzki 2011,\n2014). In addition, we report on the ability to accurately\nmodel the JWST PSF in each band and the impact on\nthe derived host galaxy properties.\nThis paper is organized as follows. Section 2 describes\nthe JWST/NIRCam data and the sample selection. Sec-\ntion 3 describes the detailed analysis method includ-\ning 2D image decomposition and careful PSF modeling,\nSED fitting, and constrcution of mock data. In Sec-\ntion 4, we present our fitting results and discuss the\nPSF effect on the results. Then, we show the evolution\nofMBH−M∗relation in Section 5 with considering the\nselection bias. Also, we discuss the possibility of scatter\nevolution in the mass relation and summarize the chal-\nlenges with 2D decomposition methods in Section 6. We\npresent the conclusion and prospects for future stud-\nies in Section 7. In this paper, all magnitude are AB\nmagnitude (Oke 1974), and we assume a standard cos-\nmology with H0= 70 km s−1Mpc−1, Ωm= 0.30, and\nΩΛ= 0.70.\n2.DATA\n2.1. COSMOS-Web\nCOSMOS-Web (PI: Jeyhan Kartaltepe and Caitlin\nCasey, GO1727, see Casey et al. 2023, for the overview)\nis a 270-hour treasury survey program in JWST Cycle 1,\ncovering 0.54 deg2with NIRCam (Rieke et al. 2023) in\nfour filters (F115W, F150W, F277W, F444W) and 0.19\ndeg2with MIRI (Bouchet et al. 2015) using F770W. The\nCOSMOS-Web field was split into three areas for obser-\nvations. In this paper, we use the January 2023 data\ncovering ∼0.022 deg2with six visits and April 2023\ndata covering ∼0.28 deg2with 77 visits. Due to the\nlarge field, the April 2023 data is separated into ten mo-\nsaic images. The January and April 2023 data together\naccount for ∼56% of the total NIRCam coverage, and\nthe remainder of the observations are scheduled in De-\ncember 2023/January 2024.4\n10h03m02m01m00m9h59m58m2°40'\n20'\n00'\n1°40'\nRight AscensionDeclination\n: Type-I AGN sample\n: This study\nCOSMOS-Web (entire)\nCOSMOS-Web (Jan23)COSMOS-Web (Apr23)\nPRIMER-COSMOS\nFigure 1. Location of the sample compared to the footprints\nof COSMOS-Web and PRIMER-COSMOS. The background\nis the mosaic image of HST/ACS F814W (Koekemoer et al.\n2007) for the COSMOS field. Red and yellow stars show the\nposition of each target AGN in the final sample and parent\ncatalog, respectively. The region enclosed by the solid red\nline shows the PRIMER-COSMOS field, and blue and green\nshaded regions correspond to the observed field of COSMOS-\nWeb (Jan23) and COSMOS-Web (Apr23), respectively. The\ngray region shows the region scheduled to be observed in the\nCOSMOS-Web in December 2023 and January 2024, increas-\ning the sample size to ∼100.\nThe data are reduced with the JWST Calibration\nPipeline1(Bushouse et al. 2023) version 1.8.3/1.10.0\nand the calibration Reference Data System version\n1017/1075 for January/April 2023 data, respectively.\nThe 5 σdepth in an aperture with a radius of 0 .′′15 has\na wide range from 26.7 to 27.5 mag in F115W and 27.5\nto 28.2 mag in F444W, depending on the number of\nintegrations (also see Section 2.1 of Casey et al. 2023).\nThe mosaic images have a resolution of 0 .′′030/pixel. De-\ntails of the image reduction process will be described in\nFranco et al.,(in preparation). In addition to NIRCam\nfour-band data, we use HST/ACS F814W data (Koeke-\nmoer et al. 2007). In this study, we do not use MIRI\ndata because it is challenging to apply the 2D decom-\nposition method (Section. 3.2) to MIRI data due to its\nlower spatial resolution and larger PSF.\n2.2. PRIMER\n1https://github.com/spacetelescope/jwstPublic Release IMaging for Extragalactic Research\n(PRIMER, PI: James Dunlop, GO1837) is a 195-hour\ntreasury program of JWST Cycle 1, targeting two equa-\ntorial HST CANDELS Legacy Fields: COSMOS and\nUDS. PRIMER-COSMOS covers 144 arcmin2with eight\nNIRCam (Rieke et al. 2023) filters (F090W, F115W,\nF150W, F200W, F277W, F356W, F410M, F444W) and\n112 arcmin2with two MIRI (Bouchet et al. 2015) filters\n(F770W and F1800W) in the COSMOS field. The pro-\ncessed COSMOS-PRIMER data consists of one mosaic\nimage. In this analysis, we use NIRCam eight-band data\nand HST/ACS F814W data. The PRIMER-COSMOS\ndata are reduced with the JWST Calibration Pipeline\n(Bushouse et al. 2023) version 1.8.3 and the calibration\nReference Data System version 1017. The 5 σdepth\nin an aperture with a radius of 0 .′′15 has a wide range\nfrom 27.9 to 28.3 mag in F090W and 28.4 to 28.9 mag\nin F444W, depending on the number of integrations,\n∼1 mag deeper than the COSMOS-Web data. The mo-\nsaic images have a resolution of 0 .′′030/pixel.\n2.3. Broad-line AGN sample\nTo evaluate the relation between M∗andMBH, we\nuse the type-I AGN sample with MBHestimates avail-\nable in Schulze et al. (2015, 2018). Schulze et al. (2015)\npresents the redshift evolution of AGN population based\non spectroscopically observed type-I AGNs from zCOS-\nMOS (Lilly et al. 2007, 2009), VVDS (Le F` evre et al.\n2005, 2013; Garilli et al. 2008), and SDSS (Schnei-\nder et al. 2010; Shen & Kelly 2012). Schulze et al.\n(2018) provides the properties of X-ray selected and\nspectroscopically-confirmed type-I AGNs in the FMOS-\nCOSMOS survey (Kashino et al. 2013; Silverman et al.\n2015). Here, we select the targets in Schulze et al.\n(2015) and Schulze et al. (2018) that are also detected by\nChandra (Chandra-COSMOS Survey; Elvis et al. 2009;\nCivano et al. 2012) or XMM-Newton (XMM-COSMOS;\nCappelluti et al. 2009; Brusa et al. 2010). The 2-10 keV\nflux sensitivity is 7 .3×10−16erg cm−2s−1for Chandra\nand 3×10−15erg cm−2s−1for XMM-Newton.\nWe have MBHestimates from spectra acquired by\nthe FMOS-COSMOS and zCOSMOS surveys with some\nAGNs having measurements from both. Considering\nthe quality of the spectroscopy, the error on MBHes-\ntimation, and the fact that H βline is used for calibrat-\ning virial mass estimators, we use the MBHfrom H β\n(FMOS-COSMOS), H α(FMOS-COSMOS), and Mg ii\n(zCOSMOS-Bright, Deep) in order of preference; e.g.,\nFMOS H βis used for an object with both FMOS H β\nand zCOSMOS Mg iiestimation. As shown in Figure 10\nof Schulze et al. (2018), there is a very good agreement\nbetween the FMOS H α- and H β-based MBHcompared5\nTable 1. Sample size for each single-epoch MBHestimation\nSurvey line CW PR zrange\nFMOS-COSMOSHα 25 3 0.69–1.7\nHβ 14 0 1.2–2.5\nzCOSMOS-bright Mg ii18 0 1.1–2.0\nzCOSMOS-deep Mg ii1 0 0.96\nTotal 58 3 0.69–2.5\nto those using MgII. We note that 39 targets in our\nsample were also observed spectroscopically in SDSS; we\ndo not use these SDSS Mg ii-based MBHmeasurements\ngiven the benefits of the FMOS and deeper zCOSMOS\nspectroscopy. The number and redshift range of each\nmeasurement are summarized in Table 1.\nFor H β, Schulze et al. (2018) used the virial mass es-\ntimation relation by Vestergaard & Peterson (2006),\nMBH(Hβ) = 106.91\u0012L5100\n1044erg s−1\u00130.5\u0012FWHM Hβ\n1000 km s−1\u00132\nM⊙,\n(1)\nwhere L5100 is continuum luminosity at 5100 ˚A and\nFWHM Hβis the full-width at half-maximum (FWHM)\nof Hβbroad line. Then, the H α-based masses are cal-\nculated as given in Schulze et al. (2017); Schulze et al.\n(2018),\nMBH(Hα) = 106.71\u0012LHα\n1042erg s−1\u00130.48\u0012FWHM Hα\n1000 km s−1\u00132.12\nM⊙,\n(2)\nLHαis the H αluminosity and FWHM Hαis the FWHM\nof the broad H αemission line. For Mg ii-based MBH\nestimation (zCOSMOS-bright Mg ii, and zCOSMOS-\ndeep Mg ii), the calibration by Shen et al. (2011) is used,\nMBH(MgII) = 106.74\u0012L3000\n1044erg s−1\u00130.62\u0012FWHM MgII\n1000 km s−1\u00132\nM⊙,\n(3)\nwhere L3000 is continuum luminosity at rest-frame\n3000 ˚A and FWHM MgIIis FWHM of Mg iibroad emis-\nsion line. These single-epoch virial mass estimations\nhave an uncertainty due to possible variability and un-\ncertainties in the modeling of broad-line regions (c.f.,\nShen 2013). In this paper, we use the MBHuncertainties\nthat also consider uncertainties from the single-epoch\nvirial mass estimation, typically ∼0.4 dex. We also\nconsider this uncertainty in generating mock data (Sec-\ntion 3.6).\nFrom the parent catalog, we select broad-line AGNs\nthat reside in the COSMOS-Web and PRIMER fields.\nThe final sample size has 61 AGNs with mass es-\ntimation summarized in Table 1. Figure 1 shows\nthe spatial location of the AGN sample within the\nCOSMOS-Web footprint. We use five broadband im-\nages from HST/ACS (F814W) and JWST/NIRCam(F115W, F150W, F277W, and F444W) for the sam-\nple residing in COSMOS-Web. Four more broadband\n(F090W, F200W, F356W) and mediumband (F410M)\nimages are available for three galaxies in the PRIMER\nfield.\nFigure 2 shows the distribution of 2–10 keV X-ray lu-\nminosity L[2−10 keV] (panel a) and MBH(panel b) as a\nfunction of redshift, where L[2−10 keV] is calculated with\nan X-ray spectral index Γ = 1 .8 (e.g., Brightman et al.\n2013). Since the sample consists of X-ray-selected ob-\njects and is flux-limited, there is a tendency for higher-\nzobjects to have larger L[2−10 keV] over the sensitiv-\nity limit. We can also see the trend of MBHincreas-\ning with redshift. The sample biases likely influence\nthis trend from the flux sensitivity and the availability\nof broad-line FWHM measurements from spectroscopic\ndata. Figure 2 (c) displays the relation between the Ed-\ndington ratio Lbol/LeddandMBHwith Eddington ratio\ndecreasing as MBHincreases. This trend is due to the\nobservational flux limitation and that Leddis propor-\ntional to MBH. While there is a correlation between\nMBHandLbol, dividing LbolbyLeddto calculate the\nEddington ratio cancels out this weaker correlation be-\ntween LbolandMBH.\nFigure 3 shows the F277W original images, i.e., before\nthe decomposition analysis, of representative AGNs in\nour sample as examples of the images before the decom-\nposition analysis. In some targets, we can recognize the\nextended components of the host galaxies far from the\ncentral PSF-like feature. However, a central AGN com-\nponent, especially those with spiky diffraction features\nin the outer part, dominates the system and buries the\nhost galaxy component. These dominant PSF compo-\nnents prevent us from obtaining host galaxy information\ndirectly and make the 2D decomposition analysis neces-\nsary.\n3.METHOD\nTo extract host galaxy components from the original\nAGN + host galaxy NIRCam images, we apply a 2D\nimage analysis tool galight (Ding et al. 2020). With\ngalight , we perform forward modeling of each image as\na superposition of a PSF component and PSF-convolved\nS´ ersic components corresponding to the light from an\nAGN and its host galaxy, respectively. We then obtain\nimages of the host galaxy, free of the AGN, by subtract-\ning the fitted PSF component from the original image.\n3.1. Comparative analysis of model PSF construction\nConsidering that the AGN can account for up to\n∼95% of the total flux (e.g., Ding et al. 2020, 2022a),\nthe results significantly depend on the accuracy of re-\nconstructing the PSF. There are different strategies to6\n0.5 1.0 1.5 2.0 2.5\nz43.043.544.044.545.0log/parenleftbig\nL[2−10keV]/ergs−1/parenrightbig\nXMM-Newton sensitivity\nChandra sensitivity(a)\n1.0 1.5 2.0 2.5\nz7.07.58.08.59.09.5log(MBH/M⊙)\nCOSMOS-WebPRIMER(b)\n7.0 7.5 8.0 8.5 9.0 9.5\nlog(MBH/M⊙)3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0log(Lbol/LEdd)\n(c)\nzCOSd\nzCOS\nFMOS_Hα\nFMOS_Hβ\nFigure 2. Characteristics of the type-I AGN sample with JWST imaging from COSMOS-Web and PRIMER: (a) Distribution\nof 2–10 keV X-ray luminosity L[2−10 keV] as a function of z. Black dashed lines indicate the F[2−10 keV] sensitivity in the sample,\n(b) Black hole mass MBHas a function of z, and (c) relation between the Eddington ratio Lbol/LeddandMBH. The color and\nshape of the points indicate the JWST survey field and the source of MBHestimation, respectively, as shown.\nCID-446\n(z=0.69)\nCID-501\n(z=0.91)\nCID-1186\n(z=0.99)\nCID-510\n(z=1.12)\nCID-361\n(z=1.18)\nCID-157\n(z=1.33)\nCID-499\n(z=1.46)\nCID-389\n(z=1.53)\nCID-1044\n(z=1.56)\nCID-36\n(z=1.82)\nCID-62\n(z=1.92)\nCID-363\n(z=1.92)\nFigure 3. Original images in F277W of some targets in\nthe order of redshift. The target IDs and the redshifts are\nshown in the up-left corner of each image. The white bars\nin the lower left part show 1′′. Depending on the host-to-\ntotal flux ratio ( H/T) that varies in the sample, a central\nAGN component dominates an entire system, burying a host\ngalactic component.\nreconstruct PSF images based on either using theoreti-\ncal PSFs (e.g., Suess et al. 2022) or stellar images (e.g.,\nNardiello et al. 2022; Ding et al. 2022a; Ding et al. 2023;\nZhuang & Shen 2023; Baker et al. 2023). The former\nuses the theoretical PSF model such as WebbPSF (Per-\nrin et al. 2012, 2014), and the latter uses natural stellar\nimages as the PSF directly or modeled PSF with tools\nsuch as PSFEx (Bertin 2011). Many previous studies con-\ncluded that the synthetic PSF simulated by WebbPSF is\nnarrower than the PSFs reconstructed with the natural\nstars (Ono et al. 2022; Ding et al. 2022a; Onoue et al.\n2023). Note that Ito et al. (2023) used the interme-\ndiate method between the former and the latter; theyused theoretical PSFs with WebbPSF and smoothed it by\ncomparing the surface brightness profile of natural stel-\nlar images. In this paper, we reconstruct the PSF using\nthree methods based on natural star images for each\nregion and filter. Then, we compare the results with\nthe different PSF reconstruction methods and discuss\nthe host galaxy characteristics with the method depen-\ndence.\n3.1.1. χ2\nν-based methods\nFirst, we follow the strategy of Ding et al. (2020,\n2022a). They first construct an empirical PSF library\nfor which each PSF is represented by the image of a sin-\ngle star. Then, the 2D decomposition analysis is run\nwith all single PSFs in the library, separately. Then,\nthey sort the results in the order of reduced chi-square\nχ2\nνand stack the PSFs with the top 3, 5, and 8 χ2\nνval-\nues. Using the single PSFs and the stacked PSFs, they\nselect the PSF with the smallest χ2\nνas the final PSF.\nThis method is based on the χ2\nν; i.e., they assumed that\nthe lower χ2\nνis indicative of a better (the closer to the\nmore accurate) PSF.\nFollowing their strategy, we apply the find PSFfunc-\ntion in galight to list PSF candidates, then select\nPSF candidates manually for each mosaic image and\nfilter. In this manual selection process, only obvious\nPSF candidates with the PSF-like complex hexagonal\nand spiky diffraction features and without a galaxy-like\nbroad component are selected. We cropped the images\nof the selected PSF candidates for the short-wavelength-\nchannel filters (F090W, F115W, F150W, and F200W)\nand long-wavelength-channel filters (F277W, F356W,\nF410M, and F444W) as squares with 150 and 240 pixels\nper side, corresponding to 4 .′′5 and 7 .′′2, respectively. Af-\nter removing neighbor objects using clean PSFfunction7\ningalight , the PSF libraries contain ∼30−50 PSF\ncandidates depending on the filter and region. Then,\nwe fit each AGN target with a superposition of a PSF-\nconvolved S´ ersic profile and each single PSF candidate in\nthe PSF library. With the fitting results of each single-\nPSF, we select PSFs with the top-5 χ2\nνand top-75%\nχ2\nνand stack them to generate an averaged PSF image.\nThese top-5 stacked and top-75% stacked PSFs are fi-\nnally used to estimate the parameters in this method.\nNote that each target has its own top-5 and top-75%\nPSFs generated from single PSFs with the lowest χ2\nν\nselected for each target.\n3.1.2. Modeling method\nZhuang & Shen (2023) compare JWST/NIRCam\nPSFs modeled with different methods ( Swarm ,\nphotutils , and PSFEx ), and concluded PSFEx recon-\nstructed PSFs provide the best performance. From\nthe 2D decomposition of simulated broad-line AGNs,\nZhuang & Shen (2023) also suggested that smaller χ2\nν\nvalues do not necessarily provide a means to distin-\nguish which PSFs are more likely to characterize the\nAGN with higher accuracy. Following the conclusion by\nZhuang & Shen (2023), we use PSFEx and compare the\nresults with χ2\nν-based selected PSFs (Section 3.1.1).\nPSFEx constructs an empirical PSF model based on\nthe output catalog of SExtractor (Bertin & Arnouts\n1996). We first run SExtractor for source detection,\nand then run PSFEx for modeling the PSF for each mo-\nsaic image and filter. PSFEx can also reconstruct local\nPSFs as a function of positions on the detector. We\ndo not use local PSFs because Zhuang & Shen (2023)\nalso concluded that a universal or global PSF usually\nshows “satisfactory” fitting results, and the sample re-\ngion (COSMOS-Web and PRIMER-COSMOS) have a\nmuch smaller number of stars than the south continu-\nous viewing zone, which Zhuang & Shen (2023) tested\nlocal PSF reconstruction.\n3.1.3. Comparing the final PSFs\nNow, we have three final PSFs for comparison; the\ntop-5 χ2\nνstacked PSF, the top-75% χ2\nνstacked PSF, and\nthePSFEx PSF (Sections 3.1.1 and 3.1.2). To compare\nPSFs, we perform, a 2D Gaussian fitting for each PSF\nimage and measure the FWHMs along the semi-major\naxis. Figure 4 compares the FWHMs of PSFs obtained\nwith each method, target, and filter. The x and y-axis\nof Figure 4 show the ratio FWHM Top−5/FWHM PSFEx\nandFWHM Top−75%/FWHM PSFEx , where FWHM is\nthe value along the semi major axis.\nFirstly, regardless of the filters, we can see that the\ndistribution extends further in the x-axis direction than\nthe y-axis. This can be attributed to greater variation\n0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20\nFWHMTop5/FWHMPSFEx0.900.951.001.051.10FWHMTop75%/FWHMPSFEx\nF115W\nF150W\nF277W\nF444WFigure 4. Comparison of FWHM (semi-major axis) for\nPSFs from different PSF reconstruction methods for all indi-\nvidual AGNs. The x-axis represents the ratio of the FWHMs\nfor the top-5 PSFs to those for the PSFEx PSFs, and the y-\naxis represents the ratio of the FWHMs for the top-75% PSFs\nto those for the PSFEx PSFs. Blue, green, orange, and red\ncolors indicate the different filters: F115W, F150W, F277W,\nand F444W. Median values and 1 σconfidence range are de-\nnoted by star symbols and error bars. The gray dashed line\nindicates y=x, i.e., the same FWHMs for the top-5 and top-\n75% PSFs. Notably, F277W and F444W exhibit a FWHM\nbias among different PSF reconstruction methods.\nin the FWHMs for the top-5 stacked PSFs. We use the\nsame PSFEx PSF for each target in the same field, and\nthe top-75% stacking in the same field uses mostly the\nsame single PSFs in the field. In contrast, top-5 stacking\nemploys only the best-fit single PSFs with the lowest\nχ2\nν. As a result, the FWHM variation for each galaxy is\nlargest for the top-5 PSF followed by the top-75% PSF\nandPSFEx PSF, and FWHM Top−5/FWHM PSFEx have\na larger scatter than FWHM Top−75%/FWHM PSFEx .\nSecondly, we focus on the FWHM bias between the\nmethods for each filter. As suggested by Zhuang &\nShen (2023), for short-wavelength filters (F115W and\nF150W), there is a significant scatter in the FWHM\nratio. On the other hand, for long-wavelength filters\n(F277W and F444W), the scatter is smaller than the\nshort-wavelength side. These results imply that, in the\nlong-wavelength filters, PSFEx PSFs are sharper than the\ntop-5 and Top-75% PSFs. The impact of these trends\non the 2D decomposition analysis is discussed in Sec-\ntion 4.2.\nNote that these trends can depend on the visual in-\nspection performed when constructing the PSF library\n(Section 3.1.1) and the settings used for Sextractor and\nPSFEx (Section 3.1.2). For example, visual inspection\ncan be biased by the hexagonal diffraction features of the\nJWST PSF, an appropriate FWHM range, and the ab-\nsence of extended structures originating from host galax-\nies. If this selection process is strongly biased by the8\nhexagonal features, it might lead to a selective choice of\nbrighter PSFs. As a result, the parameter distributions\npresented here may not necessarily match the distribu-\ntion of the actual PSF. Nonetheless, even different PSF\nreconstruction methods can result in different FWHMs.\nTherefore, when performing a 2D decomposition anal-\nysis with only one PSF reconstruction method and not\nconsidering the possibilities of other PSFs, 2D decompo-\nsition results can be biased by a specific PSF. Because\ndetermining the PSF shape perfectly is challenging, it\nis also important to discuss uncertainties by consider-\ning the results obtained with possible different PSFs.\nWe discuss how different PSF reconstruction methods\naffect the results of the 2D decomposition and the final\nM∗estimation in Section 4.2, and we perform a detailed\ncomparison of the obtained final PSFs in appendix B.\n3.2. Decomposition\nUsing galight , we fit the AGN + host galaxy images\nwith the composite model of a PSF component and a\nsingle S´ ersic profile (S´ ersic 1968) convolved by a PSF.\nNote that we do not assume the same morphology in\nevery wavelength, and the fitting is performed in each\nfilter independently. In the case there are nearby galax-\nies that can affect the fitting, those galaxies are also\nmodeled as S´ ersic components and fitted simultaneously.\nOur S´ ersic model has seven free parameters: amplitude,\nS´ ersic index n, effective radius re, coordinates of the\ncenter xc, yc, and ellipticity parameters e1, e2. The PSF\nmodel corresponding to the AGN component has three\nfree parameters: amplitude and coordinates of the cen-\nterxc, yc. Thus, the number of free parameters for PSF\n+ single S´ ersic component is ten in total. Note that the\nactual number of free parameters in the fitting changes\ndepending on the number of nearby objects also fitted\nwith a S´ ersic profile.\nTo avoid unphysical results, the range of nandreis\nconstrained to [0 .3,9] and [0 .′′06,2.′′0], respectively. Note\nthat some galaxies show clear substructures that do not\nsuit a S´ ersic profile, such as bars and spiral arms. Thus,\nusing a S´ ersic profile is a first-order approximation to\nmodel the global component of AGN host galaxies.\nIn the fitting process, first, we cut the image into\nsquare regions centered on the target and with a ra-\ndius seven times the standard deviation along the semi-\nmajor axis of the 2D Gaussian fitting with photutils\n(Bradley et al. 2023). Then, the above model is opti-\nmized with Particle Swarm Optimizer (PSO; Kennedy\n& Eberhart 1995). galight also supports Markov Chain\nMonte Carlo (MCMC) to estimate the posterior param-\neter distributions. As Ding et al. (2023) suggested, we\nalso confirm that the uncertainty estimated with MCMCis much smaller than the uncertainties from different\nPSFs. Thus, we do not use MCMC in the decomposi-\ntion analysis, and we estimate the uncertainty from the\nresults with different single PSFs (Section 3.4). We set\nthe supersampling factor relative to pixel resolution to\n3, which controls interpolation within a pixel to perform\na subpixel shift of the PSF (c.f., Ding et al. 2023).\nAs described above, for the PSF library constructed\nwith galight , we fit with every single PSF in the library\nand sort them in order of χ2\nν. Then we fit with the three\nfinal PSFs; top-5 χ2\nνstacked, top 75% χ2\nνstacked, and\nPSFEx . Figure 5 compares the fitting results with the\nfinal PSFs for the example galaxy CID-62 in the F444W.\nWe can find that 2D decomposition makes it possible to\ndetect the host galaxy initially buried under the PSF\ncomponent with all three model PSFs.\nFor targets that have n >8.5 in any band other than\nF277W, we rerun the fit while fixing nto the value\nfound for the F277W band. This filter has the low-\nest number of values hitting the upper limit on n, falls\nabove the rest-frame 4000 ˚A break, and is somewhat\ncentral to JWST wavelength coverage. This pertains to\n10, 11, and 7 sources detected in F115W, F150W, and\nF444W, respectively. From the mock test, we confirm\nthat galight can return n∼9 even if the actual value is\nmuch smaller (Appendix C). This is likely due to fitting\nwhere some of the AGN emission is attributed to a cen-\ntral stellar concentration of the host thus overestimating\nthe host galaxy flux as well.\n3.3. Detection of host galaxy\nFigure 5 shows an example where the host galaxy is\nclearly detected. However, for some galaxies, the strong\nPSF component dominates the total flux maybe due to\nnot only a low host-to-total flux ratio H/T but also com-\npact morphology, making it challenging to distinguish\nthe host signal from their PSF component. To gauge,\nin a quantitative manner, which AGNs have accurate\nhost galaxy information, we exclude cases where the host\ngalaxy is undetectable, following three strategies below.\n(1) Bayesian Information Criteria: In addition\nto the PSF + S´ ersic model (PS+SE model) described\nabove, we also fit with a model containing only a PSF\ncomponent (PS model). Then we calculate Bayesian\nInformation Criteria (BIC; Schwarz 1978) for the two\nmodels, PS+SE and PS model, as,\nBIC = −1\n2χ2+kln (n), (4)\nwhere kis the number of free parameters, and nis the\nnumber of data points. We regard that the PS+SE\nmodel provides a better description of the data than the9\nFigure 5. Example fits for CID-62 ( z∼1.92) using F444W. Each row shows results for a different PSF (top-5 stacked,\ntop-75% stacked, and PSFEx , from top to bottom). In each row, the images are as follows: original, model, data – model point\nsource (host galaxy only), and normalized residuals, from left to right. The right panel shows the 1D surface brightness profile\nwhere dashed lines indicate half-width at half-maximum (HWHM) of each PSF. In the data-point source image, we reveal a\ndisk-like host galaxy which is buried under the PSF before subtraction, regardless of the PSF reconstruction method.\nPS model when BIC PS+SE is much smaller than BIC PS,\nas\nBIC PS+SE <BIC PS−10, (5)\nWe decide the threshold value of the BIC difference of\n10 based on Kass & Raftery (1995).\n(2)S/Nof the host galaxy: To estimate the signif-\nicance of the detection of the host galaxy, we calculate\ntheS/N of the host galaxy as done in Ding et al. (2023).\nWe construct an error map of the PSF-subtracted im-\nages considering two sources of error: noise from the\nobserved images and the uncertainty propagated from\ndifferent PSF reconstructions. Due to the intrinsic vari-\nations of the PSF image even in the same FoV (Zhuang\n& Shen 2023; Yue et al. 2023) and errors in the ob-\nserved image used in PSF reconstruction, the recon-\nstructed PSFs contain uncertainties. Thus, considering\nthe uncertainty from PSF reconstruction is indispens-\nable to calculate the S/N of detected host galaxies. For\nthe PSF uncertainty, we use the pixel-by-pixel standard\ndeviation of the fitted PSF components when fitting the\nPS+SE model with each single PSF in the PSF library.Then, we calculate the final noise map as a compos-\nite of the observed noise map and the PSF uncertainty\nmap. With the final noise map, we calculate the signal-\nto-noise ratio of host galaxy S/N hostwithin a radius of\n2re. Then, we define the detection as cases with high\nS/N host, as\nS/N host>5. (6)\n(3) Manual inspection and removal: We find\nthat some objects have invalid central values in their\nF444W image and shallower surface brightness profiles\nthan any PSFs. The fitting of these galaxies fails even\nwith applying a mask in the central region. We find\ntwo cases (CID-50 and CID-208) with such features and\nlabel them as non-detections.\nAs a more challenging case, CID-166 passes the con-\nditions (1) and (2) but clearly shows PSF-like features\nnot completely removed. Since CID-166 does not show\na host-like component in the PSF-subtracted images in\nany filter, the host galaxy seems to be an extremely low-\nH/T or compact object (see AppendixC for the limita-10\ntion of detecting hosts). Therefore, we manually exclude\nCID-166 in the following discussion.\nWe also find some obvious false detections in F814W,\nwhere the image shows a dominant PSF feature and no\nextended host-like feature, and galight fit the PSF-like\nfeatures as a host galaxy with H/T∼1. We confirm\nthat the decomposition of the JWST images clearly de-\ntects a host-like extended feature. Thus, in addition to\nthe above strategies, we recognize 16 obvious false de-\ntections in F814W as non-detection cases.\nWith the above three strategies, we confidently report\nthe detection of an AGN host galaxy for those that fulfill\nboth the conditions (5) and (6) for each of the final\nPSFs; i.e., we determine whether the host is detected or\nnot for each top-5, top-75%, and PSFEx PSF, separately.\nAlso note that this decision is made for each sample and\neach filter, i.e., a galaxy detected in one filter may not be\ndetected in another filter. The number of objects which\nis detected over one (two) filters of NIRCam is 60 (55),\noccupying ∼98 (90)% out of the entire N= 61 sample\nwith the top-5 PSF (see Appendix A for the number of\ndetection in each filter).\n3.4. Photometry of host galaxies\nWe calculate the S´ ersic flux of the host galaxies con-\nsidering Galactic dust extinction (Schlegel et al. 1998)\nfor the detected cases. On the uncertainty of photom-\netry, we set a larger error of 0.2 mag, which represents\nlikely systematic uncertainties (e.g., Ding et al. 2022a;\nZhuang & Shen 2023; Zhuang et al. 2023) and errors\ndiscussed in Section 3.3, which considers both observa-\ntional errors and PSF uncertainty in a radius of 2 re. We\nuse the 3 σvalue for the undetected filters as the upper\nlimit.\nHere, we also fit with a model containing only a S´ ersic\ncomponent (SE model). For targets with BIC PS+SE >\nBIC SE−10, the SE model is better or flexible enough\nto describe the data than the PS+SE model, and we\nuse the S´ ersic photometry calculated with the SE model\ninstead of the PS+SE model. Such cases are observed\nonly in F444W and are very limited (two or three objects\ndepending on the PSF).\n3.5. SED fitting\nWe fit the photometry of the host galaxies with\nCIGALE (v2022.1 , Boquien et al. 2019; Yang et al. 2022)\nSED fitting library. For a stellar population, we use\nthe single stellar population model by Bruzual & Char-\nlot (2003) ( bc03 module) and the Chabrier initial mass\nfunction (IMF; Chabrier 2003) with the M∗cutoff of\n0.1M⊙and 100 M⊙(Bruzual & Charlot 2003). We as-\nsume a delayed- τmodel for a star-formation history(SFH), where SFR at each look-back time tis modeled\nas\nSFR (t)∝\n\n(t−tage) exp\u0010\n−t−tage\nτ\u0011\n(t > t age),\n0 ( t < t age),(7)\nwhere tageand τindicate the starting time of\nstar-formation activity and the declining timescale\nof SFR. We also consider a nebular emission\nwith nebular module and a dust attenuation with\ndustatt modified starburst module that assumes the\nmodified Calzetti et al. (2000) law. We set E(B−V),\nM∗,tage, and τas free parameters; their grid values are\ndecided basically following Zhuang et al. (2023) CIGALE\nrun and summarized in Table 2. To avoid unphysical so-\nlutions, we set the upper limit of tageto 0.95tH, where\ntHindicates the cosmic age at each redshift. Otherwise,\nstellar metallicity and the ionization parameter Uare\nfixed at Solar metallicity and log U=−2.\nWe also apply the Bayesian-based spectral energy den-\nsity (SED) fitting code, Prospector (Leja et al. 2017;\nJohnson et al. 2021), to see the M∗estimation uncer-\ntainty with different SED fitting codes. Prospector\nis based on the Flexible Stellar Population Synthesis\n(FSPS, Conroy et al. (2009); Conroy & Gunn (2010))\nto generate model SEDs of galaxies. For the compari-\nson, we use almost the same settings with the CIGALE\ndescribed above: the Chabrier IMF, a delayed- τSFH\nmodel, Solar metallicity, Calzetti et al. (2000) dust at-\ntenuation law, and nebular emission with log U=−2.\nProspector can fit with the non-parametric SFH, which\nseparates galaxy formation history into several age bins\nand assumes a constant SFR in each bin (e.g., Leja et al.\n2019). Lower et al. (2020) input cosmological hydrody-\nnamic simulation data into Prospector and concluded\nthat non-parametric SFH tends to reconstruct M∗more\naccurately than parametric SFHs. Lower et al. (2020)\nalso suggested that parametric SFH tends to underesti-\nmate M∗. However, in this study, the available photome-\ntries are limited in the number and wavelength range\n(only five/nine bands in the near-infrared wavelength\nrange for the COSMOS-Web/PRIMER-COSMOS field).\nFurthermore, the photometry derived in Section 3.4 con-\ntain uncertainties from the 2D decomposition analy-\nsis. Therefore, we choose to use the parametric SFH\n(delayed- τmodel) instead of the non-parametric assess-\nment. The parameter prior settings in the MCMC run\nare summarized in Table 2. We compare the results with\nCIGALE andProspector in Section 4.4.\n3.6. Generating mock data to consider selection effects\nAs mentioned, our sample is X-ray-flux limited (Sec-\ntion 2.3), raising the possibility of bias toward larger11\nTable 2. Major parameter settings in SED fitting\nparameter description values ( CIGALE ) prior ( Prospector )\nlog (M∗/M⊙) total stellar mass Scaled with the data Uniform: min=9, max=13\nlog (Z∗/Z⊙) stellar metallicity Fixed at 0 Fixed at 0\nlogU ionization parameter Fixed at -2 Fixed at -2\nE(B−V)/mag Color excess for the nebular lines 0, 0.001, 0.005, 0.01, 0.03, 0.05,\n0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7\nˆτdust,2 optical depth of the diffuse dust\nattenuationUniform: min=0, max=3\nτ/Gyr timescale of delayed- τSFH 0.01, 0.05, 0.1, 0.25, 0.5, 0.75,\n1.0, 1.25, 1.5, 2.0, 2.5, 3.0, 3.5,\n4.0, 4.5, 5.0, 6.0, 7.0, 8.0, 9.0, 10,\n12, 14, 16, 18, 20Uniform in log ( τ/Gyr): min=-2,\nmax=1.5\ntage/Gyr starting time of delayed- τSFH equally sampled in [1 Gyr,\n1.95tH] with the separation of\n0.1 GyrUniform: min=1, max=0.95 tH\n10 11 12\nlogM∗,mock678910logMBH,mock(a)\nMock-all\nMock-observed\n20\n 15\n 10\nlogF[2−10keV],mock0.00.51.0PDF(b)\n10 11 12\nlogM∗,mock101112log˜M∗,mock(e)2 3 4\nlogFWHMHα,mock0.00.51.01.5PDF(c)\n6 7 8 910\nlogMBH,mock678910logMvir,mock(d)\n10 11 12\nlog˜M∗,mock678910logMvir,mock(f)\nFigure 6. Procedure for generating mock galaxies at z∼1.5 with an example parameter set of α=αlocal,β=βlocal,γ= 0,\nσµ= 0.4. Orange and blue contours show the distribution of the entire sample (1 σ−5σ) and that above the detection limit\nof the observed sample (1 σ−3σ, see Sections 5.1 for the definition of the detection). The individual panels are as follows: (a)\ntrue black hole mass MBH,mock - true stellar mass MBH,mock, before adding observational uncertainties, (b) 2 −10 keV flux\n(F[2−10 keV]) ,mock), (c) H αFWHM (FWHM Hα,mock), (d) pseudo-observed virial mass Mvir,mock - true black hole mass MBH,mock\nwhere Mvir,mock is the mock black hole mass after considering the observational bias. (e) pseudo-observed stellar mass ˜M∗,mock\n- stellar mass M∗,mock where ˜M∗,mock is the stellar mass after considering the observational bias, and (f) pseudo-observed virial\nmass Mvir,mock - pseudo-observed stellar mass ˜M∗,mock. The latter is to be compared with the observed MBH−M∗plane shown\nin the left panel of Figure 12. Black dashed lines in panels (a) and (f) indicate MBH−M∗local relation obtained from the\nfitting of the local galaxies (H¨ aring & Rix 2004; Bennert et al. 2011b, , see Section 5), and gray dashed lines in panels (d) and\n(e) indicate y=xline.\nMBHor higher Eddington ratio (Lauer et al. 2007;\nSchulze & Wisotzki 2011, 2014). Due to this selection\neffect, a direct comparison of the observational results\nwith the local relation is not appropriate. Thus, in this\nstudy, we generate a mock AGN-galaxy catalog based\non the procedure in Li et al. (2021) and apply the mock\nobservation (adding selection biases and observationaleffects) to discuss the intrinsic evolution of the mass re-\nlation. The procedure for generating the mock catalogs\nis described below.\nFirst, we generate the mock redshift zmock and mock\ntrue stellar mass M∗,mock based on the COSMOS2020\nstellar mass function (SMF) by Weaver et al. (2023).\nNext, we use the M∗,mock to generate the mock true BH12\nmass MBH,mock for the mock sample. Here, we assumed\nthe local relation as\nlog\u0012MBH\nM⊙\u0013\n=αlog\u0012M∗\nM⊙\u0013\n+β, (8)\nwhere αandβindicate the slope and the intercept of\nthe local MBH−M∗plane. Then, assuming a normal\ndistribution, we calculate MBH,mock as,\nlogMBH,mock =N\u0012\nαlog\u0012M∗\nM⊙\u0013\n+β+γlog (1 + z), σ2\nµ\u0013\n.\n(9)\nHere, the parameter γindicates the strength of red-\nshift evolution of the mass relation, and σµis the in-\ntrinsic scatter of the mass relation. Figure 6 (a) shows\nMBH,mock−M∗,mock distribution with the example pa-\nrameter set of γ= 0, σµ= 0.4.\nThen, based on zmock andMBH,mock, we assign mock\nEddington ratio λEdd,mock by sampling the Eddington\nratio distribution function by Schulze et al. (2015). Us-\ning the λEdd,mock andMBH,mock, we calculate the bolo-\nmetric luminosity Lbol,mock.\nFrom Lbol,mock, we obtain L[2−10 keV] ,mock using\nthe bolometric correction by Duras et al. (2020).\nWith the calculated L[2−10 keV] ,mock, mock X-ray flux\nF[2−10 keV] ,mock is determined with the assumption of\nΓ = 1 .8, the same assumption as in Section 2.3. Fig-\nure 6 (b) shows F[2−10 keV] ,mock distribution with the\nexample parameter set of γ= 0, σµ= 0.4.\nWe also generate mock virial BH mass Mvir,mock with\ntheLbol,mock,MBH,mock, and the assumed FWHM dis-\ntribution. We first calculate mock continuum and line\nluminosity L5100,mock,L3000,mock, and LHα,mock. For\nL5100,mock andL3000,mock, we used the bolometric cor-\nrection by Netzer & Trakhtenbrot (2007) and Trakht-\nenbrot & Netzer (2012), respectively. For LHα,mock, we\nuse the L5100,mock and scaling relation between LHαand\nL5100,mock by Jun et al. (2015). These bolometric cor-\nrection and scaling relations are the same as those in\nSchulze et al. (2018). Then, we assume that FWHM\nof the emission line follows the log-normal distribution\nwith the scatter of 0.17 dex (Shen et al. 2008), and we\ngenerate the observed FWHM of H α, Hβ, and MgII\nemission lines following Equations (1), (2), and (3). To\nconsider the bias in single-epoch virial mass estima-\ntion, we add the bias with βbias= 0.6 in calculating\nFWHM. βbiasrepresents the proportion by which ∆ L,\nthe variation of luminosity from the mean luminosity\n¯L, affects the variation in FWHM. Thus, the FWHM\nis generated by a lognormal distribution with the scat-\nter of 0.17 dex and the mean value corresponding to\nthe luminosity of ¯L+β∆L(c.f. Shen 2013; Li et al.\n2021). Finally, the mock virial BH mass Mvir,mock iscalculated using the mock observed FWHM and the\nluminosity following Equations (1), (2), and (3). Fig-\nures 6 (c) and (d) show the FWHM Hα,mock histogram\nandMvir,mock−MBH,mockdistribution with the example\nparameter set of γ= 0, σµ= 0.4.\nConsidering possible systematic uncertainties from 2D\ndecomposition analysis, we added an error based on a\nnormal distribution with 0.2 dex to the M∗,mock to con-\nsider the uncertainty of M∗derived from observations,\nresulting in the mock observed stellar mass ˜M∗,mock.\nThe resulted ˜M∗,mock−M∗,mock distribution is shown\nin Figure 6 (e).\nFinally, we get the mock data set of the redshift\nzmock, observed stellar mass ˜M∗,mock, and virial BH\nmass Mvir,mock. Figure 6 (f) shows the final mock ob-\nserved mass distribution ( Mvir,mock−˜M∗,mock) with the\nexample parameter set of γ= 0,σµ= 0.4. In Section 5,\nwe discuss the mass relation, comparing our results with\nthis mock catalog.\n4.RESULTS OF AGN - HOST DECOMPOSITION\nFirstly, we examine the results obtained from the 2D\ndecomposition.\n4.1. Morphological parameters\nIn Figure 7 (a)–(c), we show the distribution of nand\nre(in the unit of pixel and kpc) for each filter using the\ntop-5 stacked PSF. The distribution of nhas a peak in\nthe distribution at n∼1−2, clearly seen in the F277W\nand F444W filters. This is similar to studies of AGN\nhosts at high redshift, which show hosts characterized\nby disk-like morphology.\nIn Figure 7 (d), we compare H/T with S/N host. We\ncan see a strong correlation between the reconstructed\nH/T andS/N host. High H/T means host galaxies dom-\ninate the AGN + host galaxy composite images, and we\ncan easily detect host galaxies with high S/N host; thus,\nour results indicate the validity of our analyses for the\nmajority of the sample. Estimated morphological pa-\nrameters and basic information for each host galaxy are\nreported in Table 3. In Appendix C, we confirm that\ngalight can reconstruct reandH/T correctly by run-\nning galight on mock galaxy images.\n4.2. Impact of different PSF models\nIn Figure 8 (a)–(c), we compare morphological pa-\nrameters, n,re, and the host-to-total flux ratio H/T,\nin F444W obtained using each final PSFs. These pa-\nrameters fall mostly along the y=xline, indicating a\nstrong correlation. Thus, we can say that different PSF\nreconstruction methods do not significantly affect the\nresults.13\n0 3 6 901020#DetectedUndetectedF115W(a)\n0 1 201020#(b)\n0.5\n 0.0 0.5 1.0 1.501020#(c)\n0 1 2 30.00.51.0H/T(d)\n0 3 6 901020#F150W\n0 1 201020#\n0.5\n 0.0 0.5 1.0 1.501020#\n0 1 2 30.00.51.0H/T\n0 3 6 901020#F277W\n0 1 201020#\n0.5\n 0.0 0.5 1.0 1.501020#\n0 1 2 30.00.51.0H/T\n0 3 6 9\nn01020#F444W\n0 1 2\nlog(re/pixel)01020#\n0.5\n 0.0 0.5 1.0 1.5\nlog(re/kpc)01020#\n0 1 2 3\nlog(S/Nhost)0.00.51.0H/T\nFigure 7. Summary of the decomposition results in each filter. Columns (a), (b), and (c) respectively show the distribution\nof the estimated n, logrein the unit of pixel and kpc. In column (d), the relation between H/T andS/N hostis shown. In each\npanel, red (gray) indicates detection (non-detection) in each filter.\nIn Figure 8 (a), the size ( re) comparison shows more\nconsistent results than n, indicating that it is minimally\naffected by different PSFs. On the other hand, in com-\nparing n(Figure 8 (b)), regardless of the PSF, we can\nfind a consistent estimation on the low nside ( n≲3).\nHowever, at larger n(≳4), the scatter increases. This\ntendency is likely because larger nimplies a compact\nS´ ersic profile that resembles the PSF, making it chal-\nlenging to distinguish from the PSF. We also find a ten-\ndency that PSFEx PSFs result in slightly larger nthan\nthe top-5 or top-75% stacked PSFs. Even when consid-\nering the other filters, we find highly consistent results,\nwith a trend of increased scatter at higher n.\nRegarding H/T, we primarily see consistent results\nwith strong correlations in Figure 8 (c). While H/T\nestimated using the top-5 and top-75% PSFs is largely\nconsistent with each other, the PSFEx PSFs tend to re-\nsult in slightly higher H/T than the other two PSFs.\nZhuang & Shen (2023) suggested that using a narrower\nPSF than an exact PSF could overestimate the host flux\nandn. Thus, the above biases in nandH/T could beexplained by the fact that PSFEx PSFs for F444W have a\nslightly narrower PSF than the other PSFs, as shown in\nFigure 4. A detailed comparison of the estimated H/T\nin other filters is summarized in Appendix B.\nIn conclusion, the decomposition results using differ-\nent PSFs are generally highly consistent with each other.\nA comprehensive discussion of the technical and prac-\ntical differences between PSF reconstruction methods\nwill also be provided in Section 6.3. Nonetheless, these\ndiscussions are based on the comparisons between es-\ntimated values, and here, we cannot definitively deter-\nmine the true exact value. Related to this, we provide\nthe result of mock tests using different final PSFs in Ap-\npendix C. Also, note that all final PSFs are not single\nstellar images but stacked or modeled PSFs based on\nmultiple stellar images. As mentioned in Section 3.1.3,\nsingle PSFs often have a wide range of sizes and shapes.\nThus, using a single stellar image without testing other\nstellar images can risk misinterpreting the PSF image\nand giving different results.14\n0 1\nTop-75%01Top-5\nρ=0.97(a) re,F444W\n0 4 8\nTop-75%048Top-5\nρ=0.91(b) nF444W\n0.0 0.5 1.0\nTop-75%0.00.51.0Top-5\nρ=0.91(c) H/TF444W\n10 11 12\nTop-75%101112Top-5\nρ=0.97(d) log(M∗/M⊙)\n0 1\nPSFEx01Top-75%\nρ=0.98\n0 4 8\nPSFEx048Top-75%\nρ=0.95\n0.0 0.5 1.0\nPSFEx0.00.51.0Top-75%\nρ=0.93\n10 11 12\nPSFEx101112Top-75%\nρ=0.98\n0 1\nTop-501PSFEx\nρ=0.97\n0 4 8\nTop-5048PSFEx\nρ=0.94\n0.0 0.5 1.0\nTop-50.00.51.0PSFEx\nρ=0.87\n10 11 12\nTop-5101112PSFEx\nρ=0.96\nFigure 8. Comparison of the decomposition results ( rein the unit of arcsec, n, and H/T) based on the F444W filter using\ndifferent PSFs (Columns a–c). Data points in red have satisfied our stringent criteria for host detection (Sec. 3.3). Column (d)\ncompares M∗estimated by the SED fitting, with colors corresponding to whether each object was detected in more than two\nbands. Spearman’s correlation coefficient ρfor each prior is shown in the lower right corner of each panel. High correlation\ncoefficients and the distribution around y=x(black dashed line) suggest consistent results among the fitting with different\nPSFs.\n4.3. Host images\nAs seen in these high-quality host galaxy images, 2D\ndecomposition analyses of JWST images open up the\npotential for a more detailed image-based galaxy anal-\nysis, such as studies traditionally conducted on inactive\ngalaxies. Figure 9 shows three-color images (F277W,\nF150W, and F115W for RGB) of each host galaxy cre-\nated by subtracting the PSF and the nearby Sersic com-\nponent.\nFirstly, thanks to the high spatial resolution, deep ob-\nservations, and meticulous decomposition analysis, we\ncan access the highest-quality AGN-host galaxy images\nup to z∼2.5, allowing us to identify substructures.\nParticularly, in the case of CID-273 ( z= 1.85) and CID-\n307 (z= 2.05), despite their redshifts ∼2, we can clearly\nidentify blue spiral arms with an overall diffuse red broad\ncomponent.\nAdditionally, galaxies such as CID-54 ( z= 0.97), CID-\n510 (z= 1.12), CID-361 ( z= 1.18), CID-445 ( z= 1.26),\nand CID-452 ( z= 1.41) show more reddish colors attheir centers than in the outer region, indicating the pos-\nsibility of having a bulge-like structure or highly dust-\nobscured region. Furthermore, there are cases with ex-\ntended red structures (e.g., CID-668; z= 0.97) which\nmay indicate the presence of dust lanes as seen in the\nX-ray obscured (type-II) AGNs (Silverman et al. 2023).\n4.4. Estimated M∗and the comparison of different\nSED fitting methods\nAs described in Section 3.5, we perform SED fitting\nof the host galaxy photometry to estimate M∗. The in-\nferred M∗for each host galaxy is reported in Table 3.\nFigure 10 shows four examples of the SED fitting re-\nsults with the residuals. Generally, for the objects with\nz≳1, the photometry or upper limit from F814W\nand F115W fall at a rest-frame wavelength shorter than\n4000 ˚A break and are important in constraining stellar\npopulation parameters of host galaxies.\nWe independently employed two distinct SED fitting\ncodes, CIGALE and Prospector , as explained in Sec-15\n1\"CID-378\n(z=0.53)\nCID-35\n(z=0.55)\nCID-446\n(z=0.69)\nCID-71\n(z=0.73)\nCID-260\n(z=0.83)\nCID-501\n(z=0.91)\nCID-668\n(z=0.97)\nCID-54\n(z=0.97)\nCID-79\n(z=0.98)\nCID-1186\n(z=0.99)\nCID-1081\n(z=1.00)\nCID-509\n(z=1.02)\nCID-356\n(z=1.03)\nCID-510\n(z=1.12)\nCID-252\n(z=1.13)\nCID-211\n(z=1.17)\nCID-369\n(z=1.17)\nCID-1111\n(z=1.17)\nCID-361\n(z=1.18)\nCID-210\n(z=1.25)\nCID-477\n(z=1.26)\nCID-445\n(z=1.26)\nCID-157\n(z=1.33)\nCID-1031\n(z=1.36)\nCID-203\n(z=1.36)\nCID-452\n(z=1.41)\nCID-1028\n(z=1.44)\nCID-499\n(z=1.46)\nCID-1078\n(z=1.48)\nCID-206\n(z=1.48)\nCID-201\n(z=1.49)\nCID-249\n(z=1.52)\nCID-389\n(z=1.53)\nCID-1174\n(z=1.55)\nCID-1044\n(z=1.56)\nCID-87\n(z=1.60)\nCID-824\n(z=1.68)\nCID-259\n(z=1.68)\nCID-59\n(z=1.78)\nCID-239\n(z=1.81)\nCID-36\n(z=1.82)\nCID-531\n(z=1.85)\nCID-273\n(z=1.85)\nCID-342\n(z=1.91)\nCID-62\n(z=1.92)\nCID-363\n(z=1.92)\nCID-1079\n(z=1.95)\nCID-307\n(z=2.05)\nCID-247\n(z=2.42)\nFigure 9. Three-color cutout PSF-subtracted images (the F277W, F150W, and F115W for RGB) of all host galaxies detected\nin all F115W, F150W, and F277W in the order of redshift. We have not performed deconvolutions based on the PSF FWHM\ndifferences between each filter, and consequently, each filter image possesses a different-sized PSF. The target IDs and the\nredshifts are shown in the up-left corner of each image. The white bars in the lower left part are 1′′in length. Thanks to the\nhigh spatial resolved deep observation of JWST and the careful decomposition analysis, we can clearly identify substructures,\nsuch as bulges, spirals, bars, and dust lanes.\ntion 3.5. Both codes are run having as similar param-\neter settings as possible. In Figure 11, we compare the\nparameters obtained from both codes. As shown in Fig-\nure 11 (a), the results exhibit a significantly high posi-\ntive correlation. However, we find an offset of approxi-\nmately ∆ log M∗= +0 .14 dex (corresponding to ∼1σ).\nThis offset is not far from a common systematic M∗\nuncertainties among SED fitting methods reported in\nPacifici et al. (2023). It remains challenging to deter-\nmine whether M∗from CIGALE orProspector is more\naccurate. In this study, we primarily use the CIGALE\nM∗in the main discussion to maintain consistency with\nprevious studies for AGN-host galaxies (e.g., Zou et al.\n2019; Ishino et al. 2020; Shen et al. 2020; Li et al. 2021;\nKoutoulidis et al. 2022; Zhuang et al. 2023; Li et al.\n2024).\nFinally, in Figure 8 (d), we compare the estimated M∗\nwith each final PSF. The estimated M∗are on the 1:1\nline and show a strong correlation, suggesting that the\nestimated M∗with different final PSFs remain highlyconsistent. This consistency can be attributed to the\nfact that, as shown in Figure 8 (d), H/T or host flux\nfrom different final PSFs is also consistent.\nWe compare our CIGALE M∗with Zhuang et al. (2023),\nwhich also utilized CIGALE on the COSMOS-Web data,\nand find that these two measurements well agree with\neach other with a scatter of ∆ log M∗= +0 .05+0.20\n−0.16.\nThis is very encouraging since their decomposition anal-\nysis is independent of our effort.\nIn Figures 11 (b)–(d), we compare results for the other\noutput SED model parameters: AV,tage, and τ. While\nAVandtageexhibit large uncertainties, they show a pos-\nitive correlation, with the median offset being close to\nzero within the range of uncertainties. Regarding τ, it\nis evident that significantly inconsistent values are ob-\nserved around log ( τ/Gyr)∼1. Considering that tage\nis generally log ( tage/Gyr)≲0.6 in our sample, this dis-\ncrepancy can be attributed to the challenge of accurately\ndetermining SFH when τ≫tage.16\n0.00.20.40.60.8fλ/10−17ergs−1cm−2Å−1\nCID-361 (z=1.18)\nlog(M∗/M⊙)=10.98±0.15\nModel SED\nModel photo.\nObserved photo.0.0 0.5 1.0 1.5 2.0 2.5λrest/µm\n0 1 2 3 4 5 61\n01(Obs−Model)/Err\n 0.00.20.40.60.8\nCID-510 (z=1.12)\nlog(M∗/M⊙)=11.31±0.100.0 0.5 1.0 1.5 2.0 2.5λrest/µm\n0 1 2 3 4 5 61\n01\n0.00.20.40.60.81.01.2fλ/10−17ergs−1cm−2Å−1\nCID-452 (z=1.41)\nlog(M∗/M⊙)=10.82±0.130.0 0.5 1.0 1.5 2.0\n0 1 2 3 4 5 6\nλobs/µm1\n01(Obs−Model)/Err\n 0.00.10.20.3\nCID-1044 (z=1.56)\nlog(M∗/M⊙)=11.32±0.110.0 0.5 1.0 1.5 2.0\n0 1 2 3 4 5 6\nλobs/µm1\n01\nFigure 10. Representative examples of SED fits to PSF-subtracted host galaxies. Source ID and their estimated M∗with\n1σconfidence range are shown in the top right corner of each plot. The inverted triangle indicates the 3 σupper limit due to\nthe non-detection in each filter. The bottom colored shades show the transmission curves of F814W, F115W, F150W, F277W,\nand F444W from left to right. The lower plot in each panel shows the difference between the observational and best-fit model\nphotometry scaled by the error of the data.\nAdditionally, we confirmed a strong negative corre-\nlation ( ρ=−0.61,p≪0.05) between ∆ log τand\n∆ log M∗. This relation is because larger τindicates\na longer-lasting SFH, i.e., star formation has been per-\nsisting more recently. Consequently, galaxies with larger\nτtend to host more young stellar populations, resulting\nin a smaller mass-to-light ratio and smaller M∗. While\nthese differences may be attributed to differences in stel-\nlar models or fitting strategies (i.e., Bayesian or non-\nBayesian), further investigation is omitted in this paper\nsince it does not impact the main results of this study.\n5.MBH−M∗RELATION\nIn the left panel of Figure 12, we plot our measure-\nments of MBHas a function of M∗. Based on the large\nsample covering a broad range in both parameters, we\nare able to fit the observational data with a linear func-\ntion and find a positive correlation with a Spearman’s\ncorrelation coefficient of ρ= 0.40. The red line repre-sents the results of a linear fit as given here.\nlogMBH=αlogM∗+β, (10)\nwith α≃0.67+0.21\n−0.21,β≃1.31+2.27\n−2.24and the orange-\nshaded region showing the 1 σconfidence interval.\nFor comparison, we also provide the linear fit to the\nlocal sample consisting of 30 inactive galaxies (H¨ aring &\nRix 2004) and 25 active galaxies (Bennert et al. 2011b).\nThe fit to these 55 galaxies results in αlocal≃0.97+0.10\n−0.11\nandβlocal≃ −2.48+1.15\n−1.11. Note that the local sample\nand our high- zsample have different selection effects.\nThus, we cannot directly compare αandβwith αlocal\nandβlocal(see Sections 5.1 and 5.2 for the discussion\nwith consideration of selection bias).\nFor investigating the redshift dependence of the mass\nrelation, we calculate ∆ log ( MBH/M⊙), the relative off-\nset of the black hole mass at given M∗from the local\nrelation:\n∆ log ( MBH/M⊙) =MBH−αlocallog (M∗/M⊙)−βlocal,\n(11)17\nTable 3. Data summary of our sample.∗\nCID XID z R.A.a\nDecl.a\nlogMBHb\nlinec\nlogM∗d\nre,F115We\nnF115We\nlogH/T F115We\n[deg] [deg] [ M⊙] [ M⊙] [′′]\n157 5404 1.33 149.6751 1.9828 8.73 2 10 .74±0.13 0.83 5.6 0.37\n162 169 2.46 149.7359 2.0276 9.30 1 11 .47±0.02 0.06 9.0 0.12\n1489 54330 1.95 149.7536 2.1256 7.86 3 10 .59±0.12 0.06 3.6 0.20\n203 341 1.36 149.8142 2.0164 8.34 3 10 .10±0.19 0.16 1.3 0.67\n307 285 2.05 149.8227 2.0897 8.93 3 11 .02±0.14 1.8 1.1 0.43\n*The full version of this catalog is to be available online which will include results based on all bands.\naJ2000\nbMBHis estimated with single epoch virial mass estimation using each line. We assume the 0.4 dex uncertainty in this study.\nc1:Hβ(FMOS-COSMOS), 2:H α(FMOS-COSMOS), 3:Mg ii(zCOSMOS-bright), 4:Mg ii(zCOSMOS-deep)\ndM∗is estimated using CIGALE with the top-5 PSF results.\neTable 3 only shows morphological parameters in F115W measured with the top-5 PSF. Parameters in the other filters and\nwith the other final PSFs are to be available in the full version.\n10 11 12\nlog(M∗,CIGALE/M⊙)101112log(M∗,Prospector/M⊙)\n∆logM∗=0.14+0.09\n−0.13ρ=0.92 (p/lessmuch0.05)(a)\n0 1 2\nAV,CIGALE/mag012AV,Prospector/mag\n∆AV=0.28+0.34\n−0.23ρ=0.73 (p/lessmuch0.05)(b)\n0 2 4 6\ntage,CIGALE/Gyr0246tage,Prospector/Gyr\n∆tage=−0.27+0.47\n−0.36ρ=0.53 (p/lessmuch0.05)(c)\n2\n 0 2\nlog(τCIGALE/Gyr)2\n02log(τProspector/Gyr)\n∆logτ=−0.06+0.48\n−0.49ρ=0.19 (p=0.126)(d)\nFigure 11. Comparison of SED fitting results with two different SED fitting codes. Panels (a) to (d) compare the estimated\nlogM∗,AV,tage, and log τ. X and Y axis correspond to the result with CIGALE and Prospector. The black dashed line\nindicates the x=yline. The median and 1 σconfidence level of the difference between the two results (defined as ∆ =\nProspector −CIGALE) and Spearman’s correlation coefficient are shown in the lower right part in each panel.\nForαlocal and βlocal, we use the above values from\nthe local samples (H¨ aring & Rix 2004; Bennert et al.\n2011b), i.e., 0 .97 and −2.48, respectively. We plot\n∆ log ( MBH/M⊙) as a function of zin the right panel\nof Figure 12. We then parameterize ∆ log ( MBH/M⊙)\nto evolve with zas:\n∆ log ( MBH/M⊙) =γlog (1 + z). (12)\nHere, we assume there is no redshift change in αandβ\nand ∆ log MBHcan be described by only the evolution\nfrom the local relation (log MBH=αlocallogM∗+βlocal).\nFitting our data with Equation (12) without consider-\ning the selection bias results in γ= 1.19+0.17\n−0.15, suggesting\npositive evolution. However, because our sample is (X-\nray) flux-limited, it’s essential to consider the impact of\nselection bias when determining the mass relation (Ding\net al. 2020; Li et al. 2021; Li et al. 2021, e.g.,). Thus, for\nthe rest of this section, we measure the intrinsic slope ofthe mass relation ( α) atz∼1.5 and the redshift evolu-\ntion parameter ( γ) by comparing our results with mock\ncatalogs as described in Section 3.6. This approach al-\nlows us to account for selection bias and measurement\nuncertainties to determine the intrinsic redshift evolu-\ntion and dispersion of the mass relation.\n5.1. Intrinsic slope ( α) of the mass relation at z∼1.5\nPast efforts to establish the evolution of the ratio be-\ntween black hole and galaxy mass assume that there\nis a linear relation at higher redshifts and the slope of\nthe relation matches the local relation (e.g., Ding et al.\n2020; Li et al. 2021). This may not necessarily be the\ncase. Here, we assess how well the parameters of a linear\nrelation can be constrained.\nIn particular, we initially assumed that αandβ, for\nconstructing the mock samples (Equation (9), have con-\nstant values independent of the redshift, and the evolu-18\n10 11 12\nlog(M∗/M⊙)78910log(MBH/M⊙)\nρ=0.36 (p=0.006)logMBH=0.67+0.21\n−0.21logM∗+1.31+2.27\n−2.24\n0.0 0.1 0.2 0.3 0.4 0.5 0.6\nlog(1+z)1\n012∆log(MBH/M⊙)\nH¨aring&Rix2004\nBennertetal.2011a\nintermediatez\nDingetal.2020\nThisstudy0.0 0.5 1.0 1.5 2.0 2.5 3.0z\nFigure 12. (left) Observed MBHandM∗distribution as shown by the red circles. A linear fit is indicated by the red line with\nthe shaded region indicating the 1 σconfidence range. The best-fit relation and Spearman’s correlation coefficient are reported\nat the top of the panel. ( right) ∆ log ( MBH/M⊙) as a function of redshift where zero corresponds to the local relation as marked\nby the gray band having a width representing the local dispersion. In both panels, samples from previous studies using HST are\nplotted: gray for low- z(H¨ aring & Rix 2004; Bennert et al. 2011b), green for intermediate- z(Jahnke et al. 2009; Bennert et al.\n2011a; Cisternas et al. 2011; Schramm & Silverman 2013) and orange indicating those from Ding et al. (2020) at z∼1.5.\n0.5 1.0 1.5 2.0 2.5\nα20\n16\n12\n8\n4\n04βα=1.54+0.31\n−0.36\nβ=−8.53+3.80\n−3.20\nαlocal=0.97+0.10\n−0.11βlocal=−2.48+1.15\n−1.11012\n0.0 0.210 12810log(MBH/M⊙)\nα=αlocal,β=βlocal\nLocal\n10 12810\nα=0.70,β=0.50\n10 12810\nα=1.30,β=−6.00\n10 12\nlog(M∗/M⊙)810log(MBH/M⊙)\nα=1.54,β=−8.53\nMedian\n10 12\nlog(M∗/M⊙)810\nα=1.80,β=−11.50\n10 12\nlog(M∗/M⊙)810\nα=2.10,β=−14.50\nFigure 13. (left) Posterior distribution in the α−βplane obtained through a comparison between mock observation and\nour sample. The posterior distributions for αandβare shown in the right and upper 1D histogram. Each contour indicates\n1, 2, and 3 σlevels. The median and 1 σconfidence levels of αandβare shown in the top-right corner of each panel. ( right)\nComparison of mock observed masses generated with each αandβatz= 1.5 with the real observed data at z= 1−2 shown\nin red circles. Contours show the distribution of mock observation, indicating 1, 2, and 3 σlevels. Each parameter set, αandβ,\nis shown in the upper left corner of each panel. As shown, we cannot reject the scenario that αdoes not evolve up to z∼2.5.\ntion is expressed simply as γlog (1 + z). However, the\nresults in the previous Section 5 may indicate a different\nαfrom the local value without considering selection bi-\nases. In this section, we examine whether the constant\nαassumption is valid by estimating the intrinsic value of\nαwhile considering selection bias and provide evidence\nfor an intrinsic relation between MBHandM∗atz >1\nfor the first time.Since the parameters α,β, and γexhibit degeneracies,\nwe constrain the zrange to z= 1−2 and perform\nfitting for αandβwith an assumption of γ= 0 over\nthis redshift range. Consequently, we cannot compare\nthe estimated βwith the values in the local relation\ndirectly. For the intrinsic dispersion of the mass relation\nσµ, we assume 0.3 dex, thus matching local studies.19\nFigure 14. Posterior distribution on the σµ−γplane obtained through a comparison between mock observation and our\nsample (the result with the top-5 stacked PSFs). Panels (a) to (c) correspond to the result using basic, Gaussian, and realistic\nprior settings, respectively. The posterior distribution for σµandγare shown in the right and upper 1D histogram. Each\ncontour (red to orange) indicates 1, 2, and 3 σfrom inner to outer. The median and 1 σconfidence levels of γandσµare shown\nin the top-right corner of each panel. Only in panel (b), we plot the contour from Li et al. (2021) in gray, and their γandσµ\nestimations are shown in the lower left corner based on SDSS quasars at 0 .3< z < 0.8 with Subaru HSC imaging. Regardless\nof the prior, the results suggest no or mild evolution at z <2.5.\nWith the mock observed data described in Section 3.6,\nwe apply the selection criteria corresponding to our sam-\nple (Section 2.3) and compare them with the observed\nresults to constrain the evolution parameters. We as-\nsume the observation thresholds as;\nF[2−10 keV] > F[2−10 keV] ,lim, (13)\nFWHM line>1000 km s−1, (14)\nwhere the first and second conditions correspond to the\ndetection limit of the X-ray observation and broad lines\nfor single epoch MBHestimation. As shown in Fig-\nure 2 (a), excluding the three objects, all sources in\nour sample have F[2−10 keV] above the XMM-Newton\nF[2−10 keV] ,lim. Note that all of the three targets with\nF[2−10 keV] smaller than XMM-Newton F[2−10 keV] ,lim\nare targets observed only with Chandra. Therefore, for\nthe 57 sources other than the three targets, we utilized\nthe XMM-Newton F[2−10 keV] ,lim, while for the remain-\ning three2, we employed the Chandra F[2−10 keV] ,lim; i.e.,\nwe changed F[2−10 keV] ,limdepending on the compared\nobservation galaxy. Because MBHof our real targets\nare based on single epoch estimation with three different\nlines (H α, Hβ, and MgII), we have three different selec-\ntion thresholds on the mock galaxy using FWHM Hα,\nFWHM Hβ, and FWHM MgII.\nFor each of our targets, we first select the mock galax-\nies with the corresponding selection bias, i.e., the selec-\n2We exclude CID-166 manually (Section 3.3), so the sum 57+3 =\n60 is smaller than our sample size 61.tion condition using the same line information used to\nestimate MBH. Furthermore, we select the mock galax-\nies with a similar redshift; |z−zmock|<0.1. Then, we\ncalculate the probability that the mock galaxies with\nthe similar zmock as the real galaxy would have the\nsame ˜M∗,mock with|∆M∗|<0.1 and Mvir,mock with\n|∆MBH|<0.1, i.e., calculate the probability pfollow-\ning\np=N|∆z|<0.1,|∆M∗|<0.1,|∆MBH|<0.1\nN|∆z|<0.1. (15)\nWe then calculate the likelihood of our sample being\nobserved for each parameter combination of αandβ.\nThereby, we estimate the probability distribution of\nthese parameters using MCMC. In the sampling, we as-\nsume a uniform prior between 0.1 to 3 for α, and -25 to\n8 for β.\nThe obtained α−βdistribution is shown on the left\npanel of Figure 13, indicating a strong anti-correlated\ndegeneracy between αandβ. Because the 1 σcontour\nincludes values ( αlocal, βlocal) corresponding to the local\nrelation, our results do not definitively reject the sce-\nnario where αandβdo not evolve compared to the\nlocal relation up to z∼2.5. However, our results have\na preference to higher αthan the local relation with\nα= 1.54+0.31\n−0.36. This result suggests high- zmass rela-\ntion is steeper than the local relation for massive ellip-\nticals, being more consistent with quasar host galaxies\natz≲0.35 (Zhuang & Ho 2023). In the right pan-\nels of Figure 13, we show the MBH−M∗distribution\nfor mock observed galaxies (similar to Figure 6f) gen-\nerated with manually sampled parameters on the ridge20\n9 10 11 12\nlog(M∗/M⊙)78910log(MBH/M⊙)\nαlocal=0.97+0.10\n−0.11\nβlocal=−2.48+1.15\n−1.11γ=0.37,σµ=0.28\n9 10 11 12\nlog(M∗/M⊙)78910log(MBH/M⊙)\nFigure 15. (left) Comparison of the observed sample and the mock observation of z∼1.5 AGN on the MBH−M∗plane. The\nmock data at z∼1.5 is constructed with the best-fit γandσµ(the result with the top-5 stacked PSFs assuming the “Gaussian”\nprior) while assuming αlocalandβlocal. Small red circles with error bars and the red dashed lines indicate the observed data\nand the linear fit results (same with Figure 12 left). Contours indicate a distribution of mock observed samples, showing 1, 2,\nand 3 σfrom inner to outer. ( right) Comparison of the intrinsic MBH–M∗relation at z∼1.5 (the red line) to the local relation\n(the gray line).\nof the α−βdegenerated relation and the real observed\ngalaxies ( α∼0.7−2.1). The plots suggest that the\nmock data exhibits a similar distribution to our sample\nwithin the comparable mass range. However, because\nthese parameters are constrained only with our results,\nmore significant differences in M∗distribution between\nthe shown parameters become evident at the lower MBH\nrange (log ( MBH/M⊙)<8). Therefore, to break this de-\ngenerate relation and improve the precision of determin-\ningαandβ, a larger sample with a wider MBHrange\nin future studies is essential.\nEven so, we demonstrate that a relation between MBH\nandM∗at high- zis realized based on having a statis-\ntical sample afforded by the COSMOS-Web data set.\nAs indicated by Figure 8 (d), the difference in the PSF\nreconstruction methods does not significantly affect the\nM∗estimation. Therefore, the posterior distribution of\nαandβthat are estimated based on M∗also shows no\nsignificant PSF dependency.\n5.2. Intrinsic evolution ( γ) of the MBH/M∗relation\nTo determine the evolution of mass relation with con-\nsideration of the selection bias and measurement uncer-\ntainties, we generate mock observed catalogs with free\nparameters of γ(evolution rate) and σµ(intrinsic dis-\npersion of the mass relation) and constrain them by\ncomparing the mock catalogs with observational data\nin a similar manner to Section 5.1. In contrast to Sec-\ntion 5.1, we fixed αandβin Equation (9) to the values\nin the local relation ( αlocal,βlocal) to consider the red-\nshift evolution. As illustrated in Section 5.1, we cannotrule out the possibility for evolution of α. However, as\ndescribed in Section 5.1, α,β, and γare strongly de-\ngenerate, and obtaining physically meaningful results is\nchallenging when all three parameters are left free for\nthe sample being considered here.\nThere is still a possibility that σµdepends on red-\nshift. Nevertheless, as discussed later, imposing strong\nconstraints on σµin our results is challenging due to the\nsample size and its uncertainties. Therefore, we set σµ\nas a constant independent of redshift in the fitting. It\nmeans σµobtained through this method is considered\nto be an averaged value over z∼0.7−2.5. Even so, this\nσµestimation has the highest statistical significance for\nsuch a study at z≳1. We discuss the redshift evolution\nofσµin Section 6.2. Finally, in this analysis, we as-\nsume no redshift-dependent parameters among the free\nparameters. Therefore, there is no need to restrict the\nredshift range within the data, as in Section 5.1.\nIn the fitting process, we assume a uniform prior dis-\ntribution for γbetween −1< γ < 1. Then, we have\nthree different prior settings for σµ: a uniform distri-\nbution between 0 .01< σ µ<1.0 (basic), a Gaussian\ndistribution with a mean of 0.3 dex and a standard de-\nviation of 0.1 dex (Gaussian), and a uniform distribution\nbetween 0 .25< σµ<1.0 with a prohibition of σµ<0.25\n(realistic).\nFigure 14 shows the estimated posterior distributions\nofγandσµusing each prior setting with the top-5\nstacked PSF results. As evident in all panels by the\norange contours, the intrinsic dispersion σµis strongly\ndegenerate with the evolution rate γwhere a smaller γ21\nresults in a larger σµas demonstrated in previous stud-\nies (Ding et al. 2020; Li et al. 2021). To reiterate, a\nsmaller value for γbiases the mass relation towards rel-\natively lower MBHvalues thus a larger σµis required to\nreproduce a certain set of observation data.\nConsidering the likelihood distribution for γ, the “ba-\nsic” prior setting shows a slight positive-to-no evolution\nwith γ= 0.34+0.37\n−0.58. Similarly, the “Gaussian” prior set-\nting results in γ= 0.37+0.35\n−0.60, just slightly closer to the\ncase of ”no evolution ( γ= 0). If we assume that the\nintrinsic dispersion should not be significantly smaller\nthan the local dispersion, we limit the allowed range for\nσµto be above 0 .25 (”Realistic” case). In this case,\nwe find γ= 0.06+0.37\n−0.50, remarkably close to the case for\nno evolution. For the latter, the intrinsic dispersion is\nslightly higher at 0 .36+0.10\n−0.07. In all cases, our results are\nconsistent with very mild or essentially a lack of evolu-\ntion with respect to the local relation.\nThen, the left panel of Figure 15 compares mock ob-\nservations using the median parameters ( γ= 0.30 and\nσµ= 0.29) under the assumption of “Gaussian” prior\nwith the actual observational MBH–M∗distribution and\nrelation. We can see that the mock data can explain\nthe observed data well. The right panel of Figure 15\ncompares the intrinsic relationship, i.e., the relation cor-\nrected for the selection bias, with the local relation.\nAgain, the resulting intrinsic relation is consistent with\nthe local relation within the range of errors.\n6.DISCUSSION\nIn Section 5.2, our findings suggest a mild or lack of\nevolution of the mass relation from the local relation\nwhen considering selection biases and measurement un-\ncertainties. In this section, we first compare the derived\nvalues of γandσµfrom Section 5.2 with previous stud-\nies. Then, we also discuss the cosmic averaging scenario\n(Peng 2007; Hirschmann et al. 2010; Jahnke & Macci` o\n2011).\n6.1. Comparison to other studies\nFirst, the conclusion of no or mild evolution with\nγ= 0.37+0.35\n−0.60is consistent with studies based on 2D\ndecomposition analysis (e.g., Ding et al. 2020; Li et al.\n2021) and studies using a SED-fitting-based decomposi-\ntion method (e.g., Sun et al. 2015; Suh et al. 2020).\nIn Figure 14, we compare our results on the estimated\nγ−σµdistribution to those of Li et al. (2021). Our\nsample has higher redshift range than Li et al. (2021),\nand the change of ∆ log ( MBH/M⊙) is also proportional\nto log (1 + z) in our model. Therefore, to reproduce the\nobservational results, when increasing γ, we need to de-\ncrease σµmore than Li et al. (2021). In other words,the slope of the γ–σµdegenerate relation is steeper in\nour study. As a result, while the sample size of this\nstudy is approximately ten times smaller than Li et al.\n(2021), the uncertainty of γis only ∼2 times larger\nthan Li et al. (2021). On the other hand, due to the\nsteep slope, imposing constraints on σµbecomes chal-\nlenging, and the uncertainty becomes ∼4 times larger\nthan Li et al. (2021). Even so, our estimated value of\nσµis 0.28+0.13\n−0.13, which is remarkably similar to Li et al.\n(2021) with σµ=0.25+0.03\n−0.04.\nIt is worth highlighting that the inference on the\nvalue of γis very close to zero for the ”Realistic” case\n(Fig. 14c) where we assume that the intrinsic scatter\n(σµ) cannot be lower than the local dispersion. Inter-\nestingly, if σµ(∼0.4−0.5) is actually higher than the\nlocal value, this would push the evolution parameter to\nnegative values ( γ∼ −0.5) thus presenting a scenario\nwhere the black holes have to catch up to their host\ngalaxies by a bit.\n6.2. Scatter ( σµ) evolution and cosmic averaging\nWhen assuming a non-casual cosmic averaging sce-\nnario (Jahnke & Macci` o 2011), major mergers average\nand equalize the mass ratio MBH/M∗through cosmic\nhistory. Thus, σµshould increase towards high redshift.\nTo test the cosmic averaging scenario with our data,\nwe generate a mock sample (20,000 parameter sets of\nMBHandM∗) atz∼1.4, the median redshift of our ob-\nservational sample. We generate M∗based on the SMF\nby Weaver et al. (2023), and calculate the MBHassum-\ning the γandσsampled in the MCMC run with “Gaus-\nsian” prior setting (Section 5.2). Each mock galaxy is\nassumed to undergo major mergers following the major\nmerger rate from Rodriguez-Gomez et al. (2015) cov-\nering z∼0−1.4. We simulate the redshift evolution\nofMBHandM∗by summing them with those of merg-\ning partners. Then, we calculate σµat each redshift to\ntrace the expected scatter evolution from the assumed\nconditions. In this simulation, we do not consider the\naccretion onto the black hole and star formation, i.e.,\nboth MBHandM∗are assumed to grow only through\nmajor mergers. We limit the mergers to those with mass\nratios within ±0.5 dex.\nFigure 16 compares the simulated redshift evolution of\nσµwith the results from this study and Li et al. (2021).\nWhen assuming the evolution only through major merg-\ners, the growth within the 1 σuncertainty range signifi-\ncantly encompasses the results of Li et al. (2021). More-\nover, our median redshift evolution is consistent with the\nresults of Li et al. (2021). Therefore, the σµdifference\nbetween the results from this study and Li et al. (2021)22\ncould be interpreted as the major merger-based scatter\nevolution.\nHowever, due to the large uncertainty in our results,\nwe cannot draw any definitive conclusions regarding the\nredshift evolution of σµ. Furthermore, as evident from\nFigure 16, our sample has a wide redshift range com-\npared to Li et al. (2021). If σµvaries with redshift, the\nsample should be binned in a narrower redshift range to\ntrace the redshift evolution. Nevertheless, as mentioned\nin Sections 5.1 and 5.2, the degeneracy relation on the\nγ−σµplane tends to steepen toward higher redshifts,\nmaking it relatively challenging to impose constraints\nonσµ. Future high- zstatistical studies will likely re-\nquire samples of a similar size to Li et al. (2021) with\nN= 584, or even larger to address the redshift evolution\nofσµ. Thus, it will be necessary to conduct compre-\nhensive surveys of high- zAGNs using next-generation\nsurvey data such as Euclid and Roman.\nFigure 16. Evolution of σµatz≲2 based on a simple\nsimulation of the cosmic averaging scenario. The vertical red\nerror bars represent the scatter in σµwithin the zrange of\nour sample using the basic prior. Faint grey lines correspond\nto each sampled parameter set while the red line and the\norange filled region represent the median and 1 σconfidence\nlevel of σµredshift evolution. The green data denotes the\nconstraint from Li et al. (2021) which is consistent within\nthe 1 σrange of our results.\n6.3. General notes on PSF reconstruction methods\nSo far, various studies have performed decomposition\nof JWST images (e.g. Ding et al. 2022a; Ding et al. 2023;\nStone et al. 2023; Yue et al. 2023; Harikane et al. 2023;\nZhuang & Shen 2023; Zhuang et al. 2023; Stone et al.2023). As discussed above or in the previous studies,\nthe results of the AGN+host galaxy 2D decomposition\ndepend significantly on the PSF reconstruction. Espe-\ncially, Zhuang & Shen (2023) discussed the effect of dif-\nferent PSF on decomposition results. Zhuang & Shen\n(2023) compared three PSF modeling methods ( SWarp ,\nphotutils , and PSFEx ), but they did not compare them\nwith χ2\nνbased methods directly. Thus, in this paper, we\nsummarize the comparison when using different PSF re-\nconstruction methods.\nIn this study, we compare three final PSFs: two ob-\ntained by χ2\nν-based methods (a top-5 stacked PSF and\na top-75% stacked PSF) and one from an empirically\nmodeling method, PSFEx . As demonstrated in Figure 4,\nwe find offsets in FWHM among different PSF recon-\nstruction methods. These PSF variations, as indicated\nby Zhuang & Shen (2023), could potentially introduce\nbiases in estimating n,re, orH/T. This is due to a\nbroader (narrower) PSF than in reality which tends to\nresult in smaller (larger) n, larger (smaller) re, and over-\nestimation of host fluxes. However, as shown in Figure 8,\nmorphological parameters such as n,re,H/T generally\nexhibit consistent relations. Besides, as shown in Fig-\nure 8 (d), different PSFs have less impact on M∗estima-\ntion than on H/T. This could be due to the fact that M∗\nis estimated from SED fitting (Section 3.5) using multi-\nband photometry that averages the uncertainty in each\nband. If so, SED fitting with a smaller number of pho-\ntometric bands (e.g., Ding et al. 2023; Yue et al. 2023),\nmay lead to more severe effects from inaccurate PSF re-\nconstruction. We also find each method has advantages\nand disadvantages from a technical aspect. Lastly, we\nsummarize below the technical comparison between each\nmethod.\nModeling method: The approach of constructing\nan empirical PSF model from numerous stars, such as\nPSFEx in this study, has the advantage of being less in-\nfluenced by noise compared to the χ2\nν-based methods, as\ndepicted in Figure 17. This method also allows flexible\nmodeling, considering PSF as a function of position or\nbrightness.\nHowever, a drawback is the requirement for many\nPSF candidates to model a local or flux-dependent PSF,\nwhich can be considered a trade-off. Furthermore, we\nalso confirm that FWHM values of PSFEx PSFs de-\npend on configuration parameters. When reconstructing\nPSFs, it is challenging to determine the best configura-\ntion parameters because the exact PSFs are not known.\nχ2\nν-based selection method: Selecting PSFs based\nonχ2\nνfrom a substantial number of stars, such as Top-5\nor Top-75% PSFs, allows easy analysis considering the\nPSF uncertainties. Also, by fitting with various differ-23\nent single PSFs, the possibility for PSF mismatches is\nminimized. Notably, Yue et al. (2023) discussed the dif-\nferences in broad-band PSFs attributed to variations in\nSED shapes between stars and AGNs. Our method us-\ning all single PSFs in the PSF library may consider this\nPSF uncertainty as a result of selecting low- χ2\nνsingle\nPSFs with a matched shape.\nHowever, using fewer PSF candidates, like the top-\n5 stacked PSF, might increase the noise of the final\nPSF, as observed in Figure 17. Note that this noise\nis generally smaller than the central main component;\nthus, it should not affect significantly except in the case\nwith small H/T. Additionally, our approach involves vi-\nsual inspection in PSF candidate selection, which might\ncause a bias. Also, as mentioned by Zhuang & Shen\n(2023), a lower χ2\nνdoes not necessarily mean more cor-\nrect PSF. Moreover, as a practical demerit, this method\nneeds more time to create a PSF library with visual\ninspection in each filter and region and more computa-\ntional cost for SED fittings with all single PSFs.\nGiven that we do not know the correct answers in\nthis study, it is challenging to discuss which method\nproduces the most accurate results. For some targets,\ndecomposition is clearly successful with one final PSF,\nwhich can then be evaluated from the residual emission\nbased on other PSFs. We also confirm that these failures\nin fitting can occur with all final PSFs. Therefore, we\nconclude that it is best to assess the impact on derived\nproperties (e.g., n,re,H/T) by varying the method of\nPSF construction (Section 3.1) and place equal weight\non assessing the uncertainties based on varying PSFs\n(Section 3.3) to obtain solid 2D decomposition results .\n7.CONCLUSIONS\nWe performed a 2D decomposition analysis of high- z\n(z∼0.7−2.5) type-I AGNs using the COSMOS-Web\n(Casey et al. 2023) and PRIMER-COSMOS surveys to\nmeasure the black hole – stellar mass relation at high-\nz. Our sample contains 61 targets that are X-ray-\nselected, broad-line AGNs with single-epoch black hole\nmass estimates (log ( MBH/M⊙)∼7.5−9.5) based on\nHβ, Hα, and Mg iifrom previous spectroscopic surveys\n(e.g., Schulze et al. 2018).\nBy utilizing HST/ACS + JWST/NIRCam imaging\nthat covers the rest-frame optical to near-infrared, we\nobtained multi-band information of the AGN host galax-\nies with unprecedented spatial resolution in which we\ncan clearly identify substructures such as bars and spi-\nrals arms (Figure 9). Since AGN–host 2D decompo-\nsition is known to be sensitive to the PSF reconstruc-\ntion methods, we compared the results with three final\nPSFs reconstructed using the modeling method PSFExand a χ2\nν-based selection method. Through a meticulous\ndecomposition analysis using various PSFs, we success-\nfully detected the host galaxies in more than two filters\nfor∼90% of the entire sample. Then, we confirmed\nthat host morphological parameters such as n,re, and\nH/T remain relatively consistent regardless of the PSF\nreconstruction method used (Figure 8). Furthermore,\ngiven the high quality of the host galaxy images, this\nstudy is expected to serve as a crucial stepping stone\nfor image-based spatially-resolved investigations of AGN\nhost galaxies, such as double-S´ ersic model fitting (de-\ncomposition fitting with AGN, bulge, and disk compo-\nnents) or parametric/non-parametric substructure anal-\nysis.\nWith AGN-subtracted photometry of the host galaxy\nin multiple bands, we estimated M∗by performing\nSED fitting and present the MBH–M∗relation at z∼\n0.7−2.5 (Figure 12). There is a positive correlation\nbetween MBHandM∗with the correlation coefficient\nofρ= 0.36 (p≪0.05). We fit the mass relation\nby a simple (log-)linear relation of log ( MBH/M⊙) =\nαlog (M∗/M⊙)+βwith consideration of selection biases\nand measurement uncertainties. Our results show that\nthe slope of the mass relation at z∼2 (α= 1.54+0.31\n−0.36)\nis slightly steeper than yet consistent with the local re-\nlation ( αlocal= 0.97+0.10\n−0.11) (Figure 13).\nAssuming the redshift evolution term of the mass re-\nlation to be γlog (1 + z), we further determine the evo-\nlution factor γand the intrinsic scatter of the mass\nrelation σµwhile considering selection biases and un-\ncertainties based comparisons to mock catalogs (Fig-\nure 14). Even though the estimated probability distri-\nbution shows strong degeneracy between γandσµ, we\nfind no or mild evolution with γ= 0.37+0.35\n−0.60. The esti-\nmated γ−σµdistribution is largely consistent with Li\net al. (2021) based on the HSC imaging of SDSS quasars\natz <0.8. Given the higher redshift range of our sam-\nple, the slope of the degeneracy relation between γand\nσµis steeper than Li et al. (2021). Therefore, despite\nthe sample being ∼10 times smaller, the estimated γ\nuncertainty is just slightly larger than Li et al. (2021).\nFurthermore, the estimated value of the intrinsic scat-\nter is σµ= 0.28+0.13\n−0.13which is consistent with the local\nrelation and the recent estimate by Li et al. (2021). We\nshow that this value σµat high-z may not be in con-\ntradiction to a cosmic averaging scenario (Figure 16) as\nrecently put forward by Li et al. (2021) and Ding et al.\n(2022b) where AGN feedback is invoked to explain the\nconstant level of dispersion with redshift. However, due\nto the small sample size, high redshift, and wide redshift\nrange, our constraints on the redshift evolution of σµare24\nweak. Thus, a larger sample size at z∼1−3 is needed,\nespecially at high- z.\nFuture large-scale surveys such as Euclid and Roman\nwill significantly augment the sample size, along with\ndeeper observations by JWST, to provide stronger con-\nstraints on SMBH and galaxy evolution. For future\nlarge imaging data sets, visual inspection for all multi-\ncomponent fits and manually exclusion of anomalous\nresults will not be feasible; thus, improvements in 2D\ndecomposition techniques or the imposition of more so-\nphisticated conditions to confirm the robustness of host\ndetection is needed. In order to address these challenges,\nit is imperative to enhance the 2D decomposition analy-\nsis by mitigating the uncertainties related to PSF recon-\nstruction. This involves conducting a meticulous analy-\nsis of AGN PSFs, determining the validity of applying\nstellar PSFs to AGN with different SEDs than stars,\nidentifying the most effective methods for accurate PSF\nreconstruction, and establishing a framework for evalu-\nating the uncertainties in reconstructed PSFs.This work is based on observations made with the\nNASA/ESA/CSA James Webb Space Telescope. The\ndata were obtained from the Mikulski Archive for Space\nTelescopes at the Space Telescope Science Institute,\nwhich is operated by the Association of Universities\nfor Research in Astronomy, Inc., under NASA contract\nNAS 5-03127 for JWST. These observations are associ-\nated with program IDs 1727 and 1837, and the specific\nobservations analyzed can be accessed via DOI. Nu-\nmerical computations were in part carried out on the\niDark cluster, Kavli IPMU. This work was made possi-\nble by utilizing the CANDIDE cluster at the Institut\nd’Astrophysique de Paris, which was funded through\ngrants from the PNCG, CNES, DIM-ACAV, and the\nCosmic Dawn Center and maintained by Stephane\nRouberol. We thank Marko Shuntov, Kei Ito, and\nMingyang Zhuang for giving us useful advice for re-\nconstructing the PSFs. We thank Mingyang Zhuang,\nYue Shen, and Junyao Li for giving us their measure-\nments for comparison. Kavli IPMU is supported by\nWorld Premier International Research Center Initiative\n(WPI), MEXT, Japan. The Cosmic Dawn Center is\nfunded by the Danish National Research Foundation\nunder grant DNRF140. TT is supported by Forefront\nPhysics and Mathematics Program to Drive Transfor-\nmation (FoPM), a World-leading Innovative Graduate\nStudy (WINGS) Program, the University of Tokyo. JS\nis supported by JSPS KAKENHI (JP22H01262) and\nthe World Premier International Research Center Ini-\ntiative (WPI), MEXT, Japan. This work was sup-\nported by JSPS Core-to-Core Program (grant number:\nJPJSCCA20210003). BT acknowledges support from\nthe European Research Council (ERC) under the Eu-\nropean Union’s Horizon 2020 research and innovation\nprogram (grant agreement number 950533) and from the\nIsrael Science Foundation (grant number 1849/19). JR\nis supported by JPL, which is run by Caltech for NASA.\nGEM and SG acknowledge the Villum Fonden research\ngrant 13160 “Gas to stars, stars to dust: tracing star for-\nmation across cosmic time,” grant 37440, “The Hidden\nCosmos,” and the Cosmic Dawn Center of Excellence\nfunded by the Danish National Research Foundation un-\nder the grant No. 140.1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\nFacilities: JWST (NIRCam)\nSoftware: astropy (Robitaille et al. 2013; Price-\nWhelan et al. 2018; Collaboration et al. 2022), CIGALE\n(Boquien et al. 2019; Yang et al. 2022), FSPS (Conroy\net al. 2009; Conroy & Gunn 2010), galight (Ding et al.\n2020), lenstronomy (Birrer & Amara 2018; Birrer et al.25\nTable 4. The number of the undetected host galaxies for each condition, PSF and filter\n(1) (2) (3)\nFilter PSF BIC S/N manual Total\nF814WTop-5 14 22 17 39\nTop-75% 14 21 17 38\nPSFEx 21 24 17 41\nF115WTop-5 2 11 1 12\nTop-75% 1 10 1 11\nPSFEx 7 10 1 12\nF150WTop-5 1 2 1 3\nTop-75% 2 3 1 5\nPSFEx 2 3 1 4\nF277WTop-5 0 0 1 1\nTop-75% 0 0 1 1\nPSFEx 0 1 1 2\nF444WTop-5 0 2 2 4\nTop-75% 0 2 2 4\nPSFEx 0 1 2 3\nTable 5. The number of host-galaxy detections for each final PSF\n# Detected filters\nPSFCOSMOS-Web PRIMER\n0 1 2 3 4 5 9\nTop-5 1 0 5 9 27 16 3\nTop-75% 1 2 4 6 28 17 3\nPSFEx 2 0 4 8 30 14 3\n2021), matplotlib (Hunter 2007), numpy (Harris et al.\n2020), photutils (Bradley 2023), psfr (Birrer et al. inprep), Prospector (Leja et al. 2017; Johnson et al. 2021),\njwst (Bushouse et al. 2023)\nAPPENDIX\nA.DETECTED NUMBERS IN EACH FILTERS\nTable 4 summarizes the number of undetected host galaxies based on the conditions described in Section 3.3 for each\nPSF and filter, while Table 5 then summarizes the number of detected hosts for each final PSF. Regarding the JWST\nfilters, we have less galaxies that are classified as non-detection due to the BIC condition than due to the S/N host\ncondition. The number of undetected hosts is lowest in F277W, followed by F150W and F444W. On the other hand,\nF115W and F814W show a larger number of undetected hosts, especially due to S/N host. This trend is because F814W\nis the HST observation with lower resolution and shallower depth than the JWST observations, and both F814W and\nF115W are in the shorter-wavelength side of the Balmer break, leading to a tendency for a smaller intrinsic H/T.\nB.DETAILED COMPARISON OF DIFFERENT PSF RESULTS\nAs mentioned in Section 3.1, 2D decomposition is significantly influenced by the differences in PSFs. In this study,\nwe compare three different PSFs and discuss the PSF dependency of the results. Figure 17 (images on the left; a–c)\nshows each final PSF image for the four NIRCam filters. We fit each final PSF and each single PSF in the PSF\nlibrary with the 2D Gaussian model, and measure the FWHM along a semi-major axis and ellipticity b/a(defined as\nFWHM minor/FWHM major). Figures 17 (d) and (e) show the distribution of FWHM and b/ain each filter for an\nexample AGN, CID-62, with zspec∼1.92. As the number of stars used increases in the order of the top-5 stacked, the26\ntop-75% stacked, and PSFEx , we can see that the background noise is correspondingly lower. Regarding the FWHM\ndistribution, the top-5 and the top-75% PSFs are consistent with the FWHM distributions of single PSFs within\nthe PSF library. PSFEx have consistent FWHMs in F115W and F150W, and slightly smaller FWHMs in F277W\nand F444W, suggesting the possibility of bias between automatic PSF selection by PSFEx and semi-automatic PSF\nselection by galight . In addition, each final PSF tends to have a higher b/athan individual single PSFs. Regardless\nof whether χ2\nν-based stacking or empirical modeling is employed, considering that the final PSF is a more reasonable\nPSF reconstruction than a single PSF, this suggests a potential bias towards a more elliptical PSF when using a single\nPSF. Next, Figure 18 compares the estimated H/T using each PSF and filter (the full-filter version of Figure 8 (c)).\nWe can find strong positive correlations for all filters, with the distribution along the y=xrelation. However, when\ncomparing the distribution with y=x, it is evident that PSFEx tends to estimate larger H/T for F277W and F444W\ncompared to top-5 and top-75% PSFs. As mentioned in Section 4.2, this trend can be interpreted by considering the\ndifferences in FWHM for each filter and PSF. As shown in Figure 4, PSFEx tends to exhibit smaller PSFs in F277W\nand F444W than the top-5 and top-75% PSFs. Zhuang & Shen (2023) suggest that different PSFs result in larger H/T\nin the 2D decomposition analysis. Therefore, it is conceivable that PSFEx shows larger H/T in F277W and F444W\nwith the smaller PSFs than the top-5 and top-75% PSFs.\n05(d) FWHM (semi-major axis)\nTop-5\nTop 75%\nPSFEx\n05(e) Ellipticity\nF115W1\"59.5 mas(a) Top-5\n56.4 mas(b) Top-75%\n57.4 mas(c) PSFEx\n024\n0.02.55.0\nF150W64.3 mas\n 62.5 mas\n 58.7 mas\n0.02.55.0\n0.02.55.0\nF277W128 mas\n 124 mas\n 120 mas\n50 100 150\nFWHM/mas0510\n0.8 0.9 1.0\nb/a0510\nF444W163 mas\n 162 mas\n 158 mas\nFigure 17. Comparison of different methods to evaluate empirical models of the PSF. Columns (a) to (c) show images of the\ntop-5 stacked PSF, the top-75% stacked PSF, and the PSFEx -PSF for an example galaxy, CID-62 ( z= 1.92), in F115W, F150W,\nF277W, and F444W, from top to bottom. FWHM along the semi-major axis measured from the 2D Gaussian fits shown in the\ntop-left corner of each image. The white bars in the lower left indicate a scale of 1′′. Columns (d) and (e) show the distribution\nof the FWHM (semi-major axis) and ellipticity b/a. Gray histograms display the distribution of each single PSF in the library\n(Section 3.1.1). The values for the top-5 stacked PSF, the top-75% stacked PSF, PSFEx -PSF are marked by the blue, green, and\nred vertical lines, respectively.27\n0.0 0.5 1.0\nH/Tst750.00.51.0H/Tst5\nρ=0.97 (p/lessmuch0.05)(a) F115W\n0.0 0.5 1.0\nH/Tst750.00.51.0H/Tst5\nρ=0.99 (p/lessmuch0.05)(b) F150W\n0.0 0.5 1.0\nH/Tst750.00.51.0H/Tst5\nρ=0.92 (p/lessmuch0.05)(c) F277W\n0.0 0.5 1.0\nH/Tst750.00.51.0H/Tst5\nρ=0.91 (p/lessmuch0.05)(d) F444W\n0.0 0.5 1.0\nH/Tpsfex0.00.51.0H/Tst75\nρ=0.98 (p/lessmuch0.05)\n0.0 0.5 1.0\nH/Tpsfex0.00.51.0H/Tst75\nρ=0.99 (p/lessmuch0.05)\n0.0 0.5 1.0\nH/Tpsfex0.00.51.0H/Tst75\nρ=0.96 (p/lessmuch0.05)\n0.0 0.5 1.0\nH/Tpsfex0.00.51.0H/Tst75\nρ=0.92 (p/lessmuch0.05)\n0.0 0.5 1.0\nH/Tst50.00.51.0H/Tpsfex\nρ=0.96 (p/lessmuch0.05)\n0.0 0.5 1.0\nH/Tst50.00.51.0H/Tpsfex\nρ=0.98 (p/lessmuch0.05)\n0.0 0.5 1.0\nH/Tst50.00.51.0H/Tpsfex\nρ=0.90 (p/lessmuch0.05)\n0.0 0.5 1.0\nH/Tst50.00.51.0H/Tpsfex\nρ=0.87 (p/lessmuch0.05)\nFigure 18. Comparison of the estimated H/T in each filter using different PSFs (Columns (a) to (d)). The gray and red\ncolors correspond to whether they were detected in each filter. Spearman’s correlation coefficient ρfor each prior is shown in\nthe lower right corner of each panel. High correlation coefficients and the distribution around y=x(black dashed line) suggest\nconsistent results among the fitting with different PSFs.\nC.GALIGHT TEST WITH MOCK IMAGE\nWe generate mock images of AGN and their host galaxies and perform 2D decomposition on these images to\nverify whether galight can accurately recover each parameter for the host galaxies. We use lenstronomy (Birrer &\nAmara 2018; Birrer et al. 2021) to generate 2,000 NIRCam/F150W mock galaxy images. These are constructed as a\nsuperposition of a PSF and a smooth analytic S´ ersic model based on randomly sampled parameters: host-to-total flux\nratio ( H/T), S´ ersic index ( n), effective radius ( re), semi-major axis orientation ( θ), and ellipticity ( b/a). The software\ntools lenstronomy andgalight have different definitions of re, with the relation re,galight = (b/a)−0.5×re,lenstronomy .\nIn this section and consistent with the main part of this paper, all reare unified under the definition in galight . The\nspecified range for each parameter is log( H/T) in [∼ −5,0],nin [1,6],rein [∼0.′′05,1.′′6],b/ain [0.2,1.0], and θin\n[0, π] (in radian). We assume the PSFEx PSF (Section 3.1.2) and that the center of host galaxy components are aligned\nwith the center of PSF components. Additionally, we add Gaussian noise equivalent to the real NIRCam/F150W\nCOSMOS-Web images. Using the generated mock images, we performed 2D decomposition with the low- χ2\nνtop-5\nPSF reconstruction strategy (Section 3.1.1). Since the PSFEx PSF is assumed when generating the mock images, the\nresults allow us to assess the performance in cases where the PSF may differ from the true PSF, similar to real data\nanalysis. Note that we cannot conclude which PSF reconstruction method is superior from this analysis. In this\ntest, we also cannot consider the effects of more complex morphological features than single S´ ersic components, such\nas double S´ ersic (disc+bulge) components, position offset between AGN and host, substructures (bars, spirals, and28\nclumps), and nearby or merging objects. Figure 19 compares the estimated major parameters ( H/T,re,n) with the\ninput values. First, focusing on H/T, we can find the distinct change in the trend around the input log ( H/T) value\nlog (H/T ans)∼ −2. For log ( H/T ans)≲−2, the estimated value log ( H/T est) remains relatively constant around −1.5,\ndeviating significantly from the y=xline. This implies that detecting weak galaxy components with log ( H/T ans)≲−2\nis practically challenging using the 2D decomposition analysis with galight . On the other hand, for log ( H/T ans)≳−2,\nlog (H/T est) exhibits a consistent trend along the y=xrelation, while slightly larger than the input values. This\ntrend is consistent with Zhuang & Shen (2023), who reported that different PSFs tend to overestimate H/T. As\nshown in Figure 7, log ( H/T est) for the real detected galaxies in this study are log ( H/T est)≳−1. If we limit the mock\nresults to log ( H/T est)≳−1, the difference between the estimation and input values (defined as estimation - input\nvalues) is ∆ log ( H/T)∼0.06±0.09 dex, suggesting sufficiently consistent estimation of H/T. Next, examining re\nandn, we find a consistent distribution along the y=xrelation, with an overall tendency to underestimate reand\noverestimate n. We can also see that some objects hit the relower limit ( re∼0.′′05) and nupper limit ( n∼9). If we\nlimit the analysis to the above realistic cases (log ( H/T est)≳−1), as shown in Figure 19, reshows consistent results\nwith ∆ r=−0.1±0.2 and less hitting to the relower limit. However, for n, many objects still exhibit the nupper\nlimit even with log ( H/T est)≳−1. If we additionally cut the upper limit object with n <8.8, the difference between\nthe estimation and the input values is ∆ n∼1.3±1.6, suggesting a consistent estimation with medium scatter. We\nconfirm that larger- nobjects tend to have larger ∆ n, indicating the increasing difficulty of accurately reconstructing\nnwhen they have larger n.\n-4 -3 -2 -1 0\nlog(H/Tans)-2.0-1.5-1.0-0.50.0log(H/Test)logH/Test>−1\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4\nre,ans0.00.40.81.21.62.0re,est\n0 1 2 3 4 5 6\nnans012345678nest\nFigure 19. Comparison of input (X-axis) and estimated (Y-axis) morphological parameters of mock galaxies. Columns (a)\nto (c) compare log ( H/T),re, and n, respectively, and the black dashed line indicate y=x. The gray and cyan circles indicate\nthe all mock galaxies and the galaxies with log ( H/T est)>−1 corresponding to the minimum H/T estfor our real galaxies (also\nsee Figures 8, 18). Only mock galaxies with log ( H/T est)>−1 are used in plotting box plots for nandre.\nREFERENCES\nBaker, W. M., Tacchella, S., Johnson, B. D., et al. 2023.\nhttp://arxiv.org/abs/2306.02472\nBeifiori, A., Courteau, S., Corsini, E. 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In\neach step of the search, an algorithm is querying a vertex q, and as a reply, it receives an answer,\nwhich either states that qis the desired target, or it gives the neighbor of qthat is closer to the\ntarget than q. A further generalization assumes that a vertex-weight function ωgives the query\ncosts, i.e., the cost of querying qisω(q). The goal is to find an adaptive search strategy requiring\nthe minimum cost in the worst case. This problem is NP-complete for general weight functions\nand one of the challenging open questions is whether there exists a polynomial-time constant\nfactor approximation algorithm for an arbitrary tree? In this work, we prove that there exist\na constant-factor approximation algorithm for trees with a monotonic cost function, i.e., when\nthe tree has a vertex vsuch that the weights of the subsequent vertices on the path from vto\nany leaf give a monotonic (non-increasing or non-decreasing) sequence S. This gives a constant\nfactor approximation algorithm for trees with cost functions such that each such sequence Shas\na fixed number of monotonic segments. Finally, we combine several earlier results to show that\nthe problem is NP-complete when the number of monotonic segments in Sis at least 4.\nKeywords: binary search, graph search, approximation algorithm, query complexity\n1 Introduction\nSearching plays a central role in computer science due to its ubiquitous real-world applications\nas well as its inherent relation to other problems. Many fundamental computational problems,\ne.g. sorting, can be formulated as searching for an element in an appropriately defined set or apply\nsearching as a subroutine. See for example the classic book [29] or a recent work on relation between\nsorting and binary search trees [5]. The search problem, both in the real world as well as in abstract\nterms, typically consists of a search space (e.g., a set, a house, a graph), a search target (vertex, or\ncar keys) and a searcher that follows some search strategy. In the engineering of data management\nsystems, searching information efficiently is essential for achieving high performance. Problem-\nsolving by searching is a well established approach in artificial intelligence [40]. In operations\nresearch, algorithmic techniques applied to searching significantly improved efficiency of searching\nfor submarines [25], to give one example. On the other hand, the works on the naval applications\ncontributed to formulation of numerous theoretical models of search games, e.g., the binary search\ngame.\n∗Some results of this work have been presented at the 47th International Symposium on Mathematical Foundations\nof Computer Science (MFCS 2022) [20].\n†Partially supported by National Science Centre (Poland) grant number 2018/31/B/ST6/00820.\n1arXiv:2401.13747v1  [cs.DS]  24 Jan 2024An important aspect of any search models is the amount of information available to the search\nagent about the potential location of the search target [40]. The uninformed search model is such\nthat no additional information is available beyond the definition of the search space. The depth-\nfirst- or breadth-first-search and related models are examples of uninformed search algorithms. On\nthe other hand, the informed search models assume that some additional information is available\nto the search agent. The additional information can be exploited to construct significantly faster\nsearch strategies. In practical setting, the additional information comes from exploiting the context\nof the search problem, e.g., an assumption that alike elements according to some similarity measure\nareclose to each other, another practical example is an assumption that elements of the search\nspace are sorted. In a theoretical setting, the additional information can be modeled as an heuristic\nfunction or an oracle.\nIn the category of the informed search models we can further characterize the search models in\nterms of when the additional information is exploited. I.e., we can first calculate a search strategy,\ne.g., in a form of a decision tree, based on the definition of the search space, which later would\nbe used to locate arbitrary elements of the search space. Using the terms of supervised machine\nlearning, we first learn the search model, which is then being inferred when searching for particular\nsearch targets. Consequently, the optimization criteria may be the time to learn the strategy, the\naccuracy and the time of locating particular search targets. In a contrary online approach, the\nsearch strategy is being refined at the same time as locating particular search targets.\n1.1 Problem statement\nWe consider the following search problem defined for a node-weighted tree T= (V, E, ω ), with a\nquery cost function ω:V(T)→R+. The goal is to design an adaptive algorithm that we call a\nsearch strategy or just strategy for short. The strategy is divided into steps and in each step it\nperforms a query by selecting some vertex qofT. As an answer it receives information that either\nqis the target or otherwise it learns which neighbor of qis closer to the target.1In the former case,\nthe search is completed. The costof the query is ω(q).\nTo formally define the search strategy, consider a node weighted tree T= (V, E, ω ) and a search\ntarget t∈V(T). If|V(T)|= 1, then search strategy A(T, t) is the element titself. Otherwise,\nletq∈V(T) be the vertex that the strategy queries first when searching in T, letCibe the i-th\nconnected component of T\\ {q}. Then, for |V(T)|>1,A(T, t) is a tuple ( q,A(C1, t), ...,A(Ck, t)),\nwhere k=|N(q)|is the number of neighbors of q. We note that this kind of search definition can\nbe understood as a form of a decision tree.\nWe denote by COST (A, t) the sum of costs of all queries performed by a strategy A, when\nfinding t. The costofAis defined as COST (A) = max t∈VCOST (A, t). The optimization criterion\nasks for a strategy of minimum cost.\nOne might think about this search process that with each query the search is narrowed to a\nsubtree of Tthat can still contain the target. The process is adaptive in the sense that the choice\nof the subsequent elements to be queried depends on the locations and answers of the previous\nqueries. Also, we consider only deterministic algorithms, where given a fixed sequence of queries\nperformed so far (and the information obtained), the algorithm selects always the same element as\nthe next query for a given tree. E.g., the optimal algorithm for the classic binary search problem on\npaths that always queries the median element is adaptive and deterministic. It can be equivalently\nstated that the algorithm calculates a strategy for the given input tree, which encodes the queries\nto be performed for each potential target.\n1Alternatively, one may think of the latter as learning which component of T−qcontains the target.\n25\n4\n1\n3\n2\n7\n6\n(a)\n1\n2\n4\n2\n1\n2\n4\n2\n7\n6\n(b)\n1\n2\n4\n2\n7\n(c)\n1\n4Figure 1: Binary search on weighted trees. The input tree (a) contains a target vertex (2), whose\nposition is unknown to the algorithm. Three queries are performed to the vertices 1, 6, and 7,\nwhich incurs the costs of 4, 2 and 1, respectively. The subtrees that contain all vertices that are\nstill candidates for the target after the first two queries are shown in (b) and (c).\nOur problem is a generalization of the classical binary search, where one considers trees instead\nof paths. While also the generalized problem can be found in the literature under the name of\nbinary search, it might be illustrative to note that answers to queries over a tree domain are not\nnecessarily binary ( leftorright ), as the set of possible answers depends on the degree of the queried\nvertex.\n1.2 Paper Organization\nThe remaining parts of the paper are organized as follows. Section 1.3 explains the motivation for\nour work. We describe practical applications of the studied search problem and several use cases for\nthe monotonic structure of the cost function. Section 1.4 provides a summary of the techniques and\napproaches used previously to tackle the problem of binary search in trees. Section 1.5 describes\nthe main results of this work, which is followed by Section 1.6 that surveys the related work.\nSubsequently, in Section 2 we introduce the notation used in the paper as well as several essential\nconcepts that are commonly used in the related body of research. In Sections 3 - 5, we develop\nseveral concepts and techniques (i.e., structured decision trees, aligned strategies and bottom-up\nprocessing). These techniques are later used in Sections 6 - 8 where we formulate and analyze\nalgorithms for searching in trees with different types of monotonic cost functions. Section 10\nconcludes the paper, where we propose final remarks and suggest potential directions for future\nresearch.\n1.3 Motivation and applications\nWe study a generalization of one of the most fundamental and well known problems in computer\nscience, i.e., binary search. Since graphs form a natural abstraction for processes in many domains,\nour generalization into the domain of graphs allows us to seek for potential applications of our algo-\nrithms and related problems for example in the domain of large networks analysis [41]. Compared\nto unordered data structures, it is known that maintaining at least a partial order over the elements\nimproves the performance of fundamental operations like search, update or insert, in terms of the\nnumber of comparisons needed [29]. In the context of data management systems, it is known that\nefficient indexing is crucial to obtain good performance. Some of the indexing schemes have the\n3form of a tree, for example the GIndex. At the same time, calculating optimal search strategies\nfor structures with a partial order is hard in general [18]. Practical approximation algorithms for\nsearching in large graphs are important for the advancement of systems focused on large graph\nanalysis.\nBinary search on trees seen as a graph ranking problem (cf. [13]) can be used to model parallel\nCholesky factorization of matrices [15], scheduling of parallel database queries [33], and VLSI\nlayouts [43]. A more general search model, i.e., binary search on graphs has been used as a\nframework for interactive learning of classifiers, clusterings or rankings [21, 22]. More applications\ncan be found in software testing [3] or mobile agent computing [6].\nThere are several practical applications of searching where the cost of a query is naturally\nhigher in some central vertex and decreases while moving towards the leaf nodes of the tree. This\noccurs naturally when considering data access times in computer systems, e.g., due to memory\nhierarchies of modern processors, characteristics of the storage devices, or distributed nature of\ndata management systems. In the literature on the binary search problem, a Hierarchical Memory\nModel of computation has been studied [1], in which the memory access time is monotonic with\nrespect to the location of a data element in the array. Another work was devoted to modeling the\nproblem of text retrieval from magnetic or optical disks, where a cost model was such that the cost of\na query was monotonic from the location of the previous query performed in the search process [36].\nWe also mention an application of tree domains in automated bug search in computer code [3]. In\nsuch case naturally occurs a possibility that the tree to be searched has the monotonicity property.\nIn particular, each query represents performing an automatic test that determines whether the\npart of the code that corresponds to the subtree under the tested vertex has an error. Thus, the\nvertices that are closer to the root represent larger parts of the code and thus may require more or\nlonger lasting tests. The scale of exa-scale software applications that may consist of several tens of\nseparate software packages motivates the use of advanced software validation strategies, in order\nto quickly identify software regressions, e.g., when updating versions of several different packages.\nAn open question has been posed several times about whether a constant-factor approximation\nalgorithm exists for arbitrary weight functions, see e.g. [2, 8, 9]. The main motivation of this\nwork is to address this question by finding natural input instances for which a constant-factor\napproximation is achievable. In particular, we give an algorithm for down-monotonic cost functions\nand motivated by the theory of parameterized complexity, study a parameter kof the input instance:\nthe maximum number of monotonic segments on any path from a central node to any leaf. The\ncase of kequal to the tree height captures the class of general cost functions. We see this as a\nstep towards solving the general problem. Our techniques lead to a constant approximation for\nany fixed k. We believe that this method can be improved to allow combining segments of different\nmonotonicity, while preserving the constant factor approximation.\n1.4 Earlier techniques\nHistorically, the transition from paths to trees had an intermediate step where one wants to search\npartial orders instead of linear ones [32, 35]. Performing comparisons in partial orders leads to a\nsimilarly defined graph problem except that one queries edges instead of vertices. However, each\ntree instance for querying edges can be solved via finding an optimal search strategy for a tree\ninstance with querying vertices [14]. For these reasons, we consider the more general vertex-query\nversion on trees.\nIt turns out that our problem reappeared several times in theoretical computer science under\ndifferent names. It has been called LIFO-search [24] in the area of graph pursuit-evasion games,\nordered coloring [28], minimum height elimination trees [7] in the context of parallel matrix factor-\n4ization, then vertex ranking [26], and more recently as tree-depth [37].\nWhile listing the main approaches to construct search strategies, we do not differentiate between\nthe edge and vertex versions of the problem as it turns out that the algorithmic techniques we\nmention are applicable to both. The majority of works, especially for the unweighted version of\nthe problem utilize the bottom-up tree processing. In this approach one usually assigns greedily\nthe lowest possible interval and argues that this leads to an optimal solution, see e.g. [3, 27, 32,\n35, 38]. We note here that by ‘interval’ we mean the following interpretation of the cost function:\nif costs are seen as query durations, then the query interval is the corresponding time period in\nwhich the query is processed. The particular contributions of these works mainly lie in various\noptimizations that bring down the computational complexity of finding optimal strategies. In case\nof non-uniform query times the problem becomes more challenging and keeping in mind that it is\nNP-complete [12] a series of works focused on finding approximate solutions. (We note that the\nunweighted case has been long known to be linear-time solvable [42]). A natural approach, for the\ncase of non-uniform costs, is to query a ‘central’ element of the search space and then computing\nthe strategy recursively for the remaining subtrees [12, 23], which for trees gives O(logn)-factor\napproximations. More involved recursive approaches allowed to break the barrier of O(logn) in\n[9, 10] giving a O(logn/log log n)-approximation. A rather involved bottom-up approach gave\nthe best approximation ratio of O(√logn) so far [14]. The main open question in this line of\nresearch is whether either the edge or vertex query model admits a polynomial-time constant\nfactor approximation for weighted trees. A possible angle at tackling this question is to consider\neither special subclasses of trees [9] or restricted weight functions [20].\n1.5 Our results\nGiven a tree T= (V, E, ω ), we say that a cost function ωisup-monotonic (respectively down-\nmonotonic ) if there exists a vertex r∈Vsuch that for any u, v∈V, ifvlies on the path between\nranduinT, then ω(u)≤ω(v) (ω(v)≤ω(u), respectively). We say that ωismonotonic if it is\neither up-monotonic or down-monotonic. For this kind of ωwe get the following algorithms.\nTheorem 1.1. There exists a linear-time 8-approximation adaptive search algorithm for an arbi-\ntrary tree with up-monotonic cost function.\nTheorem 1.2. There exists a polynomial-time 2-approximation adaptive search algorithm for an\narbitrary tree with down-monotonic cost function.\nWe say that ωisk-monotonic ,k≥1, if Tcan be partitioned into vertex disjoint subtrees\nT1, . . . , T lsuch that each of them is either up-monotonic or down-monotonic, and for some r∈V,\nany path from rto a leaf in Tintersects at most ktrees Ti,i∈ {1, . . . , l}. By combining Theorem 1.1\nand Theorem 1.2, we obtain:\nTheorem 1.3. There exists a polynomial-time 8k-approximation adaptive search algorithm for an\narbitrary tree with a k-monotonic cost function.\nThis implies a constant factor approximation algorithm for instances with fixed kand raises a\nquestion whether the problem is FPT with respect to k? We give a negative answer to this question\nby proving:\nTheorem 1.4. The problem of finding a vertex search strategy for bounded diameter spiders with\nk-monotonic cost functions is strongly NP-complete for any fixed k≥4.\nAspider is a tree with at most one vertex of degree greater than two.\n51.6 Related work\nOne can distinguish three different research directions related to our problem. We list main results\nseparately for each of those as follows.\nVertex search in trees. It is known that an optimal vertex search strategy can be computed\nin linear time for an unweighted tree and this result has been obtained independently in [42]\nand in [38]. On the other hand, the weighted version becomes (strongly) NP-complete and FPT\nwith respect to the maximum cost2[18]. For weighted paths there exists a dynamic programming\nO(n2)-time optimal algorithm [9] (see also [30]), which although covers graphs of special structure,\nis interesting as it may be viewed as the classical binary search with non-uniform comparison costs.\nFor general trees there is a polynomial-time O(√logn)-approximation algorithm [14]. The latter\nis derived from a QPTAS that for any 0 < ε < 1 gives a (1 + ε)-approximation in time nO(logn/ε2)\n[14]. We also remark on a closely related problem of finding, for an arbitrary input graph, its\nspanning tree that has the minimum search cost [34]. Another generalization that appeared in the\ncontext of parallel scheduling problems, expressed in our terminology, is to find a search strategy\nin which one is allowed to perform several queries simultaneously [45]. This line of research can\nbe placed in a wider context of analyzing adaptive algorithms versus some coding schemes and we\nrefer interested reader e.g. to a much broader survey in [39]. Another related model of searching in\ntrees is where the weight of each vertex indicates the likelihood of the vertex being a search target.\nI.e., this search model can be considered as a generalization of finding a Binary Search Tree for a\npath to the domain of trees [4, 31].\nEdge search in trees. The edge search problem on unweighted trees is essentially different\nthan the vertex search. There has been a series of papers providing polynomial-time algorithms\nwhich eventually lead, again independently under the names of edge ranking and searching in\npartial orders, to linear-time optimal algorithms [32, 35]. Moreover, although this gives the same\ncomplexity as for vertex search, the algorithms themselves were much more involved and thus the\nedge search version is more challenging to analyze for unweighted trees. Despite the fact that the\ncomplexity is resolved, upper bounds on the number of queries have been also developed [16, 23, 33].\nWe briefly refer to similar generalizations including spanning tree computations [33] or the ‘batch’\nquerying [44]. Interestingly, there is a simple edge-subdividing reduction making the weighted\nvertex search more general than the weighted edge search [14]. However, there is no such transition\nfrom non-weighted edge search to non-weighted vertex search or vice versa. The weighted edge\nsearch is NP-complete for several already restricted classes of trees [10, 12].\nVertex search in general graphs. We note that our vertex search problem has a natural\nreformulation for graphs: a query to a vertex q, whenever qis not the target gives anyneighbor\nofqthat is closer to the target than q. This version has been introduced in [23]. It turns out\nthat log2nqueries are enough to perform a search in any graph which gives a quasi-polynomial\nalgorithm [23]. The same work proves that the weighted version is PSPACE-complete. See also for\nfurther references e.g. [11, 17, 19, 21].\n2 Definitions and Preliminaries\nThe set of vertices and edges of a tree Tis denoted by V(T) and E(T), respectively. For a connected\nsubset of vertices V′⊆V(T) ofT, we denote by T[V′] the induced subtree of Tconsisting of all\nvertices from V′. For a rooted T, we denote its root by r(T), while for any v∈V(T),Tvis the\n2With an assumption of integer-valued cost functions.\n6subtree of Trooted at vwhich consists of vand all its descendants. N(v) denotes the set of\nneighbors of a vertex v.\nFor any two intervals I= [a, b) and I′= [a′, b′) we write I > I′(I < I′) when a≥b′(b≤a′).\nThus, whenever I > I′orI < I′, we have I∩I′=∅. For an interval I= [a, b) and a number c, we\nwrite I < c (I > c ), when b≤c(a≥c, respectively). Furthermore, we define the sum I+cas the\ninterval [ a+c, b+c).\nWe say that an interval I= [a, b) isminimal (maximal ) in a given set of intervals, if and only\nif for any other interval I′= [a′, b′) in the set, we have that a≤a′(respectively b≥b′). Note\nthat every finite set of intervals contains both a minimal and a maximal interval. In this work, all\nintervals we use are left-closed, right-opened, and with integer endpoints.\nWe extend the classic lexicographic order over sequences of integers, denoted by binary operator\n<lto sequences of intervals. For two interval sequences SandS′(each with pairwise disjoint\nintervals), we write S <lS′if and only if there exists an integer isuch that i+1/∈SS,i+1∈SS′\nand\n[0, i)∩[\nS= [0, i)∩[\nS′.\nWe then write S≤lS′if and only if S <lS′orS=S′.\nWe use the term sequence of queries to refer to the sequence of vertices queried by a search\nstrategy Afor a target tand we denote it by QA(T, t). The vertex queried by Ain the i-th step\nis denoted by QA,i(T, t). Given a search strategy for T, one can easily generate a search strategy\nfor any subtree T′by simply discarding the queries to the vertices outside of T′. A useful way of\nencoding search strategies is by using decision trees.\nDefinition 2.1. We define a decision tree forT= (V, E, w ) generated by a search strategy Aas a\nrooted tree T= (V,E), where V=V(T). The root of Tis the first vertex queried by A. For any\nv∈ V,vhas a child v′if and only if v′is queried by Aright after vfor some choice of the target.\nEach v∈ Vcorresponds to a subset T(v)⊂V(T) which contains all vertices that can still contain\nthe target when Aqueries v.\nIt can be seen that the root rof any decision tree corresponds to T(r) =V(T) and the leaves\ncorrespond to single vertices in V(T).\nLemma 2.2 shows a property of the decision trees with respect to the positions of vertices that\nbelong to some connected component of the input tree.\nLemma 2.2. LetTbe a decision tree for T. IfT′is a subtree of T, then there exists a vertex\nu∈V(T′), such that for every v∈V(T′)it holds v∈V(Tu).\nProof. We traverse the decision tree Tstarting at the root of Tand following a path to some leaf.\nIfr(T)∈V(T′), then the lemma follows. Otherwise, consider children viofr(T). Because in T\nthere is only one edge incident to r(T) that lies on a path to all vertices of T′, all vertices of T′\nbelong to exactly one subtree Tvi. We continue the traversal in Tviand set uas the first visited\nvertex in this process that belongs to T′.\nWe now formally define the cost of a search process, which is the parameter of Tthe algorithm\naims to minimize. In terms of the search strategy Afor an input tree T= (V, E, ω ), the sum of\ncosts of all queries performed when finding a target t∈V(T) is denoted by COST (A, t), while the\ncostofAis defined as COST (A) = max t∈VCOST (A, t).\nSometimes in our analysis, we will take one decision tree and measure its cost with different\nquery times, i.e., for different weights. This approach will allow us to artificially increase the cost\nof some queries in the decision tree, which we formalize as follows. Let Tbe a decision tree for\n7T= (V, E, ω ) and ω′an arbitrary cost function defined on V(T), potentially a different one than\nω. We denote by COST (T, ω′) the cost of the decision tree according to the cost function ω′:\nCOST (T, ω′) = max\bX\nv∈V(P)ω′(v)|Pis a path from the root to a leaf in T\t\n.\nThecostofTis then COST (T, ω) and we shorten it to COST (T). (Note that one may equivalently\ndefine the cost of Tby taking COST (T) =COST (A) where Tis generated by A.) The minimum\ncost that is achievable for Tis denoted by\nOPT (T) = min {COST (T)| Tis a decision tree for T}.\nWe say that Tis optimal if and only if COST (T) =OPT (T).\nWe will use the following simple folklore lemma whose proof is given for completeness. Lemma 2.3\nhas been previously formulated in [9] in the context of a slightly different problem of edge search\nin weighted trees.\nLemma 2.3. IfTis a decision tree for T, then for any subtree T′ofTthere exists a decision tree\nT′, such that COST (T′)≤COST (T).\nProof. We first apply Lemma 2.2 and reduce Tto its subtree rooted at u∈V(T′). Then, we\nsequentially remove from Tuall nodes that do not belong to T′. For every removed node v,\nconsider all subtrees rooted at the children of vin the current decision tree. When removing v\nwe also remove all subtrees that do not contain any vertex from T′but we connect the roots of\nall other subtrees (note that there will be only one such subtree) directly under the parent of v.\nHence, the reduced tree is still a decision tree. What is more, all paths from the root to a leaf in\nthe obtained decision tree result from corresponding paths from Tby removing zero or more nodes,\nwhich upperbounds the cost of T′by the cost of T.\nWe say that a cost function ωisrounded if the values taken by ωare powers of two (for\nanyv∈V(T),ω(v) = 2kfor some k∈N). We also say that T′is the rounding ofTif it is\nobtained from Tby rounding the cost function to the closest, greater power of two. We obtain\nthat OPT (T′)≤2OPT (T) because ω′≤2ω. On the other hand, Lemma 2.4 shows how large is\nthe overhead from applying an optimal decision tree for the input with a rounded cost function to\nsearch through the input tree with the original weights.\nLemma 2.4. LetT′= (V, E, ω′)be the rounding of T= (V, E, ω ). Let TandT′be optimal\ndecision trees respectively for TandT′. We have:\nCOST (T′, ω)≤2COST (T) = 2 OPT (T).\nProof. We have COST (T′, ω)≤COST (T′, ω′)≤COST (T, ω′)≤2COST (T, ω). The first in-\nequality holds because ω≤ω′. The second inequality holds because T′is optimal for T′. The last\ninequality holds because ω′≤2ω.\n2.1 Strategy functions\nIn case of uniform query times, an optimal search strategy can be obtained by calculating a strategy\nfunction f:V(T)→N, where for each pair of distinct vertices v1, v2∈V(T), iff(v1) =f(v2), then\neach path connecting v1andv2has a vertex v3such that f(v3)> f(v1). As outlined earlier, methods\nusing functions analogous to fcan be found in the literature on problems such as elimination trees,\n8vertex ranking [32, 42], tree-depth [37], and others. One property of the strategy function is that\nit encodes the (reversed) order of the queries. E.g., in any sequence of queries, a vertex with a\nhigher value of fis always queried before those with lower values of the strategy function. Thus,\nthe maximal value of fover the vertices of a tree is the worst-case number of queries necessary to\nsearch the tree. In order to express the variable costs of queries we extend the strategy function\ninto an extended strategy function (see also [12]).\nDefinition 2.5. A function f:V(T)→ {[a, b): 0≤a < b}, where |f(v)| ≥ω(v) for each v∈V(T),\nis an extended strategy function if for each pair of distinct vertices v1, v2∈V(T), iff(v1)∩f(v2)̸=∅,\nthen the path connecting v1andv2has a v3such that f(v3)> f(v1)∪f(v2).\nWe say that an fisoptimal forT= (V, E, ω ) if and only if supS\nv∈Vf(v) is minimal among all\nextended strategy functions for T.3Based on the time interpretation of strategy functions, optimal\nfunctions correspond to optimal strategies.\nObservation 2.6. For any T= (V, E, ω ), there exists an optimal extended strategy function f,\nsuch that |f(v)|=ω(v), for every v∈V(T).\nProof. Letf′be an optimal extended strategy function for T= (V, E, ω ), while v∈V(T), an\narbitrary vertex such that |f′(v)|> ω(v). Taking f′(v) = [ av, bv) and δv=|f′(v)| −ω(v), we\nmodify f′intofby setting f(v) = [av+δv, bv). Because supS\nv∈Vf(v) = supS\nv∈Vf′(v),fis an\noptimal extended strategy function for T.\nGiven an extended strategy function ffor a node weighted tree T= (V, E, ω ), one easily obtains\na search strategy for the tree by first locating the vertex qwith the maximal interval assigned by\nf. In a trivial case, where |V(T)|= 1, this completes the construction of A. Otherwise, let us\nfirst note that there is only one such vertex qinV(T) with a maximal interval. Assume that\nthere is a q′∈V(T),q̸=q′, such that b′=b, where f(q) = [a, b),f(q′) = [a′, b′). Then, because\nf(q)∩f(q′)̸=∅, by the definition of f, on the simple path between qandq′, a vertex pwould\nhave to exist such that f(p)> f(q)∪f(q′). Consequently, qcould not be maximal. By the same\nargument, this maximal interval has an empty intersection with intervals assigned to any other\nvertex. Let |N(q)|=kand let Tibe the i-th component of T\\ {q},i≤k. Then, for arbitrary\nsearch target t,A(T, t) is a tuple ( q,A(T1, t), ...,A(Tk, t)). Then, the construction of Acontinues\nrecursively by using frestricted to each component Ti.\nOn the other hand, given a search strategy A, we can obtain an extended strategy function f\nthat encodes the same sequences of queries as those made by A. Let qbe the vertex queried as\nfirst by A. We set f(q) = [0 , ω(q)) in the trivial case, i.e. when |V(T)|= 1. If |V(T)|>1, let\nA= (q,A(T1, t), ...,A(Tk, t)),N(q) =k. We fix the values of ffor all vertices different than qby\nrecursively repeating the process for the components Ti. Note that each v∈V(T) is referenced\nexactly once in A. This is the case, because the components Tihave disjoint vertex sets. After\nfixing fin the components, we set f(q) = [supS\nv̸=qf(v),supS\nv̸=qf(v) +ω(q)). We will now show\nthatfis an extended strategy function. Consider vertices x̸=yinV(T), such that f(x)∩f(y)̸=∅.\nConsider the sequences of queries SxandSy, leading respectively to querying xandy. Since both\nSxandSyare generated by the same strategy A, they have a common prefix of length at least one\nof identical vertices. Let vbe the last vertex in the prefix. Since x̸=y, at least one vertex out of\nxandywas set after vby our procedure before both xandy. In other words, because x̸=yonly\none of them might be v. Consequently we have that f(x)∪f(y)< f(v).\n3We note thatS\nv∈Vf(v) is an interval with the left endpoint equal to 0 for an optimal f.\n9This also shows that the interval assigned to a vertex vbyfmight encode the time interval\nfor querying v, where the ‘time’ is understood as dictated by the precedence relation we use for\nintervals.\nGiven an extended strategy function fforT, we say that a vertex uisvisible from a vertex v\nif and only if on the path that connects uandvthere is no vertex xsuch that f(u)< f(x). The\nidea of visibility has been used e.g. in [45] and also earlier under the name of ‘critical lists’ in [32].\nThe sequence of values (intervals) visible from vin descending order is called a visibility sequence\n(extended visibility sequence ) for vand is denoted by S(v). Whenever we need to point out that\nS(v) is obtained from a given fin the above way, we say that S(v) isderived from f.\nThe concept of visibility is also used when fis defined only on a subset of V(T), i.e., on all\ndescendants of some vertex v. In this context, which typically occurs during bottom-up processing\nofT, only those vertices are considered for which fhas been already defined.\nWe will decide on a strategy function f(v) for each vertex vby assigning an integer jvtov\nin a bottom-up processing of the input tree. Each jvwill determine f(v) differently for different\nvertices. However, the jvwill be computed via an appropriate use of the following operator.\nDefinition 2.7 ([38]).Letfbe a strategy function defined on all descendants of a v∈V(T). Let\nvi, 1≤i≤k, be the children of v. The vertex extension operator attributes vwith f(v), based on\nthe visibility sequences S(vi) in the following way. Let Mbe the set of integers that belong to at\nleast two distinct visibility sequences S(vi), while m= max( M∪ {− 1}). We set f(v) as the lowest\ninteger greater than mthat does not belong to any S(vi).\nThe following lemma characterizes the vertex extension operator.\nLemma 2.8 ([38]).Given the visibility sequences assigned to the children of v∈V(T), the vertex\nextension operator assigns to va visibility sequence that is lexicographically minimal (the vertex\nextension operator is minimizing). Moreover, (lexicographically) increasing the visibility sequence\nof a child of vdoes not decrease the visibility sequence calculated for v(the vertex extension operator\nis monotone).\n3 Structured decision trees\nIn Sections 3, 4 we consider rooted trees T= (V, E, ω ), in which while traversing any path from\nthe root to a leaf, the weights of the visited vertices are non-increasing. In particular, the root of\nTis selected properly according to the definition of an up-monotonic cost function. This section\ngives a foundation for the algorithm in Section 6.\nWe define a layer of a tree Tas a subgraph LofTsuch that all vertices in V(L) have the same\ncost, ω(u) =ω(v) for any u, v∈V(L). A connected component of Lis called a layer component .\nWe also denote by ω(L) the cost of querying a vertex that belongs to L, i.e., ω(L) =ω(v) for any\nv∈V(L).\nTheupper border of a layer Lis the set of roots of its layer components. The subset of V(L)\nconsisting of vertices that have at least one child that belongs to a different layer is called the lower\nborder ofL. We will also say that a layer component L′isdirectly below a layer component Lif\nand only if r(L′) has a parent in L. The toplayer component is the one that contains the root of\nTand a bottom layer component is any Lsuch that there is no component directly below L. We\nnote that we will apply the above terms to a rounded ω.\nDefinition 3.1. LetTbe a tree with a up-monotonic cost function and Ta decision tree for T.\nConsider a layer component LofT. Let v=r(L). We say that Tisstructured with respect to\n10Lif for every vertex u∈V(Tv), it holds that u∈V(Tv). We say that a decision tree TforTis\nstructured if and only if Tis structured with respect to all layer components of T.\nInformally speaking, in a structured Twe require that each vertex in the entire subtree Tvis\nbelow vinT, for each vthat is the root of a layer component. This is one of the central ideas\nused in the algorithm for up-monotonic cost functions for the following reason. While performing\nbottom-up processing of (an unweighted) Tin [38] (and also other works, e.g. [32]) each vertex\nvencodes which steps are already used for queries of the vertices from Tv. In our case, that is\nin a weighted T, we need to encode intervals and the technical problem with that is that they\nhave different durations while moving upwards from one layer component to the next. Thus, some\nintervals that are free to perform queries while moving to a parent of vmay be too short due to the\nweights in the next layer component. To deal with that, we make v=r(L) also the root of Tv. In\nthis way only one interval, i.e., the one assigned to vneeds to be taken into account while moving\nupwards.\nOur method requires the decision tree to be structured only when processing the internal com-\nponents of T. By not enforcing structuring with respect to the top layer component, a potential\nadditive cost of a single query can be avoided. (This will be reflected in line 6 of Algorithm Process-\nComponent). However, this optimization does not change the worst-case cost of the strategy. In\nresult, the decision tree generated by our algorithm is not structured in general. Although r(T), the\nfirst vertex queried by the strategy, always belongs to the top layer component of T, we may have\nthat r(T)̸=r(T). In contrast, in a structured decision tree we always have a stronger property,\nnamely r(T) =r(T).\nIt turns out in our analysis that considering only structured decision trees introduces an overall\nmultiplicative cost of 2 to the performance of the algorithm:\nLemma 3.2. LetT= (V, E, ω )be a tree with a up-monotonic and rounded cost function. There\nexists a structured decision tree T′forT, such that COST (T′)≤2OPT (T).\nProof. LetTbe any decision tree for T. Since Tis selected arbitrarily, it is enough to prove that\nCOST (T′)≤2COST (T).\nIn order to construct the structured decision tree, we traverse Tin a breadth-first fashion\nstarting at the root of T. The T′is constructed in the process.\nBy Lemma 2.2, for an arbitrary layer component LofT, there exists a vertex u′∈V(L) such\nthat all vertices from Lbelong to Tu′. Among all vertices of L,u′is visited first during the traversal.\nConsider an arbitrary node ubeing accessed during the traversal. Let Lbe the layer component\nsuch that u∈L. IfTis structured with respect to Lwe continue with the next node. Otherwise\n(which for a specific Lwill happen only once, when u=u′, as argued in the last paragraph of the\nproof), we modify Tby performing the following steps.\nLetv=r(L). We want to replace the subtree Tuwith a subtree consisting of the same nodes\nbut rooted at v. Consider a subtree T′of the input T, which can still contain the target when the\nsearch process is about to query uaccording to T. E.g., V(T′) =V(Tu). Let vhaskneighbors in\nT′. We attach under vthe decision trees created with Lemma 2.3 for the kcomponents obtained\nby removing vfrom T′. Lemma 2.3 shows that the cost of any of the kdecision trees is not greater\nthan COST (Tu). Accounting for the cost of v, structuring Lincreases COST (T) no more than\nω(L). Consider an arbitrary path Pfrom root to a leaf in T. For every node u∈Pthat triggers\nstructuring (informally, uisu′for some L) an additional node from Lis inserted in P. Thus, in\nthe worst case, the weighted length of arbitrary Pincreases 2 times when transforming Tinto\nstructured T′.\n11By starting the breadth-first traversal at the root of T, structuring with regards to one layer\ncomponent does not make Tnot structured with regards to any of the previously structured layer\ncomponents.\nBy combining Lemmas 2.4 and 3.2 one obtains Corollary 3.3, which relates the cost of an\noptimal decision tree for an input Twith the cost of a structured decision tree obtained through\nthe analysis of the rounding of T.\nCorollary 3.3. For any tree Twith a up-monotonic cost function there exists a structured decision\ntree with rounded weights, whose cost is at most 4·OPT (T).\n4 Bottom-up tree processing\nWe extend the state-of-the-art ranking-based method for searching in trees with uniform costs\nof queries [38, 42], where the algorithm calculates a strategy function f:V(T)→N, given the\ninput tree T= (V, E). In our approach, to express the variable costs of queries we will use an\ninterval-based extended strategy function.\nKeeping in mind that the cost function is the time needed to perform a query, the values of an\nextended strategy function indicate the time periods in which the respective vertices will be queried.\nNote that, e.g., due to the rounding, the time periods will be longer than the query durations and\nthis is easily resolved either by introducing idle times or by following the actual query durations\nwhile performing a search based on a strategy produced by our algorithm.\nIn our approach, informally speaking, we will partition the rounding of the input tree into\nseveral subtrees (the layer components) and for each of them, we will use the algorithm from [38].\nA useful class of extended strategy functions is one that generates decision trees that are struc-\ntured (see Definition 3.1).\nDefinition 4.1. We say that an extended strategy function f, defined on the vertices of a tree T\nwith up-monotonic cost function, is structured if for any layer component L,f(u)> f(v) for each\nv∈V(Tu), where u=r(L).\nIt follows that a structured extended strategy function represents a structured decision tree.\nAs mentioned earlier, we will process each layer component Lwith the vertex extension operator\nbut in order to do it correctly, we need to initialize appropriately the roots of the layer components\nbelow L. For that, we define the following operators. Let fbe an extended strategy function defined\non the vertices of Tv, where vis the root of a layer component L′. The structuring operator assigns\ntovthe minimal interval so that fis a structured extended strategy function on Tv. (We note that\nin our algorithm vwill have some interval assigned when the structuring operator is about to be\napplied to v. Hence, the application of this operator will assign the new required interval.) The cost\nscaling operator aligns the interval attributed to v: iff(v) = [a, b) and Lis the layer component\ndirectly above L′, the cost scaling operator assigns vthe interval [ ω(L)⌈b\nω(L)⌉ −ω(L), ω(L)⌈b\nω(L)⌉).\nWe will say that vertex visaligned to ω(L) after this modification (see the beginning of Section 5\nfor a formal definition). Informally speaking, this is done so that the children of the leaves in Lhave\nintervals whose endpoints are multiples of ω(L) so that they become ‘compatible’ with the allowed\nplacements for intervals of the vertices from L. Yet in other words, this allows us to translate\nthe intervals of the children of the leaves in Linto integers that can be treated during bottom-up\nprocessing of Lin a uniform way.\n12v\n1\n3\n5\n2\n1\n4\n1\nscr(v)\n2\n1Figure 2: Example of a screening neighborhood. Labels at the vertices show the corresponding\nvalues of some strategy function. The dashed line encloses the screening neighborhood of the\nvertex v. The (single) vertex screening vis filled in black. All remaining vertices visible from vare\nfilled with light gray. Note that the vertex with label 5 > f(v) is visible from v, although it does\nnot belong to the screening neighborhood of v. All vertices that are not visible from vare white.\nObservation 4.2. Let a layer component L′be directly below a layer component LinT= (V, E, ω ).\nIfr(L′)is assigned an interval [a, b), whose endpoints are consecutive multiples of ω(L′)(i.e. a=\nkω(L′)andb= (k+ 1)ω(L′)for some integer k), then the cost scaling operator assigns to r(L′)an\ninterval [a′, b′)such that b′≤b+ω(L)−ω(L′).\nProof. Consider a layer component LofT= (V, E, ω ) and a structured extended strategy function\nfdefined on the vertices below L. The cost scaling operator selects b′as the smallest multiple of\nω(L) that is greater or equal than b. Since a′=b′−ω(L), we have that a′< b. On the other\nhand, because b−a=ω(L′),ais the highest multiple of ω(L′) that is less than b. Subsequently,\nbecause w(L) is divisible by w(L′), we obtain that a′≤a. What follows, a′+ω(L′)≤b, and finally\nb′≤b+ω(L)−ω(L′).\n5 Aligned strategies\nIn this section we give foundations for analyzing trees with down-monotonic cost functions, i.e.\nwhere the weights of the visited vertices are non-decreasing while traversing any path from the root\nto a leaf. In other words, we consider rooted trees T= (V, E, ω ), where the root of Tis selected\nproperly according to the definition of a down-monotonic cost function. Concepts provided in this\nsection are used in Section 7, for the analysis of Algorithm Down-Monotonic.\nThe first thing the Algorithm Down-Monotonic does is to switch from the input tree to its\nrounding. Although the terms below apply to general weight functions, we should remember that\nthey will be applied to the rounded ones in our analysis.\nGiven an extended strategy function f, we say that the vertex visaligned if the endpoints of\nthe interval f(v) are multiples of ω(v). We also say that fisaligned if all vertices are aligned.\nFor an extended strategy function f, a vertex uisscreening a vertex vif and only if, on the path\nthat connects vandu,uis the only vertex such that f(v)< f(u). In other words, for any path P\nthat starts with v,u∈V(P) is screening vif it is the closest vertex to v, such that f(v)< f(u).\nObservation 5.1. Ifs, s′are distinct vertices screening some vertex, then f(s)∩f(s′) =∅.\nFurthermore, for an arbitrary vertex v, the screening neighborhood ofvis the set scr( v)⊆V(T)\ndefined as follows. We have that u∈scr(v) if and only if either uis screening vor the path that\nconnects uandvhas no vertex screening v. The subtree T[scr(v)] is denoted by Tscr(v). For an\nexample see Figure 2. By definition, we include vto be in scr( v). Some of the properties of the\nscreening neighborhood are as follows.\n13Observation 5.2. Ifu∈scr(v)is a leaf of Tscr(v), then we have that uis screening vor is a leaf\ninT(or both).\nAs stated in Observation 5.3, it may happen that a vertex is visible from v, but does not belong\nto scr( v). Consider a vertex uvisible from vwith f(v)< f(u), and a vertex u′such that ulies\non the path between vandu′andf(u)< f(u′). Even if u′was visible from v(which then implies\nf(v)< f(u′)), it would not belong to scr( v). In general, not every vertex visible from vbelongs to\nscr(v). At the same time not every vertex in scr( v) must be visible from v.\nObservation 5.3. If a vertex uis visible from vandf(u)< f(v), then u∈scr(v).\nWe will several times use an important property that for any u∈scr(v)\\{v}, either f(u)< f(v)\norf(v)< f(u). This fact basically allows to simplify arguments by reducing the number of possible\nsubcases in the following lemmas.\nObservation 5.4. For each u∈scr(v),u̸=v, it holds f(v)∩f(u) =∅.\nProof. By contradiction, assume a u∈scr(v) exists such that f(u)∩f(v)̸=∅. Since fis an\nextended strategy function, there must be an xon the path between uandv, such that f(x)> f(v).\nAmong all such vertices xconsider one with the maximum value of f(x). This choice implies that\nf(x)> f(v) and there is no vertex yon the path between vandxsuch that f(y)> f(x)∪f(v).\nThis gives that xis screening vcontradicting u∈scr(v).\nLemma 5.5. Letfbe an extended strategy function for T= (V, E, ω ). Consider an arbitrary\nv∈V(T). Let smin∈scr(v)be the vertex screening vwith the minimal value of f, among all\nvertices screening v. Consider an arbitrary function f′:V(T)→ {[a, b): 0≤a < b},|f′(v)| ≥ω(v),\nwhich satisfies the following conditions:\n1.f′(u) =f(u), for any u̸∈scr(v),\n2.f′(u) =f(u), ifu∈scr(v)is screening v,\n3.f′(u)< f(smin), for any u∈scr(v)that is not screening v. (This includes v.)\nWe have the following: if f′is an extended strategy function for Tscr(v), then f′is also an extended\nstrategy function for the entire T.\nProof. The proof is by case analysis dictated by the definition of visibility function. Consider two\nvertices x, y∈V(T) such that x̸=yandf′(x)∩f′(y)̸=∅. For each possible choice of xandy, we\nshow that on the path connecting xandy, there is a vertex zsuch that f′(x)∪f′(y)< f′(z). Note\nthat based on Observation 5.1, sminis unique in scr( v).\n•Letx, y∈scr(v). By assumption, f′is an extended strategy function for Tscr(v). Thus, on\nthe path between xandythere exists a required vertex z.\n•Letx, y̸∈scr(v). For all vertices outside of Tscr(v),f′is identical to f. Because fis an\nextended strategy function, there exists z∈V(T), such that f(x)∪f(y)< f(z).\n1. If zis outside of Tscr(v), then f′(z) =f(z) and consequently f′(x)∪f′(y)< f′(z).\n2. If z∈scr(v) and zis screening v, then also in this case f′(z) =f(z) and consequently\nf′(x)∪f′(y)< f′(z).\n14x\ny\nv\nP\nz\ns1\ns2\nP\nscr(v)Figure 3: An illustration for the case when x, y̸∈scr(v) in Lemma 5.5.\n3. If z∈scr(v) and zis not screening v, letPbe the path connecting xandy. (See Figure 3\nfor an illustration.) Because the endpoints x, y̸∈scr(v) and z∈scr(v) is not screening\nv, it follows from the definition of scr( v) that Pcontains vertices s1ands2screening v,\nsuch that the path between s1ands2contains z. Let s= argmaxs∈{s1,s2}f(s). Because\nzis on a path either between vands1orvands2, based on the definition of a screening\nvertex, we have that f(z)< f(s). Subsequently, we have that f′(x) =f(x)< f(z)<\nf(s) =f′(s). By similar arguments, f′(y)< f′(s). Hence, on the path between xand\ny,sis the required vertex separating xandy.\n•Letx∈scr(v) and y /∈scr(v). Let Pbe the path connecting xandy. Based on the definition\nof scr( v),Pcontains at least one vertex screening v(otherwise, for x∈scr(v),ywould have\nto belong to scr( v) too). Let s∈V(P) be the vertex screening vthat is closest to y. We note\nthatPcontains two vertices screening vonly when xis screening v.\n1. Let xbe screening v. We have f′(x) =f(x). Because fis an extended strategy function\nforT, there is z∈V(P), such that f(x)∪f(y)< f(z). We consider two subcases.\nLetx=s. Recall that s∈V(P) is the vertex screening vthat is closest to y. Hence,\nxis the only vertex in Pthat belongs to scr( v). Subsequently, z /∈scr(v) and we have\nthatf′(z) =f(z). What follows, f′(x) =f(x)< f(z) =f′(z).\nLetx̸=s. Ifz /∈scr(v) orzis screening v(i.e., z=s), then f′(z) =f(z) and zis\nthe required vertex separating xandy. Ifz∈scr(v) and zis not screening v, we have\nthatf(z)< f(s). What follows, f′(x) =f(x)< f(z)< f(s) =f′(s).\n2. If xis not screening v, then by definition of f′andsmin,f′(x)< f′(smin)≤f′(s). What\nfollows, sis the required vertex separating xandy.\nThe technical Lemma 5.6 shows a property of the screening neighborhood of specific unaligned\nvertices. This property is later used in the proof of Lemma 5.7.\nLemma 5.6. Consider an extended strategy function fforT= (V, E, ω ). Let v∈V(T)be an\nunaligned vertex, such that f(v)is minimal. Denote f(v) = [ xv, yv),|ω(v)|= 2k. Take a∈N\nsuch that a2k< x v<(a+ 1)2k. For each u∈scr(v)such that f(u)∩[a2k, xv)̸=∅, it holds\nf(u)⊂[a2k, xv).\nProof. We know that aexists because vis unaligned. By Observation 2.6, |f(v)|= 2k. It follows\nfrom Observation 5.4 that f(u)< f(v) for any uas in the statement of the lemma, and consequently,\nthatuis aligned.\nLetf(u) = [ xu, yu). By assumption and by f(u)< f(v), we know that a2k< y u< x v.\nSince uis aligned, we have that xu=b2k′andyu= (b+ 1)2k′, for some integer k′. Note that\n15a2k< yu< xv<(a+ 1)2kimplies k′< k. Thus, yu≥a2k+ 2k′, because uis aligned. This gives\nxu=yu−2k′≥a2k, as required in the lemma.\nWe now prove the main result of this section:\nLemma 5.7. For each instance with down-monotonic and rounded cost function, there exists an\noptimal extended search strategy function that is aligned.\nProof. In the proof, we consider an optimal extended strategy function f, such that the number\nof unaligned vertices is minimal. Based on f, we construct another optimal extended strategy\nfunction according to which, there are fewer unaligned vertices. This shows, by contradiction, that\nthere must exist an optimal extended strategy function such that all vertices are aligned.\nLetv∈V(T) be an unaligned vertex, such that the interval f(v) is minimal. Let ω(v) = 2k,\nf(v) = [xv, yv). Take a∈N, such that a2k< xv<(a+ 1)2kand ( a+ 1)2k< yv<(a+ 2)2k.\nBased on the strategy function f, we will now construct a f′with fewer unaligned vertices.\n1. For all u /∈scr(v), we set f′(u) =f(u).\n2. If s∈scr(v) is screening v, we set f′(s) =f(s).\n3. For all other vertices in scr( v):\nFor the vitself, we set f′(v) = [a2k,(a+ 1)2k).\nFor any vertex u∈scr(v) such that f(u)∩[a2k, xv)̸=∅, we set f′(u) = [xu+ 2k, yu+ 2k),\nwhere f(u) = [xu, yu).\nFor all other vertices u∈scr(v) we set f′(u) =f(u).\nFor brevity, we will refer to the vertices u∈V(T) for which f′(u)̸=f(u) asupdated vertices.\nWe will refer to all other vertices as stillvertices. A useful fact about the updated vertices is the\nfollowing:\n(*) We have f′(u)< yv, for every updated vertex u∈V(T).\nBy definition of f′, updated vertices are those whose value of fintersects with [ a2k, xv) or the v\nitself. The fact (*) holds for v. Consider u∈V(T), such that f(u)∩[a2k, xv)̸=∅. Denote f(u) =\n[xu, yu),f′(u) = [x′\nu, y′\nu). By Lemma 5.6, we have that yu≤xv. Subsequently, y′\nu≤xv+ 2k=yv.\nThis completes the proof of (*).\nLet us note that vis aligned according to f′and all vertices which were aligned according to f\nare also aligned according to f′. To see the latter, consider an updated vertex u̸=v. Ifuis aligned\naccording to f, then by the definition of f′,u∈scr(v) and f(u)∩[a2k, xv). But then, by Lemma 5.6,\nf(u)⊆[a2k, xv) which in particular means that ω(u)≤2kand hence uis indeed aligned according\ntof′. We also observe that sup f′(v)<supf(v), and for each u, such that f(u)∩[a2k, xv)̸=∅, we\nhave sup f′(u)≤supf(v). Hence, f′is optimal due to the optimality of f. The above in particular\nimplies that it remains to argue that f′is a valid extended strategy function.\nWe will now show that f′is an extended strategy function for T. Based on Lemma 5.5, it\nsuffices to show that f′is an extended strategy function for the subtree Tscr(v). Consider vertices\nu∗, v∗∈scr(v), such that u∗̸=v∗andf′(u∗)∩f′(v∗)̸=∅. We analyze all possible choices of the\npairu∗,v∗. Motivated by Observation 5.4, we categorize the choices for selecting u∗depending on\nhowf′(u∗) relates to the interval f(v) = [xv, yv).\n16Case 1: Let u∗be such that yv< f′(u∗). We argue that\nf(v)< f(v∗) (1)\nSuppose otherwise, which in view of Observation 5.4 implies f(v∗)< f(v). Since u∗is still by (*),\nf(u∗) =f′(u∗). Then, yv< f(u∗) can be written as f(v)< f(u∗). But then f(v∗)< f(v) gives\nf(v∗)< f(u∗), which contradicts f(u∗)∩f(v∗)̸=∅and implies (1). The latter and (*) prove that\nv∗is still, thus yv< f′(v∗).\nBecause fis an extended strategy function, there is a vertex z∈scr(v) on the path between u∗\nandv∗, such that f(z)> f(u∗)∪f(v∗). Because f(z)> f(u∗)> f(v), we have by (*) that zis still.\nSubsequently, f′(u∗) =f(u∗)< f(z) =f′(z) and since v∗is still f′(v∗) =f(v∗)< f(z) =f′(z),\nthus f′(z)> f′(u∗)∪f′(v∗).\nCase 2: Let u∗be such that f′(u∗)< a2k. By repeating the steps in Case 1, i.e. when yv< f′(u∗),\nand using Lemma 5.6, we obtain that f′(v∗)< a2k. Because fis an extended strategy function,\nthere is a vertex z∈scr(v) on the path between u∗andv∗, such that f(z)> f(u∗)∪f(v∗). If\nzis an updated vertex, we have that f′(z)> a2k, hence f′(u∗)< f′(z). Otherwise, we have\nf′(u∗) =f(u∗), and the claim follows from f(u∗) =f′(u∗) and f(v∗) =f′(v∗).\nCase 3: Let u∗be such that f′(u∗)∩[(a+ 1)2k, yv)̸=∅. Note that u∗̸=vbecause by definition\noff′,f′(v) = [a2k,(a+ 1)2k). Because f′(u∗)∩f(v)̸=∅, it follows from Observation 5.4 that u∗\nis updated. Subsequently, we have that f(u∗)∩[a2k, xv)̸=∅. Based on Lemma 5.6, it holds that\nf(u∗)⊂[a2k, xv) and subsequently that f′(u∗)⊂[(a+ 1)2k,(a+ 2)2k). By definition, f′(v∗)∩\nf′(u∗)̸=∅. Thus, f′(v∗)∩[(a+ 1)2k, yv)̸=∅. Following the same steps as for u∗, we obtain that\nv∗is updated and that f′(v∗)⊂[(a+ 1)2k,(a+ 2)2k).\nBecause fis an extended strategy function, there is z∈scr(v) on the path between u∗andv∗,\nsuch that f(z)> f(u∗)∪f(v∗). Because both f(u∗) and f(v∗) are subsets of [ a2k, xv), we have\nthat ω(u∗)< ω(v) and ω(v∗)< ω(v). Because ωis down-monotonic, we have that neither u∗nor\nv∗belong to Tv. What follows, z̸=v. Ifzis an updated vertex, then we have f′(z)> f′(u∗).\nOtherwise, because z̸=v,f′(z) =f(z)> yv, and we have that f′(u∗)< yv< f′(z). Following the\nsame steps for v∗, we obtain f′(z)> f′(u∗)∪f′(v∗).\nCase 4: Let u∗be such that f′(u∗)∩[a2k,(a+ 1)2k)̸=∅. We will start by showing that this\ncondition holds only when u∗=v.\nConsider the following ways we may choose u∗∈scr(v), with respect to the interval f(u∗):\n1.f(u∗)< a2k. By definition of f′, vertex u∗is still, thus f′(u∗)< a2k. What follows, such u\ndoes not exist.\n2.f(u∗)∩[a2k, xv)̸=∅. By Lemma 5.6, we have that f(u∗)⊂[a2k, xv). By definition of f′, we\nhave that ( a+ 1)2k≤f′(u∗).\n3.f(u∗)∩[xv, yv)̸=∅. Based on Observation 5.4, we have u∗=v.\n4.yv≤f(u∗). By definition of f′,u∗is still, thus ( a+ 1)2k< yv≤f(u∗) =f′(u∗).\nSubsequently, because v∗̸=u∗, we have that f′(v∗)∩[a2k,(a+ 1)2k) =∅. This shows that it is\nnot possible to find a v∗∈scr(v), such that because f′(u∗)∩f′(v∗)̸=∅.\n176 The algorithm for up-monotonic cost functions\nIn this section we propose an algorithm for trees with up-monotonic cost functions (cf. Algo-\nrithm Up-Monotonic). The algorithm starts with a pre-processing pass, which transforms the\ninput tree to its rounding and selects a root arbitrarily from the set of vertices with the highest\ncost o query (cf. definition of an up-monotonic cost function, Section 1.5). An extended strategy\nfunction fis then calculated during a bottom-up traversal over the layer components of the tree\nby applying Algorithm Process-Component. What follows, the extended strategy function for the\nvertices of a layer component Lis calculated only after it is already defined for all vertices below\nthe component. Moreover, since we use a structured f, only its values assigned to the roots of the\nlayer components below Lare important when calculating ffor the vertices of L.\nAlgorithm Up-Monotonic: (input: tree Twith up-monotonic cost function;\noutput: an extended strategy function fcalculated for T)\n1T= a rooted rounding of T\n2foreach layer component Lin bottom-up fashion do\n3 Apply Algorithm Process-Component to L\n4return f\nGiven a layer component L, Algorithm Process-Component iterates over the vertices of Lin a\ndepth-first, postorder fashion. Suppose that a structured extended strategy function fis defined\non all vertices below Land that fis aligned to ω(L) at all roots of layer components directly below\nL. To calculate ffor the vertices of L, we first calculate the strategy function f′for each vertex\nofLand then obtain ffrom f′using the formula: f(u) = [ f′(u)ω(L),(f′(u) + 1) ω(L)), where\nu∈V(L). (Recall that this is a conversion that takes us from the integer-valued strategy function\nf′to the interval-valued extended strategy function.) To provide an applicable input for the vertex\nextension operator at the leaves of L, for each child u∈V(L′) of such leaf, we calculate the integer\nf′(u) =b\nω(L′)that is derived from the extended strategy function at u, i.e. from f(u) = [a, b). Note\nthat since our decision trees are structured, the visibility sequence corresponding to each root of\na layer component below L(which includes u) consists of only one value, the one that equals the\nstrategy function at the root. What follows, deriving the value of f′only for the roots of the layer\ncomponents directly below Lis sufficient to create a valid input for the vertex extension operator.\nIn other words, we make these preparations to use the operator and the corresponding method\nfrom [38]. Once each vertex v∈V(L) obtains the corresponding extended strategy function f(v),\nwe apply for the root of Lthe structuring operator followed by the cost scaling operator. This will\nclose the entire ‘cycle’ of processing one layer component.\n6.1 Analysis\nLemma 6.1 characterizes the interval of the extended strategy function that Algorithm Process-\nComponent assigns to the root of a layer component.\nLemma 6.1. Consider a layer component LofT= (V, E, ω )and a structured extended strategy\nfunction fdefined on the vertices below L. Let the roots of the layer components directly below\nLbe aligned to ω(L). Suppose that the values of fover V(L)are obtained by a call to procedure\nProcessComponent(L,f) . We have that fis a structured extended strategy function on V(Tr(L)),\nand the lowest possible interval I, such that I > f (u), for each ubelow r(L), is assigned to r(L).\n18Algorithm Process-Component: (input: layer component L). Let fbe an\nextended strategy function defined for all descendants of vertices in Lthat do\nnot belong to L.\n1foreach root vof a layer component directly below Ldo\n2 f′(v) =b\nω(L), where bis the right endpoint of the interval f(v)\n3foreach u∈V(L) in a postorder fashion do\n4 Obtain f′(u) by applying the vertex extension operator to v\n5 f(u) = the interval derived from f′(u), see Section 4\n6ifthe root vofLis not the root of Tthen\n7 Update f(v) by applying the structuring operator to v\n8 Update f(v) by applying the cost scaling operator to v\nProof. Due to the alignment of fat the roots of the components directly below L, the strategy\nfunction f′derived from ffor these roots is consistent with the query costs in L. That is, the\nconstruction of f′done at the beginning of the procedure gives an integer-valued strategy function.\nSince fis structured, among all vertices below L, the extension operator correctly needs to consider\nonly the values of f′at the roots of components directly below L.\nBy Lemma 2.8, the visibility sequence calculated for the root of Lis (lexicographically) mini-\nmized among all possible valid assignments of f′. Hence in particular, the value of max {f′(v)|v∈\nV(Tr(L))}is minimized. It follows that if there exists such an optimal assignment of f′that is\nstructured, then this f′is calculated by the procedure is structured and consequently the fob-\ntained from f′is minimal and structured. If there exists no optimal assignments of f′overV(L)\nthat is structured, the structuring operator modifies fand assigns to r(L) the lowest interval that\nis greater than f(u) for any ubelow r(L).\nIt follows from Lemma 6.1 and Observation 4.2 that given an extended strategy function ffixed\nbelow a layer component L, Algorithm Up-Monotonic extends fto the vertices of Lin such a way\nthat the interval assigned to r(L) is not higher than the optimal interval (assuming that fis fixed\nbelow L) incremented by an additive factor ω(L′)−ω(L), where L′is a layer component directly\nabove L. IfLis the top component, the structuring transform is not applied and the additional\ncost is not incurred.\nWe now formulate a technical Lemma 6.2 that helps analyze the strategy generated by Algorithm\nUp-Monotonic.\nLemma 6.2. LetT= (V, E, ω )be a tree with a up-monotonic and rounded cost function. We denote\nbyfoptan optimal structured extended strategy function on V(T). Let Lbe a layer component of\nTwith klayer components Lj,1≤j≤k, directly below L. Let vj=r(Lj),1≤j≤k. We define\nf′as a function defined on V(T)such that for all vertices below any vj,f′is equal to fopt, while\nfor the vertices vjand above, f′is equal to fopt+ω(L). It holds that f′is a structured extended\nstrategy function on V(T).\nProof. We first show that f′is an extended strategy function. We need to show that for any pair\nx, y∈V(T),x̸=y, iff′(x) =f′(y), then there is a vertex z∈V(T) separating x, ysuch that\nf′(z)> f′(x). (See Definition 2.5.)\nWe analyze the following cases with respect to the locations of the vertices xandy. In all cases\nwe assume that x̸=y,f′(x) =f′(y), and zis the vertex separating xandyaccording to f′.\n19•x∈V(Li),y∈V(Lj) and i=j.\nLetv=r(Li). Because foptis structured, f′is also structured within the subtree Tv, since\nit assigns va higher interval than foptandf′is otherwise identical to fopt. What follows,\nbecause f′is structured and we assumed f′(x) =f′(y), we have that x̸=vandy̸=v.\nConsider the foptas defined on all vertices of Li. Because x, y∈V(Li), we have z∈V(Li).\nIfz̸=v, then f′(z) =fopt(z)> fopt(x) =f′(x), where the last equation holds because x̸=v.\nIfz=v, we also have f′(z)> f′(x), because f′is structured in Tv.\n•x∈V(Li),y∈V(Lj) and i̸=j.\nBecause f′is structured in Li, we have z=r(Li).\n•Both xandybelong to V(L).\nBecause foptis an extended strategy function in L, there is a z∈V(L) such that fopt(z)>\nfopt(x). Then, by adding ω(L) to both sides of the inequality one obtains f′(z)> f′(x).\n•x∈V(Li) for some iandy∈V(L).\nIfx̸=r(Li) then z=r(Li), because f′is structured in Li.\nOtherwise, both xandyare in the subtree T[V(L)∪ {x}]. Recall that in this subtree f′is\nderived from foptby adding a constant offset, which gives an extended strategy function.\nTo show that f′is structured in the remaining cases, we observe that for all vertices above L,\nf′is derived from the structured foptby shifting the assigned intervals by a positive offset, which\ndoes not change the structural property of the layer components’ roots.\nLemma 6.3. The cost of the decision tree generated by Algorithm Up-Monotonic is at most 2 times\ngreater than the cost of an optimal structured decision tree.\nProof. The proof is by induction, bottom-up over the layer components of the input tree. Take a\ntreeT= (V, E, ω ) with an up-monotonic and rounded cost function. We denote by c(i),1≤i≤l,\nthe query cost to a vertex in the i−thlayer, where the layers are ordered according to their cost\nof query and lis the number of layers in T. Let foptbe an optimal (structured) extended strategy\nfunction and falgbe the extended strategy function generated by Algorithm Up-Monotonic. We\ndenote by z(v) the cost of querying the parent of vor the cost of vitself if there is no vertex above\nv, i.e. when vis the root of T.\nWe want to prove the following induction claim. For any layer component Lit holds falg(v)≤\nfopt(v) +z(v), where v=r(L).\nThe cost of the decision tree generated by falgis equal to the supremum of falg(r(T)). Since the\ncost of a single query to the top layer is not more than the cost of an optimal structured decision\ntree, the lemma follows from the induction claim applied to the top layer component.\nLetLbe a bottom layer component. Based on Lemma 6.1 we know that by applying the vertex\nextension and structuring operators, Algorithm Up-Monotonic generates an optimal structured\nassignment of the extended strategy function to the vertices of L. In a case where Lis also the top\ncomponent (which is the trivial case of Tbeing a single layer component) we have falgis optimal\nand the induction claim holds. Otherwise, the cost scaling operator extends the interval assigned\ntor(L), increasing the cost of Lby (at most) an additive factor of c(2)−c(1) over the optimal\ncost. Since the added factor is less than z(r(L)), the induction claim holds for the bottom layer\ncomponent L.\n20LetLbe a layer component from the i-th layer, with klayer components Lj, 1≤j≤k, directly\nbelow L. We denote the roots of the layer components Ljbyvj. Consider a strategy function f′\nwhich is defined as in Lemma 6.2, that is for all vertices below any vj,f′is equal to fopt, while\nfor vertices vjand all vertices above, f′is equal to fopt+ω(L). Lemma 6.2 implies that f′is a\nstructured extended strategy function on V(T).\nFrom the induction hypothesis we have that for any vj,falg(vj)≤fopt(vj) +z(vj) =fopt(vj) +\nω(L). According to Lemma 6.1 and Observation 4.2, Algorithm Up-Monotonic assigns intervals to\nthe vertices of Lsuch that the interval assigned to the root of Lis the lowest possible (when Lis\nthe top component), or the lowest possible interval incremented by a positive offset of ω(L′)−ω(L),\nwhere L′is the layer component directly above L. If we sum up the additive increments due to\ncost scaling in all layer components below L, we obtainP\ni<lc(i)< c(l) =z(v) (recall that c(i)’s\nare powers of 2). Thus, falg(v)≤fopt(v) +z(v).\nBy combining Lemmas 2.4, 3.2, and 6.3 we obtain that Algorithm Up-Monotonic generates a\nsolution for the binary search problem with a cost at most 8 times greater than that of an optimal\nsolution. We note that Algorithm Up-Monotonic maintains the main structure of the linear time\nalgorithm from [38] (a single bottom-up pass over vertices of the tree.) Our method extends the\nstate-of-the-art algorithm by adding a fixed number of O1 steps, computed when visiting vertices\nduring the bottom-up traversal. See Algorithm Process-Component. What follows, Algorithm Up-\nMonotonic also runs in linear time. This proves Theorem 1.1.\n7 The algorithm for down-monotonic cost functions\nIn this section, we describe and analyze the algorithm providing Theorem 1.2. To construct an\ninterval-based extended strategy function f, our procedure adapts the greedy linear-time algorithm\nfor searching in trees with uniform query costs [38]. Let T= (V, E, ω ) be a rooted tree, such that\nωis rounded and down-monotonic. Recall that the root must be one of the vertices of minimal\nweight.\nIn our algorithm that takes Tas input, the intervals that fassigns to the vertices are calculated\nin a bottom-up fashion. Consider a vertex v∈V(T) with k >0 children v1, . . . , v kso that fis\ndefined on all descendants of v. For each vi, we denote the extended visibility sequence derived from\nfrestricted to TvibyS(vi). Let Ibe the set of all intervals Ithat belong to any S(vi), such that\nthere exists an interval I′∈S(vj),i̸=j, where I∩I′̸=∅. IfI ̸=∅, letIMbe the maximal interval\ninI. IfI=∅, then let IM= [−1,0). The minimal interval Iwith endpoints being multiples of\nω(v), such that I > I M,|I|=ω(v) and I∩I′=∅for each I′∈S(v1)∪ ··· ∪ S(vk) is called the\ngreedy interval forv. In case when vhas no children the greedy interval forvis [0, ω(v)).\nThe algorithm for a rounded down-monotonic cost function simply assigns the greedy interval\nfor each vertex, once such intervals are constructed for the children. The pseudo-code is shown as\nAlgorithm Down-Monotonic.\nAlgorithm Down-Monotonic: (input: a tree with down-monotonic cost func-\ntion).\n1Compute a rooted rounding of the input tree.\n2foreach vertex vin a bottom-up fashion do\n3 Compute the greedy interval I′forvand set f(v) =I′.\n4return the search strategy that corresponds to f.\n217.1 Analysis\nIn order to prove the correctness and efficiency, we will now introduce concepts that allow to express\nthe extended visibility sequences used in our methodology in terms of the vertex extension operator\nover integer-valued sequences that were developed for searching in unweighted trees [38].\nDefinition 7.1. LetS(v) be an extended visibility sequence for some v∈V(T). Take a d∈Nthat\ndivides ω(v). We say that the visibility sequence Sd(v) iscorresponding to S(v) with granularity d\nif and only if it lists in decreasing order all values obtained in the following way. For every interval\n[a, b)∈S(v), add to Sd(v) the valuesa\nd+ifor each i∈ {0, . . . , n −1}, where n=b−a\nd.\nOur further analysis will leverage a key property of aligned extended strategy functions, ex-\npressed by Observation 7.2, which is later used in Lemma 7.3.\nObservation 7.2. LetS1andS2be extended visibility sequences such that there exists a d∈N\nthat divides the endpoint of each interval in both S1andS2. We have that S1<lS2if and only if\nSd\n1<lSd\n2.\nAn extended visibility sequence S(v) of a vertex vislexicographically minimal if:\n•there exists an extended strategy function fdefined on Tvsuch that S(v) is derived from f,\nand\n•there exists no extended strategy function f′defined on Tv, such that S′(v)<lS(v), where\nS′(v) is the extended visibility sequence for vderived from f′.\nThe following Lemma 7.3 serves as an induction step, which when applied bottom-up over the\ninput tree, shows that Algorithm Down-Monotonic indeed returns an optimal strategy.\nLemma 7.3. Consider a tree T= (V, E, ω ), with rounded and down-monotonic ω. If the extended\nvisibility sequence of each child of some vertex vis lexicographically minimal, then the greedy interval\nforvis also lexicographically minimal.\nProof. Letv1, . . . , v kbe the children of v. Let f, defined for the subtrees Tv1, . . . , T vk, be such\nthat for each i∈ {1, . . . , k }the lexicographically minimal sequence S(vi) is derived from f. Take\nextended visibility sequences ˜S(vi) for each i∈ {1, . . . , k }that are derived from an arbitrary\nextended strategy function. By assumption, for each child vi, we have that S(vi)≤l˜S(vi). Based\non Observation 7.2, we also have that\nSdi(vi)≤l˜Sdi(vi), (2)\nwhere di=ω(vi) for each i. Note that the dimay be different for different children vi.\nWe apply the vertex extension operator (see Definition 2.7) for vusing the sequences ˜Sdi(vi).\nSince the vertex extension operator is minimizing [38], it assigns to vthe lexicographically minimal\nsequence ˜Sd(v) available for v, out of all valid visibility sequences, given the sequences assigned\nto the children. Note that since ωis rounded and down-monotonic, we have that ddivides difor\neach i∈ {1, . . . , k }. For each i, we have that (2) implies Sd(vi)≤l˜Sd(vi). Intuitively, we may\nswitch from the granularities ditodpreserving the ordering of sequences. On the other hand, the\nvertex extension operator is monotone [38] and when applied on the sequences Sdi(vi), the resulting\nsequence Sd(v) satisfies Sd(v)≤l˜Sd(v). Consider extending ftoTvby assigning to vthe greedy\ninterval to v. By the definition, the extended visibility sequence S(v) derived from such fis such\nthatSd(v) corresponds to S(v). This proves that S(v) is lexicographically minimal as required.\n22By induction over the subtrees of T, with the induction claim formulated by Lemma 7.3, one\nobtains that the search strategy returned by Algorithm Down-Monotonic is optimal for the rounding\nTof the input tree T′. On the other hand, we have OPT (T)≤2·OPT (T′). This implies that\nthe algorithm is 2-approximate for an arbitrary input tree. Since the tree has nvertices, the\nlexicographically minimal sequence for each vertex vhasO(n) intervals. Hence, the computation\nof the greedy interval takes O(n) time, as well as any other list manipulation task, proving that\nthe algorithm has polynomial running time. This completes the proof of Theorem 1.2.\n8 The algorithm for k-monotonic cost functions\nConsider a partitioning of a rounded tree T= (V, E, ω ) into subtrees T1, . . . , T las in the definition\nof ak-monotonic cost function (cf. Section 1.5). The subtree Tithat contains the root of Tis\ncalled the root subtree . In the case of Algorithm Down-Monotonic, our methodology made it easier\nto frame it as a procedure that is computing an extended strategy function. This then allows to get\nthe corresponding search algorithm as explained in Section 2. However, with the tools we have it\nis more natural to present a procedure (cf. Algorithm General-Monotonic) that gives Theorem 1.3\nas an adaptive process that instead of outputting a search strategy, actually performs the adaptive\nsearch and returns the target.\nAlgorithm General-Monotonic: (input: a tree Twith k-monotonic cost func-\ntion; output: the target).\n1LetT′be the root subtree of the rounding of the input tree.\n2Compute a search strategy AforT′.\n3Execute A.\n4ifthe target tis inT′then\n5 return t.\n6else\n7 Letvbe the child of a leaf of T′such that Tvcontains the target.\n8 Call recursively Algorithm General-Monotonic with input Tv.\nTo compute the search strategy in line 2, if T′is down-monotonic, use the 2-approximation\nAlgorithm Down-Monotonic, otherwise use the 8-approximation Algorithm Up-Monotonic. If the\ntarget tis not in T′, then let vbe the first vertex outside of T′on the path from the root of Ttot.\nNote that all queries performed within T′are then pointing towards vin their replies. The same\nholds during a query to the parent of v, which is a leaf of T′. Thus, the vertex vcan be correctly\nselected in line 7. The depth of the recursion is at most kand each strategy execution in line 3\nintroduces a cost of at most 8 ·OPT (T′)≤8·OPT (T). The recursion correctly ends when T′is\nsuch that no vertex of T′has a child that does not belong to T′(i.e., all leaves in T′are also leaves\ninT). This gives Theorem 1.3.\n9 The NP-completeness\nIt has been shown in [10] that the edge variant of binary search is NP-complete already for specific\nsmall-diameter trees. We will use this fact to argue that vertex search with k-monotonic cost\nfunctions is not FPT with respect to k.\n23The edge search problem considers as an input an edge-weighted tree T′= (V, E, c ), where c\nattributes each edge with the cost of the query. For an arbitrary search target t∈V(T′), a query\nto an edge e∈E(T′) incurs the cost of c(e) and learns which one of the two components in T′−e\ncontains t. For a given input tree, similarly as in the vertex search, the goal of the algorithm is to\nobtain a search strategy that has minimal total cost in the worst case. Note that this problem can\nbe equivalently defined as performing vertex search in the line graph of T′.\nFollowing the description provided in [14], we formulate Lemma 9.1, which shows a connection\nbetween the two problems. First, for each tree T′we define its subdivision T= (V, E, ω ) as\nfollows. Let M= 1 +P\ne∈E(T′)c(e) exceed the weight of all edges in T′. For each e∈E(T′)\nintroduce a vertex v(e) with weight ω(v(e)) = c(e). Then, V(T) =V(T′)∪ {v(e):e∈E(T′)},\nE(T) ={{x, v(e)},{v(e), y}:e={x, y} ∈E(T′)}, and ω(x) =Mfor each x∈V(T′). In other\nwords, we obtain Tby putting the weight Mon each vertex of T′and subdividing each edge\neso that the new vertex v(e) that comes from the subdivision of e‘inherits’ its weight from e.\nInformally, the idea behind this construction is that when performing a vertex search on T, one\nshould never query any vertex xinV(T′) due to the large weights. Instead, in order to learn\nwhether xis the target in T, it is favorable to query all of its neighbors.\nLemma 9.1 ([14]).For each tree T′, the edge search cost of T′equals OPT (T).\nProof. Consider algorithm Afor vertex search in Tbased on the edge search algorithm on T′.A\nqueries only the oldvertices. If the queried vertex is the target Astops. Otherwise, Acontinues\nin the subgraph that can still contain the target, until either the target vertex is queried or the\nsubtree is reduced to a single vertex.\n•If target is one of the old/original vertices, it will be found by reducing Tto a single vertex -\nthe search target. The cost in worst case is OPT(T’).\n•If target vertex is one of the new vertices, Awill continue the search until it queries the search\ntarget. Note that every edge of T’ is queried when searching for some target vertex in T’.\nThe cost of searching such target in worst case is OPT(T’).\nIt has been shown in [10] that the edge search is (strongly) NP-complete in the class of spiders\nof diameter at most 6. For each such spider T′, the diameter of its subdivision Tequals 12, and\nthe corresponding cost function ωis 4-monotonic. This proves Theorem 1.4.\n10 Conclusions\nOur first remark is on a possible complexity gap with respect to k. It is not known whether the\nproblem for 1-monotonic cost functions is NP-complete (cf. [20]). Hence an open problem remains\nwhether the value of k= 4 in Theorem 1.4 is in fact tight. We conjecture that the problem is\nNP-complete for a k <4. Our conjecture is supported by the fact that we obtained the constant\nk= 4 basically via a ‘black-box’ transformation from the edge search [10]. Thus, a more refined\nreduction might provide the right threshold for k.\nIn a wider context, our work is en route to a search for a constant factor approximation algorithm\nfor an arbitrary cost function in trees. Our choice of looking at k-monotonic cost functions may\nbe seen as the problem parametrization with respect to k. However, it is interesting if other\nparametrization methods are also natural and algorithmically promising.\n24We recall a related tree search problem, called edge search , in which one performs queries on\nedges: each reply provides information which endpoint of the queried edge is closer to the target\n[3]. One can similarly define for example an up-monotonic cost function ω:E(T)→R+for the\nedge search by requiring that there exists a choice for the root rso that for any two edges {x, y}\nand{y, z}, ifxis closer to rthan y, then ω({x, y})≥ω({y, z}). We have outlined some evidence\nthat for unweighted trees, the edge search has been more challenging to analyze than the vertex\nsearch, both when it comes to finding good upper bounds (see [42] vs [16, 23, 33]) and optimal\nalgorithms (see [42] vs [38, 32]). In view of the relation between the two problems in their weighted\nvariants (see Section 9) it may be possible that the above does not carry out to this case. 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X-ray reflectometry confirmed the formation of a monolayer, while grazing inci-\ndence small-angle X-ray scattering revealed short-range lateral correlations with a characteristic length equal\nto the mean particle size. Remarkably, our findings indicated that at zero surface pressure, the particles\norganized into submicrometer clusters, merging upon compression to form a homogeneous layer. These layers\nwere subsequently transferred to a solid substrate using the Langmuir-Schaefer technique and further char-\nacterized via scanning electron microscopy and polarized neutron reflectometry. Notably, our measurements\nunveiled a second characteristic length in the lateral correlations, orders of magnitude longer than the mean\nparticle diameter, with polydisperse particles forming circular clusters densely packed in a hexagonal lattice.\nFurthermore, our evidence suggested that the lattice constant of this mesocrystal depended on the character-\nistics of the particle size distribution, specifically the mean particle size and the width of the size distribution.\nAdditionally, we observed internal size separation within these clusters, where larger particles were positioned\ncloser to the center of the cluster. Finally, polarized neutron reflectometry measurements provided valuable\ninsights into the magnetization profile across the layer.\nI. INTRODUCTION\nOrdered arrays of magnetic nanoparticles (MNPs) of-\nfer promising opportunities for innovative material de-\nsign, allowing for the fine-tuning of both individual MNP\nproperties (such as size, magnetic moment, volume, and\nanisotropy) and the interactions between MNPs. These\nopportunities can be harnessed to enhance the density of\nbit-patterned magnetic recording media1,2, develop novel\nsensors and optoelectronic devices3–5, and design cata-\nlyst surfaces6. Furthermore, the collective behavior of\nnanoparticle assemblies holds significant relevance for the\nfundamental study of self-ordering phenomena.\nVarious methods can be employed to create ordered\nsingle-crystalline monolayers of MNPs, and the choice of\nsynthesis conditions can significantly influence the out-\ncome. In the past decade, it has been demonstrated7–11\nthat highly ordered, large-area arrays of MNPs can\nbe assembled on the surface of water using the Lang-\nmuir technique. The interactions among nanoparticles\non the water surface are primarily governed by mag-\nnetic dipole-dipole forces, steric effects, and Van der\nWaals forces. These interactions are intricately linked\nto the properties of the surfactant shell, nanoparti-\ncle volume, and interparticle distance. Recent studies\na)Electronic mail: victor.ukleev@helmholtz-berlin.de\nb)Electronic mail: avorobiev@ill.frhave shown that the ordering of monodisperse MNPs\nis strongly dependent on their size and magnetic mo-\nment. For instance, 10 nm iron oxide particles read-\nily form large, highly ordered monolayers, whereas 15\nnm and 20 nm MNPs, as well as binary mixtures\nof 10 nm and 20 nm MNPs, fail to assemble into\nstable monolayers and instead form three-dimensional\nstructures12–14.Furthermore, long-range ordering can be\nobserved in some multicomponent instances15,16, even\nwithin polydisperse two-dimensional (2D) and three-\ndimensional (3D) nanoparticle assemblies17–20. Con-\nsequently, the self-organization of initially disordered\nsystems, such as polydisperse ferrofluids, through di-\nverse pathways holds the potential to facilitate the con-\ntrolled synthesis of mesocrystals for a wide range of\nmaterials21,22.\nBulk-sensitive small-angle scattering is a valuable tool\nfor investigating the bulk structural and magnetic prop-\nerties of mesoscopic systems23,24. Conversely, surface\nX-ray and neutron scattering techniques are well-suited\nfor both in-situ and ex-situ characterization of Langmuir\nmonolayers25. By employing a combination of X-ray re-\nflectometry (XRR), grazing-incidence small-angle X-ray\nscattering (GISAXS), and spin-polarized neutron reflec-\ntometry (PNR), it becomes feasible to explore the elec-\ntronic, nuclear, and magnetic structure and dynamics of\nthese intricate systems in three dimensions25,26.\nIn our current study, we leveraged the Langmuir tech-\nnique to fabricate arrays of polydisperse iron oxide MNPs\n1arXiv:2401.13768v1  [cond-mat.mtrl-sci]  24 Jan 2024FIG. 1. The distribution of the particle size Din the samples\nIO-T30 and IO-S26 (curves are normalized to have area equal\nto 1)\non the liquid surface of water. These assemblies were sub-\njected to in-situ surface X-ray scattering analysis. Specif-\nically, a specular XRR experiment at the air/liquid inter-\nface was conducted to investigate the Langmuir film pro-\nfile in the out-of-plane direction. Additionally, GISAXS\nwas employed to examine in-plane interparticle correla-\ntions. We complemented these structural measurements\nwith surface pressure-area isotherm recording and Brew-\nster angle microscopy (BAM). Furthermore, ex-situ scan-\nning electron microscopy (SEM) was carried out to in-\nspect the resulting film after its deposition on a solid\nsubstrate by means of Langmuir-Schaefer technique. To\ngain insights into the depth distribution of the magnetic\nmoment within the thin film post-deposition, we con-\nducted a PNR experiment.\nII. EXPERIMENTAL\nThe iron oxide (IO) nanoparticles used in this study\nwere synthesized through the chemical deposition of dis-\npersed magnetite27. The fabricated MNPs were coated\nwith a layer of sodium oleate, selected by weight, and dis-\npersed in chloroform. Information about the size distri-\nbution was obtained from wide-angle synchrotron diffrac-\ntion and transmission electron microscopy (TEM)28. The\nsize distribution followed a gamma distribution with a\nmean size value of D= 10 .8 nm and a dispersion of\nσ= 3.5 nm, achieved after subsequent centrifugation.\nThis sample is denoted as IO-T30.\nThe Langmuir films were prepared using a custom-\ndesigned Langmuir trough directly installed on the sam-\nple goniometer, with the support of an active antivi-\nbration device, Halcyonics MOD2-S. The maximum and\nminimum subphase surface areas were 456 cm ²and 115\ncm², respectively. After spreading the nanoparticles, the\nsolvent was allowed to evaporate for 15 minutes before\nmeasurement. The surface pressure was continuously\nFIG. 2. (a) Surface pressure isotherms obtained for polydis-\nperse IO-T30 particles spread on water. Grey vertical line\nindicates value of a nominal area occupied by a particle of\nradius rp= 5 nm covered with a surfactant layer of thickness\nls= 2.5 nm. BAM images at (b) 1 mN/m, (c) film relaxed\nto 20 mN/m after original compression to 35 mN/m, (d) af-\nter decompression to 6 mN/m. The bright areas correspond\nto the particles, and the dark areas correspond to the water\nsurface.\nmonitored and recorded during the Langmuir film for-\nmation using a Wilhelmy plate made of Whatman paper\nand a microbalance, Model PS4 from Nima Technology\nLtd.\nFor ex-situ observation of the monolayers forming di-\nrectly on the water surface, a Brewster Angle Microscope\nequipped with a CCD camera and 10 ×and 20 ×lenses\nwas employed, along with a separate Langmuir trough.\nSEM measurements of the MNP films transferred onto\nsolid substrates via the Langmuir-Schaefer method were\nconducted using a LEO 1530 microscope at ESRF.\nThe in-situ XRR and GISAXS measurements on the\nliquid surface were conducted at the ID10B beamline at\nESRF in Grenoble, France. The beam size was 300 ×100\nµm2(horizontal ×vertical), and the beam energy was 8\nkeV, corresponding to a wavelength of λ= 1.54˚A. Two-\n2FIG. 3. (a) Experimental XRR data for IO-T30 particles\non water surface (gray symbols) together with the best fit\nmodel curves (black lines) together with corresponding refer-\nence data for water (blue symbols and curve). (b) Best fit\nmodel SLD profiles as obtained at different pressure values.\ndimensional PILATUS 300K and linear Vantec detectors\nwere utilized for the GISAXS and XRR measurements,\nrespectively.\nEx-situ PNR experiments on the IO-T30 sample,\nwhich had been transferred onto a Si substrate, were\nperformed using the neutron reflectometer Super ADAM\nat the Institut Laue-Langevin in Grenoble, France29.\nA monochromatic neutron beam with a wavelength of\nλ= 5.2˚A and the incoming polarization of P0= 99.8%\nwas employed. The intensity of the scattered neutron\nbeam was detected using a two-dimensional3He detector\nDENEX 300TN. A magnetic field of H= 7 kOe, applied\nin the sample plane, was generated by an electromagnet.\nFIG. 4. (a) Two-dimensional GISAXS pattern obtained from\nIO-T30 sample on water surface at Π = 34 mN/m (a). (b)\nEvolution of the GISAXS intensity along the cut taken at\nqz=qcin the course of compression: dashed line 10 mN/m,\nsolid line 34 mN/m (d). Note log-log scale.\nIII. RESULTS AND DISCUSSION\nPolydisperse IO-T30 nanoparticles were deposited in\nthe Langmuir trough for film formation. The pressure-\narea isotherm for this sample, recorded during the com-\npression process, is presented in Figure 2. Following the\nHarkins notation30, this curve exhibits three distinct re-\ngions: the liquid expanded and gaseous state (LE+G) at\npressures Π = 0 to 12 mN/m, the liquid condensed state\n(LC) at Π = 13 to 25 mN/m, and the solid state (S)\nbeyond 25 mN/m. The LC to S transition occurs when\nthe available area for one particle reaches the calculated\nvalue of the area occupied by a particle with the average\nsize of 10 nm.\nBAM image (Fig. 2b) depict the formation of solid\nclusters immediately after deposition (Fig. 2b), corre-\nsponding to the LE+G state. Subsequently, a homoge-\nneous coverage of the water surface is observed in the\nrelaxed ensemble at Π = 20 mN/m, following an initial\ncompression to 35 mN/m (LC) (Fig. 2c). Further relax-\n3FIG. 5. (a,b) SEM images of IO-T30 layer with different magnifications. (c) Corresponding FFT image showing two contribu-\ntions: individual polydisperse MNPs and orientationally-disordered mesocrystal of the nanoparticle clusters. (d) Magnification\nof the low- qpart of the FFT pattern. (e) Radially averaged FFT showing two characteristic correlation lengths.\nation leads to the formation of the hashed structure at\nΠ = 5 mN/m (LE) state, as shown in Figure 2d.\nSignificantly, the surface pressure isotherm exhibited\nby the polydisperse sample can be viewed as a super-\nposition, containing the distinctive features observed in\nthe isotherms of monodisperse nanoparticles as reported\nin Ref.12. Consequently, the S regime corresponds to\nsmall particles with an approximate size of 10 nm, the\nLC regime aligns with medium-sized particles of around\n15 nm, and the LE+G regime corresponds to larger parti-\ncles with an approximate size of 20 nm. This comparative\nanalysis provides valuable insights into the surface behav-\nior of polydisperse mixtures, revealing how different size\nfractions contribute to the overall self-organization dy-\nnamics, drawing parallels with the behavior observed in\nmonodisperse systems.\nThe XRR curves, along with the corresponding elec-\ntron scattering length density (SLD) profiles for the IO-\nT30 sample, obtained from the fit, are presented in Fig-\nure 3. As an example, the experimental XRR curve from\nthe pure air/water interface, fitted with the Fresnel de-\ncay function, is also included. Due to the particle size\ndistribution in the polydisperse sample, the roughness\nof the layer is of the same order of magnitude as thelayer thickness. As a result, no thickness oscillations are\nobserved in the XRR curve at the low coverage regime\n(LE+G), owing to the rapid decay of the reflected inten-\nsity induced by the roughness. The first Kissieg fringe is\nevident in the XRR curve at pressure values of Π = 24\nmN/m and 34 mN/m when the layer reaches the S state\nand becomes more homogeneous. The model incorpo-\nrates a layer with a wide distribution of electron density,\ncorresponding to the size distribution of the nanoparti-\ncles (Fig. 3b). Further compression of the layer leads to\nan increase in the SLD, indicating the elimination of gaps\nbetween the nanoparticles, while the total film thickness\nremains constant, and no transition to a bi- or multilay-\nered structure (hashing) is observed.\nIn-plane correlations of the polydisperse IO-T30 sam-\nple were studied using GISAXS (Fig. 3c) at a constant\nincident angle αi= 0.13◦, which is slightly below the\ncritical angle αc= 0.15◦, at two surface pressures of\nΠ = 10 mN/m and Π = 44 mN/m. The scattered inten-\nsity exhibits a nearly homogeneous decay along the q||\nvector without the appearance of Bragg peaks. Images\nat both pressures suggest the absence of inter-particle\ncorrelations in the Langmuir film, primarily due to the\nhigh polydispersity of the nanoparticles. To determine\n4FIG. 6. Two-dimensional experimental PNR intensity for IO-T30 particles deposited onto Si wafer as a function of the projection\nof the incoming and outgoing wavevectors normal to the sample plane ( pi= (2π/λ) sinαiandpf= (2π/λ) sinαf, where αi\nandαfare incident and scattering angles, respectively) (a). Integrated specular reflectivity intensities measured with neutron\npolarization parallel ( R+) and antiparallel ( R−) to the applied magnetic field H= 7 kOe as a function of momentum transfer\nvector qz(b). Solid lines represents the fitted curves according to the reconstructed nuclear and magnetic SLD profiles of the\nfilm shown in panel (c).\nthe mean in-plane interparticle distance parameter a, the\nKrattky representation I·q2\n||of the intensity along the\ncut taken at qz=qcwas utilized (Fig. 3d). The mean in-\nplane interparticle distance was found to be a= 14.9(1)\nnm, with an average coherent domain size of D= 230\nnm.\nFigures 5a and b depict the IO-T30 sample deposited\non a gold thiol-treated surface from the Langmuir layer at\nΠ = 34 mN/m. The SEM images reveal a non-uniform,\ndisordered layer of three-dimensional clusters of nanopar-\nticles with varying sizes. The large-scale image (Fig.\n5b) illustrates that larger MNPs aggregate into clusters\nwith an average size of 200 nm, surrounded by smaller\nparticle domains. This size separation is likely driven\nby magnetic dipolar interactions, favoring the agglom-\neration of MNPs with larger volume and, consequently,\ngreater magnetic moment. This magnetically-induced\nself-separation mechanism has been previously proposed\nfor simplified systems consisting of binary assemblies of\nmonodisperse nanoparticles13,14. Interestingly, the clus-\nters form a short-range ordered mesocrystal on the scale\nof hundreds of nanometers, visible in the real-space im-\nage (Fig. 5a) but not in the GISAXS pattern, as the in-\nstrument configuration was tuned to detect signals from\nsmaller objects at higher scattering angles. The mean in-\nterparticle distance a= 17.7 nm, derived from the radial\naverage of the fast Fourier transformed (FFT) SEM im-\nage (Fig. 5c), is in reasonable agreement with the in-situ\nexperiment results, accounting for the local nature of the\nSEM probe.\nThe manifestation of large ( >100 nm) ferromagnetic\ndomains in the ordered nanostructures of iron oxide\nnanoparticles has been recently revealed through neutron\nscattering31–34, resonant soft X-ray scattering35, and off-\naxis electron holography36. In this study, PNR was ap-plied to investigate the out-of-plane distribution of mag-\nnetization within the nanoparticle film after deposition\nonto a solid substrate. Previous reports have unequiv-\nocally demonstrated the effectiveness of this approach,\neven in the case of very thin films, such as monolayers\nof MNPs with a 10 nm diameter37. Theoretical descrip-\ntions and experimental details can be found in Ref. 38.\nThe two-dimensional PNR ( pi, pf) map obtained from\nthe IO-T30 sample is presented in Figure 6a, featuring\nthe specular and off-specular contributions of neutron re-\nflection from the film. The integrated specular reflection\ncurves R+(−)(qz), measured with opposite neutron po-\nlarizations in the saturating magnetic field H= 7 kOe\napplied in the sample plane, were fitted using the Parratt\nalgorithm within the GenX software package39,40(Fig.\n6b).The magnitude of the applied field is sufficient to\nfully saturate the magnetization in the sample plane41.\nThe PNR curves were fitted with the same model as the\nXRR. The model allowed a reduced nuclear SLD ρncom-\npared to the in-situ X-ray study in order to account for\npossible density reduction of the film during the transfer\nfrom the water subphase to the solid substrate42. The\nmagnetic SLD part ρmwas allowed to vary freely.\nAs a result of the fitting routine, nuclear and mag-\nnetic SLD profiles were obtained, as shown in Figure\n6c. Similar to the electron density profiles obtained\nin the XRR study, the nuclear and magnetic densities\nof the film are primarily concentrated in an approxi-\nmately 15 nm-thick layer above the substrate ( z= 0).\nThe round-shaped profile with a maximum at z≈6.3\nnm from the Si surface corresponds to the center of\nthe MNPs layer. The magnetic SLD is directly pro-\nportional to the in-plane net magnetization component\nM[kA/m] = 3 .5·109ρm[˚A−2]38. According to this equa-\ntion, the peak value of magnetization is M= 450 ±24\n5FIG. 7. SEM image of IO-S26 sample showing a large cluster\nof polydisperse MNPs.\nemu/cm3, matching the expected value for magnetite\n(480 kA/m3)43. Therefore, the shape of the neutron\nSLD confirms the chemical depth profile obtained from\nXRR and the planar homogeneity of the film observed\nby SEM. The magnetic depth profile reflects the original\nparticle size distribution in the ferrofluid, with magne-\ntization values enhanced compared to the monodisperse\ncase of IO-1037. The absence of any notable peaks in the\noff-specular PNR signal (Fig. 6a), which is sensitive to\nlateral correlations with sizes from a few hundred nm to\na few microns, implies that there is no long-range order\nformed by the mesoscale clusters.\nPrevious studies have shown that formation of mixed\nclusters of nanoparticles is energetically favourable in bi-\nnary mixtures of monodisperse MPNs as a result of the\ninterplay between magnetic dipolar, van der Waals in-\nteractions, and steric repulsion13,14. Similar mechanisms\nshould also apply to the polydisperse mixtures with com-\nparable mean particle sizes. Notably, within the poly-\ndisperse ensemble featuring slightly larger average parti-\ncle sizes, such as in the case of S26 (Fig. 1a), the phe-\nnomenon of size-separated mesoscale structures becomes\neven more pronounced. As illustrated in Fig. 7, the\nself-organization of micrometer-scale clusters of MNPs is\nevident, with larger particles concentrated in the cen-\ntral region and smaller ones at the periphery. We hy-\npothesize that the increased magnetic moment of this\nassembly plays a crucial role in fostering the formation\nof these more substantial clusters, distinguishing it from\nthe behavior observed in the T30 configuration. This\nobservation underscores the significant influence of both\nparticle size and magnetic characteristics on the intricate\nself-assembly processes within polydisperse nanoparticle\nensembles.\nRecent studies have unveiled that small monodis-\nperse self-assembled particles with diameters less than or\nequal to 20 nm tend to exhibit not only disordered (su-\nperparamagnetic) behavior but also anti-ferromagnetic\nordering33,35,44,45. Consequently, further exploration of\nmesoscopic magnetic textures that may emerge atop the\nstructural clusters in polydisperse samples represents an\nintriguing avenue for future research.IV. CONCLUSION\nThe observations made in this study provide valuable\ninsights into the self-organization of polydisperse iron\noxide nanoparticles on a water surface. While previ-\nous research has primarily focused on the self-ordering of\nmonodisperse nanoparticles, the behavior of polydisperse\nsystems adds complexity to the understanding of these\nprocesses. In contrast to the self-ordering observed in\nmonodisperse MNPs reported in Ref.12, where 10 nm, 15\nnm, and 20 nm MNPs form in-plane ordered structures,\nthe polydisperse mixture IO-T30 tends to assemble into a\nmore homogeneous amorphous monolayer. Furthermore,\na short-range ordered mesocrystal of larger particle ag-\nglomerates is formed due to a magnetically-driven self-\nseparation process, similar to what has been previously\ninvestigated in the case of binary mixtures13,14. The fact\nthat polydisperse particles tend to form a more homo-\ngeneous amorphous monolayer suggests that size vari-\nations play a significant role in the self-assembly pro-\ncess. This observation has practical implications for\nthe controlled synthesis of mesocrystals, as it highlights\nthe need to consider particle size distributions when de-\nsigning nanoparticle-based materials. Formation of the\nmesoscale clusters of MNPs generates another degree of\nfreedom that can be used to control the self-assembly, for\nexample, by aligning the clusters into long-range ordered\nstructures using external magnetic fields46–48.\nThese findings contribute to our understanding of self-\nassembly processes in complex nanoparticle systems. The\ninsights gained here can inform the design and synthe-\nsis of materials with tailored properties, opening up new\npossibilities for applications in areas such as magnetic\ndata storage, sensing, and catalysis.\nFurther investigations into the magnetic textures that\nmay arise within polydisperse nanoparticle assemblies\npromise exciting prospects for future research. Under-\nstanding the interplay between structural and magnetic\nproperties in these systems could lead to the development\nof novel materials with unique functionalities.\nACKNOWLEDGMENTS\nAuthors thank European Synchrotron Radiation Fa-\ncility and Institut Laue-Langevin for provided beamtime\nand technical assistance.\nAUTHOR DECLARATIONS\nConflict of Interest\nThe authors have no conflicts to disclose.\n6Author Contributions\nV.U., A.K., I.S., O.K., A.V performed experiments,\nV.U., A.V. analyzed the data and wrote the manuscript,\nV.U., O.K., and A.V. jointly conceived the project. All\nauthors read and edited the manuscript.\nDATA AVAILABILITY STATEMENT\nData is available from V.U. and A.V. upon request.\nREFERENCES\n1S. Sun, C. Murray, D. Weller, L. Folks, and A. Moser,\n“Monodisperse FePt nanoparticles and ferromagnetic\nFePt nanocrystal superlattices,” Science 287, 1989–\n1992 (2000).\n2S. Kinge, M. Crego-Calama, and D. N. 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To\nour knowledge, this is the first characterization of facet-defining sequence-independent\nliftings that are efficiently computable from the underlying cover. Finally, we demon-\nstrate via experiments that PC lifting can be a useful alternative to GNS lifting. We\ntest our new lifting techniques atop a number of novel cover cut generation routines,\nwhich prove to be effective in experiments with CPLEX.\n1 Introduction\nLifting is a technique for strengthening cutting planes for integer programs by increasing the\ncoefficients of variables that are not in the cut. We study lifting methods for valid cuts of\nknapsack polytopes , which have the form conv( P) where\nP=(\nx∈ {0,1}n:nX\nj=1ajxj≤b)\nfora1, . . . , a n, b∈Nwith 0 < a 1, . . . , a n≤b. We interpret Pas the set of feasible packings of\nnitems with weights a1, . . . , a ninto a knapsack of capacity b. Such knapsack constraints arise\nin binary integer programs from various industrial applications such as resource allocation,\n∗Computer Science Department, Carnegie Mellon University. sprasad2@cs.cmu.edu\n†Management Science & Engineering and Computer Science Departments, Stanford University.\nvitercik@stanford.edu\n‡Computer Science and Machine Learning Departments, Carnegie Mellon University.\nninamf@cs.cmu.edu\n§Computer Science Department, Carnegie Mellon University, Strategy Robot, Inc., Strategic Machine,\nInc., Optimized Markets, Inc. sandholm@cs.cmu.edu\n1arXiv:2401.13773v1  [math.OC]  24 Jan 2024auctions, and container packing. They are a very general and expressive modeling tool, as\nany linear constraint involving binary variables admits an equivalent knapsack constraint\nby replacing negative-coefficient variables with their complements. A minimal cover is a set\nC⊆ {1, . . . , n }such thatP\nj∈Caj> bandP\nj∈C\\{i}aj≤bfor all i∈C. That is, the items\ninCcannot all fit in the knapsack, but any proper subset of Ccan. The minimal cover cut\ncorresponding to Cis the inequality\nX\nj∈Cxj≤ |C| −1,\nwhich enforces that the items in Ccannot all be selected. A lifting of the minimal cover cut\nis any valid inequality of the form\nX\nj∈Cxj+X\nj /∈Cαjxj≤ |C| −1. (1)\nThe lifting coefficients αjare often computed one-by-one—a process called sequential lifting\nthat depends on the lifting order. Sequential lifting can be expensive since one must solve an\noptimization problem for each coefficient. Furthermore, one must reckon with the question\nof what lifting order to use. To lessen this computational burden, the lifting coefficients can\nbe computed simultaneously. This method is called sequence-independent lifting and is the\nfocus of this work. Our contributions include: (i) a generalization of the seminal sequence-\nindependent lifting method developed by Gu et al. [11] and a correction of their proposed\ngeneralization; (ii) the first broad conditions under which sequence-independent liftings that\nare efficiently computable from the underlying cover—via our new techniques—define facets\nof conv( P); and (iii) new cover cut generation methods that, together with our new lifting\ntechniques, display promising practical performance in experiments.\n1.1 Preliminaries on sequence-independent lifting\nWe begin with an overview of the lifting function f: [0, b]→Rassociated with a minimal\ncover C, defined by\nf(z) =|C| −1−max(X\nj∈Cxj:X\nj∈Cajxj≤b−z, xj∈ {0,1})\n.\nFori /∈C, the value f(ai) is the maximum possible coefficient αisuch thatP\nj∈Cxj+αixi≤\n|C| −1 is valid for conv( P). The lifting function has a more convenient closed form due to\nBalas [1]. First, relabel the items so C={1, . . . , t}anda1≥ ··· ≥ at. Let µ0= 0 and for\nh= 1, . . . , t letµh=a1+···+ah. Let λ=µt−b >0 be the cover’s excess weight. Then,\nf(z) =(\n0 0≤z≤µ1−λ\nh µ h−λ < z ≤µh+1−λ.\nThe lifting function has an intuitive interpretation: f(z) is the maximum hsuch that an\nitem of weight zcannot be brought into in Cand fit in the knapsack, even if we are allowed\n2to discard any hitems from C. The lifting function fmay be used to maximally lift a single\nvariable not in the cover. To lift a second variable, a new lifting function must be computed.\nThis (order-dependent) process can be continued to lift all remaining variables, and is known\nassequential lifting . Conforti et al. [6] and Hojny et al. [13] contain further details.\nSuperadditivity and sequence-independent lifting. A function g:D→Ris super-\nadditive if g(u+v)≥g(u) +g(v) for all u, v, u +v∈D. Ifg≤fis superadditive,\nX\nj∈Cxj+X\nj /∈Cg(aj)xj≤ |C| −1\nis a valid sequence-independent lifting for conv( P). This result is due to Wolsey [25]; Gu\net al. [11] generalize to mixed 0-1 integer programs. The lifting function fis generally\nnot superadditive. Gu et al. [11] construct a superadditive function g≤fas follows. Let\nρh= max {0, ah+1−(a1−λ)}be the excess weight of the cover if the heaviest item is replaced\nwith a copy of the ( h+1)-st heaviest item. For h∈ {0, . . . , t−1}, letFh= (µh−λ+ρh, µh+1−λ]\nand for h∈ {1, . . . , t −1}, let Sh= (µh−λ, µ h−λ+ρh].Shis nonempty if and only if\nρh>0. For w: [0, ρ1]→[0,1], Gu et al. define\ngw(z) =\n\n0 z= 0\nh z ∈Fh h= 0, . . . , t −1\nh−w(µh−λ+ρh−z)z∈Sh h= 1, . . . , t −1.\nGu et al. prove that for w(x) =x/ρ 1,gwis superadditive. We call this particular lifting\nfunction the Gu-Nemhauser-Savelsbergh (GNS) lifting function . Furthermore, gwis undom-\ninated, that is, there is no superadditive g′with f≥g′≥gwandg′(z′)> g w(z′) for some\nz′∈[0, b].\n1.2 Our contributions\nIn Section 2, we prove that under a certain condition, gwis superadditive for any linear sym-\nmetric function w. This generalizes Gu et al.’s result [11] for w(x) =x/ρ 1and furthermore\ncorrects an error in their proposed generalization, which incorrectly claims wcan be any\nsymmetric function. Of particular interest is the constant function w= 1/2; we call the re-\nsulting lifting piecewise-constant (PC) lifting . In Section 2.1 we give a thorough comparison\nof PC and GNS lifting. We show that GNS lifting can be arbitrarily worse than PC lifting,\nand characterize the full domination criteria between the two methods. In Section 2.2, we\nprovide a broad set of conditions under which PC lifting defines facets of conv( P).To our\nknowledge, these are the first conditions for facet-defining sequence-independent liftings that\nare efficiently computable from the underlying cover.1\nIn Section 3, we experimentally evaluate our lifting techniques in conjunction with a\nnumber of novel cover cut generation techniques. Our cut generation techniques do not solve\n1Balas [1] proved that under certain conditions, (sequential) lifting coefficients are fully determined in-\ndependent of the lifting order. This can be viewed as a set of sufficient conditions under which sequence-\nindependent (or sequential; they are one and the same here) lifting would yield a facet-defining cut. However,\nwhen sequence-independent lifting can be non-trivially performed, ours is the first such result.\n3expensive NP-hard separation problems (which has been the norm in prior research [15]).\nInstead, we cheaply generate many candidate cover cuts based on qualitative criteria, lift\nthem, and check for separation only before adding the cut. This approach is effective in\nexperiments with CPLEX.\n1.3 Related work\nCover cuts and their associated separation routines were first shown to be useful in a branch-\nand-cut framework by Crowder et al. [7]. Since then, there has been a large body of work\nstudying various computational aspects, both theoretical and practical, of cover cuts, sep-\naration routines, and lifting. The seminal work of Gu et al. [11] showed how sequence-\nindependent lifting can be performed efficiently using gwforw(x) =x/ρ 1. Gu et al. [9]\nperform a computational study of sequential lifting, and Wolter [26] presents some computa-\ntional results on the interaction between the sequence-independent lifting technique of Gu et\nal. [11] and different separation techniques. To our knowledge, this is the only computational\nstudy of sequence-independent lifting published to date. Our computational study takes a\ndifferent approach than prior work. Rather than solving separation problems exactly, which\ninvolves expensive optimization, we generate large pools of candidate cover cuts, lift them,\nand check for separation before adding cuts to the formulation. This approach proves to\nbe effective in our experiments. (The separation problem is NP-hard [10, 16], but checking\nseparation is a trivial linear time operation. More on separation can be found in Kaparis\nand Letchford [15].) Marchand et al. [21] and Letchford and Souli [17, 18] present other\nsequence-independent lifting functions based on superadditivity.\n2 New sequence-independent lifting functions: struc-\nture and properties\nWe generalize the result of Gu et al. [11] and also point out an error in their suggested\ngeneralization. Gu et al. claim that if µ1−λ≥ρ1, then gwis superadditive for any\nnondecreasing w: [0, ρ1]→[0,1] such that w(x) +w(ρ1−x) = 1. This claim is incorrect (we\nprovide counterexamples in Appendix A). We show that this claim is correct when restricted\nto linear w.\nTheorem 2.1. Fork∈[0,1/ρ1], letwk(x) =kx+1−kρ1\n2, and let gk=gwk. Ifµ1−λ≥ρ1,\ngkis superadditive and undominated.\nThe GNS lifting function is given by g1/ρ1. The proof of Theorem 2.1 follows the proof\nthat g1/ρ1is superadditive [11] with a few key modifications; we defer it to Appendix B. If\nµ1−λ < ρ 1,gkis superadditive if and only if k= 1/ρ1.\nOf particular interest is the superadditive lifting function g0, which we refer to as the\npiecewise-constant (PC) lifting function . The following result shows that the lifting obtained\nviagkis dominated by the union of the liftings obtained via g0(PC lifting) and g1/ρ1(GNS\nlifting). Thus, in order to get as close to conv( P) as possible, it suffices to study these two\nlifting functions.\n4//Figure 1: The PC lifting function g0is the piecewise constant step function depicted by the\nsolid black lines. The GNS lifting function g1/ρ1is obtained by replacing the solid lines in\nthe intervals Shwith the depicted dashed lines. If all coefficients of variables being lifted lie\nin the blue and red regions with at least three coefficients in the leftmost blue region, PC\nlifting is facet-defining and dominates GNS lifting (Theorem 2.8).\nProposition 2.2. Letk∈(0,1/ρ1). IfP\nj∈Cxj+P\nj /∈Cg0(aj)xj≤ |C| −1andP\nj∈Cxj+P\nj /∈Cg1/ρ1(aj)xj≤ |C| −1, thenP\nj∈Cxj+P\nj /∈Cgk(aj)xj≤ |C| −1.\nProof. We have gk(z) =kρ1g1/ρ1(z) + (1 −kρ1)g0(z) by direct computation, so gklifting is\na convex combination of GNS and PC lifting.\nExample 2.3. LetC={1,2,3,4}and consider a knapsack constraint of the form 16 x1+\n14x2+ 13x3+ 9x4+P\nj /∈Cajxj≤44.Cis a minimal cover with µ1= 16, µ2= 30, µ3= 43,\nµ4= 48, λ= 8,ρ1= 6,ρ2= 5,ρ3= 1, and µ1−λ≥ρ1. Fig. 1 depicts g0andg1/ρ1truncated\nto the domain [ µ1−λ, µ 3−λ] = [8 ,35].\n2.1 Comparisons between PC and GNS lifting\nThe following result shows GNS lifting can be arbitrarily worse than PC lifting.\nProposition 2.4. For any ε >0, t∈Nthere exists a knapsack constraint with a minimal\ncover Cof size tsuch that PC lifting yields\nX\nj∈Cxj+X\nj /∈C1\n2xj≤ |C| −1\nand GNS lifting is dominated by\nX\nj∈Cxj+X\nj /∈Cεxj≤ |C| −1.\nThe proof is in Appendix C. At a high level, we construct an instance where the length of\nS1, which is ρ1, is large, and consider coefficients that are at the leftmost part S1. GNS barely\nlifts such coefficients, while PC yields lifting coefficients of 1 /2. The next proposition fully\ncharacterizes the domination criteria between PC and GNS lifting. Its proof is immediate\nfrom the plots in Fig. 1.\n5Proposition 2.5. Assume µ1−λ≥ρ1. Furthermore, suppose {j /∈C:∃hs.t.aj∈Sh} ̸=∅\n(else, GNS and PC trivially produce the same cut). If, for all j /∈C,\n1.aj∈Sh=⇒ρh>ρ1\n2andaj≤µh−λ+ρh−ρ1\n2with at least one such aj∈Sh\nsatisfying aj< µ h−λ+ρh−ρ1\n2, PC strictly dominates GNS.\n2.aj∈Sh=⇒ρh>ρ1\n2andaj=µh−λ+ρh−ρ1\n2, PC and GNS yield the same cut.\n3.aj∈Sh=⇒(ρh≤ρ1\n2) or ( ρh>ρ1\n2andaj> µ h−λ+ρh−ρ1\n2), GNS strictly dominates\nPC.\n4. Otherwise, neither PC nor GNS dominates the other.\nExample 2.6. Consider the constraint 16 x1+ 14x2+ 13x3+ 9x4+a5x5+a6x6+a7x7≤44\nwith minimal cover C={1,2,3,4}.\n1. Let a5= 9,a6= 10, a7= 23. GNS yields\nx1+x2+x3+x4+1\n6x5+1\n3x6+4\n3x7≤3.\nPC yields the dominant cut\nx1+x2+x3+x4+1\n2x5+1\n2x6+3\n2x7≤3.\n2. Let a5= 11, a6= 17, a7= 24. GNS and PC both yield the cut\nx1+x2+x3+x4+1\n2x5+x6+3\n2x7≤3.\n3. Let a5= 12, a6= 13, a7= 26. GNS yields\nx1+x2+x3+x4+2\n3x5+5\n6x6+11\n6x7≤3.\nPC yields the weaker cut\nx1+x2+x3+x4+1\n2x5+1\n2x6+3\n2x7≤3.\n4. Let a5= 9,a6= 13, a7= 24. GNS yields\nx1+x2+x3+x4+1\n6x5+5\n6x6+3\n2x7≤3.\nPC yields\nx1+x2+x3+x4+1\n2x5+1\n2x6+3\n2x7≤3.\nNeither cut is dominating.\nOpen question: Gu et al. [10] and Hunsaker and Tovey [14] construct knapsack problems\nwhere branch-and-cut requires a tree of exponential size, even when all lifted cover inequal-\nities (all inequalities of the form (1)) are added to the formulation. Do there exist knapsack\nproblems where branch-and-cut requires exponential-size trees when all GNS-lifted cover\ninequalities are added, but does not when PC-lifted cover inequalities are added (or vice\nversa)?\n62.2 Facet defining inequalities from sequence-independent lifting\nWe now provide a broad set of sufficient conditions under which PC lifting yields facet-\ndefining inequalities. Our result relies on the following complete characterization of the\nfacets of the knapsack polytope obtained from lifting minimal cover cuts, due to Balas and\nZemel [2, 13], which we restate using the notation and terminology of Gu et al. [11] and\nConforti et al. [6, sec. 7.2]. Given a minimal cover Candj /∈C, leth(j) be the index such\nthatµh(j)≤aj< µ h(j)+1.\nTheorem 2.7 (Balas and Zemel [2]) .LetCbe a minimal cover. Let J={j /∈C:aj>\nµh(j)+1−λ}and let I= ({1, . . . , n } \\C)\\J. LetQ(J) ={Q⊆J:P\nj∈Qaj≤b, Q̸=∅}.\nThe inequalityX\nj∈Cxj+X\nj /∈Cαjxj≤ |C| −1\nis a facet of conv( P)if and only if αj=h(j) +δj·1(j∈J)where each δj∈[0,1]and\nδ= (δj)j∈Jis a vertex of the polyhedron\nT=(\nδ∈R|J|:X\nj∈Qδj≤f\u0010X\nj∈Qaj\u0011\n−X\nj∈Qh(j)∀Q∈Q(J))\n.\nThe characterization of facets based on Tin Theorem 2.7 does not translate to a tractable\nmethod of deriving facet-defining inequalities, since one would need to enumerate the vertices\nofT. In fact, Hartvigsen and Zemel [12] show the question of determining whether or not\na cutting plane is a valid lifted cover cut is co-NP -complete. Deciding whether or not a\ncutting plane is a facet-defining lifted cover cut is in DP(DPis a complexity class introduced\nby Papadimitriou and Yannakakis [22] to characterize the complexity of facet recognition).\nCritically, these complexity results hold when the underlying cover is given as input.\nOur result, to the best of our knowledge, provides the first broad set of sufficient con-\nditions under which sequence-independent liftings that can be efficiently computed from the\nunderlying minimal cover—via PC lifting—are facet defining. We now state and prove our\nresult.\nTheorem 2.8. LetC={1, . . . , t},a1≥ ··· ≥ at, be a minimal cover such that µ1−λ≥\nρ1>0. Suppose |{j /∈C:aj∈S1}| ≥ 3and for all j /∈C:\naj∈Sh=⇒ρh>ρ1\n2andaj≤µh−λ+ρh−ρ1\n2,\naj∈Fh=⇒aj≥µh(equivalently h(j)≥h).\nThen, the cutX\nj∈Cxj+X\nj /∈Cg0(aj)xj≤ |C| −1,\nobtained via PC lifting, defines a facet of conv( P).\nProof. First we show that J=∪h≥1{j /∈C:aj∈Sh}andI=∪h≥0{j /∈C:aj∈Fh}. Let\nj /∈Cbe such that aj∈Sh. We have aj> µ h−λ > µ h−1(asλ≤aifor any i∈C) and\n7aj≤µh−λ+ρh−ρ1\n2< µ h−λ+ρh< µ h(the final inequality follows directly from expanding\noutµhandρh). So h(j) =h−1, and as aj> µ h−λ=µh(j)+1−λ,j∈J. Next, let j /∈Cbe\nsuch that aj∈Fh. Then, aj≤µh+1−λ < µ h+1, and by assumption aj≥µh, soh(j) =h.\nTherefore aj≤µh(j)+1−λ, and so j∈I. The facts that aj∈Sh=⇒h(j) =h−1 and\naj∈Fh=⇒h(j) =hwill be used repeatedly in the remainder of the proof.\nWe now determine the constraints defining the polyhedron T⊂R|J|in Theorem 2.7. (For\nj∈I, PC lifting produces a coefficient of h(j), as required by Theorem 2.7.) Partition Jinto\ntwo sets J=J1∪J>1where J1={j /∈C:aj∈S1}andJ>1=∪h>1{j /∈C:aj∈Sh}. Each\nsingleton Q={j} ∈Q(J) yields the constraint δj≤1. Consider now Q={j1, j2} ∈Q(J).\nWe consider two cases, one where j1∈J1and the other where j1∈J>1. First, let j1∈J1.\nLethbe such that aj2∈Sh. We have aj1+aj2> a j2> µ h−λ, sof(aj1+aj2)≥h, and\naj1+aj2≤µ1−λ+ρ1\n2+µh−λ+ρh−ρ1\n2\n=a1−λ+µh−λ+ (ah+1−a1+λ)\n=µh+ah+1−λ=µh+1−λ,\nsof(aj1+aj2)≤h. Therefore f(aj1+aj2) =h, and we get the constraint δj1+δj2≤\nf(aj1+aj2)−h(j1)−h(j2) =h−0−(h−1) = 1. Suppose now that j1, j2∈J>1(if\nj2∈J1that is handled by the first case). Let h1, h2be the integers such that aj1∈Sh1and\naj2∈Sh2. We have aj1+aj2> µ h1−λ+µh2−λ= (a1+···+ah1) + (a1+···+ah2)−2λ >\n(a1+···+ah1−1) + (a1+···+ah2)−λ > µ h1+h2−1−λsof(aj1+aj2)≥h1+h2−1, and\nf(aj1+aj2)−h(j1)−h(j2)≥1. So for such pairs, we obtain a constraint δj1+δj2≤H(j1, j2),\nwhere H(j1, j2)≥1. Finally, we consider Q∈Q(J) with |Q| ≥3. For j∈Qlethjbe the\ninteger such that aj∈Shj. We claim that\nX\nj∈Qaj> µP\nj∈Qhj−⌊|Q|/2⌋−λ.\nThis claim is most succinctly proven using the quantities defined to prove Theorem 2.1\n(originally used by Gu et al. [11] to prove superadditivity of g1/ρ1). To avoid notational\nclutter, we defer the proof of this claim to Appendix D. The claim implies f(P\nj∈Qaj)≥P\nj∈Qhj− ⌊|Q|/2⌋, so the constraint induced by Qis of the formP\nj∈Qδj≤H(Q), where\nH(Q) :=f\u0010X\nj∈Qaj\u0011\n−X\nj∈Qh(j)≥X\nj∈Qhj− ⌊|Q|/2⌋ −X\nj∈Q(hj−1) =⌈|Q|/2⌉.\nWe can now write down a complete description of T:\nT=\n\nδ∈R|J|:(1)δj≤1∀j∈J\n(2)δi+δj≤1∀(i, j)∈J1×J\n(3)δi+δj≤H(i, j)∀(i, j)∈J>1×J>1\n(4)P\nj∈Qδj≤H(Q)∀Q∈Q(J),|Q| ≥3\n\n\nwhere H(i, j)≥1 for all ( i, j)∈J>1×J>1andH(Q)≥ ⌈|Q|/2⌉for all Q∈Q(J),|Q| ≥3.\nWe argue that δ= (1/2, . . . , 1/2) is a vertex of T. Constraints of type (1), (3), and (4)\n8are satisfied, and type (2) constraints are tight. Fix distinct j1, j2, j3∈J1. The set of |J|\ntype (2) constraints {δj+δj1≤1∀j∈J\\ {j1}} ∪ { δj2+δj3≤1}is linearly independent,\nand hence δ= (1/2, . . . , 1/2) is a vertex of T. PC lifting produces precisely the coefficients\nprescribed by Theorem 2.7: g0(aj) =h(j) for j∈Iandg0(aj) =h(j) +1\n2forj∈J, so we\nare done.\nFig. 1 illustrates the sufficient conditions of Theorem 2.8. While a facet-defining PC\nlifting can be efficiently obtained given a minimal cover satisfying the sufficient conditions\nof Theorem 2.8, we do not show how to find a minimal cover satisfying these conditions. We\nconjecture that finding such a cover is NP-hard.\nThe proof of (the converse direction of) Theorem 2.7 maps the binding constraints defin-\ning a vertex of Tto get nlinearly independent points at which the lifting (1) is binding.\nOne could characterize those points directly to prove Theorem 2.8, but we use Theorem 2.7\nto simplify the proof.\nExample 2.9. Consider the constraint 16 x1+14x2+13x3+9x4+9x5+10x6+11x7+23x8≤44\nwith minimal cover C={1,2,3,4}. GNS lifting yields the cut\nx1+x2+x3+x4+1\n6x5+1\n3x6+1\n2x7+4\n3x8≤3\nand PC lifting yields the strictly dominant facet-defining cut\nx1+x2+x3+x4+1\n2(x5+x6+x7) +3\n2x8≤3.\nRemark. There exist facet-defining lifted cover inequalities with half-integral coefficients that\ncannot be obtained via PC lifting. Balas and Zemel [2] provide an example of a knapsack\nconstraint and minimal cover for which δ= (1/2, . . . , 1/2) is a vertex of T, but µ1−λ≥ρ1\ndoes not hold . It is an interesting open question to investigate when such facets arise and\nhow to (efficiently) find them. The lifting procedure of Letchford and Souli [17] also produces\nhalf-integral coefficients, but it is unclear when it can yield facets (it produces cuts that in\ngeneral are incomparable with ours).\nIt would be interesting to derive similar conditions under which GNS lifting defines facets.\nWe provide some sufficient conditions, but it is likely that a stronger result could be derived.\nWe leave this as an open question.\nProposition 2.10. If for all j /∈C,aj∈Sh=⇒aj=µh−λ+ρh, GNS lifting strictly\ndominates PC lifting and defines a facet of conv( P).\nProof. The condition implies g1/ρ1coincides with the lifting function fon all aj,j /∈C,\nwhich, by sequential lifting (e.g., Prop. 7.2 in [6]), means GNS lifting is facet defining. PC\nlifting is strictly weaker due to item 3 of Prop. 2.5.\n3 Experimental evaluation\nWe evaluate our new sequence-independent lifting techniques in conjunction with a number\nof methods for generating the minimal cover cuts that are to be lifted. We describe each\ncomponent of the experimental setup below.\n9Cover cut generation\nLetPn\nj=1ajxj≤bbe a knapsack constraint, let xLPbe the LP optimal solution at a given\nnode of the branch-and-cut tree, and let I={i:xLP\ni>0}. Enumerate IasI={1, . . . , s}\n(relabeling variables if necessary). We do not include variables not in Iin any of the minimal\ncovers, since these do not contribute to the violation of xLP(though such variables may be\nassigned a nonzero coefficient in the lifted cover cut). Next, we present the five cover cut\ngeneration techniques that we use in experiments.\nContiguous covers. First, sort the variables in Iin descending order of weight; without\nloss of generality relabel them so that a1≥a2≥ ··· ≥ as. For each i∈ {1, . . . , s}, let\nj∈ {i+ 1, . . . , s}be such that C={i, i+ 1, . . . , j}is a minimal cover (if such a jexists).\nThis is the contiguous cover starting at i. We generate all such contiguous cover cuts for\neach i. (Balcan et al. [4] introduced these cover cuts, though they did not lift them nor did\nthey restrict to xLP\ni>0.)\nSpread covers. As before, sort the variables in Iin descending order of weight; a1≥\n··· ≥ as. For each i, letj∈ {i+ 1, . . . , s}bemaximal (if such a jexists) such that there\nexists k∈ {j+ 1, . . . , s}such that C={i} ∪ {j, . . . , k }is a minimal cover. Intuitively, ican\nbe thought of as a heavy “head”, and the “tail” from jtokis as light as possible. We coin\nthis the spread cover with head i. We generate all such spread cover cuts for each i.\nHeaviest contiguous cover. We define this as the contiguous cover starting at 1.\nDefault cover. Sort (and relabel) the variables in Iin descending order of LP value so\nthat xLP\n1≥ ··· ≥ xLP\ns. Let jbe minimal so that {1, . . . , j}is a cover. Then, evict items,\nlightest first, until the cover is minimal. We coin this the default cover. These cover cuts\nare loosely based on the “default” separation routine implemented by Gu et al. [9]. (Their\nroutine was for sequential lifting and does not directly carry over to our setting.) Wolter [26]\ntested similar routines.\nBang-for-buck cover. Sort (and relabel) the variables in Iin descending order of “bang-\nfor-buck” so thatc1\na1≥ ···cs\nas, where ciis variable i’s objective coefficient. Let jbe minimal\nso that {1, . . . , j}is a cover. Then, evict items, lightest first, until the cover is minimal. We\ncoin this the bang-for-buck cover.\nExample 3.1. Consider the knapsack constraint 10 x1+ 9x2+ 8x3+ 7x4+ 6x5+ 6x6+\n5x7+ 4x8≤26, suppose the LP optimal solution at the current branch-and-cut node is\nxLP= (0.1,0.8,0.7,0.4,0,1,0.2,0.8), and suppose c= (5,7,9,1,2,6,6,5). We have I=\n{1,2,3,4,6,7,8}. The contiguous minimal covers are {1,2,3},{2,3,4,6},{3,4,6,7,8}. The\nspread minimal covers are {1,4,6,7},{2,4,6,7},{3,4,6,7,8}. The heaviest contiguous cover\nis{1,2,3}. The default cover is {2,3,6,8}. The bang-for-buck cover is {2,3,6,7}.\n10Algorithm 1 Lifted cover cut generation at a node of branch-and-cut\nInput: IP data c,A,b, LP optimum at current node xLP, per-node cut limit ℓ\n1:Initialize cuts=∅.\n2:foreach knapsack constraint a⊤x≤bdo\n3: foreach cover cut cut∈CoverCuts (a, b,c,xLP)do\n4: foreach lifted cover cut liftedcut ∈Lift (cut)do\n5: ifliftedcut separates xLPthen\n6: cuts←cuts∪ {liftedcut }.\n7:Add the top min {ℓ,|cuts|}cuts in cutswith respect to efficacy.\nLifting\nWe evaluate three lifting methods. (1) GNS: The cover cut is lifted using g1/ρ1. (2) PC:If\nµ1−λ≥ρ1the cover cut is lifted using g0, and otherwise it is lifted using g1/ρ1. (3) Smart:\nIfµ1−λ≥ρ1, two liftings are generated: one with g0and one with g1/ρ1. If either lifting\ndominates the other, the weaker lifting is discarded. If µ1−λ < ρ 1, the cover cut is only\nlifted using g1/ρ1.\nIntegration with branch-and-cut\nAlgorithm 1 is the pseudocode for adding lifted cover cuts at a node of the branch-and-cut\ntree and is called (once) at every node of the tree. It uses the prescribed lifting method Lift\natop the prescribed cover cut generation method CoverCuts for each constraint, and adds\nthe resulting lifted cut to a set of candidate cuts if it separates the current LP optimum\nxLP. It adds the ℓdeepest cuts among the candidate cuts to the formulation. (The depth\norefficacy of a cut α⊤x≤βis the quantityα⊤xLP−β\n∥α2∥and measures the distance between\nthe cut and xLP.) The ℓcuts are added in a single wave/round at the current node, and\nno further cuts are generated at that node. Algorithm 1 does not solve NP-hard cover cut\nseparation problems and instead relies on cheap and fast ways of generating many candidate\ncuts through CoverCuts , and only adds those that are separating.\nExperimental results\nWe evaluated our techniques on five different problem distributions: two distributions over\nwinner-determination problems in multi-unit combinatorial auctions ( decay-decay from [23]\nandmultipaths from CATS version 1.0 [19]) and three distributions over multiple knapsack\nproblems ( uncorrelated andweakly correlated from [8] and Chv´ atal from [4]). We ran experi-\nments written in C ++using the callable library of CPLEX version 20.1 on a 64-core machine,\nand implemented Algorithm 1 with ℓ= 10 within a cut callback. We generated 100 integer\nprograms from each distribution. We set a time limit of 1 hour, allocated 16GB of RAM,\nand used a single thread for each integer program.\n11103104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\n101102103104105106107\nTree size020406080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\n103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\ncplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\n103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\ncplex/\nsmart/spread_\ngns/spread_\npc/spread_\nsmart/default_\ngns/default_\npc/default_\n200 400 600 800 1000 1200 1400 1600 1800 2000\nOverall cut limit5\n10\n15\n20Per-node cut limitweakly_correlated; pc/contiguous run-times\n300400500600700Figure 2: Illustrative experiments comparing different lifting methods. The first five plots\nare performance plots for the five different problem distributions, with various parameter\nsettings. The heatmap illustrates the effect of varying the per-node cut limit and overall cut\nlimit on run-time (avg. over first 10 weakly-correlated instances).\nWe present illustrative results in Fig. 2, focusing on settings where PC lifting had sig-\nnificant impact. The full set of experiments are in App. E. We focus on tree size as the\nperformance measure, but we provide full run-time results in App. E. Each curve is labeled\nsetting/lifting/covers orsetting/CPLEX . The setting parameter is either empty, np,\nord. Empty corresponds to all cuts, all heuristics, and presolve off. A setting of npdenotes\npresolve off and all other settings untouched. A setting of ddenotes the default settings (this\nsetting is incompatible with our lifting implementations since presolve is incompatible with\ncut callbacks). If an underscore follows the label, CPLEX’s internal cover cut generation is\nturned off; otherwise it is on. To ensure fair comparisons, in all CPLEX runs not involving\n12our new techniques (those labeled setting/CPLEX ), we register a dummy cut callback that\ndoes nothing. This disables proprietary techniques such as “dynamic search” which do not\nsupport callbacks and thus do not support experiments of the type required in this paper.\nWe now describe the performance plots (Fig. 2), which display how many instances each\nmethod was able to solve using trees smaller than the prescribed size on the x-axis within the\n1 hour time limit. Weakly correlated, contiguous covers (top left): GNS solves 81 instances,\nSmart solves 83 instances, and PC solves 88 instances. These all outperform default CPLEX\nwhich solves 80 instances and builds trees an order of magnitude larger. Uncorrelated,\ncontiguous covers (top right): GNS solves 43 instances, Smart solves 48 instances, and\nPC solves 55 instances, while default CPLEX solves 78 instances. Chv´ atal, heaviest covers\n(middle left): PC lifting dramatically outperforms the other methods. Here, all CPLEX\nparameters are turned off, so we are directly comparing our lifted cover cut implementations\nwith CPLEX’s own cover cut generation routines. PC and Smart strictly outperform GNS\nand CPLEX (GNS is the only one unable to solve all 100 problems), and the largest tree\nsize required by PC is an order of magnitude smaller than any of the other methods. This\ntranslates into a run-time improvement as well due to PC lifting (Fig. 6 in the appendix). On\nthe auction instances, there is little discernible performance difference between the lifting\nmethods. Decay-decay, spread covers (middle right): our lifting implementations with all\nother CPLEX parameters off dramatically outperform default CPLEX on problems requiring\ntrees of size >104(though default CPLEX solves all 100 problems while our methods do\nnot.) Multipaths, spread covers (bottom left): we once again directly compare our lifted\ncover cut implementations with CPLEX’s internal cover cut generation. Here, spread covers\nyield over an order of magnitude reduction in tree size relative to CPLEX, while default\ncovers perform extremely poorly. We observed that contiguous and spread covers generally\nresulted in the best performance (with heaviest contiguous covers on par), and default and\nbang-for-buck covers performed drastically worse.\nWhile our techniques often led to significantly smaller trees than CPLEX, this did not\ntranslate to significant run-time improvements. However, in most settings our techniques\nwere not too much slower than CPLEX and sometimes they were faster. Full run-time\nperformance plots are in App. E. A possibility for this is that we did not limit the total\nnumber of cuts added throughout the tree, causing the formulation to grow very large.\nWe ran an experiment to investigate the run-time impact of varying the number of cuts\nallowed (i) at each node ( ℓin Alg. 1) and (ii) throughout the entire search tree. We plot the\naverage run-time (time limit of 1 hour) over the first 10 weakly-correlated instances using\npc/contiguous , visualized as a heatmap (Fig. 2; bottom right). The best settings (limiting\nthe overall number of cuts to 800) yield an average run-time of around 300 seconds and\nsolved all 10 instances. Default CPLEX ( d/CPLEX ) averaged roughly 900 seconds and only\nsolved 9 instances to optimality.\n4 Conclusions and future research\nIn this paper we showed that PC lifting can be a useful alternative to GNS lifting. We\nproved that under some sufficient conditions, PC lifting is facet-defining. To our knowledge,\nthese are the first broad conditions for facet-defining sequence-independent liftings that can\n13be efficiently computed from the underlying cover. We invented new cover cut generation\nroutines, which in conjunction with our new lifting techniques, displayed promising practical\nperformance.\nThere are a number of important future research directions that stem from our findings.\nFirst, a much more extensive experimental evaluation of PC lifting is needed. We have\nmade several simplifying design choices, including (i) adding only one wave of lifted cover\ncuts at each node; (ii) ranking cuts solely based on efficacy (efficacy is not always the best\nquality to prioritize [4, 5, 24]); (iii) keeping ℓconstant across nodes while it could be a\ntuned hyperparameter, or even adjusted dynamically during search. Our experiments show\npromise even with these fixed choices, but a more thorough suite of tests could find how\nto best exploit the potential of our new techniques. A more comprehensive theoretical and\npractical comparison of PC and GNS lifting is needed as well. For example. PC lifting\nmight possess desirable numerical properties in a solver since it always involves half-integral\ncoefficients (as opposed to GNS). Another direction here is to use machine learning (which\nhas already been used to tune cutting plane selection policies [3, 20]) to decide when to use\nPC or GNS lifting. There might also be additional ways of determining what lifting method\nto use based on problem structure, and detecting that automatically.\nAcknowledgements\nThis material is based on work supported by the Vannevar Bush Faculty Fellowship ONR\nN00014-23-1-2876, NSF grants RI-2312342 and RI-1901403, ARO award W911NF2210266,\nNIH award A240108S001 and Defense Advanced Research Projects Agency under cooperative\nagreement HR00112020003.\nReferences\n[1] Egon Balas. Facets of the knapsack polytope. Mathematical programming , 8(1):146–164,\n1975.\n[2] Egon Balas and Eitan Zemel. Facets of the knapsack polytope from minimal covers.\nSIAM Journal on Applied Mathematics , 34(1):119–148, 1978.\n[3] Maria-Florina Balcan, Siddharth Prasad, Tuomas Sandholm, and Ellen Vitercik. Sample\ncomplexity of tree search configuration: Cutting planes and beyond. 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In Proceedings of the 2nd ACM conference on\nElectronic Commerce (EC) , pages 66–76, 2000.\n[20] Sirui Li, Wenbin Ouyang, Max B. Paulus, and Cathy Wu. Learning to configure separa-\ntors in branch-and-cut. In Thirty-seventh Conference on Neural Information Processing\nSystems (NeurIPS) , 2023.\n15[21] Hugues Marchand, Alexander Martin, Robert Weismantel, and Laurence Wolsey. Cut-\nting planes in integer and mixed integer programming. Discrete Applied Mathematics ,\n2002.\n[22] Christos H Papadimitriou and Mihalis Yannakakis. The complexity of facets (and some\nfacets of complexity). In Proceedings of the fourteenth annual ACM symposium on\nTheory of computing , pages 255–260, 1982.\n[23] Tuomas Sandholm, Subhash Suri, Andrew Gilpin, and David Levine. Winner deter-\nmination in combinatorial auction generalizations. In International Conference on Au-\ntonomous Agents and Multiagent Systems (AAMAS) , pages 69–76, 2002.\n[24] Mark Turner, Timo Berthold, Mathieu Besan¸ con, and Thorsten Koch. Cutting plane\nselection with analytic centers and multiregression. In International Conference on\nIntegration of Constraint Programming, Artificial Intelligence, and Operations Research\n(CPAIOR) , pages 52–68. Springer, 2023.\n[25] Laurence A Wolsey. Valid inequalities and superadditivity for 0–1 integer programs.\nMathematics of Operations Research , 2(1):66–77, 1977.\n[26] Kati Wolter. Implementation of cutting plane separators for mixed integer programs.\nDipolma thesis, Technische Universit¨ at Berlin , 2006.\nA Counterexample to claim of Gu, Nemhauser, and\nSavelsbergh\nGu, Nemhauser, and Savelsbergh [11] remark that if µ1−λ≥ρ1, a large family of superad-\nditive lifting functions can be constructed by considering any nondecreasing function w(x)\nofx∈[0, ρ1] with w(x) +w(ρ1−x) = 1. Consider the class of logistic functions centered at\nρ1/2 of the form\nwk(x) =1\n1 +e−k(x−ρ1/2)\nwhere k≥0. Each wkis nondecreasing on [0 , ρ1], and satisfies wk(x) +wk(ρ1−x) = 1 for all\nx. In the following example, we show that using w0.9for sequence-independent lifting can\nyield invalid cuts. Moreover, the lifting function gw0.9is not superadditive.\nmaximize 112 x1+ 108 x2+ 107 x3+ 106 x4+ 102 x5+ 84x6+ 82x7\nsubject to 112 x1+ 108 x2+ 107 x3+ 106 x4+ 102 x5+ 84x6+ 82x7≤268\nx∈ {0,1}7\nAn optimal solution is given by x∗= (0,0,0,0,1,1,1), which has an objective value\nof 268, satisfying the single knapsack constraint with equality. The set C={2,3,4}is\na minimal cover, and the corresponding minimal cover inequality is x2+x3+x4≤2.\nWe compute the relevant parameters needed for sequence-independent lifting. We have\nµ0= 0, µ1= 108 , µ2= 215 , µ3= 321 , λ= 53, and ρ0= 53, ρ1= 52, ρ2= 51. We have that\n16µ1−λ≥ρ1is satisfied. Thus, the lifting function gw(truncated to the range [0 ,213]) is\ngiven by\ngw(z) =\n\n0 0 < z≤55\n1−w(107−z) 55 < z≤107\n1 107 < z≤162\n2−w(213−z) 162 < z≤213.\nUsing gw0.9yields the lifted cover inequality\nx1+x2+x3+x4+ 0.99x5+ 0.93x6+ 0.71x7≤2.\nBut 0 .99+0 .93+0 .71 = 2 .63>2, sox∗violates this inequality, so the lifted cover inequality\nis invalid for our problem. Furthermore, gw0.9is not superadditive. We have\ngw0.9(82) + gw0.9(82) = 0 .71 + 0 .71> gw0.9(82 + 82) ≈1.00.\nOne need not look at logistic functions to derive this counterexample. The step function\nw(x) =\n\n0 x < ρ 1/2\n1/2x=ρ1/2\n1 x > ρ 1/2\ndisproves the claim as well (in that gwis not superadditive, and results in an invalid cut in\nthe above example).\nB Proof of Theorem 2.1\nThe proof that gkis superadditive closely follows the proof that g1/ρ1is superadditive by [11]\n(Lemma 1 in their paper), with a couple key modifications. As done in [11], we will establish\nsuperadditivity of a function that is defined slightly more generally.\nGiven v1>0,ui≥0, ui≥ui+1, i= 1,2, . . . , v i≥0, vi≥vi+1, i= 1,2, . . .such that\nui+vi>0 for all i, letM0= 0 and Mh=Ph\ni=1(ui+vi) for h= 1,2, . . . ,∞. Define\n˜gk(z) =\n\n0 z= 0\nh M h< z≤Mh+uh+1, h = 0,1, . . .\nh+ 1−wk(Mh+1−z)Mh+uh+1< z≤Mh+1, h = 0,1, . . .\nwhere wk(x) =kx+1−kv1\n2.\nThe superadditive lifting function gkis recovered by letting ui=ai−ρi−1fori∈\n{1, . . . , t},vi=ρifori∈ {1, . . . , t−1}. This yields Mh=µh−λ+ρhandMh+uh+1=µh+1−λ\n(see Gu et al. [11] for further details).\nLemma B.1. Letk∈[0,1/v1]. Ifu1≥v1, the function ˜gkis superadditive on [0,∞).\n17Proof. We prove that max {˜g(z1)+˜g(z2)−˜g(z1+z2) :z1, z2∈[0,∞)} ≤0, which is equivalent\nto superadditivity. We break the analysis into cases as done in [11].\nCase 1: Mh1+uh1+1< z 1≤Mh1+1andMh2+uh2+1< z 2≤Mh2+1. Then, as in [11],\nz1+z2≥Mh1+h2+uh1+h2+1. The first subcase of [11] (Case 1.1) is z1+z2≤Mh1+h2+1, that\nis,z1+z2lies on a “sloped” segment of ˜ g. We show that the assumption u1≥v1rules this\ncase out, that is, z1+z2> M h1+h2+1.\nProof that z1+z2> M h1+h2+1: First, suppose h1= 0. We have\nz1+z2> u 1+Mh2+uh2+1\n≥vh2+1+Mh2+uh2+1 (since u1≥v1≥vh2+1)\n=Mh2+1,\nas desired. The case where h2= 0 is symmetric. Thus, suppose h1, h2≥1, and without loss\nof generality let h1≤h2(the case where h2≤h1is symmetric). We have\nz1+z2> M h1+uh1+1+Mh2+uh2+1\n=h1X\ni=1(ui+vi) +uh1+1+h2X\ni=1(ui+vi) +uh2+1\n= (u1+v1) +h1X\ni=2(ui+vi) +uh1+1+h2X\ni=1(ui+vi) +uh2+1\n≥(u1+v1) +h1+h2−1X\ni=h2+1(ui+vi) +uh1+1+h2X\ni=1(ui+vi) +uh2+1\n=Mh1+h2−1+u1+v1+uh1+1+uh2+1\nNow, uh1+1≥uh1+h2+1and as h1, h2≥1,uh2+1≥uh1+h2. Furthermore, u1≥v1≥vh1+h2≥\nvh1+h2+1. Therefore, the right-hand-side is at least\nMh1+h2−1+ (uh1+h2+vh1+h2) + (uh1+h2+1+vh1+h2+1) =Mh1+h2+1,\nas desired. This completes the proof that under Case 1, z1+z2> M h1+h2+1. We now proceed\nwith the casework under Case 1, starting our numbering from Case 1.2 so that the labeling\nof the cases coincides with [11].\nCase 1.2: Mh1+h2+1< z 1+z2≤Mh1+h2+1+uh1+h2+2. Then\n˜g(z1) + ˜g(z2)−˜g(z1+z2)\n=\u0012\nh1+ 1−k(Mh1+1−z1)−1−kv1\n2\u0013\n+\u0012\nh2+ 1−k(Mh2+1−z2)−1−kv1\n2\u0013\n−(h1+h2+ 1)\n= 1−k(Mh1+1+Mh2+1−z1−z2)−(1−kv1)\n≤1−k(Mh1+1+Mh2+1−Mh1+h2+1−uh1+h2+2)−(1−kv1)\n= 1−k\u0010\n(u1+v1) +h1+1X\ni=2(ui+vi) +h2+1X\ni=1(ui+vi)−h1+h2+1X\ni=1(ui+vi)−uh1+h2+2\u0011\n−(1−kv1)\n18= 1−k(u1+v1−uh1+h2+2)−kh1+1X\ni=2((ui+vi)−(ui+h2+vi+h2))−(1−kv1)\n=−k(u1−uh1+h2+2)|{z }\n≥0−kh1+1X\ni=2((ui+vi)−(ui+h2+vi+h2))\n| {z }\n≥0\n≤0.\nCase 1.3: Mh1+h2+1+uh1+h2+2< z 1+z2≤Mh1+h2+2. Then\n˜g(z1) + ˜g(z2)−˜g(z1+z2)\n=\u0012\nh1+ 1−k(Mh1+1−z1)−1−kv1\n2\u0013\n+\u0012\nh2+ 1−k(Mh2+1−z2)−1−kv1\n2\u0013\n−\u0012\n(h1+h2+ 2)−k(Mh1+h2+2−z1−z2)−1−kv1\n2\u0013\n=−k(Mh1+1+Mh2+1−Mh1+h2+2)−1−kv1\n2\n≤ −k(Mh1+1+Mh2+1−Mh1+h2+2) (since k≤1/v1)\n≤0.\nThe remaining cases (1.4, 2, 3.1 - 3.2 from [11]) and their proofs follow [11] verbatim, so\nwe omit them. This completes the proof that ˜ gis superadditive.\nNow, the proofs that gis superadditive, maximal, and undominated are identical to those\nin [11].\nC Proof of Proposition 2.4\nWe prove Proposition 2.4, which states that for any ε > 0 and any t∈Nthere exists a\nknapsack constraint a⊤x≤bwith a minimal cover Cof size tsuch that PC lifting yieldsP\nj∈Cxj+P\nj /∈C1\n2xj≤ |C|−1 and GNS lifting is dominated byP\nj∈Cxj+P\nj /∈Cεxj≤ |C|−1.\nProof. Leta1≥ ··· ≥ atand let bbe such that a1+···+at> banda1+···+at−1≤b. Let\nλ′=a1+···+at−b. Furthermore, choose a1, . . . , a t, bso that a1−λ′���a2−a1+λ′>0.\nLetM≥1\n(a2−a1+λ′)εand consider the knapsack constraint\nM(a1x1+···+atxt) + (1 + M(a1−λ′))(xt+1+···+xn)≤Mb.\nC={1, . . . , t}is clearly a minimal cover with µ1=Ma 1,λ=Mλ′andρ1=M(a2−a1+λ′),\nand by the choice of a1, . . . , a t,µ1−λ=M(a1−λ′)≥M(a2−a1+λ′) =ρ1. Hence, PC\nlifting can be used to yield a valid lifted cut. As 1 + M(a1−λ′) = 1 + ( µ1−λ), we have\ng0(1 +M(a1−λ′)) =1\n2and\ng1/ρ1(1 +M(a1−λ′)) = 1 −µ1−λ+ρ1−(1 +µ1−λ)\nρ1=1\nρ1=1\nM(a2−a1+λ′)≤ε.\n19D Proof of claim in Theorem 2.8\nIn the proof of Theorem 2.8 we critically used the fact that given Q∈Q(J),|Q| ≥3, with\nhjsuch that aj∈Shj,X\nj∈Qaj> µP\nj∈Qhj−⌊|Q|/2⌋−λ.\nThis allowed us to ensure that the constraint induced by every such Qwas satisfied by the\npoint (1 /2, . . . , 1/2). We prove this claim here.\nProof. We use the quantities uh, vh, Mhdefined in Appendix B. We break the proof into two\ncases. We will use the observation that uh≥vhfor any hsuch that ρh>0. This is because\nfor such h,uh=u1=a1−λ, and so uh=u1≥v1≥vh.\nCase 1: |Q|= 2ℓis even. Let (without loss of generality) Q={a1, . . . , a 2ℓ}, lethjbe such\nthataj∈Shj, and let H=h1+···+h2ℓ. We have\n2ℓX\nj=1aj>2ℓX\nj=1µhj−λ=2ℓX\nj=1Mhj−1+uhj\n≥MH−2ℓ+2ℓX\nj=1uhj\n=H−2ℓX\ni=1(ui+vi) +2ℓX\nj=1uhj\n≥H−2ℓX\ni=1(ui+vi) +2ℓX\nj=ℓ+1(uhj+vhj)\n≥H−2ℓX\ni=1(ui+vi) +ℓX\nj=1(uH−2ℓ+j+vH−2ℓ+j)\n=H−ℓX\ni=1(ui+vi)\n=MH−ℓ\n=µH−ℓ−λ+ρH−ℓ\n≥µH−ℓ−λ,\nas desired.\nCase 2: |Q|= 2ℓ+ 1 is odd. Let Q={a1, . . . , a 2ℓ+1}, lethjbe such that aj∈Shj, and let\nH=h1+···+h2ℓ+1. We have\n2ℓ+1X\nj=1aj>2ℓ+1X\nj=1µhj−λ\n=2ℓ+1X\nj=1Mhj−1+uhj\n20≥MH−(2ℓ+1)+2ℓ+1X\nj=1uhj\n=H−(2ℓ+1)X\ni=1(ui+vi) +2ℓ+1X\nj=1uhj\n≥H−(2ℓ+1)X\ni=1(ui+vi) +2ℓX\nj=ℓ+1(uhj+vhj) +uh2ℓ+1\n≥H−(2ℓ+1)X\ni=1(ui+vi) +ℓX\nj=1(uH−(2ℓ+1)+ j+vH−(2ℓ+1)+ j) +uH−(2ℓ+1)+( ℓ+1)\n=H−ℓ−1X\ni=1(ui+vi) +uH−ℓ\n=MH−ℓ−1+uH−ℓ\n=µH−ℓ−λ,\nas desired.\nE Full experimental results\nWe provide the full set of experimental results. In all plots, we evaluate our methods against\nthree CPLEX settings.\n1.d/CPLEX denotes default CPLEX with all settings untouched.\n2.np/CPLEX denotes CPLEX with presolve turned off and all other settings untouched.\n3.CPLEX denotes CPLEX with all heuristics and presolve turned off. Furthermore, all\ncuts are turned off with the exception of CPLEX’s internal cover cut generation, which\nis left on.\nIn all above versions of CPLEX, we register a dummy cut callback that does nothing.\nThis disables proprietary search techniques such as “dynamic search”. This is needed for a\nfair comparison against our lifting implementations, since these require cut callbacks.\nE.1 Direct evaluation of lifted cover cut methods\nFirst, we evaluate our lifting methods atop bare-bones CPLEX settings; all CPLEX cuts,\nheuristics, and presolve settings are switched off. We toggle CPLEX’s internal cover cut\ngeneration routine on and off to measure the impact of our routines with and without the\ncover cuts added by CPLEX (turning CPLEX cover cuts off corresponds to an underscore\nlabel in our plots).\nFigures 3, 5, 7, and 9 contain the tree size and run-time performance plots for the weakly\ncorrelated, Chv´ atal, decay-decay and multipaths distributions, respectively, when CPLEX\n21cover cuts are turned on. Figures 4, 6, 8, and 10 contain the tree size and run-time perfor-\nmance plots for the weakly correlated, Chv´ atal, decay-decay and multipaths distributions,\nrespectively, when CPLEX cover cuts are turned off.\nE.2 Lifted cover cut methods atop default solver settings\nNext, we evaluate our lifting methods atop close-to-default CPLEX settings; all CPLEX\nsettings are untouched with the exception of presolve, which is turned off. Presolve, which\nrefers to a toolkit of optimizations that operate directly on the input formulation, is in-\ncompatible with user-implemented cutting plane callbacks. As before, We toggle CPLEX’s\ninternal cover cut generation routine on and off to measure the impact of our routines with\nand without the cover cuts added by CPLEX (turning CPLEX cover cuts off corresponds to\nan underscore label in our plots).\nFigures 11, 13, 15, 17, and 19 contain the tree size and run-time performance plots\nfor the weakly correlated, uncorrelated, Chv´ atal, decay-decay and multipaths distributions,\nrespectively, when CPLEX cover cuts are turned on. Figures 12, 14, 16, 18, and 20 contain\nthe tree size and run-time performance plots for the weakly correlated, uncorrelated, Chv´ atal,\ndecay-decay and multipaths distributions, respectively, when CPLEX cover cuts are turned\noff.\n22103104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n100101102103\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplex\n100101102103\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplexFigure 3: Weakly correlated, CPLEX cover cuts on, all other parameters off\n23104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n100101102103\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplex\n100101102103\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplexFigure 4: Weakly correlated, CPLEX cover cuts off, all other parameters off\n24102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n102\n101\n100101102103\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplex\n102\n101\n100101102103\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplexFigure 5: Chv´ atal, CPLEX cover cuts on, all other parameters off\n25102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n102\n101\n100101102103\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n102\n101\n100101102103\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n102\n101\n100101102103\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplex\n102\n101\n100101102103\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplexFigure 6: Chv´ atal, CPLEX cover cuts off, all other parameters off\n26102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplexFigure 7: Decay-decay, CPLEX cover cuts on, all other parameters off\n27102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplexFigure 8: Decay-decay, CPLEX cover cuts off, all other parameters off\n28101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/contiguous\ngns/contiguous\npc/contiguous\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/spread\ngns/spread\npc/spread\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/heaviest\ngns/heaviest\npc/heaviest\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/default\ngns/default\npc/default\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/bfb\ngns/bfb\npc/bfb\nnp/cplex\ncplexFigure 9: Multipaths, CPLEX cover cuts on, all other parameters off\n29101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/contiguous_\ngns/contiguous_\npc/contiguous_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/spread_\ngns/spread_\npc/spread_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/heaviest_\ngns/heaviest_\npc/heaviest_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/default_\ngns/default_\npc/default_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nsmart/bfb_\ngns/bfb_\npc/bfb_\nnp/cplex\ncplexFigure 10: Multipaths, CPLEX cover cuts off, all other parameters off\n30103104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n103104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n103104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n103104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n100101102103\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n103104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplex\n100101102103\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplexFigure 11: Weakly correlated, CPLEX cover cuts on, all other parameters but presolve on\n31103104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n103104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n104105106107\nTree size020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n100101102103\nRun-time020406080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n100101102103\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n104105106107\nTree size01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplex\n100101102103104\nRun-time01020304050607080Number of instances solvedweakly_correlated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplexFigure 12: Weakly correlated, CPLEX cover cuts off, all other parameters but presolve on\n32101102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\n101102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\n101102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\n102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\n101\n100101102103104\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\n102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\n101\n100101102103104\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplexFigure 13: Uncorrelated, CPLEX cover cuts on, all other parameters but presolve on\n33102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\n101102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\n101102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\n102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\n102103104105106107\nTree size01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\n101\n100101102103\nRun-time01020304050607080Number of instances solveduncorrelated; 35 items, 8 knapsacks\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplexFigure 14: Uncorrelated, CPLEX cover cuts off, all other parameters but presolve on\n34101102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplexFigure 15: Chv´ atal, CPLEX cover cuts on, all other parameters but presolve on\n35102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplex\n102\n101\n100101102\nRun-time020406080100Number of instances solvedchvatal; 60 items, 9 knapsacks\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplexFigure 16: Chv´ atal, CPLEX cover cuts off, all other parameters but presolve on\n36102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplexFigure 17: Decay-decay, CPLEX cover cuts on, all other parameters but presolve on\n37102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n102103104105106\nTree size020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solveddecay-decay; 100 items, 1000 bids\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplexFigure 18: Decay-decay, CPLEX cover cuts off, all other parameters but presolve on\n38101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/contiguous\nnp/gns/contiguous\nnp/pc/contiguous\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/spread\nnp/gns/spread\nnp/pc/spread\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/heaviest\nnp/gns/heaviest\nnp/pc/heaviest\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/default\nnp/gns/default\nnp/pc/default\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/bfb\nnp/gns/bfb\nnp/pc/bfb\nnp/cplex\ncplexFigure 19: Multipaths, CPLEX cover cuts on, all other parameters but presolve on\n39101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/contiguous_\nnp/gns/contiguous_\nnp/pc/contiguous_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/spread_\nnp/gns/spread_\nnp/pc/spread_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/heaviest_\nnp/gns/heaviest_\nnp/pc/heaviest_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/default_\nnp/gns/default_\nnp/pc/default_\nnp/cplex\ncplex\n101102103104105106\nTree size020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplex\n100101102103\nRun-time020406080100Number of instances solvedmultipaths; 70 items, 2000 bids\nd/cplex\nnp/smart/bfb_\nnp/gns/bfb_\nnp/pc/bfb_\nnp/cplex\ncplexFigure 20: Multipaths, CPLEX cover cuts off, all other parameters but presolve on\n40"    },    {        "title": "2401.13780v1.Effective_stick_slip_parameter_for_structurally_lubric_2D_interface_friction.pdf",        "content": "APS/123-QED\nEffective stick-slip parameter for structurally lubric 2D interface\nfriction\nJin Wang,1, 2Andrea Vanossi,3, 1and Erio Tosatti1, 2, 3, ∗\n1International School for Advanced Studies (SISSA),\nVia Bonomea 265, 34136 Trieste, Italy\n2International Centre for Theoretical Physics (ICTP),\nStrada Costiera 11,34151 Trieste,Italy\n3CNR-IOM, Consiglio Nazionale delle Ricerche - Istituto Officina dei Materiali,\nc/o SISSA, Via Bonomea 265, 34136 Trieste, Italy\n(Dated: January 26, 2024)\n1arXiv:2401.13780v1  [cond-mat.mtrl-sci]  24 Jan 2024Abstract\nThe wear-free sliding of layers or flakes of graphene-like 2D materials, important in many experi-\nmental systems, may occur either smoothly or through stick-slip, depending on driving conditions,\ncorrugation, twist angles, as well as edges and defects. No single parameter has been so far iden-\ntified to discriminate a priori between the two sliding regimes. Such a parameter, η, does exist\nin the ideal (Prandtl-Tomlinson) problem of a point particle sliding across a 1D periodic lattice\npotential. In that case η >1 implies mechanical instability, generally leading to stick-slip, with\nη=2π2U0\nKpa2, where U0is the potential magnitude, athe lattice spacing, and Kpthe pulling spring\nconstant. Here we show, supported by a repertoire of graphene flake/graphene sliding simulations,\nthat a similar stick-slip predictor ηeffcan be defined with the same form but suitably defined Ueff,\naeffandKeff. Remarkably, simulations show that aeff=aof the substrate remains an excellent\napproximation, while Keffis an effective stiffness parameter, combining equipment and internal\nelasticity. Only the effective energy barrier Ueffneeds to be estimated in order to predict whether\nstick-slip sliding of a 2D island or extended layer is expected or not. In a misaligned defect-free\ncircular graphene sliding island of contact area A, we show that Ueff, whose magnitude for a mi-\ncrometer size diameter is of order 1 eV, scales as A1/4, thus increasing very gently with size. The\nPT-like parameter ηeffis therefore proposed as a valuable tool in 2D layer sliding.\nI. INTRODUCTION\nThe contact interface between graphene or graphene-like 2D material layers and flakes or\nislands has acquired great importance in the last decade [1–5]. Owing to the great strength\nof both slider and substrate, an applied planar force can cause this interface to slide without\ndamage or wear [6, 7]. Both experiments and simulations have explored the frictional aspects\nof the sliding process, as reviewed in [8]. In particular, when a 2D island or layer is forced,\nthrough a tip or a spring, to slide on a substrate, different frictional behaviours are in\nprinciple possible, depending basically on the nature of total free energy E(r), generally\nreferred to as E(x), as a function of the relative coordinate xof the two centers of mass.\nThe first possibility, usually known as structural superlubricity, is academic and strictly\napplies only to the ideal case where both layers are of infinite size, defect free, incommen-\n∗tosatti@sissa.it\n2surate (and Aubry unpinned [9, 10]), is E(x) = const .. Because there is no energy barrier,\nsuperlubric sliding occurs for an arbitrarily weak applied force, with tiny frictional dissi-\npation – mostly due to moir´ e out-of-plane motions [11–13] – proportional to velocity. The\nsecond possibility, more realistic even if uncommon in practice, is realized when flake edges,\ndefects, or weak commensurability cause E(x) to depend on x, but the effective barrier\nUeff= max {∆E(x)}= max {E(x)} −min{E(x)}is weak relative to a hard pulling spring\nwhose stiffness Kpis large. In this case too one may still have smooth sliding, with the aver-\nage value of the washboard oscillating frictional force still growing linearly with velocity [8].\nThe third and commonest case occurs when the free energy barrier Ueffis strong, and/or the\npulling spring Kpis soft. In this case the sliding motion can only occur through a succession\nof mechanical instabitities and, as in the one-dimensional Prandtl-Tomlinson (PT) model\n[14] stick-slip will ensue. The average stick-slip friction force in this case remains large even\nat low velocities, and its growth with velocity becomes much weaker, typically logarithmic\nrather than linear [6, 8, 12, 15–17].\nWith the last two realistic situations of nonzero barrier in mind, we are concerned here\nwith understanding and possibly predicting the occurrence of either smooth sliding or stick-\nslip ahead of experiments and without recourse to simulations. A concise parameter that\ncould discriminate between two sliding states is clearly desirable. In the paradigmatic 1D\nPT model, where the total potential energy is E(x, t) =U0\n2cos(2πx\na) +Kp\n2(x−xspring)2there\nis precisely such a parameter,\nη=2π2U0\nKpa2(1)\nwhere Kpis the pulling spring stiffness, athe periodic potential spacing, and the energy\nbarrier U0is the potential magnitude [10, 18]. In this model, mechanical stability, ∂2E/∂x2>\n0, occurs for η <1, a situation verified when the barrier is weak and the spring is stiff, the\nmechanical evolution is stable and the sliding motion is smooth. For η >1, the evolution\nencounters mechanical instability and the sliding develops discontinuities, which give rise to\nstick-slip. Simple as it is this model and ηparameter describes well the transition between\nsmooth sliding and stick-slip of tip-based frictional systems, as also verified by a variety of\nexperiments [18–23].\nWe are interested here in extending this kind of parameter to 2D structurally lubric (SL)\nsystems, such as mesoscale size islands sliding on crystalline substrates in incommensurate\n3contact [12, 24–26]. A sliding flake or island is in principle a much more complex system, en-\ncompassing a larger number of degrees of freedom as opposed to just one as in the PT model\n(Fig. 1b). The strength of bonds in a 2D material however enslaves all atomic coordinates\nof the island, at least during adiabatic, quasi-static sliding, to just macroscopic coordinates,\nnamely the center-of-mass (COM) coordinate ⃗R, plus the island-substrate “twist” angle θ.\nIn many practical cases, moreover, the island is forced to slide by drivers that cannot rotate.\nWith this situation in mind, it seems natural to try to identify an ηparameter also for\nextended 2D SL contacts.\nUnsurprisingly, this kind of extension requires great caution, with many issues and com-\nplications with respect to the PT model. Even without rotations, the potential field of the\n2D surface-to-surface contact is generally vastly different from sinusoidal. It will depend\non small internal elastic distortions, both in-plane and out-of-plane, that accompany the\nCOM motion. Other features, including island size, twist angle θ, sliding direction Φ, slider\nshape, etc., will act to deform the potential field. With these caveats in mind, we may still\ntentatively submit to test a trial ηeffwith the same PT form but where all relevant param-\neters U, K, a can take effective magnitudes that differ case by case. The eventual quality of\nthis trial remains to be determined and judged by discovering what values these constituent\nparameters take in practice – a task we propose to pursue here by realistic simulations. We\nthus propose to try\nηeff=2π2Ueff\nKeffa2\neff(2)\nAs said above the effective free energy barrier (inclusive of temperature effects if present)\nisUeff= max {∆E(x)}along the chosen sliding direction. More delicate and crucial is the\ndefinition of effective substrate periodicity aeff. We propose using\naeff= 4xinst (3)\nwhere xinstis the COM coordinate where mechanical instability will occur upon sliding.\nWhen the island is displaced from its equilibrium at x= 0 to xinst, mechanical instability\noccurs when the second derivative of E(x) changes sign for at least one value of xspring.\nThat is also the point of maximum lateral force, dE/dx |x=xinst= max {dE/dx }. Finally, the\neffective stiffness Keffis generally affected by the internal elastic stiffness Ksliderof the slider,\ntypically in the spring chain form\nK−1\neff=K−1\np+K−1\nslider. (4)\n4The key problem to be answered at this stage is therefore, how predictable or unpredictable\nthese three effective parameters might be in practice. That is, how large is their difference\nfrom those that could be just guessed, e.g., by treating the whole island as a point slider.\nIn the rest of this work, by using mainly twisted graphene islands as our demonstration\nworkhorse, we employ molecular dynamics simulations to study systematically the sliding\nenergy landscape for cases with different contact areas, twist angles, sliding directions, edge\nshapes, lattice mismatches, pinning defects. Devoid of conceptual pretense, the work aims at\nproviding a practical tool that could predict the quasi-static sliding behaviour of structurally\nlubric sliders. Parameters Ueffandaeffare estimated from case to case to discover if/how\nthey may reasonably represent the sliding of a generic SL island. The influence of contact\nelasticity, absent in the PT model (and still negligible as we shall see for most nanoscale\nislands), should not in general be forgotten. Elasticity must play an important a role in\ncontacts exceeding the micron size, in which case the effective stiffness will diminish relative\nto the bare external pulling stiffness, as suggested by Eq. (4) above. Further, we will discuss\nthe impact of common defects existing in real systems, beyond the island perimetral edges\nthat provide the omnipresent sliding energy barrier. In conclusion, we will show how to\nmake use of ηeffin order to seek experimental conditions that will minimize or maximize\nfriction .\nII. SIMULATIONS: MODEL AND METHODS\nIn simulations we focus on statics of the slider-substrate interface, as appropriate to\nascertain the nature of static friction (smooth versus mechanically unstable). Kinetic friction\nsimulations of similar models can be found e.g., in ref. [8]. Our main MD simulation model\nconsists of a rigid graphene substrate (also a rigid Au(111) substrate in Section VI) with\na finite-sized graphene slider, initially rigid (in Section III to VI), then fully flexible in\nsubsequent Sections VII to IX, portrayed in Fig. 1(a, b). We focus on a circular slider shape\nwith diameter D. The effects of different shapes will be discussed in Section V. The edge of\nthe slider is passivated by H-atoms. The slider is generally rotated by a twist angle θwith\nrespect to the substrate. In order to keep this exploration at the simplest level, temperature\nwas set throughout at T=0. Because we wish to address large sliders, all that can change\n5FIG. 1. Models and protocols. (a) Schematic sketch of the simulation model. The slider (blue)\nhas a diameter Dand rotated by θdegree with respect to the rigid substrate (pink). The x-axis\nis paralleled to the zigzag direction of the substrate. (b) Side view of the model and simulation\nprotocol. (c) The sliding energy landscape and (d) the lateral force trace. The slider used here has\na diameter D= 14 nm and twist angle θ= 5◦. The lattice constant of the graphene substrate is\na=√\n3lCC, where lCC= 1.42˚A is the equilibrium C-C bond length.\natT >0 is a possible thermolubric reduction of edge- or defect-related free energy barriers\n, purely quantitative and of decreasing relevance as the island size increases.\nAll simulations are performed with the LAMMPS code [27, 28]. The interlayer and\nintralayer interaction are described by REBO and by registry-dependent ILP force fields\nrespectively [29–31]. Without attempting to mimic the actual experimental forcing, the\ncenter-of-mass of the slider is dragged by a moving spring of stiffness Kp. For simulations\nwith rigid flakes (Section III to VI), the total potential energy E(x) is scanned by rigidly\ndisplaced the slider. For convenience, here we focus on ∆ E(x) =E(x)−min{E}(shown\nin Fig. 1c). The lateral (driving) force is calculated by F=dE/dx (Fig. 1d). Effective\nsliding free energy barrier and static friction are defined by Ueff= max {∆E(x)}andFs=\n6max{F(x)}respectively. For simulations with flexible islands (Section VII to IX), a quasi-\nstatic sliding protocol is adopted (Fig. 1b). Starting from an energy minimum, the slider is\ndisplaced by a pulling spring with spring constant Kp, a parameter controlled by the external\ndriving system [20, 32]. In AFM-based experiments, its magnitude is typically within the\nrange of 1 ∼100 N/m [12, 26, 33, 34]. During the quasi-static sliding, one end of the pulling\nspring is tethered to the COM of the slider, the other end is displaced by δx= 0.02˚A in\neach step, followed by a full structural optimization with the CG+FIRE algorithm. The\nenergy and force tolerance used in optimizations are 10−15and 10−4eV/˚A respectively. To\nrestrict the global rotation of the slider, planar springs perpendicular to the sliding direction\n(with spring constant ki= 1 N/m) are tethered to each slider atom – a virtual constraint to\ncounteract the global torque [8].\nIII. SLIDER SIZE AND TWIST ANGLE\nWith the simulation protocol introduced above, we are set to discuss the influence to Ueff\nandaefffrom various factors. We begin in this section with the size and twist angles. Simula-\ntions results for the size and twist angle dependence of Ueffandaeffare shown in Fig. 2 (a-b).\nSize dependence. The values of aeffare found to be uniformly close to a, independent\nof size, and as we shall see later, approximately independent of driving details such as side\nor central pulling. The energy barrier of the island Ueff, due to the edges which even in\nthe absence of other defects break full translational symmetry, logically increases with size,\nas does the perimeter. Physically the barrier is due to the uncompensated moir´ e nodes\nentering/exiting the edge. Its size scaling for a SL system is Ueff∼Ui(D/a sli)γ, where Ui\nrepresents the per-atom sliding energy barrier, asliis the lattice constant of the slider and γ\nis a scaling exponent. For a defect-free graphene/graphene interface, the basic parameter Ui\ndetermining the edge-induced energy barrier is estimated with the present force field to be\nabout 0 .1 eV. By fitting the upper envelope of simulation results (Fig. 2a), we get γ∼1/2,\ni.e., the barrier is approximately proportional to the slider perimeter’s square root. This\nscaling, Ueff∝A1/4∝D1/2(Ais the contact area), agrees with previous studies of circular\nislands [8, 35–38]. Its meaning is that among all perimetral atoms, only the front and rear\nones dominate the friction, a fact also well established for nanoribbons [8, 39, 40]. That\n7implies as we will show in Sect V that γis shape dependent, and can generally rise from 0\nto 1.\nTwist dependence. The values of aeffremain close to a, for all twist angles θwe studied\nhere, from 2◦to 30◦. The effective barrier Ueffhas a more interesting dependence on twist\nangle. The largest value Ui∼0.1 eV is obtained for systems with small twist angles.\nThe barrier drops as θincreases (Fig. 2b), scaling as Ueff∝θγ−2, a decrease due to the\ndecreasing contribution from the moir´ e edge [35]. For a system with θ= 30◦– closest to the\nideal “superlubric” state, its Uivalue is even smaller, approximately 10−2eV.\nFIG. 2. Effective energy barrier Ueff(upper panels) and periodicity aeff(lower panels) as a function\nof (a) diameter D, (b) twist angle θ, and (c) sliding direction Φ. The power function fits and the\nscaling exponents are shown in (a) and (b).\nIV. SLIDING DIRECTION\nThe next point that distinguishes the real-world SL system from the 1D PT model is the\nsliding direction – the potential energy evolves differently as the COM of the slider moves\nalong different directions (characterized by the angle Φ with x-axis). By symmetry, both\nUeffandaeffpossess 60◦symmetry with respect to Φ.\nSimulation results of a D= 14 nm and θ= 5◦model are shown in Fig. 2 (c). For sliding\ndirections from Φ = 0◦to 60◦, both Ueffandaeffdo not differ much. Once again, for all\nthe graphene islands which we considered, with diameter Dranging from 4 to 20 nm and\nθfrom 2◦to 30◦, the effective periodicity aeffwas found to remain remarkably close to the\n8substrate (also graphene) lattice constant a. The overall shape of E(x) being very strongly\ndependent upon the sliding direction Φ, this result seems quite surprising. It is explained\nas follows.\nAssuming a weak interaction between slider and substrate [11, 36, 37, 41], the general\nE(r) can be represented by\nE(r) =−2Uslider\n9\"3X\ni=1cos(ki·r) +3\n2#\n(5)\nwhere Uslideris the barrier of the whole slider and kiis the reciprocal vector of the triangular\nlattice, with magnitude |ki|= 4π/√\n3a, and ais the lattice constant of the substrate.\nStarting from the energy minimum, i.e., r= 0, one can get the slope along an arbitrary\ndirection Φ,\nS=∇E·⃗t (6)\nwhere ⃗t= (cos Φ ,sin Φ). The steepest slope along direction Φ satisfies\ndS\ndr= 0 (7)\nIn polar coordinates where x=rcos Φ and y=rsin Φ, one gets\ndS\ndr=8π2Uslider\n27a2\b\ncos[2πr\n3a(√\n3 cos Φ −3 sin Φ)](cos Φ −√\n3 sin Φ)2\n+ cos[2πr\n3a(p\n(3) cos Φ + 3 sin Φ)](cos Φ +√\n3 sin Φ)2\n+ 4 cos(4πr√\n3acos Φ) cos2Φ\t(8)\nWith a variable substitution:\nm= cos Φ −√\n3 sin Φ\nn= cos Φ +√\n3 sin Φ(9)\nthe above formula simplifies to\ndS\ndr=8π2Uslider\n27a2\b\ncos(2πmr√\n3a)m2+ cos(2πnr√\n3a)n2+ cos[2πr√\n3a(m+n)](m+n)2\t(10)\nConsidering the 60-degree symmetry of Φ and the fact that the largest slope position must be\ninside the potential well, the cosine terms can be approximated as cos x≈1−x2/2 +x4/24.\nThe above equation further simplifies to\ndS\ndr=8π2Uslider\n27a2\b\n[m2+n2+ (m+n)2]−2π2r2\n3a2[m4+n4+ (m+n)4]\n+2π4r4\n27a4[m6+n6+ (m+n)6] +O[(r\na)6\n]\t(11)\n9Noting that\nm2+n2+ (m+n)2= 6\nm4+n4+ (m+n)4= 18\nm6+n6+ (m+n)6= 60 + 6 cos(6Φ)(12)\nSubstituting Eqs. (12) into Eq. (11), we finally conclude that\n1.dS/dr is weakly Φ-dependent (when r < a );\n2.dS/dr = 0 occurs at r≈a/4.\nIn simple words, even though the overall E(r) is strongly direction dependent, its\nquadratic growth near r= 0 is approximately independent of direction, and so is the\ninstability point of maximum slope. That implies that case-by-case corrections to aeffare\nunnecessary, this parameter being well approximated by the bare substrate lattice constant.\nGiven this weak directional dependence, simulations in the subsequent sections are all along\nthe zigzag ( x) direction.\nThe effective barrier does, unlike aeff, depend upon the sliding direction, although only\nweakly. The sixfold symmetry is confirmed in Ueff(Φ), and the relative difference of Ueff\nbetween zigzag (Φ = 0) and armchair (Φ = 30◦) sliding direction is only ≈10%, a difference\nwhich also agrees with experimental sliding of SL graphite/hBN interfaces [42].\nV. SLIDER SHAPE\nBesides circular sliders [25, 35, 42], instructive if not particularly realistic, there are many\nother candidate shapes that may help anticipate the often irregular forms encountered in 2D\nmaterial-based SL experiments [26, 43, 44]. In this section, we examine results for triangular,\nhexagonal, and mixed-shape sliders – to explore the variety of cases (Fig. 3a). In view of\ndifferent shapes, we use the number of carbon atoms in the slider Nslito characterize the\nsize of the model. A circular slider with Nsli= 6688 has diameter D= 15 nm. The edges\nof triangular and hexagonal sliders used in our simulations are along zigzag directions, a\nchoice based on the fact that zigzag direction has a slightly lower fracture toughness [45].\nNonetheless, the orientation of edges in experiments could still scatter. This means that\neven for the same shape, the orientation of the edges can further affect the result [37].\n10FIG. 3. Results for different slider shapes. (a) Simulation model with circular, hexagonal, trian-\ngular, and an irregular shape slider. This shape is built based on the shape of flake reported in\nexperiments (inset) [26]. Number of slider atom Nsliis shown in the figure. Twist angle θ= 5◦\nfor all shapes in the simulation. (b, c) Effective energy barrier Ueffand periodicity aeffas a func-\ntion of the contact size N1/2\nsli. For clarity, (b) and (c) show the results for triangular (yellow) and\nirregular-shaped (red) islands and hexagonal islands (green) respectively. Circular shape results\n(blue) are shown in background to facilitate direct comparison. The fitting and the corresponding\nscaling exponents γare shown in the upper panels of (b) and (c). The exponent for the hexagonal\nshaped case scatters, from 0 to ≈0.7.\nSimilar to previous observations, simulation results in Fig. 3(b) show that aeffremains\nclose to afor all shapes and size we studied, even for the irregular shape case (marked by red\nsquares). The effective energy barriers are smaller for triangular and hexagonal systems than\nfor circular shapes, at least for the chosen θ= 5◦and Φ = 0. A larger barrier occurs when\nmore moir´ e nodes simultaneously cross the island edge, and coincident crossings happen to\nbe less abundant in the chosen shapes compared to circular. The barrier growth with size is\nalso sublinear. Even if data are insufficient to extract an accurate scaling exponent γfrom\nUeff∼Nγ/2\nsli, the data are compatible with 0 < γ < 1 as anticipated. We note that certain\nshapes show a surprising γ <1/2, such as the triangular shape of Fig. 3(b), where γ≈0. As\nalso seen in some previous simulations [37, 38], this surprising lack of growth of the sliding\nbarrier with size is possible when polygonal islands slide along or close to a wedge direction.\nThis reinforces the concept that a choice of shapes and orientation might be crucial when\nseeking structurally lubric sliding of large size sliders.\n11VI. HETEROSTRUCTURES\nThe above results and discussion was focused on graphene homo-structures. Although\ngraphene and its interfaces are still most popular in SL research, hetero-structures are at-\ntracting increasing interest. That is because of their rich electronic properties [1–3] but also\nof their robust “superlubric” behavior – the sliding energy barrier remains low under arbi-\ntrary twist angles [12, 40, 43, 46]. In this section, we consider the aligned graphene/Au(111)\nhetero-structure (Fig. 4a, b) as a representative hetero-interface. We simulate this system\nto extract the size dependence of Ueffandaeff, giving a direct comparison to the results in\nSection III.\nFIG. 4. Models and results for graphene/Au(111) heterostructure. (a, b) Top view and side view\nof the simulation model. (c) Effective energy barrier Ueffand periodicity aeffas a function of the\ndiameter of the slider. Twist angle θ= 0◦(aligned) for all heterostructures used here. The lattice\nspacing of the Au(111) substrate is aAu(111) ≈2.885˚A.\nSimulation results in Fig. 4(c) show that the main conclusions in previous sections still\nhold for hetero-structures. Specifically, Ueffscales sublinearly with diameter ( γ= 1/2), and\naeffis very close to the lattice constant of the substrate aAu(111) =aAu/√\n2≈2.885˚A, where\naAu= 4.08˚A is the lattice constant of gold. In addition, the magnitude of the sliding\nenergy barrier of graphene/Au(111) hetero-junction is tiny – comparable to the graphene\nhomo-structure with large twists (Fig. 2), which shows that the system has exceptionally\ngood superlubric properties [31, 39, 47].\n12VII. ELASTICITY\nThe simulations in the previous four Sections are based on rigid island sliders. The\nrationale for this choice is that 2D materials such as graphene are very stiff. However,\nas size increases, or when the driving method changes, the influence of elasticity may no\nlonger be ignored. In this section, the simulated graphene island sliding on a rigid graphene\nsubstrate is flexible with D= 28 nm and θ= 5◦. To compare with the rigid island case,\nthree simulation protocols are introduced, namely, uniform drag, edge-drag and edge-push,\ncorresponding to three typical driving methods in experiments [12, 26, 39, 48, 49]. For the\nedge drag and push cases, the pulling spring is tethered to the narrow edge region (green\ncolor in Fig. 5a) instead of the COM as in the uniform case.\nFIG. 5. Models and results for different driving methods. (a) Schematic sketch of the simulation\nmodel. In edge-drag and edge-push simulation, the slider moves by dragging/pushing the rightmost\nedge (highlighted green). (b) Potential energy evolution as a function of displacement. Parameters\nused in simulations are D= 28 nm ( Nsli= 23437) and θ= 5◦.\nResults are shown in Fig. 5(b). The flexibility of the island, both in-plane and out-\nof-plane, causes the potential energy E(x) to decrease compared to the rigid case at all\npositions, and to further deviate from sinusoidal. Due to that, the relaxed aeffgrows slightly\nlarger than a. For both edge-driven cases, we find aeff= 2.72˚A, compared to a= 2.46˚A\nof the rigid case. This increase is connected with the entry and exit of the moir´ e pattern\nAA nodes – higher energy regions [8, 35, 50] – at the edge of the slider. For a rigid slider,\nthe AA node is forced to enter/exit smoothly from the edge; while for a flexible slider,\n13deformability causes the entry and exit of local AA to delay the overall COM movement\nof the slider – the AA node remains pinned at the edge for a while. This pinning cannot\nlast long, especially for 2D materials with very high in-plane stiffness – and when the next\nmoir´ e is about to approach the edge, the previous pinned moir´ e is forced to leave. As a\nresult, although elasticity causes aeff> a, that increase is always modest, well below its\nsinusoidal upper limit 2 a. The potential energies obtained by two edge-driven methods in\nour simulations overlap almost completely (Fig. 5b). This is due to the symmetry of the\ngraphene homo-structures. That is different from the hetero-structure used in previous work\n[51], where push and drag (implying respectively compression and elongation) have different\neffects on incommensurability.\nThe influence of elasticity is also reflected in the effective stiffness, as suggested by Eq. (4).\nIn nanoscale simulations, the effect of elasticity remains negligible due to the large in-plane\nstiffness of 2D materials. For a nanoscale monolayer graphene slider, in fact, the internal\nstiffness is Kslider∼Y d≈300 N/m ( Yis the Young’s modulus and dis the thickness of\ngraphene, which is approximated by the interlayer distance of graphite), much larger than the\nexternal stiffness Kp– typically on the order of 10 N/m in experiments. At the microscale,\nhowever, a thousand times larger linear size, Kslidercan decrease and become important. For\nan edge-dragged island, the internal stiffness decreases as the size Lincreases, Kslider∝L−1,\nand the internal stiffness may become comparable to the external one [52]. In addition to\nthe in-plane size, the thickness of the slider and the stacking will also affect the internal\nstiffness [53] – this is another aspect that may deserve investigation in the future.\nVIII. PINNING BY DEFECTS\nIn the previous Sections we focused on defect-free SL islands whose interface was intact\nand atomically smooth. Real systems are generally more complex than this ideal case,\nand defects will inevitably be introduced during the synthesis and preparation of samples.\nTo address that kind of situation, in this chapter we discuss the influence of two common\ndefects, vacancies and surface steps, on Ueffandaeff.\nOur simulation models containing steps and vacancies are shown in Fig. 6(a,b). One\nexternal monolayer step (green) is obtained by cutting the upper graphene layer of an AB\nstacked bilayer substrate along its armchair direction. A single vacancy (shown in inset)\n14is introduced to the substrate within the contact region. Structures for both cases are\nwell-optimized before the sliding simulation.\nIn the results of Fig. 6(c), we see that the difference between aeffandafor systems with\nand without step or vacancies is negligible, which again suggests using aeff∼ain general\nSL contacts. On the other hand, while the energy barrier Uefffor the system with a single\nvacancy is only slightly higher than the one with perfect interface, that for the system with\nexternal step is evidently higher, an increase which will obviously enhance friction. This is\nconsistent with the experimental observation that the SL graphite contacts with external\nsteps have higher friction than that of the perfect and buried step cases [54].\nFIG. 6. Simulation model with (a) external step and (b) vacancy. The step is AB stacked with\nsubstrate and colored in green. The vacancy is shown in the inset. (c) Simulation results for Ueff\nandaefffor systems without defects (perfect), with one external step, and with a single vacancy.\nOther parameters used are D= 7.5 nm and θ= 5◦.\nIX. DISCUSSION AND CONCLUSIONS\nWith full-atom quasi-static simulations, the dependence of effective sliding energy bar-\nrierUeffand periodicity aeffon size D, twist angle θ, and island sliding direction Φ have\nbeen examined for structurally lubric graphene interfaces. Based on these two parameters,\ncombined with the lateral stiffness Kpas appropriate in a given experiment or simulation,\none should be able to estimate ηeffand predict whether the sliding state will be smooth or\n15stick-slip.\nOur prediction tool is thus ready to be tested. Of course it ought to be tested in (future)\nexperiments. But it can also be tested right away by a direct “realistic” kinetic friction sim-\nulation, inclusive of temperature, sliding velocities, as well as energy dissipation. A room\ntemperature simulation with D= 7.5 nm, θ= 5◦and low velocity ( v0= 1 m/s) provides a\ngood example. The kinetic simulation model and set-ups (Fig. 7a) are similar to previous\nwork [8, 11]. To account for energy dissipation, a Langevin thermostat is applied to the\nbottom layer with temperature T= 300 K and (realistically underdamped) damping coeffi-\ncient of 0 .1 ps−1[11]. The time step and total simulation time used in kinetic simulations\naredt= 1 fs and 5 ns.\nBefore the actual kinetic sliding simulation, in order to have a sense of the difference\nbetween the fully flexible tri-layer system (Fig. 7a) and the fully rigid bi-layer system in\nSect III, we firstly perform a quasi-static simulation. Compared to the results of the rigid\nmodel aeff= 2.44˚A and Ueff= 0.298 eV in Fig. 2(a), here we have Ueff= 0.301 eV and\naeff= 2.43˚A and Ueff= 0.301 eV – the difference is negligible. In particular, elasticity\nslightly lowers the barrier, as discussed in Sect VII, but the additional bottom graphene\nlayer (required by the kinetic simulations) compensates that.\nShown in Fig. 7(b-d) are the results of kinetic friction simulations with different spring\nstiffnesses Kp. As shown in Fig. 7(b), there is a clear stick-slip when Kp= 10 and 3 N/m\n(corresponding to ηeff= 1.57 and 5 .22 respectively), as opposed to smooth sliding with\nKp= 100 N/m (corresponding to ηeff= 0.16). The simulation results very well meet the\ntheoretical predictions – for the underdamped low-temperature system (here kBT/U eff≈\n0.086), when ηeff<1, there is smooth sliding; when 1 < η eff<4.6, there is single stick-slip;\nand when 4 .6< η eff<7.8, there is double-slip [19, 21].\nThe difference between smooth sliding and stick-slip further leads to significant differences\nin the mechanical power dissipated during sliding. We show in Fig. 7(c) the accumulated\ndissipated energy Wdissas a function of time: for systems with Kp= 10 and 3 N/m, Wdiss\nincreases significantly with time, while the increase of Wdissfor the Kp= 100 N/m system\nis imperceptible. This confirms that for ηeff<1 the sliding of an island is, despite a nonzero\nbarrier, still structurally lubric.\nFor completeness, we also extracted for display the kinetic friction force of the system by\nFk=∆Wdiss\nv0∆t(the result has been verified to be equal to the time averaged lateral force ⟨F⟩).\n16FIG. 7. Kinetic friction simulations. (a) Simulation model. Parameters used are D= 7.5 nm,\nθ= 5◦,T= 300 K and v0= 1 m/s. (b) Force traces of different Kpcases. Results for Kp= 100,\n10 and 3 N/m correspond to smooth sliding (black), single stick-slip (red) and double-slip (green)\nrespectively. (c) Dissipated energy as a function of time for the three cases. Note the exceedingly\nsmall friction for ηeff<1. (d) Kinetic friction as a function of Kp(lower x-axis) and ηeff(upper\nx-axis). The average value and error bar are estimated from three independent simulations. The\ntheoretically predicted transition stiffness Kcis marked by the shaded region. Note the good\nagreement between the ηeffprediction and the actual drop of friction.\nFig. 7(d) shows clearly that for stick-slip cases, i.e., ηeff>1 (or Kp<15.7 N/m), the kinetic\nfriction is significant; while for smooth sliding cases, the friction is much smaller. For this\nnanoscale simulation system, the transition stiffness dividing the two regimes is on the order\nof 10 N/m, as marked by the shaded region.\nCoincidentally, in many AFM-based experiments [12, 26, 33, 34], the lateral stiffness\nof the system is also on the order of 10 N/m. This naturally requires us to estimate Ueff\n17in experiments – typically on microscale and with large twist angles (small twist islands\nor flakes rotate easily back to 0◦). Using size scaling Ueff∼Ui(D/a)1/2and substituting\nUi= 10−2eV from Section III, it can be estimate that Ueffof a microscale system is on the\norder of 1 eV. Assuming Kp= 10 N/m, we can qualitatively estimate that ηeff∼5>1.\nIn experimental reality, the energy barrier will be generally larger due to the presence of\nmore defects and/or contaminants. This implies that SL systems driven by AFM probes\nare likely to exhibit stick-slip motion unless the lateral stiffness is very much strengthened.\nAn observation that may be made here is that many SL friction experiments do not\ndirectly show stick-slip advancement of the slider, so much that superlubricity is claimed\nin some cases. Strictly speaking that claim seems improper, because the measured velocity\ndependence, when available, is always much weaker than linear, in fact logarithmic – and\nthat is the hallmark of stick-slip. The two elements, the absence of visible stick-slip and a\nvery sublinear velocity dependence, appear contradictory at first sight. One likely expla-\nnation might be a simple lack of experimental resolution, atomic size steps being as small\nas they are. For very large sliders, other possibilities may involve a coexistence of many\ndistributed pinning points, interfering with one another and transforming the advancement\nfrom stick-slip to apparently continuous. A common feature of these seemingly contradictory\ncases should be a large noise. Noise is actually an observable of great importance, generally\nnot reported and unduly neglected. The multi-pinned stick-slip should precisely differ from\nsmooth sliding by a large increase of frictional noise. Nevertheless, the logarithmic velocity\ndependence of friction [8] remains a safe diagnostic and an incontrovertible proof of stick-slip\nin SL sliding, which we argue will experimentally occur once our criterion ηeff>1 is verified.\nIn summary, we proposed here a single PT-like parameter ηeff= 2π2Ueff/Keffa2\neffto de-\nscribe the transition between smooth and stick-slip sliding of structurally lubric islands and\nlarge size interfaces. MD simulations show systematically how the parameters vary with size,\ntwist angle, sliding direction, lattice mismatch, elasticity, and pinning defects – all variables\nthat characterize real experiments. Firstly, the sliding energy barrier Ueffof an island has a\nsublinear size scaling and is accompanied by moir´ e-sized fluctuations. For a circular island\nUeffdecreases like θ−3/2as the twist angle θgrows, and depends weakly on sliding direc-\ntion. Interfacial pinning defects widely seen in experiments, especially external steps, can\n18significantly increase the barrier. On the other hand, the effective periodicity aeffis for the\ncases we studied, and assuming a rigid substrate, always close to the lattice constant of the\nsubstrate a. This result is attributed to the large in-plane stiffness of 2D materials leading to\na relatively direction independent energy profile close to the x= 0 minimum. Our nanoscale\nsimulations suggest that the island’s elasticity should not be ignored, specifically at micron\nor larger sizes. Elasticity reduces the intra-slider stiffness Kslider, and that in turn reduces\nthe overall driving stiffness from KptoKeff. A smaller Keffcan yield ηeff>1, leading to\nstick-slip.\nReal kinetic simulations offer a preliminary verification of the accuracy of our proposed\nηeff. Lastly, based on the analysis and extrapolation of simulation results, we believe that\nmost existing SL experiments widely satisfy ηeff>1. Although stick-slip may be generally\ndifficult to see directly in force traces, the logarithmic friction velocity dependence provides\na safe diagnostic of its presence. We believe that the analysis of noise might in the future\nbe crucial in order to further uncover the stick-slip nature of friction, when present. On\nthe other hand, our proposed ηeffparametrization shows that it is not impossible, even for\nnot so small islands (from nano to microscales), to achieve ηeff<1 and therefore smooth\nsliding and negligible absolute friction (not just differential friction coefficient [8]), despite\nthe inevitable edge-related energy barriers. For that goal, it will be instrumental to employ\nstiff drivers, sliding in directions where the slider shape has sharp wedges as opposed to\nflat edges. The engineering community interested in achieving virtually frictionless smooth\nsliding should concentrate efforts towards reaching these conditions.\n19ACKNOWLEDGMENTS\nThe author acknowledge support from ERC ULTRADISS Contract No. 834402. Support\nby the Italian Ministry of University and Research through PRIN UTFROM N. 20178PZCB5\nis also acknowledged. We are grateful for discussions with A. Khosravi and A. Silva.\n[1] A. K. Geim and I. V. Grigorieva. Van der waals heterostructures. 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Lett. , 125:026101, 2020.\n24"    },    {        "title": "2401.13811v1.Value_sharing_and_Stirling_numbers.pdf",        "content": "arXiv:2401.13811v1  [math.CV]  24 Jan 2024VALUE SHARING AND STIRLING NUMBERS\nAIMO HINKKANEN AND ILPO LAINE\nDedicated to the memory of Lawrence Zalcman\nAbstract. Letfbe an entire function and L(f) a linear differ-\nential polynomial in fwith constant coefficients. Suppose that f,\nf′, andL(f) share a meromorphic function α(z) that is a small\nfunction with respect to f. A characterization of the possibilities\nthat may arise was recently obtained by Lahiri. However, one case\nleaves open many possibilities. We show that this case has more\nstructure than might havebeen expected, and that a more detaile d\nstudy of this case involves, among other things, Stirling numbers\nof the first and second kinds. We prove that the function αmust\nsatisfy a linear homogeneous differential equation with specific co-\nefficients involving only three free parameters, and then fcan be\nobtained from each solution. Examples suggest that only rarely\ndo single-valued solutions α(z) exist, and even then they are not\nalways small functions for f.\n1.Introduction\nWe show that one case of a value sharing problem investigated by\nLahiri in [3] has more structure than might have been expected, an d\nthat a more detailed study of this case involves, among other things ,\nStirling numbers of the first and second kinds.\nWe assume that the reader is familiar with Nevanlinna’s value dis-\ntribution theory (see, for example, [2]). Unless otherwise mentione d,\nall functions are assumed to be meromorphic functions in the comple x\nplaneC. We say that functions FandGshare a function (possibly\nconstant) acounting multiplicities (CM) if F−aandG−ahave the\nexactly same zeros in C, and the multiplicities of the zeros are the same\nat each zero.\nOur starting point is formed by the papers [5] due to Wang and\nLaine and [6] by Zhang. For the convenience to the reader, we reca ll\nthe following theorem from [6]:\nTheorem 1.1. Suppose that fis a non-constant entire function, α(z)\nis a non-zero small function with respect to f, andn≥2is an integer.\nLetL(f) :=anf(n)+an−1f(n−1+···+a1f′+a0fbe a linear differential\n2010Mathematics Subject Classification. Primary 30D35; Secondary 11B73.\nKey words and phrases. Uniqueness theory, Linear differential polynomials, Stir-\nling numbers.\n12 AIMO HINKKANEN AND ILPO LAINE\npolynomial to fwith constant coefficients such that a0/ne}ationslash= 0. Iff,f′and\nL(f)shareα(z)CM, then one of the following assertions holds:\n(i)f(z) =bez, wherebis a non-zero constant.\n(ii)α(z)reduces to a constant, say a,a0+/summationtextn\nj=1ajcj−1= 1for some\nconstant c/ne}ationslash= 0, and\nf(z) =becz−a(1−c)\nc,\nwhereb/ne}ationslash= 0is a constant.\n(iii)f(z) =λecz+p(z)+a(z),wherecis a constant, p(z)anda(z)\nare two polynomials satisfying p′−cp=a−a′, anddega≤n−3.\nThe case (iii) here is a completion to Theorem 1.1 in [5]. However,\nas pointed out in a recent paper [3] by Lahiri, even the above theore m\ndue to Zhang is not complete. The key result of Lahiri is [3], Theorem\n1.1, completing these two preceding papers. For the convenience o f the\nreader, we recall this result due to Lahiri:\nTheorem 1.2. Letfbe a non-constant entire function, αbe a small\nfunction relative to f,n≥2be a positive integer and L(f) :=a0f+\na1f′+···+anf(n)be with constant coefficients, an/ne}ationslash= 0. Suppose that f\nandf′shareαCM andfandL(f)shareαCM as well, except possibly\nin a finite set. Then five possibilities may appear:\n(i)f(z) =α+exp((λecz+µ)/c)andL(f) =α+anλnexp((λecz+\nnc2z+µ)/c), whereα′/ne}ationslash=αandc/ne}ationslash= 0,λ/ne}ationslash= 0andµare constants.\n(ii)f(z) =λezandL(f) =f, whereα′/ne}ationslash=αandλ/ne}ationslash= 0is a constant.\n(iii)f(z) =λecz+α+pandL(f) = ((1−c)a0+c)f+(1−c)(1−a0)α,\nwhereαis a non-constant polynomial and λ/ne}ationslash= 0,c/ne}ationslash= 0,1are constants.\nMoreover, p=c−1\nc/parenleftBig\n−α\nc−1+/summationtextu\nj=1α(j)\ncj/parenrightBig\nis a polynomial, u= degαand\np′−cp=α−α′.\n(iv)f(z) =λecz+(1−1/c)αandL(f) =α+((1−c)a0+c)(f−α),\nwhereαis a non-zero constant and λ/ne}ationslash= 0,c/ne}ationslash= 0,1are constants.\n(v)fsatisfies the differential equations f′=λeczf+(1−λecz)αand\nL(f) =anλnenczf+(1−anλnencz)α, whereαis non-constant, α′/ne}ationslash=α,\nandc,λare non-zero constants. Moreover, a0= 0andan−1/ne}ationslash= 0. Ifn=\n2, thena1+a2c= 0, andifn≥3, then/summationtextn\nk=1(−1)k−1ak/parenleftBig\n2an−1\nn(n−1)an/parenrightBigk−1\n=\n0.\nIn this paper, we analyze the case (v) of the above theorem in more\ndetail, trying to demonstrate that the fifth case actually may appe ar\nin some special cases only.\nWe denote the Stirling numbers of the first kind by s(n,k) and those\nof the second kind by S(n,k). We will recall the definitions and basic\nproperties of these numbers in Section 3. We obtain the following\nresult.VALUE SHARING AND STIRLING NUMBERS 3\nTheorem 1.3. Suppose that f,n,L(f), andαare as in case (v) of\nTheorem 1.2, so that (1−λecz)αis an entire function. Then a0= 0\nand\naj=ancn−js(n,j) (1.1)\nfor1≤j≤n. Furthermore, α(z)satisfies the linear homogeneous\ndifferential equation\n0 =/parenleftBigg\n1−ann−1/summationdisplay\np=0cn−p−1λpepcz(−1)n−p−1(n−p−1)!/parenrightBigg\nα\n−ann−2/summationdisplay\nk=1α(k)cn−1−k/parenleftBigg\ns(n,k+1)+n−k/summationdisplay\np=1(λ/c)pepcz(s(n−p,k+1)−cs(n−p,k))/parenrightBigg\n−an(1−λecz)α(n−1)(1.2)\nof order n−1. When n= 2, the middle line in (1.2) represents an\nempty sum, which we understand to be equal to zero. Finally, fcan be\nsolved for from the equation\nf′(z) =λeczf(z)+(1−λecz)α(z). (1.3)\nTheorem 1.3 leaves open various questions. If n≥2 is given, for\nwhich choices of the non-zero complex parameters λ,c, andan, does\nthe equation (1.2) have a non-trivial solution α(z) that is meromorphic\nin the complex plane? When such a solution αexists, when is the\nfunction fobtained from (1.3) entire, and if it is, when is αa small\nfunction for f? We will discuss these questions for n= 2 and n= 3.\nThe general case remains open.\nThis paper has been organized as follows. After some elementary\nremarks (mainly for the case n= 2) in Section 2, we recall some basic\nfacts of Stirling numbers in Section 3. These facts will then be ex-\ntensively used in the next Section 4 to construct expressions for t he\nderivatives f(n)off. The next two sections will then be devoted to\nthe differential equation satisfied α(z), first constructing the equation\nitself in Section 5, and then considering the possibilities to solve the\nequation in Section 6.\n2.Initial remarks on Lahiri’s work\nWe first proceed to considering the case n= 2. In this case, the\ndifferential equations satisfied by fmay be written as follows:\nf′(z) =λeczf(z)+(1−λecz)α(z) (2.1)\nand\nL(f)(z) =a2λ2e2czf(z)+(1−a2λ2e2cz)α(z).(2.2)4 AIMO HINKKANEN AND ILPO LAINE\nThe solution to equation (2.1) may be written as\nf(z) =Cexp((λ/c)ecz)+exp((λ/c)ecz)/integraldisplay\nexp(−(λ/c)ecz)(1−λecz)α(z)dz,\n(2.3)\nCbeing a constant of integration. Obviously, we now have\nf′−α\nf−α=λecz,\nso the CM-condition for f,f′is satisfied. Observe that (1 −λecz)α(z) in\nthe expression of f(z) is entire by (2.1), hence the integration in (2.3)\nbrings no problems (although it usually cannot be expressed in explicit\nform). To verify the CM-condition for fandL(f), we may proceed\nby recalling a1=−a2cfrom Theorem 1.2, differentiating (2.1), and\nsubstituting this and (2.1) into\n−a2cf′(z)+a2f′′(z)−a2λ2e2czf(z) = (1−a2λ2e2cz)α(z).\nWe obtain, after simplification, that\na2(1−λecz)α′(z)−(1+a2c−a2λecz)α(z) = 0.(2.4)\nIt is now immediate to verify that\nα(z) =/tildewideCe(1+1\na2c)cz(λecz−1)−1+1\nc−1\na2c. (2.5)\nRecalling that\n(1−λecz)α(z) =−/tildewideCe(1+1\na2c)cz(λecz−1)1\nc−1\na2c\nhas to be an entire function, we must have\n1\nc−1\na2c=s∈ {0}∪N. (2.6)\nMoreover, we must have sc/ne}ationslash= 1. Therefore, a2= 1/(1−sc), where\ns∈ {0}∪N. In particular, if s= 0, then a2= 1.\nExample. Takes= 1. Then a2=1\n1−canda1=−a2c=c\nc−1.\nIn this special case, see (2.5), we get α(z) =ez, with the constant of\nintegration being = 1. Recalling (2.3), we obtain\nf(z) = exp/parenleftbiggλ\ncecz/parenrightbigg\n+ez. (2.7)\nComputing L(f) =c\nc−1f′(z)+1\n1−cf′′(z), we get\nL(f) =ez+1\n1−cλ2e2czexp/parenleftbiggλ\ncecz/parenrightbigg\n.\nBy computation, we now get\n(f′−α)/(f−α) = (f′−ez)/(f−ez) =λecz\nand\n(L(f)−α)/(f−α) = (L(f)−ez)/(f−ez) =1\n1−cλ2e2cz=a2λ2e2cz.VALUE SHARING AND STIRLING NUMBERS 5\nObserve that in this case, we clearly have that α(z) =ezis a small\nfunction with respect to fin (2.7).\nWe next proceed to considering the cases when s= 0 ors≥2. Then\nthe integral in (2.3) cannot be expressed in explicit form. Indeed,\nsuppose that (2.5) holds where1\nc−1\na2c=sis a non-negative integer.\nThen with the notation u=λeczthe integrand on the right hand side\nof (2.3) is a sum where mruns from 0 to sand where the generic term\nis a constant multiple (with the constant depending on m) of\ne−(1/c)uum−1−s+(1/c)du.\nWithv=−(1/c)u, this is a constant multiple of\nevvm−1−s+(1/c)dv.\nThis can be integrated explicitly if m−1−s+(1/c) is a non-negative\ninteger, and hence for every mwith 0≤m≤sif 1/cis an integer with\n1/c≥s+1. Assuming that1 /c=ν≥s+1we thenhave a2=ν/(ν−s)\nso thata2is a rational number with a2≥1. In other cases the integral\ncan be expressed in terms of the incomplete gamma function.\nRemark. We next point out a couple of necessary conditions for the\nexistence of an entire function fto satisfy equations (2.1) and (2.2).\nSuppose first that αis entire. Looking at points ˜ zsuch that λec˜z= 1,\nthen it is immediate to observe that f′(˜z)−f(˜z) = 0. To consider the\ncase when αis meromorphic in general, eliminating αfrom equations\n(2.1) and (2.2), we obtain\nλecz(1−anλn−1e(n−1)cz)f+(a1(1−λecz)−(1−anλnencz))f′\n1−λecz\n=−(a2f′′+···+anf(n)). (2.8)\nThe right hand side of (2.8) is entire, hence the same applies with\nthe left hand side as well. This means that the numerator\nλecz(1−anλn−1e(n−1)cz)f+(a1(1−λecz)−(1−anλnencz))f′\nvanishes at all points ˜ zsuch that λec˜z= 1. This means that we get two\npossiblenecessary conditions: Either(1) an= 1, or(2) f′(˜z)−f(˜z) = 0.\n3.Stirling numbers\nIn what follows, it appears that Stirling numbers may be used to\nobtain a formula for the derivatives f(n)in terms of f. Indeed, from\n(2.1) we can get a formula for f(n)of the form\nf(n)=Anf+Bn (3.1)\nwhere neither AnnorBninvolves for its derivatives. We will find\nexplicit expressions for AnandBn. They will involve Stirling numbers\nof the second kind, see, e.g., [1], Chapter 8.6 AIMO HINKKANEN AND ILPO LAINE\nStirlingnumbersofthesecondkind S(n,k)aredefinedasthenumber\nof ways of partitioning a set of nelements into k(non-empty) subsets.\nFor this to make sense, we must have 1 ≤k≤n. In order to use\nS(n,k) in formulas in a more general way, it is customary to define\nS(n,k) = 0 if 1 ≤n < k; to define S(n,0) = 0 if n≥1; and to define\nS(0,0) = 1. We also define S(n,−1) = 0 for n≥1.\nWe will need the identity\nS(n,k) =S(n−1,k−1)+kS(n−1,k). (3.2)\nTo prove this, one can argue as follows. To partition {1,2,...,n}into\nksubsets, we can partition {1,2,...,n−1}intok−1 subsets and use\n{n}as thekthsubset. This gives rise to the term S(n−1,k−1). Or\nwe can partition {1,2,...,n−1}intoksubsets and for each choice,\ninclude the element nin any one of those ksubsets. This gives rise to\nthe term kS(n−1,k).\nThere are many formulas for S(n,k). We will not need all of them\nbut we mention some of them for completeness. We have\nS(n,k) =/summationdisplay\n1a1−12a2−13a3−1···kak−1(3.3)\nwhere the sum is taken over all positive integers aisuch that a1+a2+\n···+ak=n. Since each ai≥1, this obviously requires that k≤n. We\nalso have\nS(n,k) =1\nk!k/summationdisplay\nm=1(−1)k−m/parenleftbiggk\nm/parenrightbigg\nmn=k/summationdisplay\nm=1(−1)k−m\nm!(k−m)!mn.(3.4)\nWe further have\nS(n,k) =n!\nk!/summationdisplay 1\na1!a2!···ak!(3.5)\nwhere the sum is taken over all non-negative integers aisuch that\na1+a2+···+ak=n. And we have\nS(n,k) =/summationdisplay n!\na1!a2!···an!/parenleftbigg1\n1!/parenrightbigga1/parenleftbigg1\n2!/parenrightbigga2\n···/parenleftbigg1\nn!/parenrightbiggan\n(3.6)\nwhere the sum is taken over all non-negative integers aisuch that\na1+a2+···+an=kanda1+2a2+3a3+···+nan=n.\nTo explain the connection to Stirling numbers s(n,k) of the first\nkind, we begin with the definition. With the notation ( t)0= 1 and, for\nn≥1,\n(t)n=t(t−1)(t−2)···(t−n+1)\nwe have, for n≥0,\n(t)n=n/summationdisplay\nk=0s(n,k)tk(3.7)VALUE SHARING AND STIRLING NUMBERS 7\nand this defines s(n,k). Clearly s(n,0) = 0,s(n,1) = (−1)n−1(n−1)!,\nands(n,n) = 1 for all n≥1. We have already defined the numbers\nS(n,k) but they could also be defined for n≥0 by\ntn=n/summationdisplay\nk=0S(n,k)(t)k.\nWe define s(n,k) =S(n,k) = 0 when k > nands(0,0) =S(0,0) = 1.\nWe recall some known results from [1], pp. 278, 281. We have\n|s(n,k)|= (−1)n−ks(n,k).\nUsing the Kronecker symbol δm,n, defined by δn,n= 1 and by δm,n= 0\nform/ne}ationslash=n, we have\nn/summationdisplay\nr=ks(n,r)S(r,k) =δn,k, (3.8)\nn/summationdisplay\nr=kS(n,r)s(r,k) =δn,k.\nThere is the recursion formula ([1], p. 293)\ns(n+1,k) =s(n,k−1)−ns(n,k). (3.9)\nBy [1], p. 323, we have\ns(n,n−1) =−n(n−1)\n2. (3.10)\nMoreover, by [1], Corollary 8.2, s(n,n) = 1.\nIn what follows, we often make use of the well-known identity (from\nPascal’s triangle)\n/parenleftbiggn\nk/parenrightbigg\n=/parenleftbiggn−1\nk/parenrightbigg\n+/parenleftbiggn−1\nk−1/parenrightbigg\n. (3.11)\nWe recall that/parenleftbign\nk/parenrightbig\n= 0 ifk <0 or ifn < k.\n4.The formula for the derivatives of f\nLemma 4.1. Suppose that (2.1) holds. Then for each n≥1, (3.1) is\nvalid with AnandBngiven as follows. We have\nAn=n/summationdisplay\nk=1bn,kekcz(4.1)\nwhere\nbn,k=S(n,k)λkcn−k(4.2)\nand\nBn=n−1/summationdisplay\nk=0βn,kα(k)(4.3)8 AIMO HINKKANEN AND ILPO LAINE\nwhere\nβn,k=n−k/summationdisplay\nj=0(ζn,k,j−cεn,k,j)cn−k−j−1λjejcz. (4.4)\nHere for n≥1,0≤k≤n−1,0≤j≤n−k, we have\nζn,k,j=n−1−k−j/summationdisplay\nm=0jm/parenleftbiggk+m\nk/parenrightbigg\nS(n−1−k−m,j) (4.5)\nfor2≤j≤n−k−1,ζn,k,n−k= 0(henceζn,0,n= 0), and\nεn,k,j=n−k−j/summationdisplay\nm=0jm/parenleftbiggk+m\nk/parenrightbigg\nS(n−1−k−m,j−1) (4.6)\nfor2≤j≤n−k−1,εn,k,n−k= 1(henceεn,0,n= 1). The equations\nabove are valid when j≥2, while\nζn,k,1=n−k−2/summationdisplay\nm=0/parenleftbiggk+m\nk/parenrightbigg\nS(n−1−k−m,1) =n−k−2/summationdisplay\nm=0/parenleftbiggk+m\nk/parenrightbigg\n=/parenleftbiggn−1\nk+1/parenrightbigg\n(4.7)\nwhen0≤k≤n−2,ζn,n−1,1=/parenleftbign−1\nn/parenrightbig\n= 0, and\nζn,k,0= 0 (4.8)\nwhen0≤k≤n−2,ζn,n−1,0= 1and in particular ζ1,0,0= 1,\nεn,k,1=/parenleftbiggn−1\nk/parenrightbigg\n, (4.9)\nand\nεn,k,0= 0. (4.10)\nNote that if we were to apply (4.5) with j=n−k, we would get\na sum where mgoes from 0 to −1. Interpreting this as an empty\nsum, hence equal to zero, we also obtain ζn,k,n−k= 0, which we noted\nseparately. For completeness, we also define ζ0,k,j= 0 andε0,k,j= 0 for\nallkandj.\nBefore proceeding, we obtain some identities for the quantities ζn,k,j\nandεn,k,jdefined in Lemma 4.1.\nLemma 4.2. Ifn≥1,1≤k≤n,0≤j≤n+1−k, we have\nζn+1,k,j=jζn,k,j+ζn,k−1,j (4.11)\nand\nεn+1,k,j=jεn,k,j+εn,k−1,j (4.12)\nwith the convention that ζn,k,j= 0andεn,k,j= 0ifk≥nor ifj≥\nn−k+1. We further have\nζ1,0,0= 1, ε1,0,0= 0, ζ1,0,1= 0, ε1,0,1= 1 (4.13)\nand\nζn+1,0,j=jζn,0,j+S(n,j), ε n+1,0,j=jεn,0,j+S(n,j−1).(4.14)VALUE SHARING AND STIRLING NUMBERS 9\nProof of Lemma 4.2. The equation (4.13) follows directly from\nthe definitions (4.7)–(4.10).\nForj= 0, (4.12) follows from (4.10) since all quantities vanish.\nForj= 0, (4.11) follows from (4.7) since it implies that\nζn+1,k,0−ζn,k−1,0= 0−0 = 0\nif 1≤k≤n−1, while if k=nthen the line after (4.7) implies that\nζn+1,k,0−ζn,k−1,0=ζn+1,n,0−ζn,n−1,0= 1−1 = 0.\nForj= 1, (4.12) follows from (4.9) since it implies that\nεn+1,k,1−εn,k,1−εn,k−1,1=/parenleftbiggn\nk/parenrightbigg\n−/parenleftbiggn−1\nk/parenrightbigg\n−/parenleftbiggn−1\nk−1/parenrightbigg\n= 0\nby (3.11).\nForj= 1, (4.11) follows from (4.7) since it implies that\nζn+1,k,1−ζn,k,1−ζn,k−1,1=/parenleftbiggn\nk+1/parenrightbigg\n−/parenleftbiggn−1\nk+1/parenrightbigg\n−/parenleftbiggn−1\nk/parenrightbigg\n= 0\nby (3.11).\nTo prove (4.11) for j≥2, note that by (4.5), we have\nζn+1,k,j−ζn,k−1,j\n=n−k−j/summationdisplay\nm=0jmS(n−k−m,j)/parenleftbigg/parenleftbiggk+m\nk/parenrightbigg\n−/parenleftbiggk−1+m\nk−1/parenrightbigg/parenrightbigg\n=n−k−j/summationdisplay\nm=0jmS(n−k−m,j)/parenleftbiggk+m−1\nk/parenrightbigg\nby (3.11). If m= 0, we have/parenleftbigk+m−1\nk/parenrightbig\n= 0, so that in the sum, mstarts\natm= 1. We set µ=m−1 and obtain\nζn+1,k,j−ζn,k−1,j=jn−k−1−j/summationdisplay\nµ=0jµS(n−k−µ−1,j)/parenleftbiggk+µ\nk/parenrightbigg\n=jζn,k,j\nas required. This proves (4.11) for j≥2.\nThe proof of (4.12) for j≥2 is similar to our proof of (4.11) above\nand is omitted.\nWe finally consider (4.14). If j≥2, from (4.5) we get\nζn+1,0,j−jζn,0,j=n−j/summationdisplay\nm=0jmS(n−m,j)−jn−1−j/summationdisplay\nm=0jmS(n−1−m,j).\nIn the first sum, we take separately the term corresponding to m= 0,\nwhich isS(n,j), and in the remaining sum we set µ=m−1 so that µ\ngoes from 0 to n−j−1. We see that the two remaining sums cancel\nout, and we obtain (4.14). The proof of (4.14) for εn+1,0,jwhenj≥2\nis similar.10 AIMO HINKKANEN AND ILPO LAINE\nIfj= 1, then (4.14) reads\nζn+1,0,1=ζn,0,1+S(n,1) =ζn,0,1+1, ε n+1,0,1=εn,0,1+S(n,0) =εn,0,1.\nSince by (4.7) and (4.9) we have\nζn,0,1=/parenleftbiggn−1\n1/parenrightbigg\n=n−1\nand\nεn,0,1=/parenleftbiggn−1\n0/parenrightbigg\n= 1,\nwe obtain (4.14) for j= 1.\nIfj= 0, then (4.14) holds by (4.8) and (4.10) since S(n,0) = 0 and\nwe interpret S(n,−1) = 0.\nProof of Lemma 4.1. Differentiating (3.1) and using (2.1), we\nobtain, writing E=λeczfor brevity,\nf(n+1)=A′\nnf+Anf′+B′\nn= (A′\nn+AnE)f+(An(1−E)α+B′\nn),(4.15)\nso that\nAn+1=A′\nn+AnE, B n+1=An(1−E)α+B′\nn.(4.16)\nBy (2.1), we have A1=E=λeczwhile (4.1) and (4.2) state that\nA1=b1,1ecz=S(1,1)λecz=λecz\nsinceS(1,1) = 1. This proves (4.1) when n= 1.\nWe now prove (4.1) in general by induction on n. We assume that\n(4.1) is valid for a certain n≥1. Then (4.16) gives\nAn+1=A′\nn+AnE=n/summationdisplay\nk=1bn,kkcekcz+n/summationdisplay\nk=1bn,kλe(k+1)cz\nand if we write this in the form\nn+1/summationdisplay\nk=1bn+1,kekcz\nthen we must have, for 1 ≤k≤n+1,\nbn+1,k=kcbn,k+λbn,k−1\nwith the convention that bn,0= 0. We are assuming that for our nthe\nequation (4.2) holds, and using it above gives\nbn+1,k=kcS(n,k)λkcn−k+λS(n,k−1)λk−1cn−k+1\n=λkcn−k+1(kS(n,k)+S(n,k−1)) =λkcn−k+1S(n+1,k)\nwhere for the last step we have used (3.2). Thus (4.1) and (4.2) are\nvalid with nreplaced by n+1 and this completes the induction proof\nforAn.\nFrom (2.1) we obtain B1= (1−E)αwhile (4.3)–(4.6) state that\nB1=β1,0α= (ζ1,0,0−cε1,0,0+(ζ1,0,1−cε1,0,1)c−1E)α.VALUE SHARING AND STIRLING NUMBERS 11\nThese two expressions for B1are identical if, and only if,\nζ1,0,0= 1, ε1,0,0= 0, ζ1,0,1= 0, ε1,0,1= 1,\nwhich is true by (4.13).\nTo prove the formula for Bnby induction on n, assume now that it\nis valid for a certain n≥1. We already have the formula for An. By\n(4.16) and our assumptions,\nBn+1=An(1−E)α+B′\nn\n= (1−E)αn/summationdisplay\np=1bn,pepcz+n−1/summationdisplay\nk=0/parenleftBiggn−k/summationdisplay\nj=0(ζn,k,j−cεn,k,j)cn−k−j−1λjejcz/parenrightBigg\nα(k+1)\n+n−1/summationdisplay\nk=0α(k)/parenleftBiggn−k/summationdisplay\nj=0(ζn,k,j−cεn,k,j)cn−k−j−1jcλjejcz/parenrightBigg\nand if we write this in the form\nn/summationdisplay\nk=0βn+1,kα(k)\nthen we must have\nβn+1,0= (1−E)n/summationdisplay\np=1bn,pepcz+n/summationdisplay\nj=0(ζn,0,j−cεn,0,j)cn−j−1jcλjejcz,\nβn+1,k=n−k+1/summationdisplay\nj=0(ζn,k−1,j−cεn,k−1,j)cn−k−jλjejcz\n+n−k/summationdisplay\nj=0(ζn,k,j−cεn,k,j)cn−k−j−1jcλjejcz(4.17)\nfor 1≤k≤n−1, and\nβn+1,n=1/summationdisplay\nj=0(ζn,n−1,j−cεn,n−1,j)c−jλjejcz. (4.18)\nWe treat these three cases separately:\n(a) We may write the expression for βn+1,0in the form\nn+1/summationdisplay\np=1γpλpepcz\nwhere for 1 ≤p≤nwe have (again with bn,0= 0))\nγp=bn,pλ−p−bn,p−1λ1−p+(ζn,0,p−cεn,0,p)cn−p−1pc\n=S(n,p)cn−p−S(n,p−1)cn−p+1+(ζn,0,p−cεn,0,p)cn−p−1pc\nwhile\nγn+1=−bn,nλ−n=−S(n,n) =−1.12 AIMO HINKKANEN AND ILPO LAINE\nAccording to (4.4), we should have\nβn+1,0=n+1/summationdisplay\nj=0(ζn+1,0,j−cεn+1,0,j)cn−jλjejcz.(4.19)\nThis will be the case if, and only if,\nζn+1,0,0=εn+1,0,0= 0 (4.20)\nand for 1 ≤j≤n,\nζn+1,0,j−cεn+1,0,j=S(n,j)−S(n,j−1)c+(ζn,0,j−cεn,0,j)j.(4.21)\nForj=p=n+1 we require that\n−1 = (ζn+1,0,n+1−cεn+1,0,n+1)c−1,\nthat is,\nεn+1,0,n+1= 1, ζn+1,0,n+1= 0, (4.22)\nwhich has been noted in the definitions in Lemma 4.1. The equation\n(4.21) is valid if, and only if,\nζn+1,0,j=S(n,j)+jζn,0,j, εn+1,0,j=S(n,j−1)+jεn,0,j.(4.23)\nBut these are the formulas (4.14) that we have proved already.\n(b)Wenext consider βn+1,n, whichaccording to(4.4)shouldbeequal\nto\n1/summationdisplay\nj=0(ζn+1,n,j−cεn+1,n,j)c−jλjejcz.\nThis agrees with (4.18) if, and only if, for j= 0 and for j= 1 we have\nζn+1,n,j=ζn,n−1,j, εn+1,n,j=εn,n−1,j.\nThis might seem unlikely but it is valid since we have for all n≥1,\nζn,n−1,0= 1, ζn,n−1,1= 0, ε n,n−1,0= 0, εn,n−1,1= 1,(4.24)\nas noted in the definitions in Lemma 4.1.\n(c) It remains to consider βn+1,kfor 1≤k≤n−1. According to\n(4.4) we should have\nβn+1,k=n+1−k/summationdisplay\nj=0(ζn+1,k,j−cεn+1,k,j)cn−k−jλjejcz.\nComparing this to (4.17) we find that we need to prove that for each j\nwith 0≤j≤n+1−k, we have\nζn+1,k,j−cεn+1,k,j= (ζn,k−1,j−cεn,k−1,j)+(ζn,k,j−cεn,k,j)j\nwhere the second term on the right is present only when j≤n−k.\nThis means that we need to have, for 0 ≤j≤n−k,\nζn+1,k,j=ζn,k−1,j+jζn,k,j, εn+1,k,j=εn,k−1,j+jεn,k,j.VALUE SHARING AND STIRLING NUMBERS 13\nBut these are the formulas (4.11) and (4.12) that we have proved a l-\nready. When j=n−k+1, we need to have\nζn+1,k,n−k+1=ζn,k−1,n−k+1, ε n+1,k,n−k+1=εn,k−1,n−k+1.\nThese equations are valid (they read 0 = 0 and 1 = 1) since by the\ndefinitions in Lemma 4.1 we have ζn,k,n−k= 0 and εn,k,n−k= 1.\nThis completes the proof of Lemma 4.1.\n5.The differential equation for α(z)\nWe first note that by Theorem 1.2 due to Lahiri, in the case (v) we\nare considering, we must have\nL(f) =anλnenczf+(1−anλnencz)α (5.1)\nwhere the initial definition of L(f) is\nL(f) =n/summationdisplay\nj=0ajf(j).\nHowever, we also assume that an/ne}ationslash= 0 and by Lahiri, we must have\na0= 0 and, if n= 2, then a1+a2c= 0, and if n≥3, then\nn/summationdisplay\nk=1(−1)k−1ak/parenleftbigg2an−1\nn(n−1)an/parenrightbiggk−1\n= 0.\nNow by Lemma 4.1,\nL(f) =n/summationdisplay\nj=1ajf(j)=n/summationdisplay\nj=1aj(Ajf+Bj) =fn/summationdisplay\nj=1ajAj+n/summationdisplay\nj=1ajBj.\nIn view of this, (5.1) reads\nC1f=C2 (5.2)\nwhere\nC1=−anλnencz+n/summationdisplay\nj=1ajAj (5.3)\nand\nC2= (1−anλnencz)α−n/summationdisplay\nj=1ajBj. (5.4)\nWe next write (2.1) in the form\n(exp(−(λ/c)ecz)f)′= exp(−(λ/c)ecz)(1−λecz)α(z).(5.5)\nIntegrating, we obtain\nexp(−(λ/c)ecz)f(z) =e−λ/cf(0)+/integraldisplayz\n0exp/parenleftbig\n−(λ/c)ecζ/parenrightbig\n(1−λecζ)α(ζ)dζ.\n(5.6)\nThus, whenever αis known, we get fin this way. Recall that the\nfunction (1 −λecz)α(z) is entire by (1.3) since fis entire.14 AIMO HINKKANEN AND ILPO LAINE\nBy Lemma 4.1, we have\nC1=−anλnencz+n/summationdisplay\nj=1ajj/summationdisplay\nk=1S(j,k)λkcj−kekcz\n=n−1/summationdisplay\np=1λpepcz/parenleftBiggn/summationdisplay\nj=pajS(j,p)cj−p/parenrightBigg\n. (5.7)\nIfC1(and hence C2) does not vanish identically and if we assume\nthatαis a small function with respect to f, then (5.2) implies after\nsome work that the Nevanlinna characteristic function of fsatisfies\nT(r,f) =K1r+o(r)\nfor some positive constant K1. But then (5.2) can be written in the\nform\nα= (D0)−1/parenleftBigg\nC1f−n−1/summationdisplay\nk=1Dkα(k)/α/parenrightBigg\nwhere each Dkis a linear combination of functions of the form epczwith\nconstant coefficients. This implies that m(r,α) =K2r+o(r) for some\nK2>0. Since N(r,α)≤N(r,1/(1−λecz)) =O(r), we obtain that\nT(r,α) =Kr+o(r) for some K >0, which is a contradiction with the\nassumption that αis a small function with respect to f. We conclude\nthatC1is identically zero, so that by (5.7) we have\nn/summationdisplay\nj=pajS(j,p)cj−p= 0 (5.8)\nfor each pwith 1≤p≤n−1. This can be used to solve for each aj\nfor 1≤j≤n−1 in terms of an.\nWhenp=n−1, (5.8) yields\nan−1S(n−1,n−1)+anS(n,n−1)c= 0.\nNowS(n−1,n−1) = 1 and S(n,n−1) =/parenleftbign\n2/parenrightbig\n=n(n−1)/2, so that\nthis reads\nan−1+n(n−1)\n2can= 0.\nWhenn= 2, this gives a1+ca2= 0, which Lahiri obtained already.\nWhenp=n−2≥1, (5.8) yields\nan−2S(n−2,n−2)+an−1S(n−1,n−2)c+anS(n,n−2)c2= 0.\nSubstituting here an−1=−n(n−1)\n2canwe get\n−an−2=anc2(−(n−1)(n−2)\n2n(n−1)\n2+S(n,n−2))\nso that, as a calculation shows,\nan−2=anc2n(n−1)(n−2)(3n−1)\n24. (5.9)VALUE SHARING AND STIRLING NUMBERS 15\nAftertheseexamples, welookfor akintheform ak= (−1)n−kancn−kdk\nfork≥1. We set dn= 1 so that this equation is valid also when k=n.\nIt will turn out that dk>0 for all k. We will show that the dkare\ngiven by the absolute values of the Stirling numbers of the first kind.\nFirst note that (5.8) reads (since S(p,p) = 1)\ndp= (−1)n−p+1S(n,p)−n−1/summationdisplay\nj=p+1(−1)j−pS(j,p)dj. (5.10)\nWhenp=n−1, the sum is empty, and we get dn−1=S(n,n−1).\nThe equation (5.10) together with dn= 1 determines the numbers dk,\n1≤k≤n, uniquely for each fixed n≥1. If we can show that the\nnumbers |s(n,k)|satisfy the same equations in place of dk, it will follow\nthatdk=|s(n,k)|. Thusdn=s(n,n) =|s(n,n)|. Withdkreplaced by\n|s(n,k)|= (−1)n−ks(n,k), (5.10) reads\n(−1)n−ps(n,p) = (−1)n−p+1S(n,p)−n−1/summationdisplay\nj=p+1(−1)j−pS(j,p)(−1)n−js(n,j)\nor\ns(n,p) =−S(n,p)−n−1/summationdisplay\nj=p+1S(j,p)s(n,j).\nOn the other hand, (3.8) gives\nδn,p=n/summationdisplay\nj=ps(n,j)S(j,p) =n−1/summationdisplay\nj=p+1s(n,j)S(j,p)+s(n,p)S(p,p)+s(n,n)S(n,p).\nUsingS(p,p) =s(n,n) = 1 and noting that p/ne}ationslash=n, soδn,p= 0, we get\ns(n,p) =−S(n,p)−n−1/summationdisplay\nj=p+1s(n,j)S(j,p),\nas required. This proves that for each fixed n≥2, we have dk=\n|s(n,k)|= (−1)n−ks(n,k) for 1≤k≤n.\nThus\naj= (−1)n−jancn−jdj= (−1)n−jancn−j(−1)n−js(n,j) =ancn−js(n,j)\n(5.11)\nfor 1≤j≤n. This proves (1.1) of Theorem 1.3.\nNowthatwehave foundtheformulasforthenumbers aj, weconsider\nthe constraint obtained by Lahiri for n≥3, that is,\nn/summationdisplay\nk=1(−1)k−1ak/parenleftbigg2an−1\nn(n−1)an/parenrightbiggk−1\n= 0.\nWe have\n2an−1\nn(n−1)an=−c16 AIMO HINKKANEN AND ILPO LAINE\nso we get\nn/summationdisplay\nk=1ck−1ak= 0\nand hence\nn/summationdisplay\nk=1(−1)n−kancn−kdkck−1= 0\nand so\nn/summationdisplay\nk=1(−1)kdk= 0. (5.12)\nWhenn= 2,d2= 1 and d1=2(2−1)\n2= 1 so that (5.12) holds. When\nn= 3,d3= 1,d2=3(3−1)\n2= 3, and by (5.9), d1=3(3−1)(3−2)(3·3−1)\n24= 2,\nso that again (5.12) holds.\nIn view of our formula for dk, the equation (5.12) reads\nn/summationdisplay\nk=1(−1)k(−1)n−ks(n,k) = 0,\nor\nn/summationdisplay\nk=1s(n,k) = 0,\nwhich follows for n≥2 from (5.12) by setting t= 1 and noting that\ns(n,0) = 0.\nWe have in general a1=ancn−1s(n,1) anda2=ancn−2s(n,2) so\nthat ifa1a2/ne}ationslash= 0, then a1/a2=cs(n,1)/s(n,2) =−c|s(n,1)/s(n,2)|.\nWhenn= 2, we have s(2,2) = 1 and |s(2,1)|= (2−1)! = 1, so that\na1/a2=−cas obtained by Lahiri, also when viewed in this way. For\nn≥3 we would not have a1/a2=−c.\nTheequation C2= 0, where C2isasin(5.4), isadifferential equation\nforαof ordern−1. We may now write (5.4) in the form\n0 = (1−anλnencz)α−n/summationdisplay\nj=1ancn−js(n,j)j−1/summationdisplay\nk=0βj,kα(k)(5.13)\nand further\n0 = (1 −anλnencz)α\n−n/summationdisplay\nj=1ancn−js(n,j)j−1/summationdisplay\nk=0α(k)j−k/summationdisplay\np=0(ζj,k,p−cεj,k,p)cj−k−p−1λpepcz\n= (1−anλnencz)α−cn−1ann−1/summationdisplay\nk=0Xkα(k)(5.14)VALUE SHARING AND STIRLING NUMBERS 17\nwhere\nXk=n/summationdisplay\nj=k+1j−k/summationdisplay\np=0s(n,j)c−k−p(ζj,k,p−cεj,k,p)λpepcz.\nWe can write\nXk=c−kn−k/summationdisplay\np=0(λ/c)pepczn/summationdisplay\nj=max{k+1,p+k}s(n,j)(ζj,k,p−cεj,k,p).\nThis can be expressed as\nXk=c−kn/summationdisplay\nj=k+1s(n,j)(ζj,k,0−cεj,k,0)\n+c−kn−k/summationdisplay\np=1(λ/c)pepczn/summationdisplay\nj=p+ks(n,j)(ζj,k,p−cεj,k,p).(5.15)\nIn view of the definitions in Lemma 4.1, according to which εj,k,0= 0,\nζj,k,0= 0 for 0 ≤k≤j−2, andζj,j−1,0= 1, we have\nXk=c−ks(n,k+1)\n+c−kn−k/summationdisplay\np=1(λ/c)pepczn/summationdisplay\nj=p+ks(n,j)(ζj,k,p−cεj,k,p).(5.16)\nThe next lemma, whose proof we will give after some remarks, shows\nhow one can simplify these expressions.\nLemma 5.1. Ifn≥1,0≤k≤n−1, we have\nn/summationdisplay\nj=p+ks(n,j)ζj,k,p=s(n−p,k+1) (5.17)\nwhen0≤p≤n−k−1, and\nn/summationdisplay\nj=p+ks(n,j)εj,k,p=s(n−p,k) (5.18)\nwhen0≤p≤n−k.\nIfp=n−k, then\nn/summationdisplay\nj=p+ks(n,j)ζj,k,p=s(n,n)ζk+p,k,p= 0. (5.19)\nWe first make some remarks. By Lemma 5.1, we have\nXk=c−k/parenleftBigg\ns(n,k+1)+n−k/summationdisplay\np=1(λ/c)pepcz(s(n−p,k+1)−cs(n−p,k))/parenrightBigg\n(5.20)18 AIMO HINKKANEN AND ILPO LAINE\nwith the convention that s(m1,m2) = 0 when m2> m1. In particular,\nsinces(n,n) = 1,s(n−1,n) = 0, and s(n−1,n−1) = 1, we have\nXn−1=c1−n(1−λecz) (5.21)\nso that the complete term in (5.14) involving α(n−1)is\n−an(1−λecz)α(n−1). (5.22)\nThis agrees with the term involving α′in (2.4) where n= 2 (note that\nwith this normalization, (5.14) ends up being (2.4) multiplied through\nby−1). Also\nX0=n/summationdisplay\np=0(λ/c)pepcz(s(n−p,1)−cs(n−p,0)) (5.23)\n=−c(λ/c)nencz+n−1/summationdisplay\np=0(λ/c)pepczs(n−p,1) (5.24)\n=−c(λ/c)nencz+n−1/summationdisplay\np=0(λ/c)pepcz(−1)n−p−1(n−p−1)!(5.25)\nsinces(n,1) = (−1)n−1(n−1)!. Hence in (5.14), the complete term\ninvolving αbut not its derivatives is\n/parenleftBigg\n1−ann−1/summationdisplay\np=0cn−p−1λpepcz(−1)n−p−1(n−p−1)!/parenrightBigg\nα. (5.26)\nWhenn= 2, this term is equal to\n(1−a2(−c+λecz))α\nas already given in (2.4) (note that with this normalization, (5.14) end s\nup being (2.4) multiplied through by −1).\nProof of Lemma 5.1. We first consider (5.17). Suppose that k≥0\nandp≥0. Then (5.17) can be considered for all nwithn≥k+p+1.\nWe prove (5.17) by a double induction. First, for each non-negative\nvalueκofk+p, we prove (5.17) by induction on nforn≥κ+ 1 =\nk+p+ 1. The non-negative values of κare handled by induction on\nthis value.\nWe begin with the value κ=k+p= 0 so that k=p= 0. Then, for\nn=κ+1 = 1, we have s(n−p,k+1) =s(1,1) = 1, and the left hand\nside of (5.17) is (since s(1,0) = 0 and s(1,1) = 1, and by the line after\n(4.8))\n1/summationdisplay\nj=0s(1,j)ζj,0,0=s(1,0)ζ0,0,0+s(1,1)ζ1,0,0=ζ1,0,0= 1.(5.27)\nThus (5.17) holds in this case. Still with κ=k+p= 0, suppose that\n(5.17) holds for a certain n≥1. Then the left hand side of (5.17) withVALUE SHARING AND STIRLING NUMBERS 19\nnreplaced by n+1 is equal to\nn+1/summationdisplay\nj=0s(n+1,j)ζj,0,0=n+1/summationdisplay\nj=1s(n+1,j)ζj,0,0\nsinces(n+ 1,0) = 0. When j≥2, we have ζj,0,0= 0 by (4.8). Thus\nwe only consider the single term s(n+ 1,1)ζ1,0,0. Nowζ1,0,0= 1 by\nthe line after (4.8). So we finally get s(n+1,1). But in our situation\n(k=p= 0) we have s(n+ 1−p,k+ 1) =s(n+ 1,1) so that holds\n(5.17) with nreplaced by n+1. This completes the base step for the\ninduction on κ.\nSuppose now that for a certain value of κ≥1, we have proved (5.17)\nwhenever k+p≤κ−1 andn≥k+p+1. We wish to prove that for\nall values of k≥0 andp≥0 such that k+p=κ, (5.17) holds for all\nn≥κ+1 =k+p+1. We prove this by induction on n. So suppose\nthatk≥0 andp≥0 are given such that k+p=κ, and begin with\nthe base step when n=k+p+1.\nSuppose that n=k+p+1. Then (5.17) reads\nk+p+1/summationdisplay\nj=p+ks(k+p+1,j)ζj,k,p=s(k+1,k+1) = 1.(5.28)\nOn the other hand, when j=k+p, we have\ns(k+p+1,j)ζj,k,p=s(k+p+1,k+p)ζk+p,k,p= 0\nsince by the definition in Lemma 4.1, we have ζn,k,n−k= 0. When\nj=k+p+1, we have\ns(k+p+1,j)ζj,k,p=s(k+p+1,k+p+1)ζk+p+1,k,p=ζk+p+1,k,p.\nBy the definition (4.5) in Lemma 4.1 (only the term m= 0 in the sum\nremains), we have\nζk+p+1,k,p=S(p,p) = 1\nwhenp≥2. Ifp= 1, then by (4.7)\nζk+p+1,k,p=ζ(k+2,k,1) =/parenleftbiggk+1\nk+1/parenrightbigg\n= 1.\nIfp= 0, then by the line after (4.8),\nζk+p+1,k,p=ζk+1,k,0= 1.\nThis proves (5.17) when n=k+p+1.\nStill keeping our fixed kandp, suppose that (5.17) holds for some n\nwithn≥k+p+1. Then the left hand side of (5.17) with nreplaced\nbyn+1 is equal to\nn+1/summationdisplay\nj=p+ks(n+1,j)ζj,k,p.20 AIMO HINKKANEN AND ILPO LAINE\nBy the recursion formula (3.9) s(n+1,k) =s(n,k−1)−ns(n,k) we\nhave (since s(n,n+1) = 0 and ζp+k,k,p= 0)\nn+1/summationdisplay\nj=p+ks(n+1,j)ζj,k,p=n+1/summationdisplay\nj=p+k(s(n,j−1)−ns(n,j))ζj,k,p\n=s(n,p+k−1)ζp+k,k,p+n/summationdisplay\nj=p+ks(n,j)ζj+1,k,p\n−nn/summationdisplay\nj=p+ks(n,j)ζj,k,p−ns(n,n+1)ζn+1,k,p\n=n/summationdisplay\nj=p+ks(n,j)ζj+1,k,p−nn/summationdisplay\nj=p+ks(n,j)ζj,k,p\n=n/summationdisplay\nj=p+ks(n,j)ζj+1,k,p−ns(n−p,k+1)\nwhere we have used our assumption (5.17). The equation (4.11) fro m\nLemma 4.2 implies that ζj+1,k,p=pζ(j,k,p)+ζ(j,k−1,p). Thus\nn/summationdisplay\nj=p+ks(n,j)ζj+1,k,p=n/summationdisplay\nj=p+ks(n,j)ζ(j,k−1,p)+pn/summationdisplay\nj=p+ks(n,j)ζ(j,k,p)\n=s(n−p,k)+ps(n−p,k+1).\nThe equation/summationtextn\nj=p+ks(n,j)ζ(j,k−1,p) =s(n��p,k) follows from our\ninduction assumption involving k+psince here the counterpart of k+p\nis (k−1)+p=κ−1 (regarding the term j=p+k−1 that may seem\nto be missing here, note that ζ(p+k−1,k−1,p) = 0 by the line after\n(4.5)). Hence\nn+1/summationdisplay\nj=p+ks(n+1,j)ζj,k,p=s(n−p,k)+ps(n−p,k+1)−ns(n−p,k+1)\n=s(n−p,k)−(n−p)s(n−p,k+1) =s(n+1−p,k+1)\nby the recursion formula (3.9). Thus (5.17) holds, whenever k+p=κ,\nwithnreplaced by n+1. This completes the proof of (5.17).\nWe prove (5.18) using induction on κ=k+p≥0 and on n≥\nκ=k+pin the same way as above. Assume first that κ= 0 so\nthatk=p= 0. Since we are assuming that n≥1, the first case for\nthe induction on nisn= 1. Then the right hand side of (5.18) is\ns(n−p,k) =s(1,0) = 0 and the left hand side of (5.18) is\n1/summationdisplay\nj=0s(1,j)εj,0,0= 0VALUE SHARING AND STIRLING NUMBERS 21\nby the definition of εj,0,0= 0 in (4.10) of Lemma 4.1. Thus (5.18) holds\nin this case.\nRemaining with k=p= 0, suppose that (5.18) has been proved for\nsomen≥1. Then, when nis replaced by n+ 1, the right hand side\nof (5.18) is s(n+1−p,k) =s(n+1,0) = 0 and the left hand side of\n(5.18) is\nn+1/summationdisplay\nj=0s(n+1,j)εj,0,0= 0\nsinceεj,0,0= 0 for all j. This completes the induction step for nand\nproves (5.18) for all n≥1 whenk=p= 0.\nSuppose then that κ≥1 and that for all k≥0 andp≥0 with\nk+p=κ−1, the equation (5.18) has been proved for all nsuch that\nn≥1 andn≥κ−1. Now fix arbitrary k≥0 andp≥0 withk+p=κ.\nWe need to prove by induction on nforn≥κthat (5.18) is valid. We\nbegin with the base step where n=κ=k+p. Then (5.18) reads (since\nthere is only one term in the sum)\ns(p+k,p+k)εp+k,k,p=s(n−p,k) =s(k,k) = 1.\nNows(p+k,p+k) = 1 and εp+k,k,p= 1 by the line after (4.6). Hence\n(5.18) holds in this case.\nKeeping our fixed κ, for the induction step on n, suppose that (5.18)\nholds for a certain nwithn≥κ. Then (5.18) with nreplaced by n+1\nreads\nn+1/summationdisplay\nj=p+ks(n+1,j)εj,k,p=s(n+1−p,k).\nUsing again the recursion formula (3.9), which gives s(n+ 1,j) =\ns(n,j−1)−ns(n,j), we obtain\nn+1/summationdisplay\nj=p+ks(n+1,j)εj,k,p=n+1/summationdisplay\nj=p+k(s(n,j−1)−ns(n,j))εj,k,p\n=s(n,p+k−1)εp+k,k,p+n/summationdisplay\nj=p+ks(n,j)εj+1,k,p\n−nn/summationdisplay\nj=p+ks(n,j)εj,k,p−ns(n,n+1)εn+1,k,p\n=s(n,p+k−1)+n/summationdisplay\nj=p+ks(n,j)εj+1,k,p−nn/summationdisplay\nj=p+ks(n,j)εj,k,p\n=s(n,p+k−1)+n/summationdisplay\nj=p+ks(n,j)εj+1,k,p−ns(n−p,k+1)22 AIMO HINKKANEN AND ILPO LAINE\nsinceεp+k,k,p= 1. By (4.12) of Lemma 4.2, we have\nεj+1,k,p=pεj,k,p+εj,k−1,p\nso that\nn/summationdisplay\nj=p+ks(n,j)εj+1,k,p=n/summationdisplay\nj=p+ks(n,j)εj,k−1,p+pn/summationdisplay\nj=p+ks(n,j)εj,k,p\n=n/summationdisplay\nj=p+k−1s(n,j)εj,k−1,p−s(n,p+k−1)εp+k−1,k−1,p+ps(n−p,k+1)\n=s(n−p,k)−s(n,p+k−1)+ps(n−p,k+1)\nsinceεp+k−1,k−1,p= 1. We conclude that\nn+1/summationdisplay\nj=p+ks(n+1,j)εj,k,p\n=s(n,p+k−1)+s(n−p,k)−s(n,p+k−1)\n+ps(n−p,k+1)−ns(n−p,k+1)\n=s(n−p,k)+ps(n−p,k+1)−ns(n−p,k+1)\n=s(n−p,k)−(n−p)s(n−p,k+1) =s(n+1−p,k)\nas required, where we have used the recursion formula (3.9) again. It\nfollows that (5.18) holds with nreplaced by n+1, and the induction\nproof is complete. This proves (5.18) in all cases considered.\nTheequation(5.19),thatis, s(n,n)ζk+p,k,p= 0,isvalidsince ζk+p,k,p=\n0 by the line after (4.5). This completes the proof of Lemma 5.1.\nIn view of Lemma 5.1, the differential equation (5.14) for αcan be\nwritten as\n0 =/parenleftBigg\n1−ann−1/summationdisplay\np=0cn−p−1λpepcz(−1)n−p−1(n−p−1)!/parenrightBigg\nα (5.29)\n−ann−1/summationdisplay\nk=1α(k)cn−1−k/parenleftBigg\ns(n,k+1)+n−k/summationdisplay\np=1(λ/c)pepcz(s(n−p,k+1)−cs(n−p,k))/parenrightBigg\n.\nTaking into account (5.22), we may also write\n0 =/parenleftBigg\n1−ann−1/summationdisplay\np=0cn−p−1λpepcz(−1)n−p−1(n−p−1)!/parenrightBigg\nα\n−ann−2/summationdisplay\nk=1α(k)cn−1−k/parenleftBigg\ns(n,k+1)+n−k/summationdisplay\np=1(λ/c)pepcz(s(n−p,k+1)−cs(n−p,k))/parenrightBigg\n−an(1−λecz)α(n−1), (5.30)VALUE SHARING AND STIRLING NUMBERS 23\nwhich is a linear differential equation for αof ordern−1. When n= 2,\nthemiddlelinein(5.30)representsanemptysum, whichweunderstan d\nto be equal to zero. This completes the proof of Theorem 1.3.\n6.Solving the differential equation for α\nThe solutions to the differential equation (5.30) for αhave already\nbeen discussed when n= 2.\nWe proceed to considering the case n= 3. Recall that we now have\nto see under what circumstances, in general, the equation (5.30) h as a\nnon-trivial meromorphic solution αso that (1 −λecz)α(z) is entire.\nEquation (5.30) now reads\n0 =/parenleftBigg\n1−a32/summationdisplay\np=0c2−pλpepcz(−1)2−p(2−p)!/parenrightBigg\nα\n−a3α′c/parenleftBigg\ns(3,2)+2/summationdisplay\np=1(λ/c)pepcz(s(3−p,2)−cs(3−p,1))/parenrightBigg\n−a3(1−λecz)α���′\n=/parenleftbig\n1−a3/parenleftbig\n2c2−cλecz+λ2e2cz/parenrightbig/parenrightbig\nα\n−a3α′c/parenleftbig\ns(3,2)+(λ/c)ecz(s(2,2)−cs(2,1))+(λ/c)2e2cz(s(1,2)−cs(1,1))/parenrightbig\n−a3(1−λecz)α′′\n=/parenleftbig\n1−a3/parenleftbig\n2c2−cλecz+λ2e2cz/parenrightbig/parenrightbig\nα\n−a3/parenleftbig\n−3c+λecz(1+c)−λ2e2cz/parenrightbig\nα′\n−a3(1−λecz)α′′. (6.1)\nRetaining only the last form for clarity, we have\n0 =/parenleftbig\n1−a3/parenleftbig\n2c2−cλecz+λ2e2cz/parenrightbig/parenrightbig\nα\n−a3/parenleftbig\n−3c+λecz(1+c)−λ2e2cz/parenrightbig\nα′\n−a3(1−λecz)α′′. (6.2)\nWe may now write our equation for αin the form\nα′′+−3c+(1+c)λecz−λ2e2cz\n1−λeczα′−1/a3−2c2+cλecz−λ2e2cz\n1−λeczα= 0.\n(6.3)\nWe write this briefly as α′′+a1(z)α′+a0(z)α= 0, where\na1(z) =−3c+(1+c)λecz−λ2e2cz\n1−λecz\nand\na0(z) =−1/a3−2c2+cλecz−λ2e2cz\n1−λecz.24 AIMO HINKKANEN AND ILPO LAINE\nAs is well-known, see, e.g., [4], p. 74, we may transform equation\n(6.3) into the form\nB′′+A(z)B= 0,\nwhere\nA(z) =a0(z)−1\n4a1(z)2−1\n2a′\n1(z)\nand\nB(z) =α(z)exp/parenleftbigg/integraldisplay/parenleftbigg1\n2a1(z)/parenrightbigg/parenrightbigg\n= (λecz−1)α(z)e−3cz/2eλ\n2cecz.\nBy elementary computation, we obtain\nB′′(z)+/parenleftbigg\n−1−c2\n4−λecz−λ2\n4e2cz+/parenleftbigg1\na3−1/parenrightbigg1\nλecz−1/parenrightbigg\nB(z) = 0.\n(6.4)\nThen, of course,\n(λecz−1)α(z) =e3cz/2e−λ\n2ceczB(z) (6.5)\nand\nf′(z) =λeczf(z)−e3cz/2e−λ\n2ceczB(z). (6.6)\nSincef(z) is entire, it follows from (6.6) that B(z) has to be entire as\nwell and so, from (6.5), that (1 −λecz)α(z) is entire.\n(1) Ifa3= 1, then equation (6.4) takes the form\nB′′(z)+/parenleftbigg\n−1−c2\n4−λecz−λ2\n4e2cz/parenrightbigg\nB(z) = 0.(6.7)\nThen it is classically well-known that all solutions to (6.7) are entire\nfunctions, forming a two-dimensional linear space. Observe that w e\nmay use Mathematica to solve (6.7):\nB(z) = (ecz)√\n1+4/c2/2e−λ\n2cecz/parenleftBigg\nC1U/parenleftBigg\n1\n2(1+/radicalbig\n1+4/c2)+1\nc,1+/radicalbig\n1+4/c2,λ\ncecz/parenrightBigg\n+\n+C2L√\n1+4/c2\n−(2+c+c√\n1+4/c2)/(2c)/parenleftbiggλ\ncecz/parenrightbigg/parenrightBigg\n, (6.8)\nwhereU(a,b,z), resp.Lβ\nα(z), stands for the Tricomi’s confluent hyper-\ngeometric function, resp. the generalized Laguerre function.\nThere are explicit series expansions for the functions\nU/parenleftBig\n1\n2(1+/radicalbig\n1+4/c2)+1\nc,1+/radicalbig\n1+4/c2,λ\ncecz/parenrightBig\nand\nL√\n1+4/c2\n−(2+c+c√\n1+4/c2)/(2c)/parenleftbigλ\ncecz/parenrightbig\n, suggesting that their growth, invery rough\nterms, is similar to the growth of the function eλ\ncecz. Consideration of\n(6.6) now shows that fcannot grow essentially faster, so that αis\nnot a small function of f. We do not attempt to give a more careful\nargument here.VALUE SHARING AND STIRLING NUMBERS 25\nIn special cases, we may obtain a more explicit solution. To this\nend, take 2 c=−3. In this case, the square-root in (6.8) simplifies and\ntaking now, for simplicity, C1= 1 and C2= 0 in (6.8) and ignoring\nunimportant multiplicative constants, the solution B(z) reduces to\nB(z) = (λe−3z/2−1)e5z\n4+λ\n3e−3z/2,\nandα(z) toα(z) =e−zexp((2λ/3)e−3z/2). Solving now for f(z) from\n(2.1), we obtain, in terms of the incomplete gamma function Γ[ a,z] =/integraltext∞\nzta−1e−tdt,\nf(z) =e−(2λ/3)e−3z/2/parenleftBigg\nC+(4·61/3(−λ)2/3)−1×\n×/parenleftbig\n4 Γ[2/3,−(4/3)λe−3z/2]+3Γ[5/3,−(4/3)λe−3z/2]/parenrightbig/parenrightBigg\nwhereCis any complex constant. Here it is easier to verify directly\nthatαis not a small function of f. We omit the details.\n(2) Ifa3/ne}ationslash= 1, then all poles of\nA(z) =−1−c2\n4−λecz−λ2\n4e2cz+/parenleftbigg1\na3−1/parenrightbigg1\nλecz−1\naresimple (and indeed at such points z, whereλecz= 1). Then all non-\ntrivial meromorphic solutions to (6.4), if such solutions exist at all, ar e\nlinearly dependent. Similarly, all possible solutions α(z) to (6.2) are\nlinearly dependent, hence this applies to\n(λecz−1)α(z) =e3cz/2e−λ\n2ceczB(z)\nas well.\nWriting (6.2) as a differential equation for the function g(z) = (1−\nλecz)α(z) that is required to be entire, and considering the equation at\nthe points zwhere 1−λecz= 0, we see that at such points ( −1/a3+\n1)g(z) = 0 so that g(z) = 0. This shows that the function α=g/(1−\nλecz) isentire (if a non-trivialsolution thatis meromorphic intheplane\nexists at all for (6.2)).\nNow writing (6.2) in the form\n0 =α′′\nα+/parenleftbigg\n(λecz−c)+2c\nλecz−1/parenrightbiggα′\nα+/parenleftbigg\n−1−(λecz−c)+A0\nλecz−1/parenrightbigg\n,\n(6.9)\nwhereA0= 1/a3−1 +c−2c2, and considering it on the sub-arcs\nof{z:|z|=r}where|ecz|is large, we find by two approximate in-\ntegrations that log α′(z) and then α(z) must be of the same order of\nmagnitude as −λeczande−(λ/c)ecz, respectively, on certain sub-arcs of\nthese sub-arcs, the number of such sub-arcs being comparable t or.\nHence by (6.6) fcannot grow essentially faster than α, and soαis not26 AIMO HINKKANEN AND ILPO LAINE\na small function of f. Again we do not attempt to give a more detailed\nproof here.\nFor larger values of n≥4, the reasoning applied in the cases n= 2,3\ncannot be used directly. However, it seems reasonable to conject ure\nthat there are entire solutions αandfto the original problem only for\nsome special values of the parameters involved.\nReferences\n[1] Charalambides, C.A., Enumerative combinatorics , Chapman & Hall/CRC,\n2002.\n[2] Hayman, W.K., Meromorphic functions , Oxford University Press, Oxford,\n1964.\n[3] Lahiri, I., An entire function that shares a small function with its deri vative\nand linear differential polynomial , Comput. Method. Funct. Theory 23(2023),\n393–316.\n[4] Laine, I., Nevanlinna Theory and Complex Differential Equations , Walter de\nGruyter, Berlin, 1993.\n[5] Wang, J. and Laine, I., Uniqueness of entire functions and their derivatives ,\nComputational Methods and Function Theory 8(2008), 327–338.\n[6] Zhang, J., Entire functions sharing a small function with their deriva tives,\nComputational Methods and Function Theory 9(2009), 379–389.\nDepartment of Mathematics, University of IllinoisUrbana- Champaign,\nUrbana, IL 61801 USA\nEmail address :aimo@illinois.edu\nDepartment of Physics and Mathematics, University of Easte rn Fin-\nland, Joensuu, 80100 Finland\nEmail address :ilpo.laine@uef.fi"    },    {        "title": "2401.13814v1.Illuminating_the_Dark_Sector__Searching_for_new_interactions_between_dark_matter_and_dark_energy.pdf",        "content": "Illuminating the Dark Sector\nSearching for new interactions between dark matter and dark\nenergy\nA thesis submitted to The University of Sheffield for the degree of\nDoctor of Philosophy\nElsa Maria Campos Teixeira\nSchool of Mathematics and Statistics\nSupported by Fundação para a Ciência e a Tecnologia\nSupervised by Professor Carsten van de Bruck\nJanuary 2024arXiv:2401.13814v1  [astro-ph.CO]  24 Jan 2024i\nTudo no mundo começou com um sim. Uma molécula\ndisse sim a outra molécula e nasceu a vida. Mas antes da pré-história\nhavia a pré-história da pré-história e havia o nunca e havia o sim.\nSempre houve. Não sei o quê, mas sei que o Universo jamais começou.\n—Clarice Lispector A Hora da Estrela\nAll the world began with a yes. One molecule\nsaid yes to another molecule and life was born. But before prehistory\nthere was the prehistory of prehistory and there was the never and there was the yes.\nIt was ever so. I don’t know why, but I do know that the Universe never began.\n—Clarice Lispector inThe Hour of the StarPreface\nThis dissertation is presented to the School of Mathematics and Statistics of the University\nof Sheffield for the degree of Doctor of Philosophy. This project has been supervised by Prof.\nCarsten van de Bruck, and the contents report on the results of the research work conducted over\nthe course of the four years of my PhD, supported by the Fundação para a Ciência e a Tecnologia\n(FCT) through the grant SFRH/BD/143231/2019. The Chapters in Part II are based partially\non ongoing work and on articles published in refereed journals, as the result of collaborations\nwith Carsten van de Bruck, Noemi Frusciante, Bruno J. Barros, Vasco M.C. Ferreira, Richard\nDaniel, Gaspard Poulot and Cameron Thomas. I list the published work below:\n1.Elsa M. Teixeira , Richard Daniel, Noemi Frusciante, and Carsten van de Bruck\n“Forecasts on interacting dark energy with standard sirens” ,\nPhys.Rev.D 108 (2023) 8, 084070, arXiv:2309.06544 [astro-ph].\n2. Carsten van de Bruck, Gaspard Poulot, and Elsa M. Teixeira ,\n“Scalar field dark matter and dark energy: a hybrid model for the dark\nsector” ,\nJCAP07 (2023), 019, arXiv:2211.13653 [hep-th].\n3.Elsa M. Teixeira , Bruno J. Barros, Vasco M. C. Ferreira, and Noemi Frusciante,\n“Dissecting kinetically coupled quintessence: phenomenology and observa-\ntional tests” ,\nJCAP11 (2022), 059, arXiv:2207.13682 [gr-qc].\n4. Carsten van de Bruck, and Elsa M. Teixeira ,\n“Dark D–Brane Cosmology: from background evolution to cosmological\nperturbations” ,\nPhys.Rev.D 101 (2020) 8, 083506, arXiv:2007.15414 [gr-qc]\nThis thesis is structured as follows:\n•Chapter 1 provides an introduction to the main concepts relevant for modern cosmology,\nincluding the theory of general relativity and the underlying cosmological principles.\n•Chapter 2 is an account of the fundamentals of the standard model of cosmology - the\nΛCDM model - including its formulation, assumptions, background and perturbed equa-\ntions, and a brief history of the Universe.\n•Chapter 3 consists of a review of the concepts in observational cosmology that are relevant\nfor the work included in the thesis. It covers the main observables and probes pf the dark\niiiiv\nsector, followed by an account of the parameters of the ΛCDM model, and a comment on\nthe current crisis it faces.\n•Chapter 4 provides a summary of the statistical methods necessary for the numerical analy-\nsis and the assessment of the support for the alternative models considered. It also includes\na short introduction to the numerical tools and data sets for that purpose.\n•Chapter5introducesthemotivationandformulationofmodelsbeyondthestandard ΛCDM\nmodel, with particular focus on models with a coupling in the dark sector.\n•Chapter 6 reports on the ongoing work with Cameron Thomas and Carsten van de Bruck.\nCameron Thomas is the author of part of the numerical work conducted for Section 6.2.\n•Chapter 7 draws from published work in the Journal of Cosmology and Astroparticle\nPhysics (JCAP) [1] in collaboration with Bruno J. Barros, Vasco M. C. Ferreira and Noemi\nFrusciante. Bruno J. Barros is credited with the original formulation of the model.\n•Chapter 8 is an account on collaborative work with Richard Daniel, Noemi Frusciante, and\nCarsten van de Bruck, which has been published in Physical Review D and is available at\n[2]. Richard Daniel is the author of the simulated standard siren catalogue on which we\nreport in Appendix B.\n•Chapter 9 is based partially on published work in Physical Review D [3] in collaboration\nwith Carsten van de Bruck and on ongoing work for the data analysis.\n•Chapter 10 reports on work published in the Journal of Cosmology and Astroparticle\nPhysics (JCAP) [4] in collaboration with Gaspard Poulot and Carsten van de Bruck.\nGaspard Poulot is credited with part of the theoretical background work and the fluid\napproximation.\n•Chapter 11 concludes with a summary of the work presented in this dissertation, with an\noutlook on coupled and interacting quintessence models, including some general thoughts\non the relevance of the research outlined in this manuscript and thoughts on the future of\nthe field.Acronyms\nBAOBaryon Acoustic Oscillations\nBBBig Bang\nBBNBig Bang Nucleosynthesis\nCDMCold Dark Matter\nCLConfidence Level\nCMBCosmic Microwave Background\nDDBDark D-Brane\nDDMDisformal Dark Matter\nDEDE Dark Energy Dominated Epoch\nEHEinstein-Hilbert\nEoSEquation of State\nETEinstein Telescope\nFLRW Friedman-Lemaître-Robertson-Walker\nGRGeneral Relativity\nGWGravitational Wave\nHBBHot Big Bang\nLISALaser Interferometer Space Antenna\nLSSLarge Scale Structure\nMDEMatter Dominated Epoch\nQFTQuantum Field Theory\nRDERadiation Dominated Epoch\nSH0ES Supernovae H0 for the Equation of State of Dark Energy\nSMStandard Model\nSNSupernova\nWIMPs Weakly Interacting Massive Particles\nvAbstract\nOs factos são sonoros mas entre os factos há um sussurro.\nÉ esse sussurro que me impressiona.\n—Clarice Lispector A Hora da Estrela\nThe facts are sonorous but between the facts there’s a whispering. It’s the whispering that astounds me. —\nClarice Lispector in The Hour of the Star\nThe current standard model of cosmology - the ΛCDM model - is appropriately named after\nits controversial foreign ingredients: a cosmological constant ( Λ) that accounts for the recent\naccelerated expansion of the Universe and cold dark matter needed to explain the formation and\ndynamics of large scale structures. Together, these form the dark sector, whose nature remains a\nmystery. After 25 years of withstanding confirmation and support for the ΛCDM model, enough\nto bypass some of its unclear theoretical issues, this paradigm is facing its biggest crisis yet. The\nrapid advent of technology has brought cosmology to an unprecedented observational era, with\nincreased technical precision and the emergence of independent measures, including probes of\nphenomena that were thought impossible to detect or even exist, such as the gravitational ripples\nthat propagate in the spacetime. However, such precision has unveiled cracks in the porcelain\nofΛCDM, with pieces that seem glued together and difficult to reconcile. Particularly worrying\nis the apparent lack of compatibility between measurements of the Universe’s present expansion\nrate based on local measurements and those based on phenomena that occurred far in the early\nUniverse and that can only be translated into present quantities through physical propagation\nunder a cosmological model. In this dissertation, we delve into extensions to the standard\nmodel that consider alternatives to the mysterious nature of the dark sector and any possible\nnew interactions therein. We analyse these alternative models, hoping to identify measurable\nobservational signatures of extra degrees of freedom in the dark sector. For this purpose, we\nfocus on Markov Chain Monte Carlo sampling techniques by exploiting recent observational\ndata and their combination to constrain and assess the validity of the model’s parameter space.\nThe extended frameworks investigated are only found to be marginally or not at all favoured\nover the ΛCDM model, ultimately falling short of resolving the tensions present in the latter.\nWe conclude with a forecast on the constraining power of upcoming gravitational wave detectors\nas standard sirens, independently and combined with current background data. By generating\nsimulated standard siren event catalogues following the specifications predicted by the proposed\nmissions,wediscusshowanemergingavenueforindependentdatamaybedecisiveinconstraining\nalternative theories of gravity and shed light on the nature of the dark sector.\nviiviii\nUnits\nUnless stated otherwise, in this dissertation I will always adopt natural units in which the speed\nof light, Planck’s constant and Boltzmann’s constant are unity, c=ℏ=kB= 1. This translates\ninto an equivalence of units for energy, mass and momentum as an inverse unit of length and\ntime:\n[Energy] =[Mass] =[Temperature] =[Length]−1=[Time]−1. (0.0.1)\nThroughout this manuscript we denote derivatives with respect to physical time tby an upper\ndot and derivatives with respect to conformal time τby a prime,\nt=cosmic time =⇒dX\ndt≡˙X , (0.0.2)\nτ=conformal time =⇒dX\ndτ≡X′. (0.0.3)\nSpatial 3-vectors are denoted an upper arrow, such as ⃗ x, and four-dimensional spacetime vectors\nare denoted in terms of their components as x= (xµ). We adopt the convention for the signature\nof the metric as (−,+,+,+).Contents\nPreface iii\nAbstract vii\nAcknowledgements xvii\nPart I: Foundations of Gravity and Cosmology\n1 Introduction 3\n1.1 Introduction to Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 A Comment on Observational Cosmology . . . . . . . . . . . . . . . . . . . . . . . . 6\n1.3 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n1.4 Homogeneity and Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n1.5 The Hubble-Lemaître Law: Cosmic Expansion . . . . . . . . . . . . . . . . . . . . . 15\n1.6 The Friedmann-Lemaître-Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . 15\n1.7 Spacetime Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n1.8 The Hot Big Bang Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n2 The Standard ΛCDM Model: Successes and Controversies 21\n2.1 Fundamentals of the Background Cosmology . . . . . . . . . . . . . . . . . . . . . . 21\n2.2 Linear Cosmological Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\n2.3 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41\n2.4 A Brief History of the Universe in Four Epochs . . . . . . . . . . . . . . . . . . . . . 43\n2.5 The Theoretical Problems with Λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49\n3 Cosmological Observational Tests 53\n3.1 The Matter Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54\n3.2 The Matter Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57\n3.3 The Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 58\n3.4 Other Cosmological Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67\n3.5 The Cosmological Parameters of the ΛCDM Model . . . . . . . . . . . . . . . . . . . 75\n3.6 Cosmological Puzzles and Observational Tensions . . . . . . . . . . . . . . . . . . . . 81\n4 Methodology 89\n4.1 Statistical Methods and Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . 89\n4.2 Einstein-Boltzmann Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97\n4.3 Baseline Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99\n5 Beyond the Standard Model: Coupled Dark Sector 101\n5.1 Beyond ΛCDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101\nixx Contents\n5.2 Scalar Field Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103\n5.3 Universal Couplings to the Matter Sector . . . . . . . . . . . . . . . . . . . . . . . . 109\n5.4 Non-Universal Couplings and Interactions in the Dark Sector . . . . . . . . . . . . . 111\nPart II: Research Results\n6 Coupled Quintessence in Flat and Curved Geometries 115\n6.1 Models and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117\n6.2 Update on Flat Coupled Quintessence . . . . . . . . . . . . . . . . . . . . . . . . . . 123\n6.3 Coupled Quintessence in Non-Flat Geometries . . . . . . . . . . . . . . . . . . . . . . 133\n6.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148\n7 Kinetically Coupled Dark Energy 157\n7.1 Theoretical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157\n7.2 Background Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160\n7.3 Evolution of Linear Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162\n7.4 Observational Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166\n7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171\n8 Forecasts on Coupled Quintessence with Standard Sirens 173\n8.1 Standard Sirens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174\n8.2 Methodology and Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175\n8.3 Forecasts Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176\n8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194\n9 Dark D-Brane Cosmology 199\n9.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201\n9.2 Background FLRW Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204\n9.3 Linear Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208\n9.4 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216\n9.5 Observational Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228\n9.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231\n10 A Hybrid Model for the Dark Sector 237\n10.1 The Hybrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238\n10.2 Fluid Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244\n10.3 Observables and Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249\n10.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255\n11 Conclusions 257Contents xi\nA Scalar Field Inflation 259\nB Mock Data Set Simulation 263\nC String Theory and the DBI Action 270\nBibliography 275List of Figures\n1.1 The Planck2018 temperature anisotropies map . . . . . . . . . . . . . . . . . . . 13\n1.2 The SDSS map of galaxies and quasars in the observable Universe . . . . . . . . . 14\n2.1 The redshift evolution of the energy densities of the components of the Universe . 30\n3.1 The Planck2018 CMB temperature–temperature power spectrum . . . . . . . . . 60\n3.2 Contributions to the CMB power spectrum . . . . . . . . . . . . . . . . . . . . . 64\n3.3 The Planck2018 CMB gravitational lensing power spectrum . . . . . . . . . . . . 68\n3.4 The distance modulus measurements from Pantheon . . . . . . . . . . . . . . . . 70\n3.5 BAO distance and expansion rate data from SDSS . . . . . . . . . . . . . . . . . 73\n3.6 The impact of varying each of the six ΛCDM parameters on the CMB temperature\npower spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77\n3.7 The impact of varying each of the six ΛCDM parameters on the matter power\nspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78\n3.8 Tension between estimates of H0from different probes . . . . . . . . . . . . . . . 83\n3.9 Tension between estimates of S8from different probes . . . . . . . . . . . . . . . 85\n6.1 1D marginalised posterior distributions in the flat M 1models . . . . . . . . . . . 124\n6.2 2D marginalised posterior distributions in the flat M 1models . . . . . . . . . . . 127\n6.3 1D marginalised posterior distributions in the flat M 2models . . . . . . . . . . . 129\n6.4 2D marginalised posterior distributions in the flat M 2models . . . . . . . . . . . 132\n6.5 1D marginalised posterior distributions in the flat M 3−model . . . . . . . . . . . 134\n6.6 2D marginalised posterior distributions in the flat M 3−model . . . . . . . . . . . 134\n6.7 1D marginalised posterior distributions in the curved M 1models . . . . . . . . . 138\n6.8 2D marginalised posterior distributions in the curved M 1models . . . . . . . . . 141\n6.9 1D marginalised posterior distributions in the curved M 2models . . . . . . . . . 144\n6.10 2D marginalised posterior distributions in the curved M 2models . . . . . . . . . 145\n6.11 1D marginalised posterior distributions in the curved M 3models . . . . . . . . . 149\n6.12 2D marginalised posterior distributions in the curved M 3models . . . . . . . . . 150\n6.13 Whisker plot of the 68%CL constraints on H0for the flat and curved models . . 151\n6.14 Whisker plot of the 68%CL constraints on S8for the flat and curved models . . 153\n6.15 Whisker plot of the 68%CL constraints on Ωmfor the flat and curved models . . 154\n6.16 Whisker plot of the 68%CL constraints on ΩKfor the flat and curved models . . 155\n7.1 Evolution of relative energy densities Ωi, the ratio ρc/ρϕand variations of the\nHubble rate ∆H/H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162\n7.2 The matter power spectrum and the lensing potential . . . . . . . . . . . . . . . . 165\nxiiList of Figures xiii\n7.3 The CMB temperature and lensing power spectra . . . . . . . . . . . . . . . . . . 166\n7.4 2D marginalised posterior distributions in the ΛCDM and Kinetic models . . . . 169\n7.5 2D marginalised posterior distributions in the Kinetic model . . . . . . . . . . . . 170\n8.1 Triangle plot of 2D marginalised posterior distributions for conformal coupled\nquintessence with ET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180\n8.2 Triangle plot of 2D marginalised posterior distributions for conformal coupled\nquintessence with LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181\n8.3 Triangle plot of 2D marginalised posterior distributions for conformal coupled\nquintessence with ET+LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182\n8.4 Triangleplotof2Dmarginalisedposteriordistributionsforkineticcoupledquintessence\nwith ET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185\n8.5 Triangleplotof2Dmarginalisedposteriordistributionsforkineticcoupledquintessence\nwith LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186\n8.6 Triangleplotof2Dmarginalisedposteriordistributionsforkineticcoupledquintessence\nwith ET+LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187\n8.7 Triangle plot of 2D marginalised posterior distributions for disformal coupled\nquintessence with ET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189\n8.8 Triangle plot of 2D marginalised posterior distributions for disformal coupled\nquintessence with LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190\n8.9 Triangle plot of 2D marginalised posterior distributions for disformal coupled\nquintessence with ET+LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191\n8.10 Triangle plot of 2D marginalised posterior distributions for mixed conformal-\ndisformal coupled quintessence with ET . . . . . . . . . . . . . . . . . . . . . . . 193\n8.11 Triangle plot of 2D marginalised posterior distributions for mixed conformal-\ndisformal coupled quintessence with LISA . . . . . . . . . . . . . . . . . . . . . . 194\n8.12 Triangle plot of 2D marginalised posterior distributions for mixed conformal-\ndisformal coupled quintessence with ET+LISA . . . . . . . . . . . . . . . . . . . 195\n9.1 Evolution of the coupling parameter . . . . . . . . . . . . . . . . . . . . . . . . . 217\n9.2 Evolutionofrelativeenergydensities,theHubblerateandtheeffectivedarkenergy\nEoS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218\n9.3 Evolution of the effective gravitational constant . . . . . . . . . . . . . . . . . . . 222\n9.4 The CMB temperature and lensing angular power spectra and the matter power\nspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224\n9.5 Evolution of the lensing potential and its derivative . . . . . . . . . . . . . . . . . 227\n9.6 Marginalised 1D and 2D constraints for the Dark D-Brane model . . . . . . . . . 232\n9.7 2D marginalised posterior distributions in the ΛCDM and Dark D-Brane models 234xiv List of Figures\n10.1 Hybrid Inflation Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239\n10.2 Evolution of the relative energy densities, the ratio ρϕ/ρΛand the Hubble rate . 250\n10.3 Evolution of relative energy densities and effective gravitational constant . . . . . 251\n10.4 Evolution of growth factor and matter power spectrum . . . . . . . . . . . . . . . 252\n10.5 Evolution of the lensing potential and its derivative . . . . . . . . . . . . . . . . . 254\n10.6 The temperature and polarisation CMB angular power spectra . . . . . . . . . . 254\nB.1 ET and LISA mock data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269List of Tables\n3.1 Estimates of cosmological parameters in the six-parameter ΛCDM model derived\nfromPlanck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76\n4.1 The Jeffreys scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97\n6.1 Coupled quintessence models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121\n6.2 Priors on ΛCDM and coupled quintessence parameters . . . . . . . . . . . . . . . 123\n6.3 Observational constraints for flat ΛCDM . . . . . . . . . . . . . . . . . . . . . . . 124\n6.4 Observational constraints for the flat M 1+model . . . . . . . . . . . . . . . . . . 125\n6.5 Observational constraints for the flat M 1−model . . . . . . . . . . . . . . . . . . 126\n6.6 Observational constraints for the flat M 2+model . . . . . . . . . . . . . . . . . . 130\n6.7 Observational constraints for the flat M 2−model . . . . . . . . . . . . . . . . . . 131\n6.8 Observational constraints for the flat M 3−model . . . . . . . . . . . . . . . . . . 133\n6.9 Observational constraints for the curved ΛCDM model . . . . . . . . . . . . . . . 135\n6.10 Observational constraints for the curved M 1+model . . . . . . . . . . . . . . . . 139\n6.11 Observational constraints for the curved M 1−model . . . . . . . . . . . . . . . . 140\n6.12 Observational constraints for the curved M 2+model . . . . . . . . . . . . . . . . 143\n6.13 Observational constraints for the curved M 2−model . . . . . . . . . . . . . . . . 144\n6.14 Observational constraints for the curved M 3+model . . . . . . . . . . . . . . . . 148\n6.15 Observational constraints for the curved M 3−model . . . . . . . . . . . . . . . . 149\n7.1 Priors on the model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167\n7.2 Observational constraints for the ΛCDM model . . . . . . . . . . . . . . . . . . . 168\n7.3 Observational constraints for the Kinetic model . . . . . . . . . . . . . . . . . . . 170\n8.1 Priors on cosmological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177\n8.2 Observational constraints for conformal coupled quintessence . . . . . . . . . . . 178\n8.3 Observational constraints for kinetic coupled quintessence . . . . . . . . . . . . . 183\n8.4 Observational constraints for disformal coupled quintessence . . . . . . . . . . . . 184\n8.5 Observational constraints for mixed conformal-disformal coupled quintessence . . 192\n9.1 Example solutions of the Dark D-Brane model . . . . . . . . . . . . . . . . . . . . 216\n9.2 Priors on the model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229\n9.3 Observational constraints for the reference ΛCDM . . . . . . . . . . . . . . . . . 232\n9.4 Observational constraints for the Dark D-Brane model . . . . . . . . . . . . . . . 233\nxvAcknowledgements\nA culpa é minha ou\nA hora da estrela ou\nEla que se arranje ou\nO direito ao grito ou\nQuanto ao futuro ou\nLamento de um blue ou\nEla não sabe gritar ou\nUma sensação de perda ou\nAssovio no vento escuro ou\nEu não posso fazer nada ou\nRegistro dos fatos antecedentes ou\nHistória lacrimogênia de cordel ou\nSaída discreta pela porta dos fundos\n—Clarice Lispector A Hora da Estrela\nIt’s all my fault - or - The hour of the star - or - Let her deal with it - or - The right to scream - or - As for\nthe future - or - Singing the blues - or - She doesn’t know how to scream - or - A sense of loss - or - Whistling in\nthe dark wind - or - I can’t do anything - or - Account of the preceding facts - or - Cheap tearjerker - or - Discreet\nexit through the back door —Clarice Lispector in The Hour of the Star\nI have to start by expressing my enormous gratitude for having been so lucky to have Carsten as a\nsupervisor. During these past four years, almost half spent during a health emergency lockdown\nand hundreds of miles apart, you were always available to help me and keep me motivated. Your\nexcitement and curiosity for the world are definitely not restricted to science and maths and I’m\ngrateful to have shared that with you. I have enjoyed all the non-academic times we shared,\nfrom pandemic socially distanced walks to insisting on keeping your word and taking your two\nPhD ‘siblings’ to Woolsthorpe Manor. I am also thankful to Sabina for sharing some of these\nmoments with us and for all the wild berries and mushroom picking.\nDespite being affected by the pandemic, I was still lucky to travel abundantly during my PhD. I\nam grateful to the School of Mathematics and Statistics of the Universe of Sheffield for providing\nme with multiple opportunities to do so, never denying any of my funding requests, to the Euro-\npean Consortium for Astroparticle Theory (EuCAPT) and the EU COST Action CosmoVerse,\nCA2113. Thanks to the University of Rome’ La Sapienza’ for hosting my STSM, ‘Università\ndi Napoli’ Federico II & INFN Sezione di Napoli for hosting the EuCAPT exchange grant, and\nLUPM, Université Montpellier for the introductory visit. Special thanks to Noemi Frusciante,\nxviixviii Nomenclature\nAlessandro Melchiorri, Giuseppe Fanizza, and Vivian Poulin for welcoming me.\nI am indebted to all the members of the CRAG research group, past and present, who made it\nsuch a fun, friendly, and stimulating environment to work in. I thank Steffen Gielen, Sam Dolan,\nEleonora di Valentino, and Elizabeth Winstanley for being academic role models and introducing\nme to various corners of the Universe. I am incredibly thankful to have shared this experience\nwith Richard, from the first day in the lift, stealing your desk and our crazy little matchbox\noffice. I will never forget the times we spent in that squared metre, whose value is greater than\nthe amount of tea bags consumed. To Ale for always having the greatest ground-lifting (literally)\nhug when it was (and also when it was not) needed. A particular mention has to be made to\nLisa, who has been my absolute role model and best of friends these last two years, which feel\nlike a decade of companionship. Thank you for always being there, for your kindness, and the\noccasional slice of cake. I cannot thank you enough for reading and scrutinising every line in this\nthesis. I am also grateful to have crossed academic paths with some wonderful people. Thank\nyou Rita, Matilde, Ruchika, William, Jéssica, Kerkyra, Vitor, Davide, João and Léo, and all the\nwonderful people that made attending conferences, workshops and seminars such an enriching\nexperience. I want to thank my collaborators with whom I learned so much: Noemi Frusciante,\nBruno J. Barros, Vasco M.C. Ferreira, Richard Daniel, Gaspard Poulot and Cameron Thomas.\nA special mention should be made to Ana Nunes, Nelson Nunes, Noemi Frusciante, Eleonora Di\nValentino and Danielle Leonard for all the advice and mentoring, and to Bruno for everything\nwe learned together and for remaining (in)sane with me during 1.5 years of lockdown.\nI cannot overstate the equally great debt I owe to my home friends. I am especially grateful\nto Rita and Joana for spreading roots and sharing this adventure with me, ensuring we never\ntook ourselves too seriously, and always sending me off with a full bag of laughs and heartfelt\nunderstanding. You have truly shown me the sun behind the English clouds. I thank Inês for\nbeingmycompanioninthis10-yearjourneythroughphysics,mathematicsandphilosophy,always\nwith amazement and creativity. I cannot wait to keep exploring the wonders of the Universe\nwith you. Most of all, I owe all of it to my family, who has always supported me unconditionally\nwith complete freedom to flourish. To my parents, who have taught me the most important\nthings of all: never take myself too seriously, always be kind, and, above all, do what makes me\nhappy. And in particular to my sister and niece for understating my physical absence in their\nlives. I dedicate this work to them, who inspire and motivate me daily. To my pardalinho , in\nyour fairy-tale homonymous words, I can only hope to inspire you always to remain\n“Curiouser and curiouser!”\n—Lewis Carroll Alice’s Adventures in WonderlandPart I\nFoundations of Gravity and Cosmology1Introduction\nComo começar pelo início, se as coisas acontecem antes de acontecer?\nSe antes da pré-pré-história já havia os monstros apocalípticos?\nSe esta história não existe, passará a existir. Pensar é um ato.\nSentir é um fato. Os dois juntos – sou eu que escrevo o que estou escrevendo.\n—Clarice Lispector A Hora da Estrela\nHow do you start at the beginning, if things happen before they happen? If before the pre-prehistory there\nwere already the apocalyptic monsters? If this story doesn’t exist now, it will. Thinking is an act. Feeling is a\nfact. Put the two together — I am the one writing what I am writing. —Clarice Lispector in The Hour of\nthe Star\n1.1 Introduction to Cosmology\nCosmology is a unique field of scientific inquiry that seeks to understand the nature and origins\nof the Universe. On what might be the only personal note in this dissertation, I like to think that\ncosmology parallels the work of archaeologists piecing together the fragments of ancient civilisa-\ntions. Both disciplines aspire to reconstruct a larger, coherent picture from fragmented pieces\n- whether it be artefacts buried in the earth or distant sources of light reaching our telescopes\nfrom the vast deepness of the Universe. Each cosmic signal we analyse is a relic from a long-past\nevent, afossil of light , offering clues about the history and structure of the ultimate civilisation\n- our Universe. This presents an epistemological challenge unique to cosmology: working to\nunderstand an entire, singular Universe based on isolated data points, inherently confined by\nthe limits of causality and our observational reach. The task has an almost philosophical un-\ndertone; it grapples with questions about the very nature of evidence and existence, especially\ngiven that we have only one Universe to study. This singularity not only imposes a limit on our\nunderstanding but also introduces an inherent bias into any laws or theories we might derive.\nThus, cosmology exists at an intersection of science and philosophy, endeavouring to build a nar-\nrative of the Universe that is both empirical and conceptually profound. It confronts us with the\nstark reality that our efforts to understand the cosmic journey are both an ambitious scientific\nundertaking and a quest that treads on deeply abstract grounds.\nMetaphysics aside, as an established scientific field, cosmology aspires to comprehend and artic-\nulate the fundamental characteristics and evolution of the Universe. It devises meticulous and\noften speculative narratives for the composition of the Universe, delving further into the journey\nof those constituents as they mature into complex structures such as galaxies. It also exam-\n34 1.1. Introduction to Cosmology\nines the dynamics and interplay of those structures within an ever-changing landscape. These\noverarching frameworks are referred to as cosmological models . Unlike the word modelmight\nsuggest, contemporary cosmological models consist mainly of abstract mathematical concepts,\noften devoid of physical intuition or tangible form.\nWhile commonly referred to as the birth of the Universe, the Big Bang might be more aptly\nportrayed as its conception. The period preceding the creation of the first atoms, culminating\nin the unimpeded propagation of light, was a vastly unfamiliar environment, an obscure cosmic\nfog of elementary particles and their extremely energetic interactions. When the Universe was\napproximately 380 000 years old, it grew more familiar. It gradually unfolded into the config-\nuration we recognise today, seemingly set on a perpetual path of expansion, according to the\nprevailing cosmological model.\nAt its core, Cosmology stands as a field deeply rooted in ancient human contemplation. It\nis impossible to disentangle from the quest to understand our place in the world, its origins,\nand ontological meaning. The foundations of modern cosmology were established in 1917 with\nAlbert Einstein’s seminal article on his theory of General Relativity (GR), titled Cosmological\nConsiderations in the General Theory of Gravity [5]. BuildinguponEinstein’swork, cosmological\nmodels were further developed by Dutch mathematician Willem de Sitter [6–8], German physicist\nKarl Schwarzschild [9, 10], and Russian mathematician Alexander Friedmann [11, 12]. These\nstudies demonstrated that general relativity could accommodate the concept of an expanding\nUniverse,atheoreticalhypothesisthatcametobeconfirmedbytherealisationthatgalaxiesstood\nas rulers to trace astronomical distances in space and which were, in fact, receding from our own\nMilky Way in all directions. In 1929, Edwin Hubble’s landmark discovery, now known as the\nHubble law, revealed a linear relation between the recessional speed of a galaxy and its distance,\nculminating in the hypothesis that the Universe is expanding. This idea was not unexpected and\nhad been largely predicted from a theoretical perspective. Indeed, it was qualitatively reinforced\nby redshift measurements and notoriously explored during the 1920s.\nAsitstandsatthemomentofwritingthisdissertation, thestandardcosmologicalmodel-referred\nto as the ΛCDM model based on its novel components - is constructed upon six foundational\nprinciples, summarised in the following equation and which will be discussed below:\nΛCDM Model =GR+FLRW +Standard Matter +Initial Conditions +CDM + Λ.\nThe GR, FLRW, and Standard Matter triad represent well-established and tested hypotheses\nbased on local empirical evidence. On the other hand, although highly supported by cosmological\nand astrophysical observations, the last three components infuse new physics into the picture\nand still lack more concrete and direct confirmation. In unison, these six hypotheses orchestrate\na comprehensive paradigm for understanding the Universe on large scales, as listed below:\n1. GR:Thetheoryofgravityisessentialforunderstandingcosmologicalphenomena,asgravity1Introduction 5\ndominates the Universe on large scales and is at the heart of cosmic evolution.\n2. Friedman-Lemaître-Robertson-Walker (FLRW) metric: GR admits only a few analytical\nsolutions as a metric theory regulated by non-linear partial differential equations. There-\nfore, deriving physical implications requires a simple mathematical description from the\nmetric, with the FLRW metric being a natural choice assuming maximal spatial symmetry.\n3. Standard Matter (from the standard model of particle physics): the directly detected\nforms of matter in the Universe - baryons (protons, neutrons, including the less massive\nelectrons), photons, and neutrinos - whose dynamics and interactions are based on robust\nphysical principles - must be included in the cosmological model to recount the expansion\nhistory of the Universe successfully.\n4. Initial Conditions (from inflation): an early period of rapid cosmic expansion, inflation is\ninvoked to resolve causality problems and account for the observed spatial curvature of the\nUniverse. Remarkably, it also provides a mechanism to explain the Universe’s initial con-\nditions, including generating perturbations around the background metric and supporting\nthe choice of the homogeneous and isotropic metric ansatz.\n5. Cold Dark Matter (CDM): a form of pressureless matter that does not interact with light,\nneeded to accurately describe the dynamics and process of formation of large-scale struc-\ntures, such as galaxies and galaxy clusters. The simplest resolution is to introduce a new,\nweakly interacting particle species.\n6. Cosmological Constant ( Λ): an additional degree of freedom that accounts for the observed\naccelerated expansion of the Universe, which in the simplest scenario is a cosmological\nconstant, i.e., some energy density contribution that remains constant over the expansion.\nFor completeness of the work, we first provide a general overview of the relevant concepts of\ncosmology, focusing mainly on the Universe’s expansion history and leading towards the physics\nof the late Universe. The main goal is to introduce and contextualise the original work presented\nin Part II of this manuscript. In that sense, Part I will consist of well-established topics in the\nfield. Nevertheless, the organisation of the discussion and the storytelling should be original,\nconnecting the ideas and guiding the reader through the relevant concepts. We will start by\npresenting a brief overview of the dynamics of general relativity. This lays out the basis for\nthe introduction of cosmology, focusing mainly on the large-scale structure of spacetime and the\ndynamics of the expansion history. To understand the full scope of the changes introduced in\neach model, a complete account of the physically relevant epochs of the Universe is needed. The\nseeds for the evolution of structures lie in the primordial perturbations produced during an initial\nperiod of inflation. These are imprinted in the Cosmic Microwave Background (CMB) radiation\nand are a fundamental cosmological probe.6 1.2. A Comment on Observational Cosmology\n1.2 A Comment on Observational Cosmology\nThe methods of scientific pursuit must be adapted to the limitations of the field, particularly\nin the realms of physical cosmology and extragalactic astronomy, where empirical assessment is\nlimited to distant observations without direct interaction or manipulation of the objects of study.\nMoreover, in the vast expanse of the cosmos, we cannot replicate experiments or observations.\nInstead, we must rely on searching for the clues and evidence left behind in fossils from the\nancient past. Some of these relics are found nearby, such as extragalactic hints on the Earth\nor the Moon, in our Solar System or even in stars within our galaxy. Even though these local\nremnants give important insight, they are just pieces of a large jigsaw puzzle. Nevertheless, here,\nthe laws of physics and the principles of general relativity play in our favour: our vision can\nreach far beyond, as the light from distant objects carries information from earlier epochs of the\nUniverse, based on the fact that observed objects at increasing distances effectively corresponds\nto looking back in time, witnessing the earlier stages of cosmic evolution. Our view of the cosmos\nis shaped by our past light cone, representing the collection of all the light that has reached us\nthroughout history. While the light cone of human existence encompasses an incredibly thin\nslice of the Universe’s history, the information travels towards us from all directions and epochs,\nspanning a vast range of time and space. Collecting the remnant pieces of cosmic history and\ndrawing them together is just like adding the pieces of the jigsaw puzzle, except that there may\nbe infinite pieces, some inaccessible or lost forever in the vastness of the cosmos, others waiting\nto be found (and most likely there are no corner pieces to start with!).\nThe debates and discussions surrounding cosmological issues can be met with an intensity and\nfervour not matched by the weight of evidence. One reason is that the observational evidence\nneeded to settle these questions often appears to be beyond our grasp or even an impossibility.\nFurthermore, the study of such fundamental questions is deeply entangled with an overarch-\ning interest in understanding the nature of our own existence and the world we inhabit. It is\ninevitable to ponder the Universe’s origins and potential fate, and it is hard not to develop a\nphilosophical preference for certain scientific outcomes or perspectives.\nThe era of precision measurements has drastically changed the scientific method. Big collabo-\nrative teams have developed to address the need to deal with unprecedentedly large amounts of\ndata, with the natural gap between theory and observations being transposed with numerical\nsimulations. This shift has led to heightened scrutiny of research endeavours and the need for\ninquiries to be more rigorously justified or supported.\nThe standard model of cosmology today has reached the status of an empirical science; that is,\nit is based on and tested according to what can be observed or measured. However, scientific\nadvancements are often made on a theoretically intuitive basis since, while being well-tested and\ninvestigated, a cosmological theory is always incomplete and subject to unpredictability. The\ncurrently accepted Λ-Cold-Dark-Matter ( ΛCDM) model relies on the extrapolation of a cosmo-1Introduction 7\nlogical history from imprints left in the observables that resemble those predicted by the model.\nThe theory is then a useful though incomplete approximate picture of the history of the cosmos,\nwith remaining cracks and hypotheses on the borderline of established physics. Nevertheless,\nthe abundance and diversity of empirical tests that the standard model has undergone main-\ntain outstanding confidence that any alternative theory that attempts to glue those cracks will\nportray a Universe that behaves much like ΛCDM. For this reason, the first natural step in\nthat direction is to consider small deviations or extensions to the minimal ΛCDM model while\nremaining conscious that this may not be enough and the answer may require a new paradigm\ndisconnected from the ΛCDM assumptions.\n1.3 General Relativity\nIn this section, we provide a summary of general relativity, with no intention of exhaustively\nintroducing or reviewing its full extent. For this purpose, we refer to e.g. [13, 14].\nAlbert Einstein’s theory of General Relativity, formulated in 1915 [15], presents an exceptional\nframework where gravitational interactions are related to the geometry of spacetime [16–18].\nThe central element is the spacetime metric1degree of freedom g, which determines physical\ndistances as\nds2=gµνdxµdxν, (1.3.1)\nand is treated as a fully dynamic variable.\nThe time coordinate interval between two events, ∆t, is not the same as perceived by different\nobservers, motivating the introduction of the more convenient invariant proper-time interval, ∆τ.\nConsidering the events AandBwhich are separated by a time-like interval ( ds2<0), the curve\nwhich connects the two is denoted by γand parameterised by λ, such that e.g.γ(λA) =Aand\nγ(λB) =B. The proper-time interval ∆τbetween AandBis then given by\n∆τ≡Z\nγp\n−ds2=ZλB\nλAdλr\n−gµνdxµ\ndλdxν\ndλ. (1.3.2)\nMassivefree-fallingparticlesmovinginacurvedspacetimefollowa geodesic trajectory ,Xα(τ), de-\nfined as the time-like trajectory that extremises the proper-time interval ∆τin Equation (1.3.2).\nThe geodesic trajectory is the one that minimises the action\nS=mZ\ndτ, (1.3.3)\nwhere mis the rest-frame mass of such particles. The trajectory of these particles follows the\n1Throughout this manuscript, we adopt Einstein’s summation convention where repeated indices are summed\nover. The metric signature is (−,+,+,+)and we use natural units with c=ℏ=kB= 1. Greek indices range\nfrom 0to3, while Latin indices denote spacetime components only, 1to3.8 1.3. General Relativity\ngeodesic equation :\nd2Xα\ndτ2+ Γα\nµνdXµ\ndτdXν\ndτ= 0, (1.3.4)\nwhere\nΓα\nµν≡1\n2gαλ[∂µgλν+∂νgλµ−∂λgµν], (1.3.5)\nis the torsion-free ( Γα\nµν= Γα\nνµ) connection (or Christoffel symbols) that gives a metric-compatible\ncovariant derivative ∇µ(·)\n∇µYλ...\nν...=∂µYλ...\nν...+ Γλ\nµαYα...\nν...−Γα\nµνYλ...\nα...+...−...Equation (1.3.5)− − − − − − − − − − → ∇ αgµν= 0,(1.3.6)\nwhere Yλ...\nν...is some arbitrary-rank tensor.\nEinstein’s gravitational theory establishes a connection between the dynamics of the metric and\nmatter through the curvature of spacetime, with the latter embodied by the Riemann tensor :\nRα\nβµν≡∂µΓα\nνβ−∂νΓα\nµβ+ Γσ\nβνΓα\nµσ−Γσ\nβµΓα\nνσ. (1.3.7)\nAn action for the metric is required to endow this formulation with dynamics. Remarkably, the\nsimplest nontrivial action that can be constructed out of a scalar from gµνleads to second-order\nequations in the effective field theory and reads:\nS=Sgrav+Smat, (1.3.8)\nwhere\nSgrav=Z\nd4x√−gR\n16πG, (1.3.9)\nis the Einstein-Hilbert (EH) action. The action Smatencodes the matter sector, and gis the\ndeterminant of the metric. The main player in the EH action is the Ricci (or curvature) scalar ,\ndefined in terms of the trace of the Ricci tensor :\nR≡Rµ\nµ=Rµνgµνwith Rµν≡Rα\nµαν. (1.3.10)\nBy employing the principle of least action, the variation of the action in Equation (1.3.8) with\nrespect to the metric yields the Einstein field equations :\nGµν≡Rµν−1\n2gµνR= 8πGT µν, (1.3.11)\nwhere Gis Newton’s gravitational constant, 8πG=κ2=M−2\nPl, and M Pl(κ) is the (reduced)\nPlanck mass. The left-hand side (LHS) is enclosed in the Einstein tensor Gµν. The source term\non the right-hand side (RHS), the energy-momentum tensor , is obtained from the variation of the\nmatter action with respect to the metric and is covariantly conserved on account of the Bianchi1Introduction 9\nidentities :\n∇µGµν= 0Equation (1.3.11)− − − − − − − − − − → ∇ µTµν= 0with Tµν≡2√−gδSmat\nδgµν.(1.3.12)\nBy definition, Tµνis always symmetric. The ν= 0component of ∇µTµνrepresents energy\nconservation, while the three remaining space-like components correspond to momentum con-\nservation. The Einstein equations epitomise the foundation of general relativity: matter and\ngeometry are cut from the same fundamental cloth, with the geometry of spacetime prescribing\nthe dynamics of matter. At the same time, massive bodies set the curvature of spacetime itself.\nTheseeminglycompactsetofEinstein’sfieldequations,inEquation(1.3.11),governtheevolution\nof the Universe and all gravitating bodies, with the concrete realisation depending on its matter\ncontent and the spacetime metric based on physical considerations.\nTo derive the dynamics of the Universe from GR, we can start with the example of Einstein’s\noriginal proposal, based upon two main assumptions, reflecting the impossibility of filling up an\ninfinite space with matter, which must itself be finite in nature [19]:\n•The Universe is finite;\n•The Universe is static.\nThe first assumption aims to ensure a metric completely described by the stress-energy tensor.\nThe second one is more subtle but reasonable in the scientific context in which Einstein first\nproposed a cosmological solution to his equations in 1917 [5]. By extrapolating the assumption\nof a spatially closed Universe to the very largest scales, a uniform distribution of matter can only\nbe achieved if the curvature of space is constant. This is in line with Einstein’s intention, which\nappears to have been to depict a cosmological model wherein a finite density of matter shaped\nthe structure of the world. During that period, only stars with minor peculiar velocities had\nbeen detected, and it was not until 1922 that nebulae were acknowledged as independent stellar\nsystems beyond our galaxy [12, 20, 21], supporting the existence of a reference frame in which\nmatter is at rest [22]. However, the two postulates above are not compatible with the Einstein\nfield equations, given in Equation (1.3.11), according to which a finite density of matter collapses\non itself due to the gravitational attraction, invalidating the static Universe assumption. Einstein\nresolved this issue by simplyintroducing a constant term, Λ, in his equations to counterbalance\nthe gravitational pull of matter, consistent with the principles of General Relativity and with\nthe Bianchi identities ∇µTµν= 0, leading to a new set of field equations [23]:\nGµν= 8πGT µν+ Λgµν. (1.3.13)\nWith this relation, the Λterm stands as a negative effect against gravity, so both effects compen-\nsate. In the context of Einstein’s static Universe, and with the solution proposed by Friedmann-10 1.3. General Relativity\nLemaître-Robertson Walker (which we assume for now but will be discussed in detail in Sec-\ntion 1.6), these equations imply\n8πGρ =1\na2R2\nE= 3Λ , (1.3.14)\nwhere ρstands for the total mass density of the Universe, and the parameter REis its curvature\nradius1, for a closed spherical geometry. Hence, the cosmological constant Λplays the role of\nsome repulsive force counteracting the gravitational collapse of the Universe. It could be included\nin the theory as a fundamental constant that determines the density of matter and the radius of\ncosmic space. However, the solution for a steady Universe is unstable because a change in the\nvalue of Λin Equation (1.3.14) (or equivalently a local rearrangement of the mean mass density)\nwould cause the Universe to either expand or collapse, like a ball that is carefully balanced on\nthe top of a mountain, thus invalidating the static assumption. Remarkably, in 1922, Friedmann\nproposed a new cosmological model that abolished the static hypothesis and replaced Einstein’s\nassumptions with [12]:\n•The Universe is homogeneous;\n•The Universe is isotropic;\n•The Universe is expanding.\nIt was only in 1929 that evidence for the expansion of the Universe came to light with Hubble’s\ndiscovery of the recession of galaxies. This led to the cosmological constant being cast aside,\nonly to be revived at the end of the millennium. Einstein himself was determined to dismiss the\ncosmological constant, as the goal for its introduction had been to represent a finite density of\nmatter in the Universe. When Friedmann showed that this could be achieved, either with or\nwithout the Λ-term - provided that the cosmic distance scale was time-dependent, which was\nfurther substantiated by Hubble’s empirical confirmation that cosmic distances indeed change\nwith time - Einstein found himself able to reject the term. He maintained that this constant\nundermined the theoretical elegance and logical simplicity of his original formulation. In his own\nwords [24]:\nThe introduction of such a constant implies considerable renunciation of the logical simplicity\nof theory, a renunciation which appeared to me unavoidable only so long as one had no reason\nto doubt the essentially static nature of space. After Hubble’s discovery of the \"expansion\" of\nthe stellar system and since Friedmann’s discovery that the unsupplemented equations involve the\npossibility of the existence of an average (positive) density of matter in an expanding Universe,\nthe introduction of such a constant appears to me, from the theoretical standpoint, at present\nunjustified.\n1The radius of curvature of space in Einstein’s static Universe is also referred to as Einstein’s radius. In the\ncontextoftheFLRWsolution(seeSection1.6), thedefinitionof REemergesfromsettingboththefirstandsecond-\ntime derivatives of the scale factor, ˙aand¨a, to zero in the Friedmann equations, arriving at aRE= 1/√4πGρ,\nwhere Gis Newton’s gravitational constant and ρrepresents the spatial density of this Universe.1Introduction 11\nPerhaps one of the most vocal critics of Einstein’s rejection of the cosmological constant was\nArthur Eddington. He fundamentally disagreed with Einstein’s view, especially regarding the\ninterpretation of cosmic expansion, seeing it as a logical consequence of the physical theory\n[25]. Eddington maintained that the general relativistic field equations expressed that length\nmeasurements should be made relative to a finite constant radius of curvature. As he famously\nwrote [20],\nFrom this point of view it is inevitable that the constant Λcannot be zero; so that empty space\nhas a finite radius of curvature relative to familiar standards. An electron could never decide how\nlarge it ought to be unless there existed some length independent of itself for it to compare itself\nwith.\nThis disagreement between Einstein and Eddington was not rooted in scientific differences but\nrather philosophical ones, as both relied on the same empirical evidence to support their views.\nEinstein believed that the time-varying spatial scale introduced by the expansion in Hubble’s law\nrenderedthe Λ-termsuperfluousandthathistheorywouldreducetoasimplerstatementwithout\nit. On the other hand, Eddington advocated that not only did the cosmological constant simplify\nthe theory’s formulation, but this reformed version also aligned with the observed expansion,\nconfirming the need for a significant cosmic repulsion on large scales.\nEddington’s view was exalted by the Nobel Prize-winning observation of the acceleration of the\ncosmic expansion in 1998 [26, 27] (which will be discussed in Chapter 2), which validated the\ncosmological constant through its predictive power. Indeed, in retrospect, the evidence gathered\nsince Einstein first proposed Λseems more aligned with Eddington’s interpretation, who believed\nthat the natural gravitational repulsion embodied by Λcould set the stage for the understanding\nof the Universe’s evolution, referring to the cosmic expansion as Einstein’s almost inadvertent\nprediction [25].\n1.4 Homogeneity and Isotropy\nModern cosmology emerged from Albert Einstein’s investigation of how his theory of general\nrelativity could be employed to understand the large-scale nature of the Universe. Einstein\nbelieved that a philosophically reasonable Universe should have the same properties everywhere\nand in all directions (therefore justifying the proposal for a uniform distribution of matter at\nthe boundary), with only minor inhomogeneities, such as the agglomerates of matter that make\nup planets or stars [5]. This opposed the standard scientific approach of examining structures\nseparately and at specific hierarchical levels based on a progressive narrative, such as the study of\nmolecules, atoms in molecules, nuclei in atoms and so on [28]. The same premise can be applied\non larger scales: the study of the nature of planets around stars is followed by stars in galaxies,\ngalaxies in clusters, and so the story proceeds.\nEinstein’s vision was that the hierarchical structure of the Universe culminates in a new ideal for12 1.4. Homogeneity and Isotropy\nmodern science: large-scale homogeneity . According to this argument, the Universe is assumed\nto be homogeneous and isotropic on large scales, meaning it is the same on average regardless of\nthe observer’s position or orientation. Einstein’s large-scale homogeneity opened the possibility\nof considering and testing a theory that describes the Universe as a whole rather than focusing\non theories for specific levels of structure. The power of this hypothesis is that observations from\nour position in the Universe can be taken to represent what the Universe looks like from any\nother place. Nevertheless, just like any hypothesis, it requires empirical evidence. The concept of\nlarge-scale homogeneity grew from philosophical reasoning, intuition, collaboration, and perhaps\neven some wishful thinking. It was not initially rooted in empirical evidence but in the pursuit\nof a unified, coherent framework to describe the Universe.\nIn more precise terms, the cosmological principle is the backbone of the standard cosmological\nmodel. It states that on large scales (greater than ∼100Mpc), the Universe is isotropic (ro-\ntational invariance) and homogeneous (translational invariance). This assumption implies no\nprivileged positions or directions in the Universe, limiting the vast array of possible cosmolog-\nical models. Isotropy1has been supported on intermediate and large scales through various\nobservations [30], discussed in Chapter 3, including:\n•The distribution of galaxy clusters and super-clusters;\n•Observations of large scale structure and of the interstellar medium in radio and X-ray\nwavelengths;\n•The uniformity of the CMB radiation temperature, with ∆T/T∼10−5, indicating a\nstrongly isotropic Universe at the time of radiation emission (approximately 300 thousand\nyears after the Big Bang).\nThe cosmological principle is also a modern formalisation of the old Copernican principle , in\nthe sense that not only do we not hold a particular position in the Solar System, but also the\nposition of the Milky Way in the Universe should not be statistically distinguishable from the\nspatial distribution of other galaxies. While the Universe contains inhomogeneous structures like\ngalaxies and galaxy clusters on smaller scales, at early times, it looked roughly the same at every\npoint and in all directions. This is best illustrated through Figure 1.1, where the impressively\ndetailed map of the temperature distribution of the CMB as measured by the Plancksatellite\nis shown. The CMB is the monument marking the largest (and hence most ancient) scales that\ncan be directly observed, depicting remarkable homogeneity.\nWe have just seen the picture of a homogeneous, isotropic and expanding Universe on large\nscales. However, which scales can be classified as sufficiently large? A standard distance unit in\nCosmology is the Megaparsec, 1Mpc = 3.2615×106light-years = 3.0856×1024cm. This is the\ntypical distance between nearby galaxies, e.g.the distance between Andromeda and the Milky\n1A general theorem of geometry ensures that isotropy around every point implies homogeneity [29].1Introduction 13\nWay is around 0.7Mpc. Determining the exact scale for which the distribution of galaxies can\nbe considered to be homogeneous is not a trivial endeavour. Even though large-scale structure\nobservations [31] show that the galaxy density inhomogeneities are still of the per cent order\non scales of ∼150Mpc, the geometry of the Universe hints that these deviations must be small\nalready on the Mpc scale. The geometry can be tested through gravitational lensing, the peculiar\nmotion of galaxies and, more importantly, the CMB [32].\nUltimately, the cosmological principle ensures that the laws of physics governing the Universe at\nlarge scales are identical for all observers, making it comprehensible to us.\nFigure 1.1 : The sky map of temperature anisotropies in the Cosmic Microwave Background as\nmeasured by the PlanckCollaboration (taken from the PlanckLegacy Archive) [33].\nCombined with the Einstein equations and a specified energy-matter distribution, the Cosmo-\nlogical Principle provides a framework for understanding how the Universe evolves. However,\nthis simplistic view of the homogeneous model has inherent limitations. Specifically, it cannot\naccount for the formation and filament-like distribution of galaxies and galaxy clusters, such as\ndepicted in Figure 1.2, nor for general variations in the matter distribution (a perfectly homo-\ngeneous and isotropic Universe would not even allow for our existence to discuss it in the first\nplace!). Likewise, the distribution of these inhomogeneities and the variations in the temperature\nand polarisation of the CMB photons are intimately connected. One can employ an approach\nbased on linear perturbation theory to address these limitations. If one first solves for the evolu-\ntion of the background quantities ( i.e.non-perturbative), that information can be incorporated\ninto the equations that govern the changes in these small-scale variations. The details of this14 1.4. Homogeneity and Isotropy\nFigure 1.2 : A map of galaxies and quasars in the observable Universe found by the Sloan\nDigital Sky Survey (SDSS) from 2000 to 2020 [31]. This particular wedge of the full survey\ncovers around 10 degrees. It encompasses about 200 000galaxies and quasars reaching from\nour observable position (not special according to the cosmological principle) up to a look-back\ntime of 12 billion years and cosmological redshift 5, with redness indicating increasing redshift\nand distance. Almost every dot in the nearby lower part of the illustration represents a galaxy,\nand the upper part represents a distant quasar. The cosmic web structure is clearly illustrated,\nwith the gravitational attraction between the nearby galaxies leading to a local Universe more\ncondensed and filamentary than the distant Universe. The edge is the CMB temperature map as\nmeasured by the PlanckCollaboration [33]. Visualisation created at John Hopkins University:\nhttps://mapoftheUniverse.net.\nmethod and how it connects with observations will be the focus of Chapters 2 and 3.\nNevertheless, this perturbative approach is not without its limitations. During the period when\nmatter dominates, structure formation starts taking place, invalidating the assumption of small\ndensity perturbations. Over this regime non-linear perturbation theory must be employed, a\ntopic which falls outside the scope of this dissertation. Nevertheless, the linear analysis of matter\nfluctuations and CMB anisotropies remains valuable. Its strength resides exactly in the fact that\nvarious methodologies can be complementary and mutually reinforcing.1Introduction 15\n1.5 The Hubble-Lemaître Law: Cosmic Expansion\nIt was Edward Arthur Milne who recognised the power of homogeneity in formulating a cosmo-\nlogical scenario and named it Einstein’s cosmological principle [34]. He showed that independent\nof general relativity, this principle explains the relation between the recession velocity vof a\ngalaxy and its distance r, known as Hubble’s law :\nv=H0r . (1.5.1)\nThe proportionality constant, H0, is known as Hubble’s constant, and the subscript is meant to\nindicate that it is a measure of the present rate of expansion of the Universe. In an evolving\nUniverse, the expansion rate is a function of time, H(t). This proposal was based on the observa-\ntion that light from distant galaxies is shifted towards longer wavelengths (redshifted), indicating\ntheir recession from us. A simple way to picture this in Euclidean geometry is to imagine two\ngalaxies at arbitrary points and write their velocities as the vector relation ⃗ v=H0⃗ r. Then an\nobserver on galaxy asees galaxy bmoving away at velocity\n⃗ vb−⃗ va=H0(⃗ rb−⃗ ra). (1.5.2)\nThis demonstrates that regardless of their position, all observers witness a consistent pattern of\ngalaxies receding from one another, as expected in a homogeneous Universe. For nonrelativistic\nrecession velocities, the cosmological redshift is defined as the ratio z=v/c, akin to a Doppler\nshift, where cis the speed of light.\nThisredshift-distance relation has recently been coined with a new name, the Hubble-Lemaître\nlaw, in acknowledgement of Lemaître’s theoretical prediction of the expanding Universe. How-\never, the recognition could be extended to the significant contribution of other players. Vesto\nMelvin Slipher’s redshift measurements1[35] and Henrietta Leavitt’s discovery of the Cepheid\nperiod-luminosity relation from variable stars in the Magellanic Clouds [36] were indispensable to\nthe production of Hubble’s famous redshift-distance plot [37], while Milton Humason’s redshift\nmeasurements [37, 38] and Milne’s cosmological considerations[34] in the 1930s were instrumental\nin providing compelling and precise evidence for the nature of this result.\n1.6 The Friedmann-Lemaître-Robertson-Walker Metric\nTo extract cosmological implications from general relativity, we need to specify the components\nof the spacetime metric in Equation (1.3.1). There are no general analytical solutions to the\n1In fact Slipher measured the shift of spectral lines of galaxies, from which it was possible to infer their\nvelocities using the Doppler shift. This led him to the groundbreaking conclusion that the nebulae being measured\nhad to be outside of the Milky Way.16 1.6. The Friedmann-Lemaître-Robertson-Walker Metric\nEinstein field equations, Equation (1.3.11), but, amongst others, Einstein realised that a co-\nherent physical description of the Universe could only be achieved under specific assumptions.\nThe fundamental underlying assumption is that the Universe is expanding and encapsulates the\ncosmological principle. That is to say that the metric needs to be such that it describes a time-\nvarying Universe which is, at each time, spatially homogeneous and isotropic. Assembling all\nthe geometric and symmetry arguments, the most generic expanding spacetime for a homoge-\nneous and isotropic Universe with matter uniformly distributed as a perfect fluid is given by the\nFriedmann-Lemaître-Robertson-Walker (FLRW) metric [11, 12, 21, 39–43] and has the following\nform:\nds2=−dt2+a2(t)γijdxidxj=−dt2+a2(t) dl2, (1.6.1)\nwhere γijare the spatial components of the metric. The spatial part of the line element, dl2,\ncan be expressed in traditional Cartesian coordinates or, more informatively, in spherical polar\ncoordinates (x1, x2, x3) = (r, θ, φ )as\ndl2=dr2\n1−Kr2/R2\n0+r2dθ2+r2sin2(θ) dφ2, (1.6.2)\nin which case\nγ11=\u0000\n1−Kr2/R2\n0\u0001−1, γ 22=r2, γ 33=r2sin2(θ). (1.6.3)\nThe scale factor a(t)describes the time evolution of the expansion scale of the Universe. It is\ndimensionless and typically normalised at present as a(t0) = 1.rrepresents the radial comoving\ncoordinate, while θandφare the comoving angular coordinates. The whole term in Equa-\ntion (1.6.2) corresponds to the line element of a maximally symmetric 3D space with a constant\ncurvature parameter Kand curvature scale R0. The derivation of the FLRW solution can be\nfound in [44, 45]. Under the assumption of homogeneity and isotropy, the intrinsic curvature\nmust be constant, and its character is encoded in K:\n•Flat space ( K= 0): the simplest case corresponding to a three-dimensional Euclidean\nspace. In such a space, parallel lines never intersect.\n•Positively curved space ( K= +1): known as spherical space as it can be thought of as a\nthree-sphere embedded in a four-dimensional Euclidean space. This is the space in which\nall parallel lines will eventually meet.\n•Negatively curved space ( K=−1): called the hyperbolic space as it can be represented\nby a hyperboloid embedded in a four-dimensional Lorentzian space. In this space, parallel\nlines will always diverge.\nIt is readily apparent that the time and space coordinates are not scaled in the same way in the\nFLRW metric, Equation (1.6.1), since the scale factor a(t)only multiplies the spatial sector. For1Introduction 17\nthat purpose, the cosmic time tmay be replaced by the conformal time τ, which is particularly\nuseful for the study of the propagation of photons (like in the CMB), which is defined as\ndτ≡dt\na(t). (1.6.4)\nMoreover, without assuming a particular curvature parameter, it is convenient to redefine the\nradial component as dχ≡dr/p\n1−Kr2/R2\n0and the metric on the unitary two-sphere,\ndΩ2= dθ2+ sin2(θ) dφ2, (1.6.5)\nsuch that the FLRW metric in Equation (1.6.1) becomes\nds2=a2(τ)\u0002\n−d2τ+\u0000\ndχ2+S2\nk(χ) dΩ2\u0001\u0003\n, (1.6.6)\nwhere a(τ)is now the conformal scale factor and\nSk(χ)≡R0\n\nχ/R 0 K= 0,\nsin(χ/R 0)K= +1 ,\nsinh( χ/R 0)K=−1,(1.6.7)\nexpresses the difference between randχfor non-flat geometries. In general we will adopt the\nconventions in which ris normalised to R0, which vanishes from the equations. Finally, it is\nworth noting that the FLRW metric encapsulates all the degrees of freedom of the spacetime\nthrough a scaling function of time, a(τ), and the constant K.\n1.7 Spacetime Geometry\nHaving introduced the FLRW metric, we can now compute the Einstein tensor by evaluating the\nleft-hand side of Equation (1.3.11).\n1.7.1 Connections\nSubstituting gµν= diag( −1, a2γij)from Equation (1.6.1) into the definition in Equation (1.6.1),\nit is straightforward to compute the Christoffel symbols. All the components with two time\nindices vanish, that is, Γµ\n00= Γ0\n0ν= 0. The non-vanishing components are:\nΓ0\nij=a˙aγij,Γi\n0j=˙a\naδi\nj,Γi\njk=1\n2γil(∂jγkl+∂kγjl−∂lγjk). (1.7.1)\naccording to Equation (1.3.5) and the Christoffel symbols are symmetric under the exchange of\nthe lower indices. In spherical polar coordinates, under the normalisation R0= 1over rand18 1.8. The Hot Big Bang Model\nremembering that H= ˙a/a, this becomes\nΓ0\nij=a2Hγij,Γi\n0j=Hδi\nj,Γ1\n11=Kr\n1−Kr2,Γ1\n22=−r(1−Kr2), (1.7.2)\nΓ1\n33=−r(1−Kr2) sin2θ ,Γ2\n12=1\nr,Γ2\n33=−sinθcosθ ,Γ3\n13=1\nr,Γ3\n23= cot θ .\n1.7.2 Ricci Tensor\nThe time-space components vanish according to the isotropy of the spacetime: R0i=Ri0= 0.\nThe non-zero components are:\nR00=−3¨a\na=−3\u0010\nH2+˙H\u0011\n, (1.7.3)\nRij=a2\"\n¨a\na+ 2\u0012˙a\na\u00132\n+ 2K\na2#\nγij=a2\u0014\n3H2+˙H+ 2K\na2\u0015\nγij. (1.7.4)\n1.7.3 Ricci Scalar\nThe Ricci scalar is computed from the components of the Ricci tensor:\nR= 6\"\n¨a\na+\u0012˙a\na\u00132\n+K\na2#\n= 6h\n˙H+ 2H2i\n. (1.7.5)\n1.7.4 Einstein Tensor\nThe non-zero components of the Einstein tensor, Equation (1.3.11), in the form Gµ\nν≡gµλGνλ\nare:\nG0\n0=−3\"\u0012˙a\na\u00132\n+K\na2#\n=−3\u0014\nH2+K\na2\u0015\n, (1.7.6)\nGi\nj=−\"\n2¨a\na+\u0012˙a\na\u00132\n+K\na2#\nδi\nj=−\u0014\n2˙H+ 3H2+K\na2\u0015\nδi\nj, (1.7.7)\nwhere again isotropy ensures that the mixed components vanish.\n1.8 The Hot Big Bang Model\nWe close with a note on the remarkable observational evidence for what exactly is meant by\ntheBig Bang mentioned at the beginning of this chapter. The advent of General Relativity\n(GR) sparked a paradigm shift, revolutionising our understanding of the Universe’s evolution.\nGeorges Lemaître was the first to propose that Einstein’s equations yielded a solution describing\nan expanding Universe that could be followed back in time to a single point [46], since ρ→ ∞1Introduction 19\nasa→0. Subsequently, Edwin Hubble’s experimental discovery of the Universe’s expansion [37]\nculminated in the development of the successful Hot Big Bang (HBB) model for the observable\nUniverse (for a more detailed description and historical context about the HBB model, we refer\nthe reader to the review in Ref. [47]).\nThe standard HBB model has provided a theoretical basis for explaining various experimental\nobservations [29], namely:\n1. Predicting the existence and the shape of the spectrum of the cosmic microwave back-\nground;\n2. Explaining the mechanism behind the primordial abundances of light elements through\nnucleosynthesis;\n3. Describing the expansion of the Universe and providing a framework to understand the\ngravitational collapse of matter, leading to the formation of galaxies and other large-scale\nstructures observed today.\nThe Hubble parameter H(t) = ˙a/aplays a crucial role as a physical observable, describing the\nexpansion rate of an FLRW Universe and setting its characteristic time and distance scales.\nThe Hubble time tH=H−1and the Hubble radius dH=cH−1(in units where c= 1we have\ndH=tH) define the time and length scales of the FLRW spacetime, respectively, and are strictly\nlocal quantities ( i.e.they are defined by the instantaneous expansion rate at a given time t).2The Standard ΛCDM Model: Successes and\nControversies\nE – e não esquecer que a estrutura do átomo não é vista mas sabe-se dela.\nSei de muita coisa que não vi. E vós também. Não se pode dar\numa prova da existência do que é mais verdadeiro, o jeito é acreditar.\n—Clarice Lispector A Hora da Estrela\nAnd — and don’t forget that the structure of the atom cannot be seen but is nonetheless known. I know about\nlots of things I’ve never seen. And so do you. You can’t show proof of the truest thing of all, all you can do is\nbelieve.—Clarice Lispector in The Hour of the Star\nIn this chapter, we build upon the gravitational framework introduced in Chapter 1 to introduce\nthe current standard model of cosmology, the ΛCDM model, by setting forth its underlying\ntheoretical assumptions and predictions. Assuming the Friedmann-Lemaître-Robertson-Walker\nbackground metric as the starting point, we will explore the equations governing the Universe’s\nexpansion history in Section 2.1. Moreover, we examine the linear perturbations produced in\nthe early Universe in Section 2.2, which serve as the initial inhomogeneities that are the seeds\nof structure formation. Bringing everything together, we summarise the most relevant epochs in\nthe Universe’s evolution and the physics involved at the relevant scales in Sections 2.3 and 2.4.\nWe conclude by contemplating the need for the cosmological constant, the simplest form of dark\nenergy, while also discussing the conceptual challenges that arise from its inclusion in Section 2.5.\n2.1 Fundamentals of the Background Cosmology\nIn the framework of the Hot Big Bang model for the early Universe, tracing back the evolution\nuntil the beginning of time leads to a problematic point known as a singularity, a fundamental\nlimitation of standard general relativity. However, it is widely expected that there exists a\nmore comprehensive theory that resolves this singularity issue, with the concept of cosmological\ninflation being a prominent avenue of investigation. While inflation is a significant topic, this\nchapter primarily focuses on the subsequent cosmic evolution following inflation and the events\nin the very early Universe, deliberately setting aside the open question of what preceded (or\nmore precisely, what was) the Big Bang (BB).\n2122 2.1. Fundamentals of the Background Cosmology\n2.1.1 Observable Universe\nLet us start by clarifying what is meant by the observable Universe. According to GR, the speed\nof light c(and therefore the speed of any detectable signal carrying information) is finite. It\nfollows that not every spacetime region is within our reach. Adopting the Big Bang scenario\nfurther implies the existence of an intrinsic limit set by the age of the Universe on the maximum\ndistance from which a point in space could have received information. This limit is known as\ntheparticle horizon (defined as the time integral of 1/a(t)), and its size keeps evolving. This\nmeans that whenever we make considerations about the Universe, these are based on the portion\nof the Universe that is causally connected to us and which lies within our particle horizon.\nAnything beyond this region is inaccessible and, more importantly, will not directly impact\nour observations. Nevertheless, two different spacetime regions can be accessible and causally\nconnectedtosomeobserverwhileremainingdisconnectedfromeachother. Allittakesisforthese\ntwo regions to be contained inside the observer’s particle horizon while their individual particle\nhorizons do not overlap. It is intuitively understandable that any causally connected regions\ncan exhibit homogeneity (without an extremely high level of fine-tuning) if they have had the\nchance to interact and reach some thermal equilibrium. On the other hand, if both regions are\nnot causally connected, then local interactions cannot justify the observed homogeneity, which\nmust only be related to some initialhomogeneity of the Universe itself.\nOur observed Universe demonstrates a remarkable level of uniformity on scales larger than an-\nticipated, given its age and the problems posed above. This challenge is known as the horizon\nproblem [48] and was first formalised in the 1960s. One possible resolution would be to consider\nthat the Universe is significantly older. However, this would imply substantial deviations from\nwhat is estimated by observations by several orders of magnitude [45]. Consequently, under the\ncurrent cosmological model, this uniformity is better explained by the inherent particularities of\nits initial conditions - in a brief moment at the beginning of the Universe which cosmologists\nbelieve happened and refer to as inflation, and which will be further discussed in Section 2.3.\n2.1.2 Invisible Universe\nOneofthemostrevolutionaryproposalsofthestandardmodelwastheassertionthatthemajority\nof matter and energy in the Universe is in some form invisible to our detectors (non-luminous),\nmeaningthatitcanonlybeobservedthroughgravitationalinteractionsorotherunknownindirect\nsignals. More precisely, the standard matter we can see and interact with amounts to around 5%\nof the total share. The remaining 95%are in the form of the mysterious dark sector, split into\ntwo components with rather distinct nature and purpose: dark matter (DM) and dark energy\n(DE).\nThe term dark matter was coined by Fritz Zwicky in 1933 when he observed that galaxies in the\nComa cluster moved faster than anticipated in order to remain bounded together [49] given the2The Standard ΛCDM Model: Successes and Controversies 23\nmeasure of visible mass within them. This implied that either the clusters were flying apart, or\nthey had to be held together by the gravitational attraction of mass outside the luminous parts\nof the cluster, or otherwise, something had to be wrong with our understanding of physics. To\nexplain the stability of the cluster, he proposed the existence of an additional form of invisible\nmatter. Later in the 1970s, Vera Rubin and colleagues measured the rotation speed of gas in the\nouter regions of galaxies [50], also reporting exceptionally high speeds, with nearly flat rotation\ncurves. These and subsequent observations, based on X-rays, radio emission, Big Bang Nucle-\nosynthesis (BBN) or the developing studies on Large Scale Structure (LSS), could be explained\nby hypothesising halos of some excess non-luminous matter surrounding the galaxies and in clus-\nters of galaxies, further supporting the need for this weakly interacting matter. Nevertheless,\nthe most robust indirect evidence for dark matter so far comes from the gravitational lensing of\nthe CMB. As the CMB photons travel across the Universe, they are gravitationally deflected by\nthe nearby large-scale structure, creating a pattern of distortions in the hot and cold spots of\nthe CMB. This phenomenon has been mapped in great detail by the Plancksatellite [51], which\nconfirmed that its magnitude relates directly to the total amount of matter in the Universe.\nObservations of the abundance of light elements suggest that ordinary baryonic matter exists\nin relatively less abundance. The exact amount of dark matter is also necessary to explain the\nmeasured rate of gravitational clustering since the density fluctuations generated in the early\nUniverse rely on dark matter to grow at such a rapid and efficient pace [52, 53].\nIn the 1980s, even after the proposal of DM, cosmology was still at a turning point. The\nextrapolated age for a Universe populated only by matter appeared to be shorter than the ages\noftheoldeststarswithinit. Moreover, observationsofthelarge-scalestructurewereincompatible\nwith the predictions from inflationary cosmology of a quasi-flat Universe, suggesting that only\n30%of the critical density (the total energy-density contribution in the Universe to ensure spatial\nflatness) could be accounted for by the total matter density. As discussed in Section 1.3, it was\nonce suggested that Einstein’s cosmological constant, introduced to create a static Universe\nsolution, or some other form of dark energy (a term that appears to have first been coined by\nHuterer and Turner [54]), might provide a solution to these issues. However, lacking empirical\nsupport, this idea remained a theoretical hypothesis. Later came the simultaneous proposal by\nRatra and Peebles [55] and Wetterich [56] in 1988, positing that the cosmological constant may\nhave been gradually approaching its natural value - zero - over an extended period.\nHowever, this question was ultimately settled by the unanticipated discovery of the accelerating\ncharacter of the Universe by two separate teams: the Supernova Cosmology Project [27], and\nthe High-z Supernova Search Team [26], by studying standard distant supernova events and\nmeasuring the distances and recession speeds of galaxies, analogous to Hubble’s analysis using\nlocal sources. These studies revealed that the expansion rate was decelerating in the early stages\nwhen the Universe was dominated by matter and transitioned to an accelerating phase quite\nrecently (in cosmological terms). It turns out that the simplest way to account for the accelerated24 2.1. Fundamentals of the Background Cosmology\nexpansion within the framework of Einstein’s theory of gravity is if the Universe has just entered\na phase dominated by Einstein’s cosmological constant Λ; more precisely, a source with an energy\ndensity that does not dilute with the expansion and which exerts a negative pressure symmetric\nto its energy density. Remarkably, the cosmological constant hypothesis accommodates the\noldest stars in the estimated age of the Universe and fulfils the critical density as required in the\ninflation paradigm, with detailed studies of the CMB anisotropies map placing the dark energy\nfractional density to the missing 70%of the total energy in the present-day Universe [32].\nThe addition of these two dark sector components is the genesisof the standard model of\ncosmology, the Lambda-Cold-Dark-Matter ( ΛCDM) model. However, and as will be one of\nthe main focuses of this dissertation, despite addressing the problems above and fitting the data\nwith outstanding accuracy, the ΛCDM model now faces a new crisis. On the fundamental side,\nthe cosmological constant cannot be reconciled with predictions for the vacuum energy from\nquantum field theory, with several orders of magnitude of difference between the predicted and\nobserved values. On the other hand, the increase in independent precision probes has brought\nto light observational incompatibilities between model-dependent measurements from the early\nUniverse and studies within the local Universe. The origin of these inconsistencies remain one\nof the most significant unresolved problems in cosmology and fundamental physics [57].\n2.1.3 Expanding Universe\nThe Hubble expansion rate is one of the protagonists of modern cosmology and is defined as:\nH≡˙a\na, (2.1.1)\nwith the fact that it provides information about the Universe’s expansion rate now made explicit.\nGenerally, His a function of time, and its value measured at present, denoted as H0, is referred\nto as the Hubble constant or parameter. As was the case for defining the conformal time,\nEquation (1.6.4), it is crucial to differentiate between the comoving distance r- which is fixed to\nthe expansion and thus oblivious to its effect - and the physical distance rphys, which increases\nproportionally to the expansion. An observer studying an object at a fixed comoving distance r\nwill measure a physical distance to it that depends on time. Therefore, the physical velocity of\na point must also account for the added contribution of the expansion and is given by:\nrphys≡a(t)rd/dt− − − → vphys≡drphys\ndt=a˙r+r˙a≡vpec+Hrphys. (2.1.2)\nThe first term, vpec, is thepeculiar velocity , that is, the velocity of the object relative to the\ncoordinate grid. The second term, Hrphys, is theHubble flow resulting from the expansion of\nthe coordinate grid itself. For galaxies moving in the Hubble flow, sufficiently distant for their\npeculiar velocities to be insignificant in comparison to the velocity arising from the expansion,2The Standard ΛCDM Model: Successes and Controversies 25\nEquation (2.1.2) simplifies to:\nvphys≃Hrphys, (2.1.3)\nwhich we have encountered before as the Hubble-Lemaître law, valid for local distances (in\ncomparison to H), in which case H≈H0. In other words, the recession of galaxies away from\nus happens at speeds proportional to their physical distance and can be solely attributed to the\nexpansion of the Universe. The proportionality constant measures the present rate of expansion,\nH0. Edwin Hubble’s reported observation of his homonym law in 1929 [37] by plotting vagainst\nrleading to an estimated H0= 500km/s/Mpc, was the onset of evidence of the Universe’s\nexpansion and provided strong support for the hot Big Bang model. During the past century,\nthe endeavour of measuring and independently confirming the value of the Hubble constant\nprogressed. At the moment, H0is typically inferred from Cosmic Microwave Background (CMB)\nmeasurements assuming a specific cosmological model or estimated directly in the local Universe,\nsuch as the luminosity patterns of Type Ia supernovae, in a model-independent fashion.\nAsisthecaseforthe H0estimates, mostobservationalprobesincosmologyrelyondetectinglight\nemitted by astrophysical and cosmological sources. In an expanding Universe, the momentum\nof particles that transverse it, as perceived by comoving observers, decays with the expansion at\nthe rate 1/a, resulting in an intrinsic effect in the light we observe. If the light is emitted with a\nspecific wavelength λ, that wavelength will be stretched by the expansion proportionally to the\nscale factor a. Consequently, a photon emitted at time tis detected on Earth at time t0with a\nwavelength given by:\nλ(t0) =a(t0)\na(t)λ(t). (2.1.4)\nIn an expanding Universe, the scale factor increases with time, meaning that a(t0)> a(t),\nimplying that the wavelength at detection is larger than the wavelength at emission. This leads\nto the introduction of the redshift parameter1, defined as the fractional change in the wavelength\nof a photon emitted at time tand detected today on Earth:\nz≡λ(t0)−λ(t)\nλ(t), (2.1.5)\nand which quantifies how much the wavelength has been stretched. Employing Equation (2.1.4),\nwe find 1 +z=a(t0)/a(t), which together with the conventional normalisation a(t0) = 1, yields:\n1 +z≡1\na(t). (2.1.6)\nGiven its direct relation with the time-dependent scale factor, the redshift parameter zis fre-\nquently preferred for the time-keeping of specific events during the history of the Universe.\n1We had previously defined the redshift as z=v/c. This is valid as long as the recessional velocity of an\nobject is much smaller than candλ0≈(1 +v/c)λfrom the Doppler effect.26 2.1. Fundamentals of the Background Cosmology\nThe Hubble parameter is often expressed in terms of the dimensionless quantity h:\nH0= 100 hkm/s/Mpc = 2.1332h×10−42GeV, (2.1.7)\nand can be used to define the average cosmological density in the Universe, also called the critical\ndensity:\nρcrit,0≡3H2\n0\n8πG= 1.88h2×10−29gcm3. (2.1.8)\nLastly, the Hubble rate defines a Hubble time\ntH≡1\nH0= 9.78×109h−1years , (2.1.9)\nwhich is a crude measure of the age of the Universe, and the present Hubble radius\nDH≡c\nH0= 2998 h−1Mpc, (2.1.10)\na fair approximation to the largest scales that can currently be observed. The following two\nsections will address other observationally meaningful parametrisations of time and distance.\n2.1.4 Time Parametrisations\nBefore moving on to the details of the standard model, it is essential to summarise the various\nclocks that refer to the notion of time in different epochs and contexts depending on the physical\nsystem under consideration. The relation between the time t(and equivalently τ) and the\nredshift z(and equivalently a) encoded in Equation (2.1.6) is especially relevant when dealing\nwith relativistic components due to the direct redshift-dependent correspondence between energy\nat emission and detection, encoding a valuable measure of energy loss over time which would not\nbe evident for the proper time. Thus, t,τ,a, and zare all relevant and distinctively informative\ndefinitions of time.\nHowever, in the very early Universe, when particle physics processes play a crucial role, the\nconcept of temperature Tconstitutes a trigger for different events. For photons with energy\nE, there is a direct correspondence between temperature and redshift since 1 +z∝E∝T\n(normalised to the temperature of the Universe today: T0= 2.7255K≃2.348×10−4eV).\nFinally, and as will be extensively discussed in the following sections, the tiny energy fluctuations\nin the early-Universe evolve according to a wave-like behaviour and can be decomposed into their\nFourier modes k, related to their wavelength λask= 2πa/λ. Assuming the separate Universe\napproximation, i.e.that each mode evolves independently, kbecomes a proxy for the scales of a\ngiven event. At early times, when the Universe was smaller, only modes with small wavelengths\ncontributed significantly to the evolution, as those with much larger wavelengths were minimally2The Standard ΛCDM Model: Successes and Controversies 27\naffected by processes occurring at small scales. The Fourier modes convey an idea of the size of\nthe Universe, which, in turn, is related to the time elapsed since the beginning of its expansion.\nTherefore, all these quantities will be used interchangeably throughout the text depending on\nthe context and the cosmological epoch.\n2.1.5 Cosmological Distances\nIn the same way, as for the time parametrisations considered above, different rulers can be used\nto measure distances, depending on which quantity is being used as a proxy of distance. Much\nof the evidence for dark energy comes from measurements of cosmological distances. In order\nto understand how present observations relate to other processes happening in the Universe, it\nis essential to understand photon propagation. Since photons have no mass, they traverse null\ngeodesics:\nds2=−dt2+a2(t) dχ2= 0⇔dχ2=a−2(t) dt2, (2.1.11)\nwith the radial component χas the direction of the photons’ propagation. The last expression\ncan be integrated to find the line-of-sight comoving distance dctravelled by the photon in some\ntime interval [ti, tf]:\ndc(t) =χ(t) =Zti\ntfcdt\na(t)dt=da/(aH)=−dza/H− − − − − − − − − − − − − − → dc(z) =Zzf\nzicdz\nH(z). (2.1.12)\nAnother important observable in cosmology is the absolute luminosity of astrophysical objects\nL, defined in terms of the observed flux F, over a sphere of radius χin an expanding spacetime.\nThis sphere has an area\n4πa2(t0)d2\nMwhere dM≡Sk(χ), (2.1.13)\nandiscalledthe metric ortransversecomoving distance ,with Sk(χ)asdefinedinEquation(1.6.6),\nwhich reflects the fact that in a curved spacetime dM̸=χ.dcin Equation (2.1.12) is often used\nwhen referring to the spatially flat geometry, in which case both quantities are equivalent. The\nluminosity distance is then expressed in terms of Fas,\nd2\nL=L\n4πFexpanding space− − − − − − − − − − → dL(z) = (1 + z)dM(z), (2.1.14)\nwhere the multiplicative redshift factor accounts for the energy loss experienced by the photons\nand the reduction in their number due to cosmic expansion since\nLi\nLf=∆Ei\n∆Ef∆tf\n∆ti= (1 + z)2. (2.1.15)\nThis will be important in Chapter 3 where we will discuss how distance measurements can be\nused to infer the value of cosmological parameters such as H0.28 2.1. Fundamentals of the Background Cosmology\nFor objects of some physical size ∆xthat are observed to subtend an angle in the sky ∆θ\n(perpendicular to the line of sight and ∆θ≪1in radians for cosmological distances), it is more\nconvenient to define the angular diameter distance :\ndA=∆x\n∆θand ∆x=a(t)Sk(χ)∆θexpanding space− − − − − − − − − − → dA(z) =dM(z)/(1 +z),(2.1.16)\nwhere we have used geometrical considerations assuming ∆θ≪1, and the dividing redshift\nfactor is now included to reflect the Universe’s expansion since the light was emitted until it is\nobserved in the sky. In summary, we have defined three different distance observables which are\nrelated as:\ndM(z) = (1 + z)dA(z),and dL(z) = (1 + z)2dA(z). (2.1.17)\nThe last expression is the distance-duality or Etherington relation [58], which holds for general\nmetrics under photon number conservation.\n2.1.6 Friedmann Equations\nAssuming perfect fluids coupled to Einstein’s field equations, Equation (1.3.11), we can gather\nall the ingredients and derive the Friedmann equations that describe the evolution of an FLRW\nUniverse:\n\u0012˙a\na\u00132\n=H2=8πG\n3ρ−K\na2, (2.1.18)\n¨a\na=˙H+H2=−4πG\n3(ρ+ 3p), (2.1.19)\nwhere ρandpare the generic contributions of all the fluids in the Universe, remembering that\nthe constant Krepresents the spatial curvature of the Universe. It is worth remarking that\nEquation (2.1.19) is also known as the Raychaudhuri equation and is a curvature-independent\nmeasure of the expansion, as it does not depend directly on K.\n2.1.7 Cosmic Ingredients\nThecosmologicalprincipledictatesthatthebackgroundenergy-momentumtensorresemblesthat\nof a perfect fluid [45], expressed as\nTµν= (ρ+p)uµuν+pgµν, (2.1.20)\nwhere ρandprepresenttheenergydensityandpressureintherestframeofthefluid, respectively,\nanduµis the four-velocity relative to a comoving observer, under the constraint uµuµ=−1,\nwhich we take to be\nuµ= (a,⃗0)and uµ= (a−1,⃗0), (2.1.21)2The Standard ΛCDM Model: Successes and Controversies 29\nfor all cosmological components. The Bianchi identities, Equation (1.3.12), imply conservation\nof energy from the time component, leading to the conservation or continuity equation:\n∇µTµν= 0ν=0− − → ˙ρ+ 3H(ρ+p) = 0 . (2.1.22)\nThe second Friedmann equation, introduced in Equation (2.1.19), is equivalent to taking the first\nFriedmannequation, Equation (2.1.18), togetherwiththecontinuityequation, Equation(2.1.22).\nAssuming a Universe dominated by a single species with an equation of state p=wρwith\nconstant w(whichwillnolongerhold, forinstance, fordynamicaldarkenergy), Equation(2.1.22)\ncan be integrated to obtain\na∝(t−ti)2\n3(1+w)and ρ∝a−3(1+ w), (2.1.23)\nwhere the subscript denotes the initial time. This result embodies how different cosmological\nfluids are diluted by cosmic expansion, which we list below for the three ingredients that play\nessential roles in the evolution of the Universe. It also shows how spatial curvature contributes\nto the first Friedmann equation, Equation (2.1.18), as a fluid with ρ∝a−2.\nThe critical energy density for a flat spatial geometry ( K= 0), defined in Equation (2.1.8) at\npresent, can be expressed as a time-evolving function as\nρcrit=3H2\n8πGwith ρcrit,0≈5×10−6GeVcm−3, (2.1.24)\nwhere the label 0denotes present time, which we often omit when discussing the current matter\ndistributions, and it is implied from the context. This can be used to express the relative energy\ndensity of the fluid components fin a dimensionless manner:\nΩf=ρf\nρcritfor f=m, r, Λ, ... , (2.1.25)\nwhere the subscripts (m, r)denote the collective matter and radiation components, whose evolu-\ntionwillbedetailedbelow, and Λgivesthecontributionfromthecosmologicalconstant. Defining\nΩK≡ −K/(aH0)2(in which case ΩK<0actually corresponds to K > 0andvice versa ), the\nFriedmann equation, Equation (2.1.18), can be rewritten as the Friedmann constraint for the\ncurvature density. This can be expressed according to the measured relative energy densities at\npresent time:\nΩ0≡Ω0\nm+ Ω0\nr+ Ω0\nΛa(t0)≡1− − − − − → Ω0\nK= 1−Ω0. (2.1.26)\nIf the Universe’s expansion is decelerating ( i.e.,¨a <0), then the absolute value of the curvature\nterm|ΩK|will continue to rise. This is except in the particular case where the Universe has been\nprecisely flat ( K= 0) from its early stages. According to the Planck 2018 data [32], this quantity\nis contrained to be Ω0\nK=−0.044+0.018\n−0.015at present at 68%Confidence Level (CL). For |ΩK|to30 2.1. Fundamentals of the Background Cosmology\ndecrease during the Universe’s evolution, a phase of cosmic acceleration ( ¨a >0) is necessary,\nor otherwise, the Universe must have emerged from an initial state extremely close to flat. To\naccount for the current observed level of the Universe’s flatness, a phase of cosmic inflation is\nneeded before the radiation-dominated era, during which the scale factor must have increased\nby a factor greater than e60[47].\nNext, according to Equation (2.1.23), we list the Universe’s constituents by their properties,\nderiving the corresponding evolution for the approximated single-component case. The evolution\nof each comic fluid with redshift is depicted in Figure 2.1.\n100101102103104105106\nRedshift (1+z)1050\n1047\n1044\n1041\n1038\n1035\n1032\n1029\n1026\nDensities (i)\nr\ncdm\nb\nFigure 2.1 : Evolution of the energy densities ρof radiation r(grey dotted line), baryons b\n(green dashed line), cold dark matter cdm (red filled line), and the cosmological constant Λ\n(blue dot-dashed line), as a function of redshift zin the ΛCDM model.2The Standard ΛCDM Model: Successes and Controversies 31\nMatter\nThe broad heading matterrefers to fluids with negligible pressure compared to the energy density\n(non-relativistic particles), such that p≪ρ⇒w≈0, resulting in the dilution rate\na∝(t−ti)2/3and ρm∝a−3∝t−2H=˙a/a− − − − → H=2\n3t. (2.1.27)\nOne way to interpret this result is to imagine that the volume of a given region of three-\ndimensional space increases as V∝a3. In contrast, the number of particles (and therefore\nthe energy in that region) remains constant. The relevant non-relativistic matter components\nfor the Universe’s evolution are:\n•Baryons: refers to ordinary matter, including the less massive electrons (fermions), thor-\noughly comprehended by physicists as an integral part of the standard model of particle\nphysics.\n•Dark Matter: approximately five times more abundant than baryons, the exact character-\nistics of dark matter remain a mystery, as was discussed in Section 2.1.2. In the standard\nmodel, it is cold dark matter (CDM). Other potential candidates include Weakly Interact-\ning Massive Particles (WIMPs), which have yet to be detected.\nRadiation\nRadiation is a gas composed of relativistic particles characterised by p=ρ/3⇒w= 1/3\naccording to statistical mechanics. As derived from Equation (2.1.23), this results in\na∝(t−ti)1/2and ρr∝a−4∝t−2H=��a/a− − − − → H=1\n2t. (2.1.28)\nThe energy redshift of relativistic particles, expressed as E∝a−1, adds to the dilution due to\nthe expansion.\nThe non-relativistic components are\n•Photons: relativistic massless particles. Photons were the dominant contribution during\nnucleosynthesis(formationofthefirstatoms)andarenowdetectedasthecosmicmicrowave\nbackground.\n•Lightparticles: IntheearlyUniverse, alltheparticlesinthestandardmodelhadaradiation\ncharacter due to the high kinetic energy relative to their mass. They behaved like matter\nonce the temperature dropped enough for their masses to play a role.\n•Neutrinos: contributed to the early Universe radiation domination, and only recently have\ntheir small masses become relevant and started to act like matter.\nAt present, both photons and neutrinos only exist in minimal quantities.32 2.2. Linear Cosmological Perturbations\nDark Energy\nInordertohavecosmicacceleration, TheFriedmannequationsrequire ¨a >0inEquation(2.1.19),\nwhich implies p <−ρ\n3, or equivalently, w <−1\n3. When w=−1andp=−ρ, the relation in\nEquation (2.1.23) only holds if ρis a constant. This is the case for the cosmological constant,\nwhich (in a flat Universe) implies that His also constant, in which case the scale factor evolves\nexponentially:\na∝eHΛtand ρΛ=Λ\n4πG∝a0H=˙a/a− − − − → H=√\n8πG\n3ρΛ. (2.1.29)\nThe cosmological constant cannot be responsible for inflation in the early Universe, as that would\nmean the accelerated expansion would never cease, contrary to the current cosmic acceleration,\nwhich may continue indefinitely.\nThecosmologicalconstantproposalcanbegeneralisedunderthebroadercategoryofdarkenergy,\nreferring to any fluid either with a constant wDE≈ −1, or a time-varying equation of state\nparameter, wDE(t), in which case Equation (2.1.23) no longer holds.\nBringing all the ingredients together for the ΛCDM model, the first Friedmann equation, Equa-\ntion (2.1.18), can be decomposed into\nE2(z) =H2\nH2\n0= Ω0\nra−4+ Ω0\nma−3+ Ω0\nKa−2+ Ω0\nΛ, (2.1.30)\nwhere E(z)is the normalised Hubble rate.\n2.2 Linear Cosmological Perturbations\nUp to this point, our treatment of the Universe assumed perfect homogeneity. However, to com-\nprehend the Universe’s large-scale structure, it becomes necessary to introduce inhomogeneities\nand study their evolution. As long as these perturbations remain relatively small, we can analyse\nthem using linear perturbation theory.\nIn practice, the quantities that can be determined observationally for a given model are not\ntypically the exact values of the perturbation variables, subject to quantum fluctuations, but\nrather realisations of expectation values. However, this methodology has an inherent problem\nsince we cannot precisely measure expectation values. The usual statistical considerations fail\nbecause we only have access to one Universe , one sample of the stochastic process that generates\nthese fluctuations. Therefore, when attempting to determine the mean square fluctuation on a\ngiven scale λ, the best approach is to average over multiple disjoint patches of size λand take this\nspatial averaging as an ensemble averaging based on an ergodic hypothesis analogue. If the scale\nλis much smaller than the Hubble horizon 1/H(our observational limit), then we can compute\naverages over many independent volumes, and this approximation works well. However, as we2The Standard ΛCDM Model: Successes and Controversies 33\napproach scales of the order of the horizon, λ≈ O(1/H0), the statistical approximation breaks\ndown due to cosmic variance .\nThe perturbations are included in the Einstein equations by taking slight deviations - denoted by\nδ- to the background quantities - denoted by an upper bar - in the metric (the Einstein tensor)\nand the matter distribution (the energy-momentum tensor):\ngµν(t, ⃗ x) = ¯gµν+δgµν(t, ⃗ x), (2.2.1)\nTµν(t, ⃗ x) =¯Tµν+δTµν(t, ⃗ x). (2.2.2)\nWe can then expand the Einstein and conservation equations to linear order in the perturbations.\nFor simplicity, we will focus on flat spacetimes only. The same derivation, including the terms\ndependent on the curvature scalar, can be found e.g.in [29, 47, 59].\nFor the study of the dynamics of the linear perturbations, it is convenient to define each relevant\nquantity in terms of its Fourier transform, where k= 2π/λare the Fourier modes. Taking any\nperturbed quantity f(τ, ⃗ x), the Fourier transform is\nf(τ,⃗k) =Z\ne−i⃗k·⃗ xf(⃗ x) d⃗ x3, (2.2.3)\nwhile the inverse reads\nf(τ, ⃗ x) =1\n(2π)3/2Z\ne−i⃗k·⃗ xf(⃗k) d⃗k3, (2.2.4)\nwhere f(⃗k)are the Fourier components of f(τ, ⃗ x)and cosmic time has been replaced by the\nconformal time τ, related by dt=adτand˙(·)7→(·)′/a. This implies the mapping ∇2→ −k2\nwhen the spatial derivatives are applied to the relevant quantities.\nThe general perturbed metric containing only scalar perturbations1is subject to a non-unique\nchoice of coordinates or gauge choice . Hence, different choices of coordinates can introduce\nsignificant changes to the values of perturbed variables. Some non-physical perturbations may\narise associated with that choice, the so-called gauge modes , which can be eliminated with more\nconvenient coordinates. Equivalently, a physical perturbation can be dropped, for instance, in\ncomoving coordinates, but these will never truly vanish; they will instead be incorporated into\nadditional perturbations in the metric. Therefore, a more physical approach to perturbation\ntheory is one which remains invariant under changes in coordinates, allowing for a more robust\nand reliable study of the true perturbed cosmological degrees of freedom in cosmology. This can\nbe achieved via the so-called Bardeen variables [60], which encode the true spacetime perturba-\n1In general, one can perform a scalar-vector-tensor decomposition of the metric perturbations, such that the\nEinstein equations for scalars, vectors and tensors are independent at linear order and can be treated separately.\nWe will only deal with scalar fluctuations and the corresponding density perturbations, so vectors and tensors\nmay be disregarded at linear order. Vector perturbations are not produced by standard inflation but would, in any\ncase, decay quickly over the expansion. Tensor perturbations are an important test of inflation but not significant\nfor the scales considered in this work.34 2.2. Linear Cosmological Perturbations\ntionsthat cannot be erased by a gauge transformation . More details on gauge invariance and\ntransformations can be found, for example, in [61–64]. It is often convenient to fix the gauge\ndepending on the scenario in question, at the cost of carefully keeping track of all the perturba-\ntions in the metric and the matter (there are no free lunches!). For the cosmological scenarios\nthis dissertation considers, we will focus on two common gauge choices, primarily for consistency\nand cross-checking reasons: the conformal Newtonian and synchronous gauges.\n2.2.1 Newtonian Gauge\nFor this dissertation, we will only be interested in cosmological scenarios well-described by the\nFLRW background metric ¯gµν, as defined in Equation (1.6.6). In the context of dealing with the\nEinstein and Friedmann equations, the scalar metric perturbations are most simply expressed in\ntheNewtonian or longitudinal gauge :\nds2=a2(τ)\u0002\n−(1 + 2Ψ) d2τ+ (1−2Φ)δijdxidxj\u0003\n. (2.2.5)\nThe metric perturbations ΦandΨare also called the Bardeen potentials [60], as they are gauge\ninvariant and hence of physical interest. The Newtonian gauge is usually preferred for studying\nstructure formation exactly because of its closest resemblance to the Newtonian analysis, with\nΦplaying the role of the gravitational potential.\nMetric Perturbations\nThis lays out the metric of the spacetime as\ngµν=a2\n−(1 + 2Ψ) 0\n0 (1 −2Φ)δij\n, (2.2.6)\nfrom which we can compute the evolution equations for the metric perturbations encoded in\nthe perturbed Einstein equations. Namely, starting from the perturbed line element, Equa-\ntion (2.2.5), we can compute the inverse metric and, through the definition of the connections,\nfurther calculate the perturbations of more complex objects, such as the Ricci tensor, δRµν, and\nthe Ricci scalar, δR. These are needed to write the linear order Einstein equations in terms of\nthe perturbed Einstein tensor δGµν:\nδGµν=δRµν−1\n2(δRg µν+Rδg µν) = 8 πGδT µν. (2.2.7)\nFor the sake of completion, in what follows, we list the main results for the perturbed quantities\nrelevant to the Einstein equations, considering only the linear terms. Further details and a\nthorough derivation can be found e.g.in Refs. [45, 63, 65].2The Standard ΛCDM Model: Successes and Controversies 35\nInverse Metric\ngµν=a−2\n−1 + 2Ψ 0\n0 (1 + 2Φ) δij\n, (2.2.8)\nConnections\nδΓ0\n00= Ψ′, δΓ0\ni0=∂iΨ, δΓ0\nij=−\u0002\nΦ′+ 2H(Φ + Ψ)\u0003\nδij, (2.2.9)\nδΓi\n00=∂iΨ, δΓi\nj0=−Φ′δi\nj, δΓi\njk=−2δi\n(j∂k)Φ +δjk∂iΦ. (2.2.10)\nRicci Tensor and Ricci Scalar\nThe perturbed Ricci tensor is calculated from the connections as:\nδRµν=∂αδΓα\nµν−∂νδΓα\nµα+δΓα\nµνΓβ\nαβ+ Γα\nµνδΓβ\nαβ−δΓα\nµβΓβ\nαν−Γα\nµβδΓβ\nαν. (2.2.11)\nThe Ricci Scalar follows from contractions of the Ricci tensor and the metric, yielding:\nδR=δgµαRαµ+gµαδRαµ. (2.2.12)\nEinstein Tensor\nWe now have all the ingredients to write explicitly the different components of the perturbed\nEinstein tensor in Equation (2.2.7):\nδGµν=δRµν−1\n2δgµνR−1\n2gµνδR , (2.2.13)\nor, equivalently, with one of the indices raised to facilitate the comparison with the matter sector:\nδGµ\nν=δgµαGαν+gµαδGαν. (2.2.14)\nWritingtheperturbedtermsexplicitly, andaccordingtotheperturbedmetricinEquation(2.2.5),\nwe arrive at the different components of the perturbed Einstein tensor:\nδG0\n0= 2a−2\u0002\n3H(HΨ + Φ′)− ∇2Φ\u0003\n, (2.2.15)\nδG0\ni=−2a−2∂i\u0000\nΦ′+HΨ\u0001\n, (2.2.16)\nδGi\nj= 2a−2\u0002\n(H2+ 2H′)Ψ +HΨ′+ Φ′′+ 2HΦ′\u0003\nδi\nj+a−2\u0002\n∇2(Ψ−Φ)δi\nj+∂i∂j(Φ−Ψ)\u0003\n.\n(2.2.17)36 2.2. Linear Cosmological Perturbations\nMatter Perturbations\nFor the purpose of this study, we assume that the background perfect fluids remain perfect fluids\nwhen perturbed, in which case there is no anisotropic stress. Matter perturbations, that is,\nperturbations to the energy-momentum tensor of each species, can first be defined in terms of a\nperfect fluid:\nT0\n0=−δρ , (2.2.18)\nT0\ni= (1 + w)¯ρvi, (2.2.19)\nTi\nj=δpδi\nj, (2.2.20)\nwhere viis the so-called scalar bulk velocity, the matter peculiar velocity with respect to the\nexpansion, vi=dxi\ndτ. The different terms in the energy-momentum tensor from each species f\nneed to be added together to make up the perturbation of the total contribution Tµν=P\nfTf\nµν:\nδρ=X\nfδρf, δp =X\nfδρf. (2.2.21)\nThe density and velocity perturbations are often expressed in terms of the dimensionless density\ncontrast and the velocity divergence\nδ≡δρ/¯ρ , (2.2.22)\nθ≡∂ivi. (2.2.23)\nOnce δbecomes of the order of unity, linear perturbation theory no longer applies.\nThe perturbed four-velocity takes the form\nuµ≡¯uµ+δuµ=a−1(1−Ψ, vi), u µ≡¯uµ+δuµ=a(−(1 + Ψ) , vi). (2.2.24)\nThis leads to the perturbed energy-momentum tensor of a perfect fluid:\nδTµ\nν=ρ\u0002\nδ(1 +c2\ns)¯uµ¯uν+ (1 + w)(δuµ¯uν+ ¯uµδuν) +δc2\nsδµ\nν\u0003\n. (2.2.25)\nThis equation introduces the sound velocity, c2\ns≡δp/δρ. For the barotropic fluids, pis only a\nfunction of ρ, even after being perturbed, implying\nc2\ns≡δp\nδρ=dp\ndρ=˙p\n˙ρ, (2.2.26)\nwhere the last step only holds for the FLRW metric and represents the adiabatic sound speed,\ndiscarding any other possible internal degrees of freedom of the fluid.2The Standard ΛCDM Model: Successes and Controversies 37\nEinstein Equations\nBringing together the expressions for the perturbed Einstein and energy-momentum tensors gives\nrise to the perturbed Einstein equations.\nThe time-time component is\nδG0\n0=−2a−2\u0002\n∇2Φ−3H(Φ′+HΨ)\u0003\n, (2.2.27)\nfrom which it follows that\n∇2Φ−3H(Φ′+HΨ) = 4 πGa2δρF.T.− − → k2Φ + 3H\u0000\nΦ′+HΦ\u0001\n=−4πGa2δρ ,(2.2.28)\nin Fourier space. The second derivative of the potential hints at the fact that this is a relativistic\ngeneralisation of the Poisson equation. Inside the Hubble radius ( k≫ H), the second term of\nthe left-hand side becomes negligible, and Equation (2.2.28) reduces to the Poisson equation in\nthe Newtonian limit: ∇2Φ≈4πGa2δρ. Only on scales comparable to the Hubble radius do the\nGR corrections start to play an important role.\nFor the trace-free part of the spatial components, we get the following:\n∇2(Φ−Ψ) = 0F.T.− − → k2(Φ−Ψ) = 0 , (2.2.29)\nand the approximation Φ≈Ψholds under the assumption of vanishing anisotropic stress con-\nsidered for the scope of this work, such that:\n∂i(2Φ′+HΨ) = 4 a2πG(¯ρ+ ¯p)viF.T.− − → k2\u0000\nΦ′+HΨ\u0001\n= 4πGa2(¯ρ+ ¯p)θ , (2.2.30)\nwhere (¯ρ+ ¯p)θ≡P\nfρf(1 +wf)θf.\nFinally, the trace of the space-space Einstein equation translates into the following evolution\nequation for the metric potential\nΦ′′+1\n3∇2(Ψ−Φ) +H(Ψ′+ 2Φ′) + Ψ\u0000\nH2+ 2H′\u0001\n= 4πGa2δp , (2.2.31)\nandδp≡P\nfρfδfc2\ns,fis the total pressure perturbation. Assuming the approximation Ψ≈Φ,\nEquation (2.2.29) becomes:\nEquation (2.2.31)F.T.− − → Φ′′+ 3HΦ′+ Φ\u0000\nH2+ 2H′\u0001\n= 4πGa2δp . (2.2.32)\nThis gives a closed equation for the evolution of Φin terms of δp, and the relation between δρand\nΦis encoded in the Poisson-like equation, Equation (2.2.28). Combined with the conservation\nequations we derive next, this leads to a closed system of linearised equations.38 2.2. Linear Cosmological Perturbations\nAltogether, the general linearly perturbed Einstein equations in Fourier space are\nk2Φ + 3H\u0000\nΦ′+HΨ\u0001\n=−4πGa2ρ , (2.2.33)\nk2\u0000\nΦ′+HΨ\u0001\n= 4πGa2ρ(1 +w)θ , (2.2.34)\nΦ′′+ 2HΦ′+HΨ′+\u0000\nH2+ 2H′\u0001\nΨ = 4 πGa2δρc2\ns, (2.2.35)\nΦ≈Φ, (2.2.36)\nrelating the metric perturbation Φto the perturbed matter content. For simplicity, we have\nomitted the sums over each fluid f, running over the matter species for a generic equation of\nstate, wf= ¯pf/¯ρf, and sound speed, c2\ns,f=δpf/δρf.\nConservation Equations\nAdditionally, the perturbed evolution equations for each component are derived from the corre-\nsponding conservation relations for the energy-momentum tensor:\n∇µTµ\nν=∂µTµ\nν+ Γµ\nµσTσ\nν−Γσ\nµνTµ\nσ= 0. (2.2.37)\nFrom the expression above, employing the definitions for matter perturbations from Section 2.2.1\nandtheChristoffelsymbolsintheNewtoniangauge, Equations(2.2.9)and(2.2.10), wecanderive\nthe following general perturbed conservation equations, before particularising for each fluid.\n•Perturbed continuity equation: Taking the ν= 0component of Equation (2.2.37) we find:\nδρ′\nf=−3H(δρf+δpf)−(¯ρf+ ¯pf)(θf−3Φ′), (2.2.38)\nfor the evolution of the density perturbation of the fluid f. The different effects can be\nidentified on the right-hand side of the equation: firstly, we have the dilution with the\nbackground expansion; secondly, the contributions from the local fluid flow; and lastly, a\npurely relativistic effect due to perturbations to the local expansion rate itself, encoded in\nΦ. Employing the definition of the density contrast, δ≡δρ/¯ρ, and transforming to Fourier\nspace:\nδ′\nf+ 3H\u0000\nc2\ns,f−wf\u0001\nδf= (1 + wf)\u0000\n3Φ′−θf\u0001\n, (2.2.39)\nrecalling that c2\ns,f=δpf/δρfandwf= ¯pf/¯ρf.\n•Euler equation : If, on the other hand, we take the spatial components ν=i, we obtain:\nvf\ni′=− \nH+¯p′\nf\n¯ρf+ ¯pf!\nvf\ni−1\n¯ρf+ ¯pf∂iδpf−∂iΨ. (2.2.40)\nOn the right-hand side, we have a correction to the momentum components from the2The Standard ΛCDM Model: Successes and Controversies 39\nvelocity flow and contributions from pressure and gravity, respectively. Once again, using\nthe relation θ=∂iviand taking the derivative of Equation (2.2.40), leads to an equivalent\nequation for the velocity divergence in Fourier space:\nθ′\nf+\"\nH(1−3wf) +w′\nf\n1 +wf#\nθf=k2\"\nc2\ns,f\n1 +wfδf+ Ψ#\n. (2.2.41)\nTheconservationequationsdependontheindividualcharacteristicsofthefluid fwhoseevolution\nwe wish totrack. Weremarkbelowthespecialcasesofmatterandradiationatsub-horizonscales\n(k≫ H), for which the equations greatly simplify:\n•Matter: Taking pm= 0the continuity and Euler equations reduce to:\nδ′\nm=−θm+ 3Φ′, (2.2.42)\nθ′\nm=−Hθm+k2Ψ. (2.2.43)\nCombining derivatives of these two equations, it is possible to write a single clustering\nevolution equation for the matter perturbations:\nδ′′\nm+Hδ′\nm=−k2Ψ + 3\u0000\nΦ′′+HΦ′\u0001\n. (2.2.44)\nThe second term on the left-hand side accounts for the expansion and, for this reason, is\noften called the Hubble friction. The −k2Ψaccounts for the scale-dependent time-varying\ngravitational potential that the matter particles feel.\n•Radiation: Forpr=ρr/3and no viscosity, the continuity and Euler equations become:\nδ′\nr=−4\n3θr+ 4Φ′, (2.2.45)\nθ′\nr=k2δr\n4+k2Ψ. (2.2.46)\nLikewise, we can compute the second-order evolution equation for δr:\nδ′′\nr+k2δr\n3=−k24\n3Ψ + 4Φ′′. (2.2.47)\nA crucial difference from the relativistic matter case is that the radiation perturbations\nare not subject to the Hubble friction but, instead, feel an induced pressure contribution\nfrom the second term on the left-hand side. The interplay between the effects in both these\nequations, Equations (2.2.44) and (2.2.47), is the genesis of the acoustic oscillation in the\nprimordial plasma, which will be discussed in more detail in Chapter 3.\nSince the equations are linear, the decomposition in Fourier modes ensures that each plane wave40 2.2. Linear Cosmological Perturbations\nfollows the same equations but with a different comoving wavenumber k. During the phase of\nlinear evolution, the physical scale λp=2π\nkaof the perturbation expands along with the cosmic\nexpansion. However, this linear treatment becomes inadequate if the perturbation enters the\nnon-linear regime. At that point, the perturbation decouples from the Hubble expansion and\nstarts to collapse gravitationally.\nLastly,itshouldbenotedthatthistreatmentoftheconservationequationssupposesnoadditional\ncouplings between the individual fluids. For example, suppose interactions of dark matter with\nother components are allowed. In that case, its density perturbations will evolve according to a\nmodifiedversionofEquation(2.2.44),includingparticulartermstoaccountforsuchnon-standard\ninteractions, e.g.for interacting dark energy models which we will introduce in Chapter 5.\nThe gravitational interaction of the different fluids is implicitly accounted for by the terms\nproportional to the expansion Hand the metric potentials ΦandΨ.\n2.2.2 A Few Words on the Synchronous Gauge\nThe synchronous gauge is a historically significant gauge, first introduced by Lifshitz during\nhis groundbreaking work on cosmological perturbation theory [66]. It played a vital role in the\ndevelopment of early CMB codes, as will be introduced in Section 4.2. However, a drawback of\nthe synchronous gauge is the emergence of spurious gauge degrees of freedom, which were at the\norigin of considerable confusion in the initial stages of cosmological perturbation theory. This\nwas one of the driving forces behind Bardeen’s development of a gauge-invariant approach [60].\nIn the previous section, we worked on the Newtonian gauge to simplify the analysis. However,\nthe synchronous gauge is also useful to study the evolution of perturbations. The transformation\nbetween both gauges can be achieved trivially [64], so it also serves as a cross-check of results.\nTherefore, for completeness, in this section, we provide the analogous perturbation equations in\nthe synchronous gauge, in which the line element is written as\nds2=a2(τ)\u0002\n−dτ2+ (δij+hij) dxidxj\u0003\n, (2.2.48)\nwhere hijrepresents the metric perturbation. In what follows, we will adopt Ma &Bertschinger’s\n[64] notation. However, we use hfor the scalar metric perturbation instead of the original hto\navoid confusion with other quantities. Once more, the metric perturbation can be decomposed\ninto scalar, vector and tensor modes:\nhij=hδij+h∥\nij+h⊥\nij+hT\nij. (2.2.49)\nThe parallel and perpendicular components, h∥\nijand h⊥\nij, can still be expressed in terms of a\nscalar ηand a divergence-free vector ⃗A,2The Standard ΛCDM Model: Successes and Controversies 41\nh∥\nij=\u0012\n∂i∂j−1\n2δij∇2\u0013\nη , (2.2.50)\nh⊥\nij=∂iAj+∂jAi. (2.2.51)\nTherefore, the scalar mode of the metric perturbations is fully characterised by handη, while\nthe vector and tensor modes Aiand hT\nijcan be discarded for the purpose of this study. In what\nfollows, we write down the relevant equations in Fourier space in terms of the Fourier transforms\nh(k, τ)andη(k, τ).\nAnalogously to the treatment for the Newtonian gauge, we derive the perturbed Einstein equa-\ntions:\nk2η−1\n2Hh′=−4πGa2X\nδρf, (2.2.52)\nk2η′= 4πGa2X\nρf(1 +wf)θf, (2.2.53)\nh′′+ 2Hh′−2k2η=−24πGa2X\nδpf, (2.2.54)\nh′′+ 6η′′+ 2H\u0000h′+ 6η′\u0001\n−2k2η= 0. (2.2.55)\nThe perturbed continuity and Euler equations for the uncoupled baryonic and radiation fluids\nare derived directly from the perturbation of the conservation relations ∇µTµ\nν= 0and are given\nby\nδ′\nf+ 3H\u0000\nc2\ns,f−wf\u0001\nδf=−(1 +wf)\u0012h′\n2+θf\u0013\n, (2.2.56)\nθ′\nf+\"\nH(1−3wf) +w′\nf\n1 +wf#\nθf=k2c2\ns,f\n1 +wfδf. (2.2.57)\nTheadvantagesofusingthesynchronousgaugebecomeevidentfromwhen f=cdm. Inthiscase,\nbecauseforcolddarkmatter w=p= 0, Equation(2.2.57), impliesthatthefluidvelocityremains\nzero as long as it has a vanishing initial condition ( θcdm(τi) = 0), removing the gauge freedom\nwithout loss of generality. Nevertheless, this remarkable simplification is lost by introducing a\ncoupling between cold dark matter and the other matter components (we will focus on the dark\nenergy case), and the fluid velocity becomes non-vanishing in general.\n2.3 Inflation\nThe first cosmologically relevant epoch in the expansion history of the Universe is the initial\ninflation period, which came to a halt at around a= 10−28(or equivalently about 10−32seconds).\nThe mechanism of inflation was first proposed by Guth in 1980 [48] as an explanation for the42 2.3. Inflation\nabsence of magnetic monopoles and further developed by Linde [67] and Albrecht and Steinhardt\n[68]. Inshort, itisaphaseintheveryearlyUniverseinwhichthecomovingHubbleradius (aH)−1\ndecreased dramatically through accelerated expansion, which drives Ω0in Equation (2.1.26) to\nunity, ensuring spatial flatness, which is preserved up to the present.\nAlthough there are some theoretical gaps and a lack of experimental proof for this inflation\nmechanism, it is currently accepted in the community as it addresses several fundamental issues.\nNamely, it provides a simple and effective solution to the horizon problem concerning the origin\nof the observed homogeneity and isotropy of the Universe, enabling a full causal connection of\nphotons in the early Universe that leads to the thermal equilibrium needed to explain the CMB\nmeasurements. Remarkably, this setting also predicts the quantum fluctuations needed to seed\nthe small inhomogeneities that grow into the cosmic structure distribution that we observe at\npresent.\nEssentially, inflation addresses the origin of the isotropy in the distribution of photons at large\nangular distances. Indeed, we observe that all the photons emitted from the last scattering\nsurface,i.e.when photons began to freely traverse the Universe without interacting with other\nmatter, possess the same temperature up to minor variations. If inflation had not occurred and\nthe Universe had expanded gravitationally from an epoch dominated by radiation, then most of\nthese photons would have been separated by distances greater than their comoving horizon at\nthe time of their last scattering, meaning that prior to their detection, they could never have\nbeen in causal contact and thereby still exhibit the same temperature today.\nThe problems presented above (further discussed in Appendix A) are related to causality and,\ntherefore, can be effectively addressed by postulating a period of extremely fast inflation before\nthe hot Big Bang to generate the homogeneity state of the Universe and the correlation of the\nCMB fluctuations.\nThe reason why this inflation solves the causality problems is that, keeping in mind the impor-\ntance of the Hubble radius during the standard Big Bang period, this would be equivalent to\nconsidering an initial phase of the early Universe during which the comoving Hubble radius is\ndecreasing:\nd\ndt(aH)−1<0⇔¨a >0, (2.3.1)\ngiven that H≡˙a/aand the notion of an accelerated expanding period becomes evident in\nthe last expression. The Hubble radius (a present particlecommunication limit) becomes much\nsmaller than the particle horizon (an absolute causality and particle communication limit), then\nthis means that even though particles (or patches of the sky) are disconnected today, they could\nstill have been in causal contact early on. Of course, this is simply an intuitive picture of how\ninflation can address the causality problems, and detailed calculations of how this is achieved for\nparticular frameworks can be found in [45].\nA possible approach to realise inflation lies in the introduction of one or more scalar fields known\nas the inflaton(s) φ, which dominate at the earliest stages and possess a property denoted as2The Standard ΛCDM Model: Successes and Controversies 43\nwφ<−1/3, thereby triggering a phase of accelerated expansion in the early Universe. The entire\nmodel’s degrees of freedom are determined by the shape of its potential V(φ). As the field moves\nalong its potential, it releases significant kinetic energy. This energy may lead to the generation\nof Standard Model (SM) particles through couplings with the inflaton, akin to Yukawacouplings,\npotentially giving rise to all the particles found in the standard model, a process referred to as\nreheating . For inflation to occur and persist for an adequate duration, the inflaton field must\nexhibitslow-roll behaviour. To achieve this, its potential energy must dominate over kinetic\nenergy ( ˙φ3≪V), and the acceleration must be slow, expressed as as ¨φ≪3H˙φduring inflation.\nThese conditions ensure that the energy density of the Universe remains nearly constant with\nρφ≃Vduring inflation, resulting in exponential growth of the scale factor, similar to the Λ-\ndominated case. However, the balance between kinetic and potential energy must eventually\nchange, causing inflation to cease and reheating to commence. When treating the inflaton as\na quantum field, it becomes possible to predict the perturbations generated by its quantum\nfluctuations. These primordial quantum fluctuations, stretched to macroscopic scales by the\nUniverse’s expansion, constitute the initial seeds for the formation of large-scale structures.\nEmploying a perturbative approach, we can anticipate the evolution of deviations δφof the field\nfrom homogeneity. These deviations evolve as a harmonic oscillator with a time-varying mass,\nenabling a quantum treatment resembling that of a quantum oscillator. For a more detailed\nintroduction to slow-roll inflation see Appendix A.\nThe remarkable consistency with observations and the support for its role in the early Universe\nhas granted inflation a place as an ingredient in the standard cosmological model, even without\nconsensus on a particular inflationary model.\nIn summary, the physical predictions of inflation include:\n•Negligible Curvature: Inflation requires that the spatial curvature of the Universe be\nextremely close to zero. Observations of the CMB, combined with lensing effects and\nBaryon Acoustic Oscillations (BAO) measurements, put tight bounds on the curvature,\nΩk= 0.0007±0.0019[32], in agreement with the inflationary predictions.\n•Adiabatic, Gaussian, and Nearly Scale Invariant Initial Perturbations: while the specific\ndetails may vary depending on the inflationary model, observations of the CMB provide\nstrong constraints on departures from these assumptions [69].\nIn Appendix A a more detailed account of the problems addressed by inflation is provided, along\nwith the illustrative example of a scalar field model of slow-roll inflation.\n2.4 A Brief History of the Universe in Four Epochs\nBringing together the information in the previous sections, we can reconstruct the historical\nepochs of the Universe by tracing its expansion backwards in time [70]. Since the Universe is44 2.4. A Brief History of the Universe in Four Epochs\nexpanding, the scale factor would approach zero at earlier times. Under such circumstances, the\nUniverse would have been in a drastically different physical state, significantly hotter, with its\nbaryonic matter ionised and photons unable to propagate freely in this primordial plasma due\nto the extreme density of the environment.\nBeginning with this early time, the Universe has undergone various epochs as part of its cooling\nand expansion process, in a hierarchy of gradually occuring phase transitions. We will also\nexplore the subsequent era marked by structure formation and followed by the more recent phase\nof accelerated expansion. Nevertheless, all the processes described are continuous, meaning there\nis no absolute division between each epoch. A concise overview of the evolution of each cosmic\nfluidduringtheseperiods, asdescribedinSection2.1.7, isillustratedinFigure2.1, forthe ΛCDM\nmodel.\n2.4.1 Early Universe: The Hot Big Bang Epoch\nIndeed, according to the Hot Big Bang model, the early Universe was a much denser and high-\ntemperature environment filled with elementary particles. As we venture back in time, we en-\ncounter periods beyond the reach of particle physics and reproduction with our present-day\nparticle colliders. Rather than relying on these tools, we can look for the Universe’s thermal evo-\nlution relics, providing an unparalleled laboratory for particle physics. Assuming conservation of\nentropy from thermodynamic, and as discussed briefly in Section 2.1.4, the events in this epoch\ncan also be pinpointed according to the cooling down temperatures:\nT(t) =T0a0\na(t). (2.4.1)\nA more detailed overview can be found in [65].\n•The Planck Epoch ( t∼10−43sorT∼1019GeV): At the earliest conceivable moment after\nthe Big Bang, the nature of the Universe was profoundly different, dominated by extreme\nenergy conditions ( EPl∼1019GeV is the Planck energy ). The Universe was hot, dense and\ngoverned by quantum gravity effects. The fundamental forces of electromagnetism, gravi-\ntation, weak nuclear interaction, and strong nuclear interaction are thought to have been\nunified, although a complete theory remains elusive. Conventional Big Bang cosmology\npredicts a gravitational singularity prior to this time, reflecting our inherent ignorance on\nthe \"origins\" of the Universe. General relativity is assumed to break down at such scales\ndue to the quantum effects, requiring a coherent theory of quantum gravity [71–73] or at\nleast some quantum compatible classical treatment.\n•The Grand Unification Epoch ( 10−43s≲t≲10−36sor1019GeV≳T≳1015GeV): The\ngravitational force separated from the other three fundamental forces. The Universe was\nstill too hot and dense for particles as we know them to exist. This is the era where2The Standard ΛCDM Model: Successes and Controversies 45\nGrand Unified Theories (GUTs) aim to explain unfamiliar physics. The transition to the\ninflationaryepochispredictedtogenerateavastnumberofmagneticmonopoles[74], which\nare conspicuously absent in current observations. This discrepancy was one of the problems\naddressed and resolved by the introduction of the theory of cosmic inflation [48, 67].\n•Inflationary Epoch ( 10−36s≲t≲10−32sor1015GeV≳T≳1013GeV): Described in\nSection 2.3 and in Appendix A, during this brief moment the Universe underwent rapid\nexponential expansion, smoothing itself out. This explains the observed large-scale unifor-\nmity of the CMB radiation. Cosmologists generally believe that inflation happened and\ncould be explained through a scalar field slowly rolling down a potential, called the infla-\nton. Inflation led to a vast thinning of the elementary particles, which were regenerated\nthrough the symmetry breaking as the inflaton field dropped to the minimum of its poten-\ntial energy, refilling the Universe with a hot and dense plasma of elementary particles in a\nprocess called reheating .\n•The Electroweak Epoch ( 10−32s≲t≲10−12sor1013GeV≳T≳103GeV): The Uni-\nverse cooled to approximately 1015GeV and the strong nuclear force separated from the\nelectroweak force, leading to distinct electromagnetic and weak nuclear forces. The Higgs\nfield also gave particles mass in this period [75]. Around this period, a process known as\nbaryogenesis occurred ( T∼1012GeV), accounting for the observed predominance of matter\nover anti-matter in the present-day Universe.\n•Quark Epoch ( 10−12s≲t≲10−6sor103GeV≳T≳1GeV): The Universe was still\ntoo hot for quarks to combine into protons and neutrons. It was filled with a quark-gluon\nplasma, a hot soup of quarks and gluons.\n2.4.2 Radiation Dominated Epoch\nThe Radiation Dominated Epoch (RDE) is the period extending from 10−10s≲t≲50kyrs\nand encompasses significant sub-phases, each characterised by cooling temperatures, ultimately\nreaching an energy level of approximately 102GeV. The RDE is when light elements form. Since\nradiation decays faster with the expansion than non-relativistic matter, this epoch will eventually\ncease and give way to a matter-dominated regime. The radiation-dominated epoch is depicted\nin grey in Figure 2.1.\n•Electroweak Symmetry Breaking ( t≲10−6sor1GeV≳T): For temperature around\n100GeV the forces have solidified into their current forms, and particles acquire masses\nthrough the Higgs mechanism.\n•Neutrino Decoupling and Hadron Epoch ( 10−6s≲t≲1sor1GeV≳T≳1MeV): The\ntemperature dropped enough for quarks to bound to create hadron and anti-hadron pairs46 2.4. A Brief History of the Universe in Four Epochs\nwhich annihilate as the Universe cools, leaving a residue of protons and neutrons. Dark\nmatter is expected to decouple from the primordial plasma by the end of the hadron epoch\n(t∼10−5s orT∼200MeV for WIMPs), while neutrinos start to decouple around this\ntime ( t∼0.2s orT∼1MeV).\n•Lepton Epoch ( 1s < t < 200sor1MeV≳T≳0.1MeV): the temperatures are still high\nenough for positron-electron pairs to form. For T≈me∼0.5MeV, the photons are no\nlonger energetic enough to keep this equilibrium and the annihilation process results in\nenergy transfer to the thermal bath.\n•BBN Epoch ( 200s≲t≲300s or0.1MeV≳T≳0.05MeV): nuclear fusion starts and\nprotons and neutrons combine into atomic nuclei. Free neutrons start fusing with protons\natT≈0.1MeV. Around three minutes after the Big Bang, the light elements were formed,\nstarting with deuterium, which rapidly transformed into4He. As the Universe’s tempera-\nture and density quickly decrease to the point where nuclear fusion is no longer possible,\nwe enter the Matter Dominated Epoch (MDE). All the neutrons have been incorporated\ninto helium nuclei, leaving a mass ratio of roughly three times more hydrogen than4He,\nwith only trace quantities of other nuclei.\n•Photon Epoch ( 300s≲t≲teqor0.05MeV≳T≳1eV): For the rest of the radiation\nepoch, the Universe consisted of a dense hot plasma filled with nuclei, electrons, and\nphotons. This epoch extends through the initial stages of the MDE when t > teq.\nMatter-Radiation Equality\nAt some point, the matter density dominates over radiation, which dilutes faster. In particular,\nthe RDE ended when the densities equalled:\nρm\nρr=ρm,0\nρr,0a4\na3=ρm,0\nρr,0(1 +zeq)−1= 1, (2.4.2)\nand thus, the redshift at the equivalence between matter and radiation was zeq∼3400or\nteq∼50 000years (for fiducial values as measured by Planck[32]). This is represented by the\nline dividing the grey and red sectors in Figure 2.1.\n2.4.3 Matter Dominated Epoch\nAt the beginning of the MDE, the Universe was still dominated by photons. Although neutral\natoms had not formed yet, the Universe was becoming increasingly transparent. The MDE is\ndepicted in red in Figure 2.1.\nShortly after equality, the Universe’s temperature dropped to approximately 1eV, equalling the\nbinding energy of the hydrogen atom. For the first time since the Big Bang, the Universe started2The Standard ΛCDM Model: Successes and Controversies 47\nemancipation from the dense primordial plasma of photons, electrons, and protons. Nevertheless,\nit remained virtually opaque to electromagnetic radiation since photons could travel only short\ndistances before colliding with a charged particle through Compton scattering. In this epoch it\nbecomes relevant to also keep track of time in terms of the redshift, which for matter domination,\nevolves approximately as:\nz−z0≈(t−t0)−2/3. (2.4.3)\n•Recombination and Decoupling Epoch ( 18kyrs≲t≲380kyrs or 6000≳z≳1100):\nAs the Universe continued to expand and cool down, the formation of neutral hydrogen\nbecame energetically favoured and ionised hydrogen and helium atoms captured the free\nelectrons. Shortly after this transitional phase, Compton scattering became inefficient, and\nphotons decoupled from matter, leading to the Cosmic Microwave Background Radiation\nwe can detect today. This event defines the last scattering surface . Until this point,\nphotons and baryons had been tightly coupled, oscillating in acoustic waves generated by\nthe compression/rarefaction cycle due to the balance between the baryons’ gravitational\nattraction and the photons’ radiation pressure. Once the two species decouple, the baryons\nare effectively released from the drag of the photons. This decoupling marks the end of\nthedrag epoch and allowed photons to travel unimpeded through the Universe without\ninteracting with matter, marking the earliest epoch observable today - the last scattering\nsurface. On the other hand, the baryons are frozen in the acoustic wave pattern at a\nscale determined by the size of the horizon at the drag epoch. It should be emphasised\nthat photon decoupling and recombination are distinct events, albeit closely related. The\nphotons present at the time of decoupling have since freely streamed through the Universe\nand are the same ones that we observe today in the CMB radiation, albeit significantly\ncooled by the expansion. By the time this period ended, the Universe’s composition had\nshifted to a dark fog containing roughly 75%of hydrogen and 25%of helium, with only\nsmall amounts of lithium.\n•Dark Ages ( 380kyrs< t < 150Myrs or 1100≳z≳60): With no stars, the Universe was\nin a dark state, as photons were not effectively being produced yet and hence there was no\nvisible light. Matter consisted of a gas of atoms that kept being dragged into denser regions\nby the pull of gravity alone, but the first stars had not yet ignited. Slowly, the gravitational\nattraction pulled matter together, and overdensities started growing into clumps as dark\nmatter started to dominate the total energy density of the Universe.\n•Cosmic Dawn ( 150Myrs < t < 200Myrs or 60≳z≳20): As matter started to clump\nsubjected to gravity’s rule, the Jeans length λJ=q\nπc2s\n10Gρ0, which determines the smallest\nstructures that can form, began to decrease and the density perturbations started to grow\nin amplitude. Cold dark matter became the dominating component, setting the stage for48 2.4. A Brief History of the Universe in Four Epochs\ngravitational collapse to amplify the slight inhomogeneities left by cosmic inflation. This\nprocess made dense areas denser and sparse regions more rarefied. Once the overdensities\novercometheJeansmassdictatedbytheJeanslength, theybegintospontaneouslycollapse\nand the first stars form\n•Star Formation and Reionisation ( 200Myrs < t < 1Gyrs or 20≳z≳6): The process\nof formation of the first starts heated the surrounding medium, once again ionising the\nhydrogen in the majority of the Universe. This moment is known as reionisation and\nhappened approximately one billion years after the Big Bang, marking the last phase\ntransition in the Universe and the end of the dark ages since the star-light across much\nof the electromagnetic spectrum was finally able to travel unimpeded through the cosmos,\neventually revealing the Universe as we see it today.\n•Large Scale Structure Formation Epoch ( 1Gyrs < t < 5Gyrs or 6≳z≳1): Subsequently,\nstars start assembling into galaxies, galaxy clusters, super-clusters and all the large-scale\nstructures. The process of structure formation is dictated by non-linear gravitational ef-\nfects, which thread the galactic cosmic web , leaving behind large empty regions called\ncosmic voids . This epoch extends to the present day, even after dark energy took over the\nevolution at the dark energy-matter equality.\nDark Energy-Matter Equality\nLikewise, we can estimate the redshift value at which the transition from a MDE, decelerating\nexpansion during which large-scale structures form, to a new regime of accelerating expansion\noccurs. Neglecting radiation and for K= 0we have that ¨a= 0when\nρm\n2ρΛ=ρm,0\n2ρΛ,01\na3=Ωm,0\n2ΩΛ,0(1 +zacc)3= 1, (2.4.4)\nwhich corresponds to a redshift zacc∼0.7, assuming fiducial Planckvalues for ΛCDM. This\nequality epoch is depicted as the line dividing the red and blue regions in Figure 2.1.\n2.4.4 Dark Energy Dominated Epoch\nIn the ΛCDM model, due to the cosmological constant’s premise of non-dilution with the ex-\npansion, Λwill eventually take over the Universe’s evolution from matter, whose density decays\nwith a−3, giving rise to the current Dark Energy Dominated Epoch (DEDE). The Universe will\nkeep expanding faster and faster, leading to a potential Big Freeze , where stars burn out and the\nUniverse becomes dark and cold.\n•Late Universe ( t≳5Gyrs or z≲1): The Sun forms in the Milky Ways and the Solar\nSystem emerges in an orchestrated dance around t∼9Gyrs. Around the 10Gyrs mark,2The Standard ΛCDM Model: Successes and Controversies 49\nthe primordial forms of life on Earth emerge, while our modern human ancestors came to\nbe only about 20 0000years ago.\n•Present time ( t∼13.8Gyrs and z= 0): After billions of years of cosmic evolution, the\nUniverse is as we observe it today, filled with galaxies, stars, and a variety of complex\nstructures. We start studying the vastness of the cosmos, which now spans a radius of 46.5\nbillion light-years ∼14Gpc∼1026m, larger than the naive guess from the age of the\nUniverse since the light is carried along with the expansion of the Universe.\n2.5 The Theoretical Problems with Λ\nThroughout its twenty-year lifespan, the ΛCDM model described above has successfully with-\nstood numerous tests under its inherent simplicity. Based on our current understanding of\nfundamental physics, this framework provides a detailed account of most of the Universe’s ex-\npansion history with only 6 free parameters to be fixed by the observational data. More intricate\nmodels rarely manage to achieve statistical preference from the data, as any extra model param-\neters must provide a sufficiently better fit to the data to justify their inclusion (see Section 4.1).\nHowever, the cosmological constant hypothesis still faces longstanding challenges, from motiva-\ntion in quantum field theory to its role in an expanding cosmology. Moreover, more recently,\nthe observational tensions in the parameters of ΛCDM, discussed in detail in Section 3.6, have\ngranted extended models an extra layer of encouragement and support.\nFor the moment, and for what concerns the core of the standard model, we will discuss the most\nprominent challenges: the cosmological constant and the cosmic coincidence problems.\n2.5.1 The Cosmological Constant Fine Tuning Problem\nIn order to realise the cosmic acceleration today, according to Equation (2.1.29), we require that\nthe cosmological constant Λis of the order of the square of the present Hubble parameter H0,\nwhose value was estimated in Equation (2.1.7). This corresponds to the following approximate\nvalue for the energy density (with M Pl≈1019GeV):\nρΛ≈10−47GeV4≈10−123M4\nPl. (2.5.1)\nThis is incompatible with the vacuum energy scales predicted for Quantum Field Theory (QFT),\nas we will illustrate next. This issue was identified long before the discovery of the late-time\nacceleration or the cosmological constant proposal. At that time, the prevailing belief was that\nthe cosmological constant was precisely zero, and much effort was devoted to explaining why\nthis should be the case. The vanishing of a constant in physics often hints at some underlying\nsymmetry.50 2.5. The Theoretical Problems with Λ\nInitially referred to as the vacuum catastrophe [76], the issue lies in the QFT predictions of an\ninexplicably large value for the vacuum energy, arising from the sum of the zero-point energies of\neach degree of freedom in all quantum fields within the Universe, up to a particular cut-off scale.\nSuch a large value should manifest itself viasignificant gravitational effects which have never\nbeen observed. Observational constraints on the upper limit of this vacuum energy completely\ncontradict the QFT result, with differences reported up to an outrageous level of 120orders of\nmagnitude [77]. For example, the vacuum energy estimated for the extreme cut-off scales of\nthe order of M Plor of the scales predicted in particle physics (such as ≈0.1GeV for quantum\nchromodynamics (QCD)), the vacuum energy density can be estimated in limit values as:\nρMPlvac≈1074GeV4and ρQCD\nvac≈10−3GeV4. (2.5.2)\nThis shows how, even while selecting a different cut-off scale for the cumulative zero-point energy\nmay somewhat alleviate this problem, reducing the discrepancy to around forty or sixty orders\nof magnitude [78], such a solution remains highly insufficient. The observed value for Λis much\nsmaller than all particle physics scales, endowing this problem with a fundamental character.\nThe ultimate resolution of the cosmological constant problem remains uncertain, and whether\nit will emerge from QFT or cosmological arguments is unclear. The former could reflect our\nincomplete understanding of how to estimate the zero-point energy of quantum fields in curved\nspacetimes, while the latter could introduce some barecosmological constant to cancel out the\ncontributions from all fields except the observed cosmological constant. Another alternative is to\nstick to the pattern of explaining physical mysteries by postulating a scalar field (like the Higgs\nfield [75] or the inflaton [79]). In this context, a potential approach, as we will further dissect, is\nto discard the cosmological constant altogether and promote it to a scalar field (or some similar\neffect from modified gravity) to account for the late-time acceleration [80, 81].\n2.5.2 The Cosmic Coincidence Problem\nThe coincidence problem embodies the improbable fact that the observed values of the dark\nmatter and dark energy densities are of the same order of magnitude today, given that these\nspecies evolve at entirely different rates throughout the history of the Universe, as discussed\nin Section 2.4. This issue is generally less consensual than the cosmological constant problem\n[82], often attributed to more philosophical or metaphysical considerations, out of the scope of\ncosmological physical theories [83]. This debate reflects the lack of consensus in the community\nover the relevance of anthropic arguments [84]. In essence, the anthropic principle suggests that\nthe values of physical quantities in the current state of the Universe are what they are because\nsuch conditions are necessary for the existence of observers in our condition, and it is not up to\nus, inevitable inhabitants of the Universe, to decide on the statistical likelihood of the present\ndesign of the Universe [83]. This doctrine can be endorsed with the idea that such statistics2The Standard ΛCDM Model: Successes and Controversies 51\nimplied by the terms coincidence orlikelihood do not make sense, to begin with, since cosmic\nvariance implies that we have only one realisation of the Universe at our disposal [85].\nHowever, the anthropic principle does not easily align with the cosmological principle, on whose\npillars observational cosmology stands, and which implies that we do not occupy any preferential\nposition in the Universe. Therefore, if we accept this incompatibility, the coincidence problem\ncan serve as additional motivation to dismiss the cosmological constant and devise alternative\ndark energy models. In particular, dynamical dark energy models, where the energy density\nof dark energy varies over time or there is some interaction in the dark sector, stand out due\nto their ability to explain the seemingly coincidental similarities between the matter and dark\nenergy densities simply through their natural dynamical character. Further discussion on this\ntype of models will be the focus of Chapter 5.3Cosmological Observational Tests\nEstou me interessando terrivelmente por fatos: fatos são pedras duras.\nNão há como fugir. Fatos são palavras ditas pelo mundo.\n—Clarice Lispector A Hora da Estrela\nI’m becoming terribly interested in facts: facts are hard rocks. You can’t escape. Facts are words spoken by\nthe world. —Clarice Lispector in The Hour of the Star\nCosmology differs fundamentally from most branches of physics by being an observational science\nrather than an experimental one. Given that the early Universe events played out without an\naudience (as much as we are aware!), and the dawn and subsequent evolution of the Universe\ncannot be replicated (we have only one sample!), our understanding is primarily based on mea-\nsuring the spatial correlations between cosmological structures at late times. The fundamental\nendeavour of modern cosmology is no longer to map the night sky into constellations that track\nthe seasons but rather to formulate a coherent narrative for the history of the Universe that pro-\nvides motivated explanations and is compatible with these observed correlations. In this sense,\ncosmology is a wide-ranged effort to reconcile various observational data with predictions from\ntheoretical models based on coherent physical principles.\nThe study of high-redshift Type Ia supernovae (SNe) provided the first compelling evidence for\nthe late-time accelerated expansion of the Universe [26, 27]. These supernovae, which serve as\nstandardisable candles based on the fixed luminosity at which they occur, appeared fainter than\nexpected in a spatially flat, matter-dominated Universe. Subsequent observations from other\nprobes further supported this picture, with the additional assumptions that make up the ΛCDM\nmodel described in Chapter 2.\nAfter the success of previous seminal probes and ground-based experiments [86], the Planck\nsatellite has given the most precise measurement of the anisotropies in the Cosmic Microwave\nBackground (CMB) [33]. This is one of the most valuable and comprehensive sources of cosmo-\nlogical information from the initial perturbations that led to the formation of cosmic structures\nviagravitational collapse. The CMB has also been instrumental in constraining cosmological\nparameters within the standard cosmological model and its extensions. Simultaneously, galaxy\nsurveys such as the Sloan Digital Sky Survey [87] have extensively mapped the distribution of\nthe Large Scale Structure across the Universe, identifying the distinct pattern known as the\nBaryon Acoustic Oscillation (BAO) scales within the matter correlation function. By leveraging\nthese findings, a standard ruler is established, whose length can be quantified across various\ncosmological epochs, enabling scientists to monitor and trace the expansion of the Universe.\n5354 3.1. The Matter Correlation Function\nThe precision and compatibility level between these observations have put the standard cosmo-\nlogical model under increasingly rigorous scrutiny. Verifying the standard model’s hypotheses\nthrough independent means solidifies its validity, and observational support can be established\nfrom multiple sources. Upcoming missions like the space-based interferometer LISA [88] and the\nEuclid satellite mission [89] will offer highly detailed maps of the cosmic microwave background\nand large-scale structures. These missions promise even finer measurements of the BAO scales\nacross different cosmological epochs. At the same time, additional insight will be extracted from\nphenomena such as supernovae, both weak and strong gravitational lensing, galaxy clustering,\nand gravitational wave events, among others. These advancements will enrich our understand-\ning of the large-scale homogeneity and any deviations therein, providing extended tests for the\nvalidity of general relativity.\nThis chapter is structured as follows. In Section 3.1 we introduce the matter correlation function,\nneeded to define the matter power spectrum in Section 3.2. These two introductory sections are\nbased on the Chapter 3 of [90]. Section 3.3 provides a description of the main properties of the\ncosmic microwave background and how to relate its features to the cosmological framework. In\nSection3.4wereportonotherrelevantprobesofthephysicsoftheUniverse, inparticularfordark\nenergy signatures. Section 3.5 provides a complete account of the impact of the six-parameters\nof the ΛCDM model on the matter power spectrum and the CMB spectrum of temperature\nanisotropies. Finally, in Section 3.6 we list the cosmological tensions and anomalies that have\nbrought the standard model to a crisis and which motivate the exploration of alternative models.\n3.1 The Matter Correlation Function\nAs elaborated in Chapter 2, the Universe’s large-scale structure is not randomly distributed.\nInstead, cosmic objects are arranged into a tangled cosmic web with galaxies anchored along\ndense filaments and clustering at their intersections, surrounded by vast empty regions known\nas cosmic voids. The mapping of these structures encodes information about the initial state\nof the hot Big Bang when the small overdensities that grew into clusters are thought to have\nformed and the subsequent governing physical processes. Different cosmological models predict\ndifferent patterns of structure formation, allowing us to infer the best-fitting values for specific\ncosmologicalparametersfromthestatisticalcharacteristicsoftheUniverse’slarge-scalestructure.\nWe rely on indispensable statistics tools to assess the validity of any theoretical model and its\nconnection to the real world. Because we have only one realisation of the observable Universe,\nspanning a wide range of physical scales, the statistical properties of the distribution of events\nin the sky are as vital to cosmologists as flour to a baker. Statistics methods in cosmology are\ngenerally categorised into two main branches: descriptive statistics , which aims to cast the data\ninto a compact and physically understandable way, and estimation or inferential statistics , deals\nwith the problem of extracting information about the model and its parameters. This chapter3Cosmological Observational Tests 55\nwill focus on the former and how observational insight can be derived from the background and\nperturbation theory introduced in Chapter 2. The latter will be deferred to Chapter 4.\nThe initial step is to consider a random particle distribution, which may correspond to various\nastrophysical objects such as galaxies or quasars. For Npoints contained within a volume V,\nthe primary statistical descriptor is the average numerical density ρ0=N/V. However, this is\ninsufficient to distinguish between, for example, Npoints clustered in one location and Npoints\nuniformly scattered throughout the volume V. If, instead, we consider an infinitesimal volume\ndVrandomly selected within V, then the term ρ0dVrepresents the mean number of particles\nin this infinitesimal volume. If dNab=⟨nanb⟩is the average number of pairs in the infinitesimal\nvolumes dVaanddVb, separated by a distance rab, the next significant statistical descriptor is\nthetwo-point correlation function ξ(rab), defined as:\ndNab=⟨nanb⟩=ρ2\n0dVadVb[1 +ξ(rab)],with rab>0. (3.1.1)\nInourprecedingdiscussion, theimplicitassumptionisthat rab>0, meaningthetwoinfinitesimal\nvolumesdonotoverlap. Inthiscontext, thenotionofaverage, denotedby ⟨·⟩, conveysthevolume\norsample average1and considers pairs at different points within a single realisation, separated by\nthe same distance rab. If the points are sufficiently far apart to be uncorrelated, the results would\nbe similar to taking an average over many realisations of the distribution. Despite the limitation\nof being confined to observations from a single Universe, the correlation function remains a\npractical and meaningful descriptor. For a completely random distribution of Nparticles, dNab\nshould be independent of specific locations, leading to a vanishing correlation function ξ. In\ncontrast, a non-vanishing ξindicates spatial correlation among the particles. The correlation\nfunction can then be expressed as a spatial average of the product of density contrasts at two\ndistinct points:\nξ(rab) =dNab\nρ2\n0dVadVb−1 =⟨δ(ra)δ(rb)⟩with δ(ra) =na/(ρ0dVa)−1, (3.1.2)\nwhere we used the fact that ⟨δ(ra)⟩=⟨δ(rb)⟩= 0. For a sample average, ξcan be computed\nover all possible positions as:\nξ(⃗ r) =1\nVZ\nδ(⃗ y)δ(⃗ y+⃗ r) dVy. (3.1.3)\n1Although we have previously introduced this notation as the average over random initial conditions (like the\nones generated from inflation), the large-scale cosmic structure we observe in practice is essentially one realisation\nof this random process. To reconcile this with theoretical models, we rely on the principle of ergodicity, according\nto which, as volume approaches infinity, ensemble averages converge to spatial averages. In this context, several\ndifferent patches of the Universe with various sizes can be considered separate realisations of the same random\nprocess. Thus, averaging measurements over a large volume mimics averaging over various realisations of the\nUniverse. However, our observations are inherently limited to a finite volume, introducing what is known as\nsample variance . In the case of the CMB, where this finite volume is the single whole observable Universe, this\nlimitation is appropriately called cosmic variance .56 3.1. The Matter Correlation Function\nIn practical terms, calculating the correlation function is often simplified by selecting a volume\ndVasuch that ρ0dVa= 1is the average particle at a distance rfrom a test particle. In that case,\nthe count of particle pairs reduces to the number of particles contained within the volume dVb:\ndNb=ρ0dVb[1 +ξ(rb)]. (3.1.4)\nThus, the correlation function is typically computed as:\nξ(r) =dN(r)\nρ0dV−1 =⟨ρc⟩\nρ0−1, (3.1.5)\nyieldingtheaveragenumberofparticlesatadistance rfromanygivenparticle, normalisedbythe\nexpected number in a uniformly distributed system. This is sometimes termed the conditional\ndensity contrast. Moreover, considering a bounded volume with N=ρ0Vparticles, imposes an\nintegral constraint on ξ(r):\nZ\nξ(r) dV=1\nρ0ZdN\ndVdV−V=N\nρ0−V= 0. (3.1.6)\nA positive correlation ξ(r)signals more particles than expected in a uniform distribution. In such\ninstances, thedistributionistermedas positively clustered . Generally, oneismoreconcernedwith\nthe dependency on the magnitude r, leading to the selection of a shell of distance rsurrounding\neach particle.\nAs introduced in Chapter 2, the decomposition in Fourier modes of perturbation variables is\nuseful for practical purposes because these modes evolve independently at the linear level. Given\nthat the average of a perturbation variable is, by definition, zero, the next statistically meaning\nquantity is the average of the pairs. When expressed in Fourier space, ( k= 2π/λ) these yield\nthepower spectrum :\nPδ(k) =V|δk|2, (3.1.7)\nwhere δkare the Fourier coefficients corresponding to the density contrast, and Vis the power\nspectrum’s normalisation factor. In cases where the quadratic form involves two distinct vari-\nables, such as |δkθk|,P(k)is said to be a cross-correlation power spectrum.\nThe power spectrum is the most widely used measure for describing clustering in linear and\nmildly non-linear regimes and is one of the most important observables in the field. It serves as\nthe Fourier-space counterpart of the correlation function, expressed as:\nP(⃗k) =1\nVZ\nξ(⃗ r)e−i⃗k·⃗ rdVand ξ(⃗ r) =V\n(2π)3Z\nP(⃗k)ei⃗k·⃗ rdk . (3.1.8)\nThe Fourier volume factors of 1/VandVinP(⃗k)andξ(⃗ r), respectively, are conventionally\nadded to the Fourier transformation to guarantee that both functions have the same dimensions.3Cosmological Observational Tests 57\nWhen assuming spatial isotropy, that is, when the correlation function depends solely on the\nmagnitude r=|⃗ r|, the spectrum likewise depends only on k=|⃗k|.\nIn the following sections, we will see in more detail how the power spectrum decomposition can\nbe used to study density fluctuations in the matter distributions.\n3.2 The Matter Power Spectrum\nAs mentioned in Section 2.3, the theory of inflation is currently the leading explanation for\nthe first generating moments of the Universe through a rapidly expanding phase in its early\nstages. During this period, the initial anisotropies are believed to have been generated through\nthe quantum fluctuations of a scalar field called the inflaton. This exponential expansion is\nessential as it amplifies the density perturbations responsible for the observed anisotropies in\nthe CMB and the formation of large-scale structures such as galaxies and galaxy clusters in the\nUniverse. Therefore, it is helpful to focus on what is happening at different scales to study the\npatterns in this matter distribution resulting from inflation. This is achieved exactly through the\npower spectrum decomposition of the density perturbations. The density contrast δ(k), which\ncharacterises the deviation of the density of a Fourier scale kfrom the background density, is the\nFourier counterpart of Equation (2.2.22).\nThe large-scale structure in the Universe is not distributed randomly but has interesting corre-\nlations between spatially separated points. By definition, the mean value of the density pertur-\nbations is zero, ⟨δ⟩= 0. Therefore, the first non-trivial statistical measure of the density field\n(at a fixed time) is exactly the two-point correlation function introduced in Section 3.1.\nTreating these density contrasts as variations arising from primordial quantum fluctuations,\nwe can now formally write the power spectrum P(k)ink-space originating from the matter\ndistribution seeds as follows:\n⟨δ(⃗k)δ∗(⃗k′)⟩=2π2\nk3P(k)δ3(⃗k−⃗k′), (3.2.1)\nnow following the convention of defining the power spectrum with the normalisation factor\n2π2/k3, rendering P(k)dimensionless. The modes are uncorrelated via the Dirac delta func-\ntionδ3(·). When considering scalar perturbations, we can derive the matter power spectrum for\nboth the pre-and post-radiation-matter equality epochs, approximately resulting in [45]:\nP(k)∝\n\nkns, k < k eq,\nkns−4, k > k eq,(3.2.2)\nwhere nsis thespectral index of the power spectrum produced during inflation and keq=\naeqH(aeq)isthescaleparametrisingthedistinctionbetweenlargeandsmallscales, corresponding\nto the mode that entered the Hubble radius exactly at the radiation–matter equality. On large58 3.3. The Cosmic Microwave Background\nscales, P(k)grows for increasing k, while it is a decreasing function of kon small scales. This\nimplies a peak in the power spectrum, directly related to the scale keq, which is sensitive to the\nbackground and perturbed expansion history. The shape of the power spectrum changes with\nparameter variations is depicted for the six ΛCDM parameters in Figure 3.7. Furthermore, there\nis a subtle oscillatory signature in the tail of the spectrum at small scales (large k), which results\nfrom the baryonic acoustic oscillations (BAO) imprinted by cosmic evolution.\nThe comparison between the theoretical power spectrum of dark matter and the observed power\nspectrumofthegalaxydistributionposesasignificantchallenge. Assessinghowthesetwospectra\nshould be related is not trivial, an issue commonly referred to as biasing. It is commonly assumed\nthat both spectra differ only by a constant factor, based on the premise that, on sufficiently large\nscales, both have their evolution dominated by the gravitational effects. This hypothesis is not\nnecessarily universally valid as it can vary depending on the details of galaxy formation and\nevolution, and is assessed ultimately through N-body simulations and observations [91].\nThis chapter introduces the observational probes relevant to this dissertation, categorises them\nand discusses the cosmological information they offer. In particular, we focus on characteristic\nsignatures and effects that can be decisive in distinguishing between different models, consider-\ning the underlying assumptions and the potential for model-independent analysis. While this\ndiscussion provides a tailored overview, more comprehensive information is available in recent\nreviews such as [92] and the sources cited within each subsection.\n3.3 The Cosmic Microwave Background\nMore than any other cosmological probe, observations of the Cosmic Microwave Background\n(CMB) spectrum, mapping both intensity and polarisation, have converted cosmology into a\nprecision science, providing a comprehensive understanding of the Universe’s geometry, evolution\nand composition. Measurements of the temperature anisotropies over the last three decades have\nplayed a crucial role in establishing the standard cosmological model, as described in Chapter 2.\nThese observations have confirmed the existence of primordial fluctuations, which are the seeds\nof structure formation consistent with the inflationary period. The power of the CMB as a\ncosmological probe lies in the fact that it captures these fluctuations at an early stage when\nthey were still small and aptly described using linear perturbation theory, unlike the complex\nand non-linear structure formation process. The evolution of small fluctuations in the primordial\nplasma can be understood from fundamental physics enclosed in well-defined equations, allowing\nfor precise theoretical predictions of the expected temperature anisotropies’ distribution in the\nCMB.\nAs discussed in Section 2.4, the pivotal moment in which light started to propagate freely and\nthe Universe became transparent to light occurred at redshift z≈1090, when the Universe\nwas around 380000years old and the effective temperature of the free streaming photons was3Cosmological Observational Tests 59\n∼3000K, is termed photon decoupling . Since then, the expansion of the Universe has redshifted\nand diluted these photons, according to the relation T∝(1 + z)discussed in Section 2.1.4.\nThey are detected today as the CMB radiation, with a characteristic average temperature of\nT0≈2.725K. These photons carry imprints of their cosmic journey across the expansion history,\nfrom the early Universe until today, including the influence of gravitational wells near massive\nclusters. This information is encoded at different scales as deviations to this average temperature\ndistribution. Because these spatial fluctuations break the isotropy of the CMB, they are referred\nto as the CMB anisotropies . Being one of the most robust and complete cosmological probes,\nnumerous experiments have been conducted to measure the temperature variations in the CMB\nradiation. Additionally, the anisotropies can be modelled for different theories rigorously and\nsystematically using modern Einstein-Boltzmann codes, which will be the focus of Section 4.2.\nThese solve the Boltzmann equation, which expresses how the distribution function of some\nparticle species evolves, assuming Gaussian random fields.\nFigure 1.1 presents the map of the CMB anisotropies as measured by the Planckcollaboration.\nThis is a captivating image of the so-called last scattering surface of the Universe, with the colour\nscale encoding variations in the temperature of the CMB photons, which, in turn, correspond\nto density fluctuations at the time of photon decoupling. The study of the CMB temperature\nfluctuations distribution follows precise statistical techniques to find the correlations between\nhot and cold spots as a function of their angular separation. This results in an angular power\nspectrum decomposed in multipoles ℓ, as depicted in Figure 3.1. The figure also includes the\ntheoretical prediction of the CMB spectrum in the ΛCDM model when fitted to the observed\ndata. The striking agreement between the theoretical prediction and the observed data highlights\nthe remarkable success of the standard cosmological model. For a more detailed overview of the\nphysics of the CMB, we refer the reader to Ref. [86, 93].\n3.3.1 The TT-Spectrum: Measuring the Anisotropies\nAs observers, we are limited to a realisation of the CMB map at a fixed point in space and time\n(⃗ x=⃗ x0andτ=τ0), leading to the evolution equation for the temperature:\nTCMB(⃗ x, ⃗ p, τ ) =TCMB ,0(τ) [1 + Θ( ⃗ x, ⃗ p, τ )], (3.3.1)\nwith TCMB∝(1 + z)and⃗ pis the direction of propagation of the photons. Next, we will\ndiscuss how to compare the observed temperature distribution function against the theoretical\npredictions.\nBeing a sky map, the distribution of CMB anisotropies is better expressed in the projection of\nthe photon direction in the celestial sphere, i.e.in polar coordinates (θ, ϕ). Since our fields are\nassumed to be statistically homogeneous, the averages over an ensemble will be independent of\nthe position. Together with the assumption of isotropy of the initial conditions of the fluctua-60 3.3. The Cosmic Microwave Background\n100101102103\n0100020003000400050006000DTT [K2]\nFigure 3.1 : Figure of the power spectrum of the CMB anisotropies (temperature). The mea-\nsured spectrum DTT\nℓ=ℓ(ℓ+ 1)CTT\nℓ/(2π)has been scaled to make the smaller scale features\nvisible. The large-scale data ( ℓ >30) is binned up by multipole intervals. The blue curve is the\nΛCDM prediction for the Planck2018bestfitvalues, produced with the CLASScode [94, 95]. The\nblack markers represent the Planckdata [32] and corresponding error bars. The data is collected\nfrom the Planck legacy archive http://pla.esac.esa.int/pla/.\ntions, this implies that the distribution of the temperature anisotropies will be the same for all\ndirections. To compare the theoretical predictions for the temperature anisotropies with CMB\nobservations, thetemperatureperturbation, Θobs= ∆T/T, isexpandedintermsofsphericalhar-\nmonic functions, Yℓm, which form an orthonormal basis for the points on a sphere, particularly\nsuited for describing the pixel patches of the sky map:\nΘobs(θ, ϕ)≡∆T\nT(θ, ϕ) =∞X\nℓ=1+ℓX\nm=−ℓaT\nℓmYℓm(θ, ϕ), (3.3.2)\nwhere ℓandmare conjugate to a unit vector in real space, representing the direction of prop-\nagation of the incoming photons. In particular, the relation θ=π/ℓin radians holds, meaning\nthat larger scales correspond to lower multipoles. The corresponding coefficients aℓmencode\nthe dependence on the direction of observation and hence are essential for parametrising the\nshape and overall distribution of the anisotropies in the CMB map in Figure 1.1. The monopole3Cosmological Observational Tests 61\ntemperature is a00and is typically omitted from the spectra along with the dipole terms1. Be-\ning statistically independent, the average value of the single harmonic coefficients vanishes, i.e.\n⟨aℓm⟩= 0. For the two-point correlator, similarly to the Fourier decomposition of random fields\nin space, the off-diagonal correlators of the expansion coefficients aℓmof the expansion vanish,\nensuring orthogonality. The intrinsic variance of the coefficients is defined as:\n⟨aℓma∗\nℓ′m′⟩=δℓℓ′δmm′Cobs\nℓ. (3.3.3)\nThe temperature-temperature (TT) spectrum is conventionally scaled in terms of the multipoles,\nresulting in\nDTT\nℓ=ℓ(ℓ+ 1)\n2πCTT\nℓ, (3.3.4)\nsuch as the example depicted in Figure 3.1. It is also possible to compute the three-point non-\nGaussian correlations , a piece of information directly encoded by the primordial non-Gaussianity\ninduced by some models of inflation, which CMB experiments have constrained [96, 97] but goes\nbeyond the scope of this work.\nThis method is so robustly studied according to fundamental physics that most of the statis-\ntical features had been predicted long before they were effectively observed. In particular, the\nTT-spectrum has been accurately mapped up to ℓ∼4000and it is largely believed that most\nof the available cosmological information has already been derived from it. In fact, CMB obser-\nvations encode potential access to six different power spectra: temperature-temperature (TT),\nE-mode-E-mode polarisation (EE) and B-mode-B-mode polarisation (BB) autocorrelations and\ntemperature-E-mode (TE), temperature-B-mode (TB), E-mode-B-mode (EB) cross-correlations.\nThe BB spectra have much lower power compared to TT, and are still quite suppressed in com-\nparison to the EE polarisation case (around 2 orders of magnitude below). So, for simplicity\nreasons, the work reported in Part II of this dissertation only considers CMB data from TT,\nEE, and TE spectra. Nevertheless, the B-mode spectra carry unique information, and their\ndetection and imprints are an independent research line [98]. This is because while scalar pertur-\nbations can only generate E-mode polarisations, primordial B-modes would unequivocally imply\nthe existence of tensor perturbations, most likely generated during inflation. Even though the\nobservational prospect for primordial B-modes still faces technical limitations due to secondary\nB-mode signals, there is a lot of potential for sensitivity improvements and their detection would\nsignificantly enrich our understanding of the early Universe.\nThis work will focus on scalar perturbations only, uncorrelated with the vector and tensor modes.\n1The monopole term essentially represents a uniform temperature across the sky. It encoded the average\ntemperature of the CMB, but it does not provide any information about fluctuations or anisotropies. The dipole\nanisotropy is primarily due to our motion with respect to the CMB rest frame. It has been extensively mapped\nbut is not particularly interesting for cosmological purposes. The dipole does not contain information about the\nearly Universe but about the present motion. On this basis, it is usually discarded to unveil intrinsic anisotropies\nof cosmological interest.62 3.3. The Cosmic Microwave Background\nThe vector perturbations decay, meaning their contribution is negligible (assuming that the ini-\ntial conditions were set very early, as is the case in inflation), and the tensor perturbations are\nconstant on super-horizon scales, subject to damped oscillations after entering the horizon, be-\ncoming suppressed compared to scalar perturbations. In a broader context, scalar perturbations\ncan be compared to sound waves propagating in the primordial plasma, leading to areas of com-\npression and rarefaction. These variations produce the pattern of peaks and troughs in the CMB\npower spectrum, as seen in Figure 3.1, where the spectrum has been rescaled by ℓ(ℓ+ 1)/(2π).\nHence, the specific positions of these peaks depend non-trivially on the composition and evolu-\ntion of the Universe and represent a valuable probe for testing signatures of alternative models\ntoΛCDM.\n3.3.2 Modelling the Anisotropies\nThe line of sight integration method provides an informative, straightforward way to study the\nphysical effects behind the CMB anisotropies [99]. For a spatially flat Universe and adopting the\nconformal Newtonian gauge Equation (2.2.5), the linear temperature anisotropy Θ≡δT/Tdue\nto the Fourier mode kand projected on the multipole ℓtoday can be written as [86]\nΘ(s)\nℓ(k, τ0) =Zτ0\n0dτS(s)\nT(k, τ)jℓ(k(τ0−τ)). (3.3.5)\nThe integrand in Equation (3.3.5) is a convolution between the ℓ-th spherical Bessel function,\ndenoted by jℓ, which gives the projection from Fourier to harmonic space. The S(s)\nTterms in the\nintegrand are the scalar sources:\nS(s)\nT(k, τ)≡g(Θ0+ Ψ) +\u0000\ngk−2θb\u0001′+ e−τop\u0000\nΦ′+ Ψ′\u0001\n+polarisation terms .(3.3.6)\nThe term\nτop(τ) =Zτ0\nτneσTadτ′, (3.3.7)\nrepresents the optical depth (not to be confused with conformal time), determined by the free\nelectron fraction neand the Thompson scattering cross section σT. An optically thin (thick)\nmedium is characterised by τop≪1(τop≫1), with τop= 1defining the optical transition. Fur-\nthermore, g≡ −τ′\nope−τopcorresponds to the visibility function, which quantifies the probability\ndistribution that a photon last scatters at a given redshift1. In addition to the gravitational po-\ntentials ΦandΨ, the source terms include the temperature monopole, the baryon velocity θband\nsubdominant polarisation modes, all dynamical functions of τandk. The line-of-sight approach\ndramatically simplifies the computation while isolating the different contributions to the total\nanisotropy. The terms in the first and second brackets in Equation (3.3.6) are proportional to\n1For simplicity, we omit the terms from the polarisation spectrum from different sources since the class of\nmodelsconsidereddoesnottypicallyaffecttheCMBpolarisation, unlessthereissomespecificcouplingtophotons.3Cosmological Observational Tests 63\nthe visibility function, which peaks around the recombination epoch and are known as primary\nanisotropies since their effect is mainly imprinted at z∼1090when the photons decouple. In\nmore detail, the contributions are the following:\n•Primary Monopole Θ0(density perturbations): anisotropy proportional to the local photon\nenergy density fluctuation at the last scattering surface. Overdensities within the early\nprimordial plasma are anticipated to exhibit higher temperatures, whereas underdensities\nare expected to be colder. These variations in over- and under-densities inherited from\nthe inflationary phase, will undergo an evolutionary process governed by the perturbation\nequations described in Section 2.2. This evolution contimues until the epoch of decoupling,\nultimately leaving a distinctive imprint in the CMB.\n•Sachs-Wolfe (SW) Effect Ψ(gravitational perturbations): redshift of the photons as they\nclimbed up the initial potential wells when emitted. Regions linked to overdensities are\ncorrespondingly linked to more robust gravitational fields. Photons traversing within these\nperturbed gravitational fields will undergo a gravitational redshift, as elucidated by Sachs\nand Wolfe in [100]. Unlike adiabatic perturbations, this effect is anticipated to generate\ncooler over-densities and warmer under-densities.\n•Doppler Shift (velocity perturbations): effects induced by the baryon velocity θbandθ′\nb.\nThe third bracket terms are proportional to e−τop, which is active since the last scattering. This\ngives rise to secondary anisotropies :\n•Integrated Sachs-Wolfe (ISW) Effect Φ′+Ψ′: accounts for how the photons travelling from\nthe last scattering surface are affected by the potential wells they encounter. Comparable\nto the SW effect, the ISW occurs on a larger scale when photons traverse through the vast\nstructures in the Universe. This results in an overall shift in the photon’s frequency, arising\nfrom the cumulative influence of the mass encountered along its path. If the potential\nwells were static, the blueshift gained when entering would be compensated by the redshift\nnecessarytoescape. Nevertheless,ifthepotentialsevolveintime,thereisanetcontribution\nto the temperature anisotropy.\nThe polarisation terms can be classified as tertiary contributions and discarded for simplicity.\nGravitational lensing of the CMB provides an additional secondary contribution to the tem-\nperature fluctuation in Equation (3.3.6). However, lensing effects require bending the photon\ntrajectories due to inhomogeneities, which are already linear effects. Hence, they constitute a\nhigher order correction to Equation (3.3.6).\nThe contribution of each effect to the temperature power spectrum defined by Equation (3.3.5)\nis depicted in Figure 3.2.64 3.3. The Cosmic Microwave Background\n101102103\n0100200300400500600700800DTT [K2]\nTotalLensed\nTotal\nSachsWolfe\nEarlyISW\nLateISW\nDopplerShift\nPolarisation\nFigure 3.2 : The individual main contributions to the CMB temperature power spectrum\nas detailed in Section 3.3.2, namely the total lensed and unlensed spectra, the Sachs-Wolfe,\nIntegrated Sachs Wolfe, Doppler effect and the contributions from polarisation. This plot was\ngenerated through the Boltzmann code CLASS[101, 102] and has no multipole scaling compared\nto Figure 3.1.\nThe measured value Cobs\nℓin Equation (3.3.3) can be directly compared to the the theoretical\nangular power spectrum C(s)\nℓ, in terms of the temperature field Θ(s)\nℓ(k), as the multipole ℓcross-\ncorrelation function in Fourier space k:\nC(s)\nℓ=2\nπZ∞\n0dkk2\f\f\fΘ(s)\nℓ(k)\f\f\f2\n, (3.3.8)\nwith\nΘl≡1\n2(−i)ℓZ1\n−1dµPℓ(µ)Θ(µ), (3.3.9)\nin terms of the Legendre polynomials Pℓ(µ), where µ=ˆk·ˆpgives the direction between the\ndirection in which the photon is propagating ˆpand its wavenumber ˆk.\n3.3.3 Scale Dependence of the CMB\nScalar perturbations can be envisioned as sound waves echoing through the primordial plasma.\nThese waves correspond to regions of rarefaction and compression, resulting in the pattern of\npeaks and valleys in the CMB power spectrum, which are the dominant contribution at small3Cosmological Observational Tests 65\nscales, ℓ≳100. The exact location of these peaks depends on the Universe’s composition and\nevolution, which are an important probe of dark energy. Initial perturbations within the baryon-\nphotonfluid canonly travela finitedistance untilrecombination1. The extent ofthis propagation\ndistance experienced by an acoustic wave depends on its speed of sound:\ncs=1√\n3\u0012\n1−3ρb\n4ργ\u0013−1/2\n, (3.3.10)\nfor the oscillations imprinted in the CMB anisotropies with frequency ωk(τ) =krs(τ), where rs\nis this limit distance, known as the sound horizon,\nrs(zdec) =Z(1+zdec)−1\n0cs(a)\na2H(a)da . (3.3.11)\nThe redshift at decoupling zdeccan be deduced by intricate formulae [103] or by taking the\nentire background history through fitting numerical codes, which is particularly important for\nnon-standard scenarios with changes to the early Universe dynamics.\nAs detailed in [90], an approximate relation for the comoving wavelength of the acoustic peaks\nisλc= 2π/k= 2rs/n, where nare integer numbers. The first peak is found at an angular scale\nofθ= 1degree which corresponds to ℓ= 180(according to θ=π/ℓ). It is helpful to define a\ncharacteristic angular location for these peaks’ positions in terms of observable quantities:\nθA=rs(zdec)\ndc\nA(zdec). (3.3.12)\nThis acoustic scale is a quasi-universal observable, as it can easily be defined across any sta-\ntistically isotropic or spherically symmetric cosmology. dc\nAis the comoving angular diameter\ndistance, defined from Equation (2.1.14) as\ndc\nA(z) = (1 + z)dA(z). (3.3.13)\nThe multipole associated with the angle θA, is expressed as\nℓA=π\nθA=πdc\nA(zdec)\nrs(zdec). (3.3.14)\nFurthermore, the comoving angular diameter distance can be parametrised by\ndc\nA(zdec) =c\nH0Rp\nΩm,0, (3.3.15)\n1To be more precise, during recombination, most electrons combined with the atomic nuclei formed during\nBig Bang nucleosynthesis. However, photons kept being scattered by the remaining free electrons. This phase\nshould not be confused with the drag epoch, where photons fully decoupled from the remaining electrons and\ncould finally free stream throughout the Universe, a process which occurred about z= 80later.66 3.3. The Cosmic Microwave Background\nwhere Rstands for the CMB shift parameter related to the sound horizon and given by\nR=H0p\nΩm,0Zzrec\n0dz\nH(z), (3.3.16)\nimplying ℓA∝ R, which can be bounded by CMB measurements, which also correspond to\nestimates of the sound horizon rs(zdec). The CMB shift parameter depends directly on the\nUniverse’s expansion history from recombination to the current epoch, defining the acoustic\npeaks’ position. This can be appreciated in Figure 3.1 depicting a comparison between the\ntheoretical prediction of the CMB temperature power spectrum for ΛCDM against the 2018\nobservations from Planck. The predictions of peak locations and magnitudes in a Universe with\ndark energy, fully defined by a cosmological constant, align closely with the observed data.\nThe TT spectrum’s highest amplitudes are attributed to two main components: the ℓ= 0\ncomponent, representing the monopole (the mean value of the blackbody signal), and the ℓ= 1\ncomponent, corresponding to the dipole as discussed in Section 3.3.2. Moving beyond these\ncomponents, the predicted and observed higher modes ( ℓ≥2) are displayed in Figure 3.1,\nexhibiting a characteristic oscillatory pattern. Further insight into the shape of the spectrum\nis provided in Figure 3.2, which decomposes the contributions from the sources of anisotropies\nintroducedinSection 3.3.2. Thebehaviourof thisspectrum atdifferent scalescanbesummarised\nas follows:\n•Forℓ <100, the Sachs-Wolf plateau emerges. These angular scales exceed the size of the\ncausal horizon at the time of the last scattering surface. Consequently, regions separated by\nsuch scales had limited interaction and co-evolution since the end of inflation. The Sachs-\nWolf plateau quantifies the primordial perturbations, primarily arising from mechanisms\noutlined in Section 3.3.2, particularly the Sachs-Wolf effect. These perturbations exhibit a\ndistinctscale-invariantbehaviour, manifestingasaflat DTT\nℓ=ℓ(ℓ+1)CTT\nℓ/(2π), consistent\nwith inflationary predictions.\n•Intherangeof 100≤ℓ≤1000,weentertherealmoftheacousticpeaks. Scalessmallerthan\nthe horizon at the last scattering surface capture causal interactions within the primordial\nplasma prior to decoupling. The intricate wavelike structure in this region results from\nthe oscillations of the coupled photon/baryon fluid within the gravitational field, primarily\ninfluenced by the massive and non-interacting dark matter. Here, gravitational forces\ncompete with fluid motion driven by photon radiation pressure. The acoustic peaks are\ncharacterised by a prominent peak around ℓ≃200, followed by two smaller peaks. These\nvisible perturbations in the plasma resemble sound waves, justifying the term acoustic.\n•Forℓ≤1000, we encounter the damping tail. In this region, the spectrum exhibits rapid\noscillations with a progressively diminishing amplitude. This behaviour arises due to the\nthickness of the last scattering surface, averaging over multiple small-scale behaviours that3Cosmological Observational Tests 67\ntend to cancel each other out. This averaging process gradually erases structures at scales\nsmaller than the thickness of the last scattering surface.\n3.3.4 CMB Lensing\nIn the context of general relativity, massive objects serve as gravitational lenses that bend the\npaths of photons. The large-scale structure between the last scattering surface and the observer\ngravitationally lenses both the temperature and polarisation anisotropies of the CMB. These\nimprints can be reconstructed as a map of the lensing potential, whose gradient determines the\nlensing deflections. For example, in propagating through a large, over-dense clump of matter in\nthe line of sight, angular structures in the CMB get magnified, appearing more extensive in the\nsky. Essentially, by looking at how the typical size of hot and cold spots in the CMB temperature\nmap vary across the sky, we can reconstruct the lensing deflections and, hence, the integrated\ndistribution of dark matter. This lensing map provides a new cosmological observable, similar\nto maps of cosmic shear estimated from the shapes of galaxies1. Its power spectrum, as depicted\nin Figure 3.3, provides access to cosmological parameters from the CMB alone that affect the\nlate-time expansion and geometry of the Universe and the growth of structure - parameters that\nhave only degenerate effects in the primary CMB anisotropies.\nThis lensing effect offers a unique window into the Universe’s structure, including its dark matter\ndistribution. It affects various cosmological observables, such as blurring the acoustic peaks in\nthe CMB and altering polarisation modes.\nStrong lensing phenomena, often involving quasars lensed by single galaxies, offer direct ways\nto study cosmological parameters like the rate of the Universe’s expansion. On the other hand,\nweak gravitational lensing, usually featuring distant galaxies lensed by closer galaxy clusters,\noffers insights into the growth and density of cosmic structures. Both strong and weak lensing\nserve as powerful tools for cosmological study, each providing complementary data that can help\nresolve current tensions in our understanding of parameters like the matter density and structure\ngrowth, as we will discuss in more depth in Section 3.6.2.\n3.4 Other Cosmological Probes\nDespite the fact that the CMB provides a powerful and robust measure of the cosmological pa-\nrametersofthe ΛCDMmodel, thesecanbereinforcedandindependentlyconfirmedbycombining\nmultiple surveys, ideally probing the Universe at different temporal and spatial scales. In fact,\nthis is what allows for the breaking degeneracies in the model parameters inferred from the CMB,\nas will be discussed in Section 3.5. Below we detail the various observational probes relevant for\nthe context of this dissertation which correspond mostly to probes of dark energy. Nevertheless,\n1Cosmic shear is induced by weak lensing as a consequence of the overall mass distribution, including non-\nluminous matter. Compared to galaxy counting, the advantage is that it is not affected by bias uncertainty.68 3.4. Other Cosmological Probes\n100101102103\n0.00.51.01.52.0D [K2]\nFigure 3.3 : The power spectrum of the CMB gravitational lensing Dϕϕ\nℓ=\n107[ℓ(ℓ+ 1)]2Cϕϕ\nℓ/(2π)factor. The blue curve is the ΛCDM prediction for the\nPlanck 2018bestfitvalues, produced with the CLASScode [94, 95]. The black\nmarkers represent, as an example, data from the Planck collaboration [51], PO-\nLARBEAR 2019 [104], SPTpol 2019 [105], SPT/Planck 2017 [106] and ACTpol 2016\n[107], and corresponding error bars. The data is part of a compilation from\nhttps://lambda.gsfc.nasa.gov/education/lambda_graphics/more/lensing_power_source.html.\na multitude of other emerging independent cosmological probes could be added to this picture,\nsuch as the 21cm and Lymann- αlines, quasars, cosmic chronometers, measurements of redshift\ndrift, surface brightness fluctuations, to name only a few. A complete review can be found in\n[92].\n3.4.1 Standard Candles: Cepheid Variables and Type IA Supernovae\nAs we have briefly mentioned in the context of the seminal measurements of the accelerated\nexpansion of the Universe [26, 27], an alternative method of estimating the expansion rate draws\non the knowledge associated with two particular astrophysical objects: a distance calibrator and\nsome objected with well-known standard luminosity. The most conventional pair corresponds to\nCepheid variables as calibrators for Type Ia supernovae1.\n1The explosion of supernovae results in an intense burst of radiation, making them exceedingly luminous\nevents. These supernovae can be categorised based on the presence of chemical element absorption lines in their\nspectra. Specifically, if the spectrum of a supernova displays a hydrogen absorption line, it is labelled as a Type3Cosmological Observational Tests 69\nCepheid variables are a vastly studied type of pulsating stars characterised by a well-established\ncorrelation between their luminosity and the period of their pulses. We can infer their intrin-\nsic luminosity from the observed pulsation period by categorising Cepheids as Population I or\nPopulation II stars. By measuring the differences in the magnitude (a measure of a luminous\nobject’s brightness, related to the photon flux), it is possible to establish a connection with the\nluminosity distance , defined in Equation (2.1.14).\nMore precisely, defining the distance modulus, µ=m−M, simply as the difference between the\napparent m(measure of an object’s brightness when observed from the Earth) and absolute M\nmagnitudes, we find the key relation\nµ\n5+ 1 = log\u0012c(1 +z)\nH0Zz\n0dz′\nE(z′)\u0013\n, (3.4.1)\nwhere the expansion rate E(z)is defined as H2(z) =H2\n0E2(z), scaled by the Hubble constant H0\ninkms−1Mpc−1, andassumingaflatgeometry( ΩK= 0anddM(z) =dc(z)). Sincethecorrected\npeak magnitude Mis the same for all the supernovae ( M≈ −19for type Ia), the luminosity\ndistance can be directly obtained from m. This methodology allows us to establish a connection\nbetween the apparent magnitudes of Cepheid variables and supernovae and the Hubble constant.\nThe redshift associated with the supernova can be determined independently by analysing the\nwavelength of the light emitted and the shifts in its characteristic spectral absorption lines. With\na significant sample size of supernova observations, one can establish the relationship between\nthe observed luminosity and redshift. Large catalogues like Pantheon (1048 supernovae [109])\nand its recent update Pantheon+ (1550 supernovae [110]) compare these observations against\ncalibrated distance-redshift measurements from the wavelength of the observed light based on\nthe spatially flat ΛCDM model. From this comparison, key cosmological parameters such as\nΩmandΩΛcan be inferred, leaving only one free parameter: the Hubble constant. The SH0ES\ncollaboration estimates H0employing Cepheid measurements as calibrators analysed using the\nHubbleSpace Telescope (HST) and the Gaiamission [111]. Planned surveys such as the Rubin\nLegacy Survey of Space and Time (Rubin LSST) are expected to provide catalogues of various\ntransient calibrators, increasing the number of known Supernova (SN) Ia by a factor of ∼10\n[112].\nHowever, the measured values of the Hubble constant obtained according to this method keep\nshowing increasing tension with estimations derived from the cosmic microwave background\nangular distance as the data accumulates and the results become more precise. This tension\nII supernova. In the absence of such a line, it is called Type I. Further sub-classifications exist within Type I\nsupernovae: Type Ia contains an absorption line of singly ionised silicon, Type Ib features a helium line, and Type\nIc lacks silicon and helium absorption lines. The explosion characteristic of Type Ia supernovae is triggered when\na white dwarf in a binary system crosses the Chandrasekhar mass limit [108] due to the accretion of gas from\nits companion star. One remarkable feature of Type Ia supernovae is the near-constancy of their peak absolute\nluminosity. Consequently, these supernovae serve as a form of standardisable candles, enabling the observational\nmeasurement of luminosity distance by assessing their apparent brightness.70 3.4. Other Cosmological Probes\nhas been highlighted in the latest paper from the Pantheon+ catalogue analysis by the SH0ES\ncollaboration [111]. Section 3.6 will investigate this issue more thoroughly.\n0.0 0.5 1.0 1.5 2.0 2.5\nz14161820222426Distance Moduli\nFigure 3.4 : The distance modulus µ(z)as a function of redshift zfor the Pantheon dataset\n(black dots), along with the expected µ(z)predicted values according to the ΛCDM model (blue\ncurve). The error bars show the uncertainties in the 1048data points used to construct the\nPantheon dataset [109].\n3.4.2 Standard Rulers: Baryonic Acoustic Oscillations\nBefore the recombination epoch, photons and baryons were tightly coupled, meaning that the\nsound waves that resulted in the temperature anisotropies in the CMB also left signatures in the\nbaryon perturbations. The drag epoch refers to the era when photons were liberated from the\nbaryons’ drag due to Compton scattering. The oscillations produced by the acoustic waves in the\nCMB, as they interact with baryonic matter, also leave discernible signatures within the matter\npower spectrum, as it plays an instrumental role in determining the distribution of galaxies. The\nassociated large-scale redshift-space correlation function displays a peak at the Baryon Acoustic\nOscillations (BAO) scale at the sound horizon size rs. TheBAO peak was initially observed at\n∼100h−1Mpc from a spectroscopic sample of luminous red galaxies from the Sloan Digital Sky\nSurvey (SDSS) [113]. The precise location of this peak provided further evidence supporting\nthe existence of a dominant dark energy component in the late Universe. The sound horizon at\nz=zdrag, which marks the baryons’ release from the Compron drag of photons, can be defined\nas:3Cosmological Observational Tests 71\nrs(τdrag) =Zτdrag\n0cs(τ) dτdτ=da/(a2H(a))− − − − − − − − − − → rs(zdrag) =Z(1+zdrag)−1\n0cs(a)\na2H(a)da ,(3.4.2)\nwhere cs(τ)denotes the sound speed of the photon–baryon plasma, given by:\nc2\ns=δpγ\nδργ+δρb, (3.4.3)\nwith the subscripts (γ, b)denoting photon and baryon quantities, respectively, and τis the\nconformal time.\nThe power spectrum in redshift space, decomposed into wavenumbers parallel and perpendicular\nto the line of sight, provides the observed redshift and angular distributions of galaxies within\nredshift space. From this power spectrum, the following quantities can be obtained [90], in\nanalogy to Equation (3.3.12) for the acoustic peak in the CMB:\nθs=rs(zdrag)\n(1 +z)dA(z), (3.4.4)\nδzs=rs(zdrag)H(z)\nc, (3.4.5)\nwith dA(z)as previously defined in Equation (2.1.17). The angle θscorresponds to observations\nperpendicular to the line of sight, and δzscorresponds to observations made along the line of\nsight.\nAlthough current BAO data is still not sufficiently abundant to provide independent measure-\nments of these two distance estimators, a combined distance scale ratio can be obtained from\nthe spherically averaged power spectrum [90]:\n\u0002\nθ2\nsδzs\u00031/3≡rs(zdrag)\n\u0002\n(1 +z)2d2\nA(z)c/H(z)\u00031/3, (3.4.6)\nor the related effective distance [113]:\ndV(z)≡\u0014\n(1 +z)2d2\nA(z)cz\nH(z)\u00151/3\n. (3.4.7)\nThus, predictions for the distance scale dV(z)can be compared to observational data. These\nobservations indeed reveal an amplified number of galaxies separated by a distance of roughly\n150Mpc or simply 100h−1Mpc, which, in turn, provides a method for calculating the expansion\nrateH0or its dimensionless counterpart h(H0= 100 hkms−1Mpc−1).\nThis probe focuses on the BAO peak in galaxy clustering, which differs from the acoustic os-\ncillations observed in the CMB, as it primarily reflects the impact of low-redshift phenomena.72 3.4. Other Cosmological Probes\nThe data collected from the Sloan Digital Sky Survey (SDSS) [114–118], amongst other sources,\nwas instrumental in measuring this BAO peak. A recent collection of BAO measurements is\ndepicted in Figure 3.5, along with the corresponding distance and expansion rate theoretical\nΛCDM curves. It is an important probe for breaking degeneracies in the CMB data. The large\nscale structures of the Universe can also be retrieved from the reciprocal weak lensing effect,\nwhich measures the response of photons to a gravitational potential, thereby mapping the mass\ndistribution in the Universe through its effect on deflecting the light of background galaxies. For\nthis purpose, large photometric and spectroscopic surveys are under development, including the\n18th data release (DR18) from the fifth generation of the SDSS survey (SDSS-V) [119] and the\nDark Energy Spectroscopic Instrument (DESI) [120]. During the process of writing this thesis\n(July 2023), ESA’s Euclid satellite was launched. This cosmological survey mission was designed\nto map the extragalactic sky, exploring the expansion history and the formation of structures,\nwith the main aim of providing new insight on the nature and properties of dark energy and\ndark matter on universal scales [89]. I am an active member of the consortium since 2022, as\npart of modified gravity and dark energy theory working groups.\n3.4.3 Standard Sirens: Gravitational Wave Detection\nIn the context of GR, and assuming propagation in a vacuum, gravitational waves follow the\nwave equation\nh′′\nP(τ, k) + 2Hh′\nP(τ, k) +k2hP(τ, k) = 0 , (3.4.8)\nin which hP(τ, k)are the Fourier modes of the gravitational wave amplitude, and the index P=\n+,×runs over the two possible polarisation states of the Gravitational Wave (GW) [121, 122].\nThegroundbreakingdetectionofGWssourcedbythemergeroftwoblackholeswasfirstachieved\nin 2015 by the LIGO-Virgo collaboration [123]. This provided compelling evidence for the ex-\nistence of gravitational waves, as remarkably predicted by GR. However, it can be argued that\nan equally noteworthy event for cosmology occurred two years later with the observation of a\nsignal pointing to a binary neutron star merger along with a counterpart detection in the electro-\nmagnetic spectrum [124, 125]. This immediately allowed the establishment of notably stringent\nconstraints on the speed of the tensor modes, thereby ruling out theories of modified gravity that\npredict a speed of GWs different from the speed of light [88].\nNamely, these signals can serve as standard sirens , which means they provide a method of\nconstructing a Hubble diagram and derive estimations of the value of H0, independent of the\ntraditional distance ladder method [121]. This complementary approach is possible because the\namplitude of gravitational waves is practically inversely proportional to the luminosity distance\ndL(z)to the source event from which it originates [126]. The electromagnetic counterpart signal\nprovides redshift measurements, meaning that H0can be estimated using the relation between3Cosmological Observational Tests 73\n0.5 1.0 1.5 2.0 2.5\nz510152025Distance/(rdz)\nCDM\nDV(z)/(rsz)\nDM(z)/(rsz)\nzDH(z)/(rsz)\nFigure 3.5 : BAO distance and expansion rate data, derived from the SDSS collaboration’s\nfinal results that span eight different redshift intervals, improved over two decades [31]. The\ndata includes isotropic BAO measurements denoted by DV(z)/rs, where DV(z)is the spher-\nically averaged volume distance and rsis the sound horizon at baryon drag, both defined in\nEquations (3.4.2) and (3.4.7). It also comprises anisotropic BAO measurements, expressed\nasDM(z)/rsandDH(z)/rs, where DM(z)is the comoving angular diameter distance and\nDH(z) =c/H(z)is the Hubble distance, both elaborated in Equations (2.1.10) and (2.1.17).\nThese measurements are detailed in Table III of [31] (see references therein). The compilation\nof data to produce this plot was provided by William Giarè.\nthe comoving distance and the luminosity distance in Equation (2.1.14).\nStandard sirens cannot yet compete with other methods of estimating H0, with the current\nconstrained value subjected to significant errors: H0= 70.0+12.0\n−8.0km/s/Mpc [127]. However, the\nprospects for future observations are promising, as next-generation surveys can yield dozens or\neven hundreds of additional measurements of H0using standard sirens.\nPresently, we have second-generation (2G) ground-based operational detectors, which include\nVirgo [128], LIGO [129], and the Kamioka Gravitational Wave Detector (KAGRA) [130], which\nwill be joined by the Indian Initiative in Gravitational-wave Observations (IndIGO) [131], planed\nfor around 2030. The investment in maintaining and developing more detectors augments the\npotential of GW astronomy, increasing the number of detected events (covering a larger cosmic\nvolume) and enhancing sky localisation (better triangulation to the source), which, in turn, fa-\ncilitates counterpart searches. However, their third-generation (3G) successors aim for increased\nsensitivity, precision, and a broader frequency range. The Einstein Telescope (ET) warrants par-74 3.4. Other Cosmological Probes\nticular attention, as it is predicted to boost the present sensitivity tenfold [132]. Moreover, ET\nis projected to widen the redshift horizon, for instance, z∼5for binary black holes (BBHs) in\ncontrast to the z∼0.5threshold of 2G instruments [133]. Forecasts suggest that the number of\ntraceable multi-messenger events might touch the count of tens of thousands of standard sirens\n[134]. While ground-based detectors might span a frequency spectrum of 1≲f≲103Hz [135],\nforthcoming space-based 3G detectors, with particular focus on the Laser Interferometer Space\nAntenna (LISA) [136], could reach peak sensitivities close to 10−3Hz and identify GW events\nbeyond z= 20. Other 3G GW detectors have been planned, such as the DECi-hertz Interferom-\neter Gravitational Wave Observatory (DECIGO) [137]. In Chapter 8, we will explore a related\napplication of standard sirens centred on ET and LISA for comparison and joint analysis of\nground and space-based experiments. More precisely, we will forecast their constraining power\nfor different models in direct comparison with current data from standard candles and rulers.\n3.4.4 Redshift Space Distortions\nAs we have seen in the previous sections, the measurements of distances to galaxies rely on\nmeasuring their redshift. If the Universe were perfectly homogeneous, there would be a straight-\nforward mapping from the radial distance (real space) to the redshift space [138]. However, we\nhave just seen that the Universe is not homogeneous1. The growth of inhomogeneous structures\nleads to peculiar velocities in galaxies moving in clusters that add to the expansion redshift and\nmanifest through distortions in the mapping to redshift space. These distortions result in an\napparent enhancement of large-scale clustering in the radial direction compared to the trans-\nverse direction, and the redshift distribution of galaxies appears disfigured. These redshift space\ndistortions (RSD) are a probe for the linear growth of structures and have been extensively\nreviewed in the literature, e.g.in [139, 140].\nThe linear growth rate f(z, k)of structures is a differential measure of the evolution of the matter\nfluctuations:\nf(z, k) =dlnδm(z, k)\ndlna=1\nHδ′\nm(z, k)\nδm(z, k), (3.4.9)\nwhere the subscript mstands for the collective matter component (baryons+CDM) [141]. As the\nname indicates, f(z, k)is a measure of the rate at which the matter perturbations grow, highly\ncorrelated with the anisotropic clustering observed in redshift space distortions. Therefore, RSDs\nserve as a probe for the combined quantity fσ8, which is the product of the growth rate and the\nroot mean square mass fluctuation amplitude for spheres of size 8h−1Mpc, and which can be\nused as a normalisation factor in the matter power spectrum at that scale [31]. More precisely,\nσ8is defined as\nσ2\n8=Z∞\n0W2(kR)P(k)\nkdk , (3.4.10)\n1We have assumed that there is a homogeneous distribution on cosmological scales, which is a different\nstatement.3Cosmological Observational Tests 75\nwhere Wis a top hat filter function in Fourier space k\nW(kR) = 3\u0014sin(kR)\n(kR)3−cos(kR)\n(kR)2\u0015\n. (3.4.11)\nHere, P(k)denotes the matter power spectrum as defined in Equation (3.2.1), and Rrepresents\ntheσ8scale radius 8h−1Mpc [47]. The combination fσ8becomes then\nfσ8(z, kσ8) =σ8(0, kσ8)\nHδ′\nm(z, kσ8)\nδm(0, kσ8), (3.4.12)\nwith kσ8= 0.125hMpc−1, and σ8is defined as\nσ8(z, kσ8) =σ8(0, kσ8)δm(z, kσ8)\nδm(0, kσ8). (3.4.13)\nFor observational and degeneracy-breaking purposes, a related parameter is defined as S8=\nσ8p\nΩm/3.S8is conventionally employed to express the tension in measurements of σ8for weak\nlensing and CMB probes, and their correlation with measurements of Ωm, and will be the focus\nof Section 3.6.\n3.5 The Cosmological Parameters of the ΛCDM Model\nThe anisotropies of the CMB are dependent on the values of various parameters. The stan-\ndard approach typically relies on six free and independent parameters chosen to avoid degen-\neracies and to improve the convergence of the model fit to the data. These parameters are\n{ωb, ωc, h, τreio, As, ns}, which play a significant role in describing the primordial power spectrum\nof fluctuations, the reionisation epoch, the expansion rate, and the constituents of the Universe.\nIn this section, we delve into the specifications of these parameters, exploring the implications of\ntheir individual variation on the power spectrum of the CMB for the ΛCDM model by keeping\nthe remaining parameters fixed to Planck2018 fiducial values [32], given in Table 3.1. The cor-\nresponding CMB temperature power spectrum change for variation of the six ΛCDM parameters\nis shown in Figure 3.6 and detailed below.\n3.5.1 Baryon and Cold Dark Matter Densities: ωbandωc\nHavingfundamentallydifferentproperties,eachpresentmatterdensityinfluencestheanisotropies’\nimprint in distinct ways.\nThe baryon density, denoted as ωb= Ωbh2, alters the drag experienced by baryons within the\ntightly coupled baryon-photon fluid. Increasing the baryon abundance increases the drag, re-\nducing the sound speed of the acoustic oscillations described in Section 3.4.2, as depicted in\nthe upper left panel of Figure 3.6. Consequently, larger overdensities will be generated by the76 3.5. The Cosmological Parameters of the ΛCDM Model\nParameters of ΛCDM from Planck\nParameter Estimated value\nBaryon density ωb 0.02236 ±0.00015\nCDM density ωc 0.1202±0.0014\nDimensionless Hubble constant h 0.6727±0.0060\nOptical depth τreio 0.0544+0.0070\n−0.0081\nScalar tilt ns 0.9649±0.0044\nScalar amplitude ln\u0000\n1010As\u0001\n3.045±0.016\nMatter density Ωm 0.3166±0.0084\nΛdensity ΩΛ 0.6834±0.0084\nProxy angular scale of the sound horizon 100θMC 1.04090 ±0.00031\nNormalisation of matter power spectrum σ8 0.8120±0.0073\nσ8(Ωm/0.3)1/2S8 0.834±0.016\nAngular scale of the sound horizon 100θ∗ 1.04109 ±0.00030\nSound horizon at recombination r∗[Mpc] 144.39±0.30\nDamping scale kD[Mpc−1]0.14090 ±0.00032\nReionisation redshift zreio 7.68±0.79\nHelium mass fraction YBBN\nP 0.246716+0.000062\n−0.000055\nAge of the Universe t0[Gyr] 13.800±0.024\nTable 3.1 : Estimates of the ΛCDM model cosmological parameters (upper section). The re-\nmaining parameters are either compatible with zero or can be derived (lower section) from the\nstandard six parameters above [32]. The results are obtained from the combination of all the\nPlanckspectra. The bounds on the parameters correspond to 1σerrors ( 68%CL).\ncold dark matter potential wells, enhancing the odd-numbered peaks of the acoustic oscillations\nrelative to the even-numbered ones associated with compression due to the gravitational poten-\ntial. Moreover, while more sensitive to the total matter density ωb+ωc, baryons induce changes\nin the sound speed, which impact the size of the sound horizon, shifting all the peaks towards\nlarger multipoles. While also influencing the free electron fraction Xe1, higher values of ωbare\nassociated with an increase in the thickness of the last scattering surface, leading to a minor sup-\npression of the diffusion damping at small multipoles (related to time variations in the optical\ndepth, ˙τreio).\n1Xe=ne/nHrepresents the redshift dependence of the ionization/free electron fraction.3Cosmological Observational Tests 77\n2505007501000DTT [K2]\nb\n2505007501000c\n200400600800DTT [K2]\nh\n200400600800\nreio\n101102103\n050010001500DTT [K2]\nAs\n101102103\n050010001500ns\n0.4\n 0.2\n 0.0 0.2 0.4\np/p\nFigure3.6 : Figure with CMB temperature power spectrum variations from variations δpof the\nstandard ΛCDM cosmological parameters p. The ΛCDM fiducial case is depicted as a dashed\nline for reference. Variations in ωbandωcimplicitly hold hfixed, while variations in hhold the\ncold dark matter and baryon densities fixed. Figure produced using CLASSand inspired by [142].\nIn the upper left panel of Figure 3.7, we also identify an overall suppression (enhancement) of the\nmatter power spectrum for higher (lower) values of the baryon density. Higher baryon densities78 3.5. The Cosmological Parameters of the ΛCDM Model\n103104P(k)[Mpc/h]3 [K2]\nb\n103104c\n103104105P(k)[Mpc/h]3 [K2]\nh\n103104105\nreio\n103\n102\nk[h/Mpc]103104105P(k)[Mpc/h]3 [K2]\nAs\n103\n102\n101\nk[h/Mpc]103104105\nns\n0.4\n 0.2\n 0.0 0.2 0.4\np/p\nFigure 3.7 : Figure with variations in the matter power spectrum from variations δpof the\nstandard ΛCDM cosmological parameters p. The ΛCDM fiducial case is depicted as a dashed\nline for reference. Variations in ωbandωcimplicitly hold hfixed, while variations in hhold the\ncold dark matter and baryon densities fixed. Figure produced using CLASS.3Cosmological Observational Tests 79\ncan amplify the large-scale BAO features in the matter power spectrum. Because baryons are\nmore affected by pressure effects from structure formation or radiation compared to dark matter,\nit does not cluster as efficiently on small scales and higher values of ωbinduce a suppression of\nthe amplitude of the spectrum. Finally, a higher baryon density effectively increases the Jeans\nlength (below which they do not collapse to form structure) and thus further suppresses structure\nformation on small scales.\nEven though dark matter does not interact electromagnetically, it is responsible for the gravita-\ntional wells in which baryons and photons oscillate. This is illustrated in the upper right panel\nof Figure 3.6, by an enhancement of the BAO features at large multipoles with the increase of\nωc= Ωch2. CDM has an opposite effect on variations of τreiocompared to baryons. An increase\ninωcshifts the epoch of matter-radiation equality to larger redshifts. Since the cold dark mat-\nter’s influence surpasses the photon-baryon fluid contribution to the gravitational potential after\nequality, this implies that the radiation pressure plays a less pivotal role after equality ( τ > τeq).\nBeing the motor of the baryon acoustic oscillations, the attenuation of the radiation’s influence\nweakens the CMB anisotropies, particularly in the first peak, as the relative abundance of dark\nmatter is increased [93], a phenomenon known as radiation driving . Dark matter is the primary\ndriver of structure formation, with the variations in dark matter density becoming imprinted on\nthe CMB through the ISW effect at low multipoles.\nThisisalsoreflectedinthematterpowerspectrumdepictedintheupperrightpanelofFigure3.7,\nas a higher density of CDM will lead to more structure on all scales, effectively increasing its\namplitude. Consequently, a higher ωcis associated to larger values of σ8. Higher CDM density\nresults in faster structure growth, leading to more power on smaller scales and the peak of the\nmatter power spectrum is shifted to smaller scales\n3.5.2 The Hubble Constant: H0\nThe Universe’s expansion rate can be directly extrapolated from the peaks of the CMB TT-\nspectrum. The angular scale of the first peak, denoted as θrec, sets the horizon size at the time\nof recombination and is parametrised as\nθ∗=s∗\nrs,∗≈τrec\nτ0, (3.5.1)\nwhere s∗is the sound horizon at recombination, approximately given by\ns∗=Zτrec\n0cs(τ) dτ≈τrec√\n3. (3.5.2)\nThefollowingpeaksarelocatedatanangularscalefollowingthesamepattern, thatis, θ∗,n≈nθ∗,\nplacing the higher harmonics in the decomposition. A higher value of H0(and equivalently\nh) makes the angular diameter distance smaller, which means that the same physical scale80 3.5. The Cosmological Parameters of the ΛCDM Model\ncorresponds to a larger angle on the sky. This shifts the acoustic peaks to larger angular scales\n(i.e., smaller multipoles), as illustrated in the central left panel of Figure 3.6. The value of H0\nindirectly affects early-Universe physics through its relation to other cosmological parameters.\nThe ISW effect, relevant at large angular scales (low ℓ), is sensitive to the rate of expansion\nrelated to the critical density of the Universe, which in turn determines the redshift at which\nmatter and radiation densities become equal, with higher values of H0leading to an enhancement\nof the low- ℓtail of the spectrum.\nAnalogously, in the central left panel of Figure 3.7, we identify an enhancement of the amplitude\nof the matter power spectrum for higher values of h, associated with a slight shift of the peak\ntowards lower k-modes. There are different effects at play, but, more importantly, a faster\nexpansion can suppress the growth of structure, leading to less clustering on large scales. At the\nsame time, it also changes the maximum size of the cosmic structures by modifying the Hubble\nhorizon, shifting the turnover peak to larger scales.\n3.5.3 Reionisation Optical Depth: τreio\nParameters like the optical depth to reionisation τreiocapture the degree and timing of reioni-\nsation. A larger optical depth damps the CMB anisotropies on all scales because some fraction\nof the CMB photons have last scattered with free electrons not at the last scattering surface\nbut more recently. This overall damping is a multiplicative effect across all multipoles ℓand is\ndepicted in the central right panel of Figure 3.6. More precisely, increasing the optical depth\nleads to an overall suppression of the CMB power spectrum. This effect is highly degenerate\nwith the one generated by the scalar power spectrum parameter As, as will be discussed next.\nThis degeneracy is only broken by considering measurements of CMB polarisation [32].\nIn the central right panel of Figure 3.7, we see that τreiohas a negligible impact on the matter\npower spectrum for the scales considered. This is because the matter power spectrum reflects\nthe distribution of matter at later times, long after reionisation has ceased. By this time, most\nof the imprints of this epoch have been washed out or are secondary to other physical processes.\n3.5.4 Amplitude and Tilt of the Primordial Power Spectrum: Asandns\nThe matter power spectrum, introduced in Section 3.2, is sourced by the primordial (scalar)\npower spectrum, which in turn is characterised by the parameters Asandns, defined at a pivot\nscale of k∗= 0.05Mpc−1forPlanckobservations. These determine the strength and the scale\ndependence of the initial density perturbations, respectively. A higher amplitude ( As) means\nmore pronounced density fluctuations in the matter distribution, leading to stronger clustering\nof matter and thus a more peaked matter power spectrum, as depicted in the lower left panel of\nFigure 3.6. The amplitude is directly related to σ8and directly scales the entire matter power\nspectrum, as shown in the lower left panel of Figure 3.7.3Cosmological Observational Tests 81\nThe parameter nsaccounts for the tilt of the power spectrum: for ns>1, there is more power\non smaller scales; for ns<1, there is more power on larger scales, as depicted in the lower right\npanel of Figure 3.6. Changing the tilt alters the relative height of the peaks and the angular\nscales at which they appear. Since the tilt nsdetermines directly how the clustering power is\ndistributed across different length scales, in the lower right panel of Figure 3.7, we see that larger\nvalues of nslead to more power on smaller scales, making the matter distribution more clumpy\nat small scales, greatly suppressing the power on larger scales.\n3.5.5 Derived Parameters\nIn addition to the standard six parameters mentioned above, other parameters embody more\ncomplex changes to the physics of the CMB anisotropies. These can be parameters of the\nstandard model, which are usually kept fixed - such as the number of relativistic degrees of\nfreedom Neff, the fraction of primordial helium YHor the mass of massive neutrinos mν- or\nparameters that account for the introduction of new physical degrees of freedom in alternative\ncosmological models . The effect of some examples of both classes of modifications in the CMB\nanisotropies will be the focus of Part II. The estimated constraints on some of the relevant\ncosmological and derived parameters of ΛCDM for the Planck2018 analysis are reported in\nTable 3.1.\n3.6 Cosmological Puzzles and Observational Tensions\nAlbeit initially yielding great consistency confirmation in favour of ΛCDM, the recent advent of\nobservational precision and techniques has revealed unexpected irreconcilable predictions from\ndifferent probes, bringing to light a significant crisis for the standard model of cosmology [57,\n77, 143–145]. These tensions arise from discrepancies between observations of the early and late\nstages of the Universe, such as those of the model-dependent CMB data analysis [33, 146, 147]\nandvarieddistance-laddersurveys[148–152],respectively. Althoughsomeoftheseinconsistencies\ncould be attributed to data errors and systematics, the statistical significance of roughly a 5σ\ntension suggests flaws in the standard model itself, with many extensions being proposed in the\nliterature to ease this problem. Exploring alternative models may reveal new insights into the\nenigmatic nature of the dark sector, the physics of the early Universe, or even the fundamental\nassumptions of ΛCDM [153].\nThe level of concordance or discordance quantifies the tension TP(if any) on a parameter P\nbetween the data sets iandj. Considering the tension in the estimate of a single parameter with\na posterior distribution that is approximately Gaussian, the difference between the mean values\nin the posterior PiandPjdivided by the respective quadrature sum of the standard deviation82 3.6. Cosmological Puzzles and Observational Tensions\nof the respective data sets, σPiandσPj, represents a robust tension metric:\nTP≡Pi−Pjq\nσ2\nPi+σ2\nPj. (3.6.1)\nBut if we want to assess a tension in two or more parameters it can become inadequate, as not\nall of the information can be captured with one-dimensional projections. There is no unique and\nwidely accepted metric for quantifying the tensions and for simplicity reasons we will always use\nthe metric defined in Equation (3.6.1) as a first approximation. Nevertheless we refer to [154] for\na more detailed account of this problem, including alternative metrics and illustrative examples.\n3.6.1 The Hubble Tension\nInbroaderterms, theHubbletensionoftenreferstothe ≈5.0σdisagreementbetweenthevalueof\ntheHubbleconstantpredictedbythe Planckcollaboration[32], HPlanck\n0 = (67 .27±0.60)km/s/Mpc\n(red squared data point in the whisker plot of Figure 3.8) at 68%confidence level (CL), and the\nlatest constraint from the Supernovae H0 for the Equation of State of Dark Energy (SH0ES)\ncollaboration R22 [111], HR22\n0= (73 .04±1.04)km/s/Mpc at 68% CL (blue crossed data point in\nthe whisker plot of Figure 3.8), based on the supernovae calibrated by Cepheids. A CMB analysis\nby the Atacama Cosmology Telescope (ACT), combining data from the WMAP satellite, finds\na value consistent with and independent from Planck:HACT+WMAP\n0 = (67 .26±1.1)km/s/Mpc\n[146]. Nevertheless, it is crucial to recognise that there are multiple sets of measurements and\nestimates for H0. These can be divided into two categories that are in agreement within their\ngroups, reflecting the persistence of the H0tension: (i)indirect model-dependent estimates at\nearly times (such as CMB and Baryon Acoustic Oscillation experiments) that assume a ΛCDM\ncosmology, and (ii)direct late-time model-independent measurements (such as distance ladder\nand strong lensing). That is why the tension is said to arise between the CMB data and the\ndirect local measurements. This is depicted in the whisker plot of Figure 3.8 for a collection of\nrecent H0estimates, showing the consistent trend between early and late time probes.\n3.6.2 The S8Tension\nRecent observations of the large-scale structure in the Universe have allowed us to place con-\nstraints on the clustering strength of matter. However, these constraints also differ from those\nreported by probes of the early Universe. More precisely, there exists a tension in the matter\nclustering power between the primary anisotropies of the CMB, as measured by the Plancksatel-\nlite, and lower redshift probes such as weak gravitational lensing and galaxy clustering. [156–174]\nat the level of 2−3σ. This tension is typically quantified in terms of the parameter S8, defined as\nS8≡σ8p\nΩm/0.3, which regulates the magnitude of weak lensing measurements [175, 176]. The\nS8parameter is closely related to fσ8(z= 0)measured by redshift space distortions (see Sec-3Cosmological Observational Tests 83\n60 70 80\nH0[km/s/Mpc]Abbott et al. (2017),70.0+12.0\n8.0\nMukherjee et al. (2020),67.6+4.3\n4.2\nGayathri et al. (2020),73.4+6.9\n10.7\nMukherjee et al. (2022),67.0+6.3\n3.8\nde Jaeger et al. (2020),75.8+5.2\n4.9\nBlakeslee et al. (2021),73.3±2.5Schombert et al. (2020),75.1±2.8Kourkchi et al. (2020),76.0±2.6Pesce et al. (2020),73.9±3.0Huang et al. (2019),73.3±4.0Li et al. (2021),71.5±1.9Freedman et al. (2021),69.8±1.9Anand et al. (2022),71.5±1.8Jones et al. (2022),71.2±3.8Anderson et al. (2023),71.8±1.5Scolnic et al. (2023),73.22±2.06Burns et al. (2018),73.2±2.3Breuval et al. (2020),72.8±2.7Riess et al. (2022),73.04±1.04Balkenhol et al. (2022),68.3±1.5Aiola et al. (2020),67.9±1.5Abbot et al. (2018),67.2+1.2\n1.0\nAghanim et al. (2020),67.9±0.8Aghanim et al. (2020),67.36±0.54Aghanim et al. (2020),67.27±0.6\nEarly Universe\nLate Universe\nFigure 3.8 : Various recent estimates of H0from different probes, as detailed in Section 3.6.1.\nThe red and blue vertical bars illustrate the 68%CL regions for the values measured by the\nPlanck[32] and SH0ES [111] collaborations. The dashed line separates measurements from\nprobes based on the early and late Universe. The colours group the type of estimates: red for\nCMB with Planck, dark red for CMB without Planck, blue for SNIa-Cepheid, navy for SNIa-\nTRGB, sea green for SNIa-Miras, dark green for Masers, green for Tully Fisher, charcoal for\nsurface brightness fluctuation, yellow for SNII, and orange for GW related measurements. Plot\nadapted from https://github.com/lucavisinelli/H0TensionRealm and based on [57, 155].\ntion3.4.4)andlensingprobes, with f= [Ω m(z)]0.55beingagoodproxyforthegrowthrateinGR,\nbased simply on the matter density parameter Ωmat a given redshift z. In particular, lower red-\nshift probes tend to favour a lower value of S8compared to the high redshift CMB estimates (red\ninthewhiskerplotofFigure3.9). Asanexample, thelatestcosmicshearanalysisofKiDS(KiDS-\n1000) and DES (DES-Y3) set SKiDS-1000\n8 = 0.759+0.024\n−0.021[157] and SDES-Y3\n8 = 0.759+0.025\n−0.023[160, 161]\n(blue crossed data point in the whisker plot of Figure 3.9), respectively, consistent with the 2−3σ\ntension with the fiducial Planckanalysis, which gives SPlanck\n8 = 0.834±0.016. The addition of84 3.6. Cosmological Puzzles and Observational Tensions\nsecondary anisotropies from CMB lensing narrows the constraint to SPlanck +lens\n8 = 0.832±0.013\n(red squared data point in the whisker plot of Figure 3.9). The agreement with high and other\nlow redshift measurements is recovered from the CMB lensing data alone. On the other hand,\na combination of the high- ℓdata from ACT and the low- ℓmeasurements of WMAP [146] is\ncompatible with the Planckresults SACT+WMAP\n8 = 0.840±0.030, even though the error bars are\nmore accommodating.\nIt is important to note that unlike locally measured quantities such as the Hubble constant,\nS8relies on a specific cosmological model for its interpretation. In the cases reported in Fig-\nure 3.9, the underlying model is always the standard flat ΛCDM model, which, despite providing\na reasonable fit to the different data sets, yields a lower rate of structure formation compared\nto what observations seem to suggest. Therefore, to arrive at a fair and accurate comparison of\nS8values, it becomes imperative to reassess the analysis of weak lensing data while assuming a\ncosmological model beyond ΛCDM (as will be the focus of Part II), before quantifying any ten-\nsion. This holds particularly true for cosmological models that exhibit variations in the growth\nof structure compared to the standard scenario. While not necessarily a trivial task, by conduct-\ning this reanalysis under different cosmological frameworks, we can gain a more comprehensive\nunderstanding of how S8is affected and whether the observed tension persists across various\nmodels. This approach promotes a more refined assessment of the discrepancies in S8values,\ndisentangling the contributions of different cosmological parameters to the overall tension.\n3.6.3 Lensing Excess\nThere is a potential indication of a systematic error in the Planckdata, known as the Alens\nanomaly. The parameter Alenswas introduced in Ref. [177] as a rescaling factor for the effects of\ngravitational lensing on the CMB angular power spectra. The term anomaly is employed since\nit is anunphysical parameter computed through smoothing the peaks in the CMB damping tail.\nWhen Alens= 0, there is no lensing effect, while Alens= 1corresponds to the value expected in\ngeneral relativity. In Ref. [32], an analysis of ΛCDM+ Alensfinds Alens= 1.180±0.065at68%\nCL. In fact, the PlanckCMB power spectra show a preference for Alens>1at more than 95%\nCL for both the standard Plik[178] and the alternative CamSpec [179, 180] likelihoods, resulting\nin a significant improvement of the statistical fit. When BAO data is included, the evidence\nforAlens>1strengthens to more than and approximately 99%CL for Plikand for CamSpec,\nrespectively.\nRemarkably, the inclusion of the Alensparameter in the analysis introduces a slight alleviation of\nthe cosmological tensions; namely, it reduces the Hubble tension from 5σto3.9σand mitigates\ntheS8tension with weak lensing experiments to less than 2σ, but is still insufficient to resolve the\ncosmic tensions fully. If Alensis attributed to systematic errors in the Planckdata analysis, it is\nstill valuable to understand how such systematics affect the constraints on H0,S8, and ultimately\nthe cosmological tensions. Nevertheless, it should be noted that ground-based CMB experiments3Cosmological Observational Tests 85\n0.6 0.7 0.8 0.9 1.0\nS8 8 m/0.3\nLesci et al. (2021),0.78±0.04Abbott et al. (2020),0.65±0.05Bocquet et al. (2019),0.739±0.041Costanzi et al. (2019),0.79+0.05\n0.04\nAde et al. (2016),0.782±0.048Lange et al. (2023),0.771±0.027Contarini et al. (2023),0.813+0.093\n0.068\nAlam et al. (2020),0.835±0.066Beutler et al. (2013),0.791±0.103Avila et al. (2022),0.732±0.115Said et al. (2020),0.655±0.034Boruah et al. (2020),0.776±0.033Madhavacheril et al. (2023),0.84±0.028White et al. (2022),0.73±0.03Li et al. (2021),0.784±0.015Sugiyama et al. (2023),0.775+0.043\n0.038\nHeymans et al. (2021),0.766+0.02\n0.014\nMandelbaum et al. (2013),0.72±0.05Abbott et al. (2022),0.776±0.017Asgari et al. (2020),0.759+0.024\n0.021\nSecco et al. (2022),0.759+0.025\n0.023\nDutcher et al. (2021),0.779±0.041Aiola et al. (2020),0.84±0.03Hinshaw et al. (2013),0.792±0.053Aghanim et al. (2020),0.832±0.013\nEarly Universe\nLate Universe\nFigure 3.9 : Various recent estimates of S8from different probes, as detailed in Section 3.6.2.\nTheredandblueverticalbarsillustratethe 68%CLregionsforthevaluesmeasuredbythe Planck\n[32] and DES-Y3 [160, 161] collaborations. The dashed line separates measurements from probes\nbased on the early and late Universe. The colours group the type of estimates: red for CMB with\nPlanck,darkredforCMBwithout Planck,blueforcosmicshear,navyforgalaxyweaklensing,sea\ngreenforCMBweaklensing, greenforpeculiarvelocities, charcoalforRSD,andorangeforcluster\nabundance measurements. Plot adapted from https://github.com/lucavisinelli/H0TensionRealm\nand based on [57, 155].\nsuch as ACT and SPT, which are unaffected by the lensing excess, still find disagreement in the\nvalues of H0andS8.\nThe evidence for Alens>1can also be interpreted as evidence for missing physics in the the-\noretical frameworks. It may require considering a closed Universe, which poses challenges to\nseveral observational measurements and to the simplest inflationary models [181–183], or hint at\nalternative cosmological theories that modify GR [32, 184, 185]. If the Alensanomaly is otherwise\nassociated with some subtle and unobserved systematic error persistent through all releases of86 3.6. Cosmological Puzzles and Observational Tensions\nthePlanckdata [32, 186–188] then its impact on the cosmological tensions needs to be com-\nprehended. Recent measurements from ACT, SPT-3G, and the combination of PlanckTT and\nSPTPol data all exhibit consistency with the standard lensing effect predicted in the ΛCDM\nmodel,e.g.AACT\nlens= 1.01±0.11[146]. This suggests that the lensing anomaly is unique to the\nPlanckdata at small scales and not present in other CMB experiments.\n3.6.4 Hints for a Closed Universe from PlanckData\nTheenhancedlensingamplitudesignal( Alensanomaly)observedintheCMBpowerspectrum[32,\n177], as indicated by various data releases from the Planckcollaboration, defies the standard\nΛCDM model paradigm. One of the explanations that have been put forward is the possibility of\nhavingaclosedUniverse1, whichisfavouredovertheflatscenarioat 3.4σlevel[32,181,182]. This\npreference for a closed Universe is ascribed to the strong degeneracy between the Alensparameter\nand the spatial curvature parameter ΩK. Moreover, a closed Universe could potentially address\nthe tension between the low and high multipoles of the angular CMB power spectrum [181, 189,\n190]. Nevertheless, both effects could be manifestations of hidden systematics in the Planckdata\nor even just statistical fluctuations.\nHowever, while Planck2018 [32] reports ΩK=−0.044+0.018\n−0.015, implying evidence for a closed\nUniverse at about 3.4σ, this evidence is reduced by replacing the baseline Pliklikelihood [178]\nwith the alternative CamSpec [179, 180]. That being said, the estimated value ΩK=−0.035+0.018\n−0.013\nstill shows evidence above the 99%CL. Other studies have also shown that taking a closed\nUniverse in the context of ΛCDM leads to an increase in the dark matter abundance, which in\nturn exacerbates both the Hubble and S8tensions, leading to a drastically small prediction of\nH0∼55km/s/Mpc [32, 181, 182] from Planck.\nHowever, alternative data combinations have been explored in the literature, leading to varying\ndegrees of preference (or lack thereof) for a closed Universe. For example, the recent results from\nthe ground-based experiment ACT, in an analysis including data from the WMAP experiment,\nare fully compatible with a flat Universe, ΩACT\nK=−0.001+0.014\n−0.010[146]. Bringing together the ACT\ndata with a fraction of the Planckdata recovers some slight preference for the closed Universe\nscenario [146]. On the other hand, the combination of Planckand BAO [114, 117, 118, 191],\nPlanckand CMB lensing [51] or Planckand Pantheon data [109], favour a flat Universe in all\ncases. However, the robustness of this analysis is questionable since the combined data sets are\nin disagreement at more than 3σwhen the curvature is left to vary [181, 182, 192].\nMoreover, while the spatial curvature prediction appears to be unaffected by local measurements\nof the Hubble constant ( H0) from the Cepheid distance ladder [193], its variation exacerbates\nboth the H0andS8tensions [181]. Therefore, adding the curvature degree of freedom to the\nstandard model cannot effectively account for the observed tensions and anomalies. Surveys such\n1Or something that mimics the same signal and is being expressed through the variation of the spatial\ncurvature parameter.3Cosmological Observational Tests 87\nas the recently launched Euclid[89] have the potential to provide tighter constraints on ΩK-\noptimisticallyoftheorderof ∼10−3at95%CL-andhelpbreakparameterdegeneracies[194,195]\nand solve the anomalies in the Planckdata [196].4Methodology\nVivemos exclusivamente no presente pois sempre e eternamente é o dia de hoje\ne o dia de amanhã será um hoje, a eternidade é o estado das coisas neste momento.\n—Clarice Lispector A Hora da Estrela\nWe live exclusively in the present because it is always eternally today and tomorrow will be a today, eternity\nis the state of things at this very moment. —Clarice Lispector in The Hour of the Star\nThis section outlines the relevant statistical methods and tools used in cosmology for analysing\nobservational data, focusing on estimating model parameters. Any particular model can only\nmakestatistical predictions about the Universe’s properties, meaning that we need to optimise\nthe amount of information that can be extracted from the available data to evaluate the validity\nand predictions of these models effectively. For a more detailed overview of statistical methods\nin cosmology, we refer to [197–199].\nIn Section 4.1 we provide a brief summary of the theory and methodology for statistical analysis\nand inference of parameters, including tests to assess the goodness of fit of a particular model to\nthe data and its evidence compared with the standard ΛCDM case. In Section 4.2 we present\na brief review of the Einstein-Boltzmann codes in cosmology and their connection with MCMC\ncodes for model sampling. We close with a detailed account of the data sets considered for the\nparameter estimates in this work.\n4.1 Statistical Methods and Bayesian Inference\nCosmology is a unique science that fosters a strong synergy between theoretical model develop-\nment, numerical analysis, and the planning and treatment of observational data. As new theories\nare put forth daily, data analysis in cosmology has become essential to test and possibly falsify\ntheir statistical support. This is part of the quest to find a framework that can handle vast\namounts of data and address existing incompatibilities while navigating complexities such as\ntheoretical biases, model independence, overfitting due to excessive parameters, and parameter\nspace degeneracies. Ultimately, developing and improving current statistical methods linking\nprecision data with accurate theoretical predictions is paramount. Due to its nature, analysis\nproblems are inherently inverse problems, i.e.from collected data, we aim to deduce something\nabout the underlying physical processes that generated the data. For instance, we may use\nPlanck CMB data to fit the ΛCDM model and then infer e.g.the relative amount of dark matter\nor baryons from these observations.\n8990 4.1. Statistical Methods and Bayesian Inference\nHence, we may categorise the data analysis process into three classes:\n1.Bayesian inference is the tool used to transform observations into constraints on theoretical\nmodels, which should balance complexity for data explanation and simplicity for concrete\nprediction.\n2.Parameter estimation aims to constrain the values of the model’s parameters and their\nuncertainties, considering errors from experimental and inherent statistical uncertainty\nfrom physical process randomness. A prominent example is the limited predictability of\nthe CMB pattern due to quantum processes during inflation, which can nonetheless still\nbe statistically modelled and taken into account along with measurement uncertainties.\n3.Model selection involves selecting among multiple models proposed to explain the obser-\nvational data, each representing different physical processes and offering a unique set of\nparameters to be varied for data fitting.\n4.1.1 Bayesian Inference\nBayesian inference is the dish of the day in the statistical analysis methods menu. Other subjects\nhave traditionally favoured the frequentist approach, which relies on the repeatability of the\nexperiment. The essence of Bayesian analysis lies in assigning probabilities to all variables,\nhandling them based on a set of rules that embody Bayes’ theorem. This theory aims to keep\nupdating our understanding as new data comes to light. This means that while all subsequent\nsteps follow some algorithm, a significant aspect of this approach is the need to quantify our base\nknowledge before gathering new data. This is called the prior probability , which can be subject\nto different views among researchers.\nTheposterior probability , the probability that the model’s varied parameter will adopt specific\nvaluesafterthedataiscollected, isrepresentedas p(θ|d). Here, θrepresentsthemodel’sunknown\nparameter, while drepresents the observed data. Using this formulation, one can determine the\nexpected values of the parameters and their associated uncertainties.\nUsually, it is easier to calculate the reverse probability, p(d|θ). For example, we consider a\nGaussian model with mean µand variance σ2. This model includes two parameters, θ= (µ, σ),\nand the probability of a single variable din relation to these parameters is described by\np(d|θ) =1√\n2πσexp\u0014\n−(d−µ)2\n2σ2\u0015\n, (4.1.1)\nwhich is not what we were initially looking for. Nonetheless, we can connect this with p(θ, d)by\napplying the Bayes’ Theorem :\np(θ|d) =p(d|θ)p(θ)\np(d), (4.1.2)4Methodology 91\nwhich expresses the confidence degree in the values of θaccording to the information given\nby the data d, or in other words, the probability that the parameters θin the theory can be\nexplained by the data d. Here, p(θ|d)is theposterior probability of the parameters, while p(d|θ)is\nreferred to as the likelihood , also symbolised by L(d;θ). Theprior probability ,p(θ), expresses our\nknowledge about the parameters before performing the experiment. This knowledge can come\nfrom previous experiments or theoretical understanding. In cases where no previous information\nexists, the prior is usually assumed to be a constant, also known as a flat prior .\nTheevidence, represented as p(d), is used to normalise the probabilities:\np(d) =Z\np(d|θ)p(θ) dθ . (4.1.3)\nFor estimating parameters, the evidence does not depend on the relative probabilities of the\nparameters; hence, it is often overlooked. Nevertheless, it is worth noting that the evidence\nplays a significant part in model selection , relating what we learn about θafter seeing the data\n(the posterior) to what we knew about θbefore accessing the data (the likelihood and the prior).\nTherefore, in some way, it measures how much we have learned or how much our knowledge has\nbeenupdatedfromthepriortotheposterior. Thisisespeciallyrelevantwhenmultipletheoretical\nmodels are being evaluated, and there is a need to distinguish between them, regardless of their\nrespective parameters (this will be the focus of Section 4.1.3).\nAssuming flat priors greatly simplifies the computation of the posterior:\np(θ|d)∝ L(d;θ). (4.1.4)\nEven though we might possess the complete probability distribution for the parameters p(θ),\noften it suffices to use the peak of the distribution as the parameter estimate. This method is\nknown as a maximum likelihood estimate. However, if the priors are not flat, the posterior peak\nmay not coincide with the maximum likelihood estimate.\nWe are generally interested in calculating the expectation value of a parameter, or a set of derived\nparameters, f(θ), from the posterior distribution, also called the mean value off(θ):\n⟨f(θ)⟩=Z\nf(θ)p(θ|d) dθ . (4.1.5)\nWhen dealing with multi-parameter models, θ=θ1, ..., θ i, ..., θ n(forn >1parameters), it may\nbe necessary to extract from the multivariate posterior distribution the posterior of a subset\nof parameters. This reduction of the posterior probability to a lower-dimensional subspace is\naccomplished via a process called marginalisation :\nf(θi) =Z\n...Z\np(θi|d) dθ1...dθi−1dθi+1...dθn, (4.1.6)92 4.1. Statistical Methods and Bayesian Inference\nwhere θirepresents one of the elements of the n-dimensional subspace of parameters θ. This is\nespecially important when there are multiple nuisance parameters related to the experiment and\nthe method itself, which are not informative for the cosmological analysis.\nThe set of parameters most favoured by the data can be determined by finding the parameters\nthat maximise the posterior probability p(θ|d):\nf(θ)|m=|max θp(θ|d)|. (4.1.7)\nThis is usually referred to as the best-fit, although strictly speaking, the term is used for the\nparameter values that maximise the likelihood and it applies strictly to f(θ)|min cases of\nuniform priors. Alternatively, the mean parameter values can be estimated by computing the\nposterior mean, also known as the marginalised mean .\nLastly, the confidence level for the derived parameters can also be computed as follows:\nf(θi) =Z\nR(r)p(θi|d) dθn=r . (4.1.8)\nIn this equation, ris a fraction associated with the likelihood that the parameters will fall\nwithin a given parameter area R. The fractions often analysed are r= 0.683,0.954,0.997, which\ncorrespond respectively to the 1σ,2σand3σconfidence levels.\n4.1.2 Parameter Estimation\nThe integral in Equation (4.1.8) can be computed numerically, limited to the maximum precision\npermittedbythecomputingsysteminuse. Forcaseswherethenumberofparameters, alsoknown\nas dimensions, n, exceeds one, the most straightforward approach is to discretise the problem\nand sum over a uniformly distributed grid in the parameter space. This grid must have a large\nenoughvolumetocoveralltheregionsinwhich f(θ)p(θ|d)isnon-zero, withsomeparticularwidth\nand resolution in each direction, depending on the smoothness of f(θ)andp(θ|d). Nevertheless,\nwhen dealing with a parameter space of high dimensionality, evaluating the likelihood on a grid\nquickly becomes computationally expensive as the number of grid points exponentially increases\nwith the dimension. Fortunately, there are more efficient ways to sample the likelihood surface,\nsuch as focusing on areas with a high likelihood. A popular approach in cosmology is the use of\nMarkov Chain Monte Carlo (MCMC) methods [199, 200].\nMonte Carlo methods use random sampling, guided by some proposed distribution and accep-\ntance criteria until the desired result is reached. In this case, we will obtain a chain of posterior\nsamples with an expected number density proportional to the posterior. A notable subclass is the\nMCMC methods, which are characterised by their sequential nature, where each step depends\nsolely on the one before. This sequence is known as a Markov chain . Each step corresponds to\na specific value of the parameters for which the likelihood is evaluated.4Methodology 93\nThe elements of the Markov chain are structured to represent random samples from the pos-\nterior parameter distribution, meaning that each chain element shows the probability of its\ncorresponding parameter values being correct. A straightforward method that accomplishes this\nis theMetropolis-Hastings algorithm [201, 202], described in Algorithm 4.1\nAlgorithm 4.1 Metropolis-Hastings Algorithm\nStep 1. Take a starting point in the parameter space randomly selected from the prior density.\nStep 2. Propose a random candidate iteration based on some probability distribution for the\nlength and direction of the jump, as long as it fulfils the detailed balance condition , which\nstatesthatgoingbacktothestartingpointisaslikelyastheiterationawayfromit. Thiscan\nbe accomplished using a symmetric proposal function , for example, a multivariate Gaussian\nabout the current point. Calculate the likelihood at the new point and, consequently, the\nposterior by multiplying by the prior.\nStep 3. If the probability at the iteration candidate is higher, accept the jump. If it is lower,\naccept the jump with a probability determined by the ratio of the posterior probabilities\nat the new and old points. If the jump is not accepted, remain at the same point, adding\na duplicate to the chain.\nStep 4. Repeat the procedure from Step 2 until the probability distribution is accurately\nmapped. This can be checked by comparing chains starting from different points and\nusing convergence statistics tests.\nThe possibility of moving to a point with a lower probability ensures that the algorithm thor-\noughly explores the shape of the posterior near its maximum and does not get stranded in a local\nmaximum. The algorithm generally begins in a region with low likelihood and migrates toward\nregions with high likelihood. The initial phase, known as the burn-in, depends on the starting\npoint and must be eliminated during the chain’s analysis. Once near the peak, most potential\njumps are in areas of lower probability, and the chain roams around the peak, mapping its shape.\nThe proposal function should be adjusted according to the scale of variation of the likelihood\nnear its maximum. A Gaussian function is commonly chosen, and its axes should ideally align\nwith the principal directions of the posterior to quickly navigate along parameter degeneracies.\nThus, a short initial run is often performed to map out the posterior distribution roughly, and\nthis mapping is used to optimise the proposal function for the actual computation. It can be\nshown that the chain should sample the target distribution fairly once it has reached a stationary\ndistribution, i.e.once there are no significant differences between consecutive steps.\nInpractice,manychainsaregenerated,startingfromarbitrarilychosenpositionsintheparameter\nspace, which should be sufficiently separated from each other. This will generate a random\nwalk, following an algorithm that assigns higher jumping probabilities to iterations in which\nthe parameters contribute considerably to the distribution. On the other hand, it also implies94 4.1. Statistical Methods and Bayesian Inference\nthat this algorithm is quite inefficient if the distribution is multi-modal, in which case other\nsampling methods may be necessary. An alternative is the Nested Sampling algorithm [203].\nThis approach simplifies the problem by not sampling from the entire likelihood distribution but\nrather by just sampling within progressively more significant regions. Instead of constructing\nchains based on a random point from which we sample, we uniformly sample within some hard\nbound on the likelihood. As we sample these livepoints, they are used to define a yet-to-be-\ndetermined contour of potential likelihood regions. We aim to generate a sample within this\nregion, repeat the process based on the new sample, and continue this contraction cycle. The\nessence of the method is that instead of generating a single chain that we sample across the entire\ndistribution, we aim to segment the distribution into numerous discrete levels or chunks. We\naddress a simplified problem within each segment that does not sample from the full likelihood\ndistribution but only within this unspecified region using a certain prior transform function.\nThe idea is that once the chains converge, the values deduced from each chain, like the sample\nmean and variance, should align. More specifically, this method compares the variance (whether\non a single parameter or all of them) inside a chain with the variance across multiple chains.\nThis requires running multiple chains, looking at the variance of the parameter away from its\nmean within each chain, and comparing it to the variance in the mean of that parameter across\nchains. Exactly how the Gelman-Rubin diagnosis handles these variances is complex. However,\nthe important question for convergence is whether, as the chains get longer, these two variances\nasymptotically reach stable values and whether those two values agree. This is measured as the\n\"r−1value\" ( ris expected to be 1for full convergence of the chains), and a usual criterion is\nto assume the MCMC has converged if max( r−1)≈10−2across all the sampled parameters.\nWe will impose this condition for all the analyses in this dissertation.\n4.1.3 Model Selection Methods\nMany of the questions that we wish to answer in cosmology go beyond parameter inference and\nfall into the model comparison realm. Examples of questions we will be asking in this dissertation\nare:\n•Does the Universe have a non-vanishing spatial curvature?\n•Is dark energy dynamic?\n•Is there evidence for a coupling in the dark sector?\nIt is important to highlight the fundamental difference between model fitting and selection. The\nmodel fitting process relies on assuming that a particular model is the true model and extracting\nthe constraints that provide the best possible fit to the available data. On the other hand, in\nmodel selection, we are interested in assessing the level of compatibility of each model with the\ndata. In exceptionally simple cases, we can select between models by comparing the maximum4Methodology 95\nlikelihood value, but this is not generally valid. A model with more degrees of freedom will\nalways give an equal or larger maximum likelihood, irrespective of the data: model complexity\nis increased, then the maximum likelihood value will generally increase, often at the cost of\nover-fitting the data. This means an accurate model selection method should account for and\npenalise for the over-fitting present in models with increased complexity. The key difference is\nin comparing posteriors instead of likelihoods.\nModel Fit to Data\nIf the distribution functions are Gaussian, assuming flat priors, then maximising the likelihood of\na given model is essentially equivalent to the least square method of minimising the χ2(best fit\nof parameters), which according to Equation (4.1.1) corresponds to the exponent of the Gaussian\ndistribution\nL(d|θ)∝exp(−χ2/2), (4.1.9)\nwhich is proportional to the peak in the posteriors for the flat-prior case. For non-Gaussian\ndistributions ( e.g.aPoisson distribution), minimising the χ2does not necessarily maximise the\nlikelihood that the fitted parameters reflect the data. In those cases the minimum χ2might still\nyield a good first approximation but the best-fitting parameters will generally be found through\nthe maximum likelihood estimation ∇θL(d|θ) = 0instead [204].\nFor a simpler analysis, we can consider information criteria for approximate model comparison,\nkeeping in mind that these make use of considerably strong assumptions about the posterior\ndistribution:\n•Akaike Information Criterion: AIC≡χ2\neff+ 2k;\n•Bayesian Information Criterion: BIC≡χ2\neff+kln(n);\n•Deviance Information Criterion: DIC≡χ2\neff+ 2\u0002\n¯χ2\neff−χ2\neff\u0003\n;\nwhere kis the number of fitted parameters, nis the number of data points, the terms\nχ2\neff=χ2\nmin≡ −2 ln(Lmax), (4.1.10)\ngive the χ2of the best-fit, and the upper bar denotes quantities computed at the average of the\nposterior distribution.\nIn this case, the best model, the one that provides the best fit to the data, must minimise\nthe AIC/BIC/DIC. However, each criterion penalises models differently, with the AIC being\nthe simplest measure, only accounting for the extra parameters. The BIC includes a stronger\npenalty for models with a larger number of free parameters kwhen the number of data points n\nis sufficiently large and gives a better approximation to the full Bayesian evidence in the large\nnlimit. On the other hand, the DIC also considers whether information about the parameters96 4.1. Statistical Methods and Bayesian Inference\nis gained by evaluating the difference according to the average of the posterior distributions.\nBecause of all these subtleties, it is always preferable to calculate the full Bayesian evidence\n(next section), although this may be non-trivial for more complex cases, in which the information\ncriteria already provide some useful insight.\nModel Comparison\nA unique combination of variable parameters and a prior distribution for these parameters dis-\ntinguishes each model in our study. A crucial aspect of Bayesian analysis is that it differentiates\nbetween a model where a quantity is fixed to a specific value and a more flexible model in which\nthat parameter is allowed to vary, even if occasionally assuming that particular value. To ex-\nemplify this, we shall consider two competing models, labelled as MandN, with Nbeing a\nsimpler model containing fewer parameters ( nN< nM). In addition, we assume that model Nis\nnestedwithin model M, meaning the nNparameters of model Nare also present in M, which\nhas a total of p=nM−nNadditional parameters. In model N, these additional parameters\nare set to reference values. We use dto denote the data vector, and θMandθNto denote the\nparameter vectors (of lengths nMandnNrespectively). Bayes’ theorem determines the posterior\nprobability of each model:\np(M|d) =p(d|M)p(M)\np(d), (4.1.11)\nwhere p(M)represents the prior information on the model Mand a similar equation applies\nfor model N. The term p(d|M)is theevidence which we have already encountered in Equa-\ntion (4.1.3), where the |Mbit was discarded since we were focusing on one model only. Ac-\ncordingly, it can be written in terms of its marginalisation over the parameter space, leading\nto:\np(d|M) =Z\ndθMp(d|θM;M)p(θM|M), (4.1.12)\nwhich is a multi-dimensional integration. The ratio of the posterior probabilities for the two\nmodels can be written as:\np(N|d)\np(M|d)=p(N)\np(M)R\ndθNp(d|θN;N)p(θN|N)R\ndθMp(d|θM;M)p(θM|M). (4.1.13)\nIf we do not have any preference towards either model, i.e.,p(N) =p(M), this ratio simplifies\nto the ratio of the evidence, which is referred to as the Bayes Factor [205],\nBN;M=R\ndθNp(d|θN;N)p(θN|N)R\ndθMp(d|θM;M)p(θM|M). (4.1.14)\nThe more complex model Minevitably yields a higher likelihood than the simpler nested model4Methodology 97\nN. However, the evidence tends to favour the simpler model if the fit is nearly as good due to the\nsmallerprior volume . If we assume uniform, and therefore separable, priors for each parameter,\nwe can express the volume as p(θM|M) = (∆ θM\n1...∆θM\nnM)−1and\nBN;M=R\ndθNp(d|θN;N)R\ndθMp(d|θM;M)(∆θM\n1...∆θM\nnM)\n(∆θN\n1...∆θNnN). (4.1.15)\nWe must integrate the likelihood across the parameter space to derive the evidence. While this\nis a standard mathematical problem, its complexity is ascribed to the most likely case that the\nintegrand is extremely sharply peaked, and we cannot predict where this peak will occur in the\nparameter space. Moreover, the multi-dimensional parameter space makes individual likelihood\nevaluations computationally expensive. Bayesian model selection techniques rely on efficient\nalgorithms capable of handling this type of integral. It is useful to establish a reference scale to\nevaluate differences in the evidence. The Jeffreys scale [206] is a standard criterion that measures\nthe strength of evidence favouring a model. A revised version proposed by Kass and Raftery\n[198, 207] is presented in the following table:\n|lnBN;M|Fractional Odds Model’s Probability Evidence\n<1.0 <3 : 1 <0.750 Inconclusive\n1.0to2.5 <12 : 1 0.923 Weak to Moderate\n2.5to5.0 <150 : 1 0.993 Moderate to Strong\n>5.0 >150 : 1 >0.993 Very strong or decisive\nTable 4.1 : Jeffreys scale to evaluate the strength of the evidence of a model N over another\nmodel M, in terms of the absolute value of lnBN;M, with a positive (negative) value indicating\nsupport for model N(M).\nIn summary, the Bayes factor strikes a balance between fit quality and additional model com-\nplexity. It rewards highly predictive models whilst penalising models with unnecessary extra\nparameters. This principle is often referred to as Occam’s razor.\n4.2 Einstein-Boltzmann Codes\nIn Section 2.2, we have introduced the equations governing the evolution of gravity and the\nmatter fields for the inhomogeneous Universe, namely the evolution of the fluctuations produced\nin the early Universe and that grow over time into the large-scale structures pattern we observe.\nHowever, the dynamics of this system of matter perturbations have a quite intricate character,\nrequiring specialised numerical tools to analyse their evolution.\nVarious numerical codes, or Boltzmann solvers, have been developed to facilitate this complex\ntask, starting with COSMICS [208], followed by CMBFAST [99], CMBEASY [209], and more recently,98 4.2. Einstein-Boltzmann Codes\nCAMB[210, 211] and CLASS[95, 101]. These codes are engineered to solve the Boltzmann and fluid\nequationsofmotionforeverytypeofmatterintheUniverse. TheworkpresentedinPartIIofthis\ndissertation relies on a modified version of CLASS, which includes an implementation of general\ncoupled dark energy. While both CLASSand CAMBaim to accomplish the same results, their\nimplementations differ in several ways. CLASSis the more recent of the two, launched publicly in\n2011, compared to CAMB, which has been public since 2000. Its design is more modular than\nCAMB’s original Fortran 90 version, making it easier to modify. Furthermore, CLASSis written in\nC, which some users might find more approachable than Fortran. Nevertheless, both codes now\ninclude a user-friendly Python wrapper, and the choice between the two boils down to individual\npreference.\nCLASS1(Cosmic Linear Anisotropy Solving System) was originally developed on request of the\nPlanckscience team as a tool independent from CAMBto check for possible bias in parameter\nestimation introduced by the code. Ultimately, the CLASS-CAMBcomparison has led to progress in\nthe accuracy of both codes, with the agreement established at 0.01%level for CMB observables,\nusing highest-precision settings in both codes. CLASScalculates various cosmological observables\nsuch as CMB anisotropy power spectra, matter power spectra, and the primordial power spec-\ntrum. ItisespeciallytailoredforsolvingEinstein-Boltzmannequations-theequationsdescribing\ncosmological perturbations’ evolution — and can simulate the evolution across numerous cos-\nmological scenarios and particle types. The software features a modular architecture. It starts\nby reading and initialising user-specified input parameters, followed by a one-time calculation\nof all vital background quantities, thereby avoiding redundant computations. These quantities\nare stored for future use. Next, the code calculates thermodynamic aspects of the Universe,\nincluding free electron fraction and matter temperature evolution, which are solely dependent\non the background evolution. Subsequent modules calculate the primordial power spectrum and\nevolve perturbation equations to compute the power spectra requested by the user. If neces-\nsary, non-linear scales can be modelled using different approaches. All output data are saved in\ndesignated files. We refer to lecture notes available at [212] for more in-depth information.\nThe CLASScode has been adapted for interacting dark energy for the purpose of this dissertation,\nincluding all the models considered in Part II. This involved several changes to existing modules,\noptimised for ΛCDM. Therefore, considerable adjustments had to be made to the input, back-\nground, thermodynamics and perturbation modules. Although still a private patch, the version\nmodified from scratch during this PhD accommodates any coupling in the dark sector, including\nthe more general disformal case (discussed in Chapters 8 and 9), and also different types of scalar\nfields for both dark energy, as introduced in Section 5.2, and is flexible enough to allow for the\nfuture incorporation of similar effects, like universal coupling to the whole matter sector.\nCLASScan be directly interfaced with the sampler MontePython2[213, 214], a widely used tool for\n1https://github.com/lesgourg/class_public\n2https://github.com/brinckmann/montepython_public4Methodology 99\nMCMC analysis in cosmology based on Python. It enables sampling various model parameter\nspaces, essentially analysing any cosmological data. Theoretically, observables are calculated\nthrough CLASSviathe Python wrapper. MontePython supports different sampling algorithms,\nsuch as the Metropolis-Hastings and Nested Sampling, and various convergence tests, including\ntheGelman-Rubin diagnosis [215]. This and other statistical measures can be calculated with the\naid of the Python GetDist1[216] package that takes in the chains generated by MontePython .\nIn order to determine the goodness-of-fit for a particular model, we will report the difference\nbetween the value of the minimum χ2test in a model Mwith respect to the ΛCDM model,\n∆χ2\nmin=χ2\nmin,M−χ2\nmin,ΛCDM, in the tables of parameter constraints. Additionally, we report\non the Bayesian evidence model comparison analysis, for which we used the public MCEvidence2\ncode [217].\n4.3 Baseline Data Sets\nThe baseline data set used throughout this dissertation consists of particular combinations of\nexisting data:\n1. ThePlanck2018 temperature and polarisation (TT TE EE) likelihood, which includes low\nmultipole data ( ℓ <30) [32, 33, 178]. We refer to this as Planck2018 orPl18.\n2. ThePlanck2018lensinglikelihood, constructedfrommeasurementsofthelensingpotential\npower spectrum [51]. We refer to this as Planck lensing orPl18len.\n3. Baryon Acoustic Oscillations measurements, collected from the 6dFGS [118], SDSS MGS\n[117], and BOSS DR12 [114] surveys. We denote this simply as BAO.\n4. Type Ia Supernovae (SNIa) distance moduli measurements, taken from the Pantheon sam-\nple [109]. We term this data set as Pantheon orSN.\nFor thePlanckdata, we consider both the high-multipole likelihood (including multipoles 30≲\nℓ≲2500for the TT spectrum and 30≲ℓ≲2000for TE and EE spectra) and the low-E\npolarisation likelihood (covering the multipole range 2≤ℓ≤30for the EE spectrum). We\ncan derive constraints on the cosmological and model-specific parameters by analysing Planck\ntemperature anisotropies and polarisation measurements.\nWe also test the differences in the constraining power in adding the Plancklensing measurement,\nas introduced in Section 3.3.4, the most significant detection of CMB lensing to date [51], which\nhelps improve constraints on cosmological parameters, with a particular focus on parameters\nthat affect late-time expansion and the background geometry.\n1https://github.com/cmbant/getdist\n2https://github.com/yabebalFantaye/MCEvidence100 4.3. Baseline Data Sets\nWhilePlancklensing measurements partially break the geometric degeneracy in the H0andΩm\nparameters, incorporating BAO measurements from galaxy surveys proves to be a much more\neffective way to shatter this degeneracy in the geometrical sector, in particular in the relative\nabundance of matter. The BAO measurements, being relatively simple geometric measurements\nunaffected by non-linear physics, offer a robust geometrical test of cosmology due to the large\nscale of its acoustic peak. We use the standard combination of measurements of DV/rdfrom the\n6dF survey at an effective redshift z= 0.106, the SDSS Main Galaxy Sample at z= 0.15, and\nthe final BOSS DR12 data with separate constraints on H(z)rdandDM/rdin three correlated\nredshift bins at z={0.38,0.51,0.61}.5Beyond the Standard Model: Coupled Dark\nSector\nMas o vazio tem o valor e a semelhança do pleno. Um meio de obter é não procurar,\num meio de ter é o de não pedir e somente acreditar\nque o silêncio que eu creio em mim é resposta a meu – a meu mistério.\n—Clarice Lispector A Hora da Estrela\nBut the emptiness has the value and the appearance of plenty. One way of getting is not looking, one way\nof having is not asking and only believing that the silence I believe to be inside me is the answer to my — to my\nmystery. —Clarice Lispector in The Hour of the Star\nIn this chapter, we introduce the formulation of alternative theories of gravity, with a particular\nfocus to models where dark matter and dark energy experience some non-standard interaction,\nas alluded to in Section 5.1. We present some examples of different dark energy models and their\ncosmological consequences in Section 5.2, in which a dynamically evolving scalar field replaces\nthe cosmological constant. Section 5.3 reports on models where there is a universal coupling\nof dark energy to the entire matter sector through conformal and disformal transformations of\nthe metric tensor. We close with Section 5.4 with an introduction to models of non-universally\ncoupled dark matter and dark energy, which will be the central focus of the work reported in\nPart II.\n5.1 Beyond ΛCDM\nAs presented in Chapter 2, the resurrection of the need for a cosmological constant was prompted\nby the use of the type Ia SNe distance-redshift relation to estimate the value of the Hubble\nconstant, unveiling a Universe expanding at an accelerating rate [26, 27]. This observation was\nfurther cemented by simultaneous CMB anisotropy measurements, which independently found\na preference for ΛCDM cosmology with a low mass content [218]. Nevertheless, as was also\ndiscussed in Sections 2.5 and 3.6, the concordance model is plagued by theoretical inconsistencies\nand, arguably more grievous, observational tensions. In an attempt to address these issues, a\nwide range of alternatives to the cosmological constant have been put forward in the literature\n[153], typically replacing the cosmological constant with another mechanism for accelerating the\nUniverse. While providing a more elucidatory theoretical framework or vaster phenomenology,\nthese approaches come at the cost of introducing modifications to one or more of the assumptions\nunderlying the standard model, as briefly described in Chapter 3. While these proposals can take\n101102 5.1. Beyond ΛCDM\nseveral forms1, the results reported in Part II all rely on some extension to the dark sector in the\nstandard ΛCDM action based on GR with a matter sector, and which we write again explicitly\nto make the modifications clear later in the chapter:\nSΛCDM =Z\nd4x√−g\"\nM2\npl\n2\u0010\nR(gµν)−2Λ\u0011\n+LM(gµν, ψM)#\n. (5.1.1)\nThe first term is the Einstein-Hilbert action, as defined in Equation (1.3.9), including the cos-\nmological constant Λ, and LMcollectively denotes the matter Lagrangian for the fields ψM,\nincorporating the standard model matter and dark matter.\nThe models studied in Part II rely on the extensions to the action in Equation (5.1.1) listed\nbelow, and how such extensions are motivated will be the focus of the referenced upcoming\nsections:\n1. Introducing a scalar field dark energy component ϕthat replaces the cosmological constant\nΛ, and can have various natures, as introduced in Section 5.2 - Chapters 6 to 10;\n2. Considering a non-universal coupling in the dark sector only, as motivated in the context of\nconformal and disformal transformations for universal couplings in Section 5.3 - Chapters 6\nand 7;\n3. Providing a joint scalar field origin for the dark sector through a joint geometrical or\nfluid description for dark matter with a scalar field origin (with the dynamics as detailed\nin Section 5.2), that is approximated by a coupling in the dark sector as described in\nSection 5.4 - Chapters 9 and 10.\nThe points listed above are encapsulated in the following extended effective action for aninter-\nacting dark sector\nSIDS=Z\nd4x√−g\"\nM2\npl\n2R(gµν) +Lϕ(gµν, ϕ,∇µϕ) +LSM(gµν, ψSM) +LDM(gµν, ϕ,∇µϕ, ψDM)#\n,\n(5.1.2)\nwhere ϕis the dark energy scalar field, ψSM/DMdenote the uncoupled standard model fields\nand the coupled dark matter sector (which in Chapter 10 is approximated from a scalar field\ndescription), respectively. Effective field theories of this kind have the advantage of being easy to\nimplement while preserving the general covariance of the theory. We formulate toy models based\non some simplistic solutions to cosmological puzzles, aiming to reproduce the effective behaviour\nof a more fundamental (but also more complicated) underlying dynamics. Ultimately, these\nstand for cosmologists as lighthouses do to sailors, signalling any effects beyond the standard\n1Extensions to GR are conventionally categorised into three classes: i) added dimensions to the spacetime; ii)\nhigher-order derivatives of the curvature or other related invariants; iii) introduction of extra fields.5Beyond the Standard Model: Coupled Dark Sector 103\nmodel. As will become clear, the advantage of modifying only the dark sector is twofold: first,\nthe theory can still be formulated at the Lagrangian level and relying only on the addition of one\nscalar degree of freedom; second, interactions are only allowed in the dark sector [219], whose\nnatureisstillnotwellunderstood, andismostlyprobedcosmologicallythroughthecharacteristic\ngravitational signatures left in the large scale structure and the CMB. This means that the local\nprecision tests of general relativity can be evaded [125, 220–223], since the fifth-force mediated by\nϕwill only influence the component that is not heavily restricted by the SM of particle physics.\nMoreover, the interaction would also influence the dynamics of dark energy, possibly contributing\nto explain some of the theoretical issues that plague the dark sector, as discussed in Section 2.5.\n5.2 Scalar Field Dark Energy\nOriginally, thebasicpremisebehindextensionstoGRwastheideathatthevalueof Λismeasured\nto be so small because it has been progressively converging to its natural vanishing state, i.e.\nΛ→0, over an extended period through the cosmic history. This evolution provides a tentative\nresolution of the theoretical problems discussed in Section 2.5. The earliest expression of this\nthought can be traced back to Dolgov [224], who proposed a variation of the theory of Brans and\nDicke [225], in which both Λand the strength of the gravitational interaction are evolving to\nzero. In that context, Reuter and Wetterich [56, 226] explored possible formulations yielding field\nequations where a dynamical Λgradually vanishes. Around the same time, Peebles and Ratra\n[227] showed how the energy density can be driven towards zero by a scalar field self-interacting\npotential, mimicking an asymptotically decreasing Λ, under the assumption that the quantum\nvacuum energy density vanishes. The name quintessence was introduced by Caldwell, Davé,\nand Steinhardt [228–230] to refer to this dynamically changing effective cosmological constant.\nInaugural considerations on the cosmological implications of this concept were further elaborated\nin [55] by Ratra and Peebles.\nA natural fundamental candidate for a dynamically changing entity in modern physics is a scalar\nfield. Relevant examples are the recently detected Higgs field [75], responsible for the mechanism\nof providing mass to the particles of the standard model of particle physics, or the inflaton,\nconsidered to be the scalar field that drives inflation [231]. Both these scalar fields play a\ncritical role in models of fundamental physics. The inflaton, in particular, gives rise to dynamics\nsimilar to dark energy since both have to be responsible for accelerated cosmic expansion periods.\nAdditionally, there is a precedent of solving problems related to missing energy by hypothesising\na new particle or field, as was the case with neutrinos and dark matter, with the latter still\nawaiting direct detection. Scalar field-based theories had been explored in the literature long\nbefore the discovery of the accelerated expansion, as viable alternatives to GR. Inspired by Weyl\nand Dirac’s studies leading to a slowly varying gravitational constant [232–236], in 1959 Jordan\nshowed that by describing gravitational couplings through a scalar field, the extra degree of104 5.2. Scalar Field Dark Energy\nfreedom behaves like a matter field obeying conservation relations [237]. These considerations\ndeveloped into a complete gravitational theory in 1961, developed by Brans and Dicke (BD)\n[225], in which the gravitational interaction is ascribed to the metric with the addition of one\nscalar degree of freedom, which came to be known also as the Jordan-Fierz-Brans-Dicke theory of\ngravity - in recognition of Fierz’s and Jordan’s seminal works on the physical understanding and\nformulation of this framework [237, 238] - and is widely considered to be the inaugural alternative\nto GR [239].\nUnder these considerations, the quintessence model has been generalised to include a dynamical\nscalar field playing the dark energy role, varying slowly along some potential V(ϕ). According to\nEquation (2.1.26), this evolving scalar field accounts for the missing energy needed to maintain\nthe Universe’s geometrical flatness. While bearing similarities to the slow-roll inflation period\nin the early Universe [47], the late-time acceleration mechanism differs in that non-relativistic\nmatter (dark matter and baryons) cannot be ignored for a complete understanding of the dy-\nnamics of dark energy. In principle, this evolving scalar field could interact with other species\ndirectly through a fifth-force , allowing for a DE component that naturally self-adjusts to repro-\nduce the inferred present energy density of Λ. For instance, this mechanism can be realised\nwith attractor-like DE solutions, which reproduce the present energy densities for a vast range\nof initial conditions.\nAn alternative approach is to consider parameterisations that can be compared against observa-\ntions, disregarding the fundamental or physical origin of such extensions, hoping to understand\nthe direction hinted at by the data. A famous example, analysed and constrained by the Planck\ncollaboration [32] as well, is the wCDM model , consisting of a constant equation of state for\nthe dark energy fluid, that deviates from w=−1, as assumed in the ΛCDM model. The value\nofwis estimated from observational data and consistency relation checks. Alternatively, the\nparameterisation can be generalised according to some redshift-dependent evolution w(z), which\nprovides insight into dark energy but is limited to handpicking a particular subset of functions\nthat determine the phenomenology of the theory [240]. The opposite side of the coin is recon-\nstructing the free function(s) in w(z)without prior restrictions or considerations [241]. While\nparameterisationsarehelpfultocapturephysicalfeaturesinasomewhatmodel-independentman-\nner, self-consistent and predictive frameworks are needed to comprehend and test the model’s\nassumptions across different regimes.\nThe remainder of this section focuses on prominent scalar field candidates for dark energy com-\nmonlyconsideredintheliteratureandwhosecharacteristicnatureleadstodistinctfeaturesinthe\ncosmological evolution that could help alleviate the standard model’s cosmological tensions and\nanomalies. Other possibilities have been proposed but will not be discussed in detail here, and\nthe interested reader is referred to reviews on alternative theories of gravity such as [90, 153, 242].5Beyond the Standard Model: Coupled Dark Sector 105\n5.2.1 Quintessence Field\nIn the framework of the FLRW cosmological background introduced in Section 2.1, a cosmolog-\nical constant term represents a source with a constant equation of state parameter, wΛ=−1.\nA similar behaviour at late times can be reproduced by a dynamically evolving scalar field,\nminimally coupled to gravity, of which the simplest example is the canonical scalar field ϕas\na QFT generalisation of the non-relativistic particle. This scalar dark energy source has been\nappropriately named quintessence in a bow to Aristotle’s fifth natural element. The crucial as-\npect of this framework is that the equation of state parameter is no longer constant and evolves\naccording to ϕand ˙ϕ, naturally addressing the conceptual issues of the cosmological constant\ndiscussed in Section 2.5, such as the incompatibility between the small value required for the\ncosmological constant to match the observations and the vacuum energy predictions from QFT.\nIn fact, in a homogeneous Universe accurately described by GR, the second Friedmann equa-\ntion, Equation (2.1.19), implies that cosmic acceleration can be achieved so long as the total\nenergy-momentum tensor components meet p <−ρ/3. From this, a new cosmic fluid can drive\nthe acceleration at the cost of violating the strong energy condition [56, 227]. The quintessence\nscalar field is a traditional example of such scenarios, and can be incorporated directly in the\ngravitational action alongside all the matter sources (in Sm), with a standard kinetic term and\na self-interacting potential V(ϕ)yielding\nS=Z\nd4x√−g\u0014M2\nPl\n2R−1\n2(∇ϕ)2−V(ϕ)\u0015\n+Sm, (5.2.1)\nwhere ϕis the quintessence field which has dimensions of mass and, once more, the first term\nis simply the Einstein-Hilbert action with M2\nPl= 1/(8πG), assuming Λ = 0. The term (∇ϕ)2=\ngµν∂µϕ∂νϕencodes the dependence on the kinetic evolution of the field.\nThe contribution of the scalar field to the total energy-momentum tensor is given by the variation\nof the action in Equation (5.2.1) according to the metric and reads\nTϕ\nµν=∂µϕ∂νϕ−gµν\u00141\n2gαβ∂αϕ∂βϕ+V(ϕ)\u0015\n. (5.2.2)\nIn the context of the flat FLRW background metric, Equation (1.6.6), the dynamical evolution\nof the scalar field is extracted through variation of the action in Equation (5.2.1) with respect\ntoϕitself and results in the Klein-Gordon equation in an expanding Universe:\n□ϕ−dV\ndϕ= 0FLRW− − − − → ¨ϕ+ 3H˙ϕ+V′(ϕ) = 0 . (5.2.3)\nThe energy density and pressure of the scalar field are computed from Equation (5.2.2):\nρϕ=−T0\n0=1\n2˙ϕ2+V(ϕ),and pϕ=Ti\ni=1\n2˙ϕ2−V(ϕ). (5.2.4)106 5.2. Scalar Field Dark Energy\nTherefore, in the case of a single-component Universe ruled by this scalar field, the Friedmann\nequations track its evolution as follows:\n3M2\nPlH2=1\n2˙ϕ2+V(ϕ),and 3M2\nPl¨a\na=−˙ϕ2+V(ϕ). (5.2.5)\nAs we can see from Equation (5.2.5), the field yields a positive accelerating contribution for ¨a >0\nas long as ˙ϕ2< V(ϕ); therefore, in order to explain the accelerated phase without invoking a\ncosmological constant, the scalar field dynamics must be designed in such a way to fulfil this\ncondition. More precisely, the equation of state parameter is given by:\nwϕ=pϕ\nρϕ=˙ϕ2−2V(ϕ)\n˙ϕ2+ 2V(ϕ). (5.2.6)\nIt can be readily appreciated that there are two limits to this equation: if the potential is\nsufficiently flat at late times such that ˙ϕ2≪V(ϕ), then wϕ≃ −1, leading to a slow-roll phase of\ncosmological constant-like accelerated expansion; if otherwise the field dynamics is dominated by\nits kinetic energy, that is ˙ϕ2≫V(ϕ), then wϕ≃1. The former is what makes a canonical scalar\nfield the simplest dynamical candidate for dark energy with different models distinguished by the\nform of their potential V(ϕ), which determines how ϕportrays the expansion of the Universe.\nNevertheless, quintessence fields have little impact on the growth of cosmological perturbations\nand are therefore challenging to probe with large-scale structure observations, producing only\nminor modification on the large-scale CMB through the change in the expansion history (see\nSection 3.3). In addition, the potential can be chosen to satisfy practically any expansion history\n(with w≥ −1), making the general case very unpredictable without further requirements on V.\nA sub-class of quintessence models is the tracker models. These theories exhibit a particular\nevolution of the energy density ρϕas it tracks the dominant component of the Universe. In this\nway, the coincidence problem is mitigated as the other components determine the evolution of\nρϕ. A potential that produces such an evolution, traditionally named the Ratra-Peebles potential\n[55, 80], is\nV(ϕ) =V0\nϕα, (5.2.7)\nwhere α > 0is a constant power left as a free parameter [243–246]. In this scenario, ρϕwill\nmimic the energy density of the dominant background component ρBin such a way that the\nequation of state parameter becomes\nwϕ=αwB−2\nα+ 2. (5.2.8)\nImposingtheaccelerationcondition ¨a >0fromEquation(2.1.19)toEquations(5.2.6)and(5.2.8)5Beyond the Standard Model: Coupled Dark Sector 107\nimplies [80]\n˙ϕ2<2\n3V(ϕ),and α <2\n2wB+ 1. (5.2.9)\nAccordingly, if the dominant contribution comes from matter, wB=wm= 0, accelerated expan-\nsion is achieved simply through the condition α < 2. Without further modifications, tracking\nsolutions lead to an overall enhancement of the Hubble rate at all times, indirectly reducing\nthe clustering rate of ordinary matter and possibly leading to a significant early dark energy\ncontribution in the early Universe [247].\nAnother standard class of quintessence potentials is the exponential case [248–250], which leads\nto a power-law expansion and takes the form:\nV(ϕ) =V0exp\u0012\n−λϕ\nMPl\u0013\n, (5.2.10)\nwhere V0is still the scale of the potential, and the constant λgives the steepness of the expo-\nnential. In this simple case, the field evolves as ϕ∝lntand accelerated expansion is achieved\ndepending on the value of λ, namely λ2<2is required. Arguably more important is the fact that\nthe exponential potential admits cosmological scaling solutions [248, 251], in which the field’s\nenergy density ( ρϕ)scalesproportionally to the background fluid’s energy density ρB[252–257],\nthat is\nρϕ\nρB=r , (5.2.11)\nwhere ris a nonzero positive constant, and the field can mimic the background even if it is\nsub-dominant over the radiation and matter-dominating eras. In this case, as long as the scaling\nsolution is the attractor, then for any generic initial conditions, the field will sooner or later enter\nthe scaling regime, thereby introducing a strategy to address the fine-tuning problem of dark\nenergy. It is worth mentioning that scaling solutions live on the border between acceleration\nand deceleration. We also note that the system needs to exit from the scaling regime defined\nin Equation (5.2.11) to bring about an epoch of accelerated expansion. For this purpose, the\nfield’s energy density must catch up to that of the fluid, provided that the potential is shallow\nrelative to the one corresponding to the scaling solutions. For instance, it has been shown that\nthe following type of double exponential potential can account for this transition [251]:\nV(ϕ) =V0\u0014\nexp\u0012\n−λϕ\nMPl\u0013\n+ exp\u0012\n−µϕ\nMPl\u0013\u0015\n, (5.2.12)\nwhere λandµare positive constants. The requirement is that the λ-exponential term provides\nthe scaling during the radiation- and matter-dominated eras before the µ-exponential term takes\nover. The scalar field is then driven out of the scaling regime and towards a solution in which\nit dominates. An important advantage of the double exponential potential is that the scaling\nregime occurs for a wide range of initial conditions, followed by an accelerated expansion phase108 5.2. Scalar Field Dark Energy\nonce the potential becomes shallow, evading the general fine-tuning of initial conditions of the\nscalar field in most quintessence models. For a cosmological analysis of quintessence with other\npotentials see, for example, [249, 251, 255, 258–268].\nQuintessence models will be the main focus of Chapter 6. Beyond the technical details, it is\nessential to highlight the resemblance between the action of the quintessence model and that of a\ngeneral scalar-tensor theory. The boundary between these two theories is somewhat blurred, and\nit has been shown that quintessence models can be recast into a Scalar-Tensor theory analogous\nto the Jordan-Brans-Dicke formulation by applying a Weyl rescaling of the metric [153, 269].\n5.2.2 K-Essence Models\nA further extension of quintessence is the k-essence model [270–272], in which a canonical field\ndoes not drive the accelerated expansion via a slowly varying potential, but by non-standard\nmodifications to the kinetic energy of the scalar field instead, i.e.\nS=Z\nd4x√−g\u0014M2\nPl\n2R+LK(ϕ, X)\u0015\n+Sm. (5.2.13)\nwhere the Lagrangian density LKencapsulates the most general scalar field action, which is a\nfunction of both ϕand its kinetic energy X=−1\n2(∇ϕ)2, and of which quintessence is a particular\ncase. While k-essence fields have a significantly increased number of phenomenological features\nwhen compared with their quintessence ancestors, the trade-off is the fact that the action in\nEquation (5.2.13) is not guaranteed to be free of pathological or unstable behaviour, such as\nghost degrees of freedom or superluminal propagation [273]. K-essence models are motivated by\nlow-energy effective string theory where an L-like term of the form in Equation (5.2.13) appears\nfor the dilaton field [274]. The possibility of achieving an accelerated expanding period from the\naction in Equation (5.2.13) was suggested in the context of inflation [274], with the extension to\nthe late-time dark energy scenario first proposed in [270], with later generalisations establishing\nthe concept of k-essence.\nThe scalar field action LKis made more tractable by assuming that it is separable in ϕand\nX, and in particular, the simple case of the ghost condensate model is often assumed, in which\nLK(ϕ, X) =f(ϕ)(X2/M4−X), where Mis a mass scale and f(ϕ)is an arbitrary non-singular\nfunction of ϕ. In these models, the effects of the quintessence potential are reproduced by the\nnon-standard kinetic term, which leads to the equation of state\nwϕ=1−X/M4\n1−3X/M4, (5.2.14)\nwhere in order to have an accelerated expansion phase, that is, wϕ<−1/3, we must require\nX < 2M4/3. The expression above also clearly illustrates the underlying assumption of k-\nessence: if Xdoes not vary in time, and there is no kinetic dynamics, then wϕremains a constant.5Beyond the Standard Model: Coupled Dark Sector 109\nIn particular, for this simple k-essence scenario, the cosmological constant case corresponds to\nX=M4/2. This analysis has been extended in multiple works, and it can be shown to address\nthe coincidence problem of dark energy [271, 272]. In Chapter 7, we will explore a generalised\nk-essence scalar field scenario derived from the framework proposed in [275].\n5.3 Universal Couplings to the Matter Sector\nIn the context of GR, gravity is interpreted as a fundamental geometrical property of spacetime\noriginating from the matter sources included therein. This notion aligns with Mach’s equivalence\nprinciple, which states that inertia is a product of the interaction between bodies [276, 277]. In\nthe proposal of Brans and Dicke [225], this principle was incorporated in the form of a varying\ngravitational coupling according to the dynamical character of the mass distribution itself. This\nformalism relied on the introduction of a scalar field such that Newton’s gravitational constant\nGis actually the local value of the gravitational coupling and the weak equivalence principle -\nstating that in a gravitational field, all point masses follow geodesics of the same metric - still\nholds [13, 225, 242]. Brans and Dicke chose a matter-scalar coupling acting universally on all\ntypesofmatter, accordingtothesamegravitationalmetric. Thisensuredthatmatterisperceived\nas free-falling in this gravitational metric, in agreement with the weak equivalence principle.\nInterestingly, while this scalar field was intentionally detached from the matter Lagrangian to\nmaintain the universality of free-fall, it inevitably couples with matter in the field equations,\nwhichdependdirectlyonthetraceoftheenergy-momentumtensorofmatter. Theoriesofgravity\nof this nature build upon the solid groundwork of GR plus a scalar field which is introduced in\na non-trivial way through what’s called as non-minimal universal coupling .\nTwo particularly relevant descriptions with an intuitive physical interpretation arise:\n•TheJordan frame in which the scalar field only features explicitly in the gravitational\nsector (given by the the Einstein-Hilbert action in Equation (1.3.9)) and the matter fields\nare said to be minimally coupled toϕ, while there is a non-minimal coupling between the\nscalar field and the curvature term. This means that the individual energy-momentum\ntensors of each source are still covariantly conserved and test particles follow the geodesics\nof the spacetime metric.\n•TheEinstein frame which denotes the opposite case, where the scalar field is only ex-\nplicitly included in the matter sector, which becomes non-minimally coupled , while the\ngravitational sector is now minimally coupled, meaning that the EH action remains unal-\ntered. Likewise, the set of field equations resemble the Einstein equations in GR, ensuring\nsecond-order field equations, with the caveat of losing covariant conservation of the energy-\nmomentum tensor of the matter fields:\n∇µTM\nµν=Qνand∇µTϕ\nµν=−Qν. (5.3.1)110 5.3. Universal Couplings to the Matter Sector\nTheQνterm encodes the non-minimal coupling between matter and the scalar field. The\nscalar degree of freedom is transformed into a matter source contribution, an imprint of its\ngravitational role in the Jordan frame. This comes with the trade-off that the matter fields\nno longer follow geodesics of the gravitational metric gµν, but rather of some transformed\nmetric including the effect of the coupling:\ngµν7→˜gµν=C(ϕ)gµν, (5.3.2)\nwhere C(ϕ)is called the conformal function.\nConformal transformations such as the one defined in Equation (5.3.2) are a conventional and\nseemingly powerful tool, first exploited in the prototype Brans-Dicke (BD) model in 1961 [225],\nwhich also clearly illustrated the physical meaning of such a metric transformation. The advan-\ntage is that if the theory has been shown to be invariant under these transformations, then the\naction can be recast between different frames (much like a change of reference frame) in which\nits physical meaning may be clearer or its field equations less involved1such as the BD theory\nand its generalisation into the broader class of scalar-tensor theories of modified gravity [278].\nIt should, however, be emphasised that the definition of these two frames is only valid when\nauniversal transformation is considered, that is, when the scalar field couples non-minimally\nto thewhole matter sector , and the equivalence principle still holds. While in the past there\nwas significant debate and division in the community on whether these two frames represent\nphysically equivalent theories, the general consensus at the moment of writing this thesis, is that\nconformally related theories are indeed distinct representations of the same physical scenario,\nespecially at the classical level [278, 279].\nWithout introducing additional scalar degrees of freedom, this formulation can be extended by\nallowing the conformal factor to depend on the derivatives of the scalar field as well, through its\nkinetic term X≡ −(1/2)gµν∇µϕ∇νϕ. This transformation is more general than the traditional\nconformal case, introducing a dependence of the metric in the conformal factor, and a related\nexample will be the focus of Chapter 7.\nA further extension was proposed by Bekenstein in 1992 [280] while investigating general cou-\nplings of matter to the gravitational metric in the context of Finsler geometry [281]. This was\naccomplished through the introduction of distinct geometries for each sector, restricted to be\nrelated through the so-called disformal transformation of the metric. Disformal transformations\nhave gained particular relevance in cosmology since it was shown that Horndeski theories2remain\n1The conformal function/factor C(ϕ)must be a smooth non-vanishing function of the spacetime coordinates.\nBy applying a conformal transformation, a mapping of the cosmological description is enforced, implying a\nmodification of the structure of the spacetime. In contrast, the coordinate separations dxµare fixed to the\nspacetime itself and remain unaltered. Nevertheless, the distance between two spacetime points accompanies the\nstructural change, meaning that a conformal transformation applies an isotropic scaling of the length and norm\nof time-like and space-like intervals and vectors alike.\n2TheHorndeskiLagrangianisadirectgeneralisationoftheBrans-Dicketheory, basedonthepremiseofhaving5Beyond the Standard Model: Coupled Dark Sector 111\nstructurally invariant under the particular class of disformal transformations which depend only\nonϕ[284]:\ngµν7−→˜gµν=C(ϕ)gµν+D(ϕ)∂µϕ∂νϕ , (5.3.3)\nwhere C(ϕ)andD(ϕ)are the conformal and disformal factors, and the conformal transformation\nis recovered by setting D= 0, while the minimally coupled case requires C= 1. It has been\nshown that theories in which CandDdepend on X[285–287] may be plagued by Ostrogradski’s\ninstabilities1[288–290].\nJust on a qualitative basis, by exiting the conformal realm, the metric transformation can no\nlonger be interpreted as a simple field redefinition. Disformal transformations do not represent\na change in coordinates but a local change in the geometry instead, encoding a stretching (or a\ncompression) of the metric specified according to the gradient of the scalar field, resulting in a\ndistortion of both angles and lengths. There is a translation along the direction of variation of\nthe field, implying that the new metric will also depend directly on the changing properties of\nthe field through spacetime.\n5.4 Non-Universal Couplings and Interactions in the Dark Sector\nFinally, one may ask whether this dark energy source encodes any non-minimal interaction, that\nis, besides the standard gravitational coupling, to the other sources in the Universe. The idea of\naninteractive dark energy scalar field ϕ, coupled to a matter component, and its subsequent cos-\nmologicalimplications, werefirstexaminedin[291–294]. Szydlowskiintroducedasimilarconcept\nof interaction between dark matter and dark energy [295], followed by a method of examination\nthat was formulated later in [296]. There’s a wealth of studies exploring the dynamical impli-\ncations of interactions between dark energy and matter [297–308]. Delving into the particular\nscenarios where dark energy is portrayed by a canonical scalar field, we come across the cou-\npled quintessence models , as direct generalisations of the models presented in Section 5.2. These\nwere initially presented and studied in [309, 309, 310], based on Luca Amendola’s non-minimally\ncoupled theories [311]. Various distinct models have been considered, each characterised by the\nuniversality and form of the coupling [219, 312–334].\nFurthermore, instead of imposing the coupling directly by an ad-hocfield-dependence in the DM\nLagrangian, theideasofconformal/disformalinvariancediscussedinSection5.3canberecovered,\nthe most general scalar-tensor extension to GR yielding second-order equations of motion both for the metric and\nthe scalar field in four-dimensional spacetime. This solution was put forward by Horndeski in the 1970s [282] but\nremained practically unnoticed for many years and was only brought to light recently in the context of Galileon\ntheory [283]. The disformal transformation represents a symmetry of the Horndeski action, just as the conformal\ntransformations are with respect to the Brans-Dicke action.\n1Ostrogradski’s theorem states that the Hamiltonian of any non-degenerate Lagrangian, depending on more\nthan first-order time derivatives, is unbounded from below, leading to instability issues that result in an unphysical\ntheory with arbitrarily high negative energy. This no-go theorem limits the space of possible scalar-tensor theories\nlike Horndeski’s. Whenever there are time derivatives of the fields of order higher than one in the Lagrangian,\nthe theory becomes unstable and unphysical.112 5.4. Non-Universal Couplings and Interactions in the Dark Sector\ne.g.assuming that the coupling between the scalar field and the SM species is suppressed or\nhidden by means of a screening mechanism [221, 335–342]. In these mechanisms the coupling\nbetween the scalar field and matter weakens and limits the gravitational influence of the scalar\nfield itself in dense, small-scale areas like the Earth and the solar system, where GR has been\nrigorously tested. Yet, on larger scales, like those generally associated with dark energy, the\nscalar field’s gravitational effects may re-surge, in a context where the theory is less constrained.\nThis subject is still an active research area, but is out of the scope of this thesis.\nConsidering an FLRW Universe, Equation (1.6.6), the continuity equations, Equation (2.1.22),\nfor the scalar field and the coupled matter fluid include a symmetric extra term Qaccounting for\nthe interaction and preserving the covariant conservation of energy, corresponding to the ν= 0\ncomponent of Equation (5.3.1), and take the form:\n˙ρDM+ 3HρDM(1 +wDM) =−Q,and ˙ρϕ+ 3Hρϕ(1 +wϕ) =Q,(5.4.1)\nThe sign of Qin Equation (5.4.1), which is (in principle) a function of the cosmological variables,\nestablishes the direction of the flow of energy transfer, and can itself be a dynamical quantity.Part II\nResearch Results6CoupledQuintessenceinFlatandCurvedGe-\nometries\nCada coisa é uma palavra. E quando não se a tem, inventa-se-a.\n—Clarice Lispector A Hora da Estrela\nEach thing is a word and when there is no word it is invented. —Clarice Lispector in The Hour of the\nStar\nAs discussed in Part I of this dissertation, the quest to comprehend the recent accelerated expan-\nsion of the Universe is among the most pressing open questions in the cosmological community\nand perhaps in the scientific endeavour. According to their inherently attractive gravitational\nnature, the matter fields accounted for in the standard model of particle physics cannot be the\ndriving force behind this acceleration. When considering general relativity at the largest observ-\nable scales, this behaviour becomes important and must be accommodated by the theory, such\nas some form of cosmic fluid that exhibits a strongly negative pressure-to-energy density ratio —\noften termed dark energy — capable of balancing out the gravitational pull of ordinary matter\nand powering the acceleration, fitting into the observational evidence.\nOver the course of Chapters 1 and 2, we have introduced a positive cosmological constant as\nthe most straightforward form of dark energy, a cornerstone of the well-accepted ΛCDM model.\nInterestingly, this constant has only recently begun to dominate the Universe’s energy content.\nIndeed, Chapter5builtuptheideaofmoredynamicvarietiesofdarkenergysupportedbycurrent\ndata. For example, Section 5.2.1 introduced quintessence models, where a scalar field drives the\nUniverse’s accelerated expansion. In these scenarios, the field’s potential energy, rather than its\nkinetic energy, becomes the dominant factor over time, effectively mimicking the cosmological\nconstant value.\nAdditionally, most studies often assume that dark energy and dark matter - classed together as\nthedark sector - have independent origins and do not share any exotic interactions. However,\ngiven the enigmatic nature of these species, there is no fundamental justification for dismissing\npotential effective couplings between the two. For example, while solar system tests [222] and\nexperimental data severely limit any interaction between dark energy and ordinary matter, such\nconstraints do not necessarily apply to dark matter. This subject was the focus of Sections 5.3\nand 5.4 and has been thoroughly explored in the literature [90, 309, 309, 334, 343–355].\nThis chapter will explore various types of coupled quintessence models and their predictive power.\nNamely, their potential to address evidence for a closed Universe found in the CMB data.\n115116\nThePlancksatellite’s observations of the cosmic microwave background temperature fluctua-\ntions and polarisation angular power spectra [33, 178] largely support the concordance ΛCDM\ncosmological model [32]. However, as thoroughly discussed in Section 3.6, with the remarkable\nrise in the precision of cosmological probes in the last decade, certain anomalies require further\nscrutiny. Namely, the Planckdata leads to the prediction of a lensing amplitude above the\nnormalised value, Alens= 1, at about 2.8standard deviations [183, 356]. In minimal extensions,\nthis so-called lensing excess implies more cold dark matter, favouring a closed Universe. This is\nsupported by both the baseline Pliklikelihood and the alternative CamSpec likelihood, with the\nlatter even supported at a 99%confidence level. These findings question the flatness assump-\ntion in the conventional ΛCDM cosmological model and have sparked significant debates. This\nhypothesis also accounts for other large-scale data irregularities, such as the reduced amplitude\nin quadrupole and octupole modes [357]. As a result, evidence pointing to a closed Universe has\nescalated to a compelling 3.4standard deviations [181, 182], surpassing the anomaly reported\nfor the lensing amplitude, sparking significant debate.\nThis finding is rather surprising given that other astrophysical measurements, like BAOs [116,\n118, 118], are found to conflict with the notion of a closed Universe when combined with the\nPlancklikelihood [181]. Moreover, the now well-established and extensively investigated discrep-\nancy between the Hubble constant as determined by the SH0ES project using Cepheid stars as\ncalibrators for estimating Type Ia supernovae luminosity distances [111] ( H0= 73±1km/s/Mpc)\nand the Plancksatellite’s CMB-based estimate including lensing measurements [32] ( H0=\n67.4±0.5km/s/Mpc) have risen above the distressful 5-sigma threshold (see also [358], where the\ndivergence reaches 5.3σ), essentially diminishing the chance of this disagreement being attributed\nto a statistical anomaly.\nLastly, itisworthmentioningthetensionintheparameter S8≡σ8p\nΩm/0.3betweenPlanckand\nweak lensing measurements like KiDS-1000 [164] or DES-Y3 [161]. Again, these discrepancies\ndwell at a 2−3σlevel under a ΛCDM assumption. Hence, there is strong support for the\nneed for more precise measurements beyond those from Planckfor gaining clearer insights into\nthese cosmological tensions and anomalies that allow for independently testing the fundamental\nassumptions behind the curtains of the modern cosmological stage.\nWe will introduce the coupled quintessence model for a general coupling in Section 6.1 along\nwith the methodology followed in this work. We include a summary of the equations governing\nits evolution at the background and linear perturbative levels for general spatial curvature and\ndiscuss the repercussions of this dark sector coupling on the CMB’s TT spectrum.. In Section 6.2\nwe present an update on the constraints for the flat models differing in the choice of the coupling\nand the potential functions. In Section 6.3, we will shift our focus to the changes introduced by\ndropping the flatness assumption and taking the curvature of the Universe to be a cosmological\nparameter that can be constrained and compared with other data sets in order to shed light on\nthe cosmological tensions. Finally, we comment on the evidence for a closed Universe in coupled6Coupled Quintessence in Flat and Curved Geometries 117\nquintessence models and summarise the main conclusions and discuss the results in Section 6.4.\n6.1 Models and Methodology\n6.1.1 Background Cosmology\nConsistent with the framework outlined in Chapter 5, the interaction within the dark sector\nin these models is achieved through a conformal non-universal coupling . As introduced in Sec-\ntion 5.1, this setting is embodied by the following action\nS=Z\nd4x\u0014√−g\u0012M2\nPl\n2R−1\n2gµν∂µϕ∂νϕ−V(ϕ)\u0013\n+√−gLSM(g,Ψi) +p\n−˜gLDM(˜g, χ)\u0015\n.\nHere, Ris the Ricci scalar, in terms of the metric degree of freedom gµν, with gbeing its deter-\nminant. We recall that MPl= (8πG)−1/2is the Planck mass and Gis Newton’s gravitational\nconstant. The quintessence scalar field enters the action as a canonical kinetic term plus a self-\ninteracting potential V(ϕ). The SM Lagrangian LSMconsists of both radiation (relativistic) and\nbaryonic (non-relativistic) components, encapsulated by the fields Ψi, which follow geodesics\ndefined by the gravitational metric gµν. Lastly, the dark matter Lagrangian LDMyields the\ndescription of the dark matter field χ, which is coupled to dark energy viaaϕ-dependent metric\ntransformation. This results in the dark matter following geodesics dictated by this modified,\ndark-energy-dependent metric ˜gµν, related to the gravitational metric gµνvia a conformal trans-\nformation:\n˜gµν=C(ϕ)gµν, (6.1.1)\nwhere C(ϕ)is the conformal function. We will see that, in practical terms, this is equivalent to an\ninteraction with dark matter, expressed as a modified effective Newton’s constant, a dynamical\nfunction of the local scalar field value, appropriately known as conformal coupling.\nWe start by recalling the general Friedmann-Lemaître-Robertson-Walker line element with cur-\nvature signature K:\nds2=a2(τ)\u0014\n−dτ2+dr2\n1−Kr2+r2\u0000\ndθ2+ sin2(θ) dφ2\u0001\u0015\n, (6.1.2)\nin terms of the conformal time τand where a(τ)is the scale factor for the expansion of the\nUniverse and a prime will denote derivatives with respect to τthroughout this chapter. The\nconstant Kis the curvature parameter, whose encodes the spacetime geometry: K= 0is the\nstandard flat case, as assumed in the standard six-parameter ΛCDM model constrained by the\nPlanckCollaboration, while K= +1andK=−1correspond to the positively (closed geometry)118 6.1. Models and Methodology\nand negatively (open geometry) curved cases. This is expressed in Cartesian coordinates as:\nds2≡gµνdxµdxν=a(τ)2\u0002\n−dτ2+γijdxidxj\u0003\n, (6.1.3)\nγij=δij\u0014\n1 +1\n4K\u0000\nx2+y2+z2\u0001\u0015−2\n. (6.1.4)\nThe Friedmann equations accounting for curvature, DE, DM and a radiation ( r) and baryonic\n(b) sectors is:\nH2=1\n3M2\nPla2(ρr+ρb+ρDM+ρϕ)−K , (6.1.5)\nH′+H2=−1\n6M2\nPla2(2ρr+ρb+ρDM+ρϕ+ 3pϕ). (6.1.6)\nThe Friedmann closure relation now must account for the extra contribution of curvature, en-\ncoded in the term ΩK≡ −K/H2\n0:\nΩ0=ρi\n3M2\nPlH2Equation (6.1.5)− − − − − − − − − − → Ωr+ Ω b+ Ω DM+ Ω ϕ+ Ω K= 1.(6.1.7)\nOwing to the energy exchange, the DM conservation equation now reads\nρ′\nDM+ 3HρDM=βρDMϕ′/MPl, (6.1.8)\nwhere ρDMis the energy density of DM. The coupling βis related to the conformal coupling\nfunction C(ϕ)via\nβ=MPl\n2d lnC\ndϕ. (6.1.9)\nThe energy density of dark matter particles no longer follows the scaling behaviour of a pres-\nsureless fluid. Instead, it scales as ρDM∝p\nC(ϕ)/a3. It’s worth noting that, when interpreted\ndirectly from the action, the mass of DM particles is field-dependent (assuming conservation of\nthe number of particles) and scales as m(ϕ)∝p\nC(ϕ). The evolution of the dark energy scalar\nfield is also influenced by this coupling, leading to a modified form of the Klein-Gordon equation,\nfirst introduced in Equation (5.2.3):\nϕ′′+ 2Hϕ′+a2V,ϕ=−a2βρDM/MPl. (6.1.10)\n6.1.2 Cosmological Perturbations\nWe consider scalar perturbations in the conformal Newtonian gauge (introduced in Section 2.2.1)\nds2=a2(τ)\u0002\n−(1 + 2Ψ) dτ2+ (1−2Φ)γijδijdxidxj\u0003\n, (6.1.11)6Coupled Quintessence in Flat and Curved Geometries 119\nwith γijas defined in Equation (6.1.4).\nAccording to Section 2.2.1, we can write the perturbed continuity and Euler equations for the\nDM component:\nδ′\nDM=−\u0000\nθDM−3Φ′\u0001\n−βδϕ′, (6.1.12)\nθ′\nDM+HθDM=k2Ψ +βϕ′θDM−β\nM2\nPlδϕ . (6.1.13)\nTheseequationsdescribetheclusteringofmatterthatleadstotheprocessofstructureformation.\nThe perturbed KG equation reads\nδϕ′′+ 2Hδϕ′+\u0000\nk2+a2V,ϕϕ\u0001\nδϕ= (6.1.14)\n\u0000\nΨ′+ 3Φ′\u0001\nϕ′−2a2V,ϕΨ−a2βδDMρDM−2a2βρDMΨ. (6.1.15)\nAdditionally, in the subhorizon limit ( k≫ H), the equation for the DM density contrast must\naccount for the gravitational effects introduced by the coupling and becomes:\nδ′′\nDM+Heffδ′\nDM−3\n2H2Geff\nGΩDMδDM≃3\n2H2(Ωbδb+ Ω rδr), (6.1.16)\nwhere Heff=H+βϕ′is the effective Hubble parameter.\nDue to the coupling in the dark sector, a long-range attractive fifth-force between DM particles\nis introduced, mediated by the scalar field ϕ. As a result, DM particles interact according to an\nenhanced effective gravitational constant, expressed as:\nGeff=G\u0000\n1 + 2 β2\u0001\n. (6.1.17)\n6.1.3 Models\nThis study explores three specific toy models characterised by their distinct conformal coupling\nand scalar field potential functions, C(ϕ)andV(ϕ).\nThe first model, referred to as M 1, employs an exponential form for both the conformal coupling\nfunction and the potential [348], more precisely:\nC(ϕ) =e2αϕ/M Pl,and V(ϕ) =V4\n0e−λϕ/M Pl, (6.1.18)\nwhere the conformal coupling parameter αand the slope of the scalar field potential λ > 0\nare dimensionless constants, and V0is a constant mass scale of the potential. The coupling\nbetween dark energy and dark matter in this model is simply a constant, β=α, resulting in a\nconstant fifth-force experienced by DM particles, expressed through Geff/G= 1+2 α2, according120 6.1. Models and Methodology\nto Equation (6.1.17). Such an interaction is at the heart of the original coupled quintessence\nmodel, thoroughly explored in the literature for various scalar field potentials. Conventionally,\nstudies focus on the general case where βcan take either positive or negative values. However,\nin this work, we choose to consider both regimes separately, distinguishing between instances\nwhere α >0, and the model is labelled as M 1+, and where α <0, and it is termed M 1−.\nThe second model we consider, denoted M 2, is characterised by the same exponential conformal\ncoupling function as in M 1, but adopts an inverse power-law form for the potential:\nC(ϕ) =e2αϕ/M Pl,and V(ϕ) =V4\n0\u0012ϕ\nMPl\u0013−µ\n. (6.1.19)\nIn this case, the conformal coupling constant αand the power of the potential µ >0are also\ndimensionless constants, and the scale of the potential V0is still a constant with dimensions\nof mass. We have already encountered this type of potential in Section 5.2.1 in the context of\nquintessence scalar fields. It has the attractive feature of admitting tracker solutions, which can\nhelp alleviate the cosmic coincidence problem. Analogously, when α >0, this model is referred\nto as M 2+, and M 2−when α <0.\nLastly, we examine a less conventional scenario, which we designate as M 3. In contrast to the\nconstant coupling present in M 1and M 2, M3exemplifies the case of a field-dependent dynamical\ncoupling parameter β. Indeed, a quintessence model with a dynamical coupling remains an\nintriguing and relatively uncharted avenue for research. A subset of conformal couplings offering\nthistime-varyingfifth-forceinvolvesthosewithminimumpointsoftheeffectivepotentialgivenby\nthe combined effects of CandV. The dark sector interaction vanishes when the field sits at this\nminimum, denoted as ϕ∗. On the other hand, if the field shifts away from the minimum - possibly\nat later stages, influenced by the scalar field potential — a fifth-force emerges, coinciding with\nthe onset of dark energy domination. In this work, we investigate such a scenario, accomplished\nthrough the same exponential potential as in model M 1and a quadratic exponential conformal\nfunction:\nC(ϕ) =eγ(ϕ−ϕ∗)2/M2\nPl,and V(ϕ) =V4\n0e−λϕ/M Pl. (6.1.20)\nHere, γandλare dimensionless constants and ϕ∗is a constant with dimensions of mass. The\nconformal coupling function C(ϕ)reaches its minimum at ϕ∗when γ > 0, and the model is\ntermed M 3+, or its maximum when γ <0, termed M 3−. As we will see, this difference will be\ncrucial to understand and differentiate between the two cases. This setting gives the varying\ndark sector coupling β=γ(ϕ−ϕ∗), implying that the fifth-force in the dark sector only switches\non when the field is displaced from the minimum/maximum ϕ∗, set to ϕ∗= 2MPlwithout loss\nof generality.\nNumerically, the mass scale V0for each model is determined through a shooting algorithm to\navoid degeneracies in the parameter space. The initial scalar field value, ϕini=ϕ(aini)with6Coupled Quintessence in Flat and Curved Geometries 121\naini= 10−14, is generally allowed to vary. However, in M 1and M 2, it is known not to affect\nthe evolution equations (which we check for M 2), and, as we will discuss in further detail,\nit leads to unphysical results for M 3+, since the model wants to sit at the minimum of the\npotential, originating an ill-defined behaviour in the sampling method. Consequently, M 1and\nM2introduce two additional parameters {α, λ}and{α, µ}(even though we also look at sampling\nϕini), respectively, while M 3introduces three extra parameters {γ, λ, ϕ ini}. In Table 6.1, we\nsummarise the differences in the three models.\nFurthermore, we are interested in considering extensions to all the models with the addition of\none extra degree of freedom of GR that is conventionally fixed in ΛCDM and extensions to it: the\ncurvature background geometry, parametrised by the curvature density parameter ΩK≡1−Ω0,\nas defined in Equation (2.1.26). This implies relaxing the assumption of a flat Universe supported\nby most inflationary models [359], such that negative (positive) values stand for a spatially closed\n(open) Universe. Nevertheless, while the vast majority of inflationary models naturally predict\nspatial flatness ( i.e.ΩK= 0), inflation in a curved Universe has been extensively explored in\nthe literature (see, e.g., Refs. [48, 360–370]). Regardless of the evidence reported for a closed\nUniverse in ΛCDM for the Planck data [181], for the sake of completeness, we will sample for\nboth positive and negative values of ΩK, following the procedure considered, for instance, in\nRef. [371] for interacting dark energy models with different phenomenological couplings.\nV(ϕ)C(ϕ)\ne2αϕ/MPleγ(ϕ−ϕ∗)2/M2\nPl\nV4\n0e−λϕ/MPl M1 M3\nV4\n0(ϕ/MPl)−µM2 —\nTable 6.1 : Summary of the models considered in this work, distinguished by the conformal\nfactor C(ϕ)and the potential V(ϕ).\n6.1.4 Methodology and Observational Data\nTo explore the extent and viability of the background and linear perturbation predictions of\nthe scenarios presented in Section 6.1.3, we follow the strategy and numerical tools outlined\nin Chapter 4. We employ the baseline data sets listed in Section 4.3, which, for clarity, we\nenumerate once more:\n•Pl18: the Cosmic Microwave Background TTTEEE+lowE likelihood from the most recent\nPlanck2018 data release [32] in all the data set combinations considered in this study.\nSpecifically, this includes the data on the CMB temperature (TT) and polarisation (EE)\nanisotropies and their cross-correlations (TE) at both small (highlTTTEEE) and large\nangular scales (lowlTT+lowEE).122 6.1. Models and Methodology\n•BAO: a compilation of baryon acoustic oscillations distance and expansion rate measure-\nments from BOSS DR12 [116], the SDSS Main Galaxy Sample [117], and 6dFGS [118],\naligning with the data used by the Planck2018 Collaboration [32] in their analysis.\n•SN:thedistancemodulimeasurementsfromTypeIaSupernovaegatheredbythePantheon\nteam [109]. This compilation offers 1048 data points for luminosity distance in the redshift\nrange z∈[0.01,2.3], fromwhichinformationonthelatetimeexpansionratecanbederived.\nWe analyse the incremental effects of adding the CMB-independent BAO and SN astrophysical\ndata alongside the Planck likelihood and the complete combination of these data sets. Our ulti-\nmate goal is to perform a Bayesian Monte Carlo Markov Chain analysis to establish observational\nconstraints on the theories considered and compare their evidence against the ΛCDM model, as\nsummarisedinTable6.3-Table6.15. WerecallthatalargerBayesfactor BM,ΛCDMindicatesmore\nsubstantial evidence of the model Mrelative to ΛCDM. This numerical factor can be translated\ninto a qualitative assessment of the level of statistical support for each extended model against\nΛCDM through the use of Jeffreys scale [207], as per the criteria listed in Table 4.1. Moreover, if\nlnBM,ΛCDM <0, there is no evidence of support for that model over ΛCDM for a given data set,\nwhile the opposite holds if lnBM,ΛCDM >0. We report the Bayes factor lnBM,ΛCDMand the\n∆χ2\neff, defined in Equations (4.1.10) and (4.1.15) to assess the support and goodness of fit in each\ntable of constraints of the corresponding models (including ΛCDM). Given the increased number\nof free parameters we choose to run the MCMC chains with convergence criterion based on the\nGelman-Rubin statistics of |r−1|≲0.02, as introduced in Chapter 4. We present the results\nfor each model, concluding by comparing the different scenarios and assessing their viability in\nlight of our analysis.\nWe vary the conventional six ΛCDM parameters (see Section 3.5 for more details): the reduced\nbaryon ωb= Ω bh2and dark matter ωcdm= Ω cdmh2energy densities, the ratio between the\nsound horizon and the angular diameter distance at decoupling θs, the reionisation optical depth\nτreio, the scalar spectral index ns, and the scalar amplitude of the primordial power spectrum\nAs, where his the reduced Hubble constant defined by H0= 100 hkm s−1Mpc−1. Additionally\nwe also sample for the respective parameters of each coupled quintessence model: {α, λ}for\nM1,{α, µ, ϕ ini}for M 2, and{γ, λ, ϕ ini}for M 3. From these primary parameters, we can derive\nand report constraints on other relevant derived quantities such as the current mass fluctuation\namplitude for spheres of size 8h−1Mpc, σ8, the total matter density at present time, Ωm, the S8\nparameter defined as S8≡σ8p\nΩm/0.3, and the Hubble expansion constant, H0. In Section 6.2\nwe will focus on the standard six-parameter ΛCDM and the corresponding coupled quintessence\nextensions. These results will then be compared to the case where the flatness assumption is\nrelaxed and ΩKis let to vary, as reported in Section 6.3. The range of the flat priors for the\nsampled cosmological and model parameters is listed in Table 6.2.6Coupled Quintessence in Flat and Curved Geometries 123\nModel Parameter Prior\nAllΩbh2[0.005,0.1]\nΩcdmh2[0.001,0.99]\n100θs [0.5,10]\nτreio [0.01,0.8]\nns [0.7,1.3]\nlog\u0000\n1010As\u0001\n[1.7,5.0]\nAll with ΩK= 0andΩK̸= 0 ΩK [−0.3,0.3]\nM1+,M1−,M3+andM3−λ [0,10]\nM2+andM2−µ [0,10]\nM1+,M1−,M2+andM2−α [0,2]and[−2,0]\nM3+andM3−γ [0,2]and[−2,0]\nM3+andM3−ϕini [0,4]\nTable 6.2 : Flat priors on the cosmological and model parameters sampled in this work, as\ndiscussed in Section 6.1.4.\n6.2 Update on Flat Coupled Quintessence\nThis section systematically reviews and interprets the findings based on the approach described\nin Section 6.1.4 for the flat ΩK= 0case. More precisely, we allocate individual subsections\nfor each model examined in this study. Each subsection offers a summary table detailing the\ncosmological parameters at a 68%confidence level (CL), a graphical representation illustrating\nthe68%and95%CL marginalised probability contours, and one-dimensional marginalised pos-\nterior distributions. The primary purpose of this analysis is to provide an update on coupled\nquintessence models and to set the stage for the comparison with the extended models in which\nthe curvature parameter is free to vary and not fixed to the flat case ΩK= 0.\nThe results of our data analysis are listed in Table 6.3-Table 6.8 where we report the parameter\nconstraints for the ΛCDM, M 1+and M 1−, M2+and M 2−, and M 3−models according to the\nPl18, Pl18 + BAO, Pl18 + SN, and Pl18 + BAO + SN data set combinations for each model.\nIn Figures 6.1, 6.3 and 6.5, we show the 1D marginalised posterior distributions for the mode-\nspecific parameters in each case, with the left and right panels corresponding to the positive and\nnegative coupling parameter cases, respectively. In Figures 6.2, 6.4 and 6.6, we display the 2D\nmarginalised posterior distributions for the conformal coupling parameter in each model (the124 6.2. Update on Flat Coupled Quintessence\nabsolute value) where once again the positive and negative coupling cases are displayed in the\nleft and right panels, respectively. We present the numerical predictions for the ΛCDM model\nin our methodology in Table 6.3 for comparison purposes and provide the main conclusions in\nbullet points in each section.\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nΩbh20.02237 ±0.00015 0 .02242 ±0.00013 0 .02239 ±0.00014 0 .02243 ±0.00013\nΩcdmh20.1202±0.0014 0 .1193±0.0010 0 .1199±0.0013 0 .11923 ±0.00097\n100θs 1.04188 ±0.00029 1 .04197 ±0.00028 1 .04192 ±0.00029 1 .04196 ±0.00028\nτreio 0.0547+0.0071\n−0.0080 0.0557+0.0072\n−0.0081 0.0551±0.0079 0 .0560±0.0079\nns 0.9654±0.0044 0 .9673±0.0038 0 .9660±0.0042 0 .9676±0.0037\nln\u0000\n1010As\u0001\n3.046±0.016 3 .046±0.016 3 .046±0.016 3 .047±0.017\nσ8 0.8118±0.0075 0 .8094±0.0072 0 .8109±0.0075 0 .8093±0.0073\nΩm 0.3160±0.0084 0 .3107±0.0060 0 .3141±0.0079 0 .3101±0.0058\nS8 0.833±0.016 0 .824±0.013 0 .830±0.016 0 .823±0.013\nH0 67.33±0.60 67 .70±0.44 67 .47±0.57 67 .75±0.43\nTable 6.3 : Observational constraints at 68%confidence level on the sampled and derived cos-\nmological parameters for different data set combinations under the flat ΛCDMmodel.\n6.2.1 M1 Flat: Exponential Conformal Factor and Exponential Potential\n0.000.040.080.12\n0123\nModel 1+\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\n0.000.030.060.09\n||\n0.00.51.01.52.0\nModel 1\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\nFigure 6.1 : 1D marginalised posterior distributions of the coupling parameter, α, and slope of\nthe potential, λ, in the M 1+model (upper panel) and M 1−model (lower panel) using the Pl18\n(red), Pl18+BAO (blue), Pl18+SN, (green) and Pl18+BAO+SN (yellow) data set combinations.6Coupled Quintessence in Flat and Curved Geometries 125\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nΩbh20.02235 ±0.00015 0 .02239 ±0.00015 0 .02236 ±0.00015 0 .02238 ±0.00014\nΩcdmh20.1193+0.0026\n−0.0015 0.1190±0.0011 0 .1182+0.0022\n−0.0015 0.1188±0.0010\n100θs 1.04185 ±0.00030 1 .04190 ±0.00029 1 .04187 ±0.00030 1 .04190 ±0.00029\nτreio 0.0548+0.0071\n−0.0081 0.0551+0.0073\n−0.0083 0.0553±0.0080 0 .0555±0.0080\nns 0.9662±0.0047 0 .9670±0.0040 0 .9667±0.0042 0 .9667±0.0040\nln\u0000\n1010As\u0001\n3.047+0.015\n−0.016 3.046±0.017 3 .047±0.016 3 .047±0.017\nλ < 1.26 0 .81+0.41\n−0.60 <0.510 <0.477\nα < 0.0464 0 .035±0.018 0 .038+0.017\n−0.030 0.029±0.016\nσ8 0.801+0.036\n−0.024 0.807+0.018\n−0.016 0.826+0.011\n−0.022 0.818+0.011\n−0.015\nΩm 0.333+0.022\n−0.045 0.3190+0.0098\n−0.018 0.301+0.018\n−0.011 0.3068+0.0084\n−0.0074\nS8 0.839±0.022 0 .831+0.014\n−0.016 0.826±0.015 0 .827±0.013\nH0 65.8+3.8\n−2.3 66.8+1.8\n−0.94 68.59+0.88\n−1.6 68.02+0.62\n−0.74\n∆χ2\nmin +0.80 +0 .12 +0 .88 +0 .54\nlnBM1+,ΛCDM −5.51 −5.31 −6.60 −6.40\nTable 6.4 : Observational constraints at 68%confidence level on the sampled and derived cos-\nmological parameters for different data set combinations under the flat M 1+model, studied in\nSection 6.2.1.\n•Coupling: In Tables 6.4 and 6.5, we observe that while Pl18 data on its own accommodates\na vanishing coupling constant αin the 68%CL region for the M 1models, incorporating\nadditional individual background data sets, BAO or SN in this case, yields higher support\nfor the dark sector interaction in both M 1+and M 1−models ( i.e.|α|>0in the 1σ\nregion), associated with larger mean values in general. The exception is in the M 1−model\nwhen adding SN data (Pl18+SN in Table 6.5), which results in tighter bounds. For the\nM1+model, we report bounded constraints for αat1σwith all combinations involving\nadditional background data. For M 1−, we find similar bounds for |α|with Pl18+BAO\nand Pl18+BAO+SN. Figure 6.1 illustrates this trend through the marginalised posterior\ndistributions, where additional datasets result in a non-zero peak for |α|, with BAO being\nthe decisive driver of the peak away from zero, possibly by setting tighter constraints\nonΩm. More precisely, with the Pl18 data set alone the following upper bounds are\nfound at 68%CL:α <0.0464for the M 1+model and |α|<0.0322for M 1−. For the full\nPl18+BAO+SNdatasetcombinationwereportat 1σ:α= 0.029±0.016fortheM 1+model\nand|α|= 0.030±0.016for M 1−. In Figure 6.2, we depict the 2D marginalised posterior126 6.2. Update on Flat Coupled Quintessence\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nΩbh20.02235 ±0.00015 0 .02239 ±0.00015 0 .02237 ±0.00015 0 .02239 ±0.00015\nΩcdmh20.1185+0.0027\n−0.0015 0.1178+0.0018\n−0.0012 0.1179+0.0026\n−0.0015 0.1179+0.0016\n−0.0011\n100θs 1.04187 ±0.00030 1 .04191 ±0.00029 1 .04189 ±0.00030 1 .04190 ±0.00028\nτreio 0.0550±0.0080 0 .0554±0.0081 0 .0547+0.0071\n−0.0080 0.0554±0.0077\nns 0.9656±0.0044 0 .9671±0.0039 0 .9666±0.0043 0 .9670±0.0040\nln\u0000\n1010As\u0001\n3.047±0.016 3 .047±0.017 3 .046±0.016 3 .047±0.016\nλ < 0.829 <0.659 <0.364 <0.408\nα > −0.0322 −0.028+0.022\n−0.013 >−0.0409 −0.030±0.016\nσ8 0.808±0.023 0 .811±0.015 0 .820+0.010\n−0.018 0.817+0.011\n−0.015\nΩm 0.321+0.021\n−0.024 0.3110+0.0094\n−0.011 0.304+0.016\n−0.010 0.3052+0.0087\n−0.0079\nS8 0.834±0.018 0 .825±0.013 0 .825±0.016 0 .824±0.013\nH0 66.5+2.2\n−1.8 67.3+1.1\n−0.80 68.14+0.77\n−1.3 67.98±0.69\n∆χ2\nmin +0.34 +0 .18 +1 .14 +0 .24\nlnBM1−,ΛCDM −6.03 −6.03 −6.97 −6.77\nTable 6.5 : Observational constraints at 68%confidence level on the sampled and derived cos-\nmological parameters for different data set combinations under the flat M 1−model, studied in\nSection 6.2.1.\ncontours for αagainst relevant cosmological and model parameters: {H0, σ8,Ωm, λ}. For\nboth the M 1+and M 1−models, positive correlations are evident in the {|α|, H0}and\n{|α|, σ8}planes, while a negative correlation is observed in the {|α|,Ωm}plane. The scalar\nfield parameters |α|andλare positively correlated for positive and negative coupling cases,\nexcept in the Pl18-only case in M 1−, where it is unclear whether any correlation exists.\n•Potential: In Tables 6.4 and 6.5, we also report on the slope of the potential λ, which\nis consistently constrained only from above in the M 1models, except when using the\nPl18+BAO dataset for the M 1+model, in which case we find a bounded 1σconstrained\nregion. Incorporation of background datasets brings λcloser to zero, with higher values\nofλallowed in the M 1+model in general. This effect is depicted in Figure 6.1 and seems\nmostly influenced by the SN data. For Pl18 we report λ < 1.26andλ < 0.829at68%\nCL for M 1+and M 1−, respectively, while the full Pl18+BAO+SN data set combination\nrestricts λto<0.477for M 1+and<0.408for M 1−.\n•H0Tension: The positive correlation between |α|andH0aids in mitigating the H0tension\nof∼4.8σinΛCDMwhenconsideringPl18dataalone. Thetensionisapparentlyattenuated6Coupled Quintessence in Flat and Curved Geometries 127\n556575\nH00.000.050.100.15\n0.70.80.9\n8\n0.20.30.40.5\nm\n0123\nModel 1+\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\n606570\nH00.000.050.10||\n0.750.800.85\n8\n0.260.300.340.38\nm\n0.00.51.01.52.0\nModel 1\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\nFigure 6.2 : 2D marginalised posterior distributions of parameters in the M 1+model (upper\npanel) and M 1−model (lower panel) using the Pl18 (red), Pl18+BAO (blue), Pl18+SN, (green)\nand Pl18+BAO+SN (yellow) data set combinations. We plot the 2D marginalised posterior\ndistributions of the conformal coupling parameter, α, against the slope of the potential, λ,\nthe Hubble constant in units km s−1Mpc−1,H0, the present-day mass fluctuation amplitude\nin spheres of radius 8h−1Mpc,σ8, and the total matter density parameter, Ωm. The shaded\ncontours indicate the 1σand2σconfidence limits.\nto∼2.5σin M 1+and to ∼2.9σin the M 1−model. It is worth noting that the mean\nvalues of H0in the M 1models, as reported in the second columns of Tables 6.4 and 6.5 for\nPl18, are smaller than in the ΛCDM model. The tension alleviation is attributed to the\nbroader error bars for H0, a direct consequence of the added number of parameters that\nprevent H0from being as precisely constrained as in the ΛCDM case.\n•S8Tension: While |α|exhibitsapositivecorrelationwiththeparameter σ8, themeanvalues\nofσ8reported for the M 1models turn out to be smaller than their ΛCDM counterpart for\nPl18 data. Nonetheless, the 68%CL region is broader in the M 1models. A suppressed\nσ8is offset by an enhanced Ωmwhen considering the Pl18 data, balancing the constraints\nderived for the S8parameter. For M 1+the1σregion for S8value is found at 0.839±0.022\nand at 0.834±0.018for M 1−. These values do not deviate far from the ΛCDM prediction\nof0.833±0.016.\n•Model Evidence: To conclude, the logarithm of the Bayes factor, lnBM1,ΛCDMis the128 6.2. Update on Flat Coupled Quintessence\ndecisive criteria for any potential support in the extended models. We find a negative\nvalue for all the data combinations, suggesting no statistical evidence for the M1 coupled\nquintessence models over ΛCDM. M 1+has a marginally less negative Bayes factor than\nM1−, indicating a slight but unimportant preference for positive values of αover negative\nones. The ∆χ2\nminis positive in all the cases considered, suggesting a worse fit to the data,\nwhich adds to the preference over a smaller parameter space in ΛCDM.\n6.2.2 M2 Flat: Exponential Conformal Factor and Inverse Power-law Poten-\ntial\n•Coupling: As evident from Table 6.6, for the M 2+model, a non-zero coupling at 68%CL is\ndetected when BAO and BAO+SN data set combinations are added to the Pl18 data, more\nprecisely α= 0.025+0.012\n−0.020andα= 0.026+0.014\n−0.019, respectively. For the M 2+model, reported\nin Table 6.7, αis bounded at 68%CL with the addition of the BAO and BAO+SN data set\ncombinationsaswell, namely α=−0.025+0.021\n−0.011andα=−0.027±0.015, respectively. While\ntheuncoupledcase α= 0iscontainedinthe 1σregionwhenPl18dataisconsideredbyitself\nand for the Pl18+SN case, the introduction of BAO and BAO+SN is again responsible for\nexcluding a zero coupling at 1σ. Close inspection of Figure 6.3 shows how the inclusion of\nthe BAO data originates a non-null peak for |α|. This trend is maintained in both M 2+and\nM2−when SN data is included. The magnitude of the coupling parameter is comparable\nin both cases. Figure 6.4 shows 2D marginalised posterior distributions of the conformal\ncoupling parameter αagainst {H0, σ8,Ωm, µ}, just as we did for the M 1models,. Both M 2\nmodels exhibit a positive correlation in the {|α|, H0}and{|α|, σ8}planes, while showing\na negative correlation in the {|α|,Ωm}plane instead. Unlike the M 1models, the coupling\nand potential parameters are not seemingly correlated.\n•Potential: The power of the inverse power potential, µ, is only bounded from above with\n68%CL in M 2models across any data set combinations. As shown in Table 6.7, adding\nbackground data sets scarcely tightens the constraint on µin the M 2models. Pl18 data\nalone limits µto be larger than 0.459, while the full Pl18+BAO+SN data set sets this\nlower limit at 0.463, both at 1σ. A parallel constraint of µ <0.502andµ <0.451at1σ\nis observed for the M 2+model with Pl18 and Pl18+BAO+SN. In Figure 6.3, the similar\nbehaviour of µdistributions between the two M 2models is noticeable. The values of |α|\nandµfall within similar ranges in both models when considering the full Pl18+BAO+SN\ndata set.\n•H0Tension: The H0tensionisalleviatedtosomeextentbythepositivecorrelationbetween\n|α|andH0, similar to the M 1models. Using Pl18 data, the M 2+and M 2−models lessen\nthe tension from approximately 4.8σto∼4.0σand2.8σ, respectively, which is a less\nsignificant reduction compared to the M 1models. Notably, contrary to the M 1models,6Coupled Quintessence in Flat and Curved Geometries 129\nthe mean H0value in the M 2−model is higher, yet with smaller error bars, making it less\neffective in reducing the tension, according to Equation (3.6.1).\n•S8Tension: In contrast to M 1, higher average values for the σ8parameter are obtained in\nthe M 2models compared to ΛCDM for all data set combinations. We also report consis-\ntently lower mean values of Ωm. The increase in σ8and the decrease in Ωmapproximately\nbalance out again, resulting in an S8value of 0.828±0.018at1σfor the M 2+model, and\n0.827±0.018for M 2−, both not significantly different from the ΛCDM value assuming\nPl18 data.\n•ModelEvidence: Tables6.6and6.7revealthatthelogarithmoftheBayesfactor, lnBM2,ΛCDM,\nis negative for all the data sets, suggesting a lack of evidence supporting the M 2models\nover the ΛCDM model. As was the case for the M 1models, there is an overall trend of pos-\nitive values of ∆χ2\nmin(except for Pl18 in M 2+and Pl18+SN and Pl18+BAO+SN in M 2−)\nwhich is less pronounced and, consistently, the fit to the data is not sufficiently improved to\njustify the addition of extra parameters. Nevertheless, it should be noted that the results\nreported in Tables 6.6 and 6.7 include the sampling of ϕini, which could in principle be\nfixed without loss of generality, decreasing the prior volume.\n0.000.030.060.09\n0.00.51.01.52.02.5\nModel 2+\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\n0.000.030.060.09\n||\n0.00.51.01.52.0\nModel 2\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\nFigure 6.3 : 1D marginalised posterior distributions of the coupling parameter, α, and slope of\nthe potential, µ, in the M 2+model (upper panel) and M 2−model (lower panel) using the Pl18\n(red), Pl18+BAO (blue), Pl18+SN, (green) and Pl18+BAO+SN (yellow) data set combinations.\n6.2.3 M3 Flat: Coupling with Minimum/Maximum and Exponential Poten-\ntial\n•Initial Conditions: The M 3model differs more significantly from M 1and M 2due to the\nnon-trivial form of the coupling, which is no longer a constant but depends directly on the\nvalue of the scalar field. For this reason, we chose to leave the initial condition for the130 6.2. Update on Flat Coupled Quintessence\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nΩbh20.02236 ±0.00015 0 .02238 ±0.00014 0 .02237 ±0.00015 0 .02238 ±0.00015\nΩcdmh20.1185+0.0028\n−0.0016 0.1190±0.0011 0 .1185+0.0021\n−0.0014 0.1188±0.0011\n100θs 1.04186 ±0.00030 1 .04190 ±0.00029 1 .04189 ±0.00029 1 .04191 ±0.00029\nτreio 0.0546±0.0079 0 .0550+0.0073\n−0.0082 0.0546±0.0082 0 .0554±0.0079\nns 0.9660±0.0045 0 .9663±0.0038 0 .9664±0.0041 0 .9666±0.0039\nln\u0000\n1010As\u0001\n3.046±0.016 3 .047+0.015\n−0.017 3.046±0.017 3 .047±0.016\nµ < 0.502 <0.500 <0.450 <0.451\nα < 0.0396 0 .025+0.012\n−0.020 <0.0387 0 .026+0.014\n−0.019\nσ8 0.823+0.014\n−0.021 0.818+0.010\n−0.014 0.8232+0.0086\n−0.018 0.8195+0.0091\n−0.014\nΩm 0.305+0.020\n−0.013 0.3068±0.0083 0 .303+0.016\n−0.0093 0.3051+0.0079\n−0.0070\nS8 0.828±0.018 0 .827±0.013 0 .826±0.015 0 .826±0.013\nH0 68.3+1.0\n−1.7 68.05±0.70 68 .44+0.67\n−1.4 68.19+0.52\n−0.67\n∆χ2\nmin −0.68 0 .12 0 .66 0 .52\nlnBM2+,ΛCDM −6.55 −6.63 −6.45 −6.71\nTable 6.6 : Observational constraints at 68%confidence level on the sampled and derived cos-\nmological parameters for different data set combinations under the flat M 2+model, studied in\nSection 6.2.2.\nscalar field ϕinias a free parameter. This field dependence originates a minimum and a\nmaximum of the effective potential at ϕ∗for M 3+and M 3−, respectively. For the sampling\nprocess, this means that the scalar field will prefer to sit at the minimum of the potential\nin M3+, leading to an unphysical highly peaked posterior distribution for ϕiniwhich spoils\nthe results, namely through an unreasonably large steep peak for the coupling posterior of\nγ. For this reason, we have opted for presenting the results only for M 3−with varying ϕini,\nlisted in Table 6.8. This means that we only consider the case in which the potential has a\nmaximum and the scalar field does not get stuck into a minimum.The features introduced\nby the varying initial conditions can be better appreciated by comparing both models. In\nFigure 6.5, we see a clear peak close to the maximum, which is not symmetric, with values\nϕini< ϕ∗= 2MPlbeing favoured. This motivates setting ϕini= 1MPlfor M 3+in furter\nstudies.\n•Coupling: InTable6.8, weverifythatallthedatacombinationscanaccommodateavanish-\ning coupling constant γin the 68%CL region for the M 3−model, with only upper bounds\nfor|γ|being reported. Figure 6.3 illustrates this trend through the marginalised posterior6Coupled Quintessence in Flat and Curved Geometries 131\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nΩbh20.02237 ±0.00015 0 .02238 ±0.00014 0 .02236 ±0.00015 0 .02238 ±0.00014\nΩcdmh20.1183+0.0030\n−0.0014 0.1188±0.0012 0 .1185+0.0021\n−0.0013 0.1186+0.0012\n−0.0011\n100θs 1.04187 ±0.00030 1 .04191 ±0.00028 1 .04187 ±0.00029 1 .04191 ±0.00028\nτreio 0.0550+0.0072\n−0.0084 0.0553±0.0081 0 .0551±0.0076 0 .0547±0.0074\nns 0.9663±0.0045 0 .9663±0.0040 0 .9662±0.0041 0 .9661±0.0038\nln\u0000\n1010As\u0001\n3.047±0.016 3 .047±0.016 3 .047±0.016 3 .046±0.015\nµ < 0.459 <0.484 <0.458 <0.463\nα > −0.0385 −0.025+0.021\n−0.011 >−0.0379 −0.027±0.015\nσ8 0.8240+0.0083\n−0.021 0.8185+0.0096\n−0.013 0.8229+0.0092\n−0.018 0.8194+0.0092\n−0.015\nΩm 0.303+0.022\n−0.010 0.3063±0.0084 0 .304+0.016\n−0.0086 0.3045+0.0085\n−0.0073\nS8 0.827±0.018 0 .827±0.013 0 .827±0.016 0 .825±0.013\nH0 68.42+0.72\n−1.8 68.05±0.71 68 .32+0.62\n−1.3 68.21+0.58\n−0.71\n∆χ2\nmin +1.14 +0 .06 −0.32 −2.12\nlnBM2���,ΛCDM −6.61 −6.51 −9.88 −3.57\nTable 6.7 : Observational constraints at 68%confidence level on the sampled and derived cos-\nmological parameters for different data set combinations under the flat M 2−model, studied in\nSection 6.2.2.\ndistributions, wherethereisonlyasmallhintofBAOdatadrivingthepeakawayfromzero.\nMore precisely, with the Pl18 data set alone, the following upper bound is found at 68%\nCL:|γ|<0.0683for the M 3−model. For the full Pl18+BAO+SN data set combination, we\nreport at 1σ:|γ|<0.0549. In Figure 6.6, we depict the 2D marginalised posterior contours\nfor|γ|against relevant cosmological and model parameters: {H0, σ8,Ωm, λ}. Due to the\nadded number of parameters, there is less evidence of correlations, with arguably slightly\npositive correlations in the {|γ|, H0},{|γ|, σ8}and (less robust) in the {|γ|, λ}plane, and\nslightly negative in the {|γ|,Ωm}plane.\n•Potential: In Table 6.8, we also report on the slope of the potential λ, which, just as in\nthe M 1models, is consistently constrained only from above. Incorporation of background\ndatasets yields tighter constraints on λcompared to Pl18. This effect is depicted in Fig-\nure 6.5, where there seems to be a minute shift of the peak away from zero with the addition\nof BAO data. For Pl18, we report λ <0.925at68%CL, while the full Pl18+BAO+SN\ndata set combination is more restrictive, namely λ <0.406.\n•H0Tension: The H0tension is alleviated to some extent by the enlarged error bars in M 3−,132 6.2. Update on Flat Coupled Quintessence\n60657075\nH00.000.050.10\n0.760.820.88\n8\n0.220.260.300.34\nm\n0.00.51.01.52.02.5\nModel 2+\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\n65687174\nH00.000.050.10||\n0.800.840.88\n8\n0.250.300.35\nm\n0.00.51.01.5\nModel 2\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\nFigure 6.4 : 2D marginalised posterior distributions of parameters in the M 2+model (upper\npanel) and M 2−model (lower panel) using the Pl18 (red), Pl18+BAO (blue), Pl18+SN, (green)\nand Pl18+BAO+SN (yellow) data set combinations. We plot the 2D marginalised posterior\ndistributions of the conformal coupling parameter, α, against the power of the potential, µ,\nthe Hubble constant in units km s−1Mpc−1,H0, the present-day mass fluctuation amplitude\nin spheres of radius 8h−1Mpc,σ8, and the total matter density parameter, Ωm. The shaded\ncontours indicate the 1σand2σconfidence limits.\nas no significant correlation seems to exist between |γ|andH0, while a positive relation\nhad been identified in the M 1and M 2models. Using Pl18 data, the M 3−model lessens\nthe tension from ∼4.8σto∼3.2σwhich is a less significant reduction compared to the\nM1and M 2models since the mean value of H0is also consistently smaller, making it less\neffective in reducing the tension.\n•S8Tension: In analogy to the M 1models, slightly smaller values of σ8are reported for\nM3−compared to ΛCDM for all data set combinations, compensated by consistently lower\nmean values of Ωm. Again, the interplay between σ8andΩmapproximately balances out,\nresulting in a S8value of 0.839+0.018\n−0.021at1σfor the M 3−, not significantly different from the\nΛCDM value assuming Pl18 data.\n•Model Evidence: Table 6.7 reveals that the logarithm of Bayes factor, lnBM3,ΛCDM, is once\nagain negative for all the data sets, implying a lack of evidence supporting the M 3−model6Coupled Quintessence in Flat and Curved Geometries 133\noverΛCDM. There is a minute overall trend of positive values of ∆χ2\nmin, with the exception\nbeing the Pl18+SN, which when compared with the Bayesian evidence, is not sufficient to\njustify the addition of extra parameters.\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nΩbh20.02235 ±0.00015 0 .02239 ±0.00015 0 .02239 ±0.00014 0 .02242 ±0.00014\nΩcdmh20.1192+0.0023\n−0.0015 0.1187+0.0015\n−0.0011 0.1187+0.0019\n−0.0013 0.1183+0.0016\n−0.00099\n100θs 1.04184 ±0.00030 1 .04193 ±0.00029 1 .04191 ±0.00029 1 .04191 ±0.00031\nτreio 0.0552±0.0079 0 .0555±0.0079 0 .0552±0.0078 0 .0545±0.0074\nns 0.9657±0.0044 0 .9666±0.0042 0 .9663±0.0042 0 .9673±0.0039\nln\u0000\n1010As\u0001\n3.047±0.017 3 .047±0.016 3 .046±0.016 3 .045±0.016\nλ < 0.925 <0.797 <0.369 <0.406\nγ > −0.0683 >−0.0645 >−0.0459 >−0.0549\nϕini 1.64+0.61\n−0.87 1.69+0.67\n−0.74 1.81±0.87 1 .92±0.84\nσ8 0.805+0.029\n−0.019 0.810±0.017 0 .8164+0.0092\n−0.014 0.8143+0.0098\n−0.014\nΩm 0.328+0.012\n−0.032 0.315+0.011\n−0.016 0.308+0.013\n−0.0086 0.3067+0.0092\n−0.0062\nS8 0.839+0.018\n−0.021 0.830+0.014\n−0.017 0.827±0.015 0 .823+0.014\n−0.013\nH0 66.1+2.8\n−1.0 67.1+1.5\n−0.79 67.89+0.66\n−1.0 67.91+0.54\n−0.73\n∆χ2\nmin +1.84 +0 .5 −0.2 +0 .32\nlnBM3−,ΛCDM −6.34 −5.94 −7.29 −7.27\nTable 6.8 : Observational constraints at 68%confidence level on the sampled and derived cos-\nmological parameters for different data set combinations under the flat M 3−model, studied in\nSection 6.2.3.\n6.3 Coupled Quintessence in Non-Flat Geometries\nWhile large-scale observations have historically suggested that our Universe is nearly homoge-\nneous, isotropic, and spatially flat, recent data has called this into question. Studies on the CMB\ntemperature and polarisation spectra, particularly from the Planck2018 collaboration, hint at a\nclosed geometric structure for the Universe. Thus, observations implying a minimal prediction\nfor the spatial curvature of the Universe cannot be taken as irrefutable evidence for spatial flat-\nness. In this section, we aim to thoroughly assess the validity of coupled quintessence models by\ntreating the Universe’s curvature as a variable parameter and how it influences the cosmological\ntensions, compared with the flat case studied in Section 6.3.134 6.3. Coupled Quintessence in Non-Flat Geometries\n0.00.10.20.3\n||\n0123\n01234\nini\nModel 3\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\nFigure 6.5 : 1D marginalised posterior distributions of the coupling parameter, γ, and slope of\nthe potential, λ, in the M 3−model using the Pl18 (red), Pl18+BAO (blue), Pl18+SN, (green)\nand Pl18+BAO+SN (yellow) data set combinations.\n5560657075\nH00.00.10.20.30.4||\n0.70.80.9\n8\n0.250.350.45\nm\n0123\nModel 3\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\nFigure 6.6 : 2D marginalised posterior distributions of parameters in the M 3−model using\nthe Pl18 (red), Pl18+BAO (blue), Pl18+SN, (green) and Pl18+BAO+SN (yellow) data set\ncombinations. We plot the 2D marginalised posterior distributions of the conformal coupling\nparameter, γ, against the slope of the potential, λ, the Hubble constant in units km s−1Mpc−1,\nH0, the present-day mass fluctuation amplitude in spheres of radius 8h−1Mpc,σ8, and the total\nmatter density parameter, Ωm. The shaded contours indicate the 1σand2σconfidence limits.\nIn the following sections, we analyse the different extensions of the curved ΛCDM model. We\nmake use of the same data set combinations as in the flat cases for all the extensions. In the\nTables 6.9 to 6.15 we provide the bounds at 68%CL (1σ) and comment on the corresponding\n95%(2σ) and 99%(3σ) CL in the text when relevant for the discussion.\n6.3.1 ΛCDM + Ω K\nTable 6.9 presents the constraints on the ΛCDM + ΩKscenario for various datasets. The second\ncolumn of Table 6.9 shows the constraints using CMB data alone, and we remark a strong\nevidence for a closed Universe at over 99%confidence level: ΩK=−0.052+0.041\n−0.053. The Hubble\nconstant is significantly lower in this case ( H0= 52.8+3.2\n−4.1km/s/Mpc at 68%CL), thereby greatly6Coupled Quintessence in Flat and Curved Geometries 135\nexacerbating the Hubble tension. Additionally, due to a strong correlation among H0,ΩK, and\nΩm, the matter density Ωmtakes extremely large values, also increasing the estimate of S8in\nthis scenario.\nWhen BAO data are combined with CMB data, as shown in the third column of Table 6.9, ΩK\naligns with a spatially flat Universe, and the Hubble constant rises to the standard flat ΛCDM\nPlanckvalue. However, this concordance comes from a dataset combination that is in disagree-\nment at over 3σ[181, 182, 192], calling into question the reliability of the Pl18+BAO results.\nThe last two columns of Table 6.9 report on the outcomes for Pl18+SN and Pl18+BAO+SN\nwhich are fairly similar, except for a slightly lower mean value of H0in the Pl18+SN case\n(H0= 65 .7+1.9\n−3.3km/s/Mpc at 68%CL), but with larger error bars. In both instances, spatial\nflatness is consistent with the data.\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nωb 0.02261 ±0.00017 0 .02239 ±0.00015 0 .02246 ±0.00016 0 .02240 ±0.00015\nωcdm 0.1180±0.0015 0 .1199±0.0014 0 .1192±0.0015 0 .1197±0.0014\n100θs 1.04206 ±0.00031 1 .04193 ±0.00030 1 .04196 ±0.00031 1 .04194 ±0.00030\nτreio 0.0484±0.0082 0 .0556±0.0078 0 .0556±0.0078 0 .0550±0.0079\nns 0.9714±0.0049 0 .9662±0.0045 0 .9678±0.0048 0 .9664±0.0045\nln\u0000\n1010As\u0001\n3.028±0.017 3 .047±0.016 3 .046±0.016 3 .046±0.016\nΩK −0.052+0.020\n−0.017 0.0009±0.0020 −0.0051+0.0057\n−0.0078 0.0010±0.0019\nσ8 0.769±0.016 0 .8118±0.0082 0 .8060+0.0094\n−0.011 0.8110±0.0082\nΩm 0.514+0.065\n−0.073 0.3099±0.0067 0 .332+0.030\n−0.021 0.3086±0.0065\nS8 1.003+0.054\n−0.047 0.825±0.013 0 .846+0.035\n−0.022 0.822±0.013\nH0 52.8+3.2\n−4.1 67.92±0.69 65 .7+1.9\n−3.3 68.03±0.67\nTable 6.9 : Observational constraints at 68%confidence level on the sampled and derived cos-\nmological parameters for different data set combinations under the curved ΛCDM + Ω Kmodel,\nstudied in Section 6.3.1.\n6.3.2 M1 + Ω K: Exponential Conformal Factor and Exponential Potential\nThe first extension of the non-flat ΛCDM framework that we explore incorporates Model M 1\nas introduced in Section 6.1.3, split into the positive and negative coupling parameters variants:\nM1++ Ω Kand M 1−+ Ω K. The derived parameter constraints are presented in Tables 6.10\nand 6.11 and Figures 6.7 and 6.8 display the 1D and 2D marginalised posterior probability\ndistributions, respectively.136 6.3. Coupled Quintessence in Non-Flat Geometries\nIn summary, when only relying on the Planck satellite’s CMB temperature and polarisation data\n(Pl18), a preference for a curved cosmological space emerges at over 99%CL, constraining the\ncurvature parameter to ΩK=−0.059+0.039\n−0.039andΩK=−0.059+0.045\n−0.062at95%CL and 99%CL for\nM1++ΩKandΩK=−0.069+0.041\n−0.044andΩK=−0.069+0.053\n−0.051at95%CLand 99%CLforM 1−+ΩK.\nThis strong preference for a closed Universe vanishes when combining the Planck data with BAO,\nfavouring spatial flatness within one standard deviation. Concerning the current expansion rate\nH0, as depicted in Figure 6.8, the mild positive correlation with the coupling parameter results in\neven lower preferred H0values, yielding H0= 50.9±5.2km/s/Mpc and H0= 48.5+4.5\n−5.8km/s/Mpc\nfor M 1++ΩKand M 1−+ΩKat68%CL, which sharply contrasts with the late-time independent\nmeasurements of H0, diverging at approximately 4.2σand4.7σ.\nWhen adding the background data sets, the statistical tension persists. For instance, the H0\nvalue inferred from the Pl18 and BAO combination (third column in Tables 6.10 and 6.11) is\n66.5+2.0\n−1.5km/s/Mpc and 67.5+1.2\n−1.0km/s/Mpc at 68%CL for M 1++ ΩKand M 1−+ ΩK, closely re-\nsemblingtheflat ΛCDMresult. Also, thematterdensityparameter Ωmnegativelycorrelateswith\nH0, amplifying the tension in the S8parameter. Compared to Pl18-only and Pl18+BAO+SN\ndatasets, this tension reaches roughly 3−4σ. Interestingly, the Pl18+SN case is consistent with\na closed Universe at more than 95%CL.\nRegarding the coupling parameter, the Pl18 data alone offers an upper limit of α < 0.100\nand|α|<0.0934at95%CL for M 1++ Ω Kand M 1−+ Ω K, which becomes less stringent,\nα= 0.068+0.052\n−0.060and|α|<0.110at95%CL for M 1++ Ω Kand M 1−+ Ω Kwhen BAO and SN\ndata are also included. This is in contrast to the flat case, where the upper limit is |α|<0.0547\nand|α|<0.0560for Pl18+BAO+SN at 95%CL for M 1+and M 1−. Thus, assuming spatial\nflatness imposes more restrictive constraints on the traditional coupled quintessence model.\nFor comparison purposes, we list below a more detailed account of each effect.\n•Coupling: Adding the varying curvature parameter to the M 1 + Ω Kmodels, results in\na non-vanishing prediction for αat1σfor all the data sets except Pl18 in M 1++ Ω K\nand for Pl18+SN and Pl18+BAO+SN in M 1−+ Ω K, as can be confirmed in Tables 6.10\nand 6.11. For instance, for the full combination we report α= 0.068+0.033\n−0.028andα=\n−0.058±0.32for M 1++ ΩKand M 1−+ ΩK, respectively. On the other hand, for the Pl18\ndata alone the conditions are relaxed to upper bounds: α <0.0515and|α|<0.0537for\nM1++ Ω Kand M 1−+ Ω K, respectively. Hence, the magnitude of the coupling parameter\nis comparable in both cases. As was the case in the flat scenario, the background data\nbrings the peak in the |α|marginalised posterior distribution away from zero, as depicted\nin Figure 6.7, with SN having the most defining impact, especially in the M 1−+ Ω Kcase,\nheavily suppressing the zero-peaked tail introduced by BAO, which pushes the maximum\nback for Pl18+BAO+SN. In Figure 6.8 we find the 2D marginalised posterior distributions\nofαagainst {H0, σ8,Ωm, λ,ΩK}. Both M 1 + Ω Kmodels exhibit a positive correlation in\nthe{|α|, λ}(except for the uncorrelated Pl18 case) and {|α|, σ8}planes. There is also a6Coupled Quintessence in Flat and Curved Geometries 137\nreduced positive correlation for {|α|, H0}with Pl18 only in M 1++ΩK. On the other hand,\nthere is a minor negative correlation for {|α|,ΩK}in both cases excluding the Pl18-only\ncase. This is possibly associated with the larger error bars in Ωm, which is constrained\ntowards more conservative values when the background data is added and does not appear\nto be correlated with α. Therefore, a more negative value of ΩKmust be compensated by\nan increased coupling, sourcing the DE component. The remaining combinations seem to\nnot be correlated.\n•Potential: The slope of the exponential potential λis only bounded from above with 68%\nCL in the Pl18 and Pl18+SN combinations in M 1++ΩKand across all cases in M 1−+ΩK.\nAs reported in Tables 6.10 and 6.11, the background data tightens the constraint on λin\nboth models, in particular, due to the constraining power of BAO data on Ωm, directly\ninfluenced and positively correlated with λ. Pl18 data alone limits λto be smaller than\n2.21and2.49, while the full Pl18+BAO+SN set brings this upper limit down to 0.54(with\nzero not included at 1σ) and 0.435, for M 1++ Ω Kand M 1−+ Ω K, respectively, both\nat1σ. In Figure 6.7, we confirm the constraining power of BAO over λ, responsible for\nsuppressing the left-hand side tail of the distribution, with SN pushing the peak towards\nzero. The values of |α|andλfall within similar ranges for all the data set combinations in\nboth models, with broader error bars for Pl18+SN.\n•H0Tension: The H0tension is mildly alleviated by the positive correlation between |α|\nandH0, similar to the flat M 1case. Using Pl18 data, the M 1++ΩKand M 1−+ΩKmodels\nlessen the tension from approximately 5.3σto∼4.2σand4.7σ, respectively, which is still\nalmost double the corresponding values for ΩK= 0. As happened in the flat case, the\nmean H0value in the M 1 + Ω Kmodels is lower than in ΛCDM + ΩK, yet with larger error\nbars, resulting in an apparent alleviation of the tension due to the larger posterior space.\nAdding background data (mostly the effect of BAO) brings the H0value closer to the more\nreasonable values reported in the flat cases. This is consistent with these data sets’ power\nin bringing ΩKclose to zero. Nevertheless, it should be noted that this represents yet\nanother tension between the CMB and background data sets, which give predictions that\nare incompatible at 99%CL.\n•S8Tension: Even though the prediction of a closed Universe when considering Pl18 alone\nhelps bring the value of σ8down, the S8tension remains due to the large values predicted\nforΩmin both the ΛCDM + Ω Kand M 1 + Ω Kmodels. By introducing the background\ndata, both σ8andΩmare restored to values compatible with the flat case. For Pl18 this\nresults in an S8value of 1.014+0.055\n−0.043at1σfor the M 1++ Ω Kmodel, and 1.026+0.052\n−0.037for\nM1−+ Ω K, both not significantly different from the ΛCDM + Ω Kvalue S8= 1.003+0.054\n−0.047.\n•Curvature: In line with what happens in the ΛCDM + Ω Kcase, there is a prediction138 6.3. Coupled Quintessence in Non-Flat Geometries\nfor a closed Universe at more than 99%CL for the M 1 + Ω Kmodels with Pl18 and\nPl18+SN. In particular, at 68%CL we find ΩK=−0.059+0.023\n−0.018for M 1++ Ω KandΩK=\n−0.069+0.024\n−0.022for M 1−+ Ω K, in contrast with ΩK=−0.052+0.020\n−0.017forΛCDM + Ω K. These\nvalues are themselves in tension with the Pl18+BAO+SN counterparts, which report ΩK=\n−0.0068+0.0071\n−0.0047for M 1++ ΩKandΩK=−0.0048+0.0069\n−0.0042for M 1−+ ΩK. This is in contrast\nwith ΩK= 0.0010±0.0019forΛCDM +ΩK, all consistent with ΩK= 0but with the mean\nvalue for ΛCDM actually depicting an open Universe. This shows the incompatibility of\npredictions between the different data sets in tension at ∼2.9σforΛCDM + Ω K,∼2.5σ\nfor M 1++ Ω Kand∼2.7σlevel for M 1−+ Ω K, already admitting that the data sets have\nbeen combined. This shows how care should be taken when deriving conclusions from their\ncombination.\n•Model Evidence: The lower sections of Tables 6.10 and 6.11 report on the ∆χ2\nminvalue\nfor the goodness of fit comparison and the logarithm of the Bayes factor, lnBM1,ΛCDMfor\nthe curved cases. We see that both model comparison criteria are significantly negative\n(except for ∆χ2\nmin>0for M 1−+ Ω Kwith Pl18+BAO+SN), yielding no support for the\nM1 + Ω Kmodels over ΛCDM + Ω K, but hinting at a better fit to the data.\n0.000.050.100.15\n012345\nModel 1++k\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\n0.000.050.100.15\n||\n012345\nModel 1+k\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\nFigure 6.7 : 1D marginalised posterior distributions of the coupling parameter, α, and slope of\nthe potential, λ, in the M 1++ Ω Kmodel (upper panel) and M 1−+ Ω Kmodel (lower panel)\nusing the Pl18 (red), Pl18+BAO (blue), Pl18+SN, (green) and Pl18+BAO+SN (yellow) data\nset combinations.\n6.3.3 M2 + Ω K: Exponential Conformal Factor and Inverse Power-law Po-\ntential\nIn Tables 6.12 and 6.13, we compile the parameter constraints for the M 2 + Ω Kmodels, and\nFigures 6.9 and 6.10 illustrate the corresponding marginalised 1D and 2D posterior distributions.\nObserving the Pl18 data in isolation, evidence for a closed Universe manifests at more than6Coupled Quintessence in Flat and Curved Geometries 139\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nωb 0.02261 ±0.00017 0 .02241 ±0.00016 0 .02255 ±0.00016 0 .02241 ±0.00016\nωcdm 0.1163+0.0041\n−0.0019 0.1137+0.0062\n−0.0040 0.1060+0.0049\n−0.0056 0.1141+0.0051\n−0.0034\n100θs 1.04205 ±0.00030 1 .04191 ±0.00030 1 .04197 ±0.00030 1 .04191 ±0.00030\nτreio 0.0488±0.0084 0 .0553±0.0079 0 .0521±0.0083 0 .0550±0.0078\nns 0.9729±0.0051 0 .9718+0.0059\n−0.0067 0.9774±0.0057 0 .9704+0.0053\n−0.0059\nln\u0000\n1010As\u0001\n3.029±0.017 3 .047±0.016 3 .038±0.017 3 .046±0.016\nΩK −0.059+0.023\n−0.018 −0.0083+0.0094\n−0.0066 −0.030±0.012 −0.0068+0.0071\n−0.0047\nλ < 2.21 1 .05+0.64\n−0.37 <0.549 0 .54+0.28\n−0.39\nα < 0.0515 0 .078+0.046\n−0.031 0.104+0.026\n−0.016 0.068+0.033\n−0.028\nσ8 0.754+0.061\n−0.049 0.845+0.034\n−0.048 0.893±0.037 0 .852+0.025\n−0.038\nΩm 0.559+0.079\n−0.15 0.310±0.021 0 .322±0.023 0 .296+0.014\n−0.011\nS8 1.014+0.055\n−0.043 0.856+0.021\n−0.033 0.923±0.039 0 .845+0.017\n−0.023\nH0 50.9±5.2 66 .5+2.0\n−1.5 63.5+2.2\n−2.6 68.10±0.79\n∆χ2\nmin −1.60 −3.68 −7.66 −1.36\nlnBM1+,ΛCDM −4.32 −4.2 −4.04 −5.89\nTable 6.10 : Observational constraints at 68%confidence level on the sampled and derived\ncosmological parameters for different data set combinations under the curved M 1++ Ω Kmodel,\nstudied in Section 6.3.2.\n99%CL, namely ΩK=−0.061+0.048\n−0.053andΩK=−0.062+0.047\n−0.054for M 2++ Ω Kand M 2−+ Ω K,\nrespectively. The coupling parameter is bounded to similar values across both models and all\ndata sets; namely, we report α <0.0563and|α|<0.0566at68%CL for Pl18 data alone, looser\nthan in the corresponding flat scenario. The H0tension in the CMB data is maintained at the\nsame level as for ΛCDM +ΩKand M 1+Ω K, displaying mean values of H0= 51.6+4.2\n−2.8km/s/Mpc\nandH0= 51.9+3.7\n−4.7km/s/Mpc at 68%CL. Compared with the flat case, we conclude that this\nworsening of the tension is attributed to the curvature effects.\nIncorporating BAO data into the analysis considerably shifts the constraints due to their con-\nflicting implications in a curved Universe, drawing this model closer to the spatially flat case.\nNevertheless, the combination of Pl18+SN yields intriguing outcomes. We detect a closed Uni-\nverse at above 95%CL (ΩK=−0.030±0.023andΩK=−0.031±0.026) with a coupled dark\nsector ( α= 0.099+0.047\n−0.061and|α|= 0.098+0.046\n−0.054at95%CL). The Hubble constant is suppressed in\nrelation to the Pl18+BAO case, leading to an enlarged Ωmdue to its correlation with H0.\nLastly,consideringthePl18+BAO+SNcombination(seethelastcolumnofTables6.12and6.13),\nthe BAO data’s tension with Planck results reinforces spatial flatness. Meanwhile, αbecomes140 6.3. Coupled Quintessence in Non-Flat Geometries\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nωb 0.02261 ±0.00017 0 .02239 ±0.00016 0 .02252 ±0.00017 0 .02239 ±0.00015\nωcdm 0.1130+0.0060\n−0.0028 0.1148+0.0062\n−0.0030 0.1045+0.0063\n−0.0073 0.1134+0.0068\n−0.0038\n100θs 1.04204 ±0.00030 1 .04191 ±0.00030 1 .04197 ±0.00029 1 .04190 ±0.00030\nτreio 0.0485+0.0084\n−0.0074 0.0556±0.0081 0 .0526±0.0081 0 .0552±0.0077\nns 0.9730±0.0049 0 .9686+0.0049\n−0.0057 0.9768±0.0063 0 .9696±0.0053\nln\u0000\n1010As\u0001\n3.029±0.017 3 .047±0.017 3 .039±0.017 3 .047±0.016\nΩK −0.069+0.024\n−0.022 −0.0028+0.0067\n−0.0036 −0.028+0.012\n−0.014 −0.0048+0.0069\n−0.0042\nλ < 2.49 <0.653 <0.305 <0.435\nα > −0.0537 >−0.0627 −0.100+0.018\n−0.032 −0.058±0.032\nσ8 0.736+0.058\n−0.051 0.829+0.022\n−0.041 0.887±0.041 0 .843+0.022\n−0.039\nΩm 0.60+0.10\n−0.15 0.303+0.020\n−0.014 0.315±0.023 0 .294+0.018\n−0.011\nS8 1.026+0.052\n−0.037 0.831+0.014\n−0.018 0.909+0.045\n−0.035 0.834+0.015\n−0.018\nH0 48.5+4.5\n−5.8 67.5+1.2\n−1.0 63.7+2.2\n−2.7 68.11±0.76\n∆χ2\nmin −2.46 −1.60 −6.38 1 .66\nlnBM1−,ΛCDM −4.24 −5.61 −4.31 −6.77\nTable 6.11 : Observational constraints at 68%confidence level on the sampled and derived\ncosmological parameters for different data set combinations under the curved M 1−+ Ω Kmodel,\nstudied in Section 6.3.2.\nmore constrained and aligns with an uncoupled and a cosmological constant scenario ( α= 0and\nµ= 0) at95%CL. It is noteworthy that H0sees a mild enhancement compared to the Pl18 flat\nand curved ΛCDM predictions, yet the tension with distance ladder measurements persists at\n4−5σ.\nBelow we list the results in more detail, with a focus on comparing the effects of the introduction\nof spatial curvature.\n•Coupling: In contrast with the flat case, a non-vanishing prediction for αat68%can be\nderived for all data set combinations except Pl18 in both M 2 + Ω Kmodels, as reported in\nTables 6.12 and 6.13. For instance, for the full combination we report α= 0.049+0.026\n−0.035and\nα=−0.050+0.040\n−0.023for M 2++ Ω Kand M 2−+ Ω K, respectively. On the other hand, for the\nPl18 data alone, these constraints are once again relaxed to upper bounds: α <0.0563and\n|α|<0.0566for M 2++ΩKand M 2−+ΩK, respectively. Hence, the magnitude of the cou-\npling parameter is comparable in both cases, in agreement with the symmetry found in the\nconstant coupling case for change of signs in the pair {α,˙ϕ}, according to Equation (6.1.8).6Coupled Quintessence in Flat and Curved Geometries 141\n40506070\nH00.00.10.2\n0.60.81.0\n8\n0.30.60.9\nm\n012345\n0.12\n0.06\n0.00\nk\nModel 1++k\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\n40506070\nH00.00.10.2||\n0.60.81.0\n8\n0.30.60.9\nm\n012345\n0.12\n0.06\n0.00\nk\nModel 1+k\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\nFigure6.8 : 2D marginalised posterior distributions of parameters in the M 1++ Ω Kmodel (up-\nperpanel)andM 1−+ Ω Kmodel(lowerpanel)usingthePl18(red), Pl18+BAO(blue), Pl18+SN,\n(green) and Pl18+BAO+SN (yellow) data set combinations. We plot the 2D marginalised pos-\nterior distributions of the conformal coupling parameter, α, against the slope of the potential,\nλ, the Hubble constant in units km s−1Mpc−1,H0, the present-day mass fluctuation amplitude\nin spheres of radius 8h−1Mpc,σ8, and the total matter density parameter, Ωm. The shaded\ncontours indicate the 1σand2σconfidence limits.\nIn Figure 6.9, we find a similar pattern to the flat case, with the background data driving\nthe peak in the |α|marginalised posterior distribution away from zero. SN data has the\ngreatest impact in suppressing the low- αtail of the distribution for Pl18+SN, which is\nrestored with the introduction of BAO into the combination. In Figure 6.10 we show the\n2D marginalised posterior distributions of αagainst {H0, σ8,Ωm, µ,ΩK}. Both M 2 + Ω K\nmodels exhibit a positive correlation in the {|α|, σ8}. For Pl18 there is also a slight positive\ncorrelation for {|α|, H0}and negative for {|α|,Ωm}, and both vanish once background data\nsets are included. On the other hand, there is a negative correlation for {|α|,ΩK}for all\ncases excluding Pl18 alone. This is once again associated with the larger error bars in Ωm,\nwhich is constrained towards more conservative values when the background data is added\nand does not appear to be correlated with α. Therefore, a more negative value of ΩKmust\nbe compensated by an increased coupling, sourcing the DE component. The remaining\ncombinations seem to be negligibly correlated. We find the scalar field parameters {|α|, µ}142 6.3. Coupled Quintessence in Non-Flat Geometries\nto be generally uncorrelated. We highlight the consistent results between the positive and\nnegativecouplingparametercasesduetothemodel’ssymmetryandshowaclearpreference\nfor the trend set by the data on the parameters.\n•Potential: The power of the inverse potential µis only bounded from above with 68%CL\nfor all the data combinations in both M 2 + Ω Kmodels. These are reported in Tables 6.12\nand 6.13, with the background data consistently shortening the 1σparameter region and,\nconsequently, the upper bound. Indeed, this is illustrated in Figure 6.9, with the back-\nground data leading to the collapse of the high- µtail of the distribution that is present in\nboth cases for Pl18, bringing the peak close to zero. Pl18 data alone limits µto be smaller\nthan 2.17and1.74, while the full Pl18+BAO+SN sets brings this upper limit down to\n0.499for M 2++ Ω Kand0.484for M 2−+ Ω K, respectively at 1σ. The symmetry of the\nmodel ensures that the values of |α|andµfall within similar ranges in both models for the\nsame data set combinations.\n•H0Tension: The H0tensionisalleviatedtosomeextentbythepositivecorrelationbetween\n|α|andH0, similar to the M 1 + Ω Kmodels. In the Pl18-only case, the M 2++ Ω Kand\nM2−+ Ω Kmodels negligibly reduce the tension in ΛCDM + Ω Kfrom approximately 5.3σ\nto∼4.6σand4.9σ, respectively, which is still considerably aggravated with respect to\nthe corresponding values for M 2with ΩK= 0. Analogously to the M 1 + Ω Kcase, the\nmean H0value in the M 2 + Ω Kmodels is lower than in ΛCDM + ΩK, yet with larger error\nbars. Once more, adding background data (dominated by the effect of BAO) brings the\nH0value closer to the ones reported in the flat cases, consistent with the constraints set\nonΩKclose to zero. Nevertheless, it stresses once more the incompatibility of the CMB\nand background data sets when ΩKis added as a free parameter.\n•S8Tension: We see that considering the inverse power law potential does not change the\npredictions significantly compared with the exponential case M 1 + Ω K. The evidence for\na close Universe when considering Pl18 alone drives the value of σ8down. However, the\nS8parameter remains too high when compared with the one reported by studies of weak\nlensing predictions, as a consequence of the large values predicted for Ωmin both the\nΛCDM + ΩKand M 2 + Ω Kmodels. By introducing the background data, both σ8andΩm\nare restored to values compatible with the flat case. For Pl18 this results in an S8value\nof1.021+0.057\n−0.040at1σfor the M 2++ Ω Kmodel, and 1.019+0.056\n−0.038for M 2−+ Ω K, both not\nsignificantly different from the ΛCDM + Ω Kvalue S8= 1.003+0.054\n−0.047.\n•Curvature: WereportevidenceforacurvedUniverseatmorethan 99%CLfortheM 2+Ω K\nmodels with CMB data and with the addition of SN. In particular, at 68%CL we find\nΩK=−0.061±0.020for M 2++ Ω KandΩK=−0.062±0.019for M 2−+ Ω K, in contrast\nwith ΩK=−0.052+0.020\n−0.017forΛCDM + Ω K, with similar sizes for the CL regions. These6Coupled Quintessence in Flat and Curved Geometries 143\nvalues are consistently in tension with the Pl18+BAO+SN counterparts, which report\nΩK=−0.0037+0.0058\n−0.0033for M 2++ Ω KandΩK=−0.0038+0.0059\n−0.0034for M 2−+ Ω K, also in\ncontrast with ΩK= 0.0010±0.0019forΛCDM + ΩK, all including ΩK= 0in the 68%CL\nregion. This corroborates the incompatibility between the data sets of different nature in\ntension at ∼2.9σforΛCDM + ΩK,∼2.8σfor M 2++ ΩKand∼3.0σlevel for M 2−+ ΩK,\nalready after the data sets have been combined.\n•Model Evidence: The lower sections of Tables 6.12 and 6.13 report on the ∆χ2\nminvalue for\nthe goodness of fit comparison and the logarithm of the Bayes factor, lnBM2,ΛCDM. We\nsee that both model comparison criteria are significantly negative (except for the negligible\n∆χ2\nmin>0for the M 2 + Ω Kmodels with Pl18+BAO+SN), yielding no support for the\nM2 + Ω Kmodels over ΛCDM + Ω Kbut revealing a better fit to the data.\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nωb 0.02264 ±0.00017 0 .02240 ±0.00015 0 .02255 ±0.00016 0 .02240 ±0.00016\nωcdm 0.1150+0.0045\n−0.0025 0.1146+0.0061\n−0.0034 0.1057+0.0056\n−0.0064 0.1159+0.0047\n−0.0026\n100θs 1.04208 ±0.00030 1 .04191 ±0.00030 1 .04198 ±0.00030 1 .04191 ±0.00030\nτreio 0.0478+0.0086\n−0.0076 0.0553±0.0078 0 .0523±0.0083 0 .0550±0.0078\nns 0.9739±0.0051 0 .9697+0.0052\n−0.0060 0.9770±0.0059 0 .9685+0.0048\n−0.0054\nln\u0000\n1010As\u0001\n3.026±0.018 3 .047±0.016 3 .038±0.017 3 .046±0.016\nΩK −0.061±0.020 −0.0054+0.0076\n−0.0044 −0.030±0.012 −0.0037+0.0058\n−0.0033\nµ < 2.17 <0.626 <0.525 <0.499\nα < 0.0563 0 .059+0.031\n−0.045 0.098+0.030\n−0.018 0.049+0.026\n−0.035\nσ8 0.774+0.041\n−0.036 0.846+0.023\n−0.043 0.887±0.040 0 .838+0.017\n−0.032\nΩm 0.533+0.078\n−0.12 0.297+0.018\n−0.014 0.323±0.022 0 .298+0.014\n−0.0085\nS8 1.021+0.057\n−0.040 0.841+0.015\n−0.024 0.919±0.039 0 .835+0.015\n−0.019\nH0 51.6+4.2\n−4.8 68.1+1.3\n−0.85 63.3+2.1\n−2.3 68.29±0.75\n∆χ2\nmin −2.04 −3.12 −5.62 0 .04\nlnBM2+,ΛCDM −5.14 −5.36 −4.26 −6.33\nTable 6.12 : Observational constraints at 68%confidence level on the sampled and derived\ncosmological parameters for different data set combinations under the curved M 2++ Ω Kmodel,\nstudied in Section 6.3.3.144 6.3. Coupled Quintessence in Non-Flat Geometries\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nωb 0.02265 ±0.00017 0 .02240 ±0.00015 0 .02254 ±0.00017 0 .02239 ±0.00015\nωcdm 0.1137+0.0052\n−0.0025 0.1146+0.0062\n−0.0031 0.1052+0.0058\n−0.0072 0.1152+0.0054\n−0.0027\n100θs 1.04208 ±0.00031 1 .04191 ±0.00030 1 .04199 ±0.00030 1 .04191 ±0.00030\nτreio 0.0478±0.0085 0 .0549±0.0079 0 .0522±0.0082 0 .0553±0.0081\nns 0.9741±0.0050 0 .9691+0.0048\n−0.0059 0.9771±0.0062 0 .9688±0.0050\nln\u0000\n1010As\u0001\n3.027±0.018 3 .046±0.016 3 .038±0.017 3 .047+0.015\n−0.017\nΩK −0.062±0.019 −0.0045+0.0069\n−0.0037 −0.031±0.013 −0.0038+0.0059\n−0.0034\nµ < 1.74 <0.450 <0.496 <0.484\nα > −0.0566 −0.053+0.046\n−0.022 −0.099+0.017\n−0.032 −0.050+0.040\n−0.023\nσ8 0.781+0.033\n−0.040 0.844+0.022\n−0.042 0.887±0.041 0 .839+0.019\n−0.033\nΩm 0.520±0.090 0 .295+0.019\n−0.011 0.323+0.021\n−0.024 0.297+0.015\n−0.0094\nS8 1.019+0.056\n−0.038 0.836+0.015\n−0.019 0.920±0.041 0 .834+0.014\n−0.017\nH0 51.9+3.7\n−4.7 68.35±0.94 63 .1±2.3 68 .25±0.76\n∆χ2\nmin −1.06 −0.94 −6.30 0 .28\nlnBM2−,ΛCDM −4.84 −5.56 −3.51 −6.72\nTable 6.13 : Observational constraints at 68%confidence level on the sampled and derived\ncosmological parameters for different data set combinations under the curved M 2−+ Ω Kmodel,\nstudied in Section 6.3.3.\n0.000.050.100.15\n0369\nModel 2++k\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\n0.000.050.100.15\n||\n0369\nModel 2+k\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\nFigure 6.9 : 1D marginalised posterior distributions of the coupling parameter, α, and slope of\nthe potential, µ, in the M 2++ Ω Kmodel (upper panel) and M 2−+ Ω Kmodel (lower panel)\nusing the Pl18 (red), Pl18+BAO (blue), Pl18+SN, (green) and Pl18+BAO+SN (yellow) data\nset combinations.6Coupled Quintessence in Flat and Curved Geometries 145\n40506070\nH00.00.10.2\n0.60.81.0\n8\n0.30.60.9\nm\n0246810\n0.12\n0.06\n0.00\nk\nModel 2++k\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\n40506070\nH00.00.10.2||\n0.60.81.0\n8\n0.30.50.7\nm\n02468\n0.12\n0.06\n0.00\nk\nModel 2+k\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\nFigure6.10 : 2DmarginalisedposteriordistributionsofparametersintheM 2++ Ω Kmodel(up-\nperpanel)andM 2−+ Ω Kmodel(lowerpanel)usingthePl18(red), Pl18+BAO(blue), Pl18+SN,\n(green) and Pl18+BAO+SN (yellow) data set combinations. We plot the 2D marginalised pos-\nterior distributions of the conformal coupling parameter, α, against the power of the potential,\nµ, the Hubble constant in units km s−1Mpc−1,H0, the present-day mass fluctuation amplitude\nin spheres of radius 8h−1Mpc,σ8, and the total matter density parameter, Ωm. The shaded\ncontours indicate the 1σand2σconfidence limits.\n6.3.4 M3 + Ω K: Coupling with Minimum/Maximum and Exponential Po-\ntential\nIn Tables 6.14 and 6.15, we summarise the observational limits on both free and derived model\nparameters for the M 3 + Ω Kmodels. Figures 6.11 and 6.12 display the 1D and 2D marginalised\ndistributions for selected relevant parameters.\nWe find that when considering only the Pl18 data, there is a preference for a closed Universe\nat over 99% CL. Namely, we find ΩK=−0.067+0.052\n−0.051andΩK=−0.066+0.051\n−0.057at99%CL. The\nmean H0value in this scenario is lower than in the ΛCDM case ( H0= 49.0+4.2\n−5.7km/s/Mpc and\nH0= 49.0+4.8\n−6.0km/s/Mpc at 68% CL), but the larger error bars ensure a similar value for the H0\ntension of approximately 4−5σ. The coupling and potential parameters are compatible with an\nuncoupled and cosmological constant-like scenario at 68%CL).\nWhen incorporating BAO data, the potential for a flat Universe is recovered at 68%CL. With146 6.3. Coupled Quintessence in Non-Flat Geometries\nthe addition of SN data, we once more reaffirm the non-zero curvature at over 68%CL for\nM3++ Ω K(ΩK=−0.033+0.028\n−0.027) and 95%CL for M 3−+ Ω K(ΩK=−0.033+0.028\n−0.027), with only a\nnon-vanishing prediction for λbeing recovered with Pl18+BAO at 68%CL (λ= 0.47±0.28for\nM3++ Ω Kandλ= 1.23±0.63for M 3−+ Ω K). The coupling parameter γis more constrained\nin the M 3−+ ΩKcase, with only non-zero values predicted at 1σfor Pl18+SN. For the full joint\nanalysis using Pl18+BAO+SN, the data break most of the degeneracies and slightly improve the\nagreement for H0(H0= 68.00±0.73andH0= 68.00±0.74km/s/Mpc at 68%CL). Both γand\nλare consistent with non-dynamic behaviour at 95%CL.\n•Initial Conditions: Just as in the flat case, the field dependence in the coupling introduces\na minimum and a maximum of the effective coupling at ϕ∗= 2MPlfor M 3++ Ω Kand\nM3−+ Ω K, respectively. In the flat case, we came across an unphysical highly peaked\nposterior distribution for ϕini, which wants to sit at the minimum of the potential in M 3+.\nWhen the curvature is allowed to vary, this feature becomes less sharp, with the larger\nerror bars and freedom introduced in the model implying a posterior distribution for ϕini\nthat is still peaked around ϕ∗but much less stringently. We see in Figure 6.11 that there\nis a seemingly symmetric distribution around the peak in the ϕiniposterior distribution for\nM3++ Ω K. At the same time, the same peaked pattern but skewed to the right (mainly\nby the effect of SN) is recovered in M 3−+ Ω Kjust as in the flat case.\n•Coupling: The non-trivial expression for the coupling in the M 3models makes this scenario\nmore distinguishable from M 1and M 2. In all the cases considered the uncoupled scenario\n|γ|= 0is accommodated by the 68%CL region at 1σ, with the exception of the Pl18+SN\ncombination in the model with a maximum, M 3−+ Ω K, for which γ=−0.077+0.035\n0.023, as\nreportedinTables6.14and6.15. ForjustthePl18dataset, theconstraintsareupperlimits,\nγ <0.357and|γ|<0.0741, for the respective models. The coupling parameter’s size is\nquitedifferentinbothcases,consistentlymoreconstrainedtowardszeroforM 3−+ΩK. This\nfeature is depicted in Figure 6.11, in which there is a similar peak for the γmarginalised\nposterior distribution across all the data sets for M 3++ Ω K, with only the Pl18+SN peak\nbeing considerably deviated from zero. In contrast, the background data pushes the peak\nin the |γ|posterior away from zero in M 3−+ Ω K, with SN most effectively narrowing\nthe lower- γtail for Pl18+SN, consistent with the bounded 68%CL region derived in\nthis case only. Adding BAO data preserves the peak in the distribution but reverses the\nnarrowing of the left-hand side tail. Figure 6.12 reveals the 2D marginalised distributions\nofγagainst {H0, σ8,Ωm, λ,ΩK}. No significant correlations are identified for M 3++ Ω K\ndue to the very peaked distribution around the minimum of the coupling and |ΩK| ≪1\nfor the background data, apart from a small positive correlation in {|γ|, σ8}for Pl18. For\nM3−+ Ω Kwe identify an overall positive correlation in the {|γ|, σ8}plane and in {|γ|, λ}\nfor the background data. A negative correlation is identified in {|γ|,ΩK}and{|γ|, H0}6Coupled Quintessence in Flat and Curved Geometries 147\nwhen the background data is included, more pronounced for Pl18+SN in both cases.\n•Potential: InbothM 3+Ω Kmodels, thesteepnessofthepotential, λ, isonlyupper-bounded\nat68%CL for all datasets, except for the Pl18+BAO combination, for which we find\nλ= 0.47±0.28andλ= 1.23±0.63for M 3++ΩKand M 3−+ΩK, as outlined in Tables 6.14\nand 6.15. The tighter bounds are set by SN data, as corroborated in Figure 6.11. We see\nthat the integration of background data in M 3++ Ω Kbrings the marginalised posterior\ntowards smaller values, with a minor non-zero peak for Pl18+BAO, while in M 3−+ Ω K\nlarger deviations are allowed for the more pronounced peak in Pl18+BAO. Once again,\nthe inclusion of background data shrinks the high- λtail present for the Pl18 dataset.\nWhen considering only Pl18, the upper limits for λare2.52and2.60. In comparison, the\nfull Pl18+BAO+SN sets reduce the limits down to 0.445and0.604, for M 3++ Ω Kand\nM3−+ Ω K, respectively, at 1σ.\n•H0Tension: A minor mitigation of the H0tension occurs even though no clear correlation\nbetween |γ|andH0has been identified. Even if the mean values reported for H0in the\nM3+Ω Kmodels are lower than the ΛCDM +ΩKcase, the increased size of the CL regions\nhas a minor influence on the tension, which decreases from around 5.3σto approximately\n4.8σand4.4σin M 3++ Ω Kand M 3−+ Ω K, respectively. Consistently with what was\nfound for the M 1 + Ω Kand M 2 + Ω Kcases, the addition of background data restores the\nvalue of H0to the order reported in the flat cases, coinciding with ΩKvanishing in the 1σ\nregion, and reflecting the incompatibility of the different data sets.\n•S8Tension: Even though the prediction for a closed Universe for the Pl18-only data set\nbringsthevalueof σ8significantlydown, comparedwithboththeflatcaseandthe ΛCDM +\nΩKcases, thisisaccompaniedbyanincreaseinthemeanvalueof Ωm, andthe S8parameter\nactually becomes larger. For Pl18 this results in an S8value of 1.041+0.058\n−0.043at1σfor the\nM3++ Ω Kmodel, and 1.027+0.056\n−0.038for M 3−+ Ω K, the highest values of all the models\nconsidered, with a similar size for the CL region.\n•Curvature: We report firm evidence for a closed Universe at more than 99%CL for the\nM3 + Ω Kmodels with CMB data and a less pronounced but still present support with the\naddition of SN. In particular, at 68%CL we find ΩK=−0.067±0.021for M 3++ Ω Kand\nΩK=−0.066+0.024\n−0.021for M 3−+ Ω K, in contrast with ΩK=−0.052+0.020\n−0.017forΛCDM + Ω K,\nwith similar sized CL regions. On the contrary, no significant prediction for curvature\nis present for the Pl18+BAO+SN counterparts, which yield ΩK=−0.0020+0.0047\n−0.0027for\nM3++ΩKandΩK=−0.0035+0.0066\n−0.0032for M 3−+ΩK, in contrast with ΩK= 0.0010±0.0019\nforΛCDM +ΩK, allaccommodating ΩK= 0inthe 68%CLregion. Datasetinconsistencies\nare further confirmed, showing tensions at approximately the 3σlevel, more precisely of\n∼2.9σforΛCDM + Ω K,∼3.0σfor M 3++ Ω Kand∼2.7σfor M 3−+ Ω K, already after148 6.4. Summary and Discussion\nthe data sets have been combined.\n•Model Evidence: The Bayes factor lnBM3,ΛCDMpoints to no support for the M 3 + Ω K\nmodels over the ΛCDM + Ω Kmodel. Nevertheless, the considerably negative values of\n∆χ2\nminindicate a better fit to the data in all cases.\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nωb 0.02264 ±0.00017 0 .02240 ±0.00015 0 .02249 ±0.00017 0 .02241 ±0.00015\nωcdm 0.1186+0.0022\n−0.0036 0.1187+0.0030\n−0.0018 0.1151+0.0047\n−0.0030 0.1177+0.0032\n−0.0018\n100θs 1.04203 ±0.00032 1 .04186 ±0.00031 1 .04191 ±0.00031 1 .04189 ±0.00030\nτreio 0.0478+0.0087\n−0.0074 0.0556±0.0077 0 .0547±0.0079 0 .0556±0.0078\nns 0.9741+0.0050\n−0.0056 0.9691+0.0049\n−0.0059 0.9741+0.0061\n−0.0074 0.9698+0.0049\n−0.0062\nln\u0000\n1010As\u0001\n3.027+0.018\n−0.016 3.048±0.016 3 .045±0.016 3 .048±0.016\nΩK −0.067±0.021 −0.00099+0.0042\n−0.0023 −0.0138+0.011\n−0.0082 −0.0020+0.0047\n−0.0027\nλ < 2.52 0 .47±0.28 <0.348 <0.445\nγ < 0.357 <0.384 <0.215 <0.229\nϕi 1.90+0.61\n−0.49 1.96+0.66\n−0.59 2.15+1.4\n−0.90 2.03±0.84\nσ8 0.738+0.060\n−0.039 0.824+0.015\n−0.027 0.841+0.024\n−0.036 0.830+0.015\n−0.027\nΩm 0.610+0.096\n−0.16 0.310+0.011\n−0.0096 0.329±0.022 0 .304+0.010\n−0.0078\nS8 1.041+0.058\n−0.043 0.837+0.014\n−0.020 0.880±0.036 0 .836+0.014\n−0.021\nH0 49.0+4.2\n−5.7 67.64±0.89 64 .9±2.3 68 .00±0.73\n∆χ2\nmin −2.96 −1.76 −5.08 −0.56\nlnBM3+,ΛCDM −2.79 −3.67 −5.42 −6.39\nTable 6.14 : Observational constraints at 68%confidence level on the sampled and derived\ncosmological parameters for different data set combinations under the curved M 3++ Ω Kmodel,\nstudied in Section 6.3.4.\n6.4 Summary and Discussion\n6.4.1 H0Tension\nThe whisker plot in Figure 6.13 summarises the constraints derived on the Hubble constant H0\nat68%CL for the Pl18-only (left) and full Pl18+BAO+SN (right) for the various cosmological\nmodels considered in this chapter, showing the particular trends for how H0varies across the\nflat (top) and curved (bottom) cases. The first thing we can appreciate, discussed throughout6Coupled Quintessence in Flat and Curved Geometries 149\nParameter Pl18 Pl18 + BAO Pl18 + SN Pl18 + BAO + SN\nωb 0.02262 ±0.00017 0 .02243 ±0.00017 0 .02254 ±0.00016 0 .02240 ±0.00015\nωcdm 0.1152+0.0050\n−0.0019 0.1118+0.0090\n−0.0054 0.1022±0.0083 0 .1157+0.0054\n−0.0025\n100θs 1.04206 ±0.00030 1 .04194 ±0.00030 1 .04200 ±0.00031 1 .04192 ±0.00029\nτreio 0.0483±0.0082 0 .0554±0.0078 0 .0517±0.0081 0 .0547+0.0069\n−0.0078\nns 0.9722±0.0048 0 .9698+0.0050\n−0.0059 0.9747±0.0054 0 .9679±0.0048\nln\u0000\n1010As\u0001\n3.028±0.017 3 .046±0.016 3 .036±0.017 3 .045±0.016\nΩK −0.066+0.024\n−0.021 −0.0104+0.013\n−0.0080 −0.032+0.013\n−0.015 −0.0035+0.0066\n−0.0032\nλ < 2.60 1 .23±0.63 <0.494 <0.604\nγ > −0.0741 >−0.128 −0.077+0.035\n−0.023 >−0.0757\nϕi 1.86±0.76 1 .46+0.55\n−0.50 1.37+0.24\n−1.1 1.66+0.70\n−0.97\nσ8 0.738±0.059 0 .868+0.042\n−0.079 0.909±0.053 0 .838+0.017\n−0.041\nΩm 0.60+0.10\n−0.16 0.311±0.020 0 .316±0.023 0 .300+0.015\n−0.0079\nS8 1.027+0.056\n−0.038 0.882+0.028\n−0.070 0.931+0.050\n−0.041 0.838+0.012\n−0.027\nH0 49.0+4.8\n−6.0 65.9+2.5\n−1.7 63.1+2.3\n−2.6 68.00±0.74\n∆χ2\nmin −1.28 −8.24 −10.5 −0.98\nlnBM3−,ΛCDM −4.32 −5.28 −6.98 −6.76\nTable 6.15 : Observational constraints at 68%confidence level on the sampled and derived\ncosmological parameters for different data set combinations under the curved M 3−+ Ω Kmodel,\nstudied in Section 6.3.4.\n0.00.51.01.5\n012345\n01234\nini\nModel 3++k\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\n0.00.10.20.3\n||\n012345\n01234\nini\nModel 3+k\nPl18\nPl18+BAOPl18+SN\nPl18+BAO+SN\nFigure 6.11 : 1D marginalised posterior distributions of the coupling parameter, γ, and slope\nof the potential, λ, in the M 3++ Ω Kmodel (upper panel) and M 3−+ Ω Kmodel (lower panel)\nusing the Pl18 (red), Pl18+BAO (blue), Pl18+SN, (green) and Pl18+BAO+SN (yellow) data\nset combinations.150 6.4. Summary and Discussion\n40506070\nH00.00.51.01.52.0\n0.60.81.0\n8\n0.30.60.9\nm\n012345\n0.12\n0.06\n0.00\nk\nModel 3++k\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\n40506070\nH00.00.10.20.30.4||\n0.60.81.0\n8\n0.30.60.9\nm\n012345\n0.12\n0.06\n0.00\nk\nModel 3+k\nPl18 Pl18+BAO Pl18+SN Pl18+BAO+SN\nFigure6.12 : 2DmarginalisedposteriordistributionsofparametersintheM 3++ Ω Kmodel(up-\nperpanel)andM 3−+ Ω Kmodel(lowerpanel)usingthePl18(red), Pl18+BAO(blue), Pl18+SN,\n(green) and Pl18+BAO+SN (yellow) data set combinations. We plot the 2D marginalised pos-\nterior distributions of the conformal coupling parameter, γ, against the slope of the potential,\nλ, the Hubble constant in units km s−1Mpc−1,H0, the present-day mass fluctuation amplitude\nin spheres of radius 8h−1Mpc,σ8, and the total matter density parameter, Ωm. The shaded\ncontours indicate the 1σand2σconfidence limits.\nthe text, is the inherent degeneracy between H0and the curvature parameter, with a clear trend\nseparating the flat and curved cases. In particular, a closed Universe drives the value of H0\ntowards smaller values compared with the ΛCDM Pl18 case (red point and vertical bar) and,\nconsequently, leads to an increase in the tension between the mean values and the R22 estimate\n(blue cross and vertical bar), which is superficially softened by the larger error bars.\nFor all the extensions to the ΛCDM + Ω Kmodel (green square and vertical bar), we reported\nan indication for a closed Universe at a confidence level exceeding 99% when only Pl18 data\nis considered. The positive correlation between H0andΩKimplies that, as ΩKdeviates from\nzero towards negative values, H0consequently decreases, exacerbating the tension with R22 as\ndepicted in Figure 6.13. This tension intensifies exclusively due to the Pl18-only evidence for a\nclosed Universe. In the case of Pl18+BAO and Pl18+BAO+SN combinations, the H0estimates\nalign closely with Planck’s (flat) ΛCDM results as ΩKapproaches zero, and the mean value of\nH0is marginally higher, thus superficially easing the R22 tension. For Pl18+SN, a prediction6Coupled Quintessence in Flat and Curved Geometries 151\nfor a closed Universe at more than 68%for model M 3++ Ω Kand95%for all the others bring\nthe estimated H0values towards systematically lower values, and we do not observe any effective\nalleviation of the H0tension in any of these six extended scenarios.\nThe Pl18-only analysis also yields values of H0consistent at 1σfor all the flat and curved cases\nseparately between themselves. There is an overall trend of models M 2yielding larger mean\nvalues of H0, balanced by a reduction in the size of the 1σregions, thereby not introducing\nsignificant benefits in addressing the tension. The larger tension reduction is present in the M 1\nflat models, predominantly due to the enlarged error bars.\nAdding the background data to the Pl18 analysis brings the H0estimate in the curved models\nin agreement with the flat case at 1σ, in line with the tension in BAO data for a closed Universe.\nFinally, it is noteworthy that coupled quintessence models introduce new physics in comparison\ntoΛCDM, mostly at low-redshifts, once DE starts to dominate. This feature prevents these\nmodels from resolving the H0tension once BAO and SN data are considered, as evidenced by\nRefs. [372–374], unless other new exotic DE physics is considered [375, 376].\n40 50 60 70 80\nH0[km/s/Mpc]M3 + k, 49.0+4.8\n6.0\nM3+ + k, 49.0+4.0\n5.9\nM2 + k, 51.7+4.0\n4.6\nM2+ + k, 51.5+4.3\n4.9\nM1 + k, 48.5+4.3\n6.0\nM1+ + k, 50.9+5.4\n4.9\nCDM + k, 52.8+3.1\n4.1\nM3, 66.1+2.8\n1.0\nM3+, 67.72±0.75M2, 68.42+0.72\n1.8\nM2+, 68.3+1.0\n1.7\nM1, 66.5+2.2\n1.8\nM1+, 65.8+3.8\n2.3\nCDM, 67.33±0.6\nR22, 73.04±1.04\nFlat\nCurvedPl18\n65 70 75\nH0[km/s/Mpc]M3 + k, 68.0±0.73\nM3+ + k, 68.0±0.73\nM2 + k, 68.25±0.76\nM2+ + k, 68.29±0.75\nM1 + k, 68.11±0.76\nM1+ + k, 68.1±0.79\nCDM + k, 68.03±0.67\nM3, 67.91+0.54\n0.73\nM3+, 67.76±0.45M2, 68.21+0.58\n0.71\nM2+, 68.19+0.52\n0.67\nM1, 67.98±0.69\nM1+, 68.02+0.62\n0.74\nCDM, 67.75±0.43\nR22, 73.04±1.04\nFlat\nCurvedPl18 + BAO + SN\nFigure 6.13 : Whisker plot with the 68%CL constraints on the Hubble parameter H0obtained\nfor the cosmological models explored in this study for the Pl18 CMB data (left) and the combi-\nnation Pl18 + BAO + SN (right), as detailed in Section 4.3. Circle and squared markers denote\nthe flat and curved models. The red and green vertical bars illustrate to reference ΛCDM flat\n(ΩK= 0) and curved ΛCDM + ΩKvalues, respectively. The blue bar corresponds to the Kilo-\nDegree Survey (KiDS-1000) value [157], very close to one reported by the Dark Energy Survey\n(DES-Y3) [160], as well, both in tension with the prediction for the six-parameter flat ΛCDM\nfrom the Planckcollaboration 2018 data release [32], as explained in a Section 3.6.1.152 6.4. Summary and Discussion\n6.4.2 S8Tension\nIn Figure 6.14, the constraints on the S8parameter from the extended cosmological models are\ndepicted at 68%CL for Pl18-only (left) and Pl18+BAO+SN (right). For reference, we also show\nthe similar estimated S8values from Kilo-Degree Survey (KIDS-1000) [164] and the Dark Energy\nSurvey (DES) Year 3 (DES-Y3) [161] (in blue), and the Planck [32] ΛCDM flat (red) and curved\n(green) cases with Pl18 data, all under the assumption of a baseline ΛCDM framework.\nFigure 6.14 reveals two key observations. Firstly, for data involving just CMB, the S8values in\nall extended models deviate significantly from those of KIDS-1000, DES-Y3, and Planck when a\nstandard ΛCDM model is assumed. For the curved scenarios, the estimated values of S8escalate\nsharply, largely due to the firm prediction of a closed Universe at more than 99%CL. The\nonly models that seem to move toward lower predictions of S8are the M 2flat models, even if\nmarginally. Nevertheless, according to the discussion in Section 3.6, it should still be emphasised\nthat any conclusions regarding the S8tension require a new analysis of the weak lensing data\nfor the cosmological model in question.\nSecondly, estimations of S8using the background data sets closely resemble the prediction for\nthe concordance ΛCDM with Planck. This might suggest that the S8tension is significantly\nenhanced in the extended coupled quintessence models that point toward a closed Universe. As\nwas the case for the H0parameter, the apparent ease of the tension may occur in the curved\nmodels simply due to enlarged error bars and not through a more compatible mean value.\n6.4.3 Matter Density\nFor the matter density parameter Ωmin Figure 6.15 we compiled once more the finding of the\ncosmological models considered and for the Pl18 (left) and Pl18+BAO+SN (right) combinations\nof datasets. We depict both the flat cases (top) and the curved ones (bottom), for which a trend\nis readily identified.\nFor the Pl18-only case, we see that the addition of spatial curvature brings Ωmtoward consider-\nably large values. Even though all the models predict values consistent within their categories,\nwe see that the curved coupled quintessence models have enlarged error bars compared with\nΛCDM which, in some cases, allows to accommodate for more reasonable values. Once more,\nthe addition of the background data brings the constraints closer to the flat prediction, in striking\ndisagreement with the Pl18-case. The error bars for the curved coupled quintessence models ate\nconsiderably larger in comparison with both flat and curved ΛCDM. This gives more flexibility in\npredicting lower values of Ωmwhich could yield a better fit to the data, at the cost of increasing\nthe disagreement with the Pl18-only cases.6Coupled Quintessence in Flat and Curved Geometries 153\n0.7 0.8 0.9 1.0 1.1 1.2\nS8M3 + k, 1.027+0.055\n0.038\nM3+ + k, 1.041+0.058\n0.043\nM2 + k, 1.019+0.056\n0.038\nM2+ + k, 1.021+0.058\n0.039\nM1 + k, 1.026+0.051\n0.038\nM1+ + k, 1.014+0.054\n0.043\nCDM + k, 1.003+0.054\n0.047\nM3, 0.839+0.018\n0.021\nM3+, 0.96±0.12M2, 0.827±0.018\nM2+, 0.828±0.018M1, 0.834±0.018\nM1+, 0.839±0.022CDM, 0.833±0.016\nDES-Y3, 0.759+0.025\n0.023\nKiDS-1000, 0.759+0.024\n0.021\nFlat\nCurvedPl18\n0.70 0.75 0.80 0.85 0.90\nS8M3 + k, 0.838+0.012\n0.027\nM3+ + k, 0.836+0.014\n0.021\nM2 + k, 0.834+0.014\n0.017\nM2+ + k, 0.835+0.015\n0.018\nM1 + k, 0.834+0.015\n0.018\nM1+ + k, 0.845+0.017\n0.023\nCDM + k, 0.822±0.013\nM3, 0.823+0.014\n0.013\nM3+, 0.849+0.016\n0.034\nM2, 0.825±0.013\nM2+, 0.826±0.013M1, 0.824±0.013\nM1+, 0.827±0.013CDM, 0.823±0.013\nDES-Y3, 0.759+0.025\n0.023\nKiDS-1000, 0.759+0.024\n0.021\nFlat\nCurvedPl18 + BAO + SN\nFigure 6.14 : Whisker plot with the 68%CL constraints on the parameter S8≡σ8p\nΩm/0.3\nderived for the cosmological models explored in this study for the Pl18 CMB data (left) and the\ncombination Pl18 + BAO + SN (right), as detailed in Section 4.3. Circle and squared markers\ndenote the flat and curved models. The red and green vertical bars illustrate the reference\nΛCDM flat ( ΩK= 0) and curved ΛCDM + ΩKvalues, respectively. The blue bar corresponds\nto the model-independent R22 value, reported in Ref. [111], in tension with the prediction for\nthe six-parameter flat ΛCDM from the Planckcollaboration 2018 data release [32], as explained\nin Section 3.6.2.\n6.4.4 Curvature\nThe findings related to the curvature density parameter, ΩK, are compiled in Figure 6.16 for the\ncosmological models considered and for the Pl18 (left) and Pl18+BAO+SN (right) combinations\nof datasets. We depict the flat cases (top) as dots in the plot for scale and comparison with the\ncurved ones (bottom).\nIn the baseline ΛCDM model using Planck data, the indication for a closed Universe is evident,\nillustrated by the green bar, and persists across all the non-flat extended models examined in\nthis work. However, this tendency varies in statistical significance due to the large error bars\nassociated with expanding the parameter space or adopting alternative parametrisations for\nthe potential and the coupling, even though this effect is not dramatic. This decrease in the\nconstraining power is mainly attributed to the strong geometrical degeneracy between different\nparameters, most notably between the coupling parameter and the curvature density parameter\nΩK, as seen by the lack of correlation in Figures 6.8, 6.10 and 6.12.\nBreaking such degeneracies changes the predictions drastically, as it is evident for recovering\nthe flatness condition in the full Pl18+BAO+SN combination. Introducing BAO large-scale\nstructure data always wipes out the evidence for a closed Universe. Regardless of the coupled\nquintessence model under consideration, constraints on ΩKtend to hover around zero, indicating154 6.4. Summary and Discussion\n0.2 0.4 0.6 0.8\nm\nM3 + k, 0.6+0.1\n0.16\nM3+ + k, 0.61+0.098\n0.16\nM2 + k, 0.524+0.081\n0.1\nM2+ + k, 0.536+0.08\n0.12\nM1 + k, 0.6+0.1\n0.15\nM1+ + k, 0.559+0.079\n0.15\nCDM + k, 0.514±0.069\nM3, 0.328+0.012\n0.032\nM3+, 0.311±0.01M2, 0.303+0.022\n0.01\nM2+, 0.305+0.02\n0.013\nM1, 0.321+0.021\n0.024\nM1+, 0.316+0.022\n0.045\nCDM, 0.316±0.0084\nFlat\nCurvedPl18\n0.26 0.28 0.30 0.32 0.34\nm\nM3 + k, 0.3+0.015\n0.008\nM3+ + k, 0.304+0.01\n0.0079\nM2 + k, 0.297+0.015\n0.0094\nM2+ + k, 0.298+0.014\n0.0087\nM1 + k, 0.294+0.018\n0.011\nM1+ + k, 0.296+0.014\n0.011\nCDM + k, 0.3086±0.0065\nM3, 0.3067+0.0092\n0.0062\nM3+, 0.311±0.0063M2, 0.3045+0.0085\n0.0073\nM2+, 0.3051+0.0079\n0.007\nM1, 0.3052+0.0087\n0.0079\nM1+, 0.3068+0.0084\n0.0074\nCDM, 0.3101±0.0058\nFlat\nCurvedPl18 + BAO + SN\nFigure 6.15 : Whisker plot with the 68%CL constraints on the matter density parameter Ωm\nobtained for the cosmological models explored in this study for the Pl18 CMB data (left) and the\ncombination Pl18 + BAO + SN (right), as detailed in Section 4.3. Circle and squared markers\ndenotethe flatandcurvedmodels. Theredand green verticalbars illustrate thereference ΛCDM\nflat ( ΩK= 0) and curved ΛCDM + ΩKvalues, respectively.\nspatial flatness within one standard deviation or slightly more. However, it is important to note\nthat BAOs are generally in tension with Planck when curvature is a free parameter, making this\ndata combination less robust. On the other hand, for Pl18 combined with the distance moduli\ndata from Pantheon (Pl18+SN), there is a systematic preference for a spatially closed geometry\nfor coupled quintessence models.\n6.4.5 Discussion\nIn this study, we explored whether the recently identified anomalies in CMB experiments, which\nare independent of Planck data, hold up when contrasted with other non-CMB observations,\nparticularly when we extend the parameter space. In other words, when we shift to more com-\nprehensive cosmological models allowing for additional variable parameters, does the evidence\nfor a flat Universe with a cosmological constant and no interactions in the dark sector endure?\nTo make the analysis more robust we have considered the full Plancklikelihood, with standard\nand nuisance parameters. Resolving these issues, as argued for in Section 3.6, is essential to\ngive physical meaning to the origin of the cosmological discrepancies and formulate coherent\nalternative frameworks.\nWe find consistent discrepancies in the coupled quintessence models when compared to the\nconcordance case which, while generally yielding a better fit to the data due to the increase\nparameter space, an analysis based on comparing the Bayesian evidence shows that the addition\nof the degrees of freedom is not justified. This leads us to conclude that the tensions in the data6Coupled Quintessence in Flat and Curved Geometries 155\n0.10\n 0.05\n 0.00 0.05\nk\nM3 + k, -0.066+0.024\n0.021\nM3+ + k, -0.067+0.02\n0.023\nM2 + k, -0.062±0.02\nM2+ + k, -0.061±0.02\nM1 + k, -0.069+0.024\n0.022\nM1+ + k, -0.059+0.023\n0.018\nCDM + k, -0.052+0.02\n0.017\nM3, 0.0\nM3+, 0.0M2, 0.0\nM2+, 0.0M1, 0.0\nM1+, 0.0CDM, 0.0\nFlat\nCurvedPl18\n0.02\n 0.01\n 0.00 0.01\nk\nM3 + k, -0.0035+0.0067\n0.0025\nM3+ + k, -0.002+0.0048\n0.0024\nM2 + k, -0.0038+0.006\n0.003\nM2+ + k, -0.0037+0.006\n0.0029\nM1 + k, -0.0048+0.0072\n0.0038\nM1+ + k, -0.0068+0.0074\n0.0043\nCDM + k, 0.001±0.0019\nM3, 0.0\nM3+, 0.0M2, 0.0\nM2+, 0.0M1, 0.0\nM1+, 0.0CDM, 0.0\nR22, 73.04±1.04\nFlat\nCurvedPl18 + BAO + SN\nFigure 6.16 : Whisker plot with the 68%CL constraints on the curvature parameter ΩKob-\ntained for the cosmological models explored in this study for the Pl18 CMB data (left) and the\ncombination Pl18 + BAO + SN (right), as detailed in Section 4.3. Circle and squared markers\ndenote the flat and curved models. The red and green vertical bars illustrate to the reference\nΛCDM flat ( ΩK= 0) and curved ΛCDM + ΩKcases, respectively, illustrating the evidence for\na closed Universe in the Planck data, as explained in Section 3.6.4.\nseem to genuinely favour the diverging patterns.\nGiven this, if we choose to maintain faith in the ΛCDM model, we must believe and prove\nthat overlooked systematics could be skewing the data. Statistical fluctuations might explain\nthe discrepancies related with the self-consistency between data collected based on the early-\nUniverse (mostly CMB) and the distance ladder methods in the late Universe. Indeed, within\ntheΛCDM model, the CMB experiments are largely in agreement about the expansion rate,\nleaving the conflict with local measurements unresolved. The second scenario involves taking the\ndata at face value, suggesting that solutions to current tensions could lie beyond the standard\nΛCDM model. Given our limited understanding of the nature of the dark sector, this option\ndeserves serious consideration. Yet, no combination of extra parameters entirely harmonises\nthe conflicting data, hinting that a more radical overhaul of cosmological paradigms may be\nrequired. While our analysis doesn’t offer definitive answers about these tensions and anomalies,\nit highlights unresolved discrepancies in the standard model that remain present even for more\naccommodating models like coupled quintessence. The forthcoming generation of high-precision\nCMB and large-scale structure measurements might provide more conclusive constraints and\ninsights on the origins of the cosmic tensions.7Kinetically Coupled Dark Energy\nDava-se melhor com um irreal cotidiano, vivia em câmara leeeenta, lebre puuuuulando no aaaar\nsobre os ooooouteiros, o vago era o seu mundo terrestre, o vago era o de dentro da natureza.\n—Clarice Lispector A Hora da Estrela\nShe could deal better with her daily unreality, living in sloooow motion, hare leeeeaping through the aaaair over\nhiiiill and daaaale, vagueness was her earthly world, vagueness was the insides of nature. —Clarice Lispector\nin The Hour of the Star\nIn this chapter, we study a particular extension in which dark energy is portrayed by a canonical\nscalar field, coupled to dark matter through an interaction term in the action, as proposed in\nRef. [275], and which we review in Section 7.1. In Sections 7.2 and 7.3, we perform a thorough\nnumericalanalysisofthedynamicsofthemodelunderconsideration, bothatthebackgroundand\nlinear perturbative levels. The constraints for the cosmological and model-specific parameters\naccording to different data sets are reported in Section 7.4 from which we find that although the\ncosmic tensions persist in the best-fitrealisation of the model, there is a non-zero prediction for\ndeviations from the standard model encoded by the coupling parameter. Computation of the\nBayesian evidence indicates no significant preference for the Kinetic model. We conclude with a\ndiscussion of the results in Section 7.5. This work has been published in JCAP and can be found\nin Ref. [1].\n7.1 Theoretical Motivation\nWebeginbyconsideringaphenomenologicaltheorywheredarkenergyisadynamicalquintessence\nfield, denoted as ϕ, minimally coupled to gravity. The DE source portrays a non-universal cou-\npling to the dark matter component as expressed in the following action [275]:\nS=Z\nd4x√−g\u0014M2\nPl\n2R+X−V(ϕ) +f(X)˜Lc(ζ, gµν) +LSM(ψi, gµν)\u0015\n. (7.1.1)\nHere, grepresents the determinant of the metric tensor gµν,Ris the curvature scalar, and\nM2\nPl= (8πG)−1correspondstothePlanckmassinunitswhere c= 1,with GbeingtheNewtonian\nconstant. The second and third terms in the action represent the Lagrangian of the scalar field,\nwhere X=−gµν∂µϕ∂νϕ/2is the kinetic term of ϕ, and V(ϕ)is the self-interacting potential of\nthe scalar field. The standard model is further extended by the introduction of a purely kinetic\nfunction f(X)that multiplies the Lagrangian of cold dark matter [275], leading to a coupling\nbetween ϕand the dark matter fields ζ.\n157158 7.1. Theoretical Motivation\nVariation of the action in Equation (7.1.1) with respect to the metric gµνyields the following\nfield equations\nM2\nPlGµν=T(ϕ)\nµν+T(c)\nµν+T(b)\nµν+T(r)\nµν, (7.1.2)\nwith Gµνbeing the Einstein tensor and T(i)\nµνthe energy-momentum tensor for the ith species,\ndefined as:\nT(i)\nµν=−2√−gδ(√−gLi)\nδgµν, (7.1.3)\nwhere i=ϕ, c, b, randcdenotes the cold dark matter, bthe baryons, and rthe radiation. Let\nus note that, for the previous definition to be valid for all the fluids present in theory, we define\nan effective dark matter Lagrangian as follows [275, 377, 378],\nLc≡f(X)˜Lc, (7.1.4)\nincorporating the effect of the coupling.\nThe theory’s matter components can be modelled as perfect fluids, with energy density ρi,\npressure pi,andEquationofState(EoS)parameter wi=pi/ρi. Therefore,theenergy-momentum\ntensor of each ith species becomes fully defined in terms of the fluid variables:\nT(i)\nµν=ρih\n(1 +wi)u(i)\nµu(i)\nν+wigµνi\n, (7.1.5)\nwith u(i)\nµbeing the 4-velocity vector associated with the ith species, under the individual con-\nstraint gµνu(i)\nµu(i)\nν=−1. Regarding the EoS parameter, we have wr= 1/3for radiation and\nwb=wc= 0for baryons and cold dark matter, respectively. Given these considerations, the\ndark matter Lagrangian takes the particular form [377, 379],\nLc=−ρc. (7.1.6)\nThe scalar field admits a perfect fluid description as well [380], provided that\nu(ϕ)\nµ=−∂µϕ√\n2X, (7.1.7)\nandX > 0, where the energy density and pressure associated with the quintessence field are\ngiven by:\nρϕ=X+V , (7.1.8)\npϕ=X−V . (7.1.9)7Kinetically Coupled Dark Energy 159\nand the scalar field EoS parameter is wϕ=pϕ/ρϕ.\nThe equation of motion for the quintessence field, or simply the Klein-Gordon equation, is\nobtained through variation of the action in Equation (7.1.1) with respect to ϕand reads:\n□ϕ−V,ϕ=−Q , (7.1.10)\nwith V,ϕ=dV\ndϕ. The term on the right-hand side of Equation (7.1.10) includes the interaction\nin the dark sector in terms of f(X)[275], and may be expressed generally in terms of fand its\nderivatives\nQ=−Lc\u001af,X\nf\u0014\n□ϕ+∂µϕ\u0012∇µLc\nLc+f,X\nf∂αϕ∇µ∂αϕ\u0013\u0015\n−f,XX\nf∂µϕ∂αϕ(∇µ∂αϕ)\u001b\n.\n(7.1.11)\nwhere f,X≡d f\ndXandf,XX≡d2f\ndX2. The uncoupled case ( Q= 0) is naturally recovered when f\nis a constant function. Let us note that Equation (7.1.10) could likewise be found through the\ncontracted Bianchi identities, yielding the following conservation relations,\n∇µT(c)µν=−∇µT(ϕ)µν=Q∇νϕ . (7.1.12)\nThese equations illustrate clearly the energy transfer between the scalar field and DM when f\nis not a constant, meaning that the dark components are not individually conserved. However,\nsince radiation and baryons remain non-interacting, i.e.,\n∇µT(r)µν=∇µT(b)µν= 0, (7.1.13)\nthen, consistently, the overall energy-momentum tensor of the theory is conserved, rendering the\ntotal action covariant.\nIn particular, we focus on the most straightforward power-law kinetic interaction, as motivated\nin Ref. [275], described by the function:\nf(X) =\u0000\nM−4\nPlX\u0001α=⇒ Q=−ρcα\nX\u0012\n□ϕ+∂µϕ∂νϕ∇µ∂νϕ\nX+∂µϕ∂µρc\nρc\u0013\n,(7.1.14)\nwhere αis a dimensionless constant governing the strength of the coupling Qin the dark sector,\nandρcrepresents the energy density of the cold dark matter. We assume a simple exponential\npotential for the field:\nV(ϕ) =V0exp\u0012\n−λϕ\nMPl\u0013\n, (7.1.15)\nwhere V0represents the energy scale of the potential, and λcharacterises its steepness. These\nchoices for the coupling function and potential are motivated by the desire to have a scaling\nregime [275] at early times, followed by an accelerated expansion driven by the scalar field. The\nexponential potential drives the system out of the scaling solution and towards the late-time160 7.2. Background Dynamics\nattractor, as described in Section 5.2.1.\nIt should be noted that the action described by Equation (7.1.1) is mathematically equivalent\nto a scalar-tensor theory in the Einstein frame with a conformal coupling in the DM action\nSc[˜gµν(X), ζ]. The coupling arises viathe conformal function C(X) =f2(X)in the metric\ntransformation ˜gµν=C(X)gµν, with a fundamentally different physical interpretation.\n7.2 Background Dynamics\nWe assume cosmological dynamics in a flat Friedmann-Lemaître-Robertson-Walker (FLRW)\nbackground, expressed by the scale factor of the Universe a(τ)in conformal time τ, as defined\nin Equation (1.6.6). The equations governing the background evolution are obtained through\nvariation of the action according to the metric and the scalar degree of freedom, namely the\nmodified Friedmann equation and conservation relations:\n3M2\nPlH2=a2(ρc+ρb+ρr+ρϕ), (7.2.1)\nϕ′′+ 2Hϕ′+a2V,ϕ=a2Q , (7.2.2)\nρ′\nc+ 3Hρc=−Qϕ′, (7.2.3)\nρ′\nb+ 3Hρb= 0, (7.2.4)\nρ′\nr+ 4Hρr= 0, (7.2.5)\nrespectively, and with′≡d/dτandH=a′\nais the conformal Hubble rate. The coupling Qgiven\nin Equation (7.1.14) becomes:\nQ= 2αρc3Hϕ′+a2V,ϕ\n2αa2ρc+ (1 + 2 α)ϕ′2. (7.2.6)\nWe can further define the energy density and pressure of the ϕfield at the background level\nthrough Equations (7.1.8) and (7.1.9), as\nρϕ=ϕ′2\n2a2+V, (7.2.7)\npϕ=ϕ′2\n2a2−V. (7.2.8)\nrespectively. Therefore Equation (7.2.2) can be written as:\nρ′\nϕ+ 3H(1 +wϕ)ρϕ=Qϕ′. (7.2.9)\nEquations (7.2.3) and (7.2.9) imply that when Qϕ′>0, energy is transferred from the cold dark\nmatter source to the scalar field, and vice versa when Qϕ′<0.7Kinetically Coupled Dark Energy 161\nAt the classical level, the energy exchange in the dark sector may be interpreted as a mass varia-\ntion for dark matter particles since mc=a3ρc, assuming conservation of the number of particles,\ni.e.Nc=Nc(τ0), with τ0being the present conformal time. Integration of Equation (7.2.3)\nyields an expression for the total energy density of coupled dark matter,\nρc=ρc(τ0)a−3exp\u0012\n2αZτ\nτ0Qϕ′\nρcdτ\u0013\n, (7.2.10)\nthat can be expressed equivalently in terms of the mass of the dark matter particles:\nmc(τ) =mc(τ0) exp\u0012\n2αZτ\nτ0Qϕ′\nρcdτ\u0013\n. (7.2.11)\nFinally, let us note that the modified Friedmann equation, Equation (7.2.1), can be cast to the\nform of the well-known Friedmann constraint:\n1 = Ω ϕ+ Ω m+ Ω r, (7.2.12)\nwhere we have defined a collective matter density ρm=ρc+ρb, and the fractional density\nparameter of the i-th species Ωi=ρia2\n3M2\nPlH2. Equation (7.2.12) can be rewritten in the form of a\nconstraint on the present scalar field fractional density, Ω0\nϕ= 1−Ω0\nm−Ω0\nr, where “0” stands for\nquantities evaluated at present, Ω0\ni=ρ0\ni\n3M2\nPlH2\n0, and H0is the Hubble parameter. For numerical\npurposes, V0, implicitly entering the definition of Ω0\nϕ, is used to perform a shooting method\nthat yields the fiducial value of Ω0\nϕfulfilling the constraint relation in Equation (7.2.12), while\nsimultaneously avoiding degeneracies. As such, V0will no longer be considered a free parameter\nof the model, leaving {λ, α}as the model free parameters. The viable parameter space has been\nstudied in Ref. [1] according to dynamical and stability conditions, along with motivation for the\nparticular initial conditions for the scalar field used in the simulations. We conduct numerical\nsimulations using a modified version of the Einstein-Boltzmann solver CLASS[95, 101, 102] to\nstudy the predictions of the model over the expansion history for different {λ, α}combinations,\nusing standard Planck2018 [32] reference values for the cosmological parameters.\nIn the left panel of Figure 7.1, we present the evolution of the relative energy densities Ωi=\nρia2/(3M2\nPlH2)for each species with respect to redshift ( 1 +z). As expected, the introduction\nof the coupling leads to the emergence of an early scaling regime during the radiation-dominated\nepoch, where the energy density of the scalar field is proportional to that of dark matter, ap-\nproximately following the relation ρc/ρϕ= 1/α, as shown in the upper right panel of Figure 7.1.\nEventually, the field exits this scaling regime and transitions towards the future attractor solu-\ntion, causing its energy density to continuously dilute as ρϕ∝a−λ2. For the values considered,\nthe coupling strength Qϕ′/ρcremains positive at all redshifts, implying energy transfer from the\ndark matter fluid to the scalar field. It should be noted that by fixing the present-day values of\nfluid densities, the energy density of cold dark matter is relatively higher at earlier times due to162 7.3. Evolution of Linear Perturbations\nits contribution in supplying energy to the scalar field at later times. This effect becomes more\npronounced for larger values of α. As the DM energy density decreases over time, the additional\nenergy transferred to the scalar field compensates for this effect compared to the uncoupled case.\nThis behaviour is illustrated in the left panel of Figure 7.1. Consequently, we also see in the\nfigure that the matter-radiation equality occurs at earlier times as the value of αincreases. Fur-\nthermore, the ϕfield starts acquiring energy at a rate proportional to its energy density, and the\nequality between matter and dark energy occurs earlier. In the lower right panel of Figure 7.1,\nwe depict deviations in the Hubble rate for the Kinetic model and the uncoupled case ( α= 0)\ncompared to ΛCDM. We remark that no significant variations in Hare observed during the\nradiation-dominated epoch, as interactions between the dark and radiation sectors are excluded.\nHowever, as the matter contribution becomes significant, around z≈106, the Kinetic models\nexhibit an increased value of H, which becomes more pronounced for higher values of α.\n0 1 2 3 4\nlog(1+z)0.00.20.40.60.8ΩΩc(α=0.01)\nΩb(α=0.01)\nΩr(α=0.01)\nΩφ(α=0.01)Ωc(α=0.03)\nΩb(α=0.03)\nΩr(α=0.03)\nΩφ(α=0.03)\n04812log(ρc/ρφ)ΛCDM\nα=0.00\nα=0.01\nα=0.03\n0 1 2 3 4\nlog(1+z)0102030∆H\nH[%]\nFigure 7.1 :Left panel : Evolution of the relative energy densities Ωiwith redshift, 1 +z, of\nthe scalar field (green), dark matter (red), baryons (yellow) and radiation (grey) for ΛCDM\n(solid lines), α= 0.01(dashed lines) and α= 0.03(dotted lines). Upper right panel : Ratio of\nthe energy densities of cold dark matter and dark energy for ΛCDM (solid black), the uncoupled\ncaseα= 0(red dashed), α= 0.01(yellow dashed-dotted) and α= 0.03(green dotted). Lower\nright panel : Fractional deviation of the conformal Hubble expansion rate for the same examples.\n7.3 Evolution of Linear Perturbations\nWe describe the perturbed FLRW metric in the Newtonian gauge according to the standard\nline element, as defined in Equation (2.2.5). The linearised Einstein equations describe the\nevolution of perturbations for different scales in terms of independent Fourier modes and extra\ncontributions from the dark sector coupling.7Kinetically Coupled Dark Energy 163\nIn particular, from Equations (7.1.8) and (7.1.9), we derive the perturbations for the energy\ndensity and pressure of the scalar field:\nδρϕ=ϕ′\na2δϕ′−ϕ′2\na2Ψ +V,ϕδϕ , (7.3.1)\nδpϕ=ϕ′\na2δϕ′−ϕ′2\na2Ψ−V,ϕδϕ . (7.3.2)\nThe perturbations of the energy-momentum tensor, Equation (7.1.5), for each species and at\nfirst order, reads\nδT(i)µ\nν= (δρi+δpi)u(i)µu(i)ν+δpiδµ\nν+ (ρi+pi)\u0010\nδu(i)µu(i)ν+u(i)µδu(i)ν\u0011\n,(7.3.3)\nwhere δu(i)µis the perturbation on the four-velocity vector of the ith species, i.e. u(i)µ=\na(−1, v(i)j), with vjbeing the peculiar velocity. As introduced in Section 2.2.1, the perturbed\nEinstein equations assuming no shear are expressed as\nk2Φ + 3H\u0000\nΦ′+HΨ\u0001\n=−4πGa2X\niδρi, (7.3.4)\nk2\u0000\nΦ′+HΨ\u0001\n= 4πGa2X\niρi(1 +wi)θi,(7.3.5)\nΦ′′+H\u0000\nΨ′+ 2Φ′\u0001\n+ Ψ\u0000\nH2+ 2H′\u0001\n+k2\n3(Φ−Ψ) = 4 πGa2X\niδpi, (7.3.6)\nΦ = Ψ . (7.3.7)\nThe first equation, corresponding to the time-time component, provides the constraint on the\nenergy density. Equation (7.3.5), which is derived from the time-space components of the per-\nturbed Einstein equations, specifies the momentum constraint. Here, we adopt the definition of\nthe velocity divergence as θi=∇ ·v(i). The trace of the spatial components results in Equa-\ntion (7.3.6), and finally, Equation (7.3.7) relates to the propagation of shear in the absence of\nanisotropic stress. This is expected because of the absence of a non-minimal coupling in the\naction in Equation (7.1.1). The governing equations for the evolution of perturbations in each\nfluid can be obtained via the conservation relations, specifically Equations (7.1.12) and (7.1.13),\nat the first order of perturbation. For species that do not interact, namely baryons and radiation,\nthese equations are as follows:164 7.3. Evolution of Linear Perturbations\nδ′\ni+ 3H\u0012δpi\nδρi−wi\u0013\nδi+ (1 + wi)\u0000\nθi−3Φ′\u0001\n= 0, (7.3.8)\nθ′\ni+\u0014\nH(1−3wi) +w′\ni\n1 +wi\u0015\nθi−k2\u0012\nΨ +δpi\nδρiδi\n1 +wi\u0013\n= 0. (7.3.9)\nFor the coupled DM, these become\nδ′\nc+θc−3Φ′=Q\nρc\u0000\nϕ′δc−δϕ′\u0001\n−ϕ′\nρcδQ , (7.3.10)\nθ′\nc+Hθc−k2Ψ =Q\nρc\u0000\nϕ′θc−k2δϕ\u0001\n, (7.3.11)\ndescribing the evolution of the density contrast, denoted as δi=δρi/ρiand the velocity diver-\ngence θi=∇ ·v(i). The perturbed coupling term is defined as\nδQ=2αρc\n2αa2ρc+ (1 + 2 α)ϕ′2\b\n−3Φ′ϕ′−ϕ′θc+\u0002\n3Hϕ′+a2(V,ϕ−Q)\u0003\nδc+\u0000\n2k2+a2V,ϕϕ\u0001\nδϕ\n−\u0002\n3Hϕ′+ 2a2(V,ϕ−Q)\u0003δϕ′\nϕ′+ 2a2Ψ (Q−V,ϕ)\u001b\n,(7.3.12)\nand we remark the explicit dependence of δQon the velocity potential θc, which is unusual\ncompared to other coupled dark energy models [3, 344]. Finally, the linearisation of the Klein-\nGordon equation gives:\nδϕ′′+ 2Hδϕ′+\u0000\na2V,ϕϕ+k2\u0001\nδϕ−\u0000\nΨ′+ 3Φ′\u0001\nϕ′+ 2a2ΨV,ϕ=a2δQ+ 2a2QΨ.(7.3.13)\nIncluding the coupling leaves an imprint on important cosmological observables that can be\nprobed against different data, particularly the matter power spectrum and the temperature-\ntemperature (TT) and lensing angular power spectra of the cosmic microwave background\n(CMB). Again, assuming standard Planck2018 values for the shape of the scalar primordial\npower spectrum [32] as well, it is possible to single out the model-specific signatures by taking\ndifferent sets of {λ, α}relevant for the scales under consideration.\nIn the left panel of Figure 7.2, we present the linear matter power spectrum at present up to the\nscale kmax= 0.1hMpc−1along with the fractional differences for the Kinetic model in contrast\nwith ΛCDM. We identify an overall suppression at intermediate scales ( 10−3hMpc−1≲k≲\n3×10−2hMpc−1), followed by an enhancement at smaller scales. This effect is primarily related\nto the global deviation of the turnover in the matter power spectrum to higher kvalues due to\nthe shift of the radiation-matter equality era to earlier times, as identified in the study of the\nbackground evolution. The positive exchange of energy from cold dark matter to dark energy at\nlate times inevitably suppresses the growth of matter perturbations at intermediate scales and\nan enhancement at smaller scales. The highest deviations are observed for larger values of α.7Kinetically Coupled Dark Energy 165\nConsequently, the amplitude of the matter power spectrum at present and a scale of 8 h−1Mpc,\ndenoted by σ8, is expected to be larger for the Kinetic model.\nThe inclusion of the coupling is also reflected in the evolution of the gravitational potentials,\nparametrised in terms of the lensing potential ϕlens= (Ψ+Φ) /2for particular scales, as depicted\nin the right panel of Figure 7.2. We identify an overall suppression in ϕlenswhich in turn leads\nto a suppression of the lensing power spectrum as well, as demonstrated in the right panel of\nFigure 7.3, with this effect becoming more pronounced for larger values of α. On the other\nhand, the time variation of ϕlensis directly related to the integrated Sachs-Wolfe effect (ISW),\nimprinted in the shape of the temperature-temperature (TT) power spectrum as a contribution\nto the radiation transfer function, and is illustrated in the left panel of Figure 7.3 as a function of\nthe angular multipole ℓ. This effect comprises two contributions: an early-time term occurring\nduring the transition from the radiation- to matter-dominated epochs, shifted to earlier times\nin the Kinetic model, and a late-time term associated with the dynamics of the dark energy\ncomponent. We observe an apparent overall enhancement compared to the reference case for\nℓ≲300, with milder differences around the plateau at ℓ <10and the most significant deviations\naround ℓ∼50. Furthermore, there is an apparent increase in the amplitude of the first peak\naccompanied by a broadening of its shape and slight variations between the peaks and troughs at\nhigher multipoles. The validity of these combined deviations can be assessed with cosmological\ndata from background observations and the large-scale structure.\n103104P(k)[Mpc/h]3\nΛCDM\nα=0.000α=0.001\nα=0.002\n10-310-210-1\nk[h/Mpc]30\n03060∆P(k)\nP(k)[%]\n0.20.40.60.81.01.21.4Φ+Ψ\nΛCDM\nα=0.000\nα=0.001\nα=0.002\n0 1 2 3 4\nlog(1+z)90\n60\n30\n0∆(Φ+Ψ)\n(Φ+Ψ)[%]\nFigure 7.2 :Left panel : The matter power spectrum as function of k, for the uncoupled case\n(dashed red line), α= 0.001(yellow dot-dashed line), α= 0.002(green dotted line) and ΛCDM\n(black solid line), along with the percentage deviations from the ΛCDM case (lower panel).\nRight panel : Evolution of the sum of the gravitational potentials as a function of the redshift at\nk= 0.01Mpc−1for the same examples.166 7.4. Observational Constraints\n02468/lscript(/lscript+1)CTT\n/lscript/2π×1010\nΛCDM\nα=0.000\nα=0.001\nα=0.002\n101102103\n/lscript0153045∆CTT\n/lscript\nCTT\n/lscript[%]\n0.00.51.01.52.0[/lscript(/lscript+1)]2Cφφ\n/lscript/2π×107\nΛCDM\nα=0.000α=0.001\nα=0.002\n101102103\n/lscript100\n0100200∆Cφφ\n/lscript\nCφφ\n/lscript[%]\nFigure 7.3 :Left panel : TT power spectrum of anisotropies as a function of the angular scale\nℓ, for the uncoupled case (dashed red line), α= 0.001(yellow dot-dashed line), α= 0.002(green\ndotted line) and ΛCDM (black solid line) for reference, along with percentage deviations with\nrespect to ΛCDM (lower panel). Right panel : Lensing angular power spectra for the same\nexamples and relative difference between the predictions for each model and for ΛCDM.\n7.4 Observational Constraints\n7.4.1 Data Sets\nThe baseline data set considered is the CMB Planck2018 (Pl18) [32] henceforth to which we\nadd BAO data from the Sloan Digital Sky Survey (SDSS) DR7 Main Galaxy Sample [117], SDSS\nDR12 consensus release [116] and the 6dF Galaxy Survey [118], in combination with distance\nmodulimeasurements of type Ia Supernova (SN) data from Pantheon [109](Pl18+BAO+SN). Fi-\nnally, we included CMB lensing potential data from Planck2018 [51, 178] (Pl18len+BAO+SN).\nFurther details are provided in Section 4.3. We use a set of free sampling parameters consisting\nof the baseline ΛCDM cosmological parameters (Ωbh2,Ωch2, zreio, θs, As, ns)as detailled in Sec-\ntion 3.5, as well as the two free parameters associated with the Kinetic model (α, λ). We consider\nflat priors for all the parameters and provide the specific range of values in Table 7.1. Our analy-\nsis also yields derived constraints on H0andS8=σ8p\nΩm/0.3. The latter is also known to be in\ntension with cosmic shear measurements [57, 145, 164] for the standard model, with CMB data\nfavouring higher values. Finally, to produce the Monte Carlo Markov Chain (MCMC) samples,\nwe follow our modification of the Einstein Boltzmann solver CLASS[95, 101, 102] interfaced with\ntheMontePython sampler [213, 214], following the Metropolis-Hastings algorithm. Subsequently,\nwe analyse the MCMC chains and produce the results reported in Ref. [1], with the aid of the\nGetDist Python package [216].7Kinetically Coupled Dark Energy 167\nParameter Prior\nΩbh2[0.005,0.1]\nΩch2[0.001,0.99]\n100θs [0.5,10]\nτreio [0.01,0.8]\nns [0.7,1.3]\nlog\u0000\n1010As\u0001\n[1.7,5.0]\nλ [0,2]\nα [0,1]\nTable 7.1 : Flat priors on the cosmological and model parameters sampled in this work.\n7.4.2 Results\nIn Tables 7.2 and 7.3, we present the constraints on the sampled parameters for the Kinetic and\nΛCDM models, respectively. These results are illustrated in Figure 7.4 where the corresponding\n2D marginalised contour plots for all the considered combinations of data sets are displayed. Al-\nthough the constraints on the cosmological parameters of the Kinetic model are compatible with\nthose of the ΛCDM case within the uncertainties, the latter yields higher and lower mean values\nforH0andΩm, respectively, across all the three data combinations considered. This tendency\nis consistent with the dominating effects of the coupling in the TT power spectrum as depicted\nin Figure 7.3, following the analysis provided in Section 3.5. Furthermore, Figure 7.4 depicts\ncontour plots for the constraints in the H0−ΩmandS8−Ωmplanes. The Kinetic model under\nthe Pl18 data predicts H0= 64.0+3.3\n−1.8, with a lower mean value ( 67.31) than in ΛCDM, thereby\napparently worsening the tension with late-time distance-ladder measurements, as detailed in\nSection 3.6. However, this discrepancy is attenuated by larger error bars ( 0.61inΛCDM), lead-\ning to an artificial reduction of the H0tension from ∼4.8σto∼3.3σ. The compatibility with\nΛCDM is restored when considering other data sets, indicating tensions between the BAO and\nSN data within this framework, just as was found in Chapter 6. A similar situation in a Galileon\nmodel [381] suggests a potential bias towards ΛCDM-like models in the BAO data [382]. With\nonly the Pl18 data, we find S8= 0.921+0.044\n−0.034at a68%confidence level (CL) in the Kinetic model,\nwith a higher mean value than in the corresponding ΛCDM case, but also accompanied by larger\nerror bars, with the same tendency persisting across all the data set combinations. Even though\na lower value is expected for mitigating the discordance with cosmic shear measurements in the\nstandard model ( S8= 0.833±0.016), a full reanalysis of the weak lensing data under the Kinetic\nmodel would be needed to evaluate the tension.168 7.4. Observational Constraints\nThe results for the specific parameters of the Kinetic model can be appreciated in the 2D contour\nplotsofFigure7.5, whereweseethatthecouplingparameter αisconsistentlyconstrainedtobeof\nthe order 10−4, regardless of the data set combination. When considering only the Planckdata, a\nhighermeanvalueof αispreferred,primarilytoaccommodatetheTTlikelihoodbetter. However,\nincorporating BAO and SN data slightly reduces the mean value of α. Furthermore, adding CMB\nlensing data shifts the peak of the posterior distribution for αto an even lower central value.\nThis behaviour can be traced to the reported lensing excess by the Planckcollaboration [32, 383].\nIn the Kinetic model, the lensing power spectrum is always suppressed compared to the ΛCDM\nscenario, with higher values of αcorresponding to lower amplitudes. Consequently, a lower mean\nvalue of αis favoured to match the CMB lensing data better. Including BAO and SN data leads\nto narrower constraints on Ωm, resulting in tighter constraints on other parameters such as H0,\nS8, and λ.\nParameter Pl18 Pl18 + BAO + SN Pl18len + BAO + SN\nΩbh20.02236 ±0.00015 0 .02242 ±0.00013 0 .02243 ±0.00013\nΩcdmh20.1202±0.0014 0 .11937 ±0.00091 0 .11935 ±0.00092\n100θs 1.04188 ±0.00030 1 .04195 ±0.00028 1 .04196 ±0.00029\nτreio 0.0542±0.0078 0 .0554+0.0071\n−0.0079 0.0573±0.0074\nns 0.9652±0.0044 0 .9672±0.0036 0 .9672±0.0037\nln\u0000\n1010As\u0001\n3.046±0.016 3 .046±0.016 3 .050±0.015\nσ8 0.8115±0.0075 0 .8093±0.0071 0 .8110±0.0061\nΩm 0.3149±0.0085 0 .3095±0.0054 0 .3093±0.0055\nS8 0.831±0.016 0 .822±0.012 0 .824±0.011\nH0 67.31±0.61 67 .68±0.40 67 .71±0.42\nTable 7.2 :68%CL bounds on the cosmological parameters for the ΛCDM model for the three\ndifferent combinations of data sets: Planck2018,Planck2018 combined with BAO and SN, and\ntheir full combination with CMB lensing.\n7.4.3 Model Selection Analysis\nLastly, we aim to evaluate whether the Kinetic model is favoured over the ΛCDM case, using dif-\nferent statistical indicators for comparison purposes. First, we consider the effective χ2-statistics,\ncorresponding to the maximum likelihood, denoted as χ2\neff, which allows us to determine how well\nthe Kinetic model fits the data when compared to ΛCDM. This is accomplished by calculating7Kinetically Coupled Dark Energy 169\n58626670\nH00.300.350.40m\n0.80.91.0\nS8Pl18 Pl18+BAO+SN Pl18len+BAO+SN\nKinetic\n6566676869\nH00.300.320.34m\n0.780.820.860.90\nS8CDM\nFigure7.4 : 68% and 95% CL 2D contours derived for the parameter combinations H0-Ωm(left\npanels) and S8-Ωm(right panels) in the Kinetic model (upper panels) and ΛCDM model (lower\npanels) for the Planck2018 data (green), the Planck2018, BAO and SN combination (yellow),\nand their combination with CMB lensing (red).\n∆χ2\neff=χ2\neff,Kinetic−χ2\neff,ΛCDM, where a negative value indicates support for the Kinetic model,\nwhile a positive value suggests otherwise. On the other hand, to statistically compare the level\nof support for one model against the other, we calculate the Bayes factor of the Kinetic model\nrelative to ΛCDM, as introduced in Equation (4.1.3). The greater the evidence for the Kinetic\nmodel relative to ΛCDM, the larger the Bayes factor ratio lnBK,ΛCDMwill be. The numerical\nBayes factor may be translated into a qualitative statement about the strength of evidence for\nan extended model against ΛCDM through the Jeffreys scale in Table 4.1.\nIn the last two rows of Table 7.3, we present the values for both the ∆χ2\neffand the lnBK,ΛCDM,\nalong with the corresponding individual values. From the analysis, we find that when considering\nonly the Pl18 data, the Kinetic model shows a slightly better fit to the data compared to the170 7.4. Observational Constraints\n606570\nH00246104\n0.80.91.0\nS80.300.350.40\nm\n0.00.51.01.52.0\nPl18 Pl18+BAO+SN Pl18len+BAO+SN\nFigure 7.5 :68%and95%CL contours obtained in the Kinetic model for the Planck2018 data\n(green), the Planck2018, BAO and SN combination (yellow), and their combination with CMB\nlensing (red).\nParameter Pl18 Pl18 + BAO + SN Pl18len + BAO + SN\nΩbh20.02226 ±0.00016 0 .02234 ±0.00015 0 .02236 ±0.00014\nΩcdmh20.1195±0.0015 0 .1186±0.0011 0 .11911 ±0.00098\n100θs 1.04196 ±0.00030 1 .04203 ±0.00028 1 .04200 ±0.00029\nτreio 0.0534±0.0079 0 .0555±0.0079 0 .0573±0.0074\nns 0.9667±0.0047 0 .9688±0.0040 0 .9675±0.0038\nln\u0000\n1010As\u0001\n3.039±0.017 3 .041±0.017 3 .052±0.015\nλ 1.11±0.48 0 .45+0.18\n−0.21 0.43+0.18\n−0.20\n104α 1.88±0.95 1 .57+0.79\n−1.00 1.14+0.55\n−0.92\nσ8 0.858±0.042 0 .884+0.042\n−0.038 0.869+0.033\n−0.043\nΩm 0.347+0.019\n−0.037 0.3102±0.0065 0 .3128±0.0061\nS8 0.921+0.044\n−0.034 0.899+0.043\n−0.039 0.887+0.036\n−0.045\nH0 64.0+3.3\n−1.8 67.41+0.59\n−0.50 67.26+0.55\n−0.47\n∆χ2\neff−0.9 0 .7 1 .0\nlnBK,ΛCDM −4.2 −5.4 −6.8\nTable 7.3 :68%CL bounds on the cosmological and model parameters for the Kinetic model\nfor the three different combinations of data sets: Planck,Planckcombined with BAO and SN,\nand their full combination with CMB lensing.\nΛCDM model, with a decrease in the chi-squared value of ∆χ2=−0.9. However, this preference7Kinetically Coupled Dark Energy 171\ndisappears when other data sets are included. This is mainly due to the BAO and SN data affect-\ning the fit to the temperature-temperature (TT) likelihood, which worsens after incorporating\nthe CMB lensing data. The Kinetic model predicts a suppressed lensing amplitude, whereas the\nCMB lensing data indicates an excess of power. However, any support for the Kinetic model be-\ncomes insignificant when considering the Bayesian evidence, for which we report negative values\nacross all the data sets. This is directly related to the increased complexity of the Kinetic model,\nincluding two additional parameters resulting in either reduced or no improvement in the data\nfitting. Therefore, we can conclude that this analysis does not affirm statistical evidence for the\nKinetic model.\n7.5 Discussion\nThisstudyinvestigatedaKineticmodelconsistingofacoupledquintessencetheorywithapower-\nlaw kinetic interaction. We explored its impact on the evolution of the background and linear\nperturbations in the Universe. We analysed the relevant cosmological observables and derived\nconstraints on the model parameters using various data sets, namely CMB, CMB lensing, BAO,\nand SN data.\nWe first performed a numerical study of the model’s predictions to assess the valid parameter\nspace with physical interest. We used a modified version of the publicly available Einstein-\nBoltzmann code CLASS. At the background level, we found that a non-zero value of αallows\nfor a scaling regime during the radiation-dominated epoch, where the ratio of the densities of\ncold dark matter and the scalar field approximately scale with α. We identified that only energy\ntransfer from DM to DE can be realised in such scenarios. Additionally, the radiation-matter\nequality is shifted towards earlier times, with direct consequences on the matter power spectrum,\nnamely a suppression of small-scale power and an enhancement of large-scale growth compared\nto the ΛCDM model. Consequently, we observed an overall suppression of the lensing potential\nand modifications in the ISW effect, which alter the shape of the temperature power spectrum\nof anisotropies at large angular scales.\nUsing an MCMC method for cosmological parameter extraction, we applied the theoretical in-\nsight gained in the numerical study to constrain the model. We found that even though there\nis an apparent reduction of the H0tension with the Planckdata only, this is due to the worse\nconstraining power (larger error bars), and not an actual higher mean value of H0. Regardless\nof the combination of data sets, the parameter αwas consistently constrained to be of the order\nof10−4. We also reported constraints on the other parameter of the Kinetic model, λ, with\nthe most robust bounds obtained when combining BAO and SN data. This is primarily caused\nby the solid constraining power of BAO data on Ωm, which indirectly affects the bounds on λ.\nFinally, we presented a model selection study using the effective χ2\neffand the Bayesian evidence,\nwith the latter indicating a statistical preference for ΛCDM over the Kinetic toy-model related172 7.5. Discussion\nto the increased dimension of the parameter space. In conclusion, we highlight the importance\nof considering the Kinetic model and other variations for future investigations, especially with\nthe availability of new probes from upcoming surveys. Further studies using high-precision data\nwill help resolve tensions and establish a definitive preference for one model, possibly hinting at\nmore complicated shapes for the kinetic function.8ForecastsonCoupledQuintessencewithStan-\ndard Sirens\nEsse não-saber pode parecer ruim mas não é tanto porque ela sabia muita coisa\nassim como ninguém ensina cachorro a abanar o rabo e nem a pessoa a sentir fome;\nnasce-se e fica-se logo sabendo. Assim como ninguém lhe ensinaria um dia a morrer:\nna certa morreria um dia como se antes tivesse estudado de cor a representação\ndo papel de estrela. Pois na hora da morte a pessoa se torna brilhante estrela de cinema,\né o instante de glória de cada um e é quando como no canto coral se ouvem agudos sibilantes.\n—Clarice Lispector A Hora da Estrela\nThat not-knowing might seem awful but it’s not that bad because she knew lots of things in the way nobody\nteaches a dog to wag his tail or a person to feel hungry; you’re born and you just know. Just as nobody one day\nwould teach her how to die: yet she’d surely die one day as if she’d learned the starring role by heart. For at the\nhour of death a person becomes a shining movie star, it’s everyone’s moment of glory and it’s when as in choral\nchanting you hear the whooshing shrieks. —Clarice Lispector in The Hour of the Star\nIn the past few years, we have witnessed the rise of gravitational wave (GW) astronomy as a new\nindependent probe of gravitational effects [126]. An accurate redshift-luminosity relation can be\nconstructed when GW events are combined with an electromagnetic (EM) counterpart multi-\nmessenger signal. In 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO)\nand Virgo collaborations made the first direct detection of a gravitational wave signal from\nthe inspiral, merger, and ringdown of a binary black hole system (GW150914) [123]. This\ninaugural measurement marked only the beginning, with the Virgo-LIGO catalogue of events\ngrowing ever since. Two years later, one of these events (GW170817) opened up a new door\nfor multi-messenger astronomy by being matched with an electromagnetic counterpart for a\nbinary neutron star merger event (GRB170817A), detected by the International Gamma-ray\nAstrophysics Laboratory (INTEGRAL)-Fermi collaboration [124, 125]. Akin to BAO and SN\ndata, conventionally used as standard rulers and candles, gravitational wave events were now\nestablished as standard sirens [121]. This single combined detection had a strong impact on the\nallowed modifications to the gravitational interaction by ruling out many proposals [384–388]\nwith many other models further constrained [88, 389–392]. The significance of these observations\nfor cosmology is twofold. First, GW150914 confirmed the existence of gravitational waves, as\npredicted by GR. Second, GW170817 established that the tensor degrees of freedom seemingly\npropagate at the speed of light [125], imposing severe constraints on modified gravity theories.\nFurthermore, the latter is why GW events are standard sirens for independent measurement\n173174 8.1. Standard Sirens\nof the Hubble parameter H0[393]. Although this method does not yet rival the traditional\ndistance ladder measurements [394], it shows how future gravitational wave telescopes could be\ninstrumental in addressing the present cosmological tensions.\nIn this chapter, we wish to investigate the constraining power of combined distance-redshift mea-\nsurementsbasedongeometric, electromagneticandgravitationalsourcesontheclassesofcoupled\nquintessence models discussed throughout this thesis. These forecasts are conducted by simulat-\ning mock catalogues of gravitational wave standard sirens (SS) from next-generation detectors,\nnamely the Einstein Telescope (ET) and the Laser Interferometer Space Antenna (LISA), along\nwith current data from Type Ia supernovae (SN), and baryon acoustic oscillations (BAO). This\nstudy aims to provide insight into the ability of upcoming SS missions to constrain extended\ntheories of gravity while simultaneously offering supplementary constraints on H0, potentially\nshedding light on the cosmic tensions. We analyse four distinct models, each characterised by\nparticular coupling functions between dark matter and dark energy, arising from a non-universal\nmetric transformation, as discussed in Section 5.4. More specifically, these are a conformally\ncoupled quintessence model, featuring a constant coupling stemming from a conformal function,\nwhich is an exponential function of the scalar field (as explored in Chapter 6); a kinetic model,\nwith the conformal function as a power law of the kinetic term of the scalar field (as investi-\ngated in Chapter 7); a purely disformally coupled quintessence field, with a constant disformal\nfunction; and a mixed disformally coupled quintessence, extending the previous model with an\nexponential conformal coupling.\nThis chapter is organised as follows. We start by giving a brief introduction to the physics of\nstandard sirens in Section 8.1. The supplementary details on the simulation of the standard siren\nevents developed for this study are provided in Appendix B. Section 8.2 provides an overview\nof the methodology used as well as a brief account of the data set combinations considered. We\noutline the criteria for particular catalogue choices, and discuss the sampling method employed\nfor the forecasts. In Section 8.3 we introduce each of the four models under study and present\nthe results of our analysis, emphasising their significant implications. Lastly, in Section 8.4, we\nsummarise our results and outline our concluding thoughts and future prospects.\nThis work has been published in Physical Review D and is available in [2].\n8.1 Standard Sirens\nCurrent GW detectors, (advanced) Virgo [128], (advanced) LIGO [129] and the Kamioka Grav-\nitational Wave Detector (KAGRA) [130], are second generation (2G) ground-based detectors,\nwith another one under planning (2030), the Indian Initiative in Gravitational-wave Observa-\ntions (IndIGO) [131]. The increasing number of detectors will boost the capabilities of GW\nastronomy both in the number of confirmed events (a larger volume of the Universe is covered)\nand sky localisation (a better triangulation of the source), which will also aid in the search for8Forecasts on Coupled Quintessence with Standard Sirens 175\na counterpart. However, 2G detectors are limited in their sensitivity and future ground-based\ndetectors are designed to become more sensitive, precise and capable of probing a larger range\nof frequencies. Special emphasis should be given to the Einstein Telescope (ET), a proposed\nground-based triangular interferometer designated as a third-generation gravitational wave de-\ntector, which is expected to improve the current sensitivity by a factor of 10 [132]. ET will also\nextend the redshift range, e.g. z∼5for binary black-holes (BBHs) compared to z∼0.5for\n2G detectors [133]. The number of detectable multi-messenger events is expected to reach tens\nof thousands of standard sirens [134]. Contemporary terrestrial detectors, such as LIGO and\nVirgo, grapple with limitations at low frequencies due to seismic and thermal noise. In contrast,\nthe ET will likely alleviate these through underground positioning and cryogenic cooling of its\ninterferometer mirrors. These enhancements will empower the ET and other third-generation de-\ntectors to make incredibly sensitive gravitational wave measurements, propelling standard siren\ndetections into precision cosmology’s forefront [395].\nWhiletheseground-baseddetectorswillcoverafrequencybandintherange 1≲f≲103Hz[135],\nthe upcoming space-based 3G detectors, such as the Laser Interferometer Space Antenna (LISA)\n[136] will have a peak sensitivity near 10−3Hz and will be able to detect GW events beyond\nz= 20, probing a wide range of targets. There are many proposals of 3G GW observatories, such\nas DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) [137]. However, we\nhave opted to focus our analysis on ET and LISA covering ground and space-based experiments.\n8.2 Methodology and Data Sets\nGiven the main objective of this study, we create simulated data that forecasts the potential\nfutureobservationsofstandardsirenevents. Specifically, wefocusonthosethatcouldbedetected\nby ET and LISA. Below, we provide a concise overview of the samples we have generated along\nwith the methodology and the data combinations used. The detailed account of the simulations\nof the mock data is found in Appendix B.\nToassessthefitofthemockdatatothecoupledquintessencemodelsexploredinthisstudy,weuse\nthe Markov Chain Monte Carlo (MCMC) technique, as described in Section 4.1. We generate the\nsamples through our private modified branch of the Einstein-Boltzmann code CLASS, interfaced\nwith the MontePython sampler [213, 214]. Moreover, we opt for the Nested Sampling algorithm\nimplemented in MultiNest1[203, 396, 397] and PyMultiNest2[398], instead of the conventional\nMetropolis-Hastings algorithm. This choice is due to the latter’s limitations in dealing with\ncomplex degeneracies between parameters and handling multi-modal distributions, as discussed\nin Section 8.3. For this reason, we recommend caution when employing more straightforward\nmethodslikeMetropolis-Hastingsandemphasisetheimportanceofconsideringmultiplesampling\n1https://github.com/farhanferoz/MultiNest\n2https://github.com/JohannesBuchner/PyMultiNest176 8.3. Forecasts Results\nstrategies. Additionally, this underscores the value of relying on various types of observations,\nwhich can break the degeneracy. The MCMC chains are then analysed and processed using the\nGetDist1Python package [216].\nWe integrate a new likelihood module into MontePython to assess the constraints imposed by the\nupcoming SS surveys. The luminosity distances for gravitational waves at each sampled point\nin the parameter space, dSS(z), are computed from CLASSaccording to Equation (2.1.14). More\nprecisely, we constructed a likelihood function for our simulated dataset of SS events based on\nan effective Gaussian distribution:\nlnLSS=−1\n2nX\ni=1\"\nd(obs)\nSS(zi)−dSS(zi)\nσdL,i#2\n, (8.2.1)\nwhere d(obs)\nSS(z)is the observed luminosity distance, which in this case corresponds to the simu-\nlated data, with total error σdLas described in Appendix B, for nobserved events.\nIn summary, we investigate the potential of constraining coupled quintessence models with up-\ncomingstandardsirendataprobedbyETandLISA,eitherasasubstitutefor currentbackground\ndataor inconjunctionwith them. Thisallows fora direct assessmentof theoriginof theimprove-\nment in the constraints on {Ωm, H0}and the model-specific parameters affecting the background\nevolution. In particular, we include BAO data from the Sloan Digital Sky Survey (SDSS) DR7\nMain Galaxy Sample [117], SDSS DR12 consensus release [116] and the 6dF Galaxy Survey\n[118], in combination with distance modulimeasurements of 1048 SN data from Pantheon [109].\nThis combined data set is referred to throughout this chapter as SN+BAO. Despite our focus\non extended theories of gravity, we adopt flat ΛCDM fiducial values for the parameters specified\nin Section 3.5, intentionally overlooking any gravity modifications, accounted for by the model\nparameters, for which we consider as fiducial value their ΛCDM limit. The models discussed in\nSection 8.3 reduce to ΛCDM in the following limits: λ= 0andβ= 0for Section 8.3.1; λ= 0\nandα= 0for Section 8.3.2; λ= 0andD0= 0for Section 8.3.3; λ= 0,β= 0andD0= 0for\nSection 8.3.4. We adopt flat priors for all the parameters, as detailed in Table 8.1.\n8.3 Forecasts Results\nFollowing the methodology and data sets elaborated in Section 8.2, we explore the predicted\npower of standard sirens probed by LISA and ET on constraining the cosmological parameters\n{Ωm, H0}and the model-specific conformal and disformal coupling parameters and the steepness\noftheself-interactingpotential. Specifically,wediscussfourdistinctcoupledquintessencemodels:\na canonical coupled quintessence model in Section 8.3.1, as introduced in Section 6.2.1; a model\nwith a kinetic coupling in Section 8.3.2, previously explored in Chapter 7; a constant disformal\nmodel in Section 8.3.3 and a hybrid conformal-disformal model in Section 8.3.4, both particular\n1https://github.com/cmbant/getdist8Forecasts on Coupled Quintessence with Standard Sirens 177\nModel Parameter Prior\nAllΩbh2[0.018,0.03]\nΩch2[0.1,0.2]\nh [0.6,0.8]\nλ [0,2]\nSections 8.3.1 and 8.3.4 β [0,2]\nSection 8.3.2 α [0,0.001]\nSections 8.3.3 and 8.3.4 D0/meV−1[0,2]\nTable 8.1 : Flat priors on the cosmological and model parameters sampled in Section 8.3.\ncases of the disformally coupled quintessence model [347, 399]. We provide a brief account of the\ntheoretical framework of each scenario before presenting the forecast results, derived according\nto the theoretical and numerical specifications discussed in Section 8.2 and in Appendix B.\nEach subsection also contains the derived 1D and 2D marginalised posterior distributions and\ncontours for {H0,Ωm}and any of the particular model-dependent parameters of the model in\nquestion, as specified in Table 8.1. These plots comprise both ET and LISA data, individually\nand combined, and feature an overlay of SN+BAO data and standalone SN+BAO results for\nreference. We provide a summary of the results in a table with the corresponding 1σvalues,\ndenoted in the text as {σp}, where pis the corresponding model parameters. Additionally, we\ndefine F(i,j)\np={σ(j)\np/σ(i)\np}, where iandjrepresent distinct data sets, to denote the effective\nchange in the parameter p.\n8.3.1 Conformal Coupling\nThe first scenario we explore is the conformal coupling model introduced in Section 6.2.1. We\nrecall that this model is characterised by the following conformal and potential functions C(ϕ)\nandV(ϕ):\nC(ϕ) =e2βϕ\nMPl, V (ϕ) =V0e−λϕ\nMPl, (8.3.1)\nwith C(ϕ)laying out the coupling in the dark sector according to Equation (5.3.2). The param-\neters βandλare dimensionless constants, while V0is a constant with dimensions of (mass)4\nrepresenting the energy scale of the potential1.\nIn these models, the mass of the dark matter particles becomes dependent on ϕ, as the DE field\nmediates a long-range force between DM particles. This translates into an effective gravitational\n1To avert degeneracies and for numerical robustness, V0is not a free parameter but rather serves as a shooting\nparameter to satisfy the flatness condition for Ω0\nϕ.178 8.3. Forecasts Results\nConformal Coupled Quintessence\nData sets Ωm σΩm H0 σH0β σβ λ σλ\nSN+BAO 0.3019+0.0088\n−0.0059 0.0074 73.2+4.7\n−3.5 4.10.085+0.055\n−0.043 0.049 0.42+0.20\n−0.360.28\nET 0.307+0.011\n−0.0050 0.0080 67.49+0.39\n−0.34 0.370.115+0.060\n−0.079 0.070 0.50+0.26\n−0.380.32\nET+SN+BAO 0.3046+0.0099\n−0.0051 0.0075 67.37±0.360.360.063+0.033\n−0.045 0.039 0.49+0.26\n−0.350.31\nLISA 0.3039+0.0093\n−0.0049 0.0071 67.50+0.50\n−0.44 0.470.167+0.085\n−0.11 0.098 0.33+0.15\n−0.320.24\nLISA+SN+BAO 0.3028+0.0065\n−0.0036 0.0051 67.52±0.370.370.048+0.025\n−0.037 0.031 0.33+0.15\n−0.290.22\nET+LISA 0.3079+0.0061\n−0.0034 0.0048 67.56±0.260.260.178+0.099\n−0.081 0.090 0.30+0.13\n−0.270.24\nET+LISA+SN+BAO 0.3044+0.0063\n−0.0032 0.0048 67.45±0.280.280.052+0.028\n−0.038 0.033 0.35+0.17\n−0.300.24\nTable 8.2 : Marginalised constraints on cosmological and model parameters for the Conformal\nCoupled Quintessence model at 68% CL.\ncouplingexpressedas Geff=G(1+2 β2)[292,400,401]. Forthisanalysis, wearemainlyconcerned\nwith the slope of the potential λand the coupling constant βas free parameters. Constraints on\nthis model have been previously reported in Ref. [344] using only background data ( H(z), BAO,\nand supernova Union 2.1), leading to upper limits β <0.193andλ <1.27. Further stringent\nlimits, β <0.0298andλ <0.6at1σ, have been derived in Ref. [346] using Planck, BAO, and\nSN data, also in agreement with Ref. [348] .\nBased on the results illustrated in Figures 8.1 to 8.3 and summarised in Table 8.2, we dis-\ncuss the parameter constraints from gravitational wave (GW) observations in comparison to\nSN+BAO data sets for the parameters {Ωm, H0, β, λ}. Specifically, with ET standard sirens,\nthe set of parameters is constrained at 1σwith accuracy {0.0080,0.37,0.0070,0.32}for ET\nand{0.0075,0.36,0.039,0.31}for ET+SN+BAO. This results in a change in error quantified\nbyF(ET,ET+SN+BAO )\nΩm,H0,β,λ={0.94,0.97,0.56,0.97}. Thus, adding the background data, compared\nto ET alone, increases accuracy in all parameters shown by the reduction in σ. A similar\ntrend is present for the LISA data set, with 1σregions of {0.0071,0.47,0.098,0.24}, while for\nLISA+SN+BAO,theybecome {0.0051,0.37,0.031,0.22}. Thisleadstoanoverallerrorreduction\ngiven by F(LISA,LISA+SN+BAO )\nΩm,H0,β,λ={0.72,0.79,0.32,0.92}.\nFor the standalone SN+BAO data set, the accuracy is {0.0074,4.1,0.049,0.28}. When compared\nto the data sets, including SS data, ET+SN+BAO and LISA+SN+BAO, a reduction in σis\nobserved for all the parameters except for σΩm, which nominally increases in the ET+SN+BAO\ncase.\nWhen evaluating the error performance of ET and LISA in comparison to SN+BAO, we report\nonly minor variations in the constraining capability for Ωm. Specifically, ET shows a slight\ndegradation, while LISA demonstrates marginal improvement. A parallel trend occurs for the\nparameter λ, where ET’s performance is nominally worse, whereas LISA’s is better than with the8Forecasts on Coupled Quintessence with Standard Sirens 179\nbackground data. Of particular note is the substantial reduction in σH0when comparing ET and\nLISA against SN+BAO. This results in an error reduction by factors of F(SN+BAO,ET )\nH0= 0.090\nandF(SN+BAO,LISA )\nH0= 0.11. Such forecasts suggest that GWs will be instrumental in addressing\nthe Hubble tension. Conversely, the parameter βexperiences an increase in error, quantified by\nF(SN+BAO,ET )\nβ= 1.4andF(SN+BAO,LISA )\nβ= 2.0. However, this degradation is alleviated when\nET and LISA are combined with the background data, improving to F(ET,ET+SN+BAO )\nβ= 0.56\nandF(LISA,LISA+SN+BAO )\nβ= 0.32.\nThe results shown in Figure 8.3 show that ET and LISA offer comparable constraining power\nover the cosmological parameters. A noteworthy feature observed is ET’s superior constraining\nability for H0, ascribed to its abundance of low-redshift data points as depicted in Figure B.1.\nBy bringing the GW data from both LISA and ET together, which implies more data points also\ncovering a broader redshift spectrum, we predict an improved constraining ability on {H0,Ωm}\nrelative to SN+BAO. Specifically, F(SN+BAO,ET+LISA )\nΩm,H0={0.65,0.063}. For the model-specific\nparameters βandλ, the constraints are modified to F(SN+BAO,ET+LISA )\nβ,λ={1.8,0.86}. By com-\nbining ET+LISA with SN+BAO, only negligible changes in the constraining power for Ωm, H0, λ\narise. Only βshows a significant tightening in its constraint, reducing the 1σerror by nearly\none-third.\nIn the context of our analysis, we find improvements in the 1σupper limits of the model pa-\nrameters when compared to the existing background constraints stated earlier for the following\ncases: β <0.14andλ <0.62(SN+BAO); β <0.175andλ <0.76(ET); β <0.096andλ <0.75\n(ET+SN+BAO); λ <0.48(both LISA and LISA+SN+BAO) and β <0.073(LISA+SN+BAO);\nλ <0.43(ET+LISA); λ <0.52andβ <0.08(ET+LISA+SN+BAO).\n8.3.2 Kinetic Conformal Coupling\nInamoreintricateexampleofacoupledquintessencemodel, weturnourattentiontoaconformal\nfunction that depends on the kinetic term of the scalar field ϕ, expressed as X=−∂µϕ∂µϕ\n2. This\nis called the kinetically coupled model, proposed in [275] (more details in the references cited\ntherein). Our discussion will focus on the specific instance of a power law form for C(X), as\nexplored in Chapter 7, based on Ref. [1]. Although originating from a Lagrangian formulation\nLDM→\u0010\nX\nMPl4\u0011α\nLDM, the model at the background level is equivalent to a kinetic-dependent\nconformal transformation ¯gµν=C(X)gµν, where:\nC(X) =\u0012X\nMPl4\u00132α\n,and V(ϕ) =V0e−λϕ\nMPl. (8.3.2)\nHere, αisadimensionlessconstant, andtheassumptionofasimpleexponentialpotentialremains\nin line with the previous case, implying similar considerations for λandV0.\nIn summary, an investigation based on the Plancklikelihood and SN+BAO background data, as\noutlinedinRef.[1], revealsthesubstantialinfluenceofBAOdatainconstraining Ωm, intrinsically180 8.3. Forecasts Results\n62687480\nH00.00.10.20.3\n0.00.51.01.5\n0.260.290.32m\n0.260.290.32\nm\n00.5 11.5\n00.10.20.3\nConformal Coupling\nET\nSN + BAO\nET + SN + BAO\nFigure 8.1 : 68% and 95% CL 2D contours and 1D marginalised posterior distributions for the\nparameters {H0,Ωm, λ, β}in the conformal coupled quintessence model with the ET mock data\n(green filled line), SN+BAO data (yellow dotted line) and their combination (red dashed line).\nThe dotted lines depict the fiducial values for the mock data {Ωm, H0}={0.3144,67.32}.\ncorrelated to the steepness of the potential λ. The coupling constant αis constrained to the order\n10−4. The results derived for the cosmological parameters were compatible with those of ΛCDM\nwithin the error margins. Additionally, we highlight a direct correlation between H0andΩm.8Forecasts on Coupled Quintessence with Standard Sirens 181\n62687480\nH00.00.10.20.30.4\n0.00.40.81.2\n0.280.300.32m\n0.28 0.30 0.32\nm\n00.4 0.8 1.2\n00.10.20.30.4\nConformal Coupling\nLISA\nSN + BAO\nLISA + SN + BAO\nFigure 8.2 : 68% and 95% CL 2D contours and 1D marginalised posterior distributions for\nthe parameters {H0,Ωm, λ, β}in the conformal coupled quintessence model with LISA mock\ndata (green filled line), SN+BAO data (yellow dotted line) and their combination (red dashed\nline). The scale is the same as in Figure 8.1 for comparison purposes, with the SN+BAO\ncontours standing as the reference. The dotted lines depict the fiducial values for the mock data\n{Ωm, H0}={0.3144,67.32}.\nInitially ascribed to the behaviour of linear perturbations when αis non-zero, this correlation\npersists even for the standard siren background data sets.182 8.3. Forecasts Results\n66676869\nH00.00.10.20.3\n0.00.51.01.5\n0.280.300.32m\n0.28 0.30 0.32\nm\n00.5 11.5\n00.10.20.3\nConformal Coupling\nET\nLISA\nET + LISA\nFigure 8.3 : 68% and 95% CL 2D contours and 1D marginalised posterior distributions for\nthe parameters {H0,Ωm, λ, β}in the conformal coupled quintessence model with ET mock data\n(green filled line), LISA mock data (yellow dotted line) and their combination (red dashed line).\nThe dotted lines depict the fiducial values for the mock data {Ωm, H0}={0.3144,67.32}.\nFrom the findings presented in Figures 8.4 to 8.6, and summarised in Table 8.3, we investi-\ngate constraints on the parameters {Ωm, H0, λ,104α}using to the same data sets as in the\nSection 8.3.1. Comparison of the errors obtained from the ET standard sirens catalogue to8Forecasts on Coupled Quintessence with Standard Sirens 183\nKinetic Coupled Quintessence\nData sets Ωm σΩm H0 σH0104α σ104αλ σλ\nSN+BAO 0.3016+0.0075\n−0.0057 0.0066 70.4±3.1 3.15.1±2.92.9 0.34+0.16\n−0.290.23\nET 0.3067+0.0093\n−0.0046 0.0070 67.45±0.360.364.8±2.92.9 0.41+0.20\n−0.310.26\nET+SN+BAO 0.3062+0.0074\n−0.0043 0.0059 67.36±0.330.335.0±2.92.9 0.37+0.19\n−0.280.24\nLISA 0.2997+0.0079\n−0.0041 0.0060 67.30±0.390.394.9±2.92.9 0.34+0.16\n−0.300.23\nLISA+SN+BAO 0.3024+0.0058\n−0.0035 0.0047 67.47±0.360.365.0±2.92.9 0.29+0.13\n−0.260.20\nET+LISA 0.3040+0.0058\n−0.0031 0.0045 67.42±0.260.265.1±2.92.9 0.31+0.15\n−0.260.21\nET+LISA+SN+BAO 0.3040+0.0058\n−0.0031 0.0045 67.42±0.270.274.9±2.92.9 0.29+0.14\n−0.250.20\nTable 8.3 : Marginalised constraints on cosmological and model parameters for the Kinetic\nModel at 68% CL.\nthe SN+BAO data reveals that, with the exception of the H0parameter, ET’s 1σbounds\nare comparable in magnitude. The constraint in H0improves by a factor of ten, quanti-\nfied by F(SN+BAO,ET )\nΩm,H0,β,λ={1.1,0.12,1.0,1.1}. Combining the SS and background data sets\n(ET+SN+BAO), the 1σconstraints tighten for all parameters when set against ET alone, ex-\npressed by F(ET, ET+SN+BAO )\nΩm,H0,α,λ={0.84,0.92,1.0,0.92}.\nFor the LISA SS scenario, it becomes apparent that every cosmological and model parameter is\neither better or equally constrained when using LISA alone, compared to SN+BAO data, with\nF(SN+BAO,LISA )\nΩm,H0,α,λ={0.91,0.13,1.0,1.0}. Pairing LISA with SN+BAO improves upon constraints\ncompared to SN+BAO alone, and LISA+SN+BAO even surpasses LISA alone in constraint\npower, particularly for Ωm, demonstrated by F(LISA,LISA+SN+BAO )\nΩm,H0,α,λ={0.78,0.92,1.0,0.87}.\nRegarding both ET and LISA, H0accuracy improves by an order of magnitude (0.36 for ET and\n0.39 for LISA) relative to SN+BAO ( 3.1), corroborating the findings of Section 8.3.1. It is worth\nnoting that the errors in the model-specific parameters remain predominantly unaltered across\nall data sets and combinations, with only minor changes reported for λwithin the 1σregion,\nwhile αremains effectively unaltered.\nIn comparing the constraining power of ET and LISA against their combination, ET+LISA, we\nhighlightthatthelatteryieldsbetterresultsforthecosmologicalparametersrelativetoanyofthe\ndata sets analysed. For model parameters, negligible variations in accuracy are observed between\nindividual and combined ET and LISA data sets. However, compared to SN+BAO, ET+LISA\nexhibits better accuracy with F(SN+BAO,ET+LISA )\nΩm,H0,α,λ={0.68,0.084,1.0,0.91}. Incorporating all\ndata sets, ET+LISA+SN+BAO, results in a negligible change in parameter constraints when\ncompared against ET+LISA.\nFinally, when assessing the constraints derived for the Kinetic model in Ref. [1] against CMB\nPlanck2018,PlanckCMB lensing, BAO, and SN, we observe that the αparameter is constrained184 8.3. Forecasts Results\nmore tightly by CMB and its combinations by an order of magnitude, relative to our data sets\nthat rely solely on background evolution. Specifically, σ104αvalues are reduced in Ref. [1] to\n0.95 (Pl18), 0.84 (Pl18+SN+BAO), and 0.7 (Pl18len+SN+BAO) as opposed to 2.9 in our cases.\nFuture ET and LISA data will provide constraints on λcomparable to PlanckCMB ( σλ= 0.48\nwith Pl18 and σλ= 0.2with Pl18+SN+BAO and Pl18len+SN+BAO). Moreover, this analysis\nsuggests that standard siren data will offer superior H0constraints by an order of magnitude\ncompared to Pl18 ( σH0= 2.5). CMB lensing enhances this constraint by an order of magnitude,\nachieving σH0= 0.6, comparable to ET and LISA, even though standard sirens yield better\nrelative errors with σH0<0.4for all considered combinations.\n8.3.3 Disformal Coupling\nIn what follows, we focus on a model featuring only a disformal coupling, given by\nC= 1, D =D4\n0and V(ϕ) =V0e−λϕ/M Pl. (8.3.3)\nIn this scenario, the conformal contribution vanishes, leaving D≡D4\n0as a constant with units\nof (mass)−4as per Equation (5.3.3). The functional form of V(ϕ)is maintained relative to the\nprevious scenarios. Prior studies on this model’s constraints have been performed in [344] and\n[346]. In particular, constraints derived solely from background data (including H(z), BAO, and\nSupernova Union2.1 data) report D0>0.07meV−1andλ <1.56at95%confidence level [344].\nWhen incorporating CMB data with lensing effects, along with BAO, SN, cosmic chronometers,\ncluster abundances, and H0priors, stringent upper limits for D0andλare set at D0<0.2500\nmeV−1andλ <0.6720at68%CL [346].\nConstant Disformal Coupled Quintessence\nData sets Ωm σΩm H0 σH0D0/meV−1σD0λ σλ\nSN+BAO 0.315±0.017 0.017 70.5±3.1 3.11.20+0.65\n−0.38 0.520.87+0.59\n−0.76 0.68\nET 0.290+0.011\n−0.013 0.012 67.58+0.36\n−0.27 0.321.06±0.510.511.06±0.580.58\nET+SN+BAO 0.298+0.011\n−0.014 0.013 67.45±0.310.311.15+0.66\n−0.44 0.550.92±0.580.58\nLISA 0.320±0.012 0.012 67.43±0.330.331.22+0.64\n−0.38 0.510.87±0.580.58\nLISA+SN+BAO 0.317±0.012 0.012 67.52±0.340.341.24+0.64\n−0.36 0.500.86+0.65\n−0.77 0.71\nET+LISA 0.3094+0.0087\n−0.0099 0.0093 67.49±0.220.221.23+0.63\n−0.36 0.500.88±0.580.58\nET+LISA+SN+BAO 0.3100+0.0092\n−0.0100 0.0096 67.47±0.250.251.24+0.63\n−0.36 0.500.88+0.68\n−0.77 0.73\nTable 8.4 : Marginalised constraints on cosmological and model parameters for the Constant\nDisformal Coupled Quintessence Model at 68% CL\nIn Figures 8.7 to 8.9, and in Table 8.4, we summarise the results for the same data sets as\npreviously discussed in Sections 8.3.1 and 8.3.2. For the ET catalogue alone, improved accuracy8Forecasts on Coupled Quintessence with Standard Sirens 185\n62687480\nH004812104\n0.00.30.60.9\n0.280.300.32m\n0.28 0.30 0.32\nm\n00.30.60.9\n04812\n104\nKinetic Conformal Coupling\nET\nSN + BAO\nET + SN + BAO\nFigure 8.4 : 68% and 95% CL 2D contours and 1D posterior distributions for the parameters\n{H0,Ωm, λ,104α}in the kinetic conformal coupled quintessence model with ET (green filled\nline), SN+BAO (yellow dotted line) data and their combination (red dashed line). The dotted\nlines depict the fiducial values for the mock data {Ωm, H0}={0.3144,67.32}.\nis observed across all cosmological and model parameters {Ωm, H0, D0, λ}, relative to SN+BAO,\nnamely F(SN+BAO,ET )\nΩm,H0,D0,λ={0.71,0.10,0.98,0.85}. The joint ET+SN+BAO data set performs\nbetter than SN+BAO, although only modest changes in parameter accuracy are seen relative to186 8.3. Forecasts Results\n62687480\nH004812104\n0.00.30.60.9\n0.280.300.32m\n0.280.300.32\nm\n00.30.60.9\n04812\n104\nKinetic Conformal Coupling\nLISA\nSN + BAO\nLISA + SN + BAO\nFigure 8.5 : 68% and 95% CL 2D contours and 1D posterior distributions for the parame-\nters{H0,Ωm, λ,104α}in the kinetic conformal coupled quintessence model with LISA mock\ndata (green filled line), SN+BAO (yellow dotted line) data and their combination (red dashed\nline). The scale is the same as in Figure 8.4 for comparison purposes, with the SN+BAO con-\ntours standing as the reference. The dotted lines depict the fiducial values for the mock data\n{Ωm, H0}={0.3144,67.32}.\nET alone, according to F(ET,ET+SN+BAO )\nΩm,H0,D0,λ={1.1,0.97,1.1,1.0}.\nFor the LISA standard sirens, we find an analogous trend in parameter accuracy compared to the8Forecasts on Coupled Quintessence with Standard Sirens 187\n666768\nH004812104\n0.00.30.60.9\n0.280.300.32m\n0.28 0.30 0.32\nm\n00.30.60.9\n04812\n104\nKinetic Conformal Coupling\nET\nLISA\nET + LISA\nFigure 8.6 : 68% and 95% CL 2D contours and 1D marginalised posterior distributions for the\nparameters {H0,Ωm, λ,104α}inthekineticconformalcoupledquintessencemodelwithETmock\ndata (green filled line), LISA mock data (yellow dotted line) and their combination (red dashed\nline). The dotted lines depict the fiducial values for the mock data {Ωm, H0}={0.3144,67.32}.\nbackgrounddata, F(SN+BAO,LISA )\nΩm,H0,D0,λ={0.71,0.11,0.98,0.85}. ThisextendstotheLISA+SN+BAO\ncombination, encapsulated by F(LISA,LISA+SN+BAO )\nΩm,H0,D0,λ={1.0,1.0,0.98,1.22}. The variation in pa-\nrameter accuracy is minimal compared to LISA alone, except for λ, which displays a larger 1σ188 8.3. Forecasts Results\nregion with σλ= 0.71.\nRegardlessofthedatacombinationused, theparameters {Ωm, D0, λ}areconstrainedatthesame\nlevel, with λshowing only a minor improvement in both ET and LISA scenarios ( σλ= 0.58as\nopposed to σλ= 0.7for SN+BAO). The accuracy in the H0parameter improves by one order\nof magnitude for ET and LISA compared to SN+BAO.\nThe model parameters for ET and LISA exhibit no significant variations, leading to con-\nsistent constraints for ET+LISA. Nonetheless, there is an increased accuracy for the cos-\nmological parameters. Following the observed trend for both ET and LISA against\nSN+BAO, the combined ET+LISA data set improves constraints further, according to\nF(SN+BAO,ET+LISA )\nΩm,H0,D0,λ={0.55,0.071,0.96,0.85}. There is negligible change in accuracy when\ncombining all data sets (ET+LISA+SN+BAO), except for λ, which sees a decrease in accuracy\nasF(ET+LISA,ET+LISA+SN+BAO )\nΩm,H0,D0,λ={1.0,1.1,1.0,1.3}.\nContrasted with the results reported in Ref. [344], we highlight the ability of the SS data to\nimpose constraints at the 68%CL, and not only at 95%CL, for all the model parameters.\nParticularly, the error in H0significantly reduces from σH0≈2.2toσH0≈0.3with standard\nsirens. When compared with results in Ref. [346] for CMB, CMB lensing and additional data,\nwe find lower and upper bounds for both λandD0at68%CL, contrasting with prior works that\nreported only upper bounds in particular with more accommodating upper bounds, considering\nthat this analysis includes only background data. Consequently, the error in H0is brought to\nthe same order of magnitude with σH0≈0.9, though still about three times larger than the one\nreported in this analysis.\n8.3.4 Mixed Conformal-Disformal Coupling\nLastly, we explore a mixed coupling, incorporating both conformal and disformal components.\nSpecifically, the model is defined by\nC(ϕ) =e2βϕ/M Pl, D (ϕ) =D4\n0and V(ϕ) =V0e−λϕ/M Pl. (8.3.4)\nSimilar to the disformal-only scenario in Section 8.3.3, constraints for this model have been\npresented in Refs. [344] and [346]. For the same background data, Ref. [344] reported constraints\nofD0>0.102,β < 0.453, and λ < 1.59at the 95%confidence level. In contrast, the work\nin [346], which incorporates CMB data, yields β≲0.17andλ≲0.35at1σ, with the exact\nconstraints varying depending on the specific data sets employed. The disformal coupling D0is\nnot always well constrained for this case, with lower limits of D0≳0.35found in particular data\nset combinations.\nIn Figures 8.10 to 8.12, and Table Table 8.5, the estimated values of the parameters\n{Ωm, H0, β, D 0, λ}are presented using the same data sets as in Sections 8.3.1 to 8.3.3. Consider-\ning the ET catalogue alone, we note an improvement in accuracy in the cosmological parameters8Forecasts on Coupled Quintessence with Standard Sirens 189\n62687480\nH00.00.81.62.4D00.00.81.62.4\n0.260.300.34m\n0.260.300.34\nm\n00.8 1.6 2.4\n00.8 1.6 2.4\nD0Disformal Coupling\nET\nSN + BAO\nET + SN + BAO\nFigure 8.7 : 68% and 95% CL 2D contours and 1D posterior distributions for the parameters\n{H0,Ωm, λ, D 0}in the constant disformal coupled quintessence model with ET (green filled line),\nSN+BAO (yellow dotted line) data and their combination (red dashed line). The dotted lines\ndepict the fiducial values for the mock data {Ωm, H0}={0.3144,67.32}.\nΩmandH0whencomparedtotheSN+BAOdataset; specifically, F(SN+BAO,ET )\nΩm,H0={0.61,0.088}.\nSimilar accuracy levels are maintained upon combining the ET, SN, and BAO data sets, with\na marginal enhancement relative to the ET-only case. As for the model-specific parame-190 8.3. Forecasts Results\n62687480\nH00.00.81.62.4D00.00.81.62.4\n0.260.300.34m\n0.260.300.34\nm\n00.8 1.6 2.4\n00.8 1.6 2.4\nD0Disformal Coupling\nLISA\nSN + BAO\nLISA + SN + BAO\nFigure 8.8 : 68% and 95% CL 2D contours and 1D posterior distributions for the parameters\n{H0,Ωm, λ, D 0}in the constant disformal coupled quintessence model with LISA (green filled\nline), SN+BAO (yellow dotted line) data and their combination (red dashed line). The scale\nis the same as in Figure 8.7 for comparison purposes, with the SN+BAO contours standing\nas the reference. The dotted lines depict the fiducial values for the mock data {Ωm, H0}=\n{0.3144,67.32}.\nters{β, D 0, λ}, the constraining ability of ET is nearly identical to that of SN+BAO, with\nF(SN+BAO,ET )\nβ,D0,λ={1.0,0.92,1.0}. Nevertheless, it is worth mentioning that the combined data8Forecasts on Coupled Quintessence with Standard Sirens 191\n666768\nH00.00.81.62.4D00.00.81.62.4\n0.260.300.34m\n0.26 0.30 0.34\nm\n00.8 1.6 2.4\n00.8 1.6 2.4\nD0Disformal Coupling\nET\nLISA\nET + LISA\nFigure 8.9 : 68% and 95% CL 2D contours and 1D marginalised posterior distributions for the\nparameters {H0,Ωm, λ, D 0}in the constant disformal coupled quintessence model with ET mock\ndata (green filled line), LISA mock data (yellow dotted line) and their combination (red dashed\nline). The dotted lines depict the fiducial values for the mock data {Ωm, H0}={0.3144,67.32}.\nset yields an increase in the β-error, with σβ= 0.71.\nWhen considering LISA’s standard sirens, there is an improved accuracy for the cosmologi-\ncal parameters, akin to the ET results, with F(SN+BAO,LISA )\nΩm,H0={0.72,0.088}. However, an192 8.3. Forecasts Results\nMixed Conformal-Disformal Coupled Quintessence\nData sets Ωm σΩm H0 σH0β σβ D0 σD0λ σλ\nSN+BAO 0.308+0.021\n−0.015 0.018 71.2±3.3 3.301.01±0.570.571.23+0.59\n−0.430.510.98±0.570.57\nET 0.286+0.010\n−0.012 0.011 67.65±0.290.290.85±0.580.581.27+0.58\n−0.350.471.03±0.580.58\nET+SN+BAO 0.294+0.011\n−0.013 0.012 67.50±0.300.300.92+0.66\n−0.76 0.711.32+0.53\n−0.350.440.97±0.580.58\nLISA 0.310+0.017\n−0.0087 0.013 67.55+0.27\n−0.31 0.290.97±0.560.561.15+0.63\n−0.440.541.01±0.560.56\nLISA+SN+BAO 0.310+0.016\n−0.010 0.013 67.59±0.330.331.01±0.580.581.25+0.53\n−0.430.480.98±0.570.57\nET+LISA 0.302+0.0120\n−0.0058 0.0089 67.54±0.200.200.92±0.550.551.09+0.76\n−0.420.591.05+0.71\n−0.56 0.64\nET+LISA+SN+BAO 0.304+0.0120\n−0.0089 0.011 67.53±0.240.240.98±0.570.571.27+0.53\n−0.410.470.97±0.570.57\nTable 8.5 : Marginalised constraints on cosmological and model parameters for the Mixed\nConformal-Disformal Coupled Quintessence Model at 68% CL.\nobservable accuracy drop for H0is evident when combining LISA with other data sets, as\ngiven by F(LISA, LISA+SN+BAO )\nH0={1.1}. Regarding model parameters {β, D 0, λ}, LISA, in\ncontrast to ET, shows heightened accuracy over SN+BAO, except for D0, represented by\nF(SN+BAO,LISA )\nβ,D0,λ={0.98,1.1,0.98}. The composite data sets maintain a comparable level of\naccuracy to the LISA-only case, indicated by F(LISA,LISA+SN+BAO )\nβ,D0,λ={1.0,0.89,1.0}.\nSimilarly to what was discussed in Sections 8.3.1 and 8.3.3, the combined GW data sets,\nET+LISA yield a marked improvement in the accuracy of Ωmand H0, when compared\nto SN+BAO alone, with F(SN+BAO, ET+LISA )\nΩm,H0={0.49,0.061}. The model-specific param-\neters exhibit only minor changes, with both D0and λbeing somewhat less constrained,\nF(SN+BAO, ET+LISA )\nβ,D0,λ={0.96,1.2,1.1}. When all data sets are considered, the cosmological\nparameters ΩmandH0display enhanced constraints compared to SN+BAO, while the model pa-\nrameters remain largely unchanged, F(SN+BAO, ET+LISA+SN+BAO )\nΩm,H0,β,D 0,λ={0.58,0.073,1.0,0.92,1.0}.\nRegardless of the data set combinations, the constraints on {Ωm, β, D 0, λ}closely resemble those\nderived from SN+BAO. Furthermore, the accuracy in constraining H0is improved by an order\nof magnitude for ET and LISA relative to SN+BAO.\nMuch like the comparison in Section 8.3.3, the main improvement predicted in our analysis in\ncomparison to Ref. [344] is the possibility of constraining all the model parameters at 68%CL.\nThe upper bounds on λfor standard sirens at 1σare consistent with the 2σlimits stated for\nearlier studies. Moreover, the precision of H0significantly improves from σH0≈2.1for the\nbackground data to σH0≈0.3in all the cases including SS data. Compared to Ref. [346], which\ntakes into account CMB, CMB lensing and additional data and which either does not provide\nconstraints for D0or only finds a lower limit for λandβ, the inclusion of standard sirens in\nour analysis yields solid constraints for all three model-parameters at 68%CL. This represents\na great achievement that could be improved by combining the background data with the CMB\nobservations. Furthermore, including CMB data brings the error in H0to the same order of\nmagnitude, σH0≈0.6, although still approximately two times larger than the values reported in8Forecasts on Coupled Quintessence with Standard Sirens 193\n607080\nH00.00.71.42.1D00.00.71.42.1\n0.00.71.42.1\n0.220.280.34m\n0.220.280.34\nm\n00.71.42.1\n00.71.42.1\n00.71.42.1\nD0Mixed Conformal-Disformal Coupling\nET\nSN + BAO\nET + SN + BAO\nFigure 8.10 : 68% and 95% CL 2D contours and 1D posterior distributions for the parameters\n{H0,Ωm, λ, β, D 0}in the mixed conformal-disformal coupled quintessence model with ET (green\nfilled line), SN+BAO (yellow dotted line) data and their combination (red dashed line). The\ndotted lines depict the fiducial values for the mock data {Ωm, H0}={0.3144,67.32}.\nthis analysis.194 8.4. Discussion\n62687480\nH00.00.71.42.1D00.00.71.42.1\n0.00.71.42.1\n0.220.280.34m\n0.22 0.28 0.34\nm\n00.71.42.1\n00.71.42.1\n00.71.42.1\nD0Mixed Conformal-Disformal Coupling\nLISA\nSN + BAO\nLISA + SN + BAO\nFigure 8.11 : 68% and 95% CL 2D contours and 1D posterior distributions for the parame-\nters{H0,Ωm, λ, β, D 0}in the mixed conformal-disformal coupled quintessence model with LISA\n(green filled line), SN+BAO (yellow dotted line) data and their combination (red dashed line).\nThe scale is the same as in Figure 8.10 for comparison purposes, with the SN+BAO con-\ntours standing as the reference. The dotted lines depict the fiducial values for the mock data\n{Ωm, H0}={0.3144,67.32}.\n8.4 Discussion\nIn this work, we have investigated the potential of upcoming gravitational wave detectors, specif-\nically ET and LISA, to improve constraints on conformal and disformal couplings between dark8Forecasts on Coupled Quintessence with Standard Sirens 195\n66.867.668.4\nH00.00.71.42.1D00.00.71.42.1\n0.00.71.42.1\n0.260.300.34m\n0.26 0.30.34\nm\n00.71.42.1\n00.71.42.1\n00.71.42.1\nD0Mixed Conformal-Disformal Coupling\nET\nLISA\nET + LISA\nFigure8.12 : 68% and 95% CL 2D contours and 1D marginalised posterior distributions for the\nparameters {H0,Ωm, λ, β, D 0}inthemixedconformal-disformalcoupledquintessencemodelwith\nET mock data (green filled line), LISA mock data (yellow dotted line) and their combination\n(red dashed line). The dotted lines depict the fiducial values for the mock data {Ωm, H0}=\n{0.3144,67.32}.\nenergy and dark matter. Four theoretical frameworks were considered: a conformal coupled\nquintessence, a kinetic model, a constant disformal coupled quintessence, and a mixed conformal-196 8.4. Discussion\ndisformal model. We also included a self-interacting exponential potential for each case and\nlooked for constraints in its slope parameter λ.\nWe have simulated mock catalogues of standard siren events in accordance with the ET and\nLISA specifications, which were subsequently used to conduct an MCMC analysis. In particular,\nwe compared the predictions for the individual SS data and also in combination with the current\nbackground SN and BAO data. Keeping in mind the assumptions that go into simulating data\nand performing forecasts, we summarise the main results for each model below:\n•Conformal Coupled Quintessence: Both LISA+SN+BAO and ET+SN+BAO combina-\ntions improve the constraints on λand the conformal coupling parameter β. Moreover,\nintegrating ET+LISA with SN+BAO reduces the error in βby one-third.\n•Kinetic Model: The ET and LISA catalogues independently, when taken independently,\nfall short of improving the constraints on λand on the conformal exponential parameter\nα. Combining LISA with SN+BAO yields a marginal improvement for λand the matter\ndensity Ωm.\n•Constant Disformal Coupled Quintessence: All combined data sets effectively constrain\nthe disformal parameter D0at the 1σlevel and the same order of magnitude, with a small\nimprovement for LISA+SN+BAO. ET, LISA and their combination perform better than\nSN+BAO in constraining λ. The combination of ET+LISA with SN+BAO allows for the\nerror in Ωmto be reduced.\n•Mixed Conformal-Disformal Coupled Quintessence: ET and LISA, when analysed sepa-\nrately, do not significantly enhance the precision of the model parameters. Only a marginal\nreduction in the 1σuncertainty of D0is noted for the full combination scenarios. The pa-\nrameter Ωmexperiences a slight decrease in the associated error when combining ET and\nLISA.\nIndependent of the model considered, we find that the accuracy of the H0parameter is con-\nsistently enhanced by an order of magnitude at 1σ, relative to the SN+BAO data set. This\nholds promise for providing further insight into addressing the H0tension and contributes to the\noverall improvement in model parameter estimates when the full combinations are considered,\nas we have just reviewed. Ultimately, our results show that upcoming third-generation gravita-\ntional wave detectors stand to enrich our understanding of dark energy-dark matter interactions\nsubstantially and may provide insights into the H0tension.\nIt should be noted that the forecast results presented in Section 8.3 reveal a considerable devi-\nation in the constraints derived for the cosmological parameters in the extended models, from\nthe expected ΛCDM fiducial values used to simulate the SS data. Even if the mean values found\nforH0andΩmdiffer from the ones used in the simulated cosmology, these are still expected8Forecasts on Coupled Quintessence with Standard Sirens 197\nto be recovered within the corresponding errors. Several factors might contribute to this bias\nobserved in the forecasts. One plausible explanation is the presence of parameter degeneracies,\nwhere the variations in one parameter could compensate for changes in another, leading to devi-\nations from the fiducial ΛCDM values. Additionally, the combination of realBAO and SN data\nwithsimulated GW data introduces complexities, especially when the fiducial cosmology for the\nGW simulations does not align exactly with the best-fit SN+BAO cosmology. Furthermore,\nthe implicit dependence of the GW simulations on the cosmological parameters can introduce\nuncertainties that influence the forecasted values. In addition to the observed bias in the fore-\ncasted values away from the fiducial ΛCDM values, another noteworthy aspect of the results\nis the occurrence of certain parameters reaching their prior bounds. For instance, Figure 8.12\ndemonstrates that the confidence regions for parameters such as λandβtouch on the specified\nlow boundaries of the prior at zero. While the particular symmetries of the mixed conformal-\ndisformal model should guarantee that the same dynamics, and therefore the same predictions,\nare expected for symmetric values of these parameters, this phenomenon further supports po-\ntential degeneracies or limitations in the parameter estimation process. This observation also\nunderscores the importance of carefully selecting and defining prior distributions for parameters\nin cosmological analyses to ensure that the parameter space is adequately explored, and that\nresults are not unduly influenced by the prior assumptions. Understanding and addressing these\npotential sources of bias is crucial for refining the accuracy and reliability of future cosmological\npredictions and parameter estimations.9Dark D-Brane Cosmology\nCéu é para baixo ou para cima? Pensava a nordestina. Deitada, não sabia.\n—Clarice Lispector A Hora da Estrela\nIs the sky above or below? Wondered the northeastern girl. Lying there, she didn’t know.\n—Clarice Lispector in The Hour of the Star\nThough the ΛCDM model continues to be considered the most plausible account of cosmological\nphenomena, the physical nature of the dark sector remains unfamiliar. Cold dark matter is\ngenerally posited as a particle, predicted by many extensions to the standard model, but has\nnot been directly observed. Moreover, the magnitude of the cosmological constant remains\nincompatible with traditional quantum field theories, as detailed in Section 2.5 [77, 144, 402].\nThese theoretical considerations motivate the investigation of alternate paradigms, such as the\nquintessence field introduced in Section 5.2.1 and which was the main focus of Chapter 6. A\nscalar field can emulate cosmological constant-like behaviour and explain the apparent high\ncoincidence in the present value of Λin a more effortless dynamical way [55, 56, 81, 227, 228].\nThough relatively few studies consider a unified origin for both dark matter and dark energy\n[403–406], various coupled dark sector models have been proposed in the literature, which is also\nthe main focus of this dissertation (literally in the title!) [80, 90, 291, 292, 309, 313].\nInthischapter,wewillbuildonthetheoreticalframeworkpresentedinAppendixC,inwhichdark\nmatter and dark energy, despite being distinct entities, share a higher-dimensional geometrical\norigin, as proposed in [407]. In particular, the dark matter sector arises from the matter on a\nD-brane moving in a higher-dimensional spacetime. In this framework, dark energy parametrises\nthe brane’s position, whose higher-dimensional dynamics are encoded in the low-energy effective\naction’s kinetic and potential energy terms. This scenario hinges on the concept of hidden sector\nbranes in string theory, which experience no interaction with the D-branes accountable for the\nvisible sector (the standard model of particle physics). This means that the setup imposes that\ndark matter can only interact gravitationally with standard model fields [70, 408], with gravity\nbeingthefundamentalinteractionpropagatinginthehigher-dimensionalbulk. Ultimately, inthis\nframework, while still separate components, dark energy and dark matter emerge as geometrical\nproperties of the D-brane, becoming inevitably coupled. Moreover, because the dark energy\nparametrisesthedynamicsofthe Dark D-brane , thekineticterminactionadoptsanon-canonical\nDirac-Born-Infeld(DBI)form, commonlyconsideredincosmology(seeAppendixC),particularly\nconcerning inflationary [409–413] and dark energy models [414–426]. In this chapter, we focus\non such a framework in which dark matter and dark energy are inherently coupled and, more\n199200\nprecisely, due to differing metrics in the D-branes laying out the geodesics for particles of the\nstandard model and dark matter [220–223]. This non-universal coupling of the scalar field to\nmatter evades conflicts with Solar System tests and the stringent constraints on the speed of\ngravitational waves [125]. The relation between these two metrics is actually a realisation of the\ndisformal transformation [280] introduced in Section 5.3, which we recall is generally described\nas\n¯gµν=C(ϕ)gµν+D(ϕ)∂µϕ∂νϕ . (9.0.1)\nThe conformal and disformal factors, C(ϕ)andD(ϕ), are related and encapsulate information\nabout the curvature of the extra dimensions.\nDisformal relations have been previously investigated in the context of brane-world cosmol-\nogy [70, 427–430] and have found various applications in a range of cosmological studies [431–\n438]. They have also been instrumental in the formulation of inflationary settings [439, 440], and\ndisformal quintessence [441, 442]. Various models of disformally coupled dark energy have beem\nexplored [284, 287, 345, 355, 399, 437, 443–448], whose results can be contrasted with the work\nderived here.\nInthischapter,wereportonanextensiontotheworkpresentedinRef.[407],wherethemodelwas\nfirst formulated, along with an account of the background evolution through a dynamical systems\nanalysis. We not only present a precise numerical exploration by evolving the equations in an\nEinstein-Boltzmann code but also derive and study the linearly perturbed equations, allowing\nfor a direct comparison between this dark D-brane scenario and other theories of dark energy\nwith disformal couplings [346, 347, 355, 399, 446]. Armed with this study we can derive and\nbetter understand constraints for the relevant cosmological parameters against current data sets.\nThischapterisstructuredasfollows. Section9.1providesathoroughexpositionofthetheoretical\nunderpinnings and general equations of motion. Section 9.2 focuses on cosmological evolution\nwithin a flat FLRW spacetime. Section 9.3 is devoted to discussing cosmological perturbations,\nwhere we present the governing equations in the Newtonian gauge and append the equations\nin synchronous gauge. We also evaluate the growth of the gravitational constant in this dark\nD-brane cosmological setting and provide predictions for CMB temperature fluctuations and\nmatter power spectra. In Section 9.4, we present the main cosmological observable features of\nthe model through a numerical study. In Section 9.5, we list the cosmological data constraints\non the parameters of the model according to a Bayesian data analysis. Finally, we conclude in\nSection 9.6.\nThis chapter is based on work published in Physical Review D [3] and ongoing work.9Dark D-Brane Cosmology 201\n9.1 The Model\nIn the framework explored in this chapter, the dark sector emerges from a hidden D3-brane,\nmoving in a higher-dimensional spacetime. This sector includes two distinct degrees of freedom:\nthe matter fields that populate the brane and the brane’s radial position1. The cold dark matter\nis identified with the particles living on the so-called dark D-brane , which does not intersect with\nthe D-brane(s) from which the standard model fields emerge, explaining why the dark D-brane\nis said to be effectively hidden. Consequently, any interactions between dark matter and the\nstandard model fields are purely gravitational in the low-energy regime. Dark energy is ascribed\nto the scalar degree of freedom that parametrises the brane’s position in the extra dimensions,\nnaturally inducing the interaction in the dark sector. The geometry of this higher-dimensional\nspacetime is encoded in the warp factor, a function of the radial coordinate. For practical\npurposes, we will focus on AdS 5×S5warped regions.\nThe theoretical structure behind the Dark D-Brane (DDB) was proposed in [407], constructed\nfrom a warped flux compactification of Type IIB String Theory. The corresponding low-energy\n4D effective action takes the form (keeping with the convention in which c=ℏ= 1and the\nmetric signature is ( −+ ++)):\nS=1\n2κ2Z\nd4x√−gR+Z\nd4x√−g\u0014\nh−1(ϕ)\u0012\n1−q\n1 +h(ϕ)∂µϕ∂µϕ\u0013\n−V(ϕ)\u0015\n+X\niZ\nd4x√−gLSM(gµν, ψi, ∂µψi) +X\njZ\nd4x√−¯gLc(¯gµν, χj, ∂µχj).(9.1.1)\nThe first term is the conventional Einstein-Hilbert action, where κ2=M−2\nPl= 8πGrepresents\nthe reduced Planck mass, Gstands for Newton’s gravitational constant, gis the determinant\nof the metric tensor gµν, and Ris the Ricci scalar. The metric gµνprescribes geodesics for the\nstandard model fields. The second term is the Dirac-Born-Infeld action [410, 411] for a D3-brane,\nin which the scalar ϕparametrises a canonical normalisation of the radial position rof the D3-\nbrane in terms of the tension of the brane T3:ϕ≡√T3r. The warp factor is also a function\nofϕ, namely h(ϕ)≡T−1\n3h(r), holding the geometrical information of the warped throat region\nin the compactified space. The term V(ϕ)is a potential function. The last two terms denote\nthe actions for the standard model fields, ψi, and the dark D3-brane, where the matter fields χj\nreside. In the latter, these fields follow geodesics determined by the induced metric on the brane\n¯gµν, connected to gµνthrough a disformal transformation as given by Equation (9.0.1). For this\nstudy, we assume that the matter fields within the hidden D-brane are Disformal Dark Matter\n(DDM), inherently coupled to the scalar field ϕ.\n1We restrict our investigation to a single species living on the hidden brane, presumed to be pressureless and\nthus serving as a suitable cold dark matter candidate.202 9.1. The Model\nBefore moving on to the cosmological implications of the described setting, we remark on the\nassumptions underlying the low-energy effective action [407]. The dark D-brane was implicitly\nassumed to be a probe brane , meaning that its presence does not induce back-reaction effects on\nthe background geometry. This implies that any extra degrees of freedom that might emerge due\nto the propagation of the brane in the higher-dimensional space are neglected in this treatment.\nWe will study the evolution of cosmological perturbations to the background cosmology within\nthis model, which are minute and only treated in linear perturbation theory. In addition, the\nbrane never enters a highly relativistic state, rendering any resulting back-reaction on the bulk\ngeometryinconsequential. Hence, forthisinvestigation, theactiongiveninEquation(9.1.1)offers\na satisfactory description of disformally coupled dark matter in this model. It is worth noting,\nhowever, that when studying the spectrum of non-linear cosmological perturbation, corrections\nto the action in Equation (9.1.1) may become indispensable due to emergent degrees of freedom.\nIn the following discussions, we will address the particular scenario emerging from this framework\nin which the conformal and disformal functions are not independent and depend on the warp\nfactor h(ϕ). Nevertheless, for completeness, we present the general equations for C(ϕ)andD(ϕ).\nThis will elaborate on how the metrics embody the phenomenological interactions between the\nDBI scalar field and the disformal dark matter, recalling that standard model particles remain\ndecoupled from the dark sector.\nFrom the action in Equation (9.1.1), the Einstein equations must be derived as\nGµν≡Rµν−1\n2gµνR=κ2\u0010\nTϕ\nµν+Tc\nµν+TS\nµν\u0011\n, (9.1.2)\nwhere each component of the energy-momentum tensor is defined accordingly as\nTϕ\nµν=−2√−gδ(√−gLϕ)\nδgµν, Tc\nµν=−2√−gδ(√−¯gLc)\nδgµν, TS\nµν=−2√−gδ(√−gLS)\nδgµν.(9.1.3)\nDue to the coupling in the dark sector, the energy-momentum tensors for both the scalar field\nand dark matter are not individually conserved. Nevertheless, to preserve general covariance,\nthe total energy-momentum tensor must still be conserved, as dictated by the Bianchi identities,\nexpressed as\n∇µ\u0010\nTµν\nϕ+Tµν\nc\u0011\n= 0,∇µTµν\nS= 0. (9.1.4)\nThe equation of motion for the scalar field, derived from variation of the action in Equa-\ntion (9.1.1), is generally given by\n∇µ(γ∂µϕ)−V,ϕ+h,ϕ\nh2γ\n2\u0000\nγ−1−1\u00012=∇µ\u0014D\nCTµα\nc∂αϕ\u0015\n−1\n2\u0014C,ϕ\nCTc+D,ϕ\nCTµν\nc∂µϕ∂νϕ\u0015\n,(9.1.5)9Dark D-Brane Cosmology 203\nwhere the subscript ϕindicates differentiation with respect to the scalar field, Tc≡gµνTµν\ncis\nthe trace of the dark matter energy-momentum tensor, and\nγ=1p\n1 +h(ϕ)∂µϕ∂µϕ, (9.1.6)\nis a Lorentz factor for the brane’s motion, quantifying its relativistic character, which is always\nreal. When γ→1, the canonical kinetic term is restored, and the scalar field ϕresembles the\nquintessence field dynamics. On the other hand, the framework enters a purely relativistic regime\nin the limit γ→ ∞.\nUntil now, the equations were derived for general conformal and disformal functions. We will\nnow converge to the scenario where the disformal metric in Equation (9.0.1) is the induced metric\non a probe D3-brane moving in a warped higher-dimensional space-time. In this context, the\nfunctions CandDin the transformation have been shown to be expressed in terms of the warp\nfactor of the brane h(ϕ)[407]:\nC(ϕ) = [T3h(ϕ)]−1/2, D (ϕ) = [h(ϕ)/T3]1/2. (9.1.7)\nIn this case, Equation (9.1.5) reduces to\n∇µ(γ∂µϕ)−V,ϕ+h,ϕ\n2h2γ\u0000\nγ−1−1\u00012=∇µ[h(ϕ)Tµν\nc∂νϕ]−Tµν\nc\n4\u0014\n−h,ϕ\nhgµν+h,ϕ∂µϕ∂νϕ\u0015\n.(9.1.8)\nUnder the assumption of a homogeneous and isotropic Universe, all the matter species in the\ntheory can be accurately modelled as perfect fluids. For cold dark matter, this implies an energy-\nmomentum tensor expressed as\nTc\nµν=ρcuc\nµuc\nν, (9.1.9)\nwhere uc\nµis the four-velocity of the fluid for a comoving observer and ρcis the energy density.\nFrom Equation (9.1.3), defining the energy-momentum tensor for the scalar field, and from the\nDBI action in Equation (9.1.1), we find\nTϕ\nµν=\u00121−γ−1\nh−V\u0013\ngµν+γ∂µϕ∂νϕ . (9.1.10)\nFor the dark energy fluid, the perfect fluid form implies\nuϕ\nµ=∂µϕ√−∂νϕ∂νϕ, (9.1.11)\nfor the scalar field’s four-velocity, and204 9.2. Background FLRW Cosmology\nρϕ=γ−1\nh+V ,and pϕ=1−γ−1\nh−V , (9.1.12)\nfor the energy density and pressure, respectively. Additionally, the EoS parameter for the scalar\nfield is defined as\nwϕ=pϕ\nρϕ=(γ−1)/hγ−V\n(γ−1)/h+V. (9.1.13)\nThe conservation relations given in Equation (9.1.4) can be manipulated to yield an explicit\nexpression for the coupling Qbetween the dark sector fluids:\n∇µTµν\nϕ=Q∂νϕ , (9.1.14)\nwith\nQ=∇αh\nh(ϕ)Tαβ\nc∂β��i\n−h,ϕ\n4hTαβ\nc[−gαβ+h∂αϕ∂βϕ]. (9.1.15)\nThe interaction term Qaccounts for the energy exchange between the dark energy and dark\nmatter sectors and is central to the discussions in this work.\n9.2 Background FLRW Cosmology\nTo study the cosmological implications of the DDB model we consider a spatially flat Friedmann-\nLemaître-Robertson-Walker (FLRW) background, as defined in Equation (1.6.1).\nThe scalar field ϕis considered to be homogeneous, simply written as ϕ=ϕ(τ). In the dark\nD-brane disformal frame, in which the dark matter geodesics are defined, the line element is\nrelated to Equation (1.6.1) as\nd¯s2=Ca2(τ)\u0000\n−Z2dτ2+ dx2+ dy2+ dz2\u0001\n, (9.2.1)\nwhere Zrepresents the disformal scalar, related to the Jacobian of the metric transformation.\nIn particular, with CandDgiven by Equation (9.1.7), the disformal scalar is found to be\nZ≡r\n1−2XD\nC=r\n1−h(ϕ)ϕ′2\na2=1\nγ, (9.2.2)\nwhere Xis the standard kinetic term for the scalar field, defined as X=−1\n2gµν∂µϕ∂νϕ=ϕ′2\n2a2.\nThe implications of Equation (9.2.1) are that while the conformal factor influences the entire line\nelement and thus dilutes dark matter, the disformal factor impacts the time component only,\naltering the light cones of dark matter particles.\nWith Tµν\ncas provided in Equation (9.1.9), the coupling term in Equation (9.1.15) can be recast\nas9Dark D-Brane Cosmology 205\nQ=ρc\n2Ca2\u0014\nϕ′2D,ϕ+a2C,ϕ−2ϕ′2DC,ϕ\nC+ 2D(ϕ′′+Hϕ′+ρ′\nc\nρcϕ′)\u0015\n. (9.2.3)\nFor the particular definitions of the conformal and disformal functions for the dark D-brane in\nEquation (9.1.7), Equation (9.2.3) reduces to\nQ=hρca−2\u00143\n4ϕ′2h,ϕ\nh−a2h,ϕ\n4h2+ϕ′\u0012\n2H+ρ′\nc\nρc\u0013\n+ϕ′′\u0015\n. (9.2.4)\nThe equation of motion for the scalar field, given by Equation (9.1.5), in conjunction with\nEquation (9.2.3), results in\nϕ′′− H\u0000\n1−3γ−2\u0001\nϕ′+a2h,ϕ\n2h2\u0000\n1−3γ−2+ 2γ−3\u0001\n+a2γ−3(V,ϕ+Q) = 0 , (9.2.5)\nwhere Qis as specified in Equation (9.2.4). Given that the brane’s warp factor h(ϕ)is always\nnon-negative, it is always warranted that γ≥1. The nonrelativistic regime is achieved when\na−2hϕ′2≪1andγ→1. In this non-trivial regime, which depends on ϕ(through h) and ϕ′, the\nDBI Lagrangian in Equation (9.1.1) simplifies to the standard quintessence case.\nIt should also be emphasised that while the DBI action in Equation (9.1.1) can be thought\nof as a specific instance of the k-essence action presented in Section 5.2.2 [270, 272], the DBI\nscalar field’s pressure can only become negative for a non-vanishing potential V(ϕ). In the limit\nV(ϕ)→0, from Equation (9.1.13), we note that wϕ→1/γ, which is invariably non-negative.\nOn the other hand, in the slow-roll limit where the kinetic term’s influence is negligible, the\nscalar field mimics a cosmological constant, with wϕ≈ −1. Thus, the coupled DBI model is\na comprehensive framework that embodies various models previously explored, such as coupled\nquintessence (see Chapter 6) [346, 355, 399] and coupled tachyonic dark energy [316, 449]. This\nis reflected in the form of the equation governing the motion of the DBI scalar field in Equa-\ntion (9.2.5), which is considerably more complex than its canonical counterpart, to which it\nreduces to in the appropriate limit.\nFrom the Einstein field equations, the Friedmann equations can be derived, describing the evo-\nlution of each i-th fluid component as\nH2=κ2a2\n3X\niρi=κ2a2\n3(ρr+ρb+ρc+ρϕ), (9.2.6)\nand\nH′+H2=−κ2a2\n6X\ni(ρi+ 3pi) =−κ2a2\n6(ρr+ 4pr+ρb+ρc+ρϕ+ 3pϕ),(9.2.7)\nwhere we include the standard model’s matter sectors of the Universe: baryons and radiation, de-\nnoted by subscripts bandr. As they are non-interacting, their evolution follows the conventional206 9.2. Background FLRW Cosmology\nconservation laws\nρ′\nr+ 4Hρr= 0,and ρ′\nb+ 3Hρb= 0, (9.2.8)\nwhere pb= 0andpr=1\n3ρrfor each respective fluid. Instead, from Equations (9.1.4) and (9.1.14),\nthe continuity equations for the interacting disformally coupled fluids are\nρ′\nϕ+ 3Hρϕ(1 +wϕ) =−Qϕ′, (9.2.9)\nwith wϕas in Equation (9.1.13), and\nρ′\nc+ 3Hρc=Qϕ′. (9.2.10)\nIn the non-relativistic regime, γ→1 +hϕ′2\n2a2, and the EoS parameter for quintessence is recovered\nonce more. Nevertheless, what distinguishes this setting is precisely the impact and signatures\nof the non-canonical relativistic behaviour. Using Equations (9.2.4) and (9.2.5), the expression\nforQcan be rewritten to include only first-order derivatives of the scalar field\nQ=−\"\nh\u0000\nV,ϕ+ 3a−2Hγϕ′\u0001\n+h,ϕ\nh\u0000\n1−3\n4γ\u0001\nγ+hρc#\nρc. (9.2.11)\nThe sign of Qin Equations (9.2.9) and (9.2.10) indicates the direction of energy transfer in the\ndark sector. The coupling yields distinct interpretations for each fluid. For the DBI scalar field,\nthe coupling can be combined with the self-interacting potential to create an effective scalar\nfield potential, Veff(ϕ, ϕ′). The coupling can also be viewed as a local change in the geometry,\nencoded in ¯g, which governs the geodesics along which dark matter propagates. To facilitate the\ncomparison with the canonical case, we introduce an effective coupling β, defined as\nβ=−Q\nκρc, (9.2.12)\nsuch that, when ϕ′<0, positive values of βcorrespond to dark energy granting energy to dark\nmatter and, conversely, when βis negative, the dark energy field is being sourced by dark matter.\nTo compare with cosmological constraints assuming a non-interacting dark sector, we introduce\nan effective dark energy EoS parameter wϕ,eff[450]. This can be thought of as a map from\nthe coupled model to an uncoupled one at the background level, where the dark matter does\nnot interact with dark energy, but all the coupling effects are transported into an effective dark\nenergy fluid,\nρϕ,eff=ρϕ+ρc−ρc,0a−3, (9.2.13)\nwhere a subscript 0indicates present values.9Dark D-Brane Cosmology 207\nThe Friedmann equation can be reformulated according to Equation (9.2.13) as\nH2=κ2a2\n3\u0000\nρr,0a−4+ρb,0a−3+ρc,0a−3+ρϕ,eff\u0001\n, (9.2.14)\nwhere ρr,0,ρb,0, and ρc,0represent present-day energy densities for radiation, baryons, and dark\nmatter, respectively. Taking the derivative of Equation (9.2.13), we obtain\nρ′\nϕ,eff+ 3Hρϕ,eff(1 +wϕ,eff) = 0 , (9.2.15)\nconsistent with a standard, uncoupled model. Comparison with Equations (9.2.9) and (9.2.10)\nyields\nwϕ,eff=pϕ\nρϕ,eff, (9.2.16)\nwith pϕas definedin Equation(9.1.12). This formulation allows fora straightforward comparison\nbetween the background cosmological evolution of the disformally coupled dark sector in this\nmodel and empirical observations.\n9.2.1 Qualitative Dynamics and Initial Conditions\nIn the present study, we focus on an AdS 5throat model for the warp factor, endowed with a\nquadratic scalar field potential, expressed as\nh(ϕ) =h0\nϕ4,and V(ϕ) =V0ϕ2\nκ2, (9.2.17)\nwhere h0, V0>0are dimensionless parameters in this definition. Four parameters govern the\ncosmological evolution in this model: the scales of the warp factor and the potential, namely h0\nandV0, and the initial conditions for the scalar field and its rate of change, denoted by ϕiand\nϕ′\ni. As discussed in [407], the geometrical scales can be encapsulated in a single, dimensionless\nparameter\nΓ0≡V0\n˜h0,with ˜h0=1\nh0. (9.2.18)\nThis combined parameter plays a crucial role in determining the dynamics near fixed-point\nsolutions of the dynamical system of equations and has a significant impact on the ultimate\nfate of the Universe. The disformally coupled framework under consideration here has the same\nnumber of parameters as its standard DBI uncoupled counterpart [416, 417, 421]. This implies\nthat the conformal and disformal effects are intertwined, and the uncoupled scenario cannot be\nrecovered from the limit values of these parameters.\nA dynamical system analysis of the background cosmological evolution for different values of the\nparameters has been conducted in Ref. [407] (see [451] for a review on dynamical systems applied208 9.3. Linear Perturbations\nto cosmology). This methodology allows us to qualitatively understand the Universe’s evolution\nthrough the fixed point solutions1, without the need for the full numerical evolution. As elabo-\nrated in Chapter 5, a particularly compelling aspect of introducing couplings into the dark sector\nis the potential emergence of scaling fixed point solutions. In such solutions, both components\ndilute at an identical rate, resulting in a consistent ratio of their fractional energy densities. This\noffers a more intuitive explanation for the observed distribution of energy in the Universe. The\nauthors of Ref. [407] reported that when Γ0>1, interesting particular cosmological solutions\nappear, including a saddle point with a scaling behaviour, Equation (5.2.11), between dark en-\nergy and dark matter. In this regime, dark matter could be driving the cosmic expansion, and\nthe observed acceleration of the expansion could be initiated during the matter-dominated era,\nfuelled by the non-minimal coupling. Regardless of the initial conditions, owing to the repelling\nnature of the saddle point, the DDM starts diluting away, and the system eventually transitions\nto a standard accelerating solution dominated by the evolution of the DBI scalar field.\nTheinitialconditionsfor ϕandϕ′playaninstrumentalroleindeterminingtheonsetandduration\nof the scaling and attractor regimes, respectively, more relativistic scenarios corresponding to a\nlonger transition. However, if this scaling is happening at present, then the effects of ϕ′\niare not\ncurrently observable. In contrast, ϕisignificantly influences the background and perturbation\nevolution. The system remains in a frozen latent state with γ≈1andwϕ≈ −1during matter\nand radiation-dominated periods. Taking larger values of Γ0, both the scaling and dark energy\ndominated solutions converge to a de Sitter-like state, with dynamics dominated by the potential,\nwhich leads to wϕ→ −1, making the Universe increasingly similar to a ΛCDM cosmology near\nthe fixed points, even though this may no longer hold at the level of the perturbations. For this\nstudy, we limit our attention to scenarios where ϕi>0andϕ′\ni<0to maintain consistency with\nthe extra-dimensional interpretation of the field ϕas the radius as the brane moves down the\nthroat, and to reproduce cosmologies that allow for the scaling regime.\nIn the following sections, we will present a numerical examination of the cosmological implica-\ntions of this model, focusing on scenarios that yield a slowly evolving cosmological constant-like\nbehaviour in the early Universe [452]. Since disformal dark matter can contribute to the ac-\ncelerated expansion, we expect unique signatures compared to standard ΛCDM and coupled\nquintessence models.\n9.3 Linear Perturbations\nIn this section, we explore the dynamics of cosmological perturbations. Based on the background\nanalysis, we anticipate that the dark D-brane model will exhibit a non-trivial and rich linear-level\nphenomenology, which we will connect with predictions according to existing observational data\n1Each realisation of the model maps to a specific trajectory in the phase space, encapsulating the Universe’s\nevolutionary dynamics. The fixed points serve as the asymptotic scenarios of this evolution, illustrating distinct\nepochs in the cosmological history of the Universe.9Dark D-Brane Cosmology 209\nin Section 9.5.2. We show particular predictions for both the temperature anisotropy spectrum\nof the CMB and the matter power spectrum. The equations presented in this chapter are tailored\nfor the Newtonian gauge and are specific to the model under discussion. Nevertheless, for a more\ncomprehensive perspective, we provide the equations for the synchronous gauge, along with those\nfor generic conformal and disformal functions, C(ϕ)andD(ϕ).\n9.3.1 Conformal Newtonian Gauge\nWe first focus on scalar perturbations in the conformal Newtonian gauge, as described in Sec-\ntion 2.2.1, where the perturbed line element takes the form\nds2=a2(τ)\u0002\n−(1 + 2Ψ) d τ2+ (1−2Φ)δijdxidxj\u0003\n, (9.3.1)\nwhere Ψ(τ, xi)and Φ(τ, xi)are the scalar metric perturbations. From Equations (9.1.2)\nand(9.3.1), thecomponentsoftheperturbedEinsteinequationsarereadilyobtainedandequated\nas\nδGµ\nν=κ2δTµ\nν, (9.3.2)\nwhere δGµ\nνandδTµ\nνare the perturbed Einstein and energy-momentum tensors, respectively.\nFor each fluid, the distinct components of δTµ\nν,fare given by\nδT0\n0,f=−δρf, (9.3.3)\nδT0\ni,f= (ρf+pf)∂ivf, (9.3.4)\nδTi\n0,f=−(ρf+pf)∂ivf, (9.3.5)\nδTi\nj,f=δpfδi\nj+ Πi\nj,f, (9.3.6)\nwhere fis an index running over each fluid. Consistently, δρf,δpf,vf, and Πi\nj,fdenote the\nperturbed energy density, pressure perturbation, peculiar velocity potential, and anisotropic\nstress tensor for the fluid f, respectively. Given a specific matter source, the perturbed energy-\nmomentum tensor is defined correspondingly. For the model in focus, the Fourier space repre-\nsentations of the perturbed Einstein equations are as follows:210 9.3. Linear Perturbations\nk2Φ + 3H\u0000\nΦ′+HΨ\u0001\n=−4πGa2X\nfδρf, (9.3.7)\nk2\u0000\nΦ′+HΨ\u0001\n= 4πGa2X\nfρf(1 +wf)θf,(9.3.8)\nΦ′′+H\u0000\nΨ′+ 2Φ′\u0001\n+ Ψ\u0000\nH2+ 2H′\u0001\n+k2\n3(Φ−Ψ) = 4 πGa2X\nfδpf, (9.3.9)\nk2(Φ−Ψ) = 12 πGa2X\nfρf(1 +wf)σf.(9.3.10)\nThese equations establish a relationship between the scalar potentials ΦandΨand the perturba-\ntions in the matter fluids. Fourier expansion has been applied to the metric potentials, effectively\nreplacing spatial derivatives with Fourier modes characterised by the wave number k. Addition-\nally, the velocity potential has been implicitly defined as θf=∇i∇ivf. For direct comparison\npurposed with the canonical case we introduce the anisotropic stress perturbation given by σf,\naccording to [347]. As a first approximation, we focus on the scenario where σfvanishes for all\nfluids. According to the fourth Einstein perturbation equation, Equation (9.3.10), this leads to\nΨ = Φ. Moving forward, we replace the perturbed energy density δρfwith the density contrast\nδf=δρf\nρf. We further assume that each fluid features an adiabatic speed of sound c2\ns,f=δpf\nδρf.\nThe perturbed conservation equations derive from the zero-divergence condition on the energy-\nmomentum tensor, ∇νTµν\nf= 0, which must now be distinguished for uncoupled and coupled\nscenarios:\n∇µδTµ\nν,u+δΓµ\nµβTβ\nν,u−δΓβ\nµνTµ\nβ,u= 0, (9.3.11)\n∇µδTµ\nν,c+δΓµ\nµβTβ\nν,c−δΓβ\nµνTµ\nβ,c=−Q∂νϕ , (9.3.12)\nwhere f={u, c}was introduced to specify uncoupled and coupled fluids relative to the scalar\nfield. The perturbations of the Christoffel symbols are denoted by δΓµ\nνβ. From Equations (9.3.3)\nto (9.3.6), we can rewrite Equations (9.3.11) and (9.3.12) for the evolution of the density contrast\nδfand the velocity potential θffor each fluid. Specifically, the baryonic and radiation fluids\nremain uncoupled from the scalar field and are described by the following conservation equations:\nδ′\nu+ 3H(c2\ns,u−wu)δu= (1 + wu)(3Φ′−θu), (9.3.13)\nθ′\nu+H(1−3wu)θu+w′\nu\n1 +wuθu=k2\"\nc2\ns,u\n1 +wuδu+ Ψ#\n−k2σu, (9.3.14)9Dark D-Brane Cosmology 211\nwhere u={b, r}andwu=pu/ρuis the EoS parameter. The first equation represents the\nperturbed continuity equation, and the second one is the Euler equation, which emerges from\nthe temporal and spatial components of the energy conservation equation, respectively.\nFor the disformal dark matter component, which is the only species coupled to the scalar field in\nthis setting, the continuity and Euler equations transform as follows, considering wc=cs,c= 0:\nδ′\nc=−(θc−3Φ′)−Q\nρcϕ′δc+Q\nρcδϕ′+δQ\nρcϕ′, (9.3.15)\nθ′\nc+Hθc=k2Ψ−Qϕ′\nρcθc+k2Q\nρcδϕ . (9.3.16)\nThe perturbation of the DBI scalar field evolves according to the perturbed Klein-Gordon equa-\ntion:\nδϕ′′+\u00143h,ϕ\nh\u0000\n1−γ−1\u0001\nϕ′− H\u0000\n7−9γ−2\u0001\n−3h(V,ϕ+Q)γ−1ϕ′\u0015\nδϕ′+\u0014\n−3h,ϕ\nhH\u0000\n1−γ−2\u0001\nϕ′\n+a2V,ϕϕγ−3+h,ϕϕ\n2h2a2\u0000\n1−3γ−2+ 2γ−3\u0001\n−3\n2h,ϕ\nha2(V,ϕ+Q)\u0000\nγ−1−γ−3\u0001\n(9.3.17)\n+h2\n,ϕ\n2h3a2\u0000\n1−3γ−1+ 3γ−2−γ−3\u0001#\nδϕ+\u0014\n6H\u0000\n1−γ−2\u0001\nϕ′−h,ϕ\nh2a2\u0000\n2−3γ−1+γ−3\u0001\n+a2(V,ϕ+Q)\u0000\n3γ−1−γ−3\u0001\u0003\nΨ−ϕ′Ψ′−3γ−2ϕ′Φ′−γ−2∂i∂iδϕ+a2γ−3δQ= 0.\nWe will present the perturbed terms stemming from the introduction of the coupling in this\nsetting, directly dependent on the warp factor h(ϕ). Nevertheless, we derive first the general\nexpression of the perturbed disformal coupling δQfor general conformal and disformal functions\nin the Newtonian gauge for completeness and as it may be relevant for extensions to this setting\nscenarios.\nδQ=a−2ρc\nC−D\nh(1−γ−2) +Dρcγ−3\u0000\nQ1δc+Q2Φ′+Q3Ψ +Q4δϕ′+Q5δϕ\u0001\n,(9.3.18)\nwith212 9.3. Linear Perturbations\nQ1=1\n2a2C,ϕ\u0012\n1−3δpc\nδρc\u0013\n−3DHδpc\nδρcϕ′−DC,ϕ\nCϕ′2+D,ϕ\n2ϕ′2(9.3.19)\n−a2D\n2h,ϕ\nh2\u0012\n1−3\n2γ−2\u0013\n+a2DQ\nhρcγ−2,\nQ2=3D\u0000\nγ−2+w\u0001\nϕ′, (9.3.20)\nQ3=3DH\u0000\n1 +γ−2+ 2w\u0001\nϕ′−a2D,ϕ\nh\u0000\n1−γ−2\u0001\n+3\n4a2Dh,ϕ\nh2\u0000\n1−γ−2\u0001\n+ 2a2D\nhC,ϕ\nC\u0000\n1−γ−2\u0001\n+a2D\nhQ\nρc\u0000\n1−γ−2\u0001\n,\nQ4=3DH\u0000\n2−3γ−2−wc\u0001\n−2DC,ϕ\nCϕ′+D,ϕϕ′−3Dh,ϕ\nh\u0000\n1−γ−1\u0001\nϕ′(9.3.21)\n+ 3DHγ−1(V,ϕ+Q)ϕ′+ 2DQ\nρcϕ′,\nQ5=−k2D\u0000\nγ−2+wc\u0001\n+ 3H\u0012DC,ϕ\nC−D,ϕ\u0013\nwcϕ′−a2\n2C2\n,ϕ\nC(1−3wc) + 2DC,ϕ\nCϕ′2\n−3\n2C,ϕD,ϕ\nCϕ′2−3\n2Dh,ϕ\nhH\u0000\n1−γ−2\u0001\nϕ′+a2\n2DC,ϕ\nCh,ϕ\nh2\u0012\n1−3\n2γ−2\u0013\n(9.3.22)\n−a2\n2D,ϕh,ϕ\nh2\u0012\n1−3\n2γ−2\u0013\n+a2\n2Dh2\n,ϕ\nh3\u00125\n4−21\n4γ−2+ 4γ−3\u0013\n+1\n2a2C,ϕϕ(1−3wc)\n−DC,ϕϕ\nCϕ′2+D,ϕϕ\n2ϕ′2−a2\n2Dh,ϕϕ\nh2\u0000\n1−3γ−2+ 2γ−3\u0001\n−a2Dγ−3V,ϕϕ\n+Q\nhρc\u0014\na2D,ϕ−a2DC,ϕ\nC−3\n2Dh,ϕϕ′2\u0015\n,\nfor general conformal and disformal functions, C(ϕ)andD(ϕ), respectively.\nFinally, the perturbation of the coupling Qas introduced in Equation (9.2.11) for the dark\nD-brane model we are investigating, reduces to:\nδQ=a−2ρc\nγ−2+hρcγ−3\u0000\nQ1δc+Q2Φ′+Q3Ψ +Q4δϕ′+Q5δϕ\u0001\n, (9.3.23)\nwith the following perturbed coefficients:9Dark D-Brane Cosmology 213\nQ1=a2Q\nρcγ−2+ 3hδpc\nδρc\u0012\na2h,ϕ\n4h2− Hϕ′\u0013\n, (9.3.24)\nQ2=3h\u0000\nγ−2+w\u0001\nϕ′, (9.3.25)\nQ3=3hH\u0000\n1 +γ−2+ 2w\u0001\nϕ′−a23\n4h,ϕ\nh\u0000\n1−γ−2\u0001\n+a2Q\nρc\u0000\n1−γ−2\u0001\n, (9.3.26)\nQ4=3hH\u0000\n2−3γ−2−w\u0001\n+ 3h2(V,ϕ+Q)γ−1ϕ′+ 2hQ\nρcϕ′−3\n2h,ϕ\u0000\n1−2γ−1\u0001\nϕ′,(9.3.27)\nQ5=−k2h\u0000\nγ−2+w\u0001\n+a2h2\n,ϕ\n2h2\u00123\n4−15\n4γ−2+ 4γ−3−3\n2w\u0013\n+a23\n4h,ϕϕ\nh\u0012\nγ−2−4\n3γ−3+w\u0013\n−3\n2h,ϕH\u0000\n1−γ−2+ 2w\u0001\nϕ′−a2hV,ϕϕγ−3−a2h,ϕ\n2hQ\nρc\u0000\n1−3γ−2\u0001\n. (9.3.28)\nOur analysis confirms that in the limit hϕ′2≪a2andγ→1, we retrieve the disformal\nquintessence scenario as previously investigated in [347, 399]. Additionally, the form of δQ\nillustrates that disformal couplings introduce a direct dependence on the Fourier scale k, namely\nthrough the first term of Q5in Equation (9.3.28). This scale-dependence is a characteristic\ntrait of disformal couplings [347, 399, 446], and we will show how it reflects in the growth and\ndistribution of perturbations through the matter power spectrum.\nFor the DBI scalar field, and focusing on adiabatic perturbations, the sound speed c2\ns,ϕis defined\naccording to\nc2\ns,ϕ≡\u0012∂p\n∂X\u0013\u0012∂ρ\n∂X\u0013−1\n=1\nγ2≤1. (9.3.29)\nThis quantity is always positive, ensuring the stability of the perturbations. The fact that the\nDBI scalar field can accommodate for c2\ns,ϕ̸= 1is correlated with its unique features, imprinted\nin the cosmic microwave background temperature fluctuations and the matter power spectrum -\ntwo key observables we will explore in the numerical study.\n9.3.2 Synchronous Gauge\nIn this study, we have considered the equations in the synchronous gauge, as introduced in\nSection 2.2.2 for the purpose of cross-checking the equations and the cosmological predictions.\nTherefore, to offer a comprehensive view, this section presents the perturbation equations in\nthe synchronous gauge, covering general conformal C(ϕ)and disformal D(ϕ)coupling functions,\nbefore particularising to the scenario examined in this work, where both C(ϕ)andD(ϕ)are tied\nto the warp factor h(ϕ).\nIn a parallel manner to the derivation for the Newtonian gauge, we compute the perturbed\nEinstein equations:214 9.3. Linear Perturbations\nk2η−1\n2Hh′=−4πGa2X\nδρf, (9.3.30)\nk2η′= 4πGa2X\nρf(1 +wf)θf, (9.3.31)\nh′′+ 2Hh′−2k2η=−24πGa2X\nδpf, (9.3.32)\nh′′+ 6η′′+ 2H\u0000h′+ 6η′\u0001\n−2k2η=−24πGa2X\nρf(1 +wf)σf, (9.3.33)\nwith θf,σfandvi\nfas previously defined.\nThe perturbed continuity and Euler equations for the uncoupled baryonic and radiation fluids\nbecome\nδ′\nu+ 3H\u0000\nc2\ns,u−wu\u0001\nδu=−(1 +wu)\u0012h′\n2+θu\u0013\n, (9.3.34)\nθ′\nu+\u0014\nH(1−3wu) +w′\nu\n1 +wu\u0015\nθu=k2c2\ns,u\n1 +wuδu−k2σu, (9.3.35)\nwith u={b, r}. For DDM, the coupled shear-free versions of these equations are:\nδ′\nc+ 3H\u0000\nc2\ns,c−wc\u0001\nδc=−(1 +wc)\u0012h′\n2+θc\u0013\n−Q\nρcδcϕ′+Q\nρcδϕ′+δQ\nρcϕ′,(9.3.36)\nθ′\nc+\u0014\nH(1−3wc) +w′\nc\n1 +wc\u0015\nθc=k2c2\ns,c\n1 +wcδc−Q\nρcϕ′θc+k2 Q\nρc(1 +wc)δϕ . (9.3.37)\nThe perturbed Klein-Gordon equation in the synchronous gauge reads\nδϕ′′+h′\n2γ−2ϕ′+\u0014\n2H+3\n4h,ϕ\nhϕ′+ 3Q\nρcϕ′\u0015\nδϕ′+\u0014\nk2γ−2+3\n2h,ϕ\nhH\u0000\n1−γ−2\u0001\nϕ′\n−a2\n2h2\n,ϕ\nh3\u00125\n4+ 4γ−3−21\n4γ−2\u0013\n+a2\n2h,ϕϕ\nh2\u0000\n1 + 2 γ−3−3γ−2\u0001\n+a2V,ϕϕγ−3#\nδϕ+a2γ−3δQ= 0,\n(9.3.38)\nwith the general perturbation of the coupling function Qexpressed as\nδQ=a−2ρc\nC−D\nh(1−γ−2) +Dγ−3ρc\u0000\nQ1δc+Q2h′+Q3δϕ′+Q4δϕ\u0001\n, (9.3.39)\nwith9Dark D-Brane Cosmology 215\nQ1=1\n2a2C,ϕ\u0012\n1−3δpc\nδρc\u0013\n−3DHδpc\nδρcϕ′−DC,ϕ\nCϕ′2+D,ϕ\n2ϕ′2−a2D\n2h,ϕ\nh2\u0012\n1−3\n2γ−2\u0013\n,\n+a2DQ\nhρcγ−2+a2DQ\nhρcγ−2(9.3.40)\nQ2=−D\n2γ−2(1 +wc)ϕ′, (9.3.41)\nQ3=3DH\u0000\n2−3γ−2−wc\u0001\n−2DC,ϕ\nCϕ′+D,ϕϕ′−3Dh,ϕ\nh\u0000\n1−γ−1\u0001\nϕ′\n+ 3DHγ−1(V,ϕ+Q)ϕ′+ 2DQ\nρcϕ′, (9.3.42)\nQ4=−k2D\u0000\nγ−2+wc\u0001\n+ 3H\u0012DC,ϕ\nC−D,ϕ\u0013\nwcϕ′−a2\n2C2\n,ϕ\nC(1−3wc) + 2DC,ϕ\nCϕ′2\n−3\n2C,ϕD,ϕ\nCϕ′2−3\n2Dh,ϕ\nhH\u0000\n1−γ−2\u0001\nϕ′+a2\n2DC,ϕ\nCh,ϕ\nh2\u0012\n1−3\n2γ−2\u0013\n(9.3.43)\n−a2\n2D,ϕh,ϕ\nh2\u0012\n1−3\n2γ−2\u0013\n+a2\n2Dh2\n,ϕ\nh3\u00125\n4−21\n4γ−2+ 4γ−3\u0013\n+1\n2a2C,ϕϕ(1−3wc)\n−DC,ϕϕ\nCϕ′2+D,ϕϕ\n2ϕ′2−a2\n2Dh,ϕϕ\nh2\u0000\n1−3γ−2+ 2γ−3\u0001\n−a2Dγ−3V,ϕϕ\n+Q\nhρc\u0014\na2D,ϕ−a2DC,ϕ\nC−3\n2Dh,ϕϕ′2\u0015\n,\nfor general C(ϕ)andD(ϕ). If we now particularise to C(ϕ) = ( T3h(ϕ))−1/2andD(ϕ) =\n(h(ϕ)/T3)1/2, these become\nδQ=a−2ρc\nγ−2+hρcγ−3\u0000\nQ1δc+Q2h′+Q3δϕ′+Q4δϕ\u0001\n, (9.3.44)\nwith\nQ1=a2Q\nρcγ−2+ 3hδpc\nδρc\u0012\na2h,ϕ\n4h2− Hϕ′\u0013\n, (9.3.45)\nQ2=−h\n2\u0000\nγ−2+w\u0001\nϕ′, (9.3.46)\nQ3=3hH\u0000\n2−3γ−2−w\u0001\n+ 3h2(V,ϕ+Q)γ−1ϕ′+ 2hQ\nρcϕ′−3\n2h,ϕ\u0000\n1−2γ−1\u0001\nϕ′,(9.3.47)\nQ4=−k2h\u0000\nγ−2+w\u0001\n+a2h2\n,ϕ\n2h2\u00123\n4−15\n4γ−2+ 4γ−3−3\n2w\u0013\n+a23\n4h,ϕϕ\nh\u0012\nγ−2−4\n3γ−3+w\u0013\n−3\n2h,ϕH\u0000\n1−γ−2+ 2w\u0001\nϕ′−a2hV,ϕϕγ−3−a2h,ϕ\n2hQ\nρc\u0000\n1−3γ−2\u0001\n. (9.3.48)\nWe will now proceed to investigate the phenomenology of these intricate equations.216 9.4. Numerical Study\n9.4 Numerical Study\n9.4.1 Implementation and Background Results\nModel M ϕi/Γ0Γ0ϕi(Mpl)γ wϕGeff/G σ8\nM3/1.51.5 31.01-0.63 0.581.40\nM3/5 531.01-0.70 0.461.24\nM3/10 10 31.02-0.73 0.401.16\nM1.5/1.51.51.51.65-0.66 0.300.79\nM1.5/551.51.49-0.76 0.410.77\nM1.5/10101.51.42-0.82 0.730.77\nTable9.1 : Summaryofparameterchoicesfortheillustrativemodelsexaminedinthisstudy. All\nmodels use an initial scalar field velocity of ϕ′\ni=−10−25Mpl/Mpc and all unspecified parameters\nare fixed to fiducial ΛCDMPlanckcosmological parameters [32], as detailed in Table 3.1. The\nmodels are illustrative and may not be cosmologically viable. Key present-day quantities are\nindicated with a superscript 0. The Lorentz factor γquantifies current deviations from standard\nquintessence. The DBI EoS parameter wϕmeasures deviations from cosmological constant-like\nbehaviour. The effective gravitational constant G0\neff/Gand the predicted clustering amplitude\nσ8values are also included, with σ8≈0.85for the reference ΛCDM simulation.\nWe have conducted the numerical analysis using a modified version of the Boltzmann code CLASS\n[95, 101] tailored to have dark energy portrayed by a DBI scalar field and a coupling to the dark\nmatter component, as described in Section 4.2. From this modified patch, we can thoroughly\ninvestigatethecosmologicalimplicationsofthetheoreticalmodelandinfertherangeofphysically\nviable parameters. Moreover, the code is adapted to handle both the background and linear\nperturbations associated with a DBI scalar field. For the simulations, we employed Planck2018\ncosmological parameters [32] assuming a spatially flat ΛCDM cosmological background, listed in\nTable 3.1.\nThe scale of the potential V0, defined in Equation (9.2.17), is treated as a shooting parame-\nter, which is adjusted for each instance of the code to match the Friedmann constraint, Equa-\ntion (2.1.26), under the assumption that the system is currently near the scaling regime. This\nchoice is motivated by the argument in Ref. [407] that the true dynamics of the system is fully\nspecified by the degenerate combination Γ0=h0V0.\nOur study focuses on understanding different regimes of the model, differentiated by their initial\nconditions and Γ0values. The sets of parameters used for illustrative purposes are listed in\nTable 9.1, along with some present-day quantities that show the impact of varying the model\nparameters. Two categories of coupled models are considered, each marked by a distinct initial9Dark D-Brane Cosmology 217\n0 1 2 3\nlog(1+z)0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0βφi=3MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n0 1 2 3\nlog(1+z)0.2\n0.00.20.40.60.8βφi=1.5MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\nFigure 9.1 : Background evolution of the effective coupling function β, defined in Equa-\ntion (9.2.12), plotted as functions of the redshift z. The left and right panels correspond to\nModels M 3/1.5-M3/10and M 1.5/1.5-M1.5/10in Table 9.1, respectively. We clearly identify two\nregimes of the theory: one in which the effective coupling is always negative throughout the cos-\nmic evolution, depicted on the left panel, and another one for which the coupling may start out\nas being negative but eventually starts to increase towards positive values at present, pictured\non the right panel.\nscalar field value and thus different characteristics of the effective coupling, as illustrated in\nFigure 9.1. The first class, embodied by models M 3/1.5-M3/10in the upper section of Table 9.1,\nassumes ϕi= 3Mpland features a consistently negative effective coupling, with an earlier onset\nand larger absolute magnitude of the effective coupling for smaller Γ0values. Conversely, the\nsecond class, exemplified by models M 1.5/1.5-M1.5/10in the lower section of Table 9.1, with\nϕi= 1.5MPl, maintainsapositivecouplingatpresent, eventhoughitcanstartbybeingnegative,\nas is the case in model M 1.5/1.5. The coupling value increases for higher Γ0values in this latter\ncategory.\nOur findings suggest a non-trivial relationship between Γ0andϕi. Owing to the warp factor’s\ninversedependenceonthescalarfieldvalue( h(ϕ)∝ϕ−4), higher ϕivaluescorrelatewithlarger h0\nvalues (and correspondingly lower V0) to reproduce the fiducial cosmology at present maintaining\nthe same Γ0.\nWe turn our attention to the redshift evolution of relevant background quantities in the six\nillustrative cosmological models: the effective coupling β, as defined in Equation (9.2.12); the\nfractional energy densities Ωifor each fluid i, defined as Ωi≡κ2ρi/3H2; the Hubble rate H,\naccording to Equation (9.2.14), and the DBI and effective DE equations of state, articulated\nin Equations (9.1.13) and (9.2.16). These quantities are illustrated in Figures 9.1 and 9.2. We\nemphasise that they encapsulate the Universe’s background evolution and also hold implications\nfor the linear growth rate of structures, deeply entangled with the background evolution.218 9.4. Numerical Study\n0 1 2 3 4\nlog(1+z)0.00.20.40.60.8Ωφi=3MPl\nΩc(Γ0=5.0)\nΩb(Γ0=5.0)\nΩr(Γ0=5.0)\nΩφ(Γ0=5.0)Ωc(Γ0=10)\nΩb(Γ0=10)\nΩr(Γ0=10)\nΩφ(Γ0=10)\n0 1 2 3 4\nlog(1+z)0.00.20.40.60.8Ωφi=1.5MPl\nΩc(Γ0=5.0)\nΩb(Γ0=5.0)\nΩr(Γ0=5.0)\nΩφ(Γ0=5.0)Ωc(Γ0=10)\nΩb(Γ0=10)\nΩr(Γ0=10)\nΩφ(Γ0=10)\n135log(ρφ/ρΛ)φi=3MPl\nΓ0=1.5\nΓ0=5.0Γ0=10.0\n0 1 2 3 4\nlog(1+z)0102030∆H\nH[%]\n1.01.52.02.5log(ρφ/ρΛ)φi=1.5MPl\n0 1 2 3 4\nlog(1+z)5\n0510∆H\nH[%]Γ0=1.5\nΓ0=5.0Γ0=10.0\n0 1 2 3 4\nlog(1+z)1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00weffφi=3MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n0 1 2 3 4\nlog(1+z)1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00weffφi=1.5MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\nFigure 9.2 : Background evolution of the fractional energy densities Ωi=k2ρi/3H2, for each\nlabelled i-th fluid (top), of the energy density of ϕrelative to Λand the Hubble expansion rate\nH(middle), defined in Equation (9.2.14), and the DBI effective equation of state parameters, as\ndefined in Equation (9.2.16) (bottom), as functions of the redshift, z. In the middle panel, we\nalso present the deviations of the Hubble expansion rate in each model, ∆H, when compared\nto the concordance model. The left and right panels correspond to Models M 3/1.5-M3/10and\nM1.5/1.5-M1.5/10in Table 9.1, respectively.9Dark D-Brane Cosmology 219\nThe left panels of Figures 9.1 and 9.2 depict the models M 3/1.5-M3/10from Table 9.1, with\nconsistently negative effective coupling, corresponding to a transfer of energy from DM to DE.\nNevertheless, as the system nears the scaling solution, characterised by 1/γ→0, we anticipate\nβturning positive. Conversely, the right panels depict models M 1.5/1.5-M1.5/10, where βremains\npositive at late times, and energy is being transferred from dark energy to dark matter. Notably,\nmodel M 1.5/1.5exhibits a unique behaviour for β, which starts by being negative, where it peaks\natz≈0.1(β≈ −0.17) before rising to become positive at z= 0(β≈0.5). Such behaviour hints\nat scenarios where the dark sector coupling, albeit minimal today, could have been substantial in\nthe past. Our simulations indicate that for larger ϕivalues (left panel), the coupling is activated\nsooner for the same value of Γ0. On the other hand, keeping ϕifixed, for negative (positive)\nβ, higher Γ0values prompt later (earlier) onsets of β, culminating in lower (higher) absolute\nmagnitudes for the coupling today. This also aligns with cosmologies nearing (or deviating\nfrom) a ΛCDM-like background evolution during the matter-dominated epochs, underlining the\ndistinction between the two regimes. Nevertheless, based on the dynamical systems study, we\nforesee that for models M 3/1.5-M3/10, the coupling will eventually become positive, irrespective\nof the initial conditions, featuring a characteristic turning point akin to the one observed in\nM1.5/1.5.\nThe top panels of Figure 9.2 illustrate the time evolution of fractional energy densities. The\ncosmological epochs occur in the sequence previously discussed: a radiation-dominant phase\ngives way to a DDM-dominant phase, culminating in dark energy becoming relevant in the\ncurrent epoch as the system gravitates towards the scaling solution. We observe that the fact\nthat the present cosmology has been fixed for all the simulations results in an artificial shift\nof the matter-radiation equality epoch. At later stages, we identify deviations from ΛCDM in\nthe fractional energy densities, with higher (lower) dark matter abundance in the presence of\nnegative (positive) couplings, reflecting the energy flow between the dark fluids. Concerning Ωϕ,\nwe note how the curves corresponding to the highest value of Γ0(filled thicker lines) approximate\nΩΛin either regime most closely, as is also illustrated in the upper-middle panels of Figure 9.2.\nThe current value of the Hubble rate is fixed at the Planckfiducial value. Nonetheless, different\nprofiles for H(z)emerge across the examined models, an effect that is entangled with the differ-\nences in the evolution of the fractional energy densities of the uncoupled species, namely baryons\nand radiation. The variations to the evolution of H(z)are depicted in the lower-middle panels\nof Figure 9.2, where generally enhanced (suppressed) expansion rates for most of the evolution\nare associated with negative (positive) effective couplings, increasingly closer to the ΛCDM ref-\nerence case for larger (smaller) Γ0values. Notably, at redshifts z≲0.5, we observe ∆H(z)>0\nfor all models, hinting at higher Hubble rates than ΛCDM benchmarks. At higher redshifts,\nfor the models with positive β,H(z)drops below the concordance model, whereas for negative\nβ, it is the higher Γ0values that bring the evolution closer to the ΛCDM curve. Variations in\nthe evolution of H(z), coupled with modifications in the effective gravitational interaction, give220 9.4. Numerical Study\nrise to distinct imprints in the linear growth of cosmic structures, as illustrated in Figures 3.6\nand 3.7.\nOverall, while the outcomes align with expectations from dynamical studies and qualitative\nconsiderations, we anticipate unique signatures at the linear cosmological perturbation level, on\nwhich we will focus next.\n9.4.2 The Growth of perturbations and the Effective Gravitational Constant\nIn this section, we step aside for a moment to examine the evolution of the density contrast\nfor DDM in the sub-horizon regime, where k≫ H. We adopt the quasistatic framework where\nthe temporal variations of the gravitational potential arise mainly from the matter and field\nfluctuations. Consequently, terms involving time derivatives, such as Φ′in the Einstein equations\norδϕ′′andδϕ′in the Klein–Gordon equation, are considered negligible relative to other terms.\nAs a result, Equations (9.3.15) and (9.3.16) become\nδ′\nc≈ −θc−Q\nρcϕ′δc+δQ\nρcϕ′, (9.4.1)\nθ′\nc≈ −H θc+k2Ψ−Qϕ′\nρc+k2Q\nρcδϕ . (9.4.2)\nWe arrive at an approximate form for the Poisson-like equation governing the gravitational field\nfrom the perturbed Einstein equations. Here, we neglect baryons and radiation, which represent\nminor contributions at the current epoch. It follows that:\nk2Ψ≈ −4πGρ cδc. (9.4.3)\nFinally, the perturbed Klein–Gordon equation can be expressed as\nAΨ +\u0000\nk2γ−2+a2m2\neff\u0001\nδϕ+a2γ−3δQ≈0, (9.4.4)\nwhere Awas defined as\nA=\u0014\n6H\u0000\n1−γ−2\u0001\nϕ′−h,ϕ\nh2a2\u0000\n2−3γ−1+γ−3\u0001\n+a2(V,ϕ+Q)\u0000\n3γ−1−γ−3\u0001\u0015\n,(9.4.5)\nand the effective mass meffis given by9Dark D-Brane Cosmology 221\na2m2\neff=−3h,ϕ\nhH\u0000\n1−γ−2\u0001\nϕ′+h,ϕϕ\n2h2a2\u0000\n1−3γ−2+ 2γ−3\u0001\n−3\n2h,ϕ\nha2(V,ϕ+Q)\u0000\nγ−1−γ−3\u0001\n+a2V,ϕϕγ−3+h2\n,ϕ\n2h3a2\u0000\n1−3γ−1+ 3γ−2−γ−3\u0001\n. (9.4.6)\nIn applying these approximations to the expression for the coupling perturbation, Equa-\ntion (9.3.23), we obtain\nδQ≈a−2ρc\nγ−2+hρcγ−3(Q1δc+Q5δϕ). (9.4.7)\nConsidering the coefficient Q5, as described by Equation (9.3.28), we find that the k2-term is the\ndominantcomponent. ByadoptingthesameanalyticalapproachasforderivingEquation(9.4.7),\nin the sub-horizon limit, we obtain the simplified relation\nδQ≃Qδc, (9.4.8)\na result that has been confirmed through numerical analysis and also holds in other alternative\ntheories featuring conformal and disformal couplings [347, 399, 446].\nTaking the time-derivative of Equation (9.4.1) and employing Equations (9.4.2) to (9.4.4), we\narrive at the following approximation to the evolution of δc:\nδ′′\nc+Heffδ′\nc≈4πGeffρcδc, (9.4.9)\nwhere we have introduced the effective Hubble parameter as\nHeff=H+Qϕ′\nρc, (9.4.10)\nand the effective gravitational constant, given by\nGeff=G\u0012\n1 + 2 β2 1\nγ(1 +a2m2\neff/k2γ2)−AQγ2\nρck2\u0013\n. (9.4.11)\nIn the limits k2≫a2m2\neff/γ2andk2≫AQγ2/ρc,Geffsimplifies to\nGeff≈G(1 + 2 β2/γ), (9.4.12)\nwhere βis as defined in Equation (9.2.12). Geffrepresents the effective gravitational interac-\ntion between two dark matter particles, accounting for both standard gravitational and scalar\nfield-mediatedlong-rangeforces. OurfindingsagreewithRef.[453]forscalar-tensorgravitymod-\nels with a conformal coupling, generalised here to include a disformal coupling and a distinct\nfunctional form for β.222 9.4. Numerical Study\n0 1 2 3\nlog(1+z)0.00.10.20.30.40.50.60.70.8Geff/G−1φi=3MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n0 1 2 3\nlog(1+z)0.00.10.20.30.40.50.60.70.8Geff/G−1φi=1.5MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\nFigure 9.3 : Evolution of the effective gravitational constant, defined in Equation (9.4.12),\nas function of the redshift z. The left and right panels correspond to Models M 3/1.5-M3/10\nand M 1.5/1.5-M1.5/10in Table 9.1, respectively, and the line styles used are the same as in the\nbackground evolution, depicted in Figures 9.1 and 9.2.\nThe evolution of Geff/Gfor the models summarised in Table 9.1 is illustrated in Figure 9.3. We\nobserve that deviations of Gefffrom Garise at lower redshifts, while it converges to Gin the early\nUniverse. This is consistent with the background study for β, showing that smaller initial values\nofϕiresult in higher present values of Geff. Interestingly, because the onset of the coupling is\nalso delayed in these cases, we expect that this should be reflected as smaller variations in both\nCMB anisotropies and the matter power spectrum as compared to ΛCDM. However, there should\nbe a balance between both effects. This observation aligns with the findings in Ref. [346] for\ndisformally coupled quintessence models. Nevertheless, the earlier activation of the deviations of\nGefffrom Gpresents a potential challenge to the model’s consistency with observational data.\nHowever, we have also intentionally depicted a scenario where Geffinitially rises (corresponding\nto a period of negative β) but ultimately declines (the Γ0= 1.5case in the right panel of\nFigure 9.3), only to rise again at later times (as βbecomes positive). This suggests the existence\nof parameter combinations for which Geff≈Gtoday, yet the structure formation is distinctly\nimpacted by the scalar-field mediated forces at intermediate redshifts. The varying dynamics\nforGeffacross different models will inevitably result in diverse outcomes for the matter density\nfluctuations, which we will attempt to capture in the following section.\n9.4.3 Cosmological Observables\nThe CMB anisotropy power spectrum and the matter power spectrum within the Dark D-brane\nframework can also be computed from the modified version of the CLASS code [95, 101]. The\nresults for the cases outlined in Table 9.1, alongside the fiducial ΛCDM case for reference, are9Dark D-Brane Cosmology 223\nillustrated in Figure 9.4. Here, the CMB temperature anisotropies are shown in the top panels,\nthe lensing CMB power spectrum in the middle panels, and the matter power spectra for DDM\nand baryons at z= 0are depicted in the bottom panels. We assumed standard adiabatic initial\nconditions, specifying an amplitude As= 2.215×10−9and a pivot scale kpivot= 0.05Mpc−1.\nIn contrast, the initial perturbations for the scalar field are set to zero, δϕi=δϕ′\ni= 0, without\nloss of generality. Mirroring the background cosmology, the evolution of the linearly perturbed\nquantities also exhibits a significant sensitivity to changes to the model parameters. This is\ncorroborated in Figure 9.4, where different values of Γ0, with ϕiheld constant, lead to distinct\nshapes in the power spectra. On the other hand, in the left and right panels of Figure 9.4 we\nalso show solutions with the same Γ0values: one with ϕi= 3MPl(models M 3/1.5-M3/10in the\nleft panel) and another with ϕi= 1.7MPl(models M 1.5/1.5-M1.5/10in the right panel), both\ncharacterised by the same line styles for direct comparison. As anticipated, varying the initial\ncondition for the scalar field yields distinct imprints on the linear perturbations, contrary to\nwhat was found in Ref. [407] from a dynamical background study.\nBy contrasting the left and right panels of Figure 9.4, we observe that the sign of the effective\ncoupling, β, which is directly related to the value of both Γ0andϕi, markedly influences the\nevolution of the perturbations. In the context of the CMB temperature power spectrum, for\nβ <0(upper-left panel), there is a marked enhancement for low multipoles ℓand suppression\nfor medium and high ℓvalues, compared to the ΛCDM prediction. On the other hand, for β >0\n(upper-right panel), there is a mild enhancement at intermediate and large multipoles. Moreover,\nhigher (lower) values of Γ0lead to fewer deviations from the ΛCDM framework when β < 0\n(β >0). It is worth noting that although the present-day effective coupling (as demonstrated in\nFigure9.1)inmodelsM 3/1.5-M3/10intheleftpanelisofthesameorderofthatinmodelsM 1.5/1.5-\nM1.5/10in the right panel, the deviations from ΛCDM are considerably more pronounced. This\ndiscrepancy is ascribed to the onset of the coupling at higher redshifts, resulting in an earlier\ndeparture from the standard model.\nThe predicted signatures for the CMB temperature anisotropies across the models can be ex-\nplained on two fronts. Firstly, the background cosmology differs, as elaborated in Section 9.2.\nSpecifically, the ratio Ωb/Ωcat the time of recombination is generally not kept constant, and\nits high-redshift evolution depends on the chosen parameters. This ratio is the primary driver\nof the shifts in multipole and changes in amplitude of the peaks and valleys in the spectrum\n(see the discussion is Section 3.5); we can see that there is a tendency for the peaks to become\nnarrower (wider) for models M 3/1.5-M3/10with β <0(M1.5/1.5-M1.5/10with β >0) relative to the\nΛCDM scenario. This introduces a degeneracy between the effective gravitational coupling and\nthe Hubble parameter, as the latter has the greatest impact on the magnitude and position of\nthe first peak. This non-trivial degeneracy is why we observe an overall enhancement of the low- ℓ\ntail of the spectrum in both cases considered, which depends on the balance between the effects.\nSecondly, the time evolution of Geff, intimately linked to the coupling βand the γ-factor, impacts224 9.4. Numerical Study\n02468/lscript(/lscript+1)CTT\n/lscript/2π×1010\nφi=3MPl\nΛCDM\nΓ0=1.5\nΓ0=5.0\nΓ0=10.0\n101102103\n/lscript60\n30\n03060∆CTT\n/lscript\nCTT\n/lscript[%]\n02468/lscript(/lscript+1)CTT\n/lscript/2π×1010\nφi=1.5MPl\nΛCDM\nΓ0=1.5\nΓ0=5.0\nΓ0=10.0\n101102103\n/lscript30\n15\n01530∆CTT\n/lscript\nCTT\n/lscript[%]\n0.5\n0.00.51.01.52.0[/lscript(/lscript+1)]2Cφφ\n/lscript/2π×107\nφi=3MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n101102103\n/lscript0100200∆Cφφ\n/lscript\nCφφ\n/lscript[%]\n0.5\n0.00.51.01.52.0[/lscript(/lscript+1)]2Cφφ\n/lscript/2π×107\nφi=1.5MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n101102103\n/lscript25\n10\n5∆Cφφ\n/lscript\nCφφ\n/lscript[%]\n103104P(k)[Mpc/h]3φi=3MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n10-310-210-1\nk[h/Mpc]60\n20100180∆P(k)\nP(k)[%]\n103104P(k)[Mpc/h]3φi=1.5MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n10-310-210-1\nk[h/Mpc]30\n15\n015∆P(k)\nP(k)[%]\nFigure 9.4 : The top and middle panels show the CMB temperature and lensing angular power\nspectra, CTT\nℓandCϕϕ\nℓ, plotted as a function of the angle scale ℓ. The bottom panels depict\nthe matter power spectra Pkfor different Fourier scales (wavenumbers) k. For each observable,\nwe also provide the ΛCDM predictions (black) and the relative deviation of each model from\nthe standard model. The left and right panels correspond to Models M 3/1.5-M3/10and M 1.5/1.5-\nM1.5/10in Table 9.1, respectively, and the line types used are the same as in Figures 9.1 and 9.2\nfor the background evolution.9Dark D-Brane Cosmology 225\nthe DDM fluid’s perturbation growth. In models M 3/1.5-M3/10,Geffmay start departing from\nzero as early as redshift z≈1.1for the given Γ0values, while for models M 1.5/1.5-M1.5/10,Geff\nonly starts growing at around a redshift of z≈0.3or later. The late-time surge in the effective\ncoupling affects the late-time Integrated Sachs–Wolfe (ISW) effect, as described in Section 3.3.\nThis effect alters the distance to the last scattering surface, suppressing (enhancing) the sound\nhorizon at the baryon-drag epoch (see Section 3.5), shifting the peaks and troughs to the left\n(right) for β <0in the left panel ( β >0in the right panel). This shift is more pronounced for\nthe cases with the largest deviations from ΛCDM in the background.\nIn the middle panels of Figure 9.4, we depict the lensing power spectrum, directly related to the\nevolution of the lensing potential ϕlen= Φ + Ψ , depicted in the top panels of Figure 9.5, along\nwith the corresponding time derivatives in the lower panel, at an intermediate scale of k= 0.01\nMpc−1. The most pronounced deviations from ΛCDM manifest from around z≲103during the\nmatter-dominatedepochwhenchangesinthedisformaldarkmatterevolutionbecomeimportant.\nFrom this point on, there is an overall enhancement of ϕlenfor the cases with β <0(left panel)\nand a suppression for β >0(right panel), with the largest departures identified for the lower and\nhigher values of Γ0in each case, consistent with the considerations for the temperature power\nspectrum. The changes to the lensing potential arise from the energy flow in the dark sector and\nreflect directly in the lensing power spectrum Cϕϕ\nℓ, depicted in the middle panels of Figure 9.4,\nthrough their role in the line-of-sight integration presented in Section 3.3. We observe a similar\nbut opposite trend compared with CTT\nℓforCϕϕ\nℓ, with an overall enhancement (suppression) of\nthe spectrum at intermediate and high multipoles for β <0in the left panel ( β >0in the right\npanel). The most significant deviations appear once again in the models with the earlier onset of\nthe coupling, corresponding to the lowest (highest) values of Γ0forβ <0(β >0). This overall\ntrend could help address the lensing excess reported in the Plancktemperature data of CMB\nanisotropies [32, 356], as discussed in Section 3.6.3.\nThe evolution of the lensing potential is also imprinted in the CMB temperature spectrum of\nanisotropies in the upper panels of Figure 9.4. The largest contribution to the modifications\ncomes from the integrated Sachs Wolfe (ISW) effect, which depends directly on ˙ϕlen(see Sec-\ntion 3.3), depicted in the lower panels of Figure 9.5. The contributions to the ISW effect can be\nseparated into early and late-time components. In the early case, the ISW effect amplifies (re-\nduces) the time derivatives of the gravitational potentials for β <0(β >0), directly associated\nwith an earlier (later) matter-radiation equality. Conversely, late-time ISW effects are primarily\ndictated by changes in the CMB lensing large-scale structure due to the modified dynamics of\nthe dark sector. This culminates in a late-time attenuation of Φ′+ Ψ′, as demonstrated in the\nlower panels of Figure 9.5.\nThe predictions for the matter power spectrum P(k), depicted in the bottom panels of Fig-\nure 9.4 for for Fourier scales 10−3hMpc−1< k < 10−1hMpc−1, are influenced by several factors.\nBesides the differing background evolution, the interplay between the effective gravitational cou-226 9.4. Numerical Study\npling Geff, as defined in Equation (9.4.12), and the effective Hubble expansion rate, defined in\nEquation (9.4.10), play a crucial role in shaping the dark matter density contrast, as elaborated\nin the previous section. We observe that for models M 3/1.5-M1.5/1.5(M1.5/5and M 1.5/10), on large\nscales k≲kpeak, the dynamical contributions of the dark sector lead to a suppression (enhance-\nment) in structure growth compared to the ΛCDM model. On smaller scales, k≳kpeak, the\nopposite effect is observed, except for model M 1.5/1.5which still experiences a slight suppression\ndue to the unique balance and sign change in the evolution of β. In agreement with the CMB\npower spectrum discussion, the deviations from the standard model are more accentuated in\nmodels M 3/1.5-M3/10, owing to the earlier redshift onset of the coupling.\nFurthermore, atleastforparticularperiodsintheevolution, theperturbationinthecoupling, δQ,\nis scale dependence viathe coefficient Q5, as presented in Equation (9.3.28), consistent with the\nchange in the overall trend of suppression/enhancement before and after the peak of the spectra.\nWith the chosen normalisation of the power spectrum, for the models with β <0(β >0), the\namplitude of the primordial perturbations must be greater (lower) to match the current structure\ncount. This is in line with the feature of these models exhibiting slightly enlarged dark energy\ndensity fractions at late times, compared to the ΛCDM predictions (red curves in the upper-left\npanel of Figure 9.2), and explains why the peak of the spectrum is shifted to the right (left).\nDeviations from ΛCDM surpass 100%in the models depicted in the left panels of Figure 9.4,\npeaking at k≈10−1hMpc−1, a point where the linear approximation is expected to break down,\nand non-linear effects take over.\nThe ISW effect is correlated with the growth, with the behaviour at larger scales (smaller k)\ncontributing to the enhancement/suppression trend in the low- ℓtail of the CMB anisotropy\nspectrum depicted in the top panels of Figure 9.4. Another potential contributor to the ISW\neffect could be a phase of early dark energy. However, unlike other dark energy models with\ndisformal couplings, such as in Ref. [446], we do not witness any relevant early dark energy\nsignatures in the Dark D-brane framework, given the form of the potential adopted here.\nWe remark that, from Table 9.1, the toy models M 3/1.5-M3/10with β <0(M1.5/1.5-M1.5/10with\nβ >0) predict values for σ8that are higher (lower) than the ΛCDM counterpart, σΛCDM\n8 ≈0.85.\nThe DDM scenario shares certain traits with other disformal models discussed in the literature\n[346, 347, 399, 446]. Specifically, the effective gravitational coupling between dark matter parti-\ncles is not constant. In the Dark D-brane setting, the coupling is negligible in the early Universe,\nsuppressed by the denominator in Equation (9.2.12). This aspect further motivates research\nexploring violations of the equivalence principle in the dark sector at later cosmological times.\nIn the following section, we will focus on the impact of this time-evolving coupling on structure\nformation by testing the DDB model against existing cosmological data.9Dark D-Brane Cosmology 227\n0.60.81.01.21.4Φ+Ψφi=3MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n0 1 2 3 4\nlog(1+z)5\n51525∆(Φ+Ψ)\n(Φ+Ψ)[%]\n0.60.81.01.21.4Φ+Ψφi=1.5MPl\nΛCDM\nΓ0=1.5Γ0=5.0\nΓ0=10.0\n0 1 2 3 4\nlog(1+z)10\n5\n05∆(Φ+Ψ)\n(Φ+Ψ)[%]\n10\n8\n6\n4\n2\n0Φ/prime+Ψ/prime(×104)φi=3MPl\nΛCDM\nΓ0=1.5\nΓ0=5.0\nΓ0=10.0\n0 1 2 3 4\nlog(1+z)900\n600\n300\n0300∆(Φ/prime+Ψ/prime)\n(Φ/prime+Ψ/prime)[%]\n10\n8\n6\n4\n2\n0Φ/prime+Ψ/prime(×104)φi=1.5MPl\nΛCDM\nΓ0=1.5\nΓ0=5.0\nΓ0=10.0\n0 1 2 3 4\nlog(1+z)120\n60\n060∆(Φ/prime+Ψ/prime)\n(Φ/prime+Ψ/prime)[%]\nFigure 9.5 : Evolution of the sum of the gravitational potentials, Φ + Ψ, (top panels ) and the\ncorresponding derivative with respect to conformal time, Φ′+ Ψ′(lower panels ) for k= 0.01\nMpc−1, plotted as a function of the redshift z. The left and right panels correspond to Models\nM3/1.5-M3/10and M 1.5/1.5-M1.5/10in Table 9.1, respectively, and the line styles used are the same\nas in Figures 9.1 and 9.2 for the background evolution.228 9.5. Observational Constraints\n9.5 Observational Constraints\n9.5.1 Methodology and Data\nAfter reviewing the main predictions of the Dark D-Brane Model (DDB) at the background and\nlinear perturbative level, we wish to study the constraints on the model’s parameters imposed\nby incremental combinations of the data sets defined below. For this purpose, we rely once more\non the MontePython code [213, 214], an MCMC sampler of the cosmological parameter space\nbased on our modification of the CLASScode, employing the Metropolis-Hastings algorithm with\na Gelman-Rubin [215] convergence criterion of R−1<10−2for all the chains, as described in\nChapter 4. From this method we derive the corresponding posterior distribution for the sampled\nparameters. In the following section, we discuss the main results of this analysis.\nThe baseline data set for this study is as described in Section 4.3. More precisely, we consider\nthe full TTTEEE+lowE CMB likelihood from the Planck2018 data release,composed of data\non the CMB temperature (TT) and polarisation (EE) anisotropies, including their joint cross-\ncorrelations (TE), at both low and high multipoles. More precisely, this corresponds to the\nhigh- ℓPliklikelihood for TT in the range 30≤ℓ≤2508and TE and EE for 30≤ℓ≤1996,\ncombined with the low- ℓ(2≤ℓ≤29) TT and EE likelihoods based on the Commander algorithm\nand the SimAlllikelihood. This data set is abbreviated as Pl18 throughout the text. We also\ninvestigatetheeffectofincludingtheCMBlensinglikelihood[51]. Wealsostudythecombination\nof Planck with other background data sets. In this case, we include baryon acoustic oscillations\n(BAO) distance and expansion rate measurements from the 6dF Galaxy Survey [118], the Sloan\nDigital Sky Survey (SDSS) DR7 Main Galaxy Sample [117] and SDSS DR12 [116], as employed\nby thePlanckcollaboration as well. Finally, we include Pantheon’s 1048 supernova (SN) data\nsample [109]. The combination of this background data is referred to throughout this chapter as\nBAO+SN.\nFor each data set combination, we sample the standard cosmological parameters, namely the\nreduced dark matter and baryon energy densities, Ωch2andΩbh2; the ratio between the sound\nhorizon and the angular diameter distance at decoupling, θs; the amplitude and power of the\nscalar primordial power spectrum As(expressed as ln\u0000\n1010As\u0001\n) and ns; and the reionisation\nredshift, zreio. In addition to these, we also consider the scalar field parameters ˜h0= 1/h0, as\ndefined in Equation (9.2.18), and the scalar field’s initial condition ϕi. We employ flat priors\non the sampled parameters, including the nuisance parameters associated with each likelihood.\nThe velocity of the scalar field is considered to be fixed and set to ϕ′\ni=−10−25(Mpl Mpc−1)\nas its influence on the dynamics was found to be negligible for this study. As briefly referred,\nthe potential scale V0is used as a derived shooting parameter to account for degeneracies in\nthe parameter space and ensure the Friedmann flatness condition. It can be constrained as a\nderived parameter and combined with ˜h0to express the results regarding the physical parameter\nΓ0. The priors for the sampled parameters are listed in Table 9.2 and the results of the study9Dark D-Brane Cosmology 229\nare summarised in Tables 9.3 and 9.4 for the ΛCDM reference case and for the Dark D-Brane\nmodel.\nThe sampled MCMC chains obtained from MontePython are analysed using the GetDist Python\npackage [216]. The statistical significance compared to the concordance model is inferred through\nthe Bayes factor lnBijin Equation (4.1.3), an estimator of the Bayesian Evidence, computed\nfrom the publicly-available MCEvidence code [217]. A negative value indicates preference over\nthe concordance model, with the opposite holding for positive values and the level of support is\nassessed by the Jeffreys scale criteria presented in Table 4.1.\nParameter Prior\nΩbh2[0.005,0.1]\nΩch2[0.001,0.99]\n100θs [0.5,10]\nzreio [0.,20.]\nns [0.7,1.3]\nlog\u0000\n1010As\u0001\n[1.7,5.0]\n˜h0 [10−20,5]\nϕi [0,20]\nTable 9.2 : Flat priors on the cosmological and model parameters sampled in this work.\n9.5.2 Results\nIn Tables 9.3 and 9.4, we report the statistical study results for the sampled cosmological and\nmodelparametersforthevariousdatacombinationsconsideredfortheDarkD-Braneand ΛCDM\nmodels at 68%confidence level. This allows for a direct comparison of the results, even if the\nfact that the Dark D-Brane model does not possess a well-defined uncoupled and ΛCDM limits\non the parameters makes this comparison non-trivial. As we have seen in the numerical analysis\nof the toy models, the case Γ0→ ∞seems to represent a ΛCDM limit at the background level,\nexplaining the apparent preference for high values of Γ0seen in Table 9.4. The results are also\nrepresented in Figure 9.6 as a 2D rectangular plot for marginalised distributions in the dark\nD-brane model and as the 1D marginalised distribution curves, including a comparison with\nΛCDM for the cosmological parameters.\nIn the first column of Tables 9.3 and 9.4, we present the constraints obtained considering Planck\ndata only. We find a non-zero and bounded prediction at 1σforΓ0, roughly quantifying the\ndeviationfromthecanonicalcase,evenifthisisabroaderconstraintcomparedtothecombination230 9.5. Observational Constraints\nwith other data sets. On the other hand, the fact that we obtained non-vanishing constraints\nforϕiat68%CL even with Planckdata alone highlights its importance as a cosmological\nparameter with impact over the dynamics, especially at the perturbative level. We verify that\nthese considerations are maintained at 95%and99%CLs, driven exactly by the model’s unique\nfeatures which cannot be trivially related to ΛCDM.\nWe also look at the effect of adding the BAO+SN background data, followed by the lensing CMB\ndata on top of the baseline Planck. We see that the inclusion of the CMB lensing data does not\nsignificantly impact the results, apart from a considerable reduction in the constraining power\nonϕi, which is related with a slight better convergence for the remaining parameters, illustrated\nby the yellow contours in Figure 9.6. On the other hand, the addition of the SN and BAO\ndata significantly improves the constraints on Ωm, leading to tighter constraints for the other\nbackground parameters as well, namely {H0,Γ0, ϕi}, and bringing Γ0to considerably higher\nvalues, closer to the ΛCDM-like limit. The latter effect is also connected to the enhancement\nin the lensing power spectrum for higher values of Γ0, which better accommodates the CMB\nlensing data under the lensing excess reported by Planck. The addition of the background data\nsets also brings the value of H0considerably closer to the ΛCDM prediction. This discussion\nshould be taken with a grain of salt as is a well-known effect, reported in other models as well\n[57, 183, 381, 454–457], possibly related to a bias in the BAO data towards ΛCDM, which could\nrequire some corrections to be properly compared with other models. More importantly, with\nthe inclusion of all the data sets, there is still evidence for a non-vanishing coupling (bounded\nΓ0andϕi) at more than 1σwithout considerably worsening the tensions with the low-redshift\ndata, present in the Planckonly case.\nIn the upper panel of Figure 9.6 we see the 2D marginalised correlations between Γ0and the\nparameters {H0, S8,Ωm, ϕi}. We find a distinct positive correlation between the coupling pa-\nrameter Γ0andH0and a negative correlation between Γ0andΩm. In particular, we see that the\ncontours are more accommodating in the Pl18-only case. The introduction of the background\ndata, namely the BAO data, greatly constrains Ωmtowards lower values, which results in larger\nvalues for Γ0andH0as well. We see that the addition of the lensing data contributes to an\nimprovement on the constraints, especially in the value of ϕi. In the lower panel of Figure 9.6 we\ndepict the 1D marginalised posterior distributions for {H0, S8,Ωm,Γ0, ϕi}for the DDB model,\nincluding the reference ΛCDM curves for {H0, S8,Ωm}as dashed lines. We see that the param-\neters are consistently constrained in ΛCDM, with H0taking higher values, while ΩmandS8are\nconstrained towards lower values. Although this seems to push the results in the opposite direc-\ntion than what is necessary to address the cosmic tension in H0andS8in the standard model,\nwe point to the enlarged error bars in the curves for those parameters, and emphasise once more\nthat the weak lensing data should be consistently reanalysed in the context of the DDB model in\norder to quantify any tension in S8. The H0tension of ∼4.8σinΛCDM when considering Pl18\ndata alone is maintained at ∼4.9σin the DDB model. While this could still hint at the need for9Dark D-Brane Cosmology 231\nnew physics, e.g.a Dark D-Brane formulation in a less simplified setting, we conclude that this\nparticular framework alone does not lead to a significant ease of the tensions between early- and\nlate-time probes. The comparison in the constraints for the Dark D-Brane and ΛCDM cases is\ngiven in Figure 9.7 for the 2D marginalised contours for the parameters directly involved in the\ntensions reported above.\nWefindthat ϕiisconstrainedtobeoftheorderofunity, withsimilarandconsistentresultsforall\nthe data combinations at 68%CL, considering that we have shown that its value greatly impacts\nthe dynamics, even possibly altering the direction of energy flow in the dark sector interaction.\nFor this range of values, the numerical study points at higher values of Γ0, leading to a greater\ntransfer of energy from DE to DM ( β >0), associated with an overall enhancement of the CMB\ntemperature-temperature angular power spectrum, as discussed over the previous sections and\ndepicted in the right-upper panel of Figure 9.4. This is consistent with the positive correlation\nbetween Γ0andH0, illustrated in the upper panel of Figure 9.6. It is worth noting that the\ncontour plots show a clear saturation point for Γ0≳20, corresponding to the case in which the\ncosmological realisations become too close to ΛCDM and the effect of Γ0becomes degenerate,\nwith increasing values just approaching more closely the ΛCDM predictions. Nevertheless, we see\nthat the constraints on ϕiare relatively tight and not significantly correlated with Γ0, showing\nhow well the data can constrain ϕiin that range of values independently of Γ0. Indeed, in the\nprevious sections we have confirmed how the two DBI parameters are not degenerate, given that\ntheyplayconsiderablydifferentrolesinthecosmologicalevolutionandinshapingtheobservables.\nIn the last two rows of Table 9.4 we present the results for the effective ∆χ2and for the Bayesian\nevidencemodelcomparisoncriteria,accordingtoSection4.1.3. WeseethattheBayesianevidence\nis considerably negative for all the data combinations, implying no support for the Dark D-\nbrane model over ΛCDM. On the other hand, the non-negligible values of ∆χ2\nminindicate an\nimprovement in the fit. However, this arises as a result of having introduced more parameters,\nmaking the model naturally more accommodating, with the Bayesian evidence confirming that\nthe increased parameter space is not justified. Even if we were tempted to say that this might\npointtoanarguableevidencefortheDarkD-Branemodelover ΛCDM,fromanobjectivepointof\nview, wehaveshownthat, asformostinteractingdarkenergymodelswithcouplingsproportional\nto the density of DM, this model does not address the cosmological tensions.\n9.6 Discussion\nIn this Chapter, we have explored the dynamics and phenomenology of the dark D-brane model.\nAs presented in previous works, it is well known that the Dark D-Brane model alleviates some\nof the theoretical problems associated with ΛCDM by introducing a theoretical motivation and\ngeometrical interpretation for the dark sector. This study analysed a simple toy model to realise\ntheDDBmodelinAdS 5×S5warpedregionscorrespondingtoaquadraticpotentialandaninverse232 9.6. Discussion\nParameter P18 P18 + BAO + SN P18len + BAO + SN\nωb 0.02236 ±0.00015 0 .02243 ±0.00014 0 .02245 ±0.00014\nωc 0.1202±0.0014 0 .11921 ±0.00098 0 .11923 ±0.00094\n100θs 1.04188 ±0.00030 1 .04197 ±0.00028 1 .04196 ±0.00029\nτreio 0.0542±0.0080 0 .0558±0.0078 0 .0565±0.0074\nns 0.9652±0.0043 0 .9676±0.0038 0 .9677±0.0037\nln 1010As 3.045±0.016 3 .046±0.016 3 .048±0.015\nσ8 0.8115±0.0076 0 .8091±0.0072 0 .8105±0.0061\nS8 0.833±0.016 0 .822±0.012 0 .824±0.011\nΩm 0.3163±0.0085 0 .3100±0.0059 0 .3100±0.0057\nH0 67.31±0.61 67 .76±0.44 67 .77±0.43\nTable 9.3 : Observational constraints at a 68%confidence level on the independent and derived\ncosmological parameters using different data set combinations for the ΛCDM model.\n62646668\nH0010203040500\n0.40.81.21.62.0\nS80.300.350.400.45\nm\n012345\ni\nPl18 Pl18+BAO+SN Pl18len+BAO+SN\n62646668\nH00.40.81.21.62.0\nS80.250.350.45\nm\n01020304050\n0\n012345\ni\nPl18 Pl18+BAO+SN Pl18len+BAO+SN\nFigure9.6 : 2D marginalised 68%and95%CL contours (top panel) and 1D marginalised curves\n(lower panel) obtained in the Dark D-Brane model under consideration for the Planck2018 data\n(red), the Planck2018, BAO and SN combination (green), and their combination with CMB\nlensing (yellow). The dashed lines correspond to the ΛCDM reference case.9Dark D-Brane Cosmology 233\nParameter P18 P18 + BAO + SN P18len + BAO + SN\nωb 0.02241 ±0.00015 0 .02251 ±0.00014 0 .02253 ±0.00014\nωc 0.142+0.012\n−0.0086 0.1327+0.0053\n−0.0041 0.1333+0.0030\n−0.0043\n100θs 1.04191 ±0.00029 1 .04204 ±0.00028 1 .04205 ±0.00029\nτreio 0.0533±0.0080 0 .0553±0.0080 0 .0531±0.0081\nns 0.9662±0.0045 0 .9699±0.0038 0 .9704±0.0038\nln 1010As 3.043±0.017 3 .043±0.016 3 .038±0.015\n˜h0 0.146+0.075\n−0.14 0.038+0.015\n−0.029 0.036+0.012\n−0.022\nϕi 1.14+0.16\n−0.45 1.24+0.18\n−0.56 1.11+0.13\n−0.28\nσ8 0.98+0.11\n−0.20 1.14+0.19\n−0.27 1.12+0.14\n−0.20\nS8 1.12+0.16\n−0.24 1.22+0.23\n−0.28 1.20+0.16\n−0.21\nΩm 0.392+0.039\n−0.035 0.347±0.016 0 .3467+0.0096\n−0.013\nH0 64.9±1.3 67 .08+0.55\n−0.50 67.20±0.57\nΓ0 8.9+2.5\n−7.3 22±8 23+7\n−8\n∆χ2\nmin −5.42 −4.24 −2.48\nlnBi,ΛCDM −5.72 −8.25 −7.40\nTable 9.4 : Observational constraints at a 68%confidence level on the independent and derived\ncosmological parameters using different data set combinations for the Dark D-Brane model.234 9.6. Discussion\n62646668\nH00.300.350.400.45m\n0.40.81.21.62.0\nS8Pl18 Pl18+BAO+SN Pl18len+BAO+SN\nDark D-Brane\n6566676869\nH00.300.320.34m\n0.780.820.860.90\nS8CDM\nFigure 9.7 : 68% and 95% CL 2D contours derived for the parameter combinations H0-Ωm\n(left panels) and S8-Ωm(right panels) in the Dark D-Brane model (upper panels) and ΛCDM\nmodel (lower panels) for the Planck2018 data (red), the Planck2018, BAO and SN combination\n(green), and their combination with CMB lensing (yellow).\nquartic warp factor.\nWe performed a numerical study of the background and linearly perturbed equations. We found\nthat the model has a non-trivial complex behaviour, with initial scalar field conditions signifi-\ncantly influencing the late-time cosmological evolution. Specifically, we categorised two distinct\nregimes based on the behaviour of the effective coupling at varying redshifts. These regimes yield\ndifferent cosmological outcomes, highlighting the model’s richness.\nAt the background level, we showed that the Universe evolves toward a scaling fixed point, where\nthe fractional energy densities of DE and DM maintain a constant ratio. For higher values of\nthe parameter Γ0, the evolution closely resembles that of the ΛCDM model. We emphasised\nthat the initial condition of the field ϕiplays a significant role, notably affecting the direction of\nenergy exchange between dark energy and dark matter. The coupling is consistently negligible\nat earlier times and its onset redshift is determined by the DBI parameters Γ0andϕi. By fixing9Dark D-Brane Cosmology 235\nΓ0we gather that, by taking initial conditions for the field that lead to higher present values of\nβ, the coupling starts to grow much earlier, during the matter dominated epoch, leaving distinct\nsignatures at the level of the perturbations. Therefore, we conclude that the difference in the\ndirection of the energy exchange between the dark fluids introduces distinct features at the level\nof the background, potentially shifting the matter-radiation equality and the time of transition\nfrom matter to dark energy dominated epochs.\nOn the perturbation front, we derived equations for both the Newtonian and synchronous gauges,\nuncovering diverse implications for the CMB and matter power spectrum. Particularly, we noted\ndeviations from the ΛCDM model that are consistent with the coupling behaviour. Moreover, the\nnumerical results in this work imply that the Dark D-brane model has the potential to produce\nunique observational features, especially in the CMB spectra.\nIn our numerical study, we consistently considered the scalar field’s initial value close to the\nPlanckmass. Within the framework of higher-dimensional theories, this corresponds to the\nD-brane being positioned far from the tip of the AdS throat. On the other hand, the study in\nRef.[407]examinedanalternativelimitwhere ϕismuchsmallerthanthePlanckmassandtheD-\nbrane is close to the tip of the AdS throat. In this setting, to account for the present-day vacuum\nenergy, the scalar field mass, mϕ, is subject only to a lower bound—making it fundamentally\ndifferent from conventional quintessence theories where a small mass is often assumed. We found\nthat numerically simulating this case is non-trivial, primarily due to the reduced initial value\nneeded for the scalar field and the need to consider high values of Γ0. Future detailed studies on\nthis specific scenario would be interesting, although we anticipate that the overarching physics\npresented in our current work should remain relevant in such contexts.\nTo test the model against observational data we have performed an MCMC analysis. We list\nbelow some of the main findings of the study:\n•Cosmic Tensions: The Dark D-Brane model gives a lower mean value of H0across all data\nsets, compared to ΛCDM. While having slightly enlarged error bars it does not address the\ntension between different H0measurements. The model accommodates for larger values of\nΩm, consequently leading to higher S8values, associated with an increased 1σregion.\n•Constraints on Model Parameters: The mean value of Γ0is always constrained at 68%CL\nand predicted to be greater than one, while ϕiis on the order of 1MPlfor all data combi-\nnations. There seems to be no correlation or degeneracy between the two DBI parameters,\nowing to their fundamentally different impact over the cosmological evolution as shown in\nthe numerical study.\n•Data Dependence: Inclusion of BAO and SN data results in tighter constraints on Ωm, and\nconsequently on H0,S8, and ϕi. These datasets also impact the degeneracies associated\nwith H0andS8but do not clearly resolve the existing tensions. A slightly higher central236 9.6. Discussion\nvalue of Γ0is favoured with the lensing data, possibly accommodating for the lensing excess\nin thePlanckdata.\n•Statistical Evidence: ∆χ2\neffandlnBi,ΛCDMstatistical tests were used to assess model\npreference. While the Bayesian evidence is significantly negative to show preference for\nΛCDM, we find that the ∆χ2\neffalso takes considerable negative values across all the data set\ncombinations, indicating a better fit to the data in the Dark D-Brane model. Nevertheless,\nwhen the two metrics are considered together, we conclude that the better fit to the data\nis not sufficient to justify the added complexity of the Dark D-Brane model.\nIn summary, while the Dark D-Brane model introduces intriguing dynamics and constraints,\nit does not conclusively resolve existing tensions in cosmological parameters. Future data and\nanalyses are crucial for a more definitive understanding of these issues. Nevertheless, it is still\na very rich phenomenological framework, which could be extended or considered at different\nregimes.10A Hybrid Model for the Dark Sector\nQuando eu era pequeno pensava que de um momento para outro eu cairia para fora do mundo.\nPor que as nuvens não caem, já que tudo cai? É que a gravidade é menor que a força do ar que\nas levanta. Inteligente, não é? Sim, mas caem um dia em chuva. É a minha vingança.\n—Clarice Lispector A Hora da Estrela\nWhen I was a little boy I thought that from one minute to the next I could fall off the face of the earth. Why\ndon’t clouds fall, since everything else does? Because gravity is less than the strength of the air that keeps them\nup there. Clever, right? Yes, but one day they fall as rain. That is my revenge. —Clarice Lispector in The\nHour of the Star\nThe mysterious components that dominate the present Universe constitute the dark sector of\ncosmology. Unveilingtheirintrinsicpropertiesandoriginsisakeyprobleminthefield. Numerous\nphenomenological dark matter candidates have been put forth, inspired by extensions to the\nstandard model of particle physics. These range from weakly interacting massive particles to\nlighter scalar fields ( e.g., [458–467]). Dark energy, however, is often treated as a separate matter\nfrom DM. In this study, we propose that both DE and DM originate from two coupled scalar\nfields governed by a common potential energy V(see also [468–475]). We focus on the potential\nenergy model commonly employed in hybrid inflation [476] which introduces a mass hierarchy\nbetween DE and DM, and which will be detailed further in this chapter. Here, the heavier\nfield is identified as DM, and its mass is determined by the DE field expectation value, which\naligns with the flat direction of the potential. This framework strongly restricts the rate at\nwhich the DE field can vary, accommodating a constant-like behaviour but implying that the\nobserved accelerated cosmic expansion is only temporary. Ultimately, both fields will settle at\nthe minimum of the potential, and other players, such as spatial curvature, will determine the\nUniverse’s future evolution.\nThis chapter is structured as follows: the model is introduced in Section 10.1, with the theoretical\nimplications of its particular parameters detailed in Section 10.1.1. A fluid approximation for the\nDM field is developed in Section 10.2, followed by an account of the influence of the scalar-field\nparameters in the cosmic evolution and its implications for CMB anisotropies and large-scale\nstructures in Section 10.2.2. Finally, we discuss the results in Section 10.3 and present our\nconclusions in Section 10.4.\nThis work has been published in JCAP and can be found in Ref. [4].\n237238 10.1. The Hybrid Model\n10.1 The Hybrid Model\nIn this section, we motivate and introduce the scalar field scenario proposed and studied in this\nwork. Due to its well-studied dynamics, the framework we consider here is an extension of the\nhybrid inflation model [476], populated by two scalar fields and with the conventional matter\nfields of the standard model. The action that describes this setting is\nS=Z\nd4x√−g\u00141\n2M2\nPlR−1\n2(∇ϕ)2−1\n2(∇χ)2−V(ϕ, χ)\u0015\n+SSM. (10.1.1)\nThe field ϕis identified as the DE field, while the field χis the DM field, and the dynamics of\nthe standard model fields are encapsulated in SSM. The potential term V(ϕ, χ)comprises the\neffective interaction in the dark sector and is akin to the potential in the hybrid inflation model,\nas is illustrated in Figure 10.1, expressed as\nV(ϕ, χ) =λ\n4(M2−χ2)2+g2\n2ϕ2χ2+µ2\n2ϕ2=V0−λM2χ2\n2+λχ4\n4+g2ϕ2χ2\n2+µ2ϕ2\n2,(10.1.2)\nwhere Mandµare mass scales, gandλare dimensionless coupling constants, and V0≡λM4/4\nsets the potential scale. For ϕandχto be suitable candidates for DE and DM, the parameters\nmust be carefully selected, and this question will be elaborated on in the next section. The\npotential has a global minimum at χ=±Mandϕ= 0, where the potential energy vanishes (see\nFigure 10.1). It should be noted that another model that diverges from ours has been proposed\nin [477], with the reverse role for DM and DE, leading to different phenomenology, dynamics,\nand valid parameter choices.\nThe effective masses m2\nχandm2\nϕfor the DM and DE fields, χandϕ, respectively, are given by\nthe second derivatives of the potential as\nm2\nχ≡∂2V\n∂χ2=g2ϕ2−λM2+ 3λχ2, (10.1.3)\nand\nm2\nϕ≡∂2V\n∂ϕ2=g2χ2+µ2, (10.1.4)\naccordingly.\nWe focus on a spatially flat cosmological model with the background given by the Friedmann–\nLemaître–Robertson–Walker (FLRW) metric. The line element in Cartesian coordinates is\nds2=−dt2+a2(t)δijdxidxj, (10.1.5)\nwhere a(t)denotes the scale factor and, as in previous chapters, H= ˙a/arepresents the cosmic10A Hybrid Model for the Dark Sector 239\n01234\n2\n1\n012V\n0.000.250.500.751.00Hybrid Inflation Potential\nFigure 10.1 : Illustrative example of the potential in the hybrid inflation model proposed in\n[476], following Equation (10.1.2) with m= 0.2,M= 1.4,λ= 0.5andg= 0.8.\nexpansion rate, and over-dots indicate derivatives of cosmic time t. The field equations of motion\nfor each scalar field are\n¨ϕ+ 3H˙ϕ=−(g2χ2+µ2)ϕ ,and ¨χ+ 3H˙χ=−λχ3+ (λM2−g2ϕ2)χ . (10.1.6)\nThe Friedmann equations take the form\nH2=ρ\n3M2\nPl,and ˙H=−1\n2M2\nPl(ρ+p), (10.1.7)\nwhere ρandpdenote the total energy density and pressure, encompassing contributions from\nboth scalar fields, baryons, and radiation:\nρ=1\n2˙ϕ2+1\n2˙χ2+V(ϕ, χ) +ρb+ρr,and p=1\n2˙ϕ2+1\n2˙χ2−V(ϕ, χ) +pr.(10.1.8)\nFor practical purposes, we decompose the energy densities of each scalar field with different\ncontributions from the hybrid potential:240 10.1. The Hybrid Model\nρχ=1\n2˙χ2−λM2χ2\n2+λχ4\n4+g2ϕ2χ2\n2, (10.1.9)\nρϕ=1\n2˙ϕ2+V0+µ2ϕ2\n2. (10.1.10)\nIt should be noted that this decomposition is purely a matter of choice and does not impact\nthe inherent physics of the model, provided the sum of both components matches the complete\nenergy density of the scalar fields. The configuration in Equation (10.1.9) groups all oscillating\nterms (i.e.terms involving χ) to achieve an effective pressureless behaviour required for structure\nformation in the matter domination era, as will become clear further on. While ϕis expected to\nact like a cosmological constant in the late Universe, its early evolution may differ significantly\ndue to interactions with the χ-field.\nIn this framework, if ϕdeviates significantly from the origin, χwill experience oscillations around\nzero. A sudden shift in the potential shape occurs as the effective dark matter mass, specified in\nEquation(10.1.3), transitionsfromapositivetoanegativevalue. AccordingtoEquation(10.1.3),\nand assuming χoscillates around 0, this transition occurs when ϕcrosses a critical threshold,\ngiven by\n|ϕc| ≈√\nλM\ng. (10.1.11)\nWhen ϕ > ϕ c,χacts as dark matter, and ϕis a dynamic dark energy component that slowly\nrolls down its potential. The dynamics of ϕare primarily dictated by the dominant constant\nscale in the potential, V0, and its interaction with χ. However, as ϕnears the critical value ϕc,χ\ndrops abruptly and starts oscillating around χ=±M. Simultaneously, V(ϕ, χ)collapses to zero,\nsignalling a rapid decay of dark energy and suggesting that the era of dark energy domination\ncomes to a halt and, therefore, is just a transient phenomenon in this model.\n10.1.1 Conditions on Model Parameters\nBefore moving on to the model dynamics, we must examine the conditions imposed by the\nmodel considerations discussed above. For that purpose, we must derive constraints for the free\nparameters g,M, and λsuch that the assumptions remain valid. For the ϕ-field to act as DE,\ntwo conditions must be met: the field must evolve slowly, and its potential energy has to be of\nthe order of the current DE density ρDE,0. This translates into the requirement\nV0=1\n4λM4≈10−47GeV4. (10.1.12)\nAnycontributionfromthe µ2-terminEquation(10.1.2)shouldnotexceedthisvalueasitmodifies\nthe DE density. Hence, the mass parameter Mis of the order 10−3eV, as expected in standard10A Hybrid Model for the Dark Sector 241\nDE models.\nConversely, for χto mimic the dark matter behaviour, the field must oscillate in a quadratic\npotential from the early Universe onward [478]. First, to avoid any damping effects in Equa-\ntion (10.1.6) that smooth out the oscillations, it is required that mχ≈gϕ≫H. Second, for the\nquadratic term to dominate over the quartic term in Equation (10.1.2), the following condition\nmust be fulfilled\ng2ϕ2−λM2≫1\n2λχ2. (10.1.13)\nThe DE ϕ-field is only changing very slowly throughout the cosmological evolution, and its\npresent-day value is large ( ϕ0≳10M2\nPl), implying that the mass of χmust also be large unless\ngis remarkably small, according to mχ=gϕ.\nAt some time in the early Universe, ti, the Hubble rate will become of the same order as mχ\n(H≈mχ), triggering rapid oscillations in the χ-field as the expansion rate drops below its mass.\nTo estimate the temperature of this transition, we assume a radiation-dominated epoch after\ninflation, such that [29]\nH2=1\n3M2\nPlπ2\n30g∗(T)T4, (10.1.14)\nwhere g∗(T)is the effective number of relativistic degrees of freedom at temperature T(of the\norder of several hundred in theories beyond the standard model). Therefore, an estimate for the\noscillation temperature is\nT≈1015\u0010g\n10−7\u00111/2\u0012ϕi\n10M2\nPl\u00131/2\u0010g∗\n100\u0011−1/4\nGeV. (10.1.15)\nThis essentially confirms the assumption that the field will start to oscillate very early in the\nradiation-dominated epoch, almost immediately after a period of inflation in this framework.\nFrom here, we solve for the evolution of the χ-field Equation (10.2.1) to estimate its initial\namplitude χiin the early Universe. Considering that ρDM,0≈g2ϕ2\n0χ2\n0≈4×10−47GeV4and\nχ(t) =χi(ai/a)3/2=χi(T/T i)3/2, we find\nχi\nGeV≈1.4×106\u0010g\n10−7\u0011−1/4\u0012ϕ0\n10M2\nPl\u0013−1/4\u0010g∗\n100\u0011−3/8\n. (10.1.16)\nThis is the required initial amplitude for χafter inflation if it is to be responsible for the observed\namount of dark matter (emphasising again that we assume that the field χis responsible for all\nDM).\nAs currently framed, the ϕ-field remains light during inflation. The only requirement is a large\nfield excursion, ϕ≳10MPl, ensuring that the mass of the χ-field remains substantial in the\nradiation-dominated epoch, and the coupling between χandϕremains sufficiently small. Being242 10.1. The Hybrid Model\nlight during inflation, ϕis susceptible to quantum fluctuations of magnitude Hinf/2π, where Hinf\nis the inflationary expansion rate. Given that ϕrepresents an (almost) flat direction and remains\nsubdominant through the radiation and matter-dominated phases, these quantum fluctuations\nwill not generate large isocurvature perturbations in the DE sector.\nHowever,thesituationconcerningthe χ-fieldislessstraightforward. If gϕ < H infduringinflation,\nin which case χis light, then the quantum fluctuations of χare also of order Hinf/2π, potentially\nresulting in substantial isocurvature modes. The amplitude of these modes is given by [479]\nAI=(H2\ninf/M2\nPl)\nπ2(χ2\ninf/M2\nPl), (10.1.17)\nwhere χinfis the field value during inflation, expected to be around 106GeV for g≈10−7.\nTwo scenarios are considered as a mechanism to circumvent the isocurvature constraint (just\nlike in the extensively studied axion-like fields [464]). First, the χ-field could be heavy during\ninflation ( gϕ > H inf), which would suppress the isocurvature modes. The challenge here lies\nin setting an adequate field amplitude at the end of inflation for χto behave like DM. Second,\nχcould exhibit non-standard dynamics during inflation, either through gravitational coupling\nas proposed in Ref. [480] or by direct coupling to the inflaton field. For the remainder of the\nchapter, we focus on the post-inflation era, assuming that the isocurvature perturbations can be\nrestrained.\nFor the ensuing numerical investigation, initial conditions are chosen at zi= 1014such that\nϕi≫ϕc, or equivalently, gϕi≫√\nλMas dictated by Equation (10.1.11). Given that mχ≫H,\nthe following constraint emerges\ngϕi≫H , (10.1.18)\nwhere the subscript idenotes quantities evaluated at ziin the numerical simulations. Moreover,\ntheϕ-field must evolve slowly, requiring m2\nϕ≪H2. Assuming that µis small compared to gχ,\nwe obtain from Equation (10.1.4)\ng2χ2≪H2. (10.1.19)\nIn the matter-dominated epoch, the χ-field takes the leading role. As discussed in further detail,\nthe relative energy density of dark matter will eventually start decreasing as the Universe keeps\nexpanding. This dilution sets the stage for the ϕ-field, which was slowly evolving until it may\nfinally become the director of the cosmic evolution, driven by its potential energy. Initially, the\ncondition\n1\n2µ2ϕ2+V0≪1\n2g2ϕ2χ2, (10.1.20)\nmust be met to ensure a matter-dominated phase. This condition implies an energy density of10A Hybrid Model for the Dark Sector 243\nχevolving as\nρχ=1\n2˙χ2+1\n2m2\nχχ2≃m2\nχχ2, (10.1.21)\nwhere we have implicitly assumed mχ≃gϕand that the χ-field, exhibiting rapid oscillations,\nis essentially pressureless when averaged over multiple oscillation periods. Given how slowly ϕ\nevolves, the effective DM mass is also virtually constant. Solving Equation (10.1.21) for χ2and\nreplacing into Equation (10.1.19), we obtain\ng2ρχ\nm2χ≪H2. (10.1.22)\nDuring the matter-dominated epoch, with χas the primary contributor, the Friedmann equation\nsimplifies to\nH2≃ρχ\n3M2\nPl. (10.1.23)\nConsequently, from Equation (10.1.22), we derive\n1≪1\n3\u0012ϕ\nMPl\u00132\n, (10.1.24)\nwith mχ≃gϕ. To satisfy this inequality, ϕmust be trans-Planckian , that is to say, that\nϕ≫MPl. Ultimately, to meet the constraints in Equation (10.1.24), ϕmust at least take on\nvalues comparable to the Planck mass. Once more, unless gis remarkably small, this implies\na substantially large DM mass, contrasting sharply with models with ultralight and light scalar\nfield DM candidates [463, 481, 482]. On a more practical note, we remark that this is potentially\nconsistent with the WIMPzilla framework as examined in [462, 483].\nOne additional criterion for the viability of χas a dark matter candidate is its stability over\ncosmological time scales. Phenomenologically, even though a decay of χintoϕis kinematically\nallowed, it is effectively negligible due to the disparity in their mass scales. The decay rate for\nχχ→ϕϕis quantified by [484]\nΓ(χχ→ϕϕ) =g4⟨χ2⟩\n8πmχ, (10.1.25)\nwhere ⟨·⟩denotes an average over one oscillation period. The decay rate, Γ, needs to be sub-\ndominant to the Hubble parameter, H,i.e.,Γ< H. As elaborated in the next section, the\ndecay rate Γscales as a−3throughout the expansion history, owing to its direct dependence on\ntheχ-field coupled to the slow variation of ϕ. In contrast, Hscales as a−3/2anda−2during\nmatter- and radiation-dominated epochs, respectively. The overall result is that Γdecreases\nmore abruptly than Has the Universe expands, thereby ensuring the DM field’s stability for\nreasonable values of g <1.244 10.2. Fluid Approximation\nFinally, we address possible quantum corrections to the potential that might invalidate the\nconsiderations above. Generally, quantum corrections to the tree-level potential are expected\nat the order of M2\nPl, with MPlserving as the natural cut-off scale. Nonetheless, in the context\nof supersymmetric models, the corrections are logarithmic in nature, ln(ϕ/M Pl)[485]. These\nlogarithmic corrections can be suppressed if the coupling constants are relatively small, which is\na natural assumption for the model under investigation. Consequently, the hybrid model shares\nthe same challenges as other models of a similar kind.\n10.2 Fluid Approximation\nTo mitigate the computational difficulties associated with the oscillations of the χ-field, we seek\nreasonable approximations that facilitate the study of its cosmological behaviour. Specifically, we\nreformulate the problem as an interacting quintessence model by adopting a fluid representation\nfor the DM field χ. A key feature of this setup is that the mass of χevolves in parallel with the\nslow rolling of the DE field. Previous investigations into scalar fields oscillating in a quadratic\npotentialhaveestablishedthatthedynamicscanbeapproximatedbyanoscillatingenvelopewith\namplitude A(t)∝a−3/2[478]. Employing the WKB approximation and using the constraints\nderived previously ( gϕ≫Hand ˙ϕ/ϕ≪1), we obtain a solution given by\nχ(t) =χi\u0012ϕi\nϕ\u00131/2\u0010ai\na\u00113/2\nsin (gϕ(t−ti)). (10.2.1)\nHere, χiistheinitialamplitudeof χ. Duetotheslowvariationof ϕ, theterm ϕi/ϕremainsnearly\nconstant, implying that χbehaves like a pressureless fluid with ρχ∝χ2∝a−3. Using Equa-\ntion (10.1.21) to replace ρχ, the oscillation-averaged energy density of DM can be approximated\nto\n⟨ρχ⟩ ≈ρχ,i\u0012ϕ\nϕi\u0013\u0010ai\na\u00113\n, (10.2.2)\nwhere ρχ,i=1\n2g2ϕ2\niχ2\nirepresents the initial energy density of χ.\nWe will henceforth neglect the bracket notation for simplification, as the quantities under consid-\neration are always oscillation-averaged. The averaged density in Equation (10.2.2) shows a linear\ndependence on ϕ. This leads us to the following continuity equation for the oscillation-averaged\ninteracting fluid:\n˙ρχ+ 3Hρχ=˙ϕ\nϕρχ. (10.2.3)\nAccordingly, the equation of motion for the DE field is reformulated as\n¨ϕ+ 3H˙ϕ=−1\nϕρχ. (10.2.4)10A Hybrid Model for the Dark Sector 245\nThis equation is fully in line with the usual DE continuity equation, assuming a perfect fluid\napproximation for the field, given by ρϕ≈˙ϕ2\n2+V0:\n˙ρϕ+ 3H(ρϕ+Pϕ) =−˙ϕ\nϕρχ, (10.2.5)\nensuring the conservation of the total energy density for both ϕandχ, as required by the\ngeneral covariance of Einstein’s equations. In line with the approximation in Equations (10.2.2)\nand (10.2.4) simplifies to\n1\na3d\ndt\u0010\na3˙ϕ\u0011\n=−ρχ,i\nϕi\u0010ai\na\u00113\n, (10.2.6)\nwhich can be integrated and solved for ˙ϕ:\n˙ϕ=\u0010ai\na\u00113\u0012\nKi−ρχ,i\nϕit\u0013\n, (10.2.7)\nwhere Ki≡˙ϕi+ρχ,i\nϕitiis an integration constant and ˙ϕidenotes the initial velocity of the\nfield, when a=ai. Provided that the relation between aandtis known, ϕ’s behaviour is\ncompletely determined. As demonstrated earlier, the fluid approximation is valid immediately\nafter inflation, given that the χ-field starts oscillating around zero as soon as this era comes to\na halt, and radiation becomes the dominant cosmic component at t=ti. Therefore, by solving\nEquation (10.2.7) in the radiation-dominated epoch, during which a(t)∝t1/2, we obtain\nϕ(t) =ϕi+Ci−Ai\u0012t\nti\u00131/2\n−Bi\u0012t\nti\u0013−1/2\n, (10.2.8)\nwith Ci≡2\u0010ρχ,i\nϕit2\ni+Kiti\u0011\n,Ai≡2ρχ,i\nϕit2\ni, and Bi≡2Kitias integration constants. During this\nepoch, ρϕ’s scaling behaviour is dictated by\nρϕ∝˙ϕ2∝a−1. (10.2.9)\nEquation(10.2.8)fullydescribesthetimevariationofthefielduptothematter-radiationequality\natteq. Upon entering the matter-dominated era, a(t)∝t2/3, resulting in the following solution\nto Equation (10.2.7):\nϕ(t) =ϕeq+Ceq−Aeqln\u0012t\nteq\u0013\n−Beq\u0012t\nteq\u0013−1\n, (10.2.10)\nwhere Ceq≡teqKeq,Aeq≡t2\neqρχ,eq\nϕeq, and Beq≡teqKeqare constants that depend on the initial\nconditions specified at the point of matter-radiation equivalence, labelled by the subscript \"eq\".\nThe constant Keqis conceptually analogous to Kifrom Equation (10.2.7), with each term being\nevaluated at teqinstead of ti. Given the slow-rolling nature of the field, it is fair to consider246 10.2. Fluid Approximation\n˙ϕeq≪teqρχ,eq/ϕeq, thus leading to Beq≈teqAeq.\nWe note that Equation (10.2.10) suggests that ˙ϕ∝a−3/2since t∝a3/2in the matter-dominated\nperiod. Furthermore, given that the coupling to dark matter is the main dynamical influence in\nthe field dynamics and ˙ϕ2≫V0, we derive\nρϕ∝˙ϕ2∝a−3. (10.2.11)\nIn this regime, the dark energy component scales analogously to ordinary and cold dark matter.\nThis property is not typical among interacting dark energy models with a constant potential.\nWe have confirmed that this is, in fact, the only specific formulation that yields this unique\nbehaviour, which is pivotal for tackling the cosmic coincidence problem in ΛCDM described in\nSection 2.5, that concerns the remarkable similarity between the value of the energy densities of\ndark energy and cold dark matter at present times [345, 449, 451, 486]. In the next section, we\nwill present numerical simulations that illustrate the dynamics within this regime.\nThe form of the interaction term on the right-hand side of Equation (10.2.3) suggests that ϕ\nshould remain significantly large through the expansion history up until the present. This aligns\nwith the considerations made in Section 10.1.1 and is an indispensable condition to avoid drastic\ndeviations from the ΛCDM paradigm. Though it may appear contradictory, the framework\nis sustained by the assumption that ϕis slowly rolling, according to˙ϕ\nϕH≪1, a condition\nthat has been verified numerically, and which corroborates that ϕ > M Plin agreement with\nEquation (10.1.24).\n10.2.1 Background Dynamics\nMathematically speaking, the fluid approximation used in our analysis mimics a fifth force me-\ndiated by the dark energy scalar field. As reviewed through the course of Chapter 5, this arises\nin theories with a non-universal conformal rescaling of the metric ˜gµν, which in this case would\ndictate the geodesics of the dark matter particles, and is expressed as:\n˜gµν=C(ϕ)gµν, (10.2.12)\nwhere gµνis the gravitational metric, i.e., the one according to which the Einstein-Hilbert retains\nits form. For this approximation to hold in our setting, conformal factor C(ϕ)must be given by:\nC(ϕ) =ϕ2\nM2\nPlfor|ϕ|>|ϕc|. (10.2.13)\nThe transformation remains invertible as long |ϕ|>|ϕc|. It should be noted that the fluid\napproximation will cease to be valid long before ϕcould reach zero. Further, C(ϕ)>0ensures\nthat the Lorentzian nature of the metric is preserved, evading any instabilities related to metric10A Hybrid Model for the Dark Sector 247\nsingularities. Hence, from the considerations above, the functional form of the coupling term in\nthe fluid approximation is reinstated as:\nQ=−Cϕ\n2Cρχ=−ρχ\nϕ. (10.2.14)\nUnder this approximation, both components of the dark sector are considered perfect fluids,\nallowing us to recast the relevant equations for laying out the dynamics. For numerical purposes,\nwe use conformal time, which we recall is defined as dτ=dt\na, with derivatives denoted by a\nprime and the Hubble rate scaled as H=aH. The conservation equations for the DM and DE\nfluids are given by\nρ′\nχ+ 3Hρχ=−Qϕ′=ϕ′\nϕρχ, (10.2.15)\nand,\nρ′\nϕ+ 3H(ρϕ+pϕ) =Qϕ′=−ϕ′\nϕρχ. (10.2.16)\nThe interaction in the dark sector means that the right-hand side of these equations is no longer\nzero but composed of symmetric terms. The direction of the energy flow depends solely on the\nsign ofϕ′\nϕ: when the ratio is positive, DE feeds the DM component, whereas a negative ratio\nindicates energy transfer from ϕto the DM fluid. Independent of the initial conditions, ϕand\nϕ′always have opposite signs, implying a unidirectional energy transfer from the χ-fluid to the\nϕ-field. The same physics is encapsulated in the modified Klein-Gordon equation:\nϕ′′+ 2Hϕ′=−a2\nϕρχ=a2Q , (10.2.17)\nwhich can be numerically integrated to obtain specific realisations of the evolution under a given\nset of cosmological parameters. For that purpose, the only free parameters specific to the model\nare the initial conditions for the DE field, ϕi=ϕ(τi)andϕ′\ni=ϕ′(τi), along with the scale of the\nhybrid potential, V0. It should be emphasised that the remaining parameters in the potential,\ngiven in Equation (10.1.2), are irrelevant to the numerical solutions, so long as they meet the\nconstraints outlined in Section 10.1.1. We set µ= 0for simplicity as it does not affect current\ndynamics. As for all the cases studied in this thesis, V0is determined using a shooting method\nfor a fiducial value of the present DE relative energy density Ω0\nϕ=ρ0\nϕ\n3M2\nPlH2\n0. Furthermore, as ϕ′\ni\ndoes not influence the dynamics due to the rapid stabilisation of the scalar field at the minimum\nof its potential in the radiation-dominated epoch, we set ϕ′\ni= 0without loss of generality. This\nnarrows our study to the effects of varying the single free model parameter, the initial condition\nϕi. We exclusively consider scenarios where ϕi>0, based on the symmetry of the potential over\nthe dynamics.248 10.2. Fluid Approximation\n10.2.2 Cosmological Perturbations\nWe map the cosmological perturbations into an interacting dark energy framework by expanding\nupon the background evolution. The aim is to identify the changes in the gravitational inter-\nactions compared to the standard ΛCDM model and to evaluate the observational signatures\nleft by the background fluid approximations. To this end, we consider once more the conformal\nNewtonian gauge for perturbations, as outlined in Section 2.2.1, for which we present the metric\nonce again\nds2=a2(τ)\u0002\n−(1 + 2Ψ) d τ2+ (1−2Φ)δijdxidxj\u0003\n, (10.2.18)\nwhere Ψ(τ, ⃗ x)andΦ(τ, ⃗ x)are the Bardeen potentials and, accordingly, δdesignates perturbed\nquantities and the correspondence ∇2→ −k2holds in Fourier space. For the scalar field system,\nthere is no anisotropic stress. The perturbed variables δϕandδρχevolve according to\nδϕ′′+ 2Hδϕ′+k2δϕ= (Ψ′+ 3Φ′)ϕ′+ 2a2QΨ +a2δQ , (10.2.19)\nand\nδ′\nχ=−(θχ−3Φ′) +Q\nρχϕ′δχ−Q\nρχδϕ′−θ′\nρχδQ , (10.2.20)\nwhere δχ=δρχ/ρχis the density contrast and δQis defined as\nδQ=ρχδϕ−ϕδρχ\nϕ2. (10.2.21)\nIn Equation (10.2.20), both the equation of state wχ=pχ/ρχand the sound speed c2\ns=δpχ/δρχ\naresettozero. Thebackgrounddynamicssupporttheformerassumption, whilethelatterfollows\nnaturally from the equation for the sound speed of an oscillating scalar field [463],\nc2\ns=k2/(4m2\nχa2)\n1 +k2/(4m2χa2). (10.2.22)\nWhile this assumption holds for an uncoupled scalar field, it still captures the relevant physics, as\nthe coupling is restricted to be very small. Since we have demanded mχto be considerably large\nfor the background approximations to hold, and given that we are considering scales k≪2mχa,\nEquation (10.2.22) implies c2\ns≈0which we nevertheless verified numerically.\nThe impact of the coupling on the density matter perturbations can be primarily assessed by\nlooking at the sub-horizon regime k≫ Hunder the quasi-static approximation when the matter\nand field perturbations are the main drivers of the linear evolution. In practical terms, this\nconsists of neglecting the time derivatives of the perturbations and metric potentials, greatly\nsimplifying the equation of motion for δχin Equation (10.2.20) [3, 399],\nδ′′\nχ+Heffδ′\nχ≃4πGeffρχδχ, (10.2.23)10A Hybrid Model for the Dark Sector 249\nwhere the subdominant contribution of baryons has been neglected, and the effective Hubble\nterm is defined as\nHeff=H\u0012\n1 +Q\nρχϕ′\nH\u0013\n, (10.2.24)\nconsisting of an additional friction contribution that accounts for the changes in the coupled\nbackground evolution. On the other hand, the effective gravitational constant in the small-scale\nlimit (large k) becomes important as it is expressed as\nGeff��G\u0000\n1 + 2 β2\u0001\n,with β≡MPlQ\nρχ, (10.2.25)\nas expected, considering the general results for scalar-tensor gravity models under a conformal\ntransformation [347, 401, 453].\n10.3 Observables and Phenomenology\nIn this section, we investigate the evolution of the hybrid model for different realisations of\nthe coupling in the dark sector and its impact on cosmological observables, compared with\nthe standard ΛCDM model. As anticipated, the general features of the model are consistent\nwith canonical coupled quintessence frameworks featuring stable, effective interactions [292, 309],\njust as the ones considered in Chapter 6 (see also [32, 184, 334, 344, 346, 348, 349, 355, 487–\n490] for recent works). Nevertheless, this is not to say that there will not be any distinct\nsignatures arising from the scalar field’s slow-roll evolution and the ϕ-dependent coupling. For\nillustration purposes, we explore four scenarios, parameterised by different initial field values,\nnamely ϕi/MPl={8,10,15,20}, recalling that ϕisets the initial scalar field dynamics as well as\nthe strength of the coupling in the dark sector, meaning that its effect over the dynamics can\nbe non-trivial. We set the initial condition for the field velocity at ϕ′\ni= 0, as we demonstrate\nlater that it has a negligible effect on the evolution. The cosmological parameters are kept at the\nPlanck2018 fiducial values for a ΛCDM cosmology [32]: H0= 67.56km/s/Mpc, Ωbh2= 0.022,\nandΩch2= 0.12, where h=H0/100. The scale of the potential is fully fixed viathe Friedmann\nconstraint for a flat cosmology. The same applies to the analysis of the perturbations. We assume\nGaussian adiabatic initial conditions and a scalar power spectrum characterised by an amplitude\nof curvature fluctuation of As= 2.215×10−9at a pivot scale of kpiv= 0.05Mpc−1, and a spectral\nindex ns= 0.962. Without loss of generality, we also consider vanishing initial conditions for\nthe perturbations in the scalar field and its corresponding velocity, i.e., δϕi=δϕ′\ni= 0. The\nbackground and cosmological perturbations are calculated using a version of the open-source\nCLASScode1[95, 101, 102] modified for this purpose.\n1https://github.com/lesgourg/class_public250 10.3. Observables and Phenomenology\n0 1 2 3 4\nlog(1+z)0.00.20.40.60.8ΩΩχ(φi=8)\nΩb(φi=8)\nΩr(φi=8)\nΩφ(φi=8)Ωχ(φi=10)\nΩb(φi=10)\nΩr(φi=10)\nΩφ(φi=10)\n258log(ρφ/ρΛ)φi=8\nφi=10φi=15\nφi=20\n0 1 2 3 4\nlog(1+z)0369∆H\nH[%]\nFigure 10.2 :Left panel: Redshift evolution of the relative energy densities Ωiof the dark\nmatter fluid χ(yellow), baryons (green), radiation (sea blue) and the scalar field ϕ(red) for\ntheΛCDM model (thin solid lines), ϕi= 8MPl(dashed/dotted-dashed lines) and ϕi= 10MPl\n(thick solid lines). Right panel: Ratio of the dark energy density ( top panel ) and fractional\ndeviations in the Hubble rate ( bottom panel ) in the hybrid coupled model with respect to the\nstandard model as a function of redshift 1 +z, for ϕi={8,10,15,20}MPl(solid red, dashed\nyellow, dotted-dashed green and dotted sea blue lines).\n10.3.1 Background Evolution\nIn the fluid approximation, where the potential V(ϕ, χ)remains constant, the field’s dynamics\nwould be indistinguishable from a cosmological constant in the absence of the coupling (which\nwouldalsoimply ϕ′≡0orϕ→ ∞). Thecouplingwiththedarkmatterfluidistheprimarydriver\nof the energy density of dark energy at early times, as depicted in the left panel of Figure 10.3.\nOnce this coupling becomes significant after the radiation-dominated era and when DM starts\nplaying a dominant role, ϕbegins to slowly evolve until its energy density mimics that of DM,\nthe component to which it couples. During this period of the evolution, the scalar field acts as an\neffective pressureless fluid, and this scaling phase ceases once the kinetic and potential energies\nbecome comparable and ϕ′2≲a2V0.\nThe end of the scaling phase is directly influenced by the initial conditions set for ϕiandρχ(see\nEquation (10.2.7)). Higher ϕivalues lead to late-time behaviour that resembles ΛCDM more\nclosely, implying an earlier exit from the scaling regime, as seen in the right panel of Figure 10.2\nand the left panel of Figure 10.3. In contrast, ϕ′quickly adjusts as the scalar field is driven\ntowards the minimum of the effective potential, irrespective of its initial value, implying that\nϕ′\niregulates the onset of the scaling phase., which starts earlier for the largest initial velocities.\nThis effect is negligible as it occurs early in the radiation-dominated phase when the scalar field’s\ncontribution is insignificant. Similar behaviour has been reported in [491], driven by a significant\nacceleration of the scalar field instead. When the ϕ-field exits the matter-scaling regime, it\nsettles at a cosmological constant-like attractor solution, where it will remain diluting with the10A Hybrid Model for the Dark Sector 251\n0 1 2 3\nlog(1+z)-48-46-44-42-40-38-36logρ[GeV4]ρΛ\nρχ\nρbρr\nρφ(φi=8)\nρφ(φi=15)\n0 1 2 3\nlog(1+z)01234567Geff/G−1[%]ΛCDM\nφi=8\nφi=10φi=15\nφi=20\nFigure10.3 :Left panel: Evolution of the energy densities ρof the dark matter fluid χ(yellow),\nbaryons (green), radiation (sea blue) and the scalar field ϕ(filled red) for ϕi= 8MPl. To\nappreciate the differences, we also include ρϕfor the ϕi= 15MPlcase (dotted-dashed red line)\nandρΛfor the standard model (thin black solid line) for completeness. Right panel: Percentage\ndeviations of the effective gravitational constant, as defined in Equation (10.2.25), with respect\nto the standard G(thin black solid line) for ϕi={8,10,15,20}MPl(solid red, dashed yellow,\ndotted-dashed green and dotted sea blue lines).\nexpansion until the fluid approximation breaks down.\nThe coupling also manifests as an amplification of ρϕ, particularly in scenarios with stronger\ncoupling (smaller ϕi), which naturally induces an earlier matter-DE equality. This shift is an\nartificial result of fixing the present cosmology. As DM loses energy to DE, it must be more\nabundant early on, leading to higher initial amplitudes for the DM energy density to maintain\nequilibrium and match the present value. On the other hand, this effect goes in hand with a\nmore significant contribution from the coupling to the scalar field dynamics, as evident from\nEquation (10.2.17). This also reflects as an earlier matter-radiation equality with decreasing ϕi\nvalues, as depicted in the left panel of Figure 10.2.\nIn the lower right panel of Figure 10.2, we show the relative deviations in the Hubble rate H(z)\nwhen comparing the hybrid coupled model with the standard ΛCDM model. Even though H0\nis fixed at the current value for both models, H(z)is enhanced by up to 9%during the matter-\ndominated phase for the model with the smallest ϕiwhile remaining negligible during radiation\ndomination. This is correlated with an enhancement of ρϕandρχand will be instrumental for\nunderstanding the scale-dependent growth of matter perturbations, which we will discuss in the\nfollowing section.252 10.3. Observables and Phenomenology\n1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00\n1+z0.20.30.40.50.60.70.80.9fσ8\nΛCDM\nφi=8\nφi=10φi=15\nφi=20\n103104P(k)[Mpc/h]3\nΛCDM\nφi=8φi=10\nφi=15\n10-310-210-1\nk[h/Mpc]0306090∆P(k)\nP(k)[%]\nFigure 10.4 :Left panel: Evolution of the cosmological observable fσ8(defined in Equa-\ntion (10.3.3)) with redshift 1 +zfor the hybrid coupled model with ϕi={8,10,15,20}MPl\n(solid red, dashed yellow, dotted-dashed green and dotted sea blue lines) and for the ΛCDM case\n(thin black solid line). The RSD data points and corresponding error bars (solid grey) correspond\nto the compilation presented in [492]. Right panel: The matter power spectrum as a function\nof Fourier scales k(top panel ) and corresponding percentage deviations ( bottom panel ), for the\nhybrid coupled model with ϕi={8,10,15,20}MPl(solid red, dashed yellow and dotted-dashed\ngreen lines) with respect to the ΛCDM case (thin black solid line).\n10.3.2 Cosmological Perturbations\nThe linear growth rate f(a)of the total matter perturbation - comprising both baryonic and\ndark matter and denoted as δm- quantifies this phenomenon as\nf(z, k) =1\nHδ′\nm(z, k)\nδm(z, k), (10.3.1)\nwith\nδm(z, k) =Ωbδb+ Ω DMδDM\nΩb+ Ω DM. (10.3.2)\nThe changes to the evolution of f(z), compared to the ΛCDM model, correlate with the modifi-\ncations to the onset of the matter-dominant era, at which point the dark sector coupling becomes\nsubstantial. The combined quantity fσ8is directly connected to observations, as was introduced\nin Section 3.4.4, defined as\nfσ8(z, kσ8) =σ8(0, kσ8)\nHδ′\nm(z, kσ8)\nδm(0, kσ8), (10.3.3)\nwhere σ8is the root mean square mass fluctuation amplitude for spheres of size 8h−1Mpc (or\nequivalently for scales kσ8= 0.125hMpc−1), generally used to set the amplitude of the matter\npower spectrum at present σ0\n8≡σ8(0, kσ8).\nIn Figure 10.4 (left panel), we display the redshift evolution of fσ8for the examined models,10A Hybrid Model for the Dark Sector 253\nas defined in Equation (10.3.3). We identify an overall enhancement in the linear growth of\nmatter perturbations. The most notable deviations from ΛCDM occur in the model with the\nminimal ϕias a direct consequence of changes to the expansion history. The numerical analysis\nreveals that σ0\n8increases across all models to match the same amplitude As, with higher values\nϕiprogressively approaching the ΛCDM prediction. We also incorporate RSD observational\ndata1from the compilation in Ref. [492] which encompasses measurements reported by multiple\nsurveys. The plots suggest an enhanced linear growth in the hybrid model, parametrised by a\nlarger fσ8, and associated with the higher σ8values as well when assuming the same spectrum\nof initial primordial perturbation amplitudes As. We conclude that, in these specific conditions,\nthe models appear not to resolve the S8tension [57, 145], although a more rigorous investigation\nis needed to affirm this conclusion, as the data interpretation is based on ΛCDM assumptions.\nIn Figure 10.4 (right panel), we depict the power spectrum of matter density fluctuation P(k)\nand the corresponding relative variations for Fourier scales 10−3hMpc−1< k < 10−1hMpc−1.\nWe report a mild suppression for larger scales (low k) and a significant amplification for in-\ntermediate to small scales (high k). Deviations reach up to 81%for the smallest value of ϕiat\n10−1hMpc−1, a point where the linear approximation is expected to no longer hold, as non-linear\neffects become dominant. This trend for large kwas expected on the basis of the fifth force,\nwhich is more significant for lower values of ϕi, having a considerable influence on the growth\nof perturbations according to Equation (10.2.25). We expect that this should play an important\nrole when comparing the model with existing observational data.\nThe slight suppression for larger scales (smaller kvalues) is ascribed to the changes in the\nbackground cosmic expansion, specifically through the friction term in Equation (10.2.24), which\ninhibits the growth of matter density perturbations. This effect overcomes the fifth-force only at\nthe largest scales, peaking at 4%for the models considered, and is essentially negligible for the\nmodels with larger ϕi, for which Geff≈G, as displayed in Figure 10.3 (right panel). Furthermore,\nwe note the shift in the turnover of the power spectrum towards smaller scales, in comparison\nwith the ΛCDM case, due to the change in the radiation-matter equality epoch towards larger\nredshifts.\nIn the left panel of Figure 10.5, we illustrate the evolution of the gravitational potentials ΦandΨ\n(top) at an intermediate scale of k= 0.01Mpc−1, along with the corresponding percentage differ-\nences relative to ΛCDM. The most evident deviations occur at z≲103, in the matter-dominated\nepoch, where alterations in the DM evolution become significant, and the energy density of the\nscalar field is scaling with the matter. The most important variations are consistently associated\nwith the smallest ϕivalues. The modifications to the lensing gravitational potential ϕlen= Φ+Ψ\nare mainly ascribed to the energy transfer from DM to the DE sector. This potential is the rel-\nevant term in the line-of-sight integration for the lensing power spectrum Cϕϕ\nℓ, as presented in\nSection 3.3, and is depicted in the left panel of Figure 10.6. We identify an overall enhancement\n1CosmoBolognaLib254 10.3. Observables and Phenomenology\n0.60.81.01.21.4Φ+Ψφi=8\nφi=10φi=15\nΛCDM\n0 1 2 3 4\nlog(1+z)0369∆(Φ+Ψ)\n(Φ+Ψ)[%]\n10\n8\n6\n4\n2\n0Φ/prime+Ψ/prime(×104)\nφi=8\nφi=10\nφi=15\nΛCDM\n0 1 2 3 4\nlog(1+z)40\n20\n020∆(Φ/prime+Ψ/prime)\n(Φ/prime+Ψ/prime)[%]\nFigure 10.5 : Redshift evolution of the sum of the gravitational potentials, Φ + Ψ, (top left\npanel) and the corresponding derivative with respect to conformal time, Φ′+Ψ′(top right panel )\nfor the hybrid coupled model with ϕi={8,10,15}MPl(solid red, dashed yellow and dotted-\ndashed green lines) and for the ΛCDM case (thin black solid line), including the percentage\ndeviations from the standard model ( bottom panels ).\n0.00.51.01.52.0[/lscript(/lscript+1)]2Cφφ\n/lscript/2π×107\nΛCDM\nφi=8φi=10\nφi=15\n101102103\n/lscript04080120∆Cφφ\n/lscript\nCφφ\n/lscript[%]\n02468/lscript(/lscript+1)CTT\n/lscript/2π×1010\nΛCDM\nφi=8\nφi=10\nφi=15\n101102103\n/lscript40\n20\n02040∆CTT\n/lscript\nCTT\n/lscript[%]\nFigure 10.6 : Lensing ( top left panel ) and TT ( top right panel ) power spectra as a function of\nthe angular scale ℓfor the hybrid coupled model with ϕi={8,10,15,20}MPl(solid red, dashed\nyellow, dotted-dashed green and dotted sea blue lines) and for the ΛCDM case (thin black solid\nline), along with the fractional deviations from the standard model ( bottom panels ).10A Hybrid Model for the Dark Sector 255\nofCϕϕ\nℓacross all angular scales, with the most substantial deviations from ΛCDM occurring\nfor the lowest ϕicase. This could potentially explain the lensing excess observed in the Planck\ntemperature data of CMB anisotropies [32, 356], which was the focus of Section 3.6.3. Likewise,\nthis enhancement aligns with the amplified effective gravitational interaction for DM particles.\nWe note that the matter density contrast δmfollows a broadly consistent trend across most of\nthe Fourier scales depicted, mirroring the increased effective gravitational constant, as presented\nin Figure 10.3.\nThe same effects can also be dug out of CMB anisotropies’ temperature-temperature (TT) power\nspectrum, presented in the right panel of Figure 10.6. The most evident modifications arise from\nthe integrated-Sachs-Wolfe (ISW) effect, proportional to ˙ϕlen(see Section 3.3), depicted in the\nright panel of Figure 10.5. This effect can be dissected into the early and late-time contributions\nto the ISW. The early ISW effect amplifies the time derivatives of the potentials due to an\nearlier transition from radiation- to matter-dominated eras. Most importantly, the late-time ISW\neffects stem from changes to the CMB lensing by large structures, owing to modified dynamics\nin the dark sector. This leads to a late-time suppression of Φ′+ Ψ′, as shown in the right\npanel of Figure 10.5. The most apparent changes in the TT power spectrum are a decrease\nin amplitude and a narrowing of the peaks and troughs, tied to a reduction in the ρb/ρDM\nratio during recombination (see Section 3.5) [350, 488]. This induces a degeneracy between the\neffective coupling and the Hubble parameter, as the latter mainly affects the first peak’s location\nand magnitude. Likewise, the shift in the acoustic peaks to higher multipoles is also linked to\nchanges in the cosmic expansion history, which modify the distance to the last scattering surface\nand thus reduce the sound horizon at the baryon-drag epoch (see Section 3.5). The more this\nshift is pronounced, the more significant the deviations from ΛCDM at the background level.\nFrom both panels of Figure 10.6, we observe that an enhanced Hubble rate drives the CMB\nspectra towards smaller angular scales (higher multipoles). Lastly, the lensing power spectrum\namplification correlates with an increased ISW tail at large angular scales, although this is\ntypically a secondary effect.\n10.4 Discussion\nIn this study, we have introduced a hybrid model for the dark sector, in which DM and DE are\nascribed to two interacting scalar fields. We started from a formulation based on an interacting\npotentialfrequentlyencounteredinhybridinflation, whichwetransportedtotheDM-DEsystem.\nWe performed a thorough examination of the cosmology implications of such a setting. With\nthe onset of a rapid oscillatory behaviour deep within the radiation-dominated epoch, the heavy\nscalar field essentially acts as pressureless DM. We have shown that, under appropriate approx-\nimations, the scalar fields can be treated approximately as a DM fluid coupled to a gradually\nevolving DE field through this oscillatory pattern.256 10.4. Discussion\nIn conclusion, we list the key predictions of the model proposed in this chapter:\n•The DE field is required to have a sizeable present value ( ϕ >MPl) for the fluid description\nto hold. Consequently, the coupling in the dark sector is currently highly suppressed,\nand DM is considerably massive, akin to the WIMPZilla framework. The energy scales\nin the interacting potential are markedly below the Planck scale; for instance, the scale\nMis approximately on the eV scale, as it depends on the coupling constant λ. The\nrequirement for super-Planckian excursions of the field, reminiscent of inflationary models,\nposes challenges for model building in theories beyond the standard model. Nonetheless,\nit is noteworthy that the current model combines two seemingly disconnected mass scales:\nthe minute mass scale Mand the large field excursions for the DE field ϕ.\n•Another exciting feature of the model is the transient nature of the dark energy-dominated\nepoch. The DM field is predicted to become light and cease to behave as a pressureless\nfluid. Both fields will ultimately settle at the actual minimum of the potential ( ϕ= 0and\nχ=±M). The fate of the Universe will then hinge on its spatial curvature. Should it\nbe closed, the expansion will cease, and the Universe will collapse, paving the way for a\npotential bounce far in the future.\nAs demonstrated, if the coupling is large enough, unique imprints are expected on the\ntemperature-temperature power spectrum of CMB anisotropies and the growth of structures\nencoded in the matter power spectrum. Both existing and forthcoming observational datasets\ncan test such characteristic signatures. Drawing upon diverse independent data sets and probing\nthe remaining relevant cosmological variables, an in-depth analysis is ongoing work.11Conclusions\nMeu Deus, só agora me lembrei que a gente morre. Mas, mas eu também?!\nNão esquecer que por enquanto é tempo de morangos. Sim...\n—Clarice Lispector A Hora da Estrela\nMy God, I just remembered that we die. But — but me too?! Don’t forget that for now it’s strawberry season.\nYes...— Clarice Lispector in The Hour of the Star\nThis project aimed to enhance our understanding of the properties of dark matter and dark\nenergy in the context of cosmology, focusing specifically on the possibility of an interaction in\nthe dark sector and its observational signatures. We have touched on the remarkable richness\nof independent probes, with particular emphasis on measurements of the physics of the cosmic\nmicrowavebackgroundanisotropies, invaluablearchaeologicalsitesofburiedknowledge. Wehave\ndelved deeply into the enigmatic realms of coupled quintessence and interacting dark energy,\nmuch like cosmic archaeologists excavating layers of ancient expansion history, piecing together\nshards of pottery from various cosmological data in an attempt to understand the intricate\nnature of the dark sector. Nevertheless, it is fascinating to think that the more we uncover, the\nmore we realise the scope of what remains hidden. With each model we explore, independent of\nthe results, we enrich our understanding of the relics and artefacts that reveal something while\nhinting at larger structures yet undiscovered, constantly suggesting new avenues of research.\nWe report on the main conclusions from each investigation included in this dissertation, which,\nwhile just a tiny overdensity contribution to the literature, may contribute to the gravitational\nclustering of the ceaseless quest to unravel the mysteries of our cosmic habitat.\n•In Chapter 6, we have analysed the standard coupled quintessence model by assuming a\nflat cosmology and letting the spatial curvature vary as a free parameter. We found no\nsignificant improvement regarding the evidence for a closed Universe in the Planckdata\nassuming ΛCDM. Nevertheless, we were able to show that the disagreement between CMB\ndata and other geometrical probes is maintained in other scenarios.\n•In Chapter 7, we have studied a coupled quintessence model in which the interaction\ndepends on the kinetic energy of the scalar field. We showed the particular observational\nsignatures of such a framework. We found no statistically significant evidence for either of\nthe models, which could be a hint that forthcoming data might be able to address these\ndifferences.\n•In Chapter 8, we assessed the potential of constraining coupled quintessence models with\n257258\nnext-generationgravitationalwavedetectors. Weshowedthatstandardsirensareapromis-\ning alternative to supernova catalogues and baryon acoustic oscillation observations. We\nfound that the accuracy in the H0parameter is consistently enhanced by an order of\nmagnitude at 1σ, which is promising for providing further insight into addressing the H0\ntension.\n•InChapter9, weperformedathoroughstudyofmodelsmotivatedbystringtheoryinwhich\nthe dark sector has a fundamental geometrical higher-dimensional origin. Even though this\nis a more intricate scenario and a preliminary analysis does not seem to suggest evidence for\nthe particular framework considered, this analysis motivates further studies with functions\nthat might be more appropriate for a cosmological description.\n•In Chapter 10, we introduced a hybrid model for the dark sector, in which DM and DE are\nascribed to two interacting scalar fields. We have shown how the DM scalar field dynamics\ncan be approximated to a fluid description that holds similarities with coupled quintessence\nmodels. We found exciting novel predictions for the nature of this DM source and a unique\ntransient nature for the dark energy-dominated epoch. The numerical analysis of the\nobservablesshowspromisingcosmologicalsignatureswithoutintroducingextraparameters,\nand comparing this setup against data is ongoing work.AScalar Field Inflation\nInflation refers to an epoch of accelerated expansion in the early Universe, needed in order to\naddress the problems listed below:\n•The horizon problem: the observed homogeneity implies extraordinary smoothness of the\nearly Universe even if most of the Universe does not appear to have been in causal contact\nin the hot Big Bang paradigm. In comoving coordinates, the size of a causally connected\nregion of space is defined according to the maximum distance that light can travel and still\nbe detected today. This threshold defines the particle horizon :\ndh(τ) =τ−τi=Zt\ntidt\na(t)=Za\naida\na˙a=Zlna\nlnai(aH)−1d lna , (A.0.1)\nwith τstanding for the comoving time, which corresponds to the comoving particle horizon\nunder the convention that τi= 0since ai≡0corresponds to the Big Bang singularity.\nThe last equality of Equation (A.0.1) shows how causality relates to the comoving Hubble\nradius\ndH=H−1= (aH)−1. (A.0.2)\nThe∆τbetween the BB singularity and the last-scattering surface is much smaller than\nthe present conformal age of the Universe, implying that most of the CMB is composed\nof non-causally connected patches. This can be shown with simple examples, such as in\nRef. [45]. Then the question of how can the CMB be homogeneous at scales much larger\nthan the particle horizon at the time when the CMB photons were released remains.\n•The flatness problem: the fact that the spatial curvature of the Universe is so small implies\na fine-tuning of the initial conditions. Relying on current constraints for the time-evolving\ncurvatureparameteroftheUniverse, |Ωk,0|<0.005[32], itispossibletoextrapolatebounds\non the same parameter evaluated at the time-scales of the early Universe, resulting in even\ntighter constraints such as |Ωk,BBN|<10−16[45]. This implies that the curvature scale at\nthat epoch must have been larger than the Hubble radius by many orders of magnitude.\nAs we have seen above, during the standard hot Big Bang, the particle horizon is well\napproximated by the Hubble radius, suggesting again a large fine-tuning degree over many\ncausally disconnected patches of space.\n•The problem of superhorizon correlations: the Universe contains density fluctuations cor-\nrelated over apparently acausal distances. For comoving coordinates, the wavelength of a\nfluctuation λremains fixed while the Hubble radius (aH)−1keeps increasing due to the\nexpansion. This results in every fluctuation inside the Hubble radius today ( subhorizon\nregime) being outside the Hubble radius at sufficiently early times ( superhorizon regime).\n259260\nScales larger than the particle horizon (approximately the same as the Hubble radius at\nearly times) at recombination would not have been inside the horizon before the emission\nof the CMB photons. However, the CMB fluctuations are not random, containing char-\nacteristic correlations on scales much larger than this apparent horizon. This is akin to\na modern version of the horizon problem, corroborated by the CMB map of temperature\nfluctuations - the CMB is not just homogeneous on apparently acausal scales, as stated\nby the horizon problem. However, its fluctuations are also strongly correlated with such\nscales.\nTo grasp the dynamics of inflation, we first define the slow-roll parameter ε:\nd\ndt\u0002\n(aH)−1\u0003\n=−1\na(1−ε)with ε≡ −˙H\nH2. (A.0.3)\nComparing this condition with Equation (2.3.1), it follows that ε <1, or in other words, that\nthe fractional change in the Hubble parameter ( ˙H/H) per Hubble time ( H−1) must be small.\nThe limit ε= 0such that H=const. ⇒a(t) =eHtcorresponds to quasi-de Sitter expansion.\nWhile this approximation is true for the cosmological constant, inducing accelerated expanding\nbehaviour at late times, it cannot hold for the inflationary epoch because any early period of\naccelerated expansion must eventually end in a finite amount of time. Nevertheless, to address\nthe early Universe issues of the Hot Big Bang model, inflation must last for a sufficiently long\nperiod, typically measured in the number of e-folds, denoted as N= ln a. This relies on ε\nremaining small for a substantial number of Hubble times and is embodied by the second slow-\nroll parameter :\nη≡˙ε\nHε. (A.0.4)\nAnalogously, inflation proceeds while |η|<1and the fractional change in εper Hubble time is\nsmall.\nThe simplest models of inflation incorporate the time-dependent dynamics of inflation into a\nscalar field φ(t, ⃗ x)called the inflaton, with a respective potential energy density V(φ)and stan-\ndard kinetic energy. In this way, it is possible to assess under which conditions the inflaton can\ndrive inflation. The Lagrangian for such a scalar field minimally coupled to gravity is given by:\nLφ=1\n2gµν∂µφ∂νφ−V(φ). (A.0.5)\nThe standard assumption in the inflationary paradigm is that, at the start of inflation, the\ninflaton is displaced sufficiently far from the minimum of its potential, which is nearly flat, to\nmaintain the slow-roll conditions.\nThe equation of motion for a homogeneous configuration of the field φ(t)is\n¨φ+ 3H˙φ+dV\ndϕ= 0. (A.0.6)AScalar Field Inflation 261\nThis is the familiar Klein-Gordon equation with an extra term, 3H˙φ, called the Hubble friction ,\nwhich plays a crucial role during inflation and reflects the influence of the expansion. Assuming\nthat this scalar field dominates the Universe, we can infer how it impacts the expansion through\nthe Friedmann equations sourced by a perfect fluid with energy density and pressure:\nρφ=1\n2˙φ2+V(φ),and pφ=1\n2˙φ2−V(φ), (A.0.7)\nleading to\nH2=1\n3M2\nPl\u00141\n2˙φ2+V(φ)\u0015\n,and\u0012¨a\na\u0013\n=1\n3M2\nPl\u0002\nV(φ)−˙φ2\u0003\n, (A.0.8)\nwhich in the limit ˙φ→0becomes\nH2≈1\n3M2\nPlV(φ)≈const. , (A.0.9)\ncorresponding to the exponential expansion, a(t)∝eHt. If instead the field is slowly varying,\nsuch that ˙φ2/2≪V(φ), the equation of motion can be approximated by\n3H˙φ∼ −V′(φ), (A.0.10)\nand the first Friedmann equation reads\nH2(t)∼V[φ(t)]\n3M2\nPl. (A.0.11)\ntaking the field evolution to be encapsulated by the Hubble friction such that the scale factor\nevolves as a(t)∝e−N(t). This will be verified as long as the potential is sufficiently flat, V′(φ)≪\nV(φ). It is possible to write the equation of state of the field in the slow roll-approximation as\np=\u00142\n3ε(φ)−1\u0015\nρ , (A.0.12)\nwhere now, substituting Equations (A.0.10) and (A.0.11) into the definition of the Hubble slow-\nrolling parameters in Equations (A.0.3) and (A.0.4) yields the potential slow-rolling parameters\nεV≈ε=M2\nPl\n2\u0012V′(φ)\nV(φ)\u00132\n,and ηV≈2ε−1\n2η=M2\nPlV′′(φ)\nV(φ). (A.0.13)\nand the slow roll approximation remains valid as long as both parameters are small, i.e.εV≪1\nand|ηV| ≪1. The number of e-folds can be expressed as\nN= lna(te)\na(ti)=−Za(te)\na(ti)Hdt∼M2\nPlZφi\nφeV(φ)\nV′(φ)dφ , (A.0.14)262\nwiththesubscripts iandelabellingthebeginningandendofinflation,respectively. Aspreviously\nmentioned, the minimum duration for inflation to resolve the HBB problems is around 60e-folds.\nHowever, inflation is typically so rapid that most models can accommodate this condition very\neasily. In summary, during inflation, the rapid exponential expansion of the Universe ensures\nflatness and homogeneity. As the potential becomes steeper, inflation eventually ends, and the\nscalar field starts oscillating around its vacuum state at the minimum of the potential.\nTo transition to a radiation-dominated Universe after inflation, the inflaton field must decay\ninto standard model particles, a process commonly referred to as reheating [79]. The specifics of\nthis process depend on the particular inflationary model. To ensure that the Universe becomes\nradiation-dominated and in thermal equilibrium during primordial nucleosynthesis, which occurs\nat temperatures around the MeV scale, it is often assumed that the reheating temperature falls\nwithin the range of 1TeV to 1016GeV which sets the minimum bound of Nmin∼[45,60]e-folds\nof inflation [29, 79].BMock Data Set Simulation\nThis Appendix details the methodology followed to create the mock catalogue of standard sirens\nused in Chapter 8.\nThe inspiral and merger events of compact objects - such as black holes and neutron stars\n- generate gravitational waves (GW) that propagate through spacetime. Interferometers are\nsensitive to the strain, h(t)produced by a GW event, which in the transverse-traceless gauge is\ndescribed as [122]\nh(t) =F×(θ0, ϕ0, ψ)h×(t) +F+(θ0, ϕ0, ψ)h+(t). (B.0.1)\nThe gravitational wave tensor perturbation hµνis decomposed into two independent propagating\nmodes, h×(t)andh+(t), which in general are functions of cosmic time t. The corresponding\nantenna beam pattern functions F×andF+depend on the angular location of the source relative\nto the detector in polar coordinates, (θ0, ϕ0), and the polarisation angle, ψ. We adopt a random\nsampling method in the range [0,2π]forθ0and[0, π]for both ϕ0andψ. The factors F×,+are\ndefined as:\nF(1)\n×=√\n3\n2\u00141\n2(1 + cos2(θ)) cos(2 ϕ) cos(2 ψ) + cos( θ) sin(2 ϕ) cos(2 ψ)\u0015\n, (B.0.2)\nF(1)\n+=√\n3\n2\u00141\n2(1 + cos2(θ)) cos(2 ϕ) cos(2 ψ)−cos(θ) sin(2 ϕ) cos(2 ψ)\u0015\n. (B.0.3)\nThe superscript indicates the specific interferometer under consideration. For instance, LISA\ncomprises only two interferometers, hence F(3)= 0. The detectors are spatially distributed in an\nequilateral triangle formation, implying that the other two antenna pattern functions are located\nwith respect to F(1)\n×,+as\nF(2)\n×,+(θ, ϕ, ψ ) =F(1)\n×,+(θ, ϕ+2π\n3, ψ)and F(3)\n×,+=F(1)\n×,+(θ, ϕ+4π\n3, ψ). (B.0.4)\nLISA, sensitive to lower frequencies, can observe GWs from long-lasting merger events. The rela-\ntive positions of the interferometer and the event change over time, a fact that is accommodated\nusing the approach described in [493]. Since LISA can probe lower frequencies and equivalently\nlarger masses, it can detect GWs from long-lasting merger events. The relative positions of the\ninterferometer and the event change over time, a fact that is accommodated following the method\ndescribed in [493], assuming a timescale of the event of\nt=tc−5(8πf)−8/3M−5/3\nc. (B.0.5)\nHere, tcdenotes the merger time, tindicates the time at which LISA detects the merger, fis the\n263264\nfrequency of the GW, and Mcis the chirp mass. The location angles are updated accordingly:\nθ= cos−1\"\n1\n2cos(θ0)−√\n3\n2sin(θ0) cos\u00122πt\nT−ϕ0\u0013#\n, (B.0.6)\nϕ=2πt\nT−tan\"√\n3 cos( θ0) + sin( θ0) cos\u00002πt\nT−ϕ0\u0001\n2 sin( θ0) cos\u00002πt\nT−ϕ0\u0001#\n,\nwhich, in turn, are used to update the beam pattern functions. Here, we have specified the\nperiod, T, as the orbit around the Sun.\nWhile the modelling degeneracies prevent the distinction between the individual masses of the\nobjects, GW detectors are sensitive to the chirp mass, a collective mass quantity related to the\nfrequency evolution of the signal emitted before the merger, during the inspiral phase of the\nbinary [494], defined as\nMc= (1 + z)\u0012(m1m2)3\nm1+m2\u00131/5\n, (B.0.7)\nwhere (1 +z)is a conversion redshift factor from the physical to the observational chirp mass.\nThe Fourier transform of the strain h(t)based on the stationary phase approximation [495],\nwhich neglects changes in the orbital frequency averaged over one period during the inspiral,\nreads:\nH=Af−7/6eiΨ(f), (B.0.8)\nwhere Ψ(f)is the phase of the waveform. Note that when replacing Hinto Equation (B.0.12),\nthe exponential term will vanish, and the Ψ(f)factor may be discarded in this analysis. Ais\nthe Fourier amplitude of the waveform signal, given by\nA=M5/6\nc\ndGW\nL(z)π−2/3r\n5\n96×q\nF2\n+(1 + cos2(l))2+ 4F2\n×cos2(l), (B.0.9)\nwhere dGW\nL(z)is the luminosity distance from the merger, and lis the inclination angle of the\norbital angular momentum with respect to the line of sight, which we have sampled randomly\nbetween [0◦,20◦], as that is the maximum detection inclination range. We can see from Equa-\ntion (B.0.9) that measuring the amplitude of gravitational wave signals allows the derivation of\nestimates of the luminosity distances of the associated mergers.\nDesignedforlow-frequencydetection,aslowas fmin= 1×10−4Hz,LISAisparticularlypromising\nforprobingextrememassratioinspirals(EMRI)andbinarymassiveblackhole(BMBH)mergers.\nFor the simulations of LISA, two quantities are responsible for determining the upper bound\nfrequency: the structure of LISA itself and the last stable orbit of the merging system. LISA\ncan detect frequencies up to fmax=c(2πL)−1, where Lis the length of LISA’s interferometer\narm, taken to be 2.5Gm and cis the speed of light. Moreover, the total mass of an orbiting\nsystem is inversely proportional to its measured frequency, implying that even though massiveBMock Data Set Simulation 265\nmergers give rise to large detection amplitudes, the frequency will fall below fmin. Therefore if\nthe last stable orbit frequency, fLSO= (63/22πMobs)−1, with Mobsbeing the observed total mass,\nis lower than fmin, such an event is ignored. Otherwise, if it lies between fminandfmax, then\nfLSOdefines the new maximum frequency for that event.\nB.0.1 Simulated cosmology\nThe simulation of the GW catalogue requires the following cosmological parameters: the redshift\nat which the merger occurred, z, the value of the Hubble rate at that redshift, H(z), its comoving\nand luminosity distance, dc(z)anddGW\nL(z)respectively, as well as the cosmic time between the\nmergeranditsobservation, t. Forthispurpose, weusethepublicEinstein-Boltzmanncode CLASS\ncode1[95, 101, 102], which we have modified to accommodate a broad range of interacting dark\nenergy models. With this new patch, we generate a mock Universe adopting the flat ΛCDM\nas the fiducial model, based on the best-fit cosmological parameters of the Planck2018 data\nrelease [32]. These include the current value of the Hubble parameter, H0= 67.32km/s/Mpc,\nthe baryon density parameter, Ωbh2= 0.022383(with h=H0/100) and the density of cold\ndark matter, Ωch2= 0.12011. Furthermore, we are also concerned with the derived quantity\nΩm= Ω b+ Ω c, fixed to Ω0\nm= 0.3144for the fiducial Planckcase.\nArmed with the background cosmology, we simulate the merger events to establish the redshift-\nluminosity relation. First, we construct a redshift distribution of events weighted by a probability\ndistribution. The characteristics of these events, such as the chirp mass, are simulated based on a\nuniform distribution. Although each simulation run produces a unique data set, the overarching\nconclusions remain unaltered, as the fiducial parameters constrain them. Once the mergers have\nbeen modelled, we emulate the observational process, accounting for errors associated with each\nevent. Events are excluded if they yield a signal-to-noise ratio below a given threshold.\nB.0.2 Distribution of simulated merger events\nThe Einstein Telescoped is engineered to explore a range of frequencies, f, akin to those of\nLIGO, thereby probing merger events of nearby compact objects. These include binary neutron\nstars (BNS) with mass ranges of [1,2],[1,2]M⊙, and black hole neutron star binaries (BHNS)\nwith [3,10],[1,2]M⊙, respectively, with the [·,·]notation indicating the uniformly distributed\nmass ranges considered. According to the advanced LIGO proposal, the ratio of BHNS to BNS\nmergers is approximately 0.03[496]. The probability distribution for the redshift of these events\nfollows:\nP∝4πd2\nc(z)R(z)\n(1 +z)H(z), (B.0.10)\n1https://github.com/lesgourg/class_public266\nwith the comoving distance and the Hubble rate at various redshifts sourced from CLASS.R(z)\nrepresents the rate of binary mergers, which, at a linear approximation level, is [497]\nR=\n\n1 + 2 zifz <1,\n3\n4(5−z)if1≤z <5,\n0 otherwise .(B.0.11)\nIn contrast, LISA focuses on lower frequencies than other proposed 3G detectors, making it more\nsensitive to higher mass binary systems since f∝M−1. Therefore, we consider simulations of de-\ntected events from extreme mass ratio inspirals (EMRI) and binary massive black holes (BMBH)\nin the ranges [1−30],[104−108]M⊙[498] and [104−108],[104−108]M⊙[499], respectively. The\nnumber of detected BMBH to EMRI events is assumed to follow a 2 : 1ratio according to the\nmission’s proposal [500, 501].\nAlthough, in principle, LISA will also be able to probe mergers of binary intermediate-mass black\nholes (IMBH) and binary compact objects, we have excluded these from the simulations due to\nthe lack of definitive observational proof of IMBH. Furthermore, events from binary compact\nobjects are only expected to be observed at redshifts z≈3[502] irrelevant for our cosmological\naim since we are interested in the higher range of redshifts which LISA can reach.\nFocusingsolelyonBMBHevents,webasetheirredshiftprobabilitydistributionsonthehistogram\noutlined in the L6A2M5N2 mission specification [499], which explores three different mechanisms\nfor BMBH formation. Our study zeroes in on the light seed model, or the pop III model , which\nsuggests that BMBHs originate from the remnants of population III stars around a redshift\nrange of z= 15−20. In [499], two other scenarios for massive black hole creation were discussed,\nnamelydelayandno-delay scenarios. These alternative routes involve the gravitational collapse\nof gas at the centre of a galaxy in the same redshift range, z= 15−20, resulting in a black hole\nformed through a heavy seed mechanism. The critical difference between them is the timing -\nwith and without delay - of the black hole’s formation relative to the merger of its host galaxy.\nMore details about these scenarios are given in Refs. [503].\nIn our investigation, we provided mock data and obtained forecasts for both the delay and no\ndelay cases but found no significant improvement in constraining power from variations in these\nmodels compared to the pop III case. Consequently, we restrict our focus to the pop III model,\nas it proves sufficient to forecast the constraining power of LISA.\nB.0.3 Simulation of measurements and errors\nWe follow the methodologies of [122, 493, 495, 497] to model the errors associated with the GW\nstandard siren catalogue. An event is considered genuinely detected only if its combined signal-\nto-noise ratio (SNR), ρ, exceeds the particular threshold ρ >8. The SNR is calculated takingBMock Data Set Simulation 267\ninto account the specifications of the interferometer in use:\nρ2\n1,2,3= 4Zfmax\nfmindf|H(f)|2\nSh(f), (B.0.12)\nwhere the numbers label the interferometer, the definition of Hwas provided in Equation (B.0.8)\nandShis the noise power spectral density. This SNR weighting function reflects the unique\ncharacteristics of the instruments in use. For ET, in particular, Shis according to\nS(ET)\nh=S0 \nxp1+a1xp2+a21 +P6\nn=1bnxn\n1 +P4\nm=1cmxm!\n, (B.0.13)\nwhere, x=f/200Hz−1,S0= 1.449×10−52Hz,p1=−4.05,p2=−0.69,a1= 185 .62,a2=\n232.56,bn={31.18,−64.72,52.24,−42.16,10.17,11.53},cm={13.58,−36.46,18.56,27.43}, as-\nsuming a lower cutoff at f= 1Hz.\nOn the other hand, for LISA, Shrelies on three noise components: the instrumental (or short)\nnoise, Sinst, the noise from low-level acceleration, Sacc, and the confusion background noise, Sconf\n[503]:\nS(LISA )\nh=20\n34Sacc+Sinst+Sconf\nL2\u0014\n1 +\u0012fL\n0.81c\u0013\u0015\n, (B.0.14)\nwhere Sacc= 9×10−30/(2πf)4(1 + 10−4/f),Sinst= 2.22×10−23andSconf= 2.65×10−23.\nTherefore, combining Equation (B.0.8) with either Equation (B.0.13) or Equation (B.0.14) allows\nus to determine the total SNR contribution for ET and LISA respectively, as expressed by\nρtot=q\nρ2\n1+ρ2\n2+ρ2\n3. (B.0.15)\nThe instrumentally induced error in the luminosity distance is computed using the Fisher Matrix\napproach,\nσdGW,inst\nL≈\u001c∂H\n∂dGW\nL,∂H\n∂dGW\nL\u001d−1\n2\n, (B.0.16)\nfollowing [493]. This expression simplifies due to the proportionality H ∝ 1/dGW\nL:\nσdGW,inst\nL≈2dGW\nL\nρ, (B.0.17)\nwhere the factor of 2accounts for the symmetry in the inclination angle, which ranges from\n−20◦to20◦. There is a correlation between the gravitational wave luminosity distance and the\ninclination of the source to the observer. For a single detector, dGW\nL(z)andlare completely\ndegenerate with each other and the antenna patterns F×,+. However, this degeneracy can be\nbroken with multiple detectors and sensitivity to both polarisations. The maximum effect of268\nthis degeneracy on the signal-to-noise ratio is a factor of two, between the source being face-on\n(inclination l= 0) and edge-on ( l=π/2).\nGravitational lensing introduces another layer of error, represented by\nσdGW,len\nL=dGW\nL\n2×0.066h\n4(1−(1 +z)1/4)i1.8\n, (B.0.18)\nand it is reduced by half to account for the event’s merger and ringdown.\nBeing space-based, LISA is also subject to a peculiar velocity-associated error of the GW sources\n[504]:\nσdGW,pec\nL=dGW\nLp\n⟨v2⟩\nc\u0014\n1 +c(1 +z)\nHdGW\nL\u0015\n, (B.0.19)\nwith an estimate of the peculiar velocity of the host galaxy with respect to the Hubble flow ofp\n⟨v2⟩= 500kms−1.\nThe total error in luminosity distance combines all these individual errors in Equation (B.0.17)-\nEquation (B.0.19):\nσdGW\nL=q\n(σdGW,inst\nL)2+ (σdGW,len\nL)2+ (σdGW,pec\nL)2. (B.0.20)\nThe simulation allows us to interpolate any number of events over a continuous redshift distri-\nbution in the range 0< z≲5for ET and 0< z≲10for LISA. However, the number of mergers\ndetected by the ET will depend on factors such as operational costs and the complementary with\nother experiments [122]. ET is anticipated to document over 104mergers per year. However,\ndue to the scarcity of EM counterpart signals, the predicted number of detectable mergers with\nan actual EM counterpart over 10years is brought down to approximately 200[505]. According\nto [499], LISA is expected to detect around 56 events over a 10-year mission.\nTo account for uncertainties in the luminosity distance of each merger, we employ a Gaussian\ndistribution centred around the background cosmology, with a standard deviation determined by\nthe calculated errors, σdGW\nL. This introduces an artificial randomness factor around each merger,\nleading to a higher deviation from ΛCDM for LISA than ET, essentially because LISA covers\na more extensive redshift range with corresponding larger errors, resulting in a broader data\nspread. This is visually illustrated in Figure B.1.BMock Data Set Simulation 269\n0 1 2 3 4 5 6 7 8\nz02468dGW\nL [104× Mpc]\nCDM Fiducial\nET data\nLISA data\nFigure B.1 : Luminosity distance as a function of redshift for the mock data points from ET\n(red) and LISA (blue). The data is generated from the fiducial model, ΛCDM, shown by the\nblack dotted line. The error bars represent the 1σtotal errors in the luminosity distance.CString Theory and the DBI Action\nThis Appendix provides a brief overview of string theory applied to cosmology and the main\nconcepts needed to understand the formulation of the Dark D-Brane Model studied in Chapter 9.\nFor more details on this and other cosmological applications of string theory, we refer the reader\nto the review in [70].\nMany cosmological models proposed in the literature have motivations based on string theory.\nInstead of treating particles as point-like entities, string theory posits that the basic building\nblocks of the Universe are one-dimensional stringsthat vibrate at different frequencies. These\nvibrations give rise to various particle types and their properties, such as mass and charge. String\ntheory also suggests the existence of more than the conventional four dimensions of GR, poten-\ntially providing a unified description of all four fundamental forces of physics in a single, coherent\npicture. The process of starting from a higher-dimensional framework and then obtaining a 4-\ndimensional effective theory is known as compactification and has been studied in great detail\n[70]. The extra dimensions are typically compactified ,i.e.mathematically folded up so small\nthat they are effectively invisible at the scales we can currently probe, which is why we do not\nreadily observe them. This has profound implications for cosmology, particle physics, and our\nunderstanding of the fundamental nature of reality, although it is yet to be empirically verified.\nThese strings can be attached to branes, mathematical objects that generalise the notion of\nphysical membranes to ndimensions, which can propagate in space and time and can be char-\nacterised by physical properties, such as tension and charge. D-branes are an important class of\nbranes that arise when one considers open (two-endpoints) strings whose endpoints must lie on\na D-brane (with the D standing for Dirichlet boundary conditions). Therefore, D-branes can be\nregarded as surfaces (which could be 0-dimensional points, 1-dimensional strings, 2-dimensional\nmembranes, and so on) to which open strings can attach their endpoints. While strings are\n1-dimensional objects, D-branes can exist in various dimensions and are essential in describing\nnon-gravitational forces like electromagnetism and nuclear forces in this context.\nD-branes play a crucial role in various string theory dualities and phenomenology. A spacetime-\nfilling D-brane would be an object that fills all the dimensions of the spacetime in which it is\nembedded. The throatrefers to a region of this extra-dimensional space shaped like a funnel\nor a tube. The term warpedrefers to the specific type of geometry that this throat can have,\noften characterised by a metric tensor that varies along the length of the throat. In this setup,\nthe D-branes can move along the throat, often essential for various phenomena such as brane\ninflation. Therefore, we will be concerned with D-branes that are dynamically embedded within\na region of that spacetime characterised by a warped geometry. Lastly, a D3-brane is a specific\ntype of D-brane that is a 3-dimensional hypersurface embedded in a higher-dimensional space-\ntime. They provide a setting for the famous Anti-de-Sitter/Conformal Field Theory (AdS/CFT)\ncorrespondence [506], a form of the holographic principle. This duality allows physicists to relate\n270CString Theory and the DBI Action 271\na theory of gravity in a higher-dimensional bulkspacetime (described by the geometry around\nthe D3-brane) to a conformal field theory on the brane itself.\nThis leads to the idea of brane cosmological models, where the visible, four-dimensional Universe\nis restricted to a D-brane (or a stack of branes) contained inside a compact higher-dimensional\nspacetime, called the bulkor the hyperspace. The standard model particles and their interactions\nlive on the visible branes . There can also be hidden dark sector D-branes , from which the dark\nsector stems, and which can only interact with the visible sector branes gravitationally, with\ngravity being the force that propagates throughout the bulk. Interactions with the bulk, and\npossibly with other branes, may leave observational imprints on our visible brane. Hidden sector\nD-branes could be relevant for various theoretical constructions or even new forces and fields\nthat have not yet been observed. They can interact with the visible sector via gravitational\ninteractions or other weaker-than-Planckian couplings, making them candidates for explaining\nphenomena not accounted for by the standard model of particle physics. For example, if dark\nmatter arises from the hidden sector, it would interact very weakly with the particles of the\nvisible sector, making it difficult to detect directly.\nThe Dirac-Born-Infeld (DBI) action is a type of action functional in theoretical physics that\ndescribes D-branes’ dynamics, including their interactions with other fields. Initially, the Born-\nInfeld action was formulated to address some of the infinities that arose in classical electrody-\nnamics and the context of string theory, and it has been adapted to describe the dynamics of\nD-branes. The DBI action is non-linear and includes higher-order terms in the field strengths.\nUnlike the standard actions in quantum field theory, which typically include quadratic terms in\nthe field or its derivatives, the DBI action has a square root of the determinant of the induced\nmetric on the brane, making it a non-canonical action. This results in a more complex set of\nequations of motion that capture a more comprehensive range of phenomena, including some\nthat simpler models do not cover, encapsulated in the following action:\nS=Z\nd4x\"\nT(ϕ) \n1−s\n1 +(∇ϕ)2\nT(ϕ)!\n−V(ϕ)#\n, (C.0.1)\nwhere ϕrepresents the D-brane’s position modulus and T(ϕ)andV(ϕ)are its tension and\npotential, respectively. The functional form of T(ϕ)andV(ϕ)depend on the dimensionality of\nthe brane and determine the phenomenology of the scenario [507–509]. A phase of accelerated\nexpansion can be achieved by the non-canonical form of the kinetic terms, given constraints on\nthespeedofthemovingD3-brane. However, constructingDBIinflationwithinaconsistentstring\ncompactification is challenging [510]. Nonetheless, this scenario serves as an essential example\nof an inflationary mechanism relying on the symmetries of an ultraviolet theory, motivating\nfurther investigations of its phenomenology and leading to the observational search for specific\ncosmological signatures.\nFor D-branes moving in the internal compact space, the scalar field(s) associated with the open272\nstring representing, e.g. the radial position of a D-brane along a warped throat, couples disfor-\nmally to matter living on such a brane [427]. Indeed, the induced metric on the brane takes\nthe form presented in Equation (5.3.3). Since a critical constraint on quintessence candidates is\nalways their interactions with standard model matter resulting in fifth forces, a natural place to\nlook for string theory quintessence candidates is from hidden sector D-branes, as they may be\ncoupled to visible sector D-branes with weaker-than-Planckian couplings. The simplest case is of\na D3-brane moving in a 5-dimensional AdS space, describing the mid-region of a warped throat,\nT(ϕ)∝ϕ4.\nThis opens up a new avenue for exploring the dark Universe, providing a fresh perspective that\ngoes beyond the limitations of the standard cosmological model. These dark D-branes can give\nrise to dark matter and dark energy candidates through their vibrations and interactions with\nthe visible sectors.\nFor the late-time acceleration, quintessence candidates need to have highly suppressed interac-\ntions with standard matter (photons and baryons) to avoid any undetected fifth forces, highly\nconstrained by local measurements in the Solar System [221–223]. Further observational sup-\nport for a non-universal or highly suppressed coupling to standard matter comes from the recent\nconstraints on the speed of gravitational waves inferred from observations of the optical coun-\nterpart of a binary neutron star merger [125]. The cosmological viability of the DBI action in\nthe sense of allowing for quintessence attractor tracker solutions for the scalar field representing\nthe brane’s position has been the focus of Refs. [415, 417]. In this scenario, the dynamics of the\nfield are constrained by the causality of the gravity side of the AdS/CFT correspondence [511].\nThe DBI action effectively describes a probe D3-brane moving in the radial direction of the AdS 5\nspacetime, with a string-inspired phenomenological action given by\nS=Z\nd4x\u0014\nh(ϕ)−1\u0012\n1−q\n1 +h(ϕ)∂µϕ∂µϕ\u0013\n−V(ϕ)\u0015\n. (C.0.2)\nThe DBI field ϕhas mass dimensions and stands as the dark energy component. The brane’s\nwarp facto rh(ϕ)≡T(ϕ)−1encodes the geometry of the higher-dimensional space, in terms of\nthe DBI field and V(ϕ)represents the potential term arising from possible interactions with the\nbulk and other brane stacks.\nThis cosmological context allows for accelerated expansion even when the Lorentz-like factor\nin the action in Equation (C.0.2), γ= 1/q\n1−h(ϕ)˙ϕ2, is much larger than unity, meaning\nthat h(ϕ)˙ϕ2≈1. The γfactor measures the relativistic motion of the D3-brane, and when\nit approaches unity, the canonical kinetic term is recovered, and ϕbehaves like a standard\nquintessence field. On the other hand, when γ→ ∞, the purely relativistic limit is approached.\nThis factor acts as a speed limit, allowing the field to evolve relatively slowly, even in steep\npotentials. The application of DBI theories to dark energy has been seminally discussed in\n[415, 416].CString Theory and the DBI Action 273\nIn Chapter 9, we study a DBI- essencescenario featuring a coupling to dark matter, which\nnaturally arises in a stacked D-brane system, where dark energy emerges from open strings\nrepresenting the radial position of a D-brane along a warped throat, and dark matter arises from\nopen strings representing matter on the same hidden sector brane. The resulting theory, referred\nto as the Dark D-Brane Model , was constructed in [407], in which the action for the dark energy\nscalar takes the form of Equation (C.0.2), while the action for the dark matter describes N\nparticles on the moving D3-brane at low energies. Further details are provided in Chapter 9, in\nthe particular case of an AdS warp factor, h(ϕ)∝ϕ−4, together with a mass term for the scalar\npotential, V(ϕ)∝ϕ2.Bibliography\n[1] E.M. Teixeira, B.J. Barros, V.M.C. Ferreira, and N. Frusciante. 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Phys. , 2:231–252, 1998."    },    {        "title": "2401.13828v1.Asymptotic_solutions_for_self_similar_fault_slip_induced_by_fluid_injection_at_constant_rate.pdf",        "content": "arXiv:2401.13828v1  [physics.flu-dyn]  24 Jan 2024Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 1 of 17\n24 Jan 2024\n1 Abstract\nWe examine the circular, self-similar expansion of frictio nal rupture due to fluid injected at a\nconstant rate. In a prior problem [ Viesca, 2021], the author examined ruptureof a planar frictional\nfault driven by line source of fluid injected at constant pres sure. Here, fluid injection occurs at\nconstant volumetric rate at a point on the fault and fluid migr ation occurs within a thin, relatively\npermeable layer containing and parallel to the fault plane. For the particular case whenthe Poisson\nratioν= 0, self-similarity of the fluid pressure profile implies tha t fault slip will also evolve in an\naxisymmetric, self-similar manner, reducing the three-di mensional problem for the spatiotemporal\nevolution of fault slip to a single self-similar dimension. The rupture radius grows as λ/radicalbig4αhyt,\nwheretis time since the start of injection, αhyis the hydraulic diffusivity of the pore fluid pressure,\nandλis a prefactor determined by a single parameter, T, which depends on the pre-injection stress\nstate and injection conditions. The prefactor has the range 0< λ <∞, where the lower and upper\nlimits of λcorrespond to, respectively marginal pressurization of th e fault and critically stressed\nconditions, in which the fault-resolved shear stress is clo se to the pre-injection fault strength.\nIn both limits of λ, we derive asymptotic solutions for the slip by perturbatio n expansion, to\narbitrary order. For the critically stressed limit, the sol ution for slip contains an interior boundary\nlayer and an outer solution. We perform a matched asymptotic expansion to provide a composite\nsolution with uniform convergence over the entire rupture. The leading order of these solutions\nwas recently used by Saez et al. [2022] to verify three-dimensional boundary-element solutions i n\nthe limit ν= 0.\n2 Problem formulation\nFor a circular shear fracture of radius R(t) undergoing relative slip δ(r,t) in a single direction,\nthe shear traction on the crack faces τ(r,t), opposing the direction of slip, is given by [ Salamon\nand Dundurs , 1971, 1977; Viesca and Bhattacharya , 2019]\nτ(r,t) =τb+µ\n2π/integraldisplayR(t)\n0/parenleftbiggE[k(r/s)]\ns−r+K[k(r/s)]\nr+s/parenrightbigg∂δ(s,t)\n∂sds (1)\nwherek(u) = 2√u/(1+u), andEandKare the complete elliptical integrals of the second and first\nkind. The background shear stress resolved on the plane of th e fracture in the absence of slip, τbis\nhere taken to be uniform. The relation (1) presumes that the P oisson ratio ν= 0. This expression\nis equivalent to that relating relative crack opening and cr ack-face tractions for a circular mode-I\ncrack. In that case, this expression holdsfor any value of th e Poisson ratio, providedthat µabove is\nreplaced by µ/(1−ν). The inversion of this equation for the relative displacem entδwas done in the\ncontext of mode-I equation [ Sneddon, 1951 ], which we adapt here for our particular mixed-mode\nII/III case with ν= 0\nδ(r,t) =4\nπR(t)\nµ/integraldisplay1\nρη/radicalbig\nη2−ρ2/integraldisplay1\n0x∆τ(xηR,t)√\n1−x2dxdη (2)\nwhere\n∆τ(r,t) =τb−τ(r,t) (3)\nandρ=r/R(t).Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 2 of 17\n24 Jan 2024\nWe may rearrange this inversion to isolate the contribution of a crack-tip singularity to the slip\ndistribution. To do so, we first denote the innermost integra l\nf(η) =/integraldisplay1\n0x∆τ(xηR,t)√\n1−x2dx\nwhere for brevity and without loss of generality, we have sup pressed explicit mention of the time\ndependence of f. The inversion (2) reduces to\nδ(r,t) =4\nπR(t)\nµ/integraldisplay1\nρη/radicalbig\nη2−ρ2f(η)dη (4)\nIntegration of (4) by parts yields\nδ(r,t)\n4R(t)/(πµ)=f(1)/radicalbig\n1−ρ2−/integraldisplay1\nρ/radicalbig\nη2−ρ2df\ndηdη (5)\nwhere we identify that\nf(1) =/integraldisplay1\n0x∆τ(xR,t)√\n1−x2dx (6)\nis a stress intensity factor as it is a pre-factor for the rela tive-displacement field behind the crack\ntip,δ∼f(1)√1−rcorresponding to a singular stress field ahead of the crack ti pτ∼f(1)/√r−1.\nThis stress intensity factor is identically zero for proble ms in which the fault has finite strength.\nWe now look to consider a particular form of ∆ τcorresponding to the injection of fluid at a\nconstant volumetric rate Qinto a thin fault zone whose friction coefficient remains at a c onstant\nvalue,f. The strength of the fault is given by\nτs=f(σ−p(r,t)) (7)\nwherepis the local fluid pressure at a point on the fault, which is pre sumed to have an initial,\nuniform value poplus a perturbation due to injection, given by the solution t o the axisymmetric\ndiffusion equation governing fluid pressure in the plane of the thin fault zone\n∂p\n∂t=αhy1\nr∂\n∂r/parenleftbigg\nr∂p\n∂r/parenrightbigg\nwhereαhy=k/(ηfβ) is the hydraulic diffusivity kis the fault-zone permeability, ηfis the fluid\nviscosity and βis the composite compressibility of the porous matrix and flu id. The corresponding\nsolution of interest is that subject to the injection condit ion\n−2πrk∂p\n∂r=Q\nh\nand is given by\np(r,t) =po+∆pE1[r2/(αt)] (8)\nwhereE1is the elliptic integral, α= 4αhy, and ∆p=Q/(4πkh). The thinness of the fault zone is\nrelevant when considering that the fault zone as a permeable , poroelastic conduit confined within\nan elastic medium, which may have a stiffness dissimilar to the fault zone. Here we consider the\ncase where the fault-zone thickness his much smaller than the diffusive lengthscale√\nαt. Under\nthis condition, there is no change in the fault-normal stres s over distances ∼√\nαtdue to localized,Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 3 of 17\n24 Jan 2024\nconstrained swelling of the fault zone around the injection source, and fluid pressure is decoupled\nfrom volumetric straining of the fault zone and follows the a utonomous diffusion equation [ Marck\net al., 2015 consider the injection scenario here; see also Jacquey and Viesca , 2023, for similar\nresults under plane-strain conditions].\nSelf-similarity of the pore-fluid pressure suggests self-s imilarity of the rupture front and slip\nprofile. In this case, the rupture radius may be written as\nR(t) =λ√\nαt (9)\nwhereλis a pre-factor to be determined. The non-singular stress co ndition that f(1) = 0 provides\na condition to solve for the pre-factor λin terms of other problem parameters. This condition is\nimposed as the finite-frictional strength of the fault precl udes a stress singularity. To impose this\nnon-singular condition, we note that the frictional streng th condition requires fault shear stress\nto equal fault shear strength, τ(r,t) =τs(r,t), when and where sliding occurs. Combining this\nequation with (3), and (6)–9), and rearranging,\nT=/integraldisplay1\n0xE1/bracketleftbig\n(λx)2/bracketrightbig\n√\n1−x2dx (10)\nwhere the dimensionless parameter\nT=σ′\n∆p/parenleftbigg\n1−τb\nτp/parenrightbigg\n(11)\nis a measure of distance of the fault stress state from critic ally stressed conditions, τb→τp, in\nwhichT→0 and the pre-injection strength of the fault is τp=fσ′, and also where σ′=σ−pois\nthe pre-injection effective fault normal stress. When the ini tial stress τbis a finite distance away\nfromτp, but the injection-induced pressure coefficient ∆ pis small compared to σ′, the fault is\nmarginally pressurized and T→ ∞.\nWe may anticipate the behavior of λin these two end-member regimes: i.e., we expect that\nthe rupture will progress quickly relatively to pore-fluid p ressure diffusion ( λ≫1) when the fault\nis critically stressed such that and relatively slowly ( λ≪1) when marginally pressurized. We can\nderive the relation between Tandλin these limits by asymptotically expanding the integral (1 2).\nFor large values of λ, we apply the change of variable u=xλ\nT=1\nλ2/integraldisplayλ\n0uE1/bracketleftbig\nu2/bracketrightbig\n/radicalbig\n1−(u/λ)2du (12)\nfollowed by the Taylor expansion\n1/radicalbig\n1−(u/λ)2≈1+(u/λ)2\n2+...+1√π[(n−1)/2]!\n(n/2)!(u/λ)n(13)\nfor even n, and\nT≈∞/summationdisplay\nn=0,2,...1\nλ2+n1√π[(n−1)/2]!\n(n/2)!/integraldisplayλ\n0un+1E1/parenleftbig\nu2/parenrightbig\ndu (14)\nPassing the limit of the integral to infinity without loss of g enerality of the power-law expansion,\nowing to the exponentially fast decay of the integrand, we ma y first evaluate the integral by parts,Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 4 of 17\n24 Jan 2024\nsecond by subsequently using another change of variable w=u2, and third by using the definition\nof the gamma function Γ( z) =/integraltext∞\n0sz−1e−sds\n/integraldisplay∞\n0un+1E1/parenleftbig\nu2/parenrightbig\ndu=2\nn+2/integraldisplay∞\n0un+1e−u2du\n=1\nn+2/integraldisplay∞\n0wn/2e−wdw=Γ(n/2+1)\nn+2=(n/2)!\nn+2(15)\nCombining (14) with (15), and changing the index of summatio nn= 2m, we find the expansion\nofTfor large λin the critically stressed limit\nT≈∞/summationdisplay\nm=01√π[(2m−1)/2]!\n2m+21\nλ2+2m=1\n2λ2+1\n8λ4+1\n8λ6+... (16)\nFor small values of λ, we may evaluate (12) using the asymptotic expansion of the e xponential\nintegral for small values of its argument [e.g., Bender and Orszag , 1999]\nE1/bracketleftbig\n(λx)2/bracketrightbig\n≈ −γ−2ln(λx)−∞/summationdisplay\nn=1(−1)n(λx)2n\nn·n!(17)\nwhereγ= 0.57721...is the Euler-Maraschoni constant, such that\nT≈ −γ/integraldisplay1\n0x√\n1−x2dx−2/integraldisplay1\n0ln(λx)√\n1−x2dx−∞/summationdisplay\nn=1(−1)nλ2n\nn·n!/integraldisplay1\n0x2n+1\n√\n1−x2dx\n≈ −γ+2−2ln(2λ)−∞/summationdisplay\nn=1(−1)nλ2n\nn·n!/parenleftbigg\nn!√π/2\n(n+1/2)!/parenrightbigg\n≈ −γ+2−2ln(2λ)−∞/summationdisplay\nn=1(−1)n√π/2\nn·(n+1/2)!λ2n\n≈ −γ+2−2ln(2λ)+2\n3λ2−2\n15λ4+8\n315λ6−...(18)\nin which the first integral is readily evaluated, the second l ikewise but by parts, and the evaluation\nof the last integral is found in Appendix C, where it occurs fo r another problem and we identify it\nasI2n+1.\nWe are interested in deriving an asymptotic expansion for sl ip in the limits of large and small\nλ. In the marginally pressurized limit, λ≪1, the rupture radius lags far behind the fluid diffusive\nlengthscale√\nαt. In the critically stressed limit, where, λ≫1, the rupture radius is much larger\nthan the diffusive lengthscale, R(t)≫√\nαt, and with this scale separation, there will be an outer\nsolution valid over distances ∼Rand an inner solution relevant over distances ∼√\nαtnearr= 0.\nThis inner solution is in-effect, an interior boundary layer. In what follows, we present in section\n3 the asymptotic expansion of the inner and outer solutions f or the critically stressed case, as well\nas their asymptotic matching and the construction of a compo site solution; and we show in section\n4 the asymptotic expansion of slip in the marginally pressur ized limit in section. Details of the\nsolution procedure will be relegated to appendices A and B fo r the inner and outer solutions, and\nappendix C for the marginally pressurized solution.Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 5 of 17\n24 Jan 2024\n3 Critically stressed faults: inner and outer solutions\nFor small values of the stress parameter T(largeλ), the critically stressed fault is highly\nsensitive to the injection of fluids and rupture outpaces the migration of injected fluids. We\nperform a perturbation expansion of the self-similar slip p rofile in terms of the small parameter\n1/λ. Below, we provide the behavior of slip on an interior bounda ry layer about the origin in\nsection 3.1. This solution captures behavior near the origi n and will asymptotically decay with\naway from it. Subsequently, in section 3.2, we provide an out er solution that captures behavior at\ndistances greater than the diffusive length away from the orig in and up to the rupture front. The\nouter solution’s approximation breaks down at the origin, w here that solution is singular. We will\nfind that the manner of the asymptotic decay of the inner solut ion matches the singularity of the\nouter solution and that the two overlap over distances inter mediate to the diffusive lengthscale and\nrupture front position. In section 3.3, we construct a compo site solution as the sum of the inner\nand outer solutions, less the common overlap.\n3.1 Inner solution\nTo examine the variation of slip over distances of the order o f the diffusive lengthscale√\nαt\naway fromthe origin, we usethe scaled radial position s=r/√\nαt, which has the domain 0 ≤s≤λ.\nWhile the details of the solution are deferred to appendix A, the final result is that the asymptotic\nexpansion for slip in the large λlimit,δin, is given by\nδin(s,t)√\nαtf∆p/µ=δ0(s,t)+4\nπ2∞/summationdisplay\nj=01\nλ2j+11\n2j+1/bracketleftBiggj/summationdisplay\nk=0s2kΓ(k−1/2)\nΓ(k+1)Γ(3/2+j−k)\n2(j−k)+1/bracketrightBigg\nwhere the leading, zeroth-order term is\nδ0(s,t) = 2√πe−s2/2/bracketleftbig\n(1+s2)I0(s2/2)+s2I1(s2/2)/bracketrightbig\n−4s\nandI0andI1are modified Bessel functions of the first kind. The subsequen t, higher-order terms\nare polynomial and, to fifth-order in the small parameter 1 /λ, the expansion is\nδin(s,t)√\nαtf∆p/µ=δ0(s,t)−1\nλ4\nπ−1\nλ32\n3π(1−s2)−1\nλ51\n5π/parenleftbig\n3−s2−s4/2/parenrightbig\n+O(λ−7) (19)\n3.2 Outer solution\nWe now examine behavior over radial distances comparable to the current rupture radius R(t),\nrelatively much larger than the diffusive lengthscale√\nαt. We rescale radial distance as ρ=r/R(t),\nand find the asymptotic expansion of the outer solution for sl ip to be\nδout(ρ,t)\nRf∆p/µ=4\nπ2∞/summationdisplay\nn=01\nλ2n+21\nρ2n+1Γ(n+3/2)\n2n+1Γ(n+1/2)\nΓ(n+2)\n·/bracketleftBigg/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+2√π(1−ρ2)3/2\nρn/summationdisplay\nk=1ρ2k\n2k+2Γ(k+2)\nΓ(k+1/2)/bracketrightBigg(20)Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 6 of 17\n24 Jan 2024\nwhere we have deferred discussion of the solution details to appendix B. This expansion, to sixth-\norder in the small parameter 1 /λwhen scaled as such, reduces to\nδout(ρ,t)\nRf∆p/µ=1\nλ22\nπ/parenleftbiggacos(ρ)\nρ−/radicalbig\n1−ρ2/parenrightbigg\n+1\nλ41\n4π/bracketleftbiggacos(ρ)\nρ3−/radicalbig\n1−ρ2/parenleftbigg\n2−1\nρ2/parenrightbigg/bracketrightbigg\n+1\nλ63\n16π/bracketleftbiggacos(ρ)\nρ5−/radicalbig\n1−ρ2/parenleftbigg8\n3−2\n31\nρ2−1\nρ4/parenrightbigg/bracketrightbigg\n+O(λ−8)\n3.3 Matched asymptotic expansion and composite solution\nThe inner and outer expansions can be used to construct a comp osite asymptotic solution,\nwhich will provide uniform convergence over the entire doma in [e.g.,Hinch, 1991]. The composite\nsolution is given by\nδcomp(ρ,t)√\nαtf∆p/µ=δin(λρ,t)+δout(ρ,t)−δoverlap(ρ,t)√\nαtf∆p/µ(21)\nwhereδoverlapis the common asymptotic behavior of the inner and outer solu tions in an interme-\ndiate region corresponding to the outer region ( s→ ∞) of the inner solution, and the inner region\n(ρ→0) of the outer solution. This overlap is subtracted to avoid double-counting contributions in\nthis intermediate region when summing the inner and outer so lutions.\nThe expression for the inner solution is given as before,\nδin(s,t)√\nαtf∆p/µ= 2√πe−s2/2/bracketleftbig\n(1+s2)I0(s2/2)+s2I1(s2/2)/bracketrightbig\n−4s\n−4\nπλ−2\n3πλ3/parenleftbig\n1−s2/parenrightbig\n−1\n5πλ5/parenleftbigg\n3−s2−s4\n2/parenrightbigg\n+O(λ−7)\nRegarding the outer solution, we must carefully note that th e above scaling for the composite and\ninner solutions do not match that of the prior expression of t he outer solution (20): their scalings\ndiffer by a factor of R(t)/√\nαt=λ. Rescaling the outer solution in the manner of (21),\nδout(ρ,t)√\nαtf∆p/µ=1\nλ2\nπ/parenleftbiggacos(ρ)\nρ−/radicalbig\n1−ρ2/parenrightbigg\n+1\nλ31\n4π/bracketleftbiggacos(ρ)\nρ3−/radicalbig\n1−ρ2/parenleftbigg\n2−1\nρ2/parenrightbigg/bracketrightbigg\n+1\nλ53\n16π/bracketleftbiggacos(ρ)\nρ5−/radicalbig\n1−ρ2/parenleftbigg8\n3−2\n31\nρ2−1\nρ4/parenrightbigg/bracketrightbigg\n+O(λ−8)\nwe may now evaluate the so-called overlap of the inner and out er solutions, here both expanded to\nfifth-order in 1 /λwhen scaled as such.\nThe overlap of these two solutions, δoverlap, is determined by comparing the outer behavior of\ninner solution, i.e., the asymptotic behavior of δin(s→ ∞,t) with the inner behavior of the outerRobert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 7 of 17\n24 Jan 2024\nsolution, i.e., the asymptotic behavior of δout(ρ→0,t). The latter can be determined to be\nδout(ρ→0,t)√\nαtf∆p/µ=1\nλ/parenleftbigg1\nρ−4\nπ+2\n3ρ2\nπ+1\n10πρ4+O(ρ6)/parenrightbigg\n+1\nλ3/parenleftbigg1\n8ρ3−2\n3π+ρ2\n5π+ρ4\n28π+O(ρ6)/parenrightbigg\n+1\nλ5/parenleftbigg3\n32ρ5−3\n5π+3ρ2\n14π+ρ4\n24π+O(ρ6)/parenrightbigg\n+O(λ−7)\nwhere we again note here that the scaling of δouton the left hand side is different from the scaling\nof slip in (20), in which R(t) =λ√\nαtwas used in place of√\nαt. In the equation above, we scale\nslip with√\nαtso that we may correctly compare this inner behavior with the outer behavior of the\ninner solution, which is\nδin(s→ ∞,t)√\nαtf∆p/µ=/parenleftbigg1\ns+1\n8s3+3\n32s5+O(s−7)/parenrightbigg\n−1\nλ/parenleftbigg4\nπ/parenrightbigg\n−1\nλ3/parenleftbigg2\n3π(1−s2)/parenrightbigg\n−1\nλ5/parenleftbigg1\n5π(3−s2−s4/2)/parenrightbigg\nRecognizing that s=ρλ, we can identify the common, overlapping terms for the two ex pansions,\nwhich we color in red, blue, and green to facilitate comparis ons. This overlap is clearly\nδoverlap(ρ,t)√\nαtf∆p/µ=1\nλ/parenleftbigg1\nρ−4\nπ/parenrightbigg\n+1\nλ3/bracketleftbigg1\n8ρ3−2\n3π/parenleftbig\n1−(ρλ2)/parenrightbig/bracketrightbigg\n+1\nλ5/bracketleftbigg3\n32ρ5−1\n5π/parenleftbigg\n3−(ρλ)2−(ρλ)4\n2/parenrightbigg/bracketrightbigg\n+O(λ−7)(22)\nFor example, the composite solution to fifth order in the smal l parameter 1 /λis\nδcomp(ρ,t)√\nαtf∆p/µ= 2√πe−(λρ)2/2/braceleftbigg\n[1+(λρ)2]I0/bracketleftbigg(λρ)2\n2/bracketrightbigg\n+(λρ)2I1/bracketleftbigg(λρ)2\n2/bracketrightbigg/bracerightbigg\n−4λρ\n+1\nλ2\nπ/parenleftbiggacos(ρ)\nρ−/radicalbig\n1−ρ2/parenrightbigg\n−1\nλρ\n+1\nλ31\n4π/bracketleftbiggacos(ρ)\nρ3−/radicalbig\n1−ρ2/parenleftbigg\n2−1\nρ2/parenrightbigg/bracketrightbigg\n−1\n(2λρ)3\n+1\nλ53\n16π/bracketleftbiggacos(ρ)\nρ5−/radicalbig\n1−ρ2/parenleftbigg8\n3−2\n31\nρ2−1\nρ4/parenrightbigg/bracketrightbigg\n−3\n(2λρ)5(23)\nwhere the first two rows are the composite to first order and the first three are that to third order.\n4 Fault slip under marginally pressurized conditions\nIn the limit of large T(smallλ), the self-similar fluid pressureprofile stretches faster t han crack\ngrowth: rupture lags the injected fluid. In this case, we may p erform a perturbation expansion for\nthe slip profile in powers of the small parameter λ. The details of the solution may be found in\nappendix C, but the perturbation to arbitrary order is given by\nδmp(ρ,t)\nR(t)f∆p/µ=8\nπ/parenleftBig/radicalbig\n1−ρ2−ρacos(ρ)/parenrightBig\n+2/parenleftbig\n1−ρ2/parenrightbig3/2∞/summationdisplay\nn=1λ2n\n2n+11\nΓ(n+3/2)n/summationdisplay\nk=1Γ(k)\nΓ(k+1/2)ρ2k−2Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 8 of 17\n24 Jan 2024\nTo fourth order in λ, the asymptotic solution is\nδmp(ρ,t)\nR(t)f∆p/µ=8\nπ/parenleftBig/radicalbig\n1−ρ2−ρacos(ρ)/parenrightBig\n+λ216\n9π(1−ρ2)3/2+λ432\n225π(1−ρ2)3/2(3+2ρ2)+O(λ6)\nReferences\nBhattacharya, P., and R. C. Viesca (2019) Fluid-induced ase ismic fault slip outpaces pore-fluid\nmigration, Science, 364, 464–468, doi:10.1126/science.aaw7354\nBender, C. M. and S. A. Orszag (1999) Advanced Mathematical M ethods for Scientists and Engi-\nneers,Springer-Verlag , doi:10.1007/978-1-4757-3069-2\nHinch, E.J.(1991), PerturbationMethods, Cambridge University Press , doi:10.1017/CBO9781139172189\nJacquey, A. B. and R. C. Viesca (2023) Nucleation and arrest o f fluid-induced aseismic slip, Geo-\nphys. Res. Lett. , 50, e2022GL101228, doi:10.1029/2022GL101228\nMarck, J., A. A. Savitski, and E. Detournay (2015), Line sour ce in a poroelasic layer bounded by\nan elastic space, Int. J. Numer. Anal. Meth. Geomech , 39, 1484–1505, doi:10.1002/nag.2405\nSalamon and Dundurs (1971) Elastic fields of a dislocation lo op in a two-phase material, J. Elas-\nticity, 1(2), 153–164, doi:10.1007/BF00046466\nSalamon and Dundurs (1977) A circular glide dislocation loo p in a two-phase material, J. Phys. C\nSolid State , 10, 497–507, doi:10.1088/0022-3719/10/4/007\nS´ aez, A., B. Lecampion, P. Bhattacharya, R. C. Viesca (2022 ) Three-dimensional fluid-drive stable\nfrictional ruptures, J. Mech. Phys. Solids. , 160, 104754, doi:10.1016/j.jmps.2021.104754\nSneddon, I. N. (1951) Fourier Transforms, McGraw-Hill .\nAppendices\nA Asymptotic solution details of inner solution for critically stressed conditions\nHere we derive the boundary-layer solution for slip near r= 0 valid over distances comparable\nto the diffusion lengthscale√\nαt. However, we note that in our original inversion for slip (2) and\n(4), radial distances are scaled by the current rupture radi usR. Here we revisit the problem of\nsolving for slip, but now scale radial distances by√\nαt, denoting\ns=r/√\nαt\nSubstituting the variable of integration in (4) with η=ξ/λ, we may thus rewrite (4) as\nδ(s,t) =−4\nπ√\nαt\nµ/integraldisplayλ\ns/radicalbig\nξ2−s2df\ndξdξ (24)\nwhere\ndf\ndξ=d\ndξ/integraldisplay1\n0x∆τ(xξ√\nαt)√\n1−x2dx (25)Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 9 of 17\n24 Jan 2024\nin which we have suppressed explicit time dependence of ∆ τ, except through the term√\nαt. That\nwe may do so is apparent considering that the particular expr ession for ∆ τof interest follows the\ncombination of (3) and (6)–9) and the requirement that τ=τs,\n∆τ(r,t) =τb−f/bracketleftbig\nσ′−∆pE1(r2/αt)/bracketrightbig\nevaluating\n∆τ(xξ√\nαt,t) =τb−f/parenleftbig\nσ′−∆pE1[(xξ)2]/parenrightbig\nIn this case, we can simplify ∆ τ(xξ√\nαt,t)⇒∆τ(xξ) and do so in our evaluation of (25),\ndf\ndξ=d\ndξ/integraldisplay1\n0x∆τ(xξ)√\n1−x2dx\n=/integraldisplay1\n0x√\n1−x2d∆τ\ndwdw\ndξdx\n=−2f∆p1\nξ/integraldisplay1\n0xe−(xξ)2\n√\n1−x2dx\nwhere in the above w=xξandd∆τ(w)/dw= 2exp[−(xξ)2]/(xξ). We now label the integral\nF(ξ) =/integraldisplay1\n0xe−(xξ)2\n√\n1−x2dx\nsuch that (24) becomes\nδ(s,t) =√\nαt\nµf∆p8\nπ/integraldisplayλ\ns/radicalbig\nξ2−s2F(ξ)\nξdξ\nor more simply\nδ(s,t)√\nαtf∆p/µ=8\nπ/integraldisplayλ\ns/radicalbig\n1−(s/ξ)2F(ξ)dξ (26)\nWe now look to asymptotically evaluate the expression (26), which is exact, in the limit of large\nλto arrive to our expansion approximation of the inner soluti on for critically stressed conditions.\nTo do so, we will first rewrite (26) as:\nδ(s,t)√\nαtf∆p/µ=8\nπ/bracketleftbigg/integraldisplay∞\ns/radicalbig\n1−(s/ξ)2F(ξ)−/integraldisplay∞\nλ/radicalbig\n1−(s/ξ)2F(ξ)/bracketrightbigg\n(27)\nwhere the leading-order term, O(1), is\n8\nπ/integraldisplay∞\ns/radicalbig\n1−(s/ξ)2F(ξ)dξ= 2√πe−s2/2/bracketleftbig\n(1+s2)I0(s2/2)+s2I1(s2/2/bracketrightbig\n−4s(28)\nwhereI0andI1modified Bessel functions of the first kind. We turn our attent ion to the second\nintegral in (27) and note that, for large λ, the integration variable ξtakes on only large values\nwarranting a Taylor expansion of the integrand’s square-ro ot term\n−/integraldisplay∞\ns/radicalbig\n1−(s/ξ)2F(ξ)dξ≈ −/integraldisplay∞\ns/parenleftbigg\n1−1\n2(s/ξ)2−1\n8(s/ξ)4+.../parenrightbigg\nF(ξ)dξ (29)\nWe look to evaluate the series of integrals of (29), which are the remaining O(1/λ) terms of (27).\nWe start with the first integral\nI(λ) =−/integraldisplay∞\nλF(ξ)dξRobert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 10 of 17\n24 Jan 2024\nTo expand Ifor large λ, we note that\ndI\ndλ=F(λ)\nand seek the expansion of\nF(λ) =/integraldisplay1\n0xe−(λx)2\n√\n1−x2dx (30)\nSince (λx)2has a minimum at x= 0, we use Laplace’s method for asymptotic expansion, denot ing\ng(x) =x/√\n1−x2:\nF(λ)≈/integraldisplay∞\n0/bracketleftbigg\ng(0)+g′(0)x+g′′(0)\n2x2+g′′′(0)\n6x3+.../bracketrightbigg\ne−(λx)2dx\nwithg(0) = 0,g′(0) = 1,g′′(0) = 0,g′′′(0) = 1/2, etc., or more compactly written as\nF(λ)≈/integraldisplay∞\n0/bracketleftBigg∞/summationdisplay\nm=0/braceleftBigg\n2m\nπΓ(1+m/2)Γ(m/2)\nm!xmmodd\n0 meven/bracketrightBigg\ne−(λx)2dx (31)\nThis requires the evaluation of/integraldisplay∞\n0xme−(λx)2dx\nwhich can be done with the substitution u= (λx)2\n1\n2λm+1/integraldisplay∞\n0um\n2−1e−udu=1\n2λm+1Γ/parenleftbiggm+1\n2/parenrightbigg\n(32)\nwhere we recall the definition of the gamma function Γ( z) =/integraltext∞\n0uz−1e−zdz= (z−1)!. Combining\n(31) and (32)\nF(λ) =∞/summationdisplay\nm=1,3,...1\n2λm+12m\nπΓ(1+m/2)Γ(m/2)Γ(m/2+1/2)\nm!\nwhichcanbefurthersimplifiedusingtheLegendreduplicati onformulaΓ( m) = 2m−1Γ(m/2)Γ(m/2+\n1/2)/√π\nF(λ) =∞/summationdisplay\nm=1,3,...1√π1\nλm+1Γ(1+m/2)Γ(m)\nm!\nRecognizing that Γ( m) = (m−1)!\nF(λ) =∞/summationdisplay\nm=1,3,...1√π1\nλm+1Γ(1+m/2)\nm\nand changing the summation index m= 2n+1 with n= 0,1,2,...\nF(λ) =∞/summationdisplay\nn=01√π1\nλ2n+2Γ(n+3/2)\n2n+1(33)\nwe may finally calculate the integral\nI(λ) =−/integraldisplay∞\nλF(λ)dλ=−1√π∞/summationdisplay\nn=01\nλ2n+1Γ(n+3/2)\n(2n+1)2Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 11 of 17\n24 Jan 2024\nWe now look to evaluate all terms in the expansion (29), which can be written in summation\nform as\n−/integraldisplay∞\nλ/radicalbig\n1−(s/ξ)2F(ξ)dξ≈ −/integraldisplay∞\nλ/bracketleftBigg\n−∞/summationdisplay\nk=0(k−3/2)!\n2√πk!/parenleftbiggs\nξ/parenrightbigg2k/bracketrightBigg\nF(ξ)dξ\nwhere the summation in brackets is the Taylor series of/radicalbig\n1−(s/ξ)2abouts/ξ= 0. Passing\nfactorials to Gamma functions, via the identity Γ( n+1) =n!, and rearranging we have\n−/integraldisplay∞\nλ/radicalbig\n1−(s/ξ)2F(ξ)dξ=∞/summationdisplay\nk=01\n2√πΓ(k−1/2)\nΓ(k+1)s2k/integraldisplay∞\nλF(ξ)\nξ2kdξ (34)\nwe denote the integral\nJ(λ) =/integraldisplay∞\nλF(ξ)\nξ2kdξ\nsuch that\ndJ\ndλ=−F(λ)\nλ2k=−1√π1\nλ2k\n∞/summationdisplay\nm=1,3,...1\nλ2n+2Γ(n+3/2)\n2n+1\n\nin which we have substituted (33) and may subsequently integ rate to find\nJ(λ) =−/integraldisplay∞\nλdJ\ndλdλ=1√π∞/summationdisplay\nn=01\nλ2(n+k)+11\n2(n+k)+1Γ(n+3/2)\n2n+1\nSubstituting this expression for Jinto (34)\n−/integraldisplay∞\nλ/radicalbig\n1−(s/ξ)2F(ξ)dξ=1\n2π∞/summationdisplay\nk=0∞/summationdisplay\nn=0s2kΓ(k−1/2)\nΓ(k+1)1\nλ2(n+k)+11\n2(n+k)+1Γ(n+3/2)\n2n+1(35)\n=1\n2π∞/summationdisplay\nk=0∞/summationdisplay\nj=ks2kΓ(k−1/2)\nΓ(k+1)1\nλ2j+11\n2j+1Γ(3/2+j−k)\n2(j−k)+1(36)\n=1\n2π∞/summationdisplay\nj=01\nλ2j+11\n2j+1/bracketleftBiggj/summationdisplay\nk=0s2kΓ(k−1/2)\nΓ(k+1)Γ(3/2+j−k)\n2(j−k)+1/bracketrightBigg\n(37)\nwhere, in passing from (35) to (36) we replaced the summation indexnwith another jchosen such\nthat 2j+1 = 2(n+k)+1, or n=j−k, and in passing from (36) to (37) we swapped the order of\nsummation. We thus find that remaining term in (27) is asympto tically expanded as\n8\nπ/parenleftbigg\n−/integraldisplay∞\nλ/radicalbig\n1−(s/ξ)2F(ξ)dξ/parenrightbigg\n≈ −1\nλ4\nπ−1\nλ32\n3π(1−s2)−1\nλ51\n5π/parenleftbig\n3−s2−s4/2/parenrightbig\n+O(λ−7) (38)\nWe therefore arrive to the asymptotic expansion of slip of th e inner solution for slip in the critically\nstressed limit:\nδin(s,t)√\nαtf∆p/µ=δ0(s,t)−1\nλ4\nπ−1\nλ32\n3π(1−s2)−1\nλ51\n5π/parenleftbig\n3−s2−s4/2/parenrightbig\n+O(λ−7) (39)\nwhere\nδ0(s,t) = 2√πe−s2/2/bracketleftbig\n(1+s)I0(s2/2)+s2I1(s2/2/bracketrightbig\n−4sRobert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 12 of 17\n24 Jan 2024\nthe outer behavior of which as s→ ∞is\nδin(s→ ∞,t)√\nαtf∆p/µ=/parenleftbigg1\ns+1\n8s3+3\n32s5+O(s−7)/parenrightbigg\n−1\nλ4\nπ−1\nλ32\n3π(1−s2)−1\nλ51\n5π/parenleftbig\n3−s2−s4/2/parenrightbig\n+O(λ−7)\n(40)\nB Asymptotic solution details of outer solution for critically stressed conditions\nHere we turn our attention to the solution for slip for distan cesr∼R(t) whenR(t)≫√\nαt,\ni.e., when λis large. We return to the inversion for slip (5), applying th e non-singular condition\nf(1) = 0:\nδ(r,t) =−4\nπR\nµ/integraldisplay1\nρ/radicalbig\nµ2−ρ2d\ndη/integraldisplay1\n0x∆τ(xηR,t)√\n1−x2dx (41)\nwhere we recall that\n∆τ(r,t) =τb−f[σ′−∆pE1(r2/αt)]\nWhen evaluating\n∆τ(xηR(t),t) =τb−f/parenleftbig\nσ′−∆pE1[(xηλ)2]/parenrightbig\nwenotethatwecansimplifythelefthandside∆ τ(xξ√\nαt,t)⇒∆τ(xηλ)anddosoinourevaluation\nof (41), as was done similarly for the inner solution in Appen dix A:\nd\ndη/integraldisplay1\n0x∆τ(xηλ)√\n1−x2dx=/integraldisplay1\n0x√\n1−x2d∆τ\ndwdw\ndζdx\n=−2f∆p1\nη/integraldisplay1\n0xe−(xηλ)2\n√\n1−x2dx\nwhere now w=xηλ, and (41) becomes\nδ(r,t) =Rf∆p\nµ8\nπ/integraldisplay1\nρ/radicalbig\nη2−ρ2\nη/integraldisplay1\n0xe−(xηλ)2\n√\n1−x2dxdη (42)\nWe identify the innermost integral, following (30), as F(ηλ) which has the known asymptotic\nexpansion for large values of its argument (33). We substitu te this expansion in (42) to find the\nasymptotic expansion of the outer solution, δout\nδout(ρ,t) =Rf∆p\nµ8\nπ∞/summationdisplay\nn=01\nλ2n+21√πΓ(n+3/2)\n2n+1/integraldisplay1\nρ/radicalbig\nη2−ρ2\nη2n+3dη\nTo evaluate the integral, we use the trigonemetric variable substitution cos θ=ρ/ηon\n/integraldisplay1\nρ/radicalbig\nη2−ρ2\nη2n+3dη=1\nρ2n+1/integraldisplayacos(ρ)\n0cos2nθsin2θdθ\nand to evaluate this last integral in turn, we’ll consider th e general integral\nI(m,p,ρ) =/integraldisplayacos(ρ)\n0cosmθsinpθdθRobert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 13 of 17\n24 Jan 2024\nwhose solution is as follows. We begin by rearranging the int egrand\nI(m,p,ρ) =/integraldisplayacos(ρ)\n0/parenleftbig\ncosm−1θ/parenrightbig\n(sinpθcosθ)dθ\n=ρm−1(1−ρ2)p+1\n2\np+1+m−1\np−1/integraldisplayacos(ρ)\n0cosm−2θsinp+2dθ\n=ρm−1(1−ρ2)p+1\n2\np+1−m−1\np+1/integraldisplayacos(ρ)\n0cosmθsinpθdθ+m−1\np+1/integraldisplayacos(ρ)\n0cosm−2θsinpθdθ\nand, in the above, we pass from the first to second line using in tegration by parts, and from the\nsecond to third via sin2θ= 1−cos2θ. We identify the first integral as I(m,p,ρ) and the second\nasI(m−2,p,ρ). Solving for I(m,p,ρ), we find the recursion relation:\nI(m,p,ρ) =ρm−1(1−ρ2)p+1\n2\nm+p+m−1\nm+pI(m−2,p,ρ) (43)\nReturning to the specific case m= 2nandp= 2, we find the recursion relation for which we need\nonly calculate the case n= 0 to find all others n >0:\nI(0,2,ρ) =1\n2/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\nI(2,2,ρ) =1\n8/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+ρ(1−ρ2)3/2\n4\n=1\n8/bracketleftBigg/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+(1−ρ2)3/2\nρ2ρ2/bracketrightBigg\nI(4,2,ρ) =1\n16/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+ρ(1−ρ2)3/2\n8+ρ3(1−ρ2)3/2\n6\n=1\n16/bracketleftBigg/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+(1−ρ2)3/2\nρ/parenleftbigg\n2ρ2+8\n3ρ4/parenrightbigg/bracketrightBigg\nFrom this recursion, we may deduce an explicit expression fo rI(m,2,ρ), recognizing from the\nsequence above that such an expression will have the form\nI(m,2,ρ) =am\n/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+(1−ρ2)3/2\nρm/summationdisplay\nk=2,4,...bkρk\n\nwhereamandbkare to be determined. From the factor ( m−1)/(m+ 2) in the recursion, we\nrecognize that\nam=(m−1)\n(m+2)·(m−3)\nm·(m−5)\n(m−2)·...·7\n10·5\n8·3\n6·1\n4·1\n2(44)Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 14 of 17\n24 Jan 2024\nSubstituting m= 2n\na2n=(2n−1)(2n−3)(2n−5)...\n(2n+2)(2n)(2n−2)...(45)\n=(2n−1)(2n−3)(2n−5)...\n2n+1(n+1)(n)(n−1)...(46)\n=(2n−1)!/[2n−1(n−1)!]\n2n+1(n+1)!(47)\n=(2n−1)!\n22n(n+1)!(n−1)!(48)\nPassing to the gamma function n! = Γ(n+1)\na2n=Γ(2n)\n22nΓ(n)Γ(n+2)\nUsing the Legendre duplication identity Γ(2 z)/[22zΓ(z)] = Γ(z+1/2)/(2√π)\na2n=1\n2√πΓ(n+1/2)\nΓ(n+2)\nam=1\n2√πΓ/parenleftbigm\n2+1\n2/parenrightbig\nΓ/parenleftbigm\n2+2/parenrightbig\nRegarding the coefficients of the series bk, we can immediately deduce that b0= 0, and may find\nfrom the recursion (43) and the factoring of amthat, for k≥2\nbk=1\nak1\nk+2\nWe may now write the asymptotic expansion for the outer solut ion under critically stressed\nconditions:\nδout(ρ,t)\nRf∆p/µ=8\nπ∞/summationdisplay\nn=01\nλ2n+21√πΓ(n+3/2)\n2n+11\nρ2n+1a2n/bracketleftBigg/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+(1−ρ2)3/2\nρn/summationdisplay\nk=1b2kρ2k/bracketrightBigg\nδout(ρ,t)\nRf∆p/µ=4\nπ2∞/summationdisplay\nn=01\nλ2n+21\nρ2n+1Γ(n+3/2)\n2n+1Γ(n+1/2)\nΓ(n+2)\n·/bracketleftBigg/parenleftBig\nacos(ρ)−ρ/radicalbig\n1−ρ2/parenrightBig\n+2√π(1−ρ2)3/2\nρn/summationdisplay\nk=1ρ2k\n2k+2Γ(k+2)\nΓ(k+1/2)/bracketrightBigg(49)Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 15 of 17\n24 Jan 2024\nC Asymptotic solution details for marginally pressurized conditions\nWe pursue the asymptotic expansion of the solution in the mar ginally pressurized limit, λ≪1.\nWe return to the integral expression for slip due to the const ant-injection-rate source\nδ(r,t) =8\nπRf∆p\nµ/integraldisplay1\nρ/radicalbig\nη2−ρ2\nη/integraldisplay1\n0xe−(xηλ)2\n√\n1−x2dx\nwhere we again identify the innermost inegral as F(ζλ) and look to expand it now for ηλ≪1. In\nthe above, we can substitute the Taylor expansion\ne−(xηλ)2=∞/summationdisplay\nn=0(−1)m\nn!(xηλ)n\nand subsequently evaluate integrals of the form\nI2n+1=/integraldisplay1\n0x2n+1\n√\n1−x2dx\nforn= 0,1,2...Lettingm= 2n+1 and x= sinθwe transform the integral to\nIm=/integraldisplay1\n0xm\n√\n1−x2dx=/integraldisplayπ/2\n0sinmθdθ\n=/integraldisplayπ/2\n0sinm−1θsinθdθ=/parenleftbig\n−sinm−1θcosθ/vextendsingle/vextendsingleπ/2\n0+(m−1)/integraldisplayπ/2\n0sinm−2θcos2θdθ\n= (m−1)/parenleftBigg/integraldisplayπ/2\n0sinm−2θdθ−/integraldisplayπ/2\n0sinmθdθ/parenrightBigg\n= (m−1)(Im−2−Im)\nand arrive to the recursion relation\nIm=m−1\nmIm−2\nGiven that I1= 1,\nI2n+1=(2n)·(2n−2)·...·6·4·2\n(2n+1)·(2n−1)·...·7·5·3=2nn!\n(2n+1)!/(2nn!)(50)\nwhich we may rewrite using the Gamma function\nI2n+1= 4n[Γ(n+1)]2\nΓ(2n+2)\nand using the Legendre duplication formula we can further si mplify\nI2n+1=√π\n2Γ(n+1)\nΓ(n+3/2)\nand arrive to an expression for the expansion of Ffor small values of its argument\nF(ηλ)≈√π\n2∞/summationdisplay\nn=0(ηλ)2n(−1)n\nΓ(n+3/2)Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 16 of 17\n24 Jan 2024\nδ(ρ,t)\nRf∆p/µ=8\nπ/integraldisplay1\nρ/radicalbig\nη2−ρ2\nηF(ηλ)dη=8\nπ∞/summationdisplay\nn=0√π/2\nΓ(n+3/2)λ2n/integraldisplay1\nρ/radicalbig\nη2−ρ2η2n−1dη\nWe denote the integrals within the sum as\nI2n−1(ρ) =/integraldisplay1\nρ/radicalbig\nη2−ρ2η2n−1dη\nforn= 0,1,2...\nForn= 0, we may evaluate I−1using the substitution ρ/η= cosθand integration by parts\nI−1(ρ) =/integraldisplay1\nρ/radicalbig\nη2−ρ2\nηdη=ρ/integraldisplayacos(ρ)\n0sin2θ\ncos2θdθ\n=ρ/bracketleftBigg/parenleftbiggsinθ\ncosθ/vextendsingle/vextendsingle/vextendsingle/vextendsingleacos(ρ)\n0−/integraldisplayacos(ρ)\n0dθ/bracketrightBigg\n=/radicalbig\n1−ρ2−ρacos(ρ)\nThe remaining integrals in the sum ( n >0) can be found by recursion. Letting m= 2n−1\nand performing integration by parts\nIm(ρ) =/integraldisplay1\nρ/radicalbig\nη2−ρ2ηmdη\n=1\n3(η2−ρ2)3/2ηm−1/vextendsingle/vextendsingle/vextendsingle1\nρ−m−1\n3/integraldisplay1\nρ(η2−ρ2)3/2ηm−2dη\n=1\n3(1−ρ2)3/2−m−1\n3/integraldisplay1\nρ(η2−ρ2)1/2ηm(η2−ρ2)η−2dη\n=1\n3(1−ρ2)3/2−m−1\n3/parenleftbig\nIm(ρ)−ρ2Im−2(ρ)/parenrightbig\nwhich leads to the recursion relation\nIm(ρ) =(1−ρ2)3/2\nm+2+m−1\nm+2ρ2Im−2(ρ)\nWe may solve this recurrence relation to have an explicit exp ression for Im(ρ). Given the form of\nthe recursion, we posit a solution of the form\nIm(ρ) =(1−ρ2)3/2\nm+2m/summationdisplay\nk=1,3,5...akρk−1\nand substitute this into the recursion relation to find that\nm/summationdisplay\nk=1,3,...akρk−1= 1+m−1\nmm/summationdisplay\nk=1,3,...akρk+1Robert C. Viesca\nTufts UniversityAsymptotic solutions for self-similar fault slip\ninduced by fluid injection at constant ratePage 17 of 17\n24 Jan 2024\nExpanding\na1+a3ρ2+a5ρ4+...= 1+m−1\nm/parenleftbig\na1ρ2+a3ρ4+.../parenrightbig\nand comparing terms, we find that a1= 1 and subsequent terms follow the recursion\nam=m−1\nmam−2\nor\nam=(m−1)\nm·(m−3)\n(m−2)·(m−5)\n(m−4)·...·4\n3·2\n1·a1\nSubstituting m= 2q+1 with q= 1,2,3...\na2q+1=(2q)·(2q−2)·(2q−4)·...·6·4·2\n(2q+1)·(2q−1)·(2q−3)·...·5·3·1·a1\nwhich we recognize as the series (50), which reduces to\na2q+1=√π\n2Γ(q+1)\nΓ(q+3/2)\nor\nam=√π\n2Γ/parenleftbigm\n2+1\n2/parenrightbig\nΓ/parenleftbigm\n2+1/parenrightbig\nTherefore,\nIm(ρ) =(1−ρ2)3/2\nm+2m/summationdisplay\nk=1,3,...√π\n2Γ/parenleftbigk\n2+1\n2/parenrightbig\nΓ/parenleftbigk\n2+1/parenrightbigρk−1\nor reverting to m= 2n−1, withn= 1,2,3,...\nI2n−1(ρ) =(1−ρ2)3/2\n2n+1n/summationdisplay\nk=1,2,...√π\n2Γ(k)\nΓ(k+1/2)ρ2k−2\nSuch that the asymptotic expression for slip in the marginal ly pressurized limit is\nδ(ρ,t)\nR(t)f∆p/µ=8\nπ/integraldisplay1\nρ/radicalbig\nη2−ρ2\nηdη+8\nπ∞/summationdisplay\nn=1λ2n√π/2\nΓ(n+3/2)/integraldisplay1\nρ/radicalbig\nη2−ρ2η2n−1dη\n=8\nπ/parenleftBig/radicalbig\n1−ρ2−ρacos(ρ)/parenrightBig\n+8\nπ/parenleftbig\n1−ρ2/parenrightbig3/2∞/summationdisplay\nn=1λ2n\n2n+1√π/2\nΓ(n+3/2)n/summationdisplay\nk=1√π\n2Γ(k)\nΓ(k+1/2)ρ2k−2\n=8\nπ/parenleftBig/radicalbig\n1−ρ2−ρacos(ρ)/parenrightBig\n+2/parenleftbig\n1−ρ2/parenrightbig3/2∞/summationdisplay\nn=1λ2n\n2n+11\nΓ(n+3/2)n/summationdisplay\nk=1Γ(k)\nΓ(k+1/2)ρ2k−2"    },    {        "title": "2401.13839v2.Edge_coloring_sparse_graphs_with__Δ__colors_in_quasilinear_time.pdf",        "content": "Edge-coloring sparse graphs with ∆colors in quasilinear time\nŁukasz Kowalik∗\nFebruary 15, 2024\nAbstract\nIn this paper we show that every graph Gof bounded maximum average degree mad( G)\nand with maximum degree ∆can be edge-colored using the optimal number of ∆colors\nin quasilinear expected time, whenever ∆≥2 mad( G). The maximum average degree is\nwithin a multiplicative constant of other popular graph sparsity parameters like arboricity,\ndegeneracy or maximum density. Our algorithm extends previous results of Chrobak and\nNishizeki [10] and Bhattacharya, Costa, Panski and Solomon [4].\n1 Introduction\nAlgorithms for edge-coloring is a classic research topic which has become active again recently.\nLet us recall that edge-coloring of graph Gis a function π:E(G)→Nwhich assigns different\nvalues (called colors) to incident edges. By k-edge-coloring we mean an edge coloring which\nuses at most kcolors. The minimum number of colors that suffice to color graph Gis called the\nchromatic index ofGand denoted χ′(G). For a maximum degree of Gdenoted ∆(G)it is clear\nthat χ′(G)≥∆(G), while the classic theorem of Vizing states that χ′(G)≤∆(G) + 1. For\ngeneral graphs, determining whether χ′(G) = ∆( G)is NP-complete, as shown by Holyer [20].\n1.1 Algorithms for general graphs\nVizing’s proof can be easily transformed to an algorithm for edge-coloring graphs in ∆ + 1\ncolors running in time O(nm), where nandmdenote the number of vertices and edges,\nrespectively, of the input graph (we use this notation throughout the whole paper). Gabow,\nNishizeki, Kariv, Leven, and Terada [18] were the first to show an algorithmic progress: they\ndesigned two algorithms, one running in time O(m√nlogn)and another in time O(m∆ log n).\nBoth these algorithms saw an improvement recently: Sinnamon [25] obtained deterministic\ntime O(m√n), while Bernshteyn and Dhawan [2] randomized time ∆O(1)m. Another direcion\nof research is improving the running time at the price of an increased number of colors: very\nrecently, Elkin and Khuzman [16] presented an algorithm that uses at most (1 +ϵ)∆colors\nfor any ϵ≥1\n∆and works in time O(mϵ−1logn), see also Duan et al. [15] for an earlier result\nof this kind.\n∗Institute of Informatics, University of Warsaw, Poland ( kowalik@mimuw.edu.pl ). This work is a part of\nproject BOBR that has received funding from the European Research Council (ERC) under the European\nUnion’s Horizon 2020 research and innovation programme (grant agreement No. 948057).\n1arXiv:2401.13839v2  [cs.DS]  14 Feb 2024NumberGraph class Time Referenceof colors\n(1 +ϵ)∆ general O(mϵ−1logn)Elkin and Khuzman [16]\n∆ + 2 α−2 arboricity α O(mlog ∆) Christiansen et al. [7]\n∆ + 1 general O(m√n) Sinnamon [25]\n∆ + 1 general O(m∆ log n) Gabow et al. [18]\n∆ + 1 general O(m∆18) Bernshteyn and Dhawan [2]\n∆ + 1 arboricity α O(mαlogn) Bhattacharya et al. [4]\n∆ + 1 arboricity α ˜O(m√nα\n∆) Bhattacharya et al. [4]\nχ′(G) bounded treewidth O(n) Zhou et al. [32]\n∆ bipartite O(mlog ∆) Cole, Ost and Shirra [13]\n∆ bounded genus, ∆≥19 O(n) Chrobak and Yung [11]\n∆ bounded genus, ∆≥9 O(nlogn) Chrobak and Nishizeki [10]\n∆ planar, ∆≥9 O(n) Cole and Kowalik [12]\n∆ ∆≥2 mad( G) O(mlognmad( G)3) This work\nTable 1: Summary of edge-coloring algorithms. Note that α= Θ(mad( G)).\n1.2 Algorithms for graph classes\nRestricting the input graph may allow for algorithms with improved performance. By the clas-\nsical Kőnig’s Theorem, for bipartite graphs only ∆colors suffice, and Cole, Ost and Shirra [13]\nprovided an O(mlog ∆)-time algorithm. Zhou, Nakano and Nishizeki [32] gave an algorithm\nthat uses χ′(G)colors and runs in linear time for graphs of bounded treewidth. Chrobak and\nYung [11] gave a linear time algorithm that uses only max{∆,19}colors and works for graphs\nof bounded genus. Graphs from this class can be also colored using only max{∆,9}colors by\nan algorithm of Chrobak and Nishizeki [10] that runs in time O(nlogn). For the special case\nof planar graphs, Cole and Kowalik [12] improved that to linear time.\n1.3 Uniformly sparse graphs and our result\nMaximum average degree is defined as mad( G) = max Sad(S), where the maximum is over all\nnonemptysubgraphsof Gandad(S) =1\nsP\nv∈V(S)d(v)istheaveragedegreeof S. Hence, when\nmad( G)is low, graph Gis uniformly sparse. It is easy to see that mad( G)equals exactly twice\nthemaximum density ofG, i.e., max S|E(S)|/|V(S)|, which is close to the arboricity defined as\nα(G) = max S,|V(S)|>1⌈|E(S)|/(|V(S)|−1)⌉. Indeed, all these three parameters (we could add\nalso degeneracy), are within a small constant of each other (see e.g. [21]). In particular, when\none of them is bounded by a constant, so are all the others. Moreover, planarity, bounded\ntreewidth or bounded genus all imply bounded maximum average degree, arboricity, etc.\nVery recently, Bhattacharya, Costa, Panski and Solomon [4] obtained two algorithms that\nuse∆ + 1colors and work in randomized time O(m√n∆/α(G))andO(mα(G) logn). In\nparticular, the latter one is quasilinear for bounded arboricity, and both algorithms improve\nover the results of Gabow et al. [18] when ∆(G) =ω(α(G)). Another recent result is an\nalgorithm of Christiansen, Rotenberg and Vlieghe [7] that uses ∆ + 2 α(G)−2colors and runs\nin deterministic time O(mlog ∆).\n2In this work we continue this line of research with the goal of improving the number of\ncolors used . Note that the quasilinear algorithm of Chrobak and Nishizeki [10] uses only ∆\ncolors for the special case of graph Gof bounded genus gand such that ∆≥9. Since then\nmad( G)≤6 +O(g/n)it is natural to ask if it is possible to generalize it to any class of\nuniformly sparse graphs. We answer this question in the affirmative by showing the following\nmain result.\nTheorem 1. Every graph Gwithnvertices and medges such that ∆(G)≥2 mad( G)andG\nhas no isolated vertices can be ∆(G)-edge-colored in expected time O(mlognmad( G)3). The\nrunning time bound holds also with high probability.\nIn particular, whenever mad( G) =O(1), the algorithm works in quasilinear time. Here,\nbyhigh probability we mean probability at least 1−n−k, for an arbitrary constant k >0.\nLet us comment on the assumption ∆(G)≥2 mad( G). Clearly, some kind of assumption\nis needed because there are sparse graphs of chromatic index ∆(G) + 1. Vizing [29] (see\nalso [26]) conjectured1that for χ′(G) = ∆( G)it suffices that mad( G)≤∆−1. While\nthis conjecture is still open, there has been a substantial progress. Fiorini [17], Haile [19]\nand Sanders and Zhao [24] obtained sufficient conditions of the form mad( G)≤1\n2∆ +o(∆).\nNext, it was improved to mad( G)<2\n3(∆ + 1)by Woodall [30, 31], and further to mad( G)<\nmin{3\n4∆−2,0.738∆−1.153}by Cao, Chen, Jiang, Liu and Lu [5]. All these results are\nin fact constructive and correspond to polynomial-time algorithms (but notquasilinear2).\nHowever, while the proof of Sanders and Zhao [24] relies on a simple procedure for extending\na partial coloring called Vizing Adjacency Lemma (VAL, see Theorem 2 below), the later\nproofs result in considerably more complicated algorithms. In fact we show that the constant\n2 is optimal if we restrict ourselves to algorithms that color the graph edge by edge using VAL\n(see Theorem 5 for a formal statement). Since we prefer to keep this article short, we leave\nthe task of improving the assumption as an open problem.\n1.4 Sketch of our approach\nLetDbe the number of available colors. Let us recall the following classical result (usually\nstated in terms of critical graphs). Here, for a statement Pthe expression [P]equals 1ifP\nholds and 0otherwise.\nTheorem 2 (Vizing Adjacency Lemma, VAL [28]) .LetGbe a simple graph and let e=xy\nbe an edge such that xhas at most D−d(y) + [d(y) =D]neighbors of degree D. Then any\npartial D-edge-coloring of Gwhich colors a subset Ecof edges of G,e̸∈Ec, can be extended\nto a partial D-edge-coloring that colors Ec∪ {e}.\nIn this work, an edge xythat satisfies the assumption of VAL is called (D, x)-weak, or\nsimply x-weakwhen Dis known. Moreover, xyisweakif it is x-weak or y-weak.\nNote that for D= ∆( G) + 1, every edge is weak, just becuse then every vertex has\nno neighbors of degree D. Hence, in a simple implementation of standard proofs of Vizing\nTheorem, we color the input graph Gedge by edge, using an algorithm originating from the\nVizing’s proof of VAL. This algorithm uses so-called fans and alternating paths. Given an\n1The original conjecture states that in every critial graph (in a sense, a minimal graph with chromatic index\n∆ + 1) we have |E(G)| ≥1\n2((∆(G)−1)|V(G)|+ 3).\n2The algorithm corresponding to the proof of Sanders and Zhao [24] can be implemented in time O(nm);\nfor the other the worst time complexity is at least that large.\n3edge e=xy, ane-fan is roughly a certain sequence of neighbors of x(equivalently, incident\nedges) — a precise definition will be given later. An alternating path is just a maximal path\nwith edges colored alternately in two colors. Of course the algorithm behind VAL may need\nto recolor some of the already colored edges. However, it turns out that it always suffices to\nrecolor some edges of an xy-fan and a single alternating path Pxy.\nWhen we aim at a quasilinear algorithm, even for D= ∆( G)+1, the approach above poses\nthe following problem: even if coloring edge xytakes just O(1)time per a recolored edge, our\nbound on the number of those edges is deg(x) +|Pxy|, and the sumP\nxy∈E(G)(deg( x) +|Pxy|)\ncan be as large as Θ(nm), forn:=|V(G)|,m:=|E(G)|. The problem with long alternating\npaths has been resolved in an elegant way by Sinnamon [25]: he shows, roughly, that if we\npick arandomuncolored edge xyout of all ℓuncolored edges, then the probability that a fixed\nalternating path is chosen is O(1\nℓ). Then he argues that the totallength of alternating paths\nis only O(∆m), soE(|Pxy|) =O(1\nℓ∆m). This means that when ℓis large, i.e., most of the\ntime, the path is usually short, and the total expected length of the alternating paths used by\nthe algorithm is O(Hℓ∆m), where Hℓ=O(logℓ)is the ℓ-th harmonic number. This idea was\nfurther extended by Bhattacharya et al. [4], who obtained only O(αmlogn)time for graphs\nof arboricity α, by a small modification of the algorithm (we build an xy-fan of edges incident\nto the endpoint of smaller degree), and a clever use of the following result.\nTheorem 3 (Chiba and Nishizeki [6]) .LetGbe a graph of arboricity α. Then,\nX\nuv∈E(G)min{d(u), d(v)} ≤2α|E(G)|.\nNote that Theorem 3 gives also the bound of O(α(G)m)for the total length of all fans\nused, so the expected total number of recolored edges is O(α(G)mlogn)in the algorithm of\nBhattacharya et al.\nNow we turn to the setting of this work, where D= ∆( G)and∆(G)≥2 mad( G). For\nsimplicity assume that mad( G)is bounded. The first problem we encounter is that in our case\nnot all edges are weak. However, Sanders and Zhao [24] show that there is always at least\none weak edge. Their proof can be transformed to an algorithm that finds a weak edge xy,\nremoves it, colors the graph recursively, and then extends the coloring. Unfortunately, then\nthe path Pxymay be long.\nInordertomakeuseoftheSinnamon-Bhattacharya-et-al.-trick,weshowthatwhenaverage\ndensity is bounded, then at least a constant fraction of edges are weak, thus generalizing a\nsimilar lower bound of Chrobak and Nishizeki [10] for bounded genus graphs. This results in\nexpected length of all alternating paths in a single recursive call of O(mlogn)using the trick,\nand the number of calls is O(logn).\nThe last problem we had to fight with is the fact that it may happen that a weak edge xy\nisx-weak but not y-weak. Then, unlike in the algorithm of Bhattacharya et al. we cannot\npick the endpoint of smaller degree and we cannot use Theorem 3 to get a small bound on\nthe total size of all the fans3. Our solution here is to use a slightly non-standard notion of\nthe fan (somewhat inspired by the multi-fan in Stiebitz et al. [26]), which still can be used to\n3We remark here that the problem of large fans was solved in yet another way in the work of Chrobak and\nNishizeki [10]. Namely, they mainly deal with graphs of maximum degree bounded by 18, while for ∆≥19\nthey use the linear-time algorithm of Chrobak and Yung [11]. However, the latter algorithm relies not on VAL\nbut on finding so-called bundles(2-vertices with common neighbors). As far as we know, this works only for\ngraphs of maximum average degree at most 3 +o(1). For that reason we could not adapt that approach here.\n4prove VAL, but we can bound its length by min{d(x), d(y)}, regardless of whether its edges\nare incident with xor with y. Actually, just bounding the size of the fan was not the only\nchallenge here: we want to keep the time of building the fan close to O(min{d(x), d(y)}).\nIn this process we need to perform various queries involving the sets of unused colors in the\nvertices of the fan, which is not trivial because the sets can be of size Ω(D), and Dcan be\nmuch larger than d(x)ord(y). Fortunately, we show that these queries, implemented carefully,\nneed only O(min{d(x), d(y)})calls to appropriate standard dictionary data structures.\n1.5 Organization of the paper\nThis paper is organized as follows. In Section 2 we show the lower bound for the number of\nweak edges for graphs with ∆≥2 ad(G). At the end of the section we will also prove that\nthe constant 2 in the assumption cannot be improved if we want to get any positive lower\nbound. Next, in Section 3 we present the variant of the fan notion we use and we show an\nefficientimplementationoftheVizingAdjacencyLemmainwhichan xy-fanhaslengthatmost\nmin{d(x), d(y)}. In Section 4 we describe the complete algorithm and we prove Theorem 1.\nFinally, in Section 5 we summarize the paper and suggest some open problems. In Appendix A\nwe provide details required to get a high probability guarantee for the running time.\n2 Lower bound for the number of weak edges\nBy a d-vertex (resp. d-neighbor) we mean a vertex (resp. neighbor) of degree d. For an integer\ni, letdi(v)be the number of i-neighbors of v. Let Dbe an integer. In what follows, we call\nan edge uv(D, u)-weakifdD(u)≤D−d(v) + [d(v) =D]. Moreover, uvisD-weak if it is\n(D, u)-weak or (D, v)-weak. Note that for D≥∆(G)+1, every edge is D-weak. For simplicity,\nwhenever Dis known, we write that an edge is u-weakorweakwhen it is (D, u)-weak or weak,\nrespectively. An edge uvisstrongif it is not weak, i.e., it is neither u-weak nor v-weak. A\nvertex is called weakif it is incident to at least one weak edge, otherwise it is called strong.\nIn this section we focus on the case D= ∆( G)and we show that for bounded average\ndegree, if ∆(G)is large enough, then at least a constant fraction of edges are weak.\nTheorem 4. LetG= (V, E)be a nonempty graph with no isolated vertices and such that\n∆(G)≥2 ad(G). Then the number of ∆(G)-weak edges in Gis at least\n|E|\n2 ad(G)2.\nProof.For brevity, we denote ˜d= ad( G). We use the discharging method [14], in a similar\nspirit as Sanders and Zhao [24], with a crucial difference that here we need a linear lower\nbound, and not just an existence of a weak edge. We assign an initial chargeto each vertex v\nas follows:\nch(v) =d(v)−˜d.\nNote that the total charge is zero:\nX\nv∈Vch(v) =X\nv∈Vd(v)− |V|X\nv∈Vd(v)/|V|= 0.\n5Next, we move the charge as follows: every vertex xof degree d(x)≥˜d+1\n2, sends\nch(x)−1\n2\nd(x)−∆ +d(y)−1\nunits of charge to every neighbor yof degree <˜d+1\n2such that edge xyis strong. We claim\nthat the expression above is always well defined and positive. Indeed, since xyis strong,\nd(x)≥[d(y)<∆] + d∆(x)≥[d(y)<∆] + ∆ −d(y) + [d(y) = ∆] + 1 = ∆ −d(y) + 2. This\nalso implies that d(y)≥∆−d(x) + 2≥2, which we state below as a claim for later use.\nClaim 1. If a vertex receives charge, then it has degree at least 2.\nFor a vertex v, let ch′(v)denote the charge of vafter moving the charge. Since the charge\nis only being moved between vertices,P\nv∈Vch′(v) =P\nv∈Vch(v) = 0. Our goal is to show\nthat\n(i)for every strong vertex the final charge is at least1\n2, and\n(ii)for every weak vertex the final charge is at least 1−˜d.\nThen, denoting the set of weak vertices by Vw, we have\n0 =X\nv∈Vch′(v)≥ |Vw| ·(1−˜d) + (|V| − |Vw|)·1\n2,\nwhich implies (˜d−1\n2)|Vw| ≥ |V|/2. Then, since Ghas no isolated vertices, ˜d≥1, and hence\n|Vw| ≥ |V|/(2(˜d−1\n2))>|V|/(2˜d). Since every weak vertex is incident with at least one weak\nedge, we get at least |V|/(4˜d) =|E|/(2˜d2)weak edges, as required. Hence, indeed it suffices\nto show (i)and(ii).\nClaim 2. For any vertex vof degree d(v)≥˜d+1\n2we have ch′(v)≥1\n2.\nProof.Since ch (v)≥1\n2, the claim holds trivially when vdoes not send charge. Hence in what\nfollows assume vsends charge. Then vis incident to at least one strong edge. Let wbe a\nneighbor of vsuch that vwis strong and d(w)is minimum among such neighbors. Note that\nd(w)<˜d+1\n2, for otherwise vdoes not send charge to any vertex. Then vsends at most\nd(v)−˜d−1\n2\nd(v)−∆ +d(w)−1\nto each neighbor uofvsuch that vuis strong and d(u)<˜d+1\n2, and does not send to other\nvertices. Note that vdoes not send to ∆-vertices, because ˜d+1\n2≤1\n2∆+1\n2= ∆−1\n2(∆−1)≤∆.\nSince vwis strong, d∆(v)≥∆−d(w) + 1. Hence vhas at most d(v)−∆ +d(w)−1neighbors\nof degree smaller than ∆and this is also an upper bound for the number of neighbors it sends\ncharge to. It follows that in total vsends at most d(v)−˜d−1\n2charge, and hence it keeps at\nleast1\n2of charge, as required. ♢\nClaim 3. For any strong vertex vof degree d(v)<˜d+1\n2we have ch′(v)≥3\n2.\n6Proof.Letwbe a neighbor of vof minimum degree. Since vwis strong, d(w)≥d∆(w)≥\n∆−d(v) + 1 >2˜d−(˜d+1\n2) + 1 = ˜d+1\n2. It follows that all neighbors of vsend charge to v,\nand\nch′(v) =d(v)−˜d+∆X\ni=d(w)ni·i−˜d−1\n2\ni−∆ +d(v)−1,\nwhere niis the number of i-neighbors of v. Since ∆−d(v) + 1 >2˜d−(˜d+1\n2) + 1 = ˜d+1\n2,\nwe infer thati−˜d−1\n2\ni−∆+d(v)−1= 1 +∆−d(v)+1−\u0010\n˜d+1\n2\u0011\ni−∆+d(v)−1is decreasing with increasing i, so\nch′(v)≥d(v)−˜d+d(v)·∆−˜d−1\n2\nd(v)−1.\nSince d(v)≥2by Claim 1 and ∆≥2˜d, we get ch′(v)≥3\n2, as required. ♢\nTo finish the proof, note that (i)follows from claims 2 and 3. For (ii), a weak vertex of\ndegree at least ˜d+1\n2has final charge at least1\n2by Claim 2, which is more than 1−˜dsince\n˜d≥1(no isolated vertices). Finally, a weak vertex vof degree <˜d+1\n2does not send charge,\nso it has final charge ch′(v)≥ch(v) =d(v)−˜d≥1−˜d, as required. This ends the proof.\nWe conclude this section by showing that the assumption ∆(G)≥2 ad(G)in the formu-\nlation of Theorem 4 is the best possible.\nTheorem 5. For every ϵ >0there are arbitrarily large graphs Gwith no isolated vertices,\nwith no weak edges and such that ∆(G)≥(2−ϵ) ad(G).\nProof.Letdbe an integer such that d≥max{3\nϵ−1,3}and set ∆ = d(d−1). Set also\nnd= (d−1)(∆−d+ 2)andn∆=d(∆−d+ 2). Construct graph Gas follows. Begin with\n∆−d+ 2copies of the complete bipartite graph Kd−1,d(note that since d≥2,∆−d+ 2 =\nd(d−2) + 2 ≥5). Let Vdbe the set of the vertices of the resulting graph of degree d, and\nletV∆be the remaining vertices, i.e., the vertices of degree d−1. Note that |Vd|=ndand\n|V∆|=n∆. Next, we partition V∆arbitrarily into ddisjoint sets of size (∆−d+ 2)and\nfor each such set we add all edges between its verties so that it induces the clique K∆−d+2.\nObserve that then every vertex in V∆has degree d−1+∆−d+1 = ∆. Note that since d≥2,\nwe have ∆≥dand hence Ghas maximum degree ∆. This completes the construction of G.\nThen, we have\nad(G) =dnd+ ∆n∆\nnd+n∆=∆z}|{\nd(d−1) +∆ d\nd−1 +d=d+ 1\n2d−1∆.\nIt follows that\n∆ =2d−1\nd+ 1ad(G) =\u0012\n2−3\nd+ 1\u0013\nad(G)≥(2−ϵ) ad(G).\nNow it suffices to check that Ghas no weak edges. Let uvbe an edge such that u∈Vdand\nv∈V∆. Then uvis not u-weak because d∆(u) =d−1>1 = ∆ −d(v) + [d(v) = ∆]. Also, uv\nis not v-weak because d∆(v) = ∆ −d+ 1>∆−d= ∆−d(u) + [d(u) = ∆]. Finally, consider\nan edge uvbetween two vertices of V∆. Then d∆(u) = ∆ −d+1>1 = ∆ −d(v)+[d(v) = ∆],\nso by symmetry uvis not weak.\n7x y1 1,2y23,4\n2y35\n3y46,7\n4\ny58 6\n8x y11 2y22,3\n4y35\n3y44,7\n6\ny56 8\nFigure 1: Rotating a fan. The numbers next to vertices denote free colors.\n3 Efficient Vizing Adjacency Lemma\nThe goal of this section is to provide an efficient algorithm which implements the Vizing\nAdjacency Lemma (Theorem 2).\nThroughout this section by Dwe denote the number of available colors. Of course in\nthis paper we are mostly interested in the case when Dis the maximum degree ∆, but it is\nconvenient to state results also for the case D > ∆, since our algorithm colors subgraphs of\nthe input graph recursively and at some point the maximum degree drops. A partial edge-\ncoloringof graph Gis a function π:E(G)→ {1, . . . , D }∪{⊥}such that it is an edge-coloring\nwhen restricted to edges colored by {1, . . . , D }. For a vertex xwe define π(x)as the set of\ncolors used at edges incident with x, i.e., π(x) ={π(xy)|y∈NG(x)} \\ {⊥}. Moreover, ¯π(x)\ndenotes the set of colors free at x, i.e., ¯π(x) ={1, . . . , D } \\π(x). An (a, b)-alternating path is\na maximal path in Gsuch that its edges are colored alternately in aandb.\nFans, developed by Vizing [27] are useful structures in edge-colorings. The notion appears\nisseveralvariations. Hereweneedanewone, whichdiffersfromthestandardfanbymodifying\ncondition(F3)andaddingcondition(F4)below. Thedefinitionissomewhatinspiredbymulti-\nfans in the book of Stiebitz et al. [26].\nLetG= (V, E)be a graph and let πbe a partial edge-coloring of G. Then, a fanis a\nsequence F= (x, y1, . . . , y k)such that\n(F1) y1, . . . , y kare different neighbors of x,\n(F2) xy1is uncolored by π,\n(F3) for every i= 2, . . . , kedge xyiis colored by πandπ(xyi)∈Si−1\nj=1¯π(yj),\n(F4) for every i= 2, . . . , k,d(yi)< D.\nThe number kis called the sizeof fan F. Since in this work we we do not use other variants of\nfan, we do not give it a special name. The notion of a fan is motivated by the fact that there\nis a simple procedure, called rotating the fan, which, given a color c∈¯π(x)∩¯π(yk)recolors\nsome edges of the form xyiso that edge xy1can be colored with a free color, see Algorithm 1\nand Figure 1. Of course, the definition of a fan does not guarantee that such a color cexists.\nLemma 6. Given graph G, its partial coloring π, a fan F= (x, y1, . . . , y k)and color c∈\n¯π(x)∩¯π(yk), Algorithm 1 finds a new partial coloring, which colors the same edges as πand\nadditionally xy1.\n8Algorithm 1: Rotating a fan.\ninput:a fan F= (x, y1, . . . , y k), color c∈¯π(x)∩¯π(yk).\n1i←k\n2while i≥1do\n// Invariant: for some j= 1, . . . , i we have c∈¯π(x)∩¯π(yj)\n3ifc∈¯π(yi)then\n4 c′←π(xyi)\n5 π(xyi)←c\n6 c←c′\n7 i←i−1\nProof.Itsufficestoobservethatatthebeginningoftheloopthefollowinginvariantissatisfied:\nfor some j= 1, . . . , iwe have c∈¯π(x)∩¯π(yj).\nWe call a fan F= (x, y1, . . . , y k)activewhen one of the two conditions below hold:\n(A1) there is i= 1, . . . , ksuch that ¯π(yi)∩¯π(x)̸=∅,\n(A2) there are i, j= 1, . . . , ksuch that i̸=jand¯π(yi)∩¯π(yj)̸=∅.\nIf additionally F′= (x, y1, . . . , y k−1)is not active, we call Faminimal active fan .\nWe will prove a series of simple lemmas. In all of them we work with a graph Gand its\npartial coloring π.\nLemma 7. If a fan F= (x, y1, . . . , y k)is not active, then\n\f\f\f\f\fk[\ni=1¯π(yi)\f\f\f\f\f≥D−d(y1) +k.\nProof.Since xy1is uncolored, |π(y1)| ≤d(y1)−1. Then, we have |¯π(y1)|=D− |π(y1)| ≥\nD−(d(y1)−1)and by (F4), |¯π(yi)| ≥1fori= 2, . . . , k. It follows that\n\f\f\f\f\fk[\ni=1¯π(yi)\f\f\f\f\f(A2)=kX\ni=1|¯π(yi)| ≥D−(d(y1)−1) + ( k−1)·1 =D−d(y1) +k.\nThe corollary below is useful for time complexity analysis of algorithms that use fans.\nCorollary 8. Consider aa fan F= (x, y1, . . . , y k)IfFis notactive, then k≤min{d(x), d(y1)}.\nIfFis a minimal active fan, then k≤1 + min {d(x), d(y1)}.\nProof.Assume Fis not active. The inequality k≤d(x)follows trivially from (F1), while\nk≤d(y1)by plugging D≥\f\f\fSk\ni=1¯π(yi)\f\f\fto Lemma 7. The claim for a minimal active fan\nfollows immediately.\nLemma 9. Letxy1be an x-weak edge. If a fan F= (x, y1, . . . , y k)is not active, then it can\nbe extended to another fan F′= (x, y1, . . . , y k+1).\n9Proof.Consider the set Z={z∈NG(x)|π(xz)∈Sk\ni=1¯π(yi)}. Then,\n|Z|(A1)=\f\f\f\f\fk[\ni=1¯π(yi)\f\f\f\f\fLemma 7\n≥ D−d(y1) +k.\nAsxy1isx-weak, we have dD(x)≤D−d(y1) + [d(y1) =D]. Since y1̸∈Z, because xy1is not\ncolored, we know that Zcontains at least D−d(y1) +k−(D−d(y1)) =kvertices of degree\n< D. It follows that Z\\ {yi|i= 2, . . . , k }has at least one vertex of degree < D, call it yk+1.\nIt is easy to check that (x, y1, . . . , y k+1)satisfies conditions (F1)-(F4).\nNow we turn to lemmas that describe algorithms. These algorithms use the following data\nstructures that represent the input graph and a partial coloring of it:\n•adjacency lists, i.e., array Nsuch that N[v]is a list of neighbors of v;\n•small degree adjacency lists, i.e., array N<Dsuch that N<D[v]is a list of neighbors of\nvof degree smaller than D;\n•for every v∈Vwe keep an associative array Mvthat maps every color c∈π(v)to\nthe edge incident with vcolored with c. The array is also used to test membership in\nπ(v). It is implemented using hashing with lookup and updates in expected constant\ntime (e.g. using universal hashing with chaining).\n•for every v∈Vsuch that d(v)≥D/2we keep a dynamic array Tvof all the colors in\n¯π(v)together with an array Pvof size Dwith pointers to elements of Tv.\n•for every v∈Vsuch that d(v)< D/ 2we keep a dynamic array Tvof all the colors in\n¯π(v)∩ {1, . . . , d (v)}together with an array Pvof size d(v)with pointers to elements of\nTv.\nNote that the total size of the data structures listed above is O(n+m)for an input graph\nwith nvertices and medges. We begin with the following simple observation.\nLemma 10. For any vertex v, arrays TvandPvcan be updated in O(1)worst-case time after\ncoloring or uncoloring an edge incident to v. Moreover, a random color from ¯π(v)can be\nfound in O(1)expected time and a color from ¯π(v)can be found in O(1)worst-case time.\nProof.This proof is inspired by [4], though they have O(d(v))time for sampling.\nWhen an edge xygets colored by c, ifd(x)≥D/2orc≤d(x)we replace cinTxby the\nlast color c′inTx(if any) and decrease the size of Txby one. The index of cinTxis found\nusing the pointer Px[c]; next Px[c′]andPx[c]are updated accordingly. The arrays TyandPy\nare updated analogously. When an edge xygets uncolored we proceed similarly.\nTo return a color from ¯π(v)we just pick the first color of Tx, if any (note that Txis\nnonempty when ¯π(v)is nonempty, even for d(v)< D/ 2).\nSampling a color from ¯π(v)is done differently depending on its degree. If d(v)≥D/2we\njust sample an element of Tv. Ifd(v)< D/ 2we keep sampling a color cfrom 1, . . . , Duntil\nc̸∈π(v). Since Pr(c∈π(v))≤d(v)/D < 1/2, we need two repetitions in expectation. Recall\nthat the condition c̸∈π(v)is checked in constant expected time using hash table Mv. Since\nthe random variables of the lookup time in Mvand the number of repetitions are independent,\nthe total expected time is constant.\n10Lemma 11. For an edge xy, one can find a color in ¯π(x)∩¯π(y), or report that it does not\nexist, in expected time O(min{d(x), d(y)}).\nProof.By symmetry, w.l.o.g. we assume d(x)≤d(y). For every c∈¯π(y)we check if c̸∈π(x)\ninO(1)expected time per color c. When c̸∈π(x), we stop and return c. If the loop ended\nwithout returning anything, we report that ¯π(x)∩¯π(y) =∅. For all iterations of this loop but\nthelastone c∈π(x), sothenumberofiterationsisatmost 1+|π(x)| ≤1+d(x)−1 =d(x).\nThe lemma below is rather standard (similar statements, but for different variants of the\nfan notion, can be found for example in Sinnamon [25] or Bhattacharya [4]).\nLemma 12. Given an uncolored x-weak edge xy1, one can find a minimal active fan F=\n(x, y1, . . .)in expected time O(d(x)). The algorithm also returns the color that makes (A1) or\n(A2) satisfied.\nProof.Our algorithm is as follows. We begin with the fan F1= (x, y1). Then we extend it\nvertex by vertex, until we get an active fan. Let us describe a single step of this process. We\nwill use a set Sof colors, initially (i.e., before all steps) empty, implemented using hashing.\nWe also use a queue Qstoring a subset of S. Let Fi= (x, y1, . . . , y i)be a fan built so far. We\nknow that i= 1orFi−1is not active (otherwise yiwould not be added). We maintain the\ninvariant that Sconsists of all the colors free at any yj,j < i. First we check if Fiis active.\nTo this end, for each color c∈¯π(yi)we check if c̸∈π(x)orc∈SinO(1)expected time per\ncolor c. If one of these checks is positive (A1) or (A2) holds, respectively, so we stop (without\nchecking further colors of ¯π(yi)) and return Fi. Otherwise we add cboth to SandQand\nproceed to the next c(or finish the loop). Let us call the loop described above as the activity\nchecking loop . When this loop finishes without returning a fan, we know that Fiis not active.\nAt this point we know by Lemma 9 that Fican be extended by a vertex yi+1. Our algorithm\nfinds yi+1as follows. We remove a color cfrom Q, and find the edge xzcolored with cusing\nMx. Ifd(z)< Dwe set z=yi+1, and otherwise we remove another color from Qand so on.\nThis loop will be called the extension loop .\nNow let us analyze the time complexity. Let F= (x, y1, . . . , y k)be the fan returned.\nClearly, each iteration of each of the two loops takes expected constant time, so it suffices to\nbound the total number of iterations. For the activity checking loop, note that by (A1) each\nbut the last iteration corresponds to a different color that is in π(x)(it is not the case for the\nlast iteration only if Fwas returned at this iteration). Hence the number of these iterations is\nat most |π(x)| ≤d(x)−1. Finally, the number of iterations in the extension loop is bounded\nby the number of colors inserted to Q. However, we insert to Qonly in the activity checking\nloop, one element per iteration. Hence the total number of insertions to Qis bounded by\nd(x)−1. It follows that the total number of iterations in both loops is O(d(x)), and each of\nthem is done in expected constant time, as required.\nNow we are going to show an algorithm which realizes the same goal as Lemma 12, but it\nis efficient if d(y1)is small, rather than d(x). As far as we know this is new, and perhaps a\nbit surprising.\nLemma 13. Given an uncolored x-weak edge xy1, one can find a minimal active fan F=\n(x, y1, . . .)in expected time O(d(y1)). The algorithm also returns the color that makes (A1)\nor (A2) satisfied.\n11Proof.Our algorithm is as follows. We begin with the fan F1= (x, y1). First the algorithm\nchecks if F1is active. Clearly, (A2) cannot hold, so (A1) is verified. This is done in time\nO(d(y1))using Lemma 11. If (A1) holds, we return F1. Hence in what follows we assume\n¯π(x)∩¯π(y1) =∅.\nThen we extend F1vertex by vertex, until we get an active fan. Let us describe a single\nstep of this process. Let Fi= (x, y1, . . . , y i)be a fan built so far. We maintain the invariant\nthatFiis not active. Such a step has two phases: (a) extend Fiby a vertex yi+1(by Lemma 9\nwe know that such a vertex exits) and (b) check if the new fan Fi+1is active.\nLet us focus on phase (a) now. We want to find a color c∈Si\nj=1¯π(yj)\\ {π(xyj)|j=\n2, . . . , i}such that there is an edge xzand colored with cwith d(z)< D; then we set yi+1=z.\nThe problem here is that both |Si\nj=1¯π(yj)|and the number of (< D)-neighbors of xcan be\nlarge, so we need to be careful. First, we use colors from ¯π(y1). More precisely, we use an\niterator zwhich iterates over the list N<D[x](initialized to the first vertex on that list, if any,\nonly once, before extending F1). We increment zuntil π(xz)̸∈π(y1). Note that the total\nnumber these increments is at most |π(y1)| ≤d(y1)−1; since one membership test is done in\nO(1)expected time using My1, this costs O(d(y1))in total.\nIf the above fails (iterator zreaches the end of the list) we know that for every (< D)-\nneighbor zofv, ifπ(vz)∈¯π(y1), then zwas used used to extend the fan earlier, i.e., z∈\n{y2, . . . , y i}. Hence we need to find yi+1among the endpoints of the edges colored by a color\nfromSi\nj=2¯π(yj). To this end, we use a queue of these colors Q(initially, i.e., before finding\nF2, empty). We remove a color cfrom Q, and find the edge xzcolored with cusing Mx. If\nd(z)< Dwe set z=yi+1, and otherwise we remove another color from Qand so on. Clearly,\nthe total expected time of this part is linear in the size ofSk−1\nj=2¯π(yj), where kis the size of\nthe minimal active fan Fkreturned by the algorithm. However, since Fk−1is not active, by\n(A2) we have |Sk−1\nj=2¯π(yj)| ≤D− |¯π(y1)|=|π(y1)| ≤d(y1)−1. This finishes the description\nof phase (a).\nNow we describe phase (b), i.e. checking if the new fan Fi+1is active. We will use a\nsetSof colors (initially, i.e., before finding F2, empty), implemented using hashing. We\nmaintain the invariant that S=Si\nj=2¯π(yj). For each color c∈¯π(yi+1)we check if c∈¯π(x)\norc∈Si\nj=1¯π(yj). If one of these checks is positive (A1) or (A2) holds, respectively, so we\nstop and return Fi+1. Note that these checks can be done in expected O(1)time: to check\nifc∈¯π(x)we just check the equivalent c̸∈π(x)using Mx, and the test if c∈Si\nj=1¯π(yj)is\ndone using two constant time queries: c̸∈π(y1)using My1andc∈S. If the check does not\nresult in returning Fi+1, we add cboth to SandQand proceed to the next c(or finish the\nloop).\nNow let us analyze the total time complexity of phase (b). Let F= (x, y1, . . . , y k)be the\nfan returned. Each iteration of the loop takes expected constant time, so it suffices to bound\nthe total number of iterations. Note that each but the last iteration corresponds to a different\ncolor in {1, . . . , D } \\¯π(y1) =π(y1), so the total number of iterations is bounded by d(y1), as\nrequired.\nBy combining Lemmas 12 and 13 we get the following.\nCorollary 14. Given an uncolored x-weak edge xy1, one can find a minimal active fan F=\n(x, y1, . . .)in expected time O(min{d(x), d(y1)}). The algorithm also returns the color that\nmakes (A1) or (A2) satisfied.\n12Now we are ready to prove the main theorem of this section. Its proof is basically an\nalgorithmization of the proof of Theorem 2.1 from Stiebitz et al. [26], using the results from\nthis section, summarized in Corollary 14.\nTheorem 15 (Efficient VAL) .LetGbe a simple graph and let e=xybe an x-weak edge.\nThen any partial D-edge-coloring πofGwhich colors a subset Ecof edges of G,e̸∈Eccan be\nextended to a partial D-edge-coloring that colors Ec∪{e}in expected time O(min{d(x), d(y)}+\n|P|), where Pis an alternating path with an endpoint in x. If needed, one can specify the color\nc′∈¯π(x)that alternates in P.\nProof.Lety1=y. We apply Corollary 14. Let F= (x, y1, . . . , y k)be the returned minimal\nactive fan. Then it satisfies (A1) or (A2). In the former case, we have a color c∈¯π(yk)∩\n¯π(x). Then we rotate Fusing Algorithm 1, which clearly runs in expected time O(k)since\nthe condition in line 3 is checked in expected constant time using My1. Recall that k=\nO(min{d(x), d(y)})thanks to Corollary 8. By property (F3), this results in the desired partial\nedge-coloring.\nNow assume Fdoes not satisfy (A1) but it does satisfy (A2), i.e., we have a color cthat\nis free both in ykandyifor some i= 1, . . . , k −1. Such yican be found simply by testing if\nc∈π(yj)for every j= 1, . . . , k −1: this takes expected time O(k) =O(min{d(x), d(y)}).\nLetc′be any color free at x(it exists since xyis uncolored, and if c′is not given, it can be\nfound in constant time using Lemma 10). Find a maximal (c, c′)-alternating path Pstarting\natx. This can be easily done in O(|P|)time using the associative arrays M∗. Then there is\nj∈ {i, k}such that the other endpoint of the path Pis not yj. We swap the color candc′on\nPinO(|P|)time, obtaining another partial coloring with the same set of colored edges, but\nwith c∈¯π(x)∩¯π(yj). Then we rotate Fjusing Algorithm 1.\nWe conclude this section with the following remark in case the results of this section turn\nuseful beyond our application in this work (which uses randomization also in other parts of\nthe coloring algorithm).\nRemark 1. All randomized algorithms in this section use randomization only in the imple-\nmentation of the associative arrays Mv. They can be made deterministic at the additional cost\nO(log log D)in the running of time, by using van Emde Boas trees instead of hashing.\n4 The main algorithm\nIn this section we present our main result. We will use the following lemma from [4].\nLemma 16 (Claim 3 in Bhattacharya et al. [4]) .LetGbe a simple graph and with a partial\nedge-coloring πwhich colors all but ℓedges of G. Let Ecbe the set of colored edges. Let ebe a\nrandom uncolored edge, let xbe an arbitrary endpoint of eand let c0be a random color of π(x).\nLetc1be an arbitrary color different than c0. Consider the unique maximal (c0, c1)-alternating\npath starting at x. Then, E(|Pr|) =O(1 +1\nℓP\nuv∈Ecmin{d(u), d(v)}).\nThe following is an easy consequence of the Nash-Williams Theorem on arboricity.\nLemma 17 (Folklore) .For any graph Gits arboricity is O(mad( G)).\nNow we are ready to prove our main result.\n13Theorem 1. Every graph Gwithnvertices and medges such that ∆(G)≥2 mad( G)andG\nhas no isolated vertices can be ∆(G)-edge-colored in expected time O(mlognmad( G)3). The\nrunning time bound holds also with high probability.\nProof.The algorithm is recursive: it invokes itself recursively for a subgraph of the given\ngraph. Consider a single recursive call and let H⊆Gbe the graph passed to the current\ncall. We have D= ∆( G)colors. We begin by removing all isolated vertices from H. If\nwe end up with the empty graph, we return the empty coloring. Otherwise, we find the set\nEwof weak edges of H. This can be easily done in linear time, by first computing, for each\nvertex vits degree d(v)and∆-degree d∆(v); then checking if an edge is weak directly from\ndefinition takes O(1)time. Next, we form a graph H′=H−Ewin time linear in the size of\nH. Then, if E(H′)̸=∅, we color H′recursively. Thus we obtain a partial coloring πofH.\nFinally, we color edges of Ewin random order, using Theorem 15 (the color c′mentioned in\nthe theorem statement is chosen randomly using Lemma 10). This completes the description\nof the algorithm.\nLet us analyze the time complexity of the single level of the recursion. By Theorem 15\nand Lemma 16, when ℓ= 1, . . . ,|Ew|edges are still uncolored, coloring a random uncolored\nedge xyofEwtakes expected time\nO \nmin{d(x), d(y)}+1\nℓX\nuv∈Ecmin{d(u), d(v)}!\n, (1)\nwhere Ecis the set of currently colored edges of H.\nSince each edge is colored once, by Theorem 3 and Lemma 17, the total contribution of\nthe first term, on all the levels of the recursion, is O(|E(G)|mad( H)) =O(mmad( G)), so we\ncan focus on the second term.\nAgain by Theorem 3 and Lemma 17, the total contribution of the second term of (1) to a\nsingleleveloftherecursionis O(H|Ew|P\nuv∈E(H)min{d(u), d(v)}) =O(H|Ew||E(H)|mad( H)),\nwhere H|Ew|is the|Ew|-th harmonic number. Since H|Ew|<ln|Ew|+1 = O(logn)we get the\nbound O(|E(H)|lognmad( G)). We see that it dominates the O(|E(H)|)time needed to find\nEwandH′. To sum up, a single level of the recursion takes time O(|E(H)|lognmad( G)).\nNote that if ∆(H)<∆(G) =D, then every edge of His weak and there are no more\nrecursive calls. Otherwise, we have D= ∆( H)and∆(H) = ∆( G)≥2 mad( G)≥2 ad(H), so\nby Theorem 4 we have |E(H′)| ≤r|E(H)|for\nr= 1−1\n2 ad(H)2≤1−1\n2 mad( G)2.\nHence, all the Llevels of the recursion take expected time\nO L−1X\ni=0ri|E(G)|lognmad( G)!\n.\nSincePL−1\ni=0ri<P∞\ni=0ri= 2 mad( G)2, the total time complexity is O(mlognmad( G)3)in\nexpectation, as required.\nObtaining the high probability guarantee for the running time is fairly standard, we move\nit to Appendix A.\n145 Conclusion and further research\nWe showed that every graph Gcan be edge-colored using the optimal number of ∆colors in\nquasilinear expected time, whenever ∆≥2 mad( G). Some of tools we developed may be of\nindependent interest, e.g., an efficient implementation of the Vizing Adjacency Lemma. We\nconclude with some open problems as follows.\n1. Is it possible to get a deterministic quasilinear time in Theorem 1? To this end, it is\nsufficient to find a different way of controlling the total length of augmenting paths.\n2. Isitpossibletorelaxourassumptionto ∆≥cmad( G)forc <2? AsshowninTheorem5\nthis would involve something more than the VAL in the coloring procedure. Since the\ngraphs constructed in Theorem 5 have arbitrarily large minimum degree, it is also not\nsufficient to find reducible configurations with vertices of bounded degree, as it was the\ncase in the work of Cole and Kowalik [12].\n3. Is it possible to design a data structure for dynamic ∆-edge-coloring graphs of bounded\nmaximum average degree, assuming ∆≥cmad( G)for a constant c, with updates in\nsublinear time? Or in polylogarithmic time? Note that the dynamic edge-coloring is an\nactive area of research recently, see e.g. [15, 8, 3, 9].\n4. Is it possible to implement a version of Theorem 1 in the LOCAL model using only\npoly(∆ ,logn)rounds? For some recent related results see e.g. [9, 1].\nAcknowledgements\nThe author wishes to thank Bartłomiej Bosek, Jadwiga Czyżewska, Konrad Majewski and\nAnna Zych-Pawlewicz for helpful discussions on related topics.\nReferences\n[1] Anton Bernshteyn. A fast distributed algorithm for (∆ + 1)-edge-coloring. J. Comb.\nTheory, Ser. B , 152:319–352, 2022. URL: https://doi.org/10.1016/j.jctb.2021.10.\n004,doi:10.1016/J.JCTB.2021.10.004 .\n[2] Anton Bernshteyn and Abhishek Dhawan. Fast algorithms for Vizing’s theorem on\nbounded degree graphs. CoRR, abs/2303.05408, 2023. URL: https://doi.org/10.\n48550/arXiv.2303.05408 ,arXiv:2303.05408 ,doi:10.48550/ARXIV.2303.05408 .\n[3] Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. Arboricity-\ndependent algorithms for edge coloring. CoRR, abs/2311.08367, 2023. URL: https:\n//doi.org/10.48550/arXiv.2311.08367 ,arXiv:2311.08367 ,doi:10.48550/ARXIV.\n2311.08367 .\n[4] Sayan Bhattacharya, Martín Costa, Nadav Panski, and Shay Solomon. 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CoRR, abs/1907.03201,\n2019. arXiv:1907.03201 .\n[26] MichaelStiebitz, DiegoScheide, BjarneToft, andLeneM.Favrholdt. Graph edge coloring:\nVizing’s theorem and Goldberg’s conjecture . Wiley Hoboken, NJ, Hoboken, NJ, 2012.\n[27] Vadim G. Vizing. On the estimate of the chromatic class of a p-graph.Diskret. Analiz ,\n3:25–30, 1964.\n[28] Vadim G. Vizing. Critical graphs with a given chromatic number. Diskret. Analiz , 5:9–17,\n1965.\n[29] Vadim G. Vizing. Some unsolved problems in graph theory (in Russian). Uspeki Mat.\nNauk., 23:117–134, 1968. Englishtranslationin[ Russian Mathematical Surveys 23(1968),\n125-141].\n[30] Douglas R. Woodall. The average degree of an edge-chromatic critical graph II. Journal\nof Graph Theory , 56(3):194–218, 2007. doi:https://doi.org/10.1002/jgt.20259 .\n[31] Douglas R. Woodall. Erratum: The average degree of an edge-chromatic critical graph\nII.Journal of Graph Theory , 92(4):488–490, 2019. doi:https://doi.org/10.1002/jgt.\n22501.\n[32] Xiao Zhou, Shin ichi Nakano, and Takao Nishizeki. Edge-coloring partial k-trees.Journal\nof Algorithms , 21(3):598–617, 1996. URL: https://www.sciencedirect.com/science/\narticle/pii/S0196677496900619 ,doi:https://doi.org/10.1006/jagm.1996.0061 .\n17A High probability guarantee for the running time\nThere are three different randomized mechanisms in the algorithm:\n(i)choosing a random weak edge to color, which influences the length of an alternating\npath that is found and swapped.\n(ii)hash tables that implement the associative arrays Mvand sets Sin lemmas 12 and 13,\n(iii)sampling a free color using Lemma 10,\nBegin with (i). To this end, we focus on the second term of the expression (1), which\ncorresponds to the expected length of the relevant alternating path in the iteration when there\nare still ℓweak edges uncolored — call the corresponding random variable Xℓ,ℓ= 1, . . . ,|Ew|.\nFirst consider the case |V(H)| ≤4n1/2ad(H). Then, at this and all deeper levels of the\nrecursion every alternating path has length at most |V(H)| −1. Since the number of levels is\nat most |E(H)|=O(n1/2ad(H)2)we get that total length of alternating paths on this and\ndeeper levels is O(nad(H)3)with probability 1.\nNow assume |V(H)| ≥4n1/2ad(H). We proceed as Bernshteyn and Dhawan [2], Section\n4. We will need a version of multiplicative Azuma’s Inequality, as follows.\nTheorem 18 (Corollary 6 in Kuszmaul and Qi [22]) .LetX1, . . . , X n∈[0, c]be real-valued\nrandom variables with c > 0. Suppose that E(Xi|X1, . . . , X i−1)≤aifor all i. Let µ=Pn\ni=1ai. Then for any δ >0,\nPr X\niXi≥(1 +δ)µ!\n≤exp\u0012\n−δ2µ\n(2 +δ)c\u0013\n.\nNote that in fact for i= 1, . . . ,|Ew|we have E(Xi|X1, . . . , X i−1)≤aiforai= Θ(1 +\n1\ni|E(H)|mad( G))andXitakesvaluesin [0,|V(H)|]. Set µ=P|Ew|\ni=1ai= Θ( H|Ew|·|E(H)|mad( G)) =\nΘ((log |Ew|)· |E(H)|mad( G)). By Theorem 18, we get that for any δ >0,\nPr\n|Ew|X\ni=1Xi≥(1 +δ)µ\n≤exp\u0012\n−δ2µ\n(2 +δ)|V(H)|\u0013\n≤exp(−Ω(δlog|Ew|)) =1\n|Ew|Ω(δ).\nSince by Theorem 4, |Ew| ≥ | E(H)|/(2 ad( H)2) =|V(H)|/(4 ad( H))≥n1/2, we get that\nPr\u0010P|Ew|\ni=1Xi≥(1 +δ)µ\u0011\n≤n−Ω(δ).It follows that with high probability the sum of all alter-\nnating path lengths that were found and swapped is O(µ), as required.\nThe mechanisms (ii)and(iii)are used for coloring a weak edge. According to expres-\nsion (1), the total time of this processes can be divided into (a) finding and rotating a fan,\nwhich takes O(P\nxy∈Emin{d(x), d(y)}) =O(mad( G)m)expected time in total and (b) finding\nalternating paths and swapping colors on them, which dominates the running time and takes\ntime O(mlognmad( G)3)w.h.p., as we have shown in the proof of Theorem 1.\nObserve that since (a) is much cheaper, we can actually afford for implementing the as-\nsociative arrays Mvand sets Sin lemmas 12 and 13, using balanced BST trees in O(logn)\nworst-case time for update or lookup. When the BST tree for Mvthat stores the colors of\nπ(v)is additionally augmented so that every node stores also the size of the subtree, it is easy\n18to sample a free color in O(logn)worst-case time: just sample an integer ifrom 1to|¯π(v)|\nand find the i-th free color O(logn)using the tree Mvby standard methods.\nThus we have replaced the mechanisms (ii)and(iii)by deterministic procedures, so that\npart (a) takes only O(mlog mad( G))total worst-case time. We are left with part (b), which\nusesO(mlognmad( G)3)lookups and updates to Mv. If we just aim for quasilinear time and\nwe do not mind another O(logn)factor we can just use the BSTs; using van Emde Boas trees\n(see Remark 1) improves this to O(log log n). However, the slow down can be avoided as\nfollows. Note that when we swap colors at an internal vertex vof the path, we do not change\nthe keys in Mv, we only change (swap) the values. Hence, there are only two updates of the\nkeysper augmenting path (at the endpoints), and since we use at most one path for a weak\nedge that is being colored, the total number of these updates is O(m). However, there are still\nO(mlognmad( G)3)lookups and updates of values in the associative arrays Mv. A solution is\nto implement Mvusing cuckoo hashing [23]; indeed, it performs lookups and updates of values\ninO(1)worst-case time, while key updates are done in time which is O(1)in expectation and\nO(logn)w.h.p.4. Then the total time of processing the internal vertices of the augmenting\npathsisworst-case O(mlognmad( G)3)conditionedontheeventthatthisisthetotallengthof\nthe augmenting paths, while the time of part (a) becomes O(mad( G)mlogn)w.h.p. (instead\nof worst-case when using only BSTs.) Thus the issues with (ii)and(iii)are resolved.\n4The latter assuming that the structure takes Ω(n)space, so instead of having a separate array Mvfor\nevery vertex v, we should rather use a single cuckoo-hashing hash table Mwhich maps pairs (v, c)to an edge\nincident with vand colored c. Note also that for fast sampling we still need the augmented BSTs as well.\n19"    },    {        "title": "2401.13847v1.Quantum_weight.pdf",        "content": "Quantum weight\nYugo Onishi1and Liang Fu1\n1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n(Dated: January 26, 2024)\nWe introduce the concept of quantum weight as a fundamental property of insulating states of\nmatter that is encoded in the ground-state static structure and measures quantum fluctuation in\nelectrons’ center of mass. We find a sum rule that directly relates quantum weight—a ground\nstate property—with the negative-first moment of the optical conductivity above the gap frequency.\nBuilding on this connection to optical absorption, we derive both an upper bound and a lower bound\non quantum weight in terms of electron density, dielectric constant, and energy gap. Therefore,\nquantum weight constitutes a key material parameter that can be experimentally determined from\nX-ray scattering.\nIt is well known that quantum states of matter with\nan energy gap have vanishing dc longitudinal conductiv-\nity at zero temperature, while the optical conductivity\nis generally nonzero at frequencies above the gap. Inter-\nestingly, the ground state property of an insulating state\nstill has important bearings on its optical conductivity.\nConsider the real part of longitudinal optical conductiv-\nityReσxx(ω), which determines the amount of optical\nabsorption in the medium. It is well known that its ze-\nroth moment known as the optical spectral weight is re-\nlated to the electron density in the system through the\nf-sum rule [1]. Higher order moments of optical conduc-\ntivity are much less studied. The negative-second mo-\nment is directly related to electric susceptibility through\nthe Kramers-Kronig relation [2]. Recently, we employed\nthe negative-first moment of both longitudinal and Hall\nconductivities to derive a universal upper bound on the\nenergy gap of (integer or fractional) Chern insulators [3].\nWhile the Hall conductivity only exists in the presence\noftimereversalsymmetrybreaking, thelongitudinalcon-\nductivity is present in all systems. In a pioneering early\nwork[4],Souza,WillkinsandMartin(SWM)showedthat\nthe negative-first moment of σxx(ω)is related to a ba-\nsic property of quantum insulators that was expressed\nin terms of the quantum metric of many-body ground\nstates over twisted boundary condition. This quantity\nwas recently termed “quantum weight” because it con-\nnects quantum metric and optical weight [3]. Despite its\nubiquitous presence in solids, the quantum weight has\nnot received adequate attention, and to our knowledge,\nits value has not been experimentally determined for any\nmaterial.\nIn this work, we provide a new and general definition\nof quantum weight for all insulators as the quadratic co-\nefficient of the ground-state static structure factor Sqat\nsmall q. For periodic systems, Sqat any qother than\nreciprocal-lattice points vanishes in the classical limit\n(ℏ= 0), and is nonzero only because of quantum fluc-\ntuation in electron position. Therefore, the new defini-\ntion of quantum weight manifests its purely quantum-\nmechanical origin, and allows it to be experimentally\nmeasured by X-ray scattering.Usingfluctuation-dissipationtheorem[5],weshowthat\nas a ground state property, the quantum weight defined\nhere is directly related to the negative-first moment of\noptical conductivity through the Planck constant. This\nestablishes the equivalence between the old and new defi-\nnition of quantum weight in terms of optical conductivity\nand static structure factor respectively.\nNext, we derive lower and upper bounds on the quan-\ntum weight, in terms of the electron density, the energy\ngap, and the dielectric constant. These bounds apply to\nany insulating system, and represent a universal relation\nbetween ground state property (quantum weight), op-\ntical response, thermodynamic response (dielectric con-\nstant), and the energy gap of the system. Remarkably,\nour bounds can provide a good estimate of the quantum\nweight of real materials.\nWe start by considering generalized optical weights Wi\nforinsulatingstates[3]. Wiisdefinedasthenegative i-th\nmomentoftheabsorptivepartoftheopticalconductivity\nσ(ω),\nWi\nαβ≡Z∞\n0dωσabs\nαβ(ω)\nωi, (1)\nwhere σabs\nαβ≡(σαβ+σ∗\nβα)/2is composed of the real\npart of longitudinal optical conductivity and the imagi-\nnary part of optical Hall conductivity. For isotropic two-\ndimensional systems, σabs\nαβ= Re( σxx)δαβ+iIm(σxy)ϵαβ.\nThroughout this work, we consider gapped systems so\nthat the absorptive part of optical conductivity vanishes\nat low frequency:\nσabs(ω) = 0forℏω≤Eg. (2)\nThis condition defines the optical gap Egwhich must be\nequal to or greater than the spectral gap ∆. We will\nconsider three optical weights: W0, W1andW2, and fo-\ncus on their real part, which is associated with optical\nlongitudinal conductivity and is present in any solids.\nReW0andReW2are related to the charge density\nand the electric susceptibility, respectively:\nReW0=πne2/(2m), (3)\nReW2= (π/2)χ. (4)arXiv:2401.13847v1  [cond-mat.str-el]  24 Jan 20242\nEq.(3) is the well-known fsum rule [6] that relates the\nfull optical spectral weight to the electron density nand\nmass m. Eq.(4)wasrecentlyderivedfromtherelationbe-\ntween conductivity and polarizability using the Kramers-\nKronig relation [2]. Here, χdescribes the polarization\ninduced by a static external electric field, and is directly\nrelated to the dielectric constant χ=ϵ0(ϵ−1)with ϵ0\nthe vacuum permittivity.\nOur focus in this work is on the real part of the\nnegative-first moment, ReW1. As we will show, the op-\ntical weight ReW1is directly related to the quantum\nfluctuation in electron position, even though this quan-\ntity itself can be defined even for classical systems. To\nmotivate the discussion, let us first consider a system of\nhighly localized electrons arising from strong potential\nand interaction effects. In the classical limit ( ℏ= 0), we\ncan treat electrons as point charges and the ground state\nis obtained by minimizing the sum of the potential en-\nergy and the electron-electron interaction energy (which\nonly depend on electron position):\nHc=X\niV(xi) +X\ni,jU(xi−xj), (5)\nwith Vthepotentialand Uthetwo-bodyinteraction. On\nthe other hand, the kinetic energy\nH0=X\nip2\ni\n2m, (6)\nleads to quantum fluctuation in electron position, which\nwe treat below by quantizing electron’s motion around\nthe ground state configuration.\nWe expand Hcup to the second order in the dis-\nplacement of the electrons from the ground state posi-\ntion, which yields a spring constant matrix k:Hc=\nEc+P\ni,jkijδxiδxj/2. Diagonalizing k, we obtain the\nnormal modes x′\nα=P\nicαiδxi(withP\nicαicβi=δαβ)\nand the spring constant kα:Hc=P\nαkαx′\nα2/2. Corre-\nspondingly, we can rewrite the total kinetic energy HK\nwith the momentum conjugate to the normal modes,\np′\nα=P\nicαipi, asP\nαp′\nα2/(2m). Then we obtain a col-\nlection of independent harmonic oscillators, one for each\nnormal mode,\nH=H0+Hc\n≈X\nαp′\nα2\n2m+1\n2kαx′\nα2+. . . . (7)\nThe frequency for each mode is ωα=p\nkα/m, leading to\nthe energy gap of ℏωαafter quantization.\nFor this system of strongly localized electrons, a uni-\nform electric field couples to the center of mass displace-\nmentP\niδxi=P\nαZαx′\nα, and the real part of the optical\nconductivity is well known from that of the harmonic os-cillator:\nReσ(ω) =1\nLX\nαπZ2\nαe2\n2mδ(ω−ωα), (8)\nwith Zαe=P\nicαiethe effective charge of the nor-\nmal mode α. Here, for simplicity we consider a one-\ndimensional system with length Lto illustrate the essen-\ntial physics. It then follows\nReW1=1\nLX\nαπZ2\nαe2\n2mωα. (9)\nOne can readily verify thatP\nαZ2\nα/L=nwith the elec-\ntron density n, as expected from the f-sum rule.\nInterestingly, ReW1isrelatedthroughthePlanckcon-\nstant to the quantum fluctuation in the polarization of\nour system, which arises from electron’s zero-point mo-\ntion and is obtained by quantization: p′\nα→ − iℏ∂′\nα.\nNoting that the total polarization is given by δP=\neP\niδxi=eP\nαZαx′\nαandD\nx′\nαx′\nβE\n=δαβℏ/(2mωα), we\ncan rewrite ReW1as\nℏ\nπReW1=1\nL\n(δP)2\u000b\n, (10)\nwhere ⟨. . .⟩is the expectation value in the ground state.\nThis relation Eq. (10) is noteworthy for two reasons.\nThe optical weight ReW1in the left-hand side is finite\neven for classical systems, while the polarization fluctua-\ntion in the right-hand side is purely quantum-mechanical\nin nature and vanishes in ℏ→0limit. The left hand side\ninvolves optical conductivity at all frequencies, while the\nright hand side is a ground state property.\nIn order to properly define polarization fluctuation for\ninsulators in general, let us consider the static struc-\nture factor that measures equal-time density-density cor-\nrelation in the ground state: Sq≡(1/V)(⟨ˆρqˆρ−q⟩ −\n⟨ˆρq⟩⟨ˆρ−q⟩). In particular, we focus on the quadratic co-\nefficient of Sqat small nonzero q:\nSq=e2\n2πKαβqαqβ+. . . , (11)\nwith ˆρq=R\ndre−iq·rˆρ(r)the charge density operator\nwith wavevector qandVthe volume of the system. We\ncall the quadratic coefficient Kdefined by Eq. (11) quan-\ntum weight . Importantly, Kcan be nonzero only because\nof quantum fluctuation. In the classical limit ( ℏ= 0), the\nground state of a periodic system is a periodic array of\nelectron point charges, and correspondingly, the static\nstructure factor is composed of δfunctions centered at\nreciprocal lattice vectors and vanishes everywhere else,\nleading to K= 0. Hence quantum weight measures the\ndegree of “quantumness” in the insulating ground state.\nTo make the connection between polarization fluctu-\nation and quantum weight, note that polarization is re-\nlated to charge density through ∇ ·P=−ρ. There-\nfore, one may identify −iqδP =ρqand then quantum3\nweight defined by Eq. (11) represents the polarization (or\ncenter-of-mass position) fluctuation, in the ground state:\nK=\n(δP)2\u000b\n/(2πe2V). In contrast to our treatment\nbased on static structure factor, previous works treated\npolarization fluctuation in terms of the second cumulant\nmoment of the position operator for systems with open\nor twisted boundary conditions [4, 7].\nMotivated by the case of strongly localized electron\nsystems discussed above, we now establish a general re-\nlation between the optical weight ReW1and the quan-\ntum weight Kencoded in ground-state static structure\nfactor Sq, which captures quantum fluctuation in elec-\ntrons’ center of mass of all insulators. Let us consider\nthe response of an insulator to a time-dependent peri-\nodic potential Vextwith wavevector qand frequency ω.\nThe induced change in the density and the current re-\nsponse are characterized by the density-density response\nfunction Π(q, ω)and the conductivity tensor σ(q, ω)re-\nspectively:\nρ(q, ω) =−Π(q, ω)Vext(q, ω), (12)\nj(q, ω) =σ(q, ω)E(q, ω), (13)\nwhere E(q, ω) =−iqVext(q, ω)is the external electric\nfield. Due to the continuity equation ∂ρ/∂t+∇ ·j= 0,\nσandΠare directly related:\nΠ(q, ω) =iqαqβσαβ(q, ω)\nω. (14)\nIt is important to note that there is no singularity at\nsmall ωand at small qin Eq. (14) for insulators [8].\nCorrespondingly, the negative-first moment of the real\npart of optical conductivity at q= 0is related to the\nimaginary part of density response Π(q, ω)at small q\nintegrated over the frequency:\nZ∞\n0dωIm Π(q, ω) =qαqβRe(W1\nαβ) +. . . . (15)\nBy the fluctuation-dissipation theorem [5], Im Π(q, ω)is\ndirectly related to the dynamical structure factor:\nIm Π(q, ω) =S(q, ω)/(2ℏ), (16)\nwhere the dynamical structural factor is defined as\nS(q, ω) =1\nVR∞\n−∞dt eiωt⟨ˆρq(t)ˆρ−q(0)⟩. Here, ˆρq(t) =R\ndre−iq·rˆρ(r, t)is the density operator with wavevec-\ntorqin the Heisenberg picture and Vis the volume of\nthe system.\nCombining Eq. (14), (15) and (16), we obtain a general\nrelation between optical conductivity and ground state\nstatic structural factor:\nqαqβZ∞\n0dωReσabs\nαβ(q, ω)\nω=1\n2ℏZ∞\n−∞dω S(q, ω)\n=π\nℏSq (17)where we have used S(q, ω) = 0forω <0at zero tem-\nperature (for more details see Supplemental Materials).\nTaking q→0limit, we obtain a relation between optical\nweight ReW1atq= 0and the quantum weight defined\nabove as the quadratic coefficient of Sq:\nReW1=Z∞\n0dωReσabs(ω)\nω=e2\n2ℏK.(18)\nEq. (17), together with its q→0limit Eq. (18), is a key\nresult of our work. It constitutes a new optical sum rule\nrelating the negative first moment of longitudinal optical\nconductivitytoground-statestructurefactorthroughthe\nPlanck constant, which is a generalization of the fsum\nrule relating the optical spectral weight to the electron\ndensity.\nThe static structure factor Sqcan be experimentally\nobtained from X-ray scattering experiments, while the\noptical weight ReW1can be determined from the ex-\nperimentally measured optical conductivity. The ratio\nbetween Sq/q2andReW1yields a fundamental physical\nconstant, the Planck constant. This provides a way of\ndetermining the Planck constant by optical spectroscopy\nmeasurements of basic material properties.\nWe have shown that quantum weight is both a funda-\nmental ground state property of insulators and an impor-\ntantmaterialparameterrelatedtoopticalproperties. We\nnow further derive lower and upper bounds on quantum\nweight in terms of common material parameters: elec-\ntron density, energy gap and dielectric constant. First,\nnoting that the real part of the longitudinal optical con-\nductivity is always non-negative and can be finite only\nfor|ω| ≥Eg/ℏ, the following inequality among optical\nweights always holds:\nReWi+1\nαα≤ℏReWi\nαα/Eg. (19)\nCombining Eq. (19) with i= 0,1and the expressions for\noptical weights W0, W1, W2shown in Eq. (3), (4) and\n(18), we obtain the bound on the quantum weight:\nπ\ne2Egχαα≤Kαα≤2ℏ2\ne2W0\nαα\nEg. (20)\nHere, αis the principal axis of the material. The stan-\ndard f-sum rule, W0=πne2/(2m)further leads to a\nuniversal bound on the quantum weight as\nπ\ne2Egχαα≤Kαα≤πnℏ2\nmEg. (21)\nThis inequality should hold for any system regardless of\nthe dimensionality of the system. Both the lower and\nupper bounds are saturated when optical conductivity is\nnonzero only at single frequency ω=±Eg/ℏ, which we\nshall call single frequency absorption. We also note that\nthe inequality (21) remains valid even when the optical4\nFIG. 1. The bound on the quantum weight Kfor real materi-\nals. As the energy gap, we used the gap obtained from optical\nmeasurements. The details of the used parameters are given\nin Ref. [2].\ngapEgis replaced with the spectral gap, although the\nbounds would generally become less tight.\nWe calculated the bound on the quantum weight for\nreal materials, and the results are shown in Fig. 1. Our\nbound (21) gives a fairly good estimate of the quantum\nweight only from the electric susceptibility (or equiva-\nlently the dielectric constant), the energy gap, and the\nelectron density. The most remarkable case is cubic\nboron nitride (c-BN). c-BN is an indirect gap insulator\nwith direct gap 14.5 eVand the dielectric constant ϵ=\n4.46[9]. From the dielectric constant, the electric suscep-\ntibility χ=ϵ0(ϵ−1)is given by χ= 3.1×10−11F m−1.\nThe cubic unit cell of c-BN has a lattice constant 3.62Å\nand contains 4 boron and nitrogen atoms, each having\n5 and 7 electrons, hence the total electron density is\nn= 1.02×1024cm−3. Then the full spectral weight\nis given by W0\naa= 4.5×1022m−1s−1Ω−1. Since K\nandχare isotropic in c-BN, we find that the quantum\nweight Kaais bounded to a remarkably narrow interval:\n0.9Å−1≤Kaa≤1.7Å. It should be emphasized that\nour analysis applies to anyelectronic systems, includ-\ning disordered and/or interacting systems. This result\ndemonstratesthatouranalysisispowerfulinunderstand-\ning quantum materials.\nFinally, for the sake of completeness, we discuss the\nrelation of the quantum weight Kdefined by static struc-\nture factor to quantum geometry. As first shown by\nSWM, the first-negative moment of optical conductiv-\nityReW1is related to the many-body quantum metric\ndefined over the twisted boundary condition [4]. Then,\nbased on the relation Eq. (18), the quantum weight K\nis also related to the many-body quantum metric. Here,\nwe generalize the work of SWM which considers systems\nwith a unique ground state and introduce the many-body\nquantum metric Gfor the general case of r-fold degener-\nated ground states (related to each other by spontaneoussymmetry breaking or topological order):\nGαβ(θ)≡1\nrrX\ni=1Re⟨∂αΨiθ|(1−Pθ)|∂βΨiθ⟩.(22)\nHere, |Ψiθ⟩is the i-th ground state under the twisted\nboundary condition specified by θ:Ψiκ(r1, . . . ,rn+\nLµ, . . . ,rN) =eiθµΨiκ(r1, . . . ,rn, . . . ,rN)where Lµ=\n(0, . . . , L µ, . . . , 0)with Lµthe system size in µ-direction,\nandPθ=|Ψθ⟩⟨Ψθ|is the projector operator associated\nwith the ground state.\nAssuming that the optical conductivity in the thermo-\ndynamic limit is insensitive to the choice of boundary\ncondition θor the particular ground state, one can show\n[3] that the optical weight ReW1and hence the quantum\nweight Kare related to Gas\nK= 2πZddθ\n(2π)dG(θ). (23)\nFor noninteracting band insulators, the ground state is\nalways unique and takes the form of a Slater determinant\nof occupied states over all wavevectors kin the Brillouin\nzone. Then, the many-body quantum metric Greduce\nto an integral in k-space:\nKαβ= 2πZddk\n(2π)dgαβ(k). (24)\nwhere g(k)is a quantum metric in k-space defined as:\ngαβ(k) = Re[ ⟨∂αΨ(k)|(1−P(k))|∂βΨ(k)⟩],(25)\nwith P(k) =|Ψ(k)⟩⟨Ψ(k)|the projection operator onto\nthe Slater determinant of occupied states at wavevector\nk:|Ψ(k)⟩=|u1(k). . . u s(k)|. Equivalently, g(k)is equal\nto the trace of non-Abelian quantum metric of occupied\nbands [10]. This quantity appears as the gauge-invariant\nterm in the localization functional of the Wannier func-\ntions [11]. Thus, Kis related to the degree of localization\nof occupied electron states, consistent with our result\non the quantum weight for strongly localized electron\nsystems ( ℏ→0) given above. We also note a relation\nbetween electron localization length and energy gap for\nnoninteracting disordered systems in one dimension [12],\nwhich is a special case of our general relation between the\nquantum weight and the energy gap for all insulators.\nTo conclude, we have established quantum weight, de-\nfined through static structure factor, as a key material\nparameter that is connected to a variety of physical ob-\nservables. The quantum weight represents the quantum\nfluctuation in electrons’ center of mass. We derived its\ngeneral relation to optical conductivity, dielectric con-\nstant, quantum geometry, and energy gap. Our results\napply to all insulators, including strongly correlated sys-\ntems. Experimental determination by X-ray scattering\nas well as first-principles calculation of quantum weight\nfor real materials [13] are called for.5\nThis work was supported by National Science Founda-\ntion(NSF)ConvergenceAcceleratorAwardNo. 2235945.\nYO is grateful for the support provided by the Funai\nOverseas Scholarship. LF was partly supported by the\nDavid and Lucile Packard Foundation.\n[1] R. Kubo, Statistical-Mechanical Theory of Irreversible\nProcesses. I. General Theory and Simple Applications\nto Magnetic and Conduction Problems, Journal of the\nPhysical Society of Japan 12, 570 (1957).\n[2] Y. Onishi and L. Fu, Universal relation between en-\nergygapanddielectricconstant(2024),arXiv:2401.04180\n[cond-mat].\n[3] Y. Onishi and L. Fu, Fundamental bound on topological\ngap (2023), arXiv:2306.00078 [cond-mat].\n[4] I. Souza, T. Wilkens, and R. M. Martin, Polarization and\nlocalization in insulators: Generating function approach,\nPhysical Review B 62, 1666 (2000), publisher: American\nPhysical Society.\n[5] H. B. Callen and T. A. Welton, Irreversibility and Gen-\neralized Noise, Physical Review 83, 34 (1951), publisher:\nAmerican Physical Society.\n[6] R. Kubo, M. Yokota, and S. Nakajima, Statistical-MechanicalTheoryofIrreversibleProcesses.II.Response\nto Thermal Disturbance, Journal of the Physical Society\nof Japan 12, 1203 (1957).\n[7] R. Resta, Polarization Fluctuations in Insulators and\nMetals: New and Old Theories Merge, Physical Review\nLetters 96, 137601 (2006), publisher: American Physical\nSociety.\n[8] We note that the static structure factor for metallic\nsystems at small qis dominated by |q|-linear term, in\ncontrast to insulators where the leading order term is\nquadratic.\n[9] O. Madelung, Semiconductors: data handbook (Springer\nScience & Business Media, 2004).\n[10] S. Peotta and P. Törmä, Superfluidity in topologically\nnontrivial flat bands, Nature Communications 6, 8944\n(2015), number: 1 Publisher: Nature Publishing Group.\n[11] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and\nD. Vanderbilt, Maximally localized Wannier functions:\nTheory and applications, Reviews of Modern Physics 84,\n1419 (2012), publisher: American Physical Society.\n[12] S. Kivelson, Wannier functions in one-dimensional dis-\nordered systems: Application to fractionally charged\nsolitons, Physical Review B 26, 4269 (1982), publisher:\nAmerican Physical Society.\n[13] B. Ghosh, Y. Onishi, S.-Y. Xu, H. Lin, L. Fu, and\nA. Bansil, Probing quantum geometry through opti-\ncal conductivity and magnetic circular dichroism (2024),\narXiv:2401.09689 [cond-mat].\nSupplemental Material\nQuantum weight in atomic insulators\nTo gain insight into the quantum weight, let us consider a solid made of an array of atoms that are far away\nfrom each other, so that the hopping between the atoms is negligible leading to trivial flat bands. In such atomic\ninsulators, optical absorption comes from electric dipole transitions between occupied and unoccupied energy levels\nwithin individual atoms. It is straightforward to show that the quantum weight Kis given by the ratio of the\nintra-atomic position fluctuation to the area of the unit cell A0:\nKAI\nαβ=2π\nA0⟨Φ|(rα− ⟨rα⟩)(rβ− ⟨rβ⟩)|Φ⟩ (26)\nwhere |Φ⟩denotes the ground state of an atom, ⟨rα⟩=⟨Φ|rα|Φ⟩is the expectation value of rαin the ground state,\nandrα=P\nirα,iis the sum of the position of each electron. Notably, the right-hand side of Eq. (26) is precisely the\npolarization (=center-of-mass) fluctuation in an array of atoms, as discussed in the main text.\nDetailed derivation of the relation between the quantum weight and the optical weight\nHere we derive the following relation between the quantum weight Kand the negative-first moment of optical\nconductivity W1for general insulators:\nReW1=e2\n2ℏK, (27)\nwhere W1is defined as\nW1=Z∞\n0dωσabs(ω)\nω(28)6\nwith the absorptive part of the conductivity tensor σabs(ω) = (σ(ω) +σ(ω)†)/2and the quantum weight is defined as\nthe quadratic coefficient in qof the static structure factor Sq= (1/V)⟨ρqρ−q⟩as\nSq=e2\n2πKαβqαqβ+. . . . (29)\nTo show Eq. (27), we use the dissipation fluctuation theorem:\nIm Π(q, ω) =S(q, ω)/(2ℏ), (30)\nwhere Π(q, ω)describes the density response to the external potential Vext, as defined in the main text. We will also\nuse the following relation between S(q, ω)andS(−q,−ω):\nS(q, ω) =eβℏωS(−q,−ω). (31)\nThis relation can be shown as follows:\nS(q, ω) =1\nVZ∞\n−∞dt eiωt⟨ˆρq(t)ˆρ−q(0)⟩ (32)\n=1\nVX\nn,mZ∞\n−∞dt eiωte−βεn\nZ⟨n|ˆρq(t)|m⟩ ⟨m|ˆρ−q(0)|n⟩ (33)\n=1\nVX\nn,mZ∞\n−∞dt ei(ω+(εn−εm)/ℏ)te−βεn\nZ⟨n|ˆρq|m⟩ ⟨m|ˆρ−q|n⟩ (34)\n=1\nVX\nn,m2πδ(ω+εn−εm\nℏ)e−βεn\nZ⟨n|ˆρq|m⟩ ⟨m|ˆρ−q|n⟩ (35)\n=1\nVX\nn,m2πδ(ω−εn−εm\nℏ)e−βεm\nZ⟨m|ˆρq|n⟩ ⟨n|ˆρ−q|m⟩ (36)\n=eβℏω1\nVX\nn,m2πδ(−ω+εn−εm\nℏ)e−βεn\nZ⟨n|ˆρ−q|m⟩ ⟨m|ˆρq|n⟩ (37)\n=eβℏωS(−q,−ω). (38)\nHere, β= (kBT)−1is the inverse temperature, Zis the partition function, |n⟩is the n-th energy eigenstate with\nenergy εn. In particular, Eq. (31) implies S(q, ω) = 0forω <0at zero temperature.\nDue to the continuity equation ∂ρ/∂t+∇ ·j= 0,Πandσsatisfies the following relation:\nΠ(q, ω) =iqαqβσαβ(q, ω)\nω. (39)\nNoting that only the symmetric part of σcontribute to Π, Eq. (30) and Eq. (39) yield\nqαqβReσabs\nαβ(q, ω)\nω=S(q, ω) +S(q,−ω)\n2ℏforω >0 (40)\nwhere we have used S(q,−ω) = 0. Integrating over frequencies from 0to∞, we will get\nZ∞\n0dωqαqβReσabs\nαβ(q, ω)\nω=Z∞\n−∞dωS(q, ω)\n2ℏ=π\nℏSq. (41)\nExpanding in qaround q= 0, we obtain\nqαqβReW1\nαβ=π\nℏe2\n2πKαβqαqβ, (42)\nwhich implies Eq. (27)."    },    {        "title": "2401.13857v2.Comparison_of_Magnetized_Thick_Disks_around_Black_Holes_and_Boson_Stars.pdf",        "content": "Comparison of Magnetized Thick Disks around Black Holes and Boson Stars\nKristian Gjorgjieski∗and Jutta Kunz†\nDepartment of Physics, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany\nPetya Nedkova‡\nDepartment of Theoretical Physics, Sofia University, Sofia 1164, Bulgaria\n(Dated: February 6, 2024)\nBoson stars are considered as promising candidates for black hole mimickers. Similar to other\ncompact objects they can form accretion disks around them. The properties of these disks could\npossibly distinguish them from other compact objects like black holes in future observations. Ret-\nrograde thick disks around boson stars and the influence of strong magnetic fields on them were\nalready studied and it was shown that they can harbor very distinct features compared to black\nhole disks. However, the case of prograde thick disks is mostly unexplored, since they may appear\nmuch more similar to black hole disks. In this work we will investigate similarities and differences\nregarding prograde thick disks around non-selfinteracting rotating boson stars and rotating black\nholes. We assume thereby a polytropic equation of state and a constant specific angular momentum\ndistribution of the disks. We classify the various conceivable boson star and black hole solutions\nby a dimensionless spin parameter aand compare their corresponding disk solutions. The influence\nof toroidal magnetic fields on the disks is analyzed by selected disk properties, as the rest-mass\ndensity distribution and the Bernoulli parameter. Disk solutions are characterized by their degree\nof magnetization represented by the magnetization parameter βmc. We found that strong magnetic\nfields can strengthen the differences of disk solutions or oppositely even lead to a correlation in disk\nproperties, depending on the spin parameter of the boson star and black hole solutions. We iden-\ntify the vertical thickness of the boson star disks as the main differentiating factor, since for most\nsolutions the vertical density distribution is far more outreaching for boson star disks compared to\nblack hole disks.\n1. INTRODUCTION\nAccretion disks are objects composed of gas and plasma particles orbiting around compact objects. They form\nwhen attracted matter takes stable orbits around a central object, where then further phenomena like friction and\nmagnetohydrodynamical effects induce instabilities, leading to an inwards spiral of matter into the central object.\nThe acting gravitational forces are compressing the disk matter and the resulting friction is heating the disk up. The\ngenerated heat is further transformed into radiation energy and leaves the disk in form of electromagnetic radiation,\nwhose frequency range depends on the properties of the disk and the central object. Since the energy emitted by the\ndisk scales with the gravitational field, accretion disks around the centers of galaxies are among the brightest objects\nobserved in the universe and are therefore also a topic of interest in the study of high energy physics.\nIn general the properties of accretion disks depend on a complex regime of magnetohydrodynamical and gravitational\neffects [1–5]. One of the main crucial aspects in their formation is the acting gravitational field and therefore the local\nspacetime background, which depends on the central object, since in most cases the self-gravitational interaction of\nthe disk is small compared to the gravitational interaction with the central object and therefore negligible. Thus, the\nstudy and observation of accretion disks is closely related to the study of the compact objects forming them [6–11].\nDue to the recent progress in the observation of distant objects, especially the progress of the Event Horizon telescope\n[12–14], it is expected that a deeper observational investigation of accretion disks will be possible in the near future.\nThis will eventually allow us to learn more about the compact objects themselves, around which they form, and about\nthe high-energy processes occurring inside the disks.\nA broad spectrum of astrophysical objects are capable of forming accretion disks around them, ranging from proto-\nstars to neutron stars and from stellar black holes to supermassive galactic nuclei. In theory, there exist many more\nhypothetical objects, yet to be actually observed, which are capable of forming accretion disks around them. Some\nof these objects are described by theoretical predictions in such a way, that they could appear similar to black holes\n∗kristian.gjorgjieski@uol.de\n†jutta.kunz@uni-oldenburg.de\n‡pnedkova@phys.uni-sofia.bgarXiv:2401.13857v2  [gr-qc]  3 Feb 20242\nfor a distant observer. One promising candidate for such black hole mimickers are boson stars. For a distant observer\nthey could resemble black holes in particular due to their negligible radiation profile and their strong gravitational\nfield. Boson stars originate from the works of John A. Wheeler about geons [15], where he developed the idea of\nself-gravitating electromagnetic fields. These ideas where further developed by considering a complex scalar field,\nminimally coupled to gravity, instead of electromagnetic fields [16–18]. This approach led to stable solutions, which\nare now known as boson stars. In general they are composed of spin-0 particles in the form of compact scalar field\ncondensates, which form an equilibrium state after a self-gravitational collapse of the massive scalar field. The coun-\nteracting force originates from the uncertainty principle, which prevents further gravitational collapse and stabilises\nthe boson star [19–21]. Depending on the model (i.e. described by the chosen potential and the chosen symmetries),\nboson stars may have different properties and a broad range of various configurations have been already described\nand analyzed [22–34]. Their size and mass can range from the atomic scale to the galactic scale. However, it should\nbe mentioned that they do not have a sharp surface, but exhibit an exponential decay asymptotically. To determine\ntheir mass and radius, a definition by compactness is usually chosen. Due to their negligible radiation profile, they are\nnot only potential black hole mimickers, but also promising candidates for dark matter [35, 36]. One way to detect\nthem would be by indirect observation of the accretion disks that could form around them, in case that one could\ndistinguish these accretion disks from those that would form around other objects like black holes. In that regard,\nthey may possess bound circular orbits which also extend into the inner regions with high density of the scalar field\ndistribution in contrast to black holes [37]. Additionally, the formation of a shadow in the strict sense would be\nprevented since they are horizonless, but nevertheless (in the presence of accretion disks) a central dark region will\nbe observed similar to that of black holes, which corresponds to the lensed image of an accretion disk’s inner edge\n[38–41].\nIn this work we will compare magnetized thick accretion disks of rotating uncharged black holes with those of rotating\nboson stars without scalar field self-interaction. The thick disk model offers a simple but realistic analytical description\nof a (non-selfgravitating) accretion disk [42–44]. In this model the accretion disk is modelled as perfect fluid orbiting\nthe compact object and the fluid particles obey a polytropic equation of state, where the pressure is a function of the\nrest-mass density. The simplest form of thick disks solutions are called Polish doughnuts , where the electromagnetic,\nviscosity and radiation terms are neglected and only the fluid part enters the equations. The resulting disk solutions\ncan be examined by their corresponding effective potential, which is linked through the relativistic Euler equations\nto thermodynamic quantities and offers therefore a general description of the disk. This effective potential depends\nonly on the metric coefficients and the specific angular momentum of the fluid particles. The thick disk model\nrepresents therefore a simple model for making qualitative statements about the accretion disks and the accretion\nprocess and is in this respect a suitable complement to computationally complex numerical methods. Over the last\ndecades the properties of thick disks and thick disk accretion onto black holes were studied in various works [45–49].\nThe purely hydrodynamical Polish doughnut model can be extended by adding a toroidal magnetic field to obtain an\nastrophysically more accurate model [50–53], since it is widely believed that the dynamics of astrophysical accretion\ndisks are strongly influenced by magnetic fields. We will use this magnetized model and furthermore we will assume\na constant specific angular momentum distribution of the disk, where the specific angular momentum enters the\nequations as a free parameter. Solutions with a constant specific angular momentum play a fundamental role, since\ndespite their simplicity they already capture the main qualitative features of more complex configurations. The disk\nsolutions will be classified by a magnetization parameter βmc, in order to distinguish between the effects of different\ndegrees of magnetization on the disk properties.\nRetrograde Polish doughnuts around boson stars without self-interaction have already been studied in [54], and it\nwas shown that they can harbour unique features like two-centered disk solutions and static surfaces. Magnetic fields\ncan greatly influence these properties and alter the unique features of retrograde thick disks as it was shown in [55].\nSince these retrograde disks can already be very distinct compared to retrograde magnetized thick disks around black\nholes, there is less of a need for a deeper comparison between them (we will address them briefly in a later section).\nTherefore our focus will be on prograde disks, which appear in principle much more similar to black hole disks and\nwhich have not been examined deeply so far. In particular we will conduct a qualitative comparison between boson\nstar and black hole disks, where we try to isolate the influence of the magnetization as much as possible, by comparing\nboson star and black hole solutions with the same spin parameter aand the same specific angular momentum ℓ0of\ntheir disks.\nIn the following section we will describe the black hole and boson star solutions which we will consider, afterwards\nsection 3 reviews the theory of magnetized thick disks and their construction procedure. Section 4 presents the\ndisk comparison and section 5 briefly focuses on retrograde disks. Throughout this work we will use the Einstein\nsummation convention and ( −,+,+,+) as the metric signature.3\n2. ROTATING BOSON STARS AND BLACK HOLES\nSpeaking of astrophysical black holes, it is generally believed that they may possess a non-zero angular momentum\nand negligible charge. Therefore we consider uncharged rotating black holes (BHs) described by the Kerr metric,\nwhich can be written in Boyer-Lindquist coordinates as,\nds2=−\u0012\n1−2Mr\nΣ\u0013\ndt2−4M2˜arsin2θ\nΣdtdφ +Σ\n∆dr2+ Σdθ2+\u0012\nr2+M2˜a2+2M3˜a2rsin2θ\nΣ\u0013\nsin2θdφ2,(1)\n∆≡r2−2Mr+M2˜a2; Σ≡r2+M2˜a2cos2θ, (2)\nwhere Mis the mass and ˜ a=J\nM2is the dimensionless spin parameter defined by the mass and the angular momentum\nJ. Further on we will work with the dimensionless spin parameter, it is therefore convenient to omit the tilde and\njust use afor the dimensionless spin parameter. In the case of boson stars (BSs) we consider non-self-interacting BSs\nobtained from the action S,\nS=Z√−g\u0012R\n16πG− Lm\u0013\nd4x, (3)\nwhere gis the metric determinant, Ris the Ricci scalar, Gis the gravitational constant and Lm=|∂µϕ|2+m2|ϕ|2\nis the Lagrangian, with ϕas the complex scalar field and mas the mass of the boson particle. Variation of eq. (3)\nleads to a coupled set of Einstein-Klein-Gordon equations,\nRµν−1\n2Rgµν= 8πGT µν, (4)\n\u0000\n□−m2\u0001\nϕ= 0, (5)\nwith Tµν= (∂µϕ∂νϕ∗+∂νϕ∂µϕ∗)−Lmgµνas the stress-energy tensor and □as the covariant d’Alembert operator. It\nshould be noted that the action (3) is invariant under global U(1) transformations, giving rise to a conserved Noether\ncurrent jµand Noether charge Q,\njµ=i(ϕ∂µϕ∗−ϕ∗∂µϕ) ;Q=Z\njt√−gdx3. (6)\nConsidering now rotating solutions, the metric should be axisymmetric and stationary, which gives rise to two Killing\nvector fields, ∂tand∂φ. The complex scalar field must nevertheless depend on all four coordinates. If the complex\nscalar field would not depend on the time coordinate, then the conserved charge would be zero (eq. (6)) leading to\ntrivial vacuum solutions. Furthermore the stress-energy tensor must depend on the azimuthal coordinate in order\nto describe rotating solutions. Consequently, the complex scalar field must depend on the time and the azimuthal\ncoordinate in a way that the metric remains unaffected. This can be ensured by decomposing the scalar field into a\nreal profile function and a complex phase,\nϕ(r, t) =ϕ0(r, θ)ei(ωt−kφ), (7)\nwhere ϕ0is the real valued profile function, ωis the angular frequency of the scalar field rotation and kis the azimuthal\nwinding number. The azimuthal winding number counts the nodes of the real and imaginary parts of the scalar field\nalong the azimuthal direction and indicates therefore the strength of the angular excitations. Due to single-valuedness\nof the scalar field, ϕ(φ= 0) = ϕ(φ= 2π),kmust be an integer. The total angular momentum of the BS is given\nby the quantisation relation J=kQ. In this work we will focus on the fundamental branch of solutions with k= 1.\nMaking use of the axial symmetry and stationarity, the line element of the metric can be written in the general form\nas\nds2=−α2dt2+A2(dr2+r2dθ2) +B2r2sin2θ(dφ+βφdt)2, (8)\nwhere αis the lapse function, βφis the shift function and where the functions A, B, α andβφonly depend on r\nandθ. Substitution of the harmonic ansatz (eq. (7)) into the field equations leads to a coupled set of differential\nequations which can be solved numerically. Solutions only exist for a set of angular frequencies ω. The solutions\nare asymptotically flat and the scalar field decays exponentially proportional to e−√\nm2−ω2r/rasymptotically. The\nequations possess a scaling symmetry, which we use to introduce dimensionless quantities by scaling with the boson\nparticle mass m, i.e. ˜ r=rm, ˜ω=ω\nmand ˜M=Mm. Further on we will omit the tilde for brevity. The solutions4\nrange from the minimum frequency at ω= 0.646 to ω= 1, featuring several branches, with the fundamental branch\nranging from the vacuum ( ω= 1) to the BS with maximum mass. Solutions below ω= 0.665 possess an ergoregion.\nHere we will focus only on BS solutions without ergoregions [56].\nFig. 1: Modulus of the scalar field in the equatorial plane vs. the scaled radial coordinate on the left side and BS mass vs. the\nangular frequency on the right side. Solutions exist only for a set of ωvalues, some of which are marked in the plot by circles.\nThe dashed vertical line marks the onset of solutions possessing an ergoregion.\nThe main difference regarding the spacetime environment of rotating BSs and rotating BHs lies in the toroidal shape\nof the BSs. As seen in Fig. 1 the maximum of the scalar field (and therefore the maximum of the energy-density) is\nlocated off-center resulting in a toroidal shape of the scalar field, which moves closer to the center with decreasing\nω. Therefore the torus formed by the scalar field is more strongly localized towards the BS center for lower values of\nω. Due to this toroidal energy density distribution the spacetime near the BS is curved in an unusual way, featuring\ndistinctive orbits as compared to BHs [54, 57].\n3. MAGNETIZED THICK DISKS\nGeometrically thick disks can be constructed by linking thermodynamic quantities to kinematic quantities through\nthe relativistic Euler equations [43]. In this procedure the disk is modelled as a perfect fluid, where the fluid particles\nmove on circular orbits around a given compact object. To get equilibrium solutions axial symmetry and stationarity\nof the background spacetime are assumed. Furthermore, as a necessary integrability condition for the relativistic\nEuler equations, the fluid must obey a barotropic equation of state where the pressure of the fluid can be expressed\nthrough its rest-mass density. As a consequence the von Zeipel theorem holds, which states that the angular velocity\nΩ and specific angular momentum ℓof a fluid particle can be expressed as a function of each other and the metric\ncomponents,\nℓ(r, θ) =−Ωgφφ+gtφ\nΩgtφ+gtt; Ω( r, θ) =−ℓgtt+gtφ\nℓgtφ+gφφ. (9)\nAt the center of the accretion disk the acting forces cancel each other, the flow of the fluid particles at the center follows\ntherefore a geodesic. Their corresponding specific angular momentum ℓ0is then identical to that of a Keplerian circular\norbit, ℓK. The general disk properties in the equatorial plane are therefore determined by the equatorial Keplerian\nspecific angular momentum of the spacetime. Assuming a constant specific angular momentum distribution of the\ndisk we can make the following statements: If ℓ0̸=ℓK(r)∀r, then no disk solution exists. If ∃!r:ℓ0=ℓK(r), then a\ndisk solutions exists where the disk center is located at the value of rwhich satisfies the equation. If there is more\nthan one intersection, the slope of ℓKdetermines if it is a center (stable geodesic motion) or a cusp (unstable geodesic\nmotion), the former being given by a positive slope and the latter by a negative. A cusp marks the intersection of\nequi-pressure (and equi-density) surfaces. We will use the specific angular momentum ℓ0as a free parameter which\ndetermines the general solution type (depending on the Keplerian specific angular distribution).\nThe purely hydrodynamical model can be extended by adding a toroidal magnetic field in order to construct magne-\ntized thick disks. We follow here the procedure described in [50, 51, 55]. The magnetic part enters the stress-energy5\ntensor only additively, and the contravariant stress-energy tensor for the fluid can then be expressed as\nTµν≡(ρh+b2)uµuν+ (pth+pmag)gµν−bµbν, (10)\nwhere ρis the rest-mass density, his the specific enthalpy, uµis the four-velocity, pthis the thermodynamic pressure,\npmag=b2\n2≡1\n2bµbµis the magnetic pressure, gµνis the contravariant metric tensor and bµ= (0,B) is the magnetic\nfield with Bbeing the co-moving three-dimensional magnetic field in the fluid frame. The relativistic Euler equations\ncan be derived by inserting the stress-energy tensor in the fundamental conservation laws of general relativistic\nmagnetohydrodynamics. The conservation laws are given by,\n∇µ(ρuµ) = 0 , (11)\n∇µTµν= 0, (12)\n∇µ(∗Fµν) = 0 , (13)\nwhere∗Fµν=uµbν−uνbµis the dual Faraday tensor. In order to solve this system of equations we assume axial\nsymmetry and stationarity. As a consequence all partial derivatives regarding the time and azimuthal coordinate ( ∂t\nand∂φ) vanish. Furthermore we assume circular radial motion for the fluid ( ur=uθ= 0) and make use of the toroidal\nmagnetic field distribution ( br=bθ= 0), which leaves the continuity equation (eq. (11)) and the Maxwell equation\n(eq. (13)) trivially satisfied. To solve eq. (12), it can be split into one equation containing the energy conservation and\nthree equations containing the momentum conservation by applying the orthogonal projection tensor hα\nβ=δα\nβ+uαuβ,\nwhich leads to\n(ρh+b2)uν∂µuν+∂µ(pth+pmag)−bν∂µbν= 0, (14)\nwhere µ=t, φare trivially satisfied. Simplifying the equation by rewriting it in terms of the specific angular\nmomentum and the angular velocity of the fluid leads to [50],\n∂µ(ln|ut|)−Ω∂µℓ\n1−ℓΩ+∂µpth\nρh+∂µ(Lpmag)\nLρh= 0, (15)\nwhere L ≡ g2\ntφ−gttgφφ. By assuming a uniform specific angular momentum distribution, the term containing the\nspecific angular momentum vanishes,\n∂µ(ln|ut|) +∂µpth\nρh+∂µ(Lpmag)\nLρh= 0. (16)\nIn order to integrate eq. (16) we assume a polytropic equation of state, where the pressure is a function of the density.\nThe thermodynamic and magnetic pressure can then be written as pth=KρΓandpmag=KmLq−1(ρh)q, with K, K m\nas the polytropic constants and Γ , qas the polytropic exponents. The polytropic exponents will here be chosen as\nfor relativistic degenerate matter, Γ = q=4\n3. Using the substitutions ˜ pmag=Lpmagand ˜ω=Lρheq. (16) can be\nwritten in integral form as\nln|ut|+Zpth\n(pth)in1\nρhdp′\nth+Zepmag\n(epmag)in1\neωd˜p′\nmag=C. (17)\nThe subscript inis referring to the inner edge of the disk. From the equation above we identify the leading term as\nthe effective potential of the disk, W= ln|ut|. This effective potential is the combination of the gravitational and\ncentrifugal potential. The integration constant Cis determined by the boundary conditions at the inner edge of the\ndisk. Since ( pth)in= (˜pmag)in= 0, the integration constant is given by the effective potential at the inner edge of the\ndisk, C= ln|(ut)|in=Win. Integration of eq. (17) yields\nW − W in+ ln( h) +q\nq−1Km(Lρh)q−1= 0. (18)\nSolving this equation for the rest-mass density is only numerically possible since it is transcendental. In order to fix\nthe gauge, the rest-mass density will be normalized at the center of the disk to ρc= 1. Rewriting eq. (18) with respect\nto the disk center and expressing the specific enthalpy in terms of the rest-mass density is leading to\nWc− W in+ ln\u0012\n1 +ΓK\nΓ−1ρΓ−1\nc\u0013\n+q\nq−1KρΓ\nc\nβmc\u0010\nρc+KΓρΓc\nΓ−1\u0011= 0, (19)6\nwhere we introduce the magnetization parameter βmc=pthc\npmagcas the ratio between thermodynamic and magnetic\npressure at the center of the disk. A high magnetization parameter ( βmc≫1) corresponds to an essentially non-\nmagnetized disk, whereas a low magnetization parameter ( βmc≪1) corresponds to a strongly magnetized disk. A\nmildly magnetized disk is represented by βmc= 1, where the thermodynamic pressure equals the magnetic pressure.\nIn our computations we will use βmc= 105,βmc= 1 and βmc= 10−5for representing non-magnetized, mildly\nmagnetized and highly magnetized disks. Besides the magnetization parameter, which we will use as our variable\nparameter, the only unknown quantity is the polytropic constant K. After computation of Kfrom eq. (19), the\neq. (18) can be solved for the rest-mass density. As a numerical solving algorithm we use the bisection method, with\nan absolute convergence error of ϵ <10−15between the last and second to last iteration step.\n4. DISK COMPARISON\nIn order to conduct a qualitative comparison between the BS and Kerr disks, we aim to isolate the effects of magnetic\nfields on the disks as much as possible. Therefore we fix the other solution and disk parameters and vary the\nmagnetization of the disk. We employ the dimensionless spin parameter a≡J\nM2for the classification of different BS\nand BH solutions, which are then compared to each other. The BS solution range for the spin parameter is depicted in\nFig. 2. Its lower bound is min( a) = 0 .798, corresponding to the ω= 0.774 solution, and it diverges in the limit ω= 1.\nSince JandMare non-monotonic with respect to ω, neither is a. This leads to a twofold degeneracy of the considered\nsolutions in the interval a∈[0.798,0.861]. We will only focus on BS solutions in the interval a∈[0.798,1), since for\na >1 naked singularities occur for the Kerr solutions. With the aim to examine qualitatively the various solutions\nclassified by the spin parameter, we have selected three values of a, one of which represents the upper solution range,\none the middle range and one the lower range, namely a= 0.995, a= 0.908 and a= 0.825. The spin parameter\na= 0.825 has two corresponding BS solutions, which were therefore both examined for the sake of completeness.\nFig. 2: The left plot shows the BS spin parameter aversus ω. The dashed black line marks the Kerr limit a= 1, the dotted\nblack line marks the value for abelow which the spin parameter is not monotonic for solutions without ergo-regions. The black\ncircles highlight the chosen exemplary solutions which were examined more closely. The right plot shows aversus the BS mass\nM.\nSince we are comparing different spacetime geometries to each other, it is suitable to choose coordinates with a similar\nphysical interpretation. Therefore we will use the proper radial distance Rras the radial coordinate for all following\nillustrations in the equatorial plane. Rris given by\nRBS\nr=Zr\n0r\ngrr\u0010\nr′, θ=π\n2\u0011\ndr′;RKerr\nr =Zr\nrHr\ngrr\u0010\nr′, θ=π\n2\u0011\ndr′, (20)\nwhere grris the corresponding radial metric component and rHis the Kerr radial coordinate of the outer event horizon\nin the equatorial plane. It should be noted that Rris giving the proper radial distance to the coordinate origin for\nthe BSs, whereas for the BHs Rris giving the proper radial distance to the outer event horizon. The origin in the\nfollowing plots corresponds therefore to the horizon in the Kerr plots. For vertical and two-dimensional plots we will7\nuse pseudo-cylindrical coordinates with the proper radial distance Rr(θ) as the radial coordinate, which is then for a\nfixed radial value θ-dependent,\nRBS\nr(θ) =Zr\n0p\ngrr(r′, θ)dr′;RKerr\nr(θ) =Zr\nrHp\ngrr(r′, θ)dr′. (21)\nSince all solutions possess axial symmetry, the vertical and two-dimensional plots can be done in the x-z-plane, where\nxandzare then defined as\nx(r, θ) =Rr(θ) sinθ;z(r, θ) =Rr(θ) cosθ. (22)\nA general analysis of the possible types of disk solutions can be done by examining the Keplerian specific angular\nmomentum distribution, as it determines the existence of a cusp (and possibly the number of disk centers). Fig. 3\npresents the equatorial Keplerian specific angular momentum distribution for the different BS and Kerr solutions,\nclassified by their spin parameter.\nFig. 3: Prograde Keplerian specific angular momentum distribution in the equatorial plane for the various BS solutions (on\nthe left) and Kerr black hole solutions (on the right) classified by their spin parameter a. The black curve sections in the Kerr\nplot correspond to unbound orbits.\nSince ℓ+\nKis strictly monotonic for the BSs, all disk solutions are composed of cuspless one-centered disks. Furthermore\nthere are no marginally stable and marginally bound orbits for the BSs. This makes disk solutions with a very small\nangular momentum possible in contrast to BH disks. These low angular momentum solutions are also called fat\ntoriand differ from solutions with a higher specific angular momentum in their much further extending vertical\nextension. For the Kerr solutions ℓ+\nKhas a minimum and solutions exist only for ℓ0≥min(ℓ+\nK) =ℓ+\nK(rms)≡ℓms,\nwhere the subscript msis referring to the marginally stable orbit. One-centered solutions with a cusp exist for ℓ0\nwith ℓms< ℓ0≤ℓ+\nK(rmb)≡ℓmb, where the subscript mbis referring to the marginally bound orbit. In these solutions\nthe disk center is located in the vicinity of the event horizon. For ℓ0> ℓmbonly solutions without a cusp are possible.\nThe specific angular momentum of thick disks in the vicinity of the inner edge is believed to be close to ℓmb[1],\ntherefore ℓ0will be set to ℓ0=ℓKerr\nmb in all following disk calculations. This corresponds to a Kerr disk solution,\nwhere the equi-density surface through the cusp closes at spatial infinity. Regarding the BS disk calculations, the\nspecific angular momentum value is set to the same value, in order to compare disks with the same specific angular\nmomentum. Hence the effects of magnetic fields are isolated as much as possible, since the spin parameter and the\nspecific angular momentum are set to the same relative value for BS and Kerr solutions. In order to represent all\nphysical feasible solutions, the effective potential at the inner edge of the disk is set to Win= 0 in all calculations.\nFollowing the described procedure the specific angular momentum of the a= 0.995 solutions is set to ℓ0=ℓKerr\nmb =\n2.142, for the a= 0.908 solutions to ℓ0=ℓKerr\nmb = 2.607 and for the a= 0.825 solutions to ℓ0=ℓKerr\nmb = 2.836. The\ncorresponding effective potentials in the equatorial plane are shown in Fig. 4.8\nFig. 4: Effective potential Win the equatorial plane with the high spin parameter solutions on the left to the lower spin\nparameter solutions on the right. The dashed and dotted curves mark the effective potential for the BSs and the solid curves\nfor the BHs. The minimum of the curves marks the location of the disk center, where the motion is geodesic.\nFor the high and mid range spin parameter solutions a test particle at the BH disk center is more energetically bound\ncompared to the BS disks, since the minimum of the effective potential is lower for the BHs. Regarding the lower\nspin parameter case the opposite applies for both BSs solutions. In all three spin parameter cases WBSis evolving\nsimilarly to WKerr with increasing radial distance, the biggest differences are located close to the disk center.\nIn the following we will first discuss the influence of magnetic fields on the disk solutions for the high spin parameter\ncase, then we will discuss the solutions for the mid spin parameter case and finally we will discuss the small spin\nparameter case. For a suitable comparison between BS and BH disks we will focus primarily on the density distribu-\ntions and on dimensionless quantities of the disk, which are defined and presented in Tables I - III. We also examine\nthe equatorial Bernoulli parameter of the solutions. It is a quantity particularly important for dynamical simulations,\nsince the time evolution of the Bernoulli parameter in numerical simulations tends to be controlled by the Bernoulli\nparameter of the initial solution [58]. For magnetized tori it can be written as,\nBe=\u0012\n1 +U+pth+b2\nρ\u0013\nut−1, (23)\nwith Ubeing the internal energy. By using the specific internal energy ϵ=U\nρand the magnetic pressure pmag=b2\n2,\nthe equation can be rewritten in terms of the magnetization parameter βm,\nBe=\u0012\n1 +KρΓ−1\u0012Γ\nΓ−1+2\nβm\u0013\u0013\nut−1, (24)\nwhere βmdenotes the ratio between thermodynamic and magnetic pressure, with βm=βmcat the disk center. The\nBernoulli parameter is representing the sum of the potential and kinetic energy with the gas enthalpy of the disk. It\ncan be used as a suitable quantity to analyze the stability of disk solutions, since for steady state solutions and in the\nabsence of viscosity it is conserved alongside streamlines. For gas at infinity Be≥0 applies, since the gravitational\npotential is approaching zero and the kinetic energy as well as the gas enthalpy are always positive. As a consequence\nany stream of disk particles with Be≥0 could potentially reach infinity and is therefore classified as unbound.\nIn contrast to that, a flow with Be < 0 is called bound. Physically feasible equilibrium torus solutions are bound\nsolutions and should therefore have a Bernoulli parameter with Be < 0, since unbound flows are in general more likely\nto cause outflows, which may affect the stability of the disk.9\n4.1. High spin parameter a= 0.995(ωBS= 0.921, ℓ0= 2.142)\nFor the high spin parameter solutions the equatorial and vertical rest-mass density distributions are shown in Figs. 5\nand 6, and some of their characteristic properties are evaluated in Table I for different magnetic fields. The density\nof the BS disk decreases less steeply compared to the BH disk, which is also reflected by the compactness parameter\nRout−Rρmaxin Tab. I. For strongly magnetized disks this difference in the density distributions increases. The density\nmaximum of the BS disk is located further from the origin and also further from the inner edge of the disk compared\nto the BH disk. The proper radial distance of the density maximum in the equatorial plane Rρmaxis more than two\ntimes higher for the BS than for the Kerr black hole in the non-magnetized case. With increasing magnetization the\ndensity maximum shifts towards the inner edge in both cases, with this shift being greater for the BH disks. The\nstrongly magnetized BS disks are less compactified compared to the strongly magnetized BH disks, since the increase\nof the maximum density is greater for the Kerr disks and their vertical and radial extensions are far less outreaching.\nThe vertical density distribution appears in both cases to be similarly affected by magnetization as the equatorial\ndensity distribution, with the greater impact on the Kerr solutions. Thus, the vertical thickness of the BH disk is far\nsmaller and most of the mass is tightly distributed around the center in the equatorial plane, while the BS disk has\na greater vertical thickness and in general is less effected by strong magnetic fields. This is also showcased in Fig. 7,\nwhich illustrates the meridional cross-section of the density distributions. The radial extension of the disk can be\nquantified by evaluating the difference in the proper radial distance between the location of the density maximum in\nthe equatorial plane Rρmaxand the equatorial cross-section of the iso-density contour which represents 10 % of the\nmaximal density, which we denote by Rout. We see that in the non-magnetized case the radial extension of the disk\nis more than 2 times higher for the BS than for the Kerr black hole and the deviation increases with magnetization.\nAs a measure of the fluid dynamics we evaluate the ratio of the angular velocity of the accreting fluid at the density\nmaximum and the angular velocity at the inner edge of the disk. We see that the angular velocity ratio for the Kerr\nblack hole is with 64 % higher than for the BS in the non-magnetized disks. Increasing the magnetization the angular\nvelocity increases for the Kerr black hole while staying approximately constant for the BS. Thus, highly magnetized\ndisks around the Kerr black hole reach with 75 % higher rotation rate than those around the BS. Since a high\nrotation rate is associated with improved stability against convection [58, 59], the thick disks in the Kerr spacetime\nare considerably more stable compared to BSs.\nThe Bernoulli parameter Bein the equatorial plane is shown in Fig. 8. It is for the non-magnetized disks nearly\nconstant in both cases, while for the magnetized solutions Beis highly affected and develops a clear minimum, with\na lower peak for the BH disk. This implies that gas and plasma flows are more energetically bound in the BH disk\nand are less affected by magnetic fields in the BS disk, which is in accordance to the density distributions being less\naffected by magnetization. Furthermore we conclude that the vertical profile and the far greater equatorial range are\nthe main distinctive features between the BS and BH disks.\nFig. 5: Equatorial density distribution of the Kerr and BS disks plotted versus the proper radial coordinate with the non-\nmagnetized, mildly magnetized and highly magnetized disks presented from left to right. The constant specific angular mo-\nmentum of the disks is set to ℓ0=ℓKerr\nmb = 2.142.10\nFig. 6: Vertical density distribution at the density maximum of the Kerr and BS disks, with the non-magnetized, mildly\nmagnetized and highly magnetized disks presented from left to right. The vertical lines mark the projection of the equatorial\nproper radial distance of the density maximum.\nβmc= 105ρmaxRρmaxRout−RρmaxRin\nRρmaxΩρmax\nΩin\nKerr 13.474 6.245 0.285 0.663\nBS 0.921 18.106 13.883 0.464 0.240\nβmc= 1 ρmaxRρmaxRout−Rρmax∆RrelRin\nRρmaxΩρmax\nΩin\nKerr 1.236 2.798 3.446 0.195 0.354 0.776\nBS 0.921 1.065 7.197 9.546 0.112 0.523 0.232\nβmc= 10−5ρmaxRρmaxRout−Rρmax∆RrelRin\nRρmaxΩρmax\nΩin\nKerr 2.424 2.288 2.408 0.341 0.433 0.856\nBS 0.921 1.281 6.467 7.650 0.202 0.582 0.218\nTab. I: Selected properties of the Kerr and BS disks in the equatorial plane, where ρmaxis the maximum density, Rρmaxis\nthe location of the density maximum, Rout−Rρmaxis the equatorial distance from the density maximum to the effective\nouter edge of the disk Rout:={Rr:ρ\u0000\nRr, θ=π\n2\u0001\n= 0.1·ρmax}, ∆Rrelis the relative difference in the location of the density\nmaximum between magnetized and non-magnetized disk,Rin\nRρmaxis the ratio between inner edge and density maximum location\nandΩρmax\nΩinis the ratio between the angular velocity at the density maximum and the inner edge of the disk.\n(a)βmc= 105\n(b)βmc= 1\n (c)βmc= 10−5\nFig. 7: Meridional cross section plots of the density distribution in pseudo-cylindrical coordinates. The upper half shows the\nBS torus, while the lower half shows the Kerr torus. The upper bound of the density scale is set to the maximum density\nbetween the BS and Kerr solution for the highest magnetization, log10ρmax= 0.385. The lower bound of the density scale is\nset to log10ρ=−5 in all plots. Black curves mark equi-density surfaces.11\nFig. 8: Bernoulli parameter of the BS and Kerr disks in the equatorial plane for different degrees of magnetization, with the\nnon-magnetized to the highly magnetized solutions from left to right. For the non-magnetized disks a zoom-in plot is inserted\nfor a better resolution, since Beis nearly constant for both disks and close to 0, with the maximum difference to 0 being on\nthe order of 10−6.12\n4.2. Medium spin parameter a= 0.908(ωBS= 0.895, ℓ0= 2.607)\nThe equatorial and vertical rest-mass density distributions are depicted in Figs. 9 and 10, and some characteristic\nquantities are given in Table II. Compared to the high spin parameter solutions, the slope of the equatorial density\ncurves are more similar, the difference in the density decrease close to the center is less than an order of magnitude for\nall degrees of magnetization. As for the high spin parameter solutions the density curve has a less steep slope for the\nBS disks for all magnetization parameters and the density maximum of the BH disks is located closer to the origin.\nThe relative change in the location of the density maximum is greater for the BHs with increasing magnetization,\nand in general the BH disks get more compactified resulting in a denser center. For the non-magnetized disk the\nproper radial distance of the density maximum is with 38 % higher for the BS compared to the Kerr black hole.\nIncreasing the magnetization the deviation grows reaching 65 % for highly magnetized disks. The vertical thickness\nof the BS disk is greater for all cases, since the vertical density curve for the BH disks has again a steeper slope. This\ndifference increases for the highly magnetized disks, where the BS vertical density profile is largely unaffected and\nthe BH vertical profile drops slightly sharper compared to the non-magnetized disk. These differences in the density\ndistribution are less profound than in the high spin parameter case, as the cross-section plots in Fig. 11 illustrate.\nThe thick disks around black holes rotate faster than those around boson stars irrespective of magnetization. In the\nnon-magnetized case the relative angular velocity at the density maximum is with 27 % higher for BHs than for BSs.\nIncreasing the magnetization the angular velocity for the Kerr black hole grows while for BSs it remains approximately\nconstant. As a result, for highly magnetized disks the deviation between the two spacetimes reaches 52 %.\nThe Bernoulli parameter in the equatorial plane (Fig. 12) behaves similarly as in the high spin parameter solutions.\nFor the non-magnetized disks it is nearly constant and for the magnetized disks it has a clear minimum, with a lower\npeak for the BS disk. Since Bebehaves similarly for BS and Kerr, the disks are comparably energetically bound. We\nconclude that the BS and BH disks remain similar even under the influence of strong magnetic fields, whereby the\nmagnetized BH disks are again characterized by a higher degree of compactification and the magnetized BS disks by\ntheir largely unaffected vertical thickness.\nFig. 9: Equatorial density distribution of the Kerr and BS disks plotted versus the proper radial coordinate with the non-\nmagnetized, mildly magnetized and highly magnetized disks presented from left to right. The constant specific angular mo-\nmentum of the disks is ℓ0=ℓKerr\nmb = 2.60713\nFig. 10: Vertical density distribution at the density maximum of the Kerr and BS disks, with the non-magnetized, mildly\nmagnetized and highly magnetized disks presented from left to right. The vertical lines mark the projection of the equatorial\nproper radial distance of the density maximum.\nβmc= 105ρmaxRρmaxRout−RρmaxRin\nRρmaxΩρmax\nΩin\nKerr 14.919 11.292 0.324 0.416\nBS 0.895 16.769 11.364 0.513 0.304\nβmc= 1 ρmaxRρmaxRout−Rρmax∆RrelRin\nRρmaxΩρmax\nΩin\nKerr 1.150 4.041 6.197 0.179 0.394 0.527\nBS 0.895 1.053 6.155 7.604 0.091 0.564 0.303\nβmc= 10−5ρmaxRρmaxRout−Rρmax∆RrelRin\nRρmaxΩρmax\nΩin\nKerr 1.727 3.481 4.348 0.292 0.458 0.616\nBS 0.895 1.222 5.680 6.035 0.161 0.611 0.298\nTab. II: Selected properties of the Kerr and BS disks in the equatorial plane, where ρmaxis the maximum density, Rρmaxis the\nlocation of the density maximum, Rout−Rρmaxis the equatorial distance from the density maximum to the effective outer edge\nof the disk, ∆ Rrelis the relative difference in the location of the density maximum between magnetized and non-magnetized\ndisk,Rin\nRρmaxis the ratio between inner edge and density maximum location andΩρmax\nΩinis the ratio between the angular velocity\nat the density maximum and the inner edge of the disk.\n(a)βmc= 105\n(b)βmc= 1\n (c)βmc= 10−5\nFig. 11: Meridional cross section plots of the density distribution in pseudo-cylindrical coordinates. The upper half shows the\nBS torus, while the lower half shows the Kerr torus. The upper bound of the density scale is set to the maximum density\nbetween the BS and Kerr solution for the highest magnetization, log10ρmax= 0.238. The lower bound of the density scale is\nset to log10ρ=−4 in all plots. Black curves mark equi-density surfaces.14\nFig. 12: Bernoulli parameter of the BS and Kerr disks in the equatorial plane for different degrees of magnetization, with the\nnon-magnetized to the high magnetized solutions from left to right. For the non-magnetized disks a zoom-in plot is inserted\nfor a better resolution, since Beis nearly constant for both disks and close to 0, with the maximum difference to 0 being on\nthe order of 10−7.15\n4.3. Lower spin parameter a= 0.825(ωBS={0.694,0.841}, ℓ0= 2.836)\nFor the lower spin parameter there are two BS solutions with ωBS= 0.694 and ωBS= 0.841. The former represents\na highly relativistic boson star while the higher value of the angular frequency ωBScorresponds to a more mildly\nrelativistic solution (see Fig. 1). The equatorial and vertical rest-mass density distributions are depicted in Figs. 13\nand 14, while some of their properties are evaluated in Table III.\nThe maximum density increases comparably for both BS solutions as well as for the Kerr solution. In contrast to\nthe high and mid spin parameter solutions the equatorial density drops steeper for the BS disks for all degrees of\nmagnetization. Thus, in this case we obtain more compact disks for the BS spacetimes than for the Kerr black hole.\nWe can attribute this effect to the fact that the BS solutions become more relativistic when their angular frequency\nωBSdecreases, which correlates with a decreasing spin parameter according to Fig. 1. We see, that for the mildly\nrelativistic BS with ωBS= 0.841 the disk density behaves in a very similar way to the Kerr disk. The BS 0.694 disk\ngets more compactified with increasing magnetization. The location of the density maxima is largely unaffected by\nmagnetization for the BS disks, since the center location is already close to the inner edge for the non-magnetized\ndisks. For the BS 0.841 disk the deviation of the location of the density maximum from the Kerr disk is small\nfor all the degrees of magnetization ranging from ≈20% for non-magnetized disks to ≈10% for highly magnetized\ndisks. It is more substantial for the BS 0.694 disks, where the values correspond to ≈40−50% (see also Fig. 17).\nRegarding the vertical density profile, the density curves of the BH and BS 0.694 disks behave more similarly for a\nhigh magnetization in contrast to the equatorial density profile. Regarding the vertical profile, the density of the BS\n0.841 disks behaves similarly to the high and mid spin parameter solutions and drops less sharply, while the vertical\ndensity of the BS 0.694 solution behaves more similar to the BH disks, having a steep decline for all magnetization\nparameters. Fig. 15 presents the cross section plots, which showcase the greater extent of the Kerr disks compared\nto both BS disks through all degrees of magnetization. The radial extension Rout−Rρmaxof the non-magnetized BH\ndisk is with ≈40% higher than the BS 0.841 disk as the difference decrease to ≈20% for high magnetization. For the\nBS 0.694 solutions the deviation of the disk extension from the Kerr BH is ≈80% irrespective of the magnetization.\nAs for the previous spin parameters that we considered, the angular velocity for the BSs depend extremely weakly\non the magnetization, maintaining almost constant values. On the other hand, for the Kerr BH it increases, reaching\nalmost 2 times higher values for highly magnetized disks compared to the non-magnetized case. As a result the highly\nmagnetized BS 0.694 disks possess a similar rotation rate as the Kerr disks with a difference in the angular velocity\nratio of ≈10%.\nIn accordance to the density distribution, the BS disks appear to be more energetically bound, as the Bernoulli\nparameter is more affected by magnetization compared to the BH disks (Fig. 16), for the strong magnetic fields the\nBS 0.694 disk has the lowest Bepeak. We conclude that in contrast to the high and mid spin parameter solutions,\nthe BS disks are more affected by magnetization than the BH disk.\nFig. 13: Equatorial density distribution of the Kerr and BS disks for the non-magnetized, mildly magnetized and highly\nmagnetized presented case from left to right. The constant specific angular momentum of the disks is ℓ0=ℓKerr\nmb = 2.836.16\nFig. 14: Vertical density distribution at the density maximum of the Kerr and BS disks, with the non-magnetized, mildly\nmagnetized and highly magnetized disks depicted from left to right. The vertical lines represent the location of the respective\ndensity maximum in the equatorial plane.\nβmc= 105ρmaxRρmaxRout−RρmaxRin\nRρmaxΩρmax\nΩin\nKerr 15.856 14.216 0.334 0.360\nBS 0.841 14.796 7.786 0.549 0.373\nBS 0.694 12.677 3.921 0.627 0.601\nβmc= 1 ρmaxRρmaxRout−Rρmax∆RrelRin\nRρmaxΩρmax\nΩin\nKerr 1.134 4.855 7.737 0.171 0.403 0.460\nBS 0.841 1.047 4.382 5.200 0.086 0.601 0.376\nBS 0.694 1.049 2.492 2.533 0.069 0.674 0.609\nβmc= 10−5ρmaxRρmaxRout−Rρmax∆RrelRin\nRρmaxΩρmax\nΩin\nKerr 1.634 4.214 5.439 0.280 0.464 0.545\nBS 0.841 1.198 4.121 4.06 0.141 0.639 0.376\nBS3 0.694 1.205 2.349 1.908 0.123 0.715 0.613\nTab. III: Selected properties of the Kerr and BS disks in the equatorial plane, where ρmaxis the maximum density, Rρmaxis the\nlocation of the density maximum, Rout−Rρmaxis the equatorial distance from the density maximum to the effective outer edge\nof the disk, ∆ Rrelis the relative difference in the location of the density maximum between magnetized and non-magnetized\ncase,Rin\nRρmaxis the ratio between inner edge and density maximum location andΩρmax\nΩinis the ratio between the angular velocity\nat the density maximum and the inner edge of the disk.17\n(a) BS: 0.841, βmc= 105\n(b) BS: 0.841, βmc= 1\n (c) BS: 0.841, βmc= 10−5\n(d) BS: 0.694, βmc= 105\n(e) BS: 0.694, βmc= 1\n (f) BS: 0.694, βmc= 10−5\nFig. 15: Meridional cross section plots of the density distribution in pseudo-cylindrical coordinates. The upper half shows the\nBS torus, while the lower half shows the Kerr torus. The upper bound of the density scale is set to the maximum density\nbetween the BS and Kerr solution for the highest magnetization, log10ρmax= 0.238. The lower bound of the density scale is\nset to log10ρ=−3 for the BS 0.841 plots and to log10ρ=−5 for the BS 0.694 plots. Black curves mark equi-density surfaces.\nFig. 16: Bernoulli parameter of the BS and Kerr disks in the equatorial plane for different degrees of magnetization, with the\nnon-magnetized to the high magnetized solutions from left to right. For the non-magnetized disks a zoom-in plot is inserted\nfor a better resolution, since Beis nearly constant for both disks and close to 0, with the maximum difference to 0 being on\nthe order of 10−6.18\n4.4. Magnetization influence on the relative difference of disk quantities\nThe relative difference of a BS disk quantity Qto its corresponding BH disk quantity ˜Qcan be expressed as δQ=\n|˜Q−Q|\n˜Q. If a disk quantity is affected by the degree of magnetization, its relative difference can be expressed as a\nfunction of the magnetization parameter βmc. The function δQ(βmc) can therefore be used as a direct measure of the\nextent to which magnetic fields influence the similarities and differences between BS and Kerr disks. Fig. 17 shows an\nanalysis of the relative difference of selected disk quantities as a function of βmcfor the previously shown high spin,\nmid spin and low spin solutions.\n(a)a= 0.995\n (b)a= 0.908\n(c)a= 0.825,ωBS= 0.841\n (d)a= 0.825,ωBS= 0.694\nFig. 17: Relative difference of selected disk quantities versus the magnetization parameter βmcfor the different BS and Kerr\nsolutions classified by their spin parameter. δRρmaxis the relative difference of the density maximum location, δρmaxis the\nrelative difference of the maximum density, δΩρmaxis the relative difference of the angular velocity which a particle would\nhave at the density maximum, δpth\nmaxis the relative difference of the maximum thermodynamic pressure, δpmag\nmaxis the relative\ndifference of the maximum magnetic pressure and Rout−maxis the relative difference of the equatorial distance to the effective\nouter edge Rout. The highly magnetized disks are represented by a small value of the magnetization parameter and are therefore\nrepresented by the left side of the plots.\nFor the high and mid range spin parameter solutions the relative difference of every quantity is the highest for the\nstrongly magnetized disks, as all quantities increase monotonically with a falling magnetization parameter. The\nlower spin parameter solutions showcase a different behavior, since the difference of the thermodynamic and magnetic\npressure and also the difference of the density maximum location as well as the difference of the compactness parameter\nRout−maxdrop for highly magnetized disks. Regarding the BS 0.694 solution the only quantity difference which\nincreases for highly magnetized disks is the density maximum. Overall the smallest differences in most of the quantities\noccur for the low spin and mildly relativistic BS 0.841 solution and the largest deviations for the high spin BS solution19\nwith a= 0.995. We conclude that strong magnetic fields increase the difference in disk quantities for high and mid\nspin parameter solutions, whereas for low spin parameter solutions the opposite can occur for most of the general\ndisk quantities. Strong magnetic fields could therefore lead to a correlation in accretion disk properties for some disk\nsolutions in the lower spin parameter range.\n5. RETROGRADE DISKS\nIn the case of retrograde disks, there are major differences between the possible BS and Kerr disk solution types. The\npossible types for Kerr solutions are similar to the prograde disks, with cusp solutions possible for a lower absolute\nvalue of the Keplerian specific angular momentum and cuspless solutions for higher absolute values (Fig. 18). The\nKeplerian specific angular momentum for BSs is non-monotonic for retrograde motion in contrast to prograde motion\n(Fig. 18). This imposes a variety of possible disk types, including two-centered solutions.\nFig. 18: Retrograde Keplerian specific angular momentum in the equatorial plane for the BS solutions on the left and Kerr\nsolutions on the right. The BS and Kerr solutions are classified by their spin parameter a. Curve sections colored black mark\nintervals where only unbound orbits exist.\nFor high spin parameter BS solutions ℓ−\nKhas a maximum and ℓ0can have possibly three intersections with ℓ−\nK.\nDepending on the second intersection these solutions mark either two-centered solutions, where the inner and outer\ncenter are connected by a cusp located at the second intersection, or a cuspless two-centered solution, where the disk\nis composed of an inner and outer torus. If the Keplerian specific angular momentum at the second intersection\ncorresponds to an unbound orbit, the disk solution is cuspless. Under the influence of strong magnetic fields the\ndensity and pressure distributions of these kind of solutions can change to solutions with one density maximum,\nwhere the outer density maximum can get suppressed for magnetized disks. A more profound analysis can be found\nin [55]. Naturally there are more possible differences between retrograde disks for BSs and BHs, especially if the disk\ncenter is located close to the origin, since the angular momentum of the disk would then be of small absolute value.\nWith increasing distance the specific angular momentum is increasing in absolute value monotonically for BHs as well\nas for BSs. For higher absolute values of the specific angular momentum there exist therefore cuspless one-centered\nsolutions for BSs and BHs. These solutions would be located further away from the origin and they would be more\ncomparable to each other in contrast to lower absolute angular momentum solutions, but astrophysical accretion disks\nare expected to have a specific angular momentum close to ℓmbandℓms. In general, these solutions would therefore\nbe of less interest than solutions where the disk center is located close to the origin (with a small absolute specific\nangular momentum of the disk). It might therefore be less difficult in the future to possibly distinguish retrograde\nBS disks from retrograde BH disks as compared to prograde disks.\n6. CONCLUSION\nWe classified BS and Kerr solutions by a dimensionless spin parameter ato compare their magnetized prograde\nthick disk solutions to each other. The influence of magnetization on the disk properties was isolated by fixing all20\nother solution parameters besides the magnetization parameter βmc. Disk solutions with the same spin parameter\nand the same specific angular momentum were compared to each other. For the specific angular momentum we\nassumed a uniform distribution throughout the disk and set its value to the Keplerian specific angular momentum\nvalue corresponding to the marginally bound orbit in the Kerr solution. The classification of BS and Kerr solutions\ndefined a spin parameter range with its lower boundary given by the lowest spin parameter of the BS solutions,\na= 0.798, and the upper boundary given by the Kerr limit a= 1. The lower half of the spin parameter range\nis composed of a BS solution possessing a twofold degeneracy, where two different BS solutions (classified by their\nangular frequency ω) could have the same spin parameter. For an exemplary analysis of the whole spin parameter\nsolution space, we picked solutions from the high, medium and low spin parameter range, namely a= 0.995,a= 0.908\nanda= 0.825, whereby two BS solutions were corresponding to the a= 0.825 solution. Each of these exemplary\nsolutions is capturing qualitatively the possible disk solution types of its represented spin parameter range. In order\nto undertake a qualitative comparison of accretion disks in different spacetime geometries we defined proper physical\ncoordinates for the comparison, defined by the proper radial distance Rr. We computed and analyzed the properties\nof BS and BH disks for different degrees of magnetization, from non-magnetized disks ( βmc≫1) to mildly magnetized\ndisks ( β∼1) to highly magnetized disks ( βmc≪1) and compared them to each other. The density distribution acted\nas the main analyzed quantity, which was presented in the equatorial plane as well as for the vertical profile at the\ndensity maximum and also in a meridional cross-section plot. Besides other presented disk quantities the Bernoulli\nparameter was used as a further quantity to analyze the influence of magnetization on the disk stability.\nWe found that magnetic fields strengthen the already present differences in non-magnetized disks for high spin\nparameter solutions such as the greater extent of the BS disks, especially in their vertical profile. The vertical thickness\nof the BS disks appears to be their main distinctive feature for the high spin parameter solutions, their properties are in\ngeneral less affected by a high magnetization compared to the BH disks, which get more compactified with increasing\nmagnetization. This is also reflected by the Bernoulli parameter, which is more affected by the magnetization for\nthe BH disks. Therefore the high magnetized BS disks appear to be less energetically bound compared to the BH\ndisk. In general, the properties of the high spin parameter BS disks are more resilient to changes in the degree of\nmagnetization.\nRegarding the mid spin parameter solutions a similar behavior can be observed, with increasing magnetization the\ndifferences between BS and BH disks increase also. However, these differences are of less impact compared to the high\nspin parameter solutions, since the non-magnetized disks are already similar to each other and, furthermore, the disk\nsolutions are affected with a comparable influence by magnetization (with a more compact Kerr solution again). Also\nthe Bernoulli parameter is similarly influenced in both cases by βmc, both disks are energetically similarly bound.\nNevertheless, like for the high spin parameter solutions, the greater vertical thickness can be identified as a main\ndistinctive feature of the BS disks compared to the Kerr disks.\nIn case of the lower spin parameter an opposite behavior compared to the high and mid spin parameter solutions can\nbe observed. With increasing magnetization the differences between BS and BH disks decrease. The equatorial extent\nof the BH disk is now greater compared to both BS disks for all degrees of magnetization, but as in the high and mid\nspin parameter solutions, the vertical density for the BH disk is stronger affected by magnetization. All three disk\nsolutions are compactified similarly, with the highly relativistic (lower angular frequency) BS solution being the most\ninfluenced by strong magnetic fields.\nWe summarize, BS and BH disks get both compactified with increasing magnetization and tend to be more bound,\nsince their equatorial Bernoulli parameter is developing a clear minimum for the highly magnetized disks. The\ndifferences in the main disk properties between BS and BH disks may increase for highly magnetized disks or even\ndecrease, depending on the spin parameter of the BS and Kerr solutions. For high and mid range spin parameter\nsolutions magnetic fields strengthen the differences and for low spin parameter solutions the differences may decrease,\ndepending on the BS solution. Strong magnetic fields could lead therefore to a correlation in disk properties of\naccretion disks around different types of compact objects, such as BS and BHs, as shown. Our results demonstrate\nthat highly magnetized disks around fast rotating boson stars deviate most strongly from the Kerr black hole. Thus,\nastrophysical systems with the corresponding properties could be a suitable laboratory for detecting boson stars. On\nthe other hand, highly magnetized disks around slowly rotating mildly relativistic boson stars resemble closely the\ncorresponding disks around Kerr black holes. Therefore, a boson star with the described physical conditions could\nrepresent a good Kerr black hole mimicker. We identify the vertical profile of BS disks as the main distinctive feature\ncompared to BH disks for high and mid spin parameter solutions and their smaller equatorial extent for low spin\nparameter solutions.21\nACKNOWLEDGMENTS\nWe would like to thank Lucas Collodel and Matheus Teodoro for discussions and BS data. JK gratefully acknowledges\nsupport by DFG project Ku612/18-1. 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Soc. 408, 752 (2010)."    },    {        "title": "2401.13862v1.Second_Laplacian_eigenvalue_on_real_projective_space.pdf",        "content": "arXiv:2401.13862v1  [math.SP]  25 Jan 2024UPPER BOUND ON THE SECOND LAPLACIAN EIGENVALUE\nON REAL PROJECTIVE SPACE\nHANNA N. KIM\nAbstract. In this paper, we prove an upper bound on the second non-zero La pla-\ncian eigenvalue on n-dimensional real projective space. The sharp result for 2-\ndimensions was shown by Nadirashvili and Penskoi and later by Karpu khin when\nthe metric degenerates to that of the disjoint union of a round pro jective space and\na sphere. That conjecture is open in higher dimensions, but this pap er proves it\nup to a constant factor that tends to 1 as the dimension tends to in finity. Also, we\nintroduce a topological argument that deals with the orthogonality conditions in a\nsingle step proof.\n1.Introduction and results\nForn-dimensional real projective space RPn, consider a round metric gand its\nconformally equivalent metrics wgwherewis the conformal factor. Throughout\nthe paper, the metric is normalized so that the volume Vol( RPn,wg)≡Voln(w) is\nequal to the volume Vol( RPn,g) of the round projective space. The Laplace–Beltrami\noperator −∆wgon (RPn,wg) has a discrete sequence of eigenvalues:\n0 =λ0(w)<λ1(w)≤λ2(w)≤ ··· → ∞ .\nThis paper provides an upper bound for the second nonzero eigenv alueλ2(w) for\nallRPn,n≥2. The strategy of the proof is to find n+ 1 trial functions, which we\nconstruct using Veronese embeddings to map projective spaces t o higher dimensional\nspheres followed by a fold and a M¨ obius transformation. We rely on t opological\narguments to show that the trial functions are orthogonal to th e first two modes.\n1.1.Results on the second eigenvalue. In this section, we review isoperimetric\ninequalities of the Laplacian eigenvalues related to spheres and proj ective spaces, and\nstate our main result.\nTheorem 1 (Li and Yau [11] for n= 2 and El Soufi and Ilias [3] for n≥2; first\neigenvalue on RPn).Assume that the metric wgis normalized so that Vol n(w) =\nVoln(1). Then the first nonzero eigenvalue of −∆wgsatisfies,\nλ1(w)≤λ1(1) = 2n+2.\nThe upper bound is attained when the metric is round on RPn. For the next higher\neigenvalue on 2-dimensional projective space, we have the following result.\n2020Mathematics Subject Classification. Primary 35P15. Secondary 58C40, 58J50.\nKey words and phrases. Spectral theory, shape optimization.\n12 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nTheorem 2 (Nadirashvili and Penskoi [13]; second eigenvalue on RP2).Assume that\nthe metric wgis normalized so that Vol 2(w) =Vol2(1)onRP2. Then the second\nnonzero eigenvalue of −∆wgsatisfies,\nλ2(w)≤10.\nThe upper bound can be obtained by a sequence of metrics approac hing to that\nof the disjoint union of a round projective space and a sphere havin g ratio of radii√\n6 :√\n2. A natural conjecture is that the upper bound can be extended to all\ndimensions.\nConjecture 3 (Second eigenvalue on RPn,n≥3).Assume that the metric wgis\nnormalized so that Vol n(w) =Voln(1)onRPn. Then the second nonzero eigenvalue\nof−∆wgsatisfies,\nλ2(w)<((2n+2)n/2+2nn/2)2/n. (1)\nThe upper bound can be obtained by a sequence of metrics approac hing to that of\ndisjoint union of a round projective space and a round sphere havin g ratio of radii√2n+2 :√n.\nWe prove the following bound on λ2(w), with a right hand side that is larger than\nin the conjecture.\nTheorem 4 (Second eigenvalue on RPn,n≥3).Assume that the metric wgis\nnormalized so that Vol n(w) =Voln(1)onRPn. Then the second nonzero eigenvalue\nof−∆wgsatisfies,\nλ2(w)<22/n(2n+2). (2)\nIndimension2, theabovetheoremgives λ2(w)<12, whichisweaker thanthesharp\nbound 10 in Theorem 2. In Lemma 19, we confirm that Conjecture 3 is stronger than\nTheorem 4, and that the theorem is asymptotically sharp as ntends to ∞because\nthe ratio between the two upper bounds approaches 1.\nTo conclude this introduction, we summarize some related literature .\nHigher eigenvalues on the 2-dimensional real projective sp aceRP2.For the\nhigher eigenvalues on RP2, it is known by Karpukhin [6] that k-th eigenvalue has\na sharp upper bound when a sequence of metrics converges to the disjoint union of\na real projective space and k−1 identical round spheres, where the ratio of radii\nbetween the projective space and the spheres is√\n6 :√\n2.\nFirst and second eigenvalue on Sn.For the first eigenvalue λ1, Hersch in 2-\ndimensions [5] and El Soufi and Ilias [3] in higher dimensions proved that the sharp\nupper bound of the first nonzero eigenvalue λ1is attainedby the round metric (within\nthat conformal class). They found trial functions orthogonal t o the constant by\ncomposing the eigenfunctions of the round sphere (which are the m+ 1 coordinate\nfunctions) with a M¨ obius transformation to move the center of th e mass to the origin.\nFor maximizing λ2on the 2-sphere, Nadirashvili [12] and later Petrides [16] showed\nthat there exists a maximizing sequence of metrics degenerating to a disjoint union\nof two equal round spheres. In my work [9], the analogous result fo r the secondSECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 3\neigenvalueλ2on the higher-dimensional sphere Snis proved. Inspired by the prior\nworks on constructing trial functions, the method in this paper re lies on building trial\nfunctions satisfying the orthogonality conditions by composing the eigenfunctions of\nthe round sphere with a fold map and a M¨ obius transformation.\nHigher eigenvalues on 2-sphere. Nadirashvili and Sire [14] confirmed the anal-\nogous result for the third nonzero eigenvalue λ3. For all eigenvalues, by using the\nwork of Petrides [17], Karpukhin, Nadirashvili, Penskoi and Polterov ich [7, Theorem\n1.2] showed that the k-th eigenvalue on S2is maximal when a sequence of metrics\ndegenerates to the disjoint union of kidentical round spheres.\n2.Overview of the proof\nLet us preview the trial functions, which are the components in Rm(n)of the vector\nvalued map:\n(Y◦T−c◦FH◦Φ)(y), y∈RPn, (3)\nwhereT−cis a M¨ obius transformation on the ball Bm(n),FHis a fold map on Sm(n)−1,\nΦ≡Φn:RPn→Sm(n)−1is a generalized Veronese map and Y(y) =yis the identity\nmap on the sphere. Note that each component Yj(y) =yjis an eigenfunction for the\nround sphere. We drop the identity map “ Y” in the work that follows since its role\nin (3) is mainly to emphasize that the trial functions are the m+ 1 components of\nthe vector field in (3).\nIn this paper, we generalize Veronese embedding to all dimensions by induction,\nbuilding on the work of Zhang [21]. We adapt the conformal factors s o that the\nimages of Veronese embeddings lie on higher dimensional spheres of r adius 1. The\nconstruction of trial functions relies on my earlier work [9]. We build m(n)-number\nof trial functions that satisfy the conditions for the variational c haracterization of\nthe second nonzero eigenvalue: orthogonality to the constant an d to the first excited\nstate on RPnwith the metric wg.\nLet us mention again some relavant previous works. For RP2, Li and Yau [11]\nused the well-known Veronese embedding, which is a minimal immersion o fRP2to\nS4, to show that the round metric maximizes the first eigenvalue. El So ufi and\nIlias [3] generalized the result to RPnfor alln≥2, showing that since RPncan be\nminimally immersed to a higher dimensional sphere by its first eigenfunc tions, the\nstandard metric induces the sharp upper bound. We combine this ide a of mapping\nreal projective space to a higher dimensional sphere with the meth od of constructing\ntrial functions from my previous paper [9].\n3.Veronese embedding of projective space into a sphere\nOur goal in this section is to construct a “generalized Veronese” ma p Φn:RPn→\nSm(n)−1that is a conformal embedding. We first define Φ n:Rn+1→Rm(n), and show\nthat when the map is restricted on the real projective space, the image lies in a higher\ndimensional sphere.4 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nWe denote m(n) as the multiplicity of the first eigenvalue of the round RPn,\nm(n) =n(n+3)\n2\nforalln≥1 (see [20, Corollary7.4.3]). Andnote thatthefirst eigenvalue ofthe round\nRPnis 2n+2. Readers may wish to skip this section since the explicit formulas ar e\nnot needed and we only need the conformal embedding property in P roposition 6.\nDefinition 5. Forn= 1, define Φ 1:R2→R2by\nΦ1(x1,x2) =/parenleftbig\n2x1x2,x2\n1−x2\n2/parenrightbig\n.\nForn= 2, define Φ 2:R3→R5by\nΦ2(x1,x2,x3) =√\n3/parenleftbigg\nx1x2,1\n2(x2\n1−x2\n2),x1x3,x2x3,1\n2√\n3(x2\n1+x2\n2−2x2\n3)/parenrightbigg\n.\nForn≥2, define Φ n:Rn+1→Rm(n)inductively by\nΦn(x1,···,xn+1) =an/parenleftbigg1\nan−1Φn−1(x1,...,xn),x1xn+1,... ...,x nxn+1,\n1\nnan/parenleftbig\nx2\n1+x2\n2+···+x2\nn−nx2\nn+1/parenrightbig/parenrightbigg\n,(4)\nwhere the constant is\nan=/parenleftbigg2n+2\nn/parenrightbigg1/2\n.\nAs a remark, when n= 1, one can see that the image consists of coordinates of the\nsquare of the complex number z=x1+ix2. Note that when n= 2, the map Φ 2is the\nwell-known Veronese conformal embedding. The inductive definition of Φnforn≥2\ninvolves a normalized Φ n−1, the cross terms xixn+1and then the final term. Notice\nthat Φn(−x) = Φn(x) and so Φ nis well-defined on RPn.\nProposition 6. If the domain of Φnin (4) is restricted from Rn+1to the projective\nspaceRPn, then\nΦn:RPn→Sm(n)−1\nis a conformal embedding with the conformal factor as an.\nThe proposition follows by combining the next two lemmas.\nLemma 7. The Veronese map Φn:Rn+1→Rm(n)satisfies |Φn(x)|=|x|2. In\nparticular, the image of the map Φnrestricted to the projective space RPnlies in the\nunit(m(n)−1)-sphere and the map Φnis injective on RPn.\nProof.One can check that the result holds easily for n= 1. Now, assume that the\nlemma holds for n−1≥1. We show that it also holds for n.SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 5\nWe have |Φn−1(x1,...,xn)|=|(x1,...,xn)|2by the induction hypothesis. The\nsquare of |Φn|is written as follows:\n|Φn(x1,...,xn+1)|2=2n+2\nn/parenleftbiggn−1\nλn−1(x2\n1+···+x2\nn)2+(x2\n1+···+x2\nn)x2\nn+1\n+1\nn(2n+2)(x2\n1+...···+x2\nn−nx2\nn+1)2/parenrightbigg\n.\nAfter replacing x2\n1+···+x2\nnwith|x|2−x2\nn+1and simplifying the above, one finds\n|Φn(x1,...,xn+1)|2=|x|4, wherex= (x1,...,xn+1)Tis a column vector.\nOne can easily see that the map is injective on RP1since on the complex plane, the\nmap can be written as Φ 1(z) =i¯z2which is injective on RP1. Using a short induction\nargument, we find Φ nis injective on RPn.\n/square\nNext, we compute the properties of DΦnand show the Veronese map Φ nis confor-\nmal onRPn.\nLemma 8. Forn≥1,\n(DΦn(x))TDΦn(x) =2(n+1)\nn|x|2I+2(n−1)\nnxxT,x∈Rn+1.\nIn particular, Φnrestricted to the projective space RPnis a conformal map into an\n(m(n)−1)-sphere with conformal factor an.\nProof.We prove by induction. For n= 1,\nDΦT\n1DΦ1= 4/parenleftbigg\nx2x1\nx1−x2/parenrightbigg/parenleftbigg\nx2x1\nx1−x2/parenrightbigg\n= 4/parenleftbigg\nx2\n1+x2\n20\n0x2\n1+x2\n2/parenrightbigg\n= 4|x|2I,\nwhere the conformal factor is a1= 2. Assume that the lemma holds for n−1≥1.\nRecall that an=/radicalbig\n2(n+1)/nand define bn=/radicalbig\n2(n−1)/n. By the induction\nhypothesis,\n(DΦn−1(xn))TDΦn−1(xn) =a2\nn−1|xn|2In+b2\nn−1xnxT\nn,\nwhereInis ann×nidentity matrix and xn= (x1,...,xn)T∈Rn. Letx=\n(xn,xn+1)T∈Rn+1. The derivative of Φ nis written as a 3 ×2 block matrix,\nDΦn(x) =an\na−1\nn−1DΦn−10\nxn+1In xn\ndnxT\nn−bn+1xn+1\n,\nwheredn=/radicalbig\n2/n(n+1). We write DΦT\nnDΦnas a 2×2 block matrix,\nDΦT\nnDΦn=a2\nn\na−2\nn−1DΦT\nn−1DΦn−1+x2\nn+1In+d2\nnxnxT\nn(1−bn+1dn)xn+1xn\n(1−bn+1dn)xn+1xT\nn xT\nnxn+b2\nn+1x2\nn+1\n.\n(5)6 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nNext, we use the induction hypothesis to expand DΦT\nn−1DΦn−1on the upper left term\nin (5), having\na−2\nn−1DΦT\nn−1DΦn−1+x2\nn+1In+d2\nnxnxT\nn=|xn|2In+a−2\nn−1b2\nn−1xnxT\nn+x2\nn+1In+d2\nnxnxT\nn\n=|x|2In+(a−2\nn−1b2\nn−1+d2\nn)xnxT\nn.\nHence we can write the matrix in (5) as follows,\nDΦT\nnDΦn=a2\nn|x|2In+1+a2\nn\n(a−2\nn−1b2\nn−1+d2\nn)xnxT\nn(1−bn+1dn)xn+1xn\n(1−bn+1dn)xn+1xT\nn(b2\nn+1−1)x2\nn+1\n.(6)\nWe only need to check all four coefficients of the second matrix in (6) are equal to\nb2\nn/a2\nn, which is straightforward. So, the conclusion of the Lemma holds fo r alln.\nFinally, weshowthatthemapisconformalon RPn. Letx∈Sn. Forallu,v∈Rn+1\nthat are tangential to the sphere at x,\nuTDΦT\nnDΦnv=uT/parenleftbigg2(n+1)\nn|x|2I+2(n−1)\nnxxT/parenrightbigg\nv\n=a2\nnuTv,\nwhere|x|2= 1 and the term including uTxandxTvvanished since the vectors u,v\nof the tangent space are orthogonal to the point xon the sphere. This proves that\nthe generalized Veronese map Φ nonRPnpreserves angles and scales distances by the\nconformal factor an.\n/square\n4.Trial Functions and Orthogonality\nRelying on the techniques from Freitas and Laugesen [4] for domains in hyperbolic\nspace, I constructed the trial functions on the sphere in my prev ious paper [9]. In\nthis paper, we adapt the method by first composing the Veronese e mbedding, since\nthe embedding takes RPnintoSm(n)−1. Then we can apply a similar argument from\nthe sphere: fold the image across some spherical cap and compose it with a suitable\nM¨ obius transformation. Now, we reintroduce the definitions of th ese maps to help\nthe reader’s understanding.\n4.1.M¨ obius transformations. WriteBmfor the unit ball in RmandSm−1=∂Bm.\nDefine the M¨ obius transformations on the closed ball [19, eq.(2.1.6) ], parametrized\nbyx∈Bm, as\nTx:Bm→Bm,\nTx(y) =(1+2x·y+|y|2)x+(1−|x|2)y\n1+2x·y+|x|2|y|2, y∈Bm. (7)\nAs a remark, T0becomes just the identity map on the ball. Also, Tx(0) =x,T−x=\n(Tx)−1, andTxmapsSm−1toSm−1, fixing the points y=±x/|x|.SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 7\n4.2.Spherical caps, reflection, and folding. The following material is drawn\nfrom [9] and is included here for the reader’s convenience. We descr ibe spherical\ncaps. For any unit vector pon the sphere, the closed hemisphere can be written as\nHp={y∈Sm−1:y·p≤0}, p∈Sm−1.\nNext, define the spherical caps as the image of the hemisphere Hpunder some M¨ obius\ntransformation:\nH≡Hp,t=Tpt(Hp), p∈Sm−1, t∈[0,1).\nHere, the M¨ obius transformation sends boundaries to boundarie s, that is,Tpt(∂Hp) =\n∂Hp,t. Then we can write the spherical cap explicitly as\nHp,t=/braceleftbigg\ny∈Sn:y·p≤2t\n1+t2/bracerightbigg\n,\nwhich can be easily calculated by (7). Note that as tapproaches to 1, the spherical\ncapHp,tcovers almost all the sphere toward p.\nGivenb∈Rm\\ {0}, the reflection Rbin the hyperplane through the origin and\nperpendicular to the vector bis defined as,\nRb(y) =y−2(y·b)\n|b|2b, y∈Sm−1.\nBy conjugation, we can define a reflection map across the boundar y of the general\nspherical cap Hp,t: let\nRH≡Rp,t=Tpt◦Rp◦(Tpt)−1:Sm−1→Sm−1.\nNote thatRHp,t(p) =−p, which says that the reflection map sends pto its antipodal\npoint. Lastly, we define a “fold map” that reflects the complement o f the spherical\ncap across the boundary:\nFH(y)≡Fp,t(y) =/braceleftigg\ny, y ∈H,\nRH(y), y∈Sm−1\\H.\nObserve that the image of the boundary of the spherical cap ∂His itself. For simplic-\nity, we use different notations FHandFp,tto denote the same map when it is clear\nfrom the context (similarly for RH).\n4.3.Center of mass and trial functions. A pointc∈Bmis the (hyperbolic)\ncenter of mass of a Borel measure µon the sphere if\n/integraldisplay\nSm−1T−c(y)dµ(y) = 0. (8)\nThe trial functions are defined to be the components in Rm(n)of the vector valued\nmap:\ny/ma√sto→(T−c◦FH◦Φn)(y), y∈RPn, (9)8 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nwhereT−cis a M¨ obius transformation on the ball Bm(n),FHis a fold map on Sm(n)−1,\nand Φ≡Φn:RPn→Sm(n)−1is the generalized Veronese map defined in Section 3.\nHow the center of mass cand spherical cap Hare chosen is explained below.\nNext, we present two different topological proofs that the trial f unctions satisfy the\northogonality conditions. The first argument is based on the idea of Hersch [5] and\nthe reflection symmetry lemma by Petrides [16]. The second argumen t is essentially\nfrom a result of Karpukhin and Stern [8, Lemma 4.2]. We give a new proo f.\n4.4.Orthogonality of the trial functions by two-step proof. Based on the\nconstruction above, we need to show that the trial functions are orthogonal to the\nconstant and the first eigenfunction with respect to the metric wg, so that we can use\nthe variational characterization of the second eigenvalue. The fir st proof of orthogo-\nnality of trial functions (9) proceeds similarly to to my previous work [9, p.3506], as\nwe now explain.\nDefine a push-forward measure µonSm(n)−1byµ= (FH◦Φ)∗vwgwherevwgis the\nvolume measure on RPnwith respect to the metric wg. A center of mass c=cHin\n(8) exists by Hersch’s lemma; see Laugesen [10, Corollary 5] for the precise statement.\nThe assumptions of the corollary are satisfied by extending µoutsideFH(Φ(RPn))\nto all of the sphere with zero, noting that this extended pushforw ard measure is a\nfinite Borel measure and 0 = µ({y})<1\n2µ(Sm(n)−1) for ally∈Sm(n)−1. The result\nof the corollary gives the existence and uniqueness of the center o f masscH=cp,t,\nand the center of mass depends continuously on the parameters o f the spherical cap,\n(p,t)∈Sm(n)−1×[0,1).\nWe later need the continuity of the center of mass as t→1. The M¨ obius transfor-\nmation and the fold map do not extend continuously when t= 1, but nonetheless, the\ncenter of mass cp,tconverges to a point c(w) which depends only on the measure and\nis independent of p. See [9, p.3506]. The underlying point is that the push-forward\nmeasure is weakly convergent as t→1.\nThe orthogonality of trial functions to the first excited state fof−∆wgonRPn\nrequires that the following vector field vanishes at some point ( p,t)∈Sm(n)−1×[0,1):\nV(p,t) =/integraldisplay\nRPnT−cH(FH(Φ(y)))f(y)dvwg(y), (10)\nwhereH=Hp,t. Note that from the continuous dependence on the parameters, the\nvector field (10) is continuous. Since the image of the Veronese map is in the sphere,\nthe argument in my previous paper [9, Section 2.8], which is based on th e topological\nargument by Petrides [16, claim 3], applies with obvious changes. We sk ip the details\nand state the result below from [9, Proposition 8].\nProposition 9 (Vanishing of the vector field) .V(p,t) = 0for somep∈Sm(n)−1and\nt∈[0,1].\nThis result finishes the first argument for the orthogonality condit ions. Before we\nproceed to the second argument, I would like to introduce different approaches to the\northogonality conditions for trial functions. Petrides [16] showed that a map with\nreflection symmetry has nonzero degree, leading to the two-step argument above.SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 9\nFreitas and Laugesen [4] gave a new proof of Petrides’s Lemma by a g lobal approach\nusing the de Rham definition of the degree.\nKarpukhin and Stern [8, Lemma 4.2] relied instead on the Lefschetz– Hopf fixed\npoint theorem, obtaining both orthogonality conditions in a one-ste p proof. In the\nnext section, we give a proof similar to their lemma by adapting a metho d for proving\nthe Borsuk–Ulam theorem. A third approach by Bucur, Martinet an d Nahon [2]\nrelies on the index theorem to give a one-step proof in a closely relate d orthogonality\nsituation.\n4.5.Orthogonality ofthetrial functions by one-step proof. WerelyonTheorem 11\nin Section 5 to prove that for some choice of parameters, the trial functions are or-\nthogonal to both the constant and the first excited state. Comp ared to the previous\nvector field defined in (10), we assume now that the center of mass is an independent\nvariable in the following vector field,\nV(x,p,t) =/parenleftbigg/integraldisplay\nRPnT−x(FH(Φ(y)))dvwg,/integraldisplay\nRPnT−x(FH(Φ(y)))f(y)dvwg/parenrightbigg\n,\nwhich is a map from Bm(n)×Sm(n)−1×[0,1) toR2m(n). HereH=Hp,t. Now,\nwe extend Vcontinuously to t= 1 and |x|= 1. When t= 1, the vector field is\ncontinuous and independent of p, by a similar argument as in the previous section.\nAsx→˜x∈Sm(n)−1, the vector field can be extended continuously by the dominated\nconvergence similarly and then the vector field becomes independen t ofpandt. In\nfact, the vector field at x= ˜xis\nV(˜x,p,t) = Voln(w)(−˜x,0)\nsince the M¨ obius transformation degenerates and/integraltext\nRPnf(y)dvwg= 0. In the next\nproposition, we again use the fact that the degree is a homotopy inv ariant to show\nthat the vector field vanishes at some choice of parameters.\nProposition 10 (Vanishing of the vector field) .V(x,p,t) = 0for some (x,p)∈\nBm(n)×Sm(n)−1andt∈[0,1].\nProof.Letm=m(n). We endow the following equivalence relation on Bm×Sm−1:\nFor anyx∈∂Bmandp,q∈Sm−1, we say (x,p)∼(x,q). Define a homeomorphism\nBm×Sm−1/∼ →S2m−1as (x,p)/ma√sto→(/radicalbig\n2−|x|2x,(|x|2−1)p) = (a,b). (This home-\nomorphism makes the boundary points of the ball collapse onto the p oints in the\nsphere such that the second part of the coordinates are zero.) T he inverse parameters\nare\nx(a) =−a/radicalbig\n1+|b|andp(b) =−b\n|b|.\nWhenb/\\e}atio\\slash= 0,x(a) only depends on asince|b|2= 1− |a|2. Whenb= 0, we have\nx(a) =−aandp(b) is undefined.\nNext, we precompose V with the inverse of the homeomorphism and d enote the\nmap as\n/tildewideV(a,b,t) =V(x(a),p(b),t),(a,b)∈S2m−1,10 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nwhere forb= 0, we have |a|= 1,x(a) =−a,|x(a)|= 1, and so /tildewideV(a,0,t) =\nVoln(w)(a,0).\nSupposeV(x,p,t) does not vanish, so that /tildewideVdoes not vanish. We obtain a con-\ntradiction later. For each t, letWt(a,b) :=/tildewideV(a,b,t)/|/tildewideV(a,b,t)|be a map from S2m−1\nto itself. When b= 0, noteWt(a,0) = (a,0).\nWhent= 1, the fold map becomes identity on all of the sphere except at one point,\nso we have\nW1(a,b) =/parenleftbigg/integraldisplay\nRPnT−x(a)(Φ(y))dvwg,/integraldisplay\nRPnT−x(a)(Φ(y))f(y)dvwg/parenrightbigg\n.\nThe mapW1is not surjective onto S2m−1since it is smooth and the right side depends\nonly on the mdimensional parameters a, while 2m−1>m. Hence,W1is homotopic\nto a constant map and so has degree zero, which implies that W0also has degree zero.\nWhent= 0, the map ( a,b)/ma√sto→W0(a,b) satisfies the reflection symmetry condition\n(11) below, by the following calculation. When b/\\e}atio\\slash= 0, we calculate that\n/tildewideV(Rb(a),−b,0)\n=/parenleftbigg/integraldisplay\nRPnT−x(Rb(a))(Fp(−b),0(Φ(y)))dvwg,/integraldisplay\nRPnT−x(Rb(a))(Fp(−b),0(Φ(y)))f(y)dvwg/parenrightbigg\n=/parenleftbigg/integraldisplay\nRPnT−Rb(x(a))(RbFp(b),0(Φ(y)))dvwg,/integraldisplay\nRPnT−Rb(x(a))(RbFp(b),0(Φ(y)))f(y)dvwg/parenrightbigg\nsincex(Rb(a)) =Rb(x(a)) by linearity of Rb, andFp(−b),0=RbFp(b),0\n=/parenleftbigg\nRb/integraldisplay\nRPnT−x(a)(Fp(b),0(Φ(y)))dvwg,Rb/integraldisplay\nRPnT−x(a)(Fp(b),0(Φ(y)))f(y)dvwg/parenrightbigg\nby the property TRba(Rby) =RbTa(y) in [9, p. 3507]\n= (Rb×Rb)/tildewideV(a,b,0).\nNow the reflection symmetry condition for W0follows by dividing each side by its\nnorm. When b= 0,W0(a,0) = (a,0). Thus (11) holds for W0.\nHence, the map W0has nonzero degree by Theorem 11, which is a contradiction.\n/square\n5.Calculation of the degree of the self-maps\nIn this section, we calculate the degree of the continuous maps bet ween odd-\ndimensional spheres with reflection symmetry. Recall the following f acts about the\ntoplogical degree of continuous maps on a sphere. See Outerelo an d Ruiz [15, Chapter\nIV.4]. Letϕ:Bn+1→Rn+1be a continuous map such that ϕ(Sn)⊂Rn+1\\{0}. De-\nfine a continuous map φ:Sn→Snasφ(x) =ϕ(x)/|ϕ(x)|forx∈Sn. It is well-known\nthat for any point p∈Sn,d(ϕ,Bn+1,0) = deg(φ,Sn,p). Moreover, since the degree of\nφis consistent on any p∈Sn, we may write deg( φ) = deg(φ,Sn,p). The main goal of\nthis section is to show the following theorem.SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 11\nTheorem 11. Letφ:S2n+1→S2n+1be a continuous map and assume that the map\nsatisfies the following reflection symmetry property:\n(Rb×Rb)φ(a,b) =φ(Rb(a),−b)whenb/\\e}atio\\slash= 0,\nφ(a,0) = (a,0) whenb= 0,(11)\nfor alla,b∈Rn+1with(a,b)∈S2n+1. Thendeg(φ) = 1whennis odd, and deg(φ)is\nodd whennis even.\nAs a remark, Karpukhin and Stern [8, Lemma 4.2] showed that the de gree is odd\nwith a similar reflection symmetry condition.\nThe strategy of the proof is derived from an extension proof of th e well-known\nBorsuk–Ulam theorem. See [15, Theorem 5.2]. The goal is to extend t he map on\nthe sphere to the one on the closed ball while preserving continuity a nd the reflection\nsymmetry. We focus on the fact that on the hyperplane R2n+1={(a,b) :a∈\nRn+1,b∈Rn×{0}}, the reflection map Rbcan be written as a Cartesian product of\nthe reflection on Rnand an identity on R. Considering this hyperplane allows us to\nextend the map step-by-step by induction on dimension of the doma in. In particular,\nthe dimension of the parameter of the reflection map determines th e inductive step.\nThe extension of the map to all of B2n+2consists of an identity map on an ǫ-ball at\nthe origin and a continuous map with reflection symmetry on the rema ining domain.\nAfter proving these extension lemmas, we calculate the degree of t he extended map\nusing the reflection symmetry.\nFor simplicity, iis used for an inclusion map or identity map depending on the\ncontext. We use Rninstead of R2n+1in the next lemma to keep the notation simple\nfor now, and return to using R2n+1again after the next lemma.\nDefinition 12. LetD⊂Rnbe a set and kbe a fixed integer such that 1 ≤k≤n/2.\nConsider (a,b)∈Rnwherea∈Rn−kandb∈Rk. The domain is called k-symmetric\nif (a,b)∈D⇐⇒((Rb×i)(a),−b)∈D, whenever b/\\e}atio\\slash= 0. In the special case\nwhennis even and k=n/2,D⊂Rnbeingn/2-symmetric means (a,b)∈D⇐⇒\n(Rb(a),−b)∈D.\nLet’s first extend a map when the dimension of its k-symmetric domain is lower\nthan that of the codomain.\nLemma 13. LetD⊂Rn(n≥3) be bounded, open and k-symmetric with fixed\n1≤k < n/2such that 0/∈D. Letφ:∂D→Rm\\ {0},n < m(mis even) be a\ncontinuous map with the following reflection symmetry prope rty:\n(Rb×i×Rb×i)φ(a,b) =φ((Rb×i)(a),−b)whenb/\\e}atio\\slash= 0,\nφ(a,0) = (a,0)∈Rmwhenb= 0,(12)\nfor all(a,b)∈∂Dwitha∈Rn−kandb∈Rk. Then there exists a continuous map\nϕ:D→Rm\\{0}which extends φand satisfies the properties (12) for (a,b)∈D.\nHere are a few remarks about the statement to illustrate the lemma . The reason\nfor assuming n < mandmis even is due to the definition of reflection symmetry12 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nproperty: on the left hand side of (12), Rbacts onkdimensional space and iacts\non (m/2−k)-dimensional space; on the right side, iis an identity map acting on\n(n−2k)-dimensional space.\nProof.Defineφ(a,0) = (a,0) for alla∈Rn−k, noting this extended φis still continu-\nous. We begin induction on nwith the case when n= 3. Here, we only consider the\ncase whenk= 1, since 1 ≤k <3/2:a∈R2andb∈R. For convenience, we identify\nthe plane R2with the hyperplane R2×{0}inR3. Letφ1:∂D∪(D∩R2)→Rmbe\ndefined as\nφ1=/braceleftigg\nφon∂D,\nionD∩R2.\nNote that the map is well-defined and continuous, satisfying the refl ection symmetry.\nThe image does not contain 0 since φdoes not vanish on ∂Dand 0/∈D. Let\nD+:={(a,b)∈D:b>0}andD−:={(a,b)∈D:b<0}. We can rewrite Das\nD=∂D∪D+∪D−∪(D∩R2).\nBy the continuous extension lemma [15, Lemma 5.1 (1)] and since ∂D+is a compact\nset, there exists a continuous extension of φ1|∂D+toφ2:D+→Rmthat is nowhere\nzero. Using the fact that Rbis simply the map N:R→R,N(x) =−x, whenb/\\e}atio\\slash= 0,\nletϕ:D→Rmbe defined as\nϕ(a,b) =/braceleftigg\nφ2(a,b) when ( a,b)∈D+,\n(N×i×N×i)φ2((N×i)(a),−b) when ( a,b)∈D\\D+.\nOne can check that the map is continuous, nowhere-zero and satis fies the reflection\nsymmetry.\nSuppose for induction that the lemma holds for domains in Rn−1,n≥4. We prove\nthe result for D⊂Rn. Consider kto be any integer such that 1 ≤k < n/2. Let\na∈Rn−kandb∈Rk. Denote the last coordinate of basbkand identify Rn−1with\n{(a,b)∈Rn:bk= 0}.\nBy the induction hypothesis, we may extend the map from ∂D∩Rn−1toD∩Rn−1,\nasfollows. Notethatwhen bk= 0, thereflection map RbcanbewrittenasaCartesian\nproduct of the reflection map and identity or inclusion map on the last coordinate.\nWe first check the case when k >1. By the induction hypothesis with k−1 and\nn−1, we can extend the map on ∂D∩Rn−1to the map on all of D∩Rn−1so that it\nis continuous, nowhere-zero and satisfies the reflection symmetr y. Whenk= 1, the\nmap can simply be extended as φ(a,0) = (a,0)∈RmonD∩Rn−1sinceb∈R. In\neither case, we denote the extension map of φ|∂D∩Rn−1asφ1:D∩Rn−1→Rm.\nThe extension from previous step allows us to define φ2: (D∩Rn−1)∪��D→Rm\nas\nφ2=/braceleftigg\nφon∂D,\nφ1onD∩Rn−1.SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 13\nLetD+:={(a,b)∈D:bk>0}andD−:={(a,b)∈D:bk<0}. We can rewrite D\nasD=∂D∪D+∪D−∪(D∩Rn−1).Similarly, by the continuous extension lemma\n[15, Lemma 5.1 (1)], there exists an extension of φ2|∂D+toφ3:D+→Rmthat is\nnowhere-zero. Define ϕ:D→Rmas\nϕ(a,b) =/braceleftigg\nφ3(a,b) when ( a,b)∈D+,\n(Rb×i×Rb×i)φ3((Rb×i)(a),−b) when ( a,b)∈D\\D+.\nOne can check that the map is nowhere-zero and satisfies the refle ction symmetry,\nand is continuous where bk/\\e}atio\\slash= 0. Showing continuity where bk= 0 requires a careful\nargument. We only need to check the direction from D−since the other direction is\neasy due to the continuity of φ3. Whenk= 1, the reflection map Rb=Ndoes not\ndepend onband the continuity from D−follows easily. When k/\\e}atio\\slash= 1, we want to show\nthat as (a,b)→(˜a,˜b) wherebk>0 and˜bk= 0, we have ( Rb×i×Rb×i)φ3((Rb×\ni)(a),−b)→φ3(˜a,˜b). Note that when ˜b/\\e}atio\\slash= 0,Rb→R˜bcontinuously and so the norm\nof the difference\n|(Rb×i×Rb×i)φ3((Rb×i)(a),−b)−φ3(˜a,˜b)|\n=|φ3((Rb×i)(a),−b)−(Rb×i×Rb×i)φ3(˜a,˜b)|\napproaches 0 as ( a,b)→(˜a,˜b) by the reflection symmetry of φ2onD∩Rn−1. When\nb= 0, it is enough to consider a sequence {(aj,bj)}∞\nj=1inD−converging to (˜ a,0).\nBy the extension φ3(˜a,0) = (˜a,0) for alla∈Rn−k, we may write the norm of the\ndifference as\n|(Rbj×i×Rbj×i)φ3((Rbj×i)(aj),−bj)−(˜a,0)|\n=|φ3((Rbj×i)(aj),−bj)−((Rbj×i)(˜a),0)|\n=|φ3((Rbj×i)(aj),−bj)−φ3((Rbj×i)(˜a),0)|. (13)\nWe want to show that this quantity approaches 0. Note that the las t line holds even\nthough ((Rbj×i)(˜a),0) might not be in D+, since we extended φ3to all ofb/\\e}atio\\slash= 0.\nMoreover, we have that the quantity\n|((Rbj×i)(aj),−bj)−((Rbj×i)(˜a),0)|=|(aj,bj)−(˜a,0)|\napproaches 0 as j→ ∞. By the uniform continuity of φ3on compact sets, the\nquantity (13) approaches 0 as j→ ∞. /square\nThe next lemma shows that the map can be extended from the bound ary to the\ninterior when the dimensions of the domain and the codomain are equa l. For conve-\nnience, we identify R2n+1with the hyperplane R2n+2∩{bn+1= 0}.\nLemma 14. LetD⊂R2n+2(n≥1) be bounded, open and (n+1)-symmetric set such\nthat0/∈D. Letφ:∂D→R2n+2\\{0}be a continuous map with the reflection sym-\nmetry property (12). Then φcan be extended to ϕ:D→R2n+2which is continuous\nwith reflection symmetry and ϕ/\\e}atio\\slash= 0onD∩R2n+1.14 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nProof.LetD+:={(a,b)∈D:bn+1>0}andD−:={(a,b)∈D:bn+1<0}. By\nLemma 13, extend φ|∂D∩R2n+1toφ1:D∩R2n+1→R2n+2so that it is continuous,\nnowhere-zero and satisfies the reflection symmetry. Define φ2: (D∩R2n+1)∪∂D→\nR2n+2by\nφ2=/braceleftigg\nφon∂D,\nφ1onD∩R2n+1.\nLet/hatwideD=∂D∪(D∩R2n+1)∪D+. Next, we use the Tietze extension theorem to extend\nthe mapφ2|∂D∪(D∩R2n+1)toφ3:/hatwideD→R2n+1since∂D+⊂/hatwideDare both compact sets.\nAs a remark, the extension theorem used here is different from the one used earlier\nsince the dimensions are now the same between the domain and the co domain, so the\nnowhere-zero property only holds at D∩R2n+1instead ofD. Letϕ:D→R2n+2be\ndefined as\nϕ(a,b) =/braceleftigg\nφ3(a,b) when ( a,b)∈/hatwideD,\n(Rb×Rb)φ3(Rb(a),−b) when ( a,b)∈D−.\nIt is easy to check that the map ϕis nowhere-zero in D∩R2n+1and satisfies the\nreflection symmetry. Continuity follows by a similar argument to the p revious lemma.\n/square\nNext, for Lemma 16 we rely on the following proposition from [15, Prop osition 3.2].\nProposition 15. Letf:D→Rn+1be a continuous mapping whose restriction to D\nisC1and leta∈Rn+1\\f(∂D)be a regular value of f|D. Thenf−1(a)is finite and\nd(f,D,a) =/summationdisplay\nx∈f−1(a)signdet(Jf(x)).\nFor the next lemma, we use the same notations D+andD−for the sets with a cer-\ntain reflection relationship (not necessarily depending the sign of th e last coordinate).\nThey are more general sets compared to the ones in the previous e xtension lemmas.\nLemma 16 (Change of Variables of the Degree) .LetD+⊂R2n+2be a bounded and\nopen set such that for (a,b)∈Rn+1, we haveb/\\e}atio\\slash= 0whenever (a,b)∈D+. Define\nD−:={(Rb(a),−b) : (a,b)∈D+}and letφ:D−→R2n+2be a continuous map with\nφ(a,0) = (a,0)whenever (a,0)∈D−, and0/∈φ(∂D−). If the map Ψ :D+→R2n+2\nis defined as\nΨ(a,b) = (Rb×Rb)φ(Rb(a),−b),forb/\\e}atio\\slash= 0,\nΨ(a,0) = (a,0), forb= 0,\nthen the degrees are related by d(φ,D−,0) = (−1)nd(Ψ,D+,0).\nNote that whenever b= 0 onD+, the point lies on the boundary ∂D+, similarly\nforD−. Moreover, 0 /∈Ψ(∂D+) due to the assumption 0 /∈φ(∂D−).SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 15\nProof. Step 1 – Continuity of the maps.\nTo show that Ψ is continuous on D+, we only need to check boundary points where\nb= 0, due to continuity of φand the assumption that b/\\e}atio\\slash= 0 whenever ( a,b)∈D+.\nThe continuity of Ψ follows by arguing like in the proof of Lemma 13.\nFor the next step, define ψ:D+→R2n+2as\nψ(a,b) =φ(Rb(a),−b)\nAsaremark, ψmightnotbeextendedtotheclosurebutisintroducedforconvenie nce.\nWe writeφ(a,b) = (φ1(a,b),φ2(a,b)) and similarly for ψand Ψ.\nStep 2 – The degree of Ψin terms of ψ.\nWe first assume that φ∈C(D−)∩C1(D−) and that 0 is a regular value. By\nProposition 15 applied to Ψ, and using that b/\\e}atio\\slash= 0 onD+,\nd(Ψ,D+,0) =/summationdisplay\n(a,b)∈Ψ−1(0)signdet(JΨ(a,b))\n=/summationdisplay\n(Rb×Rb)ψ(a,b)=0signdet\n∂aRbψ1(a,b)∂bRbψ1(a,b)\n∂aRbψ2(a,b)∂bRbψ2(a,b)\n\n=/summationdisplay\nψ(a,b)=0signdet\nRb(∂aψ1(a,b))Rb(∂bψ1(a,b))\nRb(∂aψ2(a,b))Rb(∂bψ2(a,b))\n,(14)\nas follows. In the first column of the matrix in (14), the reflection ma pRbis inde-\npendent of the variable a. For the second column, the terms including ψ(a,b) vanish\nbecause we evaluate the derivatives at ψ(a,b) = 0; this allows us to take Rboutside\nthe partial derivatives.\nStep 3 – The degree of ψin terms of φ.\nWe calculate the sign of the determinants inside the sum in (14),\nsigndet\nRb(∂aψ1(a,b))Rb(∂bψ1(a,b))\nRb(∂aψ2(a,b))Rb(∂bψ2(a,b))\n\n= signdet\nRb0\n0Rb\n\n∂aψ1(a,b)∂bψ1(a,b)\n∂aψ2(a,b)∂bψ2(a,b)\n\n= signdetJψ(a,b), (15)\nwhereRbis the matrix for the reflection map Rband the determinant of the block\nmatrix is 1 since det Rb=−1. Next, we calculate the Jacobian Jψ(a,b) in (15):16 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nJψ(a,b) =\n∂a(φ1(Rb(a),−b))∂b(φ1(Rb(a),−b))\n∂a(φ2(Rb(a),−b))∂b(φ2(Rb(a),−b))\n\n=Jφ(Rb(a),−b)\nRb∂bRb(a)\n0−I\n, (16)\nby the chain rule. The determinant of the second matrix in (16) is ( −1)n+2since\nthe determinant of the block matrix can be calculated as for a 2 ×2 matrix when the\nlower left submatrix is 0. So we can conclude that d(φ,D−,0) = (−1)nd(Ψ,D+,0).\nStep 4 – Smooth approximation of continuous φ.\nWe now assume that φis just continuous on D−and 0 is not necessarily a regular\nvalue.\nFirst suppose there is no point of the form ( a,0) inD−. That is, if ( a,b)∈D−\nthenb/\\e}atio\\slash= 0. Choose a C1(D−)-map˜φhomotopic to φsuch that 0 is a regular value\nwith 0/∈˜φ(∂D−). To recall briefly about the existence of such functions, we use t he\nextension theorem to extend φoutside the domain D−and then we use mollification.\nWecan choose ˜φclose enoughto φso thatthe imageofthe boundarydoes notcontain\n0. Also, if 0 is not a regular value, by Sard’s theorem, there exists a c lose enough\nvaluepnear 0 that is a regular value. We shift the map by −pand hence make 0 a\nregular value of ˜φ. Due to the assumption that there is no point of the form ( a,0) in\nD−, the condition φ(a,0) = (a,0) does not need to be considered for mollification.\nDefine˜Ψ :D+→R2n+2from˜φsimilarly to how Ψ is constructed from φ. Note\nthat this map is C1(D+) and continuous on D+. We conclude that d(˜φ,D−,0) =\n(−1)nd(˜Ψ,D+,0) by repeating step 2 and 3.\nNext, based on the fact that Rbis a linear isomorphism when b/\\e}atio\\slash= 0, we show that\n˜Ψ is homotopic to Ψ. Consider a continuous homotopy ht(a,b) defined on D−×[0,1]\nwhereh0(a,b) =˜φandh1(a,b) =φ. Wemaychoosethat0 /∈ht(∂D−)forallt∈[0,1].\nDefinekt(a,b) := (Rb×Rb)ht(Rb(a),−b). We want to show that ktis a homotopy\nfrom˜Ψ to Ψ and 0 /∈kt(∂D+) for allt∈[0,1]. The continuity of ktfollows since\nhtis continuous and Rb×Rbis continuous when b/\\e}atio\\slash= 0. We show 0 /∈kt(∂D+)\nby contradiction. Suppose 0 ∈kt(∂D+), saykt(a,b) = 0 for some ( a,b)∈∂D+.\nThenht(Rb(a),−b) = 0, which is impossible since ( Rb(a),−b)∈∂D−. Hence, 0 /∈\nkt(∂D+). We have d(φ,D−,0) =d(˜φ,D+,0) and similarly for Ψ and ˜Ψ, so we have\nthe conclusion in Step 3.\nNext suppose there does exist a point of the form ( a,0) inD−, define a set D−,ǫ⊂\nD−where|b|>ǫfor allD−,ǫsuch that 0 /∈φ(D−\\D−,ǫ). Such an ǫexists since φis\ncontinuous and 0 /∈φ(∂D−). By the excision property [15, p. 44], d(φ|D−,ǫ,D−,ǫ,0) =\nd(φ,D−,0) andd(Ψ|D+,ǫ,D+,ǫ,0) =d(Ψ,D+,0). Note that D−,ǫwas explained above\nas the case when b/\\e}atio\\slash= 0. By combining the degree relationship between D−andD−,ǫ,\nand the previous case, we have the conclusion of the lemma. /squareSECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 17\nAs a remark, we are not assuming any reflection symmetry for φin the previous\nlemma since we only want to understand the relationship of the degre es between φ\nand Ψ.\nProof of Theorem 11. Fix 0< ǫ <1 and define U:=B(0,ǫ)⊂B2n+2andD1:=\nB2n+2\\U.\nDefineφ1:U∪S2n+1→R2n+2by\nφ1=/braceleftigg\nionU,\nφonS2n+1.\nThenφ1is a continuous function on ∂D1=S2n+1∪∂Uthat is the identity near\nthe origin and φ1satisfies the reflection symmetry. By Lemma 14, there exists a\ncontinuous extension φ2ofφ1|∂D1such thatφ2:D1→R2n+2with the reflection\nsymmetry and φ2/\\e}atio\\slash= 0 onD1∩R2n+1.\nDefine the following continuous map φ3onB2n+2=U∪D1,\nφ3=/braceleftigg\nionU,\nφ2onD1.\nIt is easy to check that the map is well-defined, continuous, and sat isfies the reflection\nsymmetry.\nWe first show that d(φ2,D1,0) is 0 when nis odd and even when nis even. Let\nD+\n1:={(a,b)∈D1:bn+1>0}andD−\n1:={(a,b)∈D1:bn+1<0}. Using the\nadditivity property [15, Proposition 2.5], we deduce\nd(φ2,D1,0) =d(φ2,D+\n1,0)+d(φ2,D−\n1,0),\nsince 0/∈φ2(D1\\(D+\n1∪D−\n1). Next, using Lemma 16, we conclude\nd(φ2,D1,0) =/braceleftigg\n0 when nis odd,\n2d(φ2,D+\n1,0) when nis even.\nDenote int as the interior of a set. By applying the additivity propert y again, we\nhave\ndeg(φ) =d(φ3,B2n+2,0)\n=d(φ3,B2n+2\\U,0)+d(φ3,intU,0)\n=d(φ2,D1,0)+d(i,intU,0)\n=d(φ2,D1,0)+1,\nwhere the additive property holds since 0 /∈φ3(B2n+2\\(D1∪intU)). Hence, deg( φ) =\n1 whennis odd and deg( φ) is odd when nis even, which finishes the proof of\nTheorem 11. /square18 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\n6.Estimating the Rayleigh Quotient — proof of Theorem 4\nRecall that the Veronese map Φ maps RPnintoSm−1wherem(n) is written as m\nfor simplicity. We apply the trial functions constructed in Section 4 t o the Rayleigh\nquotient, to estimate the second eigenvalue from above.\nThe variational characterization states that\nλ2(RPn,w)≤/integraltext\nRPn|∇wgu|2\nwgdvwg/integraltext\nRPnu2dvwg, (17)\nwhereuis a trial function in H1(RPn) which satisfies the orthogonality condition\nto the constant and to the first excited state. By substituting th e definition of trial\nfunctions (3) into (17), we have\nλ2(RPn,w)≤/integraltext\nRPn|∇wg(T−c◦FH◦Φ)j|2\nwgdvwg/integraltext\nRPn(T−c◦FH◦Φ)2\njdvwg, j= 1,...,m. (18)\nFor eachj, we multiply by the denominator on both sides of the inequality (18)\nand sum for all j’s. Note that ( T−c◦FH◦Φ)(y) is a point on the unit sphere. This\nimplies that the sum of the denominators is equal to the volume of RPnwith the\nmetricwg, that is,/integraltext\nRPn/summationtext\nj(T−c◦FH◦Φ)2\njdvwg= Vol(RPn,wg). Recall that by our\nnormalization, Vol( RPn,wg) = Voln(1). Hence, the inequality in (18) can be written\nas,\nλ2(RPn,w)Voln(1)≤/integraldisplay\nRPnm(n)/summationdisplay\nj=1|∇wg(T−c◦FH◦Φ)j|2\nwgdvwg.\nBy applying H¨ older’s inequality to the right hand side,\nλ2(RPn,w)Voln(1)\n≤\n/integraldisplay\nRPn\nm(n)/summationdisplay\nj=1|∇wg(T−c◦FH◦Φ)j|2\nwg\nn/2\ndvwg\n2/n\nVoln(1)1−2/n.\n(As a remark, the application of H¨ older is not needed when n= 2.) We introduce a\nnew notation: for an n-dimensional manifold ( M,g) and a map F:M→Sm−1, we\ndenote|∇gF|g=/radicalig/summationtextm\nj=1|∇g(F)j|2g. The inequality becomes\nλ2(RPn,w)n/2Voln(1)≤/integraldisplay\nRPn|∇wg(T−c◦FH◦Φ)|n\nwgdvwg\n=/integraldisplay\nRPn|∇g(T−c◦FH◦Φ)|n\ngdvg,\nby changing wgto the round metric gonRPnand using conformal invariance. We\nagain change the variables from RPnto Φ(RPn), which is an embedded submanifold\nofSm−1. Denoten-dimensional Hausdorff measure on the submanifold as dHnandSECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 19\nwrite∇for the gradient on the embedded submanifold. Since Φ is conformal by\nProposition 6,\nλ2(RPn,w)n/2Voln(1)≤/integraldisplay\nΦ(RPn)|∇(T−c◦FH)|ndHn\n=/integraldisplay\nΦ(RPn)∩H|∇(T−c)|ndHn+/integraldisplay\nΦ(RPn)∩Hc|∇(T−c◦RH)|ndHn,\nwhereHc=Sm−1\\H.\nBy changing variable in each part and using the fact that the reflect ionRHand\nM¨ obius transformation T−care conformal, we have the following inequality:\nλ2(RPn,w)n/2Voln(1)\n≤/integraldisplay\nT−c(Φ(RPn)∩H)|∇y|ndHn(y)+/integraldisplay\nT−c(RH(Φ(RPn))∩Hint)|∇y|ndHn(y) (19)\nwhereHint=RH(Hc) is the interior of H. Recall that the Veronese surface is embed-\nded intoSm−1. At each point ˜ yin the surface, the tangent space has dimension n, and\nafterrotatingthecoordinatesystem, wemaysuppose ˜ y= (0,...,0,1)andthetangent\nspace canbewritten asthespanof {∂y1,...,∂yn}wherey= (y1,...,yn,yn+1,...,ym).\nAt ˜ywe compute\n|∇y|n=/parenleftiggn/summationdisplay\nj=1|∇yj|2/parenrightiggn/2\n=nn/2. (20)\nNow substitute (20) into (19). Then we have\nλ2(RPn,w)n/2Voln(1)\n≤nn/2/parenleftbig\nVoln(T−c(Φ(RPn)∩H))+Voln/parenleftbig\nT−c(RH(Φ(RPn))∩Hint/parenrightbig/parenrightbig\n<nn/2/parenleftbig\nVoln(T−c(Φ(RPn)))+Vol n(T−c(RH(Φ(RPn)))/parenrightbig\n, (21)\nwhere we dropped the intersections with HandHint. Note that the last inequality\n(21) is strict, as follows. Either Φ( RPn)∩Hchas positive n-volume or else Φ( RPn)∩\nHdoes, or both, and so after reflecting the second set with RHwe deduce that\nΦ(RPn)∩HcorRH(Φ(RPn))∩(Hint)cor both have positive n-volume. Hence when\nwedroptheintersections with HandHintin(21), atleast oneofthevolumes becomes\nstrictly larger. This last line is bounded above by 2 nn/2times the (m−1)-conformal\nvolume of Φ (see El Soufi and Illias [3, p.259]), since T−candT−c◦RHare conformal\ndiffeomorphisms of the sphere Sm−1. The (m−1)-conformal volume of Φ equals\n(an)nVoln(1) by [3, Corollary 2.3 and p.267], and so\nλ2(RPn,w)n/2Voln(1)<2nn/2(an)nVoln(1)\n= 2Voln(1)(2n+2)n/2,\nwhich finishes the proof of Theorem 4.20 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\n7.Remark about Conjecture 3\nThe conjecture is based on the hope that in (21), the sum of the vo lumes is max-\nimized when |c|<1,p∈Φ(RPn), andt→1. In that limit, Φ( RPn)∩H(non-\nreflected set) tends to the image of n-projective space under the Veronese map and\nRH(Φ(RPn))∩Hint(reflected set) tends toan n-sphereembedded inthehigherdimen-\nsional sphere, as explained below. This kind of “bubbling” is motivated by the proof\nof the conjecture in 2-dimension by Nadirashvili and Penskoi [13]. Als o, Petrides [18]\nproved that under certain Palais–Smale condition, the k-th eigenvalue for the com-\npactn-dimensional manifold is maximal with a sequence of metrics that conv erges to\nthat of “bubbling”.\nWe have not been able to show that the sum of the volumes in (21) is ma ximal in\nthe bubbling situation. In this section, though, we estimate the volu me when the fold\nmapdegenerates ( t→1)andseparately when the M¨ obius transformationdegenerates\n(|c| →1). Both cases support our conjecture.\nFirst, we calculate that as the fold map Fp,tdegenerates when t→1, the volume\napproaches at most the volume of the projective space and a sphe re.\nProposition 17 (Fold map degenerates) .Given anN-dimensional imbedded smooth\nsurfaceγ:BN→SM,1≤N <M, we have\nlimsup\nt→1HN(Fp,t◦γ(BN))≤ HN(SN)+HN(γ(BN)), (22)\nwherep∈SMis fixed.\nHere, we use the closed ball BNfor its compactness, and the fold map Fp,tacts on\nSM(that is,Mis the “m−1” in Section 4.2).\nProof.Our main strategy is to decompose the image γ(BN) into two pieces, inside\nthe capHp,tand outside. Inside the cap, the fold map is the identity and so that\npart ofFp,t◦γ(BN) has volume at most HN(γ(BN)). We will show that the fold of\nthe piece outside has volume at most HN(SN) ast→1. For the rest of the proof, we\nmay suppose the image γ(BN) containsp, since otherwise the image will lie entirely\ninside the cap when tis close to 1 and there is nothing to prove.\nLet us consider when the image Fp,t◦γ(BN) is mapped onto the Euclidean space\nRMunder the stereographic projection Π, with pmapping to the origin and ∂Hp,t\nmapping to the sphere of radius ǫcentered at the origin, where ǫ→0 ast→1. Write\nρ:= Π◦γ:BN→RM. After some appropriate reparametrization of γ, we may\nassume that ρ(0) = 0 and its Jacobian Dρ(0) =/parenleftbigI0/parenrightbigT, which is an M×Nmatrix.\nWriteBM(ǫ) as the image of the smaller spherical cap (the piece outside Hp,t)\nprojected to the Euclidean space RM. The part of the surface intersecting the ǫ-ball\nis written as ρ: Ωǫ→BM(ǫ) where Ω ǫ⊂BN.\nLet the smooth surface Rǫ◦ρ: Ωǫ→RM\\BM(ǫ) be the fold of ρunder the map\nRǫ:RM→RMthat reflects (inverts) the points in RMacross the boundary of BM(ǫ).SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 21\nIt can be written as,\nRǫ◦ρ(z) =ǫ2\n|ρ(z)|2ρ(z)\nwherez∈Ωǫ. Note that Rǫ◦ρ= Π◦Fp,t◦γon ΩǫsinceRǫ◦Π = Π◦Rp,t. We want\nto estimate the limit of the weighted volume of Rǫ◦ρ(Ωǫ) from above,\nlimsup\nǫ→0/integraldisplay\nRǫ◦ρ(Ωǫ)/parenleftbigg2\n1+|z|2/parenrightbiggN\ndHN≤ HN(SN), (23)\nwhereHNis a Hausdorff measure for N-dimensional surface and 2 /(1+|z|2) is the\nconformal factor for stereographic projection.\nThe idea of the following proof is that the smooth surface near 0 can be approxi-\nmatedasaflat N-planeandso Rǫappliedtothatsurfaceisapproximatelyan N-plane\noutside the ball of radius ǫ.\nWriteR=R1for inversion in the unit sphere, so that Rǫ=ǫ2RandR(z) has a\nconformal factor of 1 /|z|2. Hence, the integral (23) becomes,\n/integraldisplay\nρ(Ωǫ)/parenleftbigg2\n1+|ǫ2R(z)|2/parenrightbiggN/parenleftbiggǫ\n|z|/parenrightbigg2N\ndHN\n=/integraldisplay\nΩǫ/parenleftbigg2\n1+|ǫ2R(ρ(z))|2/parenrightbiggN/parenleftbiggǫ\n|ρ(z)|/parenrightbigg2N\n|Dρ(z)|dHN\n=/integraldisplay\nΩǫ/parenleftbigg2ǫ2/|ρ(z)|2\n1+ǫ4/|ρ(z)|2/parenrightbiggN\n|Dρ(z)|dHN, (24)\nwhere|Dρ(z)|:RN→Ris the Jacobian determinant of ρ. By a change of variable\nwithz=ǫ2ζ, the integrand in (24) becomes\n/parenleftbigg2ǫ2/|ρ(ǫ2ζ)|2\n1+ǫ4/|ρ(ǫ2ζ)|2/parenrightbiggN\n|Dρ(ǫ2ζ)|ǫ2N=/parenleftbigg2ǫ4\n|ρ(ǫ2ζ)|2+ǫ4/parenrightbiggN\n|Dρ(ǫ2ζ)|.(25)\nNote the domain Ω ǫ/ǫ2approaches all of RNwhenǫ→0 since the points in Ω ǫhave\nnorm at most O(ǫ). The linear approximation of ρ(z)≈(z,0) at 0 implies that there\nexists an lower bound such that |ρ(z)|2≥ |z|2/2 for small enough ǫ, withz∈Ωǫ.\nHence we can find an integrable dominator on RNfor the right hand side of (25) as,\n1Ωǫ/ǫ2(ζ)/parenleftbigg2ǫ4\n|ρ(ǫ2ζ)|2+ǫ4/parenrightbiggN\n|Dρ(ǫ2ζ)| ≤/parenleftbigg2ǫ4\n|ǫ2ζ|2/2+ǫ4/parenrightbiggN\n2\n≤/parenleftbigg2\n|ζ|2/2+1/parenrightbiggN\n2 where ζ∈RN.(26)\nBy using the upper bound (26), we apply the dominated convergenc e to (25) as\nǫ→0. To find the limit of the integrand, we expand with the Taylor expans ion\nρ(ǫ2ζ) =ǫ2ζ+O(ǫ4) whenǫis close enough to 0 and ζis fixed. Then we evaluate the22 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nlimit (25) as ǫ→0 as,\n/integraldisplay\nRN/parenleftbigg2\n|ζ|2+1/parenrightbiggN\ndHN (27)\nby dominated convergence. The last integral is equal to HN(SN).\nNote that RN\\Ωǫapproaches to all of RNasǫ→0 by the weak convergence of\nthe measure in Section 4.4. The volume of this part becomes HN(γ(RN)), and added\nwith (27), we have (22) as ǫ→0. /square\nNext we show that as the M¨ obius transformation degenerates, t he volume of the\nimage of the surface is at most that of Sm. Recall that any point y∈SMunder the\nM¨ obius transformation Tx(y) approaches ˜ xasx→˜xwith|˜x|= 1, except at ˜ y=−˜x.\nThis implies that for any N-dimensonal submanifold where N≤MinSM, the image\nunderTxcollapses to a point as xapproaches to the boundary unless the submanifold\ncontains ˜y, in which case the image stretches out to an N-sphere asx→˜x.\nProposition 18 (M¨ obius transformationdegenerates) .Given anN-dimensional em-\nbedded smooth surface ω:BN→SM,1≤N≤M, we have the following upper bound,\nlimsup\n|x|→1HN(Tx◦ω(BN))≤ HN(SN). (28)\nProof.We write the south pole of SMassand its north pole as n. Without loss\nof generality, we may assume that x→n. Then first assume that s∈ω(BN). We\ndeal with the case when s/∈ω(BN) later. We may parametrize the surface so that\nω(0) =sandDω(0) =/parenleftbigI0/parenrightbigT. The left side of (28) becomes\n/integraldisplay\nTx◦ω(BN)dHN=/integraldisplay\nω(BN)/parenleftbigg1−|x|2\n1+|x|2+2x·s/parenrightbiggN\ndHN(s) (see [19, 2.1.7] and [1, (30)])\n=/integraldisplay\nBN/parenleftbigg1−|x|2\n1+|x|2+2x·ω(z)/parenrightbiggN\n|Dω(z)|dHN(z), (29)\nwhere|Dω(z)|is the Jabobian determinant of ω. Note that the integrand is similar\nto the Poisson kernel. We want to show that the integral in (29) is bo unded above by\nthe volume of the sphere as x→n. Note that the factor in (29) can be rewritten as\n1−r2\n1+r2−2rcosθ(y,ω(z))\nin terms of r=|x|,y=−x/|x|, and the angle θbetweenyandω(z). Sincex→n,\nwe havey→sand soy·s→1 andr→1.\nThe main idea of the proof is that when xis close to n, we divide BNinto two\nparts: the ball at 0 with some radius δand the rest of BN. We prove the following\nstatement: For fixed δ >0, there exist parameters C(δ) andα(δ) such that C(δ)→1SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 23\nandα(δ)→1 asδ→0 and\nlimsup\nx→n/integraldisplay\nBN\\BN(δ)/parenleftbigg1−r2\n1+r2−2rcosθ(y,ω(z))/parenrightbiggN\n|Dω(z)|dHN(z) = 0, (30)\nlimsup\nx→n/integraldisplay\nBN(δ)/parenleftbigg1−r2\n1+r2−2rcosθ(y,ω(z))/parenrightbiggN\n|Dω(z)|dHN(z)≤/parenleftbiggC(δ)\nα(δ)/parenrightbiggN\nHN(SN),\n(31)\nwherey=−x/|x| →sandr→1 asx→n. Hence, the limsup of (29) as x→nis\nat most (C(δ)/α(δ))NHN(SN), which tends to HN(SN) asδ→1.\nTo show (30), we use the fact that the angle θ(y,ω(z)) is bounded below away from\n0 whenyis nearsand|z| ≥δ. Hence, the integrand in (30) tends to 0 uniformly as\nx→n.\nTo show (31), the first step is to show that the distance on the sph ereθ(y,ω(z))\ncan be approximated by the Euclidean distance from z(y) toz, where for yclose tos,\nz(y) is the unique point z∈BN(δ) such that |y−ω(z)|is minimal. That is, ω(z(y)) is\na closest point to yon the surface, and such that z(y) is unique for ynear the surface.\nWe claim that there exists α(δ)>0 such that\nθ(y,ω(z))\n|z(y)−z|≥α(δ) (32)\nfor allynearsandz∈BN(δ). We show that α(δ)→1 asδ→0. Since spherical\ndistance is greater than Euclidean distance,\nθ(y,ω(z))\n|z(y)−z|≥|y−ω(z)|\n|z(y)−z|\n≥β(δ)|y−ω(z)|\n|ω(z(y))−ω(z)|,\nwhere the constant satisfies β(δ)→1 asδ→0, sinceβ(δ) is a lower bound of\n|ω(z(y))−ω(z)|/|z(y)−z|andDω(0) =/parenleftbigI0/parenrightbigT. Consider the triangle with vertices\naty,ω(z(y))andω(z). Writethesidelengths as a=|y−ω(z(y))|,b=|ω(z(y))−ω(z)|,\nandc=|y−ω(z)|, so that the last quantity displayed is β(δ)c/b. By the law of sines,\nc2≥b2sin2η,\nwhere the angle ηat pointω(z(y)) is close to π/2 because the side from ω(z(y))toyis\nnormal to the surface and the vector from ω(z(y)) toω(z) is nearly tangential to the\nsurface. Inthis case, our lower bound α(δ) isdefined as β(δ)c/b≥β(δ)|sinη|:=α(δ),\nwhereα(δ)→1 becauseβ(δ)→1 andη(δ)→π/2 asδ→0. This finishes the proof\nof (32).24 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\nBased on the lower bound (32), the second step is to show that the integral in (31)\nhas the following upper bound,\n/integraldisplay\nBN(δ)/parenleftbigg1−r2\n1+r2−2rcosθ(y,ω(z))/parenrightbiggN\n|Dω(z)|dHN\n≤/integraldisplay\nBN(δ)/parenleftbigg1−r2\n1+r2−2rcos(α(ǫ)|z−z(y)|)/parenrightbiggN\n|Dω(z)|dHN\n≤C(δ)N/integraldisplay\nBN(δ)/parenleftbigg1−r2\n1+r2−2rcos(α(ǫ)|z−z(y)|)/parenrightbiggN\ndHN,\nwhereC(δ) denotes the maximum of |Dω(z)|onBN(δ). NoteC(δ)→1 asδ→0.\nNext, we change the variable with α(δ)(z−z(y)) = (1−r)ζto write the upper bound\nas follows,\n/integraldisplay\nBN(δ)/parenleftbigg1−r2\n1+r2−2rcosθ(y,ω(z))/parenrightbiggN\n|Dω(z)|dHN\n≤/parenleftbiggC(δ)\nα(δ)/parenrightbiggN/integraldisplay\nBN(2δα(δ)/(1−r))/parenleftbigg1−r2\n1+r2−2rcos(1−r)|ζ|/parenrightbiggN\n(1−r)NdHN.\nThe domain approaches to all of RNasr→1. By considering the Maclaurin series,\nwe use dominated convergence and find the limit of the upper bound. Note that\ncos(1−r)|ζ| ≤1−(1−r)2|ζ|2/2+(1−r)4O(|ζ|4), since (1 −r)|ζ| ≤2δα(δ) is small.\nThen the last integral has the following upper bound:\n/integraldisplay\nBN(2δα(δ)/(1−r))/parenleftbigg(1−r)2(1+r)\n1+r2−2r(1−(1−r)2|ζ|2/2+(1−r)4O(|ζ|4))/parenrightbiggN\ndHN(ζ)\n=/integraldisplay\nBN(2δα(δ)/(1−r))/parenleftbigg(1−r)2(1+r)\n(1−r)2+r(1−r)2|ζ|2−(1−r)4O(|ζ|4)/parenrightbiggN\ndHN(ζ)\n=/integraldisplay\nBN(2δα(δ)/(1−r))/parenleftbigg1+r\n1+|ζ|2(r−(1−r)2O(|ζ|2))/parenrightbiggN\ndHN(ζ)\n→/integraldisplay\nRN/parenleftbigg2\n1+|ζ|2/parenrightbiggN\ndHN(ζ)\nasr→1, by dominated convergence. The last integral is equal to HN(SN).\nLastly, assume that south pole s/∈ω(BN). In case when s/∈ω(BN), there exists\nδsmall enough so that the volume vanishes in the limit. In case when s∈ω(∂BN),\nconsider a bigger ball containing BNto enlarge the surface ω(BN). The south pole\nlies in the interior of this bigger ball and so the volume satisfies the con clusion (28)\nby our work above.\n/squareSECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE 25\nAppendix A.Calculations\nHere, we check that ratio of the right-hand sides of (1) and (2) ap proaches 1 as the\ndimensionntends to ∞.\nLemma 19. Forn≥2, let us denote An= ((2n+ 2)n/2+ 2nn/2)2/nandBn=\n22/n(2n+2). ThenAn/Bnis strictly smaller than 1 and approaches 1 as n→ ∞.\nProof.Note thatAn/Bn<1 for alln≥2 because\nAn=/parenleftbig\n(2n+2)n/2+2nn/2/parenrightbig2/n\n</parenleftbig\n2(2n+2)n/2/parenrightbig2/nsince 2≤2n/2and 2n<2n+2\n= 22/n(2n+2) =Bn.\nFurther,An/Bn≥2−2/n→1 asn→ ∞, since\nAn=/parenleftbig\n(2n+2)n/2+2nn/2/parenrightbig2/n\n>/parenleftbig\n(2n+2)n/2+(n+1)n/2/parenrightbig2/nby the inequality 22/nn>n+1 forn≥2\n=/parenleftbig\n2n/2+1/parenrightbig2/n(n+1)>2(n+1) = 2−2/nBn.\n/square\nAcknowledgements\nI am grateful for support from the University of Illinois through th e Campus Re-\nsearch Board award RB22004 (to Richard Laugesen), the Lois M. L ackner fellowship\nfrom the Department of Mathematics, and the National Science Fo undation award\n#2246537 (to Richard Laugesen).\nReferences\n[1] V. Ahlfors. M¨ obius Transformations in Several Dimensions. Univ ersity of Minnesota, 1989.\n[2] D.Bucur, E.MartinetandM.Nahon. Sharp inequalities for Neumann eigenvalues on the sphere.\nArXiv:2208.11413.\n[3] A. El Soufi and S. Ilias. Immersions minimales, premi` ere valeur propre du laplacie n et volume\nconforme. Math. Ann. 275 (1986), no. 2, 257–267.\n[4] P. Freitas and R. S. Laugesen. Two balls maximize the third Neumann eigenvalue in hyperbol ic\nspace.Ann. Sc. Norm. Super. Pisa Cl. Sci., 23(3), 1325-1355.\n[5] J. Hersch. Quatre propri´ et´ es isop´ erim´ etriques de membranes sph´ eriques homog` enes. C. R. Acad.\nSci. Paris S´ er. A-B 270 (1970), A1645-A1648.\n[6] M. Karpukhin. Index of minimal spheres and isoperimetric eigenvalue ineq ualities. Invent.\nMath. 223 (2020), 335-377.\n[7] M. Karpukhin, N. Nadirashvili, A. Penskoi and I. Polterovich. An isoperimetric inequality for\nLaplace eigenvalues on the sphere. J. Diff. Geometry 118 (2021), no. 2, 313-331.\n[8] M. Karpukhin and D. L. Stern. Min-max harmonic maps and a new characterization of con-\nformal eigenvalues. J. Eur. Math. Soc., to appear. ArXiv:2004.04086.\n[9] H. N. Kim Maximization of the second Laplacian eigenvalue on the sphe re.Proc. Amer. Math.\nSoc. 150 (2022), 3501-3512.26 SECOND LAPLACIAN EIGENVALUE ON REAL PROJECTIVE SPACE\n[10] R. S. Laugesen. Well-posedness of Hersch–Szeg˝ o’s center of mass by hyperb olic energy mini-\nmization. Ann. Math. Qu´ e., 45(2) (2021), 363-390.\n[11] P. Li and S. T. Yau. A new conformal invariant and its applications to the Willmo re conjecture\nand the first eigenvalue of compact surfaces. Invent. Math. 69 (1982), 269-291.\n[12] N. Nadirashvili. Isoperimetric inequality for the second eigenvalue of a sph ere.J. Differential\nGeom. 61 (2002), no. 2, 335-340.\n[13] N. Nadirashvili and A. Penskoi. An isoperimetric inequality for the second non-zero eigenv alue\nof the Laplacian on the projective plane. Geom. Funct. Anal. 28 (2018) 1368-1393.\n[14] N. Nadirashvili and Y. Sire. Isoperimetric inequality for the third eigenvalue of the\nLaplace–Beltrami operator on S2.J. Differential Geom. 107 (2017), 561-571.\n[15] E. Outerelo and J. M. Ruiz. Mapping Degree Theory. Graduate S tudies in Mathematics, 108.\nAmerican Mathematical Society, 2009.\n[16] R. Petrides. Maximization of the second conformal eigenvalue of spheres .Proc. Amer. Math.\nSoc. 142 (2014), no. 7, 2385-2394.\n[17] R. Petrides. On the existence of metrics which maximize Laplace eigenval ues on surfaces. Int.\nMath. Res. Notices. 14 (2018), 4261–4355.\n[18] R. Petrides. Maximizing one Laplace Eigenvalue on N-dimensional Manifolds.\nArXiv:2211.15636.\n[19] M. Stoll. Harmonic and Subharmonic Function Theory on the Hype rbolic Ball. London Math-\nematical Society Lecture Note Series. Cambridge, Cambridge Unive rsity Press, 2016.\n[20] M. Taylor. Partial Differential Equations II: Qualitative Studies of Linear Equations. Springer\nNew York, 2010.\n[21] Y. Zhang. Isometric embeddings of real projective spaces into Euclid ean spaces. Differential\nGeom. Appl. 27 (2009), 100-103.\nDepartment of Mathematics, University of Illinois, Urbana –Champaign, IL 61801,\nU.S.A.\nEmail address :nekim2@illinois.edu"    },    {        "title": "2401.13864v1.Lorentzian_quantum_wells_in_graphene__the_role_of_shape_invariance_in_zero_energy_states_trapping.pdf",        "content": "arXiv:2401.13864v1  [cond-mat.mes-hall]  25 Jan 2024Lorentzian quantum wells in graphene: the role of\nshape invariance in zero-energy states trapping\nFrancisco Correaa, Luis Inzunzaband V´ ıt Jakubsk´ yc\naDepartamento de F´ ısica\nUniversidad de Santiago de Chile, Av. Victor Jara 3493, Sant iago, Chile\nbInstituto de Ciencias F´ ısicas y Matem´ aticas\nUniversidad Austral de Chile, Casilla 567, Valdivia, Chile\ncNuclear Physics Institute\nCzech Academy of Science, 250 68 ˇReˇ z, Czech Republic\nAbstract\nConfining Dirac fermions in graphene by electrostatic fields is a challenging task. Electric quantum\ndots created by a scanning tunneling microscope (STM) tip ca n trap zero-energy quasi-particles. The\nLorentzian quantum well provides a faithful, exactly solva ble, approximation to such a potential,\nhosting zero-energy bound states for certain values of the c oupling constant. We show that in this\ncriticalconfiguration, thesystemcanberelatedtothefree particlemodelbymeansofasupersymmetric\ntransformation. The revealed shape invariance of the model greatly simplifies the calculation of the\nzero modes and naturally explains the degeneracy of the zero energy.\n1 Introduction\nIt is difficult to control Dirac fermions in graphene by electrostatic fi elds because the Klein tunneling\nallows the quasi-particles to escape from the electric traps. An elec tric field localized in one direction can\nform waveguides for Dirac quasi-particles with non-vanishing [ 1] or vanishing energy [ 2–4]. Such localized\nelectric dots can trap quasi-particles of non-vanishing energy only for finite time [ 5], but they can host\nbound states of zero energy [ 1,6,7].\nAn axially symmetric electric field can be generated by an STM tip positio ned close to the graphene\nsheet. This configuration can be very well approximated by the Lor entzian well potential [ 6,8–10]. It\nallows the existence of analytical solutions of the radial Dirac equat ion [6] in terms of hypergeometric\nfunctions. Moreover, the Lorentzian well potential can host squ are integrable eigenstates of zero energy,\nthe zero modes, provided that its strength takes certain discret e values. For these values of the coupling\nconstant, the hypergeometric functions are truncated into poly nomials that render the wave functions\nsquare integrable. As a result, the zero energy degeneracy is alwa ys an even number and depends on the\nmagnitude of the critical coupling constant. In the current article , we provide an alternative insight into\nthe existence of zero modes in the Lorentzian well based on the sup ersymmetric transformations [ 11,12].\nThe supersymmetric (also called Darboux) transformations can be used to construct new solvable\nquantum Hamiltonians by specific modification of the known ones, and are also useful in the context of\nsoliton theory [ 13]. Such transformations can be represented by a differential oper ator which provides a\nmappingbetweentheeigenstatesoftheoriginalandthenewHamilto nian. TheDarbouxtransformationfor\none-dimensional Dirac-type operators [ 14] has been applied e.g. in the context of deformed nanotubes [ 15],\nintegrable models [ 16] or optical settings [ 17] . Zero modes in graphene have already been analyzed using\nsupersymmetric methods in [ 18–20].\nAny initial (solvable) Dirac system can be modified in an infinite number o f ways using the supersym-\nmetric transformations and can be illustrated with the simplest exam ple: a free particle system. Applying\nthis technique to free Dirac systems, one can derive multi-paramet ric families of new Hamiltonians with\n1novel spectral and scattering properties [ 16]. These new systems differ substantially from the original one\nin that they have non-trivial potentials and host bound states. Ho wever, they also partially inherit the free\nparticle dynamics. Since they show trivial scattering with pure tran smission, they are called reflectionless\nsystems. These quantum models are related to the existence of so liton solutions in fermion field theories\nsuch as Gross-Neveu and Nambu-Jona-Lasinio [ 16,21,22].\nThere is another interesting aspect of the supersymmetric trans formation worth mentioning in the\npresent context. The potential of the transformed system may coincide with the potential of the original\nHamiltonian up to a different value of the coupling constant. If this is t he case, the energy levels as well\nas the corresponding eigenstates can be found algebraically. Such systems are called shape-invariant [ 23].\nThe P¨ oschl-Teller, Scarf model as well as the harmonic oscillator ar e prominent examples of such non-\nrelativistic models [ 11]. In this article, we will show that these phenomena can also be found in relativistic\nmodels of Dirac fermions confined by Lorentzian wells.\nThe article is organized as follows. In the next section, we review the exact solution of the stationary\nDirac equation with Lorentzian well for zero energy. We clarify which are the critical values of the coupling\nparameter that support the existence of the bound states. In t he section 3, we review the main features\nof the supersymmetric transformation for a generic Dirac operat or described by 2 ×2 matrices. In the\nsection 4, we apply the Darboux transformation to the Lorentzian well potential and we show its shape\ninvariant property and that the bound states can be generated b y the transformation.\n2 The Lorentzian well\nConsider a two-dimensional massless fermion subject to a scalar Lo rentzian well potential. The stationary\nDirac equation reads as\nHℓψ=−i(σ1∂x+σ2∂y)ψ+Vℓ(x,y)σ0ψ= 0, Vℓ(x,y) =−2ℓ\n1+x2+y2, ℓ∈R, ℓ>0.(2.1)\nHere, we consider only the coupling ℓas positive since the Hamiltonian with ℓ<0 can be obtained by a\nunitary transformation σ3Hℓσ3=−H−ℓ. The matrices σi,i= 1,2,3, are the usual Pauli matrices and σ0\nis the identity matrix. The equation is separable in polar coordinates d ue to its rotational symmetry and\ncan be solved explicitly [ 6].\nThe regular solutions at the origin with positive angular momentum mcan be written as\nψm,ℓ=eimϕrm\n(1+r2)m+1\n2F1/parenleftBig\nm+1−ℓ,m+1+ℓ,1+m;r2\nr2+1/parenrightBig\niℓreiϕ\n1+m2F1/parenleftBig\nm+1−ℓ,m+1+ℓ,2+m;r2\nr2+1/parenrightBig\n, m= 0,1,2,...,(2.2)\nJψm,ℓ=/parenleftbigg\nm+1\n2/parenrightbigg\nψm,ℓ, J=−i∂ϕ+1\n2σ3. (2.3)\nThe wavefunctions are given in terms of the hypergeometric funct ion2F1(a,b,c;z) with parameters a,b\nandc[24]. The case of solutions having negative values of the angular moment ummcan be obtained by\nthe action of the symmetry operator Tσ2, [Hℓ,Tσ2] = 0, where Tis the complex conjugation operator,\nJ(Tσ2ψm,ℓ) =−/parenleftbigg\nm+1\n2/parenrightbigg\n(Tσ2ψm,ℓ). (2.4)\nThe states ( 2.2) show a square integrable behavior provided that the hypergeome tric functions are\ntruncated to the Jacobi polynomials [ 24]. This is only possible if either their first or the second argument\nis a negative integer. Remembering that mis integer-valued ( 2.2), this implies that the coupling constant\nℓmust also be an integer number. In addition, square integrability implie s that the angular momentum\nmcan only take values from the restricted set m= 0,1,...,ℓ−1. Thus, the physical meaning of the\ncomponents of ψm,ℓcan be assessed from their asymptotic behavior at the origin and at the infinity,\nr→0, ψm,ℓ→/parenleftbiggrm\nr1+m/parenrightbigg\n, r → ∞, ψm,ℓ→/parenleftbigg1\nr2+m\n1\nr1+m/parenrightbigg\n. (2.5)\n2It is learned from here that ψm,ℓare normalizable except for m= 0, where logarithmic divergence emerges\nfor larger. In conclusion, zero modes can only exist for integer values of ℓ. The degeneracy of the zero\nenergy states is an even number equal to 2(ℓ−1)with the corresponding zero modes ψm,ℓandTσ2ψm,ℓ,\nwherem= 1,...ℓ−1.\n3 Supersymmetric (Darboux) transformations in 2D Dirac sys -\ntems\nDarboux transformations make it possible to map a differential equa tion associated with a Hamiltonian\nsystem into another one while preserving the form of the kinetic ter m. In this way, an original Dirac\n(Schr¨ odinger) Hamiltonian is transformed into a new Dirac (Schr¨ o dinger) one with a modified potential\nterm. The solutions of the new system can be easily derived from the solutions of the original model. This\ntype of transformation was originally studied in the context of differ ential equations by G. Darboux [ 25],\nusing Schr¨ odinger-type operators, but before the advent of q uantum theory. Later on, the Darboux trans-\nformationfoundtheirapplicationinsupersymmetricquantummecha nics[11,12]. Itisthereforeoftencalled\na supersymmetric transformation. Here we briefly review the cons truction of Darboux transformations for\na Dirac-type two-dimensional operator. For further details, one can refer to the one-dimensional case [ 14],\nwhich have also been applied in two dimensions for the study of omnidire ctional Klein tunneling [ 26].\nLet us suppose that we start with the original equation of the follow ing form,\nhψ= (γ1∂x+γ2∂y+V(x,y))ψ= 0, (3.1)\nwhere both γaandV(x) are 2×2 matrices. The γamatrices have constant entries whereas the entries of\nV(x) can be arbitrary functions in general. At this point, the operator hdoes not have to be hermitian.\nWe assume that two solutions χ=/parenleftbigχ1χ2/parenrightbig\nandξ=/parenleftBig\nξ1\nξ2/parenrightBig\nofhof (3.1) are known. Then, we construct the\nmatrixUwhose columns correspond to χandξ,\nhχ=hξ= 0, U= (χ,ξ) =/parenleftbigg\nχ1ξ1\nχ2ξ2/parenrightbigg\n, hU= 0. (3.2)\nNow, we define two additional operators Land/tildewidehin terms of the matrix U,\nL=∂x−Σ,/tildewideh=h−[γ,Σ],Σ = (∂xU)U−1. (3.3)\nBy construction, the “seed” solutions χandξare annihilated by L,Lχ=Lξ= 0. The operators h,/tildewidehand\nLsatisfy the following formal equality\nLh=/tildewidehL. (3.4)\nThis is the so-called intertwining relation between the operators hand/tildewideh, mediated by the differential\noperatorL, which represents the Darboux transformation. It is an immediate consequence from ( 3.4) that\nthe solution of hψ= 0 can be transformed into the solution /tildewideψ,/tildewideh/tildewideψ= 0,\nhψ= 0⇒/tildewideh/tildewideψ= 0,/tildewideψ= (Lψ). (3.5)\nIt is worth noticing that both Land/tildewidehare uniquely defined in terms of the solutions χandξ. When\ncompared to h, the operator /tildewidehhas an additional potential term [ γ,Σ] that is a function of χandξ, see\n(3.3). So far there are no specific requirements for the solutions and t hus for the resulting Hamiltonian\n/tildewideh. However, it is clear that the explicit choice of the seeds χandξhas a large impact on the form of\n/tildewideh. A common scenario is to construct new systems devoid of singularit ies compared to V(x,y), but this\ndepends on the physics of the problem to be considered.\nTherecan be constructed an “inverse”intertwining operator /tildewideL, which mediates the inverseintertwining\nrelation\n/tildewideL/tildewideh=h/tildewideL. (3.6)\n3In the case that both hand/tildewidehare hermitian operators, we can write /tildewideL=L†. In the generic case, they can\nbe formally written as /tildewideL=/tildewideU∂x/tildewideU−1, where/tildewideUis a specific eigenmatrix of /tildewideh,\n/tildewideh/tildewideU= 0. (3.7)\nThe columns /tildewideχand/tildewideξof the matrix /tildewideU= (/tildewideχ,/tildewideξ) are sometimes called missing states in the literature. The\nname comes from the fact that, unlike /tildewideψin (3.5), they cannot be formally obtained by applying the L\noperator to the eigenstates ψof the original Hamiltonian h. They are defined in terms the seed states χ\nandξand may correspond to bound state of /tildewidehthat have no analog in the original equation ( 3.1).\n4 Confining Lorentzian well via SUSY transformations\nWe will show that the Lorentzian well model is shape invariant and tha t it can be mapped to the massless\nfree particle system using supersymmetric transformations. This mapping is only possible when the\ncoupling parameter of the Lorentzian well takes integer values, th e same values that allow the confinement\nofzero-modes. Despite the fact that the Lorentzianwell hasbee n studied before [ 6,8–10], up to the authors\nknowledge, its shape invariant property has not been revealed bef ore. We also show how the zero modes\nappear due to the mapping between the two systems.\nIn order to proceed, let us multiply the equation in ( 2.1) by iσ2. As a result, we obtain a manifestly\nnon-hermitian Hamiltonian operator hℓ,\nhℓ= iσ2Hℓ, h ℓψ=/parenleftbigg\n∂y−iσ3∂x−iσ22ℓ\n1+x2+y2/parenrightbigg\nψ= 0. (4.1)\nComparing hℓwithhin (3.1), we can identify γ1≡ −iσ3andγ2=σ0. The new operator hℓpreserves\nthe symmetry represented by Tσ2while its the solutions of ( 4.1) are the zero modes of the Lorentzian\nwell. It is worth noting that the Hamiltonian of the form hℓ, belongs to the Davey-Stewartson II class of\nintegrable systems [ 13].\nAswediscussedabove, the Darbouxtransformisfixedby thematr ixU, whosecolumnsareeigenvectors\nofhℓ. Here, sinceweareinterestedindesigningaspecifictransformatio nwhosepropertieswillsoonbecome\nclear, we fix Uin the following way\nUℓ= (ωℓ,Tσ2ωℓ), hℓUℓ= 0, (4.2)\nwhere\nωℓ=/parenleftBigg\n−(x+iy)−ℓ−1/parenleftbig\nx2+y2+1/parenrightbigℓ\ni(x+iy)−ℓ/parenleftbig\nx2+y2+1/parenrightbigℓ/parenrightBigg\n, h ℓωℓ= 0. (4.3)\nOnce we have the matrix U, following ( 3.3) we can construct both the intertwining operator Lℓand the\noperator/tildewidehℓ,\n/tildewidehℓ=hℓ−i/parenleftBiggx−iy\nx+iy0\n0x+iy\nx−iy/parenrightBigg\nσ22\n1+x2+y2, (4.4)\nLℓ=∂x−1\n1+x2+y2/parenleftBiggx2+y2−(ℓ+1)(x2+y2−1)\n(x+iy)ix+2(ℓ+1)y−y\n(x+iy)\nix−2(ℓ+1)y+y\n(x−iy)x2+y2−(ℓ+1)(x2+y2−1)\n(x−iy)/parenrightBigg\n, (4.5)\nLℓhℓ=/tildewidehℓLℓ. (4.6)\nThe new potential appearing in /tildewidehℓhas the shape of the Lorentzian well up to the multiplicative diagonal\nmatrix. We can transform out this matrix by a similarity transformat ion in terms of the matrix Sas\nfollows,\nS/tildewidehℓS−1=hℓ−iσ22\n1+x2+y2=hℓ+1,/tildewidehℓ=S−1hℓ+1S, S =/parenleftbiggx+iy0\n0x−iy/parenrightbigg\n.(4.7)\n4When substituting /tildewidehℓ=S−1hℓ+1Sinto (4.6), we get a new intertwining relation,\nLℓhℓ=hℓ+1Lℓ. (4.8)\nHere, the operator Lℓis a non-singular modified intertwining operator that satisfies [ Tσ2,L] = 0 and can\nbe written explicitly,\nLℓ=SLℓ=/parenleftBigg\n(x+iy)∂x+ℓ−2ℓ+1\n1+x2+y2 −ix+(1+2ℓ)y\n1+x2+y2\n−ix−(1+2ℓ)y\n1+x2+y2 (x−iy)∂x+ℓ−2ℓ+1\n1+x2+y2/parenrightBigg\n. (4.9)\nThe intertwining relation ( 4.8) indicates that the supersymmetric transformation ( 3.3) of the (non-\nhermitian) Hamiltonian hℓproduces exactly the same operator hℓ+1with the coupling constant shifted by\none. Therefore, the operator hℓis shape-invariant under the supersymmetric transformations generated\nbyLℓ.\nUp to now, we have derived the intertwining relations and revealed th e shape invariance nature for the\nHamiltonian hℓin its non-hermitian form. Let us show what are the corresponding im plications for the\nphysically relevant Dirac Hamiltonian Hℓ. To derive the analogous intertwining relation for Hℓ, we can\nmultiply ( 4.8) from the left by σ2,\nσ2Lℓσ2Hℓ=Hℓ+1Lℓ. (4.10)\nIn contrast to ( 4.8), this is an asymmetric intertwining relation for HℓandHℓ+1, i.e. the intertwining\noperator has a different form on the left side and on the right side. S uch asymmetric intertwining relations\nhave already appeared in the analysis of zero modes in [ 26,27].\nThe Hamiltonian Hℓhas 2ℓ−2 zero modes ψm,ℓandTσ2ψm,ℓform= 1,2,...ℓ−1. The intertwining\nrelation can be used to map them to those of Hℓ+1,\nHℓψm,ℓ= 0 ⇒Hℓ+1Lℓψm,ℓ= 0, m= 1,2,...ℓ−1, (4.11)\nwhile the states given by Tσ2ψm,ℓcan be mapped in the same manner. In this way, we obtain the 2( ℓ−1)\nzero modes for the Hamiltonian Hℓ+1. However, since the degeneracy of its zero energy is 2 ℓ, there are\ntwo more missing zero modes to be found. To do this, we conjugate ( 4.8) and multiply the relation by i σ2\nin order to convert hℓintoHℓ,\nL†\nℓHℓ+1=Hℓσ2L†\nℓσ2. (4.12)\nThe two states annihilated by σ2L†\nℓσ2will be zero energy eigenstates of Hℓ. If they are square integrable,\nthey will represent the missing bound states. The intertwining oper atorσ2L†\nℓσ2can be written in the\nfollowing form\nσ2L†\nℓσ2=−σ2(U−1\nℓ)†∂xU†\nℓS†σ2, (4.13)\nwhere we used ( 4.9). It annihilates the matrix /tildewideUℓ+1that is defined as follows,\n/tildewideUℓ+1=σ2(S†)−1(U†\nℓ)−1= (ψℓ,ℓ+1,Tσ2ψℓ,ℓ+1). (4.14)\nAs the columns ψℓ,ℓ+1andTσ2ψℓ,ℓ+1are square integrable, see ( 2.2) they correspondto the two additional\nzero modes of Hℓ+1.\nLet us summarize our results so far. There is a mapping between HℓandHℓ+1that can be used to\nobtain 2ℓ−1 zero modes of Hℓ+1from those of Hℓ. The additional two zero modes are also fixed by the\nmapping, see ( 4.14). This also explains the fact that the zero energy has even degene racy, as the zero\nmodes appear in pairs when the strength of the confining Lorentzia n well coupling is increased from ℓto\nℓ+1. In particular, the Hamiltonian H1has no normalizable zero modes that could be transformed into\nthose ofH2. Nevertheless, the formula ( 4.14) makes it possible to recover the pair of the zero modes,\nnamelyψ1,2andTσ2ψ1,2. If we now perform the Darboux transformation on H2, we get the Hamiltonian\nH3. It inherits two zero modes from H2and the other two are generated by ( 4.14), covering the expected\nfourfold degeneracy of the zero energy.\n5The action of the intertwining operators on the zero modes can be o btained in a straightforward\nway using the Gauss contiguous relations for hypergeometric func tions [24]. They provide the following\nrelations\nLℓψm,ℓ= (ℓ+m+1)ψm,ℓ+1,/tildewideLℓψm,ℓ= (ℓ−m−1)ψm,ℓ−1, (4.15)\nwhere we denoted /tildewideLℓ=σ2L†\nℓ−1σ2the intertwining operator that maps HℓintoHℓ−1. These equations\nreveal that the Darboux transformations acts as a ladder opera tors with respect to the index ℓ. The fact\nthat the index mis preserved by Lℓis due to the following relations\n[J,Lℓ] =−Sσ2Hℓ,[J,/tildewideLℓ] =Sσ2Hℓ. (4.16)\nSince we are interested in the action of Lℓin the subspace of zero modes, the right-hand sides of the\nequalities cancel out, and we find that LℓandJeffectively commute in this subspace.\nThe relations( 4.15) arevalid forany integervalueof m, including for the non-physicalstates. In partic-\nular, the non-physical spinor ψℓ,ℓofHℓtransforms into ψℓ,ℓ+1. This state coincides with the first missing\nstate in ( 4.14). Thus, the non-physical state is regularized into a physical one b y the transformation.\nLet us now discuss the relation of Hℓto the free particle system. If we define a chain of Darboux\ntransformations given by the operator,\nLℓ=LℓLℓ−1...L0, (4.17)\nthe Hamiltonian Hℓare intertwined with the free particle Hamiltonian H0by the following relations\nLℓH0=Hℓ+1σ2Lℓσ2, H0L†\nℓ=σ2L†\nℓσ2Hℓ+1. (4.18)\nThe kernel of the operator Lℓconsists of 2 ℓstates. To find them, let us consider the non-physical solutions\nofHℓ,\nφm,ℓ=(1+x2+y2)ℓ\n(x+iy)m/parenleftbigg−m\nx+iy2F1/parenleftbig\nℓ,ℓ−m,−m;−x2−y2/parenrightbig\niℓ2F1/parenleftbig\nℓ+1,ℓ−m,1−m;−x2−y2/parenrightbig/parenrightbigg\n, m=ℓ,ℓ+1,... (4.19)\nwhereφℓ,ℓ=ℓωℓis just the initial seed state, see ( 4.3). Using the ℓladder operators, we find that\nφℓ,ℓ∝ Lℓ−1...L0φℓ,0,Lℓφℓ,ℓ= 0. (4.20)\nThe stateφℓ−1,0is annihilated by Lℓ−1...L0φℓ,0, while the state φℓ−1,0vanishes after the action of\nLℓ−2...L0φℓ,0, and so on. In this way, the kernel of Lℓcan be written as\nKer(Lℓ) = span{φ0,0,Tσ2φ0,0,φ1,0,Tσ2φ1,0...Tσ2φℓ,0}, φ ℓ,0=ℓ/parenleftbigg−(x+iy)−ℓ−1\ni(−1)ℓ(x−iy)ℓ/parenrightbigg\n.(4.21)\nTherefore, equation ( 4.18) establishes the relation between the free particle zero energy so lutions and\nthose of the Lorentzian well. Although it is similar to the intertwining re lation between the free particle\nHamiltonian and the energy operators of reflectionless models, see [16,28], the fact that the relation ( 4.18)\nis asymmetric restricts its use to the subspace of zero energy solu tions.\n5 Conclusion\nWe showed that the Dirac Hamiltonian of the Lorentzian well is shape in variant with respect to Darboux\ntransformation ( 4.12). In the configuration where it possesses zero modes, it is intertw ined with the\nfree-particle Hamiltonian via asymmetric intertwining relation ( 4.18). Existence of this mapping provides\nexplanation for the 2( ℓ−1)-fold degeneracy of the energy levels. It also allows to construc t the zero modes\nfrom the states of free particle in a straightforward manner.\nThe derivation of the shape invariance of the Lorentzian well Hamilto nian relied on analyzing its non-\nhermitian counterpart hℓ, see (4.1). The standard supersymmetric transformation ( 3.3) fails to recover\n6the shape invariance of hℓbecause the transformed potential differs from the Lorentzian w ell. The key\nobservation of the present result is the existence of a similarity tra nsformation that can be incorporated\ninto the definition of Lℓso that we can obtain the shape invariance of the non-hermitian ope rators (4.8).\nThe non-hermitian operator hℓappears as one of the linear auxiliary equations related to the nonline ar\nintegrable Davey-Stewartson II equation [ 13]. In fact, the Lorentzian potential well can be understood in\nthis context as a solitonic solution of the corresponding hierarchy. It is interesting to explore this relation\nin more detail, as well as the role of solitonic solutions from the point of view of fermionic models.\nThe intertwining relations ( 3.4) and (3.6) for the non-hermitian hℓandhℓ+1can be rewritten in terms\nof the commuting extended matrix operators,\nHℓ=/parenleftbigg\nhℓ0\n0hℓ+1/parenrightbigg\n,Qℓ=/parenleftbigg\n0σ2L†\nℓσ2\nLℓ0/parenrightbigg\n,[Hℓ,Qℓ] = 0. (5.1)\nThe commutation relations follow from ( 4.12) and the conjugate relation with substitution h†\nℓ=−σ2hℓσ2.\nIt is remarkable that the diagonal operator Q2\nℓcommutes with the extended Hamiltonian Hℓby means of\nthe relations\n[hℓ,σ2L†\nℓσ2L] = 0,[hℓ+1,Lℓσ2L†\nℓσ2] = 0. (5.2)\nTheconstructionoftheextendedobjects( 5.1)isinspiredbyWitten’smodelforthespontaneousbreakdown\nof supersymmetry [ 11], where the diagonal operator Hℓrepresents the supersymmetric Hamiltonian while\nthe anti-diagonal operator represents the supercharge Qℓ. In contrast to the toy model of Witten, the\nsquare of the supercharge operator does not coincide with the su persymmetric Hamiltonian, Q2\nℓ∝ne}ationslash=Hℓ. In\nour scenario, where the operators are also two-dimensional, the o peratorσ2L†\nℓσ2Lℓcannot be proportional\nto a polynomial in hℓ, since there is no kinetic term of the form ∂y. Despite the fact that this discussion\nholds for the non-hermitian system, we believe that it is an indication t hat hidden integrals for the\nhermitian system should somehow exist, and it would be interesting to find a way to reveal them. It is\nworth mentioning in this context a series of papers where peculiar pr operties of physical systems have\nbeen related to the existence of pure quantum hidden symmetry op erators constructed under a similar\napproach [ 29,30].\nFinally, the current article shows that the standard approach of s upersymmetric transformations ap-\nplied to non-standard (non-hermitian operators hℓ) can provide relevant information on physical systems\nsuch as the Dirac fermion in the presence of Lorentzian wells. The ex ploration of this line of research in\na wide class of models will be followed in the forthcoming publications.\nReferences\n[1] T. Y. Tudorovskiy, A. V. Chaplik, “Spatially inhomogeneous states of charge carriers in grap hene,”\nJETP Lett. 84, 619 (2007).\n[2] C. A. Downing, M. E. Portnoi, “One-dimensional Coulomb problem in Dirac materials,”\nPhys. Rev. A 90, 052116 (2014) [arXiv:1411.5983 [cond-mat.mes-hall]] .\n[3] R. R. Hartmann, N. J. Robinson, and M. E. 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Thies, “Time-Dependent Hartree-Fock Solution of Gross-Neveu mod els:\nTwisted Kink Constituents of Baryons and Breathers,” Phys. Rev. Lett. 111, no.12, 121602 (2013)\n[arXiv:1306.4007 [hep-th]] .\n[23] R. Dutt, A. Khare and U.P. Sukhatme, “Supersymmetry, shape invariance, and exactly solvable\npotentials,” American Journal of Physics 56, 163 (1988) .\n[24] F. W. Olver, D. W. Lozier, R. F. Boisvertand C. W. Clark, NIST handbook of mathematical functions\n(Cambridge university press, 2010).\n[25] G. Darboux, “Sur une proposition relative auxequations lineaires,” Compt. Rend. Acad. Sci. (Paris)\n94, 1456-1459 (1882).\n[26] A. Contreras-Astorga, F. Correa and V. Jakubsky, “Super-Klein tunneling of Dirac\nfermions through electrostatic gratings in graphene,” Phys. Rev. B 102, no.11, 115429 (2020)\n[arXiv:2006.08207 [cond-mat.mes-hall]] .\n[27] M. V. Ioffe, D. N. Nishnianidze, E. V. Prokhvatilov, “New solutions for graphene with scalar potentials\nby means of generalized intertwining,” Eur. Phys. J. Plus 134, 450 (2019) .\n[28] M. S. Plyushchay, “Exotic Nonlinear Supersymmetry and Integrable Systems,”\nPhys. Part. Nucl. 51, no.4, 583-588 (2020) [arXiv:2001.04133 [hep-th]] .\n8[29] L. Inzunza and M. S. Plyushchay, “Hidden symmetries of rationally deformed superconformal me-\nchanics,” Phys. Rev. D 99(2019), 025001 [arXiv:1809.08527 [hep-th]] .\n[30] L. Inzunza and M. S. Plyushchay, “Klein four-group and Darboux duality in conformal mechani cs,”\nPhys. Rev. D 99(2019), 125016 [arXiv:1902.00538 [hep-th]] .\n9"    },    {        "title": "2401.13884v1.Constant_Stepsize_Q_learning__Distributional_Convergence__Bias_and_Extrapolation.pdf",        "content": "Constant Stepsize Q-learning: Distributional Convergence, Bias\nand Extrapolation\nYixuan Zhang, Qiaomin Xie∗\nDepartment of Industrial and Systems Engineering\nUniversity of Wisconsin-Madison\nAbstract\nStochastic Approximation (SA) is a widely used algorithmic approach in various fields, including\noptimization and reinforcement learning (RL). Among RL algorithms, Q-learning is particularly popular\ndue to its empirical success. In this paper, we study asynchronous Q-learning with constant stepsize, which\nis commonly used in practice for its fast convergence. By connecting the constant stepsize Q-learning to a\ntime-homogeneous Markov chain, we show the distributional convergence of the iterates in Wasserstein\ndistance and establish its exponential convergence rate. We also establish a Central Limit Theory for\nQ-learning iterates, demonstrating the asymptotic normality of the averaged iterates. Moreover, we\nprovide an explicit expansion of the asymptotic bias of the averaged iterate in stepsize. Specifically, the\nbias is proportional to the stepsize up to higher-order terms and we provide an explicit expression for the\nlinear coefficient. This precise characterization of the bias allows the application of Richardson-Romberg\n(RR) extrapolation technique to construct a new estimate that is provably closer to the optimal Q function.\nNumerical results corroborate our theoretical finding on the improvement of the RR extrapolation method.\n1 Introduction\nStochastic Approximation (SA) is a fundamental algorithmic paradigm in various fields, including optimization,\nmachinelearning, stochastic controland filtering, andreinforcementlearning(RL).SAusesrecursivestochastic\nupdates to solve fixed point equations. One prominent example is the widely-used stochastic gradient descent\n(SGD) algorithm for finding the optimal solution to an objective function [ 69]. In the context of RL, well-\nknown algorithms such as Q-learning and TD-learning can be viewed as SA algorithms for solving Bellman\nequations [ 8]. Classical SA theory suggests using diminishing stepsize, ensuring asymptotic convergence\nto the desired solution [ 16]. However, SA with constant stepsize is commonly used in practice due to its\nsimplicity and faster convergence. In this case, SA iterates can be viewed as a time-homogeneous Markov\nchain. Adopting this perspective, a growing line of recent work establishes weak convergence of constant\nstepsize SA and characterizes the stationary distribution [12, 42, 55, 38, 116].\nIn this paper, we investigate constant stepsize Q-learning, which is an important instance of nonsmooth\nSA with Markovian noise. Q-learning is a popular RL algorithm that has played a significant role in the\nempirical success of RL [ 83]. It directly learns the optimal action-value function (or Q-function) from sample\ntrajectories. At each iteration, Q-learning asynchronously updates a single state-action pair based on one\ntransition from the trajectory. Consequently, the iterations inherently involve Markovian noise resulting from\nthe sampling process of a Markov chain under the behavior policy. Q-learning’s asymptotic convergence\nand finite-sample guarantees have been extensively studied [ 105,18,104,47,27,77]. These non-asymptotic\nresults provide upper bounds on either the mean squared error (MSE) E[∥qk−q∗∥2\n∞\u0003\nor high probability ℓ∞\nerror∥qk−q∗∥∞.\nThe main goal of this paper is to gain a more comprehensive understanding of the behavior of constant\nstepsize Q-learning and its error decomposition. In the discounted setting, Q-learning aims to solve the\n∗Emails: yzhang2554@wisc.edu ,qiaomin.xie@wisc.edu\n1arXiv:2401.13884v1  [stat.ML]  25 Jan 2024fixed point equation involving the Bellman operator, which is contractive in the nonsmooth ℓ∞norm. Hence\nQ-learning can be viewed as an instance of SA with a nonsmooth operator and Markovian noise. Recently,\nnon-asymptotic analysis of Markovian SA has been gaining attention [ 11,100,25,27,55]. However, these\nresults either concern linear SA or provide upper bounds on the error.\nIn this work, we study the Q-learning iterates through the lens of Markov chain theory. We provide a\nmore precise characterization of the MSE in terms of the decomposition\nE∥qk−q∗∥2≍\r\rEqk−Eq(α)\n∞\r\r2\n|{z }\noptimization error+\r\rEq(α)\n∞−q∗\r\r2\n|{z }\nasymptotic bias+ Var ( qk)|{z}\nvariance(1)\nwhere the random variable q(α)\n∞denotes the limit of the Q-learning iterate qkwith stepsize α. Our main\nresults characterize the behavior of the three terms above, as summarized in the following.\n•By casting the constant stepsize Q-learning as a time-homogeneous Markov chain, we establish the\nconvergence of the iterates to a unique random vector in W2, the Wasserstein distance of order 2.\nMoreover, {qk}k≥0converges to the limit q(α)\n∞exponentially fast due to the use of a constant stepsize.\nThis result leads to an upper bound on the optimization error ∥Eqk−Eq(α)\n���∥, which also decays\nexponentially in k. The convergence rate cannot be obtained from the existing upper bound on\nE∥qk−q∗∥2or∥qk−q∗∥∞, which does not vanish as k→ ∞ .\n•We show that the variance Var (qk)is of order O(1). By law of large numbers, one can use the averaging\nprocedure to eliminate the variance. Specifically, the Polyak-Ruppert tail-averaged iterate has a variance\nof order O(1/k). Consequently, for large k, the deviation between the averaged iterate and the optimal\nsolution q∗for large kis dominated by the asymptotic bias Eq(α)\n∞−q∗.\n•We present an explicit expansion of the asymptotic bias Eq(α)\n∞−q∗in the stepsize α:\nEq(α)\n∞−q∗=αB+eO(α2), (2)\nwhere Bis a vector independent of the stepsize αandeOdenotes the variant of Othat ignores logarithmic\norder. Importantly, the bias characterization is exact, as equation (2) is an equality rather than an\nupper bound. Consequently, one can use Richardson-Romberg extrapolation technique to reduce the\nbias and obtain an estimate closer to q∗with order-wise smaller bias eO(α2).\nOn the technical side, to deal with the nonsmooth operator, we employ a local linearization of the operator\nin the neighborhood of the optimal solution q∗. While local linearization has been explored in nonlinear\nSA literature, they mainly consider the asymptotic regime with diminishing stepsizes [ 72,79,80,51]. We\ngeneralize this approach to characterize the dependence on the constant stepsize. It is worth noting that while\nthe linear approximation component resembles similar behavior as linear SA [ 55], a precise characterization of\nthe bias requires a careful analysis of the linear approximation error to establish a proper higher order of α.\n1.1 Related Work\nHere we discuss closely related work and defer other related work to Section A in supplementary materials.\nQ-learning. An increasing volume of recent work has been dedicated to understanding finite-time guarantees\nof Q-learning in various scenarios. There are two types of results on the distance between the estimate\nqkand the optimal Q-function q∗: high probability bounds and mean (square) error bounds. For classical\nasynchronous Q-learning, as considered in this paper, [ 6] provide the first result on MSE with constant\nstepsize and [ 27] improve the result by at least a |S||A|factor. The work by [ 76] presents the best known high\nprobability sample complexity. It is worth noting that these two types of bounds are not directly comparable,\nas discussed in [ 27].Importantly, these results are achieved either by rescaled linear stepsize αk=a/(b+k)\n2[94,27] or by a carefully chosen constant stepsize based on the target accuracy [ 27,76]. Contrasting with\nthese findings, our results provide a precise characterization of the convergence rate as well as the bias induced\nby constant-stepsize α, for any αin a given range. Our explicit characterization enables the application\nof RR technique, leading to an estimate with reduced bias, while simultaneously enjoying the exponential\nconvergence of the optimization error.\nSomerecentworkalsostudiesPolyak-RuppertaveragedQ-learning. [ 114]and[79]proveafunctionalcentral\nlimit theorem for the averaged iterates of synchronous Q-learning with constant stepsize and diminishing\nstepsize, respectively. In contrast, we focus on asynchronous Q-learning involving Markovian data.\nStochastic approximation. There is a growing interest in investigating general SA with constant stepsize.\nMost work along this line considers i.i.d. or martingale difference noise, and establishes finite-time guarantees\nfor contractive/linear SA [ 25,84,44] or SGD [ 38,116]. Recent work investigates constant-stepsize SA with\nMarkovian noise, motivated by applications in RL. For linear SA, the work by [ 100] provides finite-time upper\nbounds on the MSE. [ 85] study LSA with PR averaging and presents instance-dependent MSE upper bounds\nwith tight dimension dependence. The work by [ 45] shows a finite-time upper bound for the p-th of LSA\niterate on general state space. The paper [ 70] shows that LSA with Markovian noise admits a biass that can\nnot be eliminated by averaging. The work [ 55] establishes the distributional convergence of LSA iterates, and\nprovides an explicit asymptotic expansion of the bias in stepsize. Going beyond LSA, the work [ 30] considers\ncontractive SA under a strong monotone condition and provides finite-time upper bound on the MSE.\nOur results have some similarities to [ 38, Proposition 2], [ 44, Theorem 3] and [ 55], in that we also study\ninstances of SA with constant stepsizes through Markov chain theory. However, our setting is different from\n[44, Theorem 3] as the sampling process in RL naturally induces Markovian noise, whereas they consider i.i.d.\ndata. While the work [ 55] also considers Markovian noise, their focus is on linear SA. In contrast, Q-learning\ninvolves nonsmooth update, which brings additional challenges on the analysis of convergence and bias. In\nparticular, for convergence proof, the difference between two coupled LSA iterates can be reformulated as an\nLSA; however, this is not the case for Q-learning, which requires a novel analysis for the coupled iterates.\nFor the bias analysis, we employ a local linearization method to decompose the Q-learning operator into a\nlinear term and a remaining approximation term. While the technique for LSA [ 55] can be used to analyse\nthe linear part, it is highly nontrivial to show the remaining term is of higher order dependence on α. We\nestablish this result by analyzing the fourth moment of the iterates. Our techniques may be of independent\ninterest and have the potential to be applied to the analysis of other nonsmooth/nonlinear SA algorithms.\n2 Preliminaries\nConsider a discounted Markov decision process (MDP) defined by the tuple (S,A, T, r, γ ),where SandA\nare the (finite) state space and action space, T:S × A → ∆(S)is the transition kernel, r:S × A → [0, rmax]\nis the reward function, and γ∈(0,1)is the discounted factor. At time t∈ {0,1, . . .}, the system is in\nstate st∈ S; upon taking action at∈ A, the system transits to st+1∈ Swith probability T(st+1|st, at)and\ngenerates a reward rt=r(st, at).\nAstationarypolicy π:S → ∆(A)mapseachstatetoadistributionovertheactions, where π(a|s)represents\nthe probability of taking action agiven state s. For each policy π, the action-value function (Q-function) is de-\nfined as follows: ∀s∈ S,∀a∈ A, qπ(s, a) =E\u0002P∞\nt=0γtr(st, at)|s0=s, a0=a\u0003\n,where ak∼π(·|sk)for all k≥\n0.An optimal policy π∗is the policy that maximizes qπ(s, a)for all s∈ Sanda∈ Asimultaneously. It is\nwell known that such an optimal policy exists [ 10] and we denote the associated Q functions as q∗≡qπ∗.\nNotably, given q∗,one can obtain the optimal policy π∗(s)∈arg max a∈AQ∗(s, a).\nBehavior policy. The goal of reinforcement learning (RL) is to learn the optimal policy based on\ntransition data from the system, without the knowledge of the MDP model (T, r).In this paper, we consider\noff-policy setting, where we have access to a sample trajectory {sk, ak, rk}k≥0generated by the MDP under a\nfixedbehavior policy eπ.\nDefine X:=S × A × S ,and let xk= (sk, ak, sk+1).Note that when eπis stationary, {xk}k≥0forms a\n3time-homogeneous Markov chain. We use P= (pij)to denote the corresponding transition matrix. In this\npaper, we impose the following assumption on the behavior policy.\nAssumption 1. {xk}k≥0is an irreducible and aperiodic Markov chain on a finite state Xwith stationary\ndistribution µX. Also, the distribution of the initial state x0isµX.\nAssumption 1 is equivalent to assuming that Markov chain {sk, ak}k≥0induced by the behavior policy\neπis uniformly ergodic with a unique stationary distribution µS[27]. This assumption is standard for\nanalyzing off-policy Q-learning [ 77,27,94]. Assumption 1 implies that {xk}k≥0mixes geometrically fast to\nthe stationary distribution µX[73], and there exist c≥0andρ∈(0,1)s.t.\nmax\nx∈X∥pk(x,·)−µX(·)∥TV≤cρk, (3)\nwhere pk(x,·)denotes the distribution of xkgiven x0=x.\nTo quantify how fast {xk}k≥0mixes to a specified precision, we define the mixing time below.\nDefinition 1. ∀δ >0, define tδ:= min {k≥0 : max x∈X∥pk(x,·)−µX(·)∥TV≤δ}.\nUnder Assumption 1, we have tα≤log(c/ρ)+log(1 /α)\nlog(1/ρ), which implies limα→0αm1tαm2= 0,∀m1, m2>0. We\nassume that x0∼µXto simplify some presentation. This assumption is not essential and can be relaxed by\nadapting our result after the Markov chain {xk}k≥0has almost mixed. We remark that the same assumption\nis considered in many previous works [11, 55, 85].\nQ-learning. The Q-learning algorithm [ 111] is an iterative method for estimating the function Q∗based\non the sample trajectory {sk, ak, rk}k≥0. It generates a sequence of Q-function estimate {qk:S × A → R}k,\naccording to the following recursion:\nqk+1=qk+αkF(xk, qk), (4)\nwhere αkis the stepsize. Here the operator F:X ×R|S∥A|7→R|S∥A|, known as empirical Bellman\noperator, is defined as: ∀(s, a)̸= (sk, ak),[F(x, q)](s, a) = 0;\n[F(x, q)](sk, ak) =r(sk, ak) +γmax\nvqk(sk+1, v)−qk(sk, ak).\nIn this paper, we focus on constant stepsize αk≡α >0. We use superscript q(α)\nkto emphasize the dependence\non the stepsize α,but omit it when it is clear from the context.\nWe record a few basic properties of Q-learning. (1) By the boundedness of reward, there exists a constant\nqmaxsuch that ∥qk∥∞≤qmax,∀k.(2) Denote the expected operator of Fby¯F(q) :=Ex∼µX[F(x, q)]. It has\nbeen shown that ¯F(q) +qis aβ-contraction mapping w.r.t. ∥ · ∥∞[27], where\nβ= 1−(1−γ) min (s,a)µS(s, a). (5)\nRecall that µSis the stationary distribution of Markov chain {sk, ak}k≥0[27]. By Assumption 1, we have\nmin(s,a)µS(s, a)>0, thus β <1. (3) Crucially, the iterates {qk}generated by Q-learning is not a Markov\nchain. On the other hand, we can see that the joint process {xk, qk}k≥0is a Markov chain on the state\nspaceX ×R|S|×|A|.In particular, employing a constant stepsize αinduces a time-homogeneous Markov chain\n{xk, qk}k≥0.\nPart of our results on Q-learning (cf. Theorem 3) requires the following assumption.\nAssumption 2. The optimal policy πis unique. That is, ∃∆>0such that for ∀s∈ S,q∗(s, a∗\ns)−q∗(s, a)≥\n2∆,∀a̸=a∗\ns,where a∗\ns:= arg max\naq∗(s, a)denotes the optimal action for each state s.\nSimilar conditions have been considered in prior work on the analysis of Q-learning variants [ 37,79].\nAssumption 2 implies that the operator in (4)can be approximated by local linearization around q∗and\nhigh-order approximation error, which leads to our precise characterization of the bias induced by constant\nstepsize.\nAdditional notations. Part of our analysis uses the reversed Markov chains. By Assumption 1, the\nMarkov chain {xk}k≥0is irreducible. An implication is that the chain {xk}k≥0running backward in time is\nalso a Markov chain [88], with transition kernel ˆP= (ˆpij)given by µX(j)ˆpji=µX(i)pij.\n43 Main Results\nIn this section, we present our main results. In Section 3.1, we show that joint data Q-learning iterates\n{xk, qk}k≥0, viewed a time-homogeneous Markov chain, converges to a unique limit distribution. We also\nestablish explicit convergence rates. Moreover, we show a central limit theorem (CLT) for the iterates {qk}k≥0\nin Section 3.2. We then precisely characterize the relationship between the limit and the stepsize in Section\n3.3. Furthermore, we investigate the implication of these results for Polyak-Ruppert tail averaging and\nRichardson-Romberg extrapolation in Section 3.4.\n3.1 Stationary Distribution and Convergence Rate\nNote that the Q-learning iterate {qk}k≥0is not a Markov chain by itself, as its dynamic depends on the\nMarkovian data {xk}k≥0. To show the distributional convergence of {qk}k≥0and quantify its convergence\nrate, we consider the joint process {xk, qk}k≥0,which can be cast as a time-homogeneous Markov chain.\nWe will analyze the convergence of this Markov chain using the Wasserstein 2-distance. The Wasserstein\n2-distance between two distributions µandνinP2(Rd)is defined as\nW2(µ, ν) = inf\nξ∈Π(µ,ν)\u0012Z\nRd∥u−v∥2\n∞dξ(u, v)\u00131\n2\n= inf\u001a\u0010\nEh\n∥θ−θ′∥2\n∞i\u00111\n2:L(θ) =µ,L(θ′) =ν\u001b\n,\nwhere P2(Rd)denotes the space of square-integrable distributions on Rd,L(θ)denote the distribution\nofθandΠ(µ, ν)is the set of all joint distributions in P2(Rd×Rd)with marginal distributions µandν.\nTo analyze the Markov chain {xk, qk}k≥0, we define the extended Wasserstein 2-distance as in [ 55]. Let\nd0(x, x′):=1{x̸=x′}and ¯d((x, θ),(x′, θ′)):=q\nd0(x, x′) +∥θ−θ′∥2\n∞, which defines a metric on X ×Rd.\nThen, the extended Wasserstein 2-distance for two distributions ¯µand¯νinP2(X ×Rd)with respect to the\nmetric ¯dis as below:\n¯W2(¯µ,¯ν) = inf\u001a\u0010\nEh\n¯d(z, z′)2i\u00111/2\n:L(z) = ¯µ,L(z′) = ¯ν\u001b\n. (6)\nWe show that the Markov chain {xk, qk}k≥0converges in ¯W2to a unique stationary distribution, geomet-\nricallyfast, as stated in the following Theorem.\nTheorem 1. Suppose that Assumption 1 holds, and the stepsize αfor Q-learning (4)satisfies\nαtα≤c0(1−β)2\nlog(|S∥A| )for some constant c0. (7)\n1.Under all initial distribution of q0, the sequence {xk, qk}k≥0converges in ¯W2to a a unique limit (x∞, q∞)∼\n¯µ. Moreover, we have\nVar(q∞)≤cQlog(|S∥A| )\n(1−β)2αtα, (8)\nwhere cQ= 912 e(3∥q∗∥∞+rmax).\n2.¯µis the unique stationary distribution of the Markov chain {xk, qk}k≥0.\n3. Let µ:=L(q∞)be the second marginal of ¯µ. For all k≥tα, we have\nW2\n2(L(qk), µ)≤24ηk−tα\u0000\nE\u0002\n∥q0∥2\n∞\u0003\n+E\u0002\n∥q∞∥2\n∞\u0003\u0001\n, (9)\nwhere η= 1−(1−β)α/2.\n5Theorem 1 states that the Markov chain {xk, qk}k≥0admits a unique stationary distribution, given that\nthe constant stepsize αsatisfies condition (7). Recall that under Assumption 1, for all m1, m2>0, we have\nlimα→0αm1tαm2= 0. Therefore, there always exists a small enough stepsize αsuch that condition (7)holds.\nWe remark that the resulting limit distribution ¯µis not generally a product distribution of its marginals µX\nandµ.\nNote that the convergence results stated in Theorem 1 cannot be obtained from existing error bounds\non Q-learning. For example, the sharpest high probability bound on ℓ∞error scales as ∥qk−q∗∥∞≲\n(1−ρ)k∥q0−q∗∥∞+O(√α), where ρ∈(0,1)[77]. Another type of upper bound is on the MSE that scales as\nE[∥qk−q∗∥2\n∞]≲(1−(1−β)α/2)k−tα∥q0−q∗∥2\n∞+O(αtα)[27]. Both upper bounds imply that the sequence\neventually falls in a neighbor of the optimal solution q∗and the initial condition is forgotten exponentially\nfast. However, these result does not imply the distributional convergence of the sequence {qk}k≥0or its\nconvergence rate in W2metric.\nWe would like to highlight the techniques employed to prove Theorem 1. A standard method for proving\nthe positive recurrence of a Markov chain is to verify irreducibility and Lyapunov drift condition [ 82], as used\nin prior work on SA [ 15] and SGD [ 116]. However, this method requires strong condition on the randomness\nof the Markov chain dynamics, which is difficult to verify in Q-learning. Instead, we draw inspiration from\nrecent work on LSA [ 55], and prove weak convergence by showing the convergence in W2distance through\ncoupling arguments. Wasserstein distance has recently been used [ 38,55,45]. We remark that the coupling\nargument in our proof is more involved due to the nonsmooth nature of the update operator F. We sketch\nthe proof outline of Theorem 1 in Section 4.1. The complete proof is deferred to Section B.\nA direct consequence of the convergence in W2metric is the convergence of the first two moments. We\ncan also obtain explicit convergence rates from Theorem 1, as detailed in the following corollary. The proof is\nprovided in Section B.5.\nCorollary 1. Under the setting of Theorem 1, for all k≥tα,\n∥E[qk−q∞]∥2\n∞≤C·\u0000\n1−(1−β)α/2\u0001k−tα,\n\r\rE\u0002\nqkq⊤\nk\u0003\n−E\u0002\nq∞q⊤\n∞\u0003\r\r\n∞≤C′·\u0000\n1−(1−β)α/2\u0001k−tα\n2.\nwhere CandC′are constants independent of αandk.\n3.2 Central Limit Theorem\nBuilding on the convergence result, we establish a CLT for {qk}k≥0. Here we define Sn=Pn−1\nk=0\u0000\nqk−E[q∞]\u0001\nandYn(t) =n−1\n2S⌊nt⌋.LetD=D[0,1]denote the Skorokhod space, which is a separable and complete\nfunction space under some proper metrics [93].\nTheorem 2. Under the setting of Theorem 1, Σ:=limn→∞1\nnEπ\u0000\nSnS⊤\nn\u0001\nexists, and for ¯µ-almost every point\n(x0, q0)the sequence {n−1\n2Sn}n≥0converge in distribution to the |S||A|-dimensional Gaussian distribution\nN(0,Σ). Furthermore, the process (Yn(t))0≤t≤1converges weakly to\u0010\nΣ1\n2B(t)\u0011\n0≤t≤1on the Skorokhod space\nD[0,1], where B= (B(t))t≥0is the standard Brownian motion.\nTheorem 2 states that the average of Q-learning iterates is asymptotically normally distributed around the\nexpected value of the unique stationary distribution. Establishing such a CLT is important for uncertainty\nquantification and statistical inference [ 79]. Similar result has been established for synchronous Q-learning\nwith constant stepsize [ 114]. It is worth highlighting that one key step in [ 114] uses the Kantorovich–Rubinstein\ntheorem [ 46] defined on a Wasserstein distance with single-step contraction. However, such result does not\nhold in our setting due to Markovian data. To this end, we use the result in Theorem 1 and ergodicity of\n{xk}k≥0to establish CLT. The detailed proof is provided in Section C.\n63.3 Bias Expansion\nUnder constant stepsize α, Theorem 1 asserts that the convergence of q(α)\nktoq(α)\n∞, which is of distribution µ.\nTherefore, the estimates q(α)\nkof Q-learning with constant stepsize do not converge to a point, but oscillate\naround the mean E[q(α)\n∞]. Here we would like to quantify the bias, i.e., the deviation of the mean E[q(α)\n∞]from\nthe optimal q∗.One of our main contributions is to provide an explicitasymptotic expansion in the step-size\nαof the bias E[q(α)\n∞]−q∗.\nTheorem 3. Suppose that Assumptions 1 and 2 hold and α≤α0for some α0.Then the following holds for\na vector Bthat is independent of α.\nE[q∞] =q∗+αB+O(α2+α2t2\nα2). (10)\nwhere B=B(r, γ, P )is explicitly given in the appendix.\nA few remarks are in order.\nThe above theorem states that the asymptotic bias of Q-learning can be decomposed into a linear term\nand a higher order term of αunder Assumption 2. We emphasize that our bias characterization of the linear\ndependence on αisexact. As discussed in the previous sub-section, existing results are typically in the form\nof an upper bound on the bias. Specifically, the high probability upper bound on ℓ∞error [77] implies an\nbias of O(√α).In contrast, our analysis reveals a refined result with αB+eO(α2)bias.\nOne key step in the proof of Theorem 3 is to calculate E[F(x∞, q∞)|x∞=i],∀i∈ X. For linear SA,\nthis step is straightforward. However, for asynchronous Q-learning, operator Fis not linear, and not even\nsmooth, which makes the analysis more complicated. In our proof, we provide a local linearization method\nwhich can bridge the gap between nonlinear SA and LSA. We outline the proof of Theorem 3 in Section 4.2.\nThe complete proof is provided in Section D.\nWe remark that the coefficient Bof the linear term is independent of α. It only depends on the underlying\nMDP and the behavior policy. One can find an explicit expression of Bin the proof (cf. Equation (37)).\nImportantly, for the special case where the associated data sequence {xk}k≥0is i.i.d., we have B=0.\nHowever, the bias term O(α2+α2t2\nα2)still remains, due to nonlinearity of Q-learning operator. This should\nbe contrasted with the LSA where the bias vanishes with i.i.d. data [ 55]. In general, the existence of\nbias implies that the mean of the sequence {qk}k≥0limit deviates from the optimal solution q∗.Therefore,\naveraging the iterates qkdoes not eliminate the bias. However, thanks to the independence of Bonα,we\ncan leverage an extrapolation technique to reduce the bias.\n3.4 Tail Average and Richardson-Romberg extrapolation\nWe now utilize the bias expansion result Theorem 3 to study the behavior of Q-learning when combined with\nPolyak-Ruppert average and Richardson-Romberg extrapolation.\n3.4.1 Polyak-Ruppert Averaging\nThe celebrated Polyak-Ruppert averaging procedure [ 96,92] can reduce the variance of the estimate and\naccelerate the convergence rate. In this paper, we consider the tail-averaging variant of PR-averaging [ 58],\nwhich is defined as follows with a burn-in period k0:\n¯qk0,k:=1\nk−k0Xk−1\nt=k0qt,fork≥k0+ 1. (11)\nThe following corollary provides non-asymptotic results for the first and second moments of ¯qk0,k. The\nproof is provide in Section E.\n7Corollary 2. Under the setting of Theorem 3, the tail-averaged iterates (11)satisfy the following: ∀k >\nk0≥tα2:\nE[¯qk0,k]−q∗=αB+O(α2+α2t2\nα2) +O\u00121\nα(k−k0)exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n,(12)\nEh\n(¯qk0,k−q∗) (¯qk0,k−q∗)⊤i\n=α2B′+O(α3+α3t2\nα2)| {z }\nasymptotic squared bias+O\u00121\n(k−k0)α\u0013\n| {z }\nvariance\n+O\u00121\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n| {z }\noptimization error, (13)\nwhere BandB′are independent of α.\nFor simplicity, let us consider the case k0=k/2and discuss the mean squared distance between the\naveraged-iterate ¯qk/2,kandq∗. From the analysis, we note the MSE can be decomposed into three parts: the\nasymptotic squared bias term ∥E[¯q∞/2,∞−q∗]∥2that is independent of kand averaging; the second part for\nthe variance of ¯qk/2,kthat scales as 1/k; and the optimization error ∥E[¯q∞/2,∞−¯qk/2,k]∥2that decays to 0\ngeometrically fast. Importantly, the larger the stepsize αis, the faster the variance and optimization error\ndecay. This finding also justifies the benefit of using constant-stepsize.\n3.4.2 Richardson-Romberg Extrapolation\nGiven the explicit expansion of the bias in stepsize α(cf. Theorem 3), we can leverage the Richardson-Romberg\n(RR) extrapolation technique from numerical analysis [50] to reduce the bias. Specifically, consider running\ntwo Q-learning recursions using the samedata stream {xk}k≥0, but with different stepsizes αand2α.Denote\nby¯q(α)\nk0,kand¯q(2α)\nk0,kthe corresponding tail-averaged iterates. The corresponding RR extrapolated iterates are\ngiven by\neq(α)\nk0,k= 2¯q(α)\nk0,k−¯q(2α)\nk0,k. (14)\nWith k0, k→ ∞, Theorems 1 and 3 imply that eq(α)\nk0,kconverges to 2q(α)\n∞−q(2α)\n∞, which has a bias\n2Eq(α)\n∞−Eq(2α)\n∞−q∗= 2\u0000\nαB+O(α2+α2t2\nα2)\u0001\n−\u0000\n2αB+O(α2+α2t2\nα2)\u0001\n=O(α2+α2t2\nα2).\nNote that compared with q(α)\n∞andq(2α)\n∞,the extrapolated sequence provides a new estimate that reduces\nthe bias by a factor of α.We formally state the result in the following corollary, which quantifies the\nnon-asymptotic behavior of the first two moments of extrapolated sequence {eq(α)\nk0,k}k≥0.The proof is provided\nin Section F.\nCorollary 3. Under the setting of Theorem 3, the RR extrapolated iterates (14)with stepsizes αand2α\nsatisfy the following for all k > k 0≥tα2:\nEh\neq(α)\nk0,ki\n−q∗=O(α2+α2t2\nα2) +O\u00121\nα(k−k0)exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n, (15)\nE\u0002\u0000\neq(α)\nk0,k−q∗\u0001\u0000\neq(α)\nk0,k−q∗\u0001⊤\u0003\n=O\u0000\nα4+α4t4\nα2\u0001\n+O\u00121\n(k−k0)α\u0013\n+O\u00121\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n.\n(16)\n8Let us compare the MSE bounds (13)on the PR-averaged iterates and extrapolated sequence (16). It\nis important to note that the asymptotic squared bias is reduced from O(α2)to roughly O(α4)by RR\nextrapolation! Meanwhile, RR extrapolation still enjoys similar decaying rates of variance and optimization\nerror. We remark that the RR procedure involves the computation of two parallel Q-learning iterates, which\ncan use either the same or different data sequences. This makes the RR procedure inherently parallelizable,\noffering potential performance improvements when implemented on parallel computing architectures.\n4 Proof Outline\n4.1 Proof Outline for Theorem 1 on Convergence\nThis sub-section is devoted to the discussion of the proof outline for the existence of the limit distribution,\nwhich is the most challenging part.\nWe first note that the space P(X ×R|S|×|A|)endowed with our extended Wasserstein 2-distance ¯W2is a Pol-\nishspace,asindicatedbyTheorem6.18in[ 108]. IfwecanestablishthatP∞\nk=0¯W2\n2(L(xk, qk),L(xk+1, qk+1))<\n∞, then the sequence {xk, qk}k≥0forms a Cauchy sequence. This, in turn, allows us to prove the existence of\nthe limit distribution, using the fact that all Cauchy sequences converge in a Polish space.\nThe next step involves coupling through the construction of two Markov chains, {x[1]\nk, q[1]\nk}k≥0and\n{x[2]\nk, q[2]\nk}k≥0,which share the same underlying data stream {x[1]\nk}k≥0={x[2]\nk}k≥0={xk}k≥0. We observe\nthat the iterates difference wk:=q[1]\nk−q[2]\nkexhibits the following dynamic, leading to the subsequent\nProposition: wk+1(sk, ak) = (1 −α)wk(sk, ak) +αγ\u0000\nmax\naq[1]\nk(sk+1, a)−max\naq[2]\nk(sk+1, a)\u0001\n.\nProposition 1. Under Assumption 1, if αtα≤c0(1−β)2\nlog(|S∥A| ), the following bound holds for all k≥tα,\nEh\n∥wk∥2\n∞i\n≤12E\u0002\n∥w0∥2\n∞\u0003\u0012\n1−(1−β)α\n2\u0013k−tα\n.\nThe key idea behind the proof of Proposition 1 is to exploit the fact that the difference between\ntwo max operators can be lower bounded by the minimum of the difference and upper bounded by the\nmaximum of the difference. This inpsires us to construct two new dynamics that serve as lower and\nupper bounds on {wk}k≥0. We prove that the lower/upper bound sequences decay geometrically fast to\n0, which immediately implies a geometric decay rate of {wk}k≥0for all initial distribution of q[1]\n0andq[2]\n0.\nThen, by carefully choosing the initial distribution of q[2]\n0, we can ensure that (xk, q[2]\nk)d=(xk+1, q[1]\nk+1).\nConsequently, ¯W2\n2\u0000\nL\u0000\nxk, qk\u0001\n,L\u0000\nxk+1, qk+1\u0001\u0001\n→0geometrically fast. This result can be applied to showP∞\nk=0¯W2\n2\u0000\nL\u0000\nxk, qk\u0001\n,L\u0000\nxk+1, qk+1\u0001\u0001\n<∞, which establishes the existence of limit distribution.\n4.2 Proof Outline for Theorem 3 on Bias Expansion\nIn the proof of Theorem 3, a crucial technique employed is the linearization of the non-smooth operator\nF(x, q). Specifically, for a fixed x, we linearize F(x, q)around the optimal solution q∗as per the following\nproposition.\nProposition 2. There exists a function F′\nq∗:X 7→R|S∥A|×|S∥A|s. t. for any (x, q)∈ X × R|S∥A|\nF(x, q) =F(x, q∗) + (Gq∗(x)−Id)(q−q∗) +R(x, q), (17)\nwithd=|S∥A|and∥R(x, q)∥∞=O\u0000\n∥q−q∗∥4\n∞\u0001\n.Furthermore, Ex∼µXGq∗(x)doesn’t have an eigenvalue of\n1.\nWe next provide a finite-time upper bound on the fourth moment of the error, which shows the remaining\nterm R(x, q)is of a higher order of αby Proposition 2. We remark that existing non-asymptotic results for\nQ-learning are limited to the first moment and second moment of the error.\n9Proposition 3. Suppose that Assumption 1 holds. Consider Q-learning (4)with constant stepsize α.There\nexists a constant α0>0such that ∀α∈(0, α0)and∀k≥tα2,\nE[∥qk−q∗∥4\n∞]≤b1(1−α(1−γ)2)k−tα2+b2α2+b3α2t2\nα2, (18)\nwhere b1,b2andb3are constants independent of α.\nNote that the first term on RHS of equation (18)decays geometrically in k, whereas the remaining two\nterms are independent of k. Consequently, as k→ ∞, the upper bound is of order O\u0000\nα2+α2t2\nα2\u0001\n.\nThe proof is inspired by the work [ 27] that constructs a Generalized Moreau Envelope (GME) M(·)to\nanalyse ∥·∥2\n∞. By some nice property of GME [ 24,27], we can derive the bound for M(·)2, which equivalently\nprovides the bound for ∥ · ∥4\n∞.We defer the complete proof of Proposition 2 and 3 to Section D.1.\nTherefore, the RHS of equation (17)can be viewed as a combination of a linear operator and a high-order\nremaining term R(x, q). Assume we are in the limit (x∞, q∞). Recall that Theorem 3 has established a\nnon-asymptotic bound for the fourth moment. Then, by fatou’s lemma,\nE[∥q∞−q∗∥4\n∞]≤lim inf\nk→∞E[∥qk−q∗∥4\n∞] =O\u0000\nα2+α2t2\nα2\u0001\n.\nWe then can analyze the dynamic of {xk, qk}k≥0as a linear SA combined with a remaining term of order\nO\u0000\nα2+α2t2\nα2\u0001\n.\n5 Numerical Experiments\nWe consider two MDPs: the first example is a 1×3Gridword with two actions (left/right); the second one is a\nclassical 4×4Gridworld with the slippery mechanism in Frozen-Lake, and four actions (left/up/right/down).\nFor both MDPs, the discounted factor is γ= 0.9and the Markovian data {xk}k≥0is generated from a\nuniformly random behavior policy. We defer details of reward function and transition kernel for the MDPs to\nSection G.\nForMarkoviandatacase, werunQ-learningwithinitialization q(α)\n0=q∗+10andstepsize α∈ {0.1,0.2,0.4}.\nWe also consider two diminishing stepsizes: a rescaled linear stepsize αk= 1/\u0000\n1 + (1 −γ)k\u0001\nas suggested\nby prior work [ 94,24] and a polynomial stepsize αk= 1/k0.75. The simulation results for the two MDPs\nare illustrated in Figure 1(a) and 1(b). We plot the ℓ1-norm error ∥¯q(α)\nk/2,k−q∗∥1for the tail-averaged (TA)\niterates ¯q(α)\nk/2,k, the RR extrapolated iterates eq(α)\nkwith stepsizes αand2α, and iterates with diminishing\nstepsizes.\nWe first observe that the larger the stepsize α,the faster it converges, as implied by Corollary 2. We\nnote that the final TA error, which corresponds to the asymptotic bias, is approximately proportional to the\nstepsize, as indicated by the roughly equal space between three TA lines in the log scale plots. Moreover, RR\nextrapolated iterates reduce the bias, which can be observed by comparing, e.g, the solid orange line (TA with\nα= 0.2) and the dotted red line (RR with α= 0.2and0.4). These results are consistent with Corollary 3.\nFurthermore, the TA and RR-extrapolated iterates with constant stepsizes enjoy significantly faster initial\nconvergence than those with diminishing stepsizes. Particularly for the more complicated the MDP, as shown\nin Figure 1(b), iterates with diminishing stepsize converge slowly, while TA and RR-extrapolated iterates\nconverge quickly and then saturate. A general choice of diminishing stepsize is of the form αk=a/(b+kc),\nwhere a, bandcare hyper-parameters. Tuning the best hyper-parameters for diminishing stepsize is generally\nmore challenging than a single parameter for constant stepsize.\nWe also perform a similar set of experiments for MDPs with linear function approximation. We observe\nsimilar behaviors of the TA iterates and RR extrapolated iterates as the tabular case. Due to space constraint,\nwe defer the details to Section G.\n10(a)1×3Gridworld.\n (b)4×4Gridworld.\nFigure 1: The errors of tail-averaged (TA) iterates and RR extrapolated iterates with different stepsizes.\n6 Conclusions\nIn this work, we provide a more comprehensive study of asynchronous Q-learning with constant stepsizes,\nthrough the framework of Markov chain theory. We establish the distributional convergence of the iterates,\ncharacterize the convergence rate, and prove a central limit theorem for the averaged iterates. Our convergence\nresults lead to a refined characterization of the error. In particular, the explicit expansion of the asymptotic\nbias w.r.t. stepsize αallows one to use the RR extrapolation for bias reduction. There are a few interesting\ndirections one can take to extend our work. 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Wortman\nVaughan, editors, Advances in Neural Information Processing Systems , volume 34, pages 1230–1242.\nCurran Associates, Inc., 2021.\nA More Related Work\nQ-learning Earlier work established the asymptotic convergence of Q-learning algorithm with diminishing\nstepsize [ 105,104]. Over the past few years, an increasing volume of work has been dedicated to understanding\nfinite-time guarantees of Q-learning in various scenarios. from tabular setting [ 6,27,94,109,76] to function\napproximation [ 30,115,41,22]. In this paper we focus on the classical asynchrounous Q-learning. There is\nanother variant of Q-learning that concerns an synchronous setting, where all state-action pairs are updated\nsimultaneously at each step. This setting requires access to a simulator, which generates independent samples\nfor each state-action pair. For synchronous Q-learning, the best-known sample complexity for mean error\nbound is eO(SA(1−γ)−5ϵ−2)[109,24]. The paper [ 75] provides the state-of-art high probability sample\ncomplexity eO(SA\n(1−γ)4ϵ2). In this paper, we focus on the classical asynchronous Q-learning which updates only\na single state-action pair upon each observation. The Markovian noise inherited in the asynchronous model\nmakes it considerably more challenging to analyze than the synchronous case.\nWe also note that there are other lines of work focusing on Q-learning variants that aim to accelerate\nconvergence and improve sample complexity, such as variance-reduced Q-learning [ 77,110,98], speedy\nQ-learning [ 3] and double Q-learning [ 113]. Another direction considers Q-learning with sophisticated\nexploration strategies, with an emphasis on regret bound [ 59,5]. Regret is a metric fundamentally different\nfrom finite-sample bounds, and techniques for these two types of guarantees are quite different. A comparison\nwith these results is beyond the scope of this paper.\nStochastic approximation . There is a rich literature on the study of SA. Classical SA theory mainly\nfocuses on the asymptotic convergence [ 64,16,18,14], typically assuming a diminishing stepsize sequence.\n18More recent studies have shifted the focus to non-asymptotic results. In particular, there is a growing interest\nin investigating general SA and SGD algorithms with constant stepsize. Most work along this line considers\nSA or SGD with i.i.d. or martingale difference noise, and establishes finite-time bounds. The paper [ 25]\nconsiders contractive SA and presents an upper bounds on the MSE. [ 68] analyzes linear SA (LSA) and\nestablishes finite-time upper and lower bounds on the MSE. The work [ 84] refines these results, providing\ntight bounds with the optimal dependence on problem-specific constants as well as a central limit theorem\n(CLT) for the averaged iterates. There are also some recent studies developing new bounds on random matrix\nproducts to analyze LSA: [ 44] establishes tight concentration bounds of LSA, and [ 43] extends these bounds\nto LSA with iterate averaging. In the context of SGD, the work in [ 38] considers strongly convex and smooth\nfunctions. They prove that the iterates converge to a unique stationary distribution by Markov chain theory.\nSubsequent work generalizes this result to non-convex and non-smooth functions with quadratic growth\n[116], and proves asymptotic normality of the averaged SGD iterates. The work [ 29] exams the limit of the\nstationary distribution as stepsize goes to zero. All these results are established under the i.i.d. noise setting.\nAdditionally, [ 12] explores SGD for non-smooth non-convex functions with martingale difference noise, and\nestablishes the weak convergence of the iterates to the set of critical points of the objective function.\nB Proof of Theorem 1\nIn this section, we provide the proof of Theorem 1. The first part of the proof, Section B.1, involves coupling\nthrough the construction of two iterates of Q-learning. Using the result of this step, we then establish the\nexistence and uniqueness of the stationary distribution for the joint Markov chain (xk, qk)k≥0(part 1 and 2\nof Theorem 1) in Section B.2. We prove the convergence rate (part 3 of Theorem 1) in Section B.3.\nB.1 Coupling and Geometric Convergence\nWe construct a pair of coupled Markov chains, (xk, q[1]\nk)k≥0and(xk, q[2]\nk)k≥0, defined as\nq[1]\nk+1(sk, ak) =q[1]\nk(sk, ak) +α\u0010\nr(sk, ak) +γmax\naq[1]\nk(sk+1, a)−q[1]\nk(sk, ak)\u0011\n,\nq[2]\nk+1(sk, ak) =q[2]\nk(sk, ak) +α\u0010\nr(sk, ak) +γmax\naq[2]\nk(sk+1, a)−q[2]\nk(sk, ak)\u0011\n.(19)\nHere (q[1]\nk)k≥0and(q[2]\nk)k≥0are two iterates generated by the Q-learning algorithm, coupled by sharing the\nunderlying data stream (xk)k≥0. We assume that the initial iterates q[1]\n0andq[2]\n0may depend on each other\nand on x0, but are independent of (xk)k≥1given x0.\nDefine the iterates difference as wk:=q[1]\nk−q[2]\nk. Note that the dynamic for {wk}k≥0can be formulated\nas follows:\nwk+1(sk, ak) = (1 −α)wk(sk, ak) +αγ\u0010\nmax\naq[1]\nk(sk+1, a)−max\naq[2]\nk(sk+1, a)\u0011\n.\nWe can exploit the dynamic of {wk}k≥0to establish its convergence rate, as stated in Proposition 1. The\nproof of Proposition 1 is deferred to Section B.4.\nWhen αtα≤c0(1−β)2\nlog(|S∥A| ), we can apply Proposition 1 to bound the square of W2distance between q[1]\nk\n19andq[2]\nkas follows: for all k≥tα,\nW2\n2\u0010\nL\u0010\nq[1]\nk\u0011\n,L\u0010\nq[2]\nk\u0011\u0011(i)\n≤¯W2\n2\u0010\nL\u0010\nxk, q[1]\nk\u0011\n,L\u0010\nxk, q[2]\nk\u0011\u0011\n(ii)\n≤E\u0014\r\r\rq[1]\nk−q[2]\nk\r\r\r2\n∞\u0015\n=Eh\n∥wk∥2\n∞i\n(iii)\n≤12E\u0002\n∥w0∥2\n∞\u0003\u0012\n1−(1−β)α\n2\u0013k−tα\n,(20)\nwhere the inequality (i)follows from the definition of W2and ¯W2; the inequality (ii)holds as the ¯W2is\ndefined by an infimum as in equation (6); the inequality (iii)follows from applying Proposition 1.\nTherefore, W2\n2\u0010\nL\u0010\nq[1]\nk\u0011\n,L\u0010\nq[2]\nk\u0011\u0011\ndecays geometrically. We will use this result in the next sub-section\nto prove that (xk, qk)k≥0converges to a unique stationary distribution.\nB.2 Existence and Uniqueness of Stationary Distribution\nNote that equation (20)always holds for any joint distribution of initial iterates (x0, q[1]\n0, q[2]\n0). After fixing an\narbitrarily chosen distribution of (x0, q[1]\n0), we need to carefully choose the conditional distribution of q[2]\n0to\nensure that (xk, q[2]\nk)d=(xk+1, q[1]\nk+1)holds for all k≥0, whered=denotes equality in distribution. Recall that\nˆPrepresents the transition kernel for the time-reversed Markov chain of (xk)k≥0, and the initial distribution\nofx0is assumed to be mixed already. Given a specific x0, we sample x−1from ˆP(· |x0). Additionally, we use\nq[2]\n−1to denote a random variable that satisfies q[2]\n−1d=q[1]\n0and is independent of (xk)k≥0. Finally, we set q[2]\n0as\nq[2]\n0=q[2]\n−1+αF(x−1, q[2]\n−1). (21)\nBy the property of time-reversed Markov chains, we have (xk)k≥−1d=(xk)k≥0.Given that q[2]\n−1d=q[1]\n0\nandq[2]\n−1is independent with (xk)k≥−1, we can prove (xk, q[2]\nk)d=(xk+1, q[1]\nk+1)for all k≥0by comparing the\ndynamic of (q[1]\nk)k≥0and(q[2]\nk)k≥0as given in equations (19) and (21).\nWe thus have for all k≥tα:\n¯W2\n2\u0010\nL\u0010\nxk, q[1]\nk\u0011\n,L\u0010\nxk+1, q[1]\nk+1\u0011\u0011\n=¯W2\n2\u0010\nL\u0010\nxk, q[1]\nk\u0011\n,L\u0010\nxk, q[2]\nk\u0011\u0011\n≤12E\u0002\n∥w0∥2\n∞\u0003\u0012\n1−(1−β)α\n2\u0013k−tα\n,\nwhere the second inequality follows from equation (20). It follows that\n∞X\nk=0¯W2\n2\u0010\nL\u0010\nxk, q[1]\nk\u0011\n,L\u0010\nxk+1, q[1]\nk+1\u0011\u0011\n≤tα−1X\nk=0¯W2\n2\u0010\nL\u0010\nxk, q[1]\nk\u0011\n,L\u0010\nxk+1, q[1]\nk+1\u0011\u0011\n+ 12E\u0002\n∥w0∥2\n∞\u0003∞X\nk=0\u0012\n1−(1−β)α\n2\u0013k\n<∞,\nwhere the last step holds since(1−β)α\n2∈(0,1). Consequently, (L(xk, q[1]\nk))k≥0forms a Cauchy sequence with\nrespect to the metric ¯W2. Since the space P2(X ×Rd)endowed with ¯W2is a Polish space, every Cauchy\n20sequence converges [ 108, Theorem 6.18]. Furthermore, convergence in Wasserstein 2-distance also implies\nweak convergence [ 108, Theorem 6.9]. Therefore, we conclude that the sequence (L(xk, q[1]\nk))k≥0converges\nweakly to a limit distribution ¯µ∈ P2(X ×Rd).\nNext, we show that ¯µis independent of the initial iterate distribution of q[1]\n0, when x0is initialized from\nits unique stationary distribution µX. Suppose there exists another sequence (xk,eq[1]\nk)k≥0with a different\ninitial distribution that converges to a limit eµ. By triangle inequality, we have\n¯W2(¯µ,˜µ)≤¯W2\u0010\n¯µ,L\u0010\nxk, q[1]\nk\u0011\u0011\n+¯W2\u0010\nL\u0010\nxk, q[1]\nk\u0011\n,L\u0010\nxk,eq[1]\nk\u0011\u0011\n+¯W2\u0010\nL\u0010\nxk,eq[1]\nk\u0011\n,˜µ\u0011k→∞−→0.\nNote that the last step holds since ¯W2\u0010\nL\u0010\nxk, q[1]\nk\u0011\n,L\u0010\nxk,eq[1]\nk\u0011\u0011k→∞−→0by equation (20). We thus have\n¯W2(¯µ,eµ) = 0 ,which implies the uniqueness of the limit ¯µ.\nMoreover, we will show that the unique limit distribution µis also a stationary distribution for the Markov\nchain (xk, qk)k≥0, as stated in the following lemma.\nLemma 1. Let(xk, qk)k≥0and(x′\nk, q′\nk)k≥0be two trajectories of Q-learning iterates, where L(x0, q0)=¯µ\nandL(x′\n0, q′\n0)∈ P2(X ×Rd)is arbitrary. Under Assumption 1 we have\n¯W2\n2(L(x1, q1),L(x′\n1, q′\n1))≤ρ¯W2\n2(L(x0, q0),L(x′\n0, q′\n0)),\nwhere the quantity ρ:=max\u0000\n1 + 2( αRmax+αγqmax)2,2(1 + αγ)2\u0001\nis independent of L(x′\n0, q′\n0). In particular,\nfor any k≥0, if we set L(x′\n0, q′\n0) =L(xk, qk), then\n¯W2\n2(L(x1, q1),L(xk+1, qk+1))≤ρ¯W2\n2(¯µ,L(xk, qk)).\nProof [Proof of Lemma 1] We prove this lemma by coupling the two processes (xk, qk)k≥0and(x′\nk, q′\nk)k≥0\nsuch that¯W2\n2(L(x0, q0),L(x′\n0, q′\n0)) =E\u0002\nd0(x0, x′\n0) +∥q0−q′\n0∥2\n∞\u0003\nand\nxk+1=x′\nk+1ifxk=x′\nk,∀k≥0.\nSince ¯W2is defined by infimum over all couplings, we have\n¯W2\n2(L(x1, q1),L(x′\n1, q′\n1))≤E\u0002\nd0(x1, x′\n1) +∥q1−q′\n1∥2\n∞\u0003\n.\nRecall the definition of the discrete metric d0(x′\n0, x0) :=1{x′\n0̸=x0}. We denote by e(s,a)∈R|S∥A|the\none-hot vector with only one “1” in the location of (s, a). We then have\n∥q1−q′\n1∥∞=∥q0−q′\n0−αe(s0,a0)q0(s0, a0) +αe(s′\n0,a′\n0)q′\n0(s′\n0, a′\n0)\n+αe(s0,a0)r(s0, a0)−αe(s′\n0,a′\n0)r(s′\n0, a′\n0)\n+αγe(s0,a0)max\naq0(s1, a)−αγe(s′\n0,a′\n0)max\naq′\n0(s′\n1, a)∥∞\n≤ ∥q0−q′\n0−αe(s0,a0)q0(s0, a0) +αe(s′\n0,a′\n0)q′\n0(s′\n0, a′\n0)∥∞\n+α∥e(s0,a0)r(s0, a0)−e(s′\n0,a′\n0)r(s′\n0, a′\n0)∥∞\n+αγ∥e(s0,a0)max\naq0(s1, a)−e(s′\n0,a′\n0)max\naq′\n0(s′\n1, a)∥∞\n≤ ∥q0−q′\n0∥∞+αd0(x′\n0, x0)qmax+αrmaxd0(x′\n0, x0) +αγ∥q0−q′\n0∥∞+αγqmaxd0(x′\n0, x0)\n= (1 + αγ)∥q0−q′\n0∥∞+ (αrmax+α(γ+ 1)qmax)d0(x′\n0, x0).\nTherefore, we obtain\nE\u0002\nd0(x1, x′\n1) +∥q1−q′\n1∥2\n∞\u0003\n=E[d0(x1, x′\n1)] +E\u0002\n∥q1−q′\n1∥2\n∞\u0003\n≤E[d0(x0, x′\n0)] + 2(1 + αγ)2E\u0002\n∥q0−q′\n0∥2\n∞\u0003\n+ 2(αrmax+α(γ+ 1)qmax)2E[d0(x0, x′\n0)]\n≤ρ¯W2\n2(L(x0, q0),L(x′\n0, q′\n0)),\n21with ρ= max\u0000\n1 + 2( αrmax+α(γ+ 1)qmax)2,2(1 + αγ)2\u0001\n.\nBy the triangle inequality of extended Wasserstein 2-distance, we obtain\n¯W2(L(x1, q1),¯µ)≤¯W2(L(x1, q1),L(xk+1, qk+1)) + ¯W2(L(xk+1, qk+1),¯µ)\n≤ρ¯W2\n2(¯µ,L(xk, qk)) + ¯W2(L(xk+1, qk+1),¯µ)\nk→∞−→0,(22)\nwhere the second inequality holds by Lemma 1 and last step comes from the weak convergence result.\nTherefore, we have proved that (xk, qk)k≥0converge to a unique stationary distribution ¯µ.\nNext, we restate a variant of Theorem 3.1 in [ 27] as follows without the assumption that rmax≤1, which\ncan be proved by Theorem 2.1 and 3.1 in [27].\nTheorem 4 (Theorem 3.1 in [ 27]).Under Assumption 1, and αtα≤c0(1−β)2\nlog(|S∥A| )(c0is a constant), for all\nk≥tα, we obtain\nEh\n∥qk−q∗∥2\n∞i\n≤cQ,1\u0012\n1−(1−β)α\n2\u0013k−tα\n+cQlog(|S∥A| )\n(1−β)2αtα, (23)\nwhere cQ,1= 3\u0000\n∥q0−q∗∥∞+∥q0∥∞+rmax\n3\u00012andcQ= 912 e(3∥q∗∥∞+rmax).\nWe remark that c0is the same numerical constant as cQ,0appearing in Theorem 3.1 in [27].\nFinally, we establish the following lemma to bound the variance of the limit random vector q∞,Var (q∞).\nLemma 2. Under Assumption 1, and αtα≤c0(1−β)2\nlog(|S∥A| )(c0is a constant), we obtain\nVar (q∞)≤cQlog(|S∥A| )\n(1−β)2αtα\nand\n(E[∥q∞∥∞])2≤E[∥q∞∥2\n∞]≤2cQc0+ 2∥q∗∥2,\nwhere cQ= 912 e(3∥q∗∥∞+rmax).\nProof [Proof for Lemma 2] We have shown that the sequence (qk)k≥0converges weakly to q∞inP2(Rd).\nIt is well known that weak convergence in P2(Rd)is equivalent to convergence in distribution and the\nconvergence of the first two moments. As a result, we have\nE\u0002\n∥q∞−q∗∥2\n∞\u0003\n= lim\nk→∞E\u0002\n∥qk−q∗∥2\n∞\u0003\n. (24)\nTaking k→ ∞on the both sides of equation (23) and combining with equation 24 yields\nE[∥q∞−q∗∥2\n∞]≤cQlog(|S∥A| )\n(1−β)2αtα.\nNote that q∗is a deterministic quantity. We thus have\nVar (q∞)(i)\n≤max\ns,aVar (q∞(s, a))≤E[∥q∞−q∗∥2\n∞]≤cQlog(|S∥A| )\n(1−β)2αtα,\nwhere the inequality (i) means an upper bound on elementwise ℓ∞norm for the covariance matrix Var (q∞).\n22In addition, we have\n(E[∥q∞∥∞])2≤E[∥q∞∥2\n∞]\n≤E[(∥q∞−q∗∥∞+∥q∗∥∞)2]\n≤2E(∥q∞−q∗∥2\n∞) + 2∥q∗∥2\n∞\n≤2cQlog(|S∥A| )\n(1−β)2αtα+ 2∥q∗∥2\n∞\n≤2cQc0+ 2∥q∗∥2\n∞.\nTherefore, we have proved parts 1 and 2 of Theorem 1.\nB.3 Convergence Rate\nSo far we have established that the Markov chain (xk, qk)k≥0converges to a unique stationary distribution\n¯µ∈ P2(X ×R|S||A|). As a result, (qk)k≥0converges weakly to µ∈ P2(R|S||A|), where µis the second marginal\nof¯µoverR|S||A|. We next focus on the convergence rate of (qk)k≥0.\nLet us consider the coupled processes defined as equation (19)in Section B.1. Suppose that the initial\niterate (x0, q[2]\n0)follows the stationary distribution ¯µ, thus L(xk, q[2]\nk) =¯µandL(q[2]\nk) =µfor all k≥0. By\nequation (20), we have for all k≥0 :\nW2\n2\u0010\nL(q[1]\nk), µ\u0011\n=W2\n2\u0010\nL(q[1]\nk),L(q[2]\nk)\u0011\n≤¯W2\n2\u0010\nL(xk, q[1]\nk),L(xk, q[2]\nk)\u0011\n≤12Eh\n∥q[1]\n0−q[2]\n0∥2\n∞i\u0012\n1−(1−β)α\n2\u0013k−tα\n≤24\u0012\n1−(1−β)α\n2\u0013k−tα\n·\u0010\nEh\n∥q[1]\n0∥2\n∞i\n+Eh\n∥q[1]\n∞∥2\n∞i\u0011\n.(25)\nHere the last step follows from the fact that (x0, q[2]\n0)follows the stationary distribution, and thus\nEh\n∥q[2]\n0∥2\n∞i\n=Eh\n∥q[2]\n∞∥2\n∞i\n=Eh\n∥q[1]\n∞∥2\n∞i\n.\nWe have completed the proof of Theorem 1.\nB.4 Proof of Proposition 1\nTo analyze the convergence rate of wk, we construct two new sequences {wk}k≥0and{¯wk}k≥0that satisfy\nthe following recursion:\nwk+1(sk, ak) = (1 −α)wk(sk, ak) +αγ\u0010\nmin\na′wk(sk+1, a′)\u0011\n,\n¯wk+1(sk, ak) = (1 −α) ¯wk(sk, ak) +αγ\u0010\nmax\na′¯wk(sk+1, a′)\u0011\n.\nLetw0=w0=¯w0. We then prove that wkand ¯wkprovide a lower bound and upper bound for wk,\nrespectively.\nLemma 3. For all k≥0and all (s, a)∈ S × A,wk(s, a)≤wk(s, a)≤¯wk(s, a).\n23Proof [Proof of Lemma 3] We use an inductive argument to prove this lemma.\nFork= 0,w0=w0= ¯w0by definition.\nNow assume for k=k0,wk0≤wk0≤¯wk0. For k=k0+ 1, we consider the following two cases:\nFor(s, a)̸= (sk0, ak0), we have\nwk0+1(s, a) =wk0(s, a)≤wk0(s, a) =wk0+1(s, a)≤¯wk0(s, a) = ¯wk0+1(s, a).\nFor(s, a) = (sk0, ak0), we have\nwk0+1(s, a) = (1 −α)wk0(s, a) +αγ\u0010\nmax\na′q[1]\nk0(sk0+1, a′)−max\na′q[2]\nk0(sk0+1, a′)\u0011\n≤(1−α)wk0(s, a) +αγmax\na′\u0010\nq[1]\nk0(sk0+1, a′)−q[2]\nk0(sk0+1, a′)\u0011\n= (1−α)wk0(s, a) +αγmax\na′(wk0(sk0+1, a′))\n≤(1−α) ¯wk0(s, a) +αγmax\na′( ¯wk0(sk0+1, a′)) = ¯wk0+1(s, a).\nwk0+1(s, a) = (1 −α)wk0(s, a) +αγ\u0010\nmax\na′q[1]\nk0(xk0+1, a′)−max\na′q[2]\nk0(sk0+1, a′)\u0011\n≥(1−α)wk0(s, a) +αγmin\na′\u0010\nq[1]\nk0(sk0+1, a′)−q[2]\nk0(sk0+1, a′)\u0011\n= (1−α)wk0(s, a) +αγmin\na′(wk0(sk0+1, a′))\n≥(1−α)wk0(s, a) +αγmin\na′\u0000\nwk0(sk0+1, a′)\u0001\n=wk0+1(s, a).\nBy induction, we complete the proof of Lemma 3.\nNotice that {−wk}and{¯wk}can be viewed as the iterates generated by the Q-learning algorithm with\nr(s, a) = 0for all (s, a). Then, for both {−wk}and{¯wk}, we obtain the following bound for the second\nmoment of wkand¯wkby Theorem 4 with the special case of q∗= 0andrmax= 0.\nEh\n∥wk∥2\n∞i\n≤12E\u0002\n∥w0∥2\n∞\u0003\u0012\n1−(1−β)α\n2\u0013k−tα\n,\nEh\n∥¯wk∥2\n∞i\n≤12E\u0002\n∥w0∥2\n∞\u0003\u0012\n1−(1−β)α\n2\u0013k−tα\n.\nBy Lemma 3, the same bound can also be applied to Eh\n∥wk∥2\n∞i\n.We thus have\nEh\n∥wk∥2\n∞i\n≤12E\u0002\n∥w0∥2\n∞\u0003\u0012\n1−(1−β)α\n2\u0013k−tα\n.\nB.5 Proof of Corollary 1\nLemma 2 states that the second moment of q∞is bounded by a constant, which is E\u0002\n∥q∞∥2\n∞\u0003\n=O(1).\nCombining this bound with equation (9) in Theorem 1, we obtain\nW2\n2(L(qk), µ)≤C(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013k−tα\n,\nwhere C(r, γ, P )is a numerical constant that only depends on the reward function r, discounted factor γ,\nand stationary distribution for Markov chain (xk)k≥0.\n24By [108, Theorem 4.1], there exists a coupling between qkandq∞such that\nW2\n2(L(qk), µ) =E[∥qk−q∞∥2\n∞].\nBy the above bounds and applying Jensen’s inequality twice, we obtain that\n∥E[qk−q∞]∥2\n∞≤(E[∥qk−q∞∥∞])2\n≤E[∥qk−q∞∥2\n∞]\n≤C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k−tα\n.\nWe thus have for all k≥tα,\n∥E[qk]−E[q∞]∥∞≤E[∥qk−q∞∥∞]≤C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k−tα\n2\n.\nFor the second moment, we notice that\n\r\rE\u0002\nqkq⊤\nk\u0003\n−E\u0002\nq∞q⊤\n∞\u0003\r\r\n∞\n=\r\r\rEh\n(qk−q∞+q∞) (qk−q∞+q∞)⊤i\n−E\u0002\nq∞q⊤\n∞\u0003\r\r\r\n∞\n=\r\r\rEh\n(qk−q∞) (qk−q∞)⊤i\n+Eh\nq∞(qk−q∞)⊤i\n+E\u0002\n(qk−q∞)q⊤\n∞\u0003\r\r\r\n∞\n≤\r\r\rEh\n(qk−q∞) (qk−q∞)⊤i\r\r\r\n∞+\r\r\rEh\nq∞(qk−q∞)⊤i\r\r\r\n∞+\r\rE\u0002\n(qk−q∞)q⊤\n∞\u0003\r\r\n∞\n≤Eh\r\r\r(qk−q∞) (qk−q∞)⊤\r\r\r\n∞i\n+Eh\r\r\rq∞(qk−q∞)⊤\r\r\r\n∞i\n+E\u0002\r\r(qk−q∞)q⊤\n∞\r\r\n∞\u0003\n≤Eh\n∥qk−q∞∥2\n∞i\n+ 2E\u0002\r\rq⊤\n∞(qk−q∞)\r\r\n∞\u0003\n≤Eh\n∥qk−q∞∥2\n∞i\n+ 2\u0010\nEh\n∥qk−q∞∥2\n∞i\nEh\n∥q∞∥2\n∞i\u00111/2\n.(26)\nMeanwhile, we have\nEh\n∥qk−q∞∥2\n∞i\n≤C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k−tα\nand Eh\n∥q∞∥2\n∞i\n=O(1).\nSubstituting the above bounds into the right-hand side of inequality (26) yields\n\r\rE\u0002\nqkq⊤\nk\u0003\n−E\u0002\nq∞q⊤\n∞\u0003\r\r\n∞≤C′(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k−tα\n2\n,\nthereby completing the proof for Corollary 1.\nC Proof of Theorem 2\nDefine f:X ×R|S||A|→R|S||A|, such that f(x, q) :=q−E(q∞). Consider {(xk, qk)}k≥0with x0∼µXand\nq0∼¯µ(· |x0).\n25\r\r\r\r\rn−1X\nk=0Pkf\r\r\r\r\r\n∞,L2\n¯µ=vuutE(x0,q0)∼¯µ∥n−1X\nk=0E[f(xk, qk)|x0, q0]∥2∞\n=vuutE(x0,q0)∼¯µ∥n−1X\nk=0E[qk|x0, q0]−nE(q∞)∥2∞\n≤vuutE(x0,q0)∼¯µ∥n−1X\nk=0E[qk|x0, q0]−nE(q∞)∥2\n2\n=vuutE(x0,q0)∼¯µn−1X\ni,j=0E[qi−E(q∞)|x0, q0]TE[qj−E(q∞)|x0, q0].\nDefine gk(x, q) :=E[qk−E(q∞)|(x0, q0) = ( x, q)], we then give the following Lemma 4 to uniformly\nbound gk(x, q)for all (x, q)∈ X × R|S||A|. The proof of Lemma 4 is given at section C.1.\nLemma 4. For all (x, q)∈ X × R|S||A|, when k≥tα, there exist two constant λ0, λ1such that\n∥gk(x, q)∥2≤λ0·λk\n1,\nwhere λ0>0and0< λ1<1.\nBy Lemma 4, we obtain\n\r\r\r\r\rn−1X\nk=0Pkf\r\r\r\r\r\n∞,L2\n¯µ≤vuutE(x0,q0)∼¯µn−1X\ni,j=0E[qi−E(q∞)|x0, q0]TE[qj−E(q∞)|x0, q0]\n≤vuutE(x0,q0)∼¯µn−1X\ni,j=0∥gi(x0, q0)∥2∥gj(x0, q0)∥2\n≤s\nλ2\n0\n(1−λ1)2=O(1).\nBy Lemma 2, we can observe thatR\n∥f(x, q)∥2\n∞¯µ(d(x, q))<∞andR\nf(x)¯µ(d(x, q)) =0. Therefore, by\nTheorem 2.1 in [114], we complete the proof for Theorem 2.\nC.1 Proof of Lemma 4\nRecall that the Markov chain {xk}k≥0mixes geometrically fast to the stationary distribution µX, and there\nexist c≥0andρ∈(0,1)s.t.\nmax\nx∈X∥pk(x,·)−µX(·)∥TV≤cρk,\n26When k≥tα, we have\ngk(x, q) =X\nx′∈XZ\nq′∈R|S||A|E[qk−E(q∞)|(x⌊k\n2⌋, q⌊k\n2⌋) = (x, q)]dP\u0010\n(x⌊k\n2⌋, q⌊k\n2⌋) = (x′, q′)|(x0, q0) = (x, q)\u0011\n=X\nx′∈XZ\nq′∈R|S||A|gk−⌊k\n2⌋(x′, q′)dP\u0010\n(x⌊k\n2⌋, q⌊k\n2⌋) = (x′, q′)|(x0, q0) = (x, q)\u0011\n=X\nx′∈XZ\nq′∈R|S||A|gk−⌊k\n2⌋(x′, q′)P\u0010\nx⌊k\n2⌋=x′|(x0, q0) = (x, q)\u0011\n| {z }\np(x′)dP\u0010\nq⌊k\n2⌋=q′|x⌊k\n2⌋=x′,(x0, q0) = (x, q)\u0011\n| {z }\nη(q′|x′)\n=X\nx′∈XZ\nq′∈R|S||A|gk−⌊k\n2⌋(x′, q′)µX(x′)dη(q′|x′)\n| {z }\nT1+X\nx′∈XZ\nq′∈R|S||A|gk−⌊k\n2⌋(x′, q′)(p(x′)−µX(x′))dη(q′|x′)\n| {z }\nT2.\nBy Corollary 1, when x0∼µX, for all k≥tαand arbitrary q0we have\n∥E[qk]−E[q∞]∥∞≤C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k−tα\n2\n.\nTherefore, we obtain\n∥T1∥2≤p\n|S||A|∥ T1∥∞\n=p\n|S||A|∥ Ex′∼µX,q′∼η(q′|x′)gk−⌊k\n2⌋∥∞\n≤p\n|S||A| C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k−⌊k\n2⌋−tα\n2\n≤ \np\n|S||A| C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013−tα\n2!\u0012\n1−(1−β)α\n2\u0013k\n2\n.\nNote that ∥qk∥∞≤qmax,∥gk(x, q)∥∞≤2qmax,we have\n∥T2∥2≤p\n|S||A|∥ T2∥∞≤p\n|S||A| cρ⌊k\n2⌋|S|2|A|=|S|5\n2|A|3\n2cρ⌊k\n2⌋≤ |S|5\n2|A|3\n2cρ−1ρk\n2.\nTherefore, we have\n∥gk(x, q)∥2=∥T1+T2∥2\n≤ ∥T1∥2+∥T2∥2\n≤  \np\n|S||A| C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013−tα\n2!\n+|S|5\n2|A|3\n2cρ−1! \nmax(s\u0012\n1−(1−β)α\n2\u0013\n,√ρ)!k\n.\nLetλ0=\u0012\u0012p\n|S||A| C(r, γ, P )\u0010\n1−(1−β)α\n2\u0011−tα\n2\u0013\n+|S|5\n2|A|3\n2cρ−1\u0013\nandλ1=max\u001ar\u0010\n1−(1−β)α\n2\u0011\n,√ρ\u001b\n,\nwe complete the proof of Lemma 4.\nD Proof of Theorem 3\nIn this section, we prove Theorem 3 on the characterization of the bias E(q∞)−q∗. The proof consists of five\nsteps, which are given in the following five sub-sections.\n27D.1 Step 1: Local linearization of Operator F\nUnlike linear SA, the operator Fin the update rule of Q-learning (cf. equation (4)) is nonlinear and\nnonsmooth, which makes the analysis considerably more challenging. To address this issue, we employ the\nlocal linearization of the operator Faround the optimal solution q∗,with a higher order remaining term as\nstated in Proposition 2 and 3. We provide complete proof here.\nProof [Proof of Proposition 2] Recall that we define the unique optimal action with respect to the optimal\nQ-function q∗as\na∗\ns:=argmax\naq∗(s, a).\nWe define a function Gq∗:X →R|S||A|×|S||A|as follows: for each x= (s0, a0, s1)∈ X,\n[Gq∗(x)] [(s, a),(¯s,¯a)] =\n\n1,(s, a) = (¯s,¯a)̸= (s0, a0)\nγ,(s, a) = (s0, a0),(¯s,¯a) = (s1, a∗\ns1)\n0,otherwise .\nNote that the operator F(x,·)is nonsmooth and does not admit any gradient. On the other hand, by the\nuniqueness of the optimal policy π∗,we can locally linearize F(x,·)around q∗. In particular, Gq∗(x)−Id\nserves as an approximate \"gradient\" of the operator F(x,·)around q∗. Define\nR(x, q) =F(x, q)−F(x, q∗)−(Gq∗(x)−Id)(q−q∗).\nWe can observe that for ∀(s, a)̸= (s0, a0),[R(x, q)] (s, a) = 0 .For(s, a) = (s0, a0), we have\n[R(x, q)] (s0, a0) =γ\u0010\nmax\naq(s1, a)−q(s1, a∗\ns1)\u0011\n≥0.\nIf∥q−q∗∥∞<∆, by Assumption 2, for any action a̸=a∗\ns1, we have\nq(s1, a∗\ns1)> q∗(s1, a∗\ns1)−δ\n≥q∗(s1, a) +δ\n> q(s1, a).\nThus,\n[R(x, q)] (s0, a0) =γ\u0010\nmax\naq(s1, a)−q(s1, a∗\ns1)\u0011\n= 0.\nIf∥q−q∗∥∞≥∆, we have\n|[R(x, q)] (s0, a0)|=γ|max\naq(s1, a)−q(s1, a∗\ns1)|\n=γ|max\naq(s1, a)−max\naq∗(s1, a) +q∗(s1, a∗\ns1)−q(s1, a∗\ns1)|\n≤2γ∥q−q∗∥∞\n≤2γ\n∆3∥q−q∗∥4\n∞.\nCombining the two situations considered above, we finally obtain that\n∥R(x, q)∥∞≤2γ\n∆3∥q−q∗∥4\n∞.\nwhich proves the first part of Proposition 2.\n28For the second part, we can multiply the Gq∗(x)by an arbitrary nonzero vector H∈R|S∥A|. Let\n(sh, ah) =arg max\n(s,a)∈S×AH(s, a)andph=µS(sh, ah). By Assumption 1, ph>0. Without loss of generality,\nwe can assume H(sh, ah)>0, otherwise we can replace Hwith−H. We then have\nE\u0002\nG′\nq∗(x)H\u0003\n(sh, ah) =γphE\u0000\nH(s1, a∗\ns1)|s0=sh, a0=ah\u0001\n+ (1−ph)H(sh, ah)\n≤γphH(sh, ah) + (1 −ph)H(sh, ah)\n< H(sh, ah),\nwhere the second step hold as the definition of (sh, ah)uses the maximum.\nWe thus have\nE[Gq∗(x)]H=E[Gq∗(x)H]̸=H.\nAsHis an arbitrary vector, we conclude that E(Gq∗(x))does not have an eigenvalue of 1, thereby completing\nthe proof for Proposition 2.\nProof [Proof of Proposition 3:]\nLetf(z) =1\n2∥z∥2\n∞andg(z) =1\n2∥z∥2\n2. Note that g(·)is a convex, differentiable, and 1-smooth function. In\nProposition 3, we work with a finite demensional space R|S∥A|. By Cauchy-Schwarz Inequality,1√\n|S∥A|∥·∥2≤\n∥ · ∥∞≤ ∥ · ∥ 2. We construct the Generalized Moreau Envelope of f(·)with respect to g(·)as follows:\nMη,g\nf(z) = min\nu∈R|S∥A|\u001a\nf(u) +1\nηg(z−u)\u001b\n,\nwhere η >0.For the ease of exposition, we use M(·)to denote Mη,g\nf(·). We restate Lemma 2.1 in [ 24] below\non the properties of M(·).\nLemma 5 (Lemma 2.1 in [ 24]).For given η >0. Then M(·)constructed above has the following properties:\n1. (Smoothness) M(·)is convex,1\nη-smooth with respect to ∥ · ∥ 2.\n2.There exists a norm ∥ · ∥ msuch that M(z) =1\n2∥z∥2\nm. Furthermore, there exist lm, um>0, such that\nlm∥ · ∥m≤ ∥ · ∥ ∞≤um∥ · ∥m. Specifically, we can let lm= (1 +η\n|S∥A|)1/2,um= (1 + η)1/2.\nTherefore, M(·)serves as a smooth approximation of the non-smooth function f(·).Then, we have for\n∀k≥0 :\n29M2(qk+1−q∗)\n(a)\n≤\u0012\nM(qk−q∗) +⟨∇M(qk−q∗), qk+1−qk⟩+1\n2η∥qk+1−qk∥2\n2\u00132\n(b)=\u0012\nM(qk−q∗) +α⟨∇M(qk−q∗), F(xk, qk)⟩+α2\n2η∥F(xk, qk)∥2\n2\u00132\n= (M(qk−q∗) +α⟨∇M(qk−q∗),¯F(qk)⟩\n+α⟨∇M(qk−q∗), F(xk, qk)−¯F(qk)⟩+α2\n2η∥F(xk, qk)∥2\n2)2\n(c)\n≤M2(qk−q∗) + 2αM(qk−q∗)⟨∇M(qk−q∗),¯F(qk)⟩| {z }\nT1\n+ 2αM(qk−q∗)⟨∇M(qk−q∗), F(xk, qk)−¯F(qk)⟩| {z }\nT2+α2\nηM(qk−q∗)∥F(xk, qk)∥2\n2\n| {z }\nT3\n+ 3α2⟨∇M(qk−q∗),¯F(qk)⟩2\n| {z }\nT4+3α2⟨∇M(qk−q∗), F(xk, qk)−¯F(qk)⟩2\n| {z }\nT5\n+3α4\n4η2∥F(xk, qk)∥4\n2\n| {z }\nT6,(27)\nwhere (a)follows from the smoothness of M(·)in Lemma 5, (b)follows from the update rule of qkin(4), and\n(c)holds by the inequality (x+y+z)2≤3(x2+y2+z2).\nNext we derive an upper bound on M(·)2by bounding each term of T1−T6.\nLemma 6. For all k≥0,T1≤ −4α(1−γ)2M2(qk−q∗).\nProof [Proof of Lemma 6] By Proposition 2.1 in [24], we have that\n⟨∇M(qk−q∗),¯F(qk)⟩ ≤ − 2\n1−γ \n1 +ηp\n|S∥A|\n1 +η!1\n2\nM(qk−q∗).\nWe can always choose a sufficiently small ηsuch that\u0012\n1+η√\n|S∥A|\n1+η\u00131\n2\n≤2−γbecause γ < 1, which is\nequivalent to\nη≤(2−γ)2−1p\n|S∥A| − 1. (28)\nSince M(·)is non-negative, we complete the proof by multiplying 2αM(qk−q∗)on both sides.\nBy Lemma 6, T1can give us a desired negative drift term of order −O(α).\nBy Cauchy-Schwarz Inequality, we can bound T2by two terms. One term is proportional to M2(qk−q∗)\nbut still keep the negative drift generated by T1and the other term is proportional to T5:\nT2≤α(1−γ)2M2(qk−q∗) +α(1−γ)−2⟨∇M(qk−q∗), F(xk, qk)−¯F(qk)⟩2\n=α(1−γ)2M2(qk−q∗) +α(1−γ)−2T5.\n30Then, we can simplify equation (27) as follows when 3α≤(1−γ)−2:\nM2(qk+1−q∗)≤(1−3α(1−γ)2)M2(qk−q∗) +T3+T4+ 2α(1−γ)−2T5+T6.\nBy Cauchy-Schwarz Inequality and Lemma A.5 in [27], T3can be bounded as follow\nT3=α2\nηM(qk−q∗)∥F(xk, qk)∥2\n2\n≤α2\nηM(qk−q∗)\u0000\n36u2\nm|S∥A| M(qk−q∗) + 2|S∥A| (3∥q∗∥∞+rmax)2\u0001\n=36u2\nm|S∥A| α2\nηM2(qk−q∗) +2|S∥A| α2\nηM(qk−q∗)(3∥q∗∥∞+rmax)2\n≤36u2\nm|S∥A| α2\nηM2(qk−q∗) +α(1−γ)2M2(qk−q∗) +|S|2|A|2α3\n(1−γ)2η2(3∥q∗∥∞+rmax)4.\nThe term T4can be directly bounded as follow:\nT4= 3α2⟨∇M(qk−q∗),¯F(qk)⟩2\n≤3α2\u0000\n∥∇M(qk−q∗)∥2∥¯F(qk)∥2\u00012\n= 3α2\u0000\n∥∇M(qk−q∗)− ∇M(q∗−q∗)∥2∥¯F(qk)−¯F(q∗)∥2\u00012\n≤3α2 p\n|S∥A|\nη∥qk−q∗∥2∥¯F(qk)−¯F(q∗)∥∞!2\n≤3α2\u00122\nη|S∥A|∥ qk−q∗∥2\n∞\u00132\n≤12u4\nm|S|2|A|2α2\nη2∥qk−q∗∥4\nm\n=48u4\nm|S|2|A|2α2\nη2M2(qk−q∗).\nBy Cauchy-Schwarz Inequality, we bound T5by the following three parts:\nT5≤3⟨∇M(qk−q∗)− ∇M(qk−tα2−q∗), F(xk, qk)−¯F(qk)⟩2\n| {z }\nT51\n+ 3⟨∇M(qk−tα2−q∗), F(xk, qk)−F(xk, qk−tα2) +¯F(qk−tα2)−¯F(qk)⟩2\n| {z }\nT52\n+ 3⟨∇M(qk−tα2−q∗), F(xk, qk−tα2)−¯F(qk−tα2)⟩2\n| {z }\nT53.\nBy Lemma A.3 in [27], for all k≥tα2with αsatisfying αtα2≤1\n12:\nT51≤3\u0012144u2\nm|S∥A| αtα2\nηM(qk−q∗) +8|S∥A| αtα2\nη(3∥q∗∥∞+rmax)2\u00132\n≤124416 u4\nm|S|2|A|2α2t2\nα2\nη2M2(qk−q∗) +384|S|2|A|2α2t2\nα2\nη2(3∥q∗∥∞+rmax)4,\nT52≤3\u0012576u2\nm|S∥A| αtα2\nηM(qk−q∗) +32|S∥A| αtα2\nη(3∥q∗∥∞+rmax)2\u00132\n≤1990656 u4\nm|S|2|A|2α2t2\nα2\nη2M2(qk−q∗) +6144|S|2|A|2α2t2\nα2\nη2(3∥q∗∥∞+rmax)4.\n31For term T53, we use the conditional expectation as follow:\nE[T53|xk−tα2, qk−tα2]\n=3E\u0002\n⟨∇M(qk−tα2−q∗), F(xk, qk−tα2)−¯F(qk−tα2)⟩2|xk−tα2, qk−tα2\u0003 (29)\nLetH=∇M(qk−tα2−q∗)· ∇M(qk−tα2−q∗)⊤. Equation (29) can be reformulated as follows:\nE[T53|xk−tα2, qk−tα2]\n= 3E\u0002\n(F(xk, qk−tα2)−¯F(qk−tα2))⊤H(F(xk, qk−tα2)−¯F(qk−tα2))|xk−tα2, qk−tα2\u0003\n= 3E\u0002\nF(xk, qk−tα2)���HF(xk, qk−tα2)−¯F(qk−tα2)⊤H¯F(qk−tα2)|xk−tα2, qk−tα2\u0003\n−6E\u0002\n(F(xk, qk−tα2)−¯F(qk−tα2))⊤H¯F(qk−tα2)|xk−tα2, qk−tα2\u0003\n= 3 X\nx∈X\u0000\nPtα2\u0000\nxk−tα2, x\u0001\n−µX(x)\u0001\nF\u0000\nx, qk−tα2\u0001⊤HF\u0000\nx, qk−tα2\u0001!\n−6 X\nx∈X\u0000\nPtα2\u0000\nxk−tα2, x\u0001\n−µX(x)\u0001\nF\u0000\nx, qk−tα2\u0001⊤H¯F\u0000\nqk−tα2\u0001!\n(a)\n≤6α2∥F(fx0, qk−tα2)∥2\n2∥H∥2+ 12α2∥F(fx1, qk−tα2)∥2∥H∥2∥¯F(qk−tα2)∥2\n≤18α2|S|2|A|2\nη2(2∥qk−tα2∥∞+rmax)2∥qk−tα2−q∗∥2\n∞\n≤18α2|S|2|A|2\nη2(2∥qk−tα2−q∗∥∞+ 2∥q∗∥∞+rmax)2∥qk−tα2−q∗∥2\n∞\n≤18α2|S|2|A|2\nη2(2∥qk−tα2−qk∥∞+ 2∥qk−q∗∥∞+ 2∥q∗∥∞+rmax)2·(∥qk−tα2−qk∥∞+∥qk−q∗∥∞)2\n≤18α2|S|2|A|2\nη2(2(∥qk∥∞+rmax\n3) + 2∥qk−q∗∥∞+ 2∥q∗∥∞+rmax)2·((∥qk∥∞+rmax\n3) +∥qk−q∗∥∞)2\n≤18α2|S|2|A|2\nη2(6∥qk−q∗∥∞+ 6∥q∗∥∞+ 2rmax)2(3∥qk−q∗∥∞+ 3∥q∗∥∞+rmax)2\n=72α2|S|2|A|2\nη2(3∥qk−q∗∥∞+ 3∥q∗∥∞+rmax)4\n≤186624 α2|S|2|A|2\nη2M2(qk−q∗) +576α2|S|2|A|2\nη2(3∥q∗∥∞+rmax)4.\nwhere (a) follows with some fx0,fx1∈ X. Here we use the facts thatP\nx∈X|Ptα2\u0000\nxk−tα2, x\u0001\n−µX(x)| ≤2α2\n(by Definition 1 of mixing time) and ∥qk−tα2−qk∥∞≤ ∥qk∥∞+rmax\n3, which has been proved in Lemma A.2\nin [27].\nBy putting these three terms together, we obtain the following bound for E[T5]:\nE(T5)≤|S|2|A|2(2115072 u4\nmα2t2\nα2+ 186624 α2)\nη2M2(qk−q∗)\n+|S|2|A|2(6528 α2t2\nα2+ 576 α2)\nη2(3∥q∗∥∞+rmax)4.\nBy Lemma A.5 in [27], we have\nT6≤3α4\n4η2\u0000\n36u2\nm|S∥A| M(qk−q∗) + 2|S∥A| (3∥q∗∥∞+rmax)2\u00012\n≤1944u4\nm|S|2|A|2α4\nη2M2(qk−q∗) +6|S|2|A|2α4\nη2(3∥q∗∥∞+rmax)4.\n32Using the above bounds for T1−T6, we can finally bound E[M2(qk+1−q∗)]by following:\nE[M2(qk+1−q∗)]≤(1−3α(1−γ)2)E[M2(qk−q∗)]\n+36u2\nm|S∥A| α2\nηE[M2(qk−q∗)] +α(1−γ)2E[M2(qk−q∗)]\n+|S|2|A|2α3\n(1−γ)2η2(3∥q∗∥∞+rmax)2+48u4\nm|S|2|A|2α2\nη2E[M2(qk−q∗)]\n+|S|2|A|2(4230144 u4\nmα3t2\nα2+ 373248 α3)\nη2(1−γ)2E[M2(qk−q∗)]\n+|S|2|A|2(13056 α3t2\nα2+ 1152 α3)\nη2(1−γ)2(3∥q∗∥∞+rmax)4\n+1944u4\nm|S|2|A|2α4\nη2E[M2(qk−q∗)] +6|S|2|A|2α4\nη2(3∥q∗∥∞+rmax)4\n≤(1−α(1−γ)2)E[M2(qk−q∗)]\n+|S|2|A|2(374007 α3+ 13056 α3t2\nα2)\nη2(1−γ)2(3∥q∗∥∞+rmax)4,\nwhere there exists a α0>0such that the last step always hold for all α≤α0.\nThen, we obtain that for all k≥tα2:\nE[M2(qk−q∗)]≤E[M2(qtα2−q∗)](1−α(1−γ)2)k−tα2\n+|S|2|A|2(374007 α2+ 13056 α2t2\nα2)\nη2(1−γ)4(3∥q∗∥∞+rmax)4.\nWe can choose η=(1−γ)2√\n|S∥A|satisfying equation (28)and by [27, Theorem A.1], we obtain the following\nbound for E[∥qk−q∗∥4\n∞]:\nE[∥qk−q∗∥4\n∞]≤4u4\nmE[M2(qk−q∗)]\n≤4u4\nmE[M2(qtα2−q∗)](1−α(1−γ)2)k−tα2\n+ 4u4\nm|S|3|A|3(374007 α2+ 13056 α2t2\nα2)\n(1−γ)8(3∥q∗∥∞+rmax)4\n≤u4\nm\nl4mE[∥qtα2−q∗∥4\n∞](1−α(1−γ)2)k−tα2\n+4u4\nm|S|3|A|3(374007 α2+ 13056 α2t2\nα2)\n(1−γ)8(3∥q∗∥∞+rmax)4\n≤u4\nm\nl4mE((∥qtα2−q0∥∞+∥q0−q∗∥∞)4)(1−α(1−γ)2)k−tα2\n+4u4\nm|S|3|A|3(374007 α2+ 13056 α2t2\nα2)\n(1−γ)8(3∥q∗∥∞+rmax)4\n≤u4\nm\nl4m(∥q0∥∞+∥q0−q∗∥∞+rmax\n3)4(1−α(1−γ)2)k−tα2\n+4u4\nm|S|3|A|3(374007 α2+ 13056 α2t2\nα2)\n(1−γ)8(3∥q∗∥∞+rmax)4.\n33By Lemma 5, we can let lm= (1 +η\n|S∥A|)1/2,um= (1 + η)1/2. Define\nb1=(1 +(1−γ)2√\n|S∥A|)2\n(1 +(1−γ)2\n(|S∥A| )3\n2)2(∥q0∥∞+∥q0−q∗∥∞+rmax\n3)4,\nb2=374007 ×4(1 +(1−γ)2√\n(|S∥A| ))2|S|3|A|3\n(1−γ)8(3∥q∗∥∞+rmax)4,\nb3=13056 ×4(1 +(1−γ)2√\n(|S∥A| ))2|S|3|A|3\n(1−γ)8(3∥q∗∥∞+rmax)4.\nWe have for all k≥tα2,\nE[∥qk−q∗∥4\n∞]≤b1(1−α(1−γ)2)k−tα2+b2α2+b3α2t2\nα2.\nThis completes the proof of Proposition 3.\nD.2 Step 2: Basic Adjoint Relationship\nWe first derive a recursive relationship for the following quantities\nz(i) :=E[q∞|x∞=i], i∈ X.\nRecall that (xk)k≥0is a time-homogeneous Markov chain with transition probability matrix P= (pij)\nand a unique stationary distribution µX. Theorem 1 shows that (xk, qk)k≥0converges in distribution to a\nlimit (x∞, q∞)∼¯µ, with marginal q∞∼µandx∞∼µX. Given (x∞, q∞), letx∞+1be a random variable\nwith conditional distribution P(x∞+1=j|x∞=i) =pij, and q∞+1=q∞+αF(x∞, q∞).\nSince (x∞, q∞)is in the stationary, (x∞+1, q∞+1)also follows the stationary distribution ¯µ. Let d=|S∥A|.\nTherefore, for any test function f:X ×Rd7→Rdthat satisfies ∥f(x, q)∥∞≤C(1 +∥q∥2\n∞)for some C∈R,\nthe following relationship holds [108, Theorem 6.9]\nE[f(x∞, q∞)] =E[f(x∞+1, q∞+1)],\nwhich is called Basic Adjoint Relationship (BAR).\nConsider the test function f(i),i∈ X, defined as\nf(i)(x, q) =q·1{x=i}.\nSubstituting f=f(i)into BAR gives\nE[q∞·1{x∞=i}] =E[q∞+1·1{x∞+1=i}]. (30)\nTo simplify the presentation, we denote by ν(i) :=µX(i)the probability of the Markov chain (xk)k≥0being\nin state i∈ Xwhen in stationary. The LHS of equation (30) can be written as follows\nE[q∞·1{x∞=i}] =ν(i)·E[q∞|x∞=i] =ν(i)z(i).\n34Recall that ˆP= (ˆpij)is the transition kernel of the time-reversal of the Markov chain (xk)k≥0. The RHS\nof equation (30) can be reformulated as\nE[q∞+1·1{x∞+1=i}] =ν(i)E[q∞+1|x∞+1=i]\n=ν(i)E[q∞+αF(x∞, q∞)|x∞+1=i]\n=ν(i)X\nj∈XˆpijE[q∞+αF(x∞, q∞)|x∞=j, x∞+1=i]\n=ν(i)X\nj∈XˆpijE[q∞+αF(j, q∞)|x∞=j].\nThe last step follows from the fact that condition on xk, qkis conditionally independent of xk+1for all k≥1.\nBy Proposition 2, we can further rewrite the above equation as\nE[q∞+1·1{x∞+1=i}]\n=ν(i)X\nj∈XˆpijE[q∞+α(F(j, q∗) + (Gq∗(j)−Id)(q∞−q∗) +R(j, q∞))|x∞=j]\n=ν(i)X\nj∈Xˆpij[z(j) +α(F(j, q∗) + (Gq∗(j)−Id)(z(j)−q∗) +E(R(j, q∞)|x∞=j))].\nWe thus obtain the following recursive relationship for {z(i)}i∈X:\nz(i) =X\nj∈Xˆpij[z(j) +α(F(j, q∗) + (Gq∗(j)−Id)(z(j)−q∗) +E(R(j, q∞)|x∞=j))]\n=X\nj∈Xˆpij[z(j) +α(F(j, q∗) + (Gq∗(j)−Id)(z(j)−q∗))] + αE[R(x∞, q∞)|x∞+1=i].(31)\nNote that the second term of the RHS of equation (31) can be bounded as\nE[R(x∞, q∞)|x∞+1=i] =1\nν(i)E[R(x∞, q∞)1{x∞+1=i}]\n(i)\n≤1\nν(i)E[R(x∞, q∞)]\n(ii)=O(α2+α2t2\nα2),\nwhere (i) holds because R(x, q)is always positive, as shown in the proof of Proposition 2; (ii) follows from\nProposition 2, Proposition 3 and Hölder’s inequality.\nLetA(x) =Gq∗(x)−Idandb(x) =F(x, q∗)−(Gq∗(x)−Id)q∗. Let Ddenote the operator given by\n(Df)(x) =A(x)f(x)for each x∈ X. We thus can simplify equation (31) by\nz=ˆP(z+α(Dz+b)) +O(α3+α3t2\nα2). (32)\nD.3 Step 3: Setting up System of δ\nDefine the difference\nδ(i) :=z(i)−µXzfor each i∈ X,\nwhere µXz:=P\ni∈Xν(i)z(i). Let Π = 1 ⊗µX. Then, by applying the operator (ˆP−Π)to both side of above\nequation we obtain\n(P∗−Π)z= (P∗−Π)δ.\n35Subtracting Πzfrom both sides of equation (32), we obtain\nδ= (ˆP−Π)z+αˆP(Dz+b) +O(α3+α3t2\nα2)\n= (ˆP−Π)δ+αˆP(Dz+b) +O(α3+α3t2\nα2).(33)\nApplying µXto both sides of equation (32), we obtain\nµX(Dz+b) =O(α2+α2t2\nα2). (34)\nSubtracting equation (34) from equation (33), we obtain\nδ= (ˆP−Π)δ+α(ˆP−Π)(Dz+b) +O(α3+α3t2\nα2).\nThen, we have\n(I−ˆP+ Π)δ=α(ˆP−Π)(Dz+b) +O(α3+α3t2\nα2).\nIt is well-known that (I−ˆP+ Π)−1exist. Therefore, we obtain\nδ=α(I−ˆP+ Π)−1(ˆP−Π)(Dz+b) +O(α3+α3t2\nα2). (35)\nD.4 Step 4: Establishing δ=O(α)\nIn this sub-section, we show that ⃗δ=O(α), as stated in the following Lemma.\nLemma 7. Under Assumption 1, and αtα≤c0(1−β)2\nlog(|S∥A| ), we have\n∥⃗δ∥∞≤α·B′′(r, γ, P )\nfor some number B′′(r, γ, P )∈Rthat is independent of α.\nProof [Proof of Lemma 7] Recalling the definition for z(i), we have\nz(i) =E[q∞|x∞=i] =E[q∞1{x∞=i}]\nν(i).\nThen by Lemma 2 and the fact that ν(i)>0, we have\n∥z(i)∥∞≤E[∥q∞∥∞]\nν(i)≤1\nνmin·q\n2cQc0+ 2∥q∗∥2∞,\nwhere νmin:= min\niν(i)>0.\nBy equation (35), we conclude that\n∥⃗δ∥∞≤α·B′′(r, γ, P )\nfor some number B′′(r, γ, P )that is independent of α.\n36D.5 Step 5: Expansion of the bias\nBy definition, ¯F(q∗) = 0andR(x, q∗)≡0. Define ¯A=EµXA(x)and¯b=EµXb(x). Then, we have\n¯Aq∗+¯b= 0. From Proposition 2, ¯Ais a non-singular matrix. Define ¯Dbe the normalized Dsuch that\n(¯Df)(x) =¯A−1A(x)f(x). Therefore, we obtain\nq∗=−¯A−1¯b\n=−¯A−1µXb\n=µX¯Dz+O(α2+α2t2\nα2),\nwhere the last inequality holds by equation (34).\nBecause δ=z−Πz, we can further obtain\nq∗=µX¯Dδ+µXz+O(α2+α2t2\nα2).\nThen,\nµXz=q∗−µX¯Dδ+O(α2+α2t2\nα2).\nz(i) =δ(i) +µXz=δ(i) +q∗−µX¯Dδ+O(α2+α2t2\nα2).\nTherefore, we obtain\nz=q∗+ (I−Π¯D)δ+O(α2+α2t2\nα2). (36)\nSubstituting equation (36) into equation (35), we obtain\nδ=α(I−ˆP+ Π)−1(ˆP−Π)(Dz+b) +O(α3+α3t2\nα2)\n=α(I−ˆP+ Π)−1(ˆP−Π)(Aq∗+b)| {z }\nv\n+α(I−ˆP+ Π)−1(ˆP−Π)D(I−Π¯D)| {z }\nΞδ\n+O(α3+α3t2\nα2)\n=αv+αΞδ+O(α3+α3t2\nα2).\nTherefore, we can finally obtain\nE(q∞) =µXz\n=q∗−µX¯Dδ+O(α2+α2t2\nα2)\n=q∗−αµX¯Dv−αµX¯DΞδ+O(α2+α2t2\nα2)\nLetB=−µX¯Dv. By Lemma 7, we have µX¯DΞδ=O(α)\nTherefore, we have\nE(q∞) =q∗+αB+O(α2+α2t2\nα2)\nwith\nB=−µX¯D(I−ˆP+ Π)−1(ˆP−Π)(Aq∗+b). (37)\nWe complete the proof of Theorem 3.\n37E Proof of Corollary 2\nIn this section, we provide the proof of the first and second moment bounds in Corollary 2.\nE.1 First Moment\nFirst, we have\nE[¯qk0,k]−q∗= (E[q∞]−q∗) +1\nk−k0k−1X\nt=k0E[qt−q∞]\n|{z }\nT1.\nBy Corollary 1, we have that for k≥tα,\n∥E[qk]−E[q∞]∥∞≤C(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013k−tα\n2\n.\nThen, when αtα≤1, we have the following bound for T1,\n∥T1∥∞=\r\r\r\r\rk−1X\nt=k0E[qt−q∞]\r\r\r\r\r\n∞≤k−1X\nt=k0∥E[qt]−E[q∞]∥∞\n≤C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k0−tα\n2 1\n1−q\n1−(1−β)α\n2\n≤C(r, γ, P )\u0012\n1−(1−β)α\n2\u0013k0−tα\n2 4\n(1−β)α\n(i)\n≤C(r, γ, P ) exp\u0012\n−(1−β)α(k0−tα)\n4\u00134\n(1−β)α\n≤C′′(r, γ, P )·1\nα·exp\u0012\n−α(1−β)k0\n4\u0013\n,\nwhere (i)follows from the inequality that (1−u)m≤exp(−um)for0< u < 1.\nTogether with Theorem 3, we have\nE[¯qk0,k]−q∗=αB(r, γ, P ) +O(α2+α2t2\nα2) +O\u00121\nα(k−k0)exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n,\nthereby finishing the proof of equation (12) for the first moment.\n38E.2 Second Moment\nWe first derive the bound for the second moment of the tail-averaged iterate. Note that\nEh\n(¯qk0,k−E[q∞]) (¯qk0,k−E[q∞])⊤i\n=1\n(k−k0)2E\n k−1X\nt=k0(qt−E[q∞])! k−1X\nt=k0(qt−E[q∞])!⊤\n\n=1\n(k−k0)2k−1X\nt=k0Eh\n(qt−E[q∞]) (qt−E[q∞])⊤i\n| {z }\nT1\n+1\n(k−k0)2k−1X\nt=k0k−1X\nl=t+1\u0010\nEh\n(qt−E[q∞]) (ql−E[q∞])⊤i\n+Eh\n(ql−E[q∞]) (qt−E[q∞])⊤i\u0011\n| {z }\nT2.\nFor the term T1, we have the following decomposition,\nEh\n(qt−E[q∞]) (qt−E[q∞])⊤i\n=E\u0002\nqtq⊤\nt−qtE\u0002\nq⊤\n∞\u0003\n−E[q∞]q⊤\nt+E[q∞]E\u0002\nq⊤\n∞\u0003\u0003\n=E\u0002\nqtq⊤\nt\u0003\n−E[qt]E\u0002\nq⊤\n∞\u0003\n−E[q∞]E\u0002\nq⊤\nt\u0003\n+E[q∞]E\u0002\nq⊤\n∞\u0003\n=\u0000\nE\u0002\nqtq⊤\nt\u0003\n−E\u0002\nq∞q⊤\n∞\u0003\u0001\n+\u0000\nE\u0002\nq∞q⊤\n∞\u0003\n−E[q∞]E\u0002\nq⊤\n∞\u0003\u0001\n−\u0000\nE[qt]E\u0002\nq⊤\n∞\u0003\n+E[q∞]E\u0002\nq⊤\nt\u0003\n−2E[q∞]E\u0002\nq⊤\n∞\u0003\u0001\n=\u0000\nE\u0002\nqtq⊤\nt\u0003\n−E\u0002\nq∞q⊤\n∞\u0003\u0001\n+ Var ( q∞)−E[qt−q∞]E\u0002\nq⊤\n∞\u0003\n−E[q∞]Eh\n(qt−q∞)⊤i(38)\nCorollary 1 and Lemma 2 imply the following bounds for k≥tα,\nE[∥qt−q∞∥∞]≤C(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013t−tα\n2\n(39)\n\r\rE\u0002\nqtq⊤\nt\u0003\n−E\u0002\nq∞q⊤\n∞\u0003\r\r\n∞≤C′(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013t−tα\n2\nE[∥q∞∥∞]≤C′′(r, γ, P ),\nVar (q∞)≤C′′′(r, γ, P )·αtα. (40)\nSubstituting these bounds into equation (38), we have\nEh\n(qt−E[q∞]) (qt−E[q∞])⊤i\n=O \u0012\n1−(1−β)α\n2\u0013t−tα\n2\n+αtα!\n.\n39Therefore, we can bound T1as follows,\nT1=1\n(k−k0)2k−1X\nt=k0Eh\n(qt−E[q∞]) (qt−E[q∞])⊤i\n=1\n(k−k0)2k−1X\nt=k0O \u0012\n1−(1−β)α\n2\u0013t−tα\n2\n+αtα!\n=O\u00121\nα(k−k0)2exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n+O\u0012αtα\nk−k0\u0013\n=O\u00121\nα(k−k0)2exp\u0012\n−α(1−β)k0\n4\u0013\n+αtα\nk−k0\u0013\n.\nRegarding the term T2, notice that for l > t, we have\nEh\n(qt−E[q∞]) (ql−E[q∞])⊤i\n=Eh\nEh\n(qt−E[q∞]) (ql−E[q∞])⊤|qtii\n=Eh\n(qt−E[q∞])E[ql−E[q∞]|qt]⊤i\n=Eh\n(qt−E[q∞]) (E[ql|qt]−E[q∞])⊤i\n.\nNote that for any y∈Rd, it holds that\n∥E[ql|qt=y]−E[q∞]∥=∥E[ql−t|q0=y]−E[q∞]∥ ≤C(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013l−t−tα\n2\n,\nwhere the second inequality holds since Corollary 1 holds for all initial value of q0.\nTherefore, when l > t, we have\nEh\r\r\r(qt−E[q∞]) (E[ql|qt]−E[q∞])⊤\r\r\r\n∞i\n≤E[∥qt−E[q∞]∥∞∥E[ql|qt]−E[q∞]∥∞]\n≤E[∥qt−E[q∞]∥∞]· \nC(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013l−t−tα\n2!\n≤(E[∥qt−q∞∥∞] +E[∥q∞−E[q∞]∥∞])· \nC(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013l−t−tα\n2!\n(i)\n≤\u0010\nE[∥qt−q∞∥∞] + ( tr(Var( q∞)))1/2\u0011\n· \nC(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013l−t−tα\n2!\n(ii)\n≤ \nC(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013t−tα\n2\n+C′(r, γ, P )√αtα!\n· \nC(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013l−t−tα\n2!\n=C2(r, γ, P )·\u0012\n1−(1−β)α\n2\u0013l−2tα\n2\n+C′′′′(r, γ, P )·√αtα·\u0012\n1−(1−β)α\n2\u0013l−t−tα\n2\n,\nwherein (i)tr(·)denotesthetraceoperatorandweusethefactthat E[∥q∞−E[q∞]∥∞]≤r\nEh\n∥q∞−E[q∞]∥2\n∞i\n=\ntr(Var( q∞))1/2; in(ii)we use the bounds in equations (39) and (40).\n40In addition, note that\n1\n(k−k0)2k−1X\nt=k0k−1X\nl=t+1O \u0012\n1−(1−β)α\n2\u0013l−2tα\n2!\n≤1\n(k−k0)2∞X\nt=k0∞X\nl=t+1O \u0012\n1−(1−β)α\n2\u0013l−2tα\n2!\n≤1\n(k−k0)2\u00124\n(1−β)α\u00132\nO\n\u0012\n1−(1−β)α\n2\u0013k0−2tα\n2\n\n=O\u00121\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n,\nand\n1\n(k−k0)2k−1X\nt=k0k−1X\nl=t+1O \u0012\n1−(1−β)α\n2\u0013l−t−tα\n2!\n≤1\n(k−k0)2k−1X\nt=k0∞X\nl=t+1O \u0012\n1−(1−β)α\n2\u0013l−t−tα\n2!\n=O\u00121\n(k−k0)α\u0013\n.\nPutting together, we obtain the following upper bound for T2,\nT2=1\n(k−k0)2k−1X\nt=k0k−1X\nl=t+1O \u0012\n1−(1−β)α\n2\u0013l−2tα\n2\n+√αtα\u0012\n1−(1−β)α\n2\u0013l−t−tα\n2!\n=O\u00121\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013\n+√αtα\n(k−k0)α\u0013\n.\nCombining the above bounds for T1andT2, we obtain\nEh\n(¯qk0,k−E[q∞]) (¯qk0,k−E[q∞])⊤i\n=O\u00121\nα(k−k0)2exp\u0012\n−α(1−β)k0\n4\u0013\n+αtα\nk−k0\u0013\n+O \n1\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013\n+p\ntα/α\n(k−k0)!\n=O p\ntα/α\n(k−k0)+1\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013!\n.(41)\nNow we are ready to bound the LHS of equation (13). First, we have the following decomposition\nEh\n(¯qk0,k−q∗) (¯qk0,k−q∗)⊤i\n=Eh\n(¯qk0,k−E[q∞] +E[q∞]−q∗) (¯qk0,k−E[q∞] +E[q∞]−q∗)⊤i\n=Eh\n(¯qk0,k−E[q∞]) (¯qk0,k−E[q∞])⊤i\n+Eh\n(E[q∞]−q∗) (¯qk0,k−E[q∞])⊤i\n+Eh\n(¯qk0,k−E[q∞]) (E[q∞]−q∗)⊤i\n+Eh\n(E[q∞]−q∗) (E[q∞]−q∗)⊤i\n. (42)\n41For the second term of RHS of equation 42, we have\nEh\n(¯qk0,k−E[q∞]) (E[q∞]−q∗)⊤i\n=1\nk−k0 k−1X\nt=k0E[qt−q∞]!\n(E[q∞]−q∗)⊤\n=O\u00121\nα(k−k0)exp\u0012\n−α(1−β)k0\n4\u0013\u0013\u0000\nαB(r, γ, P ) +O(α2+α2t2\nα2)\u0001\n=O\u00121\nk−k0exp\u0012\n−α(1−β)k0\n4\u0013\u0013\n.\nSimilarly, we have the same bound for the third term of equation (42). For the last term of RHS of\nequation (42), we have\nEh\n(E[q∞]−q∗) (E[q∞]−q∗)⊤i\n= (E[q∞]−q∗) (E[q∞]−q∗)⊤\n=\u0000\nαB(r, γ, P ) +O(α2+α2t2\nα2)\u0001\u0000\nαB(r, γ, P ) +O(α2+α2t2\nα2)\u0001⊤\n=α2B′(r, γ, P ) +O(α3+α3t2\nα2).\nCombining all these bounds, we obtain\nEh\n(¯qk0,k−q∗) (¯qk0,k−q∗)⊤i\n=α2B′(r, γ, P ) +O(α3+α3t2\nα2)\n+O p\ntα/α\n(k−k0)+1\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013!\n=α2B′+O \nα3+α3t2\nα2+p\ntα/α\n(k−k0)+ +1\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013!\n.\nthereby completing the proof of Corollary 2.\nF Proof of Corollary 3\nIn this section, we give the proof of the first and second moment bounds in Corollary 3.\nF.1 First Moment\nWe have\nEh\neq(α)\nk0,ki\n−q∗=\u0010\n2¯q(α)\nk0,k−¯q(2α)\nk0,k\u0011\n−q∗\n=2\u0010\n¯q(α)\nk0,k−q∗\u0011\n−\u0010\n¯q(2α)\nk0,k−q∗\u0011\n(i)=2\u0012\nαB(r, γ, P ) +O(α2+α2t2\nα2) +O\u00121\nα(k−k0)exp\u0012\n−α(1−β)k0\n4\u0013\u0013\u0013\n−\u0012\n2αB(r, γ, P ) +O(α2+α2t2\nα2) +O\u00121\nα(k−k0)exp\u0012\n−α(1−β)k0\n2\u0013\u0013\u0013\n=O(α2+α2t2\nα2) +O\u00121\nα(k−k0)exp\u0012\n−α(1−β)k0\n4\u0013\u0013\nwhere (i)follows from Corollary 2.\n42F.2 Second Moment\nWe first introduce the following short-hands:\nu1:= ¯q(α)\nk0,k−Eh\nq(α)\n∞i\n, u 2:= ¯q(2α)\nk0,k−Eh\nq(2α)\n∞i\nand v:= 2Eh\nq(α)\n∞i\n−Eh\nq(2α)\n∞i\n+q∗.\nWith these notations, ˜qk0,k−q∗= 2u1−u2+v. We then have the following bound\n\r\r\r\rE\u0014\u0010\neq(α)\nk0,k−q∗\u0011\u0010\neq(α)\nk0,k−q∗\u0011⊤\u0015\r\r\r\r\n∞≤\r\r\r\rE\u0014\u0010\neq(α)\nk0,k−q∗\u0011\u0010\neq(α)\nk0,k−q∗\u0011⊤\u0015\r\r\r\r\n2\n=\r\r\rEh\n(2u1−u2+v) (2u1−u2+v)⊤i\r\r\r\n2\n≤Eh\n∥2u1−u2+v∥2\n2i\n≤3E∥2u1∥2\n2+ 3E∥u2∥2\n2+ 3∥v∥2\n2.\nBy equation (41), we have\nE∥u1∥2\n2= Tr\u0000\nE\u0002\nu1u⊤\n1\u0003\u0001\n=O p\ntα/α\n(k−k0)+1\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013!\n.\nSimilarly, we have\nE∥u2∥2\n2=O p\ntα/α\n(k−k0)+1\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n2\u0013!\n.\nBy Theorem 3, we have ∥v∥2\n2=O\u0000\nα4+α4t4\nα2\u0001\n.\nCombining these bounds together, we have\nEh\n(˜qk−k0−q∗) (˜qk−k0−q∗)⊤i\n=O\u0000\nα4+α4t4\nα2\u0001\n+O p\ntα/α\n(k−k0)+1\n(k−k0)2α2exp\u0012\n−α(1−β)k0\n4\u0013!\n.\nG Experiment Details\nTabular case. Wwe consider two MDPs for our numerical experiments.\nThe first example is a 1×3Gridword with S={0,1,2}andA={−1,1}. For each step, the agent can\nwalk in two directions: left or right. If the agent walks out of the space, the agent would get a reward of -4\nand stay at the same state. Otherwise, the agent can walk to the next state with probability of 0.95 or still\nstay at the same state with probability of 0.05. For the case that the agent does not exceed the space, the\nreward function is determined by the current state r(s, a) =r(s)with r(0) = 0 , r(1) = 10 andr(2) = 0 .5.\nThe discounted factor is set as γ= 0.9.\nThe second example is a classical 4×4Gridworld combined with the slippery mechanism in Frozen-Lake.\nFor each step, the agent can walk in four directions: left, up, right or down. Specially, there are two state\nA and B in which the agent can only intend to move to A′andB′. After the action is selected by the\nbehavior policy, the agent will walk in the intended direction with probability of 0.9 else will move in either\nperpendicular direction with equal probability of 0.05 in both directions. If the agent walks out of the\nspace, the agent would get a reward of -1 and stay in the same state. Otherwise, the reward function is also\ndetermined by the current state with r(A) = 10,r(B) = 5andr(s) = 0fors̸=A, B.The discounted factor\nis set as γ= 0.9.\n43Linear function approximation. Our second set of experiments consider Q-learning with linear function\napproximation. More specifically, we consider approximating the Q-function by a linear subspace spanned by\nbasisvectors ϕ= (ϕ1, . . . , ϕ d)⊤:S×A → Rd. Thegoalistofind θ∗suchthat ˜qθ∗:= Φθ∗bestapproximatesthe\noptimal Q function q∗, where Φdenotes the feature matrix Φ =\u0002ϕ(s1, a1)··· ϕ(s|S|, a|A|)\u0003⊤∈R|S||A|× d.\nWe assume that Φhas a full column rank, which is standard in literature [ 8,30,80]. Note that θ∗can be\ncalculated by projected value iteration algorithm.\nIn this case, the Q-earning algorithm reduces to updating the parameter θ∈Rdas follows [8]:\nθk+1=θk+αϕ(sk, ak)\u0010\nrk+γmax\na′ϕ(sk+1, a′)⊤θk−ϕ(sk, ak)⊤θk\u0011\n, (43)\nwhere (sk, ak, rk, sk+1)is the sample generated by the behavior policy at time step k.\nFor the MDP and feature vectors, we consider a similar setup as the work [ 30, Appendix D.1]. We provide\nthe detail description here for completeness. We consider an MDP with |S|= 20states and |A|= 5actions.\nWe generate the rewards and transition probabilities as follows: for each (s, a)∈ S × A,\n•The reward r(s, a)is drawn uniformly in [0,1].\n•For the transition probability T(·|s, a),we first obtain |S|numbers by uniformly sampling of [0,1], and\nthen normalize these |S|numbers by their sum to make it a valid probability distribution.\nAs for the feature matrix, we consider d= 10 .For each (s, a),each element of ϕ(s, a)is drawn from\nBernoulli distribution with parameter p= 0.5,and then we normalize the features to ensure ∥ϕ(s, a)∥ ≤1.\nWe repeat this process until the matrix Φhas a full column rank.\nWe set the discounted factor to be γ= 0.5and the Markovian data {xk}k≥0is generated from a uniformly\nrandom behavior policy.\nWe run Q-learning with linear function approximation (43)with initialization θ(α)\n0=θ∗+ 10and stepsize\nα∈ {0.1,0.2,0.4}. We also consider two diminishing stepsizes: αk= 1/\u0000\n1 + (1 −γ)k\u0001\nandαk= 1/k0.75as we\nused in tabular Q-learning. The simulation results for the Q-learning with linear function approximation are\nillustrated in Figure 2. We plot the ℓ1-norm error ∥¯θ(α)\nk/2,k−θ∗∥1for the tail-averaged (TA) iterates ¯θ(α)\nk/2,k,\nthe RR extrapolated iterates eθ(α)\nkwith stepsizes αand2α, and iterates with diminishing stepsizes.\nFigure 2: The Q-learning with linear function approximation errors of tail-averaged (TA) iterates and RR extrapolated\niterates with different stepsizes.\nWe can observe some similar results as tabular Q-learning’s:\n•The larger the stepsize α,the faster it converges.\n44•The final TA error, which corresponds to the asymptotic bias, is approximately proportional to the\nstepsize.\n•RR extrapolated iterates reduce the bias.\n•The TA and RR-extrapolated iterates with constant stepsizes enjoy significantly faster initial convergence\nthan those with diminishing stepsizes.\n45"    },    {        "title": "2401.13895v1.Monte_Carlo_simulations_of_the_capture_and_cooling_of_alkali_metal_atoms_by_a_supersonic_helium_jet.pdf",        "content": "Monte-Carlo simulations of the capture and cooling of alkali-metal atoms by a\nsupersonic helium jet\nJeremy Glick, William Huntington, Michael Borysow, Kevin Wen, Daniel Heinzen\nDepartment of Physics, University of Texas at Austin, Austin, TX 78712\nJacek K los and E. Tiesinga\nJoint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD 20899\n(Dated: January 26, 2024)\nWe present three-dimensional Monte-Carlo simulations of the capture of 1000 K7Li or 500 K\n87Rb atoms by a continuous supersonic4He jet and show that intense alkali-metal beams form with\nnarrow transverse and longitudinal velocity distributions. The nozzle creating the4He jet is held at\napproximately 4 K. These conditions are similar to those in the cold7Li source developed by some of\nus as described in [Phy. Rev. A 107, 013302 (2023)]. The simulations use differential cross-sections\nobtained from quantum scattering calculations of7Li or87Rb atoms with4He atoms for relative\ncollision energies between k×1 mK to k×3000 K, where kis the Boltzmann constant. For collision\nenergies larger than ≈k×4 K the collisions favor forward scattering, deflecting the7Li or87Rb\natoms by no more than a few degrees. From the simulations, we find that about 1 % of the lithium\natoms are captured into the4He jet, resulting in a lithium beam with a most probable velocity of\nabout 210 m/s and number densities on the order of 108cm−3. Simulations predict narrow yet\nasymmetric velocity distributions which are verified by comparing to fluorescence measurements of\nthe seeded7Li atoms. We find agreement between simulated and experimentally measured seeded\n7Li densities to be better than 50 % across a range of4He flow rates. We make predictions for\ncapture efficiency and cooling of87Rb by a supersonic4He jet. The capture efficiency for87Rb is\nexpected to be similar to7Li.\nI. INTRODUCTION\nSupersonic jets of typically noble gases serve as a valu-\nable tool for producing intense cold beams. Seeding these\ninert sources with other atoms and molecules can also cre-\nate cold beams of these particles [1, 2]. In fact, these sys-\ntems have long been used for studies in molecular physics,\nfluid mechanics, and molecular spectroscopy [3–7]. Su-\npersonic expansions of4He can reach milliKelvin temper-\natures in the moving frame of the jet, due to the adia-\nbatic expansion of the carrier gas [8–10] and thus seeded\nspecies can be cooled to similarly low temperatures.\nHistorically, seeding of non-condensible atoms or\nmolecules generally occurs by mixing into the carrier gas\nwell in advance of the jet nozzle. Condensible species\ncan be seeded just before or immediately after the noz-\nzle; this arrangement is known as a Smalley source [7, 11].\nIn Ref. [10], however, an experimental apparatus is de-\nscribed in which7Li atoms are efficiently captured by\na4He jet after the jet has significantly expanded and\nhas cooled to temperatures below 10 mK from its initial\ntemperature of 4.2 K at the cryogenically cooled nozzle.\nPost-nozzle seeding then eliminates the heat load on the\nnozzle and condensation that may otherwise occur with\na continuous lithium source and relatively small nozzle\ndiameters. Modeling the capture and thermalization of\nthe seeded species is essential for optimizing the source’s\nperformance. Over the years, researchers have examined\npre-nozzle seeded jets [1, 12, 13]; to our knowledge, how-\never, modeling of the post-nozzle capture of an effusive\nbeam into a supersonic jet has not been carried out.\nDetermining conditions for efficient post-nozzle seedingrequires knowledge of the collisional properties between\nalkali-metal atoms and noble gas atoms. Such proper-\nties have been extensively studied over the past 60 years.\nFor example, the collision-energy dependence of the elas-\ntic scattering cross-sections for collisions between7Li and\nnoble gas atoms were measured in the early 1970s [14, 15].\nDiffusion coefficients describing the propagation of den-\nsity gradients of trace amounts of alkali-metal atoms\nin noble-gas buffer gasses have been computed as well\nas measured [16–19]. Potential energy surfaces as well\nas collisional rates coefficients have been computed as\nwell [20–22]. The single bound state for the7Li4He sys-\ntem, for example predicted in Ref. [23], was observed in\n2013 [24]. Ultra-cold, µK and mK samples of7Li and\n87Rb atoms are used to measure pressure in the ultra-\nhigh vacuum (UHV) regime [25–32]. These devices rely\non precise knowledge of thermalized elastic rate coeffi-\ncients between the ultracold alkali-metal atoms and the\nambient-temperature atomic and molecular gasses in the\nUHV vacuum. To validate these devices known pressures\nof ambient temperature background gases, often noble\ngases, are deliberately added into the vacuum chamber.\nTo model the seeding process, it is crucial to under-\nstand the velocity-phase-space distribution of supersonic\nexpansions. Models of supersonic expansions have been\nperformed using various numerical methods as a means\nof finding approximate solutions to the Boltzmann equa-\ntion for the phase space density of the atomic species\nand molecules. Early research utilized the method of\nmoments approach [33–35] enabling studies of the de-\ncoupling of the transverse and longitudinal temperatures\nwhen the jet transitions from continuum or fluid-like flowarXiv:2401.13895v1  [physics.atom-ph]  25 Jan 20242\nto free molecular flow. Modern simulations are often\nperformed using computational fluid dynamics (CFD)\nsolvers [36, 37] or direct-simulation Monte-Carlo (DSMC)\nsimulations [38–40]. Both CFD and DSMC can model\nturbulent flow and shock front formation as well as eval-\nuate the role of nozzle geometry on beam brightness.\nCFD solvers rely on finding numerical solutions to the\nNavier-Stokes or related differential equations. DSMC\nmethods, pioneered by G. Bird [41], are often favored for\nrarefied gas flows. Particles move in classical straight-\nline paths interspersed by particle-particle collisions that\nchange the directions and velocities of atom or molecular\npairs based on sampling from differential scattering cross-\nsections at the relevant relative kinetic energy. Macro-\nscopic gas quantities are then determined by statistical\naverages. Accurate differential scattering cross-sections,\nhowever, are not always available for every molecular sys-\ntem and phenomenological cross-section models must be\nemployed.\nHere, we report on three-dimensional (3D) Monte\nCarlo simulations of the capture of7Li and87Rb by a su-\npersonic helium jet. We quantify the fraction of lithium\nand rubidium atoms that become entrained in the jet over\na wide range of helium flow rates using accurate theoret-\nical differential cross-sections for7Li-4He and87Rb-4He\nscattering. Simulated density and velocity profiles are\npresented and analyzed. We compare data from7Li ex-\nperiments to validate our approach.\nII. EXPERIMENTAL SETUP AND\nTHEORETICAL TREATMENT\nWe model the dynamics of seeding and cooling ener-\ngetic7Li or87Rb atoms into a supersonic helium jet. A\nschematic of the simulated apparatus and the definition\nof our coordinate system are shown in Fig. 1. The helium\nsonic nozzle is directed along the (0 ,0,1) or ˆ zdirection\nwith its tip located at the origin ⃗ xN= (0,0,0) of our coor-\ndinate system. The helium nozzle diameter dN= 0.020\ncm. Here and elsewhere system parameters are moti-\nvated by the designs of Ref. [10]. The alkali-metal source\ncreating an effusive beam of atoms is oriented along the\n(0,sinθS,cosθS) direction with angle θS= 50◦from the ˆ z\naxis, where the opening or aperture in front of the oven\nis located at ⃗ xS= (0,−5.6,−3.0) cm. This source has\na conical nozzle with an aperture diameter of dS= 0.10\ncm. The nozzle diameters are thus much smaller than\n|⃗ xN−⃗ xS|. Moreover, the geometrical intersect or seeding\ndistance between the helium jet and alkali-metal beam di-\nrections is 1.7 cm downstream from the helium nozzle. A\nskimmer extracts the cold alkali-metal atoms. The skim-\nmer is located 16 cm from the helium nozzle and has a\ncircular opening with a diameter of 2.54 cm. The line\nconnecting the nozzle and the opening of the skimmer\ndefines the center line of the helium jet.\nThe orientation and location of the alkali-metal source\nrelative to the helium jet are by no means unique.\nFIG. 1. Schematic of the simulated apparatus showing\nlithium or rubidium atoms (red region) created in a hot\natomic oven, captured by a supersonic cryogenic helium jet\n(cyan region), and then extracted as cold atoms with a skim-\nmer. The lower left corner shows the Cartesian xyzcoordi-\nnate system used in the simulations. The yandzaxes are\nin the plane of this image, while the origin is at the tip of\nthe cryogenic4He sonic nozzle. System dimensions and he-\nlium jet and oven parameters can be found in the text. In\nthe experimental realization, a charcoal adsorption pump is\nused to remove background helium to avoid the formation of\nshock fronts. A camera collects fluorescence emitted in the\nx-direction that is induced by a probe laser beam. The probe\nlaser can be directed either vertically through the window la-\nbeled “optical access” or horizontally along the center-line of\nthe helium jet.\nFor specificity, however, our simulations closely match\nthe physical design constraints of the implementation in\nRef. [10]. In principle, this restricts our ability to opti-\nmize the design. In practice, we find that optimizing the\nflow rate of the4He jet is sufficient for creating intense\ncold alkali-metal atom beams.\nA. Static helium jet\nThe motion of all helium and alkali-metal atoms can,\nin principle, be modeled with the DSMC method. The\nefficient capture of the alkali-metal (AM) atoms, how-\never, requires a dense helium jet. Our expected stagna-\ntion helium number densities, n0, are a few times 1019\ncm−3with a stagnation temperature of the helium atoms\nof about T0= 4.2 K. The stagnation conditions are the\nconditions in the reservoir before the nozzle. Based on\nthese conditions, the mean free path between4He-4He\ncollisions is a few microns using the4He-4He cross-section\nfrom Ref. [42]. This small mean free path puts the ex-\npansion well into the continuum regime and simulating\nthe helium expansion directly with a DSMC method is\nchallenging. Furthermore, we cannot take advantage of\nthe two-dimensional (2D) axial symmetry of the helium\njet and a three-dimensional (3D) simulation is required3\nto model the injected alkali-metal atom beam.\nWe therefore introduce approximations based on the\nrealization that the number densities of the alkali-metal\natoms are orders of magnitude smaller than those in the\nhelium jet. Firstly, we realize that the mean-free path\nbetween AM-AM collisions is larger than the centime-\nter size of the system and we only need to follow one\nalkali-metal atom at a time sampled from their phase-\nspace density at the aperture of the atomic oven. Sec-\nondly, the value for the mean-free path between AM-4He\ncollisions lies between those for4He-4He and AM-AM\ncollisions. Then the number density of the4He jet and\nthe relationships among the mean free paths have several\nimplications. The heat added to the helium jet as the\nalkali-metal atoms slow can be neglected. Consequently,\nwe can assume that the helium jet is expanding adiabat-\nically, is in the collisional regime, and is locally always in\nthermal equilibrium. In other words, we can treat the jet\nas a static background flow and only need to determine\nlithium trajectories as they collide with4He atoms.\nFrom Refs. [43, 44], it can be shown that the unit-\nnormalized position and velocity phase-space probability\ndistribution of the axially symmetric expansion of the he-\nlium atoms in the jet starting from position a few nozzle\ndiameters away is well described by\nPHe(⃗ r,⃗ v) = (1)\nc0\u0012mHe\n2πkT 0\u00133/2\nζ(ˆr) exp\u0014\n−1\n2mHe(⃗ v−vHe,tvˆr)2\nkTHe(r)\u0015\n,\nwhere ˆ r=⃗ r/ris the orientation of ⃗ rin the Cartesian\ncoordinate system defined in Fig. 1. In spherical coordi-\nnates ˆ ris specified by polar angle θ∈[0, π] and azimuthal\nangle φ∈[0,2π]. In addition, the position-dependent he-\nlium temperature and radial terminal or streamline ve-\nlocity are,\nTHe(r) =c1T0\u0012dN\nr\u00134/3\n(2)\nand\nvHe,tv=r\n5kT0\nmHe, (3)\nwhere mHeis the mass of the4He atom and kis the\nBoltzmann constant. Here, coefficients c0= 1.62 and\nc1= 0.287. Finally, the angular function ζ(ˆr) is given by\nζHe(ˆr) =1\nZ(tHe)cos2(tHeθ), (4)\nfor 0 ≤θ < π/ (2tHe) and zero otherwise.\nWe have tHe>1 and Z(tHe) is defined such thatRπ\n0sinθdθR2π\n0dφ ζHe(ˆr) = 1. For our jet tHe= 1.15 and\nπ/(2tHe) = 78◦. Locally, after integrating PHe(⃗ r,⃗ v) over\nall velocities and using Eq. (2), we find that\npHe(⃗ r) =Z\nd3⃗ v PHe(⃗ r,⃗ v) =c0c3/2\n1\u0012dN\nr\u00132\nζHe(ˆr),(5)and, similarly, that the root-mean-square velocity in the\nmoving frame of the helium gas is\nvHe,rms(⃗ r) =s\n3kTHe(r)\nmHe. (6)\nThe helium number density is\nnHe(⃗ r) =n0pHe(⃗ r) =c0c3/2\n1n0\u0012dN\nr\u00132\nζHe(ˆr),(7)\nwhere\nn0= 0.513r\n1\n2mHe\nkT0\u00124\nπd2\nN\u0013\n˙N0 (8)\nand ˙N0are the helium number density and number flow\nrate at the sonic nozzle, respectively [44]. Typical flow\nrates are between 2 ×1019atoms/s and 1 ×1020atoms/s\n[10].\nThe phase-space distribution defined in Eqs. (1)\nthrough (4), as stated before, breaks down for radii\nsmaller than a few nozzle diameters. That is, the local\nhelium temperature THe(r) should approach T0for small\nr. Equation (1) also breaks down for large radii r, where\nthe4He-4He collision rate is insufficient to maintain local\nthermal equilibrium.\nIn the experiments of Ref. [10] and in our simulations,\nT0= 4.2 K and the largest allowed radius rfor which\nEq. (1) is expected to be valid is about 10 cm for the low-\nest helium reservoir densities which we consider. From\nEq. (2), the initial helium temperature implies that the\nhelium temperature drops to 7 mK approximately 1 cm\naway from the sonic nozzle. The terminal velocity for\n4He atoms is ≈210 m/s. The terminal kinetic energy is\nmHev2\nHe,tv/2 = 5 kT0/2≈k×11 K. As a general feature of\nsupersonic expansion, we realize that vHe,tv≫vHe,rms(r)\nforr≫dN.\nThe form for ζHe(ˆr) in Eq. (4) is based on findings\nfrom early measurements in low-density wind tunnels [45]\nas well as, more recently, in supersonic molecular jets\nusing linear Raman Spectroscopy [46]. Strictly speaking,\nthe form was only observed for θ <20◦as shock fronts\noccurred in these experiments. In our model, we can\nassume that the background pressure in the apparatus is\nsufficiently low that the helium smoothly transitions to\nfree molecular flow without the formation of shocks. In\nthe experimental apparatus, a charcoal adsorption pump\nis used to remove background helium atoms and minimize\nthe possibility of shock waves forming.\nWe have run simulations with other functional forms\nforζ(ˆr), including the more complex profiles from\nRefs. [45, 46]. They differ mainly for θ > 20◦and we\nfound negligible differences in capture rates of the alkali-\nmetal atoms, as few AM-4He collisions occur for large\nangles θ. The explanation is that AM-4He collisions in\nthe periphery of the helium expansion are early on in\nthe alkali-metal-atom slowing process so that collisions4\noccur at large, ∼k×100 K collision energies nearly in-\ndependent of the low helium jet temperature. Moreover,\nrelatively few collisions occur in the periphery due to the\nlow helium density and small AM-4He cross sections at\nlarge collision energies.\nFinally, we note that in more accurate descriptions of\nthe helium jet the transverse, xandy, and longitudi-\nnal,z, temperature of the helium atoms need not be the\nsame [33, 47]. The effects of orientation-dependent he-\nlium temperatures should be negligible for alkali-metal\natoms entering the jet as the relative velocity is domi-\nnated by the alkali-metal-atom velocity. It might become\nimportant once the alkali-metal-atom’s velocity becomes\ncomparable to the local helium velocity. This, however,\nonly occurs once an alkali-metal atom is already captured\nin the jet and does not affect the capture efficiency. We\nhave not included orientation-dependent helium temper-\natures in our simulations.\nB. Collision Cross-Sections\nThe Monte-Carlo simulations rely on an accurate\nknowledge of the differential cross-section\ndσAM−He\ndΩ, (9)\nfor collisions of7Li and87Rb with4He, each in its2S or1S\nelectronic ground state, as a function of relative collision\nenergy Eand polar and azimuthal collision angles θcand\nφc. We will also use the total elastic cross-section\nσAM−He(E) =Zπ\n0sinθcdθcZ2π\n0dφcdσAM−He\ndΩ.(10)\nFormal discussions of scattering theory describing cross-\nsections and collision angles but also partial waves, cen-\ntrifugal barriers, and Wigner threshold laws can be found\nin Refs. [48–50].\nWe rely on the recent evaluations of the relevant X2Σ+\npotential energy curves as functions of the interatomic\nseparation Rdefined as a distance between point-like\natomic nuclei and numerical solutions of the Schr¨ odinger\nequation for the relative motion of the atoms using these\nisotropic potentials by some of us in Refs. [32, 51, 52].\nAn isotropic potential only depends on Rand not the\norientation of the interatomic axis. We use the reduced\nmass µ=mAMmHe/(mAM+mHe) computed from atomic\nmasses in the relative kinetic energy operator.\nIn the simulation, the atomic oven produces7Li or\n87Rb atoms at temperatures of several hundreds of Kelvin\nand no external magnetic field is applied. Hence, we can\nassume that the eight hyperfine states of the electronic\nground state of7Li and87Rb are equally populated. For\nour purposes, we can also ignore the R-dependence of the\nhyperfine energies of the alkali-metal atoms. Hence, the\ndifferential cross-sections are the same for all hyperfine\nstates of the alkali-metal atom.\n02 4 6 8 10 0.010.11dσ/dΩ (10-12 cm2)0\n2 4 6 8 10 θ\nc (degrees)0.010.11(a) 7Li-4HeE/k=300 K\n10 K\n5 mK1 K\n500 K(b) \n87Rb-4HeFIG. 2. Differential cross sections d σAM−He/dΩ for (a)\n7Li+4He and (b)87Rb+4He as functions of collision angle\nθcfor five relative kinetic energies E. The collision energies\nfor the colored curves are the same in the two panels.\nFigure 2 shows d σAM−He/dΩ as functions of collision\nangle θcup to 10◦for selected collision energies between\nE/k = 5 mK and 500 K. Panel (a) of the figure shows\ndata for the7Li-4He collision, while panel (b) shows data\nfor87Rb-4He. For an isotropic potential, the differen-\ntial cross section is independent of φc. The relative un-\ncertainty of differential cross-sections is below 5 % for\nE/k > 40 K. For E/k < 0.1 K it is larger than 20 %.\nWe observed that for the largest collision energies\nshown in Fig. 2 the differential cross-section drops off\nrapidly with angle θc. For example, for E≥k×10\nK the half width at half maximum is less than a few de-\ngrees. For much smaller collision energies, the differential\ncross-section is independent of θc. In other words, during\nthe initial cooling process the alkali-metal atoms undergo\nonly small deflections when colliding with helium atoms.\nThe angular dependence of d σAM−He/dΩ can mostly\nbe understood from the long-range dispersion or van-\nder-Waals −C6/R6behavior of the X2Σ+potential,\nwhere the positive C6is the van-der-Waals dispersion\ncoefficient. The dispersion interaction introduces nat-\nural length and energy scales. These are the van-der-\nWaals length x6=4p\n2µC6/ℏ2and van-der-Waals energy\nE6=ℏ2/(2µx2\n6). With the dispersion coefficients from\nRef. [53], we find x6= 21 .4a0and 28 .0a0for7Li+4He\nand87Rb+4He, respectively, where a0= 0.0529177 nm\nis the Bohr radius. Similarly, E6/k= 74 mK and 29 mK,\nrespectively.\nFor collision energies E≪E6, only a single partial5\nwave, the so-called swave, contributes to d σAM−He/dΩ\nand we enter the Wigner threshold regime. Then\ndσAM−He/dΩ→a2is independent of θc, where ais the\nscattering length. For collision energies E≫E6, a semi-\nclassical approximation [49] gives a qualitative expression\nfor the differential cross-section at small scattering angles\nθc≪π. In fact, we have\ndσAM−He\ndΩ=f0(E/E 6)3/5n\n1−f1(E/E 6)4/5(θc/2)2o\nx2\n6,\n(11)\nwith f0= 0.363 046 ···andf1= 2.018 179 ···corre-\nsponding to a sharply peaked differential cross-section.\nStill, a comparison with the quantum results in Fig. 2,\nnot shown, finds that the semi-classical model overesti-\nmates d σAM−He/dΩ at θc= 0 by a factor two, while\nit underestimates the half width at half maximum by a\nsimilar factor.\nC. Monte Carlo Simulation of the cooling of7Li or\n87Rb atoms\nThe Monte-Carlo simulations begin by generating a\n7Li or87Rb particle at the center of the aperture in front\nof the alkali-metal source (AMS). The particle’s velocity\n⃗ vAMis sampled from a Maxwell-Boltzmann distribution\nfor an effusive source at temperature TAM[43, 54] with\nprobability distribution for its speed vAM=|⃗ vAM|given\nby\nfLi(v) =1\n2\u0012mAM\nkTAM\u00132\nv3exp\u0014\n−1\n2mAMv2\nkTAM\u0015\n,(12)\nwithR∞\n0dv fLi(v) = 1 and angular-velocity probability\ndistribution given by ξ(θ, φ) = cos( θ)/N(tAMS) for 0 ≤\nθ≤π/(2tAMS) and zero otherwise with respect to the\n(0,sinθS,cosθS) direction. Here, mAMis the mass of\nthe alkali-metal atom and N(tAMS) is defined such thatRπ\n0sinθdθR2π\n0dφ ξ(θ, φ) = 1. For our alkali-metal source\ntAMS =π/0.18 or 2 tAMS/π= 0.09. As the diameter of\nthe aperture of the atomic source is small compared to\nthe distance an alkali-metal atom travels before it enters\nthe4He jet, we can omit sampling over the initial location\nof the alkali-metal atom.\nOnce the initial position and velocity of the7Li or87Rb\natom is generated, the particle is propagated through the\nhelium jet with variable time steps ∆ tthat are small frac-\ntions of the local mean-free time for collisions with4He\natoms. For much of the slowing process, the alkali-metal-\natom’s kinetic energy is large compared to mHev2\nHe,tv/2\nandkTHe(r) of the helium atoms. This allows us to es-\ntimate the local mean-free time for an alkali-metal atom\nat position ⃗ rAMand velocity ⃗ vAMin this regime with-\nout the need for sampling the thermal distribution of the\nhelium atoms and computing the local rate coefficient\nK(⃗ rAM,⃗ vAM) =⟨vrelσAM–He (E)⟩, where the brackets in-\ndicate an average over the velocity distribution of the4He\natoms only, the relative velocity ⃗ vrel=⃗ vAM−⃗ vHe, andE=µv2\nrel/2. The mean free time is then well approxi-\nmated by\nτ(⃗ rAM,⃗ vAM)≈1\nnHe(⃗ rAM)vapprox σAM–He (Eapprox ),\n(13)\nwhere\nvapprox =|⃗ vAM−vHe,tvˆrAM|, (14)\nandEapprox =µv2\napprox /2. Once the velocity of an alkali-\nmetal atom approaches the ≈210 m/s terminal velocity\nof the helium atoms, Eq. (13) loses accuracy. Hence, for\n|⃗ vAM−vHe,tvˆrAM|< vcutoff, where vcutoff = 50 m/s, the\nmean-free time at ⃗ rAMand⃗ vAMis evaluated exactly with\nτ(⃗ rAM,⃗ vAM) =1\nnHe(⃗ rAM)K(⃗ rAM,⃗ vAM),(15)\nand\nK(⃗ rAM,⃗ vAM) =Z\nd3vHevrelσAM–He (E)PHe(⃗ rAM,⃗ vHe)\npHe(⃗ rAM)\n=K(|⃗ vAM−vHe,tvˆrAM|, THe(rAM)),(16)\nwhere\nK(v, T) = 4 π\u0010mHe\n2πkT\u00113/2\ne−mHev2/(2kT)(17)\n×Z∞\n0v2\nreldvrel[vrelσAM–He (E)]\n×e−mHev2\nrel/(2kT)sinh[mHevrelv/kT ]\nmHevrelv/kT.\nHere, we realize that K(⃗ rAM,⃗ vAM) is only a function of\nthe local speed of the alkali-metal atom in the frame mov-\ning along with velocity vHe,tvˆrAMand the local helium\ntemperature THe(rAM). For efficient Monte-Carlo sim-\nulations, the rate coefficient K(v, T) is precomputed on\na grid of Tfrom 0.1 mK to 100 mK and vfrom 0 m/s\ntovcutoff. A two-dimensional interpolator is then used to\ndetermine K(v, T) at any vandTwithin the boundary of\nthe grid. The grid size is chosen such that the difference\nbetween the interpolated value and that from numerical\nintegration of Eq. 16 is less than 1 %. Moreover, the dif-\nference between Eqs. (13) and (15) is less than 2 % for\nv > v cutoff andTHe<100 mK, but Eq. (13) is an order\nof magnitude faster to compute.\nWe use an acceptance-rejection procedure to deter-\nmine whether an alkali-metal atom, located at phase-\nspace point ( ⃗ rAM,⃗ vAM), collides with a4He atom. This\nprocedure starts by computing the local mean-free time\nτland equate time step ∆ tto the preliminary value sτl\nwith 0 < s≪1. Our value for sis discussed below.\nThe time step ∆ tis adjusted if one of two conditions is\nmet. First, if vAM∆t >∆Sfor distance ∆ Sdiscussed\nbelow, we set ∆ t= ∆S/vAM. We then compute τn, the\nlocal mean-free time at the “next” phase-space position\n(⃗ rAM+⃗ vAM∆t,⃗ vAM) and if τl≥2τn, we half ∆ tand the6\nprocess of adjusting ∆ trepeats. The process halts when\nneither inequality is met and we accept ∆ t. We observe\nthat sτlis the upper bound to the accepted time step.\nThe constraint vAM∆t <∆Sis required as alkali-metal\natoms entering the jet far from the jet nozzle are in re-\ngions of relatively low4He number density. Using only\nthe local mean-free time to determine ∆ tcan incorrectly\nresult in a particle traveling through the jet without un-\ndergoing collisions.\nFinally, the probability of a collision between an alkali-\nmetal atom and a4He atom is pAM= (1/τl+ 1/τn)∆t/2.\nIn our simulations, we use s= 0.1 and ∆ S= 1 mm.\nThese values for sand ∆ Sensure that the likelihood\nof a collision is always less that 15 % and that we can\ncompute sufficiently accurate statistical averages within\na reasonable amount of time on the computational re-\nsources available to us.\nA random number Pbetween 0 and 1 is now generated\nfrom the uniform probability distribution. If P ≥ pAM\nthen the alkali-metal atom moves from ( ⃗ rAM,⃗ vAM) to\n(⃗ rAM+⃗ vAM∆t,⃗ vAM). IfP< p AMa AM-4He collision\noccurs at ( ⃗ rAM,⃗ vAM). We then generate a helium veloc-\nity⃗ vsampled from distribution PHe(⃗ rAM,⃗ v) in Eq. (1) for\nvelocities |⃗ vAM−vHe,tvˆrAM|> vcutoff, while for other ve-\nlocities we sample from vrelσAM−He(E)PHe(⃗ rAM,⃗ v). This\nlatter sampling technique is the DBRC algorithm de-\nscribed in Ref. [55] and “favors”4He velocities where\nvrelσAM−He(E) is large.\nCollisions change the velocities of the atoms conserv-\ning the center-of-mass velocity ⃗ vcmand the length of the\nrelative velocity vrel=|⃗ vrel|, while keeping the atoms at\nthe same location ⃗ rAM. The final velocities ⃗ wAMand⃗ w\nfor the alkali-metal atom and4He, respectively, are most\nconveniently evaluated in center-of-mass and relative co-\nordinates with scattering angle θc∈[0, π] sampled from\nsinθcdσAM−He/dΩ at the relative collision energy E=\nµv2\nrel/2 and azimuthal angle φcuniformly sampled from\n[0,2π]. Some thought then shows that the final veloc-\nity of the alkali-metal atom is ⃗ wAM=µ⃗ wrel/mAM+⃗ vcm,\nwhere the final relative velocity\n⃗ wrel= (sin θccosφcˆx′+sin θcsinφcˆy′+cos θcˆz′)vrel(18)\nwith unit vector ˆ z′= ˆvrel= (nrel,x, nrel,y, nrel,z) par-\nallel to the initial relative velocity and unit vector\nˆx′= (−nrel,y, nrel,x,0)/q\nn2\nrel,x+n2\nrel,yperpendicular to\nˆz′. Finally, unit vector ˆ y′= ˆz′׈x′so that the three unit\nvectors form a positively oriented orthonormal basis.\nThe steps of the simulation are then repeated until the\nparticle leaves our spatial bounds. The typical bounds of\nthe simulation are x∈[−6 cm,6 cm], y∈[−6 cm,6 cm]\nandz∈[−4 cm,10 cm]. These bounds are motivated by\nthe4He jet density profile and the locations of the atomic\noven and skimmer. The net result is an ensemble of com-\nputed trajectories from which observable quantities can\nbe calculated. The number of computed trajectories N\nvaries from a few million to 100 million for converged re-\nsults. A larger Nis required for smaller spatial regions\nFIG. 3. (Top panel) Representative trajectory of seeded7Li\natoms (red curve) from our 3D Monte-Carlo simulations over-\nlaid on a 2D cut through the number density profile of the4He\njet at 200 sccm for the (0 , y, z) plane. The sonic nozzle of the\n4He jet is located at ( y, z) = (0 ,0). The trajectory has been\npost-selected to lie within planes x=−0.1 cm and x= 0.1\ncm. The dashed lines indicate the solid angle within which\nmost7Li atoms from the atomic oven enter our simulation\nregion. (Bottom panels labelled L jwith j= 1, 2, 3, 4, 5, and\n6)7Li number probability distributions as functions of the\n7Li kinetic energy in a moving frame of the4He jet for vari-\nous small areas L jalong but perpendicular to the trajectory\nshown in the top panel. The vertical axes of the six panels\nare on different scales and should not be compared.\nor velocity intervals. The simulation is written in Python\nand utilizes the Numba library for optimized performance\n[56].\nIII. RESULTS\nA. Seeding with7Li\nThe representative trajectory of seeded7Li atoms\nwhich exit the skimmer projected onto the yzplane is\nshown in the top panel of Fig. 3 for a4He flow rate of 200\nsccm. Here, a4He flow rate of 1 sccm, an abbreviation\nfor standard cubic centimeters per minute, corresponds\nto˙N0= 4.48×1017atoms/s. (The term standard in\nunit sccm reflects standard conditions for temperature\nand pressure of 273.15 K and 101.325 kPa, respectively.)\nPlaces along the trajectory with a larger curvature imply\nplaces with a higher rate of collisions with helium atoms.\nThe six panels at the bottom of Fig. 3 show distributions\nof the kinetic energy E=mLi(⃗ vLi−vHe,tvˆr)2/2 of7Li\natoms in the frame moving along with the local terminal\nvelocity of helium atoms in the jet. Each panel shows7\nFIG. 4. Contour and line graphs of 2D and 1D cuts through\n3D7Li velocity distributions for two positions along the center\nline of the jet at a helium flow rate of 200 sccm. The velocity\ndistributions are relative to the terminal velocity of the4He\natoms at these positions. We have combined data from all\ntrajectories passing within a 1 mm radius circle perpendicular\nto the center line. The top row is for a distance of 2 cm from\nthe sonic nozzle while the bottom row is for a distance of\n10 cm. Simulated7Li velocity distributions (black dots) and\nGaussian profiles fit to the FWHM of the central peak (red\ncurves) as functions of velocity components vz,vy, and vx\nfor the left, middle, and right columns, respectively. The top\nrow of panels shows velocity distributions taken along the\ncenter line of the helium jet at a distance of 2 cm from the\nnozzle, while the bottom row is at a distance of 10 cm. For\neach line graph, we specify the fitted FWHM velocity vfitas\nTfit=mLiv2\nfit/(2k).\na distribution of kinetic energies of atoms as they pass\nthrough a rectangular region perpendicular to and cen-\ntered on points along the trajectory in the top panel of\nFig. 3. The region extends by ±0.1 cm into the xdirec-\ntion. Its length in the y-zplane is 0.4 cm.\nThe distribution of kinetic energies as the7Li atoms\npass through area L1 and begin to enter the jet is a\nMaxwell-Boltzmann distribution with a temperature of\n800 K, equal to the effusive source temperature. For L2,\nthe7Li atoms have started to cool as the distribution\nbecomes more pronounced for smaller E. At the third\nregion, a small second peak at E/k≈10 mK has ap-\npeared indicating that7Li atoms are becoming trapped\nin the jet. This second peak becomes more pronounced\nfor the last three positions along the trajectory with its\npeak position shifting to a smaller E. The captured7Li\natoms are getting colder. It should be recognized that as\nthe L jhave a finite area, the increasing prominence of\nFIG. 5. Percentage of simulated7Li atoms with energy in the\nmoving frame up to the value specified on the x-axis. Results\nare at a distance of 10 cm from the sonic nozzle with the solid\nangle defined relative to the location of the sonic nozzle. (a)\nResults for various solid angles with a helium flow rate of 200\nsccm. (b) Results for various helium flow rates and a solid\nangle of 0.02 sr.\nthe peak near E/k≈1 mK relative to those at higher\nkinetic energies cannot used to estimate the jet capture\nefficiency. For example, for L6, an area nearly perpen-\ndicular to the center line of the jet, we do not account for\nmany of the7Li atoms that fail to be captured. Those\natoms will have a large kinetic energy.\nFigure 4 shows several cuts through simulated three-\ndimensional7Li velocity distributions 2 cm and 10 cm\naway from the sonic nozzle along the center line of the\njet and with a4He flow rate of 200 sccm. First, not un-\nexpectedly given the complex device geometry in Fig. 1,\nthe velocity distributions are asymmetric, can have large\nwings, and are different along the three directions. Sec-\nond, the lithium atoms velocity distributions become nar-\nrower for larger distances away from the sonic nozzle and,\nfinally, the central peak is more prominent relative to the\nwings.\nAs the7Li kinetic energy and velocity profiles do not\nfollow a Maxwell-Boltzmann distribution, extracting a\ntemperature is dubious. Nevertheless, we can define an\neffective temperature for the captured atoms by fitting8\nthe central peaks to Gaussian distributions as shown in\nFig. 4. At a distance of 10 cm from the nozzle, results\nindicate similar temperatures as that of the helium. It\nis desirable to know what percentage of the injected7Li\natoms cool to these effective temperatures. As this is\ncomplicated by the asymmetry of the velocity distribu-\ntions we instead quantify the cumulative fraction of total\nsimulated particles with a kinetic energy in the moving\nframe less than a specified value E. We define the seed-\ning efficiency η(E,Ω) as this fraction of particles which\nare also directed within a solid angle Ω = πr2/L2; i.e.\nthrough a circle of radius rcentered on the beam axis at\na distance Lfrom the nozzle.\nIn Fig. 5, the seeding efficiency is given for a variety\nof flow rates and solid angles at a distance of L= 10 cm\nfrom the nozzle. A distance of 10 cm is chosen as most\nof the cooling has occurred by this point. Additional\ndiscussion of cooling versus distance from the nozzle can\nbe found in the appendix. For our source geometry, the\nskimmer can extract atoms within a solid angle of ≈0.02\nsr. As simulations show that the seeding efficiency in-\ncreases with solid angle, a larger extractable flux could\nbe obtained with a different skimmer geometry. However,\nthis also results in a larger flux of helium atoms leaving\nthe skimmer. For reference, with a solid angle of ≈0.02\nsr, a helium flow rate of 200 sccm, and the density distri-\nbution given by Eq. 7, the gas load of helium leaving the\nskimmer is ≈0.033 Torr L/s. In practice, careful con-\nsideration should be taken to ensure acceptable vacuum\npressures outside of the cryogenic region.\nB. Comparison with7Li experiments\nIn the experimental realization, studied in Ref. [10],\nlocal7Li density and velocity distributions were deter-\nmined with fluorescence spectroscopy on transitions be-\ntween magnetic sublevels of the F= 2 hyperfine state\nof the electronic 1s22s(2S) ground state to three of the\nfour hyperfine components F′of the excited 1s22p(2P3/2)\nstate. This is the D 2line in7Li. We measure laser-\ninduced fluorescence as a function of laser frequency ν\neither with a linearly polarized probe laser parallel to\nthe center line of the jet and propagation direction −ˆz\n(laser beam ρ=∥) or a linearly polarized probe laser per-\npendicular to the center line and propagation direction\n+ˆy(laser beam ρ=⊥). The laser perpendicular to the\njet is located 4.1 cm from the sonic nozzle.\nIn our simulations, the laser spatial intensity profile\nIρ(⃗ r) of lasers ρ=||and⊥, propagating parallel and\nperpendicular to the center line of the jet, respectively,\nare modeled as rectangles with a width along xequal to\ntwice the beam waist, w0= 0.41 cm, along their prop-\nagation direction. The height of the rectangle along y\nis generally chosen to be small such that variations in\nthe laser intensity can be neglected. Inside the rectangle,\nthe intensity is constant while outside it is zero. More-\nover, the lasers operate at sufficiently low intensities thatoptical pumping effects are negligible.\nFrom the ensemble of trajectories, we compute the dis-\ntribution PAM(x, y,⃗ v AM) of positions xandyat which\nparticles pass through a surface perpendicular to the jet\naxis at a distance Lfrom the nozzle, with ⃗ vAMtheir ve-\nlocity at that position.\nThe simulated spectral profile for laser beam ρis\nSρ(∆) = κZ\nAdxdyZ\nd3⃗ vAMPAM(x, y,⃗ v AM) (19)\n×Wρ(⃗ vAM)3X\nF′=1DF′Iρ(⃗ rAM)\n1 + 4[∆ −⃗kρ·⃗ vAM/h−∆hf(F′)]2/γ2,\nwhere κis a constant of proportionality. Since we\ndo not simulate absolute signal levels, we adjust κsuch\nthatR\nSρ(∆) d∆ = 1 for both experiment and simula-\ntion. ∆ = ν−ED2/his the laser detuning from the\naverage transition frequency from F= 2 to F′= 1,2,3\nweighted by the respective transition strength. The in-\nterrogation area Ais a rectangle with a height equal to\n0.60(2) mm. The width is equal to the laser beam waist.\nWe have found that results are insensitive to variations\nin the width of the rectangle indicating negligible varia-\ntions in the velocity profiles vyandvzalong x. As such\nwe do not account for the laser intensity profile along x\nfor computing simulated spectral profiles. Here, his the\nPlanck constant. Moreover, the sum is over the optically\nallowed hyperfine levels F′of the 1s22p(2P3/2) state, and\n∆hf(F′) = +9 .619 MHz, +3 .730 MHz, and −5.656 MHz\nare hyperfine frequency shifts for the F′= 1, 2, and\n3 levels of the7Li 1s22p(2P3/2) state, respectively [57].\nThe relative strengths DF′are 1 /20, 1 /4, and 7 /10 for\nF′= 1, 2 and 3, respectively, to account for the different\nline strengths of the F= 2 to F′transitions. The natural\nlinewidth of the7Li 1s22p(2P3/2) state is γ= 5.87 MHz\n[58].\nThe inner product ⃗kρ·⃗ vAMin the denominator of the\nLorentzians in Eq. (19) accounts for Doppler shifts. Here,\nunit vector ˆkρ=⃗kρ/kρgives the direction of laser ρ\nand wavenumber kρis given by the dispersion relation\nhν=ℏckρwith speed of light in vacuum cand re-\nduced Planck constant ℏ. It is worth remembering that\nthe temperature of captured alkali-metal atoms is much\nsmaller than mAMv2\nHe,tv/(2k) and for laser beam ρ=||\nand7Li atoms, k||vHe,tv/h= 310 MHz. Weight function\nWρ(⃗ vAM) in Eq. (19) accounts for the reduced number of\nfluorescence photons emitted from faster-moving atoms\nthat spend less time in the laser beam. For both paral-\nlel and perpendicular lasers, we use Wρ(⃗ vAM) = 1 /vAM\nas the detected volume 4.1 cm from the sonic nozzle is\nsmall.\nLine-shapes for both experiment and simulation are\ngiven in Fig. 6. Line-shapes at this distance show a clear\nasymmetry. Simulated profiles reproduce the measured\nasymmetry while also predicting a narrower central peak.\nFor the longitudinal profiles, we estimate an optical den-\nsity of ≈0.3 over the distance of our chamber ( ≈2 m).9\nFIG. 6. Experimental7Li fluorescence spectra (black markers) and simulated spectra (red curves) as functions of the laser\ndetuning ∆ from the average transition frequency of the F= 2 to F′= 1,2,3 transitions of the D 2line of7Li weighted by the\nrespective transition strength. Transverse spectra taken 4 .1 cm downstream from the nozzle are shown along the top row (a),\nwhile longitudinal spectra are shown along the bottom row (b). From left to right spectra are taken for six, increasing4He flow\nrates, indicated above the top row in units of sccm.\nThe non-negligible absorption effects may in part explain\nthe larger discrepancy for the longitudinal profiles.\nOne additional source of broadening that we have not\nincluded is laser frequency noise. The FWHM of the\nlaser line-width over the integration time of the atoms is\nabout 2 MHz. As this is small compared to the other\nbroadening mechanisms, there is no apparent change in\nthe simulated spectral profiles when Sρ(∆) is convolved\nwith the laser noise profile.\nA point of interest in our apparatus is maximizing\nseeded lithium that is extracted by a 2.54 cm diameter\nskimmer located 16 cm from the nozzle as shown in Fig.\n1. Simulations indicate that flow rates around 200 sccm\nfor a seeding distance of 1.7 cm maximize extractable\nlithium. To compare to experiment, the injected lithium\nflux is measured with zero helium flow. This allows us to\ncompute the average simulated alkali-metal atom num-\nber density that is within the projected area Aof the\nskimmer using the following:\nnsim=Φ\nNANcX\ni1\nvi, (20)\nwhere Φ is the measured experimental alkali-metal-atom\nflux entering the cryogenic region, Nis the total number\nof simulated alkali-metal particles, viis the speed of sim-\nulated particles within the capture region, and Ncis the\nnumber of simulated particles within A. We compare\nthe simulated and measured total density at all veloci-\nties. This comparison is made because the experimental\nmeasurements cannot distinguish if an atom is within a\nnarrow velocity range in both the transverse and longi-\ntudinal directions.Results using our transverse probe for a variety of he-\nlium flow rates at a distance of 4.1 cm from the nozzle\nare given in Fig. 7. At this distance, the projection of\nthe skimmer corresponds to a circle of radius ≈0.33 cm.\nIn Fig. 7(a), the number density for a circle centered\nat (x, y) = (0 ,0) is given with the simulation correctly\npredicting a maximum average density at a helium flow\nrate of 150 sccm. This result may appear to contradict\nthe seeding efficiencies given in Fig. 5, however, it is im-\nportant to note that the simulated and experimentally\nmeasured density includes all velocity classes extending\nwell past the values shown in Fig. 5. While a flow rate\nof 150 sccm resulted in a larger density within A, the\nsimulation predicts that a flow rate of 200 sccm results\nin more seeded atoms with lower energies in the moving\nframe.\nShown in Fig. 7 (b) is the number density but with the\ncircle centered at the location of the peak density along y\nwith x= 0. For flow rates below ≈150 sccm the location\nof the peak number density occurs above the center-line\nof the jet while for flow rates above ≈150 sccm it occurs\nbelow the center-line. Since the helium density along yis\nlargest at y= 0, it may initially seem surprising that the\nlocation of the peak7Li density is not necessarily located\naty= 0. This effect of helium flow rate on the position of\nthe extractable lithium atoms is best visualized by exam-\nining the position of simulated particles. The positions\nof all particles which pass the xy plane 4 .1 cm in front\nof the nozzle are given by Fig. 8 (a). At the highest flow\nrates particles are being deflected by the jet while at the\nlowest flow rate, the shadow of particles exiting the aper-\nture in front of the oven is visible. In Fig. 8 (b), we show10\nFIG. 7. Measured (black circles) and modeled (red triangles)\naverage7Li number densities nLiat 4.1 cm from the nozzle\nwithin a circle of radius ≈0.33 cm as a function of helium\nflow rate. Results in (a) correspond to a circle centered at\n(x, y) = (0 ,0) while (b) corresponds to a circle centered at\nthe location of the peak number density along ywith x= 0.\nThe terminal velocity of the lithium beam is 210(2) m/s for a\nhelium flow rate below 220 sccm and 217(2) above 220 sccm.\nThe simulated7Li number densities follow from the measured\nflux leaving the lithium source of 1 .4(2)×1014s−1at a lithium\noven temperature of 800 K.\nthe position of only particles with energies ≤k×50 mK\nin the moving frame. Notably, if one wishes to maximize\nthe extraction of thermalized lithium, it would be advan-\ntageous to not extract at the center-line of the jet but\nbeneath it.\nFor all simulated density results, we find that the un-\ncertainties of the capture rates obtained with our MC\nresults are larger than those of the differential cross-\nsections. This simply reflects the small theoretical un-\ncertainties in the differential cross section for E > 10\nK.\nDue to the large reservoir of helium, we assume no ap-\npreciable heating of the jet and expect this to hold true\nfor sufficiently low lithium flux. For reference, with a\nseeded flux on the order of 1014s−1, there are approx-\nimately 500,000 helium atoms per lithium atom for 150\nsccm of helium flow. For an oven temperature of 800 K,\nthe average energy of a lithium atom entering the jet is\n≈k×1070 K. If this energy were uniformly distributed\namong the helium atoms it would deposit only k×2 mK\nFIG. 8. 2D histogram of the positions of simulated7Li atoms\nthat pass the xy plane 4 .1 cm from the nozzle. The red circle\nindicates particles that are within the projection of the skim-\nmer. The numbers above each plot indicates the4He flow\nrate. (a) Results for all7Li energies in the moving frame. (b)\nResults for7Li energies ≤k×50 mK in the moving frame.\nper helium atom. To further examine how energy is de-\nposited in the jet we record the location as well as the\npre and post-collision velocity of the lithium and helium.\nA map of the energy deposition can then be constructed\nand is given by Fig 9 (a), which shows that most of the\nenergy deposited by the lithium is deposited before the\nlithium reaches the center of the jet.\nNaively it might seem like a large fraction of energy\nthat is deposited in the periphery of the jet would re-\nmain there as the jet expands. This would limit heating\neffects along the center of the expansion. However, we\nhave examined this further and found that it is only par-\ntially true. Depending on the energy transferred from the\nlithium atom and scattering angles, an energetic helium\natom can have a significantly larger mean free path rela-\ntive to the local mean free path of less energetic helium\natoms.\nTo quantify this effect, the simulation is repeated us-\ning the location and post-collision velocity of the helium\nwithin the blue square in Fig 9 using the4He-4He cross-\nsection from Ref [42]. The helium atoms then travel un-\ntil they leave the simulation bounds or undergo a single11\nFIG. 9. (a). Location and relative magnitude of energy de-\nposited by7Li atoms from x=−0.1 cm to x= 0.1 cm. (b)\nRelative magnitude of energy deposited for the next He-He\ncollision for helium atoms which underwent a collision with a\nlithium atom in the blue square.\ncollision. Fig. 9 (b) shows the energy deposition for this\ngroup of helium atoms. Helium atoms containing approx-\nimately 30% of the energy deposited in the blue square\nleave the simulation bounds without undergoing further\ncollisions. While this method does not allow us to de-\ntermine how the remainder of the energy is ultimately\ndistributed in the jet, it illustrates that a non-negligible\namount of energy is effectively removed. It also suggests\nthat heating of the center of the jet can come from en-\nergy that is initially deposited in the periphery. For suf-\nficiently high lithium flux there will be heating of the\njet and the assumptions made for the jet density and\nvelocity distributions will no longer be valid. To date,\nwe have been able to seed the jet with a lithium flux on\nthe order of 1015s−1and extract a lithium beam with\na longitudinal temperature of 7(3) mK and a transverse\ntemperature <20 mK [10].\nC. Seeding with87Rb\nAs our method is intended to be applicable for a vari-\nety of species, there is interest in examining the efficiency\nfor seeding heavier particles into the jet. Here we present\nsimulations of seeding87Rb into the helium jet. We as-\nsume the same solid angle as our lithium source but at\nan oven temperature of 550 K due to rubidium’s signif-\nicantly higher vapor pressure. Due to having a higher\ninitial momentum, seeding becomes more challenging as\nmore collisions are required to thermalize the species into\nthe jet.\nUnder these conditions, optimal seeding efficiency is\nfound when the collisional thickness is increased by\nchanging the seeding position to 1 cm and increasing\nthe helium flow rate to 300 sccm [59]. This results in\nconsiderably more collisions for rubidium as compared\nto lithium as shown in Fig. 10. In Fig. 11, the cooling\nFIG. 10. Probability distributions for the total number of\n4He-7Li and4He-87Rb collisions for atoms at a distance of 10\ncm from the nozzle and within a solid angle of 0.02 sr. The\n4He flow rate is 200 sccm for7Li and 300 sccm for87Rb.\nprocess of rubidium injected into the helium jet is shown.\nAdditionally, the position of all particles that pass the xy\nplane 4.1 cm in front of the nozzle are shown in Fig. 12.\nAs with lithium, a large distribution of kinetic energies\nis observed far from the nozzle. However, as the cooling\nprocess is slower with respect to distance from the nozzle,\na clear central peak at k×mK energies is not observed.\nThis is further reflected in the velocity distributions and\nvelocity phase space plots given by Fig. 13 where the\nasymmetry in vzandvyprevents reliable fitting to a cen-\ntral peak. The qualitative difference in the shape of the\ndistributions as compared to lithium can be in part un-\nderstood due to the lower initial speed of the rubidium\natoms. The mean speed for a rubidium source at 550 K\nis 435 m/s, compared to 1800 m/s for a lithium source at\n800 K. A rubidium atom that is entrained and partially\nthermalized along the center-line of the jet can remain\nalong the center-line longer than a lithium atom with a\nhigher velocity.\nDifferences in the velocity distributions of vx,vy, and\nvzcan be understood by considering geometric con-\nstraints. The velocity distributions are analyzed over a\n1 mm radius circle centered at ( x, y) = (0 ,0). As the\natoms are injected from an oven beneath the jet centered\natx= 0, this sets a limit on the maximum value of vxfor\na given initial speed. The same constraint does not apply\nforvz. It is interesting that the distribution for vyhas a\nsharp cutoff compared to the other two velocity profiles.\nAll rubidium atoms initially have a large vyin the mov-\ning frame. Since the mass of87Rb is ≈20 times larger\nthan that of4He, this ensures that a87Rb atom deflected\ndownwards has essentially thermalized with the helium.\nThis results in the apparent sharp cutoff and asymmetry\nas there is a larger distribution of partially thermalized\nrubidium atoms.\nThe seeding efficiency for rubidium for a variety of flow\nrates and solid angles is shown in Fig. 14 for a distance of\nL= 10 cm from the nozzle. If a velocity filter was used to\nremove faster moving rubidium, then it is predicted that\nat 300 sccm of helium flow, ≈1% of the seeded rubidium12\nFIG. 11. (Top panel) Representative trajectory of seeded\n87Rb atoms (red curve) from our 3D Monte-Carlo simulations\noverlaid on a 2D cut through the number density profile of the\n4He jet at 300 sccm for the (0 , y, z) plane. The sonic nozzle\nof the4He jet is located at ( y, z) = (0 ,0). The trajectory\nhas been post-selected to lie within planes x=−0.1 cm and\nx= 0.1 cm. The dashed lines indicate the solid angle within\nwhich most87Rb atoms from the atomic oven enter our sim-\nulation region. (Bottom panels labelled L jwith j= 1, 2, 3,\n4)87Rb number probability distributions as functions of the\n87Rb kinetic energy in a moving frame of the4He jet for var-\nious small areas L jalong but perpendicular to the trajectory\nshown in the top panel. The vertical axes of the four panels\nare on different scales and should not be compared.\nFIG. 12. 2D histogram of the position of simulated87Rb\natoms that pass the xy plane, 4.1 cm from the nozzle. The\nred circle indicates particles that are within the projection of\nthe skimmer at this distance.\nwould leave the skimmer with energies less than 75 mK\nin the moving frame of the helium.\nIV. CONCLUSION\nWe have simulated post-nozzle seeding into a super-\nsonic helium jet. Conditions that maximize seeded flux\nFIG. 13. Contour and line graphs of 2D and 1D cuts through\n3D87Rb velocity distributions for two positions along the\ncenter line of the jet at a helium flow rate of 300 sccm. The\nvelocity distributions are relative to the terminal velocity of\nthe4He atoms at these positions. We have combined data\nfrom all trajectories passing within a 1 mm radius circle per-\npendicular to the center line. The top row is for a distance\nof 2 cm from the sonic nozzle while the bottom row is for a\ndistance of 10 cm. The top row of panels shows velocity dis-\ntributions taken along the center line of the helium jet at a\ndistance of 2 cm from the nozzle, while the bottom row is at\na distance of 10 cm.\nagree with experimental measurements as well as pro-\nvide reasonable agreement to measured spectral profiles.\nResults show that captured lithium and rubidium atoms\nhave energies in the moving frame of the jet on the or-\nder of single to tens of k×mK. Since efficient seeding of\nrubidium is indicated, it may be possible to seed other\nheavier species such as molecules. To achieve efficient\nseeding the species would need to have collision cross\nsections with helium which increase drastically at low\ncollision energies. All of the alkali atoms have these prop-\nerties but it is less clear if this is true for other species.\nIf so, then this method could be used to form an in-\ntense beam of cold atoms and molecules by extracting\nthe seeded species with a magnetic or electrostatic lens\n[10, 60]. Such beams can be useful for studies of cold\ncollisions, atom optics, and precision measurements. Al-\nternatively, cold collision studies could be carried out in\nthe region close to the nozzle.\nThe simplified model of the jet, while not capturing\nall relevant features, provides a suitable starting point\nfor understanding seeding dynamics. A 3D Direct Simu-\nlation Monte Carlo (DSMC) approach that simulates the13\nFIG. 14. Percentage of simulated87Rb atoms with energy\nin the moving frame up to the value specified on the x-axis.\nResults are at a distance of 10 cm from the sonic nozzle with\nthe solid angle defined relative to the location of the sonic\nnozzle. (a) Results for various solid angles with a helium flow\nrate of 300 sccm. (b) Results for various helium flow rates\nand a solid angle of 0.02 sr.\nhelium jet and lithium beam would accurately capture\nthe jet dynamics but would be significantly more com-\nputationally expensive [38]. It would however allow for a\ndetailed study of heating dynamics. Additionally, adding\ninelastic collisions to the model would allow for the study\nof rotational cooling of seeded molecules [13, 61]. If ef-\nficient rotational cooling is possible, then our approach\ncould enable cooling similar to what is achieved in buffer\ngas-based sources but with considerably better transla-\ntional cooling [62].\nV. ACKNOWLEDGEMENTS\nWe gratefully acknowledge the financial support of\nthis work by Fondren Foundation and the Army Re-\nsearch Laboratory Cooperative Research and Develop-\nment Agreement number 16-080-004.\nVI. APPENDIX\nWe have compared our static model for the helium jet\nto direct simulation Monte Carlo (DSMC) calculations of\nFIG. 15. Angular number density profile of a4He expan-\nsion with simulated results (markers) and analytic profile (red\ncurve) as a function of angle θ. Results are for4He densities\nat the nozzle opening with diameter dN= 0.2 mm and nozzle\nopening temperatures of 4 ×1019cm−3and 300 K (brown tri-\nangles), 2 ×1019cm−3and 100 K (black circles), and 1 ×1019\ncm−3and 200 K (blue squares).\nhelium expansions. These are performed using the DS2V\nsoftware, a direct simulation Monte Carlo program writ-\nten by G. Bird [63]. For the computational resources\navailable to us, simulations of the helium expansion are\nlimited to source conditions which correspond to sub-\nstantially lower collision rates than what is expected in\nour system. Furthermore, the simulations begin at the\nopening of the nozzle where the source conditions such\nas temperature, density, and velocity are specified at the\nnozzle opening and not equal to the stagnation conditions\nwell inside the nozzle reservoir. Nevertheless, the colli-\nsion rate and mean free path are such that the flow begins\nwell into the continuum regime. The criteria for contin-\nuum flow can be described using the Knudsen number\n(Kn), given by\nKn =λmfp\nL, (21)\nwhere λmfpis the mean free path and Lis a characteristic\nlength scale. Generally, the condition for continuum flow\nis satisfied when Kn <0.01 [41]. The Knudsen numbers\nfor simulations of the helium jet at the opening of the\nnozzle, where Lis the nozzle diameter, are such that\nKn<0.005.\nFor a given nozzle shape and opening nozzle diameter,\nthe angular density profile is expected to be independent\nof the stagnation density and temperature so long as the\nflow begins in the continuum regime. As such, the he-\nlium expansion can be simulated to large angles from the\ncenterline to compare with Eq. (4). Results comparing\nthe analytic profile to simulations for various nozzle tem-\nperatures and densities are shown in Fig. 15. Agreement\nwith Eq. (7) is generally within 6% for angles θup to 30\ndegrees with inaccuracies of up to 30% at larger angles\naround 55 degrees.14\nFIG. 16. Simulated4He number density profile (a) and flow\nspeed profile (b) for a helium expansion with a nozzle opening\ntemperature of 100 K and nozzle diameter of 0.2 mm.\nDensity and speed profiles are presented in Fig. 16 for\na sonic nozzle with an opening diameter of 0.2 mm. The\nflow speed decreases from the expected terminal veloc-\nity for large angles with the speed dropping by 5% from\nθ= 0 to θ= 70 degrees. At large angles the collision\nrate is not sufficient to maintain local thermal equilib-\nrium and the conversion of random thermal motion into\nforward kinetic energy ceases after a short distance. As\nour source has a greater collision rate, the variations in\nflow speed in Fig. 16 are expected to be an upper bound\nfor variations we might expect. Nevertheless, the dis-\nagreement with Eq. (3) at large angles is negligible as\nthe AM-4He relative velocities are dominated by the AM\nvelocity at large angles.\nIn the lithium seeding simulation, the calculation of\nthe helium jet parameters relies on the assumption that\nour region of interest is within the continuum-flow regime\nof the jet. To examine the validity of this assumption, we\ninvestigated the transition of a helium expansion from the\ncontinuum-flow regime to the free-molecular flow regime\nusing the DS2V software. The ratio T⊥/T∥of transverse\nto longitudinal temperature along the centerline of the jet\nis commonly used as a parameter to quantify the level of\nthermal equilibrium as the jet transitions from contin-\nuum to free-molecular flow [33, 47]. It has been shown\n[41] that the continuum flow assumptions are valid if the\ncollision rate Γ is much larger than the proportional rate\nof change of density nHe, such that\nΓ≫1\nnHednHe\ndt. (22)\nThis motivated [64] the definition of a non-equilibrium\nparameter Pdefined as\nP=1\nΓ\f\f\f\fd(lnnHe)\ndt\f\f\f\f, (23)\nwhich serves as an approximate universal non-\nequilibrium parameter when investigating the behavior\nFIG. 17. Temperature ratio plotted as a function of the non-\nequilibrium parameter Pfor various conditions at the opening\nof the nozzle. DSMC simulated results (markers) are fit to an\ninverse power law (solid lines). (a) Nozzle opening density\nfixed at 2 ×1019cm−3with varying starting temperatures.\n(b) Nozzle opening temperature fixed at 200 K with varying\nnozzle opening densities.\nFIG. 18. Value of the non-equilibrium parameter Pfor when\nT⊥/T∥= 0.5 for various starting conditions in the DSMC\nsimulation. Results for a nozzle opening density of 2 ×1019\ncm−3with varying nozzle opening temperature are show in\n(a) while varying nozzle opening densities for a nozzle opening\ntemperature of 200 K are shown in (b).\nofT⊥/T∥for various jet conditions [65, 66]. As we are\ninterested in near-continuum flow, Eq. (23) can be ex-\npressed as\nP=vsλmfp\nvnHe\f\f\f\fdnHe\ndr\f\f\f\f, (24)\nwhere vsis the flow speed in the lab frame and vis the\nmean velocity in the moving frame. Recognizing from15\nFIG. 19. Pparameter versus distance from the helium nozzle\nfor various helium flow rates. The Pparameter is calculated\nassuming an adiabatic expansion of the helium.\nEq. (7) that\n1\nnHe\f\f\f\fdnHe\ndr\f\f\f\f=2\nr, (25)\nPcan now be defined as\nP=2vsλmfp\nvr. (26)\nUsing the DS2V software, Pis computed using Eq. (26)\nfor various densities and temperatures at the opening of\nthe nozzle. The transverse and longitudinal velocity dis-\ntributions, and thus T⊥andT∥, are also computed. Plots\nofT⊥/T∥vsPare shown in Fig. 17. We consider the lo-\ncation where T⊥/T∥= 0.5 as a reasonable estimate for\nwhen the probability distribution of the expansion can\nno longer be well approximated by Eq. (1). This occurs\nwhenP= 0.60(2) as shown in Fig. 18.\nFrom Eq. (2), (3), and (7), along with the4He-4He\ncross-section from Ref. [42], the parameter Pcan be cal-\nculated for our source conditions as a function of distance\nfrom the helium nozzle. Results for Palong the center-\nline of the expansion for various helium flow rates are\nshown in Fig. 19. The values are calculated assuming an\nadiabatic expansion and will become inaccurate as the\nvalue of Pbecomes large. However, at small to interme-\ndiate values of P, the approximate location where the he-\nlium collisions begin to turn off can be estimated. Exam-\nining the location where P= 0.6, it can be seen that the\njet is expected to remain collisional well past the region\nwhere seeding occurs. In fact, at the highest flow rate of\n300 sccm, the helium is expected to continue to cool past\n20 cm from the nozzle. This would amount to a temper-\nature on the order of 100 µK. It is unlikely, however, that\nthese temperatures will be reached due to cluster forma-\ntion. At high helium flow rates, cluster formation of the\nhelium can occur releasing heat of condensation, setting\na limit on the cooling. Some of us describe in Ref. [10]\nthat at our higher helium flow rates, we are likely in a\nregime where some cluster formation is occurring. While\ndirect observation of clustering in our jet has not been\nobserved, it has been observed in prior experiments with\nFIG. 20. Seeding efficiency of7Li atoms with energies in the\nmoving frame up to and including (a) k×1 mK, (b) k×5 mK,\nand (c) k×10 mK versus distance from the nozzle. For each\nsubplot, the seeding efficiency is shown for three models of the\nhelium expansion where the minimum helium temperature\nis set to 0 mK, 1 mK, and 3 mK. The seeding efficiency is\nspecified for a solid angle of 0.02 sr and a helium flow rate of\n200 sccm.\npure helium expansions. In these experiments, the low-\nest temperatures that have been achieved are typically\naround 0.5 mK to 1 mK [9, 67–69]. As a result, it seems\nreasonable to conclude that the minimum temperature\nof the4He jet without the heat load of the alkali atoms\nis likely comparable to past works.\nWhile cluster formation is not included in the simula-\ntion to determine the ultimate limit on the helium tem-\nperature, it is still desirable to explore how variations in\nthe helium temperature affect the seeding efficiency. To\ndo this, an artificial temperature floor in the expansion is\nintroduced such that the helium temperature decreases\naccording to Eq. (2) until it reaches a minimum value\nafter which the temperature is constant. This is not\na physically accurate way to model heating from clus-\nter formation or decoupling as the jet transitions from\ncontinuum to free molecular flow. It does, however, al-\nlow us to explore with a simple model how modifications\nto the helium velocity distributions affect seeding effi-\nciency. Three different minimum temperatures of 0 mK,\n1 mK, and 3 mK are used in the simulations. Results for\nthe seeding efficiency versus distance from the nozzle for16\nFIG. 21. Seeding efficiency of87Rb atoms with energies in\nthe moving frame up to and including (a) k×10 mK, (b)\nk×50 mK, and (c) k×100 mK versus distance from the nozzle.\nFor each subplot, the seeding efficiency is shown for three\nmodels of the helium expansion where the minimum helium\ntemperature is set to 0 mK, 1 mK, and 3 mK. The seeding\nefficiency is specified for a solid angle of 0.02 sr and a helium\nflow rate of 300 sccm.these minimum temperatures are shown in Fig. 20 and\nFig 21. Here, the seeding efficiency is displayed as the\npercentage of injected alkali-metal atoms within a solid\nangle of 0.02 sr and with energies up to and including\nk×1 mK k×5 mK and k×10 mK for7Li and k×10\nmK, k×50 mK and k×100 mK for87Rb. Unsurpris-\ningly, variations in the helium temperature only have a\nsubstantial impact on alkali-metal atoms with low en-\nergies in the moving frame. Since this occurs after en-\ntrainment, the source conditions that maximize seeding\nefficiency are unaffected by these variations in the helium\ntemperature. Interestingly, a higher helium temperature\nresults in a high seeding efficiency for87Rb at low ener-\ngies. A consequence of the4He-87Rb resonance at around\n10 mK. 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Postal 70-543, 04510 Mexico DF, Mexico\nNovember 2023\nAbstract\nWe consider the propagation of photons in the background of a Bose-Einstein (BE) condensate of a\ncharged scalar field, by extending a method recently propose d to treat the propagation of fermions in a\nBE condensate. We determine the dispersion relations of the collective modes of the system, as well as the\nphoton polarization tensor and the dielectric constant tha t result after the symmetry breaking associated\nwith the BE condensation in the model. Two modes correspond t o the transverse photon polarizations,\nand their dispersion relations have the usual form of the tra nsverse photons in a plasma. The other\ntwo modes, which we denote as the ( ±) modes, are combinations of the longitudinal photon and the\nmassive scalar field. The dispersion relation of the ( −) mode decreases as a function of the momentum\nin a given range, and the corresponding group velocity is neg ative in that range. We also determine the\nwavefunctions of the ( ±) modes, which can be used to obtain the corrections to the dis persion relations\n(e.g., imaginary parts due the damping effects) and/or the eff ects of scattering, due to the interactions\nwith the excitations of the system. The results can be useful in various physical contexts that have been\nconsidered in the literature involving the electrodynamic s of a charged scalar BE condensate.\n1 Introduction and motivation\nMany extensions of the standard electroweak theory depend on t he presence of additional scalar particles,\nboth neutral and charged, that interact with the standard mode l particles, in particular with the neutrinos\nand photons. Such interactions typically have implications in the cont ext of nuclear, astrophysics, and\ncosmological plasmas that may contain a background of the scalar p articles. As is well known, the intrinsic\nproperties of neutrinos and photons propagating in such a medium, such as the dispersion relations and/or\nelectromagnetic properties, are modified in ways that lead to remar kable and observable effects.\nIn previous works we have carried out a systematic calculation of th e neutrino dispersion relation, in-\ncluding the damping and decoherence effects in models in which the neu trinos interact with a scalar ( φ)\nand fermion ( f) via a coupling of the form ¯fRνLφ, or just with neutrinos themselves ¯ νc\nRνLφ(see Ref. [1]\nand references therein). Couplings of those forms produce addit ional contributions to the neutrino effective\npotential when the neutrino propagates in a background of φandfparticles, or a pure neutrino background,\nwhich can effects in various neutrino physics contexts. More recen tly, we have considered the problem of\ndetermining the corresponding quantities (e.g, effective potential and/or dispersion relation and damping) of\n∗nieves@ltp.uprrp.edu\n†sarira@nucleares.unam.mx\n1a fermion that propagates in a thermal background that contains a scalar Bose-Einstein (BE) condensate[2].\nThe method can be applied to the case of a neutrino propagating in su ch a background, and the problem of\nfermions propagating in such backgrounds can be relevant in other applications in astrophysical, heavy-ion\ncollisions and dark-matter contexts[3, 4, 5, 6, 7, 8, 9].\nOur motivation in the present work is to consider, in a similar vein, the p ropagation of photons in a\nthermal background that contains a BE condensate of a charged scalar. The problem of the propagation of\nphotons in a medium is a classic subject which in the language of plasma p hysics can be treated within the\nformalism of the kinetic Boltzmann equation, and in other contexts a nd/or more generally using the frame-\nwork of thermal field theory (TFT). Various aspects and applicatio ns of the problem of photon propagation\nin a BE condensate of the charged scalars have been studied in prev ious recent works, in various contexts\nand using different approaches, for example the electrodynamics o f fermions and charged scalar bosons[10],\nCherenkov effect[11], exciton-polaritons[12], the BE condensation of photons[13, 14] and the reduction of the\nphoton velocity in atomic gases[15].\nOur goal in the present work is to apply the same technique used in Re f. [2], to the case in which the\nscalaris electromagneticallychargedand study the propagationof photons in the BE condensate background\nof such scalars. The field theoretical method we used in Ref. [2] to t reat the BE condensate, and apply here,\nhas been discussed by various authors[16, 17, 18]. As we will see, t he implementation proposed here to treat\nthe propagation of photons allows us to determine succinctly the dis persion relations of the collective modes\nthat result after the symmetry breaking associated with the BE co ndensation in the model. We are also able\nto derive the polarization tensor and the dielectric constant. The m ethod also allows us to determine in a\nvery efficient way the mixing parameters between the longitudinal ph oton (plasmon) and the physical scalar\nfield that make up the propagating modes involving the plasmon. Thes e are relevant for the calculation of\nthe thermal corrections to the dispersion relations (e.g., imaginary parts due to damping effects) and the\nother physical quantities due to interactions with the thermal exc itations of the BE condensate. The results\ndescribed above are obtained in a general way, within the context o f the model, but not tied to any specific\napplication. In this way they can be useful and pave the way for fur ther development in various contexts\nand applications such as those considered in the references mentio ned above.\nThe plan of the presentation describing the present work is as follow s. In Section 2 we summarize the\nessential features and the symmetry breaking mechanism of the s calar BE condensation model discussed in\nRef. [2], which serves as the basis and motivation for the extension w e present in Section 3 to the charged\nscalar case. In Section 3 we discuss various qualitative properties o f the model that are useful for the\ninterpretation, such as current conservation, symmetry break ing and the spectrum. The propagating modes\nand the corresponding dispersion relations are determined in Sectio n 4. As we show, two modes correspond\nto the transverse photon polarizations, and their dispersion relat ions have the usual form of the transverse\nphotons in a plasma. The other two modes, which we denote as the ( ±) modes, are combinations of the\nlongitudinal photon and the massive scalar field. Various distinguishin g aspects of the results obtained are\ndiscussed in Section 5. A particularly noticeable one is the fact that t he dispersion relation of the ( −) mode\ndecreases as a function of the momentum in a given range, and the c orresponding group velocity is negative\nin that range. In Section 6 we obtain the expression for the photon polarization tensor, or self-energy. This\nprovides an alternative way to obtain the dispersion relations, but m ore importantly it has also the virtue\nthat it provides an expression for the dielectric function of the sys tem, which is useful on its own merits, for\nexample for studying also the response of the system to external fields. Finally in Section 7 we determine the\nmixing parameters of the spinors of the ( ±) modes. These parameters can be used to obtain the corrections\nto the dispersion relations (e.g., imaginary parts due the damping effe cts) and/or the effects of scattering,\ndue to the interactions with the excitations of the system. In Sect ion 8 we summarize our conclusions and\noutlook.\n2 Scalar BE condensate model\nAs a preamble to the presentation of the model for the charged sc alar coupled to the photon, we first\nsummarize the essential features of the scalar BE condensation m odel discussed in Ref. [2], as well as its\njustification.\n2In Ref. [2] we used the following model for the φ,\nLφ= (dµφ)∗(dµφ)−Vφ, (2.1)\nwhere\ndµ≡∂µ−iµuµ, (2.2)\nand\nVφ=m2φ∗φ+λ(φ∗φ)2. (2.3)\nµis a parameter, to be identified with the chemical potential of the φanduµ, which is to be identified with\nthe velocity four-vector of the medium, is\nuµ= (1,/vector0), (2.4)\nin the medium’s own rest frame. To simplify the notation, we write\nvµ=µuµ. (2.5)\nExpanding the dterm in Eq. (2.1), the Lagrangian is\nLφ= (∂µφ)∗(∂µφ)+i[φ∗(v·∂φ)−(v·∂φ∗)φ]−U(φ), (2.6)\nwhere\nU≡Vφ−µ2φ∗φ=−(µ2−m2)φ∗φ+λ(φ∗φ)2. (2.7)\nIfm2> µ2, thisUcorresponds to a standard massive complex scalar with mass m2−µ2. On the other hand,\nifµ2> m2, the minimum of the potential is not at φ= 0, and therefore φdevelops a non-zero expectation\nvalue and the U(1) symmetry is broken.\nWe assume the second option,\nµ2> m2, (2.8)\nand proceed accordingly. Namely, we put\nφ=1√\n2(φ0+φ1+iφ2), (2.9)\nwhere\n/angbracketleftφ/angbracketright ≡1√\n2φ0, (2.10)\nis chosen to be the minimum of\nU0=−1\n2(µ2−m2)φ2\n0+1\n4λφ4\n0. (2.11)\nThus,\nφ2\n0=µ2−m2\nλ. (2.12)\nSubstituting Eqs. (2.9) and (2.12) in Eq. (2.1) we obtain the Lagrang ian for the scalar excitations φ1,2,\nLφ=1\n2/bracketleftbig\n(∂µφ1)2+(∂µφ2)2/bracketrightbig\n+φ2v·∂φ1−φ1v·∂φ2−U(φ), (2.13)\nwhere\nU(φ) =−1\n2(µ2−m2)[(φ0+φ1)2+φ2\n2]+1\n4λ[(φ0+φ1)2+φ2\n2]2. (2.14)\nUsing Eq. (2.12), the quadratic part of Uinvolves only φ1,\nU2=1\n2m2\n1φ2\n1, (2.15)\nwhere\nm2\n1= 2(µ2−m2), (2.16)\n3whileφ2does not appear. However, φ1andφ2are mixed by the vµterm. Therefore the next step is to find\nthe propagator matrix of the φ1,2complex and find the modes that have a definite dispersion relation.\nA convenient way to do it is to use matrix notation,\nˆφ=/parenleftbiggφ1\nφ2/parenrightbigg\n. (2.17)\nThe Lagrangian, in momentum space, can then be written in the form\nL(2)\nφ(k) =1\n2ˆφ∗(k)∆−1\nφ(k)ˆφ(k), (2.18)\nwhere\n∆−1\nφ(k) =/parenleftbiggk2−m2\n12iv·k\n−2iv·k k2/parenrightbigg\n. (2.19)\nThe classical equations of motion are then\n∆−1\nφ(k)ˆφ= 0. (2.20)\nThe dispersion relations of the eigenmodes are given by the solutions of\nD= 0, (2.21)\nwhereDis the determinant of ∆−1\nφ,\nD= (ω2−κ2)(ω2−κ2−m2\n1)−4µ2ω2, (2.22)\nwhere we have used Eq. (2.5). The dispersion relations obtained in th is way can be written in the form\nω2\n±(κ) =κ2+1\n2m2\n+±/radicalbigg\n1\n4m4\n++4µ2κ2, (2.23)\nwhere\nm2\n+=m2\n1+4µ2. (2.24)\nThe zero mass mode, i.e., ω−(0) = 0, is the realization of the Goldstone mode associated with the b reaking\nof the global U(1) symmetry.\nAs discussed in Ref. [2], the motivation for considering this model is ba sed on the following result:\nthe calculation of the effective potential Veff(T,µ) forφcan be carried out in TFT using µ= 0 in the\npartition (and/or distribution) function, but using the µ-dependent Lagrangian Lφdefined in Eq. (2.1).\nThen neglecting the T-dependent terms (that is, at zero temperature), Veff(0,µ) is simply the Upotential\ngiven in Eq. (2.7). In other words, calculations with the φpropagators, φLagrangian and partition function\nforφ,\nZ=e−βH+αQ, (2.25)\ngive the same result as calculating them with the Lagrangian for φ′obtained by using ∂µ→Dµ=∂µ−iµuµ\nand using\nZ=e−βH′. (2.26)\nThe simple way to obtain this result is the following. The evolution equat ion forφis\ni∂tφ=−[H,φ]. (2.27)\nIf now calculate the time derivative of φ′≡eiµtφwe get\n∂tφ′=iµφ′+eiµt∂tφ\n=iµφ′+eiµti[H,φ]\n=iµφ′+i[H,φ′]\n=−i[µQ,φ′]+i[H,φ′]\n=i[H−µQ,φ′]. (2.28)\n4Therefore\ni∂tφ′=−[H−µQ,φ′], (2.29)\nwhich is the result stated above, with the identification of H′. On the other hand, when we express the\npartition function in terms of the field φ′, then\nZ≡e−βH+αQ=e−β(H−µQ)=e−βH′. (2.30)\nThat is, in the calculations involving φ′we use the partition function with H′and zero chemical potential.\nThe operator proof we have presented seems to be applicable to an y field (e.g., a fermion field), not\ninvolving Lagrangian dynamics arguments, and therefore the resu lt shown above holds for any field and not\njust for a scalar field. On the basis of this result, the Lagrangian fo rmulation that we have presented is\njustified, which is the most efficient way to do the dynamics of the φ′, that is finding the evolution equation\nto obtain the dispersion relations and other properties (e.g., the pr opagators) of the eigenmodes.\n3 Charged scalar BE condensate model\n3.1 Lagrangian\nWe extend the model by assuming that the φfield is electrically charged and interacts with electromagnetic\nfield in the usual way. The Lagrangian is then\nL=−1\n4F2+LφA, (3.1)\nwhereFµνis the electromagnetic field tensor and\nLφA= (ˆDµφ)∗(ˆDµφ)−Vφ, (3.2)\nwith[19]\nˆDµ=dµ+iqAµ=∂µ−ivµ+iqAµ, (3.3)\nandVφgiven as before in Eq. (2.3). vµis defined in Eq. (2.5) and we will sometimes write\nˆDµ=∂µ−iωµ, (3.4)\nwhere\nωµ=vµ−qAµ. (3.5)\n3.2 Current\nIt will be useful for later purposes to identify the electromagnetic current, in particular its zeroth component\nor the charge density. The electromagnetic current, defined as\nj(em)\nµ=−∂LφA\n∂Aµ, (3.6)\nis then given by\nj(em)\nµ=iqJµ, (3.7)\nwhere\nJµ=φ∗ˆDµφ−(ˆDµφ)∗φ. (3.8)\nFor consistency we verify that it is conserved as a consequence of the equation of motion.\nFor the equation of motion we first get\n∂LφA\n∂φ∗=iωµ(ˆDµφ)−∂Vφ\n∂φ∗, (3.9)\n5and∂LφA\n∂(∂µφ∗)=ˆDµφ. (3.10)\nThe Lagrange equation\n∂µ∂LφA\n∂(∂µφ∗)−∂LφA\n∂φ∗= 0, (3.11)\nthen gives the equation of motion\n∂µ(ˆDµφ)−iωµ(ˆDµφ)+∂Vφ\n∂φ∗= 0, (3.12)\nor\nˆD2φ+∂Vφ\n∂φ∗= 0. (3.13)\nTo verify\n∂·J= 0, (3.14)\nfrom Eq. (3.8) we get\n∂·J= (∂φ)∗·ˆDφ+φ∗∂·ˆDφ−(∂·ˆDφ)∗φ−(ˆDφ)∗·∂φ. (3.15)\nPutting\n∂µ=ˆDµ+iωµ, (3.16)\nthe four terms with the iωµcancel in pairs, the two terms of the form ( ˆDφ)∗(ˆDφ) cancel as well, and we are\nleft with\n∂·J=φ∗ˆD2φ−(ˆD2φ)∗φ, (3.17)\nwhich is zero by the equation of motion.\n3.3 Spectrum of the model\nTo determine the spectrum of the model we adopt the unitary gaug e, which is the convenient one to use for\nthis purpose. Thus we parametrize φin the form\nφ=1√\n2(φ0+ρ)eiθ/φ0. (3.18)\nThe field θdoes not appear in Vφ, and by a gauge transformation it disappears also from the kinetic t erm.\nTo be clear that we are employing the unitary gauge we will denote by Vµthe transformed vector potential\nVµ=Aµ+1\nqφ0∂µθ. It then follows that\nLφA=1\n2(∂ρ)2+1\n2(v−qV)µ(v−qV)µ(φ0+ρ)2−Vφ, (3.19)\nor, expanding the various factors,\nLφA=1\n2(∂ρ)2+1\n2(qφ0)2V2+1\n2q2V2ρ2+q2φ0ρV2−qv·V(φ0+ρ)2−Uφ, (3.20)\nwith\nUφ=−1\n2(µ2−m2)(φ0+ρ)2+1\n4(φ0+ρ)4. (3.21)\nProceeding as in Section 2, we assume\nµ2> m2, (3.22)\nand choose φ0such that Uφhas the minimum at ρ= 0, thus,\nφ2\n0=µ2−m2\nλ. (3.23)\n6From Eq. (3.20), the bilinear part of L, including the F2term, is then\nL(2)=−1\n4F2+1\n2m2\nVV2+1\n2(∂ρ)2−1\n2m2\n1ρ2−2mV(v·V)ρ, (3.24)\nwhere\nmV=qφ0,\nm2\n1= 2(µ2−m2). (3.25)\nThe physical picture that emerges is this: the field θbecomes the longitudinal component of V, and we end\nup with two fields, Vandρ. The longitudinal component of Vis mixed with the scalar field ρ(the term\nρv·V). What remains is to find the dispersion relationsof the propagating modes, which appearto be similar\nthat the dispersion relations of the φ±of the BE condensate model discussed in Section 2. As we show the\nconsequence is that the propagating modes with definite dispersion relations involve a superposition of the\nlongitudinal component of Vand theρ.\nThe following comment is in order before proceeding. In Eq. (3.20) th ere is a term linear in Vµofthe form\n2qφ0ρv·V= (2qµφ2\n0)u·V. This term is a tadpole, meaning that the ground state has a net con tribution to\nthe charge due to φ0. Indeed from the Eq. (3.7), it follows that the symmetry breaking m echanism produces\na net charge density given by\nj(em)0=qn0, (3.26)\nwith\nn0= 2µφ2\n0. (3.27)\nPhysically, the total charge of the system must be zero, which is as sured by assuming that the φbackground\nis superimposed on an oppositely charged background to render th e total system electrically neutral. Thus\nfor practical purposes we can disregard that term.\n4 Eigenmodes and the dispersion relations\n4.1 The evolution equations\nWe write it in momentum space by making the correspondence[20],\n−1\n4F2→ −V∗µk2˜gµνVν,\n1\n2(∂ρ)2→k2ρ∗ρ,\n1\n2m2\nVV2→m2\nVV∗µVµ,\n1\n2m2\n1ρ2→m2\n1ρ∗ρ,\n(u·V)ρ→(u·V)∗ρ+c.c, (4.1)\nand therefore\nL(2)(k) =−V∗µ[k2˜gµν−m2\nVgµν]Vν+ρ∗(k2−m2\n1)ρ−2µmV[(u·V)∗ρ+c.c], (4.2)\nwhere\n˜gµν=gµν−kµkν\nk2. (4.3)\nIf it were not for the last term, this is the usual free Lagrangian fo r a vector with squared mass m2\nVand a\nscalar with squared mass m2\n1. However the last term mixes the scalar with the longitudinal compon ents of\nVµ.\n7We decompose Vµin the form\nVµ=Vµ\nT+VLeµ\n3+kµ\n√\nk2Vk, Vµ\nT=/summationdisplay\ni=1,2Vieµ\ni (4.4)\nwhere\neµ\n1,2= (0,/vector e1,2),\neµ\n3=˜uµ\n√\n−˜u2, (4.5)\nwith\n˜uµ= ˜gµνuν=uµ−(k·u)kµ\nk2. (4.6)\nIn particular,\n˜u·V=−/radicalbig\n−˜u2VL,\nk·V=√\nk2Vk, (4.7)\nand therefore\nu·V= ˜u·V+(k·u)(k·V)\nk2=−/radicalbig\n−˜u2VL+(k·u)Vk√\nk2. (4.8)\nThus the quadratic part of the Lagrangian is\nL(2)(k) =−(k2−m2\nV)V∗\nT·VT+(k2−m2\nV)V∗\nLVL+m2\nVV∗\nkVk+(k2−m2\n1)ρ∗ρ\n+2µmV/radicalbig\n−˜u2[V∗\nLρ+c.c.]−2µmV(k·u)√\nk2[V∗\nkρ+c.c]. (4.9)\nSince the component Vkdoes not have a kinetic energy term, and therefore it is not a dynam ical variable,\none way to proceed is to eliminate it from the Lagrangian by using its La grange equation. The equation is\nm2\nVVk−2µmV(k·u)√\nk2ρ= 0, (4.10)\nand therefore\nVk=2µ(k·u)\nmV√\nk2ρ. (4.11)\nSubstituting this back in L(2)(k) we then get\nL(2)(k) =−(k2−m2\nV)V∗\nT·VT+(k2−m2\nV)V∗\nLVL+/parenleftbigg\nk2−m2\n1−4µ2(k·u)2\nk2/parenrightbigg\nρ∗ρ\n+2µmV/radicalbig\n−˜u2[V∗\nLρ+c.c.]. (4.12)\nThe alternative way is to find the equations for VLandρand eliminate Vkusing Eq. (4.11). The final\nequations for VLandρshould be the same. As a verification, the Lagrange equations from Eq. (4.9) are,\n(k2−m2\nV)VL+2µmV/radicalbig\n−˜u2ρ= 0,\n(k2−m2\n1)ρ+2µmV/radicalbig\n−˜u2VL−2µmV(k·u)√\nk2VL= 0, (4.13)\nand using Eq. (4.11) they become\n(k2−m2\nV)VL+2µmV/radicalbig\n−˜u2ρ= 0,\n2µmV/radicalbig\n−˜u2VL+/bracketleftbigg\nk2−m2\n1−4µ2(k·u)2\nk2/bracketrightbigg\nρ= 0, (4.14)\nwhich is the same that is obtained from Eq. (4.12). The picture that e merges is that the transverse modes\ndevelop a standard degenerate dispersion relation for the two pola rizations, while the longitudinal mode VL\nmixes with the scalar ρ. Our next task is to solve the evolution equations we have obtained, to find the\ndispersion relations and the corresponding mixing parameters VLandρeigenmodes.\n84.2 Dispersion relations\nAtκ= 0 (remembering that ˜ u2=−κ2/k2), the equations in Eq. (4.14) decouple,\n(ω2−m2\nV)VL= 0,\n(ω2−m2\n1−4µ2)ρ= 0, (4.15)\nso we can identify the plasma frequency as\nω2\np0=m2\nV=q2φ2\n0=q2(µ2−m2)\nλ. (4.16)\nTheρmode has an effective mass\nω2\nρ0=m2\n1+4µ2=m2\n+, (4.17)\nwhich coincides with the mass of the “+” mode in the un-gauged model (see Eq. (2.24)).\nForκ/negationslash= 0 the dispersion relations are the solutions of\n(k2−m2\nV)/bracketleftbigg\nk2−m2\n1−4µ2(k·u)2\nk2/bracketrightbigg\n−4µ2m2\nVκ2\nk2= 0, (4.18)\nor\n(k2−m2\nV)(k2−m2\n1)−4µ2\nk2/bracketleftbig\n(k·u)2(k2−m2\nV)+m2\nVκ2/bracketrightbig\n= 0. (4.19)\nWe now use the definition of κ,\n(u·k)2=k2+κ2, (4.20)\nto rewrite the term in square brackets,\n(k·u)2(k2−m2\nV)+m2\nVκ2=k2(k2−m2\nV+κ2), (4.21)\nso that the equation becomes\n(k2−m2\nV)(k2−m2\n1)−4µ2/bracketleftbig\nk2−m2\nV+κ2/bracketrightbig\n= 0, (4.22)\nwhich write it as\n(k2)2−(m2\nV+m2\n+)k2+m2\nVm2\n+−4µ2κ2= 0, (4.23)\nwherem2\n+is defined in Eq. (2.24) (which coincides with the squared mass of the “ +” mode in the un-gauged\nmodel and with effective squared mass of the ρas indicated in Eq. (4.17)). This is a quadratic equation for\nk2, with solutions\nω2=κ2+1\n2(m2\nV+m2\n+)±/bracketleftbigg1\n4(m2\nV−m2\n+)2+4µ2κ2/bracketrightbigg1\n2\n. (4.24)\n5 Discussion\nFor the following discussion it is convenient to normalize the various pa rameters as follows,\nx2=κ2\n2µ2,\ny2=ω2\n2µ2,\nM2= =m2\nV+m2\n+\n4µ2,\n∆ = =|m2\nV−m2\n+|\n4µ2. (5.1)\n91.61.822.22.42.62.833.23.4\n0 0.5 1 1.5 2y\nxy+,∆ = 0.2\ny−,∆ = 0.2\ny+,∆ = 0.5\ny−,∆ = 0.5\ny+,∆ = 1\ny−,∆ = 1\nFigure 1: Sample plots of the dispersion relations as written in Eq. (5.2 ). We have set M= 2, and ∆ as\nindicated. Theupper branchescorrespondto ω+andthe loweronesto ω−. The lowestbranch, corresponding\ntoω−with ∆ = 1, is well approximated by a formula of the form given in Eq. (5.7 ) for small values of κ.\nSee the discussion in the text.\nIn terms of them, the dispersion relations given in Eq. (4.24) are exp ressed in the form\ny2\n±=x2+M2±/radicalbig\n∆2+2x2. (5.2)\nPlots of the dispersion relations are shown in Fig. 1 for sample values o f the mass parameters. It must be\nkept in mind that numerically ∆ < M2, and also it is important to recognize that\nM2>1. (5.3)\nEq. (5.3) follows from the formulas for m2\nVandm2\n+in terms of µ(e.g. Eqs. (4.16) and (4.17)), that give\nM2= 1+1\n4µ2/bracketleftbiggq2(µ2−m2)\nλ+2(µ2−m2)/bracketrightbigg\n. (5.4)\nRemembering Eq. (2.8), this implies Eq. (5.3).\nOne particular feature worth mentioning is the result for the non-r elativistic (NR) limit of the ω−branch\nin the case that ∆ ≃1. For sufficiently small values of κ, the dispersion relations are\n(y2\n±)NR= (M2±∆)+/parenleftbig\n1±∆−1/parenrightbig\nx2∓x4\n2∆3. (5.5)\nThus, in particular,\n(y2\n−)NR= (M2−∆)+/parenleftbig\n1−∆−1/parenrightbig\nx2+x4\n2∆3, (5.6)\nand in the case that ∆ ≃1, the dispersion of the ( −) branch is\n(y2\n−)NR≃(M2−1)+1\n2x4. (5.7)\nIn other words, the κ2does not appear, which is reminiscent of the dispersion relation that has been obtained\ninNRmodelsoftheBEcondensationofchargedscalars[21]. However ,inourcase,the( ±)dispersionrelations\nare different, and this occurs only for the ω−dispersion relation. More generally, the presence of a O(κ2)\n101.781.81.821.841.861.881.91.921.941.96\n00.20.40.60.811.2y\nxy−,∆ = 0.2\ny−,∆ = 0.5\ny−,∆ = 0.8\nFigure 2: Sample plots of the dispersion relation for ω−illustrating the cases in which it has a miminum\nfor certain range of values of the mass parameters. We have set M= 2, and ∆ as indicated.\nterm (even in the T= 0 regime that we are considering) is mentioned in Ref. [22], as well as the fact that\nthe coefficient is negative; which in our case can occur for ω−if ∆<1.\nAnother noteworthy feature is the fact that if ∆ <1, thenω−decreases in a certain range of κ, and\nreaching a minimum value. This is illustrated in Fig. 1 in the plots of ω−for ∆ = 0 .2,0.5, and more\nemphatically in Fig. 2. In any case, the expression for ω2\n−, as given by Eq. (5.2), remains positive for all\nvalues of κ. To show this let us note the following. From Eq. (5.2) it follows that y2\n−becomes zero at x=x±,\nwhere\nx2\n±= (1−M2)±/radicalbig\n(1−M2)2−(M4−∆2). (5.8)\nSinceM2>1, as shown in Eq. (5.3), there is no solution of the equation\ny2\n−= 0, (5.9)\nandω2\n−remains positive for all values of κ.\nA related feature, in the case that ∆ <1, is that the group velocity of the ω−branch becomes negative\nin the range of κin which ω−decreases. The group velocity for each branch is\nvg=∂ω\n∂κ=x\ny/bracketleftbigg\n1±1√\n∆2+2x2/bracketrightbigg\n. (5.10)\nPlots of vgare shown in Figs. 3 and 4 for the ω±branches, respectively. Each figure also displays the\ncorresponding plot of v±≡κ/ω±for reference. In each branch, vdoes not change appreciably for the\ndifferent values of ∆. The group velocity of the ω−branch becomes negative in the range of κin which ω−\ndecreases. In fact it is zero at\nx=/radicalbigg\n1−∆2\n2, (5.11)\nand it is negative (positive) for κsmaller (larger) than this value, respectively.\nThe results discussed in this section indicate that the BE condensat e medium has non-trivial optical\nproperties. For example, as we have observed above, the group v elocity of the ω−mode can be reduced, zero\nor negative, depending on the value of κ. In fact, it has been experimentally shown that the speed of light\ncan be reduced drastically in a BE condensate[15], and the results ca n have consequences in other contexts.\nFor example, if the dark matter has a BE condensate component, s uch situation may arise that due to the\ndispersive nature of the dark matter medium, the properties of th e propagating photons may be drastically\naltered and maybe even that no electromagnetic signal may be obse rved within a certain range of κ.\n1100.10.20.30.40.50.60.70.8\n00.20.40.60.811.2v+\nxv+,∆ = 0.2\nv+,∆ = 0.5\nv+,∆ = 0.8\nvg+,∆ = 0.2\nvg+,∆ = 0.5\nvg+,∆ = 0.8\nFigure 3: Sample plots of the group velocity vg+for theω+branch as written in Eq. (5.10). We have set\nM= 2, and ∆ as indicated.\n−0.2−0.100.10.20.30.40.50.60.7\n00.20.40.60.811.2v−\nxv−,∆ = 0.2\nv−,∆ = 0.5\nv−,∆ = 0.8\nvg−,∆ = 0.2\nvg−,∆ = 0.5\nvg−,∆ = 0.8\nFigure 4: Sample plots of the group velocity vg−for theω−branch as written in Eq. (5.10). We have set\nM= 2, and ∆ as indicated.\n126 Polarization tensor and dielectric constant\nAn alternative way to obtain the dispersion relations is through the p hoton polarization tensor, or self-\nenergy. This method has also the virtue that it provides us with an ex pression for the dielectric function of\nthe system, which is useful on its own merits. The starting point is th e bilinear part of the Lagrangian for\nVµandρgiven in Eq. (4.12). The idea is to integrate out the variable ρ, which is the tree-level analog of\ncalculating the 1-loop self-energy, leaving only the dynamical variab les ofV, and identify the polarization\ntensor by comparing the result with\nLeff=−V∗µ/bracketleftbig\n(k2−πT)Rµν+(k2−πL)Qµν/bracketrightbig\nVν, (6.1)\nwhereRµνandQµνare defined by\nQµν=˜uµ˜uν\n˜u2,\nRµν= ˜gµν−Qµν. (6.2)\nIn terms of the variables VT,Lwe have defined in Eq. (4.4),\nLeff=−(k2−πT)V∗\nT·VT+(k2−πL)V∗\nLVL. (6.3)\nFrom Eq. (4.12), the equation for ρis precisely the equation given in the second line ofEq. (4.14). Solving\nit forρand substituting the solution back in Eq. (4.12) gives\nL(2)(k) =−(k2−m2\nV)V∗\nT·VT+(k2−m2\nV)V∗\nLVL−(2µmV√\n−˜u2)2\nk2−m2\n1−4µ2(k·u)2\nk2V∗\nLVL. (6.4)\nComparing Eqs. (6.3) and (6.4), we then identify\nπL=m2\nV+(2µmV√\n−˜u2)2\nk2−m2\n1−4µ2(k·u)2\nk2, (6.5)\nwhich can be written in the form\nπL=m2\nV+4µ2m2\nVκ2\nk2(k2−m2\n1)−4µ2(k·u)2, (6.6)\nremembering that\n˜u2=−κ2\nk2. (6.7)\nIt is simple to verify that the dispersion relation equation\nk2−πL= 0, (6.8)\ngives again Eq. (4.18), as it should be. While this is reassuring, the virt ue of Eq. (6.6) is that it gives us an\nexpression for the longitudinal dielectric function, as function of ωandκ,\nǫℓ= 1−πL\nk2, (6.9)\nwhich allows to study also the response of the system to external fi elds.\n7 Mixing\nBesides the dispersion relation and the polarization tensor, anothe r important factor is the ρ−VLmixing\nin the eigenmodes. Once the solutions ω±are given, these are determined from Eq. (4.14). We write the\n1300.20.40.60.81\n012345678\nκ/2µ|VL+|2\n|ρ+|2\nFigure 5: Plots of the mixing coefficients ρ+,VL+defined in Eq. (7.1), as functions of κ. We have set\nM= 2, and ∆ = 1.\n00.20.40.60.81\n012345678\nκ/2µ|VL−|2\n|ρ−|2\nFigure 6: Plots of the mixing coefficients ρ−,VL−defined in Eq. (7.1) as functions of κ. We have set M= 2,\nand ∆ = 1.\n14elements of the mode eigenvectors in the form\n/parenleftbigg\nρ\nVL/parenrightbigg\n−=1/radicalbig\nN−\nΛ\nmVγ−\n,\n/parenleftbigg\nρ\nVL/parenrightbigg\n+=1/radicalbig\nN+\n−mVγ+\nΛ−γ2\n+\n, (7.1)\nwhere\nγ±=2µκ/radicalBig\nk2\n±,\nk2\n±=ω2\n±−κ2=1\n2(m2\nV+m2\n+)±/bracketleftbigg1\n4(m2\nV−m2\n+)2+4µ2κ2/bracketrightbigg1\n2\n,\nΛ =1\n2(m2\nV−m2\n+)+/bracketleftbigg1\n4(m2\nV−m2\n+)2+4µ2κ2/bracketrightbigg1\n2\n, (7.2)\nand\nN+= (Λ−γ2\n+)2+(mVγ+)2,\nN−= Λ2+(mVγ−)2. (7.3)\nWith this convention, when mV> m+in the limit κ= 0 these solutions correspond to\n/parenleftbiggρ\nVL/parenrightbigg\n−=/parenleftbigg1\n0/parenrightbigg\n, (7.4)\n/parenleftbigg\nρ\nVL/parenrightbigg\n+=/parenleftbigg\n0\n1/parenrightbigg\n. (7.5)\nPlotsofthemixingcoefficients |ρ±|2and|VL±|2isshownin Fig.5andFig.6. Fortheplotswetake M2and∆\nasindicatedinthecaption,andweconsiderthecase mV> m+sothat, inparticular,( mV/2µ)2=1\n2(M2+∆).\nThe plots do not vary significantly for other values of M2and ∆.\nAs already indicated above, at κ= 0 theω−solution corresponds to a pure ρ,\n/parenleftbiggρ\nVL/parenrightbigg\n−=/parenleftbigg1\n0/parenrightbigg\n(κ= 0). (7.6)\nOn the other hand, the dispersion relation ω−satisfies\nk2\n−= 0, (7.7)\nforκ=κ0, where\nκ2\n0=m2\nVm2\n+\n4µ2. (7.8)\nIt is easy to see from Eq. (7.2), that γ−increases relative to Λ as k2\n−→0, and therefore from Eq. (7.1) it\nfollows that /parenleftbigg\nρ\nVL/parenrightbigg\n−=/parenleftbigg\n0\n1/parenrightbigg\n(κ=κ0). (7.9)\nMoreover, k2\n−is positive for κ < κ0, but it is negative for κ > κ0. Therefore, for κ > κ0, we have\nγ−=−2iµκ/radicalBig\n|k2\n−|, (7.10)\n15and then the ω−eigenspinor is of the form\n/parenleftbiggρ\nVL/parenrightbigg\n−=1/radicalbig\nN−\nΛ\n−imV|γ−|\n(κ > κ0). (7.11)\nThe situation here resembles the circular polarization of a propagat ingphoton, or more precisely in this case,\nelliptic polarization,\ne±=/parenleftbigga\n±ib/parenrightbigg\n. (7.12)\nThus in the present case, if κ > κ 0, theω−mode oscillates between the two flavors(VLandρ), as it\npropagates, and at some spacetime points it is a pure VLand at other points a pure ρ.\n8 Conclusions and Outlook\nWe have presented a model for the propagation of photons in a med ium that contains a Bose-Einstein\n(BE) condensate of a charged scalar field. The idea is based on the m odel recently proposed to treat the\npropagation of fermions in a BE condensate, extended by coupling t he scalar field with the electromagnetic\nfield in the usual way. Our goal has been to study the effects of the symmetry breaking on the the photon\ndispersion relations and the optical properties of the system. To t his end, the propagating modes were\nidentified, and the corresponding dispersion relations were determ ined, in Section 4. As we have shown\nthere, two of the modes correspond to the transverse photon p olarizations, and their dispersion relations\nhave the usual form of the transverse photons in a plasma. There are two other modes, which we denote as\nthe (±) modes, which are combinations of the longitudinal photon and the m assive scalar field. Some salient\naspects of the results obtained were discussed in Section 5. A part icularly distinguishable one is the fact\nthat the dispersion relation of the ( −) mode decreases as a function of the momentum in a given range, an d\ntherefore the correspondinggroup velocityis negativein that ran ge, which showsthat the BE condensatecan\nhave some peculiar and non-trivial properties. As a complement, in S ection 6 we gave the expression for the\nphoton polarization tensor. On one hand, this provides an alternat ive way to obtain the dispersion relations,\nbut more importantly it has also the virtue that it provides an expres sion for the dielectric function of the\nsystem, which is useful on its own merits, e.g. for studying also the r esponse of the system to external fields.\nAnother useful result is the determination of the mixing parameter s of the ( ±) modes presented in Section\n7, which can be used to obtain the corrections to the dispersion rela tions (e.g., imaginary parts due the\ndamping effects) and/or the effects of scattering, due to the inte ractions with the excitations of the system.\nIn summary, the optical properties of a medium that contains a BE c ondensate have particular features that\nmotivate further exploration. The results of the present work ca n be useful to study the effects they may\nhave in specific applications in the various contexts considered in the references cited.\nThe work of S. S. is partially supported by DGAPA-UNAM (Mexico) PAP IIT project No. IN103522.\nA Justification of Eq. (4.1)\nThe justification of Eq. (4.1) is the following. For any field we write\na(x) =/integraldisplayd4k\n(2π)4a(k)e−ik·x. (A.1)\nIfa(x) is real (Hermitian), then\na(−k) =a∗(k). (A.2)\nNow consider something like (an action) involving two real fields\nS=/integraldisplay\nd4x/bracketleftbigg1\n2a2(x)+1\n2b2(x)+αb(x)a(x)/bracketrightbigg\n. (A.3)\n16What is the Lagrangian in momentum space? Using the transformatio n ofaandb,\nS=/integraldisplayd4k\n(2π)4/integraldisplayd4k′\n(2π)4/bracketleftbigg1\n2a(k′)a(k)+1\n2b(k′)b(k)+αb(k′)a(k)/bracketrightbigg\nδ(k+k′)\n=/integraldisplayd4k\n(2π)4/bracketleftbigg1\n2a(−k)a(k)+1\n2b(−k)b(k)+αb(−k)a(k)/bracketrightbigg\n. (A.4)\nBut we would not obtain the correct equations if we just use\nL(k) =1\n2a(−k)a(k)+1\n2b(−k)b(k)+αb(−k)a(k), (A.5)\nwhich is equal to\nL(k) =1\n2a∗(k)a(k)+1\n2b∗(k)b(k)+βb∗(k)a(k). (A.6)\nIn particular that this L(k) is not real (Hermitian). The correct procedure is to notice that, since the first\nterm mixes a(k) witha(−k), and similarly with b(k), we must include both also in the αterm. Therefore\nwe must write\nS=/integraldisplay\nk>0d4k\n(2π)4/bracketleftbigg1\n2a(−k)a(k)+1\n2b(−k)b(k)+αb(−k)a(k)+(k↔ −k)/bracketrightbigg\n, (A.7)\nand therefore the correct Lagrangian density to use in momentum space is\nL(k) =a∗(k)a(k)+b∗(k)b(k)+α(b∗(k)a(k)+c.c), (A.8)\nwith the understanding that the complete action is\nS=/integraldisplay\nk>0d4k\n(2π)4L(k). (A.9)\nReferences\n[1] Jose F Nieves and Sarira Sahu, Neutrino effective potential in a fermion and scalar backgro und in the\nresonace region , Phys. Rev. D 105, 095022 (2022), [arxiv: 2201.04661].\n[2] Jose F Nieves and Sarira Sahu, Model for the propagation of fermions in a Bose-Einstein con densate,\nPhys. Rev. D 107, 116012 (2023), [arXiv:2302.14285].\n[3] Gordan Baym, Christopher Pethick, Neutron Stars , Annu. Rev. Nucl. Sci. 25, 27 (1975).\n[4] Vesteinn Thorsson, Madappa Prakash, J. M. Lattimer Composition, structure and evolution of neutron\nstars with kaon condensates , Nucl. Phys. A, 572, 693 (1994).\n[5] A. Schmitt, Dense Matter in Compact Stars, A Pedagogical Introduction , Lect. Notes Phys. 811\n(Springer, Berlin Heidelberg 2010) [DOI 10.1007/978-3-642-12].\n[6] G. Q. Li, C. -H. Lee, G. E. Brown, Kaons in dense matter, kaon productionin heavy-ion collisi ons, and\nkaon condensation in neutron stars , Nucl. Phys. A, 625, 372 (1997).\n[7] Maria Cr˘ aciun, Tiberiu Harko, Testing Bose–Einstein condensate dark matter models with t he SPARC\ngalactic rotation curves data , EPJC 80, 735 (2020).\n[8] P.Sikivie and Q.Yang Bose-Einstein Condensation of Dark Matter Axions Phys.Rev. Lett. 103, 111301\n(2009).\n[9] Ji Ji Fan, Ultralight repulsive dark matter and BEC , Physics of the Dark Universe 14, 84 (2016).\n[10] Alexander D. Dolgov, Angela Lepidi and Gabriella Piccinelli, Electrodynamics at non-zero temperature,\nchemical potential and Bose condensate , JCAP 02, 027 (2009), [arxiv: 0811.4406].\n17[11] Yurii Slyusarenko and Andrii Sotnikov, Propagation of relativistic charged particles in ultracol d atomic\ngases with Bose-Einstein condensates , Phys. Rev. A 83, 023601 (2011).\n[12] J. Kasprzak et al., Bose-Einstein condensation of exciton polaritons , Nature 443, 409 (2006).\n[13] Alex Kruchkov and Yurii Slyusarenko, Bose-Einstein condensation of photons in an ideal atomic ga s,\nPhys. Rev. A 88, 013615 (2013).\n[14] J. T. Mendon¸ ca and H. Ter¸ cas, Bose-Einstein condensation of photons in a plasma , Phys. Rev. A 95,\n063611 (2017).\n[15] Lene Vestergaard Hau, S. E. Harris, Zachary Dutton and Cyr us H. Behroozi, Light speed reduction to\n17 metres per second in an ultracold atomic gas , Nature, 397594 (1999).\n[16] H. ArthurWeldon, Chemical potentials in real-time thermal field theory , Phys.Rev.D76, 125029(2007).\n[17] Antonio Filippi, Inclusion of Chemical Potential for Scalar Fields , Imperial/TP/96-97/37\n[arxiv:hep-ph/9703323v1].\n[18] See Appendix A in Ref. [5].\n[19] The presence of µin the Lagrangian can be eliminated by letting\nAµ=A′\nµ−1\nqvµ. (A.10)\nHowever, doing this transformation of Aµwould restore the µin the partition function, which defeats\nthe purpose of using the transformed field φ′. Of course this is nothing more than the statement of\ninvariance under a gauge transformation\nφ=eiqΛφ′,\nAµ=A′\nµ−∂µΛ. (A.11)\n[20] See Appendix A.\n[21] A.S.AlexandrovandW.H.Beere, Collective excitations and screening properties of a conde nsed charged\nBose gas , Phys. Rev. B 51, 5887 (1995).\n[22] B. Davoudi1 and M. P. Tosi, Single-particle and collective excitations in a charged Bo se gas at finite\ntemperature , Phys. Rev. B 72, 134520 (2005).\n18"    },    {        "title": "2401.13900v2.3d_gravity_from_Virasoro_TQFT__Holography__wormholes_and_knots.pdf",        "content": "MIT-CTP/5691\n3d gravity from Virasoro TQFT:\nHolography, wormholes and knots\nScott Colliera,b, Lorenz Eberhardtc,dand Mengyang Zhange\naPrinceton Center for Theoretical Science, Princeton University, Princeton, NJ 08544,\nUSA\nbCenter for Theoretical Physics, Massachusetts Insitute of Technology, Cambridge, MA\n02139, USA\ncInstitute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA\ndInstitute for Theoretical Physics, University of Amsterdam, PO Box 94485, 1090 GL\nAmsterdam, The Netherlands\neJoseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA\nE-mail: sac@mit.edu ,l.eberhardt@uva.nl ,mengyang@princeton.edu\nAbstract: We further develop the description of three-dimensional quantum grav-\nity with negative cosmological constant in terms of Virasoro TQFT formulated in\nour previous paper [1]. We compare the partition functions computed in the Vira-\nsoro TQFT formalism to the semiclassical evaluation of Euclidean gravity partition\nfunctions. This matching is highly non-trivial, but can be checked directly in some\nexamples. We then showcase the formalism in action, by computing the gravity parti-\ntion functions of many relevant topologies. For holographic applications, we focus on\nthe partition functions of Euclidean multi-boundary wormholes with three-punctured\nspheres as boundaries. This precisely quantifies the higher moments of the structure\nconstants in the proposed ensemble boundary dual and subjects the proposal to thor-\nough checks. Finally, we investigate in detail the example of the figure eight knot\ncomplement as a hyperbolic 3-manifold. We show that the Virasoro TQFT partition\nfunction is identical to the partition function computed in Teichm¨ uller theory, thus\ngiving strong evidence for the equivalence of these TQFTs. We also show how to\nproduce a large class of manifolds via Dehn surgery on the figure eight knot.arXiv:2401.13900v2  [hep-th]  12 Mar 2024Contents\n1 Introduction 1\n2 Structural properties of Virasoro TQFT 3\n2.1 Volume conjecture 4\n2.2 The volume of hyperbolic tetrahedra 4\n2.3 Volume conjecture for handlebodies 6\n2.4 Mutations of hyperbolic manifolds 9\n2.5 Consistency conditions on the crossing kernels 12\n3 Holographic examples 14\n3.1 Cyclic defect wormholes 16\n3.2 Four-boundary non-Gaussianity wormhole 19\n3.3 Many-boundary wormholes and higher non-Gaussianities 25\n3.4 Handle wormholes 34\n3.5 Twisted I-bundles 36\n4 The figure eight knot complement 38\n4.1 Direct computation 39\n4.2 Comparison to Teichm¨ uller TQFT 41\n4.3 Computation via the Seifert surface 43\n4.4 Semiclassical expansion 44\n4.5 Dehn surgery 48\nA Semiclassical expansion of the double sine function 51\n1 Introduction\nThree-dimensional quantum gravity with negative cosmological constant has proven\nto be one of the most interesting and productive toy models of quantum gravity. The\nmajor outstanding problems are to fully solve the theory from first principles and\nfirmly establish a holographic correspondence for the theory. There has been major\nprogress on both fronts over the last few years; see [1–7] and [8–14], respectively.\nIn our previous paper [1], we developed a formalism that computes the gravity\npartition function algorithmically on a background of fixed (on-shell) topology. This\nfixes the contributions of hyperbolic three-manifolds to the gravitational path integral\nand represents a large step towards a complete solution of the theory directly from\n– 1 –the bulk. The next step would involve performing the sum over all three-dimensional\ntopologies that appear in the gravitational path integral, which may also require suit-\nable non-perturbative or off-shell contributions. While our previous paper developed\nthe formalism in terms of the Virasoro TQFT, this paper gives several interesting\napplications that exemplify its practical utility and should be viewed as a natural\ncontinuation of [1].\nOn the holographic side, a consistent picture is emerging that the gravitational\npath integral computes certain universal statistical features in a putative ensemble of\nholographic 2d CFTs. While the full non-perturbative definition of such an ensemble\nis still not settled, this perspective makes very concrete predictions that can be\nquantitatively matched between the bulk and boundary. In this paper we will study\npartition functions of Virasoro TQFT on multi-boundary wormholes to exemplify the\nextent to which the gravitational path integral precisely captures universal statistics\nof CFT data, transcending the Gaussian approximation of [10].\nWe will assume that the reader is acquainted with the concepts introduced in\n[1], but now recall some key features. As suggested by holography, the Hilbert space\nof 3d quantum gravity is spanned by the left- and right-moving Virasoro conformal\nblocks on the spatial surface Σ. This factorization of the Hilbert space allows one to\nconsider, say, only the left-movers as a fundamental building block.1A key ingredient\nin the proposal of [1] is an explicit form of the inner product on this conformal block\nHilbert space. Important for the consistency of this structure is the fact that the\nconformal blocks transform among each other under crossing transformations and\nas such the Hilbert space carries a unitary action under crossing transformations.\nThere are remarkably explicit expressions for the crossing transformation in terms\nof the Ponsot-Teschner fusion kernel Fand the modular crossing kernel S[15–18].\nSince the Hamiltonian in gravity vanishes, the theory can be viewed as a TQFT on\na background topology. This data completely specifies the TQFT that we called\nVirasoro TQFT in [1]. The TQFT partition function on a fixed topology can be\ncomputed via surgery techniques similarly to Chern-Simons theory. The Virasoro\nTQFT partition function then immediately leads to the full 3d gravity partition\nfunction via the following formula, valid for all hyperbolic three-manifolds\nZgrav(M) =X\nγ∈Map( ∂M)/Map( M,∂M )|ZVir(Mγ)|2. (1.1)\nHere we sum over all images of the manifold Munder the boundary mapping class\ngroup Map( ∂M), which is part of the sum over topologies, modulo the bulk mapping\nclass group Map( M, ∂M ), which is gauged in gravity.\n1The left- and right-movers are entangled only by the sum over topologies in the gravitational\npath integral.\n– 2 –We start in Section 2 by analyzing the gravity partition functions as computed\nin Virasoro TQFT and their relation to the semiclassical evaluation of the gravita-\ntional path integral. Comparing the two expressions leads to the (refined) volume\nconjecture that we already mentioned in [1] and discuss further here. We also discuss\nthe existence of non-isomorphic hyperbolic manifolds with identical Virasoro TQFT\npartition functions. This in particular implies that the gravitational path integral is\nnot powerful enough to detect the topology of hyperbolic manifolds.\nWe then discuss examples that are relevant for the holographic description of\n3d gravity in Section 3. We focus on a class of manifolds obtained by removing\nthree-punctured spheres from S3and connecting the boundaries appropriately with\nWilson lines. They compute holographically the higher moments of the structure\nconstants in the proposed ensemble description of the boundary dual. We find that\nthe partition functions may be computed using diagrammatic rules that are simply\ntheq-deformations of the rules for the computation of disk partition functions in JT\ngravity + matter [19–21]. When projecting the Wilson lines on a disk, one associates\na Virasoro 6j-symbol to every crossing of lines and one integral to every loop formed\nby the internal lines, see eqs. (3.63) and (3.64) for the precise formulae. We also use\nVirasoro TQFT to compute the gravity partition function on a class of contributions\nto the single-boundary gravitational path integral that are not handlebodies. These\nnon-handlebody instantons are formed by quotients of the two-boundary Euclidean\nwormhole and we find that the gravity partition function is related to the partition\nfunction of Liouville CFT on a particular non-orientable surface.\nFinally, we consider the example of the figure eight knot complement in Section 4,\nwhich constitutes one of the simplest examples of a hyperbolic manifold with no\nasymptotic boundary. We compute its Virasoro TQFT partition function from a\nvariety of perspectives and demonstrate that it agrees with the partition function\ncomputed in an a priori different TQFT known as Teichm¨ uller TQFT. This lends\nstrong credence to the equivalence of the two theories, even though Virasoro TQFT\nprovides a far more convenient framework for holographic applications. We also\nillustrate the procedure of Dehn surgery on the figure eight knot, which leads to\nthe gravity partition function on a whole family of hyperbolic three-manifolds whose\nvolume accumulates to that of the figure eight knot complement.\n2 Structural properties of Virasoro TQFT\nWe start by discussing the relation of the formulation of gravity in terms of Virasoro\nTQFT and the semiclassical gravity path integral. Comparing the two leads to the\nvolume conjecture and we discuss various consequences for the volumes of hyperbolic\n3-manifolds, conformal blocks and one-loop determinants. We then also explain some\nof the consistency conditions of the Virasoro TQFT. Such consistency conditions are\n– 3 –all implied by the consistency of the mapping class representation on the initial value\nsurface, but often the three-dimensional viewpoint is much more powerful.\n2.1 Volume conjecture\nWe already stated the (refined) volume conjecture in [1], but it will play a much\nmore prominent role in the present paper. By comparing the usual metric approach\nof 3d gravity and the Virasoro TQFT approach, one obtains the following prediction\nfor the semiclassical expansion of partition functions:\n|ZVir(M)|2= e−c\n6πvol(M)\"Y\nγ∈P∞Y\nm=21\n|1−qm\nγ|2+O(c−1)#\n. (2.1)\nHere we used that the gravity tree-level action isc\n6πvol(M), where vol( M) is the\nvolume of the hyperbolic manifold. We also used the explicit form of the one-loop\ndeterminant as computed in [3]. This explicit formula for the one-loop determinant\nis valid for hyperbolic manifolds without defects that can be written as H3/Γ for\na so-called Kleinian group Γ. In case Mhas defects, the volume conjecture should\nstill hold, but there is no known general formula for the one-loop determinant. We\nrecall that Pdenotes the set of all primitive geodesics on the three-manifold in\nquestion. Alternatively, we can think of Pas the set of primitive conjugacy classes\nin the Kleinian group Γ (i.e. conjugacy classes that are not powers of other conjugacy\nclasses) and also identify the conjugacy class of γwith the conjugacy class of γ−1,\nsince this corresponds to orientation reversal of the corresponding geodesic. We\ncould of course extend the matching to higher loop order, but will restrict here to\nthe tree-level and one-loop piece.\nWe refer to this equation as the refined volume conjecture, since the classical\nvolume conjecture is the corresponding statement for the tree-level term in the1\nc-\nexpansion [22]. The relation (2.1) should also hold in the presence of boundaries, in\nwhich case the volume of the hyperbolic manifold is the renormalized volume [23].\n2.2 The volume of hyperbolic tetrahedra\nLet us explain one of the simplest non-trivial instances of the volume conjecture in\nmore detail. Consider a single hyperbolic tetrahedron as in Figure 1 with dihedral\nangles θispecifying the angle between the two faces meeting at the edge.\nThe dihedral angles have to satisfy rather complicated conditions for such a\nhyperbolic tetrahedron to exist. We can take two identical such hyperbolic tetrahedra\nand identify them along the corresponding faces. This leads to a topological three-\nsphere with conical defects running in the form of the tetrahedron through it. The\nhyperbolic tetrahedron is specified by the dihedral angles θj∈(0, π) spanned by the\ntwo faces meeting at an edge. Upon gluing two tetrahedra, the conical defect angle\n– 4 –θ6θ5θ4θ3\nθ1 θ2\nFigure 1 . Tetrahedron with dihedral angles specified.\nbecomes 2 π−2θj. Semiclassically, the relation between defect angles αjand Liouville\nmomentum reads2[24]\nPj=iαj\n4πb∼iQ\n2−iθj\n2πb(2.3)\nThis is the hyperbolic three-manifold that we use for the volume conjecture. Some\nextra care is necessary to correctly normalize the vertices. We observe that the\nrenormalized volume of the Euclidean wormhole of the form Σ 0,3×Iexactly vanishes.\nSince it evaluates to the Liouville structure constant C0(P1, P2, P3) in the Virasoro\nTQFT, this means that for the purposes of the volume conjecture, we should define\na juncture with a normalization constant C0(P1, P2, P3)−1\n2as follows\n1p\nC0(P1, P2, P3)×P1\nP3P2 . (2.4)\nWe compute the TQFT partition function ZVirof this tetrahedral configuration\nin Section 3.2. The result is given by\nZVir\n\n=ρ0(P6)−1C0(P1, P2, P3)C0(P3, P4, P5)FP3,P6\u0014P4P2\nP5P1\u0015\n(2.5)\n=p\nC0(P1, P2, P3)C0(P1, P5, P6)C0(P2, P4, P6)C0(P3, P4, P5)\n×\u001aP1P2P3\nP4P5P6\u001b\n. (2.6)\n2Here we are adopting the standard notation from Liouville theory for the central charge and\nconformal weights:\nc= 1 + 6( b+b−1)2= 1 + 6 Q2,∆j=c−1\n24+P2\nj. (2.2)\n– 5 –The symbol {P1P2P3\nP4P5P6}is the crossing kernel for sphere four-point function conformal\nblocks in the Racah-Wigner normalization [17], which we also call the Virasoro 6j-\nsymbol. It has the correct tetrahedral symmetry as required by the picture, where\nthe vertices of the tetrahedron are formed by ( P1, P2, P3), (P1, P5, P6), (P2, P4, P6)\nand ( P3, P4, P5).\nThus the prediction of the volume conjecture is now that\n2 vol( η1, η2, η3, η4, η5, η6) =−2πlim\nb→0b2log\u001aiQ\n2−iη1\n2πbiQ\n2−iη2\n2πbiQ\n2−iη3\n2πb\niQ\n2−iη4\n2πbiQ\n2−iη5\n2πbiQ\n2−iη6\n2πb\u001b\n,(2.7)\nwhere the volume on the left hand side is the volume of the hyperbolic tetrahedron\nspecified by the dihedral angles ηj.\nOne can evaluate the integral in the defining formula for the crossing kernel\nvia saddle-point approximation in this limit and confirm that it agrees with the\nvolume formula for a hyperbolic tetrahedron. This was done in [17], but the volume\nconjecture gives a conceptual derivation of that fact.\nWe also mention that the Virasoro crossing kernel has the following Regge sym-\nmetry [25, 26]\nFP3,P6\u0014P4P2\nP5P1\u0015\n=FP3,P6\u00141\n2(P2+P4+P5−P1)1\n2(P1+P2+P4−P5)\n1\n2(P1+P4+P5−P2)1\n2(P1+P2+P5−P4)\u0015\n. (2.8)\nThis implies via the volume conjecture that the volume of a hyperbolic tetrahedron\nis invariant under the replacement\nθ1→1\n2(θ1+θ2+θ4−θ5), θ 5→1\n2(−θ1+θ2+θ4+θ5), (2.9)\nθ2→1\n2(θ1+θ2+θ5−θ4), θ 4→1\n2(θ1−θ2+θ4+θ5), (2.10)\nwith θ3andθ6unchanged. This property is very non-trivial to see geometrically and\ngiving a direct proof of it is rather hard.\n2.3 Volume conjecture for handlebodies\nSemiclassical vacuum blocks. Let us apply the volume conjecture in the form\n(2.1) to a handlebody. Recall that the Virasoro TQFT partition function on a genus- g\nhandlebody SΣgevaluates to the vacuum Virasoro conformal block,\nZVir(SΣg) =11 1\n, (2.11)\nwhere we drew a genus-2 surface for concreteness. As such the volume conjecture\n(2.1) gives the semiclassical expansion of vacuum blocks,\n11 1\n∼e−c\n12πvol(SΣg)Y\nγ∈P(Γg)∞Y\nm=21\n1−qm\nγ. (2.12)\n– 6 –Here, vol( SΣg) is the in general complex volume of the handlebody.3Such a semiclas-\nsical expansion of the conformal blocks is familiar from 2d CFT, where the leading\nterm is called the semiclassical conformal block [27–29], but to our knowledge there is\nno general CFT derivation of the one-loop determinant, and even direct derivations\nof the leading term are somewhat limited. The group Γ g⊂PSL(2 ,C) that appears\nin the one-loop determinant is the Schottky group of the corresponding handlebody.\nThe Virasoro TQFT approach gives a simple derivation of this fact. It also shows\nthat the semiclassical block is nothing else than the volume of the corresponding\nhandlebody. It was shown in [30] that this volume is identified with the on-shell\nvalue of the Liouville action as defined by Takhtajan and Zograf [31],\nSL(Σg) =−4 Re vol( SΣg). (2.13)\nDefining the on-shell Liouville action requires one to pick a conformal block chan-\nnel. The Virasoro TQFT also makes a prediction about the order one term in the\nsemiclassical expansion.\nOne-loop determinant. Let us recall the formula derived in [32] for the holomor-\nphic factorization of the Laplacian on a Riemann surface. We have\ndet′∆2\ndetN2=cge−13\n12πSL(Σg)\f\f\f\f(1−q1)2(1−q2)Y\nγ∈P(Γg)∞Y\nm=2(1−qm\nγ)2\f\f\f\f2\n. (2.14)\nfor some constant cgindependent of the moduli. It depends on the renormalization\nscheme used to define the determinant det′∆2. Here ∆ 2is the Laplacian acting on\nholomorphic quadratic differentials on the surface Σ and the prime indicates that we\nremoved the zero modes. det N2is the determinant of ⟨φj|φk⟩and{φj}j=1,...,3g−3\nis a natural basis of holomorphic quadratic differentials as defined in [32]. We also\ndenoted qj=qγjwith γ1, . . . , γ gthegfree generators of the Schottky group. The\nperhaps unnatural seeming factor (1 −q1)2(1−q2) appears because of the specific\nway in which φjis defined and is a result of fixing the PSL(2 ,C) conjugacy freedom\nfor the Schottky group. Thus we have\n\f\f\f\f\f\f\f\f11 1\f\f\f\f\f\f\f\f2\n∼c′\nge−c−13\n6πvol(SΣg)\np\ndet′∆2× |1−q1|2|1−q2|p\ndetN2.\n(2.15)\nThis tells us that the one-loop partition function is exactly the inverse square root of\nthe partition function of a bc-ghost system with a particular choice of ghost insertions.\n3In general to write this formula only for a chiral half (which goes beyond the volume conjecture\n(2.1)), we also need to assign an imaginary part to the volume which is known as the Chern-Simons\ninvariant.\n– 7 –Explicit check. Here we explicitly check the one-loop refinement of the volume\nconjecture for handlebodies in a simple example, perturbatively in the moduli of the\nRiemann surface in an expansion about a pinching limit. Consider for concreteness\na genus-two Riemann surface formed by plumbing two two-holed disks D1andD2:\nD1={z1∈C|r1<|z1|< r3,|z1−1|> r2}, (2.16a)\nD2={z2∈C|˜r1<|z2|<˜r3,|z2−1|>˜r2}. (2.16b)\nGluing the boundaries of the disks according to the following inversion map prepares\na disk with three holes\n|z2|= ˜r3:z2∼1\np1z1, with|p1|=1\nr3˜r3. (2.17)\nThe remaining identifications are\n|z2|= ˜r1:z2∼p3\nz1, with|p3|=r1˜r1, (2.18a)\n|z2−1|= ˜r3:z2−1∼p2\nz1−1, with|p2|=r2˜r2. (2.18b)\nThe complex plumbing parameters piparameterize the moduli of the Riemann sur-\nface, with the pi→0 limit a pinching locus in which the surface is realized by gluing\ntwo spheres along long narrow tubes. The corresponding Virasoro conformal blocks\nmay then straightforwardly be computed as an expansion in powers of the plumbing\nparameters pi, see for example [33] for details.\nThis parameterization of the genus-two Riemann surface is clearly equivalent to\nthe Schottky parameterization, in which one realizes the Riemann surface Σ gas a\nquotient of the form\nΣg= (C∪ {∞} − Λ)/Γ. (2.19)\nHere Γ = ⟨γ1, . . . , γ g⟩is the Schottky group, which is a free group generated by the\nloxodromic elements γ1, . . . , γ gof PSL(2 ,C), and Λ is the limit set of the action of\nΓ. The generators γiact on the Riemann sphere by M¨ obius transformation. In our\nexample of the genus-two Riemann surface formed by plumbing two-holed disks as\nabove, the generators of the Schottky group may be taken to be\nγ1(z) =p1p3z , γ 2(z) =(1−p2)z−1/p1\nz−1/p1. (2.20)\nEach generator γis conjugate to diag( q1/2\nγ, q−1/2\nγ), with |qγ|<1. Here we have\nqγ1=p1p3, q γ2=1−p1+p1p2−p\n1−2p1(1 +p2) +p2\n1(1−p2)2\n1−p1+p1p2+p\n1−2p1(1 +p2) +p2\n1(1−p2)2.(2.21)\nWe are now in a position to directly compare the perturbative expansion of\nthec→ ∞ limit of the genus-two Virasoro vacuum block as parameterized in the\n– 8 –plumbing frame above4with that of the gravity one-loop determinant on the genus-\ntwo handlebody (2.12). We find\n11 1\f\f\f\f\f\f\f\f\nc→∞=Y\nγ∈P(Γ)∞Y\nm=21\n1−qm\nγ(2.22)\n= 1 + p2\n1p2\n2+p2\n2p2\n3+p2\n3p2\n1+ 4(p3\n1p2\n2+p3\n2p2\n3+p2\n2p3\n3) +. . . (2.23)\nOn the left-hand side we evaluate the genus-two identity block in the plumbing frame\nperturbatively in the moduli by brute force, and on the right-hand side we evaluate\nthe one-loop determinant by taking the product over primitive conjuguacy classes of\nthe Schottky group. We have verified the agreement between these two expressions\nup to total degree 12 in the expansion in the plumbing parameters.\n2.4 Mutations of hyperbolic manifolds\nOne may ask whether ZViris a perfect invariant of a hyperbolic three-manifold, or,\nin other words, is ZVirpowerful enough to distinguish any two hyperbolic manifolds?\nAs in Chern-Simons theory, the answer to this question is negative. There exist non-\nisometric hyperbolic three-manifolds M1andM2with ZVir(M1) =ZVir(M2). The\nreason for this is a general operation known as mutation.\nThere are different kinds of mutations, these are all relatively subtle operations\nthat go undetected by most knot invariants, including the Virasoro TQFT partition\nfunction ZVir.5Let us first explain the classical example of knot mutations. Consider\nFigure 2 . The Conway knot and its mutant, the Kinoshita-Teresaka knot. They are not\nequivalent, but the value of ZViris the same.\na region of a knot in which two strands enter and two strands exit. The path integral\n4As explained in [33], in the plumbing frame the c→ ∞ limit of the Virasoro blocks is actually\nfinite; in other words, the corresponding Liouville action vanishes. The c→ ∞ limit of the vac-\nuum block as computed in (2.12) plays an important role in determining the seed of the recursive\nrepresentation of arbitrary Virasoro blocks.\n5In Virasoro TQFT, we think of a knot as a defect inserted in the three-sphere S3that is knotted\nappropriately.\n– 9 –over this region gives a state in the Hilbert space of the four-punctured sphere where\nall the four labels are identical (since we considering a knot associated to a single\nVirasoro representation). Thus, whatever the resulting state is, it can be expanded\nin terms of s-channel conformal blocks. However, any s-channel conformal block\nwith identical external labels is invariant under a Z2×Z2symmetry group generated\nby rotations around the xandyaxis as follows:\nP0\nP0P0\nP0P, (2.24)\nthe composition of which yields a rotation by 180 degrees. It thus follows that the\nVirasoro TQFT partition function on the excised four-punctured sphere is invariant\nunder the same symmetry operations. In particular, this means that one can cut the\nfour-punctured sphere with a tangle inside, apply one of these symmetry operations,\nand then reglue the tangle. This leads in general to an inequivalent knot, but the\ndifference is not detectable by computing ZVir. A famous example of a mutant\nhyperbolic knot pair is the Conway knot and the Kinoshita-Terasaka knot shown in\nFigure 2. In particular, since for this example, the mapping class group of both knot\ncomplements is trivial, the gravitational path integral on the Conway knot and the\nKinoshita-Terasaka knot is exactly the same and the gravitational path integral is\nhence not a sufficiently refined observable to be able to detect the topology of all\nhyperbolic three-manifolds.\nVia the volume conjecture (2.1), this implies in particular that mutant knot\ncomplements have the same hyperbolic volume. This result is known in the math\nliterature [34], but the present discussion makes it tautological. More surprisingly,\nthe refined volume conjecture (2.1) also implies that the corresponding manifolds\nhave the same one-loop determinants.\nUsing the same techniques of Virasoro TQFT, one can also show that the\ngeodesics fully inside or outside the cutting surface have the same length.6Thus\nthe length spectra of two mutant manifolds partially coincide. However, the length\nspectrum in general differs as one can see by an explicit computation using the soft-\nware SnapPy [35]. We display in Table 1 the low-lying length spectrum on the Conway\nknot and the Kinoshita-Teresaka knot. Thus even though the geodesic length spec-\ntrum determines the one-loop determinant (we have qγ= e−ℓγ) and the one-loop\ndeterminants agree, the length spectrum is in general different.\nThere are other versions of mutations. We can consider any embedded surface\ninMwith a special symmetry such as the four-punctured sphere above. Cutting\nMalong such a surface, applying the symmetry and regluing leads to a mutated\n6This is based on the observation that inserting Wilson lines with degenerate Virasoro represen-\ntations measure the geodesic length in the classical limit [26].\n– 10 –Conway Kinoshita-Teresaka\n1.044 + 2 .327i 1.044 + 2 .327i\n1.152−2.266i 1.152−2.266i\n1.384 + 2 .840i 1.384 + 0 .508i\n1.756−2.011i 1.530 + 2 .037i\n1.831−0.095i 1.831−0.095i\n1.907 + 2 .521i 1.907 + 2 .521i\n1.938−2.402i 1.938−2.402i\n2.011 + 0 .738i 2.031 + 2 .934i\n2.184−1.327i 2.097 + 2 .938i\n2.230−1.770i 2.183−1.425i\n2.233−1.893i 2.233−1.893i\nTable 1 . The low-lying length-spectrum of primitive geodesics on the complement of\nthe Conway and the Kinoshita-Teresaka knot. The imaginary part encodes the holonomy\naround the geodesic. The length spectrum partially agrees. All geodesics have multiplicity\n1 since the two manifolds have trivial isometry groups.\nmanifold. For example, we can cut along a genus 2 surface without punctures and\nuse the Z2hyperelliptic involution. Every genus 2 conformal block is invariant under\nthe corresponding Z2symmetry acting by a rotation around the xaxis as follows:7\nP1\nP2P3\n. (2.25)\nThus any partition function of Virasoro TQFT on a hyperbolic three-manifold with\nonly a genus 2 boundary must have the same property. In particular, we can produce\ntwo hyperbolic three-manifolds by cutting along a genus 2 surface and applying such\na rotation. This yields in general non-equivalent hyperbolic three-manifolds, but\nwith the same value of ZVir. Via the volume conjecture (2.1), this implies again that\nsuch a pair of manifolds has the same hyperbolic volume and the same value of the\ninfinite product appearing as the one-loop determinant in (2.1).\nAs a concrete example that is perhaps more familiar and directly relevant to\nholography, consider the Euclidean wormhole of the form Σ 2×[0,1]. Since the genus\n2 surface is hyperelliptic, we can perform the hyperelliptic involution on one side,\nwhich formally leads to a different manifold, but with identical partition function. As\nexplained in [1], the Virasoro TQFT partition function on the Euclidean wormhole is\nsimply given by the partition function of Liouville CFT ZLiouville (Σ2,0|m1,m2), where\n7This would fail at genus 3 since we have in general two different Liouville momenta on the\nbottom and top of the middle loop, which get exchanged by this operation. Correspondingly, not\nevery genus 3 surface is hyperelliptic.\n– 11 –P0P1 P2\nFigure 3 . The network of Wilson lines used to derive the relation between the modular\ncrossing kernel Sand the sphere crossing kernel F.\nthe left-moving moduli m1are associated to the left boundary and the right-moving\nmoduli m2to the right-moving boundary. However, the Liouville partition function\nis already invariant when we apply the hyperelliptic involution to only m1, which\nmeans that the partition function of this twisted wormhole also equals the Liouville\npartition function.\n2.5 Consistency conditions on the crossing kernels\nThe crossing kernels FandSon the four-punctured sphere and the once-punctured\ntorus, respectively, are subject to a number of constraints known as the Moore-\nSeiberg consistency conditions [36]. We listed them in the Appendix of [1], see also\n[18]. They express consistency of the projective representation of the 2d mapping\nclass group on the space of conformal blocks. For example, the modular crossing\nkernelShas to satisfy the SL(2 ,Z) relations together with the Dehn twist Ton the\nonce-punctured torus.8\nThese relations are also necessary for the consistency of the three-dimensional\ntheory. However, they can often be seen much easier from the three-dimensional\nperspective. We explain here one simple example that shows that Scan be fully\nexpressed in terms of Fthat we also use later in the paper. We should mention that\nthis construction is standard in the context of modular tensor categories which can\nbe viewed as the rational counterpart of Virasoro TQFT [37].9\nConsider S3with a network of Wilson lines as in Figure 3. We recall that a\njuncture of Wilson lines was defined as follows in [1]:10\nP1\nP3P2 ≡1\nC0(P1, P2, P3)×P1\nP3P2 . (2.26)\nOn the right hand side, we excise a spherical boundary around the puncture. The\npath integral then creates a state in the boundary Hilbert space, which is one-\ndimensional and hence can be canonically identified with Cby fixing the standard\nnormalization of the three-point function on the sphere.\n8More precisely, the crossing kernels give rise to a projective representation of SL(2 ,Z).\n9We thank Sahand Seifnashri for explaining the MTC computation to us.\n10C0(P1, P2, P3) is the universal Liouville three-point function, see [1, eq. (2.17)].\n– 12 –The main point is now that the value of the partition function on the network\nof Wilson lines in Figure 3 can be computed in two different ways as follows.\nLet us first consider the Heegaard splitting into two once-punctured tori. The two\nonce-punctured tori are homeomorphic to tubular neighborhoods of the Wilson lines\nP1andP2, respectively. The normalization of the juncture in (2.26) is chosen such\nthat the Virasoro TQFT path integral on the once-punctured tori leads precisely to\nthe respective conformal blocks on the boundary torus. The two once-punctured tori\nare interlocking and hence we have to apply an S-modular transformation. Being\nmore careful about the definition of the S-modular transformation actually shows\nthat we need the inverse of the modular crossing kernel S. Since Ssquares to eπi∆0,11\nthe inverse of Sdiffers from Sonly by the phase e−πi∆0. In the end, we obtain for\nthe partition function of the Wilson line network M,\nZVir(M) = e−πi∆0Z\ndP′\n1SP1,P′\n1[P0]\u001cP0P2\f\f\f\fP0P′\n1\u001d\n(2.27)\n=e−πi∆0SP1,P2[P0]\nρ0(P2)C0(P0, P2, P2), (2.28)\nwhere we applied [1, eq. (2.21)] for the evaluation of the inner product between\nconformal blocks. It reads\n⟨FC\ng,n(⃗P)|FC\ng,n(⃗P′)⟩=δ3g−3+n(⃗P−⃗P′)Q\ncuffs a ρ0(Pa)Q\npair of pants ( i,j,k)C0(Pi, Pj, Pk). (2.29)\nHere, FC\ng,nare the genus- g n-point Virasoro blocks in a particular OPE channel\nC,C0(P1, P2, P3) is the Liouville three-point function, and ρ0(P) the inverse of the\ntwo-point function. In other words, conformal blocks in a channel Care orthogonal\nwith a density given by the inverse OPE density of Liouville theory. See [1] for our\nconventions for the Liouville structure constants.\nWe can alternatively compute the partition function by a Heegaard splitting\nalong a four-punctured sphere containing the Wilson line P0and the stubs of the\nWilson lines P1andP2. We have in hopefully obvious notation\nZVir(M) =Z\ndPFP0,P\u0014P2P1\nP2P1\u0015\nZVir \nPP1 P2!\n(2.30)\n=Z\ndPFP0,P\u0014P2P1\nP2P1\u0015\ne2πi(∆−∆1−∆2)ZVir \nPP1 P2!\n(2.31)\n=Z\ndPFP0,P\u0014P2P1\nP2P1\u0015e2πi(∆−∆1−∆2)\nC0(P, P 1, P2), (2.32)\n11See [1, eq. (A.5a)].\n– 13 –where we used the braiding move twice in the second line. In the last line we recognize\nthe Euclidean wormhole with two three-punctured sphere on both ends, which eval-\nuates to the Liouville three-point function. Taking into account the normalization\nin eq. (2.26), we get an inverse structure constant.\nComparing (2.28) and (2.32) then expresses the modular crossing kernel fully in\nterms of the sphere crossing kernel,\nSP1,P2[P0] =Z\ndPρ0(P2)C0(P0, P2, P2)\nC0(P, P 1, P2)eπi(2∆+∆ 0−2∆1−2∆2)FP0,P\u0014P2P1\nP2P1\u0015\n.(2.33)\nUsing the explicit form of the sphere crossing kernel given e.g. in eq. (2.42a) of [1],\none can use various known identities of the involved integrals of special functions to\nderive the known expression of the modular fusion kernel from this integral formula\n[18, 26].\nThis identity can also be derived from a two-dimensional point of view by requir-\ning the consistency of the representation of the mapping class group on the space of\nconformal blocks on the two-punctured torus. It is in fact a special case of the cor-\nresponding Moore-Seiberg relation. However, the corresponding derivation is much\nmore complicated and subtle than the three-dimensional point of view.\n3 Holographic examples\nWe now move on to holographic applications of the Virasoro TQFT formalism. We\nwill mostly focus on multi-boundary wormholes that have direct implications for the\ndescription of the holographic dual of 3d gravity in terms of an ensemble of CFT\ndata. In order to set the stage for this discussion, let us briefly recapitulate the\nensemble description of AdS 3gravity.\nIn [10] it was shown that averaged products of CFT observables in a Gaussian\nensemble for the CFT data defined by\ncijk= 0, (3.1a)\ncijkc∗\nℓmn=C0(Pi, Pj, Pk)C0(¯Pi,¯Pj,¯Pk)\n×\u0000\nδiℓδjmδkn+ (−1)ℓi+ℓj+ℓkδiℓδjnδkm+ 4 permutations\u0001\n, (3.1b)\ntogether with a Cardy spectrum of heavy states, agree with the on-shell actions\n(and in certain cases, the one-loop determinants) of suitable Euclidean wormholes in\nsemiclassical AdS 3gravity coupled to massive point particles. Here ℓi=P2\ni−¯P2\niis\nthe spin of the corresponding primary. The averaged CFT quantities are computed\nby performing a simultaneous conformal block decomposition of the observables and\ncomputing Wick contractions of the structure constants using (3.1). The gravity\ncomputations were mostly restricted to two-boundary wormholes with topology Σ ×\n[0,1], with Σ a (possibly punctured) Riemann surface, corresponding to two-copy\n– 14 –averaged observables on the CFT ensemble side. Indeed one may view (3.1) as being\ndetermined by an explicit computation of the 3d gravity partition function on a\nEuclidean wormhole with the topology of a three-punctured sphere times an interval\n[10]; see figure 4.\ni\nj\nki\nj\nk\nFigure 4 . The Euclidean wormhole with the topology of a three-punctured sphere times\nan interval that contributes to the variance of the structure constants in the ensemble\ndescription of the dual of AdS 3gravity.\nIn [1] the correspondence between two-boundary Euclidean wormhole partition\nfunctions and averaged products of CFT observables was extended to finite central\ncharge using Virasoro TQFT. In particular the TQFT partition function on the\nEuclidean wormhole was computed, with the result\nZVir(Σ×[0,1]|m1,m2) =ZLiouville (Σ|m1,m2), (3.2)\nwhere m1,m2collectively denote the moduli of the Riemann surfaces at the two\nboundaries, and ZLiouville (Σ) is the correlation function on Σ in Liouville CFT.\n|ZVir(Σ×[0,1])|2agrees with the the averaged CFT computations performed in\nthe Gaussian ensemble (3.1), and its large- cexpansion agrees with the semiclassical\ngravity saddle-point computations in [10].\nExcept in certain very special cases it is not clear how to compute the grav-\nity path integral on configurations with more than two asymptotic boundaries in the\nmetric formalism. Such configurations in particular encode non-Gaussian corrections\nto the ensemble formulation of the boundary theory defined in (3.1), which are known\nto be needed for the internal consistency of the ensemble description from a variety\nof points of view [11, 21, 38]. For example, the existence of a Gaussian contraction\noften depends on the specific choice of channel in the conformal block decomposition\nof the CFT observables; crossing symmetry then requires non-Gaussian statistics in\nthe dual channel in order to reproduce the result in the channel where the Gaussian\ncontraction exists. Hence non-Gaussian corrections, which are necessary for an inter-\nnally consistent description of the boundary ensemble, are not presently accessible\nin the metric formulation of AdS 3gravity.\nIn the remainder of this section we will study Euclidean wormholes in AdS 3\ngravity with more than two asymptotic boundaries using Virasoro TQFT. We will\nmostly focus on wormholes with more than two three-punctured sphere boundaries,\nsince these determine the leading contributions to higher moments of the structure\nconstants in the ensemble description of the holographic dual. We will see in some\n– 15 –examples that the resulting non-Gaussian statistics precisely affirm the consistency\nof the results computed in the Gaussian ensemble.\nThere may also be non-Gaussian corrections to (3.1) associated to Euclidean\nwormholes with two three-punctured sphere boundaries but with higher topology in\nthe bulk. We will not study such corrections here, but let us briefly mention that\nwe have already encountered such a correction associated with a higher-topology\nwormhole. In section 2.5 we studied a configuration of Wilson lines equivalent to the\nfollowing two-boundary wormhole with linked Wilson lines\nM=i\nj\nji\nk\nk. (3.3)\nThe TQFT partition function on this wormhole may be computed as described in\nsection 2.5. One finds the following for the wormhole partition function\nZVir(M) =C0(Pi, Pj, Pj)SPjPk[Pi]\nρ0(Pk), (3.4)\ncorresponding to the following averaged product of structure constants that would\notherwise vanish (in the case that j̸=k) in the Gaussian ensemble12\ncijjc∗\nikk=|ZVir(M)|2=\f\f\f\fC0(Pi, Pj, Pj)SPjPk[Pi]\nρ0(Pk)\f\f\f\f2\n. (3.5)\n3.1 Cyclic defect wormholes\nA simple class of examples that demonstrate the practical utility of the TQFT re-\nformulation of 3d gravity is provided by multi-boundary wormholes with defects\nconnecting the sphere boundaries. For concreteness, consider the case where each\nboundary is a four-punctured sphere with defects connected in a pairwise cyclic way.\nSee figure 5 for a depiction of such a wormhole with three boundaries. In [10] it was\nargued that such on-shell wormholes contribute to the following averaged product of\nfour-point functions\n⟨O1O2O3O4⟩⟨O 3O4O5O6⟩···⟨O 2k−1O2kO1O2⟩ (3.6)\nin the ensemble of CFT data dual to semiclassical 3d gravity.13The Gaussian en-\nsemble hence makes a specific prediction for the gravitational partition function of\n12Strictly speaking, we get a different bulk manifold when we exchange the two ends of the\nWilson line labelled by k. Exchanging them leads to a braiding phase eπi∆j. Summing over both\npossibilities imposes that the spin of ihas to be even.\n13Wormholes of this sort also contribute to the Renyi entropies of certain coarse-grained states\nin 2d CFT [39].\n– 16 –these wormholes [10]:\nZgrav(Mk)?=\f\f\f\f\fZ∞\n0dP ρ 0(P)C0(P1, P2, P)C0(P3, P4, P)···C0(P2k−1, P2k, P)\n×P1\nP2P4\nP3P(z1)P4\nP3P5\nP6P(z2)···P2k\nP2k−1P1\nP2P(zk)\f\f\f\f\f2\n.(3.7)\nHere we have used the notation Mkto refer to the k-boundary sphere four-point\nwormhole, the stick diagrams are shorthand for the conformal blocks as usual, and zi\nrefers to the cross-ratio of the defect insertions on the ithboundary. The effect of the\nGaussian average is to set all the internal weights equal in this particular conformal\nblock decomposition of the wormhole partition function.\n1\n2\n4312\n65\n436\n5\nFigure 5 . The three-boundary sphere four-point wormhole M3.\nIt is not at all clear how to compute the wormhole partition function of the\nk-boundary sphere four-point wormhole in the metric formalism of 3d gravity, even\nin the semiclassical limit. Here we will describe how this wormhole partition func-\ntion may be straightforwardly computed in the Virasoro TQFT, reproducing the\nexpectation from the Gaussian ensemble (3.7).\n6\n51\n23\n4\nFigure 6 . The three-boundary sphere four-point wormhole as a compression body.\nFor concreteness and brevity of the equations we consider here the case k= 3,\nbut emphasize that the generalization to higher kis completely straightforward.\n– 17 –The idea is to view the wormhole as a compression body as indicated in figure 6. To\ncompute the partition function on the corresponding compression body we insert a\ncomplete set of states in the Hilbert space of the inner boundaries. This produces a\nparticular state in the Hilbert space of the outer boundary. Proceeding in this way\nwe have\nZVir(M3) =Z∞\n0dPadPbρ0(Pa)ρ0(Pb)C0(P1, P2, Pa)C0(P3, P4, Pa)C0(P3, P4, Pb)\n×C0(P5, P6, Pb)P1\nP2P4\nP3PaP4\nP3P5\nP6PbP5\nP6P2\nP1Pb PaP4\nP3.\n(3.8)\nWe have temporarily restored the sphere boundaries in representing the conformal\nblocks in order to emphasize that the last conformal block should be interpreted as\na state in the Hilbert space of the outer sphere boundary with a loop of Wilson lines\nin the interior, notas a higher-genus conformal block. In particular we can remove\nthe loop by recalling the TQFT identity [1]\nPa PbP4\nP3=δ(Pa−Pb)\nρ0(Pa)C0(P3, P4, Pa)Pa, (3.9)\nwhich leads us to\nZVir(M3) =Z∞\n0dPaρ0(Pa)C0(P1, P2, Pa)C0(P3, P4, Pa)C0(P5, P6, Pa)\n×P1\nP2P4\nP3PaP4\nP3P5\nP6PaP5\nP6P2\nP1Pa.(3.10)\nThe generalization to the case of kfour-punctured sphere boundaries follows im-\nmediately by viewing Mkas a compression body with k−1 inner boundaries and\nrepeated application of the identity (3.9). Upon squaring the TQFT partition func-\ntion to obtain the 3d gravity partition function, we hence verify (3.7), the prediction\nfrom the averaged product of ksphere four-point functions in the Gaussian ensem-\nble. Much like the case of the two-boundary Euclidean wormhole revisited in [1],\nthe correspondence between the averaged CFT quantities and the gravity partition\nfunction on a fixed topology persists beyond the semiclassical limit.\n– 18 –3.2 Four-boundary non-Gaussianity wormhole\nConsider a wormhole with four three-punctured spheres as asymptotic boundaries,\nwith defects threading the bulk of the wormhole in the following tetrahedral config-\nuration\nM=\n33\n22t\nt\ns s44\n11\n(3.11)\nThe gravity path integral on this wormhole should compute the following connected\npart of the fourth moment of structure constants in the dual description of 3d gravity\nin terms of an ensemble of CFT data\n|ZVir(M)|2↔c12scs34c14tct32. (3.12)\nComputation via Heegaard splitting. It straightforward to apply the Heegaard\nsplitting technique described in detail in [1] to compute the Virasoro TQFT partition\nfunction on the four-boundary wormhole. For instance, we can cut Malong a four-\npunctured sphere through the bulk of the wormhole as pictured in figure 7. This cuts\nthe four-boundary wormhole into two generalized compression bodies M1andM2.\nEach compression body has an outer boundary given by a four-punctured sphere and\ntwo three-punctured sphere inner boundaries. The Virasoro TQFT path integral on\neach compression body prepares a state in the Hilbert space of the four-punctured\nsphere, and the inner product of these states computes the TQFT partition function\non the four-boundary wormhole. Using (2.26) to write the three-punctured sphere\nboundaries in terms of trivalent Wilson line junctions, the TQFT partition functions\non the compression bodies are given by\n⟨ZVir(M1)|=C0(P1, P2, Ps)C0(P3, P4, Ps)*P1\nP2P4\nP3Ps\f\f\f\f\f, (3.13a)\n|ZVir(M2)⟩=C0(P1, P4, Pt)C0(P2, P3, Pt)\f\f\f\f\fP1\nP2P4\nP3Pt+\n. (3.13b)\n– 19 –33\n22t\nt\ns s44\n11 M1=3\n42\n1s\nM2=1 4\n2 3t\nFigure 7 . The Heegaard splitting of the four-boundary wormhole Minto two generalized\ncompression bodies M1andM2, each with the topology of a three-ball with two three-balls\ndrilled out in its interior and with Wilson lines connecting the two-sphere boundaries as\nshown in the figure. The TQFT path integral on each of M1andM2prepares a state in\nthe Hilbert space of the four-punctured sphere.\nUp to the C0factors, the compression body partition functions are given by individual\nsphere four-point conformal blocks in the s- and the t-channel. The inner product of\nthese states is proportional to the Virasoro fusion kernel essentially by definition:\n*P1\nP2P4\nP3Ps\f\f\f\f\fP1\nP2P4\nP3Pt+\n=Z\ndP′\nsFPtP′s\u0014P1P2\nP4P3\u0015*P1\nP2P4\nP3Ps\f\f\f\f\fP1\nP2P4\nP3P′\ns+\n(3.14)\n=FPtPs\u0014P1P2\nP4P3\u0015\nρ0(Ps)C0(P1, P2, Ps)C0(P3, P4, Ps)(3.15)\n=FPsPt\u0014P1P4\nP2P3\u0015\nρ0(Pt)C0(P1, P4, Pt)C0(P2, P3, Pt). (3.16)\nIn the penultimate line we computed the inner product by expanding the t-channel\nblock in a complete basis of s-channel blocks using the Ponsot-Teschner fusion kernel\n[15, 16], and in the last line we did the reverse. The equivalence of these two expres-\nsions is not a priori obvious without appealing to consistency of the conformal block\n– 20 –inner product, but it is guaranteed by for example a special case of the pentagon\nidentity, which is one of the Moore-Seiberg consistency conditions satisfied by the fu-\nsion kernel. In fact, this combination has a tetrahedral symmetry inherited from the\nbulk Wilson line configuration that is obscured by this presentation. Indeed, it can\nbe rewritten in a manifestly tetrahedrally symmetric form in terms of the Virasoro\n6jsymbol in the Racah-Wigner normalization [17] as follows\n*P1\nP2P4\nP3Ps\f\f\f\f\fP1\nP2P4\nP3Pt+\n=\u001aP1P2Ps\nP3P4Pt\u001b\np\nC0(P1, P2, Ps)C0(P3, P4, Ps)C0(P1, P4, Pt)C0(P2, P3, Pt).(3.17)\nThe upshot is that the Virasoro TQFT partition function on the four-boundary\nwormhole can be expressed in terms of the Virasoro 6 jsymbol via the following\ninner product in the Hilbert space of the four-punctured sphere\nZVir(M) =⟨ZVir(M1)|ZVir(M2)⟩ (3.18)\n=C0(P1, P2, Ps)C0(P3, P4, Ps)C0(P1, P4, Pt)C0(P2, P3, Pt)\n×*P1\nP2P4\nP3Ps\f\f\f\f\fP1\nP2P4\nP3Pt+\n(3.19)\n=p\nC0(P1, P2, Ps)C0(P3, P4, Ps)C0(P1, P4, Pt)C0(P2, P3, Pt)\u001aP1P2Ps\nP3P4Pt\u001b\n.\n(3.20)\nConsistency with boundary ensemble description. The gravity partition\nfunction Zgrav(M) =|ZVir(M)|2on the four-boundary wormhole (3.20) makes a\nconcrete prediction for the connected contribution to the fourth moment of structure\nconstants in the description of 3d gravity in terms of an ensemble of CFT data:14\nc12scs34c14tct32⊃ |ZVir(M)|2=q\nc2\n12sc2\ns34c2\n14tc2\nt32\f\f\f\f\u001aP1P2Ps\nP3P4Pt\u001b\f\f\f\f2\n(3.21)\nThis represents the leading correction to the Gaussian ensemble elucidated in [10].\nWe say that the fourth moment contains this contribution (rather than being literally\n14This fourth moment has previously appeared in [11] where it was argued for by requiring that\nthe variance of the crossing equation vanish, and in [38] where it followed from genus-three modular\ninvariance (using similar logic as that which shows that the variance should be given by the C0\nformula).\n– 21 –equal to it) because there may be corrections to (3.21) associated with wormholes\nwith the same boundaries but with higher topology in the bulk. It is expected that\nin the semiclassical limit such contributions are parametrically suppressed and hence\nthat (3.21) represents the leading contribution to the fourth moment.\nHere we will see that this non-Gaussian correction in fact exactly ensures the\ninternal consistency of the results predicted by the Gaussian ensemble.\nTo illustrate the point, consider the two-boundary Euclidean wormhole with\nthe topology of a (possibly punctured) Riemann surface Σ times an interval. The\ngravity path integral on the Euclidean wormhole is given by the square of (3.2),\nthe corresponding observable in Liouville CFT with the moduli on the two sides\npaired. This agrees with the averaged product of CFT observables in the Gaussian\nensemble (3.1). However the computation in the Gaussian ensemble often relies on\nthe choice of a specific channel in the conformal block decomposition; this is obviously\ninconsistent with crossing symmetry of the ensemble. Associativity of the OPE then\nrequires non-Gaussian statistics in order to reproduce this result in other channels.\nRelatedly, while the Gaussian ensemble is crossing symmetric on average, higher\nmoments of the crossing equation do not vanish; this has recently been emphasized\nin [11].\nFor concreteness, consider in particular the averaged product of four-point func-\ntions of local operators Oi. In the Gaussian ensemble, we have\n⟨O1(0)O2(z,¯z)O3(1)O4(∞)⟩⟨O 1(0)O2(z′,¯z′)O3(1)O4(∞)⟩∗\n=X\ns,s′c12sc34sc∗\n12s′c∗\n34s′\f\f\f\f\fP1\nP2P4\nP3Ps(z)P1\nP2P4\nP3P′\ns(z′)\f\f\f\f\f2\n(3.22)\n=\f\f\f\f\f\fZ\ndPsρ0(Ps)C0(P1, P2, Ps)C0(P3, P4, Ps)P1\nP2P4\nP3Ps(z)P1\nP2P4\nP3Ps(z′)\f\f\f\f\f\f2\n(3.23)\n=|ZLiouville (P1, P2, P3, P4|z, z′)|2. (3.24)\nHere we expanded the four-point functions in the same OPE channel, and performed\nthe Gaussian contractions in the third line using eq. (3.1). If we had instead expanded\none four-point function in the S-channel and the other in the T-channel, we would\nhave gotten zero in the Gaussian ensemble since\nc12scs34c14tct32|Gaussian = 0 (3.25)\nfor distinct external operators.15This is obviously inconsistent with basic principles\nof conformal field theory. The result for the averaged product of four-point functions\n15We thank Vladimir Narovlansky for asking a question that raised this point.\n– 22 –in terms of the four-point function in Liouville CFT is equal to the partition function\nof 3d gravity coupled to point particles on the Euclidean wormhole with the topology\nof a four-punctured sphere times an interval, so we seek a correction to the Gaussian\nensemble that preserves (3.24). If we supplement the Gaussian ensemble with the\nfourth moment (3.21) as computed by the four-boundary wormhole, we instead have\n⟨O1(0)O2(z,¯z)O3(1)O4(∞)⟩⟨O 1(0)O2(z′,¯z′)O3(1)O4(∞)⟩∗\n=X\ns,tc12scs34c∗\n41tc∗\nt23\f\f\f\f\fP1\nP2P4\nP3Ps(z)P1\nP2P4\nP3Pt (z′)\f\f\f\f\f2\n(3.26)\n=\f\f\f\f\fZ\ndPsdPtρ0(Ps)C0(P1, P2, Ps)C0(P3, P4, Ps)FPsPt\u0014P1P4\nP2P3\u0015\n×P1\nP2P4\nP3Ps(z)P1\nP2P4\nP3Pt (z′)\f\f\f\f\f2\n(3.27)\n=|ZLiouville (P1, P2, P3, P4|z, z′)|2, (3.28)\nin agreement with the previous computation and with the wormhole partition func-\ntion.\nOn braiding and the u-channel. In the discussion so far we have suppressed\nan important subtlety. In 2d CFT, the structure constants are not strictly invariant\nunder permutations of the three operators. For example, swapping a pair of operators\nleads to a sign that depends on the sum of the spins of the three operators\ncikj= (−1)ℓi+ℓj+ℓkcijk. (3.29)\nThis is inherited from reality properties of the structure constants: they are real\nif the sum of spins is even and imaginary if the sum of spins is odd, and the swap\ncomplex-conjugates the structure constants, cikj=c∗\nijk. Similarly, in the computation\nof wormhole partition functions with bulk Wilson lines via Heegaard splitting, there\nmay be crossings of lines that need to be undone via braiding operations. These\nbraidings introduce phases that depend on the conformal weights.\nIn general, we can read off the ordering of the structure constants from a bulk\nmanifold by fixing a cyclic ordering and reading the labels around three-punctured\nboundaries cyclically. The same applies in CFT computations, where we read off\nthe labels of the structure constants cyclically around every vertex in the conformal\nblocks.16\n16The overall cyclic direction does not matter since every label appears twice and thus cancels if\nwe reverse the overall cyclic direction.\n– 23 –As a simple example of a wormhole computation for which such braidings are\nessential, consider the following four-boundary wormhole:\nM=1\n2 s\ns 3\n4 42 uu 31\n, (3.30)\nwhich is essentially the same as (3.11). The boundaries of this wormhole are three-\npunctured spheres corresponding to the structure constants that appear in the s-\nandu-channel conformal block decompositions of the sphere four-point function\n⟨O1O2O3O4⟩.\nWe compute the TQFT partition function as before by splitting along a four-\npunctured sphere in the bulk. Undoing the crossing of the Wilson lines and com-\nputing the inner product in the Hilbert space of the splitting surface leads to the\nfollowing result for the TQFT partition function on this four-boundary wormhole\nZVir(M) =p\nC12sCs34C31uCu42\u001aP1P2Ps\nP4P3Pu\u001b\neπi(P2\n1+P2\n4−P2\ns−P2\nu). (3.31)\nHere we have introduced the shorthand\nCijk≡C0(Pi, Pj, Pk), (3.32)\nWe notice the presence of an additional phase compared to (3.20). This result fol-\nlows from taking the inner product between an s- and a u-channel Virasoro conformal\nblock, and hence this phase may be understood in terms of the crossing transforma-\ntion that relates s- and u-channel blocks. This crossing transformation is given by\n1 423\ns=Z∞\n0dPueπi(P2\n1+P2\n4−P2\ns−P2\nu)FPsPu\u0014P1P3\nP2P4\u0015\n1 423\nu.(3.33)\nThe combination that appears on the right-hand side is sometimes referred to as\nthe “R-matrix.” The semiclassical near-extremal limit of the R-matrix governs the\nout-of-time-order four-point function in the Schwarzian theory [19].\n– 24 –Hence for the following fourth moment of CFT structure constants we find\nc12scs34c31ucu42⊃ |ZVir(M)|2= (−1)ℓ1+ℓ4+ℓs+ℓuq\nc2\n12sc2\ns34c2\n13uc2\nu24\f\f\f\f\u001aP1P2Ps\nP4P3Pu\u001b\f\f\f\f2\n.\n(3.34)\nThis is exactly consistent with the previous result (3.21) upon relabeling t→u,\n4↔3 and making use of the exchange property (3.29). It is also consistent with the\naveraged product of sphere four-point functions in the Gaussian ensemble, where we\nexpand one four-point function in the s-channel and the other in the u-channel.\n3.3 Many-boundary wormholes and higher non-Gaussianities\n3.3.1 A simple six-boundary example\nConsider the following wormhole with six three-punctured sphere boundaries\nM=a\na11\n22\n5\n54\n43\n3b b\n6\n6c c. (3.35)\nAs indicated by the diagram, it contributes to the following sixth moment of CFT\nstructure constants\n|ZVir(M)|2↔c12ac2b3c3c4c45ac5c6c6b1. (3.36)\nThere are several Heegaard splittings that one could employ to compute the Virasoro\nTQFT partition function on this wormhole, but the simplest is indicated in figure 8:\nwe cut the wormhole through the bulk along a three-punctured sphere. This divides\nMinto two generalized compression bodies M1andM2, each of which is itself a four-\nboundary wormhole of the type described in the previous subsection. The partition\nfunction of Virasoro TQFT on the generalized compression bodies was computed in\n(3.20) as\n|ZVir(M1)⟩=p\nC12aC2b3C6b1Ca36\u001aP1P2Pa\nP3P6Pb\u001b\n(3.37a)\n|ZVir(M2)⟩=p\nC3c4C45aC5c6Ca36\u001aP3P4Pc\nP5P6Pa\u001b\n(3.37b)\n– 25 –a\na11\n22\n5\n54\n43\n3b b\n6\n6\nc c6 3\nab\n1 2\n3 6\nac\n4 5M=M1=\nM2=\nFigure 8 . A Heegaard splitting of the wormhole with six three-punctured sphere bound-\naries. Each constituent compression body is equivalent to the four-boundary wormhole\nstudied in section 3.2.\nThen the Virasoro TQFT partition function on the six-boundary wormhole is given\nby the following inner product between these states in the Hilbert space of the shared\nthree-punctured sphere boundary\nZVir(M) =⟨ZVir(M1)|ZVir(M2)⟩ (3.38)\n=p\nC12aC2b3C3c4C45aC5c6C6b1\u001aP1P2Pa\nP3P6Pb\u001b\u001aP3P4Pc\nP5P6Pa\u001b\n. (3.39)\nNotice that here the only effect of the inner product is to divide by the extra factor\nofC0(P3, Pa, P6).\nThis particular Heegaard splitting of the six-boundary wormhole is far from\nunique: for example, we could have cut it through a five-punctured sphere, or along\nthree four-punctured spheres. In all cases, the corresponding splittings yield the\nsame result (3.39) for the TQFT partition function.\nThis wormhole partition function implies that the corresponding sixth moment\nof CFT structure constants is given by\nc12ac2b3c3c4c45ac5c6c6b1⊃q\nc2\n12ac2\n2b3c2\n3c4c2\n45ac2\n5c6c2\n6b1\f\f\f\f\u001aP1P2Pa\nP3P6Pb\u001b\u001aP3P4Pc\nP5P6Pa\u001b\f\f\f\f2\n.\n(3.40)\nConsistency with boundary ensemble description. Much like the fourth mo-\nment of the structure constants inferred from the four-boundary wormhole of section\n3.2, the sixth moment (3.40) is needed for consistency of the description of the bound-\nary theory in terms of an ensemble of CFT data. There are a variety of ways to see\n– 26 –this. Roughly, for each Heegaard splitting of the wormhole, there is a correspond-\ning product of CFT observables for which consistency of the ensemble description\nrequires that the appropriate moment of CFT data is correctly computed by the\nwormhole.\nFor concreteness, consider the average of the following product of five-point func-\ntions\n⟨ObO1OaO4Oc⟩⟨O bO1OaO4Oc⟩∗. (3.41)\nThis is associated with splitting the wormhole (3.35) along a five-punctured sphere\nin the bulk. The average (3.41), which corresponds to the two-boundary sphere five-\npoint function wormhole, is given by the corresponding five-point function in Liouville\nCFT as in (3.24). In the Gaussian ensemble this however requires that we expand\nthe two five-point functions in aligned channels when taking the ensemble average.\nOf course we are free to expand the five-point functions in different channels, in\nwhich case we need to invoke the non-Gaussian statistics. The combination of OPE\nchannels that is associated to the particular Heegaard splitting is determined by\nthe combination of sphere three-point boundaries that appear in each compression\nbody of the Heegaard splitting. For example, if we compute the averaged product of\nsphere five-point functions by expanding in the following channel where there is not\na Gaussian contraction\n⟨ObO1OaO4Oc⟩⟨O bO1OaO4Oc⟩∗\n=X\nO2,3,5,6c12ac2b3c3c4c∗\n4a5c∗\n56cc∗\n61b\f\f\f\f\nb1 a\n4\nc32(m1)\nb1a 4\nc65(m2)\f\f\f\f2\n(3.42)\n=\f\f\f\fZ∞\n0dP3dP6ρ0(P3)ρ0(P6)C0(P1, Pb, P6)C0(P6, Pa, P3)C0(P3, Pc, P4)\n×b\n631a4\nc(m1)b\n631a4\nc(m2)\f\f\f\f2\n(3.43)\n=|ZLiouville (Pb, P1, Pa, P4, Pc|m1, m2)|2, (3.44)\nthen making use of the sixth moment (3.40) and the fact that the 6 jsymbols im-\nplement crossing transformations on the conformal blocks, we reproduce exactly the\nresult from the Gaussian contraction, the sphere five-point function in Liouville the-\nory. Here micollectively denote the moduli of each five-point function.\nWe could have considered other Heegaard splittings, corresponding to averaged\nCFT observables that receive contributions from this combination of structure con-\nstants in a particular OPE channel. For example, the following averaged product of\n– 27 –three four-point functions\n⟨O6O1O2O3⟩⟨O 1O2O4O5⟩⟨O 6O5O4O3⟩ (3.45)\nreceives contributions from the sixth moment (3.40) in a specific OPE channel that\nprecisely reproduce the result (3.10) for the averaged product in the Gaussian en-\nsemble.\n3.3.2 A more nontrivial six-boundary example\nHere we consider another wormhole with six three-punctured sphere boundaries, but\nwith the defects arranged slightly differently between the boundaries\nM=a\na1\n1\n6\n6\n5\n5443322\nc\nb\nbc(3.46)\nThis contributes to a different sixth moment of the structure constants\n|ZVir(M)|2↔c1a2c2b3c3c4c4a5c5b6c6c1. (3.47)\nWe could of course compute the TQFT partition function on this wormhole by a\nstraightforward Heegaard splitting, for example along three four-punctured spheres.\nIn this case it turns out to be most convenient to replace the three-punctured sphere\nboundaries with trivalent junctions as in (2.26) and hence regard the wormhole as a\nnetwork of Wilson lines embedded in S3:\nZVir(M) =C12aC23bC34cC45aC56bC61cZVir a\nacb\nbc1\n6\n5 432\n!\n(3.48)\n≡C12aC23bC34cC45aC56bC61cZVir(M′). (3.49)\nHere M′is the network of Wilson lines depicted on the right-hand side of (3.48)\nembedded in S3. Braiding the Wilson lines and applying a fusion transformation,\n– 28 –the TQFT partition function may then be simplified as follows17\nZVir(M′) = (BPaP5\nP4BPcP1\nP6BP3Pc\nP4)−1ZVir 1\n6\n5 43b\nac2!\n(3.51)\n= (BPaP5\nP4BPcP1\nP6BP3Pc\nP4)−1Z\ndPdFP3Pd\u0014P2Pc\nPbP4\u0015\nZVir 1\n6\n5 4db\nac2!\n.\n(3.52)\nWe then recognize the following Wilson line identity (see [1, eq. (3.44)])\n2 3t\n14\ns=r\nC23t\nC12sC34sC14t\u001aP1P2Ps\nP3P4Pt\u001b\n2 3t\n, (3.53)\nwhich allows us to recast the TQFT partition function as\nZVir(M′) = (BPaP5\nP4BPcP1\nP6BP3Pc\nP4)−1Z\ndPdρ0(Pd)r\nC2cdCa6d\nC2b3C3c4C4a5C5b6\n×\u001aP2P3Pb\nP4PdPc\u001b\u001aP4P5Pa\nP6PdPb\u001b\nZVir 1\n6 d\nac2!\n.(3.54)\nFinally, we undo the crossings by braiding the Wilson lines and recognize the remain-\ning configuration as the four-boundary wormhole studied in section 3.2 to arrive at\nZVir(M) =p\nC12aC23bC34cC45aC56bC61ceπi(P2\n1+P2\n3+P2\n5−2P2\n2−2P2\n4−2P2\n6)\n×Z\ndPdρ0(Pd) e3πiP2\nd\u001aP6P1Pc\nP2PdPa\u001b\u001aP2P3Pb\nP4PdPc\u001b\u001aP4P5Pa\nP6PdPb\u001b\n.(3.55)\n17Here\nBPjPk\nPi=eπi(P2\ni−P2\nj−P2\nk−Q2\n4)(3.50)\nis the braiding phase.\n– 29 –Once the dust has settled, as in previous examples the wormhole partition function is\ngiven by factors of√C0for each sphere three-point boundary together with a suitable\ncombination of Virasoro 6 jsymbols associated with the Wilson line crossings.\nThis wormhole partition function implies that the corresponding sixth moment\nfor the CFT structure constants receives the following contribution\nc12ac23bc34cc45ac56bc61c⊃q\nc2\n12ac2\n23bc2\n34cc2\n45ac2\n56bc2\n61c(−1)ℓ1+ℓ3+ℓ5\n×\f\f\f\fZ\ndPdρ0(Pd)e3πiP2\nd\u001aP6P1Pc\nP2PdPa\u001b\u001aP2P3Pb\nP4PdPc\u001b\u001aP4P5Pa\nP6PdPb\u001b\f\f\f\f2\n.(3.56)\nAs in previous examples, this sixth moment precisely affirms the internal consistency\nof the description in terms of an ensemble of CFT data. Indeed, if one expands for\nexample the product of two sphere five-point functions or three sphere four-point\nfunctions in certain OPE channels where there is not a Gaussian contraction, this\nleads to a result consistent with the computation in the Gaussian ensemble. For\nconcreteness, consider the following averaged product of three four-point functions,\nall expanded in the u-channel\n⟨O2OcOaO6⟩⟨O 6OaObO4⟩⟨O 4ObOcO2⟩\n=X\nO1,3,5c1a2c2b3c3c4c4a5c5b6c6c1\f\f\f\f\f2 6ca\n16 4ab\n54 2bc\n3\f\f\f\f\f2\n(3.57)\n=\f\f\f\f\fZ\ndP1ρ0(P1) dP3ρ0(P3) dP5ρ0(P5)p\nC1a2C2b3C3c4C4a5C5b6C6c1eπi(P2\n1+P2\n3+P2\n5)\n×e−2πi(P2\n2+P2\n4+P2\n6)Z\ndPdρ0(Pd) e3πiP2\nd\u001aP6P1Pc\nP2PdPa\u001b\u001aP2P3Pb\nP4PdPc\u001b\u001aP4P5Pa\nP6PdPb\u001b\n×2 6ca\n16 4ab\n54 2bc\n3\f\f\f\f\f2\n(3.58)\n=\f\f\f\f\fZ\ndPdρ0(Pd)C2cdC4bdC6ad2 6ca\nd6 4ab\nd4 2bc\nd\f\f\f\f\f2\n.(3.59)\nSo we see that applying the statistics (3.56) precisely reproduces the result (3.10)\nanticipated from the Gaussian ensemble.\nNotice that in this case the corresponding sixth moment receives contributions\nfrom configurations in which the Wilson lines have a different pattern of over- and\nunder-crossings in the bulk, in addition to those with higher topology in the bulk. In\nprinciple, we could consider contributions from the manifolds formed by cutting M\n– 30 –along a six-punctured sphere in the bulk and gluing in another six-punctured sphere\nwith any tangle formed by three strands in the bulk. As a simple example, we could\nhave considered the following six-boundary wormhole\nM=a\na1\n1\n6\n6\n5\n5443322\nc\nb\nbc. (3.60)\nThe TQFT partition function on this wormhole differs from (3.55) in a subtle way\nZVir(M) =p\nC1a2C2b3C3c4C4a5C5b6C6c1eπi(−P2\n1+P2\n3+P2\n5−2P2\n4)\n×Z\ndPdρ0(Pd) eπiP2\nd\u001aP6P1Pc\nP2PdPa\u001b\u001aP2P3Pb\nP4PdPc\u001b\u001aP4P5Pa\nP6PdPb\u001b\n.(3.61)\nThe only difference from (3.55) are the phases, particularly that which appears in the\nintegral over the intermediate Liouville momentum Pd. Although both contribute to\nthe corresponding sixth moment of the structure constants, between (3.55) and (3.61)\nis not a priori obvious which Wilson line configuration dominates in the semiclassical\nlimit.\n3.3.3 Diagrammatic rules for multi-boundary wormholes and CFT statis-\ntics\nAlthough the intermediate details of the computations were nontrivial, there is an\nunderlying simplicity to the previously discussed results for the Virasoro TQFT par-\ntition functions of wormholes with three-punctured sphere boundaries and trivial\ntopology in the bulk, and hence for the leading contributions to the non-Gaussian\nstatistics of CFT data in the boundary ensemble description of 3d gravity. In all\ncases, the wormhole partition function involves a factor of√C0for each three-\npunctured sphere boundary, together with a suitable combination of Virasoro 6 j\nsymbols. Here we describe diagrammatic rules that straightforwardly reproduce\nthese results and that enable the computation of more nontrivial wormhole parti-\ntion functions. These rules will turn out to be a slight generalization of the disk\nFeynman rules in JT gravity coupled to matter (see e.g. [21]).18\n18SC is grateful to Baur Mukhametzhanov for discussions on this. Baur also independently\nobserved that higher moments of CFT data required for internal consistency of the ensemble de-\n– 31 –It is simplest to describe the situation in which the sphere boundaries are con-\nnected in a cyclic way, as in (3.30) and (3.46); the CFT statistics in other configura-\ntions may be obtained from the results in these cases by application of the swapping\nrule (3.29).\nThe idea is the following. Starting from a wormhole configuration with the\nboundaries connected in a cyclic way, replacing the punctured sphere boundaries\nwith a trivalent vertex as follows\n1\n2 3−→\n31\n2(3.62)\nproduces a disk diagram with lines that may cross in the interior of the disk, such\nas that drawn in (3.48). It is important to keep track of the way that the lines over-\nand under-cross in the projection to a two-dimensional disk diagram. The TQFT\npartition function associated with this disk diagram is then computed according to\nthe following simple Feynman rules:\n•Each trivalent vertex contributes a factor of√C0:\n31\n2=p\nC0(P1, P2, P3). (3.63)\n•Each closed region in the interior of the disk is associated with a Liouville\nmomentum Pthat is integrated with the measure ρ0(P)dP.\n•Each crossing of a pair of lines in the interior of the disk contributes a Virasoro\n6jsymbol\n1 2\n3 4ts\n=\u001aP1P2Ps\nP3P4Pt\u001b\neπi(P2\n1+P2\n3−P2\n2−P2\n4). (3.64)\nHere the labels 1, 2, 3 and 4 are associated to the four faces delineated by the\nWilson lines sandt.\nThe Virasoro 6 jsymbol plays the role of a quartic vertex in these diagrammatic rules,\ndressed with a phase that keeps track of the way that the Wilson lines over- and\nscription of 3d gravity were reproduced by generalizations of the disk Feynman diagrams in JT\ngravity coupled to matter [40].\n– 32 –under-cross. This reproduces the partition function on the four-boundary wormhole\n(3.31) essentially by design.\nAs a simple example, consider the six-boundary wormhole studied in section\n3.3.2. The two-dimensional projection of this configuration involves three crossings\nof Wilson lines and one closed region in the interior of the disk, so the TQFT partition\nfunction involves a single integral of three 6 jsymbols. Indeed, a straightforward ap-\nplication of these rules immediately reproduces the TQFT partition function (3.55).\nThe Virasoro 6 jsymbol obeys many identities that facilitate the consistency of\nthis description. For instance, it is often the case that there is an ambiguity of how\nto arrange the Wilson line crossings in the interior of the disk. The TQFT partition\nfunction as computed from these rules should be independent of such choices. For\nexample, we should have\na\nb2\n1 3\n4=a\nb2\n1 3\n4, (3.65)\nwhich is guaranteed by idempotency of the Virasoro 6 jsymbol\nZ\ndPsρ0(Ps)\u001aP4P1Pb\nP2PsPa\u001b\u001aP2P3Pa\nP4PsPb\u001b\n=δ(P1−P3)\nρ0(P1). (3.66)\nThere is also a Yang-Baxter equation, which facilitates moving a line over a crossing\nas follows,\na b\n12\n3\n4\n56c=12\n3\n4\n56a b\nc\n. (3.67)\nIn equations, this translates to\nZ\ndPdρ0(Pd)\u001aP1P2Pa\nP3PdPb\u001b\u001aP5P6Pb\nP1PdPc\u001b\u001aP3P4Pc\nP5PdPa\u001b\neπi(P2\nd+P2\n2+P2\n4+P2\n6)\n=Z\ndPeρ0(Pe)\u001aP6P1Pc\nP2PePa\u001b\u001aP2P3Pb\nP4PePc\u001b\u001aP4P5Pa\nP6PePb\u001b\neπi(P2\ne+P2\n1+P2\n3+P2\n5).(3.68)\nThis identity follows from the consistency of braiding on the sphere. Indeed, the\nR-matrix also appears as the braiding matrix of conformal blocks as in eq. (3.33).\nThe Yang-Baxter equation then corresponds to the fundamental relation in the braid\n– 33 –group as follows:\n12 34a b c\n=\n12 34a b c\n. (3.69)\nUsing (3.33) to unbraid the left- and right-hand side and comparing the result leads\nto the Yang-Baxter equation (3.68).\nThese diagrammatic rules for wormhole partition functions are structurally iden-\ntical to the disk Feynman rules for JT gravity coupled to matter, as described for\nexample in [21]. The only differences are that here the trivalent vertex is given by√C0, the quartic vertex is given by the Virasoro 6 jsymbol rather than the SL(2 ,R)\n6jsymbol, and one must keep track of the over- and under-crossings of the Wilson\nlines in the bulk, leading to extra phases in the quartic vertex. Indeed, these rules\nprecisely reduce to the JT gravity + matter disk Feynman rules in the semiclassical\nnear-extremal limit of [5, 41]. In this limit one takes\nc= 1 + 6( b+b−1)2, P ext=bsext, P int=i\n2(b+b−1−2bhint), b→0 (3.70)\nfixing sextandhintin the semiclassical limit. Here Pextare the Liouville momenta of\nthe Wilson lines forming the perimeter of the disk, Pintare the Liouville momenta of\nthose in the interior of the disk, and this limit corresponds to sending the external\nWilson lines very near extremality while assigning the internal Wilson lines a fixed\nconformal weight hint. With all external Wilson lines near extremality, the extra\nphase in the quartic vertex (3.64) cancels and we no longer need to keep track of the\nover- and under-crossing of the Wilson lines in the semiclassical limit.\nIt is likely that these diagrammatic rules may be derived directly from the tensor\nmodel for AdS 3gravity recently introduced in [11], with tensor model diagrams\ncorresponding to specific wormhole topologies. However we will not pursue this any\nfurther here.\n3.4 Handle wormholes\nIn [10], the on-shell action of a class of wormholes contributing to certain single-\nboundary observables was constructed. These wormholes admitted an elegant inter-\npretation in terms of the Coleman-Giddings-Strominger mechanism [42–44], whereby\nthe existence of Euclidean wormholes induce random bulk couplings in the low-energy\neffective theory. Here we demonstrate that the gravity partition function on these\n– 34 –single-boundary “handle wormholes” is straightforward to compute using Virasoro\nTQFT.\nFor concreteness, consider the sphere four-point function of pairwise identical op-\nerators ⟨O1O2O2O1⟩. Suppose there is a third species of defect, dual to the operator\nO3. Naively, the trivalent coupling λ123in the bulk low-energy effective field theory\nvanishes since in the Gaussian ensemble the averaged structure constant vanishes\nc123= 0. However, there is a two-boundary wormhole that computes the variance\nc2\n123̸= 0, so the conclusion that the defects are entirely non-interacting in the bulk\ncannot quite be correct. In particular, we expect a topology that corresponds to the\nexchange of O3in the O1× O 2OPE and hence contributes to the bulk-dual of the\nfour-point function ⟨O1O2O2O1⟩.\nConsider the following topology discussed in [10]\nM=1\n21\n23. (3.71)\nIt is constructed by starting with a compression body whose outer boundary is a\nfour-punctured sphere and two three-punctured sphere inner boundaries, and then\nidentifying the two inner boundaries as shown in (3.71). The Wilson lines corre-\nsponding to O1andO2traverse the resulting wormhole and that corresponding to\nO3forms a closed loop in it. The TQFT partition function on the compression body\n(without the identification among the inner boundaries) is simply proportional to\nthe corresponding sphere four-point conformal block\nZVir\u00121\n21\n23\u0013\n=C0(P1, P2, P3)21\n21\n23. (3.72)\nTo implement the identification of the inner boundaries, we first view the partition\nfunction on the compression body as a state in the tensor product Hilbert space\nassociated with the inner and outer boundaries H0,4⊗ H 0,3⊗ H 0,3. Taking the inner\nproduct between the states in the three-punctured sphere Hilbert spaces implements\nthe identification between the inner boundaries and leaves us with the following state\ninH0,4:\nZVir(M) =C0(P1, P2, P3)1\n21\n23. (3.73)\nSquaring the TQFT partition function leads to the expected contribution to the\ngravity path integral corresponding to the exchange of O3in the O1× O 2OPE,\n– 35 –with squared OPE coefficient given by the corresponding variance in the Gaussian\nensemble\nZgrav(M) =|C0(P1, P2, P3)|2\f\f\f\f1\n21\n23\f\f\f\f2\n. (3.74)\nThis is precisely the result that was computed semiclassically in [10].\n3.5 Twisted I-bundles\nLet us discuss another interesting example which has appeared before in the litera-\nture on AdS 3gravity known as a twisted I-bundle. It was studied in [45] as a simple\nexample of a non-handlebody saddle-point contribution to the 3d gravity path inte-\ngral with a single higher-genus boundary. The name stems from the fact that these\nthree-manifolds are constructed as a non-trivial I-bundle over a Riemann surface,\nwhere Iis an interval. Consider a hyperbolic Riemann surface Σ together with an\norientation-reversing (i.e. anti-holomorphic) fixed-point free involution Φ : Σ →Σ.\nWe can then consider a quotient of the Euclidean wormhole Σ ×[0,1] as follows:\nMΦ= (Σ×[0,1])/{(z, x)∼(Φ(z),1−x)}. (3.75)\nThis identification is again orientation-preserving and thus we get an orientable hy-\nperbolic manifold with a single boundary Σ, where the hyperbolic structure is inher-\nited from the Euclidean wormhole.\nΦ induces an involution on the boundary Teichm¨ uller space which we also call\nΦ and hence the boundary moduli are constrained to lie on the fixed point set TΦ.\nBy the uniformization theorems of three-dimensional hyperbolic manifolds that we\nreviewed in the Appendix of [1], we are however guaranteed that the manifold with\nthe same topology can also be defined away from the real locus in Teichm¨ uller space.\nThe construction then proceeds by taking a quotient of a quasi-Fuchsian wormhole,\nwhere the moduli of the left boundary are the image under Φ of the moduli of the\nright boundary.\nFrom the TQFT point of view, it is very simple to determine the Virasoro TQFT\npartition function on these manifolds. Indeed, we could squash the manifold to the\nsurface Σ × {1\n2}and the quotient by Φ simply produces eΣ× {1\n2}. Here,\neΣ = Σ /{z∼Φ(z)} (3.76)\nis the non-orientable surface obtained from quotienting Σ. Given that the Virasoro\nTQFT partition function on the Euclidean wormhole is simply the Liouville partition\nfunction, we see that Φ acts precisely by an orientifold projection. In other words, the\npartition function on the twisted I-bundle is simply the Liouville partition function\non the non-orientable surface eΣ.\nTo see that this makes sense, recall that the conformal block expansion on a non-\norientable surface involves a single conformal block on the doubled surface Σ which\n– 36 –hence defines a state in the boundary Hilbert space of the twisted I-bundle. Let us\nmake this more concrete by recalling the precise construction of Liouville theory on\na non-orientable surface. We can construct a non-orientable surface by including a\nnumber of cross-caps on an orientable surface.19E.g. on a torus with one puncture\nand a cross-cap, we have\n= (3.77)\nThe orientifold acting reflects the right hand side of the picture to the left side and\nsimultaneously rotates by 180 degrees around the dashed horizontal line. This map\nhas no fixed point and the quotient indeed leads to the crosscap state. On the level of\nthe conformal blocks, this means that the conformal block of the Liouville partition\nfunction on this surface takes the form\nZL \nP0!\n=Z∞\n0dP1dP2dP3ρ0(P1)ρ0(P2)Γ(P3)C0(P0, P1, P2)\n×C0(P1, P2, P3)P0 P0P1\nP2P2\nP1P3,(3.78)\nwhere the picture represents the ordinary conformal block. The only new ingredient\nis the normalization of the crosscap state given by Γ( P3). It is fully determined by\nrequiring consistency with the bootstrap. It is in general given by [46]\nΓ(P) =P1,Pp\nS1,P×p\nρ0(P) =P1,P. (3.79)\nHere, the first factor is the general result when the two-point function of the theory is\ncanonically normalized. We then multiply byp\nρ0(P) to account for our normaliza-\ntion of the two-point function. The P-matrix describes the modular transformation\nof the M¨ obius strip characters:\nP=T1\n2ST2ST1\n2. (3.80)\nIt is simple to work this out explicitly:\nPP1,P2= 8Z\ndPeπi(P2\n1+P2\n2+4P2−1\n4)cos(4 πP1P) cos(4 πP2P) = 2 cos(2 πP1P2).(3.81)\n19Since two crosscaps are equivalent to a handle in the presence of another crosscap, one can\nrestrict to one or two crosscaps.\n– 37 –Thus we have\nΓ(P) =P1,P=PP1=i(b2+1)\n2b,P+PP1=i(b2−1)\n2b,P= 4 cosh( πbP) cosh( πb−1P).(3.82)\nThe + sign comes from a careful treatment of the factor T1\n2in the definition of the\nP-matrix; more physically, it comes because the orientifold projection acts by a factor\n(−1)Non a level Ndescendant. This is the same result as obtained in [47, 48] after\ntranslating to our conventions. This fully specifies the Liouville partition function on\nany non-orientable surface and hence directly gives the value of ZViron any twisted\nI-bundle.\nFinally, the gravity partition function is given by applying eq. (1.1),\nZgrav(MΦ) =X\nγ∈Map(Σ) /Map(eΣ)|ZL(eΣγ)|2, (3.83)\nwhere we used that the bulk mapping class group is the mapping class group of the\nnon-orientable surface ˜Σ under which the Liouville partition function is invariant by\ncrossing symmetry.\n4 The figure eight knot complement\nIn this section, we look at one particular hyperbolic 3-manifold in detail and illustrate\nsome features of the theory at this example. The manifold in question is the figure\neight knot complement, i.e. S3with a Wilson line inside forming a figure eight knot.\nThis manifold is known to admit a hyperbolic metric. The figure eight knot is the\nhyperbolic knot with the smallest possible volume and the only knot with the crossing\nnumber 4, as demonstrated in Figure 9. Thus it is usually denoted as 4 1. There are\nFigure 9 . A visualization of the figure eight knot.\ntwo approaches to calculate the Virasoro TQFT partition function of the figure eight\nknot complement. One way to compute the partition function is via the Heegaard\nsplitting procedure. The other way is to consider the surface bundle construction\n– 38 –of the figure eight knot, and to use the mapping torus technique introduced in [1].\nThese two approaches will lead to different integral expressions as the final results.\nWe check that these two expressions agree and both have the same semiclassical\nexpansions as expected.\n4.1 Direct computation\nLet us first compute the partition function by successively undoing the over- and\nunder-crossings in a particular projection of the knot.\nWe start by computing the partition function via surgery. We embed the above\nknot configuration into a three-sphere to create the figure eight knot complement.\nIn the TQFT setup, we consider the knot as a tangled Wilson loop with associated\nconformal weight ∆ 0=Q2\n4, i.e. the cusp, although we will keep the label of the\nWilson loop generic for most of the discussion. If we slice the above figure 9 into\nhalves along the equatorial S2, we obtain two manifolds M1, M2with boundaries\nas four-punctured sphere. The path integral over each half prepares a state in the\nHilbert space HΣ0,4, and the partition function is the inner product between these two\nsphere 4-point conformal blocks. Here the Wilson lines inside each component have\nnontrivial braidings. Before evaluating the inner product, we want to untangle the\nWilson lines. For this purpose, we need to apply the crossing and braiding operations\non the boundary surface Σ 0,4.\nTo make the crossing and braiding explicit, we firstly specify the intermediate\nchannels in the figure eight knot. In the diagram, we have identity operators prop-\n1\n1\nFigure 10 . Heegaard splitting of the figure eight knot complement.\nagate in the intermediate channels corresponding to the contractible cycles in the\nbulk. We can use the fusion kernel Fto transform the diagram 10 into the other\nchannel\n– 39 –ZVir 1\n1P0!\n=Z\ndPsdPtF1,Ps\u0014P0P0\nP0P0\u0015\nF1,Pt\u0014P0P0\nP0P0\u0015\n×ZVir Pt\nPsP0!\n,(4.1)\nwhere P0labels the conformal weight of the Wilson loop, i.e. P0= 0 for ∆ = Q2/4.\nAfter transforming the figure eight knot diagram into the other channel, we can\nuntangle the knot at each trivalent node via the braiding move Bas follows\nZVir(41) =Z\ndPsdPtF1,Ps\u0014P0P0\nP0P0\u0015\nF1,Pt\u0014P0P0\nP0P0\u0015\n(BP0,P0\nPs)2\n×(BP0,P0\nPt)−2ZVir Pt\nP0 P0Ps!\n.(4.2)\nThe fusion kernel Fcorresponding to the exchange of the identity operator can be\nwritten in terms of ρ0andC0as follows\nF1,P\u0014P0P0\nP0P0\u0015\n=ρ0(P)C0(P, P 0, P0). (4.3)\nMeanwhile, we recognize the remaining contraction as the four-boundary wormhole\ndiscussed in Section 3.2 for which we can use the result (3.20), normalized by inverse\nstructure constants to account for the normalization of the junctures.\nIn the end, we obtain an integral expression of the figure eight knot partition\nfunction\nZVir(41) =Z\ndPsdPtρ0(Ps)ρ0(Pt) (BP0,P0\nPs)2(BP0,P0\nPt)−2\u001aP0P0Ps\nP0P0Pt\u001b\n. (4.4)\nThere are two momentum integrals in the above formula (4.4), and we can reduce\nthe number of integrals by one by using the relation (2.33) between the fusion kernel\nFand the modular S-matrix S. We hence get\nZVir(41) =Z∞\n0dPρ0(P)\nρ0(P0)eπiQ2\n4−3πiP2SP0,P0[P] (4.5)\n=Z∞\n0dPρ0(P) e3πiQ2\n8−5πiP2\n2\nSb(Q\n2+iP)Z∞\n−∞dxe−4πixP 0Sb(Q\n4+iP\n2±iP0±ix),\n(4.6)\n– 40 –where we inserted the explicit expression for the modular crossing kernel in the second\nline [18].\nOf course, this expression is dependent on the framing that we implicitly chose\nin this computation. For the figure eight knot complement, a nice way to fix the\nframing anomaly is by requiring that the partition function should be real. Indeed,\ncomplex conjugation corresponds to orientation reversal, but since the figure eight\nknot is invariant under orientation reversal (this property is called amphichirality),\nwe can choose the partition function to be real.\nOne can easily check, for example numerically, that this is the case if we multiply\nthe above expression with e4πiP2\n0= e4πi(∆0−c\n24), which is part of the ambiguity from\nframing. We hence have\nZVir(41) =Z∞\n0dPρ0(P) e3πiQ2\n8+4πiP2\n0−5πiP2\n2\nSb(Q\n2+iP)Z∞\n−∞dxe−4πixP 0Sb(Q\n4+iP\n2±iP0±ix),\n(4.7)\nwhich is the formula we will use from now on.\nChoice of contour. There is one additional subtlety with this formula. As it\nstands, the integral over Pis actually not convergent. Indeed, using the asymptotics\nof the double sine function, see e.g. [26, eq. (B.53)] and using that the integral over\nxis dominated for small x, we see thatZ∞\n−∞dxe−4πixP 0Sb(Q\n4+iP\n2±iP0±ix)∼1√\n2eπiP2\n2−πPQ\n2+πiQ2\n24−πi\n12. (4.8)\nCombining this with the asymptotics of the rest of the integrand, we see that the\nintegrand behaves for large Re( P) as\nintegrand( P)∼e3πQP\n2−5πiP2\n2× O(order 1 in P). (4.9)\nThus the integral in (4.7) doesn’t converge for Pon the real axis. However, we see\nthat we could have improved convergence by taking Pto run along a contour starting\natP= 0 and asymptoting for large Pthe line R−ia, where the shift ahas to be at\nleast a >3Q\n10to ensure convergence. Shifting the contour in this way doesn’t cross\nany poles and is hence a generally harmless operation. Thus it is understood that\nthe integral over Pin (4.7) actually follows this modified contour.\n4.2 Comparison to Teichm¨ uller TQFT\nThe figure eight knot partition function can also be obtained in Teichm¨ uller TQFT\ndeveloped in [49–52]. Translating to our conventions, the expression for the Te-\nichm¨ uller TQFT partition function is20\nZTeich(41) =√\n2Z\nR−i0+dx Sb(ix±2iP0), (4.10)\n20Teichm¨ uller TQFT depends on a parameter ℏ, which, following the conventions of [50], we\nidentify as ℏ=−iπb2. This expression does not literally match the one given in [49–52]. We are\n– 41 –Here the integral runs slightly below the real axis to avoid the poles at x=±2P0.\nThis formula can be obtained by realizing the figure eight knot complement as a\ngluing of two tetrahedra. Each tetrahedron gives rise to one double sine function\nand the gluing to the integral (modulo some constraints).\nAs we already conjectured in our previous paper [1], we expect that Virasoro\nTQFT is equivalent to Teichm¨ uller TQFT and thus the two expressions should\nmatch,\nZVir(41)!=ZTeich(41). (4.11)\nThis equality turns out to be quite hard to prove analytically. However, we checked\nnumerically for various values of bandP0that the two expressions agree.\nFigure 11 . Numerical comparison of the Virasoro TQFT and the Teichm¨ uller TQFT\npartition function of the figure eight knot complement. The plotted data points are for the\nTeichm¨ uller expression (4.10), but are indistinguishable from the Virasoro data points.\nThe numerical evaluation is in principle straightforward. We restricted our at-\ntention to rational values of b2, since in this case, there is a simple way to express the\ndouble sine function through the Barnes G-function for which we can use efficient\nimplementations, for example in Mathematica ,\nSb(z) = (2 π)√mnz−m+n\n2m−1Y\nk=0n−1Y\nℓ=0G\u0000k+1\nm+ℓ+1\nn−z√mn\u0001\nG\u0000k\nm+ℓ\nn+z√mn\u0001. (4.12)\nIt is then simple to compute the required integrals in (4.7) over a converging contour\nand compare with the simpler expression (4.10). We computed the partition functions\nunsure whether this is a typo in the previous literature. In any case, the semiclassical expansion\nthat we discuss below does match previous expressions, which gives us a lot of confidence in the\ncorrectness of (4.10). We thank Boris Post and Davide Saccardo for discussions about this.\n– 42 –forb= 1, b=√\n2 and b=√\n3 for P0= 0,0.02, . . . , 0.2. To the precision we have\ncomputed, all values agree to seven decimal places, thus showing the equality (4.11)\nbeyond reasonable doubt. The data points are plotted in Figure 11.\nFrom this discussion, it may seem that the Teichm¨ uller TQFT always produces\nsimpler expressions than Virasoro TQFT, but this is not the case. The expressions\nin Teichm¨ uller TQFT become more complicated when the 3-manifold in question\nrequires more tetrahedra to form a triangulation, while this is not necessarily so in\nVirasoro TQFT. It is in general quite hard to recognize when two integral repre-\nsentations of the partition function agree since there are an enormous number of\nnon-trivial integral identities relating them.\n4.3 Computation via the Seifert surface\nLet us explain a completely different way to compute the partition function that will\nlead to an inequivalent integral for the partition function.\nThe figure eight knot admits a genus 1 Seifert surface. This means that we can\nrealize the knot as the boundary of a one-holed torus embedded in S3, so that the\nboundary of the one-holed torus coincides with knot. This is depicted in Figure 12.\nHowever, even more is true. One can slightly deform the Seifert surface and obtain\nFigure 12 . The Seifert surface of the figure eight knot. One can easily verify that the\nboundary of the Seifert surface traces out a figure eight knot.\na foliation of the knot complement in terms of one-holed tori. The figure eight knot\ncomplement is in fact a surface bundle over a circle, i.e. it is of the form\n[0,1]×Σ1,1/∼, (4.13)\nwhere Σ 1,1is the one-holed torus and we identify\n(0, z)∼(1, ϕ(z)) (4.14)\nwith ϕ=ST3being the corresponding mapping class group element in SL(2 ,Z)\ngenerated by SandT.\n– 43 –This might let one suspect that we can compute the partition function of the\nfigure eight knot complement as\nZVir(41)?= trH1,1(S[P0]T3), (4.15)\nbut this is not quite correct yet. Indeed, taking the trace over the Hilbert space\nH1,1of conformal blocks on the once-punctured torus would lead to the partition\nfunction of the three-dimensional manifold where the Wilson line runs along the\nthermal circle S1. This is not what we want, since the Wilson line bounds the Seifert\nsurface, which forms the meridian of the boundary torus of the manifold. This means\nthat the correct expression is obtained by applying the S-modular transformation in\nthe external parameter P0. So we conclude that we should have\nZVir(41) =Z∞\n0dP′\n0SP0,P′\n0[1] trH1,1(S[P′\n0]T3). (4.16)\nWe can easily plug in the explicit expressions for the modular crossing kernel and get\nan alternative expression for the partition function of the figure eight knot comple-\nment. This expression is even more unwieldy then the previous ones, since it involves\nthree integrals, one from the definition of S, one from the trace, and one from the\nintegral over P′\n0. This pushes our numerical capabilities a bit too far. Instead, we\nwill check below that the first two terms in the semiclassical expansion agree with\nthe semiclassical expansion of the previous expression.21\nWe note that this expression makes reality of the partition function manifest,\nwhile it was obscured in the expression (4.7) that we discussed above. Indeed, one of\nthe Moore-Seiberg relations states that TSTST =Sas operators (see [1, eq. (A.5b)])\nand thus\ntr(ST3)∗= tr(T−1S−1T−2) = tr(STS−1T−1) = tr(ST2STS−1) = tr(ST3),(4.17)\nand so after Fourier transformation we still get a real function.\n4.4 Semiclassical expansion\nWe now write down the semiclassical expansion of the Virasoro TQFT partition\nfunction in the form (4.10) and check the volume conjecture explicitly. This was\nalready done before in the context of Teichm¨ uller TQFT [49, 50] and hence we shall\nbe rather brief.\nThe key identity is the semiclassical expansion of the double sine function,\nlogSb(x0+x) =∞X\nn=0(2πib2)n−1\n2n!\n×\u0010\nLi2−n\u0000\ne−2πibx 0\u0001\n−(−1)nLi2−n\u0000\ne2πibx 0\u0001\u0011\nBn\u0000\n1−x\nb\u0001\n.(4.18)\n21We also checked that the corresponding expressions for the figure eight knot partition function\nin SU(2) kChern-Simons theory agree where all the integrals are just finite sums.\n– 44 –In this identity we think of x0as being of order O(1\nb), while xis of order O(1). This\nidentity is standard for the quantum dilogarithm to which the double sine function\nis closely related, see e.g. [52, Proposition 6]. For completeness, we have included a\nshort derivation in appendix A.\nWe now apply this expansion as follows. In the semiclassical limit, the argument\nof the double sine function in eq. (4.10) becomes large and we write P0=η0\nb. We\ncan then evaluate the integral via saddle point approximation. We write xto leading\norder asx0\nb. Then the saddle-point equation is\n0 =−1\n2π∂x0X\n±\u0000\nLi2(e2π(x0±2η0))−Li2(e−2π(x0±2η0))\u0001\n(4.19)\n= logY\n±,±\u0000\n1−e2π(±x0±2η0)\u0001\n. (4.20)\nThe solution to this saddlepoint equation takes the form\nx0=1\n2πlog\u0010\ncosh(4 πη0)±11\n2±2q\u0000\ncosh(4 πη0)±11\n2\u00012−1\u0011\n+ni , n ∈Z.(4.21)\nThe steepest descent contour runs through the saddle point at\nx0=1\n2πlog\u0010\ncosh(4 πη0)−1\n2−q\u0000\ncosh(4 πη0)−1\n2\u00012−1\u0011\n(4.22)\nand hence only that one is relevant for our analysis. For this to be valid, we should\nassume that\n|η0|<−1\n2πlog\u0010√\n5−1\n2\u0011\n, (4.23)\nsince otherwise x0becomes real and the saddle-point evaluation is different. We\nobtain the semiclassical expansion\nZVir(41) =√\n2 e−1\n2πb2vol(4 1)\n∆1\n4exp\u0010∞X\nn=1Snb2n\u0011\n, (4.24)\nwhere\n∆ =−X2+ 2X+ 1 + 2 X−1−X−2, X = e4πη0, (4.25)\nand\nvol(4 1, η0) =−i\n2\u0010\nLi2\u0000X−1+X−1+i√\n∆\n2X\u0001\n+ Li 2\u0000X−1+X−1+i√\n∆\n2X\u0001\n−Li2\u0000X−1+X−1−i√\n∆\n2X\u0001\n−Li2\u0000X−1+X−1−i√\n∆\n2X\u0001\u0011\n.(4.26)\n– 45 –The first few orders for the higher loop corrections are given by\nS1=−π\n12∆3\n2(X3−X2−2X2+ 15−2X−1−X−2+X−3), (4.27a)\nS2=2π2\n∆3(X3−X2−2X+ 5−2X−1−X−2+X−3), (4.27b)\nS3=π3\n90∆9\n2(X8−4X7−128X6+ 36X5+ 1074 X4−5630X3+ 5782 X2\n+ 7484 X1−18311 + 7484 X−1+ 5782 X−2−5630X−3+ 1074 X−4+ 36X−5\n−128X−6−4X−7+X−8). (4.27c)\nNot surprisingly, this reproduces the semiclassical expansion given in [49–51]. Also\nnoticed there, the one-loop determinant equals the Reidemeister torsion of the figure\neight knot, which can also be derived from computing the functional analytic one-\nloop determinants appearing in 3d gravity. Thus this shows the validity of the volume\nconjecture (2.1) for the figure eight knot.\nExpression from Seifert surface. We now reproduce the semiclassical expansion\nfrom the expression that we got from the computation via the Seifert surface as\ndescribed in section 4.3. This gives strong evidence that the expression (4.16) is in\nfact equal to the simpler expression given by eq. (4.10).\nBy using the explicit formula of the Virasoro crossing kernel shown in [1], we\nrewrite the integral formula (4.16) in terms of double-sine functions\nZVir(41) = 2√\n2Z∞\n0dP′\n0cos 4πP0P′\n0Z∞\n0dP ρ0(P)\n×Z∞\n−∞dξeπi∆′\n0\n2−6πi(P2−1\n24)−4πiξP\nSb(Q\n2+iP′\n0)Sb(Q\n4+iP′\n0\n2±iP±iξ).(4.28)\nFor simplicity, in the following computation, we consider P0= 0 which sets the\nconformal weight of the knot to be ∆ =Q2\n4. As we will see later, the saddle-point\nequation in the semiclassical approximation will be simplified in this case. In general,\nwe can also compute the partition function for the knot with a generic conformal\nweight, while the complexity of solving the saddle-point equations increases. Once\nwe consider the semiclassical limit of this expression, we similarly rescale P′\n0=η0\nb,\nP=x\nbandξ=η\nb. Then we apply the expansion formula of the double sine function\nto write the integrand into a expansion in 1 /b2.\nZVir(41) =Zdη0dxdη\nb3eP∞\nn=0S(n)b2(n−1). (4.29)\nInb→0 limit, we can approximate this integral by saddle-point. The leading order\n– 46 –contribution is proportional to 1 /b2with the coefficient\nS(0)=πi\n8+ 2πx+πiη2\n0\n2−6πix2−4πiηx−i\n4π\u0000\nLi2(e−2πη0+iπ)−Li2(e2πη0−iπ)\u0001\n+i\n4πX\n±,±\u0000\nLi2(e−2π(η0\n2±x±η)+πi\n2)−Li2(e2π(η0\n2±x±η)−πi\n2)\u0001\n.(4.30)\nThis leads to three saddle-point equations.\nSinceS[P′\n0] only depends on the conformal weight ∆′\n0=P′2\n0+Q2\n4, the function\ntr(S[P′\n0]T3) is even in P′\n0. This observation implies that η0= 0 will be a saddle-point\nand we can reduce one saddle-point equation with respect to η0. When η0is set to\nbe 0, we have the saddle-point equations of ηandxrespectively as follow\n0 =−4πix+i\n2log\"\u0012cosh (2 πx) +isinh (2 πη)\ncosh (2 πx)−isinh (2 πη)\u00132#\n, (4.31a)\n0 =−12πix+ 2π−4πiη+i\n2log\"\u0012cosh (2 πη) +isinh (2 πx)\ncosh (2 πη)−isinh (2 πx)\u00132#\n. (4.31b)\nThe first equation can be solved by taking 2 πiη= arcsin (sinh (2 πx)). By plugging\nthis relation between ηandxinto the second equation, we solve for xand obtain the\nfollowing saddle-point of S(0)\nx=1\n4πlog \n−1−3√\n3i−p\n−42 + 6√\n3i\n4!\n. (4.32)\nWe also explicitly check that∂S(0)\n∂η0is vanishing when η0= 0 and x, ηtake the given\nsaddle-point values. Therefore, η0= 0 is indeed the saddle-point along η0direction\nas we justified before. By evaluating the S(0)at the saddle point, we recover the\nhyperbolic volume of the figure eight knot as expected\nS(0)=−vol(4 1)\n2π. (4.33)\nIn order to compare the semiclassical result with the refined volume conjecture\n(2.1), we should also study the higher-loop corrections. Using the expansion of\ndouble-sine functions in (4.18), we can compute the partition function to all orders\nperturbatively in b2. Here we focus on the order one factor in the expansion\nZ(1)=1\n2s\n−(2π)3\ndet(Hess S(0))eS(1), (4.34)\nsince this factor is closely related to the one-loop determinant in the 3d gravity\ncalculation. The prefactor comes from the Gaussian integral around the saddle point.\nThe additional factor of1\n2appears because the integral is restricted to P′\n0>0, while\n– 47 –the minus sign inside the square root originates from the fact that the Gaussian\nintegral has the form e1\nb2S0. The three factors of bget cancelled against the three b’s\nfrom the Jacobian in (4.29). We collect all order-one terms in the expansion\neS(1)=4isinh(2 πx)\nsinh( π(i\n4−η0\n2±x±η))1\n4, (4.35)\nwhich upon inserting the saddlepoint value simplifies to\neS(1)= 4√\n2isinh(2 πx). (4.36)\nWe then take the Gaussian integral contribution to (4.34) into account, we obtain\nthe order-one correction to the partition function\nZ(1)=2√\n2isinh(2 πx)q\n7i+ 5 cosh(2 πx)p\n6−2 cosh(4 πx)−5icosh(4 πx)=√\n2\n31\n4. (4.37)\nThis result matches with the order-one term in the expression (4.24) with the Rei-\ndemeister torsion√\n∆ =√\n3 atP0= 0.\nNote that in the refined volume conjecture (2.1), we write the semiclassical ex-\npansion of the partition function in terms of the central charge c, while we have\ntheb2expansion in this part of calculation. The central charge cis defined as\nc= 1 + 6( b+1\nb)2= 13 +6\nb2+ 6b2. Therefore, strictly speaking, the one-loop de-\nterminant from the gravity calculation is not equal to Z(1). Instead, we need to\nrenormalize Z(1)to obtain the one-loop determinant\nZone-loop =Z(1)e13\n12πvol(4 1), (4.38)\nwhich should be compared with the calculations performed in [3].22\n4.5 Dehn surgery\nAs final application to the figure eight knot computation, we discuss an example of\nDehn surgery. Consider the figure eight knot and excise a small tubular neighborhood\naround the knot. We can then glue back a torus, but twisted by an SL(2 ,Z) element.\nSuch an element is specified by a two coprime integers ( p, q) specifying the slope of\nthe meridian (the contractible curve).\nThe Virasoro TQFT partition function on a solid torus gives simply the vacuum\ncharacter χ1in the appropriate channel, while it gives a generic Virasoro character\nχPwith the inclusion of a Wilson line of momentum P. We can write23\nZVir(41, P0) =⟨ZVir(4◦\n1)|χP0⟩, (4.39)\n22The computation in [3] is not directly applicable to the figure eight knot case because of the\npresence of the cusp, in which case the relevant Kleinian groups has parabolic elements.\n23As explained in [1], the normalization of the inner product on the torus is somewhat ambiguous,\nbut this ambiguity will cancel out of the calculation.\n– 48 –where 4◦\n1is the figure eight knot complement with a tubular neighborhood around the\nknot removed and we emphasize the P0-dependence of the Virasoro TQFT partition\nfunction.\nThus the partition function of a manifold obtained by Dehn surgery from the\nfigure eight knot is given by\nZVir(41(p, q)) =⟨ZVir(4◦\n1)|U(p, q)|χvac⟩ (4.40)\n=Z∞\n0dPU(p, q)1,P⟨ZVir(4◦\n1)|χP⟩ (4.41)\n=Z∞\n0dPU(p, q)1,PZVir(41, P) (4.42)\nwhereU(p, q) is the representation of the SL(2 ,Z) modular transformation on the\nVirasoro characters. It takes the explicit form (see e.g. [53])\nU(p, q)1,P=ε(p, q)r8\nqe−2πi\nq(p∗Q2\n4−pP2)\u0012\ncosh\u00102QPπ\nq\u0011\n−e2πip∗\nqcosh\u00102ˆQPπ\nq\u0011\u0013\n.\n(4.43)\nHere ˆQ=b−b−1,ε(p, q) is a P-independent 24-th root of unity coming from the\ntransformation behaviour of the Dedekind η-function and p∗is the modular inverse of\np,pp∗≡1 mod q. This leaves an ambiguity in the expression which can be absorbed\nin the framing ambiguity. For the figure eight knot, we should also notice that\nbecause of amphichirality, the Dehn surgeries ( p, q) and ( −p, q) are equivalent and\nwe can focus on p, q≥0.\nIt is in particular simple to evaluate the hyperbolic volume of this class of man-\nifolds via saddle point approximation. Set P=η\nbas before. Then the action is\nS= vol(4 1, η) +π2i\nq(p∗−4pη2)±4π2η\nq. (4.44)\nSince we focus on the volume, we can omit the purely imaginary part involving p∗.\nThe sign choice of the last term is also immaterial, since we can send η→ −η. We\nhence find that\nvol(4 1(p, q)) = Re\u0012\nvol(4 1, η) +4π2η(1−piη)\nq\u0013\f\f\f\f\nη=η∗, (4.45)\nwhere we plug in the saddle-point value η∗and the volume is given by (4.26). The\nsaddle-point equation is transcendental and doesn’t admit a closed form solution.\nHowever, it is straightforward to compute the volumes of various examples numeri-\ncally, see Table 2. We compared them to the volumes as computed by the program\nSnapPy . It is also simple to compute the volumes in a large pandqexpansion, since\nfor large porq, the saddle point η∗→0 and the volume converges to the volume of\n– 49 –pq 1 2 3 4 5 6 7 8 9\n1 0 1.3985 1.7320 1.8581 1.9186 1.9521 1.9725 1.9858 1.9950\n2 0 1.7371 1.9195 1.9727 1.9951\n3 0 1.4407 1.8634 1.9210 1.9732 1.9862\n4 0 1.7571 1.9231 1.9738 1.9955\n50.9813 1.5295 1.7714 1.8735 1.9557 1.9745 1.9870 1.9958\n61.2845 1.9287 1.9754\n71.4638 1.6496 1.8058 1.8871 1.9321 1.9591 1.9882 1.9965\n81.5832 1.8243 1.9358 1.9776 1.9970\n91.6678 1.7521 1.9027 1.9397 1.9789 1.9897\nTable 2 . The volumes of manifolds obtained from Dehn surgery from the figure eight knot.\nZero entries indicate that the corresponding manifolds do not admit a hyperbolic metric.\nThe other two exceptional cases that do not admit a hyperbolic metric are ( p, q) = (1 ,0)\nand (0 ,1), see also [54, Theorem 4.7].\nthe figure eight knot. We find to the first few orders\nvol(4 1(p, q)) = vol(4 1)−2√\n3π2\np2+ 12q2+4π4(p4−72p2q2+ 144 q4)√\n3(p2+ 12q2)4\n−8π6(23p8−8904p6q2+ 302400 p4q4−1620864 p2q6+ 767232 q8)\n45√\n3(p2+ 12q2)7+···(4.46)\nThe correction to the figure eight knot volume is always negative as required by\ngeneral theorems about Dehn surgery [54, Theorem 6.5.6.]. This expansion is a known\nresult, see [55]. This case of Dehn surgery exemplifies the existence of accumulation\npoints in the spectrum of three-manifolds. We discussed their implications for the\ngravitational path integral in our previous paper [1].\nAcknowledgements\nWe would like to thank Alex Belin, Jeevan Chandra, Tom Hartman, Daniel Jafferis,\nDiego Liˇ ska, Alex Maloney, Baur Mukhametzhanov, Boris Post, Sahand Seifnashri,\nSteve Shenker, Julian Sonner, J¨ org Teschner and Ka Ho Wong for useful discus-\nsions. While at the IAS, L.E. was supported by the grant DE-SC0009988 from the\nU.S. Department of Energy. This material is based upon work supported by the\nU.S. Department of Energy, Office of Science, Office of High Energy Physics of U.S.\nDepartment of Energy under grant Contract Number DE-SC0012567 (High Energy\nTheory research), DOE Early Career Award DE-SC0021886 and the Packard Foun-\ndation Award in Quantum Black Holes and Quantum Computation.\n– 50 –A Semiclassical expansion of the double sine function\nIn this appendix, we will derive the semiclassical expansion of the double sine function\n(4.18). We start from the integral representation\nlogSb(x0+x) =1\n4Z\nR+i0+dt\ntsinh\u0000\n(Q\n2−x−x0)t\u0001\nsinh(bt\n2) sinh(t\n2b)(A.1)\n=1\n4Z\nR+i0+dt\nte(b2+1\n2−bx−bx0)t−e−(b2+1\n2−bx−bx0)t\n(eb2t\n2−e−b2t\n2) sinh(t\n2)(A.2)\n=1\n4Z\n(R+i0+)∪(R+i0−)dt\nte(b2+1\n2−bx−bx0)t\n(eb2t−1) sinh(t\n2). (A.3)\nHere we rescaled tand put t→ − tin the second expression to have the integrand\nhave the same form. We can now use the definition of the Bernoulli polynomials and\nget as formal expansion\nlogSb(x0+x) =∞X\nn=0b2n−2\n4n!Bn(1−x\nb)Z\n(R+i0+)∪(R+i0−)dttn−2e(1\n2−bx0)t\nsinh(t\n2). (A.4)\nThe remaining integral can be computed for example by pulling off the contour off\nand summing over the residues at t= 2πim. This gives\nZ\nR+i0+dttn−2e(1\n2−bx0)t\nsinh(t\n2)=∞X\nm=12(2πim)n−2e−2πimbx 0= 2(2 πi)n−2Li2−n(e−2πibx 0).\n(A.5)\nWe similarly evaluate the contribution from the other contour R+i0−which then\nrecovers (4.18).\nReferences\n[1] S. Collier, L. Eberhardt and M. Zhang, Solving 3d Gravity with Virasoro TQFT ,\n2304.13650 .\n[2] X. Yin, Partition Functions of Three-Dimensional Pure Gravity ,Commun. Num.\nTheor. Phys. 2(2008) 285 [ 0710.2129 ].\n[3] S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity ,\nJHEP 08(2008) 007 [ 0804.1773 ].\n[4] J. Cotler and K. Jensen, A theory of reparameterizations for AdS 3gravity ,JHEP 02\n(2019) 079 [ 1808.03263 ].\n[5] H. 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This, combined with the existing results in [ 29,33], allows us to resolve partially\nProblem 1.8, Problem A.1 and Conjecture A.4 regarding the op timal Lusin type constant and\nthe characterization of martingale type in a recent remarka ble work due to Xu [ 48] .\nSeveral difficulties originate from the arbitrariness of X, which excludes the use of vector-\nvalued Calder´ on-Zygmund theory. To surmount the obstacle s, we introduce the novel vector-\nvalued Hardy and BMO spaces associated with sectorial opera tors; in addition to Mei’s duality\ntechniques and Wilson’s intrinsic square functions develo ped in this setting, the key new\ninput is the vector-valued tent space theory and its unexpec ted amalgamation with these ‘old’\ntechniques.\n1.Introduction\nMotivated by Banach space geometry [ 34,35] and Stein’s semigroup theory [ 37], the investiga-\ntion of the vector-valued Littlewood-Paley-Stein theory has star ted with Xu’s Poisson semigroup\non the unit circle[ 44], andthen wascontinuedin [ 29,45,46] forsymmetric Markoviansemigroups.\nAfterwards, Betancor et aldeveloped this theory in some special cases which are not Markovian\n(cf. [3,5–7]), such as Schr¨ odinger, Hermite, Laguerre semigroups etc., see also [ 1,2,4,22,24,33,39]\nforrelatedresults. In arecentremarkablepaper[ 48], Xu investigatedforthe firsttime the vector-\nvalued Littlewood-Paley-Stein inequalities for semigroups of regular contractions te´tLutą0on\nLppΩqfor afixed1ăpă 8. That is, for a Banach space Xof martingale cotype q(2ďqă 8),\nhe showed the Lusin cotype of Xrelative to te´t?\nLutą0, in other words, there exists a constant\nCą0 such that\n}Gq,?\nLpfq}pďC}f}LppXq,@fPLppΩq bX, (1.1)\n2020 Mathematics Subject Classification: Primary: 42B35, 47D03. Secondary: 46F10, 46E30.\nKey words: Littlewood-Paley-Stein inequalities; Hardy and BMO space s; vector-valued tent spaces; holomor-\nphic functional calculus; martingale type and cotype; intr insic square function.\n1BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 2\nwhere\nGq,?\nLpfqpxq “ˆˆ8\n0›››t?\nLe´t?\nLpfqpxq›››q\nXdt\nt˙1\nq\n.\nMoreimportantly,bydeeplyexploringholomorphicfunctionalcalculu s,Fendler’sdilation,Calder´ on-\nZygmund theory and Wilson’s intrinsic square functions, he was able t o obtain the sharp bounds\ndepending on the martingale cotype constant, and the latter in tur n enables him to resolve an\nopen problem posed by Naor and Young [ 32]. More precisely, let L?\nL\nc,q,ppXqbe the least constant\nCin (1.1)—the Lusin cotype constant of X, andMc,qpXqthe martingale cotype qconstant of\nX, he obtained\nL?\nL\nc,q,ppXq/lessorsimilarmax!\np1\nq,p1)\nMc,qpXq (1.2)\nwith the order max!\np1\nq,p1)\nbeing sharp. We refer the reader to Section 6for the definition of\nMc,qpXqand the martingale type constant Mt,qpXq.\nBy duality, the converse inequality of ( 1.1) also holds under the condition that Xis of mar-\ntingale type q(1ăqď2)\n}f´Fpfq}LppXqďC}Gq,?\nLpfq}p,@fPLppΩq bX,\nwhereFis the obvious vector-valued extension of the projection from LppΩqonto the fixed point\nspace of te´tLutą0, and the resulting typebounds satisfy\nL?\nL\nt,q,ppXq/lessorsimilarmax!\np,p11\nq1)\nMt,qpXq. (1.3)\nNevertheless the order max!\np,p11\nq1)\nis now very likely to be suboptimal suggested by the special\ncaseL“∆—the Laplacian on Rd,q“2 andX“C, where\nMt,2pCq “1,and?p/lessorsimilarL?\n∆\nt,2,ppCq/lessorsimilarp, (1.4)\nsee for instance [ 47, Theorem 1]. The sharpness of ( 1.4) whenpÑ1 is essentially equivalent to\nthe fact that L1pRdq-norm of the classical g-function controls that of the Lusin square function,\nwhich dominates in turn L1pRdq-norm of the function itself; this involves the deep theory of\nHardy/BMO spaces. Other than this special case, the problem of d etermining the optimal order\nofL?\nL\nt,q,ppXqwhenpÑ1in(1.3)hasbeenleftopenwidelyeveninthecase L“∆, seee.g. Remark\n1.3, Problem 1.8 and Problem 8.4 in the aforementioned paper [ 48]. For the other endpoint-side,\nthe optimal order of L?\nL\nt,q,ppXqaspÑ 8has been determined in [ 49] for all symmetric Markovian\nsemigroups. However it seems much harder to consider the corres ponding problem for a fixed\nsemigroup, and actually the special case L?\n∆\nt,2,ppCqremains open (cf. [ 47, Problem 5]).\nIn the present paper, we will determine the optimal order of LL\nt,q,ppXqaspÑ1 in (1.3) for a\nlargeclass of approximationidentities te´tLutą0onRd, and thus answer the questions mentioned\nin [48, Remark 1.3 and Problem 1.8]. Moreover, our result will assert that t he Lusin type of\nXrelative to this class of approximation identities implies the martingale t ype ofX, and thus\npartially resolves [ 48, Problem A.1 and Conjecture A.4].\nLetLbe a sectorial operator of type α(0ďαăπ{2) onL2pRdq, and thus it generates a\nholomorphic semigroup e´zLwith 0 ď |Argpzq| ăπ{2´α. Partially inspired by [ 16, Section\n6.2.2], the kind of approximation identity te´tLutą0that we will be interested in in the present\npaper is assumed to have kernel Kpt,x,y qsatisfying the following three assumptions: there exist\npositive constants 0 ăβ,γ ď1 andcsuch that for any tą0,x,y,h PRd,BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 3\n(1.5) |Kpt,x,y q| ďctβ\npt` |x´y|qd`β,\n(1.6) |Kpt,x`h,yq ´Kpt,x,y q| ` |Kpt,x,y `hq ´Kpt,x,y q| ďc|h|γtβ\npt` |x´y|qd`β`γ\nwhenever 2 |h| ďt` |x´y|, and\n(1.7)ˆ\nRdKpt,x,y qdx“ˆ\nRdKpt,x,y qdy“1.\nOne may find these concepts in Section 2. Then it is well-known(see e.g. [ 48]) that the semigroup\nte´tLutą0extends to LppRd;Xq(1ďpď 8), whereLppRd;Xqis the space of all strongly\nmeasurable functions f:RdÑXsuch that }fpxq}XPLppRdq. The resulting semigroup is still\ndenoted by te´tLutą0without confusion.\nLet 1 ăqă 8, theq-variant of Lusin area integral associated with Lis defined as follows:\nforfPCcpRdq bX,\nSq,Lpfqpxq “˜ˆ8\n0ˆ\n|y´x|ăt››tLe´tLpfqpyq››q\nXdydt\ntd`1¸1\nq\n.\nOur main result reads as below.\nTheorem 1.1. LetLbe a sectorial operator of type α(0ďαăπ{2) onL2pRdqsatisfying (1.5),\n(1.6)and(1.7). Let1ďpă 8and1ăqă 8. ForanyBanach space XandfPCcpRdq bX,\nthere hold\np´1\nq}Sq,?\n∆pfq}p/lessorsimilarγ,β}Sq,Lpfq}p/lessorsimilarγ,βp1\nq}Sq,?\n∆pfq}p, (1.8)\np´1\nq}Sq,Lpfq}p/lessorsimilarγ,β}Gq,Lpfq}p/lessorsimilarγ,βp1\nq}Sq,Lpfq}p. (1.9)\nMoreover, the orders in both (1.8)and(1.9)areoptimalaspÑ1.\nWhenX“Candq“2, the equivalence ( 1.8) in the case 1 ăpă 8without explicit orders\nfollows from the classical Littlewood-Paley theory which in turn relies on Calder´ on-Zygmund\ntheory; while the case p“1 is deduced from the holomorphic functional calculus, Calder´ on-\nZygmund theory and the theory of Hardy and BMO spaces associat ed with differential operators\n(cf. [16, Theorem 6.10]). Our estimate ( 1.8) for any Banach space X, any 1 ďpă 8and any\n1ăqă 8goes much beyond this and its proof provides a new approach to the mentioned\nscalar case with optimal orders as pÑ1. Indeed, the arbitrariness of Xpresents a surprise\nand usually one expects certain property of Banach space geomet ry to be imposed on the square\nfunction inequalities. For the technical side, the arbitrariness of Xprevents us from the use of\n(vector-valued) Calder´ on-Zygmund theory. Instead, we will ma ke use of vector-valued Wilson’s\nintrinsic square functions as a media to relate ∆ and L, and then exploit the vector-valued tent\nspace theory such as interpolation, duality as well as atomic decomp osition. Even though both\nof these two tools have been developed or applied in the literature, t hey need to be taken care\nof in the present setting. For instance, because our L’s are usually not translation invariant or\nof scaling structure, we have to introduce Wilson’s intrinsic square f unctions via nice functions\nof two variables satisfying ( 4.1), (4.2) and (4.3); to avoid the use of Calder´ on-Zygmund theory\nto deal with Wilson’s intrinsic square functions (cf. [ 41,47]), we prove the boundedness of a\nlinear operator Kon vector-valued tent spaces (see Lemma 3.5); last but not the least, since our\ninterestedXis arbitrary, one cannot establish the basic theory of vector-valu ed tent space usingBEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 4\nCalder´ on-Zygmund theory as in [ 23,26–28], and we shall adapt the classical arguments (cf. [ 11]),\nsee Section 3for details.\nAfter all the preparing work, the equivalence ( 1.8) will be an immediate consequence of Theo-\nrem4.1, where we collect all the intermediate estimates involving vector-va lued Wilson’s square\nfunctions.\nRegarding another equivalence ( 1.9), in the special situation X“Candq“2 andL“?\n∆,\nthe equivalence for 1 ăpă 8without optimal orders comes from the classical Littlewood-Paley\ntheory while the case p“1 constitutes one essential part of the famous real variable theo ry on\nHardy spaces (cf. [ 17–19] ); in particular the upper estimate of ( 1.9) follows from harmonicity of\nPoisson integrals or Calder´ on-Zygmund theory. Again, the arbitr ariness ofXexcludes the use\nof vector-valued Calder´ on-Zygmund theory and there is an obvio us lack of harmonicity related\nto generalL. To surmount these difficulties, in addition to the application of Theor em4.1—\nWilson’sintrinsic squarefunctions, we will fully developthe duality theo rybetweenvector-valued\nHardy and BMO type spaces in Section 5; the latter is inspired by Mei’s duality arguments [ 31]\n(see also [ 43,47]). In turn, part of the theory of vector-valued Hardy and BMO sp aces will be\ndeduced from vector-valued tent spaces, and the projection πL(see Lemma 3.6) will play a key\nrole in passing from the results about tent spaces to those on Hard y/BMO spaces.\nTogether with the related results in [ 29,33] where the authors showed the Lusin type qof a\nBanach space Xrelative to te´t?\n∆utą0is equivalent to the martingale type qofX(see Section\n6), our vector-valued tent space theory and Theorem 1.1imply the following result, resolving\npartially [ 48, Problem 1.8, Problem A.1 and Conjecture A.4] (see Remark 6.4).\nTheorem 1.2. LetLbe a sectorial operator of type α(0ďαăπ{2) onL2pRdqsatisfying (1.5),\n(1.6)and(1.7). Let1ăqď2. The followings are equivalent\n(i)Xis of martingale type q;\n(ii)Xis of Lusin type qrelative to/visualspace\ne´tL(\ntą0. Moreover, we have the following estimate for\nthe corresponding Lusin type constant,\nLL\nt,q,ppXq/lessorsimilarγ,βpMt,ppXq,1ăpă 8.\nCombining the main result in [ 33], a much stronger result than Theorem 1.2involving the\ncasep“1,8will be presented in Corollary 6.3.\nThe paper is organized essentially as described above with a rigorous introduction of vector-\nvalued tent space, Hardy spaces and BMO spaces in the next sectio n.\nNotation: Inthe followingcontext, Xwill beanarbitraryfixed Banachspacewithout further\nelaboration. X˚denotes the dual Banach space of X. Additionally, the positive real interval\nR`“ p0,8qis equipped with the measure d t{twithout providing additional explanations.\nWe will use the following convention: A/lessorsimilarB(resp.A/lessorsimilarαB) means that AďCB(resp.\nAďCαB) for some absolute positive constant C (resp. a positive constant Cαdepending only\non a parameter α).A«BorA«αBmeans that these inequalities as well as their inverses hold.\nWe also denote by } ¨ }pthe norm } ¨ }LppRdqand by } ¨ }LppXqthe norm } ¨ }LppRd;Xqp1ďpď 8q.\n2.Preliminaries\n2.1.Functional calculus. We start with a brief introduction of some preliminaries around the\nholomorphicfunctional calculus (cf. [ 30]). Let 0 ďαăπ. Define the closed sectorin the complex\nplaneCas\nSα“ tzPC:|argz| ďαu,BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 5\nandS0\nαis denoted as the interior of Sα. Letγąαand denote by HpS0\nγqthe space of all\nholomorphic functions on S0\nγ. Define\nH8pS0\nγq “/visualspace\nbPHpS0\nγq:}b}8ă 8(\n,\nwhere }b}8“sup/visualspace\n|bpzq|:zPS0\nγ(\nand\nΨpS0\nγq “/visualspace\nψPHpS0\nγq:Dsą0 s.t. |ψpzq| ďc|z|sp1` |z|2sq´1(\n.\nA densely defined closed operator Lacting on a Banach space Yis called a sectorial operator of\ntypeαif for eachγąα,σpLq ĂSγand\nsup/visualspace\n}zpzId´Lq´1}BpYq:zRSγ(\nă 8,\nwhere } ¨ }BpYqdenotes the operator norm and Id the identity operator.\nAssume that Lis a sectorial operator of type α. Let 0 ďαăθăγăπand Γ be the\nboundary of Sθoriented in the positive sense. For ψPΨpS0\nγq, we define the operator ψpLqas\nψpLq “1\n2πiˆ\nΓψpzqpzId´Lq´1dz.\nBy Cauchy’stheorem, this integralconvergesabsolutelyin BpYqand it isclearthat the definition\nis independent of the choice of θ. For every tą0, denote by ψtpzq “ψptzqforzPS0\nγ, we have\nψtPΨpS0\nγq. Set\nhpzq “ˆ8\n0ψptzqdt\nt, z PS0\nγ.\nOne gets that his a constant on S0\nγ, hence by the convergence lemma (cf. [ 13, Lemma 2.1]),\nhpLqx“ˆ8\n0ψptLqxdt\nt“cx, x PDpLq XimpLq.\nBy applying a limiting argument, the above identity extends to impLq. In particular, take\nψpzq “z2e´2z, then\n(2.1)ˆ8\n0´tLe´tLp´tLe´tLqxdt\nt“1\n4x, x PimpLq,\nwhich will be useful later. We refer the readerto [ 21] for more information on functional calculus.\n2.2.Main assumptions. Throughout the paper, we assume Lis a sectorial operator of type α\n(0ďαăπ{2) onL2pRdqsuch that the kernels tKpt,x,y qutą0ofte´tLutą0satisfy assumptions\n(1.5), (1.6) and (1.7) withβą0,0ăγď1. It is well-known that such an Lgenerates\na holomorphic semigroup e´zLwith 0 ď |Argpzq| ăπ{2´α(cf. [21, Chapter 3, 3.2]). Let\ntkpt,x,y qutą0be the kernels of t´tLe´tLutą0and it is easy to see\nkpt,x,y q “tBtKpt,x,y q.\nThe following lemma is justified in [ 16, Lemma 6.9]\nLemma 2.1. LetLbe an operator satisfying (1.5)and(1.6)withβą0,0ăγď1. Then\n(i)there exist constants 0ăβ1ăβ,0ăγ1ăγandcą0such that\n(2.2) |kpt,x,y q| ďctβ1\npt` |x´y|qd`β1,\nand\n|kpt,x`h,yq ´kpt,x,y q| ` |kpt,x,y `hq ´kpt,x,y q| ďc|h|γ1tβ1\npt` |x´y|qd`β1`γ1BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 6\nwhenever 2|h| ďt` |x´y|;\n(ii)forαăθăπ{2, there exist positive constants 0ăβ2ăβ,0ăγ2ăγandcą0such\nthat for any |argz| ăπ{2´θ,\n(2.3) |Kpz,x,y q| ďc|z|β2\np|z| ` |x´y|qd`β2\nand\n|Kpz,x`h,yq ´Kpz,x,y q| ` |Kpz,x,y `hq ´Kpz,x,y q| ďc|h|γ2|z|β2\np|z| ` |x´y|qd`β2`γ2\nwhenever 2|h| ď |z| ` |x´y|.\nRemark 2.2.By [12, Lemma 2.5], the estimate ( 2.3) implies that for all kPN,tą0 and almost\neverywhere x,yPRd,\n(2.4)ˇˇtkBk\ntKpt,x,y qˇˇďctβ2\npt` |x´y|qd`β2.\nConvention. To simplify notation, we will write below γ,βinstead ofγ1, β1andγ2, β2\nappearing in Lemma 2.1, and it should not cause any confusion.\nOne can verify that/visualspace\ne´tL(\ntą0is a family of regular operators on LppRdqfor 1 ďpď 8. Then\nit is well-known (see e.g. [ 48]) that the semigroup te´tLutą0extends toLppRd;Xq(1ďpď 8),\nwhich is the space of all strongly measurable functions f:RdÑXsuch that }fpxq}XPLppRdq.\nThe resulting semigroup is still denoted by te´tLutą0without causing confusion. To well define\nthe vector-valued BMO type spaces, we need more notations. For εą0, define\nNε“\"\nfPL1\nlocpRd;Xq:Dcą0 such thatˆ\nRd}fpxq}X\np1` |x|qd`εdxďc*\n,\nequipped with norm defined as the infimum of all the possible constant c. ThenNεis a Banach\nspace (cf. [ 16]). For a given generator L, let Θ pLq “suptβ2ą0 : (2.3) holds u. Then we define\nN“#\nNΘpLq, if ΘpLq ă 8;Ť\n0ăεă8Nε,if ΘpLq “ 8.\nIt is clear that LppRd;Xq ĂNfor all 1 ďpď 8. Moreover, By the definition of Nand Remark\n2.2, we know that the operators e´tLandtLe´tLare well-defined on N.\nDenote by FLthe fixed point space of te´tLutą0onN, namely\nFL“/visualspace\nfPN:e´tLpfq “f,@tą0(\n.\nIt is well-known that FLcoincides with the null space of ttLe´tLutą0, and the resulting quotient\nspace is defined as NL:“N{FL. For 1 ďpă 8, the fixed point subspace of LppRd;Xqis\nFLXLppRd;Xq “ t0u(see [16, Theorem 6.10]); in other words, the projection from LppRd;Xq\nto the fixed point subspace for all 1 ďpă 8is 0. See e.g. [ 29,48] for more information on this\nprojection.\nRemark 2.3.LetL˚be the adjoint operator of L. ThenL˚is also a sectorial operator with the\nkernels of te´tL˚utą0satisfying ( 1.5), (1.6) and (1.7) (cf. [16, Theorem 6.10]).\n2.3.Vector-valued tent, Hardy and BMO spaces. In this subsection, we introduce several\nspaces including vector-valuedtent spaces, vector-valuedHard y and BMO spaces associatedwith\na generator L.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 7\n2.3.1.Vector-valued Tent spaces. We first introduce vector-valued tent spaces. We denote by\nRd`1\n`the usual upper half-space in Rd`1i.e.Rdˆ p0,8q. Let Γ pxq “ tpy,tq PRd`1\n`:|y´x| ătu\ndenote the standard cone with vertex at x. For any closed subset FĂRd, define RpFq “Ť\nxPFΓpxq. IfOĂRdis an open subset, then the tent over O, denoted by pO, is given as\npO“`\nRpOCq˘C.\nFor any strongly measurable function f:Rd`1\n`ÑX, we define two operators as follows:\nAqpfqpxq “˜ˆ\nΓpxq}fpy,tq}q\nXdydt\ntd`1¸1\nq\n,Cqpfqpxq “sup\nxPBˆ1\n|B|ˆ\npB}fpy,tq}q\nXdydt\nt˙1\nq\n,\nwhere the supremum runs over all balls BinRd.\nDefinition 2.4. Let 1 ďpă 8and 1 ăqă 8. The vector-valued tent space Tp\nqpRd`1\n`;Xqis\ndefined as the subspace consisting of all strongly measurable func tionsf:Rd`1\n`ÑXsuch that\n}f}Tp\nqpXq:“ }Aqpfq}pă 8,\nandT8\nqpRd`1\n`;Xqis defined as the subspace of all strongly measurable functions g:Rd`1\n`ÑX\nsuch that\n}g}T8qpXq:“ }Cqpgq}8ă 8.\nLetCcpRd`1\n`q bXbe the space of finite linear combinations of elements from CcpRd`1\n`qand\nX. The following density follows from the standard arguments (see e.g . [23]), and we omit the\ndetails.\nLemma 2.5. LetXbe a Banach space and 1ăqă 8. ThenCcpRd`1\n`q bXis norm dense in\nTp\nqpRd`1\n`;Xqfor1ďpă 8, and weak- ˚dense in`\nT1\nq1pRd`1\n`;X˚q˘˚.\n2.3.2.Vector-valued Hardy spaces. Given a function fPNL, theq-variant of Lusin area integral\nfunction of fassociated with Lis defined by\nSq,Lpfqpxq “˜ˆ\nΓpxq}tLe´tLpfqpyq}q\nXdydt\ntd`1¸1\nq\n;\nand theq-variant of Littlewood-Paley g-function is defined by\nGq,Lpfqpxq “ˆˆ8\n0}tLe´tLpfqpxq}q\nXdt\nt˙1\nq\n.\nDefinition 2.6. Let 1 ďpă 8and 1 ăqă 8. We define the vector-valued Hardy space\nHp\nq,LpRd;Xqassociated with Las\nHp\nq,LpRd;Xq “/visualspace\nfPNL:Sq,Lpfq PLppRdq(\n,\nequipped with the norm\n}f}Hp\nq,LpXq“ }Sq,Lpfq}p.\nIt is easy to check that Hp\nq,LpRd;Xqis a Banach space from the definition of NL. The\nspaceHp\nq,LpRd;Xqhas deep connection with the vector-valued tent space, namely, a strongly\nmeasurable function fPNLbelongs to Hp\nq,LpRd;Xqif and only if Qpfq PTp\nqpRd`1\n`;Xqwhere\nQpfqpx,tq “ ´tLe´tLpfqpxq. Moreover,\n}f}Hp\nq,LpXq“ }Qpfq}Tp\nqpXq.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 8\n2.3.3.Vector-valued BMO spaces.\nDefinition 2.7. Let 1 ďpď 8and 1 ăqă 8. We define the vector-valued BMO space\nBMOp\nq,LpRd;Xqassociated with Las\nBMOp\nq,LpRd;Xq “ tfPNL:}CqpQpfqq}pă 8u\nequipped with the norm\n}f}BMOp\nq,LpXq“ }CqpQpfqq}p.\nIn particular, for p“ 8, we denote it by BMO q,LpRd;Xqfor short.\nIt is easy to verify that BMOp\nq,LpRd;Xqequipped the the norm } ¨ }BMOp\nq,LpXqis a Banach\nspace from the definition of NL.\nThe vector-valued Hardy and BMO spaces enjoy the similar relations hip as the scalar-valued\nones (see e.g. [ 11]). We collect them below with a brief explanation.\nLemma 2.8. LetXbe any fixed Banach space and 1ăqă 8. One has for fPCcpRd`1\n`q bX,\n(2.5) }Cqpfq}p/lessorsimilarˆp\np´q˙1\nq\n}Aqpfq}p, q ăpď 8,\nand\n(2.6) }Aqpfq}p/lessorsimilarqp\nq}Cqpfq}p,1ďpă 8.\nTherefore, we have for 1ďpďq,\nBMOp\nq,LpRd;Xq ĂHp\nq,LpRd;Xq\nand forqăpă 8,\nHp\nq,LpRd;Xq “BMOp\nq,LpRd;Xq\nwith equivalent norms.\nProof.Given anX-valued function fdefined on Rd`1\n`, we consider the scalar-valued function\nrfpx,tq “ }fpx,tq}X. Then one may apply ( 2.5) and (2.6) in the case X“Cforrf(see e.g. [ 11,\nTheorem 3]) to obtain ( 2.5) and (2.6) for general X. Thus by using the operator Qand the\ndensity in Lemma 2.5, for anyfPBMOp\nq,LpRd;Xq(1ďpďq), we get\n}f}Hp\nq,LpXq“ }AqpQpfqq}p/lessorsimilarqp\nq}CqpQpfqq}p“ }f}BMOp\nq,LpXq,\nand the same argument works for qăpă 8. /square\nRemark 2.9.In particular, BMO q,LpRd;Xqis closely related to the Carleson measure. Recall\nthat a scalar-valued measure µdefined on Rd`1\n`is a Carleson measure if there exists a constant\ncsuch that for all balls BinRd,\n|µppBq| ďc|B|,\nwherepBis the tent over B. The norm is defined as\n}µ}c“sup\nB|B|´1|µppBq|,\nwhere the supremum runs over all the balls in Rd.\nFor a vector-valued function fPNL, we define a measure µq,fas\nµq,fpx,tq “}Qpfqpx,tq}q\nXdxdt\nt.\nThenfbelongs toBMO q,LpRd;Xqif and only if µq,fis a Carleson measure, and moreover\n}f}BMO q,LpXq“ }µq,f}1\nq\nc.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 9\n3.Theory of vector-valued tent spaces and two key linear opera tors\nIn thissection, wewill first presentthe basictheoryofvector-va luedtent spacessuchasatomic\ndecomposition, interpolation and duality, and then introduce two imp ortant linear operators K\nandπLwhich enable us to exploit the basic theory of tent spaces to investig ate in later sections\nvector-valued Wilson’s square functions and Theorem 1.1.\nNote that if the underlying Banach space Xhas some geometric property such as UMD, then\nthe vector-valued tent space theory have been established in the literature [ 26–28]. In the present\npaper, we observe that the theory of vector-valued tent space holds for any Banach space; and\nthis is quite essential for the applications in the present paper.\n3.1.Basic theory of vector-valued tent spaces. We begin this subsection by presenting\nthe atomic decomposition of tent space in the context of vector-v alued context. It has been\nestablished in [ 27, Theorem 4.5], for the completeness of this article, we will attach th e proof.\nRecall that a strongly measurable function a:Rd`1\n`ÑXis called an pX,qq-atom if\n(1) suppaĂpBwhereBis a ball in Rd;\n(2)´´\nRd`1\n`}apx,tq}q\nXdxdt\nt¯1\nqď |B|1\nq´1.\nLemma 3.1. LetXbe any fixed Banach space and 1ăqă 8. For eachfPT1\nqpRd`1\n`;Xq, there\nexists a sequence of complex numbers tλkukě1and pX,qq-atomsaksuch that\nf“ÿ\nkě1λkak,}f}T1qpXq«ÿ\nkě1|λk|.\nProof.Letabe an pX,qq-atom and supp aĂpBwhereB“BpcB,rBqwith center cBand radius\nrB. If Γ pxq XpB‰ H, there exists py,tq PΓpxq XpB. Then we have |x´cB| ď |x´y| ` |y´cB| ă\nt`rBă2rB. By H¨ older’s inequality and Fubini’s theorem,\n}a}T1qpXq“ˆ\n2B˜ˆ\nΓpxq}apy,tq}q\nXdydt\ntd`1¸1\nq\ndx/lessorsimilar|2B|1´1\nq˜ˆ\nRd`1\n`}apy,tq}q\nXdydt\nt¸1\nq\n/lessorsimilar1.\nTherefore any pX,qq-atom belongs to T1\nqpRd`1\n`;Xq.\nLet 0 ăλă1{2. We define two sequences of open sets tOkukPZand tO˚\nkukPZas\nOk“/visualspace\nxPRd:Aqpfqpxq ą2k(\n, O˚\nk“/visualspace\nxPRd:Mp /BDOkqpxq ą1´λ(\n,\nwhereMp /BDOkqis the centered Hardy-Littlewood maximal function. It is clear that bothOkand\nO˚\nkhave finite measure. Additionally, the following properties hold: Ok`1ĂOk,O˚\nk`1ĂO˚\nk\nand |O˚\nk| ďCλ|Ok|(see e.g. [ 11]).\nWe follow a similar construction as in [ 27]. The Vitali covering lemma and [ 28, Lemma 4.2]\nassert that for each O˚\nk, there exist disjoint balls Bj\nkĂO˚\nk(jě1) such that\npO˚\nkĂď\njě1x5Bj\nk,ÿ\njě1|Bj\nk| ď |O˚\nk|.\nWith this setup, we proceed to define a family of functions χj\nkby the partition of unity:\n0ďχj\nkď1,ÿ\njě1χj\nk“1 onpO˚\nkand suppχj\nkĂx5Bj\nk.\nTherefore\nf“ÿ\nkPZfk“ÿ\nkPZÿ\njě1χj\nkfk“ÿ\nkPZÿ\njě1λj\nkaj\nk,BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 10\nwhere\n(3.1) fk“f /BDpO˚\nkzpO˚\nk`1, λj\nk“ |5Bj\nk|1\nq1˜ˆ\n5Bj\nkAqpfkqqpxqdx¸1\nq\n, aj\nk“χj\nkfk\nλj\nk.\nNow we only need to show that each aj\nkis an pX,qq-atom and\nÿ\nkPZÿ\njě1|λj\nk|/lessorsimilar}f}T1qpXq.\nIt is clear that supp aj\nkĂx5Bj\nk. Furthermore,\n}a}q\nLqpRd`1\n`;Xqď |5Bj\nk|1´q}Aqpfkq /BD5Bj\nk}´q\nq˜ˆ\ny5Bj\nk}fkpy,tq}q\nXdydt\nt¸\nď |5Bj\nk|1´q}Aqpfkq /BD5Bj\nk}´q\nq˜ˆ\n5Bj\nkpAqpfkqpxqqqdx¸\n“ |5Bj\nk|1´q.\nHence each aj\nkis an pX,qq-atom.\nAccordingto [ 11, Lemma 5], it is known that Aqpfkqis supported in O˚\nkzOk`1, then we deduce\nthatAqpfkqpxq ď2k`1by definition. Thus\nÿ\nkPZÿ\njě1|λj\nk| ďÿ\nkPZÿ\njě1|5Bj\nk|1\nq12k`1|5Bj\nk|1\nqďÿ\nkPZ2k`1|O˚\nk| ďÿ\nkPZ2k`1Cλ|Ok|.\nHowever, Aqpfqpxq ą2pk`mqonOk`m, then\n2k|Ok| “ˆ\nOk2kdx“8ÿ\nm“0ˆ\nOk`mzOk`m`12kdxď8ÿ\nm“02´mˆ\nOk`mzOk`m`1Aqpfqpxqdx.\nHence\nÿ\nkPZ2k`1Cλ|Ok| ď8ÿ\nm“0ÿ\nkPZ2´m`1Cλˆ\nOk`mzOk`m`1Aqpfqpxqdx/lessorsimilar}f}T1qpXq.\nWe complete the proof. /square\nRemark 3.2.From the atomic decomposition of T1\nqpRd`1\n1;Xq—Lemma 3.1, one may conclude a\nmolecule decomposition of the correspondingHardy space. This migh t have further applications,\nand we include it in the Appendix.\nThe following lemma is the complex interpolation theory of vector-valu ed tent spaces.\nLemma 3.3. LetXbe any fixed Banach space, 1ăqă 8and1ďp1ăpăp2ă 8such that\n1{p“ p1´θq{p1`θ{p2with0ďθď1. Then\nrTp1qpRd`1\n`;Xq,Tp2qpRd`1\n`;Xqsθ“Tp\nqpRd`1\n`;Xq,\nwith equivalent norms, where r¨,¨sθis the complex interpolation space. More precisely, for fP\nCcpRd`1\n`q bX, one has\n}f}Tp\nqpXq/lessorsimilar}f}rTp1qpRd`1\n`;Xq,Tp2qpRd`1\n`;Xqsθ/lessorsimilarp1\nq}f}Tp\nqpXq.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 11\nProof.For the interpolation theory, we introduce two important operato rs, which allow us to\nrelateTp\nqpRd`1\n`;XqwithLppRd;EqwithEbeing the Banach space LqpRd`1\n`;Xqequipped with\nthe measure d xdt{td`1. The first operator is defined as\nipfqpx,y,t q “ /BDΓpxqpy,tqfpy,tq,\nforfPTp\nqpRd`1\n`;Xq. Then it is clear that }ipfq}LppEq“ }f}Tp\nqpXq. Denote by rTp\nqthe range of\nthe operator i. Now we introduce another operator Ngiven by\nNpfqpx,y,t q “ /BDΓpxqpy,tq1\nwdtdˆ\n|z´y|ătfpz,y,t qdz,\nwherewdis the volume of the d-dimensional unit ball. It is known that Nis a continuous\nprojection from LppRd;Eqonto itself with range rTp\nqfor 1 ăpă 8(cf. [23]). Consider the\nmaximal operator\nM1pfqpx,y,t q “sup\nxPB1\n|B|ˆ\nB}fpz,y,t q}Xdz,\nwhere the supremum is taken over all balls BinRd. It is known from the maximal inequalities\n(see e.g. [ 38, Chapter II]) that M1is bounded on Lp`\nRd;LqpRd`1;dydt{td`1q˘\nfor 1 ăpă 8;\nin particular, we view }f}Xas a scalar-valued function in Lp`\nRd;LqpRd`1;dydt{td`1q˘\n, then\n}M1pfq}LppRd;LqpRd`1\n`,dydt\ntd`1qq“ }M1p}f}Xq}LppRd;LqpRd`1\n`,dydt\ntd`1qq/lessorsimilarp1\nq}f}LppEq, q ďpă 8.\nThen we deduce from the definition of Nthat\n}Npfqpx,y,t q}Xď /BDΓpxqpy,tq1\n|Bpy,tq|ˆ\nBpy,tq}fpz,y,t q}XdzďM1pfqpx,y,t q.\nTherefore\n}Npfq}LppEqď }M1pfq}LppRd;LqpRd`1\n`;dydt\ntd`1qq/lessorsimilarp1\nq}f}LppEq, q ďpă 8.\nWe denote by Fthe Banach space Lq1pRd`1\n`;X˚qequipped with the measure d xdt{td`1. Then\nit is clear that FĂE˚andFis norming for E. For 1 ăpăq, we have\n}Npfq}LppEq“sup\ngˇˇˇˇˇˆ\nRdˆ\nRd`1\n`xNpfqpx,y,t q,gpx,y,t qyXˆX˚dydt\ntd`1dxˇˇˇˇˇ\n“sup\ngˇˇˇˇˇˆ\nRdˆ\nRd`1\n`xfpx,y,t q,Npgqpx,y,t qyXˆX˚dydt\ntd`1dxˇˇˇˇˇ\nď }f}LppEq}Npgq}Lp1pFq/lessorsimilarp11\nq1}f}LppEq}g}Lp1pFq,\nwhere the supremum is taken over all gin the unit ball of Lp1pRd;Fq. We conclude\n(3.2) }Npfq}LppEq/lessorsimilarmax!\np1\nq,p11\nq1)\n}f}LppEq,1ăpă 8.\nNow we turn to the interpolation theory. The proof of the case 1 ăp1ăp2ă 8follows\nfrom [23] by virtue of the immersion iand the projection N.\nFor the case p1“1, we adapt the classical argument as in [ 11, Lemma 4, Lemma 5]. Since\nthe immersion iis an isometry, the exactness of the exponent θof complex interpolation functor\nreads that\n}ipfq}rL1pRd;Eq,Lp2pRd;Eqsθ\nď }i}1´θ\nT1qpRd`1\n`;XqÑL1pRd;Eq}i}θ\nTp2qpRd`1\n`;XqÑLp2pRd;Eq}f}rT1\nqpRd`1\n`;Xq,Tp2qpRd`1\n`;Xqsθ\nď }f}rT1qpRd`1\n`;Xq,Tp2qpRd`1\n`;Xqsθ.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 12\nBy the interpolation theory of vector-valued Lpspaces, (see e.g. [ 9]), we have\n}ipfq}rL1pRd;Eq,Lp2pRd;Eqsθ“ }ipfq}LppEq“ }f}Tp\nqpXq.\nThus\n}f}Tp\nqpXqď }f}rT1qpRd`1\n`;Xq,Tp2qpRd`1\n`;Xqsθ.\nFor the reverse direction, let fPTp\nqpRd`1\n`;Xqand }f}Tp\nqpXq“1. By taking into account the\natomic decomposition of T1\nqpRd`1\n`;Xq—Lemma 3.1, we define the interpolation functor Fas\nFpzq “ÿ\nkPZ2kpαpzqp´1qfk,\nwhereαpzq “1´z`z{p2andfkis defined in ( 3.1). We have Fpθq “f. Then the proof can be\nthen conducted in the same way as in [ 11, Lemma 5], we omit the details. /square\nWe now provide a characterization of Tp\nqpRd`1\n`;Xq-norm. It belongs to the norming subspace\ntheory of vector-valued Lp-spaces, see e.g. [ 15, Chapter II, Section 4]. The proof is in spirit the\nsame as the scalar-valued case (cf. [ 23, Theorem 2.4] and [ 11, Theorem 1], but we include a proof\nhere to provide explicit orders for later applications.\nLet pΩ,F,µqbe a measure space. Recall that a Banach space Xhas theRadon-Nikod´ ym\nproperty with respect to pΩ,F,µqif for eachµ-continuous vector-valued measure ν:FÑXof\nbounded variation, there exists gPL1pΩ;Xqwith respect to the measure µsuch that\nνpEq “ˆ\nEgdµ, @EPF.\nIn the following context, we call a Banachspace has the Radon-Niko d´ ymproperty for short when\nthere is no ambiguity. We refer readers to [ 14, Chapter III] for more details.\nLemma 3.4. LetXbe any fixed Banach space and 1ăqă 8. The space Tp1\nq1pRd`1\n`;X˚qis\nisomorphically identified as a subspace of the dual space of Tp\nqpRd`1\n`;Xq. Moreover, it is norming\nforTp\nqpRd`1\n`;Xqin the following sense,\n}f}Tp\nqpXq/lessorsimilarmax!\np1\nq,p11\nq1)\nsup\ngˇˇˇˇˇˆ\nRd`1\n`xfpx,tq,gpx,tqyXˆX˚dxdt\ntˇˇˇˇˇ,1ăpă 8, (3.3)\nwhere the supremum is taken over all gPCcpRd`1\n`qbX˚such that }Aq1pgq}p1ď1; and similarly,\n}f}Tp\nqpXq/lessorsimilarˆppq´1q\nq´p˙1\nq1\nsup\ngˇˇˇˇˇˆ\nRd`1\n`xfpx,tq,gpx,tqyXˆX˚dxdt\ntˇˇˇˇˇ,1ďpăq, (3.4)\nwhere the supremum is taken over all gPCcpRd`1\n`q bX˚such that }Cq1pgq}p1ď1. Furthermore,\nifX˚has the Radon-Nikod´ ym property, then\nTp1\nq1pRd`1\n`;X˚q “`\nTp\nqpRd`1\n`;Xq˘˚,1ďpă 8.\nProof.We adopt the maps iandNused in the proof of Lemma 3.3. We first prove the estimate\n(3.3).\nFor anygPCcpRd`1\n`q bX˚andfPTp\nqpRd`1\n`;Xq, we denote by\ngpfq “ˆ\nRd`1\n`xfpx,tq,gpx,tqyX��X˚dxdt\nt.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 13\nThus we have\n|gpfq| “ˇˇˇˇˇˆ\nRd`1\n`C\nfpy,tq,gpy,tq˜\nw´1\ndˆ\n|x´y|ăt1dx¸G\nXˆX˚dydt\ntd`1ˇˇˇˇˇ\n“w´1\ndˇˇˇˇˇˆ\nRdˆ\nRd`1\n`xipfqpx,y,t q,ipgqpx,y,t qyXˆX˚dydt\ntd`1dxˇˇˇˇˇ.(3.5)\nSinceipfq PrTp\nq, we haveNpipfqq “ipfq. Then we deduce that\n}f}Tp\nqpXq“ }ipfq}LppEq“sup\ngˇˇˇˇˇˆ\nRdˆ\nRd`1\n`xipfqpx,y,t q,gpx,y,t qyXˆX˚dydt\ntd`1dxˇˇˇˇˇ\n“sup\ngˇˇˇˇˇˆ\nRdˆ\nRd`1\n`xNpipfqqpx,y,t q,gpx,y,t qyXˆX˚dydt\ntd`1dxˇˇˇˇˇ\n“sup\ngˇˇˇˇˇˆ\nRdˆ\nRd`1\n`xipfqpx,y,t q,Npgqpx,y,t qyXˆX˚dydt\ntd`1dxˇˇˇˇˇ(3.6)\nwhere the supremum is taken over all gin the unit ball of Lp1pRd;Fq. Notice that Npgq “\ni`\ni´1pNpgqq˘\nand by (3.2)\n}i´1pNpgqq}Tp1\nq1pX˚q“ }Npgq}Lp1pFq/lessorsimilarmax!\np1\nq,p11\nq1)\n}g}Lp1pFq.\nConsequently, combining ( 3.5) and (3.6), we obtain\n}f}Tp\nqpXq/lessorsimilarmax!\np1\nq,p11\nq1)\nsup\ng|gpfq|,1ăpă 8,\nwhere the supremum is taken over all gPCcpRd`1\n`q bX˚such that }Aq1pgq}p1ď1, and we\nactually exploit a limiting argument: since not only the subset of CcpRd`1\n`q bX˚with norm\n}Aq1pgq}p1ď1 is contained in the unit ball of Tp1\nq1pRd`1\n`;X˚q, but also its closure contains the\nunit sphere, and thus one concludes that this subset is still norming forTp\nqpRd`1\n`;Xq.\nNow we deal with the estimate ( 3.4) in the case 1 ăpăq. LetgPLp1pRd;Fq. By definition\nwe have\n}i´1pNpgqqpy,tq}q1\nX˚ď˜\n1\n|Bpy,tq|ˆ\nBpy,tq}gpz,y,t q}X˚dz¸q1\nď1\n|Bpy,tq|ˆ\n|z´y|ăt}gpz,y,t q}q1\nX˚dz.\nFor a ballBinRd, we observeˆ\npB}i´1pNpgqqpy,tq}q1\nX˚dydt\nt/lessorsimilarˆ\npBˆ\n|z´y|ăt}gpz,y,t q}q1\nX˚dzdydt\ntd`1\nďˆ\n2Bˆ\nRd`1\n`} /BDpBpy,tqgpz,y,t q}q1\nX˚dydt\ntd`1dz\n“ˆ\n2BHq1pzqdz,\nwhere\nHpzq “˜ˆ\nRd`1\n`} /BDpBpy,tqgpz,y,t q}q1\nX˚dydt\ntd`1¸1\nq1\n.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 14\nThen we have\nCq1“\ni´1pNpgqq‰\npxq/lessorsimilar´\nMpHq1qpxq¯1\nq1,\nwhereMis the Hardy-Littlewood maximal operator. Therefore when q1ăp1ď 8, we obtain\n}Cq1“\ni´1pNpgqq‰\n}p1/lessorsimilar}MpHq1q1\nq1}p1/lessorsimilarˆppq´1q\nq´p˙1\nq1\n}H}p1ďˆppq´1q\nq´p˙1\nq1\n}g}Lp1pFq.\nThus we observe\n}f}Tp\nqpXq/lessorsimilarˆppq´1q\nq´p˙1\nq1\nsup\ng|gpfq|,1ăpăq,\nwith the supremum being taken over all gPCcpRd`1\n`q bX˚such that }Cq1pgq}p1ď1.\nForthe endpoint case p“1of (3.4), because ofthe failureofvector-valuedCalder´ on-Zygmund\ntheory, the above arguments adapted from [ 23, Theorem 2.4] do not work any more. Instead,\nby using the atomic decomposition of T1\nqpRd`1\n`;Xq—Lemma 3.1, one may carry out the classical\narguments as in [ 11, Theorem 1] in the present vector-valued setting, and we leave th e details to\nthe interested reader.\nWhen the Banach space X˚has the Radon-Nikod´ ym property, one gets F“E˚(cf. [25,\nTheorem 1.3.10]). Then the duality follows from then an analogous arg ument in [ 23] for 1 ăpă\n8. Again, the duality in the case p“1 can be deduced as in the scalar-valued case [ 11, Theorem\n1], and we leave the details to the interested reader. /square\n3.2.The two linear operators KandπL.LetK:Rd`1\n`ˆRd`1\n`ÑRbe a reasonable real-\nvalued function such that for fPCcpRd`1\n`q bX, the linear operator Kis well defined as below,\nKpfqpx,tq:“ˆ\nRd`1\n`Kt,spx,yqfpy,sqdyds\ns.\nLemma 3.5. LetXbe any fixed Banach space and 1ăqă 8. Assume that the kernel Kt,spx,yq\nsatisfies the following estimation: there exist positive co nstantsκ,ε,Csuch that\n(3.7) |Kt,spx,yq| ďCmin/visualspaces\nt,t\ns(εmin/visualspace1\nt,1\ns(d\n`\n1`min/visualspace1\nt,1\ns(\n|x´y|˘d`κ.\nThen the linear operator Kinitially defined on CcpRd`1\n`qbXextends to a bounded linear operator\nonTp\nqpRd`1\n`;Xqfor1ďpă 8. More precisely,\n}Kpfq}Tp\nqpXq/lessorsimilarε,κp1\nq}f}Tp\nqpXq,@fPTp\nqpRd`1\n`;Xq,1ďpă 8.\nFurthermore, for any fPCcpRd`1\n`q bX, we have\n}CqpKpfqq }p/lessorsimilarε,κ}Cqpfq}p,1ďpď 8.\nProof.FixfPCcpRd`1\n`q bX. Without loss of generality, we can assume κăεfrom (3.7). We\nfirst deal with the case p“q. By H¨ older’s inequality, we have\nˆ\nRd`1\n`}Kpfqpx,tq}q\nXdydt\nt“ˆ\nRd`1\n`›››››ˆ\nRd`1\n`Kt,spy,wqfpw,sqdwds\ns›››››q\nXdydt\nt\nďˆ\nRd`1\n`˜ˆ\nRd`1\n`|Kt,spy,wq|dwds\ns¸q\nq1\n¨˜ˆ\nRd`1\n`|Kt,spy,wq|}fpw,sq}q\nXdwds\ns¸\ndydt\nt.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 15\nWe obtain that\nˆ8\n0ˆ\nRd|Kt,spy,wq|dwds\nsďˆt\n0ˆ\nRdCsεt´εt´d\np1`t´1|y´w|qd`κdwds\ns\n`ˆ8\ntˆ\nRdCtεs´εs´d\np1`s´1|y´w|qd`κdwds\ns\n/lessorsimilarε,κˆ\nRdCt´d\np1`t´1|y´w|qd`κdw`ˆ8\nttεs´ε´1ds\n/lessorsimilarε,κ1.\nIt is clear that in the assumption of Kt,spy,wq,pw,sqplays the same role as py,tq. Thus\n(3.8) }Kpfq}q\nLqpRd`1\n`;Xq/lessorsimilarε,κˆ\nRd`1\n`}fpw,sq}q\nXdwds\ns“ }f}q\nLqpRd`1\n`;Xq.\nThen the case p“qis done since }f}Tq\nqpXq« }f}LqpRd`1\n`;Xq. Moreover,from the proof we observe\nthatKis always bounded on LppRd`1\n`;Xqfor 1 ďpď 8.\nFor 1 ďpăq, by the interpolation—Lemma 3.3, it suffices to show the case p“1. By the\natomic decomposition—Lemma 3.1, It suffices to show that\n(3.9) }Kpaq}T1\nqpXq/lessorsimilarε,κ1,\nwhereais an pX,qq-atom with supp aĂpBandB“BpcB,rBq. One can write\n}AqrKpaqs}1“ˆ\n4BAqrKpaqspxqdx`ˆ\np4BqCAqrKpaqspxqdx\n“I`II.\nFrom (3.8) we obtain\n(3.10) }AqrKpaqs}q\nq/lessorsimilarε,κˆ\nRd`1\n`}apw,sq}q\nXdwds\nsď |B|1´q.\nThen we can estimate the term I:\n(3.11) Iď |4B|1\nq1}AqrKpaqs}q/lessorsimilarε,κ1.\nNow we handle the second term II. By H¨ older’s inequality, we observe\npAqrKpaqspxqqqďˆ8\n0ˆ\n|y´x|ătˆˆ\npB|Kt,spy,wq|q1dwds\ns˙q\nq1\n¨ˆˆ\npB}apw,sq}q\nXdwds\ns˙dydt\ntd`1\n/lessorsimilar|B|1´qˆ8\n0ˆ\n|y´x|ătˆˆ\npB|Kt,spy,wq|q1dwds\ns˙q\nq1dydt\ntd`1\n“ |B|1´qˆrB\n0ˆ\n|y´x|ătˆˆ\npB|Kt,spy,wq|q1dwds\ns˙q\nq1dydt\ntd`1\n` |B|1´qˆ8\nrBˆ\n|y´x|ătˆˆ\npB|Kt,spy,wq|q1dwds\ns˙q\nq1dydt\ntd`1\n“:J1`J2.\nWhenxP p4BqC, wPB, we have\nrBă |x´w| ď |x´y| ` |y´w| ăt` |y´w|,BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 16\nhence\n|x´cB| ď |x´w| ` |w´cB| ă2pt` |y´w|q ď2pmax tt,su ` |y´w|q.\nTherefore we observe from ( 3.7) that\n|Kt,spy,wq|/lessorsimilarε,κmin/visualspaces\nt,t\ns(εmin/visualspace1\nt,1\ns(d\npmax tt,su ` |y´w|qd`κmin/visualspace1\nt,1\ns(d`κ\n/lessorsimilarε,κmin/visualspaces\nt,t\ns(εmin/visualspace1\nt,1\ns(´κ\n|x´cB|d`κ“mintsεtκ´ε,tεsκ´εu\n|x´cB|d`κ.\nThen\nJ1/lessorsimilarε,κ|B|1´q|B|q\nq1\n|x´cB|qpd`κqˆrB\n0ˆˆrB\n0min!\nsq1εtq1pκ´εq,tq1εsq1pκ´εq)ds\ns˙q\nq1dt\nt\n“1\n|x´cB|qpd`κqˆrB\n0ˆˆt\n0tq1pκ´εqsq1εds\ns`ˆrB\ntsq1pκ´εqtq1εds\ns˙q\nq1dt\nt\n/lessorsimilarε,κrqκ\nB\n|x´cB|qpd`κq.\nForJ2, sincetěrBěs, and |x´cB| ă2pt` |y´w|q,\nJ2/lessorsimilarε,κ|B|1´q|B|q\nq1\n|x´cB|qpd`κqˆ8\nrBˆˆrB\n0sq1εtq1pκ´εqds\ns˙q\nq1dt\nt\n/lessorsimilarε,κrqκ\nB\n|x´cB|qpd`κq.\nThus\nAqrKpaqspxq/lessorsimilarε,κrκ\nB\n|x´cB|d`κ, x P p4BqC.\nSince\nˆ\np4BqCrκ\nB\n|x´cB|d`κdx“8ÿ\nm“2ˆ\n2m`1Bz2mBrκ\nB\n|x´cB|d`κdxď8ÿ\nm“2ˆ\n2m`1Brκ\nB\n2mpd`κqrd`κ\nBdx\n/lessorsimilar8ÿ\nm“2p2m`1rBqd\n2mpd`κqrd\nB/lessorsimilar8ÿ\nm“22´mκ/lessorsimilarκ1,\nwe obtain\n(3.12) II/lessorsimilarε,κ1.\nFor the case qăpă `8, we denote by K˚the adjoint operator. It is clear that the kernel\nofK˚has the same estimation as that of K. ForfPCcpRd`1\n`q bX, we obtain from Lemma 3.4\nthat\n}Kpfq}Tp\nqpXq/lessorsimilarp1\nqsup\ngˇˇˇˇˇˆ\nRd`1\n`xKpfqpx,tq,gpx,tqyXˆX˚dxdt\ntˇˇˇˇˇ\n“p1\nqsup\ngˇˇˇˇˇˆ\nRd`1\n`xfpx,tq,K˚pgqpx,tqyXˆX˚dxdt\ntˇˇˇˇˇ\nďp1\nq}f}Tp\nqpXq}K˚pgq}Tp1\nq1pX˚q/lessorsimilarp1\nq}f}Tp\nqpXq,BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 17\nwhere the supremum is taken over all gPCcpRd`1\n`q bXsuch that }Aq1pgq}p1ď1. Consequently,\nwe observe that Kextends to a bounded linear operator on Tp\nqpRd`1\n`;Xqfor 1 ďpă 8. More\nprecisely,\n}Kpfq}Tp\nqpXq/lessorsimilarε,κp1\nq}f}Tp\nqpXq,@fPTp\nqpRd`1\n`;Xq.\nNow we prove the second assertion of this lemma. Fix fPCcpRd`1\n`q bX, take a ball BinRd,\nwe can write\nˆˆ\npB}Kpfqpx,tq}q\nXdxdt\nt˙1\nq\n“sup\ngˇˇˇˇˆ\npBxKpfqpx,tq,gpx,tqyXˆX˚dxdt\ntˇˇˇˇ\n“sup\ngˇˇˇˇˆ\npBxfpx,tq,K˚pgqpx,tqyXˆX˚dxdt\ntˇˇˇˇ\nďsup\ng}K˚pgq}Lq1ppB;X˚qˆˆ\npB}fpx,tq}q\nXdxdt\nt˙1\nq\n,\nwhere the supremum is taken over all gin the unit ball of Lq1ppB;X˚q. From (3.8) we know that\n}K˚pgq}Lq1pRd`1\n`;X˚q/lessorsimilarε,κ}g}Lq1ppB;X˚q.\nThus for any xPRd,\npCqrKpfqs pxqqq“sup\nxPB1\n|B|ˆ\npB}Kpfqpx,tq}q\nXdxdt\nt\n/lessorsimilarε,κsup\nxPB1\n|B|ˆ\npB}fpx,tq}q\nXdxdt\nt“ pCqpfqpxqqq.\nTherefore we obtain\n}CqrKpfqs }p/lessorsimilarε,κ}Cqpfq }p,1ďpď 8.\nMoreover, from ( 2.5) we also observe\n}CqrKpfqs }p/lessorsimilarε,κˆp\np´q˙1\nq\n}Aqpfq}p, q ăpď 8.\nThe proof is completed. /square\nNow we come to the second important linear operator, which will relat e the tent space\nTp\nqpRd`1\n`;Xqto the Hardy space Hp\nq,LpRd;Xq.\nRecallthe operator Qpfqpx,tq “ ´tLe´tLpfqpxq. Define the operator πLacting onCcpRd`1\n`qb\nXas\nπLpfqpxq “ˆ8\n0Qpfp¨,tqqpx,tqdt\nt,@xPRd.\nIt is easy to verify that πLis well-defined. The following lemma asserts that πLextends to a\nbounded linear operator from Tp\nqpRd`1\n`;XqtoHp\nq,LpRd;Xq. We will denote it by πLas well.\nLemma 3.6. LetXbe any fixed Banach space and 1ăqă 8. The operator πLinitially defined\nonCcpRd`1\n`q bXextends to a bounded linear operator from Tp\nqpRd`1\n`;XqtoHp\nq,LpRd;Xqfor\n1ďpă 8. More precisely,\n}πLpfq}Hp\nq,LpXq/lessorsimilarβp1\nq}f}Tp\nqpXq,@fPTp\nqpRd`1\n`;Xq,1ďpă 8.\nFurthermore, for any fPCcpRd`1\n`q bX, we have\n}πLpfq}BMOp\nq,LpXq/lessorsimilarβ}Cqpfq}p,1ďpď 8.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 18\nProof.LetfPCcpRd`1\n`q bX. Recall that kpt,x,y qis the kernel of the operator Q, then\nQrπLpfqspx,tq “ˆ\nRdkpt,y,z qπLpfqpzqdz\n“ˆ\nRdkpt,y,z q˜ˆ\nRd`1\n`kps,z,w qfpw,sqdwds\ns¸\ndz\n“ˆ\nRd`1\n`ˆˆ\nRdkpt,y,z qkps,z,w qdz˙\nfpw,sqdwds\ns.(3.13)\nWe denote by\nΦt,spy,wq “ˆ\nRdkpt,y,z qkps,z,w qdz.\nNote that kpt,¨,¨qis the kernel of the operator Q“ ´te´tL, thus Φ t,sp¨,¨qis the kernel of\n´tLe´tL˝ p´sLe´sLq “tsL2e´pt`sqL. On the other hand, B2\nrpe´rLq|r“t`s“L2e´pt`sqLwhich\nhas the kernel B2\nrKpr,¨,¨q|r“t`s. Then by ( 2.4), we obtain\n|Φt,spy,wq|/lessorsimilard,βts\npt`sq2´βpt`s` |y´w|qd`β/lessorsimilarβmin/visualspaces\nt,t\ns(\nmin/visualspace1\nt,1\ns(d\n`\n1`min/visualspace1\nt,1\ns(\n|x´y|˘d`β.\nDenote by\nP“4Q˝πL. (3.14)\nFrom Lemma 3.5, we conclude that Pinitially defined on CcpRd`1\n`q bXextends to a bounded\nlinear operator on Tp\nqpRd`1\n`;Xq. Moreover,\n}Ppfq}Tp\nqpXq/lessorsimilarβp1\nq}f}Tp\nqpXq.\nTherefore\n}πLpfq}Hp\nq,L“4´1}Ppfq}Tp\nqpXq/lessorsimilarβp1\nq}f}Tp\nqpXq,\nwhich is the desired assertion.\nFor the second part, we obtain the desired assertion from Lemma 3.5immediately. /square\nRemark 3.7.One can verify that P˝P“P, thusPserves as a continuous projection from\nTp\nq`\nRd`1\n`;X˘\nonto itself. Indeed, we can also obtain this lemma under the assumpt ion thatLis\na sectotrial operator satisfying only ( 2.3).\n4.vector-valued intrinsic square functions\nIn this section, we begin with the introduction of vector-valued intr insic square functions,\noriginally presented by Wilson in [ 42] in the case of convolution operators. We then proceed to\ncompare them with the q-variant of Lusin area integral associated with a generator L.\nRecall that Lis assumed to be a sectorial operator of type α(0ďαăπ{2) satisfying\nassumptions ( 1.5), (1.6) and (1.7) withβą0,0ăγď1. Define Hγ,βas the family of functions\nϕ:RdˆRdÑRsuch that\n(4.1) |ϕpx,yq| ď1\np1` |x´y|qd`β,\n(4.2) |ϕpx`h,yq ´ϕpx,yq| ` |ϕpx,y`hq ´ϕpx,yq| ď|h|γ\np1` |x´y|qd`β`γBEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 19\nwhenever 2 |h| ď1` |x´y|and\n(4.3)ˆ\nRdϕpx,yqdx“ˆ\nRdϕpx,yqdy“0.\nForϕPHγ,β, defineϕtpx,yq “t´dϕpt´1x,t´1yq.\nLetfPCcpRdq bX. We define\nAγ,βpfqpx,tq “sup\nϕPHγ,β››››ˆ\nRdϕtpx,yqfpyqdy››››\nX,@ px,tq PRd`1\n`.\nThen the intrinsic square functions of fare defined as\nSq,γ,βpfqpxq “˜ˆ\nΓpxqpAγ,βpfqpy,tqqqdydt\ntd`1¸1\nq\n,\nand\nGq,γ,βpfqpxq “ˆˆ8\n0pAγ,βpfqpx,tqqqdt\nt˙1\nq\n.\nTheorem 4.1. LetXbe any fixed Banach space, 1ăqă 8and1ďpă 8. LetLbe any fixed\nsectorial operator Lsatisfying (1.5),(1.6)and(1.7). For anyfPCcpRdq bX, we have\nSq,γ,βpfqpxq «γ,βGq,γ,β pfqpxq, (4.4)\nSq,Lpfqpxq/lessorsimilarSq,γ,βpfqpxq, Gq,Lpfqpxq/lessorsimilarGq,γ,β pfqpxq, (4.5)\nand\n}Sq,γ,βpfq}p/lessorsimilarγ,βp1\nq}Sq,Lpfq}p. (4.6)\nRemark 4.2.The following g-function version of ( 4.6) holds also\n}Gq,γ,βpfq}p/lessorsimilarγ,βp2\nq}Gq,Lpfq}p. (4.7)\nBut its proof is much more involved and depends in turn on Theorem 1.1that will be concluded\nin the next section.\nAs in the classical case [ 41], the assertions ( 4.4) and (4.5) can be deduced easily from the\nfollowing facts on Hγ,β.\nLemma 4.3. LetϕPHγ,β. The following properties hold:\n(i)iftě1, thent´d´γϕtPHγ,β;\n(ii)if|z| ď1,tě1, then p2tq´d´γ´β`\nϕpzq˘\ntPHγ,β, whereϕpzqpx,yq “ϕpx´z,yq.\nProof.The proofissimilarto the caseofWilson [ 41], while the presentsettingis non-convolutive,\nlet us give the sketch. The claim (i) is easy by definition. For the claim (ii) , notice that\n2´1p1` |x´y|q ď1` |px´zq ´y| ď2p1` |x´y|q.\nBy definition, we have\n|ϕpzqpx,yq| “ |ϕpx´z,yq| ď1\np1` |px´zq ´y|qd`βď2d`β\np1` |x´y|qd`β.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 20\nand\n|ϕpzqpx`h,yq ´ϕpzqpx,yq| “ |ϕpx´z`h,yq ´ϕpx´z,yq|\nď|h|γ\np1` |px´zq ´y|qd`β`γ\nď2d`β`γ|h|γ\np1` |x´y|qd`β`γ.\nThe same H¨ older continuity estimation holds for the variable y. Thus we obtain 2´d´β´γϕpzqP\nHγ,β. Then the claim (ii) follows from the claim (i). /square\nWith Lemma 4.3, the assertions ( 4.4) and (4.5) will follow easily. The most challenging part\nof Theorem 4.4lies in (4.6). In addition to the interpolation and duality theory on the (vector -\nvalued) tent space that have been built in Section 3, the following pointwise estimate is another\ntechnical part in the proof of estimate ( 4.6).\nRecall that kpt,x,y qis the kernel of the operator Q. LetθPHγ,β, define\nLθ\nt,spy,wq “ˆ\nRdθtpy,zqkps,z,w qdz.\nLemma 4.4. Letν“2´1mintγ,βuandζ“ pd`2´1βqpd`βq´1, then\nsup\nθPHγ,β|Lθ\nt,spy,wq|/lessorsimilarγ,βmin/visualspaces\nt,t\ns(p1´ζqνmin/visualspace1\nt,1\ns(d\n`\n1`min/visualspace1\nt,1\ns(\n|y´w|˘d`1\n2β.\nProof.To estimate the kernel Lθ\nt,spy,wq, we follow a similar argument presented in [ 20, Chapter\n8, 8.6.3].\nLetθPHγ,β, we have\n(4.8) |θtpy,zq| ďt´d\np1`t´1|y´z|qd`β,@y,zPRd, tą0.\nFor 2 |z´z1| ăt` |y´z|, we havet´1|z´z1| ă1`t´1|y´z|, then\n|θtpy,zq ´θtpy,z1q| ďt´d´γ|z´z1|γ\np1`t´1|y´z|qd`β`γďmin/visualspace\npt´1|z´z1|qγ,`\n1`t´1|y´z|˘γ(\ntdp1`t��1|y´z|qd`β`γ\n/lessorsimilarmin/visualspace\n1,pt´1|z´z1|qγ(\ntd.\nFor 2 |z´z1| ět` |y´z|, we havet´1|z´z1| ě1{2, then\n|θtpy,zq ´θtpy,z1q| ď |θtpy,zq| ` |θtpy,z1q| ď2t´d/lessorsimilarmin/visualspace\n1,pt´1|z´z1|qγ(\ntd.\nHence\n|θtpy,zq ´θtpy,z1q|/lessorsimilarmin/visualspace\n1,pt´1|z´z1|qγ(\ntd,@y,z,z1PRd, tą0.\nOn the other hand, Lemma 2.1asserts that there exists a positive constant Cksuch that\nC´1\nkpkps,¨,¨qqs´1PHγ,β(see also the Convention afterwards). Thus, one gets for all w,z,z1P\nRd, są0,\n|kps,z,w q|/lessorsimilarCks´d\np1`s´1|z´w|qd`β,|kps,z,w q ´kps,z1,wq|/lessorsimilarCkmin/visualspace\n1,ps´1|z´z1|qγ(\nsd.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 21\nNow we start to deal with the kernel Lθ\nt,spy,wq. By symmetry, it suffices to handle the case\nsďt. First we observe the following estimate,\nˆ\nRds´dmin/visualspace\n1,pt´1|u|qγ(\np1`s´1|u|qd`βdu“ˆ\n|u|ăts´dpt´1|u|qγ\np1`s´1|u|qd`βdu`ˆ\n|u|ąts´d\np1`s´1|u|qd`βdu\nďˆ\n|v|ăt{s´s\nt¯γ |v|γ\np1` |v|qd`βdv`ˆ\n|u|ątsβ|u|´d´βdu\n“:J1`J2\nTakingν“2´1mintγ,βu, and we have |v|γă pt{sqγ´ν|v|ν. Then we obtain\nJ1ď´s\nt¯νˆ\nRd|v|ν\np1` |v|qd`βdv/lessorsimilarγ,β´s\nt¯ν\n.\nForJ2, we have\nJ2/lessorsimilarˆ8\ntsβr´β´1dr/lessorsimilarβ´s\nt¯β\nď´s\nt¯ν\n.\nThus forsďt, by the vanishing property ( 4.3) ofkps,¨,wq, one gets\n|Lθ\nt,spy,wq| ˇˇˆ\nRdrθtpy,zq ´θtpy,wqskps,z,w qdzˇˇˇˇ\nďCkˆ\nRdmin/visualspace\n1,pt´1|z´w|qγ(\ntds´d\np1`s´1|z´w|qdz\n/lessorsimilarγ,βt´d´s\nt¯ν\nďmin\"1\nt,1\ns*d\nmin\"s\nt,t\ns*ν\n.\nOn the other hand,\n|Lθ\nt,spy,wq| ďˆ\nRd|θtpy,zq||kps,z,w q|dz/lessorsimilarβmin/visualspace1\nt,1\ns(d\n`\n1`min/visualspace1\nt,1\ns(\n|y´w|˘d`β.\nLetζ“ pd`2´1βqpd`βq´1, we then get\n|Lθ\nt,spy,wq| “ |Lθ\nt,spy,wq|1´ζ|Lθ\nt,spy,wq|ζ/lessorsimilarγ,βmin/visualspaces\nt,t\ns(p1´ζqνmin/visualspace1\nt,1\ns(d\n`\n1`min/visualspace1\nt,1\ns(\n|y´w|˘d`1\n2β.\nIt is clear that the estimation of Lθ\nt,spy,wqis independent of the choice of θ, and thus the desired\nestimate is obtained. /square\nNow let us prove Theorem 4.1.\nProof.The pointwise estimate ( 4.4) follows from Lemma 4.3(ii). Indeed, for |x´y| ăt, let\nw“ px´yq{t; then for any ϕPHγ,β, we have 2´d´β´γϕpwqPHγ,β. Hence\nAγ,βpfqpx,tq “sup\nϕPHγ,β››››ˆ\nRdϕtpx,zqfpzqdz››››\nX\nď2d`β`γsup\nϕpwqPHγ,β››››ˆ\nRd´\nϕpwq¯\ntpx,zqfpzqdz››››\nX\n“2d`β`γAγ,βpfqpy,tq.\nExchanging xandyand taking ´win place ofw, the reverse inequality is also true. Then ( 4.4)\nfollows immediately.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 22\nNow we turn to the pointwise estimates ( 4.5). Lemma 2.1asserts that there exists a positive\nconstantCksuch thatC´1\nkpkpt,¨,¨qqt´1PHγ,β(see also the Convention afterwards). Conse-\nquently, for all xPRd, tą0, we have\n(4.9)}Qpfqpx,tq}X“››››ˆ\nRdkpt,x,y qfpyqdy››››\nX“Ck››››ˆ\nRd`\nC´1\nkpkpt,x,y qqt´1˘\ntfpyqdy››››\nX\nďCksup\nϕPHγ,β››››ˆ\nRdϕtpx,yqfpyqdy››››\nX“CkAγ,βpfqpx,tq.\nThen the estimates ( 4.5) follows trivially.\nBelow we explain the proof of ( 4.6). LethPCcpRd`1\n`q bX, we have\nAγ,βrπLphqspy,tq “sup\nθPHγ,β›››››ˆ\nRdθtpy,zq˜ˆ\nRd`1\n`kps,z,w qhpw,sqdwds\ns¸\ndz›››››\nX\n“sup\nθPHγ,β›››››ˆ\nRd`1\n`ˆˆ\nRdθtpy,zqkps,z,w qdz˙\nhpw,sqdwds\ns›››››\nX\n“sup\nθPHγ,β›››››ˆ\nRd`1\n`Lθ\nt,spy,wqhpw,sqdwds\ns›››››\nX\nďˆ\nRd`1\n`˜\nsup\nθPHγ,β|Lθ\nt,spy,wq|¸\n}hpw,sq}Xdwds\ns\n“:Lp}h}Xqpy,tq,\nwhere the linear operator Lhas the kernel\nLt,spy,wq “sup\nθPHγ,β|Lθ\nt,spy,wq|.\nThen by Lemma 4.4and Lemma 3.5in the case X“C, one obtaines\n}Lp}h}Xq}Tp\nqpCq/lessorsimilarγ,βp1\nq}}h}X}Tp\nqpCq“p1\nq}h}Tp\nqpXq,1ďpă 8.\nTherefore\n}Aγ,βrπLphqs}Tp\nqpCq/lessorsimilarγ,βp1\nq}h}Tp\nqpXq,1ďpă 8.\nLetfPCcpRdq bX, then we have Qpfq PTp\nqpRd`1\n`;Xq; moreover from the formula ( 2.1) and\nthe fact that the fixed point subspace of LppRd;Xqis 0 (see the statement before Remark 2.3),\nthe following Calder´ on identity holds\n(4.10) f“4ˆ8\n0QrQpfqp¨,tqs p¨,tqdt\nt.\nTherefore, one has that for 1 ďpă 8,\n}Sq,γ,βpfq}p“ }Aγ,βpfq}Tp\nqpCq“4}Aγ,βrπLpQpfqqs }Tp\nqpCq\n/lessorsimilarγ,βp1\nq}Qpfq}Tp\nqpXq“p1\nq}Sq,Lpfq}p,\nwhich is the desired inequality. /square\nRemark 4.5.For anyfPCcpRdq bX, by Lemma 3.5, we also obtain\n}Aγ,βpfq}T8qpCq“4}Aγ,βrπLpQpfqqs }T8qpCq/lessorsimilarγ,β}Qpfq}T8qpXq“ }f}BMO q,LpXq.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 23\nTogether with the pointwise estimate ( 4.9), one gets the BMO–version of Theorem 1.1: LetL\nbe a generator as in Theorem 1.1, then\n(4.11) }f}BMO q,LpXq«γ,β}f}BMOq,?\n∆pXq.\n5.Proof of the main Theorem\nAs pointed out in the introduction, the equivalence ( 1.8) in Theorem 1.1is an easy conse-\nquence of Theorem 4.1; but for another equivalence ( 1.9), we need to develop fully Mei’s duality\narguments between vector-valued Hardy and BMO type spaces [ 31]. This will be accomplished\nin the present section by combining the theory of vector-valued te nt spaces and vector-valued\nWilson’s square functions—Theorem 4.1.\nFirst of all, based on the duality between tent spaces—Lemma 3.4, the boundedness of the\nprojectionπL—Lemma 3.6—yields the following vector-valued Fefferman-Stein duality theorem .\nTheorem 5.1. LetXbe any fixed Banach space and 1ăqă 8. LetLbe as in Theorem\n1.1. Both the spaces BMOp1\nq1,L˚pRd;X˚qandHp1\nq1,L˚pRd;X˚qare isomorphically identified as\nsubspaces of the dual space of Hp\nq,LpRd;Xq. Moreover, they are norming for Hp\nq,LpRd;Xqin the\nfollowing sense,\n}f}Hp\nq,L/lessorsimilarβmax!\np1\nqp11\nq1,p12\nq1)\nsup\ngˇˇˇˇˆ\nRdxfpxq,gpxqyXˆX˚dxˇˇˇˇ,1ăpă 8,\nwhere the supremum is taken over all gPCcpRdqbX˚such that }g}Hp1\nq1,L˚pX˚qď1, and similarly,\n}f}Hp\nq,L/lessorsimilarβˆppq´1q\nq´p˙1\nq1\nsup\ngˇˇˇˇˆ\nRdxfpxq,gpxqyXˆX˚dxˇˇˇˇ,1ďpăq,\nwhere the supremum is taken over all gPCcpRdq bX˚such that }g}BMOp1\nq1,L˚pX˚qď1. Further-\nmore, ifX˚has the Radon-Nikod´ ym property. Then\nBMOp1\nq1,L˚pX˚q “´\nHp\nq,LpRd;Xq¯˚\n,1ďpăq;\nHp1\nq1,L˚pX˚q “´\nHp\nq,LpRd;Xq¯˚\n,1ăpă 8.\nRemark 5.2.Indeed, we can also obtain this duality theorem under the assumptio n thatLbe a\nsectotrial operator satisfying only ( 2.3), see Remark 3.7.\nThe more essential auxiliary result is the following duality property, w hich is inspired by [ 31,\nTheorem 2.4] (see also [ 43,47]).\nProposition 5.3. LetXbe any fixed Banach space and 1ďpăq. LetLbe any fixed sectorial\noperator satisfying (1.5),(1.6)and(1.7). Then for any fPCcpRdq bXandgPCcpRdq bX˚,\none has\n(5.1)ˇˇˇˇˆ\nRdxfpxq,gpxqyXˆX˚dxˇˇˇˇ/lessorsimilarγ,β}Gq,Lpfq}p\nq\np}Sq,Lpfq}1´p\nq\np}g}BMOp1\nq1,L˚pX˚q.\nProof.FixingfPCcpRdqbXandgPCcpRdqbX˚, we consider truncated versionsof Gq,Lpfqpxq\nas follows:\nGpx,tq:“ˆˆ8\nt}Qpfqpx,sq}q\nXdxds\ns˙1\nq\n, x PRd,tą0.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 24\nBy approximation, we can assume that Gpx,tqis strictly positive. The operator ´tL˚e´tL˚is\ndenoted by Q˚. By the Calder´ on identity—( 4.10), we have\nˇˇˇˇˆ\nRdxfpxq,gpxqyXˆX˚dxˇˇˇˇ“4ˇˇˇˇˇˆ\nRd`1\n`xQpfqpx,tq,Q˚pgqpx,tqyXˆX˚dxdt\ntˇˇˇˇˇ\n“4ˇˇˇˇˇˆ\nRd`1\n`A\nGp´q\nqpx,tqQpfqpx,tq,Gq´p\nqpx,tqQ˚pgqpx,tqE\nXˆX˚dxdt\ntˇˇˇˇˇ\n/lessorsimilar˜ˆ\nRd`1\n`Gp´qpx,tq}Qpfqpx,tq}q\nXdxdt\nt¸1\nq\n¨˜ˆ\nRd`1\n`Gq´p\nq´1px,tq}Q˚pgqpx,tq}q1\nX˚dxdt\nt¸1\nq1\n“I¨II.\nThe termIis estimated as below,\nIq“ ´ˆ\nRdˆ8\n0Gp´qpx,tqBtpGqpx,tqqdtdx\n“ ´qˆ\nRdˆ8\n0Gp´1px,tqBtGpx,tqdtdx\nď ´qˆ\nRdˆ8\n0Gp´1px,0qBtGpx,tqdtdx\n“qˆ\nRdGppx,0qdtdx“q}Gq,Lpfq}p\np,\nsinceGpx,tqis decreasing in t, andGpx,0q “Gq,Lpfqpxq.\nFor the term II, we introduce two more variants of Sq,γ,βpfq(cf. [47]). The first is defined\nsimilarly to Gp¨,tq:\nSpx,tq “˜ˆ8\ntˆ\n|y´x|ăs´t\n2pAγ,βpfqpy,sqqqdyds\nsd`1¸1\nq\n, x PRd, tą0.\nTo introduce the second one, let Dkbe the family of dyadic cubes in Rdof side length 2´k, that\nis,\nDk“#\n2´kdź\nj“1rmj,mj`1q:mjPZ,kPZ+\n.\nDenotecQas the center of a cube Q. Then, we define\nSpx,kq “˜ˆ8\n?\nd2´kˆ\n|y´cQ|ăspAγ,βpfqpy,sqqqdyds\nsd`1¸1\nq\n,ifxPQPDk, kPZ.\nBy definition, we have the following properties,\n(i)Sp¨,kqis increasing in k,\n(ii)Sp¨,kqis constant on every cube QPDk,\n(iii)Spx,´8q “0 andSpx,8q “Spx,0q “Sq,γ,βpfqpxq.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 25\nIfsětě?\nd2´kandxPQPDk, thenBpx,s´t\n2q ĂBpcQ,sq, whereBpx,tqdenotes the ball\nwith center xand radius t. This implies\nSpx,tq ďSpx,kq, x PQPDkwhenevertě?\nd2´k.\nUsing (4.4) and (4.5) we have\nGq,Lpfqpxq/lessorsimilarGq,γ,βpfqpxq «γ,βSq,γ,βpfqpxq,\nand similarly,\n(5.2) Gpx,tq/lessorsimilarγ,βSpx,tq.\nNow we proceed to estimate the term Bbased on these observations. Applying ( 5.2) toII, we\nhave\nIIq1/lessorsimilarγ,βˆ\nRd`1\n`Sq´p\nq´1px,tq}Q˚pgqpx,tq}q1\nX˚dxdt\nt\n“8ÿ\nk“´8ÿ\nQPDkˆ\nQˆ?\nd2´k`1\n?\nd2´kSq´p\nq´1px,tq}Q˚pgqpx,tq}q1\nX˚dt\ntdx\nď8ÿ\nk“´8ÿ\nQPDkˆ\nQˆ?\nd2´k`1\n?\nd2´kSq´p\nq´1px,kq}Q˚pgqpx,tq}q1\nX˚dt\ntdx\n“ˆ\nRd8ÿ\nk“´8kÿ\nj“´8Dpx,jqˆ?\nd2´k`1\n?\nd2´k}Q˚pgqpx,tq}q1\nX˚dt\ntdx\nwhereDpx,jq “Sq´p\nq´1px,jq ´Sq´p\nq´1px,j´1q. ThenDpx,jqis constant on every cube QPDj.\nThus\nIIq1/lessorsimilarγ,βˆ\nRd8ÿ\nj“´8Dpx,jq˜8ÿ\nk“jˆ?\nd2´k`1\n?\nd2´k}Q˚pgqpx,tq}q1\nX˚dt\nt¸\ndx\n“8ÿ\nj“´8ÿ\nQPDjˆ\nQDpx,jqˆ?\nd2´j`1\n0}Q˚pgqpx,tq}q1\nX˚dt\ntdx\n“8ÿ\nj“´8ÿ\nQPDjDpx,jq /BDQpxqˆ\nQˆ2?\ndℓpQq\n0}Q˚pgqpx,tq}q1\nX˚dt\ntdx,\nwhereℓpQqdenotesthe lengthof Q. Thereexistsaball BsuchthatQĂB,Qˆp0,2?\ndℓpQqs ĂpB\nand |B|/lessorsimilar|Q|. Then we deduce that\nˆ\nQˆ2?\ndℓpQq\n0}Q˚pgqpx,tq}q1\nX˚dt\ntdxďinf\nyPBtCq1rQ˚pgqs pyquq1\n|B|/lessorsimilarinf\nyPQtCq1rQ˚pgqs pyquq1\n|Q|.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 26\nTherefore\nIIq1/lessorsimilarγ,β8ÿ\nj“´8ÿ\nQPDjDpx,jq /BDQpxqinf\nyPQtCq1rQ˚pgqs pyquq1\n|Q|\nď8ÿ\nj“´8ÿ\nQPDjˆ\nQDpx,jq pCq1rQ˚pgqs pxqqq1\ndxďˆ\nRd8ÿ\nj“´8Dpx,jq pCq1rQ˚pgqs pxqqq1\ndx\n“ˆ\nRdSq´p\nq´1px,8q pCq1rQ˚pgqs pxqqq1\ndx“ }Sq´p\nq´1\nq,γ,βpfq}r›››pCq1rQ˚pgqsqq1›››\nr1\n“ }Sq,γ,βpfq}q´p\nq´1p}Cq1rQ˚pgqs }q1\np1,\nwhere 1 {r“1´q1{p1“ pq´pq{pqp´pq.\nCombining the estimates of IandIIwith Theorem 4.1, we get the desired assertion. /square\nFinally, we arrive at the proof of our main theorem.\nProof of Theorm 1.1.The first part ( 1.8) of Theorem 1.1is a consequence of Theorem 4.1. In-\ndeed, suppose Lbe a generator such that the kernels of the generating semigroup satisfy (1.5) ,\n(1.6) and (1.7) with 0 ăβ,γ ď1, then the classical Poisson semigroup generated by?\n∆ satisfy\nobviously the same assumptions. Then\n}Sq,Lpfq}p/lessorsimilar}Sq,γ,βpfq}p/lessorsimilarγ,βp1\nq}Sq,?\n∆pfq}p,1ďpă 8.\nSimilarly we obtain\n}Sq,?\n∆pfq}p/lessorsimilarγ,βp1\nq}Sq,Lpfq}p,1ďpă 8.\nAs for another part ( 1.9), one side is easy by Theorem 4.1,\n}Gq,Lpfq}p/lessorsimilar}Gq,γ,βpfq}p«γ,β}Sq,γ,βpfq}p/lessorsimilarγ,βp1\nq}Sq,Lpfq}p,1ďpă 8.\nFor the reverse direction, by Theorem 5.1and Proposition 5.3, we have for 1 ďpă p1`qq{2,\n}f}Hp\nq,LpXq/lessorsimilarˆppq´1q\nq´p˙1\nq1\nsup\ngˇˇˇˇˆ\nRdxfpxq,gpxqyXˆX˚dxˇˇˇˇ\n/lessorsimilarγ,βsup\ng}Gq,Lpfq}p\nqp}Sq,Lpfq}1´p\nqp}g}BMOp1\nq1,L˚pX˚q\n/lessorsimilarγ,β}Gq,Lpfq}p\nqp}Sq,Lpfq}1´p\nqp,\nwhere the supremum is taken over all gPCcpRdq bX˚such that its BMOp1\nq1,L˚pX˚q-norm is not\nmore than 1. Hence\n}Sq,Lpfq}p/lessorsimilarγ,β}Gq,Lpfq}p,1ďpă1`q\n2.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 27\nNow we deal with the case p1`qq{2ďpă 8. LetfPCcpRdq bX, we deduce from Theorem\n5.1that\n}f}Hp\nq,LpXq/lessorsimilarmax!\np11\nq1p1\nq,p12\nq1)\nsup\nhˇˇˇˇˆ\nRdxfpxq,hpxqyXˆX˚dxˇˇˇˇ\n/lessorsimilarp1\nqsup\nhˇˇˇˇˇˆ\nRd`1\n`xQpfqpx,tq,Q˚phqpx,tqyXˆX˚dxdt\ntˇˇˇˇˇ\n/lessorsimilarp1\nqsup\nh}Gq,Lpfq}p}Gq1,L˚phq}p1\n/lessorsimilarγ,βp1\nqp11\nq1sup\nh}Gq,Lpfq}p}Sq1,L˚phq}p1\n/lessorsimilarp1\nq}Gq,Lpfq}p,\nwhere the supremum is taken over all hPCcpRdq bX˚such that its Hp1\nq1,L˚pX˚q-norm is not\nmore than 1.\nCombining the estimations above we conclude that\np´1\nq}Sq,Lpfq}p/lessorsimilarγ,β}Gq,Lpfq}p/lessorsimilarγ,βp1\nq}Sq,Lpfq}p,1ďpă 8.\nWe complete the proof. /square\n6.Applications\nIn this section, we first recall the previous related results in [ 29,33]. These, together with the\ntent space theory and Theorem 1.1, will enable us to obtain the optimal Lusin type constants\nand the characterization of martingale type. In particular, this re solves partially Problem 1.8,\nProblem A.1 and Conjecture A.4 in the recent paper of Xu [ 48].\nSome notions and notations need to be presented. We first introdu ce the vector-valued atomic\nHardy space H1\natpRd;Xq. A measurable function aPL8pRd;Xqis called an X-valued atom if\nsuppaĂB,ˆ\nRdapxqdx“0,}a}L8pXqď |B|´1,\nwhereBis a ball in Rd. The atomic Hardy space H1\natpRd;Xqis defined as the function space\nconsisting of all functions fwhich admits an expression of the form\nf“8ÿ\nj“1λjaj,8ÿ\nj“1|λj| ă 8,\nwhereajis anX-valued atom. The norm of H1\natpRd;Xqis defined as\n}f}H1\natpXq“inf#8ÿ\nj“1|λj|:fpxq “8ÿ\nj“1λjajpxq+\n.\nThis is a Banach space.\nThe BMO space BMO pRd;Xqis defined as the space of all fPL1\nlocpRd;Xqequipped with\nthe semi-norm\n}f}BMO pXq“sup\nB1\n|B|ˆ\nB}f´fB}Xdxă 8,\nwhere the supremum runs over all the balls in RdandfBrepresents the average of foverB.\nBMO pRd;Xqis a Banach space modulo constants.\nIt is well-known that BMO pRd;X˚qis isomorphically identified as a subspace of the dual\nspace ofH1\natpRd;Xq(cf. [10]) and it is norming in the following sense\n}f}H1\natpXq«sup/visualspace\n| xf,gy |:gPBMO pRd;X˚q,}g}BMO pXqď1(BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 28\nwithuniversalconstants. Furthermore,iftheBanachspace X˚hastheRadon-Nikod´ ymproperty,\nthen (cf. [ 8])\npH1\natpRd;Xqq˚“BMO pRd;X˚q, (6.1)\nwith equivalent norms.\nWe recall the following definitions on the geometric properties for Ba nach spaces. A Banach\nspaceXis said to be of martingale type q(with 1 ăqď2) if there exists a positive constant c\nsuch that every finite X-valuedLq-martingale pfnqně0, the following inequality holds\nsup\nně0E}fn}q\nXďcqÿ\nně1E}fn´fn´1}q\nX,\nwhereEdenotestheunderlyingexpectation; andthe leastconstant ciscalledthe martingaletype\nconstant, denoted as Mt,qpXq. WhileXis said to be of martingale cotype q(with 2 ďqă 8)\nif the reverse inequalities holds with c´1in place of cand the corresponding martingale cotype\nconstantisdenotedby Mc,qpXq. Pisier’sfamousrenormingtheoremshowsthat Xisofmartingale\ncotype (respectively, type) qif and only if Xadmits an equivalent q-uniform convex(respectively,\nsmooth) norm. We refer the reader to [ 34–36] for more details.\nLetXbe a Banach space and 1 ăqď2. The authors in [ 29] showed that the assertion that\nXis of martingale type qis equivalent to the one that for any 1 ăpă 8, there exists a constant\ncpsuch that for any fPCcpRdq bX,\n}f}LppXqďcp}Sq,?\n∆pfq}p. (6.2)\nLateron, in[ 33]theauthorsinvestigatedtherelationshipsbetween H1\natpRd;XqandH1\nq,?\n∆pRd;Xq\nas well as the ones between BMO pRd;XqandBMOq,?\n∆pRd;Xq, and provided insights into the\ngeometric properties of the underlying Banach space X.\nTheorem 6.1. LetXbe a Banach space and 1ăqď2. The followings are equivalent\n(i)Xis of martingale type q;\n(ii)there exists a positive constant c such that for any fPCcpRdq bX,\n}f}H1\natpXqďc}Sq,?\n∆pfq}1;\n(iii)there exists a positive constant c such that for any fPCcpRdq bX,\n}f}BMO pXqďc}f}BMOq,?\n∆pXq.\nMoreover, the constants in (ii)and(iii)are majored by Mt,qpXq.\nThefollowingtheoremfollowsfromtheinterpolationtheorybetween vector-valuedtentspaces—\nLemma3.3—and the boundedness ofthe projection πL—Lemma 3.6. See for instancethe general\ninterpolation theory of complemented subspaces (cf. [ 40, Section 1.17]), and we omit the details.\nTheorem 6.2. LetXbe any fixed Banach space, 1ăqă 8and1ďp1ăpăp2ă 8such\nthat1{p“ p1´θq{p1`θ{p2with0ďθď1. LetLbe as in Theorem 1.1. Then\nrHp1\nq,LpRd;Xq,Hp2\nq,LpRd;Xqsθ“Hp\nq,LpRd;Xq,\nwith equivalent norms, where r¨,¨sθis the complex interpolation space. More precisely, for fP\nCcpRdq bX, one has\n}f}Hp\nq,LpXq/lessorsimilar}f}rHp1\nq,LpRd;Xq,Hp2\nq,LpRd;Xqsθ/lessorsimilarp2\nq}f}Hp\nq,LpXq.\nNow we are at the position to give the applications.\nCorollary 6.3. LetXbe a Banach space and 1ăqď2. LetLbe as in Theorem 1.1. The\nfollowings are equivalentBEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 29\n(i)Xis of martingale type q;\n(ii)for anyfPCcpRdq bX,\n}f}H1\natpXq/lessorsimilarγ,βMt,qpXq}Gq,Lpfq}1;\n(iii)for any1ăpă 8andfPCcpRdq bX,\n}f}LppXq/lessorsimilarγ,βpMt,qpXq}Gq,Lpfq}p;\n(iv)for anyfPCcpRdq bX,\n}f}BMO pXq/lessorsimilarγ,βMt,qpXq}f}BMO q,LpXq.\nProof.(i)ô(ii). This follows immediately from Theorem 1.1and Theorem 6.1.\n(iii)ñ(i). This is deduced from Theorem 1.1and (6.2).\n(i)ñ(iii). In the case 1 ăpăq, by Theorem 1.1, it suffices to show that\n}f}LppXq/lessorsimilarMt,qpXq}Sq,?\n∆}p. (6.3)\nKeeping in mind ( 6.2) and Theorem 6.1(ii), we consider\nrH1\nq,?\n∆pRd;Xq,Hq\nq,?\n∆pRd;XqsθĂ rH1\natpRd;Xq,LqpRd;Xqsθ;\nthen combining Theorem 6.2with the interpolation between H1\natpRd;XqandLqpRd;Xq(cf. [9,\nTheorem A]), one gets for any fPCcpRdq bX, 1{p“1´θ`θ{q,\n}f}LppXq/lessorsimilar}f}rH1\natpRd;Xq,LqpRd;Xqsθ/lessorsimilarMt,qpXq}f}rH1\nq,?\n∆pRd;Xq,Hq\nq,?\n∆pRd;Xqsθ/lessorsimilarMt,qpXq}f}Hp\nq,?\n∆pXq.\nThis is the desired ( 6.3). Combining it with the related result for qďpă 8in [48], we conclude\n}f}LppXq/lessorsimilarγ,βpMt,qpXq}Gq,Lpfq}p,1ăpă 8.\n(i)ô(iv). This follows from Remark 4.5and Theorem 6.1(iii). /square\nRemark 6.4.(1). Taking L“?\n∆ in the assertion (iii), we get\nL?\n∆\nt,q,ppXq/lessorsimilarpMt,qpXq,1ăpă 8,\nwhere the order is optimal as ptends to 1. This solves partially [ 48, Problem 1.8].\n(2). The implication (iii) Ñ(i) says that a Banach space Xwhich is Lusin type qrelative\ntote´tLutą0implies the martingale type qfor a large class of generators L. This answers\npartially [ 48, Problem A.1 and Conjecture A.4].\n7.Appendix\nFrom the atomic decomposition of T1\nqpRd`1\n`;Xq, we derive the following molecular decompo-\nsition forH1\nq,LpRd;Xqfor any Banach space X, which might have further applications.\nTheorem 7.1. LetXbe any fixed Banach space and 1ăqă 8. For anyfPH1\nq,LpRd;Xq,\nthere exist a sequence of complex numbers tλjujě1and corresponding molecules αj“πLpajqwith\najpx,tqbeing an pX,qq-atom such that\nf“ÿ\njě1λjαj,}f}H1\nq,LpXq«ÿ\njě1|λj|.\nProof.LetfPH1\nq,LpRd;Xq. It follows that Qpfq PT1\nqpRd`1\n`;Xq. HenceQpfqadmits an atomic\ndecomposition by Lemma 3.1. More precisely, there exist a sequence of complex numbers tcjujě1\nand pX,qq-atomsajsuch that\nQpfq “8ÿ\nj“1cjaj,}f}H1\nq,LpXq“ }Qpfq}T1qpXq«8ÿ\nj“1|cj|.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 30\nThen by Lemma 3.6, it follows that πLpajq “αjPH1\nq,LpRd;Xqfor alljě1. Recall below the\nCalder´ on identity—( 4.10),\nfpxq “4ˆ8\n0QrQpfqp¨,tqspx,tqdt\nt.\nThis further deduce that\nfpxq “48ÿ\nj“1cjˆ8\n0Qrajp¨,tqspx,tqqdt\nt“48ÿ\nj“1cjαjpxq,\nand thus we obtain the desired molecular decomposition. /square\nAcknowledgments. The subject of this article was proposed by Professor Quanhua Xu when\nhe was completing his paper [ 48]. The authors are very grateful to his suggestions, encour-\nagement, support as well as his careful reading over the whole pap er. We also would like to\nthank Professor ´Eric Ricard for his comments and suggestions. G. Hong was support ed by Na-\ntional Natural Science Foundation of China (No. 12071355, No. 12 325105, No. 12031004) and\nFundamental Research Funds for the Central Universities (No. 2 042022kf1185).\nReferences\n[1] I. Abu-Falahah, P. Stinga, and J. Torrea. Square functio ns associated to Schr¨ odinger operators. Stud. Math. ,\n2(203):171–194, 2011.\n[2] J. J. Betancor, A. J. Castro, J. C. Fari˜ na, and L. Rodr´ ıg uez-Mesa. UMD Banach spaces and square functions\nassociated with heat semigroups for Schr¨ odinger, Hermite and Laguerre operators. Math. Nachr. , 289(4):410–\n435, 2016.\n[3] J. J. Betancor, A. J. Castro, and L. Rodr´ ıguez-Mesa. Cha racterization of uniformly convex and smooth\nBanach spaces by using Carleson measures in Bessel settings .J. Convex Anal. , 20:763–811, 2013.\n[4] J. J. Betancor, J. C. Fari˜ na, V. Galli, and S. M. Molina. 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Soc. , 376(01):371–389, 2023.BEST CONSTANTS IN THE VECTOR-VALUED LITTLEWOOD-PALEY-STE IN THEORY 32\nInstitute for Advanced Study in Mathematics, Harbin Institut e of Technology, Harbin 150001,\nChina\nEmail address :gxhong@hit.edu.cn\nLaboratoire de Math ´ematiques, Universit ´e de Bourgogne Franche-Comt ´e, 25030 Besanc ¸on Cedex,\nFrance\nEmail address :xu.zhendong@univ-fcomte.fr\nDepartment of Mathematics, University of Illinois Urbana-Ch ampaign, USA\nEmail address :hzhang06@illinois.edu"    },    {        "title": "2401.13941v2.AC_Driven_Series_Elastic_Electrohydraulic_Actuator_for_Stable_and_Smooth_Displacement_Output.pdf",        "content": " \nAC\n-\nDriven \nSeries\n \nElastic\n \nElectro\nhydraulic \nActuator\n \nfor Stable and Smooth Displacement\n \nOutput\n \n \nQ\nuan Xiong, \nStudent\n \nMember\n,\n \nIEEE\n,\n \nXuanyi Zhou, Dannuo Li and Raye Chen\n-\nHua Yeow, \nMember\n, \nIEEE\n \n \n\n \nAbstract\n—\nsoft \nelectrohydraulic actuators known as HASEL \nactuators have attracted \nwidespread \nresearch interest due to \ntheir \noutstanding dynamic performance and \nhigh output power\n.\n \nHowever, the displacement of electrohydraulic actuators \nusually\n \ndecline\ns\n \nwith time\n \nunder constant\n \nDC\n \nvoltage\n, which hampers its \nprospective application\n.\n \nA\n \nmathematical model \nis firstly \nestablished to not only \nexplain\n \nthe decrease \nin\n \ndisplacement \nunder DC voltage \nbut also predict the \nrelatively stable \ndisplacement \nwith \noscillation under AC square wave\n \nvoltage\n. \nT\nhe \nmathematical model is validated since the \nactual displacement\n \nconfirms\n \nthe trend \nobserved\n \nby \nour\n \nmodel\n.\n \nTo \nsmooth\n \nthe \ndisplacement oscillation\n \nintroduced by AC voltage\n, a seri\nal\n \nelastic \ncomponent is \nincorporated\n \nto form a SE\n-\nHASEL actuator. \nA \nfeedback control with \na \nproportion\n-\nintegration algorithm enables \nthe SE\n-\nHASEL actuator to eliminate the \nobstinate \ndisplacement \nhysteresis. \nOur\n \nresults reveal\ned\n \nthat\n, through our methodology,\n \nthe SE\n-\nHASEL \nactuator can \ngive\n \nstable and smooth \ndisplacement \nand \nis \ncapable of absorbing external impact disturbance\n \nsimultaneously.\n \nA\n \nrotary joint based on the SE\n-\nHASEL actuator \nis developed to\n \nreflect\n \nits \npossibility to generate a common rotary \nmotion \nfor wide\n \nrobotic application\ns\n.\n \nMore importantly, t\nhis \npaper also proposes a highly\n \naccurate needle biopsy robot \nthat \ncan be utilized\n \nin \nMRI\n-\nguide surgical procedures\n. \nOverall, we \nhave achieved AC\n-\ndriven series elastic \nelectrohydraulic actuators\n \nthat can exhibit\n \nstable and smooth displacement output.\n \n \nIndex Terms\n—\nSoft robotics\n, \nelectrohydraulic actuators\n,\n \nelectrostatic adhesion\n, \nsoft actuators\n,\n \nneedle biopsy robots\n.\n \n \nI.\n \nINTRODUCTION\n \nO\nFT\n \nactuators \ndemonstrated tremendous \npotential\n \nin \nextensive\n \nrobotic appli\ncations\n, such as grasping delicate \nobjects with\n \nlow risk of \ndamage \n[1]\n–\n[4]\n, interacting \nsafely with human\ns\n \nfor assistance and rehabilitation \n[5]\n–\n[7]\n, \nand maneuvering in highly confined environments\n \n[8]\n–\n[10]\n. \nCompared with conventional rigid actuators like \ngeared \nmotors, \nsoft actuators\n \npossess \na \ncompliant and deformable \nfeature that\n \nbrings about\n \nhigh safety\n \nand \nadaptability\n \nto soft \nrobotic systems\n \n[11]\n–\n[13]\n.\n \nCurrently, \nsoft \np\nneumatic actuators \nas the mainstream soft actuators have been \ndeveloped \nfor\n \n \nThis work was supported by \nNational Robotics Programme\n–\nRobotics\n \nEnabling Capabilities and Technologies (W202\n5d0243)\n \nand A*STAR \nIndustry Alignment Fund \n–\n \nPre\n-\nPositioning \n(A20H8A0241)\n.\n \n(C\norresponding \nauthor: \nRaye Chen\n-\nHua Yeow\n). \n \nQuan Xiong, Xuanyi Zhou\n,\n \nDannuo Li\n \na\nnd Raye Chen\n-\nHua Yeow are \nwith \nthe \nEvolution Innovation Lab\n,\n \nDepartment of Biomedical Engineering\n,\n \nNational University of Singapore, Singapore,\n \n117583 Singapore (e\n-\nmail: \ne0788090@u.nus.edu\n; \nzhou447837285@gmail.com\n;\n \nldannuo@gmail.com\n;\n \nrayeow@nus.edu.sg\n).\n \nRa\nye Chen\n-\nHua Yeow\n \nis\n \nalso \nwith \nComputer Science & Artificial \nIntelligence Laboratory, \nMassachusetts Institute of Technology, 02139, \nUnited States.\n \nvarious\n \nsoft\n \nrobots\n \nlike prosthetic robots, surgical robots, \nexoskeletons and locomotion robots \n[2], [14]\n–\n[17]\n.\n \nHowever, \nthe\nse require \nindispensable gas pressure or vacuum \nelectromagnetic motor\n-\nbased \npumps \noften\n \nlimit\ning\n \ntheir\n \nperformance and \nportability\n \n[18], [19]\n.\n \nThe e\nmerging \nflexible\n \nelectrostatic adhesion (EA) \ntechnolog\ny\n \nhas\n \nbeen introduced into soft robotics in these \nyears\n \n[20]\n–\n[23]\n. For example,\n \nsoft grippers made of \nmembranous EA\n \ncan promptly stick to objects with low power \nconsumption and \nlow\n \ncontacting\n \nstress to objects\n \nand unstick\n \nfrom\n \nobjects by just power\ning\n \noff \n[24]\n–\n[26]\n.\n \nThe\n \nflexible EA\n \ntechnologies\n \nwere\n \nalso integrated into haptic devices that can \ngenerate desired feedback force to human fingers by \nregulating applied voltage\ns\n \n[27]\n–\n[29]\n. \n \nRecently,\n \nt\nhe soft \nactuators based on \nflexible \nEA\n \nattracted \nS\n \nFig. \n1\n.\n \nThe\n \nsoft \nelectrohydraulic actuator.\n \n(a) \nThe principle \nof electrostatic adhesion. (b) The principle of \nelectrohydraulic actuators. (c) The leakage model of EA. \n(d) The \nactual\n \nelectrohydraulic actuators\n \nmade of \npolyimide films and dielectric oil\n.\n \n \nmuch attention\n \nfrom researchers since \nthey\n \npave\nd\n \na \nprospective\n \npath\n \nto\n \nportable\n \nand soft actuators\n \nwithout \ntethered connection to source\ns of pressure or vacuum fluid\n \n[18], [30]\n.\n \nDielectric elastomer actuator\ns\n \n(DEA) \nmake \nuse\n \nof\n \nEA force to squeeze the dielectric elastomer and \ngenerate \nmechanical \ndeform\nation\n. \nAlthough DEAs have\n \nwidespread\n \nsoft robotic applications\n \nlike micro flapping robots and \ncrawling robots\n \n[8], [31]\n–\n[33]\n, the main \ndisadvantage\n \nlies \nwithin\n \ntheir low tolerance to electric breakdown which can \nresult in \nsevere\n \ndamage. Moreover, their pre\n-\nstretched \ndielectric elastomer films also \nmake\n \nthe\nir \nfabrication\n \nprocess\n \ncomplicated\n.\n \nTo surmount those above challenges, \nelectrohydraulic actuators known as \nhydraulically amplified \nself\n-\nhealing electrostatic (HASEL) actuators were \npresented\n \n[18], [34], [35]\n.\n \nA dielectric liquid bladder is substituted for \nthe dielectric elastomer in HASEL actuators\n, and the\n \nEA \nbetween two electrodes\n \ndeforms the bladder\n \nand actuate\ns\n \nthe \nload.\n \nThey displayed excellent performance including high \npower/weight ratio, fast response\n,\n \nand low\n-\ncost manufacturing \nprocess.\n \nMo\nre\n \nimportant\nly\n, \nc\nonventional pneumatic actuators\n \nwith \npump\ns\n \nand electromagnetic motor\ns\n \nare inadequate in \nmagnetic\n-\nsensitive scenarios\n, such as surgeries or medical \nexaminations \nwithin\n \nmagnetic resonance imaging (MRI)\n \ndevices\n \n[36]\n–\n[38]\n. \nThe strong \nelectromagnetic\n \nfields will \naffect\n \nthe imaging quality\n, or even damage the MRI device\n. \nNevertheless, EA\n-\nbased actuators\n \nwithout permanent magnets\n \ncan work as normal in the MRI environment\n \n[39]\n, \nsince they \nonly \ngenerate negligible magnetic \ninterference\n \ndue to \ntheir\n \nextreme\n-\nlow current (\n\nA\n \nlevel\n).\n \nC\nonstant \nDC voltage\ns\n \nhave been\n \nwidely \napplied to \nelectrohydraulic\n \nactuator\ns \n[35], [40], [41]\n.\n \nIt is a common \nphenomenon that DC operating voltage\n \ncauses\n \ngradual\n \nEA \nforce \ndrop\ns\n \nwith time\n \n[42], [43]\n. \nThis phenomenon \nmore or \nless \noccurs in many EA devices\n \nwith a variety of dielectric \nmaterials (\nlike \nBiaxially\n-\noriented Polypropylene\n \n(BOPP)\n, \npolyethylene terephthalate\n \n(PET)\n, \npolyimide\n \n(PI)\n, \nMylar, \nPolyvinylidene Fluoride\n \n(PVDF)\n, Luxprint\n, et al.\n)\n, \nsuch as\n \nEA \nclutches\n \n[27], [44], [45]\n \nand \nelectrohydraulic\n \nactuators\n \n[40], \n[46]\n–\n[48]\n.\n \nDespite\n \nthe\n \npossib\nility of\n \nselect\ning\n \nappropriate \ndielectric materials with less EA force loss since t\nhe EA \ndecreasing rates\n \nunder DC voltage\n \nare different \nin\n \nmaterials\n, \nit \nmay sacrifice other significant\n \nattributes\n, like mechanical\n \nbehavior\n \nand\n \ntemperature \nvariations\n.\n \nI\nn this \nwork\n,\n \nwe \nproposed to use \nlow\n-\nfrequency \nAC square \nwave voltage to drive the electrohydraulic actuator\ns\n.\n \nUsing\n \nAC operating voltage \ncan\n \neliminate the EA force decrease\n \nand \nproduce stable displacement output. \nH\nowever\n,\n \nit\n \nalso led to \nperiodical \noscillation \nwhich is \nundesirable.\n \nWe then connected \nan elastic component in series with the electrohydraulic \nactuators to isolate this \noscillation\n, i.e., series elastic \nHASEL\n \n(SE\n-\nHASEL)\n \nactuator\ns\n.\n \nEven though the\n \nresultant\n \ndisplacement of SE\n-\nHASEL actuators is stable and smooth, \nobvious \nhysteresis\n \ndue to large\n \nmagnitude\ns\n \nof \napplied AC \nvoltage\n \nwas observed in our \nfollowing \nexperiments\n, which \ndims the prospects for model\n-\nbased\n \nopen\n-\nloop\n \ncontrol\n.\n \nThus, \nwe adopted\n \nto implement \nfeedback control \non\n \nour actuators.\n \nWe characterized the \noutput \ndisplacement decrease under DC \nvoltage \nusing\n \ndifferent electrohydraulic actuators and the \namplitude\ns\n \nof displacement \noscillation\n \nunder \ndifferent \nfrequencies of \nAC voltage\n \nand loads\n.\n \nThe \noscillation \nisolation \neffect w\nith our presented \nSE\n-\nHASEL actuators\n \nwas also \nverified by experiments. \nFurthermore,\n \nexperimental results\n \nindicated\n \nthat\n \nour actuators with feedback control can \novercome hysteresis\n \nand \nachieve \naccurate, stable and smooth \ndisplacement output\ns.\n \nWe \nalso \ndeveloped\n \na rotary \nrobotic joint \nand a \nprecise \nneedle biopsy robot \ndriven by SE\n-\nHASEL \nactuators\n, which could potentially be deployed in MRI\n-\nguided \nsurgical procedures\n.\n \nT\nhe \nmain \ncontributions of this article are as follow\ns\n.\n \nFirst, \nwe built a mathematic\nal\n \nmodel to explain the EA force decline \nand\n \npredict\n \nthe \noscillation\n \nresult\ning\n \nfrom\n \nAC square wave \nvoltage based on \na\n \nleakage model.\n \nSecond, we \nproposed \nthe \nSE\n-\nHASEL actuators\n \nwith AC\n \noperating voltage \ncapable of\n \noutputting \nstable and smooth displacement\n. Third\n, we \nreported \na n\novel\n \nfabrication process for electrohydraulic actuators made \nof polyimide\n \n(PI)\n \nmembranes\n \nwhich have \nhigh \nmechanical\n \nstrength and good temperature tolerance\n. Fourth,\n \na\n \nrobust \nfeedback controller\n \nis \nimplemented\n \nto \nimprove the control \naccuracy\n.\n \nLast, we are the first to deploy electrohydraulic \nactuators for medical biopsy.\n \nII.\n \nM\nODELING\n \nA typical EA device is composed of two electrodes and a \nsandwiched layer of dielectric material. Applying high voltage \nto the electrodes, positive charges are induced to the positive \nelectrode while negative charges accumulate to the negative \ncounterpart (see Fig. 1a). The Maxwell stress generated by \nCoulomb force of charges\n \nwith different polarity\n \n(Electrostatic \nadhesion)\n \nvertically squeeze\ns\n \nthe two electrodes.\n \nThe EA can \nbe calculated based on the parallel capacitor model\n \nas follow\ns\n,\n \n2\n0\n2\n( )\n1\n2\nr o\nAu t\nF\nd\n \n\n,                \n  \n       \n \n  \n(\n1\n)\n \nwhere \nu\no\n(t)\n \nis the \nactual \napplied voltage, \nε\nr\n \nis the\n \nrelative\n \npermittivity \nof the insulation layer, \nε\n0\n \nis the vacuum \npermittivity, \nA\n \nis the overlap area of the two electrodes, and \nd\n \nis the thickness of the dielectric layer. \n \nT\nhe electrohydraulic actuator\n \nreplaces the dielectric layer \nwith\n \na deformable dielectric bladder made of a dielectric \nmembranous shell and inner dielectric liquid. The \nflexible \nelectrodes \ncompress\n \nthe bladder\n \nby EA\n \nand cause its\n \ndeform\nation which \nlifts\n \nmechanical\n \nload\ns\n \n(see Fig. 1\n \nb\n \nand d\n)\n.\n \nA\n. \nElectrostatic \nA\ndhesion\n \nunder DC Voltage\n \nThe decline of EA under constant DC applied voltage \nwas \nindicated\n \nin many\n \npieces of research\n.\n \nA\nn\n \nelectric \nleakage \nmodel\n \n[49]\n \nwas built to explain the decline of \nE\nA in haptic \napplication\ns\n \n(see Fig. 1c)\n. \nWe also use\n \nthis model\n \nfor\n \nour \nelectrohydraulic actuators.\n \nA\n \nleakage resistance\n,\n \nR\n \nis \nconnected \nin parallel \nto the two electrodes (capacitor \nC\n2\n) of \nthe electrohydraulic actuator, and another capacitor \nC\n1\n \nin the \ncable of input voltage \nu\ni\n \n(see Fig. 1c)\n.\n \nThe actual applied \nvoltage to the electrohydraulic actuator is the \noutput voltage  \nu\no\n.\n \nTo build the mathematical model of the leakage model, we \nfirst derive \na \ndifferential equation by Kirchhoff's law,\n \n1 2\n( )\ni o o o\nd u u du u\nC C\ndt dt R\n\n \n.                       \n \n  \n(2)\n \nT\nhen we get the transfer function by Laplace \ntransformation\n,\n \n1\n1 2\n( )\n( )\n( ) ( ) 1\no\ni\nU s\nRC s\nT s\nU s R C C s\n \n \n.                   \n \n  \n(3)\n \nConsidering the constant DC input voltage\n \nas the unit step \nsignal 1(\nt\n)\n,\n \nwe can \ncalculate the input voltage\n \nU\ni\n \nand the \noutput voltage\n \nU\no\n \nin \nthe \nLaplace domain,\n \n1\ni\nU\ns\n\n,\n   \n                                 \n \n  \n(4)\n \n1\n1 2\n( )\n(C C ) 1\no i\nRC\nU T s U\nR s\n \n \n.\n                \n \n  \n(5)\n \nT\nhe actual applied voltage in \nthe \ntime domain is also \ncalculated by inverse Laplace transformation,\n \n( ) 1( )\nPt\no\nu t Ke t\n\n.                  \n        \n  \n(6)\n \n1\n1 2\nC\nK\nC C\n\n\n,                             \n \n  \n(7)\n \n1 2\n1\n( )\nP\nR C C\n\n\n\n,\n \n         \n \n                   \n(8)\n \nWhere \nK\n \nand \nP\n \nare constants. \nHence, the actual applied \nvoltage reduces \nto 0 V \nwith time \nunder\n \na constant DC voltage\n \nas shown in Fig. 2a\n.\n \n \nB. \nElectrostatic \nA\ndhesion\n \nunder \nAC S\nquare\n \nWave\n \nVoltage\n \nWe \nthen establish the mathematical model of the actual \noutput voltage under AC square wave \ninput \nvoltage\n \n(see Fig. \n2b)\n. \nWe\n \nassume th\ne magnitude of the AC voltage is 1\n \nV\n, and \nthe frequency \nis \nf\n. The input voltage described in \nthe \ntime \ndomain is\n \n1\n( ) 1( ) 2 ( 1) 1( )\nn\nk\ni\nk\nu t t t ka\n\n   \n\n,                 \n \n  \n(\n9\n)\n \n      [0, )    \nt t na t a n N\n    \n,                (\n10\n)\n \n1\n2\na\nf\n\n,                                   \n \n  \n(\n11\n)\n \nwhere \nt\n \nis the actual time, \na\n \nis the time interval when the input \nvoltage remains constant, and \nN\n \nrepresents the natural \nnumbers.\n \nThe input voltage\n \nU\ni\n \nand output voltage\n \nU\no\n \ncan be \ncalculated in Laplace domain,\n \n1\n1 2\n( 1)\nn\nk kas\ni\nk\nU e\ns s\n\n\n  \n\n,   \n                    \n(1\n2\n)\n \n1\n1\n1 2\n( ) (1 2 ( 1) )\n( ) 1\nn\nk kas\no i\nk\nRC\nU T s U e\nR C C s\n\n\n   \n \n\n. \n \n  \n(1\n3\n)\n \nWe can inversely transfer them to time domain functions,\n \n( )\n1\n( )\n( ) 2 ( 1)\n1\n       ( 2 2 ( 1) )\n1\nn\nPt k P t ka\no\nk\nPt Pt n P t na\nPa\nu t Ke K e\nKe Ke K e\ne\n\n\n\n  \n    \n\n\n.\n \n(\n14\n)\n \nIf time is infinite\n \n(\nn\n \nis infinite)\n, \naccording to (10) and (14) , \nthe \noutput voltage is\n \n1\n( ) ( 1) 2\n1\nn P t\no\nPa\nn\nu t Ke\ne\n\n\n \n\n.                   \n \n  \n(15)\n \nNoted that\n \nthe \nabsolute value of \noutput voltage \noscillate\ns\n \nperiodically with time\n \n(at twice the frequency of AC input \nvoltage)\n, but remains rough\nly\n \nstable with time\n,\n \naccording to \n(1\n5\n)\n, as shown in Fig.2b,\n \nwhere \nK\n1\n \nand \nK\n2\n \nare the upper and \nlower limitations respectively,\n \n1\n2\n1\nPa\nK\nK\ne\n\n\n,                \n  \n   \n  \n          \n \n  \n(16)\n \n2\n2\n1\nPa\nPa\nKe\nK\ne\n\n\n.              \n    \n               \n \n  \n(17)\n \nThis \nvoltage \noscillation\n \noccurs when the polarity of the input \nvoltage is reversed, and\n \nmeanwhile\n \nresults in\n \nthe \nsudden \nchange\n \nof \nEA\n \nforce\n \nbased on (1)\n.\n \nC. \nDisplacement of Electrohydraulic\n \nA\nctuators\n \nA mathematical model is \nalso \nrequired\n \nto describe the \nrelationship between the\n \nEA and \nthe displacement output of \nelectrohydraulic actuators.\n \nApplying a high voltage to the \nelectrodes, part electrodes \nzip by EA, and squeeze the \ndielectric liquid to the other side.\n \nMeanwhile, the movement \nof the liquid \nlifts the load \nM\n.\n \nThe section view of the activated \nelectrohydraulic actuator is demonstrated in Fig. 3.\n \nGenerally, \nthe stiffness of thin dielectric films and flexible electrodes \n(micro\n-\nmeter \nscale\n) are neglected and the liquid is \nincompressible. The volume of liquid is constant and the \ndielectric membranes on both sides of the bladder are circular \nso that we can derive,\n \n2\n0 0\n( ( 2) )\nHS H L V\n\n \n.                       (\n18\n)\n \nWhere \nH\n \nis the displacement output of \nthe \nelectrohydraulic \n \nFig. \n2\n.\n \nThe \ninput voltage and output voltage based on \nleakage model\n. \n(a)\n \nThe response under constant DC \nvoltage. (b) The response under AC square wave voltage.\n \n \nFig. \n3\n.\n \nThe \npart section view of the deformed bladder of \nelectrohydraulic actuator\n.\n \n \nactuator, \nS\n \nis the contacting length, \nL\n0\n \nand \nV\n0\n \nare constants, \nthat are the width of \nthe \nelectrohydraulic actuator and the \nvolume of liquid respectively. Given the equilibrium \nconditions, \nthe fluidic pressure is related to the load,\n \n0\ne\nL PS Mg\n\n,                             \n \n  \n(\n19\n)\n \nwhere \nP\ne\n \nis the pressure of the \ndielectric liquid\n \nand \nMg\n \nis the \nload\n.\n \nWe can calculate the displacement,\n \n2\n0\n0 0 0\n2 2\n( )\ne e\nV\nMg Mg\nH\nL P L L P\n\n \n  \n.       \n \n \n       \n(20)\n \nConsidering that\n \nthe electrodes \nare not\n \ncompletely\n \nzipped\n \nwith \nload\n, we can acquire another \nequilibrium \nrelationship \nbetween \nthe liquid pressure and EA stress, \nas follow\ns\n,\n \n2\n0\n2\n( )\n1\n2\nfilm o\ne\nfilm\nu t\nP\nd\n \n\n,\n                         \n \n  \n(\n2\n1)\n \nw\nhere\n \nε\nfilm\n \nis the \nrelative permittivity\n \nof the \ndielectric \nbladder \nshell and \nd\nfilm\n \nis the total thickness of the double\n-\nfold dielectric \nfilms\n \n(\nthe \nzipped bladder)\n. \nC\nombin\ned\n \nwith the oscillation of \nthe output voltage due to AC input voltage\n \nand th\nis\n \ndisplacement model, we can predict the output displacement \noscillation \nof \nthe \nelectrohydraulic actuator under AC square \nwave voltage.\n \nIII.\n \nF\nABRICATION AND \nS\nYSTEM \nA\nSSEMBLY\n \nA. \nF\nabrication of electrohydraulic actuators \n \nP\nolyimide\n \n(PI)\n \nmaterials \nwith\n \na high relative permittivity\n \n(3.4 \nwhich\n \nis approximately 50% higher than BOPP \nmaterials)\n, \nlow cost, good \nmechanical strength and \ntemperature tolerance, have been widely used in \nelectric\n \nindustries\n, like \nFPC (\nflexible printed circuits\n) based on PI \nmembranes\n.\n \nHowever, it is hardly used in electrohydraulic\n \nactuators\n, mainly because of its complicated\n \nand expensive\n \nadhesive \nbonding \ntechn\nique\n \nfor enclosed dielectric bladders\n.\n \nHere, we proposed a low\n-\ncost adhesive bonding method\n \nby \nusing\n \nPVC double\n-\nadhesive tapes\n \nto fabricate the \nelectrohydraulic actuators.\n \nFirst, \nwe \ncut the PVC tape \nsandwiched by two release papers to a box \nshape\n \n(6\n\n6 cm\n \nof \nthe inner square and \n7\n\n7 cm\n \nof the outer square)\n \nby a cutting \nmachine (\nSilhouette Cameo\n, \nSilhouette\n).\n \nSecond, we stick the \ncut \nPVC tape to a PI film\n \n(50 \n\nm thick)\n \nand \nstick\n \na \nnarrow\n \ntape\n \n(3 mm wide)\n \nin\n \nthe middle\n \nto divide \nit \ninto two chambers\n. \nThird, another PI film covers the other side but leave\ns\n \na \nnarrow channel\n \nby inserting a scrap of release paper\n \nfor liquid \nimport\n. Fourth, we use \nan\n \niron (200 ℃) to heat the PVC tape \narea except\n \nfor\n \nthe \ninlet\n \nchannel\n \n(see Fig. 4 \ni\n \nto \niv\n)\n \nto \nbond\n \ntwo \n \nFig. \n4\n.\n \nThe \nfabrication of electrohydraulic actuator\ns\n \nmade \nof polyimide films.\n \n \nFig. \n5\n.\n \nThe s\nystem composition\n.\n \n(a) The mechanical \ndesign and prototype of SE\n-\nHASEL actuator. (b) The \nfeedback control diagram. (c) The electronic block \ndiagram.\n \n \nPI films\n.\n \nFifth, \nwe put two copper tapes \nin\n \nthe middle of \nthe \nPI \nbladder as electrodes.\n \nSixth, 3.2 mL dielectric oil is imported \ninto the bladder\n \nby a peristaltic pump (\nYX\n-\nLP01\n-\n3\n, \nYan\nxiao\n)\n.\n \nLast, we \nseal\n \nthe import channel \nwith\n \nthe heating iron \nin\n \nthe \nfourth step\n \nand then tape\nd\n \nwires to electrodes\n \n(see Fig. 4 \nv\n \nto \nviii\n)\n.\n \nB\n. \nSE\n-\nHASEL\n \nSystem \n \nT\no smooth the output displacement of electrohydraulic \nactuators, \na spring is connected with a stack of \nelectrohydraulic actuators in series\n \n(SE\n-\nHASEL actuator)\n. \nWe \nused two vertical sliders to constrain the direction of \ndisplacement (see Fig. 5\na\n). \nA plate mounted on the lower \nslider is \nin contact\n \nwith electrohydraulic actuators, and the \nother plate mounted on the upper slider holds the load. \nThe \ndamping of sliders, the spring and the mass compose a typical \nsecond\n-\norder system, which acts as a \npassive \nlow\n-\npass filter to \nrestrain the \noscillation \npulse\n \nfrom\n \nelectrohydraulic actuators \n(\noscillation \nisolation).\n \nC\n. Feedback Control\n \nA Proportion\n-\nIntegration controller\n \nand \nthe \nfeedback control \nstrategy\n \n(see Fig. \n5b\n) \nare\n \napplied to overcome the hysteresis of \noutput displacement. The \nProportion\n-\nIntegration \nalgorithm \nreal\n-\ntime calculates the magnitude of the AC square wave \nvoltage,\n \ni p i\nu K e K edt\n \n\n.                      \n \n  \n(22)\n \nwhere \nK\np\n \nis the proportion parameter, \nK\ni\n \nis the integration \nparameter and \ne\n \nis the error value between the target and \nactual output displacement of \nthe \nSE\n-\nHASEL\n \nactuator\n \n(the \nupper plate in Fig. 5\na\n)\n. The frequency of the AC applied \nvoltage is pre\n-\nset. \nK\np\n \nand \nK\ni\n \nare manually pre\n-\nadjusted.\n \nD. Electric System\n \nT\nhe \nelectronic block diagram\n \nis indicated in Fig. \n5c\n.\n \nA laser \nsensor\n \n(\nBL\n-\n100NMZ\n, \nBOJKE\n)\n \nis mounted over the \nupper \nplate (see Fig. 5\na\n) to measure the actual output displacement \nof the SE\n-\nHASEL actuator. A \nd\nata acquisition device\n \n(USB6211, NI\n-\nDAQ\n)\n \nacquires this displacement signal\n \nfrom \nthe laser sensor\n \nand tran\nsfer\ns\n \nit to the LabVIEW\n \nplatform\n \n(LabVIEW 2018)\n \no\nn \na \nlocal desktop \ncomputer\n. Our \nProportion\n-\nIntegration \ncontrol algorithm\n \nembedded in \nthe \nLabVIEW\n \nplatform \ncalculates the \nneeded\n \nmagnitude of AC \nsquare wave voltage\n \nin real\n-\ntime\n. Then, the LabVIEW \nplatform \nmodulates this magnitude value to an AC square \nwave and \nreturn\ns\n \nthe \nmodulated AC\n \nvoltage signal to the DAQ\n \nwhich outputs the AC voltage to a voltage amplifier (Trek \n \nFig. \n6\n.\n \nThe d\nisplacement \nd\necrease under DC \nv\noltage\n. (a) \nThree different electrohydraulic actuators\n. (b) \nThe actual \noutput \ndisplacements of three electrohydraulic actuators made of three different materials under a 6 kV constant DC voltage.\n \nThe \nsnapshots of electrohydraulic actuators\n: (c)\n \nU\nnder constant DC voltage. (d) Under AC square wave voltage.\n \n \n10/40, Advanced Energy)\n. \nThe AC voltage is amplified by \n1000 times and then supplied to the SE\n-\nHASEL actuator.\n \nIV.\n \nE\nXPERIMENTS\n \nAND \nD\nISCUSSION\n \nA. \nDisplacement\n \nDecrease under DC Voltage\n \nWe experimentally explored the drop of output displacement \nof electrohydraulic actuators\n \n(stack\ning\n \n1\n1\n \nactuators)\n \nunder \nconstant \nDC voltage.\n \nFirst, we built a vertical linear slider \nsetup to \nvisually \nobserve the output displacement. A 20 g \nweight \nwa\ns fixed on the plate which \nwa\ns mounted on the \nsliding block (30 g).\n \nWhile\n \ns\nupplying a constant 6 kV DC \nvoltage to the actuators, the output displacement leap\ned\n \nup\n \nrapidly\n \nat the beginning and then decline\nd\n \ngradually (see Fig. \n6c\n \nand \nvideo \nS1).\n \n \nT\no quantitively characterize the\n \neffect \nof\n \nusing different \ndielectric\n \nmaterials on displacement,\n \nwe \nmanufactured\n \nthree \nelectrohydraulic actuators\n \n(3\n \nstacked electrohydraulic\n \nactuators)\n \nwith 25 \n\nm \nthick \nPI films, 25 \n\nm\n \nthick\n \npre\n-\ncoated \nBOPP films\n \n[50]\n \nand 50 \n\nm \nthick \nPI films\n \nwith the same size\n \nand the same volume of liquid\n \nas described in \nS\nection III\n.\nA\n \n(see Fig. \n6a\n)\n.\n \nWe \napplied a constant 6 kV voltage to the three \nactuators and \nused \na\n \nlaser sensor to measure the \nactual \noutput \ndisplacement.\n \nThe rate\n \nat which displacement decreased\n \nvarie\nd\n \ngreatly\n \nwith \nthe type of \ndielectric materials\n \nused \nas \nillustrated\n \nin Fig. \n6b\n. \nThe actuators made of 25 \n\nm PI \nfilm\ns \nremained relatively stable with time\n \nas \ntheir\n \ndisplacement \nreduced by \nonly \n6.5\n% in 80 seconds\n \nfrom applying the voltage\n.\n \nNevertheless, t\nhe \ndisplacement of \npre\n-\ncoated \nBOPP \nfilms\n \ndropped dramatically by 96.3% in 10 seconds.\n \nThe \ndisplacement of 50 \n\nm PI films showed an intermediate drop \n(71% in 10 seconds).\n \nThus, w\ne used \nthe 50 \n\nm thick PI film\n \nmaterial \nin our following experiments\n.\n \nOverall, t\nh\ne\n \ndisplacement drop can be explained by our mathematical \nmode\nl\n \n(6).\n \nB. \nOscillation\n \nof Displacement Output under AC \nV\noltage\n \nT\no keep the EA force stable, we adopted the AC square wave \napplied \nvoltage\n \nto our electrohydraulic actuators\n.\n \nIn contrast with \nthe displacement output under DC voltage, it \nalways \nmaintained\n \nan\n \nalmost \nconstant \nvalue\n \nwith visible oscillation\n \nunder \nan\n \nAC \nsquare wave voltage\n \nwith 6 kV magnitude\n \n(see Fig. \n6d\n \nand \nvideo \nS1)\n.\n \nThen we altered the frequency\n \nof the applied voltage\n \n(\nDC\n, \n0.2 Hz, 2 Hz\n \nand\n \n10 Hz\n)\n, and measured the actual displacements \nby the displacement sensor.\n \nThe experimental results are shown \nin Fig. \n7a\n.\n \nWith the AC square wave applied voltage, the \ndisplacement output \nwa\ns generally stable over time\n \nand \noscillating around the stable value\n \nat twice the frequency of AC \napplied voltage\n \n(clearly observed in the red curve in Fig. \n7a\n)\n, \nwhich was \nexactly \nrevealed\n \nby our mathematical model as \nshown in Fig. 2b\n, (20) and (21)\n.\n \nWith \na \nfrequency from 0.2 Hz to \n2 Hz, the stable value under AC applied voltage increased \nobviously, \nbut\n \nit \nkept\n \nnearly unchanged when the frequency was \nover 2 Hz.\n \n \nW\ne further investigated the displacement oscillation by testing \nthe displacement amplitude with different frequencies from 0.1 \nHz to 10 Hz and three \ndifferent loads (50 g, 80 g and 130 g).\n \nWe \ncollected the displacement data and manually calculate\nd\n \nthe \namplitude \nusing\n \nthe peak\n-\npeak value at the stable stage\n \n(60 s\n \nafter applying voltage\n)\n.\n \nThe results show\ned\n \nthat the amplitude\ns\n \ndecline\nd\n \nwith the frequency of the applied voltage (see Fig. \n7b\n)\n, \nwhich \nwas\n \nconsistent with the trend \npredicted\n \nby \nour \nmathematical model \nas shown in (\n1\n6) and (\n1\n7).\n \nBy increasing the \ntime \ninterval\n \na\n \n(reducing frequency)\n \nof AC voltage, the \ndifference between the upper limit \nK\n1\n \nand the lower limit \nK\n2\n \nrise\ns \nalong with\n \nthe amplitude\n.\n \nIn addition, the load also influenced \nthe amplitude. The amplitude of \nthe \n80 g \nwas \nobserved to be\n \nhigher than \nthe\n \n50 g and 130 g loads. \n \nNoted that,\n \nthe decline\n \nof amplitude with the frequency \nlevels \noff\n, as the frequency passes 1 Hz\n. Hence, we can apply an AC \nvoltage with a relative\nly\n \nlower frequency (2 Hz) to our SE\n-\nHASEL actuators. There are \nfour\n \nmain \nadvantages\n \nas follows. \nFirst,\n \na 2 \nH\nz AC voltage \nis\n \nsufficient to\n \nmaintain a stable \ndisplacement output with acceptable oscillation\n. Second, the \npower consumption augments with the frequency. Third, \na \nhigher frequency\n \nof voltage\n \nis more hazardous \nto\n \nthe \nhuman\n \nbody\n \n[51]\n.\n \nLast, \nthe higher frequency increases the \nhardware \nrequirement\ns\n \nand \nthe \ncost of electronic components for portable \n \nFig. \n7\n.\n \nDisplacement under \nAC\n \nVoltage\n. (a) \nThe actual \noutput displacements under the AC square wave voltages \nwith different \nfrequencies and constant DC voltage\n. (b) \nThe \noscillation\n \namplitude\n \nof the output displacement\n \nunder \nthe AC square wave voltage with different frequencies\n \nand \ndifferent loads\n.\n \n \ndevice development\n \n(e.g., the faster switching response of high \nvoltage relays and optocouplers)\n.\n \n \nC. \nDisplacement Output of \nthe \nSE\n-\nHASEL Actuator\n \nAlthough the \noscillation can be reduced to a \nrelatively \nlow \nlevel \nby increasing frequency, \nthe periodical vibration\n \nis\n \nalso \nundesirable\n,\n \nespecially in some delicate \nrobotic \noperations\n.\n \nTo \nachieve \nsmoother\n \ndisplacement output, we used a \nseries \nspring\n \n(14 N/m stiffness)\n \nto isolate the oscillation from the \nelectrohydraulic actuators\n \n(SE\n-\nHASEL actuator)\n.\n \nWe applied a 2 \nHz AC square voltage\n \nwith \n6 kV magnitude\n \nto the \nSE\n-\nHASEL \nactuator\n \nwith a 20 g weight on the upper plate\n \nand \nrecorded\n \nthe \nactual displacement. Then, we substituted the spring with a rigid \npole and re\n-\nconducted the testing. Furthermore, with a 100 g \nweight, we also \ntested the actual displacement with a spring and \na rigid pole\n.\n \nThe results indicated that SE\n-\nHASEL actuators can \noutput a smooth displacement without obvious impulse\n \n(see Fig. \n8a\n)\n.\n \nWe\n \nalso\n \nexperimentally characterized hysteresis\n \nbehavior by \nvarying\n \nthe magnitude of AC applied voltage\n \nfrom 0 to 8 kV\n. \nSE\n-\nHASEL actuators cannot eliminate the output \ndisplacement hysteresis\n \ncompletely\n. \nWe tested the output \ndisplacement with three loads (60 g, 160 g and 260 g), and \ncalculated the strain (dividing the output displacement by the \ntotal height of the stack of electrohydraulic actuators)\n \nas \nshown in Fig. \n8b\n. \nWith a 60 g load, the maximum \nhysterical\n \nstrain (MHS) \nis about 15% \nat 3 kV applied AC voltage.\n \nThe \nmaximum output strain (MOS) is about 50 %. The ratio of \nMHS to MOS is 30%.\n \nT\nhe hysteresis \nsubsides\n \nand the MOS \nalso declines with heavier load. However, the ratio of MHS to \nMOS \nwas \nroughly \nmaintained\n \nat \n30% with different loads.\n \nD. Displacement Output with Feedback Control \n \nW\ne applied the feedback control to our SE\n-\nHASEL actuator \n(11 \nstacked \nelectrohydraulic actuators)\n \nwith a \nProportion\n-\nIntegration\n \ncontroller\n.\n \nThe control cycle \nwas\n \nset to \n1 ms in \nthe \nLabVIEW platform, and the controller parameters \nK\np\n \nand \nK\ni\n \nwere \nmanually adjusted\n \nto 0.8 and 0.005.\n \nWe put a 50\n \ng \nweight on the SE\n-\nHASEL actuator, and then \nimplemented\n \nthe \nfeedback control to follow the target waves (\n0.05 Hz \nsquare \nwaves and sine waves). The displacement and applied voltage \nwere recorded in Fig. \n9a\n.\n \nFor the square waves\n \nwith a constant \noffset\n, \nthe displacement of \nour SE\n-\nHASEL actuator can \nstably \nand smoothly follow the target with less than \n10 \n% overshoot\n, \nand the root\n-\nmean\n-\nsquare error (RMSE) is 0.077 mm\n. At the \nfalling edges, the actual displacement usually lag\ns\n \nbehind the \ntarget, mainly because the electrohydraulic actuator can only \ngenerate the unidirectional force (upward).\n \nThe sinking of the \nload was only dominated by its gravities.\n \n \nOur SE\n-\nHASEL actuator with PI feedback control can also \nfollow the sine waves with slight fluctuation\n \nand \nstick\n-\nslip \nphenomenon \nwhich is probably due to the nonlinear friction\n \nduring movement\n. \nHowever\n, t\nhe following performance at low \ntarget values is \npoor\n \nresulting \nin\n \nthe total RMSE (0.211 mm)\n \nris\ning\n \nto \n3 times that of \nsquare waves\n. \nT\nhe \nmagnitude of \napplied \nAC \nvoltage \nwas \neffectively \nmodulated by the \nProportion\n-\nIntegration\n \ncontroller\n \nto correct the error between \nthe actual value and the target value\n \n(see Fig. \n9b\n)\n.\n \nT\nhen target waves with higher frequencies\n \n(0.1 Hz, 0.2 Hz \nand 0.5 Hz)\n \nwere also tested\n \nrespectively\n \nas shown in Fig. \n9 c \nand d\n.\n \nOverall, t\nhe\n \nactual displacement can follow the under \n0.2 Hz target waves without attenuation\n \nand phase lag\n. For the \ns\nqu\nare waves, \nthe \nmaximum \nactual value\n \novershooted the \ntarget value by 1\n3 %\n, and the actual displacement at the falling \nstage also lagged behind the target.\n \nThe occasional\nly large\n \novershoot at the falling edge is mainly because of the sudden \nrelease of \nthe \ngravitational potential energy of the lifted \nload.\n \nThe output displacement cannot reach the up and down \nplateaus of the 0.5 Hz square wave.\n \nFor the sine waves, less \novershoot was observed\n, but the control precision at low target \nvalues also needed to improve.\n \nWhen the frequency \nwas\n \nto 0.5 \nHz, the\n \nactual displacement\n \nattenuate\nd\n, but \nits\n \nphase \nanchored \nthe target waves\n.\n \nT\no demonstrate the vibration isolation of the SE\n-\nHASEL \nactuator, we \nobtained\n \nthe actual displacemen\nt\n \nof the upper and \nlower plates (Plate \nA\n \nand Plate B\n, see Fig 1\n0\n)\n \nunder \na \nfeedback\n \ncontrol process in real\n-\ntime\n.\n \nWe st\nuck\n \ntwo right \npaper plate\ns\n \nto Plate A and Plate B, and t\nwo laser sensors were \nused to \nmeasure the displacement simultaneously.\n \nThe \ndisplacement of Plate A \nwhich was isolated by the series \nspring \n(the output \ndisplacemen\nt\n \nof \nthe \nSE\n-\nHASEL actuator) \nwas smoother and controlled to a target value. However, the \n \n \nFig. \n8\n.\n \nDisplacement \no\nutput of the SE\n-\nHASEL \na\nctuator\n.\n \n(a) \nThe displacement output of the electrohydraulic \nactuators and our proposed SE\n-\nHASEL actuator.\n \n(b) The \ndisplacement hysteresis with the magnitude of AC applied \nvoltage.\n \n \ndisplacement of Plate B which was directly \nin \ncontact with the \nelectrohydraulic actuators oscillated periodically\n \n(sharp peaks \nin the curve, see Fig. \n10\n \na\nnd \nvideo \nS2\n)\n.\n \n \n \nE. \nImpact Absorption \nof the SE\n-\nHASEL Actuator\n \nT\nhe \nimpact, \ndisturbance\n,\n \nor variation of the load \nwill \ndramatically \ninfluence the output displacement of \nelectrohydraulic\n \nactuator\ns\n, and even destabilize the system\n.\n \nHere, we conducted a\nn\n \nimpact disturbance \ntest.\n \nFirst, the \noutput \ndisplacement\n \nof \nthe \nSE\n-\nHASEL \nactuator\n \nwas controlled \nto\n \n5 mm. Then, we applied a\nn instantaneous\n \n50 g weight on \nPlate A\n \nto simulate \nan impact disturbance\n.\n \nAfter \nthe \ndisplacement stabilized\n, we removed the load. The actual \ndisplacement\n \nand the applied voltage\n \nare\n \nshown in Fig. 1\n1\n \nand \nvideo S3\n.\n \nAfter applying\n \nthe\n \nload, the\n \nabsolute\n \ndisplacement \ndropped rapidly to almost 1 mm and then recovered to 5 mm \nin 1 second \nwhen \nthe magnitude of AC voltage\n \nwas increased\n.\n \nUpon removing\n \nthe load, \nthe displacement soared to 11 mm \n \nFig. \n1\n0\n.\n \nThe\n \nactual displacement of Plate A and Plate B\n \nin \nthe SE\n-\nHASEL actuator.\n \n \nFig. \n9\n.\n \nThe \nDisplacement \no\nutput with \nf\needback \nc\nontrol\n. \nThe SE\n-\nHASEL actuator following targets waves and its actual \napplied voltage. (a) The \nsquare waves. (b) The sine waves\n;\n \nThe SE\n-\nHASEL actuator following targets square and sine waves \n(0.1 Hz, 0.2 Hz and 0.5 Hz frequency). (\nc\n) The square waves. (\nd\n) The sine waves.\n \n \nand \nquickly \ndescended to 5 mm with slight fluctuation around \nthis value.\n \n \nThe\n \nresults revealed that the\n \nelastic component in the\n \nSE\n-\nHASEL actuator\n \ncan not only \nisolate \nthe vibration from the \nelectrohydraulic actuators to the load but also \nbuffer \nthe \nexternal \nimpact or \ndisturbance from the load to the \nelectrohydraulic actuators inversely.\n \nIt allows the control \nsystem to \nunhurriedly\n \nand stably\n \nrespond\n \nto \nthe \nimpact or \ndisturbance.\n \n \nF\n. \nApplications\n \n1\n) \nRotary Robotic Joint\n \nT\no \ndemonstrate\n \nthe application potential, we developed a \nrotary\n \nrobotic\n \njoint based on the SE\n-\nHASEL actuator. \nWe \ndesigned a link\n-\nslider mechanism to trans\nmit\n \nthe linear \ndisplacement to a rotation movement (see Fig. \n12\n).\n \nWe \nmanually increased the target value from 0 mm to 8 mm by a 2 \nmm increment, and then inversed the process back to 0 mm. \nThe snapshots in \nFig\n. 1\n2\na and video S\n4\n \nexplicitly \ndisplay the \nrotation\n \nmovement \n(\nrotation angle \nrange\n: \n48.1\n \ndegrees\n) \nof the \njoint. \n \nThough there are some sharp peaks\n \nin the output \ndisplacement curve (see Fig. 1\n2\nb)\n, probably \ndue to\n \nthe large \nmoment of inertia of the joint and friction, t\nhe\n \ncontrol \nsystem \nwas generally\n \nstable and the\n \nactual displacement of \nthe \nSE\n-\nHASEL actuator \nmatched\n \nthe \narbitrary \ntarget\n \nwith\n \nacceptable \nerror (\n0.\n3\n86 mm\n \nRMSE\n)\n.\n \nAfter \nthe \n50s, the \ndisplacement \ncannot return to 0 rapidly, mainly because\n \nof\n \nthe \nunidirectional actuation property of electrohydraulic actuators. \nAt this moment, the applied voltage is \nalready \n0\n \nV and only\n \ngravity\n \ncannot overcome the\n \nslight\n \nplastic deformation\n \nof \nelectrohydraulic actuators\n.\n \n2)\n \nNeedle Biopsy Robot\n \nThe needle biopsy is a\nn\n \nimportant\n \nmedical \nmethod\n \nfor \ncancer diagnosis and pathological \nanalysis\n \n[52]\n, which \nattracts \nmany researchers to\n \nd\nive into \nneedle biopsy robot\nic\ns\n.\n \nAn \nalternative technique (MRI) can provide cross\n-\nsectional \nimages of \nthe \nhuman body\n \nthat can\n \nguide the \nbiopsy \nrobot and \nimprove the penetration accuracy for needle biopsy\n \n[53]\n–\n[55]\n. \nHere, w\ne showcased a needle biopsy robot developed by our \nSE\n-\nHASEL actuator, to clarify its \nparticular compatibility \nwith\n \nMRI scenario\ns\n \nwhere it is quite challenging for \nconventional electromagnetic motor\n-\nbased needle biopsy \nrobots.\n \n \nW\ne designed a needle fixture to replace Plate A\n \nin Fig. 1\n0\n, \nand \nvertically mounted a syringe needle\n \non this fixture (Fig. \n13\na). \nO\nver the needle head, \nwe used a piece of chicken breast \n(about 2 mm thickness) and a piece of pork liver to simulate \nhuman \nmuscle and visceral organ\n.\n \nA 3 mm thick plate was \nsandwiched by the chicken and live\nr leaving only \na \n0.9 mm air \ngap due to the deformation of \nthe \nchicken and liver\n \nto simulate \nother human tissue \n(see Fig. \n13\n \na and b)\n.\n \nThere is a \n6 mm \nthick\n \nbreach\n \nin this plate for needle penetration\n \nand \nobservation\n.\n \nA 50 g weight was also fixed on the needle \nfixture to supply a recovering force\n \nfor pushing the needle out\n \n(see the side view in Fig. \n13\nb)\n.\n \nFirst, we adjust\ned\n \nthe position of \nthe \nneedle head\n \nto the \nbottom of the muscle (chicken),\n \nand manually measured the \nactual distance from the needle head to our target position\n,\n \nwhich is \nsimilar \nto\n \nthe \nguiding \nfunction of MRI.\n \nWe then set \nthe target displacement of \nthe \nSE\n-\nHASEL actuator to achieve \ndifferent biopsy samples: \nthe \nliver \ntissue \nand \nthe \ngap\n \n'\ntissue\n'\n.\n \nFor liver tissue sampling, we first set the target to 2 mm and \nthen increase\nd\n \nthe target to 3.5 mm. The needle \npenetrated the \nmuscle tissue, following the liver tissue (see video S5\n \nand Fig. \n13\nd\n). Th\ne actual displacement as shown in Fig. \n13\nc can follow \nour targets with only 0.067 mm RMSE.\n \nFor liver tissue \nsampling,\n \nwe directly set the target to 2.5 mm. The needle \n \nFig. \n1\n2\n.\n \nThe\n \nrotary joint developed based on the SE\n-\nHASEL actuator.\n \n(a) The snapshots of the \nrotary motion of \nthe joint\n. (b) The displacement of SE\n-\nHASEL actuator.\n \n \nFig. \n1\n1\n.\n \nThe\n \nimpact absorption\n \nof the SE\n-\nHASEL actuator \nwith feedback control.\n \n \npenetrated the muscle tissue, but stay\ned\n \nwithin the gap \n'\ntissue\n'\n \n(see video S5 and Fig. \n13\nd). The actual displacement can also \nfollow the target precisely (\nonly \n0.051 mm RMSE).\n \nThe \nsnapshots\n \nand the microscopic views of liver tissue \nin Fig. \n13\nd \nclearly display the difference in the deepest needle position \nbetween these two samplings.\n \nOur SE\n-\nHASEL actuator\n \nwith \nfeedback control can output accurate, smooth and stable \ndisplacement, which \ncan\n \navoid additional damage to the \nhuman body and \nis crucial for medical \nrobotic \napplications.\n \nV.\n \nC\nONCLUSION\n \nIn this article, our primary \nnovelty\n \nrevolved around \nusing \nlow\n-\nfrequency AC square wave voltage to drive \nelectrohydraulic actuators, effectively mitigating the gradual \nEA force decay observed under constant DC voltage. To \nfurther enhance their stability and precision, we introduced \nSE\n-\nHASEL actuators, which seamlessly integrated an elastic \ncomponent to achieve smooth, oscillation\n-\nfree displacement \noutput. The implementation of feedback control overcome\ns\n \nhysteresis and ensure\ns\n \naccurate, stable, and seamless \ndisplacement outputs under various conditions. This control \nstrategy, combined with our \nnovel\n \nactuator designs, offered a \npowerful toolset for harnessing the full potential\n \nof \nelectrohydraulic actuators\n \nthrough\n \nstable and smooth \ndisplacement output\n. In addition to our practical findings, we \npresented a mathematical model that elucidated the underlying \nprinciples governing the behavior of electrohyd\nraulic actuators \nunder different voltage inputs. Our contributions extended \nbeyond theoretical frameworks, as we introduced a cost\n-\neffective fabrication method for electrohydraulic actuators \nusing polyimide (PI) materials, renowned for their mechanical \nrobustness and thermal tolerance. This approach further \nstreamlined the production process, making these actuators \nmore accessible for broader applications. The experimental \nvalidation of our research findings provided compelling \nevidence of the feasibility a\nnd effectiveness of our proposed \n \nFig. \n13\n.\n \nThe\n \nneedle biopsy robot by the SE\n-\nHASEL actuator. (a) The \ndesign of the needle biopsy robot. (b) The actual \nassembled needle biopsy \nrobot. (c) The actual displacement of SE\n-\nHASEL actuator. (d) The needle biopsy for sampling the \nliver tissue and the gap tissue.\n \nFor sampling liver tissue, the needle penetrates to the liver\n \nwith a pierced hole\n.\n \nFor sampling \ngap tissue, the needle only penetrates to the gap.\n \n \nsolutions. Through a series of experiments, we demonstrated \nthe behavior of electrohydraulic actuators under varying \nvoltage inputs, loads, and control strategies. These \nexperiments underscored the practical applicability of our \ntechnology and control methods.\n \nLastly, our study showcased \nthe versatility and potential of SE\n-\nHASEL actuators by \nintroducing a rotary joint that effectively converted linear \ndisplacement into rotational movement\n \nand a \nprecise \nneedle \nbiopsy robot actuated by our SE\n-\nHASEL actuator.\n \nThese two \nrobotic applications \ncreated\n \na \nnovel\n \nway\n \nfor\n \nMRI\n-\ncompatible \nmedical robotics\n.\n \nCurrently\n, \nthe \nmathematical model's \ncapacitance and \nresistance \nparameters are unknow\nn\n, resulting in only \nqualitative analysis of the behavior of electrohydraulic \nactuators. \nIn subsequent works\n, \nwe will \ndetermine\n \nthe \ncapacitance and the resistance parameters\n \nquantitatively\n \nby \nfitting the theoretical input current\n \nderived through\n \n(2) and (6)\n \nand the actual\n \ninput current\n \nacquired from the high\n-\nvoltage \nsupply device\n \nunder \na\n \nconstant\n \nDC input voltage.\n \nAdditionally\n, \naccording to the mathematical model of displacement \nmentioned \nin Section \nII\n.C, \nwe will \ncomprehensively optimize \nthe design parameters of the SE\n-\nHASEL actuator, such as the \nvolume of dielectric liquid and the size of electrodes.\n \nFor the \nneedle biopsy robot, we will substitu\nt\ne all the \nferromagenetic\n \ncomponents (sliders, need\nle\n, spring) with components made of \nmagnetic\n-\ncompatible materials, and test with \na \nreal MRI \nmechanie\n \nin the future.\n \nA\nCKNOWLEDGMENT\n \nT\nhe authors acknowledge the assistance of \nthe \nDepartment \nof Mechanical Engineering, National University of Singapore.\n \nA\nPPENDIX\n \nT\nhe \nlegends of supporting videos:\n \nVideo S1: The displacement output of the electrohydraulic \nactuators under DC voltage and AC voltage\n.\n \nVideo S2: \nThe motion of Plate A and Plate B of the SE\n-\nHASEL actuator.\n \nVideo S3: \nImpact \na\nbsorption of the SE\n-\nHASEL \na\nctuator\n \nVideo S4: \nThe r\notary \nr\nobotic \nj\noint\n \nby the SE\n-\nHASEL actuator\n \nVideo S5: \nThe n\needle \nb\niopsy \nr\nobot \nby the SE\n-\nHASEL actuator\n \n \nR\nEFERENCES\n \n[1]\n \nN. 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Desai, “Design, \ndevelopment, and evaluation of an MRI\n-\nguided SMA spring\n-\nactuated neurosurgical robot,” \nInt. J. Rob. Res.\n, vol. 34, no. 8, pp. \n1147\n–\n1163, 2015.\n \n "    },    {        "title": "2401.13950v1.AM_SORT__Adaptable_Motion_Predictor_with_Historical_Trajectory_Embedding_for_Multi_Object_Tracking.pdf",        "content": "AM-SORT: Adaptable Motion Predictor with\nHistorical Trajectory Embedding for\nMulti-Object Tracking\nVitaliy Kim[0009−0000−2031−7599], Gunho Jung[0000−0002−1143−9663], and\nSeong-Whan Lee[0000−0002−6249−4996]\nDepartment of Artificial Intelligence, Korea University, Seoul, Republic of Korea\n{vitaliy, gh_jung, sw.lee}@korea.ac.kr\nAbstract. Many multi-object tracking (MOT) approaches, which em-\nploy the Kalman Filter as a motion predictor, assume constant velocity\nand Gaussian-distributed filtering noises. These assumptions render the\nKalman Filter-based trackers effective in linear motion scenarios. How-\never, these linear assumptions serve as a key limitation when estimating\nfuture object locations within scenarios involving non-linear motion and\nocclusions. To address this issue, we propose a motion-based MOT ap-\nproach with an adaptable motion predictor, called AM-SORT, which\nadapts to estimate non-linear uncertainties. AM-SORT is a novel exten-\nsion of the SORT-series trackers that supersedes the Kalman Filter with\nthe transformer architecture as a motion predictor. We introduce a his-\ntorical trajectory embedding that empowers the transformer to extract\nspatio-temporal features from a sequence of bounding boxes. AM-SORT\nachieves competitive performance compared to state-of-the-art trackers\non DanceTrack, with 56.3 IDF1 and 55.6 HOTA. We conduct extensive\nexperiments to demonstrate the effectiveness of our method in predicting\nnon-linear movement under occlusions.\nKeywords: Multi-object tracking ·Adaptable motion predictor ·Non-\nlinear motion ·Historical trajectory embedding.\n1 Introduction\nMotion-based multi-object tracking (MOT) approaches [2,3,15,24,26,31] uti-\nlize a motion predictor to extract spatio-temporal patterns and estimate object\nmotion in future frames for subsequent object association. The original Kalman\nFilter [12] is widely employed as a motion predictor, which operates under as-\nsumptions of constant velocity and Gaussian-distributed noises in the prediction\nand filtering stages, respectively [2]. Constant velocity postulates that object\nspeed and direction remain consistent over a short period, and Gaussian dis-\ntributions assume constant error variance in both estimations and detections.\nWhile these assumptions result in resource efficiency for the Kalman Filter by\nsimplifying mathematical modeling, they are only valid for a specific scenario\nwhere the object displacement remains linear or consistently small at each timearXiv:2401.13950v1  [cs.CV]  25 Jan 20242 V. Kim et al.\nFig. 1.Results on dancetrack0004 sequence from DanceTrack for (a) OC-SORT and\n(b) AM-SORT (Ours). The object, marked in yellow, moves to the left and becomes\noccluded in the middle frame. Then, the yellow object changes the movement direction\nto the right after occlusion, and OC-SORT does not capture this sudden directional\nshift, causing an ID-switch from 13 to 10.\nstep [26]. Due to the neglect of scenarios with non-linear motion and occlusions,\nthe Kalman Filter inaccurately estimates object locations in complex situations.\nTo address the limitations of the original Kalman Filter, alternative esti-\nmation algorithms were proposed, such as Extended Kalman Filter (EKF) [21]\nand Unscented Kalman Filter (UKF) [11]. EKF linearizes object motion model-\ning, and UKF estimates non-linear transformations by employing the first and\nthird-order Taylor series expansions, respectively. However, both methods are\nstill conditioned on linear approximations for non-linear systems and assume\nGaussian-distributed noises. On the other hand, particle filters [10] avoid lin-\nearization by utilizing a set of discrete particles to handle non-linearity and\nnon-Gaussian noises, yet require expensive computational resources. Recent OC-\nSORT [3] improved the original Kalman Filter by placing a greater emphasis on\nobservationsratherthanestimationstoreducenoisesinmotionprediction.While\nthis approach allows for tracking objects with linear motion during occlusions,\nOC-SORT still faces challenges with non-linear motion. When the lack of obser-\nvations occurs caused by non-linear motion or occlusions, OC-SORT relies on\nits linear estimations, formulated upon the linear assumptions inherent to the\nKalman Filter. Consequently, this linear assumption-based modeling accumu-\nlates errors in motion prediction leading to significant trajectory deviations.\nWe argue that the linear assumptions inherent to the Kalman Filter lead to\ninaccurate motion estimations and false identity matches when objects involve\nnon-linearuncertaintiescharacterizedbysuddenspeedchanges,directionalshifts\nand occlusions. Due to these assumptions, the accumulated errors in motion\nestimations restrict the Kalman Filter-based approaches in handling non-linear\nuncertainties. Fig. 1 shows tracking results in a non-linear motion scenario underAM-SORT: AM Predictor with Historical Trajectory Embedding for MOT 3\nFig. 2.Comparison of (a) conventional transformer-based MOT and (b) our frame-\nworks. The key difference lies in the input feature level: typical transformer-based ap-\nproaches take frames as input and primarily utilize appearance information, whereas\nAM-SORT processes bounding boxes and solely relies on motion information.\nocclusion using (a) OC-SORT and (b) our AM-SORT. As illustrated in Fig.\n1(a), the identity switch occurs for the yellow object after an occlusion event.\nThe linear motion assumptions in the Kalman Filter cause directional errors in\nmotion estimations that the yellow object continues moving to the left. As a\nresult, the Kalman Filter relies on these linear estimations with accumulated\ndirectional errors, failing to predict the directional shift to the right.\nIn this paper, we propose an adaptable motion predictor with historical tra-\njectory embedding for MOT that addresses the limitations of the linear as-\nsumptions inherent to the Kalman Filter. The adaptation ability releases the\nmotion predictor from the constraints of linear assumptions, allowing it to es-\ntimate uncertainties related to non-linear motion. Inspired by transformer ar-\nchitectures [5,9,25], known for their ability to capture complex dependencies in\nsequence data, we explore the utilizing of a transformer encoder as an adapt-\nable motion predictor. In contrast to conventional transformer-based MOT ap-\nproaches, we leverage the transformer to encode only motion information with-\nout visual features for object association, as shown in Fig. 2. Utilizing bounding\nboxes as input features provides a limited object representation compared to\nappearance information but significantly reduces computational complexity. To\nmaintain simplicity and resource efficiency comparable to the Kalman Filter, we\nfocus on the transformer encoder to learn object discrimination features exclu-\nsively from object trajectories.\nFurthermore, our adaptable motion predictor derives benefits from analyzing\nand observing longer object trajectories compared to the Kalman Filter, which\npredicts object motion solely based on estimations from the previous time step.\nTo enhance the representation of long object trajectories, we present historical\ntrajectory embedding that encodes the spatio-temporal information from the se-4 V. Kim et al.\nquence of bounding boxes. Consequently, we concatenate the embedded bound-\ning boxes with a prediction token that functions as an embedded bounding box\nof the current frame. The encoder extracts the spatio-temporal features from\nthe historical trajectory embedding, enabling the prediction token to estimate\nthe bounding box in the current frame. Notably, AM-SORT utilizes sequences of\nbounding boxes as input, omitting the visual features of objects, which enables\nthe model to process with low computational cost.\nOur contributions are summarized as follows:\n–We propose a novel SORT-series tracker with an adaptable motion predic-\ntor, called AM-SORT, which provides non-linear motion estimations without\nlinear assumptions;\n–We introduce historical trajectory embedding to effectively capture motion\nfeatures from a sequence of bounding boxes;\n–The qualitative results show that AM-SORT accurately predicts the non-\nlinear changes in object motion, demonstrating its competitiveness with\nstate-of-the-art approaches.\n2 Related Work\n2.1 Motion-Based Methods in Multi-Object Tracking\nDanceTrack [22] reveals the limitations of appearance-based MOT methods in\ndistinguishing objects that share highly similar visual features. This motivates\nthe development of motion-based and hybrid methods that leverage both ap-\npearance and motion information. [1–3,24,26,31] propose trackers that solely\nemploy motion features without appearance information. CenterTrack [31] in-\ntroduces an efficient tracker that represents each object as a single point and\npredictstheirassociationswithminimalinputasdetectionsfromapairofframes.\nPermaTrack [24] addresses the limitations of CenterTrack in recovering objects\nafter occlusions. It assumes object permanence under occlusions and contin-\nues modeling the spatio-temporal movement of lost objects. LGM [26] proposes\na motion-based model for the vehicle tracking task, leveraging both local and\nglobal motion consistencies to track and recover vehicles after occlusions. How-\never, its applicability is limited to tracking vehicles with linear motion, lacking\nrobustness in handling objects with non-linear motion.\nAlong with these works, the SORT-series trackers [2,3,27,30] utilize the\nBayesian estimation [16] as a motion model. For instance, SORT [2] employs the\noriginal Kalman Filter [12] with linear assumptions for object motion estimation\nand the Hungarian matching algorithm [14] to match predictions and detections.\nHowever,asmotionfeaturesaloneofferlimitedinformation,Deep-SORT[27]and\nByteTrack [30] propose a hybrid method by incorporating the visual features\nwith the Kalman Filter predictions to enhance object discrimination. On the\nother hand, OC-SORT [3] improves robustness in handling occlusions without\nappearanceinformationbyprioritizingobservationsinsteadoflinearestimations,\nbut still struggles in recovering lost objects under non-linear motion and long-\nterm occlusions.AM-SORT: AM Predictor with Historical Trajectory Embedding for MOT 5\nFig. 3.IllustrationoftheAM-SORToverallpipeline.Thehistoricaltrajectoryoflength\nTis fed into the transformer encoder to estimate the track predictions Pt. Through\nutilizing an off-the-shelf detector, detections Dtare obtained. Subsequently, the Hun-\ngarian matching algorithm associates DtwithPt, resulting in the final output tracks.\n2.2 Transformers in Multi-Object Tracking\nIn scenarios involving non-linear motion and occlusions, transformers demon-\nstrate promising results for their inherent power to model complex interactions\nandadaptivelyprocesssequentialinformation.AsshowninFig.2(a),theexisting\ntransformer-based MOT approaches learn object queries to capture mainly ap-\npearance information [7,18,23,29,32]. In particular, TransTrack [23] utilizes the\ntransformer to extract the object-level appearance features and learn the aggre-\ngated visual embedding of each object for subsequent IoU-based matching. Since\nappearance information is sensitive to occlusions, TrackFormer [18], MOTR [29]\nand MeMOTR [7] jointly model both motion and appearance by representing\neach object as an autoregressive track query and recurrently propagating them\nto associate with identical instances across subsequent frames.\nIn contrast, AM-SORT only leverages motion information, employing sim-\nple and lightweight bounding boxes. To the best of our knowledge, AM-SORT\nstands out as the first successful application of transformers in purely motion-\nbased methods. We believe that AM-SORT will encourage further research on\nadaptable motion predictors.\n3 Proposed Method\nAM-SORT leverages motion cues to robustly track objects with non-linear mo-\ntion patterns. Our primary focus is on achieving accurate estimations of non-\nlinear uncertainties by introducing an adaptable motion predictor based on the\ntransformerencoderwhichsupersedestheKalmanFilter.Fig.3showstheoverall\npipeline of AM-SORT. Specifically, we input the historical trajectory of an indi-\nvidual object containing a sequence of bounding boxes in the previous frames,6 V. Kim et al.\nFig. 4.Illustration of our historical trajectory embedding in the motion predictor. The\nhistorical trajectory embedding encodes a comprehensive representation of a bounding\nbox sequence by jointly considering spatio-temporal information.\ndenotedas Bt−T:t−1={bt−T, . . . ,bt−2,bt−1},where Tisthepre-definedhistor-\nical trajectory length. The bounding boxes are represented as b= (cx, cy, w, h ),\nwhere (cx, cy)is the center coordinate of the object in the image plane, wandh\nstand for width and height, respectively. The transformer encoder produces the\nrefined prediction token, which is subsequently converted into a bounding box\nˆbtthrough the prediction head. The estimated bounding boxes generate a set of\ntrack predictions for the current frame, denoted as Pt. Subsequently, detections\nin the corresponding frame, referred to as Dt, are associated with Ptbased on\nIntersection-over-Union (IoU) using the Hungarian matching algorithm [14].\n3.1 Historical Trajectory Embedding\nHistorical trajectory embedding jointly encodes the spatial and temporal infor-\nmation from a sequence of bounding boxes and consists of three operations: spa-\ntial embedding, prediction token concatenation, and temporal embedding. Fig.\n4 illustrates the structure of our historical trajectory embedding in the motion\npredictor.\nFor spatial embedding, we utilize the sinusoidal positional encoding [25] to\ntransform low-dimensional bounding boxes into a high-dimensional space to fa-\ncilitate a fine-grained representation of each bounding box as follows:\nxt−T= PE spat(bt−T), (1)\nwhere PEspat:R4→RDrepresents the spatial embedding operation, Dis an\nembedding dimension and xt−Tdenotes the spatial embedding of the bounding\nbox.AM-SORT: AM Predictor with Historical Trajectory Embedding for MOT 7\nSubsequently, a prediction token is concatenated with the spatial embeddings\nat the end of the entire sequence. This prediction token is a learnable embedding\nthat functions as a bounding box in the current frame t. The mathematical\nformulation is as follows:\nXt−T:pred = Concat( xt−T, . . . ,xt−1,xpred), (2)\nwhere Xt−T:preddenotes the spatial embedding of the historical trajectory, ob-\ntained through the concatenation Concat( ·)of spatial embeddings and the pre-\ndiction token xpred.\nFor temporal embedding, we employ positional encoding similar to spatial\nembeddings. In contrast, we encode natural numbers which assign serial num-\nbers to each spatial embedding in the sequence in reverse order from T+ 1to1,\nstarting from the last element. This ensures that the model prioritizes the termi-\nnal part of the historical trajectory embedding, even for objects with historical\ntrajectory lengths less than T. Thus, the historical trajectory embedding is as\nfollows:Zt−T:pred =Xt−T:pred + PE temp(NT+1:1), (3)\nwhere Zt−T:predrepresents our historical trajectory embedding, PEtemp:R→\nRDdenotesthetemporalembeddingand NT+1:1isasequenceofnaturalnumbers\nfrom T+ 1to1.\nNotably, in the context of bounding box prediction where object localization\nis crucial, we enrich the historical trajectory embedding with additional spatial\ninformation before passing it through each encoder layer.\n3.2 Adaptable Motion Predictor\nWe utilize the transformer encoder as an adaptable motion predictor, which\ncontains multi-head self-attention (MHSA) [25] layers and feed-forward neural\nnetworks. MHSA facilitates interactions among each bounding box within the\nhistoricaltrajectoryextractingtheirnon-linearrelationships.Thisprocessrefines\nthe prediction token with sufficient information for precise localization of the\nobject bounding box in the current frame, formulated as:\nˆZt−T:pred = Enc( Zt−T:pred), (4)\nwhere Enc(·)represents the transformer encoder operations, with ˆZt−T:predde-\nnoting the refined historical trajectory embedding. The prediction head receives\nonly the prediction token ˆ zpred, which is the last element in the refined historical\ntrajectory embedding, and utilizes it to generate the bounding box coordinates\nas follows:\nˆbt= Head( ˆ zpred), (5)\nwhere Head(·)denotes the prediction head and ˆbtis the estimated bounding\nbox in the current frame. The prediction head is composed of three linear layers\neach accompanied by a ReLU activation function, and the last layer utilizes\na Sigmoid activation function to convert the bounding box coordinates in the\nrange between 0 and 1.8 V. Kim et al.\n3.3 Training\nWe train our adaptable motion predictor by comparing the predicted bounding\nboxes with the ground truth. We extract all the trajectories in an entire tracking\nvideo and segment them into bounding box sequences of length T+ 1. The\nbeginning bounding box sequence of each trajectory segment is utilized as a\nhistorical trajectory to estimate ˆbat the frame T+ 1, while the last bounding\nboxbin the segment is considered as the ground truth. We adopt the L1 loss\nfunction as the prediction loss to enhance robustness to outliers, such as errors\nin object detection and track prediction. Specifically, the estimated attributes\n(ˆcx,ˆcy,ˆw,ˆh)of bounding box ˆbare compared to the respective attributes of\nground truth bwith L1 loss and our total prediction loss Lpredis computed as\nthe mean value:\nLpred(ˆb,b) =1\n4X\ni|ˆbi−bi|, i∈(cx, cy, w, h ). (6)\nMasked Tokens. We employ masked tokens as an augmentation strategy to\nsimulatetheeffectofnon-linearmotionandocclusions.Wemaskboundingboxes\nwithin historical trajectories with a probability p. Subsequently, the masked\nbounding boxes are replaced by masked tokens to prevent the encoding of their\nspatial information. These masked tokens are represented as learnable embed-\ndings, that are initialized with random values and optimized during training. In\nthis manner, we enhance our model to gain a clear comprehension of missing\ntrajectory segments. Our augmentation strategy with masked tokens facilitates\neffective masking operations, ensuring robust training in complex scenarios.\nAdditionally, we utilize masked tokens to handle padding in historical tra-\njectory embeddings during inference. We fill the historical trajectory embedding\nwith masked tokens to maintain the constant length for newborn objects with\npast bounding boxes fewer than T.\n4 Experiments\n4.1 Dataset and Evaluation Metric\nWe provide experimental results on DanceTrack [22], MOT17 [19] and MOT20\n[4]. DanceTrack mainly consists of dance videos featuring objects with similar\nappearances. DanceTrack provides scenarios characterized by non-linear object\nmotion and occlusions, thereby posing significant challenges for motion-based\ntracking approaches. MOT17 and MOT20 contain pedestrian tracking videos in\npublic spaces, where object motion is represented by slow and smooth move-\nments, approximately linear. However, these datasets are still challenging due to\nhighly crowded scenes with dense object populations.\nWeusetheevaluationmetricsincludingHOTA(HigherOrderTrackingAccu-\nracy) [17], AssA (Association Accuracy) [17], DetA (Detection Accuracy) [17],\nIDF1 [19] and MOTA (Multi-Object Tracking Accuracy) [19]. HOTA offers a\nbalanced evaluation of both detection and association accuracy, in contrast to\nMOTA or DetA which is biased toward measuring detection. IDF1 and AssA\nare used to demonstrate the association performance.AM-SORT: AM Predictor with Historical Trajectory Embedding for MOT 9\nTable 1. Tracking results on the DanceTrack test set.\nTracker Appear. Motion HOTA ↑IDF1↑MOTA ↑AssA↑DetA↑\nDeepSORT [27] ✓ ✓ 45.6 47.9 87.8 29.7 71.0\nByteTrack [30] ✓ ✓ 47.3 52.5 89.5 31.4 71.6\nMOTR [29] ✓ ✓ 54.2 51.5 79.7 40.2 73.5\nMeMOTR [7] ✓ ✓ 68.5 71.2 89.9 58.4 80.5\nTransTrack [23] ✓ 45.5 45.2 88.4 27.5 75.9\nGTR [32] ✓ 48.0 50.3 84.7 31.9 72.5\nQDTrack [6] ✓ 54.2 50.4 87.7 36.8 80.1\nGHOST [20] ✓ 56.7 57.7 91.3 39.8 81.1\nCenterTrack [31] ✓ 41.8 35.7 86.8 22.6 78.1\nTraDes [28] ✓ 43.3 41.2 86.2 25.4 74.5\nSORT [2] ✓ 47.9 50.8 91.831.2 72.0\nOC-SORT [3] ✓ 54.6 54.6 89.6 40.2 80.4\nAM-SORT (Ours) ✓ 55.6 56.3 89.6 40.480.3\n4.2 Implementation Details\nWe train our adaptable motion predictor on the corresponding tracking datasets\nwithout incorporating extra samples from other datasets. To ensure a fair com-\nparison, we utilize the publicly accessible YOLOX [8] detector weights developed\nby ByteTrack [30] for object detection following the baselines. The transformer\nencoder is comprised of 6 layers with the multi-head self-attention employing\n8 heads. The embedding dimension Dis set to 512. We use the Adam [13] to\noptimize the network with a learning rate of 0.0001 for 50 epochs and set the\nbatch size to 512. The historical trajectory embedding length Tis predefined\nas 30. The masking probability pis selected as 0.1. Analysis of the choice of T\nandpcan be found in Section 4.5. All experiments were conducted on a single\nNVIDIA TITAN XP.\n4.3 Benchmark Results\nTable 1 shows the benchmark results on the DanceTrack test set. AM-SORT\nachieves competitive performance compared to the appearance-based and hybrid\ntrackers, and state-of-the-art results among motion-based MOT approaches. It\nobtains 56.3 IDF1 and 55.6 HOTA, outperforming the baselines. It is impor-\ntant to note that a significant gain of 1.7 is observed for IDF1, which measures\nassociation performance and re-identification accuracy.\nTable 2 shows the tracking performance on the MOT17 and MOT20 test sets\nto verify the generalizability covering linear object motion. AM-SORT achieves\nhigher results compared to state-of-the-art MOT approaches. As mentioned\nearlier, MOT17 and MOT20 are designed for tracking pedestrians, where mo-\ntion patterns are generally linear and do not contain non-linear scenarios. De-\nspitethesedifferentconditions,AM-SORTstilldemonstratesconsistentimprove-\nments, even though it does not align with primary issues.10 V. Kim et al.\nTable 2. Tracking results on the MOT17 and MOT20 test sets under private detection\nprotocols.\nMOT17 MOT20\nTracker HOTA ↑IDF1↑MOTA ↑AssA↑HOTA ↑IDF1↑MOTA ↑AssA↑\nHybrid:\nMOTR [29] 57.2 68.4 71.9 55.8 57.8 68.6 73.4 /\nMeMOTR [7] 58.8 71.5 72.8 58.4 54.1 66.1 63.7 55.0\nDeepSORT [27] 61.2 74.5 78.0 59.7 57.1 69.6 71.8 55.5\nByteTrack [30] 63.1 77.3 80.3 62.0 61.3 75.2 77.8 59.9\nAppearance-based:\nQDTrack [6] 53.9 66.3 68.7 52.7 60.0 73.8 74.7 58.9\nTransTrack [23] 54.1 63.9 74.5 47.9 48.9 59.4 65.0 45.2\nGTR [32] 59.1 71.5 75.3 57.0 / / / /\nGHOST [20] 62.8 77.1 78.7 / 61.2 75.2 73.7 /\nMotion-based:\nSORT [2] 34.0 39.8 43.1 31.8 36.1 45.1 42.7 35.9\nCenterTrack [31] 52.2 64.7 67.8 51.0 / / / /\nTraDes [28] 52.7 63.9 69.1 50.8 / / / /\nPermaTrack [24] 55.5 68.9 73.8 53.1 / / / /\nOC-SORT [3] 63.2 77.5 78.063.4 62.1 75.9 75.5 62.0\nAM-SORT (Ours) 63.3 77.8 78.0 63.5 62.0 76.1 75.5 61.3\n4.4 Qualitative Results\nFig. 5 shows a qualitative comparison of OC-SORT and AM-SORT. These ex-\namples illustrate identity switches of the yellow-marked object in OC-SORT. In\nFig. 5 Row 1, due to the linear assumptions inherent to the Kalman Filter, OC-\nSORTestimatesathin-shapedboundingboxforthemarkedobjectinthemiddle\nframe. It is unable to predict the sudden change in a wide-shaped bounding box\nleading to a false match. Similarly, the linear assumptions prevent the capture\nof the directional shift to the right after occlusion in Fig. 5 Row 2. In contrast,\nAM-SORT maintains consistent identities under these non-linear object motion\nand occlusions.\n4.5 Ablation Study\nAssociation Cost in Hungarian Matching Algorithm. During inference,\nthe SORT-series trackers utilize the Hungarian matching algorithm for object as-\nsociation.ToshowtheimpactoftheassociationcostsintheHungarian matching\nstep, we compare OC-SORT and AM-SORT at different combinations of asso-\nciation costs including IoU, motion direction difference ∆θand L1 distance.\nMotion direction difference calculates the direction similarity between existing\ntracks and new observations. AM-SORT with IoU alone as in Table 3 Row 1\noutperforms OC-SORT with IoU alone as in Table 3 Row 1 by 3.2 IDF1 and\nachieves an increase of 0.2 IDF1 compared to OC-SORT with the best settingsAM-SORT: AM Predictor with Historical Trajectory Embedding for MOT 11\nFig. 5.Qualitative comparison of OC-SORT and AM-SORT (Ours). The first row\nshows the tracking results in the scenario with non-linear changes of the bounding box\nfordancetrack0010 sequence; the second row in the scenario with non-linear object\nmovement during occlusions for dancetrack0019 .\nTable 3. Analysis at various association cost matrices.\nIDF1↑\nIoU ∆θL1 OC-SORT AM-SORT (Ours)\n✓ 51.1 54.3\n✓ ✓ 54.1 52.1\n✓ ✓ 53.4 56.5\n✓ ✓ ✓ 53.2 53.2\nas in Table 3 Row 2. On the other hand, motion direction difference degrades\nthe tracking performance of our model. The reason is that the motion direction\ncue, which is incorporated in OC-SORT to compensate for the approximately\nestimated bounding boxes in non-linear scenarios, is not suitable for AM-SORT.\nOur adaptable motion predictor already captures non-linear directional shifts\nin the prediction step making location-based matching sufficient. Furthermore,\nincorporating location-based association costs, IoU and L1 distance as in Table\n3 Row 3, gains an extra 2.5 IDF1 compared to OC-SORT with the best settings.\nReliabilityofBoundingBoxPredictions. Toverifythereliabilityofbound-\ning box predictions, we evaluate OC-SORT and AM-SORT at progressively in-\ncreased IoU thresholds. The higher IoU threshold requires a larger overlap to as-\nsociate detections with predictions. Table 4 demonstrates that AM-SORT with\nIoUaloneoutperformsOC-SORTwiththebestsettingsatIoUthresholdsgreater\nthan 0.4, while AM-SORT with the best settings achieves superior performance\nacross all IoU threshold values. The higher IDF1 demonstrates that AM-SORT\nhas a larger number of positively matched tracks with the ground truth. This in-\ndicates that our adaptable motion predictor captures more accurately the object\narea, which serves as strong evidence for the higher reliability of the bounding\nbox predictions.\nImpactofHistoricalTrajectoryEmbeddingLength. Todemonstratehow\ntracking performance varies with the historical trajectory embedding length,12 V. Kim et al.\nTable 4. Analysis on prediction reliability at varying IoU thresholds.\nIDF1↑\nIoU threshold 0.3 0.4 0.5 0.6 0.7 0.8\nOC-SORT 54.1 51.5 46.9 38.7 25.5 15.8\nAM-SORT (IoU) 54.3 51.2 48.4 40.4 27.0 16.4\nAM-SORT (IoU+L1) 56.5 53.6 50.7 42.2 28.9 17.0\nTable 5. Impact of the historical trajectory embedding length T.\nIDF1↑\nT 5 10 20 30 40 50\nAM-SORT (Ours) 52.9 53.5 55.2 56.555.0 54.3\nTable 6. Impact of masked tokens at varying probabilities p.\nIDF1↑\np 0 0.05 0.1 0.2 0.3 0.4\nAM-SORT (Ours) 55.1 56.1 56.554.7 53.2 51.9\nwe evaluate AM-SORT at different Tvalues. Table 5 shows that performance\nincreases with a longer historical trajectory and decreases when Tis greater than\n30. We conclude that a longer historical trajectory provides more comprehensive\nspatio-temporal information about object motion up to T= 30. In contrast,\noverly old bounding boxes of the historical trajectory can provide noises and\nthus negatively impact the overall tracking performance. We set T= 30, which\ncovers 1.5 seconds of object trajectory in a 20 FPS video.\nImpact of Masked Tokens. To show the effectiveness of utilizing masked\ntokens during training, we provide the tracking results for mask probabilities p\nranging from 0 to 0.4. Table 6 demonstrates that employing masked tokens with\na probability of p= 0.1results in a 1.4 increase in IDF1 compared to training\nwithout masked tokens at p= 0. Conversely, masking with probability p≥0.2\nslightly drops the performance. We suggest that utilizing moderate masking of\np= 0.1provides robust training to occlusions.\n5 Conclusion\nIn this paper, we propose AM-SORT, a motion-based tracker with an adaptable\nmotion predictor, that effectively addresses non-linear motion and occlusion. We\nintroducehistoricaltrajectoryembeddingtoencodespatio-temporalinformation\nin bounding box sequences for a comprehensive representation of object trajec-\ntory.Weleveragethe abilityof transformerstomodellong-range dependenciesin\nobjecttrajectory,enablingourmotionpredictortoadapttocomplexmotionpat-\nterns. As a result, AM-SORT achieves competitive performance compared with\nstate-of-the-art methods and outperforms existing motion-based approaches.AM-SORT: AM Predictor with Historical Trajectory Embedding for MOT 13\nAcknowledgments. This work was supported by the Institute of Information & com-\nmunications Technology Planning & Evaluation (IITP) grant funded by the Korea gov-\nernment (MSIT) (No. 2019-0-00079, Artificial Intelligence Graduate School Program\n(KoreaUniversity)).Furthermore,weextendoursincereappreciationtoHo-JoongKim\nfor his invaluable support and feedback on the current research.\nReferences\n1. 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In:\nProceedingsoftheIEEE/CVFConferenceonComputerVisionandPatternRecog-\nnition. pp. 8771–8780 (2022)"    },    {        "title": "2401.13971v1.Stochastic_Weakly_Convex_Optimization_Beyond_Lipschitz_Continuity.pdf",        "content": "Stochastic Weakly Convex Optimization Beyond Lipschitz Continuity\nWenzhi Gao1and Qi Deng2\n1gwz@stanford.edu , ICME, Stanford University\n2contact.qdeng@gmail.com , Shanghai University of Finance and Economics\nJanuary 26, 2024\nAbstract\nThis paper considers stochastic weakly convex optimization without the standard Lipschitz continuity\nassumption. Based on new adaptive regularization (stepsize) strategies, we show that a wide class of\nstochastic algorithms, including the stochastic subgradient method, preserve the O(1/√\nK)convergence\nrate with constant failure rate. Our analyses rest on rather weak assumptions: the Lipschitz parameter can\nbe either bounded by a general growth function of ∥x∥or locally estimated through independent random\nsamples. Numericalexperimentsdemonstratetheefficiencyandrobustnessofourproposedstepsizepolicies.\n1 Introduction\nThis paper studies the following stochastic optimization problem\nmin\nx∈Rnψ(x):=f(x) +ω(x):=Eξ∼Ξ[f(x, ξ)] +ω(x). (1)\nHere, f(x, ξ)is a continuous function in x, with ξbeing a random sample drawn from a particular distribution\nΞ. The function ω(x)is lower-semicontinuous, and its proximal mapping is easy to evaluate. We assume both\nf(x, ξ)andω(x)are weakly convexfunctions. A function gis defined as λ-weakly convex if g+λ\n2∥·∥2is convex,\nfor some λ≥0. When λis unspecified, gis called weakly convex. Weak convexity has found many important\napplications, including phase retrieval, robust PCA, reinforcement learning, and many others [13, 3, 29]. And\nrecent years witnessed a surge in interest regarding weakly convex optimization, leading to a substantial body\nof work on efficient algorithms with finite time complexity guarantees [4, 7, 10, 8, 24]. In particular, under\nthe global Lipschitz continuity assumption, Davis et al. [8] develop a model-based approach and analyzes the\nconvergence of several stochastic algorithms under a unified framework.\nWhile this global Lipschitz assumption is valid for many problems, such as piece-wise linear functions, it can be\noverly restrictive. To illustrate, consider the weakly convex function ψ(x) =|ex+e−x−3|whose subgradient\nψ′(x)explodes exponentially as ∥x∥grows (Figure 1 ). Hence, in algorithm design, treating the Lipschitz con-\nstant as any fixed constant can lead to highly unstable iterations and, potentially, to the algorithm divergence.\nTo address this issue, a straightforward strategy is to impose an explicit convex compact set constraint, such\nas{x:∥x∥ ≤B}to address this issue. However, it introduces extra parameter tuning and may lead to a\nsignificantly overestimated Lipschitz constant. The latter phenomenon is evident in the toy example, where\nthe Lipschitz constant grows exponentially with the domain set’s diameter. To deal with non-globally Lips-\nchitz settings, one research direction is to shift from standard Euclidean geometry to Bregman divergence. Lu\n[23] shows that when convex non-Lipschitz functions exhibit “relative” Lipschitz continuity under a carefully\nchosen divergence kernel, mirror descent still obtains the desired sublinear rate of convergence to optimality\n1arXiv:2401.13971v1  [math.OC]  25 Jan 2024−1.5−1−0.5 0.5 1 1.50.511.5\nxy\nFigure 1: f(x) =|ex+e−x−3|exhibits exponential growth as ∥x∥ →+∞\ndefined in the sense of Bregman divergence. However, there are trade-offs to consider. Compared to SGD, a\nmirror descent update is more expensive, often involving a nontrivial root-finding procedure. Additionally,\nchoosing the right kernel is a nuanced and critical task, heavily reliant on an in-depth understanding of the\nsubgradient’s growth dynamics ([6, 32]).\nAlternatively, recent works aim to develop new algorithms/analyses under relaxed Lipschitz assumptions.\nFor example, Asi and Duchi [2] show that using stochastic proximal point update, algorithmic dependency\non the global Lipschitz constant can be relaxed to E[∥f′(x⋆, ξ)∥2], which only relies on the optimal solution\nx⋆. Mai and Johansson [25] show that, for stochastic convex optimization with quadratic growth, subgradient\nmethodsincorporatingaclippingstepsizestillensuresconvergence, evenifLipschitzconstantexhibitsarbitrary\ngrowth. In weakly convex optimization, Li et al. [21] shows that when the Moreau envelope of objective has\na bounded level-set, local Lipschitz continuity alone is sufficient to ensure convergence of the subgradient\nmethod. Nevertheless, extending their analysis to stochastic optimization remains challenging. Grimmer [15]\nhave established the convergence of SGDin convex optimization, without Lipschitz continuity. In [33], the\nauthors propose a relaxed subgradient bound for weakly convex optimization:\nE[∥f′(x, ξ)∥2]≤c0+c1∥x∥, c0, c1≥0, (2)\nwhich naturally induces a bound on the local Lipschitzness as a function of ∥x∥. Whether SGDstill converges\nin case of arbitrary non-Lipschitzness, especially those not conforming to the bounded assumption in (2),\nremains an open area of investigation. The major difficulty in analyzing stochastic optimization without the\nstandard Lipschitz assumption stems from stability issues. In this paper, a stochastic algorithm is considered\nstable if it produces iterations in a bounded set with probability greater than 0. Unlike the deterministic case,\nestablishing stability in the face of randomness is not straightforward, especially when dealing with non-convex\nfunctions. This challenge motivates the exploration of an appropriate definition of non-Lipschitzness and the\ndevelopment of efficient algorithms for stochastic weakly convex optimization in this non-standard setting.\n1.1 Contributions\nWeshowthatcarefullychosenadaptivestepsizescaneffectivelydealwitharbitrarynon-Lipchitznessinstochas-\ntic model-based weakly convex optimization. Our contributions are as follows:\n1) When the Lipschitz constant is not uniformly bounded above but instead depends on a general growth\nfunction G(·), we design a novel adaptive stepsize strategy such that stochastic weakly convex optimization\nachieves the O(1/√\nK)convergence rate with a constant failure probability. Our analysis does not assume\nany specific form of G, such as those implied by (2). To our knowledge, this is the first result of stochastic\nweakly convex optimization for arbitrary non-Lipschitz objectives. Our analysis applies to a broad class of\nmodel-based algorithms [4, 10], including SGDas a special case.\n22) Even if the growth function Gis unknown, we show that it is still possible to achieve the same convergence\nguarantee. To this end, we introduce a new adaptive stepsize based on the concept of “reference Lipschitz\ncontinuity”, which allows us to estimate the Lipschitz parameter of a stochastic model function using local\nsamples. Our algorithm is highly flexible, and it can be applied to most of the weakly convex problems\nof interest. Moreover, our analyses can be extended to solving convex stochastic optimization without\nLipschitz continuity. A more detailed discussion is left to Section D .\nOther related works Adaptive stepsize strategy and gradient clipping are two important tools adopted\nin our algorithm framework. On the one hand, stepsize selection has been an important topic in stochastic\noptimization, and it has been justified that adaptive stepsize benefits stochastic first-order methods both in\ntheory and in practice [12, 18, 22, 16, 9, 17, 26]. On the other hand, gradient clipping [31] will be employed\nas a technique in the paper. In theory, gradient clipping was initially identified as a tool to solve problems\nwith generalized Lipschitz smoothness condition [20, 30, 31]. Recent works [14, 19] show that gradient clipping\ncan effectively deal with problems with heavy-tail noise. It is also observed that gradient clipping improves\nrobustness and stability of SGD[24] in stochastic convex optimization. In our analysis, a generalized version\nof gradient clipping is developed to alleviate the instability arising from stochastic noise.\n2 Preliminaries\nNotations Throughoutthepaper ∥·∥and⟨·,·⟩denotetheEuclideaninnerproductandnorm. Subdifferential\noffis given by ∂f(x):={v:f(y)≥f(x) +⟨v, y−x⟩+o(∥x−y∥), y→x}andf′(x)∈∂f(x)is called a\nsubgradient. A growth function G(·)is a continuous non-decreasing function mapping from R+toR+.\nEnvelope smoothing For weakly convex optimization, our analysis adopts the Moreau envelope as the\npotential function. Let fbe aλ-weakly convex function. Given ρ > λ, the Moreau envelope and the associated\nproximal mapping of fare given by\nf1/ρ(x):= min\ny\b\nf(y) +ρ\n2∥x−y∥2\t\nand proxf/ρ(x):= arg min\ny\b\nf(y) +ρ\n2∥x−y∥2\t\n.\nThe Moreau envelope can be interpreted as a smooth approximation of the original function. f1/ρ(x)is\ndifferentiable and its gradient is ∇f1/ρ(x) =ρ(x−proxf/ρ(x)). If∥∇f1/ρ(x)∥ ≤ε, then xis in the proximity\nof a near stationary point proxf/ρ(x)[4]. An important observation is that, existence of f1/ρ(x)only relies on\nweak convexity, instead of Lipschitz continuity of f.\nModel-based optimization Our main algorithm will be presented in a “model-based” fashion, which en-\ncompasses several first-order methods, including the most widely used (proximal) subgradient method. Model-\nbased optimization [4] contains two components: a stochastic model function and a stepsize (regularization)\nparameter. In each iteration, we can construct a local approximation of f(x)based on random sample ξkand\nthe current iterate xk. The stochastic function, denoted by fxk(·, ξk) +ω(x), is called a model function. Then\nwe take parameter γkand minimize this local approximation under quadratic regularizationγk\n2∥x−xk∥2to\nobtain the next iterate xk+1. Typical models include\n•(Sub)gradient. fx(y, ξ) =f(x, ξ) +⟨f′(x, ξ), y−x⟩,E[f′(x, ξ)] =f′(x)∈∂f(x)\n•Prox-linear [11]. f(x, ξ) =h(c(x, ξ))andfx(y, ξ) =h(c(x, ξ) +⟨∇c(x, ξ), y−x⟩)\n•Truncated [2]. fx(y, ξ) = max {f(x, ξ) +⟨∇f(x, ξ), y−x⟩, ℓ}, where ℓis a known lower-bound of model\nAlgorithm 1 summarizes model-based optimization.\n3Algorithm 1: Stochastic model-based optimization\nInput x1\nfork =1, 2,...do\nSample data ξkand choose the proximal regularizer weight γk>0\nxk+1= arg min\nxn\nfxk(x, ξk) +ω(x) +γk\n2∥x−xk∥2o\n. (3)\nend\nWe see that 1)model function fx(·, ξ);2)regularization parameter γkare two core components for our\nalgorithm design. Throughout this paper, we show how properly chosen γkimproves convergence beyond\nLipschitz continuity. We start by making assumptions.\nAssumptions. We make the following assumptions across the paper.\nA1:It is possible to generate i.i.d. samples {ξk}.\nA2: ω(x)isκ-weakly convex and Lω-Lipschitz continuous for all x∈domω.\nA3:Eξ[fx(x, ξ)] = f(x)for all x∈domωandEξ[fx(y, ξ)−f(y)]≤τ\n2∥x−y∥2for all x, y∈domω.\nMoreover, fx(y, ξ)is convex for all x, y∈domωandξ∼Ξ.\nA4: ψ1/ρ(x) = min z{ψ(z) +ρ\n2∥z−x∥2}is lower bounded by −Λ≤0.\nA5:Thev-level-set of ψ1/ρ(x):Lv={x:ψ1/ρ(x)≤v}has a bounded diameter diam (Lv)≤Bv<∞.\nRemark 1.The first four assumptions are general for weakly convex optimization. A5is satisfied when ψis\ncoercive [21], and it will be invoked to deal with non-Lipschitzness. Although A5excludes some special cases\nsuch as interpolation, often these cases can be handled by exploiting different analyses [21]. In A3, while we\nadopt a convex model function to simplify the analysis, it can be extended to weakly convex model functions.\nRemark 2.GivenA1toA3, it follows that f(x)isτ-weakly convex and ψ(x)is(τ+κ)weakly convex.\nStructure of the paper Section 3 discusses the convergence of weakly convex optimization with the\nstandard Lipschitz condition, which serves as a benchmark to provide sufficient intuition for the algorithm\ndesign in more challenging scenarios. Section 4 and5discuss two cases where standard Lipschitz condition\nfails to hold. Section 6 conducts numerical experiments to verify our results.\n3 Weakly convex optimization under standard Lipschitzness\nIn this section, we consider the standard case as a benchmark and assume that\nB1: fx(x, ξ)−fx(y, ξ)≤Lf(ξ)∥x−y∥for all x, yandξ∼Ξ.E[Lf(ξ)2]≤L2\nf.\nThis assumption is common in nonsmooth optimization, and the following descent property is known.\nLemma 3.1. Suppose that A1toA3as well as B1holds, then given ρ > κ +τ, γk> ρ,Ek[ψ1/ρ(xk+1)]≤\nψ1/ρ(xk)−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+2ρL2\nf\n(γk−ρ)(γk−κ),where Ek[·]:=E[·|ξ1, . . . , ξk]denotes the conditional expectation\ntaken over ξ1, . . . , ξk.\nTaking into account that γkis generally taken to be much larger than the other constants in the algorithm,\nLemma 3.1 reveals the following relation\nEk[ψ1/ρ(xk+1)]≤ψ1/ρ(xk)− O(γ−1\nk)∥∇ψ1/ρ(xk)∥2+O(L2\nfγ−2\nk), (4)\n4where O(L2\nfγ−2\nk)characterizes the error from both stochastic noise and nonsmoothness. Taking γk≡ O(√\nK)\nand telescoping the relation of Lemma 3.1 , we get the following convergence result.\nTheorem 3.1. Under the same conditions as Lemma 3.1 , if we take γk≡ρ+κ+α√\nK, then we have\nmin1≤k≤KE[∥∇ψ1/ρ(xk)∥2]≤2ρ\nρ−τ−κ\u0002ρD\nK+1√\nK\u0000\nαD+2ρL2\nf\nα\u0001\u0003\n,where D=ψ(x1)−infxψ(x).\nTheorem 3.1 is standard in the literature [4, 10]. One important intuition we want to establish is that, the\nchoice of γk≡ O(√\nK)is a consequence of the following trade-off: suppose we telescope over (4) directly, then\n1\nKKX\nk=1O(γ−1\nk)E[∥∇ψ1/ρ(xk)∥2]≤ O\u00101\nK\u0011\n+1\nKKX\nk=1O(L2\nfγ−2\nk).\nFirst, we need large, in other words, conservative γk, such that the error of potential reduction O(P\nkL2\nfγ−2\nk)is\nproperlybounded. Butthecostofbeingconservativeisthereducedamountofreduction O(γ−1\nk)∥∇ψ1/ρ(xk)∥2.\nNow that due to B1,Lfis exactly bounded by constant. Finally, the optimal trade-off finally gives the choice\nofγk≡ O(√\nK), and O(1/√\nK)rate of convergence. As we will show in the following sections, in face of\nnon-Lipschitzness, the error O(L2\nfγ−2\nk)cannot be bounded by choosing some constant γk. What we do is find\nsuitably large γk, adaptively, to reduce this error. Using A5and a probabilistic analysis, we manage to do\nthis without compromising the convergence rate.\n4 Weakly convex optimization under generalized Lipschitzness\nBefore achieving our goal of solving non-Lipschitz weakly convex optimization problems, we start from a less\nchallenging case of non-Lipschitzness characterized as follows.\nC1: fx(y, ξ)−fx(z, ξ)≤Lf(ξ)G(∥x∥)∥y−z∥for all x, y, z;ξ∼ΞandE[Lf(ξ)2]≤L2\nf; Recall that growth\nfunction Gis monotonically increasing.\nThis assumption implies that our model function is globally Lipschitz, but the Lipschitz constant has a known\ndependency on the norm of the expansion point x. Our analysis applies if we can estimate an upper bound of\nG. But for the brevity of analysis, we take this upper bound to be Gitself. Many real-life applications have\nthis structure, especially if the source of non-Lipschitzness is a high-order polynomial.\nExample 4.1 (Phase retrieval) .Consider f(x, ξ) =|⟨a, x⟩2−b|, a∈Rn, b∈R+. The subgradient model\nfx(y, ξ) =⟨f′(x, ξ), y−x⟩=⟨2·sign(⟨a, x⟩ −b)⟨a, x⟩a, y−x⟩satisfies\nfx(y, ξ)−fx(z, ξ)≤2∥a∥2∥x∥ · ∥y−z∥=Lf(ξ)G(∥x∥)∥y−z∥,\nwhere Lf(ξ) = 2∥a∥2,G(∥x∥) =∥x∥.\nExample4.2 (Subgradientmethod) .SGDcorresponds to the model function fx(y, ξ) =f(x, ξ)+⟨f′(x, ξ), y−x⟩.\nThen,C1is satisfied when ∥f′(x, ξ)∥ ≤Lf(ξ)G(∥x∥). It follows that E[∥f′(x, ξ)∥2]≤L2\nfG2(∥x∥). One can\nreadily see that relation (2)corresponds to the special case of G2(·)being a linear function.\nOne direct consequence of C1is that the Lipschitz constant of fx(·, ξ)can go to ∞when∥x∥ → ∞. Moreover,\nwe cannotdirectly relyon Lipschitzness of f(x). Taking subgradientupdate asanexample, thisimplies f′(x, ξ)\nhas a large norm, leading to a higher chance of divergence. From the perspective of convergence analysis, the\nerror term O(L2\nfγ−2\nk)in (4) becomes hard to bound, and Lemma 4.1 quantifies this hardness.\nLemma 4.1. SupposeA1toA3as well as C1holds, then given ρ > κ +τ, γk> ρ,\nEk[ψ1/ρ(xk+1)]≤ψ1/ρ(xk)−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+ρ(G(∥xk∥)Lf+Lω)2\n2γk(γk−κ),\nwhere γkis independent of ξk.\n5According to Lemma 4.1 , the error of potential reduction involves the norm of the current iteration, which\nmakes the previous constant stepsize analysis invalid, since we cannot assume G(∥xk∥)is bounded. As a\nnatural fix, we can take\nγk=O\u0000\nG(∥xk∥)√\nK\u0001\nto reduce the error. However, according to the trade-off we previously mentioned, unless G(∥xk∥)is bounded\nby some constant independent of K, the reduction in the potential function can be arbitrarily small, and we\nstill cannot obtain a valid convergence rate. To resolve this issue, we essentially need to show the boundedness\nof{∥xk∥}, and our solution is to associate boundedness of {∥xk∥}with another bounded quantity during the\nalgorithm: E[ψ1/ρ(xk)]. Intuitively, since E[ψ1/ρ(xk)]is reduced during the algorithm, it remains bounded\non expectation, and from A5we know that boundedness of {ψ1/ρ(xk)}implies boundedness of ∥xk∥, giving\nbounded G(∥xk∥)and the O(1/√\nK)rate we want. The following asymptotic result confirms our intuition.\nTheorem 4.1. Under the same conditions as Lemma 4.1 andA4,A5, ifγk=ρ+κ+ (G(∥xk∥) + 1) kζ, ζ∈\n(1\n2,1), then as k→ ∞,{∥xk∥}is bounded with probability 1; Moreover, the sequence {infj≤k∥∇ψ1/ρ(xj)∥}\nconverges to 0almost surely.\nWhile it is relatively easy to show convergence asymptotically, to obtain a convergence rate, we need a more\ncareful analysis of the algorithm behavior. One major difficulty is that boundedness of E[ψ1/ρ(xk)]is unable\ndirectly to give us information of ∥xk∥, since this relation holds only on expectation. To deal with this issue,\nwe resort to probabilistic tools. And we establish a new probabilistic argument in the following subsection.\n4.1 Stability of the iterations\nIn this subsection, we aim to analyze the stability of the iterates of a stochastic algorithm on a non-Lipschitz\nfunction. Theintuitionisverysimple: ifastochasticalgorithmreducessomepotentialfunctionwithabounded\nlevel-set, then the iterates will stay in a bounded region with high probability. We provide the basic sketch of\nproof and leave a more rigorous argument to the appendix.\nOur analysis relies on two simple facts that we gain from the adaptive stepsize.\nLemma 4.2 (Informal) .Under the same conditions as Lemma 4.3 , ifγk=O\u0000\n(G(∥xk∥) + 1)√\nK\u0001\n, then we\nhave, for all k= 2, . . . , K, that\nE[∥xk+1−xk∥]≤ O(1√\nK) (5)\nE[ψ1/ρ(xk)]≤ O(1) (6)\nRelation (5) says that with our adaptive stepsize strategy, we are very careful exploring the feasible region\nand at each iteration we only take a tiny step of O(1/√\nK). The second relation (6) comes directly from\nLemma 4.1 . Indeed, with γk=O\u0000\n(G(∥xk∥) + 1)√\nK\u0001\nwe haveρ(G(∥xk∥)Lf+Lω)2\n2γk(γk−κ)=O(1/K). And if we\ntelescopeLemma 4.1 and take expectation over all the randomness for k= 2, . . . K, we have\nE[ψ1/ρ(xk)]≤Pk\nj=1O(1/K) =O(1), k = 2, . . . , K.\nEach of the two relations alone may not tell us useful information, since they both hold on expectation. How-\never, when they are combined, a more useful result is available. Our argument is as follows:\nConsider the event “ ∥xk∥is large” and we wish to upper-bound its probability. We have the following facts:\n1. If∥xk∥is large, ∥xk+1∥ ≥ ∥ xk∥ − O (1/√\nK)by triangle inequality and ∥xk+1∥is also large.\n2. If∥xk+1∥is large, then ψ1/ρ(xk+1)is large by A5.\n3.E[ψ1/ρ(xk+1)]is bounded by some constant.\n6In other words, conditioned on the event “ ∥xk∥is large”, to ensure that E[ψ1/ρ(xk+1)]is still bounded, either\ncase 1)the event happens with low probability, or case 2) xk+1has to immediately jump back to a bounded\nregion of smaller radius. However, since our adaptive stepsize restricts the “jump” between two consecutive\niterations, case 2) cannot happen. Therefore, it is unlikely that “ ∥xk∥is large”.\nThis argument brings us the following tail-bound characterizing the behavior of ∥xk∥as a random variable.\nLemma 4.3. Under the same conditions as Lemma 4.1 as well as A4,A5, if we take γk=ρ+κ+τ+\nα(G(∥xk∥) + 1)√\nK, then the tail bound\nPn\n∥xk∥ ≥Ba∆+4(Lf+Lω)\nα√\nKo\n≤2∆\na∆ + Λ,\nholds for all 2≤k≤K, where ∆ =ψ1/ρ(x1) + Λ +ρ(Lf+Lω)2\nα2 >0and recall that diam(La∆)≤Ba∆.\nLemma 4.3 provides a useful characterization of the tail probability on the norm of the iterations. Now that\nthe bound holds for all xk, we can immediately condition on the event that Θ(K)iterations lie in the bounded\nset, to retrieve an O(1/√\nK)convergence rate.\nTheorem 4.2. Under the same conditions as Lemma 4.3 , given δ∈(0,1/4), at least with probability\n1−p, p∈(2δ,1),(1−2p−1δ)Kiterations will lie in the ball with radius R(δ) =Bδ−1∆+4(Lf+Lω)\nα√\nKand\nmin\n1≤k≤KE[∥∇ψ1/ρ(xk)∥2]≤p\np−2δ·2ρ\nρ−τ−κh\nD+ρ\n2α2(Lf+Lω)2i\u0012ρ+τ+κ\nK+α(Gδ+ 1)√\nK\u0013\n,\nwhere Gδ:= max z≤R(δ)G(z).\nTheorem 4.2 shows that, with constant probability, we can retrieve O(1/√\nK)convergence rate after K\niterations. This probability argument can be further improved, for example, by running the algorithm inde-\npendently multiple times [5]. The analysis in this section serves as a step-stone for the next section, where we\ndeal with non-Lipschitz optimization even without knowing G(·).\n5 Weakly convex optimization under unknown Lipschitzness\nWhile the analysis in Section 4 already extends the solvability of weakly convex optimization to non-Lipschitz\nfunctions,itreliesontheknowledgeofanexplicitgrowthfunction G(∥x∥),whichboundsthelocalLipschitzness.\nHowever, inmanyrealscenarios, eitheraccessto G(∥x∥)isnotviable, ortheboundlacksapredefinedfunctional\nform. In these scenarios, we assume that the growth function is unknown a priori.\nD1:For all fixed x∈domω,fx(z, ξ)−fx(y, ξ)≤Lip(x, ξ)∥z−y∥for all y, z;ξ∼Ξ.\nAlthough D1andC1look similar, they are very different in nature. The most direct consequence is that\nLip(x, ξ)issample-dependent,whichmeansanystepsizestrategybasedonitwillintroducebiasinthestochastic\nalgorithm. To resolve this issue, our new stepsize policy relies on constructing an estimator of Lip(x, ξ). We\nintroduce the property of “allowing a good estimation of Lipschitz constant”, which we call the reference\nLipschitz continuity .\n5.1 Reference Lipschitz continuity\nDefinition1 (ReferenceLipschitzcontinuity) .A stochastic model function fx(y, ξ)satisfies reference Lipschitz\ncontinuity if\n•Given an anchor x,fx(·, ξ)is globally Lipschitz with a Lipschitz constant Lip(x, ξ)\n7•Given ξ, ξ′∼Ξ,Eξ,ξ′[|Lip(x, ξ)−Lip(x, ξ′)|2]≤σ2<∞..\nThe first property is determined by the stochastic model function, and is common since most model functions\nare compositions of Lipschitz functions and linear expansions. The second property states that the expected\ndifference between models’ Lipschitzness is bounded by noise parameter σ. One direct outcome of reference\nLipschitz continuity is that we can cheaply estimate Lip(x, ξ)based on Lip(x, ξ′), where ξandξ′are two\nindependent samples drawn from Ξ.\nExample 5.1 ((Sub)gradient) .fx(y, ξ) =f(x, ξ) +⟨∇f(x, ξ), y−x⟩\nThe model is globally Lipschitz with Lip(x, ξ) =∥∇f(x, ξ)∥. IfE[∥∇f(x)− ∇f(x, ξ)∥2]≤σ2, then\nE[|Lip(x, ξ)−Lip(x, ξ′)|]≤E[∥∇f(x, ξ)− ∇f(x, ξ′)∥]≤2σ.\nEven if fis nonsmooth, the property may still hold. One example is the composition problem f(x, ξ) =\nh(c(x, ξ)), where hisLh-Lipschitz continuous and cis smooth. Then ∂f(x, ξ) =∇c(x, ξ)∂h(c(x, ξ)). If we\nestimate the Lipschitz constant with Lh∥∇c(x, ξ)∥, then E[|Lip(fx(·, ξ))−Lip(x, ξ′)|]≤2Lhσ.\nExample 5.2 (Proximal linear) .fx(y, ξ) =h(c(x, ξ) +⟨∇c(x, ξ), y−x⟩)\nWhen hisLhLipschitzcontinuous, themodelisgloballyLipschitzwith Lip(x, ξ) =Lh∥∇c(x, ξ)∥. IfE[∥∇c(x)−\n∇c(x, ξ)∥2]≤σ2, then E[|Lip(x, ξ)−Lip(x, ξ′)|]≤LhE[���∇c(x, ξ)− ∇c(x, ξ′)∥]≤2Lhσ.\nExample 5.3 (Truncated model) .fx(y, ξ) = max {f(x, ξ) +⟨∇f(x, ξ), y−x⟩, ℓ}.\nThe model is ∥∇f(x, ξ)∥-Lipschitz and the reasoning of reference Lipschitz continuity is the same as in Ex-\nample 5.1 . Note that the truncated model encompasses stochastic Polyak stepsize as a special case [28].\nIn this section, we would assume that our model satisfies the reference Lipschitz continuity.\nD2:The stochastic model fx(·, ξ)satisfies the reference Lipschitz continuity with noise parameter σ.\n5.2 Algorithm design and analysis\nAs we did in Section 4 , before getting down to the algorithm design, we first need to see what happens to\nour potential reduction in this new setting. Firstly, Lemma 5.1 characterizes the descent property of our\npotential function under the assumption that γkis independent of ξk.\nLemma 5.1. SupposeA1toA3as well as D1,D2hold, then given ρ > κ +τ,\nEk[ψ1/ρ(xk+1)]≤ψ1/ρ(xk)−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+Ekhρ\n2γk(γk−κ)(Lip(xk, ξk) +Lω)2i\n,\nwhere γkis chosen to be independent of ξkand is considered deterministic here.\nCompared to Lemma 4.1 , bounding the error termρ(Lip(fxk(·,ξ))+Lω)2\n2γk(γk−κ)becomes more challenging due to the\nlack of information about its growth. However, thanks to the reference Lipschitz continuity, by sampling ξ′, an\nindependent copy of ξk, we can utilize Lip(xk, ξ′)as a surrogate for Lip(xk, ξk). The preference of Lip(xk, ξ′)\nover Lip(xk, ξk)is driven by the fact that Lip(xk, ξk)is correlated with xk+1, which significantly complicates\nthe analysis [1]. To mitigate the impact of large noise σon the accuracy of our estimation, we clip the estimator\nby a threshold α >0:max{Lip(xk, ξ′), α}. Consequently, we set\nγk=O(max{Lip(xk, ξ′), α} ·√\nK).\nRemark 3.When fx(y, ξ) =f(x, ξ) +⟨∇f(x, ξ), y−x⟩andξ=ξ′, we retrieve the gradient clipping technique.\nOur approach can be seen as a generalization of gradient clipping to the proximal setting.\nThe next theorem establishes the asymptotic result and confirms our intuition.\n8Theorem 5.1. Under the same conditions as Lemma 5.1 ,A4,A5, ifγk=ρ+κ+τ+max {Lip(xk, ξ′), α}kζ,\nζ∈(1\n2,1), ask→ ∞,{∥xk∥}is bounded with probability 1;{infj≤k∥∇ψ1/ρ(xj)∥}converges to 0almost surely.\nTo obtain a non-asymptotic result, we again apply the probabilistic analysis to obtain the tail bound.\nLemma 5.2. Under the same conditions of Lemma 5.1 as well as A4,A5, if we take γk=ρ+τ+κ+\nmax{Lip(fxk(·, ξ′)), α}√\nK, then the tail bound\nPn\n∥xk∥ ≥Ba∆+4(α+σ+Lω)\nα√\nKo\n≤2∆\na∆ + Λ,\nholds for all 2≤k≤K, where ∆ =ψ1/ρ(x1) + Λ +ρ\nα2(α+σ+Lω)2>0.\nTheorem 5.2. Assuming the conditions of Lemma 5.2 hold, then given δ∈(0,1/4), with probability at least\n1−p, p∈(2δ,1),(1−2p−1δ)Kiterations will lie in the ball with radius R(δ) =Bδ−1∆+4(α+σ+Lω)\nα√\nKand\nmin\n1≤k≤KE[∥∇ψ1/ρ(xk)∥2]≤p\np−2δ·2ρ\nρ−τ−κh\nD+ρ\nα2(α+σ+Lω)2i\u0012ρ+τ+κ\nK+α+Gδ√\nK\u0013\n,\nwhere Gδ:= max xsupξ∼ΞLip(x, ξ),∥x∥ ≤R(δ).\n6 Experiments\nIn this section, we perform numerical experiments to demonstrate the effectiveness of our proposed methods.\nWe consider the following robust non-linear regression problem:\nmin\nx1\nmmX\ni=1|r(x, ai)−bi|=:1\nmmX\ni=1f(x, ξi), (7)\nwhere, given observations {ai}from A∈Rm×n, regression model r(x, a)and target label bi, we aim to fit\nthe model coefficient xgiven problem data. The following table summarizes our tested regression models and\ntheir Lipschitz properties. These three functions exhibit increasing non-Lipschitzness.\nTable 1: Nonlinear regression models. r(a, x) =⟨a, x⟩2represents the standard robust phase retrieval problem;\nr(a, x) =⟨a, x⟩5+⟨a, x⟩3+1is a high-order polynomial of ⟨a, x⟩; while e⟨a,x⟩+10exhibits exponential growth.\nLoss r(x, a) ∂f(x, ξ) G(∥x∥) Lip(fx(y, a))\nr1 ⟨a, x⟩2sign(⟨a, x⟩2−b)·2⟨a, x⟩a ∥x∥ 2|⟨a, x⟩| · ∥a∥\nr2⟨a, x⟩5+⟨a, x⟩3+ 1 sign( r2−b)·(5⟨a, x⟩4+ 3⟨a, x⟩2)a5(∥x∥4+∥x∥2) 5(∥a∥4∥x∥4+∥a∥2∥x∥2)\nr3 e⟨a,x⟩+ 10 sign( e⟨a,x⟩−b)·e⟨a,x⟩·a eA∥x∥(A= 3.0) e⟨a,x⟩· ∥a∥\n6.1 Experiment setup\nDataset . We let m= 300 , n= 100. Data generation is consistent with [10], where, given condition number\nparameter κ≥1, we compute A=QD, Q ∈Rm×n. Here each element of Qis drawn from N(0,1);D=\ndiag(d), d∈Rn, di∈[1/κ,1]for all i. Then a true signal ˆx∼ N(0, I)is generated, giving the measurements b\nby formula bi=r(x, ai). We randomly perturb pfail-fraction of the measurements with N(0,25)noise added\nto them to simulate data corruption.\n1) Dataset . We follow [10] and set set κ∈ {1,10}andpfail∈ {0.2,0.3}.\n92) Initial point . We generate x′∼ N(0, In)and start from x1=10x′\n∥x′∥forr1,x1=x′\n∥x′∥forr2, r3.\n3) Stopping criterion . We run algorithms for 400 epochs ( K= 400 m). Algorithms stop if f≤1.2f(ˆx).\n4) Stepsize . We let γk=θ·√\nKfor vanilla algorithms; γk=θ·G(∥xk∥)√\nKfor adaptive stepsize with known\ngrowth condition; γk=θ·max{Lip(xk, ξ′), α}√\nKfor adaptive stepsize with unknown growth condition.\nθ∈[10−2,101]serves as a hyper-parameter.\n5) Clipping . Clipping parameter αis set to 1.0.\n6) Mirror descent . For experiments on mirror descent, we construct kernels for r1, r2according to [6].\n6.2 Comparing different stepsizes\nWe conduct experiments to investigate the number of iterations for each stepsize to converge, under different\nchoices of θ. As the experiments suggest, when the function exhibits low-order growth, our adaptive choices\ntend to be conservative. However, when the function exhibits high-order growth, our adaptive stepsize tends\nto converge within a reasonable range of stepsizes. It is worth noticing that for problem r2,SGDdiverges for\nθ∼108, while our proposed approaches work robustly. Moreover, we notice that our adaptive stepsize based\non reference Lipschitz property never diverges in practice, although it is sometimes conservative on problems\nwhere function growth is mild (such as r1).\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\n10-210-1100101024681012104\nSPL\nSPL-G\nSPL-R\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\n10-210-1100101024681012104\nSPL\nSPL-G\nSPL-R\nFigure 2: Problem r1. Left two: (κ, p fail) = (10 ,0.2); Right two: (κ, p fail) = (10 ,0.3). x-axis: parameter θ;\ny-axis: number of iterations. SGDdenotes vanilla SGD; SGD-Gdenotes SGD adaptive to known Lipschitzness;\nSGD-Rdenotes SGD adaptive to unknown Lipschitzness. The same applies to SPL.\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\n10-210-1100101024681012104\nSPL\nSPL-G\nSPL-R\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\n10-210-1100101024681012104\nSPL\nSPL-G\nSPL-R\nFigure 3: Problem r2. Left two: (κ, p fail) = (1 ,0.2); Right two: (κ, p fail) = (10 ,0.3). x-axis: parameter θ;\ny-axis: number of iterations.\n6.3 Comparison with mirror descent\nLast, we compare our proposed method with the commonly adopted mirror descent approach for non-Lipschitz\nproblems. We test both mirror descent and our proposed SGD-based approaches.\nAsFigure 5 suggests, we see that mirror descent indeed often exhibits more stable performance compared to\nvanilla SGD. However, we see that our approaches still exhibit superior convergence performance. Moreover,\n1010-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\n10-210-1100101024681012104\nSPL\nSPL-G\nSPL-R\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\n10-210-1100101024681012104\nSPL\nSPL-G\nSPL-RFigure 4: Problem r3. Left two: (κ, p fail) = (1 ,0.2); Right two: (κ, p fail) = (1 ,0.3). x-axis: parameter θ; y-axis:\nnumber of iterations.\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\nMirror\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\nMirror\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\nMirror\n10-210-1100101024681012104\nSGD\nSGD-G\nSGD-R\nMirror\nFigure 5: Left two: Problem r1,(κ, p fail) = (1 ,0.3); Right two: Problem r2,(κ, p fail) = (1 ,0.3). x-axis:\nparameter θ; y-axis: number of iterations.\neach mirror descent iteration involves a root-finding subproblem, which is far more complicated than our\napproaches.\n7 Conclusions\nWe develop novel adaptive stepsize (regularization) strategies and show that for weakly convex objectives\nwithout Lipschitz continuity, stochastic model-based methods can still converge at the desirable O(1/√\nK)rate\nwith constant failure probability. To our knowledge, this is achieved under the least restrictive assumptions\nknowntodate. 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A unified analysis for the subgradient methods minimizing\ncomposite nonconvex, nonsmooth and non-lipschitz functions. arXiv preprint arXiv:2308.16362 , 2023.\n13Appendix\nTable of Contents\nA Proof of results in Section 3 15\nA.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\nA.2 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\nB Proof of results in Section 4 16\nB.1 Proof of Lemma 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\nB.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\nB.3 Proof of Lemma 4.2 and 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\nB.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\nC Proof of results in Section 5 21\nC.1 Auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\nC.2 Proof of Lemma 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\nC.3 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\nC.4 Proof of Lemma 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\nC.5 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\nD Stochastic convex optimization beyond Lipschitz continuity 25\nD.1 Convex optimization under standard Lipschitzness . . . . . . . . . . . . . . . . . . . . . . . 26\nD.2 Convex optimization under generalized Lipschitzness . . . . . . . . . . . . . . . . . . . . . . 26\nD.3 Convex optimization beyond Lipschitzness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\nD.4 Proof of results in Subsection D.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27\nD.5 Proof of results in Subsection D.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\nD.6 Proof of results in Subsection D.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\nStructure of the appendix The appendix is organized as follows. In Section A ,Section B andSec-\ntion C, we respectively prove the results for weakly convex optimization under different Lipschitz continuity\nassumptions. In Section D and its subsections, we extend our results to convex optimization.\n14A Proof of results in Section 3\nA.1 Proof of Lemma 3.1\nBy the optimality conditions of the proximal subproblems (3), we have, for any ξk∼Ξ, that\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(ˆxk, ξk) +ω(ˆxk) +γk\n2∥ˆxk−xk∥2−γk−κ\n2∥xk+1−ˆxk∥2\nf(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk∥2≤f(xk+1) +ω(xk+1) +ρ\n2∥xk+1−xk∥2\nSumming over the above two relations, we deduce that\nγk−ρ\n2∥xk+1−xk∥2−γk−ρ\n2∥ˆxk−xk∥2+γk−κ\n2∥xk+1−ˆxk∥2\n≤f(xk+1)−fxk(xk+1, ξk) +fxk(ˆxk, ξk)−f(ˆxk)\nConditioned on ξ1, . . . , ξk−1and taking expectation with respect to ξk, we have\nγk−ρ\n2Ek[∥xk+1−xk∥2]−γk−ρ\n2∥ˆxk−xk∥2+γk−κ\n2Ek[∥xk+1−ˆxk∥2]\n≤Ek[f(xk+1)−fxk(xk, ξk)] +Ek[L(ξk)∥xk+1−xk∥] +τ\n2∥ˆxk−xk∥2(8)\n=Ek[f(xk+1)]−f(xk) +Ek[L(ξk)∥xk+1−xk∥] +τ\n2∥ˆxk−xk∥2(9)\n≤LfEk[∥xk+1−xk∥] +Ek[L(ξk)∥xk+1−xk∥] +τ\n2∥ˆxk−xk∥2(10)\nwhere (8) uses Lf(ξ)-Lipschitzness of fxk(x, ξ); (9) uses quadratic bound from A3and (10) applies Lf-\nLipschitzness of f(x)[4]. Re-arranging the terms, we have\nγk−κ\n2Ek[∥xk+1−ˆxk∥2]\n≤γk−ρ+τ\n2∥ˆxk−xk∥2−γk−ρ\n2Ek[∥xk+1−xk∥2] +Ek[(L(ξk) +Lf)∥xk+1−xk∥]\n≤γk−ρ+τ\n2∥ˆxk−xk∥2+2L2\nf\nγk−ρ(11)\n=γk−κ\n2∥ˆxk−xk∥2−ρ−τ−κ\n2∥ˆxk−xk∥2+2L2\nf\nγk−ρ\nwhere (11) uses the relation −a\n2x2+bx≤b2\n2aandEξ[(L(ξ) +Lf)2]≤4L2\nf. Dividing both sides byγk−κ\n2,\nEk[∥xk+1−ˆxk∥2]≤ ∥ˆxk−xk∥2−ρ−τ−κ\nγk−κ∥ˆxk−xk∥2+4L2\nf\n(γk−ρ)(γk−κ)\n15and the potential function is reduced by\nEk[ψ1/ρ(xk+1)] = min\nx{f(x) +ω(x) +ρ\n2∥x−xk+1∥2}\n≤f(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk+1∥2\n≤f(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk∥2−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+2ρL2\nf\n(γk−ρ)(γk−κ)\n=ψ1/ρ(xk)−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+2ρL2\nf\n(γk−ρ)(γk−κ),\nwhich completes the proof.\nA.2 Proof of Theorem 3.1\nGiven fixed stepsize γk≡γ=ρ+κ+α√\nK, where we have, after telescoping, that\nρ(ρ−τ−κ)\n2(γ−κ)KX\nk=1E[∥ˆxk−xk∥2]≤ψ1/ρ(x1)−E[ψ1/ρ(xK+1)] +2ρL2\nfK\n(γ−ρ)(γ−κ).\nRe-arranging the terms and summing over k= 1, . . . , K, we have\nmin\n1≤k≤KE[∥∇ψ1/ρ(xk)∥2]≤2ρ\nρ−τ−κ\u0014(γ−κ)D\nK+2ρL2\nf\nγ−ρ\u0015\n≤2ρ\nρ−τ−κ\u0014ρD\nK+αD√\nK+2ρL2\nf\nα√\nK\u0015\n,\nwhere D=ψ1/ρ(x1)−infxψ(x)≥ψ1/ρ(x1)−E[ψ1/ρ(xK)]and this completes the proof.\nB Proof of results in Section 4\nFor brevity of notation, in the proof we define\nGk:=G(∥xk∥) (12)\nand use them interchangeably in this section. We also note that Gkis a random variable whose randomness\ncomes from samples from previous iterations ξ1, . . . , ξk−1.\nB.1 Proof of Lemma 4.1\nFirstly, we still use optimality condition to get, for a given ξk, that\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(ˆxk, ξk) +ω(ˆxk) +γk\n2∥ˆxk−xk∥2−γk−κ\n2∥xk+1−ˆxk∥2\nf(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk∥2≤f(xk) +ω(xk)\n16Summing over the above two relations, we deduce that\nγk\n2∥xk+1−xk∥2−γk−ρ\n2∥ˆxk−xk∥2+γk−κ\n2∥xk+1−ˆxk∥2\n≤f(xk)−fxk(xk+1, ξk) +fxk(ˆxk, ξk)−f(ˆxk) +Lω∥xk+1−xk∥ (13)\n≤f(xk)−f(xk, ξk) +fxk(ˆxk, ξk)−f(ˆxk) + (GkLf(ξk) +Lω)∥xk+1−xk∥, (14)\nwhere (13) applies Lω-Lipschitz continuity of ω(x); (14) applies C1. Dividing both sides byγk−κ\n2and re-\narranging the terms, we have\n∥xk+1−ˆxk∥2≤γk−ρ\nγk−κ∥ˆxk−xk∥2−γk\nγk−κ∥xk+1−xk∥2\n+2\nγk−κ[f(xk)−f(xk, ξk) +fxk(ˆxk, ξk)−f(ˆxk) + (GkLf(ξk) +Lω)∥xk+1−xk∥]\nNext conditioned on ξ1, . . . , ξk−1, taking expectation with respect to ξk, and recalling that G(∥xk∥), and\ntherefore γkis fixed given ξ1, . . . , ξk−1, we have\nEk[∥xk+1−ˆxk∥2]\n≤γk−ρ+τ\nγk−κ∥ˆxk−xk∥2+2\nγk−κEk[(GkLf(ξk) +Lω)∥xk+1−xk∥ −γk\n2∥xk+1−xk∥2](15)\n≤γk−ρ+τ\nγk−κ∥ˆxk−xk∥2+(GkLf+Lω)2\nγk(γk−κ)(16)\n=∥ˆxk−xk∥2−ρ−τ−κ\nγk−κ∥ˆxk−xk∥2+(GkLf+Lω)2\nγk(γk−κ),\nwhere (15) applies f(xk)−Ek[f(xk, ξk)] = 0 ,Ek[fxk(ˆxk, ξk)]−f(ˆxk)≤τ\n2∥xk−ˆxk∥2; (16) applies the relation\n−a\n2x2+bx≤b2\n2aandE[Lf(ξ)]2≤E[Lf(ξ)2]≤L2\nf. Now we can deduce reduction of the potential function by\nEk[ψ1/ρ(xk+1)] = min\nx{f(x) +ω(x) +ρ\n2∥x−xk+1∥2}\n≤f(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk+1∥2\n≤f(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk∥2−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+ρ(GkLf+Lω)2\n2γk(γk−κ)\n=ψ1/ρ(xk)−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+ρ(GkLf+Lω)2\n2γk(γk−κ)\nand this completes the proof.\nB.2 Proof of Theorem 4.1\nFirst we introduce the following lemma.\nLemma B.1 (Robbins-Siegmund [27]) .LetAk, Bk, CkandVkbe nonnegative random variables adapted to the\nfiltration Fkand satisfying E[Vk+1|Fk]≤(1+Ak)Vk+Bk−Ck. Then on the event {P∞\nk=1Ak<∞,P∞\nk=1Bk<\n∞}, there is a random variable V∞such that Vka.s.−→V∞andP∞\nk=0Ck<∞almost surely.\nNow we get down to the proof. Recall that in Lemma 4.1 we have shown that\nEk[ψ1/ρ(xk+1)]≤ψ1/ρ(xk)−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+ρ(GkLf+Lω)2\n2γk(γk−κ)\n17and we can boundGkLf+Lω\nγk−κ=GkLf+Lω\nρ+τ+kζ(Gk+ 1)≤GkLf+Lω\nkζ(Gk+ 1)≤Lf+Lω\nkζ(17)\nto get\nEk[ψ1/ρ(xk+1) + Λ] ≤[ψ1/ρ(xk) + Λ] −ρ(ρ−κ−τ)\n2(γk−κ)∥ˆxk−xk∥2+ρ\n2k2ζ(Lf+Lω)2\nThen we invoke Lemma B.1 , plugging in the relation\nAk= 0, B k=ρ(Lf+Lω)2\n2k2ζ, C k=ρ(ρ−κ−τ)\n2(γk−κ)∥ˆxk−xk∥2, V k=ψ1/ρ(xk) + Λ≥0(18)\nThen with ζ∈(1\n2,1)P∞\nk=1Bk=ρ(Lf+Lω)2\n2k2ζ<∞we know that {ψ1/ρ(xk) + Λ} →ψ1/ρ(x∞) + Λ <∞and\nthatP∞\nk=1ρ(ρ−κ−τ)\n2(γk−κ)∥ˆxk−xk∥2<∞. ByA5,∥xk∥is bounded with probability 1 and Gkis bounded almost\nsurely. FinallyP∞\nk=11\nγk−κ=∞ ⇒ infj≤k∥ˆxj−xj∥ → 0almost surely and this completes the proof since\n∥ˆxk−xk∥=ρ−1∥∇ψ1/ρ(xk)∥.\nB.3 Proof of Lemma 4.2 and 4.3\nFollowing (17), we first we bound the error of potential reduction byρ(GkLf+Lω)2\n2γk(γk−κ)≤ρ(Lf+Lω)2\nα2K.\nThen a telescopic sum gives, for all 2≤k≤Kthat\nE[ψ1/ρ(xk) + Λ] ≤ψ1/ρ(x1) + Λ +ρ(Lf+Lω)2\nα2=: ∆.\nHere ∆is a constant that only depends on the initialization of the algorithm. Next we consider one step of\nthe algorithm\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(xk, ξk) +ω(xk)\nand a re-arrangement gives\nγk\n2∥xk+1−xk∥2≤fxk(xk, ξk)−fxk(xk+1, ξk) +ω(xk)−ω(xk+1)\n≤(Lf(ξk)Gk+Lω)∥xk+1−xk∥,\nwhere we use C1and Lipschitz continuity of ω(x). Dividing both sides by ∥xk+1−xk∥, we have\n∥xk+1−xk∥ ≤2(Lf(ξk)Gk+Lω)\nγk≤2(Lf(ξk)Gk+Lω)\nα(Gk+ 1)√\nK.\nConditioned on ξ1, . . . , ξk−1and taking expectation with respect to ξk, we get\nEk[∥xk+1−xk∥]≤2Eξk[Lf(ξk)]Gk+ 2Lω\nα(Gk+ 1)√\nK≤2LfGk+ 2Lω\nα(Gk+ 1)√\nK≤2(Lf+Lω)\nα√\nK.\nThis completes the proof of Lemma 4.2 . By Markov’s inequality, we know that, for any 2≤k≤K, the\nfollowing bound holds\nPξk∼Ξn\n∥xk+1−xk∥ ≤4(Lf+Lω)\nα√\nK\f\fξ1, . . . , ξk−1o\n≥1\n2\nand without loss of generality we let Z=4(Lf+Lω)\nα√\nK, and clearly Z=O(1/√\nK) =O(1).\nThis relation says “it’s likely that xkandxk+1are close”, and we leverage this intuition to derive a tail-bound\n18on∥xk∥. So far, we have the following properties in hand:\n1.E[ψ1/ρ(xk)]is bounded by a constant ∆−Λfor all k\n2. If∥xk∥ ≥Bv, then ψ1/ρ(xk)≥v\n3. If∥xk∥ ≥Bv, it’s likely that ∥xk+1∥ ≥Bv− O(1/√\nK).\nOur reasoning is as follows: given large a >1, conditioned on ∥xk∥ ≥Ba∆, then it is likely that ∥xk+1∥ ≈Ba∆\nsince∥xk+1−xk∥is likely to be small. And it implies E[ψ1/ρ(xk+1)]≥a∆>∆. However, we know that\nE[ψ1/ρ(xk+1)]≤∆, and this will therefore reversely bound the probability that ∥xk∥ ≥Ba∆+O(1/√\nK). We\nformalize the proof as follows.\nFirst recall that by A5,∥xk∥ ≥ Bvimplies ψ1/ρ(xk)≥v. Taking v=a∆, a > 1, we have ∥xk∥ ≥ Ba∆⇒\nψ1/ρ(xk)≥av. Now we consider the event ∥xk∥ ≥Ba∆+Zand apply law of total expectation to get\n∆≥E[ψ1/ρ(xk+1) + Λ]\n=E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z]·P{∥xk∥ ≥Ba∆+Z}\n+E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≤Ba∆+Z]·P{∥xk∥ ≤Ba∆+Z}\n≥E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z]·P{∥xk∥ ≥Ba∆+Z}, (19)\nwhere (19) uses ψ1/ρ(x) + Λ≥0for all x. Next we consider the expectation\nE\u0002\nψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z\u0003\n, (20)\nand successively deduce that\nE[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z]\n=E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z,∥xk+1−xk∥ ≤Z]·P{∥xk+1−xk∥ ≤Z}\n+E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z,∥xk+1−xk∥ ≥Z]·P{∥xk+1−xk∥ ≥Z}\n≥1\n2E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z,∥xk+1−xk∥ ≤Z] (21)\n≥1\n2(a∆ + Λ) , (22)\nwhere (21) is by Markov’s inequality P{∥xk+1−xk∥ ≤Z} ≥0.5and (22) uses the triangle inequality\n∥xk+1∥=∥xk+1−xk+xk∥ ≥ ∥ xk∥ − ∥ xk+1−xk∥ ≥ ∥ xk∥ −Z≥Ba∆\nand we recall that ∥xk+1∥ ≥Ba∆implies ψ1/ρ(xk+1)≥a∆. Chaining the above inequalities, we arrive at\n∆≥\u0000a∆+Λ\n2\u0001\n·P{∥xk∥ ≥Ba∆+Z}\nDividing both sides bya∆+Λ\n2gives the desired tail bound\nPn\n∥xk∥ ≥Ba∆+Zo\n≤2∆\na∆ + Λ.\nSince Λ≥0, the bound is nontrivial when a >2, and this completes the proof.\n19B.4 Proof of Theorem 4.2\nGiven δ∈(0,1/4), we know, from Lemma 4.3 that for 2≤k≤K,\nP\b\n∥xk∥ ≥Bδ−1∆+Z\t\n≤2∆\nδ−1∆ + Λ≤2δ.\nThen denote Ik=I\b\n∥xk∥ ≤Bδ−1∆+Z\t\n, and we havePK\nk=1E[1−Ik]≤2δK. By Markov’s inequality, given\np∈(2δ,1),\nP(KX\nk=1(1−Ik)≥2δK\np)\n≤2δK\n2p−1δK=p\nand a re-arrangement gives P\bPK\nk=1Ik≥K(1−2p−1δ)\t\n≥1−p. Now we telescope Lemma 4.1 and get\nKX\nk=1E\u0014ρ(ρ−κ−τ)\n2γk∥ˆxk−xk∥2\u0015\n≤ψ1/ρ(x1)−E[ψ1/ρ(xK+1)] +ρ\n2α2(Lf+Lω)2(23)\n≤ψ1/ρ(x1)−inf\nxψ(x) +ρ\n2α2(Lf+Lω)2, (24)\n=D+ρ\n2α2(Lf+Lω)2\nwhere (23) uses the previously established bound Ekh\nρ(GkLf+Lω)2\n2γk(γk−κ)i\n≤ρ(Lf+Lω)2\nα2K. (24) uses the fact that\nψ1/ρ(xK+1)≥infxψ(x). Next we notice, conditioned on the eventPK\nk=1Ik≥K(1−2p−1δ), defining Gδ:=\nmax zG(z), z≤Bδ−1∆+Z, that\nKX\nk=1E\u0014ρ(ρ−κ−τ)\n2γk∥ˆxk−xk∥2\u0015\n=X\nk∈{j:Ij=0}E\u0014ρ(ρ−κ−τ)\n2γk∥ˆxk−xk∥2\u0015\n+X\nk∈{j:Ij=1}E\u0014ρ(ρ−κ−τ)\n2γk∥ˆxk−xk∥2\u0015\n≥X\nk∈{j:Ij=1}E\u0014ρ(ρ−κ−τ)\n2γk∥ˆxk−xk∥2\u0015\n≥X\nk∈{k:Ij=1}ρ(ρ−κ−τ)\n2(ρ+κ+τ+α(Gδ+ 1)√\nK)E[∥ˆxk−xk∥2] (25)\n≥ρ(ρ−κ−τ)(1−2p−1δ)K\n2(ρ+κ+τ+α(Gδ+ 1)√\nK)min\nk∈{j:Ij=1}E[∥ˆxk−xk∥2], (26)\nwhere (25) applies the fact that if Ij= 1,∥xj∥ ≤Gδ; (26) utilizes the fact that |{k:Ij= 1}| ≥K(1−2p−1δ).\nPutting the inequalities together, we have\nmin\nk∈{j:Ij=1}E[∥ˆxk−xk∥2]≤2(ρ+κ+τ+α(Gδ+ 1)√\nK)\nρ(ρ−κ−τ)(1−2p−1δ)K\u0014\nD+ρ(Lf+Lω)2\n2α2\u0015\n=2(D+ρ(Lf+Lω)2\n2α2)\nρ(ρ−κ−τ)(1−2p−1δ)\u0014ρ+κ+τ\nK+α(Gδ+ 1)√\nK\u0015\n,\n20Recalling that ∥ˆxk−xk∥=ρ−1∥∇ψ1/ρ(xk)∥, at least with probability 1−p,\nmin\n1≤k≤KE[∥∇ψ1/ρ(xk)∥2]≤ min\nk∈{k:Ik=1}E[∥∇ψ1/ρ(xk)∥2]\n≤p\np−2δ·2ρ\nρ−τ−κ\u0014\nD+ρ(Lf+Lω)2\n2α2\u0015\u0012ρ+τ+κ\nK+α(Gδ+ 1)√\nK\u0013\nand this completes the proof.\nC Proof of results in Section 5\nFor brevity of expression, we define Lk\nf:=Lip(xk, ξk), L′\nf:=Lip(xk, ξ′)(kis hidden when clear from the\ncontext) and use them interchangeably.\nC.1 Auxiliary lemmas\nLemma C.1. Given independent nonnegative random variables XandY. IfEX,Y[|X−Y|2]≤σ2,then\nEX,Yh\nX2\nmax{Y2,α2}i\n≤\u0000σ+α\nα\u00012,EX,Yh\nX\nmax{Y2,α2}i\n≤σ\nα2+1\nα,EX,Yh\nX\nmax{Y,α}i\n≤σ\nα+ 1\nwhere α >0.\nProof.For the first relation, we successively deduce that\nEX,Yh\nX2\nmax{Y2,α2}i\n=EX,Yh\n(X−Y+Y)2\nmax{Y2,α2}i\n=EX,Yh\n(X−Y)2+Y2+2Y(X−Y)\nmax{Y2,α2}i\n=EX,Yh\n(X−Y)2\nmax{Y2,α2}i\n+EYh\nY2\nmax{Y2,α2}i\n+EX,Yh\n2Y(X−Y)\nmax{Y2,α2}i\n≤σ2\nα2+ 1 + EX,Yh\n2Y|X−Y|\nmax{Y,α}·max{Y,α}i\n(27)\n≤σ2\nα2+2σ\nα+ 1 =\u0000σ+α\nα\u00012,\nwhere (27) usesa\nmax{b,c}≤a\ncand the last inequality applies\n2Y|X−Y|\nmax{Y,α}·max{Y,α}=2Y\nmax{Y,α}·|X−Y|\nmax{Y,α}≤2·|X−Y|\nα. (28)\nSimilarly we can deduce that\nEX,Yh\nX\nmax{Y2,α2}i\n≤EX,Yh\n|X−Y|\nmax{Y2,α2}i\n+EYh\nY\nmax{Y2,α2}i\n≤σ\nα2+1\nα,\nEX,Yh\nX\nmax{Y,α}i\n≤EX,Yh\n|X−Y|\nmax{Y,α}i\n+EX,Yh\nY\nmax{Y,α}i\n≤σ\nα+ 1,\nwhich completes the proof.\nC.2 Proof of Lemma 5.1\nBy the optimality condition, we have\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(ˆxk, ξk) +ω(ˆxk) +γk\n2∥ˆxk−xk∥2−γk−κ\n2∥xk+1−ˆxk∥2\n21f(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk∥2≤f(xk) +ω(xk)\nand summation over the two relations gives\nγk\n2∥xk+1−xk∥2−γk−ρ\n2∥ˆxk−xk∥2+γk−κ\n2∥xk+1−ˆxk∥2\n≤f(xk)−fxk(xk+1, ξk) +fxk(ˆxk, ξk)−f(ˆxk) +Lω∥xk+1−xk∥\n≤f(xk)−f(xk, ξk) +fxk(ˆxk, ξk)−f(ˆxk) + (Lk\nf+Lω)∥xk+1−xk∥, (29)\nwhere (29) applies D1. Fixing ξ′, we divide both sides byγk−κ\n2and get\n∥xk+1−ˆxk∥2≤γk−ρ\nγk−κ∥ˆxk−xk∥2−γk\nγk−κ∥xk+1−xk∥2\n+2\nγk−κ[f(xk)−f(xk, ξk) +fxk(ˆxk, ξk)−f(ˆxk) + (Lk\nf+Lω)∥xk+1−xk∥]\nConditioned on ξ1, . . . , ξk−1and taking expectation with respect to ξk, we have\nEk[∥xk+1−ˆxk∥2]\n≤γk−ρ+τ\nγk−κ∥ˆxk−xk∥2+2\nγk−κEk[(Lk\nf+Lω)∥xk+1−xk∥ −γk\n2∥xk+1−xk∥2] (30)\n≤γk−ρ+τ\nγk−κ∥ˆxk−xk∥2+Ekh1\nγk(γk−κ)(Lk\nf+Lω)2i\n(31)\nwhere (30) again uses f(xk)−Ek[f(xk, ξk)] = 0 ,Ek[fxk(ˆxk, ξk)]−f(ˆxk)≤τ\n2∥xk−ˆxk∥2; (31) uses the relation\n−a\n2x2+bx≤b2\n2a. Now we have\nEk[∥xk+1−ˆxk∥2]≤ ∥ˆxk−xk∥2−ρ−τ−κ\nγk−κ∥ˆxk−xk∥2+Ekh1\nγk(γk−κ)(Lk\nf+Lω)2i\n.\nIn view of our potential function, we have\nEk[ψ1/ρ(xk+1)] = min\nx{f(x) +ω(x) +ρ\n2∥x−xk+1∥2}\n≤f(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk+1∥2\n≤f(ˆxk) +ω(ˆxk) +ρ\n2∥ˆxk−xk∥2−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+Ekhρ\n2γk(γk−κ)(Lk\nf+Lω)2i\n=ψ1/ρ(xk)−ρ(ρ−τ−κ)\n2(γk−κ)∥ˆxk−xk∥2+Ekhρ\n2γk(γk−κ)(Lk\nf+Lω)2i\nand this completes the proof.\nC.3 Proof of Theorem 5.1\nOur reasoning is the same as in Theorem 4.1 , and we start by bounding Eξ′Ekh(Lk\nf+Lω)2\nγk(γk−κ)i\n.\nFor brevity we omit Ek[·]and notice that, for γk≥2κ, that,\n(Lk\nf+Lω)2\nγk(γk−κ)≤2(Lk\nf+Lω)2\nγ2\nk=(Lk\nf)2\nγ2\nk+2Lk\nfLω\nγ2\nk+L2\nω\nγ2\nk\n22so that we can bound the three terms respectively using\nEξ′h(Lk\nf)2\nγ2\nki\n≤Eξ′h(Lk\nf)2\nmax{(L′\nf)2,α2}k2ζi\n≤\u0010α+σ\nα\u001121\nk2ζ\nwhere we invoked Lemma C.1 and take X=Lk\nf, Y=L′\nfand we recall that by reference Lipschitz property\nD2:E[|Lk\nf−L′\nf|2]≤σ2. Similarly, we can deduce that\nEξ′h2Lk\nfLω\nγ2\nki\n≤Eξ′h2Lk\nfLω\nmax{αL′\nf,α2}k2ζi\n≤\u0010α+σ\nα2\u00112Lω\nk2ζ\nandL2\nω\nγ2\nk≤L2\nω\nα2k2ζsince γk≥αkζ. Putting the things together, we have\nEξ′h(Lk\nf+Lω)2\nγ2\nki\n≤(α+σ)2+ 2Lω(α+σ) +L2\nω\nα2k2ζ=(α+σ+Lω)2\nα2k2ζ.\nand\nEk[ψ1/ρ(xk+1)]≤ψ1/ρ(xk)−Eξ′\u0014ρ(ρ−τ−κ)\n2(γk−κ)\u0015\n∥ˆxk−xk∥2+ρ(α+σ+Lω)2\nα2k2ζ,\nor\nEk[ψ1/ρ(xk+1) + Λ] ≤[ψ1/ρ(xk) + Λ] −Eξ′\u0014ρ(ρ−τ−κ)\n2(γk−κ)\u0015\n∥ˆxk−xk∥2+ρ(α+σ+Lω)2\nα2k2ζ.\nInvokingLemma B.1 , plugging in the relation Ak= 0, Vk=ψ1/ρ(xk) + Λ ≥0,Bk=ρ(α+σ+Lω)2\nα2k2ζand\nCk=Eξ′\u0002ρ(ρ−τ−κ)\nγk−κ\u0003\n∥ˆxk−xk∥2. Note that since Eξ′\u0002ρ(ρ−τ−κ)\nγk−κ\u0003\nis determined by xk, we can view Ck=\ng(∥xk∥)∥ˆxk−xk∥2for some function gand the rest of reasoning is the same as in Lemma 4.1 .\nC.4 Proof of Lemma 5.2\nSimilar to Lemma 4.3 we first bound the error of potential reduction Ekhρ(Lk\nf+Lω)2\n2γk(γk−κ)i\n, and according to the\nproof ofLemma 5.1 ,\nEξ′Ek\"\nρ(Lk\nf+Lω)2\n2γk(γk−κ)#\n≤ρ(α+σ+Lω)2\nα2K.\nTelescoping the relation E[ψ1/ρ(xk+1)]≤ψ1/ρ(xk) +ρ(α+σ+Lω)2\n2α2Kgives\nE[ψ1/ρ(xk) + Λ] ≤ψ1/ρ(x1) + Λ +ρ(α+σ+Lω)2\nα2=: ∆.\nNext we show that Ek[∥xk+1−xk∥]is bounded. By the optimality condition we have\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(xk, ξk) +ω(xk)−γk−κ\n2∥xk+1−xk∥2\nand\nγk\n2∥xk+1−xk∥2≤fxk(xk, ξk)−fxk(xk+1, ξk) +ω(xk)−ω(xk+1)\n≤(Lk\nf+Lω)∥xk+1−xk∥.\nDividing both sides by ∥xk+1−xk∥, we get ∥xk+1−xk∥ ≤2(Lk\nf+Lω)\nγk≤2Lk\nf\nγk+2Lω\nα√\nK.Taking expectation on\n23both sides, we have Eξ′Ek[Lk\nf/γk]≤α+σ\nα√\nK, where we invoke Lemma C.1 with X=Lk\nf, Y=L′\nfagain. Now\nEk[∥xk+1−xk∥]≤2(α+σ+Lω)\nα√\nK.\nThe rest of the reasoning is consistent with Lemma 4.3 up to difference in constants. And we still present\nthem for completeness. Applying Markov’s inequality, we know that\nPξk,ξ′∼Ξ{∥xk+1−xk∥ ≤4(α+σ+Lω)\nα√\nK\f\fξ1, . . . , ξ k−1} ≥1\n2.\nByA5,∥xk∥ ≥Bvimplies ψ1/ρ(xk)≥v. Taking v=a∆, we have ∥xk∥ ≥Ba∆⇒ψ1/ρ(xk)≥av. Without\nloss of generality, let Z=4(α+σ+Lω)\nα√\nK>0, and we condition on the event ∥xk∥ ≥Ba∆+Zto deduce that\n∆≥E[ψ1/ρ(xk+1) + Λ]\n=E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z]·P{∥xk∥ ��Ba∆+Z}\n+E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≤Ba∆+Z]·P{∥xk∥ ≤Ba∆+Z}\n≥E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z]·P{∥xk∥ ≥Ba∆+Z}, (32)\nwhere (32) uses ψ1/ρ(x)+Λ≥0for all x. Next we consider the expectation E[ψ1/ρ(xk+1)+Λ|∥xk∥ ≥Ba∆+Z]\nand we successively deduce that\nE[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z]\n=E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z,∥xk+1−xk∥ ≤Z]·P{∥xk+1−xk∥ ≤Z}\n+E[ψ1/ρ(xk+1) + Λ|∥xk∥ ≥Ba∆+Z,∥xk+1−xk∥ ≥Z]·P{∥xk+1−xk∥ ≥Z}\n≥\u0010a∆ + Λ\n2\u0011\n·P{∥xk+1−xk∥ ≤Z}, (33)\nwhere (33) is by P{∥xk+1−xk∥ ≤Z} ≥0.5and that, conditioned on ∥xk+1−xk∥ ≤Z,\n∥xk+1∥=∥xk+1−xk+xk∥ ≥ ∥ xk∥ − ∥ xk+1−xk∥ ≥ ∥ xk∥ −Z≥Ba∆.\nChaining the above inequalities, we arrive at\n∆≥\u0010a∆ + Λ\n2\u0011\n·P{∥xk∥ ≥Ba∆+Z}\nDividing both sides bya∆+Λ\n2gives the desired tail bound.\nC.5 Proof of Theorem 5.2\nThe proof again follows the clue of Theorem 4.2 . Recall that Z=4(α+σ+Lω)\nα√\nKand given δ∈(0,1/4),\nP{∥xk∥ ≥Bδ−1∆+Z} ≤2∆\nδ−1∆ + Λ≤2δ.\nDenoting Ik=I{∥xk∥ ≤ Bδ−1∆+Z}, we havePK\nk=1E[1−Ik]≤2δKand by Markov’s inequality, given\np∈(2δ,1), we have\nPnKX\nk=1(1−Ik)≥2δK\npo\n≤2δK\n2p−1δK=p\n24andP{PK\nk=1Ik≥K(1−2p−1δ)} ≥1−p. Now we telescope over Lemma 5.1 and get\nKX\nk=1E\u0014ρ(ρ−τ−κ)\n2γk∥ˆxk−xk∥2\u0015\n≤ψ1/ρ(x1)−E[ψ1/ρ(xK+1)] +ρ(α+σ+Lω)2\n2α2\n≤ψ1/ρ(x1)−inf\nxψ(x) +ρ\n2α2(α+σ+Lω)2\n=D+ρ\n2α2(α+σ+Lω)2.\nNext define\nGδ:= max\nxsup\nξ∼ΞLip(x, ξ)subject to ∥x∥ ≤Bδ−1∆+Z,\nand we have, conditioned on the eventPK\nk=1Ik≥K(1−2p−1δ),\nKX\nk=1E\u0014ρ(ρ−τ−κ)\n2γk∥ˆxk−xk∥2\u0015\n=X\nk∈{j:Ij=0}E\u0014ρ(ρ−τ−κ)\n2γk∥ˆxk−xk∥2\u0015\n+X\nk∈{j:Ij=1}E\u0014ρ(ρ−τ−κ)\n2γk∥ˆxk−xk∥2\u0015\n≥X\nk∈{j:Ij=1}E\u0014ρ(ρ−τ−κ)\n2γk∥ˆxk−xk∥2\u0015\n≥X\nk∈{j:Ij=1}E\u0014ρ(ρ−τ−κ)\n2(ρ+κ+τ+ (α+Gδ)√\nK)∥ˆxk−xk∥2\u0015\n=ρ(ρ−τ−κ)\n2(ρ+κ+τ+ (α+Gδ)√\nK)X\nk∈{j:Ij=1}E[∥ˆxk−xk∥2]\n≥ρ(ρ−τ−κ)(1−2p−1δ)K\n2(ρ+κ+τ+ (α+Gδ)√\nK)min\nk∈{j:Ij=1}E[∥ˆxk−xk∥2]\nRe-arranging the terms, we have, at least with probability 1−p, that\nmin\n1≤k≤KE[∥∇ψ1/ρ(xk)∥2]≤ min\nk∈{k:Ik=1}E[∥∇ψ1/ρ(xk)∥2]\n≤p\np−2δ·2ρ\nρ−τ−κh\nD+ρ\nα2(α+σ+Lω)2i\u0012ρ+λ\nK+α+Gδ√\nK\u0013\nand this completes the proof after re-arrangement.\nD Stochastic convex optimization beyond Lipschitz continuity\nIn this section, we consider applying the above mentioned ideas to convex optimization. When dealing with\nconvex optimization problems, instead of relying on Moreau envelope smoothing, we have a better potential\nfunction ∥x−x⋆∥directly relevant to distance to optimal set X⋆. This turns out greatly simplifies our\nassumptions.\nE1:f(x, ξ)is convex for all ξ∼Ξ.λ=κ= 0andτ= 0.\nIt is rather straight-forward to extend our results to convex optimization. And we remark that our analysis is\ndifferent from [25], where the authors focus on subgradient method and assume quadratic growth condition.\n25D.1 Convex optimization under standard Lipschitzness\nLemma D.1. Suppose that A1toA3,E1as well as B1holds, then given γk>0\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(Lf+Lω)2\nγ2\nk, (34)\nwhere x⋆∈ X⋆is any optimal solution.\nTheorem D.1. Under the same assumptions as Lemma D.1 , if we take γk≡γ=α√\nK, then\nmin\n1≤k≤KE[ψ(xk)−ψ(x⋆)]≤1\n2√\nKh\n∥x1−x⋆∥2α+(Lf+Lω)2\nαi\n,\nwhere x⋆∈ X⋆is an optimal solution.\nRemark 4.We observe the same trade-off as in weakly convex optimization, where we have, given telescopic\nsum of (34), that\n1\nKKX\nk=1O(γ−1\nk)E[ψ(xk)−ψ(x⋆)]≤ O\u00101\nK\u0011\n+1\nKKX\nk=1O(L2\nfγ−2\nk).\nCompared with weakly convex case\n1\nKKX\nk=1O(γ−1\nk)E[∥∇ψ1/ρ(xk)∥2]≤ O\u00101\nK\u0011\n+1\nKKX\nk=1O(L2\nfγ−2\nk),\nthis resemblance implies our previous analysis for weakly convex optimization is immediately applicable.\nD.2 Convex optimization under generalized Lipschitzness\nLemma D.2. SupposeA1toA3,E1as well as C1holds, then given γk>0,\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(G(∥xk∥)Lf+Lω)2\nγ2\nk,\nwhere x⋆∈ X⋆is an optimal solution.\nTheorem D.2. With the same conditions as Lemma D.2 , ifγk= (G(∥xk∥) + 1) kζ, ζ∈\u00001\n2,1\u0001\n, then as\nk→ ∞,{∥xk∥}is bounded with probability 1 and {infj≤kf(xj)−f(x⋆)}converges to 0 almost surely.\nLemma D.3. Under the same conditions as Lemma D.2 , if we take γk=α(G(∥xk∥) + 1)√\nK, then the tail\nbound\nP\u001a\n∥xk−x⋆∥ ≥2(Lf+Lω)\nα√\nK+a\u001b\n≤2∆\na,\nholds for all 2≤k≤K, where ∆ =∥x1−x⋆∥+Lf+Lω\nα.\nTheorem D.3. Under the same conditions as Lemma D.2 , given δ∈(0,1/4), p∈(2δ,1),(1−2p−1δ)K\niterations will lie in the ball centered around x⋆with radius R(δ) =δ−1∆ +2(Lf+Lω)\nα√\nKand\nmin\n1≤k≤KE[ψ(xk)−ψ(x⋆)]≤p\np−2δ·Gδ+ 1\n2√\nK\u0014\n∥x1−x⋆∥2α+(Lf+Lω)2\nα,\u0015\n(35)\nwhere Gδ:= max zG(z),∥z−x⋆∥ ≤δ−1∆ +2(Lf+Lω)\nα√\nK.\nRemark 5.We note that x⋆is actually arbitrary. Therefore we can take it to be a minimum norm optimal\nsolution to get a tighter bound.\n26D.3 Convex optimization beyond Lipschitzness\nLemma D.4. Suppose that A1toA3,E1as well as D1,D2hold, then given γ >0,\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +Ek\u0014(Lip(xk, ξk)) +Lω)2\nγ2\nk\u0015\n, (36)\nwhere γkis chosen to be independent of ξkand is considered deterministic here.\nTheorem D.4. With the same conditions as Lemma D.4 , ifγk= max {Lip(xk, ξk)), α}kζ, ζ∈(1\n2,1), then\nask→ ∞,{∥xk∥}is bounded with probability 1 and {infj≤kf(xj)−f(x⋆)}converges to 0 almost surely.\nLemma D.5. Under the same conditions as Lemma D.4 , if we take γk= max {Lip(f(xk, ξk), α}√\nK, then\nthe tail bound\nP\u001a\n∥xk−x⋆∥ ≥2(α+σ+Lω)\nα√\nK+a\u001b\n≤2∆\na,\nholds for all 2≤k≤K, where ∆ =∥x1−x⋆∥+α+σ+Lω\nα.\nRemark 6.Now that in convex optimization our potential function ∥x−x⋆∥2itself already defines a bounded\nset. The proof of D.5can also be done based on a conditional probability argument.\nTheorem D.5. Under the same conditions as Lemma D.4 , given δ∈(0,1/4), p∈(2δ,1),(1−2p−1δ)K\niterations will lie in the ball centered around x⋆with radius R(δ) =δ−1∆ +2(α+σ+Lω)\nα√\nKand\nmin\n1≤k≤KE[ψ(xk)−ψ(x⋆)]≤p\np−2δ·Gδ+α\n2√\nK\u0014\n∥x1−x⋆∥2α+(α+σ+Lω)2\nα,\u0015\n(37)\nwhere Gδ:= max xsupξ∼ΞLip(x, ξ),∥x−x⋆∥ ≤δ−1∆ +2(α+σ+Lω)\nα√\nK.\nD.4 Proof of results in Subsection D.1\nD.4.1 Proof of Lemma D.1\nLetx⋆∈ X⋆be an optimal solution to the problem. Then by three-point lemma, we have\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2\n≤fxk(x⋆, ξk) +ω(x⋆) +γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−x⋆∥2.\nRe-arranging the terms, we deduce that\nγk\n2∥xk+1−x⋆∥2\n≤γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−xk∥2+fxk(x⋆, ξk) +ω(x⋆)−fxk(xk+1, ξk)−ω(xk+1)\n≤γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−xk∥2+ (Lf(ξk) +Lω)∥xk+1−xk∥ (38)\n+fxk(x⋆, ξk) +ω(x⋆)−f(xk)−ω(xk),\nwhere (38) applies B1to get fxk(xk+1, ξk)−fxk(xk, ξk)≤Lf(ξk)∥xk+1−xk∥. Dividing both sides byγk\n2,\n∥xk+1−x⋆∥2≤ ∥xk−x⋆∥2− ∥xk+1−xk∥2+2(Lf(ξk) +Lω)\nγk∥xk+1−xk∥\n+2\nγk[f(x⋆, ξk) +ω(x⋆)−f(xk)−ω(xk)] +2\nγk[fxk(x⋆, ξ)−f(x⋆, ξ)].\n27Conditioned on x1, . . . , xkand taking expectation with respect to ξk, we successively deduce that\nEk[∥xk+1−x⋆∥2]\n≤ ∥xk−x⋆∥2−Ek[∥xk+1−xk∥2] +Ek[2γ−1\nk(Lf(ξk) +Lω)∥xk+1−xk∥]\n+2\nγk[f(x⋆) +ω(x⋆)−f(xk)−ω(xk)] (39)\n≤ ∥xk−x⋆∥2+(Lf+Lω)2\nγ2\nk+2\nγk[f(x⋆) +ω(x⋆)−f(xk)−ω(xk)] (40)\n=∥xk−x⋆∥2+(Lf+Lω)2\nγ2\nk+2\nγk[ψ(x⋆)−ψ(xk)],\nwhere (39) applies E1to get Eξk[fxk(x⋆, ξ)−f(x⋆, ξ)]≤0; (40) uses −a\n2x2+bx≤b2\n2aand that E[Lf(ξ)2]≤L2\nf.\nRe-arranging the terms, we arrive at\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(Lf+Lω)2\nγ2\nk\nand this completes the proof.\nD.4.2 Proof of Theorem D.1\nTaking γk≡γ=α√\nKand telescoping from 1, . . . , K, we have\nmin\n1≤k≤KE[ψ(xk)−ψ(x⋆)]≤1\nKKX\nk=1E[ψ(xk)−ψ(x⋆)]\n≤1\n2√\nKh\n∥x1−x⋆∥2α+(Lf+Lω)2\nαi\nand this completes the proof.\nD.5 Proof of results in Subsection D.2\nD.5.1 Proof of Lemma D.2\nDefine Gk:=G(∥xk∥). Let x⋆∈ X⋆be an optimal solution to the problem. Similarly, we have\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2\n≤fxk(x⋆, ξk) +ω(x⋆) +γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−x⋆∥2.\nRe-arranging the terms, for ξk∼Ξ, we deduce that\nγk\n2∥xk+1−x⋆∥2\n≤γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−xk∥2+fxk(x⋆, ξk) +ω(x⋆)−fxk(xk+1, ξk)−ω(xk+1)\n≤γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−xk∥2+ (GkLf(ξk) +Lω)∥xk+1−xk∥ (41)\n+f(x⋆, ξk) +ω(x⋆)−f(xk)−ω(xk),\n28where (41) applies convexity. Dividing both sides byγk\n2, we have\n∥xk+1−x⋆∥2≤ ∥xk−x⋆∥2− ∥xk+1−xk∥2+2(GkLf(ξk) +Lω)\nγk∥xk+1−xk∥\n+2\nγk[f(x⋆, ξk) +ω(x⋆)−f(xk)−ω(xk)]\nNext, conditioned on x1, . . . , xkand taking expectation with respect to ξk, we successively deduce that\nEk[∥xk+1−x⋆∥2]\n≤ ∥xk−x⋆∥2−Ek[∥xk+1−xk∥2] +Ek[2γ−1\nk(GkLf(ξk) +Lω)∥xk+1−xk∥]\n+2\nγk[f(x⋆) +ω(x⋆)−f(xk)−ω(xk)]\n≤ ∥xk−x⋆∥2+(GkLf+Lω)2\nγ2\nk+2\nγk[f(x⋆) +ω(x⋆)−f(xk)−ω(xk)] (42)\n=∥xk−x⋆∥2+(GkLf+Lω)2\nγ2\nk+2\nγk[ψ(x⋆)−ψ(xk)],\nwhere (42) again uses −a\n2x2+bx≤b2\n2aand the assumption Eξ[Lf(ξ)2]≤L2\nf. Re-arranging the terms, we get\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(GkLf+Lω)2\nγ2\nk(43)\nand this completes the proof.\nD.5.2 Proof of Theorem D.2\nNow that our recursive potential reduction is changed into\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(GkLf+Lω)2\nγ2\nk\nand we boundGkLf+Lω\nγk=GkLf+Lω\n(Gk+ 1)kζ≤Lf+Lω\nkζ,\ngiving\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(Lf+Lω)2\nk2ζ.\nInvokingLemma B.1 with Ak= 0, Vk=∥xk−x⋆∥2≥0, Bk=(Lf+Lω)2\nk2ζandCk=2\nγk[ψ(xk)−ψ(x⋆)], we\ncomplete the proof with the same argument as in Theorem 4.1 .\nD.5.3 Proof of Lemma D.3\nOur proof is a duplicate of Lemma 4.3 using a different potential function. We start by bounding the error\nof potential reduction by(GkLf+Lω)2\nγ2\nk≤(Lf+Lω)2\nα2K.Then telescoping of (43) gives us, for all 2≤k≤K, that\nE[∥xk−x⋆∥]2≤E[∥xk−x⋆∥2]≤ ∥x1−x⋆∥2+(Lf+Lω)2\nα2≤\u0010\n∥x1−x⋆∥+Lf+Lω\nα\u00112\n=: ∆2\n29andE[∥xk−x⋆∥]≤∆. Next consider, by optimality condition, that\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(xk, ξk) +ω(xk)−γk\n2∥xk+1−xk∥2.\nA re-arrangement gives\nγk∥xk+1−xk∥2≤(Lf(ξk)Gk+Lω)∥xk+1−xk∥.\nDividing both sides by ∥xk+1−xk∥,\n∥xk+1−xk∥ ≤Lf(ξk)Gk+Lω\nγk=Lf(ξk)Gk+Lω\nα(Gk+ 1)√\nK.\nTaking expectation, we get Ek[∥xk+1−xk∥]≤Lf+Lω\nα√\nK.Then by Markov’s inequality,\nPξk∼Ξ\u001a\n∥xk+1−xk∥ ≤2(Lf+Lω)\nα√\nK|ξ1, . . . , ξk−1\u001b\n≥1\n2\nand without loss of generality, we let Z=2(Lf+Lω)\nα√\nK>0. Then we successively deduce that\n∆≥E[∥xk+1−x⋆∥]\n=E[∥xk+1−x⋆∥|∥xk−x⋆∥ ≤Z+z]·P{∥xk−x⋆∥ ≤Z+z}\n+E[∥xk+1−x⋆∥|∥xk−x⋆∥ ≥Z+z]·P{∥xk−x⋆∥ ≥Z+z}\n≥E[∥xk+1−x⋆∥|∥xk−x⋆∥ ≥Z+z].\nNext, consider the expectation E[∥xk+1−x⋆∥|∥xk−x⋆∥ ≥Z+z]and we successively deduce that\nE[∥xk+1−x⋆∥|∥xk−x⋆∥ ≥Z+z]\n=E[∥xk+1−x⋆∥|∥xk−x⋆∥ ≥Z+z,∥xk+1−xk∥ ≤Z]·P{∥xk+1−xk∥ ≤Z}\n+E[∥xk+1−x⋆∥|∥xk−x⋆∥ ≥Z+z,∥xk+1−xk∥ ≥Z]·P{∥xk+1−xk∥ ≥Z}\n≥z\n2·P{∥xk+1−xk∥ ≥Z} (44)\nwhere (44) is by P{∥xk+1−xk∥ ≤Z} ≥0.5and that, conditioned on ∥xk+1−xk∥ ≤Z,\n∥xk+1−x⋆∥=∥xk+1−xk+xk−x⋆∥ ≥ ∥ xk−x⋆∥ − ∥ xk+1−xk∥ ≥z.\nChaining the above inequalities, we arrive at\n∆≥z\n2·P{∥xk−x⋆∥ ≥Z+z}.\nDividing both sides byz\n2and taking z=agives the desired tail bound.\nD.5.4 Proof of Theorem D.3\nGiven δ∈(0,1/4), we have\nP{∥xk−x⋆∥ ≥Z+δ−1∆} ≤2∆\nδ−1∆≤2δ.\n30Denote Ik=I{∥xk−x⋆∥ ≤δ−1∆ + Z}. We havePK\nk=1E[1−Ik]≤2δKand by Markov’s inequality, given\np∈(2δ,1),\nP(KX\nk=1Ik≥K(1−2p−1δ))\n≥1−p.\nNow we telescope over Lemma D.3 and deduce that\nKX\nk=1E\u00142\nγk(ψ(xk)−ψ(x⋆))\u0015\n≤ ∥x1−x⋆∥2−E[∥xK+1−x⋆∥2] +(Lf+Lω)2\nα2\n≤ ∥x1−x⋆∥2+(Lf+Lω)2\nα2.\nNext we condition on the eventPK\nk=1Ik≥K(1−2p−1δ), define Gδ:= max zG(z),∥z−x⋆∥ ≤δ−1∆ + Z, and\nsuccessively deduce that\nKX\nk=1E\u00142\nγk(ψ(xk)−ψ(x⋆))\u0015\n=X\nk∈{j:Ij=0}E\u00142\nγk(ψ(xk)−ψ(x⋆))\u0015\n+X\nk∈{j:Ij=1}E\u00142\nγk(ψ(xk)−ψ(x⋆))\u0015\n≥X\nk∈{j:Ij=1}E\u00142\nγk(ψ(xk)−ψ(x⋆))\u0015\n≥X\nk∈{j:Ij=1}E\u00142\nα(Gδ+ 1)√\nK(ψ(xk)−ψ(x⋆))\u0015\n≥2(1−2p−1δ)√\nK\nα(Gδ+ 1)min\nk∈{j:Ij=1}E[(ψ(xk)−ψ(x⋆))], (45)\nwhere (45) follows from the eventPK\nk=1Ik≥K(1−2p−1δ). Putting the results together, we have\nmin\n1≤k≤KE[ψ(xk)−ψ(x⋆)]≤ min\nk∈{j:Ij=0}E[ψ(xk)−ψ(x⋆)] (46)\n≤p\np−2δ·Gδ+ 1\n2√\nK\u0014\n∥x1−x⋆∥2α+(Lf+Lω)2\nα\u0015\n(47)\nand this completes the proof.\nD.6 Proof of results in Subsection D.3\nIn this section, we again define Lk\nf:=Lip(xk, ξk), L′\nf:=Lip(xk, ξ′)to simplify notation.\nD.6.1 Proof of Lemma D.3\nBy the optimality condition we have\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(x⋆, ξk) +ω(x⋆) +γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−x⋆∥2.\n31Re-arranging the terms, we get\nγk\n2∥xk+1−x⋆∥2\n≤γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−xk∥2+fxk(x⋆, ξk) +ω(x⋆)−fxk(xk+1, ξk)−ω(xk+1)\n≤γk\n2∥xk−x⋆∥2−γk\n2∥xk+1−xk∥2+ (Lk\nf+Lω)∥xk+1−xk∥\n+f(x⋆, ξk) +ω(x⋆)−f(xk)−ω(xk).\nDividing both sides byγk\n2,\n∥xk+1−x⋆∥2≤ ∥xk−x⋆∥2− ∥xk+1−xk∥2+2(Lk\nf+Lω)\nγk∥xk+1−xk∥\n+2\nγk[f(x⋆, ξk) +ω(x⋆)−f(xk)−ω(xk)]\nConditioned on x1, . . . , xk, taking expectation with respect to ξk, we have\nEk[∥xk+1−x⋆∥2]\n≤ ∥xk−x⋆∥2−Ek[∥xk+1−xk∥2] +Ek[2γ−1\nk(Lk\nf+Lω)∥xk+1−xk∥]\n+2\nγk[f(x⋆) +ω(x⋆)−f(xk)−ω(xk)] (48)\n≤ ∥xk−x⋆∥2+(Lk\nf+Lω)2\nγ2\nk+2\nγk[f(x⋆) +ω(x⋆)−f(xk)−ω(xk)]\n=∥xk−x⋆∥2+(Lk\nf+Lω)2\nγ2\nk+2\nγk[ψ(x⋆)−ψ(xk)],\nwhere (48) use the fact that γkdoes not inherit randomness from ξk. Re-arranging the terms, we get\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +Ekh(Lk\nf+Lω)2\nγ2\nki\nand this completes the proof.\nD.6.2 Proof of Theorem D.4\nWe start by bounding Ekh(Lk\nf+Lω)2\nγ2\nki\nand notice that\n(Lk\nf+Lω)2\nγ2\nk=Lk\nf\nγ2\nk+2Lk\nfLω\nγ2\nk+L2\nω\nγ2\nk\nso that we can bound\nEξ′hLk\nf\nγ2\nki\n=Eξ′h(Lk\nf)2\nmax{(L′\nf)2,α}k2ζi\n≤\u0010α+σ\nα\u001121\nk2ζ,\nwhere we invoked Lemma C.1 and take X=Lk\nf, Y=L′\nf, and we recall that by D2,E[|Lk\nf−L′\nf|2]≤σ2.\nSimilarly, we can deduce that\nEξ′h2Lk\nfLω\nγ2\nki\n≤Eξ′h2Lk\nfLω\nmax{L′\nf,α}2i\n≤Eξ′h2Lk\nf\nαL′\nfi\n≤\u0010α+σ\nα2\u00112Lω\nk2ζ\n32Eξ′h\nL2\nω\nγ2\nki\n≤L2\nω\nα2k2ζ.\nPutting the bounds together,\nEξ′h(Lk\nf+Lω)2\nγ2\nki\n≤(α+σ+Lω)2\nα2k2ζ.\nThen we have\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(α+σ+Lω)2\nα2k2ζ\nand we complete the proof by invoking Lemma B.1 .\nD.6.3 Proof of Lemma D.5\nWe start by bounding Ekh(Lk\nf+Lω)2\nγ2\nki\n≤(α+σ+Lω)2\nα2K. Telescoping the relation\nEk[∥xk+1−x⋆∥2]≤ ∥xk−x⋆∥2−2\nγk[ψ(xk)−ψ(x⋆)] +(α+σ+Lω)2\nα2K\ngives, for all 2≤k≤K, that\nE[∥xk−x⋆∥2]≤ ∥x1−x⋆∥2+(α+σ+Lω)2\nα2≤\u0010\n∥x1−x⋆∥+α+σ+Lω\nα\u00112\n=: ∆2.\nandE[∥xk−x⋆∥]≤∆. Next consider\nfxk(xk+1, ξk) +ω(xk+1) +γk\n2∥xk+1−xk∥2≤fxk(xk, ξk) +ω(xk)−γk\n2∥xk+1−xk∥2\nand re-arrangement gives γk∥xk+1−xk∥2≤(Lk\nf+Lω)∥xk+1−xk∥.Dividing both sides by ∥xk+1−xk∥, we\nget∥xk+1−xk∥ ≤Lk\nf+Lω\nγk.Taking expectation, Ek[∥xk+1−xk∥]≤α+σ+Lω\nα√\nK.Then by Markov’s inequality,\nPξk,ξ′∼Ξn\n∥xk+1−xk∥ ≤2(α+σ+Lω)\nα√\nK|ξ1, . . . , ξk−1o\n≥1\n2. and without loss of generality, let Z=2(α+σ+Lω)\nα√\nK>0.\nBy the same reasoning, we arrive at ∆≥z\n2·P{∥xk−x⋆∥ ≥Z+z}and dividing both sides byz\n2gives the\ndesired tail bound.\nD.6.4 Proof of Theorem D.5\nFollowing the same argument as Theorem D.4 , we have, for δ∈(0,1/4), that P{∥xk−x⋆∥ ≥Z+δ−1∆} ≤2δ.\nDefine Ik=I{∥xk−x⋆∥ ≤δ−1∆ +Z}. We have, by Markov’s inequality, that P{PK\nk=1Ik≥K(1−2p−1δ)} ≥\n1−p. Then telescoping over Lemma D.5 gives\nKX\nk=1E\u00142\nγk(ψ(xk)−ψ(x⋆))\u0015\n≤ ∥x1−x⋆∥2−E[∥xK+1−x⋆∥2] +(α+σ+Lω)2\nα2\n≤ ∥x1−x⋆∥2+\u0010α+σ+Lω\nα\u00112\n.\nConditioned onPK\nk=1Ik≥K(1−2p−1δ), we get\nKX\nk=1E\u00142\nγk(ψ(xk)−ψ(x⋆))\u0015\n≥2(1−2p−1δ)√\nK\nα+Gδmin\nk∈{j:Ij=1}E[ψ(xk)−ψ(x⋆)]\nwhere Gδ:= max xsupξ∼ΞLip(x, ξ),∥x−x⋆∥ ≤δ−1∆ +Z. Combining two inequalities completes the proof.\n33"    },    {        "title": "2401.13982v1.Buffer_layer_controlled_Nickeline_vs_Zinc_Blende_Wurtzite_type_MnTe_growths_on_c_plane_Al2O3_substrates.pdf",        "content": "Buffer -layer -controlled Nickeline vs Zinc -Blende/Wurtzite -type MnTe growths \non c-plane Al 2O3 substrates  \nDeepti Jain1, Hee Taek Yi1, Alessandro R.  Mazza2,3, Kim Kisslinger4, Myung -Geun Han5, Matthew \nBrahlek2 and Seongshik Oh1,6,*     \n  \n1 Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA   \n2 Materials Science and Technology  Division, Oak Ridge National Laboratory, Oak Ridge,  TN 37831, U SA \n3 Present address:  Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New \nMexico 87545, USA  \n4 Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA  \n5 Condensed Matter Physics and Materials  Science, Brookhaven National Laboratory, Upton, NY  11973, US A \n6 Center for Quantum Materials Synthesis, Rutgers, The State University of New Jersey, Piscataway, NJ 08854, USA   \n \n* Correspondence should be addressed to ohsean@physics.rutgers.edu  \n \n \nAbstract  \nIn the recent past, MnTe  has proven to be a crucial component of the intrinsic magnet ic topological insulator \n(IMTI) family [MnTe] m[Bi 2Te3]n, which hosts a wide range of magneto -topological properties depending \non the choice of m and n. However, bulk crystal growth allow s only a few combinations of m and n for \nthese IMTIs  due to the strict limitations of the thermodynamic growth conditions. One way to overcome \nthis challenge is  to utilize atomic  layer -by-layer molecular beam epitaxy (MBE) technique , which allows \narbitrary sequence s of [MnTe] m and [Bi 2Te3]n to be formed beyond the thermodynamic limit . For such MBE \ngrowth, finding optimal growth template s and conditions  for the parent building block, MnTe , is a key \nrequirement . Here, we report that two different hexagonal phases of MnTe  - nickeline (NC) and zinc -blende /wurtzite  (ZB-WZ) structures , with distinct in-plane lattice constants of 4.20 ± 0.04 Å and 4.39  ± \n0.04 Å, respectively  – can be selectively grown on c -plane Al2O3 substrates  using different buffer layers \nand growth temperatures.  Moreover, we provide the first comparative studies of different MnTe phases \nusing atomic -resolution scanning transmission electron microscopy and show that ZB and WZ -like stacking \nsequences can easily alternate between the two. Surprisingly, In2Se3 buffer layer , despite its l attice constant \n(4.02 Å) being closer to  that of the NC phase , fosters the ZB-WZ instead, whereas Bi2Te3, sharing the same  \nlattice constant (4.39 Å) with the ZB -WZ phase, fosters the NC phase. These discoveries suggest that lattice \nmatching is not always the most critical factor determining the preferred phase during epitaxial growth . \nOverall, this will  deepen our understanding of epitaxial growth modes for chalcogenide materials and \naccelerate progress toward new IMTI phases as well as other magneto -topological applications.  \n \nI. Introduction  \n MnTe has been studied extensively in the past few decades  due to its innate property of being an \nantiferromagnetic semiconductor, making it valuable for memory devices  [1,2] , optoelectronic s [3,4]  and \nspintronics  [5-7] to name a few.  It is known to exist in three  phases; nickeline (NC , Fig. 1(a,b) ), wurtzite \n(WZ , Fig. 1(c,d) ), and  zinc blende (ZB, Fig. 1(e,f))  [8]. In bulk crystals, which are governed by \nthermodynamics, MnTe crystallizes only in hexagonal NC structure. Consequently, NC -structure MnTe \nphase has been most comprehensively investigated, and its structural, optical, electronic and magnetic \nproperties are well established  [7,9-16]. Due to its high Neel temperature of 307 -310K  [9,10] , it is a \ncandidate for antiferromagnetic spintronic s applications that can be operated at room temperature. It is \nworth noting that while NC MnTe has always been identified as an  antiferromagnet, it was recently \npredicted  to belong to a new and distinct class of magnetic materials known as  altermagnet s [17,18] . The \nother phases  of MnTe , however, are metastable; they cannot be grown via equilibrium growth methods and \nrequire different modes of growth to materialize.  Interest in the ZB MnTe phase stemmed from its role in some diluted magnetic semiconductors and \nit was found that it has a lower Neel temperature (~65 K) [19,20]  and a wider optical bandgap (~3 eV) [21-\n24] compared to the NC phase (1.26 -1.5 eV) [14,15] .  While majority of ZB MnTe has been grown via \nMBE, with Mn being deposited under excess Te, the re have  also been a few reports in which the ionized \ncluster beam method was used, with NC MnTe polycrystals as the source material . Cubic ZB MnTe has \nmostly been achieved by epitaxial stabilization on ZB -structure substrates such as GaAs, CdTe and InSb, \noften utilizing additional buffer layers like ZnTe and CdTe between the substrate and the film  [19,21,25 -\n27]. When ZB -structure is viewed along the (111) direction, the arrangement of atoms on the surface has \nsix-fold symmetry , as can be seen from Fig. 1 (e), and the hexagonal -like ZB MnTe(111) film has also been \nstabilized on GaAs(001) [25,26] , BaF 2(111)  [24], SrTiO 3(001) [28], mica  [29] and Al2O3(0001) with \nCdTe(111) buffer  [20]. On the other hand, intrinsically hexagonal WZ MnTe phase  [Fig. 1( c)], with a wide \nbandgap of 2.4 -3 eV, has been so far grown only in a polycrystalline form on amorphous substrates like \nglass and indium -zinc-oxide, for optoelectronic applications  [2,4,30,31] . Sometimes, multi -phases of MnTe \nhave been observed depending  on the choice of substrate temperature and Te:Mn flux ratio , and they  usually \ninvolve the stable phase, NC MnTe , and either of the metastable phases, ZB or WZ MnTe  [4,28 -30]. \nHowever, c o-exist ing ZB and WZ MnTe phases have never been reported , despite their structural \nsimilarities . \n  One of the active research areas  where  MnTe  plays a critical role is  the newly discovered intrinsic \nmagnetic topological insulator (IMTI) family, [MnTe] m[Bi 2Te3]n [32-35], a class of materials that ha s been \npredicted to host exotic topological phases such as axion insulators  [36,37] , magnetic Weyl semimetals  \n[33,38] , and high temperature quantum anomalous Hall effect s [32,39] . Despite some progresses, there are \nstill multiple  challenges hindering the realization of its  many  possibilities. So far, m ajority of studies on \n[MnTe] m[Bi 2Te3]n have been performed on bulk crystals and thin flakes exfoliated from them. Inherently, \nbulk crystal growth relies on macroscopic diffusion of constituent elements along all the three directions \nuntil they reach thermodynamically the most stable configuration. In such a growth mode , it becomes extremely difficult to form highly layered structures with large unit cell sizes, as in the case of \n[MnTe] m[Bi 2Te3]n compounds with large m and n values. Additionally, among the compounds that are \nthermodynamically stable i.e., m = 1, n ≥ 1, the ones with higher n exist within a  very narrow range of \ntemperatures.  As a consequence, so far only m = 1, n ≤ 7 phases have successfully been grown  [34]. These \nshortcomings can be overcome by growing the films with atomic -layer -by-layer molecular beam epitaxy \n(MBE) technique, which can potentially enable finely tuned, atomic scale engineering of this family of \nmaterials for all possible values of m and n. The main motivation behind the following work is creating a \nfoundation to grow thin films of these IMTIs, by focusing on finding a suitable template to grow one of its \nbuilding blocks first, i.e., MnTe. Here, we report that on Al 2O3(0001) substrates, which  are both economical \nand successfully used for various high quality topological thin film growths  [40-44], NC and ZB -WZ phases \nof MnTe can be selectively grown with relative ease, using two distinct buffer layers combined with \ndifferent growth temperatures.   \nII. Methods  \n  On Al 2O3(0001) substrates, o ur initial goal was to find optimal growth conditions  for NC MnTe \nphase because it is the phase found in the bulk crystals, and the atomic -sequence  of Te atoms  surrounding \nthe Mn layer in bulk crystals of [MnTe][Bi 2Te3]n is equivalent to that of the NC structure  [45]. Prior to any \ndeposition , the substrate s were cleaned ex -situ by UV  generated ozone followed by in -situ heating up to \n750 oC under oxygen pressure of 1 × 10−6 Torr.  This step helps get rid of any organic contaminants on the \nsurface of the substrate.  Thickness of the films were determined by quartz crystal microbalance (QCM) and \nRutherford backscattering spectroscopy (RBS) , and the growth was monitored in -situ using  reflection high-\nenergy electron diffraction  (RHEED) . For all the films described below, adsorption -controlled growth mode \nis used with several times more tellurium or selenium fluxes than those of the metal elements. Each element \nis evaporated from a standard effusion cell.  \nThe lattice constant of NC MnTe, aNC = 4.14 Å, is not a good match with that of the sapphire \nsubstrate , aAl2O3  = 4.76 Å. A common solution for such a lattice mismatch problem is to introduce a structurally compatible buffer layer between the film and the substrate . In this case, the buffer layer chosen \nwas insulating In 2Se3, with a lattice constant of a In2Se3 = 4.02 Å, which is closer to that of NC MnTe. We \nhave previously grown high-quality , single -phase In2Se3 on Al2O3, involv ing a multi -step recipe  [42]. \nAccordingly, first , a seed layer of 3  QL Bi 2Se3 is grown at 135  °C and an additional 7  QL is deposited at \n300 °C. This serves as a good template to grow In 2Se3 (5 nm), also at 300  °C. When this layered structure \nis annealed to 600  °C, the Bi 2Se3 layer diffuses through the In 2Se3 and evaporates away, leav ing behind \nIn2Se3 directly on  Al2O3. Once the buffer layer is ready, the substrate is cooled down to 450  °C and the \nMnTe film is grown on top [Fig. 2(a)]. The final growth temperature was chosen  after multiple trials,  based \nupon the temperature range in which a single crystalline, 2D surface could be observed with RHEED.   \n Contrary to our expectation, the phase of MnTe grown on In 2Se3 was not NC. Fig. 3(a) shows \nsnapshots of RHEED pattern s of the In2Se3 buffer layer and the MnTe  film taken during growth : the sharp \nand localized streaks imply a good epitaxial growth for both, the buffer and the film. Based on the RHEED \nstreak spacing, the in -plane lattice constant of the MnTe film is found to be 4.39 ± 0.04 Å, quite different \nfrom the expected 4.14  Å of NC MnTe . Additionally, on comparing the distance between RHEED streaks \nfor the two high symmetry directions, it can be seen that the distance ratio is √3, whi ch is an indication that \nit has six -fold in -plane symmetry. According to the literature, it could be either ZB(111) or WZ (0001)  \nphase.  \n Since we were unable to obtain NC MnTe on In2Se3, we tried using a different buffer layer : Bi2Te3. \nAs can be inferred from the existence of [MnTe] m[Bi 2Te3]n, Bi 2Te3 and NC MnTe are compatible with each \nother. However, Bi 2Te3 is conducting; hence, we would need a very thin layer of it, so that it does not \ninterfere with the transport properties of MnTe. Based on our previous reports, Bi 2Te3 has poor adhesion  to \ninert Al 2O3 substrate at its optimal growth temperature, and inserting a less inert  layer like Cr 2O3 between \nthem can help it stick better  [44]. Following this recipe, 1 nm Cr 2O3 was deposited on the substrate  at 700  \n°C under oxygen pressure of 1 × 10−6 Torr and after that  1 QL Bi 2Te3 was grown a t 300  °C. We finally \ngrew MnTe on this template at 300 °C [Fig. 2(b)] . We can see from Fig. 3(b) that the in -plane lattice constant of MnTe  grown on this buffer is 4.20 ± 0.04 Å, much closer to that of NC MnTe. In Fig. 3(c) and \n(d), we can see that while both types of MnTe phases  start growing with a lattice constant close to that of \ntheir respective  buffer layer,  they gradually relax to the final lattice constant s of 4.39 Å and 4.20 Å, \nrespectively,  after 10 ~20 monolayers .  \n \nIII. Results and Discussion  \n To probe further, x-ray diffraction ( XRD ) was carried out on the two films , using a Panalytical \nX’Pert Pro and a monochromated Cu K α1 source.  Fig. 4 shows the 2 θ scans for both MnTe  films. Signature \n(0003n) peaks belonging to the Al 2O3 substrate and a small peak from the Se capping layer c an be seen in \nthe patterns of both films. All the peaks of the MnTe  film with lattice constant 4.20 ± 0.04 Å can be \nidentified with the (0002n) peaks of NC MnTe phase  in the literature , as shown in Fig. 4(c) [46]: this implies  \nthat this is in fact the NC MnTe phase. On the other hand, t he XRD pattern of the MnTe  film grown on \nIn2Se3 buffer reveals the coexistence of both ZB and WZ phases. It can be seen in Fig. 4(a) that each \nprominent peak is a superposition of two peaks; ZB(nnn) and WZ(00 02n). Fig. 4(b) shows an enlarged \nsection of the XRD pattern where the ZB(111) and WZ(00 02) peaks have been resolved, and are consistent \nwith similar studies conducted previously  [28,31] . The presence of these two peaks also implies that the c -\naxis lattice constant is not uniform for the entire film. Transport studies involving temperature dependence \nof longitudinal resistance of a 30 nm NC MnTe  film are shown in Fig. 4(d). The shape indicates \npredominantly semiconducting behavior, similar to previous reports on NC MnTe. A small, elongated hump \ncan be seen around 250 -300 K consistent with  an AFM transition temperature associated with interaction \nbetween itinerant electrons  and localized Mn spins  [47]. While we could not perform similar measurements \non the ZB -WZ MnTe phase, we carried out I -V measurements and found the two -point resistance of the \nZB-WZ phase to be ~20 GΩ at room temperature . This is not surprising since the reported bandgap s for \nZB MnTe and WZ MnTe are ~3 eV and 2.4 -3 eV respectively, compared to 1.26-1.5 eV  for NC MnTe.    High-angle annular dark -field scanning tunneling  electron microscopy (HAADF -STEM)  was \ncarried out to gain insight into the atomic structure of these films. The cross -sectional STEM sample was \nprepared using a FEI Helios G5 UX focused ion beam system with final Ga+ milling performed at 2 keV.  \nThen, t he HAADF -STEM was performed with a JEOL ARM 200CF equipped with a cold field emi ssion \ngun and  spherical aberration correct ors, which was operated at 200 kV . The detection angles for HAADF \nimaging were ranging fr om 68 to 280 mrad.  Figure s 5(a) and (e) show clear boundaries between the films \nand the buffer layers. Upon closer inspection, it can be seen that there is a difference in arrangement of Mn \nand Te atoms in both films. In Fig. 5(f), the Mn atoms fall in a straight line along the (00 01) direction while \nthe Te atoms form a zigzag pattern . This AcBc AcBc  sequence is characteristic of the N C structure due to \ninterpenetrating primitive hexagonal lattice of the Mn atoms and close packed hexagonal lattice of Te atoms  \n[Fig. 1( b)]. In contrast, the Mn ato ms do not align along the direction  of growth  in ZB -WZ MnTe  [Fig. \n5(b)]. Additionally, on comparing the positions of Mn and Te atoms relative  to one another in the STEM \nimage of the ZB -WZ phase [ Fig. 5(b) ], Fig. 1(d) and Fig. 1(f), it can be seen that the directions of growth \nare (00 0͞1) and (͞1͞1͞1) for the WZ and ZB phase respectively.  Fig. 5(c) and ( d) further illustrate the \narrangement of atoms in ZB -WZ MnTe . It can be seen that the initial growth of ZB -WZ MnTe  follows an \nABCABC sequence, correspond ing to the (͞1͞1͞1) growth mode of ZB structure  illustrated in Fig. 1(f) . \nHowever, a s highlighted by the yellow dotted lines in Fig 5(c), stacking faults are very common and the \nsequence sometimes changes to ABAB, which corresponds to the WZ stacking  shown in Fig. 1(d) . As the \ngrowth progresses , the stacking changes between ZB and WZ quite randomly [Fig . 5(d)]. It can be inferred  \nfrom the STEM images that ZB MnTe(111) and WZ MnT e have similar formation energ ies. Nonetheless, \ndespite frequent switching between  ZB and WZ stacking along the direction  of growth , it is notable that the \nin-plane lattice constant as judged from the RHEED approaches the stable value of 4.39 ± 0.04 Å, which is \nidentical to that of Bi 2Te3 as shown in Fig. 3(a,c). Here, it is important to note that this ZB -WZ-sequence \nmixed phase is very different from other  common mixed phases , in that the only difference between ZB \nand WZ phases is just the stacking sequence . In other words, from the viewpoint of the topmost  layer, this \nmixed -sequence structure is just like a single -crystalline 2D lattice structure with a well -defined lattice constant of 4.39 ± 0.04 Å, slightly smaller than the effective lattice constants ( ~4.5 Å) [21,48]  of pure ZB \nor WZ phases. Accordingly, this ZB -WZ mixed -sequence platform could provide its own unique \napplications distinct from the pure ZB or WZ platform.  \n \nIV. Conclusion  \n Our study provides a detailed structural analysis of NC and ZB -WZ phases of MnTe, grown \nselectively on Al 2O3(0001) substrates  with two different buffer layers . Surprisingly, In2Se3 buffer layer, \ndespite its lattice constant (4.02 Å) being closer to that of the NC phase, fosters the ZB/WZ  phase , whereas \nBi2Te3, sharing the same lattice constant (4.39 Å) with the ZB/WZ phase, fosters the NC phase.  This \nsuggests that lattice matching is not always the most critical factor determining the preferred phase during \nepitaxial growth. Furthermore, we have provided the first atomic -resolution STEM studies of ZB -WZ \nMnTe phase, showing that stacking sequences of ZB(111) and WZ(0001) MnTe can easily alternate from \neach other, which suggests that the formation energ ies of ZB(111) and WZ(0001) MnTe are extremely \nclose to each other. The absence of literature for the coexistence of these two phases, despite their  similar \nformation energies , could be due to two possibilities . First, the few reports of ZB(111) MnTe growth in the \nliterature may have some  portions of WZ-like stacking, and vice versa , but could not be confirmed due to \nlack of detailed STEM studies . Second,  the In2Se3 buffer layer  may play a role in stabilizing both of  these \nmeta-stable phases.  Although it is an open question whether it is possible to achieve pure ZB or WZ \nsequences  on the In 2Se3 buffer layer,  having MnTe thin films with ZB -WZ mixed sequences  on the In2Se3 \nbuffer is an unexpected yet significant finding, considering the proximity of its in-plane lattice constant \n(4.39 ± 0.04 Å) to that (4.39 Å) of Bi 2Te3 and [MnTe] m[Bi 2Te3]n. Due to th is perfect  lattice match , it can be \nused as a n optimal foundation to grow these IMTIs  with minimal interfacial defect s. As demonstrated \nrecently by some of us, this also opens the possibility of  combining superconducti ng Fe(Te,Se) with IMTIs  \ntoward novel topological superconductivity , via hybrid symmetry epitaxy  [49]. On the other hand, t he NC \nphase with Bi 2Te3-Cr2O3 buffer layer can also be tailored to enable growth of [MnTe] m[Bi 2Te3]n on Al2O3(0001) with different interfacial  conditions, as will soon be published in a follow -up work. These \nstudies shed light on the critical role of buffer layers in stabilizing selected phases and will open many  \navenues in topological and spintronic applications.  \n \nAcknowledgements  \nThis work is supported by National Science Foundation’s DMR2004125, Army Research Office’s \nW911NF2010108,  and MURI W911NF2020166 . The work at Oak Ridge National Laboratory  is supported  \nby the U. S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials \nSciences and Engineering Division.  The scanning transmission electron microscop y work performed at \nBrookhaven National Laboratory is sponsored by the US Department of Energy, Basic Energy Sciences, \nMaterials Sciences and Engineering Division, under contract no. DE -SC0012704.  This research used \nthe Electron  Microscopy  resources  (the Helios  G5 FIB)  of the Center  for Functional  Nanomaterials  (CFN),  \nwhich  is a U.S. Department  of Energy  Office  of Science  User  Facility,  at Brookhaven  National  Laboratory  \nunder  Contract  No. DE-SC0012704 . We acknowledge  Hussein  Hijazi  for RBS  measurements.  \n \n \n  \n \nFigure 1.  Crystal structures of different phases of MnTe. (a,c,e) Nickeline, wurtzite and zinc blende \nphases of MnTe  respectively . The hexagonal -like (111) plane which is the  surface in (111) growth mode of  \nthe zinc blende phase  is illustrated in (c) . (b,d,f) Differences in  stacking sequences for Mn and Te atoms in \nthe nickeline, wurtzite and zinc blende phase respectively : the crystallographic directions are chosen to \nmatch those in Fig. 5.   \n \n \n \nFigure 2. Growth schematic . A schematic illustrating the buffer layered growth of (a) ZB-WZ MnTe and \n(b) NC MnTe on Al 2O3(0001) substrates.  \n \n \n \n \n \n \n \nFigure 3. RHEED analysis  of the two different phases . (a,b) RHEED patterns showing real time growth \nof epitaxial (a)  ZB-WZ MnTe and (b) NC MnTe. Both lattice constants are highlighted with respect to that \nof their buffer layers. Additionally, by comparing the streaks for the high symmetry directions, it can be \nseen that the in -plane structure s for both films are hexagonal , as indicated by the geometric ratios . (c,d)  \nEvolution of growth for (c) ZB -WZ MnTe and (d) NC MnTe as seen via RHEED spacing in pixels, \ninversely proportional to the lattice constant. Due to the discrete nature of pixels, the error bar for the lattice \nconstant is ~0.04  Å. \n \n \n \n  \n \nFigure 4. X-Ray Diffraction patterns and R vs T .  (a) XRD pattern of ZB -WZ MnTe with ZB(nnn) and \nWZ(00 02n) peaks highlighted. 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Oh, Nano Letters 22, 7522 \n(2022).  \n \n "    },    {        "title": "2401.14050v1.Thermal_and_Magnetoelastic_Properties_of_the_van_der_Waals_Ferromagnet_Fe___3_δ__GeTe__2___Anisotropic_Spontaneous_Magnetostriction_and_Ferromagnetic_Magnon_Excitations.pdf",        "content": "Thermal and Magnetoelastic Properties of the van der Waals Ferromagnet Fe 3−δGeTe 2:\nAnisotropic Spontaneous Magnetostriction and Ferromagnetic Magnon Excitations\nReinhard K. Kremer1,∗and Eva Br¨ ucher1,†\n1Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany\n(Dated: January 26, 2024)\nBy determining the lattice parameters as a function of temperature of the hexagonal van der\nWaals ferromagnet Fe 2.92(1)Ge1.02(3)Te2we obtain the temperature dependence of the spontaneous\nin-plane magnetostriction in the ferromagnetic and the linear thermal expansion coefficients in the\nparamagnetic state. The spontaneous magnetostriction is clearly seen in the temperature depen-\ndence of the in-plane lattice parameter a(T), but less well pronounced perpendicular to the planes\nalong c. Below TCthe spontaneous magnetostriction follows the square of the magnetization and\nleads to an expansion of the hexagonal layers. Extrapolating to T→0 K we obtain a spontaneous\nin-plane saturation magnetostriction of λsp,a(T→0)≈ −220×10−6. In the paramagnetic state\nthe linear thermal expansion coefficients amount to 13.9(1) ×10−6K−1and to 23.2(2) ×10−6K−1for\nthe in-plane and out-of-plane direction, respectively, indicating a linear volume thermal expansion\ncoefficient of 50.8(4) ×10−6K−1which we use to estimate the volume thermal expansion contri-\nbution to the heat capacity determined at constant pressure. A Sommerfeld-type linear term in\nthe low-temperature heat capacities can be quantitatively ascribed to 2dim ferromagnetic magnon\nexcitations.\nINTRODUCTION\nTwo-dimensional (2dim) van der Waals (vdW) ferro-\nmagnets with sufficiently high Curie temperatures lately\nhave attracted special attention with respect to their\napplicability in modern nanoelectronic and spintronic\ndevices. Among a number of prominent systems, the\nternary iron germanium telluride, Fe 3−δGeTe 2(δ≈0.1)\n(FGT), with a Curie temperature, TC, close to 220 K has\nbeen intensively investigated. After the initial synthe-\nsis of polycrystalline specimen and the basic structural\nand magnetic characterization,[1] accessibility of larger\ncrystalline samples enabled a broad variety of experi-\nments with a focus on mono- and multilayer samples.\nThese proved FGT to be a highly promising platform\nfor studying the complex interplay of magnetic and elec-\ntronic properties in reduced dimensions up to the point\nof monolayer devices.\nFGT exhibits a number of intriguing magneto-\nelectronic properties: Below TCFGT is an itinerant\nferromagnet.[1, 2] Though, an antiferromagnetic transi-\ntion at ∼150 K has also been conjectured.[3] Using den-\nsity functional calculations Jang et al. concluded that\nFe defects and hole doping are the key to drive FGT into\nthe ferromagnetic phase, whereas they proposed that sto-\nichiometric FGT to be antiferromagnetic.[4]\nA large anomalous Hall effect already at small polar-\nizing external magnetic fields has been found for bulk\nsamples.[5–8] Exfoliated monolayers still show uniax-\nial anisotropy and robust ferromagnetism, however with\nTCreduced to ∼130 K.[9, 10] Broadest interest at-\ntracted the electrolyte gating experiments by Deng et\nal.. They showed that already small gate voltages ap-\nplied to flakes of FGT with Li+intercalated in between\nthe Te double layers can tune TCfrom 100 K up to roomtemperature.[11] Laser-induced spin and charge pho-\ntocurrents in single-layer FGT were predicted by first-\nprinciples calculations suggesting applications of FGT in\nopto-spintronics.[12] N´ eel- or Bloch-type skyrmions have\nbeen reported to occur in FGT monolayers.[13–17] Xu et\nal.argued that unusual terms in the Hamiltonian con-\nnecting four spins can account for these observations.[18]\nRecently, successful generation and manipulation of ter-\nahertz spin-current has been reported by Chen et al. .[19]\nDespite the booming interest in the magnetic and\nmagnetoelectric properties of FGT, the lattice proper-\nties and especially magnetoelastic coupling still remained\nscarcely explored. By using density-functional theory\nZhuang et al. found that the orbital moment of the Fe\natoms is sizeable, causing a large magnetic anisotropy en-\nergy increasing with tensile strain, and a large in-plane\nmagnetostrictive coefficient of -559 ×10−6for monolayer\nFGT.[9] In Raman spectroscopy measurements Milosavl-\njevi´ c et al. observed fingerprints of spin-phonon coupling\nat around 152 K and also anomalous behavior of the Ra-\nman frequencies and mode linewidths at the ferromag-\nnetic transition.[20]\nFGT crystallizes in the hexagonal crystal system with\nlattice parameters a≈3.99˚A and c≈16.34 ˚A. The crys-\ntal structure of FGT contains slabs of Fe 3Ge sandwiched\nby vdW bonded Te double layers.[1] The Fe atoms occupy\ntwo different crystallographic sites. Notably, the Fe2 site\n(Wyckoff position 2c) in samples prepared under stoi-\nchiometric conditions is not fully occupied (typically δ∼\n0.1), whereas the Fe1 site, within error bars, exhibits no\ndeficiency. Increasing δ, i.e. enlarging the Fe2 deficiency\nreduces TC. The in-plane lattice parameter adecreases\nwith increasing δ, whereas the out-of-plane lattice param-\netercincreases for larger Fe2 deficiency.[21] Strong uniax-\nial anisotropy aligns the Fe magnetic moments along thearXiv:2401.14050v1  [cond-mat.str-el]  25 Jan 20242\nceasy axis making FGT an auspicious platform for mag-\nnetic data storage devices.[9, 22–24] The ordered mag-\nnetic moments of the Fe atoms for δ= 0.1 amount to\n2.2µBfor the Fe1 and 1.5 µBfor the Fe2 crystallo-\ngraphic sites, respectively.[21] For δ∼0.25 both ordered\nmoments converge to a common value of ∼1.4µB.[9, 21]\nHere, we report on measurements of the zero-field\nspontaneous magnetostriction when FGT enters into the\nferromagnetic state. Magnetostriction, i.e. the deforma-\ntion of the geometrical shape of a ferromagnetic spec-\nimen during the magnetization process can be evoked\ne.g. by an external magnetic field. Especially magnetic\nfield induced magnetostriction is of great technological\nimportance but also fundamental to model the magne-\ntization process itself and the formation of the domain\nstructure. Magnetostriction induced by an external mag-\nnetic field depends on the orientation of the external\nmagnetic field with respect to the orientation of domain\nmagnetization and the direction of the exchange inter-\naction between the magnetic moments.[25] Spontaneous\nmagnetostriction where the shape change, λ=δl(T)/l(T),\nis initiated by the increase of the spontaneous magneti-\nzation M(T < T c, H= 0) below TCin vanishing ex-\nternal magnetic field Hi.e. by the internal ferromag-\nnetic domain formation can provide very valuable infor-\nmation, e.g. about spontaneous reorientation processes\nand changes in the domain structure. Technologically\nmagnetostrictive effects may become important for the\ninteraction of FGT monolayers deposited on substrates.\nIn addition, we review preceding heat capacity exper-\niments which have found large linear terms in the low\nand high temperature specific heats which the authors at-\ntributed to magnetic contributions.[2] Especially in the\nparamagnetic regime, the published heat capacity data\nsubstantially overshoot the Dulong-Petit limit, suggest-\ning a critical reconsideration of the thermal properties.\nBy using the linear thermal volume expansion coefficients\nin the paramagnetic state to determine the lattice ex-\npansion contribution to the heat capacities measured at\nconstant pressure, Cp, we revise these findings.\nThe Sommerfeld-like linear term in the low-\ntemperature heat capacities can be quantitatively\nattributed to 2dim ferromagnetic magnon excitations.\nA comparison with the spin stiffness constants found by\ninelastic neutron scattering gives quantitative agreement.\nEXPERIMENTAL\nSamples of FGT were prepared from stoichiometric\nmixtures of powders of the elements, Fe (Alfa Aesar,\npurity 99.998%), Ge (Thermo Fisher, purity 99.999%)\nand Te (Thermo Fisher, purity 99.999%) using a minute\namount of iodine as mineralizer. The starting materials\nwere thoroughly mixed and reacted in evacuated quartz\nglass tubes in a two-zone furnace heated to temperaturesbetween 750◦C and 650◦C. Phase purity and compo-\nsition of the samples was checked by energy-dispersive\nspectroscopy (EDX) employing a Tescan Vega4 LMU\nSEM equipped with an Oxford X-MaxNN20 detector and\nby x-ray powder diffraction (XRPD) using Mo Kα1and\nCuKα1radiation. The magnetic properties of the sample\nwere determined by powder and single crystal dc mag-\nnetization (MPMS, Quantum Design) and specific heat\nmeasurements (PPMS, Quantum Design).\nThe lattice parameters of a polycrystalline sample of\nFGT were determined in zero external magnetic field as a\nfunction of temperature from Rietveld profile analysis[26]\nof XRPD patterns collected on a Bruker D8 Discovery x-\nray diffractometer (Bragg-Brentano scattering geometry)\nusing Cu Kα1radiation. A PHENIX (Oxford Cryosys-\ntems) closed cycle cooling cryostat was used to set the\ntemperature of the sample. Each XRPD pattern was\ncollected at stabilized temperature. The particle size of\nthe powder was adjusted to 63 µm or less by straining the\npowder through sieves of the respective mesh size. The\npowder sample was thermally anchored with ApiezonN\nvacuum grease to the sample holder platform, equipped\nwith an inlay of a Si wafer specially oriented to suppress\nbackground scattering from the sample holder.\nRESULTS AND DISCUSSION\nSample Characterization\nFigure 1 displays a SEM picture of a typical crystal an-\nalyzed with energy dispersive x-ray spectroscopy. Three\nspots on an a-bsurface were tested and very good agree-\nment was found for the element concentration, proving\nhomogeneity of the element concentration. EDX analy-\nses carried out on crystals of several other FGT sample\npreparations indicated a spread of the Fe2 concentration\nof 0.82 ≤(1−δ)≤0.92.\nXRPD on crushed crystals and polycrystalline samples\nof FGT proved phase purity of our specimen. Figure 2\ndisplays a typical room-temperature XRPD pattern an-\nalyzed with a Rietveld profile refinement assuming the\nstructure model proposed by Deiseroth et al. (space\ngroup P63/mmc , no. 194).[1] At room temperature the\nrefined lattice parameters amount to a= 3.99939(8) ˚A\nandc= 16.3225(4) ˚A, in good agreement with the val-\nues reported by Deiseroth et al. . In the refinements\nthe Ge and Fe1 lattice sites were assumed to be fully\noccupied, whereas for Fe2 the occupancy converged to\n0.918(11),in good accord with the EDX result. At room\ntemperature the z-positional parameter of the Te and Fe1\natoms (Wyckoff positions 4f and 4e, respectively) were re-\nfined to 0.5902(2) and 0.6702(3), also in good agreement\nwith the values obtained from single crystal structure\ndetermination.[1]\nThe Curie temperatures of the FGT samples were de-3\nFIG. 1. SEM image of a crystal of our sample with the rect-\nangles marking the areas where EDX element analysis was\nperformed. The variation of element concentration across the\nanalyzed areas amounted to less than 0.5%, 1.4% and 0.07%\nfor Fe, Ge and Te, respectively. The lower inset shows the\nSEM spectrum collected at spot no. 1.\n0 20 40 60 80 100 120-20000200040006000Intensity (arb. units)\n2Q (deg)\nFIG. 2. (color online) XRD pattern of FGT (Cu Kα1radi-\nation) collected at 295 K. The red circles represent the ex-\nperimental data, the solid black line show the result of the\nRietveld profile refinement and the vertical green bars mark\nthe positions of the Bragg reflections used to calculate the\nrefined pattern.\ntermined from heat capacity and magnetization measure-\nments. Figure 3 summarizes the magnetic susceptibili-\nties and the specific heat of the sample with composi-\ntion Fe 2.92(1)Ge1.00(3)Te2. The magnetization data ex-\nhibit the characteristic splitting of the zero-field cooled\n(zfc) and field - cooled (fc) branches reported before.[1]\nThe zfc - fc hysteresis closes with increasing magnetic\nfield. The steepest descent of the magnetization is found\nat∼218 K (see inset (b) in Figure 3) where also the\nλ-type anomaly in the specific heat is observed (Figure\n3(d)). At room temperature the heat capacity linearly\napproaches a value of ∼156 J/molK, consistent with the\nDulong-Petit value of 5.92 ×3R, where Ris the molar gas\nconstant, but substantially lower than the findings re-\nported by Bin Chen et al. .[2] The linear increase of the\nheat capacity will be discussed in detail below. At low\ntemperatures in the ferromagnetic state the heat capac-ities follow a power law (Figure 3(e))\nCmol/T=γ+βT2, (1)\nwith γ= 109(1) mJ/molK2, close to what has been ob-\nserved by Bin Chen et al. .[2] The slope β= 1.25(6)\nmJ/molK4implies a Debye temperature, Θ Deb(T→\n0 K) of 210(2) K. The majority of the linear contribution\nto the heat capacity can be ascribed to two-dimensional\nferromagnetic magnons, as will be analyzed in detail be-\nlow.\n0 100 200 3000510152025\n0 1 2 3 4 5 600.511.5Moment per Fe  ( mB)\nm0H (T)\n200 210 220 230 240-1.5-1-0.50dcmol/dT (cm3/K mol)\nTemperature (K)cmol (cm3/mol)\nTemperature (K)     100 Oe\n   1000 Oe\n 10000 Oe\n020406080100120140160Cp (J/molK)(a)\n0 50 100 150 200 25000.10.20.30.4Cp/T (J/molK2)\nTemperature2 (K2)\n(b)\n(c)\n(d)(e)\nFIG. 3. (color online) (upper panel, black circles) (a) Molar\nheat capacity of a crystal of Fe 2.92(1)Ge1.02(3)Te2. The blue\ndots exemplify the reduction of the Curie temperature for a\nsample with element composition Fe 2.85GeTe 2(data down-\nshifted by 10 J/molK for better comparison). (b) displays\nthe power law behavior at low temperatures.(lower panel) (c)\nzfc and fc magnetic susceptibilities (indicated by the arrows)\nof the FGT sample of composition Fe 2.92(1)Ge1.02(3)Te2mea-\nsured with external magnetic fields of 100 Oe, 1 kOe and\n10 kOe. The inset (d) displays the derivative of the magneti-\nzation measured with an external field of 100 Oe as a function\nof temperature. The (upper) inset (e) shows the magnetic mo-\nment per Fe atom. The (red) solid line is a guide to the eye.\nThe Curie temperature TCdepends on the Fe2 con-\ntent. An increase of δleads to a reduction of the Curie\ntemperature (see blue dots in Figure 3(a)). Figure 4(a)4\ndisplays the variation of the Curie temperature as a func-\ntion of the composition around δ≈0.1. A decrease of\nTCis paralleled by a decrease of the volume of the crys-\ntallographic unit cell (Fig. 4(b)), the latter induced by\nthe decrease of the in-plane lattice parameter a.\n0.05 0.1 0.15227227.2227.4227.6227.8Cell volume ( Å3)\nd\n216.5217217.5218218.5219219.5TC (K)(a)\n(b)\nFIG. 4. (color online) (a) Curie temperature, TCas a function\nof the Fe2 deficiency, δof our samples. δwas determined from\nthe Rietveld refinements of the XRPD patterns collected using\nMoKα1radiation and from EDX analyses. (b) Volume of the\ncrystallographic unit cell as a function of δ. The (red) dashed\nlines are guides to the eye.\nSpontaneous Magnetostriction\nA fraction of the same sample with composition\nFe2.92(1)Ge1.02(3)Te2which had never been exposed to an\nexternal field was subsequently used for the temperature\ndependent XRPD measurements. Figure 5 displays the\nreduced lattice parameters a∗(T)=a(T)/a(295 K) and\nc∗(T)=c(T)/c(295 K), relative to their room temperature\nvalues, as derived from the Rietveld profile refinements\nof the temperature dependent XRPD patterns. With de-\ncreasing temperature the relative contraction of the lat-\ntice parameters perpendicular to the layers is about a\nfactor of two larger than the in-plane contraction. In\nthe paramagnetic regime the cell volume and the lat-\ntice parameters decrease linearly with temperature withrates ( dVcell/dT)/Vcell(295 K) = -50.8(4) ×10−6K−1, and\n-13.9(2) ×10−6K−1, and - 23.2(2) ×10−6K−1, for aand\nc, respectively.\n0 50 100 150 200 250 3000.9950.9960.9970.9980.9991\nTemperature (K)x(T)/x(295 K )\na\nc\nFIG. 5. (color online) Temperature dependence of\nthe lattice parameters a∗(T)=a(T)/a(295K) and of\nc∗(T)=c(T)/(295 K).\nFigure 6 displays the temperature dependence of the\nz-positional parameters of the Te and the Fe1 atoms.\nWhereas z(Fe1) shows a faint increase with decreasing\ntemperature, still within error bars, the zpositional pa-\nrameter of the Te atoms decreases and levels off at low\ntemperatures.\nFigure 7 compiles the volume of the hexagonal unit cell\nand the lattice parameters a(T) and c(T) as a function\nof the temperature. Whereas the in-plane lattice param-\netera(T) exhibits a noticeable shoulder near T∼TC,\nc(T) bents away from the high-temperature linear behav-\nior before leveling off at low temperatures. A noticeable\nresponse to the onset of ferromagnetic order is not seen\nfor the out-of-plane lattice parameter c. The shoulder in\na(T) is ascribed to a negative spontaneous in-plane mag-\nnetostriction, i.e. an expansion of the hexagonal planes,\ninduced by ferromagnetic ordering.\nExtrapolating the linear behavior of a(T) from the\nparamagnetic regime and subtracting from the data be-\nlowTC, reveals a spontaneous in-plane magnetostriction\nstarting below ∼223 K, which saturates below ∼50 K.\nThe temperature dependence of the spontaneous magne-\ntostriction (see Figure 8) is reminiscent of a continuous\nphase transition with a critical temperature of 223(1) K,\nmatching TC, as determined from the magnetization and\nspecific heat experiments. A tentative extrapolation of\nthe spontaneous magnetostriction to T→0 K (see the\nblack dashed line in Fig. 8) indicates a spontaneous in-\nplane saturation magnetostriction, λsp,a(0)), of\nλsp,a(T→0)≈ −2.2×10−4.5\n0.66980.670.67020.67040.67060.67080.6710.6712z-Fe1\n0 50 100 150 200 250 3000.58980.58990.590.59010.59020.5903z-Te\nTemperature (K)(a)\n(b)\nFIG. 6. zpositional parameters of the (a) Fe1 and (b)\nthe Te atoms (Wyckoff positions 4e and 4f, respectively) of\nFe2.92(1)Ge1.02(3)Te2as a function of temperature.\nAttempts to model the temperature dependence of the\nlattice parameters to a Debye function over the whole\ntemperature range lead to unstable fits and were not\nmeaningful. For the same reason the out-of-plane spon-\ntaneous magnetostriction was difficult to ascertain since\nthere is no visible anomaly in the temperature depen-\ndence of the c-lattice parameter at TC.\nIn order to relate the spontaneous magnetostriction\nto the temperature dependence of the magnetization we\ncarried out isothermal magnetization measurements on\na thin crystal platelet of Fe 2.92(1)Ge1.02(3)Te2with the\nmagnetic field aligned along the c-axis. After a correction\nfor the demagnetizing field, we derived the temperature\ndependence of the zero-field magnetization from modified\nArrott Belov plots[27, 28] (see inset in Fig. 9). The zero-\nfield magnetization can be very well fitted to a critical\npower law (see Fig. 9) given by\nM(T) =M0(1−T/T C)β, (2)\nwith a critical exponent β= 0.32(1), consistent with find-\nings reported earlier by Liu et al..[29, 30]\nFig. 10 displays the temperature dependence of the\nspontaneous magnetostriction as a function of the mag-\nnetization. The log-log plot reveals a power law behavior\n0 50 100 150 200 250 300224224.5225225.5226226.5Vcell (Å3)\nTemperature (K)\n3.993.99544.005a (Å)Fe2.93(2) Ge1.02(3) Te2\n16.2616.2816.316.3216.34c (Å)(a)\n(b)\n(c)FIG. 7. (color online) Temperature dependence of the lat-\ntice parameters a(a), c(b), and the cell volume (c) of\nFe2.92(1)Ge1.02(3)Te2. Error bars are of the size of the sym-\nbols. The solid (red) lines represent linear fits between 235 K\nand 295 K of the temperature dependence of the respective\nquantities. The relative slopes are given in the text.\nof the spontaneous magnetostriction according to\nλsp,a(T) = [M0(1−T/T C)β]1.5(1). (3)\nTaking into consideration the critical exponent β= 0.32\nfor the magnetization, eq.(3) indicates a critical exponent\nfor the spontaneous in-plane magnetostriction of ∼0.48,\ni.e. close to the critical exponent expected for mean field\nbehavior.6\n0 50 100 150 200 250 30000.511.522.53104 ´ (a(T) - p(T)) / a(T)\nTemperature (K)\nFIG. 8. (color online) Spontaneous in-plane magnetostriction\nof Fe 2.92(1)Ge1.02(3)Te2as a function of temperature. The red\nsolid line is a fit to a critical power law, τβwith the reduced\ntemperature τ= 1 - ( T/T C) with a critical temperature, TC\n222.4(8) K and a critical exponent β= 0.48(2). The black\ndashed line represents an extrapolation to T→0 K.\n120 140 160 180 200 22000.10.20.3Saturation Moment (Tesla)\nTemperature (K)Tc= 216.9(1) K\nb= 0.32(1)\n0 2 4 6 8 10\n00.010.020.030.040.050.06\nM1/b (T1/b)(m0Hint/M)1/g\n 170 K      210 K\n 190 K      214 K\n 200 K      216 K\n 206 K      218 K\nFIG. 9. (color online) Temperature dependence of the zero-\nfield spontaneous magnetization of Fe 2.92(1)Ge1.02(3)Te2de-\nrived from modified Arrott Belov plots displayed (lower inset)\nadopting critical exponents β= 0.33 and γ= 1.38. The main\nframe shows a fit of the magnetization with a power-law tem-\nperature dependence (solid red) line according to eq. (2).\nThermal Lattice Expansion Contribution to the\nSpecific Heat\nIn a first investigation of the heat capacities of FGT\nBin Chen et al. had found large linear contributions\nat low and high temperatures, hitherto no conclusively\n0.06 0.1 0.3 0.60.10.30.613104 ´ (a(T) - p(T)) / a(T)\nM (T) = M0 (1-T/Tc)bFIG. 10. (color online) Log-log plot of the spontaneous in-\nplane magnetostriction as a function of the magnetization.\nThe blue dashed line corresponds to a power law with an\nexponent 1.5.\nexplained.[2] At room temperature they found a value\nof∼200 J/molK and a stark linear increase above\nTC, which they attributed to electronic and magnetic\ncontributions.[2]\nIn contrast to these earlier findings our heat capacity\ndata measured at constant pressure, Cp, (see Fig. 3(d))\ntend to a value of ∼156 J/molK, only moderately ex-\nceeding the Dulong-Petit limit for the heat capacity at\nconstant volume, Cv(T→ ∞ )×3NR = 148 J/molK,\nwith N= 5.92 being the number of atoms per formula\nunit and Rthe molar gas constant. At 350 K our Cp\ndata attain a value of ∼158 J/molK, substantially lower\nthan that reported by Bin Chen et al. .\nIn the following, using our thermal expansion data for\nT > T C, we estimate the electronic and the lattice ther-\nmal expansion contribution to the heat capacities at high\ntemperatures. The difference between the heat capacities\nmeasured at constant pressure and at constant volume is\ngiven by\nCp−Cv=α2\nv(T)BVmolT, (4)\nwhere αv(T) is the temperature dependent coefficient of\nthe volume thermal expansion, Bis the bulk modulus\nandVmolthe molar volume.\nIn the paramagnetic regime above TCand up to room\ntemperature the lattice expands approximately linearly\nwith increasing temperature. The linear thermal expan-\nsion coefficients amount to αa= 13.9(1) ×10−6K−1and\ntoαc= 23.2(2) ×10−6K−1for the in-plane and out-of-\nplane direction, respectively, resulting in a linear vol-\nume thermal expansion coefficient of αv= 50.8(4) ×10−6.\nThe in-plane linear thermal expansion coefficient of\nFe2.92(1)Ge1.02(3)Te2in the paramagnetic state compares7\n200 250 300 350 400146148150152154156158160162164Cp (J/molK)\nTemperature (K)\nFIG. 11. (color online) High-temperature heat capacities at\nconstant pressure, Cp, of Fe 2.92(1)Ge1.02(3)Te2. The solid (red)\nline is a fit to eq.(5) with parameters given in the text.\nwell with that of elementary iron for which an ex-\npansion coefficient of 11.6 ×10−6was observed at room\ntemperature.[31]\nIn order to fit the heat capacity at high temperatures\ndisplayed in Figure 11 we used a series expansion for\nthe Debye contribution[32] and included a term linear in\ntemperature, E T,\nCp(T) = 5 .92×3R(1−1\n20Θ2\n∞\nT2+1\n560Θ4\n∞\nT4) +E T, (5)\nwith Rbeing the gas constant, and Θ ∞the Debye tem-\nperature. E T comprises the thermal expansion contri-\nbution given by Eq. (4) but also a linear electronic (Som-\nmerfeld) term γ. A fit of the heat capacities of several\nruns (Fig. 11) using data for T >225 K yields\nE= 37(2) mJ /molK2,\nand\nΘ∞= 222(3) K ,\nthe latter being very close to the Debye temperature\nΘDebye (T→0 K) which amounts to 210 K.\nThe Sommerfeld-term, γ, from the heat capacity of the\nconduction electrons can be estimated from\nγ=π2k2\nB\n3N(EF), (6)\nwith the Boltzmann constant kBand the electronic den-\nsity of states N(EF) at the Fermi level. The electronic\ndensities of states for two spin directions at EFper for-\nmula unit obtained from density functional calculations\nhas been reported to 6(2) states/eV.[33, 34] Using thisvalue and Eq. (6) one arrives at a conduction electron\nterm of 14 mJ/molK2and at a value of 23 mJ/molK2for\nthe thermal expansion contribution. Adopting the vol-\nume thermal expansion coefficient αVol= 50.8 ×10−6K−1\nand the molar volume Vmol= 6.785 ×10−5m3mol−1[1]\nresults in a Bulk modulus Bof\nB∼130(40)GPa ,\nwherein the sizeable error bar arises from the uncer-\ntainty of the electronic density of states at the Fermi\nenergy. Our finding for the bulk modulus is by a fac-\ntor of two larger than the value of ∼70 GPa reported\nin literature[34, 35] The difference between our value\nand the reported results might be partially due to the\nnegligence of possible electron-phonon enhancement of\nthe electronic Sommerfeld term. We also note that for\ntheir high-pressure M¨ ossbauer measurements O’Hara et\nal.used FGT samples with a TCof 155 K, indicating a\nsubstantial deficit on the Fe2 site which may also affect\ntheir value for the bulk modulus.[35]\nEven accepting the full value of E≈37 mJ/molK2as\ndue to the electron contribution, the excess heat capac-\nity of almost 50 J/molK at room temperature over the\nDulong-Petit limit reported by Bin Chen et al. [2] is far\ntoo large to be reconciled with our findings.\n2dim Ferromagnetic Magnon Heat Capacity\nWe finally comment on the linear contribution to the\nlow temperature heat capacity which amounts to ∼109\nmJ/molK2, again too large to be attributed to the con-\nduction electron contribution. However, the increase lin-\nearin temperature, also reported by Bin Chen et al.\nwith a similar magnitude, can be readily reconciled if we\ntake ferromagnetic magnon excitations in a 2dim lattice\ninto account. Adopting a quadratic dispersion for long-\nwavelength magnons in a 2dim ferromagnet the magnon\ncontribution to the heat capacity at low temperatures\nvaries linearly with temperature.[36] From inelastic neu-\ntron scattering data, Song Bao et al. have derived spin\nwave stiffness constants for FGT between 57 and 69\nmeV˚A2for the in-plane spin waves.[37] For the magnon\nheat capacity, Cmag, of a 2dim honeycomb lattice, Grosu\net al. arrived at[38]\nCmag=R3√\n3π2a2\n24DT, (7)\nwhere Ris the molar gas constant, a≈4˚A the in-plane\nlattice parameter and Dthe spin wave stiffness constant.\nWith D≈63 meV ˚A2one estimates a magnon contribu-\ntion linear in temperature of\nCmag/T= 123 mJ /molK2.\nAdding the conduction electron contribution of\n∼14 mJ/molK2, this finding is in fair agreement8\nwith our experimental observation of 109 mJ/molK2\nfound from the Sommerfeld plot at low temperatures\n(see Fig. 3).\nSUMMARY AND CONCLUSION\nUsing temperature dependent XRPD measurements\nwe have determined the hexagonal lattice parameters of\nthe 2dim van der Waals ferromagnet Fe 2.92(1)Ge1.02(3)Te2\nwhich exhibits a Curie temperature of ∼217 K. Spon-\ntaneous magnetostriction induced by the transition into\nthe ferromagnetic state leads to an expansion of the vdW\ncoupled layers, whereas a noticeable effect of the sponta-\nneous magnetostriction perpendicular to the layers is not\nseen in the c-lattice parameters. For T→0 K, the in-\nplane spontaneous magnetostriction can be extrapolated\nto∼-220×10−6K−1by more than a factor of two smaller\nthan the ab-initio calculations Zhuang et al. .\nUsing the linear volume thermal expansion coefficient\nin the paramagnetic regime we estimate the difference\nof the specific heats determined at constant pressure and\nconstant volume. The majority of the linear Sommerfeld-\ntype contribution to the low-temperature heat capac-\nity can be quantitatively ascribed to 2dim ferromagnetic\nmagnon excitations.\nIn summary, our experimental results evidence a sub-\nstantial in-plane magnetostriction for a sample of FGT\nwith composition Fe 2.92(1)Ge1.02(3)Te2having a Curie\ntemperature of 217 K. A careful reconsideration of the\nheat capacities of such sample enables us to correct and\nunderstand previously reported linear contributions to\nthe low- and high-temperature heat capacity data of\nFGT. Particularly noteworthy is the low-temperature\nmagnon heat capacity contribution, linear in tempera-\nture, which can be quantitatively attributed to the lay-\nered ferromagnetic character of FGT.\nAcknowledgments We thank V. Duppel for performing\nthe SEM/EDS measurements and Hj. Deiseroth for valu-\nable suggestions as to the preparation of FGT. A useful\ncommunication with Houlong L. Zhuang is gratefully ac-\nknowledged.\n∗R.Kremer@fkf.mpg.de\n†E.Bruecher@fkf.mpg.de\n[1] H.-J. Deiseroth, K. Aleksandrov, C. Reiner, L. Kienle,\nand R. K. Kremer, Fe 3GeTe 2and Ni 3GeTe 2– two new\nlayered transition-metal compounds: Crystal structures,\nhrtem investigations, and magnetic and electrical prop-\nerties, Eur. J. Inorg. Chem. 2006 , 1561 (2006).\n[2] B. Chen, J. Yang, H. Wang, M. 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This settles a conjecture of Breu illard and Green,\nand recovers and improves - with completely different method s - a recent result\nof Jing–Tran–Zhang corresponding to the case G= SO3(R). As is often the\ncase, the above doubling inequality is but a special case of g eneral product-set\nestimates. We prove that for all ǫ >0 and for any pair of sufficiently small\nmeasurable subsets A,Ba Brunn–Minkowski-type inequality holds:\nµG(AB)1\ndG−dH≥(1−ǫ)/parenleftbigg\nµG(A)1\ndG−dH+µG(B)1\ndG−dH/parenrightbigg\n.\nGoing beyond the scope of the Breuillard–Green conjecture, we prove a\nstability result asserting that the only subsets with close to minimal doubling\nare essentially neighbourhoods of proper subgroups i.e. of the form\nHδ:={g∈G:d(g,H)< δ}\nwhereHdenotes a proper closed subgroup of maximal dimension, ddenotes a\nbi-invariant distance on Gandδ >0.\nOur approach relies on a combination of two toolsets: optima l transports\nand its recent applications to the Brunn–Minkowski inequal ity, and the struc-\nture theory of compact approximate subgroups.\n1.Introduction\nLetGbe a connected Lie group of dimension dGandµGbe the Haar probability\nmeasure i.e. the unique G-invariant Borel probability measure on G. Thedoubling\nconstant of a measurable subset A⊂Gis defined as the ratio\nµG(A2)\nµG(A)\nwhereA2:={a1a2:a1,a2∈A}is the Minkowski product. The doubling constant\nis an important object of study in both additive combinatorics [4] and geometric\nanalysis [24]. The minimal value it can take in a given group Ghas been a long-\nestablished subject of research. Without any assumption on the a mbient group\nthe only known result dates back to the foundational work of Kemp erman [37].\nETH Zurich, 101 Ramistrasse, 8006 Zurich, Switzerland\nE-mail address :smachado@ethz.ch .\nDate: March 13, 2024.\n12 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nIn Euclidean spaces and nilpotent Lie groups of dimension dG, it can be easily\nestablished that the doubling constant takes value at least 2dG[30]. Recent work\nof Jing, Tran and Zhang [34] yields similar inequalities in many non-compa ct Lie\ngroups.\nThis paper is concerned with proving doubling inequalities in the more ch alleng-\ning situation of compact Lie groups. If Gis a compact group then taking G=A\nimplies that the doubling constant takes value 1. This somewhat trivia l example\nindicates that any meaningful lower bound on the doubling thus requ ires dealing\nwith subtle global phenomena absent in other setups. This motivate d Breuillard\nandGreentoaskand investigatedoublingincompactLiegroups. The yconjectured\nthat doubling for small sets in SO 3(R) must be at least 4 −ǫ. In higher-dimension,\ntheir conjecture becomes:\nConjecture 1.1 (Breuillard–Green conjecture for compact Lie groups, [26, 33]) .\nLetGbe a compact Lie group of dimension dGand letdHbe the maximal dimension\nof a proper closed subgroup H. Then the minimal value of the doubling constant\nfor subsets of small measure should be about 2dG−dH.\nWhenGadmits anabelian factor, the conjecturefollowsfromKemperman’s [37].\nLittle progresswas made on the Breuillard–Greenconjecture beyo nd that case until\narecentbreakthroughofJing,TranandZhang[33]buildinguponpr eviousworkand\nmodel-theoretic tools. They solved there the original Breuillard–Gr een conjecture\n(corresponding to G= SO3(R) above) but their method does not generalise.\nThe main result of this paper is a proof of the general case of this co njecture\nalong with sharp error terms:\nTheorem 1.2 (Minimal doubling in compact Lie groups) .LetGbe a compact\nconnected Lie group. For all A⊂Gcompact we have\nµG(A2)≥/parenleftBig\n2dG−dH−CµG(A)2\ndG−dH/parenrightBig\nµG(A)\nwheredGdenotes the dimension of G,dHdenotes the maximal dimension of a\nproper closed subgroup and C:=C(dG)>0denotes a dimensional constant.\nThe presence of the correcting term CµG(A)2\ndG−dHis necessary, sharp (e.g. [33,\nFact 2.7]) and new even in the case G= SO3(R). The constant Cis not explicit\nhowever (see §8.3).\nAs is often the case, the doubling inequality (Theorem 1.2) stems fro m a much\nmore general lower bound on the measure of product-sets. The lo wer bound we\nobtain takes the form of a Brunn–Minkowski-type inequality:\nTheorem 1.3 (Brunn–Minkowski in compact Lie groups) .LetGbe a compact\nconnected Lie group. Let α >0. There is ǫ >0such that if A,Bare compact\nsubsets ofGwithµG(A),µG(B)≤ǫ, then\nµG(AB)1\ndG−dH≥(1−α)/parenleftBig\nµG(A)1\ndG−dH+µG(B)1\ndG−dH/parenrightBig\nwheredGdenotes the dimension of GanddHdenotes the maximal dimension of a\nproper closed subgroup.\nThe Brunn–Minkowski-type inequality is sharp and the presence of t he correct-\ning termαis also necessary. We can also deduce an estimate of αof the formMINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 3\nCmax(µG(A),µG(B))c, but this is far from optimal when µG(B) becomes small in\nfront ofµG(A). So we do not pursue this.\nEven when specialised to SO 3(R), our proof provides a new approach to the\nresult of [33] based on geometric ideas related to Jordan’s theorem , the structure of\napproximate groups and the optimal transport approach to the B runn–Minkowski\ninequality. Thisnewapproachprovidesfurtherinsightintothestru ctureofsetswith\ndoublingclosetotheoptimum. Recallthattherelevanceofthecons tant2dG−dHwas\nexplained in [26, 33] by the fact that neighbourhoods of maximal sub groups should\nhave close to optimal doubling and have doubling about 2dG−dH. The additional\ninsight brought by our method reverses this observation. We show that, in general,\nneighbourhoods of maximal subgroups are the onlyexplanation for doubling close\nto the optimum:\nTheorem 1.4 (Stability of the doubling inequality) .LetGbe a semi-simple con-\nnected compact Lie group of dimension dGand letdHbe the maximal dimension of\na proper closed subgroup. Let δ>0. There are ǫ,α>0such that if µG(A)<ǫand\nµG(A2)≥/parenleftbig\n2dG−dH+α/parenrightbig\nµG(A)\nthen there are a proper closed subgroup Hof maximal dimension and δ′>0such\nthatA⊂Hδ′:={g∈G:d(g,H)<δ′}andµG(Hδ′\\A)≤δµG(Hδ′).\nWhenG=SOn(R),n/\\e}a⊔io\\slash= 4, Theorem 1.4 thus asserts that the only small subsets\nwith close to minimal doubling are - up to a small error - inverse images o f small\nballs of the sphere Sn−1seen as the symmetric space SO n(R)/SOn−1(R). This\nis part of a larger framework concerning stability results related to the Brunn–\nMinkowski inequality - see [20, 21] for Euclidean spaces, [48, 11] for tori and [32]\nfor groups with an abelian factor - and is the first result of its kind in t he context\nof non-commutative groups.\n1.1.Historical background. The first results on the doubling constant can be\ntraced back to fundamental papers of Brunn [7], Minkowski [44] an d Lusternik [39]\nregarding Minkowski sums in RdG. Their works unveiled a natural relationship\nbetween minimal doubling and dimension: the logarithm of the minimal do ubling\nconstant equals the dimension dG. This tells us that minimal doubling is intimately\nlinked to the local behaviour of the set considered. These inequalitie s and their\ngeneralisations have been a fundamental tool of geometric analys is in relation to\nisoperimetric inequalities. We refer to the survey [24] for this and mo re.\nAs a counterpart, the seminal work of Freiman [23] showed that do ubling is also\nlinked to global phenomena. Freiman’s work showed quantitatively th at the only\nsubsets of Zexhibiting small doubling are arithmetic progressions - for which the\nsmall doubling is also 2 to the power of a dimensional constant. Many w orks have\nimproved upon Freiman’s results. They were extended to other abe lian and non-\nabelian groups (e.g. [27, 29]) culminating in a general classification res ult for finite\napproximate subgroups due to Breuillard, Green and Tao [6]. The qua ntitative\naspects of Freiman’s results were also improved towards the polyno mial Freiman\nconjecture [45, 25].\nThe interaction between these two fields recently gained traction f ollowing the\nseminal work of Figalli and Jerison on the quantitative stability proble m of the\nBrunn–Minkowski inequality [20]. They observed there that the one -dimensional\nstability could be seen as a consequence of Freiman’s 3 k−4 theorem [22] and used4 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nit to kickstart their proof. Following this, further quantitative sta bility results were\nobtained [19, 8] culminating in the recent sharp bounds obtained by F igalli, van\nHintum and Tiba [21]. Keevash, van Hintum and Tiba [50] also provided a d iscrete\ncounterpart to the Brunn–Minkowski inequality and the relevant s tability result by\nsolving a conjecture of Ruzsa establishing a Brunn–Minkowski type inequality in\nZn- we also mention in that direction the recent work of Gowers, Green , Manners\nand Tao [25] on the polynomial Freiman–Ruzsa conjecture over finit e fields. We\nshow in this article that some of the techniques and ideas developed in that field\nare also decisive in solving the Breuillard–Green conjecture for comp act Lie groups\n(Theorem 1.2) and the related stability result (Theorem 1.4), where they were\nperhaps less expected. We also mention [17] which appeared at the s ame time as\nthe first version of this article and studies product-set estimates for large subsets\nof compact Lie groups.\nFinally, we point the reader interested in further considerations an d open prob-\nlems to the last section ( §8) of this paper.\n1.2.A look at the methods. The proofs of Theorems 1.2, 1.3 and 1.4 have the\nsame overall structure. They are the combination of two parts: a result about the\nglobal structure and a counterpart about local doubling.\nIn the caseofTheorems1.2and 1.3, the local resultis anextensiono fthe Brunn–\nMinkowski inequality to subsets in small enough neighbourhoods of t he identity:\nTheorem 1.5. LetGbe a Lie group of dimension dGanddbe a bi-invariant\ndistance such that (G,d)has diameter 1. For allǫ>0, there isρ>0such that for\nanyA,B⊂Bd(e,ρ)compact we have\nµG(AB)1\ndG≥(1−ǫ)/parenleftBig\nµG(A)1\ndG+µG(B)1\ndG/parenrightBig\n.\nMoreover, we can choose ǫ<C(dG)ρ2for some dimensional constant C(dG)>0.\nTo establish this result we adapt a strategy described in particular in [19, p.7]\nproving Brunn–Minkowski-type inequalities thanks to optimal trans port.\nTheorem 1.5 combined with a simple double counting argument yields The orem\n1.2 for subsets contained in a neighbourhood of a proper subgroup ; Theorem 1.3\nfor those same subsets follows along similar lines with the simple addition of the\noptimisation of a parameter. That we can reduce to such a situation is guaranteed\nby our next result concerning the global structure of approximat e subgroups:\nProposition 1.6 (Approximate subgroups of small measure are close to sub-\ngroups).LetΛbe an open K-approximate subgroup contained in a compact semi-\nsimple Lie group G. For allδ >0there isǫ>0such that if µG(Λ)≤ǫthen there\nis a proper subgroup Hsuch that Λis covered by C(K)translates of Hδ.\nProposition 1.6 is a coarse version of Jordan’s seminal theorem stat ing that a\nfinite subgroup is contained in finitely many cosets of a proper (abelia n) connected\nsubgroup. Jordan’s result extends to finite approximate subgrou ps [5] and a fun-\ndamental result of Carolino shows that compact approximate subg roups admit a\n“finite + convex” decomposition (see Theorem 4.6). Our approach is then in-\nformed by the following idea: a convex subset of small measure must be contained\nin a small neighbourhood of a subgroup, while a finite one is understoo d thanks to\nthe extension [5] of Jordan’s result.MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 5\nThe proof of Theorem 1.4 follows along the same principle. We observe that our\nproof of Theorem 1.5 shows the following surprising fact: for some ρ>0 small, we\ncan prove the following inequality\nµG(AAg,ρ)≥(2dG−dH−ǫ)µG(A)\nwhereAg,ρdenotes the intersection of Awith the ball of radius ρaboutg. We\ncan use this insight to show that, in fact, if µG(A2)≤(2dG−dH+α)µG(A), then\nµG(Ah,ρ)≃µG(Ah′,ρ) for allh,h′in a closed subgroup HofGof maximal dimen-\nsion. This is used to reduce our stability question to a local stability on e which we\nsolveby provingthe following local stability result under a non-degen eracyassump-\ntion (see Proposition 6.4 for a precise statement and §8.4 for a discussion about\nhow to waive non-degeneracy):\nProposition (Proposition 6.4) .Letδ >0. There are ρ,α >0such that if A⊂\nBG(e,ρ)is a compact non-degenerate subset ofGsatisfying\nµG(A2)≤(2dG+α)µG(A)\nthen there is a convex subset C⊂gsuch thatµG(exp(C)∆A)≤δ(µG(A) +\nµG(exp(C))).\nThe proof of local stability relies on ideas recently introduced in [33] r elated to\nthe so-called density functions . This essentiallyreducesto the study ofconvolutions\n1A∗1B(e,ρ′)whereBG(e,ρ′) denotes a ball of radius ρ′at the identity and ρ′>0\nis a judiciously chosen scale, see §8.4.\nTo conclude the proof of the global stability from the local stability r equires two\nadditional pieces of information: that in our case Ccan be taken invariant under\nthe adjointactionof Handwell-knownstabilityresultsdue toKazhdan[35]applied\nto an almost-1-cocycle of Hgiven as the “centre of mass” of the Ah,ρ’s.\nThis outline only provides qualitative statements due to the reliance o n the\nstructure theory of approximate subgroups as well as ultraprod ucts. We finally\nobtain the quantitative estimate on the correcting term of Theore m 1.2 from a\nbootstrap argument using stability in §8.1. There we also discuss directions in\nwhich one could improve our results, further open problems related to doubling\ninequalities at large and related questions.\n1.3.Outline of the paper. We start in §3 by establishing the local Brunn–\nMinkowski (Theorem 1.5) thanks to an optimal transport argumen ts. We follow\nup in §4 by showing that small approximate subgroups are contained in neig hbour-\nhoods of proper subgroups (Proposition 1.6). We conclude in §5 the proof of our\nfirst main results (Theorems 1.2 and 1.3) by relying on the previous tw o sections\nand a double counting argument.\nWe start our way towards the stability result (Theorem 1.4) by esta blishing in\n§6 stability for the local Brunn–Minkowski inequality. We then conclud e the proof\nof Theorem 1.4 in §7.\nWe end this paper in §8 with a proof of the estimate of the correcting term of\nTheorem 1.2, a number of remarks and open questions.\n1.4.Notation. Given two subsets X,Yof a groupGwe denote by XYthe set of\nproducts {xy:x∈X,y∈Y}and we write X−1:={x−1:x∈X}. For alln∈N\ndefine inductively X0={e},X1=X,X2=XXandXn+1=XXn.6 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nThroughoutthisarticle, Gwilldenoteacompactsemi-simpleLiegroupofdimen-\nsiondG. We say a Lie group is semi-simple if it is connected and has no non-trivia l\nnormal connected abelian subgroup. Compact Lie groups can be eq uipped with\na bi-invariant (i.e. invariant by right and left translations) distance dgivingG\ndiameter 1 as well as a Haar probability measure µGwhich is a bi-invariant Borel\nprobability measure, see [46] for this and more.\nA subgroup H⊂Gis maximal if it is proper ( H/\\e}a⊔io\\slash=G) and for all subgroups L\nwithH⊂L⊂Gwe haveL=GorL=H. Given a subgroup Handδ >0 write\nHδtheδ-neighbourhood of H. It is defined by\nHδ:={g∈G:∃h∈H,d(h,g)<δ}=Bd(e,δ)H=HBd(e,δ)\nwhereBd(e,δ) denotes the ball of radius δcentred ate.\nThroughout the paper, we will deal with several parameters and c onstants that\ndepend one upon the other. When ambiguouswe will indicate in betwee n parenthe-\nses dependence i.e. a constant C(dG)>0 is a constant depending on the parameter\ndG. We will usually keep track of the exact value of each constant with u p to one\nexception. In many steps of our proofs constants depending exc lusively on the di-\nmension of the ambient Lie group Gwill appear. We will denote all these constants\nbyC(dG) even though they might differ. We do however keep reference to t he\ndimension to emphasise the dependence. As a rule of thumb, αandǫwill denote\nsmall quantities, δandρwill denote radii of certain balls and λwill be a Lebesgue\nmeasure on a vector space.\n2.Acknowledgements\nI am deeply grateful to Yifan Jing and Chieu Minh Tran for many fascin ating\ndiscussions and their encouragement. I am indebted to Alessio Figalli, Federico\nGlaudo and Lauro Silini for sharing their insights on optimal transpor t and the\nBrunn–Minkowski inequality. It is my pleasure to thank Peter van Hin tum and\nNicolas de Saxc´ e for many insightful comments on a first version of this work.\n3.Local Brunn–Minkowski via optimal transport\n3.1.Preliminaries to the proof.\n3.1.1.Baker–Campbell–Hausdorff formula. TheBaker–Campbell–Hausdorff(BCH)\nformula is a well-known formula relating product in the Lie group struc ture with\naddition and Lie brackets in the Lie algebra. Namely:\nProposition 3.1 (§5.6, [28]) .There is a neighbourhood of the identity Vingsuch\nthat for every X,Y∈Vwe have\n(1) logexp XexpY=X+Y+1\n2[X,Y]+...\nwhere the remaining terms range over all higher Lie brackets made using XandY.\n3.1.2.Haar-measure in exponential coordinates. Using the BCH formula (1) one\ncan show how the Haar measure on the Lie group and the Lebesgue m easure on\nthe Lie algebra relate through exponential coordinates.\nProposition 3.2 (Haar measure in exponential coordinates) .LetGbe a Lie group\nandgits Lie algebra. There is a neighbourhood of the identity U⊂gsuch that theMINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 7\nexponential map exp :U→Gis an homeomorphism onto its image with inverse\nlogand\nlog∗µG(deX) =H(X)λ(dX)\nwhereHis continuous, H(0)>0andH(X) =H(0)+O(|X|2)for any norm |·|.\nProof.Proposition 3.2 is a direct consequence of the fact that the expone ntial\nmap is a diffeomorphism and that its differential at 0 is the identity (e.g. [46,\n§4.1.2]). Upon choosing the right normalisation of λwe may always assume that\nH(0) = 1. Estimates for H(X) are standard but we sketch a proof exploiting\nthe symmetries of the Haar measure. Since the Haar measure is left -invariant,\nwe find by a change of variable that Hcan be expressed as a jacobian. Namely,\nH(X) =|Jac0(expX·)|−1H(0). UsingtheBCHformula,weseethatthedifferential\nofY/ma√s⊔o→expX·expYat 0 is equal to Id+[X,·]\n2+O(|X|2|Y|+|Y|2|X|). Since [X,·]\nis skew-symmetric with respect to any bi-invariant scalar product [3 8, Prop. 4.24],\none has|Jac0(expX·)|= 1+O(|X|2) which establishes the claim. /square\nOne can be much more precise and show that H(X) :=|detsinhX/2\nX/2|.We will\nnot need the exact formula for Hhowever.\n3.1.3.Optimal transport map. We use the following result of Brenier:\nTheorem 3.3 (Brenier, [3]) .Letµ1,µ2be two probability Borel measures on Rn\nthat are absolutely continuous with respect to the Lebesgue measure. There exists a\nconvex function f:Rn→Rsuch thatT=∇fsatisfiesµ1=T∗µ2.\nNote that the convex function fconsidered above is not assumed to be differen-\ntiable. Convex functions are however always differentiable almost ev erywhere and\nwe refer to [52] for this and the precise definition of ∇freferred to in the above\n[52, Def. 10.2].\n3.1.4.Change of variable. Our strategy relies on a general change of variable theo-\nrem that can be found in [52]. It uses a notion of approximate differen tiability and\nwe refer to [52, Def. 10.2] for a definition.\nTheorem 3.4 (Change of variable, Thm 11.1, [52]) .Letf∈L1(RdG)be a non-\nnegative function and let T:RdG→RdGbe a Borel map. Define dµ(x) =f(x)dλ(x)\nandν=T∗µ. Assume that:\n•There exists a measurable set Σ⊂RdGsuch thatf= 0almost everywhere\noutside of ΣandTis injective on Σ;\n•Tis approximately differentiable almost everywhere on Σ.\nLet˜∇Tbe the approximate gradient of Tand letJTbe defined almost everywhere\nonΣbyJT(x) :=|det˜∇T(x)|. Thenνis absolutely continuous with respect to\nthe Lebesgue measure if and only if JT>0almost everywhere. In that case νis\nconcentrated on T(Σ)and its density gis determined by the equation\n(2) f(x) =g(T(x))JT(x).\nEquation (2) is often called the Monge–Amp` ere equation and it plays a key role\nin our approach.8 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\n3.2.Proof of the local Brunn–Minkowski theorem. Our strategy follows the\none outlined in [19, p.7].\nProof of Theorem 1.5. By inner regularity of the Haar measure, it is enough to\nprovethe resultforcompactsubsets. Let U⊂gbe aneighbourhoodofthe originon\nwhich the exponential map exp is defined, exp : U→Ginduces a diffeomorphism\nonto its image and the BCH formula (Proposition 3.1) holds for all x,y∈U.\nLetA,B⊂U⊂gbe two compact subsets and suppose that A,Bare moreover\ncontained in a ball of radius ρ>0 (with respect to some fixed norm). Let µAand\nµBbe the restrictions of the Lebesgue measure λtoAandBnormalised to total\nmass 1. Let T=∇fdenote the optimal transport map given by Brenier’s theorem\n(Theorem 3.3). We have µB=T∗µAandTis convex. Finally, in this proof c(ρ)>0\ndenotes a constant that might change line to line but that will remain OdG(ρ) asρ\ngoes to 0.\nWe will consider the map\nF=id·T\nwhere·denotes the (partial) multiplicative law g×g→Gdefined by the BCH\nformula (Proposition 3.1). According to the BCH formula (Propositio n 3.1), we\nhave\n||F(x)−F(y)|| ≥ ||x+T(x)−y−T(y)||−c(ρ)||x−y||.\nBut\n||x+T(x)−y−T(y)|| ≥ ||x−y||\nsinceTis the gradient of a convex map. Thus,\n||F(x)−F(y)|| ≥(1−c(ρ))||x−y||.\nSoFis injective as soon as ρis sufficiently small. Since Tis approximately differ-\nentiable everywhere [52, Thm 14. 25], we know that Fis as well, so we can apply\nthe change of variable formula (Theorem 3.4).\nMoreover, det ˜∇T(x) =λ(B)\nλ(A)as a consequence of Theorem 3.4. Indeed, we know\nthatT∗µA=µBand the density of µBis equal to1\nλ(B)1Bwhile the density of µA\nis1\nλ(A)1Aby the conclusion of Theorem 3.4 we see that indeed |det˜∇T(x)|=λ(B)\nλ(A)\nalmost everywhere.\nNow, we follow a sketch in [19] to prove a bound on the Jacobian of F. Using\nthe BCH formula ˜∇Fis equal almost everywhere to\nId+˜∇T(x)+A′\nρ,x+A′′\nρ,x◦˜∇T(x)\nwhereA′\nρ,x−ad(T(x))\n2andA′′\nρ,x−ad(x)\n2have all their coefficients bounded above by\nOdG(ρ2). According to [38, Prop. 4.24], for all Y∈g,ad(Y) is a skew-symmetric\nmatrix with respect to any orthonormal basis.\nHence,Id+A′′\nρ,xhas determinant at least (1 −c(ρ)2) in absolute value. In\nparticular, Id+A′′\nρ,xis invertible for ρsufficiently small. Therefore,\n|det˜∇F(x)| ≥(1−c(ρ)2)−1|det/parenleftBig\nId+˜∇T(x)+Aρ,x/parenrightBig\n|\nwhereId+Aρ,x= (Id+A′′\nρ,x)−1(Id+Aρ,x). Thus, in any orthonormal basis Aρ,xis\nthe sum of a skew-symmetric matrix and a matrix that has all its entr ies bounded\nabove in modulus by OdG(ρ2).MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 9\nButTis the gradient of a convex function so ˜∇T(x) is a positive definite ma-\ntrix [52, Thm 14.25]. Denote by λ1,...,λ n>0 its eigenvalues. There is Oxan\northonormal change of basis such that Ox˜∇T(x)O−1\nxis diagonal. Therefore,\n|det/parenleftBig\nId+˜∇T(x)+Aρ,x/parenrightBig\n| ≥ |det/parenleftBig\nId+Ox˜∇T(x)O−1\nx+OxAρ,xO−1\nx/parenrightBig\n|\n≥dG/productdisplay\ni=1(1+λi−c(ρ)2)\n−/summationdisplay\nJ⊂[1;dG],|[1;dG]\\J|≥2/productdisplay\nj∈J(1+λj+c(ρ))c(ρ)dG−|J|\n≥(1−c(ρ)2)dG/productdisplay\ni=1(1+λi).\nwhere we have used the triangle inequality, the positivity of eigenvalu es and the\ndecomposition of OxAρ,xO−1\nxinto a skew-symmetric matrix and a matrix with\ncoefficients bounded in O(ρ2) to go from the first to the second line. The positivity\nofeigenvalues is then used to gofrom the second to the last line. Usin g the AM-GM\ninequality [18, Proof of Thm 1] we see now that\n|det˜∇F(x)| ≥(1−c(ρ)2)/parenleftBigg\n1+n/productdisplay\ni=1λ1\ndG\ni/parenrightBiggdG\n(3)\n≥(1−c(ρ)2)/parenleftBigg\n1+/parenleftbiggλ(B)\nλ(A)/parenrightbigg1\ndG/parenrightBiggdG\n. (4)\nSo Theorem 3.4 implies\nλ(A·B)\nλ(A)≥/integraldisplay\nA|det˜∇F(x)|1\nλ(A)dλ(x)≥(1−c(ρ)2)/parenleftBig\nλ(A)1\ndG+λ(B)1\ndG/parenrightBigdG\n.\nFinally, using Proposition 3.2 we obtain\nµG(expAexpB)≥(1−c(ρ)2)/parenleftBig\nµG(expA)1\ndG+µG(expB)1\ndG/parenrightBigdG\n.\n/square\n4.Approximate subgroups of small measure\nThe goal of this section is to establish Proposition 1.6 concerning the global\nstructure of approximate subgroups of small measure.\n4.1.Preliminaries on approximate subgroups.\n4.1.1.Local groups. The notion of a local group is instrumental to our approach.\nWe now define a local group as well as a normal sublocal group. We re fer the reader\nto [6, Appendix B] for additional background.\nDefinition 4.1 (Local groups, B.1, [6]) .Alocal group Gis a topological space\ntogether with a distinguished element e∈G, an involutive inversemap()−1:G→\nGand a partially defined product map·: Ω→Gsuch that:\n(i)Ωis a neighbourhood of G×{e}∪{e}×GinG×G;\n(ii) (Continuity) the maps ()−1and·are continuous on GandΩrespectively;10 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\n(iii) (local associativity) if g,h,k∈Gare such that g·(h·k)and(g·h)·kare\nwell-defined, then g·(h·k) = (g·h)·k;\n(iv) (identity) for all g∈G,e·g=g·e=g;\n(v) (invertibility) for all g∈G,(g,g−1)and(g−1,g)belong to Ωandg·g−1=\ng−1·g=e.\nRecall that a K-approximate subgroup Λ of a local group Gis a subset such that\nΛ100is well-defined, that is symmetric (i.e.e∈Λ and Λ = Λ−1) and for which\nthere isF⊂Gof size at most Ksuch that Λ2⊂FΛ.\nWe will also need to pass to sublocal groups:\nDefinition 4.2 (Sublocal group, B.11, [6]) .Given two local groups GandH, we\nsay thatHis asublocal group ofGifHis the restriction of Gto a symmetric\nsubset containing the identity, and there is an open neighbo urhood of the identity\nV⊂Gsuch that whenever the product of g,h∈His well-defined and g·h∈V,\ntheng·h∈H. We callVanassociated neighbourhood .\nThe sublocal group is moreover said normalif there exists an associated neigh-\nbourhoodVwith the additional property that whenever g∈G,h∈Hare such that\ng·h·g−1is well-defined and lies in V, theng·h·g−1∈H.\nGiven a local group Gand a normal sublocal group H, one can define a notion\nof quotient G/H. We use this notion in what follows and we refer to [6, B.12] for\ndefinitions and background.\n4.1.2.Small doubling. Approximate subgroups naturally appear in relation with\nsmall doubling . At the lowest level, this is hinted at by the following:\nLemma 4.3 (Ruzsa’s covering lemma, Lem. 3.6, [47]) .LetA,B⊂Gbe two\ncompact subsets. Suppose that µG(AB)≤KµG(B). Then there is a subset F⊂G\nof size at most Ksuch thatA⊂FBB−1.\nProposition 4.4 (Prop. 4.5 and Thm. 4.6, [47]) .LetA,B⊂Gbe two mea-\nsurable subsets such that AB:={ab:a∈A,b∈B}is measurable. Write\nK:=µG(AB)\nµG(A)1/2µG(B)1/2. Then there is a KO(1)-approximate subgroup Λ⊂Gsuch\nthatAis covered by KO(1)left-translates of ΛandBis covered by KO(1)right\ntranslates of Λ.\nMoreover, there is a universal constant C >0such that for all m≥0,\nµG(AΛmB)≤KCmµG(A)1\n2µG(B)1\n2\n4.1.3.An elementary result. In the spirit of Ruzsa’s covering lemma, we have:\nLemma 4.5. LetA,Bbe subsets of a group Gwithe∈Bandn>0be an integer.\nIf there isFfinite such that A⊂FB, then there are F′⊂Fandm≤ |F|such\nthat\nA⊂/unionsqdisplay\nf∈F′fBnm\nandf1Bnm+1∩f2Bnm+1=∅for allf1/\\e}a⊔io\\slash=f2∈F′.\n4.1.4.Convex + finite decomposition. Acrucialtoolarisesfromastructuretheorem\nof Carolino’s for compact approximate subgroups [9]. It concerns a connected\ncomponent of sorts of an approximate subgroup. Namely, if A⊂Gwe define its\nstar-connected component as\nA⋆:={expX:∀t∈[0;1],exptX∈A}.MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 11\nTheorem 4.6 (Main theorem, [9]) .LetΛbe a compact approximate subgroup in a\nconnected semi-simple Lie group. There are two approximate subgroups Λ0⊂Λ1⊂\nΛ2⊂Λ4such that:\n(i)Λ1is aC(K)approximate subgroup C(K)translates of which cover Λ;\n(ii) the set of commutators [Λ1,Λ1]is contained in Λ1;\n(iii)Λ0is contained in (Λ4)⋆;\n(iv)Λ0is a normal sublocal group with reference neighbourhood Λ2andΛ1/Λ0\nis aCK-approximate subgroup of a local group Λ2/Λ0.\nIn the notation of [9, Thm 9.5], Λ 2isAǫ, Λ1isAǫ/6C0K4and Λ 0= exp(Bδ)\nwhereAis a strong approximate subgroup contained in Λ4provided by [9, Cor.\n7.4]. Note that all these subsets are C(K)-approximate subgroups and Bδis in fact\n(almost) a convex subset.\n4.2.Large star-shaped component.\nProposition 4.7. Letρ,δ >0. LetΛbe aK-approximate subgroup in a compact\nsemi-simple Lie group G. Suppose that Λis not contained in the δneighbourhood\nof any proper subgroup and that Λ⋆contains an element all of whose projections to\na simple factor of Ghave length ρ. Then\nµG(Λ)≥C(ρ,δ,G,K )>0.\nProof.The main step of our proof will be establishing the following claim:\nClaim4.8. LetΛbe aK-approximate subgroupnot contained in the δ-neighbourhood\nof a proper subgroup. Let V=V1⊕...⊕Vrbe a finite dimensional sum of irre-\nducible representations of G- with the action of Gdenoted by ·- equipped with\naG-invariant scalar product (·,·). Choosev∈Vany unit vector whose projec-\ntion toVihas norm at least δfor alli >0. Then there are C(δ,K,V)>0and\nv1=v,...,v dwithvi∈Ai·vand|(vi\n||vi||,V<i)| ≤1−C(δ,K,V)for alliwhere\nV<i:=span(v1,...,v i−1).\nHere, when v∈Vis a vector and W⊂Vis a subspace, we denote by ( v,W)\nthe norm of the orthogonal projection of vtoW.\nLet us explain how one can conclude from Claim 4.8. Take v∈gsuch that\nexpv∈Λ⋆and all of its projections to simple factors have length ρ. Apply Claim\n4.8 toG,gandvand letv1=v,...,v dbe the vectors obtained. Since for each\ni≥0,vi∈Ai·vthen\n{expt1v1···exptdvd:t1,...,td∈[0;1]} ⊂Λd2.\nThe map\nD: (t1,...,td)∈[0;1]dG/ma√s⊔o−→log(expt1v1···exptdvd)∈RdG\nadmits the linear map\n(r1,...,r d)∈RdG/ma√s⊔o−→r1v1+...+rdvd\nas differential at 0. But ( vi,V<i)≤α2(1−C(δ,ρ,K,V )) for alliand||vi||=α. So\n|detD| ≥C(δ,ρ,K,G )>012 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nwhere the explicit value of C(δ,ρ,K,G ) can be obtained by an analysis of the\nGram–Schmidt matrix. Thus, by the above discussion and Propositio n 3.2, we\nhave\nµG({expt1v1···exptdvd:t1,...,td∈[0;1]})≥C(δ,ρ,K,G )>0.\nIt remains to establish the claim. We will use multiple times the following\nobservation: let W⊂Vbe a subspace of dimensions dWandv∈Va unit vector\nsuch that |(v,W)| ≤1−cfor somec >0, letB⊂Wdenote the unit ball, then\nthe convex hull of B∪{−v;v}contains the ball of radius r(dW)cwherer(dW)>0\nonly depends on dW(and is at least1\ndW). We leave the simple proof to the reader.\nWe build inductively vectors vi∈Λi·vsatisfying |(vi,V<i)| ≤1−c(i,δ,V) for\nsomec(i,δ,V)>0 and, in addition, that the convex hull of v1,...,v icontains the\nball of radius r(i,δ)>0 centred at 0 in Vi:=V<i+1. Ifi= 1, this is obvious.\nSuppose that vjis constructed up to ismaller than the dimension of V. Since Λ is\nnot contained in the δ-neighbourhood of the stabiliser of Vi, there areλ∈Λ and a\nunit vector v′∈Visuch that |(λ·v′,Vi)| ≤1−c(δ,V) for somec(δ,V)>0. Define\nv′\ni+1=r(i,δ)λ·v′and notice that r(i,δ)v′belongs to the convex hull of v1,...,v i\nby the induction hypothesis. So r(i,δ)v′belongs toCo(Λi·v) by induction. In turn,\nv′\ni+1∈Co(Λi+1·v). Butv′is a convex combination of λ1·v,...,λ k·vfor some\nλ1,...,λ k∈Λi. There is thus l≤ksuch that |(λλl·v,Vi)| ≤1−r(i,δ)c(δ,V). Set\nvi+1:=λλl·v∈Λi+1·v. By the observation at the beginning of this paragraph,\nthe convex hull of v1,...,v i+1contains a ball of radius r(i+1,δ). This ensures that\nvi+1is as desired and that our induction process can go on one more step . So the\nclaim is proved. /square\n4.3.Small star-shaped component.\nProposition 4.9. For everyδ>0, there isρ>0such that the following is true:\nLetΛbe a compact K-approximate subgroup of a semi-simple Lie group G. Sup-\npose that (Λ8)⋆is contained in the ball of radius ρcentred ate. ThenΛis contained\nin theδ-neighbourhood of some proper subgroup.\nRemark 4.10. A proper subgroup may be trivial, and so we are in particular d one\nif we can show that Λis contained in a small neighbourhood of the identity.\nProof.Fixρ>0 and suppose that (Λ8)⋆is contained in the ball of radius ρcentred\nate. Let Λ′= Λ2∩BwhereBis the ball centred at eand of radius1\n4. Then Λ′\nis aC(dG)K6-approximate subgroup and Λ is covered by C(dG) translates of Λ′\n[49, Prop. 2.6.5]. Moreover, for any two elements g,h∈Bwe have by a standard\ncomputation that d(e,[g,h])≤1\n2d(e,g) [6, Proof of Lem. 2.2]. Let Λ 0,Λ1,Λ2be\nas in Theorem 4.6 applied to Λ′. By our assumption Λ 0is contained in the ball\nof radiusα. By Theorem 4.6 Λ 0is a normal sublocal group of Λ 2with reference\nneighbourhood Λ 0.\nTakeλ∈Λ1of size at most3\n2infλ′∈Λ1\\Λ0d(e,λ′). Then for every λ′∈Λ1we\nhave [λ,λ′]∈Λ1and\nd(e,[λ,λ′])≤1\n2d(e,λ)≤3\n4inf\nλ′∈Λ1\\Λ0d(e,λ′).\nHence, [λ,λ′]∈Λ0. Therefore, the image of λin Λ1/Λ0is central. By repeating\nthe above argument we find that Λ 1/Λ0is a nilpotent local C(K)-approximate\nsubgroup. By the structure of approximate subgroups [6] and up on shrinking Λ 1,MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 13\nwe find that there is a normal subgroup N⊂Λ1/Λ0such that the quotient has\nrank at most C(K). Suppose that there is λ∈Λ1projecting to an element nofN\nand such that d(e,λ)≥10ρ. Thenλ2Λ0∩Λ1/\\e}a⊔io\\slash=∅as it contains the inverse image\nofn2. Assuming that d(e,λ)<1\n2there isλ2∈Λ1projecting to an element of N\nsuch thatd(e,λ2)≥19ρ. Repeating this process enough times, we find λ′∈Λ1\nsuch thatd(e,λ′)≥1/2. Butλ′∈B4, a contradiction. So the inverse image of\nNis contained in the ball of radius 10 αcentred ate. LetZdenote the centre of\n(Λ1/Λ0)/N.\nSuppose that there is an element of Λ 1projecting to Zwithd(e,λ1)≥√ρ.\nThen for all λ∈Λ1we have [λ,λ1]∈Λ1and projects to an element of Nin\nΛ1/Λ0. Therefore, d(e,[λ,λ1])≤10ρ. In other words, d(λ1,λλ1λ−1)≤10ρ. Since\nd(e,λ1)≥√ρwe findδ′(ρ)>0 withδ′(ρ) =O(ρC(dG)) asρ→0 such that\nλ∈Hδ′(ρ)whereHis simply the centraliser of λ1.\nIf there is no such λ1and provided ρis sufficiently small, we can repeat the\nprocess and look for an element of size at least4√ρprojecting to an element of the\ncentre of ((Λ 1/Λ0)/N)/Z. We can do so inductively until we have exhausted all\nelements of Λ 1. But (Λ 1/Λ0)/Nhas rank at most C(K) so this process must stop\nafterC(K) steps at most. Thus, either Λ 1is contained in a ball of radius2C(K)√ρ\ncentred ateor Λ1is contained in the δ′′(ρ)-neighbourhood of a proper subgroup\n(the stabiliser of some non-trivial element of g) withδ(ρ) =O(ρC) asρgoes to 0.\nSettingδ(ρ) = max{2C(K)√ρ,δ′′(ρ)}concludes. /square\n4.4.Proof of Proposition 1.6.\nProof of Proposition 1.6. Fixδ >0. Letǫ >0 to be chosen later and suppose\nµG(A)<ǫ, chooseρ=ρ(ǫ)>0 as the minimal value such that C(ρ,ρ,G,K )>K7ǫ\nwhereC(ρ,ρ,G,K ) is the constant given by Proposition 4.7 applied to Λ8. By\nProposition 4.7, ρexists as soon as ǫis small enough and goes to 0 as ǫgoes to 0.\nSuppose that Λ is not contained in the δ-neighbourhood of a proper subgroup. By\nProposition 4.7, if (Λ8)⋆contains an element of length ρ, then Λ8has measure at\nleastK7ǫ, contradicting our hypothesis. So (Λ8)⋆is contained in the ball of radius\nρcentred at e. According to Proposition 4.9 now, if ǫ- and, hence, ρ- is small\nenough, there is an approximate subgroup Λ′contained in a δ-neighbourhood of a\nproper closed subgroup such that Λ is covered by C(K) translates of Λ′. So Λ itself\nis covered by C(K) translates of the δ-neighbourhood of a proper subgroup. /square\n5.Expansion of small subsets\nInthissection,weproveTheorem1.3and,thus, Theorem1.2withou tanestimate\nonthecorrectingterm. Thefirstpartofthe proofisastraightfo rwardapplicationof\nProposition1.6whichenablesustoassume A,Bcontainedinasmallneighbourhood\nof a proper subgroup. The second part consists in leveraging the lo cal Brunn–\nMinkowski (Theorem 1.5).\n5.1.Doubling of balls. An essential role will be played by a special case of The-\norem 1.2 concerning the doubling of balls. This doubling is controlled by t he di-\nmension:14 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nLemma 5.1. LetGbe a connected Lie group of dimension dGanddbe a bi-\ninvariant Riemannian distance. We have:\nµG(Bd(e,2ρ))\nµG(Bd(e,ρ))= 2dG(1−Sρ2+o(ρ2))\nwhereS >0is a dimensional constant proportional to the scalar curvat ure.\nThis can be seen from the formula for the volume of balls from scalar c urvature\n(e.g. [33, Fact 2.7]) and the fact that bi-invariant metric on compact Lie groups\nhave strictly positive scalar curvature at every point [43, Thm 2.2].\n5.2.Double counting. Chooseδ,ρ>0,H⊂Ga proper connected subgroup and\nletX⊂Hδbe a compact subset. Let BH(h,ρ) (resp.BG/H(g,δ)) denote the ball\nof radiusρath(resp. radius δatg) with respect to the restriction of dtoH(resp.\nthe projection of dtoG/H). Choose a cross-section Tδ⊂BG(e,δ) ofBG/H(e,δ)\ni.e. a measurable subset such that the restriction of the projectio nG→G/His a\nbijection from TδtoBG/H(e,δ). For instance, write hthe Lie algebra of H,h⊥its\northogonal with respect to the Killing form, and |·|the norm on garising from the\nRiemannian metric. Define Tδas the image through the exponential map of the\nball centered at 0 of radius δinh⊥with respect to |·|. Then\nLemma 5.2.\n(5)/integraldisplay\nHµG(X∩TδBH(h,ρ))dµH(h) =µG(X)µH(BH(e,ρ)).\nThe proof of (5) relies and the quotient formula for the Haar measu re [14, Ch.\n1]\n(6) µG(X) =/integraldisplay\nG/HµH(g−1X∩H)dµG/H(gH)\nProof of Lemma 5.2.\n/integraldisplay\nHµG(X∩TδBH(h,ρ))dµH(h) (7)\n=/integraldisplay\nH/integraldisplay\nG/HµH(t−1X∩BH(h,ρ))dµG/H(tH)dµH(h) (8)\n=/integraldisplay\nG/H/integraldisplay\nHµH(t−1X∩BH(h,ρ))dµH(h)dµG/H(tH) (9)\n=/integraldisplay\nG/HµH(t−1X∩H)µH(BH(e,ρ))dµG/H(tH) (10)\n=µG(X)µH(BH(h,ρ)) (11)\nwhere for all cosets tH,tis a representative chosen in Tǫ. In the series of equalities\nabove, we have used the quotient formula to obtain (8), the fact t hat the map:\nY⊂H→/integraldisplay\nHµH(Y∩BH(h,ρ))dµH(h)\nis a Haar measure of total mass µH(BH(e,ρ)) to go from (9) to (10), and the\nquotient formula again to conclude. /squareMINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 15\n5.3.Multiplicative properties of rectangles. The rectangles introduced in the\nprevious section will play a central role throughout this paper. With notations as\nabove, write R(h,δ,ρ) :=TδBH(h,ρ). Now, the map\nh⊥×h−→G\n(h1,h2)/ma√s⊔o−→eh1eh2\nis a local diffeomorphism with differential at eequal to the identity. It sends\nBh⊥(e,δ)×Bh(e,ρ) toR(e,δ,ρ) diffeomorphically as soon as δandρare sufficiently\nsmall.\nAbove,Bh⊥(e,δ) (resp.Bh(e,ρ)) denote the ball of radius δ(resp.ρ) about 0\ninh⊥(resp.h) with respect to the restriction of the Ad(G) invariant norm. In\nparticular, we find that R(e,δ,ρ) is invariant under conjugation by an element of\nH. We have in particular:\nLemma 5.3. Leth1,h2∈H. There is a dimensional constant c=c(dG)>0such\nthat as soon as δ,ρare sufficiently small\nR(h1,δ,ρ)R(h2,δ,ρ)⊂R(h1h2,2δ,2ρ+cδ2)2.\nProof.We haveR(h1,δ,ρ) =h1R(e,δ,ρ).So\nR(h1,δ,ρ)R(h2,δ,ρ) =h1R(e,δ,ρ)h2R(e,δ,ρ) =h1h2R(e,δ,ρ)R(e,δ,ρ)\nsince the subsets R(e,δ,ρ) are invariant under conjugation by elements of H.\nThus, ifh1,h′\n1∈Bh⊥(e,δ) andh2,h′\n2∈Bh(e,ρ) we have that\neh1eh2eh′\n1eh′\n2=eh1eAd(eh2)h′\n1eh2eh′\n2.\nSinceh⊥isstableundertheadjointactionof H,Ad(eh2)h′\n1∈h⊥. Writeeh1eAd(eh2)h′\n1=\neh′′\n1eh′′\n2with (h′′\n1,h′′\n2)∈Bh⊥(e,10(δ+ρ))×Bh(e,10(δ+ρ)). Sinceh1+Ad(eh2)h′\n1∈\nh⊥,|h′′\n2|=O(δ2) according to the BCH formula 1. Since in addition HδHδ=H2δ,\n|h′′\n1| ≤2δ. Now,eh2,eh′\n2,eh′′\n2∈Hso\neh1eh2eh′\n1eh′\n2∈R(e,2δ,2ρ+cδ2).\n/square\n5.4.Lower bound on expansion close to a subgroup. We start with a dou-\nbling estimate:\nProposition 5.4. Letǫ>0andH⊂Gbe a proper closed subgroup of dimension\ndH. There isδ>0such that if A,B⊂Gare two compact subsets with A,B,AB ⊂\nHδ, then\nµG(AB)≥/parenleftbig\n2dG−dH−ǫ/parenrightbig\nmin{µG(A),µG(B)}.\nMoreover,ǫcan be chosen smaller than C(dG)ρ2for some dimensional constant\nC(dG)>0.\nProof.Letρ>0 to be chosen later. Choose h0such thatµG(B∩TδBH(h0,ρ)) is\nmaximal. Withoutlossofgenerality,wecanalsoassumethat µG(B∩TδBH(h0,ρ))≥\nµG(A∩TδBH(h,ρ)) for allh∈H. Then Theorem 1.5 asserts that\n(2dG−ǫ\n2)µG(A∩TδBH(h,ρ))≤µG((A∩TδBH(h,ρ))(B∩TδBH(h0,ρ)))16 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nwhen max(δ,ρ) is sufficiently small. So, by Lemma 5.2,\n(2dG−ǫ\n2)µG(A)µH(BH(e,ρ)) = (2dG−ǫ\n2)/integraldisplay\nHµG(A∩TδBH(h,ρ))dµH(h) (12)\n≤/integraldisplay\nHµG((A∩TδBH(h,ρ))(B∩TδBH(h0,ρ)))dµH(h) (13)\n≤/integraldisplay\nHµG(AB∩TδBH(hh0,2ρ+cδ2))dµH(h) (14)\n=µG(AB)µH(BH(e,2ρ+cδ2)) (15)\nwhere we have used the inclusion AB⊂Hδto go from (13) to (14) and (5) to\nconclude (16). So\nµG(AB)≥(2dG−ǫ\n2)µH(BH(e,ρ))\nµH(BH(e,2ρ+cδ2))µG(A).\nTherefore, choosing ρ=δ, Lemma 5.1 yields\nµG(AB)≥(2dG−dH−ǫ)µG(A)\nforδsufficiently small.\nThe moreover part is a consequence of the estimate in Theorem 1.5. /square\nThe subsets of the form\nX∩TδBH(h,ρ)\nforX⊂Hδ,h∈Handρ >0 play an important role in the end of the proof of\nTheorem 1.2 as well as in the proof of Theorem 1.4. While we have refra ined from\nusing it until now for the sake of completeness, we will often use the shorthand\nnotation:\nXh,ρ:=X∩TδBH(h,ρ) =X∩R(h,δ,ρ)\nfrom now on.\nOur proof of Theorem 1.4 will be based on the series of inequalities exh ibited in\nthe proof of Proposition 5.4. We can in fact improve upon the above s trategy and\nestablish a Brunn–Minkowksi-type inequality:\nProposition 5.5. Letǫ>0andH⊂Gbe a proper closed subgroup of dimension\ndH. There is δ >0such that if A,B⊂Gare two compact subsets such that\nA,B,AB ⊂Hδ, then\nµG(AB)1\ndG−dH≥(1−ǫ)/parenleftBig\nµG(A)1\ndG−dH+µG(B)1\ndG−dH/parenrightBig\n.\nProof.The proof is almost identical to the proof of Proposition 5.4. Let ρ,c>0 to\nbe chosen later. Recall that for h∈Handρ′>0, we write Ah,ρ′:=A∩TδBH(h,ρ′)\nand we define similarly Bh,ρ′and (AB)h,ρ′. Notice moreover that Proposition 5.5\nis trivial when max/parenleftbigg/parenleftBig\nµG(A)\nµG(B)/parenrightBig1\ndG,/parenleftBig\nµG(B)\nµG(A)/parenrightBig1\ndG/parenrightbigg\n≤ǫ.So we assume from now on\nmax/parenleftbigg/parenleftBig\nµG(A)\nµG(B)/parenrightBig1\ndG,/parenleftBig\nµG(B)\nµG(A)/parenrightBig1\ndG/parenrightbigg\n≥ǫ.\nChooseh0such thatµG(Bh0,cρ) is maximal. Suppose as we may that\nµG(Bh0,cρ)\nµG(B)µH(BH(e,cρ))≥sup\nh∈H/parenleftbiggµG(Ah,ρ)\nµG(A)µH(BH(e,ρ)),µG(Bh,cρ)\nµG(B)µH(BH(e,cρ))/parenrightbigg\n.MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 17\nNotice that above we are implicitly comparing the values of µG(Ah,ρ) ashranges in\nHwith the expected value µG(A)µH(BH(e,ρ)) if we were to draw h∈Huniformly\nat random (and similarly for B). Again, by Theorem 1.5 we have\n(16) (1 −ǫ\n2)/parenleftBig\nµG(Ah,ρ)1\ndG+µG(Bh0,cρ)1\ndG/parenrightBig\n≤µG(Ah,ρBh0,cρ)1\ndG.\nBy our assumption on h0we have\nµG(Bh0,cρ)\nµG(Ah,ρ)≥µG(B)µH(BH(e,cρ))\nµG(A)µH(BH(e,ρ)).\nTogether with (16) this then yields\nµG(Ah,ρBh0,cρ)≥(1−ǫ\n2)dGµG(Ah,ρ)/bracketleftBigg\n1+/parenleftbiggµG(B)µH(BH(e,cρ))\nµG(A)µH(BH(e,ρ))/parenrightbigg1\ndG/bracketrightBiggdG\n.\nFollowing now inequalities (13) to (16) above, we have\nµG(Ah,ρBh0,cρ)≥(1−ǫ\n2)dGµG(A)µH(BH(e,ρ))\nµH(BH(e,(1+c)ρ+c(dG)δ2))\n×/bracketleftBigg\n1+/parenleftbiggµG(B)µH(BH(e,cρ))\nµG(A)µH(BH(e,ρ))/parenrightbigg1\ndG/bracketrightBiggdG\n.\nThis can be rewritten\n(1−ǫ\n2)dG/parenleftBigg\n(µG(A)µH(BH(e,ρ)))1\ndG+(µG(B)µH(BH(e,cρ)))1\ndG\nµH(BH(e,(1+c)ρ+c(dG)δ2))1\ndG/parenrightBiggdG\n≤µG(AB).\nWhich yields, when ρ=δandδis sufficiently small,\n(1−ǫ)(µG(A)ρdH)1\ndG+(µG(B)(cρ)dH)1\ndG\n((1+c)ρ)dH/dG≤µG(AB)1\ndG\nwhere we have used Lemma 5.1 to estimate the volume of balls in H. We now get:\n(1−ǫ)(µG(A))1\ndG+(µG(B)cdH)1\ndG\n(1+c)dH/dG≤µG(AB)1\ndG.\nSettingc:=/parenleftBig\nµG(B)\nµG(A)/parenrightBig1\ndG−dH(which is optimal) we find\n(1−ǫ)/parenleftBig\nµG(A)1\ndG−dH+µG(B)1\ndG−dH/parenrightBig\n≤µG(AB)1\ndG−dH.\n/square\nWe found the correct value of cfrom a simple variational argument over c. The\nBrunn–Minkowski inequality is therefore the optimal inequality one c ould get by\nintroducingtheparameter c. Itisthereforereassuringtoobservethatthisinequality\nis in fact sharp.\nLet us end this paragraph with a simple remark. Given the simplicity of t he\nproof of the local Brunn–Minkowski (Theorem 1.5), it is tempting to adapt the\nsame strategy in Gto obtain a global Brunn–Minkowski inequality. A potential\napproach would go as follows: pick the correct notion of cost on Gequipped with\na left-invariant metric, find T:A→Ban optimal transport map relative to\nthis cost, prove that F:x/ma√s⊔o→a·T(x) sendsAonto a set of measure at least18 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\n(1−ǫ)/parenleftBig\nµG(A)1\ndG+µG(B)1\ndG/parenrightBigdG\n. While this strategy works perfectly in Rn, it\nfails in compact simple groups already in the case A=B. Indeed, the map Tis\nthen the identity and F(x) =x2. But the differential of Fis not invertible at any\nx/\\e}a⊔io\\slash=esatisfyingx2=e. So no lower bound on the volume of A2can be obtained\nfromF(A). This simple observation shows once again that doubling in compact\nnon-abelian Lie groups offers different and new challenges compared to its abelian\ncounterpart. The double-counting argument (Lemma 5.2) we explo it above is our\nway of circumventing this difficulty.\n5.5.The Brunn–Minkowski inequality in compact groups.\nProof of Theorem 1.3. Letǫ >0. Letδ >0 to be determined later. By inner\nregularityoftheHaarmeasure, itsufficestoproveTheorem1.2for compactsubsets.\nSo letA,B⊂Gbe two compact subsets of Gand suppose µG(A)≥µG(B). Write\nα:= min{µG(A),µG(B)}. Notice that ǫ≥µG(B)\nµG(A)then the inequality is trivial. We\nsuppose from now on that τ:=µG(B)\nµG(A)≥ǫ.\nSuppose as we may that\nK:=µG(AB)\nmin{µG(A),µG(B)}≤22(dG−dH)\nτ\nwheredHis the dimension of a proper subgroup Hof maximal dimension. By\nLemma 4.4 there is a C(dG,τ)-approximate subgroup Λ and two finite subsets\nF′′\n1,F′′\n2of size at most C(dG,τ) such that A⊂F′′\n1Λ andB⊂F′′\n2Λ. According\nto Proposition 1.6, if α >0 is sufficiently small, then there is a proper subgroup,\ndenotedby H, andF′\n1,F′\n2⊂Gofsizeat most C(dG,τ) suchthat A⊂F′\n1H2−C(dG,τ)δ\nandB⊂H2−C(dG,τ)δF′\n2.\nWe can now find δ′≤δandF1⊂F′\n1,F2⊂F′\n2such thatA⊂F1Hδand\nB⊂Hδ′F2and the subsets ( fH2δ′)f∈F1(resp. (H2δ′f)f∈F2) are pairwise disjoint\n(seeLemma4.5fordetails). Choosenow δsothatProposition5.4holdsin H2δ′⊂G\nfor this choice of ǫ.\nChoosef0∈F2such that\nµG(B∩Hδ′f0) = sup\nf∈F2µG(B∩Hδ′f).MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 19\nSuppose thatµG(B∩Hδ′f0)\nµG(B)≥supf∈F1µG(A∩fHδ′)\nµG(A).We have\nµG(AB)≥µG(A(B∩Hδ′f0))(17)\n=/summationdisplay\nf∈F′\n2µG((A∩fHδ′)(B∩Hδ′f0)) (18)\n≥(1−ǫ)/summationdisplay\nf∈F′\n2/parenleftBig\nµG(A∩fHδ′)1\ndG−dH+µG(B∩Hδ′f0)1\ndG−dH/parenrightBigdG−dH\n(19)\n≥(1−ǫ)/summationdisplay\nf∈F′\n2µG(A∩fHδ′)/parenleftBigg\n1+/parenleftbiggµG(B∩Hδ′f0)\nµG(A∩fHδ′)/parenrightbigg 1\ndG−dH/parenrightBiggdG−dH\n(20)\n≥(1−ǫ)/summationdisplay\nf∈F′\n2µG(A∩fHδ′)/parenleftBigg\n1+/parenleftbiggµG(B)\nµG(A)/parenrightbigg 1\ndG−dH/parenrightBiggdG−dH\n(21)\n= (1−ǫ)/parenleftBig\nµG(A)1\ndG−dH+µG(B)1\ndG−dH/parenrightBigdG−dH\n. (22)\nwhere we have used pairwise disjointness of the family the subsets ( fH2δ′)f∈F′\n1to\ngo from (18) to (19), the assumption thatµG(B∩Hδ′f0)\nµG(B)≥supf∈F1µG(A∩fHδ′)\nµG(A)to go\nfrom (21) to (22) and Proposition 5.4 to go from (19) to (20).\nIfµG(B∩Hδ′f0)\nµG(B)≤supf∈F1µG(A∩fHδ′)\nµG(A), then a symmetric proof with Aplaying\nthe role ofByields\nµG(AB)≥(1−ǫ)/parenleftBig\nµG(A)1\ndG−dH+µG(B)1\ndG−dH/parenrightBigdG−dH\n.\nSo Theorem 1.2 is proven. /square\nProof of Theorem 1.2. Theorem 1.2 without the estimate on the correcting term is\nsimply Theorem 1.3 when A=B. We refer to §8.1 for the missing estimate. /square\nBefore we move on let us remark a simple consequence of Theorem 1.3 . Namely\na version of [6, Lemma 10.4] for compact semi-simple Lie groups:\nCorollary 5.6. LetGbe a semi-simple compact Lie group and K≥0. There is\nǫ(K)>0such that the following holds. Let A⊂Gbe aK-approximate subgroup\ncontaining a neighbourhood of the identity. Then either Λ4contains a subgroup of\ncodimension O(log(K))orµG(A)≥ǫ(K).\n6.A stability result for the local Brunn–Minkowksi\nBeyond generalising to all dimensions, our approach also brings infor mation re-\ngarding stability questions. We start with stability results concernin g certain types\nof sets close to the identity in Lie groups. We will deduce this argumen t by tak-\ning limits of subsets and relating the qualitative stability statement to equality\nof the Brunn–Minkowski inequality in Euclidean spaces. Compared fo r instance\nto the limiting process found in Christ, our approach requires only mu ch weaker\nboundedness assumptions on the sets considered. Namely, we only ask only have\nto know a priori that the sets considered are approximate subgroups, following in\nthe footsteps of [6, 33, 30].20 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nAn important difference from these latter works is that we do retain precise\ninformation on the space we obtain at the limit - the so-called model. It is indeed\nobtained by a suitable rescaling procedure that we describe now.\n6.1.Models for small convex neighbourhoods. In this section, we identify a\nneighbourhood of the identity U⊂Gand its image in the Lie algebra g. Therefore,\nthe underlying set of Uis a neighbourhood of the origin of a real vector space\nequipped with (partially defined) vector space addition + and scalar m ultiplication\n(λ,g)/ma√s⊔o→λg, the Lie bracket [ ·,·] and the group multiplication ·.\nGiven a compact subgroup HofG,H∩Uis identified (as in the previous para-\ngraph) with its Lie algebra. Define H⊥the exponential of the subspace orthogonal\n- with respect to the Killing form - to H. Ifx∈H⊥∩Uandy∈H∩U, then\n[y,x]∈H⊥.\nForδ,ρ>0, define\nR(e,ρ,δ) :=H∩B(0,ρ)+H⊥∩B(0,δ).\nForh∈H, write moreover\nR(h,ρ,δ) :=hR(e,ρ,δ) =R(e,ρ,δ)h.\nThese sets are approximate subgroups and, as a family, they admit a natural\nlimit asδandρgo to 0. Note also that although they are close to the subsets\nstudied in §5.3, they are not exactly the same. Consider ρn≥δn→0 and for every\nsequence of elements xn∈R(e,ρn,δn) define\nφ((xn)) =F −limxn,1\nρn+xn,2\nδn\nwherexn,1istheHcomponentof xn,xn,2istheH⊥componentof xnandFissome\nfixed ultrafilter on N(see [6, App. A] and [33, §4] for background on ultrafilters\nand ultraproducts in the context of approximate groups). The ma pφconstructed\nthat way exhibits good properties:\nProposition6.1. Letρn≥δn,φandFbe as above. Then φ:/a\\}b∇acke⊔le{⊔/producttextR(e,ρn,δn)/F/a\\}b∇acke⊔∇i}h⊔ →\nRdGis a surjective group homomorphism for the group law induced from the group\nlaw ofGand addition on RdG.\nRecall here that the set/producttextR(e,ρn,δn)/Fis defined as\n/productdisplay\nR(e,ρn,δn)/∼\nwhere (xn)n∼(yn)nif{n∈N:xn=yn} ∈ F. The set/producttextR(e,ρn,δn)/Fis now\na subset of the group/producttextG/Fand can be considered as an ”algebraic” limit of the\nsubsetsR(e,ρn,δn). Once again, we refer to [6, App. A] for background on these\nobjects in the context of approximate subgroups (see also §6.3 below).\nProof.Surjectivity of the map is straightforward. If xn,yn∈R(e,ρn,δn), then\n[xn,yn] = [xn,1,yn,1]+[xn,1,yn,2]+[xn,2,yn,1]+[xn,2,yn,2]\nwherexn=xn,1+xn,2andyn=yn,1+yn,2are the decompositions with respect\ntoHandH⊥.\nSince\n[xn,1,yn,1]∈HMINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 21\nand by bilinearity of the Lie bracket, we have [ xn,yn]∈R(e,Cρ2\nn,Cρnδn) for some\nC≥2 depending on Gonly. Soφ(([xn,yn])n) = 0. According to the Baker–\nCampbell–Hausdorff formula (Proposition 3.1), the same argument f urthermore\nprovesφ(xn·yn) =φ(xn+yn). This establishes our claim. /square\nBefore we move on, let us further comment on the above construc tion. First of\nall, notice that the kernel of φis made of those sequences ( xn)n∈Nsuch thatxn∈\nR(e,ρ′\nn,δ′\nn) with (ρ′\nn)nand (δ′\nn)nnegligible in front of ( ρn)nand (δn)nrespectively.\nMoreover, given a norm ||·||′ongobtained as the sum of a norm ||·||onhand\na norm||·||⊥onh⊥the formula\n|φ((xn)n)|=F −lim||xn,1||\nρn+||xn,2||⊥\nδn\ndefines a norm |·|onRdGfor which the balls are images through φof ultraproducts\nof subsets of R(e,ρn,δn). This second observation will come in handy in the next\nsection.\n6.2.Sets near equality are approximate subgroups. As mentioned in the\nintroduction to §6, we will only be able to take limits under the assumptionthat the\nsets considered are approximate subgroups. Fortunately, we ar e in that situation:\nLemma 6.2. There isα>0depending only on dGsuch that the following holds.\nLetU⊂Gbe a neighbourhood of the identity, A⊂U⊂Gbe a measurable subset\nsuch thatµG(A2)≤2dG(1+α)µG(A)and suppose that for all B,C⊂Umeasurable\nµG(BC)1\ndG≥(1−α)/parenleftBig\nµG(B)1\ndG+µG(C)1\ndG/parenrightBig\n,\nthenAis contained in a C(dG)-approximate subgroup of measure ≤C(dG)µG(A).\nBefore proving this, let us remark that the notion of boundedness and regularity\nproved for instance by Christ in [10] to allow for the limiting procedure to work\n(see also the discussion in [20, Appendix B]) would be a much stronger a ssumption.\nThe approach we offer here thus allows for more flexibility. As a trade -off, the\nconclusions drawn from our limiting procedure require more work to b e exploited.\nProof.Let Λ be the approximate subgroup provided by Proposition 4.4 and le t\nF1,F2be two subsets of size at most C(dG) such that A⊂F1Λ andA⊂ΛF2.\nUpon shrinking F1andF2and considering ΛC(dG)instead of Λ, we may assume\nthat for all f/\\e}a⊔io\\slash=f′∈F1(resp.f/\\e}a⊔io\\slash=f′∈F2) we havefΛ2∩f′Λ2=∅(Lemma 4.5).\nSuppose - the other case can be treated symmetrically - that ther e isf2∈F2\nsuch that\nµG(A∩Λf2)≥max/parenleftBigg\nsup\nf∈F2µG(A∩Λf),sup\nf∈F1µG(A∩fΛ)/parenrightBigg\n.\nNote that\nµG(A∩Λf2)≥1\nC(dG)µG(A).22 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nNow,\nµG(A2)≥µG(A(A∩Λf2))\n≥/summationdisplay\nf∈F1µG((A∩fΛ)(A∩Λf2))\n≥/summationdisplay\nf∈F1(1−α)/parenleftBig\nµG(A∩fΛ)1\ndG+µG(A∩Λf2)1\ndG/parenrightBigdG\n≥/summationdisplay\nf∈F12dG(1−α)µG(A∩fΛ)\n≥2dG(1−α)µG(A).\nSinceµG(A2)≤2dG(1+α)µG(A), we therefore find that for all f∈F1,\n(1−α)/parenleftBig\nµG(A∩fΛ)1\ndG+µG(A∩Λf2)1\ndG/parenrightBigdG\n−2dG(1−α)µG(A∩fΛ)≤2dG+1αµG(A).\nBy subadditivity of x/ma√s⊔o→x1\ndGwe have\nµG(A∩fΛ)≥/parenleftBigg\n1\nC(dG)−4/parenleftbiggα\n1−α/parenrightbigg1\ndG/parenrightBiggdG\nµG(A).\nSo forαsmaller than some constant C(dG) depending on the dimension only,\nµG(A∩fΛ)≥1\nC(dG)µG(A).\nBut by Proposition 4.4,\nµG(Λ(A∩fΛ))≤µG(AΛA)≤C(dG)µG(A).\nByRuzsa’scoveringlemma(Lemma4.3), Λisthereforecoveredby C(dG)-translates\nof (A∩fΛ)(A∩fΛ)−1⊂fΛf−1. Since this is true for all f∈F1, we see that\nAΛ∪ΛA−1is aC(dG)-approximate subgroup containing Aof measure at most\nC(dG)µG(A). /square\nWe have furthermore:\nLemma 6.3. WithA⊂U⊂Gas in Lemma 6.2. Suppose that B⊂Uis such\nthatµG(A)≥(1−α)µG(B)and\nµG(AB)≤2dG(1+α)µG(B).\nIfαis sufficiently small, depending on dGonly, thenBis contained in one right\ntranslate of AC(dG).\nProof.By Lemma 4.4, Bis contained in C(dG) right translates of the stabiliser\nSofA. There are therefore a constant C(dG) and a subset Fsuch thatB⊂\n⊔f∈FΛC(dG)fwith the (ASC(dG)f)f∈Fpairwise disjoint and |F| ≤C(dG) (Lemma\n4.5). Write Bf:= ΛC(dG)∩Bf−1MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 23\nThus,\n2dG(1+α)µG(B)≥µG(AB) (23)\n≥/summationdisplay\nf∈FµG(ABff) (24)\n≥(1−α)/summationdisplay\nf∈F/parenleftBig\nµG(A)1\ndG+µG(Bf)1\ndG/parenrightBigdG\n(25)\n≥(1−α)/summationdisplay\nf∈F2dGµG(Bf) (26)\n≥(1−α)2dGµG(Bf). (27)\nSo for allf∈F,\n/parenleftBig\nµG(A)1\ndG+µG(Bf)1\ndG/parenrightBigdG\n−2dGµG(Bf)≤2dG+1α\n1−αµG(B).\nWhich implies\nµG(Bf)≥/parenleftBigg\n(1−α)1\ndG−4/parenleftbiggα\n1−α/parenrightbigg1\ndG/parenrightBiggdG\nµG(B).\nForαsufficiently small, depending on dGonly,µG(Bf)>1\n2µG(B). Since this is\ntrue for all f∈Fand/summationtextµG(Bf) =µG(B),|F|= 1. /square\n6.3.Limits of approximate subgroups and density functions. Our goal is\nnow to prove local stability under the assumption that the subsets we are consider-\ningdonotessentiallyliveinaspaceoflowerdimension. Ourassumptionw illinvolve\nthe group homomorphism φconstructed above as a limit along an ultrafilter.\nProposition 6.4. LetAnbe subsets such that µG(A2\nn)/µG(An)→2dG. Suppose\nthat there are αn≥ǫnwithαngoing to 0asngoes to∞and such that An⊂\nR(e,αn,ǫn)andφ(AA−1)has non-empty interior. Then there is a convex set Cn\nsuch thatµG(Cn∆An)→0andφ(C) =φ(A).\nFix a sequence of compact subsets Anas in Proposition 6.4 and an ultrafilter\nF. WriteA:=/producttext\nnAn/Fand Γ the group it generates. Let m≥0 be an integer\nandBn⊂/parenleftbig\nAnA−1\nn/parenrightbigmbe a measurable subset for all n≥0. Denote by B:=/producttext\nn≥0Bn/F, such a set is called an internal subset of Γ. A useful feature of\nultraproducts is that the internal set Bshares many of the algebraic properties of\nthe setsBn. In particular, if the Bn’s are allK-approximate subgroups, then Bis\naK-approximate subgroup.\nOne can also define a measure of Bas follows\nµ(B) =F −limµG(Bn)\nµG(An).\nInternalsubsets areclosed under finite intersectionand this can be used to extend µ\nto a Γ-invariant σ-additive measure on the σ-algebra generated by internal subsets,\nsee [33, §4] where these elementary properties are checked thoroughly. B y Lemma\n6.2,µmoreover takes bounded values on internal subsets - hence µisσ-finite.\nBy Theorem 1.5 we find that for any two internal subsets B,Cwe have\n(28) µ(BC)1\ndG≥µ(B)1\ndG+µ(C)1\ndG.24 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nAs a direct corollary, any ascending union of internal sets also satis fies (28). Note\nin particular that if x∈RdGandBrdenotes the open ball of radius r>0 centred\natxfor the norm |·|, thenx+Bris an ascending union of internal sets.\nWe can now study the so-called ”density function” of Aand its doubling which\nwill enable us to relate the algebraic limit given by the ultraproduct Aand the\nrescaling procedure given by the map φfrom§6.1. Let us recall a result from [33,\n§8]:\nLemma 6.5. LetB⊂Γbe an internal subset covered by finitely many translates of\nAA−1. There is a measurable function fBdefined on RdGsuch that for all X⊂RdG\nmeasurable we have /integraldisplay\nXfB(x)dλ(x) =µ(B∩φ−1(X)).\nMoreover, for any positive real number r, letBrbe ball centred at 0for the norm\n|·|inRdG. Furthermore,\nfB(x) = lim\nr→0µ(B∩φ−1(x+Br))\nλ(Br)\nfor almost all x∈RdG([33, Prop. 8.5]).\nProof.According to Lemma 6.2, ( AA−1)4is an approximate subgroup. By [30,\nThm 4.2] (see also [33, §6] for a detailed account), it has a locally compact model\nψ:/a\\}b∇acke⊔le{⊔A/a\\}b∇acke⊔∇i}h⊔ →L. By [33, Prop. 8.3] there is a measurable function hB:L→Rsuch\nthat for all X⊂Lmeasurable we have/integraldisplay\nXhB(x)dλ(x) =µ(B∩φ−1(X)).\nAccording to [30, Thm 4.2] again, there is a continuous group homomo rphism\nϕ:L→RdGsuch thatφ=ϕ◦ψ. Define the map\nfB(x) :=/integraldisplay\nkerϕhB(x′h)dµkerϕ(h)\n-wherex′isanyelementin thefibre of ϕabovex-whenpossible, andset fB(x) = 0\notherwise. By the quotient formula (6), fBsatisfies the hypothesis. /square\nIn ourcase, wecan provethat such density functions satisfy a st rikinginequality:\nLemma 6.6. LetB,C⊂Γtwo internal subsets. We have:\n(1)∀t∈[0;1], for almost every x,y∈RdG,\nfBC(x+y)1\ndG≥1fB(x)fC(y)>0/parenleftBig\n(1−t)fB(x)1\ndG+tfC(y)1\ndG/parenrightBig\n;\n(2) for almost every x∈RdG,fA2(2x) =fA(x).\nProof.Remark that it suffices to prove (1) for t∈(0;1). We know that for all\nx,y∈RdGand all open subsets X,Y⊂RdGthat\nµ(BC∩φ−1(x+y+X+Y))1\ndG≥µ((B∩φ−1(x+X))(C∩φ−1(y+Y)))1\ndG(29)\n≥µ((B∩φ−1(x+X))1\ndG+µ(C∩φ−1(y+Y))1\ndG (30)\nas a consequence of (28).\nNow, forr,s>0,MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 25\n/parenleftBigg\nµ(BC∩φ−1(x+y+Br+s)\nλ(Br+s)/parenrightBigg1\ndG\n≥/parenleftbiggλ(Br)\nλ(Br+s)/parenrightbigg1\ndG/parenleftBigg\nµ((B∩φ−1(x+Br))\nλ(Br)/parenrightBigg1\ndG(31)\n+/parenleftbiggλ(Bs)\nλ(Br+s)/parenrightbigg1\ndG/parenleftBigg\nµ((C∩φ−1(y+Bs))\nλ(Bs)/parenrightBigg1\ndG\n. (32)\nThis becomes\n/parenleftBigg\nµ(BC∩φ−1(x+y+Br+s)\nλ(Br+s)/parenrightBigg1\ndG\n(33)\n≥r\nr+s/parenleftBigg\nµ((B∩φ−1(x+Br))\nλ(Br)/parenrightBigg1\ndG\n+s\nr+s/parenleftBigg\nµ((C∩φ−1(y+Bs))\nλ(Bs)/parenrightBigg1\ndG\n. (34)\nThus, taking r= (1−t)2−nands=t2−nwe get (1) when n→ ∞.\nLet us show (2). We have fA2(2x)≥fA(x) for almost every x∈RdGaccording\nto (1). But\nµ(A) =/integraldisplay\nRdGfA(x)dλ(x) (35)\n≤/integraldisplay\nRdGfA2(2x)dλ(x) (36)\n= 2−d/integraldisplay\nRdGfA2(x)dλ(x)≤µ(A). (37)\nHence, equality holds almost everywhere. /square\nWe therefore find:\nCorollary 6.7. The function fAis a scalar multiple of an indicator function of a\nconvex set.\nMoreover, if Bis such that µ(AB)≤2dGµ(B)andµ(B) =µ(A),fBis a\ntranslate of fA.\nProof.By combining (1) and (2) of Lemma 6.6 we find that for all x,y∈RdGand\nt∈[0;1] we have\nfA/parenleftbiggx+y\n2/parenrightbigg1\ndG≥1fA(x)fA(y)>0/parenleftBig\n(1−t)fA(x)1\ndG+tfA(y)1\ndG/parenrightBig\n.\nTakingt= 1 we have\nfA/parenleftbiggx+y\n2/parenrightbigg1\ndG≥1fA(x)fA(y)>0fA(y)1\ndG.\nChooseǫ >0 and letAǫbe the subset of φ(A) such that fA(x)≥(1−ǫ)supfA\nwhenx∈Aǫ. Define by A>0the essential support of fA. By the above inequality,\nwe have thatA>0+Aǫ\n2⊂Aǫ. Hence,Aǫis co-null in A>0andA>0is co-null in its\nconvex hull by the equality case of the Brunn–Minkowski inequality. Takingǫ→0\nwe see that fA= supfA·1A>0almost everywhere. Finally, we can see that A>0is\nsimplyφ(A) up to a null subset.26 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nA similar proof yields the result for fB. /square\nCorollary 6.7 will be used via the following observation. Let φn:g→RdGbe\ngiven byφn(x) =x1\nρn+x2\nδn. Then ifA,BandCare as in the statement of Corollary\n6.7 we have for all U⊂Rnopen,\n(38) F −limλ(U∩φn(An))\nλ(φn(An))=λ(U∩C)\nλ(C).\nThis is particularly useful in combination with the equality\n(39) F −limλ(U∩φn(An))\nλ(φn(An))=F −limµG(φ−1\nn(U)∩An)\nµG(An)\nwhich is a consequence of the uniqueness of the Lebesgue measure up to a scalar\nand the fact that µGis well approximated by λarounde(Lemma 7.3).\n6.4.Typical intersections with one-parameter subgroups. Asexplainedear-\nlier, the result of the limiting procedure is not usable “out of the box” . The density\nfunctions considered in §6.3 only explain how the mass spreads in a convex set,\nrelative to the total mass of the sets Anconsidered. It does not, however, assert\nthat the mass of the sets Anstays comparable to the mass of the convex set C.\nWe establish this thanks to a simple combinatorial argument - assumin g first\ne∈A- harnessing the one-dimensional stability results by considering int ersections\nwith one-parameter subgroups. It asserts that a set with near- minimal doubling\nintersects the rays going in ”most directions” in a large subset of a s omewhat long\ninterval.\nLemma 6.8. LetAbe a subset of the unit ball of Rdsuch that 0∈A,λ(A˜+A\n2\\A)≤\ncλ(A)where˜+denotes addition restrictedtoeach ray starting at the orig in. Suppose\nthat there are moreover ρ,δ>0such that,\nλ(A\\B(0,4ρ))≥(1−δ)λ(A).\nLet1\n100> ǫ >0. Ifc < ǫ4ρd−1δ, then choosing a∈Auniformly at random we\nhave with probability 1−4δthat the set R≥0a∩Ais a subset of an interval I(a)of\nlength at least ρandλ(I(a)\\A)≤ǫλ(I(a)).\nThemainideaiselementary: wewritetheLebesguemeasureinpolarco ordinates\nto relate the measure of Ato the measure of its intersection with rays starting at 0.\nWe choose a∈Auniformly at random and consider the intersection of Awith the\nray [0;a). Successive applications of the Markov inequality then show that t hese\nintersections alsosatisfy a small doubling assumption, and they mus t be sufficiently\nlong. Combining these with well-known stability results for Brunn–Mink owski in\ndimension one yields Lemma 6.8.\nProof.The Lebesgue measure can be written in polar coordinates in the for m\nλ(f) =/integraldisplay\nθ∈Sd−1/integraldisplay\n[0;1]rd−1f(rθ)drdθ.\nGivenx∈Rdwe will denote by λxthe Lebesgue measure on the ray starting at 0\nand going through xappearing in the above decomposition. We therefore find that\ncλ(A)≥λ/parenleftBig\n1A˜+A\n2\\A/parenrightBig\n≥/integraldisplay\nθ∈Sd−1/integraldisplay\n[0;1]rd���11A˜+A\n2\\A(rθ)drdθ.MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 27\nLet us now draw a∈Auniformly at random and let θ(a)∈Sd−1denote the direc-\ntion ofa. The law of the random variable θ(a) has density/parenleftBig/integraltext\n[0;1]rd−11A(rθ)dr/parenrightBig\ndθ.\nBy the Markov inequality, with probability at least 1 −δ\n2we therefore have the in-\nequality\n/integraldisplay\n[0;1]rd−11A˜+A\n2\\A(rθ(a))dr≤2δ−1c/integraldisplay\n[0;1]rd−11A(rθ(a))dr.\nIn particular,\n/integraldisplay\n[ρ;1]1A˜+A\n2\\A(rθ(a))dr≤2ρ1−dδ−1c/integraldisplay\n[0;1]1A(rθ(a))dr.\nDenotingA(a,ρ) the intersection of Awith the segment [ ρ;1] in the direction θ(a),\nthe Markov inequality again implies that with probability 1 −δ\n2we have\nλa(A(a,4ρ))≥1\n2λa(A(a)).\nThus, with probability 1 −δ, we find\nλa/parenleftbiggA(a,ρ)+A(a,ρ)\n2\\A(a,ρ)/parenrightbigg\n≤4ρ1−dδ−1cλa(A(a,ρ)).\nBy the stability in dimension one [20, Thm 1.1], we find that\nλa(Co(A(a,ρ))\\A(a,ρ))≤4ρ1−dδ−1cλa(A(a,ρ))\nwhereCo(A(a,ρ)) denotes the convex hull of A(a,ρ).\nSince 0∈A, we have that λ/parenleftBig\n0˜+A\\B(0,4ρ)\n2\\A/parenrightBig\n≤cλ(A). Proceeding as above,\nwe find that with probability 1 −δas well\nλa/parenleftbiggA(a,4ρ)\n2\\A/parenrightbigg\n≤4ρ1−d\n2δ−1cλa(A(a,4ρ)).\nButA(a,4ρ)\n2∩A⊂A(a,ρ) and all elements ofA(a,4ρ)\n2∩Ahave size at mostmaxA(a)\n2.\nTherefore, since λa(A(a,4ρ)\n2) =1\n2λa(A(a,4ρ)),A(a,ρ) contains an element of size at\nmostmaxA(a)\n2as soon asc≤ρd−1δ\n16.\nFinally, with probability at least 1 −δ>0,A(a,4ρ) is not empty, meaning that\nA(a,ρ) contains an element of size at least 4 ρ. Putting all of this together, with\nprobability at least 1 −4δ,Co(A(a,δ)) contains an interval of length at least 2 ρ\nand\nλa(Co(A(a,ρ))\\A(a,ρ))≤4ρ1−dδ−1cλa(A(a,ρ)).\n/square\nLemma 6.8 can in particular be applied to understand the doubling of a s ubset\nAin the neighbourhood of eof some Lie group. Indeed, the raywise sum of log A\nis contained in log A2. Thus, ifAhas close to optimal doubling, Lemma 6.8 shows\nthat intersections with most one-parameter subgroups going thr oughAlook like a\nlong interval. In fact, the simplex spanned by any number of these o ne-parameter\nsubgroups has a large intersection with A- that is the idea we exploit now.28 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nLemma 6.9. Suppose (An)n≥0andρn,δn>0are as in the statement of Proposi-\ntion 6.4. Let φnbe as in §6.1. Then there is an open subset U⊂Csuch that\nF −limµG(φ−1\nn(U))\nµG(R(e,ρn,δn))>0\nand\nF −limµG(φ−1\nn(U)∩An)\nµG(φ−1n(U))= 1.\nProof.Recallφn:g→RdGis defined by φn(x1+x2) =x1\nρn+x2\nδnforx1∈h\nandx2∈h⊥. LetCbe the convex subset given by Corollary 6.7 applied to the\nultraproduct of the An’s.\nChoosedGelementsc1,...,c dGintheinteriorof Cthatspan RdG. Chooseρ0>0\nsuch that ˜ci+Bρ0⊂Cfor alliand for every ˜ c1∈c1+Bρ0,...,˜cdG∈cdG+Bρ0,\nthe simplex spanned by ˜ c1,...,˜cnhas volume at least some constant v>0. Choose\nρ>0 such that λ(C\\B4ρ)≥/parenleftBig\n1−λ(Bρ0)\n4/parenrightBig\nλ(C).\nDefine by induction the map ϕk:Gk→Gfork= 1,...,d Gbyϕ1(g) =gand\nϕk+1(g,g) =ϕk(g)g\n2whereg∈Gk,g∈G. We see by induction that there is a\nmeasurable subset Ω n⊂AdGnsuch thatϕdG(Ωn)⊂Anand\nlimλ⊗dG(Ωn)\nλ(An)dG= 1\nwhereλ⊗dGis thedG-fold power of the Lebesgue measure i.e. a d2\nG-dimensional\nLebesgue measure.\nTherefore, according to the polar coordinates for the Lebesgue measure and the\nMarkov inequality, drawing ( a1,...,a dG)∈AdGnuniformly at random we have with\nprobability at least 1 −δn,\nλ⊗dG(An(a1)×···×An(adG)∩Ωn)≥(1−δn)λ⊗dG(An(a1)×···×An(adG))\nfor someδn→0. In the above, we are using λ⊗dGto denote the product of the\nLebesgue measures λaiappearing in the polar decomposition of the relevant ray in\nthe direction ai, see the proof of Lemma 6.8.\nWe will now apply the above Lemma 6.8 to the sets φn(An). Notice that the\nray-wise addition φn(An)˜+φn(An) is contained in φn(A2\nn). Whenδn<λ(Bρ0)\n4we\ncan findci,n∈ci+Bρ0∩φn(An) fori= 1,...,d Gsuch that\nλci,n(Co(An(ci,n))\\An(ci,n))\nλci,n(An(ci,n))−→0\nandCo(An(ci,n)) is a segment of length at least 2 ρ(Lemma 6.8). Denote also\nmi,nandMi,nthe infimum and supremum respectively of Co(An(ci,n)). Upon\nconsidering a subsequence, we may assume that mi,nandMi,nconverge to mi∈g\nandMi∈grespectively for all i. Moreover, |Mi,n−mi,n| ≥ρ2\n2.\nNow, the set we are looking for is the Gromov–Haudorff limit Sof\nSn:=φn[ϕdG(Co(An(c1,n))×···×Co(An(cd,n)))].\nFromtheBCHformula S⊂gisthesimplexspannedbyM1\n2dG,...,MdG\n2and/summationtextdG\ni=1mi\n2i.\nAnother application of the BCH formula to compute the differential o fϕdGyieldsMINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 29\nmoreover\nλ(φn[ϕdG(An(˜c1)×···×An(˜cdG)∩Ωn)]∆S)\nλ(SdG)→0.\nSinceϕdG(An(˜c1)×···×An(˜cdG)∩Ωn)⊂An, this yields the result. /square\n6.5.Proof of local stability.\n6.5.1.The casee∈A.We will now apply Lemma 6.8 in combination with results\nof section §6.3.\nProof of Proposition 6.4 when e∈A.Let (An)n≥0, (δn)n≥0and (ρn)n≥0be as in\nthe statement of Proposition 6.4. Let fAbe the density function of Astudied in\n§6.3. Then fAis a scalar multiple of the indicator function of a convex subset\nC⊂RdG(Corollary 6.7).\nLetSbe given by Lemma 6.9. We know by (38) that\nF −limλ(S∩φn(An))\nλ(φn(An))=λ(S)\nλ(C).\nButF −limλ(Sd∩φn(An))) =λ(Sd). So\nF −limλ(φn(An))) =λ(C).\nSinceF −limλ(˜An\\C)\nλ(φn(An)))= 0 by (38), the result is established. /square\n6.5.2.The general case. In the general case, we have to reduce to the case e∈A.\nThis will be a consequence of Corollary 6.7.\nLemma 6.10. LetAbe as above. Then for every a∈A,µ(aA∆Aa) = 0.\nProof.The setsφ(aA) andφ(Aa) are both contained in φ(a)+φ(A). According to\nCorollary 6.7, λ(φ(a) +φ(A)) = 2−dGλ(φ(A2)) sinceφ(A) is convex. By Corollary\n6.7 again,\nµ/parenleftbig\nA∩φ−1(φ(a)+φ(A))/parenrightbig\n= 2−dµ(A2) =µ(A).\nWriteB:=A∩φ−1(φ(a)+φ(A)). We have aA,Aa⊂Bandµ(aA) =µ(Aa) =\nµ(B) =µ(A). This proves our result. /square\nWe can now conclude the proof of Proposition 6.4 in the general case .\nProof of Proposition 6.4. Choosea∈Asuch thatφ(a) lies at the barycentre of\nφ(A). According to Lemma 6.10 we know that µ(aA∆Aa) = 0. Equivalently,\nµ(a−1A∆Aa−1) = 0. Now, the set ˜A=a−1A∩Aa−1is internal, contains the\nidentity, has doubling 2dGand is a co-null subset of A. So we are done according\nto the case e∈A. /square\n7.Global stability\nWe finally show how the local stability results imply stability in G.30 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\n7.1.Sets with small doubling are contained in one neighbourhood of a\nsubgroup. We first reduce to subsets contained in a neighbourhood of a prope r\nclosed subgroup (whereas Proposition 1.6 asserts that an approx imate subgroup is\ncontained in a finite collection of translates of such a neighbourhood ).\nLemma 7.1. Letδ >0. There is ǫ >0such that if A⊂Gis a compact subset\nof measure at most ǫandµG(A2)≤/parenleftbig\n2dG−dH+ǫ/parenrightbig\nµG(A), then there is a compact\nsubgroupHof dimension dHsuch thatA⊂Hδ.\nProof.Fixρ >0. According to the first part of this paper (Proposition 1.6) for\nδ′>0 there are a closed subgroup H,ǫ1>0 andC(dG) such that if ǫ≤ǫ1\nthenAis covered by C(dG) left-translates of Hδ′andBis covered by C(dG) right-\ntranslates of Hδ′. ChooseHto be a maximal proper subgroup, in particular H\nis the normaliser of its connected component of the identity. Choos eF1,F2⊂G\nsuch thatA⊂F1Hδ′andA⊂Hδ′F2. By Lemma 4.5 we may assume that for\nallf,f′∈F1,fH2δ′+ρ∩f′H2δ′+ρ=∅and a symmetric property for f,f′∈F2.\nSuppose as we may that supf∈F2µG(A∩Hδ′f)≥supf∈F1µG(A∩fHδ′) and choose\nf0∈F2so thatµG(A∩Hδ′f0) is maximum.\nAccording to Proposition 5.5 for all α>0 we find as soon as ǫ>0 is sufficiently\nsmall:\nµG(A2)≥µG(A(A∩Hδ′f0)) (40)\n≥/summationdisplay\nf∈F1µG((A∩fHδ′)(A∩Hδ′f0)) (41)\n≥/summationdisplay\nf∈F1(1−α)/parenleftBig\nµG(A∩fHδ′)1\ndG−dH+µG(A∩Hδ′f0)1\ndG−dH/parenrightBigdG−dH\n(42)\n≥(1−α)2dG−dHµG(A)+/summationdisplay\nf(1−α)Rf. (43)\nwhereRfis defined as\n/parenleftBig\nµG(A∩fHδ′)1\ndG−dH+µG(A∩Hδ′f0)1\ndG−dH/parenrightBigdG−dH\n−2dG−dHµG(A∩fHδ′).\nSinceµG(A∩Hδ′f0)≥µG(A∩fHδ′),Rfis positive for all f∈F1. SoRf≤\n(ǫ+2dG−dHα)\n1−αµG(A). SinceµG(A∩Hδ′f0)≥µG(A)\n|F2|≥µG(A)\nC(dG)we find that for all\nα′>0 ifǫ(and, hence, α) is sufficiently small, then for all f∈F1\n1−α′≤µG(A∩fHδ′)\nµG(A∩Hδ′f0)≤1.\nAll in all, for all α′>0 we find\nµG((A∩fHδ′))(A∩Hδ′f0))≤2dG−dHµG(A∩fHδ′)+α′µG(A∩fHδ′)\nas soon as ǫis sufficiently small. Write Ah,ρ,f:=A∩fTδ′BH(h,ρ). Forǫ,δ′,α′\nsufficiently small we find by Lemma 7.5 that\n(44) µG(Ah,ρ,f)≥µG(A∩fHδ′)µH(BH(e,ρ))\n2\nfor allh∈H,f∈F1.\nThus,fH⊂ABG(e,δ′+ρ)⊂H2δ′+ρF2. WriteH0the connected component\nofH. Then there is f′∈F2such thatfH0⊂H2δ′+ρf′according to the firstMINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 31\nparagraph. For δ′,ρsufficiently small, this implies that fis contained in the δ-\nneighbourhood of the normaliser of H0i.e. inHδ. Since this is true for all f∈F1,\nF1⊂Hδ. In turn, this implies A⊂Hδ+δ′. /square\n7.2.A coarse stability result. Considerδ,ǫ>0andassumefromnowonthatwe\nare givenA,B,AB ⊂Hδfor some proper closed subgroup Hof maximal dimension\nsuch that\nµG(AB)≤/parenleftbig\n2dG−dH+ǫ/parenrightbig\nmin(µG(A),µG(B)).\nOur goal in this section is to prove that the mass of such A’s andB’s is spread\nevenly around H. To do so we study the subsets TδBH(h,ρ)∩B,TδBH(h,ρ)∩A\nandTδBH(h,ρ)∩ABfor varying ρ>0 andh∈H. We write\nAh,ρ:=TδBH(h,ρ)∩A, B h,ρ:=TδBH(h,ρ)∩B,\n(AB)h,ρ:=TδBH(h,ρ)∩AB.\nThe main result of this section is:\nProposition 7.2. Letρ,α>0. There are δ,ǫsuch that the following holds. Write\nm(ρ) := suph∈Hmax(Ah,ρ,Bh,ρ).For everyh∈H,\n(1−α)m(ρ)≤Ah,ρ≤m(ρ).\nMoreover,\nµH(BH(e,ρ))µG(A)≤m(ρ)≤(1−α)−1µH(BH(e,ρ))µG(A).\nOur proof of Proposition 1.2 relied on a series of inequalities we recall n ow. For\nallα0,α1>0, we had for all ρ>0 small enough and all ǫ,δ>0 sufficiently small:\nµG(A)µH(BH(e,ρ)) =/integraldisplay\nHµG(Ah,ρ)dµH(h) (45)\n≤(2dG−α0)−1/integraldisplay\nHµG(Ah,ρBh0,ρ)dµH(h) (46)\n≤(2dG−α0)−1/integraldisplay\nHA,ρµG((AB)hh0,2ρ+cδ2)µH(h) (47)\n=(2dG−α0)−1/parenleftbig\nµG(AB)µH(BH(2ρ+cδ2)) (48)\n−/integraldisplay\nH\\HA,ρh0µG((AB)h,2ρ+cδ2)dµH(h)). (49)\nwhereHA,ρis the subset of those h∈Hsuch thatAh,ρ/\\e}a⊔io\\slash=∅andh0is chosen so\nthatµG(Bh0,ρ) is at least (1 −α1)m(ρ) (note that by a symmetry argument we can\nalways assume existence of at least one such h0).\nWe supposegiven fromnow on A,Bas abovewith respect to parameters δ,ǫ>0.\nWe will establish below a series of results holding whenever δ,ǫ>0 are sufficiently\nsmall. When doing so, we will also implicitly mean that α0,α1are also chosen\njudiciously - we leave to the reader the verification that this is always possible.\nSince all the inequalities above are close to equality, we can show that the as-\nsumption on the choice of h0is justified:\nLemma 7.3. Letα >0. There is ρ0>0such that for all ρ0> ρ >0the\nfollowing holds for all δ,ǫ >0sufficiently small: for all hin a subset of measure\nat least(1−α)µG(HA,ρ)ofHA,ρwe haveµG(Ah,ρ)≥(1−α)m(ρ)and a similar\nstatement holds for B.32 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nThis will be a simple consequence of the Markov inequality and the follow ing\nobservation:\nLemma 7.4. Letα,C >0. LetX,Ybe subsets of Gsuch thatµG(XY)1\ndG≥\n(1−α)/parenleftBig\nµG(X)1\ndG+µG(Y)1\ndG/parenrightBig\n. Suppose that µG(XY)≤Cmin(µG(X),µG(Y)).\nThen\n/parenleftBigg\nC1\ndG−1+α\n1−α/parenrightBiggdG\nmin(µG(X),µG(Y))≥max(µG(X),µG(Y))\nProof.SupposeµG(X)≤µG(Y). We have\nC1\ndGµG(X)1\ndG≥(1−α)/parenleftBig\nµG(X)1\ndG+µG(Y)1\ndG/parenrightBig\nwhich yields\n(C1\ndG−1+α)µG(X)1\ndG≥(1−α)µG(Y)1\ndG.\n/square\nProof of Lemma 7.3. We will assume for notational simplicity that the parameters\nα0,α1are smaller than α. For allh∈HA,ρwe have\nµG((AB)hh0,2ρ+c(dG)δ2)≥µG((Ah,ρ)(Bh0,ρ)) (50)\n≥(2dG−α)µG(Ah,ρ). (51)\nBut we have\n(2dG−dH+α)µG(A)≥µG(AB)\n≥/integraltext\nHA,ρµG((AB)hh0,2ρ+c(dG)δ2)\nµH(BH(e,2ρ+c(dG)δ2))\nand\n(2dG−α)µG(A) =/integraltext\nHA,ρ(2dG−α)µG(Ah,ρ)\nµH(BH(e,ρ))\nWritepµG(HA,ρ) for somep∈[0;1] the measure of the set of those h∈HA,ρsuch\nthat\n(2dG+cα)µG(Ah,ρ)≥µG((AB)hh0,2ρ+c(dG)δ2). (52)\nWe see that by the Markov inequality, we have\np≥1−/parenleftbig\n2dG−dH+α/parenrightbigµH(BH(e,2ρ+c(dG)δ2))\nµH(BH(e,ρ))−(2dG−α)\n(1+c)α. (53)\nButµH(BH(e,2ρ+c(dG)δ2))\nµH(BH(e,ρ))≤2dHforδsufficiently small (Lemma 5.1). So\np≥1−/parenleftbig\n2dG−dH+α/parenrightbig\n2dH−(2dG−α)\n(1+c)α(54)\n= 1−(2dH+1)\n(1+c). (55)\nBy Lemma 7.4, for any such hwe have moreover\n/parenleftbigg1+cα\n1−cα/parenrightbiggd\nµG(Ah,ρ)≥µG(Bh0,ρ).MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 33\nSo we are done (e.g. with a choice c=α−1\n2). A symmetric argument shows the\nestimate for µG(Bh,ρ).\n/square\nAs a first consequence, we can prove that AandBcannot avoid a large chunk of\nHδ. This will be crucial later on when we showthat sets near equality are contained\ninHδ- and not merely in a collection of translates of Hδ.\nLemma 7.5. Letρ >0. There are ǫ,δ >0such that for every h∈H,Ah,ρand\nBh,ρare non-empty.\nThe idea is to relate this to doubling in Hand apply Kemperman’s inequality.\nProof.Letα>0, and suppose ǫ0is given by Lemma 7.3. Recall that\nm(ρ) := sup\nhmax(µG(Ah,ρ),µG(Bh,ρ)).\nDefine the subset\nHα\nA,ρ:={h∈HA,ρ:µG(Ah,ρ)≥(1−α)m(ρ)}.\nFor anyh1∈Hα\nA,ρ,h2∈HB,ρwe have\nµG(TδBH(h1h2,2ρ+cδ2)∩AB)≥µG(Ah1,ρBh2,ρ)≥µG(Ah1,ρ)≥(1−α)m(ρ).\nFurthermore, according to Lemma 5.2,\nm(ρ)≥µH(BH(e,ρ))µG(A)\nµH(HA,ρ).\nSo (49) implies\nµG(AB)≥/parenleftbig\n2dG−α0/parenrightbigµH(BH(h,ρ))\nµH(BH(h,2ρ+cδ2))\n×/parenleftbigg\n1+(1−α)µH(Hα\nA,ρHB,ρ\\h0HB,ρ)\nµH(HA,ρ)/parenrightbigg\nµG(A).\nBut\nµG(AB)≤(2dG−dH+ǫ)µG(A).\nSuppose that α<1\n2. Then, according to Lemma 5.1, for δ<<ρandαsufficiently\nsmall, we have\nµH(Hα\nA,ρHB,ρ\\h0HB,ρ)≤4dGǫµH(HA,ρ).\nBy Kemperman’s inequality [37] in Hwe have\nµH(Hα\nA,ρHB,ρ\\h0HB,ρ)≥min/parenleftbig\nµH(Hα\nA,ρ),1−µH(HB,ρ)/parenrightbig\n.\nFor anyα′∈(0;1\n2), ifδ,ǫare sufficiently small\n(1−α′)µH(HA,ρ)≤µH(Hα\nA,ρ)\naccording to Lemma 7.3. So\nmin(µH(HA,ρ),1−µH(HB,ρ))≤(1−α′)−14dGǫµH(HA,ρ)≤4dG+1ǫµH(HA,ρ),\nand, hence,\n1−µH(HB,ρ)≤4dG+1ǫ\nas soon asǫ<4−dG−1. /square34 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nProof of Proposition 7.2. Recall the following formula valid for all ρ′>0,h′∈H:\n(56) µG(Ah′,ρ) =/integraltext\nHµG(Ah′,ρ∩TδBH(h,ρ′))dµH(h)\nµH(BH(e,ρ′))\nwhich was already used in the proof of Lemma 5.2. For any ρ′>0, ifδ,ǫ >0\nare sufficiently small, we know that Ah,ρ′is not empty for all h∈H(Lemma\n7.5). Thus, for any α′>0, as soon as δ,ǫ >0 are sufficiently small we have by\nLemma 7.3, that there is a subset X⊂Hof measure at least 1 −α′,such that\nµG(Ah,ρ′)≥(1−α′)m(ρ′) for allh∈X. Hence:\nµH(BH(e,ρ′))µG(Ae,ρ) =/integraldisplay\nHµG(Ae,ρ∩TδBH(h,ρ′))dµH(h) (57)\n≥/integraldisplay\nBH(e,ρ−ρ′)µG(Ah,ρ′)dµH(h) (58)\n≥/integraldisplay\nBH(e,ρ−ρ′)∩XµG(Ah,ρ′)dµH(h) (59)\n≥(1−α′)(µH(B(e,ρ−ρ′))−α′)m(ρ′). (60)\nBut (56) implies that m(ρ)≤m(ρ′)µH(B(e,ρ))\nµH(BH(e,ρ′)). In turn, this yields:\nµG(Ae,ρ)≥(1−α′)µH(B(e,ρ−ρ′))−α′\nµH(B(e,ρ))m(ρ).\nAnd the first inequality follows once we choose α′,ρ′,δ,ǫsufficiently small relative\nto one another.\nNow, the moreover part follows from the above and the equality\n(61) µG(A) =/integraltext\nHµG(Ah,ρ′)dµH(h)\nµH(BH(h,ρ′)).\n/square\n7.3.Approximate subgroup at the identity and commensurability under\nrotations. Wekeepthenotationsof §7.2. Inthissection, wefocusmorespecifically\non the case A=Bi.e. assuming that we are given a compact subset with A,A2⊂\nHδand\nµG(A2)≤/parenleftbig\n2dG−dH+ǫ/parenrightbig\nµG(A).\nLemma 7.6. Letǫ′>0. There isρ0>0such that for all ρ0>ρ>0the following\ninequality holds for all δ,ǫ>0sufficiently small.\nµG(A2\ne,ρ)≤(2dG+ǫ′)µG(Ae,ρ).\nFurthermore, Ae,ρis aC(dG)-approximate subgroup.\nProof.Letρ > ρ′>0,α >0 to be chosen later. According to Proposition 7.2, if\nδ,ǫare sufficiently small, then\n(1−α)µH(BH(e,ρ′))µG(A)≤µG(Ah,ρ′)≤(1−α)−1µH(BH(e,ρ′))µG(A).MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 35\nNotice moreover that for all ρ0,α′>0, as soon as ρ′,δ,α >0 are sufficiently\nsmall,\nµG(A2\\TδBH(e,ρ0))≥/integraltext\nH\\BH(e,ρ0+2ρ′+cδ2)µG((A2)h,2ρ′+cδ2)dh\nµH(BH(e,2ρ′+cδ2))(62)\n≥/integraltext\nH\\BH(e,ρ0+2ρ′+cδ2)µG(Ah,ρ′Ae,ρ′)dh\nµH(BH(e,2ρ′+cδ2))(63)\n≥(2dG−α′)/integraltext\nH\\BH(e,ρ0+2ρ′+cδ2)µG(Ah,ρ′)dh\nµH(BH(e,2ρ′+cδ2))(64)\n≥µH(BH(e,ρ′))(2dG−α′)µG(A\\Ae,ρ0+2ρ′+cδ2))\nµH(BH(e,2ρ′+cδ2)). (65)\nIndeed, Lemma 5.2 was applied in the first line and to obtain the last line; Theorem\n1.5 and the first inequality of the proof then enable us to go from the second to the\nthird line. This becomes\nµG(A2\\TδBH(e,ρ0))≥(2dG−dH−α′)µG(A\\TδBH(e,ρ0+ρ′))\naccording to Lemma 5.1.\nBut for any ρ>0,\nµG(A2\ne,ρ)+µG(A2\\TδBH(e,2ρ+cδ2))≤µG(A2)≤(2dG−dH+ǫ)µG(A)\nso if 2ρ+cδ2=ρ0,\nµG(A2\ne,ρ)≤2dG−dHµG(Ae,ρ0+cδ2+2ρ′)+α′′µG(A)\nwhereα′′=ǫ+2dG−dH−(2dG−α′)µH(BH(e,ρ′))\nµH(BH(e,2ρ′+cδ2)). By Lemma 5.1, if ρ′>δare\nsufficiently small, then α′′≤ǫ+2−dHα′+C(dG)ρ′2.\nNow, for any ρ′′>ρ′,\nµH(BH(e,ρ′))µG(Ae,ρ′′) =/integraldisplay\nHµG(Ae,ρ′′∩TδBH(h,ρ′))dµH(h) (66)\n≥µH(BH(e,ρ′′−ρ′))(1−α)m(ρ′) (67)\nby Lemma 7.3. And, similarly,\nµH(BH(e,ρ′))µG(Ae,ρ′′) =/integraldisplay\nHµG(Ae,ρ′′∩TδBH(h,ρ′))dµH(h) (68)\n≤µH(BH(e,ρ′′+ρ′))m(ρ′). (69)\nTherefore, (69) with ρ′′=ρ0+cδ2+2ρ′implies\nµ(A2\ne,ρ)≤(2dG−dH)µH(BH(e,ρ0+cδ2+3ρ′))\nµH(BH(e,ρ′))µ(Ae,ρ′)+α′′µG(A).\nApplying (69) with ρ′′= 1 we find in addition\nµ(A2\ne,ρ)≤(2dG−dH)µH(BH(e,ρ0+cδ2+3ρ′))\nµH(BH(e,ρ′))µ(Ae,ρ′)+α′′m(ρ′)\nµH(BH(e,ρ′)).\nWhich by (67) yields\nµ(A2\ne,ρ)≤(2dG−dH)µH(BH(e,ρ0+cδ2+3ρ′))\nµH(BH(e,ρ−ρ′))µG(Ae,ρ)+α′′µG(Ae,ρ)\nµH(BH(e,ρ−ρ′)).36 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nSo by Lemma 5.1 and our estimate of α′′above , we have that if α′,ρ′,δare\nsufficiently small\nµ(A2\ne,ρ)≤(2dG+ǫ′)µG(Ae,ρ).\nThe “furthermore” statement is now a direct application of Lemma 6 .2. /square\nLemma 7.7. Letǫ′>0. There isρ0>0such that for all ρ0>ρ>0the following\nholds for all δ,ǫ>0sufficiently small.\nFor allh1,h2∈H,\nµG(Ah1,ρAh2,ρ)≤(2dG+ǫ′)minµG(Ah1,ρ),µG(Ah2,ρ).\nMoreover, for all h∈Hthe approximate subgroup Ae,ρA−1\ne,ρis commensurated\nby an element of Bg(h,C(dG)ρ).\nProof.The proof of the inequality is obtained exactly as in the proof of Lemm a 7.6.\nLet us prove the moreover part. According to Lemma 6.2, for all h∈Hthere isg∈\nGsuchthatAh,ρg−1⊂(Ae,ρA−1\ne,ρ)C(dG). Inparticular, hg−1∈Bg(e,C(dG)ρ). Since\nin addition µG(Ah,ρg−1gAe,ρ)≤(2dG+ǫ′)minµG(Ah1,ρ),µG(Ah2,ρ) we deduce\nusing Proposition 4.4 - as in the proof of Lemma 6.2 - that gcommensurates\nAeA−1\ne. /square\n7.4.Proof of stability. We conclude now the proof of the stability result.\nProof of Theorem 1.4. We will prove qualitative stability by a compactness argu-\nment. Suppose that we are given a sequence An⊂GwithµG(An)→0 and\nµG(A2\nn)\nµG(An)→2dG−dHand letδn>0 be the least real such that there is a proper closed\nsubgroupHn⊂Gsuch thatAn⊂(Hn)δn. Upon considering conjugates of the\nAn’s, we can suppose that Hn=Hfor some fixed subgroup. By Lemma 7.1, we\nhave thatδn→0.\nChoosenow ρn,αn>0goingto0as ngoesto∞andsuchthat forall h1,h2∈H,\nµG((An)h1,ρn)≥(1−αn)m(ρn)\nand\nµG((An)h1,ρn(An)h2,ρn)≤(2dG−dH+αn)m(ρn).\nChoosemoreover ρ′\nn>0 such thatρ′\nn\nρn→0 and (An)h,ρ′nis non-empty for all h∈H.\nSuchαn,ρn,ρ′\nnexistbyProposition7.2andLemma7.7. Tosimplifynotationswrite\nfor allh∈H,An,h:=TδnBH(h,ρn)∩A.\nOur first goal is to prove\nClaim 7.8. An,esatisfies the conditions of Proposition 6.4.\nThe proof of this claim will appeal to the theory of compact approxim ate sub-\ngroups and ideas reminiscent of Lemma 7.1.\nProof of Claim 7.8. Usingthenotationsof §6.1,onecancheckthat An,e⊂R(e,ρn+\ncρnδn,δn) for some constant c >0. Letφbe the limit map for a choice of non-\nprincipal ultrafilter and let Adenote the ultraproduct of the An,efor the same\nultrafilter. The image of φcan be identified with gusing a map constructed as\nfollows. Take x∈gand writex=x1+x2withx1∈handx2∈h⊥. De-\nfineϕn(x) := (ρn+cρnδn)x1+δnx2. Define now ϕ(x) :=φ((ϕn(x))n≥0). Then\nϕ:g→RdGis a linear isomorphism and we use ϕto identify the range of φwithMINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 37\ng. Remark from this construction that if h∈Handxn∈R(e,ρn+cρnδn,δn) for\nalln≥0, thenφ((hxnh−1)n≥0) is well defined and equal to Ad(h)(φ(x)).\nNow let us show that φ(A) satisfies the conditions of Proposition 6.4. First of\nall, asρ′\nnis negligible compared to ρn, the projection of φ(A) tohis dense in\nthe projection of φ((R(e,ρn+cρnδn,δn)) toh. Moreover, by the choice of δn, the\nprojection of φ(˜A) toh⊥is not reduced to a point. Suppose that it is not finite.\nThe approximate subgroup Λ = φ(AA−1)2intersects a vector subspace V⊂gin a\nneighbourhood of the origin [40, thm 1.4] and Vcontains has a proper subspace.\nBy [40, Thm 1.4], Vis moreoverinvariantunder anylinear map commensurating Λ.\nBut Λ is commensurated by the elements of the group Hacting on hvia the adjoint\nrepresentation (Lemma 7.7). The restriction of the adjoint repre sentation to h⊥-\ntheso-calledisotropyrepresentation-isirreducibleas Hisaproperclosedsubgroup\nofmaximaldimension. This canbe checkedfrom the tables [15] - see t he table in §A\nbelow - and the classification from [53] of isotropy irreducible homoge neous spaces.\nChecking again the tables from [15] (or, §A), the only subrepresentation for the\nadjoint action of Hongthat projects surjectively to both handh⊥isg. SoV=g.\nIt remains to check that the projection Λ to h⊥cannot be finite. Suppose\notherwise, then there are r≥2 of minimal size and h1,...,h rsuch thatφ(A)⊂/unionsqtext\ni(hi+h)∩φ(A). DefineAi:=φ−1(hi+h)∩Afor alli= 1,...,r. ThenAiis\ninternal and non-empty. Choose i0such thatAi0has maximal µ-measure. We find\nµ(A2)≥r/summationdisplay\ni=1µ(AiAi0) (70)\n≥2dGµ(A)+r/summationdisplay\ni=1µ(Ai)/bracketleftBigg/parenleftbigg\n1+µ(Ai0)\nµ(Ai)/parenrightbiggdG\n−2dG/bracketrightBigg\n(71)\n≥2dGµ(A). (72)\naccording to Theorem 1.5. But µ(A2) = 2dGµ(A). SoAAi0has full measure in\nA2andµ(Ai0) =µ(Ai) for alli= 1,...,r. In turn, this implies that AAihas full\nmeasure in A2for alli= 1,...,r. Projecting to h⊥we obtain that {hi+hj}has\nsizer. By Kemperman’s inequality, we thus have r= 1. A contradiction. /square\nWe can thus apply the local stability result (Proposition 6.4). We thus have a\nconvex subset C⊂gsuch that\nµG(An,e∆ϕn(C)) =o(µG(An,e)).\nHere,ϕndenotesthelinearendomorphismalreadystudiedinthe proofofClaim 7.8.\nAs in the proof of Claim 7.8, Lemma 7.7 and Corollary 6.7 also implie invarianc e\nofCunder the adjoint action of H. Sinceϕncommutes with this action for all\nh∈H,ϕn(C) is also invariant under this action.\nBy Corollary 6.7 and Lemma 7.7 again, we know that there is n0such that for\nalln≥n0, allh∈Hthere isgh,n∈R(e,0,δn) such that we have\nµG(An,h∆ϕn(C)hgn,h) =o(µG(An,e)).\nNow, forh,h′∈Hwe have\nµG(Ae,nAhh′,n∆Ah,nAh′,n) =o(µG(Ae,n))\nby Lemma 7.7. So d(ghh′,ghhgh′h−1) =o(δn) i.e.h/ma√s⊔o→ghis aδn-1-cocycle in\nthe language of [35]. By [35, Lem. 2] there is continuous group homom orphism38 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nfn:H→Gsuch thatdg(fn(h),hgn,h) =o(δn) for allh∈H. Thus,\nµG(ϕn(C)hgn,h∆ϕn(C)fn(h)) =o(µG(An,e))\nwhich implies\nµG(An,h∆ϕn(C)fn(h)) =o(µG(An,h)).\nRemark that therefore,\nµG/parenleftbig\nfn(h)−1ϕn(C)fn(h)∆ϕn(C)/parenrightbig\n=o(µG(An,e)).\nNow,µG(ϕn(C)fn(H)∆fn(H)δn) =o(µG(A′\nn))byinvarianceunder Hofφn(C)and\nbecaused(h,fn(h))≤δn. In addition, µG(An∆A′\nn) =o(µG(An)) by the previous\nparagraph. Hence, µG(An∆fn(H)δn) =o(µG(An)).\nSo our result follows. /square\n8.Closing remarks\n8.1.From qualitative stability to quantitative results. The above considera-\ntions are focused on establishing a sharp bound for small doubling at the first order\nand for small subsets. Many of our arguments exploit non-quantit ative results -\nsome for inherent reasons and others for incidental ones. We men tion among those\nthe compactness argument used in §6 and Carolino’s structure theorem (Theorem\n4.6).\nObtaining some amount of quantitative informationis howevernot ho peless with\nour techniques. We can for instance exploit the stability result (The orem 1.4) to\noperate a bootstrap argument yielding the proof of the quantitat ive estimate in\nTheorem 1.2.\nProof.The argument is simple. According to Theorem 1.4, there is ǫ >0 such\nthat ifµG(A)≤ǫandµG(A2)≤2dG−dHµG(A), then there is δ >0 such that\nµG(A)⊂HδandµG(A)≥µG(Hδ)\n2. Soδ=O(µG(A)1\ndG−dH) (e.g. [33, Fact 2.17]).\nBy Proposition 5.4 now, µG(A2)≥/parenleftBig\n2dG−dH−C(dG)µG(A)2\ndG−dH/parenrightBig\nµG(A) ./square\nRemark 8.1. Since we do not have at our disposal a stability result for A/\\e}a⊔io\\slash=B, we\ndo not get a similar estimate for the Brunn–Minkowski inequa lity. Tracking down\nestimates along the proof we however obtain ǫ≤C(dG)µG(A)C(dG).\n8.2.The quest for minimisers. A combination of the above and the stability\ntheorem 1.4 corroborate a conjecture first raised in [33]. Namely:\nConjecture 8.2. The subset Hδminimises the doubling constantµG(A2)\nµG(A)forA\nranging through measurable subsets with µG(A) =µG(Hδ).\nIt might appear odd at first that stability results are known without knowledge\nof such minimisers. While this contrasts with results around the Brun n–Minkowski\ninequality in Euclidean spaces, we believe that this situation is the norm rather\nthan the exception for results in additive combinatorics in non-comm utative groups\nand/ordiscretegroups. Wemention in that directiontheworkofKe evash–Lifshitz–\nMinzer [36] on product-free sets, the polynomial Freiman Ruzsa conjecture [45, 25]\nand doubling minimisers in the Heisenberg group (resp. nilpotent Lie gr oups) [42].\nA similar state of affairs can be found in non-compact simple Lie groups , where\nKunze–Stein phenomena provide lower bounds of (almost) the right magnitude,MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 39\nbut no minimiser of the doubling constant is known (knowledge of such a min-\nimiser could imply improvements of [31, 12]). We hope our work will provid e new\nideas towards finding doubling minimisers in non-commutative Lie group s. More\nprecisely:\nProblem 8.3. LetGbe a connected Lie group and α >0. What is the structure\nof a minimiser of the ratioµG(A2)\nµG(A)forAranging through measurable subsets with\nµG(A) =α?\nConcrete groups Gof particular interest are SO 3(R), SL2(R). Note howeverthat\nsuch minimizers never exist in the Heisenberg group H3(R) [42].\n8.3.Babai’s conjecture for compact Lie groups. Babai’s conjecture is an\ninfluential open statement that predicts the rate of growth in finit e simple groups\n[1]. It has served as motivation for many formidable works related to product\ntheorems, expansion and random walks. Large parts of Babai’s con jecture are\nnow known, mostly focusing on groups with bounded rank. For unbo unded rank,\nBabai’s conjecture is known for random generators picked accord ing to a measure\nwith support of exponential size, see the introduction of [16] and r eferences therein.\nInarecentpaperEllis, Kindler,LifshitzandMinzer[17]putforwarda continuous\nanalogue to Babai’s conjecture in the unbounded rank setting:\nConjecture 8.4. LetGbe a compact simple group and rthe dimension of one\nof its maximal tori. Let A⊂Gbe a measurable set. We have Am=Gfor\nm=O(logµG(A)\nrG)where the implied constants do not depend on G.\nThey proved in [17] the conjecture for sets of at least exponentia l size. Our work\nprovides a counterpart for sets of ”infinitesimal” size:\nProposition 8.5. LetGbe a compact simple Lie group. There is ǫ(G)>0such\nthat for all A⊂Gwe haveµG(Am)≥ǫ(G)form≃logµG(A)−1\ndG−dH.\nIn combination with results from [17], one would need to provide the es timate\nǫ(G)≥ecrGfor somec>0 to prove Conjecture 8.4. In particular, it would follow\nfrom a positive answer to the following problem:\nProblem 8.6. Letc >0is thereC >0such that if A⊂Gis a measurable\nexp(c(dG−dH))-approximate subgroup of measure at least exp(−C(dG−dH)), then\nAis contained in H1\n2for some proper closed subgroup H?\nSuch a result would be optimal as considering H1/2shows. Any progress on this\nproblem would be significant and need novel ideas. Refinements of ou r method\nof proof of Proposition 1.6 using mutliscale analysis - in the spirit of de S axc´ e’s\nproduct theorem [13] - suggest that one could answer the weaker problem where\ndG−dHis replaced with ( dG−dH)O(1)with theO(1) constant being potentially\nlarge. This makes the dimensional constants appearing in our work c omputable, in\nprinciple.\n8.4.Local Brunn–Minkowski improved. The local Brunn–Minkowski inequal-\nity (Theorem 1.5) and the relevant stability result (Proposition 6.4) p roved in this\npaper are used as tools in the proof of our two main theorems. Thes e results are\nnonetheless interesting in and of themselves and we believe they cou ld be improved\nsubstantially.40 MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS\nFirst of all, Proposition 6.4 could be turned into a quantitative statem ent. Let\nA⊂R(e,δ,ρ), to establish quantitative results, one could replace the study of\ndensity functions with the study of functions of the form 1A∗1R(e,δα,ρα)for 1>\nα>0. These maps also satisfy functional equalities reminiscent of Lemm a 6.6 and\nrelated to [51], but their study becomes more technical and tedious . One could\nthen obtain a quantitative stability result by following the same strat egy as above.\nIn another direction, the non-degeneracy assumption from Proposition 6.4 could\nbe removed. This could perhaps be achieved by adopting another wa y of rescaling\n(i.e. changing the rescaling map φin§6.1). Another way one could do so is sug-\ngested by considering the John ellipsoid of the convex set provided b y Carolino’s\n”finite + convex” decomposition. The Lie model φobtained mimicking section\n§6.1 would most probably become nilpotent - hence less tractable. Usin g as addi-\ntional input McCrudden’s work on the (lack of) equality case for Bru nn–Minkowski\ninequality in nilpotent Lie groups [42] might be a key to conclude.\n8.5.Quantitative and asymmetric stability. By utilising the quantitative lo-\ncalstabilityapproachmentionedabove,onecouldpotentiallyshowt hat inTheorem\n1.4δ1=C(dG)αcfor some constant 0 <c<1 depending on G. When comparing\nwith stability results in Euclidean spaces, this raises the following ques tion:\nProblem 8.7. Can one obtain a dependency δ1=C(dG)αcwithcindependent of\nGin Theorem 1.4?\nBut the most obvious open problem remains the asymmetric stability r esult:\nProblem 8.8. Is it possible to prove a stability result for two subsets A,B⊂G\nsuch thatµG(AB)1\ndG−dH≤(1+ǫ)/parenleftBig\nµG(A)1\ndG−dH+µG(B)1\ndG−dH/parenrightBigdG\n?\nThe only part of our method that would have to be replaced is Lemma 6 .8.\nOne would be tempted to adapt proofs of stability from the Euclidean spaces case\n[20, 21], but some non-trivial challenges arise in such attempts.\n8.6.The use of ultrafilters. Seeing that our method indicates what seems to be\na clear path to quantitative stability results, one could be led to wond er why we\ndecided to use ultrafilters in our proof. The reason is threefold.\nFirst of all, ultrafilters allow to streamline the proof and several ste ps become\nrelatively simple consequences of general results from the theory of approximate\nsubgroups. Secondly, the ultrafilter method allows for more versa tility. It allows\nus to reduce the stability problem to an equality problem in some other Lie group,\nLsay. When L=Rnthis might appear to be a small simplification. But different\nstability results would require to deal with other L’s. See the third paragraph of\n§8.4 where we would require Lto be nilpotent. Finally, it seems that the ideas\ndeveloped in §6.3 show great promise in establishing existence of minimisers for the\ndoubling ratio in vast generality.\nAppendix A.Subgroup of maximal dimension\nWe collectin this appendix the Lie algebras gofcompactsimple Lie groupsalong\nwith the Lie algebra hMof the proper closed subgroup of maximal dimension. This\ntable follows from the classification of simple Lie algebras and is copied f rom [2,\nTable 2.1] which is based on [41, §4].MINIMAL DOUBLING FOR SMALL SUBSETS IN COMPACT LIE GROUPS 41\ng dimgdimg/hMhM\nar(r/\\e}a⊔io\\slash= 3)r(r+2)2r R×ar−1\nbr(r≥3)r(2r+1)2r dr−1\ncr(r≥2)r(2r+1)4(r−1)a1×cr−1\ndr(r≥3)r(2r−1)2r−1br−1\ne6 78 26 f4\ne7 133 54 R×e6\ne8 248 112 a1×e7\nf4 52 16 b4\ng2 14 6 a2\nReferences\n[1] Laszlo Babai. 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ISSN 0001-5962. doi:\n10.1007/BF02394607."    },    {        "title": "2401.14084v1.Can_radial_temperature_profiles_be_inferred_using_NH3__1__1__and__2__2__observations_.pdf",        "content": "MNRAS 000, 1–15 (2023) Preprint 26 January 2024 Compiled using MNRAS L ATEX style file v3.0\nCan radial temperature profiles be inferred using NH 3(1, 1) and (2, 2)\nobservations?\nRobert Estalella,1,2,3⋆Aina Palau,4and Gemma Busquet1,2,3\n1Departament de Física Quàntica i Astrofísica (FQA), Universitat de Barcelona (UB), Martí i Franquès 1, 08028 Barcelona, Spain\n2Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), Martí i Franquès 1, 08028 Barcelona, Spain\n3Institut d’Estudis Espacials de Catalunya (IEEC), Gran Capità, 2-4, 08034 Barcelona, Spain\n4Instituto de Radioastronomía y Astrofísica, Universidad Nacional Autónoma de México, Antigua Carretera a Pátzcuaro 8701, Ex-Hda. San José de la Huerta,\n58089 Morelia, Michoacán, México\nAccepted 2024 January 18. Received 2024 January 11 ; in original form 2023 March 08\nABSTRACT\nA number of works infer radial temperature profiles of envelopes surrounding young stellar objects using several rotational\ntransitions in a pixel-by-pixel or azimuthally-averaged basis. However, in many cases the assumption that the rotational tem-\nperature is constant along the line of sight is made, while this is not the case when a partially resolved envelope, assumed to be\nspherically symmetric, is used to obtain values of temperature for di fferent projected radii. This kind of analysis (homogeneous\nanalysis) is intrinsically inconsistent. By using a spherical envelope model to interpret NH 3(1,1) and (2,2) observations, we\ntested how robust it is to infer radial temperature profiles of an envelope. The temperature and density of the model envelope are\npower laws of radius, but the density can be flat for an inner central part. The homogeneous analysis was applied to obtain radial\ntemperature profiles, and resulted that for small projected radii, where the optical depth of the lines is high, the homogeneous\ntemperature can be much higher than the actual envelope temperature. In general, for larger projected radii, both the temperature\nand the temperature power-law index can be underestimated by as much as 40%, and 0.15, respectively. We applied this study to\nthe infrared dark cloud G14.225 −0.506 for which the radial temperature profile was previously derived from the dust emission\nat submillimeter wavelengths and the spectral energy distribution. As expected, the homogeneous analysis underestimated both\nthe temperature and the temperature power-law index.\nKey words: ISM: clouds – ISM: molecules – radiative transfer – techniques: spectroscopic\n1 INTRODUCTION\nCharacterizing the envelopes surrounding young stellar objects is\nimportant to correctly infer their properties and compare them to\ntheoretical models. A typical assumption used to describe these en-\nvelopes is to consider that their density and temperature decrease\nwith radius following power-laws. This has widely been determined\nfrom observations (e.g., Beuther et al. 2002; Mueller et al. 2002;\nHatchell & van der Tak 2003; Williams et al. 2005; Butler & Tan\n2012; Giannetti et al. 2013; Palau et al. 2014, 2021; Gieser et al.\n2021) and is also predicted from theoretical models (e.g., Larson\n1969; Shu 1977; Gómez et al. 2021).\nThe density and temperature structure of the envelopes surround-\ning young stellar objects have been determined in many cases us-\ning the radial intensity profiles from the continuum emission. Even\nthough it is widely known that molecular clouds are largely filamen-\ntary, it is also true that the most active star-forming sites are located\nat the centers of converging filaments, giving rise to the so-called\nhub-filament systems (e.g., Dewangan et al. 2020; Wang et al. 2020;\nKumar et al. 2022; Liu et al. 2023). It has been shown that the hubs\nare reasonably reproduced by models of spherical symmetry, such\n⋆E-mail: robert.estalella@gmail.comas those studied in Beuther et al. (2023). Actually, Kainulainen et al.\n(2013) find that fragmentation of molecular clouds at spatial scales\n0.5–10 pc is in agreement with the fragmentation expected for a self-\ngravitating cylinder, and therefore a filamentary structure, while at\nscales<0.5 pc the fragmentation is in agreement with spherical\nJeans fragmentation. In addition, it has been shown that assuming\nspherical geometry is reasonable for cores embedded within fila-\nmentary structures for aspect ratios up to 2 (Roy et al. 2014). Thus,\neven in the era of filamentary molecular clouds, assuming spherical\nsymmetry to model the hubs where active star formation is taking\nplace seems a reasonable first approximation.\nIn the case of spherical symmetry, and if one assumes optically\nthin emission, the Rayleigh-Jeans approximation, and an envelope\nwith infinite radius, the intensity as function of the projected dis-\ntance to the source, b, follows a power-law with index 1 −(p+q),\nwhere pandqare the power-law indices of the density and tempera-\nture (e.g., Beltrán et al. 2002; Beuther et al. 2002; Palau et al. 2014).\nTherefore, given an observed radial intensity profile, one needs to\nmake a specific assumption about either the density or the tempera-\nture to infer the other.\nMolecular line observations, including transitions of NH 3, H2CO,\nCH 3CN, HCOOCH 3, and complex organic molecules, of partially\nresolved envelopes have been used to derive the temperature radial\nprofile of the envelope (e.g., Ahmadi et al. 2018; Beltrán et al. 2018;\n©2023 The AuthorsarXiv:2401.14084v1  [astro-ph.GA]  25 Jan 20242 R. Estalella, A. Palau, &G. Busquet\nFigure 1. Example of NH 3(J,K)=(1,1) (top) and (2 ,2) (bottom) lines in\nG34.43 +0.24. The (1 ,1) transition shows the main and four (two inner and\ntwo outer) satellite quadrupole hyperfine lines, while the satellite lines of the\n(2,2) transition are too faint to be observed.\nGieser et al. 2019, 2021; Busch et al. 2022). The standard procedure\nto infer the temperature from rotational diagrams, for instance, of\nNH 3, assumes that, along the line of sight, the medium is homoge-\nneous and in particular that the rotational temperature is constant.\nHowever, the derived values of Trotfor di fferent projected radii are\ninterpreted as the radial profile of temperature of the envelope. This\nkind of analysis is inconsistent because the gas is assumed to be\nhomogeneous along the line of sight and, at the same time, inhomo-\ngeneous in the radial direction.\nIn this paper, we aim at studying whether the determination of the\nradial temperature profiles from the NH 3(1,1) and (2,2) data are a\nreasonable approach to the actual temperature profile. Since the col-\nlisional coe fficients for the quadrupole-hyperfine transitions of NH 3\nare not available in the literature, the existing numerical radiative\ntransfer models are of limited usefulness. Thus, we developed a nu-\nmerical radiative transfer calculation for the NH 3(1,1) and (2,2)\nhyperfines, which is described in Appendix A. A comparison with\na full radiative transfer calculation is also included. The structure\nof the paper is as follows: in Section 2 we present the basics of\nthe standard NH 3analysis, in Section 3 we present the limitations\nof this analysis, in Section 4 the spherical model used here is de-\nscribed, in Section 5 the optically thin case is considered, in Section\n6 a radiative transfer calculation is used for di fferent parameters of\nthe envelope, in Section 7 the results found here are applied to the\nG14.225−0.506 case and in Section 8 we summarize our main con-\nclusions.\n2 NH 3(J,K)=(1,1)AND (2,2)ANALYSIS BASICS\nThe intensity of the NH 3(J,K)=(1,1) and (2,2) lines can be ex-\npressed as\nTL(j,k)=ηB[Jν(Tex)−Jν(Tbg)](1−exp[−τ(j,k)]), (1)Table 1. Frequencies in temperature units, spontaneous emission coe ffi-\ncients, and optical depth ratio of the main (m), inner satellite (s) of NH 3\n(1,1) and (2,2) to the total optical depth of the transition (Osorio et al. 2009;\nMangum & Shirley 2015).\nhνjk/k A jk Rjkm= Rjks=\nLine (K) (10−7s−1)τ(j,k; m)/τ(j,k)τ(j,k; s)/τ(j,k)\n(1,1) 1.13716 1.66838 1 /2 5 /36=0.1389\n(2,2) 1.13850 2.23246 43 /54=0.7963 7/135=0.0519\nwhereηBis the beam filling-factor, Jν(T) is the Plank correction to\nthe Rayleigh-Jeans approximation,\nJν(T)=hνjk/k\nexp(hνjk/kT)−1, (2)\nνjkis the frequency of the transition, Texis the excitation temper-\nature, Tbgis the background temperature, and τ(j,k) is the optical\ndepth of the transition. Since we are dealing with the observation of\nan angularly resolved envelope, in the following we will consider\nthe beam filling factor ηB=1.\nThe standard procedure to analyze observations of the (1 ,1) and\n(2,2) inversion transitions of ammonia consists in fitting the in-\ntensity of the main and inner-satellite quadrupole-hyperfines of the\n(1,1) line (see Fig 1), TL(1,1; m) and TL(1,1; s). Since the op-\ntical depth ratio of the main and inner satellite lines is known\nRjkm/Rjks=18/5=3.6 (see Table 1), the optical depth of the main\nline,τ(1,1; m) is obtained from the equation\nTL(1,1; m)\nTL(1,1; s)=1−exp[−τ(1,1; m)]\n1−exp[−τ(1,1; m)/3.6]. (3)\nThis approach is valid if the magnetic hyperfines are blended, that\nis, the hyperfines line-width is large enough so that the quadrupole\nhyperfines profiles show no substructure and are nearly Gaussian.\nA more robust method is to fit the magnetic hyperfine structure\nof the (1,1) line using a procedure like CLASS (NH 3method). The\noutputs of the fit include the intensity and optical depth of the main\nline, TL(1,1; m) andτ(1,1; m). For the (2 ,2) line, the satellite lines\nare usually unobservable (see Fig. 1), and a single Gaussian is fitted\nto the main line, giving the main line intensity, TL(2,2; m). Since the\nexcitation temperature of the (1 ,1) and (2,2) transitions are assumed\nto be the same, and the frequencies of the two transitions are very\nclose, the intensity ratio can be used to derive the optical depth of\nthe (2,2) main line, from the relation\n1−exp[−τ(2,2; m)] =TL(2,2; m)\nTL(1,1; m)\u00001−exp[−τ(1,1; m)]\u0001.(4)\nAlternatively, the magnetic hyperfine structure of the (1 ,1) and\n(2,2) lines can be fitted simultaneously using a procedure like HfS\n(Estalella 2017), and the optical depth of the (2 ,2) main line is ob-\ntained directly, with the same assumption of equal excitation temper-\nature for the (1 ,1) and (2,2) lines (see for instance Sepúlveda et al.\n2020).\nOnce the opacities are known, Eq. 1 is used to derive the excitation\ntemperature Tex. This temperature is not well determined since it\ndepends on the correct calibration of telescope output, and on the\ncoupling of the telescope beam with the source (the beam filling\nfactor). The usual assumption is to take ηB=1. However, in the\nRayleigh-Jeans limit and the optically thin case, the NH 3column\ndensity can be derived independently of the value of Tex(see Section\n5).\nFrom these data, the column density of ammonia molecules in the\nMNRAS 000, 1–15 (2023)NH 3spherical model 3\nTable 2. Degeneracies and energies above the (1 ,1) level of the lower\nmetastables levels of NH 3(Mangum & Shirley 2015).\n(Ejk−E11)/k\n(J,K)gjk (K)\n(0,0) 4−22.64\n(1,1) 12 0 .00\n(2,2) 20 40 .99\n(3,3) 56 99 .76\n(1,1) and (2,2) levels, N11andN22, can be derived (see for instance,\nEstalella 2017),\nNjk=r\nπ\n4 ln 28πν3\njk\nc3Ajkexp(hνjk/kTex)+1\nexp(hνjk/kTex)−1τ(j,k; m)\nRjkm∆v, (5)\nwhere Ajkis the spontaneous emission coe fficient (Table 1), Rjkm=\nτ(j,k; m)/τjk, is the ratio of main line to total optical depth, given in\nTable 1, and ∆vis the full-width at half-maximum of the hyperfine\nlines.\nTo obtain the rotational temperature, Trot, and the total NH 3col-\numn density, N(NH 3), it is necessary to determine the fraction, fjk,\nof ammonia molecules that are in the ( J,K) rotational level,\nnjk=fjkn(NH 3), (6)\nfrom which the total hydrogen gas density is derived adopting an\nammonia fractional abundance, X, assumed to be constant,\nn(NH 3)=X n(H2). (7)\nThe usual assumption for moderate temperatures is that only the\nmetastable levels ( J=K) of ammonia are populated, and that they\nare populated according to a unique excitation temperature, called\nthe rotational temperature Trot. The rotational temperature is given\nby the Boltzmann equation,\nn22\nn11=g22\ng11exp[−(E22−E11)/kTrot], (8)\nwheregjkis the degeneracy and Ejkthe energy of the level ( J,K) (see\nTable 2). The latter equation is used routinely to derive the rotational\ntemperature by assuming that N22/N11≃n22/n11,\nTrot=(E22−E11)/k\nln g22\ng11N11\nN22!. (9)\nThe rotational temperature can also been expressed as a function of\nthe optical depths of the (1 ,1) and (2,2) main lines by using Eq. 5,\nTrot=(E22−E11)/k\nln A22R22mg22\nA11R11mg11τ(1,1; m)\nτ(2,2; m)!, (10)\nresulting, in practical units, in\n\u0014Trot\nK\u0015\n=40.99\nln \n3.544τ(1,1; m)\nτ(2,2; m)!. (11)\nThe rotational temperature is a good approximation to the kinetic\ntemperature if the radiative transitions up the K-ladder are negligi-\nble, and only collisional transitions between the metastable levels\noccur. The rotational temperature can be corrected to estimate the\nkinetic temperature (Tafalla et al. 2004). An improved version of the\ncorrection is given in Estalella (2017),\nTrot=Tk\n1+Tk\n40.99ln\u00021+0.73 exp(−16.26/Tk)\u0003. (12)\n00.20.40.6nJK/n(NH3)0.448\n0.264\n5/23 = 0.217\n3/23 = 0.130\n23.9 K 73.0 K11\n22\n0 20 40 60 80 100\nTrot (K)00.51n22/n115/3 = 1.333Figure 2. Fractional abundance of NH 3(1,1) (blue) and (2 ,2) (red) (top)\nand ratio n22/n11(bottom) as a function of rotational temperature.\nThe last equation has to be solved iteratively to obtain Tk.\nIn order to obtain the total NH 3column density, we can con-\nsider that for moderate values of Trot(<∼100 K), only the levels\n(J,K)=(0,0), (1,1), (2,2), and (3,3) are populated, so that the\npartition function Q(Trot) is\nQ(Trot)=3,3X\njk=0,0gjke−(Ejk)/kTrot. (13)\nThe fractional abundances of the (1 ,1) and (2,2) levels, fjk=\nnjk/n(NH 3), are given by\nf11=g11\nQe−(E11/kTrot)=g11\n3,3X\njk=0,0gjke−(Ejk−E11)/kTrot, (14)\nf22=g22\nQe−(E22/kTrot)=g22\ng11e−(E22−E11)/kTrotf11. (15)\nThe maximum of f11(f11≃0.448) occurs for Trot≃23.9 K, and\nfor high temperatures f11→3/23=0.130. The maximum of f22\n(f22≃0.264) occurs for Trot≃73.0 K, and for high temperatures\nf22→5/23=0.217 (Fig. 2, top). The ratio f22/f11=n22/n11is a\nsimple function of Trot(Eq. 15) and is shown in Fig. 2 (bottom). It is\nan increasing function of Trot, and for high temperatures f22/f11→\n5/3=1.333.\nOnce Trotis known, the fractional abundance f11can be calcu-\nlated, and we further assume that\nN11\nN(NH 3)≃n11\nn(NH 3)=f11, (16)\nMNRAS 000, 1–15 (2023)4 R. Estalella, A. Palau, &G. Busquet\nso the the hydrogen column density is\nN(H2)=N11\nX f11. (17)\n3 LIMITATIONS OF THE STANDARD NH 3ANALYSIS\nThere is an implicit assumption in the approximations made to de-\nrive the expressions derived so far: that the gas is homogeneous and,\nin particular, that the rotational temperature is constant along the line\nof sight. This has no importance as long as the clump of gas analyzed\nis assumed to be homogeneous. However, this is not the case when\nwe analyze ammonia data of a partially resolved envelope, assumed\nto be spherically symmetric, and use Eq. 9 to obtain values of Trot\nfor di fferent projected radii (the radial profile of temperature of the\nenvelope). This kind of analysis is intrinsically inconsistent. On the\none hand, Eq. 9 assumes that the gas is homogeneous along the line\nof sight, and on the other hand, these results are used to infer a radial\ndependence of Trot, implying that Trotis not constant along the line\nof sight.\nIn addition, the assumption of homogeneity along the line of sight\ncan be incompatible with the observed intensities of the lines when\nobserving a region with strong gradients of density or temperature\nalong the line of sight. Let us examine the three equations used to\nderiveτ(1,1; m) (Eq. 3), τ(2,2; m), (Eq. 4), and Trot(Eq. 11). For\nsimplicity, in this discussion we will call τ1m=τ(1,1; m),τ2m=\nτ(2,2; m), T1m=TL(1,1; m), T1s=TL(1,1; s), T2m=TL(2,2; m),\nandr21the ratio of (2 ,2) to (1,1) intensities, r21=T2m/T1m.\nLet us call r11the intensity ratio of main to satellite (1 ,1) lines,\nr11=T1m/T1s. Equation 3 used to derive the optical depth of the\n(1,1) line requires that\n1<r11<R11m\nR11s=3.6. (18)\nForr11→1 we haveτ1m→+∞, while for r11=3.6, the optical\ndepth is zero. If r11is less than 1, no value of the optical depth τ1m\ncan be derived, and if it is greater than 3.6, Eq. 3 gives a negative\noptical depth.\nThe derivation of the rotational temperature (Eq. 11) requires that\nτ2m<aτ1m, (19)\nwhere a=(A22R22mg22)/(A11R11mg11)=3.544. The relation be-\ntween T1m,T2m,τ1m, andτ2m(Eq. 4) can be written as\n1−exp(−τ1m)\n1−exp(−τ2m)r21=1. (20)\nBy substituting Eq. 19, we obtain a condition on the intensity ratio\nr21andτ1m,\nrτ=1−exp(−τ1m)\n1−exp(−aτ1m)r21<1. (21)\nThe values of rτfor the limiting values of τ1marerτ→r21/afor\nτ1m→0, and rτ→r21forτ1m→∞ .\nIn order to fulfill Eq. 19, the intensity ratio r21and (1,1) optical\ndepthτ1mcan not be arbitrary. The intensity ratio must be r21<a=\n3.544. If r21<1 (that is, (2 ,2) intensity lower than (1 ,1)), the optical\ndepth can have any value. However, if r21>1,τ1mmust be below a\nmaximum value τmax, solution of the equation, for r21>1,\n1−exp(−τmax)\n1−exp(−aτmax)r21=1. (22)\nForr21→1, the solution of the equation gives τmax→∞ , while for\nr21→a, we haveτmax→0 (see Fig. 3).\n0 1 2 3 4\nr21 = T2m/T1m0.010.1110100τ1m\nSolution for Trotτmax\n3.544τ2m < τ1mτ2m > τ1m\nNo solution for TrotFigure 3. Plot ofτmax(Eq. 22), as a function of the intensity ratio r21=\nTL(2,2; m)/TL(1,1;m), for 1<r21<3.544 (see text). A value of Trotcan\nonly be obtained for r21<1, or for r21>1 andτ(1,1; m)< τ max(hatched\narea).\nIn summary, in order to obtain a value of the rotational tem-\nperature, the observational parameters, r11=T1m/T1s, and rτ=\nr21[1−exp(−τ1m)]/[1−exp(−aτ1m)], where r21=T2m/T1mand\na=3.544, must fulfill the two conditions:\n(1,1) line only: 1≤r11≤3.6 , and\n(1,1) and (2,2) lines: rτ≤1. This condition is fulfilled either if\nr21<1 or, if 1<r21<a, forτ1m< τ max, whereτmaxis the solution\nof Eq. 22.\nIn the following we will call the standard NH 3analysis, under\nthe assumption of gas homogeneity, homogeneous analysis , and the\nkinetic temperature derived from Eqs. 9 and 12, homogeneous tem-\nperature ,Thom\nk. We will estimate Thom\nk, in a consistent way, for a\nspherically symmetric envelope, and its dependence on the density\nand temperature structure of the envelope.\n4 SPHERICAL ENVELOPE MODEL\nLet us assume a spherically symmetric envelope, with a density that\nat large radii is a power-law of radius with index −p,\nn(r)=n0(r/r0)−p, (23)\nbut that may have a flat inner part, the so-called Plummer-like den-\nsity, with a characteristic radius rc,\nn(r)=nch\n1+(r/rc)2i−p/2. (24)\nThe central density, at r=0, is nc. For small radii, r≪rc, the\ndensity is flat, n≃nc, while for large radii, r≫rc, the density\nfollows a power law, n≃nc(r/rc)−p. We can consider the Plummer-\nlike density as a generalization of the power-law density. This can\nbe seen by writing the Plummer-like function as\nn(r)=n0h\n(rc/r0)2+(r/r0)2i−p/2. (25)\nThis expression of the Plummer-like function is useful because for\nrc=0 we obtain directly a power-law function, with the reference\nMNRAS 000, 1–15 (2023)NH 3spherical model 5\nlog rlog n\nlog rc1log nc1\nlog n0\nlog r0log nc2\nlog rc2rc < r0, nc > n0\nrc > r0, nc < n0n=n0(r/r0)-p\nFigure 4. Plummer-like density as a function of radius for the same values\nofr0,n0, and p, and two di fferent values of rc, one lower (red lines) and the\nother higher (blue lines) than r0. The red and blue dots indicate the actual\ndensity of the envelope at r0,n(r0), while the black dot indicates the density\nparameter n0(see text).\nradius r0and the density parameter n0well defined, while on the\ncontrary, the central density nc→∞ . In the small and large radii\nlimits we obtain\n(r≪rc) n(r)≃n0(rc/r0)−p,\n(r≫rc) n(r)≃n0(r/r0)−p. (26)\nThus, the central density ncandn0are related through\nnc=n0(rc/r0)−p. (27)\nThe density parameter n0is not an actual density. It can be inter-\npreted as the density at the reference radius r0of the extrapolation\nof the Plummer-like function for r≫rc(see Fig. 4). In the case\nrc>r0,n0is even higher than the maximum density of the envelope,\nnc.\nWe consider that the Plummer-like law for the density is valid up\nto the envelope radius, taken as the radius for which the density is a\ngiven minimum value, nmin, the ambient gas density.\nRegarding the temperature of the envelope, we consider a a power\nlaw of the radius, with a power-law index −q,\nT(r)=T0(r/r0)−q, (28)\nwhere T0is the temperature at the reference radius r0. The gas tem-\nperature has also a minimum value, Tmin, which corresponds to a\nradius such that for larger radii the envelope temperature is taken\nconstant, equal to Tmin. The temperature power-law index for a cen-\ntrally heated envelope, with the dust and the gas well coupled, de-\npends on the dust opacity index β,q=2/(4+β). For the usual\nvalues ofβ, 0< β < 4, the temperature power-law index results in\n0.25<q<0.5.\nIn the following we will derive the homogeneous temperature,\nThom\nk, for a spherical envelope as a function of the projected radius,\nb. Two di fferent approximations will be considered:\n(i) Rough approximation, optically thin case, power-law density.\n(ii) Radiative transfer calculation, with an excitation temperature\ngiven by the two-levels model, Plummer-like density.\nz axisz\nobserver\nrdrdz\nbb axisFigure 5. Geometry of a spherically-symmetric envelope.\n5 OPTICALLY THIN CASE, POWER-LA W DENSITY\nLet us assume that the emission is optically thin, that the Rayleigh-\nJeans approximation is valid, that Tex≫Tbg, and that Trot≃Tk. In\nthis case, the intensities of the (1 ,1) and (2,2) main lines are given\nby\nTL(j,k; m)=Texτ(j,k; m). (29)\nEquation 5 giving the column density as a function of the optical\ndepth, in the optically thin case, can be simplified to\nNjk=r\nπ\n4 ln 216πkν2\njk\nhc3AjkRjkmTexτ(j,k; m)∆v, (30)\nso that, in the optically thin case, the column densities are propor-\ntional to the line intensities, and their ratio can be obtained directly\nfrom the ratio of line intensities,\nN11\nN22=A22R22m\nA11R11mTL(1,1; m)\nTL(2,2; m). (31)\nThus, the homogeneous temperature can be simply derived from the\nratio of intensities of the (1 ,1) and (2,2) lines. From Eqs. 31 and 9,\nwe have\n\"Thom\nk\nK#\n=40.99\nln \n3.544T(1,1; m)\nT(2,2; m)!. (32)\nLet us estimate, for a given projected radius b, the column densi-\ntiesN11andN22,\nNjk(b)=Z\nline of sightnjk(r)dz, (33)\nwhere dzis the elementary length along the line of sight, and\nr=√\nb2+z2(see Fig. 5). The function that is integrated, the num-\nber density njk(r), depends on the fractional abundance fjk, which\ndepends on the envelope temperature (Eq. 14), and the envelope den-\nsity,n(r),\nnjk(r)=fjk(T(r))X n(r). (34)\nIf we assume that the density is given by a simple power law, the\nmaximum of njkoccurs at the minimum value of r,rmin=b, with a\nvalue njk(b), and we can roughly approximate the integral along the\nMNRAS 000, 1–15 (2023)6 R. Estalella, A. Palau, &G. Busquet\nline of sight by\nZ\nline of sightnjk(r)dz≃njk(b) 2b. (35)\nThus, with this approximation, the column density ratio for a pro-\njected radius bequals the fractional density radio for r=b,\nN11(b)\nN22(b)=n11(r=b)\nn22(r=b). (36)\nSince the envelope temperature gives the fractional density ratio\nf11/f22(Eq. 15), and the same expression for the column density\nratio gives the homogeneous temperature (Eq. 9), both temperatures\ncoincide.\nIn conclusion, in the optically thin case and with the rough ap-\nproximation made for the integral along the line of sight, and for a\npower-law density, the homogeneous temperature coincides with the\nenvelope temperature,\nThom\nk(b)=T(b)=T0(b/r0)−q. (37)\nHowever, the NH 3(1,1) emission is in many cases partially thick,\nand the optically thin approximation does not hold. In addition, even\nfor a power-law density, the maximum NH 3(1,1) volume density\nn11is not necessarily found at the minimum distance of the line of\nsight to the envelope center, because for temperatures greater than\n∼20 K, the density n11decreases with temperature (Fig. 2), and the\napproximation used to estimate the column density does not hold.\n6 RADIATIVE TRANSFER CALCULATION\nA radiative transfer code was developed to derive the intensities of\nthe NH 3(1,1; m), (1,1; s), and (2 ,2; m) lines for a spherical enve-\nlope, and calculate the homogeneous optical depths τhom(1,1; s) and\nτhom(2,2; s), and the homogeneous temperature Thom\nkderived from\nthe homogeneous analysis. The description of the method is given\nin Appendix A.\n6.1 Results\nA summary of results for di fferent envelope parameters is shown in\nFigs. 6 and 7. In Fig. 6 we show the actual temperature profile and\nthe homogeneous temperature profile obtained for envelopes with\npower-law density profiles ( rc=0), and in Fig. 7 with Plummer-\nlike density profiles ( rc>0). The parameters of the envelopes, at a\ndistance of 1 kpc, are the same for all the panels of the figures, unless\nstated otherwise, reference radius r0=1 kau, density parameter n0=\n107cm−3, density power-law index p=2, temperature at r0,T0=50\nK, temperature power-law index q=0.5, line width ∆v=0.5 km s−1,\nminimum density nmin=103cm−3, minimum temperature Tmin=8\nK, and NH 3abundance X=10−8. For each panel of Figs. 6 and 7 a\nsingle parameter of the envelope, indicated in the panel, is changed.\nIn Fig. 6 (power-law density, rc=0), the first panel shows the\nresults for di fferent densities, n0=106cm−3(resulting renv=0.15\npc), 107cm−3(renv=0.5 pc) and 108cm−3(renv=1.5 pc). The\nsecond panel shows di fferent values of the density power-law index,\np=1.5 (resulting renv=2.2 pc), p=2 (renv=0.5 pc) and p=\n2.5 (renv=0.08 pc). The third panel shows di fferent values of the\nenvelope temperature, T0=20, 50, and 100 K. The fourth panel,\ndifferent values of the temperature power-law index, q=0.25, 0.33,\nand 0.5. Finally, the bottom panels shows di fferent values of the line\nwidth, ∆v=0.2, 0.5, 1 km s−1. For the three bottom panels the\nresulting envelope radius is the same, renv=0.5 pc.\n10100T (K)log n0=6\nlog n0=7\nlog n0=8\n10100T (K)p=1.5\np=2\np=2.5\n10100T (K)T0=20\nT0=50\nT0=100\n10100T (K)q=0.25\nq=0.33\nq=0.5\n1 10\nb (kau)10100T (K)∆v=0.2\n∆v=0.5\n∆v=1Figure 6. Radial profiles of the homogeneous temperature Thom\nk(solid lines)\nfor di fferent parameters of an envelope with a power-law density profile. The\ndashed lines indicate the radial profiles of the actual kinetic temperature of\nthe envelope. For all panels, except for the parameter indicated in each panel,\nthe envelope is at a distance of 1 kpc, with n0=107cm−3,p=2,rc=0,\nT0=50 K, q=0.5, and ∆v=0.5 km s−1.Top panel: Envelope densities\nn0=106, 107, and 108cm−3.Second panel: Density power-law indices\np=1.5, 2, and 2.5. Third panel: Envelope temperatures T0=20, 50, and\n100 K. Fourth column: Temperature power-law indices q=0.25, 0.33, and\n0.5.Bottom column: Line widths ∆v=0.2, 0.5, and 1 km s−1.\nMNRAS 000, 1–15 (2023)NH 3spherical model 7\n10100T (K)\nrc=0,   log nc=∞\nrc=1,   log nc=7\nrc=3,   log nc=6\nrc=10, log nc=5\nrc=30, log nc=4\n1 10\nb (kau)10100T (K)\nrc=0,   log10 nc=∞\nrc=1,   log10 nc=8\nrc=3,   log10 nc=7\nrc=10, log10 nc=6\nrc=30, log10 nc=5log10 n0 = 7\nlog10 n0 = 8\nFigure 7. Radial profiles of the homogeneous temperature Thom\nk(solid lines)\nfor an envelope with a Plummer-like density profile with di fferent values of\nthe radius of the inner flat region, rc=0, 1, 3, 10, and 30 kau. The top panel\nis for a density n0=107cm−3, and the bottom panel for n0=108cm−3.\nThe dashed lines indicate the radial profiles of the actual kinetic temperature\nof the envelope. For both panels the envelope is at a distance of 1 kpc, with\np=2,T0=50 K, q=0.5, and ∆v=0.5 km s−1. The values of rc(kau) and\nthe resulting central densities nc(cm−3) are indicated in each panel. In the\nbottom panel, the profiles for rc=0, 1, and 3 are nearly coincident.\nIn Fig. 7 we show the results for Plummer-like density profiles for\ndifferent values of the radius of the inner flat region, rc=0, 1, 3, 10,\nand 30 kau. The top panel is for a density parameter n0=107cm−3,\ngiving an envelope radius renv≃0.5 pc, and the bottom panel is for\nn0=108cm−3, corresponding to renv≃1.5 pc. The central densities,\nnc, are indicated for each value of rc.\nAs can be seen in Figs. 6 and 7, in some cases for most sets of\nparameters there is a critical value of bbelow which no value of\nThom\nkcan be calculated. This happens when for small projected radii\nthe lines become optically thick, and some of the conditions of Eqs.\n18, 19, and 21 are not fulfilled, making impossible the calculation\nofThom\nk, or of both τ(2,2; m) and Thom\nk, or even of all three param-\netersτ(1,1; m),τ(2,2; m), and Thom\nk. Near the critical value Thom\nk\nincreases sharply with decreasing projected radius. This behaviour,\nas explained above, is an optical depth e ffect. For small projected\nradii the optical depth of the lines is high. However, the (2 ,2) line\nis optically thinner than the (1 ,1), and thus it has more contribution\nfrom the the inner, hotter part of the envelope. This increase of the\nintensity of the (2 ,2) line is interpreted by the homogeneous analysis\nas a higher Thom\nk.\nFar from the critical value, and for projected radii larger that the\nradius of the inner flat region, the homogeneous temperature is a fair\nestimate of the envelope radial profile of temperature, although it\nsystematically underestimates the envelope temperature. A possible\nexplanation is that for optically thin (1 ,1) and (2,2) lines, the mate-\n-50%-40%-30%-20%-10%0%(T0h-T0)/T0\nlog n0=6\nlog n0=7\nlog n0=8-0.15-0.10-0.050.00\nqh-q\n-50%-40%-30%-20%-10%(T0h-T0)/T0\np=1.5\np=2\np=2.5-0.15-0.10-0.050.00\nqh-q\n-50%-40%-30%-20%-10%(T0h-T0)/T0\nT0=50\nT0=80\nT0=100-0.15-0.10-0.050.00\nqh-q\n-50%-40%-30%-20%-10%(T0h-T0)/T0\nq=0.25\nq=0.33\nq=0.5-0.15-0.10-0.050.00\nqh-q\n0.1 1 10\nrc (kau)-50%-40%-30%-20%-10%(T0h-T0)/T0\n∆v=1\n∆v=0.5\n∆v=0.2\n0.1 1 10 100\nrc (kau)-0.20-0.15-0.10-0.050.00\nqh-qFigure 8. Comparison of the power-law radial profile of the envelope tem-\nperature, T=T0(r/r0)−q, and of Thom\nkobtained from the homogeneous\nanalysis, for the projected radii for which it can be fitted by a power law,\nThom\nk≃T0h(b/r0)−qh. The results obtained are shown as a function of the\nradius of the inner flat region rc. Each rows has two panels. The left panel\nshows the normalized di fference of temperatures, ( T0h−T0)/T0, and the right\npanel, the di fference in power-law indices, qh−q. For each row, the three\nlines (black, red, and blue) correspond to di fferent values of the parameter\nindicated in the right panel of the row. The rest of parameters are the same as\nthose used in Figs. 6 and 7.\nrial along the line of sight contributes to the (2 ,2) emission only if\nits temperature is high enough, which occurs near the center of the\nenvelope, which is not the case for the (1 ,1) emission. This deficit\nof (2,2) emission is interpreted by the homogeneous analysis as a\nlower Thom\nk.\nHowever, a much more remarkable feature of the homogeneous\nanalysis is that for Plummer-like density profiles, the radial profile\nofThom\nkcan be much flatter than the temperature profile of the en-\nvelope. This is especially noticeably for projected radii less than the\nradius of the flat inner region, b≲rc(see Fig. 7). It appears that\nthe flattening of the density profile is erroneously interpreted by the\nhomogeneous analysis as a flattening of the temperature profile.\nIn order to see the e ffect of the radius of the inner flat region and\nthe rest of envelope parameters on the di fference between Thom\nkand\nthe envelope temperature, we calculated for several values of rcand\ndifferent physical parameters of the envelope the radial profile of\nThom\nkas a function of rc, and fitted a power law for the range of\nprojected radii where the profile could be approximated well by a\npower law,\nThom\nk≃T0h(b/r0)−qh, (38)\nMNRAS 000, 1–15 (2023)8 R. Estalella, A. Palau, &G. Busquet\nobtaining values of T0handqh. These were compared with the val-\nues of T0andqof the envelope. The results are shown in Fig. 8. The\nresults are shown as a function of rc, ranging from 0 to 30 kau. The\nranges of the physical parameters of the envelope examined were\n106cm−3≤n0≤108cm−3(top row of Fig. 8), 1 .5≤p≤2.5 (sec-\nond row), 50 K≤T≤100 K (third row), 0 .25≤q≤0.5 (fourth\ncolumn), and 0 .2≤∆v≤1 km s−1(bottom row). These ranges in-\nclude the typical values derived for massive dense cores and clumps\nin previous works (e.g., Beuther et al. 2002; Williams et al. 2005; Gi-\nannetti et al. 2013; Palau et al. 2014, 2021; Gieser et al. 2021). Two\npanels are shown for each row, the left panel with the normalized\ndifference of temperatures, ( T0h−T0)/T0, and the right panel with\nthe power-law indices di fference, qh−q. For each row the three lines\nand dots correspond to di fferent values of the parameter indicated in\nthe same row. As can be seen in the figure, for small values of rcthe\ndifference ( T0h−T0)/T0is typically−20% to−30%, and qh−qof the\norder of−0.05. However, these di fferences increase dramatically for\nhigher values of the inner flat region. The temperature di fference can\nbe as high as−40% to−50%, and the power-law index di fference can\nreach−0.10 or−0.15. The di fferences also increase with increasing\ndensity parameter n0, density power-law index p, and temperature\npower-law index q.\n6.2 E ffect of beam smoothing\nUp to now we have not considered the e ffect of beam smoothing on\nthe observed profiles. Beam smoothing will produce a flattening of\nthe observed radial profiles at the center of the envelope, in a region\nwith a size of the order of the beam size.\nIn order to model the e ffect of a finite beam size, the radial profiles\nof the line intensities TL(1,1; m), TL(1,1; s), and TL(2,2; m) were\nconverted to 2D-maps. The maps were then 2D-convolved with a\nGaussian beam pattern, and the resulting maps were ring-averaged\nto obtain the beam-smoothed radial profiles of the line intensities.\nFrom these, the homogeneous analysis was performed to obtain the\nradial profile of homogeneous temperature Thom\nk.\nIn Fig. 9 we show the results for an envelope with a power-law\ndensity ( rc=0), at a distance of 1 kpc, for di fferent beam sizes: no\nbeam smoothing (top panel), and half-power beam widths (HPBW)\nof 1′′, 3′′, 10′′, and 30′′(second to fifth panels). As can be seen in the\nfigure, the e ffect of beam smoothing is not noticeable for beam sizes\nless than 3′′. The reason is that for the parameters of density and\ntemperature of the envelope, the homogeneous temperature could\nnot be derived for projected radii less than ∼2′′, where the e ffect of\nbeam smoothing is important. However, the e ffect of beam smooth-\ning is very apparent for beam sizes of 10′′and 30′′, for which the\nradial profile of Thom\nkbecomes flat for projected radii less than ap-\nproximately the HPBW.\nThe result of beam smoothing for the same envelope, but with a\nPlummer-like density profile is shown in Fig. 10. In all three panels\nthe beam is the same, HPBW =3′′, but the radius of the inner flat\nregion is di fferent: rc=3 kau (top panel), 10 kau (center), and 30 kau\n(bottom). As can be seen, the discrepancy between the homogeneous\ntemperature and the envelope temperature becomes important for\nvalues of rclarger than the beam size.\n110100T (K)\n110100T (K)\n110100T (K)\n110100T (K)\n1 10\nb (kau)0.1110100T (K)beam = 0\"\n1\"\n3\"\n10\"\n30\"Figure 9. Effect of beam smoothing on the radial profiles of the line in-\ntensities TL(1,1; m) (red), TL(1,1; s) (orange), TL(2,2; m) (blue), and homo-\ngeneous temperature Thom\nk(black continuum line). The temperature of the\nenvelope is indicated with a black dashed line. The beam HPBW (in arcsec)\nis indicated in each panel, and the beam profiles, with an arbitrary height at\nthe origin of 100, are shown as green dotted lines. The parameters of the en-\nvelope are n0=107cm−3,p=2,T0=50 K, q=0.5, and ∆v=0.5 km s−1\nThe distance is 1 kpc. At this distance, 1 arcsec corresponds to a length of 1\nkau.\n7 APPLICATION TO G14.225 −0.506\n7.1 G14.2 Hub N and S\nA well-suited observational case to apply the study presented above\nis the infrared dark cloud G14.225 −0.506 (hereafter G14.2), located\nat a distance of 1.98 kpc (Xu et al. 2011). The cloud consists of two\nMNRAS 000, 1–15 (2023)NH 3spherical model 9\n110100T (K)\n110100T (K)\n1 10\nb (kau)110100T (K)rc = 3 kau\n10 kau\n30 kau\nFigure 10. Effect of beam smoothing for a Plummer-like density profile for\ndifferent values of the radius of te inner flat region, rc=0.3, 1, and 3 kau. The\nradial profiles of the line intensities are shown as a red line for TL(1,1; m), an\norange line for TL(1,1; s), and a blue line for TL(2,2; m). The homogeneous\ntemperature Thom\nkis shown as a black solid line, and the temperature of the\nenvelope is indicated with a black dashed line. The beam HPBW is 5′′for all\npanels and the beam profile, with an arbitrary height at the origin of 100, is\nshown as a green dotted line. The parameters of the envelope are n0=107\ncm−3,p=2,T0=50 K, q=0.5, and ∆v=0.5 km s−1The distance is 1 kpc.\nAt this distance, 1 arcsec corresponds to a length of 1 kau.\nmain hubs, called Hub N and Hub S, being the center of a num-\nber of filamentary structures. This cloud has been deeply studied in\nNH 3(1,1) and (2,2) using a combination of single-dish and interfer-\nometer. In particular, the cloud has been observed in NH 3(1,1) and\n(2,2) with high angular resolution and with no missing short spac-\nings, allowing to perform a reasonable study of its structure. A de-\ntailed description of the main observational results on G14.2 can be\nfound in Busquet et al. (2013), Busquet et al. (2016), Santos et al.\n(2016), Ohashi et al. (2016), and Chen et al. (2019).\n7.2 G14.2 data description\nThe NH 3(J,K)=(1,1) and (2,2) observations were carried out with\nthe Very Large Array and the E ffelsberg radio telescope. The data\nwere combined, resulting in a final synthesized beam of 8 .′′17×7.′′00,\nP.A.=−15.◦1, and maps with a pixel size of 2′′. A full description\nof the observations and data reduction can be found in Busquet et al.\n(2013).\nThe (1,1) line was observed with 63 channels 0.62 km s−1wide,\nwith a velocity range between 2.8 and 41.2 km s−1, encompassing\nthe main line, the inner satellite lines, and one of the outer satellite\nlines. For the (2 ,2) line, the number of channels was 15, with the\n00.10.20.3TL (K)\n10 20 30 40\nVLSR (km s-1)00.10.2TL (K)\n10 20 30 40\nVLSR (km s-1)G14 Hub N G14 Hub S NH3(1,1) NH3(1,1)\nNH3(2,2) NH3(2,2)Figure 11. G14.2: NH 3(1,1) (top) and (2 ,2) (bottom) spectra at the position\nof the center of Hub N (left) and Hub S (right), averaged within a diameter\nof 4′′.\nsame spectral resolution, and a velocity range between 16.5 and 25.2\nkm s−1, encompassing the main line (see Fig. 11).\n7.3 G14.2 data analysis\nThe NH 3(1,1) and (2,2) lines in the two hubs were analyzed using\nthehfs_nh3_cube utility of the HfS tool (Estalella 2017). For ev-\nery pixel, the spectra of the pixels inside a circle of 2′′radius were\naveraged and the resulting (1 ,1) and (2,2) spectra were fitted simul-\ntaneously, obtaining the parameters of the lines: line width, central\nvelocities and intensities of the (1 ,1) and (2,2) lines, and optical\ndepths of the (1 ,1) and (2,2) main lines. From these, the standard ho-\nmogeneous analysis provided the physical parameters pixel to pixel:\nexcitation, rotational, kinetic temperatures, and column density of\nNH 3.\nWe show for Hub N (Fig. 12) and Hub S (Fig. 13), the in-\ntegrated intensity, corrected for opacity, Aτm∆v, of the (1,1) line\n(top panel) and (2 ,2) line (middle panel), where Ais the amplitude\nA=ηB[Jν(Tex)−Jν(Tbg)],ηBis the beam filling factor, τmis the main\nline optical depth, and ∆vis the line width. The figure also shows the\npositions of the millimeter dust continuum sources detected with the\nSubmillimeter Array (SMA) by Busquet et al. (2016).\nThe maps of the homogeneous temperature, Thom\nk, for Hub N and\nHub S are shown in the bottom panels of the same Figs. 12 and\n13. The NH 3(2,2) integrated intensity are also shown as contours.\nThese maps were used to average the values of Thom\nkin rings, 2′′\nwide, centered on the position of the emission peaks in the hubs\nN and S at 350 µm observed with the Caltech Submillimeter Ob-\nservatory (CSO) reported by Busquet et al. (2016). The CSO po-\nsitions were chosen as center of the rings because the CSO angu-\nlar resolution (9′′) is very similar to that of the NH 3observations,\nand thus both observations are sensitive to similar structures at the\nsame scale. These positions are shown as small circles in Figs. 12\nand 13. The positions coordinates are α(J2000) =18h18m12.s53,\nMNRAS 000, 1–15 (2023)10 R. Estalella, A. Palau, &G. Busquet\nFigure 12. G14.2, Hub N. Top and middle panels: NH 3(1,1) and (2,2) main\nline integrated intensities, corrected for opacity, Aτm∆v(see main text in §\n7.3). The color scale is in units of K km s−1.Bottom panel: homogeneous\ntemperature Thom\nk. The color scale for Thom\nkgoes from 10 to 21 K. The con-\ntours are the integrated NH 3(2,2) emission, starting at 0.1 K km s−1with\nsteps of 0.05 K km s−1. For all panels the crosses mark the positions of the\nSMA mm sources MM1 and MM3, in order of increasing RA, and the circle\nmarks the position of the 350 µm emission peak (Busquet et al. 2016). The\nbeam is shown in the bottom-left corner of each panel.\nFigure 13. G14.2, Hub S. Top and middle panels: NH 3(1,1) and (2,2) main\nline integrated intensities, corrected for opacity, Aτm∆v(see main text in §\n7.3). The color scale is in units of K km s−1.Bottom panel: homogeneous\ntemperature Thom\nk. The color scale for Thom\nkgoes from 10 to 21 K. The con-\ntours are the integrated NH 3(2,2) emission, starting at 0.1 K km s−1with\nsteps of 0.05 K km s−1. For all panels the crosses mark the positions of the\nSMA mm sources MM2 to MM9, in order of increasing right ascension, and\nthe circle marks the position of the 350 µm emission peak (Busquet et al.\n2016). The beam is shown in the bottom-left corner of each panel.\nMNRAS 000, 1–15 (2023)NH 3spherical model 11\n2 4 6 8 10 10 20 30 40\nb (kau)1012141618202224262830T (K)1 2 4 6 8 10 20Rinn (arcsec)\nHub N\nHub SG14.2   Tkhom\nFigure 14. G14.2: Thom\nkradial profile obtained from the maps of Figs. 12\nand 13 (small circles and error bars), for Hub N (red) and Hub S (blue), in a\nlog–log scale. The error bars indicate the rms dispersion of the values of Tk\naveraged in each ring. The red and blue solid lines are the best fit to the radial\nprofiles, with the 2D-convolution of a Gaussian beam of 7 .′′6 with a power\nlawThom\nk=T0h(r/r0)−qh. The dashed lines are the Thom\nkpower-law profiles\ndeconvolved from the Gaussian beam.\nTable 3. G14.2 Hub N and S. Results from the fit of the observed Thom\nkradial\nprofiles with the 2D-convolution of a power-law T=T0h(r/r0)−qhwith a\nGaussian beam of 7 .′′56.\nT0haRmaxbPA rangeb\nG14.2 (K) qhχrc(arcsec) (deg)\nHub N 28 0.19 0.33 26 20 – 120\nHub S 31 0.25 0.14 12 −60 – 120\naTemperature at r0=1 kau.bMaximum radius and range of position\nangles used to obtain the Thom\nkradial profile.cχr=p\nχ2/(n−1).\nδ(J2000) =−16◦49′28.′′2 (Hub N), and α(J2000) =18h18m13.s06,\nδ(J2000) =−16◦57′19.′′2 (Hub S). The radial profiles obtained for\nThom\nkare shown in Fig. 14 as small circles and error bars. The er-\nror bars are the rms dispersion of the values of Thom\nkaveraged in\neach ring. The maximum radius and range of position angles (PA)\nused are given in Table 3. They were chosen to avoid contamination\nfrom other dense clumps in the region, which produce an increase\nin temperature to the southeast of the center of the rings in both\nhubs. By constraining the range of PAs, we assure that the profiles\nare obtained in smooth regions una ffected by feedback from nearby\nsources.\n7.4 Homogeneous analysis results\nTheThom\nkradial profiles for Hub N and S are similar (Fig. 14), show-\ning a flat profile for small radii, less than the beam radius, and a de-Table 4. G14.2 Hub N and S. Comparison of the power-law radial profile of\ntemperature, T=T0(r/r0)−q, from the dust emission and from the homoge-\nneous analysis.\nFrom dustaFrom NH 3b\nT0 T0h\nG14.2 (K) q (K) qh (T0h−T0)/T0qh−q\nHub N 51 0.34 28 0.19 −45%−0.15\nHub S 45 0.34 31 0.25 −30%−0.09\naBusquet et al. (2016).bFrom Table 3.\ncline for large radii. We fitted the observed radial profiles with the\n2D-convolution of a power law T0h(r/r0)−qhwith a Gaussian with\na HPBW of 7 .′′56. The best-fit values of the temperature at r0=1\nkau, T0h, and the power-law index, qh, are shown in Table 3. The\npower-law homogeneous temperature obtained from the fit, decon-\nvolved from the Gaussian beam, are shown as dashed lines in Fig.\n14. As can be seen, although the results are similar for both hubs,\nthe temperature of Hub S appears to be a bit steeper and with lower\nvalues than that of Hub N.\nBusquet et al. (2016) obtain the temperature radial profile from the\nsubmillimeter dust emission and spectral energy distribution (SED)\nfitting. Their results are shown in the left columns of Table 4. As\ncan be seen, the NH 3homogeneous analysis underestimates signif-\nicantly the temperature value and power-law index of the envelope.\nPart of the di fferences can be attributed to the lack of dust-gas cou-\npling in the lower-density region of the hubs (see, for instance Doty\n& Neufeld 1997). Indeed, Urban et al. (2009) find, for densities of\n105cm−3and dust temperatures of 50 K, that the gas temperature\ncan be lower by∼20 K. However, the di fferences are compatible\nwith the results obtained in Section 6 shown in Fig. 8. As reported in\nBusquet et al. (2016), both hubs have a flat inner region with a radius\nrc≃20 kau, and for such a density distribution, we can expect T0\ndifferences of the order of −40%, and power-law index di fferences\nof up to∼−0.15, in accordance with the results obtained from the\ncomparison with dust emission.\n8 DISCUSSION AND CONCLUSIONS\nThe spherical envelope model, using a radiative transfer calculation\nof the intensities of the NH 3(1,1; m), (1,1; s), and (2 ,2; m) lines, al-\nlowed us to compare the kinetic temperature derived from the stan-\ndard NH 3analysis, the homogeneous temperature Thom\nk, with the ac-\ntual temperature of the envelope.\nTests were performed with di fferent characteristics of the enve-\nlope, similar to the typical values derived for massive dense cores\nand clumps. In many cases we found that there is a critical value of\nthe projected radius below which no value of Thom\nkcould be calcu-\nlated, when the lines become optically thick enough. Near the criti-\ncal value, the homogeneous temperature can be much higher than the\nenvelope temperature. For larger projected radii, far from the critical\nvalue, the homogeneous temperature found from the standard NH 3\nanalysis was similar to the envelope temperature, but systematically\nunderestimates the actual temperature T0and temperature power-\nlaw index q. In the case of a power-law density, n∝r−p,T0is under-\nestimated by∼20%, and qby∼0.05. However, for a density with a\ncentral flat region, that is, a Plummer-like density, n∝(r2\nc+r2)−p/2,\nthe discrepancy between the homogeneous temperature and the ac-\ntual envelope temperature can be much higher, up to ∼40% in T0\nand∼0.15 in q, forrc≃10 kau.\nMNRAS 000, 1–15 (2023)12 R. Estalella, A. Palau, &G. Busquet\nWe applied this study to the infrared dark cloud G14.2. We used\nthe NH 3(J,K)=(1,1) and (2,2) data obtained with the Very Large\nArray and the E ffelsberg radio telescope for Hub N and Hub S. The\ndata were analyzed using the homogeneous analysis, that is, stan-\ndard NH 3analysis with the assumption of homogeneity along the\nline of sight, providing a radial profile of homogeneous temperature,\nThom\nk. The radial profiles were fitted by the convolution of a power\nlawTh0(r/r0)−qhwith a Gaussian beam of 7 .′′56. The homogeneous\ntemperatures obtained from the G14.2 data were compared with the\ndust temperatures of Busquet et al. (2016). The lack of dust-gas cou-\npling in the lower-density parts of the hubs could contribute to the\nobserved discrepancies ( ∼40% in the temperature T0and∼0.1 in\nthe temperature power-law index q), but the discrepancies are in ac-\ncordance with the expected results for envelopes with a central flat\nregion.\nSeveral conclusions can be drawn from this work. Regarding the\nhomogeneous analysis of NH 3(1,1) and (2,2) radial profiles, two\nresults are relevant. Firstly, for small projected radii, where the op-\ntical depth of the lines is high, the homogeneous temperature can\nbe much higher then the actual envelope temperature. Secondly, for\nlarger projected radii, the temperature values and power-law indices\nderived from the homogeneous analysis could be seriously underes-\ntimated, depending on the density profile of the envelope. For power-\nlaw density profiles, with a very high density at small radii, charac-\nteristic of very young, Class 0 objects, the temperature can be under-\nestimated by∼20%, and the power-law index by ∼0.05. However,\nfor density profiles with a flat central region, characteristic of more\nevolved, Class I and beyond objects, the e ffect is more severe. The\ntemperature can be underestimated by as much as ∼40%, and the\npower-law index by ∼0.15.\nA similar study for other commonly used dense gas tracers such\nas H 2CO and CH 3OCHO should be carried out to properly assess\nthe validity of the temperature radial profiles inferred from dense\ngas thermometers.\nACKNOWLEDGEMENTS\nWe thank the anonymous referee for his /her comments, helping to\nimprove the paper. R. E. and G. B. acknowledge financial support\nfrom the grants PID2020-117710GB-I00 and CEX2019-000918-M\nfunded by MCIN /AEI/10.13039 /501100011033. A. P. acknowl-\nedges financial support from the UNAM-PAPIIT IN111421 and\nIG100223 grants, the Sistema Nacional de Investigadores of CONA-\nCyT, and from the CONACyT project number 86372 of the ‘Ciencia\nde Frontera 2019’ program, entitled ‘Citlalcóatl: A multiscale study\nat the new frontier of the formation and early evolution of stars and\nplanetary systems’, México.\nDATA A V AILABILITY\nNo new data were analysed in support of this research. The data used\nwere already published in Busquet et al. (2013).\nREFERENCES\nAbramowitz M., Stegun I. A., 1972, Handbook of Mathematical Functions\nwith Formulas, Graphs, and Mathematical Tables. New York: Dover\nAhmadi A., et al., 2018, A&A, 618, A46\nBeltrán M. T., Estalella R., Ho P. T. P., Calvet N., Anglada G., Sepúlveda I.,\n2002, ApJ, 565, 1069Beltrán M. T., et al., 2018, A&A, 615, A141\nBeuther H., Schilke P., Menten K. M., Motte F., Sridharan T. 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D., 2009, ApJ, 698, 1341\nWang J.-W., Koch P. M., Galván-Madrid R., Lai S.-P., Liu H. B., Lin S.-J.,\nPattle K., 2020, ApJ, 905, 158\nWilliams S. J., Fuller G. A., Sridharan T. K., 2005, A&A, 434, 257\nXu Y ., Moscadelli L., Reid M. J., et al. 2011, ApJ, 733, 25\nAPPENDIX A: RADIATIVE TRANSFER CALCULATION\nIn the following we will derive the intensities of the NH 3(1,1; m),\n(1,1; s), and (2 ,2; m) lines for a spherical envelope by integration\nof the radiative transfer equation, and calculate the homogeneous\noptical depths τhom(1,1; s) andτhom(2,2; s), and the homogeneous\ntemperature Thom\nkderived from the homogeneous analysis.\nA1 The radiative transfer equation\nThe radiative transfer equation is evaluated for every projected ra-\ndius b. Let us consider the slab of geometrical width ∆z, and opti-\ncal depth ∆τ, located at position zalong the line of sight (see Fig.\n5), with z2+b2=r2. For this elementary slab with optical depth\n∆τν, the incoming intensity at zis attenuated a factor exp( −∆τν),\nand is increased by the average source function of the slab, times\n[1−exp(−∆τ)],\nIν(z+ ∆z)=Iν(z) exp(−∆τ)+Sν(¯z)[1−exp(−∆τ)], (A1)\nMNRAS 000, 1–15 (2023)NH 3spherical model 13\nwhere ¯ zcan be approximated by the center of the slab, ¯ z=z+ ∆z/2.\nBy subtracting Iν(z) on both sides of the equation and rearranging\nterms the equation can be written as\n∆Iν=[Sν(¯z)−Iν(z)][1−exp(−∆τ)]. (A2)\nThe source function Sνis defined by the excitation temperature Tex\nSν=2hν3\nc21\nexp(hν/kTex)−1=2kν2\nc2Jν(Tex), (A3)\nEquation A2 can be written in terms of the brightness temperature\nTb(giving the intensity) and the excitation temperature Tex(giving\nthe source function),\n∆Jν(Tb)=[Jν(Tex)(¯z)−Jν(Tb)(z)][1−exp(−∆τ)]. (A4)\nA2 Excitation temperature Tex\nAssuming that the excitation temperature of the (1 ,1) transition is\ngiven by the two-levels model, Texcan be expressed as a weighted\naverage of the kinetic and brightness background temperatures, Tk\nandTbg,\nJν(Tex)=Jν(Tk)+yJν(Tbg)\n1+y, (A5)\nwhere the weight factor yis\ny=A11\nnγ111\n1−e−hν/kTk, (A6)\nA11is the spontaneous emission coe fficient, nis the hydrogen den-\nsity,n(H2), andγ11is the de-excitation collisional coe fficient,\n\u0014γ11\ns−1cm3\u0015\n=2.27×10−11\u0014Tk\nK\u00151/2\n. (A7)\nThus, for the slab of material between zandz+dzwe consider\nTk=T(r), and a background temperature given by the bright-\nness temperature of the incident radiation from the preceding slab,\nTbg=Tb(z). Then, the excitation temperature of the (1 ,1) transition\nfor this slab is obtained from Eq. A5. For the (2 ,2) transition we take\nthe same excitation temperature.\nA3 Optical depth τ\nOnce the excitation temperature is known, we can estimate the el-\nementary optical depths of the (1 ,1; m), (1,1; s), and (2 ,2; m) lines\nfor the slab,\n∆τ(j,k;l)=r\n4 ln 2\nπc3AjkRjkl\n8πν3\njk∆vexp (hνjk/kTex)−1\nexp (hνjk/kTex)+1njk∆z,(A8)\nwith the volume densities n11andn22given by\nn11=f11(T(r))X n(r) (A9)\nn22=f22(T(r))X n(r) (A10)\nwhere the fractional abundances f11and f22are given by Eqs. 14\nand 15, with the rotational temperature calculated from the kinetic\ntemperature (Eq. 12).\nA4 Iterative scheme\nFor any given projected radius b, the integral along the line of sight\nhas to be evaluated from −zmaxtozmax, where zmax=p\nr2\nenv−b2.\nThe initial conditions at z=−zmaxareτ(1,1; m) =τ(1,1; s) =\n0102030T (K)Tk\nTex (n0 = 108)\nTex (n0 = 107)\nTex (n0 = 106)\n0.1110Optical depthτ(1,1:m)\nτ(1,1;s)\nτ(2,2;m)\n-50 -40 -30 -20 -10 0 10 20 30 40 50\nz (kau)110Jν(Tb) (K)T(1,1;m)\nT(1,1;s)\nT(2,2;m)Figure A1. Examples of integration along the line of sight. The observer is\nlocated rightwards. The distance is 1 kpc, and the projected radius bis 5′′\n(5000 au, or 5 kau). The envelope parameters are p=2,rc=0,T0=50\nK,q=0.5,∆v=0.5 km s−1,nmin=103cm−3,Tmin=8 K, NH 3abun-\ndance X=10−8and three values of the density, n0=108(continuum\nlines), 107(dashed lines), and 106cm−3(dash-dotted lines). Top: Temper-\nature of the envelope, Tk, and excitation temperatures Tex.Middle: Optical\ndepthsτ(1,1; m) (red), τ(1,1; s) (orange), and τ(2,2; m) (blue). Bottom: Line\nintensities Jν(Tb)(1,1; m) (red), Jν(Tb)(1,1; s) (orange), and Jν(Tb)(2,2; m)\n(blue).\nτ(2,2; m) =0,Tb(1,1; m) =Tb(1,1; s) =Tb(2,2; m) =Tbg=\n2.752 K.\nFor a given z, we have determined from the preceding step the\nvalues at zof the optical depths τ(1,1; m),τ(1,1; s),τ(2,2; m),\nand the brightness temperatures of the lines, Tb(1,1; m), Tb(1,1; s),\nTb(2,2; m).\nFor this step, from ztoz+ ∆z, we take ¯ z=z+ ∆z/2 (this is\nequivalent to the second order Runge-Kutta method, Abramowitz\n& Stegun 1972), r=√\nb2+¯z2, Eq. A5 is used with Tk=T(r),\nTbg=Tb(1,1; m), and n=n(r) to calculate Tex. The densities n11and\nn22are calculated, njk=fjkX n(r), and Eq. A8 is used to estimate\n∆τ(1,1; m), ∆τ(1,1; s), and ∆τ(22; m). Finally, Eq. A4 is used to\ncalculate ∆Jν(Tb(1,1; m)), ∆Jν(Tb(1,1; s)), and ∆Jν(Tb(2,2; m)).\nAn example of the integration along the line of sight is shown in\nFig. A1, for a power-law density with three di fferent values of the\ndensity. For n0=106cm−3the three lines, (1 ,1; m), (1,1; s), and\n(2,2; m) are optically thin; for n0=107cm−3the (1,1; m) line is\nmoderately optically thick; and for n0=108cm−3the three lines are\noptically thick, the (1 ,1; m) line being very thick (middle panel). In\nthe latter case we see that the three lines present self-absorption from\nthe colder gas of the part of the envelope facing the observer, for\nz>0 (bottom panel). For the (1,1:m) line, optically thicker than the\nothers, the self-absorption is higher, resulting in an intensity lower\nthan those of the (1 ,1; s) and (2,2; m) lines.\nMNRAS 000, 1–15 (2023)14 R. Estalella, A. Palau, &G. Busquet\n10100T (K)TkThomT11mT11sT22m\n0.1110Optical depthτ11m\nτ11s\nτ22m\n1 10\nb (kau)0123rr11\nrτ\nFigure A2. Example of radial profiles, obtained for an envelope at a distance\nof 1 kpc, with n0=107cm−3,p=2,rc=0,T0=50 K, q=0.5,∆v=0.5\nkm s−1,nmin=103cm−3,Tmin=8 K, and NH 3abundance X=10−8.\nTop panel: Temperature of the envelope (black dashed line), line intensities\nTL(1,1; m) (red), TL(1,1; s) (orange), TL(2,2; m) (blue), and homogeneous\ntemperature Thom\nk(black solid line), for the values of bfor which it could be\ncalculated. Central panel: Optical depths τ(1,1; m) (red), τ(1,1; s) (orange),\nandτ(2,2; m) (blue). The dashed lines are the values obtained from the inte-\ngration along the line of sight. The continuum lines are the values obtained\nfrom the homogeneous analysis. Bottom panel: r 11=T1m/T1s(red), and\nrτ=[1−exp(−τ1m)]/[1−exp(−aτ1m)]r12(blue), where r12=T2m/T1mand\na=3.544 (see text). The horizontal dashed lines at 1 and 3.6 indicate the\nminimum and maximum values of r11allowed for the homogeneous analysis\n(see Section 3). The vertical dashed line at rτ=1 indicates the minimum\nprojected radius bfor which the homogeneous temperature can be obtained.\nThe final step is to derive the intensities of the lines, TL(1,1; m) =\nJν(Tb(1,1; m))−Jν(Tbg),TL(1,1; s)=Jν(Tb(1,1; s))−Jν(Tbg), and\nTL(2,2; m) =Jν(Tb(2,2; m))−Jν(Tbg). Once the three line intensities\nare known, the standard analysis of Sec. 2 is used to derive the homo-\ngeneous estimates of the optical depths τhom(1,1; m),τhom(1,1; s),\nandτhom(2,2; m), the homogeneous estimate of the rotational tem-\nperature, and finally the homogeneous temperature, Thom\nk.\nA5 Results\nIn Fig. A2 we show an example of results obtained for an envelope\nwith a power-law density profile, at a distance of 1 kpc. For this\ndistance, an angular size of 1′′corresponds to a linear size of 1 kau =\n1000 au. The parameters of the envelope are a reference radius r0=\n1 kau, density parameter n0=107cm−3, density power-law index\np=2, radius of the inner flat region rc=0, temperature at r0,\nT0=50 K, temperature power-law index q=0.5, line width ∆v=\n10100T (K)TkThT1mT1sT2m\n10100T (K)TkThT1mT1sT2m\n1 10\nb (kau)110100T (K)\n1 10\nb (kau)TkThT1mT1sT2mLIME This work n0=106 cm-3\nn0=107 cm-3\nn0=108 cm-3Figure A3. Comparison of LIME (left) and radiative transfer of this\nwork (right) of the radial profiles of the temperature of the envelope\n(black dashed line), line intensities TL(1,1; m) (red), TL(1,1; s) (orange),\nTL(2,2; m) (blue), and homogeneous temperature Thom\nk(black solid line).\nThe envelope parameters are a reference radius r0=1 kau, density parame-\ntern0=106cm−3(top), 107cm−3(middle), and 108cm−3(bottom), density\npower-law index p=2, radius of the inner flat region rc=0, temperature at\nr0,T0=50 K, temperature power-law index q=0.5, line width ∆v=0.5\nkm s−1, minimum density nmin=103cm−3, minimum temperature Tmin=8\nK, and NH 3abundance X=10−8.\n0.5 km s−1, minimum density (setting the envelope radius) nmin=\n103cm−3, minimum temperature Tmin=8 K, and NH 3abundance\nX=10−8. The top panel shows the radial profiles of temperature of\nthe envelope (dashed line), line intensities, TL(1,1; m), TL(1,1; s),\nTL(2,2; m), and homogeneous temperature Thom\nk. The middle panel\nshows the radial profiles of optical depths of the lines, τ(1,1; m),\nτ(1,1; s), andτ(2,2; m), calculated from the integration along the\nline of sight (dashed lines), and estimated from the homogeneous\nanalysis (continuum lines).\nAs can be seen in the top panel, for small values of the projected\nradius the homogeneous temperature Thom\nkcould not be derived. The\nreason is that for bbelow a critical value of 1.94 kau, the intensity\nof the (2,2) line is too high to fulfil the condition of Eq. 21, and\nτhom(2,2; m) could not be calculated (see bottom panel of Fig. A2).\nThe failure of the homogeneous analysis is a consequence of the\n(1,1; m) line being optically thick for small b, and thus tracing the\ncold part of the envelope facing the observer, while the (2 ,2; m) line,\nwith a lower optical depth, is tracing hotter gas near the center of the\nenvelope. For values of the projected radius near the critical value\ngiven above, the optical depth of the (2 ,2) line derived from the\nhomogeneous analysis is much higher than the actual optical depth,\nand the homogeneous temperature can be very di fferent from the\nenvelope temperature. For larger projected radii, far from the critical\nvalue, the radial profile of the homogeneous temperature is similar\nto, but systematically below that of the envelope temperature.\nA6 Comparison with LIME\nThe Line Modelling Engine (LIME) (Brinch & Hogerheijde 2010)\nwas used to compare the spherical envelope model with a full radia-\ntive transfer calculation. LIME assumes that the gas and dust tem-\nMNRAS 000, 1–15 (2023)NH 3spherical model 15\nperatures are the same. Unfortunately, the collision coe fficients of\nthe quadrupole-hyperfine transitions of NH 3are not available in the\nliterature.\nLIME was run using the molecular data for para-NH 3from the\nLeiden Atomic and Molecular Database (LAMDA) (Schöier et al.\n2005), which contains energy levels, transition frequencies, Einstein\ncoefficients of the inversion transitions, and the collision rates with\npara-H 2. The output of LIME provided us the radial profiles of the\nintensities and optical depths of the inversion transitions NH 3(1,1)\nand (2,2),TL(j,k) andτ(j,k). The intensity of the main and inner\nsatellite quadrupole-hyperfine lines was then estimated as\nTL(j,k; s)=TL(j,k)1−exp[−Rjksτ(j,k)]\n1−exp[−τ(j,k)], (A11)\nTL(j,k; m) =TL(j,k)1−exp[−Rjkmτ(j,k)]\n1−exp[−τ(j,k)].\nFrom these intensities, the standard NH 3analysis was performed to\nobtain the homogeneous temperature radial profile.\nThe LIME calculation assumes that the five quadrupole-hyperfine\nlines of each inversion transition are completely blended, with an op-\ntical depth that is the total optical depth of the inversion transitions.\nSo, the inversion optical depths calculated by LIME are higher than\nthe actual optical depths of the hyperfine lines. As a consequence,\nfor partially thick emission, the region contributing to the LIME in-\ntensity is closer to the observer (that is, farther from the envelope\ncenter, and so cooler and less dense) than for the hyperfine intensity.\nThus, we expect discrepancies between LIME calculations and the\nradiative transport calculations of this work, which could be more\nimportant in the optically thick regime.\nIn the left panels of Fig. A3 we show the line intensities\nTL(1,1; m), TL(1,1; s), TL(2,2; m), and homogeneous temperature\nThom\nkfor three di fferent density parameters, n0=106, 107, and 108\ncm−3calculated with LIME. In all three cases the homogeneous tem-\nperatures obtained from the the LIME calculation, at the projected\nradii where it can be calculated (solid black line), are near the enve-\nlope temperature.\nIn the right panels of Fig. A3 we show, for comparison, the same\nresults calculated with the spherical envelope model of this work.\nAs can be seen, at low densities ( n0=106cm−3) there is a good\nagreement between the line intensities calculated by both methods,\nwhile at higher densities the discrepancies are most probably due\nto the overestimation of the optical depths made by LIME, as dis-\ncussed above. Thus, despite the shortcomings of the LIME calcula-\ntions, there is a general agreement in the line intensities and, more\nimportant, a good agreement between the homogeneous temperature\nThom\nkderived from both methods. In conclusion, we can say that the\nradiative transfer calculation used in the present work is correct.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–15 (2023)"    },    {        "title": "2401.14089v1.GQHAN__A_Grover_inspired_Quantum_Hard_Attention_Network.pdf",        "content": "1\nGQHAN: A Grover-inspired Quantum Hard\nAttention Network\nRen-Xin Zhao, Member, IEEE, Jinjing Shi*,Member, IEEE, and Xuelong Li, Fellow, IEEE\nAbstract —Numerous current Quantum Machine Learning (QML) models exhibit an inadequacy in discerning the significance of quan-\ntum data, resulting in diminished efficacy when handling extensive quantum datasets. Hard Attention Mechanism (HAM), anticipated to\nefficiently tackle the above QML bottlenecks, encounters the substantial challenge of non-differentiability, consequently constraining its\nextensive applicability. In response to the dilemma of HAM and QML, a Grover-inspired Quantum Hard Attention Mechanism (GQHAM)\nconsisting of a Flexible Oracle (FO) and an Adaptive Diffusion Operator (ADO) is proposed. Notably, the FO is designed to surmount\nthe non-differentiable issue by executing the activation or masking of Discrete Primitives (DPs) with Flexible Control (FC) to weave\nvarious discrete destinies. Based on this, such discrete choice can be visualized with a specially defined Quantum Hard Attention Score\n(QHAS). Furthermore, a trainable ADO is devised to boost the generality and flexibility of GQHAM. At last, a Grover-inspired Quantum\nHard Attention Network (GQHAN) based on QGHAM is constructed on PennyLane platform for Fashion MNIST binary classification.\nExperimental findings demonstrate that GQHAN adeptly surmounts the non-differentiability hurdle, surpassing the efficacy of extant\nquantum soft self-attention mechanisms in accuracies and learning ability. In noise experiments, GQHAN is robuster to bit-flip noise in\naccuracy and amplitude damping noise in learning performance. Predictably, the proposal of GQHAN enriches the Quantum Attention\nMechanism (QAM), lays the foundation for future quantum computers to process large-scale data, and promotes the development of\nquantum computer vision.\nIndex Terms —Machine learning, Quantum machine learning, Grover’s algorithm, Hard attention mechanism, Grover-inspired quantum\nhard attention Network, Quantum neural network, Quantum circuit.\n✦\n1 I NTRODUCTION\nINrecent years, QML has developed tremendously [1–3].\nHowever, many current QML models treat each quantum\ndata equally and neglect the value of the intrinsic con-\nnections between data. This not only mandates substantial\nquantum storage resources for comprehensive information\nretention, but also constitutes a threat to the prospective\nlarge-scale quantum data processing on forthcoming quan-\ntum computers. The above urgent issues can be effectively\naddressed by HAM.\nHAM was first proposed in 2015 [4], as a discrete com-\npetitive model that adheres to the winner-take-all law which\nonly concentrates on the most vital parts and overlooks the\nrest. This unique computing mechanism decreases the cost\nof data acquisition [5] for designing interpretable [6], com-\nputationally effective [7] and scalable [8] models, mitigating\ncatastrophic forgetting to a certain extent [9], consequently\nresulting in widespread applications in computer vision\n[10], natural language processing [11] and video processing\n[12, 13], etc. One of the striking cases is that HAM achieves\ntwo impressive accuracies of 95.83% and 99.07% for safe\ndriving recognition and driver distraction detection, respec-\n•Ren-Xin Zhao is with the School of Computer Science and Engineering,\nCentral South Univerisity, China, Changsha, 410083. Jinjing Shi is with\nthe School of Electronic Information, Central South Univerisity, China,\nChangsha, 410083.\n•Xuelong Li is with the Institute of Artificial Intelligence (Tele AI), China\nTelecom Corp Ltd, 31 Jinrong Street, Beijing 100033, P . R. China.\n•Jinjing Shi is the corresponding author.\n•E-mails: renxin zhao@alu.hdu.edu.cn, shijinjing@csu.edu.cn, li@nwpu.e\ndu.cn.\nManuscript received XXXX; revised XXXX.tively, in addition to a 38.71% reduction in runtime [14]. Al-\nthough very effective, HAM is stuck in a non-differentiable\ndilemma that hinders its optimization due to the discrete na-\nture of information selection processing [15]. To conquer this\ndilemma, various strategies such as reinforcement learning\n[11, 12] and Gumbel-softmax based straight-through esti-\nmator [16, 17] have been proposed. Nevertheless, formulat-\ning a proficient reward function aligned with the task in\nreinforcement learning proves challenging, and misguided\nreward specifications may engender suboptimal attention\npatterns. Straight-through estimator schemes may also re-\nquire a trade-off between computational efficiency and ac-\ncuracy. Hence, it is inherently more practical and convenient\nto make HAM compatible with gradient systems. To achieve\nthis goal, a quantum scheme inspired by Grover’s algorithm\nis attempted to compensate for the above shortcoming.\nGrover’s algorithm is a quantum algorithm composed\nof an oracle and a diffusion operator for unstructured\nsearch [18, 19], where the oracle selects specific target items\nthrough phase flipping. The diffusion operator amplifies the\namplitudes of the chosen ones and suppresses the ampli-\ntudes of the other items. Obviously, the working mechanism\nof Grover’s algorithm is somehow similar to HAM, as it\nalso involves discretely selecting specific targets. However,\nthe application of Grover’s algorithm alone is insufficient\nto surmount the non-differentiable nature. Simultaneously,\nwhen the number of target terms is unknown or dynami-\ncally changing, the performance and adaptivity of Grover’s\nalgorithm are constrained by the inability to ascertain the\nappropriate number of iterations [20–23]. Thus a dramatic\nmodification of this algorithm is necessary to create a differ-\nentiable quantum hard attention mechanism, which inducesarXiv:2401.14089v1  [quant-ph]  25 Jan 20242\nTab. 1: Quantum Gates\nName of Quantum Gate Mathematical Notation Matrix Representation Symbol of Quantum Gate\nPauli X gate X\u0014\n0 1\n1 0\u0015\nPauli rotating X gate RX(θ)\u0014\ncos(θ/2) −isin(θ/2)\n−isin(θ/2) cos( θ/2)\u0015\nHadamard gate H1√\n2\u0014\n1 1\n1−1\u0015\nControlled Y gate CRY(θ)\n1 0 0 0\n0 1 0 0\n0 0 cos( θ/2)−sin(θ/2)\n0 0 sin( θ/2) cos( θ/2)\n\nMulti-controlled Z gate MCZ\n1 0 0 · · · 0\n0 1 0 · · · 0\n...............\n0 0 0 · · · − 1\n\nDiscrete primitive C−Λ(b) Eq. (14)\nthe three main propositions of this paper:\n1. Can current QML models be equipped with a quan-\ntum HAM to distinguish the intrinsic importance of quan-\ntum data?\n2. How can a differentiable GQHAN be constructed by\ncombining the Gover’s algorithm with HAM?\n3. How does GQHAN remain efficient when the target\nterm is unknown or dynamically altering?\nTo this end, the contribution of this paper lies in address-\ning these pertinent inquiries and presenting novel insights\ninto the potential of GQHAN:\n•GQHAM is posited to discern the significance of quan-\ntum data, thereby augmenting the efficacy of QML\nmodel processing.\n•FO and ADO are proposed as solutions for addressing\nthe discrete non-differentiable predicament inherent in\nGQHAM.\n•Based on GQHAM, a GQHAN is constructed on Pen-\nnyLane for Fashion MNIST binary classification experi-\nments, achieving an impressive accuracy of no less than\n98% and lower convergence values, which indicates\nthat it has surpassed the two quantum soft attention\nmechanisms in terms of classification accuracy and\nlearning ability.\nThe subsequent sections of this paper are organized\nas follows: Section 2 contains a brief overview of QAM,\nquantum computing and Grover’s algorithm. In Section 3,\nthe mathematical mechanism of GQHAM is elaborated.The\nGQHAN framework and its workflow are described in\nSection 4. Section 5 delineates the experimental details and\nseveral meaningful conclusions are drawn. Eventually, a\nsummary is presented.2 R ELATED WORKS\nIn this section, a succinct overview of quantum computing\nfoundation, QAM, and Grover’s algorithm is presented.\n2.1 Quantum Computing Foundation\nQubits and quantum gates are the figurative embodiment of\nquantum theory. In this context, a qubit |ψ⟩=α|0⟩+β|1⟩is\nthe smallest unit that carries information, where |0⟩= [1,0]T\nand|1⟩= [0 ,1]Tdenote the ground state and the ex-\ncited state respectively. αandβare amplitudes satisfying\n|α|2+|β|2= 1 [28]. In stark contrast to classical com-\nputation, |ψ⟩is in a superposition of |0⟩and|1⟩, thereby\nconferring upon it the remarkable ability for exponential\ndata representation. Moreover, the linear evolution of qubits\nis contingent upon quantum gates whose mathematical\nessence is the unitary matrices. The quantum gates used in\nthis paper are shown in Tab. 1, including Pauli X gate, Pauli\nrotating X gate, Hadamard gate, controlled Y gate, multi-\ncontrolled Z gate and DP , where DP is one of the innovations\nin this paper.\n2.2 Quantum Attention Mechanisms\nIn QML, QAM attempts to boost the performance of QML\nmodels by discriminating the significance of quantum data.\nEarly researches on QAM are inspired by quantum physics\nprinciples. For example, a parameter-free QAM utilizing\nweak measurements was introduced in 2017 to enhance\nbidirectional LSTM sentence modeling, demonstrating su-\nperior performance compared to classical attention mech-\nanism [24]. Nonetheless, these investigations exclusively3\nFig. 1: Quantum Amplitude Encoding [32]\nFig. 2: Flexible Oracle\nFig. 3: Adaptive Diffusion Operator\ndepended on classical computer simulations owing to the\nabsence of an ansatz framework. QAM with ansatz struc-\nture has not appeared until 2022. In 2022, a probabilistic\nfull-quantum self-attention network with an exponentially\nscalable self-attention representation realized completely on\na quantum computer was proposed [25]. Its learning ability\noutperforms hardware-efficient ansatz and QAOA ansatz\nin a comparable configuration. In the same year, a quan-\ntum self-attention network was designed by a Baidu team\nto calculate quantum self-attention scores using hardware-\nefficient ansatz with remarkable success in text categoriza-\ntion [26]. In 2023, a quantum self-attention network based\non the quantum kernel method and deferred measurement\nprinciple was introduced, which achieved impressively high\naccuracy with few parameters [27]. While QAM has exhib-\nited considerable promise in domains like quantum ma-\nchine vision, its theoretical underpinnings and practical\napplications necessitate further augmentation owing to the\nconstraints posed by quantum hardware.\n2.3 Grover’s Algorithm\nGrover’s algorithm has undergone iterative enhancements\nin multiple aspects such as phase and initial state optimiza-\ntion, since its initial proposal, where phase optimization\ninvolves strategically adjusting the phase flip in the oracle.\nIllustratively, in a fixed phase rotation Grover’s algorithm,\nthe rotation phase shifts from πto1.91684 π, thereby ele-\nvating the success probability to 99.59% [29]. Initial state\noptimization allows for arbitrary initial states, markedly am-\nplifying the generality of Grover’s algorithm. Some reasons\nfor this are that the mean and variance of the amplitudedistribution of the initial state affects the measurement of\nthe optimal time and the maximum probability of success,\nbut the measurement of the optimal time is independent\nof the initial state which is an arbitrary pure state [30, 31].\nAlthough numerous efforts have been conducted to boost\nthe efficiency of Grover’s algorithm, it has not been explored\nto incorporate trainable parameters to further strengthen its\nflexibility and applicability. In other words, in this paper,\nan attempt is presented to convert the untrainable Grover’s\nalgorithm into a trainable QML model to distinguish it from\nprevious improvement strategies.\n3 G ROVER -INSPIRED QUANTUM HARD ATTEN -\nTION MECHANISM\nIn this section, GQHAM, which successfully mitigates the\nnon-differentiability issue, is presented and defined as fol-\nlows:\nDefinition 3.1 (Grover-inspired Quantum Hard Attention\nMechanism) .\nGQHAM :=UDU′\nS|In⟩2, (1)\nis composed of an input quantum state |In⟩2, an ADO UD,\nand an FO U′\nSthat contains three novel concepts of DP , FC\nand QHAS.\nThe tenets of the FO and the ADO are elucidated em-\nphatically in the ensuing subsections.\n3.1 Flexible Oracle\nThe classical input is set as\nIn= [ab]l−1\nb=0∈R1×l⊆ X, (2)4\nwhere abindicates an element. lis the total number of ele-\nments. Xdenotes the space where the input data is located.\nSubsequently, Eq. (2) is converted to an input quantum state\n|In⟩2=l−1X\nb=0αb|b⟩2+2n−1X\nb=l0|b⟩2⊆ H (3)\nby quantum amplitude encoding UAin Fig. 1 [32], where\nthe subscript 2 indicates that Eq. (3) is situated in the\nsecond quantum register. |b⟩2refers to the basis vector. H\nis the Hilbert space. n=⌈log2l⌉is the amount of qubits.\nPl−1\nb=0α2\nb= 1.αb=ab/qPl−1\nd=0a2\nd.ab, ad∈In. Then m\ntarget basis vectors\nM={|cb⟩2}m−1\nb=0⊆ {|b⟩2}2n−1\nb=0(4)\nfrom Eq. (3) are randomly selected and linearly combined\nwith the opposite values of their corresponding probability\namplitudes into the target term\n|focus⟩2=m−1X\nb=0βb|cb⟩2. (5)\nThe formation of Eq. (5) is carried out by an oracle\nUS=\n(−1)f(|0⟩)\n(−1)f(|1⟩)\n...\n(−1)f(|2n−1⟩)\n(6)\nwhich is mathematically a 2n×2ndiagonal unitary matrix,\nwhere\nf(x) =\u001a1x∈ M\n0else. (7)\nTo be specific,\nUS|b⟩2=\u001a−|b⟩2|b⟩2∈ M\n|b⟩2 else, (8)\nwhich articulates that Eq. (4) are capable of self-labeling by\nphase inversion when acted on by Eq. (6). From a holistic\npoint of view, if Eq. (6) is updated once, the conditional\njudgments in Eq. (7) must be performed several times. This\nnot only falls into the trap of discrete non-differentiability\nbut also makes the problem inaccessible because it is diffi-\ncult to directly reproduce such multiple complex operations\nas Eq. (6) and Eq. (7) with quantum gates. However, from\nanother perspective, Eq. (6) can be disassembled into\nUS=2n−1Y\nb=0λb, (9)\nwhere any element\nλb=\n1\n...\n(−1)f(|b⟩)\nb\n...\n1\n=\u001aΛ(b)b∈ M\nI else.(10)Λ(b) =\n1\n...\n−1b\n...\n1\n. (11)\nEq. (10) (or Eq. (11)) represents a diagonal unitary matrix\nwhose b-th element on the diagonal is (−1)f(|b⟩)\nb(or−1)\nand the others are 1. All symbols Icollectively stand for the\nidentity matrix throughout this article.\nDeconstructing a intricate problem through perspective\nshifts proves more lucid and facile than contemplating it\nholistically in advance, since it is now only necessary to\nconsider how to switch the two forms of Eq. (10), Eq. (11)\nandI, with continuous parameters. Besides, it is worth\nnoting that when λb=I, it is equivalent to applying no\noperation to the current quantum circuit. Guided by the\nabove idea, a set of DPs\n{C−Λ(b)}2n−1\nb=0(12)\nis introduced.\nDefinition 3.2 (Discrete Primitive) .In Eq. (12), an arbitrary\nDP\nC−Λ(b) =I⊗ |0⟩1⟨0|1+ Λ(b)⊗ |1⟩1⟨1|1 (13)\nas shown in Tab. 1 is a controlled quantum gate with one\ncontrol bit and ntarget bits, which means that Eq. (11) is ap-\nplied on the first quantum register only when the controlled\nqubit is |1⟩1. In Eq. (13), ⊗represents the tensor symbol. ⟨0|1\nand⟨1|1correspond to the conjugate transpositions of |0⟩1\nand|1⟩1, respectively.\nCorollary 3.1. Mathematically, Eq. (13) is a unitary diagonal\nmatrix.\nProof.\nC−Λ(b) =I⊗ |0⟩1⟨0|1+ Λ(b)⊗ |1⟩1⟨1|1\n=I⊗\u00141 0\n0 0\u0015\n+ Λ(b)⊗\u00140 0\n0 1\u0015\n=\u0014I 0\n0 Λ( b)\u0015\n2n+1×2n+1(14)\nsatisfies C−Λ(b)C−Λ(b)†=C−Λ(b)†C−Λ(b) =Iand is\ntherefore a unitary diagonal matrix, where C−Λ(b)†is the\nconjugate transpose of C−Λ(b).\nIn adherence to Definition 3.2, all elements in Eq. (12)\nfunction when the controlled qubit is |1⟩1. Leveraging this\nuniqueness, quantum gates containing parameters are used\nto determine whether the controlled qubit is |1⟩1or not, thus\nselectively activating and shielding certain elements in Eq.\n(12). For this purpose, the concept of FC is proposed.\nDefinition 3.3 (Flexible Control) .Suppose that the quantum\ncontrol bit of Eq. (13) is initialized to |0⟩1. Subsequently\nPauli rotating X gates RX(θ)are posed on |0⟩1to form the\nflexible control\nRX(θ)|0⟩1=\u001a|1⟩1 θ= (4k+ 1)π, k∈Z\nRX(θ)|0⟩1else.(15)\nEq. (15) shows that the control bit is |1⟩1only when the\ncontinuous variable θis located at (4k+ 1)π, k∈Z.5\nProof. In quantum computing, the interconversion between\nground state |0⟩1and excited state |1⟩1can be realized by a\nPauli X gate X. Therefore, RX(θ)is equal to the form of X\nonly whenθ\n2=π\n2+ 2kπ, k∈Z. That is,\n\u0014cos\u0000π\n2+ 2kπ\u0001−isin\u0000π\n2+ 2kπ\u0001\n−isin\u0000π\n2+ 2kπ\u0001cos\u0000π\n2+ 2kπ\u0001\u0015\n=−i\u00140 1\n1 0\u0015\n,\nwhere the global phase −ican be neglected.\nNaturally, the definition of hard attention scores can be\nderived from Definition 3.3:\nDefinition 3.4 (Quantum Hard Attention Score) .\nQHAS :=\u001a1θ= (4k+ 1)π, k∈Z\n0else(16)\ncan be used to visualize the choice of FC in training.\nBased on Eq. (12) and Eq. (15), FO can be created.\nDefinition 3.5 (Flexible Oracle) .The FO\nU′\nS=2n−1Y\nb=0RX(θb)⊗I×C−Λ(b), (17)\nas shown in Fig. 2, replaces Eq. (6) by applying flexible\ncontrol to each DP in Eq. (12), thus coming to select and\ncombine different DPs and create new discrete operations,\nultimately freeing GQHAM from the confinement of dis-\ncrete non-differentiability.\n3.2 Adaptive Diffusion Operator\nEq. (17) focuses attention on important M, but the am-\nplitudes of these Mmay be so small that they cannot\nbe thoroughly sorted out, which requires amplifying their\nquantum amplitudes while shrinking the amplitudes of the\nirrelevant terms by the diffusion operator. The diffusion\noperator in the original Grover algorithm [18] is defined as\nUD=H⊗nX⊗n(MCZ )X⊗nH⊗n. (18)\nAnother geometric interpretation is that Eq. (18) is equiva-\nlent to\nUD= 2|In⟩⟨In| −I, (19)\nwhere |In⟩is the conjugate transpose of ⟨In|.Iis the iden-\ntity matrix. This implies that the symmetry axis of Grover’s\nalgorithm is |In⟩for each iteration, which loses flexibility.\nThus, Eq. (18) is improved and the ADO is proposed.\nDefinition 3.6 (Adaptive Diffusion Operator) .The ADO\nUD=U1(MCZ )U†\n1, (20)\nis shown in Fig. 3, where\nU1=n−1Y\nb=0CRY(θb)[b, f]n−1⊗\nb=0H[b]. (21)\nU†\n1=n−1⊗\nb=0H[b]n−1Y\nb=0CRY(θb+n)[b, f]. (22)\nf=\u001ac+ 1 c̸=n−1\n0 c=n−1. (23)Eq. (21) and Eq. (22) use a quantum coordinate represen-\ntation [25]. Eq. (20) replaces Xin Eq. (18) with CRY(θ)to\nmake it trainable, which indicates that this axis of symmetry\nis no longer fixed and has flexibility.\nAt present, the GQHAM mechanism stands elucidated\ncomprehensively. The ingenuity of GQHAM resides in the\nactivation or masking of DPs through continuous param-\neters, thereby amalgamating diverse discrete choices with-\nout transgressing the tenets of gradient backpropagation.\nThis innovation ultimately surmounts the challenge of non-\ndifferentiability. Furthermore, enhancements have been in-\nstituted in the original expansion operator to augment the\nflexibility of GQHAM.\n4 G ROVER -INSPIRED QUANTUM HARD ATTEN -\nTION NETWORK\nIn this section, GQHAN in Fig. 4 is constructed based on\nGQHAM and its workflow is elaborated.\nFig. 4: Grover-inspired Quantum Hard Attention Network\nAs depicted in Fig. 4, GQHAN is segregated into dual\ncomponents: the ansatz segment in the yellow box and\nthe classical optimizer module in the blue box. Among\nthem, the ansatz segment is constituted by n+ 1 qubits,\nencompassing an ancillary qubit q0in the first quantum\nregister and ninput qubits q1∼qnin the second quantum\nregister. In addition, UAin Fig. 1, U′\nSin Fig. 2 and UDin\nFig. 3 sequentially are quantum amplitude encoding, FO\nand ADO. Ultimately, the data is extracted via quantum\nmeasurement, depicted within the gray box in Fig. 4, and\nsubsequently transmitted to the classical optimizer for fur-\nther processing. Employing classical optimizers, such as the\nquantum natural simultaneous perturbation stochastic ap-\nproximation optimizer [34] or the quantum natural gradient\noptimizer [35], the parameters undergo iterative refinement\nuntil convergence of the loss function is achieved. It is\nnoteworthy that in this paper, the Nesterov momentum\noptimizer [36], a variant amalgamating momentum com-\nponents with gradient descent, is utilized to incorporate\nhistorical gradients into the optimization process.\nIn accordance with the framework depicted in Fig. 4, the\nprecise workflow of GQHAN unfolds as follows.\nStep 1: The initial quantum state\n|Ψ⟩1⊗ |Φ⟩2=|0⟩1⊗ |0⊗n⟩2 (24)\nis prepared in the first and second quantum registers.\nStep 2: Eq. (2) are embedded into GQHAN by quantum\namplitude encoding UA:\n|Ψ⟩1⊗ |Φ⟩2UA− − → | 0⟩1⊗ |In⟩2. (25)6\nStep 3: Eq. (17) is used to adjust the continuous param-\neter during training to partition Eq. (3) into its salient and\nnegligible components, |focus⟩2and|focus⟩2:\n|Ψ⟩1⊗ |Φ⟩2U′\nS− − → | ϕ⟩1⊗(|focus⟩2+|focus⟩2), (26)\nwhere |ϕ⟩1is determined by Eq. (15).\nStep 4: Apply Eq. (20) to dynamically amplify the ampli-\ntudes in Eq. (5) while reducing the other amplitudes to end\nup as\n|Ψ⟩1⊗ |Φ⟩2UD− − → | ϕ⟩1⊗ |^focus⟩2. (27)\nStep 5: For specific Fashion MNIST binary classification,\nthe expectation value\nE=⟨T|P|T⟩ (28)\non the last qubit qnis measured as the predicted label, where\n|T⟩=|Ψ⟩1⊗ |Φ⟩2.Pis the projection operator.\nStep 6: Cost function is\nf(In,θ) =1\nmmX\ni=1[yi−sgn(E)]2, (29)\nwhere sgn (·)is the sign function. mis the number of terms.\nyistands for real label. θare the trainable parameters. The\noptimization rules for the Nesterov momentum optimizer\nare as follows:\n(\nθ(t+1)=θ(t)−a(t+1)\na(t+1)=γ·a(t)+η· ∇f(D,D′,θ(t)−γ·a(t))(30)\nwhere the superscript trepresents the t-th iteration and t+\n1stands for the next iteration. The momentum term γis\nadjustable and generally takes the value 0.9. ηis the learning\nrate.a(t+1)anda(t)are accumulator terms. ∇f(D,D′,θ(t)−\nγ·a(t))denotes the gradient.\nIn the end, all the above steps are repeated until the\ncost function converge. To sum up, by analyzing Step 1\nto 6 above, the operation mechanism of GQHAN in the\nclassification problem is revealed, which lays a theoretical\nfoundation for the experiment.\n5 E XPERIMENT\nIn this section, GQHAN is implemented on the PennyLane\nplatform to conduct binary classification experiments on\nFashion MNIST. Precisely, the experiments are categorized\ninto the subsequent segments.\n•The performance of GQHAN, QSAN [25] and QKSAN\n[27] is evaluated and compared in the Fashion MNIST\nbinary classification task under a uniform classical op-\ntimizer configuration and no noise.\n•A visualization of QHAS is performed to represent the\nresults before and after training.\n•To elucidate the performance on a real quantum com-\nputer, the impact of bit-flip error and amplitude damp-\ning error on GQHAN is scrutinized.5.1 Dataset\nFashion MNIST [38], the recognized and widely adopted\nbenchmark datasets in machine learning, consists of 10,000\ntest images and 60,000 training images, respectively, each\ncontaining 28 by 28 pixel points. In this experiment, 550\nimages labeled 0 and another 550 tagged 1 are taken as the\ndataset from Fashion MNIST. 500 images from each class of\ntags are randomly sampled and assembled into a training\nset with the rest as a test set. Moreover, acknowledging the\nexisting constraint on the number of public qubits, exem-\nplified by the limited provision of 5 to 7 free-to-use qubits\nby IBM, one strategy is to compress the dimensionality of\nall images in the dataset to 8 using the principal component\nanalysis algorithm.\n5.2 Experimental Setting\nTab. 2 delineates the specific experimental configurations\nof GQHAN, QSAN [25] and QKSAN [27], detailing key\nparameters of both the ansatz and classical optimizers. In\nthe ansatz, paramount metrics encompass parameter counts\nin trainable layers, layer quantity, and requisite qubits. Clas-\nsical optimization scrutinizes factors such as learning rate,\nloss function type, batch size, maximum step size, optimizer\ntype, and relevant parameters. The type and configuration\nof the classical optimizer, along with the number of ansatz\nparameters, are maintained as consistently as possible to en-\nsure and underscore equitable comparisons between quan-\ntum models. Additionally, a substantial learning rate is\ndeliberately chosen to expedite convergence.\nTab. 2: Experimental Configuration\nIndicatorsModels\nGQHAN QKSAN [27] QSAN [25]\nparameters 14 14 9\nlayers 7 6 26\nqubits 4 8\nlearning rate 0.09\nloss function square loss\nbatch size 30\nstep 120 steps\noptimizer Nesterov Momentum Optimizer: γ= 0.9\nTab. 3: Critical Data\nIndicatorsModels\nGQHAN QKSAN [27] QSAN [25]\ntest accuracy of the last 10 steps 98.59% 98% 96.8%\ntrain accuracy of the last 10 steps 98.65% 97.22% 96.77%\nnumber of step to start convergence 19 25 55\nconvergence value of the loss function 0.219 0.323 0.42\n5.3 Experimental Analysis\n5.3.1 Classification Experiment\nThe outcomes of the comparative analysis involving\nGQHAN and two quantum soft self-attention mechanisms,7\nFig. 5: Comparison of Fashion MNIST Binary Classification for GQHAN, QKSAN and QSAN.\n-1.79762857 -2.89409402  0.41817778 -2.18221055 -2.54826743  1.30317891 1.73537963 0.98563166\n-1.46214254 -3.00919078 0.1326493 -0.98292757 3.12689563 1.45345429 0.25868764 0.98563166Final Quantum Hard Attention ScoreInitial Quantum Hard Attention Score\nFig. 6: Visualization of Quantum Hard Attention Score\nFig. 7: Noise Experiments of GQHAN\nnamely QKSAN and QSAN, are depicted in Fig. 5. In Fig.\n5, the abscissae of all three subplots represent the training\nsteps, while the ordinates denote the test accuracy, train\naccuracy, and loss function sequentially. The blue, orange\nand green curves delineate GQHAN, QKSAN, and QSAN\ncorrespondingly. According to Fig. 5, the critical data are\nenumerated in Tab. 3. Finally, the following conclusions can\nbe drawn.\n•By averaging the accuracy over the last 10 steps,\nGQHAN attains a test accuracy of 98.59%, surpassing\nQKSAN at 98% and QSAN at 96.8%. Regarding training\naccuracy, it outperforms QKSAN by 1.42% and QSAN\nby 1.87%.\n•In terms of convergence speed, GQHAN initiates con-\nvergence around step 20, slightly lagging behind QSAN\nby approximately 5 steps but significantly outpacing\nQKSAN.\n•Concerning convergence values, GQHAN converges\nto 0.219, compared to 0.323 for QSAN and 0.42 for\nQKSAN, indicating a superior learning capability.\nIn summary, the analysis and comparison from three\nperspectives demonstrates that GQHAN is able to slightly\nsurpass the two quantum soft self-attention mechanisms in\nterms of performance in the Fashion MNIST classification.5.3.2 Visualization of Quantum Hard Attention Score\nThe visualization results of QHAS under the experimental\nconditions of the previous subsection are shown in Fig. 6.\nAll values in Fig. 6 represent the parameters θinRX(θ).\nHere, green color is used to indicate that they are not\nselected, while red color has the opposite meaning of green\ncolor. Since the parameter initialization are random, this\nbatch of parameters does not meet the requirement of Eq.\n(16). Instead, in the last round of training, the parameter\n3.12689563 is marked red and is very close to π, at which\npoint it can be considered as being discretely selected in\nengineering.\n5.3.3 Noise Experiments\nThe amplitude damping and bit-flip noise experiments for\nGQHAN are illustrated in Fig. 7. Except for the blue line\nwhich indicates the experimental results in the absence of\nnoise, Fig. 7 depicts three sets of experiments conducted for\nboth amplitude damping and bit-flip noise, each occurring\nwith probabilities of 0.1, 0.2, and 0.3. Likewise, the mean\nmetrics derived from the final 10 steps upon completion of\nthe training regimen were employed as pivotal assessment\ndata, as delineated in Tab. 4. According to the data presented\nin Tab. 4, discernible conclusions can be derived.8\nTab. 4: Critical Data for Noise Experiments\nModelsIndicators\ntest accuracy train accuracy convergence value\nnoiseless 98.59% 98.65% 0.219\namplitude damping 0.1 97.6% 92.42% 0.269\namplitude damping 0.2 96% 91.24% 0.348\namplitude damping 0.3 92% 89.14% 0.459\nbit flip 0.1 98% 93.27% 0.297\nbit flip 0.2 98% 92.68% 0.416\nbit flip 0.3 96% 91.58% 0.576\n•In assessing the diminution of test and training accu-\nracy, the impact of amplitude damping noise exhibits a\nmore conspicuous manifestation.\n•With equiprobability of noise emergence, the bit-flip\nperturbation induces a more pronounced escalation in\nconvergence values, thereby exacerbating the learning\nefficacy degradation of GQHAN.\nIn total, GQHAN exhibits a degree of resilience against\nboth types of noise. Concerning accuracy, the impact of\nbit-flip noise manifests as a comparatively minor decline\nin intuitive values. However, it is noteworthy that the\ndegradation in its effect on learning performance is more\nconspicuous.\n6 C ONCLUSION\nA quantum HAM, GQHAM, is proposed to compensate for\nthe inability of many current QMLs to recognize the signif-\nicance of quantum data. In addition, FO and ADO are de-\nsigned to overcome the nativity of non-differentiability due\nto discrete selection. A QHAS visualization definition is also\nprovidedon this premise. Ultimately, GQHAN, which can\nbe deployed on a quantum computer, is constructed based\non GQHAM. In binary classification experiments on Fash-\nion MNIST on the PennyLane platform, GQHAN achieves\n98.59% and 98.65% test and train accuracies, respectively,\nand outperforms the two quantum soft self-attention in\nterms of learning ability. In the noise experiments, con-\ntrasted with amplitude damping noise, the impact of bit\nflip noise on GQHAN is relatively diminished in precision\nbut looms larger in learning capabilities. 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Park et al., “Configurable sublinear\ncircuits for quantum state preparation,” Quantum Infor-\nmation Processing, vol. 22, no. 2, pp. 123, 2023.\n[33] A. Kandala, A. Mezzacapo et al., “Hardware-efficient\nvariational quantum eigensolver for small molecules\nand quantum magnets,” Nature, vol. 549, no. 7671, pp.\n242-246, 2017.\n[34] J. Gacon, C. Zoufal et al., “Simultaneous perturbation\nstochastic approximation of the quantum Fisher infor-\nmation,” Quantum, vol. 5, pp. 567, 2021.\n[35] J. Stokes, J. Izaac et al., “Quantum natural gradient,”\nQuantum, vol. 4, pp. 269, 2020.\n[36] Y. Nesterov, “A method for unconstrained convex\nminimization problem with the rate of convergence\nO(1/k2),” in Doklady an ussr, pp. 543-547, 1983.\n[37] Y. Lecun, L. Bottou et al., “Gradient-based learning\napplied to document recognition,” Proceedings of the\nIEEE, vol. 86, no. 11, pp. 2278-2324, 1998.\n[38] H. Xiao, K. Rasul et al., “Fashion-MNIST: A novel\nimage dataset for benchmarking machine learning al-\ngorithms,” arXiv preprint arXiv:1708.07747, 2017.\nRen-Xin Zhao (Member, IEEE) received his B.S. degree from the Col-lege of Automation, Hangzhou Dianzi University, Hangzhou, China in\n2017 and his M.S. degree from the College of Electrical and Information\nEngineering, Hunan University, Changsha, China in 2020. He is now\na PhD student in the School of Computer Science and Engineering\nat Central South University, Changsha, China. His interests include\nquantum machine learning, quantum neural networks, and design and\noptimization of quantum circuits.\nJinjing Shi (Member, IEEE) is now a professor\nin the School of Electronic Information of Cen-\ntral South University. She received her B.S. and\nPh.D. degrees in the School of Information Sci-\nence and Engineering, Central South University,\nChangsha, China, in 2008 and 2013, respec-\ntively. She was selected in the ”Shenghua liey-\ning” talent program of Central South University\nand Special Foundation for Distinguished Y oung\nScientists of Changsha in 2013 and 2019, re-\nspectively. Her research interests include quan-\ntum computation and quantum cryptography. She has presided over\nthe National Natural Science Foundation Project of China and that of\nHunan Province. There are 50 academic papers published in important\ninternational academic journals and conferences. She has received the\nsecond prize of natural science and the outstanding doctoral dissertation\nof Hunan Province in 2015, and she has received the Best Paper Award\nin the international academic conference MSPT2011 and Outstanding\nPaper Award in IEEE ICACT2012.\nXuelong Li (M’02-SM’07-F’12) is with the Institute of Artificial Intelli-\ngence (Tele AI), China Telecom Corp Ltd, 31 Jinrong Street, Beijing\n100033, P . R. China"    },    {        "title": "2401.14091v2.Rapid_evolution_of_the_recurrence_time_in_the_repeating_partial_tidal_disruption_event_eRASSt_J045650_3_203750.pdf",        "content": "Astronomy &Astrophysics manuscript no. main ©ESO 2024\nMarch 5, 2024\nLetter to the Editor\nRapid evolution of the recurrence time in the repeating partial tidal\ndisruption event eRASSt J045650.3 −203750\nZhu Liu1, Taeho Ryu2, A. J. Goodwin3, A. Rau1, D. Homan4, M. Krumpe4, A. Merloni1, I. Grotova1, G. E. Anderson3,\nA. Malyali1, and J. C. A. Miller-Jones3\n1Max-Planck-Institut für extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching, Germany\ne-mail: liuzhu@mpe.mpg.de\n2Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany\n3International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia\n4Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany\nReceived xxx xx, xxxx; accepted xxx xx, xxxx\nABSTRACT\nIn this letter, we present the results from further X-ray and UV observations of the nuclear transient eRASSt J045650.3 −203750\n(hereafter J0456−20). We detected five repeating X-ray and UV flares from J0456 −20, making it one of the most promising repeating\npartial tidal disruption event ( pTDE) candidates. More importantly, we also found rapid changes in the recurrence time Trecurof the\nX-ray flares by modelling the long-term X-ray light curve of J0456 −20.Trecurfirst decreased rapidly from about 300 days to around\n230 days. It continued to decrease to around 190 days with an indication of a constant Trecurevidenced from the latest three cycles.\nOur hydrodynamic simulations suggest that, in the repeating pTDE scenario, such rapid evolution of Trecurcould be reproduced if the\noriginal star is a 1 M ⊙main-sequence star near the terminal age and loses nearly 80–90% of its mass during the initial encounter with\na supermassive black hole (SMBH) of mass around 105M⊙. The inferred mass loss of 0.8–0.9 M ⊙is higher than the estimated value\nof around 0.13 M ⊙from observation, which could be explained if the radiation e fficiency is low (i.e. ≪0.1). Our results indicate that\nrepeating pTDEs could be e ffective tools to explore the dynamics around supermassive black holes beyond our own Galaxy.\nKey words. X-rays: individuals: eRASSt J045650.3 −203750 – Accretion, accretion disks – Galaxies: nuclei – Black hole physics\n1. Introduction\nTidal disruption events (TDEs) are typically considered one-o ff\nevents where a star is completely destroyed by a supermassive\nblack hole (SMBH) at the first pericenter passage. However, the-\noretical calculations and numeric simulations have shown that a\npartial TDE ( pTDE) can also occur (e.g. Guillochon & Ramirez-\nRuiz 2013; Ryu et al. 2020). In a pTDE, the star loses only a frac-\ntion of its mass and survives its first encounter with the SMBH.\nIf the star is initially in a bound orbit with low eccentricity, it is\nexpected to generate repeating flares (e.g. Hayasaki et al. 2013;\nRyu et al. 2020; Nixon & Coughlin 2022; Cufari et al. 2022,\n2023; Melchor et al. 2024). Because the encounter cross sections\nofpTDEs are generally larger than or comparable to those of full\nTDEs, the rate of pTDEs is expected to be larger or comparable\nto full TDEs (Krolik et al. 2020; Bortolas et al. 2023). Repeat-\ningpTDEs are particularly interesting as they may be e ffective\nprobes to explore stellar dynamics around SMBHs beyond our\nown Galaxy and as they provide ideal laboratories to study the\naccretion processes in SMBHs.\nOnly a few repeating pTDE candidates have been reported\nso far (e.g. ASASSN-14ko, Payne et al. 2021; HLX-11, Webb\net al. 2023; eRASSt J045650.3 −203750, hereafter J0456 −20,\nLiu et al. 2023; RX J133157.6 −324319.7, Malyali et al. 2023;\nAT2018fyk, Wevers et al. 2023). Among these, J0456 −20 is one\nof the best-studied repeating nuclear transients discovered in a\n1Note that HLX-1 is believed to be an intermediate mass black hole\n(IMBH) with mass of ∼104−5M⊙(Davis et al. 2011; Webb et al. 2012)quiescent galaxy ( z=0.077). Liu et al. (2023) reported the de-\ntection of three repeated X-ray and UV flares from J0456 −20. In\nparticular, the profiles of the X-ray flares are similar and can be\ncharacterised by four distinctive phases: an X-ray rising phase\n(PX, rise ) leading into an X-ray plateau phase ( PX, plat ), which is\nterminated by a rapid X-ray drop phase ( PX, drop ), and followed\nby an X-ray faint state ( PX, faint ). These results provide strong\nevidence that J0456 −20 is a repeating nuclear transient, making\nJ0456−20 one of the most promising pTDE candidates.\nQuasi-period eruptions (QPEs) are a class of recurring X-ray\nflares found in galactic nuclei, with periods less than one day\n(Miniutti et al. 2019; Giustini et al. 2020; Arcodia et al. 2021).\nTheir origin remains elusive, though recent studies indicate a\npotential link to pTDEs (e.g. GSN 069, Miniutti et al. 2023;\nRX J133157.6−324319.7, Malyali et al. 2023). The discovery of\nSwift J0230, which exhibits QPE-like behaviours with a period\nof around 22 days (Evans et al. 2023; Guolo et al. 2024), further\nstrengthens the link between repeating pTDEs and QPEs. It is\nthus interesting to study the evolution of Trecurof the flares for re-\npeating pTDEs. However, substantial evolution in Trecurhas only\nbeen reported in ASASSN-14ko and HLX-1. A period derivative\nof−0.0026±0.0006, with a period of 115 .2+1.3\n−1.2days, has been\nreported for ASASSN-14ko (Payne et al. 2022). HLX-1 initially\nshows quasi-periodic X-ray outbursts spaced by about 1 yr be-\ntween 2009 and 2012. The Trecurthen started to increase until\n2018, after which no X-ray outbursts were detected (Godet et al.\n2014; Webb et al. 2023). Long-term multi-wavelength observa-\ntions on a larger sample of repeating pTDEs are required to fully\nArticle number, page 1 of 13arXiv:2401.14091v2  [astro-ph.HE]  4 Mar 2024A&A proofs: manuscript no. main\nunderstand the evolution of Trecurin these objects and potentially\nprovide further evidence to confirm the connection between re-\npeating pTDEs and QPEs.\nIn this letter, we report the discovery of the rapid evolution\nofTrecurin J0456−20. A detailed analysis of the latest X-ray and\nUV data reveals five X-ray and UV flares in J0456 −20 (marked\nas cycle 1–5 in Fig. 1). These results confirm that J0456 −20 is\nindeed a repeating nuclear transient. In addition, they also sug-\ngest an initially rapid decrease in Trecurby more than two months\nbetween cycles 1 and 2, slowing to around 40 days between cy-\ncles 2 and 3. Trecurreaches an almost constant value of around\n190 days in the latest three cycles. This paper is structured as\nfollows. In Sect. 2, we present the multi-wavelength data reduc-\ntion. X-ray spectral and light curve modelling are presented in\nSect. 3. Finally, we discuss and summarise our results in Sects. 4\nand 5. Throughout this paper, we adopt a flat ΛCDM cosmology\nwith H0=67.7 km s−1Mpc−1andΩm=0.308 (Planck Collab-\noration et al. 2020). Therefore z=0.077 corresponds to a lu-\nminosity distance of Dld=360 Mpc. All magnitudes will be re-\nported in the AB system (not corrected for Galactic extinction).\nAll the quoted uncertainties correspond to the 90% confidence\nlevel, unless specified otherwise.\n2. Observations and Data Reduction\nIn this paper, we analysed all observations after MJD 59720. The\nreader should refer to Liu et al. (2023) for the details of data\nanalysis before this date.\n2.1. XMM-Newton\nA pre-approved XMM-Newton target of opportunity (ToO) ob-\nservation was performed on 2022 Aug 23 (hereafter X5). In ad-\ndition, two XMM-Newton Director’s Discretionary Time (DDT)\nobservations were performed on 2023 Feb 10 and Mar 4 (here-\nafter X6 and X7, respectively). J0456 −20 was detected in all new\nXMM-Newton observations.\nThe Observation Data Files (ODFs) for each observation\nwere reduced using the XMM-Newton Science Analysis System\nsoftware (SAS, version 19.1 , Gabriel et al. 2004) with the latest\ncalibration files. For each observation, the SAS tasks emchain\nandepchain tasks were used to generate the event lists for the\nEuropean Photon Imaging Camera (EPIC) MOS (Turner et al.\n2001) and pn (Strüder et al. 2001) detectors, respectively. High\nbackground flaring periods were identified and filtered from the\nevent lists. For all the EPIC images, a circular region with a radii\nof 30,20,25′′was chosen as the source region for X5, X6, and\nX7, respectively. A source-free annulus region with an inner ra-\ndius of 50′′and an outer radius of 100′′was chosen as the back-\nground region for all MOS observations. The background for the\npn camera was extracted from a circular region with a radius of\n60′′centred at the same CCD read-out column as the source po-\nsition for all observations. X-ray events with pattern ≤12 for\nMOS and≤4 for pn were selected to extract the X-ray spectra.\nWe used the tasks rmfgen andarfgen to generate the response\nmatrix and ancillary files, respectively. The X-ray spectra were\nrebinned to have at least 1 count per bin.\n2.2. Swift observations\nThe XRT online data analysis tool2(Evans et al. 2009) was used\nto check whether the source was detected for each observation.\n2http://www.swift.ac.uk/user_objectsIt was also used to generate the X-ray spectra for observations in\nwhich J0456−20 was detected and to calculate the 3 σcount rate\nupper limits for non-detections. The X-ray spectra were rebinned\nto have at least 1 count in each bin.\nTheSwift/UVOT data were reduced using the UVOT analy-\nsis pipeline provided in HEAS oft(version 6.31 ) with UVOT\ncalibration version 20201215 . Source counts were extracted\nfrom a circular region with a radius of 5′′centred at the source\nposition. A 20′′radius circle from a source-free region close to\nthe position of J0456 −20 was chosen as the background region.\nThe task uvotsource was used to extract the photometry.\n2.3. NICER observations\nTheNICER data were analysed using HEAS oftwith the NICER\ndata analysis software (version 10) and calibration files (ver-\nsion20221001 ). The nicerl2 task is used to generate cleaned\nX-ray events. Events with overshoot higher than 1.5 or under-\nshoot larger than 300 were removed. The nicerl3-spec task\nwas then used to generate the X-ray spectra for each NICER ob-\nservation. The X-ray spectra were then rebinned to have at least\none count per bin. We adopted the SCORPEON model to gener-\nate background models for each observation. The nicerarf and\nnicerrmf tasks were used to generate the response matrix and\nancillary file for each observation, respectively. The same pro-\ncedures were also adopted to re-analyse the NICER observations\ntaken during the first two X-ray rising phases (i.e. observation\ncarried out during MJD 59418-59448 and 59600-59641).\n2.4. Chandra observations\nWe requested Chandra DDT observations of J0456 −20, which\nwere performed on 2023 Mar 18, Apr 11, and April 14 with\nthe Advanced CCD Imaging Spectrometer (ACIS). We used\nthe CIAO (Fruscione et al. 2006, version 4.15 ) software pack-\nage to reduce the Chandra data with calibration files CALDB\nversion 4.10.4 . We reprocessed the Chandra data using the\nCIAO script chandra_repro . The CIAO task dmextract was\nused to extract the source and background spectra. We ex-\ntracted the source spectra using a circular region with a ra-\ndius of 2′′. The background spectra were extracted using an\nannulus (concentric with the source) region with an inner and\nouter radius of 6′′and 20′′, respectively. The response files were\ngenerated using the mkacisrmf andmkarf tasks. The posi-\ntion of the X-ray flare measured from Chandra is (RA, Dec) =\n(04:56:49.81 ,−20◦37′47.98′′) with a 2σuncertainty of 0 .54′′\n(Appendix A), consistent with the centre of the host galaxy.\n2.5. ATCA radio observations\nWe observed the coordinates of J0456 −20 nine times with the\nAustralia Telescope Compact Array (ATCA) between 2022 Aug\nand 2023 Mar, in addition to the five observations between 2021\nMar and 2022 Apr reported in Liu et al. (2023). We observed\nthe target during the X-ray outburst phase as this was previously\nwhen radio emission had been detected and therefore observed\nthe target with the array in various configurations. In each ob-\nservation, we used the dual 5.5 GHz and 9 GHz receiver, placing\nthe 2x2 GHz of bandwidth split into 2048x1 MHz channels at a\ncentral frequency of 5.5 GHz and 9 GHz. Data were reduced in\nthe Common Astronomy Software Application (CASA, verison\n5.6.3 , CASA Team et al. 2022) using standard procedures, in-\ncluding flux and bandpass calibration with PKS 1934-638 and\nArticle number, page 2 of 13Liu et al.: Rapid recurrence time evolution in J0456 −20\n1014\n1013\n1012\n1011\nfunabs,rest,0.22.0keV(ergs1cm2)\ncycle 1\n cycle 2 cycle 3 cycle 4 cycle 5NICER Swift eROSITA XMM-Newton ChandraPlateau Rapid drop Faint phase Rising phase\n21\n22\n23mAB\n59100 59300 59400 59500 59600 59700 59800 59900 60000 60100\nMJD\nATCA Radio UVOT/UVM2\n17\n16\n15\nMAB104210431044Lunabs,rest,0.22.0keV(ergs1)\nFig. 1. Long-term X-ray and UV light curves for J0456 −20. The coloured regions represent the four phases: the plateau phase ( PX, plat, light cyan),\nthe rapid drop phase ( PX, drop , light blue), the faint phase ( PX, faint , light yellow), and the rising phase ( PX, rise, light green). Upper panel: Red points\nwith error bars are the unabsorbed rest-frame 0 .2−2.0 keV X-ray light curve from eROSITA (hexagons), Swift/XRT (squares), NICER (circles),\nXMM-Newton (stars), and Chandra (diamonds). The error bars indicate 90% uncertainties. The points with downward arrows represent the 3 σ\nflux/luminosity upper limits. The points with blue colour are the data used to model the profiles of the PX, rise phase. The green dashed lines show\nthe best-fitting power-law model for the five X-ray rising phases. The grey shaded regions mark the 1 σuncertainty of the model. Bottom panel:\nUV light curve from Swift/UVOT UVM2 (red squares). The error bars mark the 1 σuncertainties. Squares with downward arrows indicate 3 σ\nupper limits. The vertical lines mark the dates of the ATCA radio observations (black dashed-dotted: non-detections; blue dashed: detections).\nphase calibration with PKS 0454-234. Additionally, we carried\nout one round of phase-only self-calibration of the target field\nat both 5.5 and 9 GHz, with a typical solution interval of 2 min-\nutes, to produce a good quality image due to a bright AGN in\nthe field. Images of the target field were created with the CASA\ntasktclean and where a source was visible at the location of\nJ0456−20, the flux density was extracted with the CASA task\nimfit by fitting a Gaussian the size of the synthesised beam.\n3. Data analysis and results\n3.1. X-ray spectral modelling\nThe X spec software (version 12.13.0 , Arnaud 1996) was used\nto fit all X-ray spectra using the Cash statistic (Cash 1979,\nCstat in X spec). As mentioned in Liu et al. (2023), a power-\nlaw model (i.e. TBabs*zashift*cflux*powerlaw , hereafter\nMpl) is preferred for observations taken at relatively high X-\nray flux (i.e. the rest-frame unabsorbed 0.2–2.0 keV flux, fX, soft ,\n≳5×10−13erg cm−2s−1). We thus first fitted all the new X-ray\nspectra with the M plmodel. The Galactic column density is fixed\nat 3.3×1020cm−2.\nForNICER data, we first fitted the total X-ray spectra over\nthe 0.25–10.0 keV range with the backgrounds model gener-\nated using SCORPEON. We then re-fitted the data by adding\nthe M plmodel. A 3 σupper limit for fX, soft was estimated\nfor observations in which the fit did not improve significantly\n(i.e.∆Cstat<11.8) after adding the M plmodel. A strong\noxygen Kαline is presented in some of the NICER data.\ncycle[1-2] cycle[2-3] cycle[3-4] cycle[4-5]180200220240260280300Recurrence time (day)\n∆M=0.13M⊙\n∆M=0.05M⊙\n∆M=0.02M⊙∆M=0.017M⊙TrecurfromPX,risephases\nTrecurfromPX,dropphasesFig. 2. Evolution of the recurrence time. The purple square (blue circle)\npoints represent Trecurmeasured from the PX, rise (PX, drop ) phases. The\nerror bars mark the 1 σuncertainties. The upward arrow indicates the 3 σ\nlower limit. The estimated total mass loss from the star is also marked.\nTheniscorpv22_swcxok_norm parameter in the background\nmodel was thus left free to properly model the oxygen K αline\nin those observations. We fitted the background subtracted spec-\nArticle number, page 3 of 13A&A proofs: manuscript no. main\ntra with the M plmodel for the data from the other missions. The\nSwift/XRT X-ray spectra were fitted over the 0.3–5.0 keV energy\nrange, while the 0.5–5.0 keV energy band was used for Chandra\nobservations. For XMM-Newton data, we jointly fitted the data\nfrom the three EPIC cameras over the 0.2–5.0 keV energy range\nfor X5 and X7 (0.2–2.0 keV for X6). The best-fitting values and\nthe 90 percent uncertainties were calculated for the fX, soft and\nphoton index parameters. In addition, the 68 percent uncertain-\nties for the fX, soft were also estimated for observations taken dur-\ning the PX, rise phase. The 3 σupper limits of fX, soft were calcu-\nlated for non-detections using either the M plor a disk model (see\nbelow). The details of the fitting results are listed in Table C.2.\nLiu et al. (2023) noticed that the UV-to-X-ray SEDs of\nJ0456−20 can be described by a multi-colour disk model\n(TBabs*zashift*cflux*diskbb , hereafter M mcd) when the\nX-ray flux is low. Thus, the M mcd model was also used\nto fitted the X-ray spectra at the early stage of the PX, rise\nphase, that is, observations taken between MJD 59600 and\n59620 in cycle 3 and X6 in cycle 5. J0456 −20 was de-\ntected only on MJD 59619 ( NICER , ObsID: 4595020126, here-\nafter N26) and on MJD 59999 (X6). The best-fitting Tinand\nfX, soft are 193+43\n−25(78+24\n−18) eV and 2 .0+0.5\n−0.3×10−14(2.3+1.2\n−0.5×\n10−13) erg cm−2s−1for X6 (N26), respectively. The M plmodel\nresulted in a higher fX, soft of 3.3+1.0\n−0.6×10−14(4.6+1.1\n−1.6×\n10−13) erg cm−2s−1with photon index of 3 .3+0.3\n−0.4(5.3+0.7\n−0.8) for X6\n(N26). Both the M mcdand the M plmodel fitted the X6 and N26\nspectra well, though the M plmodel gives a slightly better fit\n(Cstat/d.o.f =136/144, compared to 152 /144) for X6.\nIn this work, we used the results from the M mcdmodel\nfor X6 and N26. The best-fitting Tinof N26 is comparable to\nthat obtained from the eRASS2 and Swift observations (in the\nrange of∼50−100 eV , Liu et al. 2023) at similar fX, soft (i.e.\n≈4×10−13erg cm−2s−1), while Tinis much higher during the\nX6 observation when J0456 −20 was in a historically low X-ray\nflux. This suggests a potential change in Tinduring the PX, rise\nphase. For this reason, di fferent values of Tin(listed in Table C.2)\nwere used to calculate the 3 σflux upper limits for NICER obser-\nvations before N26 during cycle 3.\n3.2. Modelling the X-ray rising phase\nThePX, rise phases during cycles 2, 3, 4 and 5 were captured by\nour follow-up observations. We also assumed that J0456 −20 is\nin the PX, rise phase of cycle1 during the eRASS2 observations.\nThis is justified by the duration of the PX, drop phase being much\nshorter than the PX, rise phase, and by the spectral property of\neRASS2 (i.e. best described by the M mcdmodel) being similar\nto that at the early stage of the PX, rise phase.\nWe jointly fitted the five PX, rise phases with a power law\nfunction frs,i(t)=A∗(t−ti)β, where i=1,2,3,4,5. We assumed\nthat the normalisation Aand power law index βare the same\nfor all the five PX, rise phases. The lmfit package is adopted to\nfit the data with the least squares method. To take into account\nupper limits and to estimate the uncertainties of the parameters\nin the model, we generated 105realisations of the fX, soft in the\nfivePX, rise phases from Gaussian distributions for observations\nof which J0456−20 is detected and uniform distributions with\na lower limit of zero (excluded) for non-detections. The best-\nfitting fX, soft and the 1σuncertainties obtained from X-ray spec-\ntral modelling of each observation are used as the means and\nstandard deviations of the Gaussian distributions, respectively.\nThe 3σfX, soft upper limits for non-detections are used as the up-\nper limits for the uniform distributions. We obtained 105valuesTable 1. Results of the X-ray rising phase modelling and estimations of\nTrecur.\nParameters Cycles[1,2,5] All cycles Trecur Trecur, drop\nMJD MJD days days\nt1 59070+7\n−859065+8\n−9299+6\n−6227±3t2 59367+5\n−559363+6\n−6227+4\n−3188±3t3 — 59591+6\n−5188+3\n−3178±3t4 — 59778+7\n−7193+3\n−3>189t5 59976+3\n−459972+4\n−5\nlogA−16.2+0.6\n−0.8−17.2+1.0\n−1.2\nβ 2.6+0.4\n−0.33.1+0.6\n−0.5\nNotes. Parameters : name of the parameter in the power-law model; Cy-\ncles[1,2,5] : results by jointly fitting the PX, rise phases of cycles 1, 2, and\n5;All cycles : results by jointly fitting the PX, rise of all the five cycles;\nTrecurandTrecur, drop are the recurrence time estimated using the PX, rise\nandPX, drop phases, respectively.\nfor each parameter in the model by fitting the 105datasets using\nthe least squares method. We estimated the best-fitting values us-\ning the median values for each parameter and estimated the 1 σ\nconfidence intervals using the 16 and 84 percentiles of fitting re-\nsults. Similarly, the recurrence time Trecurfor cycle iandi+1, the\nlower, and the upper intervals of the 1 σconfidence intervals are\nestimated using the median, the 16, and 84 percentile of a sample\ncalculated using ti+1−ti. To test whether the results could be af-\nfected by the inclusion of flux upper limits, we applied the same\nprocedures to cycles 1, 2, and 5 only (i.e. cycles without upper\nlimits in the PX, rise phases). We found the results to be consis-\ntent within 1 σuncertainties (see Table 1). The fitting results and\nthe estimated Trecurare listed in Table 1. We also calculated the\nrecurrence time using the PX, drop phase (Appendix B). The esti-\nmated values of Trecurare also listed in Table 1. It is clear from\nFig. 2 that the Trecurof the X-ray flares in J0456 −20 show rapid\nchanges. The mass loss for each cycle marked in Fig. 2 are cal-\nculated using the method outlined in Sect. 3.3.\n3.3. Estimation of the mass loss\nFollowing Liu et al. (2023), a cycle is defined as the time be-\ntween the start of two consecutive PX, rise phases. The total en-\nergy released in each cycle was estimated by\nEtot=Lfa∆tfa+Lpl∆tpl+Z∆trs\n0Lrs(t) dt, (1)\nwhere Lrsis the bolometric luminosity during the PX, rise\nphase, which is calculated using the best-fitting power law\n(see Sect. 3.2), that is, Lrs(t)=κ4πD2\nldfrs(t), whereκ=\nLbol/LX, res 0.2-2.0 keV is the bolometric correction factor and Dld\nis the luminosity distance. LplandLfais the average bolometric\nluminosity during each of the PX, plat andPX, faint phases, respec-\ntively. ∆tfa,∆tpl, and ∆trsare the duration of the PX, faint ,PX, plat ,\nandPX, rise phases, respectively. The PX, faint phase in cycle 5 was\nnot covered by our follow-up observations. In this work, we as-\nsumed ∆tfa=70 days for cycle 5. Liu et al. (2023) estimated the\nκto be in the range of 3 −20 during the PX, rise phase, calculated\nby modelling the UV to X-ray data using either the M mcd(when\nJ0456−20 is X-ray faint) or Comptonized M mcdmodel. The Tin\nof the M mcdmodel is around 40 −60 eV and does not change\nsignificantly (see also Fig. 10 in Liu et al. 2023). Following Liu\nArticle number, page 4 of 13Liu et al.: Rapid recurrence time evolution in J0456 −20\n0.01 0.1 1.0 10\nEnergy (keV)105\n104\n103\nE2F(E)(keV2Photonscm2s1keV1)\nAbsorbed model Unabsorbed model Unabsorbed diskbb\nSwift5\nXMM-Newton/X3\nFig. 3. UV to X-ray SEDs at di fferent X-ray flux levels. The solid circles\nare the unfolded spectra for Swift5 (blue) and XMM-Newton /X3 (pur-\nple). The dotted lines are the absorbed models. The solid and dashed\nlines represent the unabsorbed intrinsic model and the diskbb compo-\nnent, respectively.\net al. (2023), we adopted a value of κ=15 to calculate Lrsdur-\ning the PX, rise phases. The high quality X-ray and UV data taken\nduring the PX, plat phase of cycle 2 (the third XMM-Newton ob-\nservation, hereafter X3) can be best-fitted with a multi-color disk\n(with Tinaround 60 eV) Comptonized by two coronae ( XMM-\nNewton /X3 in Fig. 3, see also Table 3 in Liu et al. 2023 for the\nbest-fitting results for X3). We estimated a value of κ∼3 using\nthis model. Both the X-ray and UV only show mild variability\nduring the PX, plat phases, indicating that κwill not change sig-\nnificantly. We thus calculated Lplassumingκ=3 for the PX, plat\nphases. Lfais poorly constrained. We thus estimated Lfausing\nthe M mcdmodel ( Tin∼45 eV) obtained by modelling the UV\nand X-ray emission from the first Swift observation in cycle 2\n(Swift5 in Fig. 3). This model resulted in a bolometric luminos-\nity of Lbol, sw =9.0×1043erg s−1. Considering that the peak UV\nmagnitude in PX, faint phases is ≲0.6 mag brighter than that dur-\ning the PX, rise phases, we thus conservatively estimated Lfaby\nmultiplying Lbol, sw by a factor of 1.8, which leads to a value of\n1.6×1044erg s−1.\nThe values of the parameters used to calculate Etotfor each\ncycle can be found in Table 2. To calculate the total mass loss\nin each cycle, we assumed that half of the tidally disrupted de-\nbris returns to the SMBH and that the radiation e fficiency is 0.1.\nWe estimated a total released energy (mass loss) of 1.12 (0.13),\n0.46 (0.05), 0.18 (0.02), 0.15 (0.017), and ≳0.12×1052erg\n(≳0.014 M⊙) for cycles 1, 2, 3, 4, and 5, respectively. A lower\nlimit is given for cycle 5 as the PX, plat phase could be longer then\nthe value quoted in Table 2.\n3.4. Radio variability\nJ0456−20 was mostly undetected after 2022 Aug, with a detec-\ntion at 9 GHz on 2022 Oct 05 and at 5.5 GHz on 2022 Oct 07\nduring the PX, plat phase of cycle 4 (see Fig. 1) with a significance\nof 7 and 9σ, respectively. In order to improve the sensitivity, we\nadditionally stacked the observations from October 2-7 and de-\ntected a faint point source at the coordinates of J0456 −20. The\nradio observations reported here indicate that the transient radioTable 2. Values of the parameters used to calculate the total release\nenergy ( Etot) and mass loss ( ∆M) in each cycle.\ncycle 1 cycle 2 cycle 3 cycle 4 cycle 5\nts(MJD) 59065 59363 59591 59778 59972\n∆trs(day) 100 84 51 39 38\n∆tpl(day) 120 65 57 61 ≳70\n∆tfa(day) 61 71 68 85 70\nLpl(1044erg s−1) 4.6 1.1 0.70 0.27 0.13\nEtot(1052erg) 1.12 0.46 0.18 0.15 ≳0.12\n∆M(M⊙) 0.13 0.05 0.02 0.017 ≳0.014\nNotes. tsis the start date of the cycle. ∆trs,∆tpl, and ∆tfaare the duration\nfor the PX, rise,PX, plat, and PX, faint phases, respectively. The estimated\nvalues for Etotand∆Mare also listed.\nsource associated with the X-ray outbursts in 2022 Mar and Apr\nhas also been fading, consistent with the decrease in the peak X-\nray flux of each cycle (see Fig. 1). A summary of the ATCA radio\nobservations of J0456 −20 is given in Table C.1. The long-term\nradio light curve is shown in Fig. C.1.\n4. Discussion\nWe have detected five repeating X-ray flares in J0456 −20 us-\ning the latest data. In addition, repeating transient radio emission\nhas also been detected in J0456 −20. These results provide fur-\nther evidence that J0456 −20 is a repeating nuclear transient, and\nmake it one of the most promising repeating pTDE candidates.\nMore importantly, our results also revealed rapid evolution of the\nrecurrence time Trecur, measured from the PX, rise phases, of the\nX-ray flares. Trecurdecreased by more than 2 months between\ncycles 1 and 2. It continued to decrease by roughly 40 days be-\ntween cycles 2 and 3. Such a rapid change is likely terminated\nas suggested by an almost constant Trecur(∼190 days) measured\nfrom the latest three cycles. Evidence for evolution of Trecurare\nalso found using the values measured from the PX, drop phases\n(Appendix B and Table 1). Though, the changes in Trecur are\nless dramatic than those measured from the PX, rise phases, which\nmay attributed to the changes in the duration of the other phases\n(see Table 2). In this work, Trecurderived from the PX, rise phase\nwere used.\nThe evolution of Trecurhas been reported only in a few repeat-\ningpTDE candidates. For instance, ASASSN-14ko showed a de-\ncrease in the periods with a period derivative of around −0.0026\n(Payne et al. 2022; Huang et al. 2023), which is much shorter\nthan that found in J0456 −20 (≲−0.2). The X-ray flares in HLX-\n1 initially showed a quasi-periodic Trecurof around 1 yr. Trecur\nthen increased by about 1 month in 2013 (Godet et al. 2014),\nand continued to increase until 2018, after which no X-ray out-\nbursts were detected (Webb et al. 2023). Unlike in HLX-1, we\nfound no evidence for an increase in Trecurin J0456−20 as of\nnow. As suggested by Godet et al. (2014), the evolution of Trecur\ncan put strong constraints on the initial mass, the mass loss, and\nthe orbital parameters of the star.\nA repeating pTDE is favoured to explain the long-term multi-\nwavelength light curve of J0456 −20 (Liu et al. 2023). We thus\nperformed simulations to test if the changes in Trecur,∆P, can\nbe explained by pTDEs. We made a grid of hydrodynamics sim-\nulations, using the moving-mesh code AREPO (Springel 2010;\nWeinberger et al. 2020; Pakmor et al. 2016), where we exam-\nined the change in the orbital period of remnants produced in\npTDEs of main-sequence stars by BHs with a mass of MBH=\nArticle number, page 5 of 13A&A proofs: manuscript no. main\n105M⊙. We considered Solar metallicity main-sequence stars\nwith masses of M⋆=1, 2, and 3 M ⊙, and a core Hydro-\ngen mass fraction of 0.01 (terminal-age) and 0.3 (middle-age),\nevolved using the 1D stellar evolution code MESA (Paxton et al.\n2013, 2015, 2019, 2011), imported into AREPO with 0.5M cells3.\nWe also considered a wide range of the pericenter distance rp,\n0.1≲rp/rt≲1.2 (tidal radius rt=(MBH/M⋆)1/3R⋆), encom-\npassing scenarios from the full disruption to no mass loss. We\nvaried the pericenter distance for a given M⋆. Meanwhile, we\nfixed the orbital period of the original orbit to be 300 days, so\nthe stellar orbit in each simulation has a di fferent eccentricity\n(e≳0.99). The initial separation between the black hole and\nthe star is 5 rt. We followed the evolution of the remnant af-\nter the first pericenter passage of the original star, using the\nHelmholtz equation of state (Timmes & Swesty 2000) until the\npost-disruption orbital parameters do not evolve, which occurred\nwhen the separation between the black hole and the remnant is\n≳5rt. We verified that the total energy in all simulations was\nconserved within a fractional error of ≲10−5. We note that a\nrapid decrease in Trecurrequires a BH mass of around 105M⊙\nin our simulations which ran with a limited range of parameter\nspace. Our simualtions with a higher BH mass of 106M⊙cannot\nreproduce Trecurobserved in J0456 −20. The required BH mass\nof 105M⊙is much smaller than the value quoted in Liu et al.\n(2023), which is around 107M⊙. However, as cautioned in Liu\net al. (2023), the values of MBHmeasured from the MBH−σ⋆\nrelation and the σ2\nrms−MBHrelation exhibit notable di fferences\nand have large uncertainties. Thus a MBHon the order of 105M⊙\ncould still be possible for J0456 −20.\nFig. 4 depicts ∆Pas a function of the original stellar mass\nM⋆and the fractional mass loss ∆M/M⋆. The general trend is\nthat∆Pdecreases as ∆M/M⋆increases until the fractional mass\nloss exceeds a critical value ∆mc, roughly≈0.7−0.8. Above\nthe critical mass loss, ∆Pstarts to increase with ∆M/M⋆. This\ntrend was observed for parabolic pTDEs in Ryu et al. (2020).\nHowever, the values of ∆Pstrongly depend on the internal struc-\nture of the star (i.e. mass and age). For pTDEs of middle-age\nstars, ∆Pis negative (positive) when ∆M/M⋆is below (above)\n∆mc, meaning the remnants become more (less) bound than that\nof the original star before the TDE. For this case, ∆Pis at most\n≃−10 days. However, ∆Pis negative across a wider range of\n∆M/M⋆and can be as large as −90 days for pTDEs of terminal-\nage stars. Most notably, the large ∆P(i.e.∼−70 days) observed\nin J0456−20 between cycles 1 and 2 can be reproduced if the\noriginal star is a 1M ⊙terminal-age star and loses nearly 80-\n90 % of its mass. While we have not explored the entire pa-\nrameter space of pTDEs, our simulations suggest that the ob-\nserved decrease in Trecurmay be explained by a severe pTDE of\na main-sequence star near the terminal age. The inferred mass\nloss (0.8−0.9 M⊙) is much higher than that estimated for cycle\n1 (∼0.13 M⊙, Sect. 3.3 and Table 2) of J0456 −20. We note that\nthis discrepancy can be alleviated if the radiation e fficiency in\nJ0456−20 is much lower than the assumed value of 0.1. It is im-\nportant to also emphasise that our simulation result merely sug-\ngests pTDE is a plausible mechanism for generating a transient\nlike J0456−20 and does not rule out other potential mechanisms.\nFor instance, Linial & Quataert (2024) summarized several or-\ngins of observed period evolution in repeating nuclear transients.\nThey suggested that the period evolution in ASASSN-14ko is\nconsistent with orbital decay induced by hydrodynamic drag as\nthe star passes through an accretion disk (see also Zhou et al.\n3We confirmed that simulations with di fferent resolutions between\n0.25M and 1M cells yield converging ∆P.\nFig. 4. Change in the orbital period ∆P(in days) for remnants in pTDEs\nof main-sequence stars with mass of M⋆=1, 2, and 3 M ⊙in hydrody-\nnamics simulations, as a function of the fractional mass loss ∆M/M⋆.\nThe original stars are Solar metallicity main-sequence stars with a core\nHydrogen mass fraction of XH≃0.01 (Terminal-age, solid lines) and\n0.3 (Middle-age, dashed lines). The simulations suggest that ∆Pin se-\nvere pTDEs of terminal-age main-sequence stars can be compatible\nwith the change in Trecurbetween cycles 1 and 2 ( ≃ −70 days, grey\nhorizontal line) of J0456-20.\n2024). The repeating flares in ASASSN-14ko could then be pow-\nered primarily through dragging-induced stripping of mass from\nthe star. This scenario, however, is unlikely to be the dominant\nprocess in J0456−20, as it requires an accretion disk with mass\nlarger than 10 M ⊙(Eq. 20 in Linial & Quataert 2024, assuming\nMBH=105M⊙) around rpto explain the observed changes in\nTrecurin J0456−20.\n5. Summary\nIn this letter, we analysed new multi-wavelength observations\nfor the promising repeating pTDE candidate J0456 −20. We have\ndetected five repeating X-ray flares and repeating transient ra-\ndio emission from J0456 −20, providing additional strong evi-\ndence that J0456−20 is a repeating nuclear transient. In addi-\ntion, the latest X-ray data also reveals changes in the recurrence\ntime Trecurof the X-ray flare. Trecurinitially decreased rapidly\nfrom∼300 days to∼230 days, it continues to decrease by\nroughly 40 days per cycle with an indication of constant val-\nues of∼190 days in the latest cycles. Our hydrodynamic sim-\nulations show that the large decrease in Trecurcan be explained\nin the pTDE scenario, providing that the original star is a 1 M ⊙\nterminal-age star with an initial fractional mass loss of around\n80-90%. Our results suggest that precise estimations of Trecurin\nrepeating pTDEs can provide additional constraints on the ini-\ntial mass of the disrupted star and mass loss during each pas-\nsage. They also indicate that repeating pTDEs could be e ffective\ntools to probe the stellar /gas dynamics around SMBHs beyond\nour own Galaxy.\nAcknowledgements. ZL is grateful to the XMM-Newton ,Swift , and NICER teams\nfor approving the ToO /DDT requests and arranging the follow-up observations.\nThe hydrodynamics simulations were conducted using computational resources\n(and/or scientific computing services) at the Max-Planck Computing & Data Fa-\ncility. DH acknowledges support from DLR grant FKZ 50 OR 2003. MK is sup-\nported by DFG grant KR 3338 /4-1 and DLR grant 50 OR 2307. 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A., Barret, D., Godet, O., et al. 2023, Astronomische Nachrichten, 344,\neasna.20230051\nWeinberger, R., Springel, V ., & Pakmor, R. 2020, ApJS, 248, 32\nWevers, T., Coughlin, E. R., Pasham, D. R., et al. 2023, ApJL, 942, L33\nZhou, C., Huang, L., Guo, K., Li, Y .-P., & Pan, Z. 2024, arXiv e-prints,\narXiv:2401.11190\nArticle number, page 7 of 13A&A proofs: manuscript no. main\n4h56m50.0s49.8s49.6s-20°37'44\"\n46\"\n48\"\n50\"\n52\"\nRight Ascension (J2000)Declination (J2000)\n1''GAIA\nATCA\nSwift/UVOT\nChandra\nXMM-Newton\neRASS3\nFig. A.1. Updated multi-wavelength positions of J0456 −20. The black\ndot marks the position given in the Gaia EDR3. The crosses represent\nthe positions measured from di fferent instruments, and the circles indi-\ncate the 2σpositional uncertainties: Swift/UVOT (green, 2 σ=0.52′′),\nChandra (red, 2σ=0.54′′), ATCA (white, 2 σ=1.0′′),XMM-Newton\n(royalblue, 2 σ=1.6′′), and eRASS3 (orange, 2 σ=2.0′′).\nAppendix A: Chandra astrometric correction\nWe ran the wavdetect tool on the Chandra data to generate a\nsource list for each observation. J0456 −20 was detected in all\nthe three Chandra observations. We used the source positions\nprovided in the Legacy Survey DR10 catalogue as a reference.\nThe overall 90 per cent absolute astrometry uncertainty of Chan-\ndrais∼1.11 arcsec4. We thus cross-matched the positions of\nthe X-ray source measured from the Chandra observations with\nthe source positions listed in the DR10 catalogue using a ra-\ndius of 1.1 arcsec. We selected sources within 3 arcmin of the\naimpoint of the telescope. This resulted in three sources (ex-\ncluding J0456−20, see A.1) that are detected in both X-ray and\nDR10 in the second Chandra observation (only two for the other\ntwo observations). The CIAO task wcs_match and wcs_update\nwere then used to correct and update the aspect ratio. The resid-\nual rms scatter in the corrected X-ray positions of the LS10\nsources is 0.28 arcsec, which corresponds to a 2 σposition er-\nror of≈0.54 arcsec (assuming Rayleigh distribution). The astro-\nmetric corrected Chandra position is consistent with the centre\nof the host galaxy (see Fig. A.1).\nAppendix B: Estimation of the recurrence time\nusing the X-ray drop phase\nThe recurrence time can also be determined using the PX, drop\nphases. The start dates of the PX, drop phases, ts, drop , can be\nwell constrained in a model-independent way due to their short\nduration (i.e. fX, soft drops by a factor of ≳10 within one\nweek). We estimated ts, drop in the range of MJD [59285 ,59291],\n[59512,59518], and [59699 ,59706], and [59878 ,59883] for cy-\ncles 1, 2, 3, and 4, respectively. The PX, drop phase was not ob-\nserved in cycle 5 because J0456 −20 was blocked by the Sun.\n4http://cxc.cfa.harvard.edu/cal/ASPECT/celmonTable A.1. Astrometric correction for Chandra observation.\nObject Coordinates /Seperations\nLS DR10J0456−20 04:56:49.80, -20◦37′47.99′′\nRef1 04:56:57.39, -20◦35′12.75′′\nRef2 04:56:58.64, -20◦35′57.25′′\nRef3 04:56:58.70, -20◦37′47.65′′\nChandraJ0456−20 04:56:49.81, -20◦37′47.98′′\nRef1 04:56:57.39, -20◦35′13.25′′\nWith correction Ref2 04:56:58.66, -20◦35′57.49′′\nRef3 04:56:58.73, -20◦37′48.58′′\nChandraJ0456−20 04:56:49.82, -20◦37′48.54′′\nRef1 04:56:57.38, -20◦35′12.69′′\nNo correction Ref2 04:56:58.64, -20◦35′56.94′′\nRef3 04:56:58.71, -20◦37′48.03′′\nSeperationJ0456−20 0.07′′\nRef1 0.18′′\nRef2 0.32′′\nRef3 0.41′′\nNotes. Ref1–3 are the reference sources used for performing astromet-\nric correction for Chandra . The Separations are calculated using the\nastrometric corrected Chandra positions and the LS DR10 positions.\n59400 59600 59800 60000\nMJD501001000Flux density (Jy)\n5.5 GHz\n9.0 GHz\n5.0 GHz\n6.0 GHz\n2.1 GHz\n17.0 GHz\n21.0 GHz5.5 GHz\n9.0 GHz\n5.0 GHz\n6.0 GHz\n2.1 GHz\n17.0 GHz\n21.0 GHz\nFig. C.1. Evolution of the radio flux density. The error bars mark the\n1σuncertainties. The downward arrows represent the 3 σupper limits.\nThe black circle shows the flux density at 5 .5 GHz measured from the\nstacked observation from 2023 Mar. The light cyan regions indicate the\nPX, plat phase in each cycle.\nHowever, a lower limit of >MJD 60072 can be given. We gen-\nerated 105realisations of ts, drop for each cycle in cycles 1–4,\nassuming uniform distributions of ts, drop in the estimated MJD\nranges. Trecurand the 1σintervals can be estimated using the me-\ndian and the 16 and the 84 percentiles of the di fference between\nconsecutive ts, drop for cycles 1–4. We calculated a 3 σupper limit\nof 189 days for Trecur between cycles 4 and 5. The estimated\nTrecurare listed in Table 1.\nAppendix C: Details of the radio and X-ray\nobservations\nTable C.1 lists the ATCA radio observations for J0456 −20.\nJ0456−20 was detected only in the PX, plat phase during cycles\n3 and 4. The long-term radio variability is shown in Fig. C.1.\nArticle number, page 8 of 13Liu et al.: Rapid recurrence time evolution in J0456 −20\nTable C.1. ATCA radio observations of J0456 −20\nDate Frequency Array Flux Density\n(GHz) config. ( µJy)\n2021-03-215.56D<78\n9 <66\n2021-04-265.56D<36\n9 <47\n2021-08-075.5EW367<125\n9 <57\n2022-03-225\n6A257±11\n6 278 ±11\n9 311 ±12\n2022-04-062.1\n6A<738\n5.5 61 ±18\n9 103 ±23\n17 117 ±22\n21 <198\n2022-08-285.56D<53.7\n9 <33.6\n2022-09-215.56D<82.2\n9 <67.2\n2022-09-295.56D<41.1\n9 <34.8\n2022-10-025.56D<57.3\n9 <43.5\n2022-10-055.56D<46.5\n9 75 ±10\n2022-10-075.56D47±5\n9 <40.3\n2022-10-155.5H214<243\n9 <140\n2023-03-145.5750C<114\n9 <66\n2023-03-315.5750C<76.5\n9 <57\n2022-10-02,\n5.5 6D 35 ±5 2022-10-05,\n2022-10-07\n2022-10-02,\n9 6D 28 ±5 2022-10-05,\n2022-10-07\n2023-03-14,5.5 750C 51 ±132023-03-31\n2023-03-14,9 750C <582023-03-31\nNotes. Upper limits are reported at 3 σ.\nThe details of the X-ray observations and the X-ray spectral\nfitting results are listed in Table C.2.\nArticle number, page 9 of 13A&A proofs: manuscript no. main\nTable C.2. Log of X-ray observations.\nMJD Texp Tin/Γ fX, soft ObsID Mission X-ray phase cycle\n58919.15 325 3.0 <0.22 eRASS1 eROSITA — —\n59101.73 340 64+29\n−180.51+0.51\n−0.26eRASS2aeROSITA PX, rise 1\n59272.63 194 2 .5+0.2\n−0.27.76+0.84\n−0.76eRASS3 eROSITA PX, plat 1\n59281.86 123 2 .8+0.2\n−0.211.30+1.61\n−1.40eRASS3 eROSITA PX, plat 1\n59284.89 1683 2 .5+0.1\n−0.111.64+1.66\n−1.4500014135001 Swift P X, plat 1\n59290.94 1664 3.0 <0.19 4604010101 NICER P X, drop 1\n59291.53 9941 3.0 <0.13 4604010102 NICER P X, drop 1\n59292.21 6394 3.0 <0.14 4604010103 NICER P X, drop 1\n59298.15 4387 3.0 <0.18 4604010201 NICER P X, drop 1\n59300.20 10040 99+120\n−400.03+0.06\n−0.020862770201 XMM-Newton P X, drop 1\n59305.09 2735 3.0 <0.20 4604010301 NICER P X, faint 1\n59320.20 3254 3.0 <0.28 00014135002 Swift P X, faint 1\n59322.59 1808 3.0 <0.30 4604010501 NICER P X, faint 1\n59323.49 4525 3.0 <0.21 4604010502 NICER P X, faint 1\n59325.27 230 3.0 <0.35 4604010504 NICER P X, faint 1\n59328.31 420 3.0 <0.24 4604010507 NICER P X, faint 1\n59329.73 3342 3.0 <0.24 00014135003 Swift P X, faint 1\n59340.45 1402 3.0 <0.54 00014135004 Swift P X, faint 1\n59391.16 7669 78 0 .18+0.18\n−0.1100014135005aSwift P X, rise 2\n59399.00 6406 4 .3+1.3\n−1.10.47+0.70\n−0.2700014135006aSwift P X, rise 2\n59405.13 1793 5 .1+1.9\n−1.61.65+4.34\n−1.1400014135007aSwift P X, rise 2\n59413.03 1641 2 .6+0.7\n−0.70.68+0.53\n−0.2900014135008aSwift P X, rise 2\n59422.51 4304 3 .5+0.1\n−0.12.45+0.21\n−0.164595020102a,bNICER P X, rise 2\n59423.35 2589 3 .5+0.1\n−0.12.45+0.21\n−0.164595020103a,bNICER P X, rise 2\n59427.10 1631 2 .9+0.6\n−0.51.21+0.74\n−0.4500014135009aSwift P X, rise 2\n59428.95 788 3 .6+0.2\n−0.53.05+0.84\n−0.914595020105aNICER P X, rise 2\n59432.96 764 3 .7+0.3\n−0.23.37+1.00\n−0.644595020107a,bNICER P X, rise 2\n59433.12 999 3 .7+0.3\n−0.23.37+1.00\n−0.644595020108a,bNICER P X, rise 2\n59434.71 1968 3 .5+0.5\n−0.53.07+1.45\n−0.9600014135010aSwift P X, rise 2\n59441.95 514 2 .9+0.1\n−0.14.55+0.28\n−0.324595020109a,bNICER P X, rise 2\n59442.05 224 2 .9+0.1\n−0.14.55+0.28\n−0.324595020110a,bNICER P X, rise 2\n59445.50 2121 3 .2+0.1\n−0.16.15+0.19\n−0.184595020111aNICER P X, rise 2\n59447.51 40200 — 4 .79+1.10\n−0.900891801101 XMM-Newton P X, plat 2\n59448.38 1179 3 .2+0.1\n−0.15.06+0.23\n−0.214595020112 NICER P X, plat 2\n59450.80 3350 2 .9+0.0\n−0.04.48+0.13\n−0.134595020113 NICER P X, plat 2\n59453.89 2954 2 .8+0.2\n−0.23.20+0.72\n−0.5900014135011 Swift P X, plat 2\n59460.31 345 3 .4+0.2\n−0.25.26+0.76\n−0.67eRASS4 eROSITA PX, plat 2\n59467.45 1568 2 .8+0.3\n−0.32.39+0.42\n−0.294604010901 NICER P X, plat 2\n59468.45 445 2 .9+0.9\n−0.91.73+2.05\n−0.9100014135012 Swift P X, plat 2\n59470.50 341 4 .1+1.1\n−0.91.67+1.21\n−0.614604010902 NICER P X, plat 2\n59472.51 2542 2 .6+0.4\n−0.41.71+0.61\n−0.4400014135013 Swift P X, plat 2\n59476.22 1560 2 .9+0.4\n−0.41.23+0.24\n−0.174604010903 NICER P X, plat 2\n59479.33 929 3 .0+0.3\n−0.32.81+0.49\n−0.354604010904 NICER P X, plat 2\n59481.30 85960 — 2 .40+0.76\n−0.360891801701 XMM-Newton P X, plat 2\n59481.49 2490 2 .7+0.4\n−0.41.48+0.58\n−0.4100014135014 Swift P X, plat 2\n59482.46 4637 2 .9+0.3\n−0.32.33+0.37\n−0.264604010905 NICER P X, plat 2\n59485.04 1183 2 .7+0.3\n−0.32.03+0.31\n−0.234604010906 NICER P X, plat 2\n59488.08 1532 3 .0+0.3\n−0.32.78+0.57\n−0.384604010907 NICER P X, plat 2\n59491.05 1410 2 .7+0.3\n−0.32.94+0.49\n−0.344604010908 NICER P X, plat 2\n59494.25 811 2 .5+0.3\n−0.32.34+0.31\n−0.244604010909 NICER P X, plat 2\n59496.67 3079 2 .7+0.3\n−0.32.05+0.64\n−0.4900014135016 Swift P X, plat 2\nArticle number, page 10 of 13Liu et al.: Rapid recurrence time evolution in J0456 −20\nTable C.2. continued.\nMJD Texp Tin/Γ fX, soft ObsID Mission X-ray phase Cycle\n59497.25 1133 2 .9+0.3\n−0.32.63+0.44\n−0.314604010910 NICER P X, plat 2\n59499.80 1703 2 .5+0.3\n−0.32.19+0.31\n−0.244604010911 NICER P X, plat 2\n59500.64 3647 2 .9+0.3\n−0.32.47+0.41\n−0.284604010912 NICER P X, plat 2\n59501.35 10186 2 .9+0.2\n−0.22.13+0.31\n−0.224604010913 NICER P X, plat 2\n59502.49 7885 2 .7+0.3\n−0.31.96+0.24\n−0.184604010914 NICER P X, plat 2\n59503.48 7906 2 .9+0.2\n−0.22.48+0.34\n−0.244604010915 NICER P X, plat 2\n59504.51 4370 2 .8+0.3\n−0.32.23+0.35\n−0.244604010916 NICER P X, plat 2\n59505.52 3600 2 .7+0.3\n−0.32.36+0.35\n−0.254604010917 NICER P X, plat 2\n59506.22 826 2 .8+0.3\n−0.32.41+0.40\n−0.294604010918 NICER P X, plat 2\n59508.15 553 3 .3+0.5\n−0.51.95+0.60\n−0.394604010920 NICER P X, plat 2\n59510.21 3079 2 .8+0.3\n−0.32.31+0.38\n−0.264604010921 NICER P X, plat 2\n59510.83 2777 2 .3+0.3\n−0.31.71+0.51\n−0.3900014135017 Swift P X, plat 2\n59511.91 2492 2 .9+0.3\n−0.31.91+0.27\n−0.204604010922 NICER P X, plat 2\n59517.60 2745 3.0 <0.15 4604011001 NICER P X, drop 2\n59520.47 3000 3.0 <0.22 4604011002 NICER P X, faint 2\n59523.54 1200 3.0 <0.27 4604011003 NICER P X, faint 2\n59524.53 1371 3.0 <1.50 00014135018 Swift P X, faint 2\n59526.63 936 3.0 <0.29 4604011004 NICER P X, faint 2\n59527.63 3924 3.0 <0.24 00014135019 Swift P X, faint 2\n59529.83 636 3.0 <0.30 4604011005 NICER P X, faint 2\n59529.87 1402 3.0 <0.69 00014135021 Swift P X, faint 2\n59532.54 1143 3.0 <0.29 4604011006 NICER P X, faint 2\n59535.32 1817 3.0 <0.28 4604011007 NICER P X, faint 2\n59538.23 2041 3.0 <0.19 4604011008 NICER P X, faint 2\n59541.33 1884 3.0 <0.14 4604011009 NICER P X, faint 2\n59543.75 1996 3.0 <0.39 00014135022 Swift P X, faint 2\n59557.41 2372 3.0 <0.41 00014135023 Swift P X, faint 2\n59562.50 193 3.0 <3.60 00014135024 Swift P X, faint 2\n59566.74 614 3.0 <1.60 00014135025 Swift P X, faint 2\n59572.58 2635 3.0 <0.38 00014135026 Swift P X, faint 2\n59584.90 2886 3.0 <0.39 00014135027 Swift P X, faint 2\n59601.40 819 180 <0.27 4595020114 NICER P X, rise 3\n59602.56 3132 180 <0.05 4595020115aNICER P X, rise 3\n59603.21 1684 180 <0.11 4595020116 NICER P X, rise 3\n59604.76 432 120 <0.21 4595020117 NICER P X, rise 3\n59605.47 425 120 <0.37 4595020118 NICER P X, rise 3\n59607.27 765 120 <0.19 4595020119 NICER P X, rise 3\n59608.60 2918 120 <0.08 4595020120 NICER P X, rise 3\n59609.72 1146 80 <0.23 4595020121 NICER P X, rise 3\n59610.47 2152 80 <0.13 4595020122 NICER P X, rise 3\n59611.79 754 80 <0.29 4595020123 NICER P X, rise 3\n59612.47 758 80 <0.08 4595020124aNICER P X, rise 3\n59613.37 1443 80 <0.19 4595020125 NICER P X, rise 3\n59619.47 5808 78+24\n−180.23+0.12\n−0.054595020126aNICER P X, rise 3\n59625.99 787 3 .5+0.3\n−0.20.67+0.11\n−0.094595020127a,bNICER P X, rise 3\n59626.06 1584 3 .5+0.3\n−0.20.67+0.11\n−0.094595020128a,bNICER P X, rise 3\n59633.19 3669 3 .6+0.2\n−0.21.11+0.11\n−0.114595020129aNICER P X, rise 3\n59635.13 10129 3 .2+0.3\n−0.30.77+0.27\n−0.2000014135099aSwift P X, rise 3\n59640.03 3707 3 .1+0.1\n−0.11.43+0.10\n−0.104604010104aNICER P X, rise 3\n59640.03 3707 3 .1+0.1\n−0.11.43+0.10\n−0.104604010105aNICER P X, rise 3\n59642.13 4069 3 .2+0.1\n−0.11.40+0.08\n−0.134604010106 NICER P X, plat 3\n59643.68 2485 3 .2+0.5\n−0.51.78+0.88\n−0.5800014135033 Swift P X, plat 3\n59645.48 2460 3 .2+0.3\n−0.31.71+0.38\n−0.264604010107 NICER P X, plat 3\nArticle number, page 11 of 13A&A proofs: manuscript no. main\nTable C.2. continued.\nMJD Texp Tin/Γ fX, soft ObsID Mission X-ray phase Cycle\n59648.20 1991 3 .2+0.4\n−0.41.54+0.36\n−0.244604010108 NICER P X, plat 3\n59650.19 2977 3 .2+0.5\n−0.51.58+0.88\n−0.5600014135034 Swift P X, plat 3\n59651.90 668 2 .6+0.6\n−0.61.39+0.35\n−0.244604010109 NICER P X, plat 3\n59652.06 720 3 .6+0.6\n−0.52.12+0.91\n−0.524604010110 NICER P X, plat 3\n59654.83 1787 3 .7+0.4\n−0.41.83+0.59\n−0.374604010111 NICER P X, plat 3\n59656.16 2665 3 .0+0.4\n−0.41.48+0.68\n−0.4700014135036 Swift P X, plat 3\n59657.15 — — 1 .92+0.04\n−0.040884960601 XMM-Newton P X, plat 3\n59663.82 3172 2 .8+0.4\n−0.41.52+0.58\n−0.4200014135037 Swift P X, plat 3\n59670.56 3127 2 .5+0.4\n−0.41.22+0.44\n−0.3200014135038 Swift P X, plat 3\n59678.08 4305 2 .8+0.5\n−0.40.68+0.34\n−0.2200014135039 Swift P X, plat 3\n59684.37 2922 2 .6+0.5\n−0.50.88+0.44\n−0.2900014135040 Swift P X, plat 3\n59691.98 6391 3 .1+0.4\n−0.31.08+0.39\n−0.2800014135041 Swift P X, plat 3\n59698.54 2627 3 .5+0.8\n−0.81.18+1.06\n−0.5600014135042 Swift P X, plat 3\n59705.30 3014 3 .0+−3.0\n−3.00.12+0.13\n−0.0800014135043 Swift P X, drop 3\n59752.82 1459 3.0 <0.50 00014135044 Swift P X, faint 3\n59756.64 2618 3.0 <0.28 00014135045 Swift P X, faint 3\n59759.19 3092 3.0 <0.25 00014135046 Swift P X, faint 3\n59766.09 2668 3.0 <0.31 00014135047 Swift P X, faint 3\n59773.49 2680 3.0 <0.43 00014135048 Swift P X, faint 3\n59780.34 3377 3.0 <0.25 00014135049 Swift P X, rise 4\n59787.41 2976 3.0 <0.40 00014135050 Swift P X, rise 4\n59794.63 2720 3.0 <0.51 00014135051 Swift P X, rise 4\n59801.23 4023 3 .1+2.2\n−2.20.12+0.46\n−0.0900014135052aSwift P X, rise 4\n59806.58 2627 3 .0+−102.0\n−102.0<0.15 5202950108aNICER P X, rise 4\n59811.17 187 3 .5+1.3\n−0.90.77+0.41\n−0.395202950109aNICER P X, rise 4\n59811.48 3179 2 .3+0.9\n−0.90.24+0.24\n−0.1200014135053aSwift P X, rise 4\n59813.97 366 3 .4+0.8\n−0.70.55+0.27\n−0.245202950110aNICER P X, rise 4\n59814.03 256 2 .8+0.7\n−0.60.53+0.23\n−0.185202950111aNICER P X, rise 4\n59814.12 9662 3 .17+0.06\n−0.060.70+0.03\n−0.030884962601aXMM-Newton P X, rise 4\n59819.86 2570 2 .4+0.7\n−0.60.43+0.33\n−0.1900014135455 Swift P X, plat 4\n59825.17 3384 2 .7+0.7\n−0.70.38+0.33\n−0.1800014135056 Swift P X, plat 4\n59828.19 3303 4 .0+0.4\n−0.30.78+0.17\n−0.105202950114 NICER P X, plat 4\n59832.46 3117 2 .7+0.5\n−0.50.68+0.38\n−0.2400014135057 Swift P X, plat 4\n59839.22 2647 2 .9+0.7\n−0.70.58+0.49\n−0.2600014135058 Swift P X, plat 4\n59839.53 2717 3 .0+0.2\n−0.20.63+0.07\n−0.075202950115 NICER P X, plat 4\n59842.32 1380 4 .4+1.8\n−1.00.64+0.42\n−0.255202950116 NICER P X, plat 4\n59845.39 1529 3 .6+0.5\n−0.50.62+0.14\n−0.185202950117 NICER P X, plat 4\n59846.51 2605 3 .4+1.2\n−1.20.56+0.82\n−0.3100014135059 Swift P X, plat 4\n59847.94 841 3 .1+0.6\n−0.50.57+0.19\n−0.185202950118 NICER P X, plat 4\n59848.10 2940 2 .8+0.3\n−0.30.65+0.09\n−0.205202950119 NICER P X, plat 4\n59851.44 1555 3 .8+0.4\n−0.40.81+0.20\n−0.155202950120 NICER P X, plat 4\n59852.96 341 3 .0 <0.73 5202950121 NICER P X, plat 4\n59853.86 2640 3 .3+0.9\n−0.80.67+0.77\n−0.3500014135060 Swift P X, plat 4\n59857.19 2144 3 .7+0.5\n−0.30.75+0.20\n−0.135202950123 NICER P X, plat 4\n59860.20 1766 3 .8+0.5\n−0.30.71+0.19\n−0.125202950124 NICER P X, plat 4\n59860.82 2555 3 .4+0.8\n−0.70.82+0.79\n−0.3900014135061 Swift P X, plat 4\n59863.58 4048 3 .3+0.2\n−0.30.57+0.06\n−0.105202950125 NICER P X, plat 4\n59867.30 2122 3 .5+0.8\n−0.50.57+0.10\n−0.175202950127 NICER P X, plat 4\n59869.51 977 4 .1+0.6\n−0.50.83+0.29\n−0.205202950128 NICER P X, plat 4\n59872.21 392 3 .0 <0.56 5202950129 NICER P X, plat 4\n59874.88 2650 2 .8+0.8\n−0.70.42+0.39\n−0.2000014135062 Swift P X, plat 4\n59874.91 2416 2 .6+0.3\n−0.30.65+0.17\n−0.115202950130 NICER P X, plat 4\nArticle number, page 12 of 13Liu et al.: Rapid recurrence time evolution in J0456 −20\nTable C.2. continued.\nMJD Texp Tin/Γ fX, soft ObsID Mission X-ray phase Cycle\n59875.49 316 3.0 <0.33 5202950131 NICER P X, plat 4\n59875.49 1211 3 .4+1.4\n−1.00.24+0.15\n−0.155202950131 NICER P X, plat 4\n59876.52 1326 3 .9+0.8\n−0.60.44+0.23\n−0.125202950132 NICER P X, plat 4\n59876.52 459 3 .7+1.2\n−0.90.49+0.29\n−0.185202950132 NICER P X, plat 4\n59877.30 2150 3 .0+0.3\n−0.20.47+0.08\n−0.075202950133 NICER P X, plat 4\n59877.30 1266 3 .1+0.5\n−0.40.53+0.13\n−0.105202950133 NICER P X, plat 4\n59878.36 1955 4 .3+0.6\n−0.50.49+0.17\n−0.125202950134 NICER P X, drop 4\n59879.59 3164 3 .0 <0.55 5202950135 NICER P X, drop 4\n59879.59 1564 4 .1+0.7\n−0.60.43+0.16\n−0.125202950135 NICER P X, drop 4\n59881.55 1119 2 .3+0.7\n−0.70.33+0.08\n−0.075202950136 NICER P X, drop 4\n59881.55 3311 1 .4+0.6\n−1.40.17+0.06\n−0.095202950136 NICER P X, drop 4\n59882.80 8366 3 .0+−3.0\n−3.00.07+0.06\n−0.0414136364 Swift P X, drop 4\n59884.59 3046 3 .0+−102.0\n−102.0<0.11 5202950137 NICER P X, drop 4\n59884.59 2552 3.0 <0.14 5202950137 NICER P X, drop 4\n59887.94 2868 3.0 <0.36 00014135065 Swift P X, faint 4\n59895.56 2688 3.0 <0.35 00014135066 Swift P X, faint 4\n59920.62 3455 3.0 <0.31 00014135067 Swift P X, faint 4\n59941.75 4054 3.0 <0.32 00014135068 Swift P X, faint 4\n59962.62 3984 3.0 <0.23 00014135069 Swift P X, faint 4\n59980.96 2963 200 <0.25 00014135070 Swift P X, rise 5\n59985.53 15381 193+43\n−250.021+0.005\n−0.0030915390401aXMM-Newton P X, rise 5\n59990.59 2926 100 <0.67 00014135071 Swift P X, rise 5\n59999.79 3384 3 .4+2.4\n−2.30.26+0.89\n−0.1700014135072 Swift P X, rise 5\n60007.25 11300 3 .06+0.07\n−0.070.44+0.03\n−0.020915390501aXMM-Newton P X, rise 5\n60009.45 2690 3 .2+1.3\n−1.30.53+0.75\n−0.2700014135073 Swift P X, rise 5\n60021.39 3189 2 .2+0.9\n−0.90.27+0.25\n−0.1300014135074 Swift P X, plat 5\n60023.00 9948 2 .6+0.4\n−0.40.29+0.24\n−0.1227737 Chandra P X, plat 5\n60031.16 2680 3 .2+0.8\n−0.80.63+0.58\n−0.3000014135075 Swift P X, plat 5\n60045.09 10059 2 .6+0.5\n−0.50.25+0.27\n−0.1227738 Chandra P X, plat 5\n60048.22 9948 2 .5+0.5\n−0.50.19+0.20\n−0.0927798 Chandra P X, plat 5\n60072.18 4795 3 .0+0.9\n−0.80.28+0.32\n−0.1500014135076 Swift P X, plat 5\nNotes. MJD is the mid-date of the coverage for each observation; Texpis the e ffective exposure time in units of seconds. Exposure for XMM-Newton\nObsID 0862770201 is calculated from combined MOS data, while the pn exposures are given for the other XMM-Newton observations; Tin/Γ: the\ninner temperature of a multi-color disk or the photon index of the power law. A value without an uncertainty means the parameter is fixed at the\ngiven value during spectral fitting; fX, soft is the rest frame intrinsic 0 .2−2.0 keV flux in units of 10−12erg cm−2s−1. The 3σupper limits are given\nfor observations in which the J0456 −20 is not detected; ObsID is the observation ID for each observation; Mission : the telescopes of which the\ndata were taken; X-ray phase marks the phases of the J0456 −20 during the observations; Cycle : the cycle the X-ray flare during which the data\nwere taken. Quoted uncertainties are at the 90% confidence level. Upper limits are reported at 3 σ.\na: observations used in modelling the profile of the PX, rise phase.\nb: the X-ray spectral fitting results were from jointly fitting of the two consecutive observations.\nArticle number, page 13 of 13"    },    {        "title": "2401.14120v1.Heavy_baryon_decays_into_light_meson_and_dark_baryon_within_LCSR.pdf",        "content": "Heavy baryon decays into light meson and dark baryon within LCSR\nYu-Ji Shi1,2∗, Ye Xing3†and Zhi-Peng Xing4‡\n1School of Physics, East China University of Science and Technology, Shanghai 200237, China\n2Shanghai Key Laboratory of Particle Physics and Cosmology,\nSchool of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China\n3School of Materials Science and Physics,\nChina University of Mining and Technology, Xuzhou 221116, China\n4Department of Physics and Institute of Theoretical Physics,\nNanjing Normal University, Nanjing, Jiangsu 210023, China\nWe studied the decays of Heavy baryon into a pseudoscalar meson and a dark baryon in\nthe recently developed B-Mesogenesis scenario, where the two types of effective Lagrangians\nproposed by the scenario are both considered. The decay amplitudes of Λ0\nbare calculated\nby light-cone sum rules using its light-cone distribution amplitudes. The decay amplitudes\nof Ξ0,±\nbare related with those of Λ0\nbthrough a flavor SU(3) analysis. The uncertainties of\nthreshold parameter and the Borel parameter are both considered in the numerical calcu-\nlation. The values of effective coupling constants in the B-Mesogenesis are taken as their\nupper limits that obtained from our previous study on the inclusive decay. The upper limits\nof the decay branching fractions are presented as functions of the dark baryon mass.\nI. INTRODUCTION\nThe Standard Model of particle physics and cosmology have been proven to be successful in\ndescribing the physics of the most microscopic and macroscopic worlds. However, there is incon-\nsistency between these two models, where two of the most confusing problems are the existence of\ndark matter (DM) and the asymmetry of matter and anti-matter. Nowadays, a number of mech-\nanisms aiming to solve this puzzle have been proposed according to the Sakharov conditions for\nthe baryogenesis [1]. One of the disadvantages among these mechanisms is the existence of high\nenergy scales and extremely massive particles, which make them difficult to be tested in experi-\nments. Recently, a new mechanism called B-Mesogenesis is proposed by Refs. [2–4], which aims to\nexplain both the relic dark matter abundance and the baryon asymmetry without introducing high\nenergy scales. The B-Mesogenesis is testable at hadron colliders and B-factories [3, 5], and also\nindirectly testable at the Kaon and Hyperon factories [6, 7]. Recently, the Belle-II collaboration\nand the LHCb collaboration have started to search for the decays of Bmesons with energy missing\naccording to the B-Mesogenesis [8, 9].\nTheB-Mesogenesis proposes a new mechanism for Baryogenesis and DM production. In this\nmodel, during a late era in the history of the early universe, a certain heavy scalar particle Φ\ndecays into b,¯bquarks, which then form charged and neutral B-mesons after the universe cool\ndown. After that, the neutral mesons B0,¯B0quickly undergo CP violating oscillations, and the\n∗Email:shiyuji@ecust.edu.cn (corresponding author)\n†Email:xingye guang@cumt.edu.cn\n‡Email:zpxing@sjtu.edu.cnarXiv:2401.14120v1  [hep-ph]  25 Jan 20242\nremained mesons continue to decay into a dark sector baryon ψwith baryon number −1 and visible\nhadron states with baryon number +1. As a result, the asymmetry of the baryon and anti-baryon\nnumber from the CP violation during B0−¯B0oscillations is induced but without violating the\ntotal baryon number. Recently, there are a number of theoretical studies on the Bmeson decays\nin the B-Mesogenesis. The exclusive decay B→pψwas firstly studied by Ref.[10] using leading\ntwist light-cone sum rules (LCSR) calculation, and a higher twist contribution are calculated in\nRef.[11]. A more complete study on Bmeson decays into an octet baryon or a charmed anti-triplet\nbaryon plus ψwas given in Ref.[12]. In addition, a similar exclusive decay of Bmeson into a\nbaryon plus missing energy are studied by Ref.[13] for probing the lightest neutralino. Besides the\nexclusive decays, previously we studied the semi-inclusive decay of B→Xu/c,d/s ψusing heavy\nquark expansion (HQE), where Xu/c,d/s denotes any possible hadron states containing u/cand\nd/squarks with unit baryon number [14]. Using the experimental upper limits on the branching\nfractions of B→Xu/c,d/s ψfrom the ALEPH experiment [15–17], we predicted the upper limits on\nthe coupling constants in the B-Mesogenesis.\nUp to now, the studies on the B-Mesogenesis mainly focus on the Bdecays. However, apart from\nthe generous production of Bmeson in the LHCb and Belle experiments, Λ0\nbis the most baryons\nthat can be produced experimentally. In the B-Mesogenesis model, Λ0\nbcan also undergo baryon\nnumber violated decays such as Λ b→Mψ, where Mis a neutral pseudoscalar meson including\nπ0, K0,¯D0. Compared with B→p/nψ , the phase space of Λ0\nb→Mψis larger, which can amplify\nthe decay width to a certain extent so that offer more possibility for observing such decays in the\nexperiments. In this work, we will perform a theoretical study on the Λ0\nb→Mψ decays. The\ndecay amplitude will be calculated by LCSR with the use of the light-cone distribution amplitudes\n(LCDAs) of Λ0\nb. One disadvantage of Λ0\nb→Mψis that since the neutral final pseudoscalar meson\nmakes it difficult to be detected. In contrast, the charged decay Ξ±\nb→Mψshould be a more ideal\nchannel for experimental searching. In the flavor SU(3) symmetry limit, the decay amplitudes\nof these decay channels are related with each other. The SU(3) symmetry analysis is a powerful\ntools frequently used in heavy hadron decays [18–24], where it is generally be performed in two\ndifferent frameworks: the topological diagram amplitude (TDA) and the irreducible representation\namplitude (IRA) methods. These two methods have been proven to be equivalent in Refs.[21,\n23, 25–28]. In this work, we will choose IRA method to obtain the relations between the decay\namplitudes of Ξ±\nb→Mψ and those of Λ0\nb→Mψ, and use them to predict the decay branching\nfractions of Ξ±\nb→Mψ.\nThis article is organized as follows: Section II introduces the B-Mesogenesis scenario proposed\nby Refs. [2–4]. Section III present a detailed LCSR calculation for the Λ0\nb→Mψdecays. Section\nIV present a SU(3) analysis on the decay amplitudes, which relates Ξ±\nb→Mψ to Λ0\nb→Mψ.\nSection V present numerical calculations on the decay amplitudes and branching fractions. Section\nVI is a summary of this work.3\nII. THE B-MESOGENESIS SCENARIO\nThis section gives a brief introduction to the B-Mesogenesis scenario [2–4], which aims to\nsimultaneously explain the baryon asymmetry and the existence of dark matter in our Universe.\nIn the B-Mesogenesis scenario, the bquark is possible to decay into two light quarks and a dark\nbaryon ψ. The total baryon number in such decay process is conserved, however, due to the invisible\nψthe visible decay products exhibit baryon number non-conserving phenomenon. As proposed by\nRefs. [2, 3], this kind of baryon number violating decays can emerge from the following two types\nof effective Lagrangians:\nLI\neff=−yubϵijkY∗i¯uj\nRbc,k\nR−ycbϵijkY∗i¯cj\nRbc,k\nR−yψdYi¯ψdc,i\nR−yψsYi¯ψsc,i\nR+ h.c,\nLII\neff=−yudϵijkY∗i¯uj\nRdc,k\nR−yusϵijkY∗i¯uj\nRsc,k\nR−ycdϵijkY∗i¯cj\nRdc,k\nR−ycsϵijkY∗i¯cj\nRsc,k\nR\n−yψbYi¯ψbc,i\nR+ h.c, (1)\nwhich correspond to the type-I and II models in the B-Mesogenesis, respectively. All the quark\nfields are taken as right handed, the superscript cindicates charge conjugate and the ys are\nunknown coupling constants. Yis a charged color triplet scalar with QY=−1/3, which is assumed\nto have large mass MY. In the Type-I model the bquark couples with u, cquarks, the dark anti-\nbaryon ψcouples with d, squarks. In the Type-II model, the situations of the bandd, squarks are\ninterchanged. In fact, as proposed by Ref. [3], there should be a third type of effective Lagrangian\nwith QY= 2/3 inB-Mesogenesis, which reads as\nLIII\neff=−ybdϵijkY∗i¯bj\nRdc,k\nR−ybsϵijkY∗i¯bj\nRsc,k\nR−yψuYi¯ψuc,i\nR−yψcYi¯ψcc,i\nR+ h.c. (2)\nIn this work, we will only consider the case of QY=−1/3 to be consistent with the studies of\nexclusive Bmeson decay in B-Mesogenesis [10, 12].\nIntegrating out the heavy boson Yin Eq. (1), one can obtain the effective Hamiltonian for the\ntwo types of models as:\nHI,uq\neff=−yubyψq\nM2\nYiϵijk(¯ψqc,i\nR)(¯uj\nRbc,k\nR) + h.c.=−GI\n(uq)¯OI\n(uq)ψc+ h.c.,\nHII,uq\neff=−yψbyuq\nM2\nYiϵijk(¯ψbc,i\nR)(¯uj\nRqc,k\nR) + h.c.=−GII\n(uq)¯OII\n(uq)ψc+ h.c., (3)\nwhere for simplicity, q=s, dand we use uto denote the uorcquarks simultanously. The effective\ncoupling constant and the effective operators in the two models are defined as\nGI\n(uq)=yubyψq\nM2\nY,OI\n(uq)=−iϵijk(uiTCPRbj)PRqk,\nGII\n(uq)=yψbyuq\nM2\nY,OII\n(uq)=−iϵijk(uiTCPRqj)PRbk, (4)\nwhere PR= (1 + γ5)/2 and Cis the charge conjugate matrix. The baryon number violated decays\nΛb→πψ, Λb→Kψand Λ b→Dψare induced by HI,ud\neff,HI,us\neffandHI,cd\neffrespectively.4\nΛbMψ\nb\nFIG. 1: The Feynman diagram of the Λ b→Mψdecays. The green and gray bubbles denote the Λ band\npseudoscalar M, respectively, and the white crossed dot denotes the O(uq)vertex.\nIII. Λb→MψDECAYS IN LCSR\nA. Hadron level calculation\nIn this section, we perform a LCSR calculation on the Λ b→Mψdecays as shown in Fig.1. The\ngreen and gray bubbles denote the Λ band pseudoscalar M, respectively, and the white crossed dot\ndenotes the O(uq)vertex. Using the effective Hamiltonians given in Eq.(3), one can express the\ndecay amplitude as\niM=−G(uq)¯uc\nψ(q, sψ)⟨M(p′)|O(uq)(0)|Λb(p, sΛ)⟩, (5)\nwhere uψis the spinor of dark baryon, with momentum q=p−p′. The transition matrix element\non the right hand side above can be parameterized by two form factors:\n⟨M(p′)|O(uq)(0)|Λb(p, sΛ)⟩=PR\u0014\nF1(q2) +/q\nmΛbF2(q2)\u0015\nuΛb(p, sΛ). (6)\nIn the framework of LCSR, the calculation of the transition matrix element given above starts\nfrom the following two-point correlation function\nΠ(p, q) =iZ\nd4x eip·xqµ⟨0|T{jM\nµ(x)O(uq)(0)}|Λb(p+q)⟩, (7)\nwhere jM\nµdenotes the interpolation current for the final meson, which reads as\njπ\nµ=1√\n2(¯uγµγ5u−¯dγµγ5d), jK\nµ= ¯sγµγ5d,and jD\nµ= ¯cγµγ5d (8)\nfor the final states: M=π0, K0andD0respectively. In LCSR, the correlation function defined in\nEq.(7) will be calculated both at the hadron and quark-gluon level. The one at hadron level can be\nexpressed by the form factors defined in Eq.(6). On the other hand, the one at quark-gluon level\nwill be calculated explicitly in QCD, with the light-cone distribution amplitudes (LCDAs) of Λ b\ntaken as non-perturbative inputs. Matching of the correlation function at these two levels enables\nus to extract the transition form factors.5\nAt the hadron level, one inserts a complete set of states with the same quantum number of M\nbetween the jM\nµ(q2) current and O(uq)(0). The correlation function becomes\nΠH(p, q) =ifM\nm2\nM−p2(p·q)PR\u0014\nF1(q2) +/q\nmΛbF2(q2)\u0015\nuΛb(p+q) +Z∞\nsthdsρH(s, q)\ns−p2, (9)\nwhere the meson decay constant is defined as: ⟨0|jM\nµ(0)|M(p)⟩=ifMpµ. Here only the pole con-\ntribution from Mis expressed explicitly, while the contribution from excited state and continuous\nspectrum above threshold sthare capsuled in the integration of ρH(s, q), which will be suppressed\nby Borel transformation. Note that in the expression: p·q= (1/2)(m2\nΛb−q2−m2\nM)+(1/2)(m2\nM−p2),\nthe second term will cancel the denominator of the pole contribution in Eq.(9), so that the terms\nproportional to it will vanish under the Borel transformation on p2. As a result, the Borel trans-\nformed correlation function becomes\nB{ΠH}(p, q) =i\n2fM(m2\nΛb−q2−m2\nM)e−m2\nM/T2PR\u0014\nF1(q2) +/q\nmΛbF2(q2)\u0015\nuΛb(p+q)\n+Z∞\nsthds e−s/T2ρH(s, q), (10)\nwhere Tis the Borel parameter. Now the contributions from excited state and continuous spec-\ntrum are suppressed by the exponential term e−s/T2. Next, the same correlation function will\nbe calculated at the quark-gluon level, where the result can be generally written as a dispersion\nintegration form\nΠQCD(p, q) =1\nπZ∞\nsmdsImΠQCD(s, q)\ns−p2, (11)\nwith smbeing the quark level threshold. From the assumption of quark-hadron duality, the last\nterm of Eq.(9), denoting the excited state and continuous spectrum at hadron level, is equivalent\nwith the spectrum integration above sthat quark-gluon level. After the Borel transformation and\nsubtracting the continuous spectrum contribution, we arrive at the sum rules equation:\nB{ΠH}(p, q)pole=B{ΠQCD}(p, q) =1\nπZsth\nsmds e−s/T2ImΠQCD(s, q). (12)\nThus the form factors F1,2can be extracted through this equation as long as the imaginary part\nof ΠQCDis obtained.\nB. Quark-gluon level calculation\nNow we perform an quark-gluon level calculation for the correlation function defined in Eq.(7).\nHere we take the type-I model and the case of M=¯D0as an example to illustrate the calculation.\nUsing the quark level expression of jD\nµgiven in Eq.(8), one can write the correlation function as\nΠQCD\nΛb→D(p, q) =−ϵijkZ\nd4xeip·x\u0002\nPRCSc(0, x)/qγ5\u0003\nγα(PR)τβ\n× ⟨0|ui\nα(q2)dj\nβ(0)bk\nγ(0)|Λb(p+q)⟩, (13)6\nΛbb\np+q\npq\n0\nx\nFIG. 2: The Feynman diagram of the the correlation function defined in Eq.(7). The green bubble denotes\nthe Λ bLCDAs, the white crossed and the black dots denote the O(uq)(x) and the jM\nµ(0) vertexes, respectively.\nwhere Scis the charm quark propagator. The non-local three quark matrix element above is\nexpressed by the Λ bLCDAs, which are defined as [29, 30]\n⟨0|ui\nα(x1)dj\nβ(x2)bk\nγ(0)|Λb(v)⟩\n=1\n8f(2)\nΛbΨ2(t1, t2)(/nγ5C)αβuΛbγ(v) +1\n4f(1)\nΛbΨs\n3(t1, t2)(γ5C)αβuΛbγ(v)\n−1\n8f(1)\nΛbΨσ\n3(t1, t2)(iσnnγ5C)αβuΛbγ(v) +1\n8f(2)\nΛbΨ4(t1, t2)(/nγ5C)αβuΛbγ(v), (14)\nwhere σnn=σµνnµnνandx1=t1n, x2=t2nare on the light cone. Inversely, the light cone vectors\nn,¯ncan be expressed by the coordinates as\nnµ=xµ\nv·x,nµ= 2vµ−xµ\nv·x.\nAt the quark-gluon level, the corresponding diagram for the correlation function defined in Eq.(7)\nis shown in Fig.2. The green bubble denotes the Λ bLCDAs, the white crossed and the black dots\ndenote the O(uq)(x) at and the jM\nµ(0) vertexes, respectively.\nThe Ψ 2,Ψs\n3,Ψσ\n3,Ψ4in Eq.(14) are the LCDAs with different twists. In the momentum space,\nthe LCDAs can be characterized by the total momentum ωof the two light quarks in Λ b, and the\nmomentum fraction uof one light quark:\nΨi(t1, t2) =Z∞\n0ωdωZ1\n0due−iω(t1u+t2u)ψi(ω, u), (15)\nwith u= 1−u, and [29]\nψ2(ω, u) =ω2u(1−u)[1\nϵ4\n0e−ω/ϵ0+a2C3/2\n2(2u−1)1\nϵ4\n1e−ω/ϵ1],\nψs\n3(ω, u) =ω\n2ϵ3\n3e−ω/ϵ3,\nψσ\n3(ω, u) =ω\n2ϵ3\n3(2u−1)e−ω/ϵ3,\nψ4(ω, u) =5N−1ZsΛb\n0\nω/2dse−s/τ(s−ω/2)3, (16)7\nwhere C3/2\n2(2u−1) is the Gegenbauer polynomial, and the normalization factor Nin˜ψ4(ω, u) reads\nasN=RsΛb\n0\n0dss5e−s/τ. The parameters in these four LCDAs are ϵ0= 200+130\n−60MeV, ϵ1= 650+650\n−300\nMeV, ϵ3= 230 MeV, a2= 0.333+0.250\n−0.333,sΛb\n0= 1.2 GeV and τ= 0.4∼0.8 GeV. [29].\nUsing the LCDAs defined in Eq.(14), and the free charm quark propagator, one can express the\ncorrelation function as a convolution of the perturbative part and the LCDAs. Firstly, the twist-3s\ncontribution can be straightforwardly written as\nΠQCD (3s)\nΛb→D(p, q) =i\n4f(1)\nΛbZ2sΛb\n0\n0dω ωZ1\n0du ψ 3s(ω, u)1\n(uωv−p)2−m2cPR/q(uω/v−/p)uΛb(v).(17)\nTo extract the imaginary part of correlation function, one has to calculate its discontinuity across\nthe complex plane of p2. The denominator can be rewritten as\n(uωv−p)2−m2\nc=\u0012\n1−uω\nmΛb\u0013\n(p2−Σc),\nΣc=1\nmΛb−uω\u0002\nuω(m2\nΛb−q2) +mΛb(m2\nc−u2ω2)\u0003\n. (18)\nNow the imaginary part comes from the term\nIm\u00141\n(uωv−p)2−m2c\u0015\n=1\n2i(−2πi)mΛb\nmΛb−uωδ(s−Σc), (19)\nwhere s=p2. Expressing ΠQCD (3s)\nΛb→Das a dispersion integration in sm< s < s th, performing the\nBorel transformation, and integrating out sby the delta function in Eq.(19), one arrives at\nB{ΠQCD (3s)\nΛb→D}(T, q) =i\n4f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(Σc−sm)θ(sth−Σc)e−Σc/T2\n×ψ3s(ω, u)PR\u0014\nmΛbω/q−mΛbω\nmΛb−uωq2\u0015\n. (20)\nThe calculation for the twist-3 σcontribution is more involved. The corresponding correlation\nfunction can be firstly written as\nΠQCD (3σ)\nΛb→D(p, q) =−i\n4f(1)\nΛbZ\nd4x eip·xZ2sΛb\n0\n0dω ωZ1\n0du e−iuωv·xZd4k\n(2π)4eik·x\n×ψ3s(ω, u)PR(1−/n/v)/q1\n/kuΛb(v), (21)\nwhere the 4-velocity of Λ bismΛbv=p+q. It can be found that the term containing no /n\nabove has almost the same form as that of ΠQCD (3s)\nΛb→D. The corresponding contribution to the Borel\ntransformed ΠQCD (3σ)\nΛb→Dis\nB{ΠQCD (3σ(1))\nΛb→D}(T, q) =−i\n4f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du e−Σc/T2θ(Σc−sm)θ(sth−Σc)\n×ψ3σ(ω, u)PR\u0014\nmΛbω/q−mΛbω\nmΛb−uωq2\u0015\n. (22)8\nNext we consider the term proportional to /nin Eq.(21). Note that nµ=xµ/v·x, to eliminate\nthe 1 /v·xone can define modified LCDAs as\n¯ψi(ω, u) =Zω\n0dττψ i(τ, u), (23)\nwith i= 2,3s,3σ,4. Thus the 1 /v·xcan be eliminated through integration by part:\nZ2sΛb\n0\n0dω ωψ i(ω, u)e−iuωv·xxµ\nv·x=iuZ2sΛb\n0\n0dω¯ψi(ω, u)e−iuωv·xxµ. (24)\nHere the boundary term has been omitted since large ωin the exponential induces high frequency\noscillation under the integration of x, which suppresses its contribution. The xµcan be written as\n−i∂/∂pµ, and thus the /nterm in Eq.(21) contributes:\nΠQCD (3σ(n))\nΛb→D(p, q) =i\n4f(1)\nΛbZ2sΛb\n0\n0dω ωZ1\n0du¯ψ3σ(ω, u)\n×PR∂\n∂pµ\u00141\n(p−uω)2−m2cγµ/v/q(uω/v−/p)\u0015\nuΛb(v). (25)\nAfter dispersion integration, performing the Borel transformation, and introducing auxiliary mass\nto lower the higher power of denominators: 1 /(p2−Σc)2= (∂/∂M2)[1/(p2−Σc−M2)]|M2=0, one\nobtains the /nterm contribution as\nB{ΠQCD (3σ(n))\nΛb→D}(T, q)\n=−i\n2f(1)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du u ¯ψ3σ(ω, u)θ[(Σc+M2)−sm]θ[sth−(M2+ Σ c)]e−(Σc+M2)/T2\n×PR\u0012mΛb\nmΛb−uω\u00132\u0002\nA(p2, q2) + (2 v·q)m2\nc+B(p2, q2)/q\u0003\nuΛb(v)\f\f\f\nM2=0, p2=Σc+M2, (26)\nwhere\nA(p2, q2) = (2 v·q)(mΛb−uω)(mΛb+uω−2v·p)−2(mΛb−v·p)q2,\nB(p2, q2) =q2−(mΛb−uω)(mΛb+uω−2v·p), (27)\nand note that all the Lorentz invariants involved above should be expressed by p2, q2as\nv·q=1\n2mΛb(m2\nΛb+q2−p2), v·p=1\n2mΛb(m2\nΛb+p2−q2),\np·q=1\n2(m2\nΛb−q2−p2) (28)\nThe twist-2 and 4 contributions to the correlation function can be obtained similarly as that of\ntwist-3. The corresponding Borel transformed correlation function is\nB{ΠQCD (2)+(4)\nΛb→D}(p, q)\n=i\n4mcf(2)\nΛbZ2sΛb\n0\n0dωZ1\n0du u e−Σc/T2θ(Σc−sm)θ(sth−Σc)9\n×ψ2(ω, u)\u0012ωmΛb\nmΛb−uω\u0013\nPR\u0002\n2v·q−/q\u0003\nuΛb(v)|M2=0\n+i\n4mcf(2)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du u e−(Σc+M2)/T2θ(Σc+M2−sm)θ(sth−M2−Σc)(¯ψ2−¯ψ4)(ω, u)\n×\u0012mΛb\nmΛb−uω\u00132\nPR\u0002\n2p·q+q2−2uωv·q−(mΛb−uω)/q\u0003\nuΛb(v)\f\f\f\nM2=0, p2=Σc+M2. (29)\nIt can be found that here the twist-2 and 4 contributions are proportional to mc. In the case of\nΛb→π, K +ψdecays, the twist-2 and 4 contributions are proportional to mu=md= 0 and\nthus vanish. For the Λ b→π, K +ψdecays in the type-I model, the calculation of the correlation\nfunction in Eq.(7) is almost the same, which can be found in the Appendix A.\nGenerally, as Eq.(10) shows, the correlation function has two independent spinor structures:\n1 and /q. In the type-I model both of them exists in the correlation function. However, it can\nbe found that the /qterm is absent in the type-II model. As a result, only F1contributes to the\nΛ0\nb→π0, K0,¯D0+ψdecays, while F2vanishes up to the twist-3 LCDA contributions. On the\nother hand, in the type-II model the Λ0\nb→π0+ψdecay is forbidden due to the flavor SU(3) limit,\nwhich will be discussed in the next section. The corresponding analytical results in the type-II\nmodels are given in the Appendix B.\nIV. SU(3) ANALYSIS\nIn the above study we have only focused on the decay processes Λ0\nb→π0, K0,¯D+ψ, while the\nvarious decay channels Ξ0,−\nb→π0,−, K0,−+ψand Λ0\nb,Ξ0\nb→η+ψhave not been considered. In\nprinciple, the amplitudes of these missed processes can also be calculated as soon as the LCDAs\nof Ξ borηare known, which have not been studied as well as the LCDAs of Λ b. However, with\nthe use of flavor SU(3) symmetry, one can still able to predict the decay widths of these missed\nchannels as long as the decay widths of Λ0\nb→π0, K0+ψare known.\nAs shown in Eq.(3), the effective Hamiltonian: −GuqO(uq)contains two light quark fields u, d\noru, s. In the flavor SU(3) representation, this Hamiltonian can be represented by a rank-two\ntensor Hij, where i, jare the flavor indexes. Its non-vanishing components are H12=Gudand\nH13=Gusforb→u, dandb→u, stransitions, respectively. Note that Hijis reducible so that\ncan be further be reduced into three SU(3) irreducible representations: a symmetric and traceless\ntensor ¯S{ij}, an anti-symmetry tensor T[ij]and a trace term Iδij:\nHij=¯S{ij}+T[ij]+I{ij}, (30)\nwhere\n¯S{ij}=1\n2Hij+1\n2Hji−1\n3δijHkk, T [ij]=1\n2Hij−1\n2Hji. (31)\nThe non-vanishing components are ¯S12=¯S21=1\n2,T12=−T21=1\n2forb→u, d, and ¯S13=¯S31=1\n2,\nT13=−T31=1\n2forb→u, s. It should be mentioned that in the type-II model, as shown by Eq.4\nthe two light flavors are anti-symmetrized, so that ¯Sijvanishes in this case.10\nTABLE I: Decay amplitudes of anti-triplet bottom baryon decays into a light meson and dark baryon from\nEq.(33), where the left and right columns correspond to b→dandb→stransitions, respectively.\nChannel Amplitude Channel Amplitude\nΛ0\nb→π0ψ√\n2s1Gud Λ0\nb→K0ψ−(s1+t1)Gus\nΞ0\nb→¯K0ψ−(s1+t1)GudΞ0\nb→π0ψ1√\n2(s1−t1)Gus\nΞ−\nb→K−ψ−(s1−t1)GudΞ−\nb→π−ψ (s1−t1)Gus\nΛ0\nb→ηψ −q\n2\n3t1Gud Ξ0\nb→ηψ\u0010\n1√\n6t1−q\n3\n2s1\u0011\nGus\nOn the other hand, the initial anti-triplet baryons Tb¯3and the final light mesons Mcan also\nbe expressed by SU(3) irreducible representations as\n(Tij\nb¯3) =\n0 Λ0\nbΞ0\nb\n−Λ0\nb0 Ξ−\nb\n−Ξ0\nb−Ξ−\nb0\n\nij, Mi\nj=\nπ0√\n2+η√\n6π+K+\nπ−−π0√\n2+η√\n6K0\nK−K0−2η√\n6\n\nij. (32)\nUsing Tb¯3,Mi\nj,¯SijandTij, one can construct SU(3) invariant amplitude as\nA=s1Tij\nb¯3¯Sjk(MT)k\niψ+t1Tij\nb¯3Tjk(MT)k\niψ+t2Tij\nb¯3Tij(MT)k\nkψ. (33)\nHere we have not considered the color singlet η1in the construction of Mi\nj, thus the third term\nabove vanishes due to Mk\nk= 0. The unknown amplitudes s1, t1, t2contain all the information\nabout the strong interaction in the decays. All the decay amplitudes of anti-triplet bottom baryon\ndecays into a light meson and dark baryon are listed in Table I, where the left and right two\ncolumns correspond to b→dandb→stransitions. Note that the amplitude of Λ0\nb→π0+ψ\nis proportional to s1which is absent in the type-II model. Therefore, in the type-II model the\nΛ0\nb→π0+ψdecay is forbidden, and we have s1= 0 in Table I so that the amplitudes of all the\ndecay channels are proportional to t1. On the other hand, in the type-I model, using the decay\namplitudes of Λ0\nb→π0ψand Λ0\nb→K0ψcalculated in this work by LCSR, we can determine the\nfraction ξK/π=t1/s1as\n−1√\n2(1 +ξK/π)λs/d=A(Λ0\nb→K0ψ)\nA(Λ0\nb→π0ψ), (34)\nwith λs/d=Gus/Gud. This enables us to predict all the decay amplitudes listed in Table I.\nV. NUMERICAL RESULTS\nThe hadron masses are taken as: mΛb= 5.62 GeV, mπ= 0.135 GeV, mK= 0.498 GeV and\nmD= 1.86 GeV. The quark masses are taken as mu=md= 0 and mc= 1.1 GeV at µ= 3 GeV [31].\nτ= 0.6 GeV is taken as the center value in its range τ= 0.4∼0.8 GeV. The threshold parameters\nshould be above the lowest state while nearly below the next lowest state. The next lowest states\ncorresponding to π, D, K areπ(1300) , K(1460) , D(2550) [31] and sthcan be parameterized as\nsπ,K,D\nth= (1−λ)m2\nπ,K,D +λ m2\nπ(1300) ,K(1460) ,D(2550) , (35)11\nΛb π ψType-I\n2 4 6 8 10 12-0.04-0.03-0.02-0.010.\nT2[GeV2]F1[GeV3]Λb π ψType-I\n2 4 6 8 10120.020.040.060.080.1\nT2[GeV2]F2[GeV3]\nΛbKψType-I\n2 4 6 8 10 120.020.040.06\nT2[GeV2]F1[GeV3]ΛbKψType-I\n2 4 6 8 10120.\n-0.02\n-0.04\n-0.06\nT2[GeV2]F2[GeV3]\nΛbDψType-I\n4 8 12 16 20 24 280.0140.0180.022\nT2[GeV2]F1[GeV3]ΛbDψType-I\n4 8 12 16 20 24 280.060.070.080.09\nT2[GeV2]F2[GeV3]\nFIG. 3: The form factors F1, F2as functions of T2in the type-I model, with q2= 0. The band width denotes\nthe uncertainty of sth: 0.6< λ < 1.0.\nwhere τ= 0 and τ= 1 correspond to the lowest and the next lowest states, respectively. In this\nwork, to estimate the uncertainties from the threshold parameter, we choose the range 0 .6< λ < 1.0\nfor error analysis in numerical calculations.\nThe form factors F1, F2as functions of T2are shown in Fig.3 and Fig.4 for the type-I and II\nmodels respectively, where q2= 0. In principle, the form factors should be independent of the\nunphysical Borel parameter. Therefore, one has to choose a window for T2where the behavior of\nF1, F2are almost stable. From Fig.3, it can be found that the stable regions are at large T2, and\ntheT2windows are chosen as\nΛb→πψ, Kψ : 4 GeV2< T2<12 GeV2,12\nΛbKψType-II\n2 4 6 8 10 1200.0050.010.0150.020.025\nT2[GeV2]F1[GeV3]ΛbDψType-II\n4 8 12 16 20 24 280.080.10.12\nT2[GeV2]F1[GeV3]\nFIG. 4: The form factors F1, F2as functions of T2in the type-II model, with q2= 0. The band width\ndenotes the uncertainty of sth: 0.6< λ < 1.0.\nΛb→Dψ: 12 GeV2< T2<28 GeV2(36)\nfor the both type-I and II models.\nOn the other hand, note that the F1,2(q2) obtained by LCSR are only reliable in the small\npositive q2orq2<0 regions, instead of the physical region q2>0. Therefore, one has to fit the\nform factors in the q2<0 region by a suitable parameterization function, and extent the form\nfactors to physical region. In this work, we use the z-series formula [32] with single pole structure\nto perform the fitting, which reads as\nF1,2(q2) =F1,2(0)\n1−q2\nm2\npole\u0002\n1 +b1,2(z(q2)−z(0)) + c1,2(z(q2)−z(0))2\u0003\n, (37)\nwhere mpoleis the mass of the lowest baryon state that can be created by O†\nuqfrom the vaccum.\nFor the Λ b→π, K, D transitions, mpoleis chosen as mΛb, mΞb= 5.79GeV , mΞbc= 6.94GeV\nrespectively, where mΞbcis taken from QCD sum rules calculation [33]. The F1,2(0), b1,2andc1,2\nin Eq.(37) are three fitting parameters, and the zfunction is defined as\nz(q2) =p\nt+−q2−√t+−t0p\nt+−q2+√t+−t0(38)\nwith t±= (mΛb±mM)2andt0=t+\u0010\n1−p\n1−t−/t+\u0011\n. The fitting results of F1,2(0), b1,2andc1,2\nare listed in Table II, while the form factors extended to the physical region are shown in Fig.5\nand Fig.6 for the type I and II models, respectively.\nWith the use of form factors F1,2obtained above, we can calculate the decay widths of Λ b→Mψ\ndecays. The decay width formula reads as\nΓ[Λb→Mψ]\n=G2\n(uq)|⃗ q|\n8m2\nΛb(2π)5\"\n(m2\nΛb+m2\nψ−m2\nM) \nF2\n1(m2\nψ) +m2\nψ\nm2\nΛbF2\n2(m2\nψ)!\n+ 4m2\nψF1(m2\nψ)F2(m2\nψ)#\n,(39)13\nΛb π ψType-I0 5 10 15 20 25 30-10-8-6-4-20\nq2[GeV2]F1[GeV3]Λb π ψType-I\n0 5 10 15 20 253002468\nq2[GeV2]F2[GeV3]\nΛbKψType-I\n5 10 15 20 250.0.20.40.60.8\nq2[GeV2]F1[GeV3]ΛbKψType-I0 5 10 15 20 25-1.0-0.8-0.6-0.4-0.20.0\nq2[GeV2]F2[GeV3]\nΛbDψType-I\n2 4 6 8 10 12 14-0.03-0.02-0.010.0.010.02\nq2[GeV2]F1[GeV3]ΛbDψType-I\n2 4 6 8 10 12140.050.0750.10.1250.15\nq2[GeV2]F2[GeV3]\nFIG. 5: The form factors F1, F2as functions of q2in the type-I model, with q2= 0. The band width denotes\nthe combined uncertainties of the Borel parameter and sth: 0.6< λ < 1.0.\nTABLE II: Fitting parameters of the F1,2for Λ b→π, K, D transitions in the Type-I and II models.\nType-I F1(0) b1 c1 F2(0) b2c2\nΛ0\nb→π0−0.02±0.005−15.77 16.41 0.07±0.014 -3.81 4.43\nΛ0\nb→K00.04±0.006 -5.74 12.34 −0.04±0.006 -5.88 13.32\nΛ0\nb→¯D00.01±0.001 26.79 -35.06 0.06±0.003 -4.0 11.89\nType-II F1(0) b1 c1 F2(0) b2c2\nΛ0\nb→K00.003±0.001 6.21 -5.76 – – –\nΛ0\nb→¯D00.07±0.0 -1.01 -17.6 – – –14\nΛbKψType-II5 10 15 20 250\n-0.01\n-0.02\n-0.03\n-0.04q2[GeV2]F1[GeV3]ΛbDψType-II\n0 2 4 6 8 10 12140.080.10.120.14\nq2[GeV2]F1[GeV3]\nFIG. 6: The form factors F1, F2as functions of q2in the type-II model, with q2= 0. The band width\ndenotes the combined uncertainties of the Borel parameter and sth: 0.6< λ < 1.0.\nwhich is a function of the dark baryon mass mψ.|⃗ q|is the 3-momentum magnitude of the dark\nbaryon in the center of mass frame:\n|⃗ q|=1\n2mΛbq\n(m2\nΛb−(mψ+mM)2)(m2\nΛb−(mψ−mM)2). (40)\nThe upper limits of the unknown coupling constants G(uq)have been determined in Ref.[14] through\na study of Bmeson semi-inclusive decays into baryons and a dark baryon, which read as\nType I : G2\nud<(1.8±0.35)×10−14GeV−4, G2\nus<(3.75±0.74)×10−14GeV−4,\nG2\ncd<(1.06±0.21)×10−12GeV−4, G2\ncs<(1.63±0.33)×10−12GeV−4;\nType II : G2\nus<(1.07±0.21)×10−11GeV−4, G2\ncs<(3.62±0.72)×10−10GeV−4.(41)\nHere we take G(uq)as their upper limit values to calculate the decay width. Accordingly, the upper\nlimits of branching fractions for Λ b→Mψas functions of mψare shown in Fig.7 and Fig.8 in the\ntype I and II models, respectively. The blue and red bands denote the uncertainties from the Borel\nparameter and the G(uq)from Eq.(41).\nIn Sec.IV, a SU(3) analysis is performed to predict the amplitudes of all the anti-triplet bottom\nbaryon decays into a meson and dark baryon. In the type-I model, using Eq.(34) one can further\ndetermine ξK/πby the ratio of branching fractions:\n1\n2(1 +ξK/π)2λ2\ns/d=B(Λ0\nb→K0ψ)\nB(Λ0\nb→π0ψ), (42)\nif the phase space volume of Λ0\nb→K0ψis assumed to be the same as that of Λ0\nb→π0ψ. The\nξK/πas a function of mψis shown in the left diagram of Fig.9, where the range of mψis chosen\nas 0 < m ψ<(mΛb−mK) and Gud, Gusare taken as the center values in Eq.(41). Now the\nbranching fractions of all the decay channels listed in Table I can be expressed in the units of\nB(Λ0\nb→π0ψ) and B(Λ0\nb→K0ψ) for the type-I and II models respectively, which are shown in\nTable III. Particularly, the branching fraction of the charged decay Ξ−\nb→K−ψin the type-I model\nis shown in the right diagram of Fig.9. The branching fraction of Ξ−\nb→π−ψhas the same shape\nas Ξ−\nb→K−ψbut a factor λ2\ns/dshould be timed on it. The branching fractions of the charged\ndecays in the type-II model are simply proportional to B(Λ0\nb→K0ψ) so we do not plot them here.15\n0 1 2 3 4 501234\nmψ[GeV]B(Λbπψ)108\n0 1 2 3 4 50.00.20.40.60.81.01.2\nmψ[GeV]B(ΛbKψ)108\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.00.51.01.52.0\nmψ[GeV]B(ΛbDψ)107\nFIG. 7: The upper limits of branching fractions for Λ b→Mψas functions of mψin the type-I model. The\nblue and red bands denote the uncertainties from the Borel parameter and the G(uq)from Eq.(41).\n0 1 2 3 4 50.000.050.100.150.200.250.30\nmψ[GeV]B(ΛbKψ)109\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.00.20.40.60.8\nmψ[GeV]B(ΛbDψ)106\nFIG. 8: The upper limits of branching fractions for Λ b→Mψas functions of mψin the type-II model. The\nblue and red bands denote the uncertainties from the Borel parameter and the G(uq)from Eq.(41).\nVI. CONCLUSION\nIn this work, we have studied the decays of Heavy baryon into a pseudoscalar meson and a\ndark baryon using the recently developed B-Mesogenesis scenario, where the two types of effective\nLagrangians proposed by the scenario are both considered. The decay amplitudes of Λ0\nbhave been\ncalculated by LCSR using its light-cone distribution amplitudes. The decay amplitudes of Ξ0,±\nb16\n0 1 2 3 4 5-1.5-1.0-0.50.00.51.01.52.0\nmψ[GeV]ξK/π\n0 1 2 3 4 50.00.51.01.52.0\nmψ[GeV]B(Ξb-K-ψ)108\nFIG. 9: The ξK/πas a function of mψfrom Eq.(42), where the range of mψis chosen as 0 < m ψ<(mΛb−mK)\n(left). The branching fractions of Ξ−\nb→K−ψas functions of mψin the type-I model (right). The blue and\nred bands denote the uncertainties from the Borel parameter and the G(ud)from Eq.(41).\nTABLE III: The amplitudes of all the anti-triplet bottom baryon decays into a light meson and dark baryon\nare expressed in the units of B(Λ0\nb→π0ψ) and B(Λ0\nb→K0ψ) for the type-I and II models, respectively.\nChannel Type-I Br. Channel Type-I Br.\nΛ0\nb→π0ψ 1 Λ0\nb→K0ψ1\n2(1 +ξK/π)2λ2\ns/d\nΞ0\nb→¯K0ψ1\n2(1 +ξK/π)2Ξ0\nb→π0ψ1\n4(1−ξK/π)2λ2\ns/d\n��−\nb→K−ψ1\n2(1−ξK/π)2Ξ−\nb→π−ψ1\n2(1−ξK/π)2λ2\ns/d\nΛ0\nb→ηψ1\n3ξ2\nK/πΞ0\nb→ηψ1\n12(1−3ξK/π)2λ2\ns/d\nChannel Type-II Br. Channel Type-II Br.\nΛ0\nb→π0ψ 0 Λ0\nb→K0ψ 1\nΞ0\nb→¯K0ψ λ−2\ns/dΞ0\nb→π0ψ 1/2\nΞ−\nb→K−ψ λ−2\ns/dΞ−\nb→π−ψ 1\nΛ0\nb→ηψ2\n3λ−2\ns/dΞ0\nb→ηψ 1/6\nhas been related with those of Λ0\nbthrough a flavor SU(3) analysis. In the numerical calculation,\nthe uncertainties of threshold parameter and the Borel parameter are both considered. The values\nof effective coupling constants in the B-Mesogenesis are taken as their upper limits that obtained\nfrom our previous study on the inclusive decay. The upper limits of the decay branching fractions\nof Λ0\nb,Ξ0,±\nb→Mψ are presented as functions of the dark baryon mass, which will be tested by\nfuture experimental detections.\nAcknowledgements\nThe work of Y.J. Shi is supported by Natural Science Foundation of China under Grant\nNo.12305103, and Opening Foundation of Shanghai Key Laboratory of Particle Physics and Cos-\nmology under Grant No.22DZ2229013-2. The work of Y. Xing is supported by National Science\nFoundation of China under Grant No.12005294. The work of Z.P. Xing is supported by China\nPostdoctoral Science Foundation under Grant No.2022M72210.17\nAppendix A: Analytical Results in the Type-I model\nIn this appendix, we present the analytical results of the quark-gluon level calculation for\nthe correlation function defined in Eq.(7). In the type-I model, the twist-3s contribution to the\ncorrelation function for the Λ b→πdecay is\nB{ΠQCD (3s)\nΛb→π}(T, q) =i\n4√\n2f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(Σ−sm)θ(sth−Σ)e−Σ/T2\n×ψ3s(ω, u)PR\u0014\nmΛbω/q−mΛbω\nmΛb−uωq2\u0015\n+i\n4√\n2f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(¯Σ−sm)θ(sth−¯Σ)e−¯Σ/T2\n×ψ3s(ω, u)PR\u0014\n(¯uω(2q·v)−2p·q−q2)ωmΛb\nmΛb−¯uω+mΛbω/q\u0015\nuΛb(v),(A1)\nwhere Σ = Σ c|mc=0and¯Σ = Σ |u→¯u. The twist-3 σcontributions are\nB{ΠQCD (3σ)\nΛb→π}(p, q) =−i\n4√\n2f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(Σ−sm)θ(sth−Σ)e−Σ/T2\n×ψ3s(ω, u)PR\u0014\nmΛbω/q−mΛbω\nmΛb−uωq2\u0015\nuΛb(v)\n−i\n4√\n2f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(¯Σ−sm)θ(sth−¯Σ)e−¯Σ/T2\n×ψ3σ(ω, u)PR\u0014\n(¯uω(2q·v)−2p·q−q2)ωmΛb\nmΛb−¯uω+mΛbω/q\u0015\nuΛb(v)\n−i\n2√\n2f(1)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du¯ψ3σ(ω, u)e−(Σ+M2)/T2\n×θ((Σ + M2)−sm)θ(sth−(Σ + M2))\u0012mΛb\nmΛb−uω\u00132\n×PR[A(p2, q2) +B(p2, q2)/q]uΛb(v)\f\f\f\nM2=0, p2=Σ+ M2\n+i√\n2f(1)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du¯ψ3σ(ω, u)e−(¯Σ+M2)/T2\n×θ((¯Σ +M2)−sm)θ(sth−(¯Σ +M2))\u0012mΛb\nmΛb−¯uω\u00132\n×(p·q−¯uωv·q)PR(mΛb−¯uω−/q)uΛb(v)\f\f\f\nM2=0, p2=¯Σ+M2, (A2)\nThe twist-3s and twist-3 σcontributions to the correlation function for the Λ b→Kdecay are\nB{ΠQCD (3s)\nΛb→K}(T, q) =−i\n4f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(¯Σ−sm)θ(sth−¯Σ)e−¯Σ/T2\n×ψ3s(ω, u)PR\u0014\n(¯uω(2q·v)−2p·q−q2)ωmΛb\nmΛb−¯uω+mΛbω/q\u0015\nuΛb(v),18\nB{ΠQCD (3σ)\nΛb→K}(p, q) =i\n4f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(¯Σ−sm)θ(sth−¯Σ)e−¯Σ/T2\n×ψ3σ(ω, u)PR\u0014\n(¯uω(2q·v)−2p·q−q2)ωmΛb\nmΛb−¯uω+mΛbω/q\u0015\nuΛb(v)\n−if(1)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du¯ψ3σ(ω, u)e−(¯Σ+M2)/T2\n×θ((¯Σ +M2)−sm)θ(sth−(¯Σ +M2))\u0012mΛb\nmΛb−¯uω\u00132\n×(p·q−¯uωv·q)PR(mΛb−¯uω−/q)uΛb(v)\f\f\f\nM2=0, p2=¯Σ+M2. (A3)\nThe twist-2, 4 contributions to the correlation function for the Λ b→π, K decays vanishes in the\nchiral limit mu=md= 0.\nAppendix B: Analytical Results in the Type-II model\nIn the type-II model, the Λ b→πdecay is forbidden in the flavor SU(3) limit. The twist-3s and\ntwist-3 σcontributions to the correlation function for the Λ b→Kdecay are\nB{ΠQCD (3s)\nΛb→K}(T, q) =−i\n2f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(¯Σ−sm)θ(sth−¯Σ)e−¯Σ/T2\n×ψ3s(ω, u)ωmΛb\nmΛb−¯uωPR[¯uωv·q−p·q]uΛb(v),\nB{ΠQCD (3σ)\nΛb→K}(p, q) =i\n2f(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(¯Σ−sm)θ(sth−¯Σ)e−¯Σ/T2\n×ψ3σ(ω, u)ωmΛb\nmΛb−¯uωPR[¯uωv·q−p·q]uΛb(v)\n−2if(1)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du¯ψ3σ(ω, u)e−(¯Σ+M2)/T2\n×θ((¯Σ +M2)−sm)θ(sth−(¯Σ +M2))\u0012mΛb\nmΛb−¯uω\u00132\n×(v·p−¯uω)(p·q−¯uωv·q)PRuΛb(v)\f\f\f\nM2=0, p2=¯Σ+M2. (B1)\nThe twist-2+4 contributions to the correlation function for the Λ b→Ddecay are\nB{ΠQCD (2+4)\nΛb→D}(p, q) =i\n2mcf(2)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(Σc−sm)θ(sth−Σc)e−Σc/T2\n×ψ2(ω, u)ωmΛb\nmΛb−¯uω(v·q)PRuΛb(v)\n+i\n2mcf(2)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du(¯ψ2−¯ψ4)(ω, u)e−(Σc+M2)/T2\n×θ((Σc+M2)−sm)θ(sth−(Σc+M2))\u0012mΛb\nmΛb−¯uω\u00132\n×(p·q−uωv·q)PRuΛb(v)\f\f\f\nM2=0, p2=Σc+M2. (B2)19\nThe twist-3 contributions to the correlation function for the Λ b→Ddecay are\nB{ΠQCD (3s)\nΛb→D}(p, q) =−if(1)\nΛbZ2sΛb\n0\n0dωZ1\n0du θ(Σc−sm)θ(sth−Σc)e−Σc/T2\n×(ψ3s+ψ3σ)(ω, u)ωmΛb\nmΛb−¯uω(uωv·q−p·q)PRuΛb(v)\n+ 2if(1)\nΛb∂\n∂M2Z2sΛb\n0\n0dωZ1\n0du¯ψ3σ(ω, u)e−(Σc+M2)/T2\n×θ((Σc+M2)−sm)θ(sth−(Σc+M2))\u0012mΛb\nmΛb−¯uω\u00132\n×(v·p−uω)(p·q−uωv·q)PRuΛb(v)\f\f\f\nM2=0, p2=Σc+M2. (B3)\n[1] A. D. Sakharov, “Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe,”\nPisma Zh. Eksp. Teor. Fiz. 5, 32 (1967) doi:10.1070/PU1991v034n05ABEH002497\n[2] G. Elor, M. Escudero and A. 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Shen, W. Wang and Z. X. Zhao, “Weak decays of doubly heavy baryons:\n”decay constants”,” Chin. Phys. C 42, no.12, 123102 (2018) doi:10.1088/1674-1137/42/12/123102\n[arXiv:1711.10289 [hep-ph]]."    },    {        "title": "2401.14128v1.Homoclinic_chaos_in_a_pair_of_parametrically_driven_coupled_SQUIDs.pdf",        "content": "arXiv:2401.14128v1  [nlin.CD]  25 Jan 2024Homoclinic chaos in a pair of parametrically-driven\ncoupled SQUIDs\nM. Agaoglou1, V. M. Rothos1, H. Susanto2\n1Department of Mechanical Engineering, Faculty of Engineer ing, Aristotle University of\nThessaloniki, Thessaloniki 54124, Greece\n2Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4\n3SQ, United Kingdom\nE-mail:rothos@auth.gr\nAbstract. An rf superconducting quantum interference device (SQUID) consists of a\nsuperconducting ring interrupted by a Josephson junction ( JJ). When driven by an alternating\nmagnetic field, the induced supercurrents around the ring ar e determined by the JJ through the\ncelebrated Josephson relations. This system exhibits rich nonlinear behavior, including chaotic\neffects. We study the dynamics of a pair of parametrically-dr iven coupled SQUIDs arranged in\nseries. We take advantage of the weak damping that character izes these systems to perform a\nmultiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the\nsystem. This picture allows us to expose the existence of hom oclinic orbits in the dynamics of\nthe integrable part of the slow equations of motion. Using hi gh-dimensional Melnikov theory,\nwe are able to obtain explicit parameter values for which the se orbits persist in the full system,\nconsisting of both Hamiltonian and non-Hamiltonian pertur bations, to form so-called Silnikov\norbits, indicating a loss of integrability and the existenc e of chaos.\n1. Introduction\nWe consider a series of electrical circuits in a line and we co uld derive the following nonlinear\nlattice equation [1],\n¨fn+γ˙fn+fn+ˆβsin(2πfn)−λ(fn+1−2fn+fn−1) = 0 (1.1)\nThe coefficient of the nonlinearity, which corresponds to the so-called Josephson critical current,\nis modulated in time, which can be realized experimentally b y modulating the surrounding\ntemperature. As the temperature also influences the damping due to the unpaired electron,\nadditionally we also consider time-periodically modulate d dissipation. Compared to [5], we\nconsider softening nonlinearity as opposed to stiffening one , i.e. the nonlinearity has different\nsign.\nHere, we wish to study the possibility of homoclinic chaos ne ar resonances in typical SQUID\nlattice, which is described by eq. (2.1). In particular, we c onsider a specific dimer for the lattice\nequation (2.1) and show that the unperturbed system has a hom oclinic orbit in their collective\ndynamics. Applying the geomertical method of singular pert urbation theory near the resonaces\nand Melnikov theory of near integrable Hamiltonian system t o predict the chaotic behavior near\nresonances [3]. Moreover, we state the theorem for the exise nce of multi-homoclinic orbits nearresonance following the references [3, 2]. In Section 2, we p erform a multi-scale analysis for\nthe dimer case and obtain amplitude equations to describe th e slow dynamics of the system of\ndimer SQUID. It is in the slow dynamics that the homoclinic or bits are found. In Section 3, we\nperform the analytical method for the existence of homoclin ic orbits in the perturbedsystem. In\nparticular we calculate the Energy-difference function usin g the Melnikov integral evaluated on\nthe homoclinic solutions and applying the singular perturb ation theory we study the dynamics\nnear resonances. In Section 4, we illustrate numerically ou r analytical results for the SQUID\nmodel.\n2. Normal mode amplitude equations\nWe consider the 1D case of eq (6) in [1]\n¨fn+γ˙fn+fn+ˆβsin(2πfn)−λ(fn+1−2fn+fn−1) = 0 (2.1)\nfor n=1,2 and f0=f3= 0 we obtain\n¨f1+γ˙f1+f1+ˆβsin(2πf1)−λf2= 0\n¨f2+γ˙f2+f2+ˆβsin(2πf2)−λf1= 0 (2.2)\nWe could consider a small parameter γ0→ǫˆγ0,ˆβ=β+fpcos(ωpt) ,γ=γ0and drive amplitude\nfp=ǫh. The normal mode frequencies for the linear system are:\nω2\n1= 1+2πβ−λ,ω2\n2= 1+2πβ+λ (2.3)\nWe use multiple time scales to express the displacements:\nf1,2(t) =√\n3ǫ\n2(A1(T)eiω1t±A2(T)eiω2t+c.c)+ǫ3/2f(1)\n1,2(t)+... (2.4)\nwheref1is taken with the positive sign and f2with the negative sign, slow time T=ǫt.\nThe drive frequency ωpis related with ω2:ωp= 2ω2+ǫΩ, ω1=ω2+ 2ǫΩ1.Substituting\nmultiple time scale expression to the system of f1,f2generates secular terms that yield two\ncoupled equations for the complex amplitudes A1,2. Express the complex amplitudes using real\namplitudes and phases as\nA1(T) =a1(T)ei(x1(T)+(Ω/2−2Ω1)T), A2(T) =a2(T)ei(x2(T)+ΩT/2)(2.5)\nthe real and imaginary parts of the two secular amplitude equ ations become:\nda1\ndT=3π3β\n2ω1a1a2\n2sin(2(x2−x1))+πˆh\n2ω1sin(2x1)a1−1\n2a1ˆγ0\ndx1\ndT=−Ω\n2+2Ω1−3π3β\n2ω1(a2\n1+2a2\n2+a2\n2cos(2(x2−x1)))+πˆh\n2ω1cos(2x1)\nda2\ndT=3π3β\n2ω1a2a2\n1sin(2(x1−x2))+πˆh\n2ω2sin(2x2)a2−1\n2a2ˆγ0\ndx2\ndT=−Ω\n2−3π3β\n2ω2(a2\n2+2a2\n1+a2\n1cos(2(x1−x2)))+πˆh\n2ω2cos(2x2) (2.6)2.1. Analytical expressions for homoclinic orbits\nWe performed a multi-scale analysis [4] for the dimer case an d obtained amplitude equations to\ndescribe the slow dynamics of the system of dimer SQUID. It is in the slow dynamics that the\nhomoclinic orbits are found. The homoclinic orbits given by\nBh(T,I) =2δa2\nqcosh(2aT)+p(2.7)\nθh(T,I) = tan−1/parenleftBig/radicalBigg\nI(δ−3)+Ω 1\nI(1−δ)−Ω1tanh(aT)/parenrightBig\n(2.8)\nχh\n1(T,I) =−a(δ2−1)/radicalbig\np2−q2tan−1[−/radicalbiggp−q\np+qtanh(aT)]+(δI−Ω\n4)T+x1(0) (2.9)\nφh(T,I) =χh\n1(T,I)−θh(T,I) (2.10)\nwhereq=I(δ2−1) + 2δΩ1,p=−Ω1(δ2−4δ+ 1)−I(δ3−6δ2+ 7δ−2) anda2=\nΩ2\n1−2IΩ1(δ−2)−I2(δ−3)(δ−1).\n28.9\n28.95\n29\n29.05\n29.1\n29.15−4\n−2\n0\n2\n40510\nI φB\n∆φ\nFigure 1. Orbits homo-\nclinic to Π 0=/braceleftbig\n(x,y,I,φ) :\nx=y= 0,1< δ <3,I >\nΩ1\n3−δ/bracerightbig\n. ForI=I0(the or-\nbitinthemiddle), dφ/dT= 0\non Π0, and the orbit is het-\neroclinic, connecting fixed\npoints on Π 0that are ∆ φ\napart. For I≶I0,dφ/dT≶\n0 on Π 0. The parameters\nareδ= 1.49704, Ω = −110,\nΩ1= 27.6085.The orbits given by Eqs.2.7-2.10 are homoclinic to Π 0. In Fig.\n2 we can see some of them. At resonance, for I=I0=−Ω/4δ,\nthe orbits are heteroclinic, connecting fixed points that ar e ∆φ\napart, where ∆ φ= ∆x1−∆θ. Therefore\n∆φ=2a(δ2−1)/radicalbig\np2−q2tanh−1[/radicalbiggp−q\np+q]−2tan−1/radicalBigg\nI0(δ−3)+Ω 1\nI0(1−δ)−Ω1\n3. Homoclinic intersections in the perturbed system\nIn this section we calculate the Energy-difference function u sing\nthe Melnikov integral evaluated on the homoclinic solution s. We\ncompute that the energy-difference function is\n∆NH(φ0) = cos(2( φ0+N∆φ))−cos(2φ0)+\nsin(N∆φ)\n2\nsin∆φ\n22(sin2φ0)sin(N∆φ)−\n2ξN\nhδ(−2∆φ−∆θ−˜L\nI0) (3.1)\nWhen the dissipation parameter ξ <sinN∆φ(1+2sinN∆φ/2\nsin∆φ/2)\n2N\nhδ(−2∆φ−∆θ−˜L\nI0)\nthen the energy function admits transverse zeros.\n3.1. Dynamics near resonance\nNow applying the singular perturbation theory we study the\ndynamics near resonances. The equations that describe the\ndynamics on Mε=/braceleftbig\n(x,y,I,φ) :x=xε(I,φ),y=yε(I,φ)/bracerightbig\nnear the resonace I=I0are\ndI\ndT=εhIδsin(2φ)−εγ0I,dφ\ndT=−δI−Ω\n4+εhδ\n2cos(2φ) (3.2)According to [4] and [3] in order to study the slow dynamics, w hich is induced by the\nperturbation on Mεnear resonance, we set a slow variable I=I0+√εJinto Eqs. (3.2)\nalong with a slow time scale τ=√εT, and we obtain the Hamiltonian system in ( J,φ)\nH(J,φ) =−δJ2\n2+I0γ0φ+1\n2I0hδcos(2φ) (3.3)\n3.2. A homoclinic connection to the sink pε\nWe are in a position to show the existence of an orbit homoclin ic to the sink pε. Here we\nillustrate numerically our analytical results for the SQUI D model. In Fig.2a we show that the\nunperturbed heteroclinic orbit, which asympotes to pcasT→ ∞, returns back to a point on\nthe circle of fixed points that is inside the homoclinic separ atrix loop. In order to have Silnikov\nChaos [4] the condition\nφs< φc+∆φ < φm (3.4)\nhas to be satisfied. The φvalues of the fixed points are shown in Fig.2b along with φc+ ∆φ\nandφmfor a particular value of Ω. The parameter values for which th eseφvalues satisfy the\ncondition 3.4 are displayed in Fig.2c.\n20\n25\n30−0.500.511.50123\nφ JB\n1.5 1.55 1.6 1.65−1.5−1−0.500.511.52\nδ  \nφc\nφs\nφm\nφc+∆φ\n1 1.2 1.4 1.6 1.8 2−300−250−200−150−100−50\nδΩNo Silnikov\nNo Silnikov\nFigure 2. a) The heteroclinic orbit given by Eqs. 2.7 and 2.10 with I=I0, superimposed with\nthe phase portrait of the unperturbed scaled system on Π εnear resonance. The parameters\nareδ= 1.55,Ω =−150,Ω1= 35,ζ= 0.45,h= 1 (ξ=γ/handζ=ξ/δ). This value of ζ\nsetsφc= 0.233383 and we can see from the figure that φs< φc+∆φ < φm. b) The values of\nφs,φc,φc+∆φandφmas functions of δ, for Ω = −150. For δ >1.524 the condition is satisfied.\nc) In gray are indicated the parameter values for which the co ndition 3.4 is satisfied. In figures\nb and c: ε= 0.01\nSummary: In this work we studied chaotic dynamics of a pair of parametr ically-driven cou-\npled SQUIDs arranged in series and we found under which condi tions it can exist by using high\ndimentional Melnikov theory [4].\nAcknowledgments: The work of M.A has been co-financed from resources of the opera tional\nprogram ”Education and Lifelong Learning” of the European Social Fund and the National Strategic\nReference Framework (NSRF) 2007-2013. The work of V.R. has be en co-financed by the European Union\n(European Social Fund – ESF) and Greek national funds through t he Operational Program “Education\nand Lifelong Learning” of the National Strategic Reference Frame work (NSRF) – Research Funding\nProgram: THALES – Investing in knowledge society through the Eur opean Social Fund.\nReferences\n[1] N.Lazarides, G.P. Tsironis and N. Eleftheriou, Nonlinear Phenomena in Complex Systems ,11,250, 2008.\n[2] G. Mennon, G. Haller and V. Rothos, Phys Lett A ,263,175, 1999.\n[3] G. Haller, Chaos near Resonances , Springer, New York, 1999.\n[4] G. Kovacic and S. Wiggins, Physica D , 57, 185, 1992; G. Kovacic, SIAM J Math Anal , 26, 1611, 1995.\n[5] Eyal Kenig, Yuriy A. Tsarin, Ron Lifshitz, Phys. Rev. E 84 (2011) 016212."    }]
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